reynolds generalizado

7/24/2019 Reynolds Generalizado.

1/14

GENERALIZED REYNOLDS NUMBER

FOR NON-NEWTONIAN FLUIDS

K. Madlener, B. Frey, and H. K. Ciezki

An extended version of the generalized Reynolds number was derived tocharacterize the duct ow of non-Newtonian gelled uids of the Herschel

Bulkley-Extended (HBE) type. This number allows also estimating thetransition from laminar to turbulent ow conditions. An experimen-tal investigation was conducted with a capillary rheometer for severalnon-Newtonian gelled uids to evaluate the introduced HBE-generalizedReynolds number Regen HBE. A good correlation between the experimen-tal results and the theory could be found for laminar ow conditions. Forone of the examined gelled fuels, the necessary high Reynolds numberscould be realized so that the transition from the laminar to the turbu-lent ow regime could be measured. Because of its general description,the HBE-generalized Reynolds number can also be applied to Newtonianliquids as well as to non-Newtonian uids of the HerschelBulkley (HB),Ostwaldde-Waele (power-law, PL), and Bingham type.

NOMENCLATURE

D duct diameter, mfDarcy Darcy friction factorK prefactor of power-law, Pasn

L duct length, mm local exponential factorn global exponential factorr radial coordinate, mRe Reynolds numberReBingham Reynolds number for Bingham uidsReHBE Reynolds number for HBE-uidsReHB Reynolds number for HB-uidsReNewton Reynolds number for Newtonian uidsRePL Reynolds number for PL-uidsu(r) velocity, m/su average ow velocity, m/s

Progress in Propulsion Physics1(2009) 237-250

DOI: 10.1051/eucass/200901237

Owned by the authors, published by EDP Sciences, 2009

This is an Open Access article distributed under the terms of the Creative Commons Attribution-Noncommercial

License 3.0, which permits unrestricted use, distribution, and reproduction in any noncommercial medium, pro-

vided the original work is properly cited.

Article available athttp://www.eucass-proceedings.euorhttp://dx.doi.org/10.1051/eucass/200901237
http://www.eucass-proceedings.eu/http://dx.doi.org/10.1051/eucass/200901237http://dx.doi.org/10.1051/eucass/200901237http://www.eucass-proceedings.eu/


2/14

PROGRESS IN PROPULSION PHYSICS

Greek letters

shear rate, s1

p pressure loss, bar dynamic shear viscosity, Pas density, kg/m3

shear stress, Paw wall shear stress, Pa0 yield stress, Pa

Abreviations and subscripts

app apparentHB HerschelBulkleyHBE HerschelBulkley-extendedPL power-laww wall

1 INTRODUCTION

Gelled fuels and propellants are shear thinning non-Newtonian uids with asignicantly dierent viscosity behavior compared to Newtonian liquids [1]. Due

to their safety and performance benets, they are interesting candidates forrocket propulsion systems, see, e.g., Ciezki and Natan [2]. During storage andtransport when very low shear forces occur, their viscosity is very high so thatthey are often described as semisolids. During the feeding process from thetank through the manifolds to the injector unit, high shear forces are appliedto the uids and their viscosity is strongly decreased. Thus, gelled propellantsoer the possibility to build engines with thrust variation up to thrust cut-o and reignition like in engines with liquid propellants. At the same time,they have simple handling and storage characteristics like engines with solidpropellants. Concerning the understanding of the duct ow characteristic of

such non-Newtonian gelled uids, there are still gaps to close. The present paperoers a further small step for a better understanding of the ow characteristicsby dening an HBE-generalized Reynolds number.

To characterize or to compare the ow characteristics of uids owing throughducts, dimensionless numbers are often used. In 1883, Osborne Reynolds rstintroduced what is today known as the Reynolds number for fully developed ductow of Newtonian liquids. The denition of the Newtonian Reynolds numberReNewton is

ReNewton = Du

Newton(1)

238


3/14

LIQUID AND GELLED ROCKET PROPULSION

Figure 1 Viscosity measurements for a kerosenethixatrol-gel and theoretical ap-proach with PL (a) and HBE (b): viscosity and shear stress

whereis the uid density,D is the duct diameter, uis the average ow velocity,andNewton is the constant Newtonian viscosity.

