Operation optimization of an organic rankine cycle (ORC) heat recovery power plant

Operation optimization of an organic rankine cycle (ORC) heat recovery power plant

Applied Thermal Engineering 31 (2011) 2032e2041 Contents lists available at ScienceDirect Applied Thermal Engineering journal homepage: www.elsevier...

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Applied Thermal Engineering 31 (2011) 2032e2041

Contents lists available at ScienceDirect

Applied Thermal Engineering journal homepage: www.elsevier.com/locate/apthermeng

Operation optimization of an organic rankine cycle (ORC) heat recovery power plant Jian Sun a, *, Wenhua Li b a b

Energy Efficiency Department, TAS Ltd, 6110 Cullen Blvd, Houston, TX 77021, USA Modeling, Analysis, Simulation and Computation (MASC), Carrier Corporation, Syracuse, NY 13057, USA

a r t i c l e i n f o

a b s t r a c t

Article history: Received 7 January 2011 Accepted 10 March 2011 Available online 21 March 2011

This paper presents a detailed analysis of an organic rankine cycle (ORC) heat recovery power plant using R134a as working fluid. Mathematical models for the expander, evaporator, air cooled condenser and pump are developed to evaluate and optimize the plant performance. Computer programs are developed based on proposed models and algorithms. The effects of controlled variables, including working fluid mass flow rate, air cooled condenser fan air mass flow rate, and expander inlet pressure, on the system thermal efficiency and system net power generation have been investigated. ROSENB optimization algorithm combining with penalty function method is proposed to search the optimal set of operating variables to maximize either the system net power generation or the system thermal efficiency. The optimization results reveal that the relationships between controlled variables (optimal relative working fluid mass flow rate, the optimal relative condenser fan air mass flow rate) and uncontrolled variables (the heat source temperature and the ambient dry bulb temperature) are near liner function for maximizing system net power generation and quadratic function for maximizing the system thermal efficiency. Ó 2011 Elsevier Ltd. All rights reserved.

Keywords: Organic rankine cycle (ORC) Heat recovery Optimization Thermal efficiency

1. Introduction High grade heat recovery technologies have been widely adopted in various industry applications, while recovering low grade heat (180  C below) used to be an expensive and unacceptable approach due to environmental concerns associated with the working fluids [1]. With rapidly increasing of the energy consumption, energy shortage and greenhouse gas emissions become serious issues to all countries. In order to meet future energy demand while reducing greenhouse gas emissions and dependence on fossil fuel substantially, restructuring the energy system is unavoidable. Study shows that more than 50% total heat generated in industry is the low grade heat and has been wasted as the thermal pollution [2]. As one of the promising technologies of converting low grade heat into electricity, the low grade heatdriven organic rankine cycle (ORC) power plant has been proved to be an attractive solution. Intense research and development, particularly on two important topics: working fluid selection and thermodynamic analysis, have been conducted in recent decades.

* Corresponding author. Tel.: þ1 713 820 0189. E-mail address: [email protected] (J. Sun). 1359-4311/$ e see front matter Ó 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.applthermaleng.2011.03.012

Working fluid selection plays a critical role in utilizing the heat source efficiently to achieve high system thermal efficiency and has been reported by many studies. Hung et al., 1997 compared efficiencies of ORCs using different working fluids such as benzene, ammonia, R11, R12, R134a and R113 [2]. Their results indicated that the system efficiency increases for wet fluids and decreases for dry fluids respectively as the turbine inlet temperature is increased when operating between two isobaric curves. The isentropic fluids achieve an approximately constant value for high turbine inlet temperature and are most suitable for recovering low temperature waste heat. By investigating the effects of various working fluids on the thermal efficiency and the total heat recovery efficiency, Liu et al., 2004 found that the best total heat recovery efficiency occurs at the appropriate evaporating temperature between inlet temperature of waste heat and condensing temperature [3]. The analysis from Tchanche et al., 2009 revealed that R134a, followed by R152a, R290, R600, R600a and R290 are most suitable fluids for low-temperature solar ORC systems with heat source temperature below 90  C [4]. Maizza et al., 2001 investigated the thermodynamic and physical properties of some unconventional working fluids for use in ORC waste energy recovery systems [5]. Equally important as working fluid selection for ORC system, thermodynamic analysis is to study the theoretical thermodynamic effects of various variables (such as temperature, pressure, and

