CodeCogs

Computes the temperature inside a thin homogeneous spherical wall.

Interface

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

using namespace Engineering::Thermodynamics::Conduction;

inline double hm_sphere(double r, double d1, double d2, double t1, double t2)

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:

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

(1)

which in turn gives:

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

(2)

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:

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

(3)

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

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

Parameters

the given radius (meters)

the internal diameter of the spherical wall (meters)

the external diameter of the spherical 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 radius r within the spherical wall (degrees Celsius).

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

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

(4)

References

Dan Stefanescu, Mircea Marinescu - "Termotehnica"

Interactive Calculator

Result –