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Mathematical modeling of impingement drying of corn tortillas
Journal of Food Engineering 50 (2001) 121±128
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Mathematical modeling of impingement drying of corn tortillas Louise M. Braud, Rosana G. Moreira, M. Elena Castell-Perez * Department of Agricultural Engineering, Texas A&M University, College Station, TX 77843-2117, USA Received 27 June 2000; accepted 4 December 2000
Abstract Impingement drying of corn tortillas was modeled using governing equations for heat and mass transfer during the drying process. Mass transfer within the product was modeled as diusion-driven mass ¯ux. Heat transfer was driven according to Fourier's Law of conduction. Boundary conditions for drying in both air and superheated steam were incorporated into the model. Convective heat transfer accounted for heat ¯ow into the product at the surface. When drying in air, convective mass transfer prevailed; in superheated steam, dierences in vapor pressure between the drying medium and the product surface accounted for mass transfer. Temperature and moisture content predictions followed the experimental trends with both air and steam drying (115± 145°C). Steam condensation unaccounted for by the model resulted in underpredictions in the moisture content in steam drying at low temperatures. Product thickness and drying medium temperature had a signi®cant eect on moisture content and temperature pro®le over time. Ó 2001 Elsevier Science Ltd. All rights reserved. Keywords: Superheated steam; Dry air; Finite dierences; Heat and mass transfer
1. Introduction Impingement drying involves the use of a single or multiple gas jets releasing a stream of gas vertically onto a surface. Impinging jets are used for cooling and heating, as well as drying, because they result in higher heat and mass transfer coecients. Dry air and superheated steam are two drying media commonly used in impingement drying. With superheated steam as the drying medium, some vapor will condense on the product surface during the transient early stage of drying. This occurs as the superheated steam contacts the cold solid (Beeby & Potter, 1992). Lujan-Acosta, Moreira, and Seyed-Yagoobi (1997) studied the use of impinging air jets for drying of corn tortillas as an alternative to convection-oven baking prior to frying. The use of this technique resulted in less oil absorption during frying and, therefore, a lower-fat tortilla chip. Chips dried with impinging air had a ®nal oil content of 14% (wet-basis) as opposed to 23±30% for commercial tortilla chips (Moreira, Castell-Perez, & Barrufet, 1999). A comparison between air and superheated steam with impingement drying of corn tortillas with opposing *
Corresponding author. Tel.: +1-979-862-7645. E-mail address: [email protected] (M.E. Castell-Perez).
jets was conducted by Li, Seyed-Yagoobi, Moreira, and Yamsaengsung (1999). Stacking two tortillas between wire meshes allowed placement of thermocouples between them to measure temperature during the drying process. Average moisture content was also recorded over time. Drying at varying temperatures (115°C, 130°C, 145°C) and convective heat transfer coecients 2 (100, 130, 160 W=m K) were conducted with both air and superheated steam. As expected, results indicated that higher steam temperatures resulted in faster drying. In comparing the use of air and superheated steam, there was little dierence in the drying curves at a lower temperature (115°C); however, at a higher temperature (145°C), superheated steam provided higher drying rates. In summary, the use of impingement drying in processing corn tortillas prior to frying shows feasibility in production of a lower-fat fried chip. Although a mathematical model of this processing technique has not been developed for this product, the need for it exists to facilitate ecient process design and optimization. Thus, the main objective of this study was to develop a mathematical model to describe the heat and mass transfer mechanisms during impingement drying of corn tortillas. This was accomplished by: 1. De®ning the governing equations that describe the transport of heat and mass (moisture) during the impingement drying process.
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2. Identifying varying boundary conditions imposed by dierent drying media (e.g. air and superheated steam). 3. Developing a computer algorithm that incorporated the governing equations and boundary conditions to simulate the drying process. 4. Validating the developed mathematical model with existing experimental data (Lujan-Acosta et al., 1997; Li et al., 1999). 5. Conducting a sensitivity analysis on the model and its input parameters to determine the eects of variations in the operating conditions. 2. Methodology 2.1. Assumptions The following assumptions concerning the drying process and changes in the product were made for model development: 1. The tortilla is homogeneous on a macroscopic scale and isotropic. Properties used were the eective properties that were uniform along the radius of the tortilla. 2. Temperature and moisture concentration gradients exist along the thickness of the tortilla only. These parameters were considered uniform along the radius of the tortilla. This assumption allows the use of onedimensional in®nite plate simpli®cations since the radius is much greater than the thickness. 3. Because the steam is highly superheated, the amount of condensate was neglected in analyzing superheated steam impingement drying. 4. The tortilla shrinks in diameter during drying; however, the eects of shrinkage, other than changes in the density and thermal properties of the tortilla, were neglected. Mass diusivity of the tortilla was considered a function of drying conditions only; thus it was held constant for a single drying temperature. 2.2. Governing equations for mass (moisture) transfer The ¯ow of moisture within the tortilla occurs by diusion and is governed by the eective diusion coecient according to Fick's Law. The mass balance of moisture within the stationary tortilla is as follows (Incropera & DeWitt, 1996): o oqb M o qb M Dw ; 1 ox ox ot which states that the rate of change in the moisture content, M, is equal to the diusion of water due to the internal moisture content gradient, oqb M=ox. Moisture within the tortilla is present in three forms, namely free water, bound water, and vapor. The trans-
port of these three forms was combined into a single moisture ¯ux term, j. The equation for this ¯ux within the system is as follows (Rohsenow & Choi, 1961): j
Dw
o qb M : ox
2
