total heat recovery heat and moisture recovery from ventilation air

TOTAL HEAT RECOVERY: HEAT AND MOISTURE RECOVERY

FROM VENTILATION AIR

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TOTAL HEAT RECOVERY: HEAT AND MOISTURE RECOVERY

FROM VENTILATION AIR

LI-ZHI ZHANG

Nova Science Publishers, Inc. New York

Copyright © 2008 by Nova Science Publishers, Inc. All rights reserved. No part of this book may be reproduced, stored in a retrieval system or transmitted in any form or by any means: electronic, electrostatic, magnetic, tape, mechanical photocopying, recording or otherwise without the written permission of the Publisher. For permission to use material from this book please contact us: Telephone 631-231-7269; Fax 631-231-8175 Web Site: http://www.novapublishers.com

NOTICE TO THE READER The Publisher has taken reasonable care in the preparation of this book, but makes no expressed or implied warranty of any kind and assumes no responsibility for any errors or omissions. No liability is assumed for incidental or consequential damages in connection with or arising out of information contained in this book. The Publisher shall not be liable for any special, consequential, or exemplary damages resulting, in whole or in part, from the readers’ use of, or reliance upon, this material. Any parts of this book based on government reports are so indicated and copyright is claimed for those parts to the extent applicable to compilations of such works. Independent verification should be sought for any data, advice or recommendations contained in this book. In addition, no responsibility is assumed by the publisher for any injury and/or damage to persons or property arising from any methods, products, instructions, ideas or otherwise contained in this publication. This publication is designed to provide accurate and authoritative information with regard to the subject matter covered herein. It is sold with the clear understanding that the Publisher is not engaged in rendering legal or any other professional services. If legal or any other expert assistance is required, the services of a competent person should be sought. FROM A DECLARATION OF PARTICIPANTS JOINTLY ADOPTED BY A COMMITTEE OF THE AMERICAN BAR ASSOCIATION AND A COMMITTEE OF PUBLISHERS. LIBRARY OF CONGRESS CATALOGING-IN-PUBLICATION DATA Zhang, Li-Zhi. Total heat recovery : heat & moisture recovery from ventilation air / Li-Zhi Zhang. p. cm. Includes bibliographical references and index. ISBN 978-1-60876-275-0 (E-Book)1. Heat exchangers. 2. Ventilation. 3. Heat recovery. 4. Condensation. 5. Moisture. 6. Humidity--Control. 7. Water harvesting. 8. Buildings--Energy conservation. I. Title. TH7683.H42Z48 2009 621.402--dc22 2008036864

Published by Nova Science Publishers, Inc. New York

CONTENTS

Preface vii Chapter 1 Total Heat Recovery in Air-conditioning 1 Chapter 2 Energy Recovery Potentials 9 Chapter 3 Estimation of Sorption and Diffusion Properties of Hygroscopic

Materials 15 Chapter 4 Performance of Energy Wheels 55 Chapter 5 Heat Mass Transfer in Bended Sinusoidal Narrow Ducts 75 Chapter 6 Convective Heat Mass Transfer in Plate-fin Channels 99 Chapter 7 Effectiveness Correlations of Total Heat Exchangers 131 Chapter 8 Numerical Simulation of Total Heat Exchangers 155 Chapter 9 Novel Membranes for Total Heat Exchanger 187 Chapter 10 Heat Mass Transfer in Cross-corrugated Triangular Ducts 229 Chapter 11 Applications of Total Heat Recovery 269 Index 307

PREFACE Energy and environment are two hot topics world wide today. Energy has been described

as “that which makes things go.” It is seen clearly in the transfer of heat and work inside environmental systems. Heat occurs due to a temperature difference between the system and its surroundings, while work makes use of that difference to perform a function. The depleting nature of primary energy resources, negative environmental impact of fossil fuels and low exergetic efficiencies obtained in conventional space heating and cooling are the main incentives for developing alternative heating, ventilating and air-conditioning (HVAC) techniques which can either save energy or employ low-grade thermal energy sources. Today, air conditioning has accounted for 1/3 of the total energy use by the whole society. The percentages are still rising in the fast developing economies like China.

The design of HVAC systems for thermal comfort requires increasing attention, especially in light of recent regulations and standardization on ventilation, so that an optimal level of indoor humidity may be reached and maintained to ensure a comfortable and healthy environment and to avoid condensation damage to the building envelope and furnishings. Fresh air ventilation is necessary, not only for breathing purposes, but also for the prevention of deadly epidemic diseases like bird flu and SARs. Energy expenses from ventilation are very heavy. Ventilation air accounts for 20-40% of the cooling load for HVAC industry. The ratio can be even higher in hot and humid regions where latent load from fresh air is as heavy as 50% of the cooling load. To reduce this part of energy is very crucial for the reduction of energy consumption of the whole HVAC system.

Total heat exchangers (enthalpy exchangers, or the so-called energy recovery ventilators) could save a large fraction of energy for cooling and dehumidifying the fresh air since cool air and dryness would be recovered from the exhaust stream to the fresh air in summer. Similarly in winter, the total heat exchangers could also save energy because exhaust moisture and heat can be recovered to save heating and humidification energy for fresh air. With total heat exchangers, the efficiency of the existing HVAC systems can be improved too. When they are combined with independent air dehumidification units, some alternative cooling technologies like deep well water cooling, chilled-ceiling, phase-change material cooling storage can be used in practice. They represent a novel trend in the sustainable development of HVAC industry.

Social resources spent in energy conservation and environmental protection have been increased substantially. Scientists and engineers worldwide are in active pursuit of novel energy recovery technologies. The topic has drawn my attention, too. In fact, my research

Li-Zhi Zhang viii

career has been centered around energy recovery and air dehumidification for more than a dozen years. From these years’ work, I have the deep feeling that total heat recovery, though very significant and interesting, is not an easy task.

The first obstacle comes from the materials. The novel total heat recovery technology requires novel adsorbent materials or highly vapor-permeable membrane materials to fulfill this task. However, there has been only limited information disclosed from public sources until now. Most of the novel materials are unavailable commercially and even if they are available, their cost is a problem. The second obstacle comes from the insufficient accumulation of heat mass transfer data for the total heat exchangers. Novel materials have led to new heat mass transfer phenomenon. Simultaneous heat mass transfer is the major phenomenon for total heat exchangers. However, information on mass transfer in heat mass exchangers is still scarce. The available information in text books and references is for sensible heat transfer only. Mass transfer in the total heat exchangers, especially with the novel materials, has been not considered sufficiently. As a result, system design and performance analysis of total heat exchangers are difficult, which has prevented their market penetrations.

To overcome these problems, in these years, I have conducted many fundamental researches on novel total heat recovery systems, from materials fabrication to heat mass transfer analysis. Numerical modeling has provided me an efficient tool. These results are helpful to advance this technology from a notion to a real application. However, these data are still quite sporadic. It’s not easy to have full access to them. On the other hand, a systematic introduction on this technology is highly desired, because energy saving in the air conditioning industry has become a hot issue in these days, with oil prices skyrocketing. This background has prompted me to write this book. I hope this technology can be systematically introduced to the public through this book.

In this book, the systems and performances used for total heat recovery are introduced. Energy wheels and membrane based total heat exchangers are specially described. Heat and mass transfer modeling of the systems are performed. Influences of key material and design parameters on the system performance are discussed. Novel membranes including hydrophobic-hydrophilic composite membrane and composite supported liquid membrane are developed for total heat exchangers and are characterized. Besides material side intensification, air side intensification measures are taken as well. Plate-fin and cross corrugated triangular ducts are two important structures that are introduced. The basic transport data in these structures are provided. Convective heat and mass transfer coefficients in plate-fin ducts of finite fin conductance with various cross sections are numerical obtained. Fluid flow and heat transfer in cross-corrugated triangular ducts are estimated by considering laminar, transitional, and turbulent complex flow regimes.

Based on the fundamental heat mass transfer data, the book illustrates some examples of the applications of total heat recovery in novel HVAC systems. Chilled-ceiling combined with desiccant cooling and independent air dehumidification are two pioneering trends in air conditioning industry. They overcome the shortcomings of conventional all air systems by decoupling the treatment of latent load from sensible load. Partial or full total heat recovery are realized in combination with these novel systems, which contribute to reduced energy use with increased indoor humidity control, even in transit seasons when traditional air conditioning systems fail to control humidity. The component modeling of various key equipments like refrigeration cycle, heat pumps, regenerative wheels, heat exchangers,

Preface ix

cooling coils, are conducted to estimate their energy performance and their effects on indoor thermal and humidity performance.

The book combines theoretical analysis with engineering practices. It covers a wide range of knowledge from fundamental heat mass transfer to novel systems design and performance analysis, from materials synthesis, characterization to heat exchanger thermodynamics and fluid dynamics. I hope the book may give some insight and design guidelines for the total heat recovery oriented air conditioning energy conservation.

When writing the book, I am keeping in mind that the fundamentals and methodologies be given the priority. The goal is not only to tell engineers and students what to do, but how to do. As a basis, first priority is given to the synthesis and characterization of novel materials for total heat recovery. Another emphasis is on energy system modeling. To evaluate an energy system, detailed steps from physical model setting up, mathematical model setting up, solution method, experimental and numerical validation, and parametric studies, are illustrated. Generally, it is a cross-discipline endeavor which relies heavily on numerical heat mass transfer for system modeling. I hope peer engineers and research students could benefit from the methodologies exhibited in this analysis and extend them to the analysis of other energy systems.

At last, I would like to thank my family for their long-term support of researches on this topic. I would also like to thank the Natural Science Foundation of China (NSFC) for their continuing financial support in my researches of total heat recovery. I am indebted to my colleagues at South China University of Technology, and others all over China and throughout the worldwide who have provided suggestions and ideas which, in no small way, have contributed to the fabric of this text. I have always strived to remain conscious of student learning needs and difficulties, and I am grateful to my many research students, at South China University of Technology and elsewhere, who have provided positive reinforcement for my efforts.

Chapter 1

TOTAL HEAT RECOVERY IN AIR-CONDITIONING

ABSTRACT

In most industrialized countries, energy consumption by HVAC sector accounts for 1/3 of the total energy consumption of the whole society. Cooling and dehumidifying fresh ventilation air constitutes 20-40% of the total energy load for HVAC in hot and humid regions. Total heat recovery-heat and moisture recovery from ventilation air has become a hot topic in these years. In this chapter, the research backgrounds are introduced. A description is given to the most commonly encountered total heat recovery equipments: energy wheels, membrane stationary total heat exchangers.

1.1. INTRODUCTION Modern buildings and their heating, ventilating, and air-conditioning (HVAC) systems

are required to be more energy efficient, while considering the ever-increasing demand for better indoor air quality, performance and environmental issues. The goal of HVAC design in buildings is to provide good comfort and air quality for occupants during a wide range of outdoor conditions. There are many researches aimed at improving the HVAC systems in buildings while reducing the energy costs and environmental impacts. Some studies concentrate on control strategies and protocols like VAV (Variable Air Volume), VRV (Variable Refrigerant Volume) and others focus on the analysis of specific components like refrigerators, cooling coils, etc. In order for these systems to have the greatest impact, it is important for the energy needs of the building to be reduced as much as practical.

Energy has been described as “that which makes things go.” It is seen clearly in the transfer of heat and work inside environmental systems. Heat occurs due to a temperature difference between the system and its surroundings, while work makes use of that difference to perform a function. Energy and environment have become the major two issues around the globe. In most industrialized countries, energy consumption by HVAC sector accounts for 1/3 of the total energy consumption of the whole society. Cooling and dehumidifying fresh ventilation air constitutes 20-40% of the total energy load for HVAC in hot and humid regions. The percentage can be even higher where 100% fresh air ventilation is required [1],

Li-Zhi Zhang 2

such as kitchen, hospital, factories. To reduce this part of energy are very crucial for the reduction of energy consumption of the whole HVAC system.

Air-conditioning in hot and humid environments is an essential requirement for support of daily human activities. Humidity problems can be found in many applications including office buildings, supermarkets, art galleries, museums, libraries, electronics manufacturing facilities, pharmaceutical clean rooms, indoor swimming pools and other commercial facilities. For thermal comfort reasons, indoor air conditions around 27°C temperature and 10g/kg humidity ratio are the accepted set points. However, the Southern China and other Southeast Asia countries have a long summer season with a daily average temperature of 30°C, and humidity ratio above 20g/kg. Outdoor relative humidity often exceeds 80% continuously for a dozen days, leading to mildew growth on wall and furniture surfaces, which affects people’s life seriously. In spring in Southern China, there is a period named “Plum raining seasons” when it rains continuously for one to two months. People can not see sun for a long time and stuff from quilts to grains gets moldy easily. Consequently, mechanical air dehumidification plays a major role in air conditioning industry in these regions.

1.2. OPPORTUNITIES AND CHALLENGES Ventilation air is the major source of moisture load in air conditioning. As shown in

Figure 1.1 for a moisture load estimation of a medium size retail store, ventilation air constitutes about 68% of the total moisture load in most commercial buildings [2]. As a consequence, treatment of the latent load from the ventilation air is a difficult and imminent task for HVAC engineers, especially in hot and humid climates like South China.

Normally the water vapor content of atmospheric air is small, some tens of grams per kilo of air. Nevertheless, due to the very high heat of vaporization, the latent heat content in air conditioning is of the same order of the sensible one. Due to the fact that the fresh air latent load accounts for 20-40% of the total load for air conditioning and air conditioning accounts for 1/3 of the total energy use in society, to reduce energy use in treating fresh air a crucial step to the whole society’s sustainable development. Total heat exchangers (or the so-called energy recovery ventilators, or enthalpy exchangers) could save a large fraction of energy for cooling and dehumidifying the fresh air since cool and dryness would be recovered from the exhaust stream to the fresh air in summer. Figure 1.2 shows a schematic of an energy wheel. Figure 1.3 shows a schematic of a stationary total heat exchanger. It is a cross-flow membrane based total heat exchanger. With total heat exchangers, the efficiency of the existing HVAC systems can be improved either. The reason is that normally the fresh air is dehumidified by cooling coil through condensation followed by a re-heating process, which is very energy intensive. This part of energy can be saved if total heat exchangers are installed to reduce the dehumidification load. Besides energy conservation, the total heat exchangers have the additional benefits of ensuring sufficient fresh air supply, which is crucial for the prevention of epidemic respiratory diseases like SARS and Bird flu.

Total Heat Recovery in Air-conditioning 3

Loads (kg/h)

8.20.9

59.511.8

7.700.30.7

0 10 20 30 40 50 60 70

PeoplePermeanceVentilationInfiltration

DoorsWet surfaces

Humid materialsDomestic loads

Figure 1.1. Sources of moisture loads in a medium size retail store.

Fresh air inFresh air out

Exhaust air in

Exhaust air out


Exhaust air in

Exhaust air out

Figure 1.2. Schematic of an energy wheel.

Fresh in

Fresh out

Exhaust in

Exhaust out

Plates

Duct Sealing

Fresh in

Fresh out

Exhaust in

Exhaust out

Plates

Duct Sealing

Figure 1.3. Schematic of a stationary total heat exchanger.

The depleting nature of primary energy resources, negative environmental impact of fossil fuels and low exergetic efficiencies obtained in conventional space heating and cooling are the main incentives for developing alternative heating, ventilating and air-conditioning

Li-Zhi Zhang 4

(HVAC) techniques which can either save energy or employ low-grade thermal energy sources. Novel air conditioning systems with total heat recovery are the directions for sustainable development of HVAC industry [3-6].

Besides temperature, humidity is another important parameter influencing people’s feeling of thermal comfort. Figure 1.4 shows the comfort zone in a psychrometric chart [7]. As seen, in summer, the narrow zone between operative temperature 24°C and 27°C, humidity between 4g/kg and 20°C wet bulb are the acceptable levels of thermal comfort. People will feel uncomfortable whether it’s too dry or too humid. The design of HVAC systems for thermal comfort requires increasing attention, especially in the light of recent regulations and standardization on ventilation [8], so that an optimal level of indoor humidity may be reached and maintained to ensure a comfortable and healthy environment and to avoid condensation damage for building envelope and furnishings.

Part load is another problem. In hot and humid climates, air conditioning is an indispensable component to maintain a comfort indoor environment with lower temperature and humidity than outdoor conditions. Operating under hot and humid outdoor conditions, air conditioning has to deal with both sensible and latent loads in a space. In many cases to deal with space latent cooling load using a small HVAC system is often challenging and difficult. The air conditioning system’s design load is calculated based on the number of occupants and their level of activity, types and quantity of equipments used in space, solar irradiation experienced, heat transmission through the building materials, heat gained from infiltrated outdoor air and many other factors. In reality, the space loads are always below their design values. Under part-load conditions, the common practice is to employ control method to maintain the space temperature while allowing the space humidity to vary. In part load conditions, supply air temperature is reduced. It is still enough to extract sensible load, but is insufficient to extract latent load. Because at the rised temperature, air cannot be dehumidified by dew-point condensation. The indoor humidity is out of control. In full load seasons from June to September, humidity is controlled very well, but in other transit seasons, humidity of the space may drift towards a high value that causes human discomfort while supporting the growth of pathogenic or allergenic organisms. It is also believed that the emission rate of formaldehyde from furniture and building materials is higher when humidity rises, resulting in poor indoor air quality.

Stringent ventilation regulations make the humidity problem more serious. In modern society, people spend most of their time in built environments. More attention has been paid to indoor air quality and indoor thermal comfort. HVAC systems are necessary for almost all buildings. However, conventional air conditioning modes, such as constant air volume (CAV) systems and variable air volume (VAV) systems, face great challenges in effective outdoor air ventilation and precision indoor air humidity control. From the view points of ventilation, the main problems with conventional air conditioning systems are analyzed as follows [1,4,5].

(1) The outdoor air in conventional air conditioning systems mixes with the re-circulated

air, which causes transmission of bacterium and virus among multiple zones. In this situation, occupants are at high risk of infection when diseases breakout, like SARS and bird flu. Human’s expectation of effective ventilation with 100% outdoor air has been increasingly rising.


15

20

25

30

350 5 10 15 20

Humidity ratio (g/kg)

Ope

rativ

e te

mpe

ratu

re(°

C)

Summer

Winter

100％RH

30％

RH

50％

RH

60％

RH

18 °C WB

20 °C WB

15

20

25

30

350 5 10 15 20


Ope

rativ

e te

mpe

ratu

re(°

C)

Summer

Winter

100％RH

30％

RH

50％

RH

60％

RH

18 °C WB

20 °C WB

Figure 1.4. Comfort zone in a psychrometric chart.

(2) The indoor relative humidity tends to rise under part load operation because the air conditioning systems usually control the indoor temperature by reducing their cooling capacities. To control load, cooling-reheating processes are required, which are very much energy intensive. This problem is very serious in hot and humid regions, like Canton. To improve humidity control, the method of decoupling temperature control and humidity control has attracted much attention. To realize independent humidity control, an independent fresh air conditioning system, or known as dedicated outdoor air system, is always required.

(3) New technologies for a more comfortable and energy efficient indoor environment, such as chilled ceiling/beam, thermal storage and VRV (Variable Refrigerants Volume), require parallel independent fresh air conditioning systems to meet demands on effective ventilation and removal of latent load. However, the energy consumption for dehumidifying fresh air is huge, which often accounts for 20%-40% of the total energy for air conditioning in hot and humid areas. The unaffordable energy cost for treatment of fresh air, particularly for fresh air dehumidification seriously restricts the application of independent fresh air conditioners. Total heat recovery becomes a necessity.

Conventionally cooling coils are used to cool and dehumidify supply air. It is called the

coupled cooling since cooling and dehumidification are accomplished simultaneously in a coupled way. To dehumidify air, air temperature must be cooled to below dew point

Li-Zhi Zhang 6

temperature like 10°C. Dehumidified air of such low temperature cannot be supplied to the space directly because people may feel cold draft under the cold air stream. Reheating has been widely used in many applications behind a cooling coil to prevent this problem. However this cooling-reheating process is energy intensive. Energy is needed to overcool the air across the cooling coil and also to reheat the off-coil air to the desired space humidity and temperature. Although reheating is able to maintain a space at its design temperature and humidity during pert-load conditions, it is not often a recommended practice chiefly due to its high energy use.

To solve this problem, nowadays there is a trend to separate the treatment of sensible load from latent load. This is the so called independent humidity control [3,4]. According to this scheme, sensible load is treated by chilled-ceilings, cooling coils, or air handling units which still cools the supply air but doesn’t necessarily cool it as low as to dew point. Supply air temperature can be adjusted as sensible load requires. The latent load is accomplished by an independent dehumidification unit, which is to treat all latent load alone. How to combine these systems with total heat recovery is a challenging yet interesting work.

The novel total recovery systems can be classified into two categories: energy wheels and stationary total heat exchangers. In the following chapters, energy wheels and membrane based total heat exchangers are specially described. Heat and mass transfer modeling of the system are performed. Influences of key material and design parameters on the system performance are discussed. Sorption and diffusion of moisture in hygroscopic materials are the key parameters influencing latent heat recovery capability. Their appraisal methods are provided and improved. Novel membranes including hydrophobic-hydrophilic composite membrane and composite supported liquid membrane are developed for total heat exchangers and are characterized. Besides materials side intensification, air side intensification measures are taken as well. Plate-fin and cross corrugated triangular ducts are two important structures that are introduced. Plate-fin is compact and mechanically strong. Cross-corrugated triangular ducts are a new type of primary surface heat mass exchanger. The basic transport data in these structures are provided. Convective heat and mass transfer coefficients in plate-fin ducts of finite fin conductance with various cross sections are numerical obtained. Fluid flow and heat transfer in cross-corrugated triangular ducts are estimated by considering laminar, transitional, and turbulent complex flow regimes. Some application examples of total heat recovery in combination to novel independent air dehumidification units and chilled-ceiling panels are provided.

1.3. CONCLUSION Energy expenses in air conditioning has rise to 1/3 of the total energy expenses in the

whole society. Conditioning of ventilation fresh air accounts for 20-40% of the total energy cost in air-conditioning industry. Stringent ventilation regulations make the problem more serious. Total heat recovery from ventilation air has become one of the most important faction of HVAC energy conservation. Present technologies in total heat recovery fall into two categories: energy wheels and stationary total heat exchangers. The stationary total heat exchangers uses hygroscopic materials like paper and membranes as the heat and moisture transfer media. There are full of challenges and opportunities in this area.


REFERENCES

[1] Niu, J.L.; Zhang, L.Z.; Zuo, H.G. Energy savings potential of chilled-ceiling combined with desiccant cooling in hot and humid climates. Energy and Buildings, 2002, 34, 487-495.

[2] Zhang, L.Z.; Zhu, D.S.; Deng, X.H.; Hua, B. Thermodynamic modeling of a novel air dehumidification system. Energy and Buildings, 2005, 37, 279-286.

[3] Zhang, L.Z.; Niu, J.L. Performance comparisons of desiccant wheels for air dehumidification and enthalpy recovery. Applied Thermal Engineering, 2002, 22, 1347-1367.

[4] Zhang, L.Z.; Niu, J.L. A pre-cooling Munters Environmental Control cooling cycle in combination with chilled-ceiling panels. Energy, 2003, 28, 3, 275-292.

[5] Niu, J.L.; Kooi vd, J.; Ree, H.vd. Energy saving possibilities with cooled-ceiling systems. Energy and Buildings, 1995, 23, 147-158.

[6] Zhang, L.Z.; Niu, J.L. Indoor humidity behaviors associated with decoupled cooling in hot and humid climates. Building and Environment, 2003, 38, 99-107.

[7] ASHRAE. 2005 ASHRAE Handbook - Fundamental. Atlanta (GA): American Society of Heating, Refrigerating and Air-Conditioning Engineers, Inc. (ASHRAE), 2005.

[8] ASHRAE. ANSI/ASHRAE Standard 62-2001, Ventilation for acceptable indoor air quality. Atlanta (GA): American Society of Heating, Refrigerating and Air-Conditioning Engineers, Inc. (ASHRAE), 2001.

Chapter 2

ENERGY RECOVERY POTENTIALS

ABSTRACT

Stringent ventilation regulations make the humidity problem more serious. In modern society, people spend most of their time in built environments. More attention has been paid to indoor air quality and indoor thermal comfort. Fresh air centered HVAC systems are necessary for almost all buildings. In this chapter, a quantitative analysis is provided for the energy expenses and possible savings in fresh air ventilation. More specifically, hot and humid climates like Canton are selected as the calculating sample. Under such climates, moisture recovery is more important than sensible recovery.

2.1. INTRODUCTION Air-conditioning in hot and humid environments is an essential requirement for support

of daily human activities. Humidity problems can be found in many applications including office buildings, supermarkets, art galleries, museums, libraries, electronics manufacturing facilities, pharmaceutical clean rooms, indoor swimming pools and other commercial facilities. For thermal comfort reasons, indoor air conditions around 27°C temperature and 10g/kg humidity ratio are the accepted set points. However, the Southern China and other Southeast Asia countries have a long summer season with a daily average temperature of 30°C, and humidity ratio above 20g/kg. Outdoor relative humidity often exceeds 80% continuously for a dozen days, leading to mildew growth on wall and furniture surfaces, which affects people’s life seriously. In spring in Southern China, there is a period named “Plum raining seasons” when it rains continuously for one to two months. People can not see sun for a long time and stuff from quilts to grains gets moldy easily. Consequently, mechanical air dehumidification plays a major role in air conditioning industry in these regions.

How much energy that can be recovered? The first impression may be that it’s trivial. However, the question is meaningless unless we take a quantitative analysis of the energy uses in air conditioning. Normally the water vapor content of atmospheric air is small, some

Li-Zhi Zhang 10

tens of grams per kilo of air. Nevertheless, due to the very high heat of vaporization, the latent heat content in air conditioning is of the same order of the sensible one.

2.2. ENERGY CALCULATIONS Sensible load for fresh air can be calculated by

)( iopa TTcQs −= (2.1)

where cpa is specific heat of air (equal to 1.005 kJkg-1K-1), To is outside temperature (°C), and Ti is indoor set point temperature (°C).

Latent load for fresh air

)ω(ω iow −= LQL (2.2)

where Lw is latent heat of water evaporation (2501 kJkg-1), ωo is outside humidity ratio (kg/kg), and ωi is indoor set point humidity ratio (kg/kg).

South China is a typical sub-tropical climate, where the dry bulb temperature is high, but not as high as 40°C. However, it’s humid almost during the whole year. Table 2.1 lists the hourly mean outdoor dry bulb temperature and humidity ratio in each month for the city of Hong Kong. As seen, average relative humidity is above 70% during the whole year. The calculated sensible load and latent load for fresh air is also listed in the table. A comparison between the sensible load and latent load is given in Figure 1.1.

The set points for indoor air are: winter, 18°C DB, 0.50 RH, 6.4g/kg HR; Summer, 27°C DB, 0.50 RH, 10g/kg HR. As seen, sensible load for fresh air in winter (January, February, and December) is negative, meaning heating in these three months are required. Sensible load from March until November is positive, meaning cooling is required in this long summer period. They are for the ventilation air only.

During the whole year, the latent load for fresh air is positive. This indicates that air dehumidification is required 12 months a year in this region. Even in winter and in transient seasons like April, when it’s cool outside, but the outdoor air is very humid with relative humidities above 70%, see Table 1. Therefore air dehumidification is a necessity here. When it’s in hot and humid months from June to September, the latent load is around 25 kJ/kg, almost 5.5 times higher than sensible load. Therefore in these regions air dehumidification is more important than air cooling. Moisture load accounts for 80% of the total load of cooling and dehumidification. Sensible heat recovery is meaningless if latent heat is not recovered.

Each occupant requires 35m3/h fresh air in an air-conditioned space. Considering an office of 20m2, the mean sensible cooling load (from various sources like heat gains form surroundings, sunlight, computers, human body heat dissipation, etc) for the office building is 100W/m2. The total sensible load is 2.0kW. If there are 5 people in the office, in August, the sensible load for fresh air is 0.23kW and the latent load for fresh air is 1.53kW. As a result, of the total air conditioning load of 2.0+0.23+1.53=3.76kW, the sensible load for fresh air only accounts for 6%, while the latent load for fresh air accounts for 41%. That’s very impressive.

Energy Recovery Potentials 11

The sensible-only heat exchangers like heat pipes, run-around heat exchangers and regenerative heat exchangers [2-4] have little use now.

Table 2.1. Hourly mean sensible and latent load for fresh ventilation air

Month DB (°C)

HR (g/kg) RH Qs

(kJ/kg) QL (kJ/kg)

1 (Jan) 14.6 8.3 0.814 -3.417 4.764 2 (Feb) 15.1 7.9 0.750 -2.915 3.764 3 (Mar) 20.3 11.6 0.794 2.311 13.017 4 (April) 20.9 13.7 0.904 2.915 18.269 5 (May) 25.6 17.1 0.849 0.603 18.078 6 (Jun) 28.1 19.9 0.853 3.116 25.081 7 (July) 29.3 20.3 0.812 4.321 26.081 8 (Aug) 28.9 20.4 0.835 3.920 26.331 9 (Sep) 28.1 19.9 0.853 3.116 25.081 10 (Oct) 25.8 15.3 0.751 0.804 13.576 11 (Nov) 21.1 11.2 0.730 3.116 12.017 12 (Dec) 16.2 9.1 0.806 -1.809 6.765

Notes: DB, dry bulb temperature; HR, humidity ratio; RH, relative humidity; Qs, sensible load, minus for heating, positive for cooling; QL, latent load, minus for humidification, positive for dehumidification. Set points for indoor air: winter, 18°C DB, 0.50 RH, 6.4g/kg HR; Summer, 25°C DB, 0.50 RH, 10g/kg HR.

-5

0

5

10

15

20

25

30

1 2 3 4 5 6 7 8 9 10 11 12

Month

Load

(kJ/k

g)

QsQL

Figure 2.1. Comparison of sensible load and latent load for fresh air on each month.

Li-Zhi Zhang 12

In summary, generally, fresh air moisture load accounts for 40% of the total load for air conditioning in hot and humid regions. Considering energy consumed by air conditioning industry has accounts for 1/3 of the total energy use by the whole society, energy conservation from ventilation fresh air is very significant and effective. Yet how to save this part of energy is an interesting and difficult task.

The general idea to save sensible and latent load from ventilation air is to use a total heat exchanger. This device is also called as the enthalpy exchanger, or the energy recovery ventilator [5,6]. Figure 2 shows a schematic of a stationary total heat exchanger. As seen, the device is like a parallel-plates air-to-air heat exchanger. However, in place of common metal foils, some new materials with vapor-permeable capabilities are used as the plates. Therefore both the sensible heat and the latent heat (moisture) can be exchanged between two air flows. Due to the sensible heat and moisture exchange, heat and humidity would be recovered from the exhaust stream in winter (especially in cold climates like in Beijing) and excess heat and moisture would be transferred to the exhaust in order to cool and dehumidify the incoming fresh air in summer.

Fresh in

Fresh out

Exhaust in

Exhaust out

Plates

Duct Sealing

Fresh in

Fresh out

Exhaust in

Exhaust out

Plates

Duct Sealing

Figure 2.2. Schematic of a cross-flow parallel-plates total heat exchanger.

Fresh air

Exhaust air

Cooling coils Heater Humidifier

Total Heat Exchanger Return air

Supply air

Sensors

Fresh air

Exhaust air

Cooling coils Heater Humidifier

Total Heat Exchanger Return air

Supply air

Sensors

Figure 2.3. An air handling unit (AHU) with total heat exchanger.

Energy Recovery Potentials 13

The total heat exchanger can be used as an independent stand alone ventilator for a room. In such cases, it is often used in combination with a VRV (variable refrigerant volume) refrigeration system, where the cooling coils are used to treat the sensible load. For other traditional all-air central air conditioning systems, it can be combined to the existing air handling unit (AHU), as shown in Figure 3, to save fresh air load.

2.3. CONCLUSION A sample calculation indicates that in hot and humid regions air dehumidification is

required 12 months a year. When it’s in hot and humid months from June to September, the latent load is around 25 kJ/kg. It is almost 5.5 times higher than sensible load. Therefore in these regions air dehumidification is more important than air cooling. Moisture load accounts for 80% of the total load of cooling and dehumidification. Sensible heat recovery is meaningless if latent heat is not recovered. Energy conservation from ventilation fresh air is very significant and effective.

REFERENCES

[1] Zhang, L.Z.; Niu J.L. Energy requirements for conditioning fresh air and the long-term savings with a membrane-based energy recovery ventilator in Hong Kong. Energy, 2001, 26, 119-135.

[2] Dhital, P.; Besant, R. W.; Schoenau, G. J. Integrating run-around heat exchanger systems into the design of large office buildings. ASHRAE Trans., 1995, 101, 979-991.

[3] Johnson, A.B.; Besant, R.W.; Schoenau, G. J. Design of multi-coil run-around heat exchanger systems for ventilation air heating and cooling. ASHRAE Trans., 1995, 101, 967-978.

[4] Manz, H.; Huber, H.; Schalin, A.; Weber, A.; Ferrazzini, M.; Studer, M. Performance of single room ventilation units with recuperative or regenerative heat recovery. Energy and Buildings, 2000, 31, 37-47.

[5] Kistler, K.R.; Cussler, E.L. Membrane modules for building ventilation. Chemical Engineering Research & Design, 2002, 80, 53-64.

[6] Zhang, L.Z.; Jiang, Y. Heat and mass transfer in a membrane-based energy recovery ventilator. J. Membrane Sci., 1999, 163, 29-38.

Chapter 3

ESTIMATION OF SORPTION AND DIFFUSION PROPERTIES OF HYGROSCOPIC MATERIALS

ABSTRACT

Sorption and diffusion properties of moisture in hygroscopic materials are the basic properties for moisture recovery. This chapter introduces some methodologies for the correct and convenient estimation of moisture sorption and diffusion in novel materials. The first method is a simple one: sorption and diffusion of moisture in a thermo-hygrostat. The second method is relatively complicated, however it’s more accurate. The method uses an emission cell to measure moisture permeation through membranes. It simultaneously considers convective mass transfer resistance on membrane surfaces.

NOMENCLATURE At transfer area (m2) C shape factor for the isotherm Dva vapor diffusivity in air (m2/s) Dvm moisture diffusivity in material (m2/s) Ev local emission rate (kgm-2s-1) Hd duct height of air stream (m) K total moisture transfer coefficient (kgm-2s-1) k convective mass transfer coefficient (m/s) kp partition coefficient (kg air/kg membrane) N air exchange rate (s-1) NTU Number of Transfer Units p Pressure (Pa) r radius coordinate (m) r0 cell radius (m) rm extent of drying RH relative humidity Re Reynolds number

Li-Zhi Zhang 16

Sc Schmidt number Sh Sherwood number T temperature (K) t time (s) u velocity (m/s) um air bulk velocity (m/s) V volume of the cell (m3) Va volumetric air flow rate (m3/s) w moisture uptake in material (kg moisture/kg dry material) W total weight of membrane including moisture (kg) Wmax maximum water uptake of membrane material (kg/kg) z coordinates in thickness (m) z0 Half the desiccant sheet thickness (m)

Greek letters ν kinematic viscosity of air (m2/s) δ air slit height (m) ω humidity ratio (kg moisture/kg air) φ angle ε moisture exchange effectiveness θ dimensionless humidity ratio ρ density (kg/m3)

Superscripts * dimensionless

Subscripts i inlet L Lower chamber o outlet s surface v vapor m material

3.1. INTRODUCTION Treatment of moisture is the fundamental aspect of total heat recovery. Hygroscopic

materials provide the media for moisture absorption and removal. In a desiccant energy wheel system, the desiccant wheel rotates between the outside fresh air and the exhaust air from room. Heat and humidity would be recovered from the exhaust in winter and excess heat and moisture would be transferred to the exhaust to cool and dehumidify the process air in the

Estimation of Sorption and Diffusion Properties … 17

summer. In a membrane total heat exchanger system, moisture is adsorbed by membrane surface in fresh air side. The adsorbed moisture then permeates to the exhaust side, and desorbs from the membrane surface in exhaust side. Energy wheel and membranes use desiccant materials. They can treat moisture because they are hygroscopic. For both applications, the sorption and diffusion characteristics in the desiccant material are the basic data for system performance. This chapter introduces the basic knowledge to measure and estimate sorption and diffusion properties of desiccant materials. They are the fundamentals for heat and moisture transfer in these systems, which provide the basics for system performance analysis and optimization.

3.2. SORPTION AND DIFFUSIVITY IN A THERMO-HYGROSTAT The measurement of sorption properties of desiccant materials is well-known, but the

simultaneous measurement of sorption and diffusion properties of desiccant materials can not be easily found from the published literature. This may be due to the fact that it is only recently that the mass transfers inside the desiccant are considered with most of the previous studies confined to heat transfers. To better estimate the wheel and membrane performance, a method for the simultaneous measurement of sorption and diffusion properties for desiccant sheets, with which a new polymer material is measured, is proposed in this section.

Sorption Experiments HUTC-MEM02, a novel hydrophilic desiccant material sheet is considered for air

dehumidification since previous studies have already found that the diffusivity of this desiccant sheet is very high. Equilibrium sorption measurements of water vapor in the desiccant sheet material are performed at various temperatures and humidities in a thermo-hygrostat. An automatic thermo-hygrostat (WS-97) is designed and constructed for sorption experiments in air-conditioning applications. Its photographic view is shown in Figure 3.1. The dimension of the working chamber in the thermo-hygrostat is 500mm×500mm×800mm. The temperatures in the chamber are measured with platinum resistance sensors and the humidities are measured with chilled-mirror dew point meters. The operating parameters in the working chamber can be varied in the following ranges: Humidity, 30-90%RH; Temperature, 20-50°C (the upper limit is controlled by the characteristics of humidity sensors, while the lower limit is determined by the temperature of cooling bath water). The non-uniformity in the chamber is: Temperature, ≤±0.1°C; Humidity, ≤±2%RH. The precision of measurement is: Temperature, 0.1°C; Humidity, 2%RH. The precision of the control is: Temperature, ≤±0.1°C; Humidity, ≤±2%RH. The air velocity in the chamber is less than 0.2m/s.

The temperature control system of the thermo-hygrostat consists of temperature-constant water tank, water circulation pumps, electric heater, cooling coils, heat exchanger for the chamber, heat exchanger for the outer cavity, and electric-magnetic valves. Before the experiment, the temperature of the water tank is first heated by the electric heater. At the same time, driven by the circulation pumps, hot water in the tank circulates through the two

Li-Zhi Zhang 18

heat exchangers, to heat the chamber and the outer cavity to a set value. When the chamber temperature is greater than the prescribed value, the hot water valve closes and the hot water circulation comes to a halt. When the chamber temperature is smaller than the prescribed value, the cool-water valve opens and the cooling water begins to circulate in the cooling coil, driving the chamber temperature down to its set point. Thus the temperature in the chamber can be kept nearly constant throughout the test by such repeated heating and cooling processes.

The humidity control system consists of humidification water tank, electric heater, cooling coil placed in the humidification water tank, humidification cylinder, fans and dehumidification column. When the humidity goes below the set value, the humidification fan begins to work, which drives ambient air to flow through the humidification water tank, where it absorbs water and is humidified. Then the moist air is transported to the working chamber, after it is mixed with the circulation air from the chamber. The extent of humidification is adjusted by raising or decreasing the temperature of humidification water tank. In some cases such as when the environment is very dry, it is very difficult to obtain a high humidity only by the above humidification method. At these times, an auxiliary ultrasonic humidifying system will be applied. When the humidity reaches the set point, humidification process stops.

On the other hand, when the humidity in the thermo-hygrostat is greater than the set value, dehumidification process begins. The circulation air from the chamber is first directed to flow through a silica gel column, where it is dehumidified, before it is returned to the working chamber. The humidity in the thermo-hygrostat is kept nearly constant by these humidification and dehumidification processes. All these temperature and humidity control processes are performed by a microcomputer.

Before the experiments, the temperature in the working chamber is first set to a prescribed value. Then the humidity in the chamber is set to a start value around 20%. This process usually needs 2.5 to 3.5 hours. During this process, the desiccant sheet sample (4.2g, 0.52mm thick) is dried in a hot-wind drying chamber of 80°C until the weight of the desiccant sheet becomes unchanging. After above preparations, the desiccant sheet is placed in the working chamber and hooked to a strain gauge whose other end is connected to the chamber roof. The increase of the weight of the desiccant sheet is detected by the resistance changes of the strain gauge. The variations of the resistance are then recorded by a computer with the help of an additional circuit that converts resistance to voltage. When the difference of desiccant sheet weight is less than 0.1mg for every 0.5 hour, it is assumed that the equilibrium has been reached for the given state, and one sorption experiment is finished. Then the humidity of the chamber is increased to a new value while the temperature is kept unchanged. Another sorption test is conducted at this set temperature and the above procedures are repeated. Altogether more than five tests are needed for one sorption isotherm at a given temperature.


Figure 3.1. Photographic view of the thermo-hygrostat.

Drying Experiments

Saturated desiccant sheet samples are regenerated in a hot wind drying box manufactured

by Shanghai Instruments Corporation. The temperature in the box can be as high as 300°C. The wind speed in the chamber measured is 0.25m/s. The heating power of the drying box is 3.3KW. The fluctuation of temperature at a set point is ±1°C.

Before the regeneration process, the desiccant sheet sample is placed in a chamber of 100% Relative Humidity for 24 hours to reach saturation. Then it is moved into the drying box that has been pre-set to a given drying temperature. Since most of the water in desiccant sheet can be air dried after 25 minutes, the weight of the desiccant sheet is measured every 5 minutes during the initial 25 minutes, and every 15 minutes during the rest of time respectively, by an electronic scale to minimize errors. The Relative Humidity of the environment during the test is 45%.

Calculation of Diffusivity The activated diffusion process of water vapor in desiccant sheet follows three stages.

First, water vapor is adsorbed on the sheet surface at the side of higher vapor partial pressure; second, adsorbed water diffuses through the desiccant sheet, driven by a concentration or activity gradient; and third, water desorbs from the other side of the sheet [1]. Usually, Fick’s law applies to the diffusion of the vapor or gas flow in desiccant sheet, which provides the basis for the method of slops [2] in diffusivity calculations. Assuming a constant density in dry desiccant sheet, the unsteady Fick’s equation [3] can be expressed as

][ vm zwD

ztw

∂∂

∂∂

=∂∂

(3.1)

Li-Zhi Zhang 20

where z is the coordinate in thickness direction (m), and the zero point locates at half of the desiccant sheet thickness. The initial and boundary conditions for this equation are

t=0 0<z<z0 w = win (3.2) t > 0 z= 0 0/ =∂∂ zw (3.3) t > 0 z= z0 w = we (3.4)

where win is the initial water concentration (kg moisture/kg material), we is the equilibrium concentration, and z0 is half the desiccant sheet thickness.

The solution of the Eqs.(3.1)-(3.4) in the case of a constant diffusivity is given by Crank[3] as

⎥⎦

⎤⎢⎣

⎡ +−

+=

−−

=Ω ∑∞

=20

22

022

ein

e

4)12(

exp)12(

18z

tDnnww

ww vm

n

ππ

(3.5)

where Ω is the dimensionless moisture uptake, and w is the mean value of the moisture uptake of the desiccant sheet. In the range of the Fourier number of Fo(= 2

0/ ztDvm )>0.3, the first term (n=0) of Eq.(3.5) is far greater than the other terms. Then the above equation can be simplified as

⎥⎦

⎤⎢⎣

⎡−=

−−

=Ω 20

2

2ein

e

4exp8

ztD

wwww vmπ

π (3.6)

To apply the method of slops, the experimental drying/sorption curves (lnΩ versus t) are

compared to the theoretical diffusion curves (lnΩ versus Fo= 20/ ztDvm ) for the desiccant

sheet tested. The slops of the experimental drying/sorption curves (dΩ/dt) and the theoretic curves (dΩ/dFo) are estimated at a given moisture uptake, using numerical or graphical differentiation. The effective moisture diffusivity at a given uptake Ω is calculated from the equation,

20thexp ])d/)(lnd/()d/)(lnd[( zFotDvm ΩΩ= (3.7)

Uncertainty Analysis

The accuracy of the calculated diffusivity could not be discussed without a through

understanding of the different contributions of the accuracy of water uptake, thickness, and measuring time, since the diffusivity is calculated from these three parameters. It should be known that the accuracy of diffusivity also relates to the number of terms in the series of


Eq.(3.5). However, when Ω>0.5, the truncation errors from Eq.(3.5) to (3.6) are within 0.1%[3], so they can be neglected in error analysis. With a logarithmic transformation on both sides of Eq.(3.6), we have

20

22

48ln

ztDvmππ

−=⎟⎟⎠

⎞⎜⎜⎝

⎛ Ω (3.8)

Another logarithmic transformation of the above equation gives

⎟⎟⎠

⎞⎜⎜⎝

⎛+−+=⎟⎟

⎠

⎞⎜⎜⎝

⎛ Ω−

4lnln2lnln

8lnln

2

0vm

2 ππ ztD (3.9)

A differentiation of Eq. (3.9) suggests

0

0

vm

vm2

2

d2dd

8ln

8lnd

zz

tt

DD

−+=Ω

Ω

π

π

(3.10)

Thus the errors in the calculations of moisture diffusivity in desiccant sheet by the

method of slops can be estimated as

ΩΩ

Ω++=

d)8/ln(

1d2dd

20

0

vm

vm

πzz

tt

DD

(3.11)

This equation clearly discloses the different contributions of the measurement errors of

time, desiccant sheet thickness, and water uptake to diffusivity.

Sorption Isotherms The measured isotherms of the HUTC-MEM02 (a novel hydrophilic desiccant sheet

material) and water vapor are shown in Figure 3.2. They are typical III class adsorption isotherms. This indicates stronger sensitivities of desiccant sheet to relative humidity at higher humidities. Actually, the water uptake is an exponential function of relative humidity, as shown in Figure 3.3. This phenomenon is attributed to the stronger interaction between water molecules and adsorbed layers in the voids of desiccant sheet with an increase in water uptake, which increases the difficulties of applying either Langmuir or Dual-sorption model to the analysis of such isotherms. Nevertheless, an empirical correlation for the isotherms of desiccant sheet studied is obtained with the help of least square fit of the experimental data.

Li-Zhi Zhang 22

0

0.05

0.1

0.15

0.2

0.25

0.3

0 1000 2000 3000 4000 5000 6000 7000

Vapor partial pressure (Pa)

Wat

er u

ptak

e (k

g/kg

)t=23.5C

t=30.1C

t=38.2C

Figure 3.2. Isotherms of HUTC-MEM02 and water vapor. The solid line: calculated; discrete dots: experimental.

221 RHcRHcw += (3.12)

sppRH = (3.13)

Tbap /ln s += (3.14)

where ps is the saturation pressure of vapor at temperature T. The values in the equation are: c1=1.07908E-01; c2=1.50516E-01; a=20.5896; b=-5098.26.

0

0.05

0.1

0.15

0.2

0.25

0 0.2 0.4 0.6 0.8 1

Relative Humidity

Wat

er c

onte

nt(k

g/kg

)

ExperimentalModel

Figure 3.3. Water uptake in relation to relative humidity of air.


0

0.05

0.1

0.15

0.2

0 0.5 1 1.5 2 2.5 3 3.5 4Time (hr)

w (k

g/kg

)

23.5,75.5%RH30.1,78.0%RHCalculated

Figure 3.4. Sorption curves of the desiccant sheet at two conditions.

Sorption Curves

The sorption curves are shown in Figure 3.4. It can be seen that the moisture uptake

increases with the lapse of time. At the beginning, the sorption rates are very fast. After 2-2.5 hours, the water adsorbed amounts to more than 93% of the total moisture that could be adsorbed by the desiccant sheet when it reaches equilibrium with vapor. During the rest of the time, the sorption processes go very slowly and the desiccant sheet adsorbs only a small quantity of water.

The sorption curves under a specified temperature and vapor pressure can be expressed by the following equation

)1()1( 21 /

2/

10atat eAeAww −− −+−+= (3.15)

The constants in the equation can be calculated by the technique of least square fit of

experimental data. The values for the two conditions in Figure 4 are listed in Table 3.1.

Table 3.1. Values in the model of uptake curves

Conditions w0 A1 a1 A2 a2 Uncertainty (%)

(23.5°C, 75.5%RH)

5.49E-3 7.64E-2 0.55 7.94E-2 0.55 0.62

(30.1°C, 78.0%RH)

5.54E-3 7.91E-2 0.73 7.91E-2 0.73 0.43

Li-Zhi Zhang 24

0

0.05

0.1

0.15

0.2

0.25

0 0.5 1 1.5 2 2.5Time (hr)

w(k

g/kg

)

55607080

Figure 3.5. Regenerating curves for HUTC-MEM02 and vapor.

Regeneration Curves

The drying temperature has a considerable influence on the time for the desiccant sheet to

be regenerated, which increases by 2 times when the drying temperature changes from 80°C to 55°C. This can be clearly seen in Figure 3.5. Most of the water in desiccant sheet can be dried after 20 minutes’ drying.

A model is summarized to express the regeneration curves of desiccant sheet as

ctb

aew += (3.16) The constants in the equation are listed in Table 3.2.

Table 3.2. The constants in the model of regeneration curves

Temperature (C°)

a b c Uncertainty (%)

55 6.00E-3 1.79 0.48 7.09 60 7.67E-3 1.26 0.37 3.80 70 5.48E-3 1.26 0.33 5.54 80 4.44E-3 1.10 0.28 9.09


Table 3.3. Diffusivity and errors calculated from drying curves at two temperatures

Time 5min 10min 20min 1hr 1.5hr T=60°C Moisture uptake (kg/kg) 0.13 0.079 0.044 0.021 0.016 Diffusivity (10-8m2/s) 3.98 3.11 2.05 0.67 0.38 Uncertainty 3.32% 7.93% 9.64% 8.88% 6.47% T=80°C Moisture uptake (kg/kg) 0.093 0.051 0.028 0.0094 0.0093 Diffusivity (10-8m2/s) 4.89 3.22 1.70 0.39 0.26 Uncertainty 3.21% 7.94% 8.86% 8.37% 7.46%

Moisture Diffusivity The diffusivity of moisture in desiccant membrane is calculated with Eq.(3.7) as

previously deduced. In Table 3.3 are shown the results from two drying curves. The uncertainties of the calculated diffusivity can be estimated from Eq.(3.11), as listed in Table 3.3. It is clear that the obtained diffusivity is in the order of 10-8m2/s, and the maximum uncertainty is less than 10%.

To discuss the diffusivity more clearly, the results from four different drying curves are plotted in Figure 3.6. The discrete dots in the figure represent the diffusivity calculated at the corresponding water uptake and temperatures. It is seen that the moisture diffusivity increases with an increase in water uptake in desiccant sheet, or an increase of sorption temperature. Furthermore, when the uptake is below 0.07kg/kg, these dots are very densely plotted, which means that the influence of temperature on diffusivity is negligible. However, as the uptake increases above 0.07kg/kg, the effects of sorption temperature on diffusivity become larger. The higher the water concentration in desiccant sheet, the greater the discrepancies of diffusivity resulted from the sorption temperatures. On the other hand, when the uptake is less than 0.05kg/kg, the diffusivity increases almost linearly with increasing uptake, while at larger moisture concentrations, diffusivity becomes stable with variations of uptake. This character indicates that a constant diffusivity can be assumed for isothermal moisture transfer through a desiccant sheet, since in most cases, the water content in a hydrophilic desiccant sheet is bigger than 0.05kg/kg at normal temperatures.

The diffusivity can also be obtained from sorption curves. They are similar to those calculated from drying curves (for instance, at T=23.1°C, w=0.15kg/kg, Dvm=2.55×10-8m2/s and at T=30.5°C, w=0.16kg/kg, Dvm=3.17×10-8m2/s). The diffusivity of moisture in other materials is usually far lower than the values of this study (10-10 m2/s for polymer gel [4]; 10-

12-10-13 m2/s for methylcellulose desiccant sheet [5]; and 10-12 m2/s for Poly Vinylchloride sheet [1], to name but a few). It is no wonder that HUTC-MEM02 has very high performance in air dehumidification.

Li-Zhi Zhang 26

0

1

2

3

4

5

6

7

8

9

0 0.05 0.1 0.15 0.2 0.25

w (kg H2O/kg dry membrane)

D(1

0-8m

2 /s)

55607080

Dvm

0

1

2

3

4

5

6

7

8

9

0 0.05 0.1 0.15 0.2 0.25

w (kg H2O/kg dry membrane)

D(1

0-8m

2 /s)

55607080

Dvm

Figure 3.6. Calculated diffusivity of moisture in desiccant membrane.

A method of directly measuring the sorption, drying, and diffusion characteristics of a desiccant sheet is offered. An equation for error analysis of diffusivity is also presented. The sorption isotherms, sorption curves, drying curves, and diffusivity variations of a novel hydrophilic polymer desiccant sheet used in air dehumidification are obtained by experiments. The constants in the mathematical models are discussed.

The obtained isotherms are typical III class adsorption isotherms. The sorption curves indicate that during the first 2-2.5 hours, the desiccant sheet adsorbs most of the water that can be adsorbed at equilibrium. The drying temperature has a major effect on regeneration time. Results also show that the moisture diffusivity increases with either an increase in water uptake in desiccant sheet, or an increase in sorption temperature. At low water concentrations in desiccant sheet, the diffusivity increases almost linearly with increasing uptake, but at larger moisture concentrations, diffusivity tends to be stable. Therefore a constant diffusivity can be assumed for most of the hydrophilic desiccant sheet in air dehumidification.

3.3. FLUID FLOW AND MASS TRANSFER IN A NOVEL EMISSION CELL Measurement of mass transfer in a thermo-hygrostat is rather complicated and expensive.

To ease the job, in this section, a novel emission cell is proposed. It is very similar to a FLEC cell, but air flow velocities are one order higher than those through a FLEC cell. It is used to measure moisture diffusivity through hygroscopic material sheets.

FELC (Field and Laboratory Emission Cell) cell was in recent years used to measure VOCs emission from a material surface. It is portable and user-friendly, thus it has become a standard for emission testing in Europe [6-8]. However, our previous studies [7-9] have found that air velocities through a FLEC cell are too small. Air is easily saturated with emitted gas


as soon as it enters into the cell, which makes it difficult to measure the emissions on the whole surface. It is not appropriate to use it in the measurement of membrane permeability directly. Therefore recently, we have modified it to fit the needs in diffusivity measurements. Though cell structures are similar, the air velocities through the cell are increased by one order. A methodology similar to previous studies is used to model the mass transfer in the units.

A complete emission model should include two mechanisms: convective mass transfer on surfaces, and diffusion in solids. As an essential part of this process, in this section the convective mass transfer coefficients in the cell are estimated. In next section, the whole emission model will be set up.

The flow geometry in the cell is shown in Figure 3.7. It is composed of two parts: cap (Figure 3.8a) and lower chamber (Figure 3.8b). When testing, the planar specimen of the emission material is placed in the lower chamber and becomes an integral part of the emission cell. The upper surface of the specimen (the emission surface) and the inner surface of the cell cap form a cone-shaped cavity. The air is supplied through the air slits in the cap. It is introduced through two diametrically positioned inlets (symmetrically placed) into a circular-shaped channel at the perimeter, from where the air is distributed over the emission surface through the circular air slit. The air flows radially inward, until it exits the cell outlet in the center.

In addition to emission experiments, this chapter uses the cell to measure moisture diffusivity in desiccant plate materials. As a first step, forced convection mass transfer and fluid flow in this cavity is of great interest.

SpecimenSpecimen

Figure 3.7. A schematic showing the flow geometry of the cell.

Li-Zhi Zhang 28

(a)

Lower chamber

Emission material

(b)

Figure 3.8. A view of the cell, showing the cap (a) and lower cavity (b).

Mathematical Models

CFD simulation is employed to calculate the convective mass transfer coefficients

between the fluid flow and emission surface. For the present situation, the flow is assumed to be laminar and steady. Considering the fluid properties to be constant, the hydrodynamic and mass transfer problem can be described by Navier-Stokes equations in cylindrical coordinates as [9]


Conservation of mass

( ) 011 *

**

***

** =∂

∂+

∂∂

+∂∂

φφu

rzu

urrr

zr (3.17)

where r, z and φ are radial, axial and angle coordinates, respectively; ur, uz, and uφ are velocities in r, z and φ directions (m/s), respectively; superscript “*” in this and the following equations represents dimensionless form.

Conservation of r-Momentum

⎥⎥⎦

⎤

⎢⎢⎣

⎡−

∂

∂+

∂

∂+⎟⎟

⎠

⎞⎜⎜⎝

⎛∂∂

∂∂

+∂∂

−

=∂∂

+∂∂

+∂∂

2*

*

22*

*2

2*

*2

*

**

***

*

*

*

*

*

**

*

**

1r

u

r

u

z

uru

rrrr

p

uru

zu

uru

u

rrrr

rrz

rr

φ

φφ

(3.18)

where p represents pressure (pa).

Conservation of z-Momentum

⎥⎥⎦

⎤

⎢⎢⎣

⎡

∂

∂+

∂

∂+⎟⎟

⎠

⎞⎜⎜⎝

⎛∂∂

∂∂

+∂∂

−=∂∂

+∂∂

+∂∂

22*

*2

2*

*2

*

**

***

**

*

*

*

**

*

** 1

φφφ

r

u

z

urur

rrzpu

ru

zuu

ruu zzzzz

zz

r

(3.19) Conservation of φ-Momentum

⎥⎥⎦

⎤

⎢⎢⎣

⎡

∂

∂+

∂

∂+⎟

⎟⎠

⎞⎜⎜⎝

⎛

∂

∂

∂∂

+∂∂

−=∂

∂+

∂

∂+

∂

∂22*

*2

2*

*2

*

**

**

*

*

*

*

*

*

**

*

** 11

φφφφφφφφφφ

r

u

z

uru

rrr

pr

uru

zu

uru

u zr

(3.20) Conservation of water vapor

⎥⎦

⎤⎢⎣

⎡∂∂

+∂

∂+⎟

⎠⎞

⎜⎝⎛

∂∂

∂∂

=∂∂

+∂∂

+∂∂

2

2

2*2*

2

**

***

*

**

** 111

φθθθ

φθθθ φ

rzrr

rrScru

zu

ru zr (3.21)

where θ is the dimensionless humidity ratio. The characteristic distance is selected as two times the spacing between the emission surface and the cap at the cell perimeter. The mean velocity at the air slit is selected as the characteristic velocity. The dimensionless forms for the variables are expressed as

Li-Zhi Zhang 30

δ2* rr = (3.22)

δ2* zz = (3.23)

νδuu 2* = (3.24)

νδvv 2* = (3.25)

νδww 2* = (3.26)

where δ is the height of space between the emission surface and the cell cap at air slit (m). Contrary to FLEC where the air slit pitch is 1mm, in this system, it is 2mm; ν is the kinematic viscosity (m2/s).

The dimensionless pressure is defined as

2

2* 4

ρνδpp = (3.27)

where ρ is density (kg/m3). The dimensionless humidity ratio is given as

si

s

ωωωω

θ−−

= (3.28)

where ω is the air humidity ratio (kg vapor/ kg air); ωi represents humidity at the air slit, and ωs represents humidity at the emission surface.

The Schmidt number is

vaDSc ν

= (3.29)

where Dva is vapor diffusivity in the air mixture (m2/s).

The Reynolds number used to characterize the airflow rate is given by


ν

δm2Re u= (3.30)

where um is the mean air velocity at the slit, and it is calculated by δπ 0

m 2 rVu = (3.31)

where V is the volumetric air flow rate to the cell (m3/s); r0 is the maximum radius of the emission surface, where air is distributed from the slit (m). The Reynolds numbers for the flow in the cell chamber are very small, say, Re=42 when V=5L/min. Since Re<<2300, the flow is thought to be laminar.

The Sherwood number is

va

2DkSh δ

= (3.32)

where k is the convective mass transfer coefficient (m/s).

Now, considering a control volume in the radial direction, the mass balance has

( ) bcmbst ωωω dAukA −=− (3.33)

where ωb is the bulk humidity ratio on a cross-section at radius r. For reasons of symmetry, only half of the cell geometry is taken into account, the mass

transfer area is

rdrA π=t (3.34) The cross-sectional area is

δπ 0c rA = (3.35) Substituting Eqs.(3.34) and (3.35) into (3.33), the local mass transfer coefficient is

( ) drd

rruk b

bs

0m ωωωδ

−−= (3.36)

The local Sherwood number is

*b

*b

*0

L 2 drd

rReScrSh θθ

= (3.37)

Li-Zhi Zhang 32

by considering

Reu =*m (3.38)

Similarly, the mean Sherwood number is

b2*2*0

*0

m lnθrr

ReScrSh−

−= (3.39)

where the bulk dimensionless humidity ratio is

*m

*

b

)(

u

dAu∫=θ

θ (3.40)

The inlet and boundary conditions for mass transfer are r*=r0

*: θ=1; (3.41) z*=0: θ=0 (3.42)

where Eq.(3.42) indicates that the boundary condition on the emission surface is a uniform concentration condition. Other boundaries are adiabatic surfaces, have no mass transfer, and are expressed as

0=∂∂

nθ

(3.43)

where n is the normal direction.

Discretisation and Solution Strategy As previous studies, for reasons of symmetry, only half of the cell is selected as the

modeling domain. The Navier-Stokes equations are solved in three-dimensional cylindrical coordinates. The round channel, which has a rectangular cross-section, the air slit, and the air inlet and outlet vents are meshed as a whole simultaneously. Totally, there are 71318 hexahedral cells in the geometry. The meshes at the entrance region of the flow on the emission surface are finer than those in other locations, to reflect the drastic variations of variables in the boundary layer. The discretised meshes are shown in Figure 3.9. The graph is amplified vertically to view the meshes above the emission surface clearly. The computations are performed using the finite volume technique. The derivatives of the diffusive terms in the N-S equations are approximated by second-order central difference and the derivatives of the


convective terms by first-order upwind difference. The discretised equations are solved by an iterative procedure and in each step they are solved by the alternating direction implicit (ADI) method. The coupling between velocity and pressure is performed through the SIMPLE algorithm [9]. A relaxation factor of 0.65 is always required in iterations.

The convergence of the iterative procedure is studied following the evolution of the normalized residues. When the normalized residues for mass, velocity and vapor concentration are less than 10-5 at every node, the iterations are considered to be converged. The fact is that after 500 iterations, the solution is usually converged, regardless of Reynolds number.

The accuracy of the numerical method is determined from solutions on successively refined grids. The rms error defined by Fletcher [10], and based on the normalized velocity components, is used for that purpose:

( )2/1

2*,,

1*,,

* /)(⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛−= ∑∑∑ muuurms

i j kkjikji (3.44)

where ui,j,k

*1 and ui,j,k* represent quantities calculated with grids having m1 and m number of

nodes, respectively. With the grids mentioned above, the solutions at V=509ml/min have an rms error lower than 0.011. This value increases with flow rate and is about 0.018 at 1000ml/min.

φ

r

z

o

Figure 3.9. Discretized meshes of the calculating domain, half of the cell, amplified vertically.

Li-Zhi Zhang 34

Experimental Work The cell is circular and made of stainless steel, with a diameter of 150mm and a

maximum height of 18mm. The deepness of the lower chamber is 10mm. A more detailed description of the structure and parameters of the cell can be found in [9]. When testing, the material is placed on the bottom surface of the lower cavity and becomes an integral part of the emission cell. In this test, to ease the measurement of convective mass transfer coefficients, distilled water is used as the loading material, instead of VOCs emission material. Rather than directly measuring the emission rate profiles of VOCs from building materials, convective surface mass transfer coefficients are calculated by measuring the humidity differences between the inlet and outlet of an air stream, which flows through the cell and exchanges moisture with water on the lower surface.

To investigate the local mass transfer coefficients at different cell radial locations, 7 glass discs with thicknesses of 10mm and diameters ranging from 134 to 148mm are prepared. In each test, a disc is placed in the lower chamber and on its bottom surface. Distilled water is injected in the space between the chamber wall and the disc. When finished, the lower chamber, the water, and the glass disk are on the same horizon. Special care is given to ensure the water doesn’t wet the disc’s upper surface. With this method, the mass transfer area between the water surface in the cell and the air is controlled. A picture showing the placement of a glass disk in the cell lower chamber is shown in Figure 3.10. When the air flows through the cell, it exchanges moisture with the water and is humidified. Since the RH of water on the water surface is 100%, by calculating the humidity differences, the mean convective mass transfer coefficients across the water surface can be calculated. Different discs therefore provide the mean mass transfer coefficients at various cell radial locations. To investigate the influences of different gases, both air and O2 are used in the experiment.

