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Fenômentos de Transporte

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- VV' as VB. B y
recognizing that the volumes not common to V and V' can be written in terms
of the elemental volumes swept out by the elemental surfaces as indicated
dV, = (v a n) dS
dv, = (v .n )dS (1.6.3-5)
the second term can be written as
(1.6.3-6)
giving the result
(1.6.3-7)
We can apply the Gauss theorem to the second term
Chapter I : Essentials 69
(1.6.3-8)
If Ikl , . has a constant value for the region of the continuum under
consideration (a conserved quantity), then I'kl.. = 0. If this is to be true for
arbitrary values of the volume, the only way it can be true is for the integrand
above to be zero. We will use this argument several times in our course of study
to obtain microscopic equations from macroscopic balances.
Chapter 1 Problems
1.1 At the end of June your checkbook balance is \$356. During July you wrote
\$503 in checks and deposited \$120. What is your balance (accumulation) at the
end of July? What is your accumulation rate per day in July?
1.2 At the end of January your parts warehouse contains 35,133 parts. During
February you sold 21,316 parts from the warehouse and manufactured 3,153
which were added to the warehouse. How many parts are in the warehouse at the
end of February?
1.3 What are the conversion factors for:
Newtons (N) to pounds force (lbf)
Pascals (Pa) to pounds force per square inch (lbf/in2)
Watts (W) to horsepower (hp)
meters CUM (m3) to quarts (qt)
1.4 To convert between English Units and SI units calculate conversion factors
for the following:
1 lbf to Newtons (N)
1 lbf/in2 to Pascal (Pa)
1 BTU to Joules (J)
1 hp to Watts (W)
I quart to m3
1.5 Two infinite plates are separated 0.01 in. by a liquid with a viscosity of 1.1
centipise (cP). The top plate moves at a velocity of 0.3 ft/s. The bottom plate
is stationary. If the velocity decreases linearly from the top plate through the
fluid to the bottom plate, what is the shear stress on each plate?
70 Chapter 1: Essentials
1.6 Two infinite plates are separated 0.2 mm by a liquid with viscosity of 0.9
cP. The top plate moves at a velocity of 0.1 m/s; the bottom plate is
stationary. Assume velocity decreases linearly through the fluid to 0 at the
bottom plate. What is the shear stress on each plate?
1.7 The expression for velocity vx, for fluid flow between two parallel plates is
2B is plate separation, L is length of the two plates, (ppp2) is the pressure drop
causing flow and p is the fluid viscosity. Determine the expression for the
average velocity, <vx>.
1.8 If the coordinate system in problem 1.7 is changed to the bottom plate of
the slit and the expression for velocity is
Determine the average velocity, e x > .
1.9 The relationship
3 3 3
v - w = C i v, - C i w, = z i ( v , - w , ) = v, - w,
i = 1 i = l i = l
shows how to transform symbolic vector notation to index vector notation. Do
the same for
a. The product of a scalar and a vector s v
b. The scalar product of two vectors v w
c. Thecrossproductoftwovectors v x w
1.10 Show that
a x (b x c) = a (b c) - c(a b)
1.11 Show that
Chupter 1: Essentials 71
f : v v = v .(f v)- V. (V. ?)
1.12 The sketch shows rectangular and spherical coordinates where
Show the relations for two coordinate systems, i.e., x = x(r,8, +), y = y(r, 8,
40, z = z(r98, 0).
1.13 The sketch shows rectangular and cylindrical coordinates, where
The coordinates are related by x = r cos 8, y = r sin 8 and z = z. Apply the chain
rule and show the relationships between the derivatives.
72 Chapter I : Essentials
2
THE MASS BALANCES
We are concerned in this course of study with mass, energy, and
momentum. We will proceed to apply the entity balance to each in tun. There
also are sub-classifications of each of these quantities that are of interest. In
terms of mass, we are often concerned with accounting for both total mass and
mass of a species.
2.1 The Macroscopic Mass Balances
Consider a system of an arbitrary shape fixed in space as shown in Figure
2.1-1. V refers to the volume of the system, A to the surface area. AA is a small
increment of surface area; AV is a small incremental volume inside the system;
v is the velocity vector, here shown as an input; n is the outward normal; a is
the angle between the velocity vector and the outward normal, here greater than ~t
radians or 180° because the velocity vector and the outward normal fall on
opposite sides of AA.
V \
Figure 2.1-1 System for mass balances
73
74 Chapter 2: The Mass Balances
Using the continuum assumption, the density can be modeled functionally
by
(2.1-1)
2.1.1 The macroscopic total mass balance
We now apply the entity balance, written in terms of rates, term by term to
total mass. Since total mass will be conserved in processes we consider, the
generation term will be zero.
output input accumulation
The total mass in AV is approximately
P AV (2.1 0 1-2)
where p is evaluated at any point in AV and At.
The total mass in V can be obtained by summing over all of the elemental
volumes in V and taking the limit as AV approaches zero
(2.1.1-3)
Accumulution of mass
First examine the rate of accumulation term. The accumulation (not rate of
accumulation) over time At is the difference between the mass in the system at
some initial time t and the mass in the system at the later time t + At.
J I duringAt
Chapter 2: The Mass Balances
r
volumetric
flow rate
through AA
7.5
The rate of accumulation is obtained by taking the limit
rate of
accumulation
of total mass
in V
(2.1.1 -4)
(2.1.1 -5)
Input and output of mass
The rate of input and rate of output terms may be evaluated by considering
an arbitrary small area AA on the surface of the control volume. The velocity
vector will not necessarily be normal to the surface.
The approximate volumetric flow rate' through the elemental area AA can
be written as the product of the velocity normal to the area evaluated at some
point within AA multiplied by the area.
(2.1.1 -6)
Notice that the volumetric flow rate written above will have a negative sign for
inputs (where a > n) and a positive sign for outputs (where a c x ) and so the
appropriate sign will automatically be associated with either input or output
terms, making it no longer necessary to distinguish between input and output.
The approximate mass flow rate through AA can be written by multiplying
the volumetric flow rate by the density at some point in AA
We frequently need to refer to the volumetric flow rate independent of a control
volume. In doing so we use the symbol Q and restrict it to the ahsolute value
Q = k I ( v * n ) l d A
76 Chapter 2: The Mass Balances
(2.1.1-7)
[F] - [\$][\$I
The mass flow rate through the total external surface area A can then be written
as the limit of the sum of the flows through all the elemental AAs as the
elemental areas approach zero. Notice that in the limit it no longer matters where
in the individual AA we evaluate velocity or density, since a unique point is
approached for each elemental =a.*
mass
through A
(2.1.1 -8) = l A p (v n) dA
Substitution in the entity balance then gives the macroscopic total mass
balance
Total mass
output
rate
Total mass -1 input
rate
] + [ accumulation To:;as] = 0
We Erequently need to refer to the mass flow rate independent of a control volume. In
doing so we use the symbol w and restrict it to the ahsolute value
w = I A p l ( v - n ) l d A
For constant density systems w = p Q
Chapter 2: The Mass Balances 77
l /Ap(v-n)dA+\$lvpdV = 0
w o u t - w i n + 9 = 0
(2.1.1-9)
We can use the latter form of the balance