Modeling a Tomato Dehydration Process Using Geothermal Energy

Transcription

1 Proceeings Worl Geothermal Congress 1 Bali, Inonesia, 5-9 April 1 Moeling a Tomato Dehyration Process Using Geothermal Energy Μ. Kostoglou 1, Ν. Chrysafis an Ν. Anritsos 1 Chemistry Department, Aristotle University of Thessaloniki, Box 116, Thessaloniki, Greece Department of Mechanical Engineering, University of Thessaly, Peion Areos, , Volos, Greece kostoglu@chem.auth.gr, nanrits@mie.uth.gr Keywors: Inustrial use, vegetable rying, moeling ABSTRACT A tomato ehyration unit has been operating since 1 in N. Erasmio, 5 km south of the city of Xanthi (Thrace), an prouces sun-rie tomatoes. The unit uses low-cost geothermal water to heat atmospheric air to C through finne-tube air heater coils. This work eals with the moeling of a tomato rying process in a semicontinuous tunnel rier using low-enthalpy geothermal energy. The moeling proceure consists of two stages: The first stage refers to the moeling of a single tomato piece, whereas the secon stage is concerne with the moeling of the proceure of air rying of tomatoes by moving the trays in a tunnel at batches. The moel can be use for the esign an optimization of the continuous rying process existing in N. Erasmio an elsewhere. 1. INTRODUCTION Dehyration (or rying) of foos is efine as the application of heat uner controlle conitions aiming at removing the majority of the water normally present in a foo by evaporation (Fellows, ). Dehyration of foos is one of the olest methos use to improve foo stability, i.e. to exten the shelf life of foos by a reuction in water activity. The rying process involves the slow removal of the largest part of water containe in the fruit or vegetable so that the moisture contents of the rie prouct is below %. In the Meiterranean countries the traitional technique of vegetable an fruit rying (incluing tomatoes) is by using the sun, a technique that has remaine largely unchange from ancient times. However, on an inustrial scale, most fruit is rie uner the sun, while most vegetables are rie using continuous force-air processes. Drie fruits an vegetables can be prouce by a plethora of rying methos. The selection of the proper technique epens upon the type of foo an the type of characteristics of the final prouct (e.g. Mujumar, 1987). Tunnel-type riers are the most flexible an efficient ehyration systems an they are wiely use in rying fruits an vegetables. In general, the whole rying process can be ivie in three stages: (1) pretreatment stage (selecting, sorting, washing of fresh prouce, processing for color preservation etc.), () rying stage (mainly using a hot air environment) an (3) post-rying treatment stage (quality control of the prouct, storage, packaging). It has been emonstrate that low-meium temperature geothermal energy can be use efficiently in rying several temperature-sensitive fruits an vegetables. The scope of this paper is to moel the process of tomato rying in a tunnel-type rier operating in a semi-continuous way. As it will be iscusse below, this process using low-temperature geothermal waters has been efficiently applie in our country in the past ecae.. GEOTHERMAL DRYING Dehyration-rying of agricultural proucts comprises an interesting application of low an meium-enthalpy geothermal energy, especially when ealing with sensitive fruits an vegetables. Fresh or recycle air is force to pass through an air-water converter an to be heate to temperatures in the range 4-1 C. The hot air passes through or above trays or belts with the raw prouce, resulting in the reuction on their moisture content. In geothermal rying, electric power is only use to rive fans an pumps. Base on 5 statistics, geothermal rying of fruits, vegetable an other agricultural proucts is performe in 15 countries worlwie (Lun et al, 5). The share of rying of agricultural proucts in the total energy use of irect geothermal applications was,8%, an the total energy use in 5 exceee TJ/year. The largest rying plants are locate in New Zealan (rying of alfalfa, woo an paper pulp). The first tomato ehyration unit worlwie has been operating since 1 in the geothermal fiel of N. Erasmio, locate 5 km south-west of Xanthi (Thrace). The unit uses low-salinity geothermal water (with a temperature of 6 C) to heat atmospheric air to C in finne tube air heater coils. In fact, the plant uses the same geothermal well that uring winter provies geothermal water for soil heating aiming at off-season asparagus cultivation an for heating of a.4-ha vegetable greenhouse. The hot air passes omcethrough above trays with fresh tomatoes place in the rying tunnels. The ehyration plant covers an area of about 4 m housing two tunnel-type riers, washers, graing, cutting an packing equipment, offices an two storage rooms kept at a temperature less than 1 C. The length of each tunnel is 13.5 