Scientia Iranica B (2014) 21(2), 449{455

Sharif University of TechnologyScientia Iranica

Transactions B: Mechanical Engineeringwww.scientiairanica.com

Research Note

Numerical solution of homogeneous double pipe heatexchanger: Dynamic modeling

H. AliHosseinpoura, Y. Kazemib and M. Fattahia,c;�

a. Department of Chemical and Petroleum Engineering, Sharif University of Technology, Tehran, P.O. Box 11155-9465, Iran.b. Department of Chemical Engineering, Razi University, Kermanshah, Iran.c. Department of Chemical Engineering, Abadan Faculty of Petroleum Engineering, Petroleum University of Technology, Abadan,

Iran.

Received 11 April 2012; received in revised form 2 September 2013; accepted 19 October 2013

KEYWORDSHeat exchanger;Homogeneousmodeling;Double pipe;Structured mesh;Finite di�erence.

Abstract. Dynamic modeling of a double-pipe heat exchanger is the subject of the currentstudy. The basis of this study is the same velocity of vapor and liquid phases or, in otherwords, homogeneous phase, in the annulus part of the exchanger. The model can predictthe temperature and vapor quality along the axial pipe from the pipe inlet up to a distancewhere steady state conditions are achieved. The simulation is conducted for two modesof co- and counter- ow in a one dimensional transient system. The physical propertiesof water are estimated from empirical correlation and a saturated vapor table with cubicspline interpolation. The exchanger model, which is a set of Ordinary Di�erential Equations(ODEs), ODEs and algebraic equations, has been solved numerically. Modeling results havebeen investigated for di�erent operating times and two modes of co- and counter-current.© 2014 Sharif University of Technology. All rights reserved.

1. Introduction

Heat exchangers are widely used in industrial appli-cation processes such as power plants, gas turbines,air conditioning, refrigeration, (domestic, urban, orcentral) heating, and cryogenic systems, among manyothers. Their universal application has led to researchinto a better comprehension of their dynamic behavior,modeling, simulation, identi�cation, and control [1].Many reports on numerical and experimental simula-tions for laminar or turbulent ow and heat transfer,concerning the employment of solid extended surfaces,such as �ns and ba�es, can be found in the litera-ture [2,3]. Most of these works discuss the optimalspacing, shapes and orientations of these structures,

*. Corresponding author. Tel.: +98 918 334 8863E-mail addresses: hap a63@yahoo.com (H.AliHosseinpour); yaser kazemy@yahoo.com (Y. Kazemi);moslem.fattahi@gmail.com (M. Fattahi)

which enhance the heat transfer performance for agiven pumping power or ow rate [4]. The objective ofany such equipment is to maximize the heat transferredbetween the two uids. Therefore, a design which in-creases the heat transferred, but simultaneously keepsthe pressure drop of the uid owing in the pipesto permissible limits, is very necessary. A commonproblem in industry is to extract maximum heat from autility stream coming out of a particular process, andto heat a process stream [5]. The design of the heatexchanger might be developed by means of analyticalmethods. These methods give a quick and globalapproach to their behaviour. However, a large numberof hypotheses and simpli�cations have to be assumed.Examples are the F�factor or "�NTU methods [6].More general and accurate approaches require the useof numerical methodologies, which subdivide the heatexchanger into many elemental volumes and solve thegoverning equations for each volume. In the two-phase ow region, the governing equations (mass, momentum

450 H. AliHosseinpour et al./Scientia Iranica, Transactions B: Mechanical Engineering 21 (2014) 449{455

and energy) can be formulated in di�erent forms, de-pending on the model used. Homogeneous models [7],drift ux models [8] or two- uid models [9,10] canbe employed to solve the two-phase ow present inthe condensation and evaporation process. To get adeep understanding of the mathematical model andthe strategies for solving the governing equations, thedouble�pipe heat exchanger may be a good optionfor application, due to its relatively simple geometryfor the secondary ow. For example, numerical andanalytical results in steady and transient states havebeen compared in di�erent studies on double�pipeheat exchangers [1,11,12]. A combination of analyticalexpressions with numerical methods is used in thedouble�pipe helical heat exchanger resolution [13],where a CFD modeling, together with a "�NTUmethod, is used to solve the heat exchanger.

This paper shows a more general approach forsimulation of the two-phase ow inside the annulus andliquid inside the center tube. The uid region equationsin the heat exchanger were discretized and a simulationwas performed for one dimension. Moreover, variationof uid properties in a radial direction was neglected.First, details of the mathematical formulation andthe numerical techniques are shown. Then, numericalaspects are presented and, �nally, the results obtainedfrom comparisons of co� and counter�current operat-ing conditions will be demonstrated.