The Reynolds number can be interpreted as the ratio of inertial forces to vis-

cous forces. It is commonly used to identify dierent ow regimes such as laminaror turbulent ow. Furthermore, it is used as a criterion for dynamic similitudethat means, if two dierent ow congurations (dierent duct diameters, dier-ent ow rates, or dierent uid properties) have the same dimensionless num-bers, they are dynamically similar. As already mentioned, the Reynolds numberReNewton of Eq. (1) is only valid for uids with a constant viscosity. The gelleduids investigated in the present paper, however, are non-Newtonian uids witha more complex viscosity characteristic compared to Newtonian liquids. The di-agrams in Fig. 1 show an example of the shear-rate dependent shear stress ( ),

239


4/14


indicated with triangles, and the shear-rate dependent viscosity ( ), indicatedwith circles, for one of the investigated non-Newtonian uids. The gel based on

liquid kerosene was mixed in a Getzmann dissolver with 7.5 %(wt.) ThixatrolST and 7.5 %(wt.) 5-methyl-2-hexanon (miak). Rheological measurements wereconducted with a Haake RS1 rotational rheometer (cone-plate) and a RosandRH2000 capillary rheometer.

The duct ow of non-Newtonian uids, particulary, for uids with a viscositycharacteristic following the Ostwaldde-Waele or PL equation PL = K

n1,was investigated in the past, for example, by Dodge and Metzner [3], Ryan andJohnson [4], Mishra and Triphathi [5], Malin [6], and Bohme [7]. For the iden-tication of dierent ow regimes or the determination of dynamic similitude,Metzner and Reed [8] introduced a generalized Reynolds number Regen PL validfor pure PL uids. This number was derived from its relation to the Darcy

friction factor fDarcy and is given by:

Regen PL = Dnu2n

K((3n+ 1)/(4n))n

8n1. (2)

The uids investigated in the present study, however, have a more complexviscosity characteristic that cannot be described by the PL all over the relevantshear rate range 102 106 s1. Figure 1ashows the theoretical approachof the PL theory to the experimentally determined viscosity of the kerosene-gel.It can be seen that especially in the high shear-rate range, there is no good

agreement between the theory and the experiments. The viscosity parametersfor the PL-theory are K = 37.78 Pas and n = 0.12 (tted in the shear-raterange 101 104 s1).

For better description of that viscosity characteristic, Madlener and Ciezki [9,10] presented the HBE-equation as an extended version of the HB law. Anadditional viscosity term as the constant viscosity in the very high shear-rate range was added to the term considering the existence of a yield stress 0and to the PL term Kn1. The denition of the HBE-theory is

HBE= 0

+Kn1 + (3)

with = in general HBE=0+Kn + . (4)

In Fig. 1b, the theoretical approach to the experimental viscosity data isshown for the HBE-law. Compared to the approach with the PL-theory inFig 1a, the HBE-theory describes the experimental results over the entire relevantshear-rate range. The determined HBE-parameters for the kerosene-gel were0 = 33 Pa, n = 0.19,K= 11.76 Pas0

.19, and= 0.0036 Pas.Since the PL theory is not capable of describing the viscosity characteristic of

the uids examined here over the entire relevant shear-rate range, the PL-based

240


5/14


Reynolds number Regen PL from Eq. (2) cannot be used to characterize the ductow of such uids either. For the characterization of ow and spray regimes

of uids following the HBE-viscosity type from Eq. (3), an HBE-generalizedReynolds number Regen HBE was derived and is presented below.

