INVESTIGACiÓN REVISTA MEXICANA DE FíSICA 47 (4) 351-356 AGDST02001

Viscous dissipation oCa power law fluid in an oscillatory pipe flowJ.R. Herrera-Velarde

Instituto Tecnológico de ZacatepecAl'. Instituto Tecllológico 27, 67980 Zacatepec, Mor., Mexico

R. Zenit and B. Mena"Illsritllto de Investigación ell Materiales, Ulli\'ersidad Nacional Autónoma de MéxicoApartado postal 70.360, el!. Uni\'ersitaria. 045/0 Coyoacdll. México, D.F., Mexico

Recibido el 16 de febrero de 200 1; aceptado el 11 de mayo de 2001

Thc ftow ficld in an oscillatory pipe is studicd thcorctically fUf a gcncralizcd newlonian nuid modcl. Thc vrlocily and tempcrature fieldsare obtaincd for (he case in which lhe mean velodt)' causcd by lhe prcssure gradient is oC lhe snmc arder as lhe osciliation vclocity. Themorncntum and lhe encrgy conscrvation cqualions are sol ved and anal y tic expressions for lhc velocilY and lemperature fields are found. Thenmure of (he velocily amllempcrature profiles is explored Cora range of paramclers. Jn general, it can be concluded thmthe tcmpcrature risewithin lhe fluid increases wilh lhe specd 0'- oscillalion as the value of the power paramcler increases. An effective heatlransfer cocfficienl iscalculated and plolted as a function of (he normalized oscillalion speed. The cases of a newlonian, shear-thinning and shear-thickening fluidare analyzcd.

Keywords: Viscous dissipation; power law fluid; oscillatory flow

Se presenla un estudio teórico de un !lujo oscilatorio en una tubería para un !luido lipa ley de palencia. Las ecuaciones de momcntum yenergía se resuelven y se encuenlrun soluciones :malíticas para tos campos de velocidad y temperatura. Se obtienen resultados para el caso enque la velocidad media debido al gradiente de presión es de la misma magnilud que la velocidad de oscilación. Se exploran la naturaleza de loscampos de velocidad y temperatura como función de los parámetros dominantes. En general, concluimos que el incremento de lemperaturaen el fluido aumenla como función de la rapidez de oscilación y del parámetro de potencia. Se calcula un coeficiente de transferencia efectivoy se grafica como función de la rapidez de ascil.teión adimensional. Los líquidos newlOniano. pseudo plástico y dilatante son nnalizados.

Descriptores: Disipación viscosa; ley de potencia; tlujo oseilmorio

PAes: 47.50.+d; 47.60.+i

1. Inlroduclion

The majority of studies that involve the Oow of a viscous nuidassume an isothermal state; however, in praetice, many nowsare far from this situation. The combination of high viscosi-t¡es and large velocity gradients may result in a significant¡ncrease of the Ouid temperature resulting from viscous dis-sipation. This effect is used, for example, in extrusion pro-cesses where the temperature increase is uscd to acceleratethe melting of the material.

The heat transfer in ducts has been extensively studiedfor the case of newtonian fluids. Greatz [?J sol ved Ihe c1as-sic problem of foreed heat convection in a pipe subjccted todifferent boundary conditions, ncglecting axial conduction.More rccently,Yin [?] sol ved Greatz problem analytically tak-ing into account the axial conduction in the fluid as well as inthe pipe, concluding that the axial conduction indeed playsan important role in the entrance region. Much less workhas been devoted to study heat transfer prablems involvingnon-newtonian viscous fluids. lt is known, however, that thenon-newtonian properties of a fluid can significantly changethe hcat transfer characteristics ['1}. The coupling between theequations of motion and the energy equation can be achievedthrough the material properties of the fluid assumed, as a firstapproximation, as temperature-independentconstitutive rela-tions.

The flow of polymer solutions and melts in oscillat-ing pipes has been studied extensively by Mena and co-workers ['1, '!, '1,?}. Imposing longitudinal oscillations on aviscoelastic fluid, the velocity fields and the pressure dropchange were studied, but only considering the isothermalcase. Among muny other conclusions, these authors state thatthe oscillation of un exit die of an extrusion process altersthe mechanical properties of the extruded product and causes'In important decrease of the pressure drop aeross the die.More recently, Herrera- Velarde [?] studied theoretically theheat dissipation on oscillatory ftows considering a viscoelas-tic model.

