ANALYSIS OF CFD HEAT TRANSFER OF VACUUM FREEZE-DRYING SHELF

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1 HEFAT202 9 th Interntionl Conferene on Het Trnsfer, Fluid Mehnis nd Thermodynmis 6 8 July 202 Mlt ANALYSIS OF CFD HEAT TRANSFER OF VACUUM FREEZE-DRYING SHELF Hong-Ping Cheng *, Shin-min Tsi 2 * Professor, Deprtment of Energy nd Refrigerting Air-Conditioning Engineering, Ntionl Tipei University of Tehnology 2 Dotorl Progrm, Deprtment of Energy nd Refrigerting Air-Conditioning Engineering, Ntionl Tipei University of Tehnology E-mil: hpheng@ntut.edu.tw ABSTRACT The vuum freeze-drying tehnology is extensively used in iohemil tehnology industry, espeilly pplile to mterils tht re sensitive to temperture, esily deteriorte, nd need to remin the originl tste. The min defet of this proess is long proess time. This study used ommeril omputtionl fluid dynmis (CFD) softwre Fluent to reserh the impt of the shelf inside the vuum freezedrying mhine on the het trnsfer performne of the oet eing dried in the ourse of heting, nd used different shelving rrngements nd inlet veloities for numeril simultion of flow field nd het trnsfer properties. The fluid flowing inside the shelf ws 50% glyol wter, the inlet veloity ws 5m/s, 9.4m/s nd 20m/s, nd the shelf were stinless steel. Assuming tht the shelf surfe ws low temperture mteril lyer 233K, nd the shelf nd mterils were in vuum nd het insulted environment, the results showed tht the temperture differene etween the inlet nd outlet rehed ws the smllest when the shelf inlet veloity ws 20m/s, nd ws the highest when the shelf inlet veloity ws 5m/s. Although the het exhnge etween the fluid nd the low temperture lyer ws etter when the inlet veloity ws low, the lrge shelf surfe temperture differene ws likely to use unstle mteril qulity. The inlet veloity ould e set s 9.4m/s in onsidertion of sving the energy of pumps. INTRODUCTION Vuum freeze-drying hs een extensively used in iohemil tehnology industry nd food proessing industry, espeilly pplile to the temperture sensitive nd deteriortive foodstuff, in order to keep the primry tste of the ingredients. The entire drying proess is rried out t low temperture. The moisture is sulimed from the solid stte into the vpor stte diretly, nd removed y vuum pumping nd heting. The foodstuff is heted on the shelf, nd the heting must y uniform in order to improve the qulity nd void the foodstuff ollpsing in the freeze-drying proess. In the studies relted to the het nd mss trnsfer mehnism of primry drying, it is very importnt to keep the rtile drying temperture elow the euteti or ollpse temperture in the primry drying proess [] [2], so s to void the melting of the rtiles eing dried. Generlly, the temperture of rtiles is not ontrolled diretly, ut to ontrol the temperture of shelves nd the pressure of vuum proess hmer to ontrol the temperture of rtiles [3] [4]. At this point, the lne etween the het supplied y shelves nd the het tken wy y the moisture sulimtion inside the rtiles determines the temperture of the rtiles [5] [6]. The ie line goes down when the foodstuff is eing heted, nd the drying lyer forms rrier lyer tht mkes the moisture unlikely to sulime, therefore, the produts nnot e too thik, so s to void n overlong drying period [7]. The studies relted to seondry drying re desried elow. The seondry drying follows the primry drying losely. It desors moisture or voltile mtter out of the porous struture of drying lyer t higher temperture in vuum environment. It is generlly greed tht this proess strts until the ie line of the rtile eing dried rehes the heting plne [8], nd the time spent is pproximtely 30~50% of primry drying [9]. Besides wter, other sustnes in mediine, iotehnology nd other domins re required to e desored t this stge [0,]. It is found in the shelf temperture prmeter ontrol during primry nd seondry drying tht the sulimtion rte n e ontrolled effetively y drying t out the melting temperture of the rtile eing dried. The proess time n e shortened effetively y implementing primry nd seondry drying t the melting temperture nd sorh temperture. The temperture of vuum hmer should e ontrolled t level slightly higher thn the melting temperture in the primry drying proess, nd the temperture of vuum hmer should not e lower thn the sorh temperture in the seondry drying proess [2] [3]. The shelves designed with flt edges nd without protrusions re dvntgeous to the drying proess [4]. Therefore, this study disussed the effet of shelves inside the vuum freeze-drying mhine on the het trnsfer performne of the rtiles eing dried in the heting proess, nd used different plies of shelves nd inlet veloities for numeril simultion of flow field nd het trnsfer properties. 028

