An iterative method for modelling the air-cooled organic Rankine cycle geothermal power plant

INTERNATIONAL JOURNAL OF ENERGY RESEARCH

Int. J. Energy Res. 2011; 35:436–448

Published online 29 April 2010 in Wiley Online Library (wileyonlinelibrary.com). DOI: 10.1002/er.1706

An iterative method for modelling the air-cooledorganic Rankine cycle geothermal power plant

M. Imroz Sohel1,�,y, Mathieu Sellier2, Larry J. Brackney3 and Susan Krumdieck2

1Scion, Te Papa Tipu Innovation Park, 49 Sala Street, Rotorua, New Zealand2Department of Mechanical Engineering, University of Canterbury, Private bag 4800, Christchurch, New Zealand3Commercial Building Systems Electricity, Resources, and Building Systems Integration Center, National Renewable Energy

Laboratory (NREL) 1617 Cole Blvd, Mailstop 5202, Golden, CO 80401, U.S.A.

SUMMARY

This work presents an iterative method for modelling the effect of ambient air temperature on the air-cooledorganic Rankine cycle. The ambient temperature affects the condenser performance, and hence the performance ofthe whole cycle, in two ways. First, changing the equilibrium pressure inside the condenser, the turbine outletpressure and the turbine pressure ratio vary. Since the turbine pressure ratio is a major parameter in determiningthe power generated by a turbine, the plant output is directly affected. Second, changing the condenser outlettemperature with ambient temperature, the pump inlet and outlet conditions are changed. Thus, the vapourizerequilibrium temperature and pressure are influenced. The developed method iteratively seeks the equilibriumconditions for both the condenser and vapourizer. Two case studies based on a real plant performance have beencarried out to demonstrate the validity of the method. The developed method demonstrates robustness andconverges regardless of the initial conditions allowed by the physical properties of the working fluid. This method iseffective for cycles that use saturated vapour as well as superheated vapour under static or dynamic conditions withappropriate initial conditions and constraints. The developed method may be applied to any Rankine cycle withclosed cycle operation. Copyright r 2010 John Wiley & Sons, Ltd.

KEY WORDS

geothermal power plant; air-cooling; organic Rankine cycle; performance analysis

Correspondence

*M. Imroz Sohel, Scion, Te Papa Tipu Innovation Park, 49 Sala Street, Rotorua, New Zealand.yE-mail: [email protected]

Contract/grant sponsor: University of Canterbury

Received 17 November 2009; Revised 11 March 2010; Accepted 12 March 2010

1. INTRODUCTION

Geothermal resources, concentrated solar energy andwaste heat are being increasingly investigated for

power generation. The Kalina Cycle has been investi-gated for low-temperature resources, but limitedcommercial application has been demonstrated [1–4].The advantages of two component working fluids have

been theoretically demonstrated [5], but organicRankine cycle (ORC) power plants have been foundto be the most economic and proven technology [6].

The literature of major applications of the ORCinclude combined heat and power [7,8], waste heatrecovery [9–16], solar thermal application [17–19],

biomass heat and power plants [20,21] and geothermal[6,22–26].

The effect of ambient temperature on the gas turbine

power plant is well studied [27–29], but it differs sig-nificantly from the ORC. In a gas turbine power plant,the ambient air temperature dictates the density of air

and hence the power output. In an ORC, the ambientair temperature affects the plant performance byaffecting the heat rejection from the system.ORC geothermal power plants often use air-cooled

condensers that make these power plants more sus-ceptible to weather conditions. Generally, these plantsare designed assuming a reasonable, stable, ambient

temperature. As the ambient temperature increases,especially during the summer, the performance of anORC plant is significantly reduced [25].

