A.C Cleland

Computer subroutines for rapid evaluation of refrigerant thermodynamic properties

Abstract Computer routines developed by Chan and Haselden for evaluation of refrigerant thermodynamic properties, whilst widely used, are not suitable for inclusion in some types of refrigeration computer programs due to their large computation time requirements. An alternative in the form of curve-fitted equations with greater computation speed is proposed. A possible application is in the area of dynamic simulation where many thousands of property evaluations must be made in every program execution. The proposed equations cover R12, R22, R114, R502 and R717, and a wide range of practical conditions (but not all conditions covered by Chan and Haselden). The accuracy of the property estimates from the proposed equations should be adequate for many practical situations, but the equations should not be seen as a general replacement for the Chan and Haselden routines.

Polynomial curve-fits for refrigerant thermodynamic properties: extension to include R134a

Abstract Coefficients that extend previously published polynomial curve-fit equations thermodynamic properties of refrigerants to R134a are presented. The calculations are simple, and computationally fast; for commonly encountered refrigeration conditions predicted properties generally agree with the source data to within about ±0.4%, and differences of more than 0.6% occur only occasionally. This level of accuracy is satisfactory for applications such as dynamic simulation of refrigeration system performance in which some accuracy must be sacrificed in favour of computation speed. The coefficients to extend a simple empirical method for calculation of energy requirements for commercial and industrial refrigeration systems to R134a are also presented. The error in this simple method is less than 3% compared with full mass and energy balance calculations.

Prediction of freezing and thawing times for multi-dimensional shapes by numerical methods

Abstract An assessment of the accuracy of numerical methods used in the prediction of freezing and thawing times was made using a comprehensive set of freezing and thawing data for both regular and irregular multidimensional shapes. For regular shapes, a finite difference method gave accurate predictions with reasonable computation costs. Predictions for two finite element method formulations were not always accurate. This was due to practical constraints on the computation costs which meant that time and spatial grids could not always be made sufficiently fine to ensure that the prediction method uncertainty was insignificant compared with the other sources of imprecision. Guidelines are suggested for the use of the finite element method as a freezing or thawing time predictor. These should ensure that the prediction method error is small while keeping the computation costs reasonable.

An analysis of the influence of material structure on the effective thermal conductivity of theoretical porous materials using finite element simulations

Two-dimensional finite element simulations were used to examine the relative influences of selected porosity-related variables on effective thermal conductivity. The finite element models simulated a steady-state thermal conductivity measurement device performing measurements on theoretical materials with varying structures. The results indicated that the extent of contact between pores or particles and the designation of components as continuous or dispersed phases were more significant factors than the size or shape of individual pores or particles. The results suggested that materials with external porosity should be considered separately to materials with internal porosity for the purposes of effective thermal conductivity prediction, and that it is unrealistic to expect a model that is a function of the component thermal conductivities and volume fractions alone to provide accurate predictions for all porous materials. If an additional parameter is incorporated into an effective thermal conductivity model it should be related to the extent of contact between pores or particles.

Prediction of thawing times for foods of simple shape

Abstract A set of 104 experimental measurements of thawing time were made over a wide range of conditions for slab, infinite cylinder and sphere shapes of a food analogue material. These results were used to assess existing thawing time prediction methods. Versions of both the finite difference and the finite element numerical methods that accounted for continuously temperature-variable thermal properties gave accurate predictions. No previously published simple prediction formula was found that was both sufficiently accurate and expressed in a form suitable for it to be adopted as a general thawing time prediction method. Four accurate, but simple, empirical formulae based on Planks equation were developed. These formulae predicted thawing times that were both highly correlated with those predicted by the numerical methods and agreed with the experimental data to within ±10% at the 95% level of confidence. The agreement was more limited by uncertainties in the experimental and thermal property data than by inaccuracy in the prediction formulae. Significantly more accurate simple formulae are unlikely to be developed unless more accurate experimental data are available.

