Where Does "P.dV" Work in Brayton Combustion Stage Go?

..."The thermodynamic cycle

The cycle consists of four processes: a) quasi-static adiabatic compression in the inlet and compressor, b) constant pressure heat addition in the combustor, c) quasi-static adiabatic expansion in the turbine and exhaust nozzle, and finally d) constant pressure cooling to get the working fluid back to the initial condition.

1 - 2 Adiabatic, quasi-static compression in inlet and

compressor

2 - 3 Combust fuel at constant pressure (i.e. add heat)

3 - 4 Adiabatic, quasi-static expansion in turbine

a. take work out and use it to drive the compressor

b. take remaining work out and use it to accelerate fluid for jet

propulsion, or to turn a generator for electrical power generation.

4 - 1 Cool the air at constant pressure

Then repeat

https://ocw.mit.edu/ans7870/16/16.unified/thermoF03/chapter_5.htm#braytonperformance

https://www.engineeringenotes.com/thermal-engineering/thermodynamics/thermodynamic-work-equations-formula-pdv-work-heat-pressure-and-temperature-measurement/48918

"non flow" work is obviously accomplished during the combustion stage, yes?

Are you implying, that when accounting for "flow" work, one finds that the fluid does no net PdV work?

3. PdV-Work:

Consider a system consisting of a gas in a cylinder fitted with a piston. During the initial condition of the piston i.e., when the piston is at position (1), the pressure inside the cylinder is P1 and volume is V1. Let the gas expands as the piston moves to position (2). Then the pressure falls to P2 and volume will increase to V2.

Now, consider a small movement of the piston dl during which pressure P is assumed to be constant. If a is the cross sectional area in m2, then the force acting on the piston is given by –

F = P x a

And the small amount of work done by the gas on the piston is causing the displacement

The initial and final states of the system can be represented on PV- diagram. The work done W1-2 will be equal to the area under the curve 1-2 on the PV-diagram.

https://www.engineeringenotes.com/thermal-engineering/thermodynamics/thermodynamic-work-equations-formula-pdv-work-heat-pressure-and-temperature-measurement/48918

http://web.mit.edu/16.unified/www/SPRING/propulsion/notes/node27.html

More is happening in the combustion chamber than just adding heat, there is obviously all of that PdV work being accomplished also (the top of the graph encloses positive area too)

https://en.wikipedia.org/wiki/Brayton_cycle

Here's my understanding. Refer to the PV diagram in the center.

A parcel of air enters the compressor. Work is done compressing it, 1->2 in the PV diagram. The air is heated by combustion in the chamber at constant pressure, 2->3. The air expands pushing the turbine, 3->4. Part of the turbine energy is used for compressing incoming air and part for external work (e.g. driving a generator). Finally, the heated air is released to the environment where it contracts and releases the heat, 4->1.

In the PV diagram, P is always positive. dV is positive moving left to right, negative moving from right to left. When dV is positive, work is being done by the air (3->4). When it is negative, work is being done to the air (1->2).

Yes, that's my understanding also. So the combustion stage accomplishes positive work, of an amount not incomparable to that accomplished by the expansion stage through the turbine.

Expansion stage positive work is easy to visualize -- the expanding gases are driving the rotation of the turbine blades, turning the drive shaft (output power) and draft shaft (pressurizer power).

Combustion stage positive work...is doing what exactly? Pushing against the sidewalls of the combustion chamber doesn't count (no "dV" if the walls are stiff). The only directions in which volume could change would be forward into the turbine or backward into the pressurizer.

So, positive PdV work accomplished by the combustion stage... is driving hot gases into the turbine inlet?? How and where does one account for that (significant) amount of energy???

The compressor compresses the air to the pressure of the combustion chamber. So there is no work done by the air expanding "backward into the pressurizer". The air is heated by combustion and expands. As the air expands under constant pressure, there is only one direction for it to go --- through the turbine. The work required to compress the air to the combustion chamber pressure is already accounted for on curve 1->2 on the PV diagram.

Realize that it isn't like a gasoline engine where intake - compression - combustion - exhaust take place sequentially. The turbine engine operates at steady-state where intake, compression, combustion, and exhaust are all taking place continuously.

A less formal way of looking at it is that the expanding gas flows in one direction because the compressor and turbine are designed differently. The expanding gas can more easily escape through the turbine and the turbine turns the compressor bringing in more air as shown in the diagram below.

https://www.britannica.com/technology/gas-turbine-engine

Please describe the (large) amount of positive work accomplished by the working fluid during the combustion stage (2 -> 3)

The energy provided by the burning of fuel allows the gas to perform work expanding in volume against a constant pressure.

https://www.khanacademy.org/science/physics/thermodynamics/laws-of-thermodynamics/a/what-are-pv-diagrams

Yes, and what receives the positive PdV work from the combustion gases? If positive work energy flows out of the working fluid, the energy must flow into something else, yes?

