The operation of a diesel engine can be idealized by the cycle shown in Fig. 20–26. Air is drawn into the cylinder during the intake stroke (not part of the idealized cycle). The air is compressed adiabatically, path ab. At point b diesel fuel is injected into the cylinder and immediately burns since the temperature is very high. Combustion is slow, and during the first part of the power stroke, the gas expands at (nearly) constant pressure, path bc. After burning, the rest of the power stroke is adiabatic, path cd. Path da corresponds to the exhaust stroke. Show that, for a quasistatic reversible engine undergoing this cycle using an ideal gas, the ideal efficiency is $e=1-\frac{(V_a/V_c{)}^{-\gamma }-(V_a/V_b{)}^{-\gamma }}{\gamma \left[\right(V_a/V_c{)}^{-1}-\left(V_a/V_b{)}^{-1}\right]}$ where $V_a/V_b$ is the “compression ratio,” $V_a/V_c$ is the “expansion ratio,” and $\gamma$ is defined by Eq. 19–15. ($\gamma =\frac{C_P}{C_V}$).

An ideal heat pump is used to maintain the inside temperature of a house at Tᵢₙ = 22°C when the outside temperature is Tₒᵤₜ. Assume the heat pump does work at a rate of 1700 W. Also assume that the house loses heat via conduction through its walls and other surfaces at a rate given by ( 650 W/C°) (Tᵢₙ - Tₒᵤₜ). If the outside temperature is less than you just calculated, what happens?

How much less per year would it cost a family to operate a heat pump that has a coefficient of performance of 2.9 than an electric heater that costs $2100 to heat their home for a year? If the conversion to the heat pump costs $15,000, how long would it take the family to break even on heating costs? How much would the family save in 20 years?

A car engine whose output power is 145 hp operates at about 15% efficiency. Assume the engine’s water temperature of 85°C is its cold-temperature (exhaust) reservoir and 495°C is its thermal “intake” temperature (the temperature of the exploding gas–air mixture). What is the ratio of its efficiency relative to its maximum possible (Carnot) efficiency?

(III) Figure 20–17 is a PV diagram for a reversible heat engine in which 1.0 mol of argon, a nearly ideal monatomic gas, is initially at STP (point a). Points b and c are on an isotherm at T = 423 K. Process ab is at constant volume, process ac at constant pressure. (a) Is the path of the cycle carried out clockwise or counterclockwise? (b) What is the efficiency of this engine?

The Stirling cycle, shown in Fig. 20–27, is useful to describe some heat engines. Find the efficiency of the cycle in terms of the parameters shown, assuming an ideal monatomic gas as the working substance. The processes ab and cd are isothermal whereas bc and da are at constant volume. How does it compare to the Carnot efficiency?

The Brayton cycle, depicted in the PV diagram of Fig. 20–28, can describe a jet engine gas turbine. In process ab the air–fuel mixture undergoes an adiabatic compression. This is followed, in process bc, with an isobaric (constant pressure) heating, by combustion. Process cd is an adiabatic expansion with expulsion of the products to the atmosphere. The return step, da, takes place at constant pressure. If the working gas behaves like an ideal gas, show that the efficiency of the Brayton cycle is

$e=1-{\left(\frac{P_b}{P_a}\right)}^{\frac{1-\gamma }{\gamma }}$