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Modelling cooling kinetics of a stack of spheres during mist chilling
Journal of Food Engineering 72 (2006) 197–209 www.elsevier.com/locate/jfoodeng
Modelling cooling kinetics of a stack of spheres during mist chilling Ire´ne Allais b
c
a,b,*
, Graciela Alvarez
a,c
, Denis Flick
a,d
a UMR Ge´nie Industriel Alimentaire, France ´ Cemagref, Unite de Recherches Technologie, Syste´mes d’Information et proce´de´s pour l’Agriculture et l’Agroalimentaire, 24 Av. des Landais, BP 50 085, 63 172 Aubie´re Cedex, France Cemagref, Unite´ de Recherches Ge´nie des Proce´de´ Frigorifiques, Parc de Tourvoie, BP 121, 92185 Antony Cedex, France d INA-PG, 16 rue Claude Bernard, 75231 Paris cedex 05, France
Received 25 June 2004; accepted 23 November 2004 Available online 22 December 2004
Abstract Mist chilling, which uses air and suspended water droplets as a cooling medium can reduce weight losses in food and accelerate heat transfers compared to forced-air cooling. The results presented here deal with the experimental study and modelling of heat and mass transfer during mist chilling in a stack of spheres. A numerical model was established to predict temperature kinetics in a stack of spheres. It takes into account local air temperature, local droplet concentration, heat and mass transfer coefficients. Model predictions are in good agreement with experimental results for single and two-phase-flow for different conditions of air velocity and water mass flow rate. 2004 Elsevier Ltd. All rights reserved. Keywords: Heat transfer; Two-phase-flow; Modelling; Stack; Chilling; Food; Temperature kinetics
1. Introduction For food industry, refrigeration is an important issue to reduce biochemical reactions, to limit the growth of micro-organisms and consequently to extend the shelf life of both fresh and cooked products. Different means of cooling are available. In most of the cases, a singlephase-flow is used: air in forced-air cooling or water applied by immersion or by spraying in hydrocooling. In processes like ‘‘hydraircooling’’ or ‘‘mist chilling’’ a two-phase-flow (air and suspended water droplets) is involved. Mass flow rates and droplets diameter are generally higher (>100 lm) in hydraircooling than in mist
*
Corresponding author. Address: Cemagref, Unite´ de Recherches Technologie, Syste´mes dÕInformation et proce´de´s pour lÕAgriculture et lÕAgroalimentaire, 24 Av. des Landais, BP 50 085, 63 172 Aubie´re Cedex, France. Fax: +33 4 73 44 0697. E-mail address: [email protected] (I. Allais). 0260-8774/$ - see front matter 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.jfoodeng.2004.11.010
chilling (<10 lm). A liquid film on the surface is often observed when spray cooling or hydraircooling are applied (Abdul-Majeed, 1981). Compared to hydraircooling, mist chilling enables water supply control to prevent excess water on the surface of the product. This is particularly important for water sensitive products such as strawberries or bakery products. Compared to forced-air cooling, mist chilling reduces chilling time but also limits weight losses and thus it can limit quality decay due to dehydration. This process was already used to chill meat carcasses after slaughter on an industrial scale (Jones & Robertson, 1988; Le´tang, 1990) but it requires more knowledge to be applied to stack of food products.
2. Bibliography Phenomena involved during the cooling of stacks of products in two-phase-flow are complex. Heat transfers
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Nomenclature total area of the sphere or the body (m2) frontal surface of the body (m2) wetted area of the sphere (m2) collection coefficient (m s1) water content (kg water (kg dry air)1) specific heat at constant pressure (J kg1 K1) d mean Sauter diameter of droplets (m) D sphere diameter (m) Dv/a water vapour/air diffusivity (m2 s1) h heat transfer coefficient for dry air (W m2 K1) 0 h equivalent heat transfer coefficient for mist chilling (W m2 K1) hm mass transfer coefficient (m s1) b H specific enthalpy (J (kg dry air)1) i radial position index j row number k time index ka or kp thermal conductivity (W m1 K1) K collection number Le Lewis number (aa/Dv/a) _ M water mass flow rate in the mist flow (kg s1) _a M dry air mass flow rate (kg s1) m_ c collected water flux density (kg s1 m2) m_ v evaporated vapour flux density (kg s1 m2) ns number of spheres in one layer Nu Nusselt number (hD/ka) Nu 0 equivalent Nusselt number (h 0 D/ka) Pat atmospheric pressure (Pa) Pr Prandtl number (m/aa) q_ convective heat flux density (W m2) r position inside sphere (m) R radius of a sphere (m) Re Reynolds number (vD/n) Rem modified Reynolds number (vD/n(1 e)) A Af Aw cc Cm Cp
depend on aerodynamical and thermal properties of the two-phase-flow, the product shape and the stack arrangement (Allais & Alvarez, 2001). For single products, many experimental studies were performed, they have achieved a better understanding of the increase of heat transfer and of the influence of main factors on heat transfers: airflow characteristics (velocity, turbulence and temperature), water droplet characteristics such as droplet diameter and velocity (Bhatti & Savery, 1975; Lee, Yang, & Hsyua, 1994), water mass flow rate (Aihara, 1990; Basilico, Jung, & Martin, 1981), product characteristics: length, surface temperature and rugosity (Aihara, Fu, & Suzuki, 1990; Bhatti & Savery, 1975; Nakayama, Kuwahara, & Hirasawa, 1988).
Sh St T t v x
Sherwood number (hmD/Da/v) Stokes number (vd2ql/(18laR)) temperature (C) time (s) superficial velocity or upstream velocity of a body (m s1) vapour content (kg water/kg dry air)
Greek symbols aa or ap thermal diffusivity (m2 s1) a coefficient in Eq. (1) b coefficient in Eq. (2) DI difference of specific enthalpy between saturated air and air at the surface temperature DHv latent heat of water evaporation (J kg1) Dr space increment in radial co-ordinates DT temperature difference (K) e stack porosity / conductive heat flux density (W m2) l dynamic viscosity (Pa s) m kinematic viscosity (m2 s1) q density (kg m3) Subscripts a air l liquid water p product s surface of sphere sat saturated v vapour w water Superscripts c core o freestream s surface
Few works in the literature were dedicated to aerodynamic phenomena and heat transfers in stacks of products. Most studies dealt with heat exchangers such as automotive radiator cores (Yang & Clark, 1975), condensers (Dreyer, Kriel, & Erens, 1992) and tube banks (Nakayama et al., 1988), but relatively few of them dealt with stacks of products. They have shown that, in stacks, void fraction and location in the stack have to be considered. Indeed, even in single-phase-flow, several phenomena cause heterogeneity of heat transfer in stacks. If the stack is in a container, the void fraction is not uniform, being greater near the container wall than in the centre (Wakao & Kaguei, 1982), thus, air velocity is higher
I. Allais et al. / Journal of Food Engineering 72 (2006) 197–209
near the wall. Turbulence intensity also varies in the stack because of turbulence production in the wake of each product. Moreover, the temperature of air circulating through layers of products increases because it is heated as it passes through the layers. As a result, the heating of the air and the spatial variations in the local heat transfer coefficient induce considerable thermal heterogeneity (Alvarez & Flick, 1999; Wakao & Kaguei, 1982). In two-phase-flow, heat transfers also depend on local collected water mass flow rate. To assess the amount of collected water can be difficult. Indeed, when a two-phase-flow circulates throughout stacks of products, the amount of water collected by the products in the stack depends on droplet trajectories, which are modified by the presence of the products in the stream. Several aerodynamical studies were focused on the assessment of collected particles. Most of them were performed on a single body but studies dedicated to aerosol and filtration have shown the main mechanisms involved in bulk deposition of particles in a filter: deposition rate depends both on the characteristics of the two-phaseflow (velocity, turbulence and droplet size distribution) and on those of the stack of products (void fraction, body shape and dimensions) (Friedlander, 1977; Rosner & Tassopoulos, 1989; Tien, 1984). Heterogeneity of deposition in a stack was mostly studied in agriculture field to control the supply of liquid inputs (p.e. pesticides), it was shown that local collected liquid mass flow rate exponentially decreased with the depth in flow direction, then, less and less liquid was available for further rows (Kirk, Bouse, Carlton, Franz, & Stermer, 1992; Watterson & Nicholson, 1996). There is a general agreement on the fact that, water droplets can considerably enhances heat transfer. The increase in heat transfer is mainly due to evaporation of water at surface. Several modes of heat transfer can be defined according to surface temperature and water mass flow rate (Basilico et al., 1981). When surface temperature is below boiling temperature and water mass flow rate is high, a liquid film is formed at the surface (water in excess). Heat transfer is due to conduction between body surface and film, to evaporation and convection between water film and two-phase-flow. When water mass flow rate is low, droplets evaporate when they are closed to the surface or just after deposition and a liquid film can not be formed. This mode of heat transfer is particularly suitable for food products sensitive to excess of water (Allais, Alvarez, & Flick, 1997). As far as modelling is concerned, external heat transfers in two-phase-flow were scarcely modelled (Table 1), owing to the very complicated involved phenomena. Prediction of heat transfer in two-phase-flow would require a better knowledge of the hydrodynamics of flow inside a stack of bodies as well as the mechanisms of deposition of droplets. Nakayama et al. (1988) and
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Dreyer et al. (1992) performed empirical models. NakayamaÕs model aims to predict surface of evaporation in a tubes bank containing 32 tubes arranged in 4 rows. It is based on heat and mass balances and it takes into account flooding from the upper tubes. This approach shown a good agreement with experimental data. DreyerÕs model aims to predict heat transfers for a 122 horizontal finned tubes exchanger when there is excess of water on surface, it is a one dimensional model similar to those used for refrigeration towers. Agreement with experimental data is ±17%. Coupled external and internal transfers in a stack of spheres were only studied by Rao, Narasimham, and Krishna Murthy (1993). Each sphere is assumed to be covered by a water film (excess of water) that flows out by gravity on the lower layers. Water flows downward, air can flow upward or downward. Heat and mass transfers of water flowing by gravity are neglected. Equations of conservation are written separately for the spheres, the liquid film at the surface of the spheres, the humid air and the water droplets. These equations as well as the initial conditions and limit conditions are solved by finite differences. A sensitivity study shows that the highest layers cool more quickly, that a crosscurrent system is more efficient, that droplet concentration little vary in the stack. Speed of cooling is significantly increased by increase of water and air mass flow rates, by higher thermal conductivity of the spheres and by a decrease of the water temperature. This theoretical model was not experimentally validated. The approach presented here deals with a model to predict heat and mass transfers in a stack of spheres during mist chilling. It is based on empirical correlation to take into account local heat transfer intensity inside the stack. It calculates the air and product temperature evolution as a function of process parameters such as air velocity and water mass flow rate. For each sphere, it predicts if water is in excess or not at the surface. The model predictions are finally compared to experimental measurements.
