making the food safe for human consumption by destroying the pathogenic microorganisms.

Conventional canning processes extend the shelf life of food products and make the food safe for human consumption by destroying the pathogenic microorganisms. Natural convection induced by thermal buoyancy effects in a gravitational force field is observed in many applications. In the food industry products are thermally processed before or after packing. In the second case it is important to establish the properties of food as affected by temperature, the characteristic process parameters and to know the behavior

Modeling the Natural Convection Heating… 11

of the processed product in enclosure. A computational fluid dynamics (CFD) approach is a very useful tool broadly applied in the research on the behavior of liquid foods during thermal processing. GHANI et al. (1999) studied natural convection heating of canned liquid food using CFD methods. They detected the slowest heating zone and identified the time required to reach the sterilization temperature in this zone for vertical cylinder filled with two different liquids. GHANI et al. (2002) focuses on the same problems, but for a horizontal cylinder. There are also studies on effect of different shapes of enclosure on a natural convection heating of liquid food (VARMA MAHESH et al. 2006). RABIEY et al. (2007) studied transient temperature and fluid flow during natural convection heating of a cylindrical can containing large food particles. In all the studies mentioned above such commercial software as FLUENT, CFX and PHOENIX was used. Many other commercial packages enable performing CFD simulation of fluids. One of the advantages of COMSOL MULTIPHYSICS package (Comsol Co.) is its ability to work integrated with MATLAB package (MathWorks Inc.). This feature of COM- SOL is not found in other engineering software packages. Therefore, the aim of this research was to study the abilities of COMSOL MULTIPHYSICS package to perform numerical simulation of heat, momentum and mass transfer in food liquid during natural convection heating of a vertical cylindrical container. The sensitivity of the model to the different values of the properties of the material and to the boundary conditions setting was also studied.

Material and experimental stand

10% water solution of sucrose was heated by natural convection in a steel can using the experimental stand shown in Figure 1. The container made of stainless steel brass with a thickness of 0.2 mm was 160 mm height and 150 mm diameter. Eight J-type thermocouples were placed on the bottom, lid, and on the wall surface of the container. Additional three thermocouples were placed inside the can: the first near the geometric centre of the can, the second and third near the bottom and the lid of the can, respectively, as can be seen in Fig. 1. Signals from thermocouples were registered by computer every 10 s. Before the beginning of the experiment, container was filled with the solution and was chilled to temperature close to 3oC. Chilled can with water was placed in steered water bath. The temperature of the water bath was maintained on the level of 40 ± 1oC during the experiment. Experiment was repeated 3 times.

Konrad Nowak et al.12

40 Co

1

5

6 11

10

7

9

8

2

3

4

Fig. 1. The scheme of the experimental setup: 1 – thermostat, 2 – thermometer, 3 – heater, 4 – stirrer, 5 – thermocouples’ wires, 6 – cover, 7 – container, 8 – thermocouples, 9 – stand, 10 – converter,

11 – PC

Mathematical model

All models were defined as two-dimension problems, with axial symmetry. Balance equations of heat (1), momentum (2) and mass transfer (3) were used as follows:

ρ · cp · ∂T

+ ∇ (–λ · ∇T) = Q + qs · T – ρ · cp · u · ∇T (1)∂t

ρ · ∂u + ρ · (u · ∇) · u = ∇ · [– p · I + η · (∇u · (∇u)T) ] – 2 · η · ∇u · I] + ∂t 3

+ (ρ – ρ0) · g (2)

∂ρ + ∇(ρ · u) = 0 (3)

∂t

In order to perform the computer simulations behavior of density, viscos- ity, thermal conductivity, and thermal capacity of a liquid food should be known and the initial and boundary conditions should be formulated. In all the cases studied the initial temperature of the liquid was evaluated as uniformly distributed in the liquid and the initial condition was described with the following formula:

t = 0 → T(r,z) = T0 = const (4)

Modeling the Natural Convection Heating… 13

Four different versions of boundary conditions, marked as M1, M2, M3 and M4, were studied. The details regarding boundary conditions applied were described underneath while the details regarding the physical properties of the liquid applied during computer simulation are set in Table 1.

Table 1 Physical properties of the water solution of the sucrose applied during simulations

Model Coefficient

ρ (T) = (-0.004) · T2 + 2.12 · T + 763.43 [kg m–3] cp = 4 183 [J kg–1 K]M1 λ = 0.599 [W m–1 K] η = 1.004 · 10–3 [Pa s]

ρ (T) = (-0.004) · T2 + 2.12 · T + 763.43 [kg m–3] M2, M3, cp(T) = (-0.01367) · T2 + 8.83 · T + 2535.78 [J kg–1 K]

M4 λ (T) = 4.93 · 10-8 · T4 + (-5.84) · 10–5 · T3 + 0.026 · T2 + (-5.12) · T + 378.43 [W m–1 K] η(T) = (-52.08 + 0.21 · T)–2.5

M1. In the first case the simplest model was applied. In this model viscosity, thermal conductivity and thermal capacity of the liquid were as- sumed to be constant, while density was assumed to be dependent on tempera- ture of the liquid. The temperature of a given point P on the walls, under the lid and in the bottom of the container was assumed to be constant. The boundary conditions were described with following equation:

t > 0 → T ⎜P∈Ω = TW = const (5)

M2. In this model viscosity, thermal conductivity, thermal capacity and density of the water solution of the sucrose were temperature dependent while similarly to the model M1 the boundary conditions were described with equation (5).

M3. The third model, similarly to the previous one was characterized by temperature dependent viscosity, thermal conductivity, thermal capacity and density of the water solution of the sucrose. It was assumed that Newton’s law of cooling (6) can be applied to describe convection type heat transfer between surface of the cylinder and the liquid outside of the cylinder.

t > 0 → – λ · grad (T) ⎜P∈Ω = h(t – TW) (6)

For this purpose the heat transfer coefficient, h, was calculated based on SERWIŃSKI (1971) on the assumption about the natural convection flow of the liquid round the vertical cylinder. Temperature dependency of the heat transfer coefficient was assumed after SERWINSKI (1971):