Prandtl-Glauert Question
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Prandtl-Glauert Question
Never thought I'd be able to pose a question amongst such learned fellows, and I will perhaps expose my ignorance just a little.....but a little nagging question, that the answer will most likely be very obvious, but anyway....
Since heat does not leave the affected air mass, this is an adiabatic change of pressure, with an associated change of temperature. Under humid air conditions, the drop in temperature in the most rarified portion of the shock wave (close to the aircraft) can bring the air temperature below its dew point. At this temperature, moisture condenses to form visible microscopic water droplets. As the pressure effect of the wave is reduced by its expansion (the same pressure effect is spread over a larger radius), the vapor effect also has a limited radius.
The part I am struggling to grasp is ....,
"Since heat does not leave the affected air mass, this is an adiabatic change of pressure, with an associated change of temperature."
??Heat does not leave...but there is an associated change of temp???
Thanks in advance
Since heat does not leave the affected air mass, this is an adiabatic change of pressure, with an associated change of temperature. Under humid air conditions, the drop in temperature in the most rarified portion of the shock wave (close to the aircraft) can bring the air temperature below its dew point. At this temperature, moisture condenses to form visible microscopic water droplets. As the pressure effect of the wave is reduced by its expansion (the same pressure effect is spread over a larger radius), the vapor effect also has a limited radius.
The part I am struggling to grasp is ....,
"Since heat does not leave the affected air mass, this is an adiabatic change of pressure, with an associated change of temperature."
??Heat does not leave...but there is an associated change of temp???
Thanks in advance
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I see that your quote comes from Wikipedia.
The following link to Wikipedia might help you...
Adiabatic process - Wikipedia, the free encyclopedia
TCF
The following link to Wikipedia might help you...
Adiabatic process - Wikipedia, the free encyclopedia
TCF
isothermal v adiabatic
By definition an adiabatic change is one in which no heat energy enters or leaves the system of interest. Doing work on a gas (by compression) or having it do work (eg push a piston) results in energy entering or leaving the system (but not heat energy as such). If the system is in thermal contact with the outside the temperature change accompanying the work will result i heat entering or leaving until temperatures are equal. This would be an isothermal change. If the system is insulated (or the change happens so fast that heat cannot flow to an appreciable extent, eg sound waves or even convection in the atmosphere to a great extent), then the change is adiabatic and the temperature of the system will change and not equalise.
From memory, the pressure/volume law for the gas is different in the two cases, PV =constant for isothermal and PV^gamma =constant for adiabatic.
This should put you off the subject for good....
Pressure-Volume Diagrams - The Physics Hypertextbook
From memory, the pressure/volume law for the gas is different in the two cases, PV =constant for isothermal and PV^gamma =constant for adiabatic.
This should put you off the subject for good....
Pressure-Volume Diagrams - The Physics Hypertextbook
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I had the fortune to try and tutor my girlfriend (now wife) in Physics when we were undergrads. Learned very quickly that using big words, formulas, and assumptions is a very bad way to illustrate a point. Sorry if I insult your intelligence by simplifying too much.
Heat is energy. Temperature is often called the "average kinetic energy" of a substance, but I don't find that definition very intuitive. I prefer to think of it as "heat density" which is not a very good use of physics terms, but is accurate enough. Temperature is how much heat there is in a certain amount of matter. (The problem is that the scales don't actually relate to a heat density but are more a relative scale...enough said, doesn't help understanding).
If you compress a portion of matter into a smaller volume, you are also putting more heat energy into that volume and are increasing the "heat density" or temperature.
So the quote "Since heat does not leave the affected air mass, this is an adiabatic change of pressure, with an associated change of temperature." means "Since you aren't changing how much heat there is, but are compressing all that air into a smaller space, the temperature will rise."
Matthew.
Heat is energy. Temperature is often called the "average kinetic energy" of a substance, but I don't find that definition very intuitive. I prefer to think of it as "heat density" which is not a very good use of physics terms, but is accurate enough. Temperature is how much heat there is in a certain amount of matter. (The problem is that the scales don't actually relate to a heat density but are more a relative scale...enough said, doesn't help understanding).
If you compress a portion of matter into a smaller volume, you are also putting more heat energy into that volume and are increasing the "heat density" or temperature.
So the quote "Since heat does not leave the affected air mass, this is an adiabatic change of pressure, with an associated change of temperature." means "Since you aren't changing how much heat there is, but are compressing all that air into a smaller space, the temperature will rise."