3. Mathematical modelling 3.1. Hypothesis The model is based on heat and mass balances for air and spheres. Previously obtained empirical correlations are used to estimate the heat transfer coefficients (in case of dry air convection) and the droplets collection rate coefficients for each row of spheres. The model is able to predict local air temperature, local droplet concentration, surface and core temperature of each sphere vs. time. The following assumptions were considered in developing the model:
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Table 1 Survey of literature on modelling of heat transfer in stacks Conditions in free stream and configuration
Regime
Main assumptions and features of the model
Experimental validation
Tree et al. (1978)
Air flow direction v (m s1) Cm (%) Body type
With excess of water
•
Good agreement with experimental data
Horizontal 0.8–2.3 0.6 Finned tubes bank
Convective heat transfer coefficient is the ame as in single-phase-flow. Sensible heat is neglected
I. Allais et al. / Journal of Food Engineering 72 (2006) 197–209
Reference
_ K 1 Md _ n1 Þ Nu0 K 2 Mð1 ¼1þ ReDh Nu 36 1011 ffiffiffi K1 ¼ p 3 2ql Ae K2 ¼
DH v Dh C pa DT a la Ae
Ae: external area, Dh: hydraulic diameter of coil, ReDh : Reynolds number of coil • Water mass flow rate is calculated by assessing the wet surface to 1/8 of the total surface • K1, K2 and n are constant. They are adjusted from experimental data for a given configuration
Pawlowski and Siwon (1988)
Air flow direction v (m s1) Cm (%) Body type
Horizontal 0–12 0.26–11.6 3 aligned cylinders
With excess of water
• •
•
Linear distribution of the velocity profile The upstream surface of the cylinders is considered as fully wet whereas the downstream surface is considered as fully dry Bouncing of droplets and evaporation of liquid film are taken into account When the distance between cylinders decrease, heat transfer is increased because the surface covered by liquid film increase. Air velocity and water mass flow rate have the same influence as for a single cylinder
Good agreement with experimental data and with calculations from Hodgson and Sunderland (1968)
Nakayama et al. (1988)
Air flow direction v (m s1) Cm (%) Body type
Horizontal 1–3 0.4–9 Smooth, finned, micro-finned tube banks
With and without excess of water
• •
•
Heat and mass transfer analogy is used Mass balance takes into account the collection of water droplets and water supply flooding from the upper tubes Evaporation depends on the difference of saturated humidity at air temperature and at surface temperature For a smooth tube: _ f C pa mA s xT a ÞAt
Case I. for Aw =At < 0:5 : AAwt ¼ hðxT Aw At
0:37
m_ 0:5 < Aw =At < 0:9 : ¼ 30:1DI Case II. Aw is constant: Aw = 0.9
Air flow direction v (m s1) Cm (%) Body type
Vertical 1–4 0.5–6.43 122 horizontal finned tube
With excess of water
• • •
•
Rao et al. (1993)
Air flow direction v (m s1) Cm (%) Body type
Vertical
With excess of water
•
Packed bed of spheres •
for
0:30 0:4
v
One-dimensional model in flow direction Model based on refrigeration towers model Heat exchanger is separated in layers and a heat balance is performed on each layer Wet surface is taken into account
Agreement with experimental data: ±17%
Temperatures and relative humidity vary only along one direction, physical properties are independent of the temperature, evaporation of the product is neglected Conservation equations are written separately for spheres, air, water droplets and liquid film on the surfaceNumeric methods are used to solve equations (finite difference), a uniform network is used between layers of spheres and a non-uniform network is used for humid air and droplets between the spheres
No experimental validation—sensitivity study
I. Allais et al. / Journal of Food Engineering 72 (2006) 197–209
Dreyer et al. (1992)
Good agreement with experimental data use of the model to design exchangers
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• Flow characteristics (air velocity and temperature, droplets concentration and diameter) upstream the stack are known and they are considered as uniform. • Porosity is uniform, thus superficial velocity is constant in the stack. • Inside a sphere, the conductive heat transfer is radial: product temperature depends on time, row number and radial position. • Initial temperature of the product is known: a parabolic profile is assumed between the centre and the surface of one sphere (experimentally, the stack was first heated in an oven; from the end of the heating to the beginning of mist chilling, the temperature of the spheres slightly decreased near their surface). • Heat conduction between the spheres in the stack is neglected. • Convective heat transfer coefficient between the surface of a sphere and air (upstream the sphere) is calculated from a correlation obtained in a previous study (Allais & Alvarez, 2001), which linked Nusselt number to Reynolds and Prandtl numbers and to row number (heat transfer intensity was lower for the first and the last row of spheres in the stack). • Droplets collection rate coefficient is calculated from a correlation obtained previously (Allais & Alvarez, 2001), which linked the dimensionless collection number (K) to Stokes number (based on the mean Sauter diameter) and row number. Indeed, collection rate decreases through the stack because droplets concentration decreases and also because the mist reaching the later rows contains smaller droplets: the largest droplets of the size-distribution are more easily held back by the spheres and they preferentially impact on the first rows. • Mass vapour transfer coefficient is calculated from heat transfer coefficient according to the Chilton– Colburn analogy (Lewis number for air and water vapour is around 1: Le 1).
• The maximum wetted surface is half of the total surface of the spheres. • Air remains saturated in water vapour when it is flowing inside the stack. • Heat and mass diffusion is negligible compared to convection in the air. There is no accumulation of internal energy or water in air. • All the physical properties of air, spheres, water and vapour are constant (for temperature between 10 C and 50 C).
3.2. Model description Coupled heat and mass transfer equations are written for each layer of spheres and combined with balances of heat, vapour and droplets in air (Fig. 1). 3.2.1. Heat and mass transfer for one sphere of layer j The product temperature varies radially from core: T cp fj; tg ¼ T p fr ¼ 0; j; tg to surface: T sp fj; tg ¼ T p fr ¼ R; j; tg. The conductive heat flux density at the surface is /s{j, t} = /{r = R, j, t}. Upstream the layer j, the air temperature is Ta{j, t}, the vapour content is xa{j, t} = xsat{Ta{j, t}} (kg vapour/kg dry air), the water droplets content is Cm{j, t} (kg water/kg dry air). 3.2.1.1. Convective heat transfer between the surface of the sphere and the air. The heat flux density is proportional to the temperature difference between the surface of the sphere and the air upstream the layer j: _ tg ¼ hfjg T sp fj; tg T a fj; tg qfj; h{j} is known from
predicted properties of particles and air: ρp,Cpp, kp ρa,Cpa, ka properties of water ρl, Cpl, Cpv, ∆ Hv
Ta{j+1,t} Cm(j+1,t }
heat transfer correlation : Nu{Re, j, Pr}
Ta{j, t} Cm(j,t }
Tp{r,j, t }
droplets collection correlation K{St, j, d/D, ρl/ ρa} maximal wetted surface fraction A w /A=1/2 heat/mass transfer similitude Le∼ ∼1, Sh=Nu Ta{j=1,t}
Cm{j=1,t} v
Tp{r,j,t=0}=To(1-α(r/R)²)
given
Fig. 1. Inputs and outputs of the model.
