revista mexicana de ngeniería - scielo méxico · 2019-09-30 · technology for refrigeration and...

Revista Mexicana de Ingeniería Química

CONTENIDO

Volumen 8, número 3, 2009 / Volume 8, number 3, 2009

213 Derivation and application of the Stefan-Maxwell equations

(Desarrollo y aplicación de las ecuaciones de Stefan-Maxwell)

Stephen Whitaker

Biotecnología / Biotechnology

245 Modelado de la biodegradación en biorreactores de lodos de hidrocarburos totales del petróleo

intemperizados en suelos y sedimentos

(Biodegradation modeling of sludge bioreactors of total petroleum hydrocarbons weathering in soil

and sediments)

S.A. Medina-Moreno, S. Huerta-Ochoa, C.A. Lucho-Constantino, L. Aguilera-Vázquez, A. Jiménez-

González y M. Gutiérrez-Rojas

259 Crecimiento, sobrevivencia y adaptación de Bifidobacterium infantis a condiciones ácidas

(Growth, survival and adaptation of Bifidobacterium infantis to acidic conditions)

L. Mayorga-Reyes, P. Bustamante-Camilo, A. Gutiérrez-Nava, E. Barranco-Florido y A. Azaola-

Espinosa

265 Statistical approach to optimization of ethanol fermentation by Saccharomyces cerevisiae in the

presence of Valfor® zeolite NaA

(Optimización estadística de la fermentación etanólica de Saccharomyces cerevisiae en presencia de

zeolita Valfor® zeolite NaA)

G. Inei-Shizukawa, H. A. Velasco-Bedrán, G. F. Gutiérrez-López and H. Hernández-Sánchez

Ingeniería de procesos / Process engineering

271 Localización de una planta industrial: Revisión crítica y adecuación de los criterios empleados en

esta decisión

(Plant site selection: Critical review and adequation criteria used in this decision)

J.R. Medina, R.L. Romero y G.A. Pérez

Revista Mexicanade Ingenierıa Quımica

1

Academia Mexicana de Investigacion y Docencia en Ingenierıa Quımica, A.C.

Volumen 14, Numero 1, Abril 2015

ISSN 1665-2738

1Vol. 14, No. 1 (2015) 213-229

THERMOPHYSICAL PROPERTIES OF R744 IN SUPERCRITICAL REGIONDURING THE STARTUP OF GAS COOLING PROCESS

PROPIEDADES TERMOFISICAS DEL R744 EN LA REGION SUPERCRITICADURANTE EL PROCESO DEL ARRANQUE DEL GAS COOLER

J.F. Ituna-Yudonago and J.M. Belman-Flores∗

Department of Mechanical Engineering, Engineering Division, Campus Irapuato-Salamanca, University ofGuanajuato, Salamanca, Gto., Mexico.

Received April 19, 2014; Accepted february 23, 2015

AbstractIn this paper, the thermophysical properties of carbon dioxide (R744) in supercritical region during the startup of gas coolingprocess were widely investigated in a horizontal hairpin gas cooler. The dynamic model was discretized by the fully implicitfinite volume method. The thermophysical properties were evaluated through the REFPROP dynamic libraries linked toMatlab. The results showed that, the thermophysical properties of R744 are characterized by large variations along the gascooler during the transient period. In the pseudo-critical region, the behavior of the specific heat, thermal conductivity,density and dynamic viscosity is characterized by an instantaneous change in the amplitude. The specific heat is theproperty most affected by the transient phenomenon in the gas cooler. Its behavior has a great impact on the local heattransfer coefficient of R744.Keywords: gas cooler, carbon dioxide, thermophysical properties, transient phenomenon, transcritical system.

ResumenEn este trabajo, se investigan ampliamente las propiedades termofısicas del dioxido de carbono (R744) en la regionsupercrıtica durante el arranque del proceso de enfriamiento de gas, en un enfriador horizontal de tipo horquilla. Elmodelo dinamico fue discretizado mediante el metodo de volumenes finitos completamente implıcito. Las propiedadestermofısicas se evaluaron a traves de las bibliotecas dinamicas REFPROP vinculadas a Matlab. Los resultados mostraronque, las propiedades termofısicas del R744 se caracterizan por grandes variaciones a lo largo del enfriador de gas duranteel perıodo transitorio. En la region pseudo-crıtica, el comportamiento del calor especıfico, conductividad termica, densidady viscosidad dinamica se caracteriza por un cambio instantaneo en la amplitud. El calor especıfico es la propiedad masafectada por el fenomeno transitorio en el enfriador de gas. Su comportamiento tiene un gran impacto en el coeficiente detransferencia de calor local del R744.Palabras clave: enfriador de gas, dioxido de carbono, propiedades termofısicas, fenomeno transitorio, sistema transcrıtico.

1 Introduction

Vapor compression system, is the most widely usedtechnology for refrigeration and air conditioning.Its popularity is due to its high coefficient ofperformance (COP), compactness, and loweroperating costs. Despite these advantages, themain drawbacks of vapor compression systemsare the high energy consumption and the leakageof refrigerants, belonging to chlorofluorocarbons

(CFCs), hydrochlorofluorocarbons (HCFCs) whichhave greater Ozone Depletion Potential (ODP) andGlobal Warming Potential (GWP) (Lugo-Leyte etal., 2013; Belman-Flores et al., 2014). So, facedwith this precarious situation and according to theKyoto (Woerdman, 2000) and Montreal protocol(Dull et al., 1989), scientists began to look for newalternatives. However, several studies have focusedon the use of refrigerants which belong to the familyof hydrocarbons (Granryd, 2001), hydrofluoroolefins

∗Corresponding author. E-mail: : [email protected]

Publicado por la Academia Mexicana de Investigacion y Docencia en Ingenierıa Quımica A.C. 213

Ituna-Yudonago and Belman-Flores/ Revista Mexicana de Ingenierıa Quımica Vol. 14, No. 1 (2015) 213-229

(Tanaka and Higashi, 2010), blends (Neil and Owen,2004), and natural refrigerants (Bolaji and Huan,2013) in which there are: carbon dioxide (ASHRAE,2009), ammonia (Bjorn, 2008), water (Lachner,2007) and air (Butler et al., 2001). Unlike all thesenew alternatives, carbon dioxide has a remarkableadvantage because it is nontoxic, nonflammable,and no harmful environmental impact. Despite theadvantages offered by the carbon dioxide, its maininconveniences are the low critical temperature (31.06°C) and the high critical pressure (7.38 MPa). Witha lower critical temperature, the vapor compressionsystem operating with the R744 tends to have ahigher volumetric capacity for a given application;therefore, leads to a decrease in the COP (Sarkarand Agrawal, 2010). Furthermore, the high criticalpressure causes several effects that influence in thedesign of the components, particularly the design ofcompressors and heat exchangers. Also, it can createsecurity problems in the cycle components (Kim et al.,2004). It should be noted that the vapor compressionsystem operating with carbon dioxide in transcriticalconditions is most concerned by the instability in thethermophysical properties. This instability occursmore in the supercritical region, and precisely inthe heat exchanger called gas cooler. In this heatexchanger, a little variation in temperature or pressurecan produce a large change in the thermophysicalproperties, and then leads to significant variations inboth heat transfer and fluid flow behaviors. Thesevariations are more meaningful when the R744temperature is near the critical point (Peixue et al.,2008; Sanchez et al., 2012).

In this way, several investigations on the behaviorof the thermophysical properties of R744, the heattransfer coefficient and pressure drop during gascooling process began to be conducted on differenttypes of gas cooler such as: horizontal single in-tube (Son and Park, 2006), horizontal macro, mini ormicro channels (Huai et al., 2005; Yun et al., 2007),vertical multi-port tube or mini-channels (Liao andZhao, 2002; Duffey and Pioro, 2005; Yoon-Yeong andHwan-Yeol, 2009). Other works have been focusedon bundles and finned tubes technology (Zhao andOhadi, 2004; Ge and Cropper, 2009). In general, thesestudies were focused on the impacts of variations inthe operating pressure and/or mass flow of R744, aswell as in the diameter of tube wherein the R744 flows,on the heat transfer coefficient and pressure drop.Son and Park (2006) investigated on heat transferand pressure drop characteristics of R744 during gascooling process in a horizontal tube. Results of this

study showed that the local heat transfer coefficientdecreases when the gas cooling pressure increasesduring the process. However, in the middle stage ofgas cooling process where the refrigerant temperatureis close to the critical temperature, the heat transfercoefficient changes much with respect to the pressurevariation. But, for constant inlet pressure of gascooler, the local heat transfer coefficient increases withrespect to the mass flux. Similar results were alsoobtained by Liao and Zhao (2002). As emphasizedby Huai et al., (2005), the mass flux has a significantinfluence on the pressure drop at a fixed operatingpressure, the pressure drop increases with the massflux. But, for a fixed mass flux, the pressure dropdecreases with the operating pressure because thevariation in the thermophysical properties of R744becomes smaller as the operating pressure increases.

