heat and mass transfer with condensation in non-saturated porous media

HEAT AND MASS TRANSFER WITH CONDENSATIONIN NON-SATURATED POROUS MEDIA

Y. Zhang, X. F. Peng, and I. ConteLaboratory of Phasechange and Interfacial Transport Phenomena,Department of Thermal Engineering, Tsinghua University, Beijing, China

Heat and mass transfer with condensation in porous media was numerically studied coupling

with distributions of temperature, vapor concentration, pressure and water saturation. The

transport processes in porous media displayed two distinct stages, or early and late stages.

Different characteristics during these stages were described with the consideration of physi-

cal nature. A systematical investigation was conducted to explore the influence of different

parameters on the processes, including the temperature, the particle diameter, the porosity,

and the thermal conductivity of the particle. A modeling experiment was conducted to

partially validate the calculation.

1. INTRODUCTION

Condensation in porous media widely exists in practical applications, includingdew formation in soil, water migration near underground cold tubes, and moistureabsorption in building walls. This problem strongly couples heat and mass transferwith phase change and normally involves three phases in it, namely, solid frame,liquid condensate, and gas mixture including vapor and non-condensable gas. Whenthe temperature in porous media is not uniform, the following mechanisms functionsimultaneously. First, heat is transferred by conduction, diffusion and convection,and released as latent heat while condensation occurring. Secondly, gas is transferredby convection, diffusion and eliminated due to condensation (for vapor). Thirdly,condensate accumulates because of condensation and is transferred by convectiondriven by gravity and the gradients of capillary pressure and gas pressure. Masstransfer and heat transfer are highly influenced by phase change, effective physicalproperties, and phase equilibrium conditions. Such coupling phenomena may behavequite differently for different situations.

Many researchers made great contributions to this field, however they empha-sized on different aspects, respectively. Some investigations [1–3] were conducted tounderstand the transport phenomena induced by interface interaction that plays a

Received 17 September 2006; accepted 30 March 2007.

This research is currently supported by the National Natural Science Foundation of China (No.

50636030) and the Specialized Research Fund for the Doctoral Program of High Education (Contract

No. 20040003076).

Address correspondence to Professor X. F. Peng, Department of Thermal Engineering, Tsinghua

University, Beijing 100084, China. E-mail: [email protected]

1081

Numerical Heat Transfer, Part A, 52: 1081–1100, 2007

Copyright # Taylor & Francis Group, LLC

ISSN: 1040-7782 print=1521-0634 online

DOI: 10.1080/10407780701453800

significant role in micro pores. Some others [4–7] were about film condensation inporous media, showing interest in film geometry and heat transfer coefficient. Andmany investigations were concerned with the macro behavior of the coupled heatand mass transfer. Among the different researchers, de Vries and Philip [8] andde Vries [9–10] combined temperature with water content to model the coupled heatand moist transfer. Because of its simplicity, this model or similar models wereextensively used in a great number of investigations [11–17]. This kind of modelscontained several thermophysical properties to be determined from experiments,and would yield underestimation under fluctuating thermal conditions [11, 15],which compromises its generality and practical value. Some recent researches[18–23] tended to model conjugate processes fully coupling heat with mass transfermechanisms. Although very complicated, they have very clear physical meaningand much sound foundation. Many researchers further coupled other mechanismsbesides the afore-mentioned ones, such as sorption [24–25] and radiation [26].

Numerical methods were extensively used to investigate the transport behaviorin a fully coupling manner [17–19, 21–26]. However, the influences of differentparameters on the transfer process were scarcely covered in the available literature.And normally, the researchers often neglected gas convection while admitting gasdiffusion without any justification [19, 22]. Their simplifications would significantlyreduce calculation cost and divergence probability, but the validity is highlyquestionable.

This article aims at accomplishing the following goals by numerical investigationon coupled transport processes in a non-saturated spherical particle-packed bed.

1. Simulating transport processes with a more general and sound model andexploring the influence of the mechanisms normally neglected in the availableinvestigations.

