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Quasi-dynamic model for an organic Rankine cycle
Energy Conversion and Management 72 (2013) 117–124
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Energy Conversion and Management journal homepage: www.elsevier.com/locate/enconman
Quasi-dynamic model for an organic Rankine cycle Musbaudeen O. Bamgbopa ⇑, Eray Uzgoren Sustainable Environment and Energy Systems, Middle East Technical University, Northern Cyprus Campus, Kalkanli, Guzelyurt, TRNC, Mersin 10, Turkey
a r t i c l e
i n f o
Article history: Available online 15 April 2013 Keywords: Organic Rankine cycle Transient analysis Solar energy R245fa
a b s t r a c t When considering solar based thermal energy input to an organic Rankine cycle (ORC), intermittent nature of the heat input does not only adversely affect the power output but also it may prevent ORC to operate under steady state conditions. In order to identify reliability and efficiency of such systems, this paper presents a simplified transient modeling approach for an ORC operating under variable heat input. The approach considers that response of the system to heat input variations is mainly dictated by the evaporator. Consequently, overall system is assembled using dynamic models for the heat exchangers (evaporator and condenser) and static models of the pump and the expander. In addition, pressure drop within heat exchangers is neglected. The model is compared to benchmark numerical and experimental data showing that the underlying assumptions are reasonable for cases where thermal input varies in time. Furthermore, the model is studied on another configuration and mass flow rates of both the working fluid and hot water and hot water’s inlet temperature to the ORC unit are shown to have direct influence on the system’s response. Ó 2013 Elsevier Ltd. All rights reserved.
1. Introduction The need for sustainable/alternative energy cannot be overemphasized. In recent years, more attention is devoted to alternative sources of energy, i.e. geothermal, solar, and wind. Recent research has demonstrated organic Rankine cycles (ORCs) are good candidates for power production (energy recovery) from low temperature heat sources, compared to water vapor cycles in the same operation temperature range [1]. To ensure their wider use and acceptance, it is vital that the performance of ORC systems be continuously analyzed to identify possible rooms for improvement. As emphasized in [2], the main distinguishing factor between ORC and a conventional Rankine cycle is that ORCs utilize refrigerants as their working fluids whereas Rankine cycles use steam. The energy interactions require thermodynamic properties of the refrigerant to be known at the inlet and exit of the sub-components. The present study uses a piece-wise regression approach in numerical modeling of the thermal properties of the selected working fluid (R245fa) based on the data supplied by the fluid manufacturer. This method is computationally effective as the required properties can be obtained by defining sub-zones of interest considering the T–s diagram of the R245fa seen in Fig. 1. A typical organic Rankine cycle (ORC), as shown in Fig. 2, is formed by assembling heat exchangers (i.e. evaporator, condenser), flow inducing unit (pump), and power extraction unit (expander). As opposed to a conventional Rankine Cycle, in which steam tur⇑ Corresponding author. Tel.: +234 8174429709. E-mail address: [email protected] (M.O. Bamgbopa). 0196-8904/$ - see front matter Ó 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.enconman.2013.01.040
bines are widely used for power extraction, their alternatives have been an active research topic when power production capacity is reduced [3]. Considering the operating temperatures of a small scale ORC system (10–100 kW), volume based expanders (i.e. screw type, scroll type and reciprocal piston) are more suitable as they not only operate under lower mass flow rates and rotational speeds [4], but they can also tolerate two-phase conditions [5]. They are reported to have isentropic efficiencies in the range of 48–68% [6]. ORC’s steady state behavior has been studied by many researchers [7–11] and many focused on the cycle’s efficiency for various working fluids [12]. Some of the studies on the steady state behavior [13] ultimately assumed constant heat input in their analysis. Others [7] justify steadiness in input by considering a supplementary boiler in the system to stabilize the heat input to the ORC. In addition to those, the dynamic behavior has also been studied by [6,14] usually to develop control strategies for waste heat recovery. Especially, Quoilin et al. [6] have presented the full dynamic model for all components of the ORC utilizing a scroll type expander. The present work develops a numerical model for an ORC unit under transient operation. Particularly, the unit is considered to receive energy input through water heated by solar-thermal parabolic trough collectors (PTCs). One of the main challenges for studies analyzing solar ORC systems is the variability of the available heat input due to intermittency of solar energy where a supplementary boiler or thermal storage is not installed. For such systems, a transient model is required to investigate the system dynamics. The method can also be applicable to the case of ORCs utilized for energy recovery from internal combustion engines
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Nomenclature A cp h H _ m V_ P Pr Re T V x w
area (m2) specific heat capacity (kJ/kg K) enthalpy (kJ/kg) heat transfer coefficient (W/m2 K) mass flow rate (kg/s) volume flow rate (m3/s) pressure (kPa) Prandtl number Reynold’s number temperature (°C) volume (m3) vapor quality work per unit mass (kJ/kg)
l e g
viscosity (Pa s) effectiveness efficiency
Subscripts b boiling c condensation l liquid p pipe ref refrigerant sat saturated v vapor w water
Greek symbols q density (kg/m3)
of heat transfer in and out of the unit respectively. Contrary to some experimental ORC studies like [15,16], in which operating states are known from direct measurements, the present numerical study requires dynamic models for each heat exchanger to be integrated through the state properties of the working fluid. Dynamic models for the turbo-machines (pump and expander) are avoided due to their negligible heat transfer irreversibility compared to their mechanical interaction with the working fluid and their faster response time to varying inputs as compared to the heat exchangers. The present approach is compared to literature at various stages and results from an integrated ORC model simulation are validated against experimental data.