For a stationary product, the mass ¯ux of moisture is proportional to the diusivity of water in the tortilla, Dw , the density of the solid and liquid mixture, qb , and the moisture content gradient. 2.3. Governing equations for energy (Heat) transfer Heat transfer within the tortilla is driven by conduction as a temperature gradient develops along its thickness. An equation to describe energy transfer during impingement drying of tortillas is as follows: ocp T oq o oT oT qb cp T b k 3 cpw j : ot ox ox ox ot In this equation, moisture in its separate forms was combined into a single ¯ux term, j. The change in internal energy within the control volume over time is represented by the two terms on the left-hand side of the equation. This energy balance states that the rate of this change in internal energy is equal to the dierence between the rate at which heat is transported out of the control volume due to moisture ¯ux, cpw j oT =ox, and the net rate of heat transported into the control volume due to conduction, o=ox k oT =ox. To account for latent heat storage during vaporization, an approach applied by Chen (1996) was used. When a control volume reaches the boiling point of the water it contains, vaporization will commence. At this point, the temperature of the control volume will remain at the boiling temperature until enough heat ¯ows into the control volume to evaporate its moisture; however, moisture is continuously ¯owing through the control volume, thus changing its moisture content. To quantify the duration of vaporization within such a control volume, the amount of heat ¯owing into the control volume for each incremental time step was compared to the amount of heat required to vaporize the amount of moisture contained in the control volume at that point in time based on the latent heat of vaporization. Once the heat stored during the phase change was enough to vaporize all of the moisture within the control volume, vaporization within the element was complete and its temperature began to elevate from the boiling point. 2.4. Initial and boundary conditions At the onset of the drying process, the temperature and moisture content of the tortilla are uniform. The initial conditions in equation form are thus:
L.M. Braud et al. / Journal of Food Engineering 50 (2001) 121±128
T jx;t0 T0 ;
4
Mjx;t0 M0 :
5
Because jets impinge from both above and below the product, an impingement zone is present on both sides. The mathematical domain is bounded by the tortilla surface and the center of the stack of tortillas. Because of the symmetry, there is no temperature or moisture concentration gradient at the product's center x 0; therefore, at this boundary the following conditions exist: oM 0; 6 ox x0;t oT 0: 7 ox x0;t At the surface boundary of the tortilla, heat transfer occurs by convection; speci®cally (Chen, 1996): h T1
Ts
k
oT ox
jk cpw T1
Ts 0:
8a
In¯ow of heat is due to the temperature dierence between the surface and the drying medium T1 Ts . Heat ¯ow out at the surface occurs due to conduction into the product and the internal energy of the moisture ¯owing out at the surface, j. Some heat is also absorbed by the moisture ¯owing out to raise its temperature to that of the drying medium and, if it has not been vaporized, to supply its heat of vaporization, k. Eq. (8a) applies when the moisture reaching the surface has not reached its boiling point and is still in its liquid form. Once moisture reaching the surface is vaporized, the latent heat of vaporization term, k, is eliminated and the boundary condition at the surface becomes (Chen, 1996): h T1
Ts
k
oT ox
jcpw T1
Ts 0:
8b
Mass transfer at the surface occurs by convection, as well, for air impingement drying. The boundary condition for convective mass transfer at a ¯at surface is as follows (Chen & Moreira, 1997): Dw
o qb M hm qb Ms ox
M1 :
9
This condition implies that the rate of moisture removal due to convection is equal to the moisture ¯ow rate due to diusion. This boundary condition applies to air drying since the moisture content gradient between the surface and the drying agent is the impetus for moisture removal (Beeby & Potter, 1992). With superheated steam drying, the vapor pressure gradient drives the moisture ¯ux at the surface. The equation estimating pressure-driven mass transfer is as follows:
hm M w R
pv;wb Twb;abs
pv;1 T1;abs
123
Dw
o qb M ; ox
10
which states that the rate of moisture removal due to the pressure gradient is equal to the negative in¯ow due to diusion. This relation was modi®ed from one provided by Brooker, Bakker-Arkema, and Hall (1981) for grain drying to account for dierences in the temperature between the surface and the drying medium. With superheated steam, the vapor pressure in the drying medium is atmospheric pressure because water vapor is the only vapor present (Beeby & Potter, 1992). The same is true at the surface of the product. 2.5. Physical and thermal properties of the corn tortilla Chen and Moreira (1997) provided equations for thermal conductivity, bulk density, and speci®c heat of corn tortillas during frying. Since none of the thermal or physical properties was highly dependent on oil content, their variation with moisture content and temperature should be applicable during drying as well. The mass diusivity of the tortilla was assumed to be constant for a given process but varying with diering drying temperature. Brooker et al. (1981) suggested a method by which the mass diusivity of a product, Dw , with in®nite plane geometry can be found based on the application of the following analytical equation: " # 2 1 8 X 1 2n 1 p2 2 MR 2 X ; exp 11 p n0 2n 12 4 where MR
M t Meq ; Min Meq
12
1=2
Dw t : 13 0:5L In solving Eq. (11) for Dw , only the ®rst term of the summation is used for simpli®cation, giving the following: 2 M t Meq 8 p Dw t 2 exp : 14 p L2 Min Meq X
Based on experimental data of average moisture content over time, Dw for a given process can be estimated. This was done based on data collected by Li et al. (1999) using the Levenberg±Marquardt nonlinear regression procedure with PlotIT software (Version 3.2, Scienti®c Programming, Haslett, MI). Once Dw is found for a given value of T, it can then be found for other temperatures using an Arrhenius-type equation: C2 Dw C1 exp ; 15 Tabs
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L.M. Braud et al. / Journal of Food Engineering 50 (2001) 121±128
where C1 and C2 are constant characteristics of a given product (Brooker et al., 1981). Using nonlinear regression on Microsoft Excel 97 (Microsoft Corporation), these constants were found for impingement drying of corn tortillas at a convective heat transfer coecient of 2 160 W=m °C. For air drying, C1 and C2 were 1:6466 10 6 and 2681.1, respectively (R2 0:99). For steam drying, C1 and C2 were found to be 9:0544 10 6 and 3212.8, respectively (R2 0:99). Other tortilla properties for which impingement drying experimental data were collected were initial moisture content (55% w.b.), initial temperature (22°C) and thickness (1.85 mm).
pressure and properties of the speci®c vapors. For water vapor diusing into air, this equation becomes:
2.6. Properties of the drying media
for 350 K < Tabs < 550 K.
