IMPROVEMENT OF DOUBLE-FLASH GEOTHERMAL POWER … · 1 IMPROVEMENT OF DOUBLE-FLASH GEOTHERMAL POWER PLANT DESIGN: A COMPARISON OF SIX INTERSTAGE HEATING PROCESSES Joachim-André Raymond

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IMPROVEMENT OF DOUBLE-FLASH GEOTHERMAL POWER PLANT DESIGN:

A COMPARISON OF SIX INTERSTAGE HEATING PROCESSES

Joachim-André Raymond Sarr, François Mathieu-Potvin*

(bold font weight for family names)

Department of mechanical engineering, 1065, Avenue de la Médecine, Université Laval,

Quebec City, Quebec, Canada, G1V 0A6

Abstract

In this paper, six different modifications of the Double-Flash power plants are proposed. These

modifications are named “interstage heating” and consist of additional heat exchangers properly

located in the system. The six interstage heating designs are analysed, optimized and then compared

to an optimized Double-Flash reference power plant. The objective function is the power plant

specific output (kJ/kg), and the design variables are the separator temperatures (°C) and the split

fraction. Optimizations are performed for a wide range of reservoir temperatures (i.e., from 140 °C

to 240 °C). Results show that interstage heating processes may increase the specific output of the

plant by about 5%, decrease the liquid content in the low pressure turbine by about 50%, and

decrease the required cooling capacity of the plant by about 10%. On the other hand, the analysis

showed that the new designs proposed have negligible influence on the high pressure turbine liquid

content or on the silica saturation coefficient.

Keywords: Double-Flash, Geothermal power, Interstage heating, Superheating, Heat exchanger.

* Corresponding author : François Mathieu-Potvin, ing. jr, Ph.D. Professor Department of mechanical engineering 1065, Avenue de la Médecine, Université Laval, Quebec City, Province of Quebec, Canada, G1V0A6 Tel.: 1-418-656-2131 x 5409 Fax.: 1-418-656-7415 Email: [email protected]

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Nomenclature

Variables

ih enthalpy at state i , 1kJ kg−

Rm� mass flow rate from the geothermal reservoir, 1kg s−

iP absolute pressure at state i , kPa

Q� heat transfer rate in heat exchanger, kW

maxQ� maximum heat transfer rate in heat exchanger, kW

outQ� geothermal plant waste heat, kW

liq,maxQ� maximum heat transfer rate based on the liquid side, kW

vap,maxQ� maximum heat transfer rate based on the vapour side, kW

iS silica concentration (amorphous form) at state i , ppm

eqS silica concentration at equilibrium (amorphous form), ppm

qS silica concentration (quartz form) in reservoir, ppm

is entropy at state i , 1 1kJ kg K− −

iT temperature at state i ,°C

RT reservoir temperature, °C

S1T first separator temperature,°C

S2T second separator temperature,°C

outnetW� geothermal plant power output, kW

w geothermal power plant specific output, 1kJ kg−

ix vapour content at state i

y split coefficient

{.} bracket for identifying a thermodynamic state

Greek symbols

,sα α ideal and actual superheated states leaving heat exchanger

,sδ δ ideal and actual compressed liquid states leaving heat exchanger

ε heat exchanger efficiency

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λ moisture content

0η turbine dry efficiency

Subscripts

atm atmospheric

f, g saturated liquid and saturated vapour states

fg evaporation (change from liquid to vapour)

i thermodynamic state i

max maximized value

opt optimized value

Abbreviations

CD condenser

DF Double-Flash cycle

HX heat exchanger

LC liquid-cooling design

LS liquid-splitting design

MH mixture-heating design

PP1 pump

PP2’ conditionnal pump

RH reheating design

SH superheating design

SP1 first separator

SP2 second separator

TB1 first turbine

TB2 second turbine

VA1 first valve

VA2 second valve

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1. Introduction

Geothermal power plants of the Flash category (i.e., Single, Double or Triple-Flash) produce ~63%

of the electricity generated from geothermal resources around the world, the remaining percentage

being mainly generated by Dry Steam and Binary geothermal power plants (DiPippo, 2012). A

specific characteristic of Flash power plants is that steam is at a saturated vapour state when

entering turbines, contrarily to fossil fuel power plants in which steam may be superheated up to

temperatures imposed by turbine blade material limits (Nag, 2008). That feature (i.e., saturated

steam at the entrance of turbines) induces several difficulties in Flash power plants. For instance,

low temperature (i.e., low enthalpy) steam yields relatively small turbines power output;

furthermore, water droplets may appear during steam expansion, which reduces the turbine overall

efficiency (e.g., Leyzerovich, 2005) and increases the blade erosion rate (e.g., Ahmad et al., 2009).

A trivial solution to the problems mentioned above is to superheat the steam with a

complementary fossil fuel boiler systems (i.e., hybrid power plants (Bidini et al., 1998)). However,

when low CO2 emissions are expected, that solution is not acceptable. Conversely, a novel

alternative (that does not involve fossil fuel combustion) was proposed by DiPippo and Vrane

(1991), and consists of superheating steam by means of a heat exchanger adequately located in a

Double-Flash power plant. That process was called “interstage reheat”, and was later re-examined

by DiPippo (2013). More recently, Mathieu-Potvin (2013) proposed a design named “self-

superheating” that also uses a heat exchanger in Single-Flash and Double-Flash designs, and which

allows steam at the entrance of turbines to be superheated.

In view of the potential advantages provided by the designs proposed by Mathieu-Potvin

(2013) and DiPippo (2013), six different architectures that could improve Double-Flash power

plants are proposed in this paper. These strategies are labelled here as “interstage heating”

processes, and are thoroughly analysed and compared.

The main goal of the work presented here is to improve the Double-Flash power plant

design. More specifically, the objectives are: (i) to develop mathematical and numerical models for

six different interstage heating designs, (ii) to optimize these designs, and (iii) to compare the

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performances of these six optimized designs with a typical Double-Flash design. The objective

function is the specific output (kJ/kg), and the design variables are the separator temperatures (°C)

and the split fraction. The other parameters investigated in the text are the liquid content in turbines,

the waste heat to power output ratio, and the silica saturation coefficient.

The paper is organized as follows: first, the design of a standard Double-Flash cycle is

described, and its weakness are highlighted (Section 2); second, six interstage heating designs are

proposed and described (Sections 3, 4 and 5); third, the six interstage heating designs are optimized

and compared to the optimized Double-Flash reference design (Sections 6, 7 and 8). Then, final

discussions and conclusions are provided (Sections 9 and 10).

2. Problem statement

A brief description of the Double-Flash power plant considered in this paper and of its

corresponding optimization problem is provided in this section. This specific thermodynamic cycle

is considered as the reference design, and various methods will be proposed in Sections 3 and 4 in

order to improve its performance.

2.1. Double-Flash reference design

A classic design for Double-Flash (DF) systems is illustrated in Fig. 1a, and its corresponding

temperature-entropy ( )T s− diagram is illustrated in Fig. 1b. The working fluid in DF cycles

typically comes from deep natural geothermal reservoirs where geological conditions are

favourable for producing hot and pressurized water (state {1} in Fig. 1). Notice that although a

fluid coming from deep underground geothermal systems typically contains dissolved gases (such

as H2S (Thorsteinsson et al., 2013)), silica (Sugita et al., 2003) and calcite (Hébert et al., 2010), it

is assumed in this paper that its properties may be approximated as equal to those of pure water,

which is in line with recent literature (e.g., Jalilinasrabady et al., 2012; Pambudi et al., 2014).

Moreover, it can be seen in Fig. 1b that the thermodynamic state of the fluid in the reservoir (state

{1}) is located at the left side of the saturated line. Indeed, reservoirs considered in this paper are

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assumed to contain water as compressed liquid; reservoirs which contain water as a saturated

mixture or superheated steam could be included in further studies.

The geofluid begins its path in the reservoir (see state {1} in Fig. 1a) and flows in a well so

as to reach the well-head. Once it has attained the power plant, the geofluid enters valves (VA1

and VA2) and separators (SP1 and SP2) in which liquid and steam phases are divided. The steam

then expands in turbines (TB1 and TB2) so as to produce work, and heat is rejected to the

environment by means of a condenser (CD). A pump (PP1) is used to raise the pressure of the fluid

before it is evacuated. Moreover, for designs in which the pressure at state {7} is smaller than

atmospheric pressure, a “conditional” pump (PP2’) is installed in order to raise the pressure of the

liquid up to the atmospheric pressure (i.e., to state {7’}), which prevents any backflow of the

geofluid in the separator. Notice that the apostrophe symbol is used to identify conditional

equipment (e.g, PP2’) and conditional thermodynamic states (e.g., state {7’}). The Double-Flash

cycle considered in this paper involves a specific configuration of the turbines; for instance, steam

at state {8} and at state {5} are mixed (which brings the fluid to state {9}) before entering the

second turbine (TB2).

The two-turbines assembly illustrated in Fig. 1a is thermodynamically equivalent to a dual-

admission/single-flow turbine in which the low pressure steam (state {8}) would merge directly

with the partially expanded steam (state {5}) (e.g., Fig. 6.6 in DiPippo (2012)). However, the

configuration studied in this paper (Fig. 1) is chosen because it allows various interstage heating

strategies (by inserting interstage heat exchangers) which would not be possible with a unique dual-

admission turbine.

It should also be noted that other Double-Flash power plant architectures exist. For

examples, turbine TB1 (present in Fig. 1) could reject steam directly into the condenser (instead of

mixing state {5} with state {8}), and a fraction of the steam flow could be redirected toward

ejector/condenser apparatus (see Fig. 5 in Zarrouk and Moon (2014)). These variations would

change the thermodynamic cycle that represents the power plant. However, the design presented

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in Fig. 1 contains the most important equipment in Double-Flash cycles and will be used as a

benchmark for evaluating the performance gains brought by the new interstage heating processes.

Finally, a detailed thermodynamic analysis is required to calculate the performance of the

Double-Flash design. That development is not included in this section, but is instead consigned to

Appendix A to lighten the text.

2.2. Parameters of interest

In this paper, four parameters of interest have been chosen for comparing the designs proposed :

(i) the specific output w , (ii) the liquid content λ , (iii) the silica saturation coefficient eqS S , and

(iv) the waste heat to power output ratio netout outQ W� � . These parameters are valid for the Double-

Flash design presented in Section 2.1, but also for other designs presented later in the text.

