numerical and experimental investigation of paper drying: heat and mass transfer with phase change...

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Applied Thermal Engineering 27 (2007) 1248–1258

Numerical and experimental investigation of paper drying:Heat and mass transfer with phase change in porous media

T. Lu a,*, S.Q. Shen b

a School of Mechanical and Electrical Engineering, Beijing University of Chemical Technology, Beijing 100029, Chinab Department of Power Engineering, Dalian University of Technology, Dalian 116024, China

Received 17 August 2006; accepted 12 November 2006Available online 12 January 2007

Abstract

The physics of paper drying combines heat and mass transfer with phase change in wet porous media. The drying processes can bemodeled with the liquid and vapor mass conservation equation, the liquid–gas mixture mass conservation equation, and the energy con-servation equation. The drying model of paper includes the convection and the capillary transport of liquid and the convection and thediffusion transport of gas and vapor. Numerical simulation and experimental investigation of whole paper drying are done on an actualpaper machine. The drying parameters in boundary conditions including the temperature of outer surface of the cylinder, the temper-ature and the relative humidity of the air pocket were measured. The numerical results of paper sheet temperature of the first 26th cyl-inders and the final moisture content after 78 cylinders agree well with experimental data, which means the drying model is valid topredict the performance of paper drying.� 2006 Elsevier Ltd. All rights reserved.

Keywords: Drying; Heat and mass transfer; Phase change; Porous media

1. Introduction

Porous media is intrinsically a multi-phase system withpores in a solid skeleton and at least one fluid phase inpores having enough channels. Paper is a mixture of hard-wood and softwood fibers sometimes with other fibers suchas straw, bamboo, cotton, and chemical fibers blended inwith the wood fibers. Paper as typically a porous mediabefore entering the dryer section with moisture content inwet basis in the range of 45–65% after being pressed bypresser. The dryer section removes most of the moistureto obtain good strength. The drying target of moisture con-tent is about 8%.

The pulp and paper industry remains extremely energyintensive, ranking forth only behind chemicals, steel, andpetroleum in terms of energy consumption and is the lead-ing industry in terms of energy expended during drying

1359-4311/$ - see front matter � 2006 Elsevier Ltd. All rights reserved.

doi:10.1016/j.applthermaleng.2006.11.005

* Corresponding author.E-mail address: [email protected] (T. Lu).

operations [1]. The dryer section in a paper plant accountsfor about 1/3 of the equipment investment and operatingcost. With a large demand for paper products, the paperindustry needs to reduce energy consumption by optimiz-ing the drying process to reduce product cost and improvequality. Mathematical models provide a cheap flexiblemethod for improving the operating efficiencies of existingmachines [2] by analyzing the fundamental heat and masstransport phenomena taking place within the paper duringdrying. Detailed analysis of the physics of the drying pro-cess can be used to design efficient new dryer systems withenhanced drying efficiencies, thereby decreasing energyconsumption or improving material properties.

An experimental study of paper drying completed byJamal and Jefferson [3] provides basic drying and tempera-ture data for moist paper exposed to gas-fired infraredradiation energy. A method to determine thickness direc-tion moisture content profiles for a sheet being dried bythrough air drying was developed by Hashemi et al. [4].Due to disadvantages of local non-uniformity of moisture

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Nomenclature

c specific heat (J kg�1 K�1)D diffusivity (m2 s�1)g gravitational constant (m s�2)H thickness of paper sheet (m)h enthalpy (J kg�1)K intrinsic permeability (m2)km convective mass transfer coefficient (m s�1)kT convective heat transfer coefficient (W m�2 K�1)L axis distance between top and bottom cylinder,

mLe Lewis numberM molar mass (kg mol�1)MC moisture contentN cylinder number (dimensionless)Nu Nusselt numberc capillaryp pressure (Pa)Pr Prandtl numberR universal gas constant (J mol�1 K�1) or radius

(m)r thermal resistance (m2 K W�1)Re Reynolds numberS saturationT temperature (K)u velocity (m s�1)x thickness direction

