ME 200 – Thermodynamics I – Spring 2020

Homework 21: Reversible Processes

Part (i): Pendulum

Part (ii): Expanding Gas

Part (iii): Heat Transfer

Part (iv): Compression and Heat Transfer

Part (v): Parallel Flow Heat Exchanger

Part (vi): Counterflow Heat Exchanger

Part (i): Pendulum

Is each swing of the pendulum clock outside of ME 1130 an example of a thermodynamically reversible process? Explain why or why not.

Solution:

No, because when the pendulum swings not all of the potential energy is converted to kinetic energy due to heat loss from friction. Therefore, the pendulum will not be able to return to its initial height without mechanical work input.

Part(ii): Expanding Gas

A gas is expanding against the atmosphere with a massless piston. What is necessary for this to be a reversible process?

Solution:

In order for this process to be reversible, there must be no friction between the cylinder walls and the piston must be equally inclined to go either direction (expand or compress). This requires a balance of mechanical forces. This process must also occur in a quasi-static process, such that no non-equilibrium effects are present, and the amount of work input to return to the initial state equals the amount of work output from the expansion process.

Part(iii): Heat Transfer

What is necessary for this heat transfer process to be reversible?

Solution:

Reversible heat exchange can only occur in the absence of a temperature gradient such that $T_{Hot}$ = $T_{Cold}$. A finite heat transfer could only occur in the limit as the heat transfer area approaches infinity and must happen very slowly such that there are no non-equilibrium effects (quasi-static heat transfer).

Part(iv): Compression and Heat Transfer

A closed system process involves compression and heat transfer to the fluid. What idealized conditions are necessary for the process to be internally reversible? Do the temperature and pressure need to be constant in order for this to internally reversible? Does the piston need to be frictionless? Explain.

Solution:

The system should be defined as only the working fluid within the compression chamber. In order for this process to be internally reversible, the conditions within the cylinder need to be in equilibrium (quasi-static process) at each state in the compression process. The temperature and pressure within the system for this idealized process do not need to be constant over time. They just need to be uniform at each instant of time. Additionally, there can be no losses due to viscous dissipation within the fluid. There can be friction between the piston and cylinder wall because those losses are outside of the given system definition. Along these same lines, 100% of the compressor input work does not have to be recovered during expansion because the whole process is not necessarily reversible.

Part(v): Parallel Flow Heat Exchanger

Is it theoretically (thermodynamically) possible for a parallel flow heat exchanger with the inlet conditions shown to approach a thermodynamically reversible process? Explain why or why not? If yes, then show a system configuration that could be utilized to thermodynamically reverse the process.

Solution:

No, because there are irreversibilities associated with any heat transfer across a non-zero temperature differential. Regardless of heat exchanger efficiency or size, there will be irreversibilities immediately at the inlet due to the temperature differential of the two fluids.

Part(vi): Counterflow Heat Exchanger

Is it theoretically (thermodynamically) possible for a counterflow heat exchanger with the inlet conditions shown below to approach a thermodynamically reversible process? Explain why or why not? If yes, then show a system configuration that could be utilized to thermodynamically reverse the process.

Solution:

Yes, a reversible heat transfer can be approached as the heat transfer area goes to infinity with perfectly matched fluids such that the temperature differential between the two streams is always infinitesimal at each point from inlet to outlet. If the temperatures of the two streams are plotted together relative to position along the heat exchanger, the lines would be exactly on top of each other such that $T_{1,out}$ = 50 °C and $T_{2,out}$ = 30 °C.

From a system standpoint, two counter-flow heat exchangers of infinite surface area could be connected in a loop with reversible pumps and no frictional (pressure) losses. This is shown schematically below.