Heat Conduction
In this chapter heat conduction problems are presented. Employing the first law of thermodynamics, the problems in the rectangular Cartesian coordinates, cylindrical coordinates, and the spherical coordinates are solved. The method of treatments of the no
- PDF / 912,565 Bytes
- 55 Pages / 439.37 x 666.142 pts Page_size
- 29 Downloads / 256 Views
Heat Conduction
In this chapter heat conduction problems are presented. Employing the first law of thermodynamics, the problems in the rectangular Cartesian coordinates, cylindrical coordinates, and the spherical coordinates are solved. The method of treatments of the nonhomogeneous boundary and differential equations are given and the lumped formulation of the heat conduction problems are discussed.
22.1 Problems in Rectangular Cartesian Coordinates The general form of the governing equation of heat conduction in solids in rectangular Cartesian coordinates for the anisotropic material is ∂ ∂x
∂ ∂ ∂T ∂T ∂T ∂T kx + ky + kz = −R + ρc ∂x ∂y ∂y ∂z ∂z ∂t
(22.1)
where T is the absolute temperature and k x , k y , and k z are the coefficients of thermal conduction along the coordinate axes x, y, and z, respectively. We will consider the analytical methods of solution of this partial differential equation and we will discuss some examples.
22.1.1 Steady Two-Dimensional Problems: Separation of Variables The general form of the governing partial differential equation for a steady twodimensional problem in x and y directions is ∂ ∂x
∂ ∂T ∂T kx + ky = −R ∂x ∂y ∂y
(22.2)
where the thermal conductivities in x and y directions are assumed to be variable. M. Reza Eslami et al., Theory of Elasticity and Thermal Stresses, Solid Mechanics and Its Applications 197, DOI: 10.1007/978-94-007-6356-2_22, © Springer Science+Business Media Dordrecht 2013
573
574
22 Heat Conduction
The solution of the partial differential equation (22.2), when k x is a function of x and k y is a function of y, is obtained by the method of separation of variables, which is the most common method of solution of this partial differential equation. When the boundary conditions of a problem are specified, then according to this method, the solution is sought as the product of functions of each coordinate separately. This allows the constants of integration in each separated function to be found directly from the homogeneous boundary conditions, and the non-homogeneous boundary conditions be treated by using the concept of expansion into a series. To show the method let us assume a general form of Eq. (22.2) a1 (x)
∂2 T ∂T ∂2 T ∂T + a + b3 (y)T = 0 (22.3) + a (x) (x)T + b (y) + b2 (y) 2 3 1 2 2 ∂x ∂x ∂y ∂y
The solution of this equation may be taken in the product form as T (x, y) = X (x)Y (y)
(22.4)
where X (x) is a function of x alone, and Y (y) is a function of y alone. Upon substitution of Eq. (22.3) into Eq. (22.4) and after dividing of the whole equation by X Y , we get d2 X dX 1 d 2Y dY 1 + a3 (x)X = − b1 (y) 2 + b2 (y) + b3 (y)Y a1 (x) 2 + a2 (x) dx dx X dy dy Y (22.5) The left-hand side of Eq. (22.5) is a function of the variable x only and the righthand side is a function of y only. Therefore, we conclude that the only way that the above equation can hold is when both sides are equal to a constant, say, ±λ2 . This constant is called the separation constant. Considering this, the equation reduces to the followin
Data Loading...
