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**Chilling and freezing of foods**

Food Modelling Club Seminar 9 November 2005 Chris Kennedy NutriFreeze Ltd.

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**Content Objectives of the models Analytical models Numerical models**

Determining your input Heat transfer Thermal properties of foods

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**Objectives Hardest part is often finding the right inputs …**

Mathematical modelling of freezing or chilling processes is usually performed to meet one or both of the following objectives. 1.Residence time modelling I want a throughput of "x units per hour" … How big is the freezer I need? What type of freezer should I use? What does it cost? 2. Quality modelling Weight loss Efficiency Equilibration Bug growth and safety Hardest part is often finding the right inputs …

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**Simple models The elementary Plank model**

Most heat transfer models for foods are based on two equations, namely: Newton’s equation (for heat transfer at the surface) The Fourier equation (Internal Heat conduction)

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1-D Numerical solution The simplest geometry to start our consideration is the infinite slab. Here we assume that all of the heat transfer out of the slab is through the top and bottom surface. The slab is symmetric so we need only consider one half. Tc is the core temperature and Ts the surface temperature of the slab.

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The Plank model The Plank model gives a simple way of calculating freezing times Assumes All heat to be removed is latent heat Thermal properties are constant The final core temperature is TF Note we have rotated our slab 90o in this diagram. The distance “a” is the half-thickness of the slab. The slab started at a uniform temperature and the graph shows the temperature profile after a certain time.

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The Plank model The Plank Equation is derived from a consideration of the energy balance. As the freezing front moves a distance Dx into the slab it creates a new slice of frozen product of volume “A. Dx”. The latent heat removed across that slice is equal to the heat conducted from the slice to the surface which must also be equal to the heat removed at the surface.

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The Plank model The equations are solved analytically to give a total freezing time.

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**The Plank model More generally**

where P and R are shape factors having values .500 and .125 for infinite plates .250 and for infinite cylinders and and for spheres

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Plank extended Pham and others have extended this equation to consider sensible heat by addition of further terms such that This is Newton's Law of Cooling with the factor 1+Bij/Kj added to account for internal resistance to heat flow Bij/Kj is the ratio of the internal and surface resistances.

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Numerical solution To gain useful estimates of temperature distributions we need to use numerical methods To accurately predict freezing times, equilibration temperatures and surface temperatures we need to know the temperature gradients across the product as a function of time

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**1-D Numerical solution Let’s return to our infinite slab.**

Remember the heat flows are symmetric and we consider heat flow through the top and bottom surfaces only.

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1-D numerical solution For the numerical model we “slice” the half-slab into n layers and consider the heat flows between each layer in a series of time steps.

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**At each time step we calculate:**

1-D Numerical solution At each time step we calculate: surface T using a potential divider and the heat flow from the previous step surface heat flow from Newton's equation each of the internal heat flows the new temperature distribution Then move to the next time step

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1-D Numerical model This figure shows the temperature evolution of each layer as a function of time in a freezing tunnel.

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1-D Numerical model Alternatively we can look at the “Key” temperatures. This slide shows the same simulation but this time we are just looking at the core (pink) and surface (blue) temperatures. The third line is the equilibrated temperature (red) calculated from the total heat content. This is the temperature that the product would equilibrate to, if the process was stopped at that point.

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1-D Numerical model We can extract other useful information from this model, such as the Heat Flux at the surface, as shown here.

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**1-D Numerical model (chickens)**

This slide shows a simulation of the temperature profile across a chicken breast in a novel accelerated maturation chiller developed by Air Products plc.

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1-D numerical model The 1-D method can also be applied to cylinders and spheres Packages such as HEATSOLV (available via evitherm website) also deal with more complex shapes by addition of a shape factor Equilibration temperatures allow us to calculate accurate residence times Surface temperatures are useful for estimation of evaporative weight loss Temperature gradients allow us to deal with large or delicate products

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**Finite element model FE analysis allows modelling of 3-D heat flow**

The basis is still the Fourier equation and Newton's Law of Cooling, but now a matrix calculation Most packages are also designed for stress modelling so this is the proffered choice of model for thermal stress analysis A number of commercial packages are available, for example: ALGOR FEAT ELFEN

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**Finite element analysis**

The picture here shows the result of a Finite Element Analysis of chilling of a beef leg using the ELFEN package.

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**Finite element analysis**

Here the output is set to show the depth of crust freezing of the leg in an accelerated chilling process. The data was taken during an EU project on The Very Fast Chilling of Beef

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**Determining heat transfer coefficients**

Most models use a single value of HTC. But heat transfer coefficients are rarely/never constant in space and time.

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The Cryomole A device for mapping heat transfer coefficients in freezers and chillers. The device is manufactured by York Electronics Centre

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**The Cryomole The sensor is a known volume or surface area of copper**

Copper T and Air T are measured at, for example, 1 sec intervals HTC is then calculated assuming infinite conductivity (a good assumption)

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The Cryomole Raw data from the Cryomole showing temperatures of the air and the copper probe

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Cryomole Issues Need to be sure that the measuring device does not change the property measured Air flows around the mole etc Limited time as accuracy drops as the temperatures converge Active devices may also be possible A bit tough to use in a fluidised bed or rotary freezer!

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**Thermal properties data**

A good model requires good thermal data for the materials concerned The two main parameters required are the enthalpy v. temperature and thermal conductivity v. temperature relationships You can of course measure these yourself (?) or get some help from a number of software programs and online databases The first port of call for all of these is:

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**Sources of data (COSTHERM)**

An easy code to predict the thermal properties of foods is COSTHERM COSTHERM was developed under the EC’s COST90 Generally looked after by Paul Nesvadba of Rubislaw Consulting The software is a series of algorithms based on food composition where the user enters: Water content Protein content Fat content etc Freezing point and density

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Outputs The plots show data for heat content and thermal conductivity of product for a range of temperatures

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**Sources of data (COSTHERM)**

Although accurate modelling requires a knowledge of the composition, this plot demonstrates the large extent to which heat content is dependent on water content.

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**Sources of data (evitherm.org)**

COSTHERM (program for predicting thermal diffusivity of liquid food) CINDAS (thermophysical properties, mainly solids) eFoodSolver (has a thermal property predictor tool at the foot of the page) HEATSOLV (1-D heat equation solver: slab, cylinder, sphere and "fractal shape" such as a fish -somewhere between cylinder and slab) NELFOOD (food properties data)

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**Nelfood available via evitherm …**

NELFOOD - Physical Properties of Food Database, hosted by the National Engineering Laboratory, Scotland (NEL) The NELFOOD interactive website allows users to view, add and modify bibliographic and experimental data on the physical properties of foods. Users can search through over bibliographic references 1500 materials 1600 experiment data sets About one third of the data in NELFOOD concerns thermal properties of foods. Other categories are mechanical, electrical, diffusion/sorption and optical/colour The data sets range over 24 categories encompassing 249 subcategories and 260 physical properties. Once data is found, it can be viewed, plotted, copied, and printed out.

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Summary Simple analytical models based on Plank are often sufficient to give a good approximation of residence time For more accurate estimates and for information on surface temperatures and temperature distributions numerical methods will provide more information

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**Summary The model can only be as good as the data**

There are now numerous sources of data which cover a wide range of thermal properties (in addition to the data you need) Software is available to estimate thermal properties Best results will always be attained from actual measurements of HTC and thermal properties

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