Investigation of Energy Dissipation in an Ejector ... · ejector refrigeration cycle was modied. An...

Investigation of Energy Dissipation in an Ejector RefrigerationCycle

Christian Tischendorf 1 Denise Janotte 2 Ricardo Fiorenzano 1 Wilhelm Tegethoff 2

1 Technical University Braunschweig, Department of Thermodynamics2 TLK Thermo GmbH

Hans-Sommer-Straße 5, 38106 Braunschweig [email protected]

Abstract

The presented work focuses on the differences in en-ergy dissipation in each cycle component compared tothe energy dissipation of the whole ejector refrigera-tion cycle. With help of this analysis, improvement ofenergetic efficiency by using an ejector can be set inrelation to the potential improvement in efficiency ofother components such as heat exchangers. Informa-tion about entropy production associated with energydissipation allows for an objective estimation of theoptimization potential of each component within anejector refrigeration cycle. In addition, the improve-ment due to the specific process control of the ejectorcycle compared to the conventional heat pump cyclecan be analyzed. The energetic benefit gained using anejector depends on the refrigerant used. The refriger-ants R134a and R744 (CO2) were compared in regardto the entropy production of the heat pump system.

In order to simulate an ejector refrigerant cycle andto evaluate the energy dissipation by means of en-tropy production, existing models for cycle compo-nents were modified. Applying the second law of ther-modynamics, local distribution of entropy productionas well as the overall entropy produced in each com-ponent was determined. The analysis showed that en-tropy production is caused by two types of effects. Onepart results from real effects such as pressure drop andheat transfer, the other part is due to the modeling as-sumptions made. Thus, the investigation of energy dis-sipation leads to a deeper understanding of the model.

The simulated amount of entropy produced is sum-marized in a record, so that the results can be read eas-ily by other programs, e.g. programs that visualize en-ergy and entropy flows. In the presented investigationthe entropy flow and dissipation effects were analyzedby means of diagrams, such as Sankey diagrams.

The complete heat pump system has been simulated

using the Modelica library TIL (TLK-IfT-Library) inorder to determine the energy dissipation in each cyclecomponent. With the modified TIL models, other pro-cess controls can also be investigated. This approachoffers the opportunity to analyze the energy dissipationin detail, and differs in that sense from the commonlyused technique of integrated energy balances and COPdeterminations.

Keywords: entropy analysis; refrigeration; com-pression cycle; simulation; CO2; R134a

1 Introduction

The pressure difference between the high and lowpressures in CO2 refrigerant systems is high comparedto other refrigerants, e.g. R134a. Previous investi-gations by other authors have shown that in a con-ventional refrigeration cycle, the pressure differencescause significant throttling losses. Using an expansionvalve results in an isenthalpic throttling process, whichmeans that the kinetic energy is completely dissipatedand the evaporation enthalpy is reduced compared toan isentropic process.

Using an ejector is one way to recover part of thelost kinetic energy and to increase refrigeration capac-ity. As a result, the energetic efficiency (COP) of therefrigerant system will be improved. In addition to theimprovement caused by an ejector, the COP can beraised by optimization of other cycle components. Thekey issues are the comparative cost effectiveness of themodifications, as well as the question of which com-ponents have the most optimization potential. There-fore, an analytical technique enabling one to recognizeoptimization potential is needed in order to assess thealternative solutions. The required technique was de-veloped during the project presented in this work.

One approach to investigate the potential is to ap-

Proceedings 7th Modelica Conference, Como, Italy, Sep. 20-22, 2009

© The Modelica Association, 2009 304 DOI: 10.3384/ecp09430090

ply the second law of thermodynamics to determinethe produced entropy in each refrigeration cycle com-ponent, so that a basic analysis of the component effi-ciencies is possible. This approach was introduced in[Franke04]. In the presented work, the investigationwas carried out using the Modelica library TIL to sim-ulate the refrigeration cycles. TIL is a component li-brary for steady-state and transient simulation of fluidsystems such as heat pump, air conditioning, refriger-ation or cooling systems, developed by TLK-ThermoGmbH and TU Braunschweig, Institute for Thermo-dynamics [for a detailed description see [Richter08]].The advantage of TIL is that it has a very shallow in-heritance structure, which makes the models easy tounderstand and extend. The results of the entropy anal-ysis were visualized by bar and Sankey charts usingthe newly developed software EnergyViewer by TLK-Thermo GmbH, in order to simplify the interpretationof the effects.

