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Dynamic modeling and simulation of an Organic Rankine Cycle (ORC) system for waste heat recovery
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Applied Thermal Engineering 28 (2008) 1216–1224 www.elsevier.com/locate/apthermeng
Dynamic modeling and simulation of an Organic Rankine Cycle (ORC) system for waste heat recovery Donghong Wei *, Xuesheng Lu, Zhen Lu, Jianming Gu Institute of Refrigeration and Cryogenics, School of Mechanical and Power Engineering, Shanghai Jiao Tong University, No. 1954 Hua Shan Road, Shanghai 200030, China Received 5 January 2006; accepted 28 July 2007 Available online 10 August 2007
Abstract The paper proposes two alternative approaches for the design of a dynamic model for an Organic Rankine Cycle (ORC) to be used for the design of control and diagnostics systems. The model has been developed in Modelica language and simulated with Dymola. The two modeling approaches, based on moving boundary and discretization techniques, are compared in terms of accuracy, complexity and simulation speed. Compared to experimental data simulations evidence that while both models have good accuracy. And the moving boundary model is faster, therefore more suitable for control design applications. Ó 2007 Elsevier Ltd. All rights reserved. Keywords: Organic Rankine Cycle; Waste heat recovery; Dynamic modeling; System simulation; Moving boundary models; Discretized models
1. Introduction Organic Rankine Cycles (ORC’s) have made their appearance in the market of power generation systems as a valid substitute to steam-based bottoming cycles for the recovery of waste heat. ORC’s are particularly viable for small scale, unsupervised power generation systems which utilize waste heat from prime movers such as micro-turbines or reciprocating engines, but also from industrial processes or byproduct in landfills. Although a small ORC is characterized by rather low efficiencies (8–12%), it is particularly easy to manufacture as it can be built in large part with components derived from vapor compression chillers. Another important advantage of ORC’s is that it can utilize waste heat from low-quality exhausts or steam, which makes it suitable for a very large range of applications which include those with low temperature waste heat sources [1]. High reliability and flexibility also
*
Corresponding author. Tel./fax: +86 21 62932602. E-mail address: [email protected] (D. Wei).
1359-4311/$ - see front matter Ó 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.applthermaleng.2007.07.019
contribute at making the value proposition of ORC’s particularly attractive. Although ORC systems are rather easy to operate, particular care must be taken in controlling and monitoring the system so to avoid that unwanted conditions occur. This is especially true during transient conditions, particularly when the load demand or the quality or flow of waste heat suddenly changes. In this case, it is important that the proportion of liquid and vapor phases in the condenser and evaporator are kept within acceptable ranges to avoid stalling or temperature shocks. Such problems can be prevented at the design stage by resorting to model-based techniques to simulate the transient behavior of the ORC system and predict the occurrence of operating conditions where critical situations may occur. Such methodology must rely on the availability of a suitable model capable to correctly interpret the system behavior in both steady state and transient conditions. While models of pumps and turbines are readily available in literature, the correct representation of dynamics in evaporator and condenser is not straightforward. Indeed, careful representation of two-phase flow that accounts for the system geometry and heat transfer is needed. In this
D. Wei et al. / Applied Thermal Engineering 28 (2008) 1216–1224
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Nomenclature Superscripts 0 saturation liquid 00 saturation vapor Symbols A area (m2) D diameter (m) t time (s) l length (m) n rotate speed T temperature (K) q density (kg/m3) h specific enthalpy (J/kg) Z length coordinate (m) v velocity (m/s) Dp pressure difference (Pa) Nu Nusselt number Re Reynolds number Dp0, m0, q0 the pressure loss, mass flow rate, density at steady state a, b coefficient, dimensionless V volumetric flow W work per unit time (W) x vapor quality M total mass in volume (kg) E total energy in volume (J)
paper, we present a novel dynamic model that includes all components of an ORC, accurate enough to capture the leading dynamic behavior but simple enough to be utilized in real-time simulations and as part of controls or diagnostics algorithms. The model of evaporator and condenser can be carried out by considering the representation of a two-phase flow exchanger by means of either discretization or moving boundary approaches. While the advantage of discretization is that high accuracy correlations for heat transfer and pressure losses can be introduced as function of the spatial coordinates, the moving boundary approach leads to models with reduced order, more suitable for use in controls and diagnostics applications. When discretization approaches are adopted, the governing equations for mass, momentum and energy representing the flow in condenser and evaporator can be discretized by solves using either a finite volume or a finite difference method. While both methods are equally suitable for use, the latter is sometimes preferred as it works better when a detailed component model needs to be developed. A systematic classification of discretization approaches is presented in [2], whereas a detailed comparison among model alternatives is discussed in [3]. The discretization approach adopted in this work has been presented in [3]: in this model, the pressure dynamics
C k Q_ m_ a Mcor Pr R1, R2,
heat capacity per unit area (J/m2 K) thermal conductivity (W/m K) heat flow (W) mass flow rate per unit time (kg/s) convective heat transfer coefficient (W/m2 K) corrected mass flow rate (kg/s) Prandtl number R3 coefficient, dimensionless
Subscripts 1 sub-cooled 2 two-phase 3 superheated i inner o outer r working fluid in inlet l saturated liquid g saturated vapor n normalized w wall amb ambient ex exhaust gas out outlet
is supposed to be characterized by time scales which are very small if compared to the relevant thermal phenomena. Accordingly, the transient of pressure dynamic is assumed to be infinitely fast. As a consequence, a change in boundary conditions influencing the pressure field will thus affect the pressures in all cells of a discretized model instantaneously. The presented model differs from others as it introduces an average global momentum balance across the entire heat exchanger channel instead of local momentum balance across the entire heat exchanger channel instead of local momentum balances for each cell. This model feature results in an increase in simulation speed and robustness (see the Appendix for more details on the adopted two-phase model). Moving boundary models consider the heat exchanger channel as subdivided in zones characterized by uniform phase and separated by boundaries. As conditions in the channel dynamically change, the volumes will also expand or contract. Expansions and contractions can be represented by the position of the dividing boundaries, i.e. where boiling starts to occur and the one where the fluid is fully evaporated as saturated vapor. Different formulations of moving boundary model have been suggested. Xiang-Dong He designed different moving boundary models for use with multivariable control solutions for vapor compression refrigeration systems [4–6]. Here, LQG (Linear Quadratic
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Gaussian) controllers were based on moving boundary model for both the evaporator and the condenser. Threeregion models have been also proposed by Willatzen et al. [7] and by Bittanti et al. [8] for refrigeration systems and once-through boilers of power plants, respectively, while a dry-expansion evaporator model is presented in [3,9]. A combination of a moving boundary approach for modeling the two-phase region and a discretization approach of mass and energy equations for the superheated region of heat exchangers has been proposed by MacArthur and Grald [10] with application to heat pumps. An interesting generalization of moving boundary models has been finally presented by Jensen [3]. In this paper, a dynamic model to support simulation, diagnostics and control of an ORC fed by gaseous waste heat is presented. Both the moving boundary and discretization approaches have been adopted for the representation of the condenser and evaporator. And the comparison is from system level. Both models are base on Modelica/ Dymola platform which is based on equation system, so the modeler can focus on the physics essence, not on the solving of mathematic equations. The model has been developed as part of a collaborative research project between United Technologies Research Center and the Department of Refrigeration and Cryogenics of Shanghai Jiao Tong University, aimed at developing a library of system level dynamic models for different types of power generation and refrigeration systems. After a description of the ORC process in Section 2, the two model approaches are discussed in Section 3. Simulation of the models, implemented in the equation-based Modelica language [11,12], will be analyzed in Section 4, together with the performance and accuracy of both approaches. A comparison of model complexity derived from using the two approaches (Section 5) and some concluding remarks (Section 6) will end the paper.
