The problem of deriving a formula to calculate sterilizing values from temperature curves was examined. For the high sterilizing value range (U/f > 1), a simple algebraic formula was found that enabled the temperature approach g and the sterilizing value U to be calculated from each other to within 3%. For low sterilizing values (U/f <= 1) regression equations were presented to calculate g from U. A table was also given as an alternative to the equations. The method took into account variations in initial product temperature and cooling water temperature. Being expressed in terms of dimensionless quantities, it was more generally applicable than previous methods derived along similar lines. (MSWord file)
Explicit equations are obtained by re-arranging Colebrook's equation to solve the following common pipe flow problems in the turbulent flow region: (a) Calculation of pressure drops, (b) Calculation of pipe diameters, (c) Calculation of flow rates. (PDF file)
Planck's equation for predicting freezing times ignores sensible heat and the gradual character of the phase-change process in foodstuffs. By making certain analytical approximations to take these effects into account, a simple freezing-time prediction method is obtained that uses no empirical factor, chart or advanced algebra. The freezing process is divided into cooling and phase-change periods, the latter being calculated with Plank's equation, the former with extensions to Newton's law of cooling. The gradual character of the phase-change process is taken into account by the use of a 'mean freezing temperature', while the thermophysical properties of the material are approximated by straight-line segments. The method agrees with published data as well as or better than any previous procedure, in cluding finite-difference computations. (PDF file)
For irregular geometries, the distance from the surface to the thermal centre is not uniform. This makes the calculation of freezing time impossible. The difficulty can be overcome by defining a 'mean conducting path' Dm/2, which can then be used to calculate Biot numbers. Dm/2 is considered to be the mean length of heat conducting paths from the surface of the body to the thermal centre. For rectangular blocks, experimental data suggest that Dm is proportional to the geometric mean of the two shorter sides. An equation for calculating the effective Biot number is presented, which incorporates one empirical coefficient. For squat-shaped bodies a good estimate of Dm can be obtained by averaging the shortest and longest dimensions. Pham's freezing time prediction method applied to brick-shaped blocks yields about the same accuracy as for the basic shapes (infinite slabs, infinite cylinders and spheres), and is more accurate than previous methods, including finite-difference methods. (PDF file)
Methods previously proposed to calculate the effect of changes in environmental
conditions were re-examined. A theoretical basis was provided for the method
of adding freezing time fractions. Testing against numerical calculations was
carried out over a wide range of parameters. The results showed that the method
based on following the movement of the freezing front performed reasonably well,
especially when Pham's equation was used to calculate freezing times. However,
inaccuracies may arise when the heat transfer coefficient decreases sharply,
especially for very low cooling medium temperatures or for shapes other than
infinite slabs. The method of adding freezing time fractions performed well
for changes in cooling medium temperature but not for changes in heat transfer
coefficient. A graphical method is also provided for the general case.
(Keywords: food products; freezing; environmental conditions) (PDF
file)
In food freezing situations where heat flows at different rates from the two surfaces of a slab, a Plank-type converging-front model enables the position of the thermal center, and hence the freezing time, to be calculated. The thermal center was defined as the point where the freezing fronts meet, i.e. the last point to cross the "mean freezing temperature". Finite-difference calculations show that this model successfully accounts for the effect of asymmetry in heat transfer coefficient or coolant temperature and enables previously developed simplified prediction methods to be applied to asymmetric situations. (PDF file)
A unified analytical solution and an approximate method are presented for calculating the time to cool or heat an object of simple shape to a given mean temperature, using a fluid with finite heat capacity in batch, parallel-flow, or counterflow modes. In the approximate method, an equivalent constant fluid temperature is calculated, which would give the same log-mean temperature difference. The cooling time at this equivalent temperature is found by conventional methods, then multiplied by a correction factor calculated from a simple regression equation. ( PDF file )
By assuming that the freezing front is similar in shape to the surface, an expression is obtained for the shape factors of ellipses and ellipsoids. This expression agreed well with numerical results for the case of infinite Riot number but disagreed by up to 17% for intermediate Biot numbers. A curve-fitting expression is proposed that agrees with numerical results to within 6%. ( PDF file )
A simple method is presented for predicting the freezing time of water-rich foodstuffs. The method assumes that heat is released at a "mean freezing temperature" common to all foodstuffs. Internal resistance effects are calculated for any shape from a "mean conducting path" intermediate between the shortest and longest distance from surface to centre. ( PDF file )
Note: This method is presently used in a number of textbooks and handbooks
including
Singh and Heldman, Introduction to Food Engineering, Third Edition
(2001)
ASHRAE Refrigeration Handbook ASHRAE, (1998)
Valentas et al, Handbook of Food Engineering Practice. CRC Press, Boca
Raton, Florida.
