A Numerical Model of Commodity Moisture Loss and Temperature Distribution During Refrigerated Storage

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1 A Numericl Model of Commodiy Moiure Lo nd Temperure Diribuion During Refrigered Sorge B.R. Becker nd B.A. Fricke Mechnicl Engineering Deprmen Univeriy of Miouri-Kn Ciy 5 Rockhill Rod, Kn Ciy, MO U.S.A. Keyword: rnpirion, repirion, numericl model, bulk refrigerion, he lod Abrc A numericl model w developed o eime he len nd enible he lod, moiure lo, nd emperure diribuion during he bulk orge of wide vriey of frui nd vegeble. A porou medi pproch w uilized o model he combined phenomen of rnpirion, repirion, ir flow, nd convecive he nd m rnfer. Thi pper dicue he modeling mehodology uilized in he curren compuer lgorihm nd decribe he developmen of he he nd m rnfer model. The reul of he compuer lgorihm re compred o experimenl d ken from he lierure. INTRODUCTION Thi pper decribe numericl model which w developed o id in he deign of bulk refrigerion fciliie for frui nd vegeble. A porou medi model incorporing he combined phenomen of rnpirion, repirion, ir flow, nd convecive he nd m rnfer i ued o eime he len nd enible he lod during refrigered orge of frui nd vegeble. The model lo predic he commodiy moiure lo nd emperure diribuion. A lierure review reveled everl exiing he rnfer model for bulk commodiy refrigerion (Bkker-Arkem nd Bicker, 966; Bird nd Gffney, 976; Adre nd Hellickon, 989; Gn nd Wood, 989; Tlbo e l., 99; McKinnon nd Bilnki, 992. However, hee model do no dequely ddre he effec of rnpirion, repirion, evporive cooling nd inernl commodiy emperure grdien. In ddiion, hee model re commodiy pecific nd do no eime enible nd len he lod. Thu, he curren numericl model w developed o eime he len nd enible he lod well he moiure lo nd emperure diribuion in he bulk refrigerion of wide vriey of frui nd vegeble. MODELING METHODOLOGY A depiced in Figure, he compuionl model i bed upon one dimenionl ir flow pern wihin bulk lod of frui or vegeble. In he compuionl model, he bulk lod i repreened porou medium compoed of "commodiy compuionl cell." The condiioned ir i modeled "ir prcel" which move hrough he "commodiy compuionl cell." Clculion commence wih pecified iniil emperure nd humidiy for he bulk lod nd he ir conined wihin i. A hown in Figure, he ime-epping begin wih he fir refrigered "ir prcel" moving ino he fir "commodiy compuionl cell." A he me ime, ech of he iniil "ir prcel" move from i originl cell ino he djcen cell, while he "ir prcel" wihin he l "commodiy compuionl cell" move from he bulk lod ino he plenum of he refrigerion uni. Wihin ech "commodiy compuionl cell," he rnpirion re, m&, i clculed for he ime-ep,