I. Allais et al. / Journal of Food Engineering 72 (2006) 197–209
0.614 hfjgD Re Nu ¼ ¼ afjg Pr1=3 ka 1e a ¼ 0.558
for j ¼ 1
a ¼ 0.752
for 1 < j < jmax
a ¼ 0.659
for j ¼ jmax
where ð1Þ
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At the surface, two modes of heat transfers have to be distinguished whereas there is water in excess or not. No excess of water: all the water which is collected is evaporated after it has been heated to the surface temperature of the sphere (droplets are at the same temperature than air) _ tg /s fj; tg ¼ qfj;
3.2.1.2. Droplets collection. The water flux density corresponding to droplets collection is proportional to upstream water content: m_ c fj; tg ¼ cc fjgqa C m fj; tg
þ m_ c fj; tgDH v
where cc{j} is known from K¼
cc fjg ¼ bfjgSt0:596 with b ¼ 0:246 for j ¼ 1; and v
b ¼ 0:9222 0:1066j þ 0:0202j2 0:0018j3 for j > 1 ðfor ql =qa ¼ 1000=1:3 769 d sauter =D ¼ 8 106 =38 103 2:1 104 Þ
ð2Þ
3.2.1.3. Convective vapour transfer between the surface of the sphere and the air. The vapour flux is proportional to vapour content difference between saturated air in contact with the wetted surface and the air upstream the layer j. Only half of the surface of the sphere can be wetted, thus the maximum vapour flux density is m_ v max fj; tg ¼
Aw max qa hm fjg A
xsat fT sp fj; tgg xsat fT a fj; tgg Aw max 1 ¼ with 2 A ðfrom Le ¼ 1
and and
1 hm fjg ¼ hfjg qa C pa
ð3Þ
Sh ¼ NuÞ
3.2.1.4. Radial heat conduction. FourierÕs heat transfer equations in spherical co-ordinates qp C pp
oT p 1 o oT p ¼ 2 ðr2 /Þ with / ¼ k p r or ot or
ð4Þ
3.2.1.5. Initial condition. Parabolic temperature profile inside the product T p fj; r; t ¼ 0g
r 2 ¼ T cp fj; t ¼ 0g þ T sp fj; t ¼ 0g T cp fj; t ¼ 0g R ð5Þ
ð7Þ
Presence of excess of water: more water is collected than it can be evaporated. The collected water is heated to surface temperature, then part of it is evaporated whereas the other part remains liquid at the surface of the sphere or drips from the sphere _ tg þ m_ c fj; tgC pl T sp fj; tg T a fj; tg /s fj; tg ¼ qfj; þ m_ v max DH v ¼ hfjg þ m_ c fj; tgC pl T sp fj; tg T a fj; tg þ
1 hfjg xsat fT sp fj; tgg xsat fT a fj; tgg DH v 2 C pa ð8Þ
The boundary condition is chosen along with the following procedure: surface temperature is first calculated according to the presence of excess of water boundary condition. The calculation is valid if the collected water mass flow rate is higher than the evaporated water mass flow rate. If not, surface temperature is calculated again according to the no excess of water boundary condition. 3.2.2. Mass and enthalpy balances for mist flow through a layer of products 3.2.2.1. Mass balance. To write the mass balance for mist through a layer of ns spheres, it is assumed that only a part of the droplets are collected when mist flows through a layer of spheres. As far as the non-collected droplets are concerned, some of them remain suspended in the flow whereas some of them evaporate due to the increase of air temperature between the layers j and j + 1 and thus they keep the air saturated. Mass balance for liquid and vapour water is expressed as follows, _ a is the mass flow rate of dry air which remains where M constant through the layers of spheres. _ a ðC m fj þ 1; tg C m fj; tg þ xsat fT a fj þ 1; tgg M
3.2.1.6. Boundary conditions. Symmetry at centre oT p fr ¼ 0; j; tg ¼ 0 or
þ m_ c fj; tg C pl ðT sp fj; tg T a fj; tgÞ þ DH v ¼ hfjg þ m_ c fj; tgC pl T sp fj; tg T a fj; tg
xsat fT a fj; tggÞ ¼ ns Aðm_ c m_ v Þ ð6Þ
with m_ v ¼ minfm_ v max ; m_ c g
ð9Þ
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3.2.2.2. Enthalpy balance. The specific enthalpy (J/kg dry air) of mist is function of air temperature and water droplets content. The enthalpy balance through a layer of products is written as follows: difference between downstream and upstream enthalpy flux is due to cooling of the spheres and to enthalpy of water which is collected without being evaporated (the later term in Eq. (10) disappears when water is not in excess). _ aðH b fT a fj þ 1; tg; C m fj þ 1; tgg M b fT a fj; tg; C m fj; tggÞ H ¼ ns A /s fj; tg ns Aðm_ c m_ v ÞC pl T sp fj; tg
ð10Þ
ns A /s fj; tg _ a ðC pa þ C m fj; tgC pl þ xsat fT a fj; tggC pv ÞðT a fj þ 1; tg ¼M T a fj; tgÞ þ ns Aðm_ c m_ v ÞC pl ðT sp fj; tg _ a ðxsat fT a fj þ 1; tgg T a fj þ 1; tgÞ þ M xsat fT a fj; tggÞððC pv C pl ÞT a fj þ 1; tg þ DH v Þ
ð12Þ
_ a ðxsat fT a fj þ 1; tgg xsat fT a fj; tggÞ is the flow where M of water which evaporates in the layer j and (Cpv Cpl)Ta{j + 1, t} + DHv is the latent heat of evaporation at Ta{j + 1, t} which slightly varies with temperature. Thus, it is assumed as constant and equals to DHv. 3.2.2.3. Upstream conditions. Mass flow rate of dry air, temperature and mist production are constant and uniform upstream the stack.
with b fT a ; C m g H ¼ ðC pa þ xsat fT a gC pv þ C m C pl ÞT a þ xsat fT a gDH v
ð11Þ
Combining Eqs. (9)–(11) leads to Eq. (12) which allows to calculate Ta{j + 1, t}: energy released by the spheres is used to increase air temperature, to heat the collected water up to surface temperature of the spheres and to evaporate part of the collected water.
T a fj ¼ 1; tg ¼ T oa
C m fj ¼ 1; tg ¼ C om
ð13Þ
3.3. Numerical procedure The equations for heat conduction inside the spheres (involving Tp{r, j, t}) are discretised by the finite volume
Fig. 2. General algorithm and detail inside a sphere at row j.
I. Allais et al. / Journal of Food Engineering 72 (2006) 197–209
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method with an explicit scheme. The variables are Tp{i, j, k}, T sp fj; kg, Ta{j, k} and Cm{j, k}, where index i relates to the radial position in a sphere, j to row number, k to time. The general algorithm is shown on Fig. 2. The boundary condition in presence of water excess (Eq. (8)) becomes: T p fimax j; kg T sp fj; kg Dr=2 ¼ ðhfjg þ m_ c fj; kgC pl ÞðT sp fj; kg T a fj; kgÞ
/s fj; kg ¼ k p
þ
1 hfjg ðxsat fT sp fj; kgg xsat fT a fj; kggÞDH v 2 C pa ð14Þ
In this equation, T sp fj; kg is unknown. The equation is non-linear because of the function xsat and it is solved by the iterative NewtonÕs method, taking T sp fj; k 1g as first estimation. In the combined enthalpy and water balance in air (Eq. (12)), Ta{j + 1, k} is unknown. This equation also involves the function xsat. and it is also solved by the iterative NewtonÕs method. The saturated vapour mass fraction is derived from: 0.622e½ð17.438T =239.78þT Þþ6.41 P at e½ð17.438T =239.78þT Þþ6.41 with P at ¼ 105 Pa is the atmospheric pressure
xsat fT g ¼
ð15Þ
ðWax and Greespan, 1971Þ. The numerical procedure was transcripted in Pascal and run on a PC (Dell Latitude).
4. Experimental device and procedure In order to validate the model, an experimental device (Fig. 3) was built. It is made up of two main pieces: a vertical cylindrical duct and a removable stack (27 cm high, 30 cm diameter) containing 359 smooth celluloid spheres of 38 mm of diameter located in the middle of the duct; the stack porosity was 0.409. The spheres were filled with 3% carraghenan gel in order to simulate thermal behaviour of foods products. A fan connected to a frequency variator allows to provide air velocities between 0.35 to 2.8 m s1 measured by a constant temperature hot wire anemometer Dantec 560 AC (accuracy 1%). Pressure drop generated by the stack was measured by an inclined tube manometer (accurate to 0.5 Pa); two pressure taps were located respectively 23 cm upstream and 18 cm downstream the stack. An ultrasonic humidifier (Stulz ens 18) produced water droplets by means of a piezoelectric element, droplets mass flow rate could ranged from 0.8 to 2.5 g/s by modifying electrical supply. Droplets Sauter mean diameter measured by
Fig. 3. Experimental installation.
laser granulometer Fiber PDA Dantec was equal to 8 lm. The whole device was placed in a 50 m3 cold room where temperature could be controlled between 0 and 20 C. Temperature was measured in carraghenan spheres using ‘‘T’’ thermocouples located at the centre and at the surface of each sphere as shown on Fig. 4. Local air temperatures were measured using ‘‘T’’ thermocouples. They were protected with a perforated grid, to prevent any contact with spheres surfaces (as shown on Fig. 4). The gel used to fill the spheres was an homogeneous and isotropic material and its thermal properties were assumed to be constant with temperature (kp, q, Cp). The values for these constants were kp = 0.519 W m1 K1, qp = 1013 kg m3, Cp = 4100 J kg1 K1 (Alvarez, 1992). Experimental procedure was the following: spheres were first warmed up to 50 C in a oven, then, they were inserted in the vertical duct and thermocouples were connected to the data logger and, finally fan and droplet generator were started.
cable
central thermocouple
diameter = 3.8 cm surface thermocouple
air thermocouple protection
air thermocouple Two-phase flow
Fig. 4. Carraghenan sphere.
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temperature of the 6th row was higher than those of 1st and 3rd rows. The air was heated when passing throughout packed spheres so local air temperature at 6th row was higher than for 1st and 3rd rows. The same behaviour was observed for simulated and experimental results. Table 2 compares experimental and simulated halfcooling time, defined as the time required to divide by 2 the initial difference of temperature between air and product. It gives also the relative difference of halfcooling time between model and experiment in 4 experimental cases. Chilling time is shorter in two-phase-flow than in single-phase-flow, velocity being the same. On the
5. Results and discussion 5.1. Comparison between model and experiments Model and experiments were compared for varying conditions of velocity and water mass flow rate. Figs. 5 and 6 compare experimental and simulated temperature evolutions in the air and at the centre of the products at three rows in the stack: 1st, 3rd and 6th row. Fig. 5 was obtained in single-phase-flow with an air velocity of 1.4 m s1. Fig. 6 was obtained in two-phase-flow for the same air velocity and for water mass flow rate equal to 1.4 g s1 which corresponds to 20 g s1 m2. In both cases,
T (°C) 50 experiment
45
air
40
model
centre of spheres
row 1 centre of spheres
35
row 3 row 6
30 25 air
20 15 10 5 0 0
500
1000
1500
2000
2500
3000
time (s) Fig. 5. Comparison of predicted and measured temperature kinetics in the packed bed in single-phase-flow (v = 1.4 m s1, water mass flux density = 0 g s1 m2).
T (°C) 45 experiment air
40
model
centre of spheres
row 1
35
centre of spheres
row 3 row 6
30 25
air
20 15 10 5 0 0
500
1000
1500
2000
2500
3000
time (s) Fig. 6. Comparison of predicted and measured temperature kinetics in the packed bed in two-phase-flow (v = 1.4 m s1, water mass flux density = 20 g s1 m2).
