Assignment 3 (PDF)

Homework 3 Problems – Heat Engines and the 2nd Law
1. The idealized Otto cycle is given below
a. Show that the efficiency of the Otto cycle can be written solely in terms of the
compression ratio 𝑏 = 𝑉1 /𝑉2 as
𝑒ℎ = 1 − 𝑏1−𝛾
Do you think that the efficiency of the cycle depends only on the compression ratio
when other effects such as friction are taken into account? Would expect a real engine
to be most efficient when operating at high power or at low power? Defend your
answer.
b. Show that the efficiency of the Otto cycle is less than a corresponding Carnot cycle
acting between the highest and lowest temperatures.
c. Suppose that the compression ratio of the above Otto cycle is 𝑏 = 8.00. At the
beginning of the compression process, 500 cm3 of gas is at 100 kPa and 20.0⁰C. At
the beginning of the power stroke, the temperature is 750⁰C. Modeling the working
substance as a diatomic ideal gas, determine the energy input, energy exhaust, and net
work of the cycle.
2. A reversible heat engine operates between two reservoirs, 𝑇𝐶 = 250 𝐾 and 𝑇𝐻 = 400 𝐾. The
cold reservoir can be considered to have infinite mass, but the hot reservoir consists of a
finite amount of gas at constant volume (1 mole with a specific heat 𝑐𝑉 = 3200𝐽/𝑘𝑔 ℃). As
a result, the temperature of the hot reservoir decreases with time, and after the heat engine
has operated for some long period of time, the temperature 𝑇𝐻 is lowered to 𝑇𝐶 . Calculate the
heat extracted from the hot reservoir during this period and the amount of work performed by
the engine during this period.
3. The idealized Diesel cycle is given below
a. Show that efficiency of the Diesel cycle can be written in terms of the compression
ratio 𝑏 = 𝑉𝐴 /𝑉𝐵 and the cut-off ratio 𝑐 = 𝑉𝐶 /𝑉𝐵 as
𝑏1−𝛾 (𝑐 𝛾 − 1)
𝑒ℎ = 1 −
𝛾(𝑐 − 1)
b. Show that for a given compression ratio, the Diesel cycle is less efficient than the
Otto cycle.
c. In a Diesel engine, atmospheric air is quickly compressed to about 1/20 of its original
volume. Estimate the temperature of the air after compression, and explain why a
Diesel engine does not require spark plugs.
4. One mole of a monotonic ideal gas is carried through the Carnot cycle described below.
At point 𝐴, the pressure is 25.0 atm and the temperature is 600 K. At point C, the
pressure is 1.00 atm and the temperature is 400 K.
a. Determine the pressures and volumes at points A, B, C, and D.
b. Calculate the net work done per cycle.
c. Determine the efficiency of an engine operating in this cycle.
5. In this question, you will analyze the thermodynamics of the Stirling heat engine.
a. Based on the description of the Stirling heat engine in class, draw a PV diagram for
this idealized Stirling cycle. Label each step in the engine’s cycle.
b. Suppose that, during stage 2 of the Stirling cycle, the gas gives up heat to the cold
reservoir instead of the regenerator and, during stage 4 of the Stirling cycle, the gas
absorbs heat from the hot reservoir. Assuming an ideal gas, show that the efficiency
of this engine can be written as
1
𝑇𝐻
𝑓
=
+
𝑒 𝑇𝐻 − 𝑇𝐶 2 ln(𝑉2 /𝑉1 )
where f is the number of degrees of freedom per molecule.
c. Show that the efficiency is less than that of a Carnot engine operating between the
same temperatures. If 𝑇ℎ = 3𝑇𝐶 and 𝑉2 = 2𝑉1 and the working substance is a
monotonic ideal gas, compute the efficiency of this cycle.
d. Suppose that you place the regenerator back into the system. Argue that, if it works
perfectly, the efficiency of a Stirling engine is the same as that of a Carnot engine.
Bonus: [6 points] Because of the importance of the Carnot theorem, you are going to perform a more
formal proof of it. Let 𝐶 be a Carnot engine operating between the temperatures 𝑇ℎ and 𝑇𝑙 (𝑇ℎ > 𝑇𝑙 )
and let 𝑀 represent any other heat engine operating between the same two temperatures as shown
below:
Let 𝑄1 (𝑄1′ ) refer to the energy exchanged as heat with the reservoir at 𝑇ℎ and let 𝑄2 (𝑄2′ ) refer to the
energy exchanged as heat with the reservoir at 𝑇𝑙 . Consider a process consisting of 𝑛′ cycles of 𝑀
and 𝑛 cycles of 𝐶 in reverse.
a. Show that the net external work done by our system is 𝑊𝑡𝑜𝑡𝑎𝑙 = 𝑛′|𝑊′| − 𝑛|𝑊| =
𝑄ℎ − 𝑄𝑙 , where
𝑄ℎ = 𝑛′|𝑄1 ′| − 𝑛|𝑄1 |
𝑄𝑙 = 𝑛′|𝑄2 ′| − 𝑛|𝑄2 |
b. Use (a) and (b) and the Kelvin-Planck statement to conclude that 𝑛′|𝑄2 ′| ≥ 𝑛|𝑄2 |.
Hint: Note that to as good an approximation we desire,
𝑄1
𝑛′
| ′| =
𝑛
𝑄1
c. Use (b) to conclude that
𝑄1
𝑄1′
| | ≥ | ′|
𝑄2
𝑄2

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