The method of this paper has been implemented in a spreadsheet
(click here).
Pham, Q.T. and Smith, C.G., 1986, Review of Scientific Instruments 57:99-105.
An analytical equation is derived for the error heat flow due to thermal imbalance
between hot plate and guard in the case of an infinite strip. Finite element
calculations sow that the same equation can be used for circular plates with
negligible error. The equation can be extended to square plates if corner effects
are neglected. An expression is derived for the effect of the distortion of
heat flow lines near the gap even when thermal balance is obtained. This expression
opens the possibility of using larger gaps than previously. The use of two-material
specimens (test material over hot plate, insulant over gap and guard) will lessen
thermal imbalance errors, material requirements and edge effects. ( PDF file
)
J. Food Eng. 6:33-50
During processes involving temperature and/or humidity changes or fluctuations,
products tend to gain or lose moisture. Elementary psychrometry enables the
direction and maximal possible moisture change (in terms of percentage weight
change) to be calculated from a minimum of data. Equations and graphical methods
are presented and illustrated with examples: cooling and freezing of meat, product
temperature fluctuations due to air infiltration or heat conduction during storage,
transport by unrefrigerated containers. (PDF)
In the numerical solution of heat conduction problems with phase change by
finite differences, enthalpy methods or temperature methods can be used. The
former require either an explicit procedure with consequent convergence problems,
or iteration at each time step if implicit procedures are used. The latter are
subject to the problem of jumping the latent heat peak, necessitating the use
of small time steps to avoid underprediction of phase change time. This paper
suggests a simple method that eliminates both problems and results in a fast,
robust procedure that uses less computation time for the same level of prediction
accuracy when compared to other finite difference schemes. (PDF
file)
A number of fixed-grid finite-element methods were tested on problems involving
heat conduction with phase change. Only methods that can deal with arbitrary
enthalpy- temperature relationships were considered. Comparisons were made of
temperature gradient versus enthalpy gradient formulations, lumped versus distributed
capacitance, time-average versus space-average apparent heat capacity, and iterative
versus non-iterative methods. The apparent heat capacity methods that incorporale
lumped capacitances and Pham's correction performed best, in terms of agreement
with analytical solutions and speed of computation (as measured by the number
of matrix solutions). The best iterative method allows marginally larger time
intervals to be used and guarantees perfect heat balance, but for a given accuracy
it is usually slower than the best non-iterative methods. A further advantage
of the non-iterative methods is thal the heal balance can serve as a useful
check of convergence, a heat balance error of more than 1% generally indicating
that convergence has not been reached. (PDF file)
Recent progresses in computer hardware and in optimisation methods have considerably widened the range of problems that can be optimised and the sophistication of the mathematical models. Modern minimisation methods such as evolutionary optimisation and simulated annealing, in conjunction with faster computers, are getting better in handling uncertainties and errors, in finding global optima and in coping with large problems. As examples, three food refrigeration processes are mathematically optimised using a novel evolutionary method. A carton thawing process is designed to ensure complete thawing combined with minimal microbial risk. A beef chilling process was designed to ensure maximum tenderness and the attainment of a 7 C deep leg temperature within a fixed period, while an alternative chilling process ensures that potential microbial growth does not exceed three generations.
