International Journal of Heat and Mass Transfer 84 (2015) 282–292

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International Journal of Heat and Mass Transfer

journal homepage: www.elsevier .com/locate / i jhmt

Semi-numerical analysis of heat transfer performance of fractal basedtube bundle in shell-and-tube heat exchanger

http://dx.doi.org/10.1016/j.ijheatmasstransfer.2015.01.0380017-9310/� 2015 Elsevier Ltd. All rights reserved.

⇑ Corresponding author. Tel.: +86 25 58139956.E-mail address: zhoujianfeng@njtech.edu.cn (J.-f. Zhou).

Jian-feng Zhou ⇑, Shi-wei Wu, Yao Chen, Chun-lei ShaoJiangsu Key Laboratory of Process Enhancement and New Energy Equipment Technology, College of Mechanical and Power Engineering, Nanjing University of Technology,Nanjing 211816, Jiangsu, PR China

a r t i c l e i n f o

Article history:Received 26 December 2013Received in revised form 18 April 2014Accepted 5 January 2015

Keywords:Tube bundleShell-and-tube heat exchangerFractalNumerical simulationTemperature field

a b s t r a c t

A bundle of topologically arranged tubes based on fractal is proposed in this work to enhance the flow ofshell-side fluid. The space for arranging tubes is separated into some periodic regions and the tubes aresymmetrically arranged in these regions. The topological arrangement of tubes is in the radial directionstarting from the shell center. Fractal treatment is applied to divide each periodic region into two smallersymmetric ones. With the alternately installed disc and doughnut baffles, the shell-side fluid converges toshell center or diverges away from the center, and the uniform shell-side flow is realized. According tothe periodic characteristic of tube bundle, numerical heat transfer unit models are established and thecharacteristic temperatures in heat exchanger are obtained using the semi-numerical simulation algo-rithm. Comparing the results with the analytical solution to the outlet temperatures of shell-side andtube-side fluids based on the Bell-Delaware method, it is revealed that, even though the number densityof tubes is reduced compared to the conventional version, the new structure has a higher heat transferefficiency due to the full use of tubes. The fluid outflows from the tube near the shell center has a highertemperature, and the concurrent and countercurrent flows result in the different temperature increasingtrends of tube-side fluid as well as the different temperature decreasing trend of shell-side fluid. Thecountercurrent results in a larger decrease of shell-side fluid temperature.

� 2015 Elsevier Ltd. All rights reserved.

1. Introduction

Shell-and-tube heat exchanger is the most important equip-ment for efficient heat transfer between two fluids at differenttemperatures in process industry. Some distinct advantagesinclude the large heat transfer area per unit volume, the widerange of operation conditions, as well as the versatile materialsused in construction. As for the heat exchangers working underhigh pressure conditions, the structure composed of cylindricalshell and circular tubes is preferred undoubtedly. Many codes forthe design of shell-and-tube heat exchangers are available for engi-neers, including TEMA and PPHX in America, GB 151-1999 inChina, JIS B 8249 in Japan, BS 5500 in Britain and AD in Germany,etc. These references provide both economical and reliable struc-tures for conventional designs of shell-and-tube heat exchangers.

To achieve a higher energy efficiency ratio, heat transferenhancement techniques for both inside and outside tubes areconstantly studied with emphasis on optimizing the design and

operation of heat exchangers The conventional intensificationtechniques include tube-side enhancements with internal tubefins, twisted-tape inserts and coiled-wire inserts, and shell-sideenhancements with external tube fins and helical baffles. Combin-ing several enhancement techniques will achieve higher energysavings compared with implementing single technique [1].

Finned tubes and twisted oval tubes which can enhance turbu-lent flows of tube-side fluid are employed to substitute smooth cir-cular tubes. The fins attached on tube walls can not only enlargeheat transfer area but also enhance turbulence near the wall. Inthe twisted oval tube heat exchanger, the overall shell-side heattransfer performance was found to be affected by both twistedpitch length and aspect ratio [2]. More attentions were paid onthe technique to reinforce the turbulence intensity of shell-sidefluid. The helical baffles have better capability in disturbing shell-side fluid flow while consume lower shell-side pressure drop. Theexperimental study on shell-side thermodynamic and hydraulicsperformance of helical baffles heat exchangers revealed that helicalbaffles structure is more suitable for fluid flow as the shell side pres-sure drop per unit fluid-flow distance is smaller [3,4]. The highercoil diameter, coil pitch and mass flow rate in shell and tube canenhance the heat transfer rate in these types of heat exchangers

Nomenclature

A heat transfer area, m2

Ao cross flow area for shell-side fluid, m2

aj coefficientbj coefficientcj coefficientcp,s the heat capacity of shell-side fluiddi inner diameter of tube, mmdj coefficientdo outer diameter of tube, mmds spacing interval of tubes, mme number of holesL tube length, mmm mass flow rate, kg s�1

N total number of holesPr Prandtl numberR fouling resistance, m2 K W�1

Re Reynolds numberre radius of non-tube region, mmr1 radius of the first auxiliary circle, mmT shell-side characteristic temperature, Kt tube-side characteristic temperature, KTin shell-side inlet temperature, KTout shell-side outlet temperature, K

tin tube-side inlet temperature, Ktout tube-side outlet temperature, KDT shell-side temperature change, DT = Tout � Tin, KDt tube-side temperature change, Dt = tout � tin, KDTlm is the log-mean temperature differencexi x coordinate of hole ni, mmyi x coordinate of hole ni, mm

Greek symbolsa heat transfer coefficients, W m�2 K�1

h the first periodic angle, radk thermal conductivity, W m�1 K�1

Subscriptsc shell-side fluid flows toward shell centerd shell-side fluid flows away from shell centeri serial number of holes in tube sheet, i = 1,2,3. . .

j serial number of heat transfer unit, j = 1,2,3. . .k serial number of symmetry line, k = 1,2,3. . .

s shell-sidet tube-side

J.-f. Zhou et al. / International Journal of Heat and Mass Transfer 84 (2015) 282–292 283

[5]. The oil cooler with helical baffles was found to get lower shell-side pressure drop and higher heat transfer coefficient per unitpressure drop at a fixed volume flow rate than the oil cooler withsegmental baffles [6,7]. The combination of rod and van type spoilerwas designed to reduce flow pressure drop in shell-side [8].