2 NUMERICAL MODELS This study used the CFD usiness softwre FLUENT 6.3 to nlyze the het trnsfer inside nd outside shelves, in order to determine preferle shelf design. The most effiient shelf het trnsfer ws otined to sve the energy required y vuum freeze-drying equipments for freezing nd heting, nd to improve the drying qulity of rtiles eing dried. The shelf vuum freeze-drying mhine designed y this lortory ws used s the reserh rrier to studying the het trnsfer nd flow hnnel design for shelves, s shown in Fig.. The vuum proess hmer ontins three plies of shelves, nd the rtile eing dried is pled on the shelf. Fig. 2. shows the three-dimensionl grph nd flow hnnel of shelves. o k l m i n h g Clod trp -50 Fig.. Shemtis of shelf freeze drying system: () hek vlve; () seondry refrigernt; () vuum vity; (d) shelf; (e) ir; (f) dry-ir; (g) wet-ir; (h) pump; (i) heter; () het exhngers; (k) ondensers; (l) reeivers; (m) hek vlve; (n) expnsion vlve; (o) ompressors; (p) vuum pump f d p e stndrd k-ε turulent model eqution, nd ll the equtions re listed s follows: Continuity u = 0 x i Momentum 2 ( + ) u u u µ i + k = 0 l µ t δ ρu u p x 3 x i δ x i (2) x k Energy Γ x µ t + C p T σ τ x Turulent kineti energy pu C p T µ + + = 0 µ t k ρu k ρp ρε x l σ τ x Turulent dissiption energy x + µ Φ + ρε = 0 l v µ 2 + t ε ρεp ρε + l ρu + C + C 2 = 0 x ε σ ε k ε τ k () (3) (4) µ (5) NUMERICAL SIMULATION SETTINGS This study rried out numeril simultion of flow field nd het trnsfer properties of internl flow hnnel of shelves, inluding different plies of shelves nd inlet veloities. There re three ses, the shelf flow hnnel model is shown in Fig Fig. 2 Three-dimensionl grph nd flow hnnel of shelves Fundmentl ssumptions: The following ssumptions were mde efore the nlytil simultion of shelf het trnsfer () Stedy-stte flow field (2) Visous Model is Lminr (3) Pressure-Veloity Coupling is SIMPLE (4) Convetion-Diffusion terms use First Order Upwind (5) The foodstuff is kept t onstnt temperture 233K in the omputtionl proess (6) The shelves re in diti ondition in vuum environment Governing eqution: FLUENT 6.3 uses the ontrol volume finite differene method (CVFDM) for disretiztion of governing equtions, the governing equtions inlude ontinuity, momentum, energy nd turulent model equtions. The turulent model dopts Fig. 3 Shelf flow hnnel model 029

3 The shelves re mde of 304 stinless steel, nd the shelves nd foodstuff re ssumed to e in vuum insultion environment. The numer of grids is shown in Tle. In hypothetil simultion onditions, the involved omputing domin hs one inlet, one outlet, one in-shelf liquid, the others re solid oundries. The ses re respetively desried elow. Tle Numer of grids in ses Cse Cse2 Cse3 Numer of grids The oundry onditions of Cse to Cse 3 re set s follows. () Inlet/outlet oundry The oundry ondition of inlet is set s veloity-inlet. The inlet veloities re 5m/s, 9.4m/s nd 20m/s respetively. The inlet temperture is fixed temperture 253K. The outlet oundry is set s outflow. (2) In-shelf liquid The fluid oundry ondition settings dopt 50% ethylene glyol solution, the fluid properties re density of 085kg/m 3, speifi het of 3490J/kg-K, therml ondutivity of 0.42w/m-k nd visosity oeffiient of 0.029kg/m-s (3) Solid oundry This solid oundry is set s wll nd the mteril is set for steel. Wll surfe hs no temperture het flux=0. The top is ssumed to ontt low temperture mteril t preset temperture of 233K. SIMULATION RESULTS ANALYSIS The internl flow hnnel of the shelf of the frozen vuum dryer uses the fluid to provide het to the shelf. The shelf temperture should e onsiderly even to void exessive temperture differene tht my result in uneven drying qulity. The numeril simultion results use different inlet veloities nd shelf plies to nlyze the temperture hnges etween the seondry oolnt nd the shelf s well s the het trnsfer effets fter the ddition of low temperture mterils. The temperture field ross setions in the diretion of Z t the shelf outlet, entre nd end re used to hek the temperture hnges in outlet nd top of the shelf. The ross setion lotion is s