Modeling Simultaneous Shrinkage and Heat and Mass
`Transfer of a Thin, Nonporous Film During Drying
`
`T.J. BOWSER and L.R. WILHELM
`
`ABSTRACT
`Thin films of food products have long been dried commercially but the
`thin film drying process is not well understood. Modeling of a drying
`system is essential for understanding and improving it. A theoretical
`model for predicting the drying rate of thin films of nonporous foods
`was proposed, developed and evaluated. The model simultaneously con-
`sidered shrinkage and heat and mass transfer within thin films dried on
`a surface with given boundary conditions. A finite element formulation
`of the model was used to develop numerical solutions of two governing
`equations. Starch was selected as a representative material for drying
`tests. Experimentally determined drying curves of modified corn, potato
`and rice starch films were compared to model predictions. The technique
`was useful in explaining the complex relationships of temperature, mois-
`ture and thickness profiles of drying films.
`
`Key Words: thin film drying, starch, rice, potato, nonporous films
`
`INTRODUCTION
`THIN FILM DEHYDRATION is an important processing
`technique
`used to provide
`low cost, high quality, shelf stable foods. This
`process has been used commercially
`for many years. Thin
`film
`drying
`technology has advanced slowly and the process has not
`been well understood.
`Improvement of thin
`film drying could
`result from speeding up the process, improving
`the quality of
`final products, and reducing energy costs. Modeling of the dry-
`ing system could help achieve such improvements.
`Some observed complexities of thin
`film drying have been
`ascribed
`to product shrinkage and the variability
`of physical
`properties with changing moisture contents and temperatures
`(van Arsdel, 1947; Kozempel et al., 1986). Classical
`thermo-
`dynamics and heat transfer theory have been applied to develop
`mathematical models describing
`temperature and moisture dis-
`tributions
`in films and slabs (Philips and de Vries, 1957; de
`Vries, 1958; Luikov and Mikhailov,
`1965; Mikhailov,
`1973;
`Raats, 1975; Wbitaker, 1977, and others). Chirife
`(1983) and
`Keey (1990) questioned the use of complex mathematical mod-
`els unless
`they could be supported by experimental
`results.
`Hougen et al. (1940) and Bruin and Luyben
`(1990) stated that
`the solutions
`to classical equations in their most general forms
`were not available and they must be solved using numerical
`techniques. Our objective was to develop a mathematical model
`to describe heat and mass transfer during
`thin
`layer drying of
`nonporous
`foods. The model could be .used to derive data
`needed for thin film dryer design and process optimization based
`on variables such as film
`thickness, drying
`time and product
`temperature and moisture histories.
`
`THEORY
`
`Finite element model
`Van Arsdel (1947), Rulkens and Thijssen (1969) and Okazaki et al.
`(1974) used numerical techniques to investigate the effects of variable
`diffusivity as a function of moisture content in a drying film with a
`
`Lockwood Greene En-
`is Engineering Consultant,
`Author Bowser
`gineers,
`Inc., 1500 International
`Drive, Spartanburg,
`SC 29304.
`Author Wilhelm
`is Professor, Dept. of Agricultural
`Engineering,
`The Univ. of Tennessee, P.O. Box 1071, Knoxville, TN 37901. Ad-
`dress
`inquiries
`to Dr. L.R. Wilhelm.
`
`shrinking coordinate system. They successfully simplified the analysis
`by assuming a constant film temperature. Such is not a valid assumption
`for thin film drying. The film temperature varies; it is critical to finished
`product quality; and it also greatly affects moisture diffusivity (Fish,
`1957; Whitney and Porterfield, 1968; Zhou et al., 1992).
`Haghighi and Segerlind (1988) used a finite element formulation to
`determine the simultaneous moisture and heat diffision equations for the
`drying of an isotropic sphere (soybean kernel). Haghighi (1990) later
`improved upon the model by considering volumetric changes in addition
`to heat and moisture diffusion. Since soybean drying was a relatively
`slow process (measured in days), Haghighi could assume constant phys-
`ical properties (specific heat, density, thermal conductivity and moisture
`diffusivity) with good results. This would not be practical with a starch
`film, since the physical properties change radically during the short dry-
`ing process.
