This chapter is devoted to investigating the best way to allocate the finite cycle time of a mechanical chiller among its principal components. Namely, for a given fixed cycle time, what should the residence time of the refrigerant be in the compressor, expansion device, evaporator and condenser in order to maximize COP for a prescribed cooling rate. From the perspective of chiller manufacturers, the relative residence time in each chiller component can be viewed as a control variable.
This assertion entails an unorthodox view of precisely what constitutes a control variable at the stage of chiller design. We contend that whether a manufacturer recognizes or treats relative residence time as a control variable is not at issue. The very fact that relative residence time introduces an additional degree of freedom in the design (as opposed to the operation) of the chiller is a sufficient incentive to explore how chiller performance can be ameliorated with respect to it.
We emphasize that the type of optimization considered in this chapter ceases to exist the moment the chiller components have been selected. From the viewpoint of chiller installers and consumers, no optimization
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is implied. Neither does the optimization exercise exist if the chiller manufacturer must accept specific off-the-shelf components.
In material terms, the control variable contemplated here is actually the relative refrigerant charge of each component. At issue is exactly how the total refrigerant mass is distributed among the condenser, evaporator, compressor and expansion device. For a chiller operating at constant refrigerant mass flow rate, at steady state, the relative refrigerant charge in any component is identical to the relative residence time the refrigerant spends in that component (mass = {constant mass flow rate} · time). Relative residence time refers to the actual time the refrigerant resides in a given component, relative to the total cycle time.
The chiller may possess an accumulator that contains a substantial quantity of refrigerant, but accumulators serve to accommodate transient operation. At steady state, essentially all the refrigerant mass is accounted for by the 4 principal components noted above.
Chiller manufacturers may commonly characterize heat exchangers, for example, in terms of their overall UA values and the mass flow rates traversing them. The fluid volume may not be considered as a design variable. In this exercise, however, we take a step back in the design process, and treat the heat exchanger’s mCE product per unit mass of refrigerant as a valid control variable. This is the extra degree of freedom introduced here. Clearly, when the heat exchanger is viewed as one of several components in a chiller cycle, this control variable can be expressed as the component’s relative refrigerant charge, for a given total refrigerant charge in the chiller.
Rather than continuing to refer to the new control variable as relative refrigerant charge, and because it is rigorously equivalent to relative residence time (for a fixed chiller cycle time or, equivalently, a fixed mass flow rate and fixed total charge), we shall call this additional degree of freedom relative residence time, and denote it by the symbol Ξi for component i. Also, in light of the evolution of the discipline of finite- time thermodynamics during the past 25 years, we offer this analysis as a relatively simple but bona fide example of optimizing the thermodynamic performance of real machines with respect to how a given finite time should be apportioned among the device’s elements. In Chapter 6, we explored optimizing a chiller when its heat exchanger inventory is constrained. Here we broaden the optimization to include the finite resource of time, and modify exactly what fixed heat exchanger inventory signifies. The global optimum with the additional control variable of time will now be determined. One benefit of using the thermodynamic model developed in the previous chapters is obtaining analytic results for optimal time divisions and optimal specific (per unit
refrigerant charge) heat exchanger allocation. In developing an analytic predictive chiller model in earlier chapters, we treated a chiller as a sort of input–output device, viewed from the outside and probed only with externally-measurable parameters such as input power, cooling rate and coolant temperatures. Here, we must intrude into the compressor, throttler, condenser and evaporator, because we need to quantitatively characterize the dissipation in each component in order to perform the finite-time optimization. This is one reason why the type of extensive chiller measurements needed for such a study is not common.
Using actual chiller performance data, we’ll see that the design and construction of commercial reciprocating chillers have evolved to the optimal operating strategies calculated from finite-time thermodynamics. This reflects the empirical wisdom embodied in these constructions. (Since both finite time and finite heat exchanger inventory are the constraints of practical interest here, a more appropriate rubric might be finite-resource thermodynamics; but we retain the finite-time appellation for historical reasons.)
We will also show that, for the particular set of constraints that relates to practical designs for manufacturers, maximizing COP is equivalent to minimizing entropy production in the universe (and not just inside the chiller). This point is not trivial because maximum COP and minimum dissipation in the universe (i.e., the combination of the chiller and its surroundings) are not necessarily identical objectives.
