Scan of working fluids based on dynamic response characters for Organic Rankine Cycle using for engine waste heat recovery

Scan of working fluids based on dynamic response characters for Organic Rankine Cycle using for engine waste heat recovery

Accepted Manuscript Scan of working fluids based on dynamic response characters for Organic Rankine Cycle using for engine waste heat recovery Gequn ...

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Accepted Manuscript Scan of working fluids based on dynamic response characters for Organic Rankine Cycle using for engine waste heat recovery

Gequn Shu, Xuan Wang, Hua Tian, Peng Liu, Dongzhan Jing, Xiaoya Li PII:

S0360-5442(17)30742-9

DOI:

10.1016/j.energy.2017.05.003

Reference:

EGY 10805

To appear in:

Energy

Received Date:

27 January 2017

Revised Date:

05 April 2017

Accepted Date:

01 May 2017

Please cite this article as: Gequn Shu, Xuan Wang, Hua Tian, Peng Liu, Dongzhan Jing, Xiaoya Li, Scan of working fluids based on dynamic response characters for Organic Rankine Cycle using for engine waste heat recovery, Energy (2017), doi: 10.1016/j.energy.2017.05.003

This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

ACCEPTED MANUSCRIPT Highlights  The dynamic response of ORC systems with 14 working fluids including water are compared  Steam Rankine Cycle responds much more slowly than ORC  High temperature working fluids respond more slowly than low temperature working fluids  Critical temperature can be used to estimate the response speed of straight-chain alkanes

1

ACCEPTED MANUSCRIPT

1

Scan of working fluids based on dynamic response

2

characters for Organic Rankine Cycle using for engine

3

waste heat recovery

4

Gequn Shu, Xuan Wang, Hua Tian*, Peng Liu, Dongzhan Jing, Xiaoya Li

5

State key laboratory of engines, Tianjin University, No.92, Weijin Road, Nankai

6

District, Tianjin, 300072, China

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Contact Information (E-mail): [email protected]

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Abstract

9

Organic Rankine Cycle (ORC) is regarded as a very suitable method to recover the

10

waste heat of internal combustion engines. Engines often operate under variable

11

working conditions, so ORC systems are also under unstable state which means it is

12

important to research the dynamic response. There are many kinds of working fluids

13

and various dynamic response characters are reflected owing to their different

14

properties. The dynamic math models of ORC systems with 14 different working

15

fluids (including water) as waste heat recovery system of a natural gas engine are

16

established by Simulink in this paper. Based on these, their dynamic response

17

characters mainly reflected by rise time, settling time and time constant are compared

18

and analyzed. The results indicate that no matter under the disturbance of working

19

fluid mass flow rate or exhaust temperature, the ORC systems with low temperature

20

working fluids respond faster than those with high temperature working fluids

21

generally. Therein, RC responds the most slowly. Furthermore, the critical

ACCEPTED MANUSCRIPT 22

temperature can be used to approximately estimate dynamic response speed of ORC

23

with straight-chain alkanes. These provide useful guidance for the selection of

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working fluids based on the working condition characteristics of different engines and

25

control system design.

26 27

1. Introduction

28

There is a strong motivation in the internal combustion engines (ICEs) for

29

increasing the energy efficiency, mainly because of rising fuel prices and stricter

30

upcoming regulations. Organic Rankine Cycle (ORC) is regarded as a very suitable

31

method to recover the waste heat of ICEs [1]. There are large amount of organic

32

working fluids and the ORC system performance has great relationship with working

33

fluid property. Therefore, selecting the most suitable working fluid for the ORC

34

applied to recover engine waste heat is a complex task and the topic has received

35

significant attention in the scientific literature [2].

36

In present researches, system performance like thermal efficiency, exergy

37

destruction factor, economic factor and so on are used for working fluid selection.

38

Dariusz Mikielewicz [3] proposed a thermodynamic criterion for selecting working

39

fluid and comparatively assessed theoretical performances of a few fluids used in

40

ORC for recovering waste heat of engines in CHP (micro combined heat and power

41

units). Among the 20 fluids investigated, ethanol, R123 and R141b appeared as the

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most suitable according to the criterions. Gequn Shu et.al [4] analyzed alkanes as

43

suitable working fluids for waste heat recovery of diesel engines based on six

ACCEPTED MANUSCRIPT 44

indicators like output power, thermal efficiency and so on. The results showed that

45

Cyclic Alkanes, Cyclohexane and Cyclopentane were considered as the most suitable

46

working fluids when considering all comprehensive indicators. Ulrik Larsen [2]

47

presented a generally applicable methodology based on the principles of natural

48

selection to determine the optimum working fluid for waste heat recovery of marine

49

engines and the results suggested that at design point, the requirements of process

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simplicity, low operating pressure and low hazard resulted in cumulative reductions in

51

cycle efficiency. Jian Song [5] made analysis of ORC systems with pure

52

hydrocarbons and mixtures of hydrocarbon and retardant for engine waste heat

53

recovery.

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cyclohexane/R141b (0.5/0.5) was optimal for this engine waste heat recovery case. It

55

could increase the net output power of the system by 13.3% compared to pure

56

cyclohexane. Other researches on ORC performance with different working fluids for

57

ICE waste heat recovery can refer to [6-8].

The

simulation

results

revealed

that

the

ORC

system

with

58

However, in the researches above, the indictors used to evaluate different

59

working fluid performance (such as thermal efficiency, exergy destruction factor,

60

turbine size parameter, turbine volume flow ratio, net power output per unit mass flow

61

rate of exhaust and so on) are all under stable work conditions of ICEs, which do not

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consider the dynamic response of ORCs with different working fluids when the

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working condition varies. In fact, the ICE working conditions often need change with

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time to meet the load. Exhaust is the most important waste heat source and on

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different working conditions of ICE, exhaust gas temperature of light-duty engines

ACCEPTED MANUSCRIPT 66

varies from 500 to 900℃ and that of heavy-duty engines is in the range of 400 to 650

67

℃ [9-10]. Besides, the exhaust mass flow rate also changes a lot under different ICE

68

working conditions [11]. These mean ORC systems are demanded for lots of dynamic

69

operation and it is necessary to scan various working fluids based on dynamic

70

response characters. Actually, ORC systems with various working fluids have

71

different dynamic response performance owing to their different physical properties.