2 ?. The m frcion of wer vpor in ech "ir prcel" i hen upded o reflec he effec of rnpirion. Subequenly, wihin ech cell, he he generion due o repirion, W, he he rnfer from he commodiy nd he evporive cooling due o rnpirion re clculed for he ime-ep. Then, wihin ech cell, he commodiy emperure nd he "ir prcel" emperure re boh upded o reflec he effec of repirion, he rnfer nd evporive cooling, hu compleing he clculion for hi ime-ep. A hown in Figure b, he fir "ir prcel" move o he econd "commodiy compuionl cell" nd newly condiioned econd "ir prcel" move ino he fir "commodiy compuionl cell." Thi econd "ir prcel" encouner he previouly upded commodiy emperure in he fir "commodiy compuionl cell." A he ime-epping coninue, ech "ir prcel" rvere he enire bulk lod. The m frcion of wer vpor conined in ech "ir prcel," when i exi he bulk lod, i ued o clcule he len he lod correponding o h "ir prcel," while i emperure i ued o clcule i enible he lod. A hi lgorihm ime-ep owrd edy e, n eime of he ime hiorie of he len nd enible he lod, well moiure lo nd emperure diribuion, re obined. Figure. Compuionl model of refrigered ir flow hrough bulk lod of commodiy. MASS TRANSFER CALCULATION Trnpirion i he moiure lo proce compoed of moiure rnpor hrough he commodiy kin, evporion from he commodiy urfce nd convecive m rnpor o he urrounding. The driving force for rnpirion i he wer vpor preure defici beween he commodiy urfce nd he urrounding ir: m& = k ( P P ( The wer vpor preure he commodiy urfce, P, i he urion preure he commodiy urfce emperure. The wer vpor preure in he ir, P, i funcion of he m frcion of wer vpor in he ir. Boh P nd P re evlued he previou ime ep by uilizing pychromeric relionhip (ASHRAE, 97. Focken nd Meffer (972 ugge h he rnpirion coefficien, k, cn be modeled follow: 2

3 k = + k k (2 The ir film m rnfer coefficien, k, decribe he convecive m rnfer which occur he urfce of he commodiy. The ir film m rnfer coefficien cn be obined from he Sherwood number, Sh, vi he following Sherwood-Reynold-Schmid correlion for convecive m rnfer from phere (Genkopli, 978: k d Sh = = Re Sc (3 δ The kin m rnfer coefficien, k, decribe he kin' diffuionl reince o moiure migrion nd i dependen upon he frcion of he commodiy urfce covered by pore. A uch, i i heoreiclly difficul o deermine he kin m rnfer coefficien, nd hu, k mu be deermined experimenlly (Chu e l., 987; Gn nd Wood, 989. During he ime ep,?, he m of wer vpor in he ir of he compuionl cell incree follow: m H 2 O = mh 2O + m& (4 The upded m frcion of wer vpor in he ir of he compuionl cell, m f, hen become: m f mh 2O = m + m& Thi complee he rnpirion clculion for one compuionl cell for he curren ime ep. HEAT TRANSFER CALCULATION The bulk lod w modeled porou medium compoed of phericl commodiie wih uniform repirory he generion deermined funcion of emperure vi he Becker-Fricke correlion (Becker nd Fricke, 996. Furhermore, he emperure wihin commodiy w umed o vry only in he rdil direcion. Thu, he governing form of he rnien he equion become: (5 k 2 T T 2 r + ρw = ρc (6 r r r An explici finie difference echnique w pplied o Equion (6 by dividing commodiy ino N phericl hell. The reuling finie difference equion pplicble o he cener node i given follow: ka ( T2 cv( T T + ρ vw = (7 ρ T The reuling finie difference equion pplicble o he inerior node i given follow: ka i + ρ ( Ti Ti kai ( Ti Ti cvi ( Ti Ti + + ρ viwi = (8 A he urfce of he commodiy, convecion he rnfer, rdiion he rnfer, nd evporive cooling due o rnpirion mu be conidered. Thu, he finie difference equion he urfce become: 3