2 HERSCHELBULKLEY-EXTENDED

GENERALIZED REYNOLDS NUMBER

Metzner and Reed [8] derived their generalized Reynolds number Regen PL fromits relation to the Darcy friction factor. For laminar and fully developed ductow, a relation exists between the Reynolds number and the Darcy friction

factor which is generally given for uids independent of their viscosity charac-teristic [11]. This relation reads

Re = 64

fDarcy(5)

whereas the denition of the Darcy friction factor is

fDarcy=(p/L) D

u2/2 (6)

where pis the pressure drop over the duct lengthL, and u is the average ow

velocity. The relation between the pressure loss and the wall shear stressw iscalculated by the equilibrium of forces over the duct with w = (D/(4L))p.The Darcy friction factor from Eq. (6) can then be written as

fDarcy=8w

u2 . (7)

The wall shear stress w in Eq. (7) can be calculated by the viscosity of theexamined uid. For a Newtonian liquid, there is a constant relation between thewall shear stress w and the wall shear rate w with w = w, where is theconstant Newtonian viscosity. The wall shear rate could then be calculated from

the Newtonian velocity prole in duct ow with w = 8u/D. With that, theDarcy friction factor would yield fDarcy= (64)/(uD) which gives Eq. (5). Fora non-Newtonian uid of the HBE-type, however, the wall shear stress w hasto be calculated by the appropriate viscosity law from Eq. (4).

Since the shear rate is dened as the negative gradient of the velocity prole = du/dr, Eq. (4) yields the following expression if the velocity gradient isconsidered at the duct wall (denoted with index w):

w =0+K

du

dr

nw

+

du

dr

w

. (8)

241


6/14


Rabinowitsch [12] and Mooney [13] developed an expression for the wall shearrate w = (du/dr)w independent of uid properties and thus valid also for non-

Newtonian uids.

du

dr

w

=3

4

8u

D

+

1

4

8u

D

d ln(8u/D)

d ln(Dp/(4L)). (9)

The expression 8u/D corresponds to the wall shear rate in case of Newto-nian uid ow and is named apparent wall shear rate appw. The expressionDp/(4L) corresponds to the wall shear stress w. So, the logarithmic expres-sion in Eq. (9) can be replaced with the reciprocal value of the local gradient mwhich displays the gradient of the shear stresswat a certain apparent wall shearrate app w (Note: For a PL uid, the local gradient mwould be identical withthe global PL exponentn; for HBE-uids, this is not the case.):

d ln(8u/D)

d ln(Dp/(4L))=

d ln(app w)

d ln(w) =

1

m;

du

dr

w

=3m+ 1

4m

8u

D . (10)

Using Eqs. (8) and (10), the Darcy friction factor of Eq. (7) can be rewrittenas follows:

fDarcy= 80+K3m+ 1

4m

n

8u

Dn

+3m+ 1

4m

8u

Du2

= 64

08

D

u

n+K

3m+ 1

4m

n8n1

+3m+ 1

4m

D

u

n1u2nDn

. (11)

The generalized Reynolds number for laminar and fully developed duct owvalid for non-Newtonian uids with a viscosity characteristic following the HBE-equation can then be determined by inserting Eq. (11) in Eq. (5):

Regen HBE=

u2nDn0

8

D

u

n+K

3m+ 1

4m

n8n1

+3m+ 1

4m

D

u

n1 (12)

with

m= nK(8u/D)

n+ (8u/D)

0+K(8u/D)n

+ (8u/D)

242


7/14


wherem is the local gradient of the shear stress vs. the shear rate in a loglogdiagram. It was determined by the dierentiation of the logarithmic expression

for the HBE-equation according to:

m= d ln (w)

d ln (app w) =

d ln

0+Kn

appw+ app w

d ln (app w) ;

m=d ln

0+Ke

n ln(appw) +eln(appw)

d ln (app w)

;

m=nKnappw+ app w

0+Kn

appw+ appw=

nK(8u/D)n

+ (8u/D)

0+K(8u/D)n + (8u/D)

where0,K, n, and are the HBE uid parameters of the viscosity law fromEq. (3).It is claimed that the introduced HBE-generalized Reynolds number