Matthew.
Do a Hover - it avoids G
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Matthew
These heat/temperature/energy issues are tricky to explain - I quite agree.
As to your
I am not sure that is strictly accurate. Surely it depends o the nature of the matter? For example a polystyrene block the size of a housebrick will have much less heat in it than a standard housebrick when they are both at the same temperature.
So how about:
Temperature is about how much heat there is in a particular piece of matter?
JF
As to your
Temperature is how much heat there is in a certain amount of matter
So how about:
Temperature is about how much heat there is in a particular piece of matter?
JF
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John,
Completely agree.
I'm not a fan of simplifying things to enhance understanding at the cost of accuracy. Hence, I'm of the school that we should drop the term "gyroscopic precession" from all helicopter dynamics books.
I do like your temperature definition.
Cheers,
Matthew.
Completely agree.
I'm not a fan of simplifying things to enhance understanding at the cost of accuracy. Hence, I'm of the school that we should drop the term "gyroscopic precession" from all helicopter dynamics books.
I do like your temperature definition.
Cheers,
Matthew.
keep it classical
amount of heat = thermal capacity times temperature.
Based on equipartition of energy, amount of energy in a particle is 0.5kT per degree of freedom, so on this basis temperature relates to the amount of (dynamic) energy per particle.
Heat, like gravity, doesn't stand too much thinking about.
Based on equipartition of energy, amount of energy in a particle is 0.5kT per degree of freedom, so on this basis temperature relates to the amount of (dynamic) energy per particle.
Heat, like gravity, doesn't stand too much thinking about.
Do a Hover - it avoids G
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Mr.Optimistic
Matthew and I are chatting about how you help people comprehend concepts.
Not quite the same thing as stating facts and relationships between facts.
Teaching can be a tricky and necessarily progressive business.
Not quite the same thing as stating facts and relationships between facts.
Teaching can be a tricky and necessarily progressive business.
I was only agreeing with you and I thought tidying up a bit
but my son has made a similar point from time to time.
So for the O/P,
The part I am struggling to grasp is ....,
"Since heat does not leave the affected air mass, this is an adiabatic change of pressure, with an associated change of temperature."
??Heat does not leave...but there is an associated change of temp???
In this case the starting point I guess had probably to start with the relationship and meaning of energy/work/heat and how you cannot talk about the quantity of heat or work in a substance but you can talk about energy.
Compress a gas in an insulated container and it will increase in temperature on account of the work done on it. Repeatedly bend a paper clip till it breaks and it will be much hotter than its surroundings etc. All work to me.
So
a) energy can enter a substance without flow of heat (by 'working it')
b) adiabatic means no more than insulated to prevent heat flow
c) so if the 'adiabatic change in pressure' was a compression then work was done on the gas (eg by something pushing a piston against the resisting pressure) and it will get hotter: if the gas expands and looses pressure it will cool -it's done work by pushing other stuff out of the way to claim the extra space.
I thought that did it..........or have I lost track ?
So for the O/P,
The part I am struggling to grasp is ....,
"Since heat does not leave the affected air mass, this is an adiabatic change of pressure, with an associated change of temperature."
??Heat does not leave...but there is an associated change of temp???
In this case the starting point I guess had probably to start with the relationship and meaning of energy/work/heat and how you cannot talk about the quantity of heat or work in a substance but you can talk about energy.
Compress a gas in an insulated container and it will increase in temperature on account of the work done on it. Repeatedly bend a paper clip till it breaks and it will be much hotter than its surroundings etc. All work to me.
So
a) energy can enter a substance without flow of heat (by 'working it')
b) adiabatic means no more than insulated to prevent heat flow
c) so if the 'adiabatic change in pressure' was a compression then work was done on the gas (eg by something pushing a piston against the resisting pressure) and it will get hotter: if the gas expands and looses pressure it will cool -it's done work by pushing other stuff out of the way to claim the extra space.
I thought that did it..........or have I lost track ?
never did like thermodynamics
and am now remembering why. It shares something in common with fluid mechanics in that 'engineering' came first and science followed: I reckon that always complicates the basis. Remember a thermodynamics text book which opened with the caution that a high proportion of the subjects 'founding fathers' died by their own hand...........
Of course in 'pure physics' they change the rules as they go along (eg electromagnetic waves have momentum: so preserving the conservation of momentum, define a new quantum number if an 'allowed' transition doesn't work, and most egregious of all invent 'the ether' with convenient properties to hide the fact that the theories aren't working. We all laugh but dark matter + dark energy = the 21st century's take on the ether.
