Numerical predicting the dynamic behavior of heat

Available online at www.sciencedirect.com

ScienceDirect Energy Procedia 00 (2017) 000–000 www.elsevier.com/locate/procedia

IV International Seminar on ORC Power Systems, ORC2017 13-15 September 2017, Milano, Italy

Numerical predicting the dynamic behavior of heat exchangers for a small-scale Organic Rankine Cycle Liuchen Liu, Tong Zhu*, Yu Pan School of Mechanical and Energy Engineering, Tongji University, Shanghai 201804, China

Abstract

Dynamic modeling is a hot topic for investigations of Organic Rankine Cycle (ORC). In this paper, Moving boundary model and discretized model were used to describe the dynamic behavior of heat exchangers in a smallscale ORC using R123 as working fluid. Overall system model is assembled using dynamic models of heat exchangers combing static models of the scroll expander and the pump. The comparison made between the established models and experimental results shows that both the models can reveal the real system performance sufficiently. Moreover, the moving boundary model is numerically faster than the discretized model, and therefore more suitable for engineering applications. © 2017 The Authors. Published by Elsevier Ltd. Peer-review under responsibility of the scientific committee of the IV International Seminar on ORC Power Systems. Keywords: Organic Rankine Cycle (ORC); dynamic simulation; moving boundary model; discretized model

1. Introduction Organic Rankine Cycle (ORC) system is one of the most widely used technique for low-grade waste heat recovery. ORC’s steady state behavior has been studied by many researchers since more than 20 years ago. Recently, developing of dynamic ORC models played an increasingly important part in system performance prediction. From the point of view of dynamic simulation, critical components of an ORC system are the heat exchangers since they are the principal media of heat transfer in and out of the unit respectively. And dynamic models for the turbomachines (pump and expander) are usually avoided due to their negligible heat transfer irreversibility compared to their mechanical interaction and their relative faster response time to the heat exchangers. Through the summarization of the previous literatures, two different models were used to described the dynamic behavior of heat exchangers, i.e. the moving boundary (MB) model and discretized model (DM) [1]. The moving boundary models are low order lumped models particular useful for optimization and control purposes. In this 1876-6102© 2017 The Authors. Published by Elsevier Ltd. Peer-review under responsibility of the scientific committee of the IV International Seminar on ORC Power Systems.

2

Author name / Energy Procedia00 (2017) 000–000

model, the inside part is divided into three parts: sub-cooled liquid region, two phase region and superheated vapor region. On the other side, discretized models based on the balance equations of mass, momentum and energy form an alternative to MB-models when the spatial changes are important. An advantage using discretized models is the possibility of using high accuracy correlations for heat transfer and pressure loss taking the spatial variations into account. The above-mentioned two models can also be related to ORC simulations. Wei et al. [2] propose two alternative approaches, which are based on moving boundary and spatially distributed-parameter model, respectively, to be used for the accurate representation of evaporator and condensers dynamic behavior in an organic Rankine system. Sun and Li [3] reported a detailed model of an ORC using R134a as the working fluid. A discretized model was applied to the evaporator and condenser. The expander and the pump were modeled with two empirical correlations based on the performance maps given by the manufacturer. Affinity law was applied to obtain the pump power consumption at different rotational speeds. The optimization results reveal that the relationships between controlled variablesand uncontrolled variables are near liner function for maximizing system net power generation and quadratic function for maximizing the system thermal efficiency. Quoilin et al. [4] report a semi-empirical model for a volumetric expander used in a small scale ORC. The model is able to predicate the variables of the main characteristics such as the working fluid mass flow rate, the discharge temperature, pressure drop, the internal leakage and the mechanical losses as well as the delivered shaft power. The established model exhibits low good accuracy and robustness. In the present paper, moving boundary model and discretized model are used for describing the two-phase flow model of the heat exchangers. The heat transfer coefficient of each of the three zones in both heat exchangers is determined by relevant heat transfer correlations. To comparing the calculation value with the experimental value, two concrete configurations of shell and tube heat exchanger (condenser) and tube-fin heat exchanger (evaporator) are taken into account. Moreover, steady-state models are established for the expander and working fluid pump. The simulation results of the established model must be compared to each other with the experimental results under same operating conditions. Nomenclature A b d E f H Ks L n Nrot Pr q Re S T W x

area [m2] fin width [m] diameter [m] Enhancement factor [-] friction factor [-] fin height [m] surface roughness [-] length of pipe [m] fin density [-] rotating speed [r/min] Prandtl number [-] heat flux density [W/m2] Reynolds number [-] spacing [m] temperature [K] system power output [W] the vapor quality [-]