J. Sun, W. Li / Applied Thermal Engineering 31 (2011) 2032e2041

mass flow rate) on the ORC system power output, thermal efficiency, secondary law efficiency, irreversibility, and availability. Hung et al. (2001) did irreversibility and efficiency analysis of an ORC system using dry fluids as working fluids [6], and found that higher inlet turbine pressure leads to less irreversibility and better thermal efficiency if the heat source temperature is fixed, however the system efficiency shows the opposite trend comparing to total irreversibility. The optimal operating condition occurred at the intersection point of the efficiency curve and availability ratio curve. Yamamoto et al. (2001) developed a numerical simulation model to estimate ORC system performance under various operating conditions [7]. Their simulation results suggested that operating conditions with saturated vapor being at the turbine inlet would give the best performance. By developing thermodynamic models using EES (Engineering Equation Solver), Chen et al. (2006) investigated the performance difference of a carbon dioxide transcritical and an R123 ORC waste heat recovery power plant [8]. Their studies concluded that with same thermodynamic mean heat rejection temperature, the CO2 transcritical power plant gives a slightly higher power output than R123 ORC power plant in recovering low grade waste heat. Saleh et al. (2007) calculated thermodynamic aspects, such as thermal efficiencies and volume flow rates, of 31 potential working fluids for ORC processes and performed a pinch point analysis for external heat exchanger for optimal using the heat source [9]. A general recommendation for an optimal ORC working fluid requires consideration of the maximum temperature of working fluid. Hettiarachchi et al. (2007) proposed a cost-effective optimum design criterion for ORC plants utilizing low-temperature geothermal heat source [10]. In their study, the objective function is the ratio of the total heat exchanger area to net power output and steepest decent optimization algorithm was used to solve the evaporation temperature, condensation temperature, geothermal and cooling water velocities. DiPippo (2007) presented a basis for comparing geothermal power plant of the binary type with the idea thermodynamic cycle [11]. Mago et al., 2008 evaluated both basic ORC system and regeneration ORC system using a combined first and second law analysis with varying certain system operating parameters at various reference temperatures and pressures [12]. Their research has proven that the system thermal efficiency can be improved by increasing turbine inlet pressure. However it was a weak function of turbine inlet temperature. Dai et al. (2009) investigated the performance of ORC plants with different working fluids under optimization conditions. Parametric optimization was conducted with adapting a genetic algorithm [13]. This parametric study only considered the effects of turbine inlet pressure and temperature on the exergy efficiency. Other parameters, such as waste heat source temperature and mass flow rate, ambient temperature, evaporator pitch temperature, condensing temperature, were assumed to be constant. Cayer et al. (2009) developed a methodology to analyze the performance of a CO2 transcritical cycle using a modified LMTD method. This method was composed of four steps: energy analysis, exergy analysis, finite size thermodynamics and calculation of the heat exchangers’ surface [14]. However, only a few studies involved performance and operation optimization of the organic rankine cycle power plant. In fact, with any given flow and temperature conditions of heat source, power may be generated with different set of operating variables, only one set of operating variables results maximum power output or thermal efficiency. The optimal operation results from tradeoffs between energy generation and consumption of different components. For instance, increasing working fluid mass flow rate increases expander power output but also consumes more pump power. Reducing the condensing temperature requires more power consumption of the air cooled condenser fan, but allows increasing

2033

Air-Cooled Condenser

Expander

Evaporator (Heat Recovery Unit) Refrigeration Pump Receiver

Fig. 1. The schematic ORC heat recovery power plant.

on expander power output. The purpose of this study is to develop the mathematical models and optimization approach to simulate an ORC power plant and search the optimal operating strategies in order to achieve either the best system thermal efficiency or the most system net power generation.

2. Organic rankine cycle (ORC) heat recovery power plant As shown in Fig. 1, the ORC heat recovery power plant used in this study is composed of an expander, an air cooled condenser, an evaporator (heat recovery unit), a working fluid pump and other auxiliary equipments. Heat from heat source such as geothermal or industry process is pumped into evaporator where R134a refrigerant is evaporated, then delivered to expander inlet. The refrigerant drives the expander to generate electricity along with pressure and temperature decreasing. The low pressure and temperature refrigerant is cooled to refrigerant liquid when passing through the air cooled condenser and collected into a receiver. The working fluid pump lefts the liquid refrigerant back into evaporator to absorb the heat again. Then the above process repeats.

3. Mathematical models An ORC heat recovery power plant converts heat into electrical power through four thermodynamic processes (as shown in Fig. 2): pressure lifting process in the working fluid pump (State 1 to 2), heating working fluid in the evaporator (State 2 to 3), generating power output in the expander (State 3 to 4), and cooling working fluid in the condenser (State 4 to 1). To quantify the irreversibility in the actual ORC plant, mathematical models are developed to predict the performance of each individual component, and the system as well. Before elaborating these mathematical models, some assumptions are given as below, 1. This study only investigates the performance of ORC plant under steady state operation.

T 3

2

4

1

S Fig. 2. The schematic TeS diagram of ORC heat recovery power plant.

2034

J. Sun, W. Li / Applied Thermal Engineering 31 (2011) 2032e2041

2. It is assuming that the generator and transformer power losses are 3% and 2% respectively. Since magnitudes of these losses are almost same for all operation conditions, these power losses won’t impact the optimization results significantly. 3. For simplification, the hydraulic calculation isn’t conducted. The pressure drops across individual components are assumed to be constant at different operating conditions.

