hm_cylinder — Conduction

Computes the temperature inside a thin homogeneous cylindrical wall.

Interface

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

using namespace Engineering::Thermodynamics::Conduction;

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

Consider the case of a thin homogeneous cylindrical wall with internal diameter $d_1$, external diameter $d_2$, length $L$, satisfying the inequalities:

$$\frac{d_1}{d_2} \approx 1 \qquad\qquad \frac{L}{d_1} \gg 1$$

(1)

and having constant thermal conductivity $\lambda$ at any of its points.

The conductive heat flow may be thought of as radial, thus by the law of conduction we have the following equation using cylindrical coordinates:

$$\frac{\partial t}{\partial \tau} = a \left( \frac{\partial^2 t}{\partial r^2} + \frac{1}{r} \frac{\partial t}{\partial r} + \frac{1}{r^2} \frac{\partial^2 t}{\partial \varphi^2} + \frac{\partial^2 t}{\partial z^2}\right).$$

(2)

where a is the thermal diffusivity and r, z, $\varphi$ are cylindrical coordinates.

For a radial heat flow and considering the x-axis along the length of the wall, the next equalities hold:

$$\frac{\partial t}{\partial r} = \frac{\mathrm{d}t}{\mathrm{d}r}, \qquad \frac{\partial t}{\partial z} = \frac{\partial^2 t}{\partial z^2} = 0, \qquad \frac{\partial t}{\partial \varphi} = \frac{\partial^2 t}{\partial \varphi^2} = 0.$$

(3)

Also, assuming steady-state conditions, the temperature does not vary with time, so $\frac{\partial t}{\partial \tau} = 0$. Hence it is true that:

$$\frac{\mathrm{d}^2 t}{\mathrm{d}r^2} + \frac{1}{r} \frac{\mathrm{d}t}{\mathrm{d}r} = 0$$

(4)

which by integration and considering appropriate limit conditions, gives the formula for the temperature inside the cylindrical wall at a radius of r:

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

(5)

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

Example 1

#include <codecogs/engineering/thermodynamics/conduction/hm_cylinder.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 cylindrical wall
  double t = Engineering::Thermodynamics::Conduction::hm_cylinder
  (r, d1, d2, t1, t2);
 
  // display the result
  printf("The temperature inside the cylindrical 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 cylindrical wall is:

30.2224870072

Parameters

the given radius (meters)

the internal diameter of the cylindrical wall (meters)

the external diameter of the cylindrical 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 cylindrical wall (degrees Celsius).

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

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

(6)

References

Dan Stefanescu, Mircea Marinescu - "Termotehnica"

Interactive Calculator

Result –