The convective heat transfer coecient, h, for each medium was measured experimentally and held constant for a given process. From this value the convective mass transfer coecient, hm , was determined using a method recommended by Martin (1977) based on a relation among the dimensionless Nusselt, Prandtl, Sherwood, and Schmidt numbers. For a single round impinging jet, the following relation applies:
2.8. Properties of superheated steam
Nu Sh 0:42 ; 0:42 Pr Sc
16
where hD ; kf m Pr ; a
17
Nu
18
hm D ; DAB m : Sc DAB
Sh
19 20
For air drying, A and B are water vapor and air, respectively. When superheated steam is the drying medium, both A and B are water vapor. 2.7. Properties of air The diusion coecient for moisture in dry air, Dwa , and its kinematic viscosity, m, are functions of the air's temperature and pressure. Pang (1997) developed an equation to de®ne this function for kinematic viscosity based on air property tables: 2 mair 1:02 10 5 Tabs 3:31 10 3 Tabs
0:3157 10
5
21
for 200 K < Tabs < 500 K. Holman (1981) provides a semiempirical equation for diusivity coecients in various gases based on operating temperature and
0:39383 T 3=2 : 22 P These were used in determining the properties of dry air. Linear regression was used to develop equations for thermal conductivity and diusivity of air as functions of temperature based on tabular data from Incropera and DeWitt (1996). The resulting equations, both with R2 values of 0.99, are as follows:
Dwa
kair 6:94 10 5 Tabs 5:91 10 3 ;
23
aair 1:84 10 7 Tabs
24
5
3:5 10
For superheated steam, the kinematic viscosity was found as the quotient of dividing the dynamic viscosity by the density. Equations for these values as functions of temperature and pressure were also given by Pang (1997) and include the following: P 3 qsteam 2:166 10 25 Tabs for 323 K < Tabs < 1773 K and 10 kPa < P < 600 kPa and lsteam 3:8 10 8 Tabs
2:2 10
6
26
for 380 K < Tabs < 700 K at atmospheric pressure. Functions for thermal conductivity and diusivity were found from linear regression of tabular values at varying temperatures from Incropera and DeWitt (1996) to be the following: ksteam 7:85 10 5 Tabs
5:21 10 3 ;
asteam 1:58 10 7 Tabs
4 10
5
27 28
for 380 K < Tabs < 550 K. Both regression equations had R2 values of 0.999. Diusivity of vapor into superheated steam was calculated using a semiempirical equation provided by Holman (1981) for diusivity coecients in various gases. For given ¯uids, this value varies with temperature and pressure. Using properties of water, this function was as follows: Dws
0:51332 T 3=2 : P
29
2.9. Model solution The input parameters for the model included initial moisture content, initial product temperature, product thickness, drying medium, drying medium temperature, and convective heat transfer coecient (Li et al., 1999). The explicit ®nite-dierence technique was chosen for
L.M. Braud et al. / Journal of Food Engineering 50 (2001) 121±128
use in this study because of its straightforward determination of unknown values (Incropera & DeWitt, 1996). The ®nite dierence approximation equations were incorporated into a simulation code written in C programming language. The program was compiled and run using Microsoft Visual C++ 6.0 (Microsoft Corporation, Redmond, WA). 2.10. Grid stability requirements The criterion for heat transfer stability of a one-dimensional interior node is as follows (Incropera & DeWitt, 1996): Fo 6
1 : 2
30
At the surface, the following must be satis®ed for stability: Fo 1 Bi 6
1 : 2
125
represent the standard deviation of the three replicates recorded experimentally. 3.2. Moisture content Figs. 1 and 2 represent comparisons of air and superheated steam drying at various temperatures. The experimental drying curves for air and steam were more similar at the lowest temperature of 115°C than at the highest temperature of 145°C; however, due to inaccuracies of the predictions for steam at the lower temperature, this occurrence was not indicated by the predicted results. Underpredictions in moisture content for steam at 115°C caused the predicted drying curves at this temperature to dier more than the experimental curves diered. Fig. 2 indicates the model's ability to predict dierences between the drying curves with air
31
Because of the high external mass transfer coecient, hm , requirements at the surface element would be the limiting factor in grid size selection. To avoid these restraints on time step selection as imposed by stability requirements at the surface element, the moisture content of the surface element was set at an equilibrium value at the ®rst time step. Running the simulation within the stability requirements for mass transfer indicated that the boundary element reached this lower limit within the third time step; therefore, the error in this simpli®cation of the numerical simulation was small and acceptable. Solution of the model with constant product properties was performed and results compared with the exact solution to validate the discretization equations.
Fig. 1. Comparison of moisture content over drying time for air and steam drying T 115°C; h 160 W=m2 °C. Lines represent predicted data; symbols represent experimental data.
2.11. Sensitivity analysis Some processing parameters that could vary in production were assessed to see how they aected the predicted drying results. Included in this analysis were the following parameters: tortilla thickness and initial moisture content, drying temperature, and convective heat transfer coecient. Variations in these parameters assessed were within typical operating conditions that could exist in corn tortilla production.
3. Results 3.1. Model validation Model predictions were compared with experimental data to assess the validity of the simulation output. Error bars in the moisture ratio and temperature graphs
Fig. 2. Comparison of moisture content over drying time for air and steam drying T 145°C; h 160 W=m2 °C. Lines represent predicted data; symbols represent experimental data.
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L.M. Braud et al. / Journal of Food Engineering 50 (2001) 121±128
and steam drying at 145°C. The use of superheated steam eected a faster fall in the moisture content as compared to air at this temperature. This occurrence was also noted in air and superheated steam drying of potatoes because the surface remained softer with steam drying, allowing faster drying (Yoshida & Hyodo, 1966). At the lowest steam temperature (115°C) moisture content predictions were within 20% for the ®rst 5 min only and between 40% and 50% in the ®nal 5 min of drying. The large underprediction of moisture content with steam drying at the lower temperature is most likely due to assumption 3, neglecting the amount of condensate. The amount of condensate depends upon the degree of superheat of the drying vapor (Beeby & Potter, 1992). At equal pressures, vapor at a lower temperature will be less superheated or closer to being at saturated vapor conditions. Thus, more moisture will condense on the product surface at lower steam temperatures because the amount of heat loss for condensation to occur is lower. As a result, the product's net drying rate decreases. This was seen in the experimental data at 115°C in the ®rst minute of drying in which there was almost no net loss of moisture. Not accounting for condensation from the impinging steam, the model underpredicted the moisture content for steam drying at 115°C (Braud, 2000). 3.3. Temperature Representative temperature pro®les over time for air and steam drying are presented in Fig. 3. The center temperature increased over time until it reached the boiling point of water (approximately 100°C); after this point, the temperature remained stationary while latent heating prevented a temperature rise. This phase lasted
Fig. 3. Comparison of temperature over drying time for air and steam drying at various temperatures h 160 W=m2 °C. Lines represent predicted data; symbols represent experimental data.