Exhaust gas
Evaporator
Turbine
Pump Air
Receiver Condenser
Fig. 1. System diagram.
Fig. 2. 100 kW ORC engineering check sample.
2. System diagram The ORC system considered in this paper is characterized by a dry-expansion evaporator where heat from an exhaust gas source such as a micro turbine or an industrial process is used to heat and evaporate R245fa refrigerant. The refrigerant drives a turbine for power generation, and is then condensed into liquid in an air-cooled condenser. The liquid refrigerant is collected in a receiver, introduced to avoid pump surge, and is then pumped back to the evaporator. A simplified scheme of the system is illustrated in Figs. 1 and 2 is the test sample. Fig. 3 presents the heat transfer process of the system. 3. Model description
Fig. 3. Temperature–entropy diagram of the system.
3.1. Models of evaporator and condenser In ORC systems, the refrigerant enters the evaporator in sub-cooled liquid phase and exits as superheated vapor.
Conversely, the condenser is characterized by entering superheated vapor and either leaving saturated liquid or a mixture of vapor and liquid with a small quality (i.e.
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In Eq. (4a), f represents the resistance coefficient of turbulent flow in a tube, and is calculated according to Filonenko equation [14]: 2
ð5Þ
f ¼ ð1:82 lg Re 1:64Þ :
A correlation from Va¨rme Atlas is used for the convection heat transfer for two-phase flows in a horizontal tube [13]: ( h i2:2 0:37 0 ð1 xÞ þ 1:2x0:4 ð1 xÞðq0 =q00 Þ a¼a Fig. 4. Schematic of general MB-model.
within 0 and 0.16). As a consequence, both the evaporator and condenser can be described as composed by three zones characterized by sub-cooled liquid, saturated mixture and superheated vapor (see Fig. 4 for a scheme of the evaporator). For each zone, appropriate one-dimensional governing equations can be derived. In the case of a moving boundary model we will have the following equations [3,9]: Mass balance: @Aq @ m_ þ ¼ 0: @t @z
ð1Þ
Energy balance: _ @ðAqh ApÞ @ mh þ ¼ pDaðT w T r Þ: @t @z
ð2Þ
Differential energy balance at the wall: C w q w Aw
@T w ¼ ai pDi ðT r T w Þ þ ao pDi ðT amb T w Þ: @t
ð3Þ
Eqs. (1)–(3) are integrated along the axial coordinate for each of the three regions to generate moving boundary equations [3]. Here, it is assumed that the change in pressure along the pipe is negligible. For the discretized model, we must instead select the number of cells in which to subdivide evaporator and condenser, based on required precision. Five cells have been considered for the current model formulation. The discretized model equations are reported in Appendix. 3.1.1. Heat transfer correlation The Gnielinski equation is used for single-phase fluid (i.e. liquid or vapor) [14]: " 2=3 # ðf =8ÞðRe 1000ÞPrf d Nuf ¼ ð4aÞ ct 1þ 0:5 2=3 l 1 þ 12:7ðf =8Þ ðPrf 1Þ where for liquid field: 0:11 Prf Prf ct ¼ ; ¼ 0:05 20: Prw Prw And for vapor field: 0:11 Tf Tf ct ¼ ; ¼ 0:5 1:5: Tw Tw
ð4bÞ
ð4cÞ
a00 0:7 0:67 þ 0 x0:01 ð1 þ 8ð1 xÞ Þðq0 =q00 Þ a
2 )0:5 :
ð6Þ
3.1.2. Pressure loss correlation The pressure loss from flow resistance along the tube is represented by means of the pressure drop equation [15]: Dp ¼ qk
l v2 Di 2
ð7Þ
while there is a large variation in phase changing, a pressure loss model that is easy to adapt to measurements is sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Dp q m_ ¼ : Dp0 q0 m_ 0 In the above equation, the root function is replaced by a polynomial with finite derivative in a small region around zero flow speed. And the density dependence can be omitted for nearly incompressible fluids or small density variations [16]. 3.2. Pump, turbine and receiver The pump model is defined as the following, and is based on the availability of performance data directly from the technical documentation of the manufacturer [16]: Dpn ¼ R1 nn þ 2R2 nn V n R3 jV n jV n :
ð8Þ
Here, the design point is characterized by nominal normalized values (pn, nn, Vn) = (1, 1, 1) and represent when it operates at nominal conditions. This relationship works correctly even in the case of back flow through the pump, i.e. when surge occurs, as well as in near surge conditions with acceptable accuracy [16]. The model of turbine is derived from its map as in [17]: M cor ¼ a ðpin =pout Þ þ b
ð9Þ
where, M cor ¼ m_ ðabsðT in Þ ^ 0:5Þ=pin . As the leading dynamics of pump and turbine are much faster than those of the two heat exchangers, we can consider static, lumped models for both components, as they are supposed to evolve to steady state conditions in an infinitely fast fashion. The model of the receiver is composed by the mass and energy balances of a volume full of working fluid.
