# The Heat Transfer Analysis of an Acting-type Heat Retention Panel used in a Hot Rolling Process
## 2 Mathematical Analysis
### Governing Equations and Boundary Conditions  
This study provides an analysis of the traditional passive heat retention panel and the acting-type heat retention panel based on the theory of fluid mechanics and the theory of heat transfer. The calculation domain is composed of a fluid domain and a solid domain. The solid domain includes the transfer bars, the heat retention panel, and the rollers beneath the conveyors. The fluid domain includes the air and the transfer bars inside the heat retention panel. The transfer bars are assumed to be a moving fluid coupled with the laminar flow passing through the heat retention panel. The continuity equation, momentum equation, and energy equation are as follows:  
\[\frac{\partial\rho}{\partial t}+\frac{\partial\big{(}\rho u_{j}\big{)}}{ \partial x_{j}}=0, \tag{1}\]  
\[\frac{\partial(\rho u_{i})}{\partial t}+\frac{\partial\big{(}\rho u_{i}u_{j} \big{)}}{\partial x_{j}}=-\frac{\partial P}{\partial x_{i}}+\frac{\partial}{ \partial x_{j}}\Bigg{[}\mu\left(\frac{\partial u_{i}}{\partial x_{j}}+\frac{ \partial u_{j}}{\partial x_{i}}\right)\Bigg{]}, \tag{2}\]  
\[\frac{\partial T}{\partial t}+\frac{\partial\big{(}u_{j}T\big{)}}{\partial x_ {j}}=\alpha(T)\frac{\partial}{\partial x_{j}}\Bigg{(}\frac{\partial T}{ \partial x_{j}}\Bigg{)}-\
abla\cdot\stackrel{{\rightarrow}}{{q}} _{rad}, \tag{3}\]  
where \(\
abla\cdot\stackrel{{\rightarrow}}{{q}}_{rad}\) is the radiative heat transfer source term. The temperature field of the solid domain is governed by the following transient conduction equation:  
\[\frac{\partial}{\partial x_{j}}k\Bigg{(}\frac{\partial T}{\partial x_{j}} \Bigg{)}+q=\rho C_{p}\frac{\partial T}{\partial t}, \tag{4}\]  
where \(\rho\), \(C_{p}\), and k are the density, heat capacity, and the thermal conductivity of the solid domain, respectively. The convective heat transfer between the surroundings and the solid surface outside the heat retention panel is calculated using the following equation:  
Figure 3: Physicals models of the acting-type heat retention panel. \[q_{s}^{C}=h(T_{s}-T_{\infty}), \tag{5}\]  
where h is the convective heat transfer coefficient for the air on the solid surface. In addition, \(T_{s}\) and \(T_{\infty}\) are the temperature of the air and the solid surface, respectively. Because the effect of the radiative heat transfer dominates the temperature field within the heat retention panel, the S2S (surface-to-surface) radiation model explained in [16] is adopted for the purpose of the current study. The S2S radiation model is used to calculate the radiation exchange in an enclosure comprising gray-diffuse surfaces, for which the governing equation is described as follows:  
\[q_{out,k}=\varepsilon_{k}\sigma\Big{(}T_{k}^{4}-T_{0}^{4}\Big{)}+\rho_{k}q_{ in,k}, \tag{6}\]  
where \(\varepsilon_{k}\) is the emissivity of the solid surface, \(\sigma\) is Boltzmann's constant, and \(q_{in,k}\) is the energy flux incident on the surface from the surroundings, which can be calculated using the following equation:  
\[q_{in,k}=\sum_{j=1}^{N}F_{kj}q_{out,j}, \tag{7}\]  
where \(q_{out,j}\) is the energy flux leaving from j surface. The energy exchange between two surfaces depends in part on their size, separation distance, and orientation. These parameters are accounted for by a geometric function called view factor that can be evaluated with the equation:  
\[F_{kj}=\frac{\text{diffuse energy leaving }A_{k}\text{ directly toward and intercepted by }A_{k}}{\text{total diffuse energy leaving }A_{k}}=\frac{1}{A_{k}}\int_{A_{k}}\int_{A_{j}}\frac{\alpha\alpha\beta_{k} \alpha\beta_{j}}{\pi\tau^{2}}\delta_{kj}dA_{j}A_{k}, \tag{8}\]  
The word directly is meant to imply \"on a straight path without intervening reflections\". Additionally, the formulation of \(F_{kj}\) must ensure that all surfaces are diffuse surfaces with uniform radiosity.