2.2.1. Specific power output

The specific power output (or more concisely, the “specific output”) expresses the amount of net

mechanical energy (kJ) produced by a power plant for each quantity (kg) of geofluid that is

extracted from a geothermal reservoir. This figure of merit may typically be written as

out

net

R

net power output

reservoir mass flowrate

Ww

m= =�

� (1)

Investigation of power plants based on the specific output has been performed, for example, in

(Vetter et al., 2013). Notice that the calculation of w for the DF cycle presented in Section 2.1 is

detailed in Appendix A.

2.2.2. Liquid content at turbines outlet

When steam expands in turbines, water droplets appear, which cause a decrease of turbine

efficiencies (e.g., Leyzerovich, 2005) and premature turbine blade erosion (e.g., Ahmad et al.,

2009). As recalled in DiPippo (2013), this problem is more frequent in geothermal or nuclear power

plants because the steam entering turbines is typically saturated (i.e., not superheated). In this

paper, the significance of this phenomenon is measured by the value of the liquid content λ at the

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turbine outlets, in line with Mathieu-Potvin (2013). Referring to Fig. 1, the liquid content at the

outlet of turbine TB1 (i.e., 5λ ) and of turbine TB2 (i.e., 10λ ) may be written,

( )S2 S25 5 g@ 5 fg@1

T Tx h h hλ = − = − (2)

and

( )CD CD10 10 g@ 10 fg@1

T Tx h h hλ = − = − (3)

where x is the quality of the saturated mixture. Notice that all designs presented in this paper are

defined in a consistent manner, so that Eqs. (2) and (3) are always valid.

2.2.3. Silica saturation coefficient

It was mentioned in Section 2.1 that geofluids coming from geothermal reservoirs typically contain

silica. Indeed, when the geofluid experiences changes of temperature, silica dissolved in water may

solidify and obstruct equipment of the geothermal power plant. The tendency of silica to undergo

solidification can be quantified by a dimensionless number called the "silica saturation coefficient".

Silica saturation coefficient was taken into account in recent work (Clarke and McLeskey Jr., 2014;

DiPippo, 2013).

The silica saturation coefficient may be calculated as follow. First, the silica concentration

qS in the fluid of the geothermal reservoir is considered as being equal to the equilibrium

concentration of the quartz form (Eq. (6.25) in DiPippo (2012)) taken at the reservoir temperature,

and can be expressed as

( ) ( )

( ) ( )

2R R

4 3 7 4R

q

R

41.598 0.23932 0.011172

1.1713 10 1.9708 10

T T

T T

S

− −

=

+ −

+ × − × (4)

where RT is the reservoir temperature (°C). Then, since the silica remains only in the liquid phase

of the fluid, its concentration ( )7S at the outlet of the second separator (state {7}) may be expressed

as

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q7

2 6(1 )(1 )

SS

x x=

− − (5)

where i

x represents the vapour content of the fluid at state i . The tendency of silica to solidify at

the outlet of the second separator may be determined by comparing 7S to the equilibrium

concentration eqS of the amorphous form calculated at S2T . As mentioned in Eq. (2.3) of DiPippo

(2012), the value of eqS may be calculated as

2 5 2 9 3

S2 S2 S2( 1.34959 1.625 10 1.758 10 5.257 10 )eq 10 T T T

S− − −− + × − × + ×= (6)

Finally, the silica saturation coefficient may be determined by the expression 7 eqS S . More

specifically, when 7 eqS S is smaller than 1, no solidification should occur; when 7 eqS S is larger

than 1, solidification may occur. Hence, in this paper, the objective regarding the silica

concentration is to reduce the value of 7 eqS S .

2.2.4. Waste heat to power output ratio

For a given net power output netoutW� , geothermal power plants must have cooling towers

approximately 8 times larger in cooling capacity outQ� than that of other types of power plants (e.g.,

coal-fired, nuclear, or combined steam and gas), see p. 96 in DiPippo (2012). In other words, the

cooling system is many times more costly for geothermal power plants than for other types of

power plants (for a given value of netoutW� ) . Reducing the waste heat to power output ratio net

out outQ W� �

could provide a reduction of the cooling system initial cost; hence, the ratio netout outQ W� � of the

designs presented in this paper will be compared. To the authors’ knowledge, exhausive analysis

of geothermal power plant based on waste heat to power output ratio has not yet been performed

in literature. Finally, notice that the developed expression of netout outQ W� � for the Double-Flash design

is given in Appendix A.

2.3. Optimization problem statement for Double-Flash design

To assess the best performance of the Double-Flash reference design for a given set of external

conditions (reservoir temperature, etc.), its operation parameters have to be optimized. The main

figure of merit (objective function) in this paper was chosen to be the specific output w . This choice

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of objective function is justified by the fact that each additional kJ produced results in additional

income for the power plant operator, which is certainly one of the most important factor to ensure

economic sustainability of the power plant. Nonetheless, other parameters which take into account

available energy in the geofluid (such as exergy and utilization efficiency (DiPippo, 2015)) could

also be used as an objective function in future work about interstage heating.

The operation parameters (design variables) are the separator temperatures S1T and S2T , in

line with previous Double-Flash design analyses. This choice is justified because thermofluid

equipment (turbines, pumps, pipes, valves, separators, etc.) may be chosen and/or sized so as to

obtain desired separator temperatures. It should be noted that separator temperature values are

limited by the reservoir temperature (upper bound), by the condenser temperature (lower bound)

and by themselves (no overlap of S1T and S2T values). The values of other variables (such as

turbines dry efficiency 0η , reservoir temperature RT and condenser temperature CDT ) are

considered as fixed for a given optimization. To summarize, the optimization problem for the

Double-Flash reference design may be stated as

Problem statement:

( )

( )

[ ]

[ ]

( )

S1 S2

S1 S2 R

S2 CD S1

R R CD 0

maximize

by optimizing ,

,respecting

,

given , , ,

w

T T

T T T

T T T

T P T η

∈ ∈

(7)

The values of other parameters of interest described in Section 2 (i.e., the liquid content λ , the

silica saturation coefficient 7 eqS S , and the waste heat to power output ratio netout outQ W� � ) are used

for comparison once the optimizations have been performed.

2.4. Proposal of improvements for Double-Flash design

Unlike power plants driven by fossil fuel (oil, coal-fired, or natural gas) in which water can be

superheated as desired, geothermal Flash power plants generate steam by means of separators,

which can only yield saturated steam. As explained earlier, saturated steam leads to water droplet

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appearance in turbines, and the low enthalpy of that steam at the entrance of the turbines yields low

power output.

In this paper, six designs are proposed to improve that situation. These designs consist of

adding a heat exchanger in the reference Double-Flash power plant. The idea is to superheat the

steam entering the second turbine (TB2) by using a fraction of the heat already present in the liquid

leaving the first separator (SP1). Following a previous work of DiPippo and Vrane (1991) in which

a specific design using that idea was named “interstage reheat”, the family of improved designs

presented in this paper is referred to as “interstage heating” processes. These processes are further

divided into two categories: (i) Liquid-Cooling designs, and (ii) Liquid-Splitting designs. By using

these designs, it is expected that the value of w will be increased, while the values of λ , netout outQ W� �

and 7 eqS S will be decreased.

Notice that the six interstage designs investigated in this paper are directly inspired from

the work recently performed by Mathieu-Potvin (2013), by DiPippo (2013) and also by DiPippo

and Vrane (1991). For the record, ideas presented in Mathieu-Potvin (2013) came to the author

after an extensive research of the patent filed by Weir brothers (Weir and Weir, 1876) who invented

regenerative feed-water heating processes in Rankine cycles, and also after reading two texts about

their subsequent industrial activities (Reader et al., 1971; Weir, 2008). On the other hand, ideas

presented by DiPippo (2013) and by DiPippo and Vrane (1991) came to these authors after a careful

analysis of reheating strategies used in nuclear power plants, which have similar saturated steam

conditions at turbine inlets to those present in geothermal power plants.

3. Liquid-Cooling (LC) strategies

In this Section, it is proposed to improve the performance (i.e., increasing the specific output w)

of Double-Flash thermodynamic cycles by adding a heat exchanger in the power plant. Notice that

the common feature of the three modified cycles proposed in this section (Section 3) is that the

heating process is performed by cooling the entire mass flow rate of liquid leaving the first

separator, which explains the name chosen for this category (i.e., Liquid-Cooling (LC)).

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3.1. Double-Flash Liquid-Cooling Superheating design (DF/LC/SH)

A first modification of the Double-Flash cycle consists of adding a heat exchanger (see Fig. 2a)

that allows heat transfer from the relatively hot liquid (state {3}) leaving the first separator (SP1)

to the relatively cold steam (state {8}) leaving the second separator (SP2). This Double-Flash (DF)

cycle with Liquid-Cooling (LC) and Superheating of steam (SH) is referred to as the DF/LC/SH

cycle.

The consequence of this modification is that steam from separator (SP2) reaches state { }α

before being mixed with state {5} (see Fig. 2b), which means that the fluid at the entrance of the

second turbine (TB2) has a higher enthalpy and hence could produce more work. On the other

hand, due to the heat exchanger (HX), the enthalpy of the liquid at the entrance of the valve VA2

is reduced (from state {3} to state { }δ in Fig. 2b) which results in a smaller amount of steam

leaving the separator SP2 and entering the turbine TB2. There is thus opposing trends with this

proposed design (steam with higher enthalpy, but smaller amount of steam), and thus, the benefits

of this interstage heating process have to be assessed by a rigorous numerical analysis. The detailed

thermodynamic analysis of that design is provided in Appendix B.

3.2. Double-Flash Liquid-Cooling Reheating design (DF/LC/RH)

The reference Double-Flash design may also be modified (see Fig. 2c) by allowing a heat transfer

between the relatively cold steam (state {5}) leaving the turbine TB1 and the relatively hot liquid

(state {3}) leaving the separator SP1. This Double-Flash (DF) cycle with Liquid-Cooling (LC) and

Reheating of steam (RH) is referred to as the DF/LC/RH cycle. That cycle was first proposed by

DiPippo & Vrane (1991) and was later analysed with modern tools by DiPippo (2013). The

corresponding T s− diagram is illustrated in Fig. 2d. It was shown in DiPippo (2013) that this

strategy yields higher specific output compared to the reference Double-Flash design. However,

the conditional pump (PP2’) and the condenser pump (PP1) were not included in that reference;

moreover, the waste heat to power output ratio and the turbine outlet liquid content of this design

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were not investigated. The detailed thermodynamic analysis of that design is provided in Appendix

C.

3.3. Double-Flash Liquid-Cooling Mixture-Heating design (DF/LC/MH)

Another system proposed in this paper (see Fig. 2e) consists of merging the strategies used in the

DF/LC/SH cycle (Section 3.1) and in the DF/LC/RH cycle (Section 3.2). First, the streams leaving

the first turbine (state {5}) and the second separator (state {8}) are mixed, which yields state {9}.

