Southern Methodist University Professor David A. Willis

HW #6

Due: Tuesday, April 7, 2009

Thursday, April 9, 2009 (distance students)

1)  (ME7386 only) Bejan text problem 7.3 (modified): One way to improve the mixing-length model, that is, to achieve a smoother overlap between measurements and the empirically adjusted curves suggested by the mixing-length model (Fig. 7.4), is to do away with the assumption that the eddy diffusivity eM is zero in a layer of finite thickness yVSL. Instead, as proposed by van Driest [16], assume that the eM decays rapidly as y decreases and becomes zero strictly at the wall. Starting with the new mixing-length model,

and assume that it is valid throughout the inner region defined by the constant shear stress postulate (7.33), derive the van Driest equation for du+/dy+ listed in Table 7.1. Using the constants given in the table (A+ = 25, k = 0.4), compare the results with another equation in Table 7.1, such as the von Kármán equation, for large and small values of y+.

2) Bejan text problem 7.12: Water flows with the velocity of U∞ = 2 m/s parallel to a plane wall. The following calculations refer to the position x = 6 m measured downstream from the leading edge. The water properties can be evaluated at 20 oC.

a) A probe is to be inserted in the viscous sublayer to the position represented by y+ = 2.7. Calculate the actual spacing y (mm) between the probe and the wall.

b) Calculate the boundary layer thickness d, and compare this value with the estimate based on the assumption that the length x is covered by laminar (note the typing error in the text) boundary layer flow.

c) Calculate the heat transfer coefficient averaged over the length x.

3) Bejan text problem 7.21: During the cooling and hardening phase of its manufacturing process, a glass bead with a diameter of 0.5 mm is dropped from a height of 10 m. The bead falls through still air of temperature 20 oC. The properties of the bead material are the same as those listed for window glass in Appendix B.

a) Calculate the “terminal” velocity of the free-falling bead, that is, the velocity when its weight is balanced by the air drag force. Also, calculate the approximate time that is needed by the bead to achieve this velocity, and show that the bead travels most of the 10 m height at terminal velocity.

b) Calculate the average heat transfer coefficient between the bead surface and the surrounding air when the bead travels at terminal velocity and when its surface temperature is 500 oC. Treat the bead as a lumped capacitance and estimate its temperature at the end of the 10 m fall. Assume that its initial temperature was 500 oC.

ME 5/7386 Convection Heat Transfer Department of Mechanical Engineering