A—Adiabatic B—Isothermal

A—Adiabatic B—Isothermal

A crazy inventor patented the following (totally useless) device: Two geometrically identical cylinders (one adiabatic and the other isothermal) have rigidly interconnected pistons as shown in the figure.

The system is completely frictionless, and at the start of the experiment (State #0), the pistons are held in place so that the gases in the cylinders are in the states described in the following:

Cylinder A Cylinder B

(Adiabatic) (Isothermal)

1. Now, the pistons are free to move. At equilibrium, what is the temperature of the gas in Cylinder A? The y of the gas is 1.5.

An external device causes the pistons to oscillate back and forth 2500 times per minute. Each oscillation causes VB to go from 0.1 ms to 1ms and back to 0.1 ms.

2. How much power is necessary to sustain these oscillations?

Consider the same oscillating system as above with the difference that in each compression and each expansion 1% of the energy is lost. This does not alter the temperature of the isothermal cylinder because it is assumed that it has perfect thermal contact with the environment at 300 K. It would heat up the gas in the adiabatic cylinder that has no means of shedding heat. However, to simplify the problem, assume that a miraculous system allows this loss-associated heat to be removed but not the heat of compression (the heat that is developed by the adiabatic processes).

3. How much power is needed to operate the system?

3.22 In a Diesel cycle one can distinguish the following different phases:

Phase 1 ^ 2 An adiabatic compression of pure air from Volume Vl to Volume V2.

Phase 2 ^ 3 Fuel combustion at constant pressure with an expansion from Volume V2 to Volume Vs.

Phase 3 ^ 4 Adiabatic expansion from Volume Vs to Volume V4.

Phase 4 ^ 1 Isometric heat rejection causing the state of the gas to return to the initial conditions. This cycle closely resembles the Otto cycle, with the difference that in the Otto cycle the combustion is isometric while in the Diesel it is isobaric.

Consider a cycle in which Vl = 10-s ms, V2 = 50 x 10-6 ms, Vs = 100 x 10-6 ms, pL = 105 Pa, Tl = 300 K, and (for all phases) Y = 1.4.

1. Calculate the theoretical efficiency of the cycle by using the efficiency expression for the Diesel cycle given in Chapter 3 of the text.

2. Calculate the efficiency by evaluating all the mechanical energy (compression and expansion) and all the heat inputs. Be specially careful with what happens during the combustion phase (2 ^ 3) when heat from the fuel is being used and, simultaneously, some mechanical energy is being produced. You should, of course, get the same result from 2 and 3.

Chapter 4

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