Greek symbols

a effective mass transfer factorb effective heat transfer factor/ porosity (m3 m�3)u relative humidity (%)c cover angle (�)k thermal conductivity (W m�1 K�1)l dynamic viscosity (kg m�1 s�1)h centrifugate angle (�)q density (kg m�3)r surface tension (N K�1)s time (s)x angular velocity (s�2)

Subscripts

a airatm environmenteff effectivef filmg gas mixtureini initiall liquidv vaporw cylinder surface1 reference situation

T. Lu, S.Q. Shen / Applied Thermal Engineering 27 (2007) 1248–1258 1249

content during drying such as reducing drying effective-ness, increasing drying cost or reducing drying capacity,and decreasing product quality, Hashemi and Douglas[5] experimentally investigate the effect of drying inten-sity, initial moisture content, sheet formation, and paperbasis weight on local moisture non-uniformity. Hashemiand Douglas [6] and Hashemi et al. [7] experimentallyresearched the drying strategy of paper for higher machinespeed for two air drying techniques, through air drying andimpingement air drying.

Mathematical modeling can also be utilized to under-stand the physics to explore new drying processes. Manyefforts have been made to analyze the paper drying pro-cesses both theoretically and experimentally. However,many theoretical models [8–13] include critical assumptionssuch as uniform temperature, moisture content, or thermalconductivity through the paper thickness to simplify theheat and mass transport phenomena and the numericalsolution. Most of these models ignore the internal trans-port mechanisms affecting the drying process and are basedon global conservation equations written for the entire por-ous media. Furthermore, few models have consideredstructural changes within the porous media during drying,which also affects moisture migration and the drying pro-cess [11,13,14]. The numerous models for lumber drying[15–20] were used here to develop paper drying model.

Most drying models have been based on the volume-aver-aging approach popularized by Whitaker [21].

The dryer section is an extremely energy intensive pro-cess in the paper manufacturing, often with productionbottlenecks, which can impact the final product qualityand cost. However, the design, operation, and evaluationof paper dryer machines are mostly empirically based. Thispaper presents a mathematical model of heat and masstransfer with phase change in porous media for paper dry-ing which includes convective and capillary transport ofliquid, convective and diffusive transport of vapor and airin paper sheet based on the mass and energy balances.The drying parameters used in boundary conditions of dry-ing model such as surface temperature of cylinder, the tem-perature and relative humidity of air in gas pocket areexperimentally measured on a running paper machine.The surface temperature of paper sheet at the inlet and out-let for the first 26th cylinder is also measured as the majorcomparative parameter with the numerical temperature ofpaper sheet. The first objective of the present paper is toapprove the validity of the drying model based on the the-ory of the heat and mass transfer in porous media for thewhole drying process of paper sheet on a running papermachine not on a experimental setup in the laboratory.The significant second objective is to characterize the trans-port phenomena of paper drying by distributions of state

1250 T. Lu, S.Q. Shen / Applied Thermal Engineering 27 (2007) 1248–1258

parameter inside paper sheet, such as water saturation,temperature, and pressure.

2. System description of paper drying

The most common drying system used in the paperindustry is a multi-cylinder dryer section, which consistsof a series (20–120 units) of cylindrical 0.75–2.0 m diametercast-iron dryer drums. The two-tier paper dryer section inFig. 1 shows the four phases of the drying process. The dry-ing process for each cylinder can be broken down into thefollowing phases:

Phase I: The sheet is in contact with the outer surfaceof the cylinder but is not covered with thefelt.

Phase II: The sheet remains in contact with the cylinderand is covered on its outer surface by the felt.

Phase III: The sheet remains in contact with the cylinderbut is no longer covered with the felt (similar tophase I).

Phase IV: The sheet is no longer in contact with thedryer cylinder and is in an open draw, wheremoisture can evaporate from both sheetsurfaces.

