Thermo Lecture Notes

5.7. FINAL COMMENTS ON CONSERVATION 141

Figure 5.22: Albert Einstein (1879-1955), German theoretical physicist who de-veloped theories that explained data better than those of Newton. Image fromhttp://www-history.mcs.st-and.ac.uk/∼history/Biographies/Einstein.html.

Another consequence of Einstein’s reformulation was the remarkable results of mass-energyequivalence via the famous relation

E = mc2, (5.208)

where c is the speed of light in a vacuum. Another way of viewing Einstein’s contributionsis via a new conservation property: the mass-energy of an isolated system is constant. It isthe conservation of mass-energy that is the key ingredient in both nuclear weapon systemsas well as nuclear power generation.

CC BY-NC-ND. 01 July 2014, J. M. Powers.

142 CHAPTER 5. THE FIRST LAW OF THERMODYNAMICS


Chapter 6

First law analysis for a control volume

Read BS, Chapter 6

Problems in previous chapters have focused on systems. These systems always were com-posed of the same matter. However, for a wide variety of engineering devices, for example

• flow in pipes,

• jet engines,

• heat exchangers,

• gas turbines,

• pumps,

• furnaces, or

• air conditioners,

a constant flow of new fluid continuously enters and exits the device. In fact, once the fluidhas left the device, we often are not concerned with that fluid, as far as the performanceof the device is concerned. Of course, we might care about the pollution emitted by thedevice and the long term fate of expelled particles. Pollution dispersion, in contrast topollution-creation, is more a problem of fluid mechanics than thermodynamics.

Analysis of control volumes is slightly more complicated than for systems, and the equa-tions we will ultimately use are slightly more complex. Unfortunately, the underlying mathe-matics and physics which lead to the development of our simplified control volume equationsare highly challenging! Worse still, most beginning thermodynamics texts do not exposethe student to all of the many nuances required for the simplification. In this chapter, wewill summarize the key results and refer the student to an appendix for a more rigorousdevelopment.

We will introduce no new axioms in this chapter. We shall simply formulate our massand energy conservation axioms for a control volume configuration. A sketch of a genericapparatus for control volume analysis is given in Fig. 6.1.

143

144 CHAPTER 6. FIRST LAW ANALYSIS FOR A CONTROL VOLUME

Qcv W

cv

..

mi(h

i + v

i2/2 + gz

i)

.

me(h

e + v

e2/2 + gz

e)

.

me(h

e + v

e2/2 + gz

e)

.

me(h

e + v

e2/2 + gz

e)

.

Figure 6.1: Sketch of generic configuration for control volume analysis.

6.1 Detailed derivations of control volume equations

This section will give a summary of the necessary mathematical operations necessary to castthe conservation of mass and energy principles in a traditional control volume formulation.The analysis presented has been amalgamated from a variety of sources. Most directly, itis a specialization of course notes for AME 60635, Intermediate Fluid Mechanics.1 Basicmathematical foundations are covered well by Kaplan.2 A detailed and readable description,which has a stronger emphasis on fluid mechanics, is given in the undergraduate text ofWhitaker.3 A rigorous treatment of the development of all equations presented here isincluded in the graduate text of Aris.4 Popular mechanical engineering undergraduate fluidstexts have closely related expositions.56 However, despite their detail, these texts have someminor flaws! The treatment given by BS is not as detailed. This section will use a notationgenerally consistent with BS and show in detail how to arrive at its results.

1J. M. Powers, 2012, Lecture Notes on Intermediate Fluid Mechanics, University of Notre Dame,http://www.nd.edu/∼powers/ame.60635/notes.pdf.

2W. Kaplan, 2003, Advanced Calculus, Fifth Edition, Addison-Wesley, New York.3S. Whitaker, 1992, Introduction to Fluid Mechanics, Krieger, Malabar, Florida.4R. Aris, 1962, Vectors, Tensors, and the Basic Equations of Fluid Mechanics, Dover, New York.5F. M. White, 2002, Fluid Mechanics, Fifth Edition, McGraw-Hill, New York.6R. W. Fox, A. T. McDonald, and P. J. Pritchard, 2003, Introduction to Fluid Mechanics, Sixth Edition,

John Wiley, New York.


6.1. DETAILED DERIVATIONS OF CONTROL VOLUME EQUATIONS 145

6.1.1 Relevant mathematics

We will use several theorems which are developed in vector calculus. Here, we give shortmotivations and presentations. The reader should consult a standard mathematics text fordetailed derivations.

6.1.1.1 Fundamental theorem of calculus

The fundamental theorem of calculus is as follows

∫ x=b

x=a

φ(x) dx =

∫ x=b

x=a

(dψ

dx

)

dx = ψ(b)− ψ(a). (6.1)

It effectively says that to find the integral of a function φ(x), which is the area under thecurve, it suffices to find a function ψ, whose derivative is φ, i.e. dψ/dx = φ(x), evaluate ψat each endpoint, and take the difference to find the area under the curve.

