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Applied Thermal Engineering 65 (2014) 369e383

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Applied Thermal Engineering

journal homepage: www.elsevier .com/locate/apthermeng

A comparison of four numerical modeling approaches for enhancedshell-and-tube heat exchangers with experimental validation

Jie Yang a,b, Lei Ma a, Jessica Bock b, Anthony M. Jacobi b, Wei Liu a,*

a School of Energy and Power Engineering, Huazhong University of Science and Technology, Wuhan 430074, ChinabMechanical Science and Engineering, University of Illinois, Urbana, IL, USA

h i g h l i g h t s

� Four different modeling approaches are developed for rod baffles heat exchanger.� Heat transfer and pressure drop of computational calculations are studied.� Experimental validation is carried out to verify the numerical results.� The comparisons of four various modeling approaches are conducted.

a r t i c l e i n f o

Article history:Received 18 September 2013Accepted 19 January 2014Available online 28 January 2014

Keywords:Rod-baffle heat exchangerTurbulenceNumerical simulationExperimental validation

* Corresponding author. Tel.: þ86 27 87541998; faxE-mail addresses: jieyang2056@gmail.com, jieyang

hust.edu.cn (W. Liu).

1359-4311/$ e see front matter � 2014 Elsevier Ltd.http://dx.doi.org/10.1016/j.applthermaleng.2014.01.03

a b s t r a c t

In the present paper, 3-D numerical simulations of a rod-baffle shell-and-tube heat exchanger with fourdifferent modeling approaches are developed and validated with experimental results. The four methodsof modeling include two in which a small subsection of the heat exchanger is modeled (the unit model,and the periodic model), one in which the heat exchanger is consider as a porous medium (the porousmodel), and one in which the entire heat exchanger is modeled with CFD (the whole model). The resultsillustrate that the periodic model, porous model and whole model can have high accuracy in predictingheat transfer, while the unit model has relatively low accuracy. The porous model and whole model alsoprovide good predictions of the pressure drop, but the unit model and periodic model fail to accuratelypredict pressure drop. The porous model requires accurate heat transfer correlations for the heatexchanger, and such correlations may not be available for new designs. The whole model demandssignificant computational resources for geometric modeling, grid generation, and numerical calculation.A demonstration of different grid systems for various models is also conducted. In summary, the presentwork provides a comparison of various modeling approaches and an analysis of trade-offs betweennumerical accuracy and computational demands for models of shell-and-tube heat exchangers.

� 2014 Elsevier Ltd. All rights reserved.

1. Introduction

Shell-and-tube heat exchangers are widely used in the petro-chemical industry, manufacturing industry, food preservation,electrical power production and energy conservation systems, dueto their structural simplicity, relatively low cost and design adapt-ability. According to Master and co-workers, they account for morethan 35%w40% of the heat exchangers used in global heat transferprocesses [1]. Many novel structures, such as segmental baffles [2],double segmental baffles [3], multi-segmental baffles [4], orifice

: þ86 27 87540724.5@uiuc.edu (J. Yang), w_liu@

All rights reserved.5

baffles [5], rod baffle [6], self-braced structure with twisted tubes[7], helical baffles [8] and ring baffles [9] have been suggested toenhance heat transfer and reduce pumping power. Among theabove-mentioned enhancement techniques, the rod-baffle heatexchanger exhibits excellent thermal-hydraulic performance andwas selected for the current study. The rod-baffle design was firstproposed by Phillips Petroleum Company in 1976 to solve flow-induced vibration of tube bundles, fouling and corrosion prob-lems of shell-side [10]; the design has found broad applicationwithadvantages of higher heat transfer efficiency, lower pressureloss, larger effective heat transfer area, better anti-fouling perfor-mance and so on compared to the segmented baffle heat exchanger[11e13].

Experiments can provide highly reliable measurements ofthermal-hydraulic performance; however, experiments can be

Nomenclature

A hydraulic area (m2)cp specific heat capacity (kJ kg�1 K�1)C empirical constant (�)di inner tube diameter (m)do outer tube diameter (m)D hydraulic diameter (m)Ds inner shell diameter (m)F LMTD correction factor (�)fv overall volume porosity (�)fvo local volume porosity (�)fs surface permeability (�)G producing term (kg m�1 s�1)H heat transfer coefficient (W m�2 K�1)K turbulent kinetic energy (m2 s�2)m mass flow rate (kg s�1)n tube quantity (�)Nu Nusselt number (�)Q heat transfer capacity (J)Re Reynolds number (�)Sk additional term of k (kg m�1 s�1)S 3 additional term of 3(m3 s�1)t time (s)T temperature (K)

u velocity (m s�1)U uncertainty (�)m dynamic viscosity, kg/(m s)3 dissipation rate (m2 s�1)DP pressure drop (Pa)D LMTD (T)r density (kg m�3)l thermal conductivity (W m�1 K�1)h tube pitch (m)s empirical constant (�)

