INTRODUCTION
Physical processes occurring in different media (gases, liquids, solids) are revealed by some effects recorded by special instruments. The classical models of mathematical physics are based in the concept of a continuous (qualitatively) homogenous medium ( 1995). Leaning upon the observations (measurements) and general physical laws and relationships, one or another suitable mathematical model can be assigned to a process observed. The development and justification of mathematical models, according to ( 1995), are often called identification. A mathematical model, as an abstract means for representing approximately a real process used for its study, is a mathematical description of essential factors and their interrelations.
A set of models, differing specifically in the number of factors included and consequently, in the description accuracy, on the one hand, and in complexity, on the other, can usually be assigned to the same process. Partial differential equations are employed for the description of processes distributed in both time and space, the medium being not discrete but continuous in space (2000). Such models are effective in the problems of air and gas dynamics, mass transfer, elasticity, electrodynamics, and, to serve the purpose of this study, heat transfer.
HEAT TRANSFER
According to (2000), heat transfer ‘is the term applied to a study in which the details or mechanisms of the transfer of energy in the form of heat are primary concern’. There are many examples of heat transfer. Familiar domestic examples, as cited by (2002) include broiling a turkey, toasting bread, heating water, and in the context of this literature review, baking pizza. Industrial examples include curing rubber, heat treating steel forgings and dissipating waste heat from a power plant (2004). Since heat is contained in a substance as molecular motion, it is not surprising that this motion is transferred from a hot body to a cold body by direct contact (1998). Heat transfer, according to (1967), is energy in transit, which occurs as a result of a temperature gradient or difference. This temperature difference is thought of as a driving force that causes heat to flow.
(2000) mentions that the transfer of heat may take place by means of: (a) conduction, (b) convection, or (c) radiation. Fast-moving molecules tend to speed up their slower neighbours on collision. This method of heat transfer is called, simply, conduction (2003). Some materials are good conductors of heat; some are poor. Generally speaking, metals are excellent conductors; and the best conductors of electricity are also the best conductors of heat ( 1998). Thus aluminum, a good electrical conductor, is likewise a good conductor of heat; and aluminum pots and pans (particularly the heavy cast ones) are excellent for cooking purposes, because they heat rapidly and uniformly. Baeyer (1998) stated that on the other hand, materials like wool, sand, asbestos, cork, and still air are poor conductors of heat; hence they are valuable for insulation of our houses, our refrigerators, and our bodies against either heat or cold.
(2003) had claimed that practitioners of the thermal arts and sciences generally deal with four basic thermal transport modes: aside from the three mentioned above, there is also phase-change. According to them, the process by which heat diffuses through a solid or stationary fluid is termed heat conduction. Situations in which heat transfer from a wetted surface is assisted by the motion of the fluid give rise to heat convection, and when the fluid undergoes a liquid-solid or liquid-vapour state transformation at or very near the wetted surface, attention is focused on this phase-change heat transfer. The exchange of heat between surfaces, or between a surface and a surrounding fluid, by long-wavelength electromagnetic radiation is termed thermal heat radiation, they concluded. Heat transfer is usually transferred by a combination of the four (1994).
In gases and liquids another process of heat transfer is very effective; namely, convection, or the rising of heated fluids in accordance with Archimedes' Principle of buoyancy (1998). It has already been seen how this works in the case of drafts in chimneys, and in the shimmery appearance of the air above hot objects. Wind is usually caused by convection currents in the atmosphere, and likewise, gliders can rise to an altitude of thousands of feet and can fly many miles with the aid of updrafts in the atmosphere (1943). Such convection currents moving either up or down cause the unpleasant ‘bumps’ in the air that often annoy airplane passengers, and even make them airsick.
Heat can also be received through empty space from the sun. Evidently not all the heat in the universe is stored up in the form of atomic and molecular motion. As a matter of fact, every object--hot or cold --constantly emits long wave length infrared heat rays. This radiation is similar to visible light except that its wave length is greater. While transfer of heat by conduction or convection requires the presence of a material medium, heat radiation travels most readily through empty, evacuated space (2005). Radiation may, however, be transmitted through any medium that does not absorb it. All objects emit heat rays; but it is a matter of everyday experience that hot objects radiate more heat than do cold ones (1998). The quantity of energy radiated increases very rapidly with increased temperature. In fact, it goes up as the fourth power of the absolute temperature (1998). This means that doubling the absolute temperature results in 4, or 16, times as much emitted radiation.
As noted by Schroeder (2000), heat flows from a region of higher temperature to a region of lower temperature. In a solid medium, heat transfer takes place primarily by the process of conduction[1], and on the other hand, heat transfer in a gas results largely from the relative motion of the gas molecules and is therefore due to convection. In liquids, both conduction and convection are significant ( 1998). (1962) stated that heat transfer by the process of radiation is an electromagnetic wave phenomenon which can take place either in a vacuum or in material media. (1953) has discussed in his book the law of heat conduction. According to him, consider the heat flow taking place inside a homogeneous solid. Let u(x,y,z,t) be the temperature distribution throughout the solid.