2. Model description

Taking into account the following basic assumptions,the governing equations have been derived:

� Homogeneous phase assumes a slip ratio (G =Uv=U1) equal to unity (both phases travel at thesame velocity);

� One-dimensional ow is performed;� The exchanger is horizontal, so the gravity e�ects

are omitted;� The wall of the heat exchanger is smooth and does

not apply friction e�ects on owing uids;� The velocity pro�le is only a function of uid

direction, and radial dependency is neglected;� The system is adiabatic and insulated (i.e. heat

transfer through outer surface is neglected);� The uid owing in the inner tube is incompressible

and single phase;� The outlet vapor as well as the inlet vapor is in a

saturated state;� Pressure drops across vapor and liquid phases in the

annulus tube are the same (dPv = dPl = dP );� In the annulus tube, the temperatures of vapor and

liquid at each node are the same and are equal to

saturated temperature at node pressure (Tv = Tl =Tsat);

� Axial heat conduction inside the uid is neglected;� Working uid in the tube and the annulus sections

is water;� The inlet vapor into the annulus tube is saturated

with a quality of one and the inlet liquid in the innertube is in a saturated condition.

As noted in the above assumptions, the coolingliquid temperature increases and the steam tempera-ture decreases. Consequently, the saturated vapor inthe annulus part may provide two phases with theelevation of pressure drop. The vapor remains in asaturated condition through the exchanger, while itstemperature, pressure and quality might vary along theexchanger and/or with time.

2.1. Governing equations in annulus tubeUsually, �rst law of thermodynamics for open systems,and considering the velocity term as u = _mt=(�tAt) inkinetic terms, gives:

@@t

(�tCpAtT sat) + _mtddz

(ht +12u2)

+ 2�RoUoverall(T sat � T ) = 0: (1)

Based upon the Maxwell's relations the following equa-tions are obtained:(

dh� = T�dS� + ��dP�dhl = TldSl + �ldPl

) dhtdz

= hl�dxdz

+ CptdT sat

dz+ �t

dPdz; (2)

hl� = h� � hl: (3)

The following equation, known as the Clapeyronequation, demonstrates the relation between saturatedtemperature and pressure:

dPdT sat =

hl�T sat�l�

; (4)

�l� = �� � �l: (5)

Finally, the governing equation for the annulus pipe(Eq. (6)) would be obtained by combining Eqs. (2)to (5) with the initial and boundary conditions(Eq. (7)):

@@t

(�tCpAtT sat) + _mt(hl�dxdz

+ CptdT sat

dz

+ �thl�

T sat�l�dT sat

dz+

12du2

dz)

+ 2�RoUoverall(T sat � T ) = 0; (6)

H. AliHosseinpour et al./Scientia Iranica, Transactions B: Mechanical Engineering 21 (2014) 449{455 451

(I:C: T sat = T sat

in at (z; t = 0)B:C: T sat = T sat

in at (z = 0; t)(7)

At = �(R2t �R2

o): (8)

2.2. Governing equations in the center tube2.2.1. Governing equations in center tube for

co-current streamThe governing equations for the center pipe are valid,the same as for the annulus tube, however, the vaporquality is zero. Therefore, the equation might beexpressed as follows:

@@t

(�lCplAiT ) + _mldhldz

� 2�RoUoverall(T sat � T ) = 0; (9)

Ai = �R2i ; (10)

in which, the initial and boundary conditions are:(I:C: T = T cin at (z; t = 0)B:C: T = T cin at (z = 0; t)

(11)

2.2.2. Governing equations in the center tube forcounter-current stream

According to the system of equations presented, thecounter current condition formulation is given by:

@@t

(�lCplAiT )� _mldhldz

� 2�RoUoverall(T sat � T ) = 0; (12)

whereas, boundary and initial conditions are:(I:C: T = T cin at (z; t = 0)B:C: T = T cin at (z = L; t)

(13)

In the aforementioned equations, the initial and bound-ary conditions are the same. This means that, initially,the annulus tube is �lled with saturated vapor andin the inner tube, cooling liquid ows without heattransfer and, suddenly, the streams switch to exchangethe heat. Utilization of the above initial conditions isguaranteed to avoid division by zero at the beginningof the solution. Besides, this technique leads to theexact modeling of the equations.

3. Physical properties

The physical properties of the heat exchanger, suchas material selection, or uid properties in the innerand outer tubes, and overall heat transfer coe�cient ateach point during the time, were calculated using steamtables by cubic spline interpolation or experimentalrelations that are listed in Table 1.

