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hm sphere

Computes the temperature inside a thin homogeneous spherical wall.
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Hm Sphere

 
doublehm_spheredoubler
doubled1
doubled2
doublet1
doublet2 )[inline]
Consider the case of a thin homogeneous spherical wall with internal radius r_1, external radius r_2, and temperatures t_1, t_2 corresponding to each surface. Also let the thermal conductivity \lambda be constant at any point inside the wall.

The total conductive heat flow that passes through the surface S = 4 \pi r^2 (r_1 < r < r_2) is expressed by the following equation:

which in turn gives:

By integrating the previous equation and considering appropriate limit conditions, it follows that the temperature inside the spherical wall at a radius of r is:

In the diagram below the value of the function t(r) is shown for a particular value of r.

1/hm_sphere-378.jpg
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Example 1

#include <codecogs/engineering/heat_transfer/conduction/hm_sphere.h>
#include <stdio.h>
 
int main()
{
  // input data
  double r = 0.28, d1 = 0.5, d2 = 0.6, 
        t1 = 45.7, t2 = 20.8;
 
  // display the various input data
  printf("Input data:\n\n");
  printf(" r = %.2lf\n", r);
  printf("d1 = %.2lf\nd2 = %.2lf\n", d1, d2);
  printf("t1 = %.2lf\nt2 = %.2lf\n\n", t1, t2);
 
  // compute the temperature inside the spherical wall
  double t = Engineering::Heat_Transfer::Conduction::hm_sphere
  (r, d1, d2, t1, t2);
 
  // display the result
  printf("The temperature inside the spherical wall is:\n\n");
  printf("%.10lf\n\n", t);
 
  return 0;
}

Output

Input data:
 
 r = 0.28
d1 = 0.50
d2 = 0.60
t1 = 45.70
t2 = 20.80
 
The temperature inside the spherical wall is:
 
29.6928571429

Note

The following inequalities must always hold when passing values to the function:

References

Dan Stefanescu, Mircea Marinescu - "Termotehnica"

Parameters

rthe given radius (<i>meters</i>)
d1the internal diameter of the spherical wall (<i>meters</i>)
d2the external diameter of the spherical wall (<i>meters</i>)
t1the temperature of the heat flow at the entry surface (<i>degrees Celsius</i>)
t2the temperature of the heat flow at the exit surface (<i>degrees Celsius</i>)

Returns

The temperature at radius r within the spherical wall (<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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