I. Allais et al. / Journal of Food Engineering 72 (2006) 197–209
207
Table 2 Difference of half-cooling time between model and experiment at different layers in the stack Water mass flow rate (g/s)
Velocity (m s1)
0
0.35
2.5
0.35
0
1.4
1.4
1.4
Simulated and experimental half cooling time (s). Relative difference in chilling time (%) Row 1
Row 3
% of heterogeneity (t6 t1/t6) * 100 Row 6
Model
Exp.
Model
Exp.
Model
Exp.
Model
Exp.
908 5.4% 648 18.3% 664 12.6% 550 18.2%
860
1005 7.5% 668 17.6% 685 11.2% 547 14.2%
930
1561 9% 759 5.1% 802 12.1% 557 10.4%
1420
41.8
39.4
720
14.5
26.3
705
17.2
17.6
500
1.8
530 581 450
550 609 470
10
Exp.: experiment.
average, the half-cooling time is decreased by 30%, ranging from 20% to 50%, depending on the row and the velocity. Indeed, in mist flow, heat transfer intensity is increased compared to single-phase-flow. From the four simulated cases, the difference between model and experiment ranged between 5% and 18%. Half cooling time predicted by the model is always higher than experimental half-cooling time. Several reasons can explain differences between experimental results and model. The model was based on several simplifying assumptions such as uniform porosity, plug air flow, no flooding, negligible conduction between the spheres. Moreover, local aerodynamical phenomena such as turbulence, wake development, formation of Karman vortices and Coanda coalescence of the jets formed in the pathways between the spheres are not considered (Alvarez & Flick, 1999). Heterogeneity of cooling, i.e. relative difference of half-cooling time between rows number 1 and 6 are also shown in Table 2. This heterogeneity was higher in single-phase-flow than in two-phase-flow. This can be related to a higher increase of air temperature between the first and the sixth row in single-phase-flow than in two-phase-flow (Figs. 5 and 6). Indeed, two-phase-flow behaves as a fluid with a higher specific heat than air: dðC pa þ xsat fT a gC pv Þ 2:3 dT
near fT a g ¼ 10 C
ð16Þ
Air warms up when it flows through the stacks and this tends to decrease the relative humidity. In two-phaseflow, droplets will then evaporate to maintain the air saturated: this will absorb energy and reduce the temperature increase. 5.2. Sensitivity study The sensitivity study was focused on the influence of the operating parameters to define which ones are the most relevant on chilling time and have to be controlled. The influence of air temperature, water mass flow rate
and air velocity was studied at row 3 using the model. A reference treatment was defined according to the following conditions: initial central temperature: 40 C and initial surface temperature 5 C below, air temperature: 6 C, water mass flux density: 8.5 g s1 m2, air velocity: 1 m s1. Every treatment was compared to the reference treatment on the basis of the half-cooling time (for the reference treatment, the corresponding final central temperature is 23 C). Conditions of simulation are closed to those encountered practically in chilling of fruits and vegetables: product temperature is ‘‘imposed’’ by the harvest climatic conditions and is seldom higher than 40 C, air velocities in chilling installation are rather low (<5 m s1) and mass flow rate is low to avoid water excess on the surface of the products. 5.3. Influence of the velocity The effect of air velocity was studied from 0.25 to 6 m s1. We have observed that half-cooling time decreases to a limiting value when velocity increases (Fig. 7). The effect of the velocity could be analysed in terms of the Biot number. In the conditions of the simulation, Biot number at row number 3 ranges from 1.5 when air velocity equals to 0.25 m s1 to 10.7 when air velocity is 6 m s1. An increase in velocity from 0.25 to about 1 m s1 significantly decreases the half-cooling time but a further increase in air velocity has little effect. This can be explained by the fact that air velocity acts only by increasing external heat transfers and thus, Biot number increases. For higher Biot numbers, heat transfer by conduction in the product becomes dominant and increasing velocity does not significantly decrease cooling time at the centre of the product. 5.4. Influence of water mass flux density Sprayed water mass flux density ranging from 0 to 34 g s1 m2 was studied (Fig. 8). Compared to
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35 g s1 m2 (Allais & Alvarez, 2001). So, the slowing down of the decrease of the half cooling time when water mass flow rate increases can be mainly explained by a similar reasoning than for air velocity: external heat transfer are increased by increasing water mass flow rates but internal resistance to heat transfers becomes the limiting phenomena. Thus, the optimal value for mass flow rate can be defined as the value above which an increase in mass flow rate does not cause a significant decrease in chilling time any more.
half cooling time (s) 850 800 750 700 650 600 550 500 450
5.5. Influence of air temperature
400 0
1
1.5 2.3
3.6
4.6
2
3
5.4
7.0
4
5
6 10.7
velocity (m/s) Biot number
Fig. 7. Sensivity study—influence of air velocity.
half cooling time (s) 750
700
650
600
550
The effect of inlet air temperature was studied from 2 C to 10 C. The half-cooling time did not strongly depend on air temperature. For example, at Ta = 10 C, the half-cooling time corresponding to a final temperature at the center of the spheres of 25 C is 572 s and at Ta = 10 C, the half-cooling time corresponding to a final temperature at the center of the spheres of 21 C is 606 s. This represent only 5.5% variation and is due to a weak non-linearity of xsat vs. Ta (for singlephase-flow, half cooling time is strictly independent of Ta). Nevertheless, from a practical point of view, the temperature at the center of the sphere has to be reduced to a fixed temperature to prevent quality losses. For example, at Ta = 10 C, the time needed to decrease the temperature from 40 C to 15 C is 1128 s whereas at Ta = 2 C the time to reach 15 C is only 812 s, this represent an appreciable time reduction (28%).
500 0
5
10
15
20
25
30
35
Water mass flux density (g.s-1.m-2)
Fig. 8. Sensivity study—influence of mass flow rate.
single-phase-flow, chilling time is reduced when mist is used. Chilling time is reduced by 19% when a 8.5 g s1 m2 water mass flux density is used. Half-cooling time decreases with water mass flux density and rapidly reach a limiting value. When the water mass flux density is higher than about 10 g s1 m2, chilling time becomes almost constant. Two phenomena could explained this behaviour: the evolution of external heat transfers (between air and the surface of the product) vs. water mass flow rate and the evolution of Biot number vs. external heat transfer coefficient. It is well known that the increase of external heat transfers vs. water mass flow rate is first linear, then it slows down (Dreyer et al., 1992; Nakayama et al., 1988; Tree, Goldschmidt, Garett, & Kach, 1978; Yang & Clark, 1975). In experimental conditions similar to those of the simulation, previous measurements of external heat transfer coefficient in the stack have shown that it increased almost linearly when water mass flow rate increased from 0 to
6. Conclusion A numerical model was performed on the basis of heat and mass balances to predict temperature kinetics in a stack of spheres. It was validated for 2 values of velocity and 2 values of water mass flow rates. In view of the simplifying assumptions, it shows a relatively good agreement with experimental results The sensitivity study has shown the influence of the main operating parameters: velocity, water mass flow rate and inlet air temperature. The use of two-phase-flow can typically reduce the half cooling time by 30%. This model could be a basis to develop a control tool and to optimise the chilling process of food products. To further improve the model, some additional features could be taken into account: • Irregular shape of the product. • Variable industrial chilling conditions (air temperature, air velocity). • Non-uniform food properties for example, due to the presence of epidermis in fruits or fatty area on meat carcasses.
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Moreover, to optimise chilling process would require to take into account the quality attributes of the product such as weight losses, aspect or shelf-life.
References Abdul-Majeed, P. (1981). Analysis of heat transfer during hydrair cooling of spherical food products. International Journal of Heat and Mass Transfer, 24, 323–333. Aihara, T. (1990). Augmentation of convective heat transfer by gas– liquid mist. ASME IHTC Jerusalem, KN-29, 445–461. Aihara, T., Fu, W., & Suzuki, Y. (1990). Numerical analysis of heat and mass transfer from horizontal cylinders in downward flow of air-water mist. Transactions of the ASME, 112, 472–478. Allais, I., & Alvarez, G. (2001). Analysis of heat transfer during mistchilling of a packed bed of spheres simulating foodstuffs. Journal of Food Engineering, 49, 37–47. Allais, I., Alvarez, G., & Flick, D. (1997). Analyse du transfert thermique entre un cylindre et un e´coulement dÕair charge´ de gouttelettes dÕeau. Revue Ge´ne´rale de Thermique(4), 276–288. Alvarez, G. (1992). Etude des transferts de chaleur et de matie´ re au sein dÕun e´changeur complexe de type palette (211p). The´se de Docteur de lÕEcole de Ge´nie Rural des Eaux et Foreˆts, Paris, France. Alvarez, G., & Flick, D. (1999). Analysis of heterogeneous cooling of agricultural products inside bins. Part I: Thermal study. Journal of Food Engineering, 39, 239–245. Basilico, C., Jung, G., & Martin, M. (1981). Etude du transfert convectif entre un cylindre chauffe´ et un e´coulement dÕair charge´ de gouttelettes dÕeau. International Journal of Heat and Mass Transfer, 24, 371–385. Bhatti, M., & Savery, C. (1975). Augmentation of heat transfer in a laminar external boundary layer by the vaporization of suspended droplets. Journal of Heat Transfer, Transactions of the ASME (may), 179–184. Dreyer, A., Kriel, D., & Erens, P. (1992). Analysis of spray-cooled finned-tube heat exchangers. Heat Transfer Engineering, 13(4), 53–71. Friedlander, S. K. (1977). Smoke, dust and haze (317p). New York: Ed Wiley.