Some research (Kim et al., 2007; Dang andHihara, 2010; Hoo-Kyu and Chang-Hyo, 2010; Liet al., 2011; Fang and Xu, 2011) have focusedin the development of new correlations to calculateheat transfer coefficient, given that there were someinconsistencies between theoretical results based onsome existing correlations and experimental results.Others have addressed the dynamic analysis of gascooler. These studies were specially carried out onthe steady state behaviors of thermophysical propertiesand heat transfer coefficient. Sanchez et al., (2012)developed and validated a stationary model of a water-R744 coaxial gas cooler. This model based onthe finite volume method, was designed to analyzethe effect of the number of finite volume on themaximum error of the enthalpy and heat capacityof the gas cooler, and the influence of the variablesthat determine the gas cooler thermal effectiveness.Despite the interesting results obtained in steady state,no information about the transient state of the analyzedproperties has been given. Ge and Cropper (2009)used the distributed method for developing the steadystate model of finned-tube air-cooled R744 gas cooler,in order to predict accurately the great variation ofboth refrigerant thermophysical properties and localheat transfer coefficients during R744 gas coolingprocess. The model has been validated with the testresults from published literature by comparing the gastemperature profiles along the coil pipes from inletto outlet at different operating conditions. Resultsshowed that the temperature and the heat capacity areboth improved with the increase of heat exchangercircuit numbers. Peixue et al., (2008) investigated onthe method to obtain the local heat transfer coefficientsof carbon dioxide at supercritical pressures in vertical

214 www.rmiq.org


multi-port mini-channels and in a vertical small tubeunder cooling process. Steady state behavior of heattransfer coefficient along the gas cooler was carefullyanalyzed. The experimental results showed that thesharply variable thermophysical properties of carbondioxide have a significant effect on the local heattransfer coefficient which varies significantly along thevertical mini-channels and small tube when the R744bulk temperatures were near the critical region.

By analyzing the contents of different workreferenced above, it becomes clear that there isno work in which behaviors of all thermophysicalproperties are widely described. Most of analysesare limited in describing the behavior of properties orheat transfer coefficient depending on the temperatureof R744, in order to know how they behave withrespect to the critical or pseudo-critical temperature,without showing how each property behaves alongthe gas cooler during the process. In addition, noresearch has focused on the analysis of the transientbehavior of R744 properties, while it well known to allthat transcritical systems are very sensitive to changesin operating conditions, and the gas cooler is thecomponent in which temperature and pressure of therefrigerant are independent (Kim et al., 2004). This

paper provides detailed information on the behaviorof thermophysical properties of R744 during the gascooling process, especially in transient regime. Andalso investigates on the impact of changes in thesethermophysical properties on the local heat transfercoefficient during the transient period, with emphasison the transient behavior of the Prandtl and Nusseltnumbers, in order to have a better understanding ofR744 behavior during the startup and steady operationof the gas cooler.

2 Description of the gas coolerThe gas cooler which is studied in this paper is ahorizontal hairpin heat exchanger as shown in Fig.1a. It is designed for an experimental facility whichis under construction in the Engineering Division ofthe University of Guanajuato. This heat exchangeris constituted of nineteen inner tubes (see Fig. 1 b)in which the refrigerant (R744) flows. These tubesare assembled in an annulus (see Fig. 1c) causinga counter-flow arrangement with the secondary fluid(water). The characteristics of gas cooler are describedin Table 1, where the material of each component isspecified along with dimensions.

a) Top view of the gas cooler

b) Inlet section of the refrigerant c) Inlet section of the secondary fluid and profile view of the inner tubes

Fig. 1. Experimental gas cooler.

B A

a) Top view of the gas cooler

b) Inlet section of the refrigerant c) Inlet section of the secondary fluid and profile view of the inner tubes


B A

a) Top view of the gas cooler b) Inlet section of therefrigerant

c) Inlet section of thesecondary fluid and profileview of the inner tubes


Table 1. Characteristics of the gas cooler.

Item Material Parameter Magnitude

Inner tube Stainless steel Inner diameter (mm) 3.75Outer diameter (mm) 5.0

Number of tubes 19Annulus Stainless steel Inner diameter(mm) 30.0

Outer diameter (mm) 33.7Radius of the bend (mm) 112.0

Length (mm) 2000.0

www.rmiq.org 215


3 Mathematical modelingThis section presents the mathematical formulationbased on the mass, momentum, and energyconservation equations for both fluids circulating inthe gas cooler. Each stream is discretized by the fullyimplicit finite volume method (FVM). The choiceof this method is justified by the precision in theresulting solutions which are unconditionally stablein time and space over any group of control volumesand, of course, over the whole calculation domain(Patankar, 1980). This method is completed by thefirst order upwind differencing scheme for calculatingfluxes through cell faces, because it enables thecoefficients of the discretized equations being alwayspositive, and therefore makes the solution physicallyrealistic (Cariaga et al., 2013). In order to reduce thecomplexity in the analysis, some considerations aremade as follows:

• Refrigerant and secondary fluid flows aremodeled as a one-dimensional fluid flow.

• Axial heat conduction, viscous stress in fluidand heat radiation are negligible.

• The gas cooler is considered isolated. So all theheat removed from the refrigerant is transferredto the water.

• The cross sections of all inner tubes are constant

3.1 Refrigerant flow in the bare inner tube

The transient model of the refrigerant flow inside tubesis governed by the mass, momentum, and energyconservation equations. The first two equations areused to assess the velocity and pressure fields in theaxial direction, while the third and fourth equationsare used to evaluate the distribution of the enthalpy andwall temperature of the inner tube.

Mass conservation:∂(ρA)r∂t

+∂(ρuA)r∂z

= 0 (1)

Momentum conservation:Ar∂(ρu)r

∂t+

(ρuA)r∂ur

∂z= −

Ar∂Pr

∂z− τz pi (2)

Energy conservation of refrigerant:

∂(ρAh−AP)r

∂t+∂(mh)r

∂z= piαi (Tr −Tti) (3)

Wall energy conservation

ρACpti∂Tti

∂t= piαi (Tr −Tti) + poαo

(Ts f −Tti

)(4)

The discretization of these equations is performedby considering that each inner tube is divided intoseveral control volumes as shown in Fig. 2a where Pis the node of interest, W and E represent neighbornodes to the west and east; w and e are the westand east faces of the control volume with uW and uebeing their corresponding velocity. Furthermore, theboundary nodes are represented by A at the inlet andB at the outlet. This representation is designed for thediscretization of the conservation equations of massand energy, while the momentum equation which hasthe velocity as the main variable is discretized on thestaggered control volumes as represented in Fig. 2b,where PS , WS and ES are the staggered nodes withuwS , uPS and ueS their corresponding velocities. Inaddition, wS and eS are the west and east faces of thestaggered control volume.

However, algebraic equations resulting from thediscretization are presented in Eqs. (5-8). The firsttwo algebraic equation systems (Eqs. 5-6) are usedto determine the pressure and velocity fields by usingthe SIMPLE method (Patankar, 1980), while the twolatter (Eqs. 7-8) are used to determine the refrigerantenthalpy and the wall temperature field. Thus, otherthermophysical properties are calculated according tothe pressure and enthalpy values in each node throughthe REFPROP dynamic libraries linked to Matlab(Lemmon et al., 2007). Details of this process aregiven in the global algorithm of resolution. Parametersby which the coefficients of each matrix are computedare described in Tables 2, 3, 4 and 5.

aPs u∗Ps = aEs u∗Es + aW su∗W s + b∗Ps (5)

a) Main control volumes

b) Staggered control volumes

Fig. 2. Schematic representation of control volumes of the gas cooler.

∆zw e

∆z

euwuW P EA B

∆z

∆z

sW sEsPseuswu sPu

sw se

W P E

A B

Fig. 2. Schematic representation of control volumesof the gas cooler.

216 www.rmiq.org


Table 2. Coefficients of the algebraic equation of momentum conservation for refrigerant side.

Table 1. Characteristics of the gas cooler. Item Material Parameter Magnitude

Inner tube Stainless steel Inner diameter (mm) 3.75 Outer diameter (mm) 5.0 Number of tubes 19

Annulus Stainless steel Inner diameter(mm) 30.0 Outer diameter (mm) 33.7 Radius of the bend (mm) 112.0

Length (mm) 2000.0

Table 2. Coefficients of the algebraic equation of momentum conservation for refrigerant side.

Node spa sW

a sEa sP

b

1 ( ) ( )

tzA

,mmax,mmax

ss

s

P,r*P,r

*e,rA,r

ΔΔ+

+−

ρ

00 !!