Specifically, let the heat flow through an imaginary plane S which is normal to the x axis be considered. Let D x be the time rate of heat flow per unit area through s in the direction normal to the plane, i.e., in the x direction. Now D x is proportional to the negative temperature gradient in the x direction; thus
¶u
¶c
where k is the thermal conductivity [2] [in cgs units, k is in cal/(cm)(°C)(sec) and D is in cal/(cm2)(sec)]. The negative sign signifies that heat flows in the direction of decreasing temperature. Likewise, if the imaginary plane s were oriented so as to be normal to the y and z axis, the values of D y and D z, respectively, would be
¶u ¶u
¶y ¶z
It is assumed that the medium is homogeneous and isotropic; hence that k is the game in all coordinate directions. If D x, D y, and D z are the values at a point in the solid, they can be regarded as the components of a vector D, which will be defined as the flux density of heat[3]. Thus, at any point in the solid,
D = D x i + D y j + D z k
MATHEMATICAL MODELLING OF HEAT TRANSFER
Janna (2000) asserted that the general heat transfer problem involving all four modes (convection, conduction, radiation, phase change) can be set up and described mathematically, and it is important to develop a method for solving heat transfer problems. (1995) asserted that mathematical simulation can greatly aid in many areas of specialisation, one outstanding example is for food technology, in that it helps in understanding and optimisation of processes, for instance it can determine the minimum time for removing moisture. Solving the equations analytically, however, is not always possible (2000). In a number of cases, one mode of heat transfer is dominant. It can then be identified and modelled satisfactorily to obtain a solution to what could be an otherwise insoluble problem (1998).
According to (1995), when applied to heat conduction and heat transfer processes in engineering systems, the computational model can be expressed in the following abstract form:
Ah (w)u = ¦8
where w and u are some sets of thermophysical characteristics (given by a vector aTC), initial temperature distribution (vector T0), geometrical characteristics (F) of the body or a set of bodies, characteristics (I), entering into the boundary conditions, discretised temperature field (T) and loading actions (g). The vector ¦8 is usually composed of time-discrete measurements for selected points in space.
However, in some cases, unsteady-state diagnostics of heat transfer conditions at the solid surface requires other mathematical models, more complicated than the heat conduction equation (1998). The situation arises in experimental studies of substance injection into gaseous boundary layers and in development of heat protection systems based in transpiration cooling or on thermal breakdown of solid materials. In such cases, direct measurement of the quantities entering into the boundary conditions on the heated surfaces is very difficult or impossible at all, but temperature measurements can be taken inside capillary-porous bodies through which gaseous product is blown or percolated.
(1959) said that experiments on temperature equilibration (calorimetry) are, by themselves, consistent with the existence of a ‘heat substance’, the amount[4] of which determines the temperature and chemical state of a body; however, experiments with friction, heat engines, etc., show that mechanical (or electric, etc.) energy and ‘heat substance’ can often be inter-converted, always quantitatively. In modern (strict) terminology (2005), the concepts are therefore:
1. Internal energy, U, of a system. It represents mainly (but not always entirely) "invisible" energy, macroscopically detectable only by changing the state of the system.
2. Heat "flow", δQ, which represents energy exchanged by any non-mechanical (and non-electrical) means. It is the thermal analogue of mechanical work (rather than energy). Like δW = F · ds, it is not a perfect differential. That is, δQ by itself has no meaning; physical conditions during the integration must be specified (and then the value depends on those conditions as well as the initial and final states).
In book (2000), on the other hand, a table of conduction equations was shown, and it appeared as follows, where plane, cylindrical and spherical coordinate systems were examined:
Figure 2. Conduction Equations
CYLINDRICAL HEAT TRANSFER
According to (2000), the velocity profile and pressure distribution existing about the cylinder are important. Many experimental and numerical investigations have been performed in the subject of heat transfer to or from a cylinder in cross flow. Effects such as Reynolds number, velocity fluctuation amplitudes and scales, and free stream oscillations have been studied. One such study is cited in the book by (1994), where the study chose a cylinder in cross flow as a baseline test case to confirm the heat transfer measurement capabilities of the facility. The stagnation point of the cylinder was located 1 cylinder diameter downstream of the free stream measurement location and 45 cm from the vertical gate. The cylinder is polished acrylic, 8.9 cm in diameter and has walls that are 6.35 mm thick. The area ratio, cylinder diameter to test section height, is 0.4375. A single fast response, thin foil, K-type thermocouple was flush mounted on the cylinder surface. The thermocouple leads were passed through a 3 mm access hole to the interior of the cylinder. A reference temperature compensator, amplifier and data acquisition system were linked in series with the thermocouple to permit surface temperature histories to be recorded. The thermocouple access hole was covered and the leads were secured with thin (0.05 mm) clear tape. In addition, a pressure tap 0.8 mm in diameter was located 180° from the thermocouple location. Manual rotation of the cylinder provides the necessary angular degree of freedom and a 360° indicator and horizontal reference line permitted ±0.5 alignment accuracy.
