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Entrance region flow of the Hershel-Bulkley fluid in a circular tube
Fluid Dynamics Research 10 (1992) 39 53 North-Holland
FLU I D DYNAMICS RESEARCH
Entrance region flow of the Hershel-Bulkley fluid in a circular tube Bigyani Das Department of Mathematics and Statistics, Unit,ersity of New Mexico, Albuquerque, NM 87131, USA Received 15 October 1991
Abstract. The steady, laminar, isothermal entrance region flow of the Hershel-Bulkley fluid in a tube has been
investigated by using the momentum integral and the momentum energy integral techniques. The resulting nonlinear ordinary differential equations on the dimensionless boundary layer thickness have been solved numerically by using the Runge-Kutta method. The results for the plug core velocity, boundary layer thickness, pressure drop, entrance length and loss coefficient have been obtained for a wide range of yield numbers and the flow behaviour index. Significant effect of viscous dissipation has been observed on the velocity distribution and boundary layer growth, whereas the effect of viscous dissipation on pressure drop is negligible. In both the methods, the values of entrance length and loss coefficient have been found to be reduced with increasing values of HersheI-Bulkley number and flow behaviour index. However, in the case of the momentum energy integral method the values of entrance length and loss coefficient are appreciably higher than those obtained by using the momentum integral method.
I. Introduction The study of entrance flow has been a subject of considerable technical importance owing to its direct engineering application in various designs of chemical, biomedical and food processes in which tube flows of Newtonian and non-Newtonian fluids are encountered. Furthermore, entrance flow is encountered in most capillary rheometric measurements of Newtonian and non-Newtonian fluids and also in every industrial process involving non-Newtonian suspensions, emulsions or solutions. A number of papers have appeared in the literature for the analysis of entrance region flows of viscous fluids. Reviews have been given by Fan and Hwang (1966) aJ3d Kapur et al. (1982). A survey of the methods applied for solving the entrance region flow equations for Newtonian fluids only is discussed by Fargie and Martin (1971) and Ward-Smith (1980). For a fluid entering a tube in laminar flow through an abrupt contraction, the velocity profile of the flow changes from an initial condition close to a flat velocity profile at the inlet to a fully developed condition at a certain distance downstream. This is the developing flow and associated with this is the conversion of pressure energy into kinetic energy, thus increasing its value and an increase in viscous friction caused by high velocity gradient at the Correspondence to." B. Das, Department of Mathematics and Statistics, University of New Mexico, Albuquerque, NM 87131, USA. 0169-5983/92/$04.75 © 1992 - The Japan Society of Fluid Mechanics. All rights reserved
40
B. Das / Hershel - Bulkley f l u i d in a circular tube
walls. For flow in a long tube, a detailed knowledge of this flow development is necessary not only to calculate the additional pressure drop, but also to solve heat and mass diffusion equations. In the present work, we have analysed the flow of a Hershel-Bulkley fluid in the entrance region of a tube. The Hershel-Bulkley fluids include a wide variety of shear thinning and shear thickening fluids. They also represent the empirical combination of the Bingham plastic and power law fluids and are known as yield power law fluids. Practical examples of such materials are greases (Greenwood and Kauzlarich, 1972), colloidal suspensions (Convey and Stanmore, 1981), starch pastes (Whorlow, 1980) and blood flow through narrow tubes (Scott-Blair and Spanner, 1974; Iida and Murata, 1980; Chaturani and Ponnalgarsamy, 1988). The fully developed flow characteristics of Hershel-Bulkley fluid in a tube has been obtained by Al-Fariss and Pinder (1987). The entrance region flow of Hershel-Bulkley fluid in channels have been investigated by Lin and Shah (1978) and Batra and Kandasamy (1990). The numerical solutions of the Hershel-Bulkley fluid in the entrance region of tubes have been obtained by Lin and Shah (1978) by considering the entire region as the flow region. However, Hershel-Bulkley fluid posseses a yield value below which no flow occurs. Therefore it is necessary to consider the plug core formation region away from the walls. Also, Lin and Shah (1978) have not presented any results for the hydrodynamic part of the entrance region flow characteristics. Accordingly, in the present work, considering the boundary layer regions near the walls separated by a plug core formation region around the center, the problem has been analysed by using the momentum integral technique. To study the effect of viscous dissipation within the boundary layers, the momentum energy integral method has also been employed for the analysis. The results for boundary layer thickness, plug core velocity and pressure drop have been calculated for values of yield number H n lying between 0 and 5 and for flow behaviour index n = 0.75, 1.0 and 1.25.
2. Formulation of the problem
The constitutive equation for the Hershel-Bulkley fluid is given by (Whorlow, 1980)
~=~y+nl~l ~,
(1)
where ~- is the shear stress, ~'y is the yield stress, "q is called the coefficient of fluidity, ~ is the rate of strain and n is the flow behaviour index.
loyer
I
Vz=V --%
S Fig. 1. Geometry of the problem.
_X
B. Das / Hershel-Bulkley fluid in a circular tube
41
The geometry of the problem is shown in fig. 1. We consider a Hershel-Bulkley fluid entering from a large chamber with a uniform flat velocity V into a horizontal circular tube of radius R. We use a cylindrical polar coordinate system (r, 0, z) with origin fixed at the flow entrance and z measured along the radial direction. The flow is assumed to be steady, laminar, incompressible, axisymmetric and isothermal with constant physical properties. The equations governing the flow are av~ = O, r Or ( rvr ) + az 1 a
--
Oc~ ar
' - tr
+tL
at~ ~ az
(2)
1 ap p az
1 a p r ar
(rTrz)'
(3)
where c r, v, are velocity components along the radial and axial directions respectively, p is the pressure and p is the density of the fluid and %= is the stress component which from the relation (1) can be written as
~-. : %, + ~ I avz/ar I"
(4)
Equations (2) and (3) will be solved under the following boundary conditions: (i) The velocity components v r and t,. are zero at the solid wall, that is Ur
=
Uz
=
0
at
r = R,
z_> 0.
(5)
(ii) The velocity at the conduit entrance is uniform, that is c==V,
t'r=0
at
z=0.
(6)
(iii) The axial velocity at the centerline is a function of axial distance only, that is ,,.=U~(z)
at
r=O.
(7)
3. S o l u t i o n o f the p r o b l e m
The form of the velocity profile observed for the fully developed flow of Hershel-Bulkley fluid in a circular tube (AI-Fariss and Pinder, 1987) indicates that the m o m e n t u m integral method and m o m e n t u m energy integral method can be applied to analyse the entrance region flow of a Hershel-Bulkley fluid in a tube. These methods have been successfully applied to study the entrance region flow problems of Newtonian (Schiller, 1922; Campbell and Slattery, 1963) and time independent non-Newtonian (Gupta, 1969, 1987; Chen et al., 1970; Batra and Jena, 1990) fluid flow problems. The plug core region of the Hershel-Bulkley fluid is analogous to the uniform velocity outside the boundary layer for a Newtonian fluid. We have boundary layer regions at the walls with thickness 6 ( z ) separated by the symmetrical plug core formation about the axis moving with the velocity Uc(z). Because of the symmetrical nature of the flow it is sufficient to carry out the analysis for the upper region only. In accordance with the above considerations the boundary condition (7) takes the form t,==U~(z)
for
O
z>O,
avyar = 0
at
r = R - 6, z > 0.
(8)
B. Das / Hershel- Bulkley fluid in a circular tube
42
3.1. Momentum integral (MI) method
In this method, considering the equilibrium in the plug flow region, the relation between the pressure gradient and the plug flow velocity is given by dUc Uc dz
1 dp p dz"
(9)
Substituting eq. (9) in eq. (3) and integrating the resulting equation across the cross-section of the tube and using eq. (2) we get d(01Uc 2) dz
dUc + 02Uc
1( -
dz
p
0l.,. n ) "ry + ~7 ~ r
r=n
(10)
'
where
01=R- -atz)~ 102 =
-~ fR - a ( z ) 1--
(11)
dr,
(12)
r dr.
Following Schiller (1922), Chen et al. (1970) and Gupta (1984), the velocity profiles in the boundary layer may be expressed in the form v:/U~(z)= l
for
r
t,JUc(z)=l-[1-(R-r)/6]
t'+l'/~
for
R-6<_r<_n.