Figure 3.10. A picture of the placement of glass disk in the chamber.


Bypass

Manometer

Compressed air Bubbler

Water

Valve Pump/Flow Meter/Controller

Flec

Pressure Meter Tem/RH sensors

Cell

Bypass

Manometer


Water


Flec


Cell

Bypass

Manometer


Water


Flec


Cell

Figure 3.11. The set-up of the test apparatus.

The cell is supplied with clean and humidified air from an air supply unit. The complete test rig is shown in Figure 3.11. The supply air flows from a compressed air bottle and is divided into two streams. One of them is humidified through a bubbler immersed in a bottle of water, and then re-mixed with the other dry air stream. The humidity of the mixed air stream is controlled by adjusting the proportions of air mixing. The airflow rates are controlled by two air pumps/controllers at the inlet and outlet of the cell. To prevent outside air from infiltrating into the cell, a manometer is installed to monitor the pressure inside the cell and ensure that it is positive. The humidities and temperatures inside and outside the cell are measured by the built-in RH and temperature sensors, which are installed in the pumps/controllers. The measuring accuracies are respectively 2% for relative humidity, 0.2°C for temperature, and 2.5% for airflow rate.

Once the humidity differences are measured, the mean mass transfer coefficient is calculated by

ϖΔ

Δ=

tm

ωAVSh (3.45)

where Δω is the humidity ratio differences between the air inlet and outlet (kg/kg), At is the transfer area between the air and water surface (m2), Δϖ is the logarithmic difference of the humidity ratio between the water surface and the air in cell (kg/kg), and Shm is the mean Sherwood number.

Local and Mean Sherwood Numbers The variations in the measured outlet relative humidity from cell with disks of four

diameters are plotted in Figure 3.12. The inlet humidity is set to a prefixed value and the gas is air. As can be seen, the outlet humidity decreases as the flow rate increases. However, when the diameters are less than 135mm, the outlet humidity changes little.

Li-Zhi Zhang 36

40

50

60

70

80

90

100

100 200 300 400 500 600

V (ml/min)

Out

let R

H

148mm144mm140mm135mm134mm

40

50

60

70

80

90

100

100 200 300 400 500 600

V (ml/min)

Out

let R

H

148mm144mm140mm135mm134mm

Figure 3.12. The outlet relative humidity of air from cell with 4 disks and various flow rates.

r/r0

ShL

00.250.50.7510

0.2

0.4

0.6

0.8

186ml/min245ml/min316ml/min420ml/min509ml/min

170ml/min

r/r0

ShL

00.250.50.7510

0.2

0.4

0.6

0.8


170ml/min

Figure 3.13. Local Sherwood numbers along the cell radius for air.


r/r0

Shm

00.250.50.7510

0.2

0.4

0.6

0.8

1

1.2


170ml/min

r/r0

Shm

00.250.50.7510

0.2

0.4

0.6

0.8

1

1.2


170ml/min

Figure 3.14. Mean Sherwood numbers at various cell radii for air.

The calculated local and mean Sherwood numbers at different cell radius under various volumetric flow rates for air are shown in Figure 3.13 and Figure 3.14, respectively. The curves for nitrogen are similar to those for air. These two figures reveal the fact that the Sherwood numbers decrease as the flow progresses. They are very large in the entrance region, and they decrease as the radius decreases, asymptotically approaching the fully developed values. The higher the flow rates (Reynolds numbers), the larger the Sh numbers. For flow rates ranging from 170 to 509ml/min, the mean Sherwood numbers of the whole cell emission surface change from 0.05 to 0.2.

The mean Sherwood numbers are also obtained from the experiments. Comparisons of the mean Sherwood numbers between those calculated and experimentally obtained are plotted in Figure 3.15. It is shown that the experimentally obtained values are in good agreement with the numerical data. The largest deviation (24%) happens when the Sh is very large, namely, at the position near the air slit, where the air begins to flow on the emission surface. This phenomenon may result from the influences of the inlet flow conditions. However, more than 90% of the numerical results are within ±5.5% deviation from the experimental data.

Figure 3.16 represents the variation of the dimensionless bulk humidity against the cell radius. The bulk humidity decreases very quickly after the air begins to make contact with the water surface. This means that the convective mass transfer coefficients are very large in the first quarter of the cell radius along the flow. Under current airflow rate conditions (less than 509ml/min), the air becomes nearly saturated in the remaining 3 quarters of cell radius along the flow. This is due to the very small spacing between the emission surface and the cell cap (minimum 1mm). When the air flow rate is increased, the exhaust air becomes less saturated.

Li-Zhi Zhang 38

0

0.5

1

1.5

2

2.5

0 0.5 1 1.5 2 2.5

Experimental Sh m

Cor

rela

ted

Shm

AirN2O2

ΔΔ

Δ

0

0.5

1

1.5

2

2.5

0 0.5 1 1.5 2 2.5

Experimental Sh m

Cor

rela

ted

Shm

AirN2O2

ΔΔ

Δ

Figure 3.15. Comparisons of Sh numbers correlated and experimentally obtained.

r/r0

00.250.50.7510

0.05

0.1

0.15

0.2


170ml/min

θb

r/r0

00.250.50.7510

0.05

0.1

0.15

0.2


170ml/min

r/r0

00.250.50.7510

0.05

0.1

0.15

0.2


170ml/min

θb

Figure 3.16. Dimensionless bulk humidity ratios along the cell radius for air.

For ease of calculation, a multi-variable linear regression technique is used to analyze the local and mean Sh numbers, and the bulk humidity. Three correlations have been obtained, as follows:


6761.006790.08578.0

m 2Re8271.0

−

⎟⎠⎞

⎜⎝⎛ −

=δ

rrScSh (3.46)

834.0

0L 2

Re3259.0−

⎟⎠⎞

⎜⎝⎛ −

=δ

rrScSh (3.47)

630.0

0806.0358.0b 2

Re2126.0−

⎟⎠⎞

⎜⎝⎛ −

=δ

θrr

Sc (3.48)

where the validity is for gas with flow conditions of 0<Re≤200.

The maximum deviations between these correlations and the experimental and/or numerical values are 6.8%, 6.4% and 7.7% for mean Sh, local Sh and θb, respectively.

Flow Patterns Figure 3.17 gives an overview of the flow velocity vectors in the cell. The shapes are

similar to previous studies. The streamlines in a cross-section at φ=90° are shown in Figure 3.18. The flow inside the cell can be analyzed in three distinct regions: the impingement region, where the flow extends from the inlet to the bottom surface and changes from axial to radial due to the presence of the bottom surface; the radial flow region, where the air flows inwardly on the emission surface and exchanges moisture with it; and the exhaust region, where the air changes direction from radial to axial and is exhausted at the center of the cell.

φ

r

z

Figure 3.17. Three-dimensional velocity vectors in the cell for V=1.5L/min.

Li-Zhi Zhang 40

r

z

o

Figure 3.18. Streamlines representation in a cross-section (φ=90°) for V=1.5L/min.

Impingement Region The bottom surface imposes a shift in the air flow direction. The fluid decelerates in the

axial direction, losing kinetic energy that is converted into pressure energy. The deceleration starts at the inlet exit and intensifies on approaching the lower surface. The increased pressure is then primarily transformed into the radial momentum of the fluid, while some of it is transformed into flows to other directions. Due to the confinement of the cell walls and the small spacing between the cap and the lower surface, two vortices on both sides of the axis of the air inlet are generated. In other words, in the impingement region and in the vicinity of the inlet, the fluid is occupied by axisymmetric recirculating regions. The rotation axes of the rolls are perpendicular to the inlet flow direction. The recirculating zones diminish with increased angles from the air inlet. In this symmetry plane, the distance to the inlet is so long that no rolls are generated. As a consequence, no vortices can be observed in this figure.

Radial Flow Region

The flow is distributed radially in the space between the bottom surface and the cap. The cell is designed so that the radial-flow cross-section area is invariant with radial location in this region. Therefore, the bulk velocity changes little along the radius. However, the local velocity at the same altitude changes due to different duct heights. The streamlines and the local velocity contours in a horizontal cross-section at half the spacing are observed. When the air low rate is below 1000ml/min, the mean radial velocity above the emission surface is relatively uniform. It does not vary much with regard to angle and radial locations. However, when the air flow rate is further increased, the flow near the air inlet (in the area of φ=60-120°) becomes demonstrably higher than at other positions. This is obviously the influence of the air inlet.


Exhaust Region The air flows from radial to axial locations and is exhausted in the center. In this region,

the radial air velocity is very small, and the resulting mass transfer between the air and the emission surface is negligible.

Humidity Profiles Due to the small spacing of 2mm between the cap and the bottom surface, the air will

become saturated as it flows along the cell radius if the air flow rates are below 1000ml/min. To see the humidity profiles more clearly, the humidity contours for a larger air flow rate, namely, 1500ml/min, are discussed. The calculated humidity distribution in a horizontal plate at half the spacing is plotted in Figure 3.19. The humidity contours in a vertical cross-section at φ=90° are plotted in Figure 3.20. The lines in the figures are constant humidity ratio lines. For this case, the inlet temperature and humidity conditions are 23.4°C and 0.0035kg/kg, respectively. This figure shows that steep humidity gradients exist in the entry region on the emission surface. When the radius decreases, humidity gradients decrease drastically. This proves that the convective mass transfer coefficient has the largest value at the beginning of the flow and decreases with the flow’s progress. Vertically, humidity gradients are very steep near the emission surface. In addition, under such a flow rate, the air inlet will seriously influence the uniformities of the velocity and humidity fields. As can be seen, unlike the contours under small flow rates, the contours here are not shaped in concentric circles, but in irregular curves.

In this section, the fluid flow and convective mass transfer coefficients are experimentally and numerically investigated. Three correlations are summarized to calculate the mean Sh, local Sh and dimensionless bulk humidity along the chamber radius. For air flow rates below 1000ml/min, the influences of the inlet on velocity distribution on the emission surface is negligible. The velocity and humidity distributions are uniform and the local mass transfer coefficients are only functions of radial locations. For larger air flow rates, the influence of the inlet on the velocity and humidity fields becomes substantial. Overall, the local mass transfer coefficients are very large in the entrance region, and they decrease as the radius decreases, asymptotically approaching zero at the center of the cell. The flow inside the cell can be analyzed in three distinct regions: the impingement region, the radial flow region, and the exhaust region. The flow in the impingement region is rather complex: around the axis of the inlet, the flow is occupied by axisymmetric recirculating regions. The rotation axes of the rolls are perpendicular to the axis of the inlet. The flow in the other two regions is relatively simple. Under current design airflow conditions, the air becomes nearly saturated shortly after it begins to flow on the emission surface, due to the small spacing between the cap and the bottom surface.

These data provides the basics for moisture diffusivity measurement with the cell.

Li-Zhi Zhang 42

Figure 3.19. Humidity profiles in a horizontal cross-section at z=0.5mm for V=1.5L/min.

Figure 3.20. Humidity profiles in a cross-section (φ=90°) for V=1.5L/min.


3.4. MEASUREMENT OF MOISTURE DIFFUSIVITY WITH THE CELL It is well known that hydrophilic polymer membranes, say, poly cellulose acetate, poly

vinylidene fluoride, polyethersulfone, Nafion, and polyvinyl alcohol, are useful for the separation of water vapor from other gases in air mixture, because the water molecule is easily incorporated into the hydrophilic polymer membranes, due to the strong affinity between the water molecule and the hydrophilic polymers, which facilitates the transport of water, while impeding the permeation of other gases through the membranes.

Moisture transport properties in such hydrophilic polymer membranes are the most important parameters affecting the system performance and the proper design of the units. Traditionally, the measurements of water vapor diffusivity in membranes are conducted by two ways: transient drying experiments, as previously described, and permeation tests [11-13]. In the transient drying experiments, transient losses of membrane weight are recorded to calculate the effective moisture diffusivity, with the analytical solution of Fick’s second law of diffusion. Though popular and extensively used, this technique has inherent problems: the assumptions and the operating conditions for the analytical solution of Fick’s law are very rigorous and any deviation from this would lead to substantial errors [14]. On the other hand, the permeation tests, though directly measure the moisture transport through membranes at steady state, are rather complicated in the set-up. Most importantly, the obtained data are case-specific, since the convective resistance on both sides of the membrane usually plays an important role in the moisture transport performance, but was always neglected, which would make the results less accurate.

In previous section, the convective mass transfer characteristics in the cell are obtained. This section will extend the work to measure the moisture diffusivity through hydrophilic polymer membranes (or sheets) with the cell and the test rig. The difference between this section and [15,16] is that the convective mass transfer coefficients are the above obtained values. Though the methodologies have similarities.

The Whole Set-up The cell has been illustrated in Figures 3.7-3.8. In this test, distilled water is poured into

the lower chamber of the cell. Then a hydrophilic polymer membrane is covered on the lower chamber. Following this step, the cap of the cell is placed on the membrane to form a sandwiched structure. The membrane holding module is shown in Figure 3.21. A 2 mm gap between the water layer and the membrane tested is kept. The saturated solution in the lower chamber supplies a constant humidity ratio below the membrane lower surface. When the humidity ratio between the two sides of the membrane is different, moisture will diffuse through the membrane. Humid air is supplied from the inlets of the cap, which will exchange moisture with the membrane and the humidity ratio will change along the path. During the test, the temperature is kept constant. The relative humidity of the inlet and the outlet air streams are measured, and the moisture exchange effectiveness can be calculated.

Li-Zhi Zhang 44

Inlet InletOutlet

Distilled water

Cap

Lower chamber

Membrane

Inlet InletOutlet

Distilled water

Cap

Lower chamber

Membrane

Figure 3.21. Arrangement of the cell with tested membrane and water.

The cell is supplied with clean and humidified air from an auxiliary air supply unit of cell. The complete test rig is shown in Figure 3.22. The system is similar to Figure 3.11, however a membrane is sandwiched by the cell. The supply air flows from a compressed air bottle and is purified through an AC carbon column filter and is then split into two streams. One of them is humidified through a bubbler immersed in a bottle of distilled water, and then re-mixed with the other dry air stream. The outlet humidity from the bottle with bubbler reaches nearly 100%. The humidity of the mixed air stream is controlled by adjusting the ratios of air mixing by two needle valves on each stream. The desired relative humidity is obtained with this method. The airflow rates are controlled by two air pumps/controllers at the inlet and outlet of the cell. This kind of pump has a built-in digital controller which keeps the flow rates to the set points. To prevent outside air from infiltrating into the cell, a manometer is installed to monitor the pressure inside the cell and ensure that it is positive. The humidities to and from the cell are measured by RH sensors, which are installed after the pumps/controllers. The measuring accuracies are respectively 2% for relative humidity, and 2.5% for airflow rate. The total uncertainty is less than 7.5%. The design of the system allowed versatility of membrane replacement and the use of existing knowledge of flow in the cell.

Before each test, the humidity sensors and the flow meters are carefully calibrated with a chilled-mirror dew point meter (accuracy 0.1°C) and a floating ball flow meter (5ml/min), respectively. Then the distilled water is poured into the lower chamber. Following this step, the tested membrane is placed on the lower chamber and a sandwiched structure is formed with the cap placed on the membrane. Special care is taken to prevent the membrane be wetted by the water. Then the cell’s outlet and inlet are closed for 24 hours to let the membrane and the cell volume become fully equilibrium with the water.



Distilled Water

Valve

Flow Meter and Controller/Pump

Humidity Sensor

By pass

The cell with membrane and solution

Pressure Meter


Humidity SensorCompressed air Bubbler

Distilled Water

Valve


Humidity Sensor

By pass


Pressure Meter


Humidity Sensor

and water


Distilled Water

Valve


Humidity Sensor

By pass


Pressure Meter



Distilled Water

Valve


Humidity Sensor

By pass


Pressure Meter


Humidity Sensor

and water

Figure 3.22. Experimental equipment set-up for moisture transport tests.

Two hydrophilic polymer membranes, the mixed cellulose (MC, mixed cellulose and pyroxylin) and acetate cellulose (AC), supplied by a local company, are selected as test material. The reason for selecting them is that they are cheap, highly hydrophilic, and have a certain mechanical strength, which are the necessary virtue for industrial applications. A general isotherm equation can be fitted to represent the equilibrium water uptake with moisture air as

RHCCW

w/1

max

+−= (3.49)

where Wmax is the maximum water uptake of membrane material (kg/kg); C is a constant named the shape factor for the material; RH is air relative humidity.

The relative humidity is calculated by humidity ratio and temperature as [16]

RHeRH T

61.1106

/5294

−=ω

(3.50)

where T is in K. The second term on the right side of the equation will generally have less than a 5% effect, and it can be usually neglected. Within a certain low concentration range, Henry’s sorption law can be used to simplify the isotherm as

ωpkw = (3.51)

where kp is the partition coefficient, kg air/kg membrane.

Table 3.4 lists the physical properties for the membrane materials. Since the membrane fabrications have great influences on the membrane properties, they are measured in the laboratory after the materials are purchased.

Li-Zhi Zhang 46

Table 3.4. Physical properties of the tested materials

Properties Mixed cellulose (MC) Acetate cellulose (AC) Pore size (μm) 0.21 0.13 Porosity 0.72 0.86 Membrane thickness (μm) 115 105 Thermal conductivity (W/mK) 0.95 0.81 Density (kg/m3) 760 866 Sorption potential, Wmax (kg/kg) 0.27 0.46 Shape factor (C) 0.88 0.74

After the 24 hours equilibrium stage, the supply air is adjusted through the controlling

valves to the desired flow rate and the desired relative humidity. Then the valves before and after the cell are opened to let the conditioned air stream flow through the cell. In the cell, the air stream exchanges moisture with the membrane, and consequently exits the system. Relative humidities are recorded by a data logging system as the air stream begins to flow the cell. This process continues for 1-2 hours, long after the moisture transfer becomes fully steady and the outlet relative humidity reaches a stable value. The whole test is performed under room temperature conditions and it is controlled to within 0.5°C variations during the test. Constant inlet relative humidity and air flow rates are maintained by the air supply unit.

After each test, the inlet humidity and air flow rates are set to new values to perform the next experiment. Modifications to the air supply unit are done to supply larger volumetric flow rates than the original cell system can.

Model Development The governing equations for predicting the moisture transport in the emission cell are

developed. The schematic of the problem is represented in Figure 3.23. A control volume based mass balance method is employed to obtain the partial differential equations. To aid in the analysis, some assumptions are made as following:

(1) Fick’s law applies to the moisture diffusion in membrane. The thermo-physical

properties of membrane keep constant throughout the experiment. The process is isothermal.

(2) Moisture diffusion in the flow direction in the air stream is negligible compared to vapor convection by bulk flow. This assumption is true for Peclet number greater than 10.

(3) Vapor diffusion in membrane is one-dimensional and in thickness direction. This is valid considering the large dimensional differences in membrane geometry.

(4) The saturated solution is in equilibrium with membrane lower surface throughout the test process.


r0

r

z

δ

Air stream

OMembrane

r0

r

z

δ

Air stream

OMembrane

Figure 3.23. A schematic of the problem.

The moisture conservation in air stream gives:

ad

vm H

Er

ut ρ

ωω=

∂∂

+∂∂

(3.52)

where ω is the humidity ratio (kg moisture/kg air); t is time (s); um is air bulk velocity (m/s) along radius; r is radius coordinate (m); Ev is the local emission rate from the membrane to air (kgm-2s-1); Hd is duct height of air stream (m); ρa is density of dry air (kg/m3). The cell is specially designed that a constant um along the radius is realized.

Moisture diffusion in membrane

⎟⎠⎞

⎜⎝⎛

∂∂

∂∂

=∂∂

zwD

ztw

vm (3.53)

where w is moisture uptake in membrane (kg moisture/kg dry membrane); z is coordinates in membrane thickness (m); Dvm is moisture diffusivity in membrane (m2/s).

Local moisture emission rate

0z2m

=∂∂

−=z

vmv zwDE ρ (3.54)

where ρm is density of membrane (kg/m3).

Initial conditions: t=0, ω=ωL, w=wL (3.55) where ωL is the humidity ratio determined by the saturated NaCl solution and temperature

(kg/kg); and wL is the water uptake of membrane in equilibrium with the solution vapor. Boundary conditions for air stream: r=r0, ω=ωi (3.56)

Li-Zhi Zhang 48

r=0, Outflow (3.57) Boundary conditions for membrane: z=2z0, ω=ωs (3.58) z=0, ω=ωL (3.59)

where ωi is the set point of humidity ratio of inlet air, ωs is the humidity ratio on membrane surface. The relations between the humidity ratio on surface and in air stream are:

( )02

vmmsazzz

wDk=∂

∂−=− ρωωρ (3.60)

where k is the convective mass transfer coefficient (m/s).

The convective mass transfer coefficients are related to the fluid dynamics in the cell, which are studied in detail in a previous section. The deduced correlation of Eq.(3.47) are used to calculate convective mass transfer coefficients.

Re-arrangement of Eq.(3.60) and substituting it into Eq.(3.54) gives

pmvm

a

Lv

kDz

k

E

ρρ

ωω021

+

−= (3.61)

We define the total moisture transfer coefficient as

pmvm

a

kDz

k

K

ρρ 021

1

+= (3.62)

Then the Number of transfer units is defined by

NVKA

NTU t= (3.63)

where At is the transfer area of the membrane surface (m2); N is the air exchange rate (s-1); V is the volume of the cell (m3).

Similar to the analysis of a heat exchanger, a moisture exchange effectiveness can be defined as


iL

io

ωωωω

ε−−

= (3.64)

where subscripts o, i, refer to air outlet and inlet respectively, and L to saturated humidity of liquid water in lower chamber.

The transient equations for air stream, membrane and air-membrane interface are solved in a coupled way with finite difference techniques. Iterations are necessary to find a converged solution. At each time step, mass balances between the air stream, the membrane, and the air/membrane interface are ensured. The outlet air relative humidity can be calculated with moisture distributions in the cell and the local moisture emission rates on the membrane. With time lapsing, outlet RH first decreases and then reaches a stable value, indicating that moisture transport becomes stable. This stable value is called the steady state outlet relative humidity. With this value, the moisture exchange effectiveness is calculated. Figure 3.24 plots the calculated moisture exchange effectiveness from the distributed model with various Number of Transfer Units. When the NTU is from 0 to 6, ε rises sharply with NTU. When the NTU is greater than 6, there’s little merit in further increasing ε by an increase of NTU. When the membrane thermophysical properties and the experimental conditions are known, NTU can be calculated from this curve with the measured moisture exchange effectiveness. Finally, the diffusivity can be obtained.

A polynomial correlation can be fitted to represent this curve as

0.015 + 0.3759 + 0.0638 -0.0052 + 0.0002- = 2 34 NTUNTUNTUNTUε (3.65) R2 = 0.9998 (3.66)

0

0.2

0.4

0.6

0.8

1

0 2 4 6 8 10 12

NTU

ε

0

0.2

0.4

0.6

0.8

1

0 2 4 6 8 10 12

NTU

ε

Figure 3.24. The variations of the moisture exchange effectiveness with the number of transfer units.

Li-Zhi Zhang 50

Estimation of Moisture Diffusivity On the other hand, when the measured moisture exchange effectiveness, the membrane

thermophysical properties, and the experimental conditions are known, NTU can be estimated from this curve. Finally, the diffusivity can be estimated.

NTU is a dimensionless parameter that governs flow rates, exchange area, and transfer coefficient. It determines the moisture exchange effectiveness. It simultaneously reflects the characters of flow rates, area, surface resistance and membrane resistance.

Specifically in this test, under air flow rate 10L/min, the measured moisture exchange effectiveness are 0.67 and 0.81, for MC and AC membranes, respectively. As shown in Figure 3.25, the corresponding NTU for these two ε are 2.75 and 4.4, respectively. The resulting diffusivity can be then calculated from Eqs.(3.62), (3.63) as 3.1×10-10m2/s and 2.1×10-10m2/s for MC and AC membrane respectively.

The benefits with this approach are that it simultaneously considers the cell fluid dynamics, membrane configurations and operating conditions. The uncertainties of the measured values are related to the air flow rates through the cell. Under current flow rates, the estimated uncertainty is 7.5%.

Substituting parameters in this study into Eq.(3.63), the resistance for membrane is 46.3 m2s/kg, the convective mass transfer resistance is from 3.6 to 86.2 m2s/kg under 10L/min air flow rate, depending on the cell location. The two resistances are in the same magnitudes, therefore the convective mass transfer resistance cannot be neglected. The diffusion resistance in air gap below the membrane is 4 m2s/kg, one order lower than the other two, therefore this resistance can be neglected.

0

0.2

0.4

0.6

0.8

1

0 2 4 6 8 10 12

NTU

ε

MCAC

0

0.2

0.4

0.6

0.8

1

0 2 4 6 8 10 12

NTU

ε

MCAC

Figure 3.25. Estimation of moisture exchange effectiveness with number of transfer units.


0.6

0.7

0.8

0 0.5 1 1.5 2 2.5

t (s)

Out

let R

H

ACMC

Figure 3.26. Transient variations of outlet RH, flow rate 10L/min.

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0 0.2 0.4 0.6 0.8 1

1-r/r 0

Air

RH

ACMC

Figure 3.27. Bulk relative humidity of air stream along membrane cell radius.

Transient lapses of outlet RH with time for the two membranes are depicted in Figure 3.26. The volumetric air flow rates are kept at 10L/min. The discrete dots are the measured values, and the solid and dashed lines are calculated values. As can be seen from the figure, the model predicts the tested data reasonably well. Generally it takes around 1 second for the moisture transport becomes steady. After this transient period, stable transport of moisture from the solution to the air stream is established. Contrary to the assumption of a stepwise

Li-Zhi Zhang 52

square wave change of air humidity, in the transient period, the outlet air humidity changes on a slope manner. The transitional time and the degree of slope are co-determined by the initial conditions: moisture in the cell volume, and in the membrane. This indicates that in real situations, estimation of moisture diffusivity by Eqs.(3.5)-(3.7) is problematic.

Figure 3.27 shows the variations of local relative humidity of the air stream along cell radius when outlet RH becomes stable. As seen from this figure, air humidity rises continuously from inlet to outlet, indicating continuous moisture emissions from the membrane to air stream.

The local emission rates along the membrane radius are plotted in Figure 3.28 for the steady transport period. From inlet to outlet, the moisture emission rates decrease, almost linearly, along the membrane radius. Near the inlet, the two membranes have demonstrable different vapor diffusion rates, but near the outlet, the emission rates are almost the same for the two membranes. This indicates that near the inlet, emissions are controlled mainly by membrane itself, but when the air humidity increases, the influence from membrane will decrease.

To deeply disclose the distributed character of the emission rates, the water uptake contours in the membrane are drawn in Figure 3.29 for the two membranes for the steady state transfer period. The dimensionless thickness coordinates are defined as

02*

zzz = (3.67)

0.0E+00

2.0E-05

4.0E-05

6.0E-05

8.0E-05

1.0E-04

0 0.2 0.4 0.6 0.8 1

1-r/r 0

Ev (

kgm-2

s-1)

ACMC

Figure 3.28. Distributions of vapor emission rate on membrane along cell radius.


(a)

(b)

Figure 3.29. Water uptake contours in the membrane, (a) Mixed Cellulose and (b) Acetate Cellulose.

From this figure, the two-dimensional water uptake fields are clear. Water uptake gradients near the air inlet is higher than those near the air outlet. This also indicates that membranes at inlet contribute more to moisture transfer than those at outlet. In other words, large disparities of moisture transfer exist on different membrane locations.

3.5. CONCLUSION Sorption and diffusion parameters are the key basic parameters influencing heat and

moisture transfer in materials. How to estimate vapor diffusivity in membranes is one of the most important issues in membrane related technology. The measurement of sorption isotherms and diffusivity in a thermo-hygrostat is traditional. It is bulky and influenced by boundary and operating conditions. As a novel method, a standard filed and laboratory emission cell has been used to predict the moisture diffusivity in hydrophilic polymer membranes. The membrane thermophysical properties and the hydrodynamics in the cell geometry are considered simultaneously. Based on this step, the NTU-effectiveness method is then used to study the moisture transport characteristics in the unit.

With the model proposed in the study, distributions of air humidity and emission rates on membrane surface demonstrate a non-uniform character. The tested membrane’s diffusivity can be estimated from the response curve of outlet RH. For the mixed cellulose membrane, the measured diffusivity is 3.6×10-10m2/s. The water permeation potential can be reflected by an effectiveness-Number of Transfer Units curve.

Li-Zhi Zhang 54

REFERENCES

[1] Okuno, H.; Renzo, K.; Uragami, T. Sorption and permeation of water and ethanol vapors in poly (vinylchloride) membrane. J. Membrane Sci., 1995, 103, 31-38.

[2] Vagenas, G.K.; Karathanos, V.T. Prediction of the effective moisture diffusivity in gelatinized food systems. J. Food Engng., 1993, 18, 159-179.

[3] Crank, J. The Mathematics of Diffusion, Second Edition. London: Oxford University Press; 1975.

[4] Asako, Y.; Maeda, K.; Jin, Z.; Yamaguchi, Y. Effective moisture diffusivity of super absorbent polymer gel and pearlite-mortar with gel. Third Asia-Pacific Symposium on Fire Science and Technology, Singapore, 1998, 359-370.

[5] Debeaufort, F.; Voilley, A.; Meares, P. Water vapor permeability and diffusivity through methylcellulose edible films. J. Membrane Sci., 1994, 91, 125-133.

[6] Uhde, E.; Borgschulte, A.; Salthammer, T. Characterization of the field and laboratory emission cell - FLEC: flow field and air velocities. Atmospheric Environment, 1998, 32, 773-781.

[7] Zhang, L.Z.; Niu, J.L. Effects of substrate parameters on the emissions of volatile organic compounds from wet coating materials. Building and Environment, 2003, 38, 939-946.

[8] Zhang, L.Z.; Niu, J.L. Mass transfer of volatile organic compounds from painting material in a standard field and laboratory emission cell (FLEC). International Journal of Heat Mass Transfer, 2003, 46, 2415-2423.

[9] Zhang, L.Z.; Niu, J.L. Laminar fluid flow and mass transfer in a standard field and laboratory emission cell (FLEC). International Journal of Heat Mass Transfer, 2003, 46, 91-100.

[10] Fletcher, C.A.J. Computational Techniques for Fluid Dynamics I, Springer Series in Computational Physics. Berlin: Springer-Verlag, 1988.

[11] Ye, X.H.; Levan, M.D. Water transport properties of Nafion membranes Part I. Single-tube membrane module for air drying. Journal of Membrane Science, 2003, 221, 147-161.

[12] Scovazzo, P.; Burgos, J.; Hoehn, A.; Todd, P. Hydrophilic membrane based humidity control. Journal of membrane Science, 1998, 149, 69-81.

[13] Scovazzo, P.; Hoehn, A.; Todd, P. Membrane porosity and hydrophilic membrane based dehumidification performance. Journal of Membrane Science, 2000, 167, 217-225.

[14] Hernandez-Munoz, P.; Gavara, R.; Hernandez, R.J. Evaluation of solubility and diffusion coefficients in polymer film-vapor systems by sorption experiments. Journal of membrane Science, 1999, 154, 195-204.

[15] Zhang, L.Z. Investigation of moisture transfer effectiveness through a hydrophilic

polymer membrane with a field and laboratory emission cell. International Journal of Heat Mass Transfer, 2006, 49, 1176-1184.

[16] Zhang, L.Z. Evaluation of moisture diffusivity in hydrophilic polymer membranes: a new approach. Journal of Membrane Science, 2006, 269, 75-83.

Chapter 4

PERFORMANCE OF ENERGY WHEELS

ABSTRACT

When hygroscopic materials are in contact with humid air, moisture would be adsorbed by the materials. When they are in contact with dry air, the adsorbed moisture would be released. Based on this mechanism, desiccant wheels can be used in total heat recovery. In this chapter, detailed modeling of an energy wheel is conducted. Effects of material parameters on the sensible and latent effectiveness are discussed. The vapor sorption curves have a great influence on system performance.

NOMENCLATURE a Pore radius (m) As Transfer area (m2) Bi Biot number C Constant in sorption curve cp Specific heat (kJkg-1K-1) de Hydrodynamic diameter of a channel (m) DA Combined ordinary and Knudson diffusivity (m2s-1) DS Surface diffusivity (m2s-1) f Fraction of desiccant in the wheel material h Convective heat transfer coefficient (kWm-2K-1) hm Convective mass transfer coefficient (kgm-2s-1) H Specific enthalpy (kJ/kg) k Thermal conductivity (kWm-1K-1) L Length of a channel (m) Le Lewis number M1 Molecular weight (kg/kmol) md Mass of the wheel (kg)

gm Mass flow rate of air stream (kg/s)

N Rotary speed (rpm) NTU Number of transfer units

Li-Zhi Zhang 56

P Pressure (Pa) qst Isosteric adsorption heat (kJ/kg) SDP Specific Dehumidification Power (gkg-1s-1) T Temperature (K) t time (s) ug Velocity of air stream (m/s) V Volume (m3) w Water uptake in desiccant (kg water/kg dry desiccant) wmax Maximum water uptake of desiccant (kgkg-1) x, y Coordinates (m)

Greek Letters α Angle (rad) β, λ Coefficients θ Dimensionless temperature ε Effectiveness εt Total porosity φ Relative humidity δ Half thickness of channel (m) ω Moisture content (kg moisture/kg dry air) ρ Density (kg/m3) ζ Tortuosity factor τ Dimensionless time ξ Resistance coefficient

Superscripts * Dimensionless form of the variable

Subscripts a Air c Cooling, exhaust air d Desiccant, dehumidification g Gas h Heating, fresh air i Inlet L Latent m Moisture o Outlet opt Optimum s Surface, sensible w Water

Performance of Energy Wheels 57

x In x direction y In y direction

4.1. INTRODUCTION A schematic of a desiccant wheel is shown in Figure 4.1. The wheel is comprised of

parallel narrow ducts of hygroscopic materials like silica gel and LiCl impregnated paper. When wet air passes through the channels, moisture is adsorbed or desorbed from the solid channel walls. Another air stream, regenerating hot air, passes through wheel channels to regenerate the adsorbed moisture from the adsorbents.

Desiccant wheels can be used in two fields: air dehumidification [1,2] and enthalpy recovery [3-5]. In the first case, process air is dried after it flows through the wheel, which rotates continuously between the process air and a hot regenerative air stream, as described in the above figure. The dried air can either be used directly or be employed to make cooling following further psychrometric processes, i.e. the so-called desiccant cooling. In the latter case, the desiccant wheel rotates between the outside fresh air and the exhaust air from room. Here fresh air is hot and humid while the exhaust air is cool and dry. Heat and humidity would be recovered from the exhaust in winter and excess heat and moisture would be transferred to the exhaust to cool and dehumidify the process air in the summer. The latter case is also called as passive dehumidification. The wheel is called as an energy wheel. This chapter will focus on this topic.

Fresh air Dried air

Regenerating air

Fresh air Dried air

Regenerating air

Figure 4.1. A schematic of a desiccant wheel.

Li-Zhi Zhang 58

Due to the complexity in heat and mass transfer, mathematical modeling has been an efficient tool to evaluate system performance. There have been many works in modeling heat and moisture transfer in desiccant wheels. As far as this topic concerned, Simonson and Besant [6] adopted the one-dimensional solid heat conduction equation in their model to account for the longitudinal thermal resistance inside the solid. However, the model of Simonson and Besant neglected the internal moisture resistance in the solid. Zhang and Niu [4,5] proposed a dual-diffusion model that takes into account both the heat and the moisture resistance in two dimensions: axial and in thickness directions of the solid. Moisture transfer is expressed in two forms: surface diffusion and gaseous diffusion (Knudsen and ordinary combined). In this chapter, the model is similar to previous studies, but material is a newly developed hygroscopic material for energy wheels.

4.2. MATHEMATICAL MODEL The energy wheel is a rotating cylindrical wheel of length L and diameter dw and it is

divided into two sections: fresh air side cooling adsorption section (angle fraction α0) and exhaust air side heating desorption section (fraction 1-α0), where the wet fresh air and dry exhaust air streams are in a counter flow arrangement. The wheel generally consists of a matrix of numerous flow channels which have, depending on the manufacturing process, a rectangular, triangular or sinusoidal shape. The flow channels that are parallel to the axis of the wheel usually have a base material with a desiccant material impregnated on their surfaces. Modern manufacturing technology has made it possible for the desiccant material closely cross-linked to the base material and distributed evenly into the macro-voids in the base material. Therefore, in this model, the wheel is approximated by a flow channel of homogeneous composite material which has a desiccant content of f =0.6∼0.8. Because the cycles the wheel channels undergo in rotating are identical, wheel performance can be represented by a single channel. The model used is transient and two-dimensional. Because of symmetry, the mid-plane of a channel can be considered to be adiabatic, and a half-size channel surrounded by dashed line is used as the physical model, as shown in Figure 4.2.

Performance Index If the wheel is used for enthalpy recovery, two effectiveness is defined: Sensible effectiveness, εs

)()(ε

hicimin

cocics TTm

TTm−

−= (4.1)


o x

y

δ

Duct length, L

Air Stream

Desiccant

de

o x

y

δ

Duct length, L

Air Stream

Desiccant

de

Figure 4.2. A side view of one of the channels in the wheel.

Latent effectiveness, εL

)()(ε

hicimin

cocicL ωω

ωω−

−=

mm

(4.2)

where minm is the least value of process and exhaust mass flows.

Heat and mass conservation for the air stream

( )gspggge

gg

g

41 TTcud

hx

Tt

Tu

−=∂

∂+

∂

∂

ρ (4.3)

( )gsgge

mgg

g

41 ωωρ

ωω−=

∂

∂+

∂

∂

udh

xtu (4.4)

where ug is the velocity (m/s), Tg and ωg are temperature (°C) and humidity ratio (kg/kg) respectively, t is time (s), x is axial coordinate (m), de is the hydrodynamic diameter of the channel (m), ρg is the density (kg/m3) and cpg is the specific heat (kJ/kg/K), h and hm are the convective heat transfer (kWm-2K-1) and mass transfer (kgm-2s-1) coefficients between the air stream and the solid surface, respectively. In the equations, subscripts “s” and “g” refer to “surface” and “gas” respectively. By using heat mass transfer analogy, the relations between h and hm can be expressed as

Lechh

pgm = (4.5)

where Le is the Lewis number of air stream.

Li-Zhi Zhang 60

Air streams in the channels are fully-developed laminar flow. The convective heat transfer coefficients are calculated by the peripherally averaged Nusselt numbers for tubes of various cross-sectional shapes [8].

For cooling and adsorption section, i.e., 0≤α*<α0, the inlet conditions,

⎪⎩

⎪⎨⎧

=

=

=

=

ci0g

ci0g

ωωx

xTT

(4.6)

For heating and desorption section, i.e., α0≤α*<1:

⎪⎩

⎪⎨⎧

=

=

=

=

hig

hig

ωωLx

LxTT

(4.7)

where subscripts “c”, “h”, “i” and “o” refer to cooling air, heating air, inlet and outlet, respectively. The dimensionless angle α*=α/2π, and α is angle in the wheel (rad).

Enthalpy conservation in the desiccant

twq

yT

xTk

tTc

∂∂

+⎟⎟⎠

⎞⎜⎜⎝

⎛

∂∂

+∂∂

=∂

∂dst2

d2

2d

2

dd

totd ρρ (4.8)

where ρd is the density of dry desiccant (kgm-3), kd is the thermal conductivity of the solid (kWm-1K-1), w is the water content in the desiccant (kg/kg), y is the coordinate in the thickness (m), qst is the adsorption heat (kJ/kg), ctot is the total heat capacity of moist desiccant, which includes two parts: dry desiccant and adsorbed water and is calculated by

pwpdtot wccc += (4.9)

where cpd and cpw are the specific heats (kJkg-1K-1) of dry desiccant and liquid water respectively.

Two phases of water, namely, gas and adsorbed state, co-exist and diffuse in the pores of the solid. There are three dominant diffusion mechanisms [5]: surface diffusion, ordinary diffusion, and Knudsen diffusion. The first diffusion is in the form of adsorbed state and the latter two are in gas state. If Fick’s law is used to express the diffusion dynamics, the moisture conservation in the solid can be expressed as


⎥⎦

⎤⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

∂∂

+⎟⎠⎞

⎜⎝⎛

∂∂

∂∂

+

⎥⎦

⎤⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

∂∂

+⎟⎠⎞

⎜⎝⎛

∂∂

∂∂

=∂∂

+∂∂

ywD

yxwD

x

yD

yxD

xtw

t

SSd

AAadat

ρ

ωωρρωρε (4.10)

where εt is the total porosity of the desiccant. On the right hand side of Eq.(4.10), the first term is the moisture transfer in gas (combined ordinary and Knudsen diffusion), and the second term is in the adsorbed phase, namely, surface diffusion. DA and DS are the effective diffusivities (m2s-1) of the combined ordinary and Knudsen diffusion and surface diffusion, respectively. They are calculated by the following equations [5].

1

AkAO

tA

11−

⎟⎟⎠

⎞⎜⎜⎝

⎛+=

DDD

ζε

(4.11)

( )dst3

0S /10974.0exp1 TqDD −×−=ζ

(4.12)

a

685.1d9

AO 10735.1P

TD −×= (4.13)

2/1

1

dAK 97 ⎟⎟

⎠

⎞⎜⎜⎝

⎛=

MTaD (4.14)

where ζ is tortuosity factor that accounts for the increase in diffusional length due to the tortuous path of the real pores, DAO is the ordinary diffusivity, DAK is the Knudsen diffusivity, D0 is a constant for surface diffusion calculation, a is the pore radius of the adsorbent, Pa is the pressure in atmospheres (atm), Td is in K, M1 is the molecule weight of water.

Water content in the desiccant is governed by a general sorption isotherm as

φ/-1max

CCfww

+= (4.15)

where wmax is the maximum water content (kg/kg), C is a constant that determines the isotherm shape, φ is the relative humidity. Selecting T and ω as two independent variables, a differential form of adsorption content can be written in terms of humidity and temperature as

dTddw ϕωψ += (4.16)

Li-Zhi Zhang 62

where

T⎟⎠⎞

⎜⎝⎛

∂∂

=ω

ψ w (4.17)

ω⎟⎠⎞

⎜⎝⎛

∂∂

=Twϕ (4.18)

Using Clapeyron equation to represent the saturation vapor pressure and assuming a

standard atmospheric pressure of 101325Pa gives the relation between humidity and relative humidity [1] as

φωφ 61.110 /52946 −= − Te (4.19)

where T is in K. The second term on the right side of the equation will generally have less than a 5% effect, thus it can be neglected. Therefore

( ) 22max/52946

/110

φφψ

CCCfwe T

+−= − (4.20)

( ) 22max

2 /15294

φφφϕ

CCCfw

T +−−= (4.21)

where ψ reflects the slope of w to ω, ant it is a dimensionless variable. Partial differential ϕ reflects the slope of w to T, and its unit is K-1. For enthalpy recovery wheel, the bigger the ψ, the better the performance; for air dehumidification wheel, the greater the ϕ, the better the performance.

Introducing the dimensionless temperature

cihi

ci

-TTTT −

=θ (4.22)

and the dimensionless time

60tN

=τ (4.23)


as well as the dimensionless coordinates

Lxx =* (4.24)

δyy =* (4.25)

Energy equation (4.8) can be normalized to

τωθθ

τθ

∂∂

+∂∂

+∂∂

=∂∂ *

st2*

2*y2*

2*x q

yk

xk (4.26)

where

( )ϕρ sttotd2

d*x

60qcNL

kk−

= (4.27)

( )ϕρδ sttotd2

d*y

60qcN

kk−

= (4.28)

( )( )cihisttotd

dst*st TTqc

qq−−

=ϕρ

ψρ (4.29)

N is the rotary speed in rpm. The mass equation (4.10) can be normalized to

⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

+∂∂

∂∂

+⎟⎠⎞

⎜⎝⎛

∂∂

+∂∂

∂∂

=∂∂

+∂∂

*T**y*T**x*0 yyyxxx

θλωλβθλωλβτθβ

τω

ωω (4.30)

where

dat

cihid*0

)(ψρρε

ϕρβ

+−

=TT

(4.31)

ψρρλ SdAaω DD += (4.32)

( )cihiSdT TTD −= ϕρλ (4.33)

Li-Zhi Zhang 64

( )dat2x

60ψρρε

β+

=NL

(4.34)

( )dat2y

60ψρρεδ

β+

=N

(4.35)

Boundary conditions for the solid phase become:

01

*0

*0

****

=∂∂

=∂∂

=∂∂

=== xxy xxyθθθ

(4.36)

01

*0

*0

****

=∂∂

=∂∂

=∂∂

=== xxy xxyωωω

(4.37)

( )g1

**

θθθ−=

∂∂

−=

Biy y

(4.38)

( )gm1

*ω

T

1*

**

ωωθλλω

−=∂∂

−∂∂

−==

Biyy yy

(4.39)

where, Bi is the Biot number for heat transfer

dkhBi δ

= (4.40)

and Bim is the Biot number for mass transfer

ω

mm λ

δhBi = (4.41)

The heat and mass transfer equations for the air streams can be normalized as

( )gs*gg

1 θθθ

τθ

−=∂

∂+

∂

∂NTU

xc (4.42)


( )gsm*gg

1 ωωω

τω

−=∂

∂+

∂

∂NTU

xc (4.43)

where

g1 60u

NLc = pgg

s

cmhA

NTU =

g

smm m

AhNTU =

4.3. PERFORMANCE ANALYSIS The two dimensional heat and mass transfer equations of the desiccant are numerically

solved by means of ADI (alternating direction implicit) method [9]. Because the equations are strongly coupled and nonlinear, iterations are necessary to get converged values for each time step. Before numerical analysis can be performed, the physical domain of the problem as well as the equations must be discretized. The whole calculating domain is divided into a number of equal-step discrete elements. Each element is identified as a control volume by a nodal point. The numbers of nodes are: 40 in axial, 5 in thickness, and 120 in time (angle). The model has been validated by an experiment in [5]. In the following, performances with varying operating conditions will be discussed. For enthalpy recovery, the inlet temperature and humidity are set to: fresh air, 35 °C, 0.021 kg/kg; exhaust air 24 °C, 0.012 kgkg-1. The base properties for the simulated desiccant wheel are listed in Table 4.1. The desiccant material is a newly developed hygroscopic material. The geometry of the channels in the wheel is sinusoidal with a width of 4.35 mm and a height of 1.74 mm. The sorption curve of the wheel material is shown in Figure 4.3. As seen, it is a typical third class sorption curve. This kind of material is appropriate for energy recovery.

Table 4.1. Base properties of the energy wheel

Symbol Unit Value a m 11×10-10 C 6.0 D0 m2s-1 1.6×10-6 f 0.75 k Wm-1K-1 0.20 L m 0.1 md kg 15.0

gm kg/s 0.4

qst kJ/kg 2650 wmax kg/kg 0.92

Li-Zhi Zhang 66

Table 4.1. (Continued)

Symbol Unit Value α0 0.50 ρd kgm-3 1129 ξ 2.8 δ mm 0.1 εt 0.70

0

0.5

1

0 0.2 0.4 0.6 0.8 1

φ

w(k

g/kg

)

0

0.5

1

0 0.2 0.4 0.6 0.8 1

φ

w(k

g/kg

)

Figure 4.3. Sorption curve of the energy wheel material.

Temperature and Humidity Profiles

If the wheel is assumed stationary, then both temperature and humidity vary with wheel

angle. The profiles of temperature and humidity of air inlet, air outlet, air mean, and desiccant mean across a channel with dimensionless time (or the wheel angle) during enthalpy recovery are shown in Figure 4.4. As seen, both the mean temperature and the mean humidity of the air stream are higher than those of the desiccant in the adsorption section (τ=0.5∼1.0, 1.5∼2.0), and are lower than those of the desiccant in the desorption region (τ=0∼0.5, 1.0∼1.5), which discloses a fact that both the moisture and the sensible heat are transferred from the fresh air stream to the exhaust air stream. Contrary to in air dehumidification, in enthalpy recovery, heating section is also the adsorption section, and the cooling section is the desorption section. Furthermore, for balanced flows, when εS>0.5, the outlet temperature of the exhaust will be higher than that of the process air. When εL>0.5, the outlet humidity of the exhaust will be higher than that of the process air.


τ

T(°

C)

0 0.5 1 1.5 2 2.520

25

30

35

40

Air inAir outAir meanDesiccant mean

(a)

τ

Hum

idity

(kg/

kg)

0 0.5 1 1.5 2 2.50.01

0.02

0.03

Air inAir outAir meanDesiccant mean

(b)

Figure 4.4. Profiles of temperature (a) and humidity (b) of air inlet, air outlet, air mean, and desiccant mean across a channel with dimensionless time (wheel angle).

Effects of Rotary Speed

There exists an optimum rotary speed at which the efficiency reaches the climax. When a

desiccant wheel rotates much faster than the optimum speed, the adsorption and desorption processes are too short, which results in poor performance. On the other hand, when the rotary speed is lower than the optimum, the adsorption and desorption processes are too long and wasting more energy in sensible heating/cooling rather than in sorption processes, and therefore are less effective. The variations of performance with rotary speed are plotted in Figure 4.5 for enthalpy recovery purposes. The wall thickness is 0.2 mm. The figure indicates

Li-Zhi Zhang 68

that the optimum rotary speed for sensible and latent heat recovery is faster. In the following analysis, the wheels are operated at the optimum speeds.

0

0.2

0.4

0.6

0.8

0 10 20 30 40 50

N (rpm)

ε

εS

εL

0

0.2

0.4

0.6

0.8

0 10 20 30 40 50

N (rpm)

ε

εS

εL

Figure 4.5. Performance variations under various rotary speeds with a wall thickness 0.2mm.

Figure 4.6 shows the effects of wall thickness on optimum rotary speeds for total heat recovery. The wheel usually reaches the highest sensible effectiveness and latent effectiveness simultaneously at the same optimum speed. The resulting values for εS and εL at the optimum speed are also shown in this figure. Similar to air dehumidification, the optimum speed decreases as the wall thickness in the wheel increases. However, when the wall thickness is bigger than 2.0mm, the optimum rotary speed becomes insensitive to wall thickness. It is disclosed that only a fraction of the thickness takes part in the sorption-desorption working cycles for thick wheels at the optimum speed. In other words, there is an “active layer” for the wheel with thick channel walls. When the wheels are operated in the optimal modes, desiccant in the “inactive layer” has no big use for the air dehumidification and enthalpy recovery. Therefore, in practice, wheels with thin walls and large transfer areas are recommended. That’s the reason why the honeycomb type desiccant wheels are advocated in industry.


0

5

10

15

20

0.0 1.0 2.0 3.0 4.0 5.0

Wall thickness (2δ, mm)

Nop

t (rp

m)

0

0.2

0.4

0.6

0.8

1

S and

L

εS

εL

Nopt

ε

0

5

10

15

20

0.0 1.0 2.0 3.0 4.0 5.0

Wall thickness (2δ, mm)

Nop

t (rp

m)

0

0.2

0.4

0.6

0.8

1

S and

L

εS

εL

Nopt

ε

Figure 4.6. Optimum rotary speed for enthalpy recovery and the corresponding sensible and latent effectiveness with various wall thickness. The number of channels is fixed.

Effects of NTU

The number of transfer units, NTU, has a great influence on the system performance.

There are many ways to modify NTU. When the wheel mass and channel size are fixed, changing the wall thickness will change the channel numbers, and consequently the contact area and NTU. When the wheel volume and wall thickness are fixed, changing channel sizes will change the number of channels and the packing density, consequently the NTU. Figure 4.7 shows the influence of NTU on total heat recovery efficiencies. A NTU of 2.4 is needed for sensible and latent efficiencies higher than 0.61. It is noted that the latent effectiveness is usually smaller than the sensible effectiveness, because the moisture transfer resistance is usually larger than the heat transfer resistance. In other words, mass diffusion in the solid is far less than the thermal diffusion.

For desiccant wheels, when the total volume of the wheel is fixed, a higher specific area can be achieved by constructing with smaller size channels and thinner channel walls. With higher Av, the wheel will be more compact and both the transfer area and the heat mass transfer coefficients will become larger. These factors all contribute to an increased performance, as shown in Figure 4.8. As can be seen, the effectiveness rises almost linearly with increasing specific area. Therefore, in practice, the honeycomb type desiccant wheels which have large contact areas should be recommended. However, increased performance is

Li-Zhi Zhang 70

achieved at the price of increased pressure drop, as shown in Figure 4.9. The pressure drop rises rapidly with Av. In other words, fan power requirement will be increased for honeycomb wheels to realize higher performance. Eventually, a compromise between increased Av and increased pressure drop has to be maintained. It is obvious that NTU increases with Av.

0

0.2

0.4

0.6

0.8

0 2 4 6 8

NTU

Effe

ctiv

enes

sεS

εL

0

0.2

0.4

0.6

0.8

0 2 4 6 8

NTU

Effe

ctiv

enes

sεS

εL

Figure 4.7. Effects of NTU on performance for enthalpy recovery.

0

0.2

0.4

0.6

0.8

0 500 1000 1500 2000 2500

A v (m2/m3)

Effe

ctiv

enes

s

0

2

4

6

8

NTU

NTUεS

εLNTU

0

0.2

0.4

0.6

0.8

0 500 1000 1500 2000 2500

A v (m2/m3)

Effe

ctiv

enes

s

0

2

4

6

8

NTU

NTUεS

εLNTU

Figure 4.8. Effects of specific area on performance.


0

50

100

150

200

250

0 500 1000 1500 2000 2500

A v (m2/m3)

P (P

a)

Figure 4.9. Pressure drop with various specific area, gm = 0.45kg/s, md=16kg.

Psychrometric Cycle

The cycle the desiccant undergoes during a revolving is certainly of interest. The

evolutions of the states of desiccant mean and air outlet in relation to wheel angle are plotted in psychrometric charts, as shown in Figure 4.10. In the figure, the dashed lines are constant enthalpy lines and the inlet states of air streams are represented by points Hi for fresh air and Ci for exhaust air. The arrows indicate the directions of angle increasing or wheel revolving. Points Ho and Co represent the two average outlet air states for fresh air stream and exhaust air stream respectively. For the desiccant, three properties, namely, water content, air humidity in the pores, and temperature, are plotted in the graph simultaneously.

In Figure 4.10, the desiccant mean states evolve along the same line (ac) during the cooling process as during the heating process (ca), but with an opposite direction. The outlet states of air distribute along line ac for cooling process whose inlet is Ci and mean outlet is Co, and along line c’a’ for the heating process, where the inlet is Hi and the mean outlet is Ho, respectively. The two inlet states Hi and Ci are on the same line of ac or c’a’. During heating and desorption, both the temperature and the humidity increase, while during cooling and adsorption, both the temperature and the humidity decrease with wheel revolving. As a result, enthalpy increases for Ci→Co (cooling air inlet to outlet) and decreases for Hi→Ho (heating air inlet to outlet). The variation of water uptake in desiccant during a wheel working cycle is relatively very small. This is the reason why wheels for enthalpy recovery applications should be rotated much faster than those for dehumidification to realize optimized performances.

Li-Zhi Zhang 72

0.015

0.02

0.025

27 28 29 30 31

T (°C)

(kg

moi

stur

e/kg

air)

0.1

0.15

w (k

g w

ater

/kg

desi

ccan

t)

a

c

ω

w a

c

T (°C)

a

a

(a)

0

0.01

0.02

0.03

20 25 30 35 40

T (°C)

ω (k

g m

oist

ure/

kg a

ir)

a

c a’

c’

Hi

Ci

Ho

Co

T (°C)

a

c'

a'

(b)

Figure 4.10. Variations of the states of desiccant mean (a) and outlet air (b) across wheel angle for enthalpy recovery: ac, during adsorption; ca or c’a’, during desorption; dashed lines, constant enthalpy.

4.4. CONCLUSION

A two-dimensional, dual-diffusion transient heat and mass transfer model presented in

chapter is superior to other one-dimensional ones. The advantages of such a model lie in the


fact that it considers the heat conduction, the surface and gaseous diffusion in both the axial and the thickness directions simultaneously. The effects of the channel wall thickness can be investigated. Many other structural and operating parameters of the wheel could also be studied.

The temperature and humidity profiles in the wheel discloses a fact that unlike air dehumidification, in total heat recovery, heat and moisture are in phase, i.e., heat and moisture are transferred simultaneously from fresh air stream with high temperature and high humidity to the exhaust air stream with low temperature and low humidity. Since the operating conditions and the purposes are different, optimum rotary speeds for total heat recovery are different from air dehumidification. Total heat recovery is less sensitive to rotary speed than air dehumidification is. Besides, the optimum rotary speed for total heat recovery is much faster than those for air dehumidification. There is a strong influence of wall thickness on the system performance and the optimum rotary speed. The higher the thickness, the lower the optimum speed. When the weight of the wheel is fixed, increased thickness leads to poorer performance. For wheels with thick walls, only the “active layer” takes part in the processes at the optimum rotary speeds. In other words, the thinner the wall, the more effective is the desiccant used. An NTU of 2.5 is needed for desiccant wheels to have a good performance.

REFERENCES

[1] Zheng, W.; Worek, W.M. Numerical simulation of combined heat and mass transfer processes in a rotary dehumidifier. Numerical Heat Transfer, Part A: Applications, 1993, 23, 211-232.

[2] Tauscher, R.; Dinglreiter, U.; Durst, B.; Mayinger, F. Transport processes in narrow channels with application to rotary exchangers. Heat and Mass Transfer, 1999, 35, 123-131.

[3] Simonson, C.J.; Besant, R.W. Energy wheel effectiveness: part I -development of dimensionless groups. Int. J. Heat Mass Transfer, 1999, 42, 2161-2170.

[4] Zhang, L.Z.; Niu, J.L. Performance comparisons of desiccant wheels for air dehumidification and enthalpy recovery. Applied Thermal Engineering, 2002, 22, 1347- 1367.

[5] Niu, J.L.; Zhang, L.Z. Effects of wall thickness on the heat and moisture transfers in desiccant wheels for air dehumidification and enthalpy recovery. International Communications in Heat and Mass Transfer, 2002, 29, 255-268.

[6] Simonson, C.J.; Besant, R.W. Heat and moisture transfer in energy wheels during sorption, condensation, and frosting conditions. ASME Journal of Heat Transfer, 1998, 120, 699-708.

[7] Majumdar, P. Heat and mass transfer in composite desiccant pore structures for dehumidification. Solar Energy, 1998, 62, 1-10.

[8] Zhang, L.Z.; Niu, J.L. A Numerical study of laminar forced convection in sinusoidal ducts with arc lower boundaries under uniform wall temperature. Numerical Heat Transfer, Part A: Applications, 2001, 40, 55-72.

Li-Zhi Zhang 74

[9] Samarskii, A.A.; Vabishchevich, P.N. Computational Heat Transfer. New York: John Wiley & Sons Inc.; 1995.

Chapter 5

HEAT MASS TRANSFER IN BENDED SINUSOIDAL NARROW DUCTS

ABSTRACT

Convective heat mass transfer coefficients in narrow channels are the basic parameters for performance analysis of energy wheels. Fluid flow and heat transfer in channels of regular cross sectional shapes have existed in references for many years. However narrow channels in energy wheels are irregular shapes. Heat mass transfer coefficients in such channels are influenced by channel shapes. In this chapter, fluid flow and mass transfer in bended sinusoidal ducts, which are the common duct shapes in energy wheels, are numerically calculated. Sherwood numbers and friction coefficients under fully developed laminar flow are calculated.

NOMENCLATURE a Half duct height As Cross-sectional area b Half duct width Dh Hydraulic diameter Dva Diffusivity e Bending ratio of duct f Friction coefficient hL Local convective heat transfer coefficient J Jacobian operator Nu Nusselt number NuT Nusselt number for thermally fully developed laminar flow P Pressure Pe Perimeter of duct Pr Prandtl number Re Reynolds number Sh Sherwood number

Li-Zhi Zhang 76

Shω Sherwood number for mass fully developed laminar flow Sc Schmidt number T Temperature u Velocity x, y Dimensional transversal coordinates z Axial coordinate R Radius of lower boundary

Greek Letters τ Aspect ratio, duct height to width ratio δ Height of lower limit of duct ω Humidity ρ Density θ Dimensionless humidity μ Dynamic viscosity ξ, η Transversal coordinates in computational plane

Superscripts and Subscripts * Dimensionless b Bulk i Inlet m Mean w Wall ω humidity

5.1. INTRODUCTION Rotary desiccant wheels are the hearts of various total heat recovery systems. Therefore,

much effort has been devoted to develop wheels of high performance combined with low cost. The honeycomb type wheel, as shown in Figure 5.1, has drawn much attention due to its coherent two advantages: large contacting area (3000m2/m3) and compactness [1-3]. A honeycomb wheel is usually composed of numerous corrugated ducts where fresh air exchanges moisture and heat with the solid adsorbent. Then the solid adsorbent exchanges heat and moisture with exhaust air, with wheel revolving. In performances modeling, convective heat mass transfer in ducts under uniform wall temperature or humidity boundary conditions are calculated first. Then they are combined with heat mass diffusion in solid walls.

There are various duct cross sectional geometries. The most commonly encountered includes sinusoidal, triangular, rectangular, etc. Due to the small diameters, the Reynolds numbers fall into the regime of a laminar flow.

Heat Mass Transfer in Bended Sinusoidal Narrow Ducts 77


Exhaust air in

Exhaust air out


Exhaust air in

Exhaust air out

Figure 5.1. Configurations of a honeycomb energy wheel.

Heat transfers of laminar flow have been extensively studied for regularly shaped ducts. The work of Shah and London [4] contains a thorough review of heat transfer under developing and fully developed laminar flow in ducts of many cross sectional shapes. In recent years, with the progress on computational techniques, ducts of irregular shapes are increasingly investigated. Sherony and Solbrig [5] investigated the heat and mass transfers in a corrugated duct surrounded by a sine curve and a flat plate. Fischer and Martin [6] studied the friction factors in ducts confined by corrugated parallel walls. Ebadian and Zhang [7] studied the fluid flow and heat transfer in a crescent-shaped lumen catheter. Dong and Ebadian [8] provided a numerical analysis of thermally developing flow in elliptic ducts with internal fins. Fluid flow and convective heat mass transfer in ducts of various cross sectional geometries are the basic data for performance analysis. Heat transfer coefficients are relatively well documented, however mass transfer coefficients are less recorded.

x

y

2a

2bδ

Upper boundary

Lower boundary

x

y

2a

2bδ

Upper boundary

Lower boundary

Figure 5.2. Cross-section view and geometry of a bended corrugated duct in the wheel.

Li-Zhi Zhang 78

5.2. FLOW AND HEAT MASS TRANSFER MODEL The corrugated sinusoidal duct geometry is the most commonly used structure in

honeycomb wheels because it is advantageous in its simplicity of construction and large surface area. In such small diameter ducts, laminar flow prevails. The cross-sectional geometry of a corrugated sinusoidal duct is shown in Figure 5.2. It is not a common sinusoidal duct. The duct is like a duct bended from a regular sinusoidal duct. It is formed during the wheel making processes, when a stack of plate-fin sinusoidal channels are bended to form a wheel. It is observed that the single tube can be approximated with a sine curve for the upper portion and an arc for the lower portion [9,10]. Heat transfer coefficients in such ducts have been calculated before. This chapter will focus on the calculation of mass transfer coefficients in the ducts.

Because of the irregular geometry and the small size of the passages, it is very difficult to directly measure anything but overall time-mean performances. Numerical solution method becomes important and will supply much needed design information. To address the complexity of the duct geometry, the numerically generated boundary-fitted coordinate system is applied to discretize the computational domain. According to this technique, the governing equations can be solved with regular geometric methods by transforming the complex duct geometry to a regular square duct.

Basic Equations The problem considered here is that of a duct shown in Figure 5.2. The upper boundary

can be expressed as a sinusoidal function

⎥⎦

⎤⎢⎣

⎡⎟⎠⎞

⎜⎝⎛−= x

bay πcos1 (5.1)

where a is the half height of the sine duct, b is the half width of the duct.

