A vertical piston-cylinder device initially contains 0.12 m^3 of helium at 20^∘ C. The mass of t

step 1 So in this problem, we have a vertical piston cylinder. So we got something like, you know, wher

A vertical piston-cylinder device initially contains 0.12...

A vertical piston-cylinder device initially contains $0.12 \mathrm{m}^{3}$ of helium at $20^{\circ} \mathrm{C}$. The mass of the piston is such that it maintains a constant pressure of $200 \mathrm{kPa}$ inside. $\mathrm{A}$ valve is now opened, and helium is allowed to escape until the volume inside the cylinder is decreased by one-half. Heat transfer takes place between the helium and its surroundings at $20^{\circ} \mathrm{C}$ and 95 kPa so that the temperature of helium in the cylinder remains constant. Determine ( $a$ ) the maximum work potential of the helium at the initial state and $(b)$ the exergy destroyed during this process.

Key Concepts

Exergy Analysis

Exergy analysis involves assessing the maximum useful work that can be extracted from a system as it is brought to equilibrium with its surroundings. This concept differentiates between energy and the quality or usability of that energy, taking into account the environmental or dead state as the reference point.

Maximum Work Potential

The maximum work potential is the theoretical upper limit of work that can be obtained from a system undergoing a reversible process. It represents the ideal extractable work when all losses and irreversibilities are absent, serving as a benchmark for evaluating the efficiency of real processes.

Exergy Destruction

Exergy destruction quantifies the loss of available work potential due to irreversibilities within a process, such as friction, throttling, or unrestrained expansion. This concept is fundamental for understanding how real processes deviate from the ideal reversible case and for identifying sources of inefficiency.

Reference Environment

The reference environment, or dead state, denotes the state of the surroundings with which a system eventually comes into equilibrium. It provides the baseline conditions (such as temperature, pressure, and composition) against which the system's exergy and work potential are measured.

Isothermal Process

An isothermal process occurs at a constant temperature. In such processes, any work done by or on the system is offset by heat transfer to maintain the temperature, which simplifies the analysis of energy and work interactions.

Constant Pressure Process

A constant pressure process is one where the pressure remains unchanged throughout the transformation. This simplification allows for direct calculation of work as the product of pressure and volume change, and it is significant in processes where external mechanical constraints, like a piston with a fixed weight, maintain a steady pressure.

Transcript

00:01 So in this problem, we have a vertical piston cylinder.

00:05 So we got something like, you know, where we have a piston here, and then it's got some weight.

00:11 So creating some pressure in here.

00:14 And we have helium inside it here.

00:16 We use red for helium.

00:19 So we have this helium in here.

00:22 And this pressure, this piston is going to keep it out of constant pressure.

00:27 The initial volume is a 0 .12 cubic medium.

00:32 It has initial temperature of 20 degrees c.

00:36 The pressure is maintained at 200 kilopascals by this piston with some weight on it or something, or maybe it just weighs that much.

00:46 And after the process, we have half the volume.

00:48 So the piston has dropped down to, you know, here or so.

00:52 Surrounding temperature and pressure.

00:55 And then we are told that, let's see here, heat is transferred to the helium, helium, so the temperature and the helium remains constant.

01:07 So we have t2 is t1.

01:11 So we can look up the ideal gas constants or ideal gas properties for helium, the ideal gas constant specific heats.

01:22 We can figure out the mass of the helium.

01:26 And so let's see.

01:29 Oh, sorry, we also, we open a valve.

01:32 I forgot that part of it.

01:34 We open a valve and let's some of this stuff out.

01:41 So we have the initial mass.

01:44 We have the pressure volume and temperature initially so that we get an initial mass.

01:49 We know the final mass is going to be because the temperature is the same and the pressure is the same.

02:00 We know the final mass is just gonna be half the initial mass because the pressure and temperature are the same and the volume is half.

02:11 So we have, again, half of this in the final state.

02:16 And that means that's how much we let released.

02:24 So let's see here, we want to get the maximum work potential.

02:28 So the maximum work potential, we can get by looking at the initial exergy of the helium.

02:35 So the initial exergy is the mass of the helium times the specific exergy.

02:40 And that's the mass of the helium times the change in internal energy from the ground state to where it is now, minus t not times the change in entropy from the ground state to where it is, plus p.

02:57 Not times the change in specific volume from the ground state to where it is now.

03:04 So we can figure out the specific volume in the initial state, and that winds up being about 3 cubic meters per kilogram.

03:13 In the ground state, again, we have all this information.

03:16 That winds up being about 4 .6 cubic meters per kilogram.

03:20 So it's compressed by this piston up here from the ground state...

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