The paper is threaded around each dryer, which isheated by condensing steam with conduction as the majormode of heat transfer to the sheet. The drying felt is ahighly porous material whose main purpose is to hold thepaper sheet in close contact with the dryer cylinder toincrease the heat transfer between the dryer drum andpaper sheet, to help prevent shrinkage and deformationof the paper sheet, to enhance the stability of the paperrunning. In phases I and III, which are a small part ofthe whole drying period, one side of the paper sheet is incontact with the cylinder surface while the other is exposed

Gas

Paper

Felt

RollerFelt

Cylinder

sheet

pocket

Measurement point: A, C, E Inlet point; B, D Outlet pointCenter point of gas pocket

Free movement drying periodContacting drying period; IVPhase definition: IIF, G Top and bottom points of cylinder; H

Fig. 1. Schematic of paper drying system in a paper dryer section.

to the small region between the paper sheet and the feltwhich can easily become saturated which limits vapor dif-fusion. Therefore, in the model phases I and III are com-bined with phase II to form the contact drying period,interval AB, with interval BC as the free movement dryingperiod. As the sheet moves from C to D, the drying bound-ary conditions be reversed from interval AB. Interval DE issimilar to BC. Points A and C are defined as the cylinderinlets and points B and D are the cylinder outlets. The inletis the beginning of the contact drying period and also theend of the previous free movement drying period while out-let is the end of the contact drying period and the start ofthe next free movement drying period. The heat and masstransfer processes differ in the contact drying period fromthose in free movement drying period, resulting in gradi-ents of the drying parameters such as moisture content,temperature, and pressure in the paper sheet. The paperdrying process can be analyzed once the drying parametersdistributions at the inlet and outlet are specified for eachcylinder.

3. Mathematical model

The model is based on the conservation equations forthe heat and mass transfer in the paper sheet during drying.Whitaker’s volume-averaging approach was utilized todevelop the conservation equations. The main assumptionsare:

(1) The solid, liquid, and gas phases are in thermody-namic equilibrium.

(2) The drying process in the paper sheet is one-dimen-sional in the thickness.

(3) The shrinkage and deformation of paper sheet duringdrying are negligible.

(4) The air and vapor within the pore spaces behave asideal gases.

(5) The energy transport is by conduction in all threephases plus convection in the liquid and gas phases,but the inertia and viscous dissipation terms arenegligible.

3.1. Governing equations

For the geometry in Fig. 2, a one-dimensional coordi-nate system was used normal to the paper movement direc-tion. The conservation equations are [22,23]:

Moisture mass conservation including liquid and vapor

o /lql þ /vqv½ �os

þ o /lqlul þ /vqvuv½ �ox

¼ 0 ð1Þ

Fluid mixture mass conservation including liquid and gas

o ql/l þ qg/g

� �os

þo ql/lul þ qg/gug

� �ox

¼ 0 ð2Þ

Fig. 2. Coordinate system and force analysis on the paper.


Energy conservation

o

osð/lqlhl þ /vqvhv þ /aqaha þ /sqshsÞ

þ o

oxð/lulqlhl þ /vuvqvhv þ /auaqahaÞ

¼ o

oxkeff

oTox

� �ð3Þ

where Darcy’s law gives the gas and liquid velocities

ui ¼ �kiKli/i

opi

ox� qi g

��

i ¼ l; g ð4Þ

where

g� ¼ g cos h� gc ð5Þgc ¼ x2R ð6Þ

The forces acting the paper are shown in Fig. 2. gc is theforce due to the centrifugal acceleration while the papersheet is in the contact period, which is zero when the paperis in the free movement period. Running the AB contactdrying period, the gravitational acceleration and the cen-trifugal acceleration are in opposite directions, so Eq. (5)has the minus sign; otherwise, the positive sign is useddescribing the CD contact drying period.

For the diffusion of the air or vapor in the binary mix-ture, the air and vapor velocities are given by [24]

ui ¼ ug �qgDM aMv

qiM2g

o

oxpi

pg

!i ¼ v; a ð7Þ

The basic equations, Eqs. (1)–(4) and (7), are augmentedby the following relations:

(1) The volume of each phase is function of porosity andliquid saturation

/l ¼ /S; /v ¼ /a ¼ /g ¼ /ð1� SÞ; /s ¼ 1� /

ð8Þ
(2) The pressure of gaseous mixture is given by Dalton’s
law

pg ¼ pv þ pa ð9Þ

(3) The pressure of air is given by Clapeyron equation

pa ¼qaRTMa

ð10Þ

(4) On the assumption that paper sheet contains freewater everywhere during drying, the Clausius–Clapeyron equation for the vapor pressure is [25]� ��
pv ¼ pv1 exp �Mvhvap
R1

