The air parcel expands as it rises inside a pipe

[Aleksej]

The air parcel expands as it rises and this expansion, or work, causes the temperature of the air parcel to decrease.

Imagine that parcels of _dry air_ go up inside the pipe that goes vertically from 0 to 5000 meters.
Suppose that they go up because they are slightly warmer that the air outside.

Situation 1

The pipe internal diameter grows from bottom to the top so the area of cross-section grows proportionally to the decline of pressure outside.
In this case the speed of parcels is constant along the pipe. The parcels expand.
Which mechanical work will they do and will their temperature decrease ?

Situation 2

The pipe internal diameter is constant.
In this case the speed of parcels grows along the pipe. The parcels expand up?
Does this mean that the parcels do mechanical work of pushing upper parcels up?
Does this mean that mechanical energy of parcels grows at the expense of their internal energy ?
Will the parcels be colder then in Situation 1 ?

 

[jartsa]

Situation2: The parcel expands vertically, pushes all the parcels above upwards, while standing on the parcels below.

Situation1: The parcel expands horizontally, pushes all the parcels above upwards, while standing on the parcels below and on the pipe 'walls', which in this case are a little bit 'floors' too.

(But I think that if we want a constant speed of parcels, then the area of the cross-section of the pipe needs to grow proportionally to the decline of density of the gas inside.)

 

[Aleksej]

(But I think that if we want a constant speed of parcels, then the area of the cross-section of the pipe needs to grow proportionally to the decline of density of the gas inside.)

Yes, it is what I mean.

Do we have any extra mechanical energy in Situation 2 ?
If yes - where did it come from ?

 

[jartsa]

Yes, it is what I mean.

Do we have any extra mechanical energy in Situation 2 ?
If yes - where did it come from ?

The parcels expand same amounts in both situations, and they cool the same amount as they expand.

(If we take the point of view of the parcel, then in case2 a parcel pushes on pipe walls that do not move to the direction of the push, while on case1 a parcel pushes on pipe walls that do move to the direction of the push)

 

[Aleksej]

The parcels expand same amounts in both situations, and they cool the same amount as they expand.

(If we take the point of view of the parcel, then in case2 a parcel pushes on pipe walls that do not move to the direction of the push, while on case1 a parcel pushes on pipe walls that do move to the direction of the push)

What about extra speed of parcels in Situation 2 ?

 

[jartsa]

What about extra speed of parcels in Situation 2 ?

I don't know. In Situation1 we have a flow in an expanding pipe, I have some faint idea that according to the Bernoulli's law in such case some of the kinetic energy of the flow gets converted into some other energy ...

 

[Aleksej]

(If we take the point of view of the parcel, then in case2 a parcel pushes on pipe walls that do not move to the direction of the push, while on case1 a parcel pushes on pipe walls that do move to the direction of the push)

No move - no work. So all work is done in vertical direction in case 2
In case 1 - all additional volume is at the sides, perhaps all work is done in horizontal direction.

 

[jartsa]

No move - no work. So all work is done in vertical direction in case 2
In case 1 - all additional volume is at the sides, perhaps all work is done in horizontal direction.

The air parcel is doing work in some kind of exercise machine. That's why 'direction of work' can be said to be horizontal, or it can be said to be vertical.

Like a person working this machine does vertical work, or horizontal:
https://en.wikipedia.org/wiki/Draisine#/media/File:Drezyna_442.jpg

 

[Aleksej]

The air parcel is doing work in some kind of exercise machine. That's why 'direction of work' can be said to be horizontal, or it can be said to be vertical.

Like a person working this machine does vertical work, or horizontal:
https://en.wikipedia.org/wiki/Draisine#/media/File:Drezyna_442.jpg

Then, the work was done, energy was lost, where is that energy now ?
At some stage it was supposed to be the mechanical energy of an expanding parcel.
What happened to it next?

 

[davenn]

Then, the work was done, energy was lost, where is that energy now ?
At some stage it was supposed to be the mechanical energy of an expanding parcel.
What happened to it next?

radiated away as heat

 

[jartsa]

Then, the work was done, energy was lost, where is that energy now ?
At some stage it was supposed to be the mechanical energy of an expanding parcel.
What happened to it next?

I said earlier that the air parcel pushes down on the inclined walls of the expanding pipe. I thought that it might be important. Well it is important:

It takes energy to push downwards on a pipe, when said pipe is moving downwards.

Or it takes energy to push oneself upwards, when one is moving upwards relative to the thing that one uses as a step.

(I said earlier that from the point of view of the air parcel it does work by pushing on the walls that are moving apart horizontally. Well that is still true, from the dumb parcel's viewpoint)

 

[jartsa]

radiated away as heat

??

What radiates and why?

 

[256bits]

Then, the work was done, energy was lost, where is that energy now ?
At some stage it was supposed to be the mechanical energy of an expanding parcel.
What happened to it next?

Pipes are open to the atmosphere at the top I presume.

The problem you face with the expanding pipe area is that you have no function to describe it.
Except as mentioned as some function of change in pressure, which at the present state in the discussion is unknown.

The atmosphere can be modeled as an adiabatic expansion, or an isothermal expansion as we move up in height.
Reality is somewhere in between.

 

[davenn]

??

What radiates and why?

read all the post !

 

[sophiecentaur]

That's why 'direction of work' can be said to be horizontal,

Work is a scalar product (W= D.F) and does not have a 'direction'.

 

[jartsa]

read all the post !

The post is just 4 words long. I guess some bug deleted some of it.

 

[Aleksej]

I said earlier that the air parcel pushes down on the inclined walls of the expanding pipe. I thought that it might be important. Well it is important:

It takes energy to push downwards on a pipe, when said pipe is moving downwards.

Or it takes energy to push oneself upwards, when one is moving upwards relative to the thing that one uses as a step.

(I said earlier that from the point of view of the air parcel it does work by pushing on the walls that are moving apart horizontally. Well that is still true, from the dumb parcel's viewpoint)

Do you mean the vertical speed will grow in case 1 ?
It is like you put horizontal spring at the bottom of the conical funnel that expands from bottom to the top
and the spring extends sideways and moves up inside the funnel because it pushes on the sloped walls ? :-)

 

[Aleksej]

Pipes are open to the atmosphere at the top I presume.

Yes

The problem you face with the expanding pipe area is that you have no function to describe it.
Except as mentioned as some function of change in pressure, which at the present state in the discussion is unknown.

As we try to keep a constant speed of parcels then "the area of the cross-section of the pipe needs to grow proportionally to the decline of density of the gas inside".
The pressure is almost equal to outside so the density declines with the altitude as in "normal" atmosphere.
The walls are thin and the heat exchange is possible.

 

[Aleksej]

Work is a scalar product (W= D.F) and does not have a 'direction'.

In physics, a force is said to do work if, when acting, there is a displacement of the point of application in the direction of the force.

 

[Doc Al]

@Aleksej
Read that wiki quote carefully: While force and displacement have direction, work does not.

 

[jedishrfu]

The problem is ill defined as it stands and hence any conjectures derived to answer it have to make assumptions as to what may happen

Rather than continuing the discussion, it’s time to close the thread. Thank you all who contributed. For the OP, you have enough here to ponder your problem more and can begin to use some explicit math to come up with an answer.

Closing thread now.

Jedi