Heat and moisture transport in a paper sheet moving over a hot printsurface

Paula Andrea Marin Zapataa, Maurice Fransenb, Jan ten Thije Boonkkampc, Louis Saesd

aDepartment of Mathematics and Computer Science, Eindhoven University of Technology, 5600 MB Eindhoven, TheNetherlands

bMesoscopic Transport Phenomena, Department of Applied Physics, Eindhoven University of Technology, 5600 MBEindhoven, The Netherlands

cCentre for Analysis, Scientific computing and Applications (CASA), Department of Mathematics and ComputerScience , Eindhoven University of Technology, 5600 MB Eindhoven, The Netherlands

dOce Technologies B.V., 5914 CA Venlo, The Netherlands

Abstract

The generation of moisture-induced deformations during the final use of paper is of concern for theprinting industry. Knowing the moisture profiles during printing is a primary step in order to understandsuch deformations. In this paper we derive a mathematical model to describe the heat and moisturetransport in paper, and use this model to study the moisture gradients induced on a paper sheet by aprinter from Oce Technologies, which contains a hot print surface. The model regards paper as a porousmedium saturated with air, and it considers heat and moisture transport along the length and thicknessdirections of the sheet. Several physical parameters are determined experimentally; others are takenfrom literature. The system of coupled differential equations is solved iteratively, and the numericalresults are also used to identify the relevant driving forces in the temperature-induced moisture changesin paper.

Keywords: Porous medium; Paper; Heat transport; Moisture transport; Printer

1. Introduction

The dimensional stability of paper, i.e., the property to maintain its original dimensions while beingused, is of importance for the printing industry. During printing, heating and adding ink/toner canchange the moisture content of the paper sheets, thereby creating moisture gradients. These gradientsare undesirable since they induce local expansion or compression of the fibers, generating mechanicalinstabilities like curls and cockles [1, 2]. Such instabilities affect the quality of the prints, and since theycan cause paper jams, they also affect the productivity of the printers.

To understand the moisture-induced deformation of paper, the heat and moisture transport processesneed to be understood first. Several authors have derived moisture transport models for the final useof paper. Dano and Bourque [3] modeled the moisture diffusion through the thickness of a paper sheet,using their model to predict the formation of curls. In their work, paper was modeled as a homoge-neous medium, but as noted in [4], better results are obtained by modeling paper as a porous medium.More elaborate models considering paper as a composite of pores and fibers were presented in [5], [6],

Email addresses: pmarin@limebv.nl (Paula Andrea Marin Zapata ), m.a.l.j.fransen@tue.nl (Maurice Fransen ),tenthije@win.tue.nl (Jan ten Thije Boonkkamp), louis.saes@oce.com (Louis Saes)

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[7] and [8]. Although these models showed good agreement with experiments, they were restricted toisothermal conditions, which do not always hold true for the printing practice. Few non-isothermalmodels for moisture transport at the end use of paper have been developed. Foss et al., [9] modeled theheat and moisture transport in a paper sheet in response to variations of the relative humidity in thesurroundings of the sheet. They studied the temperature changes induced by water sorption, but theimpact of temperature on the evolution of the moisture content was not clearly established. In orderto study the moisture changes at the final use of printing paper, a model that accounts for moisturetransport in response to changes in temperature still needs to be developed.

In this paper we derive a model to describe the heat and moisture transport in a paper sheet inresponse to temperature variations. By choosing appropriate boundary conditions, the model is usedto describe the heat and moisture transport in a paper sheet inside a printer of Oce Technologies,which contains a hot print surface. In such a printer, moisture-induced mechanical instabilities can beobserved during printing at the higher range of relative humidity. In order to gain insight in the causeof these instabilities, the model is used to estimate the temperature and moisture gradients induced bythe printer, and to study how variations in physical and printing parameters affect those gradients. Theremainder of the paper is organized as follows: Section 2 presents a detailed derivation of the governingequations for heat and moisture transport in paper. Based on these equations, Section 3 introduces atwo-dimensional model of the target printing environment. Section 4 presents the experimental deter-mination of some of the model parameters. Section 5 describes the numerical solution of the model, andthe numerical results are presented and discussed in Section 6. Finally, Section 7 presents a summaryand conclusions.

2. Governing equations for heat and moisture transport in paper

Paper is a porous medium composed of cellulose fibers and often fillers and additives that areembedded in a complex solid matrix. A framework to model transport in porous media was derived byBear and Bachmat [10] with the method of Representative Elementary Volumes (REV). Following theirapproach, paper can be idealized at the macroscopic level as consisting of two overlapping continua: Asolid matrix, mainly composed of cellulose fibers, and a pore space saturated with air. The pore spacecan contain water vapor, and the solid matrix can contain liquid water in an adsorbed state, bondedby physicochemical forces to cellulose fibers. At relative humidities (RH) above 80% water might alsocondensate as bulk liquid [11], but we neglect this effect since the operation range for the target printeris restricted to RH< 80%. In this section we derive mass balances of water and heat balances for thesolid paper matrix and the pore space.