0 ( )0,mmax *e,r s!

( )

zp

APPtzu

rz

p,r*E

*PPP,r sss

Δ−

⎥⎦⎤

⎢⎣⎡ −+

ΔΔ

τ

ρ ""

( )12

−zn,...,

( ) ( )

tzA

,mmax,mmax

ss

ss

P,rP,r

*e,r

*w,r

ΔΔ+

+

°ρ

00 !!

( )0,mmax *w,r s!

( )0,mmax *e,r s!

( )

zp

APPtzu

rz

p,r*E

*PPP,r sss

Δ−

⎥⎦⎤

⎢⎣⎡ −+

ΔΔ

τ

ρ ""

zn ( ) ( )

tzA

,mmax,mmax

ss

ss

P,rP,r

*e,r

*w,r

ΔΔ+

+

°ρ

00 !!

( )0,mmax *w,r s!

0 ( )zp

APPtzu

rz

p,r*P

*WPP,r sss

Δ−

⎥⎦⎤

⎢⎣⎡ −+

ΔΔ

τ

ρ ""

Table 3. Coefficients of the algebraic equation of pressure correction for refrigerant side.


Table 4. The coefficients of the algebraic equation of energy conservation for refrigerant side.

Node cPa c

Wa cEa c

Pb

1 ( )

er

*

er*

Pr

Auc

dAzAtc

⎟⎠⎞⎜

⎝⎛+

+⎟⎠⎞⎜

⎝⎛ Δ

Δ

2

2

21

1 ρ 0 ( )

er

*

er*

Auc

dA

⎟⎠⎞⎜

⎝⎛+

+

221

ρ

( ) ( )°

⎟⎠⎞⎜

⎝⎛ Δ

Δ+

−

'P

Pr

er**

Ar**

PzAtc

AuAu

21

ρρ

( )12

−zn,...,

( )

( )

wr

*

wr*

er

*

er*

Pr

Auc

dAAuc

dAzAtc

⎟⎠⎞⎜

⎝⎛−

+⎟⎠⎞⎜

⎝⎛+

+⎟⎠⎞⎜

⎝⎛ Δ

Δ

2

2

2

21

21

1

ρ

ρ

( )

wr

*

wr*

Auc

dA

⎟⎠⎞⎜

⎝⎛+

+

221

ρ

( )

er

*

er*

Auc

dA

⎟⎠⎞⎜

⎝⎛+

+

221

ρ

( ) ( )°

⎟⎠⎞⎜

⎝⎛ Δ

Δ+

−

'P

Pr

er**

wr**

PzAtc

AuAu

21

ρρ

zn ( )

( )

wr

*

wr*

er

*

er*

Pr

Auc

dAAuc

dAzAtc

⎟⎠⎞⎜

⎝⎛−

+⎟⎠⎞⎜

⎝⎛+

+⎟⎠⎞⎜

⎝⎛ Δ

Δ

2

2

2

21

21

1

ρ

ρ

( )

wr

*

wr*

Auc

dA

⎟⎠⎞⎜

⎝⎛+

+

221

ρ

0 ( ) ( )°

⎟⎠⎞⎜

⎝⎛ Δ

Δ+

−

'P

Pr

er**

wr**

PzAtc

AuAu

21

ρρ

Node rhPa rh

Wa rhEa rh

Pb

1 ( )[ ] ( )[ ]

tzA

,mmax,mmax

iP

erwr

ΔΔ+

+−

ρ

00 !!

0 ( )[ ]0,mmax wr!− ( )[ ][ ]( )[ ] Awr

P,tiP,rii

iPPPP

h,mmaxzTTp

tzAPPh

0!

"""

+Δ−+ΔΔ+

°°

−

α

ρ

( )12

−zn,...,

( )[ ] ( )[ ]

tzA

,mmax,mmax

iP

erwr

ΔΔ+

+−

ρ

00 !!

( )[ ]0,mmax wr! ( )[ ]0,mmax er!− ( )[ ][ ] zTTp

tzAPPh

P,tiP,rii

iPPPP

Δ−+ΔΔ+

°°

−

α

ρ """

zn ( )[ ] ( )[ ]

tzA

,mmax,mmax

iP

erwr

ΔΔ+

+−

ρ

00 !!

( )[ ]0,mmax wr! 0 ( )[ ][ ] zTTp

tzAPPh

P,tiP,rii

iPPPP

Δ−+ΔΔ+

°°

−

α

ρ """

acPP′P = ac

E P′E + acW P′W + bc

P (6)

ahrP hr,P = ahr

E hr,E + ahrWhr,W + bhr

P (7)

aTtiP TtiP = aTti

E TtiE + aTtiW TtiW + bTti

P (8)

www.rmiq.org 217





Node cPa c

Wa cEa c

Pb

1 ( )

er

*

er*

Pr

Auc

dAzAtc

⎟⎠⎞⎜

⎝⎛+

+⎟⎠⎞⎜

⎝⎛ Δ

Δ

2

2

21

1 ρ 0 ( )

er

*

er*

Auc

dA

⎟⎠⎞⎜

⎝⎛+

+

221

ρ

( ) ( )°

⎟⎠⎞⎜

⎝⎛ Δ

Δ+

−

'P

Pr

er**

Ar**

PzAtc

AuAu

21

ρρ

( )12

−zn,...,

( )

( )

wr

*

wr*

er

*

er*

Pr

Auc

dAAuc

dAzAtc

⎟⎠⎞⎜

⎝⎛−

+⎟⎠⎞⎜

⎝⎛+

+⎟⎠⎞⎜

⎝⎛ Δ

Δ

2

2

2

21

21

1

ρ

ρ

( )

wr

*

wr*

Auc

dA

⎟⎠⎞⎜

⎝⎛+

+

221

ρ

( )

er

*

er*

Auc

dA

⎟⎠⎞⎜

⎝⎛+

+

221

ρ

( ) ( )°

⎟⎠⎞⎜

⎝⎛ Δ

Δ+

−

'P

Pr

er**

wr**

PzAtc

AuAu

21

ρρ

zn ( )

( )

wr

*

wr*

er

*

er*

Pr

Auc

dAAuc

dAzAtc

⎟⎠⎞⎜

⎝⎛−

+⎟⎠⎞⎜

⎝⎛+

+⎟⎠⎞⎜

⎝⎛ Δ

Δ

2

2

2

21

21

1

ρ

ρ

( )

wr

*

wr*

Auc

dA

⎟⎠⎞⎜

⎝⎛+

+

221

ρ

0 ( ) ( )°

⎟⎠⎞⎜

⎝⎛ Δ

Δ+

−

'P

Pr

er**

wr**

PzAtc

AuAu

21

ρρ

Node rhPa rh

Wa rhEa rh

Pb

1 ( )[ ] ( )[ ]

tzA

,mmax,mmax

iP

erwr

ΔΔ+

+−

ρ

00 !!

0 ( )[ ]0,mmax wr!− ( )[ ][ ]( )[ ] Awr

P,tiP,rii

iPPPP

h,mmaxzTTp

tzAPPh

0!

"""

+Δ−+ΔΔ+

°°

−

α

ρ

( )12

−zn,...,

( )[ ] ( )[ ]

tzA

,mmax,mmax

iP

erwr

ΔΔ+

+−

ρ

00 !!

( )[ ]0,mmax wr! ( )[ ]0,mmax er!− ( )[ ][ ] zTTp

tzAPPh

P,tiP,rii

iPPPP

Δ−+ΔΔ+

°°

−

α

ρ """

zn ( )[ ] ( )[ ]

tzA

,mmax,mmax

iP

erwr

ΔΔ+

+−

ρ

00 !!

( )[ ]0,mmax wr! 0 ( )[ ][ ] zTTp

tzAPPh

P,tiP,rii

iPPPP

Δ−+ΔΔ+

°°

−

α

ρ """

Table 5. The coefficients of the algebraic equation of wall energy.


Table 6. Coefficients of the discretized equation of energy for secondary fluid.

Node tiTPa tiT

Wa tiTEa tiT

Pb

1 °

Δ P,tiCpAt rP,ti!ρ1

0 0

[ ] [ ]°

Δ+

−+−

P,tiCpAt

T

TTpTTp

rP,tiP,ti

P,tiP,sfooP,tiP,rii

!!

!!!!

ρ

αα

( )12

−zn,...,

°


0 0

[ ] [ ]°

Δ+

−+−

P,tiCpAt

T

TTpTTp

rP,tiP,ti

P,tiP,sfooP,tiP,rii

!!

!!!!

ρ

αα

zn °


0 0

[ ] [ ]°

Δ+

−+−

P,tiCpAt

T

TTpTTp

rP,tiP,ti

P,tiP,sfooP,tiP,rii

!!