Power companies make the bulk of their revenuewhen demand is high. Therefore, it is very important

Copyright r 2010 John Wiley & Sons, Ltd.

for them to have high power capacity to maximize thebenefit during high demand periods. The sensitivity ofbinary cycle power plants (such as the ORC) to am-

bient conditions creates challenges for producers topredict power output accurately. A model that canpredict hourly plant performance on a daily basis

would be very useful for optimizing capacity.Both fundamental and empirical modelling methods

have been used to predict binary cycle plant perfor-mance. Investigations based on the second law of

thermodynamics are prevalent [6,25,30]. The ambientair acts as the heat sink temperature for an air-cooledcondenser plant. As the ambient temperature increases,

the Carnot efficiency decreases,

Z ¼ 1�TL

THð1Þ

In Equation (1), TL and TH are the absolute tem-peratures of the heat sink and source, respectively, and

Z is the Carnot efficiency.The plant performance may also be expressed as

an empirically derived function of various operatingparameters including ambient temperature and con-

denser working pressure and temperature. Empiricalmodelling approaches are common practice in industryand are well described in the literature [31–33]. The

plant performance is generally expressed as a functionof the turbine pressure ratio result in a set of charts orcurves. Such fitted curves are often plant specific and

only apply for a narrow range of operations, renderingthem limited for addressing the broader class of plantdesign and optimization problems.

The effect of condenser pressure on plant perfor-mance has been widely discussed in the literature[31–34]. The general consensus is that increasingcondenser pressure generally results in a decrease in

power output. A clear discussion on how ambienttemperature affects the condenser performance andhence the overall plant performance is not readily

available in the literature. Durmayaz and Sogut [35]presented an iterative method to calculate thecondenser equilibrium condition of a pressurized

water reactor nuclear power plant with cooling watertemperature affected by weather conditions. Theiterative approach allows the performance of thewhole plant to be calculated using fundamental rather

than empirical relationships.The present work introduces an iterative method

for modelling a closed ORC power plant. The heat

sink (ambient) temperature affects the condenserperformance, consequently influencing the wholecycle performance in two ways. First, changing the

equilibrium pressure inside the condenser leads tochanges in both the turbine outlet pressure and therelated turbine pressure ratio. Second, changing the

condenser outlet temperature (via the heat sink tem-perature) results in changes in both the pump inletand outlet conditions. This, in turn, influences both the

vapourizer equilibrium temperature and pressure. Themethod presented here seeks the equilibrium conditionsof both the condenser and the vapourizer based on

initial conditions (supplied to the model as the startingpoint of the calculation) and external factors. Stabilityand rate of convergence of the developed method are

important considerations and are also discussed in thispaper. Two case studies are presented that demonstratethe application of the iterative modelling method.For modelling, MatLab [36] interfaced with the thermo-

physical property database REFPROP [37] hasbeen used.

2. THE ITERATIVE METHOD

Figure 1 shows a T– s presentation of a Rankine cycle.Process 4-1 represents constant pressure heat rejectionby the condenser. Applying the first law of thermo-

dynamics (neglecting kinetic and potential energies) foran open system, we can write:

_mcycleðh1 � h4Þ ¼ _Qcon ð2Þ

where _mcycle is the mass flow rate in the cycle, h1 is the

enthalpy at state point 1, h4 is the enthalpy at statepoint 4 and _Qcon is the condenser heat load. Since,process 4-1 is an isobaric process:

p4 ¼ p1 ð3Þ

The condenser heat load, _Qcon, is a function of

inlet condition (state point 4), outlet condition (statepoint 1), heat sink temperature, mass flow rate in thecycle and design of the condenser. During steady-state

operation of a plant, the mass flow rate in the cycle isconserved. We assume that the design of a particularcondenser is fixed. Therefore, condenser inlet and

outlet conditions and the heat sink temperature arethe primary parameters influencing the condenserperformance.A comprehensive model of the condenser heat load

was developed previously using the fundamentals of

2s

4s

s

4

2

1

3

T

Figure 1. T–s diagram of a Rankine cycle.

Air-cooled ORC geothermal power plant method for modelling M. I. Sohel et al.

Int. J. Energy Res. 2011; 35:436–448 r 2010 John Wiley & Sons, Ltd.

