design and optimization of coil finned tube heat exchangers for cryogenic applications

8/8/2019 Design and Optimization of Coil Finned Tube Heat Exchangers for Cryogenic Applications

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Design and optimization of coil nned-tube heat exchangersfor cryogenic applications

Prabhat Kumar Gupta a, *, P.K. Kush a , Ashesh Tiwari b

a Cryogenics Section, Raja Ramanna Centre for Advanced Technology (RRCAT), Indore 452013, Indiab Institute of Engineering and Technology, DAVV, Indore (MP), India

Received 7 March 2006; received in revised form 14 March 2007; accepted 24 March 2007

Abstract

Coiled nned-tube heat exchangers have been used in small and medium helium refrigerators/liqueers, miniature J–T refrigerationsystems for many years. The efficiency of these cryogenic systems strongly depends on the thermal and pressure drop performanceof these heat exchangers. A considerable improvement in the performance of heat exchanger is possible by choosing an appropriategeometrical conguration for a given process requirement. In the present study, geometry of heat exchanger has been derived taking intoconsideration the clearance provided for manufacturing of the heat exchangers and an optimized geometrical congurations have beennd out. The results show the possibility of adjusting the thermal and pressure drop performance by varying the clearance. The predic-tions of four end temperatures from present design method have been compared with the actual experimental results of one of theprototypes fabricated in our lab.Ó 2007 Elsevier Ltd. All rights reserved.

Keywords: Helium (B); Heat transfer (C); Heat exchangers (E)

1. Introduction

A series of coiled nned-tube heat exchanger is used in acryogenic refrigerator/liqueer. These heat exchangers wererst used by Collins in his helium liqueer [1]. The mainrequirements of these heat exchangers are high effectivenessand low pressure drops in both of uid streams to stipulatedlimits. These parameters govern the performance of thewhole system. In fact, a cryogenic liqueer will produce

no liquid if the heat exchanger effectiveness is less thanapproximately 85% in contrast to a conventional heatexchanger, used in other process plants, with lesser effective-ness [2]. Atrey [3] has shown in his analysis that decrease inheat exchanger effectiveness from 97% to 95% reduces theliquefaction yield in helium liqueer by 12%. This necessi-tates thorough understanding of different loss contributing

mechanisms that affect the performance of heat exchangerto arrive at an optimum geometrical conguration.

One of the major issues of developing these heat exchang-ers are to ensure uniform ow distribution over the nnedtubes of heat exchanger by controlling the manufacturingclearance to achieve the higher order of magnitude of effec-tiveness. However, some diametrical clearance is providedin order to ease the assembly of the nned-tube bundle andthere is always some leakage of ow through the diametric

clearance. Hence, in addition to other losses, there is alwaysdetrimental effect on the thermal performance of heatexchanger due to leakage of ow through clearance. Thisis due to the fact that there is no heat exchange with thenned tube as the ow passes through this clearance. Onthe other hand, the pressure drop performance is improveddue to increase in available cross-section area.

The design of cryogenic heat exchangers is always sub- ject to limited pressure drop conditions. If the clearanceeffect is not considered for predictions of pressure dropsthen it may be overestimated and it will require the larger

0011-2275/$ - see front matter Ó 2007 Elsevier Ltd. All rights reserved.

doi:10.1016/j.cryogenics.2007.03.010

* Corresponding author. Tel.: +91 731 2488336; fax: +91 731 2488300.E-mail address: [email protected] (P.K. Gupta).

www.elsevier.com/locate/cryogenicsCryogenics 47 (2007) 322–332

mailto:[email protected]

mailto:[email protected]



shell diameter to keep the pressure drop with in stipulatedlimits. This will result in lower ow velocity within heatexchangers and therefore lower heat transfer coefficient inthe shell side. Hence, it will require larger unit and as a con-sequence of this there is a need to optimize the geometry of heat exchanger to

• minimize the cool down time of the system,• minimize the refrigeration loss to cool the unit,• minimize the radiation loss, and to• reduce the cost of the system.

So, fact is that the clearance provided for ease of manufac-turing can be used for adjusting the thermal and pressuredrop performance of nned-tube heat exchangers. To thebest of the knowledge of present authors, a little informa-tion has been published in open literature regarding thedesigning of coiled nned-tube heat exchangers. Geistand Lashmet [4] presented the heat transfer factor and fric-tion factor for different n geometries. Croft and Tebby [5]presented the expressions for thermal design and they have

suggested the correlations for calculation of heat transfer

coefficients for shell side and tube side ow. They used theirdesigning method for the Clarendon laboratory helium liq-ueer heat exchangers [6]. Croft and Cosier [7] alsodesigned a new form of nned-tube heat exchanger byapplying the design method described by Croft and Tebby[5]. However, design proposed by Croft et al. does not con-sider the effect of diametrical clearance on thermal andpressure drop performance.