2. Numerical model
Fig. 1. T–s diagram of R245fa (HFC 1,1,1,3,3 pentafluoropropane).
The numerical model of the ORC unit in this study is presented as a synthesis of sub-models of four main components; evaporator (ev), twin screw expander (ex), condenser (co) and pump (pu) as shown in Fig. 2 and the thermodynamic states at the inlets of the pump, the evaporator, the expander and the condenser (numbered as 1–4 in Fig. 2) are identified through models for each subcomponent. In the present study, the condenser is assumed to be capable of completing the cycle at the low pressure side. This is feasible by considering an accumulation tank between the condenser and pump as the usual case in reality to ensure sub-cooling [6]. The following sections discuss the models of each main component. 2.1. State 1-2: Across pump
Fig. 2. Representation of a typical ORC system.
and for investigating the possible control strategies of critical parameters such as evaporating temperature and mass flow rates as seen in [6]. Critical components of a solar based ORC are the heat exchangers (evaporator and condenser) since they are the principal media
As indicated in Fig. 2, the refrigerant is saturated liquid at the inlet of the pump. The pump’s influence on the fluid is described by its characteristic and performance curves which estimate the pressure rise as a function of the volumetric flow rate and pump efficiency. These curves, usually supplied by the manufacturer, can be approximately obtained by regression into the form of Eq. (1) as suggested by [17].
_ V_ þ 2R2 nV_ þ R3 n Dp ¼ R1 jVj
ð1Þ
where, p, n and V_ are pressure, rotational speed and volumetric flow rate, respectively. The coefficients, R1, R2 and R3, are specific for the pump under consideration. For a single speed pump without back
M.O. Bamgbopa, E. Uzgoren / Energy Conversion and Management 72 (2013) 117–124
flow, Eq. (1) can be represented in terms of the mass flow rate after combining coefficients and the rotational speed together, as
DP ¼ C 1
_ m
!2
qpu;in
þ C2
_ m
qpu;in
þ C3
ð2Þ
The constants in Eq. (2), C1, C2, and C3, are obtained directly by regression to the curves supplied by the pump’s manufacturer. Then, the pump efficiency can be defined as a function of a dimensionless parameter, Xpu, as proposed in Quoilin et al. [6] and given as
gpu ¼ A0 þ A1 logðX pu Þ þ A2 logðX pu Þ2 þ A3 logðX pu Þ3
ð3Þ
In Eq. (3), coefficients, A0 to A3, are also obtained using the pump data. The dimensionless parameter, Xpu, is a measure comparing the actual and the maximum flow rates, as defined in
X pu ¼
_ m _ qpu;in V pu ; max
ð4Þ
where V_ pu;max is the maximum flow rate of the selected pump. This efficiency is to be multiplied by the electromechanical efficiency, eem,pu, to yield an overall efficiency in calculating the pump power as seen in
wpu ¼ h2 h1 ¼
DP
model is required. Generally, the differential equations originating from Eq. (6) are solved numerically by finite difference/volume methods as seen in [18] or in [19]. A finite volume representation of the double pipe approximation of an actual heat exchanger (of any geometry) showing counter-flow configuration between water and refrigerant is as shown in Fig. 3. As the exit states of both fluids are not known in priori, the arising differential equations for each discrete node (seen in Fig. 3) will be solved ‘marching’ from one inlet to exit (node 0 to N + 1). The present study considers the following simplifications: (i) The main mechanism for heat transfer in fluid flow is through convection rather than conduction. (ii) Heat transfer is purely between both fluids through inner pipe as perfect insulation is assumed for outer pipe. (iii) Uniform flow without viscous effects in pipes. Therefore, the finite volume equations for the R245fa, pipe and water volumes are expressed simultaneously in explicit form in Eqs. (8)–(10), respectively;
3 t t t _ tref ;i htref ;i1 htref ;i þ m AH T T p;i ref ;i ref ;i tþ1 t 5 Refrigerant href ;i ¼ href ;i þ Dt 4 V ref qtref ;i 2
ð5Þ
eem;pu qpu;in gpu
119
ð8Þ
3 Htw A T tw;i T tp;i þAHtref ;i T tþ1 T tp;i þqtþ1 ref ;i condp;i tþ1 t 4 5 Pipe T p;i ¼ T p;i þ Dt V p qp C p 2
2.2. States 2-3 and 4-1: Evaporator and condenser Energy balance across the heat exchangers can be expressed as
ð9Þ
in
@E _ ref ðhin hout Þ ¼ Q_ ev =co þ m @t
ð6Þ
The actual heat transfer rate in the evaporator or in the condenser, Q_ ev =co , is related to the heat exchangers effectiveness, e, which defines a measure against the maximum possible heat transfer rate for the heat exchanger, Q_ max , as given in
Q_
e ¼ _ eV=co Q max
ð7Þ
Solution for Eq. (6) can be obtained through heat exchanger’s effectiveness at steady operation. However, if the conditions of the heat exchanger are changing, i.e. during startup or operating with a variable heat source such as solar energy, then a transient
2 t 4 T tþ1 w;i ¼ T w;i þ Dt
3 t _ tw C tp;w T tw;iþ1 T tw;i þHtw A T tþ1 m p;i T w;i 5 Water V w qtw;i C tp;w
ð10Þ
Momentum equation is not included since the model neglects pressure drop in the heat exchanger and the pipe’s material properties are assumed to be constant in the temperature range as seen in Eq. (9). The evaporator and condenser models can be handled in the same way except that the mechanism of heat transfer is different when boiling and condensation occurs. The heat transfer coefficient(s) in the developed heat exchanger models are estimated using correlations for single and two-phase flows.