2. Presenting the basic features of the transport behavior with condensation innon-saturated porous media.

NOMENCLATURE

Cp mass heat capacity, J=kg K

D diffusivity, m2=s

d particle diameter, m

g gravitational acceleration, m=s2

hfg latent heat of condensation, J=kg

K permeability, m2

L thickness, m

M molecular weight, kg=mol

_mm condensation rate, kg=m3 s

p pressure, Pa

R universal gas constant, J=mol K

S water saturation, -

Sir irreducible water saturation, -

T temperature, K

t time, s

v superficial velocity, m=s

x distance from the bottom, m

Y mass fraction of air or water vapor, -

e porosity, -

k thermal conductivity, W=m K

m viscosity, Pa � sq density, kg=m3

r surface tension, N=m

Subscripts

a air

c capillary

eff effective

g gas, including water vapor and air

s solid or saturated

v water vapor

w liquid water

0 initial condition

1082 Y. ZHANG ET AL.

3. Exploring the influence of different parameters on the transport processes,including temperature, particle diameter, porosity, and the thermal conductivityof solid particles.

4. Conducting a modeling experiment to partially validate the numericalsimulations.

2. FORMULATIONS

2.1. Problem Description

Consider a one-dimensional unsteady problem of heat and mass transfer withcondensation in a porous medium, which is 60 mm thick and composed of sphericalparticles. The top of the porous bed is a cold impermeable wall. Initially the porousbed was filled with saturated humid air at the temperature of T0 and the pressure ofp0 ¼ 101325 Pa. Suddenly, the bottom is exposed to hot saturated humid air at thetemperature of T1 and the pressure of p0 while the top wall temperature is kept atT0. The problem is described in Figure 1. Vapor will diffuse into the porous bedand condensate inside. As far as this investigation is concerned, the condensate doesnot accumulate so much that it would overcome interfacial force and drop off thebottom, and the bottom is treated as impermeable for condensate.

2.2. Modeling

Besides the basic assumptions for a general volume-averaging method [27],the following simplifications are introduced.

1. No gravity effect is considered for gas.2. The pore diameter is small, and the local thermal equilibrium holds true.3. Darcy’s law holds and thermal dispersion is neglected, because the velocities are

very low, as will be demonstrated later.

Figure 1. Analytical model.

TRANSFER WITH CONDENSATION IN POROUS MEDIA 1083

4. Thermal diffusion is neglected due to not very high temperature difference.5. Gas is not dissolved in liquid.

Similar assumptions were made or implicitly admitted by other researches availablein literature [18–19, 21–26]. The governing equations are specified as following.

Mass balance equations

qqt½eð1� SÞqv� þ

qqxðqvvgÞ ¼ � _mmw þ

qqx

qgDeffqYv

qx

� �ð1Þ

qqt½eð1� SÞqa� þ

qqxðqavgÞ ¼

qqx

qgDeffqYa

qx

� �ð2Þ

qqtðeSqwÞ þ

qqxðqwvwÞ ¼ _mmw ð3Þ

Momentum equations

vw ¼Kw; eff

mw

� qqxðpg � pcÞ � qwg

� �ð4Þ

vg ¼Kg;eff

mg

� qpg

qx

� �ð5Þ

Here Darcy’s law is applied. In Eq. (5), Gas flows under pressure gradient. For liquidwater migration in Eq. (4), the driving force further includes gravity and capillarypressure pc. When a porous medium is non-saturated, there exist capillary forcesor pressure drops caused by curved liquid-gas interfaces. If water is not evenlydistributed in the porous media, capillary pressure gradient may have a greatcontribution to water migration.Energy equation

qqtð1� eÞqsCpsT þ eSqwCpwT þ eð1� SÞðqaCpa þ qvCpvÞT

#"

þ qqxðqwCpwTvwÞ þ

qqx½ðqaCpa þ qvCpvÞTvg�

¼ qqx

qgCpaTDeffqYa

qx

� �þ qqx

qgCpvTDeffqYv

qx

� �þ qqx

keffqT

qx

� �þ hfg _mmw ð6Þ

Phase equilibrium

pv ¼ psðTÞ ð7Þi.e., the partial pressure of vapor is considered as the saturated pressure ps corres-ponding to the local temperature. Empirical correlation [28] was used to estimateps and the maximum relative error in the calculation is less than 2� 10�3.

