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A mathematical model has been formulated to describe the heat transfer in liquid foods flowing in circular ducts, subjected to microwave irradiations. Three types of liquids with different rheological behavior are considered: skim milk (Newtonian), apple sauce and tomato sauce as non-New-tonian fluids. Each one can flow with different velocities but always in laminar way. The temperature profiles have been obtained solving the transient momentum and heat equations by numerical resolution using the Finite Element Method. The generation term due to the microwave heating has been evaluated according to Lambert’s law. Dielectric properties are considered to be temperature dependent.

Microwave heating has been utilized since the 1940s [

Microwave heating of solid foods has been largely investigated in the last years. The energy equation is written as a conductive heat transfer with a generation term. The latter has been modeled by many authors, using two different approaches to evaluate the effects of the microwave distribution: by solving the Maxwell’s equations [

As regards fluids, there are fewer studies and the most are about batch processes [

The physical system represented in

A laminar flow at various velocities is realized by changing the difference of pressure between the inlet and the outlet sections. Only the axial component of the velocity is different from zero and it is a function of the variable r: v_{z} = v_{z}(r). The temperature, even though the microwave penetration is only radial, is also function of the axial direction z by the effect of the flow field: T = T(r, z).

In order to find the temperature profile, the following mathematical model has been constructed.

It consists of the following three differential equations in cylindrical coordinate system with their boundary conditions [

As initial condition, the fluid is stationary.

As boundary conditions, no slipping at the wall has been assumed

and symmetry on the axis has been considered.

As initial condition, the temperature of the fluid is assumed to be uniform

As boundary conditions for the radial direction, symmetry respect to the axis and heat convective flux at the wall have been considered.

while for the z direction uniform temperature in the inlet section and Danckwerts condition in the outlet one have been imposed.

Initial temperature T_{0} and input temperature T_{in} are both equal to environmental temperature T_{air} .

Heat generation due to microwaves has been modeled according to the Lambert’s law along the radial direction of a cylindrical sample [

where

The attenuation factor for each fluid has been considered as a function of temperature, calculated by interpolation starting from graphic relationships for dielectric constant and loss tangent versus temperature in a range 10˚C - 90˚C [

Such a system, with the equations and boundary conditions written before, results to be axial-symmetric.

To solve the previous partial differential equations, a Finite Elements Method (FEM) has been used. To practically implement this solution, COMSOL Multiphysics^{®} has been utilized with the following mesh features: 2049 mesh points, 3840 triangular elements, 256 boundary elements and 4 vertex elements.

Three fluid foods have been considered: skim milk, with a Newtonian behavior (constant viscosity), apple sauce and tomato sauce as non Newtonian fluids, modeled with a power law having different fluid consistency coefficient and flow behavior index. All the physical properties of the three fluids are resumed in

As ε' and ε" are temperature functions, average values have been obtained by integrating in the entire domain and in the time (range 0 - 50 s). They are fundamental for microwave heating, because they determine respectively the energy absorbed and the fraction converted in heat power.

It is possible to make a qualitative analysis of the results observing the following temperature maps reported in Figures 3-5. They have been obtained with an incident power Q_{1} = 20,000 W・m^{−2}, for each fluid, for two different velocities (2 mm/s and 4 mm/s) and for two different instants of time (30 s and 50 s). They show the temperature profiles in a rz plane section of the real physical system on the left and in a 3D plot on the right. The higher heating is obtained for longer times and lower velocities. The nature of liquid also have a great influence: skim milk reaches higher temperature with a less uniform profile.

The graphs in Figures 6-8 provide a more significant analysis since they show the temperature profiles along the radial direction, obtained with the same incident power of 20,000 W・m^{−2} and appointing the outlet section instead of the instant of time. In this way, each fluid element has the residence time due to its velocity, which is related to its position.