Subscriptsave average valuec cold waterci inlet cold waterco outlet cold water3 dissipation rate termeff effective termf fluidh hot waterhi inlet hot waterho outlet hot wateri, j tensork kinetic energy termx, y, z coordinate axis

J. Yang et al. / Applied Thermal Engineering 65 (2014) 369e383370

extremely expensive and time-consuming compared to computa-tional fluid dynamics (CFD). Unfortunately, for very complex flows,such as those prevailing in the rod-baffle shell-and-tube heatexchanger, selecting an appropriate modeling approach can bedifficult. There are complex tradeoffs between accuracy andcomputational expense. A heat exchanger with 500 heat transfertubes and 10 baffles requires at least 150 million computationalcells to resolve the geometry [14].

Research [15e29] about different models has been conducted.There are four main modeling approaches used for numericalsimulation: the unit model, the periodic model, the porous model andthe whole model. The unit model [15e17] regards the flow zone inshell-and-tube heat exchangers as a unit flow channel. As shown inFig. 1, the unit flow channel is restricted by four virtual boundarywalls and four quarter-tube walls. In most cases, the virtual wallsare set as symmetric boundary conditions and the tubewalls are setas solid-wall boundary conditions. Obviously, the unit model ig-nores the inlet and outlet nozzles effects. Dong et al. [15] simulateda flow unit duct model of rod-baffle heat exchanger and conductedcomparisons to both correlations and experiments. The resultsindicated that the model demonstrated a high precision in pre-dicting heat transfer coefficient and pressure drop, even though the

Fig. 1. The unit model.

model did not take the inlet and outlet nozzles and frictionalresistance of inner shell wall into account. Liu et al. [16] applied theunit model for a new longitudinal flow shell-and-tube heatexchanger (the rod-vane compound baffle heat exchanger) andcompared to the rod-baffle heat exchanger. The results demon-strated that the unit model is applicable for longitudinal flow heatexchanger. You et al. [17] used the unit model of a trefoil-holebaffled shell-and-tube heat exchanger to visually analyze the flowfield and one working point was quantitatively compared toexperimental data, which displayed a good accuracy in thermal-hydraulic performance. The periodic model [18e20] regards theflow as periodically fully developed by setting the inlet and outletas periodic boundary conditions. The periodic model also neglectsthe inlet and outlet effects of the shell-side. Gu et al. [18] studiedthe symmetrically periodic model for segmented-baffle and rod-baffle heat exchanger to overcome some insufficiencies of theprevious modeling methods, but the results were not validated byany correlations or experiments. Lei et al. [19] conducted simula-tions of a helical-baffle heat exchanger with different baffle incli-nation angles using a periodic model as the helically baffle heatexchangers are periodic along the flow direction. A comparison ofnumerical and experimental results was carried out by examiningthe average Nusselt number and friction factor, and the resultsindicated a good consistency. Zhang et al. [20] carried out perfor-mance predictions of a heat exchanger with middle-overlappedhelical baffles with different helix angles using the simplified pe-riodic model, and they analyzed the pressure drop inconsistencybetween simulation and experiment, mainly resulting from inletand outlet nozzles. The model with volume porosity, surfacepermeability and distributed flow resistances [21e24] regards flowzones as porous media without heat transfer tubes, so that a rela-tively coarse grid can be used. The porous model eliminates tubebundles and achieves accuracy by introducing volume porosity andsurface permeability as parameters, adding distributed heat sour-ces and distributed flow resistances that are determined fromempirical correlations. Obviously, the porous model is not suitable