2 Simulated Refrigeration Cycle

Figure 1: Object diagram of the simulated cycle

In the presented work, an ejector heat pump usedfor heating water for domestic use and floor heatingusing R744 or R134a as refrigerant was simulated.The principal functionality of a common ejectorrefrigerant cycle is described in the following liter-ature [Elbel06]. For this investigation, the commonejector refrigeration cycle was modified. An objectdiagram of the modified ejector refrigeration cycle is

shown in figure 1. The cycle consists of the followingcomponents: gas cooler (R744)/ condenser (R134a),medium pressure evaporator (MP), low pressureevaporator (LP), valve, separator, compressor andejector. In addition, a temperature controller anda super-heating controller were added in the cycle.As well, a high pressure controller was added to theejector component model. The heat pumps weresimulated at the following conditions.

heating capacity for both operation modes:5000 Wtemperature floor heating water:30 ◦C to 35 ◦Ctemperature domestic hot water:10 ◦C to 60 ◦Coverall heat transfer capability gas cooler/condenser:1400 W/Koverall heat transfer capability evaporator LP/MP:500 W/Kheat exchanger pressure drop R744 refrigerant:1 barheat exchanger pressure drop R134a refrigerant:0,2 barwater mass flow rate evaporatorequal for all simulationswater temperature of heat source10 ◦Chigh pressureset to optimize the conditions in the gas cooler/condenser(low mean driving temperature difference)

The water that serves as a heat source first flowsthrough the evaporator MP and afterward through theevaporator LP. The temperature controller controls thespeed of the compressor, such that the desired outputtemperature is achieved by a constant water mass flowrate through the gas cooler or condenser. The con-troller integrated in the ejector controls the high pres-sure level by setting the mass flow rate through thedriving nozzle. The controller determines a high pres-sure level, which can be adjusted such that optimalconditions are found in the gas cooler or condenser.The super-heating controller is used to create optimalconditions in the evaporator LP.

3 Effects Causing Entropy Produc-tion in the Cycle Components

An important distinction must be made between en-tropy production caused by numerical errors based on


© The Modelica Association, 2009 305

modeling assumptions and the entropy produced dueto real physical effects. The numerical errors dependon the mathematical model as well as on the degreeof discretization when using a discretized model. Thereal physical effects depend on the quality and the con-struction of the refrigeration cycle components andcan be influenced by the specific process control. Inaddition to this, the produced entropy is dependent onthe refrigerant used. In the presented work the follow-ing entropy producing effects generated in the cyclecomponents were investigated:

• Heat Exchangerpressure drop, heat transfer and numerical errors(modeling)

• Valvepressure drop (isenthalpic expansion)

• Compressorefficiency based model

• Ejectorspecial efficiency based model

• Separatormixing effects

3.1 Heat Exchanger

Figure 2: Illustration of cell structure of a tube

The heat exchangers used for the investigation con-sist of tubes connected via heat ports. The numberof tubes and the direction in which the medium flowsthrough them can vary. The manner of connectionspecifies the kind of heat exchanger. For this paper,counterflow heat exchangers consisting of one liquidand one refrigerant tube were used. The medium usedin the liquid tube was water. The tubes are dividedinto cells, and each tube is comprised of two types ofcells: wall cells and fluid cells (see figure 2). The fluidcells can be either liquid or refrigerant cells, depend-ing on the type of medium flowing through them. Theconnection of fluid and wall cells via heat ports allowsthe exchange of heat between the cells. The temper-ature of the connected heat ports of two cells are set

equal. The heat transfer inside the cell between themedium and the heat port is determined by the heattransfer relation and the heat transfer coefficient. Inthe cell model, equations to determine the heat floware implemented. A similar modeling of heat trans-fer and fluid flow is presented in [Patankar80]. Thenumber of wall and fluid cells in each tube determinesthe degree of discretization (finite volume approach).An introduction to the finite volume approach can befound in [Baumann06]. In order to model the entropyproduction within the heat exchangers, a hierarchicalapproach was followed. First the entropy productiondue to the aforementioned effects was determined foreach cell of the heat exchanger. The second law ofthermodynamics for a transient system yields:

d(sm)dt

=n

∑i=1

misi + SQ + SQprod + S∆p

prod + SMprod (1)

With d(sm)dt being the change of entropy of the cell.