In the present work, the expressions have been derivedtaking in to account of clearance. These presented expres-sions can be used as design charts for thermal and pressuredrop design of nned-tube heat exchanger. The methodol-ogy prescribed in the present work has been used to com-pare four end temperatures of one heat exchanger testedin our lab. The main system parameters are given in Table1. These parameters correspond to one of the units fabri-cated in our laboratory and may be suitable for mediumsize helium liqueers/refrigerators. Using these charts, theoptimization for thermal and pressure drop performancehas been carried out for the operating parameters listedin Table 1 as an example. The geometry of heat exchanger

has been optimized for the selection of

Nomenclature

As surface area (m 2)Afc free ow area of ns (m 2)Acc clearance cross-section area (m 2)

Asc total projected shell side free ow area (m2

)C heat capacity rate of uids dened by the prod-uct of mass ow rate, _m and specic heat, cc orch (W/K)

c diametrical clearance (m)d f n diameter (m)d o n root diameter (m)D e mean diameter of shell (m)Dh hydraulic diameter (m) f friction factorG mass velocity (kg/m 2s)h heat transfer coefficient (W/m 2 K)hf n height (m)J sh correction factor for shell side heat transfer

coefficientJ sp correction factor for shell side pressure dropk bypass area factor = Acc /Afc

L axial length of shell (m)l length of nned tube (m)_m mass ow rate (g/s)_mf actual mass ow rate passes through ns (g/s)n number of ns per meter lengthNTU overall number of transfer unitsP absolute pressure (bar)D P pressure drop (bar)

Q heat transfer from either uid (W)

Re Reynolds numbers perimeter of tube (m)t mean thickness of n (m)

T temperature (K)T m uid mean temperature (K)D T LMTD log mean temperature difference (K)U overall heat transfer coefficient(W/m K) as

dened in Eq. (14)

Greeksl viscosity of uid(kg/m s)q density of uid (kg/m 3)g n efficiencye effectiveness of heat exchanger

Subscriptsc cold uidh hot uidi innero outerin inletout outlett tubes shellmax maximum

Superscript* non-dimension quantity

P.K. Gupta et al. / Cryogenics 47 (2007) 322–332 323



• A suitable mean diameter and an appropriate diametri-cal clearance for given n geometries.

• An appropriate n geometry for given mean diameterand diametrical clearance.

The correction factor in respect of ideal nned-tubebundle for heat transfer coefficients and pressure drop per-formance has been presented in graphical form which canbe useful for practicenor engineers for quick estimation.The maximum allowable diametrical clearance (when50% ow passes through clearance) for different n heightand number of ns has been presented in graphical formfor quick reference.

2. Geometry for thermal and pressure drop design

In this section, we have presented the derived formulaeby considering the diametrical clearance. Fig. 1 shows thetypical geometrical parameters of coiled nned-tube heatexchangers.

The total shell side free ow area Asc , is given by

Asc ¼ p Deðd f þ cÞ Àp De½ðd f À d oÞnt þ d o ð1Þ

Free ow area offered by the ns cross-section, Afc

Afc ¼ p De½ðd f À d oÞð1 À nt Þ ð2Þ

Free ow area offered by the clearance cross-section, Acc

Acc ¼ p Dec ð3Þ

If free ow area offered by the ns is greater than free owarea offered by the clearance then maximum ow will passthrough the ns. If the free ow area offered by the ns isequal to the area offered by the clearance, then half of theow of total ow rate will be diverted through the clear-ance. In the present analysis, it is assumed that the maxi-mum area offered by the clearance should be equal orless than the actual area offered by the nned-tube cross-section. On the basis of above assumptions, the maximumallowable clearance for the thermal and pressure dropperformance, the case when 50% ow passes through clear-ance, can be given by

cmax ¼ ½ðd f À d oÞð1 À nt Þ ð4Þ

It should be noted that if there is no clearance ( c = 0) thenEq. (1) is reduced to Eq. (2). It means that the total projected free ow area Asc will be equal to the free ow areaoffered by the ns, Afc .