72

Furthermore, dynamic response characters of different working fluids can be referred

73

to make selection for recovering the waste heat of different types of ICEs, because

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working condition characteristics of ICEs are diverse. For example, automotive

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engines of which working conditions vary frequently should choose the working fluid

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with fast response speed. By contrast, working fluids with slow response speed can be

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applied in such as ship engines, due to their relatively steady working conditions.

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Dynamic simulation is an effective way to research the dynamic response of

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various working fluids [12]. Some researchers have built the dynamic models of ORC

80

system for waste heat recovery (WHR) of engines to analyze its dynamic response

81

and part load performance. Tilmann Abbe Horst [13] built a dynamic model of

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Rankine Cycle for passenger car waste heat recovery system and investigated

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dynamic operating characteristics of the evaporator based on measurements and

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simulations. Sylvain Quoilin [14] proposed a dynamic model of the ORC with R245fa

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as working fluid, focusing specifically on the time-varying performance of the heat

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exchangers, and three different control strategies were compared to find the best one.

87

Francesco Casella [16] established a specific dynamic model of ORC with benzene as

ACCEPTED MANUSCRIPT 88

working fluid for waste heat recovery and validated it by using carefully conceived

89

experiments to reproduce steady-state and dynamic measurements of key variables,

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both in nominal and in off-design operating conditions. However, there is no research

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focusing on comparison of dynamic response of various working fluids, especially the

92

effects of fluid property on its dynamic response characters.

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Since the dynamic response of ORC systems with different working fluids is very

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important as mentioned above and there is few research focusing on comparing them,

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the dynamic math models of ORC systems with 14 different working fluids including

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water as the waste heat recovery system (WHRS) of a natural gas engine are

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established by Simulink in this study. Based on this, their dynamic response

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characters are compared and analyzed, finding some regularities to provide useful

99

reference for the selection of working fluid and control design in practice engineering.

100 101

2. Used system description

102

2.1 Top system

103

As the heat balance experiments of a natural gas engine with rated power of 1000

104

kW have been done, all the ORC systems are applied to recover the exhaust waste

105

heat of this kind of engine in this study. All the systems are designed based on the

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rated working condition of the engine. The main parameters of the exhaust at rated

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work condition are shown in Table 1, from which it can be seen that the exhaust

108

temperature after turbo charger is quite high. Therefore, it is very meaningful to

109

recover the exhaust waste heat. According to the actual volume ratio of natural gas

ACCEPTED MANUSCRIPT 110

and air and assume that the fuel burns completely, the composition of the exhaust can

111

be calculated: N2 = 73.4%, CO2 = 7.11%, H2O = 14.22% (gas), O2 = 5.27%. It should

112

be noticed that in order to avoid corroding the pipe wall, the temperature of exhaust

113

out of WHRS cannot drop below acid drew point [17]. The acid drew point changes

114

with the sulphur content and it is assumed to be 110℃ in the paper. According to our

115

formal research [18], the final temperature of exhaust out of WHRS decreases as ICE

116

working condition gets down. Therefore, the design final exhaust temperature should

117

be enough higher than acid drew point at rated ICE working condition and it is

118

designed as 160℃ for all the ORC systems.

119

Table 1. Main parameters of the natural gas engine

120

Parameter

values

Number of cylinders

8

Rated power

1,000 kW

Endurance speed

600r/min

Exhaust temperature after turbo charger

540℃

Volume flow rate of intake air

1.16 m3/s

Volume flow rate of natural gas

0.089m3/s

Exhaust mass flow rate

1.56kg/s

2.2 Bottoming ORC system

121

There are many kinds of organic working fluids as mentioned above. According to

122

the slope of the saturated vapor line in the T-S diagram, the organic working fluids are

123

classified into three types: wet, isentropic and dry fluids [19]. Dry fluids have positive

ACCEPTED MANUSCRIPT 124

slope, wet fluids have negative slope, and isentropic fluids have infinitely large slope.

125

A disadvantage of wet fluids such as water is that they may condense during

126

expansion, which can lead to erosion of the turbine blades, so wet fluids need a large

127

superheat degree, while isentropic and dry fluids do not have the problem and they are

128

regarded as more suitable working fluids [3]. All of the working fluids except for

129

water in this study are dry or isentropic fluid. Besides, according to their

130

decomposition temperature they can be classified to low temperature (LT) working

131

fluids (often with low critical temperature) and high temperature (HT) working fluids

132

(often with high critical temperature) [20]. Low-temperature working fluids are

133

usually halohydrocarbon refrigerant and six kinds are selected in the paper: R236ea,

134

R114, R245fa, R245ca, R123, and R141b. Hydrocarbon, including alkanes and

135

aromatic hydrocarbon, and siloxane are common HT working fluids [20]. Therein,

136

hydrocarbon is considered as suitable high temperature working fluids for waste heat

137

recovery of engines [21], so hexane, heptane, Octane, nonane, decane, cyclohexane

138

and toluene are selected in the present work. Beside, water is a quite suitable HT

139

working fluid and researched here as well.

140

The ORC systems are used to recovery just the exhaust waste heat and the

141

fundamental principle is simple: the exhaust heats the working fluid to become vapor

142

of high temperature and pressure by several heat exchangers and then the vapor

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expands in a turbine or expander to generate power, decreasing the temperature and

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pressure. After that, the vapor is cooled into liquid form in the condenser. Finally the

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liquid working fluid is pumped to the preheater again and a new cycle starts. The

ACCEPTED MANUSCRIPT 146

system diagram and T-S diagram are shown in Fig. 1 and 2. It should be noticed that

147

although the dry fluid has a positive slope in T-S diagram, a slight superheating

148

degree should be maintained [14].

149 150

Fig. 1. The schematic diagram of ORC WHRS.

151

3. Dynamic Mathematical Model

Fig. 2. T-S diagram of ORC

152

The dynamic models of the main components are built at first and then system

153

model is created by appropriately combining each of the component models

154

according to their interrelationships. For the ORC system level modeling, the

155

physical interrelationships between each two components are shown as Fig.3.

156

Because pump and expander response much faster compared to the heat exchangers,

157

their models are usually replaced by static models [13-14, 22]. The math model is

158

established by Simulink and all the physical property parameters are obtained by

159

Refprop.

