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OCTOBER 19-20, 2012 - YMCA University of Science & Technology

OCTOBER 19-20, 2012 - YMCA University of Science & Technology

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Proceedings <strong>of</strong> the National Conference on<br />

Trends and Advances in Mechanical Engineering,<br />

<strong>YMCA</strong> <strong>University</strong> <strong>of</strong> <strong>Science</strong> & <strong>Technology</strong>, Faridabad, Haryana, Oct <strong>19</strong>-<strong>20</strong>, <strong>20</strong>12<br />

The termε∆R / h o varies from about zero for a vertical surface to about 4 o C for horizontal surface.<br />

The ratioα h o varies from 0.026 m 2 - 0 C/WS for a light colored surface to a maximum <strong>of</strong> about 0.053m 2 - 0 C/W for a darkcolored<br />

surface or any surface for which the permanent lightness cannot be anticipated.<br />

5. Numerical Analysis <strong>of</strong> the Basic Heat Equation<br />

There are various methods to treat the basic heat equations numerically. Some <strong>of</strong> the famous methods are finite<br />

element method, finite difference method, transform methods and time series method. From a computational<br />

standpoint; two methods that have been used widely are a finite difference procedure and conduction transfer<br />

function method. Besides giving surface temperatures, finite difference method gives the temperature<br />

distribution across the wall and this is one special advantage <strong>of</strong> it over conduction transfer function method. For<br />

the above mentioned reason finite difference method is selected.<br />

The finite difference method involves discrete approximations <strong>of</strong> first derivative as follows<br />

Approximations to the governing differential equations are obtained by replacing all continuous derivatives by<br />

discrete formulas such as those in above equation. The relation between the continuous, exact, solution and the<br />

discrete approximation is shown in fig.<br />

Finite difference equations will be developed with the dependent variable φ as a function <strong>of</strong> only one independent<br />

variable, x, i.eφ φx . The resulting formulas will then be used to approximate derivatives with respect to time or space. It<br />

is possible to solve the above heat conduction equation together with its boundary equations using methods<br />

like forward difference, backward difference and Crank-Nicolson. Due to numerical instabilities <strong>of</strong> forward<br />

difference method and numerical oscillation <strong>of</strong> Crank-Nickolson method m, the backward difference method is<br />

used in this work using backward difference, the PDE <strong>of</strong> heat conduction is discretized as follows.<br />

Wall discretization<br />

For all interior nodes above equation can be discretized using implicit method as<br />

Where<br />

<strong>20</strong>8

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