Czech J. Food Sci. Vol. 26, No. 3: 99 20 Prediction of the Averge Surfce Het Trnsfer Coefficient for Model Foodstuffs in Verticl Disply Cbinet Krel HOKE, Aleš LANDFELD, Jiří SEVERA 2, Krel KÝHOS, Rudolf ŽITNÝ 2 nd Miln HOUŠKA Food Reserch Institute Prgue, Prgue, Czech Republic; 2 Fculty of Mechnicl Engineering, Czech Technicl University Prgue, Prgue, Czech Republic Abstrct Hoke K., Lndfeld A., Sever J., Kýhos K., Žitný R., Houšk M. (2008): Prediction of the verge surfce het trnsfer coefficient for model foodstuffs in verticl disply cbinet. Czech J. Food Sci., 26: 99 20. Clcultions of trnsient tempertures of food products fter they re trnsferred from wrm environment into disply cbinet, require dt on the surfce het trnsfer coefficient (SHTC). There is no forced ir flow in n ordinry disply cbinet, so the energy trnsfer is chieved minly by free convection, conduction to supporting plte, nd rdition. Theoreticl nlysis of the het trnsfer to cylindricl smple demonstrtes the reltive influences of these mechnisms. This work investigtes the pprent surfce trnsfer coefficients with metl models. Heted models were plced individully (bre) in continers with nd without lids. Ech model ws surrounded by identicl continers filled with wter. These were initilly t the sme temperture s the model or t the men cbinet temperture. There were one, two, or three lyers of these wter continers. From the mesured time-temperture histories of the model nd the ir surrounding the model, the SHTCs were clculted s functions of time nd trnsformed into the dependencies between SHTC nd temperture difference. The highest SHTCs were observed when the model ws plced directly on the metl shelf of the disply cbinet. The models surrounded by cool wter continers showed lower SHTC vlues. The lowest SHTC vlues were found with the models plced in the middle of three lyers of wrm wter continers. Plcing the model on n insulting bse leds to lower SHTC. This effect confirms tht the het conduction through the substrte increses the het trnsfer from the model nd thus increses the verge vlue of the pprent SHTC. Keywords: SHTC; verticl disply cbinet; correltions; food sfety The mthemticl modelling of the temperture history of foods in the distribution chin together with the modelling of micro-orgnisms potentil growth cn contribute to the enhncement of the methods for the quntittive microbil risk ssessment nd to improvements on food sfety. In the commercil prctice, it hppens very often tht the chilled foods re not plced into the disply cbinets pre-cooled but in heted stte either from the trnsport or becuse of hving been stored out of chilled spce for some time. In order to be ble to determine the time course of the temperture of foods put into the disply cbinet in this wy nd to evlute the degree of the microbiologicl risk, it is lso necessry to know, mong other prmeters, the surfce Supported by the Czech Science Foundtion (Grnt No. 0/02/0649) nd by the Ministry of Agriculture of the Czech Republic (Project No. MZe 000270220). 99
Vol. 26, No. 3: 99 20 Czech J. Food Sci. het trnsfer coefficient (SHTC) from the foods into the chilled ir in the cbinet. In the vilble literture, prcticlly no dt re given on the SHTCs of foods put into the chilled ir of the disply cbinet under the conditions of negligible forced convection. There exists literture dt bout forced convection. Recently, Becker nd Fricke (2004) collected nd evluted gret number of equtions for cooling nd freezing the foods under the conditions of forced convection. The problem of cooling the continers with foods in thick lyer on the pllet under the conditions of forced ir convection t different velocities is lso delt with Crniol et l. (998). The uthors specified the Nusselt criterion vlues s functions of the continer lyer order nd the ir velocity. Our cse (plcing wrm food into cool ir spce with low velocity ir flow) is most similr to the solved cse of the time-vrying SHTC when storing the frozen foods (Scott & Beck 992). The uthors used the Krlsruhe test substnce in the form of flt block s model of frozen foods. They studied the trnsfer of frozen foods from the surroundings t 6 C into n environment of bout 35 C nd bck. The free convection round the model cn be chrcterised by the time-vrying SHTC within the rnge from 6 to 2 W/(m 2 K). Recently Anderson et l. (2004) delt with thwing nd freezing selected kinds of met products in home refrigertors under the conditions of free convection. They specified the men vlues of the SHTC within the rnge of 8 5 W/(m 2 K) for freezing nd 5 7 W/(m 