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hm plane

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#include <codecogs/engineering/heat_transfer/conduction/hm_plane.h>

using namespace Engineering::Heat_Transfer::Conduction;

	double	`hm_plane` (double x, double delta, double t1, double t2)[inline] Computes the temperature at a given distance within a thin planar homogeneous wall.
	Real	cc_hm_plane (Real x, Real delta, Real t1, Real t2) This function is available as a Microsoft Excel add-in.

Function Documentation

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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:

(1)

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

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

Considering

as a function

of the distance within the wall, we obtain a new expression for the conductive heat flow:

(2)

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

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

(3)

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

thus obtaining the formula which gives the value of the temperature at distance

within the wall:

(4)

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

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

is shown for a particular value of

The example code below computes the value of

for some particular input data.

Example:

#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

Parameters:

x	the distance within the planar wall (meters)
delta	the thickness of the planar wall (meters)
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 (degrees Celsius)

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

References:

Dan Stefanescu, Mircea Marinescu - "Termotehnica"

Authors:

Grigore Bentea, Eduard Bentea (September 2006)

Source Code:

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Last Modified: 12 Nov 07 @ 13:48 Page Rendered: 2008-05-14 16:27:18

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