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Engineering › Thermodynamics › Conduction ›

hm plane

Computes the temperature at a given distance within a thin planar homogeneous wall.

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Hm Plane
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hm_plane Computes the temperature at a given distance within a thin planar homogeneous wall.	<script type="text/javascript" src="http://www.codecogs.com/ic_json-395&u=4"></script>

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doublehm_plane(	double	`x`
	double	`delta`
	double	`t1`
	double	`t2`	)[inline]

For a planar homogeneous thin wall, the conductive heat flow per unit area is unidirectional and based on Fourier's Law it is given by:

$q = -\lambda\frac{\mathrm{d}t}{\mathrm{d}x} = \frac{\lambda}{\delta}(t_1 - t_2) \quad \left[\frac{W}{m^2}\right]$

(1)

where $\lambda$ is the thermal conductivity of the wall (constant at any point) and $t_1>t_2$ .

Considering $t_2$ as a function $t(x)$ of the distance within the wall, we obtain a new expression for the conductive heat flow:

$q(x) = \frac{\lambda}{x}\left[t_1 - t(x)\right] \quad \left[\frac{W}{m^2}\right].$

(2)

Next we will make the heat flow constant at any distance within the wall:

$q(x) = q, \qquad \forall x \in [0, \delta]$

(3)

thus obtaining the formula which gives the value of the temperature at distance $x$ within the wall:

$t(x) = t_1 \left(1 - \frac{x}{\delta} \right) + t_2 \frac{x}{\delta} \quad \left[ {}^{\circ} \mathrm{C} \right].$

(4)

In the left figure of the following diagram you may notice that the distance x is considered starting from the hotter side of the wall, marked in red. Also in the right figure the value of $t_x$ is shown for a particular value of $x = x_0$ .

The example code below computes the value of $t_x$ for some particular input data.

Example 1

#include <codecogs/engineering/heat_transfer/conduction/hm_plane.h>
#include <stdio.h>
 
int main()
{
  // the distance within the wall
  double x = 0.165;
 
  // the the thickness of the wall
  double delta = 0.213;
 
  // the temperature of the entry surface for the heat flux
  double t1 = 32.257;
 
  // the temperature of the exit surface for the heat flux
  double t2 = 22.421;
 
  // display the various input values
  printf("   t1 = %.3lf\n   t2 = %.3lf\ndelta = %.3lf\n", t1, t2, delta);
  printf("    x = %.3lf\n\n", x);
 
  printf("The temperature at distance %.3lf within the wall is %.5lf\n",
  x, Engineering::Heat_Transfer::Conduction::hm_plane(x, delta, t1, t2));
 
  return 0;
}

Output

t1 = 32.257
   t2 = 22.421
delta = 0.213
    x = 0.165
 
The temperature at distance 0.165 within the wall is 24.63756

Note

The distance within the wall x must be a real positive number less than or equal to the thickness of the wall $\delta$ and it must be true that $t_1 > t_2$ .

References

Dan Stefanescu, Mircea Marinescu - "Termotehnica"

Parameters

x	the distance within the planar wall (<i>meters</i>)
delta	the thickness of the planar wall (<i>meters</i>)
t1	the temperature of the heat flow at the entry surface (degrees Celsius)
t2	the temperature of the heat flow at the exit surface (degrees Celsius)

Returns

The temperature at distance x within the wall (<i>degrees Celsius</i>)

Authors

Grigore Bentea, Eduard Bentea (September 2006)

Source Code

Source code is available when you agree to a GP Licence or buy a Commercial Licence.

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