ACCEPTED MANUSCRIPT

160 Fig. 3. The physical

161

162

interrelationships between each two components

3.1.1. Heat exchanger

163

The exhaust heat exchanger (including preheater, evaporator and superheater in

164

Fig. 1) in this work is the same with that in literature [13], which is a finned tube heat

165

exchanger. The flow pattern in the heat exchanger is shown as Fig. 4 and the specific

166

structure of the heat exchanger can refer to [13]. Since the number of tube pass is

167

more than 4, this kind of flow pattern can be regard as countercurrent [23]. The

168

exchangers in this paper have been, for sake of simplicity, represented as a typical

169

counter flow straight pipe, despite it is well known that complex designs are usually

170

adopted in order to enhance heat exchange and to reduce the overall dimensions of the

171

system. This assumption simplifies the dynamic problem in a great extent and is

172

commonly adopted when heat exchanger dynamic modeling is considered [13-14, 22,

173

25-26].

ACCEPTED MANUSCRIPT

174 175

Fig. 4. Flow pattern in exhaust heat exchanger

176

The phase change occurs in the exhaust heat exchanger and convective heat transfer

177

coefficients are very different in various phases, so the moving boundary method is

178

applied to build the dynamic model. In this method, the working fluid side of the heat

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exchanger is divided into three regions: sub-cooling region, two-phase region and

180

superheating region. The lumped parameter method is used in each region. The idea

181

of the moving boundary model is to dynamically track the lengths of the different

182

regions in the heat exchanger. The modeling method requires several assumptions

183

about the fluid flow in the heat exchangers. These assumptions are as follows [24]:

184

1.

The heat exchanger is a long, thin, horizontal tube.

185

2.

The working fluid and exhaust flowing through the heat exchanger tube can be

186

modeled as a one-dimensional fluid flow.

187

3.

Axial conduction of working fluid and exhaust is negligible.

188

4.

Pressure drop along the heat exchanger tube due to momentum change in

189

refrigerant and viscous friction is negligible. Thus the equation for conservation of

190

momentum is not needed.

191

5.

192

of vapor volume to total volume, and has long been used to describe certain

193

characteristics of two-phase flow.

The assumption of mean void fraction is used. Void fraction is defined as the ratio

ACCEPTED MANUSCRIPT 194

The notations used in the moving boundary model are given in Fig. 5. Other

195

notations not appearing in the figure are α0 (the heat transfer coefficient between

196

exhaust and pipe wall), α1, α2, α3 (internal pipe heat transfer coefficient in subcooling

197

region, two-phase region and superheating region), p (the pressure in evaporator), pe

198

(the pressure of exhaust), Ai, Ao, Aw (the cross sectional area of inner pipe, outer pipe

199

and pipe wall).

200 201 202

Fig. 5. Notations used in the moving boundary model.

The general differential mass balance for the three regions is:



L1

0

203

(1)

The general differential energy balance for the three regions is:



L1

0

204

L1 m   A   dz   dz  0 0 t z

L1 mh L1   Ah  Ap   dz   dz    i Di Tw  Tr dz 0 0 t z

(2)

A simplified differential energy balance for the wall is: c pww Aw

dTw   i Di Tr  Tw    o Do Ta  Tw  dt

(3)

205

Equations (1)–(3) are integrated over the three regions to give the general three

206

region lumped models for a two-phase heat exchanger. The average densities of the

207

subcooling region and superheating region are expressed as the function of the

ACCEPTED MANUSCRIPT 208

pressure and average specific enthalpy. The average specific enthalpy is arithmetic

209

mean value of the enthalpy at inlet and outlet of sub-cooling region and superheating

210

region. In two-phase region the average density and specific enthalpy can be written

211

as:





ρs  γρ g  1  γ ρ l





ρs h s  γρ g h g  1  γ ρ l h l

(4)

(5)

212

Therein, γ is the mean void fraction which is defined as the ratio of vapor volume to

213

total volume. The void fraction γ in the two phase region is related to liquid fraction η

214

via the equation:   1

(6)

215

The same equation holds for the average valuesγ andη over the whole region

216

[27]. A slip flow model is employed to predict the average fluid state by means of the

217

average void fractionγ and the slip velocity ratio S = ug/ul between the gas and the

218

liquid velocities. The slip flow model proposed by Zivi [28] is used here because of

219

its simplicity: S  u g / ul  (l /  g )1/3  1/3

(7)

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Using this slip correlation, the average liquid fractionη in the pipe becomes a

221

function of only one variable, the density ratio µ.The specific deducing process ofη

222

can be found in Ref. [27]. Withη the average void fractionγ can be known from

223

equation (6): 1

   ( z )dz  0

1  (1/ ) 2/3 (2 / 3ln(1/ )  1) ((1/ ) 2/3  1) 2

(8)

ACCEPTED MANUSCRIPT 224

Applying Leibniz’s rule (Equation (9)) on the mass and energy balance equations

225

and simplifying the equations, the moving boundary models of the three regions can

226

be acquired. Corresponding to the three regions of working fluid, the exhaust side is

227

also divided into three regions and the general energy and mass balance equations are

228

the same with equations (1-2). The specific equations of every region of both working

229

fluid and exhaust sides are attached in the appendix.



z2

z1

dz dz f ( z , t ) d z2 dz   f ( z , t )dz  f ( z2 , t ) 2  f ( z1 , t ) 1 t dt z1 dt dt

(9)

230

The heat transfer coefficient outside the evaporator pipe can be obtained by Briggs’

231

Equation [29]. Considering the phase change in the tube, different correlations are

232

used for the convective heat transfer coefficient inside tube. For single-phase fluid,

233

the Equation (11), Sieder-Tate correlation is used. There are lots of heat transfer

234

correlations of phase change [30]. In this study, the heat transfer coefficient of two-

235

phase region is calculated as a function of the heat transfer coefficient of subcooling

236

region (αl) and superheating region (αg), the densities of saturated liquid (ρl) and

237

saturated gas (ρg) and the average gas quality (x) [31]. Table 2. Heat transfer correlations Finned tube heat exchanger

Heat transfer correlations

Exhaust side [29]

Di f

 Di Gmax  =0.1378      

238

 cp       

u f  0.027 Re f 0.8 Prf

Inside tube (single phase) [23] Inside tube (Evaporating) [31]

0.718

    l      

     1  x   1.2 x 1  x   l   g  0.4

x can be derived from the following equations:

0.37  2.2     

1

3

 

   

1

3

 Sf   Hf

 0.14   w  f

  

0.296

(11)

0.67   0.7      g x 0.01 1  8 1  x   l    l   g   





(10)

2       

0.5

(12)_

ACCEPTED MANUSCRIPT

x

havg  hl hg  hl

avg  l (1   )   g 

havg 

l hl (1   )  l hl  avg havg

(13)

(14)

(15)

239

The dynamic performance of the ORC with variable exhaust and working fluid

240

mass flow rate is the main focus of this research, while the condensation pressure can

241

be controlled by adjusting the temperature and mass flow rate of cooling water.