In order to understand the effecl of oscillations in a poiy-mer extrusion process, the non iso-thermal case must be stu-died. This paper presents the study of the pipe flow in whichthe pipe oscillates in the main dircction of the ftow. Resultsare obtained for an inelastic power law fluid. The temperatureprofilcs are obtained using the velocity profiles and by as-suming temperature independent properties. The coefficientof heat transfer is obtained and presented for a range of ftowconditions and fluid properties.

2. Problcm dcfinilion

Consider a fluid that flows in a duct with circular cross-sec-tion of radius T resulting fram uniform pressure gradient 'VP

J.R. HERRERA-VELAR DE, R. ZENIT. AND B. MENA

TofiGURE 1. Schematic of (he flow cntcring the oscillaling die witha dc\'clopcd \'clocity prolik, a mean Icmpcraturc To and subjcClcdto nn oscillaling wal! with comtan! (Cmpcralurc To.

3524 VP

aa"4- •

To -~~

- ---.- --~~-

•

a____ ....1'_

The aboye set of equations can be sol ved when a modelfor the rheological behavior is given. To obtain analytical so-lutions, a generalized newtonian model is considered.

2.1. Generalized ncwtonian model

The generalized newtonian !nodel considers a modified vis-cosity-shear rate relationship to account for the case in whichthe viseosity of the fluid depends on lhe shear rate. A we\l,known generalized model is the "power law" modcl ['!l. inwhich the viscosity of the fluid dcpends on the magnitude ofthe strain rate,

in the axial direction of lhe pipe, z. Additionally, the ductoscillatcs in lhe dircclion parJllcl lo lhe flow imposing un ud-ditianal shear stress lo lhe fluid. A schcmatic diagr~\In of thcflow is shown in Fig. 1.

Considering the consc:rvation equations for ao incolll-prcssible and uniform Iiquid,

The relevant component of the stress tensor for the prob-lem of interest is Tr:. which can be expressed simply by

where n and In are constants characteristics of a parti-cular lluid. The magnitudc of the strain rate is defined as

e = JLi Lj ejJc1jf2. Itence, the ij component of the ex-tra stress tensor is

\7. 17 = O,

(01')P75t =-VP+V'T,

pCp(l" vT) + v' (kvT) = T : V ••,

(I)

(2)

(3 )

l]=lnen-I,

- "T = mcij.

(0"_) "T = 111 ---r: 8r

(6)

(7)

(8)

4) OT/Or = O at ,. = Ofor a\l z.

3) T = To at r = a for aH z.

\\/here G= -8P fDz is the constant pressure gradient and Tr:

is the component of the stress tensor in the axial direction ofthe pipe. The boundary conditions for this equation are

(9)

(11 )

( 10)

¡.T,., =:i[ -G+piw2.4exp(iwt)].

. 2, (') _ G 1 O ( )lW [Jn exp lWt - + ~or rTr: .

For the rheological model discussed above, the velocity andtemperature field are obtained. \Ve consider the case in whichthe mean velocity causcd by the pressure gradient is of thesame order as the oscillating velocity. For simplicity, the ma-terial parameters (m, [J, k, Cp) are all considered to be of or-der lo

1'0 solve the momenlum conservation [Eq. (4)] \••..e as-sume that the axial velocity component Hz can be decom-posed in Uo = 110(1). the motion due to the periodic oscilla-tion of the walls, and IIp resulting fram the pressure gradient.

3. Rcsults

If the pressure gradicnt G is constant. lhe expressionaboye can be integratcd with respect to r to obtain the generalsolution for the component Tr: of the stress tensor. Wilh thecondition that the stress must be finite at r = O,we obtain

where w and A are Ihe frequency and amplitude of the os-ei\lation. If we lake "o :R{w.4exp(iwt)}, Eg. (4) can beexpressed as

Note that if 1t = 1 the newtonian case is recovered wherethe value of 1ft corresponds to the shear viscosity 1/.

(4 )

(5)(OT) 1 O ( OT) (Ou_)pCp", Oz = k:;: O,. "0;: + T" 0,.- .

\Ve consider that the temperalUrc profiJe of the fluid is con-slanl with To at the enlrancc of the oscillating wall. The os-cillating \val! temperature is also held constant at 1'0' Hence.lhe boundary conditions are given by.

1) ", = aRe [exp(iwl)] at ,. = (1.

2) 8",/0,' = O al,. = O.

The energy eg"alion [Eg. (3)1 for a fu\ly developed te111-pcrature field for the same llow, ncglecting the axial conduc-lion, reduces to

whcrc t7 is the vclocity vector, J> is lhe scalar pressure, f isthe dcvialoric stress tensor. p is Ihe density, Cp is lhe spccific!leal, k is the therma! conductivity and T is the tcmperature.