T� 1

T 1

ð11Þ

(5) Leerett’s formula for the capillary force is [26]ffiffiffiffir
pc¼ pg�pl¼
/K

rðT ÞJðSÞ

rðT Þ¼ r0�bT

JðSÞ¼ 0:364 1� exp½�40ð1�SÞ�f g

þ0:221ð1�SÞþ 0:005

S�0:08

ð12Þ

(6) The enthalpy–temperature linear relations is given by

hi ¼ cpiT þ Ci i ¼ a; v; l; s ð13Þwhere Ci is a constant for each phase.

(7) Effective thermal conductivity of porous media isgiven by [27]

keff ¼ ðkmg /g þ km

l /l þ kms /sÞ

1m m ¼ 0:25 ð14Þ

(8) Relative permeabilities is [17]� �
kl ¼
S � Sir

1� Sir

3

; kg ¼ 1� 1:1S ð15Þ

(9) Effective diffusivity is !
D ¼ 2:17� 10�5 101; 325
pg

T273:16

� �1:88

ð16Þ

3.2. Boundary and initial conditions

Different boundary conditions were used for the twopaper surfaces. When the sheet is in the contact drying peri-ods AB and CD, the side contacting the cylinder surface isimpermeable and on the heated surface so the moisture andair fluxes are zero:

moisture flux:

/lqlul þ /vqvuv ¼ 0 ð17Þ


air flux:

/aqaua ¼ 0 ð18Þand the cylinder surface temperature is assumed to be con-stant, so the energy boundary condition is

keff

oTox¼ T w � T

rf

ð19Þ

where rf is the gas film thermal resistance due to the air thatis drawn in the between paper and cylinder while the cylin-der is rotating.

On the exposed side pressed by the dry felt, the fluxes ofmass, energy, and pressure are continuous:

moisture flux:

/lqlul þ /vqvuv ¼akmMv

Rpv

T�

upv;sat

T atm

� �ð20Þ

heat flux:

keff

oToxþ /lqlulhvap ¼ bkTðT atm � T Þ ð21Þ

pressure:

pg ¼ patm ð22Þ

where a and b are the effective mass and heat transfer fac-tors due to the resistance of the dry felt for mass and heattransfer. Here both a and b are equal to 0.8 [28]. For tur-bulent flow, the Nusselt number was computed by [29]

Nu ¼ 0:023Re4=5Pr1=3 ð23ÞThe mass transfer coefficient at the convective surface wasgiven by [30]

km ¼kT

ðqcÞaLe�

23 ð24Þ

When the paper sheet is in the free movement periodsBC and DE, the boundary conditions on both sides ofthe sheet are the same. The boundary conditions are similarto Eqs. (20)–(22), but a and b are equal to one because bothsides of paper sheet are exposed to the air.

The initial conditions are:

T ðx; sÞjs¼0 ¼ T ini ð25Þpg

s¼0¼ patm ð26Þ

The initial moisture content distribution corresponds tothat of hydrostatic equilibrium

opc

ox

s¼0

¼ �qlg ð27Þ

Eqs. (4)–(16) can be used to express the governing Eqs.(1)–(3) together with the boundary and initial conditions asfunctions of the three state variables: the liquid saturation,S, the temperature, T, and the mixture gas pressure, pg. Theone-dimensional model is similar to that presented byTurner and Iliac [19], and Lu et al. [31,32]. The moisturecontent in wet basis during paper drying was derived by

MC ¼ /lql þ /vqv

/lql þ /vqv þ /sqs

ð28Þ

4. Numerical method

Governing equations (1)–(3) and Boundary conditionsequations (17)–(22) can be rewritten as the functions ofthe variables such as moisture content, MC, temperature,T, and gas pressure, pg, through using other relations men-tioned above. The numerical solution technique introducedto solve the system of equations was a fully implicit finitedifference scheme, which is known to yield stable solutionsfor any selection of space and time grid steps. The convec-tion terms in the energy equation were modeled using theupwind scheme with central differencing implemented forthe other terms. We choose grid number 43 in thicknessdirection and time step 0.01 s, which can ensure the solu-tion accuracy is independent of space and time steps.