between 5 and 17 min, depending on the drying medium and its temperature. The internal temperature then rose to its maximum temperature, which was approximately 15°C greater than the established temperature of the drying medium. The center temperature exceeded the temperature set for drying due to local variations within the impingement zone in actual drying conditions. Li et al. (1999) noted that the temperature within the impingement zone showed some variation. The temperature set for a process represented the average temperature while at the zone's center the temperature was maximal. The trends of the temperature predictions showed good agreement with the experimental data at all conditions. At a convective heat transfer coecient of 2 160 W=m °C, the model's predictions at each temperature had a maximum relative error of 16% or less for air drying. The relative error peaked between 31% and 43% at 0.5 min of drying for all steam conditions tested experimentally. After the ®rst minute of drying, none of the predicted temperatures at any condition had a relative error of over 12% as compared to the experimental measurements. As previously mentioned, the existence of an inversion point is commonly noted when comparing air and superheated steam at equal mass ¯ow rates. The experimental data referenced in this study were collected at equal heat transfer coecients rather than mass ¯ow rates. However, at a single heat transfer coecient of 2 160 W=m °C, superheated steam was the more ecient drying agent when the temperature was 130°C and 145°C while the two media exhibited comparable drying rates at 115°C. At both 115°C and 145°C there were dierences between air and steam drying in the duration of the latent heating phase and the amount of time required to reach the maximum temperature. In superheated steam, the latent heating phase was shorter at all temperatures and the product's center reached its maximum temperature more quickly. The latent heating phase was shorter when the amount of moisture in the steam-dried sample was less than in the air-dried sample (at 145°C) and when the air-dried and steam-dried samples had approximately equal moisture contents (115°C); this can be attributed to the fact that structural changes occurred during drying and diered between drying media. This caused the moisture in the steam-dried sample to be more easily vaporized than in the air-dried sample and is indicated by the higher diusivity values with steam drying as compared to air drying. The model's predictions of the temperature dierence at these temperatures were due to the lower predicted moisture content in steam as compared to air. Since the predicted moisture content was lower with steam drying, less time was required for latent heating.
L.M. Braud et al. / Journal of Food Engineering 50 (2001) 121±128
127
3.4. Sensitivity analysis The sensitivity analysis indicated that variations in certain parameters have signi®cant eects on required drying time and thus drier design. These parameters were ranked from most to least in¯uential based on effects seen within the analyzed ranges as follows: product thickness, drying temperature, convective heat transfer coecient, and initial moisture content. Knowledge of the parameters that do signi®cantly impact drying behavior is useful in designing drier operations to ascertain which conditions need the most sensitive control and to assess how controlling uncertainty in these parameters would reduce variation in properties of the ®nal product. 3.5. Eect of product thickness The thickness of the product being dried had a great eect on the drying curve and temperature pro®le over drying time. In commercial production, tortillas usually have a thickness less than that of the tortillas dried experimentally. The thickness typically varies from 0.89 to 1.29 mm (Moreira et al., 1999). To predict the drying behavior of thinner tortillas, the model was run at these two thicknesses for both air and steam at a temperature of 145°C and a convective heat transfer coecient of 2 160 W=m °C. A representative graph of the eect of thickness on drying behavior is provided in Fig. 4 for air drying; the trend seen was similar to the eect of varying product thickness with steam drying. As indicated in this ®gure, the predicted moisture content for the thinnest tortilla (0.89 mm) fell much more rapidly than for the experimental tortilla with thickness 1.85 mm. Fig. 5 illustrates the variations in the temperature pro®le over time for diering thicknesses when dried in air. The
Fig. 5. Predicted and experimental center temperature as a function of drying time for air impingement drying at 145°C and varying product thickness (L). Lines represent predicted data; symbols represent experimental data.
predictions indicate that the thickness aected the amount of time required for latent heating to commence as well as the duration of the latent heating phase. The thinnest product reached the boiling temperature most quickly and remained at this temperature for the shortest amount of time. As expected, the results indicate that heat was able to reach the center of the product more rapidly and moisture reached the surface more quickly in the thinner product because it had less resistance through which to travel. A similar trend was seen with steam drying. 3.6. Eect of drying temperature Variations in the temperature of the drying medium eected changes in both the drying curves and temperature pro®les over time for both air and steam within the temperature range studied. This analysis was run at a 2 heat transfer coecient of 160 W=m °C as the model was most accurate at this condition for air and steam. At lower temperatures, the moisture content fell more slowly because of the lower diusivity or higher resistance. As temperature increased, the center temperature reached the drying medium temperature more quickly, as indicated in Fig. 3 for both air and steam. This is because the higher temperature gradient at the surface and through the product resulted in a higher heat ¯ux. Since the heat ¯owed faster, the duration of the latent heating phase was shorter.
4. Conclusions Fig. 4. Predicted and experimental moisture content ratio as a function of drying time for air impingement drying at 145°C and varying product thickness (L). Lines represent predicted data; symbols represent experimental data.
The model was able to predict the trend in the temperature changes over time at the experimental conditions. This supports the validity of the governing
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L.M. Braud et al. / Journal of Food Engineering 50 (2001) 121±128
equations for energy transfer within the product during drying. The diusion-driven governing equations for mass transfer were able to predict the overall trend of the drying behavior and simulated the exponential decrease in moisture content over time when steam condensation did not impact drying. Comparison of predicted data with experimental measurements indicated that steam condensation should be incorporated into the boundary conditions when steam is the drying medium as its eects were especially evident when drying at low steam temperatures. Also, accounting for changing diusivity in both air and steam drying would improve the accuracy of the model's predictions and the usefulness of such a model as a simulation tool. Acknowledgements The authors wish to thank Dr. Jamal Seyed-Yagoobi for his assistance with use of the impingement drying facilities at the Drying Research Center, Texas A&M University. References Beeby, C., & Potter, O. E. (1992). Steam drying. In R. Toei, & A. S. Mujumdar (Eds.), Drying '92 (pp. 41±58). New York: Hemisphere Publication Corporation.