Fig. 3. Finite volume representation of a counter-flow heat exchanger.
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given by Ng et al. [25] as a function of its in-built volume ratio (rv) and pressure ratio (rp)
wex ¼
Pexp;in
qexp;in
!
K
rv rp
r1n n K¼ v "1 n # a1 a2 a3 logr p þ þ n¼ logr v logr 2v logr3v
ð18Þ ð19Þ ð20Þ
Eq. (20) suggests that for a given twin screw expander, the polytropic expansion index, n, can be defined by the regression constants a1, a2, and a3. 3. Validation studies
Fig. 4. Comparison of the boiling heat transfer coefficient correlations.
3.1. Grid independent solution For cases where the fluids exist in single state (liquid water, pure liquid and pure vapor R245fa), Gnielinski equation given in Eqs. (11) and (12) is used as defined in [20] for turbulent flow;
H¼
k 1 þ 12:7ðf =8Þ0:2 ðPr2=3 1Þ d ðf =8ÞðRe 1000ÞPr
f ¼ ð0:79 ln Re 1:64Þ2
ð11Þ ð12Þ
When the fluid is in two-phase (saturated mixture), Eqs. (11) and (12) are not adequate. For boiling in the evaporator, the Kenning– Cooper (K&C) correlation in Eq. (13) is used for simplicity as given by Sun and Mishima [21] based on their findings. Comparison of the Kenning–Cooper correlation to the widely used Chen correlation [21–23] can be visualized in Fig. 4 for a test case presented by Vaja [19]. The test case considers saturated water at atmospheric pressure, flows with 0.1 kg/s through a 0.02 m diameter heated tube at 120 °C. A correlation from Varme Atlas (used in [14]) is also included in the comparison.
Hb ¼ ½1 þ 1:8X
0:87
048 0:023Re0:8 ðk=dÞ l Pr l
ð13Þ
where k is thermal conductivity, d is the pipe diameter, and X is the Martinelli factor expressed as;
X¼
1x x
0:9
qv ql
0:5
ll lv
0:1 ð14Þ
For the condenser, Dobson and Chato correlation given in Eq. (15) (adopted from [24] for simplicity) is adopted in the present study for Rev < 35,000.
"
3 0
g ql ðql qg Þkl hlv Hc ¼ 0:555 ll ðT sat T s Þdh
#0:25
The transient evaporator model is finite volume based. Therefore, for consistency, it is expected that the accuracy of results increase with the number of discrete nodes (N) assigned for the numerical solution. On the other hand, a very high value of N would lead to avoidable computational burden beyond the desired level of accuracy to be attained. In order to find a reasonable number of discrete nodes, N, effectiveness of the evaporator is computed and compared for a range of number of nodes, with predefined conditions (inlet temperatures and flow rates) for water and R245fa. The comparison is based on the heat exchanger’s effectiveness, eN. The subscript, N, denotes the number of nodes employed in the numerical solution. Error based on N number of nodes can be expressed as in
errorN ¼
eN eexpected eexpected
where eexpected represents the expected effectiveness of the heat exchanger and is computed using a sufficiently large number of nodes. Fig. 5 illustrates the trend of error in effectiveness as obtained from the evaporator model (after steady state is achieved) for different values of N, where eexpected is computed for 160 nodes (N = 160). The inflow conditions considered in the test cases are: 10 kg/s hot water enters the evaporator at 93 °C and 1 atm. 0.75 kg/s R245fa inflow at 27 °C and 678 kPa. The tests are started from the same initial conditions, i.e. constant temperature throughout the heat exchanger, and marched in time until steady state conditions are obtained. It should be noted that error in Eq. (21) also represents the error in actual heat
ð15Þ 0
where the modified latent heat hlv is a function of latent heat hlv, given as; 0
hlv hlv þ 3=8Cpl ðT sat T p Þ
ð16Þ
For higher Rev, the Boyko and Kruzhilin correlation is considered as presented in [18], given by;
Hc ¼
0:5 kl q 0:43 1þx l 1 0:021Re0:8 l Pr l d qv
ð17Þ
2.3. State 3-4: Expander Following the considerations for over and under-expansion, the nominal polytropic work output of a twin screw expander was
ð21Þ
Fig. 5. Grid independency tests for N = 10, 20, 40, 60, 70, 75 and 80.
