91Ó°ÊÓ

First, let's find the heat transfer rate per unit length, \(q'\), without partition. To do this, we will use the convective heat transfer equation: \(q' = hA(T_{mg} - T_x)\) Where: - \(h\) is the convective heat transfer coefficient - \(A\) is the surface area of the tube per unit length - \(T_{mg}\) is the combustion gas temperature - \(T_x\) is the tube temperature The convective heat transfer coefficient, \(h\), can be found using the Nusselt number, \(Nu = \frac{hD}{k}\), where \(D\) is the diameter of the tube and \(k\) is the thermal conductivity of the gas. In this case, the gas can be assumed to have the thermophysical properties of air. We need the Reynolds number, \(Re = \frac{\rho u D}{\mu}\), to then find the Nusselt number, \(Nu =C \cdot Re^{n} \cdot Pr^{1/3}\), where \(C, n\) are constants depending on the flow regime, \(Re\) is the Reynolds number, and \(Pr\) is the Prandtl number. To find the Reynolds number, we need the gas density \((\rho)\), dynamic viscosity \((\mu)\), and velocity \((u)\). We can obtain these values from the given flow rate \(\dot{m}_e = 0.05\ kg/s\) and apply the following relations: \(u = \frac{\dot{m}_e}{\rho A}\) \(Re = \frac{\rho u D}{\mu}\) \(Nu = C \cdot Re^{n} \cdot Pr^{1/3}\) \(h = \frac{Nu \cdot k}{D}\) We can then plug the values for density, viscosity, and velocity back into the equation for the heat transfer rate per unit length to find \(q'\).

To find the partition temperature, \(T_F\), and the total heat transfer, \(q'\), for a flow rate of \(0.05\ kg/s\) and emissivities \(\epsilon_n = \epsilon_p = 0.5\), we will need to consider both convective and radiative heat transfer on both sides of the partition. For the convective heat transfer component, we will use a similar approach as in part a, with the only difference being the addition of the partition: \(q'_{conv} = hA_1(T_{mg} - T_F) + hA_2(T_F - T_x)\) Where: - \(h\) is the convective heat transfer coefficient - \(A_1\) and \(A_2\) are the surface areas of the partition per unit length on the gas and tube sides, respectively For the radiative heat transfer component, we will use the following equation for radiative heat transfer between two surfaces: \(q'_{rad} = A_1F_{1-2}\sigma(\epsilon_n(T_{mg}^4 - T_F^4) + \epsilon_p(T_F^4 - T_x^4))\) Where: - \(\sigma\) is the Stefan-Boltzmann constant - \(F_{1-2}\) is the view factor between the gas and the partition - \(\epsilon_n\) and \(\epsilon_p\) are the emissivities of the gas and partition, respectively We can then solve the following system of equations to find the partition temperature, \(T_F\), and the total heat transfer, \(q'\): \(q' = q'_{conv} + q'_{rad}\)

For each flow rate, \(\dot{m}_g = 0.02, 0.05, \) and \(0.08\ kg/s\), and equivalent emissivities, we need to compute and plot the partition temperature, \(T_p\), and heat transfer rate per unit length, \(q'\), as functions of emissivity, \(\epsilon\). To do this, follow the steps outlined in parts a and b for each flow rate and emissivity value in the specified range. To generate a plot of partition temperature and heat transfer rate as functions of emissivity, use the calculated values of \(T_p\) and \(q'\) for each flow rate and emissivity value. For the flow rate of \(0.05\ kg/s\) and equivalent emissivities, we also need to plot the convective and radiative contributions to the heat transfer rate per unit length, \(q'\), as functions of emissivity. This can be done by plotting the convective and radiative heat transfer components, \(q'_{conv}\) and \(q'_{rad}\), separately as functions of emissivity for this flow rate. To summarize, the exercise requires calculation of convective and radiative heat transfer rates for different flow rates and emissivities, determination of partition temperature and heat transfer rate with and without partition, and plotting partition temperature and heat transfer rates as functions of emissivity.