2.1. Mass balances

If we denote the pore space by the subscript p and the solid matrix by the subscript f (fibers),neglecting possible convection, and assuming the volumetric fraction of pores and fibers to be constantin time and space, the balances of water in paper are given by

η∂Cp

∂t= η∇ ·

(Deff

p ∇Cp

)+ ηΓp − Spf , (1)

(1− η)∂Cf

∂t= (1− η)∇ ·

(Deff

f ∇Cf

)+ (1− η)Γf + Spf , (2)

where Cp [kg m−3] is the concentration of water vapor in the pores, Cf [kg m−3] is the concentrationof bonded water in the fibers, 0 < η < 1 [-] is the volumetric fraction of pore space, also known as theporosity of the paper sheet, (1− η) [-] is the volumetric fraction of fibers, the diagonal tensors Deff

p and

2

Defff [m2 s−1] are the effective diffusivities of water vapor in the pores and the fibers, respectively, Γp

is the rate of production of water in the pores per unit volume of pores, Γf is the rate of production ofwater in the fibers per unit volume of fibers, and Spf [kg m−3 s−1] is the rate of mass exchange frompores to fibers per unit volume of porous medium, in this case by sorption or desorption.

Neglecting possible addition of ink leads to Γp = Γf = 0. We also assume the diffusion of bondedwater along the fibers network to be negligible. This assumption is based on the diffusivity values alongthe thickness direction of a paper sheet reported by Bandyopadhyay [5] and Foss [9] (of the order of10−14 [m2 s−1] ). For diffusivities of the order of 10−14 to 10−12 [m2 s−1] the characteristic diffusiontime would be of the order of hours for the thickness direction, or higher for the in-plane directions.These diffusion time scales are much larger than the time scales of interest during printing (in the orderof a minute).

The water sorption on paper fibers has been successfully modeled by means of a linear driving force[5, 6, 9, 12]. Following this approach, the term Spf can be expressed as

Spf = (1− η)Kmi (Cf,equ − Cf) , (3)

with Cf,equ [kg m−3] the water concentration in the fibers that would be in equilibrium with the localvapor concentration in the pores, and Kmi [s−1] an internal (pores to fibers) mass transfer coefficient.This coefficient accounts for the resistance to water vapor adsorption as well as the internal area availablefor contact between pores and fibers. An explicit relation between Cf,equ and Cp can be derived byusing a sorption isotherm, i.e., a relation between the equilibrium moisture content in a material andthe relative humidity at a given constant temperature. One of the isotherms that has been reported tofit data for paper well is the Guggenheim-Anderson-De Boer or GAB-isotherm [11, 13]. It is given by

Xequ =(XmCK)RH

(1−KRH)(1−KRH + CKRH), (4)

where RH [-] is the relative humidity; Xequ [-] is the equilibrium moisture content on a dry basis, i.e.,the mass of water per unit mass of dry paper; and C [-], K[-] and Xm[-] are the GAB-parameters. Theseparameters are temperature dependent. For a given temperature they are determined experimentally byfitting (4) to equilibrium moisture data, measured at that temperature, for different relative humidities.Since for paper the adsorption isotherms can differ from the desorption isotherms [14], the GAB-parameters also depend on the history of the sorption process. However, for simplicity, hysteresiseffects are not considered. The equilibrium moisture content Xequ is related to the water concentrationin pores and fibers by

Xequ =ηCp + (1− η)Cf,equ

ηρp + (1− η)ρf≈ Cf,equ

ρf, (5)

where ρp and ρf [kg m−3] are the density of air and dry fibers, respectively. Since ηCp ∼ 10−3,(1− η)Cf,equ ∼ 101, ηρp ∼ 10−1 and (1− η)ρf ∼ 102, the approximation in the right hand side is basedon the fact that ηCp � (1− η)Cf,equ and ηρp � (1− η)ρf .

The relative humidity RH in (4) is defined as the ratio of the partial pressure of water vapor in air tothe saturated vapor pressure of water at a given temperature. By using the ideal gas law, RH can beexpressed in terms of the water concentration in the pores as

RH =Cp

Csp

, (6)

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with Csp the saturated water vapor concentration in air. Cs

p can be obtained as a function of thetemperature from a straightforward substitution of the ideal gas law in the Arden-Buck’s relation [15]as follows

Csp =

A1

Texp

[−A2 +A3T −A4T

2

T −A5

], (7)

with T the temperature in Kelvin, and the constants A1 to A5 given by A1 = 1.32 [K], A2 = 5.42× 103

[K], A3 = 21.01 [-], A4 = 4.26× 10−3 [K−1] and A5 = 16.01 [K].

After substituting (3) in (1) and (2), and taking into account that Γp, Γf and Defff equal zero, the

model for moisture transport in paper reads

∂Cp

∂t= ∇ ·

(Deff

p ∇Cp

)− (1− η)Kmi

η(Cf,equ − Cf) , (8)

∂Cf

∂t= Kmi (Cf,equ − Cf) , (9)

where Cf,equ = Cf,equ (Cp, T ) is the nonlinear function of Cp and T given by (4), (5) and (6), withCs

p = Csp(T ) defined in (7) and K, C and Xm implicit functions of the temperature (see Table 1).