242

Besides, if the cooling water temperature is closed to the condensing temperature, the

243

condensing pressure will change slightly under different working conditions. Therefore,

244

the condenser pressure and the enthalpy of working fluid at the outlet of condenser are

245

assumed to be constant. This turned out to be a sound choice for the validation in [16]

246

and the condenser is also simplified like that in former studies [13-14, 27, 32-33]. It

247

allows avoiding a dynamic model of the condenser, and has beneficial effects of

248

reducing the computational effort [15].

249

3.1.2. Pump and Turbine

250

The pump model is defined by a simple expression for the mass flow [27]: m pump  v  pumpVcyl 

(16)

251

Where ηv is the volumetric efficiency, ρpump is the working fluid density at the pump

252

inlet, Vcyl is the cylinder volume and ω is revolution speed. In the pump, the working

253

fluid goes through a non-isentropic pumping process. The ideal enthalpy of working

254

fluid after isentropic pumping is written as hspout, hpout and hpin are the enthalpy of

ACCEPTED MANUSCRIPT 255

working fluid at the outlet and inlet of pump, respectively. ηsp is the isentropic

256

efficiency of the pump. hpout  hpin 

hspout  hpin sp

(17)

257

The turbine is simplified as a nozzle [27]. Since most turbines of ORC systems for

258

waste heat recovery of engine are relative small and too large ratio between the

259

pressure at turbine inlet and outlet leads to great difficulty in manufacture, so the

260

pressure ratios of all the systems with different working fluids are designed as 10,

261

which is greater than their critical pressure ratios. Therefore, for most of mass flow

262

rate values the fluid reaches supersonic speed, which allows to neglect the influence

263

of the outlet pressure [33]. m t  Cv out p

(18)

264

Where Cv is a coefficient, ρout is the density at the evaporator outlet, p is the

265

evaporating pressure. In the evaporator, the working fluid goes through a non-

266

isentropic pumping process. The ideal enthalpy of working fluid after isentropic

267

expanding is written as hstout, htout and htin are respectively the enthalpy of working

268

fluid at the outlet and inlet of turbine. ηst is the isentropic efficiency of the turbine.

htout  htin  (htin  hsout ) st

269

(19)

270 271

3.2. Model Validation

272

The math model in this paper is based on [27], but some improvements have been

273

made in this mathematical model. In reference [27], all the heat transfer coefficients

274

are constant, while they are added in this math model. In order to verify the model

275

with reference [27], all the heat transfer coefficients are assumed to be constant as

ACCEPTED MANUSCRIPT 276

well in the validation part. Fig. 6 shows the comparison between the transient

277

responses of the models in [27] and this paper. At 20 s the pump speed is increased by

278

5%, at 50 s the outer heat transfer coefficient is increased by 10% and at 80 s the

279

nozzle coefficient is increased by 10%. It can be seen that the model has enough

280

consistency with that in reference [27].

281 (a)

282

283 (b)

284 285

Fig. 6. Model validation. (a) The variation of evaporating pressure. (b) The variation of

286 287 288

subcooling region length

4. Results and discussion In this part, ORC systems with different working fluids and RC are designed under

ACCEPTED MANUSCRIPT 289

the rated working condition of the natural gas engine to recover the exhaust waste

290

heat. Based on the dynamic math models, their dynamic response under the

291

disturbance of exhaust temperature and working fluid mass flow rate is compared.

292

The six kinds of low-temperature working fluids are: R236ea, R114, R245fa, R245ca,

293

R123, and R141b. The seven kinds of high-temperature working fluids are hexane,

294

heptane, octane, nonane, decane, cyclohexane and toluene. Including RC, there are

295

fourteen different systems in total. Since the turbine in ORC for engine waste heat

296

recovery is relatively small, the ratio between the pressures of turbine inlet and outlet

297

cannot be too large. Otherwise, the turbine is quite difficult and expensive to

298

manufacture [20]. Therefore, all the evaporating pressures of the systems are designed

299

as 2MPa and all the condensing pressures are 200kPa, making all the pressure ratio

300

10. Except RC, the superheat degrees of all ORC systems are designed as 10K. Since

301

water is wet working fluid, a great superheat degree 150K is selected. The lowest

302

exhaust temperature at the outlet of heat exchanger is set as 433K for all the systems.

303

The geometrical parameters of the exchangers in all systems are the same, except the

304

heat transfer area.

305

4.1. Dynamic response of ORC with different working fluids

306

Fig. 7 and Fig. 8 describe the dynamic response of evaporating pressure (p) and

307

working fluid enthalpy at the end of heating (hout) under the disturbance of exhaust

308

inlet temperature and working fluid mass flow rate. In these two figures, all the

309

systems operate under the designed working condition steadily before 100th second. In

310

Fig. 6, the exhaust inlet temperature decreases by 5% at 100th second. In Fig. 7 the

ACCEPTED MANUSCRIPT 311

working fluid pump speed decreases by 3% at 100th second. From these figures, it can

312

be found that no matter under the disturbance of working fluid mass flow rate or

313

exhaust temperature, the ORC with low temperature working fluid needs less time to

314

settle down than ORC with high temperature working fluid. Therein, RC responds

315

much more slowly than the others. The dynamic response of different LT working

316

fluids or HT working fluids is also distinctive. In order to show their difference more

317

clearly, rise time, settling time and time constant which describe the dynamic

318

response speed are calculated based on the data in Fig. 7 and Fig. 8 as shown in Table

319

3 and 4. Rise time is the time required for the response to rise from x% to y% of its

320

final value. 0% to 100% rise time common for underdamped systems, 5% to 95% for

321

critically damped and 10% to 90% for overdamped systems [34]. Settling time is the

322

time required for the response curve to reach and stay within a range of certain

323

percentage (usually 5% or 2% and 2% is selected in the paper) of the final value [34].

324

The time constant is the time for the step response to reach 1-1/e≈63.2% of its final

325

value.

