the effect of neighbouring tubes on heat ansfer … · 2015-04-09 · int. j. mech. eng. & rob....

Int. J. Mech. Eng. & Rob. Res. 2015 M K Goel and S N Gupta, 2015

THE EFFECT OF NEIGHBOURING TUBES ON HEATTRANSFER COEFFICIENT FOR NEWTONIAN ANDNON-NEWTONIAN FLUIDS FLOWING ACROSS

TUBE BANKM K Goel1* and S N Gupta1

*Corresponding Author: M K Goel,[email protected]

The experimental investigation of heat transfer coefficient for Newtonian and non-Newtonianfluids, obeying Power-law, has been carried out by measuring local surface temperature ofheated tube and bulk inlet and outlet temperatures of test fluid. The effect of neighbouring tubeswas studied by placing dummy tubes around it one by one. Experiment was conducted withsingle tube and using four more tube arrangements. The following correlation has been obtainedfor heat transfer for the flow of water, 0.5% PVA and 1% and 1.5% CMC solutions across tubebank.

3/1

3/1

Re35.1Pr

ncNu

for Re < 60

3/2

3/1

Re35.0Pr

ncNu

for Re > 60

Keywords: Non-newtonian, Power-law, Tube bank, Cross-flow, Dummy

INTRODUCTIONThe flow of fluids over an array of rods orcylinder represents an idealization of severalimportant industrial processes. The powergeneration field, the chemical industry, andother technology based industries increasinglyemploy heat exchangers involving tubes incross-flow. The cross-flow over the bank oftubes represents the true picture of the flowencountered in the shell side heat exchangers.

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Int. J. Mech. Eng. & Rob. Res. 2015

1 Department of Mechanical Engineering, Hi-Tech Institute of Engineering & Technology, Ghaziabad, UP, India.

Consequently, considerable research efforthas been expended in exploring andunderstanding the convective transportprocess between moving fluids andsubmerged tube banks having different typeof arrangements.

Most of the fluids encountered in chemicaland allied processing applications do notadhere to the classical Newtonian postulateand are accordingly known as non-Newtonian.

Research Paper

13


Most particulate slurries (China clay, coal inwater, sewage sludge, multiphase mixturessuch as oil water emulsions, gas-liquiddispersions such as froths and foams andbutter) are non-Newtonian fluids. Melts andsolutions of high molecular weight naturallyoccurring and synthetic polymers are found tobe non-Newtonian. Further examples ofsystems displaying a variety of non-Newtoniancharacteristics include pharmaceuticalformulations, cosmetics and toiletries, paints,synthetic lubricants, biological fluids (blood,synuvial fluid, saliva, etc.), and food stuffs (jams,jellies, soups, marmalades, etc.).

Frequent occurrence of such fluids inindustries has created considerable interestin the study of bahaviour of these fluids invarious process equipments where these aresubjected to heating and/or cooling in heatexchangers. The design of cross flow heatexchanger is a complex task requiring theexamination and optimization of a wide varietyof design parameters such as friction factor,heat transfer, Reynolds number, etc. Heattransfer is considerably influenced by flowregime around the tube. Therefore, the needof an appropriate model describing the realflow situation with in the heat exchanger cannotbe overlooked.

Several models and techniques have beenput forward to describe the flow behaviour andheat transfer phenomena in cross flow heatexchangers which suffer from one discrepancyor the other. The results based on abovemodels are fairly inaccurate. Zukauskas(1972) proposed that the flow across a bundleof tubes is comparable to that in a parallel platechannel. The fluid flow was approximated asthat of flow through a series of parallel plates

with spacing equal to minimum space betweenthe tubes. The model was later modified toconverging-diverging parallel plate channelmodel to account for fluid contraction betweentwo adjacent tubes and expansion elsewhere.The flow pattern is assumed to follow asinusoidal path in the direction of flow andReynolds number calculation is based onequivalent hydraulic diameter and averagevelocity. It is reported by Prakash et al. (1987)that the parallel plate channel model can givebetter correlation of pressure drop data for theflow of both Newtonian and Non-Newtonianwhile the analogy between heat andmomentum is restricted to the condition of lowReynolds number range.