J. Yang et al. / Applied Thermal Engineering 65 (2014) 369e383 371

for heat exchangers where there are no correlations available. Pri-thiviraj et al. [14,21] numerically investigated a conventional shell-and-tube heat exchanger using the distributed resistance approach,which regards that every single computational cell may havemultiple tube bundles leading to a relatively coarse grid system.Sha et al. [22] improved this porous model by adding surfacepermeability and volumetric porosity. By doing this, the porousmodel can obtain higher precision while keep the computationalresources in an acceptable limit. Later He et al. [23] modified theporous model in conjunction with k- 3two-equation and extendedthe application of the porous model into three kinds of heat ex-changers (vertical baffles, helical baffles, and finned tube banks).Recently Shi et al. [24] performed a semi-porous media approachusing a large-scale sparse tubular heat exchanger and compared itto the conventional porous model with the experimental valida-tion. The results showed that the semi-porousmodel can predict airflow distribution, reduce computational cost significantly, and beeasily implemented. The whole model is applied when simplifica-tions are not appropriate and for small, simple heat exchangers[25e29]. Although the whole model can provide highly accurateheat transfer and pressure drop performance predictions, it con-sumes more computational resources to build the model, generatecells, and complete the calculations. The applications of differenthelically baffled heat exchangers with various geometries [25e28]demonstrated that high precision of thermal-hydraulic predictionsof the whole model was achieved by a large grid system. Ozdenet al. [28] numerically investigated the effects of different turbulentmodels on a small segmental baffle shell-and-tube heat exchangerwith nine tubes due to the limitations of computational resourceswith the emphasis on choosing the proper mesh density, order ofdiscretization, and turbulence model. The novelty and significanceof the above research rest in providing four different numericalapproaches, illustrating implementation process of the modelingmethods, and demonstrating wide case applications. Once a newheat transfer enhancement technique towards heat exchangersapplication is invented, certain criteria are required to evaluate theperformance and effectiveness. So far, the performance evaluationcriteria (PEC) [29e39] have been proposed and successfully utilizedto analyze and evaluate the thermal-hydraulic performance.

Although the realization of four different models is wellestablished, a criterion of how to select the most appropriatenumerical approach according to corresponding circumstanceshas not been built yet. In other words, despite much numericalresearch on shell-and-tube heat exchangers, a trade-off betweennumerical accuracy and computational resources has seldombeen considered in the above literature review. As a matter of fact,a comprehensive comparison about thermal-hydraulic perfor-mance accuracy, consumed computational resources, and appli-cability of the above-mentioned four different modelingapproaches has not been found in the open literature; such acomparison is especially important for large-scale heat ex-changers [40]. Therefore it is necessary to compare differentmodels and elucidate the advantages and disadvantages of theapproaches currently available to heat exchanger designers. Inthis paper, four different models, the unit model, the periodicmodel, the porous model, and the whole model of a rod-baffleshell-and-tube heat exchanger, are adopted for computationalstudy and an experimental system is used to validate and assessthe numerical results. The present work conducts a comparisonon numerical accuracy, grid system size, computational period,and restriction for four different models, analyzing trade-offsbetween computational accuracy and demands, providing theadvantages and disadvantages of each modeling approach to ac-ademic researchers and engineering designers, and filling theexisting gap in the open literature.

2. Physical model

2.1. Model introduction

The geometrical configuration of the rod-baffle heat exchangerfor four different models is presented in Fig. 2. Rod baffles aremanufactured from an array of support rods that arewelded at eachend to a circumferential baffle ring, and a variable number of bafflesare placed along the flow direction in alternating orientation(vertically arranged, horizontally arranged, vertically arrangedagain, etc) so that, together, they fix tubes in fixed locations. As seenin Fig. 2, the rod baffles are perpendicular to each other to brace theheat transfer tubes. For the unit model, the unit flow channel isformed by the outer walls of four tubes and a square boundary wall.Several cylinders arranged in a perpendicular way are inserted inthe flow zone as rod baffles. For the periodic model, the flowchannel is formed with one complete rod baffle placed betweentube bundles. For the porous model, the exchanger with rod baffles,the inlet and outlet of the shell-side is modeled without the pres-ence of tubes bundles. For the whole model, a detailed heatexchanger with heat transfer tubes, the inlet and outlet for theshell-side, connecting rods and rod baffles are modeled. All pa-rameters of numerical simulations are identical with the experi-mental setup. Regular ke 3equations are adopted for turbulent flowzone, and these equations will be expressed in the followingcontent.

2.2. Governing equations, grid generation and boundary conditions

2.2.1. Governing equations for regular modelsThe conservation equations for the regular computational

models are presented in the tensor form in the Cartesian coordinatesystem as the flow is steady and the fluid is incompressible [41,42].

Continuity equation:

vujvxj

¼ 0 (1)

Momentum equation:

r$v�uiuj

�vxj

¼ �vpivxi

þ v

vxj

m

vuivxj

þ vujvxi

!!(2)

Energy equation:

r$v�ujT�

vxj¼ v

vxj

l

cp

vTvxj

!(3)

k-equation:

r$vðkuiÞvxi

¼ v

vxj

"�mþ mt

sk

�vkvxj

#þ Gk � r 3 (4)

3-equation:

r$vð 3uiÞvxi

¼ v

vxj

"�mþ mt

33

�v 3

vxj

#þ C1 3 3

kGk � C2 3r

32

k(5)

where r is fluid density, l is thermal conductivity, cp is specific heatcapacity, p is pressure, m is the dynamic viscosity, k is turbulentkinetic energy, 3is turbulent dissipation rate, Gk is producing termof turbulent kinetic energy generated by mean velocity gradient,C1 3and C2 3are empirical constants, sk and s 3are Prandtl numberscorresponding to turbulent kinetic energy and turbulent dissipa-tion rate, mt is defined as follows:

Fig. 2. Geometrical diagram of four different models.

