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Chapter 15: Problem 35

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

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

Step by step solution

01

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: $T_1 = 500.0^{\circ} \mathrm{C} + 273.15 = 773.15 \mathrm{K} \newline T_2 = 27^{\circ} \mathrm{C} + 273.15 = 300.15 \mathrm{K} \newline T_3 = 47^{\circ} \mathrm{C} + 273.15 = 320.15 \mathrm{K}$
02

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 \(= 1 - \frac{T_2}{T_1} = 1 - \frac{300.15}{773.15} \approx 0.61153\)
03

Calculate new maximum efficiency with cooling towers

Similarly, we can calculate the new maximum efficiency with cooling towers using the formula: New efficiency \(= 1 - \frac{T_3}{T_1} = 1 - \frac{320.15}{773.15} \approx 0.5862\)
04

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 \(47^{\circ} \mathrm{C}\).

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Most popular questions from this chapter

A fish at a pressure of 1.1 atm has its swim bladder inflated to an initial volume of \(8.16 \mathrm{mL}\). If the fish starts swimming horizontally, its temperature increases from \(20.0^{\circ} \mathrm{C}\) to $22.0^{\circ} \mathrm{C}$ as a result of the exertion. (a) since the fish is still at the same pressure, how much work is done by the air in the swim bladder? [Hint: First find the new volume from the temperature change. \(]\) (b) How much heat is gained by the air in the swim bladder? Assume air to be a diatomic ideal gas. (c) If this quantity of heat is lost by the fish, by how much will its temperature decrease? The fish has a mass of \(5.00 \mathrm{g}\) and its specific heat is about $3.5 \mathrm{J} /\left(\mathrm{g} \cdot^{\circ} \mathrm{C}\right)$.
An engincer designs a ship that gets its power in the following way: The engine draws in warm water from the ocean, and after extracting some of the water's internal energy, returns the water to the ocean at a temperature \(14.5^{\circ} \mathrm{C}\) lower than the ocean temperature. If the ocean is at a uniform temperature of \(17^{\circ} \mathrm{C},\) is this an efficient engine? Will the engineer's design work?
A container holding \(1.20 \mathrm{kg}\) of water at \(20.0^{\circ} \mathrm{C}\) is placed in a freezer that is kept at \(-20.0^{\circ} \mathrm{C} .\) The water freezes and comes to thermal equilibrium with the interior of the freezer. What is the minimum amount of electrical energy required by the freezer to do this if it operates between reservoirs at temperatures of $20.0^{\circ} \mathrm{C}\( and \)-20.0^{\circ} \mathrm{C} ?$
A \(0.500-\mathrm{kg}\) block of iron at \(60.0^{\circ} \mathrm{C}\) is placed in contact with a \(0.500-\mathrm{kg}\) block of iron at $20.0^{\circ} \mathrm{C} .\( (a) The blocks soon come to a common temperature of \)40.0^{\circ} \mathrm{C} .$ Estimate the entropy change of the universe when this occurs. [Hint: Assume that all the heat flow occurs at an average temperature for each block.] (b) Estimate the entropy change of the universe if, instead, the temperature of the hotter block increased to \(80.0^{\circ} \mathrm{C}\) while the temperature of the colder block decreased to \(0.0^{\circ} \mathrm{C}\) [Hint: The answer is negative, indicating that the process is impossible. \(]\)
List these in order of increasing entropy: (a) 1 mol of water at $20^{\circ} \mathrm{C}\( and 1 mol of ethanol at \)20^{\circ} \mathrm{C}$ in separate containers; (b) a mixture of 1 mol of water at \(20^{\circ} \mathrm{C}\) and 1 mol of ethanol at \(20^{\circ} \mathrm{C} ;\) (c) 0.5 mol of water at \(20^{\circ} \mathrm{C}\) and 0.5 mol of ethanol at \(20^{\circ} \mathrm{C}\) in separate containers; (d) a mixture of 1 mol of water at $30^{\circ} \mathrm{C}\( and 1 mol of ethanol at \)30^{\circ} \mathrm{C}$.
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