# Improvement of Radiant Heat Efficiency of the
## 2 FE Analysis of the RT and the Strip
### Heat Transfer Model of the RT  
A pre-conceptual CAD design, including a set of honeycomb patterns printed on the RT, served as the basis for its heat transfer model of the RT. The following modes of heat transfer were considered.  
* Radiative exchange between the RT surface and the steep strip.
* Conduction through the tube thickness and the set of honeycombs printed on it.  
The heat transfer by convection was neglected due to the very small amount of fluid moving through the RTs during operation. A 200 mm long straight tube was modeled and analyzed to shorten the calculation time and overcome the limitation of the computer memory capacity. The FE model of its longitudinal direction half (\(Y\)-\(Z\) plane) is presented in Figure 1c. Hereafter, this type of FE model is called 'partial FE model' for convenience.  
Figure 1: (**a**) Appearance and size of an actual radiant tube (RT). (**b**) A set of pratt truss patterns generated on a part of straight tube surface. (**c**) A set of honeycomb patterns generated on a part of straight tube surface. (**d**) Schematic of a honeycomb and design variables. _“w’_ represents the width of honeycomb grid. \(l_{1}\) and \(l_{2}\) stands for the inner and outer length of the honeycomb.  
The radiant heat exchange between the partial FE model surface and steel strip surface is considered a cavity thermal radiation problem with a gray body, and diffuse reflections between the surfaces were assumed. A Cavity is defined in Abaqus/Standard as a collection of surfaces composed of facets [16]. Cavity radiation problems are intrinsically nonlinear due to the \(T^{4}\) dependence of the flux. Further nonlinearity can be introduced by describing the emissivity (\(\varepsilon\)) as a function of temperature. The radiative heat flux (\(q_{i}^{C}\)) on the surface of the cavities is defined by Equation (1).  
\[q_{i}^{C}=\sigma\varepsilon_{i}\sum_{j=1}^{N}\varepsilon_{j}\sum_{k=1}^{N}F_{ ij}C_{ji}^{-1}\left(\left(T_{j}-T_{i}\right)^{4}-\left(T_{i}-T_{j}\right)^{4}\right) \tag{1}\]  
where \(N\) is the number of facets forming the cavity, \(F_{ij}\) the geometrical view factor matrix, \(\sigma\) the Stefan-Boltzmann constant, and \(\varepsilon_{j}\) the emissivity. \(T_{i}\) and \(T_{j}\) denote the temperature of the facet \(i\) and \(j\), respectively. \(C_{ji}\) is the reflection matrix defined by Equation (2).  
\[C_{ji}=\delta_{ij}-(1-\varepsilon_{i})F_{ij} \tag{2}\]  
with \(\delta_{ij}\) representing the Kronecker delta function.  
The heat transfer of radiation between the opposing surfaces strongly depends on the geometry and orientation, as well as on their radiative properties and temperatures [17]. The amount of radiative heat transfer on any other two sides is defined through the view factor \(F_{ij}\)[18; 19]. Figure 2 is a schematic of the view factor, which is defined as the fraction of the radiation leaving the surface (\(dA_{i}\)) intercepted by the surface (\(dA_{j}\)). Equation (3) defines the view factor between two surfaces with a distance of R [20].  
\[F_{ij}=\frac{1}{A_{i}}\int_{A_{i}}\int_{A_{j}}\frac{\cos\theta_{i}*\cos\theta_ {j}}{\pi R_{ij}{}^{2}}dA_{i}dA_{j} \tag{3}\]  
where \(R_{ij}\) is the distance between surface \(dA_{i}\) and \(dA_{j}\), and \(\theta_{i}\) and \(\theta_{j}\) are the polar angles between \(R_{ij}\) and the normal surface. For a closed cavity problem, the view factor satisfies the reciprocity relation (\(A_{i}F_{ij}=A_{j}F_{ji}\)). The sum of the view factors calculated in each cavity is 1 (Equation (4)).  
\[\sum_{j=1}^{N}F_{ij}=1 \tag{4}\]  
Figure 2: Schematic representation of the view factor.  
The element type C3 DST (8-node thermally coupled brick, trilinear displacement, and temperature) was used for temperature-displacement coupled FE analysis. A mesh convergence test was performed to determine the proper element size. The aspect ratio of the elements used for geometric modeling over the partial FE model was 1.5 (width, \(w\)): 1.0 (length, \(l_{1}\)): 1.5 (height, \(t\)), and 100,788 elements were used.