Regen HBE from Eq. (12) is valid not only for uids with a viscosity charac-teristic of the HBE-type but also for all viscosity laws included in that equation.Those are the HB, the Ostwaldde-Waele (PL), the Bingham, and the Newtonianlaws. The reduced viscosity laws and their corresponding generalized Reynoldsnumbers are listed below:

HBE:

= 0

+Kn1

+ Regen HBE=. . . (see Eq. (12)) ;

HB:

= 0 =0

+Kn1

Regen HB= u2nDn

(0/8) (D/u)n

+K((3m+ 1)/(4m))n

8n1

withm= nK(8u/D)n

0+K(8u/D)n ;

PL:

0= 0, = 0 = Kn1

Regen PL= u2nDn

K((3m+ 1)/(4m))n

8n1

withm= n (see Eq. (2)) ;

243


8/14


Bingham:

K= 0, n= 1 = 0

+

Regen Bingham= uD

(0/8) (D/u) +(3m+ 1)/(4m)

withm= (8u/D)

0+ (8u/D) ;

Newton:

0 = 0, K= 0, n= 1 = ReNewton=uD

with m= 1 (see Eq. (1)) .

It is worth mentioning that for a PL uid, the HBE-generalized Reynoldsnumber corresponds to the generalized Reynolds number by Metzner and Reedfrom Eq. (2). In case of a Newtonian liquid, the HBE-generalized Reynolds num-ber reduces itself to the Newtonian Reynolds number from Eq. (1). Hence, theHBE-generalized Reynolds number can be applied to non-Newtonian uids withthe mentioned viscosity characteristics above, as well as to Newtonian liquids.

3 VALIDATION AND EXPERIMENTAL RESULTS

Experimental data are used for the evaluation of the introduced HBE-generalizedReynolds number Regen HBE. From the data, the Darcy friction factor fDarcycan be determined and plotted against the calculated Reynolds number. Inthe laminar ow region, the two parameters should follow the relation fDarcy= 64/Re according to Eq. (5). The determination of the Darcy friction factorand the Reynolds number requires the information about the uid properties, theduct geometry, and the measured pressure loss per volumetric ow rate. The testuids (TF1TF5) examined in this paper are two Newtonian liquids and three

non-Newtonian gels with compositions according to Table 1. The density of thekerosene-based fuels is about = 800 kg/m3, the density of the paran-basedfuels is about = 818 kg/m3.

The viscosity behavior of the test fuels can be described by the HBE-equation (3) over the entire relevant shear rate range 102 106 s1

relevant to injection processes. The corresponding tted HBE-parameters areshown in Table 2.

The experiments for the evaluation were conducted with a Rosand RH2000capillary rheometer. In the capillary rheometer, the examined uid was driven

244


9/14


Table 1 Composition of investigated test fuels in %(wt.)

Test uid Basic fuel Gellant AdditiveTF1 (Newtonian) 100% paran TF2 (Newtonian) 100% kerosene TF3 (non-Newtonian) 85% paran 7.5% thixatrol ST 7.5% miakTF4 (non-Newtonian) 96% paran 4.0% aerosil 200 TF5 (non-Newtonian) 85% kerosene 7.5% thixatrol ST 7.5% miak

Table 2 HerschelBulkley-extended parameters of the testfuels TF1TF5

Test uid 0, Pa n K, Pasn

, PasTF1 1 0.026TF2 1 0.0012TF3 45 0.38 5.07 0.026TF4 83 0.57 2.54 0.026TF5 33 0.19 11.76 0.0036

by a piston from the reservoir through a capillary. Pressure losses were messuredat several volumetric ow rates with capillaries of diameters D = 0.2 and 0.3 mm

and lengths L= 16 and 24 mm. The necessary corrections were applied to theraw data [12, 14].To demonstrate the advantage of the introduced generalized Reynolds num-

ber Regen HBE in comparison to the Newtonian Reynolds number ReNewton andthe generalized number for PL uids RegenPL, the experimental data for thekerosenethixatrol-gel (TF5) are shown as an example in three diagrams of Fig. 2.