2 K) for thwing, respectively. As the SHTC from the foods into the environment inside the disply cbinet is influenced by the shpe of the foods, the kind of pckging, loction, the temperture difference, nd the mechnisms of the het trnsfer, it is not possible to use the vlues vlid for completely different geometries nd conditions. The objective of this work is therefore to specify the time courses of SHTCs nd the correltions of SHTC with temperture difference for typicl food object in specific disply cbinet. Theoreticl nlysis The simplest wy how to determine the men SHTC experimentlly is to substitute smple of food with geometriclly identicl metllic model nd to record the time temperture profile of this model. The SHTC cn be evluted by using only two recorded tempertures: the mbient temperture nd the uniform temperture of the model. The procedure cn be illustrted by nlysis of the cooling of wrm metllic continer (height H, initil temperture T c0 ) tht is plced upon steel plte (thickness h) t temperture T 0. The continer is fully enclosed by the cooling cbinet wlls nd by ir hving the sme initil temperture T 0. As long s the therml conductivity of the contents of the continer is very high, it cn be ssumed tht its temperture is uniform the time necessry to ttin uniform temperture is quite short for the typicl size of metllic models (~ H 2 /π ~ s). The rte of temperture chnges is controlled by three bsic mechnisms: nturl convection, rdition, nd conduction, ssuming perfect contct with the supporting plte. With the im to estimte the reltive contributions of these het trnsfer mechnisms, we shll consider simplified cse, cylindricl smple of the rdius R nd height H (ρ c, c c, λ c ) plced upon n infinitely lrge plte of the thickness h (ρ, c, λ, therml diffusivity = λ/ρc). It is lwys useful to describe the problem in terms of dimensionless quntities, for exmple dimensionless temperture θ (plte), θ c (cylinder), nd dimensionless rdius θ = T T 0, θ c = T c T 0, r = r* () T c0 T 0 T c0 T 0 R It could be possible to introduce lso the dimensionless time (Fourier number), however, it is probbly better to preserve the dimensionl time due to the fct tht the chrcteristic time hs quite different mening nd different vlues in the different het trnsfer mechnisms mentioned bove. Free convection. The het trnsfer coefficient α corresponding to the free convection is usully correlted with the Ryleigh number. As soon s uniform α on the smple surfce cn be ssumed, nd ssuming thermlly insulted bottom, the rte of the recorded temperture decrese of the metllic cylinder is described by simple eqution: dθ c α = ( + 2H ) θ c (2) dt ρ c c c H R For constnt het trnsfer coefficient, the solution of Eq. (2) is n exponentil time temperture profile, however, more relistic solution corresponds to the vrying het trnsfer ccording to 200
Czech J. Food Sci. Vol. 26, No. 3: 99 20 Nu = αh = C R R m (3) λ where: R Ryleigh number m exponent being usully /4 for lminr nd /3 for turbulent flows respectively Using correltion (3), the solution of (2) cn be expressed s follows θ c (t) = ( + t/τ) m where: τ = H 2 ρ c c c (4) C R R m 0 m λ ( + 2H/R) where: Ryleigh number ws evluted t the initil temperture T 0 Rdition. Rdition het trnsfer cn be estimted for sources nd sinks tht re not very distnt from ech other using the Boltzmnn lw. Tempertures T 0 (cool cbinet wlls cting s het sinks) nd T c (cylinder, θ c dimensionless temperture of cylinder) in the cse of negligible rtio of the smple surfce to the surfce of the surrounding wlls re relted by the following eqution: 2 dθ c σ (T = 0 + T c ) (T 0 + T 2 c ) 2H (+ ) θc (5) dt ρ c c c H R ssuming reltive emissivity of the smple surfce =. This expression corresponds, in the cse of smll difference between T 0 nd T c, to the equivlent het trnsfer coefficient α = 4σT 3 0, where σ is the Stefn-Boltzmnn constnt. The vlue α = 5 W/(m 2 K), corresponding to the typicl men temperture T 0 = 280 K, is quite comprble with the vlues predicted by free convection. Conduction. A wrm metllic smple is cooled down lso by the direct therml contct with the supporting plte (shelf ) representing n dditionl het cpcity the plte cts s firmly ttched thin fin. Assuming n insulted thin stinless steel plte, the temperture in the plte depends only upon the dimensionless rdil coordinte r s described by the Fourier eqution θ α θ = (r ) (6) t R 2 r r r together with the boundry nd initil conditions θ(t, r = ) = θ c (t), θ = (t, r ) = 0, θ(t = 0, r) = 0 (7) The temperture in the insulted metllic cylinder is uniform nd depends