m, the height. m an the with 1.4 m. More information on the whole process can be foun in Anritsos et al. (3) an a schematic of one tunnel rier is epicte in Figure 1. The amount of time require to ry tomatoes epen upon the tomato variety, the temperature an humiity of incoming hot air, the air velocity, the thickness of the tomato slices an the efficiency of the rier. The resience time of the prouct (for tomatoes of roma variety) in the rier is about 3 hours, fixe after a trial-an-error proceure. The raw prouce (mainly well but not overripene tomatoes of the roma variety) is grown locally. The final sun-rie prouct is usually sol as a elicatessen foo at an ae value. The solis content of tomatoes use in the units ranges between 8 to 1% w/w an the moisture content of the final prouct is estimate to be less than %. Accoringly, the weight of the processe prouct is reuce about 1-1 times after rying. Currently the plant 1

2 geothermal water, Τ=59-6 ο C Drie prouct Air Height=. m Air, ~46 ο C Hot air ~57 ο C with 1, m Τ~55 ο C, reinjection Fans Air-water convector Length 13,5 m Fresh prouce Figure 1: Schematic of the tunnel rier for tomato ehyration. is capable of proucing more than 1 kg per ay of ehyrate tomatoes with 1% moisture from 1-1 tn of fresh prouce. The final prouct is store in conitione storage rooms. During the first eight years of operation of the ehyration plant 68 tn of high-quality rie tomatoes were prouce, sol in Greece an abroa. The annual evolution of rie tomato prouction is illustrate in Figure. The unit is capable of rying an other vegetables an fruits an inee, more that 15 kg of peppers, figs, aubergines, cherries an apples have been also rie upon eman in the past three years. Prouction (tn) Year Figure : Annual prouction of sun-rie tomatoes in metric tons. It is by coincience that the temperature of incoming air stream is C, which is consiere to be the optimum temperature for properly rying tomatoes that keep their ark re color an flavor. It is well known that at temperatures 8 C or higher, tomatoes to be rie haren on the outsie an severe oxiative amage is taken place. In a tunnel rier of the type use in N. Erasmio there are Ν trays with tomatoes along the irection of air flow. Every time perio δt, the trays are move by one position an the final tray with the rie prouct is remove an a new tray with fresh prouce is introuce in the tunnel. The locations of the trays along the tunnel are given by the subscript i (i=1,, N). 3. MODELING OF THE DRYING PROCESS The moeling proceure consists of two stages: The first is the moeling of the single tomato piece an the secon is the moeling of the particular proceure for air rying of the tomatoes by moving the trays with the prouce within the tunnel at batches. For the first stage the problem is rather to select than to evelop a particular moel since there is a plethora of rying moels for foo-stuff in the literature from quite empirical to relatively funamental ones, each one covering particular aspects of the process. The secon stage requires the evelopment of a moel base on the appropriate mass an heat balances vali uner reasonable assumptions. This is a new evelopment since the particular rying operation has not been stuie in the literature accoring to our knowlege. 3.1 Stage I. Single Tomato Drying Moel Foos are extremely complex in their structure so there is not a general acceptable way to moel their rying behavior leaing to the appearance in the literature of several moeling methoologies. The most usual are the lumpe purely empirical moels resulting by simply fitting the rying curves an offering no other type of information (e.g. Mujumar, 1987; Togrul, 5). Nevertheless sometimes it is attempte to relate the fitting parameters with parameter having a physical meaning like the iffusion coefficient (Keey, 197). Regaring the more phenomenological approaches their use epens on the nature of the prouct unergoing rying. The two major categories of the proucts are the porous an the nonporous. The actual mechanisms of rying for the two categories are ifferent; capillarity ominates in the first case an molecular iffusion in the secon case. Nevertheless, moels evelope for the one category are use for the other very usual in the literature. Even in reality, materials being nonporous at the beginning of rying, are transforme to porous at the later stages (e.g. Vallous et al, ). Let us have a look at the possible moeling approaches for the case of porous materials. The most funamental approach is the irect solution of the governing equations (using avance techniques as the Lattice-Boltzmann methos in a geometry similar to this of the material (reconstructe from two imensional images or X-ray tomography). Although this approach is feasible toay, its value for engineering type calculations is questionable. The next level of reuce sophistication is to use instea of the actual geometry of the material, a network of pores with iealize shape (Yiotis et al, 6). The literature is reach in moels of this level but although they are useful to analyze the interaction between the scales (pore vs macroscopic) their value for engineering calculations is also questionable. At the next level the geometry is greatly simplifie an the material is consiere as homogeneous