The lower boundary is an arc, which can be expressed as

δω +−= )1(sinRy (5.2)

⎟⎠⎞

⎜⎝⎛ −

=R

bxarccosω (5.3)

⎟⎟⎠

⎞⎜⎜⎝

⎛+= δ

δ

2

21 bR (5.4)

where R is the radius of the arc (m), δ is the height of the arc (m). The value of δ can be greater or less than 0. The signs of δ for two consecutive ducts in a wheel are opposite. When


δ=0, the lower boundary becomes a flat plane. The duct becomes a common flat plate sinusoidal duct [5].

Aspect ratio of the duct

ba

=τ (5.5)

Bending ratio of the duct is defined as

be δ

= (5.6)

The flow in the duct is considered to be laminar and hydrodynamically fully developed,

but thermally developing in the entrance region of the duct. The fluid is Newtonian with constant thermal properties. Additionally, a uniform wall humidity boundary condition is considered.

Momentum equation For two-dimensional fully developed laminar flow (fluid has axial velocity only), the

Navier-Stokes equations reduce to [11,12]

dzdP

yu

xu

=⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

+∂∂

2

2

2

2

μ (5.7)

where μ is dynamic viscosity (Pa.s), u is fluid velocity (m/s), P is the pressure (Pa), z is the axial coordinate (m).

Energy equation

⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

+∂∂

=∂∂

2

2

2

2

P yT

xTk

zTucρ (5.8)

where T is the fluid humidity (K), k is thermal diffusivity(kWm-1K-1), ρ is density (kgm-3) and cP is specific heat of air (kJkg-1K-1).

Mass conservation

⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

+∂∂

=∂∂

2

2

2

2

vaωωω

yxD

zu (5.9)

where Dva is vapor diffusivity in dry air (m2s-1)

This chapter only considers mass transfer. The above equations (5.7) and (5.9) can be normalizes as

Li-Zhi Zhang 80

042h

2

2*

*22

2*

*2

=+∂

∂⎟⎠⎞

⎜⎝⎛+

∂

∂Db

y

uab

x

u (5.10)

and

2*

22

2*

2

*ya

b

xzU

∂

∂⎟⎠⎞

⎜⎝⎛+

∂

∂=

∂∂ θθθ

(5.11)

where, in the above equations, the dimensionless velocity is

2h

* μ

DdzdP

uu −= (5.12)

and dimensionless humidity is

Wi

W

ωωωω

−−

=θ (5.13)

where ωi is the inlet humidity, and ωw the humidity of duct wall.

Dimensionless coordinates

bxx

2* = (5.14)

ayy

2* = (5.15)

ReScDzz

h

* = (5.16)

where the hydraulic diameter

PeAD s

h4

= (5.17)

where As is the cross section area of the duct (m2), Pe is the perimeter of the duct (m).

In Eq.(5.11), U is a coefficient defined by


2h

2

*m

* 4Db

uuU = (5.18)

where u*

m is the average dimensionless velocity on a cross section, and it is calculated by

s

**m A

dAuu ∫∫= (5.19)

The characteristics of fluid flow and mass transfer in the duct can be represented by the

product of the friction coefficient and the Reynolds number, the dimensionless bulk humidity, and the Sherwood number.

*m

hm2m

h

21

2)(

uDu

udzdP

DfRe =⎟⎟⎠

⎞⎜⎜⎝

⎛

⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜

⎝

⎛

−=μ

ρρ

(5.20)

∫∫∫∫=

dAu

dAuz *

**

b )(θ

θ (5.21)

Local Sherwood number

va

hLL D

DkSh = (5.22)

where kL is local mass transfer coefficient (m/s). As will be discussed later, the local Sherwood number decreases asymptotically from a very high value near the entrance of a tube to the fully developed value Shω at the end of mass entry length. It can be used to calculate the local convective mass transfer coefficient and to estimate the mass entry length of a tube.

Considering the mass balance in a control volume of length Δz, mass transferred through convection

( )bWL1 ωω −Δ= zPekQ ρ (5.23)

Mass change of vapor in the fluid in the control volume

bpsm2 ωρ Δ= cAuQ (5.24)

Li-Zhi Zhang 82

Since

21 QQ = (5.25)

and

vaDDuzReScDzz

2hm*

h* ρ

Δ=Δ=Δ (5.26)

thus

*b

h

va

b

bhm

bL 4

141

zDD

zDuk

ΔΔ

−=Δ

Δ−=

θθ

θρ

θ (5.27)

Substituting above equation to Eq.(5.22), and considering the control volume to be

infinitely small, then we obtain the local Sherwood number as

*b

bL 4

1dzd

Shθ

θ−= (5.28)

Average Sherwood number from 0 to z*

**

0 L*m1 dzNuz

Shz

∫= (5.29)

Substituting Eq.(5.28) to (5.29), it is obtained

b*m ln41 θz

Sh −= (5.30)

Boundary Conditions

The flow is assumed hydrodynamically developed and mass developing. This means that the cross-sectional velocity field doesn’t change with tube length, while the humidity fields vary with tube length. For the honeycomb type desiccant wheels, strictly speaking, the tube wall is neither an ideal uniform humidity nor an ideal uniform mass flux boundary condition. However, the humidity difference on the wall is relatively small, compared with fluid humidity variations [5]. Therefore, a uniform wall humidity boundary condition (ω) is considered (ωw=const). In other words,

u*=0, θ=0 on the wall of the duct (5.31)


Inlet Condition θ=1, at z*=0 (5.32) In heat transfer, Nusselt number under uniform temperature conditions is denoted as NuT.

Another boundary condition often considered is the constant heat flux (H) boundary condition. For a heat exchanger with highly conductive materials (e.g., copper, aluminum), the H condition may apply. In practice, it may be difficult to achieve this boundary condition for noncircular ducts, as discussed in the work of Shah and London [4]. It is already known that NuH is higher than NuT for all duct geometries. For sinusoidal ducts, NuH is approximately 30% higher than NuT. In mass transfer, Sherwood number under uniform humidity boundary condition is denoted as Shω. The Sherwood number under uniform mass flux boundary condition is denoted as ShH.

5.3. BOUNDARY FITTED COORDINATES The difficulty with the complex nature of the duct shape may be circumvented by a

numerically generated coordinate system. The basic idea of the boundary fitted coordinate system is to have a coordinate system such that the body contour coincides with the coordinate lines. One of the methods often used to accomplish this goal was suggested by Thompson et al [13] and Thomas and Middlecoff [14]. The transformation between the physical coordinates (x, y) and the boundary fitted coordinates (ξ, η), which is usually a square domain, is achieved by solving two Poisson equations on (x, y) domain, namely

),(2

2

2

2

ηξξξ Pyx

=∂∂

+∂∂

(5.33)

),(2

2

2

2

ηξηη Qyx

=∂∂

+∂∂

(5.34)

where P(ξ, η) and Q(ξ, η) are grid distribution inhomogeneous functions in the computational domain. These two equations may be more easily solved on the computational plane. Therefore, Thompson et al [13] inverted Eqs.(5.33) and (5.34) into the transformed domain (ξ, η), where the boundary is easy to specify. At the same time, using the method proposed by Thomas and Middlecoff [14] for selecting P, Q, Eqs. (5.33) and (5.34) are inverted into

02 2

22

2

2

=⎥⎦

⎤⎢⎣

⎡∂∂

+∂∂

+∂∂

∂−⎥

⎦

⎤⎢⎣

⎡∂∂

+∂∂

ηψ

ηγ

ηξβ

ξφ

ξα xxxxx

(5.35)

Li-Zhi Zhang 84

02 2

22

2

2

=⎥⎦

⎤⎢⎣

⎡∂∂

+∂∂

+∂∂

∂−⎥

⎦

⎤⎢⎣

⎡∂∂

+∂∂

ηψ

ηγ

ηξβ

ξφ

ξα yyyyy

(5.36)

where

22

⎥⎦

⎤⎢⎣

⎡∂∂

+⎥⎦

⎤⎢⎣

⎡∂∂

=ηη

α yx (5.37)

ηξηξβ

∂∂

∂∂

+∂∂

∂∂

=yyxx

(5.38)

22

⎥⎦

⎤⎢⎣

⎡∂∂

+⎥⎦

⎤⎢⎣

⎡∂∂

=ξξ

γ yx (5.39)

22

2

2

2

2

⎥⎦

⎤⎢⎣

⎡∂∂

+⎥⎦

⎤⎢⎣

⎡∂∂

⎥⎦

⎤⎢⎣

⎡∂∂

∂∂

+∂∂

∂∂

−=

ξξ

ξξξξφ

yx

yyxx

(5.40)

22

2

2

2

2

⎥⎦

⎤⎢⎣

⎡∂∂

+⎥⎦

⎤⎢⎣

⎡∂∂

⎥⎦

⎤⎢⎣

⎡∂∂

∂∂

+∂∂

∂∂

−=

ηη

ηηηηψ

yx

xxyy

(5.41)

The numerical value of φ at each grid point along horizontal boundary η=ηb in terms of

boundary values x, y, is computed once the differential operators are replaced by central-difference operators in Eq.(5.40). The values of φ at internal mesh points are computed by linear interpolation along the vertical mesh lines ξ=const. Similarly, the numerical values of ψ at mesh points along vertical boundary ξ=ξb are computed through central difference of Eq.(5.41). Linear interpolations along horizontal lines η=const are performed to obtain the values of ψ at internal grids. The procedure for evaluating the parameters φ and ψ insures that the grid throughout the interior of the computational domain will be governed by the grid distribution that is assigned on the boundaries, and that the transverse grid lines will be locally orthogonal to the boundaries.

Once the values of φ and ψ are calculated, the numerical solution of Eqs.(5.35), (5.36) by standard successive line over-relaxation (SLOR) [15] on a uniform, rectangular grid Δξ, Δη results in a grid point distribution throughout the physical domain that is controlled entirely


by the distribution of grid points on the boundaries. The resulting grid constructions for the corrugated ducts are shown in Figure 5.3 (a) for e>0 and (b) for e<0 respectively. The corresponding computational domain is shown in Figure 5.3 (c). The number of nodes in the figure is 21×21. The distribution of nodes on duct boundaries is pre-arranged to ensure the dynamics in the corners of the duct are well reflected. As a result, in most of the cases, the nodes on boundaries are not evenly distributed. With this technique, the unevenly distributed nodes on boundaries can be reflected on the internal nodes.

(a)

(b)

ξ

η

0 0.5 10

0.5

1

(c)

Figure 5.3. Grid configurations, (a) the physical plane for e>0; (b) the physical plane for e<0; (c) the computational domain.

Li-Zhi Zhang 86

After the set up of boundary fitted coordinates, Eqs. (5.10) and (5.11) can be transformed to computational domain as following

04112h

2**2**

=+⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

−∂∂

⎟⎠⎞

⎜⎝⎛

∂∂

+⎥⎦

⎤⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

−∂∂

∂∂

DbJuu

ab

Juu

J ξβ

ηγ

ηηβ

ξα

ξ (5.42)

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

−∂∂

⎟⎠⎞

⎜⎝⎛

∂∂

+⎥⎦

⎤⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

−∂∂

∂∂

=∂∂

ξθβ

ηθγ

ηηθβ

ξθα

ξθ 2

*11

ab

JJzJU (5.43)

where α, β, γ have the same definition as in Eqs.(5.37)-(5.41), J is the Jacobian transformation operator, which is defined as

ξηηξηξ ∂∂

∂∂

−∂∂

∂∂

=∂∂

=yxyxyxJ

),(),(

(5.44)

Equations (5.42) and (5.43) were then discretized based on a control volume shown in

Figure 5.3. After these transformations, the governing differential equations were reduced to a set of algebraic equation systems (see Appendix), which can be solved by ADI techniques [15]. It is clear that iteration is needed to obtain the solution. In each direction, a tri-diagonal matrix is solved, by treating the non-linear cross-derivative, ∂2/∂ξ∂η, as a source term and using its values of last iteration. Although momentum equation in the finite difference form is solved only in two directions for the determination of the velocity distribution, the energy equation must be solved at every step change in the flow direction to determine the humidity distribution. The proposed finite difference schemes are implicit numerical schemes with second order accuracy. They are unconditionally stable. The following convergence criterion was chosen for the study.

∀ i, j, k, 5

,,

1,,,, 10−−

≤−m

kji

mkji

mkji

F

FF (5.45)

where F refers to the dependent variable u* or θ, respectively, m stands for the mth iteration.

Finite Difference Equations Consider a control volume represented by node (i, j, k), Eq. (5.42) can be discretized to


( ) ( )

( ) ( )

044

4

4

4

,2h

2*,1

*1,1

*,1

*1,1

2

21,

*,1

*1,1

*,1

*1,1

2

21,

2

*1,

*,

2

21,

2

*,

*1,

2

21,

*1,

*1,1

*1,

*1,1

,21

*1,

*1,1

*1,

*1,1

,21

2

*,1

*,

,212

*,

*,1

,21

=+ΔΔ

−−+⎟⎠⎞

⎜⎝⎛

⎟⎠⎞

⎜⎝⎛+

ΔΔ

−−+⎟⎠⎞

⎜⎝⎛

⎟⎠⎞

⎜⎝⎛−

Δ

−⎟⎠⎞

⎜⎝⎛

⎟⎠⎞

⎜⎝⎛−

Δ

−⎟⎠⎞

⎜⎝⎛

⎟⎠⎞

⎜⎝⎛+

ΔΔ

−−+⎟⎠⎞

⎜⎝⎛+

ΔΔ

−−+⎟⎠⎞

⎜⎝⎛−

Δ

−⎟⎠⎞

⎜⎝⎛−

Δ

−⎟⎠⎞

⎜⎝⎛

−−−+−+

−

−+−+++

+

−

−

+

+

−−−++−

−

−−++++

+

−

−

+

+

jijijijiji

ji

jijijiji

ji

jiji

ji

jiji

ji

jijijiji

ji

jijijiji

ji

jiji

ji

jiji

ji

JDbuuuu

ab

J

uuuuab

J

uuab

Juu

ab

J

uuuuJ

uuuuJ

uuJ

uuJ

ηξβ

ηξβ

ηγ

ηγ

ηξβ

ηξβ

ξα

ξα

(5.46)

Equation (5.43) can be discretized to

( ) ( ) ( )

( ) ( )

ηξθθθθβ

ηξθθθθβ

ηθθγ

ηθθγ

ηξθθθθβ

ηξθθθθβ

ξθθα

ξθθαθθ

ΔΔ

−−+⎟⎠⎞

⎜⎝⎛

⎟⎠⎞

⎜⎝⎛+

ΔΔ

−−+⎟⎠⎞

⎜⎝⎛

⎟⎠⎞

⎜⎝⎛−

Δ

−⎟⎠⎞

⎜⎝⎛

⎟⎠⎞

⎜⎝⎛−

Δ

−⎟⎠⎞

⎜⎝⎛

⎟⎠⎞

⎜⎝⎛+

ΔΔ

−−+⎟⎠⎞

⎜⎝⎛+

ΔΔ

−−+⎟⎠⎞

⎜⎝⎛−

Δ

−⎟⎠⎞

⎜⎝⎛−

Δ

−⎟⎠⎞

⎜⎝⎛=

Δ

−

−−−+−+

−

−+−+++

+

−

−

+

+

−−−++−

−

−−++++

+

−

−

+

+

−

4

4

4

4

,,1,1,1,,1,1,12

21,

,,1,1,1,,1,1,12

21,

2,1,,,

2

21,

2,,,1,

2

21,

,1,,1,1,1,,1,1

,21

,1,,1,1,1,,1,1

,21

2,,1,,

,212

,,,,1

,21*

1,,,,,

kjikjikjikji

ji

kjikjikjikji

ji

kjikji

ji

kjikji

ji

kjikjikjikji

ji

kjikjikjikji

ji

kjikji

ji

kjikji

ji

kjikjiji

ab

J

ab

J

ab

Jab

J

J

J

JJzJU

(5.47)

Li-Zhi Zhang 88

where

21,1,

21,1,

, 22 ⎟⎟⎠

⎞⎜⎜⎝

⎛Δ

−+⎟⎟

⎠

⎞⎜⎜⎝

⎛Δ

−= −+−+

ηηα jijijiji

ji

yyxx (5.48)

⎟⎟⎠

⎞⎜⎜⎝

⎛Δ

−⋅⎟⎟

⎠

⎞⎜⎜⎝

⎛Δ

−+⎟⎟

⎠

⎞⎜⎜⎝

⎛Δ

−⋅⎟⎟

⎠

⎞⎜⎜⎝

⎛Δ

−= −+−+−+−+

ηξηξβ

22221,1,,1,11,1,,1,1

,jijijijijijijiji

ji

yyyyxxxx (5.49)

2

,1,12

,1,1, 22 ⎟⎟

⎠

⎞⎜⎜⎝

⎛Δ

−+⎟⎟

⎠

⎞⎜⎜⎝

⎛Δ

−= −+−+

ξξγ jijijiji

ji

yyxx (5.50)

⎟⎟⎠

⎞⎜⎜⎝

⎛Δ

−⋅⎟⎟

⎠

⎞⎜⎜⎝

⎛Δ

−−⎟⎟

⎠

⎞⎜⎜⎝

⎛Δ

−⋅⎟⎟

⎠

⎞⎜⎜⎝

⎛Δ

−= −+−+−+−+

ξηηξ 2222,1,11,1,1,1,,1,1

,jijijijijijijiji

ji

yyxxyyxxJ

(5.51)

i, j

i, j+1 i+1, j+1

i+1, j

i-1, j+1

i-1, j

i, j-1 i+1, j-1i-1, j-1

ξ

η

i, j

i, j+1 i+1, j+1

i+1, j

i-1, j+1

i-1, j

i, j-1 i+1, j-1i-1, j-1

ξ

η

(a)

i, j

i, j+1 i+1, j+1

i+1, j

i-1, j+1

i-1, j

i, j-1 i+1, j-1i-1, j-1

x*

y*

i, j

i, j+1 i+1, j+1

i+1, j

i-1, j+1

i-1, j

i, j-1 i+1, j-1i-1, j-1

x*

y* (b)

Figure 5.4. Grid numbering. (a) computational domain; (b) physical domain.


In equations (5.46) to (5.51), the subscripts (i, j, k) represent the ξ, η and z* directions in the computational domain. The grid nodes (i, j) in the computational domain (ξ, η) and in the physical domain are presented in Figure 5.4. The values at the boundaries of a control volume are obtained by interpolation of values at nodes, for example, Ji, j+1/2 are calculated from Ji, j and Ji, j+1.

5.4. FRICTION AND MASS TRANSFER COEFFICIENTS

Validation of the Procedure To assure the accuracy of the results presented, numerical tests were performed for the

duct to determine the effects of the grid size. It indicates that 21×21 grids on cross section and Δz*=0.00035 axially are adequate (less than 0.1% difference compared with 31×31 grids and Δz*=0.00025). For hydrodynamically fully developed laminar flow in ducts, (fRe) is a constant and in the mass entry region, the local Sherwood number will decrease and approach asymptotically to a lower limiting value Shω with the marching of flow.

Mass transfer coefficients for various cross sectional shapes are scarce. To validate the procedure, (fRe) and NuT for some ducts are calculated and compared with the results found in references. It should be known that the dimensionless mass conservation equation is the same form with heat conservation equation. Therefore heat transfer equation is first used to validate the model. The comparisons are listed in Table 5.1.

From this table, it can be concluded that maximum errors are less than 0.8% for (fRe) and less than 0.9% for NuT.

Table 5.1. Comparisons of (f⋅Re) and NuT of fully developed laminar flow for some ducts

from present case and those from literature

Shape τ (f⋅Re) NuT

Present case

Refs[4,5] Error (%)

Present case

Refs[4,5] Error (%)

Circular 16.151 16 0.51 3.692 3.657 0.41

Square 14.225 14.227 0.78 3.061 2.976 0.84 Elliptic 0.5 16.931 16.823 0.70 3.676 3.742 0.64 Equilateral Triangular 13.326 13.321 0.11 2.496 2.46 0.65 Isosceles Triangular 0.5 13.147 13.153 0.18 2.331 2.34 0.90 Sine 2.0 14.476 14.553 0.16 2.658 Unavailable Sine 1.5 13.964 14.022 0.63 2.614 2.6 0.54 Sine 1.0 12.912 13.023 0.77 2.463 2.45 0.53 Sine 0.75 12.326 12.234 0.75 2.317 2.33 0.56 Sine 0.50 11.163 11.207 0.30 2.155 2.12 0.71

Notes: τ, aspect ratio (duct height/duct width); for the Sine ducts listed: e=0 (flat Sinusoidal).

Li-Zhi Zhang 90

Effects of Bending Ratios For e=0, the corrugated ducts reduce to sine ducts with flat lower boundaries whose

results are listed in Table 5.1. The values of fRe and NuT are in excellent agreement with the published data. The code can be used to calculate mass transfer coefficients. In practical desiccant wheels, e would be greater or less than 0, especially when the ducts are in zones of small diameters. For these ducts, the friction and mass transfer coefficients are affected by bending ratio e. The values of (f⋅Re) and Shω for various bending ratios and aspect ratios are listed in Table 5.2.

Table 5.2 shows that the greater the e, the smaller the (f⋅Re) and the Shω. This character in return discloses that the higher the friction coefficients, the higher the Sherwood numbers. Generally speaking, e>0 has positive effects on friction coefficients, but negative effects on Sherwood numbers. On the other hand, e<0 would increase friction coefficients and Sherwood numbers simultaneously, compared to ducts with flat lower boundaries. These phenomena are attributed to the fact that the bigger the e, the larger the dead spaces in the corners of ducts. The larger the dead zones, the more inefficient of the transfer area, which results in decreased friction coefficients and Sherwood numbers.

Table 5.2. Values of (f⋅Re) and Shω with various aspect and bending ratios

τ=2.0 e -0.5 -0.25 0 0.25 0.5 (f⋅Re) 15.19 15.146 14.576 13.308 11.635 Shω 2.715 2.675 2.558 2.289 2.002 τ=1.0 e -0.5 -0.25 0 0.25 0.5 (f⋅Re) 14.738 14.209 12.922 11.281 8.298 Shω 2.966 2.735 2.363 1.895 1.228 τ=0.75 e -0.5 -0.25 0 0.25 0.5 (f⋅Re) 15.027 13.945 12.326 10.064 7.09 Shω 2.981 2.652 2.217 1.562 0.867 τ=0.65 e -0.5 -0.25 0 0.25 0.45 (f⋅Re) 15.366 13.975 12.117 9.701 6.931 Shω 3.008 2.605 2.071 1.4 0.778 τ=0.5 e -0.5 -0.25 0 0.25 0.35 (f⋅Re) 16.803 14.477 11.173 9.206 7.485 Shω 3.105 2.522 2.035 1077 0.757


τ=0.4 e -0.5 -0.25 0 0.2 (f⋅Re) 15.717 12.819 10.221 8.291 Shω 3.232 2.473 1.783 1.03

The effects of e on (fRe) are shown in Figure 5.5. and the effects on Shω are shown in

Figure 5.6. Figure 5.5 shows the variations of (fRe)/(fRe)0 (where subscript 0 means e=0) with various e for different aspect ratios. It is seen that the greater the e, the smaller the (fRe). The ratio can be as low as 0.5 when e=0.5 and as high as 1.5 when e=-0.5.

0.5

1

1.5

2

-0.5 -0.25 0 0.25 0.5

e

(fR

e)/(f

Re)

0

τ=2.0τ=1.0

τ=0.75τ=0.4

0.5

1

1.5

2

-0.5 -0.25 0 0.25 0.5

e

(fR

e)/(f

Re)

0

τ=2.0τ=1.0

τ=0.75τ=0.4

Figure 5.5. Effects of duct bending ratios on friction coefficients.

0

0.5

1

1.5

2

-0.5 -0.25 0 0.25 0.5e

Nu T

/Nu T

0

τ=2.0τ=1.0τ=0.75τ=0.4

Shω/S

h ω0

e

0

0.5

1

1.5

2

-0.5 -0.25 0 0.25 0.5e

Nu T

/Nu T

0

τ=2.0τ=1.0τ=0.75τ=0.4

Shω/S

h ω0

e

Figure 5.6. Effects of duct bending ratios on Sherwood numbers.

Li-Zhi Zhang 92

0. 25

0.25

0.25

0.25

0.25

0.75

0.75

0.75

1.25

1.25

1.25

1.75

1.75 2.25

x*

y*

0 0.25 0.5 0.75 10

0.25

0.5

0.75

1

(a) τ=0.5, e=0.14.

0.25

0.25

0.25

0.25

0.25

0.75

0.75

0.75

1.25

1.25

1.25

1.75

1.75

2.25

x*

y*

0 0.25 0.5 0.75 1-0.25

0

0.25

0.5

0.75

1

(b) τ=0.5, e=-0.14.

Figure 5.7. Fully developed velocity profile. The isoclines are lines of constant u*/u*m.

Figure 5.6 shows the variations of Shω/ Shω0 (ratio of the Sherwood number for bending duct to that for e=0 with the same aspect ratio) for different aspect ratios. The smaller the bending ratio, the higher the Sherwood number. The comparisons of Figures 5.5 and 5.6 also disclose that the higher the friction coefficients, the higher the Sherwood numbers. Generally speaking, e>0 has positive effects on friction coefficients, but negative effects on Sherwood numbers. On the other hand, e<0 would increase friction coefficients and Sherwood numbers simultaneously, compared to ducts with flat lower boundaries. To know the reason why, let’s plot velocity fields in Figure 5.7 and humidity fields in Figure 5.8. The comparisons of Figures 5.7(a), 5.7(b) and 5.8(a), 5.8(b) show that the greater the e, the larger the dead spaces in the corners. The larger the dead zones, the more inefficient of the transfer area, which


results in decreased friction coefficients and Sherwood numbers. It is interesting to note that the shapes of isotherms and iso-velocities are very similar to those of triangular ducts, however, the maximum velocity and humidity occur closer to the center than those of the triangular ducts [4] for same aspect ratios (duct height to width ratio). The shapes of isolines change from triangle with round corners near the boundary to circles at the center gradually. Furthermore, both the velocity gradients and the humidity gradients have their highest values near straight boundaries, while minima in the corners. These phenomena are also in agreements with the holographic interferometric observations of humidity fields in such ducts [16].

0.10

0.10

0.10

0.30

0.30

0.300.50

0.50

0.70

x*

y*

0 0.25 0.5 0.75 10

0.25

0.5

0.75

1

(a) e=0.14

0.10

0.10

0.10

0.10

0.30

0.30

0.30 0.50

0.50

x*

y*

0 0.25 0.5 0.75 1-0.25

0

0.25

0.5

0.75

1

(b) e=-0.14

Figure 5.8. Equal concentration curves for τ=0.5, at z*=0.1. The values are dimensionless humidity θ.

Li-Zhi Zhang 94

Local and Mean Sherwood Numbers The axial variations of bulk humidity are shown in Figure 5.9 for τ=1.0. Inspection of the

curves in this figure reveals that the bulk humidity is strongly variant with e at the entrance region of the duct. However, as the air passes through the duct, the bulk humidity is dependent on the value of e. Positive e results in a bigger bulk humidity than a negative e at the same z* position, suggesting a decreased mass transfer rate. Besides, a bigger bulk humidity would finally lead to a longer thermal entry length.

Local Sherwood numbers against z* for τ=1.0 are presented in Figure 5.10. It is seen that ShL decreases from a high value near the entrance to the fully developed value Shω at a greater axial distance. This figure also illustrates that the local Sherwood number decreases dramatically at the entrance region of the duct as the bending ratio increases. However, this variation decreases gradually until it reaches the asymptotic limiting value. Generally speaking, positive e has greater impacts than negative e do.

Figure 5.11 shows the variations of mean Sherwood numbers along the flow. The mean Sherwood numbers are higher than the local values at the same position. However, as z* increases, the mean values will decrease gradually to the limit values of Shω. The effects of bending ratio e on the mean Sherwood numbers are very similar to those on the local values.

The variations of bulk humidity, the local and mean Sherwood numbers along the tube length for other aspect ratios are similar in shape to those for τ=1.0. The differences lie in the length of the thermal entry region. The sharper the corner of a duct, the longer is the tube from the entrance to the fully developed point. Most importantly, duct shape mainly influences the cross-sectional velocity and humidity profiles in the duct, as shown in Figures 5.7 and 5.8. It is disclosed that the sharper the corner, the larger the dead space, the smaller the Sherwood number.

z*10-4 10-3 10-2 10-1 1000

0.2

0.4

0.6

0.8

1

e=-0.25e=0.0e=0.25

Figure 5.9. Axial variations of bulk humidity, τ=1.0.


z*

Nu L

10-4 10-3 10-2 10-10

5

10

15

20

e=-0.25e=0.0e=0.25

ShL

z*

Nu L

10-4 10-3 10-2 10-10

5

10

15

20

e=-0.25e=0.0e=0.25

ShL

Figure 5.10. Axial variations of local Sherwood numbers, τ=1.0.

z*

Nu m

10-4 10-3 10-2 10-10

10

20

30

40

50

e=-0.25e=0.0e=0.25

Shm

z*

Nu m

10-4 10-3 10-2 10-10

10

20

30

40

50

e=-0.25e=0.0e=0.25

Shm

Figure 5.11. Axial variations of mean Sherwood numbers, τ=1.0.

Li-Zhi Zhang 96

Besides bended sinusoidal cross sectional ducts, there are ducts of other cross sections. The friction coefficients and Sherwood numbers of developed flow in other cross sectional shapes, under uniform humidity conditions and uniform mass flux boundary conditions cab be calculated.

5.5. CONCLUSION Convective mass transfer and fluid flow in corrugated ducts confined by sinusoidal and

arc curves are analyzed numerically for various combinations of aspect and bending ratios with uniform wall humidity conditions. The boundary-fitted coordinate is used to solve the difficulty induced by the complex physical domain. The velocity and humidity fields are calculated and graphically illustrated to investigate the effects of sharp corners in the ducts. It is found that bending ratio, e, has a great influence on both the friction coefficients and the Sherwood numbers. The product of friction coefficient and Reynolds number (fRe) could drop 50% when e=0.5 and rise 50% when e=-0.5, compared to ducts with flat lower boundaries. Positive e would decrease the Sherwood number and negative e would increase the Sherwood number significantly. Besides, bending ratios other than zero could also affects the thermal entry length, local Sherwood numbers, and humidity/velocity profile shapes. All these are due to the fact that the greater the bending ratios, the larger the dead spaces, for both the fluid flow and the mass transfer. The results can be used to analyze the pressure drop and mass transfer properties in rotary energy wheels.

REFERENCES

[1] Jin, W.; Kodama, A.; Goto, M.; Hirose, T. An adsorptive desiccant cooling using honeycomb rotor dehumidifier. Journal of Chemical Engineering of Japan, 1998, 31, 706-713.

[2] Zheng, W.; Worek, W.M. and Novosel, D. Performance optimization of rotary dehumidifiers. ASME Journal of Solar Energy Engineering, 1995, 117, 40-44.

[3] Kodama, A.; Goto, M.; Hirose, T.; Kuma, T. Experimental study of optimal operation for a honeycomb adsorber operated with thermal swing. Journal of Chemical Engineering of Japan, 1993, 26, 530-535.

[4] Shah, R.K.; London, A.L. Laminar flow forced convection in ducts. New York: Academic Press Inc.; 1978.

[5] Sherony, D.F.; Solbrig, C.W. Analytical investigation of heat or mass transfer and friction factors in a corrugated duct heat or mass exchanger. International Journal of Heat Mass Transfer, 1970, 13, 145-159.

[6] Fischer, L.; Martin, H. Friction factors for fully developed laminar flow in ducts confined by corrugated parallel walls. International Journal of Heat Mass Transfer, 1997, 40, 635-639.

[7] Ebadian, M.A.; Zhang, H.Y. Fluid flow and heat transfer in the crescent-shaped lumen catheter. ASME Journal of Applied Mechanics, 1993, 60, 721-727.


[8] Dong, Z.F.; Ebadian, M.A. A numerical analysis of thermally developing flow in elliptic ducts with internal fins. International Journal of Heat and Fluid Flow, 1991, 12, 166-172.

[9] Niu, J.L.; Zhang, L.Z. Heat transfer and friction coefficients in corrugated ducts confined by sinusoidal and arc curves. International Journal of Heat Mass Transfer, 2002, 45, 571-578.

[10] Zhang, L.Z.; Niu, J.L. A numerical study of laminar forced convection in sinusoidal ducts with arc lower boundaries under uniform wall temperature. Numerical Heat Transfer, Part A: Applications, 2001, 40, 55-72.

[11] Shah, R.K. Laminar flow friction and forced convection heat transfer in ducts of arbitrary geometry. International Journal of Heat Mass Transfer, 1975, 18, 849-862.

[12] Kays, W.M. and Crawford, M.E. Convective Heat and Mass Transfer. New York: McGraw-Hill, Inc.; 1993.

[13] Thompson, J.F.; Thames, F.; Martin, C. Automatic numerical generation of body-filled curvilinear coordinate system for field containing any number of arbitrary two-dimensional bodies. Journal of Computational Physics, 1974, 24, 299-319.

[14] Thomas, P. D.; Middlecoff, J.F. Direct control of grid point distribution in meshes generated by elliptic equations. AIAA Journal, 1982, 18, 652-656.

[15] Samarskii, A.A.; Vabishchevich, P.N. Computational Heat Transfer. New York: John Wiley & Sons Inc.; 1995.

[16] Tauscher, R.; Dinglreiter, U.; Durst, B.; and Mayinger, F. Transport processes in narrow channels with application to rotary exchangers. Heat and Mass Transfer, 1999, 35, 123-131.

Chapter 6

CONVECTIVE HEAT MASS TRANSFER IN PLATE-FIN CHANNELS

ABSTRACT

Plate-fin structure is the most common structure for stationary total heat recovery. Common traditional air-to-air sensible only heat exchangers use well conductive metal foils as the heat transfer media. For total heat exchanger, novel water-permeable materials are used to simultaneously permeate heat and moisture. They are less-conductive materials both for heat transfer and mass transfer. In this chapter, the influences of finite fin conductance on heat and mass transfer in plate-fin channels were investigated.

NOMENCLATURE a Half duct height (m) Ac Cross-sectional area (m2) At Transfer area (m2) b Half duct width (m) cP Specific heat (kJkg-1K-1) Dh Hydraulic diameter (m) Dva Vapor Diffusivity in air (m2/s) Dwf Vater Diffusivity in fin material (m2/s) f Friction coefficient h Convective heat transfer coefficient (kWm-2K-1) j Chilton-Colburn j factor k Mass transfer coefficient (m/s) kp Partition coefficient L Length (m) Le Lewis number Nu Nusselt number NuT Nusselt number for thermally fully developed laminar flow with T condition

Li-Zhi Zhang 100

P Pressure (Pa) Pf Perimeter of duct (m) Pr Prandtl number q Heat flux (kWm-2) Q Total heat transfer (kW) Re Reynolds number RH Relative humidity s Tangent coordinate for fin (m) Sc Schmidt number Sh Sherwood number Shω Fully developed Sherwood number under uniform concentration condition St Stanton number T Temperature (K) u Velocity (ms-1) U Velocity coefficient W Water uptake (kg moisture/kg material) x, y Dimensional transversal coordinates (m) yf Perpendicular coordinate for fin (m) z Axial coordinate (m)

Greek Symbols δ Fin thickness (m) ρ Density (kgm-3) θ Dimensionless temperature μ Dynamic viscosity (kgm-1s-1) Ω Conductance parameter ψ Correction factor of temperature or humidity difference for cross flow λ Heat conductivity (kWm-1K-1) ηfin Fin efficiency ξ Dimensionless humidity ω Humidity ratio (kg vapor/kg dry air) τ Aspect ratio α Half apex angle

Superscripts • Dimensionless

Subscripts

a Air b Bulk e Exhaust air f Fin, fresh air

Convective Heat Mass Transfer in Plate-fin Channels 101

i Inlet L Local, lower surface, moisture (latent heat) m Mean s Sensible heat tot Total u Upper surface w Wall

6.1. INTRODUCTION Energy wheels have the ability to recover both sensible heat and moisture from

ventilation air. However, there are some inherent shortcomings with desiccant wheels. They have moving parts. The wheel rotates alternately between the fresh air and the exhaust air, which leads to crossovers between the fresh air and exhaust air. Therefore it’s inevitable that the fresh air will be polluted by the exhaust air. That fails to meet the requirements for ventilation. In addition, desiccant wheels are very expensive and their maintenance is difficult. The long-term reliability is questionable because of the cyclic nature of the system. The wheel material’s ability to adsorb moisture deteriorates steadily with time. These factors have restricted their developments in real applications.

In contrast, the stationary total heat exchangers are superior in that they are used in steady-state adsorption and permeation status. They have no moving parts and the reliability is quite high. The duct sealing is easy and the crossovers can be prevented. The difference between a total heat exchanger and a common sensible-only heat exchanger is that water-permeable plates are used in stead of metal plates. Currently there are two types of vapor-permeable materials: paper and hydrophilic polymer membranes. Paper is cheap, but its moisture efficiency is lower than 0.4. Membranes have higher latent effectiveness above 0.65. The shortcomings are that they are expensive than paper. Therefore there are many researches now to develop new membranes from cheap materials. Whether it’s paper or other vapor-permeable material, heat and moisture transfer in the enthalpy exchangers are of interest.

As other air-to-air heat exchangers, plate-fin channels are the most popular structure for total heat exchangers. Figure 6.1 shows a real photo of two plate-fin heat exchangers. Figure 6.2 shows the schematic of the plate-fin heat exchanger. In the figure, the cross-section for a single channel is sinusoidal. Other popular geometries include triangular, rectangular, etc. Usually a cross flow between fresh air and exhaust air is used, due to the convenience in duct sealing.

The reason why plate-fin ducts are selected as the basic structure lies in several facts: (1) They are easy to construct. Especially the corrugated fins are easy to manufacture in large scale by machines; (2) The mechanical strength is very high even with very thin plates; (3) Heat transfer intensification is needed on both sides of a plate, therefore equal structure and area are realized on both sides; (4) Duct sealing are easy to realize because of the multi-points contact between a fin and a neighboring plate. (5) Packing density is rather high because the channel height can be very small (1-2mm) as a result of plates separating by a corrugated fin.

Li-Zhi Zhang 102

(a) Sinusoidal duct.

(b) Rectangular duct.

Figure 6.1. A photo of two plate-fin heat exchangers.


Figure 6.2. A schematic of a sinusoidal plate-fin heat exchanger.

6.2. SINUSOIDAL DUCTS OF FINITE FIN CONDUCTANCE

Heat and mass transfer in plate-fin total heat exchangers can be separated into three steps:

(1) convective heat mass transfer from fresh air to plates and fins; (2) conductive heat and mass transfer in plates; (3) convective heat mass transfer from plates and fins to exhaust air. Of the heat mass transfer parameters, convective heat mass transfer coefficients in plate-fin duct are the key parameter to analyze performance and product design.

Convective Plate-fin ducts with traditional well-conductive metal walls have been studied by many authors. A comprehensive review of the theoretical and experimental studies on fully developed forced convection and heat transfer in ducts of various cross sections up to the 1970s had been conducted by Shah and London [1], and documented in several well known references [1-3]. The results have been regarded as the basic data for heat exchanger design for these years. Strictly speaking, plate-fin ducts are different from common ducts of uniform wall temperature. In a heat exchanger, plates contact with air streams directly, but fins don’t. Heat is transferred by conduction from plates to fins first, then it is transferred from fins to air stream by convection. It should be noted that real fins have limited conductance, which may comprise fin efficiencies. The effects of finite fin conductance on heat transfer have been investigated by several authors [4]. However, nowadays, with the application of many new heat transfer materials like paper and polymers to total heat recovery, the effects of finite fin conductance are very large. The fin conductance of such materials is very low.

The problem aforementioned will be addressed in this chapter. The difference between this chapter and previous studies is that beside heat transfer, mass transfer in fins will be considered either. To overcome the non-rectangular nature of the duct cross section, as before, a boundary-fitted coordinate technique will be used to transform the physical domain to a computational domain. Another benefit with this technique is that the program written can be easily modified to study ducts of other cross sectional shapes. Sinusoidal plate-fin duct will first be considered.

Li-Zhi Zhang 104

x

y

z

2b

2aL

Flow

sy1

x

y

z

2b

2aL

Flow

syf

A

D

C x

y

z

2b

2aL

Flow

sy1

x

y

z

2b

2aL

Flow

syf

x

y

z

2b

2aL

Flow

sy1

x

y

z

2b

2aL

Flow

syf

A

D

C

Figure 6.3. Geometry of a plate-fin sinusoidal duct.

Governing Equations

The problem considered here is that of a representing duct shown in Figure 6.3. The

geometries of the sinusoidal duct are also depicted in the figure: height 2a; width 2b; duct length, L. Fin curve can be expressed as a sinusoidal function

⎥⎦

⎤⎢⎣

⎡⎟⎠⎞

⎜⎝⎛ π

−= xb

ay cos1 (6.1)

Aspect ratio of duct

ba

22

=τ (6.2)

Duct Area

∫=b

ydxA2

0c (6.3)

Length of each fin, AD or DC

∫ =+=

b

xdydxL

0

22f )()( (6.4)

Perimeter of duct

ff 22 LbP += (6.5) Hydraulic diameter


f

ch

4PA

D = (6.6)

where Ac is the cross section area of the duct (m2), Pf is the perimeter of the duct (m). The flow in the duct is considered to be laminar and hydrodynamically fully developed, but thermally developing in the entrance region of the duct. The fluid is Newtonian with constant thermal properties. Additionally, a uniform wall temperature boundary condition is considered for the plate (AB and BC). The Nusselt numbers under uniform heat flux boundary conditions are usually 20-30% higher than those under uniform temperature conditions [1,2].

The governing equations are summarized as following [5,6]. Momentum

dzdP

yu

xu

=⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

+∂∂

μ 2

2

2

2

(6.7)

where μ is dynamic viscosity (Pa.s), u is fluid velocity (m/s), P is the pressure (Pa), z is the axial coordinate (m), P is pressure (Pa).

Energy conservation

⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

+∂∂

=∂∂

2

2

2

2

P yT

xT

zTuc aλρ (6.8)

where T is fluid temperature (K), λa is thermal conductivity of air (kWm-1K-1), ρ is density (kg/m3), cp is specific heat (kJkg-1K-1).

The above two governing equations can be normalized to

042h

2

2*

*22

2*

*2

=+∂

∂⎟⎠⎞

⎜⎝⎛+

∂

∂Db

yu

ab

xu

(6.9)

2*

22

2*

2

* yab

xzU

∂

θ∂⎟⎠⎞

⎜⎝⎛+

∂

θ∂=

∂θ∂

(6.10)

with a dimensionless velocity

2h)/(

*DdzdP

uu μ−= (6.11)

and a dimensionless temperature

Li-Zhi Zhang 106

wi

w

TTTT

−−

=θ (6.12)

where in the equations, Ti is the inlet temperature of the fluid, and Tw is the wall temperature.

Dimensionless coordinates are defined by

bxx

2* = (6.13)

ayy

2* = (6.14)

RePrDzz

h

* = (6.15)

where Re is Reynolds number and Pr is Prandtl number.

In Eq.(6.10), velocity coefficient U is defined by

2h

2

*

* 4Db

uuU

m

= (6.16)

where u*


c

**

A

dAuum

∫∫= (6.17)

The characteristics of fluid flow in the duct can be represented by the product of the

friction coefficient and the Reynolds number as

*hm

2m

h

21

2)(

muDu

udzdPD

fRe =⎟⎟⎠

⎞⎜⎜⎝

⎛μ

ρ

⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜

⎝

⎛

ρ−= (6.18)

Dimensionless bulk temperature

∫∫∫∫ θ

=θdAu

dAu*

*

b (6.19)


Nusselt number

a

hDNuλ

h= (6.20)

where h is convective heat transfer coefficient (kWm-2K-1) between fluid and wall.

An energy balance in a control volume in the duct [5] will give the equation for estimation of the local Nusselt number as

*b

bL 4

1dzd

Nuθ

θ−= (6.21)

and the mean Nusselt number from z*=0 to z* by

b*m ln41

θ−=z

Nu (6.22)

Fins

The coordinate system for two fins (AD and DC) is s, y1, and shown in Figure 6.4. Axis s is tangent to fin surface and y1 is normal to fin surface. The directions vary from point to point. At any location along the fin, there is a balance between the net conduction along the fin and the heat transfer from the surface of the fin to the fluid. Heat transfer in fin is governed by the following one-dimensional model [4,6]

Lu2f

2

fλ qqds

Td+=δ (6.23)

ufau λ ⎟⎟

⎠

⎞⎜⎜⎝

⎛∂∂

−=yTq (6.24)

LfaL λ ⎟⎟

⎠

⎞⎜⎜⎝

⎛∂∂

=yTq (6.25)

where ( )f/ yT ∂∂ is the normal gradient of fluid temperature on the lower or upper surface of fin. The heat flux at the lower surface and the upper surface are skew symmetric, as schematically depicted in Figure 6.4. The relation is mathematically expressed by

sLsqq

−=

fLu (6.26)

where Lf is the length of one curved fin and it is calculated numerically by Eq.(6.4).

Li-Zhi Zhang 108

A

Dqu

qL

Fin

Center

A

Dqu

qL

Fin

Center

Figure 6.4. Skew-symmetric heat flux distributions on the upper and lower fin fluid interfaces.

Equations (6.23)-(6.26) can be normalized to

**f

**f

*f

2f

2

ssLs yyds

d

−⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

+⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

=Ωθθθ

(6.27)

where Ωs is a dimensionless parameter named fin conductance parameter for sensible heat transfer. It is defined by

( )a2λλ

a

fs

δ=Ω (6.28)

where dimensionless fin temperature

Wi

Wff TT

TT−−

=θ (6.29)

ass

2* = (6.30)

ayy2

f*f = (6.31)

aLL2

f*f = (6.32)


Table 6.1 lists the fin conductance parameters for the commonly used materials, for a passage of duct height 2mm, fin thickness 0.1mm, and air as working fluid, calculated by Eq.(6.28). Heat conductivity data are taken from ref.[2]. As seen, for well-conductive metal materials like copper, bronze, iron, steel, and aluminium, the fin conductance parameters are larger than 100; while for low conductive non-metal materials like plywood, clay, glass, papers, and polymers, the fin conductance parameters are generally less than 5, in some cases even lower than 1.0. In summary, large differences exist in fin conductance parameters for various materials, from 0.2 for polymer to 760 for pure copper.

Boundary Conditions

The boundary conditions for fluid are u*=0, on the 3 walls of the duct (6.33) θ=0, at y*=0 (6.34) Inlet condition θ=1, at z*=0 (6.35) The boundary conditions for fins, TA=TC=TD=Tw, or θf =0, at s*=0, Lf

* (6.36) Fin-fluid coupling:

θ=θf , at fin-fluid interfaces (6.37)

Table 6.1. Values of fin heat conductance parameter for some fin materials

Fin materials λf (Wm-1K-1) Ωs Pure copper 401 762 Aluminium 237 450.3 Iron 80.2 152.4 Steel 60.5 115 Bronze 52 98.8 Carbon 1.6 3.04 Glass 1.4 2.66 Clay 1.3 2.47 Teflon 0.35 0.67 Paper 0.18 0.36 Wood 0.16 0.30 Polymer Membrane 0.14 0.25 Plywood 0.12 0.23

Li-Zhi Zhang 110

Numerical Method

Boundary-fitted Coordinates Commonly, duct of rectangular cross section is easy to solve under the x-y coordinate

system. However, the cross section in this case is a sinusoidal one. To ease the solution, a boundary fitted coordinate transformation technique is used to transfer the sinusoidal domain to a square domain. Another benefit with this methodology is that the program can be easily modified to calculate other ducts of arbitrary cross sectional shapes, as long as the grids points on boundaries are specified. This will provide a broad basis for program validation.

The basic idea of the boundary fitted coordinate system is to have a coordinate system such that the body contour coincides with the coordinate lines. The transformation between the physical coordinates (x, y) and the boundary fitted coordinates (ξ, η), which is usually a square domain, is achieved by solving two Poisson equations on (x, y) domain [7,8],

),(2

2

2

2

ηξ=∂

ξ∂+

∂ξ∂ P

yx (6.38)

),(2

2

2

2

ηξ=∂

η∂+

∂η∂ Q

yx (6.39)

where P(ξ, η) and Q(ξ, η) are grid distribution inhomogeneous functions in the computational domain. These two equations may be more easily solved on the computational plane. Therefore, Thompson et al [7] inverted Eqs.(6.38) and (6.39) into the transformed domain (ξ, η), where the boundary is easy to specify. At the same time, using the method proposed by Thomas and Middlecoff [8] for selecting P, Q, Eqs. (6.38) and (6.39) are inverted into

02 2

22

2

2

=⎥⎦

⎤⎢⎣

⎡η∂

∂ψ+

η∂∂

γ+η∂ξ∂

∂β−⎥

⎦

⎤⎢⎣

⎡ξ∂

∂φ+

ξ∂∂

αxxxxx

(6.40)

02 2

22

2

2

=⎥⎦

⎤⎢⎣

⎡η∂

∂ψ+

η∂∂

γ+η∂ξ∂

∂β−⎥

⎦

⎤⎢⎣

⎡ξ∂

∂φ+

ξ∂∂

αyyyyy

(6.41)

where

22

⎥⎦

⎤⎢⎣

⎡η∂

∂+⎥

⎦

⎤⎢⎣

⎡η∂

∂=α

yx (6.42)

η∂∂

ξ∂∂

+η∂

∂ξ∂

∂=β

yyxx (6.43)


22

⎥⎦

⎤⎢⎣

⎡ξ∂

∂+⎥

⎦

⎤⎢⎣

⎡ξ∂

∂=γ

yx (6.44)

22

2

2

2

2

⎥⎦

⎤⎢⎣

⎡ξ∂

∂+⎥

⎦

⎤⎢⎣

⎡ξ∂

∂

⎥⎦

⎤⎢⎣

⎡ξ∂

∂ξ∂

∂+

ξ∂∂

ξ∂∂

−=φyx

yyxx

(6.45)

22

2

2

2

2

⎥⎦

⎤⎢⎣

⎡η∂

∂+⎥

⎦

⎤⎢⎣

⎡η∂

∂

⎥⎦

⎤⎢⎣

⎡η∂

∂η∂

∂+

η∂∂

η∂∂

−=ψyx

xxyy

(6.46)

The numerical value of φ at each grid point along horizontal boundary η=ηb in terms of

boundary values x, y, is computed once the differential operators are replaced by central-difference operators in Eq.(6.45). The values of φ at internal mesh points are computed by linear interpolation along the vertical mesh lines ξ=const. Similarly, the numerical values of ψ at mesh points along vertical boundary ξ=ξb are computed through central difference of Eq.(6.46). Linear interpolations along horizontal lines η=const are performed to obtain the values of ψ at internal grids. The procedure for evaluating the parameters φ and ψ insures that the grid throughout the interior of the computational domain will be governed by the grid distribution that is assigned on the boundaries, and that the transverse grid lines will be locally orthogonal to the boundaries.

Once the values of φ and ψ are calculated, the numerical solution of Eqs.(6.40), (6.41) by standard successive line less-relaxation on a uniform, rectangular grid Δξ, Δη results in a grid point distribution throughout the physical domain that is controlled entirely by the distribution of grid points on the boundaries.

The resulting grid constructions for the sinusoidal duct and the corresponding computational domain are shown in Figure 6.5.

After the set up of boundary fitted coordinates, Eqs. (6.9) and (6.10) can be transformed to computational domain as following

04112h

2**2**

=+⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛ξ∂

∂β−

η∂∂

γ⎟⎠⎞

⎜⎝⎛

η∂∂

+⎥⎦

⎤⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛η∂

∂β−

ξ∂∂

αξ∂

∂DbJuu

ab

Juu

J (6.47)

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛ξ∂θ∂

β−η∂θ∂

γ⎟⎠⎞

⎜⎝⎛

η∂∂

+⎥⎦

⎤⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛η∂θ∂

β−ξ∂θ∂

αξ∂

∂=

∂θ∂ 2

*

11ab

JJzJU (6.48)

Li-Zhi Zhang 112

where α, β, γ have the same definition as in Eqs.(6.42)-(6.44), J is the Jacobian transformation operator, which is defined as

ξ∂∂

η∂∂

−η∂

∂ξ∂

∂=

ηξ∂∂

=yxyxyxJ

),(),(

(6.49)

BA C

D

•

•

• •B

A C

D

•

•

• •

(a) The physical plane.

0 0.25 0.5 0.75 10

0.25

0.5

0.75

1

ξ

η

A B

CD

0 0.25 0.5 0.75 10

0.25

0.5

0.75

1

ξ

η

0 0.25 0.5 0.75 10

0.25

0.5

0.75

1

ξ

η

A B

CD

(b) The computational plane.

Figure 6.5. Grid configurations for the duct cross section and the corresponding apexes. (a) physical plane; (b) the computational domain.


Differential area on the physical cross section is

ηξ= dJddA (6.50)

RESULTS AND DISCUSSION The meshes in fluid are three dimensional. To account for the inlet influences, grids are

denser near inlet, while relatively sparser and evenly distributed after inlet. The meshes on two fins are two dimensional and they are the same structure in z direction with the meshes of fluid.

The velocity and momentum equations (6.47) and (6.48) are further discretized. The diffusion term is discretized by central-difference scheme and the convective term is by upwind scheme. The proposed finite difference schemes are implicit numerical schemes with second order accuracy. They are unconditionally stable. The problem is a typical conjugate one. After the solution of fluid velocity and temperature, temperatures on one fin are solved on the generated grids, taking the current fluid temperature as the default boundary conditions. The values on the other fin are obtained by symmetry. The whole calculating procedure can be summarized as the following:

a. Grid generation for both the fluid and the fins. b. Solve momentum equation Eq.(6.47). Get the velocity fields and resistance data for

the duct. c. Assume initial temperature fields in the fluid. d. Taking current fluid temperature as the boundary conditions for fins. Get the

temperature fields on one fin, by the solution of Eq.(6.27). Get the temperature on the other fin because it is in axis symmetry with the first fin.

e. Taking the current values of temperature on two fins as the default values, get the temperature profiles in the fluid by solving Eq.(6.47).

f. Go to (d), until the old values and the newly calculated values of temperature at all calculating nodes are converged.

Directions of coordinate s are dictated by neighboring grid points along ξ=0 for fin AD

and along η=1.0 for fin DC respectively. Directions of coordinate y1 are determined from neighboring grid points normal to ξ=0 for fin AD and normal to η=1.0 for fin DC respectively.

To assure the accuracy of the results presented, a grid independence test was performed for the duct to determine the effects of the grid size. It indicates that 21×21 grids on duct cross section and Δz*=0.001 axially are adequate (less than 0.1% difference compared with 31×31 grids and Δz*=0.0005).

To further validate the numerical program, ordinary ducts of various cross sections are calculated under uniform temperature conditions for all walls. Namely, ducts with no fins are considered first. For hydrodynamically fully developed laminar flow in ducts, (fRe) is a constant. The local Nusselt numbers in the duct will decrease along the flow and reach stable values when the flow is thermally fully developed. The fully developed Nu values under

Li-Zhi Zhang 114

uniform temperature conditions are denoted as NuT. The calculated values of (fRe) and NuT for various cross sections and aspect ratios are listed in Table 6.2. Comparisons are made with the values form well-known references [1-3]. As seen, the current study predicts the flow well. They are in accordance with the published data. Maximum uncertainty is less than 5%.

Velocity and Temperature Profiles

After the program is validated, sinusoidal ducts with one plate of uniform temperature and two fins shown in Figure 6.1 are modeled. Velocity and temperature fields in the fluid are obtained. Figure 6.6 shows the velocity profile distribution on the duct cross section for τ=1.0. The values in the figure are dimensionless velocity u*. As seen, in the center, the contours are near-circular, but in places near the boundaries, they are near-triangles. In the corners, there are dead spaces with little fluid flowing which will affect heat transfer finally. The flow shows a same pattern as that of a common sinusoidal duct without fins.

The dimensionless temperature profiles in the fluid are shown in Figure 6.7. The results also found that at higher fin conductance parameters, the temperature contours are similar to those in ducts with three walls of uniform temperature. At lower fin conductance parameters, temperature contours show in majority a pattern of straight lines parallel to the plate, which is similar to temperature profiles in parallel-plates ducts. Besides, in the center, isotherms are in elliptical shapes. When Ω increases, the contours transform from parallel horizontal lines to triangular shapes, implicating increased heat transfer from fins to fluid.

Table 6.2. Fully developed (fRe) and Nusselt numbers for ducts of various cross sections

(fRe) NuT

Cross sections Refs This study Error

(%) Refs This study Error (%)

Circular 16.0 15.88 0.8 3.657 3.66 0.6 0.125 20.5 20.34 0.9 5.60 5.73 2.3 0.25 18.25 18.16 0.6 4.44 4.55 2.5 0.5 15.50 15.32 1.3 3.39 3.43 1.8

Rectangular 2a/2b

1.0 14.227 14.11 1.5 2.976 3.06 2.8 0.289 13.243 12.77 2.8 2.301 2.262 1.7 0.5 13.301 12.94 2.9 2.359 2.451 3.9 0.866 13.321 13.41 0.5 2.500 2.594 3.7

Isosceles Triangular 2a/2b

1.866 13.09 12.96 1.0 2.284 2.391 4.7 0.5 11.207 11.170 0.4 2.12 2.181 2.8 0.75 12.234 12.212 0.2 2.33 2.374 1.7 1.0 13.003 12.954 0.3 2.45 2.521 2.9 1.5 14.002 14.115 0.8 2.6 2.573 1.1

Sine 2a/2b

2.0 14.553 14.647 0.7 ⎯ 2.886 ⎯


0.006 0.006

0.006

0.006

0.00

6

0.00

6

0.01

2

0.012

0.012

0.012

0.01

2

0.01

2

0.018

0.018

0.018

0.018

0.01

8

0.01

8

0.023

0.02

3

0.023

0.02

3

0.02

3

0.029

0.029

0.029

0.029

0.02

9

0.035

0.035

0.035

0.03

5

0.041

0.041

0.041

0.04

1

0.047

0.047

0.047

0.04

7

0.05

3

0.0530.

053

0.058

0.058

0.05

80.064

0.0640.06

4

0.0700.070

0.076

0.07

60.

082

x*

y*

0 0.25 0.5 0.75 10

0.25

0.5

0.75

1

Figure 6.6. Dimensionless velocity profiles (u*) on duct cross section, τ=1.0.

0.050 0.050

0.0990.099

0.099

0.149

0.149

0.198

0.198

0.248

0.248

0.2970.2

97

0.297

0.347

0.34

7

0.347

0.34

7

0.396

0.396

0.396

0.39

6

0.446

0.446

0.495

0.495

0.545

0.54

5

x*

y*

0 0.25 0.5 0.75 10

0.25

0.5

0.75

1

Figure 6.7. Dimensionless temperature contours on duct cross section at z*=0.1 for τ=1.0 and Ωs=1.0.

Li-Zhi Zhang 116

0 018 0.018

0.018 0.018

0.038 0.038

0.038 0.038

0.051 0.051

0.051 0.051

0.077 0.077

0.077 0.077

0.103 0.103

0.103 0.103

0.1540.154

0.154

0.1540.154

0.2060.206

0.206

0.2060.206

0.2570.257

0.257

0.257

0.257

0.309

0.309

0.309

0.309

0.360

0.360

0.360

0.360

0.411

0.411

0.411

0.463

0.463

0.514

0.514

0.566

0.566

0.61

7

0.617

0.669

0.669

0.720

0.77

1

z*

s*

0 0.05 0.1 0.150

0.5

1

1.5

Figure 6.8. Dimensionless temperature contours on fins for ducts of τ=1.0 and Ωs=1.0.

Dimensionless temperature profiles on the fins are show in Figure 6.8. Temperatures at upper and lower boundaries are equal to the plate temperature. Temperatures at other locations vary with conductance parameters. The higher the fin conductance parameters, the more homogeneous the temperature distributions on fins are. Under infinitely large fin conductance parameters, temperatures on fins will be equal to plate temperature. In this situation, the plate-fin duct becomes a common duct with three walls of uniform temperature.

z*

Nu

0 0.05 0.1 0.150

1

2

3

4

5

6

0.2

0.4

0.6

0.8

1

θ bNuLNum

θb

z*

Nu

0 0.05 0.1 0.150

1

2

3

4

5

6

0.2

0.4

0.6

0.8

1

θ bNuLNum

θb

Figure 6.9. Variations of bulk temperature, NuL and Num along flow axis, τ=1.0, Ωs=0.1.


Nusselt Numbers Once the temperature fields in fluid has been determined, the dimensionless bulk

temperature, the local Nusselt numbers, and the mean Nusselt numbers can be calculated. Figure 6.9 shows the axial variations of bulk temperature, local and mean Nusselt numbers for a duct. The higher the fin conductance parameters, the more rapid for the fluid bulk temperature to reach the plate temperature, indicating much more heat is exchanged. The thermal entrance length zth

* is defined as the axial distance required to achieve a value of the local Nusselt number NuL, which is 1.05 times the fully developed Nusselt value NuT. The calculated thermal entry length for various aspect ratios and fin conductance parameters are around 0.09.

After the thermal entry length, the local Nusselt number will come to a stable value NuT. This fully developed value varies with aspect ratios and fin conductance parameters. Table 6.3 lists the calculated NuT for various aspect ratios and fin conductance parameters.

Table 6.3. Fully developed Nusselt numbers and fin efficiencies for plate-fin sinusoidal

ducts

τ Ωs NuT ηfin τ Ωs NuT ηfin ∞ 1.541 1.0 ∞ 2.181 1.0 25 1.525 0.982 25 2.135 0.978 10 1.211 0.774 10 2.109 0.966 5 1.043 0.666 5 1.906 0.875 2 0.873 0.556 2 1.522 0.699 1 0.794 0.510 1 1.262 0.578 0.5 0.738 0.476 0.5 1.053 0.484 0.1 0.693 0.447 0.1 0.821 0.378

0.2

0 0.643 0.413

0.5

0 0.735 0.338 ∞ 2.521 1.0 ∞ 2.886 1.0 25 2.494 0.988 25 2.712 0.940 10 2.367 0.940 10 2.653 0.918 5 2.166 0.860 5 1.922 0.666 2 1.723 0.683 2 1.513 0.523 1 1.376 0.547 1 1.193 0.413 0.5 1.083 0.429 0.5 0.924 0.321 0.1 0.752 0.298 0.1 0.631 0.218

1.0

0 0.578 0.228

2.0

0 0.356 0.124 ∞ 2.584 1.0 1 1.015 0.392 25 2.391 0.924 0.5 0.831 0.322 10 2.171 0.839 0.1 0.662 0.256 5 1.657 0.640 0 0.172 0.067

5.0

2 1.276 0.493

5.0

Li-Zhi Zhang 118

Fin efficiency is defined as the ratio of NuT at a certain fin conductance to that at Ωs=∞, or

∞=Ω

Ω=s T,

s T,finη

NuNu

(6.51)

The fin efficiency for various cases is also listed in Table 6.3.

6.3. RECTANGULAR DUCT A schematic of a plate-fin duct of rectangular cross section is shown in Figure 6.10 [9]. It

is also one of the most commonly encountered total heat exchanger structure. The duct height is 2a, and the duct width is 2b. The definition of fin heat conductance parameter is similar to a sinusoidal duct. The calculated values of (fRe) for various cross sectional aspect ratios are listed in Table 6.4.

( )aas 2λ

λ f δ=Ω (6.52)

Figure 6.10. Schematic of a compact heat exchanger comprised of rectangular plate-fin passages.

Table 6.4. Fully developed (fRe) for rectangular ducts of various aspect ratios

τ=2a/2b (fRe)

0.125 20.51 0.25 18.25 0.5 15.51 1.0 14.227 3.0 17.252

Table 6.5 lists the calculated NuT for various aspect ratios and fin heat conductance

parameters [9]. Also listed are values of thermal entry length. The thermal entrance length zth*


is defined as the axial distance required to achieve a value of the local Nusselt number NuL, which is 1.05 times the fully developed Nusselt value NuT [1]. Generally, the thermal entry length is in the range from 0.01 to 0.1. The larger the aspect ratios are, the longer the thermal entry length is. As listed in Table 6.5, the larger the fin conductance parameters, the higher the fully developed NuT. In heat exchanger design, fully developed NuT can be estimated from this table for different aspect ratios and fin conductance parameters.