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Hodgson & Sunderland (1968). Heat transfer from a spray cooled isothermal cylinder. I&EC Fundamentals, 7(4), 567–572. Jones, S. D. M., & Robertson, W. M. (1988). The effects of spraychilling carcasses on the shrinkage and quality of beef. Meat Science, 24(3), 177–188. Kirk, I., Bouse, L. F., Carlton, J. B., Franz, E., & Stermer, R. A. (1992). Aerial spray deposition in cotton. Transactions of the ASAE, 35(5), 1393–1399. Lee, S. L., Yang, Z. H., & Hsyua, Y. (1994). Cooling of a heated surface by mist flow. Journal of Heat Transfer, 116, 167–172. Le´tang, G. (1990). Brumisation des viandes. Revue Ge´ne´ rale du Froid(mai), 25–29. Nakayama, W., Kuwahara, N., & Hirasawa, S. (1988). Heat transfer from tube banks to air–water mist flow. International Journal of Heat and Mass Transfer, 11(2), 449–460. Pawlowski, M., & Siwon, B. (1988). Heat transfer between gas–liquid spray stream flowing perpendicularly to the row of the cylinders. Wa¨rme und Stoffu¨bertragung, 22, 97–109. Rao, K., Narasimham, G., & Krishna Murthy, M. (1993). Analysis of heat and mass transfer during bulk hydraircooling of spherical food products. International Journal of Heat and Mass Transfer, 36(3), 809–822. Rosner, D. E., & Tassopoulos, M. (1989). Deposition rates from polydispersed particle populations of arbitrary spread. AIChE Journal, 35(9), 1497–1508. Tien, C. (1984). Particle deposition in granular media. Annual progress report DE-ACO72-79-ER 10386 004, Syracuse University, Syracuse, New York. Tree, D., Goldschmidt, V., Garett, R., & Kach, E. (1978). Effect of water sprays on heat transfer of a fin and tube heat exchanger. In Proceedings 6th heat transfer conference (Vol. 4, HX-26, pp. 339–344). Wakao, N., & Kaguei, S. (1982). Heat and mass transfer in packed beds. New York: Gordon and Breach Science Publishers. Watterson, M., & Nicholson, K. (1996). Dry deposition and interception of 4–22 lm diameter particles to a lettuce crop. Journal of Aerosol Science, 27(5), 759–767. Wax, A., & Greespan, L. (1971). Journal of Research of the National bureau of Standards, 75A(3). Yang, W. J., & Clark, D. W. (1975). Spray cooling of air cooled compact heat exchangers. International Journal of Heat and Mass Transfer, 18, 311–317.
Modelling cooling kinetics of a stack of spheres during mist chilling Ire´ne Allais b
c
a,b,*
, Graciela Alvarez
a,c
, Denis Flick
a,d
a UMR Ge´nie Industriel Alimentaire, France ´ Cemagref, Unite de Recherches Technologie, Syste´mes d’Information et proce´de´s pour l’Agriculture et l’Agroalimentaire, 24 Av. des Landais, BP 50 085, 63 172 Aubie´re Cedex, France Cemagref, Unite´ de Recherches Ge´nie des Proce´de´ Frigorifiques, Parc de Tourvoie, BP 121, 92185 Antony Cedex, France d INA-PG, 16 rue Claude Bernard, 75231 Paris cedex 05, France
Received 25 June 2004; accepted 23 November 2004 Available online 22 December 2004
Abstract Mist chilling, which uses air and suspended water droplets as a cooling medium can reduce weight losses in food and accelerate heat transfers compared to forced-air cooling. The results presented here deal with the experimental study and modelling of heat and mass transfer during mist chilling in a stack of spheres. A numerical model was established to predict temperature kinetics in a stack of spheres. It takes into account local air temperature, local droplet concentration, heat and mass transfer coefficients. Model predictions are in good agreement with experimental results for single and two-phase-flow for different conditions of air velocity and water mass flow rate. 2004 Elsevier Ltd. All rights reserved. Keywords: Heat transfer; Two-phase-flow; Modelling; Stack; Chilling; Food; Temperature kinetics
1. Introduction For food industry, refrigeration is an important issue to reduce biochemical reactions, to limit the growth of micro-organisms and consequently to extend the shelf life of both fresh and cooked products. Different means of cooling are available. In most of the cases, a singlephase-flow is used: air in forced-air cooling or water applied by immersion or by spraying in hydrocooling. In processes like ‘‘hydraircooling’’ or ‘‘mist chilling’’ a two-phase-flow (air and suspended water droplets) is involved. Mass flow rates and droplets diameter are generally higher (>100 lm) in hydraircooling than in mist
*
Corresponding author. Address: Cemagref, Unite´ de Recherches Technologie, Syste´mes dÕInformation et proce´de´s pour lÕAgriculture et lÕAgroalimentaire, 24 Av. des Landais, BP 50 085, 63 172 Aubie´re Cedex, France. Fax: +33 4 73 44 0697. E-mail address: [email protected] (I. Allais). 0260-8774/$ - see front matter 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.jfoodeng.2004.11.010
chilling (<10 lm). A liquid film on the surface is often observed when spray cooling or hydraircooling are applied (Abdul-Majeed, 1981). Compared to hydraircooling, mist chilling enables water supply control to prevent excess water on the surface of the product. This is particularly important for water sensitive products such as strawberries or bakery products. Compared to forced-air cooling, mist chilling reduces chilling time but also limits weight losses and thus it can limit quality decay due to dehydration. This process was already used to chill meat carcasses after slaughter on an industrial scale (Jones & Robertson, 1988; Le´tang, 1990) but it requires more knowledge to be applied to stack of food products.
2. Bibliography Phenomena involved during the cooling of stacks of products in two-phase-flow are complex. Heat transfers
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Nomenclature total area of the sphere or the body (m2) frontal surface of the body (m2) wetted area of the sphere (m2) collection coefficient (m s1) water content (kg water (kg dry air)1) specific heat at constant pressure (J kg1 K1) d mean Sauter diameter of droplets (m) D sphere diameter (m) Dv/a water vapour/air diffusivity (m2 s1) h heat transfer coefficient for dry air (W m2 K1) 0 h equivalent heat transfer coefficient for mist chilling (W m2 K1) hm mass transfer coefficient (m s1) b H specific enthalpy (J (kg dry air)1) i radial position index j row number k time index ka or kp thermal conductivity (W m1 K1) K collection number Le Lewis number (aa/Dv/a) _ M water mass flow rate in the mist flow (kg s1) _a M dry air mass flow rate (kg s1) m_ c collected water flux density (kg s1 m2) m_ v evaporated vapour flux density (kg s1 m2) ns number of spheres in one layer Nu Nusselt number (hD/ka) Nu 0 equivalent Nusselt number (h 0 D/ka) Pat atmospheric pressure (Pa) Pr Prandtl number (m/aa) q_ convective heat flux density (W m2) r position inside sphere (m) R radius of a sphere (m) Re Reynolds number (vD/n) Rem modified Reynolds number (vD/n(1 e)) A Af Aw cc Cm Cp
depend on aerodynamical and thermal properties of the two-phase-flow, the product shape and the stack arrangement (Allais & Alvarez, 2001). For single products, many experimental studies were performed, they have achieved a better understanding of the increase of heat transfer and of the influence of main factors on heat transfers: airflow characteristics (velocity, turbulence and temperature), water droplet characteristics such as droplet diameter and velocity (Bhatti & Savery, 1975; Lee, Yang, & Hsyua, 1994), water mass flow rate (Aihara, 1990; Basilico, Jung, & Martin, 1981), product characteristics: length, surface temperature and rugosity (Aihara, Fu, & Suzuki, 1990; Bhatti & Savery, 1975; Nakayama, Kuwahara, & Hirasawa, 1988).