3.1. Expander The power is generated when the high pressure superheat working fluid passes through the expander, and leaves from the expander as low pressure superheat fluid. The energy balance is given by,

 ¼ mEXP hEXP

WEXP

hEXP

Ent

 hEXP



(1)

Lvg

where, WEXP is the expander power output, hEXP is the expander efficiency, mEXP denotes the working fluid mass flow rate through expander, hEXP_Ent,hEXP_Lvg denote enthalpies of the working fluid at the expander inlet and outlet. The expander power output WEXP is obtained from,

WEXP ¼ mEXP Cp;EXP hEXP TEXP

1g

Ent

1p

!

g

(2)

where, Cp,EXP is the isobaric specific heat of working fluid, TEXP_Ent is the expander inlet temperature, p denotes the pressure ratio of expander inlet and outlet, p ¼ PEXP Ent =PEXP Lvg , g is mean isentropic coefficient,g ¼ Cv;EXP =Cp;EXP . The expander efficiency can be derived from its characteristic map provided by expander manufacture. Zhang et al., 2002, developed analytical performance expression of expander to describe approximately the mass flow and efficiency characteristics for the evaluation of the pressure ratio p and expander efficiency hEXP as below [15],

GEXP GEXP

Des

rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi   ¼ a TEXP Ent Des  p2  1 =p2  1 =TEXP Ent Des (3)

Where, GEXP

hEXP

hEXP

Des

¼

2 #



1  t 1  NEXP=N 

2

EXP Des

NEXP=N

EXP Des

GEXP=G

!



NEXP=N

i

i-1

i+1 Working Fluid

T_WF_Lvg( i )

T_WF_Ent( i )

Fig. 3. Discrete segments of evaporator.

inappropriate since the assumption of constant specific heat doesn’t hold. The zone modeling method treats the evaporator as a three-zone heat transfer unit, including super-heat zone, twophase zone and sub-cool zone. Then the single node lump modeling method can be applied in each zone. A robust iterative algorithm is required to solve the heat transfer surface area of each zone. The distributed modeling method divides the evaporator into segments along the flow direction. In each segment, the single node lump modeling method, for example Ntu-3 method and LMTD method, can be adopted to calculate the heat and mass balance. Except for the relatively higher computation cost, the distributed modeling method is a robust approach with higher accuracy compare to the other two methods. This study is to utilize the distributed modeling method to simulate the evaporator and the air cooled condenser. According to the distributed modeling method, the shell-tube evaporator is simplified as a multi-segment counter flow heat exchanger. Fig. 3 shows three consecutive segments: segment i-1, i, and iþ1. Since it is reasonable and accurate enough to assume that the specific heats of both hot and cold fluids are approximately constant in each individual segment, the single node modeling method (Ntu-3 method in this study) can be utilized to establish the conservation equations. In segment i, the heat absorbed from water side results in the enthalpy increase at the working fluid side,

 qEV;WF ðiÞ ¼ mEV;WF hEV;WF

Lvg ðiÞ

 hEV;WF

Ent ðiÞ



(5)

where, qEV,WF(i) denotes the evaporator working fluid side heat transfer rate of the segment i, mEV,WF is the working fluid mass flow rate through the evaporator, hEV,WF_Ent,hEV,WF_Lvg are working fluid inlet and outlet enthalpies of the segment i. Similarly, the energy balance at the hot fluid side can be written as,

 qEV;HF ðiÞ ¼ mEV;HF Cp;EV;HF ðiÞ TEV;HF

Ent ðiÞ

 TEV;HF

Lvg ðiÞ



(6)

where, qEV,HF(i) is the evaporator hot fluid side heat transfer rate of the segment i, Cp,EV,HF is the isobaric specific heat of hot fluid, mEV,HF is the hot fluid mass flow rate through the evaporator, TEV,HF_Ent,TEV,HF_Lvg are hot fluid inlet and outlet temperatures of the segment i. The maximum possible energy transferred from the hot fluid to the working fluid is given as,

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi m NEXP TEXP Ent ¼ EXP and a ¼ 1:4  0:4 PEXP Ent NEXP Des

"

T_HF_Lvg( i )

T_HF_Ent( i )

Hot Fluid

EXP Des

GEXP=G

EXP Des

ð4Þ

EXP Des

Where, NEXP NEXP_Des are expander actual and design rotating speeds. 3.2. Evaporator The evaporator is a shell-tube heat exchanger with the liquid working fluid flowing into the shell through distribution system and moving uniformly over tubes and the hot fluid flowing inside tubes to reject heat to the working fluid. In general, three different modeling approaches can be used to predict the evaporator performance: distributed modeling method, zone modeling method and single node lump modeling method. When phase change occurs, the single node lump modeling approach is

 qEV;MAX ðiÞ ¼ CEV;MIN ðiÞ TEV;HF

Ent ðiÞ

 TEV;WF



Lvg ðiÞ

(7a)



withCEV;MIN ðiÞ¼MIN mEV;WF Cp;EV;WF ðiÞ;mEV;HF Cp;EV;HF ðiÞ

(7b)

Where: qEV,MAX(i) denotes the maximum possible heat transfer rate of the evaporator segment i. The effectiveness is defined as,

T_Air_Lvg( i )

T_Air_Ent( i )

Air

i

i-1

i+1 Working Fluid

T_WF_Lvg( i )

T_WF_Ent( i )

Fig. 4. Discrete segments of air cooled condenser.