Table 6.5. Fully developed Nusselt numbers and thermal entry length for rectangular

plate-fin ducts

2a/2b Ωs NuT zth* 2a/2b Ωs NuT zth

* ∞ 5.746 0.014 ∞ 4.537 0.013 25 5.683 0.016 25 4.476 0.029 10 5.676 0.016 10 4.463 0.030 5 5.574 0.021 5 4.375 0.041 2 5.563 0.020 2 4.364 0.037 1 5.542 0.019 1 4.345 0.038 0.5 5.536 0.018 0.5 4.3431 0.033 0.1 5.524 0.014 0.1 4.305 0.026

0.125

0 5.504 0.014

0.25

0 4.294 0.021 ∞ 3.501 0.041 ∞ 3.021 0.044 25 3.485 0.043 25 3.013 0.045 10 3.483 0.045 10 3.012 0.046 5 3.452 0.043 5 3.012 0.049 2 3.443 0.039 2 2.684 0.043 1 3.353 0.043 1 2.485 0.043 0.5 3.294 0.039 0.5 2.271 0.042 0.1 3.153 0.033 0.1 1.942 0.040

0.5

0 3.122 0.026

1.0

0 1.815 0.039 ∞ 3.500 0.041 ∞ 4.527 0.025 25 3.471 0.046 25 4.464 0.019 10 3.465 0.048 10 3.396 0.018 5 3.352 0.021 5 3.033 0.048 2 2.256 0.051 2 2.044 0.065 1 1.896 0.056 1 1.393 0.082 0.5 1.491 0.062 0.5 0.938 0.098 0.1 1.014 0.074 0.1 0.485 0.125

2.0

0 0.875 0.070

4.0

0 0.399 0.094 The results in the table show that, besides aspect ratios, the fin heat conductance

parameters have equal tremendous impact on heat transfer phenomenon in the plate-fin narrow passages. When fin conductance parameters increase from 0 to infinitely large, the

Li-Zhi Zhang 120

duct behaves from like a duct with two well-conductive wall and two adiabatic walls to a duct with four well-conductive walls. The consequent fully developed Nusselt numbers increase accordingly. For ducts with certain limited conductance parameters, the Nusselt numbers lie between these two limiting values. For fins made with low conductivity materials like paper, polymer, glass, carbon, plywood, etc, the nature of limited heat transfer from fins to fluid should be taken into account, for a better estimation of the exchanger performance. The Nusselt numbers under uniform heat flux boundary conditions are usually 20-30% higher than those under uniform temperature conditions.

Parallel-plates Channels Parallel-plates channels are a special type of rectangular ducts. The aspect ratio is

infinitely small. In this case, the finite fin conductance has no influence on convective heat transfer coefficients. The fully developed Nusselt number under uniform wall temperature is 7.54 [2]. The friction coefficients and the fully developed Nusselt numbers for rectangular ducts of other cross sectional shapes are also found in [2].

6.4. TRIANGULAR DUCT A schematic of a plate-fin duct of triangular cross section is shown in Figure 6.11 [10]. It

is one of the most commonly encountered total heat exchanger structure. The cross section is usually isosceles triangle. Sometimes, it is approximated by a sinusoidal cross section, or vice versa. Anyway, this chapter gives its exact numerical results for heat mass transfer. As ducts of other geometries, the duct height is 2a, and the duct width is 2b. For triangular duct, it’s more convenient to use the apex angle 2α to define the geometry.

2α, Apex angle, 2α, Apex angle,

Figure 6.11. Schematic of a cross-flow plate-fin heat exchanger with triangular duct geometry.


Table 6.6. Fully developed (fRe) for triangular ducts of various half apex angles

α (°) (fRe) 5 12.511 15 13.08 30 13.322 45 13.302 60 13.244 75 13.141

The fin heat conductance parameter is still defined as

( )aa 2λλ f

sδ

=Ω (6.53)

The calculated values of (fRe) for various half apex angles are listed in Table 6.6 for

triangular ducts [10]. Table 6.7 lists the calculated NuT for various half apex angles and fin conductance

parameters. The larger the fin conductance parameters, the higher the fully developed NuT.

Table 6.7. Fully developed Nusselt numbers for plate-fin triangular ducts

α Ωs NuT α Ωs NuT ∞ 2.391 ∞ 2.596 25 2.285 25 2.513 10 2.194 10 2.444 5 2.063 5 2.301 2 1.597 2 1.938 1 1.285 1 1.605 0.5 1.021 0.5 1.277 0.1 0.722 0.1 0.843

15°

0 0.653

30°

0 0.653 ∞ 2.451 ∞ 2.262 25 2.395 25 2.103 10 2.296 10 2.027 5 2.183 5 1.927 2 2.024 2 1.794 1 1.741 1 1.619 0.5 1.441 0.5 1.415 0.1 0.975 0.1 1.065

45°

0 0.656

60°

0 0.593

Li-Zhi Zhang 122

6.5. CONVECTIVE MASS TRANSFER COEFFICIENTS Convective heat transfer in plate-fin ducts with finite fin conductance has been

summarized. Duct cross sections include sinusoidal, rectangular, and triangular. The fins include common metals to non-metal materials. Besides heat transfer coefficients, to properly evaluate the heat mass exchanger performance, mass transfer coefficients are equally important. Traditionally, mass transfer coefficients in a common pipe are obtained from heat transfer coefficients by heat mass transfer analogies, of which the most frequently cited one is the Chilton-Colburn Analogy [6]. According to this methodology, following equations can be used to estimate convective mass transfer coefficients in a pipe:

Ls jj = (6.54) The Chilton-Colburn j factor for heat transfer [6]

3/2ss PrStj = (6.55)

where Pr is the Prandtl number of air. Stanton number for heat transfer

uchSt

pas ρ

= (6.56)

where h is convective heat transfer coefficients (kWm-2K-1), u is air stream bulk velocity (m/s), cp is specific heat (kJkg-1K-1), ρa is dry air density (kgm-3).

The convective heat transfer coefficient is also represented by a Nusselt number

a

h

λhdNu = (6.57)

where dh is the hydrodynamic diameter of the channel (m), λ is thermal conductivity (kWm-

1K-1). The Chilton-Colburn j factor for mass transfer

3/2LL ScStj = (6.58)

where Sc is the Schmidt number of moisture air. Stanton number for mass transfer

ukSta

L ρ= (6.59)

where k is convective mass transfer coefficient (m/s).


Mass transfer in boundary layers is also described by a Sherwood number

va

h

DkdSh = (6.60)

where Dva is vapor diffusivity in air (m2/s). Substituting Eqs.(6.55) through (6.60) into (6.54), a relation can be obtained

3/1−⋅= LeNuSh (6.61)

ScLe Pr

= (6.62)

where Le is commonly called the Lewis number. For ventilation air and vapor mixture, which is always near atmospheric states, the Lewis number varies in the range of 1.19 to 1.22 [6], therefore it is usually approximate that Sh=Nu.

If above analogy still holds in our case, then the estimation of convective mass transfer coefficients would be simple, from the previously obtained data. However, in plate-fin ducts used for non-metal total heat exchanger, the fin conductance parameters for heat and mass transfer are so different that it’s questionable that such an analogy still exists. It’s therefore imperative to obtain the convective mass transfer coefficients and find the relations between the heat and mass transfer in such plate-fin ducts.

Governing Equations Air travels in the duct while exchanging moisture with duct walls. Mass conservation in

the air stream can be expressed by

⎟⎟⎠

⎞⎜⎜⎝

⎛∂

ω∂+

∂ω∂

=∂ω∂

2

2

2

2

va yxD

zu (6.63)

where ω is humidity ratio of air (kg vapor/kg dry air).

The above equation can be normalized to

2*

22

2*

2

* yab

xzU

∂

ξ∂⎟⎠⎞

⎜⎝⎛+

∂

ξ∂=

∂ξ∂

(6.64)

where dimensionless humidity ratio

Li-Zhi Zhang 124

wi

w

ω−ωω−ω

=ξ (6.65)

where in the equations, ωi is the inlet air humidity, and ωw is the humidity on wall surface.

Dimensionless bulk humidity

∫∫∫∫ ξ

=ξdAu

dAu*

*

b (6.66)

Similarly, an mass balance in a control volume in the duct will give the equation for

estimation of the local Sherwood number as

*b

bL 4

1dzd

Shξ

ξ−= (6.67)

and the mean Sherwood number from z*=0 to z* by

b*m ln41

ξ−=z

Sh (6.68)

Similar to heat conduction in the fin in Figure 6.4, at any location, there is a mass balance

between the net water diffusion along the fin and the mass transfer from the surface of the fin to the fluid. The phenomenon is expressed by

Lu2

2

fwf mmds

WdD +=δρ (6.69)

ufavau ⎟⎟

⎠

⎞⎜⎜⎝

⎛∂∂

−=y

Dm ωρ (6.70)

LfavaL ⎟⎟

⎠

⎞⎜⎜⎝

⎛∂∂

=y

Dm ωρ (6.71)

where Dwf is diffusivity of water in fin materials (m2/s); ρf is density of fin materials (kg/m3); W is water content in fin materials (kg water/kg dry material); mu is moisture flux from upper surface of fin, and mL is moisture flux from lower surface. ( )f/ y∂∂ω is the normal gradient of vapor concentration on the lower or upper surface of fin. Due to symmetry, the mass flux at the lower surface and the upper surface are also skew symmetric. The relation is mathematically expressed by


sLsmm

−−=

fLu (6.72)

Water uptake in a hygroscopic material is a function of air relative humidity. Figure 6.12

shows the measured sorption isotherm of the material used for the investigated total heat exchanger. The discrete dots are the measured data with a thermo-hygrostat. A polynomial curve can be regressed to represent this isotherm from 0% to 100% RH. The relation between RH and humidity ratio is

ω= − TeRH /5294610 (6.73)

The temperature differences between two surfaces of a fin are quite small due to the

small fin thickness. For air conditioning industry, heat mass exchanger always works between 40%RH and 80%RH. Considering these factors, the relative humidity can be a linear expression of humidity ratio, and the sorption isotherm for the material can also be approximated by a linear equation as

ω= pkW (6.74)

where kp is defined as the partition coefficient.

Substituting Eqs.(6.74) in (6.69) to (6.72), following dimensionless equation can be found

0

0.1

0.2

0.3

0.4

0.5

0 0.2 0.4 0.6 0.8 1

Relative humidity RH

W(k

g/kg

)

0

0.1

0.2

0.3

0.4

0.5

0 0.2 0.4 0.6 0.8 1

Relative humidity RH

W(k

g/kg

)

Figure 6.12. Sorption isotherm for a material in total heat exchanger.

Li-Zhi Zhang 126

**f

**f

*f

2*f

2

LsLs yyds

d

−⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

−⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

−=Ωξξξ

(6.75)

where ΩL is defined as the dimensionless fin mass conductance parameter. It is calculated by

( )aDkD

2vaa

pwffL ρ

δρ=Ω (6.76)

where dimensionless humidity ratio in fins

Wi

Wff ω−ω

ω−ω=ξ (6.77)

Boundary Conditions The boundary conditions of humidity for fluid ξ=0, at y*=0 (6.78) Inlet condition ξ=1, at z*=0 (6.79) The boundary conditions for fins ξf=0 at s*=0, Lf

* (6.80) Fin-fluid coupling:

and ξ=ξ f at fin-fluid interfaces (6.81) It can be found that Eqs.(6.64) and (6.75) are in the same forms as Eqs.(6.10) and (6.27)

respectively. Therefore, the solution of dimensionless humidity ratio can be an analogy to the solution of dimensionless temperature, if the fin mass conductance parameters are the same values as the fin heat conductance parameters. In other words, the solution of dimensionless humidity is the same as the dimensionless temperature, if assuming same values of heat and mass fin conductance parameters. Consequently, it is only necessary to solve either the heat transfer equations or the mass transfer equations. For mass transfer properties, if assuming ΩL is the same value as Ωs, ShL will equal to NuL. Correspondingly, the fully developed Sherwood number under uniform concentration boundary conditions Shω is equal to NuT.


In a summary, convective mass transfer coefficients can be estimated in the following steps: (1) Calculate the value ΩL by Eq.(6.76); (2) Use ΩL to replace Ωs in Table 6.3, 6.5 or 6.7, depending on which shape the cross section is; (3) the value of Shω is equal to NuT in the table.

6.6. COMPARISONS OF HEAT AND MASS TRANSFER As defined by Eq.(6.28), fin heat conductance parameters are determined by duct

geometry and heat conductivity of fin materials. Table 6.3 lists the values of fin heat conductance parameters for some frequently encountered materials, including metal and non-metals, for a sinusoidal duct of height 2mm, aspect ratio 0.5, and fin thickness 0.1mm, as mentioned above. As seen, for almost all the metals, the fin heat conductance parameters are larger than 100, and the resulting fin heat efficiencies can be as high as 0.90-0.98. For such traditional metal compact heat exchangers, the influences of finite fin heat conductance on heat transfer are negligible. Direct utilization of heat transfer properties for a common duct is acceptable. For the total heat exchangers which use non-metals to simultaneously transfer heat and moisture, the fin heat conductance parameters are usually less than 1.0, and the resulting fin efficiencies for heat transfer can be as low as 0.40. Though it’s true that the fins still participate in the heat transfer enhancement, at least partially, at this stage, the effects of finite fin heat conductance on heat transfer will be substantial.

Similarly, as defined by Eq.(6.76), fin mass conductance parameters are determined by duct geometry and mass diffusivity of moisture in fin materials. Moisture diffusivity in materials is not accumulated as detail as heat conductivity. There is only sporadic data available. Moisture diffusivity in air is around 2.5×10-5m2/s [11]. A literature review found that moisture diffusivities in most non-metal hygroscopic materials are in or less than the order of 10-10 m2/s. For example, water diffusivities are from 0.5×10-12 to 1.5×10-12m2/s in a PSS-Na/Al2O3 composite silicate [12]; from 8.0 ×10-10m2/s to 5.0 ×10-12m2/s in a Nafion polymer membrane [13]; and from 1.0 ×10-13m2/s to 1.0 ×10-12m2/s in a methylcellulose film [14]. The densities of these materials are in the order of 1000kg/m3. The partition coefficients are in the order of 10, or 15-20% water uptake under 60% RH. Based on these properties, the values of fin mass conductance parameters are in the order of 10-3. With such low mass conductance parameters, the fin efficiencies for mass transfer will be below 0.1 to 0.2. Under such circumstances, nearly all the mass transfer between the two air streams will be accomplished by the plate, rather than by the fins. The fins seem to behave only like supporting materials, if excluding their role in partial participation in heat transfer. In engineering, sometimes supporting materials or spacers are necessary to separate the two streams because the plates are thin and soft.

6.7. CONCLUSION Laminar flow and heat mass transfer in plate-fin ducts with various geometries such as

sinusoidal, rectangular, triangular, parallel-plates, are investigated by considering finite fin

Li-Zhi Zhang 128

conductance in heat and mass transfer. Finite fin conductance parameters in heat and mass have determining effects in convective heat and mass transfer in the ducts.

In designing novel heat mass exchangers, it’s simple to emulate a common compact heat exchanger with such a plate-fin configuration. However, the effectiveness of fins on heat transfer will be strongly compromised. Even worse, it has little use in enhancing mass transfer. Nevertheless, this structure is still popular sue to its high mechanical strength and compactness.

REFERENCES


[2] Incropera, F.P.; Dewitt, D.P. Introduction to Heat Transfer. 3rd edn. New York: John Wiley & Sons; 1996. Chapter 8, pp. 416.

[3] Shah, R.K.; Bhatti, M.S. Laminar Convection Heat Transfer In Ducts. in: Handbook of Single-Phase Convective Heat Transfer, (Ed. S. Kakac, R.K. Shah, W. Aung). New York: Wiley; 1987.

[4] Baliga, B.R.; Azrak, R.R. Laminar fully developed flow and heat transfer in triangular plate-fin ducts. ASME Journal of Heat Transfer, 1986, 108, 24-32.

[5] Niu, J.L.; Zhang, L.Z. Heat transfer and friction coefficients in corrugated ducts confined by sinusoidal and arc curves. International Journal of Heat and Mass Transfer, 2002, 45, 571-578.

[6] Zhang, L.Z. Heat and mass transfer in plate-fin sinusoidal passages with vapor-permeable wall materials. International Journal of Heat Mass Transfer, 2008, 51, 618-629.

[7] Thompson, J.F.; Thames, F.; Martin, C. Automatic numerical generation of body-filled curvilinear coordinate system for field containing any number of arbitrary two-dimensional bodies. J. Comput. Physics, 1974, 24, 299-319.

[8] Thomas, P.D.; Middlecoff, J.F. Direct control of grid point distribution in meshes generated by elliptic equations. AIAA J., 1982, 18, 652-656.

[9] Zhang, L.Z. Thermally developing forced convection and heat transfer in rectangular plate-fin passages under uniform plate temperature. Numerical Heat Transfer, Part A-Applications, 2007, 52, 549-564.

[10] Zhang, L.Z. Laminar flow and heat transfer in plate-fin triangular ducts in thermally developing entry region. International Journal of Heat Mass Transfer, 2007, 50, 1637-1640.

[11] Kays, W.M.; Crawford, M.E. Convective heat and mass transfer, 3rd edn. New York: McGraw-Hill, Inc.; 1990.

[12] Aranda, P.; Chen, W.J. Martin, C.R. Water transport across polystyrenesulfonate/ alumina composite membranes. Journal of Membrane Science, 1995, 99, 185-195.

[13] Ye, X.H. LeVan, M.D. Water transport properties of Nafion membranes Part I. Single tube membrane module for air drying. Journal of Membrane Science, 2003, 221, 147-161.


[14] Debeaufort, F.; Voilley, A.; Meares, P. Water vapor permeability and diffusivity through methylcellulose edible films. Journal of Membrane Science, 1994, 91, 125-133.

Chapter 7

EFFECTIVENESS CORRELATIONS OF TOTAL HEAT EXCHANGERS

ABSTRACT

How to estimate the sensible and latent effectiveness of a total heat exchanger is a key issue in engineering design and optimization. Effectiveness correlations are a simple way to calculate performance. Contrary to sensible-only heat exchangers, total heat exchangers are influenced not only by geometries, but also by material properties. In this chapter, effectiveness correlations are deduced for total heat exchanger, both parallel types and plate-fin types. They can be conveniently used in engineering design.

NOMENCLATURE A Transfer area (m2) C Constant in sorption curve cp Specific heat of air (kJkg-1K-1) Dwp Diffusivity of water in plate (m2/s) Dwp

* Equivalent diffusivity of moisture in plate (m2/s) H Specific enthalpy (kJ/kg) H* Ratio of latent to sensible energy differences between the inlets of two air

streams h Convective heat transfer coefficient (kWm-2K-1) k Convective mass transfer coefficient (m/s) m Mass flow rate of air streams (kg/s)

Wm Mass flow rate of moisture flow (kgm-2s-1) NTU Number of transfer units for sensible heat NTUL Number of transfer units for latent heat Nu Nusselt number n Number of channels R Ratio for heat/mass capacity

Li-Zhi Zhang 132

Sh Sherwood Number T Temperature (°C) U Total heat transfer coefficient (kWm-2K-1) UL Total mass transfer coefficient (m/s) wmax Maximum water uptake of desiccant (kgkg-1) x, y Coordinates (m) x*, y* non-dimensional coordinates xF Length of supply channel (m) yF Length of exhaust channel (m)

Greek Letters ψ Coefficient of moisture diffusive resistance in membrane (CMDR) λ Thermal conductivity of membrane (kWm-1K-1) θ Moisture uptake in membrane (kgkg-1) ε Effectiveness r Resistance (m2K/kW for thermal and s/m for moisture) φ Relative humidity δ Thickness of membrane (m) ω Moisture content (kg moisture/kg dry air) α Ratio of diffusive to convective moisture resistance for membrane β Ratio of total number of transfer units for moisture to that for sensible heat

Subscripts a Air c Convective e Exhaust f Fresh i Inlet L Latent, moisture p plate o Outlet tot Total w Water

7.1. INTRODUCTION Stationary total heat exchangers are similar to air-to-air sensible heat exchangers, either

plate type or plate-fin type. The difference is that water vapor-permeable materials are used instead of metal foils. For common heat exchangers, ε-NTU (effectiveness-NTU) method is a simple way to predict performance and to design a heat exchanger [1,2]. How to emulate this methodology to total heat exchanger is of interest.

Effectiveness Correlations of Total Heat Exchangers 133

Heat mass transfer in a total heat exchanger is more complicated than a sensible-only heat exchanger. Detailed modeling of the system requires finite difference computations, which are hard and time consuming for common engineers. Therefore, ε-NTU method for the total heat exchanger provides an efficient yet simple tool for exchanger optimization and design. In this chapter, simple correlations that could predict the sensible and latent effectiveness will be introduced, by summarizing the couplings between the performance and the sorption characteristics of plate materials, and the operating conditions. The deduction of this chapter is more general than previous studies.

7.2. EFFECTIVENESS-NTU CORRELATIONS Consider a simple parallel-plates total heat exchanger. The plates are plain. It is a cross-

flow total heat exchanger with paper or membranes as the plates. The plate thickness is δ. This structure is common in practice. The device is shown in Figure 7.1. Two air streams- the fresh and the exhaust flow in thin, parallel, alternating membrane or paper layers, in order to transfer heat and moisture from one air stream to the other. In air conditioning, the fresh air is usually the outside air and the exhaust is the room air that needs to be discharged to the outside. The governing dimensionless equations for simultaneous heat and moisture transfer in the total heat exchangers, based on the assumptions listed in Table 7.1, are as follows [3,4]:

Fresh air

)(NTU2* fpfff TT

xT

−=∂∂

(7.1)

)ωω(NTU2*

ωfpfLf

f −=∂∂

x (7.2)

Exhaust air

)(NTU2* epeee TT

yT

−=∂∂

(7.3)

)ωω(NTU2*

ωepeLe

e −=∂∂

y (7.4)

Li-Zhi Zhang 134

Fresh in

Fresh out

Exhaust in

Exhaust out

Membrane plates

Duct Sealing

Fresh in

Fresh out

Exhaust in

Exhaust out

Membrane plates

Duct Sealing

Figure 7.1. Schematic of a cross-flow total heat exchanger.

Table 7.1. Assumptions used in governing equations 1. There is no lateral mixing of the two air streams. 2. Heat conduction and vapor diffusion in the two air streams are negligible compared

to energy transport and vapor convection by bulk flow. 3. Adsorption of water vapor and plate material is in equilibrium adsorption-state. 4. Both the heat conductivity and the water diffusivity in the plate are constants. 5. Heat and moisture transfer is one-dimensional in membrane. It is in thickness.

where

F

*xxx = ,

F

*yyy = ,

pff

totf

pff

FFfffNTU

cmAh

cmyxhn

== , pee

tote

pee

FFeeeNTU

cmAh

cmyxhn

== ,

f

totfaLfNTU

mAkρ

= , e

toteaLeNTU

mAkρ

= ,

where Tpf and Tpe are the temperature of plate in fresh side and exhaust side respectively (°C), ω is humidity ratio in air streams (kg/kg), cp is specific heat of air (kJkg-1K-1); x and y are coordinates (m), h is convective heat transfer coefficient (kWm-2K-1), k is convective mass transfer coefficient (m/s), m is mass flow rate of dry air (kg/s), xF and yF are lengths of flow channels (m), n is the number of channels for each flow. The subscript “f”,”p”, and “e” mean


“fresh air”, “plate”, and “exhaust”, respectively. Previous studies found that the temperature difference between the two sides of membrane is very small due to the small thickness of membrane [5]. So it is reasonable to assume that Tmf = Tme =0.5(Tf+Te).

From assumptions (1) and (2), it is seen that heat and moisture transfer is one-dimensional and along the flow direction. However, due to the cross-flow arrangements, the temperature and humidity distributions in the air streams are two-dimensional.

The above dimensionless parameters NTU are the commonly recognized Number of Transfer Units. They give an insight into the characteristics of heat and moisture exchange between fluids and surfaces.

Moisture flow rate through the plate:

δρ pepf

wppw

θθDm

−= (7.5)

where θpf, θpe are moisture uptake in plate at two surfaces (kg.kg-1), δ is the plate thickness, and Dwp is the water diffusivity in plate (m2s-1).

The equilibrium between the plate and moisture at its surface can be expressed with a general sorption curve as

φθ

/1max

CCw

+−= (7.6)

where wmax represents the maximum moisture content of the plate material (i.e., moisture uptake when φ=100%) and C determines the shape of the curve and the type of sorption.

The parameters θ, φ, and ω can be correlated by ideal gas state equation and psychrometric relations.

The convective heat transfer coefficients are obtained from Nusselt numbers for parallel plates [1,2] and the mass transfer in boundary layers is often described by Sherwood correlations. By using the well-known Chilton-Colburn Analogy [6]

3/1−⋅= LeNuSh (7.7)

We have

3/1

p

−= Lechk (7.8)

For ventilation air and vapor mixture, which is always near atmospheric states, the Lewis

number, Le, varies in the range of 1.19 to 1.22, see Ref. [6]. For plate-fin ducts, the convective heat transfer coefficients and convective mass transfer

coefficients can be referenced form the calculated data in previous sections. Analogous to the heat transfer effectiveness commonly used in heat exchanger analysis,

the concept of effectiveness can be applied to the heat and moisture transfer processes in a

Li-Zhi Zhang 136

membrane based enthalpy exchanger. For a constant specific heat and heat of phase change, the effectiveness is defined as

Sensible effectiveness

)()()()(

εeifiminp

fofifps TTcm

TTcm−

−= (7.9)

Latent effectiveness

)ωω()()ωω()(

εeifiminp

fofifpL −

−=

cmcm

(7.10)

Enthalpy transfer effectiveness, i.e., total energy transfer effectiveness

)()(

εeifimin

fofiftot HHm

HHm−

−= (7.11)

where H is the specific enthalpy of air, and it is calculated by [4]

)86.12501(ωp TTcH ++= (7.12)

where T is in °C.

The third term in Eq.(7.12) usually has a less than 3% effect, thus it can be neglected. Then the enthalpy effectiveness can be further simplified as

*1*εε

ε Lstot H

H++

= (7.13)

where

TTTcH

ΔΔ

≈−

−=

ω2501)(

)ωω(2501*

eisip

eisi (7.14)

where H* is essentially a ratio of latent to sensible energy differences between the inlets of two air streams flowing through the total heat exchanger. H* can in theory vary from –∞ to +∞, but varies typically from –6 to +6 for enthalpy recovery in HVAC applications. From above equation, it is clear that the total enthalpy effectiveness is not a simple algebraic average of sensible and latent effectiveness. When H*=1, εtot =(εs +εL)/2. As H* → ∞, εtot → εL; as H* → 0, εtot → εs; as H* → -1, εtot → ±∞.


An overall number of transfer units is used to reflect the sensible heat transfer in an exchanger. For the total heat exchanger that has equal area on both sides, the total number of transfer units for sensible heat is

minp

tot

)( cmUA

NTU = (7.15)

where U is the total heat transfer coefficient. Its general form is

1

ef

1λδ1

−

⎥⎦

⎤⎢⎣

⎡++=

hhU (7.16)

The term in the middle is the thermal resistance of plate, which value is around 0.005

m2K/kW. Other two terms are convective thermal resistance. Their values are in the order of 40 m2K/kW, or 8000 times larger than plate resistance. Therefore, plate resistance for heat transfer can be neglected.

The sensible effectiveness is a function of two dimensionless parameters, NTU and R1, the ratio of minimum to maximum heat capacity rate of two air streams.

maxpaminpa1 )/()( cmcmR = (7.17)

The moisture flow rate through the membrane can also be expressed as

)ωω(ρ)ωω(ρ epeepsffW −=−= kkm aa (7.18)

)(θθθθθ pfpepf

pfpf

pspe φφφ

φφ

−∂∂

+=Δ∂∂

+= (7.19)

Substituting Eq.(7.19) into Eq.(7.5), we have

)( pepfpf

wppW φφ

φθ

δρ

−⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

=D

m (7.20)

Using Clapeyron equation to represent the saturation vapor pressure and assuming a

standard atmospheric pressure of 101325Pa gives the relation between humidity and relative humidity as

φφ 61.110ω 6

)15.273/(5294

−=+Te

(7.21)

Li-Zhi Zhang 138

where the second term on the right side of the equation will generally have less than a 5% effect, thus it can be neglected. Thus the relation between φ and ω is expressed by

ω106

)15.273/(5294 +

=Teφ (7.22)

Substituting Eq.(7.22) into Eq.(7.20), we have

)ωω(10 pepf6

)15.273/(5294

pf

wppW −⎟⎟

⎠

⎞⎜⎜⎝

⎛∂∂

=+TeD

mφθ

δρ

(7.23)

The total moisture emission rate can be written as

)ωω(ρ efLaW −= Um (7.24)

where

1

ep

f

11−

⎟⎟⎠

⎞⎜⎜⎝

⎛++=

kr

kU L (7.25)

where in the equation, the first and the third term are the convective mass transfer resistance in air sides, and the middle term is plate mass transfer resistance, which is expressed by

pf

)/5294(

6

wpp

ap

10ρ

δρ

⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

=

φθTe

Dr (7.26)

Term UL can be called the overall mass transfer coefficient for the device. It is indicated

the overall moisture transfer coefficient, has an expression similar to overall heat transfer coefficient.

The differentiation of Eq.(7.6) gives

22max

)/1(θ

φφφ CCCw

+−=

∂∂

(7.27)

The Eq. (7.26) can be further simplified as

ψρ

δρ

wpp

ap D

r = (7.28)


pfmax)/5294(

226 )/1(10ψCwe

CCT

φφ+−= (7.29)

where the coefficient of diffusive resistance for plate, ψ, is co-determined by the operating conditions and the slope of sorption curves of plate material.

Sometimes the sorption curve is represented by a simple linear equation

ωθ pk= (7.30)

where kp is called partition coefficient. We can see that

ψ1

p =k (7.31)

It means that partition coefficient is not a constant. However, it is often approximated by

a constant. Similar to the definition of total number of transfer units for heat, the total number of

transfer units for moisture can be written as

min

LtotaL )(

ρm

UANTU = (7.32)

The comparison of total transfer units for moisture and sensible heat, assuming equal

specific heats for two air streams, gives

UcU

NTUNTU paLL ρ

==β (7.33)

It can be further simplified to [3]

αβ

+=

11

(7.34)

where

c

m

rr

=α (7.35)

fc

2k

r = (7.36)

Li-Zhi Zhang 140

where rc is the convective moisture transfer resistance, and α is the ratio of diffusive resistance to convective resistance for plate. As can be seen, the total number of transfer units for moisture can be estimated from the total number of transfer units for sensible heat, and ratio of diffusive to convective moisture resistance. As α→∞, NTUL→0, no moisture can be permeated through the plate. In this case, the “plate” is like a metal plate. On the other hand, as α→0, NTUL→NTU, εL=εs. If α=1, NTUL=NTU/2. Under the common operating conditions, the values of α vary from 2 to 10 [4], which implies that plate resistance for moisture transfer cannot be neglected.

Deduction of Effectiveness Correlations Considering a cross-flow membrane exchanger with only one flow channel. At any point

in the exchanger a heat and mass balance for an infinitely small volume dxdy can be written from as

( )dxdyTTUdq es −= (7.37)

( )dxdyUmd esLW ωω −= (7.38)

Equation (7.37) is a basic heat transfer equation, and Eq.(7.38) has been widely employed

as a mass permeation model through a membrane in chemical industry. Across the elements xF and yF units in length the energy and moisture balances yield

( )dxdy

xT

ycm

dq∂∂

−= s

F

spa (7.39)

( )

dxdyyT

xcm

dq∂∂

= e

F

epa (7.40)

dxdyxy

mmd W ∂∂

−= s

F

s ω (7.41)

dxdyyx

mmd W ∂∂

= e

F

e ω (7.42)

Combining Eqs. (7.37) and (7.39) and then Eqs. (7.37) and (7.40) gives

( ) ( )xTTT

cmUy

∂∂

−=− ses

spa

F (7.43)


( ) ( )yT

TTcm

Ux∂∂

=− ees

epa

F (7.44)

Similarly, combining Eqs. (7.38) and (7.41) and then Eqs. (7.38) and (7.42) gives

( )xm

yU L

∂∂

−=− ses

s

F ωωω (7.45)

( )ym

xU L

∂∂

=− ees

e

F ωωω (7.46)

Differentiating Eqs. (7.43) and (7.44) with respect to y and x and taking their sum gives

( )( )

( )( )

yxTT

yTT

cmUy

xTT

cmUx eeses

∂∂−∂

−=∂−∂

+∂−∂ )( s

2

spa

F

epa

F (7.47)

Similarly, differentiating Eqs. (7.45) and (7.46) with respect to y and x and taking their

sum gives

( ) ( )yxym

yUxm

xU eesLesL

∂∂−∂

−=∂−∂

+∂−∂ )( s

2

s

F

e

F ωωωωωω (7.48)

Let dimensionless variables

eisi

es

TTTT

−−

=1θ , eisi

es

ωωωω

θ−−

=2 , Fxxx =* ,

Fyyy =*

and substituting in Eqs.(7.47), (7.48),

( ) ( ) **1

2

*1

spa

FF*1

epa

FF

yxycmyUx

xcmyUx

∂∂∂

−=∂∂

+∂∂ θθθ

(7.49)

**2

2

*2

s

FFL*2

e

FFL

yxymyxU

xmyxU

∂∂∂

−=∂∂

+∂∂ θθθ

(7.50)

With

Li-Zhi Zhang 142

( )epa

FFa cm

yUxNTU = , ( )spa

FFb cm

yUxNTU = , e

FFLa m

yxUNTU L= ,

s

FFLb m

yxUNTU L=

Eqs.(7.49) and (7.50) become

0**1

2

*1

*1 =

∂∂∂

+∂∂

+∂∂

yxyNTU

xNTU ba

θθθ (7.51)

0**2

2

*2

*2 =

∂∂∂

+∂∂

+∂∂

yxyNTU

xNTU LbLa

θθθ (7.52)

Initial condition: 1)0,0()0,0( 21 == θθ Mason [3] obtained a solution for Eq.(7.51) in the form of an infinite series by employing

the Laplace transformation as following:

n

n

baxNTUyNTU

nyxNTUNTUeyx ba ∑

∞

=

+−⎥⎦

⎤⎢⎣

⎡=

02

**)()(**

1 )!())((),(

**θ (7.53)

Since Eqs.(7.52) and (7.51) are the same form of differential equations. They are

identical if NTUa is replaced by NTULa and NTUb by NTULb. Therefore, the solution to Eq.(7.52) can be written as

n

n

LbLaxNTUyNTU

nyxNTUNTUeyx LbLa ∑

∞

=

+−⎥⎦

⎤⎢⎣

⎡=

02

**)()(**

2 )!())((),(

**θ (7.54)

The overall heat transferred in the exchanger is the integral of Eq.(7.53)

( )∫ ∫−=1

0

1

0****

1eisiFF ),( dydxyxTTyUxQ θ (7.55)

The overall moisture transferred in the exchanger is the integral of Eq.(7.54)

( )∫ ∫−=1

0

1

0****

2eisiFFL ),( dydxyxyxUM w θωω (7.56)


( )minpa

FF

cmyUxNTU = , ( )min

FFLL m

yxUNTU =

( )( )

maxpa

minpa1 cm

cmR = ,

max

min2 m

mR =

∫ ∫=Ω1

0

1

0****

11 ),( dydxyxθ , ∫ ∫=Ω1

0

1

0****

22 ),( dydxyxθ

( ) )(ε

eisiminpas TTcm

Q−

= , )(

εeisimin

WL ωω −

=m

M

Then

1sε Ω= NTU (7.57)

2LLε Ω= NTU (7.58)

∑∞

==Ω

0ba

ba1 )()(

))((1

nNTUfNTUf

NTUNTU (7.59)

∑∞

==Ω

0LbLa

LbLa2 )()(

))((1

nNTUfNTUf

NTUNTU (7.60)

where

∑=

−−=n

m

mz

mzezf

0 !1)( (7.61)

There are two cases: • If minpaepa )()( cmcm = , then NTUa=NTU, and NTUb= R1NTU

• If maxpaepa )()( cmcm = , then NTUa= R1NTU, and NTUb= NTU

In both cases, Eq.(7.57) can be replaced by the relationship

[ ]⎭⎬⎫

⎩⎨⎧

−⎥⎦

⎤⎢⎣

⎡−= ∑∑ ∑

=

−∞

= =

−n

m

mNTUR

n

n

m

mNTU

mNTURe

mNTUe

NTUR 0

1)(

0 01s !

)(1!

)(1)(

11ε

(7.62)

Li-Zhi Zhang 144

Similarly, for moisture effectiveness, we have

[ ]⎭⎬⎫

⎩⎨⎧

−⎥⎦

⎤⎢⎣

⎡−= ∑∑ ∑

=

−∞

= =

−n

m

mNTUR

n

n

m

mNTU

mNTURe

mNTUe

NTUR 0

L2)(

0 0

L

L2L !

)(1!

)(1)(

1L2Lε

(7.63) We can see at this step that the moisture effectiveness has the same form of expression

with sensible effectiveness. The only differences are that NTU is replaced by NTUL and R1 is replaced by R2.

For heat transfer, Eq.(7.62) is too complicated, since it has infinite series. Therefore, Kays and London [1] used following empirical equation to represent the sensible effectiveness as

⎥⎦

⎤⎢⎣

⎡ −−−= − 22.0

1

78.01

s1)exp(exp1

NTURNTURε (7.64)

Similar to the deduction of Eq.(7.64) for sensible heat transfer, the correlation for latent

effectiveness can be written as [3]

⎥⎦

⎤⎢⎣

⎡ −−−= −

222.0

L

278.0

LL

1)exp(exp1ε

RNTURNTU

(7.65)

NTUNTU ⋅= βL (7.66)

maxmin2 / mmR = (7.67) The value of relative humidity of plate in fresh air side, which is determined by latent

effectiveness (permeation rate), needs to be known before the calculation of diffusive resistance for membrane rm and the diffusive to convective ratio α. Iterations are performed to obtain a converged solution for φpf.

The latent effectiveness correlations for other flow arrangements, such as concurrent flow and counter flow, can also be derived from those corresponding correlations for sensible effectiveness, using the definition of Eqs.(7.66) and (7.67).

For concurrent flow,

2

2LL 1

)]1(exp[1ε

RRNTU

++−−

= (7.68)


0

0.1

0.2

0.3

0.4

0.5

0.6

0 10 20 30 40

Effe

ctiv

enes

s

V (L/min)

sensible Latent

0

0.1

0.2

0.3

0.4

0.5

0.6

0 10 20 30 40

Effe

ctiv

enes

s

V (L/min)

sensible Latent

Figure 7.2. Calculated and experimental sensible and latent effectiveness for a cross-flow membrane enthalpy exchanger. Solid lines are calculated, discrete points are measured.

For counter flow,

)]1(exp[1)]1(exp[1

ε2L2

2LL RNTUR

RNTU−−−

−−−= (R2<1) (7.69)

L

LL 1 NTU

NTU+

=ε (R2=1) (7.70)

The sensible effectiveness can also be calculated with above equation if NTUL is replaced

by NTU for every flow arrangement.

Validation To demonstrate the suitability of the correlations in predicting the effectiveness, sensible

and latent effectiveness are calculated with the proposed correlations and compared with experimental results in [5], as shown in Figure 7.2.

Li-Zhi Zhang 146

0

0.5

1

1.5

2

2.5

3

0 0.2 0.4 0.6 0.8 1

◊

NTU

L

C=0.1C=1C=10

Figure 7.3. Variations of NTUL with relative humidity for different membranes, NTU=4.2.

From Figure 7.2, it is obvious that both the sensible and the latent effectiveness are properly represented by the correlation from the present study. The largest discrepancies between the predictions by the correlation and the experimental data result from the cases with the smallest air flow rate, where the uncertainties of the experimental data are the biggest. The average errors between the predicted and experimental results are 7.3% and 8.6% for sensible and latent effectiveness respectively.

For a given exchanger, the sensible effectiveness is a fixed value at the specified flow rates. However the latent effectiveness will be affected by the two important dimensionless factors proposed in this study: the ratio of diffusive to convective resistance (α), and the ratio of total number of transfer units of moisture to that of sensible (β). The values of α and β are in turn affected by the membrane material types and operating inlet conditions.

Figure 7.3 demonstrates the variations of NTUL, when NTU is kept constant, with different inlet humidities. The value of NTUL decreases and increases for first-type (C<1) and third-type (C>1) materials, with increasing inlet humidity respectively. The trends of resulting latent effectiveness are the same as those of NTUL, which can be deduced from Eq.(7.30), see Figure 7.4. For the linear type material (C=1), the NTUL and the latent effectiveness will not change with the outside conditions. The number of transfer units for moisture would keep at 0.45 times of that for sensible heat for this material.

The above discussions suggest that an total heat exchanger with linear type membrane cores always performs better than those with other membrane cores. For example, to obtain a latent effectiveness of 0.6 under an inlet humidity of 50%, NTUL should be at least 2.0, which means that the minimum values of NTU required for the exchanger are: 4.44 with linear type; 8 with third-type; and 11.1 with first-type membrane. A smaller NTU usually makes the total heat exchanger more compact and cheaper.


0

0,1

0,2

0,3

0,4

0,5

0,6

0 0,2 0,4 0,6 0,8 1

Relative Humidity π

Late

nt e

ffec

tiven

ess (

πL) C=0.1

C=1C=10

Figure 7.4. Latent effectiveness for three types of membranes, NTU=4.2.

It should be kept in mind that the calculation of plate moisture resistance by Eq.(7.26) is rather complicated, though it discloses many material and operating conditions. In real applications, to simplify analysis, moisture diffusivity is directly measured form permeation test. In such cases, an equivalent mass transfer coefficient is obtained. Plate mass resistance is written as

ewp

pδ

Dr = (7.71)

where Dwp

e is called the equivalent mass diffusivity (m2/s), which is calculated by humidity ratio gradients between the two side of a plate, rather than by water uptake gradients. It has already reflected properties like sorption, humidity, temperature, etc. The equivalent mass transfer coefficient 1/rp is usually directly measured by permeation tests. Then the performance of a total heat exchanger is estimated by ε–NTU correlations, by substituting the obtained mass resistance rp in Eq.(7.25). Certainly accuracies will be comprised since the operating conditions of permeation test are different from operating conditions of total heat exchangers. However it’s simple.

Comparing Eqs(7.28), (7.29) and (7.71), we have

ψρρ wp

a

pewp

DD = (7.72)

Li-Zhi Zhang 148

7.3. CORRELATIONS FOR PLATE-FIN TOTAL HEAT EXCHANGER For plate-fin total heat exchangers, heat transfer area on both sides of a plate is extended

by fins. Therefore the extended surface and changed heat transfer coefficients should be considered. Heat conduction through the plate is in equilibrium with the convective heat transfer on both sides. The equilibrium on a whole exchanger basis can be expressed by

( ) ( ) ( )pepfpp

pfff δλ

TTA

TThA −=− (7.73)

( ) ( ) ( )pepfpp

peee δλ

TTA

TThA −−=− (7.74)

where Ap is total transfer area (m2) of plates; λp is heat conductivity of plate (kWm-1K-1); δ is thickness of plate (m); (hA)f is the fresh side convective heat transfer coefficient times total fresh air side transfer area. The convective heat transfer coefficients in plate-fin ducts can be obtained from previous sections. Total transfer area includes plate and fins area.

Moisture diffusion through the plate is in equilibrium with the convective mass transfer on two surfaces. The equations can be expressed by

( ) ( ) ( )pepfwppp

pfffa θθδ

ρωωρ −=−

DAkA (7.75)

( ) ( ) ( )pepfwppp

peeea θθδ

ρωωρ −−=−

DAkA (7.76)

where Dwp is water diffusivity in plate material (m2/s). If the equivalent moisture diffusivity in plate Dwp

e is used, then the above two equations can be re-written as

( ) ( ) ( )pepf

ewpp

pfff ωωδ

ωω −=−DA

kA (7.77)

( ) ( ) ( )pepf

ewpp

peee ωωδ

ωω −−=−DA

kA (7.78)

The overall number of transfer units for sensible heat transfer and moisture transfer are

( )( )

( )( )

ep

tot

fp

tot

GcUA

GcUA

NTU == (7.79)


( )( )

( )( )e

totLa

f

totLa ρρG

AUG

AUNTU L == (7.80)

respectively, where

( ) ( ) ( ) 1e

1pp1

f1

tot

λ −−

−− +⎟⎟⎠

⎞⎜⎜⎝

⎛+= hA

AhAUA

δ (7.81)

( ) ( ) ( ) 1e

1

wpp1f

1totL

−

−

−− +⎟⎟⎠

⎞⎜⎜⎝

⎛+= kA

DAkAAU

e

δ (7.82)

Generally, the two streams have the same structure and exchanger area, therefore

finpef AAAA +== (7.83)

The equivalent moisture diffusivity in plate Dwp

e, is the same definition as Eq.(7.72). At this step, the effectiveness can be calculated from ε–NTU correlations developed for parallel-plates exchangers.

Table 7.2. Structural and physical parameters of the total heat exchanger

Name Symbol Unit Value Number of channels for each flow n 115 Half duct height a mm 1 Half duct width b mm 2.5 Hydrodynamic diameter Dh mm 1.66 Exchanger length xF, yF mm 185 Plate thickness δ μm 55 Equivalent diffusivity, paper Dwp

e m2/s 2.97e-7 Equivalent diffusivity, membrane Dwp

e m2/s 8.92e-6 Heat conductivity, paper λp Wm-1K-1 0.38 Heat conductivity, membrane λp Wm-1K-1 0.44 Heat conductivity of air λa Wm-1K-1 0.0263 Density of air ρa kg/m3 1.18 Plate density, paper ρp kg/m3 860 Plate density, membrane ρp kg/m3 800 Moisture diffusivity in air Dva m2/s 2.82×10-5 Plate partition coefficient kP 0.58 Kinematic viscosity ν m2/s 15.89e-6 Volumetric air flow rates V m3/h 150

Li-Zhi Zhang 150

7.4. AN APPLICATION EXAMPLE

Problem Considering two plate-fin total heat exchangers. The schematic is shown in Figure 4.2.

The geometries are the same and they are listed in Table 7.2. The first exchanger uses paper as the plates, and the second one uses membranes as the plates. The measured equivalent moisture diffusivity from permeation tests are: Dwp

e=2.97e-7m2/s for paper and Dwpe=8.92e-

6m2/s for membrane. The task now is to estimate the sensible and latent effectiveness of the two total heat exchangers.

Solution

Air mean velocity in the channels

smxan

Vu /98.0185.0*002.0*115*3600

150)2(3600 F

a ===

Reynolds number

7.98689.15

366.1*98.0Re ha =−

−==

eeDu

ν<<2300

Therefore the flow is laminar flow. The problem is laminar flow in plate-fin ducts. Duct aspect ratio

4.022

==baτ

Let’s first consider the paper exchanger. It uses paper as the plate and fin material. The

fin materials is the same as the plate material. Fin heat conductance parameter

( ) 40.0002.0*0263.0

6e55*38.02λδλ f

s =−

==Ωaa

Fin mass conductance parameter

( ) ( ) 04e9.2002.0*)582.2(

)797.2(*)655(2

δ2ρ

δρ

va

ewp

vaa

pwppL ≈−=

−−−

===Ωe

eeaD

DaDkD


As seen, the equivalent diffusivity considers the partition coefficient, density ratio, etc inside.

ψρρ

ρρ

a

wpp

a

pwppewp

DkDD ==

Based on the calculated aspect ratio and fin conductance parameters, by referring to

Table 4.3, we have

95.0L =Nu

705.0L =Sh ( ) 21.11Re =f

12

h

aL KWm05.1500166.0

0263.0*95.0λ −−===D

Nuh

m/s012.000166.0

582.2*705.0D

h

vaL =−

==e

DSh

k

Friction coefficient

1136.07.9821.11

==f

Pressure drop

Pa7.2898.0*18.1*00166.0

185.01136.0*2ρ2 22aa

h

F ===Δ uDx

fP

Each duct has two plates area on both sides, so the total plate area

2FFp m87.7185.0*185.0*115*22 === ynxA

For the considered duct geometry, the fin to plate area ratio 29.1=α Air side transfer area

2ef m02.18*)1( =+== pAAA α

Consequently we have

Li-Zhi Zhang 152

( ) ( ) ( )

39.7)6e55

3e38.0*87.7()02.18*01505.0(2

δλ

11

1e

1pp1

f1

tot

=−

−+×=

+⎟⎟⎠

⎞⎜⎜⎝

⎛+=

−−

−−

−− hAA

hAUA

( ) 2tot kWm135.0 −=UA

( ) ( ) ( )

78.32)6e55

7e97.2*87.7()02.18*012.0(2 11

1e

1ewpp1

f1

totL

=−

−+×==

+⎟⎟⎠

⎞⎜⎜⎝

⎛+=

−−

−

−

−− kADA

kAAUδ

( ) /sm031.0 3totL =AU

( )( ) 72.2

005.1*3600/150*18.1135.0

fp

tot ===GcUA

NTU

( )( ) 744.0

3600/150*18.1031.0*18.1ρ

f

totLaL ===

GAU

NTU

The calculated effectiveness with varying NTU are drawn in Figure 7.5 for a cross flow

heat exchanger, based on Eq.(7.37).

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

0 1 2 3 4 5 6 7

NTU

Effe

ctiv

enes

s

Figure 7.5. Effectiveness-NTU for a cross flow heat exchanger.


According to the figure, when NTU=2.72, the sensible effectiveness is

67.0ε =s When NTUL=0.744, the latent effectiveness is

40.0ε L = As seen, the latent effectiveness with paper is around 0.40. Let’s consider the second exchanger. Based on the same procedure, when membrane is

used as the plate and fin materials, the results are NTU=2.72, 67.0ε =s

NTUL=2.03, 62.0εL = Pressure drop is the same with the first exchanger. As seen, the sensible effectiveness of membrane exchanger is the same as the paper

exchanger, but the membrane exchanger’s latent effectiveness is 55% higher than paper exchanger. The main reason is that the moisture diffusivity in membranes is higher.

7.5. CONCLUSION Heat and moisture transfer coefficients in total heat exchangers have been analyzed. The

ε–NTU method has been extended to total heat exchangers. Example shows that it is an efficient yet simple way to estimate total heat exchanger performance. It also provides a design tool for total heat exchangers, either in the form of parallel-plates or plate-fin ducts.

REFERENCES

[1] Kays, W.M.; Crawford, M.E. Convective heat and mass transfer. New York: McGraw-Hill Inc.; 1993. pp.432-435.

[2] Incropera, F.P.; Dewitt, D.P. Fundamentals of heat and mass transfer, 3rd edn. New York: Wiley; 1990. pp.416-420.

[3] Zhang, L.Z.; Niu, J.L. Effectiveness correlations for heat and moisture transfer processes in an enthalpy exchanger with membrane cores. ASME Journal of Heat Transfer, 2002, 122, 922-929.

[4] Niu, J.L.; Zhang,L.Z. Membrane-based enthalpy exchanger: material considerations and clarification of moisture resistance. Journal of membrane science, 2001, 189, 179-191.

Li-Zhi Zhang 154


[6] Taylor, R..; Krishna R.. Multicomponent mass transfer. New York: John Wiley & Sons, Inc.; 1993.

Chapter 8

NUMERICAL SIMULATION OF TOTAL HEAT EXCHANGERS

ABSTRACT

Computer modeling has become en efficient tool to analyze total heat exchangers. Correlations are simple, however, they cannot disclose the insight into the mechanisms of heat and moisture transport. In total heat exchangers, heat and moisture are strongly coupled, which makes performance analysis difficult. In this chapter, a detailed mathematical modeling is provided both for the simple parallel-plates exchanger and for the complicated plate-fin structure. Special efforts are spent in the methodology in mathematical modeling.

NOMENCLATURE a half duct height (m) As cross section area (m2) b half duct width (m) C shape factor for the isotherm cp specific heat (kJkg-1K-1) D diffusivity (m2/s) Dh Hydrodynamic diameter (m) f Friction coefficient h convective heat transfer coefficient (kWm-2K-1) k convective mass transfer coefficient (m/s) K total transfer coefficient (kWm-2K-1 for heat and m/s for mass)

vm local emission rate (kgm-2s-1) NTU Number of transfer units Nu Nusselt number P pressure (Pa)

Li-Zhi Zhang 156

Pe perimeter (m) Re Reynolds number RH air relative humidity Sc Schmidt number Sh Sherwood number T temperature (K) u velocity (m/s) V volumetric flow rate (m3/s) W water uptake (kg/kg) Wmax maximum water uptake of membrane material (kg/kg) x, y, z coordinates (m)

Greek letters ε effectiveness δ plate or fin thickness (m) ρ density of dry air (kg/m3) ξ dimensionless humidity θ dimensionless temperature μ dynamic viscosity (Pas) ω humidity ratio (kg moisture/kg air) λ thermal conductivity (kWm-1K-1)

Superscripts * dimensionless ‘ exhaust air duct

Subscripts 1 fresh air side 2 exhaust air side a air b bulk e exhaust G geometric f fresh i inlet L local, latent m mass, mean, membrane o outlet p plate s surface, sensible tot total v vapor

Numerical Simulation of Total Heat Exchangers 157

8.1. INTRODUCTION Numerical simulation is an efficient tool to simulate heat mass transfer. In fact

computational modeling of membrane-related energy systems has been actively conducted recently [1-9]. In previous chapter, the effectiveness-NTU correlations have been proposed for total heat exchangers. Though it’s simple and convenient enough for engineers to evaluate system performance, detailed analysis of the local heat and mass transfer in the exchanger cannot be performed.

In this chapter, a detailed finite difference modeling of the coupled heat and moisture transfer in total heat exchangers will be addressed. The effects of non-ideal boundary conditions on the convective heat and mass transfer between the fluids and the solid surfaces will be studied. The object of interest is a membrane-based cross-flow total heat exchanger, as illustrated in Figure 8.1. As previously introduced [10], the device is like an air-to-air sensible heat exchanger. But in place of traditional metal heat exchange plates, novel hydrophilic membranes, which can exchange both heat and moisture simultaneously, are used as the heat mass transfer media. The boundaries conditions for channels are neither uniform temperature nor uniform concentration, but are naturally formed by heat mass coupling. Therefore the established Nusselt and Sherwood numbers for parallel plates-channels are not used.

8.2. PARALLEL-PLATES EXCHANGER

Governing Equations A schematic of the unit cell for model set up is shown in Figure 8.2. The fresh air and the

exhaust air flow through the passages in a cross flow arrangement. Geometries of ducts: height 2a, width 2b, membrane thickness δ, fresh duct length xF, exhaust air duct length yF. The fresh air is usually hot and humid and the exhaust air is usually cool and dry. Due to symmetry and balanced flow, the equations governing the transport phenomena are the same for the two passages. In the following, governing equations will be set up for the fresh air. Then the different coordinate system for the exhaust flow will be given.

In most applications in air conditioning, Reynolds numbers are far below 2000, therefore it can be assumed laminar flow. Other assumptions include: (1) Adsorption of water vapor and membrane material is in equilibrium adsorption-state; (2) Both the heat conductivity and the water diffusivity in the membrane are constants; (3) The heat of sorption is assumed constant and equal to the heat of vaporization; (4) It is hydrodynamically fully developed, but thermally and concentrationally developing; (5) The fluid is Newtonian with constant thermal properties. As seen, this investigation uses heat mass coupling to find the boundary conditions, rather than by assuming uniform temperature or concentration conditions.

Li-Zhi Zhang 158

Fresh inFresh out

Exhaust in

Exhaust outMembrane

Fresh inFresh out

Exhaust in

Exhaust outMembrane

Figure 8.1. A cross-flow membrane-based total heat exchanger.

Dimensionless equations governing momentum, energy and mass conservation in the duct are written by [11,12]

042h

2

2*

*22

2*

*2

=+∂

∂⎟⎠⎞

⎜⎝⎛+

∂

∂Db

yu

ab

xu

(8.1)

2*

22

2*

2

* yab

xzU

h ∂

θ∂⎟⎠⎞

⎜⎝⎛+

∂

θ∂=

∂θ∂

(8.2)

2*

22

2*

2

* yab

xzU

m ∂

ξ∂⎟⎠⎞

⎜⎝⎛+

∂

ξ∂=

∂ξ∂

(8.3)

where the dimensionless velocity

2h

*

)/( DdzdPuu μ

−= (8.4)

Dimensionless temperature

eifi

ei

TTTT

−−

=θ (8.5)

Dimensionless humidity

eifi

ei

ωωωω

−−

=ξ (8.6)


x

y

z

Fresh air in

Exhaust air in Membranex

y

z

Fresh air in

Exhaust air inx

y

z

Fresh air in

Exhaust air in Membrane

Figure 8.2. The coordinate system of the unit showing one membrane and two neighboring flow passages.

where T is temperature and ω is humidity. Subscripts “fi” and “ei” denotes fresh air in and exhaust air in respectively.


bxx

2* = (8.7)

ayy

2* = (8.8)

RePrDzz

h

*h = (8.9)

ReScDzz

h

*m = (8.10)

Hydraulic diameter

PeA

D sh

4= (8.11)

where As is the cross section area of the duct (m2)

Li-Zhi Zhang 160

, Pe is the perimeter of the duct (m). Pr is Prandtl number, and Sc is Schmidt number. The dimensionless axis z* has a different definition for heat transfer and mass transfer.

In Eqs.(8.2) and (8.3), velocity coefficient U is defined by

2h

2

*m

* 4Db

uuU = (8.12)

where u*


s

**

A

dAuum

∫∫= (8.13)

The characteristics of fluid flow in the duct can be represented by the product of the

friction factor and the Reynolds number as

*hm

2m

h 2

21

)(mu

Du

u

dzdPD

fRe =⎟⎟⎠

⎞⎜⎜⎝

⎛μ

ρ

⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜

⎝

⎛

ρ−= (8.14)

Dimensionless bulk temperature

∫∫∫∫ θ

=θdAu

dAuzh *

**

b )( (8.15)

Dimensionless bulk humidity

∫∫∫∫ ξ

=ξdAu

dAuzm *

**

b )( (8.16)

Nusselt number

λ= hhDNu (8.17)

Sherwood number

va

h

DkD

Sh = (8.18)


where h and k are convective heat transfer coefficient (kWm-2K-1) and convective mass transfer coefficient (m/s) between fluid and wall, respectively. The local Nusselt and Sherwood numbers may change from point to point on a duct surface. It is more practical to evaluate the peripherally mean local Nusselt and Sherwood numbers along the duct.

An energy balance in a control volume of length Δz [12] helps to deduce the peripherally mean local heat transfer coefficient

( ) *h

b

hbwL 4

1zD

hΔ

θΔλθ−θ

−= (8.19)

and the peripherally local Nusselt number

( ) *h

b

bwL 4

1dzd

Nuθ

θ−θ−= (8.20)

Average Nusselt number from 0 to zh

*

*h0 L*m

*1 dzNuz

Nu hz

h∫= (8.21)

Similarly, for mass balance we have

( ) *m

b

h

va

bwL 4

1zD

Dk

ΔξΔ

ξ−ξ−= (8.22)

( ) *m

b

bwL 4

1dzd

Shξ

ξ−ξ−= (8.23)

*m0 L*

mm

*m1 dzSh

zSh

z

∫= (8.24)

As will be discussed later, the local Nusselt number and the local Sherwood number

decrease asymptotically from very high values near the entrance of a tube to certain fully developed values at the end of thermal entry length. Under uniform temperature (uniform mass concentration) boundary condition, the fully developed value is denoted as NuT (ShT for mass). Under uniform heat flux (mass flux) boundary condition, it is denoted as NuH (ShH ). In this study, under the real convective flow boundary conditions, the fully developed value is denoted as NuC (ShC for mass transfer).

Inlet and boundary conditions. The governing equations for the exhaust air stream are in the same form with the fresh air, however the coordinate systems are different. If we use x’*,

Li-Zhi Zhang 162

y’*, z’ to replace the coordinates x*, y*, z, in Eqs.(8.1)-(8.3), and (8.9)-(8.10), respectively, then we get the equations for the exhaust stream. The relations of the two coordinates systems are

⎪⎩

⎪⎨

⎧

−==

=

xbzyyzx

2'''

**

**

(8.25)

Inlet conditions for fresh air

0*h =z , 1=θ (8.26)

0*

m =z , 1=ξ (8.27) Inlet conditions for exhaust air

0*h =z , 0=θ (8.28)

0*

m =z , 0=ξ (8.29) Boundary conditions of two air streams for velocity u*=0, on all walls of the duct (8.30) Adiabatic boundary conditions for fresh air

0* =x or 1* =x , 0** =∂

ξ∂=

∂θ∂

xx (8.31)

and for exhaust air

0'* =x or 1'* =x , 0'' ** =

∂ξ∂

=∂

θ∂xx

(8.32)

The boundary conditions on other two tube wall surfaces must be obtained numerically,

due to the interactions of temperature and humidity between the air streams and membrane materials.

Temperature boundary conditions on membrane surfaces, for fresh air

0* =y or 1* =y , ( ) 1m** , θ=θ Gzx (8.33)


For exhaust air

0'* =y or 1'* =y , ( ) 2m** ',' θ=θ Gzx (8.34)

where term ( )** , Gzxθ refers to air temperature adjacent to membrane surface at point (x*, zG*), subscript “m” refers to membrane, and “1” and “2” refer to fresh side and exhaust side at the same point of membrane, respectively. Variable zG* denotes membrane geometric position

. bzzG 2

* = (8.35)

Due to the small thickness in membrane (around 100μm, measured thermal conductivity

0.127Wm-1K-1), temperature differences between the two sides of a membrane are rather small. Heat released on the adsorption side of the membrane (fresh air side) could be balanced by the heat absorbed on the desorption side (exhaust side) of the membrane [13]. In fact, previous investigations disclosed that the temperature differences are in the order of 10-4 °C [14], meaning that Eqs.(8.33), (8.34) can be expressed by a single equation by

mθ=θ=θ m2m1 (8.36) The value of θm is not a fixed value. Rather, it is a result of local couplings between the

two streams on membrane surface. It should be also noted that in some cases when the solid plate is thick (such as dehumidification process with a sorbent plate), temperature difference between the two surfaces may be large. Under such conditions, heat transfer equations in the solid should be set up to account for the thickness effect and the adsorption-desorption thermal effect [15].

Mass boundary conditions on membrane surfaces, for fresh air

0* =y or 1* =y , ( ) vG mzxq =** , (8.37) For exhaust air

0'* =y or 1'* =y , ( ) vG mzxq =** ',' (8.38)

where vm (kgm-2s-1) is the moisture emission rate through membrane at point (x*, zG*), and it is determined by diffusion equation in membrane as

δ−

ρ= 21vmmv

mm WWDm (8.39)

Li-Zhi Zhang 164

where ρm and Dvm are density of membrane and moisture diffusivity in membrane, respectively. Variable W is water uptake in membrane (kg moisture/kg dry membrane), and it is expressed by a general sorption equation as

RHCCW

W/1

max

+−= (8.40)

where Wmax is the maximum water uptake of membrane material (kg/kg); C is a constant named the shape factor for the material; RH is air relative humidity. This sorption curve directly links water content to RH, a variable can be directly measured.

As seen from Eqs.(8.37), (8.38), mass transfer in the fresh air and in the exhaust air are coupled together by moisture emissions through membrane.

The relative humidity is calculated by humidity ratio and temperature as [15]

6

/5294

10

TeRH=

ω (8.41)

where T is in K.

Moisture emission on membrane surface on fresh air or exhaust air side is also calculated by

ayyDm

2,0vaav

ωρ=∂

∂−= (8.42)

Emission rate q is not a constant value either. However, the values are attainable from

couplings of Eqs.(8.37)-(8.42), plus governing equations (8.1)-(8.3).

Numerical Methods Numerical methods are always important for mathematical modeling. The objective of

the current numerical work is to find a solution for the model. To fulfill this task, two steps should be implemented: (1) the solution of the governing equations for various components, fresh air, exhaust air, and membrane; and (2) the couplings of different components to find the values on membrane boundary surfaces.