In the diagrams, the Darcy friction factor is plotted vs. the Reynolds numberof the Newtonian, PL, and HBE-denitions. The viscosity value for calculatingthe Newtonian Reynolds numbers (Eq. (1)) in Fig. 2awas chosen with Newton= 0.0036 Pas as the viscosity value of the gel in the very high shear raterange. The PL parameters for calculating the PL Reynolds number (Eq. (2)) in

Fig. 2b were determined withK= 37.78 Pas andn = 0.12 according to the PLt in Fig. 1. The HBE-parameters for calculating the HBE-Reynolds number(Eq. (12)) in Fig. 2cwere taken from Table 2.

In all three diagrams, the solid line corresponds to the ratio fDarcy= 64/Refor laminar ow, which is generally given for uids independent of their viscositycharacteristic. It can be seen that the experimental data of the friction factorvalues comply with this relation only when plotted against the generalized HBE-Reynolds number Regen HBE. The calculation of the Reynolds number with theNewtonian or the PL denitions yields signicant errors. For high Reynolds

245


10/14


Figure 2 Friction factor vs. dierent denitions of the Reynolds number for thekerosenethixatrol-gel (TF5)

246


11/14


Figure 3 Friction factor vs. HBE-generalized low (a) and high (b) Reynolds numberfor liquid (TF2) and gelled (TF5 trixatrol) propellants based on kerosene

numbers, an increase in the friction factor values in Fig. 2c can be seen whichwill be discussed later.

Figures 3 and 4 show the friction factor values of the kerosene- and theparan-based fuels plotted against the HBE-generalized Reynolds number. Theresults are presented in separate diagrams for low and high Reynoldsnumbers. The values of the Darcy friction factor were calculated from Eq. (6)taking into account the experimentally determined pressure loss for a givenow rate and capillary geometry. The corresponding HBE-generalized Reynoldsnumbers for the given ow rate were calculated with Eq. (12) taking into ac-count uid properties and viscosity parameters. For the liquid kerosene andthe kerosene-gel in Fig. 3, the relation fDarcy = 64/Re for laminar ow (in-

247


12/14


Figure 4 Friction factor vs. HBE-generalized low (a) and high (b)Reynolds numberfor liquid (TF1) and gelled (TF3 and TF4) propellants based on paran

dicated with the solid line) is successfully achieved up to Reynolds numbers ofRegen HBE 1000.

At higher Reynolds numbers, slightly lower values compared to the relationfDarcy= 64/Re occur, before in the range between 2000


13/14


HBE-uids. Dodge and Metzner [3] and Ryan and Johnson [4] made investiga-tions which, however, are only valid for pure PL uids and cannot be applied to

the uids investigated in the present paper. Since the viscosity of the kerosene-gel reaches the Newtonian plateau at high shear rates caused by high volumetricow rates, it is assumed that the transition from laminar to turbulent ow con-ditions for the kerosene-gel could occur also near the critical Reynolds numbersof Newtonian liquids (Recrit Newton 2300).

For the liquid paran and the two paran-gels, the experimental resultsare plotted in Fig. 4. Due to the higher viscosity of the paran-based uidscompared to the kerosene-based uids, lower Reynolds numbers are achieved.The data show a good agreement with the laminar relation (solid line) up toRegen HBE 200. The transition from laminar to turbulent ow could not bereached in the conducted experiments. The deviation from the solid line which

was obtained for the kerosene-based uids just before the laminar-to-turbulenttransition seems to be signicantly more pronounced for the paran-based fu-els. There is no explanation yet for this deviation. Since the (Newtonian) liquidparan shows the same negative deviation from the solid line in Fig. 4b as theparan-gels, it is assumed that this is not due to the non-Newtonian viscos-ity characteristic of the gels. All data are corrected with Bagley [14] and thenon-Newtonian gels additionally with the WeissenbergRabinowitsch correction.However, obviously, there might be eects due to the very small capillary di-ameters and the inlet geometry which are more pronounced for high-viscosityuids compared to low-viscosity uids (e.g., vena contracta). Hence, further

investigations are required.