The terms ∑ni=1 misi and SQ specify the entropy con-

veyed by mass and heat flows. SQprod , S∆p

prod , SMprod rep-

resent the entropy production rate due to heat transfer,pressure drop and modeling respectively. Heat trans-fer and pressure drop are real effects that cause en-tropy production, which can be determined by formu-las presented later. The last production term, however,is due to the modeling of the cell as a volume withconstant medium properties such as enthalpy or tem-perature. For a fluid cell this can be illustrated by theimage of an agitator stirring the medium inside the cellso that it is perfectly mixed. Because of the modelingassumptions, the enthalpy of the medium inside thecell matches the enthalpy at the outlet port of the cell.Likewise, the temperature of the medium inside thecell matches the temperature at the outlet port of thecell. If there is a temperature change, this mixing ofthe cell content produces entropy. This entropy pro-duction SM

prod is numerical error and differs from SMi,Tprod

the entropy production due to mixing of two streamsof ideal gas ma and mb with temperature Ta and Tb re-spectively which is presented in [Cerbe07] as follows:

SMi,Tprod = macma|tMi

ta lnTMi

Ta+ mbcmb|tMi

tb lnTMi

Tb(2)

with TMi being the mixing temperature and cm themean specific heat capacity of stream a or b. HenceSM

prod has to be determined via the second law. Trans-forming equation 1 and setting dm

dt = ∑ni=1 mi yields:



SMprod = m

dsdt−

n

∑i=1

mi(si− s)− SQ− SQprod− S∆p

prod (3)

The index i labels the variables of stream i flowingin or out of the cell. The variables of the medium in-side the cell do not have index labels. Each fluid cellhas only one inlet min and one outlet stream mout . Themodeling assumption yields s = sout so that the equa-tion can be simplified further.

SMprod = m

dsdt− min(sin− s)− SQ− SQ

prod− S∆pprod (4)

Without heat transfer and pressure drop, there canstill be entropy produced by continuously mixing thefluid inside the cell. In the case that the specific en-tropy at inlet and inside the cell are not equal, the lastterm in the equation does not reduce to zero. This oc-curs if the temperatures are not equal because the spe-cific entropy depends on pressure and temperature.

SMprod = m

dsdt− min(sin− s) (5)

Each fluid cell emits a heat flow Q at the heat portwith the temperature Thp. This flow conveys entropythat can be determined by the following equation.

SQ =Q

Thp(6)

Before being emitted, the heat flow is transferredfrom the medium inside the cell (temperature T ) to theheat port (temperature Thp). The temperature gradientbetween Thp and T is determined by the heat trans-fer relation and the heat transfer coefficient. In thecell model, the equations for the heat transfer phe-nomena are implemented. This heat transfer causesentropy production, which can be determined accord-ing to [Bejan88] as follows.

SQprod =

QT− Q

Thp(7)

Pressure drop between the inlet and outlet port alsoleads to entropy production. It can be determined aspresented in [Bejan88].

S∆pprod = m

∫ in

out

vT h=const

d p (8)

This formula can be linearized

S∆pprod = m

vT

∆p (9)

∆p represents the pressure drop.