For one-dimensional model, the calculation domaincontains J grid nodes. The discrete equations has the form

An � wn ¼ Bn � wn�1 þ Cn ð29Þwhere the two-dimensional coefficient matrices, An, is3J · 3J, one-dimensional coefficient matrices, Bn and Cn

are 3J. The variables are moisture content, MC, tempera-ture, T, and gas pressure, pg, written in vector form aswn ¼ ðMCn

0; Tn0; p

ng0; . . . MCn

J�1; TnJ�1; p

ngJ�1ÞT where n is the

number of time step. All coefficient matrices, An, Bn andCn are the function of wn. Therefore, Eq. (29) is non-linear.In order to solve the non-linear problem, a predictor–cor-rector method is implemented by evaluating the non-linearcoefficients at previous time step, n � 1

As�1 � ws ¼ Bs�1 � wn�1 þ Cs�1 ð30ÞLet wn�1 = ws�1 as initial value compute the coefficientmatrices, Eq. (30) become linear. The variable vector couldbe solved by Gauss elimination. Let s is the iteration step inthe n � 1 time step. Let ws as new value compute the coef-ficient matrices again, ws+1 could be obtained. Repeat theloop in n � 1 time step, we could get the predictor value,wn� of the variable wn as accuracy reaches convergence cri-terion. The convergence criterion for each state variable ineach node is

minwnþ1

j � wnj

wnj

$ % !6 1:0e� 6 ð31Þ

Using the variable predictor value, wn� , to calculate thecoefficient matrices of Eq. (29), it become linear. Therefore,the corrector value, wn, is solved. Following the loop men-tioned above, variables of n + 1 time step, wn+1, could besolved.

In the numerical simulation, the physical data for water,vapor, and air were given by Huang [24]. All of the param-eters describing the paper, and the paper machine are listedin Table 1, which are given by the laboratory of papermak-ing factory.

Table 1Parameters of paper sheet and paper machine

Type and symbol Value Unit

Moisture content (initial), MC 58 %Moisture content before coating, MC 8 %Moisture content after coating, MC 28 %Moisture content (final), MC 8 %Average temperature of paper sheet after coating, T 328 KLiquid saturation (initial), S 99 %Paper porosity, / 58 %Conductivity of solid fibers, ks 1.4 W K�1 m�1

Density of solid fibers, qs 800 kg/m3

Paper thickness, H 2.7 · 10�4 mIrreducible saturation, Sir 1 %Machine speed 330 m/minCylinder diameter, R 1.5 mDistance between two dryer axes, L 2.4 mCover angle of paper contacting the cylinder, c 240 �Total number of cylinders, N 78Coat location Before no. 68Gas film thermal resistance, rf 2.75 · 10�3 m2 K W�1

360

380

400


5. Experimental measure

The surface temperature of cylinder and the paper sheetwere measured by infrared measurement. The relativehumidity and temperature of air in gas pocket weremeasured by digital hygrothermograph. The infrared appa-ratus and digital hygrothermograph were calibrated fortemperature by a standard T type thermocouple. Digitalhygrothermograph was calibrated for relative humidityby a dry–wet bulb thermometer. All the experimental datawere measured at least three times in the same location likepoints A–H shown in Fig. 1 during steady running of papermachine. Note that we measured the surface temperatureof paper sheet from the felt side both at inlet and outlet.All the data show good reproducibility with the maximalrelative error 2%. The accuracy of the infrared apparatusFLUKE 65 made in American is 2K. The accuracy of thedigital hygrothermograph YOKOGAWA 245105 made inJapan is 1K and 1% for the temperature and the relativehumidity respectively.

The air temperature measured at center of gas pocketwas given by linear curve fit with shell temperature ofdryer, Tw,

T atm ¼ 25þ nT w n ¼ 0:502 N < 68

n ¼ 0:681 N P 68 ð32Þ

0 20 40 60 80

280

300

320

340

Tw

(K)

N (dimensionless)

Fig. 3. Experimentally measured cylinder shell temperatures.