For the gt:ometry of the flow showed in Fig. 1. the velo-city vector reduces to (llr, 110, llJ = (O,O~u:) which salis-f¡es the cguation of continuily identica\ly [Eg. (1 )j. The 1110-Illentum equation IEq. (2)J in the z-direction is reduced to

Ou, 1 Op-O = G + --O ("T,,),t r r

R,','. M •..•. Fú. ~7 (~) (2ool) 351-356

VISCOUS DISSIPATION OF A POWER LAW FLUID IN AN OSCILLATORY PIPE FLOW 353

Using this express ion in Eq. 8, we obtain "

( d") (r) ~ , l-' = - [-e+iwzpAexp(iwt)]",dr 2m

(J 2)

"

o: ~ uml

dient G is ealculated in terms of thc oscillation speed

_1~ ~1 U U ~4 ~5 ~6 ~1 ~ ~,mal poaiIJon. ,It

shear lhinning, n=1/2shear thickening, 0=2Newtonian, n=1

- - - - ----- -- :_-----.:-- ..••..-------- .......•. -

-,

-<),

lo,J o

FIGURE 2. Dimensionless velocity as a funetion of dimension-Icss position or powcr law l1uids with: a) ( -- ) shcar lhin-ning liquid, n = 1/2, b) ( - - -- ) shcar lhickening liquid.71 = 2, and e) ( - . - . _.) ncwLOnian Iiquid. n = 1. Forwp/rlo = 1, o/um = 1: and m = 1.

-e + iw20 exp(iwt) = - [lel + iwzo cxp(iwt)],

l

(_n_) (Iel + iwopexP(iwt))"1 + 1l 2m

x ( a ~ +1 - r ~+1) + ° exp( iwt), (J 3)

A non-dimensional velocity is defined as

which can be integrated with rcspcct to r to obtain lhe vc-locity profile using lhe wal! vclocity as a boundary condition,u,(r = a) =" cxp(iwt),

Wc must note that lhe paralllctcr G is indecd a negativenumber (-DP IDz) Ihal determines the mean direction of Iheflow (fonn lefl 10 right in this case), \Ve can simplify themathematical analysis if we consider that

10 avoid lhe case in which a negative number is raised lo afractional power. Adopting this sign convention does nol af-feel lhe nature of lhe solution and simplifics the analysis sig-nificantly. Hcncc.

henee

lel = 2111(°(1 :"))"fUI" +1

Figure 2 shows the non-dimensional yclocity prefiles forthrce cases of fluids: a shear thinning fluid (n < 1), a shcarthickening fluid (n > 1) and a newtonian fluid (11 = 1),

1'he energy equation [Eq, (5») can be written for a powerlaw fluid leading to

( J 5)1':.1' 1 d ( dI') (du_)"+lf!C"u,~=k-- "-/- +m -l'uZ T dr ( r (r

° exp(iwt)+ l (14)e:,,) G~D"a~+lwhere um is the maximum velocilY foc [he non-oscillatingPoiseuille pipe flow given by

u =(_" )(Iel)~a~+lm 1 + n 2m

1'0 make Ihe magnitude of Ihe oscillation veJocity be ofrhe same arder as that of the Poiseuille flow, the pressure gra-

I

where 1':.TI I':.z is considered lo be a constanl parameter, 1'heaboye equation can be integrated for the given set af bouo-dary conditions since Ihe velocity profile [Eq, (13)] is known,

I' = 1'0 + CI {J ~ (J I"u,(,-') dr') dI' - [J ~(J r'u,(r') dr') (17'] ,=J-Cz {J ~(Jr'(D¡~;~')) MI <Ir') dI' - [J ~(J r'(Duo;:") )"+1 dr') drLJ (16)

where

C _ pC" (1':.1')1 - k ~z amI

111Cz =-.

kAn anal y tic solutian can be faund for the aboye expr~s~ion resulting in." ','.-'

", ' ., (C' ~+I C C 2 ('" )] C Z (C ( 1)) ,,+1 ( )T-T,::::C' 3a." + 4 (r2_a1)_ 3rt r~+~~a:-+3 _ 2

1l311+ " "T~+ ..3_a~+3

o' " ,,,,, 4 " (1+3,,)2 .', '(1 +3,,)2 " 'I '., - : i ,1 : '\' l \ Ir 1 j , ""!.- -. .: "

(J 7)

,Rél', Me."iF,s. 47 (4J{2001) 3SI~356

354 IR. IIERRERA-VELARDE. R. ZENIT. AND B. MENA

whcrc

c" = (_"_) (IGI + ¡"'OpeXP(iWt)) ~1 + 11 2m

shear lhinnlng, n=1/2shear Ihickening, n=2Newtonian. n=l

and

,,

<07 011 09

,

,"

" •• ••

\

shear lhinning, n=1/2shear lhickening. n=2Newtonian. n=1

shear lhlnning. n=1/2shear lhiCkening. n=2Newtonian, n= 1

(b)

(a)

02 0.3 O. OS 011~ potIIu'l, ,JI

02 03

02 0.3 O. 0.$ 011 07 011 O,'--'p<II'IoOn."1

.,

..,

.,

'.

