Please include detailed calculation steps in the answer!

One of the steps in a biorefinery process plant involves the recovery of bio-diesel. The reaction mixture (just called "oil" below) is heated and transferred into a reactor for transesterification to occur. In one such plant design, cylindrical pipes are used to transfer the heated oil into the reactor. It is noticed that a part of this pipe is exposed to the ambient air (see figure below). As such, the oil is expected to cool down along this section of the pipe, which is undesirable. The length of the exposed section of the pipe is L. The pipe has an inner radius r, an outer radius ra and is made out of a material with thermal conductivity k. The heat transfer coefficient of the ambient is h, and the ambient temperature can be taken to be Tg. It can be assumed that oil is incompressible with density p, specific heat capacity Cp, heat transfer coefficient h, and enters the exposed section of the pipe with a temperature of To The oil flow through the pipe can be assumed to be fast, turbulent and at steady state with a velocity v.. ZEL z=0 11 r2 0 O Oil flow (a) Obtain a differential equation describing the temperature profile of the oil in the exposed section of the pipe. (10 points) (b) Solve the obtained differential equation using appropriate boundary condition/s. (8 points) (c) Using the values given in the table below, what is the rate of heat loss to the ambient via the exposed section of the pipe? (10 points) (d) The engineers think that rather than assuming a constant ambient temperature Ta, a more accurate description could be Twe-wz, where z is the coordinate along the pipe axis, such that the ambient temperature is Tw at z = 0, while it is Twe-wl at z = L. How does this modify your answers in (a) and (b)? (15 points) (e) Given that heat loss to the ambient is undesirable, suggest ways in which the heat loss can be prevented/minimized (No calculations required to prove your point here). (7 points) Units E EE m m Quantity L ri 12 k h T h. T. V. TW a W/m/K W/m/K kg/m3 J/kg/K W/m/K K m/s K 1/m Value 10 0.15 0.20 14.0 2.5 298 1200 2200 150 333 2.5 310 0.005 w