Shock wave formation.

[twentysixpointfive]

I understand how and why the air above a wing can go supersonic when the aircraft is subsonic, but what exactly causes the actual shockwave to form where it does? Is it caused by the decreasing camber of the wing further aft causing the air to once again decrease to subsonic speeds?

I understand why the air increases above Mach 1, but what causes the associated slowdown and shockwave? :ugh:

[Love_joy]

what exactly causes the actual shockwave to form where it does?

Its been a while since I looked at supersonic flow profiles, but from memory - the shockwave forms at the point on the structure where the airflow first, or just after, reaches M1.

A good reference for all of this gumph is 'Aerodynamics for the Naval Aviator', its ancient and new copies look like they are photocopies of photocopies, but the content is very good.

When I was in Uni looking at all this, someone once mentioned that a B747 wing creates supersonic airflow near the wing root at high speed cruise - can anyone confirm or scupper this rumor?

[Pugilistic Animus]

Yeah, that is a great basic text on the topic-good reference:ok:

PA

[gr8shandini]

Actually, the shock forms when the airflow starts to slow down again, not when it first goes supersonic. Check out Brian's post in the "I spy" thread that just popped up on the front page again. He's got a good explaination and a cool link with some animations.

[twentysixpointfive]

Thanks...although to be honest, I'm still not sure that I understand the cause of the air slowing down to below supersonic speeds. I know it does, and I know that is what causes the shockwave, but what is the cause of it slowing down?

[barit1]

The fastest flow (lowest static pressure) is over the first 10-20% of the upper surface.

But the shock wave forms, I believe, from the area rule effect (cross-section of the wing + fuselage increasing around the center section). When sitting on the shady side of the fuselage, you can often see the shadow of the shock wave on the wing surface. It dances around a bit in light turbulence. I think there are photos of same somewhere in pprune.

[gr8shandini]

Thanks...although to be honest, I'm still not sure that I understand the cause of the air slowing down to below supersonic speeds. I know it does, and I know that is what causes the shockwave, but what is the cause of it slowing down?

Well, there's two ways you can think about it. The first is that an airplane is designed to slip through the air leaving a minimal wake, right? Well, in order for that to happen, the air that passes over the upper surface of the wing has to eventually rejoin with the air that passed under the wing. So if you accelerate the air over the upper surface to produce lift, it eventually has to slow down to rejoin the flow that went under the wing relatively unchanged.

If you want to get slightly more technical, you can think of a venturi tube. The mass flow through the venturi has to be equal at all points along the tube. Mass flow can be described as the density times the area of the cross section of the tube times the velocity of the flow (rho*A*V). So in incompressible flow, if the area of the tube is decreasing, the velocity must increase to keep the mass flow constant, and vice versa. Now think of expanding this venturi wider and wider, until at some point, the air passing through the center doesn't even know it's flowing through a tube yet the air near the wall still has to accelerate / decelerate to keep the average mass flow constant. At this point, it doesn't even matter if the other side is there, right? Well that's the case with your wing.

Phew, that's a lot of setup. Anyway, keeping the venturi in mind, air is accelerating over the wing whenever the area is decreasing (i.e. the distance between the surface and the chord line is getting larger) and slowing when the area is increasing. Now things get more complicated as you go faster because the density can change and that's where the shockwaves come in. However, the air eventually has to return to the freestream condition and hence ultimately be going slower than it was over the top of the wing.

I hope that made some kind of sense to somebody. If you want to know more, pick up a copy of "Aerodynamics for Naval Aviators." It was written so that even a squid can understand so it's probably much easier to digest than what I posted.

Edit: Oops. Looks like lovejoy already suggested AfNA. In that case, I give that notion a hearty second.

[Brian Abraham]

twentysixpointfive - wiki comes in for a lot of flak from some people, but take a look here and see if it answers your question satisfactorily. If not, come back and I'll try harder.
http://en.wikipedia.org/wiki/Shock_wave

[twentysixpointfive]

Ah! The example with the venturi makes it clear as day. Thanks for taking the time to explain it all.
I've got a copy of Aerodynamics for Naval Aviators, but I couldn't really find a good explanation for my exact question in there, unless I missed it. I did however, realize where the author of Fly The Wing got all of his illustrations! ;)

[Pugilistic Animus]