J. Yang et al. / Applied Thermal Engineering 65 (2014) 369e383372

mt ¼ rCmk2

3(6)

where Cm ¼ 0.09, C1 3¼ 1.44, C2 3¼ 1.92, sk ¼ 1.0, s 3¼ 1.3 and Gk isdefined as follows:

Gk ¼ mt

vuivxj

þ vujvxi

!vuivxj

(7)

2.2.2. Governing equations for the model based on volume porosity,surface permeability and distributed flow resistances

The conservation equations for the computational model on thebasis of volume porosity, surface permeability and distributed flowresistances are presented in the tensor form in the Cartesian co-ordinate system [21e24,42].

Continuity equation:

vujvxj

¼ 0 (8)

Momentum equation:

r$v�fsuiuj

�vxj

¼ �vðfspiÞvxi

þ v

vxj

fsmeff

vuivxj

þ vujvxi

!!� fi

r

2juijui

(9)

Energy equation:

r$v�fsujT

�vxj

¼ v

vxj

fsleffcp

vTvxj

!þ Uq

cp(10)

k-equation:

r$vðfskuiÞ

vxi¼ v

vxj

"fs

�mþ mt

sk

�vkvxj

#þ fvðGk � r 3Þ þ Sk (11)

3-equation:

r$vðfs 3uiÞvxi

¼ v

vxj

"fs

�mþ mt

s 3

�v 3

vxj

#þ fv

�C1 3 3

kGk � C2 3r

32

k

�þ S 3

(12)

where fv stands for the volume porosity which equals to ratio offluid volume to fluid-plus-solid volume on a control volume, fsstands for the surface permeability which is equal to the ratio offluid area to fluid-plus-solid area on a control volume face.U standsfor the specific surface area of tube wall in the fluid-solid mixedregion, q is the internal heat source due to the absence of heattransfer tubes, meff and leff are the effective dynamic viscosity andthe effective thermal conductivity, which are equal to the sum oflaminar and turbulent counterparts. Sk and S 3 are the additional

Fig. 3. Meshes of unit model.

J. Yang et al. / Applied Thermal Engineering 65 (2014) 369e383 373

generation term of turbulent kinetic energy [42] and additionalgeneration term of turbulent kinetic energy dissipation rate [23]due to the absence of tube bundles, respectively, and defined asfollows:

Sk ¼ 12

Xi

firjuij3 (13)

S 3 ¼ 1:92 3

2k

Xi

firjuij3 (14)

The last term in Equation (9) e firjuijui/2 is the distributed flowresistance which accounts for the influence of heat transfer tubes.The friction coefficient fi [43] in the term is calculated by empiricalcorrelations as follows:

fx ¼ 4�Cxh

��hfv

h� D

�2�1� fv01� fv

�(15)

fy ¼ 4�Cyh

��hfv

h� D

�2�1� fv01� fv

�(16)

fz ¼ 4�Czh

��1� fv01� fv

�(17)

where D is the hydraulic diameter for shell-side, h is tube pitch, fvand fv0 are overall and local volume porosity factors, respectively.The local volume porosity factor [44] is defined as:

fv0 ¼ 1� p4

�doh

�2

(18)

Fig. 4. Meshes of p

where do is the outer diameter for tube and friction factors Cx, Cyand Cz [43] are given as follows:

For cross flow:

Cx ¼�0:619$Re�0:198

x Rex < 80001:156$Re�0:2647

x 8000 � Rex < 2� 105(19)

Cy ¼�0:619$Re�0:198

y Rex < 80001:156$Re�0:2647

y 8000 � Rex < 2� 105(20)

For parallel flow:

Cz ¼8<:

31=Rez Rex < 22500:131$Re�0:294

z 2250 � Rex < 250000:066$Re�0:227

z Rex � 25000(21)

where Reynolds number is based on the velocity components andthe outer diameter of tube.

2.2.3. Grid generationThe geometrical modeling and grid generation procedures are

carried out with commercial CFD preprocessor GAMBIT 2.3. Thegrid independence test was completed for each model. Taking theunit model as an example, five different grid systems with5.0 � 104, 1.9 � 105, 3.8 � 105, 5.3 � 105 and 7.5 � 105 cells areadopted for calculation. The differences in heat transfer coefficientand pressure drop between the third and fourth model are around7% and the differences between the fourth and fifth model arearound 2%. Thus, taking numerical resource cost and solution ac-curacy into consideration, the fourthmodel with 5.3�105 cells wasadopted. The grid diagrams for the unit model, periodic model,porous model and whole model are presented in Figs. 3e6. Due tothe complex structure of rod baffle, the models are discretized withhexahedral meshes for the most flow region and with tetrahedral

eriodic model.