The partial differential equations for momentum, energy, and mass transport, Eqs.(8.5)-(8.7), are discretized by means of a finite volume method. According to this technique, during the calculation values are stored only at the control volume (CV) centers (grid points). The values at all CV boundaries must be expressed. For the convective terms, an upwind differencing scheme is used, by replacing the boundary value by the values of the neighbouring upstream CV. For the Peclet numbers involved in the present problem, this approximation is close to reality. The diffusive fluxes at the CV boundary are approximated


by assuming a linear variation of variable between two grid points. With this procedure, an algebraic tri-diagonal matrix will be formed to find the values at each node.

Table 8.1. Properties of the membrane used

Properties Unit Values ρm kg/m3 856 Dvm m2/s 3.4×10-11 δ μm 105 Wmax kg/kg 0.28 C 0.85

There are totally three components: fresh air, exhaust air, and membrane, to be

calculated. When solving equations for one component, the values in other components are assumed already-known values and will give assumed boundary conditions to this component. The solution of three components is repeated until the solutions are converged, i.e., differences of old and new temperature and humidity values at each grid node are within a lower limit.

After these procedures, all the governing equations are satisfied simultaneously. The temperature and humidity fields on membrane surfaces are the obtained.

Characteristics of the membrane used for the modeled total heat exchanger are listed in Table 8.1. The nominal operating conditions: fresh air inlet 35°C and 0.025kg/kg; exhaust air inlet 25°C and 0.010kg/kg.

After the inlet and outlet temperature and humidity are calculated, the sensible exchange effectiveness and the latent exchange effectiveness can be calculated for a balanced flow

eifi

fofis TT

TT−−

=ε (8.43)

eifi

fofiL ωω

ωωε

−−

= (8.44)

The outlet temperature and humidity could also be calculated from the model. It is

obvious that the sensible effectiveness is equal to the dimensionless outlet temperature of exhaust air, while the latent effectiveness is equal to the dimensionless outlet humidity of exhaust air. It should be also noted that since there is only one membrane in the exchanger, the boundary conditions at y*=1 should be modified to have zero heat and mass fluxes.

Experimental Work An experimental set up has been built in the laboratory of SCUT to study the heat and

mass transport in a membrane based cross flow exchanger. This is a single-plate small scale total heat exchanger. The whole set-up is shown in Figure 8.3. Ambient air is humidified and

Li-Zhi Zhang 166

is driven to a heating/cooling coil in a hot/cool water bath. The cooling coil can also act as dehumidifier when necessary. After the temperature and humidity reach the set points, it is then drawn through the exchanger for heat and moisture exchange. Another flow is driven directly from ambient to the exchanger as the second flow. A Cellulose Acetate membrane is sandwiched by two stainless steel shell. Two air passages on both sides of membrane are formed, which is like a one-plate plate-and-shell heat exchanger. A schematic of the exchanger is shown in Figure 8.4. Photograph of the real two half shells is shown in Figure 8.5. In the test, a 10mm thick insulation layer is placed on the inner surface of the shell to prevent heat dissipation from the shell to the surroundings. After the exchanger and tubes are installed, additional insulation is added on the outside surfaces to minimize heat losses from the unit. The dimensions of the air passages formed by the shell inner insulation and the membrane are: 2b=10cm, 2a=2mm.

Humidifier

Vacuum Pump

Valve Heat Mass Exchanger

Hot/cold water bath

Stream 2 in

Temperature and Humidity Sensor

Stream 1 out

Stream 2 out

Stream 1 in

Ambient

Ambient

Ambient

Flow Meter

Humidifier

Vacuum Pump

Valve Heat Mass Exchanger

Hot/cold water bath

Stream 2 in


Stream 1 out

Stream 2 out

Stream 1 in

Ambient

Ambient

Ambient

Flow Meter

Figure 8.3. Experimental set-up.

A − A

2 b

A A

2 a

H a lf sh e ll

M em b ran e

2 b

A − A

2 b

A A

2 a

H a lf sh e ll

M em b ran e

2 b

Figure 8.4. Structures of the exchanger comprised by two symmetric half cells.


Figure 8.5. Photo of the two stainless steel half shells.

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0 5 10 15 20 25 30

Effe

ctiv

enes

s

V (L/min)

Sensible, calculated Sensible, experimental Latent, calculated Latent, experimental

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0 5 10 15 20 25 30

Effe

ctiv

enes

s

V (L/min)

Sensible, calculated Sensible, experimental Latent, calculated Latent, experimental

Figure 8.6. Sensible and latent effectiveness of the exchanger, experimental and calculated.

Li-Zhi Zhang 168

0.090.16

0.16

0.24

0.24

0.31

0.31

0.31

0.39

0.39

0.39

0.46

0.46

0.46

0.54

0.54

0.54

0.61

0.61

0.61

0.69

0.69

0.76

0.76

0.84

0.84

0.91

0.91

z*

x*

0 0.25 0.5 0.75 10

0.25

0.5

0.75

1

Fresh in

Exhaust in

zG*

0.090.16

0.16

0.24

0.24

0.31

0.31

0.31

0.39

0.39

0.39

0.46

0.46

0.46

0.54

0.54

0.54

0.61

0.61

0.61

0.69

0.69

0.76

0.76

0.84

0.84

0.91

0.91

z*

x*

0 0.25 0.5 0.75 10

0.25

0.5

0.75

1

Fresh in

Exhaust in

zG*

Figure 8.7. Distribution of dimensionless surface temperature on membrane surfaces.

0.160.24

0.31

0.31

0.39

0.39

0.46

0.46

0.54

0.54

0.54

0.61

0.61

0.61

0.69

0.69

0.76

0.76

0.76

0.84

0.840 .91

z1*

x*

0 0.25 0.5 0.75 10

0.25

0.5

0.75

1

Fresh in

E xhaust in

zG*

0.160.24

0.31

0.31

0.39

0.39

0.46

0.46

0.54

0.54

0.54

0.61

0.61

0.61

0.69

0.69

0.76

0.76

0.76

0.84

0.840 .91

z1*

x*

0 0.25 0.5 0.75 10

0.25

0.5

0.75

1

Fresh in

E xhaust in

zG*

Figure 8.8. Distribution of dimensionless humidity on membrane surface on fresh air side.


0.090.16

0.160.24

0.24

0.31

0.31

0.39

0.39

0.39

0.46

0.46

0.54

0.540.54

0.61

0.69

0.76

z1*

x*

0 0.25 0.5 0.75 10

0.25

0.5

0.75

1

Fresh in

Exhaust in

zG*

0.090.16

0.160.24

0.24

0.31

0.31

0.39

0.39

0.39

0.46

0.46

0.54

0.540.54

0.61

0.69

0.76

z1*

x*

0 0.25 0.5 0.75 10

0.25

0.5

0.75

1

Fresh in

Exhaust in

zG*

Figure 8.9. Distribution of dimensionless humidity on membrane surface on exhaust air side.

34.8

34.8

59.1

59.1

59.1

83.3

83.3

107.

6

131.

9156.2

180.5

204.8

277.6

zG*

x*

0 0.25 0.5 0.75 10

0.25

0.5

0.75

1

zG*

Fresh in

Exhaust in

34.8

34.8

59.1

59.1

59.1

83.3

83.3

107.

6

131.

9156.2

180.5

204.8

277.6

zG*

x*

0 0.25 0.5 0.75 10

0.25

0.5

0.75

1

zG*

Fresh in

Exhaust in

Figure 8.10. Distribution of heat flux on membrane surface (W/m2).

Li-Zhi Zhang 170

5.19E-05 5.19E-05

6.41E-05

6.41E-05

6.41

E-05

6.41E-05

7.62E-05

7.62

E-0

5

8.84E-05

8.84

E-0

5

1.00E

-04

1.13E

-041.25E-041.3

7E-0

4

1.49E-04

61E-04

22E-04

zG*

x*

0 0.25 0.5 0.75 10

0.25

0.5

0.75

1

Fresh in

Exhaust in

zG*

5.19E-05 5.19E-05

6.41E-05

6.41E-05

6.41

E-05

6.41E-05

7.62E-05

7.62

E-0

5

8.84E-05

8.84

E-0

5

1.00E

-04

1.13E

-041.25E-041.3

7E-0

4

1.49E-04

61E-04

22E-04

zG*

x*

0 0.25 0.5 0.75 10

0.25

0.5

0.75

1

Fresh in

Exhaust in

zG*

Figure 8.11. Distribution of mass flux on membrane surface (kgm-2s-1).

6

7

8

9

10

11

12

13

14

0 0.05 0.1 0.15 0.2

z *

Nus

selt

Num

bers

NuL

Num

zh*

6

7

8

9

10

11

12

13

14

0 0.05 0.1 0.15 0.2

z *

Nus

selt

Num

bers

NuL

Num

6

7

8

9

10

11

12

13

14

0 0.05 0.1 0.15 0.2

z *

Nus

selt

Num

bers

NuL

Num

zh*

Figure 8.12. Local and mean Nusselt numbers along flow direction, b/a=50.


6

7

8

9

10

11

12

13

14

0 0.05 0.1 0.15 0.2 0.25

z *

Sher

woo

d N

umbe

rs

ShL

Shm

zm*

6

7

8

9

10

11

12

13

14

0 0.05 0.1 0.15 0.2 0.25

z *

Sher

woo

d N

umbe

rs

ShL

Shm

zm*

Figure 8.13. Local and mean Sherwood numbers along flow direction, b/a=50.

During the experiment, air flow rate is changed, to have different Reynolds numbers. Humidity, temperature, and volumetric flow rates are monitored at the inlet and outlet of the exchanger. Equal air flow rates are kept for the two air streams. The uncertainties are: temperature ±0.1ºC; humidity ±1%; volumetric flow rate ±0.5%.

The calculated sensible and latent effectiveness is plotted in Figure 8.6. They are compared to the experimental data. The differences between the calculated and the tested are less than 7%. The model is validated.

Membrane Surface Values Contrary to ε–NTU correlations, with the model, besides the outlet variables, the

variables in the exchanger can also be calculated. The temperature, humidity, heat flux, and moisture flux across the membrane are calculated and plotted in Figures 8.7-8.11. From these figures, it can be shown that the variables on membrane surfaces are neither uniform values nor uniform fluxes. The temperature, humidity, heat flux, and mass flux exhibit a 2 dimensional distributed nature. The contour values of heat and mass fluxes decrease along the diagonal line like a ripple in a pool, whose center is at the point where the two flows intersect and enter the exchanger. The contour lines of temperature and humidity are parallel to diagonal line of the square membrane. The differences of humidity between the two surfaces of the membrane are demonstrably large, indicating large internal moisture resistance.

Nusselt and Sherwood Numbers Figures 8.12 and 8.13 show the local and mean Nusselt and Sherwood numbers along the

duct respectively. In the thermal and concentrational developing region, the local Nusselt and Sherwood numbers decrease sharply, from very high values to 2 lower limiting values

Li-Zhi Zhang 172

denoted by NuC and ShC. The flow is called thermally or/and concentrationally developed after that period. The calculated NuC and ShC for various aspect ratios are listed in Table 8.2. It is found that ShC≈NuC for most aspect ratios, indicating that heat mass analogy is satisfied. For comparison, the values of NuT and NuH from published references are also listed. It is found that NuH is usually 20% larger than NuT, and NuC is generally less than NuH. It is higher than NuT at larger aspect ratios, but 37% less than NuT at smaller aspect ratios. For parallel plates that have large aspect ratios, NuC could be approximated by NuT. Further, for hydrodynamically fully developed laminar flow in ducts, (fRe) is a constant. The calculated values of (fRe) for various aspect ratios are also listed in Table 8.2. As seen, the current study predicts the flow well.

It could be stressed that the moisture resistance through the membrane is larger than the internal thermal resistance, consequently sensible effectiveness is 25% larger than the latent effectiveness under current situations, though Nusselt numbers are almost equal to Sherwood numbers in the passages.

Correlations for local and mean Nusselt numbers:

( )( ) 3/2/PrRe045.01

/PrRe0668.0zDzD

NuNuh

hCL

++= (8.45)

The obtained boundary conditions are neither uniform flux nor uniform value boundary

conditions. In the thermal and concentrational developing region, the local Nusselt and Sherwood numbers decrease sharply, from very high values to 2 lower limiting fully developed values.

The fully developed Nusselt number NuC is generally less than NuH. It is higher than NuT at larger aspect ratios, but 37% less than NuT at smaller aspect ratios. For parallel plates that have large aspect ratios, NuC can be approximated by NuT.

Table 8.2. Fully developed (fRe) and Nusselt, Sherwood numbers for

various aspect ratios

b/a NuH NuT NuC ShC (fRe) (fRe) Sources Ref [7,8] Ref [7,8] (This study) (This study) Ref [7,8] (this study) 1.0 3.61 2.98 1.89 1.89 57.0 56.51 1.43 3.73 3.08 2.51 2.52 59.0 58.03 2.0 4.12 3.39 3.09 3.10 62.0 61.82 3.0 4.79 3.96 4.08 4.00 69.0 68.05 4.0 5.33 4.44 4.65 4.52 73.0 72.42 8.0 6.49 5.60 6.08 6.03 82.0 81.63 50.0 7.74 7.81 92.37 100.0 8.03 8.05 93.96 ∞ 8.23 7.54 96.0


The internal moisture diffusion resistance of the membrane is larger than the internal thermal resistance, therefore sensible effectiveness is 25% larger than the latent effectiveness, though Nusselt numbers are almost equal to Sherwood numbers in the passages.

8.3. PLATE-FIN EXCHANGER From above simulations, it is found that the fully developed Nusselt numbers and

Sherwood numbers under naturally balanced boundary conditions from heat mass transfer can be approximated by those under uniform temperature or concentration boundary conditions. That will simplify modeling in future. In this part, we will extend the numerical simulations to the more complicated plate-fin total heat exchangers. However, the boundary conditions will be simpler. The convective heat mass transfer between the membrane and air streams will use the fully developed Nusselt and Sherwood numbers developed for plate-fin ducts in Section 4.

Figure 8.14. Schematic of a cross-flow plate-fin heat exchanger with sinusoidal passages.

2b

2a

2b

2a

Figure 8.15. Geometries of a single sinusoidal plate-fin duct.

The modeled structure is shown in Figure 8.14, which is a popular plate-fin sinusoidal channels structure. Figure 8.15 shows the single duct geometry. As previously mentioned, it

Li-Zhi Zhang 174

is preferred because it has many virtues like it is stationary, compact, and easy to construct. The differences between total heat exchangers and other air-to-air heat exchangers are that in place of metal materials, current commercial total heat exchangers employ hygroscopic paper as the material for fins and plates, to transfer both sensible heat and moisture simultaneously.

Heat and Mass Transfer in Air Streams Actual heat mass transfer in the numerous channels is complicated. A meso-scopic model

is set up. Each channel cross section is represented by one temperature or humidity. The temperature and humidity vary along flow directions (x for fresh air and y for exhaust air) and the corresponding cross directions (y for fresh air and x for exhaust air) simultaneously. On each channel cross section, though temperature and humidity are two-dimensionally different locally, as studied for the parallel-plates channels, in this study for the whole exchanger, they are represented by a lumped parameter for each channel cross section. It can be considered as a semi-lumped parameter model. The two air streams, one hot and humid (fresh air), and the other cool and dry (exhaust air), exchange both sensible heat and moisture simultaneously in the exchanger in a cross flow arrangement. Two-dimensional heat mass transfer model can be set up to govern the energy and mass conservations in the two air streams [15]:

)( *f

*mfsf*

*f TTNTU

xT

−=∂∂

(8.46)

)( *e

*mese*

*e TTNTU

yT

−=∂∂

(8.47)

)ωω(ω *f

*mfLf*

*f −=

∂∂ NTU

x (8.48)

)ωω(ω *

e*meLe*

*e −=

∂∂

NTUy

(8.49)

where x is flow direction for fresh stream and y is flow direction for exhaust stream.

The dimensionless temperature and humidity are defined by

eifi

ei*

TTTT

T−−

= (8.50)

eifi

ei*

ωωωω

ω−

−= (8.51)


The dimensionless coordinates are defined by

F

*

xxx = (8.52)

F

*

yyy = (8.53)

where xF and yF are channel lengths for fresh air and exhaust air (m). Here xF=yF. The air side number of transfer units for heat and moisture are defined by

( )( )

fp

fsf Gc

hANTU = (8.54)

( )( )

ep

ese Gc

hANTU = (8.55)

( )

( )f

faLf

ρGkA

NTU = (8.56)

( )

( )e

eaLe

ρGkA

NTU = (8.57)

where k and h are air side convective mass transfer coefficient (m/s) and convective heat transfer coefficient (kWm-2s-1) respectively; G is air mass flow rate (kg/s); A is total transfer area including plates and fins (m2) for each stream; cp is specific heat (kJkg-1K-1). Subscripts “f” refers to fresh side and “e” refers to exhaust side; “s” refers to sensible and “L” refers to latent; “mf” refers to membrane surface on fresh side, and “me” refers to membrane surface on exhaust side.

The outlet temperature and humidity values are calculated by Eqs.(8.46) to (8.49). Then it’s convenient to calculate sensible and latent effectiveness from Eqs.(8.43) and (8.44). Humidity ratios can be converted to relative humidity from psychrometric chart.

The convective heat transfer coefficient and mass transfer coefficient can be calculated by

a

h

λ=

hDNu (8.58)

va

h

DkDSh = (8.59)

Li-Zhi Zhang 176

where Dva is vapor diffusivity in air (m2/s), Dh is the hydrodynamic diameter (m). For plate-fin channels of finite fin conductance, the fully developed Nusselt and Sherwood numbers are influenced by aspect ratios (a/b), and fin conductance parameters, as calculated in Chapter 6. This is quite different from the simple classical data of a sensible-only heat exchanger with infinite fin conductance [17,18]. Heat and Mass Transfer through Plates

Heat conduction through the plate is in equilibrium with the convective heat transfer on

both sides. A meso-scale approach is employed. Whether it’s paper or membrane, the equilibrium can be expressed by

( ) ( ) ( )pepfpp

pfff δλ

TTA

TThA −=− (8.60)

( ) ( ) ( )pepfpp

peee δλ

TTA

TThA −−=− (8.61)

where Ap is total transfer area (m2) of plates; λp is heat conductivity of plate (kWm-1K-1); δ is thickness of plate (m); (hA)f is the fresh side convective heat transfer coefficient times total fresh air side transfer area. The convective heat transfer coefficients in plate-fin ducts can be obtained from previous chapters. Total transfer area includes plate and fins area.

Moisture diffusion through the plate is in equilibrium with the convective mass transfer on two surfaces. The equations can be expressed by

( ) ( ) ( )pepfvppp

pfffa θθδ

ρωωρ −=−

DAkA (8.62)

( ) ( ) ( )pepfvppp

peeea θθδ

ρωωρ −−=−

DAkA (8.63)

where Dvp is moisture diffusivity in plate material (m2/s). If the equivalent moisture diffusivity in plate Dvp

e is used, then the above two equations can be re-written as

( ) ( ) ( )pepf

evpp

pfff ωωδ

ωω −=−DA

kA (8.64)

( ) ( ) ( )pepf

evpp

peee ωωδ

ωω −−=−DA

kA (8.65)

Moisture emission rate through the plate from the fresh air to the exhaust air


( )pepf

evpa

v ωωρ

−=δD

m (8.66)

In fact, the final sensible effectiveness and the latent effectiveness can be estimated from

the total number of transfer units with established correlations [7,8]. However, to know the details of heat and moisture transfer in the exchanger, detailed equations should be solved.

Boundary Conditions Fresh:

10*

f * ==xT (8.67)

1ω 0*f * =

=x (8.68)

Exhaust:

01*

e * ==xT (8.69)

0ω1

*e * =

=x (8.70)

Table 8.3. Structural and physical parameters of the total heat exchanger

Symbol Unit Value Symbol Unit Value n 116 A mm 1 xF, yF mm 185 B mm 2.4 δ μm 105 kP 0.59 Dvp

e, paper m2/s 2.97e-7 Da m2/s 2.82e-5 Dvp

e, membrane m2/s 8.92e-6 λa Wm-1K-1 0.0263 λp, paper Wm-1K-1 0.44 λp, membrane Wm-1K-1 0.42 ρa kg/m3 1.18 ρp, paper kg/m3 860 ρp, membrane kg/m3 810 RHfi 0.56 Tfi °C 35 RHei 0.52 Tei °C 25

Li-Zhi Zhang 178

Simulation Case Consider two total heat exchangers. One is paper-plate and paper fin, called unit 1.

Another one is paper fin and membrane plate, called unit 2. The structural and basic physical parameters of the two total heat exchangers are listed in Table 8.3.

Solution Procedure A finite difference technique is used to discrete the partial differential equations

developed for the air streams. The calculating domain is divided into a number of discrete nodes. Each node represents a control volume. The number of calculating node is 50 in x direction. An upstream differencing scheme is used for two air streams. The two air streams and the asymmetric membrane are closely coupled. Heat transfer and mass transfer are also related to each other. Therefore iterative techniques are needed to solve these equations. A description of the iterative procedure is as following: (1) assume initial temperature and humidity fields in the two streams. (2) Calculate the temperature and humidity values on membrane surfaces by Eqs. (8.60) through (8.65). (3) Taking the current values of temperature and humidity on membrane surfaces as the default values, get the temperature

A

A

B

B

Blowers

Nozzles & straightener

Heaters &Humidifiers

Total heatexchangerStraightener

Shells

exchanger

Sensors

Fresh air

Exhaust air

C

C

A

A

B

B

Blowers

Nozzles & straightener

Heaters &Humidifiers

Total heatexchangerStraightener

Shells

exchanger

Sensors

Fresh air

Exhaust air

C

C

Figure 8.16. Experimental set up for commercial scale total heat exchanger.


and humidity profiles in two air streams by solving Eqs.(8.46) through (8.49). (4) Go to (2), until the old values and the newly calculated values of temperature and humidity at all calculating nodes are converged.

After these procedures, all the governing equations are solved simultaneously. To assure the accuracy of the results presented, numerical tests were performed for the duct to determine the effects of the grid size. It indicates that 50 grids are adequate (less than 0.1% difference compared with 80 grids). The final numerical uncertainty is 0.1%.

When the temperature and humidity fields in the exchanger are calculated, the sensible and latent effectiveness are calculated using mean outlet values. These are numerically obtained data.

Another experimental test rig has been set up in SCUT. This is a real scale test rig for the measurement of sensible and latent effectiveness of commercial-scale total heat exchangers. The test rig is shown in Figure 8.16.

The purpose of the experiment is to measure the steady state heat and moisture transfer through the enthalpy exchanger cores, by the measurements of inlet and outlet temperature, humidity and air flow rates. The sensible and latent effectiveness and pressure drop are the performance indices.

Figure 8.17. The structure of the unit core which is inserted into the shell in the test rig in the test.

A schematic of the test-rig is shown in Figure 8.16. It shows the ducting work, air controlling units, and instrumentation. Figure 8.17 shows the unit core structure. For the test rig, two parallel air ducts with a 210×210mm cross section are assembled. Each duct is comprised of a variable speed blower, a wind tunnel, a set of nozzles, wind straightenners,

Li-Zhi Zhang 180

electric heating coils, steam humidification tubes, temperature and humidity sensors. An exchanger shell is designed to hold the core. The small converging wind tunnels produce steady, homogeneous, fully developed air flow to the exchanger. The heating power and the steam generation currents can be adjusted according to the set points temperature and humidity. After the air temperature and humidity are adjusted to the set points, the two ducts are connected to the two inlets of the exchanger shell respectively. The exchanger shell is designed to station the exchanger core and separate the cross flow two air streams. The cores, either all paper or paper-fin and membrane-plate, can be inserted into the quadrate cavity in the center of the shell. The whole test rig is built in a constant temperature and constant humidity room, so the inlet temperature and humidity can be controlled and maintained very well even under very hot and humid ambient weather conditions. A 15mm thick plastic foam insulation layer is pasted on the outer surfaces of the ducts and the shells to prevent heat dissipation from the system to the surroundings. Moisture dissipation from air stream to the surroundings is negligible since the duct and shell materials are highly hydrophobic and they could adsorb little moisture. The heat loss from the system is below 0.6%, and moisture loss is less than 0.5%.

Figure 8.18. Photo of the test rig for commercial scale total heat exchangers.


0.6

0.7

0.8

0.9

80 100 120 140 160 180 200 220

V (m3/h)

Sens

ible

effe

ctiv

enes

Unit 1Unit 2

Figure 8.19. Sensible effectiveness of the two plate-fin total heat exchangers under various air flow rates.

The completed test rig is demonstrated in Figure 8.18. A software is designed to balance the heat and moisture transfer in the whole test rig. The nominal operating conditions: fresh air inlet 35°C and 0.021kg/kg; exhaust air inlet 27°C and 0.012kg/kg. The corresponding inlet relative humidity (RH) is 59% and 54% for fresh air and exhaust air respectively. During the experiment, equal air flow rates are kept for the two ducts. The design air flow rates are 150m3/h. In the test, they are changed by variable speed blowers, to have different air velocities. Humidity, temperature, and volumetric flow rates are monitored at the inlet and outlet of the exchanger. Before and after each test, temperature and humidity sensors are calibrated with a Pt-100 temperature sensor and a chilled-mirror dew-point meter. Hot-wire anemometers that are used to measure the wind speed before and after the exchanger are compared with the air flow rates measured by nozzles. The offset is controlled to within 1% limit. Volumetric air flow rates are varied from 80m3/h to 210m3/h, corresponding to frontal air velocities from 0.27m/s to 0.78m3/h which are typical for commercial enthalpy exchangers. Air flow under such conditions is laminar, with Reynolds numbers not exceeding 200. A digital pressure differential gauge is used to measure the pressure drop across the tested core. The uncertainties are: temperature ±0.1ºC; humidity ±2%; volumetric flow rate ±1%, pressure drop ±1%. The final uncertainty is ±4.5% for sensible and latent effectiveness. Six sensors are uniformly positioned on the inlet and outlet surfaces of the shell to have a mean value of measurements. Due to the small channels in the core, outlet air from the core is quite evenly distributed. In other words, the core itself has the effect of another ideal wind straightener. Anyway, another 6 wind straighteners, which are made of plates with numerous evenly distributed small holes drilled, are installed in the ducts before and after the shell and nozzles to well distribute the wind. In addition, heat and mass balance between the fresh air and the exhaust air are checked. Their differences are controlled to be less than 0.1%. From these preparatory works, the test rig is considered to be reliable.

Li-Zhi Zhang 182

0.2

0.3

0.4

0.5

0.6

0.7

0.8

80 100 120 140 160 180 200 220

V (m3/h)

Late

nt e

ffect

iven

ess

Unit 1Unit 2

Figure 8.20. Latent effectiveness of the two plate-fin total heat exchangers under various air flow rates.

After the measurement of mean inlet and outlet temperature and humidity, the sensible and latent effectiveness are calculated. They are the experimentally obtained data. Figure 8.19 and 8.20 show the calculated and tested sensible and latent effectiveness, respectively. The membrane plate unit has a higher latent effectiveness than paper plate unit. The sensible effectiveness is the same for the two exchangers. Membrane is superior to paper.

The model can disclose details inside the channel that the experiment cannot. Temperature and humidity fields in the air streams are calculated for the nominal operating conditions. The dimensionless temperature of the plate is shown in Figure 8.21. The unit is paper-fin and membrane-plate.

As seen from these figures, the temperature of fresh air decreases along the flow in x direction, while the temperature of the exhaust increases along the flow in y direction. Because the two flows are in cross flow arrangement, the temperature profiles exhibit a two-dimensional nature. The plate temperature is almost equal to the average temperature of the fresh air and exhaust air, meaning little conductance resistance in plate.

The humidity on membrane surface in fresh air side is shown in Figure 8.22. To intensify moisture transfer in enthalpy exchangers, it’s necessary to use plate materials that have high moisture diffusivities.

The local moisture permeation rates through the membrane are shown in Figure 8.23 for the paper unit and Figure 8.24 for the membrane unit. In both cases, the highest moisture emission rates are located on the surfaces where the two air streams interconnect, since here the driving force is the highest. It is also observed that the emission rates through the membrane plate are far higher than those through the paper plate. This is just the reason that the unit 2 has 60% higher latent effectiveness than unit 1.


0.0410.087

0.087

0.132

0.176

0.176

0.223

0.223

0.268

0.268

0.310

0.310

0.356

0.356

0.356

0.398

0.398

0.398

0.438

0.438

0.438

0.478

0.478

0.478

0.518

0.518

0.518

0.55

9

0.559

0.559

0.59

9

0.599

0.599

0.63

9

0.63

9

0.63

9

0.679

0.67

9

0.67

9

0.71

9

0.71

9

0.71

9

0.75

9

0.75

9

0.79

9

0.79

9

839

0.83

9

0.83

9

0.88

0

0.88

0

0.92

0

0.92

0

0.96

0

x*

y*

0 0.25 0.5 0.75 10

0.25

0.5

0.75

1

Figure 8.21. Dimensionless temperature of membrane temperature.

0.1320.1760.223

0.268

0.268

0.310

0.310

0.356

0.356

0.356

0.398

0.398

0.438

0.438

0.438

0.478

0.478

0.478

0.518

0.518

0.518

0.559

0.559

0.559

0.59

9

0.59

9

0.63

9

0.63

9

0.63

9

0.67

9

0.67

9

0.67

9

0.71

9

0.71

9

0.75

9

0.75

9

0.79

9

0.79

9

0.83

9

0.83

9

0.88

0

0.92

0

x*

y*

0 0.25 0.5 0.75 10

0.25

0.5

0.75

1

Figure 8.22. Dimensionless humidity of membrane surface on fresh air side.

Li-Zhi Zhang 184

1.318E-05

7.598E-067.971E-06

8.343E-06

8.715E-06

9.088E-069.460E-069.833E-06

1.020E-051.058E-05

1.132E-051.207E-05

1.430E-05

x*

y*

0 0.25 0.5 0.75 10

0.25

0.5

0.75

1

Figure 8.23. Emission rate on plate surface for core 1 of paper plate (kgm-2s-1).

2.718E-05

1.390E-05

1.656E-05

2.187E-05

1.39

0E-0

5

2.187E-05

2.984E-05

4.047E-05

1.124E-05

x*

y*

0 0.25 0.5 0.75 10

0.25

0.5

0.75

1

Figure 8.24. Emission rate on plate surface for core 2 of membrane plate.


8.4. CONCLUSION Detailed mathematical models have been set up for analysis of coupled heat and moisture

transfer in total heat exchangers. Two structures are considered: parallel-plates and plate-fin ducts. Due to the couplings between heat and mass transfer, iterative numerical schemes are required. Though the procedures are complicated, they provide a tool for the detailed analysis of the inside heat mass transfer in the exchanger.

Both Temperature and humidity exhibit a two dimensional nature. The higher the moisture diffusivity is, the higher the latent effectiveness is. Simulated results consider all the material and operating conditions. They are more accurate.

REFERENCES

[1] Al-Sharqawi, H.; Lior, N. Conjugate computation of transient flow and heat and mass transfer between humid air and desiccant plates and channels. Numerical Heat Transfer Part A-Applications, 2004, 46, 525-548.

[2] Saman, W.Y.; Alizadeh, S. Modelling and performance analysis of a cross-flow type plate heat exchanger for dehumidification/cooling. Solar Energy, 2001, 70, 361-372.

[3] Yeh, H.M.; Chen, Y.K. Membrane extraction through cross-flow rectangular modules. Journal of Membrane Science, 2000, 170, 235-242.

[4] Dindore, V.Y.; Brilman, D.W.F.; Versteeg, G.F. Modelling of cross-flow membrane contactors: physical mass transfer processes. Journal of Membrane Science, 2005, 251, 209-222.

[5] Hagg, M.B. Membranes in chemical processing - A review of applications and novel developments. Separation and purification methods, 1998, 27, 51-168.

[6] Incropera, F.P.; Dewitt, D.P. Introduction to Heat Transfer, 3rd edn. New York: John Wiley & Sons; 1996. Chapter 8, pp. 416.

[7] Shah, R. K.; London, A.L. Laminar flow forced convection in ducts. New York: Academic Press Inc.; 1978.

[8] Kays, W.M.; Crawford, M.E. Convective Heat and Mass Transfer. New York: McGraw-Hill, Inc.; 1993.

[9] Mengual, J.I.; Khayet, M.; Godino, M.P. Heat and mass transfer in vacuum membrane distillation. International Journal of Heat and Mass Transfer, 2004, 47, 865-875.

[10] Zhang, L.Z.; Niu, J.L. Energy requirements for conditioning fresh air and the long-term savings with a membrane-based energy recovery ventilator in Hong Kong. Energy, 2001, 26, 119-135.

[11] Ebadian, M.A.; Zhang, H.Y. Fluid flow and heat transfer in the crescent-shaped lumen catheter. ASME Journal of Applied Mechanics, 1993, 60, 721-727.

[12] Niu, J.L.; Zhang, L.Z. Heat transfer and friction coefficients in corrugated ducts confined by sinusoidal and arc curves. International Journal of Heat Mass Transfer, 2001, 45, 571-578.

[13] Favre, E. Temperature polarization in pervaporation. Desalination, 2003, 154, 129-138. [14] Zhang, L.Z.; Jiang, Y.; Zhang, Y.P. Membrane-based Humidity Pump: performance

and limitations. Journal of membrane science, 2000, 171, 207-216.

Li-Zhi Zhang 186

[15] Niu, J.L.; Zhang, L.Z. Membrane-based enthalpy exchanger: material considerations and clarification of moisture resistance. Journal of Membrane Science, 2001, 189, 179-191.

[16] Zhang, L.Z. Heat and mass transfer in a cross flow membrane-based enthalpy exchanger under naturally formed boundary conditions. International Journal of Heat Mass Transfer, 2007, 50, 151-162.

[17] Zhang, L.Z. Thermally developing forced convection and heat transfer in rectangular plate-fin passages under uniform plate temperature. Numerical Heat Transfer, Part A-Applications, 2007, 52, 549-564.

[18] Zhang, L.Z. Laminar flow and heat transfer in plate-fin triangular ducts in thermally developing entry region. International Journal of Heat Mass Transfer, 2007, 50, 1637-1640.

Chapter 9

NOVEL MEMBRANES FOR TOTAL HEAT EXCHANGER

ABSTRACT

Stationary total heat exchangers use novel vapor-permeable membranes for simultaneous heat and moisture transfer. To improve performance, air side intensification and material side intensification should be taken simultaneously. Novel membranes that have high vapor permeability are the key factor to the success of commercial total heat exchangers. In this chapter, two novel membranes are developed for total heat exchangers. They are composite supported liquid membrane and hudrophobic-hydriphilic composite membrane. Their performances and characterization are conducted.

NOMENCLATURE A area (m2) c volumetric concentration of CO2 (m3 CO2/m3 air) C Concentration (KG/M3) D diffusivity (m2/s) Dh hydrodynamic diameter (m) dp pore diameter (m) F gas flow through a single pore (kg/s) G flow coefficient [kgm/(sPa)] Hd duct height at inlet (m) J emission rate (kgm-2s-1) k convective mass transfer coefficient (m/s) kB Boltzmann constant (1.38×10-23 J/K) Kn Knudsen number kp Henry constant (kgm-3Pa-1) K total transfer coefficient (m/s) L height of air gap (m) m molality of electrolyte (mol LiCl /kg water)

Li-Zhi Zhang 188

M molecule weight (kg/mol) N number of pores per unit area (m-1) NTU Number of Transfer Units p partial pressure (Pa) p0 Saturation pressure (Pa) P total pressure (Pa) Pe permeability (kgm-1s-1)/(kg/kg) q heat flux (kWm-2) R gas constant, 8.314 J/(mol K) r radius coordinate (m); resistance (m2s/kg, or s/m) r0 cell radius (m) Sh Sherwood number T temperature (K) ua air bulk velocity along radius (m/s) V air flow rate (L/min) v molecular diffusion volume x mass fraction of solute (kg LiCl/kg solution)

Greek Letters φ relative humidity ω humidity ratio (kg moisture/kg air) τ pore tortuosity ε porosity δ thickness (μm) λ heat conductivity (kWm-1K-1), mean free path (m) η viscosity (Pas) ψ Effectiveness σi molecular collision diameter (m) σp geometric standard deviation μp mean pore diameter (m) γ resistance (m2s/kg) ρ density (kg/m3) α selectivity

Superscripts * dimensionless

Subscripts

a air D air stream e effective i inlet

Novel Membranes for Total Heat Exchanger 189

k Knudsen l liquid L lower chamber m mean o outlet, ordinary p Poisseuille s solid sol solution v vapor w liquid water

9.1. INTRODUCTION Air dehumidification is a major task in air conditioning in hot humid regions [1,2]. With

the developments in membrane technology, polymer membranes have been used in air dehumidification. Hydrophilic polymer membranes that are permeable to vapor, but impermeable to air, have been considered. Various materials have been tested: for instance Nafion [3,4], regenerated cellulose [5], Cellulose triacetate [6], sulfonated poly(phenylene oxide) [7], polyether-polyurethane [8] siloxane-amide copolymer [9], polystyrene-sulfonate [10], polyvinylidene fluoride and polyethersulfone [11], and cellophane [12]. They are dense solid membranes.

Due to the similarity in vapor permeation mechanisms, recently, membranes that were used for air dehumidification, are used for total heat exchangers [13,14]. However, air dehumidification is driven by trans-membrane pressure gradients higher than several bars, while the trans-membrane partial pressure difference in a total heat exchanger is less than 2kPa. In other words, pressure driving force in a total heat exchanger is only 1/50 of the traditional air dehumidification. Consequently, membrane-based total heat exchangers with common dehumidification membranes have very low vapor permeability, and a low latent effectiveness. Besides, they have one shortcoming: they are expensive.

Membrane

PP Net

Membrane

PP Net

Figure 9.1. Composite hydrophobic-hydrophilic membrane structure.

Vapor diffusion in common dense membranes is rather small. Recently there have been great developments in membranes for total heat exchanger. Among the various novel inventions, composite hydrophobic-hydrophilic membrane [15] and composite supported liquid membrane [16] are two categories.

Li-Zhi Zhang 190

9.2. HYDROPHOBIC-HYDROPHILIC COMPOSITE MEMBRANE Vapor diffusion in dense layers is rather small. To increase vapor permeation rates,

composite membranes have been used. According to this scheme, a thin active layer is cast onto a thick porous PP (polypropylene) support layer or other materials. The porous support layer provided the necessary mechanical strength while the thin active layer provided the permselective separating effect. The permeation rates can be greatly improved due to the reduction in resistance.

It is accepted that vapor permeations through dense membranes are based on solution-diffusion mechanism. The more hydrophilic the material is, the more moisture it can adsorb, and more moisture can permeate through membrane. According to this theory, materials that have large quantities of hydrophilic groups such as -SO3H, -NH2, -COOH, -OH are required to have a strong hydrophilicity.

Cellulose triacetate (CTA) is a material that is very hydrophilic. It has good membrane-forming properties, good chemical and thermal stability. Furthermore, most importantly it is cheap, which is the prerequisite condition for commercial applications. On the other hand, LiCl salts are very hydrophilic since they can build hydrogen bonds with water molecules. When dispersed in the CTA membrane, they can off-set the effects of CTA cross-linking and make the membrane very hydrophilic and thus facilitate the transport of water vapor.

Membrane Preparation The composite membrane is formed by coating CTA casting solution onto the PP support

membrane. The fabrication process is comprised of the following three steps: Formulation of casting solution. CTA powder is weighted and placed into a vessel with

Acetone at about 90. The solution is heated and stirred until it is completely dissolved. It took about 2 hours.

A certain amount of cross-linking agent (eg. L-malic acid), catalyst (eg. glacial acetic acid) and additive (LiCl) is added to the solution. The solution is continuously stirred at 70 until these different compositions are completely dissolved to form a homogeneous solution. The solution is cool down and placed still for de-bubbling. It took 2 hrs.

The polymer solution is coated on the PP support membrane. The thickness of the active layer is controlled by a casting knife. The thickness of the final active layer can be from several microns to a dozen microns, depending on the gap between the knife edge and the support layer. Then the asymmetric membrane is placed into a vacuum drying oven for cross-linking at 100°C for 1 hour. The membrane is further dried at 60°C for 2 hours.

The composition of the casting solution is listed in Table 9.1. L-malic acid is the cross-linker. Glacial acetic acid is catalyst. LiCl is used as additive to increase hydrophilicity. Acetone is used as the solvent for CTA. For a comparison, same thickness of CTA layer is ensured for different membranes.


Table 9.1. The composition of the casting solution for active layer

Composition of CTA casting solution Membrane Symbol CTA (g) Acetone

(g) L-malic acid (g)

glacial acetic acid (drops)

LiCl (g)

CMB 8 92 0 0 0 CM0 8 92 4 1 0 CM1 8 92 4 1 0.5 CM2 8 92 4 1 1 CM3 8 92 4 1 1.5 CM4 8 92 4 1 2 CM5 8 92 4 1 2.5

Humidifier

Vacuum Pump

Valve

Total heat exchanger

Hot/cold water bath

Exhaust in


Fresh out

Exhaust out

Fresh in

Ambient

Ambient

Ambient

Flow Meter

Humidifier

Vacuum Pump

Valve


Hot/cold water bath

Exhaust in


Fresh out

Exhaust out

Fresh in

Ambient

Ambient

Ambient

Flow Meter

Figure 9.2. Experimental set-up for membrane vapor permeation.

Membrane Half shellMembraneMembraneMembrane Half shell

Figure 9.3. Structure of the single plate total heat exchanger comprised by two symmetric half cells.

Li-Zhi Zhang 192

Figure 9.4. Photo of the two stainless steel half shells.

Vapor Permeation Measurements

A test rig has been set up to measure the vapor permeability through the fabricated

membranes. The whole test set-up is shown in Figure 9.2. Two air streams, one humid fresh air and one dry exhsust, flow through a membrane exchanger to exchange moisture. For the humid strip, ambient air is humidified and is driven to a heating/cooling coil in a hot/cool water bath. After the temperature and humidity reach test conditions, the air is then drawn through the exchanger for moisture exchange. For the dry stream, it is driven directly from ambient to the exchanger. The two inlet temperatures are set to the same values. The composite membrane is sandwiched by two stainless steel half shells. Two parallel air passages on both sides of membrane are formed, which is like a counter-flow one-plate plate-and-shell heat exchanger. A schematic of the single plate total heat exchanger is shown in Figure 9.3. The real photo of the two shells is shown in Figure 9.4. The flow channel height is 2mm, and both the width and length are 10cm. The effective membrane area is 100cm2.

To have a balanced flow, equal air flow rates are kept for the two air streams. The uncertainties are: temperature ±0.1ºC; humidity ±1%; volumetric flow rate ±1%. The final uncertainty is ±4.5%.

The moisture transfer in the exchanger is governed by three resistances: the boundary layer resistance on humid air side, the membrane resistance, and the boundary layer resistance on dry air side. For convenience, a total mass transfer coefficient k is used to summarize the moisture transfer through the membrane. It summarizes the three resistance simultaneously. For different membranes, the boundary layer resistance is the same if the working conditions are the same, i.e., with the same air bulk velocities in the channels. The resistance from the porous support layer is the same. The thickness of the active layer is the same. As a result, the total moisture transfer coefficient is an index of vapor permeability of the fabricated CTA layer. After the inlet and outlet humidity are measured, the total mass transfer coefficients are calculated by:


( )lmt

o11ica

ωωω

Δ−

=A

AuK (9.1)

where Ac is the cross section area of air duct (m2), At is the transfer area of membrane in the cell (m2), Δωlm is the logarithmic mean humidity difference between the solution surface and air stream, and it is calculated by

( ) ( )

( )( )2i1o

2o1i

2i1o2o1ilm

ωωωω

ln

ωω-ωωω

−−

−−=Δ (9.2)

where ω represent temperature humidity ratio (kg/kg); subscripts 1 and 2 represent air stream 1 and air stream 2 respectively; subscripts i and o represent inlet and outlet respectively.

The vapor permeation rate Pe (kg/(m2·s)) is calculated by the following equation

( )t

o11icaa

AAu

Peω−ωρ

= (9.3)

where ρa is air density (kg/m3), At is the membrane transfer area (m2).

0.6 0.8 1.0 1.2 1.42.0x10-5

2.5x10-5

3.0x10-5

3.5x10-5

4.0x10-5

4.5x10-5

5.0x10-5

5.5x10-5

6.0x10-5

Pe (k

gm-2

s-1)

ua (m/s)

CM0 CM1 CM2 CM3 CM4 CM5

2.0

2.5

3.0

3.5

4.0

4.5

5.0

5.5

6.0

Pe(1

0-5kg

m-2

s-1)

0.6 0.8 1.0 1.2 1.42.0x10-5

2.5x10-5

3.0x10-5

3.5x10-5

4.0x10-5

4.5x10-5

5.0x10-5

5.5x10-5

6.0x10-5

Pe (k

gm-2

s-1)

ua (m/s)


2.0

2.5

3.0

3.5

4.0

4.5

5.0

5.5

6.0

Pe(1

0-5kg

m-2

s-1)

Figure 9.5. Effects of LiCl content on vapor permeation rates.

Li-Zhi Zhang 194

Characterization of the Membrane The fabricated membranes are characterized by Contact angle measurements (OCA20,

Dataphysics, Germany), vapor sorption experiments (Hydrosorb-1000, Quantachrome, USA), Scanning Electron Microscope (SEM) (LEO 1530VP, Germany) to investigate their physical properties. The characterization may be helpful to disclose why the LiCl additive is useful to facilitate moisture transfer.

Vapor Permeation Tests The vapor permeations between the dry and the humid air streams are measured in the

test rig shown in Figure 9.2. The moisture flux and the total mass transfer rate are shown in Figures 9.5 and 9.6 respectively. As seen, the higher the air flow rates are, the higher the moisture permeation rates are. The reason behind is that the higher the air stream velocities, the less the convective moisture transfer resistance in the two boundaries layers adjacent to membrane. Furthermore, both the moisture permeation rate and the total mass transfer coefficient increase with higher LiCl content in membranes. This proves that LiCl salts indeed can improve moisture permeation substantially. CM5 that has 2.3% LiCl has the highest moisture permeation rate. Actually, under air velocity of 1.0m/s, its vapor permeation rate is 70% higher than CM0 that has no LiCl additives. Further increase of LiCl content in casting solution is not suggested since the membrane will be difficult to get dry.

0.6 0.8 1.0 1.2 1.4 1.6 1.8

3.0x10-3

4.0x10-3

5.0x10-3

6.0x10-3

7.0x10-3

8.0x10-3

9.0x10-3

1.0x10-2

1.1x10-2

1.2x10-2

1.3x10-2

1.4x10-2

ua (m/s)


k(1

0-2m

/s)

0.2

0.4

0.6

0.8

1.0

1.2

0.6 0.8 1.0 1.2 1.4 1.6 1.8

3.0x10-3

4.0x10-3

5.0x10-3

6.0x10-3

7.0x10-3

8.0x10-3

9.0x10-3

1.0x10-2

1.1x10-2

1.2x10-2

1.3x10-2

1.4x10-2

ua (m/s)


k(1

0-2m

/s)

0.2

0.4

0.6

0.8

1.0

1.2

K

0.6 0.8 1.0 1.2 1.4 1.6 1.8

3.0x10-3

4.0x10-3

5.0x10-3

6.0x10-3

7.0x10-3

8.0x10-3

9.0x10-3

1.0x10-2

1.1x10-2

1.2x10-2

1.3x10-2

1.4x10-2

ua (m/s)


k(1

0-2m

/s)

0.2

0.4

0.6

0.8

1.0

1.2

0.6 0.8 1.0 1.2 1.4 1.6 1.8

3.0x10-3

4.0x10-3

5.0x10-3

6.0x10-3

7.0x10-3

8.0x10-3

9.0x10-3

1.0x10-2

1.1x10-2

1.2x10-2

1.3x10-2

1.4x10-2

ua (m/s)


k(1

0-2m

/s)

0.2

0.4

0.6

0.8

1.0

1.2

K

Figure 9.6. Effects of LiCl content on total mass transfer coefficient.


Table 9.2. Contact angles between the membrane surface and water droplet

Mambrane symbol CM0 CM1 CM2 CM3 CM4 CM5 LiCl content (g) 0 0.5 1 1.5 2 2.5 Contact angles (°) 85.9 82.6 79.3 65.3 57.9 54.3

SEM Studies SEM images of the surface and the cross-section of the developed composite membranes

are shown in Figure 9.7, in which, the surface of the support PP layer is shown in (a), and the surface and the cross-section of the composite membrane CM3 are shown in (b) and (c) respectively. It can be seen that the PP layer is porous and the CTA layer is dense. The porous layer provides the mechanical support and the CTA active layer provides the selective permeation of moisture. CO2 permeation tests are also conducted with one air stream of higher CO2 ratio (20%) and other air stream of low CO2 (0%). Air flow rates are kept at 1.0m/s. No CO2 permeation through the composite membranes is observed. The precision of CO2 sensor is 0.1 vol%. The calculated H2O/CO2 permeation is beyond 1000. The membranes made are assumed defects free.

(a)

Figure 9.7. (Continued on next page.)

Li-Zhi Zhang 196

(b)

PPCTA

PPCTA

(c)

Figure 9.7. The SEM graphs of (a) surface of PP, (b) surface of CM3, and (c) cross-section of CM3.

Contact Angles

The contact angles between the surface of the composite membrane and water droplets

are measured. Totally 5 locations on CTA side are measured for each membrane. The


measured values are then averaged to represent the final contact angle between the membrane and water droplets. Table 9.2 lists the contact angles for membranes with LiCl content varied from 0 to 2.5g. The compositions of other components are shown in Table 9.1. As shown in Table 9.2, the contact angle decreases as the content of LiCl increases, indicating increased hydrophilicity. The membrane without LiCl (CM0) shows the largest contact angle, implicating the least hydrophilicity. The reason behind is that when the content of LiCl is increased, the high water sorption potential of LiCl can increase hydrophilicity.

Sorption Tests The sorption isotherms between the membrane material and water vapor are measured in

a sorption analyzer Hydrosorb-1000. The adsorption (in solid line) and desorption (in dashed line) isotherms of three membranes CM0, CM2 and CM4, at 25°C are shown in Figure 9.8. The relative humidity changes from 0 to 1.0. As seen, membranes can adsorb more water vapor with increased LiCl content. LiCl is a strong hygroscopic salt. Addition of LiCl in the CTA membrane can increase its potential for moisture adsorption. At 60% relative humidity, cross-linked CTA membrane CM0 can only adsorb 0.07g/g moisture. After adding 2% LiCl, membrane CM4 can adsorb 0.4 g/g moisture. The moisture adsorption potential increases by 4.7 times. Considering the solution-diffusion mechanisms of vapor permeation in membranes, that’s no wonder vapor permeability was greatly improved. When RH approaches 1.0, the adsorbed moisture increases sharply due to multilayer, chemical adsorptions. This on the other hand implicates that the membranes are nonporous. This is in agreement with the SEM results.

It is also observed from Figure 9.8 that the higher the relative humidity is, the steeper the sorption isotherms become. Considering the driving force for vapor permeation is the moisture gradients across the membrane, therefore the higher the relative humidity, the greater the vapor permeation rates are.

The results of the present work suggest that the addition of LiCl changes the CTA membrane hydrophilicity substantially. The composite membranes fabricated with LiCl additives exhibit improved moisture permeation properties, as well as good mechanical and physical properties. They provide promising choices for total heat exchanger industry. The increased hydrophilicity is mainly due to LiCl addition.

Membrane that has 2.3% LiCl has the highest moisture permeation rate. Its vapor permeation rate is 70% higher than membrane that has no LiCl additives. The reason is that with addition of LiCl, the hydrophilicity of the membrane is improved greatly. For instance, at 60% relative humidity, addition of 2% LiCl in cross-linked CTA membrane can increase its moisture adsorption potential by 4.7 times.

Li-Zhi Zhang 198

0.0 0.2 0.4 0.6 0.8 1.0

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

x (g

/g)

φ

CM0 adsorption curve CM0 desorption curve CM2 adsorption curve CM2 desorption curve CM4 adsorption curve CM4 desorption curve

CM0

CM2

♦ CM4

0.0 0.2 0.4 0.6 0.8 1.0

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

x (g

/g)

φ

CM0 adsorption curve CM0 desorption curve CM2 adsorption curve CM2 desorption curve CM4 adsorption curve CM4 desorption curve

CM0

CM2

♦ CM4

Figure 9.8. Water vapor sorption isotherms of membranes at 298K. The solid lines represent adsorption, the dashed lines represent desorption.

Protective layerProtective layer

Figure 9.9. Concept of the composite supported liquid membrane (CSLM).


9.3. COMPOSITE SUPPORTED LIQUID MEMBRANE Moisture diffusivity in solid polymer membranes is usually very low, in the order of 10-

12~10-13m2/s. Total heat exchangers only have limited trans-membrane vapor partial pressure difference, therefore performances are quite limited with common homogeneous solid membranes.

In contrast to solid membranes, moisture diffusion in liquid membranes (~10-9m2/s [17,18], diffusivity) is several orders higher than that in solid membranes. Due to this reason and the inherent high selectivity, in recent years, there has been much effort in progressing the researches of supported liquid membranes (SLM) in various fields: air dehumidification [19], SO2/CO2 separation [20], H2S/CH4 separation [21], wastewater treatment [22], metal ions concentration (uphill transport) [23], separation of isomeric amines between two organic phases [24], to name but a few.

To improve the performances of total heat exchangers, a novel membrane, a composite supported liquid membrane (CSLM), which employs LiCl liquid solution immobilized in a porous support membrane to facilitate the transport of moisture, is prepared [25,26]. To protect the SLM, two hydrophobic PVDF (Polyvinylidene Fluoride) layers are formed on both surfaces of the SLM. The concept is shown in Figure 9.9. The sweep represents exhaust air. In this chapter, a different skin layer fabricated in our laboratory is used. In stead of PVDF layers, PES (Polyether sulphone) layers are used as the skin layers. As before, the CA (Cellulose acetate) is stilled used as the middle support layer to absorb liquid LiCl solution. PVDF are highly hydrophobic, so the adhesion between PVDF and the CA liquid layer is difficult. However, PES are easier to be glued to the CA layer.

Figure 9.10. SEM graph of the surface of PES membrane, 3000 times magnified.

Li-Zhi Zhang 200

Figure 9.11. SEM graph of the surface of the Cellulose Acetate membrane, 5000 times magnified.

Preparation of the Supported Liquid Membranes

Three types of commercial membrane were obtained from a supplier. Very hydrophilic

Cellulose Acetate (CA) membranes with nominal pore diameter 0.45μm a thickness 80μm are used as the support media to immobilize LiCl solution. Two hydrophobic PES membranes (equal nominal pore diameter 0.15μm, thickness 45μm) are used as the protective layer. Crystals of LiCl⋅H2O with laboratory class purity is used as the solute.

PES layer

CA layer

PES layer

PES layer

CA layer

PES layer

Figure 9.12. SEM graph of the cross section of the composite membrane without LiCl solution immobolized, 1000 times magnified.


PES layer

CA layer

PES layer

PES layer

CA layer

PES layer

Figure 9.13. SEM graph of cross section of the composite membrane with LiCl solution immobilized, 1000 times magnified.

Before the preparation of composite membrane, each membrane is experimented and observed for their basic micro structures. Figures 9.10 and 9.11 show the SEM (Scanning Electron Photomicrograph) graphs of the CA membrane and PES membrane, respectively.

Under room temperature, well-stirred LiCl solution with 40% mass fraction is first prepared in a closed glass container. Vacuum degassing is applied for 2 hours for the three membranes, after which, the CA membrane is dipped into the LiCl solution. After 24 hours, the CA membrane is moved from the solution and placed onto a clean glass plate which is cleaned by alcohol. Surplus LiCl solution on surfaces of CA membrane is blotted off with paper tissue. To be sure that no ionic liquid is removed from the membrane pores, the cleaning procedure is very gentle. At this stage, PVC glue is brushed on one surface of the two PES membranes, and at the same time on both surfaces of the CA membrane. After a few seconds, the two PES membranes are glued to the CA membrane and are pressed together gentlely for a few seconds. The prepared composite membrane is placed in a constant-temperature-constant-humidity chamber for another 24 hours, before experiment is performed.

For comparison, a composite membrane with no LiCl solution immobilized in the CA membrane is also made with the same procedure. The cross section SEM views of the two composite membranes are shown in Figure 9.12 and Figure 9.13 respectively. To prevent the microstructure being destroyed by knife crushing when preparing cross section samples, the membranes are first frozen in liquid nitrogen before they are broken off to see the cross sections. Before observations, they are gilded with gold.

As seen from Figure 9.12, there are some gaps between different layers. Some big cavities in the support layers are also observed, which are presumed to be imperfections in

Li-Zhi Zhang 202

membrane fabrications. However, they have no adverse effects for this study because during operation, they will be filled with liquid solution. In the preparation process, some thickness of the CA membrane is dissolved by the glue, resulting in a lesser support layer thickness than raw material. Figure 9.13 with LiCl solution shows that the CA layer and PES layer connect to each other very closely and have a dense and continuum interface. There are more big cavities in the support layer. The reason behind this may be that with LiCl solution soaked, the wetted molecular chains in CA membrane structure become more flexible and they will swell and expand to two sides. The boundaries between different layers are pressed together and linked to each other closely. The final CA layer thickness is 52 μm.

Moisture Transport Measurement Here another permeation cell is used. The cell is very sensitive. The membrane module is

a circular cell having an exchange area of 176.7cm2. It is composed of two parts: the lower chamber and the cap, as shown in Figure 9.14. When testing, the flat sheet membrane is placed on the lower chamber inside which distilled water is contained. The cap is then covered on the membrane surface and form a sandwiched structure. The membrane and the inner surface of the cap form a cone-shaped cavity. The air is supplied through the air slits in the cap. It is introduced through two diametrically positioned inlets (symmetrically placed) into a circular-shaped channel at the perimeter, from where the air is distributed over the membrane surface through the circular air slit. The air flows inward radially, until it exits the cap outlet in the center. The cap is designed that a constant axial air velocity is realized. When flowing across the membrane, the air stream exchanges moisture with the distilled water through the composite CSLM, and is humidified.

Inlet InletOutlet

Distilled water

Cap

Lower chamber

CSLM Membrane

Inlet InletOutlet

Distilled water

Cap

Lower chamber

CSLM Membrane

Figure 9.14. Schematic of the test cell for CSLM.



Distilled Water

Valve


Humidity Sensor

By pass

Cell

Pressure Meter



Distilled Water

Valve


Humidity Sensor

By pass

Cell

Pressure Meter



Distilled Water

Valve


Humidity Sensor

By pass

Cell

Pressure Meter


Humidity Sensor

Figure 9.15. The set-up of the test apparatus.

The whole experimental set up is shown in Figure 9.15. The cell is supplied with clean and humidified air from an air supply unit. The supply air flows from a compressed air bottle and is divided into two streams. One of them is humidified through a bubbler immersed in a bottle of distilled water, and then re-mixed with the other dry air stream. The humidity of the mixed air stream is controlled by adjusting the proportions of air mixing. The airflow rates are controlled by two air pumps/controllers at the inlet and outlet of the cell. The humidities and temperatures to and from the cell are measured by the built-in RH and temperature sensors, which are installed in the pumps/controllers. A detailed description of the test procedure is given in [27].

In the test, the vapor evaporation is slow, and the cell is well conductive. Therefore only moisture transfer is considered, by neglecting thermal influences.

Analysis of Transfer Resistance The schematic of the moisture transport in the cell is represented in Figure 9.16. The

variations of air humidity are depicted in Figure 9.17. The representations are: 1-2, from lower chamber solution surface to membrane lower surface; 2-3, from first layer’s (PES) lower surface to its upper surface; 3-4, from the second layer’s (liquid layer) lower surface to its upper surface; 4-5, from the third layer’s (PES) lower surface to its upper surface; 5-6, from membrane upper surface to air stream.

Moisture conservation in air stream is represented by a one-dimensional transient equation:

ad

6va

6a

6

ρω1ωω

HJ

rrD

rrru

t+⎟

⎠⎞

⎜⎝⎛

∂∂

∂∂

=∂

∂+

∂∂

(9.4)

Li-Zhi Zhang 204

z

r

Hd

O

δ3

L

uaAir Duct

Air gap

Composite SLM

Solution in cell

δ2δ1

z

r

Hd

O

δ3

L

uaAir Duct

Air gap

Composite SLM

Solution in cell

δ2δ1

Figure 9.16. Schematic of the mass transfer model in the cell.

1

2 3

45

6

L1L2 L3

ZO

1

2 3

45

6

L1L2 L3

1

2 3

45

6

L1L2 L3

ZO

Figure 9.17. Air humidity variations through the composite membrane.

where ω6 is humidity in air stream (kg/kg), t is time (s), r is radius (m), Dva is vapor diffusivity in air (m2/s), ua is air velocity (m/s) in radial direction, J is the local moisture emission rate from the membrane to air (kgm-2s-1), Hd is height of air stream at inlet (m), ρa is density of dry air (kg/m3). The upper cavity of the cell is specially designed that the radial air velocity ua keeps constant in the flow. Air duct height Hd changes with flow.

On the membrane upper surface, the moisture emission rate

( )65a ωω −ρ= kJ (9.5)

where k is the local convective mass transfer coefficient (m/s) between air stream and membrane. Convective mass transport in the channel can be represented by [27]

834.0

d

0

2Re3359.0

−

⎟⎟⎠

⎞⎜⎜⎝

⎛ −=

Hrr

ScSh (9.6)


where Sh, Re, and Sc are Sherwood number, Reynolds number and Schmidt number, respectively. They are defined as

va

d2DkH

Sh = (9.7)

νda2

ReHu

= (9.8)

vaDSc ν

= (9.9)

where ν is the kinematic viscosity of air (m2/s), r is radial coordinate (m) and r0 is the radius of the cell (m).

Initial conditions: t=0, ω=ωL (9.10)

where ωL is the humidity ratio determined by the saturated NaCl solution and temperature (kg/kg);

Boundary conditions: r=r0, ω=ωi (9.11) r=0, stream outlet (9.12) Resistance in this component

kr

aD

1ρ

= (9.13)

This resistance is the inverse of convective mass transfer coefficient in air flow side,

divided by the density of dry air. It is also called the convective resistance.

Moisture Transfer through the Composite Membrane Moisture diffusion through the composite membrane is expressed by

321

52ae

-ωδ+δ+δ

ωρ= DJ (9.14)

Li-Zhi Zhang 206

where De is the effective moisture diffusion coefficient in the composite membrane (m2/s), it is calculated by

e3

3

e2

2

e1

1

321e

DDD

Dδ

+δ

+δ

δ+δ+δ= (9.15)

where De1, De2, De3 are the effective diffusivity in the first layer, second layer and the third layer of the composite membrane.

Gas Transport in Porous Media Transport of gas through porous media has been extensively studied and theoretical

models have been developed based on the kinetic theory of gases. Various types of mechanisms have been proposed for transport of gases or vapors through microporous membranes: Knudesn model, viscous model, ordinary molecule diffusion model, often summarized by the dusty gas model. The pore size is important for elucidating the physical nature of the mass transport through the membrane. Most of the literature reports have used the average pore size to calculate the mass flux. Nevertheless, because of pore size distribution of the membranes, more than one mechanism of mass transport can simultaneously occur. In this study, pore size distribution will be considered to classify various mechanisms.

Pore Size Distribution The pore size distribution can be expressed by the probability density function (that is,

log-normal distribution) described by the following equation

0.00

0.25

0.50

0.75

1.00

10 100 1000 10000

d p (nm)

f(d p

)

M 1 M 2 M 3 M 4

0.00

0.25

0.50

0.75

1.00

10 100 1000 10000

d p (nm)

f(d p

)

M 1 M 2 M 3 M 4

Figure 9.18. Pore size distribution of four porous membranes.


( )( ) ⎥

⎥⎦

⎤

⎢⎢⎣

⎡

σ

μ−−

πσ= 2

p

2pp

2/1ppp

p

ln2

lnlnexp

)2(ln1

)()( d

dddddf

(9.16)

where dp is the pore size, μp is the mean pore size, and σp is the geometric standard deviation, f(dp) is fraction of number of pores with diameters not greater than dp. Figure 9.18 shows the pore size distributions of four PES porous membranes.

The surface porosity εs, defined as the ratio between the area of the pores to the total membrane surface area, can be calculated from following Equation

2j

1js 4dfN n

j∑

=

π=ε (9.17)

where N is the number of pores per unit area, known as pore density, and fj is the fraction of the number of pores with size dj, j=1,…, n, is the jth class of pore sizes.