only upon time dθ c = dt R 2 r r= 2φ ( θ ) (8) φ = hρc (9) Hρ c c c where the dimensionless criterion φ, the reltive het cpcity of the plte nd smple chrcterises the influence of the het conduction. The system (Eqs 6 9) of prtil nd ordinry differentil equtions for the temperture of the cylinder nd tht of the plte cn by solved nlyticlly by using Lplce trnsform (Crslw & Jeger 986). Trnsforming time to the Lplce vrible p, we cn esily derive the Lplce trnsform of solutions, expressed in terms of Bessel functions of the second kind K 0 for the temperture field in the plte ~ ~ K θ(p,r) = θ(p) 0 (qr), q = R2 p (0) K 0 (q) nd for the Lplce trnsform of the cylinder temperture ~ R θc (p) = 2 K 0 (q) q(qk 0 (q) + 2φK (q)) () The time profile of the cylinder temperture cn be obtined from Eq. () by using the inversion theorem nd by clculting residuls of Eq. (), giving the result in the following integrl form: c( t) = 2 i γ + i ut γ i (2) where: u = (uj 0 (u) 2 φj (u)) 2 + (uy 0 (u) 2 φy (u)) 2 (3) is expressed by mens of Bessel functions of the first kind J n nd Y n. The rte of temperture chnges follows immeditely from Eq. (2) nd cn be relted to the rte of temperture chnges for free convection d 8ϕ 2 2 tu / R udu c ( e conduction R u (4) H d dt ) 2 2 0 = 2 c H ( ) t ( ) 2 + ccch R convection dt ( + ) e ccch R Remrk: The integrls in Eqs. (2) nd (4) hve to be evluted numericlly. 8 e ~ ϕ c( u) du = 2 e 0 2 2 tu / R du u u 20
Vol. 26, No. 3: 99 20 Czech J. Food Sci. Combined effect of het trnsfer mechnisms. Previous prgrphs nlysed the possible het trnsfer mechnisms seprtely, giving the results (temperture profiles nd cooling rtes corresponding to the instntneous chnge in the mbient temperture) in n nlyticl form. With the im to nlyse complex cse, it is simpler to crry out numericl nlysis, for exmple by using the finite difference method. The problem ws formulted by mens of the previously introduced correltions for the rdition, Eq. (5) nd free convection Eq. (3), where C R = 0.59 nd m = /4 re the vlues recommended for lminr flow by Šesták nd Rieger (974). The effect of the het conduction ws solved not only ccording to section 2.3 (perfectly thermlly insulted plte), but lso ssuming constnt het trnsfer coefficient t the plte surfce. The following cses were evluted (Figure ). A cylindricl smple with n insulted bottom (free convection nd rdition), the sme smple cooled down lso from bottom, further on completely insulted smple connected to circulr insulted fin, nd finlly the most relistic cse when ll het trnsfer mechnisms re considered. The geometry nd properties were selected s close s possible to rel cses: R = 0.05 m, H = 0. m, h = 0.002 m, β = 0.0037 K, ν = 3 0 6 m 2 /s, = 9 0 6 m 2 /s, (ir), ρ c = 2700 kg/m 3, c c = 900 J/(kgK), λ c = 200 W/(mK), (cylinder), ρ = 7800 kg/m 3, c = 460 J/kg K, λ = 5 W/(mK), (plte), mbient temperture T 0 = 0 C, the initil temperture of the cylinder T c0 = 20 C. The results shown in Figure demonstrte tht the effect of rdition is significnt (even more importnt thn the free convection in this prticulr cse), nd tht the effect of the het conduction in the supporting plte cnnot be neglected either, nmely t the very beginning of the cooling process (Figure 2). There re other effects tht hve not been tken into ccount, for exmple the velocity field disturbnce fter the plcement of the smple into the cooling cbinet nd the effects of the ir-curtin. Generlly speking, there re severl mechnisms, chrcterised by different time constnts: the shortest one is the evolution of hydrodynmic nd therml boundry lyer on the smple surfce (nlysis indictes tht this time constnt is of the order of seconds), followed by the het conduction into the supporting plte (time constnt bout minute). The convection nd rdition re chrcterised by much longer time constnts (up to n hour), however, not exctly the sme becuse the free convection het trnsfer coefficient is decresing function of the temperture difference while the equivlent het trnsfer coefficient for rdition is lmost constnt. The reltive mgnitude of the het flow mechnisms nlysed is of the sme order nd depends upon the smple size, temperture differences, reltive het cpcities of the smple nd shelf φ, emissivity of the smple surfce nd the cbinet wlls. The procedure described bove cn be pplied only for the simplest cses i.e. simple geometry (cylinder without wrpping) nd stnd lone smple. For geometriclly more complicted smple nd first of ll for group of mutully intercting smples, other fctors strt to be importnt: the wrpping of the smple surfce, the reltive distnce between the smples, the effects of the lids, etc. 20 8 6 Conduction (insulted plte) T ( C) 4 2 0 8 Free convection (insulted bottom) 6 4 2 0 Free convection, rdition, conduction (het trnsfer lso to the supporting plte from both sides) Free convection lso t bottom Only rdition 0 600 200 800 2400 3000 360 4200 t (s) Figure. Time temperture dependencies of metllic cylinder (R = 0.05 m, H = 0. m) 202
