transport phenomena

2. You recently joined this major chemical company to build pipes to deliver water at a controlled pressure and flow rate to different scrubbers for removing toxic compounds coming out from fume vents. These pipes are cylindrical tubes of length L and radius R, which are at a slight angle to the horizontal ground, carry water upwards from a pressure P0 at the beginning and PL at the end. The pressure is always kept within the range needed for steady and laminar flow. Systematically solve the shell momentum balance for this problem using the steps we outlined in class to get an expression for the velocity and shear stress profile in the pipe. You need this expression to understand how the different variables in the setup affect the overall flow rate and drag forces in pipe which you might need control for the fume production rate that day and the amount of heating the pipe wall can withstand. Here are the steps I will be looking for: a) Choose a frame of reference (coordinate system). Where are you positioning it? (10 pts) b) Analyze the flow situation: What are the velocity components present? How does velocity vary with the spatial coordinates? Does gravity play a role? What are the non-zero components of momentum in your flow setup? (10 c) Do momentum balance over a shell. What does the differential form of the momentum balance equation look like? pts) (10 pts) d) What are the boundary conditions for this problem? (10 pts) e) What is the equation controlling the shear profile? (10 pts) f) What is the equation controlling the velocity profile? (10 pts) g) What is the average volumetric flow rate and drag stress you expect for any combination of problem variables. (10 pts) h) At what radius does the maximum velocity occur in the flow profile? (10 pts) i) What is the volumetric flow rate on a day where: (10 pts) PLL=5eN/m2;PLL=1e5N/m2,=20,L=100m;R=0.5m; Density of fluid =1000kg/m3; Viscosity of fluid = 0.001kg/(msec)