J. Sun, W. Li / Applied Thermal Engineering 31 (2011) 2032e2041

3EV ðiÞ ¼ qEV ðiÞ=q

(8)

EV;MAX ðiÞ

where: : qEV(i) denotes the heat transfer rate in the segment i: qEV ðiÞ ¼ qEV;WF ðiÞ ¼ qEV;HF ðiÞ. The evaporator effectiveness is calculated as below [16],

2035

rate through evaporator, hAC,WF_Ent,hAC,WF_Lvg are the working fluid inlet and outlet enthalpies of the segment i. Similarly, at air side, energy conservation is given by,

 qAC;Air ðiÞ ¼ mAC;Air Cp;Air ðiÞ TAC;Air

91

0:5 > > > > 2 > =

0:5  1 þ exp  NtuEV ðiÞ  1 þ Cr;EV ðiÞ 2 ðiÞ   2 1 þ Cr;EV ðiÞ þ 1 þ Cr;EV

 0:5 3EV ðiÞ ¼ > > 2 ðiÞ > > 1  exp  NtuEV ðiÞ  1 þ Cr;EV > > > > > ; > : 1  exp½  NtuEV ðiÞ 8 > > > > > <

Lvg ðiÞ

 TAC;Air



Ent ðiÞ

(14)



Where Cr;EV ðiÞ ¼ CEV;MIN ðiÞ=CEV;MAX ðiÞ, and CEV;MAX ðiÞ ¼ MAX

mEV;WF Cp;EV;WF ðiÞ; mEV;HF Cp;EV;HF ðiÞ ;

Single  Phase

(9)

Two  Phase

The number of heat transfer units NtuEV(i) is a grouping of terms defined as,

where, qAC,Air(i) denotes the evaporator air side heat transfer rate of the segment i, Cp,Air is the isobaric specific heat of air, mAC,Air is the air mass flow rate through the condenser, TAC,HF_Ent,TAC,HF_Lvg are air inlet and outlet temperatures of the segment i. The maximum possible energy transferred from the working fluid to air is given as,

NtuEV ðiÞ ¼ UAEV ðiÞ=C

 qAC;MAX ðiÞ ¼ CAC;MIN ðiÞ TAC;WF

(10)

EV;MIN ðiÞ

The correlations between the segment i and adjacent segments i-1, iþ1 can be defined with following connection equations,

4EV;HF

Ent ðiÞ

¼ 4EV;HF

Lvg ði

 1Þ

(11a)

4EV;HF

Lvg ðiÞ

¼ 4EV;HF

Ent ði

þ 1Þ

(11b)

4EV;WF

Ent ðiÞ

¼ 4EV;WF

Lvg ði

 1Þ

(11c)

4EV;WF

Lvg ðiÞ

¼ 4EV;WF

Ent ði

þ 1Þ

(11d)

where 4 denotes fluid properties such as temperature, enthalpy, density, entropy,. etc.

3AC ðiÞ ¼

8 > > > > < > > > > :

# "   exp  K  NtuAC ðiÞ  CAC;r ðiÞ  1 K  CAC;r ðiÞ 1  exp½  NtuAC ðiÞ

1  exp

Lvg ðiÞ





 with CAC;MIN ðiÞ ¼ MIN mAC;WF Cp;AC;WF ðiÞ; mAC;Air Cp;AC;Air ðiÞ 15Þ Where: qAC,MAX(i) is the maximum possible heat transfer rate of the condenser segment i. The effectiveness is defined as,

3AC ðiÞ ¼

qAC ðiÞ qAC;MAX ðiÞ

(16)

where: qAC(i) denotes the heat transfer rate of the segment i: qAC ðiÞ ¼ qAC;WF ðiÞ ¼ qAC;Air ðiÞ and,

(17)

Two  Phase

Where,Cr;AC ðiÞ ¼ CAC;MIN ðiÞ=CAC;MAX ðiÞ, CAC;MAX ðiÞ ¼ MAXfmAC;WF Cp;AVC;WF ðiÞ; mAC;Air Cp;AC;Air ðiÞg, and

K ¼ Ntu0:22 ðiÞ AC

NEV X

qEV ðiÞ; NEV is the total number of evaporator segments i¼1

 TAC;Air

Single  Phase

The total heat transfer rate QEV is the summation of heat transfer rates of all segments,

QEV ¼

Ent ðiÞ

(12)

The number of heat transfer units NtuAC(i) is calculated by,

NtuAC ðiÞ ¼ 3.3. Air cooled condenser The distributed modeling method is also used to predict the air cooled condenser performance. The air cooled condenser is divided into discrete segments as shown in Fig. 4. Ntu-3 method establishes the governing equations in the segment i as following, the heat absorbed from the working fluid results in the enthalpy increase.

UAAC ðiÞ CAC;MIN ðiÞ

(18)

The relationship between the segment i and the adjacent segments i  1, i þ 1 can be defined with following connection equations,

4AC;Air

Ent ðiÞ

¼ 4AC;Air

Lvg ði

 1Þ

(19a)

4AC;Air

Lvg ðiÞ

¼ 4AC;Air

Ent ði

þ 1Þ

(19b)

(13)

4AC;WF

Ent ðiÞ

¼ 4AC;WF

Lvg ði

 1Þ

(19c)

where, qAC,WF(i) denotes the condenser working fluid side heat transfer rate of the segment i, mAC,WF is the working fluid mass flow

