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nhm sphere

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

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Contents

Interface
Nhm Sphere
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Interface

C++

Nhm Sphere

std::vector<double>nhm_sphere(	int	`n`
	double	`t1`
	double	`t2`
	double*	`d`
	double*	`lambda`	)

For a spherical non-homogeneous wall, formed by n layers of various diameters $\inline d_1, d_2, \ldots, d_n$ and thermal conductivities $\inline \lambda_1, \lambda_2, \ldots, \lambda_n$ the total conductive heat flow is unidirectional and is given by the following formula:

$Q = 2\pi (t_1 - t_2) \left[ \sum_{i=1}^n \frac{1}{\lambda_i} \left( \frac{1}{d_{i-1}} - \frac{1}{d_i} \right) \right]^{-1}$

(2)

where $\inline d_0$ is the internal diameter of the spherical wall.

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

The temperature at the contact surface between layer $\inline i$ and layer $\inline i + 1$ will be denoted by $\inline ts_i$ , for any value of $\inline i$ between 1 and $\inline n-1$ . The value of these temperatures are obtained by considering the following equality relation between the total conductive heat flows which pass through each layer and the value of $\inline Q$ given above:

$Q_1 = Q_2 = \ldots = Q_n = Q$

(3)

where:

$Q_k = 2 \pi (t_1 - ts_k) \left[ \sum_{i=1}^k \frac{1}{\lambda_i} \left( \frac{1}{d_{i-1}} - \frac{1}{d_i} \right) \right]^{-1}.$

(4)

Hence we obtain the following formula:

$ts_k = t_1 - \frac{Q}{2 \pi} \left[ \sum_{i=1}^k \frac{1}{\lambda_i} \left( \frac{1}{d_{i-1}} - \frac{1}{d_i} \right) \right]^{-1} \qquad i = \overline{1, n-1}.$

(5)

In the diagram below you may notice the temperatures $\inline t_1, ts_1, ts_2, \ldots, ts_{n-1}, t_2$ which decrease while the heat flow passes through layers of increasing diameters and various thermal conductivities.

MISSING IMAGE!

1/nhm_sphere0-378.jpg cannot be found in /users/1/nhm_sphere0-378.jpg. Please contact the submission author.

Example 1

#include <codecogs/engineering/heat_transfer/conduction/nhm_sphere.h>
#include <stdio.h>
 
int main()
{
  // input data
  int n = 4;
  double t1 = 75.0, t2 = 20.0,
  d[5] = {0.1, 0.15, 0.2,  0.25, 0.35},
  lambda[4] = {0.75, 0.95, 1.2,  1.0};
 
  // display the various input data
  printf("Input values:\n\n");
  printf(" n = %d\nt1 = %.2lf\nt2 = %.2lf\n\n", n, t1, t2);
  printf("diameters:\n(");
  int i;
  for (i = 0; i < n; i++)
    printf("%.2lf, ", d[i]);
  printf("%.2lf)\n\n", d[n]);
  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::Heat_Transfer::Conduction::nhm_sphere
  (4, t1, t2, d, 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 = 4
t1 = 75.00
t2 = 20.00
 
diameters:
(0.10, 0.15, 0.20, 0.25, 0.35)
 
lambda:
(0.75, 0.95, 1.20, 1.00)
 
 
The temperatures at contact surfaces between layers are:
 
(45.09862, 33.29544, 27.68893)

Note

The inequality $\inline t_1 > t_2$ must always hold when passing values to the function. Also the diameters array must be given in increasing order, thus:

$d_0 < d_1 < d_2 < \ldots < d_n.$

(6)

References

Dan Stefanescu, Mircea Marinescu - "Termotehnica"

Parameters

n	the number of layers
t1	the temperature of the heat flow at the entry surface (<i>degrees Celsius</i>)
t2	the temperature of the heat flow at the exit surface (<i>degrees Celsius</i>)
d	an array with the diameters of the layers (<i>meters</i>)
lambda	an array with the thermal conductivities of the layers (<i>Watts per meter Celsius</i>)

Returns

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

Authors

Grigore Bentea, Lucian Bentea (October 2006)

Source Code

Source code is available when you buy a Commercial licence.

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