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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 + - . This page intentionally left blank 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 " + & + 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 volume added (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