4AC;WF

Lvg ðiÞ

¼ 4AC;WF

Ent ði

þ 1Þ

(19d)



qAC;WF ðiÞ ¼ mAC;WF hAC;WF

Ent ðiÞ

 hAC;WF

 Lvg ðiÞ

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J. Sun, W. Li / Applied Thermal Engineering 31 (2011) 2032e2041

where 4 denotes fluid properties such as temperature, enthalpy, density, entropy,. etc. The total heat transfer rate is the summation of heat transfer rate of all segments,

QAC ¼

NAC X

qAC ðiÞ; NAC is the total number of segments

(20)

i¼1

The power consumption of air condenser fan is calculated by,

 WFAN ¼ NFan

mFAN mFAN Des

WFAN

Des

mAC;Air ¼ NFan  mFAN

(21)

(22)

3.4. Working fluid pump

Wbhp

(23)

hp

The brake horse power is determined by the pump head H and the pump flow rate M,

SG  M  H kc

(24)

Where, kc is the units conversion constant, SG is the fluid specific gravity. In order to estimate the variable speed pump power consumption, the pump efficiency hP_Des under the design pump speed is calculated by the polynomial function as below,

hP

Des

¼ c0 þ c1 MP

Des

þ c2 MP2

Des

(25)

Where c0, c1, c2 are obtained by curve fitting with manufacture data, MP_Des denotes the design pump mass flow rate. When pump is running at different speed, pump affinity law gives,

MP RPMP ¼ MP Eqv RPMP Eqv HP ¼ HP Eqv



RPMP RPMP Eqv

Eqv

¼ a0 þ a1 MP

hth ¼

þ

(29)

WEXP  WP  WFAN QEV

a2 MP2 Eqv

Optimal operating ORC heat recovery power plant is to utilize the heat resource in the most efficient way in order to maximize out the net electricity generation and/or thermal efficiency. In general, the optimization problem can be formulated as,

X

Wj ðxi Þ or

j

hj ðxi Þ

j

ð31aÞ

  p ¼ 1; 2; /; Np

(31b)

s:t: gp ðxi Þ  0

Where, the objective function J is the summation of power generation/consumption Wj or thermal efficiency hj of each component with respect to control variables xi. The constraint function g is to define the operating limits of each control variables. Ni, Nj, Np denote the number of controlled variables, the number of power generation/consumption components, and the number of constraints. For the system (shown in Fig. 1) used in this study, the objective function J is the summation of power generation/consumption or thermal efficiency of expander, working fluid pump, and air cooled condenser fan. The controlled variables are independent variables which are manipulated by the control system to specify the plant operation. The uncontrolled variables are measurable quantities that may not be controlled. The controlled and uncontrolled variables used in this study are selected as, Controlled variables are:  Relative working fluid mass flow rate  Relative condenser fan air mass flow rate  Expander Inlet Pressure

(27)

Where a0 , a1 , a2 are obtained by curve fitting with manufacture data. Combining the above Eq. (28) and affinity law Eq. (26) and (27), the equivalent pump heat and flow can be obtained. Then Eq. (25) is

X

  i ¼ 1; 2; .; Ni ; j ¼ 1; 2; .; Nj

 Ambient dry bulb temp  Heat source temp  Heat source flow rate

(28)

(30)

4. Optimization algorithm

Uncontrolled variables:

2

Eqv

Wnet ¼ WEXP  WP  WFAN

(26)

where RPMP, RPMP_Eqv denote the pump speed and equivalent pump speed. The equivalent pump heat and flow can be found by applying the design head-flow curve as below,

HP

The system net power generation is,

Max J ¼ FðxÞ ¼

The pump performance is evaluated based on pump characteristic, system hydraulic characteristic and pump affinity law. The pump power consumption Wp is defined with dividing the brake horse power Wbhp by the pump efficiency hp, given as,

Wbhp ¼

3.5. Cycle efficiency

The system thermal efficiency is calculated by,

3

where, NFan is the number of condenser fans, mFAN, mFAN_Des denote actual and design fan air mass flow rates. WFAN, WFAN_Des denote actual and design fan power consumption. The total air condenser mass flow is the sum of all fan air mass flow rates given as,

WP ¼

used to estimate the equivalent pump efficiency to approximate the pump efficiency under operating speed RPMP.

Therefore, the optimization problem can be reformulated by,

X Max J ¼ FðxÞ ¼ Wj ðxi Þ ¼ WEXP þ WP þ WFAN   i ¼ 1; 2; .; Ni ; j ¼ 1; 2; .; Nj Or, J ¼

P

(32a)

hj ðxi Þ ¼ hEXP þ hWF þ hFAN with hj ðxi Þ ¼ Wj ðxi Þ=QEV up

s:t: xlow < xi  xi i

(32b)

Where, WEXP, WP, WFAN are expander power generation, working fluid pump power consumption and condenser fan power

J. Sun, W. Li / Applied Thermal Engineering 31 (2011) 2032e2041

2037

Description Design weather/site conditions Dry Bulb/Wet Bulb Temperature,  C/ C Elevation, m Heat source information Heat mass flow, kg/s Heat flow temperature,  C Minimum allowed discharge temperature,  C Heat fluid type ORC heat recovery power plant description System net power generation, kW System thermal efficiency, % Evaporator (Heat Recovery Unit) Working fluid type Working fluid mass flow rate, kg/s Heat transfer rate, kW Boiling temperature,  C Evaporator pitch temperature,  C Expander Expander inlet pressure, kPa Expander tip diameter, mm Expander rotation speed, rpm Expander efficiency, % Expander maximum power output, kW Air Cooled Condenser Heat transfer rate, kW Condensing temperature,  C Condenser pitch temperature,  C Qty. of/Fans Power of fan, kW/fan Air mass flow rate, kg/s per fan Working Fluid Pump Pump head, kPa Pump flow rate, kg/s Pump efficiency, % Pump driver rating, kW