Besides the improvement of structure, the modified heating andcooling medium are used to improve the performance of heatexchanger systems. Nanofluid is a new engineering fluid whichcan improve the performance of heat exchanger [9]. It wasobserved that the convective and overall heat transfer coefficientincreased with the application of nanofluids compared to ethyleneglycol or water based fluids [10]. Different parameters such as par-ticle size, shape and volume concentration affect the performanceof these systems [11]. Fitted porous media in heat exchanger canintensify the turbulent fluid flow and as a result, heat transfer per-formance is improved [12].

Since it is a large nonlinear system, the performance predictionof shell-and-tube heat exchanger depends on the algorithms whichcan establish the relationship among the construction and heattransfer capability. The artificial neural networks has been usedto predict the outlet temperature differences in each side ofshell-and-tube heat exchanger [13] and the heat transfer rates[14]. The optimization of shell-and-tube heat exchanger is amulti-objective duty comprising enlarging heat transfer area,reducing pumping power and lowering manufacturing cost, etc.[15,16]. The harmony search algorithm [17], particle swarm opti-mization technique [18] and genetic algorithm were applied tosolve the associated optimization problems [19]. High efficientalgorithms are required in optimization to evaluate the heatexchangers performances, among which the Bell-Delaware methodis often preferred [20]. As a widely used analytical method, theBell-Delaware method [21] was employed to predict the heatand mass transfer performances in a shell-and-tube heat exchan-ger. Furthermore, the imperialist competitive algorithm was suc-cessfully applied for optimal design of shell and tube heatexchangers with higher accuracy in less computational time [22].Another efficient technique to increase energy saving is to retrofitheat exchanger networks. Such intensification has been widely

studied in the process industry in recent years from the point ofview of individual heat exchangers [1].

As a dangerous pressure vessel, the shell-and-tube heat exchan-ger operating at a high pressure experiences challenges such ashard design, manufacture, installation and inspection procedures.Due to process requirements, the size of some heat exchangerexceeds the range covered by the mandatory design code, henceadditional heat transfer and stress analysis is indispensable. Sinceit is a complex fluid and structure coupled system comprising tubesheet, tube, head, shell, flange, baffle and fluid, it is very difficult toaccurately calculate the temperature, deformation and stress ofshell-and-tube heat exchanger, especially for huge heat exchangerwith large number of tubes. The finite element method and finitevolume method are widely used to aid the research on shell-and-tube heat exchanger. Some commercial software, such as ANSYSand FLUENT, are frequently used to perform the investigations intothe stress, flow and temperature fields in heat exchangers. On thebase of the advanced model building and numerical analysis tech-niques, the two- or three-dimensional thermal, fluid and thermalstress coupled analysis can be conveniently realized [23]. The baf-fle spacing, baffle cut and shell diameter dependencies of the heattransfer coefficient and the pressure drop can be measured bynumerically modeling the heat exchanger [24]. Furthermore, thegeometric optimization can also be performed based on numericalmodel [25]. Compared with numerical analysis, it is much difficultto obtain analytical results [26]. Though the numerical simulationis currently the most accurate method to carry out the analysis ofshell-and-tube heat exchanger, there exists an unavoidable trou-ble, i.e. the huge number of grids in modeling. Even if the structureis symmetric or periodic, the simulation is still time-consumingand difficult to be performed.

As the main heat transfer component, steel tube bundle playsthe most important role in shell-and-tube heat exchanger. Thearrangement of tubes in shell-and-tube heat exchanger is usuallyin the form of regular triangle or square layout, and the tube spac-ing interval is a constant. A review of recent investigation on tubebundles shows that tube spacing plays an important role in deter-mination of compactness of the heat exchanger [27]. When the

x

y

z1

z2z3 z4

f1f2f3

n2

n6

n12

n0

n1

n'2 n3

n4n5

n'6 n7

n8

n9

n10n11

n13n'12

Fig. 2. Topological diagram of holes in tube sheet.

284 J.-f. Zhou et al. / International Journal of Heat and Mass Transfer 84 (2015) 282–292

shell-side fluid flows cross the tubes in a circular shell with seg-mental baffles, it is difficult to ensure shell-side fluid flows by alltubes. Indeed the velocity field of shell-side fluid is not homoge-neous and the use ratio of tubes is not as high as expected. Enlarg-ing the design margin is the general method to make up thedeficiency. In this work, a bundle of topologically arranged tubesbased on fractal was proposed to homogenize shell-side flow andas a result, the use ratio of tubes reaches 100%. To evaluate the heattransfer performance of the new tube bundle, a semi-numericalalgorithm is proposed and the characteristic temperatures in theheat exchanger are obtained.

2. Structure of shell-and-tube heat exchanger

The heat exchanger investigated in this work is composed of acylindrical shell, tube sheets, tubes, channel ends, baffles and noz-zles. The disc and doughnut baffles are alternately installed alongshell axis at a certain spacing interval, as illustrated in Fig. 1. Theshell and baffles form a zigzag passage which directs shell-sidefluid to flow to the shell center or close to the shell wall. If theshell-side fluid inlet and the tube-side fluid inlet are at the sameside, fluids will flow in the same direction called concurrent flow;otherwise it is a countercurrent flow. The main structure parame-ters, such as thickness and diameter of shell, dimensions of tubeand number of baffles, can be determined according to the designcodes mentioned in Section 1. The difference of the new arrange-ment scheme compared with the conventional version is that themean spacing interval among tubes becomes larger and the num-ber density of tubes is reduced. The latter heat transfer analysiswill provide basis to estimate the heat transfer efficiency of thetube bundle.