4 kan cvn ( TN TN TN LmA & + ρv NWN = (9 TN TN + heff A T ρ ( ( The effecive he rnfer coefficien, h eff, include boh convecion nd rdiion: h = h + h ( eff convecion rdiion The convecion he rnfer coefficien, h convecion, cn be obined from he Nuel number, Nu, vi he following Nuel-Reynold-Prndl correlion (Genkopli, 978: Nu h convecion = = Re Pr ( k ir The rdiion he rnfer coefficien, h rdiion, in Equion ( i given by: d 2 2 h = σ ( T + T ( T + T (2 rdiion The formulion given by Equion (7, (8 nd (9 define he emperure diribuion wihin ingle commodiy. However, Equion (9 require knowledge of he emperure of he ir prcel reiden wihin he "commodiy compuionl cell," T. Thi ir emperure i deermined ech ime ep by performing n energy blnce beween he ir prcel nd h porion of he bulk lod which i conined wihin he "commodiy compuionl cell:" n h c eff A ( T T N = m c p, ( T T Since Equion (7, (8, (9 nd (3 re explici finie difference equion, hey cn be olved direcly for he upded nodl emperure. The he rnfer clculion begin he cener node of he commodiy nd proceed ouwrd o he ir prcel. Thi complee he he rnfer clculion for one compuionl cell for he curren ime ep. EXPERIMENTAL VERIFICATION OF THE NUMERICAL MODEL To verify he numericl model ccurcy, i clculed reul were compred wih experimenl d on he bulk refrigerion of frui nd vegeble, obined from he lierure. Bird nd Gffney (976 repored experimenl d ken from bulk lod of ornge. They recorded commodiy cener nd urfce emperure he ir exi of bulk lod for period of wo hour. The bulk lod of ornge w.67 m deep nd he commodiie were iniilly 32 C. The refrigered ir w emperure of -. C wih velociy of.9 m/. Figure 2 how Bird nd Gffney' experimenl d long wih he oupu from he compuer lgorihm. Comprion of he model reul wih Bird nd Gffney' d on ornge how h he lgorihm correcly predic he rend of commodiy emperure wih mximum error of.4 C. Bruewiz e l. (992 conduced (3 Figure 2. Curren numericl reul nd experimenl emperure d for forced ir cooling of ornge (Bird nd Gffney,

5 experimen o deermine moiure lo from peche during po-hrve cooling. The po-hrve cooling w performed 4 C, 92% relive humidiy in chmber wih 2 ir chnge per minue for period of four dy. Peche were picked in he morning when he mbien emperure w 6 C. Experimenl d from Bruewiz e l. how h he peche lo 2.5% of heir weigh due o moiure lo during he four dy cooling period. The numericl model prediced weigh lo of 2.53% he end of he four dy period, in good greemen wih he experimenl d. Figure 3 how he reul from he curren compuer lgorihm well he experimenl d. CONCLUSIONS Thi pper h decribed he developmen nd performnce of numericl Figure 3. Curren numericl reul nd experimenl moiure lo d for po hrve cooling of peche (Bruewiz e l., 992. model which eime he len nd enible he lod well he moiure lo nd emperure diribuion wihin bulk lod of frui or vegeble. In he compuionl model, he bulk lod i repreened porou medium compoed of "commodiy compuionl cell" nd he condiioned ir i modeled "ir prcel" which move hrough hee "commodiy compuionl cell." A m rnfer model w developed o upde he m frcion of wer vpor wihin ech "commodiy compuionl cell" ech ime ep. An explici finie difference formulion of he rnien he equion in phericl coordine w derived which ccoun for boh rdiion nd convecion he rnfer he commodiy urfce. Thi formulion yield he emperure diribuion wihin he commodiie reiden in ech "commodiy compuionl cell" ech ime ep. I lo yield he emperure of he "ir prcel" reiden wihin ech "commodiy compuionl cell" ech ime ep. To verify he ccurcy of he compuer lgorihm, i clculed reul were compred wih experimenl d obined from he lierure. The reul of hee comprion how good greemen beween he numericl reul nd he experimenl d for boh emperure nd moiure lo. Nomenclure A i urfce re of i h node A ingle commodiy urfce re A urfce re of cener node c commodiy pecific he cpciy c p, pecific he cpciy of ir d dimeer of commodiy h convecion convecion he rnfer coefficien h eff effecive he rnfer coefficien h rdiion rdiion he rnfer coefficien k commodiy herml conduciviy ir film m rnfer coefficien k k ir k k L m m f m H2O m H2O m& m & herml conduciviy of ir kin m rnfer coefficien rnpirion coefficien len he of vporizion of wer m of ir ime m frcion of wer vpor in ir ime +? m of wer vpor in ir ime m of wer vpor in ir ime +? rnpirion re per uni re of commodiy urfce rnpirion re in compuionl 5