3 whereas the localize governing equations are transforme to effective ones vali for the assume homogeneous materials. Three ifferent approaches are met in the literature for the equations that escribes the heat an mass transfer in the material unergoing rying (Kocaefe et al, 7): i) At first the oler one for the case of capillary porous meia, the moel of rying of Luikov (1975). This moel inclues couple heat an mass transfer equations. ii) The secon an simpler moel is base on simple iffusion an conuction equations couple to each other only through the bounary conitions. This moel is actually vali only for non-porous materials but in rying literature, it is use for porous material too. An extene iscussion on this inconsistency an on the exact relation of this moel to the other homogeneous material moels (i) an (iii) can be foun in Datta (6) (iii) The thir an most complicate approach is the so calle multiphase moel which inclues couple equations for the heat, mass an momentum transfer. This moel although more sophisticate than the others has the isavantage that it contains a large number of phenomenological parameters. Regaring the non-porous materials the only rying moeling approach inclues the iffusion an conuction equations with bounary coupling. The shrinkage also is very important (more for non-porous an less for porous materials) an it is taken into account using from very sophisticate to empirical approaches. Now having the above general view on what type of rying moels exists in the literature let us examine the specific case of the tomato. Tomato (solanum lycopersicum) is a non-porous material with a very inhomogeneous structure. There are regions with quite ifferent properties, obviously the rying of the internal more wet material is much easier than the material close to the surface of the tomato. Also the egree of shrinkage is extremely large. Despite all this complexity the experimental rying curves are very simple an can be fitte with a first orer moel (e.g. Kiranouis et al, 1993) or equivalently a simple iffusion moel (Krokia et al, 3). The shape of the tomato piece consiere in the present work is hemispherical but it is not appropriate to solve the iffusion equation in spherical coorinates since the evaporation of water occurs mainly from the plane surface. It is more relevant to approximate the semispherical piece with a cylinrical slice (with evaporation only from the upper sie) of the same raius an of thickness resulting from the total mass conservation. In the case of tomato with a raius R, the equivalent slice thickness,, can be erive by equating the volume of hemisphere to that of a cylinrical slice as R = (1) 3 Normally the iffusion moel requires also the solution of the conuction equation in the material but as the rying time is very large with respect to the characteristic time for conuction in the tomato, transients can be ignore an it can be consiere that the tomato has always a uniform temperature equal to this of the gas. Regaring shrinkage, it is not taken explicitly into account but it is consiere that the rying acceleration resulting from the size reuction counterbalances the eceleration ue to the reuction of the iffusivity as the water content ecreases. So the overall process can be moele as constant thickness-constant iffusivity, iffusion process. The governing equation (which is one-imensional ue to the geometry transformation escribe above) is: X X = D t x where x is the vertical istance (x= on the tray surface), t is the time an X the tomato moisture in ry basis an D is the effective iffusivity. One of the bounary conitions is X the no-water flux on the tray surface (i.e. = ). x x= The secon bounary conition shoul be the regular convection conition for mass transfer from gas to soli. The mass transfer coefficient is well-known to epen on temperature. This more funamental approach is abanone here by introucing another aspect of empiricism. A Dirichlet bounary conition is assume by setting the surface moisture equal to equilibrium moisture X e for the particular conitions of the gas (temperature T g an humiity X g ) an the anticipate effect of the gas velocity on the overall rying rate is assigne to the (gas velocity epenent) iffusivity. So the secon bounary conition takes the form X(,t)=X e (T g,x g ). The next step is to solve the equation () with its bounary conitions. The problem can be solve analytically resulting in a series solution, but