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FEAnalyst

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- How to solve this problem involving radiation in a triangular cavity?

Hi,

I'm trying to solve a problem involving radiation in a triangular cavity:

As you can see, lengths and emissivities of all surfaces are given. For two of them, the heat flux is known and the temperature has to be found while for the remaining surface it's the other way around.

I have the numerical and analytical solutions for this problem, but I want to know exactly how to obtain the latter. Here's my approach, based on the book by A. Bejan: $$Q_{1}=\frac{\varepsilon_{1} \cdot A_{1}}{1- \varepsilon_{1}} \cdot \left( \sigma \cdot T_{1}^{4}-J_{1} \right)$$ $$Q_{2}=\frac{\varepsilon_{2} \cdot A_{2}}{1- \varepsilon_{2}} \cdot \left( \sigma \cdot T_{2}^{4}-J_{2} \right)$$ $$Q_{3}=\frac{\varepsilon_{3} \cdot A_{3}}{1- \varepsilon_{3}} \cdot \left( \sigma \cdot T_{3}^{4}-J_{3} \right)$$ $$J_{1}=(1- \alpha_{1}) \cdot \left( (J_{2} \cdot F_{12})+(J_{3} \cdot F_{13}) \right) + \varepsilon_{1} \cdot \sigma \cdot T_{1}^{4}$$ $$J_{2}=(1- \alpha_{2}) \cdot \left( (J_{1} \cdot F_{21})+(J_{3} \cdot F_{23}) \right) + \varepsilon_{2} \cdot \sigma \cdot T_{2}^{4}$$ $$J_{3}=(1- \alpha_{3}) \cdot \left( (J_{1} \cdot F_{31})+(J_{2} \cdot F_{32}) \right) + \varepsilon_{3} \cdot \sigma \cdot T_{3}^{4}$$ where view factors are calculated from the following formulas: $$F_{12}=\frac{L_{1}+L_{2}-L_{3}}{2 \cdot L_{1}}$$ $$F_{13}=\frac{L_{1}+L_{3}-L_{2}}{2 \cdot L_{1}}$$ $$F_{31}=\frac{L_{3}+L_{1}-L_{2}}{2 \cdot L_{3}}$$ $$F_{21}=\frac{L_{2}+L_{1}-L_{3}}{2 \cdot L_{2}}$$ $$F_{23}=\frac{L_{2}+L_{3}-L_{1}}{2 \cdot L_{2}}$$ $$F_{32}=\frac{L_{3}+L_{2}-L_{1}}{2 \cdot L_{3}}$$

The problem is that the software that I use to solve this system of 6 equations for the 6 variables ##J_{1}##, ##J_{2}##, ##J_{3}##, ##Q_{1}##, ##T_{2}## and ##T_{3}## doesn't give any solution so I assume that something is wrong with my equations. Do you know where I made the mistake in the calculations? Maybe I misinterpreted the formulas from Bejan's book. They look like this: $$Q_{i}=\frac{\varepsilon_{i} A_{i}}{1- \varepsilon_{i}} \cdot \left( \sigma T_{i}^{4}-J_{i} \right)$$ $$J_{i}=(1-\alpha_{i}) \sum_{j=1}^{n} J_{j}F_{ij}+ \varepsilon_{i} \sigma T_{i}^{4}$$ for ##i=1,2,...,n## where ##n## is the number of surfaces in a cavity. $$F_{ij}=\frac{L_{i}+L_{j}-L_{k}}{2L_{i}}$$

I'm trying to solve a problem involving radiation in a triangular cavity:

As you can see, lengths and emissivities of all surfaces are given. For two of them, the heat flux is known and the temperature has to be found while for the remaining surface it's the other way around.

I have the numerical and analytical solutions for this problem, but I want to know exactly how to obtain the latter. Here's my approach, based on the book by A. Bejan: $$Q_{1}=\frac{\varepsilon_{1} \cdot A_{1}}{1- \varepsilon_{1}} \cdot \left( \sigma \cdot T_{1}^{4}-J_{1} \right)$$ $$Q_{2}=\frac{\varepsilon_{2} \cdot A_{2}}{1- \varepsilon_{2}} \cdot \left( \sigma \cdot T_{2}^{4}-J_{2} \right)$$ $$Q_{3}=\frac{\varepsilon_{3} \cdot A_{3}}{1- \varepsilon_{3}} \cdot \left( \sigma \cdot T_{3}^{4}-J_{3} \right)$$ $$J_{1}=(1- \alpha_{1}) \cdot \left( (J_{2} \cdot F_{12})+(J_{3} \cdot F_{13}) \right) + \varepsilon_{1} \cdot \sigma \cdot T_{1}^{4}$$ $$J_{2}=(1- \alpha_{2}) \cdot \left( (J_{1} \cdot F_{21})+(J_{3} \cdot F_{23}) \right) + \varepsilon_{2} \cdot \sigma \cdot T_{2}^{4}$$ $$J_{3}=(1- \alpha_{3}) \cdot \left( (J_{1} \cdot F_{31})+(J_{2} \cdot F_{32}) \right) + \varepsilon_{3} \cdot \sigma \cdot T_{3}^{4}$$ where view factors are calculated from the following formulas: $$F_{12}=\frac{L_{1}+L_{2}-L_{3}}{2 \cdot L_{1}}$$ $$F_{13}=\frac{L_{1}+L_{3}-L_{2}}{2 \cdot L_{1}}$$ $$F_{31}=\frac{L_{3}+L_{1}-L_{2}}{2 \cdot L_{3}}$$ $$F_{21}=\frac{L_{2}+L_{1}-L_{3}}{2 \cdot L_{2}}$$ $$F_{23}=\frac{L_{2}+L_{3}-L_{1}}{2 \cdot L_{2}}$$ $$F_{32}=\frac{L_{3}+L_{2}-L_{1}}{2 \cdot L_{3}}$$

The problem is that the software that I use to solve this system of 6 equations for the 6 variables ##J_{1}##, ##J_{2}##, ##J_{3}##, ##Q_{1}##, ##T_{2}## and ##T_{3}## doesn't give any solution so I assume that something is wrong with my equations. Do you know where I made the mistake in the calculations? Maybe I misinterpreted the formulas from Bejan's book. They look like this: $$Q_{i}=\frac{\varepsilon_{i} A_{i}}{1- \varepsilon_{i}} \cdot \left( \sigma T_{i}^{4}-J_{i} \right)$$ $$J_{i}=(1-\alpha_{i}) \sum_{j=1}^{n} J_{j}F_{ij}+ \varepsilon_{i} \sigma T_{i}^{4}$$ for ##i=1,2,...,n## where ##n## is the number of surfaces in a cavity. $$F_{ij}=\frac{L_{i}+L_{j}-L_{k}}{2L_{i}}$$