represented y the red dotted line. Cse : Figure 4 illustrtes the setionl drwing of the single-ply shelf temperture field in Z diretion. The setionl lotion s shown in the figure is represented y the red dotted line. The high temperture region on the left of Figure 4 is the inlet nd the high temperture on the right is the outlet. At the topmost of Figure 4, it is the single-ply shelf of inlet veloity t 5m/s. When the inlet veloity is ssumed s 5m/s, the fluid veloity is slower nd the het exhnge effet is reltively etter. As result, the temperture in the outlet diretion is reltively lower. In the middle of Figure 4, it is the shelf with inlet veloity t 9.4m/s. Sine the inlet veloity is reltively fst, the outlet temperture will e higher thn the single-ply shelf with veloity t 5m/s. The ottom of Figure 4 is the shelf with inlet veloity t 20m/s, whih is the single-ply shelf with fstest fluid speed. Hene, the outlet temperture will e the highest, whih is only lower thn the inlet temperture y.45k. It lso pplies to the smllest temperture differene in se of others flow veloity. Figure 4 illustrtes the setion of the temperture field in the middle of the flow hnnel. The setionl lotion is represented y the red dotted line s shown in the Figure 4. Aording to the temperture olours s shown in the figure, the flow veloity is proportionl to the stility of het provided y the fluid. When the fluid flows in the entre of the flow hnnel, the het exhnge effet with the low temperture lyer rehes ertin level. If the het is not suffiient, it will result in reltively lower temperture of the metl etween the shelf nd the flow hnnel. At the ottom of Figure 4 is the shelf with inlet veloity t 20m/s. As the flow veloity is fst, it n provide reltively more het or higher stility. Hene, in terms of temperture distriution, the flow hnnel entrl temperture in the inlet diretion of the left nd the temperture in the outlet diretion re reltively lose. Figure 4 illustrtes the frthest end of the shelf in the longest distne from the inlet, nd the setionl lotion is represented y the red dotted line on the right s shown in Figure 4. At this setionl lotion, there is no fluid ut only solid steel. Aording to the performne in se of vrious flow veloities, when the flow veloity is fster, the deep lue representing low temperture will e less. When it is t the frozen vuum drying stte, the single-ply shelf uses the inlet veloity t 20m/s. It n provide more stle het even t the end of the shelf s ompred with other flow veloities, nd the shelf temperture differene will e reltively smller. 030

4 To explore the impt of the flow hnnel quntity of het nd the low temperture lyer, in the middle of the shelf in Y diretion s shown in Figure 5, 200 points t n intervl of.345mm from left to right re seleted to nlyze the temperture distriution fter het exhnge. The red dotted line of the setion of the Z diretion represents the orresponding lotion of the setion. The green lok on the left is the inlet flow hnnel nd the lue lok on the right is the outlet flow hnnel s shown in Figure 5. The urves illustrte the temperture distriution of the shelf flow hnnel nd the surfe low temperture lyer in se of different flow veloities. When the flow veloity is 20m/s, s the green tringulr urve s shown in Figure 5, the temperture distriution from the inlet flow hnnel t the left to the outflow hnnel on the right is reltively even with very smll temperture differene. When the flow veloity is redued to 9.4m/s, s the red qudrilterl urve s shown in Figure 5 hs suggested, temperture differene ours fter the het exhnge of fluids, nd the temperture is the lowest t the flow hnnel t the fr right. The temperture differene with the flow hnnel in the inlet diretion is 0.3K. When the flow veloity is 5m/s, fter the ontinuous het exhnge of the fluid, the flow veloity will e slower nd the temperture will e highest when the quntity of het is fixed. As the lue dimond urve s shown in Figure 5 hs suggested, the flow hnnel temperture differene is 0.7K, whih is the poorest in terms of shelf surfe temperture uniformity in se of different inlet flow veloities K m/s 9.4m/s 20m/s point Figure 5 Shelf surfe temperture urve in Cse Figure 4 Temperture field of Z-diretion setion of shelf in Cse 03

