Document zdJoajzNbnDdjxeXLZx8M6V13

2 CHAPTER 1 1965 Guide And Data Book nated by subscripts 1 and % respectively, and the several terms for the energy term e are enumerated, then: f `7 iE -dQ - AdW + (4^ +AZ, -- + u, + AP,n)dm, *.. . . - W. Q( - Q. Tt -- T Q< " T, 3. The coefficient of performance of all reversible engines (cycles) operating as a refrigerator between the same.tempera- -- (-------- h AZt -- ; i-i } APtnt)dmt (6) 2g, Qt . A steadyflow system is defined as a system in which the mass, flows into and out of the.system are equal and at a,constant rate, and the propertiee ofthe system atany^.point are also constant, with respect to.time..An application of the above ..' general equation to the steady flow system (dB = 0),gives:. i j H------( +Q't -- AtW\ AZt ~ APtPi^ - (i + ~ + AZ, j- + //>,,,,) jm' (7) 4. For any engine cycle:'' <iQ ' ~ '< 0'(Clauaius inequality)--" The inequality exists for any but reversible cycles. For a reversible cycle: X).o where O' rate of heat transfer, Btu per hour. AW' " power (work rate), Btu per hour. u b average velocity of stream flow, feet per second. Z = height of stream above a oven datum,' feet. m' -- rate of stream flow, pounds per hour. g. -- acceleration of gravity at sea level. g " acceleration of gravity at height of stream. Itshould be noted that, in Equations 6 and 7, u-j- APv = h, or enthalpy. The energy transfer or heat added to the system by con* duction, radiation or convection may .also be expressed -in terms of the properties of. the system: dQ Cdt - medt (8) where C -- me--heat capacity. e -- specific heat capacity of the system. : and it follows, for a reversible process, that: . ',i is independent of the path traversed from.tHe initiaTto.the final state, so long as it is reversible. A property entropy 8 is thus defined as: /. (y).~S'~S ; From the above inequality it also follows that, for ah' irre versible adiabatic process, (S -- Si) > 0. If an engine receives heat at temperature 2\, the entropy decreaseofthe reservoir is Qi/Ti, and, if the engine gives up heat to the receiver at a lower temperature T%, then the entropy increase of these surroundings ia increased by an amount Qi/7V .............. The net entropy increase of the universe is: AiStimk - ASmrM >0 :' In applications to systems at moderate pressures, the specific heat capacity may be regarded as a function of temperature only, or as a constant. Empirical equations relating the spe cific heat capacity as'a function-of temperature are available in the literature.11 Where extremes in pressure are en countered, it may be necessary to consider the effect of pres sure on the specific heat capacity. However/ thermal data' oh specific heat capacity of high pressures is scarce,'and P-o-T datawhich maybe used-to calculate the pressure deviations are rarely of sufficient accuracy to permit'such 'calculations. In many cases, sufficient accuracy may be obtained by em ploying alternate forms of a calculation, in which values of enthalpy and entropy may be used in the appropriate relation ships for heat transfer. and the net energy, made unavailable for doing.work.due to theirreversibilities, is .the above quantity multiplied by ,te; temperature 7t of the receiver, or: . . Q. - T(AjS.iBk -- aS^O,)- . This concept is important in the. analysis of irreversible process, as indicated in Part II,-Refrigeration Cycles, EQUATIONS OF STATE An equation of state of a pure substance is a mathematical relation between toe pressure, specific volume, and the tem perature, when toe system is in thermodynamic equilibrium: t `\ f{P,v,T) - 0 SECOND LAW OF THERMODYNAMICS :- `Th& second law of thermodynamics is a statement to the effect that, if an engine operates- continuously receiving energy from a source at a-given temperature T\ and dis charging energy to a receiver at a lower temperature 7* there is a limit to the amount of work that can be obtained from-the conversion of heat into work, and t.h?R limit depends solely on the temperatures of the source and the receiver. The statement may be adapted tbinctude systems of eiore than one source or sinki ' V.; ' , A number of corollaries follow from this statement: , * 1. No engine can be more efficient than a reversible engine oper ating between these rame temperatures. 2..The efficiency of alhrevermble engines, (cycles) operating .'between these temperatures ia thesame, and: ': -- Experimental evidence indicates the above relation to be valid for any pure substance.-At present, it is not possible to theoretically develop such an equation of state from me chanics. As a result, empirical equations with varying amounts of theoretical significance have been proposed. V ' . The van der Waals equation is an equation of state having theoretical, significance. It qualitatively predicts' the be havior of a substance, but with limited accuracy. : (''+?)('-6) -RT . W The' virial equation, a power series type of expansion in terms ofthe density and temperature coefficients, ' may be used as an equation of state. P. - A +-- ....................... (10) Themio'dynamib and Refrigeration Cycles An advantage of this form is that'statistical mechanics may be used1 as -means-'of predicting toe second ana toiru cuefli- cients.' Tbennodynamiciste have recently concentrated On the partition^ or distribution- function to determine the! virial coefficients.