The surface porosity is different from the void volume, which is determined by

τε=ε s (9.18)

where τ is the pore tortuosity. Thus, the number of pores per unit area can be calculated from the following equation

∑=

πτε

= n

j

dfN

1

2jj4

/ (9.19)

if the effective membrane porosity, which takes into account the tortuosity of the membrane pores ε/τ, is known.

Mass Transfer of Vapor through a Single Membrane Pore The established theory considers three mechanisms for mass transfer in a pore as depicted

in Figure 9.19: Poiseuille flow, ordinary molecular diffusion and Knudsen diffusion, or a combination of them.

The governing quantity that provides a guideline in determining which mechanism is operative in a given pore under given operating conditions is the ratio of the pore size to the mean free path λ, which is calculated for a species i using the following expression [17],

m2i

Bi π2

λP

Tkσ

= (9.20)

Li-Zhi Zhang 208

where σi is the molecular collision diameter (m), 2.641Å and 3.711 Å for water vapor and air, respectively; kB is the Boltzmann constant, 1.38×10-23 J/K, Pm is the mean total pressure within the membrane pores (Pa), and T is the absolute temperature (K).

Poiseuille flow

Knudsen diffusion

Molecular diffusion

Poiseuille flow

Knudsen diffusion

Molecular diffusion

Figure 9.19. Diffusion mechanisms for gases in pores.

For gaseous mixtures of two components, the mean free path and the collision diameters are different from the corresponding quantities for the pure component. The following relationship can be applied for vapor-air mixtures

2va

vaσσ

σ+

= (9.21)

Under room temperature and atmospheric pressure, calculated λ for air is 0.07μm; while

under vacuum conditions, mean free path for air may be several microns to several meters. Obviously, operating conditions have a great influence on diffusion mechanisms.

Knudsen number

p

λKnd

= (9.22)


Poisseuille Flow When Kn<0.01, i.e., the pore size is large in relation with the mean free path of gas

molecules, the molecule-molecule collisions between gas molecules themselves will dominate and viscous Poisseuille flow will occur. Under this mechanism, the gas flow through a single pore FP (kg/s) is

vmv

v

4P

P1

128π p

RTpMdF Δ

τδη= (9.23)

where Mv is the molecule weight of vapor (kg/mol); R is gas constant, 8.314 J/(mol K); ηv is the vapor viscosity (Pas); τ is the tortuosity of membrane. pm is the mean partial pressure of water vapor (Pa), Δpv is the transmembrane vapor partial pressure difference (Pa).

Equation (9.23) can be re-written in the following form as

vPP1 pGF Δτδ

= (9.24)

where the flow coefficient Gp [kgm/(sPa)] is

RTpMdG mv

v

4P

P 128π

η= (9.25)

Knudsen Diffusion If the mean free path of the gas molecules is large in relation with the pore size, the

molecule-pore wall collisions are dominant over the molecule-molecule collisions and the gas transport takes place via Knudsen flow. In this case, one obtains the following relationship for the gas mass flow in a single pore,

vv

v3p

K1

π8

12π

pMRT

RTMd

F Δ=τδ

(9.26)

For a relatively small pore size, Kn≥10, Knudsen flow is assumed predominant. Vapor density (kg/m3)

RTMp vv

v =ρ (9.27)

Equation (9.26) can be written as

Li-Zhi Zhang 210

vKK1 pGF Δ=

τδ (9.28)

v

v3p

K π8

12π

MRT

RTMd

G = (9.29)

Molecule Diffusion Unless steps are taken to remove dissolved air from the feed and permeate prior to

processing, the dissolved air acts as a stagnant layer. If the dominant resistance is the molecular diffusion resistance caused by the virtually stagnant air trapped within the membrane pores, one obtains the following relationship for the gas flow in a single pore,

vvav

2p

O 4π

pD

RTMd

F Δ=τδ

(9.30)

where the ordinary diffusion coefficient of water vapor molecule in air is expressed by [17,29]

( ) av23/1

a3/1

vm

75.1a

011

MMvvP

TCD +

+= (9.31)

where Ca=3.203×10-4. Pm is the mean total pressure in pores. The terms vv and va are molecular diffusion volumes and are calculated by summing the atomic contributions: va =20.1, and vv =12.7 [17]. Mv and Ma are molecule weight of vapor and air in kg/mol. M is 0.018 kg/mol for water vapor and 0.029 kg/mol for air respectively. Under room pressure, air is trapped in membrane pores, therefore this mechanism exists.

Equation (9.30) can be re-written as

vOO1 pGF Δ=

τδ (9.32)

RTDMd

G vav2p

O 4π

= (9.33)

Combined Flow Between the two limits of Knudsen diffusion and Poisseuille flow, i.e., 0.01≤Kn<10, the

above mentioned three mechanims may coexist. The combined flow is


( )[ ] v11-

O1-

KPPKO1 pGGGF Δ++τδ

=−

(9.34)

When Kn<0.01, the Poisseuille flow is dominant, the Knudsen mechanism may be

neglected, then the combined flow is

[ ] vOPPO1 pGGF Δ+τδ

= (9.35)

When Kn≥10, the Knudsen flow is dominant, the Poisseuille mechanism may be

neglected, then the combined flow is

( ) v11

O1

KKO1 pGGF Δ+τδ

=−−− (9.36)

Total Mass Flux Across a Membrane In the case of a membrane with a pore size distribution, all the above mechanisms may

exist, but to different extents, depending on the operating conditions and membrane morphological characteristics. Finally, considering the various diameters of pores in the membrane, the moisture flux across the membrane Jm (kg/m2s) is

⎟⎟⎠

⎞⎜⎜⎝

⎛++= ∑ ∑ ∑

=

=

=

=

=

=

)1.0(

1

)100( max)(

POjjPKOjjKOjjm

λ λdm

j

dp

mj

ddn

pj

FfFfFfNJ (9.37)

where N is the number of pores per unit area (1/m2), m is the last class of pores in the Knudsen region, p is the last class of pores transition region.

Substituting Eqs.(9.34) to (9.36) into Eq.(9.37), one gets

( ) v1

jjm pGfNJn

j

Δ⎥⎦

⎤⎢⎣

⎡τδ

= ∑=

(9.38)

where

( )( )⎪

⎪⎩

⎪⎪⎨

⎧

+

++

+

=−−−

−

11O

1K

11-O

1-KP

OP

GG

GGG

GG

G (9.39)

Kn<0.01

0.1≤Kn<10

Kn≥10

Li-Zhi Zhang 212

Humidity ratio in ambient air is in the range of 0.005kg/kg to 0.035 kg/kg, therefore the above equation can be simplified to

Pp ω608.1v = (9.40)

Substituting the relations of partial pressure and humidity ratio into Eq.(9.38), one gets

δΔ

ρ= meffam

ωDJ (9.41)

where ρa is density of dry air (kg/m3) and the effective moisture diffusion coefficient is defined as (m2/s)

∑=τρ

=n

j

GfPND1

jja

eff608.1

(9.42)

Effective Diffusivity in the First and the Third Layer The two protective layers on both sides of The liquid layer are highly hydrophobic. The

established theory of gas diffusion in such membranes considers three mechanisms: Poiseuille flow, ordinary molecular diffusion and Knudsen diffusion, or a combination of them.

As discussed previously, when Kn (ratio of the pore size to the mean free path) ≥10, the Knudsen flow is dominant, the Poisseuille mechanism may be neglected. Actually, in most cases for air conditioning industry with microporous membranes, Knudsen number is larger than 10, and Poisseuille flow can be neglected, then the flow is considered to be combined Knudsen and ordinary diffusion.

In a simple form, Knudsen diffusion coefficient

v

PK π

83 M

RTdD = (9.43)

where R is gas constant, 8.314 J/(mol K).

The effective diffusivity of combined Knudsen and ordinary flow is

( ) 11-O

1-K

1-KO

−+= DDD (9.44)

Effective diffusivity in this layer

iDD KO,i

iei τ

ε= , i=1, 3 (9.45)


Resistance in these two layers

3,1a

3,13 1,

eDr

ρδ

= (9.46)

The resistance through these two layers is called the diffusion resistance, which is

calculated by diffusion distance divided by air density and moisture diffusivity in membrane.

Effective Diffusivity in the Second Layer This layer is the supported liquid layer. Water transfer in liquid layer is described by:

2

wwlq

2

2 Cδτ

ε Δ= DJ (9.47)

where Dwlq is water diffusivity in liquid membrane (m2/s), ΔCw is the difference of water concentration in liquid membrane solution (kg/m3) between the two sides of liquid membrane.

Water vapor partial pressure, temperature, and LiCl solution concentration are governed by a thermodynamic equation [25]

2v)()()(log

TmC

TmBmAp ++= (9.48)

3

32

210)( mAmAmAAmA +++= (9.49)

33

2210)( mBmBmBBmB +++= (9.50)

3

32

210)( mCmCmCCmC +++= (9.51)

where in this equation pv is in kPa, T is in K, and m is molality of the electrolyte (mol LiCl /kg water).

)1(0425.0 xxm

−= (9.52)

where x is mass fraction of solute (kg LiCl/kg solution).The constants in Eqs. (9.48) to (9.51) are given by

A0= 7.3233550, A1=−0.0623661,

Li-Zhi Zhang 214

A2= 0.0061613, A3= −0.0001042, B0=−1718.1570, B1=8.2255, B2=−2.2131, B3=0.0246, C0=−97575.680, C1=3839.979, C2=−421.429, C3=16.731, Water concentration in solution is

( ) solw 1 ρ−= xC (9.53)

where ρsol is solution density (kg/m3), and it is calculated by the following equation

i

i xx∑

=⎟⎠⎞

⎜⎝⎛

−ρρ=ρ

3

0iwsol 1

(9.54)

where ρw is pure water density at temperature T, and ρi are given below:

ρ0 =1.0, ρ1 =0.540966 ρ2 =−0.303792, ρ3 =0.100791 The thermodynamic equilibrium chart of LiCl solution dictated by Eqs.(9.48) to (9.52)

could be represented by a series of linear equations as

Table 9.3. Values of kp and Cw0 for LiCl solution

T (°C)

kp (kgm-3Pa-1)

Cw0 (kgm-3)

14 0.1428 727.8 24 0.0867 706.4 35 0.0463 700.2 45 0.0293 693.2

w0vpw CpkC += (9.55)

where kp is called the Henry coefficient (kgm-3Pa-1), and Cw0 is a constant (kg/m3). They are functions of temperature as given in Table 9.3.

Substituting Eq. (9.47) and psychrometric relation (9.40) into (9.47) gives the effective moisture diffusivity

wlq2a

2pe2

608.1D

PkD

τρε

= (9.56)


Resistance in this layer

2a

22

eDr

ρδ

= (9.57)

As seen, the resistance in this layer is similar to the resistance in the two protective

layers. It is calculated by diffusion distance divided by dry air density and the effective moisture diffusivity in this layer.

Moisture Diffusion in the Air Gap Moisture transfer below the membrane can be expressed by

LDJ 21

vaaω−ω

ρ= (9.58)

where L is the height of air gap (m).

Resistance in this component

vaa

LL D

rρ

δ= (9.59)

It is also calculated by diffusion distance divided by moisture diffucsivity and dry air

density.

Moisture Permeability Mean moisture permeability across the whole membrane surface, (kgm-1s-1)/(kg/kg), is

calculated by

( ) ( )321lmt

iOcaa δ+δ+δωΔ

ω−ωρ=

AAu

Pe (9.60)

where Ac is the cross section area of air duct (m2), At is the transfer area of membrane in the cell (m2), Δωlm is the logarithmic mean humidity difference between the solution surface and air stream, and it is calculated by

⎟⎟⎠

⎞⎜⎜⎝

⎛−−

−=Δ

oL

iL

iolm

ωωωωln

ωωω (9.61)

Li-Zhi Zhang 216

where subscripts o, i represent outlet and inlet of air stream, respectively. The permeability Pe here represents moisture transfer rate (kg/s) for unit area of

membrane under unit transmembrane humidity difference (kg vapor/kg dry air), times total thickness. It reflects the performance of membrane.

Dimensionless radius

0

* 1rrr −= (9.62)

In the simulations, it is assumed that heat effects are negligible due to slow water

evaporation rates from the solution. Further, the physical properties like moisture diffusivity are uniform and constant in the membrane.

Resistance Analysis Both experimental and numerical values of outlet RH are obtained. Table 9.4 lists the

values of operating conditions and system configurations. The tested mean permeation rate across the whole membrane surface is 1.1×10-4 kgm-2s-1, which is 3 times higher than the value obtained with a highly hydrophilic solid polymer membrane of comparable thickness (2.5×10-5 kgm-2s-1). The corresponding permeability with this supported liquid membrane is 5.2×10-6 (kgm-1s-1)/(kg/kg).

r D

23%

r L

9%

r 1

28%

r 2

12%

r 3

28%

Figure 9.20. Percentages of various resistances to total moisture transfer resistance.

Figure 9.20 shows the percentages of various resistances to total resistance. As seen, the convective moisture transfer resistance accounts for 23% of the total resistance. The two protective layers account for 28% of the total resistance each. The air gap diffusion resistance


amounts for less than 10% of the total resistance. The supported liquid layer, LiCl solution layer, only accounts for 12% of the total resistance.

Table 9.4. Parameters used in the test and analysis

Symbol Unit Value Symbol Unit Value T °C 26.0 ε1, ε3 0.66 δ1, δ3 μm 48 ε2 0.51 δ2 μm 52 τ1, τ3 2.0 dp1, dp3 μm 0.18 τ2 2.8 dp2 μm 0.23 V L/min 10.0∼30 r0 mm 75 L mm 0.1 Hd mm 1.0 Dwl m2/s (3×10-9)

Effects of Protective Layers The protective layers have great influences on membrane performance. Figure 9.21

shows the effects of two protective layer thicknesses on mean permeation rate and the permeability through the membrane. The permeation rate decreases with an increase in protective layer thickness, due to the increased moisture resistance. As for the permeability, it first decreases, and then increases with thickness increasing. The reason is that the permeability is co-determined by two contradicting factors: membrane resistance and membrane thickness. The final permeability is the result of balance between these two factors. When the thickness is less than 20μm, increased resistance with thickness is more influential, while when the protective thickness is greater than 20μm, increases of thickness is more influential. When the two protective thicknesses are reduced from 100μm to 5μm, the permeation rate can be improved by 20%.

Figure 9.22 shows the effects of protective layer porosity on mean permeation rate and the permeability. Porosity plays a big role. Both permeation rate and permeability increases with porosity, due to decreased resistance. The mean permeation rate increases 1.3 folds with a porosity increase from 0.2 to 0.8.

Figure 9.23 shows the influence of mean pore diameters of protective layers on performance. The smaller the pore sizes, the greater the resistance, and the better the performance. When the pores are greater than 0.4μm, performance improvement with larger pores becomes slower. On the other hand, with larger pores, the use with protective layers to support and stabilize the liquid layer becomes limited. Therefore membrane with 0.4 μm diameter pores is a good optimization.

Li-Zhi Zhang 218

5.0E-05

1.0E-04

1.5E-04

0 20 40 60 80 100 1205.0E-06

5.2E-06

5.4E-06

5.6E-06

5.8E-06

6.0E-06

JmPe

δ1, δ3 (μm)

J m(k

gm-2

s-1)

Pe(k

gm-1

s-1)/(

kg/k

g)

5.0E-05

1.0E-04

1.5E-04

0 20 40 60 80 100 1205.0E-06

5.2E-06

5.4E-06

5.6E-06

5.8E-06

6.0E-06

JmPe

δ1, δ3 (μm)

J m(k

gm-2

s-1)

Pe(k

gm-1

s-1)/(

kg/k

g)

Figure 9.21. Effects of protective thickness on mean moisture permeation rate and permeability.

5.0E-05

1.0E-04

1.5E-04

0 0.2 0.4 0.6 0.8 12.0E-06

3.0E-06

4.0E-06

5.0E-06

6.0E-06

JmPe

ε1, ε3

J m(k

gm-2

s-1)

Pe(k

gm-1

s-1)/(

kg/k

g)

5.0E-05

1.0E-04

1.5E-04

0 0.2 0.4 0.6 0.8 12.0E-06

3.0E-06

4.0E-06

5.0E-06

6.0E-06

JmPe

ε1, ε3

J m(k

gm-2

s-1)

Pe(k

gm-1

s-1)/(

kg/k

g)

Figure 9.22. Effects of protective layer porosity on mean permeation rate and permeability.


Effects of Liquid Layer The liquid layer has a major impact on performance, since it is not only the barrier, but

also the active layer facilitates moisture transfer. Figures 9.242 and 9.25 show the effects of liquid layer thickness and porosity on performances, respectively. As seen, permeation rate and permeability decrease with thickness increasing, due to resistance increasing. The performance increases with porosity increasing, due to resistance decreasing. The porosity has greater impacts on performance than thickness does. An increase of porosity from 0.2 to 0.8 has a 90% permeation rate improvement.

5.0E-05

1.0E-04

1.5E-04

0 0.2 0.4 0.6 0.8 12.0E-06

3.0E-06

4.0E-06

5.0E-06

6.0E-06

7.0E-06

JmPe

dp1, dp3 (μm )

J m(k

gm-2

s-1)

Pe(k

gm-1

s-1)/(

kg/k

g)5.0E-05

1.0E-04

1.5E-04

0 0.2 0.4 0.6 0.8 12.0E-06

3.0E-06

4.0E-06

5.0E-06

6.0E-06

7.0E-06

JmPe

dp1, dp3 (μm )

J m(k

gm-2

s-1)

Pe(k

gm-1

s-1)/(

kg/k

g)

Figure 9.23. Effects of nominal pore diameter of protective layer on mean permeation rate and permeability.

Li-Zhi Zhang 220

5.0E-05

1.0E-04

1.5E-04

0 20 40 60 80 1004.0E-06

5.0E-06

6.0E-06

JmPe

δ2 (μm )

J m(k

gm-2

s-1)

Pe(k

gm-1

s-1)/(

kg/k

g)

5.0E-05

1.0E-04

1.5E-04

0 20 40 60 80 1004.0E-06

5.0E-06

6.0E-06

JmPe

δ2 (μm )

J m(k

gm-2

s-1)

Pe(k

gm-1

s-1)/(

kg/k

g)

Figure 9.24. Effects of liquid layer thickness on mean permeation rate and permeability.

5.0E-05

1.0E-04

1.5E-04

0 0.2 0.4 0.6 0.8 12.0E-06

3.0E-06

4.0E-06

5.0E-06

6.0E-06

JmPe

ε2

J m(k

gm-2

s-1)

Pe(k

gm-1

s-1)/(

kg/k

g)

5.0E-05

1.0E-04

1.5E-04

0 0.2 0.4 0.6 0.8 12.0E-06

3.0E-06

4.0E-06

5.0E-06

6.0E-06

JmPe

ε2

J m(k

gm-2

s-1)

Pe(k

gm-1

s-1)/(

kg/k

g)

Figure 9.25. Effects of liquid layer porosity on mean permeation rate and permeability.


9.4. HEAT CONDUCTIVITY OF MEMBRANES

Composite Supported Liquid Membrane As illustrated in Figure 9.17, heat flux through the composite membrane

( )321

52e

δδδλ

++−

=TT

q (9.63)

where 2 and 5 denote two membrane surfaces. Heat conductivity

3

3

2

2

1

1

321e

λδ

+λδ

+λδ

δ+δ+δ=λ (9.64)

The heat conductivity in the first layer can be analyzed by [30]

( )1s1a1 1 ε−λ+ελ=λ (9.65)

where subscripts a and s denote moist air and solid material part, respectively. The effective heat conductivity of the third layer is assumed as the same to the first layer, since they are the same material.

Similarly, the heat conductivity in the second layer, where a liquid solution is stationed in the porous media, can be analyzed by

( )2s2lq2 1 ελελλ −+= (9.66)

where subscripts lq denote liquid solution.

Composite Hydrophobic-hydrophilic Membrane As illustrated in Figure 9.26, heat flux through a composite membrane is

( )21

42e

δδλ

+−

=TT

q (9.67)

where 2 and 4 denote the two surfaces of the composite membrane. Heat conductivity

Li-Zhi Zhang 222

2

2

1

1

21e

λλδδδδ

λ+

+= (9.68)

The heat conductivity in the first porous layer can be analyzed by

( )1s1a1 1 ε−λ+ελ=λ (9.69) The second layer is the dense layer. Its heat conductivity is λ2.

12

34

5L1 L2

ZO

q

12

34

5L1 L2

ZO

q

Figure 9.26. Heat transfer through a composite membrane.

9.5. MEMBRANE SELECTIVITY

An ideal membrane should let vapor permeates freely, but prevent other unwanted gases

to permeate. CO2 is a typical unwanted gas in air conditioning industry. It is an index for indoor air pollution. Usually, indoor air CO2 concentrations should be less than 0.5%. Therefore membrane selectivity is defined by permeations of H2O over CO2. The relation is expressed by

2CO

v

PePe

=α (9.70)

It indicates that selectivity is the ratio of vapor permeability to CO2 permeability. A test

rig shown in Figure 9.26 has been set up to measure the permeability and selectivity of CO2 and H2O. Membrane is sandwiched by two cells as demonstrated in Figure 9.28. Selectivity can be calculated by the ratio of permeation rates. The membrane and the inner surface of each cell form a cone-shaped cavity. Two air streams are supplied through the air slits in each


cell. Each stream is introduced through two diametrically positioned inlets (symmetrically placed) into a circular-shaped channel at the perimeter, from where the air is distributed over the whole membrane surface through the circular air slit. The air flow inward uniformly in radial direction, until they exit the cell outlets in the center. One air stream has no inlet CO2, which represents the fresh air. The other air stream has 2-4% CO2, which represents polluted exhaust air. The inlet humidity is also different for two streams, representing outdoor humid air and indoor dry air, respectively. The humidity is adjusted by two bubblers after the air sources. When flowing across the membrane, the air streams exchange moisture and CO2 simultaneously. The convective mass transfer coefficients on both sides of membrane in the cells are rather large and the resistance can be neglected. After measurement of the inlets and outlets humidity and CO2 concentrations, the permeability and selectivity can be estimated.

The moisture permeability is calculated by

( )m

fofi

ωδωω

Δ−

=A

VPev (9.71)

( ) ( )

⎟⎟⎠

⎞⎜⎜⎝

⎛−−

−−−=Δ

eifo

eofi

eifoeofim

ωωωω

ln

ωωωωω (9.72)

where V is the volumetric air flow rate of fresh air (m3/s), ω is air humidity ratio (kg vapor/kg dry air), A is membrane area (m2), Δωm is the log mean humidity difference between the two air streams. Subscripts “i", “o”, “f”, “e” denote inlet, outlet, fresh and exhaust air respectively. The unit of permeability is [mol/(m⋅s)]/[mol/m3 humidity difference], meaning vapor flux in mol/(m2s)) times the membrane thickness (m) divided by humidity difference in mol/m3.

Similarly, the permeability for CO2

( )m

fofiCO2 cA

ccVPe

Δ−

=δ

(9.73)

( ) ( )

⎟⎟⎠

⎞⎜⎜⎝

⎛

−

−

−−−=Δ

eifo

eofi

eifoeofim

cccc

xcxcc

ln

(9.74)

where c is volumetric concentration of CO2 (m3 CO2/m3 air), Δcm is the log mean CO2 concentration difference between the two air streams. The unit of permeability for CO2 is the same as for vapor. Different units of humidity and CO2 concentration are used because their conventional units are different in air conditioning industry and in measurement.

Now the newly developed composite supported liquid membrane is measured. Table 9.5 lists the values of permeability and selectivity under five operating conditions. For each test, several minutes are needed for the system to become fully steady state. After outlet RH

Li-Zhi Zhang 224

reaches the steady state, water vapor permeability and CO2 permeability are calculated. The water vapor permeability is in the order of 1.39e-7 to 3.44e-7 [mol/(m⋅s)]/[mol/m3], and the CO2 permeability is in the order of 3.19e-11 to 1.21e-10 [mol/(m⋅s)]/[mol/m3].The resulted selectivity is ranging from 2845 to 4355.

The permeability of water vapor decreases with increasing air flow rates. The reason may be that with increasing air flow rates, the operating fresh air inlet humidity increases, and the log mean humidity difference increases. But the permeation increases slowly, correspondingly, the permeability decreases a little bit. The permeability of CO2 decreases more rapidly with flow rates, resulting an increased selectivity. When the CSLM is replaced by the hydrophobic-hydrophilic composite membrane, the measured selectivity is higher than 5000. This indicates that both membranes are ideal for total heat exchanger.

Table 9.5. Permeability and selectivity for composite supported liquid membrane

V ml/min

Pev [mol/(m⋅s)]/[mol/m3]

PeCO2 [mol/(m⋅s)]/[mol/m3]

α

121 3.44e-7 1.21e-10 2845

174 3.55e-7 1.11e-10 3218 235 2.85e-7 8.09e-11 3521 335 1.87e-7 4.45e-11 4206 454 1.39e-7 3.19e-11 4355


Distilled W ater

Valve


Humidity/CO2Sensor

Membrane sandwiched by two cells

Pressure Meter

Humidity/CO2 Sensor

Compressed CO2

Compressed air

Fresh air

Exhaust air


Distilled W ater

Valve


Humidity/CO2Sensor

Membrane sandwiched by two cells

Pressure Meter

Humidity/CO2 Sensor

Compressed CO2

Compressed air

Fresh air

Exhaust air

Figure 9.27. Test rig for H2O/CO2 selectivity.


Inlet InletOutlet

Cells

Membrane

Inlet InletOutlet

Inlet InletOutlet

Cells

Membrane

Inlet InletOutlet

Figure 9.28. Two cells to sandwich membranes.

9.6. CONCLUSION

Two novel membranes are developed for total heat exchangers. One employs a thin

active layer on a support porous layer. The other one employs a liquid layer supported in one porous media. The two are highly permeable for water vapor. They provide a solution to total heat exchangers, which demand cheap, permselective, high vapor permeable membranes. Moisture diffusivity and heat conductivity are evaluated. For sensible heat transfer, membrane resistance is very small and it can be neglected. However for moisture transfer, membrane resistance is the major part of total resistance. Therefore it cannot be neglected. Novel membranes should be developed to reduce this part of resistance. It has a determining effect of latent effectiveness of a total heat exchanger.

REFERENCES

[1] Harriman, L.G.; Judge, J. Dehumidification equipment advances. ASHRAE Journal, 2002, 44, 22-29.

[2] Zhang, L.Z.; Niu,J.L. Energy requirements for conditioning fresh air and the long-term savings with a membrane-based energy recovery ventilator in Hong Kong. Energy, 2001, 26, 119-135.

[3] Reineke, C.R.; Moll, D.J.; Reddy, D.; Wessling, R.A. Functional Polymers. New York: Plenum Press; 1989.

[4] Ye, X.H.; Levan, M.D. Water transport properties of Nafion membranes Part I. Single-tube membrane module for air drying. Journal of Membrane Science, 2003, 221, 147-161.

[5] Cha, J.S.; Li, R.; Sirkar, K.K. Removal of water vapor and VOCs from nitrogen in a hydrophilic hollow fiber gel membrane permeator. Journal of Membrane Science, 1996, 119, 139-153.

Li-Zhi Zhang 226

[6] Pan, C.Y.; Jensen, C.D.; Bielech, C.; Habgood, H.W. Permeation of water vapor through cellulose triacetate membranes in hollow fiber form. Journal of Applied Polymer Science, 1978, 22, 2307-2323.

[7] Hu, H.; Jia, J. Xu, J. Studies on the sulfonation of poly(phenylene oxide) (PPO) and permeation behavior of gases and water vapor through sulfonated PPO membranes, II. Permeation behavior of gases and water vapor through sulfonated PPO membranes. Journal of Applied Polymer Science, 1995, 51, 1405-1409.

[8] Dilandro, L.; Pegoraro, M. Bordogna, L. Interaction of polyether-polyurethane with water vapor and water-methane separation selectivity. Journal of Membrane Science, 1991, 64, 229-236.

[9] Wang, K.L.; McCray, S.H. Newbold, D.D. Cussler, E.L. Hollow fiber air drying. Journal of Membrane Science, 1992, 72, 231-244.

[10] Aranda, P. Chen, W.J. Martin, C.R. Water transport across polystyrenesulfonate/ alumina composite membranes. Journal of Membrane Science, 1995, 99, 185-195.

[11] Scovazzo, P.; Hoehn, A.; Todd, P. Membrane porosity and hydrophilic membrane based dehumidification performance. Journal of Membrane Science, 2000, 167, 217-225.

[12] Morillon, V.; Debeaufort, F.; Blond, G.; Voilley, A. Temperature influence on moisture transfer through synthetic films. Journal of Membrane Science, 2000, 168, 223-233.

[13] Kistler, K.R.; Cussler, E.L. Membrane modules for building ventilation. Chemical Engineering Research & Design, 2002, 80, 53-64.


[15] Zhang, Li-Zhi; Wang, Yuan-Yuan; Wang, Cai-Ling; Xiang, Hui. Synthesis and characterization of a PVA/LiCl blend membrane for air dehumidification. Journal of Membrane Science, 2008, 308, 198-206.

[16] Zhang, L.Z. Fabrication of a Lithium Chloride solution based composite supported liquid membrane and its moisture permeation analysis. Journal of Membrane Science, 2006, 276, 91-100.

[17] Cussler, E.L. Diffusion-Mass Transfer in Fluid systems. Cambridge: Cambridge University Press; 2000.

[18] Isetti, C.; Nannei, E.; Magrini, A. On the application of a membrane air-liquid contactor for air dehumidification. Energy and Buildings, 1997, 15, 185-193.

[19] Ito, A. Dehumidification of air by a hygroscopic liquid membrane supported on surface of a hydrophobic microporous membrane. Journal of Membrane Science, 2000, 175, 35-42.

[20] Sengupta, A.; Raghuraman, B.; Sirkar, K.K. Liquid membranes for flue gas desulfurization. Journal of Membrane Science, 1990, 51, 105-126.

[21] Quinn, R.; Appleby, J.B.; Pez, G.P. Hydrogen sulfide separation from gas streams using salt hydrate chemical absorbents and immobilized liquid membranes. Separation Science and Technology, 2002, 37, 627-638.

[22] Lin, S.H.; Pan, C.L.; Leu, H.G. Equilibrium and mass transfer characteristics of 2-chlorophenol removal from aqueous solution by liquid membrane. Chemical Engineering Journal, 2002, 87, 163-169.

[23] Dreher, T.M.; Stevens, G.W. Instability mechanisms of supported liquid membranes. Separation Science and Technology, 1998, 33, 835-853.


[24] Fortunato, R.; Afonso, C.A.M.; Reis, M.A.M.; Crespo, J.G. Supported liquid membranes using ionic liquids: study of stability and transport mechanisms. Journal of Membrane Science, 2004, 242, 197-209.

[25] Zhang, L.Z. Fabrication of a Lithium Chloride solution based composite supported liquid membrane and its moisture permeation analysis. Journal of Membrane Science, 2006, 276, 91-100.

[26] Zhang, L.Z. Effects of membrane parameters on performance of vapor permeation through a composite supported liquid membrane. Separation Science and Technology, 2006, 41, 3517-3538.

[27] Zhang, L.Z. Evaluation of moisture diffusivity in hydrophilic polymer membranes: a new approach. Journal of Membrane Science, 2006, 269, 75-83.

[28] Khayet, M.; Matsuura, T. Pervaporation and vacuum membrane distillation processes: modeling and experiments. AIChe Journal, 2004, 50, 1697-1712.

[29] Tomaszewska, M.; Gryta, M.; Morawski, A.W. Mass transfer of HCl and H2O across hydrophobic membrane during membrane distillation. Journal of Membrane Science, 2000, 166, 149-157.

[30] Datta, A.K. Porous media approaches to studying simultaneous heat and mass transfer in food processes. I: problem formulations. Journal of Food Engineering, 2007, 80, 80-95.

Chapter 10

HEAT MASS TRANSFER IN CROSS-CORRUGATED TRIANGULAR DUCTS

ABSTRACT

To increase sensible and latent effectiveness of a stationary total heat exchanger with limited transfer area, heat and mass transfer intensification should be performed. There are two directions for heat mass transfer intensification: material side intensification and air side intensification. Cross corrugated triangular ducts are a structure to intensify heat mass transfer in air side. They belong to the primary surface heat exchangers. In this chapter, fluid flow and convective heat mass in the cross-corrugated triangular ducts are numerically calculated. Due to the different flow regimes under different Reynolds numbers, various momentum models from laminar, transitional, to fully turbulent are considered.

NOMENCLATURE Aci cross-sectional area at inlet or outlet of a cycle, m2 Acyc surface area of the channel, m2 cp specific heat, kJ/(kgK) Dh hydraulic diameter of the channel, m Dva vapor diffusivity in air, m2/s f friction factor h heat transfer coefficient, kW/(m2K) k turbulent kinetic energy, m2/s2 Lcyc length of a cycle in flow direction, m Nu Nusselt Number P time average pressure, Pa Pr Prandtl Numbers q heat flux, W/m2; mass flux, kgm-2s-1 Re Reynolds number, Sc Schmidt Numbers

Li-Zhi Zhang 230

T temperature, K U time average velocity, m/s Vcyc volume of channel, m3 x, y or z coordinates, m ycyc pitch of the repeated segment of the duct, m zcyc width of the repeated segment of the duct, m

Greek letters ρ density, kg/m3 ν kinematic viscosity, m2/s μ molecular viscosity, Ns/m2 τ shear stress, N/m2 ω specific dissipation, s-1 ε turbulent dissipation rate, m2/s3

Superscripts * dimensionless ‘ fluctuation

Subscripts 0 inlet i inlet m mean o outlet t turbulent w wall

10.1. INTRODUCTION Parallel plates and plate-fin have been the main structure for total heat exchangers.

Parallel-plates are simple, however their heat mass transfer capability is limited. Additional spacers are required to ensure the channel pitch not narrowed by neighboring membranes collapsing, contacting, or oscillating under air flow. Plate-fin is strong, stable and compact. However as discussed in previous chapters, due to the finite fin conductance both for sensible heat and latent heat, the fin efficiency is quite limited.

To enhance the heat and mass transfer, a structure named the cross-corrugated triangular membrane duct has been proposed [1-3]. It belongs to a type of primary surface heat exchangers, which have been used for air-to-air sensible heat exchangers. The concept is shown in Figure 10.1. Flat membrane sheets are corrugated to form a series of parallel equilateral triangular ducts. Sheets of the corrugated plates are then stacked together to form a 90 degree orientation angle between the neighboring plates, which guarantees the same flow

Heat Mass Transfer in Cross-corrugated Triangular Ducts 231

pattern for both fluids. The shaded area is blocked. The membranes are very thin and soft, which requires plastic frame cases to support them, as seen in Figure 10.2. Consequently, with a pre-designed plastic frame, triangular shaped duct walls are formed to construct the required geometry. The structure gives better heat mass transfer. This efficiency improvement is attributed to the pattern of flow that undergoes abrupt turnaround, contraction, and expansion.

Cross corrugated ducts with sinusoidal cross sections, which are mainly used for rotary regenerators, have been investigated extensively by various investigators [4-10].This chapter gives the recent research results of cross-corrugated triangular ducts. Triangular cross sections are naturally formed by the corrugations of ultra-thin materials like paper, plastic films, tinsel, and hydrophilic membranes, which are increasingly used in air conditioning industries, due to their superiorities in weight-lightness, cheapness, and abilities in selective transfer.

CFD modeling is a cost-effective means to study the flow field and mass transport phenomenon in fine yet complex geometries, where it’s usually hard to get information of local flow and mass fraction distributions, either for technical or economical reasons. With appropriate models, the flow and mass transfer can be simulated rather accurately. In fact, CFD has become the most efficient tool for design of chemical processes and equipments. Using CFD to study mass transport in heat mass exchangers is one of the fast developing technologies. In this chapter, the convective heat mass transfer coefficients and the friction factors, which are the major parameters affecting membrane total heat exchanger performance, were investigated with this efficient tool.

Fresh

Exhaust

Fresh

Exhaust

Figure 10.1. Cross-corrugated triangular ducts.

Li-Zhi Zhang 232

Spacer MembraneSpacer Membrane

Figure 10.2. The spacer/case to form the channel.

10.2. LAMINAR FLOW

Mathematical Model

A computational domain is selected as Figure 10.3. It is a representing unit cell in the

total heat exchanger. The basic set of equations that require solving comprise of equations for: • conservation of mass; • conservation of momentum, in three coordinate directions and • conservation of energy. The general form of the mass continuity equation as shown below is valid for

compressible and incompressible flows:

( ) 0=∂∂

+∂∂

ii

uxt

ρρ (10.1)

where ρ is the fluid density (kg/m3), t the time (s), u the flow velocity (m/s), subscript i denotes coordinates directions, say, x, y or z.

xz y

Air outAir in

Duct wallSymmetryx

z y

Air outAir in

Duct wallSymmetry

Figure 10.3. The single cross-corrugated channel segment for computation.


The conservation of momentum in the ith direction in an inertial reference frame is governed by

( ) ( ) ij

ijji

ji B

xuu

xu

t+

∂

∂=

∂∂

+∂∂ σ

ρρ (10.2)

where Bi is a body force in the ith direction. It includes contributions from gravitational acceleration and external body forces. The stress tensor, σij, is given by

ijk

k

i

j

j

iijij x

uxu

xu

p δμμδσ∂∂

−⎥⎥⎦

⎤

⎢⎢⎣

⎡

∂

∂+

∂∂

+−=32

(10.3)

where the final term is the effect of volume dilation (zero for an incompressible fluid). The pressure and molecular viscosity are denoted by p (Pa) and μ (Ns/m2), respectively; δij is the Kroneker operator, which equals to 1 when i=j, and 0 when i≠j.

In addition, a general scalar advection-diffusion equation for a dependent variable, φ, is given by

Sx

Γuxt i

ii

=⎥⎦

⎤⎢⎣

⎡∂

φ∂−φρ

∂∂

+∂ρφ∂ )(

(10.4)

Γ is the diffusion coefficient and S a source or sink term representing creation or

destruction of Γ. For fully compressible flow, the energy transport equation is solved for the total enthalpy,

H, according to

txTHu

xtH

ii

i ∂ρ∂

=⎥⎦

⎤⎢⎣

⎡∂∂

λ−ρ∂∂

+∂ρ∂ )(

(10.5)

where λ is the thermal conductivity [kW/(mK)] and T the temperature (K). H is expressed in terms of the static enthalpy, h, according to:

2

21 uhH += (10.6)

ρpUh ie += (10.7)

Li-Zhi Zhang 234

For weakly compressible and incompressible flow, the kinetic energy term (1/2ρu2) is assumed to be negligible compared to the internal term, Uie. The pressure work term, p/ρ, may also be safely ignored.

The above equations, also known as the Navier-Stokes equations, represent five transport equations in the seven unknown field variables ux, uy, uz, p, ρ, h, and T. The thermal equation of state provides a sixth equation relating density, ρ, to temperature T and pressure p. For air, this equation is called the idea gas equation. The seventh equation required to close the entire system is a thermodynamic relation between the state variables. For air, this equation defines the function of static enthalpy in terms of temperature and pressure, i.e., h=h(T,P). Since the fluid is assumed to be thermally perfect, the static enthalpy is a function of temperature only.

The hydraulic diameter of the channel is defined as

cyc

cych

4AV

D = (10.8)

where Vcyc and Acyc are the volume and the surface area of the channel, respectively.

The Reynolds number, Re, is

μρ hmRe Du

= (10.9)

where um is the area-weighted mean velocity through a cross-section, (m/s).

The cycle-average heat transfer coefficient is evaluated from the temperature difference between the inlet and the outlet of a cycle

( )TA

TTAcuh

Δ

−=

cyc

oicipmm

ρ (10.10)

where cp is the specific heat of fluid, kJ/(kgK); Aci is the cross-sectional area at inlet or outlet of a cycle, (m2); Ti and To are fluid mass weighted temperature at inlet and outlet of a cycle, respectively (K); ΔT is the logarithmic temperature difference between the wall and the fluid, which is calculated by

wo

wi

wowi

ln

)()(

TTTT

TTTTT

−−

−−−=Δ (10.11)

where Tw is the wall temperature (K).

The cycle-average Nusselt number, Num, is defined as


λhm

mDhNu = (10.12)

The cycle-average friction factor is calculated by

2m

hcyc

oi

m

21 u

DL

pp

fρ

−

= (10.13)

where Lcyc is the length of a cycle, (m); pi and po are pressure at inlet and outlet of a cycle, respectively, (Pa).

In a cycle, the heat transfer coefficient and friction factor have different local values. The local heat transfer coefficient along the flow is defined by

wwbL )(

1nT

TTh

∂∂

−−= (10.14)

where Tb is the bulk temperature, (K); wn

T∂∂

is the temperature gradient at the wall surface in

normal position. The local Nusselt number

λhL

LDhNu = (10.15)

The local Darcy friction factor

2m

w

i

L

8

unu

fρ

μ ⎟⎠⎞

⎜⎝⎛

∂∂

= (10.16)

where w

i ⎟⎠⎞

⎜⎝⎛

∂∂

nu

is the velocity gradient at the wall surface in normal position.


cycLxx =* (10.17)

Li-Zhi Zhang 236

0

*yyy = (10.18)

0

*zzz = (10.19)

where y0, z0 are the pitch and width of the repeated segment of the duct (m), respectively.

Solution Method and Validation The problem is solved with a commercial CFD code Fluent. The meshes on the outside

walls of a computational block are shown in Figure 10.4. The graph depicts the meshes only for the first 3 and a half cycles, to get an amplified view of the mesh structure. Totally there are 10 cycles in this block.

Boundary conditions are defined. A uniform temperature and non-slip velocity wall conditions are assumed. At the inlet, velocity is set to uniform and parallel to the corrugation of the upper wall.

The governing equations are solved by using standard finite difference methods that employ control-volume based discretization techniques along with a pressure-correction algorithm. The N-S equations are solved by SIMPLEC scheme, while the convective term in the energy equation is solved by first-order upwind implicit approximation, and the diffusive term is by second-order central difference scheme. The fluid is selected as air.

Figure 10.4. The grid distribution for the computation domain, showing 2 and a half cycles, totally 10cycles.


Table 10.1. Comparisons of (fD⋅Re) and NuD of fully developed laminar flow for some ducts from present study and those from literature

Duct geometry τ (fD⋅Re) NuD Present

study Refs Deviation

(%) Present study

Refs Deviation (%)

Circular 64.334 64.000 0.51 3.667 3.657 0.55 Square 56.470 56.908 0.78 3.011 2.976 0.84 Elliptic 0.5 67.754 67.292 0.70 3.753 3.742 0.56 Equilateral Triangular 53.334 53.284 0.11 2.476 2.46 0.65 Isosceles Triangular 0.5 52.718 52.612 0.18 2.331 2.34 0.90 Sine 2.0 58.314 58.212 0.16 2.658 Unavailable Sine 1.5 55.726 56.088 0.63 2.614 2.6 0.54 Sine 1.0 51.688 52.092 0.77 2.463 2.45 0.53 Sine 0.75 49.304 48.936 0.75 2.317 2.33 0.56 Sine 0.50 44.682 44.828 0.30 2.141 2.12 0.99 Notes: τ, aspect ratio (duct height/duct width).

The grid independency test has been done. The calculations were primarily carried out

with three different grid densities, 219985, 114992, and 429970 mesh points. The channel fully developed periodic mean pressure drop and temperature change for the two fine grids are almost the same and 10% higher than that for the coarse grid. For the finest grids, 429970, the solution time is very long, which is hard to use practically. Based on the above experience, which establishes the grid independency, the final calculations are performed for the 219985 grids and the results obtained in this paper refer to the grid geometry mentioned above.

To validate the solution procedure, the methodology has been used to calculate the friction factor and Nusselt numbers of fully-developed flows in ducts of various cross-sections. For the fully developed laminar flow in such ducts, the local values of (fL⋅Re) and NuL are constants. In such cases, these constant values are denoted as (fD⋅Re) and NuD, respectively. Table 10.1 shows the values calculated and give in refs [11,12]. From this table, it can be concluded that maximum errors are less than 0.8% for (fD⋅Re) and less than 0.99% for NuD.

Flow Distribution and Friction Factor Figure 10.5 shows the vector plot of the velocity in the x-y plane at z*=0.5 (the center in

width), for Re=1000. As seen from this figure, the flow has two distinct patterns: in the corrugation troughs of the upper wall, the flow is parallel steady flow, while in the troughs of the lower wall, fluid re-circulation or swirl flows are generated due to the reason that the fluid separates from the rear-facing facet and reattaches to the front facing facet. The shapes of flow re-circulation in the valleys become almost identical to each other, after 3-5 cycles, indicating a cyclic manner. The maximum value of the velocity occurs near the peaks of the lower wall, where the flow has the least cross section area. The velocity is the smallest where

Li-Zhi Zhang 238

the duct expansion occurs. The vortexes resulted from duct expansion and contraction are helpful for momentum transfer.

The variations of the local wall friction factor along the perimeter of the duct cross section that is perpendicular to x axis are shown in Figure 10.6 for the upper wall. A double climax pattern is found in the figure. The local wall friction factor has the maximum value at the center of the border edge of the upper wall (points B and D), where the velocity is the highest. It has the least value at the ends of the border edge of duct walls (Points A, C, E), where the velocity is also the least. Due to the symmetry of duct cross section, the local wall friction factor also distributes symmetrically.

Figure 10.5. Velocity contours in the x-y plane at z*=0.5, Re=1000.

A

B

C

D

EA

B

C

D

E

(a) The geometry of duct cross section and positions.

z*

fL*

0 1

1

A

B

C

D

E

z*

fL*

0 1

1

A

B

C

D

E

(b) The local wall friction factor.

Figure 10.6. Local wall friction factor on the cross-section plane at the center of 5th cycle, Re=1000; (a) the geometry of duct cross section and (b) the local wall friction factor distribution.


0

0.5

1

1.5

2

0 2 4 6 8 10 12

Cycles

f m

Re=1000Re=533Re=100

Figure 10.7. Mean friction factor of each cycle along the duct。

Variations of the local wall friction factor along duct length also have a cyclic manner: the local wall friction factor becomes the least at the valley of the cycle, while reaches the highest at the crest. The reason is that at the valley, flow has the least velocity; while at the crest, flow has the highest velocity. After 3-5 cycles, the behavior of local wall friction factor gets stable on a cycle-by-cycle basis.

For the cross-corrugated triangular ducts, the cyclic mean friction factor is one of the most important parameters affecting heat exchanger design. In Figure 10.7 is shown the mean friction factor of each cycle along the flow direction, for 3 different flow conditions. As shown, the cyclic mean friction factor is very high at the entrance, but it decreases rapidly in this region. After 3-5 cycles, it decreases gradually to a stable value, which is called the fully developed cyclic mean friction factor, fD. Also indicated in this figure is that the higher the Re, the greater the fD.

Figure 10.8 shows the variations of the fully developed cyclic mean friction factor with increasing Re for ducts of three cross sectional shapes: the corrugated duct in this study, the parallel flat plates duct, and a constant cross sectional area triangular duct. It should be noted that for the straight ducts of constant cross section, fD denotes the fully developed local friction factor. Generally speaking, the friction factor for a corrugated duct is greater than that for a triangular duct. When the Reynolds numbers are less than 200, fD of the corrugated duct is less than the parallel plates duct. At higher flow rates, i.e., bigger Reynolds numbers, fD of the corrugated duct becomes greater than the parallel plates duct. This characteristics disclose a fact that flow field in the corrugated duct has been enhanced, in comparison to a traditional duct, especially when at higher Re numbers, due to the existence of vortexes. The growth of steady swirl in the corrugation valleys with Reynolds numbers and the concomitant periodic

Li-Zhi Zhang 240

disruption and thinning of the boundary layer promote enhanced transport of momentum. In contrary, in the non-swirl flow regime, the enlarged surface area of the corrugated plate is solely responsible for the enhancement, which is significantly less.

A correlation has been proposed to establish the relations between the periodically mean friction factor of the cross-corrugated triangular duct with Reynolds numbers as following:

5121.0Re03.11 −=Df (10.20)

for 10≤Re≤2000

Temperature Distribution and Nusselt Number The isotherms shown in Figure 10.9 in an x-y midplane for Re=1000 clearly show a

cyclic manner. After 5-7 cycles, the thermal boundary layer has become fully developed. The temperature profiles in a y-z cross section are shown in Figure 10.10. The temperature gradients have the highest values near the center of the boundary edge of the upper wall, while have the least values on the valley of the lower wall. Flow conditions also have an influence on temperature distribution. At smaller Re numbers, temperature gradients in the valleys of the lower wall are smaller, due to the reduced intensities of swirls.

Re

f D

101 102 10310-1

100

101

CorrugatedFlat parallelTriangular

Figure 10.8. Friction factor for fully developed flow for three ducts.


Figure 10.9. Temperature contour on the x-y plane at z*=0.5, Re=1000.

Figure 10.10. Temperature contour on the z-y plane at x*=5.5 (center of the 5th cycle), Re=1000.

Figure 10.11 plots the distribution of the normalized local wall heat transfer coefficients (h*=h/hmax) in an x-y plane at z*=0.5, and Re=1000. The values shown in the figure is for the lower wall. The local wall heat transfer coefficients demonstrate a cyclic pattern: the shapes of the local wall heat transfer coefficients distribution resemble each other on a cycle-by-cycle basis. The heat transfer coefficients have the highest values at the crests of each cycle, while have the least values at the valleys of each cycle. In combining the analysis of flow fields, it can be concluded that heat transfer coefficients have the highest values where the velocity are the highest.

Li-Zhi Zhang 242

Figure 10.11. Distribution of normalized local wall heat transfer coefficient on the x-y plane at z*=0.5, Re=1000.

Figure 10.12 shows the local wall heat transfer coefficients in an y-z plane, for the upper wall. A double climax pattern is observed, similar to that observed in local friction factor analysis. The local wall heat transfer coefficients are the greatest at the center of the triangular edges (Points B, D, in Figure 10.6(a)), while they are the least neat the sharp corners of the cross section. This is due to the dead spaces in the sharp corners. After about 5-7 cycles, the magnitudes of the variations of the local heat transfer coefficients become stable, which implicates a fully developed thermal boundary condition.

Figure 10.12. Distribution of normalized local wall heat transfer coefficient in the z-y plane at x*=5.5.


Figure 10.13 shows the mean Nusselt numbers for each cycle, Num, under different flow rates. At the entrance, the cyclic mean Nusselt numbers are very high, due to very thin boundary layers at the entrance. Along the flow direction, the cyclic mean Nusselt numbers decrease rapidly in the first 5-7 cycles, and then arrive gradually at some stable values, which is denoted as the fully developed value, NuD. For parallel flat plates, NuD=7.54; for triangular straight duct, NuD=2.47 (Incropera and Dewitt, 1996). Contrary to the fully developed flow in straight ducts of constant cross sections, the fully developed Nusselt numbers, NuD, are not constants. Rather, they are variables influenced by Reynolds numbers.

0

2

4

6

8

10

12

14

16

18

0 2 4 6 8 10 12

Cycles

Nu m

Re=1000Re=533Re=100

Figure 10.13. Cyclic mean Nusselt numbers for the flow.

Re

Nu D

101 102 1030

2

4

6

8

10

12

14

16

CorrugatedFlat parallelTriangular

Figure 10.14. Comparison of fully developed Nusselt numbers for three ducts.

Li-Zhi Zhang 244

In Figure 10.14 is shown the comparisons of fully developed Nusselt numbers of three ducts: the parallel flat plates, equilateral triangular straight duct, and the proposed cross-corrugated triangular duct. As is shown, the Nusselt numbers of the corrugated duct are higher than a common straight triangular duct. In Re range between 100 and 2000, compared to a triangular straight duct, the heat transfer for the cross-corrugated duct is enhanced by 60% to 480%. As expected, the lower Nusselt numbers of the triangular straight duct is due to large dead spaces in the sharp corners.

At lower Re numbers, the Nusselt numbers of the cross corrugated duct are lower than parallel flat plates. However, at higher Reynolds numbers, i.e., 500≤Re≤2000, the Nusselt numbers of the cross-corrugated duct grow rapidly, and surpass the parallel flat plates with large margins. The reason behind this fact is that at lower flow rates, the swirls in the valleys of the cross-corrugated duct is not strong enough, and the cyclic heat transfer is mainly determined by the parallel flows in the corrugations of the upper wall, which is sacrificed by the dead spaces in the corners. However, when the flow rates are high, strong swirls are generated in the valleys, which effectively reduces the thickness of the boundary layers and offsets the influences of dead spaces. In such cases, the swirls in the valleys are the dominant factor for heat transfer. The heat transfer is enhanced by a factor of 2 when Re=2000, and is expected to rise sharply with further increases in Re.

Studies of the flow with different flow rates and several gases have helped to propose a correlation to estimate the fully developed cyclic mean Nusselt numbers as following:

333.02157.0 PrRe5528.0=DNu (10.21)

for 10≤Re≤2000. Pr is the Prandtl number.

10.3. TURBULENT FLOW When Reynolds numbers are high, laminar model is not appropriate. This section is to

study turbulent flow for Re=2000~20000. The computational domain and grid structure are the same as previous laminar flow. However the model is different.

Turbulence Models The equations describing the fluid flow and heat transfer are transport equations for the

continuity, momentum, and energy, which are developed from conservation laws of physics. The fluid flow is described by conservation of mass (the continuity equation), momentum (Navier-Stokes equations) and energy (the temperature equation for the fluid). The velocities and temperatures are time-averaged and divided into a mean and a fluctuating value,

'jjj uUu += and 'TTT += . Together with the boundary conditions, they form the

steady state governing equations for incompressible flow with negligible external and viscous forces:


( )0=

∂

∂

j

j

xρU

(10.22)

( ) ( )tijij

jiji

j xxPUU

xττρ +

∂∂

+∂∂

−=∂∂

(10.23)

⎟⎟⎠

⎞⎜⎜⎝

⎛

∂

∂+

∂∂

=i

j

j

iij x

UxU

μτ ; ''ji

tij uuρτ −= (10.24)

( ) ( )'jj

jjp

j

qqx

TUcx

+∂∂

=∂∂ ρ (10.25)

j

pj x

Tcq

∂∂

=Pr

μ; '' Tucq jp

tj ρ−= (10.26)

where ρ, μ, and cp are density, viscosity, specific heat, respectively. It would be impossible to solve these equations analytically because of non-linearity and the stochastic nature of turbulence. The extra terms that appear due to averaging the velocity and temperature are the Reynolds stress and the turbulent heat flux. Modeling these is known as the closure problem of turbulence. Various turbulence models have been proposed and totally 4 models are considered.

The definition of hydraulic diameter, Reynolds number, heat transfer coefficient, Nusselt number, and friction factor are the same as laminar flow. Velocity, temperature are selected as the time averaged mean values.

Standard k-ε Model The k-ε model is the most popular of the two-equation models and has produced

qualitatively satisfactory results for a number of complex flows. According to this concept, the turbulent shear stress in Eq.(10.23) is determined by

ρδμτ kxU

xU

iji

j

j

it

tij 3

2−⎟

⎟⎠

⎞⎜⎜⎝

⎛

∂

∂+

∂∂

= (10.27)

where ijδ is the Kronecker delta function, 1=ijδ when i=j and zero when ji ≠ . The

turbulent viscosity μt is determined by Prandtl-Kolmogorov equation [13]

ερμ μ /2kCt = (10.28)

Li-Zhi Zhang 246

where the turbulence kinetic energy k and dissipation rate ε are calculated by

ερμ

σμ

ρ−⎟

⎟⎠

⎞⎜⎜⎝

⎛

∂

∂+

∂∂

∂∂

+⎟⎟⎠

⎞⎜⎜⎝

⎛

∂∂

∂∂

=∂∂

i

j

j

i

j

it

jk

t

jjj x

UxU

xU

xk

xxkU 1

(10.29)

kC

xU

xU

xU

kC

xxxU

i

j

j

i

j

it

j

t

jjj

2

ε2ε1ε

1 ερμεε

σμ

ρε

−⎟⎟⎠

⎞⎜⎜⎝

⎛

∂

∂+

∂∂

∂∂

+⎟⎟⎠

⎞⎜⎜⎝

⎛

∂∂

∂∂

=∂∂

(10.30)

The turbulent heat transfer term in Eq.(10.26) is determined by the following equation

j

t

xTq

∂∂

σμ

=θ

' (10.31)

The constants in the above model take following values [13]: Cμ=0.09; Cε1=1.44; Cε2=1.92; σk=1.3; σε=1.3; σθ=1.3

Renormalized k-ε model The RNG-based k-ε model follows the same framework as the above two equations

model but uses Renormalization Group methods [14]. The model is to provide improved predictions of near-wall flows and flows with high streamline curvature. The momentum and energy equations can be re-written in the following form:

( )⎥⎥⎦

⎤

⎢⎢⎣

⎡

∂∂

−⎟⎟⎠

⎞⎜⎜⎝

⎛

∂

∂+

∂∂

∂∂

+∂∂

−=∂

ρ∂

k

keff

i

j

j

i

jij

ji

xu

μxU

xU

μxx

Px

UU32

eff (10.32)

( )⎥⎥⎦

⎤

⎢⎢⎣

⎡

∂∂

−⎟⎟⎠

⎞⎜⎜⎝

⎛

∂

∂+

∂∂

∂

∂+

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛

∂∂

α∂∂

=∂

ρ∂

k

keff

i

j

j

i

i

j

jeffT

jj

pj

xu

μxU

xU

μxU

xTμ

xxTcU

32

eff (10.33)

where

2

1⎥⎥⎦

⎤

⎢⎢⎣

⎡+=

εk

μC

μμ μeff (10.34)

( )⎥⎥⎦

⎤

⎢⎢⎣

⎡

∂∂

∂∂

+−∂

∂=

∂∂

jeffk

jj

jtij

jj x

kμx

εxUτ

xkU α

ρ (10.35)


Rx

μxk

CxUτ

kC

xεU

jeff

jj

itij

jj −⎟

⎟⎠

⎞⎜⎜⎝

⎛

∂∂

∂∂

+−∂∂

=∂∂ εαε

ρε

ε

2

ε2ε1 (10.36)

where R in the ε equation is given by

kβηηηηC

R 0μ 2

3

3

1

1ε

+

⎟⎟⎠

⎞⎜⎜⎝

⎛−

= (10.37)

with ε

=ηSk

, and η0=4.38, β=0.012. Other constants are [15]:

Cμ=0.085; Cε1=1.42; Cε2=1.68 The term S is the modulus of the mean rate-of-strain tensor, Sij, which is defined as

ijij SSS 2= (10.38)

where

⎟⎟⎠

⎞⎜⎜⎝

⎛

∂

∂+

∂∂

=i

j

j

iij x

UxU

S21

(10.39)

The RNG k-ε model yields an accurate description of how the effective turbulent

transport varies with the effective Reynolds number (or eddy scale). The coefficients αT, αk, αε, in Eqs. (10.34)-(10.36) are the inverse effect Prandtl number for T, k and ε respectively. They are computed using the following formula:

effμμ

=+α

+α−α

−α3679.0

0

6321.0

0 3929.23929.2

3929.13929.1

(10.40)

where α0 is equal to 1/Pr, 1.0, and 1.0, for the computation of αT, αk, αε, respectively.

Low Reynolds k-ω Model Unlike the k-ε model, it is easier to prescribe the boundary conditions in the k-ω model.

We know that k=0 on solid boundaries, and ω can be specified at the first few grid points

Li-Zhi Zhang 248

away from the wall as 2

6βyμ

=ω , (y the distance to wall). The resulting equations for k, ω,

and μt are:

ωk

t*α=ν (10.41)

The equations for kinetic energy and specific dissipation rate are given as:

( )⎥⎥⎦

⎤

⎢⎢⎣

⎡

∂∂

νσ+ν∂∂

+ωβ−∂∂

=∂∂

jtk

jj

iij

jj x

kx

kxU

τxkU * (10.42)

( )jjj

tjj

iij

tjj xx

kxxx

Uτ

xU

∂ω∂

∂∂

ωσ+

⎥⎥⎦

⎤

⎢⎢⎣

⎡

∂ω∂

νσ+ν∂∂

+ωβ−∂∂

νσ

=∂

ω∂ωω

12 22

2 (10.43)

The model constants are provided in Ref [16, 17]:

4

4

*

Re1

Re185

1009

⎟⎟⎠

⎞⎜⎜⎝

⎛+

⎟⎟⎠

⎞⎜⎜⎝

⎛+

=β

β

β

R

R

t

t

(10.44)

k

t

k

t

R

RRe

1

Re*0

*

+

+α=α (10.45)

( ) 1*0

Re1

Re

95 −

ω

ω α+

+α=α

R

Rt

t

(10.46)

β=0.075, α0=0.1, σk=0.1, σω=0.5, Rβ=8, Rk=6, α0

*=β/3, Rω=2.7


Full Reynolds Stress Modeling The Reynolds stress model (RSM) is also considered as a choice. The RSM model

equation for the transport of Reynolds stresses is given by the following equation:

( ) ( )[ ]( )

k

j

k

i

i

j

j

i

k

ikj

k

jkiji

kk

jikikjkjik

jikk

xu

xu

xu

xu

p

xU

uuxU

uuuuxx

uupuuux

uuUx

∂

∂

∂∂

−⎟⎟⎠

⎞⎜⎜⎝

⎛

∂

∂+

∂∂

+

⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

+∂

∂−⎥

⎦

⎤⎢⎣

⎡∂∂

∂∂

+

++∂∂

−=∂∂

''''

''''''

'''''''

2μ

ρμ

δδρρ

(10.47)

The summation convention is used in the above equations. Therefore, there are totally 6

equations. In addition to the Reynolds stress transport equations, the dissipation rate is modeled by the dissipation equation as in the standard k-ε model. A detailed description of the model is given by Moore et al. [18] and Rokni et al. [19]. The default constants supplied by CFD code for simulation are:

Cμ=0.09; Cε1=1.44; Cε2=1.92; C1ps=1.8

Boundary Conditions At the walls, the no-slip condition is used. For k-ε and Reynolds stress models, the two-

layer-based, non-equilibrium wall functions methods are also employed. The key elements in the non-equilibrium wall functions are as follows:

1. The log-law for mean velocity is sensitized to pressure-gradient effects. 2. The two-layer-based concept is adopted to compute the budget of turbulence kinetic

energy in the wall-neighboring cells. 3. The law of the wall for mean temperature remains the same as in the standard wall

function. The wall-neighboring cells are assumed to consist of a viscous sublayer and a fully

turbulent layer, where the logarithmic law of the wall applies. This method requires some consideration of mesh, i.e., the cell adjacent to the wall should be located to ensure that the

parameter y+ ⎟⎟⎠

⎞⎜⎜⎝

⎛≡ τ

μyρu

(uτ, friction velocity) or y* ⎟⎟⎠

⎞⎜⎜⎝

⎛≡

μykρC 1/2

p1/4μ falls into the 30-60

range. In the present study, the y+ is adapted into 35-55 range. In addition, a uniform wall temperature condition is assumed.

Li-Zhi Zhang 250

At the inlet, all dependent variables are assumed to enter the pipe with uniform profile in the direction parallel to the corrugation of the upper wall, i.e,

u=u0, T=T0, k=k0, ε=ε0 (10.48) The inlet boundary values of k and ε are computed from an estimated turbulence

intensity, I, and turbulent length scale, l, as follows:

( )200 3

2 Iuk = , l

kC

2/304/3

0 με = (10.49)

The turbulent intensity I, defined as uu /' , is equal to 5%, and the length scale l is set to

be 0.07⋅Dh in the present study, as suggested by Li et al. [15]. The exit boundary condition is treated as an outflow condition, which means that the

diffusion flux for all dependent variables are set to zero at the exit and an overall mass balance is obeyed. This outflow boundary condition is true if the flow becomes fully developed at a position far upstream from the exit because the accuracy of the exit boundary condition should not affect the flow and heat transfer fields far upstream from the exit. The results computed afterwards indicated that the flow becomes fully developed after 3-5 cycles after the inlet. Therefore, the assumption is considered valid.

Solution Method The governing equations are solved by using standard finite difference methods that

employ control-volume based discretization techniques along with a pressure-correction algorithm. The N-S equations are solved by SIMPLEC scheme, while the convective term in the energy equation is solved by first-order upwind implicit approximation, and the diffusive term is by second-order central difference scheme. The fluid in the study is selected as air.