usually only the first term of the series (vali at large time) is taken into account. The analytical solution cannot be use for cases of non-constant gas conitions, so in orer to avoi a numerical solution (inconsistent with the level of sophistication of the moel), the following approximate solution technique is propose: The moisture profile is assume to have a parabolic shape (very similar to the sinusoial profile of the first term of the analytical solution). The profile shape which fulfills both the bounary conitions is: e () X= X + c( x ) (3) Taking the integral of the equation from x= to x= results in: X X x = D x t x X X Xx = D D t x x = x= x The average moisture in the tomato will be: (4) e (5) 1 Xm = Xx = X + c 3 Now solving equation (3) with respect to c an substituting in (5) the moisture profile can be erive in terms of the average moisture. A further substitution in (4) leas to: X t m D = 3 (X m X e) (6) This equation is equivalent to the first term of the analytical solution but it has a slightly ifferent proportionality coefficient. The present result is more appropriate in that it comprises a uniform approximation to the exact solution rather than a limiting case. The approach use here is similar to the so-calle Linear Driving Force approximation which has been use extensively in asorption stuies an originally propose by Glueckauf an Coates (1947). 3

4 Equation (6) is the final equation for the single tomato moisture evolution base on a compromise between funamental mechanisms, literature an experience on one han an simplicity, flexibility an parametric epenence on the other. Accoring to this equation the gas state influences the rying process as follows: gas velocity influences the kinetics of the process (D), gas humiity influences the riving force (equilibrium X e ) an gas temperature influences both kinetics an equilibrium. 3. Stage : Drying Process Moel The rying process is shown schematically in Figure 1. The gas has an inlet temperature, T go, an inlet humiity, X go (kg water/kg air), an a mass flow rate F. The temperature an the humiity of the gas changes along the rying chamber an this is why a process moel has to be evelope in aition to the single tomato moel. In the chamber there are N trays with tomatoes along the flow. At each time step δt the line with the trays procees by one tray position so the tomatoes of the final tray are receive as rie prouct as one new tray is ae to the line. The whole process operates in a perioic steay state but before trying to analyze its ynamics let us erive the governing equations ignoring the motion of the trays. The positions of the trays along the flow are enote with the subscript i (i=1, N). It is assume that the conitions on each tray are uniform i.e. all the tomatoes on it rying at the same rate. This implies that the gas has constant properties over each tray. The region over a tray can be consiere as a perfect gas mixture. This assumption is vali when the changes over the tray are not very pronounce an it correspons to a finite volume iscretization of the gas phase equations. The evolution of the average moisture in tray i is given by: Xi 3D i (X i X ei ) t = i=1,,3,.n (7) where X ei is the equilibrium moisture corresponing to the local gas conitions (X gi, T gi ) an D i is the iffusivity corresponing to T gi. The gas humiity balance over the tray i is (assuming perfect mixing): X + = (8) F(X i g 1 X gi) mi t where m i is the mass of tomato on tray i. There is no nee for an equation for the tomato temperature since, as it has been escribe in the previous section, tomato for the slow gas temperature variation in this situation is in thermal equilibrium with the gas. Consequently, it can be assume that it always has the local gas temperature. A global heat balance in the gas phase for the tray i can be written as X + = (9) Fc i p(tg 1 T gi) mi H t The bounary conitions of the above problem are X g1 =X go =inlet gas humiity an T g1 =T go =inlet gas temperature. In case of batch operation of the process the initial conition is simply X i =X o =initial moisture of the tomato. For the case of perioic semi-continuous operation the situation is more complex. At a specific location i there is a perioicity of the tomato an gas properties associate to a perio δt. Each tray has a resience time Nδt an its motion along the tunnel is escribe in the Eulerian frame of reference employe here, by the following conitions (j=1,, N): at t = jδ t : X = X, X = X ( i =,1,..N 1) i+ 1 i f N (1) where X f is the moisture of the final prouct. To better analyze the situation the following non-imensional parameters are introuce: D = D(T ); D = D(T ) / D(T ); o go T g go 3Dot / ; B 3Dom / ( F ); = oδ τ= = p 3D t/ (11) Using this imensionless variables an some algebra the system of equations takes the form X i = (X i X ei ) τ (1) Xgi+ 1 Xgi = BD Ti(Xi