Subsidiary equationsSince the pressure is not high, ideal gas state equation can be properly used

here,

q ¼ pM

TR; for vapor and air ð8Þ


And the effective diffusivity is expressed as [29]

Deff ¼ Df ðeÞgðSÞ ¼ De1:5ð1� SÞ1:5 ð9ÞDifferent models to estimate the effective thermal conductivity of saturated porousmedia, which is filled with one fluid, were comprehensively reviewed in the availableliterature [30], but the methods for non-saturated media, which are filled with morethan one fluid, as the cases in this investigation, are not available. The volumetricaverage method is employed to give a crude estimation, or

keff ¼ keffðSÞ ¼ ð1� eÞks þ eSkw þ eð1� SÞkg ð10ÞWater effective permeability Kw,eff and the capillary pressure pc as the function ofthe water saturation are estimated using classical empirical formulas [31–33]. Theirreducible water saturation Sir, below which water can not form a continuous filmor flow in porous media, is taken as 0.09.

Initial and boundary conditions are set as mentioned above in Section 2.1.Some physical constants or properties are listed in Table 1 and remain unchangedin all simulation cases. By default, they are taken at an average temperature of50�C, and a characteristic pressure of 101325 Pa. Gas viscosity is taken as that ofthe air for convenience.

Influences of temperature T, particle diameter d, porosity e, and thermal con-ductivity of particles ks were explored by comparing the results of different cases.

Table 1. Physical properties and constants predetermined

Symbol Name Unit Value

qw Water density kg=m3 988.1

Cpw Water heat capacity J=kg K 4174

kw Water thermal conductivity W=m K 0.648

mw Water viscosity Pa � s 5.494� 10�4

rw Water surface tension N=m 6.769� 10�2

Cpv Vapor heat capacity J=kg K 1899

Cpa Air heat capacity J=kg K 1005

kg Gas thermal conductivity W=m K 0.0283

mg Gas viscosity Pa � s 1.96� 10�5

D Vapor diffusivity in air m2=s 3.63� 10�5

hfg Latent heat of vaporization J=kg 2.38� 106

Sir Irreducible water saturation – 0.09

qsCps Solid volumetric heat capacity J=m3 K 2� 106

Mw Water molecular weight kg=mol 0.018

Ma Air molecular weight kg=mol 0.0288

R Universal gas constant J=mol K 8.314

Table 2. Parameters in different cases

Reference case (default value) T0 ¼ 30�C T1 ¼ 70�C e ¼ 0.36 d ¼ 0.1 mm ks ¼ 1 W=m K

Temperature influence T0 ¼ 50�C T1 ¼ 90�C T0 ¼ 10�C T1 ¼ 50�CPorosity influence e ¼ 0.48 e ¼ 0.26

Diameter influence d ¼ 1 mm d ¼ 0.01 mm

Thermal conductivity influence ks ¼ 0.02 W=m K ks ¼ 50 W=m K


The parameters for the reference case were taken as default values, and are changedwhen studying the influence of a specified parameter. Different cases investigated inthis paper are listed in Table 2.

2.3. Numerical Techniques

Discretization. For the mass and energy equations, first-order backwarddifference was used temporally. Spatially, for inner nodes, the convection term inthe energy balance equation was discretized with the second-order upwind scheme,and other terms were discretized by the central difference. First-order scheme wasemployed for boundary nodes.

Staggered grids were used. The gird size is 2 mm and the time step is 25 s. Sincethe transport processes are very slow, further fining the grids and time step does notresult in more significant difference, except for some singular points, such as thecondensation rate at the top boundary node.

Iteration. Equations (2) and (5) were solved simultaneously, and Eqs. (3) and(4) were solve simultaneously. The coupled equations were first linearized beforebeing solved. At each time step, the following steps were repeated until the solutionsconverged.

1. Solving Eq. (6) in the whole domain to update T.2. Solving Eqs. (7) and (8) in the whole domain to update qv.3. Solving Eq. (1) in the whole domain to update _mmw.4. Solving Eqs. (2), (5), and (8) in the whole domain to update qa and vg.5. Solving Eqs. (3) and (4) in the whole domain to update S and vw.

At each step, thermal properties were instantly updated and equations wereiteratively solved.

