2.1 ONE-DIMENSIONAL HEAT CONDUCTION EQUATION
Consider heat conduction through a large plane wall such as the wall of a house, the glass of a single pane window, the metal plate at the bottom of a pressing iron, a cast iron steam pipe, a cylindrical nuclear fuel element , an electrical resistance wire, the wall of a spherical container, or a spherical metal ball that is being quenched or tempered . Heat conduction in these and many other geometries can be approximated as being one-dimensional since heat conduction through these geometries will be dominant in one direction and negligible in other directions. Below we will develop the one-dimensional heat conduction equation in rectangular, cylindrical, and spherical coordinates.
2.1.1 Heat Conduction Equation in a Large Plane Wall
Consider a thin element of thickness x in a large plane wall, as shown in Figure 7.
FIGURE 7 One-dimensional heat conduction through a volume element in a large plane wall
Assume the density of the wall is , the specific heat is C, and the area of the wall normal to the direction of heat transfer is A. An energy balance on this thin element during a minor interval ?t can be expressed as
Or
(9)
But the change in the energy content of the element and the rate of heat generation within the element can be expressed as
(10)
(11)
Sub . in eq . (9)
(12)
Deviding by A gives
(13)
(14)
since, from the definition of the derivative and Fourier’s law of heat conduction,
(15)
Noting that the area A is constant for a plane wall, the one-dimensional transient
heat conduction equation in a plane wall becomes
(16)
The thermal conductivity k of a material, in general, depends on the temperature T (and therefore x), and thus it cannot be taken out of the derivative. However, the thermal conductivity in most practical applications can be assumed to remain constant at some average value. The equation above in that case reduces to
(17)
where the property is the thermal diffusivity of the material and represents how fast heat propagates through a material. It reduces to the following forms under specified conditions
(18)
(19)
(20)
2.1.2 Heat Conduction Equation in a Long Cylinder
Now consider a thin cylindrical shell element of thickness r in a long cylinder, as shown in Figure 8. Assume the density of the cylinder is, the specific heat is C, and the length is L. The area of the cylinder normal to the direction of heat transfer at any location is where r is the value of the radius at that location. Note that the heat transfer area A depends on r in this case and thus it varies with location. An energy balance on this thin cylindrical shell element during a minor interval can be expressed as
FIGURE 8 One-dimensional heat conduction through a volume element in a long cylinder
(21)
The change in the energy content of the element and the rate of heat generation within the element can be expressed as
(22)
(23)
Substituting into Eq. 21 , we get
(24)
where You may be tempted to express the area at the middle of the element using the average radius as But there is nothing we can gain from this complication since later in the analysis we will take the limit as and thus the term will drop out. Now dividing the equation above by gives
(25)
(26)
since, from the definition of the derivative and Fourier’s law of heat conduction (27)
Noting that the heat transfer area in this case is , the one-dimensional transient heat conduction equation in a cylinder becomes
(28)
For the case of constant thermal conductivity, the equation above reduces to
(29)
where again the property is the thermal diffusivity of the material. Equation 26 reduces to the following forms under specified conditions
(30)
(31)
(32)
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 .
|