What is the correct way to calculate the efficiency of a heat exchanger?

Question

I want to evaluate the efficiency of a heat exchanger. From measurements I know all four temperatures $T_1$, $T_2$, $t_1$ and $t_2$ as well as the mass flows of both fluids $m_1$ and $m_2$ and if necessary I would also able to retrieve e.g. pressure measurements.

Now after some research it seems there are multiple ways to calculate a value to attribute to a heat exchange process.

On a very simple level I found the thermal efficiency

$$ \eta_{thermal} = \frac{T_1 - T_2}{T_1 - t_1}. $$

Then there is the so called effectiveness $$ \epsilon = \frac{q_{act}}{q_{max}} $$

where I should be able to calculate $q_{act}$ by

$$ q_{act} = \dot{m} c_p (T_1 - T_2) $$

I am not 100% sure this is correct though, and I don't know which temperature I would need to use for $c_p$ in this equation.

Then the next question is how do I calculate q_{max}? Could I use the same equation just with e.g. the inlet temperature of the hot fluid $T_1$ and the outlet temperature of the cold fluid $t_2$? But which mass flow would I then use for the calculation?

Finally there are more sophisticated methods like NTU and LMTD for which I am not sure if I actually need them, if I in my case simply want to calculate an efficiency/effectiveness of a heat exchanger.

So what is the correct way if I basically want to compare a single heat exchanger during it's operating life? Efficiency? Thermal or another equation? Or effectiveness? And do I need a method like NTU or LMTD to calculate it?

Answer 1

Effectiveness using $\epsilon = \frac{q_{act}}{q_{max}}$ seems to be most general. It assumes comparing 2 cases and I would choose comparing cases where inlet temperatures and flowrates are fixed for both of them.

When calculating $q_{act}$ for the case where you already know all the inlet/outlet temperatures, you can use enthalpy difference between inlet and outlet multiplied by mass flow instead of approach with $c_p$:

$$q_{act} = \left(h_C\left(T_{C,out}\right)-h_C\left(T_{C,in}\right)\right)\cdot \dot{m}_C = \left(h_H\left(T_{H,in}\right)-h_H\left(T_{H,out}\right)\right)\cdot \dot{m}_H$$

However, calculating $q_{max}$ might be trickier, because you need to imagine, that the heat exchange between both fluids is perfect at any point in the exchanger, and this I think can manifest in 2 ways:

• In cocurrent heat exchanger, each fluid enters with different temperature, but they will reach the same temperature at the outlet, i.e. $T_{C,out}^* = T_{H,out}^*$.
• In countercurrent heat exchanger, different inlet temperatures, $T_{C,in}$ and $T_{H,in}$, are also given at the start and the outlet temperatures will be: $T_{C,out}^* = T_{H,in}$ and $T_{H,out}^* = T_{C,in}$;

Knowing all the inlet/outlet temperatures, you can also use the enthalpy approach for calculating $q_{max}$:

$$q_{max} = \left(h_C\left(T_{C,out}^*\right)-h_C\left(T_{C,in}\right)\right)\cdot \dot{m}_C = \left(h_H\left(T_{H,in}\right)-h_H\left(T_{H,out}^*\right)\right)\cdot \dot{m}_H$$

Expressing efficiency, flowrates can cancel out, so the expression is simplified:

$$\epsilon = \frac{h_C\left(T_{C,out}\right)-h_C\left(T_{C,in}\right)}{h_C\left(T_{C,out}^*\right)-h_C\left(T_{C,in}\right)} = \frac{h_H\left(T_{H,in}\right)-h_H\left(T_{H,out}\right)}{h_H\left(T_{H,in}\right)-h_H\left(T_{H,out}^*\right)}$$

Some heat exchangers might involve different setups, but I think it is always possible to imagine situation with perfect heat transfer between the fluids.