Chapter 4: Engine and Refrigerators

Derive equation $4.10$for the efficiency of the Otto cycle.

The amount of work done by each stroke of an automobile engine is controlled by the amount of fuel injected into the cylinder: the more fuel, the higher the temperature and pressure at points 3 and 4 in the cycle. But according to equation 4.10, the efficiency of the cycle depends only on the compression ratio (which is always the same for any particular engine), not on the amount of fuel consumed. Do you think this conclusion still holds when various other effects such as friction are taken into account? Would you expect a real engine to be most efficient when operating at high power or at low power? Explain.

At a power plant that produces 1 GW10⁹ watts) of electricity, the steam turbines take in steam at a temperature of 500^o, and the waste heat is expelled into the environment at 20^o
(a) What is the maximum possible efficiency of this plant?
(b) Suppose you develop a new material for making pipes and turbines, which allows the maximum steam temperature to be raised to 600^o. Roughly how much money can you make in a year by installing your improved hardware, if you sell the additional electricity for 5 cents per kilowatt-hour? (Assume that the amount of fuel consumed at the plant is unchanged.)

A small scale steam engine might operate between the temperatures $20°\text{C}$and $300°\text{C}$, with a maximum steam pressure of $10$bars. Calculate the efficiency of a Rankine cycle with these parameters.

Use the definition of enthalpy to calculate the change in enthalpy between points 1 and 2 of the Rankine cycle, for the same numerical parameters as used in the text. Recalculate the efficiency using your corrected value of${\text{H}}_2$, and comment on the accuracy of the approximation${\text{H}}_2{\text{≈H}}_1$.

Calculate the efficiency of a Rankine cycle that is modified from the parameters used in the text in each of the following three ways (one at a time), and comment briefly on the results: (a) reduce the maximum temperature to $500°\text{C}$; (b)reduce the maximum pressure to 100 bars; (c)reduce the minimum temperature to $10°\text{C}$.

Calculate the efficiency of a Rankine cycle that is modified from the parameters used in the text in each of the following three ways (one at a time), and comment briefly on the results:

$\left(a\right)$reduce the maximum temperature to localid="1649685342874" $500°\mathrm{C};$

$\left(b\right)$reduce the maximum pressure to localid="1649685354408" $100$bars;

$\left(c\right)$reduce the minimum temperature to localid="1649685367285" $10°\mathrm{C}$.

In a real turbine, the entropy of the steam will increase somewhat. How will this affect the percentages of liquid and gas at point$4$in the cycle? How will the efficiency be affected?

A coal-fired power plant, with parameters similar to those used in the text above, is to deliver$1\text{GW}\left({10}^9\text{watts}\right)$of power. Estimate the amount of steam (in kilograms) that must pass through the turbine(s) each second.

In table 4.1, why does the entropy of water increase with increasing temperature, while the entropy of steam decreases with increasing temperature?