Water at 28°C flows in a straight horizontal pipe in which there is no exchange of either heat or work with the surroundings. Its velocity is in a pipe with an internal diameter of 2.5 cm until it flows into a section where the pipe diameter abruptly increases. What is the temperature change of the water if the downstream diameter is 3.8 cm? If it is 7.5 cm? What is the maximum temperature change for an enlargement in the pipe?

Question

What will be an ideal response?

Answer 1

Mass balance at steady state requires that the mass flow (and therefore volumetric flow for water with nearly constant density) be the same before and after the diameter change. The mass flowrate in the 2.5 cm diameter pipe is



and the mass flow rate after the expansion is



Setting these equal shows that the velocity in the outlet pipe is



So,

The change in kinetic energy per unit mass is then



Since no heat is exchanged with the surroundings, there is no change in elevation (no change in gravitational potential energy) and no non-flow work is done on the surroundings, the energy balance is simply



From which ?H = 79.6 J

This enthalpy change corresponds to a temperature increase given by



from which ?T = 0.019 K.

If the downstream diameter were 7.5 cm, then we would have = 1.556 m/s, and the change in kinetic energy per unit mass would be ?96.8 J The corresponding temperature increase would be 96.8/4180 = 0.0232 K. The maximum temperature increase would occur if the downstream velocity went to zero, so that all of the kinetic energy of the flow were converted to internal energy. In this case, we would have leading to a temperature increase of 98/4180 = 0.0234 K. For the downstream diameter of 7.5 cm, we had very nearly reached this limit.