Braud, M. L. (2000). Mathematical Modeling of Impingement Drying of Corn Tortillas. MS Thesis, Department of Agricultural Engineering, Texas A&M University, College Station, Texas, USA. Brooker, D. B., Bakker-Arkema, F. W., & Hall, C. W. (1981). Drying cereal grains. Westport, CT: The AVI Publishing Company. Chen, Y. (1996). Simulation of a deep-fat process for tortilla chips. M.S. Thesis, Texas A&M University, College Station, Texas. Chen, Y., & Moreira, R. G. (1997). Modeling of a batch deep-fat frying process for tortilla chips. Transactions of the Institution of Chemical Engineers, 70(C), 181±190. Holman, J. P. (1981). Heat transfer (5th ed.). St. Louis: McGraw-Hill. Incropera, F. P., & DeWitt, D. P. (1996). Fundamentals of heat and mass transfer. New York: Wiley. Li, Y. B., Seyed-Yagoobi, J., Moreira, R. G., & Yamsaengsung, R. (1999). Superheated steam impingement drying of tortilla chips. Drying Technology, 17(1&2), 191±213. Lujan-Acosta, J., Moreira, R. G., & Seyed-Yagoobi, J. (1997). Airimpingement drying of tortilla chips. Drying Technology, 15(3&4), 881±897. Martin, H. (1977). Heat and mass transfer between impinging gas jets and solid surfaces. In: J. P. Hartnett, & T. F. Irvine, Jr. (Eds.). Advances in Heat Transfer (Vol. 13, pp. 1±60). New York: Academic Press. Moreira, R. G., Castell-Perez, M. E., & Barrufet, M. A. (1999). Deepfat frying: fundamentals and applications. Gaithersburg, MD: Aspen Publishers. Pang, S. (1997). Some considerations in simulation of superheated steam drying of softwood lumber. Drying Technology, 15(2), 651± 670. Rohsenow, W. M., & Choi, H. Y. (1961). Heat, mass, and momentum transfer. Englewood Clis, NJ: Prentice-Hall. Yoshida, T., & Hyodo, T. (1966). Superheated vapor speeds drying of foods. Food Engineering, 38(4), 86±87.
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Mathematical modeling of impingement drying of corn tortillas Louise M. Braud, Rosana G. Moreira, M. Elena Castell-Perez * Department of Agricultural Engineering, Texas A&M University, College Station, TX 77843-2117, USA Received 27 June 2000; accepted 4 December 2000
Abstract Impingement drying of corn tortillas was modeled using governing equations for heat and mass transfer during the drying process. Mass transfer within the product was modeled as diusion-driven mass ¯ux. Heat transfer was driven according to Fourier's Law of conduction. Boundary conditions for drying in both air and superheated steam were incorporated into the model. Convective heat transfer accounted for heat ¯ow into the product at the surface. When drying in air, convective mass transfer prevailed; in superheated steam, dierences in vapor pressure between the drying medium and the product surface accounted for mass transfer. Temperature and moisture content predictions followed the experimental trends with both air and steam drying (115± 145°C). Steam condensation unaccounted for by the model resulted in underpredictions in the moisture content in steam drying at low temperatures. Product thickness and drying medium temperature had a signi®cant eect on moisture content and temperature pro®le over time. Ó 2001 Elsevier Science Ltd. All rights reserved. Keywords: Superheated steam; Dry air; Finite dierences; Heat and mass transfer
1. Introduction Impingement drying involves the use of a single or multiple gas jets releasing a stream of gas vertically onto a surface. Impinging jets are used for cooling and heating, as well as drying, because they result in higher heat and mass transfer coecients. Dry air and superheated steam are two drying media commonly used in impingement drying. With superheated steam as the drying medium, some vapor will condense on the product surface during the transient early stage of drying. This occurs as the superheated steam contacts the cold solid (Beeby & Potter, 1992). Lujan-Acosta, Moreira, and Seyed-Yagoobi (1997) studied the use of impinging air jets for drying of corn tortillas as an alternative to convection-oven baking prior to frying. The use of this technique resulted in less oil absorption during frying and, therefore, a lower-fat tortilla chip. Chips dried with impinging air had a ®nal oil content of 14% (wet-basis) as opposed to 23±30% for commercial tortilla chips (Moreira, Castell-Perez, & Barrufet, 1999). A comparison between air and superheated steam with impingement drying of corn tortillas with opposing *
Corresponding author. Tel.: +1-979-862-7645. E-mail address: [email protected] (M.E. Castell-Perez).
jets was conducted by Li, Seyed-Yagoobi, Moreira, and Yamsaengsung (1999). Stacking two tortillas between wire meshes allowed placement of thermocouples between them to measure temperature during the drying process. Average moisture content was also recorded over time. Drying at varying temperatures (115°C, 130°C, 145°C) and convective heat transfer coecients 2 (100, 130, 160 W=m K) were conducted with both air and superheated steam. As expected, results indicated that higher steam temperatures resulted in faster drying. In comparing the use of air and superheated steam, there was little dierence in the drying curves at a lower temperature (115°C); however, at a higher temperature (145°C), superheated steam provided higher drying rates. In summary, the use of impingement drying in processing corn tortillas prior to frying shows feasibility in production of a lower-fat fried chip. Although a mathematical model of this processing technique has not been developed for this product, the need for it exists to facilitate ecient process design and optimization. Thus, the main objective of this study was to develop a mathematical model to describe the heat and mass transfer mechanisms during impingement drying of corn tortillas. This was accomplished by: 1. De®ning the governing equations that describe the transport of heat and mass (moisture) during the impingement drying process.
