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# Fenômentos de Transporte

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```concentration contains (3 lbm salt)/gal of solution.
If the initial salt concentration in the tank is (1/2 lbm &)/gal, find:
a. The output concentration of salt as a function of time
assuming that the output concentration is the same as
that within the tank.
b. Plot the output salt concentration as a function of time
and find the asymptotic value of the concentration m(t) as
t + - .
c. Find the response time of the stirring tank, i.e., the time
at which the output concentration m(t) will equal 62.3
percent of the change in the concentration from time t =
0 to time t + - .
3
THE ENERGY BALANCES
3.1 The Macroscopic Energy Balances
The entity balance can be applied to forms of energy to obtain, for example,
macroscopic energy balances. Total energy, like total mass, can be modeled as
a conserved quantity for processes of interest here, which do not involve
interconversion of energy and mass by either nuclear reactions or mass moving
at speeds approaching that of light. On the other hand, neither thermal energy
nor mechanical energy is conserved in general, but each can be created or
destroyed by conversion to the opposite form - thermal to mechanical (for
example, via combustion processes in engines), or, more frequently, mechanical
to thermal (through the action of friction and other irreversible processes).
3.1.1 Forms of energy
Just as we did for mass balances, to develop the macroscopic energy
balances we consider an arbitrary system fixed in space. Since mass (because of
its position, motion or physical state) carries with it associated energy, each of
the macroscopic energy balances will therefore contain a term which corresponds
to each term in the corresponding macroscopic mass balance.
Energy associated with mass can be grouped within three classifications.
1. Energy present because of position of the mass in a field
(e.g., gravitational, magnetic, electrostatic). This energy is
called potential energy, a.
2. Energy present because of translational or rotational motion
of the mass. This energy is called kinetic energy, K.
3. All other energy associated with mass (for example,
rotational and vibrational energy in chemical bonds, energy of
113
I14 Chapter 3: The Energy Balances
Brownian motion\u2019, etc.). This energy is called internal energy,
U.
In general, U, 4, and K are functions of position and time.
In addition, however, energy is also transported across the boundaries of a
system in forms not associated with mass. This fact introduces terms in the
macroscopic (and microscopic) energy balances that have no counterpart in the
overall mass balances.
Although energy not associated with any mass may cross the boundary of
the system, within the system we will consider only energy that is associated
with mass (we will not, for example, attempt to model systems within which a
large part of the energy is in the form of free photons).
As we did with mass, we assume that we continue to deal with continua,
that is, that energy, like mass, is \u201csmeared out\u201d locally. A consequence of this is
that we group the energy of random molecular translation into internal energy,
not kinetic energy, even though at a microscopic level it represents kinetic
energy. The continuum assumption implies that we cannot see the level of
detail required to perceive kinetic energy of molecular motion, but rather simply
see a local region of space containing internal energy.
We shall denote the amount of a quantity associated with a
unit mass of material by placing a caret, (A ) above the quantity.
Thus, for example, 0 is the internal energy per unit mass (F LM) and \$I is the
rate of doing work per unit mass (F L/[M t]).
3.1.2 The macroscopic total energy balance
We now apply the entity balance, written in terms of rates, term by term to
total energy. Since total energy will be conserved in processes we consider,
the generation term will be zero.
* Even though Brownian motion represents kinetic energy of the molecules, this
kinetic energy is below the scale of our continuum assumption. It is furthermore,
random and so not recoverahle as work (at least, until someone discovers a real-life
Maxwell demon).
Chapter 3: The Energy Balances
rate of output of] - [rate of input of]
total energy total energy
rate of accumulationof = I total energy +I
Rate of accumulation of energy
The energy in AV is approximately2
(0 + 6 + E) p AV
(3.1.2-1)
(3.1.2-2)
where 6, a, K, and p are each evaluated at arbitrary points in AV and At.
The total energy in V can be obtained by summing over all of the elemental
volumes in V and taking the limit as AV approaches zero 1 total energy ; ] . 6 i i &quot; + & + R ) p A v ]
= AV lim - t o (7[(*+&+R)p*v])
= ( o + & + g ) p d V
Applying the definition of accumulation yields
(3.1.2-3)
(3.1.2-4)
(3.1.2-5)
This is a model which neglects systems that contain energy not associated with
mass that changes significantly with time, as well as neglecting any conversion of
mass into energy or vice versa, as we have stipulated earlier - it would not be suitable,
for example, in making an energy balance on the fireball of a nuclear explosion
where both large, changing amounts of radiation are involved as well as processes
that convert mass into energy.
116 Chapter 3: The Energy Balances
accumulation]
Ofenergy = I ( o + & + R ) p d V ( 1 d g A t 1 \u201d t+&
(3.1.2-6)
We then obtain the energy accumulation rate by dividing by At and taking
the limit as At approaches zero
(o+&+R)pdVI I+& -I v ( o + & + R ) p d V I t
At
(3.1.2-7)
Rates of input and output of energy
We will first consider input and output of energy during addition or
removal of mass. As each increment of mass is added to or removed from the
system, energy also crosses the boundary in the form of
kinetic, potential and internal energy associated with the
mass
. energy transferred in the process of adding or removing the
mass.
Consider for the moment the system shown in Figure 3.1.2-1. Illustrated is
the addition to a system of material which is contained in a closed rigid container
(a beverage can, if you wish).
Chapter 3: The Energy Balances
flow work
per
unit mass
117
= p V
Figure 3.1.2-1 Flow work
Note that to move the additional mass inside the system we must push -
that is, we must compress some of the material already in the system to make
room for the added material (thereby doing work on the mass already in the
system and increasing its energy).
In the typical process shown in Figure 3.1.2- 1, we do not change the energy
associated with the added mass, but we cannot either add or remove mass
without performing thisflow work on the mass already in the system.
In general, the flow work necessary to add a unit mass to a system is the
pressure at the point where the mass crosses the boundary multiplied by the
(3.1.2-8)
Therefore, the total energy associated with transferring a unit mass into a
system is the sum of the internal, potential, and kinetic energy associated with
the mass plus the flow work (whether or not the material is contained in a rigid
container)
(0 + 6 + K) + p 9 (3.1.2-9)
Thermodynamics teaches that it is convenient to combine the flow work with
the energy associated with the mass added or removed by defining the abstraction
enthalpy, H.
I18 Chapter 3: The Energy Balances
A = o + p v (3.1.2- 10)
Notice that the enthalpy combines some energy associated with mass with
some energy not associated with mass (the flow work). Definition of
enthalpy is not made for any compelling physical reason but rather for
convenience - the two terms always occur```