6 n c N Nu P P Pr r Re Sc Sh T T T T T i T i T N T N T T T v i v N cell number of commodiie in compuionl cell number of node Nuel number mbien wer vpor preure wer vpor preure evporing urfce of commodiy Prndl number commodiy rdiu Reynold number Schmid number Sherwood number ime commodiy emperure dry bulb ir emperure ir emperure ime ir emperure ime +? emperure of i h node ime emperure of i h node ime +? emperure of urfce node ime emperure of urfce node ime +? produc urfce emperure emperure of cener node ime emperure of cener node ime +? volume of i h node volume of urfce node v W W i W N W d volume of cener node re of he generion of commodiy per uni m of commodiy re of he generion of commodiy per uni m of commodiy for node i re of he generion of commodiy per uni m of commodiy for urfce node re of he generion of commodiy per uni m of commodiy for cener node coefficien of diffuion of wer vpor in ir?r lengh of node in rdil direcion? ime ep ize? deniy of commodiy Sefn-Bolzmnn conn Lierure Cied Adre, N. nd Hellickon, M.L Simulion of he rnien refrigerion lod in cold orge for pple nd per. Trncion of he ASAE 32: 38. ASHRAE. 97. Brochure on Pychromery. ASHRAE, Aln, GA. Bird, C.D. nd Gffney, J.J A numericl procedure for clculing he rnfer in bulk lod of frui or vegeble. ASHRAE Trncion 82: 525. Bkker-Arkem, F.W. nd Bicker, W.G A deep-bed compuionl cooling procedure for biologicl produc. Trncion of he ASAE 9: 834. Becker, B.R., nd Fricke, B.A Trnpirion nd repirion of frui nd vegeble. p.. In New Developmen in Refrigerion for Food Sfey nd Quliy, Inernionl Iniue of Refrigerion, Pri, Frnce. Bruewiz, G.H., Zhng, X. nd Smih, M.W Picking ime nd pohrve cooling effec on pech weigh lo, impc prmeer, nd bruiing. Applied Engineering in Agriculure 8: 84. Chu, K.V., Romero, R.A., Bird, C.D. nd Gffney, J.J Trnpirion coefficien of frui nd vegeble in refrigered orge, ASHRAE Repor 37-RP, ASHRAE, Aln, GA. Focken, F.H. nd Meffer, H.F.T Biophyicl properie of horiculurl produc reled o lo of moiure during cooling down. Journl of he Science of Food nd Agriculure 23:

7 Gn, G. nd Wood, J.L A deep bed imulion of vegeble cooling. p. 23. In: V.A. Dodd nd P.M. Grce (ed, Agriculurl Engineering, A.A. Blkem, Roerdm. Genkopli, C.J Trnpor Procee nd Uni Operion. Allyn nd Bcon, Boon. McKinnon, I.R. nd Bilnki, W.K He nd m rnfer chrceriic of frui nd vegeble prior o hipmen. SAE Technicl Pper 9262, SAE, Wrrendle, PA. Tlbo, M.T., Oliver, C.C. nd Gffney, J.J. 99. Preure nd velociy diribuion for ir flow hrough frui pcked in hipping coniner uing porou medi flow nlyi. ASHRAE Trncion 96: 46. 7

IX.1.1 The Laplace Transform Definition 700. IX.1.2 Properties 701. IX.1.3 Examples 702. IX.1.4 Solution of IVP for ODEs 704

IX.1.1 The Laplace Transform Definition 700. IX.1.2 Properties 701. IX.1.3 Examples 702. IX.1.4 Solution of IVP for ODEs 704 Chper IX The Inegrl Trnform Mehod IX. The plce Trnform November 4, 7 699 IX. THE APACE TRANSFORM IX.. The plce Trnform Definiion 7 IX.. Properie 7 IX..3 Emple 7 IX..4 Soluion of IVP for ODE 74 IX..5 Soluion