0260-8774/01/$ - see front matter Ó 2001 Elsevier Science Ltd. All rights reserved. PII: S 0 2 6 0 - 8 7 7 4 ( 0 0 ) 0 0 2 3 4 - X
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2. Identifying varying boundary conditions imposed by dierent drying media (e.g. air and superheated steam). 3. Developing a computer algorithm that incorporated the governing equations and boundary conditions to simulate the drying process. 4. Validating the developed mathematical model with existing experimental data (Lujan-Acosta et al., 1997; Li et al., 1999). 5. Conducting a sensitivity analysis on the model and its input parameters to determine the eects of variations in the operating conditions. 2. Methodology 2.1. Assumptions The following assumptions concerning the drying process and changes in the product were made for model development: 1. The tortilla is homogeneous on a macroscopic scale and isotropic. Properties used were the eective properties that were uniform along the radius of the tortilla. 2. Temperature and moisture concentration gradients exist along the thickness of the tortilla only. These parameters were considered uniform along the radius of the tortilla. This assumption allows the use of onedimensional in®nite plate simpli®cations since the radius is much greater than the thickness. 3. Because the steam is highly superheated, the amount of condensate was neglected in analyzing superheated steam impingement drying. 4. The tortilla shrinks in diameter during drying; however, the eects of shrinkage, other than changes in the density and thermal properties of the tortilla, were neglected. Mass diusivity of the tortilla was considered a function of drying conditions only; thus it was held constant for a single drying temperature. 2.2. Governing equations for mass (moisture) transfer The ¯ow of moisture within the tortilla occurs by diusion and is governed by the eective diusion coecient according to Fick's Law. The mass balance of moisture within the stationary tortilla is as follows (Incropera & DeWitt, 1996): o oqb M o qb M Dw ; 1 ox ox ot which states that the rate of change in the moisture content, M, is equal to the diusion of water due to the internal moisture content gradient, oqb M=ox. Moisture within the tortilla is present in three forms, namely free water, bound water, and vapor. The trans-
port of these three forms was combined into a single moisture ¯ux term, j. The equation for this ¯ux within the system is as follows (Rohsenow & Choi, 1961): j
Dw
o qb M : ox
2
For a stationary product, the mass ¯ux of moisture is proportional to the diusivity of water in the tortilla, Dw , the density of the solid and liquid mixture, qb , and the moisture content gradient. 2.3. Governing equations for energy (Heat) transfer Heat transfer within the tortilla is driven by conduction as a temperature gradient develops along its thickness. An equation to describe energy transfer during impingement drying of tortillas is as follows: ocp T oq o oT oT qb cp T b k 3 cpw j : ot ox ox ox ot In this equation, moisture in its separate forms was combined into a single ¯ux term, j. The change in internal energy within the control volume over time is represented by the two terms on the left-hand side of the equation. This energy balance states that the rate of this change in internal energy is equal to the dierence between the rate at which heat is transported out of the control volume due to moisture ¯ux, cpw j oT =ox, and the net rate of heat transported into the control volume due to conduction, o=ox k oT =ox. To account for latent heat storage during vaporization, an approach applied by Chen (1996) was used. When a control volume reaches the boiling point of the water it contains, vaporization will commence. At this point, the temperature of the control volume will remain at the boiling temperature until enough heat ¯ows into the control volume to evaporate its moisture; however, moisture is continuously ¯owing through the control volume, thus changing its moisture content. To quantify the duration of vaporization within such a control volume, the amount of heat ¯owing into the control volume for each incremental time step was compared to the amount of heat required to vaporize the amount of moisture contained in the control volume at that point in time based on the latent heat of vaporization. Once the heat stored during the phase change was enough to vaporize all of the moisture within the control volume, vaporization within the element was complete and its temperature began to elevate from the boiling point. 2.4. Initial and boundary conditions At the onset of the drying process, the temperature and moisture content of the tortilla are uniform. The initial conditions in equation form are thus:
L.M. Braud et al. / Journal of Food Engineering 50 (2001) 121±128
T jx;t0 T0 ;
4
Mjx;t0 M0 :
5
Because jets impinge from both above and below the product, an impingement zone is present on both sides. The mathematical domain is bounded by the tortilla surface and the center of the stack of tortillas. Because of the symmetry, there is no temperature or moisture concentration gradient at the product's center x 0; therefore, at this boundary the following conditions exist: oM 0; 6 ox x0;t oT 0: 7 ox x0;t At the surface boundary of the tortilla, heat transfer occurs by convection; speci®cally (Chen, 1996): h T1
Ts
k
oT ox
jk cpw T1
Ts 0:
8a
In¯ow of heat is due to the temperature dierence between the surface and the drying medium T1 Ts . Heat ¯ow out at the surface occurs due to conduction into the product and the internal energy of the moisture ¯owing out at the surface, j. Some heat is also absorbed by the moisture ¯owing out to raise its temperature to that of the drying medium and, if it has not been vaporized, to supply its heat of vaporization, k. Eq. (8a) applies when the moisture reaching the surface has not reached its boiling point and is still in its liquid form. Once moisture reaching the surface is vaporized, the latent heat of vaporization term, k, is eliminated and the boundary condition at the surface becomes (Chen, 1996): h T1
Ts
k
oT ox
jcpw T1
Ts 0:
8b
Mass transfer at the surface occurs by convection, as well, for air impingement drying. The boundary condition for convective mass transfer at a ¯at surface is as follows (Chen & Moreira, 1997): Dw
o qb M hm qb Ms ox
M1 :
9
This condition implies that the rate of moisture removal due to convection is equal to the moisture ¯ow rate due to diusion. This boundary condition applies to air drying since the moisture content gradient between the surface and the drying agent is the impetus for moisture removal (Beeby & Potter, 1992). With superheated steam drying, the vapor pressure gradient drives the moisture ¯ux at the surface. The equation estimating pressure-driven mass transfer is as follows:
hm M w R
pv;wb Twb;abs
pv;1 T1;abs
123
Dw
o qb M ; ox
10
which states that the rate of moisture removal due to the pressure gradient is equal to the negative in¯ow due to diusion. This relation was modi®ed from one provided by Brooker, Bakker-Arkema, and Hall (1981) for grain drying to account for dierences in the temperature between the surface and the drying medium. With superheated steam, the vapor pressure in the drying medium is atmospheric pressure because water vapor is the only vapor present (Beeby & Potter, 1992). The same is true at the surface of the product. 2.5. Physical and thermal properties of the corn tortilla Chen and Moreira (1997) provided equations for thermal conductivity, bulk density, and speci®c heat of corn tortillas during frying. Since none of the thermal or physical properties was highly dependent on oil content, their variation with moisture content and temperature should be applicable during drying as well. The mass diusivity of the tortilla was assumed to be constant for a given process but varying with diering drying temperature. Brooker et al. (1981) suggested a method by which the mass diusivity of a product, Dw , with in®nite plane geometry can be found based on the application of the following analytical equation: " # 2 1 8 X 1 2n 1 p2 2 MR 2 X ; exp 11 p n0 2n 12 4 where MR
M t Meq ; Min Meq
12
1=2
Dw t : 13 0:5L In solving Eq. (11) for Dw , only the ®rst term of the summation is used for simpli®cation, giving the following: 2 M t Meq 8 p Dw t 2 exp : 14 p L2 Min Meq X
Based on experimental data of average moisture content over time, Dw for a given process can be estimated. This was done based on data collected by Li et al. (1999) using the Levenberg±Marquardt nonlinear regression procedure with PlotIT software (Version 3.2, Scienti®c Programming, Haslett, MI). Once Dw is found for a given value of T, it can then be found for other temperatures using an Arrhenius-type equation: C2 Dw C1 exp ; 15 Tabs
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where C1 and C2 are constant characteristics of a given product (Brooker et al., 1981). Using nonlinear regression on Microsoft Excel 97 (Microsoft Corporation), these constants were found for impingement drying of corn tortillas at a convective heat transfer coecient of 2 160 W=m °C. For air drying, C1 and C2 were 1:6466 10 6 and 2681.1, respectively (R2 0:99). For steam drying, C1 and C2 were found to be 9:0544 10 6 and 3212.8, respectively (R2 0:99). Other tortilla properties for which impingement drying experimental data were collected were initial moisture content (55% w.b.), initial temperature (22°C) and thickness (1.85 mm).
pressure and properties of the speci®c vapors. For water vapor diusing into air, this equation becomes:
2.6. Properties of the drying media
for 350 K < Tabs < 550 K.