3. Topology of tubes based on fractal mechanism

Since the holes in the tube sheet should match the tube layout,the arrangement process of holes in the tube sheet can be used todescribe the topology scheme of tubes. The arrangement of holesfollows three rules: (1) the space interval of holes should not besmaller than the referenced values given in the design code; (2)the holes are evenly distributed along the circumferential directionso that a homogeneous radial flow of shell-side fluid can be gener-ated; (3) the holes present triangle layout. In the 2D Cartesiancoordinate system, the center of tube sheet locates at zero(Fig. 2). The spacing interval of tubes, ds, is usually no less than1.25 times of outer diameter of tube, do [28]. So the center of anyhole, (x,y), cannot fall within the circle defined by Eq. (1):

x� xið Þ2 þ y� yið Þ2 ¼ d2s ð1Þ

tube

Tube

-sid

e flu

id in

let

Tube

-sid

e flu

id o

utle

t

shell-side fluid inlet

Shell-sidefluid outlet

tube sheet doughnut baffle disc baffle shell

Fig. 1. Shell-and-tube heat exchanger with disc and doughnut baffles.

where xi and yi are the coordinates of hole ni, i = 1,2,3, . . . ,N, and N isthe total number of holes.

Firstly, we locate the first hole n0 at the zero point of tube sheet,i.e., x0 = 0 and y0 = 0. Then other six holes can be easily arrangedaround the hole n0 with the spacing interval, ds, among whichthe hole n1 is located on coordinate y, that is x1 = 0 and y1 = ds. Linez1 passes through the center of the hole which is adjacent to holen1. Line z2 is the symmetry axis of coordinate y and line z1. Simi-larly, line z3 is the symmetry axis of coordinate y and line z2. Theequation of coordinate y is

x ¼ 0 ð2Þ

and the equation of line zk (k = 1,2,3. . .) is given by

y ¼ x tanp2� h

2k�1

� �ð3Þ

where h is the first periodic angle. To ensure all holes possess cir-cumferential periodicity, they are located on the symmetry lines.

The position of hole n2 (x2,y2) can be determined by drawing anauxiliary circle taking the point (x1 = 0, y1 = ds) as center and ds asradius. Line z2 crosses this circle at two points, and the point whichhas the larger radius is regarded as the center of hole n2. Mathe-matically, (x2,y2) can be solved from the simultaneous Eqs. (1)and (3) when k = 2 and i = 1. The next step is to find the center ofhole n3 which locates on coordinate y. Similarly, (x3,y3) can beobtained by solving the simultaneous Eqs. (1) and (2) when i = 2.It was found that when k = 2 and i = 3, the simultaneous Eqs. (1)and (3) has only one solution, i.e., line z2 is tangential to the circle(x � x2)2 + (y � y2)2 = d2

s . Coincidently, the tangent point is just thecenter of hole n2. On this condition, hole n2 should be moved alongline z2 to the new position (x02, y02):

x02 ¼ y3 sin p2 � h

2k

� �

y02 ¼ y3 cos p2 � h

2k

� �8><>: ð4Þ

Both the centers of holes n3 and n02 are on arc f1 where the firstfractal position is, and the following step is to determine the centerof hole n4 on line z3 (k = 3) based on hole n3. In this way, holes n5,

inside arc f1

between arcs

f1 and f2

between arcs

f2 and f3

outside

arc f3

Fig. 4. Periodic models of shell-side fluid.

ms, Tinms, Tin

J.-f. Zhou et al. / International Journal of Heat and Mass Transfer 84 (2015) 282–292 285

n6, n7, n06 . . . can all be located. In brief, the position of hole ni

depends upon its previous hole ni�1. During the course of topology,if there is no solution or only one solution to the simultaneous Eqs.(1) and (3), a fractal treatment is applied, and the periodic region,where it is going to be arranged by new holes, is divided into twosmaller symmetric regions. Once a new hole ni is located based onits former hole ni�1, the center distance between holes ni and ni�2

should be computed to see whether it is larger than ds; if it is not,hole ni should be moved away from hole ni�2 along the line wherethey locate until their center distance equals ds.

The bolded circles in Fig. 2 present the mother holes which willbe mirrored and copied to fully occupy the tube sheet. The holesinside arc f1 can be divided into 6 periodic groups while the holesbetween arcs f1 and f2 can be divided into 12 periodic groups.Analogically the periodicity will be doubled when a new fractaltreatment is performed.

Considering that the main flow pattern of shell-side fluid isaxial flow when it runs around baffles, the contribution of thetubes near shell axis and wall to heat transfer is relatively smallbecause the turbulence is weakened. Sometimes the tubes nearthe shell center and wall can be left out, so that shell-side fluidcan well mix and turn around rapidly when it approaches thenotch of baffle. To implement this idea, first of all, the position ofthe first circle of holes should be determined. If the radius of thenon-tube region is re, the number of the first circle of holes, e,can be calculated by Eq. (5).

e ¼ Intre þ d=2ð Þ � 2p

ds

� �ð5Þ

and the radius of the first circle, r1, can be obtained by

r1 ¼eds

2pð6Þ

Thereby the holes can be arranged on the first circle uniformlyat the spacing interval ds. The rest is to choose two adjacent holesas the base to locate next hole.

Based on Eqs. (1)–(4), (6), a computer program was developedto draw the graph of tube sheet with all holes using the parametersdo and ds.

4. Numerical model of heat transfer unit

4.1. Periodic model of shell-side fluid

It can be seen from the topology of tubes that the spacing inter-val among tubes is not constant. That is different from conven-tional shell-and-tube heat exchanger. Therefore, it is necessary toestablish a new method to predict the heat exchange efficiencyof the tube bundle. The semi-numerical algorithm established inthis work is based on a series of numerical heat transfer unitmodels which are sampled from the entire solid and fluid coupledmodel.

Upstream

Downstream

12

34

56

Heat transfer unit

Fig. 3. Streamlines of shell-side fluid.