Hurt's text is a true classic but his wording--is very sophisticated and one topic build upon the other---it's not really the type of text you go to for quick answers---but if you read chapter one 'basic aerodynamic--he goes over EAS and compressibilty as well as the the relationship between dyanamic and static pressure--without that his later chapter on high speed aerodynamics is nearly incomprehensible---but with out his satbility and control chapter---those aspects of high speed aerodynamics is also nearly incomprehensible---you MUST READ it beginning to end--or you wont gwet it---when I say basic I don't mean superficial---I mean foundational---later courses--will give you all the Blade element theroy you want---the book is very helpful for begining aerodynamics students but---you don't get easy answers--H. H Hurt can really make it HURT:}

PA

[glad rag]

Fascinating subject and gives the chance to air this old image again!

http://home.comcast.net/%7Ebzee1a/DSC_7901a_crop.jpg

[Pugilistic Animus]

Oooh a Mach cone:8

[gr8shandini]

Oooh a Mach cone:8


Yup. But if this post were truly worthy of that geek smiley, you'd have whipped out the protractor and shock tables and told us how fast that Hornet was going.

[slip and turn]

Well I havent got my protractor or any shock tables handy but I am estimating the angle is around 60 degrees and conclude the speed must have been around M1.15 - how'd I do? If I am right then I learned it here in the Tech Forum on PPRuNe :8

[airfoilmod]

I learned almost everything I know about airflow listening, not reading, so what follows is highly suspect. First my brother was present during that pass and swears it was subsonic.

Everything about airflow comes from area ruling, as suggested above. And Bernoulli and his "inviscid flow". There is a very substantial eddy at wing root and fuselage, it can be impenetrable dependent on available thrust. When the F-102 was launched as a supersonic fighter, it disappointed. Something was keeping it from Mach 1 despite all the calculations. Turns out a "slice" through the A/C at this barrier of air was representative of more cross section than 24 very obese opera singers. What the Tunnel taught the builders of the F-106 (the new 102), was to pinch the body at the wing root (coke bottle). This allowed the "dam" of viscid air to reduce and allow the supersonic region of airflow past.

Back to the Hornet. The Wave you see is most likely the "tail event" or the first of two potential "Booms" passing through Mach1. The larger pic would likely show the "nose event" wave or #2 Boom. When there are no shock waves of note attached to A/C it is assuredly smokin' nuttin'.

Bernoulli's work suggests that air in an infinite enclosure is "Incompressible" below a speed of 267(?) knots. Above that speed and it becomes compressible, both against itself and solids (wings, tails,etc). I think it is the leading edge of this "packed" air that slows the faster air down and creates those beautiful cones.

If this is accurate, It is due to memory of Hoser and Coon talking after several beers. If it's not I'll track them down and wring their necks.

Airfoil

[Pugilistic Animus]

Gr8shandini,---
Ok, Ok, I forgot how rough the crowd here can be:}---no I'm not going to derive it---can't do good math on Pprune

but it's simply---M=CSC[mu].....mu= angular Mach come deflection simple vector math---now may I? :8

any book in advanced aero eng has this though---so don't accuse me of wiki/google---I like McGraw Hill

PA

[Pugilistic Animus]

Der, forgot to post my answer I'm estimating about 45 deg CSC45=1.17M:8:8:8


PA

[barit1]

Just to follow up on airfoilmod's post -

The area rule (http://history.nasa.gov/SP-4219/Chapter5.html) effect has been credited to Dr. Richard Whitcomb, although others paved the way. It was Whitcomb's work that solved the XF-102 transonic drag problem - implemented on production versions of the F-102 (and F-106, of course, although a bigger airframe). I had a close colleague who worked on these.

[Brian Abraham]

Oooh a Mach cone
Afraid not. The mach cone develops when the aircraft itself reaches Mach 1. What you see in the photo is the shock wave from airflow which is supersonic at that particular point on the airframe. The aircraft itself is travelling at less than Mach 1. If it were at Mach 1 or greater you would see the mach cone (which develops at the nose of the aircraft) impinging on the water.

[airfoilmod]

As I recall, the optical effect at the boundary of Hi/Lo pressure isn't to do with relative pressure, But HEAT. Also, consider this manouver is happening @ ~600knots in GROUND EFFECT (Bay effect?). At that speed and 28 feet altitude, one PUSHES on the Stick to get down to that level.The Blues are a cut above.

