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So i need to find the power of a cooling system. I have the $T_{coil}=-2^{\circ}C$. $T_{input}=6^{\circ}C$ and $T_{output}=1^{\circ}C$ . I also have mass flow rate of the fluid along with its specific heat. Now my doubt is this . in my opinion the power of the system will directly be $\dot{m}C_{p}(T_{input}-T_{output})$, which is the heat taken by cooling system in order to cool the fluid. But my doubt here is that $T_{coil}$ doesn't change the power of cooling system in any manner which doesn't seem right to me. Can someone care to explain?

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It would be easier to answer if you gave a figure of the device.

In fact, there cannot be a single temperature for the coil. Heat left by the fluid must be absorbed by the coil, hence parts of the coil are hotter than others. In particular, the surface of the coil in contact with the fluid is approximately at the same temperature as the fluid.

To increase the "cooling power" of the device, that is, the heat flux, you need to increase the temperature difference between fluid and coil by cooling down the coil even more. So cooling power indeed depends on $T_\text{coil}$, in a non-trivial way, where "depends" means "is physically determined by". But that doesn't mean $T_\text{coil}$ should appear explicitly in the balance equation you wrote for heat flux. Actually, $T_\text{output}$ is a function of $T_\text{input}$ and $T_\text{coil}$; but since you know the value of $T_\text{output}$, you can disregard the existence of this function.

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