Cooling system - Heat transfer - challenging problem!

In your textbook, find the equations for heat transfer as a function of mass flow rate. There should also be an example or examples that, when worked out individually, will enable you to grasp the concepts necessary for working out this problem.

At first glance, this looks like an Ordinary Differential Equation (ODE) boundary value problem (though I reserve the right to be wrong!).

Good luck in Transport Phenomena!

Nice illustrations in describing the problem, but understanding theory is far more important than being able to draw well.

If m is the mass flow rate in the pipe, with fluid of specific heat C_p, then m = Q₁/C_p.

It will be a bit more complicated to figure out T_in or T_out and the LMTD, because that depends on heat conductivity, distances, areas, and other such stuff.

A class in HX 101 might be in order....

" The radial heat transfer trough the pipe can be neglected, because the thickness of the pipe wall is small."

How can you write this when the heat transfer through the pipe wall will be the one to cool the internal volume ? You should have written : the thermal heat resistance of the pipe wall can be neglected due to its high conduction and low thickness. Without heat transfer through the pipe wall the internal temperature would be 180°C !!!!!! This shows that you have not clearly understood how heat is evacuated. Now for your problem you have two flows: one through the walls and the other to the cooling fluid. If the wall on which the plate is placed is insulated then the internal air temperature has to be 150°C since copper has a high conductivity and the plate will be at same temperature all over. Your balance is then : input energy due to 30°C difference = output energy via the fluid. The last depends on the heat transfer you want to neglect! Consider that the internal temperature is 150°C and that the fluid has a variation from in to outlet. You have to verify not only the different convection coefficients but also the area since what you are interested in is the GLOBAL heat transfer. Since the heat transfer in the pipe depends on the velocity and the temperature rise you should consider for the fluid you MUST make the computations for different flows and define which one is the solution. For the equations I let you write all of them and if you want a correction display them and you will get the support. I think that you are interested in the steady state if so you do not need any differential equation only an energy balance. The differential equation would be required ONLY if you would want to analyse how the internal temperature evolutes versus time.