4 SUMMARY

In the present paper, it was suggested to use the introduced HBE-generalizedReynolds number Regen HBE for characterizing the duct ow conditions of theinvestigated non-Newtonian gelled uids. The Reynolds number is based onthe HBE viscosity law. Experimental data of capillary rheometer measurementswere used to verify, whether the general relation fDarcy = 64/Re between theDarcy friction factor and the Reynolds number is fullled in the laminar ow re-

gion. The results obtained with the introduced HBE-generalized Reynolds num-ber were compared to the results obtained with the well-known denitions ofReynolds numbers for Newtonian and PL uids. It was shown for the kerosenethixatrol-gel that the relation in the laminar ow region could only be fulllled ifthe introduced HBE-generalized Reynolds number was applied to the experimen-tal data. For Reynolds numbers between 2000 < Regen HBE


14/14


assumption. For the paran-based uids, a good agreement was obtained be-tween the experimental data and the theoretical relation fDarcy= 64/RegenHBE

in the laminar ow region for low Reynolds numbers. For high Reynolds numbers,a signicant negative deviation of the friction factor from the laminar relationfDarcy = 64/Regen HBE was found. Since this eect was measured also for theliquid paran, it is assumed that it is not due to the non-Newtonian viscositycharacteristic of the gels. For very high Reynolds numbers, there are obviouslyother uid mechanical eects due to the use of very small capillary diameterswhich are more distinct for high-viscosity uids compared to low-viscosity uids.

REFERENCES

1. Brinker, C. J., and G. W. Scherer. 1990.Sol-gel science. Boston: Academic Press.

2. Ciezki, H. K., and B. Natan. 2005. An overview of investigations on gel fuels forramjet applications. ISABE Paper No. 2005-1065.

3. Dodge, D. W., and A. B. Metzner. 1959. Turbulent ow of non-Newtonian systems.AIChE J. 5(2):198204.

4. Ryan, N. W., and M. M. Johnson. 1959. Transition from laminar to turbulent owin pipes. AIChE J. 5(4):43335.

5. Mishra, P., and G. Tripathi. 1970. Transition from laminar to turbulent ow ofpurely viscous non-Newtonian uids in tubes. Chem. Eng. Sci. 26(6):91521.

6. Malin, M. R. 1998. Turbulent pipe ow of HerschelBulkley uids. Int. Commun.

Heat Mass Transfer 25(3):32130.7. Bohme. G. 2000. Stromungsmechanik nichtnewtonscher Fluide. Teubner Verlag, 2.

Auage, 2000.

8. Metzner, A. B., and J. C. Reed. 1955. Flow of non-Newtonian uids correlationof the laminar, transition and turbulent-ow regions. AIChE J. 1(4):43440.

9. Madlener, K., and H. K. Ciezki. 2005. Analytical description of the ow behaviourof extended-HerschelBulkley uids with regard to gel propellants.36th Conference(International) of ICT & 32nd Pyrotecnics Seminar (International). Karlsruhe,Germany.

10. Madlener, K., and H. K. Ciezki. 2005. Theoretical investigation of the ow be-haviour of gelled fuels of the extended HerschelBulkley type. 1st European Con-ference for Aerospace Science (EUCASS). Moscow, Russia.

11. Hutter, K. 2002.Fluid- und Thermodynamik. Auage: Springer Verlag.

12. Rabinowitsch, B. 1929. Uber die Viskositat und Elastizitat von Solen. Z. Phys.Chem. 145A.

13. Mooney, M. 1931. Explicit formulas for slip and uidity.J. Rheol. 2(2):21022.

14. Bagley, E. B. 1957. End corrections in the capillary ow of polyethylene.J. Appl.Phys. 28(5):62427.

250

reynolds generalizado

Documents