Convergence criteria. Let i denote the node number, r the residual of thediscretized equations, and C the value of the time-derivative term. For a balanceequation, if the inequality P

i jrijPi jCij

< 10�4 ð11Þ

holds, the iteration is supposed to have converged. For the air mass balanceequation, when t is very big,

Pi jCij ! 0. If the inequalityX

ijrij < 10�9 ð12Þ

holds, we also suppose a converged solution has been reached. When the iterationsfor all the mass and energy equations have converged, calculation starts for the nexttime step.

3. FUNDAMENTAL CHARACTERISTICS

3.1. General Behavior

Here only the reference case is considered to explore the characteristics ofenergy and mass transport in non-saturated porous media. The transport process


can be divided into two distinct stages, the early and the late stages, characterizedby their dominant mechanisms. During the early stage, when t < 2,000 s, the tem-perature field is far from the steady state and the rapid change of temperatureresults in remarkable variation of local condensation, as shown in Figure 2. First,vapor condenses mostly adjacent to the bottom, and no condensation is observedat the top, because no vapor is transferred there. Then the condensation rate _mmw

decreases near the bottom as the condensation area extends to the upper zone,

Figure 2. Distributions in the early stage for the reference case; (a) temperature difference; (b) liquid water

saturation; (c) condensation rate; (d) air density; (e) pressure difference, and (f) gas velocity.


see Figures 2c and 2f. After 400 s, _mmw at the top increases significantly and itgradually exceeds those at all other positions. Obviously, vapor penetrates thewhole porous bed to the top as the condensation becomes weak in the bottomregion and easily condenses at the cold wall. Furthermore, since the water satu-ration S is still lower than the irreducible saturation Sir in this early stage, the con-densate can not form a continuous liquid film to flow in the bed. S is proportionalto the integral of _mmw on time and decreases from the bottom to the top. However,at the top, there is jump because of high _mmw induced by the cold wall where thetemperature is lowest (Figure 2b).

Since vapor is continuously condensed, macro gas flow from the bottom to the topis formed to supply enough vapor (Figure 2f ). Pressure decreases along the flow direc-tion (Figure 2e). Inverse gradient of qa causes downward diffusion of air to counteractthe macro flow (Figure 2d ). The maximum pressure difference between the inside press-ure p and the initial pressure p0 is about 4 Pa, which is negligible compared with P0.Although the temperature is not very high in this case, the air density qa decreases byabout 40% at the bottom compared with the initial state (about 1.1 kg=m3).

During the late stage, when t > 2,000 s, temperature profiles at different timesnearly overlap, and the temperature gradient grad T is shown to depict the slowchange of the temperature field in detail (Figure 3a), which may be a sign of theresistance to heat conduction. Minus liquid water velocity vw means downward flow,and positive upward (Figure 3d ). Clearly, liquid water is sucked from both top andbottom to the middle, because _mmw and S at both sides are higher than that in themiddle. Where S reaches Sir, S changes sharply, resulting in the rapid changes ofmany effective thermophysical properties spatially, including effective thermalconductivity and effective diffusivity, and consequently the rapid changes of gradT and _mmw (Figures 3a–3c). As time proceeds, S increases slowly correspondingto weak condensation, and thermal properties are altered slowly, causing theredistribution of other parameters (Figures 3a–3f ).

Succinctly speaking, the early stage is characterized by the rapid developmentof T and accordant redistributions of _mmw. And the late stage is characterized by theinner water transport, corresponding to water flow, slow increase of S and associatedchange of thermal properties and redistributions of vw and _mmw. The results agree withWijersundera et al. [19] and Larbi et al. [17] (under condition D) who conducted asimilar study using a distinct model.

3.2. Influence of Water Accumulation

From Eq. (1), water accumulation or the increase of S is directly affected bytwo factors, water convection and condensation rate. Convection causes redistri-bution of condensed water and contributes to inner water transport without anychange of total water amount in the porous medium, while condensation increasesliquid water, and particularly contributes to local water accumulation.

When water content is very low, water convection will be significantly limitedby the contact with the solid particle and pore structures. Particularly if watersaturation is below the irreducible water saturations Sir, which is taken as 0.09 inthe calculation, the liquid water is assumed not to continuously flow in porousmedia, except liquid patches=films and=or droplets locally spreading or contracting.


