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Approximate analytical solution to flow of a power-law fluid in tapered ducts
243
The Chemical Engineering Journal, 22 (1981) 243 - 245 0 Elsevier Sequoia S.A., Lausanne - Printed in The Netherlands
Short Communication Approximate analytical solution to flow of a power-law fluid in tapered ducts
A. B. JARZEBSKI
and J. THULLIE
Polish Academy of Sciences, Institute of Chemical Engineering, 44-100 Gliwice, Baltycka 5 (Poland) (Received
metries are considered, viz. circular and plane-wall. The solution procedure is based on a perturbation concept and for a conical duct is.as follows.
10 November
1980)
Power-law flow in a conical duct The motion of a power-law fluid in a slightly tapered circular duct is assumed to be governed by the equations
au,
dp
1
An approximate analytical solution to flow of a power-law fluid in slightly tapered ducts is obtained using a perturbation method. Expressions derived for pressure drop allow for both viscous and inertial effects and hence provide more accurate predictions than the creeping flow solutions normally used.
pv,--$
Introduction Laminar converging flow of nonlinear inelastic fluids occurs in many industrial processes and in bioengineering. For this reason it has already been investigated by a number of researchers and a good survey of reported results is given in ref. 1’. A routine approach to the problem assumes purely radial flow, i.e. a velocity field given by the expressions v, = f(0)/r2 and v, = f(@/r for flows in conical ducts or tapered slits, respectively. If the effect of inertial forces on the motion is neglected, simple solutions of the Newtonian and power-law problem can be obtained [2,3]. However, as inertial effects grow rapidly downstream they may, at a certain distance from the inlet, become very pronounced and comparable with the viscous effects, . and could possibly even dominate the process. In such cases the existing analytical solutions prove irrelevant and the predictions they provide can be misleading. This is found to be so for markedly pseudoplastic fluids [ 4,5] . Since for everyday use analytical solutions are superior to numerical procedures, as they allow convenient analysis of the phenomena, the authors have endeavoured to provide approximate analytical solutions to flow in slightly tapered ducts. Two standard geo-
Introducing new dimensionless variables Vn, P and R, and putting
Q=27r
=-z--
7 0
rsine
7,/j sine)
%
ae
(1)
r2v, sine de
(2)
where 1 au,
Tre
n-1 1 au,
I I
=-lJo rae
Re=-
ran I V0l
l-”
(3)
-r ae
VOP
PO
and 8 for sine, eqn. (1) takes the form dP 1 Re a 2 aR (VR2)= Re+ dR (1- R)“+le
xi (I-ah In-l av, 1 ae
-8 ae
X
(4)
Initial and boundary conditions are P(R = 0) = 0,
v,(e=e,) av,/ae(e
= 0, =o)=o
(5) Following Lu’s proposal [5], Vn is assumed to be expressed by the following two-term asymptotic form: Vn = Vl + Re V2 + O(Re2)
(6)
After substituting for Vn in eqn. (4), neglecting terms of order Re and solving the resulting equation simultaneously with the appropriately transformed eqn. (2), the following expression for V, is readily obtained:
v, =
‘+“-[l-(_!J+l] (1 + n)(l - R)2
(7)
244
i.e. the well-known solution to the problem in the creeping flow approximation. Then substituting VI from eqn. (7) into eqn. (4) and omitting terms of order Re2 gives the governing equation of the form
$-&-(v12)=RedP+’ dR
a
av,
xae [I ae
+Re-
(1
ae
As the above equation is nonlinear in V2, it is linearized by neglecting the O(Re) term in the apparent viscosity. Solving the resulting linear equation gives ,:jI)..
3(1 + 3n)2 2(1 + 2n)(3 + 5n)
(9)
I
Power-law flow in a tapered slit The equation of motion is
au,
n-l 1 au,
=--porae I I --r ae
Proceeding as for the conical flow problem and solving eqn. (10) simultaneously with the appropriate integral continuity condition, one obtains the final equation P=
l 2Rene0
(““r[(l_‘,,2n nfIO
(1 + 2n)2 (12n2 + lln + 1) 2(1 + n)2 (2 + 3n)(3 + 4n)
-l]+
1
1
0.355
0.765
Numerical, Ve + 0
18
2.217
16.94
216.6
18 18
2.042 2.185
15.35 16.43
185.5 198.5
Numerical, V, z 0
125
0.414
3.312
42.10
Creep flow approx.