Sh St T t v x
Sherwood number (hmD/Da/v) Stokes number (vd2ql/(18laR)) temperature (C) time (s) superficial velocity or upstream velocity of a body (m s1) vapour content (kg water/kg dry air)
Greek symbols aa or ap thermal diffusivity (m2 s1) a coefficient in Eq. (1) b coefficient in Eq. (2) DI difference of specific enthalpy between saturated air and air at the surface temperature DHv latent heat of water evaporation (J kg1) Dr space increment in radial co-ordinates DT temperature difference (K) e stack porosity / conductive heat flux density (W m2) l dynamic viscosity (Pa s) m kinematic viscosity (m2 s1) q density (kg m3) Subscripts a air l liquid water p product s surface of sphere sat saturated v vapour w water Superscripts c core o freestream s surface
Few works in the literature were dedicated to aerodynamic phenomena and heat transfers in stacks of products. Most studies dealt with heat exchangers such as automotive radiator cores (Yang & Clark, 1975), condensers (Dreyer, Kriel, & Erens, 1992) and tube banks (Nakayama et al., 1988), but relatively few of them dealt with stacks of products. They have shown that, in stacks, void fraction and location in the stack have to be considered. Indeed, even in single-phase-flow, several phenomena cause heterogeneity of heat transfer in stacks. If the stack is in a container, the void fraction is not uniform, being greater near the container wall than in the centre (Wakao & Kaguei, 1982), thus, air velocity is higher
I. Allais et al. / Journal of Food Engineering 72 (2006) 197–209
near the wall. Turbulence intensity also varies in the stack because of turbulence production in the wake of each product. Moreover, the temperature of air circulating through layers of products increases because it is heated as it passes through the layers. As a result, the heating of the air and the spatial variations in the local heat transfer coefficient induce considerable thermal heterogeneity (Alvarez & Flick, 1999; Wakao & Kaguei, 1982). In two-phase-flow, heat transfers also depend on local collected water mass flow rate. To assess the amount of collected water can be difficult. Indeed, when a two-phase-flow circulates throughout stacks of products, the amount of water collected by the products in the stack depends on droplet trajectories, which are modified by the presence of the products in the stream. Several aerodynamical studies were focused on the assessment of collected particles. Most of them were performed on a single body but studies dedicated to aerosol and filtration have shown the main mechanisms involved in bulk deposition of particles in a filter: deposition rate depends both on the characteristics of the two-phaseflow (velocity, turbulence and droplet size distribution) and on those of the stack of products (void fraction, body shape and dimensions) (Friedlander, 1977; Rosner & Tassopoulos, 1989; Tien, 1984). Heterogeneity of deposition in a stack was mostly studied in agriculture field to control the supply of liquid inputs (p.e. pesticides), it was shown that local collected liquid mass flow rate exponentially decreased with the depth in flow direction, then, less and less liquid was available for further rows (Kirk, Bouse, Carlton, Franz, & Stermer, 1992; Watterson & Nicholson, 1996). There is a general agreement on the fact that, water droplets can considerably enhances heat transfer. The increase in heat transfer is mainly due to evaporation of water at surface. Several modes of heat transfer can be defined according to surface temperature and water mass flow rate (Basilico et al., 1981). When surface temperature is below boiling temperature and water mass flow rate is high, a liquid film is formed at the surface (water in excess). Heat transfer is due to conduction between body surface and film, to evaporation and convection between water film and two-phase-flow. When water mass flow rate is low, droplets evaporate when they are closed to the surface or just after deposition and a liquid film can not be formed. This mode of heat transfer is particularly suitable for food products sensitive to excess of water (Allais, Alvarez, & Flick, 1997). As far as modelling is concerned, external heat transfers in two-phase-flow were scarcely modelled (Table 1), owing to the very complicated involved phenomena. Prediction of heat transfer in two-phase-flow would require a better knowledge of the hydrodynamics of flow inside a stack of bodies as well as the mechanisms of deposition of droplets. Nakayama et al. (1988) and
199
Dreyer et al. (1992) performed empirical models. NakayamaÕs model aims to predict surface of evaporation in a tubes bank containing 32 tubes arranged in 4 rows. It is based on heat and mass balances and it takes into account flooding from the upper tubes. This approach shown a good agreement with experimental data. DreyerÕs model aims to predict heat transfers for a 122 horizontal finned tubes exchanger when there is excess of water on surface, it is a one dimensional model similar to those used for refrigeration towers. Agreement with experimental data is ±17%. Coupled external and internal transfers in a stack of spheres were only studied by Rao, Narasimham, and Krishna Murthy (1993). Each sphere is assumed to be covered by a water film (excess of water) that flows out by gravity on the lower layers. Water flows downward, air can flow upward or downward. Heat and mass transfers of water flowing by gravity are neglected. Equations of conservation are written separately for the spheres, the liquid film at the surface of the spheres, the humid air and the water droplets. These equations as well as the initial conditions and limit conditions are solved by finite differences. A sensitivity study shows that the highest layers cool more quickly, that a crosscurrent system is more efficient, that droplet concentration little vary in the stack. Speed of cooling is significantly increased by increase of water and air mass flow rates, by higher thermal conductivity of the spheres and by a decrease of the water temperature. This theoretical model was not experimentally validated. The approach presented here deals with a model to predict heat and mass transfers in a stack of spheres during mist chilling. It is based on empirical correlation to take into account local heat transfer intensity inside the stack. It calculates the air and product temperature evolution as a function of process parameters such as air velocity and water mass flow rate. For each sphere, it predicts if water is in excess or not at the surface. The model predictions are finally compared to experimental measurements.
3. Mathematical modelling 3.1. Hypothesis The model is based on heat and mass balances for air and spheres. Previously obtained empirical correlations are used to estimate the heat transfer coefficients (in case of dry air convection) and the droplets collection rate coefficients for each row of spheres. The model is able to predict local air temperature, local droplet concentration, surface and core temperature of each sphere vs. time. The following assumptions were considered in developing the model:
200
Table 1 Survey of literature on modelling of heat transfer in stacks Conditions in free stream and configuration
Regime
Main assumptions and features of the model
Experimental validation
Tree et al. (1978)
Air flow direction v (m s1) Cm (%) Body type
With excess of water
•
Good agreement with experimental data
Horizontal 0.8–2.3 0.6 Finned tubes bank
Convective heat transfer coefficient is the ame as in single-phase-flow. Sensible heat is neglected
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Reference
_ K 1 Md _ n1 Þ Nu0 K 2 Mð1 ¼1þ ReDh Nu 36 1011 ffiffiffi K1 ¼ p 3 2ql Ae K2 ¼
DH v Dh C pa DT a la Ae
Ae: external area, Dh: hydraulic diameter of coil, ReDh : Reynolds number of coil • Water mass flow rate is calculated by assessing the wet surface to 1/8 of the total surface • K1, K2 and n are constant. They are adjusted from experimental data for a given configuration
Pawlowski and Siwon (1988)
Air flow direction v (m s1) Cm (%) Body type
Horizontal 0–12 0.26–11.6 3 aligned cylinders
With excess of water
• •
•
Linear distribution of the velocity profile The upstream surface of the cylinders is considered as fully wet whereas the downstream surface is considered as fully dry Bouncing of droplets and evaporation of liquid film are taken into account When the distance between cylinders decrease, heat transfer is increased because the surface covered by liquid film increase. Air velocity and water mass flow rate have the same influence as for a single cylinder
Good agreement with experimental data and with calculations from Hodgson and Sunderland (1968)
Nakayama et al. (1988)
Air flow direction v (m s1) Cm (%) Body type
Horizontal 1–3 0.4–9 Smooth, finned, micro-finned tube banks
With and without excess of water
• •
•
Heat and mass transfer analogy is used Mass balance takes into account the collection of water droplets and water supply flooding from the upper tubes Evaporation depends on the difference of saturated humidity at air temperature and at surface temperature For a smooth tube: _ f C pa mA s xT a ÞAt
Case I. for Aw =At < 0:5 : AAwt ¼ hðxT Aw At
0:37
m_ 0:5 < Aw =At < 0:9 : ¼ 30:1DI Case II. Aw is constant: Aw = 0.9
Air flow direction v (m s1) Cm (%) Body type
Vertical 1–4 0.5–6.43 122 horizontal finned tube
With excess of water
• • •
•
Rao et al. (1993)
Air flow direction v (m s1) Cm (%) Body type
Vertical
With excess of water
•
Packed bed of spheres •
for
0:30 0:4
v
One-dimensional model in flow direction Model based on refrigeration towers model Heat exchanger is separated in layers and a heat balance is performed on each layer Wet surface is taken into account
Agreement with experimental data: ±17%
Temperatures and relative humidity vary only along one direction, physical properties are independent of the temperature, evaporation of the product is neglected Conservation equations are written separately for spheres, air, water droplets and liquid film on the surfaceNumeric methods are used to solve equations (finite difference), a uniform network is used between layers of spheres and a non-uniform network is used for humid air and droplets between the spheres
No experimental validation—sensitivity study
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Dreyer et al. (1992)
Good agreement with experimental data use of the model to design exchangers
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• Flow characteristics (air velocity and temperature, droplets concentration and diameter) upstream the stack are known and they are considered as uniform. • Porosity is uniform, thus superficial velocity is constant in the stack. • Inside a sphere, the conductive heat transfer is radial: product temperature depends on time, row number and radial position. • Initial temperature of the product is known: a parabolic profile is assumed between the centre and the surface of one sphere (experimentally, the stack was first heated in an oven; from the end of the heating to the beginning of mist chilling, the temperature of the spheres slightly decreased near their surface). • Heat conduction between the spheres in the stack is neglected. • Convective heat transfer coefficient between the surface of a sphere and air (upstream the sphere) is calculated from a correlation obtained in a previous study (Allais & Alvarez, 2001), which linked Nusselt number to Reynolds and Prandtl numbers and to row number (heat transfer intensity was lower for the first and the last row of spheres in the stack). • Droplets collection rate coefficient is calculated from a correlation obtained previously (Allais & Alvarez, 2001), which linked the dimensionless collection number (K) to Stokes number (based on the mean Sauter diameter) and row number. Indeed, collection rate decreases through the stack because droplets concentration decreases and also because the mist reaching the later rows contains smaller droplets: the largest droplets of the size-distribution are more easily held back by the spheres and they preferentially impact on the first rows. • Mass vapour transfer coefficient is calculated from heat transfer coefficient according to the Chilton– Colburn analogy (Lewis number for air and water vapour is around 1: Le 1).
• The maximum wetted surface is half of the total surface of the spheres. • Air remains saturated in water vapour when it is flowing inside the stack. • Heat and mass diffusion is negligible compared to convection in the air. There is no accumulation of internal energy or water in air. • All the physical properties of air, spheres, water and vapour are constant (for temperature between 10 C and 50 C).
3.2. Model description Coupled heat and mass transfer equations are written for each layer of spheres and combined with balances of heat, vapour and droplets in air (Fig. 1). 3.2.1. Heat and mass transfer for one sphere of layer j The product temperature varies radially from core: T cp fj; tg ¼ T p fr ¼ 0; j; tg to surface: T sp fj; tg ¼ T p fr ¼ R; j; tg. The conductive heat flux density at the surface is /s{j, t} = /{r = R, j, t}. Upstream the layer j, the air temperature is Ta{j, t}, the vapour content is xa{j, t} = xsat{Ta{j, t}} (kg vapour/kg dry air), the water droplets content is Cm{j, t} (kg water/kg dry air). 3.2.1.1. Convective heat transfer between the surface of the sphere and the air. The heat flux density is proportional to the temperature difference between the surface of the sphere and the air upstream the layer j: _ tg ¼ hfjg T sp fj; tg T a fj; tg qfj; h{j} is known from
predicted properties of particles and air: ρp,Cpp, kp ρa,Cpa, ka properties of water ρl, Cpl, Cpv, ∆ Hv
Ta{j+1,t} Cm(j+1,t }
heat transfer correlation : Nu{Re, j, Pr}
Ta{j, t} Cm(j,t }
Tp{r,j, t }
droplets collection correlation K{St, j, d/D, ρl/ ρa} maximal wetted surface fraction A w /A=1/2 heat/mass transfer similitude Le∼ ∼1, Sh=Nu Ta{j=1,t}
Cm{j=1,t} v
Tp{r,j,t=0}=To(1-α(r/R)²)
given
Fig. 1. Inputs and outputs of the model.