Then, the resulting fluid (state {9}) enters a heat exchanger (HX), where heat is transferred from

the relatively hot liquid (state {3}) to the relatively cold steam (state {9}). The fluids leave at states

{ }δ and { }α , respectively (see Fig. 2f).

The idea here is to transfer as much heat as possible to the steam that enters the second

turbine (TB2), by heating a mixture (i.e., Mixture-Heating) made of the steam coming from the

separator SP2 and from the turbine TB1. However, the drawback is that the large enthalpy drop of

the liquid (from state {3} to state { }δ ) results in a smaller vapour mass fraction at state {6}, and

hence, the corresponding mass flow rate in turbine TB2 is decreased. Overall, whether this system

provides higher specific output compared to the DF/LC/SH cycle (Section 3.1) or to the DF/LC/RH

cycle (Section 3.2) requires a detailed investigation. The complete method for calculating power

output and other parameters of interest for this design is given in Appendix D.

3.4. Optimization problem statement

To evaluate the benefits of Liquid-Cooling strategies, the designs described in Sections 3.1, 3.2

and 3.3 will be optimized and then compared to the optimized Double-Flash reference design. The

design variables of the three LC systems are the separator temperatures S1T and S2T . Other

parameters (such as heat exchanger efficiency, turbines dry efficiency and reservoir

thermodynamic state) are considered as fixed parameters for a given optimization. To summarize,

the optimization problem may be stated as

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Problem statement:

( )

( )

[ ]

[ ]

( )

S1 S2

S1 S2 R

S2 CD S1

R R CD 0

maximize

by optimizing ,

,respecting

,

given , , , ,

w

T T

T T T

T T T

T P T η ε

∈ ∈

(8)

The problem stated by Eq. (8) is valid for each of the three designs presented in Section 3, namely,

the DF/LC/SH, the DF/LC/RH and the DF/LC/MH cycles. The values of other parameters of

interest described in Section 2 (i.e., the liquid content λ , the silica saturation coefficient eqS S ,

and the waste heat to power output ratio netout outQ W� � ) will be used for comparison purposes once

optimizations have been performed.

4. Liquid-Splitting (LS) strategies

In this Section, it is proposed to improve the performance of Double-Flash thermodynamic cycle

by carrying only a specific fraction of a hot liquid stream to the interstage heat exchanger. This

strategy is inspired from a work performed by Mathieu-Potvin (2012). The common feature of the

three modified cycles proposed in this section (Section 4) is that the heating process is performed

by splitting the stream leaving the first separator (SP1), which explains the name that was chosen

for this category (i.e., Liquid-Splitting (LS)),

4.1. Double-Flash Liquid-Splitting Superheating design (DF/LS/SH)

It is proposed to modify the Double-Flash power plant (see Fig. 3a) by heating the steam leaving

the separator SP2 with only a fraction ( )1 y− of the liquid mass flow rate leaving the separator

SP1, while the other fraction ( )y continues its route in the thermodynamic cycle. This Double-

Flash (DF) cycle with Liquid-Splitting (LS) and Superheating of steam (SH) is referred to as the

DF/LS/SH cycle. To the authors’ knowledge, that design was never analysed previously in

literature.

Similarly to the Liquid-Cooling (LC) systems presented in Section 3, the idea with the

DF/LS/SH cycle is to increase enthalpy of steam that enters the turbine TB2, thereby increasing

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the work output during steam expansion. The difference here is that the fraction ( )1 y− may be

modulated and optimized in order to provide maximal utilization of the heat present in the fluid;

the other fraction ( )y remains at high temperature, which could provide more vapour at the outlet

of the separator SP2 and then more steam for the turbine TB2. However, once the fraction ( )1 y−

has been used for heat transfer in the heat exchanger (HX), that fluid is ejected from the cycle (see

state { }δ in Fig. 3a and 3b) and does not produce power. Whether this stratagem provides

advantages over the reference Double-Flash cycle presented in Fig. 1 will have to be assessed with

a thorough thermodynamic analysis. The method for calculating the performance of the DF/LS/SH

cycle is presented in Appendix E.

4.2. Double-Flash Liquid-Splitting Reheating design (DF/LS/RH)

The reference Double-Flash cycle may also be modified by allowing heat transfer between a

fraction ( )1 y− of the liquid leaving the separator (state {3}) and the steam at the outlet of the high

pressure turbine (state {5}), by means of a heat exchanger (HX). This Double-Flash (DF) cycle

with Liquid-Splitting (LS) and Reheating of steam (RH) is referred to as the DF/LS/RH cycle (see

Fig. 3c and 3d). The purpose of this system is the same as that of the DF/LC/RH presented in

Section 3.2, i.e., reheating state {5}. However, heating is now provided by using a fraction ( )1 y−

of the liquid leaving the separator SP1, while the DF/LC/RH cycle presented in Section 3 uses the

entire liquid flow rate leaving that separator. A detailed explanation of the steps for calculating the

performance of the DF/LS/RH cycle is given in Appendix F.

4.3. Double-Flash Liquid-Splitting Mixture-Heating design (DF/LS/MH)

The last system proposed in this paper (see Fig. 3e and 3f) consists of merging the strategies used

for the DF/LS/SH cycle (Section 4.1) and for the DF/LS/RH cycle (Section 4.2). Indeed, a fraction

( )1 y− of the liquid leaving the separator SP1 transfers its heat to the relatively cold fluid entering

the heat exchanger at the thermodynamic state {9}, where state {9} is the result of a mixing of the

steam leaving the first turbine (state {5}) and of the steam leaving the second separator (state {8}).

That system (i.e., DF/LS/MH cycle) is similar to the DF/LC/MH cycle presented in Section 3.3,

16

with the difference that heating is now provided by using only a fraction ( )1 y− of the hot liquid

leaving the separator SP1. A detailed explanation of the steps required for calculating the

performance of the DF/LS/MH cycle is given in Appendix G.

4.4. Optimization problem statement

The three Liquid-Splitting designs described in Sections 4.1, 4.2 and 4.3 will also be optimized in

this paper, but in addition to the separator temperatures, the fraction y has also to be considered

as a design variable. In practice, the desired value of y could be obtained by an appropriate choice

of thermofluid equipment (e.g., valves, etc.). The value of y is bounded between 0 (i.e., no fluid

flow in the cycle and no power output) and 1 (equivalent to the Double-Flash reference cycle),

which are two limiting scenarios. To summarize, the optimization problem for the three LS designs

may be stated as

Problem statement:

( )

( )

[ ]

[ ]

( )

S1 S2

S1 S2 R

S2 CD S1

R R CD 0

maximize

by optimizing , ,

,

respecting ,

[0,1]

given , , , ,

w

T T y

T T T

T T T

y

T P T η ε

∈

∈ ∈

(9)

The value of the other parameters of interest described in Section 2 (i.e., λ , eqS S , and netout outQ W� �

) will be used for comparisons once optimizations have been performed.

5. Summary of interstage heating strategies

For clarity purposes, the Double-Flash reference cycle presented in Section 2 and the six cycles

with interstage heating presented in Sections 3 and 4 are recapitulated in Table 1. These designs

can be distinguished: (i) by the strategy used to extract heat from the hot side, i.e., LC (the liquid

enters the heat exchanger) or LS (only a fraction of the liquid enters the heat exchanger), and (ii)

by the strategy used to increase enthalpy of the relatively cold steam, i.e., SH (heating the steam

leaving the separator), RH (heating the steam leaving the turbine), or MH (heating the steam

17

leaving both equipment). Among the designs reported in Table 1, only the Double-Flash and the

DF/LC/RH cycles have already been reported or investigated in literature, to the authors’

knowledge. The location of the designs description in the text, as well as that of their

thermodynamic analysis, are reported in the last two columns of Table 1 for the reader convenience.

Table 1. Summary of the six interstage heating designs Hot side category

Cold side category

System as a whole

Description in text

Thermodynamic analysis

LC LS SH RH MH Design label Sections Appendices

none none DFa Section 2.1 Appendix A

● ● DF/LC/SH Section 3.1 Appendix B

● ● DF/LC/RHa Section 3.2 Appendix C

● ● DF/LC/MH Section 3.3 Appendix D

● ● DF/LS/SH Section 4.1 Appendix E

● ● DF/LS/RH Section 4.2 Appendix F

● ● DF/LS/MH Section 4.3 Appendix G

a Thermodynamic cycles existing in literature (DiPippo, 2013).

6. Impact of heat exchanger efficiency

The heat exchanger efficiency ε was not considered as a design variable in the optimization

problem statements (Eqs. (8) and (9)), which simplifies the analysis. In other words, it is assumed

that the higher the value of ε , the higher the specific output of power plant. To validate this

hypothesis, the specific output was evaluated in this Section for the entire range of ε , i.e., [0,1]ε ∈

. Thermodynamic states were numerically calculated for each design by means of a thermodynamic

toolbox (Holmgren, 2006), and a programming script (MATLAB, 2010) was used to perform

calculations with different values of heat exchanger efficiency.

18

6.1. Preliminary analysis of Liquid-Cooling cycles

The thermodynamic analysis was first performed for the three Liquid-Cooling designs (i.e., those

described in Section 3) by using the operating conditions given in Table 2. These values are

representative of typical liquid-dominated high-temperature reservoirs, and of standard turbines

and condensers (e.g., p. 97 in DiPippo (2012)). Notice that it was verified that the value of RP has

negligible impact on all results presented in this paper, as long as state {1} remains at compressed

liquid state. In other words, the thermodynamic properties of state {1} depend essentially on the

value of RT .

The specific output of the LC designs is reported as a function of the heat exchanger

efficiency in Fig. 4a. The Double-Flash specific output value (i.e., 50.58 kJ/kgw = ) is also reported

in that figure by a horizontal line for the conditions given in Table 2.

Table 2. Operation parameters for investigating impact of ε

RT 180 °C S1T 150 °C

RP 3500 kPa S2T 100 °C

CDT 50 °C 0η 0.85

It can be seen in Fig. 4a that w increases monotonically with ε for the three LC designs,

which indicates that the heat exchanger should be designed so as to obtain the highest value of heat

exchanger efficiency. It should be noted that the limiting value 0ε = on the abscissa is equivalent

to the Double-Flash reference cycle (i.e., no heat exchanger), which is confirmed by the intersection

of all curves in Fig. 4a.