2.2. Heat balance

The heat balance of a porous medium saturated with one fluid (in this case air) and with a ther-moelastic solid matrix was derived in [10]. If there is no convective flow through the porous mediumand for relatively small solid grains, the fluid and the solid matrix can be assumed to be in thermalequilibrium. With this assumption, the heat balance on paper can be written as

∂

∂t

((ηρpcv + (1− η)ρfcf)T

)= ∇ · (Λ∇T ) + ηΓp,h + (1− η)Γf,h + (1− η)ν

∂εf∂tT, (10)

where cv [J kg−1 K−1] is the heat capacity at constant volume of the air inside the pores, cf [J kg−1

K−1] is the heat capacity of the fibers, ρp and ρf [kg m−3] are the density of the air inside the pores andof the fibers, respectively, the diagonal tensor Λ [J m−1 s−1 K−1] is the thermal conductivity of paper,Γh,p [J m−3 s−1] is the rate of heat production in the pores per unit volume of pores, Γh,f [J m−3 s−1]is the rate of heat production in the fibers per unit volume of fibers, εf [K−1] is the solid’s dilatation,ν [J m−3] is the thermoelastic coefficient of the fibers network, and T is the temperature of the papersheet, which is assumed to be the same for pores and fibers.

The term ηρpcv + (1 − η)ρfcf represents the volumetric heat capacity of paper. In practice, it isobtained by multiplying the density of paper with its specific heat capacity, both of them determinedexperimentally. Thus, this term can be rewritten as

ηρpcv + (1− η)ρfcf = cρ, (11)

with ρ [kg m−3] the density of paper and c [J kg−1 K−1] the specific heat capacity of paper.

Γh,p and Γh,f are heat sources associated with the adsorption and desorption of water molecules onthe fibers. Since we assumed the bonded water diffusivity to be negligible, the concentration in thefibers can only change due to water adsorption or desorption. Therefore, the heat sources can beexpressed in terms of the rate of change of Cf as follows

ηΓh,p + (1− η)Γh,f = −(1− η)∆Hads∂Cf

∂t, (12)

4

with ∆Hads < 0 [J kg−1] the specific enthalpy of adsorption of water vapor on the fibers. ∆Hads is afunction of Cf [16]. At lower Cf -values the adsorbed water molecules are mainly bound to the solid’smolecules. However, as Cf increases, water molecules also bind other water molecules creating a multi-layer. Since the binding energy for molecules bound to other water molecules is lower than for thosedirectly in contact with the solid [17], the magnitude of ∆Hads decreases with increasing Cf . A curverelating ∆Hads and Cf is presented in Section 4 together with other model parameters.

From (10) to (12), neglecting thermal expansion (ν = 0), and assuming ρ and c to be constant intime, the heat balance of paper reads

ρc∂T

∂t= ∇ · (Λ∇T )− (1− η)∆Hads

∂Cf

∂t, (13)

with ∆Hads = ∆Hads(Cf) (see Figure 2).

2.3. Dimensionless equations

The heat and mass balances are made non-dimensional by introducing the dimensionless variablesCp, Cf , T , x and t as follows

Cp =Cp

Cp,ref, Cf =

Cf

Cf,ref, T =

T

Tref, x =

x

`, t =

Drefp t

`2,

with Cp,ref , Cf,ref , Tref the highest possible values of Cp, Cf and T , respectively, ` a characteristiclength, and Dref

p a reference diffusivity. In terms of the dimensionless variables, equations (8), (9) and(13) read

∂Cp

∂t= ∇ ·

(Dp∇Cp

)−Np

(Cf,equ − Cf

),

∂Cf

∂t= Nf

(Cf,equ − Cf

), (14)

∂T

∂t= ∇ ·

(Le∇T

)−∆Hads

∂Cf

∂t,

with Np and Nf dimensionless constants, Dp the dimensionless water vapor diffusivity tensor, Le the

Lewis number tensor for thermal diffusion, and ∆Hads the dimensionless enthalpy of adsorption. Theyare given by

Nf =`2Kmi

Deffp

, Np =(1− η)Cf,ref

ηCp,refNf , Dp =

Deffp

Drefp

, Le =Λ

Drefp ρc

, ∆Hads =(1− η)Cf,ref

ρcTref∆Hads.

The remainder of this paper will refer to dimensionless variables unless stated otherwise. For ease ofnotation, the hat-signs will be omitted.

3. Mathematical model of the printing environment

A schematic representation of a paper sheet in the target printing environment is shown in Figure 1.When paper is printed, it is moved along two metallic surfaces. The first surface, defined as the printsurface (ps), is where printing takes place. It has a constant homogeneous temperature Tps and lengthLps. The second surface, defined as the drying surface (ds), is used to keep the sheet straight while the

5

z

z = Lz

z = 0

Ly

y = 0 Yps(t)Lps Lds

y

Paper sheet

Print surface Drying surface

Figure 1: Coordinate system in the printing environment.

toner dries. It has the same temperature as the bulk environment (T∞), and its length is denoted byLds.