Fig. 6. Meshes of whole model.

Fig. 5. Meshes of porous model.

J. Yang et al. / Applied Thermal Engineering 65 (2014) 369e383374

Table 1Thermo-physical properties of water.

Parameter Value

cp (J/kg K) 4182m (kg/m s) 0.001003r (kg/m3) 998.2l (W/m K) 0.6

J. Yang et al. / Applied Thermal Engineering 65 (2014) 369e383 375

meshes for the rod baffle region. After the grid independence test,the final cell numbers for the unit model, periodic model, porousmodel and whole model are 5.3 � 105, 1.4 � 106, 3.6 � 106 and2.5 � 107, respectively.

The commercial CDF software Fluent 6.3 was adopted for all thenumerical simulations. The 3-D, double-precision, pressure-basedsolver was used. The conservation equations are discretized with afinite volume formulation. In particular, for the model on the basisof volume porosity, surface permeability and distributed flow re-sistances, the influences of absence of tubes on the conservationequations of momentum, energy, turbulence kinetic energy andturbulence kinetic energy dissipation rate are treated by intro-ducing the corresponding source terms with six user definedfunctions (UDFs). The corresponding UDFs are programmed byMicrosoft Visual Cþþ, and the program is in the Appendix forconvenience.

2.2.4. Boundary conditionsThe standard wall function method is adopted for the near-wall

region, and the non-slip boundary condition is adopted on all solidsurfaces. The surfaces of the rod baffles are set as adiabatic becausethe impact caused by thermal conduction of the rod baffle can beneglected, and taking the rod baffle surface as adiabatic allows for acoarser grid. The inner wall of shell-side is also set as adiabaticbecause the heat exchanger is thermally isolated during theexperiment. The finite difference method and the second-orderupwind difference scheme are applied, and the SIMPLE algorithmis adopted for the coupling between pressure and velocity fields.

Fig. 7. Schematic diagram fo

The second-order upwind difference scheme is applied for energyand momentum computation, and the standard difference schemeis used for the pressure. For the unit model, porous model, andwhole model, the velocity-inlet boundary condition is applied forthe inlet since for incompressible fluid the velocity-inlet boundarycondition is equal to mass-flow-inlet boundary condition in Fluent,and the outflow boundary condition is applied for the outlet sincethe pressure for outlet is not given. The periodic boundary condi-tion is applied for inlet and outlet for the periodic model. Thetemperatures of the tube wall and inlet fluid are set as constants forthe unit model, periodic model and whole model. The fourboundary walls of the unit model are set as symmetry boundaryconditions. The other setting parameters adopted default settingsaccording to user’s guide in Fluent.Water is set as theworking fluid,and the corresponding parameters are presented in Table 1. Thefollowing assumptions are made to simplify numerical simulations:the thermalephysical properties of the fluid such as r, m, cp, l areconstant; the working fluid is incompressible, isotropic, Newtonianand continuous; the effect of gravity is negligible and viscousheating and thermal radiation are ignored.

3. Experimental apparatus

3.1. Experimental table and test section

The schematic for the experimental system is presented in Fig. 7.The system consists of three loops, which are hot water loop,cooling water loop and refrigerating loop. The hot water loopcontains a 58 kW electrical heater, water tank, hot water pump,flowmeters and tube-side of rod-baffle heat exchanger. The coolingwater loop contains the water side of a plate heat exchanger, watertank, cold water pump, flow meters and the shell-side of a rod-baffle heat exchanger. The refrigerating loop contains a 58 kWrefrigerating unit device, pump and the refrigerant side of a plateheat exchanger. When the experiment is in steady operation, theheat generated by electrical heater is transferred from the hotwaterto the cold water in the rod-baffle heat exchanger. Then the thermal

r experimental system.

Fig. 9. Diagram for a single rod baffle (unit: mm).

