Chemical Engineering and Processing 43 (2004) 1555–1560

Heat and mass transfer in multicylinder dryingPart II. Analysis of internal and external transport resistances

Lars Nilsson∗

Division for Chemistry, Karlstad University, Chemical Engineering 65188, Karlstad, Sweden

Received 21 July 2003; received in revised form 5 March 2004; accepted 5 March 2004Available online 10 May 2004

Abstract

The present study investigates two models—two limiting cases—for the internal heat and mass transfer in a multicylinder dryer: thefirst model assumes that complete redistribution of heat and moisture in the thickness direction occurs instantaneously. The second modelassumes that moisture transfer occurs only by vapor diffusion and that heat transfer takes place by conduction and condensation. Thetwo models are written in dimensionless form, and applied for the same set of standard paper machine data. For basis weights belowapproximately 0.05 kg d.s./m2 the results are so close that the first, simple model is sufficient. A higher limiting basis weight—approximately0.16 kg d.s./m2—was previously indicated by comparison of model predictions, assuming the simple model, to extensive sets of machine datafrom four different paper machines.© 2004 Elsevier B.V. All rights reserved.

Keywords: Biot number; Fourier number; Mathematical model; Paper drying

1. Introduction

A simulation model lacking any description of the inter-nal heat and mass transfer processes was applied to a totalof 163 sets of production data from four paper machinesproducing a wide range of basis weights[1]. For three pa-per machines producing basis weights ranging from 0.056to 0.159 kg d.s./m2 the model predictions are adequate. Onlyfor the paper machine producing the heaviest grades (0.189to 0.390 kg d.s./m2) the model predictions are flawed by asystematic error, probably caused by significant internal re-sistances to heat and mass transfer within these thick sheets.Measured sheet moisture profiles during cyclic contact dry-ing for two basis weights (0.438 and 0.212 kg/m2) showa much more pronounced moisture profile for the thickersheet.[2].

In the present study, two different simulation models areapplied to a set of standard machine data. The first model as-sumes no gradients in temperature or moisture content in thethickness direction, so that the rates of internal heat and mass

∗ Tel.: +46-54-7002099; fax:+46-54-7002040.E-mail address: lars.nilsson@kau.se (L. Nilsson).

transfer are overestimated. The second model assumes inter-nal mass transfer to occur by vapor diffusion alone. Internalheat transfer then occurs by conduction and by condensationof the evaporated vapor. The second model underestimatesthe rates of internal heat and mass transfer, since capillaryflow and surface diffusion are not taken into account. Evi-dently, actual machine data must fall in between these twomodels. By comparing the results from the two models, thepresent study aims at establishing the range of basis weightsfor which assuming instantaneous internal redistribution ofheat and mass is a good enough assumption for successfullycalculating the machine speed/final moisture content.

2. Mathematical models

In order to give the results in dimensionless form, thedimensionless groups inEqs. (1)–(4)are introduced:

BiD = kconvG

2Deρ(1)

FoD = 4Detresidenceρ2

G2(2)

0255-2701/$ – see front matter © 2004 Elsevier B.V. All rights reserved.doi:10.1016/j.cep.2004.03.002

1556 L. Nilsson / Chemical Engineering and Processing 43 (2004) 1555–1560

NiE = U

kconvρCPF(3)

NiI = k

DeρCPF(4)

The Biot number (Eq. (1)) and the Fourier number for dif-fusion (Eq. (2)) are well known dimensionless groups. Thetwo Nilsson numbers (Eqs. (3) and (4)) compare heat diffu-sivity to mass diffusivity for the transfer processes externalto the web (Eq. (3)) and internal within the web (Eq. (4)).

Concerning convective heat and mass transfer between thesurface of the sheet and the air in the hood, Lewis analogy(Eq. (5)) is assumed to hold:

kconv = hconv

ρACPA(5)

The position in the thickness direction and the time arenow expressed in dimensionless form:

z̃ = 2zρ

GBiD (6)

t̃ = t

tresidenceFoD (7)

It should be mentioned that the two heat transfer coeffi-cientsU andhconv representing heat transfer from the con-densing steam and the air in the dryer hood respectivelymight well depend on the speed of the dryer. In the presentmodels, such phenomena were not accounted for, since thestudy aims only at comparing two different assumptions withrespect to the internal transport processes.