Airfoil

[Brian Abraham]

optical effect at the boundary of Hi/Lo pressure isn't to do with relative pressure, But HEAT.
Half right, half wrong. :p In a shock wave the properties of the fluid (density, pressure, temperature, velocity, Mach number) change almost instantaneously.

[airfoilmod]

Half-Full? OK, then I'm saying at that moment, Heat is the salient factor.

Airfoil

[Brian Abraham]

Sorry airfoil, but pressure changes are the driving force behind the whole process. As an object moves through a gas, the gas molecules are deflected around the object. If the speed of the object is much less than the speed of sound of the gas, the density of the gas remains constant and the flow of gas can be described by conserving momentum, and energy. As the speed of the object approaches the speed of sound, we must consider compressibility effects on the gas. The density of the gas varies locally as the gas is compressed by the object.

For compressible flows with little or small flow turning, the flow process is reversible and the entropy is constant. The change in flow properties are then given by the isentropic relations (isentropic means "constant entropy"). But when an object moves faster than the speed of sound, and there is an abrupt decrease in the flow area, shock waves are generated in the flow. Shock waves are very small regions in the gas where the gas properties change by a large amount. Across a shock wave, the static pressure, temperature, and gas density increases almost instantaneously. The changes in the flow properties are irreversible and the entropy of the entire system increases. Because a shock wave does no work, and there is no heat addition, the total enthalpy and the total temperature are constant. But because the flow is non-isentropic, the total pressure downstream of the shock is always less than the total pressure upstream of the shock; there is a loss of total pressure associated with a shock wave. Because total pressure changes across the shock, we can not use the usual (incompressible) form of Bernoulli's equation across the shock. The Mach number and speed of the flow also decrease across a shock wave.

[Been Accounting]

If the light is in the right direction you can see a very faint shock wave on the upper surface of an A320 wing in cruise.

I assume the pressure changes cause some refraction effects which appear as a shadow of the shock wave.

[Brian Abraham]

Many of you will have seen the following photo, some thing similar either in photo or in person at an airshow. What you are seeing is not a Mach cone, but a Prandtl-Meyer expansion fan. The angle of the fan can be used to calculate the Mach of the airflow producing the fan, but not the Mach of the aircraft - important difference. When an object moves faster than the speed of sound, and there is an abrupt decrease in the flow area, shock waves are generated. If the flow area increases, however, a different flow phenomenon is observed. If the increase is abrupt, we encounter a centered expansion fan. You can see an expansion fan on top of the canopy as the flow area suddenly increases. The expansion fan on the fuselage/wing is due once again to the flow area increasing. The word "expansion" denotes that the area is increasing. There are some marked differences between shock waves and expansion fans. Across a shock wave, the Mach number decreases, the static pressure increases, and there is a loss of total pressure because the process is irreversible. Through an expansion fan, the Mach number increases, the static pressure decreases and the total pressure remains constant.
http://i101.photobucket.com/albums/m56/babraham227/f18.jpg

[airfoilmod]

You say there is no increase in energy? That wants explanation. I am limited in the extent of my mathematical prowess, but believe I understand the inviscid nature of "air" below the defined value of ~260 knots. First, an increase in pressure produces an immediate increase in temperature, (Boyle?), and pressure waves are present attached to any A/C travelling through air faster than 260 knots, well below Mach 1 (IAS). Consider that in the case of the Hornet flying over San Francisco Bay, it is manouvering in ground effect, which impacts the compressibility of the ambient air to begin with. Thirty feet above its position, the Pressure drop might well have been invisible. My simple understanding is this: As a mass transits an air mass, it compresses it, the air having less time to maintain its density (low) before establishing a high pressure Flow about the airframe. As the air re-enters the "hole" caused by the A/C in the air behind it, it accelerates and produces an immense release of the energy it was forced to contain during the brief transit of the A/C. This "Thunderclap" is one of two separate events precipitated by the energy of the passing A/C. "Shock" by definition means an abrupt change in "stasis" and is accompanied by all manner of fascinating phenomena, begging varying degrees of explanation, from simple to complex. This isn't a defense of my interpretation, merely my description of my understanding of a sonic event from my perspective. If you like, I will certainly admit that Pressure Differential caused that neat picture.

At my age, I'm stuck in the way I learn. I don't use Wiki, I think it diminishes the quality of an exchange. Besides, I have seen glaring errors in it, and stubbornly hold to my method of writing my interpretation, and then subjecting myself to the challenges of others, having to defend or acquiesce as necessary. Further, I don't hesitate to challenge data presented by others; I've had to eat humble pie more times than I care to count. This may be one of those times.

