Thermal Food Processing

440 Thermal Food Processing: New Technologies and Quality Issues
14.2.4.1 Model for a Single Particle in a Static Heater
Containing a Fluid
A model for a single solid particle in a static heater containing a fluid was developed
initially by de Alwis and Fryer 1 and further enhanced by de Alwis and Fryer 34 for
a two-dimensional system. The main purpose of the model is to predict the sterilization
effect in a practical situation, where complex food shapes are heated. It is
a finite-element model that has been designed to simulate three types of situations:
• Zero convection: This is the case of highly viscous and gel-forming
foods, where convective processes are less significant.
• Enhanced conduction: When convection cannot be neglected, one of
the convenient ways of treating the problem is to consider the existence
of an effective conductivity value (σ eff), which can be determined empirically
and approximately to replace the effects of both convection and
conduction.
• Well-stirred liquid: In previous work, 19 where an unstirred fluid was
used, no hot spots were noticed when very low viscosity mixtures were
heated, indicating that rapid convective mixing was taking place. In
this case, a well-stirred liquid condition can be applied.
To obtain the electric field distribution, Equation 14.1 was written for a twodimensional
situation, yielding
∂
⋅
∂
∂ ⎛ ⎞
⎜ ⎟ +
⎝ ⎠
∂
⋅
∂
∂
V ⎛ V ⎞
σx ⎜σy
⎟ = 0
x dx y ⎝ dy ⎠
(14.20)
where x and y are the space coordinates of the system and σ x and σ y the values of
σ in the x and y directions, respectively. The solution of this equation was obtained
using boundary conditions of (1) a uniform voltage on the electrodes, or (2) no
current flux elsewhere across the boundary.
For heat generation, a network theory approach was used in which each
triangular element was considered an isolated network, with nodal voltages known
by solution of Equation 14.20. Heat generation in the particle was thus found by
i=
3
∑i = 1
̇Q= V ⋅I
(14.21)
where i is the node number.
Finally, the temperature distribution has been found by means of an energy
balance, similar to that presented in Equation 14.13:
i i
ρ ⋅ ⋅ ∂ ∂
= ⋅
∂ ∂
∂ ⎛ ⎞
⎜ ⎟ +
⎝ ∂ ⎠
∂
⋅
∂
∂
T
⎛ ⎞
C
⎜ ⎟
t x ⎝ ∂ ⎠
k
T
x y k
T
p x y +
y
̇ Q
V sys
(14.22)