While being printed, the sheet is moved instantaneously over a distance Ystep after every time intervaltstep. This movement is modeled using the coordinates of the paper sheet as the reference frame, i.e.,the point y = 0 is always located at the left corner of the sheet, while the position of the print and thedrying surfaces change in time. In this way, the y-coordinate of the left extreme of the print surfaceYps(t) is given by

Yps(t) = −Ystep

⌊t

tstep

⌋+ Yps(0),

with b c the floor function and Yps(0) the initial position of the print surface.

Since the thickness of the paper sheet is much smaller than its width and length, the area availablefor heat and mass exchange on the lateral surfaces of the sheet is negligible compared to the area onthe top and bottom surfaces. Thus, we can assume the lateral surfaces to be isolated. Taking thisinto account, and assuming there are no other factors inducing gradients along the width of the sheet(x-direction), we can describe the heat and moisture transport in a sheet on the printing environmentwith a two-dimensional model in y and z. In terms of the dimensionless variables, this model is givenby the two-dimensional version of the system in (14), i.e.,

∂Cp

∂t= Dp,y

∂2Cp

∂y2+Dp,z

∂2Cp

∂z2−Np (Cf,equ − Cf) , (15)

∂Cf

∂t= Nf (Cf,equ − Cf) , (16)

∂T

∂t= Ley

∂2T

∂y2+ Lez

∂2T

∂z2−∆Hads

∂Cf

∂t, (17)

with Dp,y and Dp,z the in-plane and through-plane dimensionless diffusivities of water vapor in paper,respectively, and similarly for the Lewis numbers Ley and Lez. The boundary conditions at y = 0 andy = Ly are given by

∂Cp

∂y= 0,

∂T

∂y= 0 at y = 0 and y = Ly.

Due to the relative movement of the print and drying surfaces, the bottom of the paper sheet can belocally exposed to three different temperature and moisture boundary conditions: Being over the print

6

surface, being over the drying surface, or being exposed to the environment. At the top surface of thesheet, water vapor and heat are transported by convection. The boundary conditions at the bottomand top of the sheet for the variable Cp are given by

∂Cp

∂z= Bi−m(Cp − C∞p ) at z = 0,

∂Cp

∂z= Bi+m(C∞p − Cp) at z = Lz,

where C∞p is the dimensionless water vapor concentration in the bulk environment, Lz is the dimen-

sionless thickness of the sheet, and Bi−m and Bi+m are the Biot numbers for mass transfer at the bottomand top of the sheet, respectively. They are given by

Bi−m =LzK

−m

Deffp,z

, Bi+m =LzK

+m

Deffp,z

,

where K+m, with units [m s−1], is a convective mass transfer coefficient at the top of the sheet, and K−m ,

with units [m s−1], is the mass transfer coefficient at the bottom of the sheet. K−m = 0 wherever thesheet is in contact with the print or the drying surface, and we assume K−m = K+

m wherever the bottomof the sheet sheet is exposed to the environment, i.e.,

K−m(t, y) =

K+m if y < Yps(t),

0 if Yps(t) ≤ y < Yps(t) + Lps + Lds,K+

m if y ≥ Yps(t) + Lps + Lds.

The temperature boundary conditions at the bottom and top of the sheet are given by

∂T

∂z= Bi−h (T − T−) at z = 0,

∂T

∂z= Bi+h (T∞ − T ) at z = Lz,

where T∞ and T− are the dimensionless temperatures of the bulk environment and the material presentat the bottom of the sheet, respectively. T− = Tps wherever the paper sheet is in contact with the printsurface, and T− = T∞ otherwise, i.e.,

T−(t, y) =

T∞ if y < Yps(t),Tps if Yps(t) ≤ y < Yps(t) + Lps,T∞ if y ≥ Yps(t) + Lps(t).

Bi−h and Bi+h are the Biot numbers for heat transfer at the bottom and top of the sheet, respectively.They are given by

Bi−h =LzK

−h

Λz, Bi+h =

LzK+h

Λz,

where K+h , with units [W m−2 K−1], is a convective heat transfer coefficient at the top of the sheet,

and K−h , with units [W m−2 K−1], is the heat transfer coefficient at the bottom of the sheet. K−h =hc wherever the paper is in contact with the print or drying surfaces, with hc the contact thermalconductance between paper and these metallic surfaces. We assume K−h = K+

h wherever the paper isexposed on both sides to the environment, i.e.,

K−h (t, y) =

K+h if y < Yps(t),

hc if Yps(t) ≤ y < Yps(t) + Lps + Lds,K+

h if y ≥ Yps(t) + Lps + Lds.

7

The dimensionfull parameters K+h and K+

m are obtained from heat and mass transfer correlations forconvective flux. K+

h is calculated based on the following expression for the average Nusselt number(Nu) in a laminar boundary layer over a flat plate [18]:

Nu =K+

h L

Λa= 0.664Re1/2Pr1/3, (18)

with the Reynolds number (Re), the Prandtl number (Pr), and the plate length L given by

Re =ρaV

∞L

µa, Pr =

µacaΛa

, L = Lps + Lds.