BRGDS, Airfoil

[chornedsnorkack]

All the refraction effects come from the change of air refractive index.

The refractive index depends on two things - density and composition.. And density depends on temperature and pressure.

If you look at the air above an open fire, or any heated object, the column will refract light. You can see it in distortion of objects behind, as well as in a shadow which it casts in external light.

The pressure is almost the same, and when the object is heated from its heat capacity, electricity etc., the composition also is the same. Different temperature means different density and refractive index.

But when you deal with air affected by wing - well, the temperature certainly changes. In moist air, fog forms in the parts where air is rarefied and cooled, like above wing and in engine inlets. Obviously the density of air decreases - and in clear dry air, the visual distortion should also be seen. But so long as the speeds are low and airflow subsonic, the distortion is continuous and hard to see.

[Pugilistic Animus]

Afraid not. The Mach cone develops when the aircraft itself reaches Mach 1.


Brian Abraham, M 1.17---is still sonic because every part of the fighter has to reach M 1.00

I forgot the exact range but supersonic isn't until like M 1.5???...So, yeah a
Mach cone---


PA

[airfoilmod]

Woh. Now I'm right out of my comfort zone. I don't want to get into Einstein, not here, not now. Which part of the fighter is super, which part not? Are you saying that Mach 1 is a different velocity for different parts of the A/C?. Even if that is so, by definition, Mach 1 has a discrete value relative to the whole A/C, No? Boggling. Anyone for Focal Calculus?

Airfoil

[Pugilistic Animus]

Hey, Airfoilmod--not exactly--I'm saying different part become sonic at different speeds-high perf.aircraft..DC-9s, P51 Mustangs, F18's will achieve upon parts of its airframe both regions of sonic and supersonic flow...as long as they fly near sonic velocities...-- --the main reason [regarding the wing] is sweep which decreases the length of the velocity vector along the wing-----an important concept is the M-crit which is when the very first supersonic flow is encountered on the airframe--[they learn this i n the wind tunnel]---it is similar, in concept, to a propeller tip or chopper blades where [in this case ] the angular velocity is higher at the tip than the roots--as a result---speed is limited because the supersonic flow creates a good deal of drag--


Well because of this all of the defined regions have a value higher than the minimum--to ensure that the whole airframe is 'immersed' in a sonic/supersonic or hypersonic flow---


PA

[Pugilistic Animus]

Remember aerodynamics is an empirical and experimental science.... not theoretical


PA

[airfoilmod]

Pug. But you describe "regions" of airflow, of course airflow has different velocities as it transits the Airframe. The airframe has one single velocity because it is a solid, and cannot by definition have more than one. I read your "parts of the fighter" as implying there was a possibility of a plural Mach value. That is by definition ridiculous. Do I misunderstand your reply? I don't find the "plural value of Mach" addressed in it relative to the A/C, not the Airflow.

Airfoil

Aside. How can Mach 1 be "About Mach 1.5"?

[Pugilistic Animus]

I'm not takin about Mach meter indications--I'm mean the airframe parts of your ERJ-145 737/A320---are sonic at M values at an IMN less than M 1.00,

with that in mind reread my post, I'm just an aerodynamicist

Back to my my Anna Gabriel'
'Mexico Lindo'

ay ay ay ay canta y no llores, porque cantando se alelgran cielito lindo los corazones---- ese lunar que tienes cielito lindo----:ok:


Old Smokey or MFS Help!:\

PA

[airfoilmod]

The airframe parts are sonic(?) at indicated Mach Number <1.0.

Alrighty then.

A donde me llaves, Senor?

[Pugilistic Animus]

Exactamente!----te llevE aqui:}


busques en Pprune por 'critical Mach number'

also Old Smokey has written extensively--about Mach meter calibration---I can do no justice--to his posts--on that topic--- so--I wont even attempt



Cuidete

PA

[airfoilmod]

Tiene los Lastwordamentes, Pujo Animales?

Aerofolio pas nadas

No es muchos Machos nombres uno, dos.

Gracias, el "Exacto", para Exactamente! Nolo Machete, si?

