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# Fenômentos de Transporte

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simultaneously and so it is convenient to group them under the Same symbol. This makes the energy associated with transferring a unit mass into the system ( R + & + R ) (3.1.2-11) Multiplying the energy associated with transferring a unit mass into the system by the mass flow rate gives the net rate (output minus input) of energy transfer associated with mass and from flow work Now consider the remainder of the energy that crosses the boundary not associated with mass. We arbitrarily (but for sound reasons) divide this energy into two classes - heat and work. By heat we mean the amount of energy crossing the boundary not associated with mass which flows as a result of a temperature gradient. By work we mean energy crossing the boundary not associated with mass Chapter 3: The Energy Balances 119 which does not transfer as a result of the temperature gradient except for flow work. Since this is definition by exclusion, or by using a complementary set, what we classify as work needs some clarification. We normally think of work as a force acting through a distance - for example, as transmitted by a rotating shaft passing through the boundary, so- called sh@ work. Note that even though such a shaft has mass, its mass is not crossing the boundary - each part of the mass remains either within or outside the system - so the energy is not transmitted by being associated with mass crossing the boundary. The definition of work also leads to some things being classified as work which we might not usually think of as work - for example, energy transferred across the boundary via electrical leads. Even though electrical current is associated with electrons, which can be assigned a minuscule mass (although they also have a wave nature), at scales of interest to us our models do not \u201csee\u201d electrons. Therefore, such energy is lumped into the work classification by default because it is not flow work, is not associated with mass, and does not flow as a result of a temperature gradient. By convention, heat into the system and work out of the system have traditionally been regarded as positive (probably because the originators of the convention worked with steam power plants where one puts heat in and gets work out - a good memory device by which to remember the c~nvention).~ We will denote the rate of transfer of heat or work across a boundary as Q or W (units: Ir: L/t]). If we wish to refer only to the amount of heat or work (units: Ir: L]), we will use Q or W. From thermodynamics we know that heat and work are not exact differentials, i.e., they are not independent of the path, although their There is currently movement away from this convention to one in which both heat and work entering the system are regarded as positive. In this text we adhere to the older convention . We use Q both for volumetric flow rate and heat transfer. The distinction is usually clear from the context; however, it would be clearer to use a different symbol. We are faced here with the two persistent problems with engineering notation: a) traditional use, and b) the scarcity of symbols that have not already been appropriated. Since heat is almost universally designated by the letter Q or q, we will how to tradition and not attempt to find another symbol. I20 Chapter 3: The Energy Balances r = ,\%,F"Q = Q energy input rate across A difference (the internal energy - from first law) is independent of path. Therefore, we usually write small amounts of heat or work as 6Q or 6W rather than dQ or dW to remind ourselves of this fact. If we add the rates that heat and work cross the boundary for each differential area on the surface of the system we obtain energy input rate [ acrossAA ] energy output rate fromheatin = 80 with units in each case of (3.1.2- 13) (3.1.2- 14) Adding the contributions of each elemental area and letting the elemental area approach zero we have fromworkout = , \ ~ , ~ 6 ~ = I energy output rate across A (3.1.2- 15) Substituting in the entity balance both the terms for energy associated with mass and not associated with mass, remembering our sign convention for heat and work, and moving the heat and work terms to the right-hand-side of the equation Chapter 3: The Energy Balances 121 (R + 4 + R) p (v n) dA +dSV(O+&+R)pdV dt = 0-w (3.1.2-16) which is the macroscopic total energy balance. Be sure to remember that the enthalpy is associated only with the inputloutput term, and the internal energy with the accumulation term. Simplified forms of the macroscopic total energy bulance The macroscopic total energy balance often can be simplified for particular situations. Many of these simplifications are so common that it is worth examining them in some detail. For example, we know from calculus that we always can rewrite the first term as IA fi p (v * n) dA+ IA 6 p (v - n) dA+ IA R p ( v . n) dA (3.1.2- 17) We now proceed to examine ways of simplifying each of the above terms, initially examining the second term, then the third term, and then the first term. The potential energy term We observe that the imposed force fields of most interest to us as engineers are conservative; that is, the force may be expressed as the gradient of a potential function, defined up to an additive constant. This function gives the value of the potential energy per unit mass referred to some datum point (the location of which determines the value of the additive constant). 122 Chapter 3: The Energy Balances not to m l o Figure 3.1.2-2 Gravitational field of earth For example, in the case illustrated in Figure 3.1.2-2, we show a mass, m, in the gravitational potential field, a, of the earth, with reference point of zero energy at the center of the earth. In spherical coordinates the potential function @ may be written as5 (3.1.2- 18) where typically = potential energy, [ N] [ m] G = a universal constant which has the same value for any pair of particles m = mass of object, [kg] m, = mass of earth, [kg] = 5.97 x 10' [kg] Resnick, R. and D. Halliday (1977). Physics: Part One. New York, NY, John Wiley and Sons, p. 338 ff. Chapter 3: The Energy Balances 123 r = distance from the center of gravity of the earth to center of gravity of object [ m] This relation is valid only outside the earth's surface. The force produced by such a field is the negative of the gradient of the potential, and in this case lies in the r-direction because the gradient of the potential is zero in the 8- and +directions (derivatives with respect to these two coordinate directions are zero). Gmm, (F), = = -Vr@ = -- a = - = m g ar r2 (3.1.2- 19) (G me/r2) is commonly called g, the acceleration of gravity L]/( t}2. This yields the expected result; that is, the force is inversely proportional to the square of the distance from the center of the earth and directed toward the center. At sea level g is approximately 32.2 ft/sec2 or 9.81 d s 2 . We can now rewrite our potential function as (3.1.2-20) In practice, most processes take place over small changes in r, at large values of r, that is, at the surface of the earth, where (r2 - rl) is of perhaps the order of 100 meters, while r2 and rl themselves are of the order of 6.37 x 106 meters (the radius of the earth). Under such a circumstance the acceleration of gravity (G md9) is nearly constant, as we now demonstrate. Examining the value of (G md$) for such a case we find that (- 6 . 6 7 2 6 ~ lO-") (5.97 x 1024) kg - - - kg2 = -9 .81727N Gme r2 (6.37 x 106)2 rn2 kg 124 l Q p ( v . n ) d A = g l z p ( v . n ) d A Chapter 3: The Energy Balances (-6.6726~ 10-l') 9 ( 5 . 9 7 ~ 102')