!!!!

ρ

αα

Node sfTPa sfT

Wa sfTEa sfT

Pb

1 ( )[ ]( )[ ]( )

tzACp

,Cpm

,Cpmmax

oPsfsf

esfsf

Asfsf

ΔΔ+

+

−

ρ

0

0

!

!

0 ( )[ ]0,Cpmmax esfsf!−

( )[ ]

[ ]( )[ ] A,sfAsfsf

P,sfP,tioo

oP,sfP,sfP,sfP,sfP,sf

T,Cpmmax

zTTptzAPPTCp

0!

""""

+

Δ−+ΔΔ+

°°

−

α

ρ

( )12

−zn,...,

( )[ ]( )[ ]( )

tzACp

,Cpm

,Cpmmax

oPsfsf

esfsf

wsfsf

ΔΔ+

+

−

ρ

0

0

!

!

( )[ ]0,Cpmmax wsfsf!

( )[ ]0,Cpmmax esfsf!− ( )[ ][ ] zTTp

tzAPPTCp

P,sfP,tioo


Δ−+ΔΔ+

°°

−

α

ρ """"

zn ( )[ ]( )[ ]( )

tzACp

,Cpm

,Cpmmax

oPsfsf

esfsf

wsfsf

ΔΔ+

+

−

ρ

0

0

!

!


0 ( )[ ][ ] zTTp

tzAPPTCp

P,sfP,tioo


Δ−+ΔΔ+

°°

−

α

ρ """"

3.2 Secondary fluid flow in the annulus

The mathematical formulation of secondary fluid flowin the annulus is similar to that presented for therefrigerant flow but, the only difference is that inthis circuit, the pressure drops along the annular tubedue to momentum change and the viscous frictionare negligible. Therefore, the corresponding transient

model is only governed by the energy conservationequation, which has the temperature of the secondaryfluid as the main variable. The correspondingdiscretized model is presented in the following matrixsystem (Eq. 9), and the coefficients are described inTable 6.

aTs fP Ts f P = a

Ts fE Ts f E + a

Ts fW Ts f W + b

Ts fP (9)

218 www.rmiq.org





Node tiTPa tiT

Wa tiTEa tiT

Pb

1 °


0 0

[ ] [ ]°

Δ+

−+−

P,tiCpAt

T

TTpTTp

rP,tiP,ti

P,tiP,sfooP,tiP,rii

!!

!!!!

ρ

αα

( )12

−zn,...,

°


0 0

[ ] [ ]°

Δ+

−+−

P,tiCpAt

T

TTpTTp

rP,tiP,ti

P,tiP,sfooP,tiP,rii

!!

!!!!

ρ

αα

zn °


0 0

[ ] [ ]°

Δ+

−+−

P,tiCpAt

T

TTpTTp

rP,tiP,ti

P,tiP,sfooP,tiP,rii

!!

!!!!

ρ

αα

Node sfTPa sfT

Wa sfTEa sfT

Pb

1 ( )[ ]( )[ ]( )

tzACp

,Cpm

,Cpmmax

oPsfsf

esfsf

Asfsf

ΔΔ+

+

−

ρ

0

0

!

!

0 ( )[ ]0,Cpmmax esfsf!−

( )[ ]

[ ]( )[ ] A,sfAsfsf

P,sfP,tioo


T,Cpmmax

zTTptzAPPTCp

0!

""""

+

Δ−+ΔΔ+

°°

−

α

ρ

( )12

−zn,...,

( )[ ]( )[ ]( )

tzACp

,Cpm

,Cpmmax

oPsfsf

esfsf

wsfsf

ΔΔ+

+

−

ρ

0

0

!

!


( )[ ]0,Cpmmax esfsf!− ( )[ ][ ] zTTp

tzAPPTCp

P,sfP,tioo


Δ−+ΔΔ+

°°

−

α

ρ """"

zn ( )[ ]( )[ ]( )

tzACp

,Cpm

,Cpmmax

oPsfsf

esfsf

wsfsf

ΔΔ+

+

−

ρ

0

0

!

!


0 ( )[ ][ ] zTTp

tzAPPTCp

P,sfP,tioo


Δ−+ΔΔ+

°°

−

α

ρ """"

3.3 Boundary conditions

For the momentum equation, the inlet pressureis considered as boundary conditions, while forthe energy equation, enthalpy from the compressoroutlet is considered as inlet boundary conditions.In this study, no boundary condition at the exitof the gas cooler has been taken into accountin order to let the model predicts the values ofparameters at the gas cooler outlet. Data relatingto boundary conditions were taken from the studyrealized by Perez et al. (2013). Furthermore, inthe annulus, the boundary conditions include the inletwater temperature, pressure and mass flow rate thatcorrespond to the output operating conditions of theexperimental heat dissipation system. For the innertube, adiabatic boundary conditions are imposed at thelateral boundaries.

3.4 Heat transfer coefficient analysis

The behavior of the hairpin heat exchanger hasattracted the attention of many researchers, due tothe influence that has the radius of curvature onthe heat transfer coefficient (Wojtkowiak and Popiel,2000; Chen et al., 2004; Adam-Medina et al., 2013).The results of investigations performed on this heatexchanger have demonstrated that the friction factoris more influenced by the radius of curvature of thetube. However, new correlations have been developedby incorporating the radius of curvature as one of theparameters for calculating the friction factor (Chen

et al., 2004). Unfortunately, these correlations weredeveloped for refrigerants operating in subcriticalconditions. Therefore, the lack of an appropriatecorrelation for the calculation of the friction factorin hairpin or return bend gas coolers has led someresearchers to use correlations developed for straightin-tube gas coolers (Sanchez et al., 2012; Park andHrnjak, 2007). Thus, in this study we use correlationsdeveloped for the straight in-tube gas coolers.

3.4.1 Inner heat transfer coefficient

Analysis of the heat transfer coefficient of R744 insideof inner tubes is calculated by using the Gnielinski(1976) correlation (Eq. 10) which was found byRieberer (1999) to be the best for single-phase R744at supercritical conditions. This correlation haspresented interesting results in the study realized bySanchez et al., (2012), with less than 10 % error incomparison with the correlation Pitla et al., (2002).

Nui =

( fi8

)(Rei − 1000)Pri

1 + 12.7( fi

8

) 12(Pri

23 − 1

) (10)

0.5 < Pri < 2000;2300 < Rei < 5× 105

αi =Nuikr

di(11)

The friction factor is calculated from Churchill (1977)correlation which extends over all flow regimes and all

www.rmiq.org 219


relative roughness (δ):

fi = 8

(

8Rei

)12

+

2.457ln

1(7

Rei

)0.9+ 0.27δ

16

+

(37530

Rei

)16−32

112

(12)

3.4.2 Outer heat transfer coefficient

The outer heat transfer coefficient is obtained fromthe correlation developed by Petukhov et al., (1963)for water flowing in the secondary circuit (Eq.13),wherein geometric parameters such as tube length (Lt),inside diameter of annulus (Di), hydraulic (Dh) andequivalent (Deq) diameters are taken into account.

Nuo =

( fo8

)Reo ·Pro

1 + 12.7( fo

8

) 12(Pro

23 − 1

) ·1 +

(Dh

Lt

) 23·0.86·

(Deq

Di

)−0.16

(13)Rei > 104

where fo =(1.8log10 (Reo)− 1.5

)−2

αo =Nuokw

Dh(14)

3.5 Global algorithm of resolution

Fig. 3 shows the algorithm for simulatingmathematical models of gas cooler. This methodologyis subdivided into four steps: definition of inputdata, assessment of the pressure and velocity fields,resolution of the discretized energy equation, andthe evaluation of R744 and water thermophysicalproperties. In the first step, geometrical parametersof the gas cooler, initial and boundary conditionsare set. Once the data is provided to the program,the process passes to the second step which involvesthe calculation of the pressure and velocity fieldsof the refrigerant. After this step, pressure andvelocity fields of both fluids are used to compute thedependent variables (enthalpy of R744, wall and watertemperature). This computation is performed at eachnode and in several iterations until the convergencecriteria is met. If the solution does not converge,then the process is reinitiated while adjusting theguessed pressure and velocity. But, if the convergencecriteria are complied, one passes to the last stepwhich is dedicated to the evaluation of thermophysicalproperties of R744, such as: specific heat, density,dynamic viscosity, thermal conductivity, as well as

Fig. 3. Global algorithm of resolution.