5 Cse 2: In the frozen vuum drying tehnology, it my require shelf of more thn one ply. Figure 6 illustrtes the two-ply shelf used in Cse 2. In Cse 2, the min inlet nd outlet re loted etween two shelves s shown in Figure 3. Figure 6 illustrtes the temperture field setion in Z diretion nd the setionl lotion is represented y the red dotted line on the right of Figure 6. As shown in Figure 6, the inlet veloity is ssumed s 5m/s. When the numer of shelf is inresed to two plies, the low temperture mteril will inrese. Hene, when the inlet veloity is 5m/s, the upper nd lower plies will exhnge het with the low temperture mteril. Therefore, the outlet temperture will e lower thn the temperture setion of the sme lotion in Cse. As shown in the figure, the metl prt in the middle of the flow hnnel will e t low temperture due to the slow flow veloity, providing lmost no het. Figure 6 illustrtes the two-ply shelf with inlet veloity t 9.4m/s. Due to fster inlet speed nd the grvittionl fore, the inlet temperture of the upper lyer shelf will e lower thn the temperture of lower level, ut the inlet temperture is lmost the sme. At the ottom of Figure 6, it is ssumed tht the inlet veloity is 20m/s, it inreses the upper lyer flow hnnel inlet temperture, nd the inresing flow veloity n inrese outlet temperture more signifintly thn the other two veloities. Figure 6 illustrtes the setion of the temperture field in Z diretion in the middle of the flow hnnel, nd the setionl lotion is represented y the red dotted lines s shown in Figure 6. At the topmost prt of Figure 6, it is the temperture field setion in Z diretion of the shelf with inlet veloity t 5m/s. As shown, the flow hnnel temperture distriution is lower t the lotion loser to the outlet. However, the temperture distriution is lmost the sme in se of the upper ply nd the lower ply shelf. As the flow veloity is reltively slow, the temperture etween flow hnnels is lower thn the shelf in se of veloities with greter temperture differene. In the middle of Figure 6, it is the shelf of inlet veloity t 9.4m/s. In Cse, the temperture distriution of the upper nd ottom plies of shelves is similr, however, the temperture etween flow hnnels is higher thn tht of the inlet veloity t 5m/s, nd evener. The temperture t lotions lose to the outlet is similr to those in other ses s they re reltively lower. In the ottom of the Figure 6, it is the shelf of inlet veloity t 20m/s, the trend of temperture of the upper ply nd the lower ply is similr to the temperture of the lotion lose to the shelf outlet flow hnnel, eing higher thn 9.4m/s y out 0.3K. When the flow veloity is different, the flow hnnel outlet temperture will hnge due to hnging flow veloity. However, the sme temperture grdient will e produed in tempertures of the upper ply nd the lower ply s the temperture hnges of the two plies re lmost the sme. Figure 6 illustrtes the temperture field setion in the Z diretion of the two plies shelf end, nd the setionl lotion is represented y the red dotted line. The temperture t this lotion indites whether the het provided y the flow hnnel n reh the extreme of the shelf end. At the topmost of Figure 6, it is 5m/s followed y 9.4m/s, nd it is 20m/s t the ottom. As seen, the temperture t the shelf end is reltively low. The temperture of the metl etween the shelf orner nd the flow hnnel shows tht the temperture distriution in se of shelves with inlet veloity t 9.4m/s nd 20m/s is more uniform. The ple in the furthest distne n provide suffiient het for the low temperture lyer for het exhnge without muh impt from the lower temperture lyer. Figures 6 to 6 illustrte the temperture hnge from the inlet to the furthest end of the shelf. When the veloity is slow, the temperture hnnel in the inlet diretion will e higher nd the temperture will signifintly redue in the outlet diretion. Menwhile, the temperture of the metl lyer etween the shelf nd the flow hnnel will e grdiently deresing. However, the temperture grdient of flow hnnel of fster inlet veloity is smller. During the freeze-drying proess, greter shelf temperture grdient n more esily led to poor qulity. 032