V∞ [m/s], ρa [kg m−3], µa [kg m−1 s−1], ca [J kg−1 K−1], and Λa [W m−1 K−1] are the (dimensionful)bulk velocity, density, dynamic viscosity, heat capacity and thermal conductivity of the air over theprint surface, respectively. The correlation in (18) is valid for incompressible flows with uniform heatflux or temperature, and satisfying Re < 2 × 105 and 0.6 ≤ Pr ≤ 50. Although neither the heat fluxnor the temperature are uniform for this application, we use this correlation to obtain an estimate ofK+

h since no experimental value is currently available for this parameter.

Using the analogy between heat and mass transfer, K+m is obtained from (18) by replacing the Prandtl

number by the Schmidt number (Sc), and the average Nusselt number by the average Sherwood number(Sh) [18]. The resulting correlation reads

Sh =K+

mL

Da= 0.664Re1/2Sc1/3, (19)

with Da [m2 s−1] the diffusivity of water vapor in air, and Sc given by Sc = µa/ (ρaDa).

4. Model parameters and experiments

Different experiments are executed in order to determine some of the transport parameters involvedin the model. All experiments are conducted with commercial printing paper provided by Oce Technolo-gies. The specific heat (c) and the in-plane and through-plane thermal conductivities (Λy and Λz) aredetermined simultaneously with the Transient Plane Source Hot-Disc method, following the ISO/DIS22007-2.2 protocol [19]. The sorption isotherms are measured at T = 20, 25 and 35◦C by means of thewell known gravimetric method. The corresponding GAB-parameters are found by fitting (4) to theisotherms with the Least Squares method. These parameters are presented in Table 1. The enthalpyof adsorption (∆Hads = ∆Hads(Cf)) is found from the sorption isotherms at several temperatures asdescribed in [20]. The resulting curve is presented in Figure 2. The through-plane effective diffusiv-ity (Deff

p,z) is found by means of a diffusion cup experiment consisting of a minor modification of the

diffusion stack introduced in [21]. The in-plane water vapor diffusivity (Deffp,y) is estimated from Deff

p,z

and the in-plane and through-plane tortuosities, τpropxy and τprop

z , reported in [22] for n-propanol vapor

in bleached kraft paper board as Deffp,y = Deff

p,z

(τpropz /τprop

xy

). The remaining physical parameters are

obtained from literature. Table 2 presents the values of the physical parameters involved in the modeland the parameters describing the printing environment.

5. Numerical solution

The differential equations derived in Section 3 are discretized following the method of lines (MOL)approach, i.e., the spatial derivatives are discretized first, and the resulting ODE system is subsequently

8

T [oC] K C Xm

20 0.83 133.92 0.036125 0.82 29.107 0.036635 0.865 39.09 0.0329

Table 1: GAB sorption isotherm parameters for different temperatures.

60 80 100 120 140 160 1802.4

2.6

2.8

3

3.2

3.4

3.6

3.8x 10

6

-∆H

ads[J/kg]

Cf [kg/m3]

Figure 2: Specific heat of adsorption as a function of the concentration of water in the fibers.

integrated in time. The spatial derivatives are approximated using central finite differences, and the timeintegration is performed with the θ-method [24]. For a rectangular grid with Ny points in y-directionand Nz points in z-direction, the discrete version of equations (15), (16) and (17) reads, respectively,(

I− θ∆tAp

)c

n+1

p + θ∆tNp

(c

n+1

f,equ − cn+1

f

)=(I + (1− θ)∆tAp

)c

n

p (20)

−(1− θ)∆tNp

(c

n

f,equ − cn

f

)+ bp,(

1 + θ∆tNf

)c

n+1

f − θ∆tNfcn+1

f,equ =(

1− (1− θ)∆tNf

)c

n

f + (1− θ)∆tNfcn

f,equ, (21)(I− θ∆tAT

)T

n+1

+(θ∆H

n+1

ads + (1− θ)∆hn

ads

)∗(c

n+1

f − cn

f

)= (22)(

I + (1− θ)∆tAT

)T

n

+ bT.

In these equations, ∆t is the time step used for time integration, θ is an adjustable parameter varyingbetween 0 and 1, I is the identity matrix of size NyNz, and the operator * is the Hadamard or entrywisematrix product [25]. cp, cf and T are vectors containing the values of Cp, Cf and T at all grid points,respectively. The superscript n is used to indicate the value of a given vector at the nth time level, andsimilarly for n+ 1. Thus, the vectors c

n+1

p , cn+1

f and Tn+1

are the unknowns of the system. The vectorcequ is used to denote the non-linear function Cf,equ(Cp,T ) evaluated at all grid points, and similarlyfor ∆hads. The matrices AP and AT are block tri-diagonals resulting from the space discretization ofequations (15) and (17), respectively. bp and bT are vectors originating from the boundary conditions.