Gue

ssed

pre

ssur

e an

d ve

loci

ty a

djus

ted

START

Input data(Geometry, guessed values, initial and

boundary conditions, convergence criteria)

Pressure and velocity field data for secondary fluid

Solve refrigerant pressure and velocity field from

SIMPLE method

Solve discretized momentum equation

Solve pressure correction equation

Calculate correctpressure and velocity

*(i, j)ru

(i, j)rP′

(i, j)sfP (i, j)sfu; (i, j)ru(i, j)rP ;

Solve discretized energy equation for refrigerant flow, wall and

secondary fluid

(i, j)rh(i, j)sfT

Calculate R744 thermophysical properties through REFPROP linked to Matlab

Convergence ?

STOP

Yes

No

j)(i,tiT

Fig. 3. Global algorithm of resolution.

Prandtl and Nusselt numbers and the heat transfercoefficient of R744. Thus, the process stops.

4 Results and discussionThis section presents the simulation results of thegas cooler dynamic model described previously. Thesimulation was performed by considering the averagemass flux of R744 and water about 142.86 kgm−2s−1

and 898.74 kgm−2s−1, respectively. Results arepresented graphically in space and time. Withrespect to space, twenty points of observation (namelytwenty finite volumes with ∆z = 0.1 m) are provided.These correspond to twenty points of temperaturemeasurement planned in the experimental setup whichis still in construction. While over time, a maximumof 240 seconds with ∆t = 0.5s is provided which wassupposed to be sufficient for observing the transientbehavior of properties at startup.

220 www.rmiq.org


a) Pressure b) Velocity

Fig. 4. Distribution of pressure, velocity in the gas cooler.

Fig. 5. Temperatures distribution in the gas cooler

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 25000

5500

6000

6500

7000

7500

8000

8500

9000

9500

10000

Position [m]

R74

4 pr

essu

re [k

Pa]

0s30s60s90s120s150s180s210s240s

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Position [m]

R74

4 ve

loci

ty [m

s-1]

0s30s60s90s120s150s180s210s240s

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2280

300

320

340

360

380

400

420

440

460

480

500

Position [m]

R74

4 te

mpe

ratu

re [K

]

0s30s60s90s120s150s180s210s240s

Gr= 142.86 kg m-2 s-1

Pr,av= 8711 kPa

a) Pressure b) VelocityFig. 4. Distribution of pressure and velocity in the gas cooler.

Given the difficulty in reporting the results of alliterations, nine samples were selected in an intervalof thirty seconds of observation. For a betterunderstanding of transient phenomena that occur ineach property, the results are graphically presentedin three different forms. In the first type of graph,results of variations in each property depending onthe position throughout the gas cooler during the first240 seconds of gas cooling process are shown. Thisrepresentation allows visualizing both transient andsteady state of the property. In the second one,behavior of the property in each node of the finitevolume (i.e. every 0.1 m of the gas cooler length) isshown, in order to identify the area that is affected themost by the transient effects. Finally, the third typeof graph presents the effect of the R744 temperatureon each property, this graph allows us to understandhow each property behaves in the critical region. Asmentioned in the previous section, before proceedingto the computation of thermophysical properties, ananalysis of pressure, temperature and velocity fieldsis performed in order to know how these parametersare distributed along the gas cooler. Results of thisanalysis are shown in Fig. 4 where it can be observedthe transient behaviors that occur in these parametersduring startup of gas cooling process. Fig. 4a showsthe pressure distribution in the gas cooler during thefirst 240 seconds of operation, where it can observedthat there is no a significant change in pressure duringthe transitional period, but, the process is characterizedby a decrease in pressure along the gas cooler whichis estimated at 211 kPa between the inlet and outlet ofthe gas cooler. This pressure drop causes an increase invelocity (from 0.1 to 0.2 m s−1) along the gas cooler as

a) Pressure b) Velocity

Fig. 4. Distribution of pressure, velocity in the gas cooler.

Fig. 5. Temperatures distribution in the gas cooler

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 25000

5500

6000

6500

7000

7500

8000

8500

9000

9500

10000

Position [m]

R74

4 pr

essu

re [k

Pa]

0s30s60s90s120s150s180s210s240s

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Position [m]

R74

4 ve

loci

ty [m

s-1]

0s30s60s90s120s150s180s210s240s

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2280

300

320

340

360

380

400

420

440

460

480

500

Position [m]

R74

4 te

mpe

ratu

re [K

]

0s30s60s90s120s150s180s210s240s

Gr= 142.86 kg m-2 s-1

Pr,av= 8711 kPa

Fig. 5. Temperature distribution in the gas cooler.

seen in Fig. 4b. The average pressure of R744, (Pr,av)in the gas cooler is about 8711 kPa.

Temperature distribution of the refrigerant duringthe gas cooling process is shown in Fig. 5 where animportant change is occurred in the 150 s before thetemperature is stabilized. Instability in the first thirtyseconds is much greater than in the rest of the startupperiod. In addition, the temperature undergoes a sharpdrop between the inlet and 0.6 m in length. Thischange can have a significant influence on the behaviorof thermophysical properties of the refrigerant as itoperates in a supercritical area.

Transient behavior of the R744 thermophysicalproperties during the gas cooling process are shownin Fig. 6 where it can be observed the importantchange that occurs in each property during the first150 seconds of the process. During this transientperiod, the profile of each sample undergoes a change

www.rmiq.org 221


a) Specific heat vs position b) Specific heat vs time

c) Density vs position d) Density vs time

e) Dynamic viscosity vs position f) Dynamic viscosity vs time

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

2

4

6

8

10

12

14

16

18

Position [m]

R74

4 sp

ecifi

c he

at [k

J kg

-1 K

-1]

0s30s60s90s120s150s180s210s240s

Gr= 142.86 kg m-2 s-1

Pr,av= 8711 kPa

0 50 100 150 200 2500

2

4

6

8

10

12

14

16

18

Time [s]

R74

4 sp

ecifi

c he

at [k

J kg

-1 K

-1]

0.1 m0.20.51.01.51.82.0

Gr= 142.86 kg m-2 s-1

Pr,av= 8711 kPa

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

200

400

600

800

1000

1200

1400

1600

1800

2000

Position [m]

R74

4 de

nsity

[kg

m-3]

0s30s60s90s120s150s180s210s240s

Gr= 142.86 kg m-2 s-1

Pr,av= 8711 kPa

0 50 100 150 200 2500

200

400

600

800

1000

1200

1400

1600

1800

2000

Time [s]

R74

4 de

nsity

[kg

m-3]

0.1 m0.20.51.01.51.82.0

Gr= 142.86 kg m-2 s-1

Pr,av= 8711 kPa

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

0.5

1

1.5

2

2.5x 10-4

Position [m]

R74

4 dy

nam

ic v

isco

sity

[Pa

s]

0s30s60s90s120s150s180s210s240s

Gr= 142.86 kg m-2 s-1

Pr,av= 8711 kPa

0 50 100 150 200 2500

0.5

1

1.5

2

2.5x 10-4

Time [s]

R74

4 dy

nam

ic v

isco

sity

[Pa

s]

0.1 m0.20.51.01.51.82.0

Gr= 142.86 kg m-2 s-1

Pr,av= 8711 kPa





0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

2

4

6

8

10

12

14

16

18

Position [m]

R74

4 sp

ecifi

c he

at [k

J kg

-1 K

-1]

0s30s60s90s120s150s180s210s240s

Gr= 142.86 kg m-2 s-1

Pr,av= 8711 kPa

0 50 100 150 200 2500

2

4

6

8

10

12

14

16

18

Time [s]

R74

4 sp

ecifi

c he

at [k

J kg

-1 K

-1]

0.1 m0.20.51.01.51.82.0

Gr= 142.86 kg m-2 s-1

Pr,av= 8711 kPa

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

200

400

600

800

1000

1200

1400

1600

1800

2000

Position [m]

R74

4 de

nsity

[kg

m-3]

0s30s60s90s120s150s180s210s240s

Gr= 142.86 kg m-2 s-1

Pr,av= 8711 kPa

0 50 100 150 200 2500

200

400

600

800

1000

1200

1400

1600

1800

2000

Time [s]

R74

4 de

nsity

[kg

m-3]

0.1 m0.20.51.01.51.82.0

Gr= 142.86 kg m-2 s-1

Pr,av= 8711 kPa

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

0.5

1

1.5

2

2.5x 10-4

Position [m]

R74

4 dy

nam

ic v

isco

sity

[Pa

s]

0s30s60s90s120s150s180s210s240s

Gr= 142.86 kg m-2 s-1

Pr,av= 8711 kPa

0 50 100 150 200 2500

0.5

1

1.5

2

2.5x 10-4

Time [s]

R74

4 dy

nam

ic v

isco

sity

[Pa

s]

0.1 m0.20.51.01.51.82.0

Gr= 142.86 kg m-2 s-1

Pr,av= 8711 kPa





0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

2

4

6

8

10

12

14

16

18

Position [m]

R74

4 sp

ecifi

c he

at [k

J kg

-1 K

-1]

