A semi-infinite solid is an idealized body that has a single plane surface and extends to infinity in all directions.

**SEMI INFINITE AND INFINITE SOLIDS**

A semi-infinite solid
is an idealized body that has a *single plane surface* and extends to
infinity in all directions, as shown in .This idealized body is used to
indicate that the temperature change in the part of the body in which we are
interested (the region close to the surface) is due to the thermal conditions
on a single surface. The earth, for example, can be considered to be a
semi-infinite medium in determining the variation of temperature near its
surface. Also, a thick wall can be modeled as a semi-infinite medium if all we
are interested in is the variation of temperature in the region near one of the
surfaces, and the other surface is too far to have any impact on the region of
interest during the time of observation. The temperature in the core region of
the wall remains unchanged in this case.

For short periods of
time, most bodies can be modeled as semi-infinite solids since heat does not
have sufficient time to penetrate deep into the body,and the thickness of the
body does not enter into the heat transfer analysis. A steel piece of any
shape, for example, can be treated as a semi-infinite solid when it is quenched
rapidly to harden its surface. A body whose surface is heated by a laser pulse
can be treated the same way.

Consider a
semi-infinite solid with constant thermo physical properties ,no internal heat
generation,uniform thermal conditions on its exposed surface, and initially a
uniform temperature of *Ti* throughout. Heat transfer in this case occurs
only in the direction normal to the surface (the *x* direction), and thus
it is one-dimensional. Differential equations are independent of the boundary
or initial conditions, and thus for one-dimensional transient conduction in
Cartesian coordinates applies. The

depth of the solid is
large (*x* → _) compared to the depthphenomenathat h can be expressed
mathematically as a boundary condition as *T*(*x* → *, t*) _ *Ti*.

Heat conduction in a
semi-infinite solid is governed by the thermal condition simposed on the
exposed surface, and thus the solution depends strongly on the boundary
condition at *x* _ 0. Below we present a detailed analytical solution for
the case of constant temperature *Ts* on the surface, and give the results
for other more complicated boundary conditions. When the surface temperature is
changed to *Ts* at *t* _ 0 and held constant at that value at all
times, the formulation of the problem The separation of variables technique
does not work in this case since the medium is infinite. But another clever
approach that converts the partial differential equation into an ordinary
differential equation by combining the two independent variables *x* and *t*
into a single variable h, called the **similarity variable**, works well.
For transient conduction in a semi-infinite medium, it is defined as Similarity
variable.

**USE OF HEISLER CHARTS :**

There are three charts,
associated with different geometries. For a plate/wall (Cartesian geometry) the
Heisler chart

The first chart is to determine the temperature at
the center 0 *T* at a given time.

By having the temperatureat the center 0 *T* at
a given time, the second chart is to determine the temperature at other
locations at the same time in terms of 0 *T* .

The third chart is to determine the total amount of
heat transfer up to the time.

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Mechanical : Heat and Mass Transfer : Conduction : Semi Infinite and Infinite Solids |

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