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121
Table 1 Evaporator (theoretical) dimensions in Vaja [19]. L
do
d
di
5.52 m
9.5 mm
16 mm
7.9 mm
transfer rate because the maximum possible heat transfer rate, Q_ max , is the same once the above inflow conditions are fixed. It is observed in Fig. 5 that selecting number of nodes, N, at 70 gives a reasonable balance between accuracy and computational burden for the conditions given above. 3.2. Validation of evaporator model The present evaporator model is validated against the results of Vaja [19], who used similar finite volume discretization approach (including momentum equations) to predict state variables from experimental results of Takamatsu et al. [26]. The conditions are given as; 0.0279 kg/s of water inflow at 56 °C (atmospheric pressure) evaporates 0.0142 kg/s R22 inflow at 27 °C and 1140 kPa in an evaporator, for which dimensions are given in Table 1. It should be noted that the simulation of the present study determined steady state was achieved once the maximum nodal hot water temperature difference experienced from one time step to the next becomes less than 0.0001 °C. Figs. 6 and 7 compare the obtained distribution of water and pipe temperatures over the length, L, of the heat exchanger respectively while Fig. 8 also compares the vapor quality of the refrigerant with position using N = 20 and time step of 0.05 s as in [19]. Figs. 6 and 7 show that the present model overestimates the experimental results. Nevertheless, the trend of the present model is similar to the experimental one like Vaja’s result as shown in Fig. 7. The vapor quality distribution in Fig. 8 suggests that vapor quality of refrigerant is not influenced by the correlation for heat transfer coefficient used. The slight differences seen in Figs. 6 and 7 can be attributed to the pressure drop across the heat exchanger and its effect accumulated in time.
Fig. 7. Pipe temperature distribution of selected models.
3.3. System level validation with experimental data Fig. 9 illustrates the major system input parameters of the ORC model, which will be compared against the data from real transient operation. Thermal energy to the ORC is supplied by circulating hot water in the PTC loop. This heat input is characterized by the mass
Fig. 6. Hot water temperature distribution of selected models.
Fig. 8. Refrigerant’s (R22) vapor quality distribution of selected models.
flow rate and state properties of the hot water at the evaporator inlet. Similarly, heat rejection through the condenser is characterized by the mass flow rate and state properties of cold water circulating in the cooling tower loop. Although the system is not an experimental set-up, measurements of the main inputs into the ORC (indicated in Fig. 9) and physical features of its main components are known. Such parameters are fed into an ORC model integrated from the sub-models of each components presented in Section 2 and the obtained output will be compared to available data. The available data from the actual system to be used is the measurement of cold water exit temperature. During a 15 min (900 s) operation, temperature of hot and cold water inlet to the ORC varies as seen in Fig. 10 at atmospheric pressure. The mass flow rates of refrigerant, hot and cold water are known to be 1.5, 2.5 and 15 kg/s respectively. The condensing temperature (Tlow) is set to 5 °C higher than the inlet cold water temperature which is a similar practice used in simulations of Kang [27]. For reasonable initial nodal distribution to kick-start the solution of the transient equations (Section 2.2), the solution approach ran the system at the first recorded input values till
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Fig. 9. Input variables for the ORC model.
Fig. 10. Variation of inlet hot and cold water temperatures in time.
steady state before enforcing the variable input profile described in Fig. 10. The pump of the ORC is a vertical multi-stage pump with maximum flow rate of 12 m3/h and using its characteristics curves, the modeling constants (in Eqs. (2) and (3)) are presented in Table 2. For the twin screw expander, considering a maximum net power output of about 40 kW, the modeling constants defining the polytropic expansion index (Eq. (20)) are also seen in Table 2. The two stainless steel compact heat exchangers; evaporator and condenser have effective heat transfer area of 18 and 11 m2 respectively. Their approximated double pipe geometry dimensions which are used as highlighted in Table 3. Based on the findings of Section 3.1, number of discretized node, N, selected for the simulation is 60 while 0.02 s time step is used. The resulting time variation of the exit temperature of cold water from the condenser as produced by the simulation can be
Table 2 Characteristic constants for the pump and the expander. i
a
A
C
0 1 2 3
– 8.74E02 1.45E01 2.41E01
0.543 0.0722 1.566 1.2524
– 796,886 197.49 40.266
Table 3 Heat exchanger dimensions for the ORC system. L
do
d
di
80 m (50 m for condenser)
0.0686 m
0.1235 m
0.0656 m
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ORC system has been modeled via a synthesis of sub-models of the pump, evaporator, expander and condenser while the working fluid (R245fa) property calculations have been performed via a simplified regression based approach. It is seen that the heat exchanger models (evaporator and condenser) are the critical components in investigating the transient mode of operation since they are the principal media of heat transfer in and out of the ORC. With a tool such as the transient model of the solar ORC at hand, the respective influences of critical parameters such as (refrigerant and water flow rates) can be controlled. This can be applied in real-time dynamic simulation of the unit under different operation scenarios to optimize power production or ensure steady operation given any level of heat input.
Acknowledgement
Fig. 11. Comparison of simulated cold water outlet temperature with recorded data.
This work is supported by the Middle East Technical University–Northern Cyprus Campus (METU–NCC) campus research fund, Project No. BAP-FEN-11-O-10.
References
Fig. 12. Net power output.
compared with recorded data as seen in Fig. 11. Although, data for the net power output was not available, Fig. 12 presents the variation of the net power output with time during the 900 s operation. For simulation of transient operation, the main comparison to be made when validating with experimental data is not with final absolute values but with the trend the data displays with time. Fig. 12 shows the result from the model slightly differs in absolute values with recorded data of the cooling water exit temperature, however they both exhibit similar trend. Hot water inlet temperature is one of the critical parameters defining the heat input to the ORC. Therefore it is expected that the net power output shows sensitivity to the transition of hot water temperature where no control measures are enforced. Fig. 12 verifies this expectation as it shows the net power output having similar profile with the hot water inlet temperature (Fig. 10) in the investigated time frame once initial conditions have been overcome.