The convective heat transfer coecient, h, for each medium was measured experimentally and held constant for a given process. From this value the convective mass transfer coecient, hm , was determined using a method recommended by Martin (1977) based on a relation among the dimensionless Nusselt, Prandtl, Sherwood, and Schmidt numbers. For a single round impinging jet, the following relation applies:
2.8. Properties of superheated steam
Nu Sh 0:42 ; 0:42 Pr Sc
16
where hD ; kf m Pr ; a
17
Nu
18
hm D ; DAB m : Sc DAB
Sh
19 20
For air drying, A and B are water vapor and air, respectively. When superheated steam is the drying medium, both A and B are water vapor. 2.7. Properties of air The diusion coecient for moisture in dry air, Dwa , and its kinematic viscosity, m, are functions of the air's temperature and pressure. Pang (1997) developed an equation to de®ne this function for kinematic viscosity based on air property tables: 2 mair 1:02 10 5 Tabs 3:31 10 3 Tabs
0:3157 10
5
21
for 200 K < Tabs < 500 K. Holman (1981) provides a semiempirical equation for diusivity coecients in various gases based on operating temperature and
0:39383 T 3=2 : 22 P These were used in determining the properties of dry air. Linear regression was used to develop equations for thermal conductivity and diusivity of air as functions of temperature based on tabular data from Incropera and DeWitt (1996). The resulting equations, both with R2 values of 0.99, are as follows:
Dwa
kair 6:94 10 5 Tabs 5:91 10 3 ;
23
aair 1:84 10 7 Tabs
24
5
3:5 10
For superheated steam, the kinematic viscosity was found as the quotient of dividing the dynamic viscosity by the density. Equations for these values as functions of temperature and pressure were also given by Pang (1997) and include the following: P 3 qsteam 2:166 10 25 Tabs for 323 K < Tabs < 1773 K and 10 kPa < P < 600 kPa and lsteam 3:8 10 8 Tabs
2:2 10
6
26
for 380 K < Tabs < 700 K at atmospheric pressure. Functions for thermal conductivity and diusivity were found from linear regression of tabular values at varying temperatures from Incropera and DeWitt (1996) to be the following: ksteam 7:85 10 5 Tabs
5:21 10 3 ;
asteam 1:58 10 7 Tabs
4 10
5
27 28
for 380 K < Tabs < 550 K. Both regression equations had R2 values of 0.999. Diusivity of vapor into superheated steam was calculated using a semiempirical equation provided by Holman (1981) for diusivity coecients in various gases. For given ¯uids, this value varies with temperature and pressure. Using properties of water, this function was as follows: Dws
0:51332 T 3=2 : P
29
2.9. Model solution The input parameters for the model included initial moisture content, initial product temperature, product thickness, drying medium, drying medium temperature, and convective heat transfer coecient (Li et al., 1999). The explicit ®nite-dierence technique was chosen for
L.M. Braud et al. / Journal of Food Engineering 50 (2001) 121±128
use in this study because of its straightforward determination of unknown values (Incropera & DeWitt, 1996). The ®nite dierence approximation equations were incorporated into a simulation code written in C programming language. The program was compiled and run using Microsoft Visual C++ 6.0 (Microsoft Corporation, Redmond, WA). 2.10. Grid stability requirements The criterion for heat transfer stability of a one-dimensional interior node is as follows (Incropera & DeWitt, 1996): Fo 6
1 : 2
30
At the surface, the following must be satis®ed for stability: Fo 1 Bi 6
1 : 2
125
represent the standard deviation of the three replicates recorded experimentally. 3.2. Moisture content Figs. 1 and 2 represent comparisons of air and superheated steam drying at various temperatures. The experimental drying curves for air and steam were more similar at the lowest temperature of 115°C than at the highest temperature of 145°C; however, due to inaccuracies of the predictions for steam at the lower temperature, this occurrence was not indicated by the predicted results. Underpredictions in moisture content for steam at 115°C caused the predicted drying curves at this temperature to dier more than the experimental curves diered. Fig. 2 indicates the model's ability to predict dierences between the drying curves with air
31
Because of the high external mass transfer coecient, hm , requirements at the surface element would be the limiting factor in grid size selection. To avoid these restraints on time step selection as imposed by stability requirements at the surface element, the moisture content of the surface element was set at an equilibrium value at the ®rst time step. Running the simulation within the stability requirements for mass transfer indicated that the boundary element reached this lower limit within the third time step; therefore, the error in this simpli®cation of the numerical simulation was small and acceptable. Solution of the model with constant product properties was performed and results compared with the exact solution to validate the discretization equations.
Fig. 1. Comparison of moisture content over drying time for air and steam drying T 115°C; h 160 W=m2 °C. Lines represent predicted data; symbols represent experimental data.
2.11. Sensitivity analysis Some processing parameters that could vary in production were assessed to see how they aected the predicted drying results. Included in this analysis were the following parameters: tortilla thickness and initial moisture content, drying temperature, and convective heat transfer coecient. Variations in these parameters assessed were within typical operating conditions that could exist in corn tortilla production.
3. Results 3.1. Model validation Model predictions were compared with experimental data to assess the validity of the simulation output. Error bars in the moisture ratio and temperature graphs
Fig. 2. Comparison of moisture content over drying time for air and steam drying T 145°C; h 160 W=m2 °C. Lines represent predicted data; symbols represent experimental data.