Greek symbols α convective heat transfer coefficient [W/m2K] β factor [-] μ viscosity [Pa.s] λ thermal conductivity [W/m K]

a Cp e F g h K m N Nu Q R r St u Xtt

αx ρ ɛ σ

fin length [m] specific heat [kJ/kgK] heat exchanger’s effectiveness [-] fin area [m2] acceleration due to gravity [m/s2] specific enthalpy [kJ/kg] heat transfer coefficient [W/m2K] mass flow rate [kg/s] suppression factor [-] Nusselt number [-] thermal energy [kW] pipe radius [m] thermal resistance [m2K/W] Stanton number [-] velocity of flow [m/s] factor [-]

air-bubble coefficient [-] density [kg/m3] efficiency [-] surface tension [N/m]

Author name / Energy Procedia00 (2017) 000–000

γ δ Φ

latent heat [kJ/kg] fin thickness [m] correction factor [-]

Subscripts amb ambient ch charge cri critical em electromechanical eva evaporation g saturated vapor in inlet max maximum out outlet pool pool boilling s swept tp two-phase flow w wall int 2 the interface of two phase region and vapor region 2 two phase region

 η

3

mean value of void fraction efficiency [-]

c calculated cond condensation e equivalent ex exhaust gas f fin i inside l saturated liquid o outside pp pump r root su supply v vapor int 1 the interface of liquid and two phase region 1 unsaturated liquid region 3 superheat region

2. System description Figure 1 illustrate the scheme of the considered ORC system. As shown in Figure1, waste heat from the ICE exhaust gas is absorbed by refrigerant R123 as working fluid in the evaporator. The temperature of waste heat source is 373-473K, which is gained from the product manual. Then the working fluid vapor entering the scroll expander with a mass flow rate of 0.4274kg/s and drives it to generate power. Afterwards, the low-pressure vapor exits the scroll expander and is led to the condenser where it is liquefied by cooling water. The sub-cooling degree is keep to 5K and cooling water temperature is 293K. The condensed working fluid is pumped back to the evaporator and the looping would continue.

Fig.1. system diagram

Fig.2. T-S diagram of the system

Figure 2 shows the T-S diagram of the ORC system. The cycle in the present work is consist of four main thermodynamic process: the isobaric heat absorption process 5-1, the expansion process in the scroll expander 1-2, the isobaric condensation process 2-5 and the pumping process 5-6. For the ideal case, the processes 1-2s and 5-6s are the isentropic processes. In order to develops a proper mathematical model of this ORC unit, submodels for each independent components. In the present work, empirical model have been applied to describe the working fluid pump. The scroll expander has been modeled using a steady-state model which represent the relationship between working fluid flow and given

4

Author name / Energy Procedia00 (2017) 000–000

rotational speed. For the two exchangers, steady-state models with detail heat transfer correlations are used to determine the boundaries between liquid, phase-change and vapor zones. The physical bounds of the total heat transfer rate are obtained based on a pinch point analysis and only sub-critical conditions are considered. For each zone, the required surface area to achieve the heat transfer rate defined by the zone boundaries is determined through the logarithmic mean temperature difference (LMTD) method. Such choice of the steady-state model combinations rather than the numerical model is resulting from the compromise between computational efficiency and the abillity to exactly estimate the charge level in heat exchangers as well as the system performance under several part-load conditions. All above calculation process are based on Matlab2015 and the physical property of working fluid was gained from the software Refprop 9.1 developed by NIST (National Institute of Standards and Technology). 3. Model description 3.1. Governing equations According to the moving boundary model, the evaporator can be described as composed by three zones characterized by pre-heating, evaporation and superheat, while condenser can be analogously described as composed by three zones characterized by pre-cooling, condensation, and sub-cooling, respectively (see Fig. 3 for a scheme of the evaporator). In order to reduced the model complexity, the axial thermal heat conduction has been neglected. Therefore, appropriate one-dimensional governing equation for each zones can be derived as follows:

