Steady State Solution Of Two Dimensional Equation Of Heat Conduction(Excluding Insulated Edges)

STEADY STATE SOLUTION OF TWO DIMENSIONAL EQUATION OF HEAT CONDUCTION (EXCLUDING INSULATED EDGES)

TWO DIMENSIONAL HEAT FLOW EQUATIONS

Introduction

When the heat flow is along curves, instead of straight lines, the curves lying in parallel planes, the flow is called two dimensional.

The differential equation for two dimensional heat flow for the unsteady case is

Here u (x, y) is the temperature at any point (x, y) in time t, a2 is the diffusivity of the material.

Let us consider now the flow of heat in a metal-plate in the xoy plane.

Let the plate be of uniform thickness h, density p, thermal conductivity k and the specific heat c. Since the flow is two dimensional, the temperature at point of the plate is independent of the z co-ordinate.

The heat-flow lies in the xoy plane and is zero along the direction normal to the xoy plane.

Now, consider a rectangular area PQRS of the plate with sides dx and dy, the edges being parallel to the co-ordinate axes, as shown in the figure.

p

The amount of heat which flows out through the surfaces QR and RS are

Equating the two-expressions for gain of heat per sec from (1) and (2), we have

The equation (3) gives the temperature distribution of the plate in the transient state.

In the steady-state, u is independent of t, so that du/dt = 0 Hence the temperature distribution of the plate in the steady-state is

 2u=0, which is known as Laplace's equation in two-dimensions.

Note: If the stream lines are parallel to the x axis, then the rate of change  du/d of the temperature in the direction of the y-axis will be zero. Then the heat-flow equation is reduced  which is the heat-flow equation in one-dimension.

FOURIER SERIES SOLUTIONS IN CARTESIAN CO-ORDINATES

Solution of the Laplace equation in two dimensions

§ Write the different solutions of Laplace's equation in Cartesian coordinates.

Problem 1. Obtain one dimensional heat flow equation from two dimensional heat flow equation for the unsteady case.

Solution: The two dimensional unsteady state heat flow equation is

Problem 2. What is the equation to heat flow when the stream lines are non planar curves?

Solution: When the stream lines are non planar curves, the flow will be three dimensional and the heat equation will be

Problem 3. What is the steady state heat equation in two dimensions in cartesian form?

Problem 4. Write down Laplace's equation in cartesian co-ordinates.

Problem 5. Write the three dimensional Laplace equation in cartesian form.

Note:  The two dimensional heat flow equation, when steady state conditions exist is u_xx + u_yy = 0.

Type 1. Finite plate with value given in x direction :

Boundary conditions :

The suitable solution is given by

Apply condition (i), we get A = 0

Apply condition (ii), we get 

Apply condition (iii), we get D = -C

The most general solution is