Fenômentos de Transporte
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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