DOI: 10.1002/er

437

thermodynamics, heat transfer and condenser designpresented elsewhere [26].Assuming that TD is the designed heat sink tem-

perature of the condenser, the mass flow rate, _mdesign, inthe cycle at sink temperature, TD, may be expressed as:

_mdesign ¼ _mcycle ¼_Qcon

ðh1 � h4Þð4Þ

Since the condenser heat load is directly related tothe ambient temperature at constant vapour–liquidequilibrium condition, if TD increases or decreases, the

condenser heat load varies inversely resulting in alower or higher mass flow rate in the cycle. This has anadverse effect on the cycle performance as power

plants are generally optimized for a specific operatingcondition. The only way to maintain the operatingcondition is to change the vapour–liquid equilibrium

state in the condenser to cool the same amount ofworking fluid.The following method is used to find the required

vapour–liquid equilibrium condition in the condenser

to maintain constant mass flow:Step 1: Calculate _Qcon based on the heat sink tem-

perature using a condenser model [26] then a value of_mcycle is calculated from Equation (4).Step 2: If _mcycle ¼ _mdesign, the condenser is operating

at the designed vapour–liquid equilibrium condition

and no further calculation is necessary.Step 3: If _mcycleo _mdesign, the equilibrium pressure,

p1, is reduced until _mcycle ¼ _mdesign and h1 is calculatedas h1 ¼ fðp1; x ¼ 0Þ, where x represents quality.

Step 4: If _mcycle4 _mdesign, the equilibrium pressure,p1, is increased until _mcycle ¼ _mdesign and h1 is calcu-lated as h1 ¼ fðp1; x ¼ 0Þ.The back pressure of the condenser dictates the

turbine outlet pressure. For the positive flow to occur,the condenser pressure must be less than the turbine

outlet pressure (p1op4). In practice, the turbine outletpressure is slightly higher than the condenser equili-brium pressure.

Recalling Figure 1, process 1-2 represents isentropiccompression of the working fluid by the cycle pump.Owing to the fact that the pump work is relativelysmall for an ORC, it can be assumed constant [38]. The

work input to the pump may be calculated as:

_Wpump ¼ _mcycleðh2 � h1Þ ð5Þ

The enthalpy at the boiler/vapourizer outlet is de-scribed by

_mcycleðh30 � h2Þ ¼ _Qin ð6Þ

where _Qin is the heat input to the system and we havedefined a hypothetical intermediate state 30, which is atrial solution for state 3. Knowing h30 and x5 1, all

other thermodynamic properties related to state 30 maybe calculated (more complex problems such as superheating are discussed in the case study section).

Applying the first law of thermodynamics (neglect-ing kinetic and potential energies) for an open system[39] for the process 3,0 to 30, the following equation is

obtained:

du ¼ u30 � u3;0 ¼ h30 � h3;0 � ðp30v30 � p3;0v3;0Þ ð7Þ

where u is the internal energy and v is the specific

volume. The subscript 3,0 represents the initial condi-tion of state 3, which is calculated from the suppliedoperating pressure and temperature (typical) values of

the vapourizer to start the calculation.The specific volume (v) of the fluid inside the boiler/

vapourizer can change if the operating parameters

(p, T) change from one steady-state operating point toanother (i.e. from 3,0 to 30 or 30 to 3). However, if thevolume inside the boiler/vapourizer is kept unchangedby means of any control mechanism (which is normally

the case), then the specific volume (v) can be assumedconstant.In the intermediate processes (i.e. from 30 to 3), no

work is done by the system nor does any heat transfertake place. Rather, equilibrium is attained by theaddition or departure of some mass to or from the

system, which ensures constant specific volume ofthe system. The relative quantity of mass (comparedwith the holdup mass) to be added or departed to or

from the vapourizer to attain equilibrium in thevapourizer is very small. If we neglect the effect of thisadded or departed mass, then the process from 30 to 3can be assumed to be closed system operation. Now,

applying the first law of thermodynamics for a closedsystem [39] and neglecting the kinetic and the potentialenergies:

du30�3 ¼ dQ30�3 � dW30�3 ð8Þ

where dQ30�3 and dW30�3 are infinitesimally small, sodu303 should tend to zero for the hypothetical process 30

to 3 at equilibrium condition. Now, Equation (7) can

be re-arranged to identify state 3 as:

u3 � u3;0 ¼ h3 � h3;0 � ðp3 � p3;0Þv3;0 ð9Þ

Equation (9) can be solved iteratively for x5 1 by

altering p3, until the left-hand side of Equation (9)equals du of Equation (7). Knowing p3 and x5 1, therest of the thermodynamic properties associated with

state 3 may be determined.If we assume that the work done by the turbine is

done isentropically, then:

s3 ¼ s4 ð10Þ

Knowing s4 and p4, the rest of the thermodynamicproperties associated with state 4 may be determined.Work done by the turbine is:

_WT ¼ _mcycleðh4 � h3Þ ð11Þ

Equation (11) presents the ideal work done ( _WT)

by the system. However, owing to irreversibilitiesassociated with the processes (i.e. heat transfer tothe surroundings, mechanical losses, etc.), the actual

Air-cooled ORC geothermal power plant method for modellingM. I. Sohel et al.