9

Physical parametersParameter ValueLz 1.1×10−4 mLy 0.25 mρ 818 kg m−3

ρf 1500 kg m−3 [1]η a 0.47Deff

p,z 3× 10−6 m2 s−1

Deffp,y 1.5× 10−5 m2 s−1

c 1200 J kg−1 K−1

Λy 0.06 W m−1 K−1

Λz 0.51 W m−1 K−1

Kmi 0.0035 s−1 [5]hc

b 1180 W m−2 K−1

ρa 1.205 kg m−3

µa 1.82×10−5 kg m−1 s−1

ca 1005 J kg−1 K−1

Λa 0.0257 W m−1 K−1

Da 2.53×10−5 m2 s−1

aCalculated from ρ and ρfbEstimated from the data reported in [23]

for uncoated samples at a pressure of 1 atm

Printing environmentParameter ValueTps 32 ◦CLps 0.15 mLds 0.2 mystep 0.025 mtstep

c 2.5 sT∞ 20 ◦CC∞p 0.333×10−1 kg m−3 (RH = 65%)

V∞ 3 m s−1

Re 6.9×104

Pr 0.713Sc 0.597K+

h 11.48 W m−2 K−1

K+m 0.011 m s−1

cMeant to represent the printing speed of a 0.84m-wideposter

Table 2: List of parameters used in the simulations.

Equations (20) to (23) are coupled. To solve this system, we eliminate cn+1

f from (21) and substi-tute it in (20). With this substitution, and after rearranging terms, we obtain the following discreteequation for Cp

(I− θ∆tAp) cn+1

p + ∆tθNp (1−N3) cn+1

f,equ = (I + (1− θ) ∆tAp) cn

p (23)

+ ∆tNp

((N2 − 1 + θ)c

n

equ + (N1 + 1− θ)cn

f

)+ bp,

which is decoupled from (21) and implicitly coupled to the temperature equation (23) by means of the

parameters Csp, K, C and Xm involved in the definition of c

n+1

f,equ. The constants N1 to N3 in (23) aregiven by

N1 =1− (1− θ) ∆tNf

1 + θ∆tNf, N2 =

(1− θ) ∆tNf

1 + θ∆tNf, N3 =

θ∆tNf

1 + θ∆tNf.

For each time step n+ 1, equations (21) to (23) are solved iteratively as follows:

1. Take T = Tn

as initial guess for the temperature vector at time n+ 1.

2. Use T to calculate the temperature-dependent parameter vectors csp, k, c, and xm. These vectors

contain the values of Csp and the GAB-parameters K, C, and Xm, respectively, at all grid points.

csp is obtained by evaluating (7) at each element of the vector T. For a given element of T, the

isotherm parameters are estimated by linearly interpolating the curves from the temperaturesavailable in Table 1, followed by fitting relation (4) to the interpolated curve with the LeastSquares method.

10

3. Solve the system in (23) using the parameter vectors calculated in step 2 to locally define cn+1

equ .

This nonlinear system for cn+1

p is solved using Newton’s iteration method [24]. The solution

constitutes a predictor of cn+1

p , denoted by c(1)p .

4. Find cn+1

f from equation (21) using c(1)p and the parameter vectors obtained in step 2 to evaluate

the vector cn+1

f,equ. The solution constitutes a predictor of cn+1

f , denoted by c(1)f .

5. Solve the system in (23) using c(1)f to evaluate the vector ∆h

n+1

ads . ∆hn+1

ads is found by linearinterpolation of the curve presented in Figure 2. The solution obtained in this step constitutes apredictor of T

n+1

, which is denoted by T(1).

6. Take T = T(1) and repeat steps 2 to 5. The solutions obtained in this step are correctors of c(1)p ,

c(1)f and T(1), denoted by c

(2)p , c

(2)f and T(2), respectively.

7. Check if the following conditions are satisfied: ‖c(k+1)p − c

(k)p ‖2 < tol, ‖c(k+1)

f − c(k)f ‖2 < tol and

‖T(k+1) − T(k)‖2 < tol, with k an iteration counter and tol a prescribed tolerance. If at leastone of the conditions is not satisfied, take T = T(k+1), and repeat steps 2 to 5. As soon asthe three conditions are satisfied, the iteration stops and the solution of the system is given by

cn+1

p = c(k+1)p , c

n+1

f = c(k+1)f , T

n+1

= T(k+1).

6. Results and discussion

The effect of temperature changes is included in the model for moisture transport in paper by meansof the saturated vapor concentration Cs

p and the GAB-parameters K, C, and Xm. For a fixed Cp-value,Cs

p induces desorption by decreasing the relative humidity for increasing temperatures, thus displacingthe equilibrium concentration Cf,equ to lower values. Simultaneously, the GAB-parameters affect themoisture equilibrium by displacing the isotherms to lower values of Cf,equ for higher temperatures. Inorder to identify the dominant driving force for temperature-induced moisture changes, the model de-rived in Section 2 is first used to study a simple set up in which a paper sheet is heated homogeneously.The sheet is initially in thermodynamic equilibrium with an environment of 20◦C and 65% relativehumidity. After 1 minute it is placed on a water impermeable surface maintained at 32◦C. Figure 3shows the results for the evolution of the integrated moisture content of the sheet when the variationwith temperature of Cs

p and the GAB-parameters is included. This figure also shows the results wheneither Cs

p or the GAB-parameters are considered constant with temperature. From these results wecan conclude that the variation of Cs

p constitutes the main driving force for the desorption process.Nevertheless, the variation with temperature of the GAB-parameters also has a significant effect andshould be taken into account in order to model temperature-induced moisture changes in paper.