0s30s60s90s120s150s180s210s240s

Gr= 142.86 kg m-2 s-1

Pr,av= 8711 kPa

0 50 100 150 200 2500

2

4

6

8

10

12

14

16

18

Time [s]

R74

4 sp

ecifi

c he

at [k

J kg

-1 K

-1]

0.1 m0.20.51.01.51.82.0

Gr= 142.86 kg m-2 s-1

Pr,av= 8711 kPa

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

200

400

600

800

1000

1200

1400

1600

1800

2000

Position [m]

R74

4 de

nsity

[kg

m-3]

0s30s60s90s120s150s180s210s240s

Gr= 142.86 kg m-2 s-1

Pr,av= 8711 kPa

0 50 100 150 200 2500

200

400

600

800

1000

1200

1400

1600

1800

2000

Time [s]

R74

4 de

nsity

[kg

m-3]

0.1 m0.20.51.01.51.82.0

Gr= 142.86 kg m-2 s-1

Pr,av= 8711 kPa

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

0.5

1

1.5

2

2.5x 10-4

Position [m]

R74

4 dy

nam

ic v

isco

sity

[Pa

s]

0s30s60s90s120s150s180s210s240s

Gr= 142.86 kg m-2 s-1

Pr,av= 8711 kPa

0 50 100 150 200 2500

0.5

1

1.5

2

2.5x 10-4

Time [s]

R74

4 dy

nam

ic v

isco

sity

[Pa

s]

0.1 m0.20.51.01.51.82.0

Gr= 142.86 kg m-2 s-1

Pr,av= 8711 kPa


222 www.rmiq.org


g) Thermal conductivity vs position h) Thermal conductivity vs time

Fig. 6. Transient behavior of the R744 thermophysical properties along the gas cooler.

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

0.05

0.1

0.15

0.2

0.25

Position [m]

R74

4 th

erm

al c

ondu

ctiv

ity [W

m-1 K

-1]

0s30s60s90s120s150s180s210s240s

Gr= 142.86 kg m-2 s-1

Pr,av= 8711 kPa

0 50 100 150 200 2500

0.05

0.1

0.15

0.2

0.25

Time [s]

R74

4 th

erm

al c

ondu

ctiv

ity [W

m-1 K

-1]

0.1 m0.20.51.01.51.82.0

Gr= 142.86 kg m-2 s-1

Pr,av= 8711 kPa

g) Thermal conductivity vs position h) Thermal conductivity vs time

Fig. 6. Transient behavior of the R744 thermophysical properties along the gas cooler.

a) Specific heat b) Density

c) Dynamic viscosity d) Thermal conductivity

Fig. 7. Transient behavior of the R744 thermophysical properties in the supercritical region.

260 280 300 320 340 360 380 400 4200

2

4

6

8

10

12

14

16

18

R744 temperature [K]

R74

4 sp

ecifi

c he

at [k

J kg

-1 K

-1]

0s30s60s90s120s150s180s210s240s

Gr= 142.86 kg m-2 s-1

Pr,av= 8711 kPa

280 300 320 340 360 380 400 4200

200

400

600

800

1000

1200

1400

1600

1800

2000


R74

4 de

nsity

[kg

m-3]

0s30s60s90s120s150s180s210s240s

Gr= 142.86 kg m-2 s-1

Pr,av= 8711 kPa

280 300 320 340 360 380 400 4200

0.5

1

1.5

2

2.5x 10-4


R74

4 dy

nam

ic v

isco

sity

[Pa

s]

0s30s60s90s120s150s180s210s240s

Gr= 142.86 kg m-2 s-1

Pr,av= 8711 kPa

280 300 320 340 360 380 400 4200

0.05

0.1

0.15

0.2

0.25


R74

4 th

erm

al c

ondu

ctiv

ity [W

m-1 K

-1]

0s30s60s90s120s150s180s210s240s

Gr= 142.86 kg m-2 s-1

Pr,av= 8711 kPa





260 280 300 320 340 360 380 400 4200

2

4

6

8

10

12

14

16

18


R74

4 sp

ecifi

c he

at [k

J kg

-1 K

-1]

0s30s60s90s120s150s180s210s240s

Gr= 142.86 kg m-2 s-1

Pr,av= 8711 kPa

280 300 320 340 360 380 400 4200

200

400

600

800

1000

1200

1400

1600

1800

2000


R74

4 de

nsity

[kg

m-3]

0s30s60s90s120s150s180s210s240s

Gr= 142.86 kg m-2 s-1

Pr,av= 8711 kPa

280 300 320 340 360 380 400 4200

0.5

1

1.5

2

2.5x 10-4


R74

4 dy

nam

ic v

isco

sity

[Pa

s]

0s30s60s90s120s150s180s210s240s

Gr= 142.86 kg m-2 s-1

Pr,av= 8711 kPa

280 300 320 340 360 380 400 4200

0.05

0.1

0.15

0.2

0.25


R74

4 th

erm

al c

ondu

ctiv

ity [W

m-1 K

-1]

0s30s60s90s120s150s180s210s240s

Gr= 142.86 kg m-2 s-1

Pr,av= 8711 kPa



www.rmiq.org 223


in position along the gas cooler which leads to anincrease in magnitude of each property. This transienteffect is more pronounced in the specific heat whichbehaves differently in comparison with the otherproperties. The specific heat is characterized by aninstantaneous change in the profile with a peak valuethat is 5 times higher than the ideal value (see Fig.6a). By observing the profiles shown in Fig. 6b,it becomes clear that the transitional phenomenon ofspecific heat is more accentuated in the middle area ofthe gas cooler. Region in which the refrigerant flowundergoes a curvature of 180°. Therefore, the suddenand instantaneous change in the specific heat can havesignificant impacts on the performance of gas cooler,since this property is involved in the calculation of theheat transfer coefficient.

On the other hand, density, dynamic viscosity andthermal conductivity do not have significant transientphenomena during the process. Each propertyexperienced a slight increase in magnitude alongthe gas cooler as shown in figs. 6c, 6e and 6g.Although transient effects in these properties are notvery significant, they have anyway a fluctuation in thearea near the exit of the gas cooler (see figs. 6d, 6f and6h).

One of the important aspects of the analysis ofthermophysical properties of R744 is to know howeach property behaves in supercritical area. Fig.7 presents the variation of each property dependingon the local temperature of R744 while the massflow and pressure are constant. It is evident fromthis analysis that, during the gas cooling processthe thermophysical properties of R744 are closelyrelated to its temperature. This interdependence isobserved through the tendencies of sample profileswhich are similar during the process. Fig. 7a showsa similar tendency to the presented in Fig. 6a,wherein the profile of the specific heat is characterizedby a maximum value (about 7 times the nominalvalue) which occurs at 315 K. This temperatureotherwise called “pseudo-critical temperature” hasbeen subject of several studies in the literature (Hoo-Kyu and Chang-Hyo, 2010; Dang and Hihara, 2010),in which it has been demonstrated that, near thepseudo-critical temperature thermophysical propertiesof R744 undergo significant changes. Thus, bycomparing the results shown in figs. 5a, 6a and7a, it becomes clear that the pseudo-critical regionundergoes a change in position along the gas coolerduring the transient period. On the other hand, density,dynamic viscosity and conductivity present a different

behavior compared to the specific heat in the pseudo-critical area. Their profiles near the pseudo-criticaltemperature are characterized by a sharp decrease asshown in figs. 7b, 7c and 7d. However, all R744thermophysical properties are less sensitive to thechange in temperature after the pseudo-critical area.

Fig. 8 shows the transient behavior of Prandtland Nusselt numbers during the gas cooling process.As can well be observed, tendencies of both numbersare almost identical to the profiles of the specificheat (see Fig. 7a and 7b). Thereby, althoughproperties such as density, dynamic viscosity andthermal conductivity involved in the computation ofPrandtl and Nusselt numbers, their influences areinsignificant in comparison with the specific heat.

Variation in Prandtl and Nusselt numbersdepending on the local temperature of R744 is shownin Fig. 9. The great impact of the specific heaton the Prandtl number is further shown in Fig. 9a,from which it can be seen that the profile is identicalto the presented in Fig. 7a. At the pseudo criticaltemperature, the Prandtl number undergoes sustainedgrowth, which extends to 7 times its nominal value(see Fig. 9a), similarly to the specific heat. TheNusselt number shows a similar trend (see Fig. 9b),but its behavior in the pseudo-critical area is notexactly the same as the specific heat, since it not onlydepends on the Prandtl number, but also depends onthe type of correlation used.

Transient behavior of the local heat transfercoefficient during gas cooling process is shown in Fig.10. These results are based on the operating conditionssuch as pressure and mass flux described in Fig. 4 andthe Gnielinski (1976) correlation. As can be seen fromFig. 10a, the local heat transfer coefficient of R744 hasa significant transient behavior during the gas coolingprocess. Each profile is characterized by an increasefrom the input of the gas cooler which grows untilreaching a peak value (about 8 times the inlet value)and then decreases in the rest of the process up to theoutlet.