A careful observation of Fig. 4a also reveals a change of slope for the DF/LC/RH and

DF/LC/MH curves. Indeed, it was verified that, for these two curves, state {α } passes from a

saturated mixture to a superheated vapour state precisely where changes of slope are observed. In

other words, state {α } is a saturated mixture for lower values of ε , and it becomes superheated

steam for larger values of ε . Hence, the change of slope is clearly explained by the qualitative

19

change (from saturated mixture to superheated vapour) of the fluid leaving the heat exchanger. For

the DF/LC/SH design in Fig. 4a, it was verified that state {α } is superheated vapour for the entire

range of ε , which explains why there is no change of slope for that design.

Finally, considering a typical heat exchanger efficiency of 0.80 (see the circles in Fig. 4a),

the values of the increase of specific output (compared to the Double-Flash design) are 1.31%,

2.49% and 4.11%, for the DF/LC/SH, DF/LC/RH and DF/LC/MH designs, respectively.

6.2. Preliminary analysis of Liquid-Splitting cycles

The same analysis was performed for the three Liquid-Splitting designs (i.e., those described in

Section 4), by using the operating conditions given in Table 2 and by assuming a fixed value of the

split fraction, i.e., 0.9y = . Notice that other values of y have been tested by the authors, but the

tendencies remained the same. Results are reported in Fig. 4b, and the Double-Flash specific output

value ( )50.58 kJ/kgw = is also reported in the same figure.

It can be seen that the three curves associated to LS designs do not intersect with the

Double-Flash curve at 0ε = . Indeed, in this analysis, the value of y is fixed (i.e., y = 0.9) for the

LS designs. Hence, even when 0ε = , a significant fraction of the fluid (1 0.1y− = ) is neither

flashed nor used in the thermodynamic cycle, which is different from the Double-Flash design in

which all the fluid is flashed and involved in the thermodynamic cycle. This explains why the three

LS curves do not intersect with the Double-Flash curve at 0ε = .

A change of slope was also observed for the DF/LS/RH and DF/LS/MH curves in Fig 4b,

and it was verified that it was due to qualitative changes (from saturated mixture to superheated

vapour) of the fluid leaving the heat exchanger at state {α }. This trend is analog to that shown in

Fig. 4a. Finally, considering a typical heat exchanger efficiency of 0.80 (see the circles in Fig. 4b),

the increase of specific output compared to the Double-Flash design are −1.87%, 0.73% and 2.50%,

for the DF/LS/SH, DF/LS/RH and DF/LS/MH cycles, respectively.

20

6.3. Discussion

One of the objectives of the analysis performed in Section 6 was to show that the specific output

can be increased by using Double-Flash cycle with either Liquid-Cooling or Liquid-Splitting

strategies, which is clearly the case by looking at Figs. 4a and 4b. That was illustrated by varying

the value of the heat exchanger efficiency ε . (Notice that changing the value of ε is equivalent to

changing the terminal temperature difference (TTD) of the heat exchanger.)

Finally, it was shown that a heat exchanger efficiency of 100% yields the maximum specific

output (see Fig. 4) when interstage heating strategies are considered. Nonetheless, the value

80%ε = will be used as a conservative value in the upcoming optimizations (Sections 7 and 8).

7. Optimization of interstage heating designs

To assess the maximal benefit of using interstage heating strategies in Double-Flash power plants,

the modified designs presented in Sections 3 and 4 were optimized and then compared to the

optimized Double-Flash reference cycle.

More specifically, optimizations of the Double-Flash design (defined by Eq. (7)), of the

Liquid-Cooling designs (defined by Eq. (8)) and of the Liquid-Splitting designs (defined by Eq.

(9)) were performed using a commercial optimization toolbox (Matlab Optimization Toolbox

2010). Calculation of the objective function was performed with means of a thermodynamic

toolbox (Holmgren, 2006) controlled by a programming script (MATLAB, 2010). The conjugate

gradient method was selected in the optimization toolbox. The calculation of directional derivatives

with the optimization toolbox was performed by selecting a minimal step of the separator

temperatures equal to 0.01°C, and a minimal step of the split fraction equal to 410− . Optimizations

were terminated when the change of the objective function (i.e., w) was smaller than 1210− kJ/kg.

Reducing the values of these parameters didn’t yield perceptible changes of the results. The

operating conditions (i.e., fixed parameters values) of the optimizations are given Table 3. Notice

that the value of RT is fixed for a given optimization. Other numerical tools may have been used

21

for the optimization (e.g., imbedded Solver in Microsoft Excel), but we chose to use Matlab

because its reliability is well proven.

Table 3. Optimization operation parameters

RT fixed

CDT 50 °C

0η 0.85

ε 0.80

7.1. Optimization of Liquid-Cooling (LC) designs

Optimization defined by Eq. (8) was performed for the three LC designs, and optimization defined

by Eq. (7) (i.e., for the Double-Flash reference design) was performed for comparison purposes.

These optimizations were achieved for a wide range of reservoir temperatures (i.e., from

R 140 °CT = to R 240 °CT = ). The values of the maximized objective function and of the optimized

design variables are reported in Fig. 5, and the corresponding values for the parameters of interest

are reported in Fig. 6.

In Fig. 5a, it can be seen that the maximal specific output of the three LC designs increases

with the reservoir temperature. It can also be observed that the order of magnitude of the

performance enhancement agrees with the results presented in DiPippo (2013) or in DiPippo and

Vrane (1991). Furthermore, the three LC designs provided better performance than the optimized

Double-Flash reference design (see Fig. 5a). Among the three LC designs, the DF/LC/MH provided

the highest relative increase of specific output, reaching 5.12% for a reservoir temperature of 240

°C (see Fig. 5b). This is slightly superior to the DF/LC/RH design proposed by DiPippo (2013),

which provided a relative increase of specific output reaching 3.91% at the same temperature (see

Fig. 5b). Notice that the relative increase of specific output w w∆ is defined as equal to

22

( )max ma x,DF max,DFw w w− , where maxw is the maximized specific output for a design, and max,DFw is

the maximized specific output achieved with the Double-Flash reference design.

The values of the optimized design variables ( S1,optT and S2,optT ) are reported in Fig. 5c and

5d. It can be observed that the values of the optimal separator temperatures for the three LC designs

are very close (either higher or lower) to those of the Double-Flash reference design. Hence,

retrofitting a Liquid-Cooling strategy (i.e., adding an interstage heat exchanger) in an existing

Double-Flash geothermal power plant would only require very small changes of the operating

conditions (temperature and pressure) in the separators. Finally, it was verified by the authors that

for all values of S1,optT smaller than ~100 °C in Fig. 5c (i.e., 100 °C corresponds to saturation

pressure of ~1 atm), the systems included a conditional pump (PP2’).

The values of the other parameters of interest for optimized LC designs are reported in Fig.

6. For instance, it can be seen in Fig. 6a that the liquid content 5λ at the high pressure turbine outlet

of LC designs are only slightly higher than that of the Double-Flash reference design, which may

not sensibly influence turbine blade erosion rate. However, the Liquid-Cooling strategies had a

significant impact on the value of liquid content 10λ at the low pressure turbine outlet (see Fig. 6b);

for instance, the value of 10λ for the DF/LC/MH design is roughly 50% lower than that of the

Double-Flash design. Hence, the erosion rate of the last stage of the low pressure turbine may be

significantly reduced and the turbine’s lifespan may be increased by using the Liquid-Cooling

designs. Moreover, it should be noticed that all designs respected the representative liquid content

( )λ upper limit of 12% for sustainable operation of steam turbine (Nag, 2008).

The waste heat to power output ratio ( )netout outQ W� � of the optimized LC and Double-Flash

designs are reported in Fig. 6c. It can be observed in that figure that the three Liquid-Cooling

strategies may decrease by approximately 10% the required cooling capacity of the geothermal

power plant. This may results in a reduction of the size of cooling equipment (cooling towers,

condenser, heat exchangers, and associated pumps), and as a consequence, in a reduction of their

initial costs.

23

Finally, the silica saturation coefficient ( )7 eqS S in the water at state {7} was reported in

Fig. 6d. It can be seen that for all designs presented in that figure, this coefficient becomes higher

than 1 for reservoir temperature higher than ~200 °C, which may induces technical problems due

to silica solidification. More precisely, as mentioned in the literature, values of silica saturation

coefficient higher than 1.2 may be outside practical operation range (Clarke and McLeskey Jr,

2014); hence, that practical limit was identified by a dashed line in Fig. 6d. It can also be observed

that Liquid-Cooling designs provided very small decreases of the silica saturation coefficient

compared to the Double-Flash reference design, and it is certainly not enough to prevent

solidification. Hence, strategies for controlling silica precipitation should be adopted in any cases,

either with or without Liquid-Cooling processes.

7.2. Optimization of Liquid-Splitting (LS) designs

Optimization defined by Eq. (9) was performed for the three Liquid-Splitting designs. The values

of the maximized objective function (i.e., w) and of the optimized design variables (i.e., S1T , S2T ,

and y ) are reported in Fig. 7, while the other parameters of interest are reported in Fig. 8. Results

of Double-Flash design optimization (already performed in Section 7.1) are also reported in Figs.

7 and 8 for comparison purposes.

In Fig. 7a, it can be observed that the three Liquid-Splitting processes provide higher

specific output than the Double-Flash design. The best design was the DF/LS/MH, which provided

a relative increase of specific output as high as 3.63% at 240 °C (see Fig. 7b).

The optimized separator temperatures are presented in Fig. 7c and 7d. It can be seen that all

curves are close to those of the Double-Flash design; in other words, the addition of a Liquid-

Splitting process in a Double-Flash cycle has only a weak impact on the optimal values for S1T and

S2T . Again, it should be noted that for all values of S1,optT smaller than ~100 °C (this limit

corresponds to a saturation pressure of ~1 atm), the systems included a conditional pump (PP2’).

The optimal value of the third design variable (i.e., the split fraction) is presented in Fig.

7e. It can be observed that at a given reservoir temperature RT (abscissa of Fig. 7e), the difference

24

of the opty value between the three types of Liquid-Splitting designs is important. Moreover, for

each Liquid-Splitting design considered individually, the value of opty changes significantly over

the range of reservoir temperatures investigated. For instance, considering the DF/LS/MH, its y

value varies from 0.90 to 0.76, for reservoir temperatures from 140 °C to 240 °C, respectively.

Hence, considering that temperature of reservoirs may change over the lifespan of a geothermal

power plant, an appropriate system should be implanted to control the split fraction value in

accordance with the optimal values reported in Fig. 7e.

The values of the other parameters of interest for the three optimized LS designs are shown

in Fig. 8. It can be deduced from Figs. 8a and 8b that only the liquid content of the low pressure

turbine 10λ is significantly decreased by the LS strategy. For instance, the best results were obtained

with the DF/LS/MH design, which provided values of 10λ that are roughly 50% lower than those

of the Double-Flash reference design (see Fig. 8b). As discussed previously, this is advantageous

for turbines’ efficiency and lifespan. Again, it should be noticed that all designs respected the liquid

content upper limit of ~12% for sustainable operation of steam turbine (Nag, 2008).