Equation 1 has to be adapted for the determinationof the entropy production inside a wall cell. Sincethere is no medium flowing through the wall cell, thereare no terms for entropy transportation via mass flow,and no pressure drop occurs. The entropy productiondue to the modeling of the wall cell can be determinedaccording to the following equation:

SMprod = m

dsdt− SQ− SQ

prod (10)

Heat can be emitted at each of the heat ports i withthe overall entropy conveyed because of heat flow be-ing SQ = ∑

ni=1 SQ,i. Each of the summands SQ,i can be

determined according to equation 6. The heat trans-fer from the medium inside the cell to one heat portor vice versa causes entropy production. Each produc-tion term SQ

prod,i can be determined in accordance withequation 7 and has to be summed in order to deter-mine the overall entropy production due to heat trans-fer, SQ

prod = ∑ni=1 SQ

prod,i.The entropy production resulting from modeling

assumptions SMprod (see equation 10) represents only

small numerical errors. There is no mixing inside thewall cell and the term SM

prod can be neglected.In order to calculate the entropy production due toeach effect for the whole tube, the results of the en-tropy production in the cells are summed. The resultsof the refrigerant and the liquid tube are then sum-marized to determine the entropy produced within thewhole heat exchanger.

3.2 Valve

In the valve model (control valve) the refrigerant isthrottled adiabatically. It is assumed that the kineticenergy of the flowing refrigerant is completely dissi-pated. Since the valve has been modeled as a compo-nent without volume, the second law for the valve hasbe applied in the steady state form.

n

∑i=1

misi + STprod = 0 (11)

As the mass flow rate mi and the specific entropy si

at the inlet and outlet ports are known, this equation isused to determine the overall produced entropy.



STprod =−

n

∑i=1

misi (12)

3.3 Compressor

The compressor model is based on an efficiency modelwith isentropic, volumetric and effective isentropic ef-ficiency. Furthermore, a heat flow from the housingsurface to the surroundings is considered, but is setto zero in the simulation. The main entropy produc-tion in the compressor is caused by the following lossmechanisms, which result from the compression pro-cess of a reciprocating type compressor. These are:friction losses due to mechanical components such aspistons and piston rings and swash plate, throttlinglosses that take place in the valves, flow channels andchambers, the heat transfer between the different com-pressor components, e. g. the suction and dischargechamber, the leakage losses and losses caused by thecontrol valve. Since the compressor is modeled as avolumeless component, the second law for the com-pressor has to be applied in steady state form:

n

∑i=1

misi + SQ + STprod = 0 (13)

SQ specifies the entropy flow that is discharged withthe heat flow.

3.4 Ejector

When a medium is throttled in a conventional valve,friction losses occur and the kinetic energy of themedium is completely dissipated. In an ejector, how-ever, the kinetic energy of a primary flow at high pres-sure can be partly used to compress a secondary flowand thus to diminish the compression work done in thecompressor. Figure 3 illustrates the functionality of anejector.

The driving flow exits the driving nozzle with a highvelocity and carries with it a flow from the suction noz-zle. Since the cross section of the suction nozzle be-comes progressively narrower, the suction flow is ac-celerated and the pressure drops. Both the suction anddriving flows exchange momentum, are mixed in themixing tube and flow through a diffuser before leavingthe ejector. The pressure inside the diffuser increaseswith decreasing velocity.

The ejector is modeled using an analogous modelconsisting of several separate components. Because

Figure 3: Exploded view of an ejector

the exchange of momentum is not ideal and entropywill be produced, the ejector can partly recover the ki-netic energy. However, the entropy produced has notbeen determined in detail for this work. A simplifiedapproach has been used instead, and the entropy pro-duction rate for steady state has been determined as thedifference between inlet and outlet entropy flow ratesaccording to the second law in steady state.

STprod =−

n

∑i=1

misi (14)

3.5 Separator

The separator model is similar to the refrigerant cellmodel apart from the fact that no heat transfer or pres-sure loss is considered, and that the refrigerant is notmixed, but separated into liquid and gaseous phase.The entropy production rate is determined by the fol-lowing equation.

STprod = m

dsdt−

n

∑i=1

misi (15)

As there are no effects producing entropy STprod is

equal to zero.

3.6 Implementation of Entropy Productionin TIL

In the Modelica library TIL, the specific entropy s isimplemented as a function of p and h. In order to re-duce the index of the differential algebraic equationsystem, the terms in the entropy equation have to beexpressed by means of state variables used for the de-scription of the system. This implies that the termdsdt (derivative of the specific entropy with respect totime) had to be rewritten in terms of p and h. Totransform the equation Bridgeman tables, which can



be found in [Bejan88] were used. This made the im-plementation more numerically effective by not rely-ing on Dymola for the rearrangement of the equationsystem. The discussed mathematical equations to de-termine the entropy production were implemented ineach model. The collection of the results in the sum-mary records simplifies the readout by other programsused to analyze the results.

