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

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?

• heat gained (output) over heat lost (input), which will account for heat gain/loss to other than the targeted fluid. Commented Nov 13, 2023 at 17:47
• In terms of which "figure of merit" to select, that depends on what is the objective or the limiting factor in the context of your design. For example, let's say you are heating a working fluid. Is it more important to maximize temperature, or is it more important to conserve energy flow required from the other side? You may have situations where a variation in the design results in trade of one vs the other Commented Jan 22 at 14:30

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.