Because of the intensive nonlinearity and coupling features of this problem, the under-relaxation technique is applied to the iteration process to accelerate convergence. The convergence criterion of

41 10−

− ≤n

n

RR

φ

φ (10.50)

is applied for all equations, where Rφ

n refers to the maximum residual value summed over all the computation cells after nth iteration. To test the criterion independence, another convergence criterion of 10-5 is applied to a case. The difference of computed periodic Nu numbers of the two convergence criterion is within 1%.

The grid independency test has also been done. The calculations were primarily carried out with three different grid densities, 209985, 104992, and 419970 mesh points. The channel fully developed periodic mean pressure drop and temperature change for the two fine grids


are almost the same and 10% higher than that for the coarse grid. For the finest grids, 419970, the solution is too time-consuming, which is hard to use practically. Based on the above experience, which establishes the grid independency, the final calculations are performed for the 209985 grids and the results obtained in this paper refer to the grid geometry mentioned above.

Model Validation Selection of an appropriate turbulence model for numerical simulation requires

consideration of computational cost, anticipated flow phenomena, and the variables of primary interest. The Nusselt numbers in the duct are of major concern in this case. Therefore the agreements of the calculated Nu and experimental data are the main criterion for selecting turbulence models.

For validation and comparison of the 4 turbulence models used, a cross-corrugated triangular duct with geometric parameters identical to Scott and Lobato [1,2] is numerically simulated: z0=2mm; Lcyc=2mm; y0=1mm. Figure 10.15 shows the experimentally obtained (the discrete data were obtained from the experimentally obtained Sherwood correlations for a 90 degrees corrugation angle) and the calculated fully developed periodic mean Nusselt numbers. As seen from this figure, of the 4 turbulence models employed, generally, the Reynolds stress model (RSM) fits the experiment the best. The differences are from 5% to 11%, for Re ranging from 2000 to 20000. At lower Re, the low Reynolds k-ω model (LKW) agrees the best, while the standard k-ε fits poorly. At higher Re, i.e., Re≥6000, the standard k-ε model (SKE) gives the best prediction. As for the Renormalized group k-ε model (RNG KE), it seems inappropriate to use such model for the corrugated triangular duct: the differences between the calculated and experimentally found are substantial. In the low Re range, the model over-predicts the Nusselt number by 58%, while in the high Re range, it under-predicts the Nu by 21%.

In the following analysis, for Re ranging from 2000 to 2000, the RSM model is employed to investigate the fluid flow and heat transfer. The imperfection with this model is that the computational time is very long [17], due to large memory size required.

Turbulent Flow and Heat Transfer Figure 10.16 shows the vector plot of the velocity in the x-y plane at z*=0.5 (the center in

width), for Re=10000. In the figure, the flow has two distinct patterns: in the corrugation troughs of the upper wall, parallel flow is predominant, while in the troughs of the lower wall, clockwise strong swirls are generated due to the reason that the fluid turns abruptly when facing the trough walls of the lower wall. The shapes of the swirls in the valleys become almost identical to each other, after 3-5 cycles, indicating a cyclic fully developed manner. These swirls intensify the momentum transfer in the duct.

Li-Zhi Zhang 252

0

10

20

30

40

50

60

70

80

0 5000 10000 15000 20000 25000

Re

Nu d

SKERNG SKELKWRSMExperimental

Figure 10.15. Predictions of fully developed periodic mean Nusselt numbers with various turbulence models.

Figure 10.16. Velocity vectors in the x-y midplane showing 5 cycles.

Figure 10.17. Velocity vectors in the y-z plane at x*=4.5.


Figure 10.17 shows the velocity vectors in the y-z plane at x*=4.5. This plane is perpendicular to the main flow direction, and is located at the center of the 4th cycle. It can be seen that in the corner regions, there exist appreciable secondary flows. These secondary flows all exhibit the same pattern: departing from one wall, arriving at the other of the same corner and leaving a small region very close to the corner where the fluid flows are almost retarded. In the central part of the upper wall, the secondary flows are very weak and cannot be observed clearly, while in the lower wall, the secondary flows are relatively strong, even in the central part. There are semi-swirls around each corner. The interactions of these semi-swirls generate demonstrable secondary flows in the central part of the lower wall valley. These secondary flows generated swirls will intensify momentum and heat transfer.

For the cross-corrugated triangular ducts, the cyclic mean friction factor is one of the most important parameters affecting heat exchanger design. Figure 10.18 shows the calculated periodic mean friction factor along the duct length with four turbulence models. The trend is like most of the developing flows: near the inlet region, the friction factor is very high; with the flow propagates, after 3-5 cycles, it decreases gradually to the fully developed value, fd. At the location near the outlet, the mean friction factor rises somewhat, which may be attributed to the influence of outflow boundary conditions.

The corrugation usually leads to increased pressure drop. Figure 10.19 shows the calculated fully-developed periodic mean friction factor with varying Reynolds numbers. Figure 10.20 demonstrates comparisons between the friction factors in parallel flat plates passages and in corrugated ducts. Under the same flow rates, the friction factor for the cross-corrugated ducts is 3 times that for a parallel flat plates duct. Another feature with this figure is that the fd decreases with an increasing Re.

To summarize the relations between the periodic mean friction factors with Renolds numbers, a correlation has been proposed, with data obtained from the RSM model. The correlation is:

27.0Re9398.1 −=Df (10.51)

for 2000≤Re≤10000

In contrast, friction factor in parallel flat plates is correlated by [11]

( ) 264.1Reln790.0 −−=Df (10.52)

Temperature Distribution and Nusselt Number After 3-5 cycles, the thermal flow has become fully developed, and the shapes of

isotherms in different valleys are similar. It’s clear that the isotherms in the last 1-2 valleys have some distortion, which may also be attributed to the exit flow boundary conditions. The temperature gradients at the walls are very high, indicating an enhanced heat transfer, due to strong turbulence. Contrary to laminar flow, which usually has lower heat transfer on the lower walls, the turbulence flow in this case has high heat transfer both on the upper walls and on the lower walls, as a result from turbulence.

Li-Zhi Zhang 254

0

0.05

0.1

0.15

0.2

0.25

0.3

0 2 4 6 8 10 12

Cycles (x *)

f m

SKERNG SKELKWRSM

Figure 10.18. Distribution of periodic mean friction factor along flow direction.

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0 5000 10000 15000 20000 25000

Re

f d

SKERNG SKELKWRSM

Figure 10.19. Calculated fully developed periodic mean friction factor.


Re

f d

5000 10000 15000 200000

0.1

0.2

0.3

0.4

CorrugatedParallel flat plate

Figure 10.20. Comparisons of periodic mean friction factors with Reynolds numbers, cross-corrugated and parallel flat plates.

20

2530

3540

45

5055

6065

70

0 2 4 6 8 10 12

Cycles (x *)

Nu m

SKERNG SKELKWRSM

Figure 10.21. Distribution of periodic mean Nusselt numbers along flow direction.

Figure 10.21 shows the mean Nusselt numbers for each cycle, Num, along the duct length, with four turbulence models. At the entrance, the cyclic mean Nusselt numbers are very high, due to very thin boundary layers at the entrance. Along the flow direction, the cyclic mean

Li-Zhi Zhang 256

Nusselt numbers decrease rapidly in the first 3-5 cycles, and then arrive gradually at some stable values, which is denoted as the fully developed value, NuD. Using the data from RSM model and a least square curve fit technique, a correlation has been formulated, which is:

333.0599.0 PrRe2338.0=DNu (10.52)

for 2000≤Re≤20000.

In contrast, heat transfer of the turbulence flow in parallel flat plates are governed by the Dittus and Boelter equation [20]

333.08.0 PrRe023.0=DNu (10.53)

The variations of fully-developed cyclic mean Nusselt numbers with Re, which is

dictated by the above correlation, is depicted in Figure 10.22. The Nusselt numbers increase, almost linearly with Re. For comparison, the fully developed Nusselt numbers in parallel flat plates passages [20] are also plotted in this figure. As can be seen, the corrugation results in a 40-60% heat transfer enhancement. This is mainly due to the enhanced momentum transfer in the cross-corrugations.

Re

Nu d

5000 10000 15000 200000

10

20

30

40

50

60

70

80

CorrugatedParallel flat plates

Figure 10.22. Comparisons of the fully developed Nusselt numbers for the corrugated duct and parallel flat plates.


Three-dimensional turbulence flow and convective heat transfer in the entrance region of a cross corrugated triangular duct which is proposed for a novel membrane module has been studied numerically employing 4 turbulence models: standard k-ε, Renormalized group k-ε, low Reynolds k-ω, and Reynolds stress model. Comparing to available experimental data, the Reynolds stress model seems superior to others during the whole Reynolds range of 2000 to 20000. The cross-corrugation nature of the duct generates strong turbulence and secondary flows to enhance heat transfer on the duct walls. Comparing to a parallel flat plate geometry, the cross-corrugation can obtain a 40 to 60% heat transfer augmentation, however with a penalty of 2 times more friction pressure drop. In addition, both the friction factor and the heat transfer exhibit a cyclic manner, and the generated swirls rotate clockwise in the corrugation valleys. The turbulence is almost uniform in the upper corrugation, but has large variations in the lower corrugations and especially near the crests of valleys.

10.4. TRANSITIONAL FLOW The aim of this section is to examine numerically flow structure and heat transfer

characteristics in a periodically fully developed cross-corrugated triangular segment. Some correlations will be provided to estimate the cyclic fully developed friction factors and Nusselt numbers in typical transitional flow regimes for Re=100 to 6000.

The flow in the cross corrugated duct is rather complex because of periodic convergent-divergent nature. Various studies have found that transitions from laminar to turbulence occur at Reynolds numbers as low as 150-500 [5-8], much lower than conventional flat-plates ducts. Due to the small characteristic length and relative low air velocity (<5m/s), typical Reynolds numbers for a cross-corrugated plate heat mass exchanger are in the range of 100 to 6000. It is therefore not wise to use a simple laminar modeling technique. In contrary, since the flow in the duct is transitional, numerical modeling should take into account of the turbulence behaviors in the geometry.

Usually, as fully turbulent flows, transitional flows are modeled with turbulence models. This section use the same turbulence models as described in above section. A fully developed cyclic flow is considered.

Geometry For a corrugation angle of 90 degrees between two neighboring plates, the fluid flows

predominantly along the furrows, i.e. in the corrugation on each of the plates (Focke et al. [5]). The flow in this study is assumed parallel to a corrugation, and periodically fully developed. It has been found that after 3-5 cycles, periodicity will be set up. The smallest volume which is worthwhile resolving is one which represents a repeat or periodic building block. The geometry of the problem considered here is shown in Figure 10.23.

Li-Zhi Zhang 258

Air flow

x

y

z

O

Air flow

x

y

z

O

Figure 10.23. The unit cell used for CFD modeling.

Figure 10.24. The external mesh on the plate surfaces.


A computational mesh has been constructed to resolve this geometry; the external mesh on the solid plates is shown in Figure 10.24. A body fitted mesh has been adopted for this work so as to obtain good resolution of the corrugations. Due to the triangular nature of the block, the computational domain is meshed with tetrahedral grids.

Three-dimensional numerical simulations of fluid flow and heat transfer in the computational domain are conducted. The solution technique is based on a finite difference/finite volume representation, while allowing for general body-fitted grids. The SIMPLEC pressure-velocity coupling algorithm is used.

Boundary Conditions The basic fluid dynamics equations are discretised onto the mesh described above and

solved numerically. The equations are elliptic in nature and as such, require boundary conditions to be defined on all boundaries of the computational domain. At the walls, the no-slip condition is used. Both constant temperature and constant heat flux are employed as energy boundary conditions.

The flow is periodically fully developed, which means that the flow in the computational domain is identical to that in the adjacent repeat segment. The flow field entering the domain is identical to that leaving it, and the shape of the pressure distribution leaving the domain is the same as that entering it, although the absolute value will be different. This difference is required to sustain the flow, and it is in fact the pressure drop across the element. In other words, periodicity is assumed for inlet and outlet boundary conditions. More specifically, a constant mean pressure drop is imposed as a boundary condition for CFD calculations. Various values of pressure drop give results under different air flow rates or Reynolds numbers.

0

5

10

15

20

25

30

35

40

0 1000 2000 3000 4000 5000 6000 7000

Re

Nu

m

ExperimantalLKWSKERNG SKELaminar

Figure 10.25. Comparisons of 10-cycles-mean Nusselt numbers with different models.

Li-Zhi Zhang 260

The governing equations are solved by using standard finite difference methods that employ control-volume based discretization techniques along with a pressure-correction algorithm. The N-S equations are solved by SIMPLEC scheme, while the convective term in the energy equation is solved by first-order upwind implicit approximation, and the diffusive term is by second-order central difference scheme. The convergence criterion and grid independence test are the same for fully turbulence modeling.

Model Validation For validation of the turbulence model, a cross-corrugated triangular duct with geometric

parameters identical to Scott and Lobato [1,2] is numerically simulated: zcyc=2mm; Lcyc=2mm; ycyc=1mm, in another separate study. Totally 10 consecutive and repeated cycles considering inlet developing conditions are modeled. The 10-cycles-mean Nusselt numbers are calculated. Figure 10.25 shows the experimentally obtained values from the literature and the calculated mean Nusselt numbers. For comparison, other 2 turbulence models, standard k-ε (SKE), Renormalized Group k-ε (RNG KE), and a laminar model are used. For the k-ε models, wall functions are needed to treat boundary conditions. As seen from this figure, of the 4 turbulence models employed, only the LKW model fits the experiment well in the whole transitional regime. The deviations are from 5% to 12%. For Re ranging from 100 to 500, the laminar model is still acceptable. At high Re numbers, laminar prediction deteriorates drastically, especially when Re>2000. Standard k-ε and RNG k-ε models are only applicable at higher Re numbers greater than 6000. The deviations of experimental values from literature and the predicted Nusselt numbers with these two models could be as high as 65%.

Transitional Flow and Heat Transfer Figure 10.26 shows the vector plot of the velocity in the x-y plane at z*=0.5 (the center in

width), for Re=1000. In the figure, the flow has two distinct patterns: in the corrugation troughs of the upper wall, parallel flow is predominant, while in the troughs of the lower wall, clockwise strong swirls are generated due to the reason that the fluid turns abruptly when facing the trough walls of the lower wall. These swirls intensify the momentum transfer in the duct.

Figure 10.27 shows the velocity vectors in the y-z plane at x*=0.5. This plane is perpendicular to the main flow direction, and is located at the center of the unit cell. It can be seen that the flow is rather complex, but with regular and interesting patterns. There are strong double swirls in the corner regions of the lower wall, which rotate in a clockwise and a counter-clockwise manner, respectively. The secondary flows in the upper corrugation are relatively weak. In summary, in the upper corrugations, parallel flows are predominant, while in the lower corrugation, secondary swirl flows are dominant. These secondary flows generated swirls will intensify momentum and heat transfer.


Figure 10.26. Velocity vectors in x-y plane at z*=1.5.

Figure 10.27. Velocity vectors in y-z plane at x*=1.5.

Temperature contours are shown in Figure 10.28 for an x-y plane at z*=0.5 when Re=1000. The temperature gradients near the lower walls are very high, indicating an enhanced heat transfer, due to strong turbulence. The secondary flows are able to bring the main stream fluid closer to the solid surfaces of the plates and therefore increase the rate of heat transfer. Enhancing these secondary flows lead to an increase in heat transfer coefficient.

Li-Zhi Zhang 262

Figure 10.28. Temperature contours in x-y plane at z*=0.5.

Figure 10.29 shows the cyclic mean friction factors under various Reynolds numbers for the developed flow. The friction factor decreases drastically with Re when the Re is below 2000, but decreases gradually afterwards. A correlation has been formulated to reflect the f-Re relations as

421.0Re5336.6 −=f (10.54)

To make comparisons with the cross corrugated sinusoidal passages, the experimental

data from Ref [5] are also plotted in the figure. It can be seen that when Re<500, two geometries have similar friction factors. When at higher Reynolds numbers, the data for cross-corrugated sinusoidal ducts are much higher than the corrugated triangular ducts. It should be mentioned that the measurements have 5-7% uncertainties [5].

Figure 10.30 shows the cyclic mean Nusselt numbers for various Reynolds numbers. The Nusselt numbers increase, almost linearly with Re. This is mainly due to the enhanced momentum transfer in the cross-corrugations, more specifically, by secondary flows and swirls. Heat transfer at lower Reynolds numbers is relatively low since the laminar free shear layers of the separated regions are an additional resistance. Transfer across turbulent-free shear layers is, however, usually very effective owing to the absence of a restraining effect such as solid wall. The additional resistance is therefore effectively removed when the free shear layers become turbulent.

Two correlations are proposed for uniform temperature and uniform heat flux boundary conditions. They are:

333.0599.0 PrRe1922.0=Nu (10.55)

for uniform temperature, and

333.0569.0 PrRe2743.0=Nu (10.56)

for uniform heat flux conditions, respectively.


0

0.2

0.4

0.6

0.8

1

1.2

0 1000 2000 3000 4000 5000 6000

Re

f

Cross TriangularCross Sinusoidal, ref[5]

Figure 10.29. Variations of cell mean friction factor with Reynolds numbers.

0

5

10

15

20

25

30

35

40

45

0 1000 2000 3000 4000 5000 6000

Re

Nu

uniform fluxuniform temCross Sinusoidal, ref [5]

Figure 10.30. Variations of cyclic mean Nusselt numbers with Reynolds numbers.

Li-Zhi Zhang 264

The experimental data from Ref [5] for a cross-corrugated sinusoidal duct are also plotted in the figure. Generally, the Nusselt numbers for a cross-corrugated sinusoidal ducts lie between the Nusselt numbers of triangular ducts under uniform temperature and uniform heat flux boundary conditions. Certainly differences are expected due to the disparities in the two geometries, but the trends are similar.

10.5. CONVECTIVE MASS TRANSFER Besides heat transfer, mass transfer can be modeled with the same strategy. The turbulent

mass concentration conservation equation is [21]

( ) ( )tjj

jj

j

qqx

YUx

+∂∂

=ρ∂∂

v (10.57)

jj x

YSc

q∂∂μ

= v ; ''vj

tj Yuq ρ−= (10.58)

where ρ, μ, and Sc are density, viscosity, Schmidt number, respectively, and qj is mass flux in kg/m2. The turbulent mass transfer term in Eq.(10.58) is determined by the following equation

j

v

t

ttj x

YSc

q∂∂μ

= (10.59)

where the turbulent Schmidt number Sct=1.0 [21]. These equations can be combined with turbulence model to be solved.

Two correlations have been proposed to estimate the fully developed cyclic mean friction factors and cyclic mean Sherwood numbers. They are:

486.0Re665.9 −=Df (10.60)

333.0539.0

D Re266.0 ScSh = (10.61) For 100≤Re≤3000. For comparison, the experimental Sherwood correlation given in ref [1] for 90 degrees

corrugation is

333.056.0D Re268.0 ScSh = (10.62)


Table 10.2. Structural parameters of total heat exchanger

Parameters Unit Value Exchanger length mm 200.0 Exchanger width mm 200.0 Channel height mm 1.95 Number of channels for each flow 100 Exhaust air temperature °C 27 Exhaust air humidity kg/kg 0.010 Fresh air temperature °C 35 Fresh air humidity kg/kg 0.021 Plate thickness μm 105

Table 10.3. Performance comparison of four total heat exchanger configurations

Case 1 Case 2 Case 3 Case 4 Structure Parallel-plates Parallel-plates Cross-

corrugated Cross- corrugated

Plate type Solid membrane CSLM Solid membrane

CSLM

Dh (mm) 3.96 3.96 2.0 2.0 Re 249 249 126 126 NuD 7.54 7.54 4.67 4.67 ShD 7.10 7.10 4.39 4.39 Air side heat transfer coefficients (kWm-2K-1)

0.05 0.05 0.061 0.061

Air side mass transfer coefficients (m/s)

0.051 0.051 0.062 0.062

fD 0.3855 0.3855 1.1856 1.1856 Plate heat resistance (m2s/kW)

0.3125 0.3852 0.3125 0.3852

Plate moisture resistance (s/m)

114.5 61.8 114.5 61.8

Total transfer area (m2) 8.0 8.0 16.0 16.0 Number of Transfer Units for sensible heat

4.15 4.15 10.0 10.0

Number of Transfer Units for latent heat

1.08 1.65 1.42 2.29

Sensible effectiveness 0.72 0.72 0.82 0.82 Latent effectiveness 0.47 0.57 0.55 0.64 Pressure drop (Pa) 12 12 71 71

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10.6. COMBINED WITH MEMBRANES Considering four design strategies for a total heat exchanger: (1) Parallel plates duct

structure with solid polymer membrane; (2) Parallel plates duct structure with composite supported liquid membrane (CSLM); (3) Cross-corrugated triangular duct structure with solid polymer membrane; and (4) cross-corrugated triangular duct structure with CSLM. The structural parameters and operating conditions are listed in Table 10.2. For the traditional solid polymer membrane, the membrane thickness is selected as 100μm.

Table 10.3 lists the heat and mass transfer properties of the four total heat exchangers. As seen, the cross-corrugated structure intensifies air side convective heat and mass transfer substantially. Compared to a parallel-plates duct, the convective heat and mass transfer coefficients in a cross-corrugated duct increased by 22%. Besides, transfer area increased by 1 fold. These two factors make the new structure more compact and have a higher heat mass transfer capabilities. As a penalty, the pressure drop increased from 12 Pa to 71 Pa.

Compared to a common solid membrane, the composite supported liquid membrane has a 46% less membrane side mass transfer resistance. Consequently, the CSLM could increase latent effectiveness by 8%, even with the same traditional parallel-plates duct structure. The effect of CSLM on decreasing heat transfer resistance is not obvious, since the dominant resistance in the unit is in air side. However, when the CSLM is combined with the cross-corrugated triangular duct, it will have the highest sensible effectiveness of 0.82, and the highest latent effectiveness of 0.64, which is highly desired in market.

10.7. CONCLUSION Convective heat mass transfer coefficients in cross-corrugated triangular ducts are solved

with CFD codes. Laminar model and turbulence models are used. Correlations are obtained for friction factor and Nusselt numbers. Laminar model is only useful under very low Reynolds numbers. Standard k-ε model is only appropriate for high Reynolds numbers. Low Reynolds number k-ω turbulence model predicts the results well under transitional flow regime.

Cross-corrugated triangular ducts are one kind of primary surface heat exchangers. The cross-corrugation nature of the duct generates strong turbulence and secondary flows to enhance heat transfer on the duct walls. Comparing to a parallel flat plate geometry, the cross-corrugation can obtain a 40 to 60% heat transfer augmentation, however with a penalty of 2 times more friction pressure drop. In addition, both the friction factor and the heat transfer exhibit a cyclic manner, and the generated swirls rotate clockwise in the corrugation valleys. The turbulence is almost uniform in the upper corrugation, but has large variations in the lower corrugations and especially near the crests of valleys.


REFERENCES

[1] Scott, K.; Lobato, J. Mass transport in cross-corrugated membranes and the influence of TiO2 for separation processes. Industrial Engineering Chemical Research, 2003, 42, 5697-5701.

[2] Scott K., Mahmood A.J., Jachuck R.J., Hu B., 2000. Intensified membrane filtration with corrugated membranes. Journal of Membrane Science 173, 1-16.

[3] Zhang, L.Z. Convective mass transport in cross-corrugated membrane exchangers. Journal of Membrane Science, 2005, 260, 75-83.

[4] Okada, K.; Ono, M.; Tomimara, T.; Okuma, T.; Konno, H.; Ohtani, S. Design and Heat transfer characteristics of new plate heat exchanger. Heat Transfer-Japanese Research, 1972, 1, 90-95.

[5] Focke, W.W.; Zachariades, J.; Olivier, I. The effect of the corrugation inclination angle on the thermohydraulic performance of plate heat exchangers. International Journal of Heat and Mass Transfer, 28, 1469-1479, 1985.

[6] Gaiser, G.; Kottke, V. Effects of corrugation parameters on local and integral heat transfer in plate heat exchangers and regenerators. Proceedings of 9th International Heat Transfer Conference, Jerusalem, vol.5, pp.85-90, 1990.

[7] Ciofalo, M.; Stasiek, J.; Collins, M.W. Investigation of flow and heat transfer in corrugated passages-2. Numerical simulations. International Journal of Heat and Mass Transfer, 1996, 39, 165-192.

[8] Stasiek, J. Ciofalo, M.; Smith, I.K.; Collins, M.W. Investigation of flow and heat transfer in corrugated passages-I Experimental results. International Journal of Heat Mass Transfer, 1996, 39, 149-192.

[9] Muley, A.; Manglik, R.M. Enhanced heat transfer characteristics of single-phase flows in a plate heat exchanger with mixed chevron plates. Journal of Enhanced Heat Transfer, 1997, 4, 187-201.

[10] Ergin, S.; Ota, A.; Yamaguchi, H. Numerical study of periodic turbulent flow through a corrugated duct. Numerical Heat Transfer Part A-Applications, 2001, 40, 139-156.


[12] Sherony, D.F.; Solbrig, C.W. Analytical investigation of heat or mass transfer and friction factors in a corrugated duct heat or mass exchanger. International Journal of Heat and Mass Transfer, 1970, 13, 145-159.

[13] Patel, C.V.; Rodi, W.; Scheuerer, G. Turbulence models for near wall and low Reynolds number flows: a review. AIAA Journal, 1984, 23, 1308-1319.

[14] Mompean, G. Numerical simulation of a turbulent flow near a right-angled corner using the speziale non-linear model with RNG k-ε equations. Computers and Fluids, 1998, 27, 847-859.

[15] Li, L.J.; Lin, C.X.; Ebadian, M. A. Turbulent mixed convective heat transfer in the entrance region of a curved pipe with uniform wall temperature. International Journal of Heat and Mass Transfer, 1998, 41, 3793-3805.

[16] Jones, R.M.; Harvey, A.D.; Acharya, S. Two equation turbulence modeling for impeller stirred tanks. ASME Journal of Fluids Engineering, 2001, 123, 640-648.

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[17] Song, B.; Amano, R.S. Application of non-linear k-ω model to a turbulent flow inside a sharp U-bend. Computational Mechanics, 2000, 26, 344-351.

[18] Moore, E.M.; Shambaugh, R.L.; Papavassiliou, D.V. Analysis of isothermal annular jets: comparison of computational fluid dynamics and experimental data. Journal of Applied Polymer Science, 2004, 94, 909-992.

[19] Rokni, M.; Sunden, B. Calculation of turbulent fluid flow and heat transfer in ducts by a full Reynolds stress model. International Journal of Numerical Methods in Fluids, 2003, 42, 147-162.

[20] Incropera, F.P.; Dewitt, D. P. Introduction to Heat Transfer. New York: John Wiley & Sons, Chapter 8, pp. 392. 1996.

[21] Longest Jr, P.W.; Kleinstreuer, C.; Kinsey, J.S. Turbulent three-dimensional air flow and trace gas distribution in an inhalation test chamber. Journal of Fluids Engineering, 2000, 122, 403-411.

Chapter 11

APPLICATIONS OF TOTAL HEAT RECOVERY

ABSTRACT

Conditioning of fresh air constitutes 20-40% of the total load for air conditioning industry, which is expanding rapidly. How to combine the total heat recovery technology with air conditioning industry is an attracting task. In this chapter, some novel air conditioning technologies that combining fresh air total heat recovery are introduced and analyzed. System modeling is performed to have a comparison of energy savings effect. The results found that energy performance could be greatly improved with total heat recovery measures. Further, besides energy savings, indoor air quality could also be improved.

NOMENCLATURE Δp total pressure rise (Pa), ACH Air infiltration rate (h-1) COP Coefficient of performance cp Specific heat (kJkg-1K-1) h Specific enthalpy (kJ/kg)

sm Indoor moisture generation rate (kg/h) Q Energy or moisture (W or kg) q Heat (kW) T Temperature (K) Va Volumetric flow rate (m3/h) Vr Room volume (m3) x Degree of dryness

Greek Letters

ε Effectiveness τ time (s)

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η Efficiency ρ Density (kg/m3) ε Effectiveness ω Humidity (kg vapor/kg air) φ Relative humidity

Subscripts a Air stream c Condenser com Compressor e Evaporator f Refrigerant fan Fan motor Motor s isentropic

11.1. INTRODUCTION Approximately one-third of the primary energy resources are consumed in the air-

conditioning sector. The depleting nature of primary energy resources, negative environmental impact of fossil fuels and low exergetic efficiencies obtained in conventional space heating and cooling are the main incentives for developing alternative heating, ventilating and air-conditioning (HVAC) techniques which can either save energy or employ low-grade thermal energy sources. Novel air conditioning systems with total heat recovery are the directions for sustainable development of HVAC industry.

Humidity has been increasingly raised as a big issue. Besides temperature, humidity is another important parameter influencing people’s feeling of thermal comfort. Figure 11.1 shows the comfort zone in a psychrometric chart [1]. As seen, in summer, the narrow zone between operative temperature 24°C and 27°C, humidity between 4g/kg and 20°C wet bulb are the acceptable levels of thermal comfort. People will feel uncomfortable whether it’s too dry or too humid. The design of HVAC systems for thermal comfort requires increasing attention, especially in the light of recent regulations and standardization on ventilation [2], so that an optimal level of indoor humidity may be reached and maintained to ensure a comfortable and healthy environment and to avoid condensation damage for building envelope and furnishings.

Applications in Total Heat Recovery 271

15

20

25

30

350 5 10 15 20


Ope

rativ

e te

mpe

ratu

re(°

C)

Summer

Winter

100％RH

30％

RH

50％

RH

60％

RH

18 °C WB

20 °C WB

15

20

25

30

350 5 10 15 20


Ope

rativ

e te

mpe

ratu

re(°

C)

Summer

Winter

100％RH

30％

RH

50％

RH

60％

RH

18 °C WB

20 °C WB

Figure 11.1. Comfort zone in a psychrometric chart.

Air-conditioning in hot and humid environments has become an essential requirement for support of daily human activities. Humidity problems can be found in many applications including office buildings, supermarkets, art galleries, museums, libraries, electronics manufacturing facilities, pharmaceutical clean rooms, indoor swimming pools and other commercial facilities. According to Figure 11.1, for thermal comfort reasons, indoor air conditions around 25°C temperature and 10g/kg humidity ratio are the accepted set points. However, the Southern China and other Southeast Asia countries have a long summer season with a daily average temperature of 30°C, and humidity ratio above 20g/kg. Outdoor relative humidity often exceeds 80% continuously for a dozen of days, leading to mildew growth on wall and furniture surfaces, which affects people’s life seriously. In spring in Southern China, there is a period named “Plum raining seasons” when it rains continuously for one to two months. People can not see sun for a long time and stuff from quilts to grains gets moldy easily. Consequently, mechanical air dehumidification plays a major role in air conditioning industry in these regions. In many of these countries, the energy used to cool and dehumidify the ventilation air ranges from 20 to 40% of the total energy consumption for air conditioning, and can be even higher where 100% fresh air ventilation is required, such as kitchen, hospital, factories.

Part load is another problem. In hot and humid climates, air conditioning is an indispensable component to maintain a comfort indoor environment with lower temperature and humidity than outdoor conditions. Operating under hot and humid outdoor conditions, air conditioning has to deal with both sensible and latent loads in a space. In many cases to deal

Li-Zhi Zhang 272

with space latent cooling load using a small HVAC system is often challenging and difficult [3]. The air conditioning system’s design load is calculated based on the number of occupants and their level of activity, types and quantity of equipments used in space, solar irradiation experienced, heat transmission through the building materials, heat gained from infiltrated outdoor air and many other factors. In reality, the space loads are always below their design values. Under part-load conditions, the common practice is to employ control method to maintain the space temperature while allowing the space humidity to vary. In part load conditions, supply air temperature is reduced. It is still enough to extract sensible load, but is insufficient to extract latent load. The indoor humidity is out of control. In full load seasons from June to September, humidity is controlled very well, but in other transit seasons, humidity of the space may drift towards a high value that causes human discomfort while supporting the growth of pathogenic or allergenic organisms. It is also believed that the emission rate of formaldehyde from furniture and building materials is higher when humidity rises, resulting in poor indoor air quality.

Stringent ventilation regulations make the humidity problem more serious. In modern society, people spend most of their time in built environments. More attention has been paid to indoor air quality and indoor thermal comfort. HVAC systems are necessary for almost all buildings. However, conventional air conditioning modes, such as constant air volume (CAV) systems and variable air volume (VAV) systems, face great challenges in effective outdoor air ventilation and precision indoor air humidity control. From the view points of ventilation, the main problems with conventional air conditioning systems are analyzed as follows.

(1) The outdoor air in conventional air conditioning systems mixes with the re-circulated

air, which causes transmission of bacterium and virus among multiple zones. In this situation, occupants are at high risk of infection when diseases breakout, like SARS and bird flu. Human’s expectation of effective ventilation with 100% outdoor air has been increasingly rising.

(2) The indoor relative humidity tends to rise under part load operation because the air conditioning systems usually control the indoor temperature by reducing their cooling capacities. To control load, cooling-reheating processes are required, which are very much energy intensive. This problem is very serious in hot and humid regions, like Canton. To improve humidity control, the method of decoupling temperature control and humidity control has attracted much attention. To realize independent humidity control, an independent fresh air conditioning system, or known as dedicated outdoor air system, is always required.

(3) New technologies for a more comfortable and energy efficient indoor environment, such as chilled ceiling/beam, thermal storage and VRV (Variable Refrigerants Volume), require parallel independent fresh air conditioning systems to meet demands on effective ventilation and removal of latent load. However, the energy consumption for dehumidifying fresh air is huge, which often accounts for 20%-40% of the total energy for air conditioning in hot and humid areas. The unaffordable energy cost for treatment of fresh air, particularly for fresh air dehumidification seriously restricts the application of independent fresh air conditioners. Total heat recovery becomes a necessity.


Conventionally cooling coils are used to cool and dehumidify supply air. It is called the coupled cooling since cooling and dehumidification are accomplished simultaneously in a coupled way. To dehumidify air, air temperature must be cooled to below dew point temperature like 10°C. Dehumidified air of such low temperature cannot be supplied to the space directly because people may feel cold draft under the cold air stream. Reheating has been widely used in many applications behind a cooling coil to prevent this problem. However this cooling-reheating process is energy intensive. Energy is needed to overcool the air across the cooling coil and also to reheat the off-coil air to the desired space humidity and temperature. Although reheating is able to maintain a space at its design temperature and humidity during pert-load conditions, it is not often a recommended practice chiefly due to its high energy use.

To solve this problem, nowadays there is a trend to separate the treatment of sensible load and latent load. This is the so called independent humidity control. According to this scheme, sensible load is treated by chilled-ceilings, cooling coils, or air handling units which still cools the supply air but doesn’t necessarily cool it as low as to dew point. Supply air temperature can be adjusted as sensible load requires. The latent load is accomplished by an independent dehumidification unit, which is to treat all latent load alone. How to combine these systems with total heat recovery is a challenging yet interesting work.

11.2. DESICCANT WHEEL WITH CHILLED CEILING Desiccant wheels have two uses: active air dehumidification and total heat recovery.

Energy wheels are simple, but incoming air cannot be dehumidified to states drier than exhaust air. In most air-conditioning systems, incoming air should be dehumidified to be drier than room air, say, 7g/kg, to further extract the moisture load of a building. In this case, to generate dry air, hot air is required to regenerate wheels. Desiccant cooling is a cycle involving desiccant wheel that can produce dry and cool air. In a desiccant cooling system, enthalpy from exhaust air is recovered through evaporative cooling of exhaust air and the subsequent sensible heat exchange.

Almost all materials have the capacity to adsorb and hold water vapor, but commercial desiccants have significant capacity for holding the water. A commercial desiccant takes up between 10 and 1100 percent [4] of its dry weight in water vapor, depending on its type and the moisture available in the environment. Desiccants remove moisture from the surrounding air until they reach equilibrium with it. This moisture can be removed from the desiccant by heating it to temperatures between 50°C and 260°C and exposing it to a scavenging air stream. The desiccant is then cooled so that it can adsorb moisture again. The transfer of moisture is due to the difference in vapor pressure at the desiccant surface and that of the surrounding air. When the vapor pressure at the desiccant surface is less than that of air, the desiccant attracts moisture and releases it when its vapor pressure is greater than that of air.

The major advantages of desiccant cooling are: a) Only air and water are required as working fluids. Fluorocarbons are not required;

thus, there is no impact on the ozone layer.

Li-Zhi Zhang 274

b) The source of thermal energy can be diverse (i.e., solar, waste heat, natural gas). The electrical energy requirement can be less than 25 percent of conventional refrigeration systems.

c) Since desiccant systems operate near atmospheric pressure, maintenance and construction are simplified.

Because of these advantages, much effort has been spent in the researches and

applications of desiccant cooling cycles [5-7]. This cycle is very attractive since a relatively low temperature heat source, such as can be provided by solar energy, is suitable for the regeneration of the desiccant in the cycle. However, there are still two shortcomings that need to be addressed. Firstly, the supply air temperature in such systems is 4°C higher than that in a conventional all-air system, which means that a 60% greater volume of air needs to be pumped and circulated through the system. As a result, fan energy, which is already astonishingly large in a conventional all-air system, would be further aggravated. This would compromise the energy savings effect of the system. Secondly, in hot and humid regions like Canton, the effectiveness of desiccant dehumidification would be quite limited, which means that a heavier desiccant wheel and/or a higher regenerating temperature are required to realize efficient dehumidification and cooling. The system would become bulky and the initial investment would skyrocket.

Decoupled cooling, in which room sensible load is extracted by chilled-ceiling, whereas the latent load is extracted by a desiccant system, is introduced here. In the system, aluminum ceiling panels are installed under the cement ceiling. Chilled water with 17 °C flows through the metal tubes connected with metal-sheet panels, removing heat collected by the panels, mainly radiantly. Certainly the metal tubes and plates can be replaced by plastics, to reduce cost. With this desiccant cooling combined with a chilled-ceiling system, the volume of the process air becomes the minimum: the process air is also the fresh air for ventilation. Therefore, large fractions of the fan energy can be saved. Other advantages of this combined system include:

a) Chilled-ceiling systems improve thermal comfort because heating and cooling are

provided directly and more evenly to the occupants without causing drafts. b) Radiant cooling by ceiling panels results in a temperature gradient in the occupied

zone varying from 0 to 2°C/m. Consequently, chilled ceiling system creates a “cool head and warm feet” radiant environment, which is more comfortable than a “warm head and cool feet” environment created by conventional all air systems.

c) The air temperature in the room is higher at the same operative temperature due to the radiant effect, in comparison with an all-air system. This will result in energy savings.

d) The latent load is treated by a desiccant cooling cycle. The chilled water to the ceiling panels is supplied by a chiller. Without the need to dehumidify air, the evaporating temperature for the chiller could be raised from conventional around 5°C to as high as 15°C, which means large improvements in COP and large quantities of energy savings.


Conventional all-Air System To have a comparison, a conventional constant volume all-air system is first introduced.

The schematic of the system and the air treatment processes in a psychrometric chart are shown in Figures 11.2a and 11.2b respectively. Ambient air is mixed with return air first before the mixed air (state 2) flows through an AHU (Air Handling Units) cooling coils where it is cooled and dehumidified. Air leaving cooling coils at state 4 (13°C, and 95% RH in summer design conditions) is then pumped to the conditioned space through supply air ducts where air temperature rises about 1°C due to heat gains on the trip (4 to 5). After absorbing the building’s sensible and latent loads, air flows to the return air ducts to be mixed with fresh air or exhausted to the surroundings. Energy wasting process of re-heating after cooling coills are omitted and the supply air temperature is determined by sensible load. Therefore, temperature is actively controlled, while humidity is passively controlled. The result of this approach is a loss of humidity control while the temperature is maintained at setpoint. In winter and transient seasons, sensible load becomes smaller and air is not sufficiently dehumidified. As a consquence, indoor humidity becomes higher in these periods. It is believed that humidity levels above 70% RH in the supply air duct rapidly increase the effect of bacteria, viruses, fungi, and mites. The system has no total heat recovery.

Desiccant Cooling (DC) with Chilled-Ceiling This is a de-coupled cooling strategy [8]. The schematic of the system components and

the corresponding psychrometric processes are shown in Figure 11.3. Chilled-ceiling panels are equipped to extract sensible load, and dehumidified fresh air is supplied by a desiccant cooling system. The principles of the desiccant cooling cycle is well-known. Ambient outdoor air at state 1 enters the supply air duct. This air passes through a desiccant wheel and hot, dry air exits at state 2. This increase in temperature is due to the heat of sorption and some sensible heat transfer. The hot, dry supply air transfers much of this heat to the return air stream in process 2 to 3 involving a sensible heat wheel. Unlike a commonly used desiccant cooling, where air at state 3 is evaporatively cooled to state 4, in this system, the warm, dry air at state 3 is cooled by a cooling coil, to keep the dryness of supply air at state 4. The cool, dry air at state 4 is then distributed to the room. After accepting the building latent load and a small amount of sensible load, the air then returns to the desiccant system through return air ducting. This is the state of the air which corresponds to state 5. This somewhat cool, fairly dry air is evaporatively cooled to as low a temperature as possible at state 6. This cold, damp air is then preheated by the rotary sensible wheel to state 7 while cooling the supply air stream. State 7 is the state of the moist air as it enters the heating coil. Hot, humid air exits at state 8 and regenerates the desiccant wheel. Warm, very humid air at state 9 is then exhausted to the surroundings. Chilled water flows through the cooling coil and chilled ceiling panels in series, implying that only one refrigerator with an evaporating temperature as high as 15°C is needed. In this scheme, partial total heat recovery is realized by the evaporative cooling of exhaust air.

Li-Zhi Zhang 276

Room

Exhaust

Ambient

1 24

5

Cooling Coil

3

(a) Schematics.

0

0.005

0.01

0.015

0.02

0.025

0.03

10 15 20 25 30 35 40

Dry bulb temperature (°C)

Hum

idity

ratio

(kg/

kg)

φ =100% 1

2

3 4 5

(°C)

0

0.005

0.01

0.015

0.02

0.025

0.03

10 15 20 25 30 35 40


Hum

idity

ratio

(kg/

kg)

φ =100% 1

2

3 4 5

0

0.005

0.01

0.015

0.02

0.025

0.03

10 15 20 25 30 35 40


Hum

idity

ratio

(kg/

kg)

φ =100% 1

2

3 4 5

(°C) (b) Psychrometrics.

Figure 11.2. Conventional constant volume all-air system, (a) schematics and (b) psychrometrics.

Ventilation distribution is provided by displacement ventilation. Displacement ventilation brings air into the space near floor level at a low velocity and air is exhausted at the ceiling. This strategy delivers fresh air to where the occupants require it and odors and airborne contaminants are carried to the ceiling and exhausted instead of being recirculated as is common with traditional HVAC systems.

Room

Chilled Ceiling Evaporative Cooler

Cooling Coil

Gas Heater

Desiccant Wheel

Sensible Heat Wheel

Exhaust

Ambient 1 2 3

5

4

6 7 8 9

11 12

10 11

(a) Schematics.

Figure 11.3. (Continued on next page.)


0

0.005

0.01

0.015

0.02

0.025

0.03

10 20 30 40 50 60 70 80 90 100


Hum

idity

ratio

(kg/

kg) 1

234

5

6 7

8

9

φ =100%

0

0.005

0.01

0.015

0.02

0.025

0.03

10 20 30 40 50 60 70 80 90 100


Hum

idity

ratio

(kg/

kg) 1

234

5

6 7

8

9

φ =100%

(b) Psychrometrics.

Figure 11.3. A desiccant cooling combined with chilled-ceiling, (a) schematics and (b) psychrometric processes.

Pre-cooling Desiccant Cycle with Chilled-Ceiling

It was discovered that in hot and humid regions, very high regenerating temperatures are

required, otherwise the performance of the desiccant system would deteriorate seriously in extreme weather conditions. To address this problem, a modification of a desiccant cycle, a Pre-cooling desiccant cooling cycle (PCDC), has been proposed and combined with chilled ceiling panels for an office building [9]. A PCDC cycle could use lower regenerating temperatures in comparison with a DC cycle.

Figure 11.4a illustrates a schematic of this cycle, while Figure 11.4b shows this cycle on a psychrometric chart. The components are similar to a DC cycle, but an additional total heat exchanger, which could be either an energy wheel or a membrane-based total heat exchanger, is added to the system. Ambient air at state 1 first passes through a total heat exchanger where it exchanges sensible heat and moisture with the exhaust air. Air at state 2 then passes through the rotating dehumidifier matrix and hot, dry air exits at state 3. The hot, dry supply air transfers much of its heat to the return air stream in process 3-4 involving a sensible heat wheel. The warm, dry supply air at state 4 is now cooled by a cooling coil to state 5 and is supplied to the conditioned space.

After accepting the building latent load and some sensible load, the air then returns to the air ducting. This room air, which is still relatively cool and dry, is first ushered into the total heat exchanger to pre-cool and pre-dry the outside fresh air, through process 6-7. Air exited from the total heat exchanger at state 7 is evaporatively cooled to state 8 and is used to cool the supply air in the sensible heat exchanger involving process 8-9. The air at this stage is further heated by a gas heater to state 10 and is used to regenerate the desiccant wheel. After

Li-Zhi Zhang 278

passing through the regenerating desiccants matrix passages, air becomes hot and humid and is exhausted to the outside.

Room

Chilled Ceiling

12

Evaporative Cooler

Cooling coil

Gas Heater

Desiccant Wheel

Sensible Heat Wheel

Exhaust

Ambient

2 345

6 7

8 9 11

14

1

10

13 Total Heat Exchanger

13

(a) Schematics.

0

0.005

0.01

0.015

0.02

0.025

0.03

10 20 30 40 50 60 70 80 90 100


Hum

idity

ratio

(kg/

kg)

φ =100%

1

2

345

6

7

89 10

11

0

0.005

0.01

0.015

0.02

0.025

0.03

10 20 30 40 50 60 70 80 90 100


Hum

idity

ratio

(kg/

kg)

φ =100%

1

2

345

6

7

89 10

11

(b) Psychrometrics.

Figure 11.4. A Pre-cooling desiccant cooling combined with chilled-ceiling, (a) schematics and (b) psychrometric processes.


The temperature of the return air at state 9 required to regenerate the desiccant is a property of the type of desiccant used and the amount of dehumidification required in process 2-3. Many different desiccants have been considered for use in dehumidification. Some of the more common ones include silica gel, lithium chloride, lithium bromide, zeolites. Most industrial rotary dehumidifiers use silica gel or lithium chloride as the desiccant when lower regeneration temperatures are desired.

This system takes full use of the total heat recovery.

Component Modeling

Ceiling Panels and Cooling Load Thermal performance is evaluated by a special cooling load program for cooled ceiling

systems ACCURACY [8] which is developed and validated previously at Delft University of Technology in Netherlands. This is a room-energy-balance method, according to which, chilled-ceiling panels are treated as individual surfaces that exchange heat convectively with room air and radiantly with other building surfaces [10]. Heat conduction within the ceiling panels is treated in a one-dimensional way using a three-nodal point model. The program calculates not only the cooling load, but also the required supply water or air temperatures for different panel installation areas. The program also works out the respective convective and radiant heat extractions by cooling panels, and the radiant and convective heat from all the window and wall surfaces. Adopting such building dynamics simulation techniques, year round simulation of rooms with or without chilled-ceilings provides hour-by-hour data of cooling/heating loads, and temperatures of room air and other components.

Moisture balance in the room is calculated by [11],

( ) ( ) ad/desroorsrssrrr ωωωωω

mmVACHvd

dV−+−⋅+−−= ρρ

τρ

(11.1)

Moisture load in the room is calculated by,

( ) ad/desroor ωω mmVACHQL −+−⋅= ρ (11.2)

where Vr is the room volume (m3), ρ is air density (kg/m3), ω is humidity ratio (kg/kg), τ is time (hr), sv is supply air volumetric flow rate (m3/h), ACH is air infiltration rate (h-1), sm is indoor moisture generation rate (kg/h), subscripts “r”, “o”, “s” refer to “room”, “outside”, and “supply” respectively. The last term ad/dem is moisture adsorption and absorption by room surfaces and furniture (kg/h). In general, the moisture adsorption/absorption properties of building materials are less understood, therefore, overall, precision of moisture prediction lags behind that for thermal prediction. The indoor surface and furniture adsorption and desorption effects are not included. In reality, such effects may help smooth the RH fluctuations, but will not significantly affect the hourly averages. In simulations, thermal analysis is coupled with moisture analysis to take into account the interactions between thermal and moisture performance.

Li-Zhi Zhang 280

At this point, the governing equations will be developed for each of the component in the above proposed PCDC cycle combined with chilled-ceiling panels.

The Total Heat Exchanger

The total heat exchanger could be either an energy wheel, or a membrane exchanger previously described. Two effectiveness: sensible effectiveness (εS) and latent effectiveness (εL) are defined. Air state at point 2 is calculated by

( )61S12 ε TTTT −−= (11.3)

( )61L12 ωωεωω −−= (11.4)

A Sensible effectiveness of 0.8 is easily obtained with a membrane system, and the latent

effectiveness is about 0.65.

The Rotary Dehumidifier The purpose of the rotary dehumidifier is to dehumidify the supply air stream as it passes

from state 2 to 3. In doing so, the temperature of the supply air is raised. The humidity ratio of the return air is increased and the temperature decreased as the desiccant is regenerated in process 10-11.

The models for the desiccant wheel can be classified into two categories: finite difference models, and correlation models. The former models are described in detail in Chapter 3, the latter one is given in Ref [12]. Detailed finite difference modeling of desiccant wheels were conducted in Chapter 3 for energy wheels. The only difference is that fresh air is replaced by regenerating air. Exhaust air is replaced by supply air. In this chapter, a simpler model provided in [12] is used. The finite difference models are detailed, but complex and difficult to find stable solutions. The correlations, which are used in this study, are simple, yet sufficient for an energy analysis. The correlations are summarized as follows:

1si1ei

1si1sof1 ff

ff−−

=η (11.5)

2si2ei

2si2sof2 ff

ff−−

=η (11.6)

0.8624j

-1.49j1j 344.42865 wTf +−= (11.7)

0.07969j

1.49j2j 127.16360/ wTf −= (11.8)


where subscripts “s” and “e” mean “supply” and “exhaust” respectively, and “i” and “o” mean “inlet” and “outlet” respectively. Effectiveness ηf1 and ηf2 could be pre-determined by the finite difference equations, considering wheel structure and dimensions.

The desiccant dehumidifier configurations are: wheel weight, 2.5kg; duct wall thickness, 0.2mm; duct geometry, sinusoidal; material, silica gel; effective material fraction, 0.7; wheel length, 0.2m, rotary speed, 30RPH. The coefficients are: ηf1=0.29; ηf2=0.85.

The Rotary Regenerator

The purpose of the regenerative heat exchanger is to transfer the heat of sorption present in the supply air stream after dehumidification to the return air stream. As can be seen from Figure 11.4b, in sensibly cooling the supply air stream from state 3 to 4, the return air stream is preheated from state 8 to 9. The more effective this component the less energy required from the heat source to regenerate the desiccant.

The effective method can be applied to a compact heat exchanger of this type in the conventional manner. By assuming equal heat capacities for the two air flows, temperature at point 4 is calculated by:

( )83R34 ε TTTT −−= (11.9)

Based on information on the construction details of the regenerator, effectiveness values

can be estimated from previously developed models. A constant effectiveness of 0.85 is selected for this study.

The Evaporative Cooler

In the return air side and after the total heat exchanger, a rigid media air cooler is used to cool the air stream. This particular type of evaporative cooler consists of rigid, corrugated packing material which form the wetted surface. Moist air flows through the corrugations. Water enters the top of evaporative cooler and flows by gravity through the wetted surfaces. Commercially available evaporative coolers are rated according to their saturation effectiveness, εC, defined as

( ) ( )( )%100/ iwbioiC TTTT −−=ε (11.10)

where Ti is the entering air dry bulb temperature, Tiwb is the entering air wet bulb temperature, and To is the exiting air dry bulb temperature.

Therefore, exit air state at point 8 is

( )7wb7C78 TTTT −−= ε (11.11) Process 7-8 is a constant enthalpy process. Moisture content at point 8 can be calculated

from psychrometrics. Saturation effectiveness εC in the range of 0.7-0.9 is attainable.

Li-Zhi Zhang 282

The Heating/cooling Coil The air cooling coil is a conventional cross flow, water-to-air heat exchanger. Its

performance is described by the effectiveness relationship: εHC=[ΔT (minimum fluid)]/[ΔT (inlets)] (11.12)

where the minimum fluid is the one with the minimum value for heat capacity. The moisture content is constant through the heating/cooling processes. An effectiveness of 0.85 is assumed for the cooling coil.

Primary Equipments

The required electric power input for a chiller can be calculated by Chiller power=Qc/COP (11.13)

where Qc is cooling energy (W) and COP is the coefficient of performance of the water chiller. For a specific system, the COP is the function of the required chilled water temperature, the ambient temperature and humidity, as well as operation capacity.

For wheel regeneration, a boiler is needed, energy input Boiler power=Qh/ηb (11.14)

where Qh is heating energy required (W) and ηb is the boiler efficiency. Fan and pump energy is an important factor in the annual energy consumption of an

HVAC system. Fan (pump) performance can be characterized by its efficiency, which itself is dependent on operational air flow rate. Mostly, rated volumetric flow rate, pressure rise and efficiency are available from the manufacturer. Then rated power can be calculated as

Fan (pump) power=VaΔp/(3600ηf) (W) (11.15)

where Va is air (water) volumetric flow rate (m3/h), Δp is total pressure rise (Pa), ηf is fan (pump) efficiency.

Effectiveness of the main equipments relates to design and operating conditions. When the operating conditions fluctuate near design conditions, the effectiveness changes in a small range. To simplify analysis, constant effectiveness values for various equipments are assumed.

Building Configurations To compare the energy consumption and indoor humidity behaviors of the different air-

conditioning systems, a typical office room in a south-facing high-rise building in Guangzhou city, Canton, is considered. The office is 5.1m long, 3.6m wide, and 2.6m high. The thickness of the envelope is 260mm and the resulting heat transmission coefficient U for the opaque part of the facade is 2.91W/m2K. The glazing area is 2.88m2 with double-pane windows of


which center-of-glass U value is 1.31 W/m2K. Louver curtain is installed behind windows. It is determined that about 70% of the ceiling is covered by radiant ceiling panels when the radiant ceiling panel system is applied. Indoor set points are: 25°C operative temperature, 50% RH in summer and 23°C OT, 50% RH in winter.

An occupancy pattern of 2 persons with a schedule from 9 to 18h is simulated in the office. When present, each person generates 75W sensible heat and 57.6g/h moisture. Of the sensible heat generated, 50W is radiative and 25W is convective. An air infiltration rate of 0.2 ACH are modeled during the time when the ventilation system was switched off and the building is not pressurized.

Besides, constant loads of 459W of equipment and lighting with a schedule from 9 to 18h are modeled in the room. Half of these loads are considered as radiative and half convective. The operating hours of the ceiling panels and ventilation systems are also from 9 to 18h. However, in summer, to prevent water condensation on ceiling panel surfaces, the desiccant system is operated one-hour in advance of ceiling panels.

The supplied chilled water flow rate to the ceiling panels is 0.5t/h, resulting a temperature rise of about 1.6°C through the panels. The processed fresh air is supplied at a rate of 67m3/h, which complies with ASHRAE standard 62-2001 [2]. The desiccant dehumidifier configurations are: wheel weight, 2.5kg; duct wall thickness, 0.2mm; duct geometry, sinusoidal; material, silica gel; effective material fraction, 0.7; wheel length, 0.2m, rotary speed, 0.5rpm.

Operating Parameters in the Cycle Detailed computer models of the desiccant cooling system are developed and combined

with the building energy simulation code ACCURACY [10]. Weather data in Canton is used to evaluate the system performance by hour-by-hour calculations. The hourly operating states in the DC cycle and PCDC cycle are calculated. Table 11.1 and 11.2 show the operating states of the two systems at 14:00 of July 15 respectively. The total sensible load of the room in this hour is 64.6W/m2, of which, 86% is accounted for by the ceiling panels and 14% is accounted for by the desiccant cooling. The total latent load is 5.1W/m2, which is treated by the desiccant cooling system solely.

Table 11.1. Operating point for the DC cycle at 14:00, July 15

State Point Temperature (°C) Humidity ratio (kg/kg) 1 32.8 0.0201 2 77 0.0085 3 28 0.0085 4 18 0.0085 5 25.7 0.0103 6 19.4 0.0128 7 68.4 0.0128 8 92.9 0.0128 9 48.7 0.0244

Li-Zhi Zhang 284

Table 11.2. Operating point for the PCDC cycle at 14:00, July 15

State Point Temperature (°C) Humidity ratio (kg/kg) 1 32.8 0.0201 2 26.8 0.0126 3 57.7 0.0085 4 30.8 0.0085 5 18 0.0085 6 25.7 0.0103 7 31.7 0.0178 8 26.1 0.0201 9 52.9 0.0201 10 77.3 0.0201 11 46.9 0.0241

30

50

70

90

110

1 2 3 4 5 6 7 8 9 10 11 12

Months

Th (

?)

MECPMEC

(°C

)

DC

PCDC

30

50

70

90

110

1 2 3 4 5 6 7 8 9 10 11 12

Months

Th (

?)

MECPMEC

(°C

)

DC

PCDC

Figure 11.5. Monthly averaged regenerating temperatures for the DC and PCDC cycles.

From these two tables, it is seen that the heat absorbed in the gas heater for wheel regeneration are the same for the PCDC and DC cycles, however, the regenerating temperature in the former cycle is about 15.6°C lower than that in the latter one. Therefore, lower grade heat can be used in a PCDC cycle. This indicates that pre-cooling improves the performance by increasing the dehumidification efficiency.

Figure 11.5 shows the monthly-averaged regenerating temperatures in a year for the DC and PCDC cycles. August is the most humid month, therefore, highest regenerating temperatures are required in this month. Similarly, it is driest in February, and the regenerating temperatures are the lowest at this time. In winter, the two cycles have the similar regenerating temperatures, however, in summer when it’s very humid, the PCDC cycle has much lower regenerating temperatures. In other words, the more humid it is, the more superior an PCDC in comparison with a DC cycle. The distributions of annual operating


hours with various regenerating temperatures are shown in Figure 11.6. In a DC cycle, only 70% of annual operating hours are with regenerating temperatures below 80°C; while in a PCDC cycle, nearly 99% of operating hours are with less than 80°C regenerating temperatures. Lower heat source usually mean higher efficiency. This is very interesting, indicating that renewable energy such as solar energy can be more efficiently employed in a pre-cooling desiccant system.

40~60°C29.5%

100~120°C0.3%

60~80°C24.7%

80~100°C30.2%

<40°C15.3%

(a) DC cycle.

40~60°C39.9%

60~80°C46.5%

80~100°C1.2% <40°C

12.4%

(b) PCDC cycle.

Figure 11.6. Annual percentage of operating hours with regenerating temperatures, (a) DC cycle and (b) PCDC cycle.

Li-Zhi Zhang 286

Annual Primary Energy Consumptions The energy required for the primary equipments is calculated. The performances of the

primary equipments are modeled with certain constant indices [8]: the COP of the chiller is 4.39 when evaporating at 15°C and 3.31 when evaporating at 5°C; the boiler efficiency is 0.75 and fan (pump) efficiency 0.60; the fan pressure rise is 1400Pa for all-air, and 1600Pa for desiccant cooling; pump pressure rise is 0.3bar for all-air and 0.5bar for ceiling panels. In the analysis, electricity consumptions are converted into the equivalent primary energy by multiplying a factor of 3. The energy used by motors driving the regenerative wheels is categorized as auxiliary energy.

The annual primary energy consumptions per floor area are illustrated in Figure 11.7. Compared to a conventional all-air system, chilled-ceiling combined with precooling desiccant cooling (PCDC+CC) saves 71.6% of fan energy, 50% of chiller energy, while consumes 1.1 times of pump energy, and additional 90.4kWh/m2 of P.E. due to desiccant regeneration and wheel driving, resulting in a total energy saving of 30%. In an all-air system, air is supplied at 14°C, while in a desiccant system combined with chilled-ceiling system, air supply temperature is 4°C higher. This increased supply air temperature helps to eliminate thermal discomforts like cold drafts. In the combined system, air is mainly for ventilation and dehumidification purposes and its volume is only 23% of that in an all-air system. Fan energy, which is usually in the same magnitude with cooling energy, is thus drastically reduced. As a result, large amount of energy is saved.

0

100

200

300

400

All-air PCDC+CC DC+CC

Ann

ual P

.E. c

onsu

mpt

ion

(kW

h/m

2

AuxiliaryBoilerFan PumpChiller

Figure 11.7. Annual primary energy consumptions of a PCDC desiccant cooling combined with chilled-ceiling (PCDC+CC), and a DC+CC cycle, in comparison with a conventional all-air system.


The electricity consumptions for the three systems are 114.7kwh/m2, 76.1kwh/m2, and 67.6kwh/m2, for conventional all-air system, PCDC+CC, and DC+CC, respectively. A PCDC+CC system saves 33.7% of electricity compared to a conventional all-air system. This would be of particular significance since “peak-saving” is a hot issue these days and the cooling load usually is the highest during peak hours. A PCDC+CC system uses more low-grade heat, less electricity.

It could also be concluded from this figure that a PCDC+CC cycle further saves about 10% primary energy, compared to a DC+CC cycle. The largest fraction of energy saving comes from reduced boiler energy. Therefore, a PCDC+CC cycle is more efficient than a DC+CC system.

Indoor Humidity The distribution of indoor relative humidity during working hours are shown in Figure

11.8 for CAV (constant volume all-air) and PCDC+CC. The optimum zone is from 40% to 60%RH and 50% RH is ideal for building occupants to avoid the hazards of fungi, bacteria, viruses, and respiratory difficulties. The figure shows that the all-air system has less annual hours in the comfort region, and has more hours in either the dry (<40%) or the humid (>60%) regions. The chilled-ceiling combined with pre-cooling desiccant cooling controls the indoor humidity well: 90% of annual operating hours is in the optimum region.

For conventional constant volume all air system, the minimum RH is in July and the RHs are well controlled to around 50% in summer season. In winter and transient seasons, the RHs rise up. RHs greater than 70% occasionally occur in these seasons. This phenomenon is due to the fact that in a conventional all air system, temperature is intentionally controlled, while the humidity is passively controlled, because the supply air state is determined by the sensible load. In winter and transient seasons, sensible load becomes smaller and air is not sufficiently dehumidified. In contrast, PCDC +CC controls indoor temperature and humidity independently.

Therefore, it can be concluded that chilled-ceiling with desiccant cooling has better indoor humidity controls with reduced energy use than its all-air counterpart does.

Even though PCDC+chilled-ceiling system realizes independent humidity control with reduced energy use, when used in hot and humid regions, condensation on ceiling panels remains a troublesome issue. To illustrate this problem, the variations of room temperature and RH on a typical summer day are shown in Figure 11.9 for a system combining chilled-ceiling with AHU. In this situation, the air infiltration rate at night ACH is set to 0.2. The operation schedule is that both the AHU and the ceiling panels are operated from 9:00 to 18:00 working hours. The maximum RH occurs on the coldest location of ceiling panels. It is indicated that both the room temperature and room mean humidity drop to the set points in the hour when the systems are started, and stay near set points afterwards. When the systems are closed after 18:00, both the temperature and humidity rise steadily due to air infiltration, to above 27°C and 70%RH. At 9:00, because the inlet cooling water to ceiling panels is the coldest and the indoor moisture level is the highest after one night’s infiltration, maximum local RH above 100% results. In other words, condensation occurs at this moment.

Li-Zhi Zhang 288

0

1000

2000

3000

4000

<40% 40~60% >60%

RH

Ann

ual o

pera

ting

hour

s (H

rs All airPCDC+CC

Figure 11.8. Indoor humidity distributions for two systems: all air and PCDC+CC.

20

30

40

50

60

70

80

90

100

0 5 10 15 20 25

Time (Hrs)

RH

(%)

20

25

30

Roo

m T

empe

ratu

re (?

