Steady state solution of two dimensional equation of heat conduction (Heat flow equations) (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, α2 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 ρ, thermal conductivity k and the specific heat c. Since the flow is two dimensional, the temperature at any 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 δx and δy, the edges being parallel to the co-ordinate axes, as shown in the figure.

Then the quantity of heat entering the area PQRS per sec. through the surface

Similarly, the quantity of heat entering the elt PQRS per sec through the surface

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

Therefore, the total gain of heat by the rectangular element PQRS per sec. = in flow - out flow

The rate of gain of heat by the area PQRS is also given by

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 Hence the temperature distribution of the plate in the steady-state is

i.e., v2u = 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 of the temperature in the direction of the y-axis will be zero. Then the heat-flow equation is reduced to which is the heat-flow equation in one-dimension.

FOURIER SERIES SOLUTIONS IN CARTESIAN CO-ORDINATES

Solution of the Laplace equation in two dimensions

Here u is a function of x and y. So assume the solution

where X is a function of x alone and Y is a function of y alone

L.H.S. is a function of X alone and R.H.S. is a function of Y alone.

Since X and Y are independent, each quantity is equal to a constant.

Case (i) Let k be positive say k = p2, then (3) & (4) become

Solving we get

Case (3) Let k = 0

Thus the possible solutions of (1) are

• Write the different solutions of Laplace's equation in cartesian coordinates.

Solution: 

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

when the stream lines are all parallel to the x axis, the rate of change of temperature in the y direction will be zero.

So equation (1) is reduced to which is the p.d.e. of one dimensional heat flow.

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 ?

Solution: The required equation is

 [Laplace equation]

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

Solution :

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

Solution :

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

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

Boundary conditions :

The suitable solution is given by

3.5a. Finite Plates

Problems based on Finite Plates - Type 1

Example 3.5a(1): A square plate is bounded by the lines x = 0, y=0, x = l, y = l. Its faces are insulated. The temperature along the upper horizontal edge is given by u(x, l) = x(l - x) while the other three edges are kept at 0° C. Find the steady state temperature in the plate. 

Solution : 

The equation to be solved is

From the given problem, we get the following boundary conditions:

Now, the suitable solution which satisfies our boundary conditions is given by

Apply condition (i) in equation (1), we get

Apply condition (ii) in equation (2), we get

Apply condition (iii) in equation (3), we get

The most general solution is

Now we apply condition (iv) in equation (5), we get

To find Bn expand f(x) in a Fourier half range sine series

From (6) and (7), we get  Bn = bn

= 0 if n is even.

Substitute the value of An in equation (5), we get

Example 3.5a(2): Find the steady state temperature distribution in a rectangular plate of sides a and b insulated at the lateral surface and satisfying the boundary conditions

Solution: 

Applying condition (i) in equation (1), we get

Applying condition (ii) in equation (2), we get

Applying condition (iii) in equation (3), we get

Substitute the value of D in equation (3), we get

The most general form is

Apply condition (iv), we get

To find Вn expand ƒ (x) in a Fourier half range sine series

Example 3.5a(3) The boundary value problem governing the steady state temperature distribution in a flat, thin, square plate is given by

Find the steady-state temperature distribution in the plate.

Sol. The equation to be solved is

Given boundary conditions are

(i) u (0, y) = 0, 0 < y < a

The suitable solution is

Apply condition (i) in equation (1), we get

Apply condition (ii) in equation (2), we get

If we take B = 0 and already A = 0 then we get a trivial solution,

Therefore, B ≠ 0

⸫ sin p a = 0

Apply condition (iii) in equation (3), we get

Apply condition (iv) in equation (5), we get

Equating the like terms, we get

Type 2. Finite plate with value given in y direction :

The most general solution is

Apply condition (iv), we get

Problems based on Finite Plates - Type 2

Example 3.5a(4): A square plate is bounded by the lines x = 0, x=a, y = 0, y = a, of a square plane bounded by the lines x = 0, y = 0, y = a are kept at temperature 0° C. The side x = a is kept at temperature given by u (a, y) = 100, 0 < y < a.

Solution: The equation to be solved is

Applying condition (i) in equation (1), we get 

Applying condition (ii) in equation (2), we get

Applying condition (iii) in equation (3), we get

The most general solution is

Applying condition (iv) in equation (5), we get

To find Вn expand 100 in a Fourier half range sine series

Example 3.5a(5): A square plate is bounded by the lines x = 0, x=a, y = 0, y = a of a square plane bounded by the lines x = a, y = 0, y = a are kept at temperature 0° C. The side x = 0 is kept at temperature given by u (0, y) = 100, 0 < y < a. Find u (x, y).