The value of the waste heat to power output ratio ( )netout outQ W� � for the LS and Double-Flash

designs are shown in Fig. 8c. It can be observed that a reduction of netout outQ W� � on the order of 10%

can be obtained by using Liquid-Splitting designs instead of Double-Flash designs. A very similar

trend was already observed with Liquid-Cooling designs (see Section 7.1). Finally, the silica

saturation coefficient ( )7 eqS S of the Liquid-Splitting designs was observed to be almost equal to

that of Double-Flash reference design (see Fig. 8d). Hence, no advantage or disadvantage is

brought by using LS designs considering the silica saturation coefficient ( )7 eqS S . The

approximate upper limit of silica saturation coefficient for practical power plant operation (

7 eq 1.2S S = ) was identified by a dashed line in Fig. 6d.

7.3. Discussion

From the analysis performed in Section 7, it can be concluded that, compared to the Double-Flash

reference design, interstage heating strategies can potentially lead to three noteworthy benefits: (i)

25

a potential increase of ~5% of the specific output w , (ii) a decrease of ~50% of the liquid content

( )10λ in the low pressure turbine, and (iii) a decrease of ~10% of the waste heat to power output

ratio ( )netout outQ W� � .

On the other hand, the interstage heating designs do not yield significant advantages or

disadvantages compared to the DF reference cycle, when considering (i) the liquid content 5λ in

the high pressure turbine, or (ii), the silica saturation coefficient 7 eqS S . Furthermore, all designs

(either Liquid-Cooling, Liquid-Splitting, or reference Double-Flash) needed a conditional pump

(PP2’) over a similar range of reservoir temperature (i.e., when S1,optT is smaller than ~100 °C);

hence, no advantage was observed regarding that aspect.

Finally, considering the three most significant aspects identified above ( w , 10λ , and

netout outQ W� � ), the results showed that the DF/LC/MH design provided the best performances among

the Liquid-Cooling strategies (see Figs. 5 and 6), and that the DF/LS/MH design provided the best

performances among the Liquid-Splitting strategies (see Figs. 7 and 8). Hence, in the following

section (Section 8), these two designs are selected to perform further analysis about the impact of

the condenser temperature.

8. Impact of condenser temperature

Geothermal power plants have been installed in areas having different climates throughout the

world. For example, Double-Flash power plants in Iceland may operate in climates having monthly

mean temperature as low as −3 °C (Icelandic Meteorological Office, 2014), while those in

California may operate in ambient temperature of 30°C or higher (National Weather Service,

2014). Since the temperature in the condenser CDT is directly related to ambient temperature, it

means that a range of conceivable values for CDT exists. Hence, in this Section, the advantages of

the best Liquid-Cooling design (i.e., the DF/LC/MH) and of the best Liquid-Splitting design (i.e.,

the DF/LS/MH) over the Double-Flash reference design are examined for various condenser

temperatures.

26

For instance, numerical optimizations defined by Eqs. (7), (8) and (9) have been performed

for the Double-Flash, the DF/LC/MH and the DF/LS/MH designs at three different condenser

temperatures, i.e., 10, 30 and 50 °C. The values of the relative increase of specific output for the

DF/LC/MH and the DF/LS/MH designs are reported in Fig. 9a ( CD 10°CT = ), in Fig. 9b (

CD 30°CT = ), and in Fig. 9c ( CD 50°CT = ). It can be observed that for a given system (either

DF/LC/MH or DF/LS/MH) and at a given reservoir temperature RT , the value of w w∆ increases

when the condenser temperature is reduced. Hence, the benefits of using interstage heating should

be higher in cold climates. Moreover, for both interstage heating designs considered in Fig. 9, the

value of w w∆ increases with the reservoir temperature (abscissa), which means that the benefits

of using an interstage heat exchanger should always be higher in the presence of high temperature

geofluids. Finally, for any combination of reservoir temperature ( )RT and condenser temperature

( )CDT presented in Fig. 9, the DF/LC/MH design always provided better performance than the

DF/LS/MH. Hence, we conclude that, by strictly considering the figures of merit investigated in

this paper, the DF/LC/MH design should be the preferred design.

9. Further discussion

One of the main conclusions learned from the study performed in this paper is that interstage

heating processes may provide higher performances than the reference Double-Flash design,

considering parameters such as specific output, turbine liquid content and waste heat to power

output ratio.

However, it should be reminded that the geothermal power plant model used in this paper

neglects many technical aspects. For example, fouling and silica solidification phenomena in the

heat exchanger (HX) are issues that may depreciate the thermodynamic benefits of interstage

heating. Furthermore, once a heat exchanger is chosen and installed for a given set of operating

conditions, it is possible that reservoir temperature, pressure and mass flow rate change over time,

which could bring the heat exchanger in non-optimal conditions (and which may reduce its

efficiency). Moreover, the use of an additional pump after the second separator (i.e., the conditional

27

pump PP2’) may not be acceptable for economic reasons, and an optimization including a

constraint on 7P (i.e., 7P > atmP ) could eliminate the necessity of using that additional equipment.

Moreover, a careful and detailed analysis should be performed before modifying (i.e.,

retrofitting) existing Double-Flash systems, because the control system of the plant was not initially

calibrated to take into account a heat exchanger. Furthermore, addition of a heat exchanger induces

additional pressure drops and small changes of temperature/pressures everywhere in the plant,

which gets the systems in operating conditions that are not those for which it has been initially

designed/optimized. Turbines, pipes, separators and other equipment are then likely to run in off-

design conditions.

To summarize, a better understanding of the techno-economical consequences of

retrofitting an existing Double-flash power plant — or of designing a new Double-Flash power

plant with interstage heating — may require a model taking into account additional thermofluid

phenomena and economical aspects.

Nonetheless, the analysis performed in this paper provides essential results which allows

the reader to estimate the order of magnitude of the potential advantages brought by interstage

heating in Double-Flash power plants.

10. Conclusion

In this paper, six interstage heating designs are analysed, optimized and then compared to an

optimized Double-Flash power plant. Three of the interstage heating designs are classed in the

Liquid-Cooling category, and the three others are classed in the Liquid-Splitting category. The

objective function was the power plant specific output, and the design variables were the separator

temperatures and the split fraction. Optimizations were performed numerically and for a wide range

of reservoir temperatures (i.e., from 140 °C to 240 °C).

The analysis revealed that interstage heating processes increase the specific output

(objective function) of the plant by ~5%, decrease the liquid content in the low pressure turbine by

~50%, and decrease the required cooling capacity of the plant by ~10%. On the other hand, the

28

analysis showed that the new designs proposed had negligible influences on the high pressure

turbine liquid content or on the silica saturation coefficient. Among all designs investigated in this

paper, the interstage heating design named “Double-Flash with Liquid-Cooling and Mixture-

Heating” (DF/LC/MH) provided the highest benefits.

Further studies could extend the analysis performed in this paper. For example, the six

interstage designs could be compared on the basis of an equal heat exchanger area (i.e., equivalent

initial cost), instead of using the assumption of equal heat exchanger efficiency (as done in this

paper). Supplementary design variables could be included in the optimization problem, such heat

exchangers inner design (overall dimension, baffles size, etc.) (e.g., Wildi-Tremblay and Gosselin,

2007) or separators inner design (shell and inside diameters, height, etc.) (e.g., Zarrouk and

Purnanto, 2015)). Also, a more general design having characteristic of both Liquid Cooling (LC)

and Liquid Splitting (LS) strategies could be developed and optimized. Finally, the

thermodynamics analysis could take into account transient phenomena of different time scales,

such as variations of outdoor temperature (over a time scale of 24h), or variations of reservoir

temperature and pressure (over a time scale of years), which could influence optimal separator

temperature values.

Acknowledgments

The authors would like to thank Professor Louis Gosselin of Université Laval for having shared

his laboratory computational resources. This research was partly funded by the “Étude-Travail”

program of Université Laval, and by the “Démarrage Nouveau Chercheur” program sustained by

the Department of mechanical engineering.

29

Chapitre d'équation (Suivant) Section 1Appendix A: Double-Flash cycle

This appendix describes how the total specific power output and other properties of the Double-

Flash cycle (see Fig. 1) can be obtained using thermodynamic data. The known parameters are

typically the reservoir temperature RT , the separator temperatures S1T and S2T , the condenser

temperature CDT and the turbine dry efficiency 0η .

The geothermal fluid comes from the reservoir at state {1}, and enthalpy of this state may be

obtained from steam tables, i.e.,

State {1}: Steam tables

1 R1

1 R

T Th

P P

= →

= (A.1)

This liquid flows in the well and undergoes expansion in a valve,

State {2}: S1

S1

Steam tables2 f @2 1

22 S1 fg@

T T

T T

h hh hx

T T h

=

=

−= → =

= (A.2)

where 2x is the mass fraction of the fluid that is in vapour state, just after the valve. The fluid then

enters the separator, where the saturated liquid phase (state {3}) is extracted,

State {3}: S1Steam tables

3 f @3 S1

3sat. liq. 0

T Th hT T

x

=== →

= (A.3)

The vapour phase (state {4}) in the separator is saturated and is extracted separately from the liquid

phase,

State {4}: S1

S1

4 g@Steam tables

4 S14 g@

4

sat. vap1

T T

T T

h hT T

s s

x

=

=

==

→ =

=

(A.4)

The steam leaving the separator at state {4} is then transferred to the high pressure turbine TB1.

The Baumann relation is used to take into account irreversibility and droplet formation in the

turbine, which allows calculating the actual thermodynamic state of the wet steam at the high

pressure turbine outlet, i.e.,

30

State {5}: S2

S2

5

g@ 55

fg@

Baumann equation

T T

T T

h

h h

hλ

=

=

→

−=

(A.5)

The saturated liquid (state {3}) leaving the first separator undergoes an isenthalpic expansion in

the second valve,

State {6}: S2

S2

Steamtables6 f @6 3

66 S2 fg@

T T

T T

h hh hx

T T h

=

=

−= → =

= (A.6)

This fluid then enters the second separator SP2. The state of the saturated liquid leaving that

separator can be determined from:

State {7}: S2

S2

7 f @Steamtables

7 S27 f @

7

sat. liq.0

T T

T T

h hT T

s s

x

=

=

==

→ =

=

(A.7)

and the state of saturated vapor leaving that same separator is determined as follow:

State {8}: S2

S2

8 g@Steam tables

8 S28 g@

8

sat. vap.1

T T

T T

h hT T

s s

x

=

=

==

→ =

=

(A.8)

Steams from states {5} and {8} are mixed to give state {9}. Using the energy conservation

principle, we can determine 9h :

State {9}: 6 2 8 2 59

6 2 2

(1 )

(1 )

x x h x hh

x x x

− +=

− + (A.9)

The fluid at state {9} then enters the low pressure turbine TB2. The Baumann relation has to be

used to take into account irreversibility and the formation of droplets in that turbine, which leads

to the actual low pressure turbine outlet state {10},

State {10}: CD

CD

10

g@ 1010

fg@

Baumann iterative scheme

T T

T T

h

h h

hλ

=

=

→

−=

(A.10)

31

Steam leaving the low pressure turbine at state {10} then passes in the condenser and leaves at

state {11},

State {11}: CD

CD

CD

11 f @Steam tables

11 CD11 f @

11 sat@

sat. liq.