Fig.3. schematic of the moving boundary model Mass balance:

A m  0 t z

(1)

Energy balance:

   Ah  AP  t



mh  πDi i Tw  Tr  z

(2)

The mass and energy balance for each three zones are the formulated by integrating Equations (1) and (2). The explicit calculation formulas are listed in Appendix A.1. The actual heat transfer capacity of heat exchanger 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:

Author name / Energy Procedia00 (2017) 000–000

5



Q

e=

(3)



Q max 3.2. Heat transfer correlations For the evaporator, both correlations of exhaust side and working fluid side in a tube-shell type heat exchanger are taken into account. For the condenser, the Gnielinski equation is used for both the single-phase fluid and a correlation from [5] is used for the convection heat transfer for two-phase flows in a horizontal tube. The explicit calculation formulas are listed in Appendix A.2. The concrete value of parameters used for both heat exchanger models are summarized in Appendix B. Table B.1 and B.2. 3.3. Expander model The angular momentum of expander is assumed negligible compared to the dynamics of both heat exchangers. Therefore, a steady-state model is established for the volumetric expander. In consideration of non-adiabatic expansion and all other losses such as internal leakage, heat transfers loss and friction, the total output expansion work can be modified by using a lumped efficiency ηexp:

Wexp  m(h1  h2 S )exp

(4)

The relationship between a given expander rotational speed and relevant working fluid mass flow rate is represented by:

m=

FF  in,exp  Vs  N exp 60

(5)

where: FF being the filling factor which has the equivalent physical meaning of the volumetric efficiency in compressor, set to a typical value provided by [4] and [6]. Vs is the volume of the expander suction chamber. The concrete value of parameters used for expander model are summarized in Appendix B. Table B.3. 3.4. Pump model As suggested by [7], a linear correlation (6) can be used to described the mass flow rate of the pump as a function of the rotational speed.

m  aN pp  b

(6)

where the parameter a and b are specific for the pump under consideration and can be obtained directly by fitting to the product operation data. Moreover, pressure rise of the working fluid through the pump can be approximately represented in a secondorder polynomial function of the volumetric flow rate, i.e. the division of mass flow rate and fluid density at the pump inlet [7]. The parameters C1, C2 and C3 in Eq.(7) are obtained directly by regression to the product operation curves.

6

Author name / Energy Procedia00 (2017) 000–000

 m Prise =C1   in, pp 

2

   C 2 m  C3  in, pp 

(7)

The overall input power of the pump can be modified by an additional electromechanical efficiency ηem,pp:

Wpp  m(h6 S  h5 )em, pp

(8)

Table B.4 in Appendix B summarized the concrete value of parameters used for pump model. 4. Validation studies 4.1. Grid independent test For the DM model, the accuracy of results increase with the number of discrete nodes (n) assigned for the numerical solution. Appropriate number of discrete nodes should be firstly estimate. The heat exchanger’s effectiveness (e) of the evaporator is therefore computed as an indicator to compared for a range of number of nodes, with predefined conditions (inlet temperatures and flow rates) for exhaust gas and R123. Error based on n number of nodes can be expressed as in:

errorn 

e n  e150 e150

(9)

where the subscript n denotes the number of nodes used in the numerical simulation, and 150 is considered to the sufficient large number of nodes. Fig. 4 illustrates the variation of error in effectiveness as obtained from the evaporator model. It can be found that selecting number of nodes, n=75 gives a suitable balance between accuracy and computational burden for the conditions given above.

Fig.4. results of grid independency tests 4.2. Validation of evaporator and condenser models Figure 5 compare the obtained distribution of working fluid side evaporator inlet/outlet pressure, evaporator outlet temperature and condenser outlet temperature. As show in Figure 5, both the variation trend of the present models are similar to the experimental result. The deviation between model predictions and test values are

Author name / Energy Procedia00 (2017) 000–000

7

acceptable for real operating process. Nevertheless, the MB models overestimates the experimental results at most of the time, while the DM exhibits higher accuracy. It should be noted that the rapid decline of testing pressure value is caused by switching operation of the different operating conditions, which the proposed model not able to capture.