Value 18.3/7.2 1219 202 160 82 Water/steam

7 6 5 4 0.2

0.2 0.4 0.6

0.6 0.8

0.8

R134a 250 66627 95 11.0

Relative Working Fluid Mass Flow

1 1

Relative Condenser Fan Air Mass Flow

Fig. 5. Variations of system thermal efficiency with mass flow rates of working fluid and condenser fan air.

(

3447 622 5600 87.0 14920

lkþ1 j

¼



 if F xkþ1  F xki i 



< F xki if F xkþ1 i

alkj blj

k

(36)

Where a, b are constants with a > 1 and 0 < b < 0. The normalized direction vector Dki;j is calculated by,

58912 32.6 6.0 36 34 120

Bki;j r ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi Dkþ1 ¼ i;j PNi k 2 B l;j l¼1

2620 250 82.6 1127

Bki;j ¼ Aki;j 

j1 X

"

l¼1

(33)

(34)

To solve the above optimization problem, three broad categories of optimization algorithm are availbale: gradient free algorithm, gradient based algorithm and global optimization algorithm. Considering the complexity of the simulation problem described in Section 3, it is difficult to differentiate the models to find the gradient at any point. Thus, a gradient free algorithm is favorable since no derivatives are required. In this study, the ROSENB algorithm [17] is adapted to search the optimal operating variables. The ROSENB algorithm starts with a set of initial controlled variables xi and step sizes lj (j ¼ 1,2,., n). Search is made in each of the xi direction as calculated by,

  k xkþ1 ¼ xki þ lj Dki;j i ¼ 1; 2; .; Ni ; j ¼ 1; 2; .; Nj i

8

0.4

Relative condenser fan air mass flow rate rAC is operated under,

0:25  rAC  1:0

9

5263 7.9

consumption. hEXP, hP, hFAN are thermal efficiencies of expander, working fluid pump, air cooled condenser. xilow and xiup denote the low and up limits of controlled variables. In the study, relative working fluid mass flow rate rWF is operated under,

0:25  rWF  1:1

System Thermal Efficiency, %

Table 1 Design parameters for the organic rankine cycle heat recovery power plant.

(35)

Aki;j ¼

Nj X

j X n¼1

(37)

! Dkþ1  Akn;j n;l

#  Dkþ1 i;l

dkl  Dki;l

(38)

(39)

l¼j

where, di is the algebraic sum of distance moved in i direction since last iteration (k). To deal with the constraint function, the penalty function method is used. The penalty function method is to solve constrained optimization problems by constructing and solving an approximated unconstrained optimization problem. The original constrained optimization problem can be approximated by an unconstrained optimization with introducing the penalty function,

Max J ¼ FðxÞ þ x  GðxÞ

(40)

Where x is called penalty parameter and G is penalty function defined in terms of the constraint functions (g) as below,

GðxÞ ¼

X

giþ ðxÞ

(41)

where,

giþ ðxÞ ¼ maxð0; gi ðxÞÞ ¼



0 gi ðxÞ

if if

gi ðxÞ  0 gi ðxÞ > 0

(42)

5. Results and discussion

kþ1

Where, xi is the new control variables for the next iteration (kþ1). xik is the old control variables at the current iteration (k). ljk is the step size at the current iteration (k). Dki;j is the normalized direction vector at the current iteration (k). The step size ljk is determined based on whether the new objective function improves or not under the current direction. The formula is given by,

5.1. Plant performance The organic rankine cycle heat recovery power plant (as shown in Fig. 1) consists of an expander, an evaporator, an air cooled condenser, and a working fluid pump. Design parameters are listed in Table 1. A plant simulation program is developed based on the

J. Sun, W. Li / Applied Thermal Engineering 31 (2011) 2032e2041

System Net Power Generation, kW

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6000 5000 4000 3000 2000 1000 0.4 0.5 0.6 0.7 0.8 0.9

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mathematical models described in Section 3. This plant simulation is conducted to illustrate the influences of controlled variables: the working fluid mass flow rate, the air cooled condenser fan air mass flow rate and the expander inlet pressure, on the system thermal efficiency and system net power generation. The uncontrolled variables, ambient dry bulb temp, heat source temp and heat source flow, are fixed at their design values (shown in Table 1) during these simulation runs. The thermodynamic properties are calculated with REFPROP program developed by NIST [18]. The simulation results are presented in Fig. 5 to Fig. 8 respectively. As shown in Figs. 5 and 6, increasing working fluid mass flow rate improves the system thermal efficiency and net power generation at the beginning. However, continuing to increase the working fluid mass flow rate will reduce the system thermal efficiency and net power generation. The reason is that the working fluid pump power consumption is offsetting the power generation augmentation resulted by increased working fluid mass flow rate. Similar impacts can be found by varying the condenser fan air mass flow rate. Therefore, at any given ambient dry bulb temperature and heat source conditions, an optimal set of working fluid mass flow rate and condenser fan air mass flow rate can be obtained to achieve the best system thermal efficiency or the most net power generation. In addition, the simulation result also indicates that both the system thermal efficiency and the net power generation are more sensitive to the working fluid mass flow rate than to the condenser fan air mass flow rate.