'.

2 ---------

'.

o"

~lS¡'-- - -- -- -- - - -- -- -- __ ~~~

!al'

"r--~---~-----~-----~---,

(e)

FIGURE 3. Temperaturc increase as a function of dimcnsionlcssradial position for power law f1uids with: a) ( --- ) shcarthinning liquid, n = 1/2, b) ( - - -- ) shcar Ihickcning li-quid. n = 2, and e) (_. - . _. ) ncwlonian Iiquid. n = 1. Forwp/rl0 = 1, a/um = 1, ÓT / Óz = 1, and m :;::1.

( 19)

Y-Y. =C [_ C3n~+3+C4a2 + C3,,2 (a~+3)]o 1 6 3" + 1 1+4"

+ C2,,2 [C3(n + 1)] ..+1 (a~+3 ). (18)1+3"" 1+4n

figure 5 shows the temperature increase as a function ofthe ration o/um for the three fluid cases. The tempcraturcincreases with the oscillation velocity. However, for high os-cillation specds a change of trend is observed. Trends similarlo lhose obscrvcd here have also been reported by ["!) for Ihecase of non-oscilatory pipe flows.

c, = (l exp(i",t).

Figure 3 shov.'s lhe tcmperature profiles for three differenttimes wilhin lhe oscillation cycle. Clearly. lhe tcmperatureprofile changcs within lhe oscillation cycle resulting from Ihechanges in lhe velocity prefiles. The temperature fields rcachi.l maximulll when t = 1T, (half a cycle). The shear thinningfluid shows a smaller tempcrature ¡ncrease than that observcdin lhe newtonian case. On lhe other hand. the tcmperature ofthe shear lhickening fluid is higher lhal thal calculaled for thenewtonian liquido Since heating of lhe fluid results directlyfram viscous dissipalion, it is lo be expected that as lhe shcar¡ncreases, I¡quid \I..'ith constant or ¡ncrcasing viscosity withshear will attain higher temperatures. figure 4 shows the cf-fcet of varying the power parameter n on the temperature pro-file for various cases of shear thinning and shear thickeningnuids.

Now, to analyze the effect of the oscillation in the tcm.pcrature increase of the fluid using the power-Iaw model, thecondition that 11111 ~ (} is relaxed. The tcmperature profile isIhcn calculated for different values of Q keeping the value of'/lm constan!. The cffective tcmperature increase is quantificdusing the mean temperature T, defined as

- Jo"T(r, t) dI'T-~---- Jo" dI'

Hence, fmm Eq. (17), the mean temperature results in

The dimensionless heat transfer coefficient is defined as

4. Hcat transrcr cocrticicnt

r haJ\ ti = -

k Y - Towhere h is the convective heat transfer coefficient and k is lhethermal conductivity.

Thc mean lempcrature results has already been calculatedand thetcmpcraturc gradient al the wall can be obtaincd by

Rev. Mex. Fis. 47 (4) (2001) 351-356

VISCOUS DISSIPATION OF A POWER LAW FLUID IN AN OSCILLATORY PIPE FLOW 355

shear thino¡og, 0=112shear Ihinoiog, n:213shear Ihinnjng, 0:3/4Newtonian, 0=1

"shear lhinoíng, 0=1/2shear thickening, n=2Newtonian, 11=1

o.•

"00 o., o, 0.3 O. 05 06

'óa! po8<tJOn, ti_o,

..• ----------------------

(a)o., l.1 1,2 1,3 1.4 1.5

O"TlM'$ionles8 O8COIlaliOn $peed. aN m " "

FIGURE 4. Tcmpcraturc ¡ncrease as a func{ion 01'dimensionlcss ra-dial posifion fUf shcar thinning Iiquid and lhickening liquids. forvarious values 01' the power paramClcr (Uf t = 71"and t::.T / ó..z = 1,andm=l.

inj2-------_

L

"

"

,.