The surface area offered by the outer nned surface inone coil; As

As ¼ p2 n

2ðd 2f À d 2oÞ þ d oð1 À tnÞ þ d f Át Ánh i De ð5Þ

The perimeter of outer nned surface (surface area per unitaxial length), so

so ¼ p2 n

2ðd 2f À d 2oÞ þ d oð1 À tnÞ þ d f Át Ánh i De

d f ð6Þ

The perimeter of inner tube surface (surface area per unitaxial length), si

si ¼ p2 De

d f d i ð7Þ

The equations derived above considering the effect of clear-ance can be used for the heat exchanger optimization. Theheat transfer coefficients and pressure drop coefficients willbe calculated on the basis of these areas prescribed above.

3. Design fundamentals

In a coiled nned-tube heat exchanger, the high pressuregas passes through nned tube in spiral form from top tobottom and the low pressure cold gas passes over thenned tube in cross ow pattern as shown in Fig. 1. There-fore, the advantage of this design is the high cross ow heattransfer coefficient on the shell side and high overall effec-

tiveness of counter-ow globally.

Table 1Geometry of the coiled nned-tube heat exchanger and its operatingparameters

Geometry of heatexchanger

Inner tube diameter, d i 8.2 mmFinned-tube diameter, d f 13.5 mmNo. of ns per meter, n 1024Axial length, L 1000.0 mm

Mean diameter, D e 145.0 mmFin type Integrated copper low height radial nsOperating parameters Working uid Helium

Temperature range 300–90 KMass ow rate 3.0 g/sHigh pressure 15.0 barsLow pressure 1.0 bar

Fig. 1. Sectional view of heat exchanger.

324 P.K. Gupta et al. / Cryogenics 47 (2007) 322–332



The assembly of these heat exchangers consists of theinner shell, outer shell and nned tube. The nned tube iswound on the inner shell and then it is jacketed by theouter shell. The dead space between two consecutive coilshas to be lled by some cord as shown in Fig. 1. A diamet-rical clearance between the shell and tube periphery has to

be provided in order to make the assembly easy. The ther-mal and pressure drop performance of heat exchanger isinuenced by this clearance. A part of the cold streampasses through this clearance without taking part in heatexchange process, making the heat exchanger ineffective.On the other hand, the pressure drop performance will beimproved by increasing the available cross-section area.The effect of clearance on the thermal and pressure dropperformance of heat exchangers is considered in the presentstudy and optimum geometries have been found out forgiven operating parameters as given in Table 1 .

If we assume the mass ow rate of the cold stream ow-ing in the shell side is given by _mc then the actual mass owrate passing through the ns is

_mf ¼_mc

k þ 1ð8Þ

where k is bypass area factor and it is given by

k ¼Acc

Afcð9Þ

The Reynolds number will be calculated based on theactual ow passes through the ns and can be given by

Ref ¼ReWOC

k þ 1ð10Þ

where Re WOC is the Reynolds number based on the totalcross-section area available for shell side ow when thereis no clearance and can be calculated as follows:

ReWOC ¼_mc Dh

Afclð11Þ

The characteristic dimension for the Reynolds number inEq. (11) is the equivalent diameter, or the hydraulic dia-meter Dh

Dh ¼4 Asc

As= Lð12Þ

G ¼_mc

Ascð13Þ

where G is the mass ow rate per unit free-ow area andwill be used for calculating the heat transfer coefficients.

In Eqs. (12) and (13) Asc will be equal to Afc , if there isnot any clearance.

4. Assumptions

Following assumptions are made for carrying out theanalysis.

1. The pressure drops due to other effects are negligible in

comparison to the core frictional pressure drop.

2. All thermo physical properties have been calculated atthe mean temperature T m of the individual uid stream.

5. Design and optimization

5.1. Thermal design

For the usual heat exchanger design problem, where allend temperatures are given, the heat transfer area can becalculated by using the overall heat transfer coefficientbased on either hot uid heat transfer area or cold uidheat transfer area. In the present study, we will followthe design procedure described by Croft and Tebby [5].And therefore, the overall heat transfer coefficient,U (W/m K), will be based on axial length of the heatexchanger instead of the heat transfer area of either uidby applying the concept of wetted perimeter of heatexchanger per unit axial length.

Fluid properties are evaluated at mean temperaturegiven by

T m ¼T h;in þ T h;out

2or

T c;in þ T c;out

2where T h , T c are the respective inlets and outlets tempera-tures of hot and cold streams of heat exchanger.