4. Conclusion This study presents modeling approaches applicable for transient operation of solar-ORCs given variable heat input. A typical
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Contents lists available at SciVerse ScienceDirect
Energy Conversion and Management journal homepage: www.elsevier.com/locate/enconman
Quasi-dynamic model for an organic Rankine cycle Musbaudeen O. Bamgbopa ⇑, Eray Uzgoren Sustainable Environment and Energy Systems, Middle East Technical University, Northern Cyprus Campus, Kalkanli, Guzelyurt, TRNC, Mersin 10, Turkey
a r t i c l e
i n f o
Article history: Available online 15 April 2013 Keywords: Organic Rankine cycle Transient analysis Solar energy R245fa
a b s t r a c t When considering solar based thermal energy input to an organic Rankine cycle (ORC), intermittent nature of the heat input does not only adversely affect the power output but also it may prevent ORC to operate under steady state conditions. In order to identify reliability and efficiency of such systems, this paper presents a simplified transient modeling approach for an ORC operating under variable heat input. The approach considers that response of the system to heat input variations is mainly dictated by the evaporator. Consequently, overall system is assembled using dynamic models for the heat exchangers (evaporator and condenser) and static models of the pump and the expander. In addition, pressure drop within heat exchangers is neglected. The model is compared to benchmark numerical and experimental data showing that the underlying assumptions are reasonable for cases where thermal input varies in time. Furthermore, the model is studied on another configuration and mass flow rates of both the working fluid and hot water and hot water’s inlet temperature to the ORC unit are shown to have direct influence on the system’s response. Ó 2013 Elsevier Ltd. All rights reserved.
1. Introduction The need for sustainable/alternative energy cannot be overemphasized. In recent years, more attention is devoted to alternative sources of energy, i.e. geothermal, solar, and wind. Recent research has demonstrated organic Rankine cycles (ORCs) are good candidates for power production (energy recovery) from low temperature heat sources, compared to water vapor cycles in the same operation temperature range [1]. To ensure their wider use and acceptance, it is vital that the performance of ORC systems be continuously analyzed to identify possible rooms for improvement. As emphasized in [2], the main distinguishing factor between ORC and a conventional Rankine cycle is that ORCs utilize refrigerants as their working fluids whereas Rankine cycles use steam. The energy interactions require thermodynamic properties of the refrigerant to be known at the inlet and exit of the sub-components. The present study uses a piece-wise regression approach in numerical modeling of the thermal properties of the selected working fluid (R245fa) based on the data supplied by the fluid manufacturer. This method is computationally effective as the required properties can be obtained by defining sub-zones of interest considering the T–s diagram of the R245fa seen in Fig. 1. A typical organic Rankine cycle (ORC), as shown in Fig. 2, is formed by assembling heat exchangers (i.e. evaporator, condenser), flow inducing unit (pump), and power extraction unit (expander). As opposed to a conventional Rankine Cycle, in which steam tur⇑ Corresponding author. Tel.: +234 8174429709. E-mail address: [email protected] (M.O. Bamgbopa). 0196-8904/$ - see front matter Ó 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.enconman.2013.01.040
bines are widely used for power extraction, their alternatives have been an active research topic when power production capacity is reduced [3]. Considering the operating temperatures of a small scale ORC system (10–100 kW), volume based expanders (i.e. screw type, scroll type and reciprocal piston) are more suitable as they not only operate under lower mass flow rates and rotational speeds [4], but they can also tolerate two-phase conditions [5]. They are reported to have isentropic efficiencies in the range of 48–68% [6]. ORC’s steady state behavior has been studied by many researchers [7–11] and many focused on the cycle’s efficiency for various working fluids [12]. Some of the studies on the steady state behavior [13] ultimately assumed constant heat input in their analysis. Others [7] justify steadiness in input by considering a supplementary boiler in the system to stabilize the heat input to the ORC. In addition to those, the dynamic behavior has also been studied by [6,14] usually to develop control strategies for waste heat recovery. Especially, Quoilin et al. [6] have presented the full dynamic model for all components of the ORC utilizing a scroll type expander. The present work develops a numerical model for an ORC unit under transient operation. Particularly, the unit is considered to receive energy input through water heated by solar-thermal parabolic trough collectors (PTCs). One of the main challenges for studies analyzing solar ORC systems is the variability of the available heat input due to intermittency of solar energy where a supplementary boiler or thermal storage is not installed. For such systems, a transient model is required to investigate the system dynamics. The method can also be applicable to the case of ORCs utilized for energy recovery from internal combustion engines
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Nomenclature A cp h H _ m V_ P Pr Re T V x w
area (m2) specific heat capacity (kJ/kg K) enthalpy (kJ/kg) heat transfer coefficient (W/m2 K) mass flow rate (kg/s) volume flow rate (m3/s) pressure (kPa) Prandtl number Reynold’s number temperature (°C) volume (m3) vapor quality work per unit mass (kJ/kg)
l e g
viscosity (Pa s) effectiveness efficiency
Subscripts b boiling c condensation l liquid p pipe ref refrigerant sat saturated v vapor w water
Greek symbols q density (kg/m3)
of heat transfer in and out of the unit respectively. Contrary to some experimental ORC studies like [15,16], in which operating states are known from direct measurements, the present numerical study requires dynamic models for each heat exchanger to be integrated through the state properties of the working fluid. Dynamic models for the turbo-machines (pump and expander) are avoided due to their negligible heat transfer irreversibility compared to their mechanical interaction with the working fluid and their faster response time to varying inputs as compared to the heat exchangers. The present approach is compared to literature at various stages and results from an integrated ORC model simulation are validated against experimental data.