2.1. Set of standard data

For simplicity, a set of standard data was used for settingboundary conditions and intitial/final conditions regardingtemperature and moisture content for the drying process.The multi-cylinder dryer was assumed to consist of 40 cylin-ders, all of them heated. The wrap angle of the web wasassumed to be 225◦, and the relation between the time forpassing the draw and the time for contact with the cylinderwas assumed to be 0.40 (tdraw/tcontact = 0.40). The steamtemperature was assumed to be 130◦C, the temperature andmoisture content of the air in the hood was assumed to be80◦C and 15 mol%, respectively. With regard to the initialconditions, the moisture content of the sheet after the presssection was assumed to be 1.50 kg H2O/kg d.s. and the tem-perature 40◦C. After the dryer section 0.10 kg H2O/kg d.s.was assumed to remain in the web.

2.2. Model I: Instantaneous redistribution of heat andwater

Assuming now that complete redistribution of water andheat occurs in the thickness direction at every instant, the

heat and mass transfer processes during cylinder contact aremodeled inEqs. (8)–(11)below:

i− 1

nFoD ≤ t̃ ≤

(i− 1

n+ tcontact

tcontact+ tdraw

1

n

)FoD (8)

dX

dt̃= −1

2

ρV

ρBiD ln

(1 − xV,∞1 − x0

V

)(9)

dT

dt̃=

1/2NiEBiD(Tsteam− T)+ 1/2BiD(ρA/ρ)(CPA/CPF)

(T∞ − T)+ (�HV/CPF)(dX/dt̃)

1 +X(CPW/CPF)

(10)

x0V = 0.001316 exp

(18.3036− 3816.44

T + 227.03

)× exp((T + 273.15)exp(−15.03X− 1.37

√X− 3.41)

− exp(−13.53X− 2.90√X+ 2.90)) (11)

Eq. (8)states the interval of dimensionless times for whichthe sheet contacts cylinder numberi. During contact with thecylinder, evaporation occurs only at one side of the sheet,cf. Eq. (9). Heat transfer by conduction from the cylinderand by convection from the surrounding air is balanced byevaporation and accumulation of heat (Eq. (10)). The vapormole fraction of air in equilibrium with the paper materialis given inEq. (11) [3].(i− 1

n+ tcontact

tcontact+ tdraw

1

n

)FoD ≤ t̃ ≤ i

nFoD (12)

dX

dt̃= −ρV

ρBiD ln

(1 − xV,∞1 − x0

V

)(13)

dT

dt̃=

BiD(ρA/ρ)(CPA/CPF)(T∞ − T)+ (�HV/CPF)(dX/dt̃)

1 +X(CPW/CPF)(14)

While passing the free draw between two cylinders(Eq. (12)), evaporation occurs from both sides of the sheet,cf. Eq. (13). Heat transfer by convection occurs at bothsurfaces and is balanced by evaporation and accumulationof heat, cf.Eq. (14).

2.3. Model II: Internal vapor diffusion and heat conduction

In model II, differential balance equations for heat andmass are written for the drying web:

dX

dt̃= −ρV

ρBi2D

∂2

∂z̃2ln(1 − x0

V) (15)

∂T

∂t̃= 1

1 +X(CPW/CPF)(NiIBi2D(∂

2T/∂z̃2)

+ (�HV/CPF)(∂X/∂t̃)) (16)

L. Nilsson / Chemical Engineering and Processing 43 (2004) 1555–1560 1557

Assuming that water transport within the sheet takes placeby water vapor diffusion through stagnant air, the diffusionequation takes the form given inEq. (15). Heat transferwithin the sheet will occur by conduction but also by con-densation of the diffusing water vapor (Eq. (16)).

The two balance equations for water and heat must becombined with relevant boundary conditions for the phasesoccurring in the multicylinder dryer.