The total heat duty ð_QÞ of the hot uid which has to beremoved by exchanging the energy with the cold uid isexpressed as follows:_Q ¼ C hðT h;in À T h;out Þ ¼ULD T LMTD ð14Þ

where L is the axial length of heat exchanger and D T LMTDis the log mean temperature difference, given by

D T LMTD ¼D T hot end À D T cold end

ln D T hot endD T cold end

ð15Þ

where D T hot end and D T cold end are the hot and cold endtemperature differencesD T hot end ¼ T h;in À T c;out ð16Þ

andD T cold end ¼ T h;out À T c;in ð17Þ

The overall heat transfer coefficient U (W/m K) based onper unit axial length of heat exchanger can be given by

U ¼1

hi siþ

1g ho so

À1

ð18Þ

where hi and ho are the heat transfer coefficients dened asthe rate of heat transfer across unit area of separating wallfor unit temperature difference between gas stream and sep-arating wall (W/m 2 K). si and so are the perimeters (surfacearea per unit axial length) as dened in Eqs. (6) and (7)across which heat is transferred. The thermal resistanceof separating wall is omitted in Eq. (18) as it is small en-

ough as compared to uid resistances. The n efficiency g




can be assumed 100% for cupper nned tube in Eq. (18)[5,9].

The value of heat transfer coefficients for the inner andouter stream is calculated using the same correlations asdescribed by Croft and Tebby [5] and are reproduced here

hi ¼ 0:033ch _m0:8h l

0:2h d À1:8

i ð19Þ

and

ho ¼ 0:021cc _m0:8f l

0:2c AÀ1:0

sc s0:2 ð20Þ

Eq. (20) can directly be used for calculating the heat trans-fer coefficient in the shell side as used by Croft and Tebby[5] for calculating the overall heat transfer coefficient, U (W/m K). However, it can also be deduced in terms of Rey-nolds number based on the actual ow passes through thens.

Eq. (20) can be written as

ho ¼ 0:021cc _mf _mÀ0:2f l

0:2c AÀ1:0

sc s0:2 ð21Þ

The above Eq. (21) can be rearranged as follows:

ho ¼ 0:027ccG f 4 _mf

l c so À0:2

ð22Þ

Eq. (22) can be expressed in terms of Reynolds number asfollows:

ho ¼ 0:027ccG f ReÀ0:2f ð23Þ

5.2. Pressure drop design

The pressure drop design is equally important as thethermal design of heat exchanger for any cryogenic sys-tems. The tube side pressure drop across the heat exchan-ger will reduce the amplitude of high pressure streamthereby reducing the area of the expansion space in PV dia-gram and the gross refrigeration produced by the refriger-ator/liqueer.

On the other hand, pressure drop in the shell side isextremely important for any cryogenic systems such ashelium liqueer/refrigerator. In a helium liqueer/refriger-ator (helium normal boiling point 4.2 K and critical pres-sure 2.2 bar), the total pressure drop of the shell sideshould not be more than 0.2 bar because of the constraintof critical pressure of the helium.

The amplitude of the pressure drop ( D P ), either shellside or tube side, per unit working length through the heatexchanger is given by

D P ¼fG 2

2q Dhð24Þ

In Eq. (24) the value of friction factor, f , has to be calcu-lated as follows:

For the tube side :For turbulent ow inside a smooth tube of any cross-

section, the friction factor was calculated by the empirical

equation as suggested by Timmerhaus and Flynn [10]

f ¼ 0:184 ReÀ0:2 1 þ 3:5d i

De ð25Þ

For shell side ow :The shell side ow is generally laminar in the coiled

nned tube heat exchanger and the friction factor for theshell side is given by [11]

for 400 < Re < 104 f ¼ 1:904 ReÀ0:2 ð26Þ

In the present analysis, the pressure drop for the tube sideand shell side is presented as dimensionless quantity and itis dened as follows:

D P Ã ¼D P P c

ð27Þ

5.3. Optimization procedure

In any helium refrigerator/liqueer, the multiple num-bers of heat exchangers are used. These heat exchangersare used at different temperature levels. The typical work-ing temperature range of rst heat exchanger is 300–90 Kfor medium size helium refrigerator/liqueer and the typi-cal thermal size (NTU) of this heat exchanger is 21. Thestringent requirements of these heat exchangers are highereffectiveness and low pressure drops in both of the streams.These heat exchangers can be made in different geometricalcongurations such as by changing the mean diameter of heat exchanger, by opting the different inner diameter of nned tube or by choosing the different n geometries tosatisfy the xed thermal size (NTU) requirement and low

pressure drop design criteria. Furthermore, for the xedvalues of NTU, one can obtain the heat exchanger cong-urations in different sizes and weight to fulll the effective-ness and pressure drops requirements but these heatexchangers may not be suitable from system performancepoint of view due to higher thermal mass. The geometricalcongurations of these heat exchangers much depend onhow one has selected the mean diameter of heat exchangerfor the xed n geometries or for the xed mean diameter,how one has selected the nned tube geometries such asinner diameter of nned tube, n height and number of ns. Therefore, it is necessary to choose the right combina-tions of different geometrical parameters for obtaining theoptimum size of heat exchanger for the given NTU.