[Brian Abraham]

I don't use Wiki, I think it diminishes the quality of an exchange
As with all sources of information you need to be selective, and even an expert text may have errors.
M 1.17---is still sonic because every part of the fighter has to reach M 1.00Definitely not. Your Airbus or Boeing airliner never gets close to Mach 1, but there may be parts of the airframe where the airflow has reached sonic velocities, as being talked about here, the shock wave formation on the wing.
I'm just an aerodynamicist
I hope not. :p
As the air re-enters the "hole" caused by the A/C in the air behind it, it accelerates and produces an immense release of the energy it was forced to contain during the brief transit of the A/C. This "Thunderclap" is one of two separate events precipitated by the energy of the passing A/C

There are two types of booms: N-waves and U-waves. The N-wave is generated from steady flight conditions, and its pressure wave is shaped like the letter "N." N-waves have a front shock to a positive peak overpressure which is followed by a linear decrease in the pressure until the rear shock returns to ambient pressure. The U-wave, or focused boom, is generated from maneuvering flights, and its pressure wave is shaped like the letter "U." U-waves have positive shocks at the front and rear of the boom in which the peak overpressures are increased compared to the N-wave.

For today's supersonic aircraft in normal operating conditions, the peak overpressure varies from less than one pound to about 10 pounds per square foot for a N-wave boom. Peak overpressures for U-waves are amplified two to five times the N-wave, but this amplified overpressure impacts only a very small area when compared to the area exposed to the rest of the sonic boom.

The strongest sonic boom ever recorded was 144 pounds per square foot and it did not cause injury to the researchers who were exposed to it. The boom was produced by a F-4 flying just above the speed of sound at an altitude of 100 feet.

In recent tests, the maximum boom measured during more realistic flight conditions was 21 pounds per square foot. There is a probability that some damage -- shattered glass, for example, will result from a sonic boom. Buildings in good repair should suffer no damage by pressures of less than 16 pounds per square foot. And, typically, community exposure to sonic boom is below two pounds per square foot. Ground motion resulting from sonic boom is rare and is well below structural damage thresholds accepted by the U.S. Bureau of Mines and other agencies. (USAF FACT SHEET 96-03)

For students interested in aerodynamics NASA have a good site here
http://www.grc.nasa.gov/WWW/K-12/airplane/guided.htm
With the caveat that even the experts can get it wrong (I’m no expert, just an interested bystander) I think even airfoilmod will find it an acceptable source of information. If NASA can't get it right we really are in big trouble. :eek: I reference this site because of the facility to play around with numbers to see what occurs.
You say there is no increase in energy? That wants explanation
Not sure as to what you are referring exactly airfoil. I admit to be being a little slow. :p

[airfoilmod]

"The total enthalpy and the total temperature are constant."

I think it's fair to say that picking apart your post would not be helpful. I mean no disrespect, but you seem to stylishly increase complexity while attempting to characterize a simple concept. I prefer a straightforward characterization. This is a technical forum, and I may be out of place. I enjoy metaphor and global inference. Mathematics is like a boggy swamp when I want to blaze a hole in the ether. I get what you write; I can't be sure if your prose is proprietary, paraphrase or quote.

"We cannot use the incompressible form of Bernoullis equation across the Shock". Well, of course, that is precisely what I said.

"The density of the gas varies locally as the gas is compressed by the object." Again, intuitively (and textually) obvious.

I wouldn't correct you, what you say is enlightening and accurate, but I think you want me to be wrong, or somehow in disagreement with technical prose. Any Natural phenomenon can be explained in a number of ways, from the entry level to the arcane and hopelessly circuitous. I am not expert in aerodynamics by any means and made that clear at the beginning. I won't learn anything if I always have to be right or the most learned in the room.

Basically, I like the pictures more than the text, always have. Never grew up I guess.

All the Best (Is Australia as beautiful as I imagine it to be?)

Airfoil

[Brian Abraham]