125
0.294
2.210
26.70
Eqn. (11)
125
0.438
3.294
39.78
TABLE 2
Calc. method
Eqn. (9)
(10)
where 1
R
Creep flow approx. Eqn. (11)
Numerical, Vo = 0 Creep flow approx.
au, dp 1 a?,0 PV,‘a, = -dr -; ae
Tre
Re
Pressure drop P predicted for conical flow by different calculation methods (00 = 5”, n = 0.44)
-I]+ -1
Gale. method
0.082
av2
3n Rzes,j~)‘[(l
Pressure drop P predicted for converging slit flow by different calculation methods (6, = 5”, n = 1)
X
--R)“‘le
(8)
‘=
TABLE 1
Discussion and concluding remarks In order to verify the expressions developed the pressure losses predicted by eqns. (9) and (11) have been compared with those obtained from numerical solutions of the problem following the algorithm given in refs. 4 and 5 and assuming parabolic initial velocity proflies. For Newtonian slit flow some of the results are given in Table 1, and for the pseudoplastic (n = 0.44) conical problem in Table 2.
R 0.101
0.301
0.600
12 12
1.778 1.346
8.18 5.49
47.6 20.6
12
1.791
8.25
51.8
Numerical, Ve # 0 Numerical, Ve = 0 Creep flow approx.
125 125 125
0.367 0.420 0.127
2.30 2.45 0.51
22.6 25.2 1.98
Eqn. (9)
125
0.566
3.17
33.5
Analysis of results obtained can be summarized as follows. The predictions of pressure drop given by eqns. (9) and (11) agree fairly well with the numerical results. This agreement is closer for Newtonian fluids and tapered slit geometry. The first observation can be seen as a direct result of linearization which relates to nonlinear fluids exclusively, while growth of inertial forces, slower in tapered slits than in conical ducts, accounts for the second observation. In general, the creeping flow solution, identical with the viscous terms in eqns. (9) and (ll), provides lower results for pressure drop, and this discrepancy grows together with R and the pseudoplasticity of fluids. As may be seen from Table 2, for n < 1 the effect of viscous forces can easily be outweighed by that of the inertial forces, thus accounting for the small susceptibility to nonisothermal effects in accelerated flows of pseudoplastic fluids 14951.
I (11)
(1- R)2 _- ’
Re
245
Nomenclature power-law index = (pO - p)/p V& dimensionless pressure ; drop = 1 - r/r,,, dimensionless coordinate R along the flow length of duct r0 Re Reynolds number mean inlet velocity (negative) vo = u, / V. , dimensionless velocity VR Greek symbols cone (wedge) half-angle $0 consistency index PO density P
Subscripts 0 refers to the inlet
References T. H. Forsyth, Polym.-Plast. Technol. Eng., 6 (1976) 101. W. L. Harrison, Proc. Cambridge Philos. Sot., 19 (1916) 307. S. Oka, Rheol. Acta, 12 (1973) 224. A. B. Jarzebski and W. L. Wilkinson, J. NonNewtonian Fluid Mech., in press. A. B. Jarzebski and W. L. Wilkinson, J. Non-Newtonian Fluid Mech., paper submitted for publication.
The Chemical Engineering Journal, 22 (1981) 243 - 245 0 Elsevier Sequoia S.A., Lausanne - Printed in The Netherlands
Short Communication Approximate analytical solution to flow of a power-law fluid in tapered ducts
A. B. JARZEBSKI
and J. THULLIE
Polish Academy of Sciences, Institute of Chemical Engineering, 44-100 Gliwice, Baltycka 5 (Poland) (Received
metries are considered, viz. circular and plane-wall. The solution procedure is based on a perturbation concept and for a conical duct is.as follows.
