Chilling and freezing of foods

Chris KennedyNutriFreeze Ltd.

Food Modelling Club Seminar9 November 2005

Content

Objectives of the models

Analytical models

Numerical models

Determining your input

– Heat transfer– Thermal properties of foods

Objectives

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 lossEfficiencyEquilibration Bug growth and safety

Hardest part is often finding the right inputs …

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)

)( TT Ass hQ

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.

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 TFNote 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.

The Plank model

The Plank Equation is derived from a consideration of the energy balance.

As the freezing front moves a distance x into the slab it creates a new slice of frozen product of volume “A. x”. 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.

The Plank model

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

kh

aL aTTt

AF 82)(

2

0

The Plank model

More generally

kR

h

aP

L aTTt

A

F

2

0)(

where P and R are shape factors having values

.500 and .125 for infinite plates

.250 and .0625 for infinite cylinders

and .167 and .0417 for spheres

Plank extended Pham and others have extended this equation to

consider sensible heat by addition of further terms such that

j

j

A

j

i K

BiQ

TThAt 1)(

0

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.

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

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.

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.

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

1-D Numerical model

This figure shows the temperature evolution of each layer as a function of time in a freezing tunnel.

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.

1-D Numerical model

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

1-D Numerical model (chickens)

Chickens breast meat profiles in a modified Air Products FreshLine AIM Chiller set up

-10

-5

0

5

10

15

20

25

30

35

40

45

0 10 20 30 40 50 60 70

Time (Mins)

tem

p (C

)

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

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

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

Finite element analysis

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

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

Determining heat transfer coefficients

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

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

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)

The Cryomole

The Cryomole

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

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!

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:

http://www.evitherm.org

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

Outputs

Enthalpy of chicken meat

-250

-200

-150

-100

-50

0

50

100

150

-50 -40 -30 -20 -10 0 10 20 30 40

Temperature (C)

Entha

lpy (k

J/kg)

Thermal Conductivity of chicken meat

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

-50 -40 -30 -20 -10 0 10 20 30 40

Temperature (C)

Ther

mal

Cond

uctiv

ity (

W/m

K)

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

Sources of data (COSTHERM)

Enthalpy (COSTHERM) water + 20% protein

-400

-300

-200

-100

0

100

200

-50 -40 -30 -20 -10 0 10 20 30 40

Temperature (C)

Enth

alp

y (kJ/k

g)

Water + 20% protein

Water + 20% Fat

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

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)

Sources of data (evitherm.org)

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 11000 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.

Nelfood available via evitherm …

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

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