(13)
The above velocity profile satisfies the boundary conditions (8). Since the above velocity profile gives most satisfactory results for the power law fluid (Gupta, 1984) it is expected to give accurate results for the flow of Hershel-Bulkley fluid, which is an empirical combination of power law fluid and Bingham fluid. This has been used by Batra and Kandasamy (1990) for analysing the flow characteristics of Hershel-Bulkley fluid in a channel. The integral form of the continuity equation is written as foRt,.r dr = 5VR 1 2.
(14)
Substituting the expression for vZ from (13) in (14) and evaluating the integral, we get, U c * - Uc(z) _ 1 V 1 + clm + c2 m 2 '
(15)
where c I = - 2 n / ( 2 n + 1), Ce= 2 n 2 / ( 2 n + 1)(3n + 1) and m = ~ / R is the dimensionless boundary layer thickness. Now substituting the velocity expressions e z from (13) in eqs. (11) and (12), evaluating the integrals and substituting the resulting expressions in (10), carrying out the necessary differentiation we get the following nonlinear ordinary differential equation in m mnF(m, n)--
dz*
|
,,m +
B. Das / Hershel - Bulkley fluid in a circular tube
43
where
a(m) = 1 + clrn + c2 m2, S l + 2S2m
m( c I + 2c2m)( G l + G2m )
aZ(m )
a3(m)
F ( m , n)
3n2(n + 1)
n(n + 1) Sl=
(2n + 1)(3n + 2) '
3 2 =
_
(3n + 1)(4n + 2)(3n + 2 ) ' n 2
/7
G 2 = 2S 2 -
G 1=2S1+ 2 n + l '
(3n + 1)(2n + 1) '
Z Z*--
-
-
2ReR '
R e = 2pR"V2-~/71 is the Reynolds number and H, = 7yRn/~lV n is the Hershel-Bulkley number. Equation (16) has been integrated numerically by using Simpson's one-third rule to determine the dimensionless boundary layer thickness m for various values of Hershel-Bulkley number H~ and power law index n. The pressure drop from the tube inlet to any section z * = z* in the entrance region can be obtained by integrating eq. (9) and is given by
P*
~-- 1 (Po--Pe)/~P
V 2
=Uc .2
_
1.
(17)
The pressure drop in the fully developed region is obtained by considering the expression for average velocity derived from the equation for the flow rate in fully developed tube flow (Al-Fariss and Pinder, 1987) and is given by
P[~ = ( P o - P f ) / ½ P V2 = 1 6 z * / K " ( Hy, n),
(18)
where
K(Hy, n ) = ( l _ H y ) , + l / ,
(
2n3(1
+
-
n
2n2(1-Hy)
n+ 1
(n + 1)(2n + 1)
Hy) 2
(n + 1)(2n + 1)(3n + 1) and Hy is the plug core radius in the fully developed region which is related to the Hershel-Bulkley number by
U,, = H y / K ' ( Hy,
n).
The detail derivation' of eq. (18) and the relation between Hy and H n are given in the appendix. For each Hy, there exists a corresponding H,, which is called the Hershel-Bulkley number. When Hy is zero, the fluid is power law fluid. When Hy approaches unity, H,, becomes infinitely large corresponding to the total plug flow. The relation between Hy and Hn has been represented graphically in fig. 2. From eqs. (17) and (18) the correction factor for fraction loss C can be expressed as
C =PL*-- 16Le/Kn(Hy, n),
(19)
where p*L e is the value of p* at the entrance length point z * = L e, the point at which the plug core velocity Uc* approaches its fully developed value.
44
B. D a s / H e r s h e l - Bulkley f l u i d in a circular tube
100.0 n=0.75~ n=0.532
u:~ 50.0
0.0 , ~ 0.0 0.2
0.4
,
0.6
0.8
1.0
Hy Fig. 2. Hershel-Bulkley number as a function of plug core radius.
3.2. Momentum energy integral (ME1) method The effect of viscous dissipation within the boundary layer region becomes more important in the downstream region. To account for this effect, we solve the problem by using the momentum energy integral technique. This method has originally been developed by Campbell and Slattery (1963) to make corrections for the unrealistic results in the region far from the entry z* > 0.02 obtained from the momentum integral method for Newtonian fluids. In addition to the basic assumptions employed in the momentum integral method, they used the macroscopic mechanical energy balance to account for viscous dissipation within the boundary layer. In this method instead of using eq. (9) for pressure gradient, it is obtained from the mechanical energy balance. The integral form of principle of momentum can be obtained by integrating eq. (3) over the tube cross-section and using the boundary conditions on t,r and vz and is given by
e 2 d -p ( Or'" tzr dr + - -2 -dz + R ry+r/i--~r" I ) r=R =
d f0R 2
P~z
0.
(20)
Now multiplying eq. (3) by the axial velocity tL., integrating across the cross-section of the tube, using the continuity equation and the boundary conditions on t~r and t' z, we get the energy integral equation as
dp R foR( St'z" Ot'z dzfo t ' ' r d r %'+ ~-r ) - r d r = 0 ' 0 r
P dz.o d "
(21)
Eliminating pressure terms from eqs. (20) and (21) we get Ou z n
"2 dz' -
RV
Jo"v;r dr-pV-~z %+.
~
-r
--r0r
dr=0.
r=R
(22)
Substituting the velocity profiles (13) in (22) and evaluating the integrals, using eqs. (14) and (15) and carrying out the necessary differentiation, we obtain the nonlinear ordinary differential equation in the dimensionless boundary layer thickness m as given by
mn(T~ - 2T2) d m / d z * = 16( HnK,m n + K 2 ) ,
(23)
B. Das / Hershel- Bulkley fluid in a circular tube
45
where
A +2mB
3(c I + 2c2m)(1 +Am +Bm 2)
a3(m)
aa(m)
T1-
T2-
2(c I + 2c2m)(1 + A , m +B,m 2) a3(m)
A 1 + 2mB 1 a2(m)
B = 2[g(n) -f(n)]
A = 2 [ f ( n ) - 1], A, = 2[f~(n) - 1],
+ 1,
B 1 = 2[gl(n ) - f , ( n ) ]
+ 1,
n(18n 2 + 28n + 11) f(n) = 1
g(n)-
-
(4n + 3)(2n + 1)(3n + 2) '
1
n(27n 2 + 34n + 11)
2
(5n + 3)(3n + l ) ( 4 n + 2 ) '
f,(n)= l-
n(4n + 3) (3n+2)(2n+l)'
1
n+l ( n mn na(-n~) ( l - m ,) - n- ++ -l +l- 2 n
K,=I
(n+l)"
[
,,(5,, + 3)
gl(n)='2-
n+l
K 2 - n,a,+l(m ) a(m) - - -n
I
(4n + 2 ) ( 3 n + 1) '
)
(1 - m ) ~ -
n
mn
+ 3n + 1
)]
The above equation (23) has been integrated numerically by using Simpson's one-third rule to determine the values of dimensionless boundary layer thickness m for different values of n and /4,. Substituting these values of m in (15), we obtain the plug core velocity Uc* represented graphically in figs. 6, 7 and 8. The entrance length has been obtained considering the point at which the plug core velocity U~* approaches 99% of its fully developed value. Integrating eq. (20), the pressure drop expressed in the dimensionless form is given by p* = - 2 + 2 Q ( m ) - 1 6 / a0
H,, +
dz,
(24)
where
Q(m) =
1 +Aim + A z m 2 a2(m )
The integral in eq. (24) has been evaluated numerically by using Simpson's one-third rule for various values of yield number H, and for different values of n to obtain the pressure drop in the entrance region. Using the relations (18) and (24), we obtain the loss coefficient values for various values of Hershel-Bulkley number and power law index.