4 Visually Supported Analysis ofSimulation Results

The simulation results were analyzed with the help of2d-plots which show the temperature relations, andSankey diagrams which show the entropy flow be-side the production rates. Therefor the entropy flowsat the inlet and outlet port of each component weredetermined. The entropy production caused by heattransfer, pressure drop, numerical errors due to model-ing and the overall entropy production were calculatedaccording to the aforementioned equations and illus-trated by bar charts. The visualized entropy flows werenormalized for each medium such that the flow withthe lowest specific entropy is equal to zero. With thehelp of the Sankey diagrams, the entropy shift withinthe flows is visualized, and for each medium the rela-tion of the entropy flows between the components be-comes clear. Sankey diagrams are a powerful visual-ization method to analyze all kind of flows. A detaileddiscussion of the application of Sankey diagrams canbe found in [Schmidt06].

4.1 Entropy Production in Heat Exchangers

The diagrams in figures 4 and 5 illustrate the tempera-ture curves in the gas cooler (R744) and the condenser(R134a) of a steady-state ejector refrigeration cycle fordomestic hot water. It can be clearly seen that the tem-perature glide in the gas cooler (R744) leads to a lowermean driving temperature difference in the gas coolerthan in the condenser (R134a). In the evaporators, therefrigerant is evaporated at a nearly constant temper-ature and exits the heat exchangers with low super-heating, which is controlled by the controller. Thisresults in a lower mean driving temperature differenceand lower entropy production due to heat transfer inthe evaporators. See figure 7 and 9 for entropy pro-duction in the evaporators in relation to entropy pro-duction in the gas cooler / condenser. In figure 6 theprofile of entropy production due to different effectsis illustrated for the cells of the gas cooler. The cor-

Figure 4: Temperature profile of refrigerant, liquid andwall cells inside the condenser (R134a domestic hotwater)

Figure 5: Temperature profile of refrigerant, liquid andwall cells inside the gas cooler (R744 domestic hot wa-ter)

Figure 6: Profile of entropy production inside the gas-cooler (R744 domestic hot water)

responding temperature profile is shown in figure 5.Wherever high temperature differences occur, the pro-duced entropy due to heat transfer and mixing (model-ing assumption) is high. The entropy production ratedue to pressure drop is low in comparison to the pro-duction rate due to heat transfer and mixing.



Figure 7: Entropy flow and production rate in an R744heat pump cycle used to heat up hot domestic water.Entropy production rate due to heat transfer (SQ), pres-sure drop (S∆p), modeling (SM) and total entropy pro-duction rate (ST ) for each component are representedby bar charts. The entropy flow is normalized to thelowest specific entropy of each medium. Entropy pro-duction and flows are represented in different scales.

4.2 Comparison R744 ejector cycle for do-mestic hot water and floor heating

The figures 7 and 8 show the Sankey diagrams of asteady state R744 ejector refrigeration cycle for do-mestic hot water and floor heating operation respec-tively. First, looking at the gas cooler, it is shown thatthe entropy production caused by heat transfer in thecase of floor heating operation is higher than in thecase of domestic hot water operation. This is due to thehigher driving temperature difference in the gas coolerin the floor heating operation. However, the entropyproduction caused by mixing (modeling assumption)in the case of floor heating operation is lower than inthe case of domestic hot water operation, despite thegreater refrigerant and water mass flow rates. The rea-son for this is the lower temperature gradients betweenthe inlet and outlet of the refrigerant as well as liquidtubes. With the heating capacities equal in both thefloor heating and domestic hot water operation modes,in the floor heating mode, the compressor and the ejec-tor produce more entropy, although the water mass

flow rate and the refrigerant mass flow rate are greater.That shows that the R744 ejector refrigeration cycle isbetter suited to heat water up to higher temperatures.

