The rate of heat transfer to water flowing in a pipe was analyzed in Prob. 7–27. Let us approach that same problem, but now with dimensional analysis. Cold water enters a pipe, where it is heated by an external heat source (Fig. P7–75). The inlet and outlet water temperatures are T_in and T_out, respectively. The total rate of heat transfer Q̇ from the surroundings into the water in the pipe is known to be a function of mass flow rate ṁ, the specific heat c_p of the water, and the temperature difference between the incoming and outgoing water. Showing all your work, use dimensional analysis to find the functional relationship between these parameters, and compare to the analytical equation given in Prob. 7–27.

FIGURE P7–75

Data from Problem 27

Cold water enters a pipe, where it is heated by an external heat source (Fig. P7–27). The inlet and outlet water temperatures are T_in and T_out, respectively. The total rate of heat transfer Q̇ from the surroundings into the water in the pipe is

where ṁ is the mass flow rate of water through the pipe, and c_p is the specific heat of the water. Write the primary dimensions of each additive term in the equation, and verify that the equation is dimensionally homogeneous. Show all your work.

FIGURE P7–27

Transcribed Image Text:

Tin m Cp = specific heat of the water Tout