4. Results and discussions
The problem of entrance region flow of Hershel-Bulkley fluid in a circular tube has been analysed by momentum integral method and the effects of viscous dissipation in the boundary layer have been studied by using the momentum energy integral method. The entrance region
46
B. Das
Hershel
/
10
Bulkley )quid in a circular tube
-
0.0
0.5 /
-
f
o a
- - ~ - - - - -
/
0.5
// 2.0
E 0.5
~x
~
tum
-
- Momentum Parameter
0.0 0.0
n e e r g y Int. M e t h o d
Int. Method H n
0.05
0,06 z*
Fig. 3. Variation of boundary layer thickness along axial distance for n = 0.75.
flow characteristics have been found to be completely described by two parameters, the Hershel-Bulkley n u m b e r / 4 , and the power law index n. The relation between the H e r s h e l Bulkley number H,, and the fully developed plug core radius Hy is represented graphically in fig. 2. We find that H,, increases as H>. increases and it becomes infinitely large as H>. approaches unity. We also observe that, for a particular value of H>,, the value of H,, becomes more as the power law index n increases. The variation of boundary layer thickness m along the axial distance determined from the two methods is shown graphically in figs. 3, 4 and 5 respectively for different values of Hershel-Bulkley number lying between 0 and 5 and power law iDdex n = 0.75, 1.0 and 1.25. We observe that in both the cases the value of boundary layer thickness increases from a zero value at the inlet till it approaches the fully developed value. In the momentum integral method the increase is sharp whereas in the momentum energy integral method the increase is relatively slower and it approaches the fully developed value asymptotically. It has also been observed that, in case of momentum integral method, the boundary layer thickness increases as the Hershel-Bulkley number H,, increases, but the reverse trend is exhibited in case of momentum energy integral method except near the region close to the entry section. This means that viscous dissipation has significant effect on the development of boundary layer in the downstream region. We also find that in both methods the boundary layer thickness becomes more for larger values of power law index n.
1.0
'
ko-
0.5.~ ~
// //. .
2.0
/
'
o
.
o
~
0.5
~
/h
E
0.5 -....
Momentum
E n e r g y Int. M e t h o d
Momentum
Int. M e t h o d
Porometeran 0.0
i
0.0
,
,
0.03
,
I
0.08
Z W'
Fig. 4. Variation of boundary layer thickness along axial distance for n = 1.0.
B. Das / Hershel- Bulkley fluid in a circular tube
47
1.0 >~x/.
~¸
2.0 ~
E
0.5
~
z /
5.0
~
n
Fe gyInt.Method - Momenu tmInt.Method Parameter Hn
0.0 0.0
,
,
, 0.02
,
,
, 0.04
Z* Fig. 5. V a r i a t i o n
of boundary
l a y e r thickness a l o n g axial d i s t a n c e f o r n = 1.25.
0.0
1.8
0.0
."
0.5
0.5
//
1.4
~ ' / ~ e nn t ut mu
E
.....Int Int. Method m Energy Mnm#nJMrv~ln{* m#fhnr] - - Momentum Int. method
~ ~ e /I__ 1 0•~
-
-
parameter Hn 1.0 0.0
0.03
0.06 z*
F i g . 6. Variation of plug core velocity along axial distance for n = 0.75.
2.2 0.0
* u
2.0
1.6 ~
~_
M
. o ~
0.0
~ m
e
-
-
n
t
u
m
Energy Int. Method
Momentum Int. Method Parameter Hn
1.0 0.0
0.0.3
0.06
z* Fig. 7. V a r i a t i o n o f p l u g c o r e velocity along axial d i s t a n c e f o r n = 1.0.
Choosing the values of Hershel-Bulkley number H,, and the flow behaviour index n as mentioned above, the distribution of plug core velocity Uc* along the axial distance is shown graphically in figs. 6, 7, and 8 for both methods. Similar behaviour as of the boundary layer thickness has also been observed for plug core velocity with increasing values of H~ and n. The values of Uc* obtained from the momentum integral method reach the fully developed values faster and hence give smaller values of entrance length in comparison to the corre-
B. Das / Her~'hel- Bulkley f l u i d in a circular tube
48
2.2 0.0
0.0
*~
1.6
I
/ "
~
m
EnergyInt. Methoc - - MomentumInt. Method Porometer
1.0 0.0
Hn
0.02
0.04
z*
Fig. 8. Variation of plug core velocity along axial distance for n = 1.25.
0.1
.
.
.
.
.
.
.
_~.75 _
,
--
~
-
-
,
,
,
,
,
.
.
.
.
.
.
Momentum
E n e r g y Int. M e t h o d
Momentum
Int. M e t h o d
o.o5
!:.~r 1.o
0.0
0.25
0.5 Hy
0.75
1.0
Fig. 9. Entrance length of the Hershel Bulkley fluids as a function of plug core radius and power law index.
sponding values obtained from the energy integral method. The values of entrance length for different values of plug core radius Hy and power law index n are shown in fig. 9. Here the p a r a m e t e r H~. has been choosen for direct comparison with the results of Chen et al. (1970) for a Bingham fluid. The relation between the Hershel-Bulkley number H , and plug core radius H~, is depicted in fig. 2. In both cases, the entrance length decreases as Hy or the power law index increases. The value of entrance length for a Bingham fluid has been obtained by putting the value of n = 1 and is found to be in good agreement with the values obtained by Chert e t a l . (1970). The results obtained for a power-law fluid by putting H v = 0 are in qualitative agreement with those obtained by Bogue (1959) and Matras and Nowak (1983). Table 1 shows the comparison of entrance length values for some pseudoplastic fluids
Table 1 Comparison of entrance length values L,. for H,, = 0 with power law fluids. Power law
Present results
index n
M1 method
MEI method
Bogue
0.726 0.75 0.85 0.9 1.0
0.03112 [).03104 0.03042 0.02995 0.02877
0.09141 0.08911 0.07990 0.07555 0.06739
Matras and Nowak
0.03215 0.03202 0.03104 0.03036 0.02877
0.10040 0.09933 0.09450 0.09191 0.08650
49
B. Das / Hershel - Bulkley fluid in a circular tube
5.0 0.5
0.0 "
2.0 2.5
)
5.0
~
10.0
//~.~"
-
-
Momenturn
Int.
Method
Porometer H n
0.0 0.0
.
.
.
.
.
.
0.03
0.06
Z* Fig. 10. Pressure drop distribution along axial distance for n = 0.75.
with those of Bogue (1959) and Matras and Nowak (1983). Our results obtained by momentum integral method are close to the results of Bogue (1959), who has obtained results for a power law fluid by using the same method but with cubic velocity profile in the boundary layer. It proves the accuracy of the numerical code. However, the results obtained by Matras and Nowak (1983) are quite higher than our results obtained by both integral methods. For different values of Hershel-Bulkley number and power law index, the dimensionless pressure distribution along the axial distance has been shown in figs. 10, 11 and 12. From the figures we found that there is negligible effect of viscous dissipation on pressure drop. Similar results have been obtained for a Bingham fluid flow (Chen et al., 1970) and for the flow of a power law fluid (Tiu and Bhattacharya, 1973). Hence, if only the knowledge of pressure distribution in the entrance region is desired, the m o m e n t u m integral method is recommended, for the calculations involved will be less. We also find that the values of the pressure drop in the entrance region becomes larger with increasing values of H n and n. The results for special cases of Hn = 0 and n = 1 agree quite well with those of Chen et al. (1970) for a Bingham fluid and with Bogue (1959) and Matras and N o w a k (1983) for a power law fluid. The values of the loss coefficient are shown in fig. 13. We observe that the value of the loss coefficient decreases with increasing values of power law index. This type of behaviour is in qualitative agreement with all the previous works for power law fluid. The change in values with yield number is analogous to that in the case of entrance length. However, the value of the loss coefficient obtained with the m o m e n t u m energy integral method are appreciably higher than those obtained in the m o m e n t u m integral method. For power law index n = 1,
5.0
2.0 5.0 *
2.5
10.0 . Method ~ "
- .-
Momentum Int. Method Poremeter Hn
0.0 0.0
0.03
0.06
z
Fig. 11. Pressure drop distribution along axial distance for n = 1.0.
B. Das / Hersbel - Bulkley f l u i d in a circular tube
50
5.0
'
'
--
5.0
f
Porarneter Hn
0.0 0.0
.
.
.
.
. 0.02 Z*
. 0.04
Fig. 12. Pressure drop distribution along axial distance for n = 1.25.