Of course in 'pure physics' they change the rules as they go along (eg electromagnetic waves have momentum: so preserving the conservation of momentum, define a new quantum number if an 'allowed' transition doesn't work, and most egregious of all invent 'the ether' with convenient properties to hide the fact that the theories aren't working. We all laugh but dark matter + dark energy = the 21st century's take on the ether.
Since I can't resist:
from the first law dU=dw + dq...since the sum of heat and work only depends on the final and initial state although the type of work and the amount of heat used in the process are dependent upon the path taken to the final and initial state the total energy change remains constant..
so,
I can assume all the change is entirely adiabatic so the 'q' term disappears for an adiabatic change since U is a state function therefore I can use the equation for isothermal work since if 'dq'= 0 then work done adiabatically =work done isothermally wad=wIsT and since w= -pdV and adiabatic work = CvdT [or the product heat capacity at constant volume and the infinitesimal temperature change].
Equating the two terms wad =wIsT we have,
CvdT =-Pdv
From the ideal gas law p=nRT/V and dividing both sides by 'T'
we have CvdT/T =-nRdV/V
integrating both sides between final and initial state of 'T and V'
we have Cv ∫ dT/T =-nR ∫dV/V
therefore,
Cv*ln[Tf/Ti ]=-nR* ln[Vi/Vf]
since the ratio 'gamma' is given by Cp/Cv and Cp-Cv =nR; it follows that gamma = Cv/nR which is thus obtained by dividing both sides by -nR
finally,
we have gamma*ln [Tf/Ti ]= ln[Vi/Vf] raising both sides to 'e' which subsequently cancels; one obtains the adiabatic gas law [Tf/Ti]^gamma =[Vi/Vf]
this equation is used to describe the relationship between pressure [since it can be related to T and V], volume and temperature during adiabatic processes...and so the mysterious 'gamma' is finally revealed...
Here's a good thermodynamics lecture I found
Happy New Year PPRuNe....
from the first law dU=dw + dq...since the sum of heat and work only depends on the final and initial state although the type of work and the amount of heat used in the process are dependent upon the path taken to the final and initial state the total energy change remains constant..
so,
I can assume all the change is entirely adiabatic so the 'q' term disappears for an adiabatic change since U is a state function therefore I can use the equation for isothermal work since if 'dq'= 0 then work done adiabatically =work done isothermally wad=wIsT and since w= -pdV and adiabatic work = CvdT [or the product heat capacity at constant volume and the infinitesimal temperature change].
Equating the two terms wad =wIsT we have,
CvdT =-Pdv
From the ideal gas law p=nRT/V and dividing both sides by 'T'
we have CvdT/T =-nRdV/V
integrating both sides between final and initial state of 'T and V'
we have Cv ∫ dT/T =-nR ∫dV/V
therefore,
Cv*ln[Tf/Ti ]=-nR* ln[Vi/Vf]
since the ratio 'gamma' is given by Cp/Cv and Cp-Cv =nR; it follows that gamma = Cv/nR which is thus obtained by dividing both sides by -nR
finally,
we have gamma*ln [Tf/Ti ]= ln[Vi/Vf] raising both sides to 'e' which subsequently cancels; one obtains the adiabatic gas law [Tf/Ti]^gamma =[Vi/Vf]
this equation is used to describe the relationship between pressure [since it can be related to T and V], volume and temperature during adiabatic processes...and so the mysterious 'gamma' is finally revealed...
Here's a good thermodynamics lecture I found
Happy New Year PPRuNe....
Last edited by Pugilistic Animus; 11th Sep 2023 at 18:29. Reason: add 'dq' instead of 'q'
PA
Good lecture.
Was it made in an adiabiatic or isothermal environment though?
Good lecture.
Was it made in an adiabiatic or isothermal environment though?
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PA, take it you had no party invites this year ?
PA, take it you had no party invites this year ?
now I have been to some crap parties
but can't remember one with a thermodynamic theme, but maybe I was distracted by the other sex and missed the seminars. Could explain a lot.
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The three laws of thermodynamics:
You cant win
You cant break even
You cant get out of the game.
That combined with the other two fundamental engineering laws explains most of life:
F=MA and "you cant push a string" *
* unless wet and frozen.
You cant break even
You cant get out of the game.
That combined with the other two fundamental engineering laws explains most of life:
F=MA and "you cant push a string" *
* unless wet and frozen.