Czech J. Food Sci. Vol. 26, No. 3: 99 20 dt /dt ( C/s) 0.04 0.03 0.02 Free convection, rdition, conduction (het trnsfer lso to the supporting plte from both sides) Figure 2. Rte of temperture decrese t the beginning of cooling (the vlues re prcticlly proportionl to the SHTC) 0.0 0 0 0 20 30 40 50 60 t (s) The conventionl chrcteristion using generl engineering correltion between dimensionless numbers strts to fil for such complex ssembly of continers. This is becuse the complex geometry necessittes using n unmngebly lrge number of the dimensionless descriptors. In view of these complictions, we suggest n lterntive pproch of specifying n extended set of experimentlly obtined SHTC vlues. We discuss this lterntive pproch in the following sections. Mteril nd methods The mesurement of the tempertures of the models during cooling ws crried out in n industrilly mnufctured refrigerted disply cbinet, which is intended for sle of chilled perishble foods in the retil network. Description of the refrigerted disply cbinet. The disply cbinet used ws Optimer 946 (Linde Chldicí technik Ltd., Beroun-Závodí, Czech Republic) with the following dimensions: length 875 mm, bredth with lighting 895 mm, height 980 mm. The cbinet hs five horizontl shelves for chilled products. The control system uses the on/off principle within n djustble temperture rnge of the ir of 0.5 4.5 C. A stedy lod ws plced into the cbinet for simulting the rel conditions. The stedy lod ws creted by identicl continers filled with wter nd closed with lids. The continers with the wter content of 0.5 l in the number of 20 pieces were evenly plced on sides of the shelves 4. On shelf 5, the number of continers ws 30. For the mesurement itself, free spce of the bredth of 380 mm in the middle of every shelf ws left for plcing the models. Description of the mesurement. The models heted up to the room temperture were plced either individully or in group with other smples into the disply cbinet in specific positions on the individul shelves. The positions of the models plced in the disply cbinet re schemticlly shown in Figure 3. Both the model temperture nd the temperture of the ir surrounding the model on its shelf were mesured. Experimentl set-up. The tempertures of the models nd the surrounding ir were mesured by Cu-Co thermocouples from the compny Omeg (USA). The thermocouples mesuring the tem- 5 2 3 4 b Figure 3. A schemtic drwing of the disply cbinet, indiction of the plcement of the model in cbinet by letters, b, c on five shelves numbered 5, e.g. position 4b mens shelf 4, position b b b b b c 203
Vol. 26, No. 3: 99 20 Czech J. Food Sci. perture of the metl models were plced into the middle of the metl models. For mesuring the thermoelectric voltges, the multimeter HP- 34970A ws used, from which the mesured dt were trnsmitted into personl computer t regulr intervls (5 s) by mens of the progrm Dtlogger (Hewlett & Pckrd Co., USA). The thermocouples were clibrted before use with the help of the stirred wter ultrthermostt Prüfgeräte Medingen (Germny) by mens of the clibrted digitl thermometer Therm 2230- Ahlborn Messtechnik (Germny). The mximum temperture mesurement error ws ± 0. C. Description of models. Three metl models were mde for three sizes of plstic continers. Figure 4 shows the shpes of the models in the continers covered withy lids. The sizes of the continers were: the big continer: bse 58 83 mm, top prt 82 08 mm, height 98 mm; the medium continer: bse 63 88 mm, top prt 82 08 mm, height 48 mm; nd the smll continer: bse 66 9 mm, top prt 82 08 mm, height 28 mm. Identicl lids 90 6 mm, height 8 mm, fitted ll three types of continer. The continers nd their lids re mde of polypropylene, the side wlls re corrugted, the thickness of the wll nd the bottom is 0.25 mm. The influence of corrugtion on the size of the het trnsfer surfce S ws not tken into considertion (plnr fces were ssumed). The metl models were mde of n lloy of tin nd led in equl proportions (:) with the estimted het conductivity of 50.6 W/(mK), specific het cpcity of 0.77 kj/(kgk), nd density of 9325 kg/m 3, bsed on the therml properties of the pure