1.4
~
5
1.2
--
Momentum Energy Int. Method
~ •
0.0
0.0
0.25
0.5
0.75
1.0
Hy F i g , 13. L o s s c o e f f i c i e n t o f t h e H e r s h e l - B u l k l e y
fluids as a function of plug core radius and power law index.
the values reduce to those for a Bingham fluid and are in good agreement with those obtained by Chen et al. (1970). Finally, we found that the solution by energy integral method provides a valid overall description of flow characteristics of the Hershel-Bulkley fluid in the entrance region of a circular tube. However, the momentum integral method provides a reasonable description of the fluid flow problem under the conditions where the effects of viscous dissipation are negligible. Furthermore, the experimental results for entrance region flow of a Hershel-Bulkley fluid in a tube are not available in literature. The only theoretical work available is that of Lin and Shah (1978), but they have not presented any results for the hydrodynamic entry length problem. Accordingly, the accuracy of the present work has been confirmed by comparing the entrance solutions for special cases of n = 1 and H n = 0 with those obtained by Chen et al. (1970) for a Bingham fluid and by Bogue (1959) and Matras and Nowak (1983) for a power law fluid.
Acknowledgement The author is grateful to the Chairman, Department of Mathematics and Statistics, University of New Mexico for his kind permission to avail the computer facility in the department.
B. Das / Hershel- Bulkleyfluid in a circulartube
51
Appendix The derivation of eq. (18) and eq. (19). In the paper by Al-Fariss and Pinder (1987), the expression for the total flow rate Q is given by ~nR2 Q-
(
,,.)1/,,
aR---~?
(n+l)a
(aR-aRo)
2vrn2R ('rv)'/" +c~2(n+ l l ( 2 n + l) aR---~? a3(n+l)(2n+
(aR-c~Ro) 2
o~R-
1)(3n+l)
(aR-o~Ro) 3,
(25)
where a
AP
Idp/dzl
2~?L
2~1
(26)
Here L is the length of the tube, Ap is the pressure difference and R 0 is the core radius in the fully developed region. Equation (25) can be simplified to the form
Q = - ~ r R B ( a R ) '/" 1
aR~
2n3( 1 - Ro/R) 2
1-
n+ l
( n + l)(2n+ l)
1
+ (n + 1)(2n + 1)(3n + 1) 1" Now by considering the equilibrium of shear and pressure forces in the fully developed region we get Trz = ~ r
Idp/dzl.
(28)
Since the yield stress % is the value of the shear stress at R = R o (the plug core radius), we get 1 ~'y = ~R o [dp/dz I
which gives
"ry/aR'rI = Ro/R = Hy,
(29)
where Hy is the dimensionless plug core radius in the fully developed region. Now we rewrite eq. (26) as
Q = -TrR3
( ' d p / d z l R ) '/n 2~1 K(Hy, n),
(30)
where
K(H>,
l'l )
=(1-Hv)'/"[
n
tn+l
2n2(1-Hy)
(n+l)(2n+ 1)
2n3( 1 -- H>')2
t
+ (n + 1)(2n + 1)(3n + 1) )"
B. Das / Hershel- Bulkley fluid in a circular tube
52
F r o m t h e e x p r e s s i o n (30) w e o b t a i n t h e a v e r a g e v e l o c i t y V as
V = ]Q / r r R 2 1 = R ( I d p / d z
l R/2~7) l/n g ( n y , n)
(31)
w h i c h can b e w r i t t e n as
Idp/dz Since Idp/dz
I= [V/K(Hy,
I = -dp/dz
-dp/dz=
n)R]~2~7/R.
(32)
( t h e flow is d u e to fall in p r e s s u r e ) , w e c a n w r i t e eq. (32) as
2 r l V ~ / R ~ + ~ K ~ (\ H
Y,
n).
(33)
I n t e g r a t i n g t h e a b o v e e q u a t i o n w i t h r e s p e c t to z f r o m z = 0 to any p o i n t z = zf, we g e t t h e d i m e n s i o n l e s s p r e s s u r e d r o p as ( P0 - p f ) / ~ p V 1
2 = 4rlV n 2 z / p R " + l K " ( H y ,
n).
(34)
By t h e d e f i n i t i o n o f R e y n o l d s n u m b e r R e a n d t h e d i m e n s i o n l e s s axial d i s t a n c e z * , t h e a b o v e r e l a t i o n c a n b e w r i t t e n as ( P 0 - p f ) / / ~ p1r
2 = 16z*/K"(Hy,
n).
(35)
Derivation o f the relation between Hy and 14,,. T h e H e r s h e l - B u l k l e y n u m b e r H~ is d e f i n e d by
H , = ~-yR"/rlV".
(36)
F r o m eqs. (26), (29) a n d (32), w e c a n w r i t e
H,, = r y / a R ' q = "cyR"K"( Hy, n ) / V % 7 .
(37)
F r o m e x p r e s s i o n s (36) a n d (37), w e get t h e r e l a t i o n b e t w e e n Hy a n d H n as
H, = Hy/Kn(n,,
n).
(38)
References AI-Fariss, T. and K.L. Pinder (1987) Flow through porous media of a shear-thinning liquid with yield stress, Can. J. Chem. Eng. 65, 391. Batra, R.L. and B. Jena (1987) Blood flow in the entrance region of a tube, Proc. Nat. Conf. FMFP India 15, 353. Batra, R.L. and A. Kandasamy (1990) Entrance flow of Hershel-Bulkley fluids in a duct, Fluid Dyn. Res. 6, 43. Bogue, D.C. (1959) Entrance effects and prediction of turbulence in non-Newtonian flow, Ind. Eng. Chem. 51, 874. Campbell, W.D. and J.C. Slattery (1963) Flow in the entrance of a tube, J. Basic Eng. ASME Trans. 85, 41. Chaturani, P. and R. Ponnalgarsamy (1988) A study of non-Newtonian aspects of blood flow through stenosed arteries and its applications in arterial diseases, Biorheology 22, 521. Chen, S.S., L.T. Fan and C.L. Hwang (1970) Entrance region flow of the Bingham fluid in a circular pipe, AIChE J. 16, 293. Convey, G.H. and B.R. Stanmore (1981) Use of the parallel-plate plastometer for the characterization of viscous fluids with a yield stress, J. Non-Newtonian Fluid Mech. 8, 249. Fan, L.T. and C.L. Hwang (1966) Hydrodynamic entrance region flow, Kansas State Unit,. Bull. 50, 3. Fargie D. and B.W. Martin (1971) Developing laminar flow in a pipe of circular crosssection, Proc. Roy. Soc. 321, 461. Greenwood, J.A. and J.J. Kauzlarich (1972) EHD lubrication with Hershel-Bulkley model greases, ASLE Trans. 4 , 269. Gupta, R.C. (1969) Flow of power law fluids in the entry region of a circular pipe, J. M~canique 8, 207. Gupta, R.C. (1984) Developing laminar power-law fluid flow in flat duct, J. Eng. Mech. ASCE Trans. 110, 752. Iida, N. and T. Murata (1980) Theoretical analysis of pulsatile blood flow in small vessels, Biorheology 17, 377. Kapur, J.N., B.S. Bhatt and N.C. Sacheti (1982) Non-Newtonian Fluid Flows (Pragati Prakasan Publ., Meerut, India). Lin, T. and V.L. Shah (1978) Numerical solution of heat transfer to yield-power law fluids flowing in the entrance region, Proc. VI Intern. Heat Transfer Conf. 5, 317.
B. Das / Hershel- Bulkley fluid in a circular tube
53
Matras, Z. and Z. Nowak (1983) Laminar entry length problem for power-law fluids, Acta Mechanica 48, 81. Schiller, L. (1922) Die Entwichklung der laminaren Geschwindigkeitsverteilung und ihre Bedeutung fiir Z~ihigkeitmessungen, Z A M M 2, 96. Scott-Blair, G.W. and D.C. Spanner (1974) An Introduction to Biorheology (Elsevier, Amsterdam). Tiu, C. and S. Bhattacharya (1973) Flow behaviour of power-law fluids in the entrance region of annuli, Can.J. Chem. Eng. 51, 47. Ward-Smith, A.J. (1980) Internal Fluid Flow (Clarendon Press, Oxford). Whorlow, R.W. (1980) Rheological Techniques (Wiley, New York) p. 33.