438 Int. J. Energy Res. 2011; 35:436–448 r 2010 John Wiley & Sons, Ltd.

DOI: 10.1002/er

electric power (Pel) produced by the unit is less thanthe ideal:

Pel ¼ ZT _mcycleðh3 � h4Þ ð12Þ

where ZT is the turbine-generator efficiency.The heat transfer to the cycle is:

_Qin ¼ UADTm ð13Þ

where _Qin is the heat input to the cycle, U is the overall

heat transfer coefficient, A is the heat transfer area andDTm is the log mean temperature difference. A is con-stant for a heat exchanger. The overall heat transfer

coefficient is calculated as a function of geothermal fluidmass flow rate using an approximate method [40,41]:

U ¼ Urð _m= _mrÞ0:5 ð14Þ

where Ur and _mr are the reference overall heat transfer

coefficient and the reference mass flow rate. These twoparameters are determined via system identificationmethods.

With typical operation of a heat exchanger involvinga phase change, DTm remains almost unchanged [42].Therefore, Equation (13) can be reduced to:

_Qin ¼ _Qin;rð _m= _mrÞ0:5 ð15Þ

where _Qin;r is the reference heat input.An initial guess of the inlet state of the working fluid

to the condenser is provided to the model. By workingour way around the cycle, we predict a new inletcondition with an improved value. Subsequent itera-

tions around the loop yield better results and theprocess converges to a unique solution within a fewiterations. Figure 2 shows the effect of the ambient

temperature on condenser heat load and equilibriumpressure of a superheated vapour ORC unit. Thecondenser equilibrium pressure is very sensitive to theambient air temperature and that explains the strong

dependence of the performance of an air-cooledcondenser geothermal power plant on ambient airtemperature. Details of the modelling involved are

presented in later sections.

3. CONVERGENCE, STABILITY ANDUNIQUENESS

The solution based on the developed method convergesexponentially if the search parameter (equilibrium

pressure) is updated each iteration proportionally tothe relative error, similar to the Kalman filter [43, 44].Figure 3 illustrates the typical convergence of the mass

flow rate using Equation (4) to find the vapour–liquidequilibrium pressure for a typical air-cooled condenserand the typical convergence of du in Equation (9) while

searching for the vapour–liquid equilibrium pressure ina typical vapourizer. Here:

ðp1Þn11 ¼ ðp1Þn� k1ðe1Þn ð16Þ

and

ðp3Þn11 ¼ ðp3Þn� k2ðe2Þn ð17Þ

where p3 and p1 are the vapourizer and condenser

pressures, respectively, k1 and k2 are tuneable para-meters and n represents the iteration number. For thesolutions shown, k1 5 0.2 and k2 5 0.1 were used. e1and e2 are calculated according to:

ðe1Þn ¼ ð _mdesign � ð _mcycleÞnÞ= _mdesign

��

�� ð18Þ

ðe2Þn ¼ ðdu� ðdueqbrmÞnÞ=du��

�� ð19Þ

The stability of the solution lies in the choice ofvalues for the two constants k1 and k2. Larger values ofk improve the rate of convergence, but may introduce

instabilities. Lower values of k improve stability at theexpense of convergence. Appropriate tradeoffs may beachieved by suitable tuning.

Independent properties (i.e. p, T, v, h and s) arepoint functions, meaning that they are not dependenton path. Any solution obtained from the methoddescribed here must be unique. As long as the pressure

values fall within the limits allowed by the thermo-physical properties of the working fluid, one shouldobtain the same solution regardless of the initial values

of pressure used in Equations (16) and (18), and (17)

0.00E+00

5.00E+03

1.00E+04

1.50E+04

2.00E+04

2.50E+04

3.00E+04

-5 0 5 10 15 20 25 30

Ambient temperature [°C]

Co

nd

ense

r h

eat

load

[kW

]

0

20

40

60

80

100

120

Co

nd

ense

r eq

uili

bri

um

pre

ssu

re [

k P

a]

heat load pressure

Figure 2. Condenser heat load and condenser equilibrium pressure as a function of ambient temperature.