`Based upon the work of past researchers, a finite element approach
`could be used to model the short time drying of a thin film of starch.
`The model considers shrinkage and simultaneous heat and mass transfer.
`Physical properties of the starch film are evaluated as a function of in-
`stantaneous film temperature and moisture content (Fig. 1). We hypoth-
`esized this model would add to the theoretical knowledge on thin film
`drying without becoming so complicated as to have little utility in dryer
`design.
`
`Model assumptions
`Assumptions of the general model were:
`IShrinkage and gradients in temperature and diffusivity are one di-
`mensional, perpendicular to the surface of the film (“2”
`direction,
`Fig. !).
`2.All moisture movement is by diffusion.
`3.Shrinkage is due to moisture migration and thermal expansion is ne-
`glected.
`4.Free shrinkage is directly proportional to the change in moisture con-
`centration.
`5.0bserved shrinkage at any instant during drying is the cumulative
`effect of free shrinkage due to moisture loss.
`Numbers three through five are from Misra and Young (1980).
`
`Moisture diffusion
`T.K. Sherwood (193 1) was one of the first researchers to apply a
`parabolic partial differential equation to model the moisture gradient in
`drying solids. This equation, commonly known as the diffusion equation,
`(Press et al., 1989) is
`
`(1)
`
`ah4 a aM
`-=-
`D--
`at
`az az
`i
`1
`where M = moisture concentration, kg/ms; t = time, set; z = distance
`perpendicular to the thin film surface, m; D = Diffusivity of moisture,
`m%ec.
`The diffusion equation does not allow for other phenomena such as
`capillarity, porous transport and gravity (Hougen et al., 1940). Equation
`(l), however, is valid for simple materials such as solutions and gels
`(e.g. starch) when the molecular transport of water takes place by dif-
`fusion (Hougen et al., 1940; Bruin and Luyben, 1990).
`The initial condition for Eq. (1) is a uniform product moisture content.
`Boundary conditions implicit to Eq. (1) are
`
`D,
`
`(z)
`
`= h,.,A CM,, -Ma,)
`
`Volume 60, No. 4, 1995-JOURNAL
`
`OF FOOD SCIENCE-753
`
`RBP_TEVA05022462
`
`DRL - EXHIBIT 1019
`DRL001
`
`

`
`D
`p(“‘] = -
`
`i-i
`-1 1
`where [k@)] = Element stitihess matrix and L = Length of segment, m,
`and using the lumped formulation
`
`(8)
`
`L [ 1
`
`MODELING THIN NONPOROUS FILM DRYING
`
`. . .
`
`Finite element equations
`The finite element equations are formulated using the direct stiffness
`method given by Segerlind (1984) (Fig. 1).
`
`Moisture diffusion
`The element equations for the moisture diffusion equation are given
`as
`
`MOIST PRODUCT FILM
`
`bo
`
`FERuEmElAYER
`
`t
`section of a drying starch
`
`film.
`
`Fig. l-Cross
`
`Da (5)
`
`= h,, (Ma - KB)
`
`where DA = Moisture diffusivity of the film, nearest surface A, m*/sec;
`D, = Moisture diffusivity of the film, nearest surface B; mVsec; h,, =
`Mass transfer coefficient for surface A, m/set; h,, = Mass transfer co-
`efficient for surface B; m/set, M, = Moisture concentration of the air
`film nearest surface A, kg/m-‘; M, = Moisture concentration of the air
`film nearest surface B, kg/m-‘; kA = Moisture content of ambient air
`over surface A, kg/m-‘; and M,, = Moisture content of ambient air over
`surface B, kg/m-‘.
`
`Heat diffusion
`The one dimensional heat diffusion equation is an analog of the mass
`diffusion equation and was given by Bird et al. (1960) as
`
`PC~ (t$
`
`= ;(k$)
`
`where p = Density of product film, kg/m3; c,, = Specific heat of product
`ihn, WikgK; T = Temperature, K; and k = Thermal conductivity,
`w/m*K.
`The initial condition for Eq. (4) is a uniform product temperature.