T T

T T

T T

h hT T

s s

P P

=

=

=

==

→ =

=

(A.11)

Since the pressure at state {11} is smaller than atmospheric pressure, a pump is used to raise the

pressure to state {12}, i.e.,

State {12}: Steam tables

1212

12 11

atmP P

hs s

= →

= (A.12)

The specific output of the geothermal power plant is

2 4 5 6 2 2 9 10R

6 2 2 12 11

( ) [ (1 ) ]( )

[ (1 ) ]( )

Ww x h h x x x h h

m

x x x h h

= = − + − + −

− − + −

�

� (A.13)

For the specific case when the pressure at state {7} is smaller than atmospheric pressure, Eq. (A.13)

is no longer valid. The state {7’} is defined as

State {7’}: Steamtables

7 ' atm7'

7 ' 7

P Ph

s s

= →

= (A.14)

where the atmospheric pressure atmP is equal to 101.325 kPa. The corresponding specific output is

2 4 5 6 2 2 9 10R

6 2 2 12 11 2 6 7 7

( ) [ (1 ) ]( )

[ (1 ) ]( ) (1 )(1 )( )

Ww x h h x x x h h

m

x x x h h x x h h′

= = − + − + −

− − + − − − − −

�

� (A.15)

Finally, the waste-heat to power-output ratio is:

out 2 6 2 10 11net

out

[ (1 )](h h )Q x x x

W w

+ − −=

�

� (A.16)

32

Section d'équation (suivante)Appendix B: DF/LC/SH cycle.


Flash cycle with Superheating and Liquid-Cooling system (see Fig. 2a) can be obtained using

thermodynamic data. The known parameters are typically the reservoir temperature RT , the

separator temperatures S1T and S2T , the condenser temperature CDT and the turbine dry efficiency

0η . Notice that states {1} to {5}, {7}, {7’} and {8} can be calculated as in Appendix A (Double-

Flash).

To obtain the enthalpy of the fluids that leave the heat exchanger (i.e., states {δ} and {α}),

the hypothetical state {δs} for which the hot liquid reaches the minimum possible temperature in

the heat exchanger (i.e., S2T ) has to be specified. The enthalpy of that hypothetical state is

State {δs}: Steam tables

δs S2δs

δs S1

T Th

P P

= →

= (B.1)

The hypothetical state {αs} for which the cold steam reaches the maximum possible temperature

in the heat exchanger (i.e., S1T ) also has to be specified. The enthalpy of that virtual state is thus

State {αs}: Steam tables

αs S1αs

αs S2

T Th

P P

= →

= (B.2)

Then, the maximum heat transfer maxQ� that could occur within the heat exchanger is to be

determined. On the liquid side, the expression for liq,maxQ� is

Liquid side: liq,max R 2 3 δs R liq,max(1 )( )Q m x h h m q= − − ≡� � � � (B.3)

and on the vapour side, the expression for vap,maxQ� is

Vapour side: vap,max R 6 2 αs 8 R vap,max(1 )( )Q m x x h h m q= − − ≡� � � � (B.4)

The value of maxQ� can be determined by selecting the minimal value between liq,maxq� and vap,maxq� ,

i.e, max vap,max liq,maxmin[ , ]q q q=� � � and maxQ� can then be expressed as

max R maxQ m q=� � � (B.5)

33

Notice that in Eq. (B.4), 6x is unknown. As a consequence, an assumption has to be made to

determine the maximum possible heat transfer. The first assumption is:

First assumption: max liq,maxQ Q=� � (B.6)

Using the definition of the heat exchanger efficiency yields

max R 2 3 δ R liq,maxmax

(1 )( )Q

Q Q m x h h m qQ

ε ε ε= → = → − − =�

� � � � ��

(B.7)

Then, by combining the definition of liq,maxq� from Eq. (B.3) with Eq. (B.7), one obtains

State {δ}: δ 3 3 δs( )h h h hε= − − (B.8)

This liquid undergoes expansion in a valve,

State {6}: S2

S2

Steam tablesδ f @6 δ

66 S2 fg@

T T

T T

h hh hx

T T h

=

=

−= → =

= (B.9)

Using the energy conservation principle, the heat transfer from the hot liquid is equal to the heat

transfer to the cold steam, i.e.,

R 2 3 δ R 6 2 α 8(1 )( ) (1 )( )m x h h m x x h h− − = − −� � (B.10)

The variable α

h can be isolated because it is the only unknown, and the entropy α

s can then be

obtained from steam tables,

State {α}:

3 δSteam tables

α 86 α

α S2

h hh h

x s

P P

− = +

→=

(B.11)

and finally, using Eqs. (B.3) and (B.4), the value of liq,maxq� and vap,maxq� can be determined.

If liq,max vap,maxq q<� � , the first assumption is verified, but if liq,max vap,maxq q>� � , the first assumption is

not verified. For this latter case, the next step is to proceed to the second assumption. The second

assumption is:

Second assumption: max vap,maxQ Q=� � (B.12)

34

With this assumption, the definition of the heat exchanger efficiency yields

max R 6 2 α 8 R vap,maxmax

(1 )( )Q

Q Q m x x h h m qQ

ε ε ε= → = → − − =�

� � � � ��

(B.13)

Using the definition of vap,maxq� (see Eq. (B.4)) in Eq. (B.13), the expression of α

h and α

s can be

obtained,

State {α}: Steam tables

αs 8α 8

α

α S2

( )h hh hs

P P

ε= + − →=

(B.14)

Eqs. (B.9) and (B.10) are also valid when using the second assumption and they form a set of two

equations with two unknowns (δ

h and 6x ). Isolating δ

h and 6x yields

State {δ}: 3 7δ 1

h h Ah

A

+=

+where α 7

8 7

h hA

h h

−≡

− (B.15)

This liquid (at state {δ}) undergoes expansion in a valve,

State {6}: S2

S2

Steam tables6 f @6 δ

66 S2 fg@

T T

T T

h hh hx

T T h

=

=

−= → =

= (B.16)

Fluids from states {5} and {α} are mixed to give state {9}. Using the energy conservation principle,

we can determine 9h :

State {9}: 6 2 α 2 59

6 2 2

(1 )

(1 )

x x h x hh

x x x

− +=

− + (B.17)

Finally, states {10} to {12} can be calculated as shown in Appendix A (Double-Flash).


2 4 5 6 2 2 9 10R

6 2 2 12 11

( ) [ (1 ) ]( )

[ (1 ) ]( )

Ww x h h x x x h h

m

x x x h h

= = − + − + −

− − + −

�

� (B.18)

35

For the specific case when the pressure at state {7} is smaller than atmospheric pressure, Eq. (B.18)

is no longer valid. In other words, the conditional pump PP2' (see Fig. 2a) must be taken into

account, and the corresponding specific output is

2 4 5 6 2 2 9 10R

6 2 2 12 11 2 6 7 7

( ) [ (1 ) ]( )

[ (1 ) ]( ) (1 )(1 )( )

Ww x h h x x x h h

m


= = − + − + −

− − + − − − − −

�

� (B.19)


out 2 6 2 10 11net

out

[ (1 )](h h )Q x x x

W w

+ − −=

�

�

(B.20)

Section d'équation (suivante)Appendix C: DF/LC/RH cycle.


Flash cycle with Reheat and Liquid Cooling system (see Fig. 2c) can be obtained using



0η . States {1} to {5}, {7}, {7’} and {8} can be calculated as in Appendix A (Double-Flash), and

states {αs} and {δs} can be calculated as in Appendix B (DF/LC/SH).

The maximum heat transfer maxQ� that could occur in the heat exchanger can now be determined.

On the liquid side, the expression for liq,maxQ� is

Liquid side: liq,max R 2 3 δs R liq,max(1 )( )Q m x h h m q= − − ≡� � � � (C.1)


Vapour side: vap,max R 2 αs 5 R vap,max( )Q m x h h m q= − ≡� � � � (C.2)


i.e., max vap,max liq,maxmin[ , ]q q q=� � � and maxQ� can then be expressed as

max R maxQ m q=� � � (C.3)

36

Then, using the definition of the heat exchanger efficiency yields the expression

max R 2 α 5 R maxmax

( )Q

Q Q m x h h m qQ

ε ε ε= → = → − =�

� � � � ��

(C.4)

In Eq. (C.4), the variable α


s can

be obtained from steam tables,

State {α}: max

Steam tablesα 5

2 α

α S2

qh h

x s

P P

ε = +

→=

�

(C.5)



R 2 3 δ R 2 α 5(1 )( ) ( )m x h h m x h h− − = −� � (C.6)

Isolating δ

h from Eq. (C.6) yields

State {δ}: 2δ 3 α 5

2

( )1

xh h h h

x= − −

− (C.7)

This liquid at state {δ} undergoes expansion in a valve,

State {6}: S2

S2

Steamtables6 f @6 δ

66 S2 fg@

T T

T T

h hh hx

T T h

=

=

−= → =

= (C.8)



State {9}: 6 2 8 2 α

96 2 2

(1 )

(1 )

x x h x hh

x x x

− +=

− + (C.9)

and the states {10} to {12} can be calculated as shown in Appendix A (Double-Flash).


37

2 4 5 6 2 2 9 10R

6 2 2 12 11

( ) [ (1 ) ]( )

[ (1 ) ]( )

Ww x h h x x x h h

m

x x x h h

= = − + − + −

− − + −

�

� (C.10)

For the specific case when the pressure at state {7} is smaller than atmospheric pressure, Eq. (C.10)

is no longer valid. In other words, the conditional pump PP2' (see Fig. 2c) must be taken into


2 4 5 6 2 2 9 10R

6 2 2 12 11 2 6 7 7

( ) [ (1 ) ]( )

[ (1 ) ]( ) (1 )(1 )( )

Ww x h h x x x h h

m


= = − + − + −

− − + − − − − −

�

� (C.11)


out 2 6 2 10 11net

out

[ (1 )](h h )Q x x x

W w

+ − −=

�

� (C.12)

Section d'équation (suivante)Appendix D: DF/LC/MH cycle.