Snyder (1953), Zukauskas (1968), (1972),(1982), (1985) and (1987), Adams (1968),Achenbach (1975), Hwang and Yao (1986),Aiba (1976), (1980) and (1990) extensivelystudied the heat transfer and flow patternacross single tube and bundle of tubes.Zukauskas presented extensive theoretical aswell as experimental information regardingdesign parameters for flow across a tube andtube bank. The results presented byZukauskas clearly showed that the heattransfer phenomena in a tube bank is influencedby several factors like tube geometry, natureof flow, temperature of heating or coolingsurface and that of bulk of fluid, etc., He,however, focused his attention only on theaverage value of heat transfer across tubebank like many other workers and no effort hasbeen made to study the transport at anyspecified location within the tube bank for theflow of non-Newtonian fluids. It is ratherinteresting and useful to determine as to howaddition of tubes around a single tube, in the

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formation of tube bank, change the flow patternwhich in turn changes the heat transfercoefficient. Also the effects of rheologicalproperties of non-Newtonian fluids and tubediameter be found to analyse the heat transfer.In order to find the above experimentally a newmethodology has been adopted in whichdesired number of tubes at desired locationmay be arranged and for different tubegeometries the heat transfer coefficient maybe calculated for different flow rates ofNewtonian as well as non-Newtonian fluids.

THEORETICALThe visualization of the physical picture of crossflow and heat transfer pattern from thephotograph published by Lorich (1929) revealsthat in tube bank the flowing fluids is not incontinuous contact with heat transfer surfacebut experiences a short contact and aftercontacting a part of tube, find a gap and thenmoves to the next tube following either aslightly curved path (inline arrangement) or asinusoidal path (staggered arrangement).Refer Figure 1. Due to short contact for a shortperiod, the temperature distribution does notattain a fully developed profile but the gradientof the temperature with respect to directionperpendicular to the flow remains confined toa very thin region near to the surface.

Thus

v << u and 2

2

2

2

yT

xT

...(1)

Following the above assumptions, for thepresent model, the steady-state differentialEquation (1) reduces to

2

2

yT

xTu

...(2)

For a constant property fluid in laminar flowthe solution of Equation (2) may be obtainedby assuming high velocity across short contactsurface of the tubes. The velocity distributionin the boundary layer is assumed to beconfined to a very thin region over the tubesurface for laminar flow. The velocity profilethus becomes:

yyuu

w

...(3)

and consequently Equation (2) takes the form,

2

2

yT

xT

yu

w

...(4)

Applying the following boundary conditions

at x = 0, y = 0, T = Tb

at x > 0, y = 0, T = Tw

and solving we get

3/1

9893.0

x

Kh vx

...(5)

and3/1

9893.023

s

vav x

Kh a

...(6)

where

Figure 1: Periodically Converging andDiverging Parallel Plate Channel

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xv D

Un

n212

312

...(7)

and

Hv D

Un

na

123

12 ...(8)

where,

hx represents the local heat transfer at anyposition x and

hav represents the average heat transfercoefficient.

EXPERIMENTALIn the present experimental programme, theheat transfer data have been obtained bymeasuring the local surface temperature ofheat flux probe and bulk inlet and outlettemperatures of the test fluid. The effect ofneighbouring tubes on heat transfer wasstudied by placing dummy tubes around theheated probe as shown in Figure 2.

Experiments were conducted using fivedifferent tube arrangements. In order toinvestigate the effect of tube diameter on heattransfer, different tubes of diameter 3.18, 2.54and 1.91 cm have been used. A schematicdiagram of the experimental setup is given inFigure 3.

Figure 3: Schematic Diagramof Experimental Set-Up

The test section of dimension 18 cm x 14.7cm x 23.5 cm was fabricated with Perspexsheet. Three tubes of diameter 2.54 cm halfexposed, were fixed on either side walls. Thetest section could accommodate desirednumber of tubes of different diameters. Thetop portion of the test section was providedwith through holes of diameter 3.18 cm whilein the bottom portion, partial grooves wereprovided to support the base of the tube againstthe pressure of flowing fluids. The top coverplate was fastened with nuts and bolts thusmaking it removable so that, wheneverrequired, the top cover plate could be removedand required number of dummy tubes couldbe fixed in the test section. The heat flux probeor active tube was tightly fitted in the removable

Figure 2: Arrangement of Dummy TubesAround Heater Tube

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top cover plate and its location was so adjustedthat it required the central position in the testsection. Arrangement was made toaccomodate different sized tubes of threediameters that were studied in the presentwork. To accommodate the correspondingdummy tubes plastic adopters (plugs) wereused in top and bottom plates of the testsection.