`Boundary conditions implicit to Eq. (4) follow
`
`k, (g) = MT,-U
`
`where k, = Thermal conductivity of the film, nearest surface A, W/mK,
`ka = Thermal conductivity of the film, nearest surface B, W/mK; h, =
`Convective heat transfer coefficient for surface A, W/mZ*K; h, = Con-
`vective heat transfer coefficient for surface B, W/mZ*K, T, = Temper-
`ature of the film nearest surface A, K; T, = Temperature of the film
`nearest surface B, K, T,, = Temperature of ambient air over surface A,
`K; and T, = Temperature of ambient air over surface B, K.
`
`Shrinkage
`The shrinkage of a thin film could be modeled using a linear coeffi-
`cient of hydra1 shrinkage (Misra and Young, 1980). Empirically deter-
`mined shrinkage coefficients were reported by Lozano et al. (1983) and
`Suzuki et al. (1976). The film is regarded as a collection of axial mem-
`bers with individual displacements
`p=LSAM
`(7)
`where p = Displacement of member, m; B = Linear coefficient of hydra1
`shrinkage, dimensionless; AM = Change in moisture content, %; and L
`= Length of member, m.
`
`where [&)I = Element capacitance matrix and
`
`(9)
`
`where p) = Element force vector
`The first term on the right side of Eq. (10) applies to the surface of
`the film at z = 0. The second term applies to the surface of the film at
`z = b. All other terms in the force vector are zero.
`
`Heat diffusion
`Element matrices for the heat diffusion equation are similar to those
`given for the moisture diffision equation. The stiffness matrix is given
`by
`
`and again using the lumped formulation, the capacitance matrix is
`
`[C(C)] = VJ
`
`10
`01
`
`2 [ 1
`
`(12)
`
`(13)
`
`is the force vector. The first term in Eq. (13) applies only to the surface
`of the film at z = 0. The second term applies to the surface of the film
`at z = b. All other terms in the heat diffision
`force vector are set equal
`to zero.
`
`General finite element solution
`The forward difference method is selected to find the finite element
`solution in time (since boundary conditions are not known at each time
`step). The general format of the forward difference equation is
`[C] Qp, = ([Cl - At [K]) a’. + AtF
`(14)
`where [C] = The capacitance matrix, @,, = Nodal values after time step
`At, At = Time step, @‘. = Nodal values before time step At, [K] =
`Global stiffness matrix, and F = Force vector before time step.
`Equation (14) can be solved for Qb using a computer and standard
`matrix manipulation routines Press et al. (1989).
`
`Time step
`The time step size must be > zero and satisfy
`
`for the moisture diffusion equation and
`
`754-JOURNAL
`
`OF FOOD SCIENCE-Volume
`
`60, No. 4, 7995
`
`RBP_TEVA05022463
`
`DRL - EXHIBIT 1019
`DRL002
`
`

`
`Property
`(kg/m3)s
`
`Density
`
`Table
`
`l-Starch(s)
`
`and water
`
`in the model
`used
`(w) properties
`Value or equation
`
`ps = 1202 - 148 x x + 259 x exp
`X0
`
`II
`
`-15.507
`
`x ;
`
`1
`
`Diffusivity
`
`(cm*/.&
`
`Cl = DL x exp
`
`Where:
`
`content
`x = moisture
`pw = 1110.4 - 0.49437 x T (“C)
`
`x0 = initial moisture
`
`content
`
`& [ 1 DL = w[&]
`[ 1
`
`E = 9.724 - 1.179 X In mc
`P
`
`(kCal)
`
`R = 1.98E-3
`
`(SK)
`
`Sorption
`
`lsothermC
`
`Thermodynamicsd
`C, NJ/kg Cl
`
`k (W/m C)
`
`LHV (kJ/kg)
`
`Shrinkagee
`
`For m, < 16%,
`
`D25 = 6.143E-11x
`
`exp 0.5 !?Z
`
`[ 1 P
`
`for m, 2 16%,
`
`D25 = 5.0 E- 11
`
`et 25°C;
`limit et 26C; D25 = diffusivity
`Where: DL = difusivity
`E = energy barrier of diffusion; R = universal gas constant;
`m, = moisture
`content, decimal
`(wb).