4.3 Comparison R744 and R134a EjectorCycle for Domestic Hot Water

The Sankey diagrams of a steady state R744 andR134a ejector refrigeration cycle for domestic hot wa-ter mode are illustrated in figure 7 and 9. The resultsshow that entropy production caused by heat transferin the condenser (R134a) is significantly higher thanin the gas cooler (R744). This is due to the highermean driving temperature difference between the re-frigerant and the water in the condenser. The temper-ature curves in the condenser and the gas cooler areillustrated in figure 4 and 5. Because of the temper-ature glide in the gas cooler, the mean driving tem-perature difference is lower. For the same reason, theentropy production resulting from mixing (modelingassumption) is also higher in the condenser. In theR744 refrigeration cycle, production of entropy in thecompressor and ejector is higher, the reason for thisbeing the different refrigerant properties and processcontrols. Because of this, the pressure difference be-tween the high and low pressure level in the R744 re-frigeration cycle is higher. An ejector heat pump usingR744 as refrigerant is suited to supply domestic hotwater better than a heat pump using R134a, if partic-ular attention is directed to the entropy production inthe condenser or gas cooler. In addition, the energysaving potential with an ejector is higher in an R744cycle than in an R134a cycle.

5 Conclusions and Outlook

The presented work shows how an analysis of thedissipation effects in thermodynamic systems can bedone with the help of simulation. For this purposethe equations are presented which are needed to math-ematically describe the observed entropy-producingphenomena. In particular, the entropy production inthe heat exchangers is examined. Entropy productionresulting from heat transfer, pressure drop and numer-ical error due to modeling are observed. Using the ex-ample of an ejector heat pump, it is shown how theresulting simulated entropy production can be visual-ized and used in the dissipation analysis. The analy-sis is carried out using bar diagrams, Sankey diagramsand 2d-plots. Heat pumps using R134a and R744 werecompared for both domestic water heating and floor



Figure 8: Entropy flow and production rate in an R744heat pump cycle used to heat water for floor heating.

Figure 9: Entropy flow and production rate in a R134aheat pump cycle used to heat domestic hot water.

heating. The investigation shows that the heat pumpwith R744 is better suited for domestic hot water oper-ation. It is shown that this analysis method is suitablefor investigation of thermodynamic systems on the ba-sis of entropy production. In the future, it is planned

to research whether the boundaries of the system canbe altered to produce an even better analysis or not. Inaddition, it is planned to carry out a more detailed anal-ysis of entropy production within the ejector model.

References

[Baumann06] Baumann W., Bunge U., Fredrich O.,Schatz M., Thiele F. Finite-Volumen-Methodein der Numerischen Thermofluiddynamik. Tech-nische Universität Berlin, Institut für Strö-mungsmechanik und technische Akustik: Vor-lesungsmanuskript, Berlin, 2006.

[Bejan88] Bejan A. Advanced Engineering Thermo-dynamics. John Wiley & Sons, New York, 1988.

[Bejan02] Bejan A. Fundamentals of Exergy Analy-sis, Entropy Generation Minimization, and theGeneration of Flow Architecture. In: Interna-tional Journal of Energy Research vol.26 no.7p.545-565, 2002.

[Cerbe07] Cerbe G., Wilhelms, G. Technische Ther-modynamik, Hanser Verlag, München, 2007.

[Elbel06] Elbel S., Hrnjak P. Development of a Proto-type Refrigerant Ejector used as Expansion De-vice in a Transcritical CO2 System,PresentationVDA Alternative Refrigerant Winter MeetingSaalfelden, 2006.

[Franke04] Franke U. Thermodynamische Prozess-analyse: Ursachen und Folgen der Irreversibil-ität. Shaker, Aachen, 2004.

[Patankar80] Patankar S. Numerical Heat Transferand Fluid Flow. Hemisphere Publ. Co, NewYork, 1980.

[Richter08] Richter C. Proposal of New Object-Oriented Equation-Based Model Libraries forThermodynamic systems. Braunschweig, Ger-many: PhD Thesis, Department of MechanicalEngineerging, Institute of Thermodynamics, TUBraunschweig, 2008.

[Schmidt06] Schmidt M. Der Einsatz von Sankey-Diagrammen im Stoffstrommanagement.Beiträge der Hochschule Pforzheim, Nr. 124,Pforzheim, 2006.



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