326 327

(a)

(b)

ACCEPTED MANUSCRIPT

328 329

(c)

(d)

330

Fig. 7. Dynamic response under the exhaust temperature decrease. (a) Enthalpy of LT working

331

fluids. (b) Evaporating pressure of LT working fluids. (c) Enthalpy of HT working fluids. (d)

332

Evaporating pressure of HT working fluids

333 334

(a)

(b)

335 336

(c)

(d)

ACCEPTED MANUSCRIPT 337

Fig. 8. Dynamic response under working fluid mass flow rate decrease. (a) Enthalpy of LT

338

working fluids. (b) Evaporating pressure of LT working fluids. (c) Enthalpy of HT working fluids.

339

(d) Evaporating pressure of HT working fluids

340

Table 3 indicates the rise time, settling time and time constant of working fluid

341

enthalpy at the end of heating and evaporating pressure under the disturbance of

342

exhaust temperature. Table 4 indicates those under the disturbance of working fluid

343

mass flow rate. It should be noticed that when the pump speed decreases by 3%, the

344

temperature of nonane increases into a value out of the calculation range of Refprop,

345

so the data of nonane is absence in Table 4. The different working fluids are arranged

346

in the order of settling time in these two tables and the setting times of enthalpy and

347

evaporating are in the same sequence. It can be found that in general the working fluid

348

with shorter settling time has the shorter rise time and time constant as well, except

349

for a few particular cases. Shorter settling time, rise time and time constant mean

350

faster response speed, so the working fluids are also arranged in the order of response

351

speed in these two tables. No matter under the disturbance of exhaust temperature or

352

working fluid mass flow rate, the response speeds of the systems with different

353

working fluids are in the same order and they are quite different. In fact, since the

354

turbine and pump response much faster than heat exchangers and the condenser can

355

be controlled, the various dynamic response of the ORC systems is mainly reflected

356

on the exhaust heat exchangers heated by exhaust directly [13]. Moreover, the

357

difference on the heat exchangers of various ORC systems depends on the physical

358

properties of different working fluids.

ACCEPTED MANUSCRIPT 359

Table 3. The rise time,settling time and time constant when exhaust temperature decreases Working

Enthalpy at the turbine inlet

Evaporating pressure

fluid

tr [s]

τ [s]

tr [s]

ts [s]

τ [s]

ts [s]

Low temperature working fluid R236ea

21.5

18.7

9.1

9.9

21.3

4.9

R114

22.5

19.5

9.4

9.9

23.5

5.0

R245fa

22.6

20.8

10.5

9.7

27.1

5.0

R245ca

22.6

34.3

11.6

9.8

29.7

5.3

R123

24.2

35.4

11.6

10.5

30.4

5.4

R141b

25.4

36.3

12.4

10.5

32.4

5.4

High temperature working fluid

360

Hexane

24.9

41.5

13.1

12.4

34.8

6.7

Heptane

29.2

48.9

15.7

16.1

41.4

8.5

Cyclohexane

31.9

59.1

17.8

14.0

47.5

9.1

Octane

36.7

61.3

20.1

23.1

52.5

11.8

Nonane

49.2

69.7

25.9

34.8

62.6

16.2

Toluene

42.1

80.1

24.3

17.6

62.6

10.4

Decane

104.5

83.1

40.0

89.8

76.4

27.2

Water

498

622

183

30.4

325.8

16.1

tr, ts, and τ are rise time, settling time and time constant.

361 362

Table 4. The rise time, settling time and time constant when working fluid mass flow rate

363

decreases Working

Enthalpy at the turbine inlet

Evaporating pressure

fluid

tr [s]

τ [s]

tr [s]

ts [s]

τ [s]

10.5

26.4

23.4

9.5

ts [s]

Low temperature working fluid R236ea

19.0

17.8

ACCEPTED MANUSCRIPT R114

19.1

17.9

10.5

27.1

24.1

9.9

R245fa

19.9

19.2

11.2

29.3

26.7

13.1

R245ca

19.9

33.8

11.4

29.5

28.1

16.8

R123

20.3

35.1

11.4

32.2

28.6

14.2

R141b

21.2

38.4

11.9

32.6

30.3

16.1

High temperature working fluid Hexane

19.1

41.3

11.6

32.1

49.3

18.7

Heptane

19.3

47.1

12.0

36.1

51.3

22.4

Cyclohexane

22.1

55.4

13.5

39.1

69.0

24.8

Octane

20.0

55.6

12.3

42.5

72.9

27.2

Nonane

/

/

/

/

/

/

Toluene

26.1

70.2

15.9

48.4

89.7

32.6

Decane

21.0

88.1

13.3

71.2

119.7

44.3

Water

537

695

198

673

763

297

364

tr, ts, and τ are rise time, settling time and time constant.

365

4.2. Effects of fluid physical property on dynamic response

366

Some researchers have developed math models to determine the time constant of

367

the counter-flow heat exchanger without phase change [35-38]. Although there is no

368

phase change in the math model, the analytical solution of the time constant is quite

369

complicated. So if there is an analytical solution of the dynamic model of the exhaust

370

heat exchanger in this paper, it must be more intricate. As a result, it is quite hard to

371

find an obvious regularity between the system dynamic response speed and the

372

working fluid physical property. However, there must be some relationships between

373

them. Table 5 indicates some important parameters in the exhaust heat exchanger and

374

Table 6 indicates some basic physical properties of the working fluids. Working fluids

375

in Table 5 are arranged in the same order with Table 3 and 4, while they are in the

ACCEPTED MANUSCRIPT 376

sequence of critical temperature in Table 6. According to the dynamic math model of

377

the exhaust heat exchanger, mass flow rate, heat capacity, evaporating latent heat,

378

evaporating temperature, condensing temperature, density and heat transfer

379

coefficients are very important to the dynamic behavior and they are greatly decided

380

by the properties of working fluids. Therefore, the effects of them are analyzed below,

381

respectively.

382

From Table 5 it can be found that the working fluid mass flow rate has

383

relationships with the response speed. In the general trend, the working fluid with

384

smaller mass flow rate usually responds more slowly. HT working fluids have smaller

385

mass flow rate than LT working fluids and they also respond more slowly in general.