Apparently, at this time convection would contribute very little to the watermigration, and condensation plays critical role in altering the water distribution inthe medium, as predicted in Figure 2b.

Figure 3. Distributions in the late stage for the reference case; (a) temperature gradient; (b) liquid water

saturation; (c) condensation rate; (d) liquid water velocity; (e) pressure difference, and (f) gas velocity.


Where the water content S is higher than 0.09, a continuous liquid region formsand water begins to flow, and the profile of S along x is relatively flat (Figure 3b).Consequently convection driven by both capillary and gravity has a great contri-bution to water migration as along with the condensation. There is a jump ataround Sir for water distribution, which means that the bottom continuous-water-region is not jet connected with the top one. In the middle region where S < Sir

the condensation is the only way to change water content.Condensation rate achieves a very high value at the top wall (Figure 3c),

caused by the impermeable boundary and the lowest temperature. The high con-densation rate provides a water source and continuously supplies water movingto the middle part of the bed. Besides the high value at the top wall, there are alsosome peaks where S change rapidly. When S increases sharply, water increasinglyplugs the diffusion path, and the effective diffusivity decreases rapidly. And fromEq. (1), the net condensation rate will change correspondently. Besides, neglectingthe high value at the boundary and the peaks, condensation rate decreases fromthe bottom to the top, following the temperature decrease. This will be furtherexplained later.

S changes sharply at S ¼ Sir, resulting in sudden change of other parameters,like the change of temperature gradient caused by the change of effective thermalconductivity and the change of condensation rate caused by the change of effectivediffusivity. It seems there are many interesting phenomena near this active zone. Atleast it involves the formation, aggregation and spreading=contracting of the liquiddroplets, and other interfacial phenomena, which would significantly affect themacro transport characteristics in this zone.

3.3. Discussion of the Accepted Assumptions

Some researchers [19, 22] assumed constant pressure and neglected gas convectionin the mass equations and neglected the convection terms in the energy equation. Fromthe results obtained for the reference case here, first, the assumption of constant pressureis obviously reasonable because the maximum pressure difference is about 4 Pa and neg-ligible compared with p0. Secondly, if comparing the diffusive vapor mass flux att ¼ 50,000 s with the convective mass flux in Eq. (1), see Figure 4, we can find thatthe convective mass flux is about 1=4 of diffusive mass flux near the bottom. Apparently,neglect of the convection term in Eq. (1) would result in biased solutions. Thirdly, a non-dimension number Pew is introduced as

Pew ¼vwqwCpwL

keffð13Þ

representing the relative strength of convective heat transfer compared withheat conduction. A maximum value of Pew exists at x ¼ 28 mm for t ¼ 50,000 s, asshown in Figure 3d. At this position vw ¼ 9� 10�9 m/s and keff ¼ 0.68 W=(mK), andaccordingly the maximum value of Pew is 3.3� 10�3, much less than 1. As a result,the convection term of liquid water can be neglected in Eq. (6). In a similar way, gas con-vection and diffusion terms can also be reasonably neglected when considering heattransfer without compromising the correctness of the solutions.


As a conclusion, although the constant pressure assumption is reasonable, andalso the convection and diffusion terms can be reasonably neglected in the energyequation, the neglect of gas convection in the mass equation is not sound.

4. INFLUENCES OF PARAMETERS

4.1. Temperature

Temperature difference between the top and bottom is maintained at 40�C for allcases, and different values of bottom temperature T0 and top temperature T1 are speci-fied to explore their influences on the transport characteristics. As shown in Figure 5, ifboth T0 and T1 are increased by 20 K, the average S reaches about 3 times of that in thereference case, and the average _mmw is also trebled. Accordingly, S exceeds Sir everywheresee Figure 5b, and a nearly linear distribution is formed to provide enough capillarypressure gradient to balance gravity. Because of more intensive condensation, the tem-perature gradient varies in a broader range. The gas convection strengthens and theinner pressure becomes lower. All of these results demonstrate that higher temperaturescause more intensive heat and mass transfer and phase change. When both T0 and T1 aredecreased by 20 K, an inverse trend appears. This is in good agreement with the numeri-cal and experimental results of Larbi et al. [17].