In the present study, the rst heat exchanger of NTU = 21 has been chosen as an example for the optimi-zation purpose while keeping the operating parameters,as described in Table 1 , xed. The expressions derived inthe preceding sections have been used for the optimizationpurpose. The calculations are performed for different val-ues of bypass factors, mean diameters, inner tube dia-meters, n height and number of ns to obtain thecorresponding pressure drops in tube side and shell sideand surface area requirements for the xed values of NTU and mass ow rate. These calculations were carried

out to study the effect of clearance, for selecting the suitable




mean diameter for given nned tube geometries or forselection of the proper nned tube geometries for givenmean diameter.

6. Results and discussion

In this section, we apply the geometrical expressionsderived above for the design and optimization of coilnned-tube heat exchangers. The actual four end tempera-tures of one heat exchanger tested in our lab have beencompared with the predicted end temperatures from thepresent design methodology. The results of the presentanalysis show that the heat exchanger geometrical congu-ration can be optimized in respect of the thermal and pres-sure drop performance and therefore, the heat exchangercan be reduced signicantly both in terms of size andweight. In this way, we show how our optimization canbe implemented in real practice for designing such heat

exchangers.

6.1. Comparison of experimental results with the presentanalysis

Recently, Gupta et al. [8] presented detailed descriptionof experimental set-up and the results obtained from one of the prototypes designed and manufactured in our lab basedon the geometrical parameters given in Table 1 . The pres-ent analysis considers the effect of leakage of ow throughclearance provided for the ease of manufacturing and it isone of the important causes to deteriorate the performance

of coiled nned-tube heat exchangers. The predictions of four end temperatures based on the present analysis con-sidering the clearance effects, are compared with the actualresults published earlier [8].

Table 2 gives the details of experimental parameters forwhich results have been compared. The quantitative com-parison of four end temperatures of a heat exchanger withthe experimental results has shown in Table 3 . I t i sobserved from the experimental results that the presentanalysis show good agreement with theoretical results. Itcan be noted from Table 3 that the heat exchanger effec-tiveness can be increased from 91.7% to 95.8% by reducingthe clearance from 1.2 mm to 0.3 mm and establish theimportance of the effect of clearance for designing of such

heat exchangers. Hence, diametrical clearance betweeninner shell, nned surface and outer shell has to be chosencarefully for manufacturing of these heat exchangers andnecessary allowance has to be provided at the design stageonly.

6.2. Maximum allowable clearance

Fig. 2 shows the maximum allowable clearance that canbe provided for the manufacturing of a heat exchanger fordifferent n height and number of ns. The gure showsthat the maximum allowable clearance is either increasedwith the n height for given number of ns or increasedwith the lesser number of ns for given n height. Thiscan be attributed as either the larger n height or the lessernumber of the ns can be used to reduce the effect of leak-age of ow through the clearance. For an example, themaximum allowable clearance required for 2 mm n heightfor 1024 ns per meter (26 ns per inch) is 2.77 mm.Generally, 1.5–2 mm diametrical clearance is sufficient forthe assembly of a medium size heat exchanger during fab-rication. Therefore, in this case the bypass area will be lessas compared to the cross-sectional area offered by the nsand the maximum ow will be diverted through the ns.Hence, it will reduce the effect of leakage of ow on thethermal performance of the heat exchanger. In anothercase, the maximum allowable clearance required for1.5 mm n height for 1024 ns per meter (26 ns per inch)

Table 2Details of experimental parameters

Working uid NitrogenHot end temperature 296.0 KCold end temperature 151.0 KHot stream pressure 15.0 barCold stream pressure 1.0 barMass ow rate 9.5 g/sNumber of transfer units, NTU 12.0Overall heat transfer coefficient 142.2 W/m KRatio of the nned side to tube side area 4.4

Manufacturing clearance, c 1.2 mm

Table 3Comparison of theoretical predictions of present analysis with experi-mental results [8]

Methodology T h,in

(K)T h,out

(K)T c,in

(K)T c,out

(K)e (%)

Theoretical resultsc = 0.3 mm 296.0 157.00 151.00 289.10 95.8c = 0.5 mm 296.0 159.00 151.00 289.0 94.5c = 1.2 mm 296.00 163.00 151.00 285.50 91.7

Experimental results(for c = 1.2 mm)

296.00 165.00 151.00 284.00 90.3

Fig. 2. Maximum allowable clearance as a function of n height and

number of ns.




will be approximately 2 mm. In this case, if the clearancegiven for the manufacturing of a heat exchanger is equalto 2 mm, then half of the ow will be diverted throughthe clearance. This will affect the thermal performance of heat exchanger signicantly. Here, it can be recommendedthat for the given input specications, the total clearance

has to be provided less than 2 mm to reduce the effect of leakage of ow through clearance for the n height of 1.5 mm. Therefore, one has to make good engineering judgment while selecting the clearance for designing andmanufacturing of these heat exchangers.