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and steam drying at 145°C. The use of superheated steam eected a faster fall in the moisture content as compared to air at this temperature. This occurrence was also noted in air and superheated steam drying of potatoes because the surface remained softer with steam drying, allowing faster drying (Yoshida & Hyodo, 1966). At the lowest steam temperature (115°C) moisture content predictions were within 20% for the ®rst 5 min only and between 40% and 50% in the ®nal 5 min of drying. The large underprediction of moisture content with steam drying at the lower temperature is most likely due to assumption 3, neglecting the amount of condensate. The amount of condensate depends upon the degree of superheat of the drying vapor (Beeby & Potter, 1992). At equal pressures, vapor at a lower temperature will be less superheated or closer to being at saturated vapor conditions. Thus, more moisture will condense on the product surface at lower steam temperatures because the amount of heat loss for condensation to occur is lower. As a result, the product's net drying rate decreases. This was seen in the experimental data at 115°C in the ®rst minute of drying in which there was almost no net loss of moisture. Not accounting for condensation from the impinging steam, the model underpredicted the moisture content for steam drying at 115°C (Braud, 2000). 3.3. Temperature Representative temperature pro®les over time for air and steam drying are presented in Fig. 3. The center temperature increased over time until it reached the boiling point of water (approximately 100°C); after this point, the temperature remained stationary while latent heating prevented a temperature rise. This phase lasted
Fig. 3. Comparison of temperature over drying time for air and steam drying at various temperatures h 160 W=m2 °C. Lines represent predicted data; symbols represent experimental data.
between 5 and 17 min, depending on the drying medium and its temperature. The internal temperature then rose to its maximum temperature, which was approximately 15°C greater than the established temperature of the drying medium. The center temperature exceeded the temperature set for drying due to local variations within the impingement zone in actual drying conditions. Li et al. (1999) noted that the temperature within the impingement zone showed some variation. The temperature set for a process represented the average temperature while at the zone's center the temperature was maximal. The trends of the temperature predictions showed good agreement with the experimental data at all conditions. At a convective heat transfer coecient of 2 160 W=m °C, the model's predictions at each temperature had a maximum relative error of 16% or less for air drying. The relative error peaked between 31% and 43% at 0.5 min of drying for all steam conditions tested experimentally. After the ®rst minute of drying, none of the predicted temperatures at any condition had a relative error of over 12% as compared to the experimental measurements. As previously mentioned, the existence of an inversion point is commonly noted when comparing air and superheated steam at equal mass ¯ow rates. The experimental data referenced in this study were collected at equal heat transfer coecients rather than mass ¯ow rates. However, at a single heat transfer coecient of 2 160 W=m °C, superheated steam was the more ecient drying agent when the temperature was 130°C and 145°C while the two media exhibited comparable drying rates at 115°C. At both 115°C and 145°C there were dierences between air and steam drying in the duration of the latent heating phase and the amount of time required to reach the maximum temperature. In superheated steam, the latent heating phase was shorter at all temperatures and the product's center reached its maximum temperature more quickly. The latent heating phase was shorter when the amount of moisture in the steam-dried sample was less than in the air-dried sample (at 145°C) and when the air-dried and steam-dried samples had approximately equal moisture contents (115°C); this can be attributed to the fact that structural changes occurred during drying and diered between drying media. This caused the moisture in the steam-dried sample to be more easily vaporized than in the air-dried sample and is indicated by the higher diusivity values with steam drying as compared to air drying. The model's predictions of the temperature dierence at these temperatures were due to the lower predicted moisture content in steam as compared to air. Since the predicted moisture content was lower with steam drying, less time was required for latent heating.
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127
3.4. Sensitivity analysis The sensitivity analysis indicated that variations in certain parameters have signi®cant eects on required drying time and thus drier design. These parameters were ranked from most to least in¯uential based on effects seen within the analyzed ranges as follows: product thickness, drying temperature, convective heat transfer coecient, and initial moisture content. Knowledge of the parameters that do signi®cantly impact drying behavior is useful in designing drier operations to ascertain which conditions need the most sensitive control and to assess how controlling uncertainty in these parameters would reduce variation in properties of the ®nal product. 3.5. Eect of product thickness The thickness of the product being dried had a great eect on the drying curve and temperature pro®le over drying time. In commercial production, tortillas usually have a thickness less than that of the tortillas dried experimentally. The thickness typically varies from 0.89 to 1.29 mm (Moreira et al., 1999). To predict the drying behavior of thinner tortillas, the model was run at these two thicknesses for both air and steam at a temperature of 145°C and a convective heat transfer coecient of 2 160 W=m °C. A representative graph of the eect of thickness on drying behavior is provided in Fig. 4 for air drying; the trend seen was similar to the eect of varying product thickness with steam drying. As indicated in this ®gure, the predicted moisture content for the thinnest tortilla (0.89 mm) fell much more rapidly than for the experimental tortilla with thickness 1.85 mm. Fig. 5 illustrates the variations in the temperature pro®le over time for diering thicknesses when dried in air. The
Fig. 5. Predicted and experimental center temperature as a function of drying time for air impingement drying at 145°C and varying product thickness (L). Lines represent predicted data; symbols represent experimental data.
predictions indicate that the thickness aected the amount of time required for latent heating to commence as well as the duration of the latent heating phase. The thinnest product reached the boiling temperature most quickly and remained at this temperature for the shortest amount of time. As expected, the results indicate that heat was able to reach the center of the product more rapidly and moisture reached the surface more quickly in the thinner product because it had less resistance through which to travel. A similar trend was seen with steam drying. 3.6. Eect of drying temperature Variations in the temperature of the drying medium eected changes in both the drying curves and temperature pro®les over time for both air and steam within the temperature range studied. This analysis was run at a 2 heat transfer coecient of 160 W=m °C as the model was most accurate at this condition for air and steam. At lower temperatures, the moisture content fell more slowly because of the lower diusivity or higher resistance. As temperature increased, the center temperature reached the drying medium temperature more quickly, as indicated in Fig. 3 for both air and steam. This is because the higher temperature gradient at the surface and through the product resulted in a higher heat ¯ux. Since the heat ¯owed faster, the duration of the latent heating phase was shorter.
4. Conclusions Fig. 4. Predicted and experimental moisture content ratio as a function of drying time for air impingement drying at 145°C and varying product thickness (L). Lines represent predicted data; symbols represent experimental data.
The model was able to predict the trend in the temperature changes over time at the experimental conditions. This supports the validity of the governing
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equations for energy transfer within the product during drying. The diusion-driven governing equations for mass transfer were able to predict the overall trend of the drying behavior and simulated the exponential decrease in moisture content over time when steam condensation did not impact drying. Comparison of predicted data with experimental measurements indicated that steam condensation should be incorporated into the boundary conditions when steam is the drying medium as its eects were especially evident when drying at low steam temperatures. Also, accounting for changing diusivity in both air and steam drying would improve the accuracy of the model's predictions and the usefulness of such a model as a simulation tool. Acknowledgements The authors wish to thank Dr. Jamal Seyed-Yagoobi for his assistance with use of the impingement drying facilities at the Drying Research Center, Texas A&M University. References Beeby, C., & Potter, O. E. (1992). Steam drying. In R. Toei, & A. S. Mujumdar (Eds.), Drying '92 (pp. 41±58). New York: Hemisphere Publication Corporation.
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