∂(ln(1 − x0V))

∂z̃

∣∣∣∣∣z̃=0

= 0 (17)

∂(ln(1 − x0V))

∂z̃

∣∣∣∣∣z̃=2BiD

= ln

(1 − xV,∞

1 − x0V |z̃=2BiD

)(18)

∂T

∂z̃

∣∣∣∣z̃=0

= −NiENiI(Tsteam− T |z̃=0) (19)

∂T

∂z̃

∣∣∣∣z̃=2BiD

= 1

NiI

ρA

ρ

CPA

CPF(T∞ − T |z̃=2BiD) (20)

As previously,Eq. (8)states the interval of dimensionlesstimes for which the sheet contacts cylinder numberi. Thetwo surfaces of the web (z̃ = 0) and (̃z = 2BiD) will alternatein being in contact with the cylinder. For cylinderi, it isassumed that the surface withz̃ = 0 is in contact with thecylinder.Eq. (17)states that the surface in contact with thecylinder is impermeable to mass transfer andEq. (18)statesthat no accumulation of mass occurs at the surface of theweb in contact with the air in the hood.Eq. (19)states thatthe heat flux from the drying cylinder is conducted into thesheet andEq. (20)finally states that no accumulation of heatoccurs at the surface of the web in contact with the air.

∂(ln(1 − x0V))

∂z̃

∣∣∣∣∣z̃=0

= −ln

(1 − xV,∞

1 − x0V |z̃=0

)(21)

∂(ln(1 − x0V))

∂z̃

∣∣∣∣∣z̃=2BiD

= ln

(1 − xV,∞

1 − x0V |z̃=2BiD

)(22)

∂T

∂z̃

∣∣∣∣z̃=0

= − 1

NiI

ρA

ρ

CPA

CPF(T∞ − T |z̃=0) (23)

∂T

∂z̃

∣∣∣∣z̃=2BiD

= 1

NiI

ρA

ρ

CPA

CPF(T∞ − T |z̃=2BiD) (24)

The range of dimensionless times when the sheet is pass-ing the free draw between cylindersi and i + 1 was previ-ously defined inEq. (12). The boundary conditions for thefree draw will be exactly the same for each free draw, stat-ing that no accumulation of mass (Eqs. (21) and (22)) orheat (Eqs. (23) and (24)) occurs at either boundary.

3. Transport coefficients

By searching the literature, a range of possible valueswas established for each transport coefficient (seeTable 1).

Table 1Literature data for internal and external transport coefficients in multi-cylinder drying.

Minimum value Maximum value Reference

hcond (W/m2K) 800 2300 [4]hcyl (W/m2K) 1000 4000hcont (W/m2K) 300 1200 [5]kconv (m/s) 0.02 0.06k (W/mK) 0.05 0.15 [6,7]De (m2/s) 3·10−7 8·10−6 [8]

Intermediate values were chosen, so that the convective masstransfer coefficient was taken askconv = 0.04 m/s, the effec-tive diffusivity of water vapor in paperDe = 4× 10−6 m2/sand the heat conductivity of paper ask = 0.09 W/mC. Thevalue for the overall heat transfer coefficientU from the con-densing steam to the paper surface is determined by threeheat transfer coefficients:

U =(

225

360hcond+ 225

360hcyl+ 1

hcont

)−1

≈ 400 W/m2 K

(25)

The two Nilsson numbers can now be calculated asNiE =0.02 andNiI = 0.04.

The TAPPI drying rates for woodfree printing and writ-ing papers at a steam temperature of 130◦C range quite lib-erally between 12 and 23 kg/m2 h with an average value of17 kg/m2 h [9]. Using the set of parameters given above thedrying rate according to model I is 21 kg/m2 h regardless ofthe basis weight. For a basis weight of 0.1 kg d.s./m2, thedrying rate according to model II is 15 kg/m2 h. Since thedrying rates calculated from the two models encompass atleast half of the range of reported TAPPI drying rates, thechosen set of parameters is acceptable.

4. Relevant machine data

In a previous study[1], the predicted machine speeds, as-suming model I, were compared to actual machine speedsfor a total of 163 sets of data from four different paper ma-chines. The slope of the line correlating predicted to actualmachine speeds can be interpreted as the slope when plot-ting the actual Fourier number versus the Fourier numbercalculated from model I:∂νI

∂νactual≈ ∂FoD,actual

∂FoD,1(26)

These data are given inTable 2.