(a) fluid side evaporator inlet pressure

(b) fluid side evaporator outlet pressure

(c) fluid side evaporator outlet temperature

(d) fluid side condenser outlet temperature

Fig.5. comparison of the simulation results and experimental data 4.3. Variation trend of different zones in evaporator Figure 6 demonstrates the variation trends of three zones in evaporator. An obvious percentage change between the preheat zone and evaporating zone has been detected in Figure 6. As shown in Figure 6, the percentage of evaporating zone reduces from 57% to 45%, while the percentage of preheat zone increases from 41% to 54%. Meanwhile, the superheat zone is hard to observe with an percentage of ca. 1%.

8

Author name / Energy Procedia00 (2017) 000–000

Fig.6. variation trend of different zones in evaporator 4.4. Comparison of computational burden To output the calculating results of 3200 steps, the totaly time-consuming of MB model was 610s, while the DM model needs more than 900s. 5. Conclusions The present work used two alternative models to describing the heat exchangers in ORC system and governing equations as well as the heat transfer correlations for each zones in both evaporator and condenser are presented. The simulation results have been compared with the experimental test values of system level. Moreover, the variation trend of different zones in evaporator has been also illustrated. Results indicated that both models exhibit good accuracy. However, the moving boundary model show faster computational speed than the discretized model, and therefore more suitable for engineering applications. Acknowledgements This work was supported by funding coming from the National Fundamental Research Program of China 973 project (2014CB249201) "Research on the stability of complex energy system integrated natural gas and renewable energy".

Author name / Energy Procedia00 (2017) 000–000

9

Appendix A. A.1. Mass and energy balance equations used in moving boundary models Sub-cooling region:

mi =mint1

(A.1.1)

  dh1  dP  h  h1 dL AL1  1  1 e  + i A1 1 = i1πDi L1 Tw1  Tr1   m  hi  h1  2 dt  dt   2 dPeva

(A.1.2)

Two phase region:

d 2 dPeva dL dL +A   2   g  2 +A  1   g  1  mint1  mint2 dPeva dt dt dt

AL2

d g hg  dPeva  d h AL2 1    1 1    1 +A 1   dPeva dPeva   dt  mint1h1  mint 2 hg +πDi i2 L2 Tw 2  Tr2 

  1h1   g hg 

dL2 dL +A  1   g  1 dt dt

(A.1.3)

(A.1.4)

Preheat region:

AL3

d 3 dPeva dL dL +A   g   3  2 +A   g   3  1  mint 2  mo dPeva dt dt dt

  dhg  dP  dh  AL3  3  1 eva + 3 o  2 dt   2 dPe  dt dL  dL2  1  πDi i3 L3 Tw3  Tr2    m o  mint2  A g 1   ho  hg  2 dt 

(A.1.5)

(A.1.6)

A.2. Heat transfer correlations used for different zones For the working fluid side in evaporator, following equation can be used for both single-phase zone:

St  Pr 2/3  St 

  umCp

f 8

(A.2.1)

(A.2.2)

In Eq. (A.2.2), f represents the resistance coefficient of fluid flow in a tube, and is calculated according to:

10

Author name / Energy Procedia00 (2017) 000–000

64 ， Re  3000 Re

(A.2.3)

f  [2  lg（R / Ks） 1.74]2

(A.2.4)

f  0.316Re  0.25， 3000  Re  5000

(A.2.5)

f  0.184Re  0.2，

(A.2.6)

f 

Re  5000

A correlation from Ref. [5] is used for the convection heat transfer for two-phase flows in a horizontal tube:

tp  El  N pool

l  0.023

l d

(A.2.7)

Rel 0.8 Prl 0.4

(A.2.8)

 pool d pool  qd pool   207     pool  lTsat  d pool

0.745

    0.0204    g ( l   v ) 

 v     l 

0.581

Pr 0.533

(A.2.9)

0.5

(A.2.10)

where: for refrigerant β=25, σ is the surface tension.

