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Question:(II) A heat engine uses a heat source at 580°C and has an ideal (Carnot) efficiency of 22%. To increase the ideal efficiency to 42%, what must be the temperature of the heat source?

Short Answer

Expert verified

The temperature of heat source is \(1147.06\;{\rm{K}}\).

Step by step solution

01

Given Data

The temperature of engine is \({T_1} = 580^\circ {\rm{C}}\).

The Carnot efficiency is \(\eta = 22\% \).

The ideal efficiency is \({\eta _{\rm{i}}} = 42\% \).

02

Understanding Carnot efficiency

In this problem, use the Carnot efficiency to find the temperature at exhaust. Again, recalculate the temperature of the heat source with the help of new ideal efficiency.

03

Calculation of exhaust temperature of engine

The relation of Carnot efficiency is given by,

\(\eta = \left( {1 - \frac{{{T_2}}}{{{T_1}}}} \right)\)

Here, \({T_2}\) is the temperature at exhaust.

On plugging the values in the above relation.

\(\begin{aligned}{c}0.22 &= \left( {1 - \frac{{{T_2}}}{{\left( {580^\circ {\rm{C}} + 273} \right)\;{\rm{K}}}}} \right)\\{T_2} &= 665.3\;{\rm{K}}\end{aligned}\)

04

Calculation of temperature of the heat source

The relation to find temperature is given by,

\({\eta _{\rm{i}}} = \left( {1 - \frac{{{T_2}}}{{{{T'}_1}}}} \right)\)

Here, \({T'_1}\) is the temperature of heat source.

On plugging the values in the above relation.

\(\begin{aligned}{c}0.42 &= \left( {1 - \frac{{665.3\;{\rm{K}}}}{{{T_1}^\prime }}} \right)\\{T_1}^\prime &= 1147.06\;{\rm{K}}\end{aligned}\)

Thus, \({T_1}^\prime = 1147.06\;{\rm{K}}\) is the required rate of heat output.

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Most popular questions from this chapter

Question:(I) A heat engine’s high temperature\({T_{\bf{H}}}\)could be ambient temperature, because liquid nitrogen at 77 K could be\({T_{\bf{L}}}\)and is cheap. What would be the efficiency of a Carnot engine that made use of heat transferred from air at room temperature (293 K) to the liquid nitrogen “fuel”(Fig.15–25)?

Question: (II) A 1.0-L volume of air initially at 3.5 atm of (gauge) pressure is allowed to expand isothermally until the pressure is 1.0 atm. It is then compressed at constant pressure to its initial volume, and lastly is brought back to its original pressure by heating at constant volume. Draw the process on a PV diagram, including numbers and labels for the axes.

Question:(II) A four-cylinder gasoline engine has an efficiency of 0.22 and delivers 180 J of work per cycle per cylinder. If the engine runs at 25 cycles per second (1500 rpm), determine (a) the work done per second, and (b) the total heat input per second from the gasoline. (c) If the energy content of gasoline is 130 MJ per gallon, how long does one gallon last?

It has been suggested that a heat engine could be developed that made use of the temperature difference between water at the surface of the ocean and water several hundred meters deep. In the tropics, the temperatures may be 27°C and 4°C, respectively.

(a) What is the maximum efficiency such an engine could have?

(b) Why might such an engine be feasible in spite of the low efficiency?

(c) Can you imagine any adverse environmental effects that might occur?

(III) Rank the following five-card hands in order of increasing probability: (a) four aces and a king; (b) six of hearts, eight of diamonds, queen of clubs, three of hearts, jack of spades; (c) two jacks, two queens, and an ace; and (d) any hand having no two equal-value cards (no pairs, etc.). Discuss your ranking in terms of microstates and macrostates.

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