I. Allais et al. / Journal of Food Engineering 72 (2006) 197–209
0.614 hfjgD Re Nu ¼ ¼ afjg Pr1=3 ka 1e a ¼ 0.558
for j ¼ 1
a ¼ 0.752
for 1 < j < jmax
a ¼ 0.659
for j ¼ jmax
where ð1Þ
203
At the surface, two modes of heat transfers have to be distinguished whereas there is water in excess or not. No excess of water: all the water which is collected is evaporated after it has been heated to the surface temperature of the sphere (droplets are at the same temperature than air) _ tg /s fj; tg ¼ qfj;
3.2.1.2. Droplets collection. The water flux density corresponding to droplets collection is proportional to upstream water content: m_ c fj; tg ¼ cc fjgqa C m fj; tg
þ m_ c fj; tgDH v
where cc{j} is known from K¼
cc fjg ¼ bfjgSt0:596 with b ¼ 0:246 for j ¼ 1; and v
b ¼ 0:9222 0:1066j þ 0:0202j2 0:0018j3 for j > 1 ðfor ql =qa ¼ 1000=1:3 769 d sauter =D ¼ 8 106 =38 103 2:1 104 Þ
ð2Þ
3.2.1.3. Convective vapour transfer between the surface of the sphere and the air. The vapour flux is proportional to vapour content difference between saturated air in contact with the wetted surface and the air upstream the layer j. Only half of the surface of the sphere can be wetted, thus the maximum vapour flux density is m_ v max fj; tg ¼
Aw max qa hm fjg A
xsat fT sp fj; tgg xsat fT a fj; tgg Aw max 1 ¼ with 2 A ðfrom Le ¼ 1
and and
1 hm fjg ¼ hfjg qa C pa
ð3Þ
Sh ¼ NuÞ
3.2.1.4. Radial heat conduction. FourierÕs heat transfer equations in spherical co-ordinates qp C pp
oT p 1 o oT p ¼ 2 ðr2 /Þ with / ¼ k p r or ot or
ð4Þ
3.2.1.5. Initial condition. Parabolic temperature profile inside the product T p fj; r; t ¼ 0g
r 2 ¼ T cp fj; t ¼ 0g þ T sp fj; t ¼ 0g T cp fj; t ¼ 0g R ð5Þ
ð7Þ
Presence of excess of water: more water is collected than it can be evaporated. The collected water is heated to surface temperature, then part of it is evaporated whereas the other part remains liquid at the surface of the sphere or drips from the sphere _ tg þ m_ c fj; tgC pl T sp fj; tg T a fj; tg /s fj; tg ¼ qfj; þ m_ v max DH v ¼ hfjg þ m_ c fj; tgC pl T sp fj; tg T a fj; tg þ
1 hfjg xsat fT sp fj; tgg xsat fT a fj; tgg DH v 2 C pa ð8Þ
The boundary condition is chosen along with the following procedure: surface temperature is first calculated according to the presence of excess of water boundary condition. The calculation is valid if the collected water mass flow rate is higher than the evaporated water mass flow rate. If not, surface temperature is calculated again according to the no excess of water boundary condition. 3.2.2. Mass and enthalpy balances for mist flow through a layer of products 3.2.2.1. Mass balance. To write the mass balance for mist through a layer of ns spheres, it is assumed that only a part of the droplets are collected when mist flows through a layer of spheres. As far as the non-collected droplets are concerned, some of them remain suspended in the flow whereas some of them evaporate due to the increase of air temperature between the layers j and j + 1 and thus they keep the air saturated. Mass balance for liquid and vapour water is expressed as follows, _ a is the mass flow rate of dry air which remains where M constant through the layers of spheres. _ a ðC m fj þ 1; tg C m fj; tg þ xsat fT a fj þ 1; tgg M
3.2.1.6. Boundary conditions. Symmetry at centre oT p fr ¼ 0; j; tg ¼ 0 or
þ m_ c fj; tg C pl ðT sp fj; tg T a fj; tgÞ þ DH v ¼ hfjg þ m_ c fj; tgC pl T sp fj; tg T a fj; tg
xsat fT a fj; tggÞ ¼ ns Aðm_ c m_ v Þ ð6Þ
with m_ v ¼ minfm_ v max ; m_ c g
ð9Þ
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I. Allais et al. / Journal of Food Engineering 72 (2006) 197–209
3.2.2.2. Enthalpy balance. The specific enthalpy (J/kg dry air) of mist is function of air temperature and water droplets content. The enthalpy balance through a layer of products is written as follows: difference between downstream and upstream enthalpy flux is due to cooling of the spheres and to enthalpy of water which is collected without being evaporated (the later term in Eq. (10) disappears when water is not in excess). _ aðH b fT a fj þ 1; tg; C m fj þ 1; tgg M b fT a fj; tg; C m fj; tggÞ H ¼ ns A /s fj; tg ns Aðm_ c m_ v ÞC pl T sp fj; tg
ð10Þ
ns A /s fj; tg _ a ðC pa þ C m fj; tgC pl þ xsat fT a fj; tggC pv ÞðT a fj þ 1; tg ¼M T a fj; tgÞ þ ns Aðm_ c m_ v ÞC pl ðT sp fj; tg _ a ðxsat fT a fj þ 1; tgg T a fj þ 1; tgÞ þ M xsat fT a fj; tggÞððC pv C pl ÞT a fj þ 1; tg þ DH v Þ
ð12Þ
_ a ðxsat fT a fj þ 1; tgg xsat fT a fj; tggÞ is the flow where M of water which evaporates in the layer j and (Cpv Cpl)Ta{j + 1, t} + DHv is the latent heat of evaporation at Ta{j + 1, t} which slightly varies with temperature. Thus, it is assumed as constant and equals to DHv. 3.2.2.3. Upstream conditions. Mass flow rate of dry air, temperature and mist production are constant and uniform upstream the stack.
with b fT a ; C m g H ¼ ðC pa þ xsat fT a gC pv þ C m C pl ÞT a þ xsat fT a gDH v
ð11Þ
Combining Eqs. (9)–(11) leads to Eq. (12) which allows to calculate Ta{j + 1, t}: energy released by the spheres is used to increase air temperature, to heat the collected water up to surface temperature of the spheres and to evaporate part of the collected water.
T a fj ¼ 1; tg ¼ T oa
C m fj ¼ 1; tg ¼ C om
ð13Þ
3.3. Numerical procedure The equations for heat conduction inside the spheres (involving Tp{r, j, t}) are discretised by the finite volume
Fig. 2. General algorithm and detail inside a sphere at row j.
I. Allais et al. / Journal of Food Engineering 72 (2006) 197–209
205
method with an explicit scheme. The variables are Tp{i, j, k}, T sp fj; kg, Ta{j, k} and Cm{j, k}, where index i relates to the radial position in a sphere, j to row number, k to time. The general algorithm is shown on Fig. 2. The boundary condition in presence of water excess (Eq. (8)) becomes: T p fimax j; kg T sp fj; kg Dr=2 ¼ ðhfjg þ m_ c fj; kgC pl ÞðT sp fj; kg T a fj; kgÞ
/s fj; kg ¼ k p
þ
1 hfjg ðxsat fT sp fj; kgg xsat fT a fj; kggÞDH v 2 C pa ð14Þ
In this equation, T sp fj; kg is unknown. The equation is non-linear because of the function xsat and it is solved by the iterative NewtonÕs method, taking T sp fj; k 1g as first estimation. In the combined enthalpy and water balance in air (Eq. (12)), Ta{j + 1, k} is unknown. This equation also involves the function xsat. and it is also solved by the iterative NewtonÕs method. The saturated vapour mass fraction is derived from: 0.622e½ð17.438T =239.78þT Þþ6.41 P at e½ð17.438T =239.78þT Þþ6.41 with P at ¼ 105 Pa is the atmospheric pressure
xsat fT g ¼
ð15Þ
ðWax and Greespan, 1971Þ. The numerical procedure was transcripted in Pascal and run on a PC (Dell Latitude).
4. Experimental device and procedure In order to validate the model, an experimental device (Fig. 3) was built. It is made up of two main pieces: a vertical cylindrical duct and a removable stack (27 cm high, 30 cm diameter) containing 359 smooth celluloid spheres of 38 mm of diameter located in the middle of the duct; the stack porosity was 0.409. The spheres were filled with 3% carraghenan gel in order to simulate thermal behaviour of foods products. A fan connected to a frequency variator allows to provide air velocities between 0.35 to 2.8 m s1 measured by a constant temperature hot wire anemometer Dantec 560 AC (accuracy 1%). Pressure drop generated by the stack was measured by an inclined tube manometer (accurate to 0.5 Pa); two pressure taps were located respectively 23 cm upstream and 18 cm downstream the stack. An ultrasonic humidifier (Stulz ens 18) produced water droplets by means of a piezoelectric element, droplets mass flow rate could ranged from 0.8 to 2.5 g/s by modifying electrical supply. Droplets Sauter mean diameter measured by
Fig. 3. Experimental installation.
laser granulometer Fiber PDA Dantec was equal to 8 lm. The whole device was placed in a 50 m3 cold room where temperature could be controlled between 0 and 20 C. Temperature was measured in carraghenan spheres using ‘‘T’’ thermocouples located at the centre and at the surface of each sphere as shown on Fig. 4. Local air temperatures were measured using ‘‘T’’ thermocouples. They were protected with a perforated grid, to prevent any contact with spheres surfaces (as shown on Fig. 4). The gel used to fill the spheres was an homogeneous and isotropic material and its thermal properties were assumed to be constant with temperature (kp, q, Cp). The values for these constants were kp = 0.519 W m1 K1, qp = 1013 kg m3, Cp = 4100 J kg1 K1 (Alvarez, 1992). Experimental procedure was the following: spheres were first warmed up to 50 C in a oven, then, they were inserted in the vertical duct and thermocouples were connected to the data logger and, finally fan and droplet generator were started.
cable
central thermocouple
diameter = 3.8 cm surface thermocouple
air thermocouple protection
air thermocouple Two-phase flow
Fig. 4. Carraghenan sphere.