6 In ddition to the temperture nlysis of the flow hnnel, this study disusses the prt etween flow hnnel nd low temperture lyer. In ples inluding the entrl prt of the shelf, the middle of the flow hnnel nd the lower temperture lyer in the Y diretion, this study nlyzes the temperture distriutions of the flow hnnel het trnsfer s shown in Figures 7 nd 7. As shown in Figure 7, the green lok represents the inlet nd the lue lok represents the outlet. As shown in Figure 7, the dimond urve represents the inlet veloity t 5m/s, the lue line represents the first ply nd the red line represents the seond ply. The first ply shelf outlet pressure is 7.34 kp nd the seond ply shelf outlet pressure is 7.23 kp, the outlet pressure differene is only.5% etween two plies. Hene, the flow differene is not signifint nd the het quntity is lose. Therefore, the temperture urves of the first ply nd the seond ply lmost overlp; however, the temperture differene of the flow hnnel is 0.6K. The tringulr urve represents the inlet veloity t 9.4m/s, the green line represents the first ply, nd the purple line represents the seond ply. The first ply shelf outlet pressure is 5.29kP, the seond ply shelf outlet pressure is 4.37kP, nd the differentil pressure is 6%. However, it is similr to 5m/s in terms of temperture disply s the temperture of the two shelves is identil. However, the temperture differene of the flow hnnel in the inlet diretion is 0.5K. The irulr urve represents the inlet veloity t 20m/s, the light lue line represents the first ply shelf, the ornge line represents the seond ply shelf, the surfe temperture distriution is lmost identil in se of the upper ply shelf nd the lower ply shelf. The temperture differene of the mximum temperture nd the lowest temperture of the flow hnnel is the miniml t out 0.3K m/s first ply 5m/s seond ply K m/s first ply 9.4m/s seond ply m/s first ply m/s seond ply Figure 6 Temperture field of Z-diretion setion of shelf in Cse 2 point Figure 7 Shelf surfe temperture urve in Cse 2 033

7 Cse3 The generl smll type frozen vuum dryer usully uses three-ply shelves. In se of the three-ply shelves, in ddition to nlysis of the flow hnnel het provision uniformity, whether the flow hnnel tempertures of the three shelves re stle nd lose to eh other should lso e disussed. In Cse 3, the inlet is loted etween the seond ply nd the third ply of shelf. Due to the grvittionl fore, the outlet is loted etween the first ply shelf nd the seond ply shelf s shown in Figure 3. Figure 8 illustrtes the temperture field setion in Z diretion of the three-ply shelves in Cse 3. As shown in Figure 8, when the inlet veloity is 5m/s, the temperture will drop to 25K fter the fluids enter the flow hnnel for het exhnge with the low temperture lyer. However, the temperture t the inlet pipe is 253K. At the outlet, the temperture of the seond ply shelf is slightly lower thn the rest plies of shelf. In se of the shelf with inlet veloity t 9.4m/s, the inlet temperture is similr to tht of the inlet veloity t 5m/s. However, the temperture will e higher in the outlet prts. The temperture of the outlet of the middle shelf will e lower thn the other two shelves. When the inlet veloity is 20m/s, the outlet temperture is higher thn the shelves of other inlet veloities. The temperture differene is the lowest mong three ses of different inlet veloities. However, the temperture of the inlet in the middle shelf redues signifintly, nd the outlet temperture is lower thn the upper nd lower plies of shelves. Figure 8 illustrtes the temperture field setion in Z diretion in the middle of the flow hnnel, nd the setionl lotion is represented y the red dotted line s shown in Figure 8. In se of the three-ply shelf, the temperture filed distriution in the middle of the flow hnnel tends to e lose in se of inlet veloity t 20m/s nd 9.4m/s. When the inlet veloity is 5m/s, the temperture of the flow hnnel in the outlet diretion is pprently lower. Figure 8 illustrtes the furthest end of the shelf. In se of the shelf with inlet veloity t 5m/s, the low temperture region in the middle of the two flow hnnels of the first ply shelf nd the third ply shelf is reltively lrge. The flow hnnel provision of temperture nd het is pprently insuffiient. At the sme lotion of the seond ply, lmost no temperture or het is provided. The temperture distriution is lmost the sme in se of inlet veloity t 9.4m/s nd t 20m/s. The high temperture provided y the flow hnnel of the seond ply shelf hs reltively signifint depression. Figure 8 Temperture field of Z-diretion setion of shelf in Cse 3 034