Considering the shell-side fluid between the left tube sheet anddoughnut baffle (in Fig. 1), it can be presumed that all the stream-lines periodically distribute along circumferential direction andpoint to the center of tube sheet, as illustrated in Fig. 3.

The flow direction of shell-side fluid is controlled by baffles, andthe continuous shell-side and tube-side fluids can be divided intoseveral parts of the same shape by baffles. Further, smaller periodicmodels, as Fig. 4 illustrates, can be sampled from the part. Therebythe number of grids in numerical model can be reduced signifi-cantly and the simulation becomes easier. The investigationrevealed that, using their correlative boundary conditions, periodicmodels are economical in numerical simulation of small size heatexchanger. With the increasing shell diameter, the number of tubescontained in periodic model increases and the amount of computa-tion increases exponentially. The periodic numerical models arestill inconvenient for engineering application.

To reduce the computation cost, the heat transfer unit with sin-gle tube was extracted from the whole tube bundle, and the proce-dure can be understood from Fig. 5. As can be seen in Fig. 3, theflow of shell-side fluid around tubes can be described as: a streamflows through the gap composed of the tubes on center line 1, andlater the stream is split into two smaller streams by the headontube on center line 2; then the two smaller streams merge againand flow through the gap composed of the tubes on center line3. The following steps are just duplications of this process. So theheat transfer process can be considered to be a stepwise course,i.e., when we compute the heat flux flows through the tubes on

b1 b2

b6b5 b4

b3

ms, Tmidc5c6

c2 c3c1 c4

ms, T’out

a1 a2

a3a4

ms, Tout

Tube CVa

CVc

CVb

Fig. 5. Establishment of heat transfer unit with single tube.

adiabatic boundary

outlet vent (tube-side)

velocity inlet (shell-side)

286 J.-f. Zhou et al. / International Journal of Heat and Mass Transfer 84 (2015) 282–292

one circle, the boundaries of the tubes on upper and lower centerlines are assumed to be adiabatic. Under this assumption, for thetubes on different circles, their contributions of to heat transfercan be considered separately without changing the total heattransfer area.

For a tube bundle with layout of 60�, the periodic control vol-ume, CVa, with the cross-section a1-a2-a3-a4 can be extracted. Thenby mirroring the tubes about line a2-a3, two independent controlvolumes, CVb and CVc, with the cross-sections b1-b2-b3-b4-b5-b6

and c1-c2-c3-c4-c5-c6 can be defined while not changing the heattransfer area. The mean temperatures, Tin and Tout are used to char-acterize the shell-side fluid at inlet and outlet vents. The originalheat transfer process in CVa can be divided into two steps, i.e.,the shell-side fluid in control volume CVb flows around the tubeand outflows with the temperature, Tmid, and then flows into thecontrol volume CVc and outflows with the temperature, T 0out. Ifthe error between Tout and T 0out is acceptable, the assembly of heattransfer unit models can be used to predict the heat transfer per-formance of the tube bundle.

4.2. Definition of characteristic temperatures

Fig. 6 illustrates a tube bundle with two baffles and three tubeswhich locate on three adjacent center lines (in Fig. 3). The threetubes are not on one plane. The two baffles divide each tube intothree sections, and the nine tube sections are labeled by A, B, C,D, E, F, G, H and I sequentially. The arrows indicate the countercur-rent flow direction of shell-side fluid, i.e., the shell-side fluid flowsby all the tube sections from I to A. T13 represents the known inlettemperature of shell-side fluid; t11, t21 and t31 are the known inlettemperatures of tube-side fluid and t11 = t21 = t31. T41 is the outlettemperature of shell-side fluid, t14, t24 and t34 are the outlet tem-peratures of tube-side fluid, and they are to be determined.

For each tube section, the tube-side inlet and outlet tempera-tures and the shell-side upstream and downstream temperaturesrelate to its adjacent tube sections. Taking section E as an example,its tube-side inlet and outlet temperatures are t22 and t23, respec-tively, however, t22 is also the outlet temperature of section Band t23 is the inlet temperature of section H. The upstream temper-ature of section E, T32, is the downstream temperature of tube sec-tion F and the downstream temperature, T32, is the upstreamtemperature of tube section D.

4.3. Numerical model of heat transfer unit

The entire heat exchanger model is composed of many smallheat transfer unit models as marked in Fig. 3. Fig. 7 shows a 3Dmodel of heat transfer unit comprising a tube section with its outerand inner fluids. Interfaces were created between the solid andfluid domains so that coupled heat transfer analysis can be carriedout. The most important parameters in the heat transfer analysisare the heat transfer coefficients on the inner and outer wall

A

B

C

F

E

D

G

H

I

shell-side fluid

t11

t21

t31 T41

T31

T21

T11t12

t22

t32

t13

t23

t33

t14

t24

t34T42

T32

T22

T12

T43

T33

T23

T13

tube

-sid

e flu

id

shell centerline

Fig. 6. Characteristic temperatures in shell-and-tube heat exchanger.

boundaries of the tube, which are not constant at different pointson the boundaries. Using numerical simulation method, the heattransfer coefficients can be determined and the mean tempera-tures of the outlet fluids can be obtained corresponding to differentinlet temperatures of shell-side and tube-side fluids. The length ofthe tube section equals the distance between two adjacent baffles.

According to the periodic characteristic shell-side fluid, the leftand right boundaries are periodic boundaries. When the shell-sidefluid flows toward shell center, the top surface is a velocity inletboundary and the bottom is an outlet vent; but when it flows inthe radial direction, the two boundaries alternate. The inlet veloc-ities can be determined according to the known mass flow rate. Asfor the four arc boundaries in the four corners, they are the wallboundaries of adjacent tubes. In the model, the upstream flowsfrom the gap composed by the two upper tubes, then flows aroundthe center tube and finally the separated stream is assembled byanother gap composed by the lower two tubes. Though the amountof heat of shell-side fluid will be transferred by the tubes itattaches to, the four curved boundaries at four corners can beregarded to be adiabatic based on the explanation in Section 4.1.In consideration of the relative small velocity of shell-side crossflow and large baffle spacing, the two end faces of shell-side fluidmodel are set to be adiabatic walls. Furthermore, the two end facesof tube can also be regarded as adiabatic.