6.1.1.2 Divergence theorem

The divergence theorem, often known as Gauss’s7 theorem, is the analog of the fundamentaltheorem of calculus extended to volume integrals. Gauss is depicted in Fig. 6.2. While it is

Figure 6.2: Johann Carl Friedrich Gauss (1777-1855), German mathematician; image fromhttp://www-history.mcs.st-and.ac.uk/∼history/Biographies/Gauss.html.

often attributed to Gauss who reported it in 1813, it is said that it was first discovered byJoseph Louis Lagrange in 1762.8

Let us define the following quantities:

• t→ time,

7Carl Friedrich Gauss, 1777-1855, Brunswick-born German mathematician, considered the founder ofmodern mathematics. Worked in astronomy, physics, crystallography, optics, bio-statistics, and mechanics.Studied and taught at Gottingen.

8http://en.wikipedia.org/wiki/Divergence theorem.



• x → spatial coordinates,

• Va(t) → arbitrary moving and deforming volume,

• Aa(t) → bounding surface of the arbitrary moving volume,

• n → outer unit normal to moving surface, and

• φ(x, t) → arbitrary vector function of x and t.

The divergence theorem is as follows:

∫

Va(t)

∇ · φ dV =

∫

Aa(t)

φ · n dA. (6.2)

The surface integral is analogous to evaluating the function at the end points in the funda-mental theorem of calculus.

If φ(x, t) has the form φ(x, t) = cφ(x, t), where c is a constant vector and φ is a scalarfunction, then the divergence theorem, Eq. (6.2), reduces to

∫

Va(t)

∇ · (cφ) dV =

∫

Aa(t)

(cφ) · n dA, (6.3)

∫

Va(t)

(

φ∇ · c︸︷︷︸

=0

+c · ∇φ)

dV =

∫

Aa(t)

φ (c · n) dA, (6.4)

c ·∫

Va(t)

∇φ dV = c ·∫

Aa(t)

φn dA, (6.5)

c ·(∫

Va(t)

∇φ dV −∫

Aa(t)

φn dA

)

︸︷︷︸

=0

= 0. (6.6)

Now, since c is arbitrary, the term in parentheses must be zero. Thus,

∫

Va(t)

∇φ dV =

∫

Aa(t)

φn dA. (6.7)

Note if we take φ to be the scalar of unity (whose gradient must be zero), the divergencetheorem reduces to

∫

Va(t)

∇(1) dV =

∫

Aa(t)

(1)n dA, (6.8)

0 =

∫

Aa(t)

(1)n dA, (6.9)

∫

Aa(t)

n dA = 0. (6.10)



That is, the unit normal to the surface, integrated over the surface, cancels to zero when theentire surface is included.

We will use the divergence theorem (6.2) extensively. It allows us to convert sometimesdifficult volume integrals into easier interpreted surface integrals. It is often useful to usethis theorem as a means of toggling back and forth from one form to another.

6.1.1.3 Leibniz’s rule

Leibniz’s9 rule relates time derivatives of integral quantities to a form which distinguisheschanges which are happening within the boundaries to changes due to fluxes through bound-aries. Leibniz is depicted in Fig. 6.3.

Figure 6.3: Gottfried Wilhelm von Leibniz (1646-1716), German mathe-matician, philosopher, and polymath who co-invented calculus; image fromhttp://www-history.mcs.st-and.ac.uk/∼history/Biographies/Leibniz.html.

Let us consider the scenario sketched in Figure 6.4. Say we have some value of interest,Φ, which results from an integration of a kernel function φ over Va(t), for instance

Φ =

∫

Va(t)

φ dV. (6.11)

We are often interested in the time derivative of Φ, the calculation of which is complicatedby the fact that the limits of integration are time-dependent. From the definition of thederivative, we find that

dΦ

dt=

d

dt

∫

Va(t)

φ dV = lim∆t→0

∫

Va(t+∆t)φ(t+∆t) dV −

∫

Va(t)φ(t) dV

∆t. (6.12)

9Gottfried Wilhelm von Leibniz, 1646-1716, Leipzig-born German philosopher and mathematician. In-vented calculus independent of Newton and employed a superior notation to that of Newton.



I

II

AI

AII

VIIV

I

wI

wII

Va

nIIn

I

Va(t) V

a(t+Δt)

Figure 6.4: Sketch of the motion of an arbitrary volume Va(t). The boundaries of Va(t)move with velocity w. The outer normal to Va(t) is Aa(t). Here, we focus on just two regions:I, where the volume is leaving material behind, and II, where the volume is sweeping upnew material.