5. Results and discussion

5.1. Analogy with diffusion process

The differential equation for unsteady diffusion in aslab, combined with convective mass transfer at the two

1558 L. Nilsson / Chemical Engineering and Processing 43 (2004) 1555–1560

Table 2Some data regarding the four paper machines

Paper machine Range of basis weights (kg d.s./m2) Average basis weight (kg d.s./m2) BiD∂νI

∂νactual≈ ∂FoD,actual

∂FoD,I

A 0.056–0.101 0.0785 0.87 1.0180B 0.065–0.118 0.0915 1.0 0.9542C 0.102–0.159 0.1305 1.5 0.9153D 0.189–0.390 0.2895 3.2 0.7557

boundaries, has well-established analytical solutions, whichcan be given in dimensionless form. For negligible internalmass transfer resistance (corresponding to model I for themulticylinder dryer) the analytical solution is:

C̃ = e−BiDFoD (27)

The case corresponding to model II for the multicylinderdryer involves taking into account diffusion within the slabas well as convective mass transfer at the boundaries. Theanalytical solution to this problem is:

C̃ = 2 ·∞∑i=1

sin(δi) · cos(δi · (1 − 2 · z̃))δi + sin(δi) · cos(δi)

· e−δ2i FoD

δi · tan(δi) = BiD (28)

The left graph inFig. 1 illustrates the Fourier number asa function of the Biot number for diffusion in a slab. Thedimensionless concentration is 0.0667 corresponding to theset of standard data for the multicylinder dryer. The rightgraph inFig. 1 illustrates the Fourier number as a functionof the Biot number for the multicylinder dryer, assumingNiE = 0.02 andNiI = 0.04 as outlined previously. Evi-dently, the results for the multicylinder dryer are completelyanalogous to the results for the diffusion problem. If internaltransport resistances are neglected—model I in the left graph(slab diffusion) and model I in the right graph (multicylinderdryer)—the Fourier number will be inversely proportionalto the Biot number. For low enough Biot numbers it is notnecessary to take into account internal transport resistances,

Fig. 1. Both graphs show the Fourier number as a function of the Biot number, the left graph for slab diffusion and the right for multicylinder drying.Model I excludes internal transport resistances in both cases, whereas model II includes internal as well as external transport resistances in both cases.

neither for slab diffusion nor for multicylinder drying. Theconcept of a limiting basis weight below which model I isapplicable for multicylinder drying has a sound theoreticalbase. At high Biot numbers the Fourier number will take aconstant value. There is a limiting Biot number (sometimesgiven asBiD ≈ 0.1 for slab diffusion), above which inter-nal transport resistances should be taken into account. Forthe multicylinder dryer, this limiting Biot number will alsodepend on the two Nilsson numbers.

5.2. Results relevant to the multicylinder dryer

Assume that the Fourier number calculated according tomodel II (FoD,II ) is plotted as a function of the Fourier num-ber calculated according to model I (FoD,I ). The slope of thetangent to that curve is∂FoD,II /∂FoD,I . For low Biot numbers(FoD,I = FoD,II ) naturally the slope of this tangent will be 1.Fig. 2shows this model comparison for different Biot num-bers (left handy-axis) and also the data for the four papermachines (right handy-axis). For the four paper machines∂FoD,actual/∂FoD,I is given (seeTable 2). Since model I over-estimates and model II underestimates the internal transportrates it would be expected that

∂FoD,II

∂FoD,I<∂FoD,actual

∂FoD,I< 1 (29)

Comparison of model predictions to machine data pre-viously indicated that model I was acceptable for papermachine C, but not for paper machine D. Therefore, the

L. Nilsson / Chemical Engineering and Processing 43 (2004) 1555–1560 1559

Fig. 2. Establishing the limiting value for the Biot number and the basis weight in multicylinder drying.

criterion for limiting basis weight was chosen as:

∂FoD,II

∂FoD,I> 0.85. (30)

Based on comparing the two models (models I and II), thelimiting Biot number can be established as approximately0.5 leading to a limiting basis weight of 0.05 kg d.s./m2.