* Im general,-however, experimental values of the second and third'ooefficients'are still preferred. For the more many higher order terms are necessary which can neither be ^ti^aciorily predicted from theory nor be deter mined from expenmental:measurements. In general, a trun cated virial fTparimhn'of three terms is valid for densities less than one-half .of the, .value at. the critical point. For higher densities, additional terms may be used which can be deter mined empirically; as a result) the theoretical implications may be of little.value, except for predicting trends.. The van der. Waals' equation, expressed .in virial form, be comes:; ... ^ i . ; - . -- . {RTb - o) : RTV RT + -------------- + --r- + * (U) The Beattie-Bridgeman equation is an empirical equation of state,that gives excellent results for densities up to approximately.ionfrhalf of the critical density. Expressed in modified virial form it becomes: (12) * where A. AA\ -- a/e) B B.(l - h/v) c/eT* A,, a, B, b. and c are constants. The'coefficients A, B, and C are imique for each sub stance.* _, With the use of high speed computers, equations of state that are far more complex have been .used, with success-in predicting P-o-T values, even to high dehsitiesl The BeHedict- Webb-Rubih equation of. state13 and the. Martiri-Hou equa^ tion,3have had. considerable use, but generally'are limited to ftenwt.ifta less than the critical value. Strobridgeu'h!as sug- gested a modified. Benedict-We^b-Rubin relation that gives excellent results, at higher densities and may be used for a P-o-T surface that extends into the liquid phase.'; J , ,, A ample relation, the ideal gas equation, is suggested if one' plots the product of P-o against P for a'real gas at a constant temperature; as'the pressure approaches zero, toe product becomes`independent of the substance of the gas, and the first virial coefficient A in Equation 10 is a function'of tempera ture: ' --s'-''; * . ' * ' Limit Pe -- A < P--0 'l ` v' ' An ideal gas ia then-defined as: '' Limit P -- 0 --rP = 1545 foot pounds per (mol) (Rankine degree). (13) The virial equation can also be written as:' z .KPpT.(B CU, The ratio Z is the compressibility factor. A useful tool for'predicting properties of a fluid in the ab. sence of' experimental data is -the concept of corresponding states which relate the properties of similar substances by suitable reducing factors, that is, the P-o-T surface of similar fluids in a given region are assumed to .be of similar shape. According to van der. Waals' principle of corresponding states, (he critical point may be used to define the reducing parame- ters to scale the surface of. one fluid to the dimensions of another. Modifications of this principle, &s suggested- by Kamerlingh Onnes, have been successfully used to improve the correspondence at low pressures. In practice, the concepts provide useful approximations, and numerous modifications have been reported in the litera ture. More complex treatments for predicting property values which recognize the similarity of fluids and their P-o-T properties are contained in' the literature on generalized equations of state. These equations ordinarily allow for the adjustment of toe P-o-T surface by the introduction of param eters. An example of this type of generalized equation, given by Hirschfelder ef of,u allows for departures from the principle of corresponding states by using two additional correlating parameters. PART II: COMPRESSION REFRIGERATION CYCLES A refrigeration cycle is one which continually transfers thermal energy from a region of low temperature to a region of higher temperature. The region of higher temperature is usu ally the ambient atmosphere or a space occupied by eco nomically available water. The two most common methods used to accomplish this transfer of thermal energy are the vapor compression refrigeration cycle, discussed in-this sec tion, and the absorption refrigeration cycle, discussed, in Part III. By means of Second-Law analyses, the following will be demonstrated in this section: (1) for a given refrigeration load, the reversed Carnot cycle, having heat transfers through negligible temperature differences, is an ideal standard, and it is the criterion for minimum requirements of work input to a compression refrigeration' cycle,' (2) departures from this ideal standard, such as fluid friction-or heat transfer through a finite temperature'difference, will degrade available energy and will cause an increase in entropy of toe universe, (3) such departures require that the 'work input to toe refrigeration cycle be greater than the ideal minimum. It will be shown that this increased work input is equal to the product of the absolute: temperature of the'atmosphere and the increase in entropy of the universe <rrA.^.ral- which' is also equal to toe sum of the available energies that are degraded by: (1) irre versibilities that are produced by the refrigerant in its cyclical operation, and (2) heat transfers through finite temperature differences. THE IDEAL BASIC VAPOR COMPRESSION REFRIGERATION CYCLE The equipment diagram for this refrigeration cycle is shown in Fig. 1 and the p-h and T-S diagrams are presented in Figs. 2 and 3. At point 1, the refrigerant leaves the evaporator as a low pressure, low temperature, dry, saturated vapor and enters the compressor, where it is compressed reversibly and isentropically. At point 2, it leaves the compressor as a high temperature, high pressure, superheated vapor and enters the condenser, where it is desuperheated and condensed reversibly at constant pressure. At point 3, it leaves the condenser as a high pressure, medium temperature, saturated liquid and enters the expansion valve, where it is expanded irreversibly a.nH adiabatically at constant enthalpy.. At point 4, it leaves the Expansion-valve as a low pressure, low temperature, low quality vapor and enters the evaporator, where it is evapo rated -reversibly at- constant pressure to the dry, saturated state at point 1. Heat transfer to the evaporator and from the condenser both occur with neglible - temperature difference between the fluid'emitting the heat' and the fluid which