Now, we haveVa(t +∆t) = Va(t) + VII(∆t)− VI(∆t). (6.13)

Here, VII(∆t) is the amount of new volume swept up in time increment ∆t, and VI(∆t) isthe amount of volume abandoned in time increment ∆t. So we can break up the first integralin the last term of Eq. (6.12) into∫

Va(t+∆t)

φ(t+∆t) dV =

∫

Va(t)

φ(t+∆t) dV +

∫

VII (∆t)

φ(t+∆t) dV −∫

VI (∆t)

φ(t+∆t) dV,

(6.14)which gives us then

d

dt

∫

Va(t)

φ dV =

lim∆t→0

∫

Va(t)φ(t+∆t) dV +

∫

VII(∆t)φ(t+∆t) dV −

∫

VI (∆t)φ(t+∆t) dV −

∫

Va(t)φ(t) dV

∆t.(6.15)

Rearranging (6.15) by combining terms with common limits of integration, we get

d

dt

∫

Va(t)

φ dV = lim∆t→0

∫

Va(t)(φ(t+∆t)− φ(t)) dV

∆t

+ lim∆t→0

∫

VII (∆t)φ(t+∆t) dV −

∫

VI(∆t)φ(t+∆t) dV

∆t. (6.16)

Let us now further define



• w → the velocity vector of points on the moving surface Va(t),

Now, the volume swept up by the moving volume in a given time increment ∆t is

dVII = w · n︸︷︷︸

positive

∆t dAII = wII∆t︸︷︷︸

distance

dAII , (6.17)

and the volume abandoned is

dVI = w · n︸︷︷︸

negative

∆t dAI = − wI∆t︸︷︷︸

distance

dAI . (6.18)

Substituting into our definition of the derivative, Eq. (6.16), we get

d

dt

∫

Va(t)

φ dV = lim∆t→0

∫

Va(t)

(φ(t+∆t)− φ(t))

∆tdV

+ lim∆t→0

∫

AII(∆t)φ(t+∆t)wII∆t dAII +

∫

AI(∆t)φ(t+∆t)wI∆t dAI

∆t. (6.19)

Now, we note that

• We can use the definition of the partial derivative to simplify the first term on the rightside of (6.19),

• The time increment ∆t cancels in the area integrals of (6.19), and

• Aa(t) = AI + AII ,

so thatd

dt

∫

Va(t)

φ dV

︸︷︷︸

total time rate of change

=

∫

Va(t)

∂φ

∂tdV

︸︷︷︸

intrinsic change within volume

+

∫

Aa(t)

φw · n dA

︸︷︷︸

net flux into volume

. (6.20)

This is the three-dimensional scalar version of Leibniz’s rule. Say we have the special casein which φ = 1; then Leibniz’s rule (6.20) reduces to

d

dt

∫

Va(t)

dV =

∫

Va(t)

∂

∂t(1)

︸︷︷︸

=0

dV +

∫

Aa(t)

(1)w · n dA, (6.21)

d

dtVa(t) =

∫

Aa(t)

w · n dA. (6.22)

This simply says the total volume of the region, which we call Va(t), changes in response tonet motion of the bounding surface.

Leibniz’s rule (6.20) reduces to a more familiar result in the one-dimensional limit. Wecan then say

d

dt

∫ x=b(t)

x=a(t)

φ(x, t) dx =

∫ x=b(t)

x=a(t)

∂φ

∂tdx+

db

dtφ(b(t), t)− da

dtφ(a(t), t). (6.23)



As in the fundamental theorem of calculus (6.1), for the one-dimensional case, we do not haveto evaluate a surface integral; instead, we simply must consider the function at its endpoints.Here, db/dt and da/dt are the velocities of the bounding surface and are equivalent to w.The terms φ(b(t), t) and φ(a(t), t) are equivalent to evaluating φ on Aa(t).

We can also apply the divergence theorem (6.2) to Leibniz’s rule (6.20) to convert thearea integral into a volume integral to get

d

dt

∫

Va(t)

φ dV =

∫

Va(t)

∂φ

∂tdV +

∫

Va(t)

∇ · (φw) dV. (6.24)

Combining the two volume integrals, we get

d

dt

∫

Va(t)

φ dV =

∫

Va(t)

(∂φ

∂t+∇ · (φw)

)

dV. (6.25)

6.1.1.4 General transport theorem

Let B be an arbitrary extensive thermodynamic property, and β be the corresponding in-tensive thermodynamic property so that

dB = βdm. (6.26)

The product of a differential amount of mass dm with the intensive property β give adifferential amount of the extensive property. Since

dm = ρdV, (6.27)

where ρ is the mass density and dV is a differential amount of volume, we have

dB = βρdV. (6.28)

If we take the arbitrary φ = ρβ, Leibniz’s rule, Eq. (6.20), becomes our general transporttheorem:

d

dt

∫

Va(t)

ρβ dV =

∫

Va(t)

∂

∂t(ρβ) dV +

∫

Aa(t)

ρβ (w · n) dA. (6.29)

Applying the divergence theorem, Eq. (6.2), to the general transport theorem, Eq. (6.29),we find the alternate form

d

dt

∫

Va(t)

ρβ dV =

∫

Va(t)

(∂

∂t(ρβ) +∇ · (ρβw)

)

dV. (6.30)


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