components, s stted by Rznjevic (984). The metl models for the individul sizes of the continers were mde in ccordnce with Figure 4. Metl models plced into continers nd covered with lids used in the experiments the stndrd metl csting procedure. The height of the model ws djusted so tht n ir gp between the model nd the lid ws creted, which corresponds to the filling of rel food into the pckging. The heights of the big, medium, nd smll metl models were 85, 36, nd 22 mm, respectively. The outer surfces nd the msses of the metl models were s follow: the big model: S = 0.0387 m 2, m = 4.760 kg; the medium model: S = 0.0254 m 2, m = 2.256 kg, nd the smll model: S = 0.024 m 2, m =.433 kg. Severl drops of oil were put into the continers before plcing the metl model in them in order to improve the therml contct. The metl model ws plced mong geometriclly identicl continers filled with wter to the sme level s the upper edge of the metl model nd closed with lid. Arrngement of models in the disply cbinets. The mesurements were crried out for different positions nd rrngements of the smple models nd surrounding continers on the shelves of the disply cbinet. At first, the influence ws studied of the pckging on the het trnsfer during the cooling of the metl models of ll three sizes. The temperture drop of the metl model nd the temperture of the ir inside the cooling cbinet on the sme shelf ner the model (distnce bout 200 mm) were recorded simultneously. The metl model without pckging, in the continers with nd without the lid, ws tested. The ir temperture ws mesured outside the lyers. Tht temperture represents the men ir temperture on the respective shelf. In the following tests, imed t studying the effect of different configurtions of the metl models nd surrounding smples (continers filled with wter nd covered with lids), the metl model ws lwys in the continer covered with the lid. The smples with wter were initilly heted up to the room temperture or cooled down to the temperture in the disply cbinet. The following rrngements were tested: Configurtion 3 3, smll, medium, nd big models. Single lyer with the metl model plced in the middle of eight smples. Configurtion 3 3 3, smll nd medium models. Three lyers with the metl model in the middle of the centrl lyer mong 26 continers filled by wter. Configurtion 3 3 2, big models. Two lyers with the big smples (9 smples in ech lyer). Procedure of prediction of the SHTC α. On the bsis of the mesured time-temperture course of 204
Czech J. Food Sci. Vol. 26, No. 3: 99 20 the metl model nd the ir temperture in the disply cbinet mesured ner the metl model, the SHTC α ws computed ccording to mc p dt m = αs(tm T p ) (5) dt With the im to obtin the time-course of α in n nlyticl form, it ws necessry to fit the function into the time course of the tempertures. The time course of the metl model temperture ws pproximted by double exponentil regression function (, b, c, d nd e-fitted prmeters) T m = e t/d + be t/e + c (6) by using the DtFit (Okdle Engineering, USA) progrm. The temperture of the ir T p in the disply cbinet ws pproximted by liner or qudrtic polynomils nd together with Eq. (6) ws used for the evlution of the SHTC from Eq. (5). The mesurement ws terminted when the temperture chnge of the model becme smll. Only tht prt of the mesurement where the temperture chnges of the model were significnt enough ws used for the clcultion (we hve used the prt of dt chrcterised by mximum reltive error of the temperture difference lower thn 5%). Those prts of the mesured temperture courses which were influenced by the regime of defrosting the disply cbinet ( sudden rise nd drop of the ir temperture in the disply cbinet) were lso excluded. Processing of dt. Preliminry mesurements of the ir velocity in the disply cbinet indicted tht the effect of forced convection is negligible nd the het trnsfer is controlled by nturl convection, rdition, nd conduction, s discussed in the previous section nd by Hoke et l. (2006). While these theoreticl models seem to be pplicble for single continer sitution, different rrngements of mny continers of complex shpes re difficult to generlise: the correltions including the terms describing ll the het trnsfer mechnisms would involve too mny prmeters. Therefore, the mesured SHTC vlues hve been pproximted quite empiriclly by mens of simple power-lw function of the temperture difference between the metl model nd the ir in the disply cbinet tmosphere α = p T n (7) where: p, n empiricl constnts predicted by mens of the computer progrm DtFit (Okdle