4. Numerical framework

According to the heat exchanger design, only axialheat transfer is considered and the system is meshedjust in this direction. In order to simplify the calcula-tions, the meshes are assumed uniform and the meshcentered method of calculation was investigated. Thecalculation path reported here has been performed withstructured mesh. Within this scheme, the linearizationof equations was done. The algorithm of solution isillustrated in Figure 1. The resulting ordinary di�er-ential equations were solved simultaneously using the�nite di�erence technique under initial and boundaryconditions. The discrete formulation of the equationswas fully implicit. At each step, the resulting set oflinear algebraic equations was solved by the iterativeGauss-Seidel method with the relaxation factor. Solu-tion of the previous step was used as the initial guessfor the iterative procedure. A computer program usingMATLAB software (version 2010a) was developed toperform the above algorithm for numerical solutions.

The Clapeyron equation term in the annulus sideequations causes nonlinearity. In order to obviate thisrestriction, the previous step time is used to calculate

Figure 1. Algorithm of numerical solution.

452 H. AliHosseinpour et al./Scientia Iranica, Transactions B: Mechanical Engineering 21 (2014) 449{455

Table 1. Estimation of physical properties of uids.

Properties Reference

sl, s� , �l, �l, �� , �� , h� , hl [10]

�[Pa.s] = A� 10( BT�C )

[12]A = 2:414� 10�5 [Pa.s]B = 247:8 [K]C = 140 [K]

Kwater

�Wm:K

�= 1� 10�5T 2 � 0:0042T + 0:7222 [11]

Kvapor

�Wm:K

�= 8:3154� 10�5T � 7:4556� 10�3

Kst.steel

�Wm:K

�= 0:0667T � 10:2 for T � 273K [11]

h = KfluidNuDH

[11]Nu =

8>><>>:0:023Re0:8Prn; 0:6 � Pr � 100

0:0214(Re0:8 � 100)Pr0:4; 0:5 � Pr � 1:5 and 104 � Re � 5� 106

0:0120(Re0:87 � 280)Pr0:4; 1:5 � Pr � 500 and 3� 103 � Re � 106

Pr = Cp�K

Re = �uDH�

DH =

8<:2Ri; for inner tube

2(Rt �Ro); for annulus

Uoverall =�

AoAihl

+Ao ln(RoRi

)

2�Kwall+ 1

ht

��1

[11]

the temperature. Thereby, a set of linear equationswith constant coe�cients is obtained that might besolved by the iterative Gauss-Seidel method. In orderto minimize errors caused by the above simpli�cations,at each step, the equations are solved using the physicalproperties of the previous step (lagging). Then, ob-tained solutions are used for calculation of new physicalproperties and this repetitious loop, at each step, wouldbe iterated over and over again up to when the propertyapproaches a �xed amount. Based on the energybalance at each node, the vapor quality at each step,due to the formation of liquid from vapor, is computedas follows:

q = _mtCpt(T ksat � T k+1sat ) = _mcond.hlv; (14)

_mcond. =_mtCpt(T ksat � T k+1

sat )hlv

; (15)

x =_mt � _mcond.

_mt: (16)

Reduction of temperature during time steps causesthe condensation of the vapor. On the other hand,considering the same velocity assumption for vaporand liquid phases at each node, the total ow rate

is considered constant. However, the velocity at eachnode varies from others, due to di�erent densities.Time steps in this method should be selected smallenough to obtain reasonable solutions.

5. Results and discussion

In this paper, two scenarios of exchanger characteristicswere used as the boundary conditions, as presentedin Table 2. The scenarios are the same, except themass ow rate of vapor. Variations of quality andtemperature along the inner and annulus axial pipe,with two scenarios for co- and counter-current streams,are depicted in Figures 2-5.

The quality of vapor and temperature along theheat exchanger with a counter ow arrangement, underconditions of scenario I, is shown in Figure 2(a) and (b),respectively. As assumed, the exchanger is �lledwith pure and saturated vapor and a cooling liquid,which suddenly comes to exchange heat transfer. Thesaturated vapor is brought to a heat saturated liquidwith a great initial heat transfer rate, leading to anextreme reduction in vapor temperature and quality.Thereafter, steady state conditions are achieved con-tinuously. As results show, the steady state conditions

H. AliHosseinpour et al./Scientia Iranica, Transactions B: Mechanical Engineering 21 (2014) 449{455 453

Table 2. The scenarios of double pipe heat exchanger.