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dM ¼ m_ in m_ out dt dE ¼ W þ Q_ dt
ð10Þ ð11Þ
100
80 superheated.L3 two-phase.L2 liquid.L1
60
4. Validation and analysis
40
A thorough calibration and validation of ORC models has been performed on the basis of experimental data made available by UTRC (United Technology Research Center). Data refer to steady state and transient behavior of a pilot 100 kW ORC system. In Figs. 5 and 6, simulations show the change in position of each zone boundary, express in percent of the total exchanger length, for both condenser and evaporator. In Figs. 7–9, validation results are reported: experimental data are represented by solid lines, the results from moving boundary models by dash lines and those from discretized models by dash-dot lines. The simulation error is also shown. An analysis of the plots reveals that models match experimental data fairly well. Indeed, the biggest relative error is
20
70 60 liquid.L1 two-phase.L2 superheated.L3
50 40 30 20 10 0 0
500
1000
time (s)
Fig. 5. Length percentage of each region in evaporator.
0 0
500
1000
time (s)
Fig. 6. Length percentage of each region in condenser.
<4%. Moreover, simulations do not show problems such as chattering or numerical oscillations typically observed in discretized models. Somewhat larger errors observed during initial transient behavior are related to the initialization of the model, which is difficult to impose as the correct initial boundary values are unknown. We can conclude from the comparison with experimental tests that both models can accurately capture the leading system dynamics and could be used for simulation and control design. On the other hand, not all models are suitable for applications which require the modeling of the system during the phases of startup and shutdown. The characterization of these operation modes is extremely important, as most failures and undesired behavior occur far from regime operation. Moving boundary models, for example, represent well all conditions where fluid is present as liquid, mixture and vapor. This is often not true during startup and shutdown conditions, where one of the three regions may not exist. Discretized models might appear more suitable for simulation during these operation phases.
Fig. 7-1. Mass flow rate of R245fa.
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Fig. 7-2. Deviation of mass flow rate of R245fa.
Fig. 8-1. Evaporator outlet temperature.
5. Selection of thermodynamic state variables One of the key issues of thermodynamic modeling design is the selection of independent state variables which are differentiates in the model’s equations. This choice affects numerical properties such as robustness and efficiency during simulation. According to the Duhem’s theorem [18], the state of equilibrium of a thermodynamic system is completely characterized by a pair of independent variables when the mass at the initial time is known. When the conservation of momentum is formulated in differential terms, a third variable must be added as independent variable, i.e. either the fluid mass flow or its velocity. To summarize, the thee state variables are needed for each spatial coordinate point or for each lumped equation of a discretized model. Using two intensive variables chosen from the _ mg variable set {p, h, q, u, s} and one variable from fm;
Fig. 8-2. Deviation of evaporator outlet temperature.
generates a variable space with 20 possible combinations. As the momentum equation is neglected in the condenser and evaporator models, the number of possible combinations is reduced to 10. Traditionally, the pair (p, h) is adopted in dynamic models developed for refrigeration systems as the equation of state for two-phase refrigerant is explicit in (p, h). Indeed, the choice of (p, h) leads to relatively robust models within all of the liquid, two-phase and vapor regions. As a consequence, this choice is made for the models presented in this paper. As the component models of turbine and pump are lumped and static, there is no need to select the state variables. The same receiver model is adopted for both
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Fig. 9-1. Turbine inlet pressure.
dure adopted for the ORC’s condenser and evaporator (Table 1). While there are 30 state variables introduced in the model which adopts the discetization technique, those for the moving boundary model are 16 (seven for the evaporator and nine for the condenser). The extra two variables in the condenser model account for the fact that the fluid at the condenser outlet can be saturated liquid or small quality (0–0.16) two-phase fluid. For this reason, the variables dh_sat and x_out must be added to the model. This problem does not hold for the evaporator, which outlet is always characterized by superheated vapor. Then, it is easy to observe that the complexity of the moving boundary model is lower than that of the discretized one. Even with the adoption of only three cells the model order is 18, still bigger than the order of the moving boundary model. As reduced order models are far more tractable, the moving boundary formulation is recommended for use on control design applications rather than the discretized version. The simulation speed is also affected by model complexity. Indeed, for the representation of a 1400 s long transient, the simulation time of the moving boundary model is 113 s while that of the discretized model is 153 s, using the same Dymola software and hardware system. Once again, the moving boundary model is characterized by higher simulation speed, which turns out to be particularly useful for long simulations with closed control loops. 6. Conclusions
Fig. 9-2. Deviation of turbine inlet pressure.
formulations of the model, and therefore the number of state variables relative to the receiver is unchanged. In conclusion, the model complexity in terms of dimension of the state vector depends only on the modeling proce-
1. Two alternative formulations of a dynamic model for an ORC were proposed in this paper. The models, based on moving boundary and discretization model reduction techniques to represent condenser and evaporator, have been validated by comparison with experimental data from an ORC pilot plant located at UTRC. Both models correctly simulate the behavior of the system during
Table 1 State variables in each model
Evaporator
Condenser
Moving boundary model
Distributed parameter model
p L1, L2
p[1] p[5] h[1] h[5]
Pressure in each section at outlet Specific enthalpy of each section at outlet
Tw[1] Tw[5]
Wall temperature of each section at outlet 15
p[1] p[5] h[1] h[5]
Pressure in each section at outlet Specific enthalpy of each section at outlet Wall temperature of each section at outlet
h3 Tw[1] Tw[3]
Pressure Length of sub-cooled region L1, Length of two-phase region L2 Specific enthalpy of superheated region at outlet Mean wall temperature of each section
Total
7
p L2, L3
Pressure Length of superheated region L3, Length of two-phase region L2 Specific enthalpy of liquid region at outlet Mean wall temperature of each section Saturated liquid enthalpy at outlet Quality of R245a at outlet 9
h3 Tw[1] Tw[3] dh_sat x_out Total Total
16
Tw[1] Tw[5]
15 30
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transients, without showing chattering or numerical oscillations, undesired phenomena frequently occurring in this type of simulations. 2. Simulations show that the models predict the data with an accuracy of 4%. The moving boundary model is less complex than the discretized version, as it is characterized by smaller order and higher computational speed. As a result, it is more acceptable for control design applications.