Engineering, USA) One correltion reltion ws pplied to severl dt sets mesured t different positions of the experimentl set-up (different shelves, different positions on ech shelf ). Therefore, the regression eqution is the men of the results of severl repetitions of the experiment (minimum two, mximum five), e.g. Figure 6. Results nd discussion The typicl time dependencies ofthe model nd ir tempertures mesured fter plcing the metl model on shelf 3, position (medium size model in pckging plced into the middle of 8 heted smples) re given in Figure 5. This shows tht the mesured oscillting temperture of the ir on the shelf of the cbinet T p ws smoothed by the regression eqution. The model temperture T m ws fitted by Eq. (6) with high ccurcy. Temperture ( C) 25 model temperture 0 mbient temperture 9 20 8 model-regression 7 5 mbient-regression 6 SHTC 5 0 4 3 5 2 0 0 0 0.5.5 2 2.5 SHTC (W/(m 2 K)) Time (h) Figure 5. Typicl dependence of SHTC vs. time nd model temperture vs. time fter the model ws trnsferred into the disply cbinet. Medium size model in pckging mong 8 heted smples in the middle of one lyer (position 3) 205
Vol. 26, No. 3: 99 20 Czech J. Food Sci. Tble. Metl model without pckging, in continer without lid nd in continer with lid (in pckging) Model Eqution Vlidity rnge ( C) SHTC t T = 0 C (W//m 2 K)) Without pckging Smll α = 0.0620 T 2.37.2 < T < 24.2 α =.55 T.038 5.6 < T <.2 6.9 Medium α = 0.0075 T 3.36 5.3 < T < 25.0 α = 8.370 T 0.252 2.8 < T < 5.3 4.9 Big α = 9.202 T 0.46 3.3 < T < 22.9 2.9 In continer without lid Smll α = 2.29 T 0.843 2.4 < T < 23.3 5.5 Medium α = 0.026 T 2.38 5. < T < 24. α = 4.85 T 0.04 2.8 < T < 5. 6.3 Big α = 6.477 T 0.305 3.6 < T < 22.2 3. In continer with lid (in pckging) Smll α = 2.985 T 0.66 4.4 < T < 25.3 2.3 Medium α = 6.436 T 0.35 5.0 < T < 26.0 4.4 Big α = 5.29 T 0.358 4.2 < T < 25.6 2. Also presented is the clculted pprent SHTC s function of time. The clculted SHTC decresed from the mximl vlue of 8.8 W/(m 2 K) t the beginning of the experiment to the vlue of 2 W/(m 2 K) t the end of the experiment. Figure 6 shows the clculted dependencies of SHTC vs. temperture difference for the sme experimentl rrngement s shown in Figure 5. There re severl curves representing the experiments mde t different positions in the cbinet. It is pprent tht there re smll differences between different shelves nd positions. We hve omitted these differences nd predicted the prmeters of regression Eq. (7) vlid for ll positions in the cbinet. The regression equtions obtined for different model sizes nd experimentl rrngements re given in Tbles 4. In some cses, it ws necessry to predict the regression eqution for two seprte rnges of temperture difference (Tbles nd 3). The shrp decy of the pprent SHTC ws found t the initil stge of the experiments, probbly due to the enhnced conduction of het from the model to the shelf. This effect ws observed for the smll nd middle size models only, when the reltive het cpcity of the shelf φ is high (0.05 or 0.). The big Tble 2. Model in pckging mong 8 smples which were heted up to the room temperture nd cooled down to the temperture in the disply cbinet (one lyer) the model in the middle of the lyer Model Eqution Vlidity rnge ( C) SHTC t T = 0 C (W/(m 2 K)) Heted up to the room temperture Smll α = 0.747 T 0.986 2.2 < T < 22.4 7.23 Medium α =.092 T 0.697 2.5 < T < 8.3 5.44 Big α = 0.86 T 0.699 4.2 < T < 24.8 4.3 Cooled down to the temperture in the disply cbinet Smll α = 4.58 T 0.53 3. < T < 24.8 5.5 Medium α = 5.8 T 0.42 3.0 < T < 23.5 3.6 Big α = 4.88 T 0.355 4.0 < T < 24.5.0 206
Czech J. Food Sci. Vol. 26, No. 3: 99 20 SHTC (W /(m 2 K) 00 0 3 4b 5c 2 regression eqution 0 00 Temperture difference ( C) Figure 6. Medium size model in pckging in the middle of 8 heted smples (in one lyer) for vrious positions in the cbinet (see Figure 3 for definition of positions) model temperture decy is probbly not much ffected by the het conduction to the shelf (φ = 0.02). We lso studied how this effect is influenced by lyer of fom polystyrene insultion t the smple bottom (Figure 0). It is obvious tht the plcement of the model on the insultion decreses the pprent SHTC vlues, however, the shrp decy of SHTC vlues persists. The exct explntion hs not been found, the cuse my be the initil disturbnce of the velocity field or short term influence of the smll het cpcity of the insultion. The comprison of the influence of different geometry rrngements for different sizes of the models is mde in Figures 7 9. The mrks on the curves do not represent the