4.4. Semi-numerical simulation method

The heat transfer unit model can represent other tube sectionson the same auxiliary circle. But because of the difference in tubespacing intervals and periodicities, the size of each representativemodel is different from the models on other circles. For each heattransfer unit, numerical simulation can be carried out and the tem-perature change of fluids with the change of shell-side and tube-side inlet temperatures can be obtained. By linking all the heattransfer units with the relationships among inlet and outlet tem-peratures, as well as the mass conservation conditions of fluids,the characteristic temperatures can be solved.

5. Example and discussion

5.1. Design parameters

A gaseous mixture is cooled from 383 K to 323 K by water at293 K using a shell-and-tube heat exchanger. The shell-side fluidis the gaseous mixture with mass flow rate of 2.3 � 105 kg h�1.

velocity inlet (tube-side)

wall steel tube

outlet vent (shell-side)

periodic boundary

Fig. 7. Model of heat transfer unit with single tube.

Table 2Design parameters of heat exchanger.

Item Value

Inner diameter of shell (mm) 1400Tube specification (mm) U25 � 2.5Tube length 4950Spacing interval of tubes (mm) 44Number of tubes 980Spacing interval of baffles 450Number of baffles 10

10 20 30 40 50 60 70 80 900.0

0.4

0.8

1.2

1.6

2.0

2.4

2.8

3.2

3.6

Tem

para

ture

cha

nge

of tw

o flu

ids

(K)

Temparature difference between Tin and tin (K)

Tin-Tout

Tin-T'out

tout-tin

t'out-tin

Fig. 8. Temperature changes of shell-side and tube-side fluids with temperaturedifference of (Tin � tin).

Table 3Number of tubes at auxiliary circles.

Circle No. Number of tubes

1, 2, 3 124, 5, 6, 7, 8 249, 10, 11, . . ., 18, 19 4820 96

J.-f. Zhou et al. / International Journal of Heat and Mass Transfer 84 (2015) 282–292 287

The outlet temperature of water was estimated to be 309 K. Thephysical parameters of the two fluids and tube material are listedin Table 1.

The main geometrical parameters of the heat exchangerresulted from conventional design according to GB 151-1999 [28]are listed in Table 2. The required heat transfer area is 380.8 m2;the average velocity of tube-side fluid is 1.3 m s�1.

5.2. Numerical simulation of heat transfer unit

It is necessary to validate the heat transfer unit model with sin-gle tube before performing the semi-numerical simulation. Basedon the data listed in Tables 1 and 2, the models similar to thatshown in Fig. 5 were established and the temperature changes ofshell-side and tube-side fluids resulting from control volume CVa

and the assembled control volumes CVb and CVc were given byFig. 8. In this case, the inlet velocities of shell-side and tube-sidefluids equal 2 m s�1 and 1.3 m s�1, respectively. With the increas-ing temperature difference of (Tin � tin), the temperature changesof the two fluids present linear variation tendency. The moreimportant discovery is that the heat transfer results of the twomodels are similar. For the shell-side fluid, the two models resultin the very small error and the maximum error is 10%; but forthe tube-side fluid, the error increases with the increasing valueof (Tin � tin) and the maximum error is about 11%. More investiga-tions were carried out and it was revealed that when the inletvelocity of shell-side fluid changes, the temperature errors for bothshell-side and tube-side fluids are about 10% in the range of(Tin � tin) from 10 to 90 K. The comparison of the two modelsindicate that to use the assembly of heat transfer unit models topredict the temperature change of fluids is reasonable.

Using the method introduced in Section 3, the tube arrange-ment scheme was obtained. These tubes are arranged at some aux-iliary concentric circles and the numbers of tubes at each circle arelisted in Table 3. The total number of holes in tube sheet with thediameter of 1400 mm is 780. There is no tube arranged near shellcenter, and the sketch-map is similar to that shown in Fig. 1 exceptfor the seven holes at the center.

The total number of tubes is 80% of the conventional designresult in Table 2, and tube length is uncertainty. Fig. 9 shows thecross-sections of the extracted twenty heat transfer unit modelsin this case. Models (1), (2), (3), (4). . . correspond to the holes n3,n4, n5, n7. . . shown in Fig. 2, respectively. The annulus on eachcross-section presents the profile of tube.

The heat transfer unit models are regular and simple in struc-ture. So the quad element was chosen and paved on end face ofmodel and then the face mesh was coopered in the axial directionto achieve volume grids. The model pre-processing was carried outwith GAMBIT 2.2. The grid independent validation was carried outto determine the appropriate grid size. As a result, there are threelayers of grids along the tube thickness direction, as can be seen inFig. 10. The number of grids in model (5) is about 1,100,000.

During the simulation of steady flow of tube-side and shell-sidefluids, the Standard k–e model was chosen. SIMPLE algorithm wasemployed to solve the flow and pressure equations. The standard

Table 1Physical parameters of fluids and steel.

Watera

Density (kg m�3) 996.2Specific heat (kJ kg�1 K�1) 4.176Kinematic viscosity (kg m�1 s�1) 8.42 � 10�4

Coefficient of heat conductivity (W m�1 K�1) 0.614

a The listed physical parameters correspond to the mean temperatures of fluids.

method was employed in the pressure term. The second orderupwind algorithm was employed in simulating the boundary layermotion of fluid. All the governing equations were solved with pres-sure-based coupled algorithm [29]. The whole simulation workwas processed with FLUENT 6.3. Fig. 11 shows the temperaturedistributions on the inlet and outlet vents of model (5). The tem-peratures exerted on the two inlet vents are constant, and the tem-perature gradient on each outlet vent boundary was found to bevery small. Therefore, it is feasible to take the temperatures onthe two outlet vents as constants, too.