The heat flux probe was fabricated from aCopper tube of 6 mm wall thickness. A pencilheater was inserted inside to fit in the tubetightly. Four slots were cut on the outer surfaceof the tube along its length at locations 0°, 90°,180° and 270° at central axis of the tube, i.e.,diametrically opposite to each other. CopperConstantan thermocouple wires, embeddedin each slot, measured the local surfacetemperature of the heated tube which, whenaveraged, gave the average surfacetemperature of heated probe. A tube bank withlongitudinal pitch 7.0 cm and transverse pitch3.5 cm was used.

A capillary tube viscometer was used tomeasure the rheological properties ofaqueous solutions of Carboxyl MethylCellulose and Poly Vinyl Alcohol while a parallelplate apparatus was used to measure thethermal conductivities of these fluids. Sinceaqueous CMC and PVA solutions followedOstwald Power Law hence the values of flowbehaviour index (n) and consistency constant(K) were determined by plotting wall shearstress (w) versus shear rate on a log-log plot.

EXPERIMENTAL PROCEDUREFor a particular tube diameter and tube bankconfiguration the flow rate of a particular fluidwas initially set to a predetermined value. A

constant electrical flux was supplied to theheated probe. After about an interval of 15minutes when steady state was achieved thetemperature of four thermocouples, placed atfour different locations on the probe wasrecorded with the help of a multi-channeldigital temperature indicator. Simultaneouslythe inlet and outlet temperatures of the fluidwere also noted. The mass flow rate of fluidwas found by collecting the fluid for knowninterval of time. The flow rate was varied andall the above mentioned parameters weremeasured. Similar sets of observation weremade using dif ferent f luids for eachconfiguration and also for the whole tube tankconsisting of thirty effective tubes of aselected diameter at a time. The experimentwas conducted with three tubes of differentdiameters.

RESULTS AND DISCUSSIONIn case of aqueous CMC solutions, it isobserved that the flow behaviour index (n)decreases with increase in polymerconcentration resulting in increased pseudoplastic behaviour. Also no change has beenobserved in flow behaviour index (n) withtemperature variation in the range investigatedin this work. However the value of consistencyconstant (K) decreases with increase intemperature. In case of aqueous PVA solution,parameter K remains constant irrespective ofchange in temperature for the rangeinvestigated. The evaluated values ofconsistency constant (K) and flow behaviourindex (n) are given in Table 1.

Viscosity, density, thermal conductivity andspecific heat for water have been taken fromthe literature (Brown and Marko, 1958).

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The heat transfer is closely related to fluiddynamics, therefore the two phenomenon areconsidered simultaneously around the curvedsurface. These are complex processes andmainly depend upon the type of fluid andReynolds number. A laminar boundary layer isformed on the portion of circular tube facingthe flow with its thickness increasingdownstream. There is also a longitudinalpressure gradient caused by curved surface.It may be noted that the point at which shearstress acquired the zero value; the boundarylayer can separate from the surface. Theseparation point is followed by a reverse flow,where the velocity vectors of fluid masses nearthe wall are in opposite directions. The fluidassuming reverse flow contracts the boundarylayer and changes to vortex which begins torotate. The whole phenomenon of fluidbehaviour is reflected with local heat transferrate. The fluid dynamics and heat transfer arealso influenced by free stream turbulence,geometry and surface roughness etc.

The variation in the flow of fluid over acylinder in cross flow gives rise to similarvariation in local heat transfer. For a Newtonianfluid on the front part, upto separation of

boundary layer, heat transfer can be foundanalytically. For low Reynolds number flow ofa Newtonian fluid, the heat transfer on the frontpart of the cylinder is maximum whichdecreases with the developing boundary layer.However, at higher Reynolds number, the heattransfer gradually increases downstream oflaminar boundary layer separation and ismainly determined experimentally. For nonNewtonian fluids, the fluid dynamics and heattransfer phenomenon is more complex innature.

To study the effect of Prandtl number onheat transfer a graph showing the variationof NuRe-1/2 versus Pr has been plotted inFigure 4.