386

Especially, the RC mass flow rate is much less than the others and its response speed

387

is much slower than the others as well. When the state of the exhaust heat exchanger

388

becomes stable, the working fluid stored in the heat exchanger must be stable state as

389

well. If there is a lot of working fluid stored in the heat exchanger, the thermal inertial

390

will be great. Large mass flow rate contributes to the mass exchange, which

391

accelerates the update of working fluid in the heat exchanger, so the system can

392

become steady more quickly. The research in [37] also proved the contribution of

393

mass flow rate to the dynamic response speed. However, there are a few special cases

394

which broke the regularity. The dynamic response behavior is decided by a lot of

395

factors instead of any one only. Therefore, the regularity between mass flow rate and

396

response speed mentioned above is not obeyed all the time.

397

The mass flow rate is decided by the flow equation:

ACCEPTED MANUSCRIPT 398 399 400 401 402 403 404 405 406 407 408 409 410 411 412 413 414

m f  Q / (hin  hout )

(20)

Therein, Q is absorbed heat; hin and hout is the enthalpy of working fluid at the inlet and outlet of the exhaust heat exchanger. Since all of the Ranking Cycles have the same absorbed heat, the mass flow rate is decided by the heat capacity and evaporating latent heat. The larger heat capacity and evaporating latent heat the working fluid has, the less mass flow rate it has. It can be seen that water has the largest heat capacity and evaporating latent heat, so its mass flow rate is the smallest. Although some of the HT working fluids such as decane have smaller evaporating latent heat than low temperature working fluids, their heat capacities are greater. As a result, they still have larger mass flow rate. It seems that there is no obvious regularity between evaporating latent heat and response speed, while in the general trend the working fluid with slower dynamic response speed often has greater heat capacity. Large heat capacity not only contributes to decreasing working fluid mass flow rate, but also leads to great thermal inertial because the working fluid needs more energy to change temperature, which does harm to fast response speed. There are some special cases in the regular between heat capacity and response speed, it is the same reason that heat capacity is not the only factor effects dynamic response speed.

415

The evaporating temperature also effects dynamic response. For the LT working

416

fluid, the more slowly it responds, the higher evaporating temperature it has. HT

417

working fluids also obey the regularity except toluene, cyclohexane and water. High

418

evaporating temperature leads to small average temperature difference, which is

419

conductive to increasing heat exchange area. Larger heat transfer area means larger

ACCEPTED MANUSCRIPT 420

volume of heat exchanger, so there is more working fluid in the heat exchanger and

421

the thermal inertia is greater as well, leading to slower response speed. Besides, the

422

heat exchanger wall can store energy and postpone the heat transfer between fluids, so

423

small heat exchanger helps the system settle down more quickly. In Table 5 the

424

working fluid with higher evaporating temperature usually has higher condensing

425

temperature. It is the same reason that high condensing temperature is conducive to

426

reducing the heat transfer temperature difference and increasing the heat exchange

427

area. In general trend, the larger heat transfer area the system has, the more slowly it

428

responds as shown in Table 5, but there are still some particular cases, especially

429

water. Furthermore, as shown in Table 6 the working fluids with higher critical

430

temperature usually have higher evaporating and condensing temperature at the same

431

pressure. Consequently, high critical temperature goes against quick response speed.

432

Density is very important to thermal inertial. Large density leads to large mass in

433

the same volume, so more energy is needed to change the thermodynamic state, which

434

goes against fast response speed. However, although LT working fluids have larger

435

density than HT working fluids, LT working fluids respond faster. This is because

436

other factors such as evaporating temperature, mass flow rate, heat capacity take the

437

dominant role. Comparing R141b and hexane, it can be found that although R141 has

438

larger mass flow rate, smaller heat capacity, lower evaporating temperature and

439

smaller heat transfer area than hexane, its density is much greater than that of hexane.

440

As a result, their response speeds are nearly the same and the time constant of hexane

441

is even a little smaller.

ACCEPTED MANUSCRIPT 442

The heat transfer coefficient also has effects on dynamic response speed. The small

443

heat transfer coefficient goes against reducing heat transfer area as well as the volume

444

of heat exchanger. In the math models of the paper, all the exhaust heat exchangers

445

have the same geometrical parameters, except the heat transfer area, so the heat

446

transfer coefficients on the exhaust side are the same. It is 650W/m2K (equivalent heat

447

transfer coefficient with fins). Heat transfer coefficients in working fluid side are

448

shown in Table 6. Despite there is no strict regularity among them, they trend to

449

become small with slowing response speed. This is mainly because the mass flow rate

450

of working fluid becomes small, which does harm to increasing the flow speed and

451

the turbulence in the working fluid side. In fact, the whole heat transfer coefficient is

452

effected greatly by the side with weaker heat transfer ability and the heat transfer

453

coefficients on the exhaust side are much smaller than those in working fluid side. As

454

a result, the whole heat transfer coefficients of all the cycles are similar and they do

455

not have much effects on the difference of dynamic response speed of different

456

working fluids.

457

Table 5. Some important parameters in the exhaust heat exchanger Working fluid

mf [kg/s]

pe/pc [K]

k1/k2/k3 (W/m2K)

L [m]

R236ea

3.2864

384.8/297.3

1807/5437/1844

70.8

R114

3.7293

391.0/296.0

1670/5095/1787

76.6

R245fa

3.0057

395.0/306.5

1802/5929/1787

79.2

R245ca

2.8513

409.9/317.4

1659/5539/1808

84.5

R123

3.1475

420.4/321.2

1540/5051/1415

90.4

R141b

2.4378

429.3/326.1

1528/5378/1471

94.8

Hexane

1.2975

479.8/365.2

1476/4325/1877

123.9

ACCEPTED MANUSCRIPT Heptane

1.2703

518.2/396.5

1490/4085/1969

159.4

Cyclohexane

1.2683

499.6/378.4

1143/3858/1844

159.1

Octane

1.2374

553.3/425.1

1304/3341/2246

225.6

Nonane

1.2142

585.1/451.6

1330/3149/2411

320.2

Toluene

1.2918

535.8/409.0

1157/4013/1563

236.2

Decane

1.1987

614.1/476.1

1293/2692/2467

517.6

Water

0.2505

485/393.4

1250/9459/399

170.6

458

mf is the mass flow rate of working fluid. pe and pc is evaporating pressure and condensing

459

pressure. k1, k2, k3 are the heat transfer coefficients in working fluid side of sub-cooling region,

460

two phase region and superheating region, respectively. L is the total length of the pipe.