From Eq. (1), condensation is driven by qqv=qx and q2qv=qx2, and thefollowing equations can be derived

qqv

qx¼ qT

qx

dqv

dTð14Þ

q2qv

qx2¼ q2T

qx2

dqv

dTþ qT

qx

� �2d2qv

dT2ð15Þ

Figure 4. Comparison of the vapor mass diffusion with convection.


Both dqv=dT and d2qv=dT2 increase as T increases, as shown in Table 3. When thespatial derivatives of temperature do not vary very much, a high temperature leadsto a higher condensation rate, and vice versa. And this is why condensation ratedeceases from the bottom to the top, as mentioned in Section 3.2.

Figure 5. Distributions in the late stage for case T1 ¼ 90�C, T0 ¼ 50�C; (a) temperature gradient; (b) liquid

water saturation; (c) condensation rate; (d) liquid water velocity; (e) pressure difference, and (f) gas velocity.


4.2. Particle Diameter

Comparison between Figure 3f and Figure 6d reveals that particle diameter dhas little effect on superficial gas velocity. According to the empirical formula of theeffective permeability [31], the pressure difference between the inside pressure and p0

is inversely proportional to the second power of the particle diameter d. When dincreases to 10 times, the pressure difference decreases to 1%, see Figure 3e andFigure 6c. Also, capillary force attenuates when d increases [32,33]. As the capillaryforce becomes very weak compared with gravity, condensate flows downwardsthroughout the porous bed and accumulates near the bottom (Figure 5a and 5b).When d decreases, the inverse trend follows.

Table 3. Relations between density and temperature for saturated water vapor

30�C 50�C 70�C

dqv=dT (kg=m3 K) 1.64� 10�3 3.85� 10�3 7.96� 10�3

d2qv=dT2 (kg=m3 K2) 7.60� 10�5 1.51� 10�4 2.65� 10�4

Figure 6. Distributions in the late stage for case d ¼ 1 mm: (a) liquid water saturation; (b) liquid water

velocity; (c) pressure difference, and (d) gas velocity.


Actually, in the late stage, the temporal derivative term can be neglected inEqs. (1) and (2). Substituting Eq. (1) into Eq. (2) and neglecting the temporalderivative term yields

qqxðqgvgÞ ¼ � _mmw ð16Þ

Integrating Eq. (16) from a given position to the top yields

ðqgvgÞjLx ¼Z L

x

� _mmwdx ð17Þ

Since the wall is impermeable at x ¼ L, Eq. (17) evolves to

ðqgvgÞx ¼Z L

x

_mmwdx ð18Þ

which means gas flow flux exactly equals vapor condensed. Additionally, from theempirical formula, Eqs. (9) and (10), particle diameter d does not affect the effective con-ductivity or the effective diffusivity of the porous medium. In the late stage, temperaturegets close to a steady state, which is hardly affected by d. From the discussion in Section4.1. it is inferred that condensation rate is very weakly affected by d. As discussed inSection 3.3., the total gas pressure is approximately constant in the bed. Furthermore,Eqs. (7) and (8) indicate that gas density qg is solely dependent upon temperature andirrelevant with d. From Eq. (18), a conclusion is reached that gas velocity is mainlydetermined by temperature distribution and is hardly affected by d.

4.3. Porosity

Increase of e results in decrease of pressure difference and capillary force, seeFigure 7. When S reaches some value, capillary force can not counteract gravity,and the liquid water flows downward. Since increasing e facilitates gas diffusionand condensation, the condensate, which is proportional to eS, increases noticeablywhen e increases from 0.36 to 0.48. Decrease of e leads to an inverse trend.

The change of e results in the change of Deff, Deff/ e1.5 from Eq. (9), whichplays an important role in water accumulation. By similar derivation with that inSection 4.2., from Eq. (2), we can deduce that vg/Deff neglecting the minor factors,and further from Eq. (1) _mmw/Deff, and thus from Eq. (3), the average value ofeS/Deff./ e1.5. Finally S/ e1=2. This can be demonstrated from the results shownin Figures 7a and 3b, indicating that increase of e can enhance water accumulation.