8 Figure 9 illustrtes the temperture urves of the shelf nd the middle of the flow hnnel. When the inlet veloity is 5m/s, the surfe temperture distriution of the first ply shelf is lmost identil with the surfe temperture distriution of the third ply shelf. However, the temperture of the seond ply is reltively lower s the seond ply shelf outlet pressure is 2.27kP nd third ply pressure is 5.23kP with differene y 56%. As result, the temperture outlet temperture will e reltively lower. However, when the inlet veloity is 9.4m/s, the differene of pressure etween the seond ply shelf outlet pressure nd the first ply shelf is 53%, nd the differene is 46% in se of inlet veloity t 20m/s. The use of suh pressure differenes is the different heights of inlet/outlet positions. Although the outlet temperture of the shelf with inlet veloity t 20m/s is the highest, the temperture differene of the seond ply shelf nd the rest two plies of shelves is the minimum when the inlet veloity is 5m/s. However, the het quntity my e insuffiient nd it my prolong the entire proessing time K point 5m/s first ply 5m/s seond ply 5m/s third ply 9.4m/s first ply 9.4m/s seond ply 9.4m/s third ply 20m/s first ply 20m/s seond ply 20m/s third ply Figure 9 Shelf surfe temperture urve in Cse 3 CONCLUSION This study uses the ommeril CFD softwre FLUENT 6.3 to simulte the het trnsfer nlysis in se of different flow veloities nd numer of shelves with the following onlusions:. The single-ply shelf s inlet veloity will ffet the temperture distriution of the entire shelf. For exmple, in se of 20m/s inlet veloity, it n e found tht the self flow hnnel temperture is the most uniform. The mximum temperture differene ours in se of flow veloity t 5m/s. In the two-ply shelf simultion the inrese in low temperture lyer will redue to the outlet temperture. Even in se of the flow hnnel of inlet veloity t 20m/s, the temperture differene is more signifint thn the single-ply shelf. When the numer of shelves inreses to 3, the temperture differene will e the miniml when the flow veloity is 5m/s. However, the temperture differene etween the inlet nd the outlet will e the mximl. When the flow veloity is 9.4m/s, the temperture differene of the seond ply is lower thn other y 0.K, nd the inlet/outlet temperture differene is reltively smller. When the inlet flow veloity is 20m/s, the inlet nd outlet temperture differene is miniml. The temperture differene of the seond ply is smller thn other two plies y 0.3K. 2. The positions of the inlet nd outlet re the mor use of temperture differene etween shelves. Regrding the pressure differene used y positions of different heights, in se of two-ply shelf, the outlet pressure nd temperture of the two plies re lose. However, in se of the three-ply shelf, the inlet/outlet nnot e loted in the middle, resulting in lower outlet pressure nd temperture of the seond ply. 3. When the shelf inlet veloity is 20m/s, the inlet/outlet temperture is miniml. When the shelf inlet veloity is 5m/s, the inlet/outlet temperture differene is mximl. Although slower inlet veloity leds to etter het exhnge with the low temperture lyer, the surfe temperture differene will e lrger. As result, the mteril qulity will e more unstle. For onsidertion of the sving energy of the pump, we n hoose the inlet veloity t 9.4m/s. ACKNOWLEDGMENT This study ws sponsored y the Ntionl Siene Counil, Tiwn NSC( E ) REFERENCES []D. F. Dyer, nd J. E. Sunderlnd, Het nd mss trnsfer mehnism in sulimtion dehydrtion, Journl of Het Trnsfer, vol. 90, pp. 379, 968. [2]Goldlithm, L. Rey, nd W. W. Rothmyr (Eds.), Freez-drying nd Advned Food Tehnology, Ademi Press, New York, pp , 975. [3]J. D. Mellor, Fundmentls of freeze-drying, Ademi Press, New York, 978. [4]N. F. H. He, nd T. J. Rosemn, Lyophiliztion of phrmeutil inetions: theoretil physil model, Journl of Phrmeutil Siene, vol. 68, pp , 979. [5]M. L. Pikl, M. L. Roy, nd S. Shh, Mss nd het trnsfer in vil freeze-drying of phrmeutils: role of the vil, Journl of Phrmeutil Siene, vol. 73, pp , 984. [6]M. J. Pikl, Use of lortory dt in freeze drying proess design: het nd mss trnsfer oeffiients nd omputer simultion of freeze drying, Journl of Prenterl Siene nd Tehnology, vol. 39, pp. 5-38, 985. [7]N.K. Shrm,C.P. Aror,Influene of produt thikness,hmer pressure nd heting onditions on prodution rte of freeze-dried yoghurt, Interntionl Journl of Refrigertion,8(5), ,995. [8]S.A. Goldlith, L. Rey, W.W. Rothmyr, Freeze-Drying nd Advned Food Tehnology, Ademi Press: London, 975. [9]T.W. Rowe, J.W. Snowmn, Edwrds Freeze-Drying, Hndook: Crwley: Cmridge,

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