FLU I D DYNAMICS RESEARCH
Entrance region flow of the Hershel-Bulkley fluid in a circular tube Bigyani Das Department of Mathematics and Statistics, Unit,ersity of New Mexico, Albuquerque, NM 87131, USA Received 15 October 1991
Abstract. The steady, laminar, isothermal entrance region flow of the Hershel-Bulkley fluid in a tube has been
investigated by using the momentum integral and the momentum energy integral techniques. The resulting nonlinear ordinary differential equations on the dimensionless boundary layer thickness have been solved numerically by using the Runge-Kutta method. The results for the plug core velocity, boundary layer thickness, pressure drop, entrance length and loss coefficient have been obtained for a wide range of yield numbers and the flow behaviour index. Significant effect of viscous dissipation has been observed on the velocity distribution and boundary layer growth, whereas the effect of viscous dissipation on pressure drop is negligible. In both the methods, the values of entrance length and loss coefficient have been found to be reduced with increasing values of HersheI-Bulkley number and flow behaviour index. However, in the case of the momentum energy integral method the values of entrance length and loss coefficient are appreciably higher than those obtained by using the momentum integral method.
I. Introduction The study of entrance flow has been a subject of considerable technical importance owing to its direct engineering application in various designs of chemical, biomedical and food processes in which tube flows of Newtonian and non-Newtonian fluids are encountered. Furthermore, entrance flow is encountered in most capillary rheometric measurements of Newtonian and non-Newtonian fluids and also in every industrial process involving non-Newtonian suspensions, emulsions or solutions. A number of papers have appeared in the literature for the analysis of entrance region flows of viscous fluids. Reviews have been given by Fan and Hwang (1966) aJ3d Kapur et al. (1982). A survey of the methods applied for solving the entrance region flow equations for Newtonian fluids only is discussed by Fargie and Martin (1971) and Ward-Smith (1980). For a fluid entering a tube in laminar flow through an abrupt contraction, the velocity profile of the flow changes from an initial condition close to a flat velocity profile at the inlet to a fully developed condition at a certain distance downstream. This is the developing flow and associated with this is the conversion of pressure energy into kinetic energy, thus increasing its value and an increase in viscous friction caused by high velocity gradient at the Correspondence to." B. Das, Department of Mathematics and Statistics, University of New Mexico, Albuquerque, NM 87131, USA. 0169-5983/92/$04.75 © 1992 - The Japan Society of Fluid Mechanics. All rights reserved
40
B. Das / Hershel - Bulkley f l u i d in a circular tube
walls. For flow in a long tube, a detailed knowledge of this flow development is necessary not only to calculate the additional pressure drop, but also to solve heat and mass diffusion equations. In the present work, we have analysed the flow of a Hershel-Bulkley fluid in the entrance region of a tube. The Hershel-Bulkley fluids include a wide variety of shear thinning and shear thickening fluids. They also represent the empirical combination of the Bingham plastic and power law fluids and are known as yield power law fluids. Practical examples of such materials are greases (Greenwood and Kauzlarich, 1972), colloidal suspensions (Convey and Stanmore, 1981), starch pastes (Whorlow, 1980) and blood flow through narrow tubes (Scott-Blair and Spanner, 1974; Iida and Murata, 1980; Chaturani and Ponnalgarsamy, 1988). The fully developed flow characteristics of Hershel-Bulkley fluid in a tube has been obtained by Al-Fariss and Pinder (1987). The entrance region flow of Hershel-Bulkley fluid in channels have been investigated by Lin and Shah (1978) and Batra and Kandasamy (1990). The numerical solutions of the Hershel-Bulkley fluid in the entrance region of tubes have been obtained by Lin and Shah (1978) by considering the entire region as the flow region. However, Hershel-Bulkley fluid posseses a yield value below which no flow occurs. Therefore it is necessary to consider the plug core formation region away from the walls. Also, Lin and Shah (1978) have not presented any results for the hydrodynamic part of the entrance region flow characteristics. Accordingly, in the present work, considering the boundary layer regions near the walls separated by a plug core formation region around the center, the problem has been analysed by using the momentum integral technique. To study the effect of viscous dissipation within the boundary layers, the momentum energy integral method has also been employed for the analysis. The results for boundary layer thickness, plug core velocity and pressure drop have been calculated for values of yield number H n lying between 0 and 5 and for flow behaviour index n = 0.75, 1.0 and 1.25.
2. Formulation of the problem
The constitutive equation for the Hershel-Bulkley fluid is given by (Whorlow, 1980)
~=~y+nl~l ~,
(1)
where ~- is the shear stress, ~'y is the yield stress, "q is called the coefficient of fluidity, ~ is the rate of strain and n is the flow behaviour index.
loyer
I
Vz=V --%
S Fig. 1. Geometry of the problem.
_X
B. Das / Hershel-Bulkley fluid in a circular tube
41
The geometry of the problem is shown in fig. 1. We consider a Hershel-Bulkley fluid entering from a large chamber with a uniform flat velocity V into a horizontal circular tube of radius R. We use a cylindrical polar coordinate system (r, 0, z) with origin fixed at the flow entrance and z measured along the radial direction. The flow is assumed to be steady, laminar, incompressible, axisymmetric and isothermal with constant physical properties. The equations governing the flow are av~ = O, r Or ( rvr ) + az 1 a
--
Oc~ ar
' - tr
+tL
at~ ~ az
(2)
1 ap p az
1 a p r ar
(rTrz)'
(3)
where c r, v, are velocity components along the radial and axial directions respectively, p is the pressure and p is the density of the fluid and %= is the stress component which from the relation (1) can be written as
~-. : %, + ~ I avz/ar I"
(4)
Equations (2) and (3) will be solved under the following boundary conditions: (i) The velocity components v r and t,. are zero at the solid wall, that is Ur
=
Uz
=
0
at
r = R,
z_> 0.
(5)
(ii) The velocity at the conduit entrance is uniform, that is c==V,
t'r=0
at
z=0.
(6)
(iii) The axial velocity at the centerline is a function of axial distance only, that is ,,.=U~(z)
at
r=O.
(7)
3. S o l u t i o n o f the p r o b l e m
The form of the velocity profile observed for the fully developed flow of Hershel-Bulkley fluid in a circular tube (AI-Fariss and Pinder, 1987) indicates that the m o m e n t u m integral method and m o m e n t u m energy integral method can be applied to analyse the entrance region flow of a Hershel-Bulkley fluid in a tube. These methods have been successfully applied to study the entrance region flow problems of Newtonian (Schiller, 1922; Campbell and Slattery, 1963) and time independent non-Newtonian (Gupta, 1969, 1987; Chen et al., 1970; Batra and Jena, 1990) fluid flow problems. The plug core region of the Hershel-Bulkley fluid is analogous to the uniform velocity outside the boundary layer for a Newtonian fluid. We have boundary layer regions at the walls with thickness 6 ( z ) separated by the symmetrical plug core formation about the axis moving with the velocity Uc(z). Because of the symmetrical nature of the flow it is sufficient to carry out the analysis for the upper region only. In accordance with the above considerations the boundary condition (7) takes the form t,==U~(z)
for
O
z>O,
avyar = 0
at
r = R - 6, z > 0.
(8)
B. Das / Hershel- Bulkley fluid in a circular tube
42
3.1. Momentum integral (MI) method
In this method, considering the equilibrium in the plug flow region, the relation between the pressure gradient and the plug flow velocity is given by dUc Uc dz
1 dp p dz"
(9)
Substituting eq. (9) in eq. (3) and integrating the resulting equation across the cross-section of the tube and using eq. (2) we get d(01Uc 2) dz
dUc + 02Uc
1( -
dz
p
0l.,. n ) "ry + ~7 ~ r
r=n
(10)
'
where
01=R- -atz)~ 102 =
-~ fR - a ( z ) 1--
(11)
dr,
(12)
r dr.
Following Schiller (1922), Chen et al. (1970) and Gupta (1984), the velocity profiles in the boundary layer may be expressed in the form v:/U~(z)= l
for
r
t,JUc(z)=l-[1-(R-r)/6]
t'+l'/~
for
R-6<_r<_n.
