The Conduction Shape Factor

This heat flow will be the same through each section within this heat-flow lane, and the total heat flow will be the sum of the heat flows through all the lanes. If the sketch is drawn so that Ajc = Ay, the heat flow is proportional to the AT across the element and, since this heat flow is constant, the AT across each element must be the same within the same heat-flow lane. Thus the AT across an element is given by 

The accuracy of this method is dependent entirely on the skill of the person sketching the curvilinear squares. Even a crude sketch, however, can frequently help to give fairly good estimates of the temperatures that will occur in a body and these estimates may then be refined with numerical techniques discussed in Sec. 3-5. An electrical analogy may be employed to sketch the curvilinear squares, as discussed in Sec. 3-9. 

The graphical method presented here is mainly of historical interest to show the relation of heat-flow lanes and isotherms. It may not be expected to be used for the solution of many practical problems. 

In a two-dimensional system where only two temperature limits are involved, we may define a conduction shape factor 5 such that 

The values of 5 have been worked out for several geometries and are summarized in Table 3-1. A very comprehensive summary of shape factors for a 

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In the heat transfer literature the corresponding quantity, hoAjK, is sometimes called the conduction shape factor. 

A comparison of Eqs. 3-4 and 3-79 reveals that the conduction shape factor 5isrelated to the thermal resistance/ by/ = l/kSoTS= 1/i / . Thus, these two quantities are the inverse of each other when the thermal conductivity of the medium is unity. The use of the conduction shape factors is ilJustraled with Examples 3-13 and 3-14. 

Shape Factor and Thermal Resistance in Orthogonal Curvilinear Coordinates. The definition of thermal resistance of a system (total temperature drop across the system divided by the total heat flow rate) yields the following general expression for the thermal resistance R and the conduction shape factor S ... 

In Eq. (4.4-4) the factor M/N is called the conduction shape factor S, where... 

We stait this chapter with one-dimensional steady heat conduction in a plane wall, a cylinder, and a sphere, and develop relations for thennal resistances in these geometries. We also develop thermal resistance relations for convection and radiation conditions at the boundaries. Wc apply this concept to heat conduction problems in multilayer plane wails, cylinders, and spheres and generalize it to systems that involve heat transfer in two or three dimensions. We also discuss the thermal contact resislance and the overall heat transfer coefficient and develop relations for the critical radius of insulation for a cylinder and a sphere. Finally, we discuss steady heat transfer from finned surfaces and some complex geometries commonly encountered in practice through the use of conduction shape factors. 

Conduction shape factors have been determined for a number of configurations encountered in practice and are given in Table 3-7 for some common cases. More comprehensive tables are available in the literature. Once the value of the shape factor is known for a specific geometry, the total steady heat transfer rate can be determined from the equation above using the specified two constant temperatures of the two surfaces and the thermal conductivity of the medium between them. Note that conduction shape factors are applicable only when heat transfer between the two surfaces is by conduction. Therefore, they cannot be used when the medium between the surfaces is a liquid or gas, which involves natural or forced convection currents. 

Conduction shape factors Sfor several configurations for use in Q = frS(T, - T2) to determine the steady rate of heat transfer through a medium of thermal conductivity k between the surfaces at temperatures Ti and... 

C What is a conduction shape factor How is it related to the thermal resistance ... 

C What is the value of conduction shape factors in engineering ... 

When steady-state conduction occurs within and outside solids, or between two contacting solids, it is frequently handled by means of conduction shape factors and thermal contact conductances (or contact resistances), respectively. This chapter covers the basic equations, definitions, and relationships that define shape factors and the thermal contact, gap, and joint conductances for conforming, rough surfaces, and nonconforming, smooth surfaces. 

The derivation above was conducted for a spherical particle. It is easy to show that the treatment can be extended to an arbitrary geometry, as the form (shape) factor 5 is applied. Equation (9.108) can then be written in the general form... 

Conduction shape factor Dimensionless factor used to account for the geometrical effects in steady-state heat conduction between surfaces at different temperatures. 

The furnace process involves injecting low end fraction of cmde oil, eg. Bunker Euel C, into a heated chamber. The temperature, shape of the injectors of the oil, rate of injection, and other factors are controlled to produce black fillers of different particle si2e and stmcture. The particle si2e and stmcture control the reinforcing character of the carbon black. There are 30 common grades of carbon black used in the mbber industry. There are numerous specialty grades produced, and several hundred are used in plastic, conductive appHcations, and other uses. 

Conductance values will depend not only on the pressure and the nature of the gas which is flowing, but also on the sectional shape of the conducting element (e.g. circular or elliptical cross section). Other factors are the length and whether the element is straight or curved. The result is that various equations are required to fake into account practical situations. 

In a series of papers, Derby and Brown (144, 149-152) developed a detailed TCM that included the calculation of the temperature field in the melt, crystal, and crucible the location of the melt-crystal and melt-ambient surfaces and the crystal shape. The analysis is based on a finite-ele-ment-Newton method, which has been described in detail (152). The heat-transfer model included conduction in each of the phases and an idealized model for radiation from the crystal, melt, and crucible surfaces without a systematic calculation of view factors and difiuse-gray radiative exchange (153). 

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