In today's competitive atmosphere, optimisation is essential to the survival and profitability of all industrial operations. Recent progresses in computer hardware and in mathematical optimisation methods have considerably widened the range of problems that can be optimised and the sophistication of the mathematical models. Modern methods, in conjunction with cheaper and faster computers, are getting better in handling uncertainties and errors, in finding global optima and in coping with large problems.
In the food industry, many processing methods have evolved over the years through
experience and time-consuming trial-and-error, under regulatory constraints
that are not well founded scientifically. There is scope for these to be scientifically
optimised. (PDF file)
Simple equations are derived to carry out degree of freedom analyses quickly
and almost mechanically, using easily countable physical entities. It is shown
that the outlet stream freedom does not depend on the number of component substances,
and in many cases not on the number of phases. The concept of phase constraint
is introduced to facilitate the degree of freedom analysis. (PDF file)
A number of fixed grid finite element methods were tested on problems involving
heat conduction with phase change. Only methods that can deal with arbitrary
enthalpy-temperature relationships were considered. Comparisons were made of
temperature gradient vs enthalpy gradients formulations, lumped vs distributed
capacitance, time-average vs space-average apparent heat capacity, iterative
vs non-iterative methods. The apparent heat capacity methods which incorporates
lumped capacitances and Pham's correction performed best, in terms of agreement
with analytical solutions and speed of computation (as measured by the number
of matrix solutions). The best iterative method allows marginally larger time
intervals to be used and guarantees perfect heat balance, but for a given accuracy
it is usually slower than the best non-iterative methods. A further advantage
of the non-iterative methods is that the heat balance can serve as a useful
check of convergence, a heat balance error of more than 1% generally indicating
that convergence has not been reached. (PDF file)
One of the main problems in applying evolutionary optimisation methods is the choice of operators and parameter values. This paper propose a competitive evolution method, in which several subpopulations are allowed to compete for computer time. The population with the fittest members, and that with the highest improvement rate in the recent past, are rewarded.
When using identical strategies in the subpopulations, this competitive strategy
provides an insurance against unlucky runs while extracting only an insignificant
cost in terms of extra function evaluations. When using different strategies
in the subpopulations, it ensures that the best strategies are used and again
the extra cost is not great. Competitive evolution is at its best when an operator
- or the lack of it - may have a very detrimental effect which is not known
in advance. Occasional mixing of the best performing subpopulations leads to
further improvement. (PDF file)
An evolutionary method is proposed for the constrained optimization of chemical engineering processes. Apart from the classical mutation, crossover and creep (small mutation), it makes use of several novel reproductive operators: shift, smoothing, extrapolation and swapping. An adaptive mutation rate is used to guard against stalling at local peaks. The method was able to solve dynamic optimization problems involving constrained time-dependent vectors, such as those arising in process control and inverse heat transfer. In addition, the method solves reputedly difficult test problems such as Shwefel's and Grierwangk's functions better than any known previous method. (PDF file)
A computer model has been written to predict the consumption of refrigerants
for vehicle air conditioning in China, India, South Korea and South-East
Asia, their effect on ozone depletion and global warming, and their costs.
A simple logarithmic relationship between per capita income and population
growth rate is assumed. Correlations between vehicle ownership and air
conditioning usage are obtained from worldwide data. Both synthetic HFC
(134a) and natural (hydrocarbons) refrigerants are considered. Sample calculations,
assuming reasonable economic growth rates, predict that the use of hydrocarbons
will lead to significant reductions in global warming potential and large
savings in cost. The synthetic HFC option will incur costs exceeding a
billion US dollars per year after the year 2005.
Last modified on 27 Nov 2001 by T. Pham