ࡱ> CEBaM u)bjbj== ,rWWX%l0008h,D*|"$$$$$$u*w*w*w*w*w*w*$+ -*$$$$$* $$* $$$u* $u*  ,')A)$p pX=:01)A)4*0*9). .A)  5.3 Macroscopic Energy Balance The general energy balance equation has the form  EMBED Equation.3 = EMBED Equation.3 ( EMBED Equation.3 + EMBED Equation.3 (  EMBED Equation.3  Let Esys be the total energy (internal + kinetic + potential) of a system,  EMBED Equation.3 be the mass flow rate of the system input stream, and  EMBED Equation.3 be the mass flow rates of the system output stream, then  EMBED Equation.3  EMBED Equation.3  =  EMBED Equation.3  EMBED Equation.3  ( EMBED Equation.3  EMBED Equation.3  +  EMBED Equation.3  (  EMBED Equation.3  (5.3-1) where Usys = system internal energy Ek,sys = system kinetic energy Ep,sys = system potential energy  EMBED Equation.3 ,  EMBED Equation.3  = internal energies per unit mass of the system inlet and outlet streams  EMBED Equation.3 ,  EMBED Equation.3  = average velocity of the system inlet and outlet streams  EMBED Equation.3  = rate of heat added to the system  EMBED Equation.3  = rate of work done by the system The net rate of work done by the system can be written as  EMBED Equation.3 =  EMBED Equation.3  +  EMBED Equation.3  where  EMBED Equation.3  = rate of shaft work = rate of work done by the system through a mechanical device (e.g., a pump motor)  EMBED Equation.3  = rate of flow work = rate of work done by the system fluid at the outlet minus rate of work done on the system fluid at the outlet  EMBED Word.Picture.8  Rate of work = Force( EMBED Equation.3  = Force (velocity Rate of flow work done on the system fluid = PinAinVin = Pin EMBED Equation.3  Rate of flow work done by the system fluid = PoutAoutVout = Pout EMBED Equation.3  Eq. (5.3-1) becomes  EMBED Equation.3  EMBED Equation.3  =  EMBED Equation.3  EMBED Equation.3  ( EMBED Equation.3  EMBED Equation.3  +  EMBED Equation.3  (  EMBED Equation.3  + Pin EMBED Equation.3 ( Pout EMBED Equation.3  (5.3-2) The internal energy can be combined with the flow work to give the enthalpy  EMBED Equation.3  EMBED Equation.3  + Pin EMBED Equation.3  =  EMBED Equation.3  EMBED Equation.3  =  EMBED Equation.3  EMBED Equation.3  In terms of enthalpies  EMBED Equation.3  and  EMBED Equation.3   EMBED Equation.3  EMBED Equation.3  =  EMBED Equation.3  EMBED Equation.3  ( EMBED Equation.3  EMBED Equation.3  +  EMBED Equation.3  (  EMBED Equation.3  (5.3-3) The internal energy and the enthalpy can be related to the heat capacities where Cp =  EMBED Equation.3 , and Cv =  EMBED Equation.3  For constant values of Cp and Cv h = Cp(T - Tref) and u = Cv(T - Tref) For solid and liquid Cp ( Cv If the system is at steady state with one inlet and one exit stream  EMBED Equation.3  =  EMBED Equation.3  =  EMBED Equation.3 , equation (5.3-3) is simplified to hout ( hin + g(zout ( zin) +  EMBED Equation.3  =  EMBED Equation.3  (  EMBED Equation.3  (5.3-4) Let ( = (out) ( (in), and q =  EMBED Equation.3 , w =  EMBED Equation.3  be the heat added to the system and work done by the system, respectively, per unit mass flow rate. Equation (5.3-4) becomes (h + g(z +  EMBED Equation.3 (V2 = q ( w (5.3-5) This equation also applies to a system comprising the fluid between any two points along a streamline within a flow field. If these two points are only infinitesimal distance apart, the differential form of the energy equation is obtained dh + gdz + VdV = (q ( (w (5.3-6) The d() notation represents a total or exact differential and applies to the change in state properties that are determined only by the initial and final states of the properties. The (() notation represents an inexact differential and applies to the change in properties that depend upon the path taken from the initial to the final point of the properties. The forms of energy can be classified as either mechanical energy, associated with motion or position, or thermal energy, associated with temperature. Mechanical energy is considered to be an energy form of higher quality than thermal energy since it can be converted directly into useful work. Mechanical energy includes potential energy, kinetic energy, flow work, and shaft work. Internal energy and heat