kept to mark the spatial changes in the wall friction and gravitational force. The momentum flows and pressure terms are canceling out, except for the first and the last one, giving an average momentum balance for the whole pipe. The average mass flow is computed as
Acknowledgements
I_ 2 I_ nþ1 ¼ m_ 2 jw2 j m_ nþ1 jwnþ1 j:
The authors gratefully acknowledge the collaboration and financial support of United Technologies Research Center, Connecticut, USA as partners in a collaborative research project aimed at developing dynamic models of ORC’s. The experimental data have been also provided by UTRC. The authors would also like to acknowledge to the help of many UTRC researchers who provided guidance and support during the model development, particularly Brian Eisenhower, Luiz Bacellar, and Guido Poncia. Appendix A. The new discretized model [4] The new model is formulated to the set of boundary conditions as Fig. A-1, which the standard discretized models were unable to work with. And as a result, the grid is slightly changed. The pipe model contains as previously a grid with n volume cells, but the inlet mass flow m_ 1 is here a boundary condition and is not calculated from the momentum balance. The outlet mass flow is changed to a simulation result not a boundary condition as the standard discretized model. The flow cells are omitted in Fig. A-1. Instead, the location of the global momentum balance is shown. The discretization grid for the mass and energy balances is unchanged. The new model is derived by adding the momentum balance equation for cell 2 to n + 1 giving: nDz
nþ1 nþ1 X X _ dm ¼ I_ 2 I_ nþ1 þ ðp1 pnþ1 ÞA F w;i þ F g;i dt i¼2 i¼2
ðA:1Þ The wall friction for the ith flow cell is called Fw,i, and the gravitational force is called Fg,i. The flow cells do not contain any separate balance equations anymore, but are only
_ ¼ m
nþ1 1X m_ i : n i¼2
The change in momentum flow is calculated as
The information concerning the inhomogeneous part of the momentum balance is lost in the summation of the momentum balance. The new model is thus a homogeneous flow model with just one momentum balance. The frictional forces and the gravitational forces are unchanged compared to the standard discretized models. The information concerning the pressure-mass flow coupling described by the local momentum balance in standard discretized models is also lost using a global momentum balance. Consequently, a pressure distribution must be supplied. In the present analysis, a linear pressure distribution is assumed. The global momentum balance states a connection between a single pressure and the mass flow. The pressure in the first cell p1 is kept as a state variable due to its presence in the global momentum balance (Eq. (1)). The pressures in the other cells are computed assuming a linear pressure distribution between p1 and pn+1 p pnþ1 for i ¼ 2; . . . ; n: ðA:2Þ pi ¼ p1 ði 1Þ 1 n The mass balances and the energy balances are formulated for each of the cells as for the standard discretized models, with the exception that the flow is assumed to be homogeneous. The rate of change of mass and energy are given as the standard discretized models. As the pressures are determined from a static relation, only the pressure in the first cell is a state variable. Derivatives of the pressure in the other cells are calculated as the time derivative of (Eq. (2)): dpi i 1 dp1 ði 1Þ dpnþ1 ¼ 1 þ for i ¼ 2; . . . ; n n n dt dt dt The term involving pn+1 is given by the boundary conditions. This term may be difficult to calculate and may often
Fig. A-1. Nomenclature for the new discretized model. (Design boundary conditions are shown in brackets, but these might be changed as required).
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be neglected, in which case the time derivative of the pressure simplifies to dpi i 1 dp1 ¼ 1 for i ¼ 2; . . . ; n: n dt dt References [1] B. Sternlicht, Waste energy recovery: an excellent investment opportunity, Energy Conversion and Management 22 (1981) 361–373. [2] N.E. Todreas, M.S. Kazimi, Nuclear Systems I – Thermal Hydraulic Fundamentals, Taylor & Francis, 1993. [3] J.M. Jensen, Dynamic modeling of thermo-fluid systems with focus on evaporators for refrigeration, Ph.D Thesis, Technical University of Denmark, Department of Mechanical Engineering, March 2003. [4] Xiang-Dong He et al. A moving-interface model of two-phase flow heat exchanger dynamics for control of vapor compression cycle, in: Heat Pump and Refrigeration Systems Design, Analysis and Applications, AES-vol. 32, ASME, 1994. [5] Xiang-Dong He et al., Modeling of vapor compression cycles for multivariable feedback control of HVAC systems, Journal of Dynamic Systems, Measurement and Control 119 (June) (1997). [6] Xiang-Dong He et al., Multivariable control of vapor compression systems, HVAC&R Reasearch 4 (3) (1998), July. [7] M. Willatzen, N.B.O.L. Pettit, L. Ploug-Sørensen, A general dynamic simulation model for evaporators and condensers in refrigeration, International Journal of Refrigeration 21 (1998) 398–414.
[8] S. Bittanti, M. Bottinelli, Performance assessment of the control system of once-through boilers, in: Proceedings of the 13th Conference on Process Control, Slovakia Republic, June 2001. [9] J.M. Jensen, H. Tummescheit, Moving boundary models of dynamic simulations of two-phase flows, in: Proceedings of the Second International Modelica Conference, 2002, pp. 235–244. [10] J.W. MacArthur, E.W. Grald, Unsteady Compressible Two-Phase Flow Model for Predicting Cyclic Heat Pump performance and Comparison with Experimental Data, Rev. Int. Froid, vol. 12, 1989. [11] Modelica Association. Specification, Tutorials, http://www.modelica.org/. [12] Dynasim AB, Dynamic Modeling Laboratory, http://www.dymola.com/. [13] P. Melin, Measurements and modeling of convective vaporization for refrigerants in a horizontal tube, Ph.D. Thesis, Department of Heat and Power Technology, Chalmers University of Technology, Go¨teborg, Sweden, 1996. [14] Yang Shiming, Tao Wenquan, Heat Transfer, third ed., Higher Education Press, China, 1998. [15] Sheng Jingchao, Fluid Mechanics Engineering, Mechanical Industry Publishing Company, China, 1988. [16] H. Tummescheit, Design and implementation of object-oriented model libraries using modelica, Ph.D. Thesis, Lund Institute of Technology, Department of Automatic Control, August (2002), pp. 104–120. [17] Shilie Weng, Combustion Turbine, Mechanical Industry Publishing Company, China, 1989. [18] I. Prigogine, R. Defay, Chemical Thermodynamics, Longman, 1969.