The gas drives the turbine which drives the compressor and whatever load to which the turbine engine is connected. (e.g. generator).

Going through the cycle 1->2->3->4->1..., ∫PdV, the area under the curve, is negative (work done to the gas) on path 4->1 and 1->2, and positive (work done by the gas) on path 2->3 and 3->4. Subtracting the work done to the gas from the work done by the gas leaves the net amount of work the turbine engine delivers to the load, the filled area inside the curves as shown.

Yes, of course

The positive PdV work accomplished during expansion through the turbine is easy to understand.

But what about the positive PdV work "under the isobaric combustion chamber line at the top of the cycle".

You are correct that it is positive work, but what is happening physically? Heated gases are expanding in the combustion chamber... and doing positive work... on what???

dU = dQ - PdV

You have noted dQ_in, as well as the increase in dU and positive PdV work accomplished by the gas...

but while you acknowledge dQ_in, and dU (temperature increase), what happens to the PdV? It is not dQ, it is not dU, and it does not vanish as if "swept under the calculation carpet".

If the math is right, the math is reflecting real tangible reality. So positive PdV work, mathematically, is doing exactly what physically, between the walls of the combustion chamber???

It goes into (specific) kinetic energy (1/2 v²) ??

But what about the positive PdV work "under the isobaric combustion chamber line at the top of the cycle".

You are correct that it is positive work, but what is happening physically? Heated gases are expanding in the combustion chamber... and doing positive work... on what???

Let me try to explain it a little better.

At 1, air at atmospheric pressure enters the compressor. The volume is reduced and the pressure increases to point 2.

The gas expands at constant pressure as fuel is burned adding heat (q in). PV=nRT, so temperature increases from point 2 to point 3.

After heat has been added, the gas continues to expand from point 3 to point 4. As it expands through the turbine, the pressure drops to atmospheric pressure at point 4.

As it cools in the atmosphere (q out), the volume of the gas decreases at constant pressure to point 1.

Again, that's an incomplete picture

dQ = dU + dW

The heat input increases the internal energy of the gas, with "room to spare"

the rest goes somewhere, according to the chart, into positive PdV work under the top curve of the chart

You keep skipping over that fact, again, yes, there's dQ input, but part of the dQ goes into internal energy & the rest somewhere else... where. does. the. rest. of. the. energy. go. which. does. not. go. into. internal. energy.?

Again, we're upstream of the turbine inlet (point 3), so we aren't talking about PdV work in the turbine... there is some sort of PdV work accomplished. in. the. combustion. chamber. also.

Using your logic, there would be Pdv work input during the cooling process external to the engine, no? Where/how does that take place?

Yes, very true, that is the difference between actual practice (open Brayton cycles) and hypothetical theory (closed Brayton cycles).

In an open Brayton cycle, the ambient atmospheric environment does the work, compressing the cooling exhaust gases, which thereby acquire energy (dW = -PdV < 0 by the fluid, >0 on the fluid)...

which they turn right around and dump back into the atmosphere (dQ = dU + PdV > dU) as heat. So, in an open cycle, you don't have to worry about re-compressing the gas, and the only thermal energy "lost" to the environment from the engine is the internal energy dU part. (Again, the dW piece is supplied from, and immediately returned as heat to, the atmosphere.)

For whatever reason, most textbooks seem to focus on the closed cycle version, although I have no exhaustive (no pun) knowledge of every Engineering Thermo book out there.

It isn't just <...air...>; it is <...air...> and fuel.

In the combustion of fuels there is an increase in the number of molecules in almost every case apart from hydrogen, hydrogen sulphide and methane. Therefore there is an increase in pressure and/or volume using fuels pother than these in excess of what would be expected from a simple change of temperature as a result of combustion.

It is this change that is used to extract shaft work from the equipment.

The <...pressurizer compressor...[sic]> is there to force more <...air...> and fuel into the equipment so that more combustion product molecules are created in the same time and space, which results in the availability of greater shaft work compared to similar naturally-aspirated equipment. The <...pressurizer compressor...[sic]> requires shaft work in order to operate, which is derived from the expansion turbine. It is the "energy cost of energy supply"; "get it in, get it burnt, and get it out".

The limit is usually determined by the structural constraints of the equipment, above any of which the risk of catastrophic and irreversible self-disassembly rises alarmingly.

Maybe, just maybe, it goes into the output shaft of the turbine?

Good thinking, yet wouldn't that then be the PdV in the expansion phase?

The math is saying, that positive PdV work, is accomplished inside of the combustion chamber, upstream of the turbine inlet (at the top right corner of the cycle as depicted on a PV graph), and downstream of the pressurizer outlet (top left corner of the same)

Note that I personally prefer the term "pressurizer" for a pressure-ratio process, and "compressor" for a volume-ratio process, otherwise you're using one word for two different thermo processes