4.4. Particle Thermal Conductivity

When the particle thermal conductivity ks goes down to 0.02 W=m K, the tem-perature field develops very slowly, see Figure 8. The peak value of _mmw still advancesupwards with T, and after 50,000s, phase change plays an important role as innerheat source in the heat transfer, and the temperature gradient varies in a broad rangespatially, see Figure 9. The average S is higher in the upper part than the lower part,


which is different from other results. When ks reaches 50 W=m K, the temperaturefield in the bed becomes very close to the steady state within 200 s, showing analmost perfectly linear distribution, meaning conduction through the solid particlesplays a dominant role among all the heat transfer mechanisms.

Apparently, low thermal conductivity of the particles blurs the differencebetween the early stage and the late stage. First, it lengthens the early stage.Secondly, when water accumulates, the effective thermal conductivity change notice-ably as predicted by Eq. (10), and temperature field will change accordingly. Allparameters are coupled so strongly that the quasi-steady state can hardly beachieved.

4.5. Experimental Observations

Here a simple experimental investigation was conducted to investigate thefundamental behavior of heat and mass transfer in non-saturated porous mediaand verify the rationality of the numerical simulation. The experimental setupwas built following the model shown in Figure 1. The porous bed was packed bythree porous layers, each of which is 30 mm thick and tightly packed with grinded

Figure 7. Distributions in the late stage for case e ¼ 0.48. (a) liquid water saturation; (b) liquid water

velocity, and (c) pressure difference.


groundnut shells or coconut shell fibers. The porous bed was cooled from the topby ice, and the bottom was exposed to hot water vapor. The top was isolated fromthe ice by a plastic film, and the sides were thermally insulated and isolated from thesurroundings.

The porous layers were weighted every hour and water accumulation wasexpressed by

wac ¼w� w0

w0100% ð19Þ

where w0 and w are the layer weight before testing and after testing, and wac theincrease of water content in percentage. The results are shown in Figure 10. X isthe dimensionless distance from vapor.

X ¼ x

Lð20Þ

For both porous beds made of grinded groundnut shells and coconut shellfibers, water first mainly accumulates near the bottom, but in the long run, water

Figure 8. Distributions in the early stage for case ks ¼ 0.02 W=m K. (a) temperature difference; (b) liquid

water saturation, and (c) condensation rate.


Figure 9. Distributions in the late stage for case ks ¼ 0.02 W=m K: (a) temperature gradient; (b) liquid

water saturation, and (c) condensation rate.

Figure 10. Experimental results for water accumulation. (a) grinded groundnut shells and (b) coconut shell

fibers.


accumulation near the top exceeds instead. Both materials have very low thermalconductivity, close to Case of ks ¼ 0.02 W=m K. Also, the history of water accumu-lation agrees with Figure 9b quite well. These results very well demonstrate thefundamental behavior of water saturation and migration, and vapor condensationin water-unsaturated porous media simulated in this investigation, even thoughthe detail characteristics were not experimentally measured or observed.

5. CONCLUSIONS

Transport processes are numerically studied coupling with heat transfer, masstransfer, phase change, and variable physical properties as vapor condensationoccurs in non-saturated porous media. The following conclusions can be drawn.

1. Convection term in the energy equation may be neglected, but neglect of it in thegas mass balance equation is not acceptable.

2. The transport process with condensation in porous media mainly consists of twostages. The early stage is characterized by the rapid development of temperatureand accordant redistributions of condensation rate. And the late stage is charac-terized by the inner water transport, corresponding to water flow, slow increaseof water saturation and associated change of thermal properties and redistribu-tions of gas velocity and condensation rate.

3. For low water saturation, particularly below the irreducible water saturations,convection would contribute very little to the water migration, and condensationplays a critical role in altering the water distribution in a porous medium, Forwater saturation higher than the irreducible water saturations, the profile of Salong x is relatively flat, and convection has a great contribution to watermigration as along with the condensation. The zone where water saturation isnear to the irreducible water saturation behaves actively, and condensation rateand temperature gradient change sharply.

4. Increase of temperature results in stronger condensation, because vapor densitygrows rapidly with temperature. Increase of the bed particle diameter andporosity reduces pressure difference and capillary force. Higher porosity alsocauses more vapor to condense. Decrease of particle thermal conductivity blursthe boundary of the two stages and increase of it improves the linearity oftemperature distribution. A modeling experiment validates the calculation.

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heat and mass transfer with condensation in non-saturated porous media

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