(13)
The above velocity profile satisfies the boundary conditions (8). Since the above velocity profile gives most satisfactory results for the power law fluid (Gupta, 1984) it is expected to give accurate results for the flow of Hershel-Bulkley fluid, which is an empirical combination of power law fluid and Bingham fluid. This has been used by Batra and Kandasamy (1990) for analysing the flow characteristics of Hershel-Bulkley fluid in a channel. The integral form of the continuity equation is written as foRt,.r dr = 5VR 1 2.
(14)
Substituting the expression for vZ from (13) in (14) and evaluating the integral, we get, U c * - Uc(z) _ 1 V 1 + clm + c2 m 2 '
(15)
where c I = - 2 n / ( 2 n + 1), Ce= 2 n 2 / ( 2 n + 1)(3n + 1) and m = ~ / R is the dimensionless boundary layer thickness. Now substituting the velocity expressions e z from (13) in eqs. (11) and (12), evaluating the integrals and substituting the resulting expressions in (10), carrying out the necessary differentiation we get the following nonlinear ordinary differential equation in m mnF(m, n)--
dz*
|
,,m +
B. Das / Hershel - Bulkley fluid in a circular tube
43
where
a(m) = 1 + clrn + c2 m2, S l + 2S2m
m( c I + 2c2m)( G l + G2m )
aZ(m )
a3(m)
F ( m , n)
3n2(n + 1)
n(n + 1) Sl=
(2n + 1)(3n + 2) '
3 2 =
_
(3n + 1)(4n + 2)(3n + 2 ) ' n 2
/7
G 2 = 2S 2 -
G 1=2S1+ 2 n + l '
(3n + 1)(2n + 1) '
Z Z*--
-
-
2ReR '
R e = 2pR"V2-~/71 is the Reynolds number and H, = 7yRn/~lV n is the Hershel-Bulkley number. Equation (16) has been integrated numerically by using Simpson's one-third rule to determine the dimensionless boundary layer thickness m for various values of Hershel-Bulkley number H~ and power law index n. The pressure drop from the tube inlet to any section z * = z* in the entrance region can be obtained by integrating eq. (9) and is given by
P*
~-- 1 (Po--Pe)/~P
V 2
=Uc .2
_
1.
(17)
The pressure drop in the fully developed region is obtained by considering the expression for average velocity derived from the equation for the flow rate in fully developed tube flow (Al-Fariss and Pinder, 1987) and is given by
P[~ = ( P o - P f ) / ½ P V2 = 1 6 z * / K " ( Hy, n),
(18)
where
K(Hy, n ) = ( l _ H y ) , + l / ,
(
2n3(1
+
-
n
2n2(1-Hy)
n+ 1
(n + 1)(2n + 1)
Hy) 2
(n + 1)(2n + 1)(3n + 1) and Hy is the plug core radius in the fully developed region which is related to the Hershel-Bulkley number by
U,, = H y / K ' ( Hy,
n).
The detail derivation' of eq. (18) and the relation between Hy and H n are given in the appendix. For each Hy, there exists a corresponding H,, which is called the Hershel-Bulkley number. When Hy is zero, the fluid is power law fluid. When Hy approaches unity, H,, becomes infinitely large corresponding to the total plug flow. The relation between Hy and Hn has been represented graphically in fig. 2. From eqs. (17) and (18) the correction factor for fraction loss C can be expressed as
C =PL*-- 16Le/Kn(Hy, n),
(19)
where p*L e is the value of p* at the entrance length point z * = L e, the point at which the plug core velocity Uc* approaches its fully developed value.
44
B. D a s / H e r s h e l - Bulkley f l u i d in a circular tube
100.0 n=0.75~ n=0.532
u:~ 50.0
0.0 , ~ 0.0 0.2
0.4
,
0.6
0.8
1.0
Hy Fig. 2. Hershel-Bulkley number as a function of plug core radius.
3.2. Momentum energy integral (ME1) method The effect of viscous dissipation within the boundary layer region becomes more important in the downstream region. To account for this effect, we solve the problem by using the momentum energy integral technique. This method has originally been developed by Campbell and Slattery (1963) to make corrections for the unrealistic results in the region far from the entry z* > 0.02 obtained from the momentum integral method for Newtonian fluids. In addition to the basic assumptions employed in the momentum integral method, they used the macroscopic mechanical energy balance to account for viscous dissipation within the boundary layer. In this method instead of using eq. (9) for pressure gradient, it is obtained from the mechanical energy balance. The integral form of principle of momentum can be obtained by integrating eq. (3) over the tube cross-section and using the boundary conditions on t,r and vz and is given by
e 2 d -p ( Or'" tzr dr + - -2 -dz + R ry+r/i--~r" I ) r=R =
d f0R 2
P~z
0.
(20)
Now multiplying eq. (3) by the axial velocity tL., integrating across the cross-section of the tube, using the continuity equation and the boundary conditions on t~r and t' z, we get the energy integral equation as
dp R foR( St'z" Ot'z dzfo t ' ' r d r %'+ ~-r ) - r d r = 0 ' 0 r
P dz.o d "
(21)
Eliminating pressure terms from eqs. (20) and (21) we get Ou z n
"2 dz' -
RV
Jo"v;r dr-pV-~z %+.
~
-r
--r0r
dr=0.
r=R
(22)
Substituting the velocity profiles (13) in (22) and evaluating the integrals, using eqs. (14) and (15) and carrying out the necessary differentiation, we obtain the nonlinear ordinary differential equation in the dimensionless boundary layer thickness m as given by
mn(T~ - 2T2) d m / d z * = 16( HnK,m n + K 2 ) ,
(23)
B. Das / Hershel- Bulkley fluid in a circular tube
45
where
A +2mB
3(c I + 2c2m)(1 +Am +Bm 2)
a3(m)
aa(m)
T1-
T2-
2(c I + 2c2m)(1 + A , m +B,m 2) a3(m)
A 1 + 2mB 1 a2(m)
B = 2[g(n) -f(n)]
A = 2 [ f ( n ) - 1], A, = 2[f~(n) - 1],
+ 1,
B 1 = 2[gl(n ) - f , ( n ) ]
+ 1,
n(18n 2 + 28n + 11) f(n) = 1
g(n)-
-
(4n + 3)(2n + 1)(3n + 2) '
1
n(27n 2 + 34n + 11)
2
(5n + 3)(3n + l ) ( 4 n + 2 ) '
f,(n)= l-
n(4n + 3) (3n+2)(2n+l)'
1
n+l ( n mn na(-n~) ( l - m ,) - n- ++ -l +l- 2 n
K,=I
(n+l)"
[
,,(5,, + 3)
gl(n)='2-
n+l
K 2 - n,a,+l(m ) a(m) - - -n
I
(4n + 2 ) ( 3 n + 1) '
)
(1 - m ) ~ -
n
mn
+ 3n + 1
)]
The above equation (23) has been integrated numerically by using Simpson's one-third rule to determine the values of dimensionless boundary layer thickness m for different values of n and /4,. Substituting these values of m in (15), we obtain the plug core velocity Uc* represented graphically in figs. 6, 7 and 8. The entrance length has been obtained considering the point at which the plug core velocity U~* approaches 99% of its fully developed value. Integrating eq. (20), the pressure drop expressed in the dimensionless form is given by p* = - 2 + 2 Q ( m ) - 1 6 / a0
H,, +
dz,
(24)
where
Q(m) =
1 +Aim + A z m 2 a2(m )
The integral in eq. (24) has been evaluated numerically by using Simpson's one-third rule for various values of yield number H, and for different values of n to obtain the pressure drop in the entrance region. Using the relations (18) and (24), we obtain the loss coefficient values for various values of Hershel-Bulkley number and power law index.