6. Conclusions

Two models for the heat and mass transfer processes inthe multicylinder dryer were compared—the first exclud-ing and the second including models for heat and masstransfer within the sheet. For low basis weights—low Biotnumbers—both models give the same results for residencetime and drying rate. Using a set of transport coefficientsleading to realistic drying rates, the limiting basis weight isestablished as approximately 0.05 kg d.s./m2, which is lowerthan the value of 0.16 kg d.s./m2, established previouslyby analysis of a number of sets of machine data from fourdifferent paper machines. Still, the mathematical model in-cluding internal heat and mass transfer (model II) probablyunderestimates the mass transfer rate within the web sincecapillary flow and surface diffusion are not taken into ac-count. Also, the heat transfer rates within the paper sheetmight be underestimated in model II, since a representativevalue for the heat conductivity of dry paper was taken.

Appendix A. Nomenclature

C̃ dimensionless concentrationCPA specific heat of air (J/kg◦C)CPF specific heat of cellulose (J/kg◦C)

CPW specific heat of water (J/kg◦C)De effective diffusivity of vapor in paper (m2/s)G basis weight of sheet (kg d.s./m2)�HV enthalpy of vaporization (J/kg)hcont heat transfer coefficient cylinder surface—web

(W/m2 ◦C)hconv heat transfer coefficient air—web (W/m2 ◦C)hcyl heat transfer coefficient of the cylinder shell

(W/m2 ◦C)k thermal conductivity of paper (W/m◦C)kconv mass transfer coefficient air—web (m/s)MW molar mass of water (kg/mol)ρ density of the fiber matrix (kg/m3)ρA density of air (kg/m3)ρV density of water vapor (kg/m3)t time (s)tcontact time for one cylinder contact (s)tdraw time for passing a free draw (s)T temperature of web (◦C)Tsteam temperature of steam (◦C)T∞ temperature of air in the dryer hood (◦C)tresidence residence time in the paper machine (s)t̃ dimensionless timeU heat transfer coefficient steam—paper surface

(W/m2 ◦C)v machine speed (m/s)xV,∞ mole fraction of vapor of air in the dryer hoodx0

V mole fraction of vapor of saturated airX moisture ratio of web (kg H2O/kg d.s.)z thickness coordinate (m)z̃ dimensionless thickness coordinate

Dimensionless groupsBiD Biot number for diffusion, cf.Eq. (1)FoD Fourier number for diffusion, cf.Eq. (2)

1560 L. Nilsson / Chemical Engineering and Processing 43 (2004) 1555–1560

NiE external Nilsson number, cf.Eq. (3)NiI internal Nilsson number, cf.Eq. (4)

References

[1] L. Nilsson, Heat and mass transfer in multicylinder drying. Part I:Analysis of machine data, in press.

[2] J.T. Paltakari, Experimental study of the effect of dryer configurationon internal temperature and moisture profiles in the sheet during cycliccontact drying, Paper 314, in: Proceedings of the 12th InternationalDrying Symposium, Noordwijkerhout, The Netherlands, 2000.

[3] M. Karlsson, M. Soininen, The influence of the hygroscopic propertiesof paper on the transient phenomena during contact drying of paperwebs, in: J.C. Ashworth (Ed.), Proceedings of the Third International

Drying Symposium, vol. 1, Drying Research Limited, Wolverhampton,1982, pp. 494–503.

[4] A. Ali, Improving dryer performance: converting to stationary siphonswith turbulence bars, Tappi J. 82 (1999) 97–102.

[5] B. Wilhelmsson, L. Nilsson, S. Stenström, L. Fagerholm, An exper-imental study of contact coefficients in paper drying, Tappi J. 77(1994) 159–168.

[6] I. Kartovaara, R. Rajala, M. Luukkala, K. Sipi, in: Punton (Ed.),Conduction of Heat in Paper, Papermaking Raw Materials, vol. 1,1985, pp. 381–412.

[7] R.J. Kerekes, A simple method for determining the thermal conduc-tivity and contact resistance of paper, Tappi 55 (1980) 1697–1700.

[8] L. Nilsson, S. Stenström, A study of the permeability of pulp andpaper, Int. J. Mult. Flow 23 (1997) 131–153.

[9] Reese, R.A. (1988). Revised TAPPI drying rate curves, Proceedingsof the TAPPI Engineering Conference, pp. 459–472.