Gaseous mixturea Steel

90 27193.297 8711.5 � 10�5

0.0279 202.4

(1) (2) (3) (4) (5)

(6) (7) (8) (9) (10)

(11) (12) (13) (14) (15)

(16) (17) (18) (19) (20)Fig. 9. Models of heat transfer units.

Fig. 10. Mesh of end face of model (5).

v=1.3 m·s-1

tin=318 K

tout=318.22 K

v=2.69 m·s-1

Tin=323 K

Tout=322.96 K

Fig. 11. Temperature contour of inlet and outlet vents of model (5).

288 J.-f. Zhou et al. / International Journal of Heat and Mass Transfer 84 (2015) 282–292

5.3. Relationship among temperatures of shell-side and pipe-side fluids

Taking the heat transfer unit model (5) as an example, the out-let temperatures of tube-side and shell-side fluids were obtainedby numerical simulation, which indicate the linear relationshipbetween the shell-side inlet temperature, Tin, and outlet tempera-ture, Tout, when the tube-side inlet temperature, tin, is constant(Fig. 12), as well as that between the tube-side inlet temperature,

tin, and outlet temperature, tout, when the shell-side inlet temper-ature, Tin, is constant (Fig. 13).

By investigating all the twenty heat transfer unit models, it wasrevealed that the temperature increase of tube-side fluid, Dtc

(Dtc = tout � tin) is in the linear relationship with the temperaturedifference between Tin and tin, as can be expressed by Eq. (7) whichwas obtained with least square procedure:

Dtc ¼ aj T in � tinð Þ ð7Þ

where j is the serial number of model and j = 1,2,3, . . . ,20; the coef-ficient aj is constant. Similarly, Eq. (8) was obtained:

DTc ¼ bj T in � tinð Þ ð8Þ

where DTc = Tout � Tin and bj is another constant. The subscript ‘‘c’’in Eqs. (7) and (8) means the shell-side fluid flows toward shell cen-ter. When shell-side fluid flows in radial direction, the temperaturechanges, Dtd and DTd, can be obtained by Eqs. (9) and (10),respectively:

Dtd ¼ cj T in � tinð Þ ð9ÞDTd ¼ dj T in � tinð Þ ð10Þ

The values of the coefficients aj, bj, cj and dj are listed in Table 4.For the concurrent as Fig. 1 illustrates, it is logical to perform

the calculation from the upper-left tube section (corresponds tomodel (20) in Fig. 9) because both shell-side and tube-side inlettemperatures are known and the calculation sequence is in accor-dance to the flow direction of shell-side fluid. For countercurrentflow as Fig. 6 illustrates, the calculation can be also performed fromthe upper-left tube section, but here the shell-side inlet and outlettemperatures are unknown. On this condition, an assumed temper-ature, Tout, was assigned to the shell-side outlet of model (20), andthe shell-side inlet temperature, Tin, can be calculated according toEq. (11),

T in ¼Tout � djtin

1� djð11Þ

and then the tube-side outlet temperature, tout, can be obtainedwith Eq. (9). Using this method, all the characteristic temperaturescan be obtained and the calculation sequence is the reverse of theflow direction of shell-side fluid. If the obtained shell-side inlettemperature of the final tube section is equal to the given value(383 K), the assumed shell-side outlet temperature of the first tubesection is correct; otherwise the assumed shell-side outlettemperature of the first tube section should be revised and all thetemperatures calculated again.

320 330 340 350 360 370 380 390

320

330

340

350

360

370

380

390

Shel

l-sid

e ou

tlet t

empe

ratu

re, T

out (

K)

Shell-side inlet temperature, Tin (K)

tin=293K

Fig. 12. Tin vs. Tout when tin = 293 K.

290 295 300 305 310 315 320293

294

295

296

297

298

Tube-side inlet temperature, tin (K)

Tube

-sid

e ou

tlet t

empe

ratu

re, t

out

(K)

Tin=383K

Fig. 13. tin vs. tout when Tin = 383 K.

Table 4Values of coefficients aj, bj, cj and dj.

j= aj bj cj dj

1 0.0572 �0.0048 0.0645 �0.00542 0.0567 �0.0047 0.062 �0.00563 0.0547 �0.0045 0.0566 �0.00584 0.0517 �0.0044 0.0539 �0.00555 0.0498 �0.005 0.0523 �0.00586 0.0485 �0.0053 0.0505 �0.0067 0.0475 �0.0055 0.0491 �0.00628 0.0468 �0.0061 0.0466 �0.00529 0.0453 �0.0126 0.0467 �0.0167

10 0.0442 �0.0137 0.046 �0.018611 0.0433 �0.0143 0.0451 �0.018412 0.0426 �0.0142 0.0442 �0.017813 0.0419 �0.0139 0.0434 �0.017214 0.0413 �0.0136 0.0427 �0.016715 0.0409 �0.0134 0.0421 �0.016216 0.0404 �0.0132 0.0416 �0.015817 0.0401 �0.013 0.0412 �0.015518 0.0397 �0.0128 0.0408 �0.015219 0.0395 �0.0128 0.0404 �0.01520 0.0389 �0.0126 0.0397 �0.0148

J.-f. Zhou et al. / International Journal of Heat and Mass Transfer 84 (2015) 282–292 289

5.4. Results and discussion

In consideration of the convenience in manufacturing, the num-ber of baffles should be odd so that the shell-side fluid can flow inor out through the nozzles attached on shell; otherwise the inlet oroutlet nozzle will penetrate tube sheet that will complicate thestructure. In the first tentative calculation, five baffles were set inshell (tube length equals 2700 mm) and the shell-side outlet tem-perature is 322.2 K for concurrent flow and 318.8 K for countercur-rent flow. Theoretically the result has met the operationrequirement, but from an engineering perspective, the margin isnot large enough. So two more baffles were added and the tubelength was prolonged to be 3600 mm.

