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Thermodynamics: A complete undergraduate course

Andrew M. Steane

Chapter 11

Practical heat engines - all with Video Answers

Educators

Chapter Questions

01:43

Problem 1

The temperature inside the engine of a crane is $2000^{\circ} \mathrm{C}$, the temperature of the exhaust gases is $900^{\circ} \mathrm{C}$. The heat of combustion of petrol is $47 \mathrm{MJ} / \mathrm{kg}$, and the density of petrol is $0.8 \mathrm{~g} / \mathrm{cm}^3$. What is the maximum height through which the crane can raise a $10000 \mathrm{~kg}$ load by burning 0.1 litre of petrol?

Mayukh Banik
Mayukh Banik
Numerade Educator
01:11

Problem 2

The operation of a diesel engine can be modelled approximately by the following cycle: (i) adiabatic compression from $\left(p_1, V_1\right)$ to $\left(p_2, V_2\right)$, (ii) heating at constant pressure to $\left(p_2, V_3\right)$, (iii) adiabatic expansion to $\left(p_4, V_1\right)$, (iv) cooling at constant volume back to $\left(p_1, V_1\right)$. Find the efficiency in terms of the two compression ratios $V_1 / V_2$ and $V_3 / V_2$.

Mayukh Banik
Mayukh Banik
Numerade Educator
08:09

Problem 3

The Stirling cycle consists of two isothermal stages and two isochoric (constant volume) stages; see Figure 11.4. (i) Show that if the heat capacity of the working fluid is a function of temperature alone, then the amount of heat entering the fluid in one of the isochoric processes is equal to the amount leaving in the other. (ii) Hence show that the thermal efficiency is $1-T_2 / T_1$, the same as that of a Carnot engine.
It follows from (i) that the energy extracted when the fluid cools from $T_1$ to $T_2$ can be stored and used to provide the energy needed to heat the fluid from $T_2$ to $T_1$. Such an energy store is called a regenerator and is a common feature of Stirling cycle engines.
Figure 11.4 Can't copy

Saman Zulfiqar
Saman Zulfiqar
Numerade Educator
08:09

Problem 4

The Stirling cycle should not be confused with the Stirling engine. A model cycle which approximates the action of the Stirling engine, and which also approximates a regenerative gas turbine, is the Ericsson cycle. This consists of two isothermal and two isobaric stages. Find the thermal efficiency if the working fluid obeys Joule's law $(U=U(T))$.
Figure 11.4 Can't copy

Saman Zulfiqar
Saman Zulfiqar
Numerade Educator
01:37

Problem 5

A cycle consists of isothermal stages at temperatures $T_1, T_2$ and two adiathermal stages, the whole being irreversible to the extent that an entropy $\Delta S$ is generated inside the system for each cycle. Find the efficiency $\eta=W / Q_1$ in terms of $T_1, T_2, \Delta S$, and $Q_1$, where $W$ is the work obtained per cycle and $Q_1$ is the heat extracted from the hotter reservoir $\left(T_1\right)$ in each cycle.

Manik Pulyani
Manik Pulyani
Numerade Educator
13:52

Problem 6

An Otto cycle uses a volume compression ratio of 9.5, with a pressure and temperature before compression of $100 \mathrm{kPa}$ and $40^{\circ} \mathrm{C}$ respectively.
The cycle is performed 3000 times per minute, using a mass of $11.5 \mathrm{~g}$ of air per cycle, and heat input $600 \mathrm{~kJ} / \mathrm{kg}$. Determine the thermal efficiency and the power output.

Saman Zulfiqar
Saman Zulfiqar
Numerade Educator
12:21

Problem 7

The Newcomen engine was the first working steam engine with a piston; it was constructed by Thomas Newcomen through incremental improvements between 1705 and 1711 and was ultimately widely used. A large cylinder (e.g. radius 1 metre, height several metres) was closed by a piston attached to one end of a heavy balanced beam. As the beam rocked to and fro, the other end provided work, for example to drive a water pump. The working principle of the Newcomen engine is as follows. A coal fire heats a quantity of water in a boiler. During the down-stroke on the pump, which is the up-stroke on the piston, steam fills the cylinder, but no useful work is done because the steam pressure is roughly $1 \mathrm{~atm}$; the weight of the pump rod provides the force required to lift the piston. At the top of the piston stroke, a valve to the boiler is closed, and another valve to a supply of cold water under pressure is briefly opened. Some cold water sprays into the chamber. Since this water is much colder than the steam, heat flows from the latter to the former, and the steam condenses. The pressure in the chamber therefore drops considerably, 'sucking' the piston downwards-which is to say, the atmospheric pressure above the piston pushes it down. This is why such engines are called 'atmospheric' engines. This is the power stroke. When the piston reaches the bottom, the valve to the boiler is re-opened and the incoming steam pushes the cold water out of the chamber.
(i) If the cylinder radius is $1 \mathrm{~m}$, and the working stroke of the piston is $3 \mathrm{~m}$, calculate the mass of steam required to fill the chamber at $100^{\circ} \mathrm{C}, 100 \mathrm{kPa}$. Determine the heat input required to produce this amount of steam from water initially at $15^{\circ} \mathrm{C}$.
(ii) Determine the amount of water at $15^{\circ} \mathrm{C}$ that will suffice to lower the pressure to $50 \mathrm{kPa}$.
(iii) Find the work output per cycle, and hence the average power output if the machine executes 15 cycles per minute. How does this compare to the heat input in part (i)?

Sanjeev Kumar
Sanjeev Kumar
Numerade Educator