DOI: 10.1002/er

439

and (19). Figure 4 shows this to be the case by de-

monstrating the convergence of du with two differentinitial values of equilibrium pressure. The continuousline presents the solution obtained using an initial

guess of the equilibrium pressure of 1010kPa. Thedashed line shows the solution obtained when aninitial guess of 2020 kPa was used. Both solutions con-verged to a unique operating value of 1818kPa for the

equilibrium.

4. EFFICIENCY

The cycle efficiencies are calculated as follows:

Z1 ¼_Wnet

_Qin

ð20Þ

Z2 ¼_Wnet

m:De

ð21Þ

where Z1 and Z2 are first and second law (energetic andexergetic) efficiencies, respectively. _Wnet is the net workdone by the cycle, m

:is the geothermal fluid flow rate

and De is the specific exergy input to the ORC. Thebrine inlet and outlet conditions of the ORC’s arecontrolled; hence, De can be assumed constant for our

case. The value of De can be calculated as [6]:

De ¼ hin � hout � T0Ds ð22Þ

where hin and hout are the inlet and outlet enthalpies ofthe geothermal fluid, respectively. Ds is the differencebetween the inlet and outlet entropies and T0 is the

equilibrium temperature (dead state temperature). Forsimplicity, the equilibrium temperature is assumed tobe 251c.

5. CONSTRAINTS

The performance of system components (e.g. boiler/vapourizer and condenser) is constrained by the systemdesign. The maximum and minimum allowable pres-

sures and temperatures of these devices are predefined.The plant performance is dependent on these limits aswell as operator interaction to maintain operating

conditions for maximum output. Such constraintsmust be applied to the equilibrium condition obtainedby the iterative method. Therefore, all iterations

(Equations (4)–(19)) must be terminated and assignedfeasible values if extremum operating conditions arereached.

0

2

4

6

8

10

12

14

16

18

1

Iteration [-]

Ab

solu

te e

rro

r [k

g/s

]0

2

4

6

8

10

12

14

16

18

20

Ab

solu

te e

rro

r [k

J/kg

]

2 3 4 5 6

Figure 3. Convergence of mass flow using Equation (4) and convergence of du using Equation (9) to calculate the vapour–liquid

equilibrium pressure.

-40

-30

-20

-10

0

10

20

30

1

Iteration [-]

Ab

solu

te e

rro

r [k

J/kg

]

2 3 4 5 6 7 8 9

Figure 4. Convergence of du using two different initial conditions of equilibrium pressure.



DOI: 10.1002/er

6. CASE STUDY 1: SATURATEDVAPOUR ORC

A geothermal steam-driven ORC with pentane as theworking fluid has been chosen for this case study. This

unit works as a bottoming unit of a steam turbine. Thegeothermal resource temperature is 2051C for both ofthe case studies. The schematic diagram of the process

is presented in Figure 5. There are four basic processesinvolved:

� Reversible adiabatic pumping process in the

pump.� Constant pressure heat transfer in the vapourizer.� Reversible adiabatic expansion in the turbine.

� Constant pressure heat transfer in the condenser.

There is no direct measurement of mass flow rate inthe physical implementation of the cycle concerned. Anapproximation is obtained using Equation (12), Z5 1

and the generated electric power (Pel). Using theiterative method, the unit is modelled for 48 h of op-eration and compared against observed data. The ex-

perimental initial conditions are supplied to the modelas a starting point for the simulation. The efficiency ofthe turbine is estimated to be 0.76. A constant pressure

loss between the turbine outlet and condenser inlet isassumed to be the nominal observed value of 10.1 kPa.Condenser heat load, Qcon ¼ fðp4;T4;Tamb; vairÞ, is

calculated from the developed condenser model [26].Figure 6 presents the observed and modelled con-

denser outlet temperatures. The average error (absolute)

is 3.27%. Figure 7 compares the observed and modelledvapourizer outlet pressure with an average percentageerror of 2.15%. Figure 8 compares the observed and

modelled vapourizer outlet temperatures with an aver-age error of 1%. Figure 9 compares the observed andmodelled electric power output of the system with an

average error of 4.20%. The iteration is terminatedwhen the tolerance limit, |e|5o0.1, is reached. Figure 10presents the relative error (absolute value) of modelledelectric power output of the saturated vapour unit. The

relative errors largely lie within 10%. Figure 11 presentscorresponding first law and second law efficiencies. Thefirst law efficiency varies between 7 and 9% and the