1% CMC 0.803 27.0 0.02287

1% CMC 0.803 33.0 0.02016

1% CMC 0.803 40.0 0.01836

1.5% CMC 0.637 22.5 0.15988

1.5% CMC 0.637 27.0 0.14465

1.5% CMC 0.637 31.0 0.13856

0.5% PVA 1.15 25.5-35.0 6.24 x 10-4

Table 1: Experimental Value of FlowBehavior Index and Consistency Constant

Fluid n Temperature°C

ConsistencyConstant Kg/

m.sec(2-n)

Figure 4: Effect of Prandtl Numberon Heat Transfer from a Cylinder:

Plot of Nu.Re-1/2 versus Pr

The slope of the line obtained is 0.33 givinga functional relationship

3/12/1 Pr

ReCNu

The index 1/3, thus obtained for Prandtlnumber is in excellent agreement withtheoretical analysis and also with the valuesreported by Richardson (1968) and Zukauskas

18


(1987) and others. The equation for the Nusseltnumber may be derived as:

3/12/1 PrRe, nNuav

For 2 , the above equation reduces to

3/12/1 PrRencNuav

where

2, nnc

In order to exclude the effect of flowbehaviour index (n) on NuavPr1/3 a factor c(n)has been introduced as defined above. Thevalues of c(n) for different fluids under studyare given below:

Fluid c(n)

0.5% PVA Solution 0.95

Water 0.97

1% CMC Solution 0.99

1.5% CMC Solution 1.01

Table 2: Values of Factor c(n)for Different Experimental Fluids

Figure 5 through 8 show the variation of hav

with Uav for various tube arrangements- F, FR,FRL, FRLB for all the four test fluids studied inthis work. Here it is found that the heat transfercoefficient is decreasing with increasingviscosity of fluids.

Figure 9 through 12 show plots of hav versusUav for all tube arrangements for the flow of0.5% aqueous PVA solution (through 2.54 cmdiameter tube), water and 1% and 1.5%aqueous CMC solutions respectively (passingthrough 3.18 cm diameter tube). It is seen fromthese plots except for water, that the tubearrangement has no significant effect on thevalue of hav for a given flow velocity. This couldbe attributed to suppression of vortices in thewake by non-Newtonian fluids.

Figure 5: Variation of Average HeatTransfer Coefficient (hav) with AverageVelocity (Uav) for Heat Transfer from aVertical Tube Placed in an Assembly ofTubes: Effect of Flow Behaviour Index


19




Figure 9: Variation of Average HeatTransfer Coefficient (hav) with AverageVelocity (Uav) for Heat Transfer from aVertical Tube Placed in an Assembly of

Tubes: Effect of Tube Arrangement (TubeDiameter 2.54 cm Fluid 0.5% PVA)


Tubes: Effect of Tube Arrangement (TubeDiameter 3.18 cm Fluid Water)

20



Tubes: Effect of Tube Arrangement (TubeDiameter 2.54 cm Fluid 1.0% CMC)


Tubes: Effect of Tube Arrangement (TubeDiameter 3.18 cm Fluid 1.5% CMC)

Figure 13: Heat Transfer During FlowThrough Assembly of Tubes: Variation

of (NuavPr–1/3)/c(n) with

CONCLUSIONThe flow pattern around a tube in a tube bundlediffers considerably from that around anisolated tube immersed in a fluid flowingacross it as the former is influenced by thepresence of neighbouring tubes. Also changein the boundary layer, formed around a cylinderis affected by neighbouring tubes whichconsequently results into considerable changein the heat transfer characteristics. Further, theturbulence generated by leading tube, ingeneral, enhances the heat transfer and thischange is more significant at higher Reynoldsnumber. The effects on boundary layer andextent of turbulence, both are dependent ontube arrangement, spacing between adjacenttube and other geometrical parameters. Forthis purpose the experimental data for all thetube diameters, f luids and differentarrangements viz. F, FR, FRL and FRLB havebeen plotted in Figure 13.

A careful examination of this plot revealsthat experimental values show different trends

21


for different range of Reynolds number. Forlower Reynolds number (Re < 60) the slope issomewhat lower than for data in higherReynolds number range (Re > 60). For Re <60 the slope of average line is found to be 1/3while, for Re > 60 a slope of 2/3 is observed.Separate analysis of each of the above regionsresults into following equations:

3/1

3/1

Re35.1Pr

ncNu

for Re < 60

3/2

3/1

Re35.0Pr

ncNu

for Re > 60

ACKNOWLEDGMENTI take this opportunity to express my mostsincere feeling of gratitude to Hi-Tech Instituteof Engineering and Technology, Ghaziabad forproviding the facilities and financial help forthe fabrication of experimental setup andcomputational work.

REFERENCES1. Achenbach E (1989), “Heat Transfer from

a Staggered Tube Bundle in Cross Flowat High Reynolds Numbers”, Int. J. of Heatand Mass Transfer, Vol. 32, pp. 271-280.