4. Results and discussions
The problem of entrance region flow of Hershel-Bulkley fluid in a circular tube has been analysed by momentum integral method and the effects of viscous dissipation in the boundary layer have been studied by using the momentum energy integral method. The entrance region
46
B. Das
Hershel
/
10
Bulkley )quid in a circular tube
-
0.0
0.5 /
-
f
o a
- - ~ - - - - -
/
0.5
// 2.0
E 0.5
~x
~
tum
-
- Momentum Parameter
0.0 0.0
n e e r g y Int. M e t h o d
Int. Method H n
0.05
0,06 z*
Fig. 3. Variation of boundary layer thickness along axial distance for n = 0.75.
flow characteristics have been found to be completely described by two parameters, the Hershel-Bulkley n u m b e r / 4 , and the power law index n. The relation between the H e r s h e l Bulkley number H,, and the fully developed plug core radius Hy is represented graphically in fig. 2. We find that H,, increases as H>. increases and it becomes infinitely large as H>. approaches unity. We also observe that, for a particular value of H>,, the value of H,, becomes more as the power law index n increases. The variation of boundary layer thickness m along the axial distance determined from the two methods is shown graphically in figs. 3, 4 and 5 respectively for different values of Hershel-Bulkley number lying between 0 and 5 and power law iDdex n = 0.75, 1.0 and 1.25. We observe that in both the cases the value of boundary layer thickness increases from a zero value at the inlet till it approaches the fully developed value. In the momentum integral method the increase is sharp whereas in the momentum energy integral method the increase is relatively slower and it approaches the fully developed value asymptotically. It has also been observed that, in case of momentum integral method, the boundary layer thickness increases as the Hershel-Bulkley number H,, increases, but the reverse trend is exhibited in case of momentum energy integral method except near the region close to the entry section. This means that viscous dissipation has significant effect on the development of boundary layer in the downstream region. We also find that in both methods the boundary layer thickness becomes more for larger values of power law index n.
1.0
'
ko-
0.5.~ ~
// //. .
2.0
/
'
o
.
o
~
0.5
~
/h
E
0.5 -....
Momentum
E n e r g y Int. M e t h o d
Momentum
Int. M e t h o d
Porometeran 0.0
i
0.0
,
,
0.03
,
I
0.08
Z W'
Fig. 4. Variation of boundary layer thickness along axial distance for n = 1.0.
B. Das / Hershel- Bulkley fluid in a circular tube
47
1.0 >~x/.
~¸
2.0 ~
E
0.5
~
z /
5.0
~
n
Fe gyInt.Method - Momenu tmInt.Method Parameter Hn
0.0 0.0
,
,
, 0.02
,
,
, 0.04
Z* Fig. 5. V a r i a t i o n
of boundary
l a y e r thickness a l o n g axial d i s t a n c e f o r n = 1.25.
0.0
1.8
0.0
."
0.5
0.5
//
1.4
~ ' / ~ e nn t ut mu
E
.....Int Int. Method m Energy Mnm#nJMrv~ln{* m#fhnr] - - Momentum Int. method
~ ~ e /I__ 1 0•~
-
-
parameter Hn 1.0 0.0
0.03
0.06 z*
F i g . 6. Variation of plug core velocity along axial distance for n = 0.75.
2.2 0.0
* u
2.0
1.6 ~
~_
M
. o ~
0.0
~ m
e
-
-
n
t
u
m
Energy Int. Method
Momentum Int. Method Parameter Hn
1.0 0.0
0.0.3
0.06
z* Fig. 7. V a r i a t i o n o f p l u g c o r e velocity along axial d i s t a n c e f o r n = 1.0.
Choosing the values of Hershel-Bulkley number H,, and the flow behaviour index n as mentioned above, the distribution of plug core velocity Uc* along the axial distance is shown graphically in figs. 6, 7, and 8 for both methods. Similar behaviour as of the boundary layer thickness has also been observed for plug core velocity with increasing values of H~ and n. The values of Uc* obtained from the momentum integral method reach the fully developed values faster and hence give smaller values of entrance length in comparison to the corre-
B. Das / Her~'hel- Bulkley f l u i d in a circular tube
48
2.2 0.0
0.0
*~
1.6
I
/ "
~
m
EnergyInt. Methoc - - MomentumInt. Method Porometer
1.0 0.0
Hn
0.02
0.04
z*
Fig. 8. Variation of plug core velocity along axial distance for n = 1.25.
0.1
.
.
.
.
.
.
.
_~.75 _
,
--
~
-
-
,
,
,
,
,
.
.
.
.
.
.
Momentum
E n e r g y Int. M e t h o d
Momentum
Int. M e t h o d
o.o5
!:.~r 1.o
0.0
0.25
0.5 Hy
0.75
1.0
Fig. 9. Entrance length of the Hershel Bulkley fluids as a function of plug core radius and power law index.
sponding values obtained from the energy integral method. The values of entrance length for different values of plug core radius Hy and power law index n are shown in fig. 9. Here the p a r a m e t e r H~. has been choosen for direct comparison with the results of Chen et al. (1970) for a Bingham fluid. The relation between the Hershel-Bulkley number H , and plug core radius H~, is depicted in fig. 2. In both cases, the entrance length decreases as Hy or the power law index increases. The value of entrance length for a Bingham fluid has been obtained by putting the value of n = 1 and is found to be in good agreement with the values obtained by Chert e t a l . (1970). The results obtained for a power-law fluid by putting H v = 0 are in qualitative agreement with those obtained by Bogue (1959) and Matras and Nowak (1983). Table 1 shows the comparison of entrance length values for some pseudoplastic fluids
Table 1 Comparison of entrance length values L,. for H,, = 0 with power law fluids. Power law
Present results
index n
M1 method
MEI method
Bogue
0.726 0.75 0.85 0.9 1.0
0.03112 [).03104 0.03042 0.02995 0.02877
0.09141 0.08911 0.07990 0.07555 0.06739
Matras and Nowak
0.03215 0.03202 0.03104 0.03036 0.02877
0.10040 0.09933 0.09450 0.09191 0.08650
49
B. Das / Hershel - Bulkley fluid in a circular tube
5.0 0.5
0.0 "
2.0 2.5
)
5.0
~
10.0
//~.~"
-
-
Momenturn
Int.
Method
Porometer H n
0.0 0.0
.
.
.
.
.
.
0.03
0.06
Z* Fig. 10. Pressure drop distribution along axial distance for n = 0.75.
with those of Bogue (1959) and Matras and Nowak (1983). Our results obtained by momentum integral method are close to the results of Bogue (1959), who has obtained results for a power law fluid by using the same method but with cubic velocity profile in the boundary layer. It proves the accuracy of the numerical code. However, the results obtained by Matras and Nowak (1983) are quite higher than our results obtained by both integral methods. For different values of Hershel-Bulkley number and power law index, the dimensionless pressure distribution along the axial distance has been shown in figs. 10, 11 and 12. From the figures we found that there is negligible effect of viscous dissipation on pressure drop. Similar results have been obtained for a Bingham fluid flow (Chen et al., 1970) and for the flow of a power law fluid (Tiu and Bhattacharya, 1973). Hence, if only the knowledge of pressure distribution in the entrance region is desired, the m o m e n t u m integral method is recommended, for the calculations involved will be less. We also find that the values of the pressure drop in the entrance region becomes larger with increasing values of H n and n. The results for special cases of Hn = 0 and n = 1 agree quite well with those of Chen et al. (1970) for a Bingham fluid and with Bogue (1959) and Matras and N o w a k (1983) for a power law fluid. The values of the loss coefficient are shown in fig. 13. We observe that the value of the loss coefficient decreases with increasing values of power law index. This type of behaviour is in qualitative agreement with all the previous works for power law fluid. The change in values with yield number is analogous to that in the case of entrance length. However, the value of the loss coefficient obtained with the m o m e n t u m energy integral method are appreciably higher than those obtained in the m o m e n t u m integral method. For power law index n = 1,
5.0
2.0 5.0 *
2.5
10.0 . Method ~ "
- .-
Momentum Int. Method Poremeter Hn
0.0 0.0
0.03
0.06
z
Fig. 11. Pressure drop distribution along axial distance for n = 1.0.
B. Das / Hersbel - Bulkley f l u i d in a circular tube
50
5.0
'
'
--
5.0
f
Porarneter Hn
0.0 0.0
.
.
.
.
. 0.02 Z*
. 0.04
Fig. 12. Pressure drop distribution along axial distance for n = 1.25.