G

VAPORIZER

M

CONDENSATE TANK

CYCLE PUMP

STEAM

CONDENSATE

GENERATOR

12

3

44

STEAM

WORKING FLUID

AIR COOLED CONDENSER

TURBINE TURBINE

Figure 5. Schematic of the saturated vapour cycle.

20

25

30

35

40

45

50

55

60

0

Time [h]

Tem

per

atu

re [

°C]

ObservedModelled

10 20 30 40 50

Figure 6. Condenser outlet temperatures for the saturated

vapour unit over 48 h of operation.



DOI: 10.1002/er

441

second law efficiency varies between 37 and 47% de-pending on the ambient air temperature and geothermalfluid flow rate.

7. CASE STUDY 2: SUPERHEATEDVAPOUR ORC

Figure 12 presents the schematic of the superheatedvapour ORC unit used for case study 2. This cycle is

driven by separated brine from the geothermal fluid(2051C). Superheating in the cycle adds some complex-ity. However, the same iterative approach may be used

to implement the system model. From a modellingperspective, there are two basic differences between thesaturated and superheated vapour ORC cycles: super-

heating and the addition of a recuperator for heatrecovery. The maximum temperature of a superheatedvapour cycle is typically much higher than that of a

saturated cycle. Considering the second law ofthermodynamics, the effect of ambient air (heat sink)variation will be less prominent in the superheatedcycle compared with a saturated cycle [30].

The pressure loss between the vapourizer outlet andturbine inlet is assumed constant and assigned anobserved value of 50.5 kPa. The pressure loss in the

recuperator is assumed constant and fixed at 50.5 kPa(4a-4), which is consistent with the observed value.

300

350

400

450

500

550

600

650

0

Time [h]

Pre

ssu

re [

k P

a]

ObservedModelled

10 20 30 40 50

Figure 7. Vapourizer outlet pressures for the saturated vapour

unit over 48 h of operation.

60

65

70

75

80

85

90

95

100

105

0

Time [h]

Tem

per

atu

re [

°C]

ObsevedModelled

10 20 30 40 50

Figure 8. Vapourizer outlet temperatures for the saturated vapour unit over 48 h of operation.

1000

1500

2000

2500

3000

3500

4000

4500

5000

0

Time [h]

Ele

ctri

c p

ow

er [

kW]

ObservedModelled

10 20 30 40 50

Figure 9. Electric power output for the saturated vapour unit over 48 h of operation.



DOI: 10.1002/er

The condenser equilibrium condition and corre-sponding outlet temperature are derived in the samemanner as in the case of the saturated vapour ORC. If

the pump input work is assumed constant, the enthalpyof state 2 is calculated using Equation (5) and theenthalpy of state 3 is calculated using Equation (6). In

the superheated cycle, heat input is given as

_Qin ¼ _Qbrine1 _Qrecuparator ð23Þ

where _Qbrine is the heat input to the system from brine

and _Qrecuparator is the heat recovered from the turbineexhaust pentane vapour. Since the heat recovered inthe recuperator is related to the mass flow rate of

pentane (which remains relatively constant), the heatrecovered is also approximately constant. Heat inputto the system from geothermal brine is equal to

_Qbrine ¼ _Qvaporizer1 _Qseparator ð24Þ

here _Qvaporizer and _Qseparator are calculated usingEquation (15).

The state at point 4 is determined as before, as-suming isentropic expansion in the turbine. The valueof electric power output is calculated using Equation

(12) with an approximate turbine efficiency of 0.9.

