The term 'one-dimensional' is applied to heat conduction problem when:
(i) Only one space coordinate is required to describe the temperature distribution within a heat conducting body;

**ONE DIMENSIONAL STEADY STATE EQUATION
PLANE WALL ****:**

The
term 'one-dimensional' is applied to heat conduction problem when:

(i)
Only one space coordinate is required to
describe the temperature distribution within a heat conducting body;

(ii)
Edge effects are neglected;

(iii)
The flow of heat energy takes place
along the coordinate measured normal to the surface.

A plane wall is considered
to be made out of a constant thermal conductivity material and extends to
infinity in the Y- and Z-direction. The wall is assumed to be homogeneous and
isotropic, heat flow is one-dimensional, under steady state conditions and
losing negligible energy through the edges of the wall under the above
mentioned assumptions the Eq. (2.2) reduces to

d^{2}T
/ dx^{2} = 0; the boundary conditions are: at x = 0, T = T_{1}

Integrating
the above equation, x = L, T
= T_{2}

T
= C_{1}x + C_{2}, where C_{1} and C_{2} are two
constants.

Substituting
the boundary conditions, we get C_{2}
= T_{1} and C_{1} = (T_{2} –T_{1})/L The temperature

distribution
in the plane wall is given by

T
= T1 –(T1 –T2) x/L (2.3)

which
is linear and is independent of the material.

small for the same heat flow rate,"

**A Cylindrical
Shell-Expression for Temperature Distribution **

In the cylindrical
system, when the temperature is a function of radial distance only and is
independent of azimuth angle or axial distance, the differential equation (2.2)
would be, (Fig. 1.4)

d^{2}T
/dr^{2} +(1/r) dT/dr = 0

with boundary conditions: at r = r_{l}, T =
T_{1} and at r = r_{2}, T = T_{2}.

The differential equation can be written as:

upon
integration, T = C_{1} ln (r) + C_{2}, where C_{1} and
C_{2} are the arbitrary constants.

From Eq (2.5) It can be seen that the temperature
varies 10gantJunically through the cylinder wall In contrast with the linear
variation in the plane wall .

where A_{2} and A_{1} are
the outside and
inside surface areas
respectively. The term A_{m} is
called

‘Logarithmic Mean
Area' and the
expression for heat low through a cylidercal wall has the same

form as that for a plane wall.

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Mechanical : Heat and Mass Transfer : Conduction : One Dimensional Steady State Equation Plane Wall |

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