The relative humidity measured at center of gas pocketwas given by linear curve fit with cylinder number, N.

u ¼

0:2þ N300

N 6 30

0:3� N � 30

70030 < N 6 68

0:2 68 < N 6 78

8>>>>><>>>>>:

ð33Þ

6. Results and discussion

The experimentally measured outer surface tempera-tures of each cylinder on an actual paper dryer, which isone of the most important factors affecting the dryingcapability of the paper dryer together with the temperatureand relative humidity of ambient air, is shown in Fig. 3. Onthe whole, the temperature is zigzag according to technicalrequirement of paper making. The cylinder temperaturesincrease before no. 30 and then remain small amplitudechange until the coating at no. 68. Coating is a manufac-ture process in paper-making where paper sheet run intoa pool filled with an aqueous colloidal solution whichresulting in moisture content of paper sheet dramaticallyincreasing (see Table 1). Therefore moisture content has asudden increase in 68th cylinder as showing in Fig. 10.The first cylinder after coating is cooler to prevent adhesionof the paper sheet to the cylinder surface. The cylinder tem-perature of the last cylinder in the dryer section is reducedto improve paper quality.

0 10 15 20 25290

300

310

320

330

340

350

360

370

380

Numerical

Experimental

T(K

)

N (dimensionless)

Relative Error: 3%

5

Fig. 4. Numerical and experimental sheet temperatures at shell inlets.

0 10 15 20 25300

310

320

330

340

350

360

370

380

Numerical

Experimental

T(K

)

N

Relative Error: 4%

5(dimensionless)

Fig. 5. Numerical and experimental sheet temperatures at shell outlets.

0 20 40 60 80

300

320

340

360

380

400

Inlet

Outlet

T(K

)

N (dimensionless)

Fig. 6. Predicted sheet temperatures at shell inlets and outlets.

0 20 40 60 80

-30

N (dimensionless)

-20

-10

0

10

20

30

Temperature rise in contact period Temperature fall in free movement period Total temperature change

ΔT(K

)

Fig. 7. Sheet temperature changes in different periods.

0 10 20 30 40 50 60 70 80

1.00

1.02

1.04

1.06Average mixture gas pressures at shell inletsAverage mixture gas pressures at shell outlets

Pg

/Pat

m

N (dimensionless)

Fig. 8. Average gas pressures in the sheet at the shell inlets and outlets.


The numerically predicted sheet temperatures at the cyl-inder inlets and outlets were compared with the experimen-tal data for cylinders 1–27 in Figs. 4 and 5. The numericalresults shows good agreement with experimental data forall of these cylinders. Figs. 4 and 5 show that the numericalresults of paper temperatures at the cylinder inlets and out-lets are close to the experimental results with relative error3% and 4% respectively. The difference between numericalresults and experimental results is acceptable within therange of experimental error. The good agreement betweennumerical results and experimental results indicates thatthe drying model of paper drying based on the fundamentalof heat and mass transfer in porous media is valid.

The predicted sheet temperatures, average gas pressuresin the porous media, and the drying rates for the wholepaper machine are presented in Figs. 6–9. As shown inFig. 6, sheet temperatures at the cylinder outlets are higherthan those at the cylinder inlets for major of cylinders. Thechanges of temperature depicted in Fig. 7 show that tem-peratures rise during contact period while they fall duringfree movement period. This phenomenon show that thesheet is heated by the cylinder during the contact dryingperiod and cooled during the free movement drying period.

The paper receives energy from the cylinder during the con-tact drying period that is divided in two parts. One part isused to evaporate the liquid on the felt side, which con-sumed by the evaporation latent heat, while the other partheats the paper itself causing the paper sheet temperatureto increase. At the beginning of the free movement drying

0 20 40 60 80

0

10

20

30

40

50

60 Inlet

Outlet

MC

(%)

N (dimensionless)

Fig. 10. Predicted moisture content at shell inlets and outlets.

0.0 0.5 1.0 1.5 2.0 2.5 3.032

34

36

38

40

42

44 404244464850

MC

(%)

x (×104 m)

Fig. 11. Moisture content distribution after contact drying period onupper dryers.