J. Yang et al. / Applied Thermal Engineering 65 (2014) 369e383376

power in coolingwater loop transfers heat fromwater to refrigerantat the plate heat exchanger. Finally, heat is rejected to ambient air.The volume flow rates for cold and hot water are measured usingfour rotary flow meters. The temperature of the water is measuredusing four K-type thermal couples that are inserted into holes at thehot water inlet, hot water outlet, cold water inlet, and cold wateroutlet, as shown in Fig. 7. The pressure drops between inlet andoutlet for the shell and the tube-side are measured using the twopressure drop transmitters. All data of temperature and pressuredifference are transmitted in the PC system and automaticallyrecorded through a data acquisition system. Experiments are con-ducted with various volume flow rates ranging from 5.5 to 19.1 m3/h since the measuring range for the rotary flow meters is 2.0e20 m3/h. The temperature ranges from 18.2 to 27.7 �C for shell-sideand 37.4e44.4 �C for tube-side.

The test section of the rod-baffle heat exchanger is shown inFig. 8. The dimensional parameters for the exchanger and rodbaffle are presented in Fig. 9 and Table 2. All thermal physicalparameters for water such as density, thermal conductivity andviscosity are adopted at the average temperature of inlet and outletand parameters for stainless steel are adopted at the averagetemperature of cold and hot water. The heat transfer from tube-side is approximately equal to the heat transfer to the shell-sidesince the shell is thermally insulated. During the experiment, allparameters are measured when the experiment is in steadyoperation for 10 min; namely, the energy balance deviation be-tween hot and cold water is less than 5% for each measurementthat is defined as follows:

h ¼ jQh � QcjðQÞmin

� 100% � 5% (22)

The experimental parameters in steady performance are listedin Table 3.

3.2. Data reduction

The Reynolds number for the shell-side is expressed as follows:

Res ¼ rsusDms

(23)

where us is the velocity for shell-side in the cross-section wherethere are no baffles, and is expressed as follows:

Fig. 8. Test section of shell a

us ¼ ms

r$

�p$D

2s4 � p$n$d

2o4

� (24)

and D is expressed as follows:

D ¼ p$D2s � p$n$d2o

p$Ds þ p$n$do(25)

where Ds is inner shell-side diameter and n is tube quantity. TheReynolds number for the tube-side is expressed as follows:

Ret ¼ rtutdimt

(26)

where ut is the velocity for tube-side and di is the inner diameter oftube. Nusselt number for shell-side is calculated as follows:

Nus ¼ hs$Dlf

(27)

where hs is the convective heat transfer coefficient for shell-sideand lf is thermal conductivity for fluid.

nd tube heat exchanger.

Table 2Structural parameters.

Material Stainless steel

Shell outer diameter 159 mmShell inner diameter 144 mmTube outer diameter 16 mmTube inner diameter 14 mmTube effective length 1000 mmTube pitch 22 mmTube number 21Baffle number 7Baffle thickness 5 mmRod diameter for baffle 6 mmBaffle pitch 120 mmBaffle distance from head 140 mmTube arrangement Non-staggered placement

J. Yang et al. / Applied Thermal Engineering 65 (2014) 369e383 377

For experimental research, the convective heat transfer coeffi-cient for shell-side hs is calculated as follows:

hs ¼ 11K � do

2ltln do

di� htdo

di

(28)

where lt is thermal conductivity for stainless steel, di is innerdiameter for tubes, ht is the convective heat transfer coefficient fortube-side and K is the overall heat transfer coefficient. The foulingfactor is not taken into consideration because the heat exchanger isnewly manufactured. Coefficient ht is calculated as follows:

ht ¼ Nut$lfdi

(29)

where lf is thermal conductivity for the fluid, and tube-side Nusseltnumber Nut is calculated using Gnielinski equation [45]. The overallheat transfer coefficient K is calculated as follows:

K ¼ QA$Dtm$F

(30)

where A is heat transfer area based on the outer diameter of heattransfer tubes, Q is heat transfer amount and Dtm is the log-meantemperature difference, F is the log-mean temperature differencecorrection factor according to TEMA standards [46].

As the uncertainties of Qh and Qc are quantitatively close, theheat transfer amount Q is calculated as follows for simplicity:

Q ¼ Qh þ Qc

2¼ mh$ChðThi � ThoÞ þmc$CcðTco � TciÞ

2(31)

where Qh and Qc are heat transfer amount for hot and cold water,mh and mc are mass flow rate for hot and cold water, Ch and Cc areheat capacity for hot and cold water, Thi, Tho, Tci and Tco are the hotwater inlet temperature, hot water outlet temperature, cold waterinlet temperature and cold water outlet temperature, respectively.It should be noticed that the arithmetic mean heat transfer amountis not preferred if the experimental uncertainties for both streamsare not identical, as Jacobi et al. presented in Ref. [47].

Table 3Experimental parameters.