Solution : The equation to be solved is

Applying conditions (iii) in equation (3), we get

The most general solution is

Applying condition (iv) in equation (5), we get

To get Bn expand 100 in a Fourier half range sine series

Substitute An is equation (5), we get

Problems based on Finite Plates : 

Type 3. (Combination of Type 1 & Type 2)

Example 3.5a(6) : A rectangular plate is bounded by the lines_x = y = 0, x = a and y = b. It's surfaces are insulated and the temperature along two adjacent edges are kept at 100° C, while the temperature along the other two edges are at 0° C. Find the steady state temperature at any point in the plate. Also find the steady state temperature at any point of a square plate of side 'a' if two adjacent edges are kept at 100° C and the others at 0° C. 

Solution: 

Let u (x, y) be the temperature satisfying the equation

Then the boundary conditions are given by

Now, we split the solutions into two solutions.

i.e., u (x, y) = u1 (x, y) + u2 (x, y)

where u1 (x, y) and u2 (x, y) are solutions of (1) and further u1 (x, y) is the temperature at any point P with the edge BC maintained at 100° C and the other three edges at 0° C where u2 (x, y)is the temperature at P with the edge AB maintained at 100° C and the other three edges at 0° C.

Therefore, the boundary conditions for the functions u1 (x, y) and u2 (x, y) are as follows.

Clearly, both u1 and u2 satisfy the equation (1). Solving (1) aud choosing the suitable solution we have,

By applying the condition (v) on (2), we get

By applying the condition (vii) on (3), we get

By applying the condition (vi) on (4), we get

The most general solution is

By the condition (viii), we get

which is a half range sine series for f(x) = 100 defined in (0, a)

Similarly,

3.5(b). Infinite Plates

PROBLEMS ON INFINITE PLATES

Example 3.56(1): A rectangular plate with insulated surface is 10 cm wide, so long compared to its width that it may be considered infinite in length. If the temperature at the short edge y = 0 is given by u = x for 0 ≤ x ≤ 5 and 10 x for 5 < x < 10 and the two long edges x = 0, x = 10 as well as the other short edge are kept at 0° C. Find the temperature function u (x, y) at any point of the plate.

Solution: The equation to be solved is

From the given problem, we get the following boundary conditions.

Applying condition (i) in equation (1), we get 

Applying condition (ii) in equation (2), we get

Applying condition (iii) in equation (3), we get

Substitute, C = 0 in equation (3), we get

The most general solution is

Now, applying condition (iv) in equation (5), we get

To find Bn then we expand f(x) as a Fourier half range sine series in (0, 10)

Substitute, the value of Bn in equation (5), we get

Example 3.56(2): A rectangular plate with insulated surface is 10 cm wide and so long compared to its width that it may be considered infinite in length without introducing an appreciable error. If the temperature of the short edge y = 0 is given by 

and the two long edges x = 0 and x = 10 as well as the other short edge are at 0° C. Prove that the temperature, u (x, y) at any point (x, y) of the plate is given by

Proof : Let u (x, y) be the temperature at any point (x,y) in the steady state. Then u satisfies the differential equation V2 u = 0

From the given problem, we get the following boundary conditions

Now, the suitable solution which satisfies our boundary conditions is given by

The most general solution is

Apply condition (iv) in equation (5), we get

To find Bn then we expand f(x) as a Fourier half-range sine series in [0, 10]

Example 3.5b(3) : A rectangular plate with insulated surfaces is 10 cm wide and so long compared to its width that it may be considered infinite in length without introducing an appreciable error. If the temperature along are short edge y = 0 is u(x, 0) = 4 (10x − x2) for 0 < x < 10 while the two long edges as well as the other short edge are kept at 0° C, find the steady state temperature function u(x, y). 

Solution: Let u(x,y) be the temperature at any point (x, y) in the steady state. Then u satisfies the differential equation

Apply condition (iv) in equation (5), we get

To find Bn then we expand f(x) as a Fourier half range sine series in (0, 10)

Substituting the value of Bn in equation (5), we get

Horizontal plate

Example 3.5b(4) : An infinitely long rectangular plate with insulated surface is 10 cm wide. The two long edges and one short edge are kept at zero temperature, while the other short edge x = 0 is kept at temperature given by

u (0, y) =

Find the steady state temperature distribution in the plate.

