Consider a 20-L evacuated rigid bottle that is surrounded by...

Consider a 20-L evacuated rigid bottle that is surrounded by the atmosphere at $100 \mathrm{kPa}$ and $25^{\circ} \mathrm{C}$. A valve at the neck of the bottle is now opened and the atmospheric air is allowed to flow into the bottle. The air trapped in the bottle eventually reaches thermal equilibrium with the atmosphere as a result of heat transfer through the wall of the bottle. The valve remains open during the process so that the trapped air also reaches mechanical equilibrium with the atmosphere. Determine the net heat transfer through the wall of the bottle and the exergy destroyed during this filling process.

Key Concepts

Rigid Container (Constant Volume Process)

This concept refers to a container whose volume remains fixed during the entire process. In thermodynamics, a process occurring in a rigid container means that, while mass and energy may change, there is no work done due to volume change. This directly simplifies the energy balance since the boundary work term is zero.

Thermodynamic Equilibrium

Thermodynamic equilibrium is the state where a system has uniform properties (temperature, pressure, chemical composition) and no net flows of matter or energy occur within or across its boundaries. In the context of the problem, the final state of the air inside the container reaches both thermal equilibrium with the surroundings and mechanical equilibrium (equal to atmospheric pressure), ensuring that energy and exergy analyses are based on a well-defined reference state.

First Law of Thermodynamics (Energy Balance)

The First Law of Thermodynamics, known as the conservation of energy, is applied to assess the energy changes in the system. It involves accounting for energy added or removed from the system as heat or work. In processes such as air being admitted into an evacuated bottle, the energy analysis examines how heat transfer through the walls balances the changes in the internal energy of the gas, even when no boundary work is done.

Exergy Analysis

Exergy analysis evaluates the maximum useful work that can be extracted from a system as it reaches equilibrium with its environment. It measures the quality or usefulness of energy and accounts for the irreversibilities that occur during a process. This analysis helps in identifying the lost work potential (exergy destruction) due to factors like friction, unrestrained expansion, and heat transfer across a finite temperature difference.

Entropy Generation and Irreversibility

Entropy generation is a measure of irreversibility in a thermodynamic process and is directly related to exergy destruction. In any real process, departures from ideal reversible conditions lead to an increase in entropy. Calculating the entropy change due to heat transfer and mixing allows one to quantify the exergy destroyed, giving insight into the efficiency loss associated with the process.

Transcript

00:04 So this is an interesting problem that will have a little discussion about, well, no, not this problem, i guess.

00:13 Maybe it's the, yeah, it's coming up here.

00:18 It's not this one.

00:19 I thought it was a, oh yeah, it's 118, not 114.

00:25 Anyway, this is just a simple little problem.

00:28 But we have a volume, a bottle that has a volume of 0 .02.

00:35 Cubic meters.

00:37 It is empty, evacuated, vacuum inside here.

00:42 The air, the surrounding air, is at 20 degrees c and 100 kilopascals.

00:49 M1 is, of course, zero.

00:50 There's nothing in the tank to begin with.

00:55 So we open it up on the air flow in.

00:59 So the amount of air that goes in is the amount of air that's in the tank after it fills.

01:06 And let's see here.

01:07 We know then that the that the temperature after it feels and the pressure after it feels, it's equilibrated with the air and we have some heat transfer out.

01:19 So it all comes back, you know, it winds up being at surrounding temperature and surrounding pressure in the end.

01:27 So we know the energy in minus the energy out is the change in energy in the system.

01:33 So we have minus q out, the heat transfer that occurred, plus the entropy, the transfer of entropy in, and that has to equal the change in internal energy, which there was no initial internal energy, so that's just the internal energy in the end...

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