)

Average RHMax RHRoom Temperature

Figure 11.9. Variations of room temperature and RH on a typical summer day with chilled-ceiling and PCDC desiccant cooling both operated from 9:00 to 18:00.


20

30

40

50

60

70

80

90

100

0 5 10 15 20 25

Time (Hrs)

RH

(%)

ACH=0.1ACH=0.2ACH=0.4

Figure 11.10. Variations of Maximum RH with 1 hour in advance dehumidification.

To prevent condensation, room air needs to be dehumidified prior to the operation of ceiling panels. A one-hour in advance dehumidification/ventilation strategy is simulated in which ceiling panels are operated from 9:00-18:00 and the desiccant system is operated from 8:00-18:00 [11]. The variations of the maximum RH with different night air infiltration rates are plotted in Figure 11.10 under this strategy. It is clearly shown that room moisture levels are substantially pulled down during the first dehumidification hour, and when the ceiling panels begin to operate, the maximum RHs are below 90%. Therefore, water condensation is prevented. The higher the ACH is, the larger the room humidity is at night. The influences of ACH on indoor humidity in the working hours are negligible if a one-hour in advance dehumidification is implemented. On the other hand, if a building is well enclosed and the air infiltration rate is less than 0.05 at night, no condensation occurs even under simultaneous dehumidification and ceiling cooling. In a summary, condensation is the most important issue in ceiling cooling. Unless properly controlled regarding dehumidification strategy, condensation remains a problem.

Capital Cost An exact capital cost difference between the combined PCDC+CC system and the

equivalent all-air system is difficult to determine. It depends mainly on market prices and contractors, and varies considerably for different countries. In Canton, it is expected that the PCDC+CC system and the DC+CC system are 30% and 25% more expensive, respectively, than a traditional all-air system for cooling load of 60W/m2 (floor area). The increased cost mainly comes from ceiling panels and desiccant wheels. The estimated payback period for a PCDC+CC system is 15 years for a 500m2 office building. We believe the capital cost and the payback period will decrease in future, if the market penetration of the combined system accelerates.

Li-Zhi Zhang 290

11.3. INDEPENDENT AIR DEHUMIDIFICATION There are various techniques for air dehumidification [2]. Traditionally, latent load and

sensible load are treated in a coupled way, like the CAV system described before. Because air is not only for ventilation, but also a heat transfer medium, and a large quantity of air is needed to extract the sensible load, energy requirements are very high. Another problem with this technique is that in transition seasons, when it is at part load conditions, humidity control will be lost.

There is an increasing trend to separate the treatment of sensible and latent load by using an independent humidity control system. According to this concept, the latent load of a room is treated by an independent humidity control system, while the sensible load is treated with some other alternative cooling techniques such as chilled-ceiling panels, or phase-change materials. Since the circulating air is dramatically reduced, energy consumption can be reduced substantially. Another benefit is that chilled water or suction temperatures can be raised, resulting in increased equipment efficiency and decreased operating costs. It is estimated that with a new system of chilled-ceiling panels combined with independent humidity control, 30% of energy could be saved in comparison to a traditional coupled system. Nevertheless, due to the hot and humid climates in south China, energy for moisture control with an independent humidity control system still accounts for 25-40% of the total energy for air conditioning. To further reduce the energy consumption in the treatment of fresh air, energy recovery measures must be combined to an independent humidity control system.

In this section, four independent humidity control systems with heat recovery measures are compared. These systems are: System 1, Mechanical dehumidification with a heat pump; System 2, Mechanical dehumidification with a sensible heat exchanger; System 3, Mechanical dehumidification with a membrane-based total heat exchanger; and System 4: Desiccant wheel driven by a heat pump. Through hour-by-hour analysis, the annual primary energy consumption for the four systems is discussed.

As an example, an office building with 5 occupants in a 20m2 room is considered. The set points for indoor air are as follows: temperature, 25°C; relative humidity, 50%. Fresh air is supplied at 37.5m3/hr, 20°C in temperature and 7g/kg in humidity ratio. The air dehumidification systems are operated when the offices is opened, i.e., from 9:00 to 18:00. At nights, the systems are shut down, to save energy. The air flow rates are selected according to ASHRAE Standard 62-2001 [2], which is determined by several factors such as room area, building occupancy pattern, and building types. The sensible load of the room is treated by chilled-ceiling panels, as previously did. The total ventilation load includes two fractions: moisture load from fresh air and moisture load from human activities. The fresh air sensible and latent load is a variable relating to weather conditions, while the load from human activities is assumed to be kept at 50g/hr.

System Descriptions Four independent air dehumidification systems with heat recovery measures are

proposed. The operations of the four systems are controlled to satisfy the load and weather


conditions. The air flow rates are fixed. For mechanical dehumidification, both the evaporating and condensing temperatures are controlled according to the load and outside temperatures. For desiccant wheels, the regeneration temperature is adjusted to fit the load. These systems can be operated as stand-alone independent air dehumidification units.

Indoor Air

A B CEvaporator Condenser

Fresh Air Supply Air

Compressor

Valve

Condenser

D

Fan

Indoor Air

A B CEvaporator Condenser

Fresh Air Supply Air

Compressor

Valve

Condenser

D

Fan(a)

0

0.005

0.01

0.015

0.02

0.025

0.03

5 10 15 20 25 30 35 40


Hum

idity

ratio

(kg/

kg

A

B C

D

(b)

Figure 11.11. Schematics (a) and Psychrometrics (b) of System 1: Mechanical dehumidification with heat pump.

System 1: Mechanical Dehumidification with Heat Pump A schematic and the corresponding psychrometric charts are shown in Figure 11.11.

Fresh air (Point A) first flows through a cooling coil where it is dehumidified below the dew point (Point B). Then the air flows through a heating coil where it is heated to the set points of

Li-Zhi Zhang 292

supply air. The system comprises a heat pump: the cooling coil acts as the evaporator, and the heating coil acts as a condenser. The heat that should be rejected is larger than that that is required to heat the supply air, so an additional condenser is necessary to reject the surplus heat of the heat pump. Exhaust air from indoor space is pumped through this condenser to enhance the performance of the heat pump system.

System 2: Mechanical Dehumidification with Sensible Heat Exchange

This concept uses a sensible heat exchanger to recover the heat of the supply air itself after it flows through the dehumidification cooling coil. As shown in Figure 11.12, fresh air at point A flows through the sensible heat exchanger, where it is cooled down (in some cases, it also loses some water content due to condensation) to point B. Subsequently, it is dehumidified by a cooling coil and returns to the sensible exchanger at point C. After heated up, it is supplied to the room. The cooling coil can be an evaporator of a small refrigeration system. Then the incoming fresh air is dehumidified and subsequently heated by a heat pump system that is similar to System 1.

Without any use of the exhaust air, this design uses a sensible-only air-to-air heat exchanger to pre-cool and reheat the outdoor air that is dried with a mechanical dehumidifier. Heat pipes, coil run-around loops and plate type heat exchangers are used for this purpose.

System 3: Mechanical Dehumidification with a Membrane-based Total Heat Exchange

In this system, a membrane based total heat exchanger is used before the fresh air is pumped to a heat pump for air dehumidification. The total heat exchanger has a membrane core where the incoming fresh air exchanges moisture and temperature simultaneously with the exhaust air. In this manner, the total heat or enthalpy from the exhaust is recovered. The schematic and the processes are shown in Figure 11.13 for this system. This system is also relatively simple, since the membrane system has no moving parts, and is compact.

System 4: Mechanical Dehumidification with a Desiccant Wheel

This two-stage equipment uses the condenser heat from a mechanical dehumidifier to re-active a desiccant wheel. First, fresh air is pre-cooled and partially dried by the mechanical dehumidifier. Then the air is dried more deeply and also reheated by the desiccant wheel. This arrangement uses both technologies at favorable points of performance. At higher inlet humidities, the mechanical refrigeration system can operate at a higher coil temperature and suction pressure, thereby saving energy. Dehumidified air from a desiccant wheel is very hot. Therefore before it is supplied to rooms, it should be cooled down to set points first. To recover the energy from exhaust air, an evaporative cooler is used to cool down the exhaust air, which is then used to cool the supply air from desiccant wheel. Under extremely humid ambient conditions, energy required for re-activation exceeds available heat from the heat pump. In such cases, an auxiliary electric heater is used to accomplish the regeneration of the desiccant wheel. This system is relatively complex due to the rotating character of the wheel and the reactivation of the desiccants. If low grade waste heat is available, the desiccant wheel system becomes superior to other systems in energy efficiency. The system is shown in Figure 11.14. Total heat recovery is realized by evaporative cooling of room exhaust air.


Besides above four systems with partial or full total heat recovery, to have a comparison, a mechanical dehumidification system with no heat recovery shown in Figure 11.15 is also considered. It is the base system.

A

B

C

Cooling Coil

Sensible heat exchanger

Fresh Air

Supply Air

D

FanA

B

C

Cooling Coil

Sensible heat exchanger

Fresh Air

Supply Air

D

Fan

(a)

0

0.005

0.01

0.015

0.02

0.025

0.03

5 10 15 20 25 30 35 40


Hum

idity

ratio

(kg/

kg

AB

C D

0

0.005

0.01

0.015

0.02

0.025

0.03

5 10 15 20 25 30 35 40


Hum

idity

ratio

(kg/

kg

AB

C D

(b)

Figure 11.12. Schematics (a) and Psychrometrics (b) of System 2: Mechanical dehumidification with sensible heat exchange.

Li-Zhi Zhang 294

Fresh Air

A

B

E


F

C DEvaporator Condenser

Supply Air

Indoor Air

Compressor

Valve

Condenser

Fan

Fresh Air

A

B

E


F

C DEvaporator Condenser

Supply Air

Indoor Air

Compressor

Valve

Condenser

Fan

(a)

0

0.005

0.01

0.015

0.02

0.025

0.03

5 10 15 20 25 30 35 40


Hum

idity

ratio

(kg/

kg A

B

C D

E

F

(b)

Figure 11.13. Schematics (a) and Psychrometrics (b) of System 3: Mechanical dehumidification with membrane-based total heat exchanger.


Fresh Air B CEvaporator

Desiccant wheel

Indoor Air

Supply Air

Compressor

Condenser

A

F

GI H

Valve

Electric Heater

D

E

Evaporative coolerFresh Air B CEvaporator

Desiccant wheel

Indoor Air

Supply Air

Compressor

Condenser

A

F

GI H

Valve

Electric Heater

D

E

Evaporative cooler

(a)

0

0.005

0.01

0.015

0.02

0.025

0.03

5 10 15 20 25 30 35 40


Hum

idity

ratio

(kg/

kg A

BC

D

0

0.005

0.01

0.015

0.02

0.025

0.03

5 10 15 20 25 30 35 40


Hum

idity

ratio

(kg/

kg A

BC

D

(b)

Figure 11.14. Schematics (a) and Psychrometrics (b) of System 4: Mechanical dehumidification + Desiccant wheel.

Li-Zhi Zhang 296

Fresh AirEvaporator Heater

Supply Air

Compressor

Valve

Condenser


Supply Air

Compressor

Valve

Condenser

FanFresh AirEvaporator Heater

Supply Air

Compressor

Valve

Condenser


Supply Air

Compressor

Valve

Condenser

Fan

Figure 11.15. The traditional system, mechanical air dehumidification with no recovery.

Component Modeling

Energy performance of the four building dehumidification system were calculated by

thermodynamic calculations. The modeling methods of total heat exchangers, evaporative coolers, desiccant wheels, sensible only heat exchanger are the same as those described in the above section. Here gives the modeling for the newly added components.

T

sO

°

°

°°

°° 1

234

5 6

°2’

T

sO

°

°

°°

°° 1

234

5 6

°2’

Figure 11.16. T-s diagram of the refrigeration system.

Refrigeration Cycle A thermodynamic model of the refrigeration system is formulated based on the processes

of refrigerant R134a shown in Figure 11.16. Saturated R134a liquid at point 4 flows through


an expansion valve and becomes wet vapor at point 5. Refrigerant at this state flows to the evaporator (also the dehumidifier) where it chills and dehumidifies the fresh air and evaporates to point 6 and further superheats to point 1. Then the refrigerant vapor is pumped by a compressor to point 2 where the vapor is displaced to the condenser and condensates from state 2 to 4 through 3. The superheat is set to 5°C. It should be noted that the exact degree of superheating may be affected by many factors, such as evaporator, expansion valve, and compressor, and are strongly related to operating conditions, control strategies. Experiments found superheating are in the range of 2 to 16°C; and the superheating increases with air temperature in the evaporator, from 3°C to 14°C when the expansion valve is fully open. From viewpoint of energy use, too large superheating is not good. Generally, 3-14°C superheating are possible. Refrigerant superheating refers to the superheating in the evaporator. Superheating from tubes can be neglected with well tube-insulating, since superheating from such sources is harmful to energy performance and should be prevented. The effects of heating in the compressor are included in the compressor’s isentropic efficiency.

The specific enthalpy of the refrigerant at the compressor exit is calculated by

s

f1f2'f1f2 η

hhhh

−+= (11.16)

where ηs is the isentropic efficiency; h2’ is the specific enthalpy at the condensing pressure by isentropic compression from the evaporating pressure. Experimental results have shown that the isentropic efficiency is a weak function of the displacement volume, and varies linearly with the compressor speed. In this analysis, a constant isentropic efficiency of 0.75 is assumed, neglecting the rotational speed of the compressor

The specific enthalpy after the expansion valve is calculated by

f4f5 hh = (11.17) The degree of dryness at the inlet of the evaporator

'56

'55

hhhh

x−−

= (11.18)

where h5’ is the specific enthalpy of the saturated liquid refrigerant at the evaporating pressure.

The electricity consumed by the compressor

( ) motarf1f2fcom /ηhhmq −= (11.19)

where fm is the mass flow rate of refrigerant (kg/s); ηmotor is the motor efficiency, which is considered as 0.75.

Li-Zhi Zhang 298

The energy load of the evaporator

( )f5f1fe hhmq −= (11.20) Heat rejected in the condenser

( )f4f2fc hhmq −= (11.21) Refrigeration efficiency

com

e

qq

COP = (11.22)

Evaporator and Condenser

Cooling and dehumidification of the incoming fresh air are performed in the evaporator. A detailed modeling of the evaporator and the condenser are rather complicated [15,16]. Usually, they are divided into regions associated to the phase of the refrigerant. Each region constitutes a separate heat exchanger. In the case of the condenser, the superheated vapor, the condensation and subcooled liquid regions are considered, whereas for the evaporator it is divided into the evaporating and superheated vapor regions. For each region, the refrigerant side and air side convective heat transfer coefficients need be calculated from the established correlations for single-phase and two-phase flow. For the evaporator, when the average fin surface temperature is calculated to be less than the local water dew point of the air stream, moisture condensation will occur. Under these conditions, the air heat transfer coefficients can no longer be calculated as in dry conditions, and a water mass balance must be carried out. In this case, the enthalpy method, proposed by Threlkeld [17] and introduced in ASHRAE Handbook [1] was most adequate for use. According to this procedure, the driving force for heat transfer is assumed to be the difference between the saturated enthalpy of the air flowing over the fins and a fictitious saturated air enthalpy evaluated at the refrigerant temperature.

The analysis of air cooling and dehumidifying coils requires coupled, non-linear heat mass transfer relationships. While the complex heat mass transfer theory that is presented in many textbooks is often required for cooling coil design, simpler models based on effectiveness concepts are usually more appropriate for energy estimation. These techniques are resulted from basic heat and moisture transfer equations for simultaneous heat and moisture transport. Therefore, in this study, to ease the analysis, thermal performance of the heat exchangers regions is evaluated by the (ε, NTU) method. According to this procedure, the heat exchanger effectiveness is defined as

)()( cihiminp TTcmQactual

−=ε (11.23)


where Thi and Tci are the inlet temperatures of the hot and cold fluids, respectively (K). Because the operations are set to fluctuate around the design points, constant evaporator and condenser effectiveness are assumed in the simulations. Then the air states at the outlets of evaporator and condenser could be obtained.

The heat extracted in the evaporator is calculated by

( )aCaBae hhmq −= (11.24)

where am is the mass flow rate of fresh air stream (kg/s); haB and haC are the specific enthalpies of air at point B and C respectively.

Similarly, heat rejected at this portion of the condenser is governed by

( )aDaCac1 hhmq −= (11.25) It should be noted that qc1 calculated from Eq.(11.25) is only a portion of that calculated

by Eq.(11.21). The evaporating temperature is fixed to 5°C, while the refrigerant flow rate and the condensing temperature varies according to the cooling load of the evaporator and the outside weather conditions. Usually, a 10°C log mean temperature difference between the condensing refrigerant and the air flowing through it is required.

Heat Pump

The cooling coil acts as an evaporator of a heat pump, and the heating coil acts as a condenser for the heat pump. The efficiencies vary with evaporating and condensing temperatures. The heat pump efficiency is defined as

Ele2

ConHP q

q=ε (11.26)

where qCon is the heat rejected at the condenser side (kW), and qEle2 is electricity consumed by the compressor (kW). The above equation is used to calculate the electric energy to drive the heat pump, from the condensing energy required and heat pump efficiencies. Depending on the operating and condensing temperatures, the heat pump efficiencies are in the range of 3-5.

Effectiveness of the main components are related to design and operating conditions. When the operating conditions fluctuate near design conditions, the effectiveness changes only in a small range. To simplify analysis, constant effectiveness for various components is assumed.

The simulations are conducted on an hour-by-hour basis. The operating hours are from 9:00 to 18:00. Fan efficiency is selected as 0.6. For the convenience of comparison, energy consumed in the form of electricity is converted to primary energy by a factor of 3.3.

The dehumidified supply air temperature is set to and fixed at 20°C. The indoor is 25 °C. So the dehumidified air has no sensible load. Rather, it will extract a small fraction of the sensible load from the building.

Li-Zhi Zhang 300

Performance Analysis The COP varies with both the evaporating temperature and the condensing temperature.

The influence of the condensing temperatures on the COP is shown in Figure 11.17. As can be seen, the COP decreases with increasing condensing temperatures. When the condensing temperature increases from 20°C to 50°C, the system COP decreases from around 8.0 to 2.5. Following correlation could be formulated for the relation for the refrigeration evaporating at 5°C:

c0417.0041.20 teCOP −= (11.27)

The effects of the evaporating temperature on the COP are shown in Figure 11.18. The

system COP rises with increasing evaporating temperatures. In fact, when the evaporating temperature increases from -10°C to 25°C, COP is improved from 2.0 to 6.5. A correlation has been formulated for the analysis of the system performance:

6377.20746.00016.00001.0 e

2e

3e +++= tttCOP (11.28)

where the condensing temperature is fixed as 45°C.

Figure 11.19 shows the distribution of the COP of the refrigeration system during a year. As indicated, in winter, the system has higher performance, in contrast, when it’s hot in summer, the system COP decreases, which will in return deteriorate the energy requirements.

0

2

4

6

8

10

12

10 20 30 40 50 60T c (C)

CO

P

♦♦

Tc (°C)

0

2

4

6

8

10

12

10 20 30 40 50 60T c (C)

CO

P

♦♦

Tc (°C)

Figure 11.17. Variations of COP versus condensing temperature, when Te=4.5°C.


0

1

2

3

4

5

6

7

8

-10 0 10 20 30T e (C)

CO

P

♦♦

Te (°C)

0

1

2

3

4

5

6

7

8

-10 0 10 20 30T e (C)

CO

P

♦♦

Te (°C)

Figure 11.18. Variations of COP versus evaporating temperature, when Tc=46°C.

2

3

4

5

6

7

8

9

10

0 2 4 6 8 10 12

Months

CO

P

Figure 11.19. Distribution of COP of the refrigeration cycle in a year, by hourly calculations.

Annual Primary Energy Requirements

The four systems are used to treat ventilation fresh air. To make comparisons, the annual

energy requirements of the four proposed systems and a dehumidification system with no heat

Li-Zhi Zhang 302

recovery measures are computed and listed in Table 11.3. As can be seen, the total energy of the systems with heat recovery saved 30% to 43%, depending on the systems involved. In the table, the values in the column “cooling” refers the energy required in the cooling coil for air dehumidification; “Heating” refers to the energy need to heat the dehumidified air to the set points; “Electricity” refers to the converted primary energy of electricity used by compressors; “Condensing” refers to the energy rejected by the condenser of the heat pump; “Auxiliary” means the converted primary energy used in electric heating, for example, the electric heater in System 4. The column “Fan” refers to the converted primary energy used to circulate the air. All the energy values are calculated on a per-person basis.

In the analysis, the systems are used to treat the latent load solely, the sensible load of the room is around 50W/m2, which will be extracted by chilled-ceiling panels.

The comparisons of the different systems are plotted in Figure 11.20. It is shown that among the 5 systems, System 3 consumes the least energy, and System 2 consumes the most. Generally speaking, energy savings for the four systems are in the same order. This is due to the reason that all the 4 systems newly proposed take into account the energy recovery measures of the exhaust air.

Of the systems studied, three systems recover, more or less, the energy from exhaust air. They are the same use as an economizer. Outside air is fresh air that needs to be dehumidified and treated. In system 1, the indoor air is used to cool the condenser. In system 3, indoor air is used to cool and dehumidify the fresh air in a total heat exchanger. In System 4, indoor air is used to cool down the dehumidified air. System 3 is the best.

Table 11.3. Annual primary energy requirements (kJ/person) for each person under

various dehumidification strategies

Cooling Heating Electricity Condensing Auxiliary Fan Total

No recovery

5.41E+06 2.11E+06 3.61E+06 6.52E+06 9.11E+04 2.63E+05 6.22E+06

System 1

5.41E+06 1.62E+06 3.62E+06 6.61E+06 3.82E+04 3.92E+05 4.13E+06

System 2

4.13E+06 1.64E+06 3.72E+06 5.32E+06 3.33E+04 5.93E+05 4.54E+06

System 3

4.32E+06 5.83E+05 3.01E+06 5.34E+06 4.54E+04 5.92E+05 3.64E+06

System 4

3.23E+06 2.33E+06 2.32E+06 4.16E+06 9.72E+05 8.81E+05 4.18E+06


0. 0E+00

1. 0E+06

2. 0E+06

3. 0E+06

4. 0E+06

5. 0E+06

6. 0E+06

7. 0E+06

Nor ecover y

Syst em 1 Syst em 2 Syst em 3 Syst em 4

St r at egi es

Annu

al P

.E.

requ

irem

ents

(kJ

/per

son)

Figure 11.20. Annual primary energy consumptions by air dehumidification for each person with four systems proposed.

11.4. CONCLUSION

The novel pre-cooling MEC desiccant cooling cycle in combination with chilled-ceiling

panels is a new generation of HVAC system. In this system, sensible load is treated by the cooling panels and the latent load is treated by the desiccant system. The results found that compared to a conventional all-air system, the proposed system saves much fan energy due to reduced air volume, and saves much chiller energy due to raised evaporating temperatures. The total primary energy savings amount to 30% for DC+CC system and 40% for PCDC+CC system, respectively. In addition, in the combined system, the temperature and the indoor humidity are decoupled and intentionally controlled independently. As a result, more annual hours are in the comfort region.

Pre-cooling improves wheel’s dehumidification efficiency, therefore, lower regenerating temperatures can be employed with a PCDC cycle. The more humid it is, the more superior a PCDC in comparison with a DC cycle. With a PCDC cycle, nearly 99% of annual operating hours are with less than 80°C regenerating temperatures. In contrast, a common DC cycle needs 30% annual hours’ heat of higher than 80°C.

These discussions prove that the proposed PCDC system is a more efficient, energy saving system, which could be used in hot and humid regions. With this new system, the regenerating temperature could be 15°C lower than the common DC cycle.

Concerns on indoor air quality have prompted the research of novel air dehumidification techniques. The systems of mechanical dehumidification are combined with energy recovery measures like a heat pump, membrane enthalpy recovery, sensible heat exchanger and

Li-Zhi Zhang 304

desiccant wheel. An hour-by-hour simulation reveals that the independent air dehumidification with heat recovery could save 29-42% of primary energy, depending on the system involved. Of the systems proposed, the mechanical dehumidification with a sensible heat exchanger consumes the largest energy, since only a small fraction of total heat is recovered. In contrast, the mechanical dehumidification with a membrane total heat exchanger consumes the least, because a full total heat recovery is realized. Because all the four systems use recovery measures, their energy consumptions are in the same order. The annual total primary energy used for independent air dehumidification is around 4.18E+06 kJ per person.

REFERENCES

[1] ASHRAE. 2005 ASHRAE Handbook - Fundamental. Atlanta (GA): American Society of Heating, Refrigerating and Air-Conditioning Engineers, Inc. (ASHRAE), 2005.

[2] ASHRAE. ANSI/ASHRAE Standard 62-2001, Ventilation for acceptable indoor air quality. Atlanta (GA): American Society of Heating, Refrigerating and Air-Conditioning Engineers, Inc. (ASHRAE), 2001.

[3] Chua, K.J.; Ho, J.C.; Chou, S.K. A comparative study of different control strategies for indoor air humidity. Energy and Buildings, 2007, 39, 537-545.

[4] Waugaman, D.G.; Kini, A.; Kettleborough, C.F. A review of desiccant cooling systems. ASME Journal of Energy Resources Technology, 1993, 115, 1-8.

[5] Collier, R.K.; Barlow, R.S.; Arnold, F.H. An overview of open-cycle desiccant-cooling systems and materials. ASME Journal of Solar Energy Engineering, 1982, 104, 28-34.

[6] Zhang, L.Z.; Niu, J.L. Performance comparisons of desiccant wheels for air dehumidification and enthalpy recovery. Applied Thermal Engineering, 2002, 22, 1347-1367.

[7] Warren, M.L.; Wahlig, M. Analysis and comparison of active solar desiccant and absorption cooling systems: part I-model description. ASME Journal of Solar Energy Engineering, 1991, 113, 25-30.

[8] Niu, J.L.; Zhang, L.Z.; Zuo, H.G. Energy savings potential of chilled-ceiling combined with desiccant cooling in hot and humid climates. Energy and Buildings, 2002, 34, 487-495.

[9] Zhang, L.Z.; Niu, J.L. A pre-cooling Munters Environmental Control cooling cycle in combination with chilled-ceiling panels. Energy, 2003, 28, 3, 275-292.

[10] Niu, J.L.; Kooi vd, J.; Ree, H.vd. Energy saving possibilities with cooled-ceiling systems. Energy and Buildings, 1995, 23, 147-158.

[11] Zhang, L.Z.; Niu, J.L. Indoor humidity behaviors associated with decoupled cooling in hot and humid climates. Building and Environment, 2003, 38, 99-107.

[12] Jurinak JJ, Mitchell JW, Beckman WA. Open cycle desiccant air conditioning as an alternative to vapor compression cooling in residential applications. ASME Journal of Solar Energy Engineering 1984; 106(3): 252-260.

[13] Zhang, L.Z.; Zhu, D.S.; Deng, X.H.; Hua, B. Thermodynamic modeling of a novel air dehumidification system. Energy and Buildings, 2005, 37, 279-286.


[14] Zhang, L.Z. Energy performance of independent air dehumidification systems with energy recovery measures. Energy, 2006, 31, 1228-1242.

[15] Bensafi, A.; Borg, S.; Parent, D.; Cyrano: a computational model for the detailed design of plate-fin-and-tube heat exchangers using pure and mixed refrigerants. International Journal of Refrigeration, 1997, 20, 218-228.

[16] Koury, R.N.N.; Machado, L.; Ismail, K.A.R. Numerical simulation of a variable speed refrigeration system. International Journal of Refrigeration, 2001, 24, 192-200.

[17] Threlkeld, J.L. Thermal Environmental Engineering, 2nd Edn. Upper Saddle River, New Jersey: Prentice-Hall, Inc.; 1970.

INDEX

A

absorbents, 226 absorption, 16, 279, 304 access, viii accuracy, 20, 33, 44, 86, 89, 113, 179, 250 acetate, 43, 45, 199 acetic acid, 190 acid, 190, 191 activation, 292 additives, 194, 197 adhesion, 199 adiabatic, 32, 58, 120 adsorption, 21, 26, 56, 58, 60, 61, 66, 67, 71, 72,

101, 134, 157, 163, 197, 198, 279 adsorption isotherms, 21, 26 advection-diffusion, 233 agent, 190 aid, 46 air pollution, 222 air quality, 1, 4, 7, 9, 269, 272, 303, 304 alcohol, 201 algorithm, 33, 236, 250, 259, 260 alternative, vii, 3, 270, 290, 304 aluminium, 109 aluminum, 83, 274 ambient air, 18, 192, 212 amide, 189 amines, 199 anemometers, 181 application, viii, 5, 6, 73, 97, 103, 226, 272 aqueous solution, 226 Asia, 54 aspect ratio, 89, 90, 91, 92, 94, 114, 117, 118, 119,

120, 127, 150, 151, 172, 176, 237 assumptions, 43, 46, 133, 135, 157 asymptotic, 94 asymptotically, 37, 41, 81, 89, 161 atmospheric pressure, 62, 137, 208, 274

attention, vii, 4, 5, 9, 76, 270, 272 averaging, 245

B

bacteria, 275, 287 bacterium, 4, 272 barrier, 219 behavior, 226, 239 Beijing, 12 bending, 90, 91, 92, 94, 96 benefits, 2, 50 Bim, 64 bird flu, vii, 4, 272 Boltzmann constant, 187, 208 boundary conditions, 20, 32, 76, 96, 105, 109, 113,

120, 126, 157, 161, 162, 163, 165, 172, 173, 186, 244, 247, 253, 259, 260, 262, 264

boundary surface, 164 breathing, vii buildings, 1, 2, 4, 9, 13, 271, 272

C

capacity, 131, 273, 282 capital, 289 capital cost, 289 carbon, 44, 109, 120 cast, 190 casting, 190, 191, 194 catalyst, 190 catheter, 77, 96, 185 cavities, 201 cell, 15, 16, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36,

37, 38, 39, 40, 41, 43, 44, 46, 47, 48, 49, 50, 51, 52, 53, 54, 157, 188, 193, 202, 203, 204, 205, 215, 222, 232, 249, 258, 260, 263

Index 308

cellulose, 43, 45, 46, 53, 166, 189, 190, 199, 200, 226

cellulose triacetate (CTA), 190, 191, 192, 195, 196, 197, 226

cement, 274 CFD, 28, 231, 236, 249, 258, 259, 266 CH4, 199 channels, 57, 58, 59, 60, 65, 69, 73, 75, 78, 97, 99,

101, 120, 131, 134, 149, 150, 157, 173, 174, 176, 181, 185, 192, 265

chemical, 140, 185, 190, 197, 226, 231 chemical industry, 140 China, vii, ix, 2, 9, 10, 271, 290 chloride, 226, 227, 279 chlorophenol, 226 circulation, 17, 18, 237 classical, 176 classified, 6, 280 clay, 109 cleaning, 201 closure, 245 CO2, 187, 195, 199, 222, 223, 224 codes, 266 coefficient of performance (COP), 269, 274, 282,

286, 300, 301 coil, 2, 6, 13, 18, 166, 192, 273, 275, 277, 282, 291,

292, 298, 299, 302 collisions, 209 comfort zone, 4, 270 commercial, 2, 9, 174, 178, 179, 180, 181, 187, 190,

200, 236, 271, 273 complexity, 58, 78 components, 1, 33, 164, 165, 197, 208, 275, 277,

279, 296, 299 composite, viii, 6, 58, 73, 127, 128, 187, 189, 190,

192, 195, 196, 197, 198, 199, 200, 201, 202, 204, 205, 206, 221, 222, 223, 224, 226, 227, 266

composition(s), 190, 191, 197 compression, 297, 304 computation, 185, 232, 236, 247, 250 computational fluid dynamics, 268 computational modeling, 157 computer(s), 10, 18, 267, 283 concentration, 19, 20, 25, 32, 33, 45, 93, 100, 124,

126, 157, 161, 173, 187, 199, 213, 214, 223, 264 condensation, vii, 2, 4, 73, 270, 283, 287, 289, 292,

298 conditioning, vii, viii, ix, 1, 2, 3, 4, 5, 6, 9, 10, 12,

13, 17, 125, 133, 157, 185, 189, 212, 222, 223, 225, 231, 269, 270, 271, 272, 273, 282, 290, 304

conductance, viii, 6, 99, 103, 108, 109, 114, 116, 117, 118, 119, 120, 121, 122, 123, 126, 127, 128, 150, 151, 176, 182, 230

conduction, 58, 73, 103, 107, 124, 134, 148, 176, 279

conductive, 83, 99, 103, 109, 120, 203 conductivity, 46, 55, 60, 100, 105, 109, 120, 122,

132, 149, 156, 163, 221, 233 configuration, 128 confinement, 40 conservation, vii, ix, 2, 6, 12, 13, 47, 59, 60, 79, 89,

105, 123, 158, 203, 232, 233, 244, 264 constant load, 283 construction, 78, 274, 281 consumption, 1, 271, 290 contaminants, 276 continuing, ix continuity, 232, 244 contractors, 289 control, viii, 1, 4, 5, 6, 17, 18, 31, 46, 54, 65, 81, 82,

86, 89, 97, 107, 124, 128, 161, 164, 178, 236, 250, 260, 272, 273, 275, 287, 290, 297, 304

controlled, 4, 17, 34, 35, 44, 46, 52, 84, 111, 180, 181, 190, 203, 272, 275, 287, 289, 290, 303

convection, 27, 46, 73, 81, 96, 97, 103, 128, 134, 185, 186, 267

convective, 15, 27, 28, 31, 33, 34, 37, 41, 43, 48, 50, 59, 60, 75, 76, 77, 81, 103, 107, 113, 120, 122, 123, 127, 128, 132, 134, 135, 137, 138, 140, 144, 146, 148, 155, 157, 161, 164, 173, 175, 176, 187, 194, 204, 205, 216, 223, 229, 231, 236, 250, 257, 260, 266, 267, 279, 283, 298

convergence, 33, 86, 250, 260 cooling, vii, viii, 1, 2, 3, 4, 5, 6, 7, 10, 11, 13, 17, 18,

57, 58, 60, 66, 67, 71, 96, 166, 185, 192, 270, 272, 273, 274, 275, 277, 278, 279, 281, 282, 283, 284, 285, 286, 287, 288, 289, 290, 291, 292, 298, 299, 302, 303, 304

cooling process, 18, 71, 282 copolymer, 189 copper, 83, 109 correlation(s), 21, 38, 39, 41, 48, 49, 131, 133, 135,

144, 145, 146, 147, 149, 153, 157, 171, 177, 240, 244, 251, 253, 256, 257, 262, 264, 280, 298, 300

cost-effective, 231 costs, 1, 290 coupling, 33, 109, 126, 157, 250, 259 cross-linked, 58, 197 cross-linking, 190 cross-sectional, 31, 60, 78, 82, 94, 229, 234 cycles, 58, 68, 236, 237, 239, 240, 242, 243, 250,

251, 252, 253, 256, 257, 259, 260, 274, 284

D

dead zones, 90, 92

Index 309

decoupling, viii, 5, 272 deduction, 133, 144 defects, 195 definition, 86, 112, 118, 139, 144, 149, 160, 245 degree, 52, 230, 297 delta, 245 demand, 1, 225 density, 16, 19, 30, 47, 59, 60, 69, 79, 101, 105, 122,

124, 149, 151, 156, 164, 188, 193, 204, 205, 207, 209, 212, 213, 214, 215, 230, 232, 234, 245, 264, 279

dependent variable, 86, 233, 250 derivatives, 32 desorption, 58, 60, 66, 67, 68, 71, 72, 163, 197, 198,

279 destruction, 233 deviation, 37, 43 dew, 4, 5, 6, 17, 44, 181, 273, 291, 298 differential equations, 86, 142 differentiation, 20, 21, 138 diffusion, 6, 15, 17, 19, 20, 26, 27, 43, 46, 47, 50, 52,

53, 54, 58, 60, 61, 69, 72, 76, 113, 134, 148, 163, 173, 176, 188, 189, 190, 197, 199, 205, 206, 207, 208, 210, 212, 213, 215, 216, 233, 250

diffusion mechanisms, 60, 197, 208 diffusion process, 19 diffusion rates, 52 diffusivities, 61, 127, 182 diffusivity, 15, 17, 19, 20, 21, 25, 26, 27, 30, 41, 43,

47, 49, 50, 52, 53, 54, 55, 61, 79, 123, 124, 127, 129, 131, 134, 135, 147, 148, 149, 150, 151, 153, 155, 157, 164, 176, 185, 187, 199, 204, 206, 212, 213, 214, 215, 216, 225, 227, 229

dilation, 233 discipline, ix discomfort, 4, 272 discrete data, 251 discretization, 236, 250, 260 discs, 34 diseases, vii, 2, 4, 272 displacement, 276, 297 distillation, 185, 227 distillation processes, 227 distilled water, 34, 43, 44, 202, 203 distribution, 41, 83, 84, 86, 97, 110, 111, 114, 128,

206, 211, 236, 238, 240, 241, 259, 268, 276, 287, 300

draft, 6, 273 dry, 4, 10, 11, 16, 18, 19, 35, 44, 47, 55, 56, 57, 58,

60, 79, 100, 122, 123, 124, 132, 134, 156, 157, 164, 174, 192, 194, 203, 204, 205, 212, 215, 216, 223, 270, 273, 275, 277, 281, 287, 298

drying, 15, 18, 19, 20, 24, 25, 26, 43, 54, 128, 190, 225, 226

dynamic viscosity, 79, 105, 156

E

economies, vii electric energy, 299 electric power, 282 electrical, 274 electricity, 286, 287, 297, 299, 302 electrolyte, 187, 213 electronic(s), 2, 9, 19, 271 emission, 4, 15, 26, 27, 28, 29, 30, 31, 32, 34, 37, 39,

40, 41, 46, 47, 49, 52, 53, 54, 138, 155, 163, 164, 176, 182, 187, 204, 272

energy, vii, viii, ix, 1, 2, 3, 5, 6, 9, 12, 13, 16, 40, 55, 57, 58, 65, 66, 67, 73, 75, 77, 86, 96, 107, 131, 134, 136, 140, 154, 157, 158, 161, 164, 174, 185, 225, 226, 232, 233, 236, 244, 246, 250, 259, 260, 269, 270, 271, 272, 273, 274, 277, 279, 280, 281, 282, 283, 286, 287, 290, 292, 297, 298, 299, 300, 301, 302, 303, 305

energy consumption, vii, 1, 5, 272, 282, 286, 290, 303, 304

energy efficiency, 292 energy recovery, vii, 2, 12, 13, 65, 154, 185, 225,

226, 290, 302, 303, 305 engineering, ix, 127, 131 enthalpy, 60, 136 envelope, vii, 4, 270, 282 environment, vii, 1, 4, 5, 18, 19, 270, 271, 272, 273,

274 environmental, vii, 1, 3, 270 environmental impact, vii, 1, 3, 270 environmental issues, 1 environmental protection, vii epidemic, vii, 2 equilibrium, 18, 20, 23, 26, 44, 45, 46, 47, 134, 135,

148, 157, 176, 249, 273 equipment, 45, 225, 283, 290, 292 ethanol, 54 Europe, 26 evaporation, 203 evolution, 33 exchange rate, 15, 48 experimental condition, 49, 50 exponential, 21 extraction, 185

Index 310

F

fabric, ix fabrication, viii, 190 factor analysis, 242 family, ix feet, 274 fiber, 225, 226 film(s), 54, 127, 129, 226, 231 filtration, 267 financial support, ix finite volume, 32, 164, 259 finite volume method, 164 floating, 44 flow field, 54, 231, 239, 241, 259 flow rate, 16, 31, 33, 35, 36, 37, 40, 41, 44, 46, 50,

51, 55, 131, 134, 135, 137, 146, 149, 156, 171, 175, 179, 181, 182, 188, 192, 194, 195, 223, 224, 239, 243, 244, 253, 259, 269, 279, 282, 283, 290, 291, 297, 299

fluctuations, 279 flue gas, 226 fluid, ix, 27, 28, 40, 41, 48, 50, 54, 75, 77, 79, 81,

82, 96, 105, 106, 107, 108, 109, 113, 114, 117, 120, 124, 126, 157, 160, 161, 229, 232, 233, 234, 236, 237, 244, 250, 251, 253, 257, 259, 260, 261, 268, 282

fluid interfaces, 108, 109, 126 fluoride, 189, 199 foils, 12, 99, 132 food, 54, 227 formaldehyde, 4, 272 fossil fuel(s), vii, 3, 270 Fourier, 20 friction, 75, 77, 81, 90, 91, 92, 96, 97, 106, 120, 128,

160, 185, 229, 231, 235, 237, 238, 239, 240, 242, 245, 249, 253, 254, 255, 257, 262, 263, 264, 266, 267

fungi, 275, 287 furniture, 2, 4, 9, 271, 272, 279

G

gas(es), 19, 26, 34, 35, 39, 43, 59, 60, 61, 135, 187, 188, 206, 208, 209, 210, 212, 222, 226, 234, 244, 268, 277, 284

gas diffusion, 212 gauge, 18, 181 gel, 18, 25, 54, 57, 225, 279, 281, 283 generation, 97, 113, 128, 180, 269, 279, 303 Germany, 194 glass, 34, 109, 120, 201, 283

gold, 201 grains, 2, 9, 271 graph, 32, 71, 199, 200, 201, 236 gravity, 281 grids, 33, 84, 89, 110, 111, 113, 179, 237, 250, 259 groups, 73 growth, 2, 4, 9, 239, 271, 272 Guangzhou, 282 guidelines, ix

H

handling, 6, 12, 13, 273 harmful, 297 hazards, 287 head, 274 heat capacity, 60, 137, 282 heat conductivity, 127, 134, 148, 157, 176, 188, 221,

222, 225 heat exchangers, 131, 155 heat loss, 166, 180 heat pumps, viii heat transfer, viii, 6, 17, 55, 59, 60, 64, 69, 75, 77,

83, 89, 96, 97, 99, 100, 103, 107, 108, 114, 119, 120, 122, 126, 127, 128, 131, 132, 134, 135, 137, 138, 140, 142, 144, 148, 155, 160, 161, 163, 175, 176, 185, 186, 225, 229, 234, 235, 241, 242, 244, 245, 246, 250, 251, 253, 256, 257, 259, 260, 261, 264, 265, 266, 267, 268, 275, 290, 298

heating, vii, 1, 2, 3, 10, 11, 13, 18, 19, 58, 60, 66, 67, 71, 166, 180, 192, 270, 273, 274, 275, 279, 282, 291, 297, 299, 302

height, 15, 16, 30, 34, 47, 65, 75, 76, 78, 89, 93, 99, 101, 104, 109, 118, 120, 127, 149, 155, 157, 187, 192, 204, 215, 237, 265

high risk, 4, 272 high temperature, 73 homogeneous, 58, 116, 180, 190, 199 Hong Kong, 10, 13, 185, 225 horizon, 34 hospital, 2, 271 hot water, 17 human, 2, 4, 9, 10, 271, 272, 290 hydrate, 226 hydro, viii, 6, 17, 21, 25, 26, 43, 45, 53, 54, 101,

157, 189, 190, 200, 216, 221, 224, 225, 226, 227, 231

hydrodynamic, 28, 59, 122, 176, 187 hydrodynamics, 53 hydrogen bonds, 190 hydrophilic, viii, 6, 17, 21, 25, 26, 43, 45, 53, 54,

101, 157, 189, 190, 200, 216, 221, 224, 225, 226, 227, 231

Index 311

hydrophilic groups, 190 hydrophilicity, 190, 197 hydrophobic, viii, 6, 180, 189, 190, 199, 200, 212,

221, 224, 226, 227

I

images, 195 in transition, 290 inactive, 68 incentives, vii, 3, 270 incompressible, 232, 233, 234, 244 independence, 113, 250, 260 independent variable, 61 indices, 179, 286 industrial, 45, 279 industrial application, 45 industrialized countries, 1 industry, vii, viii, 2, 4, 6, 9, 12, 68, 125, 197, 212,

222, 223, 269, 270, 271 infection, 4, 272 infinite, 142, 144, 176 inhalation, 268 insight, ix, 135, 155 inspection, 94 insulation, 166, 180 intensity, 250 interaction(s), 21, 162, 226, 253, 279 interface, 49, 202 inventions, 189 investment, 274 ionic, 201, 227 ionic liquids, 227 iron, 109 irradiation, 4, 272 isothermal, 25, 46, 268 isotherms, 21, 26, 93, 114, 197, 240, 253 iteration, 86, 250

J

Jacobian, 75, 86, 112 Japan, 96 Japanese, 267 Jerusalem, 267

K

kinetic energy, 40, 229, 234, 246, 248, 249 Kolmogorov, 245

L

lamina, viii, 6, 28, 31, 60, 73, 75, 76, 77, 78, 79, 89, 96, 97, 99, 105, 113, 150, 157, 172, 181, 229, 237, 244, 245, 253, 257, 260, 262

laminar, viii, 6, 28, 31, 54, 60, 73, 75, 76, 77, 78, 79, 89, 96, 96, 97, 99, 105, 113, 127, 128, 150, 157, 172, 181, 185, 186, 229, 232, 237, 244, 245, 253, 257, 260, 262, 266, 267

Langmuir, 21 Laplace transformation, 142 law(s), 19, 43, 45, 46, 60, 244, 249 lead, 43, 94, 261 learning, ix limitations, 185 linear, 38, 84, 111, 125, 139, 146, 165, 214 linear regression, 38 links, 164 liquid nitrogen, 201 liquid water, 49, 60, 189 literature, 17, 89, 127, 206, 237, 260 lithium, 226, 227, 279 location, 40, 50, 107, 124, 253, 287 logging, 46 London, 54, 77, 83, 96, 103, 128, 144, 185, 267 long-term, ix, 13, 101, 185, 225 losses, 43 lumen, 77, 96, 185

M

machines, 101 magnetic, 17 maintenance, 101, 274 malic, 190, 191 manufacturer, 282 manufacturing, 2, 9, 58, 271 market, viii, 266, 289 market penetration, viii, 289 market prices, 289 mass transfer process, 73, 185 material surface, 26 mathematical, ix, 26, 58, 155, 164, 185 matrix, 58, 86, 165, 277, 278 measurement, 17, 21, 27, 34, 41, 53, 179, 182, 223 measures, viii, 6, 269, 290, 302, 303, 305 mechanical, 2, 9, 45, 101, 128, 190, 195, 197, 271,

291, 292, 293, 296, 303 media, 6, 16, 99, 157, 200, 227, 281 membrane permeability, 27 membranes, viii, 6, 15, 17, 43, 50, 51, 52, 53, 54,

101, 128, 133, 146, 147, 150, 153, 157, 187, 189,

Index 312

190, 192, 194, 195, 197, 198, 199, 200, 201, 206, 207, 212, 224, 225, 226, 227, 230, 231, 267

memory, 251 metal ions, 199 metals, 122, 127 methane, 226 methylcellulose, 25, 54, 127, 129 microstructure, 201 mirror, 17, 44, 181 mites, 275 mixing, 35, 44, 134, 203 modeling, viii, ix, 6, 7, 32, 55, 58, 76, 133, 155, 157,

164, 173, 227, 231, 257, 258, 260, 267, 269, 280, 296, 298, 304

models, 26, 185, 206, 229, 231, 245, 251, 252, 253, 255, 257, 259, 260, 266, 267, 280, 281, 283, 298

modern society, 4, 9, 272 modules, 13, 185, 226 modulus, 247 moisture content, 135, 282 moisture sorption, 15 molecules, 21, 190, 209 momentum, 40, 86, 113, 158, 164, 229, 232, 233,

238, 240, 244, 246, 251, 253, 256, 260, 262 morphological, 211 motors, 286

N

NaCl, 47, 205 Nafion, 43, 54, 127, 128, 189, 225 natural gas, 274 Navier-Stokes equation, 28, 32, 79, 234, 244 Netherlands, 279 New Jersey, 305 New York, 74, 96, 97, 128, 153, 154, 185, 225, 267,

268 Newtonian, 79, 105, 157 nitrogen, 37, 225 nodes, 33, 65, 85, 89, 113, 178 non-linear, 65, 86, 245, 267, 268, 298 non-linearity, 245 non-metals, 127 non-uniform, 17, 53 non-uniformity, 17 normal, 25, 32, 107, 113, 124, 206, 235 normal distribution, 206 novel materials, viii, ix, 15 number of Transfer Units (NTU), 15, 49, 50, 53, 55,

69, 70, 73, 131, 132, 133, 135, 137, 140, 143, 144, 145, 146, 147, 149, 152, 153, 155, 157, 171, 188, 298

numerical analysis, 65, 77, 97

Nusselt, 60, 75, 83, 99, 105, 107, 113, 114, 117, 119, 120, 121, 122, 131, 135, 155, 157, 160, 161, 170, 171, 172, 173, 176, 229, 234, 235, 237, 240, 243, 244, 245, 251, 252, 253, 255, 256, 257, 259, 260, 262, 263, 264, 266

O

observations, 93, 201 odors, 276 oil, viii operator, 75, 86, 112, 233 optimization, 17, 96, 131, 133 organic, 54, 199 organic compounds, 54 orientation, 230 oxide, 189, 226 ozone, 273

P

Pacific, 54 paper, 6, 57, 101, 103, 109, 120, 133, 149, 150, 153,

174, 176, 177, 178, 180, 182, 184, 201, 231, 237, 251

parameter, 4, 50, 100, 103, 108, 109, 118, 121, 126, 150, 174, 249, 270

partial differential equations, 46, 164, 178 partition, 15, 45, 125, 127, 139, 149, 151 passive, 57 pathogenic, 4, 272 payback period, 289 pearlite, 54 Peclet number, 46, 164 peer, ix penalty, 257, 266 performance, viii, ix, 1, 6, 17, 25, 43, 54, 55, 58, 62,

67, 69, 70, 73, 75, 76, 77, 103, 120, 122, 131, 132, 133, 147, 153, 155, 157, 179, 185, 187, 216, 217, 219, 226, 227, 231, 267, 269, 277, 279, 282, 283, 284, 292, 296, 297, 298, 300, 305

periodic, 237, 239, 250, 251, 252, 253, 254, 255, 257, 267

periodicity, 257, 259 permeability, 54, 129, 187, 188, 189, 192, 197, 215,

216, 217, 218, 219, 220, 222, 223, 224 permeable membrane, viii, 187, 225 permeation, 15, 43, 53, 54, 101, 140, 144, 147, 150,

182, 189, 190, 191, 193, 194, 195, 197, 202, 216, 217, 218, 219, 220, 222, 224, 226, 227

pharmaceutical, 2, 9, 271 physical properties, 45, 46, 194, 197, 216

Index 313

physics, 244 pitch, 30, 230, 236 planar, 27 plastic(s), 180, 231, 274 platinum, 17 plywood, 109 Poisson equation, 83, 110 polarization, 185 poly(phenylene oxide) (PPO), 226 poly(vinylchloride), 54 polyether, 189, 226 polymer(s), 17, 25, 26, 43, 45, 53, 54, 101, 103, 109,

120, 127, 189, 190, 199, 216, 227, 266 polymer film, 54 polymer membranes, 43, 45, 53, 54, 101, 189, 199,

227 polynomial, 49, 125 polypropylene, 190 polystyrene, 189 polystyrenesulfonate, 128, 226 polyurethane, 189, 226 polyvinyl alcohol, 43 pools, 2, 9, 271 poor, 4, 67, 272 poor performance, 67 pore(s), 60, 61, 71, 73, 187, 188, 200, 206, 207, 209,

210, 211, 212, 217, 219 porosity, 54, 56, 61, 188, 207, 217, 218, 219, 220,

226 porous, 190, 192, 195, 199, 206, 207, 221, 222, 225 porous media, 206, 221, 225 positive reinforcement, ix powder, 190 power, 19, 70, 180, 282 Prandtl, 75, 100, 106, 122, 160, 229, 244, 245, 247 prediction, 251, 260, 279 preparation, 201, 202 pressure, 19, 22, 23, 29, 30, 33, 35, 40, 44, 61, 62,

70, 79, 96, 105, 137, 155, 179, 181, 188, 189, 199, 208, 209, 210, 212, 213, 229, 233, 234, 235, 236, 237, 249, 250, 253, 257, 259, 260, 266, 269, 273, 282, 286, 292, 297

prevention, vii, 2 prices, viii probability density function, 206 procedures, 18, 165, 179, 185 product design, 103 program, 103, 110, 113, 114, 279 promote, 240 property, 279 protocols, 1 public, viii pumps, 17, 35, 44, 203

pure water, 214 purification, 185 PVA, 226 PVC, 201

R

radius, 15, 31, 36, 37, 38, 40, 41, 47, 51, 52, 55, 61, 78, 188, 204, 205, 216

range, ix, 1, 20, 45, 119, 123, 135, 212, 244, 249, 251, 257, 281, 282, 297, 299

raw material, 202 reality, 4, 164, 272, 279 recovery, viii, ix, 1, 4, 5, 6, 7, 9, 10, 13, 15, 16, 55,

57, 58, 62, 65, 66, 67, 68, 69, 70, 71, 72, 73, 76, 99, 103, 136, 269, 270, 272, 273, 275, 279, 290, 292, 293, 296, 302, 303, 304

recovery technology, viii, 269 reduction, vii, 2, 190 reference frame, 233 refrigerant, 13, 296, 297, 298, 299 refrigeration, viii, 13, 274, 292, 296, 300, 301, 305 regenerate, 57, 273, 277, 279, 281 regenerated cellulose, 189 regeneration, 19, 24, 26, 274, 279, 282, 284, 286,

291, 292 regular, 75, 78, 260 regulations, vii, 4, 6, 9, 270, 272 relationship(s), 143, 208, 209, 210, 282, 298 relaxation, 33, 84, 111, 250 reliability, 101 renewable energy, 285 research, vii, ix, 1, 231, 303 residential, 304 residues, 33 resistance, 15, 17, 18, 43, 50, 58, 69, 113, 132, 137,

138, 139, 140, 144, 146, 147, 153, 171, 172, 173, 182, 186, 188, 190, 192, 194, 205, 210, 213, 215, 216, 217, 219, 223, 225, 262, 265, 266

resolution, 259 resources, vii, 3, 270 respiratory, 2, 287 retail, 2, 3 returns, 275, 277, 292 Reynolds number, 15, 30, 31, 33, 37, 75, 76, 81, 96,

100, 106, 150, 156, 157, 160, 171, 181, 205, 229, 234, 239, 240, 243, 244, 245, 247, 253, 255, 257, 259, 262, 263, 266, 267

Reynolds stress model, 249, 251, 257, 268 room temperature, 46, 201, 208, 287, 288 rotation axes, 40, 41

Index 314

S

salt(s), 190, 194, 197, 226 sample, 9, 13, 18, 19 SARS, 2, 4, 272 saturation, 19, 22, 62, 137, 281 savings, 7, 9, 13, 185, 225, 269, 274, 302, 303, 304 scalar, 233 Scanning Electron Microscope (SEM), 194, 195,

196, 197, 199, 200, 201 schema, 276, 277, 278 Schmidt number, 16, 30, 76, 100, 122, 156, 160, 205,

264 science, 153, 185 selecting, 45, 83, 110, 251 selectivity, 188, 199, 222, 223, 224, 226 sensors, 17, 35, 44, 180, 181, 203 separation, 43, 199, 226, 267 series, 20, 142, 144, 214, 230, 275 Shanghai, 19 shape, 15, 45, 58, 61, 83, 94, 127, 135, 155, 164, 259 shear, 230, 245, 262 Sherwood number, 16, 31, 32, 35, 36, 37, 75, 76, 81,

82, 83, 89, 90, 91, 92, 94, 95, 96, 100, 123, 124, 126, 156, 157, 160, 161, 171, 172, 173, 176, 188, 205, 264

signs, 78 silica, 18, 57, 279, 281, 283 silicate, 127 siloxane, 189 similarity, 189 simulation, 28, 73, 157, 249, 251, 267, 279, 283,

304, 305 simulations, 173, 216, 259, 267, 279, 299 sine, 77, 78, 90 Singapore, 54 skin, 199 SO2, 199 society, vii, 1, 2, 6, 12 software, 181 solar, 4, 272, 274, 285, 304 solar energy, 274, 285 solid phase, 64 solid surfaces, 157, 261 solubility, 54 solutions, 33, 165, 280 solvent, 190 sorption, 15, 17, 18, 20, 21, 23, 25, 26, 45, 53, 54,

55, 61, 65, 67, 68, 73, 125, 131, 133, 135, 139, 147, 157, 164, 194, 197, 198, 275, 281

sorption curves, 20, 23, 25, 26, 55, 139 sorption experiments, 17, 54, 194 sorption isotherms, 26, 53, 197, 198

sorption process, 23, 67 Southeast Asia, 2, 9, 271 spacers, 127, 230 species, 207 specific heat, 10, 59, 60, 79, 105, 122, 134, 136, 139,

155, 175, 229, 234, 245 speed, 19, 55, 63, 67, 68, 69, 73, 179, 181, 281, 283,

297, 305 sporadic, viii, 127 square wave, 52 stability, 227 stabilize, 217 stages, 19 stainless steel, 34, 166, 167, 192 standard deviation, 188, 207 standardization, vii, 4, 270 steady state, 43, 49, 52, 179, 223, 244 steel, 109 stochastic, 245 storage, vii, 5, 272 strain, 18, 247 strategies, 1, 266, 297, 302, 304 streams, 35, 43, 44, 58, 60, 64, 71, 103, 127, 131,

133, 134, 135, 136, 137, 139, 149, 162, 163, 171, 173, 174, 178, 180, 182, 192, 194, 203, 222, 223, 226

strength, 45, 101, 128, 190 stress, 230, 233, 245, 249, 257 students, ix summer, vii, 2, 4, 9, 10, 12, 17, 57, 270, 271, 275,

283, 284, 287, 288, 300 sunlight, 10 supply, 2, 4, 5, 6, 35, 44, 46, 78, 132, 203, 272, 273,

274, 275, 277, 279, 280, 281, 286, 287, 292, 299 supported liquid membrane, viii, 6, 187, 189, 198,

199, 216, 223, 224, 226, 227, 266 surface area, 78, 207, 229, 234, 240 surface diffusion, 58, 60, 61 surplus, 292 sustainable development, vii, 2, 4, 270 symmetry, 31, 32, 40, 58, 113, 124, 157, 238 synthesis, ix synthetic, 226 systematic, viii systems, vii, viii, ix, 1, 2, 4, 5, 6, 7, 9, 13, 17, 54, 76,

86, 157, 161, 226, 270, 272, 273, 274, 276, 279, 282, 283, 287, 288, 290, 292, 293, 301, 302, 303, 304, 305

T

tanks, 267 Tc, 301

Index 315

technology, viii, 53, 58, 189 Teflon, 109 temperature gradient, 235, 240, 253, 261, 274 test procedure, 203 textbooks, 298 theoretical, ix, 20, 103, 206 theory, 136, 190, 206, 207, 212, 298 thermal, vii, ix, 2, 4, 5, 9, 58, 60, 69, 79, 94, 96, 105,

117, 118, 119, 122, 132, 137, 156, 157, 161, 163, 171, 172, 173, 190, 203, 233, 234, 240, 242, 253, 270, 271, 272, 274, 279, 286, 298

thermal analysis, 279 thermal energy, vii, 4, 270, 274 thermal properties, 79, 105, 157 thermal resistance, 58, 137, 172, 173 thermal stability, 190 thermodynamic, 213, 214, 234, 296 thermodynamic calculations, 296 thermodynamic equilibrium, 214 thermodynamics, ix three-dimensional, 32, 268 Ti, 10, 106, 234, 281 time, 2, 4, 9, 16, 17, 19, 20, 21, 23, 24, 26, 47, 49,

51, 56, 59, 62, 65, 66, 67, 78, 83, 101, 110, 133, 201, 204, 229, 230, 232, 237, 244, 245, 251, 271, 272, 279, 283, 284

time consuming, 133 TiO2, 267 tissue, 201 total energy, vii, 1, 2, 5, 6, 12, 136, 271, 272, 286,

290, 302 transformation(s), 21, 83, 86, 110, 112 transition(s), 211, 257 transmembrane, 189, 199, 209, 216 transmission, 4, 272, 282 transport, viii, 6, 43, 45, 46, 49, 51, 52, 53, 54, 128,

134, 155, 157, 164, 165, 190, 199, 203, 204, 206, 209, 225, 226, 227, 231, 233, 234, 240, 244, 247, 249, 267, 298

transport phenomena, 157 trend, vii, 6, 253, 273, 290 turbulence, 245, 246, 249, 250, 251, 252, 253, 255,

256, 257, 260, 261, 264, 266, 267 turbulent, viii, 6, 229, 230, 244, 245, 246, 247, 249,

250, 251, 257, 262, 264, 267, 268 turbulent flows, 257 two-dimensional, 53, 58, 72, 79, 97, 128, 135, 174,

182

U

ultra-thin, 231 uncertainty, 25, 44, 50, 114, 179, 181, 192

uniform, 32, 40, 41, 73, 76, 79, 82, 83, 84, 96, 97, 100, 103, 105, 111, 113, 114, 116, 120, 126, 128, 157, 161, 171, 172, 173, 186, 216, 236, 249, 250, 257, 262, 264, 266, 267

V

vacuum, 185, 190, 208, 227 validation, ix, 110, 251, 260 validity, 39 values, 4, 22, 23, 25, 37, 39, 43, 46, 50, 51, 65, 68,

84, 86, 89, 90, 93, 94, 111, 113, 114, 118, 120, 121, 126, 127, 137, 140, 146, 161, 164, 165, 171, 172, 175, 178, 179, 192, 197, 216, 223, 235, 237, 240, 241, 243, 245, 246, 250, 256, 259, 260, 272, 281, 282, 302

vapor, viii, 12, 15, 16, 19, 22, 23, 24, 30, 33, 46, 47, 52, 53, 54, 55, 62, 79, 81, 100, 101, 123, 124, 128, 129, 134, 135, 137, 156, 176, 187, 189, 190, 191, 192, 193, 194, 197, 198, 199, 203, 204, 208, 209, 210, 213, 216, 222, 223, 224, 225, 226, 227, 229, 273, 297, 298, 304

variable(s), 4, 13, 29, 32, 38, 56, 62, 141, 164, 165, 171, 179, 181, 234, 243, 251, 272, 290, 305

variation, 37, 71, 94, 165 vector, 237, 251, 260 velocity, 16, 17, 29, 31, 33, 39, 40, 41, 47, 59, 79,

80, 81, 82, 86, 92, 94, 96, 105, 106, 113, 114, 115, 122, 150, 156, 158, 160, 162, 188, 194, 202, 204, 230, 232, 234, 235, 236, 237, 238, 239, 241, 245, 249, 251, 253, 257, 259, 260, 276

ventilation, vii, 1, 2, 4, 5, 6, 9, 10, 11, 12, 13, 101, 123, 135, 226, 270, 271, 272, 274, 276, 283, 286, 289, 290, 301

ventilators, vii, 2 versatility, 44 vinylidene fluoride, 43 virus(es), 4, 272, 275, 287 viscosity, 16, 30, 76, 100, 149, 188, 205, 209, 230,

233, 245, 264 voids, 21, 58 vortices, 40

W

wall temperature, 73, 76, 97, 103, 105, 106, 120, 234, 249, 267

waste, 274, 292 wastewater treatment, 199 water, vii, 2, 9, 10, 16, 17, 18, 19, 20, 21, 22, 23, 24,

25, 26, 29, 34, 35, 37, 43, 44, 45, 47, 52, 53, 54, 56, 60, 61, 71, 99, 101, 124, 127, 131, 132, 134,

Index 316

135, 147, 148, 156, 157, 164, 166, 187, 190, 192, 195, 196, 197, 202, 208, 209, 210, 213, 216, 224, 225, 226, 273, 274, 275, 279, 282, 283, 287, 289, 290, 292, 298

water diffusion, 124 water evaporation, 10, 216 water sorption, 197 water vapor, 2, 9, 17, 19, 21, 22, 29, 43, 132, 134,

157, 190, 197, 208, 209, 210, 224, 225, 226, 273 wet, 4, 34, 54, 57, 58, 270, 281, 297 wet coating, 54 wind, 18, 19, 179, 181 wind tunnels, 180 windows, 282

winter, vii, 10, 11, 12, 16, 57, 275, 283, 284, 287, 300

working conditions, 192 working hours, 287, 289 writing, ix

Y

yield, 140

Z

zeolites, 279

total heat recovery heat and moisture recovery from ventilation air

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