The temperature change of tube-side fluid along axial directioncan be seen in Fig. 14. Certainly the temperature of tube-side fluidincreases because it is heated. But the mean temperatures of fluid indifferent tubes are different. As can been seen in Table 4, the nearerto shell center the tube, the larger the coefficient ai or ci, that resultsin the higher mean temperature of the fluid in No. 5 tube than thatin No. 15 tube. The serial number of tubes is in accordance with thatof the models in Fig. 9. Corresponding to concurrent flows, the tem-perature growth rate of tube-side fluid reduces as it travels due tothe decreasing temperature difference between two fluids; how-ever it increases corresponding to countercurrent flows.

The overall outlet temperature of tube-side fluid can beobtained by mass-averaged integral, which equal 306.3 K and307.4 K corresponding to concurrent and countercurrent flows,respectively. The velocity of shell-side fluid increases when it flowstoward shell center and the shell-side heat transfer coefficientincreases correspondingly. Therefore, for the tube nearer to shellcenter, its tube-side outlet temperature is higher, as can be seenin Fig. 15. Exerting the characteristic temperatures, which can beobtained from heat transfer unit models, on tubes, the stresses intubes and tube sheet can be calculated by establishing a periodictubes-and-tube sheet finite element model. The result is helpfulfor intensity evaluation of heat exchanger.

As the shell-side fluid travels, its temperature decreases contin-uously, as can be seen in Fig. 16. The difference between the tem-perature of concurrent flow and that of countercurrent flow is notobvious in the former 120 steps (the step means the shell-side fluidflows by a circle of tubes, seen in Section 4.2), but it is enlarged inthe later 40 steps. The final shell-side outlet temperature is 316.4 Kcorresponding to concurrent flow while 310.5 K corresponding tocountercurrent flow. It was revealed that the temperature decreaseof shell-side fluid is about 0.42 K at each flow step. In this case,

1 2 3 4 5 6 7 8 9

292

294

296

298

300

302

304

306

308

310

Tem

pera

ture

in tu

be-s

ide

fluid

(K)

Serial number of node point

in No.5 tube, concurrent in No.15 tube, concurrent in No.5 tube, countercurrent in No.15 tube, countercurrent

Fig. 14. Temperature distribution in tube-side fluid.

0 20 40 60 80 100 120 140 160300

310

320

330

340

350

360

370

380

390

Tem

pera

ture

in s

hell-

side

flui

d (K

)

Flow step of shell-side fluid

concurrent countercurrent

Fig. 16. Temperature of shell-side fluid at different flow step.

290 J.-f. Zhou et al. / International Journal of Heat and Mass Transfer 84 (2015) 282–292

countercurrent flow results in the better cooling effect for shell-side fluid.

The heat transfer analysis result indicates that the new tubebundle type is highly efficient because it implements the dutyusing only 58% of heat transfer area compared with conventionaldesign. Using the periodically arranged tube bundle can make fulluse of tubes so that the design margin can be narrowed signifi-cantly to save material. Furthermore, the design parameters canbe revised using the semi-numerical heat transfer analysis methodwhich is more precise than empirical method.

If the mass flow rates of shell-side and tube-side fluids, ms andmt, are changed, the heat transfer performance will change corre-spondingly. The simulations will be performed again on the baseof the twenty models shown in Fig. 9 to determine the coefficientsaj, bj, cj and dj in Eqs. (7)–(10). In addition, if the critical value ofspace interval, ds, and the tube dimensions, di and do, are changed,it is necessary to establish new heat transfer unit models.

5.5. Analytical heat transfer analysis for the heat exchanger

The semi-numerical heat transfer analysis for the heat exchan-ger depends on a number of numerical models; hence it is not asconvenient as analytical method. With the Bell-Delaware method,the tube-side and shell-side outlet temperatures can be obtainedby analytical calculation. The overall heat transfer coefficient basedon tube outer diameter, a, is given by Eq. (12) [21]:

a ¼ 11asþRs þ do ln do=dið Þ

2k þ Rtdodiþ 1

at

dodi

ð12Þ

where as and at are the shell-side and tube-side heat transfercoefficients, respectively, Rs and Rt are the fouling resistances onthe shell-side and tube-side, respectively, and k is the thermalconductivity of tube. In this study, Rs = 0.000086 m2 K W�1 andRt = 0.000172 m2 K W�1.

In the Bell-Delaware method, the shell-side heat transfer coeffi-cient, as, is given by

as ¼ a0sJcJlJbJsJr ð13Þ

where a0s is the ideal heat transfer coefficient, and the equationsuggested by Shah and Sekulic [21] is used, which holds

a0s ¼ jmscp;sPr�2=3s Ao ð14Þ

where ms is the mass flow rate of shell-side fluid, ms = 63.9 kg s�1,cp,s is the heat capacity of shell-side fluid, Ao is the cross flow area

0 2 4 6 8 10 12 14 16 18 20

305

306

307

308

309

310

311

312

313

Out

let t

empe

ratu

re o

f tub

e-si

de fl

uid

(K)

Serial number of tube

concurrent countercurrent

Fig. 15. Outlet temperatures of tube-side fluid in different tubes.

for shell-side fluid between two subsequent baffles taken at themid circle of tubes, Ao = 0.4 m2. The Prandtl number, Prs, is givenby Eq. (15).

Prs ¼cp;sls

ksð15Þ

In Eq. (14), j is the Colburn factor, which holds

j ¼ a11:33

Prs=do

� �a

Resð Þa2 ð16Þ

where

Res ¼msdo

lsAoð17Þ

and

a ¼ a3

1þ 0:14 Resð Þa4

The correction factor for the baffle cut and spacing, Jc, is calcu-lated by Jc = 0.55 + 0.72Fc, and Fc is the fraction of the total numberof tubes in the cross-flow section. Jl, Jb, Js and Jr are the tube-to-baf-fle and baffle-to-shell leakage correction factor, the bundle andpass partition bypass stream correction factor, the correction factorfor large baffle spacing at the inlet and outlet sections and the cor-rection factor for adverse temperature gradient build-up in laminarflows, respectively. In this study, Jl = Jb = Js = Jr = 1.

