## 1. Grey Body Factor Gebhart factor or grey body factor is the ratio of energy absorbed by the target body $A_j$ originating from a source body $A_i$ *through all possible paths including reflections of other bodies* to the total radiation emitted by the source $A_i$ and is denoted by $B_{ij}$ $B_{ij}=\frac{energy\ absorbed\ at\ A_j\ originating\ from\ A_i}{Total\ radiation\ emitted\ at\ A_i}$ $B_{ij}=\frac{\dot q_{ij}}{\dot q_i}$ ## 2. View Factor vs Grey Body Factor Grey body factor differs from [[View Factor]] because it includes the reflective paths for radiant heat to flow from source $i$ to target $j$. Heat transfer between the two bodies is given by $\dot q_{ij} = A_i\epsilon_iB_{ij}\sigma(T^4_i-T^4_j)$ If there are $n$ bodies participating in radiation, then the sum of $B_ij$ to all other surfaces should be 1 $\sum_{j=1}^n B_{ij} =1$ ### 2.1. Reciprocity The following properties are true $A_i\epsilon_iB_{ij} = A_j\epsilon_jB_{ji}$ You will have only one [[Radiation Conductance|RadK]] between every node pair. However, RadCAD will calculate both pairs and then determine which has the least error and weight towards it ## 3. Radiation Conductance (Radk) ![[Radiation Conductance]] ## 4. Calculating Grey Body Factors ### 4.1. Gebhart's Method Gebhart's method uses previously calculated [[View Factor|view factors]] to calculate grey body factors. So if there are $n$ opaque bodies, then $B_{ij} = \epsilon_j FF_{ij} + \sum_{k=1}^n((1-\epsilon_k) FF_{ik}B_{kj})$ #### 3.1.1. Example - three body system For illustration, if we consider only three bodies, we arrive at the following set of equations $\begin{alignat*}{4} B_{11} = \epsilon_1 FF_{11} +(1-\epsilon_1) FF_{11}B_{11}+(1-\epsilon_2) FF_{12}B_{21}+(1-\epsilon_3) FF_{13}B_{31}\\ B_{21} = \epsilon_1 FF_{21}+(1-\epsilon_1) FF_{21}B_{11} +(1-\epsilon_2) FF_{22}B_{21}+(1-\epsilon_3) FF_{23}B_{31}\\ B_{31} = \epsilon_1 FF_{31}+(1-\epsilon_1) FF_{31}B_{11} +(1-\epsilon_2) FF_{32}B_{21}+(1-\epsilon_3) FF_{33}B_{31} \end{alignat*} $ We can solve for them using methods for solving linear system of equations Note that $(1-\epsilon)$ term is equal to the radiation reflected by a body since transmissivity is zero (opaque bodies assumption) [[Monte-Carlo Ray Tracing method]]