00 " " o., o_~ 0.5 06<a<l<aIpoMion. ,le

(b)

shear Ihickening, 0=4shear thickening, 0:3shear Ihickening, 0=2Newtonian,0=1

FIGURE 6. Heal transfcr coelTicicnt as a function of the oscillalionvclocity for laws liquido

direct diffcrentialion of the lemperalurc profile [Eg. (17)] re-sulting in

(DT) _ C (CI(l~+2 + C4(l C3n' 2+l)- - j - ---(l "Dr r=::a 2 1+ 3n

_ C,n [C3(n+ 1)]n+la2+~ (20)1+3n n

Figure 6 shows the heat transfer coefficient as a func-tion of the oscillating velocily for three cases studied. Forthe range showed, an ¡ncrease of the heat transfer coefficientcan be observed.

Aditionally, two limiting cases can be analyzed. For lheCase where the axial temperature gradient is zero (Cl = O),the Nusselt number, that corresponds to the case of pure vis-cous dissipation, reduces to

1+ 4/1Pi uvisc =

FIGURE 5. Tempcrature ¡ncrcase T - To, as a function of (he 110[-malizcd oscillation vclocity o/um• for t = 11".

"

""'"!1"i,"'

1, "

shear thinning, 0 .•112shear Ihickening, n_2Newtonian, 0_'

1.1 1.2 1.3 1~ 15",""""ion_ <llCil1al.,.,opeed, oh", " " "

Hence, it can be inferred that the heat transfer is only a func-tion of the properties of the fluid and nol of the oscillatingcharacteristics of the flow. However, on the other hand, whenonly the convcctive contribution (C2 = O) is considered theheat transfer coefficient reduces to

which docs not depend directly on the value of the axial tem-perature gradient.ó.T /.ó.z but now depends on the oscillatingcharacteristics of the now (C3 anu C4). Hcnce, to oblain anincrease of lhe fluid temperature with ¡ncreasing oscillationspeed, the erreet of the axial temperature gradient must beincluded.

Re\'. Mex. Fi<. 47 (4) (2001) 351-356

356 J.R. HERRERA.VELARDE. R. ZENIT, AND B. MENA

5. Summary and conclusions

\Vc have studied the temperature ¡ncrease resulting fmm vis-caus dissipation in an oscillatory pipe flow. A variable vis-cosity inelastic "pO\\'cr law" moJel was used lo charactcrize(he rheology of the fluid. 130th velocity and temperature fieldswerc obtained for three fluid cases: ncwtonian, shear thinningand shcar thickening fluids. Propcrties of unitary valuc \vereconsidered to obtain numcrical results.

In general. il was found thal the tempcrature ¡ncreaseswilh the power para meter and with the speed of oscilla-tion. An effectivc hcat transfer coefficient was also ealculatedwhich was observed to ¡ncrease with the oscillation specd. 1'0our kno\\'ledge, experimental mcasurernents ofthe heal trans-

To whom corrcspondcncc should he addrcssed.1. L. Greal7.,All11. Ph)'s. Chel1l. 18 (1883) 79.2. X. Yin and H.H. Ball, J. lIeal 1"{"(1II5. 118 (1996) 482.3. S.G. Etemad and H. Mujurndar, flli. J. Heat Man TrallS. 38

(1995) 2225.4. O. ~1ancro and B. Mena, Rh('ol. Aela 17 (1977) 573.5. o. Mancro, R. Valcnwcla, and B. Mena, Rheol. Aera 17 (1978)

693.G. B. Mena, O. ~lancro, nnd D.M. Binding, J. NOII.Nl'\I'I. Fluid

Meeh 5 (t979) 427.

fer in this flow configuration do not exist. \Ve have recognizedthat there is a very small number of experimental studicsthat investigate the heal transfer processes in non-newtonianfluids.

Acknowlcdgmcnts

The support of PAPIIT-DGAPA of UNAM through its grantsIN503494 and IN 103900, "nd CONACyT through its scho-larship program and (he grant 0073-A 7026 are greatly "c-knowledged. JRH- V \\.'ishes to acknowledge the Instituto Tec-nológico de Zacatepec for its support during the completionof his graduate studics.

7. J.Casulli, J.R. Clcrmol1t, A. Von Ziegler, and B. ~1cna, J.Pol)'lII. I:"lIg. Sci. 30 (1990) 1551.

8. J.R. Herrera- Vclardc and B. Mena. Rel: .\fex. Frs. -16 (2000)(551.

9. W. Ostwald. Kolloid-Z. 3(, (1925) 99.

10. R.B. Bird, R.e. Amstrong, and O. Ibssagcr, !J)'II11lllics 01 PoI)'.lIIeric Liq/lids, Vol. 1: fluid mcchanics, (1ohn Wilcy & Sons,Ncw York, (977).

R,,'. Me,. nI. 47 (4) (200t) 351-356