Applied Thermal Engineering 28 (2008) 1216–1224 www.elsevier.com/locate/apthermeng
Dynamic modeling and simulation of an Organic Rankine Cycle (ORC) system for waste heat recovery Donghong Wei *, Xuesheng Lu, Zhen Lu, Jianming Gu Institute of Refrigeration and Cryogenics, School of Mechanical and Power Engineering, Shanghai Jiao Tong University, No. 1954 Hua Shan Road, Shanghai 200030, China Received 5 January 2006; accepted 28 July 2007 Available online 10 August 2007
Abstract The paper proposes two alternative approaches for the design of a dynamic model for an Organic Rankine Cycle (ORC) to be used for the design of control and diagnostics systems. The model has been developed in Modelica language and simulated with Dymola. The two modeling approaches, based on moving boundary and discretization techniques, are compared in terms of accuracy, complexity and simulation speed. Compared to experimental data simulations evidence that while both models have good accuracy. And the moving boundary model is faster, therefore more suitable for control design applications. Ó 2007 Elsevier Ltd. All rights reserved. Keywords: Organic Rankine Cycle; Waste heat recovery; Dynamic modeling; System simulation; Moving boundary models; Discretized models
1. Introduction Organic Rankine Cycles (ORC’s) have made their appearance in the market of power generation systems as a valid substitute to steam-based bottoming cycles for the recovery of waste heat. ORC’s are particularly viable for small scale, unsupervised power generation systems which utilize waste heat from prime movers such as micro-turbines or reciprocating engines, but also from industrial processes or byproduct in landfills. Although a small ORC is characterized by rather low efficiencies (8–12%), it is particularly easy to manufacture as it can be built in large part with components derived from vapor compression chillers. Another important advantage of ORC’s is that it can utilize waste heat from low-quality exhausts or steam, which makes it suitable for a very large range of applications which include those with low temperature waste heat sources [1]. High reliability and flexibility also
*
Corresponding author. Tel./fax: +86 21 62932602. E-mail address: [email protected] (D. Wei).
1359-4311/$ - see front matter Ó 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.applthermaleng.2007.07.019
contribute at making the value proposition of ORC’s particularly attractive. Although ORC systems are rather easy to operate, particular care must be taken in controlling and monitoring the system so to avoid that unwanted conditions occur. This is especially true during transient conditions, particularly when the load demand or the quality or flow of waste heat suddenly changes. In this case, it is important that the proportion of liquid and vapor phases in the condenser and evaporator are kept within acceptable ranges to avoid stalling or temperature shocks. Such problems can be prevented at the design stage by resorting to model-based techniques to simulate the transient behavior of the ORC system and predict the occurrence of operating conditions where critical situations may occur. Such methodology must rely on the availability of a suitable model capable to correctly interpret the system behavior in both steady state and transient conditions. While models of pumps and turbines are readily available in literature, the correct representation of dynamics in evaporator and condenser is not straightforward. Indeed, careful representation of two-phase flow that accounts for the system geometry and heat transfer is needed. In this
D. Wei et al. / Applied Thermal Engineering 28 (2008) 1216–1224
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Nomenclature Superscripts 0 saturation liquid 00 saturation vapor Symbols A area (m2) D diameter (m) t time (s) l length (m) n rotate speed T temperature (K) q density (kg/m3) h specific enthalpy (J/kg) Z length coordinate (m) v velocity (m/s) Dp pressure difference (Pa) Nu Nusselt number Re Reynolds number Dp0, m0, q0 the pressure loss, mass flow rate, density at steady state a, b coefficient, dimensionless V volumetric flow W work per unit time (W) x vapor quality M total mass in volume (kg) E total energy in volume (J)
paper, we present a novel dynamic model that includes all components of an ORC, accurate enough to capture the leading dynamic behavior but simple enough to be utilized in real-time simulations and as part of controls or diagnostics algorithms. The model of evaporator and condenser can be carried out by considering the representation of a two-phase flow exchanger by means of either discretization or moving boundary approaches. While the advantage of discretization is that high accuracy correlations for heat transfer and pressure losses can be introduced as function of the spatial coordinates, the moving boundary approach leads to models with reduced order, more suitable for use in controls and diagnostics applications. When discretization approaches are adopted, the governing equations for mass, momentum and energy representing the flow in condenser and evaporator can be discretized by solves using either a finite volume or a finite difference method. While both methods are equally suitable for use, the latter is sometimes preferred as it works better when a detailed component model needs to be developed. A systematic classification of discretization approaches is presented in [2], whereas a detailed comparison among model alternatives is discussed in [3]. The discretization approach adopted in this work has been presented in [3]: in this model, the pressure dynamics
C k Q_ m_ a Mcor Pr R1, R2,
heat capacity per unit area (J/m2 K) thermal conductivity (W/m K) heat flow (W) mass flow rate per unit time (kg/s) convective heat transfer coefficient (W/m2 K) corrected mass flow rate (kg/s) Prandtl number R3 coefficient, dimensionless
Subscripts 1 sub-cooled 2 two-phase 3 superheated i inner o outer r working fluid in inlet l saturated liquid g saturated vapor n normalized w wall amb ambient ex exhaust gas out outlet
is supposed to be characterized by time scales which are very small if compared to the relevant thermal phenomena. Accordingly, the transient of pressure dynamic is assumed to be infinitely fast. As a consequence, a change in boundary conditions influencing the pressure field will thus affect the pressures in all cells of a discretized model instantaneously. The presented model differs from others as it introduces an average global momentum balance across the entire heat exchanger channel instead of local momentum balance across the entire heat exchanger channel instead of local momentum balances for each cell. This model feature results in an increase in simulation speed and robustness (see the Appendix for more details on the adopted two-phase model). Moving boundary models consider the heat exchanger channel as subdivided in zones characterized by uniform phase and separated by boundaries. As conditions in the channel dynamically change, the volumes will also expand or contract. Expansions and contractions can be represented by the position of the dividing boundaries, i.e. where boiling starts to occur and the one where the fluid is fully evaporated as saturated vapor. Different formulations of moving boundary model have been suggested. Xiang-Dong He designed different moving boundary models for use with multivariable control solutions for vapor compression refrigeration systems [4–6]. Here, LQG (Linear Quadratic
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Gaussian) controllers were based on moving boundary model for both the evaporator and the condenser. Threeregion models have been also proposed by Willatzen et al. [7] and by Bittanti et al. [8] for refrigeration systems and once-through boilers of power plants, respectively, while a dry-expansion evaporator model is presented in [3,9]. A combination of a moving boundary approach for modeling the two-phase region and a discretization approach of mass and energy equations for the superheated region of heat exchangers has been proposed by MacArthur and Grald [10] with application to heat pumps. An interesting generalization of moving boundary models has been finally presented by Jensen [3]. In this paper, a dynamic model to support simulation, diagnostics and control of an ORC fed by gaseous waste heat is presented. Both the moving boundary and discretization approaches have been adopted for the representation of the condenser and evaporator. And the comparison is from system level. Both models are base on Modelica/ Dymola platform which is based on equation system, so the modeler can focus on the physics essence, not on the solving of mathematic equations. The model has been developed as part of a collaborative research project between United Technologies Research Center and the Department of Refrigeration and Cryogenics of Shanghai Jiao Tong University, aimed at developing a library of system level dynamic models for different types of power generation and refrigeration systems. After a description of the ORC process in Section 2, the two model approaches are discussed in Section 3. Simulation of the models, implemented in the equation-based Modelica language [11,12], will be analyzed in Section 4, together with the performance and accuracy of both approaches. A comparison of model complexity derived from using the two approaches (Section 5) and some concluding remarks (Section 6) will end the paper.