461 462

Table 6. The physical properties of working fluids Tcri

pcri

qlalent

Cp (saturated liquid

ρ(saturated liquid

[K]

[MPa]

[kJ]

at pe/pc) (kJ/kgK)

at pe/pc) (kg/m3)

R236ea

398.07

3.20

79.5

1.25/1.66

1427/1061

R114

418.8

3.26

72.2

0.988/1.37

1461/1067

R245fa

427.2

3.65

109

1.34/1.80

1315/989

R245ca

447.6

3.93

113

1.37/1.83

1336/1006

R123

456.3

3.66

96.9

1.05/1.38

1403/1052

R141b

477.5

4.21

137

1.19/1.56

1178/908

Hexane

507.8

3.03

166

2.53/3.75

589/422

Heptane

540.2

2.73

136

2.68/4.19

589/402

Cyclohexane

553.6

4.08

203

2.27/3.15

694/532

Octane

569.3

2.50

113

2.77/4.69

586/378

Toluene

591.8

4.13

214

2.10/2.82

752/576

Nonane

594.6

2.28

88.3

2.86/5.51

582/358

Decane

617.7

2.10

52.2

2.94/10.15

576/310

Water

647.1

22.06

1890

4.56/4.57

942/845

Working fluid

ACCEPTED MANUSCRIPT 463

Tcri and pcri is the critical temperature. qlalent is latent heat. Cp is heat capacity.ρis density.

464 465

The dynamic response of ORC systems with different working fluids depends on a

466

lot of factors comprehensively, so there are always some special cases in the

467

regularities mentioned above. However, for the working fluids of hexane, heptane,

468

octane, nonane and decane which belong to the straight-chain alkanes with 6 to 10

469

carbon atoms, the response speed becomes slow with increasing critical temperature.

470

The properties of straight-chain alkanes change very regularly with critical

471

temperature [20]. The higher critical temperature it has, the higher evaporating

472

temperature, higher condensing temperature, smaller mass flow rate and greater heat

473

capacity it gets. Owing to these regular variation, their dynamic response speed

474

changes quite regularly as well. Therefore, the critical temperature can be applied to

475

estimate the response speed of ORC with different straight-chain alkanes. Whereas, it

476

is not very exact for wider range of working fluids. For example, the critical

477

temperature of cyclohexane is a litter lower than that of octane, while its time constant

478

and rise time are little greater than those of octane. Because cyclohexane belongs to

479

cycloalkanes and it does not obey the property change regularity of straight-chain

480

alkanes. For the LT working fluids, they belong to different kinds of

481

halohydrocarbons and there seems to be no obvious regularity of the property

482

variation. Although the working fluid with higher critical temperature trends to

483

respond slowly, some working fluids have quite different critical temperature, while

484

their response speeds are so similar such as R114 and R236ea. Consequently, it is

ACCEPTED MANUSCRIPT 485

extremely hard to use one feature of working fluids like critical temperature to predict

486

their response speeds.

487

In a word, the dynamic response speed of ORC systems with different working

488

fluids is decided by many factors comprehensively. High working fluid mass flow

489

rate, small heat capacity, small evaporating latent heat, low critical temperature and

490

low density contributes to increasing dynamic response speed. In general the

491

properties of LT working fluids are more beneficial to fast response speed than HT

492

working fluids, so LT working fluids respond faster. RC has much larger heat

493

capacity and evaporating latent heat, so it response much more slowly.

494 495

Conclusion

496

The dynamic math models of ORC systems with 14 different working fluids

497

(including water) as the waste heat recovery system of a natural gas engine are

498

established by Simulink in this study. They are designed with the same evaporating

499

pressure and condensing pressure. Based on these, their dynamic response characters

500

reflected by rise time, settling time and time constant are compared and analyzed. The

501

results indicate that no matter under the disturbance of working fluid mass flow rate

502

or exhaust temperature, the ORC with LT working fluid usually responds faster than

503

the ORC with HT working fluid. Therein, RC responds much more slowly than the

504

others.

505

The dynamic response speed of ORC systems with different working fluids

506

depends on a lot of factors comprehensively. High working fluid mass flow rate,

ACCEPTED MANUSCRIPT 507

small heat capacity, small evaporating latent heat, low critical temperature and low

508

density are conductive to increasing dynamic response speed. It is difficult for a kind

509

of working fluid to get all the characters at the same time. Among different working

510

fluids, these characters are hard to all change at the direction beneficial to fast

511

response speed with one feature like critical temperature. So it is extremely difficult to

512

use one feature of the working fluid to predict their response speed. However, for

513

some working fluids of the same type such as straight-chain alkanes, critical

514

temperature can be used to compare their dynamic response speed. The higher critical

515

temperature the straight-chain alkane has, the faster it responds. Besides, these

516

characters are indicators to estimate the dynamic response speed. In general LT

517

working fluids have more properties beneficial to fast response speed than HT

518

working fluids, so LT working fluids respond more quickly.

519

The ORC with various working fluids for waste heat recovery of ICEs have

520

different dynamic performance and they are suitable for different working condition

521

characteristics of ICEs. LT working fluids are more suitable for automotive engines

522

which often operate under working conditions varying frequently because of their fast

523

response speed. By contrast, HT working fluids are more suitable to be applied in

524

engines which often operate relatively steadily, for example ship engines and engine-

525

generators.

526

ACCEPTED MANUSCRIPT 527

Acknowledgements This work was supported by a grant from the National Natural Science Foundation

528 529

of China (No. 51676113).

530 Nomenclature T

Temperature (K)

ρ α Cp m A t D h Re Nu Pr γ S μ u L p x ηv Vcyl ω Cv ηst

Density (kg/m3) Heat transfer coefficient (W/m2·K) Specific heat (J/kg·K) Mass flow rate (kg/s) Area (m2) Time (s) Diameter (m) Specific enthalpy (J/kg) Reynolds number Nusselt number Prandtl number void fraction (m2/s) Slip ratio Density ratio Velocity (m/s) Length (m) Pressure (Pa) Vapor quality Volumetric efficiency Cylinder volume (m3) Pump speed (rpm) Turbine coefficient Isentropic efficiency of expander

ηsp η Subscripts l g e i o w in out r avg p t

Isentropic efficiency of pump Dynamic viscosity (Pa·s) or liquid fraction or efficiency Liquid Gas Heat source (exhaust) Inside Outside Wall Inlet Outlet Working fluid Average Pump Turbine

Abbreviations ORC RC MB WHRS ICE

Organic Rankine Cycle Rankine Cycle Moving Boundary Waste Heat Recovery System Internal Combustion Engine

HT LT

High Temperature Low Temperature

531 532

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[26] Luong, D.; Tsao, T.-C. Linear quadratic integral control of an Organic Rankine Cycle for waste

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heat recovery in heavy-duty diesel powertrain. In Proceedings of the 2014 American Control

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Conference (ACC), Portland, OR, USA, 4–6 June 2014.

