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Say you are cooling liquid A with water. The rate of heat transfer is given by Q = mH2OCpH2OdTH2O = mACpAdTA, where m is the mass, Cp is the mean heat capacity and dT is the change in temperature. So, if you increase m, the mass of water, Q increases (the rate of heat transfer increases) and hence the cooling rate would increase. And if m were to be decreased, the cooling rate would decrease. Strictly speaking, it should be the mass flowrate and not the mass that would be the parameter.

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Q: How would bigger mass in the water affect the cooling rate?
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