Exhaust gas
Evaporator
Turbine
Pump Air
Receiver Condenser
Fig. 1. System diagram.
Fig. 2. 100 kW ORC engineering check sample.
2. System diagram The ORC system considered in this paper is characterized by a dry-expansion evaporator where heat from an exhaust gas source such as a micro turbine or an industrial process is used to heat and evaporate R245fa refrigerant. The refrigerant drives a turbine for power generation, and is then condensed into liquid in an air-cooled condenser. The liquid refrigerant is collected in a receiver, introduced to avoid pump surge, and is then pumped back to the evaporator. A simplified scheme of the system is illustrated in Figs. 1 and 2 is the test sample. Fig. 3 presents the heat transfer process of the system. 3. Model description
Fig. 3. Temperature–entropy diagram of the system.
3.1. Models of evaporator and condenser In ORC systems, the refrigerant enters the evaporator in sub-cooled liquid phase and exits as superheated vapor.
Conversely, the condenser is characterized by entering superheated vapor and either leaving saturated liquid or a mixture of vapor and liquid with a small quality (i.e.
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In Eq. (4a), f represents the resistance coefficient of turbulent flow in a tube, and is calculated according to Filonenko equation [14]: 2
ð5Þ
f ¼ ð1:82 lg Re 1:64Þ :
A correlation from Va¨rme Atlas is used for the convection heat transfer for two-phase flows in a horizontal tube [13]: ( h i2:2 0:37 0 ð1 xÞ þ 1:2x0:4 ð1 xÞðq0 =q00 Þ a¼a Fig. 4. Schematic of general MB-model.
within 0 and 0.16). As a consequence, both the evaporator and condenser can be described as composed by three zones characterized by sub-cooled liquid, saturated mixture and superheated vapor (see Fig. 4 for a scheme of the evaporator). For each zone, appropriate one-dimensional governing equations can be derived. In the case of a moving boundary model we will have the following equations [3,9]: Mass balance: @Aq @ m_ þ ¼ 0: @t @z
ð1Þ
Energy balance: _ @ðAqh ApÞ @ mh þ ¼ pDaðT w T r Þ: @t @z
ð2Þ
Differential energy balance at the wall: C w q w Aw
@T w ¼ ai pDi ðT r T w Þ þ ao pDi ðT amb T w Þ: @t
ð3Þ
Eqs. (1)–(3) are integrated along the axial coordinate for each of the three regions to generate moving boundary equations [3]. Here, it is assumed that the change in pressure along the pipe is negligible. For the discretized model, we must instead select the number of cells in which to subdivide evaporator and condenser, based on required precision. Five cells have been considered for the current model formulation. The discretized model equations are reported in Appendix. 3.1.1. Heat transfer correlation The Gnielinski equation is used for single-phase fluid (i.e. liquid or vapor) [14]: " 2=3 # ðf =8ÞðRe 1000ÞPrf d Nuf ¼ ð4aÞ ct 1þ 0:5 2=3 l 1 þ 12:7ðf =8Þ ðPrf 1Þ where for liquid field: 0:11 Prf Prf ct ¼ ; ¼ 0:05 20: Prw Prw And for vapor field: 0:11 Tf Tf ct ¼ ; ¼ 0:5 1:5: Tw Tw
ð4bÞ
ð4cÞ
a00 0:7 0:67 þ 0 x0:01 ð1 þ 8ð1 xÞ Þðq0 =q00 Þ a
2 )0:5 :
ð6Þ
3.1.2. Pressure loss correlation The pressure loss from flow resistance along the tube is represented by means of the pressure drop equation [15]: Dp ¼ qk
l v2 Di 2
ð7Þ
while there is a large variation in phase changing, a pressure loss model that is easy to adapt to measurements is sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Dp q m_ ¼ : Dp0 q0 m_ 0 In the above equation, the root function is replaced by a polynomial with finite derivative in a small region around zero flow speed. And the density dependence can be omitted for nearly incompressible fluids or small density variations [16]. 3.2. Pump, turbine and receiver The pump model is defined as the following, and is based on the availability of performance data directly from the technical documentation of the manufacturer [16]: Dpn ¼ R1 nn þ 2R2 nn V n R3 jV n jV n :
ð8Þ
Here, the design point is characterized by nominal normalized values (pn, nn, Vn) = (1, 1, 1) and represent when it operates at nominal conditions. This relationship works correctly even in the case of back flow through the pump, i.e. when surge occurs, as well as in near surge conditions with acceptable accuracy [16]. The model of turbine is derived from its map as in [17]: M cor ¼ a ðpin =pout Þ þ b
ð9Þ
where, M cor ¼ m_ ðabsðT in Þ ^ 0:5Þ=pin . As the leading dynamics of pump and turbine are much faster than those of the two heat exchangers, we can consider static, lumped models for both components, as they are supposed to evolve to steady state conditions in an infinitely fast fashion. The model of the receiver is composed by the mass and energy balances of a volume full of working fluid.