2. Numerical model
Fig. 1. T–s diagram of R245fa (HFC 1,1,1,3,3 pentafluoropropane).
The numerical model of the ORC unit in this study is presented as a synthesis of sub-models of four main components; evaporator (ev), twin screw expander (ex), condenser (co) and pump (pu) as shown in Fig. 2 and the thermodynamic states at the inlets of the pump, the evaporator, the expander and the condenser (numbered as 1–4 in Fig. 2) are identified through models for each subcomponent. In the present study, the condenser is assumed to be capable of completing the cycle at the low pressure side. This is feasible by considering an accumulation tank between the condenser and pump as the usual case in reality to ensure sub-cooling [6]. The following sections discuss the models of each main component. 2.1. State 1-2: Across pump
Fig. 2. Representation of a typical ORC system.
and for investigating the possible control strategies of critical parameters such as evaporating temperature and mass flow rates as seen in [6]. Critical components of a solar based ORC are the heat exchangers (evaporator and condenser) since they are the principal media
As indicated in Fig. 2, the refrigerant is saturated liquid at the inlet of the pump. The pump’s influence on the fluid is described by its characteristic and performance curves which estimate the pressure rise as a function of the volumetric flow rate and pump efficiency. These curves, usually supplied by the manufacturer, can be approximately obtained by regression into the form of Eq. (1) as suggested by [17].
_ V_ þ 2R2 nV_ þ R3 n Dp ¼ R1 jVj
ð1Þ
where, p, n and V_ are pressure, rotational speed and volumetric flow rate, respectively. The coefficients, R1, R2 and R3, are specific for the pump under consideration. For a single speed pump without back
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flow, Eq. (1) can be represented in terms of the mass flow rate after combining coefficients and the rotational speed together, as
DP ¼ C 1
_ m
!2
qpu;in
þ C2
_ m
qpu;in
þ C3
ð2Þ
The constants in Eq. (2), C1, C2, and C3, are obtained directly by regression to the curves supplied by the pump’s manufacturer. Then, the pump efficiency can be defined as a function of a dimensionless parameter, Xpu, as proposed in Quoilin et al. [6] and given as
gpu ¼ A0 þ A1 logðX pu Þ þ A2 logðX pu Þ2 þ A3 logðX pu Þ3
ð3Þ
In Eq. (3), coefficients, A0 to A3, are also obtained using the pump data. The dimensionless parameter, Xpu, is a measure comparing the actual and the maximum flow rates, as defined in
X pu ¼
_ m _ qpu;in V pu ; max
ð4Þ
where V_ pu;max is the maximum flow rate of the selected pump. This efficiency is to be multiplied by the electromechanical efficiency, eem,pu, to yield an overall efficiency in calculating the pump power as seen in
wpu ¼ h2 h1 ¼
DP
model is required. Generally, the differential equations originating from Eq. (6) are solved numerically by finite difference/volume methods as seen in [18] or in [19]. A finite volume representation of the double pipe approximation of an actual heat exchanger (of any geometry) showing counter-flow configuration between water and refrigerant is as shown in Fig. 3. As the exit states of both fluids are not known in priori, the arising differential equations for each discrete node (seen in Fig. 3) will be solved ‘marching’ from one inlet to exit (node 0 to N + 1). The present study considers the following simplifications: (i) The main mechanism for heat transfer in fluid flow is through convection rather than conduction. (ii) Heat transfer is purely between both fluids through inner pipe as perfect insulation is assumed for outer pipe. (iii) Uniform flow without viscous effects in pipes. Therefore, the finite volume equations for the R245fa, pipe and water volumes are expressed simultaneously in explicit form in Eqs. (8)–(10), respectively;
3 t t t _ tref ;i htref ;i1 htref ;i þ m AH T T p;i ref ;i ref ;i tþ1 t 5 Refrigerant href ;i ¼ href ;i þ Dt 4 V ref qtref ;i 2
ð5Þ
eem;pu qpu;in gpu
119
ð8Þ
3 Htw A T tw;i T tp;i þAHtref ;i T tþ1 T tp;i þqtþ1 ref ;i condp;i tþ1 t 4 5 Pipe T p;i ¼ T p;i þ Dt V p qp C p 2
2.2. States 2-3 and 4-1: Evaporator and condenser Energy balance across the heat exchangers can be expressed as
ð9Þ
in
@E _ ref ðhin hout Þ ¼ Q_ ev =co þ m @t
ð6Þ
The actual heat transfer rate in the evaporator or in the condenser, Q_ ev =co , is related to the heat exchangers effectiveness, e, which defines a measure against the maximum possible heat transfer rate for the heat exchanger, Q_ max , as given in
Q_
e ¼ _ eV=co Q max
ð7Þ
Solution for Eq. (6) can be obtained through heat exchanger’s effectiveness at steady operation. However, if the conditions of the heat exchanger are changing, i.e. during startup or operating with a variable heat source such as solar energy, then a transient
2 t 4 T tþ1 w;i ¼ T w;i þ Dt
3 t _ tw C tp;w T tw;iþ1 T tw;i þHtw A T tþ1 m p;i T w;i 5 Water V w qtw;i C tp;w
ð10Þ
Momentum equation is not included since the model neglects pressure drop in the heat exchanger and the pipe’s material properties are assumed to be constant in the temperature range as seen in Eq. (9). The evaporator and condenser models can be handled in the same way except that the mechanism of heat transfer is different when boiling and condensation occurs. The heat transfer coefficient(s) in the developed heat exchanger models are estimated using correlations for single and two-phase flows.