2. Adams D and Bell K J (1968), “HeatTransfer and Pressure Drop for Flow ofCarboxy Methyl Cellulose Solution AcrossIdeal Tube Banks”, Chem. Engg. Prog.Symp. Ser., Vol. 64, No. 82, p. 133.

3. Aiba S (1990), “Heat Transfer Around aTube in In-Line Tube Banks Near a PlaneWall”, J. of Heat Transfer, Vol. 112,November, pp. 933-938.

4. Aiba S and Yamazaki Y (1976), “AnExperimental Investigation of Heat

Transfer Around a Tube in a Bank”, Trans.ASME, J. of Heat Transfer, August,pp. 503-508.

5. Aiba S, Ota T and Tsuchida M (1980),“Heat Transfer of Tubes Closely Spacedin an In-Line Bank”, Int. J. of Heat andMass Transfer, Vol. 21, pp. 311-319.

6. Brown A I and Marco S M (1958),Introduction to Heat Transfer, 3rd Edition,McGraw Hill, New York.

7. Hwang T H and Yao S C (1986), “CrossFlow Heat Transfer in Tube Bundles atLow Reynolds Numbers”, J. HeatTransfer, Vol. 108, pp. 697-700.

8. Lorisch W L (1929), Mitt.Forschumgsarb, p. 322.

9. Prakash O and Gupta S N (1987), “HeatTransfer to Newtonian and Inelastic Non-Newtonian Fluids Flowing Across TubeBanks”, Haet Transfer Engg., Vol. 8,No. 1, pp. 25-30.

10. Snyder N W (1953), “Heat Transfer in Airfrom a Single Tube in a Staggered TubeBank”, A. I. Ch. E. Symp. Series, Vol. 49,No. 5, pp. 11-20.

11. Zukauskas A (1972), “Heat Transfer fromTubes in Cross Flow”, Advances in HeatTransfer, Vol. 8, pp. 93-158.

12. Zukauskas A (1982), “Convective HeatTransfer in Heat Exchangers”, p. 472,Nauka, Moscow.

13. Zukaukas A (1987), “Heat Transfer fromTube in Cross Flow”, Advances in HeatTrasfer, Vol.18, pp. 87-159.

14. Zukauskas A and Ziugzda J (1985), “HeatTransfer of a Cylinder in Cross Flow”,

22


p. 208, Hemisphere Publication Corpn.,Washington DC.

15. Zukauskas A, Makarevicius V andSlanciauskas A (1968), “Heat Transfer in

APPENDIX

Nomenclature

Ac average area of cross-section of the tube bank, m2

b relative longitudinal pitch, Sl/d

B width of the parallel place channel, m

c(n) ø(n, /2) a function

d diameter of the tube bank, m

do outside diameter of the tubes in the bank, m

D gap between two plates at any axial position in the converging-diverging parallel plate channel model, m

D1 minimum gap at x = 0, m

D2 maximum gap at x = xo/2, m

DH hydraulic diameter of the tube bank, m

g acceleration due to gravity, m2/sec

gc conversion factor, Kg-m/Kgf sec2

hav average convective heat transfer coefficient, w/m2 °C

k thermal conductivity, W/m2 °C

K power law consistency constant, Kg/m sec(2–n)

n flow behavior index

Q volumetric flow rate, m3/sec

SL longitudinal pitch or spacing of longitudinal rows, m

ST transverse pitch, m

U velocity, m/sec

U–

average velocity, m/sec

Us superficial velocity, m/sec

x axial position in x-direction, m

y axial position in y-direction, m

z height of the tube bank or exposed length of the tube in a tube bank, m

Banks of Tubes in Cross Flow of Fluids”,p. 210, Mintis, Lithuania.

23


Dimensionless Groups

Nu Nusselt number

Nuav Average Nusselt number around the tube

Pe Peclet number

Pr Prandtl number

Re Reynolds number

Re Reynolds number based on tube diameter

Greek Letters

thermal diffusivity

void fraction calculated from tube spacing and geometry

coefficient of viscosity, Kg/m sec

eff effective viscosity, Kg/m sec

angular position with respect to front stagnation point, degrees

Subscripts

av average

H hydraulic

L longitudinal

o outer

T transverse

x in x-direction

y in y-direction

APPENDIX (CONT.)

Nomenclature

the effect of neighbouring tubes on heat ansfer … · 2015-04-09 · int. j. mech. eng. & rob....

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