`For m 2 38,
`a, = 1.0
`
`For m < 38,
`
`aw = 0.5 X
`
`[
`
`mxC-Zxm-CxV+<
`
`~m2xC-2XmxCxV+4xmxV+CXV2
`k x m x (C-l)
`
`I
`
`Where: k = 0.662;
`
`C = 19.03;
`
`V = 13.27;
`
`m =
`
`g H20
`1009 solids
`
`C,, = 1.8608 + 2.4311 E-3 X T (“C)
`
`C pw = 4.1598 + 4.2091 E-4
`
`x T (“C)
`
`kg = 0.19306 + 8.4997 E-4
`
`x T (“C)
`
`k, = 0.59075 + 9.8601 E-4
`
`x T (“C)
`
`hf9 = 2502.535
`
`- 2.3858 X T (“C)
`
`ihs = 130.204 X exp(-0.098
`
`x m);
`
`LHV = ihs + hfg
`
`m=
`
`kg H20
`100 kg starch
`
`Where: LHV = Latent heat of vaporization,
`
`Hz0
`
`in starch
`
`Sb = [kb(v)
`
`+ Lblm
`
`hli
`Lb = ~
`illi
`
`- m,,
`-m,
`’
`
`kb=-’
`
`1-Q
`Illi - m,’
`
`@=
`
`(Q+
`(m, +
`
`1)Pi
`l)p,
`
`content, dry basis
`Where: mi = moisture
`m, = previous moisture
`content, dry basis
`Pi = density,
`kg/m3
`PO = previous density,
`a Lozano et al. (1983) (for ps) and Okos (1986) (for pw).
`b Fish (19571.
`Cvan den Berg et al. 11975).
`d Okos (1986) (for CP and k); Brooker (1967) (for hrs); and van den Berg et al. (1975) (for ihsl.
`e Suzuki et al. (1976).
`
`kg/m3
`
`for the heat diffusion equation. The time step restriction is necessary to
`insure that the quantity ([Cl - At[K]), from Eq. (14), remains positive
`definite (Mohtar and Segerlind, 1992; Haghighi and Segerlind, 1988;
`Misra and Young, 1979).
`
`Finite element soiution
`Simultaneous solution of Eq. (l), (4) and (7) will give the instanta-
`neous moisture concentration, temperature and thickness of a drying
`
`film. This is accomplished by numerically solving Equation (1) for a
`given time step to obtain the moisture profile. Next, the results are used
`in a numerical solution of Eq. (4) to obtain the temperature profile for
`the same time step. Equations (1) and (4) are coupled in the finite ele-
`ment formulation by means of their forcing functions, Eq. (10) and (13),
`respectively. The thermal energy required to evaporate moisture from
`the first and/or last node of the film during a drying time interval is
`added to the forcing fknction of the heat diffusion equation at the par-
`titular node(s). Finally, the moisture profile can be used in the solution
`
`Volume 60, No. 4, 1995-JOURNAL
`
`OF FOOD SCIENCE-755
`
`RBP_TEVA05022464
`
`DRL - EXHIBIT 1019
`DRL003
`
`

`
`MODELING THIN NONPOROUS FILM DRYING
`
`. . .
`
`80 I
`
`6
`a 40
`2
`.s 3.
`E
`& 20
`E
`P
`4
`
`lo-
`
`0’
`0
`
`80.0
`s 3 70.0
`; r” 60.0
`5
`;; 50.0
`s
`2 40.0
`a
`.s 30.0
`E
`3 20.0
`E
`P 10.0
`4
`
`0.0
`0
`
`Average air temp = 39 ‘C
`Initial film thickness = 3.2 mm
`+ Measured
`Predicted
`
`10
`
`20
`
`30
`Time, min
`
`40
`
`50
`
`60
`
`for modified
`and measured drying curves
`Fig. 2-Predicted
`starch on a water vapor permeable
`drying surface.
`
`corn
`
`Fig. 5-Predicted
`an impermeable
`
`Average air temp = 44 ’ C
`Initial film thickness = 1.8 mm
`-) Measured
`Predicted
`
`10
`
`20
`
`30
`Time, min
`and measured drying curves
`drying surface.
`
`40
`
`50
`
`60
`
`for rice starch on
`
`1
`
`Average air temp = 44 o C
`Initial film thickness = 2.6 mm
`+ Measured
`Predicted
`
`0
`
`10
`
`20
`
`30
`Time, min
`
`40
`
`50
`
`60
`
`and measured drying curves
`Fig. 3-Predicted
`on a water vapor permeable
`drying surface.