Prediction of freezing and thawing times for multi-dimensional shapes by simple formulae part 2: irregular shapes

Calculated and experimental data for multi-dimensional irregular shapes wer used to assess various methodologies to include the effect of shape in empirical freezing and thawing time prediction methods. The principles underlying two existing geometric factors, EHTD and MCP, were found to be valid; so there seems to be no need for other approaches. Used in conjunction with accurate slab freezing and thawing time prediction methods, the proposed empirical formulae for EHTD and MCP gave accurate predictions for all of the two-dimensional shapes and most of the three-dimensional shapes tested, except those with oval cross-sections in the third dimension. This was attributed to the lack of data for this group of shapes.

A generalised mathematical modelling methodology for design of horticultural food packages exposed to refrigerated conditions: part 1, formulation

Abstract A generalised mathematical modelling methodology to predict rates of key heat and mass transfer processes within refrigerated horticultural packages is described. A novel zoned approach with flexible geometry descriptions and flexible unifying concepts are used to ensure wide-ranging model applicability. The model component hierarchy, which treats the in-pack fluid, packaging and product as equally important components, closely aligns the modelled system and the physical system. This, together with the flexible zone definition methodology and associated definitions of sub-models for intra- and inter-zonal heat and mass transfer pathways support the use of an object-oriented simulation computer programme design. In Part 2, the heat transfer sub-models are presented, and the total model system tested against experimental data for several package-product combinations. In Part 3, the mass transfer sub-models are presented and further test results reported.

Prediction of chilling times of foods in situations where evaporative cooling is significant : Part 1. Method development

Abstract The finite difference method was used to simulate the unsteady state cooling of spheres, infinite slabs and infinite cylinders of food materials subject to both convection and evaporation at the product surface. Simulations were conducted across wide ranges of air temperature, surface heat transfer coefficient, product initial temperature, surface water activity and air relative humidity. Algebraic equations are proposed for finding three parameters—the product equilibrium temperature as time → ∞, a slope parameter of semi-log plots relating unaccomplished temperature change to time and an intercept parameter of the same plots. The first of these equations is based on psychrometric theory and the other two were derived by using non-linear regression to curve-fit the numerically simulated cooling rates. These equations allow the numerically simulated cooling times to be predicted within about ±5%. In Part 2 the accuracy of these equations as a simple chilling time prediction method is tested experimentally for model food systems. In Part 3 their application to real foods is considered.

Prediction of freezing and thawing times for foods of regular multi-dimensional shape by using an analytically derived geometric factor

Abstract Analytical solutions for transient heat conduction with phase change and general boundary conditions were used to calculate a geometric factor to take account of the effect of product geometry on freezing and thawing times for a number of regular two- and three-dimensional objects. These analytically derived formulae depend only on the Biot number and simple parameters that describe object shape. The accuracy of these formulae was demonstrated by comparisons with both large sets of experimental food freezing and thawing data and numerically calculated data. The performance of the new formulae was better than that of any previous ones and the formulae are simple to use.

Prediction of chilling times of foods in situations where evaporative cooling is significant—Part 2. Experimental testing

Abstract The algebraic model developed in Part 1 for prediction of chilling times, where cooling at the product surface is by evaporation as well as convection, was tested. Experimental measurements were made by chilling cylindrical test samples constructed of a food analogue in an air chiller. During chilling, the surface of each sample was continually wetted by a wetting agent of known water activity and the relative humidity of the air was controlled. Over the 30 trials, the proposed method predicted measured equilibrium temperature, intercept and slope parameters of semi-log unaccomplished temperature change vs time plots as accurately as could reasonably be expected taking into account data uncertainties (agreement largely within ± 6%, but at worst ± 10%). Chilling times were also predicted sufficiently accurately for many industrial applications. The simple model can be used with confidence for most chilling conditions likely to be encountered in industrial practice and including a w values below 1.0. Part 3 considers applications to food materials in which surface water activity may not be constant.