Starting with a heat balance, show that the effectiveness for a counterflow arrangement is



GIVEN

Counterflow heat exchanger

FIND

Show that the effectiveness is



ASSUMPTIONS

Heat loss to surroundings in negligible

SKETCH


A heat balance on an element of the heat exchanger yields



Rearranging





Therefore





Integrating from a to b



But, from



From which



Therefore



Substituting these into the energy balance



Solving for effectiveness







Multiplying the numerator and denominator by



yields the following result

Physics & Space Science

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