Reynolds analogy for minimum pumping power—M (C). A gas of molecular weight Mw is pumped through a cylindrical coolant duct in order to remove heat from part of a nuclear reactor. The temperature rise...

Reynolds analogy for minimum pumping power—M (C). A gas of molecular weight Mw is pumped through a cylindrical coolant duct in order to remove heat from part of a nuclear reactor. The temperature rise ΔT, the logarithmicmean temperature difference ΔTlog mean, the transfer area A, and heat load q are specified, so in the heat-balance equation: q = mcpΔT = AhΔTlog mean, (4) the only variables are the mass flow rate m and the specific heat cp of the gas. Thus, mcp is a constant, c1 for example. Use the Reynolds analogy to prove that the pumping power P = QΔp (where Q is the volumetric flow rate and Δp is the pressure drop in the duct) is lowest if a gas is selected with the largest possible value of M2 wc3 p. Assume ρ = Mwp/RT for the gas, with p, R, and T effectively constant, so that ρ = c2Mw, where c2 is another constant. 16. Turbulent mass flux—E. Consider flow in a pipe of diameter D and length L with a mean axial velocity vzm. Note that the Reynolds analogy gives the turbulent mass flux per unit area of the wall as m = τw/vzm. Let mt be the total such mass flux based on the total wall area of the pipe. Also define mc as the convective mass flow rate along the pipe. For flow in a smooth pipe of diameter D = 0.05 m at a Reynolds number Re = 104, how long would the pipe have to be so that mt = mc?