B. HOW FINITE TIME ENTERS GOVERNING PERFORMANCE EQUATIONS
Let’s revisit the derivation presented in Chapter 4 for the chiller’s thermodynamic performance.
The energy balance, Equation (4.15), remains unaltered. Consider- ation of the relative residence time of the refrigerant in each of the principal chiller components enters in the entropy balance. Equation (4.16) is modified as follows:
(7.1)
where δScomp = entropy production rate per relative residence time during compression; δSexp = entropy production rate per relative residence time during expansion/throttling;
Ξ
comp = fraction of cycletime for compression (dimensionless); and Ξexp= fraction of cycle time
for expansion.
The relative residence times Ξ will be viewed as control variables
Q Q
T
Q Q
T S S
cond condleak cond
evap evap leak
evap
comp comp exp +
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in the design of optimized chillers. Equivalently, we ask what the optimal allocation of cycle time among the various branches of the refrigerant cycle is. Again, we hasten to stress that varying the relative time allocation does not relate to the operation of a chiller that has already been built. Rather, we are adopting a pre-construction perspective of the chiller. After the chiller has been assembled, the key degrees of freedom considered here cease to exist.
The rate of internal entropy production
∆S
int is given by:∆S
int=δS
compΞ
comp+δS
expΞ
exp.
(7.2)We relate to the compression and throttling branches as adiabatic, which is an excellent approximation for throttling. Non-adiabaticity for compression is generally small, with the degree of deviation being reflected in the low experimental value of
Q
compleak .We also assign all the internal dissipation to the compressor and expansion device. In accordance with common practice in chiller analysis, internal losses in the heat exchangers are treated as negligible or lumped with the internal losses in the other components. In Chapter 12, we’ll show that internal dissipation in heat exchangers is not always negligible, and can noticeably impact diagnostic procedures. For the procedures outlined here, however, the internal dissipation in the heat exchangers of the mechanical chillers has only a small impact on the optimization and therefore is omitted.
Since it is coolant, rather than refrigerant, temperatures that are readily and non-intrusively measurable (as well as heat flows and power input), and in terms of which chiller performance equations should conveniently be cast, the energy balance on the heat exchangers is also expressed in terms of relative residence times:
Qcond =Xcond(mCE)cond¢
d
Tcond -Tcondini
(7.3)(
evap)
in evap evap evap evap (mCE) T T Q =Ξ ′ − (7.4)where
Ξ
cond = fraction of cycle time in the condenser; and Ξevap =fraction of cycle time in the evaporator. Note that the products mCE with a prime (') superscript are essentially values of mCE per relative refrigerant charge, and are equivalently viewed here as mCE values per relative residence time. The heat exchangers can be viewed as autonomous components characterized by (mCE)' values and by design control variables Ξ. The same observation pertains to the compressor
and throttler for the local entropy production rates δS and the relative residence times Ξ. The tradeoff between
Ξ
cond andΞ
evap is a principaloptimization step here in chiller design. By definition, the relative residence times are normalized – the finite-time constraint:
Ξcomp +Ξexp +Ξcond +Ξevap =1. (7.5)
We do not need to know the actual values of the residence times or even the cycle time itself to complete the optimization exercise. The relative residence times
Ξ
are sufficient. We now combine Equation (4.15) with Equations (7.1)–(7.5) above to yield the characteristic chiller curve for 1/COP as a function of 1/Qevap and all key chiller variables:[
( )]
. ) ( ) ( ) ( ) ( ) ( ) ( 1 COP 1 cond cond int evap evap evap in evap int evap evap cond cond evap evap evap int in evap evap evap int evap cond cond in cond ′ Ξ − ∆ ′ Ξ + ∆ − ′ Ξ + ′ Ξ ∆ Ξ ′ − ′ Ξ − ∆ ′ Ξ = + mCE S Q mCE T S mCE mCE Q mCE S T mCE S Q mCE T (7.6)In deriving Equation (7.6), we have neglected heat leaks. The usually small heat leaks exert a negligible influence on the optimal allocation of residence time and heat exchanger inventory. The derivation in the absence of heat leaks also results in simple analytic formulae with which fundamental functional dependences are transparent and easily evaluated.