Skim milk | Apple sauce | Tomato sauce | |
---|---|---|---|

Density, ρ [kg・m^{−}^{3}] | 1047.7 | 1104.9 | 1036.9 |

Specific heat, c_{p} [J・kg^{−1}・K^{−1}] | 3943.7 | 3703.3 | 4000.0 |

Thermal conductivity, k [W・m^{−1}・K^{−1}] | 0.5678 | 0.5350 | 0.5774 |

Viscosity, μ [Kg・m^{−1}・s^{−1}] | 0.0059 | ||

Fluid consistency coefficient | 32.734 | 3.9124 | |

Flow behavior index | 0.197 | 0.097 | |

Dielectric constant, ε' | 66.31 | 68.97 | 74.27 |

Dielectric loss, ε" | 13.26 | 5.30 | 46.42 |

In these graphs all the fluids show a minimum (cold spot) located at about half the radium. This is the sum of two effects: Lambert’s law that predicts a different heating along the radial direction and the flow field that produces different residence times inside the tube. In particular, near the axis (r = 0) the velocity is higher and so the residence times are lower, but according to the Lambert’s law, the power density is maximum by effect of the term R/r; conversely, near the wall (r = R) the velocity is lower and the times of exposition to the microwaves are higher, but the Lambert’s power density undergoes the effect of the exponential decay. These are opposite effects and their combination produces the previous profiles, with higher temperatures on the axis and on the wall.

The kind of fluid plays an important role, both for the rheological behavior and the dielectric properties, while the physical properties are quite the same for all of them. On one hand, pseudo plastics having a flatter velocity profile than Newtonian fluids, can’t fully balance the effect of Lambert’s law. On the other hand, the dielectric properties, in particular dielectric loss, determinant for the absorbed heat, are different fluid by fluid. Such dielectric properties cause a lower difference in the absorbed heat between axis and wall in the case of tomato sauce, as it can be noticed in

As the ε' is quite the same for all the fluids, they absorb about the same quantity of energy from microwaves, but tomato sauce transforms the higher fraction of this energy into heat (Q_{converted}) by a higher value of ε'' and so it shows a higher average temperature. These results are resumed in

In this paper, microwave heating of three liquid foods moving in a cylindrical duct with a laminar flow has been analyzed.

The multiphysics mathematical model considers the momentum and energy transport at unsteady state.

The analysis has been achieved with the Finite Element Method solving the mathematical model with Comsol^{®}3.5.

Contrasting effects of Lambert’s law for microwave heating in case of a cylindrical geometry and distribution of residence times of the fluids in the duct have been taken into account.

Different factors play a role in case of microwave heating of a liquid. In particular, rheological and dielectric properties of the fluids have been considered and two different operational conditions have been obtained by varying the average velocity of the fluids.

Results show that absorbed power has always a maximum on the axis, caused by the ratio R/r appearing in the radial contribution to the microwave heat source for a cylindrical geometry; however, near the wall of the tube a high quantity of absorbed heat has been found, in this case due to the long times of residence of the fluid inside the tube.

Axis QLR 10^{−6} [W・m^{−3}] | Wall QLR 10^{−6} [W・m^{−3}] | |
---|---|---|

Skim milk | 14.71 | 1.187 |

Apple sauce | 7.063 | 0.362 |

Tomato sauce | 7.790 | 2.614 |

Average QLR 10^{−6} [W・m^{−3}] | Average T [K] | |
---|---|---|

Skim milk | 1.491 | 287 |

Apple sauce | 1.108 | 285 |

Tomato sauce | 2.887 | 294 |

The final result is a double pick of absorbed heat on the axis and on the wall; between the two opposite effects the term R/r overcomes the one due to the residence time.

In any case, the absorbed powers are almost equal for the three fluids because the dielectric constants ε' are of the same order of magnitude, whereas the fraction of this energy which is dissipated into heat is different in the three cases as the dielectric losses ε" are different.

The tomato sauce gives the higher average temperature and also the more uniform temperature distribution.

We have compared these results with those obtained studying a cylindrical system with anon-symmetric microwaves exposure [

In both the cases, the temperature is higher on the axis, it decreases until about half the radius and it increases again near the wall. Furthermore, the lower difference between the hot and the cold spot in the temperature profile can be found in the tomato sauce case.