nhm_plane — Conduction

Computes the temperatures at contact surfaces between the layers of a planar non-homogeneous wall.

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Interface

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

using namespace Engineering::Thermodynamics::Conduction;

inline std::vector<double> nhm_plane(int n, double t1, double t2, double *delta, double *lambda)

For a planar non-homogeneous wall, formed by n layers of various thicknesses $\delta_1, \delta_2, \ldots, \delta_n$ and thermal conductivities $\lambda_1, \lambda_2, \ldots, \lambda_n$, the conductive heat flow per unit area is unidirectional and is given by the following formula:

$$q = (t_1 - t_2) \left( \sum_{i=1}^n \frac{\delta_i}{\lambda_i} \right)^{-1}.$$

(1)

It is assumed that the i-th layer has constant thermal conductivity $\lambda_i$ at any of its points.

The temperature at the contact surface between layer $i$ and layer $i + 1$ is denoted by $tw_i$, for any value of $i$ between 1 and $n-1$. The value of these temperatures are obtained by considering the following equality relation between the conductive heat flows per unit area which pass through each layer and the value of $q$ given above:

$$q_1 = q_2 = \ldots = q_n = q$$

(2)

where:

$$q_k = (t_1 - tw_k) \left( \sum_{i=1}^k \frac{\delta_i}{\lambda_i} \right)^{-1}.$$

(3)

Hence we obtain the following formula:

$$tw_k = t_1 - q \sum_{i=1}^k \frac{\delta_i}{\lambda_i} \qquad k = \overline{1, n-1}.$$

(4)

In the diagram below you may notice that the temperature decreases linearly while the heat flow passes through layers of various thicknesses and thermal conductivities.

Example 1

#include <codecogs/engineering/thermodynamics/conduction/nhm_plane.h>
#include <stdio.h>
 
int main()
{
  // input data
  int n = 3;
  double t1 = 22.73, t2 = -15.4,
   delta[3] = { 0.1, 0.15, 0.2},
  lambda[3] = {0.75, 0.95, 1.2};
 
  // display the various input data
  printf("Input values:\n\n");
  printf(" n = %d\nt1 = %.2lf\nt2 = %.2lf\n\n", n, t1, t2);
  printf("delta:\n(");
  int i;
  for (i = 0; i < n - 1; i++)
    printf("%.2lf, ", delta[i]);
  printf("%.2lf)\n\n", delta[n - 1]);
  printf("lambda:\n(");
  for (i = 0; i < n - 1; i++)
    printf("%.2lf, ", lambda[i]);
  printf("%.2lf)\n\n", lambda[n - 1]);
 
  // compute the temperatures at the contact surfaces between all layers
  std::vector<double> result =
  Engineering::Thermodynamics::Conduction::nhm_plane
  (3, t1, t2, delta, lambda);
 
  // display the results
  printf("\nThe temperatures at contact surfaces between layers are:\n\n");
  printf("(");
  for (i = 0; i < result.size() - 1; i++)
    printf("%.5lf, ", result[i]);
  printf("%.5lf)\n\n", result[result.size() - 1]);
 
  return 0;
}

Output

Input values:

 n = 3
t1 = 22.73
t2 = -15.40

delta:
(0.10, 0.15, 0.20)

lambda:
(0.75, 0.95, 1.20)


The temperatures at contact surfaces between layers are:

(11.62701, -1.52126)

Parameters

the number of layers

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

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

delta

an array with the thicknesses of the layers (meters)

lambda

an array with the thermal conductivities of the layers (Watts per meter Celsius)

Returns

A vector containing the temperatures at the contact surfaces between all layers (degrees Celsius).

The inequality $t_1 > t_2$ must always hold when passing values to the function.

References

Dan Stefanescu, Mircea Marinescu - "Termotehnica"