2.1.3-Heat Conduction Equation in a Sphere
Now consider a sphere with density , specific heat C, and outer radius R. The area of the sphere 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 also, and thus it varies with location. By considering a thin spherical shell element of thickness r and repeating the approach described above for the cylinder by using instead of , the one-dimensional transient heat conduction equation for a sphere is determined to be (FIGURE 9 )
FIGURE 9 One -dimensional heat conduction through a volume element in a sphere
(33)
which, in the case of constant thermal conductivity, reduces to
(34)
where again the property is the thermal diffusivity of the material. It reduces to the following forms under specified conditions
(35)
(36)
(37)
Example: A 2-kW resistance heater wire with thermal conductivity k = 15 W/m • °C, diameter
D = 0.4 cm, and length L = 50 cm is used to boil water by immersing it in water (Fig. blow). Assuming the variation of the thermal conductivity of the wire with temperature to be negligible, obtain the differential equation that describes the variation of the temperature in the wire during steady operation.
SOLUTION
The resistance wire can be considered to be a very long cylinder since its length is more than 100 times its diameter. Also, heat is generated uniformly in the wire and the conditions on the outer surface of the wire are uniform. Therefore, it is reasonable to expect the temperature in the wire to vary in the radial r direction only and thus the heat transfer to be one-dimensional. Then we will have T = T (r ) during steady operation since the temperature in this case will depend on r only. The rate of heat generation in the wire per unit volume can be determined From
Noting that the thermal conductivity is given to be constant, the differential equation that governs the variation of temperature in the wire is simply Eq. 35
which is the steady one-dimensional heat conduction equation in cylindrical coordinates
for the case of constant thermal conductivity.
Example
A spherical metal ball of radius R is heated in an oven to a temperature of 600°F throughout and is then taken out of the oven and allowed to cool in ambient air at T = 75°F by convection and radiation (Fig. blow). The thermal conductivity of the ball material is known to vary linearly with temperature. Assuming the ball is cooled uniformly from the entire outer surface, obtain the differential equation that describes the variation of the temperature in the ball during cooling.
SOLUTION
The ball is initially at a uniform temperature and is cooled uniformly from the entire outer surface. Also, the temperature at any point in the ball will change with time during cooling. Therefore, this is a one-dimensional transient heat conduction problem since the temperature within the ball will
change with the radial distance r and the time t. That is, T = T (r, t ). The thermal conductivity is given to be variable, and there is no heat generation in the ball. Therefore, the differential equation that governs the variation of temperature in the ball in this case is obtained from Eq. 33 by setting the heat generation term equal to zero. We obtain
where g. = zero
Then
which is the one-dimensional transient heat conduction equation in spherical coordinates under the conditions of variable thermal conductivity and no heat generation.
2.2. GENERAL HEAT CONDUCTION EQUATION
In the last section we considered one-dimensional heat conduction and assumed heat conduction in other directions to be negligible. Most heat transfer problems encountered in practice can be approximated as being one dimensional, and we will mostly deal with such problems in this text.
However, this is not always the case, and sometimes we need to consider heat transfer in other directions as well. In such cases heat conduction is said to be multi-dimensional, and in this section we will develop the governing differential equation in such systems in rectangular, cylindrical, and spherical coordinate systems.
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 .
|