CodeCogs

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

Interface

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

using namespace Engineering::Thermodynamics::Conduction;

inline double hm_plane(double x, double delta, double t1, double t2)

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

Parameters

the distance within the planar wall (meters)

delta

the thickness of the planar wall (meters)

the temperature of the heat flow at the entry surface (degrees Celsius)

the temperature of the heat flow at the exit surface (degrees Celsius)

Returns

The temperature at distance x within the wall (degrees Celsius)

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"

Interactive Calculator

delta

Result –