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Engineering › Thermodynamics › Conduction ›

hm sphere

Computes the temperature inside a thin homogeneous spherical wall.

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Interface
Hm Sphere
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Interface

C++

Excel

Hm Sphere

doublehm_sphere(	double	`r`
	double	`d1`
	double	`d2`
	double	`t1`
	double	`t2`	)[inline]

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

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

$Q = - \lambda S \frac{\mathrm{d}t}{\mathrm{d}r}$

(2)

which in turn gives:

$\mathrm{d}t = - \frac{Q}{\lambda S} \mathrm{d}r.$

(3)

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:

$t(r) = t_1 - (t_1 - t_2) \frac{d_2}{2r} \frac{2r - d_1}{d_2 - d_1}.$

(4)

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

MISSING IMAGE!

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

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:

$d_2 > d_1 > 0, \qquad t_1 > t_2, \qquad d_1 \leq 2r \leq d_2.$

(5)

References

Dan Stefanescu, Mircea Marinescu - "Termotehnica"

Parameters

r	the given radius (<i>meters</i>)
d1	the internal diameter of the spherical wall (<i>meters</i>)
d2	the external diameter of the spherical wall (<i>meters</i>)
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>)

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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