Properties ScenarioI

ScenarioII

Outer tube inner diameter (cm) 13 13Inner tube inner diameter (cm) 5.4 5.4Inner tube thickness (mm) 1 1Heat exchanger length (cm) 300 300Cold water mass ow rate (kg/s) 1 1Vapor mass ow rate (kg/s) 0:1 0:5Inlet water temperature (K) 300:15 300:15Inlet vapor temperature (K) 373:15 373:15

Figure 2. Vapor quality and temperature forcounter-current versus length of double pipe heatexchanger under scenario I with time step of 0:01 seconds.

in the heat exchanger would be obtained after 120seconds.

The temperature and vapor quality for thecounter ow arrangement under scenario II is shown inFigure 3(a) and (b). The di�erence between scenariosI and II is just the mass ow rate of saturated vapor.As Figure 3(a) clearly indicates, the reduction in vapor

Figure 3. Vapor quality and temperature forcounter-current versus length of double pipe heatexchanger under scenario II with time step of 0:01 seconds.

quality in scenario II is lower than in scenario I. FromFigure 3(b), in scenario II, the uid temperature ofthe inner tube increases further and, also, the vaportemperature reduction is less; in other words, thee�ciency is lower.

The quality of vapor and temperature along theexchanger, with a co-current ow arrangement underconditions of scenario I, are shown in Figure 4(a)and (b). The saturated vapor is brought to a heatsaturated liquid with a great initial heat transfer rate,leading to an extreme reduction in vapor temperatureand quality. Thereafter, steady state conditions areachieved continuously. As the results show, the steadystate conditions in the heat exchanger will be obtainedafter 120 seconds. In co-current ow close to theoutlet end of the exchanger, the temperature di�er-ence between inner and annulus tubes tends to zero.Therefore, the exchanger length could be decreased foreconomical considerations.

Figure 5(a) and (b) depict the vapor quality andtemperature along the double pipe heat exchanger for

454 H. AliHosseinpour et al./Scientia Iranica, Transactions B: Mechanical Engineering 21 (2014) 449{455

Figure 4. Vapor quality and temperature for co-currentversus length of double pipe heat exchanger underscenario I with time step of 0:01 second.

co-current ow under scenario II. Trends similar tothose of Figure 3(a) and (b) are observed. It is seenthat the temperature increased sharply along the pipeclose to the inlet end and, �nally, reached a nearlyconstant value due to a reduction in heat transferrate. Furthermore, it is observed that the vapor qualitydecreases through the exchanger.

Since inlet vapor is saturated, pressure distribu-tion at each node can be obtained simply by usingsteam tables. The reduction of vapor quality andelevation of inlet liquid temperature in the co-currentcon�guration is less than in the counter current. Thisresult shows that heat transfer e�ciency in the countercurrent arrangement is higher than in the co-currentarrangement.

6. Conclusions

The present study is a numerical analysis of heattransfer in a double pipe heat exchanger with twocon�gurations. The obtained results are exploited by

Figure 5. Vapor quality and temperature for co-currentversus length of double pipe heat exchanger underscenario II with time step of 0:01 second.

highlighting the derived equations at the inner andannulus pipes. Physical parameters were predictedfrom correlations on the structure of the �elds. Thetemperature and vapor quality along the axial pipefrom the inlet, ending at a distance where steady stateconditions are achieved, is predicted. The reduction ofvapor quality in the co-current condition is less than inthe counter current, which shows that the heat transfere�ciency of the counter current is more than that ofthe co-current. When the mass ow rate of the vapor isincreased, with the constant mass ow rate of coolant,heat transfer e�ciency decreases, as expected.

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Biographies

Hossein AliHosseinpour obtained BS and MS de-grees in Chemical Engineering from the Departmentof Chemical and Petroleum Engineering at SharifUniversity of Technology, Tehran, Iran. His researchinterests include membrane synthesis and modeling inall �elds of chemical engineering.

Yaser Kazemi obtained BS and MS degrees in Chem-ical Engineering from the Department of ChemicalEngineering at Razi University, Kermanshah, Iran, in2006 and 2009, respectively. His research interestsinclude membrane preparation and modeling in the�eld of chemical engineering.

Moslem Fattahi obtained a BS degree in Chemi-cal Engineering, in 2006, from Razi University, Ker-manshah, Iran, and his MS and PhD degrees, in2008 and 2013 from the Department of Chemicaland Petroleum Engineering at Sharif University ofTechnology, Tehran, Iran. His main research interestsinclude modeling and reaction engineering, catalystpreparation, characterization and evaluation in the�eld of chemical engineering.