6.3. Correction factors for heat transfer coefficientand pressure drop

The thermal and pressure drop performance of theseheat exchangers are affected signicantly by the clearanceprovided for the manufacturing point of view. The neces-sary design allowance has to be provided to take intoaccount the effect of clearance at the design stage only. Inthe present analysis, the correction factors for the thermaland pressure drop design have been determined withrespect to the ideal nned-tube bundle (when clearance iszero). Figs. 3 and 4 show the correction factors for the dif-ferent values of bypass area factors ( k ). It could be notedfrom these gures that the values of J hs and J ps are 1.0for the value of k = 0.0 and the values of the heat transfercoefficients and pressure drops calculated from Eqs. (20)and (24) can be used for nal design purpose. If there isclearance then some correction factor corresponding tothe value of k obtained form these gures has to be multi-

plied for getting the actual heat transfer coefficient andpressure drop. It could also be seen from Fig. 3 that if the clearance area is only 10% of the cross-section areaoffered by the ns (k = 0.1), the effect of clearance is notsignicant and the thermal performance of the heatexchanger will not be affected much. These graphs are par-ticularly useful for the quick estimation of heat transfercoefficients and pressure drop design.

6.4. Effect of mass ow rate on sizing of heat exchangers

Fig. 5 shows the effect of mass ow rate on the pressureloss of tube side and shell side of a heat exchanger. Theeffect of mass ow rate variations on the size (axial length)of a heat exchanger for the given value of temperaturerange (300–90 K) is also shown in Fig. 5. The trend of the gure shows that the tube and shell side pressure dropincrease sharply but the sizing of heat exchanger increasesonly by 15% for the variation in mass ow rate from 2.5 g/sto 4.5 g/s for given temperature range. The heat exchangersfor any helium liqueers/refrigerators are designed for lim-ited pressure drop conditions. Hence, it can be interpretedfrom the gure that the same heat exchangers can be usedfor the different capacity helium liqueers/refrigerators,

however, the pressure drops within the heat exchangersshould be in the acceptable limits.

6.5. Effect of bypass area factor (k) on sizing and pressuredrops

The effect of bypass area factor ( k ) on the non-dimen-sional frictional pressure loss of tube side, shell side acrossthe heat exchanger and on the surface area, As, requirement

Fig. 3. Heat transfer coefficient correction factor as a function of bypass

area factor.

Fig. 4. Pressure drop correction factor as a function of bypass area factor.

Fig. 5. Non-dimensional pressure drops and axial length as a function of

mass ow rate.




is plotted in Fig. 6. It can be shown from Fig. 6 that theshell side pressure drop decreases signicantly up to the

value of k = 0.5 and after that there is not much reductionin the shell side pressure drop as the value of k increases.This is due to the fact that the axial length of heat exchan-ger goes up as the clearance increases and offset the advan-tage of the reduction of total shell side pressure drop. Onthe other hand the tube side pressure drop and surface arearequirement for the given heat duty increases continuouslyas the value of the bypass area factor k increases. It can beinterpreted from the gure that the clearance provided forthe ease of manufacturing can be used for adjusting theshell side pressure drop in a heat exchanger for the specicrange only. Fig. 6 shows that the shell side pressure dropcan be reduced up to 70% at the expense of the increasein surface area and tube side pressure drop by providingsuitable clearance between the shell and nned tube. Here,it can be concluded that the designer has the choice toadjust the shell side pressure drop performance accordingto their nned-tube geometries by providing suitableclearance.