Fig. 8. Variations of system net power generation with expander inlet pressure and condenser fan air mass flow rates.

Fig. 7 and Fig. 8 indicate that higher expander inlet pressure results in both higher system thermal efficiency and higher system net power generation. Also the system thermal efficiency and the system net power generation increase linearly with the expander inlet pressure. However, the expander inlet pressure can’t be increased arbitrarily. First of all, higher expander inlet pressure demands more plant capital cost and operation cost since more expensive equipment and maintenance are required. More seriously, most working fluids have higher chemical instability at higher pressure situation. High expander inlet pressure could cause them decomposition and deterioration. Thus, ORC heat recovery power plants are usually operated at their design expander inlet pressure. 5.2. Operation optimization Based on the optimization method discussed in Section 4, an optimization program is developed to search the optimal set of the controlled variables subject to different uncontrolled variables in order to achieve the best system thermal efficiency or most system net power generation. According to the plant performance analysis in previous section, the expander inlet pressure is controlled at the design pressure. the working fluid mass flow rate and the condenser fan air mass flow rate are considered in this optimization study. Fig. 9 and Fig. 10 demonstrate the search path and 1

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Fig. 6. Variations of system net power generation with mass flow rates of working fluid and condenser fan air.

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Fig. 7. Variations of system thermal efficiency with expander inlet pressure and condenser fan air mass flow rates.

Fig. 9. Search path.

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Fig. 12. Optimal relative condenser air mass flow rate to maximize the system net power generation. Fig. 10. Optimization process.

convergence process of the optimization. In this example, ambient dry bulb temperature, heat source flow and temperature, and expander inlet pressure are also fixed at their design conditions. Initial step size is 0.1 in both directions. The constants a, b are 2 and 0.5 respectively. It took ROSENB algorithm 8 iterations (total 58 simulation runs) to converge and find the maximal net power 5302 kW at working fluid mass flow rate 250 kg/s and total condenser fan air mass flow rate 3370 kg/s. Two optimization problems are performed using the mathematical models and optimization algorithm described in previous sections: one is to maximize the system thermal efficiency, the other is to maximize the system net power generation. For both problems, the manipulated variables are the relative working fluid mass flow rate and the relative condenser fan air mass flow rate. Another controlled variable, expander inlet press, is constant at its design value 3447 kPa. Calculation results are summarized in the Figs. 11 to 14, which demonstrate the variations of the optimal relative working fluid mass flow rate and the optimal relative condenser fan air mass flow rate along with the changes of uncontrolled variables: ambient dry bulb temperature, heat source temperature. Another uncontrolled variable, heat source mass flow rate, is assumed to be constant at design flow rate. Other parameters are listed in Table 1. As shown in the Figs. 11 and 12, to maximize the system net power generation, both the optimal relative working fluid mass flow rate and the optimal condenser fan air mass flow rate are required to increase with the increasing of the heat source temperature. Higher

heat source temperature represents more available heat energy, which demands more working fluid and generates more electricity. In the meantime, more exhaust heat comes from expander and requires larger condenser capacity. It can be seen from the results that the optimal relative working fluid mass flow rate and the optimal condenser fan air mass flow rate are more sensitive to heat source temperature than the ambient dry bulb temperature. In addition, simple near linear functions are accurate enough to represent the relationship of the optimal relative working fluid mass flow rate, the optimal condenser fan air mass flow rate and the heat source temperature, the ambient dry bulb temperature. From the standpoint of maximizing the system thermal efficiency, increasing heat source temperature demands more working fluid and condenser fan air mass flow at the beginning, then less amount of working fluid and condenser fan mass flow are required with continuing to increase the heat source temperature (as shown in the Figs.13 and 14). The reasons is that although more heat are utilized to produce more electricity with increasing heat source temperature, there is a point where the increased electricity can be offset by the power consumed by the working fluid pump and condenser fan in order to utilize the heat, therefore the thermal efficiency starts to drop except for reducing the usage of heat. As shown in the Figs. 13 and 14, quadratic functions can be used to present the relationship of the optimal relative working fluid mass flow rate, the optimal condenser fan air mass flow rate and the heat source temperature, the ambient dry bulb temperature when maximizing the system thermal efficiency.

Fig. 11. Optimal relative working fluid mass flow rate to maximize the system net power generation.

Fig. 13. Optimal relative working fluid mass flow rate to maximize the system thermal efficiency.

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Fig. 14. Optimal relative condenser air mass flow rate to maximize the system thermal efficiency.