experimentl dt. They re used only for the individul regression curves identifiction. Figure 7 is devoted to the smll model. It cn be seen tht the metl model without pckging exhibits the lrgest pprent SHTC vlues. If the model is plced into the pckging (plstic continer fitting tightly on the model), the SHTC is slightly smller. Also, covering the plstic continer with lid ffects the SHTC vlues. If the model is plced in one lyer of chilled smples (the model is plced on the metl shelf ), the SHTC vlues do not differ very much from the cse vlid for the individul model. If the model is plced into the centre of three lyers of chilled smples, the SHTC vlues re much lower. Similr vlues were with the found with the model plced into one lyer of heted smples. The lowest SHTC vlues were found with the metl model plced into the centre of three lyers of heted smples (the sme continers filled with wter). Figure 8 is devoted to the medium model. The rrngements of the model without pckging nd Tble 3. Model in pckging mong 26 smples heted up to the room temperture nd cooled down to the temperture in the disply cbinet (three lyers)-the model in the middle of the centrl lyer Model Eqution Vlidity rnge ( C) SHTC t T = 0 C (W/(m 2 K)) Heted up to the room temperture Smll α =.072 T 0.83 4.2 < T < 20.0.63 Medium α =.009 T 0.5 4.2 < T < 8.6.42 Cooled down to the temperture in the disply cbinet Smll α =.339 T 0.8 2.7 < T < 20.5 8.64 Medium α = 0.043 T 2.32 4.7 < T < 20. α = 2.46 T 0.4 2.7 < T < 4.7 6.32 Tble 4. Model in pckging mong 7 smples heted up to the room temperture nd cooled down to the temperture of the disply cbinet (two lyers) the model in the middle of the upper lyer Model Eqution Vlidity rnge ( C) SHTC t T = 0 C (W/m 2 K) Heted up to the room temperture Big α =.454 T 0.097 3.9 < T < 8.9.82 Cooled down to the temperture of the disply cbinet Big α = 3.333 T 0.33 3. < T < 20.5 7.4 207
Vol. 26, No. 3: 99 20 Czech J. Food Sci. 000 smll model without pckging SHTC (W/(m 2 K)) 00 0 smll model in continer without lid smll model in continer nd lid (in pckging) smll model in pckging one lyer heted smples smll model in pckging one lyer chilled smples smll model in pckging 3 lyers heted smples smll model in pckging 3 lyers chilled smples 0 00 Temperture difference ( C) Figure 7. SHTC versus temperture difference for the smll model in vrious rrngements the model in pckging without the lid exhibit the lrgest vlues of SHTC. The curves for these two cses cross over t the temperture difference of bout 5 C. Similr but slightly lower vlues were exhibited by the rrngements of model in pckging with the lid nd the model plced in one lyer between chilled smples. Much lower vlues of SHTC were predicted for the model plced in one lyer of heted smples nd tht plced into the centre of three lyers of chilled smples. The lowest vlues of SHTC, nerly independent of temperture difference, were found for the model plced into the centre of three lyers of heted smples. 00 medium model without pckging medium model in continer without lid medium model in continer nd lid (in pckging) medium model in pckging lyer heted smples medium model in pckging lyer chilled smples medium model in pckging 3 lyers heted smples medium model in pckging 3 lyers chilled smples SHTC (W/(m 2 K)) 0 0 00 Temperture difference ( C) Figure 8. SHTC versus temperture difference for the medium size model in vrious rrngements 208
Czech J. Food Sci. Vol. 26, No. 3: 99 20 SHTC (W/(m 2 K)) 00 0 big model without pckging big model in continer without lid big model in continer nd lid (in pckging) big model in pckging lyer heted smples big model in pckging lyer chilled smples big model in pckging 2 lyers heted smples big model in pckging 2 lyers chilled smples 0 00 Temperture difference ( C) Figure 9. SHTC versus temperture difference for the big model in vrious rrngements Figure 9 is given to the big model. The tendencies re similr to those with the medium nd smll models. The highest but very similr vlues of SHTC hve been predicted for the model without pckging, in the continers without nd with the lid. Similr vlues were found for the model in pckging plced into one lyer of chilled smples. Much lower vlues of SHTC were found for the model plced between two lyers of chilled smples (the model ws plced in the second lyer nd not on the metl shelf of the cbinet). Lower vlues thn these were found with the model plced into one lyer of heted smples. The lowest vlues of SHTC were found with the model plced into two lyers of heted smples (the model ws plced in the second