but I think you want me to be wrong, or somehow in disagreement
Certainly not airfoil. Like every one here, and I hope thats why everyone is here, I'm here to learn and get the grey matter churning. I hold to the following
The peculiar evil of silencing the expression of an opinion is, that it is robbing the human race; posterity as well as the existing generation; those who dissent from the opinion, still more than those who hold it. If the opinion is right, they are deprived of the opportunity of exchanging error for truth: if wrong, they lose, what is almost as great a benefit, the clearer perception and livelier impression of truth, produced by its collision with error…….We can never be sure that the opinion we are endeavouring to stifle is a false opinion; and if we were sure, stifling it would be an evil still
I can't be sure if your prose is proprietary, paraphrase or quote
Tis quote for the most part. My fault for including terms (eg enthalpy) that you find a distraction, but I include them so that those who have an interest might find an avenue to broaden their understanding. I confess to being a bit of a geek with an interest and time on my hands to seek answers. Not that I necessarily grasp a full and meaningful understanding of what I read and it is by discussion that you can get to grips with a subject and uncover your own misunderstandings. Hope I make sense.
I like the pictures more than the text
I'm afraid I always bought Playboy for the articles. :E BTW I don't expect you to believe that. :)
Re Australia and beautiful. Depends on the viewer I guess, it does range from barren desert to tropical jungle so there should be some thing to keep most people happy - if you can put up with the flies, poisonous spiders, snakes and jellyfish.

[Pugilistic Animus]

Definitely not. Your Airbus or Boeing airliner never gets close to Mach 1,


Ooh no---I thought all airliner's move at 1000mph---almost as fast as a freight train--right---well roger tower--- ready to get cleared for TO---from the tarmac:zzz:


good book ---can't think of the author's not home---but the title is unique

try 'Engineering Supersonic Aerodynamics'---by, some :8

Warning it gets a lil rough in the partial differential equations:}

[Brian Abraham]

Your Airbus or Boeing airliner never gets close to Mach 1
Just taking a little liberty with the language as in a miss is as good as a mile Pug. Mmo of .9 of a 747 falls 10% short and gets to be how long is a piece of string sort of argument.

[Pugilistic Animus]

If you were to take an elevator to the top of a building; and your friend took the stair and you both jumped who has the greater PE? as you see it doesn't matter the path [such functions were the path determines the amount of something are called 'path functions'....but any function that only depends on the initial and final states--and can be completely characterized by the initiall and final state are called state functions---like the bottom and top of that imaginary building----state functions are always written as 'delta somthing'

delta[E[internal enery],H[enthalpy,S [entropy]---but heat and work are path functions as a result of the PE available in the molecules of air as they are heated with compression---can be specified anyway---in others words---- you can assume an isothermal compression or an adiabatic compression, both different types of work and/or heat, but since the final quantities are equal---the work done to cause compression of the atmosphere on the airfoil..can be written by either equation [the one for isothermal or adiabatic] ---so lets look at those relations between the state functions and path functions--I need to do some light algebra--for this:8---and when I write d= I do not mean differentials--- I mean delta--final minus initial---

E is the internal energy of a gas[atmosphere] if you add to that the work of expansion/compression [PV work]--you get the Enthalpy [H] of the gas

So dH= dE+PV work ---if I subtract the wasted energy that is used to create disorder--entropy dS-- [which increases with temp]--

I have my Free energy ----dG

So I finally am left with dG = [E+pv work ==dH]----dG = dH - TdS---


there's
--more but I'll leave this to digest--so I can really be understood--I'll get to the point--- but I want to make sure I'm being understandable===and I wouldn't go too far ahead of me unless you're an engineer or scientist--because it may be hard to guess where I'm going---but it takes along time for me to get Tired:}

any questions so far?:)

I'll get to how it causes buffeting and all the good stuff--which is all to do with this ---but it's not an easy path for a non-specialist and it's hard to write math on Pprune and explain nice, but I'm not trying to be confusing--that's not hard to do for a few of us here if that's the goal



PA

[Pugilistic Animus]

This has been a hard one for me cuz---I'm skipping special maths---{energy is the capacity to move stuff against a force [i.e do work]

The change in internal energy-Ef-Ei---a State function

The change in internal energy---is independent of path, but is composed of path functions--and for simplicity [in aerothermodynamics one excludes such paths as electrical work ] ---and IE is independent of volume perfect gases----

dE = Q[heat] + W [work] ---

Assuming only adiabatic [no heat lost or gained] expansion work W = - P[B]ex dV [look at the units;)]----then dE=w=-Pexdv


--the heat capacity [Cp] at constant pressure is equal = dH= dQ+Vdp[p=constant] so dH=dQ

and dE=CvdT [q=0] so [Work] adiabatic=Wad, but because of path independence ---I can say that dE [due to Wad] = work done isothermally---or electrically--if you want [dE=STATE function]

so CvdT [Wad]=-Pdv [isothermal work]


that's enuf:\--for me too---but I'll continue this it's a not easy to explain this:{ ---any questions so far:)?

PA