2.2.1 Rectangular Coordinates
Consider a small rectangular element of length ?x, width ?y, and height ? z, as shown in Figure 8. Assume the density of the body is and the specific heat is C. An energy balance on this element during a small time interval ?t can be expressed as
Or
(38)
FIGURE 10 Three-dimensional heat conduction through a rectangular volume element.
Noting that the volume of the element is V element = ? x?y?z, the change in the energy content of the element and the rate of heat generation within the element can be expressed as
………………………………………………………………………………………………….(39)
(40)
since, from the definition of the derivative and Fourier’s law of heat conduction,
Equation 40 is the general heat conduction equation in rectangular coordinates. In the case of constant thermal conductivity, it reduces to
(41)
Where the property is again the thermal diffusivity of the material. Equation 41 is known as the Fourier-Biot equation, and it reduces to these forms under specified conditions:
where the quantity ? = k/?c ( is called the thermal diffusivity of the material) .( ?) density (kg/m3),(c) specific heat of material, J/kg • ?C,( k )thermal conductivity (W/m.k)
(42)
(43)
(44)
2.2.2 Cylindrical Coordinates
The general heat conduction equation in cylindrical coordinates can be obtained from an energy balance on a volume element in cylindrical coordinates, shown in Figure 11, by following the steps just outlined. It can also be obtained directly from Eq. 40 by coordinate transformation using the following relations between the coordinates of a point in rectangular and cylindrical coordinate systems:
FIGURE 11 A differential volume element in cylindrical coordinates
After lengthy manipulations, we obtain
(45)
2.2.3 Spherical Coordinates
The general heat conduction equations in spherical coordinates can be obtained from an energy balance on a volume element in spherical coordinates, shown in Figure 12, by following the steps outlined above. It can also be obtained directly from Eq. 40 by coordinate transformation using the following relations between the coordinates of a point in rectangular and spherical coordinate systems:
FIGURE 12 A differential volume element in spherical coordinates.
Again after lengthy manipulations, we obtain (46)
Obtaining analytical solutions to these differential equations requires a knowledge of the solution techniques of partial differential equations, which is beyond the scope of this introductory text. Here we limit our consideration to one-dimensional steady-state cases or lumped systems, since they result in ordinary differential equations.
المادة المعروضة اعلاه هي مدخل الى المحاضرة المرفوعة بواسطة استاذ(ة) المادة . وقد تبدو لك غير متكاملة . حيث يضع استاذ المادة في بعض الاحيان فقط الجزء الاول من المحاضرة من اجل الاطلاع على ما ستقوم بتحميله لاحقا . في نظام التعليم الالكتروني نوفر هذه الخدمة لكي نبقيك على اطلاع حول محتوى الملف الذي ستقوم بتحميله .
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