The results from the simulation of a paper sheet in the printing environment are shown in Figure 4 and5. The numerical solution is obtained with a spatial resolution ∆z = 5× 10−6 m in the z-direction and∆y = 2.5 × 10−3 m in the y-direction. The resolution for time integration was ∆t = 0.2 s. Instead ofpresenting the results in terms of the variables Cp and Cf , we use the scaled variables relative humidityin the pores (RHp = Cp/C

sp(T )) and moisture content (X = Cf/ρf), respectively, since the latter allow

for easier interpretation and comparison with results from other authors. Figure 4 shows the tempera-ture and moisture profiles along the length of the sheet for z = Lz/2 and t = 23 s. All variables showstep-like profiles resulting from the step-wise movement of the print surface. Due to the high value ofthe thermal contact conductance between paper and the metallic surfaces, the temperature of the sheetfollows the temperature of the print and drying surfaces, which creates temperature differences of up to12◦C along the length of the sheet. Although these temperature differences generate changes of about25% in RH, changes of only 0.2% in moisture content are observed. This is a result of the short time

11

0 5 10 15 20 25 304

4.5

5

5.5

6

6.5

7

7.5

8

t [min]

IntegratedMoisture

Content×100

[%]

Complete model

Constant GAB-parameters

Constant Csp

Figure 3: Effects of the variation of Csp and GAB-parameters with temperature on the evolution of the integrated moisture

content in a paper sheet heated homogeneously.

interval for which the paper stays at the print surface. Each y-coordinate of the sheet is in contact forat most 15 s with the hot print surface, a very small time compared to the characteristic adsorptiontime, i.e., 1/Kmi = 286 s.

Figure 5 shows the moisture and temperature profiles along the thickness of the sheet for y = Ly/2and t = 17.6 s. This time corresponds to 1 second after the print surface has arrived to the middle ofthe sheet (Ly/2), when the moisture gradients at that point are the largest. Although the figure showstemperature and relative humidity gradients along the thickness, these gradients only appear shortlyafter the print surface reaches y = Ly/2, and they are smoothed out after 2 seconds. Since the char-acteristic time for the desorption process is much larger than 2 seconds, no gradients in the moisturecontent are observed along the thickness of the sheet. Small moisture and temperature gradients alongthe thickness direction were expected since the Biot numbers for heat (Bi+h = 0.02) and mass transfer(Bi+m = 0.4) are lower than 1.

The moisture profiles in Figure 4 and 5 suggest that the contact time with the hot print surface is tooshort to generate significant gradients that could cause dimensional instabilities in the printed sheet.According to experimental research performed by Oce Technologies (data not published), a paper sheetplaced over two surfaces with different temperatures only develops dimensional instabilities when theinduced difference in moisture content between those surfaces is of the order of 3% or higher. Also, whenmodeling cockling phenomena in paper, Lipponen et al., [2] found that local out-of-plane deformationonly takes place when the difference in moisture content between the top and bottom of the sheet ex-ceeds a value of about 2%. For the physical parameters listed in Table 2, differences of the order of 3% inmoisture content along the length of the sheet only appear when each y-coordinate is in contact with theprint surface for about 3 minutes (data not shown). This corresponds to a printing speed of the order of1 mm s−1, ten times slower than the real printing speed. Moreover, when the sheet is heated for longertimes, moisture content differences of 2% along the thickness of the sheet only appear when the Biotnumber for mass transfer is significantly higher, i.e., the sheet is then much thicker and/or K+

m is higher.

12

0 0.05 0.1 0.15 0.2 0.2535

40

45

50

55

60

65

y [m]

RH

p×

100

[%]

0 0.05 0.1 0.15 0.2 0.25

7.64

7.66

7.68

7.7

7.72

7.74

7.76

7.78

y [m]

Moisture

content×100

[%]

0 0.05 0.1 0.15 0.2 0.25

20

22

24

26

28

30

32

y [m]

T[◦C]

0 0.05 0.1 0.15 0.2 0.25

20

22

24

26

28

30

32

y [m]

T−

[◦C]

Drying surface(Tds)

Print surface (Tps)

Figure 4: Moisture and temperature profiles along the length of a paper sheet in the printing environment for z = Lz/2and t = 23 s.

It is important to take into account that the predicted moisture gradients might be affected by in-accuracies in the estimation of the model parameters. The model is particularly sensitive to the valueof the pores to fibers mass transfer coefficient Kmi, indicating that the evolution of the moisture contentis greatly determined by the speed of the sorption process. This coefficient was not measured for thisapplication, but obtained from the value reported in [5] for bleached kraft board. Several processesmight lead to variations in Kmi among different paper types. Differences in the manufacturing of thesheet, for example during bleaching or addition of sizings and additives, can change the affinity betweenfibers and water vapor, affecting the kinetics of the sorption process. Moreover, since Kmi also accountsfor the surface area available for contact between pores and fibers, variations in the internal surface areaof the sheet will also be reflected on the Kmi-value. Finally, the Kmi-value reported in [5] was measuredunder isothermal conditions, and no variations with temperature of this coefficient are included in ourmodel.

