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Chapter: Mechanical : Heat and Mass Transfer : Conduction

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.



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.




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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