Question

An electric power station generates steam at \(500.0^{\circ} \mathrm{C}\) and condenses it with river water at \(27^{\circ} \mathrm{C} .\) By how much would its theoretical maximum efficiency decrease if it had to switch to cooling towers that condense the steam at \(47^{\circ} \mathrm{C} ?\)

Short answer

Answer: The decrease in the theoretical maximum efficiency is approximately 0.02533 or 2.533%.

Step-by-step solution

Convert temperatures from Celsius to Kelvin

To convert the given temperatures from Celsius to Kelvin, we just need to add 273.15 to each value. Converting the temperatures, we have: $T_1 = 500.0^{\circ} \mathrm{C} + 273.15 = 773.15 \mathrm{K} \newline T_2 = 27^{\circ} \mathrm{C} + 273.15 = 300.15 \mathrm{K} \newline T_3 = 47^{\circ} \mathrm{C} + 273.15 = 320.15 \mathrm{K}$

Calculate initial maximum efficiency

Now that we have the temperatures in Kelvin, we can plug them into the formula for maximum efficiency. The initial efficiency is given by: Initial efficiency \(= 1 - \frac{T_2}{T_1} = 1 - \frac{300.15}{773.15} \approx 0.61153\)

Calculate new maximum efficiency with cooling towers

Similarly, we can calculate the new maximum efficiency with cooling towers using the formula: New efficiency \(= 1 - \frac{T_3}{T_1} = 1 - \frac{320.15}{773.15} \approx 0.5862\)

Calculate the decrease in efficiency

To find the decrease in efficiency, subtract the new efficiency value from the initial efficiency value: Decrease in efficiency \(= 0.61153 - 0.5862 \approx 0.02533\) So, the theoretical maximum efficiency of the electric power station would decrease by approximately 0.02533 or 2.533% if it had to switch to cooling towers that condense the steam at \(47^{\circ} \mathrm{C}\).