Heat Conduction in a Rectangle

Transient heat conduction or mass transfer in solids with constant physical properties (diffusion coefficient, thermal diffusivity, thermal conductivity, etc.) is usually represented by a linear parabolic partial differential equation. Steady state heat or mass transfer in solids, potential distribution in electrochemical cells is usually represented by elliptic partial differential equations. In this chapter, we describe how one can arrive at the analytical solutions for linear parabolic partial differential equations and elliptic partial differential equations in finite domains using a separation of variables method. The methodology is illustrated using a transient one dimensional heat conduction in a rectangle. 

2 Separation of Variables for Parabolic PDEs with Homogeneous Boundary Conditions 

Parabolic partial differential equations with homogenous boundary conditions are solved in this section. The dependent variable u is assumed to take the form u = XT, where X is a function of x alone and T is a function of t alone. This leads to separate differential equations for X and T. This methodology is best illustrated using an example. 

Consider heat transfer in a finite slab.[l] The dimensionless temperature profile is governed by  

We observe that in equation (7.7) the left hand side is a function of t only and the right hand side is a function of x only. Hence, both sides of equation (7.7) should be equal to a constant  

---

Example 7.4. Heat Conduction in a rectangle with an Initial Profile... 

Consider heat conduction in a rectangle with a nonhomogeneous boundary condition at x = l.[l]... 

Consider Example 7.6, heat conduction in a rectangle with radiation at the surface, [2] which is solved here using the Laplace transform technique. The dimensionless temperature profile is governed by ... 

Example 8.15. Heat Conduction in a Rectangle with a Time Dependent Boundary Condition... 

The origiiral problem. We begin by placing the first boundary-value problem for for the heat conduction equation in which it is required to find a continuous in the rectangle Dt = 0[Pg.459]

A body is called thin if one (or more) of its characteristic dimensions is much smaller than the others. A thin rod and a thin plate are examples of such bodies. We consider a boundary value problem describing a heat conduction process in a thin rod, where the ratio e of the thickness of the rod to its length is a small parameter. To simplify the presentation, we consider the problem for a planar rod, that is, in the thin rectangle (OsjfSl) X (0[Pg.166]

Above we considered the heat conduction problem in a thin rectangle. The asymptotics of the solution can be constructed similarly for the problem in a thin three-dimensional rod of constant cross-section S. In this case, by stretching the variables y and z with the coefficient e, we obtain the problem... 

---