What is the equation to determine the inside surface temperature along the length of the counter flow heat exchanger?


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The temperature equations for a counter flow exchanger are the same for a parallel flow exchanger as long as steady-state is a valid assumption, so the flow must be fully developed and the temperature profile must be fully developed. For a differential element where the flow travels along $x$ we have the conservation of heat. Properties are known about substances and geometry and we analyze only the averages across the pipe's cross sections, $P$ is perimeter, $h$ is convection coefficient, $c_p$ fluid specific heat at average temperature, $T_m$ and constant (from steady state) average mass flow rate, $\dot{m}$.

$$Q_{in,pipe}+Q_{in,conv} = Q_{out}$$ $$\dot{m} c_p T_m(x) + h P dx (T_m(x) - T_s(x)) = \dot{m} c_p T_m(x+dx)$$

By the definition of the derivative and some algebra

$$\frac{\dot{m} c_p}{h P dx} \frac{dT_m(x)}{dx} + T_m(x) = T_s(x)$$

If you have the mean temperature profile of the fluid, then you can evaluate this expression. The mean temperature is often identified with the inlet and outlet temperatures and it may be the case that you will need a set of boundary conditions that describe heat transfer from surroundings into the pipe material and then into the fluid. While these may be closed form, they might be series solutions.


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