# How important is a hot fluid temperature in a heat exchange?

Please consider an air to air heat exchanger.

On one side, we have 200°C hot air entering the heat exchanger, and leaving it at 150°C. And on the other side, we have 15°C fresh air to the inlet. Temperature is increased up to 130°C to the outlet.

So, hot air temperature difference is 50°C and cold air temperature difference is 115°C.

Considering this formula: $$\dot{Q} = \dot{m}_{hot}\cdot C_{hot}\cdot (T_{hot1}- T_{hot2}) = \dot{m}_{cold}\cdot C_{cold}\cdot (T_{cold2} - T_{cold1})$$

$$\dot{Q}$$ = heat exchange rate in W
$$\dot{m}$$ = mass flowrate in m/s
C = specific heat
T1 = temperature in
T2 = temperature out

I wonder what would be the resulting temperature of cold air on exit if hot air temperature would be 250°C in and 200°C out. This formula would give me 130°C or so in any case since only the difference between temperature in and temperature out matters.

I think that C depends of the temperature but it seems to not vary that much to make a real difference in the resulting temperature.

• I presume $m$ is mass and $T$ is temperature but what is $C$? (You should explain your variables in the question.) I imagine the efficiency of the heat exchanger is going to be significant too. Feb 11, 2022 at 22:21
• Sorry, I have edited my question. C is the specific heat of a fluid Feb 11, 2022 at 22:35
• Thanks. I'm an electrical engineer and don't work in this field but I imagine that the efficiency of the heat transfer will increase with higher $\Delta$T between the hot and cold sides. There should be some data and charts in a typical datasheet. Feb 11, 2022 at 22:44
• I would think so too, in a way that in reality, cold air temperature would be greater than 130°C and hot air temperature, lower than 200°C on the exit. But I would like someone to confirm it to me. I am asking because I have no data sheet on this heat exchanger as it is quite old now. Feb 11, 2022 at 23:16
• FWIW, mass flow is often notated $\dot{m}$, rather than $m$ Feb 12, 2022 at 0:21

The specific heat of gases does change with temperature. The higher the temperature, the higher the specific heat. But from 200°C to 250°C, there won't be a significant change in its specific heat value. So we can just stick with the formula using nominal specific heat values to find the exit temperature of cold air. Also, for gases, there are two specific heat values; specific heat at constant pressure, $$C_p$$, and specific heat at constant volume, $$C_v$$. Usually, in heat exchangers, we use $$C_p$$ to calculate heat transfer.

You can refer to this link to get an idea of how specific heat changes with temperature: Specific Heat Capacities of Air

Hope this helped.

• This helped. Though, since fluids want to reach equilibrium, I would instinctively think that increasing the hot air temperature would also increase the cold air temperature on the oulet. If this is not the case, it then means that the only important parameter for air is flowrate and that temperature does not matter that much. If hot air temperature would now be 600°C with the same flowrate, the cold air outlet temperature would not be much greater than 130°C which seems really counterintuitive to me. Feb 22, 2022 at 16:34
• @Snite Again, if we in the 600°C regime, the $C_p$ value would change significantly for that temperature. The outlet temperature of hot water and inlet temperature of cold water is important. Because that is what will define if we can take a constant $C_p$ value or not. I'm not sure how you calculated the cold water outlet temperature to be 130°C without its inlet and hot water outlet temperature. Feb 23, 2022 at 3:24

There is indeed a temperature dependence for heat capacity with air. The temperature dependence is even more significant if there is humidity in the air.

However, in this example, there is no mention of the actual flow rates that can have an effect on the in and out temperatures of a heat exchanger.

## heat exchanger and mass rates.

However, for a heat exchanger calculations, the boundary that heat is exchanged is the exchange surface area A. The heat that is lost through one fluid ($$\dot{q}_{hot}$$) and its transmitted through the area A ($$\dot{q}_{A}$$)is gained by the fluid on the other side $$\dot{q}_{cold}$$. I.e.

$$\dot{q}_{hot} = \dot{q}_{A} =\dot{q}_{cold}$$

So the heat exchange is rate (i.e. the power) is limited by the material and the design of the heat exchanger. Pumping more hot or cold fluid can affect the exit conditions.

In real life applications:

• mass flow rates for hot and cold
• input and output temperatures for hot
• input and output temperatures for cold

can all be design variables that can be used for the design and the operation of the heat exchanger.

E.g.: for an heating application, that the input temperature of the hot is known and the input and output temperature of the cold is known, changing the mass flows will affect the output temperature of the hot.

However, there is no restriction that the masses must be the same. More specifically, in a unit of time the masses involved can be vastly different.