Fig. 3. Finite volume representation of a counter-flow heat exchanger.
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given by Ng et al. [25] as a function of its in-built volume ratio (rv) and pressure ratio (rp)
wex ¼
Pexp;in
qexp;in
!
K
rv rp
r1n n K¼ v "1 n # a1 a2 a3 logr p þ þ n¼ logr v logr 2v logr3v
ð18Þ ð19Þ ð20Þ
Eq. (20) suggests that for a given twin screw expander, the polytropic expansion index, n, can be defined by the regression constants a1, a2, and a3. 3. Validation studies
Fig. 4. Comparison of the boiling heat transfer coefficient correlations.
3.1. Grid independent solution For cases where the fluids exist in single state (liquid water, pure liquid and pure vapor R245fa), Gnielinski equation given in Eqs. (11) and (12) is used as defined in [20] for turbulent flow;
H¼
k 1 þ 12:7ðf =8Þ0:2 ðPr2=3 1Þ d ðf =8ÞðRe 1000ÞPr
f ¼ ð0:79 ln Re 1:64Þ2
ð11Þ ð12Þ
When the fluid is in two-phase (saturated mixture), Eqs. (11) and (12) are not adequate. For boiling in the evaporator, the Kenning– Cooper (K&C) correlation in Eq. (13) is used for simplicity as given by Sun and Mishima [21] based on their findings. Comparison of the Kenning–Cooper correlation to the widely used Chen correlation [21–23] can be visualized in Fig. 4 for a test case presented by Vaja [19]. The test case considers saturated water at atmospheric pressure, flows with 0.1 kg/s through a 0.02 m diameter heated tube at 120 °C. A correlation from Varme Atlas (used in [14]) is also included in the comparison.
Hb ¼ ½1 þ 1:8X
0:87
048 0:023Re0:8 ðk=dÞ l Pr l
ð13Þ
where k is thermal conductivity, d is the pipe diameter, and X is the Martinelli factor expressed as;
X¼
1x x
0:9
qv ql
0:5
ll lv
0:1 ð14Þ
For the condenser, Dobson and Chato correlation given in Eq. (15) (adopted from [24] for simplicity) is adopted in the present study for Rev < 35,000.
"
3 0
g ql ðql qg Þkl hlv Hc ¼ 0:555 ll ðT sat T s Þdh
#0:25
The transient evaporator model is finite volume based. Therefore, for consistency, it is expected that the accuracy of results increase with the number of discrete nodes (N) assigned for the numerical solution. On the other hand, a very high value of N would lead to avoidable computational burden beyond the desired level of accuracy to be attained. In order to find a reasonable number of discrete nodes, N, effectiveness of the evaporator is computed and compared for a range of number of nodes, with predefined conditions (inlet temperatures and flow rates) for water and R245fa. The comparison is based on the heat exchanger’s effectiveness, eN. The subscript, N, denotes the number of nodes employed in the numerical solution. Error based on N number of nodes can be expressed as in
errorN ¼
eN eexpected eexpected
where eexpected represents the expected effectiveness of the heat exchanger and is computed using a sufficiently large number of nodes. Fig. 5 illustrates the trend of error in effectiveness as obtained from the evaporator model (after steady state is achieved) for different values of N, where eexpected is computed for 160 nodes (N = 160). The inflow conditions considered in the test cases are: 10 kg/s hot water enters the evaporator at 93 °C and 1 atm. 0.75 kg/s R245fa inflow at 27 °C and 678 kPa. The tests are started from the same initial conditions, i.e. constant temperature throughout the heat exchanger, and marched in time until steady state conditions are obtained. It should be noted that error in Eq. (21) also represents the error in actual heat
ð15Þ 0
where the modified latent heat hlv is a function of latent heat hlv, given as; 0
hlv hlv þ 3=8Cpl ðT sat T p Þ
ð16Þ
For higher Rev, the Boyko and Kruzhilin correlation is considered as presented in [18], given by;
Hc ¼
0:5 kl q 0:43 1þx l 1 0:021Re0:8 l Pr l d qv
ð17Þ
2.3. State 3-4: Expander Following the considerations for over and under-expansion, the nominal polytropic work output of a twin screw expander was
ð21Þ
Fig. 5. Grid independency tests for N = 10, 20, 40, 60, 70, 75 and 80.