0 10 20 30 40 50 60 70 800

4

8

12

16

20

24 Average drying rate during contact drying periodAverage drying rate during free movement drying period

Dry

ing

Rat

e(k

g⋅h-1

⋅m-2)

N (dimensionless)

Fig. 9. Average drying rates during the contact drying period and the freemovement drying period.


period, the paper sheet has a relatively high temperature soevaporation is accelerated which cools the sheet resulting intemperature falling. As a whole, the process of paper dry-ing is an alternation of heating during contact period andcooling during free movement period.

The average gas pressure in paper sheet at the cylinderoutlet is much greater than that at the cylinder inlet(Fig. 8). Vapor transport is resisted by the cylinder surfaceand the dry felt which increases the gas pressure during thecontact drying period. The vapor easily escapes into the airpocket during the free movement drying period, whichreduces the gas pressure.

The average drying rate during the free movement dry-ing period shown in Fig. 9 is higher than that during thecontact drying period since the paper sheet temperatureand pressure at the beginning of the free movement periodare both relatively high and twice as much area is exposedto the gas pocket, so the moisture is more easily removedduring the free movement drying period than during thecontact drying period. According to the boundary condi-tion equation (20), the drying rate is depended on the sheetpressure and temperature, ambient temperature and rela-tive humidity, so the drying rate increases with the increaseof sheet temperature and ambient temperature, and withthe decrease of relative humidity of ambient. In practicalpaper drying there is an additional infrared heating methodbesides cylinder heating after coating process in practicaldrying process of paper which resulting in heat transfercoefficient increasing. The heat transfer coefficient increasesfrom 260 W m�2 K�1 before coating to 290 W m�2 K�1

after coating resulting in increasing temperature of sheet.Thus the drying rate increases with the increasing of heattransfer coefficient after the 68th cylinder.

The numerical results for the moisture content in Fig. 10show that the moisture content at each cylinder inlet ishigher than that at the cylinder outlets, which means themoisture evaporates during the contact drying period aswell as in the free movement drying period. Although tem-peratures of paper sheet shown in Fig. 6 and ambient

described by Eq. (32) increase, the moisture contentdecreases slowly before no. 30 due to the increase of the rel-ative humidity expressed by Eq. (33) and saturation pres-sure of vapor computed by Eq. (11). The moisturecontent decreases quickly due to the drying rate increasing(see Fig. 9) after no. 30.

All of the results show that the paper coating processbefore no. 68 intensely impacts the change history of mois-ture content, temperature, gas pressure, and drying ratesdue to the dramatic increase of moisture content andabrupt decrease of average temperature of paper sheet aftercoating (Table 1).

The distributions of the drying parameters: moisturecontent, MC, the temperature, T, and the gas pressure, pg

throughout the sheet thickness at the inlets and outletson the upper cylinders from the 40th cylinder to the 50thcylinder are shown in Figs. 11–16. The moisture contentdistributions in Fig. 11 show the large moisture gradientsthroughout the paper thickness after the contact dryingperiod. As expected, the moisture migrates from the shellside to the felt side during the contact drying period. The

0.0 0.5 1.0 1.5 2.0 2.5 3.0

34

36

38

40

42

44 40 42 44 46 48 50

MC

(%)

x (×104 m)

Fig. 12. Moisture content distribution after free movement drying periodfollowing upper dryers.

0.0 0.5 1.0 1.5 2.0 2.5 3.0368

369

370

371

372

373

374

375

40 42 44 46 48 50

T(K

)

x (×104 m)

Fig. 13. Temperature distribution after contact drying period on upperdryers.

0.0 0.5 1.0 1.5 2.0 2.5 3.0361.50

361.75

362.00

362.25

362.50

362.75

363.00

363.25

40 42 44 46 48 50

T(K

)

x (×104 m)

Fig. 14. Temperature distribution after free movement drying periodfollowing upper dryers.

0.0 0.5 1.0 1.5 2.0 2.5 3.01.00

1.01

1.02

1.03

1.04

1.05

1.06

1.07

1.08

1.09

1.10

40 42 44 46 4850

P g/P

atm

x (×104 m)

Fig. 15. Gas pressure distribution after contact drying period on upperdryers.

