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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

2036

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

System Net Power Generation, kW

2038

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

Relative Working Fluid Mass Flow

1

0.9

1

0.8

0.7

0.6

0.5

0.4

0.3

0.2

5000 4000 3000 2000 1000 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1

1

0.7

0.8

0.9

0.6

0.4

0.3

0.2

Relative Condenser Fan Air Mass Flow

Relative Expander Inlet Pressure

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

Relative Working Fluid Mass Flow

10 8 6 4 2 0.65 0.7 0.75

0.2 0.8 0.85

50 00 00 40

0.9

50 00

5000

45 00

0.8

4500

4500 40 00

4000

4000

0.7 35 00

35 00

3500

0.6

0.5

30 00

3000

30 00

25 00 25 00

25 00

0.4 0.6

0.9 0.95

0.8

20 00

0.4

20 00

2000

15 00 15 00

1 Relative Expander Inlet Pressure

0.5

Relative Condenser Fan Air Mass Flow

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

System Thermal Efficiency, %

6000

Relative Condenser Fan Air Mass Flow

0.3

0.4

0.5

0.6

0.7

0.8

Relative Condenser Fan Air Mass Flow

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

Fig. 9. Search path.

0.9

1

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

2039

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.

2040

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

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

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

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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