`
`for potato
`
`starch
`
`$ 20
`E
`g 10 -
`4
`
`0'
`0
`
`Average air temp = 4L? C
`Initial film thickness = 1.9 mm
`-a- Measured -. Predicted
`
`10
`
`20
`
`30
`Time, min
`
`40
`
`50
`
`J
`60
`
`and measured drying curves
`Fig. 4-Predicted
`a water vapor permeable
`drying surface.
`
`for rice starch on
`
`. -
`
`E
`$60
`0
`g-50
`E
`s 40
`
`30
`
`5
`.v,
`2 20
`sl
`g F 10
`a
`0 -I
`0.0
`
`0.2
`
`0.4
`
`1.0
`0.8
`0.6
`Film Thickness,
`
`1.2
`mm
`
`1.4
`
`1.6
`
`1.8
`
`moisture
`Fig. GPredicted
`starch
`film during drying.
`
`distribution
`
`and shrinkage
`
`of a rice
`
`of Eq. (7) to find film shrinkage. This procedure is repeated iteratively
`until the system reaches equilibrium conditions or the desired drying
`period expires. The described model was implemented using the Pascal
`programming language. A fully commented program code listing (in-
`cluding references for physical property equations) is available in
`Bowser (1994).
`Satisfactory implementation of the finite element solution required de-
`termination of several physical property parameters as the film was
`heated and dried. Table 1 lists the relationships used for these parame-
`ters.
`
`& METHODS
`MATERIALS
`Experimental verification of the model
`Experimental verification of the finite element model was performed
`using data collected by Bowser (1994). The drying chamber was a cus-
`tom fabricated unit that provided two paths for air flow. The first path
`permitted air to flow over the surface of the drying film in a conventional
`drying technique. The drying film rested on a permeable stainless steel
`fiber media treated with a release agent to prevent product sticking and
`pore clogging. This media was supported by a macroporous copper ma-
`terial that served as the second path for air flow through the drying
`chamber. The flow of drying air above the film and below the permeable
`support of the film permitted approximately equal drying rates from both
`film surfaces.
`Room air was conditioned by dehumidification and heating and di-
`rected to the drying chamber. Immediately before entering the chamber,
`
`756-JOURNAL
`
`OF FOOD SCIENCE-Volume
`
`60, No. 4, 1995
`
`RBP_TEVA05022465
`
`DRL - EXHIBIT 1019
`DRL004
`
`

`
`the flow was divided to provide approximately equal flow rates and air
`pressures within each flow channel. Air exiting the two channels through
`the drying chamber was passed through desiccant columns filled with a
`molecular sieve desiccant. Two desiccant columns were used for each
`channel. Flow was alternated between the desiccant columns on each
`channel to permit weighing, at 4 min intervals. The desiccant columns
`were removed from the system and weighed on a scale to determine
`moisture removed during the drying process.
`Starch gel was a logical selection as the test product since it is a
`principle constituent of many food materials (Fish, 1957) and is often
`dried in thin films. Starch materials are readily available and simple to
`handle. In addition, extensive published information regarding starch
`properties is readily available (see Table 1). Three starch products were
`used: corn starch, potato starch, and rice starch. Each product was dried
`using the same procedure. A starch slurry was prepared and applied to
`the drying surface in the thinnest layer possible (about 1.9 mm for rice
`starch and 3.2 mm for others). The drying chamber was sealed, and
`drying air was forced through the two channels in the chamber. Tem-
`perature of drying air was well below that of typical commercial drying.
`This was necessary to produce a drying period long enough such that
`the time to manually apply the film to the drying surface and to start the
`dryer would not significantly affect results. Also, drying rates and other
`drying parameters could be effectively measured.
`
`RESULTS & DISCUSSION
`RESULTS OF THE MODEL’S PREDICTIONS were compared to ex-
`perimentally determined drying curves of modified corn, potato,
`and rice starch films (Figs. 2, 3 and 4). Measured and predicted
`drying curves were compared for corn, potato and rice starch
`films dried on a water vapor permeable surface. The measured
`and predicted drying curves for a rice starch film dried was also
`compared on an impermeable surface (Fig. 5).