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Flows. The second International Modelica Conference, Germany, March 18-19, 2002.

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entropy production. J Heat Trans, 86 (1964) 247–52.

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triangular pitch banks of finned tubes. Chem.Eng.Prog.Symp.Ser. 59 (1963) 1-10.

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Heidelberg New York: Springer; 1997 [German].

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ACCEPTED MANUSCRIPT 606

units during transient operation. Applied Energy, 151 (2015) 119–131.

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Based Approach. 52nd IEEE Conference on Decision and Control, December 10-13, 2013. Florence,

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8570-9.

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3721-3730.

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Transf. 121 (1999) 746-748.

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cross-flow heat exchangers under variable fluid mass flow rate for data center cooling applications.

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[38] Y. Xuan, W. Roetzel, Dynamics of shell-and-tube heat exchangers to arbitrary temperature and

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step flow variations. AIChE J., 39 (3) (1993) 413-421.

623 624 625

Appendix

626

Mass balance for the subcooling region of working fluid: A  1  l 

627

  dL1  AL1  1  p dt 

 h1

1 1 2 h1

p

dhl dp

 dp 1    AL1 1  dt 2 h1 

Energy balance for the subcooling region of working fluid:

p

dhin  m in  m 12 dt

(1)

ACCEPTED MANUSCRIPT  1 dh   1 1 dhl AL1  1 l  h1  1   p h 2 h1 dp  2 dp p 1    m in hin  m l hl  i1Di L1 Tw1  Tr1 

628

dTw1   i1Di Tr1  Tw1    o Do Te1  Tw1  dt

(3)

 d d   dp dL1 dL  A 1     l   g  2  AL2   g  1    l   m 12  m 23 dt dt dp  dt  dp

 d h  dp d h dL1 dL  A 1     l hl   g hg  2  AL2   g g  1    l l  1 dt dt dp  dp  dt  m 12 hl  m 23 hg  i 2 Di Tw 2  Tr 2  L2

(5)

Energy balance for the wall in the two-phase region: c pww Aw

632

(1)

Energy balance for the two-phase region of working fluid:

A  l hl   g hg 

631

(2)

Mass balance for the two-phase region of working fluid: A  l   g 

630

 dh  in  dt p

Energy balance for the wall in the subcooling region: c pww Aw

629

  dp  dL 1    1  A  1h1  l hl  1  AL1  1  h1 1   dt  dt 2 h1   

dTw 2   i 2 Di Tr 2  Tw 2    o Do Te 2  Tw 2  dt

(6)

Mass balance for the superheating region of working fluid: A   g  3 

 dL1 dL 1  A   g  3  2  AL3 3 dt dt 2 h31

p

  dhout  AL3  3  p dt 

 h31

1 3 2 h3

p

dhg  dp  dp  dt 

(7)

 m 23  m out 633

Energy balance for the superheating region of working fluid:

 1 dhg  dp 1    1  dhg AL3  3  h3 3  h3 3  1  AL3  3  h3 3  2 dp  dt 2  p h31 2 h3 p dp h31     dL dL   A   g hg  3 h3   1  2   m 23 hg  m out hout   i 3 Di Tw3  Tr 3  L3 dt   dt

 dh  out  dt p

(8)

ACCEPTED MANUSCRIPT

634

Energy balance for the wall in the superheating region: c pww Aw

635

 dL1  Ae L1 e1 dt pe

he1

dpe 1   Ae L1 e1 dt 2 he1

pe

dheout 1   Ae L1 e1 dt 2 he1

pe

dhe12  m e12  m eout dt

  dp 1    dL1  Ae  L1he1 e1  L1  e  Ae  e1 Le1  he1 Le1 e1   dt 2  dt pe h he1 e1     m e12 he12  m eout heout   0 Do L1 Te1  Tw1 

  dh dh   eout  e12   dt dt  pe  

(11)

Mass balance for the second region of exhaust corresponding to two-phase region:

Ae  e12  e 23   1  Ae L2 e 2 2 he 2 638

(10)

Energy balance for the first region of exhaust corresponding to subcooling region: Ae  e1he1  e12 he12 

637

(9)

Mass balance for the first region of exhaust corresponding to subcooling region: Ae  e1  e12 

636

dTw3   i 3 Di Tr 3  Tw3    o Do Te3  Tw3  dt

 dL1 dL  Ae  e 2  e 23  2  Ae L2 e 2 dt dt pe

pe

dhe12 1   Ae L2 e 2 dt 2 he 2

pe

he 2

dpe dt

dhe 23  m e 23  m e12 dt

(12)

Energy balance for the second region of exhaust corresponding to two-phase region:   dp  dL1 dL  Ae  e 2 he 2  e 23 he 23  2  Ae  L2 he 2 e 2  L2  e   dt dt dt pe h e2     dh  dh  1   Ae  e 2 L2  he 2 L2 e 2   e12  e 23   m e 23 he 23  m e12 he12   0 Do L2 Te 2  Tw 2    2 he 2 p  dt dt  e   Ae  e12 he12  e 23 he 23 

639

(13)

Mass balance for the third region of exhaust corresponding to the superheating region:

ACCEPTED MANUSCRIPT   dL dL  Ae  e 23  e3   1  2   Ae L3 e3 dt  pe  dt  1  Ae L3 e3 2 he3

pe

dhe 23 1   Ae L3 e3 dt 2 he3

pe

he 3

dpe dt

dhein  m ein  m e 23 dt

(14)

640

Energy balance for the third region of exhaust corresponding to the superheating

641

region:    dL dL  Ae  e 23 he 23  e 3 he 3   1  2   Ae  L3 he 3 e 3  dt  pe  dt   1  Ae  e 3 L3  he 3 L3 e 3  2 he 3 

642

h3

 dp  L3  e   dt 

  dh dh   e 23  ein   m ein hein  m e 23 he 23   0 Do L3 Te 3  Tw3   dt dt  pe  

(15)