6. Conclusion This paper presents a detailed analysis of an organic rankine cycle heat recovery power plant using R134a as working fluid. Mathematical models for expander, evaporator, and air cooled condenser and pump are developed to evaluate the plant performance. In order to maximize either the system new power generation or the system thermal efficiency, ROSENB algorithm with penalty function method is proposed to search the optimal set of controlled variables. Computer programs are developed based on these simulation models and optimization algorithm. The simulation and optimization results indicate,  Higher expander inlet pressure results more system net power generation and higher system thermal efficiency. However expander inlet pressure depends on the working fluid property and project cost. And linear relationships exist among the system thermal efficiency, the system net power generation and expander inlet pressure.  The working fluid mass flow rate has more influence on the system thermal efficiency and the net power generation than the condenser fan air mass flow rate.  To maximize the system net power generation, both the optimal relative working fluid mass flow rate and the optimal condenser fan air mass flow rate increase with the heat source temperature increasing.  Near linear functions are accurate enough to represent the relationships of the optimal relative working fluid mass flow rate, the optimal relative condenser fan air mass flow rate and the heat source temperature, the ambient dry bulb temperature when trying to maximize the system net power generation.  To maximize the system thermal efficiency, increasing heat source temperature demands more working fluid and condenser fan air mass flow at the beginning, then less amount of working fluid and condenser fan mass flow are required with continuing to increase the heat source temperature.  Quadratic functions can be used to represent the relationships of the optimal relative working fluid mass flow rate, the optimal relative condenser fan air mass flow rate and the heat source temperature, the ambient dry bulb temperature when trying to maximize the system thermal efficiency. Nomenclature Isobaric specific heat, kJ/kg  C Cp;EXP Cp;EV;WF Evaporator working fluid isobaric specific heat, kJ/kg  C

Cp;EV;HF Evaporator hot fluid isobaric specific heat, kJ/kg  C Cp;AC;WF Air cooled condenser working fluid isobaric specific heat, kJ/kg  C Cp;AC;Air Air cooled condenser air isobaric specific heat, kJ/kg  C hEXP Ent ,hEXP Lvg Entering/leaving expander working fluid enthalpies, kJ/kg hEV;WF Ent ,hEV;WF Lvg Entering/leaving evaporator working fluid enthalpies, kJ/kg hEV;HF Ent hEV;HF Lvg Entering/leaving evaporator hot fluid enthalpies, kJ/kg hAC;WF Ent ,hAC;WF Lvg Entering/leaving air cooled condenser working fluid enthalpies, kJ/kg hAC;Air Ent ,hAC;Air Lvg Entering/leaving air cooled condenser hot fluid enthalpies, kJ/kg Working fluid pump head, m HP Working fluid pump design head, m HP Des Working fluid pump equivalent head, m HP Eqv Expander working fluid mass flow rate, kg/s mEXP mEV;WF Evaporator working fluid mass flow rate, kg/s Evaporator hot fluid mass flow rate, kg/s mEV;HF mAC;WF Air cooled condenser working fluid mass flow rate, kg/s mAC;Air Air cooled condenser total air mass flow rate, kg/s mFAN ,mFAN Des Actual and design air mass flow rate per air cooled condenser, kg/s Working fluid pump flow rate, m3/s MP Working fluid pump design flow rate, m3/s MP Des Working fluid pump equivalent flow rate, m3/s MP Eqv NEXP ,NEXP Des Expander rotating speed/design expander rotating speed, rpm Evaporator number of heat transfer unit NtuEV Air cooled condenser number of heat transfer unit NtuAC PEXP Ent ,PEXP Lvg Entering/leaving expander working fluid pressure, Pa Evaporator working fluid side heat transfer rate, kW qEV;WF Evaporator hot fluid side heat transfer rate, kW qEV;HF Evaporator heat transfer rate, kW qEV qEV;MAX Evaporator maximum possible heat transfer rate, kW Evaporator total heat transfer rate, kW QEV Air cooled condenser working fluid side heat transfer qAC;WF rate, kW Air cooled condenser air side heat transfer rate, kW qAC;Air Air cooled condenser heat transfer rate, kW qAC qAC;MAX Air cooled condenser maximum possible heat transfer rate, kW Air cooled condenser total heat transfer rate, kW QAC SG Working fluid specific gravity TEXP Ent ,TEXP Lvg Entering/leaving expander working fluid temperature,  C TEXP Ent Des Design entering expander working fluid temperature,  C TEV;WF Ent ,TEV;WF Lvg Entering/leaving evaporator working fluid temperature,  C TEV;HF Ent ,TEV;HF Lvg Entering/leaving evaporator hot fluid temperature,  C TAC;WF Ent ,TAC;WF Lvg Entering/leaving air cooled condenser working fluid temperature,  C TAC;Air Ent ,TAC;Air Lvg Entering/leaving air cooled condenser hot fluid temperature,  C Evaporator overall heat transfer coefficients UAEV Air cooled condenser overall heat transfer coefficients UAAC Expander power generation, kW WEXP Power consumption of working fluid pump, kW WP Brake horse power of working fluid pump, kW Wbhp WFAN ,WFAN Des Actual and design air cooled condenser fan power consumption, kW hEXP ,hEXP Des Expander efficiency/design expander efficiency

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hP ,hP hth p,pDes g 3EV 3AC

Des

Working fluid pump efficiency/design working fluid pump efficiency System thermal efficiency Pressure ratio/design pressure ratio of expander inlet and outlet Mean isentropic coefficient Evaporator effectiveness Air cooled condenser effectiveness

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