Simplified Heat Transfer from Boiling water into pipe [closed]

Question

I am trying to find the heat transfer from boiling water into a pipe of cold water running through a boiler. It has been forever since I took thermodynamics and i'm a bit confused.

I have a model for finding the heat transfer through the walls of the pipe using the thermal resistance method and there are a bunch of examples with heat transfer to flowing water so I think I am good on that front.

My question is.... How do I find a Heat transfer value from the boiling water to the surface of the pipe. I have included a simple diagram below.

There are two methods I have found that I think have lead to my confusion.

First, there is the heat transfer equation:

q = mc(T2-T1)

If we were to use this equation, would the m be the mass of the boiling water? or would you turn it into a mass flow rate problem and use the mass flow rate of the water running through the pipe? Since there are 3 different temperatures in this system (the temperature of the boiling water, the cold water entering the pipe and the hotter water exiting the pipe) I am not sure if this is the right way to go about it.

The second way to find this (I think) is:

q = h2- h1

using the enthalpys of the water at different temperatures. Again, I do not think this is the correct method since it seems....too simple.

There are a lot of heat exchanger example problems out there. HOWEVER they all seem to know the heat transfer rate or they are assuming a counter or parallel flow heat exchanger.

Since only one of the fluids is moving in this heat exchanger, trying to figure out what information to use has been tricky.

There is probably something I am assuming or thinking about wrong so if anyone could help out that would be awesome! Thanks!!

Answer 1

If you know the entry $T_1$ and exit $T_2$ temperatures of the colder fluid, then the calculation is easy. The heat transferred to the colder fluid is nearly the equation you had:

$$Q=\dot m c_p (T_2-T_1) $$

where the heat transfer rate $Q$ (e.g. Watts) is a product of the temperature change times the mass flow rate $\dot m$ (e.g. in kg/s) and the specific heat of $c_p$ of the colder fluid (e.g. in J/(kg-K)). You may have to compensate for $c_p$ increasing with temperature if the temperature rise is great.

If you do not know the exit temperature or need to find how $T_2$ varies for different flow rates or boiler temperatures, that is a more advanced problem. The equation typically used is some form of:

$$Q=UA\Delta T_{lm}$$

Where $UA$ is the overall heat transfer coefficient $[\rm {J/(m}^2 \rm {K)]}$ times the overall surface area $A$ and $\Delta T_{lm}$ is the Log Mean Temp Difference, which is a function of the inlet and exit temps of both hot and cold streams. These are parameters you could perhaps estimate from Heat Exchanger literature for your system's parameters.