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Computes the radiative heat flow between two plane surfaces that are separated by a number of absorptive thin shields.

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Interface

#include <codecogs/engineering/thermodynamics/radiation/flow_shield.h>

using namespace Engineering::Thermodynamics::Radiation;

inline double flow_shield(double T1, double T2, double e1, double e2, int n, double *es)

This module calculates the radiative heat flow per unit surface between two plane surfaces, considering the case when these are separated by a number of absorptive thin shields.

On account of the Stefan-Boltzmann law, the unit radiative heat flow between the two surfaces separated by $n$ shields is given by

$q = e_{1, S_1, S_2, \ldots, S_n, 2} \,\, C_0 \left[\left(\frac{T_1}{100}\right)^4 - \left(\frac{T_2}{100}\right)^4\right] \qquad \left[\frac{W}{m^2}\right]$

(1)

where

$e_{1, S_1, S_2, \ldots, S_n, 2} = \left(\frac{1}{e_{1,S_1}} + \frac{1}{e_{S_1,S_2}} + \frac{1}{e_{S_2,S_3}} + \ldots + \frac{1}{e_{S_{n-1},S_n}} + \frac{1}{e_{S_n,2}} \right)^{-1}$

(2)

and

$e_{1, S_1} = \left(\frac{1}{e_1} + \frac{1}{e_{S_1}} - 1\right)^{-1} \qquad e_{S_n, 2} = \left(\frac{1}{e_2} + \frac{1}{e_{S_n}} - 1\right)^{-1}$

(3)

$e_{S_i, S_{i+1}} = \left(\frac{1}{e_{S_i}} + \frac{1}{e_{S_{i+1}}} - 1\right)^{-1} \qquad i = 1, 2, \ldots, n - 1$

(4)

with $e_1$ , $e_2$ the emissivity factors of the first and second surface $(0 < e_1, e_2 \leq 1)$ , $e_{S_i}$ the emissivity factor of the $i$ -th shield $(0 < e_{S_i} \leq 1)$ , $C_0$ the emissivity constant of the black body $\displaystyle \left(C_0 \approx 5.669 \left[\frac{W}{m^2 K^4}\right]\right)$ and $T_1$ , $T_2$ the corresponding absolute temperatures of the two plane surfaces.

The above expression of the unit heat flow can be further simplified to the following formula

$q = C_0 \left[\left(\frac{T_1}{100}\right)^4 - \left(\frac{T_2}{100}\right)^4\right] \left[\frac{1}{e_1} + \frac{1}{e_2} - (n + 1) + 2 \sum_{i=1}^n \frac{1}{e_{S_i}} \right]^{-1} \qquad \left[\frac{W}{m^2}\right].$

(5)

In the diagram below is shown the radiative heat transfer between two plane surfaces, separated by $n$ shields $S_1$ , $S_2$ , $\ldots$ , $S_n$ .

Example 1

The example below computes the unit radiative heat flow between an oxidated aluminium plane surface at 873.16 degrees Kelvin and an oxidated copper plane surface at 403.16 degrees Kelvin, separated by two thin shields. The first shield is made of zinc-covered iron at 297.16 degrees Kelvin, while the second is made of brass at 323.16 degrees Kelvin. You may notice that in the presence of these two thin shields, the heat flow between the given plane surfaces decreases by a factor of 72% than in the case of a non-absorptive medium, as calculated in the flow_noshield module.

#include <codecogs/engineering/heat_transfer/radiation/flow_shield.h>
#include <stdio.h>

int main()
{
  // the temperature of the oxidated aluminium surface
  double T1 = 873.16;
 
  // the temperature of the oxidated copper surface
  double T2 = 403.16;
 
  // the emission factor of the aluminium surface
  double e1 = 0.19;
 
  // the emission factor of the copper surface
  double e2 = 0.76;

  // the number of shields
  int n = 2;
 
  // the emission factors of the iron and brass shields
  double es[3] = {0.276, 0.22};

  // display radiative heat flow between the two plane surfaces
  printf("Radiative heat flow = %.5lf W per sq. meter\n",
  Engineering::Heat_Transfer::Radiation::flow_shield
  (T1, T2, e1, e2, n, es));
 
  return 0;
}

Output

Radiative heat flow = 1579.33128 W per sq. meter

Parameters

the absolute temperature of the first surface (Kelvin)

the absolute temperature of the second surface (Kelvin)

the emissivity factor of the first surface

the emissivity factor of the second surface

the number of shields

an array with the emissivity factors of the shields

Returns

the radiative heat flow between the two plane surfaces (Watt per square meter)

A table with the emissivity factors of various materials at different temperatures can be found at the following link http://www.monarchserver.com/TableofEmissivity.pdf

References

Dan Stefanescu, Mircea Marinescu - "Termotehnica"