1.4
~
5
1.2
--
Momentum Energy Int. Method
~ •
0.0
0.0
0.25
0.5
0.75
1.0
Hy F i g , 13. L o s s c o e f f i c i e n t o f t h e H e r s h e l - B u l k l e y
fluids as a function of plug core radius and power law index.
the values reduce to those for a Bingham fluid and are in good agreement with those obtained by Chen et al. (1970). Finally, we found that the solution by energy integral method provides a valid overall description of flow characteristics of the Hershel-Bulkley fluid in the entrance region of a circular tube. However, the momentum integral method provides a reasonable description of the fluid flow problem under the conditions where the effects of viscous dissipation are negligible. Furthermore, the experimental results for entrance region flow of a Hershel-Bulkley fluid in a tube are not available in literature. The only theoretical work available is that of Lin and Shah (1978), but they have not presented any results for the hydrodynamic entry length problem. Accordingly, the accuracy of the present work has been confirmed by comparing the entrance solutions for special cases of n = 1 and H n = 0 with those obtained by Chen et al. (1970) for a Bingham fluid and by Bogue (1959) and Matras and Nowak (1983) for a power law fluid.
Acknowledgement The author is grateful to the Chairman, Department of Mathematics and Statistics, University of New Mexico for his kind permission to avail the computer facility in the department.
B. Das / Hershel- Bulkleyfluid in a circulartube
51
Appendix The derivation of eq. (18) and eq. (19). In the paper by Al-Fariss and Pinder (1987), the expression for the total flow rate Q is given by ~nR2 Q-
(
,,.)1/,,
aR---~?
(n+l)a
(aR-aRo)
2vrn2R ('rv)'/" +c~2(n+ l l ( 2 n + l) aR---~? a3(n+l)(2n+
(aR-c~Ro) 2
o~R-
1)(3n+l)
(aR-o~Ro) 3,
(25)
where a
AP
Idp/dzl
2~?L
2~1
(26)
Here L is the length of the tube, Ap is the pressure difference and R 0 is the core radius in the fully developed region. Equation (25) can be simplified to the form
Q = - ~ r R B ( a R ) '/" 1
aR~
2n3( 1 - Ro/R) 2
1-
n+ l
( n + l)(2n+ l)
1
+ (n + 1)(2n + 1)(3n + 1) 1" Now by considering the equilibrium of shear and pressure forces in the fully developed region we get Trz = ~ r
Idp/dzl.
(28)
Since the yield stress % is the value of the shear stress at R = R o (the plug core radius), we get 1 ~'y = ~R o [dp/dz I
which gives
"ry/aR'rI = Ro/R = Hy,
(29)
where Hy is the dimensionless plug core radius in the fully developed region. Now we rewrite eq. (26) as
Q = -TrR3
( ' d p / d z l R ) '/n 2~1 K(Hy, n),
(30)
where
K(H>,
l'l )
=(1-Hv)'/"[
n
tn+l
2n2(1-Hy)
(n+l)(2n+ 1)
2n3( 1 -- H>')2
t
+ (n + 1)(2n + 1)(3n + 1) )"
B. Das / Hershel- Bulkley fluid in a circular tube
52
F r o m t h e e x p r e s s i o n (30) w e o b t a i n t h e a v e r a g e v e l o c i t y V as
V = ]Q / r r R 2 1 = R ( I d p / d z
l R/2~7) l/n g ( n y , n)
(31)
w h i c h can b e w r i t t e n as
Idp/dz Since Idp/dz
I= [V/K(Hy,
I = -dp/dz
-dp/dz=
n)R]~2~7/R.
(32)
( t h e flow is d u e to fall in p r e s s u r e ) , w e c a n w r i t e eq. (32) as
2 r l V ~ / R ~ + ~ K ~ (\ H
Y,
n).
(33)
I n t e g r a t i n g t h e a b o v e e q u a t i o n w i t h r e s p e c t to z f r o m z = 0 to any p o i n t z = zf, we g e t t h e d i m e n s i o n l e s s p r e s s u r e d r o p as ( P0 - p f ) / ~ p V 1
2 = 4rlV n 2 z / p R " + l K " ( H y ,
n).
(34)
By t h e d e f i n i t i o n o f R e y n o l d s n u m b e r R e a n d t h e d i m e n s i o n l e s s axial d i s t a n c e z * , t h e a b o v e r e l a t i o n c a n b e w r i t t e n as ( P 0 - p f ) / / ~ p1r
2 = 16z*/K"(Hy,
n).
(35)
Derivation o f the relation between Hy and 14,,. T h e H e r s h e l - B u l k l e y n u m b e r H~ is d e f i n e d by
H , = ~-yR"/rlV".
(36)
F r o m eqs. (26), (29) a n d (32), w e c a n w r i t e
H,, = r y / a R ' q = "cyR"K"( Hy, n ) / V % 7 .
(37)
F r o m e x p r e s s i o n s (36) a n d (37), w e get t h e r e l a t i o n b e t w e e n Hy a n d H n as
H, = Hy/Kn(n,,
n).
(38)
References AI-Fariss, T. and K.L. Pinder (1987) Flow through porous media of a shear-thinning liquid with yield stress, Can. J. Chem. Eng. 65, 391. Batra, R.L. and B. Jena (1987) Blood flow in the entrance region of a tube, Proc. Nat. Conf. FMFP India 15, 353. Batra, R.L. and A. Kandasamy (1990) Entrance flow of Hershel-Bulkley fluids in a duct, Fluid Dyn. Res. 6, 43. Bogue, D.C. (1959) Entrance effects and prediction of turbulence in non-Newtonian flow, Ind. Eng. Chem. 51, 874. Campbell, W.D. and J.C. Slattery (1963) Flow in the entrance of a tube, J. Basic Eng. ASME Trans. 85, 41. Chaturani, P. and R. Ponnalgarsamy (1988) A study of non-Newtonian aspects of blood flow through stenosed arteries and its applications in arterial diseases, Biorheology 22, 521. Chen, S.S., L.T. Fan and C.L. Hwang (1970) Entrance region flow of the Bingham fluid in a circular pipe, AIChE J. 16, 293. Convey, G.H. and B.R. Stanmore (1981) Use of the parallel-plate plastometer for the characterization of viscous fluids with a yield stress, J. Non-Newtonian Fluid Mech. 8, 249. Fan, L.T. and C.L. Hwang (1966) Hydrodynamic entrance region flow, Kansas State Unit,. Bull. 50, 3. Fargie D. and B.W. Martin (1971) Developing laminar flow in a pipe of circular crosssection, Proc. Roy. Soc. 321, 461. Greenwood, J.A. and J.J. Kauzlarich (1972) EHD lubrication with Hershel-Bulkley model greases, ASLE Trans. 4 , 269. Gupta, R.C. (1969) Flow of power law fluids in the entry region of a circular pipe, J. M~canique 8, 207. Gupta, R.C. (1984) Developing laminar power-law fluid flow in flat duct, J. Eng. Mech. ASCE Trans. 110, 752. Iida, N. and T. Murata (1980) Theoretical analysis of pulsatile blood flow in small vessels, Biorheology 17, 377. Kapur, J.N., B.S. Bhatt and N.C. Sacheti (1982) Non-Newtonian Fluid Flows (Pragati Prakasan Publ., Meerut, India). Lin, T. and V.L. Shah (1978) Numerical solution of heat transfer to yield-power law fluids flowing in the entrance region, Proc. VI Intern. Heat Transfer Conf. 5, 317.
B. Das / Hershel- Bulkley fluid in a circular tube
53
Matras, Z. and Z. Nowak (1983) Laminar entry length problem for power-law fluids, Acta Mechanica 48, 81. Schiller, L. (1922) Die Entwichklung der laminaren Geschwindigkeitsverteilung und ihre Bedeutung fiir Z~ihigkeitmessungen, Z A M M 2, 96. Scott-Blair, G.W. and D.C. Spanner (1974) An Introduction to Biorheology (Elsevier, Amsterdam). Tiu, C. and S. Bhattacharya (1973) Flow behaviour of power-law fluids in the entrance region of annuli, Can.J. Chem. Eng. 51, 47. Ward-Smith, A.J. (1980) Internal Fluid Flow (Clarendon Press, Oxford). Whorlow, R.W. (1980) Rheological Techniques (Wiley, New York) p. 33.