Flow for hot water 5.6e18.9 m3/hFlow for cold water 5.45e19.1 m3/hInlet temperature for hot water 38.9e44.4 �COutlet temperature for hot water 37.4e40.9 �CInlet temperature for cold water 18.2e26.4 �COutlet temperature for cold water 21.7e27.7 �C

The log-mean temperature difference (LMTD) between hot andcold water is calculated as follows:

Dtm ¼ ðThi � TcoÞ � ðTho � TciÞln Thi�Tco

Tho�Tci

(32)

For numerical simulations, the convective heat transfer coeffi-cient for shell-side hs is calculated as follows:

hs ¼ QA$Dtm$F

zQ

A$ðTw � TaveÞ$F (33)

where Q is heat transfer amount, A is heat transfer area, Tw istemperature for tube wall and Tave is average temperature for theworking fluid. The above four parameters may vary in differentcomputational cases. As illustrated in Refs. [48], the difference be-tween arithmetic mean temperature difference (AMTD) betweenTw � Tin and Tw � Tout is close (4%) to LMTD when (Tw � Tin)/(Tw � Tout) � 2. In fact, the results in this work using differenttemperature differences are almost identical (less than 1%).Therefore, the latter expression in Equation (33) is adopted tocalculate the heat transfer coefficients for simplicity.

3.3. Uncertainty in experimental data

The test range and accuracy of measurements are given inTable 4. The uncertainties of the experimental data are calculatedwith the method of reference [49]. It involves calculating variablesderivatives with respect to each experimental quantity and appliesalready known uncertainties, which is calculated as follows:

UR ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiXni¼1

�vRvxi

Uxi

�2vuut (34)

where R ¼ f(x1, x2, ., xn) represents a desired variable, x1, x2, ., xnare the variables that impact R, and their absolute uncertainties areexpressed by Ux1, Ux2, ., Uxn, respectively. After calculation, theuncertainties for the Reynolds number, the total heat transfer, theoverall heat transfer coefficient, the shell-side heat transfer coef-ficient, the Nusselt number, and the pressure drop for both theshell- and tube-side are 0.26%e0.74%, 0.62%e1.16%, 4.64%e5.69%,4.64%e5.69%, 4.64%e5.69% and 0.99%e5.89%, respectively, whenthe Reynolds number ranges from 14,000 to 5000 as the largeReynolds number corresponds to small uncertainty in Table 5.

4. Analysis of numerical and experimental results

In this paper, the computational simulations were conductedunder the same conditions as the experiments. The comparisons ofNusselt number for four numerical models and experiment arepresented in Fig.10. It is clearly seen that the Nusselt number trendsof all data are similar, that is, Nusselt number increases with Rey-nolds number. For the unit model, it is noted that the heat transfercoefficient is relatively smaller than the experimental data. This ismainly because the boundary is set as symmetry wall condition,thus there is no heat and mass transfer on that boundary in

Table 4Test range and accuracy of instruments.

Instruments Range Accuracy

K-type thermocouples �100 to 100 �C 0.1 �CRotary flow meters for shell-side 2.0e20.0 m3/h 0.1 m3/hRotary flow meters for tube-side 2.0e20.0 m3/h 0.1 m3/hPressure drop transmitter for shell-side 320,000 Pa 100 Pa

4000 6000 8000 10000 12000 1400050

100

150

200

250

300

350

-20%

+20%

+20%

Nu

Re

unit modelperiodical modelporous modelwhole modelexperimental test

-20%

Fig. 10. Comparison of shell-side Nusselt number for different models and experiment.

4000 6000 8000 10000 12000 140000

2000

4000

6000

8000

10000

12000

-20%

+20%

-20%

+20%

P / P

a

Re

porous model whole modelexperiment test

Δ

Fig. 12. Comparison of shell-side pressure drop for porous model, whole model andexperiment.

J. Yang et al. / Applied Thermal Engineering 65 (2014) 369e383378

numerical calculation,while in fact there is plenty of turbulence andsecondary flow leading to heat and mass transfer on the virtualboundary in the experiment. The unit model also ignores theimpingent effects on the tube bundle caused by the inlet flowon theshell-side. Therefore, the unit model is preferable to be used forvisual analysis rather than quantitative analysis to some degree [11].For the periodicmodel, it is noted that theNusselt number is smallerthan experimental values for some data. But when the Reynoldsnumber exceeds 11,000, the Nusselt number for the periodic modelis larger than that of the experiments. It is expected that the periodicmodel is more suitable for heat exchangers of relatively large lengthas the inlet and outlet impingent effects would be neglected in thatcase. For the porous model, it is noted that the variation tendenciesfor numerical and experimental Nusselt numbers are very consis-tent. Quantitatively, the deviation of Nusselt number betweencalculation and experiment is 6.5e12.4%, which shows that theporous model has a relatively high accuracy for heat transfer pre-dictions. For the whole model, the values of Nusselt number arealmost identical to experiments for relatively lowReynolds number.The deviation from experimental data increases with the Reynoldsnumber. However, the Nusselt number trends for the whole modelare still in good agreement with experiments.