Flash cycle with Mixture-Superheating and Liquid-Cooling system (see Fig. 2e) can be obtained

using thermodynamic data. The known parameters are typically the reservoir temperature RT , the


0η . States {1} to {5}, {7}, {7’} and {8} can be calculated as in Appendix A (Double-Flash), and

states {αs} and {δs} can be calculated as in Appendix B (DF/LC/SH).

Fluids from states {5} and {8} are mixed to give state {9}. Using the energy conservation principle,


State {9}: 6 2 8 2 59

6 2 2

(1 )

(1 )

x x h x hh

x x x

− +=

− + (D.1)

Note that in Eq. (D.1), the value of 6x is unknown and, as a result, the value of 9h is unknown and

will be determined subsequently.

This mixture (state {9}) is then sent to the heat exchanger HX.

38



Liquid side: liq,max R 2 3 δs R liq,max(1 )( )Q m x h h m q= − − ≡� � � � (D.2)


Vapour side: vap,max R 2 6 2 αs 9 R vap,max[ (1 )]( )Q m x x x h h m q= + − − ≡� � � � (D.3)



max R maxQ m q=� � � (D.4)

Notice that in Eq. (D.3), 6x is unknown. As a consequence, an assumption has to be made to

determine the maximum possible heat transfer. The first assumption is:

First assumption: max liq,maxQ Q=� � (D.5)

Using the definition of heat efficiency yields

max R 2 3 δ R liq,maxmax

(1 )( )Q

Q Q m x h h m qQ

ε ε ε= → = → − − =�

� � � � ��

(D.6)

Then, by combining the definition of liq,maxq� (see Eq.(D.2)) with Eq.(D.6), one obtains

State {δ}: δ 3 3 δs( )h h h hε= − − (D.7)

This liquid (state {δ}) undergoes expansion in a valve,

State {6}: S2

S2

Steam tablesf @6 δ

66 S2 fg@

T T

T T

h hh hx

T T h

δ =

=

−= → =

= (D.8)

Since 6x is known, the value of 9h can be determined by using Eq. (D.1).



R 2 3 δ R 2 6 2 α 9(1 )( ) [ (1 )]( )m x h h m x x x h h− − = + − −� � (D.9)

39

The variable α


s can then be


State {α}: 2 Steam tables

α 9 3 δ

2 6 2 α

α S2

(1 )( )

(1 )

xh h h h

x x x s

P P

− = + −

+ − →=

(D.10)

If liq,max vap,maxq q<� � , the first assumption is verified, but if liq,max vap,maxq q>� � , the first assumption is

not verified. For this latter case, the next step is to proceed to the second assumption. The second

assumption is:

Second assumption: max vap,maxQ Q=� � (D.11)

With this assumption, the definition of heat exchanger efficiency (i.e., maxQ Qε=� � ) yields

R 2 3 δ R 2 6 2 αs 9(1 )( ) [ (1 )]( )m x h h m x x x h hε− − = + − −� � (D.12)

Eqs. (D.1) and (D.8) are also valid when using the second assumption. In this case, they can be

combined with Eq. (D.12) to form a set of three equations with three unknowns (δ

h , 9h and 6x ).

Isolating these three variables yields

State {δ}:

2 αs 57 3

2δ

( )

1

1

x h hAh h

xh

A

ε −+ −

−=

+ where αs 8

8 7

( )h hA

h h

ε −≡

− (D.13)

State {6}: S2

S2

from Eq. (D.13)

Steam tablesδ f @6 δ

66 S2 fg@

T T

T T

h hh hx

T T h

=

=

−= → =

=

��

(D.14)

and,

State {9}: 6 2 8 2 59

6 2 2

(1 )

(1 )

x x h x hh

x x x

− +=

− + (D.15)

Eq. (D.9) is also valid when using the second assumption. Isolating the variable hα which is the

only unknown yields

40

State {α}: 2α 9 3 δ

2 6 2

(1 )( )

(1 )

xh h h h

x x x

−= + −

+ − (D.16)



2 4 5 6 2 2 α 10R

6 2 2 12 11

( ) [ (1 ) ]( )

[ (1 ) ]( )

Ww x h h x x x h h

m

x x x h h

= = − + − + −

− − + −

�

� (D.17)

For the specific case when the pressure at state {7} is smaller than atmospheric pressure, Eq. (D.17)

is no longer valid. In other words, the conditional pump PP2' (see Fig. 2e) must be taken into


2 4 5 6 2 2 α 10R

6 2 2 12 11 2 6 7 7

( ) [ (1 ) ]( )

[ (1 ) ]( ) (1 )(1 )( )

Ww x h h x x x h h

m


= = − + − + −

− − + − − − − −

�

� (D.18)


out 2 6 2 10 11net

out

[ (1 )](h h )Q x x x

W w

+ − −=

�

� (D.19)

Section d'équation (suivante)Appendix E: DF/LS/SH cycle


Flash cycle with Superheating and Liquid-Splitting system (see Fig. 3a) can be obtained using


separator temperatures S1T and S2T , the condenser temperature CDT , the splitting coefficient y and

the turbine dry efficiency 0η . States {1} to {5}, {7}, {7’} and {8} can be calculated as in Appendix

A (Double-Flash). Moreover, states {αs} and {δs} can be calculated as in Appendix B (DF/LC/SH).

From state {3}, a fraction ( )1 y− of the fluid is drained and undergoes temperature drop in a heat

exchanger. Furthermore, the fraction y of the fluid from state {3} (i.e., the fraction that was not

drained) undergoes expansion in a valve,

41

State {6}: S2

S2


66 S2 fg@

T T

T T

h hh hx

T T h

=

=

−= → =

= (E.1)

Then maximum heat transfer maxQ� that could occur within the heat exchanger is to be determined.

On the liquid side, the expression for liq,maxQ� is

Liquid side: liq,max R 2 3 δs R liq,max(1 )(1 )( )Q m y x h h m q= − − − ≡� � � � (E.2)


Vapour side: vap,max R 6 2 αs 8 R vap,max(1 )( )Q m yx x h h m q= − − ≡� � � � (E.3)

The value of maxQ� can be determined by selecting the minimal value between liq,maxq� and vap,maxq�

,


max R maxQ m q=� � � (E.4)

Using the definition of the heat exchanger efficiency provides the expression

max R 6 2 α 8 R maxmax

(1 )( )Q

Q Q m yx x h h m qQ

ε ε ε= → = → − − =�

� � � � ��

(E.5)

The variable α


s can be obtained

from steam tables,

State {α}: max Steam tables

α 86 2 α

α S2

(1 )

qh h

yx x s

P P

ε = +

− →=

�

(E.6)



R 2 3 δ R 6 2 α 8(1 )(1 )( ) (1 )( )m y x h h m yx x h h− − − = − −� � (E.7)

Isolating the value of δ

h yields

State {δ}: 6δ 3 α 8( )

(1 )

yxh h h h

y= − −

− (E.8)

42



State {9}: 6 2 α 2 59

6 2 2

(1 )

(1 )

yx x h x hh

yx x x

− +=

− + (E.9)



2 4 5 6 2 2 9 10R

6 2 2 12 11

( ) [ (1 ) ]( )

[ (1 ) ]( )

Ww x h h yx x x h h

m

yx x x h h

= = − + − + −

− − + −

�

� (E.10)

For the specific case when the pressure at state {7} is smaller than atmospheric pressure, Eq. (E.10)

is no longer valid. In other words, the conditional pump PP2' (see Fig. 3a) must be taken into


2 4 5 6 2 2 9 10R

6 2 2 12 11 2 6 7 7

( ) [ (1 ) ]( )

[ (1 ) ]( ) (1 )(1 )( )

Ww x h h yx x x h h

m

yx x x h h y x x h h′

= = − + − + −

− − + − − − − −

�

� (E.11)


out 2 6 2 10 11net

out

[ (1 )](h h )Q x x x

W w

+ − −=

�

� (E.12)

Section d'équation (suivante)Appendix F: DF/LS/RH cycle.


Flash cycle with Reheat and Liquid-Splitting system (see Fig. 3c) can be obtained using



the turbine dry efficiency 0η . The states {1} to {5}, {7}, {7’} and {8} can be calculated as in

Appendix A (Double-Flash), and states {αs} and {δs} can be calculated as in Appendix B

(DF/LC/SH).

43


exchanger. Furthermore, the fraction y of fluid from state {3} (i.e., the fraction that was not


State {6}: S2

S2


66 S2 fg@

T T

T T

h hh hx

T T h

=

=

−= → =

= (F.1)



Liquid side: liq,max R 2 3 δs liq,max(1 )(1 )( ) RQ m y x h h m q= − − − ≡� � � � (F.2)


Vapour side: vap,max R 2 αs 5 R vap,max( )Q m x h h m q= − ≡� � � � (F.3)



max R maxQ m q=� � � (F.4)

Using the definition of the heat exchanger efficiency yields the expression

max R 2 α 5 R maxmax

( )Q

Q Q m x h h m qQ

ε ε ε= → = → − =�

� � � � ��

(F.5)

The variable α


s can then be


State {α}: max

Steam tablesα 5

2 α

α S2

qh h

x s

P P

ε = +

→=

�

(F.6)



R 2 3 δ R 2 α 5(1 )(1 )( ) ( )m y x h h m x h h− − − = −� � (F.7)

44

Isolating δ

h from Eq. (F.7) yields

State {δ}: 2δ 3 α 5

2

( )(1 )(1 )

xh h h h

y x= − −

− − (F.8)



State {9}: 6 2 8 2 α

96 2 2

(1 )

(1 )

yx x h x hh

yx x x

− +=

− + (F.9)


The specific output of the geothermal power plant is:

2 4 5 6 2 2 9 10R

6 2 2 12 11

( ) [ (1 ) ]( )

[ (1 ) ]( )

Ww x h h yx x x h h

m

yx x x h h

= = − + − + −

− − + −

�

� (F.10)

For the specific case when the pressure at state {7} is smaller than atmospheric pressure, Eq. (F.10)

is no longer valid. In other words, the conditional pump PP2' (see Fig. 3c) must be taken into


2 4 5 6 2 2 9 10R

6 2 2 12 11 2 6 7 7

( ) [ (1 ) ]( )

[ (1 ) ]( ) (1 )(1 )( )

Ww x h h yx x x h h

m


= = − + − + −

− − + − − − − −

�

� (F.11)


out 2 6 2 10 11net

out

[ (1 )](h h )Q x yx x

W w

+ − −=

�

� (F.12)

45

Section d'équation (suivante)Appendix G: DF/LS/MH cycle.


