(Solved): A long solid cylinder is immersed in a fluid bath. The fluid is a diathermic oil. The cylinder has ...

A long solid cylinder is immersed in a fluid bath. The fluid is a diathermic oil. The cylinder has a constant surface temperature \( T_{0} \) (see Figure 3 ) and radius \( \mathrm{R}_{0} \). The cylindrical tank containing the diathermic oil has a constant wall temperature \( T_{\infty} \) and radius \( R_{\infty} \). It is of interest to study the heat transfer phenomenon in the diathermic oil surrounding the cylinder in the absence of fluid motion. The general form of energy balance equation is given by Equation 5. Figure 3 \[ \rho c_{p}\left(\frac{\partial T}{\partial t}+v_{r} \frac{\partial T}{\partial r}+\frac{v_{\theta}}{r} \frac{\partial T}{\partial \theta}+v_{z} \frac{\partial T}{\partial z}\right)=k\left(\frac{1}{r} \frac{\partial}{\partial r}\left(r \frac{\partial T}{\partial r}\right)+\frac{1}{r^{2}} \frac{\partial^{2} T}{\partial \theta^{2}}+\frac{\partial^{2} T}{\partial z^{2}}\right)+S \] (b) Assuming steady state conditions, simplify Equation 5, justifying all the steps, and develop the expression of temperature distribution for the diathermic oil surrounding the cylinder in absence of fluid motion. (c) Starting from temperature distribution, express and calculate the total energy flux at the following positions: \[ r_{1}=0.3 \mathrm{~m} ; \mathrm{r}_{2}=0.9 \mathrm{~m} \] (d) Calculate the energy transfer rate across the outer surface of the cylinder and across the wall of the tank. Assume both the cylinder and tank are \( 2.5 \mathrm{~m} \) long. DATA for Question 2 Properties of diathermic oil: \[ \rho=1000 \mathrm{~kg} \mathrm{~m}^{-3}, \quad k=0.12 \mathrm{~W} \mathrm{~m}^{-1} \mathrm{~K}^{-1}, \quad c_{p}=1700 \mathrm{~J} \mathrm{~kg}^{-1} \mathrm{~K}^{-1} \] Dimensions: \[ \begin{array}{ll} R_{0}=0.02 \mathrm{~m} ; & R_{\infty}=1 \mathrm{~m} \\ T_{0}=100^{\circ} \mathrm{C} ; & T_{\infty}=20^{\circ} \mathrm{C} \end{array} \]