In order to determine the exact magnitude of the moisture differences induced by the printing en-vironment, the experimental determination of Kmi for the specific paper used in this application as wellas the dependency of Kmi on the temperature are needed. However, further conclusions can be drawn

13

0 0.2 0.4 0.6 0.8 1

x 10−4

35

40

45

50

55

60

65

z [m]

RH

p×

100

[%]

0 0.2 0.4 0.6 0.8 1

x 10−4

7.7

7.72

7.74

7.76

7.78

7.8

z [m]

Moisture

content×100

[%]

0 0.2 0.4 0.6 0.8 1

x 10−4

20

22

24

26

28

30

32

z [m]

T[◦C]

17.6 s Nt = 89

Figure 5: Moisture and temperature profiles along the thickness of a paper sheet in the printing environment for y = Ly/2and t = 17.6s.

by studying which parameter values lead to significant moisture content differences along the lengthand thickness of the sheet. Figure 6 shows the moisture profiles obtained with different values of thepores to fibers mass transfer coefficient Kmi and the convective heat and mass transfer coefficients atthe top of the sheet, K+

m and K+h . The figure shows the results when Kmi is 10 and 100 times higher

than the value reported in Table 2. Variations in K+m and K+

h are introduced indirectly by changing thebulk velocity of air V∞. Whenever the Reynolds number exceeds 2.5×105, K+

m and K+h are calculated

from the correlation reported in [18] for the average heat transfer coefficient for turbulent boundarylayers over a flat plate.

Figure 6 shows that in order to achieve moisture content differences of the order of 3%, Kmi and K+m

need to be significantly higher than the values reported in Table 2. Such differences are only achievedwhen Kmi is 100 times higher and for air velocities above 15 m s−1. These results support the idea thatmechanical stabilities reported by Oce Technologies in the target printer are induced by other factorsbesides the movement over the hot print surface. We have found that air velocities above 15 m s−1 arenecessary to obtain large moisture content gradients, but this velocity is significantly higher than theone present in the target printer. These results also hold when the simulations are performed at higherrelative humidities.

7. Summary and Conclusions

The goal of this paper was to model heat and moisture transport at the final use of paper, when pa-per is heated by, for example, a printer. Therefore, an elaborated two-phase non-isothermal model wasderived. This model enabled us to identify the relevant driving forces behind the temperature-inducedmoisture changes in paper. It has been shown that the variation with temperature of the saturatedvapor concentration constitutes the main driving force for moisture desorption. Although to a lesserextent, variation with temperature of the GAB-parameters also affects the evolution of the moisturecontent and should be included in the model.

The model was applied to describe the heat and moisture transport in a paper sheet moving overa hot print surface. The corresponding numerical simulations predicted changes in moisture contentof only 0.2%. This suggests that the contact time with the hot print surface is too short to createmoisture gradients able to induce dimensional instabilities on the printed sheet. However, the experi-mental determination of the pores to fibers mass transfer coefficient Kmi as well as its variation withtemperature need to be investigated in order to obtain more reliable predictions. Variation in other

14

00.05

0.10.15

0.20.25 0

0.5

1

1.5

x 10−4

5.5

6

6.5

7

7.5

8

z [m]

y [m]

Moisture

content×100[%

]

V ∞ = 3, K+m = 0.011, K+

h = 11.48, Kmi = 0.035

00.05

0.10.15

0.20.25 0

0.5

1

1.5

x 10−4

5.5

6

6.5

7

7.5

8

z [m]

y [m]

Moisture

content×100

[%]

V ∞ = 3, K+m = 0.011, K+

h = 11.48, Kmi = 0.35

00.05

0.10.15

0.20.25 0

0.5

1

1.5

x 10−4

5.5

6

6.5

7

7.5

8

z [m]

y [m]

Moisture

content×100

[%]

V ∞ = 8, K+m = 0.017, K+

h = 18.8, Kmi = 0.35

00.05

0.10.15

0.20.25 0

0.5

1

1.5

x 10−4

5.5

6

6.5

7

7.5

8

z [m]

y [m]

Moisture

content×100

[%]

V ∞ = 16, K+m = 0.06, K+

h = 66.9, Kmi = 0.35

Figure 6: Moisture content profiles at t = 28 s for different values of Kmi and V∞.

model parameters like the printing speed and the bulk velocity of the air inside the printer might leadto larger moisture gradients. The model can therefore be helpful in the design phase of new printers.Furthermore, it can be used to calculate moisture profiles in any application that considers the finaluse of printing paper.

8. Acknowledgements

The authors thank the companies Oce Technologies B.V. and LIME B.V. for supporting this workand are grateful for the permission to publish this paper. We also would like to thank dr. E. vanDam (Oce Technologies), prof. A. Hirschberg (Eindhoven University of Technology) and prof. R.M.M.Mattheij (Eindhoven University of Technology) for valuable discussions. The experimental assistanceof dr. L. Pel, ir. N. Reuvers, and ir. H. Dalderop of the Eindhoven University of Technology is alsogratefully acknowledged.

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