Another example of cylindrical heat transfer was shown by Naterer (2003) in where an angular variation of convective heat transfer coefficient is observed for external flow past a cylinder. On the upstream side of the cylinder for laminar flow, the convective heat transfer coefficient decreases with angle (see Figure 1). This trend is a result of the growing boundary layer thickness with angle on the upstream side. However, on the back side of the cylinder, separation of the boundary layer from the surface, and the resulting increase of fluid missing, causes a rise in the convection coefficient with the angle. Similar trends can be observed in the case of turbulent flow. In all cases, it can be seen that the local heat transfer coefficient varies with the angle along the surface of the cylinder. If the average (or total) heat transfer from the cylinder to the air would be considered, the average value of h would be used in Newton’s law of cooling, rather than the local heat transfer coefficient shown in Figure 1.
Figure 2 is from Naterer’s ‘Heat Transfer in Single and Multiphase System’ book (2003). The figure shows the local heat transfer coefficient in a cylinder.
Figure 2. Heat Transfer from a Cylinder in a Cross-Flow.
The fluid temperature in Newton’s law must be carefully specified. For external flows, such as the precious example, the fluid temperature is usually selected to be the ambient fluid temperature (Evans 1998). However, for internal flows, (such as flow in a duct or pipe), the fluid temperature in Newton’s law of cooling is usually selected to be the mean temperature of the fluid, which typically varies with position ( 1997). These selections are closely linked with the manner by which empirical correlations for the heat transfer coefficient are defined.
Figure 3 is adapted from (2003), which are the conduction shape factors for selected two-dimensional systems, where
[ q = Sk (T1 – T2) ]
Although the presented conduction shape factor does not give the temperature distribution, Boyett, Bejan & Krauss (2003) stated that it provides a simple equation for the rate of heat transfer:
q = kS DT
where k is the thermal conductivity of the conducting medium, DT the temperature difference driving the heat flow, and S the conduction shape factor. Figure 3 provided expressions for the conduction shape factor for various two-dimensional configurations. Accordingly, (2002) cited that the conduction resistance for a two-dimensional system follows as
1
Sk
SYSTEM SCHEMATIC RESTRICTIONS SHAPE FACTOR
Figure 3. Conduction Shape Factors for Selected Two-Dimensional Systems
HEAT TRANSFER IN THE CONTEXT OF PIZZA BAKING
According to (2005), the system of bread baking is difficult, in this context, pizza baking, in which a chain of physical, chemical and biochemical changes occurs in the bread. The said changes are basically the effect of instantaneous heat and mass transfer within it. Fast heat transfer techniques could be utilised to hasten up the baking process of pizza and create new pizza properties (2000). In a study conducted by (2005), they investigated the effect of infrared radiation on crust formation of par-baked baguettes after baking. The parameters examined were crust thickness, crust colour, heating time and total water loss. The outcomes showed that infrared radiation, as against heating in a traditional household oven, augmented the speed of colour improvement of the crust and made the heating time briefer. As a conclusion, the study found out that crust was thinner for infrared-heated baguettes.
Several mathematical models have been put forward by various experts on the subject of heat transfer in the context of baking. The earliest known model is that of (1975), in where the concepts of irreversible thermodynamics and the concept of moisture transfer potential for water movement in a capillary porous body are applied. The said model assumes that the solid body is represented structurally as a porous slab with capillaries, that pressure gradients within the porous body are very small, and that external resistance to heat and mass transfer is negligible. Eight years later, (1988) developed another model, which considered evaporation – condensation in the gaseous phase and conduction in the liquid phase of simultaneous heat and mass transfer in dough and crumb (not in the bread crust). (2005) described that the model is based on a phenomenological hypothesis, which stressed the effect of air bubbles, contained both in the dough and in the crumb, on heat and mass transfer. A one dimensional finite difference method has been used to investigate heat transfer in the dough. Mass transfer is
only determined by the evaporation - condensation mechanism. (1994) did not agree with the above model, so they set up experiments to develop a phenomenological model of bread baking in a forced-convection electric oven of their own.
(1993) also proposed their own mathematical model, based on the Crank-Nicolson finite difference scheme, in where heat conduction and water diffusion are considered in one dimension in Cartesian coordinate system, and diffusion together with evaporation and condensation has been assumed to be the mass transfer mechanisms inside the dough. Lastly, a model for simultaneous heat, water, and vapour diffusion in side food during heat processing was developed by in 1999. This mathematical model was evaluated for a drying pro cess for bread crumb slabs. (1999) have also measured lo cal water content and temperature in several points in side the slab. They have proved that the simulated water content levels and temperatures conform well to the experimental values and show that the evaporation and condensation model describes well the diffusion mechanisms in a porous food.
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