For the tube-side, the heat transfer coefficient is given by

at ¼ 0:023kt

diPr1=3

t Re0:8t

lt

ltw

� �0:14

ð18Þ

where Prt ¼ cp;tlt=kt and Ret ¼ qtdiv i=lt. Fluid properties are takenat the fluid arithmetic mean temperature, except for viscosity, ltw,which is taken at the arithmetic mean outer tube wall temperature.

The system must satisfy the equation governing heat exchang-ers, defined as

Q ¼ AaDTlm ð19Þ

where Q is the heat duty, A is the heat transfer area based on theouter diameter of tube, and DTlm is the log-mean temperature dif-ference, which holds

DTlm ¼T in � toutð Þ � Tout � tinð Þ

ln T in � toutð Þ= Tout � tinð Þ½ � ð20Þ

Table 5Comparison between semi-numerical and analytical analysis results.

Semi-numerical result Analytical result

Countercurrent Concurrent L = 3,600 mm L = 6,300 mm

Heat transfer area (m2) 220.4 220.4 220.4 385.7Heat transfer coefficient (W m�2 K�1) 1806.8 1438.1 532.6 532.6Shell-side outlet temperature (K) 310.5 316.4 337.7 322.9Tube-side outlet temperature (K) 307.4 306.3 300.1 302.4Log-mean temperature difference (K) 39.8 45.2 61.8 65.0Shell-side cooling power (W) 1.59 � 107 1.43 � 107 0.95 � 107 1.34 � 107

Tube-side heating power (W) 1.73 � 107 1.52 � 107

Error in semi-numerical result 8.1% 5.9%

J.-f. Zhou et al. / International Journal of Heat and Mass Transfer 84 (2015) 282–292 291

The heat duty can be obtained from the energy balance between theshell-side and tube-side streams.

Q ¼ mscp;s T in � Toutð Þ ¼ mtcp;t tout � tinð Þ ð22Þ

For the heat exchanger sampled in Section 5.1, the heat transferarea, A, the temperatures, Tin and tin, and mass flow rates, ms andmt, are known. Hence Tout and tout can be solved by combiningEqs. (12), (19) and (22). Firstly, assume an initial value for Tout

and solve tout using Eq. (22). Secondly calculate as, at and a usingEqs. (13), (18) and (12), respectively. Thirdly, determine whetherthese values are compatible with Eq. (19), if they are not, changethe value of Tho and repeat the three steps.

Both the semi-numerical and analytical heat transfer analysisresults are listed in Table 5. For each heat transfer unit model,the heat quantity obtained by tube-side fluid is equal to that elim-inated from shell-side fluid. But in terms of the inlet and outlettemperatures of two fluids in the heat exchanger, it was found thatthe heat quantity obtained by tube-side fluid is not equal to thateliminated from shell-side fluid. Their difference is 8.1% for coun-tercurrent and 5.9% for concurrent. That is due to the accumulativeerror arising from Eqs. (7)–(10) that are used to estimate the outlettemperatures of heat transfer unit models. The analytical resultindicates that for the tube bundle with the length of 3600 mm, itcannot meet the cooling power requirement. By increasing thetube length to be 6300 mm, the outlet temperature of shell-sidefluid approaches 323 K. This result can be understood from thatthe overall heat transfer coefficient obtained by the semi-numeri-cal method is much larger than that by analytical calculation. Interms of overall heat transfer area, the analytical result is in accor-dance with the initial design results that listed in Table 2.

5.6. Error analysis

As shown in Fig. 8, separating heat transfer unit models fromtube bundle will introduce certain errors into the semi-numericalsimulation results. Furthermore, mean velocities and temperaturesare used to define the boundary setting of the numerical modelinstead of applying velocity and temperature fields. Following sim-plifications will also introduce additional error into results, includ-ing the baffles were assumed to be adiabatic; the fouling resistanceson shell-side and tube-side, the change of physical properties of flu-ids, as well as gravity, were not considered in simulation. Since it isvery difficult to obtain the complete temperature field in the heatexchanger to compare with the characteristic temperaturesobtained in this work, a quantitative error to the result was notgiven. Despite the ignored details in heat and mass transfer pro-cesses, this method still has a higher precision than the analyticalcalculation. Comparing the Bell-Delaware method with the semi-numerical method proposed in this work, the later can only beapplied in analyzing the tube bundle which has periodic

characteristic along the circumferential direction. Admittedly, toestablish a certain number of heat transfer unit models and performrelated numerical heat transfer analysis is more time-consumingthan Bell-Delaware method.

6. Conclusion

The tube bundle with periodic characteristics along the circum-ferential direction was constructed to guide the shell-side fluid toflow by all tubes in a circular shell. Relative to flow direction ofshell-side fluid, the tubes are arranged in triangular layout andthe space interval of tubes is no longer constant compared withconventional layout. A semi-numerical algorithm was put forwardto evaluate the heat transfer performance.

The periodic structure makes it easy to predict the flow direc-tion and mean velocity of shell-side fluid. According to the periodiccharacteristics of flow field, the large solid and fluid coupled entitycan be partitioned into many smaller heat transfer units. Charac-teristic temperatures in the heat exchanger are obtained by assem-bling the representative unit models using the relationships amongtheir inlet and outlet velocities and temperatures. The basic form ofthe topology of tubes is triangle and that is proportional to increasethe heat transfer coefficient. The infinite fractal dimension ensuresthat the shell diameter can be enlarged as much as desired.

The temperature of tube-side fluid outflows from the tube nearshell center was found to be higher than that from far tubes. Thecountercurrent flow of shell-side fluid results in the better heattransfer efficiency. The heat transfer analysis revealed that thenew tube bundle type is highly efficient because it implementsthe duty using only 57% of heat transfer area compared with theanalytical calculation result based on the Bell-Delaware method.

Conflict of interest

None declared.

Acknowledgments

This work was supported by the National Natural ScienceFoundation of China 51205184 and 51306087, Jiangsu NaturalScience Foundation (BK2011810 and BK2012430) and JiangsuNatural Science Foundation of Higher Education (11KJB480004).

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