Solution: The equation to be solved is

From the given problem, we get the following boundary conditions

Now, the suitable solution which satisfies our boundary conditions is given by

Applying condition (i) in equation (1), we get

Applying condition (ii) in equation (2), we get 

Applying condition (iii) in equation (3), we get

The most general solution is

Applying condition (iv) in equation (5), we get

To find Bn, expand ƒ (y) in a half range Fourier sine series in [0,10].

= 0 if n is even.

Substitute, Bn in equation (5), we get

Example 3.5b(5) : A rectangular plate with insulated surface is 8 cm wide and so long compared to its width that it may be considered infinite in length without introducing an appreciable error. If the temperature along the short edge y = 0 is given by u (x, 0) = 100 sin while the long two edges x=0 and x = 8 as well as the other short edge are kept at 0° C, find the steady state temperature function u(x, y).

Solution: The equation to be solved is

Now, the suitable solution which satisfies our boundary condition is given by

Applying condition (i) in equation (1), we get

Applying condition (ii) in equation (2), we get

Substitute the value C = 0 in eqn. (3), we get

The most general solution is

Applying condition (iv) in equation (5), we get

EXERCISE 3.5

1. A square plate is bounded by the lines x = 0, y = 0, x = 20, y = 20. Its faces are insulated. The temperature along the upper horizontal edge is given by u(x, 20) = x (20 - x) when 0 < x < 20 while the other three edges are kept at 0° C. Find the steady state temperature in the plate.

2. A square plate has its faces and the edge y=0 insulated. Its edges x = 0 and x = л are kept at 0° C and the edge у=л is kept at temperature f(x). Find the steady state temperature.

3. Find the steady state temperature at any point of a square plate whose two adjacent sides are kept at 0° C and the other two edges are kept at 100° C.

4. A rectangular metal plate is bounded by the lines x = 0, x = a, y = 0 and y = b. The three sides x = 0, x = a and y = b are insulated and the side y = 0 is kept at temperature Show that the temperature in the steady state is

5. The three sides x 0, x = a, y = 0 of a square plate bounded by the lines x 0, x = a, y = 0 and y = a are kept at temperature 0° C. The side y = a is kept at steady temperature given by u (x, a) = bx (x − a), 0 ≤ x ≤ a when b is a constant. Find the steady-state temperature u (x, y) in the plate.

6. Find the solution of the equation V2v = 0 for 0 < x <л, 0 < y <л, given u (0,y) = u(л,у) = u(x, π) 0 u (x, 0) = sin2x.

7. A rectangular plate is bounded by the lines x = 0, x = a, y = 0 and y = b and the temperature at the edges are given by 

Find the steady state temperature distribution inside the plate.

8. A long rectangular plate has its surface insulated and the two long sides as well as of the short sides are maintained at 0° C. Find an expression for the steady state temperature u (x, y) if the short side y=0 is л cm long and is kept at u°0 C.

9. An infinitely long rectangular plate with insulated surface is 10 cm wide. The two long edges and one short edge are kept at 0° C temperature while the other short edge x = 0 is kept at temperature is given by

Find the steady state temperature distribution in the plate.

10. A long rectangular. plate has its surfaces insulated and the two long sides as well as one of the short sides are kept at 0° C. While the other sides u (x, 0) = 3x and the length being 5 cm. Find u (x,y).

11. A long rectangular plate of width l cm with insulated surface has its temperature u equal to zero on both long sides one of the shorter sides so that u (0, y) = u (l, y) = u(x, ∞) = 0 and u (x, 0) = kx. Find u (x, y).

12. An infinitely long vertical uniform plate is bounded by two parallel edges and an end at right angles to them. The breadth is л. This end is maintained at a temperature uo at all points and other edges are kept at zero temperature. Determine the temperature at any point of the plate in the steady state.

13. An infinite long plate of width л with insulated surfaces has its temperature zero on both long sides and are of the short sides. The side y = 0 is maintained at a temperature 3x. Find the steady state temperature u (x, y).

14. A rectangular plate with insulated surface is 10 cm wide so long compared to its width that it may be considered infinite length. If the temperature along short edge y = 0 is given u (x, 0) = 8 πx/10 sin when 0 < x < 10, while the two long edges x = 0 and x = 10 as well as the other short edge are kept at 0°C, find the steady state temperature function u (x, y).

ANSWERS 3.5