BTPR-11-0437 Supplement
Supporting information on governing equations used to simulate concurrent phenomena of heat transfer, fluid flow and electric fields occurring during PEF processing in which the simultaneous solution of the partial different equations describing the conservation of mass, momentum, energy and electric charge have to be solved.
1
Governing equations for electrostatics:
We modelled the static electric field using the electric potential U . Combining the definition of the potential U with Gauss’ law and the equation of continuity, one can derive the classical
Poisson’s equation, which, under static conditions, is defined by:
E
 
U (1) where E is the electric field strength.
Using Eq. 1 together with the constitutive relation D = ε
0
E + P, Gauss’ law can be rewritten as Poisson’s equation:
 
 
0

U

P

 
(2) where ε
0
is dissipation rate of energy, P is the pressure and ρ is the density.
This equation holds for non-conducting media and is used in the COMSOL’s Electrostatics application mode. When handling conductive media, the governing equation for the electric potential is described by the (electro-) static form of the continuity equation:
  J   

    U  J e

 0 where J e is the density of the electrical current.
(3)
Governing equations for laminar flow and heat transfer in the system:

The thermo- and fluid-dynamic behaviour of the pressure medium is described by conservation equations of mass, momentum and energy. The development of a flow field is governed by the continuity equation (mass balance):
 
 t
   
0 (4) where
ρ
is the density of the liquid, t the time and v the velocity vector.
Due to the low flow rates used in the simulations of this study, the fluid flow was laminar flow conditions in the treatment chamber were laminar with Reynolds numbers clearly below
2,300. Hence, the momentum equation can be described by the Navier-Stokes equation:


 t
 v

 v

 v
 
P
   
 v
  g (5) where v is the average velocity, P the pressure,
η
the dynamic viscosity of the fluid, and g represents the gravity constant.
The Navier Stokes equation was coupled with the energy conservation equation for heat transfer through convection and conduction, assuming non-isothermal flow.


  C p
 t
 T

   
 v  C p
 T  Q   
 k   T

(6)
Where C
P
is the heat capacity and T the temperature. The energy source term Q arises from electric energy dissipation in the liquid and can be expressed as:
Q
    
E
2
(7)
Where
φ is a factor for the time-averaged potential (to account for the pulsed potential in a stationary solution) which can be estimated for ideal rectangular pulses as the product of pulse repetition rate f and pulse width τ .

Governing equations for motion of chemical species:
The mass balance of chemical species and thus the flux of ions in an electrical field is given by the Nernst-Planck equation:

C
 t i  


D
  
C i
 z i
 u i

C i

F
  
U


R i
(8) where C i
is the concentration of chemical species, D is the diffusion coefficient, z i
is the valence of ionic species, u i
is the mobility of chemical species, F is the Faradays constant and
R i
is the production term of species.
In batch PEF systems, the motion of chemical species is known to cause electrolysis resulting in a pH gradient towards the electrodes.
2 However, in continuous PEF systems the liquid only receives a few, very short pulses (2 µs pulse duration in this study) while it passes through the treatment zone. In the scenarios discussed in this study, the electrical potential was applied during 0.054% of the (average) residence time of the liquid in the treatment zone only. Hence, it is safe to assume that any motion of chemical species during the application of the electric field will be largely effaced by the flow of the liquid and the Nernst-Planck f
F g
J equation can be neglected in the numerical model.
Notation
C
D
E k i concentration of chemical species [mol·m 3
] diffusion coefficient [m
2 ·s -1
] electric field strength [V·m -1 ] pulse repetition rate (frequency) [Hz, s
-1
]
Faradays constant [96,485.3365 C mol
-1
] gravity constant [9.8 m·s
-2
] density of the electrical current [A·m -2
] thermal conductivity [W·m -1 ·K -1 ]

P
Q
R i
T pressure [Pa] effective specific energy input [kJ·kg -1 ] production term of species [mol·m -3 ·s -1
] temperature [K, °C] t
 v
U time [s] u i z i velocity vector [m·s -1
] voltage [V] mobility of chemical species [mol·m 2 ·J -1 ·s -1
] valence of ionic species
ρ
σ
Greek letters
φ factor for the time-averaged potential [-]
η dynamic viscosity [Pa·s] density [kg·m -3 ] electrical conductivity [S·m -1
]
Abbreviations
PEF pulsed electric field
Operators d
∂

References differential partial differential
Nabla operator (vector differential operator)

1. Buckow, R.; Schroeder, S.; Berres, P.; Baumann, P.; Knoerzer, K., Simulation and evaluation of pilot-scale pulsed electric field (PEF) processing. Journal of Food
Engineering 2010, 101, 67-77.
2. Meneses, N. s.; Jaeger, H.; Knorr, D., pH-changes during pulsed electric field treatments - numerical simulation and in situ impact on polyphenoloxidase inactivation.
Innovative Food Science and Emerging Technologies 2011, 12, 499-504.