X ei) (13) H = (14) Tgi+ 1 Tgi BD Ti(Xi X ei ) c p As it can be seen in the above equations the key esign parameters of the process are the number of trays N, the imensionless time step p an the imensionless parameter B. Fortunately, the mathematical problem has an analytical solution escribe by the following set of equations. For the time perio between jδτ an (j+1)p it hols (j=,1, N-1): τ= jp +τ (15a) X = X + (X X )e τ (15b) j+ 1 ej+ 1 jf ej+ 1 p X = X + (X X )e (15c) jf ej j 1f ej Xgj+ 1 = BD Tj(Xj X ej) + Xgj (15) H T = BD (X X ) + T (15e) gj+ 1 Tj j ej gj cp In the above equations X jf stans for the moisture of the tray j at time τ=jp an X -1f =X o. The above relations give by sequential substitutions the temporal an spatial evolution of the variables X, X g an T g. The solution is of Lagrangian type since inex j refers to both the position along the ryer an time. To test the implementation of the above solution to a special algorithm some inicative results revealing the influence of the esign parameters p an B on the rying process will be presente. A typical set of rying conitions is consiere (X o =9, X go =., T go =6 o C, N=5, p=.5). The effect of the parameter B on the rying performance is shown in Figure 3. The curves in this figure can be interprete in two ways: (i) as regular rying curves for the material of a particular tray as it moves along the ryer (Langrangian escription, time axis t/p with t= being the moment the tray enters the ryer) an (ii) as spatial istribution along the ryer (Eulerian escription, length axis where L is the length that correspons to a tray) at the time moment being an integer multiple of p (moment of changing tray position). 4

5 1 6 Tomato moisture (ry basis) B= B=.5 B=.1 B=. T g ( o C) B=.5 B= B=.1 B= i,t/p Figure 3: Moisture profile of the prouct along the rier for various values of parameter B. When B= the gas flow rate is large enough to keep the gas properties constant along the ryer. As B increases the gas humiity increases an its temperature ecreases along the ryer leaing to an increase of equilibrium moisture of the tomato an thus, to a reuce riving force for rying. This behaviour is shown in Figure 3. Also it can be notice from this Figure that the largest amount of moisture removal is achieve at the first half of the resience time. Although this is true for all rying processes, it is more pronounce in the present ryer ue to the variation of the gas properties along the flow (as time procees the rying riving force ecreases). The variation of the gas humiity along the ryer for several values of B is shown in Figure 4. As B increases the variation of gas humiity increases too. On the other han, the temperature of the gas ecreases along the ryer by an amount of cooling increasing with the value of B as it can be seen in Figure 5. From Figures 4 an 5 it is obvious that the most abrupt changes of the gas properties occur close to the entrance of the ryer Figure 5: Variation of gas temperature along the ryer for several values of B. The moisture of the rie prouct is presente against the value of B in Figure 6 for two values of the parameter p. As it is expecte the moisture of the prouct increases as B increases an p (equivalent to rying time) ecreases. Using a figure like this one, one can choose the appropriate rying time for a particular pair of esire prouct moisture an B. The equilibrium moisture X e is compute using the GAB equation which is the most wiely use for foostuffs an gives acceptable preictions in any case (e.g. Kiranouis et al, 1996): ( ) X = X Ck α (1 k α ) 1 (1 C)kα e m w w w (16) where X m is the monolayer moisture content, α w the water activity, an C, k are constants with an Arrhenius temperature epenency. The corresponing preexponential factors an activation energies, an also X m can be foun tabulate in the literature for several vegetables (with tomato among them). The water activity in the tomato is estimate as α w =RH/1 where RH is the relative humiity of the air. Air humiity (kg vapor/kg ry air) B= B=.5 B=.1 B= Final tomato moisture (ry basis).35 p=.5.3 p= B Figure 4: Variation of gas humiity along the ryer for several values of B. Figure 6: Moisture of the final rie prouct for two values of the parameter p. 5