Profiles at transient state are characterized by asharp decline after the maximum magnitude afterthe peak value in comparison with the profileat steady-state which experienced a large area ofconstant magnitude after the maximum value. Themiddle region of gas cooler is characterized by greatinstability during the transitional period as shown inFig. 10b). Similar behavior is observed in the specificheat (see Fig. 6b).

224 www.rmiq.org


a) Prandtl number vs position b) Prandtl number vs time

c) Nusselt number vs position d) Nusselt number vs time

Fig. 8. Transient behavior of the Prandtl and Nusselt numbers along the gas cooler.

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

1

2

3

4

5

6

7

8

9

10

Position [m]

R74

4 Pr

andt

l num

ber

0s30s60s90s120s150s180s210s240s

Gr= 142.86 kg m-2 s-1

Pr,av= 8711 kPa

0 50 100 150 200 2500

1

2

3

4

5

6

7

8

9

10

Time [s]

R74

4 Pr

andt

l num

ber

0.1 m0.20.51.01.51.82.0

Gr= 142.86 kg m-2 s-1

Pr,av= 8711 kPa

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

20

40

60

80

100

120

140

160

180

Position [m]

R74

4 N

usse

lt nu

mbe

r

0s30s60s90s120s150s180s210s240s

Gr= 142.86 kg m-2 s-1

Pr,av= 8711 kPa

0 50 100 150 200 2500

20

40

60

80

100

120

140

160

180

Time [s]

R74

4 N

usse

lt nu

mbe

r

0.1 m0.20.51.01.51.82.0

Gr= 142.86 kg m-2 s-1

Pr,av= 8711 kPa





0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

1

2

3

4

5

6

7

8

9

10

Position [m]

R74

4 Pr

andt

l num

ber

0s30s60s90s120s150s180s210s240s

Gr= 142.86 kg m-2 s-1

Pr,av= 8711 kPa

0 50 100 150 200 2500

1

2

3

4

5

6

7

8

9

10

Time [s]

R74

4 Pr

andt

l num

ber

0.1 m0.20.51.01.51.82.0

Gr= 142.86 kg m-2 s-1

Pr,av= 8711 kPa

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

20

40

60

80

100

120

140

160

180

Position [m]

R74

4 N

usse

lt nu

mbe

r

0s30s60s90s120s150s180s210s240s

Gr= 142.86 kg m-2 s-1

Pr,av= 8711 kPa

0 50 100 150 200 2500

20

40

60

80

100

120

140

160

180

Time [s]

R74

4 N

usse

lt nu

mbe

r

0.1 m0.20.51.01.51.82.0

Gr= 142.86 kg m-2 s-1

Pr,av= 8711 kPa



a) Prandtl number b) Nusselt number

Fig. 9. Behavior of the Prandtl and Nusselt numbers in the supercritical region.

a) Local heat transfer coefficient vs position b) Local heat transfer coefficient vs time

Fig. 10. Transient behavior of the local heat transfer coefficient along the gas cooler.

280 300 320 340 360 380 400 4200

1

2

3

4

5

6

7

8

9

10


R74

4 Pr

andt

l num

ber

0s30s60s90s120s150s180s210s240s

Gr= 142.86 kg m-2 s-1

Pr,av= 8711 kPa

280 300 320 340 360 380 400 4200

20

40

60

80

100

120

140

160

180


R74

4 N

usse

lt N

umbe

r

0s30s60s90s120s150s180s210s240s

Gr= 142.86 kg m-2 s-1

Pr,av= 8711 kPa

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

500

1000

1500

2000

2500

3000

Position [m]

R74

4 he

at tr

ansf

er c

oeffi

cien

t [W

m-2

K-1

]

0s30s60s90s120s150s180s210s240s

Gr= 142.86 kg m-2 s-1

Pr,av= 8711 kPa

0 50 100 150 200 2500

500

1000

1500

2000

2500

3000

Time [s]

R74

4 he

at tr

ansf

er c

oeffi

cien

t [W

m-2

K-1

]

0.1 m0.20.51.01.51.82.0

Gr= 142.86 kg m-2 s-1

Pr,av= 8711 kPa



www.rmiq.org 225






280 300 320 340 360 380 400 4200

1

2

3

4

5

6

7

8

9

10


R74

4 Pr

andt

l num

ber

0s30s60s90s120s150s180s210s240s

Gr= 142.86 kg m-2 s-1

Pr,av= 8711 kPa

280 300 320 340 360 380 400 4200

20

40

60

80

100

120

140

160

180


R74

4 N

usse

lt N

umbe

r

0s30s60s90s120s150s180s210s240s

Gr= 142.86 kg m-2 s-1

Pr,av= 8711 kPa

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

500

1000

1500

2000

2500

3000

Position [m]

R74

4 he

at tr

ansf

er c

oeffi

cien

t [W

m-2

K-1

]

0s30s60s90s120s150s180s210s240s

Gr= 142.86 kg m-2 s-1

Pr,av= 8711 kPa

0 50 100 150 200 2500

500

1000

1500

2000

2500

3000

Time [s]

R74

4 he

at tr

ansf

er c

oeffi

cien

t [W

m-2

K-1

]

0.1 m0.20.51.01.51.82.0

Gr= 142.86 kg m-2 s-1

Pr,av= 8711 kPa



Fig. 11. Transient behavior the local heat transfer coefficient in the supercritical region.

280 300 320 340 360 380 400 4200

500

1000

1500

2000

2500

3000


R74

4 he

at tr

ansf

er c

oeffi

cien

t [W

m-2

K-1

]

0s30s60s90s120s150s180s210s240s

Gr= 142.86 kg m-2 s-1

Pr,av= 8711 kPa

Fig. 11. Transient behavior the local heat transfercoefficient in the supercritical region.

The behavior of the heat transfer coefficient inthe supercritical region is shown in Fig. 11. Itbecomes clear that the peak value that characterizeseach profile during the transitional period (see Fig.10a) corresponds to the pseudo-critical temperature.Therefore, the conclusion reached in the analysisof the specific heat is also valid for the heattransfer coefficient. The analysis from which it isdemonstrated that the pseudo-critical area is not atthe same location during the transitional period. Butundergoes a displacement in the opposite direction tothe refrigerant flow until stabilizing at the middle ofthe gas cooler.

Conclusions

This paper presents comprehensive behavior of R744thermophysical properties during the startup of gascooling process in the supercritical region. Thestudy was performed on a horizontal hairpin gascooler. The model was developed in the Matlab andthermophysical properties were evaluated through theREFPROP dynamic libraries linked to Matlab. Themain conclusions are summarized as follows:

• The thermophysical properties of R744 manifesta significant transient behavior during the gascooling process. This behavior is characterizedby large variations in properties along the gascooler. The area of the curvature of gas cooleris the most affected by transient phenomena.

• The specific heat of R744 is the propertymost affected by the transient phenomenon inthe gas cooler. Its behavior is characterizedby an instantaneous change in the amplitudedistinguished by a peak value that occurs inthe region where the temperature of R744 ispseudo-critical.

• The pseudo-critical area is not at the samelocation during the transitional period. Butundergoes a displacement in the oppositedirection to the refrigerant flow until stabilizingat the middle of the gas cooler.

• Local heat transfer coefficient of R744 in the gascooler as well as Prandtl and Nusselt numbers

226 www.rmiq.org


behave the same as the specific heat during thetransitional period.

• The peak of the local heat transfer coefficientis evaluated at 8 times the ideal value andinstantly moves along the gas cooler during thetransitional period.

NomenclatureA area [m2]C speed of sound [m s−1]Cp specific heat [kJ kg−1 K−1]d tube diameter [mm]h specific enthalpy [kJ kg−1]G mass flux [kg m−2 s−1]k thermal conductivity [W m−1 K−1]m mass flow [kg s−1]nz total number of nodesP pressure [kPa]p perimeter of inner tubes [m]T temperature [K]t time [s]u velocity [m s−1]z axial direction [m]

Greek symbolsα convective heat transfer coefficient [W m−2

K−1]ρ density [kg m−3]µ dynamic viscosity [Pa s]τz shear stress [N m−2]∆ increment

Subscriptsi inner, cell indexj cell indexo outerr refrigerantt tubeti inner tubes f secondary fluidw water

Superscripts° previous* guesseds staggered

References

Adam-Medina M., Escobar, R.F., Juarez-Romero,D., Guerrero-Ramırez, G.V., Lopez-Zapata, B.,(2013). Fault detection in a heat exchangerusing second order sliding mode observers.