Figure 13 presents the observed and modelled con-denser outlet temperatures with an average error of6.54%. Figure 14 shows the observed and modelled

vapourizer pressures with an average error of 1.58%.Figure 15 presents the observed and modelledvapourizer outlet temperatures with an average error

of 1.56%. Figure 16 presents the observed and mod-elled electric power output of the unit with an averageerror of 3.16%. Figure 17 presents the relative error

(absolute value) of modelled electric power of the su-perheated vapour, which shows that the relative errorremained within 7%. Lastly, Figure 18 presents thecorresponding first law and second law efficiencies.

The first law efficiency varies between 13 and 15% andthe second law efficiency varies between 37 and 43%.

8. DISCUSSION

Most of the modelling works available in the literaturecompare their results against experimental dataperformed under controlled environments [9,45]. In a

real plant, uncertainty of a physical model increasesover time due to degradation of the plant [46]. In

0

2

4

6

8

10

12

14

16

1Data point [-]

Rel

ativ

e er

ror

[%]

7 13 19 25 31 37 43

Figure 10. Observed relative error in modelled electric power output for the saturated vapour unit over 48 h of operation (48 data

points).

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

1

Time [h]

Sec

on

d la

w e

ffic

ien

cy [

-]

0.05

0.06

0.07

0.08

0.09

0.1

0.11

0.12

0.13

0.14

0.15

Fir

st la

w e

ffic

ien

cy [

-]

η2

η1

6 11 16 21 26 31 36 41 46

Figure 11. First and second law efficiencies of the saturated vapour unit over 48 h of operation.



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443

VAPORIZERSEPARATOR

AIR COOLEDCONDENSER

CYCLE PUMP

TURBINE

BRINE SUPPLY

BRINE RETURN

G

M

Working fluid

Brine 12

3

4

4a

RECUPERATOR

Figure 12. Schematic of the superheated vapour cycle.

0

5

10

15

20

25

30

35

40

0

Time [h]

Tem

per

atu

re [

°C]

ObservedModelled

10 20 30 40 50

Figure 13. Condenser outlet temperatures for the superheated vapour unit over 48 h of operation.

1000

1100

1200

1300

1400

1500

1600

1700

1800

1900

2000

0

Time [h]

Pre

ssu

re [

kPa]

ObservedModelled

10 20 30 40 50

Figure 14. Vapourizer outlet pressures for the superheated vapour unit over 48 h of operation.



DOI: 10.1002/er

recent years, application of artificial neural networksand genetic methods for modelling thermal power

plant has become popular [46,47]. These modelsprovide a high degree of accuracy without complicatedphysics-based models and also address the problem ofincreased model uncertainty with age. However, such

models are very plant specific and cannot be readilyused in conceptual design and developments. Incontrast, the method developed here is based on

fundamental thermodynamics and could be very usefulin conceptual design and development.

In the saturated vapour cycle, the error lies largelywithin 10% (Figure 13). Only one data point is found to

lie above 14%. However, the external parameters, i.e.ambient temperature and geothermal fluid flow rate, forthis data point are very similar to the neighbouring datapoints. Therefore, this point can be assumed to be noise

or an ‘outlier’ [46]. Table I summarizes the average andmaximum relative errors of the saturated vapour cycleand the superheated vapour cycle. Wei and co-workers

[9] have reported a maximum relative error of 4% fortheir semi-empirical model. Quoilin and co-workers [45]

100

120

140

160

180

200

220

0

Time [h]

Tem

per

atu

re [

°C]

ObservedModelled

10 20 30 40 50

Figure 15. Vapourizer outlet temperatures for the superheated vapour unit over 48 h of operation.

0

500

1000

1500

2000

2500

3000

3500

4000

4500

5000

0 10 20 30 40 50

Time [h]

Ele

ctri

c p

ow

er [

kW]

ObservedModelled

Figure 16. Electric power output for the superheated vapour unit over 48 h of operation.

0

1

2

3

4

5

6

7

0

Data point [-]

Rel

ativ

e er

ror

[%]

10 20 30 40 50

Figure 17. Observed relative error in modelled electric power output for the superheated vapour unit over 48 h of operation

(48 data points).



DOI: 10.1002/er

445

have reported the maximum error of their model to lie

within 10% and commented that their error was aconsequence of cumulated subcomponent models in-accuracies. Smrekar and co-workers [46] have reporteda maximum relative error of 7.19% of a boiler model of

a real power plant. They have used artificial neuralnetworks for the modelling purpose. In our work, themaximum error remained at 10% and seems consistent

with the existing literature. Moreover, the efficiencies ofboth of the ORC units (energetic and exergetic) foundto be very consistent with the literature [6].