6 The functional form with respect to gas velocity an temperature of the effective moisture iffusivity in the tomato is taken by Hawlaner et al. (1991):.5 1.u D D e e o 34/T g = (17) where u is the gas velocity. In the above general parametric stuy, the parameters were imensionless so there was no nee to know the value of D o. In orer to simulate an actual ryer an to give the results in imensional units we have to estimate this value. This was one by simulating the batch experiments escribe in (Krokia et al, 3) an requiring to fin a similar rying time. The final result is D o = m /s. A particular continuous ryer will be simulate now. The total flow rate is 8 m 3 /hr, the vapor molar fraction in the inlet flow is 4% (corresponing to a relative humiity of %) an the inlet flow temperature is 58 o C. There are 3 lines of trays with each line containing N=5 trays. The time step δt equals to one hour. The spacing between the lines of trays is such that the gas velocity takes the value 1.5 m/s. The main parameter of which the effect will be stuie here is the tomato loaing of the tray (m in kg of tomatoes). The rying curve for a particular tray (or equivalently the moisture istribution along the ryer) is shown in Figure 7 for several values of the tray loaing m. In the limit of zero loaing the final moisture is 1% but for loaings of practical interest the moisture of the prouct is much larger an increases with the loaing. The only way to keep the prouct moisture low increasing the prouction rate m is to increase the air flow rate (in orer to keep constant the value of parameter B. The reason for the increase of the prouct moisture as m increases can become evient by observing Figures 8 an 9. In Figure 8 the gas humiity along the ryer for three values of m is presente. As m increases the humiity goes to larger values with the major increase to occur at the first stages of the ryer. On the opposite, as m increases the temperature goes to lower values. So the reason for large moisture of the prouct as m increase is not that the limite rying time but the shifting of the equilibrium ue to the increase humiity an ecrease temperature of the gas. The moisture of the prouct is presente against the tray loaing m in Figure 1 for several gas velocities (constant flow rate). Increasing the velocity from 1.5 m/s to m/s has just a small influence on the prouct, efinitely smaller than increasing flow rate with constant velocity. The moel evelope here after being calibrate with experimental results for a particular prouct (i.e. to fin D o ) can be use for the esign an optimization of the continuous rying process stuying in the present work. Air humiity (kg vapour/kg ry air) m=4 m=6 m= Figure 8: Gas humiity along the ryer for three values of m. 4. CONCLUDING REMARKS A moel for the semi-continuous foo rying process in a tunnel is evelope in this work after being calibrate with experimental results for a particular prouct (i.e. to fin D o ). This moel can be efficiently use for the esign an optimization of the continuous rier system for tomatoes employe in N. Erasmio using low-temperature geothermal waters. 6 Tomato moisture (ry basis) m= m=4 m=6 m=8 T g ( o C) m=4 m=6 m= i,t/p Figure 7: Moisture istribution along the rier for various tray loaings (kg/tray). 6 Figure 9: Gas temperature along the ryer for three values of m.

7 X f v=1 v=1.5 v= Figure 1: Moisture of the final prouct as a function of air velocity u (in m/s). REFERENCES Anritsos, N., Dalampakis, P., an Kolios, N.: Use of Geothermal Energy for Tomato Drying, GeoHeat Center Quarterly Bul., 4(1), (March 3) Datta, A.K. Porous meia approaches to stuying simultaneous heat an mass transfer in foo processes. I: Problem formulations. J. Foo Engineering, 8 (7) Fellows, P.J.: Foo Processing Technology: Principles an Practice, CRC Press, New York, pp (). Glueckauf, E. an Coates, J.I.: Theory of chromatography, Part 4. The influence of incomplete equilibrium on the front bounary of chromatograms an the effectiveness of separation, J. Chem. Soc. (1947) m Hawlaer, M.N.A., Uin, M.S., Ho, J.S., an Teng, A.B.W.: Drying Characteristics of Tomatoes, J. Foo Engineering, 14, (1991) Keey, R.B.: Drying, Principle an Practice, Pergamon Press, New York, 197. Kiranouis, C.T., Maroulis, Z.B., Tsami, E., an Marinos- Kouris, D.: Equilibrium Moisture Content an Heat of Desorption of Some Vegetables, J. Foo Engineering,, (1993) Kocaefe, D., Younsi, R., Poncsak, S., Kocaefe, Y.: Comparison of ifferent moelsfor the high temperature heat treatment of woo, Int. J. Thermal Sciences, 46 (7) Krokia, M.K., Karathanos, V.T., Maroulis, Z.B., an Marinos-Kouris, D.: Drying Kinetics of Some Vegetables, J. Foo Engineering, 59, (3) Luikov, A.V.: Systems of Differential Equations of Heat an Mass Transfer in Capillary-porous Boies, Int. J. Heat Mass Transfer, 18, (1975) Lun, J.W., Freeston, D.H., Boy, T.L.: Direct application of Geothermal Energy: 5 Worlwie Review, Geothermics, 34, (5) Mujumar, A.S.: Hanbook of Inustrial Drying, Marcel Dekker Inc., New York, pp (1987). Togrul, H.: Simple moeling of infrare rying of fresh apple slices, J. Foo Engineering, 71 (5) Vallous, N.A., Gavrieliou, M.A., Karapantsios, T.D. an Kostoglou, M. Performance of a ouble rum ryer for proucing pregelatinize maize starches, J. Foo Engineering, 51, ,. Yotis, A., Tsimpanogiannis, I.N., Stubos, A.K., Yortsos, Y.C.: Pore network stuy of the characteristic perios in the rying of porous materials, J. Colloi Interface Science, 97 (6),