 1  E  2.37   0.29   X tt  

0.85

(A.2.11)

N  4048X tt 0.22 Bo1.13 , X tt  1

(A.2.12)

N  2.0  0.1X tt 0.28 Bo0.33 1  X tt  5

(A.2.13)

,

Bo 

q Gm  

(A.2.14)

0.5

 1  x    v   l  X tt         x   l    v  0.9

0.1

(A.2.15)

where: x is the dryness of working fluid；ρv is the density of gas-phase；ρl is the density of liquid-phase；μl is the dynamic viscosity of gas-phase and μv is the dynamic viscosity of liquid-phase. For the exhaust gas side in evaporator, finned tube heat transfer efficiency should be taken into account.

Author name / Energy Procedia00 (2017) 000–000

11

Therefore, the heat transfer coefficient is calculate by:

1 1  r 1   ri   0  K i    0 

(A.2.16)



F2'  f F2'' F2' +F2''

(A.2.17)



F2’+F2”  d 0l

(A.2.18)

" F2'  nf Sf  d0 , F2  2[ab   a  b   ]nf

f  m

(A.2.19)

th(mH ) mH

(A.2.20)

h0U 2h0  A 

 0  0.251

  umax de    de   

0.67

(A.2.21)

 S1  d r     dr 

0.2

 S1  d r   1  S  1 

0.2

 S1  d r     S2  d r 

F2’d r  F2" F2" / (2nf ) de  F2’ F2"

0.4

(A.2.22)

(A.2.23)

For the condenser:

Nu f 

( f / 8)(Re 1000) Pr f

d   1  ( ) 2/3    1)  l 

(A.2.24)

2000  Re  6000

(A.2.25)

1  12.7( f / 8) ( Pr 0.5

f  1.82lg Re 1.64 

2

2/3 f

1/4

 mL3 L2 g    0.752    N t d0 L (tb  tw )  where: γ=hin-hout

(A.2.26)

12

Author name / Energy Procedia00 (2017) 000–000

Appendix B. Parameters used in the models Table B.1. Evaporator geometrical dimensions. Parameter Sf δf nf d0 a b

Value 3.54mm 0.8mm 230 fins/m 9.52mm 252mm 410mm

Parameter S1 S2 δ λ ri r0

Value 15.72mm 13.61mm 0.001m 400W/(m﹒K) 0.000172(m2K) /W 0.000174(m2K) /W

Parameter S1 S2 ri r0

Value 15.875mm 13.75mm 0.000172(m2K) /W 0.000174(m2K) /W

Table B.2. Condenser geometrical dimensions. Parameter d0 di λ δ

Value 0.0127m 0.01022m 400W/(m﹒K) 0.00124m

Table B.3. Expander model parameters. Parameter FF Vs

Value 0.6 73.6 cm3

Parameter Nexp,max ηexp

Value 3000 rpm 0.5

Parameter C1 C2 C3

Value -3.93651012 1.4495109 -1.3264105

Table B.4. Pump model parameters. Parameter a b ηem,pp

Value 8.910-5 0.02 0.8

References [1] Vaja I. Definition of an object oriented library for the dynamic simulation of advanced energy systems: methodologies, tools and application to combine ICE–ORC power plants. PhD dissertation; University of Parma; 2009. [2] Wei DH, Lu XS, Lu Z, Gu JM. Dynamic modeling and simulation of an Organic Rankine Cycle (ORC) system for waste heat recovery. Appl Therm Eng 2008; 28:1216-24. [3] Sun J, Li W. Operation optimization of an organic rankine cycle (ORC) heat recovery power plant. Appl Therm Eng 2011; 31:2032-41. [4] Quoilin S, Aumann R, Grill A, Schuster A, Lemort V, Spliethoff H. Dynamic modeling and optimal control strategy of waste heat recovery Organic Rankine Cycles. Appl Energ 2011;88:2183-90. [5] Qian SW. Heat Exchanger Design Manual. 1rd ed. Peking: Chemical Industry Press of China; 2006; 365-70. [6] Quoilin S, Lemort V, Lebrun J. Experimental study and modeling of an Organic Rankine Cycle using scroll expander. Appl Energ 2010;87:1260-8. [7] Ziviani D, Woodland BJ, Georges E, Groll EA, Braun JE, Horton WT, Broek M, Paepe MD. Development and a validation of a charge sensitive Organic Rankine Cycle (ORC) simulation tool. Energies 2016;9;389:1-36.