I'm trying to linearise in real time (part of a simulink simulation) radiative resistances. What I've done is the following:
Radiative law: $$P=\frac{\Delta T^4}{R_r}$$
At any given timestep, I know what the temperature difference across the radiative resistance is, so I know the power $P$ running through it. From the convective law, I then know that: $$R_{c,eq}(\Delta T)=\frac{\Delta T}{P}=\frac{R_r}{\Delta T^3}$$
However, it's not coherent with the result I get when using the tangent method:
$$\frac{1}{R_{c,eq}}(\Delta T)=\frac{dP}{d(\Delta T)}(\Delta T)=\frac{4\Delta T^3}{R_r}$$ $$\Rightarrow R_{c,eq}(\Delta T)=\frac{R_r}{4\Delta T^3}$$
There is a factor 4 between those two results, which is right and why are they different?
A radiative resistance is always between two bodies $(i,j)$ with a temperature difference. In general, the flow of heat between $(i)$ and $(j)$ has the form: $$ Q_{j\rightarrow i} = \sigma\cdot A_{i,j}\left(Tj^4-T_i^4\right) $$
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$\sigma = 5.73 \times 10^{-8} W/m^2/K^4$ (Stefan-Bolzmann constant)
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$A_{i,j} = $ factor with dimension of area $[m^2]$,
dependent on emission coefficients, radiative areas and, last but not least:
view factors .
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$T = $ temperature
There is a Wikipedia reference about all this.The heat flow can be written as the admittance $\gamma_{i,j}$ of a resistor, times the temperature difference: $$ Q_{j\rightarrow i} = \gamma_{i,j}\left(Tj-T_i\right) $$ Where the admittance - though linearized reasonably well - is still dependent on the temperatures: $$ \gamma_{i,j} = \sigma\cdot A_{i,j}\left(Tj^2+T_i^2\right)\left(Tj+T_i\right) $$ So iterations may be necessary, but the hard part is in $A_{i,j}$ and the view factors, most of the time.