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Table 1 Evaporator (theoretical) dimensions in Vaja [19]. L
do
d
di
5.52 m
9.5 mm
16 mm
7.9 mm
transfer rate because the maximum possible heat transfer rate, Q_ max , is the same once the above inflow conditions are fixed. It is observed in Fig. 5 that selecting number of nodes, N, at 70 gives a reasonable balance between accuracy and computational burden for the conditions given above. 3.2. Validation of evaporator model The present evaporator model is validated against the results of Vaja [19], who used similar finite volume discretization approach (including momentum equations) to predict state variables from experimental results of Takamatsu et al. [26]. The conditions are given as; 0.0279 kg/s of water inflow at 56 °C (atmospheric pressure) evaporates 0.0142 kg/s R22 inflow at 27 °C and 1140 kPa in an evaporator, for which dimensions are given in Table 1. It should be noted that the simulation of the present study determined steady state was achieved once the maximum nodal hot water temperature difference experienced from one time step to the next becomes less than 0.0001 °C. Figs. 6 and 7 compare the obtained distribution of water and pipe temperatures over the length, L, of the heat exchanger respectively while Fig. 8 also compares the vapor quality of the refrigerant with position using N = 20 and time step of 0.05 s as in [19]. Figs. 6 and 7 show that the present model overestimates the experimental results. Nevertheless, the trend of the present model is similar to the experimental one like Vaja’s result as shown in Fig. 7. The vapor quality distribution in Fig. 8 suggests that vapor quality of refrigerant is not influenced by the correlation for heat transfer coefficient used. The slight differences seen in Figs. 6 and 7 can be attributed to the pressure drop across the heat exchanger and its effect accumulated in time.
Fig. 7. Pipe temperature distribution of selected models.
3.3. System level validation with experimental data Fig. 9 illustrates the major system input parameters of the ORC model, which will be compared against the data from real transient operation. Thermal energy to the ORC is supplied by circulating hot water in the PTC loop. This heat input is characterized by the mass
Fig. 6. Hot water temperature distribution of selected models.
Fig. 8. Refrigerant’s (R22) vapor quality distribution of selected models.
flow rate and state properties of the hot water at the evaporator inlet. Similarly, heat rejection through the condenser is characterized by the mass flow rate and state properties of cold water circulating in the cooling tower loop. Although the system is not an experimental set-up, measurements of the main inputs into the ORC (indicated in Fig. 9) and physical features of its main components are known. Such parameters are fed into an ORC model integrated from the sub-models of each components presented in Section 2 and the obtained output will be compared to available data. The available data from the actual system to be used is the measurement of cold water exit temperature. During a 15 min (900 s) operation, temperature of hot and cold water inlet to the ORC varies as seen in Fig. 10 at atmospheric pressure. The mass flow rates of refrigerant, hot and cold water are known to be 1.5, 2.5 and 15 kg/s respectively. The condensing temperature (Tlow) is set to 5 °C higher than the inlet cold water temperature which is a similar practice used in simulations of Kang [27]. For reasonable initial nodal distribution to kick-start the solution of the transient equations (Section 2.2), the solution approach ran the system at the first recorded input values till
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Fig. 9. Input variables for the ORC model.
Fig. 10. Variation of inlet hot and cold water temperatures in time.
steady state before enforcing the variable input profile described in Fig. 10. The pump of the ORC is a vertical multi-stage pump with maximum flow rate of 12 m3/h and using its characteristics curves, the modeling constants (in Eqs. (2) and (3)) are presented in Table 2. For the twin screw expander, considering a maximum net power output of about 40 kW, the modeling constants defining the polytropic expansion index (Eq. (20)) are also seen in Table 2. The two stainless steel compact heat exchangers; evaporator and condenser have effective heat transfer area of 18 and 11 m2 respectively. Their approximated double pipe geometry dimensions which are used as highlighted in Table 3. Based on the findings of Section 3.1, number of discretized node, N, selected for the simulation is 60 while 0.02 s time step is used. The resulting time variation of the exit temperature of cold water from the condenser as produced by the simulation can be
Table 2 Characteristic constants for the pump and the expander. i
a
A
C
0 1 2 3
– 8.74E02 1.45E01 2.41E01
0.543 0.0722 1.566 1.2524
– 796,886 197.49 40.266
Table 3 Heat exchanger dimensions for the ORC system. L
do
d
di
80 m (50 m for condenser)
0.0686 m
0.1235 m
0.0656 m
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ORC system has been modeled via a synthesis of sub-models of the pump, evaporator, expander and condenser while the working fluid (R245fa) property calculations have been performed via a simplified regression based approach. It is seen that the heat exchanger models (evaporator and condenser) are the critical components in investigating the transient mode of operation since they are the principal media of heat transfer in and out of the ORC. With a tool such as the transient model of the solar ORC at hand, the respective influences of critical parameters such as (refrigerant and water flow rates) can be controlled. This can be applied in real-time dynamic simulation of the unit under different operation scenarios to optimize power production or ensure steady operation given any level of heat input.
Acknowledgement
Fig. 11. Comparison of simulated cold water outlet temperature with recorded data.
This work is supported by the Middle East Technical University–Northern Cyprus Campus (METU–NCC) campus research fund, Project No. BAP-FEN-11-O-10.
References
Fig. 12. Net power output.
compared with recorded data as seen in Fig. 11. Although, data for the net power output was not available, Fig. 12 presents the variation of the net power output with time during the 900 s operation. For simulation of transient operation, the main comparison to be made when validating with experimental data is not with final absolute values but with the trend the data displays with time. Fig. 12 shows the result from the model slightly differs in absolute values with recorded data of the cooling water exit temperature, however they both exhibit similar trend. Hot water inlet temperature is one of the critical parameters defining the heat input to the ORC. Therefore it is expected that the net power output shows sensitivity to the transition of hot water temperature where no control measures are enforced. Fig. 12 verifies this expectation as it shows the net power output having similar profile with the hot water inlet temperature (Fig. 10) in the investigated time frame once initial conditions have been overcome.
4. Conclusion This study presents modeling approaches applicable for transient operation of solar-ORCs given variable heat input. A typical
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