ADVANCED THERMODYNAMICS ENGINEERING phần 4

Wv = − d(Ecv −T0Scv) /dt + åj=1QR,j(1−T0 /TR,j)−meye + miyi − T0scv, (40) The absolute specific stream availability or the absolute specific flow or stream availability y is defined as y (T,P,T0) = eT(T,P) – T0 s(T,P) = (h(T,P) + ke + pe) – T0 s(T,P). (41) where the terms yi and ye denote the absolute stream availabilities, respectively, at the inlet and exit of the control volume. They are not properties of the fluid alone and depend upon the temperature of the environment. The optimum work is obtained for the same inlet and exit states when ˙cv = 0. In this case, Eq. (40) assumes the form Wv,opt = − d(Ecv −T0Scv) /dt + åj=1QR,j(1−T0 /TR,j)−meye + miyi. (42) where the term QR,j 1− T0 /TR,j) represents the availability in terms of the quality of heat en-ergy or the work potential associated with the heat transferred from the thermal energy reser-voir at the temperature TR,j. When the kinetic and potential energies are negligible, y = h – T0 s. (43) For ideal gases, s = s0– R ln (P/Pref), where the reference state is generally assumed to be at Pref = 1 bar. Therefore, y = y0 + R T0 ln (P/Pref), (44) and y0 = h0 –T0 s0. The enthalpy h (T) = h0 (T) for ideal gases, since it is independent of pres- sure. If the exit temperature and pressure from the control volume is identical to the envi- ronmental conditions T0 and P0, i.e., the exit is said to be at a restricted dead state, in that case ˙ cv,opt = ˙ cv,opt 0 and the exit absolute stream availability at dead state may be expressed as ye,0 = eT,e,0 (T0,P0) – T0 se,0 (T0,P0). (45) Note that eT,e,0 = h0 since ke and pe are equal to zero at dead state. 2. Lost Work Rate, Irreversibility Rate, Availability Loss The lost work is expressed through the lost work theorem, i.e., LW = I = Wv,opt −Wv = T0scv . (46) The terms ˙ cv, ˙ cv,opt > 0 for expansion processes and ˙ cv, ˙ cv,opt < 0 for compression and electrical work input processes. The lost work is always positive for realistic processes. The availability is completely destroyed during all spontaneous processes (i.e., those that occur without outside intervention) that bring the system and its ambient to a dead state. An example is the cooling of coffee in a room. 3. Availability Balance Equation in Terms of Actual Work We will rewrite Eq. (40) as d(Ecv −T0Scv) /dt = miyi + åN1QR,j(1−T0 /TR,j)−meye − Wv − I. (47) The term on the LHS represents the availability accumulation rate within the control volume as a result of the terms on the RHS which represent, respectively: (1) the availability flow rate into the c.v.; (2) the availability input due to heat transfer from thermal energy reservoirs; (3) the availability flow rate that exits the control volume; (4) the availability transfer through ac- Icv Input Stream availability d/dt [ECV-ToSCV] QRj(1-To/TRj) Figure 7: Exergy band or Sankey diagram illustrating availabilities. tual work input/output; and (5) the availability loss through irreversibilities. The Band or Sankey diagram illustrated in Figure 7 employs an accounting procedure to describe the avail-ability balance. This includes irreversibility due to temperature gradients between reservoirs and working fluids, such as water in a boiler with external gradients, the irreversibilities in pipes, turbines, etc. a. Irreversibility due to Heat Transfer We can separate these irreversibilities into various components. For instance consider the boiler component. Suppose the boiler tube is enclosed by a large Tubular TER. For a single TER the availability balance equation is given as d(Ecv −T0Scv) /dt = Qb(1− T0 /TR, )+ miyi −meye + Ib+R , (48) We have seen that an irreversibility can arise due to both internal and external processes. For instance, if the boundary AB in Figure 6 is selected so as to lie just within the boiler, and the control volume encloses the gases within the turbine, then Eq. (47) becomes d(Ecv−T0 cv ) = Qb(1− T0 /Tw b)+ miyi −meye + Ib, (49) where Tw,b denotes the water temperature just on the inside surface of the boiler (assumed uni-form) and QR,1 = Qb , the heat transfer from the reservoir to the water. The irreversibility Ib arises due to temperature gradients within the water. Subtracting (48) from (49), the irreversi-bility that exists to external temperature gradient between reservoir and wall temperature alone can be expressed as Ib+R −Ib = QbT0 1/Tw,b −1/TR,1) (50) Recall the entropy generation ˙ cv = ˙cv/T0. Thus, the entropy generated due to gradients ex-isting between a TER and a boiler tube wall s = Ib+R − Ib = Qb ( 1 − 1 ) (51) o w b R,1 4. Applications of the Availability Balance Equation We now discuss various applications of the availability balance equation. An unsteady situation exists at startup when a turbine or a boiler is being warmed, and the availability starts to accumulate. Here, d (Ecv – T0 Scv)/dt ¹ 0. If a system has a nondeformable boundary, then Wcv = Wshaft, P0dVcyl/dt = 0, ˙ u = 0 When a system interacts only with its ambient (that exists at a uniform temperature T0), and there are no other thermal energy reservoirs within the system, the