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dM ¼ m_ in m_ out dt dE ¼ W þ Q_ dt
ð10Þ ð11Þ
100
80 superheated.L3 two-phase.L2 liquid.L1
60
4. Validation and analysis
40
A thorough calibration and validation of ORC models has been performed on the basis of experimental data made available by UTRC (United Technology Research Center). Data refer to steady state and transient behavior of a pilot 100 kW ORC system. In Figs. 5 and 6, simulations show the change in position of each zone boundary, express in percent of the total exchanger length, for both condenser and evaporator. In Figs. 7–9, validation results are reported: experimental data are represented by solid lines, the results from moving boundary models by dash lines and those from discretized models by dash-dot lines. The simulation error is also shown. An analysis of the plots reveals that models match experimental data fairly well. Indeed, the biggest relative error is
20
70 60 liquid.L1 two-phase.L2 superheated.L3
50 40 30 20 10 0 0
500
1000
time (s)
Fig. 5. Length percentage of each region in evaporator.
0 0
500
1000
time (s)
Fig. 6. Length percentage of each region in condenser.
<4%. Moreover, simulations do not show problems such as chattering or numerical oscillations typically observed in discretized models. Somewhat larger errors observed during initial transient behavior are related to the initialization of the model, which is difficult to impose as the correct initial boundary values are unknown. We can conclude from the comparison with experimental tests that both models can accurately capture the leading system dynamics and could be used for simulation and control design. On the other hand, not all models are suitable for applications which require the modeling of the system during the phases of startup and shutdown. The characterization of these operation modes is extremely important, as most failures and undesired behavior occur far from regime operation. Moving boundary models, for example, represent well all conditions where fluid is present as liquid, mixture and vapor. This is often not true during startup and shutdown conditions, where one of the three regions may not exist. Discretized models might appear more suitable for simulation during these operation phases.
Fig. 7-1. Mass flow rate of R245fa.
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Fig. 7-2. Deviation of mass flow rate of R245fa.
Fig. 8-1. Evaporator outlet temperature.
5. Selection of thermodynamic state variables One of the key issues of thermodynamic modeling design is the selection of independent state variables which are differentiates in the model’s equations. This choice affects numerical properties such as robustness and efficiency during simulation. According to the Duhem’s theorem [18], the state of equilibrium of a thermodynamic system is completely characterized by a pair of independent variables when the mass at the initial time is known. When the conservation of momentum is formulated in differential terms, a third variable must be added as independent variable, i.e. either the fluid mass flow or its velocity. To summarize, the thee state variables are needed for each spatial coordinate point or for each lumped equation of a discretized model. Using two intensive variables chosen from the _ mg variable set {p, h, q, u, s} and one variable from fm;
Fig. 8-2. Deviation of evaporator outlet temperature.
generates a variable space with 20 possible combinations. As the momentum equation is neglected in the condenser and evaporator models, the number of possible combinations is reduced to 10. Traditionally, the pair (p, h) is adopted in dynamic models developed for refrigeration systems as the equation of state for two-phase refrigerant is explicit in (p, h). Indeed, the choice of (p, h) leads to relatively robust models within all of the liquid, two-phase and vapor regions. As a consequence, this choice is made for the models presented in this paper. As the component models of turbine and pump are lumped and static, there is no need to select the state variables. The same receiver model is adopted for both
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Fig. 9-1. Turbine inlet pressure.
dure adopted for the ORC’s condenser and evaporator (Table 1). While there are 30 state variables introduced in the model which adopts the discetization technique, those for the moving boundary model are 16 (seven for the evaporator and nine for the condenser). The extra two variables in the condenser model account for the fact that the fluid at the condenser outlet can be saturated liquid or small quality (0–0.16) two-phase fluid. For this reason, the variables dh_sat and x_out must be added to the model. This problem does not hold for the evaporator, which outlet is always characterized by superheated vapor. Then, it is easy to observe that the complexity of the moving boundary model is lower than that of the discretized one. Even with the adoption of only three cells the model order is 18, still bigger than the order of the moving boundary model. As reduced order models are far more tractable, the moving boundary formulation is recommended for use on control design applications rather than the discretized version. The simulation speed is also affected by model complexity. Indeed, for the representation of a 1400 s long transient, the simulation time of the moving boundary model is 113 s while that of the discretized model is 153 s, using the same Dymola software and hardware system. Once again, the moving boundary model is characterized by higher simulation speed, which turns out to be particularly useful for long simulations with closed control loops. 6. Conclusions
Fig. 9-2. Deviation of turbine inlet pressure.
formulations of the model, and therefore the number of state variables relative to the receiver is unchanged. In conclusion, the model complexity in terms of dimension of the state vector depends only on the modeling proce-
1. Two alternative formulations of a dynamic model for an ORC were proposed in this paper. The models, based on moving boundary and discretization model reduction techniques to represent condenser and evaporator, have been validated by comparison with experimental data from an ORC pilot plant located at UTRC. Both models correctly simulate the behavior of the system during
Table 1 State variables in each model
Evaporator
Condenser
Moving boundary model
Distributed parameter model
p L1, L2
p[1] p[5] h[1] h[5]
Pressure in each section at outlet Specific enthalpy of each section at outlet
Tw[1] Tw[5]
Wall temperature of each section at outlet 15
p[1] p[5] h[1] h[5]
Pressure in each section at outlet Specific enthalpy of each section at outlet Wall temperature of each section at outlet
h3 Tw[1] Tw[3]
Pressure Length of sub-cooled region L1, Length of two-phase region L2 Specific enthalpy of superheated region at outlet Mean wall temperature of each section
Total
7
p L2, L3
Pressure Length of superheated region L3, Length of two-phase region L2 Specific enthalpy of liquid region at outlet Mean wall temperature of each section Saturated liquid enthalpy at outlet Quality of R245a at outlet 9
h3 Tw[1] Tw[3] dh_sat x_out Total Total
16
Tw[1] Tw[5]
15 30
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transients, without showing chattering or numerical oscillations, undesired phenomena frequently occurring in this type of simulations. 2. Simulations show that the models predict the data with an accuracy of 4%. The moving boundary model is less complex than the discretized version, as it is characterized by smaller order and higher computational speed. As a result, it is more acceptable for control design applications.