are thermal energy forms that cannot be converted directly into useful work. For systems that involve significant temperature changes, the mechanical energy terms are usually negligible compared with the thermal terms. In such cases the energy balance equation reduces to a heat or enthalpy balance, i.e. dh = (q. Example 5.3-1. ---------------------------------------------------------------------------------- In a residential water heater, water at 60oF flows at a constant 5 GPM into the 100 gallons tank and leaves at 3 GPM. Initially the tank has 10 gallons of 75oF water in it. The tank gas heater heats the tank contents at a constant rate of 800 Btu/min. Assume perfect mixing, determine the temperature of the discharge water after 20 min. of operation. Water: Cp = Cv = 1 Btu/(lb.oF), density = 62.4 lb/ft3. Unit conversion 1 ft3 = 7.481 gal.  EMBED Word.Picture.8  Solution ------------------------------------------------------------------------------------------ Step #1: Define the system. Step #2: Find an equation that contains the temperature of the discharge water. The energy balance for the system gives the desired equation. Step #3: Apply the energy balance on the system with the reference temperature Tref = 0oF. Neglect the changes in kinetic and potential energies compared with the changes in thermal energies.  EMBED Equation.3 ((VCpT) = (FoCpTo - (FCpT +  EMBED Equation.3   EMBED Equation.3 (VT) = FoTo - FT +  EMBED Equation.3   EMBED Equation.3  = 5 - 3 = 2 => V = 10 + 2t Step #4: Specify the initial condition for the differential equation. At t = 0, T = 75oF Step #5: Solve the resulting equation and verify the solution. V  EMBED Equation.3  + T EMBED Equation.3  = FoTo - FT +  EMBED Equation.3  (10 + 2t) EMBED Equation.3  + 2T = (5)(60) - 3T +  EMBED Equation.3  (2t + 10)  EMBED Equation.3  = 395.91 - 5T  EMBED Equation.3  =  EMBED Equation.3 ( - EMBED Equation.3 ln EMBED Equation.3  =  EMBED Equation.3 ln EMBED Equation.3  395.91 - 5T = 20.91 EMBED Equation.3  => T = 79.182 - 4.182  EMBED Equation.3  at t = 20 min., T = 79.1oF ----------------------------------------------------------------------------------------------------- Equation (5.3-5) and its differential form, equation (5.3-6), are not convenient for solving engineering problems. (h + g(z +  EMBED Equation.3 (V2 = q ( w (5.3-5) dh + gdz + VdV = (q ( (w (5.3-6) We can use thermodynamics relations to convert the enthalpy term into a form that involves temperature, pressure, and density changes across the system. du = Tds ( Pd(1/() dh = du + d(P/() = Tds ( Pd(1/() + d(P/() dh = Tds ( Pd(1/() + Pd(1/() +  EMBED Equation.3  = Tds +  EMBED Equation.3  (5.3-7) For an idealized reversible process in which no energy dissipation occurs, the entropy change arises from heat transfer across the system boundaries Tds = (q In any real system, the process is irreversible and there is dissipation of energy, therefore Tds = (q + (ef = du + Pd(1/() (5.3-8) dh = (q + (ef +  EMBED Equation.3  In this equation (ef represents the thermal energy generated due to the irreversibility of the system. Substituting dh = (q + (ef +  EMBED Equation.3  into the differential energy balance, Eq. (5.3-6), gives  EMBED Equation.3  + gdz + VdV + (ef = ( (w This equation can be integrated along a streamline from the inlet to the outlet of the system to give  EMBED Equation.3  + g(zo ( zi) +  EMBED Equation.3 ((oVo2 ( (iVi2) + ef + w = 0 (5.3-9) where ef =  EMBED Equation.3  , w =  EMBED Equation.3 , and ( = kinetic energy correction factor, ( = 2 for laminar flow, ( = 1 for turbulent flow. The kinetic correction factor is due to the fact that the velocity profile is not uniform over the cross-sectional area of flow. For uniform flow, the rate of kinetic energy entering a C.V. is given as  EMBED Equation.3  =  EMBED Equation.3 VA The kinetic energy per unit mass flow rate is then  EMBED Equation.3  =  EMBED Equation.3 V2 For turbulent flow, the velocity profile is almost flat, therefore  EMBED Equation.3 (  EMBED Equation.3 V2 ( ( = 1 for turbulent flow  EMBED CorelDRAW.Graphic.10  Figure 5.3-1 Laminar velocity profile in a pipe. 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