Beer is stored in a refrigerated, pressurized barrel. The pressure driven flow delivers the beer from the refrigerated region of the tube to a long plastic tube that delivers beer to the bar. During peak hours, the flow man be assumed to be steady state and laminar. Assume a constant density and viscosity for the Newtonian fluid. $1$e pressure in the barrel is $P_i$ and the pressure exiting the tube is $P_{\text{a}\text{t}\text{m}\text{.}}.$

Conductivity coefficients are $k_b$ for the beer and $k_t$ for the tube. The coefficient of convective heat transport for the outside of the tube in the room air is $h.$

a$)$ Determine the velocity profile of the beer in the tube.

b$)$ Give a simplified differential equation describing the energy transport in the beer phase. DO NOT SOLVE

c$)$ Give a simplified differential equation describing the energy transport in the tube wall. DO NOT SOLVE

d$)$ Give the boundary conditions that are appropriate for solving these energy balances. Indicate clearly where these boundary conditions apply. DO NOT SOLVE