It should be noted here that we have compared thereal plant performance of a decade old plant with themodel. There are a minimum number of inputs require

for our model. The developed model is very genericand can be used for conceptual design, analysis andoptimization. Therefore, the method presented in thispaper can be considered reasonably accurate (in the

context of existing methods) and very useful.

9. CONCLUSION

This work has introduced an iterative method for

modelling a closed ORC. The heat sink temperature, inthis case the ambient temperature, affects the modelledcondenser performance. Consequently, it influences the

performance of the whole cycle. This occurs in twoways: (i) changes in the equilibrium pressure inside thecondenser result in a change in turbine outlet pressure

and pressure ratios and (ii) changes in the condenseroutlet temperature caused by the heat sink temperaturealso affect the pump inlet and outlet conditions as well

as the vapourizer equilibrium temperature–pressure.These are competing effects. However, changes related

to the turbine pressure ratio tend to dominate the power.Calculating the vapour–liquid equilibrium condition

of the condenser was performed by assuming that the

mass flow rate in an ORC in steady-state operationremains relatively constant. The vapour–liquid equili-brium condition of the vapourizer is found by assum-

ing that the specific volume inside the vapourizer isunchanged for steady–state operation. Termination ofthe iterative search for unique state solutions isachieved when reaching a slack equilibrium condition

within a prescribed tolerance or by meeting a con-straint. As the model essentially assumes steady-stateoperation of the power cycle, the possible unit time

where this model can be applied is bounded by the timerequired by a system to come into steady state. Thesaturated vapour cycle yielded an average error of

4.20% with a maximum error of 9.25% and thesuperheated vapour cycle yielded an average error of3.16% with a maximum error of 6.48%. The mainadvantage of using the developed method lies on the

fact that it requires a minimum number of inputs:condenser (p,T), vapourizer (p,T), condenser heat load,turbine efficiency (overall), pump work and the

extremum conditions of all the components. Theseinputs should represent typical operating conditions ofa plant. The model can predict the appropriate plant

performance depending on the system heat input(geothermal fluid flow in this case) and the heat sinktemperature. As the method is based on basic ther-

modynamics, rather than empirical or semi-empiricalapproaches, it is widely applicable. The main focus ofthis work is on the ORC but the developed method isapplicable to any closed Rankine cycle.

NOMENCLATURE

A 5 area (m2)e 5 relative error (�)h 5 specific enthalpy (kJ kg�1)

Table I. Summary of results of observed and modelled electric

power output.

Average error

(|e|) (%)

Maximum error

(|e|) (%)

Saturated vapour cycle 4.20 9.25

Superheated vapour cycle 3.16 6.48

0.2

0.25

0.3

0.35

0.4

0.45

1

Time [h]

Sec

on

d la

w e

ffic

ien

cy [

-]

0.125

0.135

0.145

0.155

0.165

0.175

0.185

Fir

st la

w e

ffic

ien

cy [

-]

η1η2

6 11 16 21 26 31 36 41 46

Figure 18. First and second law efficiencies of the superheated vapour unit over 48 h of operation.



DOI: 10.1002/er

k1 5 proportionality constant (–)k2 5 proportionality constant (–)m:

5mass flow rate (kg s-1)

p 5 pressure (kPa)Pel 5 electric power (MW)Q 5 heat transfer (MJ)_Q 5 heat transfer rate (MW)s 5 specific entropy (kJK�1 kg�1)T 5 temperature (1C)u 5 specific internal energy (kJ kg�1)

U 5 overall heat transfer coefficient(MWm�2K�1)

v 5 specific volume (m3 kg�1)

vair 5 velocity of air (m s�1)W 5work done (MJ)

W:

5work rate or power (MW)

x 5 quality (–)

Subscript

amb 5 ambientcon 5 condenserH 5 high

L 5 lowin 5 inputr 5 referenceT 5 turbine

1–4 5 states

Greek letters

Z 5 efficiency (�)

ACKNOWLEDGEMENTS

This work was funded and supported by the Universityof Canterbury.

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An iterative method for modelling the air-cooled organic Rankine cycle geothermal power plant