# All problems should be completed, but only one randomly chosen problem will be graded.

ME 410 Summer 2019 Homework #5 Due date: Jul 1, 11:59pm Eastern Homework Format 1. All problems should be completed, but only one randomly chosen problem will be graded. 2. For each problem, state what is given, list your assumptions (if appropriate), provide a sketch (if

appropriate), and provide your analysis (show your work!). Your answer MUST be clearly circled or boxed. If the answer is not clearly indicated, it will not be graded. Include units and appropriate significant digits.

3. All solutions should be combined into a SINGLE PDF and submitted to Canvas. Multiple files will NOT be graded.

4. Points will be deducted for sloppy solutions. 5. You are encouraged to study with others, but your submission should be your own work.

Problem 1: An engineer is designing a cooling fan system for a factory that produces 2024-T6 grade aluminum plate and rod stock. The parts exit an aging heat treatment furnace at 250Β°C, and must be cooled to 50Β°C in 3 minutes by a fan blowing air at 20Β°C. To start, assume that lumped capacitance analysis is reasonable. (a) For a plate with length L=50 cm, width W=50 cm, and thickness t=1 cm, what airflow velocity would be required to achieve this cooling? Assume that the airflow is along the length of the plate, and that the boundary layer is fully turbulent (i.e., πππππ₯π₯,ππ = 0). Is lumped capacitance analysis appropriate (you must justify your answer)? (b) For a cylindrical rod with diameter D=3 cm and length 50 cm, what would be the airflow velocity required? Assume that the airflow is perpendicular to the rod axis (i.e., the rod is a cylinder in crossflow). Is lumped capacitance analysis appropriate? Problem 2: An experimental nuclear core simulation device consists of a long thin-walled metallic tube with diameter D and length L. It is electrically heated to produce a sinusoidal surface heat flux distribution: ππ”π π  = ππ”πππ π π π π π οΏ½ππ

π₯π₯ 2πΏπΏ οΏ½, where x is the distance from the tube inlet. Fluid flows through the tube with a

flowrate of οΏ½ΜοΏ½π and an inlet temperature of ππππ,ππ, and can be considered turbulent and fully developed over its entire length. Using the supplied variables, develop expressions for: (a) the total rate of heat transfer (ππ) from the tube to the fluid; (b) the fluid outlet temperature, ππππ,ππ; (c) the axial distribution of the wall temperature, πππ π (π₯π₯). Problem 3: In a molten-salt nuclear reactor, radioactive liquid is flowing through a cylindrical pipe of diameter D. Fission in the liquid results in a uniform internal heat generation of οΏ½ΜοΏ½π W/m3. The pipe wall also has a heater on it that can provide a uniform surface heating rate of qβ W/m2. The radioactive liquid has a specific heat capacity of Cp J/kg-K, a mass flowrate of οΏ½ΜοΏ½π kg/s, and an inlet temperature of Tm,i. Conduction in the fluid in the x-direction can be assumed to be negligible, conduction through the pipe wall is negligible, and the flow in the pipe is steady and fully developed. Using a differential control volume in the pipe, perform an energy balance to obtain a differential equation that describes the variation of the bulk temperature (Tm) as a function of x-distance along the pipe. What is the boundary condition? DO NOT SOLVE the differential equation.

Plate Rod

Problem 1

Tm,i

x

Problem 3