Re: [Phys-l] Bernoulli, enthalpy, work/KE theorem, etc.

["LaMontagne, Bob" <RLAMONT@providence.edu>]

I have sometimes used a "back of the envelope" approach to introducing enthalpy in fluid flow.

Consider a pipe tilted upward at an angle theta. Then choose a small element of mass, m, with a volume, V, in the pipe of cross sectional area A. The net force on m is

PA - (P+dP)A - mgsin(theta) = -Adp - mgsin(theta)

Now apply the work-KE formula

[-AdP - mgsin(theta)]ds = d(1/2 mv^2)

this is

-VdP - mg[ds sin(theta)] = d(1/2 mv^2)

use dh = ds sin(theta) and rearrange

VdP + mg dh +d(1/2 mv^2) = 0

The left side of the equation can be called dH, H being a quantity (call it enthalpy) that is a constant during the motion of the little parcel. If this is compared to the energy equation dE = -PdV +mgdh + d(1/2 mv^2) for the parcel, it is clear that H = E + PV.

Bob at PC

> -----Original Message-----
> From: phys-l-bounces@carnot.physics.buffalo.edu [mailto:phys-l-
> bounces@carnot.physics.buffalo.edu] On Behalf Of John Denker
> Sent: Monday, December 06, 2010 6:15 AM
> To: Forum for Physics Educators
> Subject: [Phys-l] Bernoulli, enthalpy, work/KE theorem, etc.
>
> I would like to inject an idea into this discussion:  enthalpy.
>
> It turns out that the so-called Bernoulli head (the quantity that
> remains constant along a stream line) is essentially equivalent
> to the specific enthalpy.  If you think about it the right way,
> this just what you would expect, since we are trying to keep
> track of the energy of the parcel under conditions where the
> parcel is free to do PdV work against the surrounding fluid.
>
> This allows us to clearly understand that the energy of the parcel
> is *not* constant (except in trivial cases).
>
> Also the density, volume, and temperature are not constant.  Not
> even approximately constant.
>
> That gives us an easy way of deriving and/or remembering the form
> of Bernoulli's equation ... and understanding what it means.
>
> I wrote this up in the form of a new section here:
>    http://www.av8n.com/physics/bernoulli.htm#sec-enthalpy
>
> This includes (yet again) a discussion of why Bernoulli's principle
> applies just fine to compressible fluids.  Even the simplified
> first-order approximation applies just fine to compressible fluids.
> It already and inherently accounts for compressibility to first
> order ... which is a good thing, since there are no incompressible
> fluids.
>
> This approach has the advantage of elegance.  The disadvantage is
> that it will scare away any students who don't have a comfortable
> grasp of what enthalpy is.
>
> ==========
>
> And now let me inject another idea:  the work/KE theorem.
>
> This gives us a second way of deriving and understanding Bernoulli's
> equation.
>
> I have not yet fully written up my notes on this, but the idea
> is implicit in the equations here:
>    http://www.av8n.com/physics/bernoulli.htm#sec-force-work-ke
>
> I suspect that just mentioning the idea is enough of a hint to rev
> up the clever people on this list.
>
> Note that "work" as in P dV is not necessarily the type of "work"
> that connects to the work/KE theorem.  Refer to the diagrams at
>   http://www.av8n.com/physics/bernoulli.htm#sec-force-work-ke
>
> =================================
>
> I have been contemplating the question, what makes fluid dynamics
> so hard?
>
> If you look at the derivation of Bernoulli's equation, it's really
> pretty simple, simpler than the homework problems that are assigned
> in any thermodynamics course.  But then ... when I look around, I
> see grossly wrong derivations that outnumber correct derivations
> by some huge factor, on the order of 100 to 1.
>
> I reckon that part of the answer is that people underestimate the
> task.  The result is so simple that people think they can treat
> it as just a mechanics problem, when in fact it is inescapably a
> thermodynamics problem.
>
> The underestimation problem may be exacerbated by the fact that a
> lot of people don't know enough thermodynamics to handle this problem,
> not anywhere near enough.  So they either give up, or they "look under
> the lamp-post" (i.e. nonthermal mechanics) even though the thing they
> are looking for isn't under that lamp-post.
>
> What's even sadder is that a lot of people have learned /wrong/
> thermodynamics, such that even if they approach the problem in the
> nominally-correct way they are guaranteed to get the wrong answer.
> I have in mind specifically the case where students have been taught
> that the "FIRST LAW OF THERMODYNAMICS" is dE = -P dV + T dS.  Alas
> that is not the first law of anything.  It is not even correct in
> general, and certainly does not correctly describe a parcel of fluid
> in the context of Bernoulli's principle.
>
> I was just sitting here wondering what happens to somebody who starts
> from the premise that dE = -P dV + T dS.  They get stuck, and then
> what?  How long to they remain stuck before they realize what the
> problem is?  This equation was sold to them as the #1 most-fundamental
> starting point for the whole subject, so why should they question it?
> Yet it's nowhere near good enough to handle this problem.  Gack.  It's
> too painful to contemplate.
>
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