CBE 341 First Midterm Examination 10/14/2016 Due: Monday, October 17, 2016 at 10:00AM (before the lecture) Ground rules: Open book (WWWR) Open lecture notes (yours and mine) Open all course materials Nothing else (including calculators) Clock starts ticking when you open the envelope. Duration of the test: 2 hours. Student name: Date & time of the examination: Start time: End time: Honor code:1) (25 pts) You work for a company that manufactures and installs wind turbines. Your company has been contracted by the great state of Colorado to install 100 wind turbines on municipal land throughout the state. You are part of a team that is tasked with estimating the power that would be derived from wind turbines placed at different locations. To calculate these estimates you turn to dimensional analysis and experiments with a smaller model system. a) The power harvested from a wind turbine (P) is expected to depend on the density (ρ), viscosity (µ), and average velocity (V) of the wind, and the length (L), width (W), and angle of orientation of the blades (Θ). Using dimensional analysis, identify a set of dimensionless groups that can be used to correlate data from experiments with the model system to the full size wind turbine. (Please remember that power is measured in watts, which is joules per second). (10 pts) b) In general there are two options for placement of the wind turbines, in the plains of eastern Colorado or in the Rockies. The plains sites are at approximately 5,000 feet above sea level, whereas sites in the Rockies are at approximately 10,000 feet above sea level. Your experiments with the model system, which is 20-fold smaller than the full system, are at sea level. In the Rockies, theρ andµ of air are 20% lower than at sea level, whereas in the plains, theρ andµ are 10% lower than at sea level. The average wind velocity in the Rockies is 6 m/s, compared to the 4 m/s that was measured in the plains. What wind velocities must be used in the model system to achieve dynamic similarity with the full size systems at these two sites. (10 pts) c) What power would you expect a full size wind turbine to generate in the Rockies if it is dynamically similar to a sea level model that can generate 80 megawatts (MW) of power? (5 pts)2) (25 pts) Tubular heat exchangers are frequently used to cool hot product streams (see figure below). As the hot product stream enters the heat exchanger, it is divided into a bundle of smaller pipes to allow efficient heat transfer with the cooling solution (e.g., water). The product stream is then recombined into a single stream after being cooled. In the heat exchanger we are considering here, the pipe diameters at position 1 and 2 are D, the smaller pipe diameters (Ds) are 1/6 the size of D, there are 12 smaller pipes in the apparatus, and the product stream is incompressible and Newtonian. The density and viscosity of the product stream are not strong functions of temperature, and can be assumed to be the same at position 1 and 2. Further, the flow is steady and fully-developed at position 1 and 2, and shortly after entering the smaller pipes. a) The flow at position 1 is turbulent and the average fluid velocity is U. The Reynold’s number for this flow is 4000. Given that all of the pipes used here are smooth, will the flow of the product stream in the smaller pipes be laminar or turbulent? (10 pts) b) Considering a fully-developed region within the smaller pipes, what is the maximum velocity of the product stream fluid? Please model turbulent flow as plug flow, and if laminar flow is in this system use 2max21srVVR= −, where Rs = Ds/2. (10 pts) c) Pressure gauges at positions 1 and 2 indicate a pressure loss of ∆P (pressure at position 1 greater than position 2). Using this information please determine an expression for the force in the z-direction that the fluid exerts on this equipment between positions 1 and 2. Please assume that the viscous stresses at the inlet and outlet faces at positions 1 and 2 are negligible. (5 pts) 1 2zgClampClamp3) (50 pts) You are working for a chemical company as a process engineer. For the current project, you are tasked with identifying the pump needed to move a desired flowrate of reactant fluid, Qd (m3/s), through a straight pipe of length L into a batch reactor where a homogeneous reaction occurs. The interior of the pipe is smooth, its inner radius is R, and at a volumetric flowrate of Qd the flow is laminar. a) As a first approximation you assume that the fluid is incompressible and Newtonian, and that the flow is steady and fully developed by the time it reaches z = 0. Further, the pipe is horizontal and any effect of gravity can be considered negligible. Derive an expression for the velocity profile in the pipe. Please note that derivatives of the pressure can be left in your solution to this part of the problem. (15 pts) b) Determine an expression for the volumetric flowrate Q in such a system. From this expression and knowledge that the measured P at the outlet of the pipe (z = L) is Pout, identify an expression for what P should be at z = 0 to attain Qd. (15 pts) c) After installing a pump to achieve the P at z = 0 you calculated for (b), you realize that Q is far less than Qd. After shaking your fist in the air, you calm down and perform some measurements to test your assumptions. You find that the fluid is indeed incompressible, but that its viscosity is not constant. Specifically, you find that the viscosity of the fluid at z = 0 is less than that at z = L. This suggests that the homogeneous reaction was proceeding in the pipe prior to reaching the batch reactor, because as the reaction proceeds the fluid viscosity increases. As a second approximation to this engineering problem, you assume the same shear stress relations as illustrated on page 100 of the Lecture notes (Equation 160), except that you replace µ with µm, where0mazµµ= +, 0µ is the viscosity at z = 0, and a is a constant that you calculated from your viscosity measurements. With this modification, re-derive an expression for the velocity profile and volumetric flowrate, and identify an expression forP. Please do not attempt to apply boundary conditions to solve for the integration constant(s) in your expression for P. Also, note that the fluid remains incompressible, the flow is still steady and fully developed, and gravity remains