Flash cycle with Mixture-Heating and Liquid-Splitting system (see Fig. 3e) can be obtained using



the turbine dry efficiency 0η . Notice that states {1} to {5}, {7}, {7’} and {8} can be calculated as

in Appendix A (Double-Flash), and that states {αs} and {δs} can be calculated as in Appendix B

(DF/LC/SH).


exchanger. Furthermore, the fraction y of fluid from state {3} (i.e., the fraction that was not


State {6}: S2

S2


66 S2 fg@

T T

T T

h hh hx

T T h

=

=

−= → =

= (G.1)

Fluids from states {5} and {8} are mixed to give state {9}. Using the energy conservation principle,


State {9}: 6 2 8 2 59

6 2 2

(1 )

(1 )

yx x h x hh

yx x x

− +=

− + (G.2)

and this mixture (state {9}) is then sent to the heat exchanger HX.

The maximum heat transfer maxQ� that could occur within the heat exchanger can now be


Liquid side: liq,max R 2 3 δs R liq,max(1 )(1 )( )Q m y x h h m q= − − − ≡� � � � (G.3)


Vapour side: vap,max R 2 6 2 αs 9 R vap,max[ (1 )]( )Q m x yx x h h m q= + − − ≡� � � � (G.4)



46

max R maxQ m q=� � � (G.5)

Using the definition of the heat exchanger efficiency provides the expression

max R 2 6 2 α 9 R maxmax

[ (1 )]( )Q

Q Q m x yx x h h m qQ

ε ε ε= → = → + − − =�

� � � � ��

(G.6)

The variable α


s can then be


State {α}: max Steam tables

α 92 6 2 α

α S2

[ (1 )]

qh h

x yx x s

P P

ε = +

+ − →=

�

(G.7)



R 2 3 δ R 2 6 2 α 9(1 )(1 )( ) [ (1 )]( )m y x h h m x yx x h h− − − = + − −� � (G.8)

Isolating the value of hδ yields

State {δ}: 2 6 2δ 3 α 9

2

(1 )( )

(1 )(1 )

x yx xh h h h

y x

+ −= − −

− − (G.9)

Finally, states {10} to {12} can be calculated as shown in Appendix A (Double-Flash), and the

specific output of the geothermal power plant is

2 4 5 6 2 2 α 10R

6 2 2 12 11

( ) [ (1 ) ]( )

[ (1 ) ]( )

Ww x h h yx x x h h

m

yx x x h h

= = − + − + −

− − + −

�

� (G.10)

For the specific case when the pressure at state {7} is smaller than atmospheric pressure, Eq. (G.10)

is no longer valid. In other words, the conditional pump PP2' (see Fig. 3e) must be taken into


2 4 5 6 2 2 α 10R

6 2 2 12 11 2 6 7 7

( ) [ (1 ) ]( )

[ (1 ) ]( ) (1 )(1 )( )

Ww x h h yx x x h h

m


= = − + − + −

− − + − − − − −

�

� (G.11)

47


out 2 6 2 10 11net

out

[ (1 )](h h )Q x yx x

W w

+ − −=

�

� (G.12)

References

Ahmad, M., Casey, M., Sürken, N., 2009. Experimental assessment of droplet impact erosion resistance of steam turbine blade materials. Wear 267(9−10),1605−1618.

Bidini, G., Desideri, U., Di Maria, F., Baldacci, A., Papale, R., Sabatelli, F., 1998. Optimization of an integrated gas turbine–geothermal power plant. Energy Conversion and Management 39(16−18), 1945−1956.

Clarke, J., McLeskey Jr., J.T., 2014. The constrained design space of double-flash geothermal power plants. Geothermics 51, 31−37.

DiPippo, R., 2013. Geothermal double-flash plant with interstage reheating: An updated and expanded thermal and exergetic analysis and optimization. Geothermics 48, 121−131.

DiPippo, R., 2012. Geothermal Power Plants, Third Edition: Principles, Applications, Case Studies and Environmental Impact. Butterworth-Heinemann, Boston.

DiPippo, R., 2015. Geothermal power plants: Evolution and performance assessments. Geothermics 53, 291−307.

DiPippo, R., Vrane, D.R., 1991. A Double-Flash Plant with Interstage Reheat: Thermodynamic Analysis and Optimization. Geothermal Resources Council Transactions 15, 381−386.

Hébert, R.L., LeDésert, B., Bartier, D., Dezayes, C., Genter, A., Grall, C., 2010. The Enhanced Geothermal System of Soultz-sous-Forêts: A study of the relationships between fracture zones and calcite content. Journal of Volcanology and Geothermal Research 196(1–2), 126−133.

Holmgren, M., 2006. XSteam toolbox (IAPWS IF-97 Steam Table). Matlab script (.m), downloaded in 2014 at http://www.mathworks.com/matlabcentral/fileexchange/9817-x-steam--thermodynamic-properties-of-water-and-steam/content/XSteam.m

Icelandic Meteorological Office, 2014. Climatological data. http://en.vedur.is/climatology/data/

Jalilinasrabady, S., Itoi, R., Valdimarsson, P., Saevarsdottir, G., Fujii, H., 2012. Flash cycle optimization of Sabalan geothermal power plant employing exergy concept. Geothermics 43, 75−82.

Leyzerovich, A.S., 2005. Wet-Steam Turbines for Nuclear Power Plants. PennWell Corporation, Tulsa (Oklahoma).

48

Mathieu-Potvin, F., 2012. Process and system for Geothermal Power Generation. United States Patent and Trademark Office. Provisional patent 61/616465.

Mathieu-Potvin, F., 2013. Self-Superheating: A new paradigm for geothermal power plant design. Geothermics 48, 16−30.

MATLAB, 2010. The Mathworks Inc.

Matlab Optimization toolbox, 2010. The Mathworks Inc.

Nag, P.K., 2008. Power Plant Engineering, 3rd Edition. Tata McGraw Hill, New Delhi.

National Weather Service, 2014. http://www.nws.noaa.gov/climate/

Pambudi, N.A., Itoi, R., Jalilinasrabady, S., Jaelani, K., 2014. Exergy analysis and optimization of Dieng single-flash geothermal power plant. Energy Conversion and Management 78, 405−411.

Reader, W.J., 1971. The Weir Group : a centenary history. Weidenfeld and Nicolson, London.

Sugita, H., Matsunaga, I., Yamaguchi, T., Kato, K., Ueda, A., 2003. Silica removal performance of seed from geothermal fluids. Geothermics 32(2), 171−185.

Thorsteinsson, T., Hackenbruch, J., Sveinbjörnsson, E., Jóhannsson, T., 2013. Statistical assessment and modeling of the effects of weather conditions on H2S plume dispersal from Icelandic geothermal power plants. Geothermics 45, 31−40.

Vetter, C., Wiemer, H.-J., Kuhn, D., 2013. Comparison of sub- and supercritical Organic Rankine Cycles for power generation from low-temperature/low-enthalpy geothermal wells, considering specific net power output and efficiency. Applied Thermal Engineering 51(1–2), 871−879.

Weir, J., Weir, G., 1876. Steam engines. United Kingdom Intellectual Property Office. Patent no. 5012.

Weir, W., 2008. The Weir Group : the history of a Scottish engineering legend, 1872-2008. Profile Books Ltd, London.

Wildi-Tremblay, P., Gosselin, L., 2007. Minimizing shell-and-tube heat exchanger cost with genetic algorithms and considering maintenance. International Journal of Energy Research 31(9), 867−885.

Zarrouk, S.J., Moon, H., 2014. Efficiency of geothermal power plants: A worldwide review. Geothermics 51, 142−153.

Zarrouk, S.J., Purnanto, M.H., 2015. Geothermal steam-water separators: Design overview. Geothermics 53, 236−254.

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Figure captions

Figure 1 Double-Flash geothermal power plant. (a) Equipment architecture. (b)

Thermodynamic diagram.

Figure 2 Interstage heating designs using Liquid-Cooling. (a) Equipment architecture of

DF/LC/SH design. (b) DF/LC/SH thermodynamic diagram. (c) Equipment

architecture of DF/LC/RH design. (d) DF/LC/RH thermodynamic diagram. (e)

Equipment architecture of DF/LC/MH design. (f) DF/LC/MH thermodynamic

diagram.

Figure 3 Interstage heating designs using Liquid-Splitting. (a) Equipment architecture of

DF/LS/SH design. (b) DF/LS/SH thermodynamic diagram. (c) Equipment

architecture of DF/LS/RH design. (d) DF/LS/RH thermodynamic diagram. (e)

Equipment architecture of DF/LS/MH design. (f) DF/LS/MH thermodynamic

diagram.

Figure 4 Impact of heat exchanger efficiency ε on the power plant specific output w . (a)

Liquid-Cooling designs. (b) Liquid-Splitting designs.

Figure 5 Optimization of Liquid-Cooling designs for various reservoir temperatures. (a)

Maximized specific output. (b) Relative increase of specific output. (c) Optimized

temperature of high pressure separator. (d) Optimized temperature of low pressure

separator.

Figure 6 Parameters of interest for optimized Liquid-Cooling designs at various reservoir

temperatures. (a) Liquid content at high pressure turbine outlet. (b) Liquid content

at low pressure turbine outlet. (c) Waste heat to power output ratio. (d) Silica

saturation coefficient.

Figure 7 Optimization of Liquid-Splitting designs for various reservoir temperatures. (a)

Maximized specific output. (b) Relative increase of specific output. (c) Optimized

temperature of high pressure separator. (d) Optimized temperature of low pressure

separator. (e) Optimized split fraction.

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Figure 8 Parameters of interest for optimized Liquid-Splitting designs at various reservoir

temperatures. (a) Liquid content at high pressure turbine outlet. (b) Liquid content

at low pressure turbine outlet. (c) Waste heat to power output ratio. (d) Silica

saturation coefficient.

Figure 9 Comparison of DF/LC/MH and DF/LS/MH designs at various reservoir

temperatures for different condenser temperatures CDT . (a) CD 10 CT = ° . (b)

CD 30 CT = ° . (c) CD 50 CT = ° .

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Figures 1 to 9

Figure 1 (2-column fitting image)

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IMPROVEMENT OF DOUBLE-FLASH GEOTHERMAL POWER … · 1 IMPROVEMENT OF DOUBLE-FLASH GEOTHERMAL POWER PLANT DESIGN: A COMPARISON OF SIX INTERSTAGE HEATING PROCESSES Joachim-André Raymond

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