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Since an evaporator also a heat exchanger, how can the inlet and outlet temperatures of the HEX accepted as the same, i.e. isothermal? In this case, cold stream (inlet and outlet of the evaporator in the refrigeration cycle) does not incur any change in temperature, which requires an infinite size of the heat exchanger.

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Assuming you mean evaporator as part of a vapour compression cycle, then because the fluid enters as a liquid and leaves as a gas, which means that the temperature can be the same but the internal energy changes due to the latent heat of vapourisation.

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