Engineers testing the efficiency of an electric generator gradually vary the temperature of the hot steam used to power it while leaving the temperature of the cooling water at a constant . They find that the generator’s efficiency increases at a rate at steam temperatures near . What is the ratio of the generator’s efficiency to the efficiency of a Carnot engine?
Ratio of generator’s efficiency to the efficiency of a Carnot engine
It is a heat engine having maximum efficiency which converts heat energy into mechanical energy by extracting heat from hot source and dumping some heat energy into cold sink. The maximum efficiency is given by the following expression.
is efficiency of engine , Tc is temperature of cold sink and Th is temperature of hot source.
We may deduce from the problem that our generator's efficiency can be expressed as the efficiency of a Carnot engine multiplied by a constant, which will almost certainly be between 0 and 1. .
Differentiate it with respect to Th.
We can solve for this constant as follows:
Put the values in equation,
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