kept to mark the spatial changes in the wall friction and gravitational force. The momentum flows and pressure terms are canceling out, except for the first and the last one, giving an average momentum balance for the whole pipe. The average mass flow is computed as
Acknowledgements
I_ 2 I_ nþ1 ¼ m_ 2 jw2 j m_ nþ1 jwnþ1 j:
The authors gratefully acknowledge the collaboration and financial support of United Technologies Research Center, Connecticut, USA as partners in a collaborative research project aimed at developing dynamic models of ORC’s. The experimental data have been also provided by UTRC. The authors would also like to acknowledge to the help of many UTRC researchers who provided guidance and support during the model development, particularly Brian Eisenhower, Luiz Bacellar, and Guido Poncia. Appendix A. The new discretized model [4] The new model is formulated to the set of boundary conditions as Fig. A-1, which the standard discretized models were unable to work with. And as a result, the grid is slightly changed. The pipe model contains as previously a grid with n volume cells, but the inlet mass flow m_ 1 is here a boundary condition and is not calculated from the momentum balance. The outlet mass flow is changed to a simulation result not a boundary condition as the standard discretized model. The flow cells are omitted in Fig. A-1. Instead, the location of the global momentum balance is shown. The discretization grid for the mass and energy balances is unchanged. The new model is derived by adding the momentum balance equation for cell 2 to n + 1 giving: nDz
nþ1 nþ1 X X _ dm ¼ I_ 2 I_ nþ1 þ ðp1 pnþ1 ÞA F w;i þ F g;i dt i¼2 i¼2
ðA:1Þ The wall friction for the ith flow cell is called Fw,i, and the gravitational force is called Fg,i. The flow cells do not contain any separate balance equations anymore, but are only
_ ¼ m
nþ1 1X m_ i : n i¼2
The change in momentum flow is calculated as
The information concerning the inhomogeneous part of the momentum balance is lost in the summation of the momentum balance. The new model is thus a homogeneous flow model with just one momentum balance. The frictional forces and the gravitational forces are unchanged compared to the standard discretized models. The information concerning the pressure-mass flow coupling described by the local momentum balance in standard discretized models is also lost using a global momentum balance. Consequently, a pressure distribution must be supplied. In the present analysis, a linear pressure distribution is assumed. The global momentum balance states a connection between a single pressure and the mass flow. The pressure in the first cell p1 is kept as a state variable due to its presence in the global momentum balance (Eq. (1)). The pressures in the other cells are computed assuming a linear pressure distribution between p1 and pn+1 p pnþ1 for i ¼ 2; . . . ; n: ðA:2Þ pi ¼ p1 ði 1Þ 1 n The mass balances and the energy balances are formulated for each of the cells as for the standard discretized models, with the exception that the flow is assumed to be homogeneous. The rate of change of mass and energy are given as the standard discretized models. As the pressures are determined from a static relation, only the pressure in the first cell is a state variable. Derivatives of the pressure in the other cells are calculated as the time derivative of (Eq. (2)): dpi i 1 dp1 ði 1Þ dpnþ1 ¼ 1 þ for i ¼ 2; . . . ; n n n dt dt dt The term involving pn+1 is given by the boundary conditions. This term may be difficult to calculate and may often
Fig. A-1. Nomenclature for the new discretized model. (Design boundary conditions are shown in brackets, but these might be changed as required).
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be neglected, in which case the time derivative of the pressure simplifies to dpi i 1 dp1 ¼ 1 for i ¼ 2; . . . ; n: n dt dt References [1] B. Sternlicht, Waste energy recovery: an excellent investment opportunity, Energy Conversion and Management 22 (1981) 361–373. [2] N.E. Todreas, M.S. Kazimi, Nuclear Systems I – Thermal Hydraulic Fundamentals, Taylor & Francis, 1993. [3] J.M. Jensen, Dynamic modeling of thermo-fluid systems with focus on evaporators for refrigeration, Ph.D Thesis, Technical University of Denmark, Department of Mechanical Engineering, March 2003. [4] Xiang-Dong He et al. A moving-interface model of two-phase flow heat exchanger dynamics for control of vapor compression cycle, in: Heat Pump and Refrigeration Systems Design, Analysis and Applications, AES-vol. 32, ASME, 1994. [5] Xiang-Dong He et al., Modeling of vapor compression cycles for multivariable feedback control of HVAC systems, Journal of Dynamic Systems, Measurement and Control 119 (June) (1997). [6] Xiang-Dong He et al., Multivariable control of vapor compression systems, HVAC&R Reasearch 4 (3) (1998), July. [7] M. Willatzen, N.B.O.L. Pettit, L. Ploug-Sørensen, A general dynamic simulation model for evaporators and condensers in refrigeration, International Journal of Refrigeration 21 (1998) 398–414.
[8] S. Bittanti, M. Bottinelli, Performance assessment of the control system of once-through boilers, in: Proceedings of the 13th Conference on Process Control, Slovakia Republic, June 2001. [9] J.M. Jensen, H. Tummescheit, Moving boundary models of dynamic simulations of two-phase flows, in: Proceedings of the Second International Modelica Conference, 2002, pp. 235–244. [10] J.W. MacArthur, E.W. Grald, Unsteady Compressible Two-Phase Flow Model for Predicting Cyclic Heat Pump performance and Comparison with Experimental Data, Rev. Int. Froid, vol. 12, 1989. [11] Modelica Association. Specification, Tutorials, http://www.modelica.org/. [12] Dynasim AB, Dynamic Modeling Laboratory, http://www.dymola.com/. [13] P. Melin, Measurements and modeling of convective vaporization for refrigerants in a horizontal tube, Ph.D. Thesis, Department of Heat and Power Technology, Chalmers University of Technology, Go¨teborg, Sweden, 1996. [14] Yang Shiming, Tao Wenquan, Heat Transfer, third ed., Higher Education Press, China, 1998. [15] Sheng Jingchao, Fluid Mechanics Engineering, Mechanical Industry Publishing Company, China, 1988. [16] H. Tummescheit, Design and implementation of object-oriented model libraries using modelica, Ph.D. Thesis, Lund Institute of Technology, Department of Automatic Control, August (2002), pp. 104–120. [17] Shilie Weng, Combustion Turbine, Mechanical Industry Publishing Company, China, 1989. [18] I. Prigogine, R. Defay, Chemical Thermodynamics, Longman, 1969.