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I. Allais et al. / Journal of Food Engineering 72 (2006) 197–209
temperature of the 6th row was higher than those of 1st and 3rd rows. The air was heated when passing throughout packed spheres so local air temperature at 6th row was higher than for 1st and 3rd rows. The same behaviour was observed for simulated and experimental results. Table 2 compares experimental and simulated halfcooling time, defined as the time required to divide by 2 the initial difference of temperature between air and product. It gives also the relative difference of halfcooling time between model and experiment in 4 experimental cases. Chilling time is shorter in two-phase-flow than in single-phase-flow, velocity being the same. On the
5. Results and discussion 5.1. Comparison between model and experiments Model and experiments were compared for varying conditions of velocity and water mass flow rate. Figs. 5 and 6 compare experimental and simulated temperature evolutions in the air and at the centre of the products at three rows in the stack: 1st, 3rd and 6th row. Fig. 5 was obtained in single-phase-flow with an air velocity of 1.4 m s1. Fig. 6 was obtained in two-phase-flow for the same air velocity and for water mass flow rate equal to 1.4 g s1 which corresponds to 20 g s1 m2. In both cases,
T (°C) 50 experiment
45
air
40
model
centre of spheres
row 1 centre of spheres
35
row 3 row 6
30 25 air
20 15 10 5 0 0
500
1000
1500
2000
2500
3000
time (s) Fig. 5. Comparison of predicted and measured temperature kinetics in the packed bed in single-phase-flow (v = 1.4 m s1, water mass flux density = 0 g s1 m2).
T (°C) 45 experiment air
40
model
centre of spheres
row 1
35
centre of spheres
row 3 row 6
30 25
air
20 15 10 5 0 0
500
1000
1500
2000
2500
3000
time (s) Fig. 6. Comparison of predicted and measured temperature kinetics in the packed bed in two-phase-flow (v = 1.4 m s1, water mass flux density = 20 g s1 m2).
I. Allais et al. / Journal of Food Engineering 72 (2006) 197–209
207
Table 2 Difference of half-cooling time between model and experiment at different layers in the stack Water mass flow rate (g/s)
Velocity (m s1)
0
0.35
2.5
0.35
0
1.4
1.4
1.4
Simulated and experimental half cooling time (s). Relative difference in chilling time (%) Row 1
Row 3
% of heterogeneity (t6 t1/t6) * 100 Row 6
Model
Exp.
Model
Exp.
Model
Exp.
Model
Exp.
908 5.4% 648 18.3% 664 12.6% 550 18.2%
860
1005 7.5% 668 17.6% 685 11.2% 547 14.2%
930
1561 9% 759 5.1% 802 12.1% 557 10.4%
1420
41.8
39.4
720
14.5
26.3
705
17.2
17.6
500
1.8
530 581 450
550 609 470
10
Exp.: experiment.
average, the half-cooling time is decreased by 30%, ranging from 20% to 50%, depending on the row and the velocity. Indeed, in mist flow, heat transfer intensity is increased compared to single-phase-flow. From the four simulated cases, the difference between model and experiment ranged between 5% and 18%. Half cooling time predicted by the model is always higher than experimental half-cooling time. Several reasons can explain differences between experimental results and model. The model was based on several simplifying assumptions such as uniform porosity, plug air flow, no flooding, negligible conduction between the spheres. Moreover, local aerodynamical phenomena such as turbulence, wake development, formation of Karman vortices and Coanda coalescence of the jets formed in the pathways between the spheres are not considered (Alvarez & Flick, 1999). Heterogeneity of cooling, i.e. relative difference of half-cooling time between rows number 1 and 6 are also shown in Table 2. This heterogeneity was higher in single-phase-flow than in two-phase-flow. This can be related to a higher increase of air temperature between the first and the sixth row in single-phase-flow than in two-phase-flow (Figs. 5 and 6). Indeed, two-phase-flow behaves as a fluid with a higher specific heat than air: dðC pa þ xsat fT a gC pv Þ 2:3 dT
near fT a g ¼ 10 C
ð16Þ
Air warms up when it flows through the stacks and this tends to decrease the relative humidity. In two-phaseflow, droplets will then evaporate to maintain the air saturated: this will absorb energy and reduce the temperature increase. 5.2. Sensitivity study The sensitivity study was focused on the influence of the operating parameters to define which ones are the most relevant on chilling time and have to be controlled. The influence of air temperature, water mass flow rate
and air velocity was studied at row 3 using the model. A reference treatment was defined according to the following conditions: initial central temperature: 40 C and initial surface temperature 5 C below, air temperature: 6 C, water mass flux density: 8.5 g s1 m2, air velocity: 1 m s1. Every treatment was compared to the reference treatment on the basis of the half-cooling time (for the reference treatment, the corresponding final central temperature is 23 C). Conditions of simulation are closed to those encountered practically in chilling of fruits and vegetables: product temperature is ‘‘imposed’’ by the harvest climatic conditions and is seldom higher than 40 C, air velocities in chilling installation are rather low (<5 m s1) and mass flow rate is low to avoid water excess on the surface of the products. 5.3. Influence of the velocity The effect of air velocity was studied from 0.25 to 6 m s1. We have observed that half-cooling time decreases to a limiting value when velocity increases (Fig. 7). The effect of the velocity could be analysed in terms of the Biot number. In the conditions of the simulation, Biot number at row number 3 ranges from 1.5 when air velocity equals to 0.25 m s1 to 10.7 when air velocity is 6 m s1. An increase in velocity from 0.25 to about 1 m s1 significantly decreases the half-cooling time but a further increase in air velocity has little effect. This can be explained by the fact that air velocity acts only by increasing external heat transfers and thus, Biot number increases. For higher Biot numbers, heat transfer by conduction in the product becomes dominant and increasing velocity does not significantly decrease cooling time at the centre of the product. 5.4. Influence of water mass flux density Sprayed water mass flux density ranging from 0 to 34 g s1 m2 was studied (Fig. 8). Compared to
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35 g s1 m2 (Allais & Alvarez, 2001). So, the slowing down of the decrease of the half cooling time when water mass flow rate increases can be mainly explained by a similar reasoning than for air velocity: external heat transfer are increased by increasing water mass flow rates but internal resistance to heat transfers becomes the limiting phenomena. Thus, the optimal value for mass flow rate can be defined as the value above which an increase in mass flow rate does not cause a significant decrease in chilling time any more.
half cooling time (s) 850 800 750 700 650 600 550 500 450
5.5. Influence of air temperature
400 0
1
1.5 2.3
3.6
4.6
2
3
5.4
7.0
4
5
6 10.7
velocity (m/s) Biot number
Fig. 7. Sensivity study—influence of air velocity.
half cooling time (s) 750
700
650
600
550
The effect of inlet air temperature was studied from 2 C to 10 C. The half-cooling time did not strongly depend on air temperature. For example, at Ta = 10 C, the half-cooling time corresponding to a final temperature at the center of the spheres of 25 C is 572 s and at Ta = 10 C, the half-cooling time corresponding to a final temperature at the center of the spheres of 21 C is 606 s. This represent only 5.5% variation and is due to a weak non-linearity of xsat vs. Ta (for singlephase-flow, half cooling time is strictly independent of Ta). Nevertheless, from a practical point of view, the temperature at the center of the sphere has to be reduced to a fixed temperature to prevent quality losses. For example, at Ta = 10 C, the time needed to decrease the temperature from 40 C to 15 C is 1128 s whereas at Ta = 2 C the time to reach 15 C is only 812 s, this represent an appreciable time reduction (28%).
500 0
5
10
15
20
25
30
35
Water mass flux density (g.s-1.m-2)
Fig. 8. Sensivity study—influence of mass flow rate.
single-phase-flow, chilling time is reduced when mist is used. Chilling time is reduced by 19% when a 8.5 g s1 m2 water mass flux density is used. Half-cooling time decreases with water mass flux density and rapidly reach a limiting value. When the water mass flux density is higher than about 10 g s1 m2, chilling time becomes almost constant. Two phenomena could explained this behaviour: the evolution of external heat transfers (between air and the surface of the product) vs. water mass flow rate and the evolution of Biot number vs. external heat transfer coefficient. It is well known that the increase of external heat transfers vs. water mass flow rate is first linear, then it slows down (Dreyer et al., 1992; Nakayama et al., 1988; Tree, Goldschmidt, Garett, & Kach, 1978; Yang & Clark, 1975). In experimental conditions similar to those of the simulation, previous measurements of external heat transfer coefficient in the stack have shown that it increased almost linearly when water mass flow rate increased from 0 to
6. Conclusion A numerical model was performed on the basis of heat and mass balances to predict temperature kinetics in a stack of spheres. It was validated for 2 values of velocity and 2 values of water mass flow rates. In view of the simplifying assumptions, it shows a relatively good agreement with experimental results The sensitivity study has shown the influence of the main operating parameters: velocity, water mass flow rate and inlet air temperature. The use of two-phase-flow can typically reduce the half cooling time by 30%. This model could be a basis to develop a control tool and to optimise the chilling process of food products. To further improve the model, some additional features could be taken into account: • Irregular shape of the product. • Variable industrial chilling conditions (air temperature, air velocity). • Non-uniform food properties for example, due to the presence of epidermis in fruits or fatty area on meat carcasses.
I. Allais et al. / Journal of Food Engineering 72 (2006) 197–209
Moreover, to optimise chilling process would require to take into account the quality attributes of the product such as weight losses, aspect or shelf-life.
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