SHTC (W/(m 2 K)) 00 0 experiment without insultion experiment with insultion 0 00 Ttemperture difference ( C) Figure 0. Results with the medium size model in continer without lid showing the influence of positioning on the SHTC lyer, i.c. not on the metl shelf of the cbinet). In this cse, the pprent SHTC ws nerly independent of the temperture difference (Tble 4). Conclusions The pprent SHTC vlue α is dependent on the time elpsed fter plcing model into the chilled ir of disply cbinet. The rdition, free convection, nd conduction into the shelves re dominnt het trnsfer mechnisms in the cses studied. An intensive unstedy het trnsfer ws observed with smll nd medium size models t the initil stges of the cooling. This effect ws cused probbly by the conduction from the model to the shelf. This effect ws observed for temperture differences greter thn 5 C. The highest vlues of the pprent SHTC were received with the model without pckging plced individully on the disply cbinet shelf. The pprent SHTC is lowered only slightly by pckging. The metl models in pckging with or without the lid exhibited nerly the sme dependencies on the temperture difference s the models without pckging or the models plced in one lyer between chilled smples. Surrounding the model with other continers lowered the SHTC substntilly, both in single 209
Vol. 26, No. 3: 99 20 Czech J. Food Sci. lyer nd in the cses when the smll nd medium size models were plced in the centre of three lyers of continers. The lowest vlues of the pprent SHTC were found with smll nd medium size metl models plced in the centre of three lyers of smples. The vlues of SHTC pproch W/(m 2 K), regrdless of the temperture difference. Therefore, to lod severl lyers of heted products into the disply cbinet cn be considered s very bd prctice. In such cse, the cooling of the centrl piece cn lst even for severl hours. Obviously, the plcement of heted products directly into the disply cbinet cnnot be recommended t ll. Nomenclture therml diffusivity of plte (m 2 /s), b, c, d, e coefficients of Eq. (6) c p specific het of the metl model (J/(kgK)) h thickness of plte (m) H height of smple (m) m mss of the metl model (kg) Nu = αh/λ Nusselt number p Lplce trnsform vrible q = R2 p uxiliry vrible p, n empiricl constnts in Eq. (7) R cylinder rdius (m) r* rdil coordinte (m) r dimensionless rdil coordinte R = gβh 3 T/(ν) Ryleigh number S surfce of the metl model (m 2 ) t time (s) T m temperture of the metl model ( C) T 0 cool cbinet wlls ( C) T p temperture of ir surrounding the model or experimentl rrngement ( C) T s surfce temperture ( C) α surfce het trnsfer coefficient SHTC (W/(m 2 K)) β coefficient of therml expnsion (/K) γ rel vlue which is greter thn the rel prt of ll poles (inversion theorem) φ reltive het cpcity of the plte nd smple λ therml conductivity (W/(mK)) ν kinemtic viscosity (m 2 /s) θ dimensionless temperture of plte θ c dimensionless temperture of cylinder ρ density (kg/m 3 ) σ Stefn-Boltzmnn constnt (= 5.6697 0 8 W/(m K 4 )) τ time constnt (s) R e f e r e n c e s Anderson B.A., Sun S., Erdoglu F., Singh R.P. (2004): Thwing nd freezing of selected met products in household refrigertors. Interntionl Journl of Refrigertion, 27: 63 72 Becker B.R., Fricke B.A. (2004): Het trnsfer coefficients for forced-ir cooling nd freezing of selected foods. Interntionl Journl of Refrigertion, 27: 540 55. Crslw H.S., Jeger J.C. (986): Conduction of Het in Solids. 2 nd Ed. Oxford University, Cry. Crniol N., Alvrez G., Kondjoyn A. (998): Convective het trnsfer in try stck of food products during cooling: influence of ir velocity nd pckging. In: Fikiin K. (ed.): Proceedings Conference Advnces in the Refrigertion Systems, Food Technologies nd Cold Chin, Sofi: 284 289. Hoke K., Lndfeld A., Houšk M. (2006): Stnovení tepelných toků při chlzení kovových modelů v obchodní vitríně. Výzkumná zpráv č. /360/2006. Avilble t www.vupp.cz request emil: m.housk@vupp.cz. Rznjevic K. (984): Termodynmické tbulky. Alf, Brtislv. Scott E.P, Beck J.V. (992): Estimtion of time vrible het trnsfer coefficients in frozen foods during storge. Journl of Food Engineering, 5: 99 2. Šesták J., Rieger F. (974): Přenosové jevy I. Přenos hybnosti tepl. Vydvtelství ČVUT, Prh. Received for publiction September 22, 2006 Accepted fter corrections October 2, 2007 Corresponding uthor: Ing. Miln Houšk, CSc., Výzkumný ústv potrvinářský Prh, v.v.i., oddělení potrvinářského inženýrství, Rdiová 7, 02 3 Prh 0-Hostivř, Česká republik tel.: + 420 296 792 306, fx: + 420 272 70 983, e-mil: m.housk@vupp.cz 20