6.6. Optimization of mean diameter of shell

Fig. 7 plots the non-dimensional tube side, shell sidepressure loss across the heat exchanger and the correspond-ing surface area requirement as a function of mean diame-ter of the shell. It is apparent from Fig. 7 that the shell sidepressure drop decreases up to a mean diameter D e =185.0 mm and then it tends to become almost constant asthe mean diameter of shell increases. This can be explainedas the available cross-section area for the ow increases asthe mean diameter of shell increases and reduces the shellside pressure drop up to mean diameter of 185.0 mm.The shell side pressure drop curve becomes almost attenedas the mean diameter of shell increases further. This is dueto the fact that the axial length of heat exchanger increasesas the mean diameter increases and offset the advantage of selecting the higher mean diameter of shell for the reduc-

tion of the shell side pressure drop. On the other hand,

the tube side pressure drop is almost constant up to themean diameter of 185.0 mm and then it tends to increaseas the mean diameter increases. This is related to the reduc-tion in the shell side heat transfer coefficient due to increasein shell diameter of a heat exchanger and the requirementof the length of nned tube will be more in this case forgiven heat duty. Therefore, the total tube side pressuredrop will increase after the certain value of the shell diam-eter. The varying trend of the frictional pressure drop of shell side with the mean diameter of the shell is a reverseof varying trend of the frictional pressure drop of tube sidewith the mean diameter of the shell. From Fig. 7 it can alsobe observed that the heat transfer surface area is increaseswith the mean diameter of the shell. This may be explained

that the value of shell side heat transfer coefficientdecreases due reduction in the ow velocity with theincrease of the mean diameter of shell. Therefore, it willrequire more nned-tube surface area for the given heatduty.

Based on the above discussion, it can be concluded thatthe mean diameter of the shell can be optimized for thethermal and pressure drop performance of a heat exchan-ger for the given n geometries. The shell side pressuredrop is the most crucial for any helium liqueer/refrigera-tor and it can be observed that there is not much reductionin the shell side pressure drop after a certain value of themean diameter of shell but on the contrary the surface arearequirement and tube side pressure drop increases. Thiswill result in the larger unit for the same cryogenic systemswhich poses other problems like more heat-in-leaks to thesystem and can affect the performance of the whole system.This indicates that the choosing of an appropriate shelldiameter is one of the most important parameter of anyheat exchanger design.

6.7. Inner tube diameter optimization

Fig. 8 shows the non-dimensional shell side, tubeside pressure drop and surface area as a function of the

inner diameter of the nned tube for a mean diameter

Fig. 6. Non-dimensional pressure drops and surface area as a function of bypass area factor ( k ). Fig. 7. Non-dimensional pressure drops and surface area as a function of

mean diameter.




D e = 145.0 mm. Fig. 8 shows that the tube side pressuredrop decreases drastically as the tube diameter increases,

but shell side pressure drop and surface area required forthe given heat duty increases. This is due to the fact thatthe inner heat transfer coefficient decreases as the tubediameter increases and hence reduces the over all heattransfer coefficient. This will be resulted in the increase of heat exchanger size (axial length) and increase the shell sidepressure drop. Therefore, the best practice for any designeris to rst select the inner tube diameter of the nned tubeaccording to the acceptable pressure drop in the tube tokeep the shell side pressure drop with in stipulated limitswith minimum surface area. For an example, if we selectthe inner tube diameter d i = 8 mm for which the tube side

pressure drop is 0.5 bar and shell side pressure drop is0.05 bar approximately for the operating parameterdescribed in Table 1 . It could be noted here that 0.5 bartube side pressure drop in the medium capacity helium liq-ueer/refrigerator can be tolerated but on the other hand if emphasis is given to reduce the tube side pressure drop fur-ther by increasing the tube diameter, the surface arearequirement will increase accordingly and the systembecomes unnecessarily bulky. Therefore, the selection of the larger diameter of inner nned tube has to be avoided.

6.8. Optimizations of n height

Fig. 9 shows the non-dimensional shell side, tube sidepressure drop and surface area as a function of the nheight for a mean diameter D e = 145.0 mm and inner tubediameter d i = 8.2 mm. The gure shows that the tube sidepressure drop and shell side pressure drop decrease as then height increases up to 1.8 mm and then the reductionin the pressure drops is minimum or almost negligible asthe n height increases but the surface area requirementincreases monotonously as the n height increases. Thepressure drop for the shell side decreases due to the factthat the available cross-section area increases as the nheight increases and the ow velocity of the stream

becomes lower which reduces the pressure drop in the shell

side. Similarly, the pressure drop for tube side decreasesdue to the fact that the total nned-tube length required

for the given heat duty and tube diameter decreases andhence total pressure drop of the tube decreases. FromFig. 9, we can conclude that there is no advantage of increasing the n height beyond 1.8 mm as there is no mea-surable reduction in the pressure drops performance but if one selects the n height more than 1.8 mm, the surfacearea requirement will be more for the same operatingparameters with out gaining in pressure drop performance.

