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Chapter 15: Q59P (page 412)

A particular car does work at the rate of about\({\bf{7}}{\bf{.0}}\;{\bf{kJ/s}}\)when traveling at a steady\({\bf{21}}{\bf{.8}}\;{\bf{m/s}}\)along a level road. This is the work done against friction. The car can travel 17 km on 1.0 L of gasoline at this speed (about 40 mi/gal). What is the minimum value for\({{\bf{T}}_{\bf{H}}}\)if\({{\bf{T}}_{\bf{L}}}\)is 25掳C? The energy available from 1.0 L of gas is\({\bf{3}}{\bf{.2 \times 1}}{{\bf{0}}{\bf{7}}}\;{\bf{J}}\).

Short Answer

Expert verified

The minimum value of temperature \({T_{\rm{H}}}\) is \(86\circ {\rm{C}}\).

Step by step solution

01

Understanding the efficiency of the engine

The ratio of the work done by the engine to the heat input at high temperature is termed the efficiency of the engine.

The expression for the efficiency is given as:

\(e = \frac{W}{{{Q_{\rm{H}}}}} = 1 - \frac{{{T_{\rm{L}}}}}{{{T_{\rm{H}}}}}\) 鈥� (i)

Here, W is the work done and \({Q_{\rm{H}}}\) is the heat input at high temperature.

02

Given data

The rate of doing work is \(\frac{W}{t} = 7\;{\rm{kJ/s}}\).

The speed of the car is \(v = 21.8\;{\rm{m/s}}\).

The car can travel a distance of about \(d = 17\,{\rm{km}}\).

The energy available is \(E = 3.2 \times {107}\,{\rm{J/L}}\).

The lower temperature is \({T_{\rm{L}}} = 25\circ {\rm{C}}\).

03

Evaluation of heat input from gasoline in the engine

The relation for the rate of heat input is given by:

\(\frac{Q}{t} = E \times v\)

Substitute the values in the above expression.

\(\begin{array}{l}\frac{Q}{t} = \left( {3.2 \times {{10}7}\,{\rm{J/L}} \times \frac{{1\;{\rm{L}}}}{{17000\;{\rm{m}}}}} \right) \times \left( {21.8\;{\rm{m/s}}} \right)\\\frac{Q}{t} = 41035.2\;{\rm{W}}\end{array}\)

04

Evaluation of the temperature of the engine

The relation for the efficiency is given by:

\(\begin{array}{c}\eta = \frac{{\left( {\frac{W}{t}} \right)}}{{\left( {\frac{Q}{t}} \right)}}\\1 - \frac{{{T_{\rm{L}}}}}{{{T_{\rm{H}}}}} = \frac{{\left( {\frac{W}{t}} \right)}}{{\left( {\frac{Q}{t}} \right)}}\end{array}\)

Substitute the values in the above expression.

\(\begin{array}{c}\left( {1 - \frac{{\left( {25\circ {\rm{C}} + 273} \right)\;{\rm{K}}}}{{{T_{\rm{H}}}}}} \right) = \frac{{\left( {7 \times {{10}3}\;{\rm{J/s}}} \right)}}{{\left( {41035.2\;{\rm{W}}} \right)}}\\{T_{\rm{H}}} = 359\;{\rm{K}}\\{T_{\rm{H}}} = \left( {359 - 273} \right)\circ C\\{T_{\rm{H}}} = 86\circ {\rm{C}}\end{array}\)

Thus, the minimum value of temperature\({T_{\rm{H}}}\)is\(86\circ {\rm{C}}\).

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Most popular questions from this chapter

An ideal gas undergoes an isothermal process. Which of the following statements are true? (i) No heat is added to or removed from the gas. (ii) The internal energy of the gas does not change. (iii) The average kinetic energy of the molecules does not change.

(a) (i) only.

(b) (i) and (ii) only.

(c) (i) and (iii) only.

(d) (ii) and (iii) only.

(e) (i), (ii), and (iii).

(f) None of the above.

Question:(I) A heat engine exhausts 8200 J of heat while performing 2600 J of useful work. What is the efficiency of this engine?

A cooling unit for a new freezer has an inner surface area of\({\bf{8}}{\bf{.0}}\;{{\bf{m}}{\bf{2}}}\), and is bounded by walls 12 cm thick with a thermal conductivity of\({\bf{0}}{\bf{.050}}\;{\bf{W/m}} \cdot {\bf{K}}\). The inside must be kept at -15掳C in a room that is at 22掳C. The motor for the cooling unit must run no more than 15% of the time. What is the minimum power requirement of the cooling motor?

Question: An ideal heat pump is used to maintain the inside temperature of a house at \({T_{{\rm{in}}}} = 22{\rm{^\circ C}}\) when the outside temperature is \({T_{{\rm{out}}}}\). Assume that when it is operating, the heat pump does work at a rate of 1500 W. Also assume that the house loses heat via conduction through its walls and other surfaces at a rate given by \(\left( {650\;{{\rm{W}} \mathord{\left/

{\vphantom {{\rm{W}} {{\rm{^\circ C}}}}} \right.} {{\rm{^\circ C}}}}} \right)\left( {{T_{{\rm{in}}}} - {T_{{\rm{out}}}}} \right)\). (a) For what outside temperature would the heat pump have to operate all the time in order to maintain the house at an inside temperature of 22掳C? (b) If the outside temperature is 8掳C, what percentage of the time does the heat pump have to operate in order to maintain the house at an inside temperature of 22掳C?

Question: (a) At a steam power plant, steam engines work in pairs, the heat output of the first one being the approximate heat input of the second. The operating temperatures of the first are 750掳C and 440掳C, and of the second 415掳C and 270掳C. If the heat of combustion of coal is \({\bf{2}}{\bf{.8 \times 1}}{{\bf{0}}^{\bf{7}}}\;{{\bf{J}} \mathord{\left/{\vphantom {{\bf{J}} {{\bf{kg}}}}} \right.} {{\bf{kg}}}}\) at what rate must coal be burned if the plant is to put out 950 MW of power? Assume the efficiency of the engines is 65% of the ideal (Carnot) efficiency. (b) Water is used to cool the power plant. If the water temperature is allowed to increase by no more than 4.5 C掳, estimate how much water must pass through the plant per hour.
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