`While generally tending to overestimate drying rates, the
`model gave good approximations of experimental drying curves
`of the starch films. The initial, steep descent of the predicted
`drying curve was a result of high water vapor pressure (fully
`wetted conditions) at the surfaces of the film as estimated by
`the model of van den Berg et al. (1975). The predicted drying
`curve had a much shallower slope as the water vapor pressure
`at the surfaces began to drop and the temperature of the film
`stabilized. The measured curve did not show an initial, steep
`descent probably because some drying occurred during appli-
`cation of the film and related start up procedures (Bowser,
`1994). Removal of the wetted surface component of the model
`would result in considerably greater agreement between the
`model and the measured data. Other differences between the
`model and experimental results may be due to differences in
`physical properties of specific starches compared to the pub-
`lished values.
`Predicted values of moisture concentration and film thickness
`at indicated times were compared (Fig. 6) for a rice starch film
`dried on an experimental, water vapor permeable drying surface.
`The film thickness prediction was based on the verified model
`of Suzuki et al. (1976). The initial film and boundary conditions
`for the plot were assumed to be the same as those obtained by
`measurement during the rice starch drying experiment (Fig. 4).
`These data (Fig. 6) are based upon output of the mathematical
`model. They provide a very powerful technique for visualizing
`complicated relationships between temperature, moisture and
`thickness of a drying film. While not shown, temperature pro-
`files across the film could also be computed in a similar manner.
`The model represents a unique application linking previously
`developed models to simultaneously solve for heat transfer,
`mass transfer, and shrinkage in thin film drying. The model
`requires further testing to compare predictions to thin film dry-
`ing data obtained over much shorter drying periods (< 15 sets).
`High speed drying is required for industrial applications and
`may result in steep temperature gradients and radical, physical
`
`property changes. The model provides a convenient and low cost
`means to predict and investigate such conditions.
`
`REFERENCES
`Bird, R.B., Stewart, W.E., and Lightfoot, E.N. 1960. Transport Phenomena.
`John Wiley and Sons, New York.
`Bowser, T.J. 1994. Thin film drying on a permeable surface. Ph.D. disser-
`tation, The Univ. of Tennessee, Knoxville.
`Brooker, D.G. 1967. Mathematical model of the psychometric chart. Trans-
`actions of the ASAE, lO(4): 558-560, 563.
`Bruin, S. and Luyben, K. Ch.A.M. 1980. Drying of food materials: a review
`of recent developments. Ch. 6, In Advances in Drying, A.S. Mujumdar
`(Ed.), p. 155-215. Hemisphere Pub., Washington, DC.
`Chirife, J. 1988. Fundamentals of the drying mechanism during air dehy-
`dration of foods. Ch. 3, In Advances in Drying, A.S. Mujumdar (Ed.), p.
`23361. Hemisphere Pub., Washington, DC.
`de Vries, D.A. 1958. Simultaneous transfer of heat and moisture in porous
`media. Transactions, American Geophysical Union? 39(5): 909-916.
`Fish, B.P. 1957. Diffusion and equilibrium propertres of water in starch.
`Department of Scientific and Industrial Research, Food Investigation,
`Technical Paper No. 5. Her Majesty’s Stationery Office, London.
`Haghighi, K. and Segerlind L.J. 1988. Modeling simultaneous heat and mass
`transfer in an isotropic sphere a finite element approach. Transactions of
`the ASAE, 31(2): 629-637.
`Haghighi, K. 1990. Finite element simulation of the therm0 hydra stresses
`in a viscoelastic sphere during drying. Drying Technol. 8(3): 451464.
`Hougen! O.A., McCauley, H.J., and Marshall, W.R., Jr. 1940. Limitations of
`diffusion equations in drying. Transactions Amer. Inst. Chem. Engr. 36:
`183-206.
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`
`supported
`Research
`Knoxville, Agricultural
`
`by a USDA National Needs
`EngineerinS Dept.
`
`fellowship
`
`and
`
`the Univ. of Tennessee,
`
`Volume 60, No. 4, 1995-JOURNAL
`
`OF FOOD SCIENCE-757
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`RBP_TEVA05022466
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`DRL - EXHIBIT 1019
`DRL005