6.9. Effect of number of ns on the pressure drop performance and sizing

Figs. 10–12 show the non-dimensional shell side, tubeside pressure drop and surface area requirement as a func-tion of number of ns for the mean diameter of 120.0 mm,145.0 mm and 175.0 mm. These gures show that the tubeside pressure drop decreases with the number of ns whileshell side pressure drop increases with the number of ns.This can be explained as the more tube length is requiredfor the given heat duty for lesser number of ns which willincrease the total tube side pressure drop for given nned-tube diameter. On the other hand the shell side pressure

Fig. 8. Non-dimensional pressure drops and surface area as a function of inner tube diameter.

Fig. 9. Non-dimensional pressure drops and surface area as a function of n height.

Fig. 10. Non-dimensional pressure drops and surface area as a function of

number of ns for De = 120.00 mm.




drop will be less for the lesser number of ns as the nnedtube will offer more cross-sectional area and reduces theow velocity through the nned tube. These gures alsoshow that the surface area rst decreases with the numberof ns then increases for the given heat duty for theD e = 120.0 and 145.0 mm but as the D e increases to175.00 the surface area requirement decreases continuouslywith the number of ns.

The above analysis clearly indicates that, there exists acondition of minimum surface area for the smaller value

of the mean diameter of the shell where the shell side pres-sure drop is also on the lower side. As the value of D e

increases, the minimum surface area conditions shiftedtowards the higher number of ns and ultimately it disap-pears for the higher value of D e.

7. Conclusions

The design and optimization of coiled nned-tube heatexchangers with the consideration of clearance providedfor the ease of manufacturing have been presented. Thethermal and pressure drop performance of a coiled

nned-tube heat exchanger depend on the clearance

between the shell and nned tube in addition to other geo-metric and operating parameters. The comparison of pre-dictions of four end temperatures, obtained from presentstudy, with experimental results conrms the importanceof clearance effect on the nal design calculations. Theresults of the present study show that the effect of clearance

can be used for adjusting the thermal and pressure dropperformance of heat exchangers in many situations suchas to nd out an optimum conguration for given ngeometries. The results of the present study also indicatethat the heat exchanger can be optimized either by choos-ing the suitable mean diameter of shell with an appropriateclearance for given n geometries or by selecting the opti-mized n geometries. From the present study, followingobservations can be noted.

• For xed operating and system parameters in the pres-ent study, it was found that only a specic range of clearance will help to reduce the pressure loss in the shellside. For the case of D e = 145.0 mm, d i = 8.2 mm andn height 1.4 mm, the shell side pressure drop reducesup to only 1 mm clearance. However, the tube side pres-sure drop increases but remained with in acceptable lim-its. Therefore, the existing n geometries can beaccommodated in such a geometrical congurationwhere the heat exchanger can satisfy both thermal andpressure drop requirements for any cryogenics systems.Moreover, the heat exchanger can be optimized in sucha way that thermal and pressure drop requirement sat-isfy without making the unit bulky.

• In the present study, it is also found that only for a

range of shell mean diameter, the shell side pressuredrop decreases while the tube side pressure drop remainsnearly constant at the expense of increase in surfacearea. For the case of d i = 8.2 mm and c = 0.9 mm, theshell side pressure drop decreases up to D e = 185.0 mmand tube side pressure drop remains constant at nearly0.65 bar. This indicates that choosing the mean diametermore than 185.0 mm will make the unit bulky withoutdecreasing the shell side pressure drop in addition toincrease in tube side pressure drop.

• The present study also makes clear that the inner diam-eter of nned tube, number of ns and n height can beoptimized for the given xed parameters. The graphicalpresentations for the correction factor for heat transfercoefficient, pressure drop and maximum allowable clear-ance for manufacturing of heat exchanger can be usefulfor practicing engineers for designing such heat exchang-ers for cryogenic applications.

Acknowledgements

The authors are grateful to Dr. Parthasarathi Ghosh,Mr. Rupul Ghosh and Mr. R.C. Sharma for their usefuldiscussions, suggestions and fabrication of heat exchanger

during carrying out this work.

Fig. 11. Non-dimensional pressure drops and surface area as a function of number of ns for D e = 145.00 mm.

Fig. 12. Non-dimensional pressure drops and surface area as a function of

number of ns for D e = 175.00 mm.




References

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[4] Geist JM, Lashmet PK. Miniature Joule–Thomson refrigerationsystems. Adv Cryo Eng 1960;5:324–31.[5] Croft AJ, Tebby PB. The design of nned-tube cryogenic heat

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[8] Gupta Prabhat, Sharma RC, Kush PK. Development of cross-counter ow heat exchanger for cryogenic temperature range. In:Proceedings of national seminar and conference on cryogenics and itsfrontier application. Bengal Engineering College; Howrah, India,2004. p. 125–7.

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design and optimization of coil finned tube heat exchangers for cryogenic applications

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