A comparison of pressure drop for the unit model and periodicmodel is presented in Fig. 11. The pressure drop in Fig. 10 is

4000 6000 8000 10000 12000 140000

2000

4000

6000

8000

10000

12000

14000

-40%

+40%

-40%

+40%

P / P

a

Re

unit model perodical modelexperiment test

Δ

Fig. 11. Comparison of shell-side pressure drop for unit model and periodic model.

calculated by multiplying the pressure drop of each unit or peri-odical region by the number of repetitions. For the unit model thereare twelve complete unit flow regions for thewhole heat exchanger,and for the periodical model there are six complete periodical flowregions for the whole heat exchanger. It should be noticed that thefrictional pressure drop for inner shell wall, inlet and outlet nozzlesof shell-side is not taken into consideration for the unit model, andthe pressure drop for the inlet and outlet nozzles is not takenconsideration for the periodical model. It is seen that the pressuredrop for the unitflowchannel is approximately 20% smaller than theexperimental data. It is also seen that the pressure drop for theperiodicalmodel is approximately 40% larger than the experimentaldata. In Ref. [20], amethod is provided to calculate the total pressuredrop, that is, multiply the cycle number by pressure drop in onecycle and add the pressure drop caused by the inlet and outletnozzles according to Gaddis and Gnielinski correlations [50].

As presented in Fig.12, the trends in pressure drop for the porousmodel and whole model are similar. The pressure drop for theporous model is smaller than experimental data, while that for thewholemodel is larger. Quantitatively, the deviation of pressure dropbetween calculation and experiment is 3.6e12.1% for the porousmodel and the difference is 0.7e12.4% for the whole model.

Numerical simulations for four methods of modeling differ ingeometry, flow fields, results and computational costs. The com-parison for different models with various cell numbers is given inTable 6. It is concluded that the porous model and whole modelhave high accuracy on predicting heat transfer, pressure drop andoverall performance but the porous model requires extra codes andknown correlations for source terms; the whole model requireslarge computational resources. It is also concluded that the unitmodel has lowaccuracy on predicting thermal characteristics and isunable to predict pressure drop due to much simplifications. It isalso concluded that the periodic model has high accuracy on heat

Table 5Experiment uncertainties.

Objects Uncertainty

Reynolds number �(0.26%e0.74%)Heat transfer amount �(0.62%e1.16%)Overall heat transfer coefficient �(4.64%e5.69%)Shell-side heat transfer coefficient �(4.64%e5.69%)Nusselt number �(4.64%e5.69%)Pressure drop �(0.99%e5.89%)

Table 6Comparison for different models of simulations.

Unitmodel

Periodicmodel

Porousmodel

Wholemodel

Cell number 530,000 1,400,000 3,600,000 25,000,000Precision of heat transfer Low High High HighPrecision of pressure drop � � High HighTime cost Short Short Medium LongRestriction None None Codes and

correlationsNone

J. Yang et al. / Applied Thermal Engineering 65 (2014) 369e383 379

transfer prediction and is more suitable for exchanger of largerlength. However it is unable to directly predict pressure drop.

5. Conclusions

In the present paper, four different modeling approaches of arod-baffle heat exchanger are developed, which are the unit model,the periodic model, the porous model, and the whole model. TheCFD software GAMBIT is used for the geometrical modeling andgrid generation procedures and FLUENT is used for all computa-tional calculations. An experimental system is built to validate theresults. The present work provides a comparison of the four nu-merical methods to fill the gap in open literature, the main con-clusions of which are as follows:

Appendix

1. The experiments validate the precision of each model in pre-dicting heat transfer and pressure drop with the experimentalvalidation. It is concluded that the periodicmodel, porousmodeland whole model have a high accuracy on predicting heattransfer performance, while the unit model has a relatively lowaccuracy; it is concluded that the porous model and wholemodel has high accuracy on predicting pressure drop, while unitmodel and periodic model are unable to directly predict hy-draulic performance.

2. Trade-offs between computational resources and accuracy areanalyzed. The whole model with highest accuracy consumeslargest resources on geometrical modeling, grid generation, andnumerical calculations; the porous model with relatively highprecision consumes medium numerical resources, however, itrequires extra codes and is not applicable for new designs; theperiodic model consumes relatively low resources and the unitmodel consumes the lowest.

Acknowledgements

This work is supported by the National Basic Research Programof China (Grant No. 2013CB228302) and the National Natural Sci-ence Foundation of China (Grant No. 51036003) and the ChinaScholarship Council.

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