0.0 0.5 1.0 1.5 2.0 2.5 3.01.000

1.002

1.004

1.006

1.008

1.010

1.012 404244464850

P g/P

atm

x (×104 m)

Fig. 16. Gas pressure distribution after free movement drying periodfollowing upper dryers.


moisture content distribution shown in Fig. 12 at the endof the free movement drying period is quite uniform dueto the moisture migrates from sheet center to outside sur-

faces. The paper temperature on the shell side shown inFig. 13 is higher than on the felt side after the contact dry-ing period because paper sheet on the shell side easilyobtains heat and paper sheet on the felt side lose heatdue to evaporating while as shown in Fig. 14 the tempera-ture inside the paper is higher than on the two sides afterthe free movement drying period because of evaporatingon both sides. Figs. 15 and 16 show that the gas pressurehas the same trends as the temperature. The pressure onthe shell side shown in Fig. 15 is higher than on the felt sideafter the contact drying period because gas near the feltside easily diffuses to ambient while as shown in Fig. 16the pressure inside the paper is higher than on the two sidesafter the free movement drying period because of gas easilyescapes to ambient through both sides.

The drying parameter distributions are affected by sev-eral mechanisms. Firstly, the evaporation is zero on theshell side of the paper sheet since the cylinder shell is imper-meable, and the felt covering the other side of the papersheet resists the heat and mass transfer during the contactdrying period. The temperature and gas pressure near shell


side are higher, so the moisture moves from the shell side tothe felt side. Secondly, the evaporation accelerates athigher temperatures when the paper sheet moves into theopen draw. The latent heat of evaporation consumersdue to the phase change then reduce the paper temperature.Since the evaporation action is stronger at the two bound-aries than inside the paper, the temperatures are lower atthe two boundaries than in the middle. In addition, thegas pressure is higher in the middle than at the boundariesbecause gas diffusion is easier at the boundaries. The tem-peratures and pressures vary for the different cylinders dueto the different shell and air temperatures and the differentrelative humidities of the drying air. Note that the even cyl-inders are on the top, while the odd cylinders are on thebottom as shown in Fig. 1. The boundary conditions arereversed on the top and bottom cylinders with x = 0 onthe upper cylinders being the shell surface, while x = 0 onthe lower cylinders is the surface covered by the felt. Thevariations of all drying parameter of the paper sheet onthe bottom cylinders are similar to those on the topcylinders.

7. Conclusions

In paper industry, the heat and mass transfer duringpaper drying is generally evaluated with empirical orsemi-empirical formulas. This paper presents a model forpaper drying based on the fundamental equation for heatand mass transfer in porous media. The drying modelapplied to a running paper machine is solved by numericalmethod based on practical parameter which is measured byexperiment. The numerical and experimental results showthat:

1. The good agreement within the range of relative errorabout 4% between the numerical and experimentalresults of sheet temperature from the 1st cylinder to27th cylinder shows that the model can simulate thepaper drying process.

2. The temperature and gas pressure increase during thecontact period as the paper is heated, but the averagedrying rate is low because of the resistance of the cylin-der surface and the felt to vapor transport. The temper-ature and gas pressure fall during the free movementperiod due to the higher average drying rate.

3. Moisture content, temperature, and gas pressure gradi-ents occur in the paper sheet during drying. Duringthe contact drying period, moisture moves from the shellside to the felt side, with the temperature and gas pres-sure being higher on the shell side than on the felt side.After free movement drying period, the moisture distri-bution is uniform across the paper thickness while thetemperature and gas pressure are higher in the middlethan on the two boundaries.

The model can be used for the design of paper dryers,the adjustment of drying parameters such as shell temper-

ature, ambient temperature and relative humidity, and toevaluate paper drying performance using drying rate.Under given drying conditions and drying objective, theusing of paper drying model could check whether the num-ber of cylinder are enough or not. It is helpful for designerto optimize the number of cylinder in order to reduceinvestment of paper machine. It could predict the dryingperformance of paper machine according to changes ofdrying parameter such as shell temperature, relative humid-ity and temperature of ambient. It is guidable to guideoperator how to adjust the drying parameter to get the dry-ing objective.

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numerical and experimental investigation of paper drying: heat and mass transfer with phase change...

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