optimum work is provided by the relation Wv,opt = Wv + I = miyi − meye −d(Ecv −T0Scv) /dt. (52) For a system containing a single thermal energy reservoir (as in the case of a power plant containing a boiler, turbine, condenser and pump, (Figure 8) or the evaporation of water from the oceans as a result of heat from the sun acting as TER), omitting the subscript 1 for the reservoir, d(Ecv −T0Scv) /dt = miyi +QR 1−T0 /TR )−meye −Wv −I. (53) For a steady state steady flow process (e.g., such as in power plants generating power under steady state conditions), mass conservation implies that ˙ i = ˙ e = ˙ . Fur-thermore, if the system contains a single inlet and exit, the availability balance as-sumes the form m(yi −ye)+ åj 1QR,j(1−T0 / TR,j)− ˙ cv −I = 0. (54) On unit mass basis yi −ye + åqR,j(1−T0 / TR,j)− wcv − i = 0, (55) where qR,j = QR,j /m,wcv = Wv /m,i = I /m. When a system interacts only with its ambient at T0 and there are no other thermal energy reservoirs within the system, the optimum work is given by the relation ˙ cv,opt = miyi − meye −d(Ecv −T0Scv) /dt . (56) In case the exit state is a restricted dead state, (e.g., for H2O, dead state is liquid water at 25°C 1 bar) ˙ cv,opt = my¢ + åj 1QR,j(1−T0 /TR,j) (57) where y¢ = y– y is the specific stream exergy or specific-relative stream availability (i.e., relative to the dead state). Since y0 = ho-Toso in the absence of kinetic and po- tential energy at the dead state, as T0 ® 0, y0 ® 0, and the relative and absolute stream availabilities become equal to each other. For a system containing multiple inlets and exits the availability equation is d(Ecv −T0Scv) /dt = inlets miyi + åj=1QR,j(1−T0 /TR,j)−exits meye −Wv −I. (58) For a single inlet and exit system containing multiple components the expression can be generalized as d(Ecv −T0Scv) /dt = species mk,iyk,i + åj=1QR,j(1−T0 /TR,j) −species mk,eyk,e − ˙ cv −I (59) where yk = hk(T,P,Xk) – T0 sk(T,P,Xk) denotes the absolute availability of each com-ponent, and Xk the mole fraction of species k. For ideal gas mixtures, yk = hk – T0 (sk0 – R ln (pk/Pref)), since the partial pressure of the k–th species in the ideal gas mixture Pk = Xk P. Consider an automobile engine in which piston is moving and at the same time mass is entering or leaving the system (e.g., during the intake and exhaust strokes). In addi- tion to the delivery of work through the piston rod Wu, atmospheric work is per-formed during deformation, i.e., W0 = P0 dV/dt. Therefore,, Wv = W + P0 dVyl /dt and the governing availability balance equation is d(Ecv −T0Scv) /dt = species mk,iyk,i + åj 1QR,j(1−T0 /TR,j) −species mk,eyk,e − W − PodV/dt − I Simplifying. d(Ecv −T0Scv + PodV/dt) /dt = species mk,iyk,i + åj 1QR,j(1−T0 /TR,j) −species mk,eyk,e − ˙ u − I For steady cyclical processes the accumulation term is zero within the control vol-ume, and yi = ye. Therefore, ˙ cv,cycle + I = åj 1QR,j(1−T0 /TR,j). c. Example 3 Steam enters a turbine with a velocity of 200 m s–1 at 60 bar and 740ºC and leaves as saturated vapor at 0.2 bar and 80 m s–1.The actual work delivered during the process is 1300 kJ kg–1. Determine inlet stream availability, the exit stream availability, and the irreversibility. Solution yi = (h1 + v2/2g) – T0 s1 = 3989.2 + 20 – 298 ´ 7.519 = 1769 kJ kg–1. Likewise, ye = 2609.7 + (802¸2000) –298 ´ 7.9085 = 256 kJ kg–1. Therefore, wopt = 1769 – 256 = 1513 kJ kg–1, and I = 1513 – 1300 = 213 kJ kg . The entropy generation s = 213¸298 = 0.715 kJ kg–1 K–1. Remarks The input absolute availability is 1769 kJ kg–1. The absolute availability outflow is 256 kJ kg–1. The absolute availability transfer through work is 1300 kJ kg–1. The availability loss is 213 kJ kg–1. The net outflow is 1769 kJ kg–1. d. Example 4 This example illustrates the interaction between a thermal energy reservoir, its ambi-ent, a steady state steady flow process, and a cyclical process. Consider the inflow of water in the form of a saturated liquid at 60 bar into a nuclear reactor (state 1). The reactor temperature is 2000 K and it produces steam which subsequently expands in a turbine to saturated vapor at a 0.1 bar pressure (state 2). The ambient temperature is 25ºC. The reactor heat transfer is 4526 kJ per kg of water. Assume that the pipes and turbines are rigid. What is the maximum possible work between the two states 1 and 2? If the steam that is discharged from turbine is passed through a condenser (cf. Figure 8) and then pumped back to the nuclear reactor at 60 bar, what is the maximum possi-ble work under steady state cyclical conditions? Assume that the inlet condition of the water into the pump is saturated liquid. Solution If the boundary is selected through the reactor, for optimum work I = s = 0. Under steady state conditions time derivatives are zero, and, since the body does not deform Wu = 0, so that Wcv = Wshaft and ˙ i = ˙ e = ˙ . Therefore, m(yi −ye)+QR1, (1−T0 /TR,1)−Wv,opt = 0. (A) Dividing Eq. (A) throughout by the mass flow rate, 1 Reservoir R QR Boiler Turbine 2 Condenser 4 Pump 3 Q0 Figure 8: Schematic of diagram of a steam power plant. ... - tailieumienphi.vn