Revista Mexicana de Ingenierıa Quımica 12,327-336.

ASHRAE. (2009). Position Document on NaturalRefrigerants, Carbon Dioxide, R744, 4.Available from:http://www2.epa.gov/sites/production/files/documents/ASHRAE PDNatural Refrigerants 2011.pdf

Belman-Flores, J.M. Gallegos-Munoz, A., Riesco-Avila, J.M., Zaleta-Aguilar, A., Mendoza-Miranda, J.M., (2014). Exergetic analysis ofa refrigeration system. Revista Mexicana deIngenierıa Quımica 13, 957-968.

Bjorn, P., (2008). Ammonia in low capacityrefrigeration and heat pump systems.International Journal of Refrigeration 31, 709-715.

Bolaji, B.O., Huan, Z., (2013). Ozone depletionand global warming: Case for the use ofnatural refrigerant - a review. Renewable andSustainable Energy Reviews 18, 49-54.

Butler, D.J.G., Gigiel, A., Russell, S., (2001).Using air for cooling, Building ResearchEstablishment, BRE Publication Draft number202296.

Cariaga, E., Vergara-Fernandez, A., Levano, M.,Vergaray, N., (2013). Numerical simulation ofthe water saturation at the interface betweenhomogeneous porous medium. RevistaMexicana de Ingenierıa Quımica 12, 527-539.

Chen, I.Y., Wang, C.C., Lin, S. Y., (2004).Measurements and correlations of frictionalsingle-phase and two-phase pressure drops ofR-410A flow in small U-type return bends.International Journal of Heat and MassTransfer 47, 2241-2249.

Churchill, S.W., (1977). Friction-factor equationspans all fluid flow regimes. ChemicalEngineering Science 7, 91-92.

Dang, C., Hihara, E., (2010). Numerical study on intube laminar heat transfer of supercritical fluids.Applied Thermal Engineering 30, 1567-1573.

Duffey, R.B., Pioro, I.L., (2005). Experimental heattransfer of supercritical carbon dioxide flowinginside channels (survey). Nuclear Engineeringand Design 235, 913-924.

www.rmiq.org 227


Dull, D., Seidel, S., Wells, J., (1989). MovingForward: Key Implications of the MontrealProtocol. Studies in Environmental Science 35,775-784.

Fang, X., Xu, Y., (2011). Modified heat transferequation for in-tube supercritical CO2 cooling.Applied Thermal Engineering 31, 3036-3042.

Ge, Y.T., Cropper, R.T., (2009). Simulation andperformance evaluation of finned-tube CO2 gascoolers for refrigeration systems. AppliedThermal Engineering 29, 957-965.

Gnielinski, V., (1976). New equations for heat andmass transfer in turbulent pipe and channel flow.Int. Chem. Eng. 16, 359-68.

Granryd, E., (2001). Hydrocarbons as refrigerants- an overview. International Journal ofRefrigeration 24, 15-24.

Hoo-Kyu, O., Chang-Hyo, S., (2010). Newcorrelation to predict the heat transfercoefficient in-tube cooling of supercriticalCO2 in horizontal macro-tubes. ExperimentalThermal and Fluid Science 34, 1230-1241.

Huai, X.L., Koyama, S., Zhao, T.S., (2005). Anexperimental study of flow and heat transfer ofsupercritical carbon dioxide in multi-port minichannels under cooling conditions. ChemicalEngineering Science 60, 3337-3345.

Kim, J.K., Jeon, H.K., Lee, J.S., (2007). Walltemperature measurement and heat transfercorrelation of turbulent supercritical carbondioxide flow in vertical circular/non-circulartubes. Nuclear Engineering and Design 237,1795-1802.

Kim, M.H., Pettersen, J., Bullard, C.W., (2004).Fundamentals process and system design issuesin CO2 vapor compression systems. Progress inEnergy Combustion Science 30, 119-174.

Lachner, B.F.J., Nellis, G.F., Reindl, D.T., (2007).The commercial feasibility of the use of watervapor as a refrigerant. International Journal ofRefrigeration 30, 699-708.

Lemmon, E.W., McLinden, M.O., Huber, M.L.,(2007). REFPROP NIST Standard ReferenceDatabase 23, v. 8.0. National Instituteof Standards; Gaithersburg, Maryland 20899,USA.

Liao, S.M., Zhao, T.S., (2002). An experimentalinvestigation of convection heat transfer tosupercritical carbon dioxide in miniature tubes.International Journal of Heat and MassTransfer 45, 5025-5034.

Li., H., Kruizenga, A., Anderson, M., Corradini,M., Luo, Y., Wang, H., Li, H., (2011).Development of a new forced convection heattransfer correlation for CO2 in both heatingand cooling modes at supercritical pressures.International Journal of Thermal Sciences 50,2430-2442.

Lugo-Leyte, R., Salazar-Pereyra, M., Ruız-Ramırez,O.A., Zamora-Mata, J.M., Torres-Gonzalez,E.V. (2013). Exergoeconomic operation costanalysis to theoretical compression refrigerationcycle of HFC-134a. Revista Mexicana deIngenierıa Quımica 12, 361-370.

Neil, A.R., Owen R.C., (2004). Energy savingrefrigerant blends comprising R125, R134a,R600 or R600a. International Refrigeration andAir Conditioning Conference, 632.

Park, C.Y., Hrnjak, P., (2007). Effect of heatconduction through the fins of a microchannelserpentine gas cooler of transcritical CO2system. International Journal of Refrigeration30, 389-397.

Patankar, V., Suhas, (1980). Numerical heattransfer and fluid flow. Hemisphere PublishingCorporation. USA.

Peixue, J., Chenru, Z., Jianqiang, D., Wenxing,Z., (2008). Experimental Investigation ofLocal Heat Transfer of Carbon Dioxide atSuper-Critical Pressures in a Vertical Tubeand Multi-Port Mini-Channels Under CoolingConditions. International Refrigeration and AirConditioning Conference, Paper 892.

Perez-Garcıa, V., Belman-Flores, J.M., Navarro-Esbrı, J., Rubio-Maya, C., (2013). Comparativestudy of transcritical vapor compressionconfigurations using CO2 as refrigerationmode base on simulation. Applied ThermalEngineering 51, 1038-1046.

Petukhov, B.S., Popov,VN., (1963). Theoreticalcalculation of heat exchanger and frictionalresistance in turbulent flow in tubes ofincompressible fluid with variable physicalproperties. High Temp 1, 69-83.

228 www.rmiq.org


Pitla, S.S., Groll, E.A., Ramadhyani, S., (2002).New correlation to predict the heat transfercoefficient during in-tube cooling of turbulentsupercritical CO2. International Journal ofRefrigeration 25, 887-895.

Rieberer, R., (1999). CO2 Properties. llR Workshopon CO2 Technology in Refrigeration. HeatPump and Air Conditioning Systems, Mainz,Germany.

Sanchez, D., Cabello, R., Llopis, R., Torrella, E.,(2012). Development and validation of a finiteelement model for water - CO2 coaxial gas-coolers. Applied Energy 93, 637-647.

Sarkar, J., Agrawal, N., (2010). Performanceoptimization of transcritical CO2 cyclewith parallel compression economization.International Journal of Thermal Sciences 49,838-843.

Son, C.-H., Park, S.-J., (2006). An experimentalstudy on heat transfer and pressure dropcharacteristics of carbon dioxide duringgas cooling process in a horizontal tube.International Journal of Refrigeration 29, 539-546.

Tanaka, K., Higashi, Y., (2010). Thermodynamic

properties of HFO-1234yf (2, 3, 3, 3-tetrafluoropropene). International Journal ofRefrigeration 33, 474-479.

Woerdman, E., (2000). Implementing the Kyotoprotocol: why JI and CDM show more promisethan international emissions trading. EnergyPolicy 28, 29-38.

Wojtkowiak, J., Popiel, C.O., (2000). Effect ofcooling on pressure losses in u-type wavy pipeflow. International Communications in Heatand Mass Transfer 27, 169-177.

Yoon-Yeong, B., Hwan-Yeol, K., (2009). Convectiveheat transfer to CO2 at a supercritical pressureflowing vertically upward in tubes and anannular channel. Experimental Thermal andFluid Science 33, 329-339.

Yun, R., Hwang, Y., Radermacher, R., (2007).Convective gas cooling heat transfer andpressure drop characteristics of supercriticalCO2/oil mixture in a minichannel tube.International Journal of Heat and MassTransfer 50, 4796-4804.

Zhao, Y., Ohadi, M.O., (2004). ExperimentalStudy of Supercritical CO2 Gas Cooling in aMicrochannel Gas Cooler, ASHRAE.

www.rmiq.org 229

revista mexicana de ngeniería - scielo méxico · 2019-09-30 · technology for refrigeration and...

Documents