Radiation component for enclosed space with multiple surfaces

 Description The Enclosed Radiation component models the thermal radiation emitted in enclosed space consisting of multiple surfaces as a result of their temperatures.

Equations

Total emissive power of the blackbody is:

And, the relationship of the irradiation $G$ and the radiosity $J$  is defined with:

$J=\mathrm{ε}\cdot \mathrm{E__b}+\left(1-\mathrm{ε}\right)\cdot G$

The net energy leaving the surface is the difference between the irradiation and the radiosity:

$\frac{\mathrm{Qflow}}{A}=J-G$

Then, the heat flow rate is:

$\frac{\mathrm{Qflow}}{A}=J-\frac{J-\mathrm{ε}\cdot \mathrm{E__b}}{1-\mathrm{ε}}$

Thus, the following equation is obtained:

Finally, the equation is generalized:

......(1)

Regarding the exchange of radiant energy by two surfaces:

${\mathrm{Qflow__1−2}}^{}=\mathrm{J__1}\cdot \mathrm{A__1}\cdot \mathrm{F__12}-\mathrm{J__2}\cdot \mathrm{A__2}\cdot \mathrm{F__21}$

And,

$\mathrm{A__1}\cdot \mathrm{F__12}=\mathrm{A__2}\cdot \mathrm{F__21}$

So,

${\mathrm{Qflow__1−2}}^{}=\mathrm{A__1}\cdot \mathrm{F__12}\cdot \left(\mathrm{J__1}-\mathrm{J__2}\right)$

Based on the above equation, the generalized expression is obtained:

$\mathrm{Qflow__i}=\sum _{i=1}^{\mathrm{numNodes}}\mathrm{A__i}\cdot \mathrm{F__i,k}\cdot \left(\mathrm{J__i}-\mathrm{J__k}\right)......\left(2\right)$

As the implementation of this component, (1) and (2) are used. Additionally, to correct the heat flow rate, the correction factor $\mathrm{cor}$ is applied, if Use correction input is true.

 References [1] : J. P. Holman. "Heat Transfer Ninth Edition", McGraw-Hill Higher Education.

Variables

 Symbol Units Description Modelica ID $\mathrm{E__b__i}$ $\frac{W}{{m}^{2}}$ Total emissive power of the blackbody - $\mathrm{T__i}$ $K$ Temperature of ith port port[i].T $\mathrm{J__i}$ $\frac{W}{{m}^{2}}$ Radiosity of ith port J[i] $G$ $\frac{W}{{m}^{2}}$ Irradiation - $\mathrm{Q__flow__i}$ $W$ Heat flow rate of ith port Q_flow[i] $\stackrel{}{\mathrm{σ}}$ $\frac{w}{{m}^{2}\cdot {K}^{4}}$ Stefan-Boltzmann constant 5.670373e-8 sigma

Connections

 Name Units Condition Description Modelica ID $\mathrm{port}\left[i\right]$ - - Thermal port, a port[i] $\mathrm{cor}\left[i\right]$ - if use correction input is true. Input signal of the correction factor for ${Q}_{\mathrm{flow}}$ cor[i]

Parameters

 Symbol Default Units Description Modelica ID $\mathrm{ε}$ $\left[0.8,0.8\right]$ - Emissivity of nodes eps $A$ $\left[1,1\right]$ ${m}^{2}$ Surface area of nodes A $F$ $0.1$ $-$ View factor for Radiation F $\mathrm{Nodes}$ $2$  Number of nodes numNodes $\mathrm{false}$  If true, input of correction for Gr_act is valid use_correction