10 November
1980)
Power-law flow in a conical duct The motion of a power-law fluid in a slightly tapered circular duct is assumed to be governed by the equations
au,
dp
1
An approximate analytical solution to flow of a power-law fluid in slightly tapered ducts is obtained using a perturbation method. Expressions derived for pressure drop allow for both viscous and inertial effects and hence provide more accurate predictions than the creeping flow solutions normally used.
pv,--$
Introduction Laminar converging flow of nonlinear inelastic fluids occurs in many industrial processes and in bioengineering. For this reason it has already been investigated by a number of researchers and a good survey of reported results is given in ref. 1’. A routine approach to the problem assumes purely radial flow, i.e. a velocity field given by the expressions v, = f(0)/r2 and v, = f(@/r for flows in conical ducts or tapered slits, respectively. If the effect of inertial forces on the motion is neglected, simple solutions of the Newtonian and power-law problem can be obtained [2,3]. However, as inertial effects grow rapidly downstream they may, at a certain distance from the inlet, become very pronounced and comparable with the viscous effects, . and could possibly even dominate the process. In such cases the existing analytical solutions prove irrelevant and the predictions they provide can be misleading. This is found to be so for markedly pseudoplastic fluids [ 4,5] . Since for everyday use analytical solutions are superior to numerical procedures, as they allow convenient analysis of the phenomena, the authors have endeavoured to provide approximate analytical solutions to flow in slightly tapered ducts. Two standard geo-
Introducing new dimensionless variables Vn, P and R, and putting
Q=27r
=-z--
7 0
rsine
7,/j sine)
%
ae
(1)
r2v, sine de
(2)
where 1 au,
Tre
n-1 1 au,
I I
=-lJo rae
Re=-
ran I V0l
l-”
(3)
-r ae
VOP
PO
and 8 for sine, eqn. (1) takes the form dP 1 Re a 2 aR (VR2)= Re+ dR (1- R)“+le
xi (I-ah In-l av, 1 ae
-8 ae
X
(4)
Initial and boundary conditions are P(R = 0) = 0,
v,(e=e,) av,/ae(e
= 0, =o)=o
(5) Following Lu’s proposal [5], Vn is assumed to be expressed by the following two-term asymptotic form: Vn = Vl + Re V2 + O(Re2)
(6)
After substituting for Vn in eqn. (4), neglecting terms of order Re and solving the resulting equation simultaneously with the appropriately transformed eqn. (2), the following expression for V, is readily obtained:
v, =
‘+“-[l-(_!J+l] (1 + n)(l - R)2
(7)
244
i.e. the well-known solution to the problem in the creeping flow approximation. Then substituting VI from eqn. (7) into eqn. (4) and omitting terms of order Re2 gives the governing equation of the form
$-&-(v12)=RedP+’ dR
a
av,
xae [I ae
+Re-
(1
ae
As the above equation is nonlinear in V2, it is linearized by neglecting the O(Re) term in the apparent viscosity. Solving the resulting linear equation gives ,:jI)..
3(1 + 3n)2 2(1 + 2n)(3 + 5n)
(9)
I
Power-law flow in a tapered slit The equation of motion is
au,
n-l 1 au,
=--porae I I --r ae
Proceeding as for the conical flow problem and solving eqn. (10) simultaneously with the appropriate integral continuity condition, one obtains the final equation P=
l 2Rene0
(““r[(l_‘,,2n nfIO
(1 + 2n)2 (12n2 + lln + 1) 2(1 + n)2 (2 + 3n)(3 + 4n)
-l]+
1
1
0.355
0.765
Numerical, Ve + 0
18
2.217
16.94
216.6
18 18
2.042 2.185
15.35 16.43
185.5 198.5
Numerical, V, z 0
125
0.414
3.312
42.10
Creep flow approx.
125
0.294
2.210
26.70
Eqn. (11)
125
0.438
3.294
39.78
TABLE 2
Calc. method
Eqn. (9)
(10)
where 1
R
Creep flow approx. Eqn. (11)
Numerical, Vo = 0 Creep flow approx.
au, dp 1 a?,0 PV,‘a, = -dr -; ae
Tre
Re
Pressure drop P predicted for conical flow by different calculation methods (00 = 5”, n = 0.44)
-I]+ -1
Gale. method
0.082
av2
3n Rzes,j~)‘[(l
Pressure drop P predicted for converging slit flow by different calculation methods (6, = 5”, n = 1)
X
--R)“‘le
(8)
‘=
TABLE 1
Discussion and concluding remarks In order to verify the expressions developed the pressure losses predicted by eqns. (9) and (11) have been compared with those obtained from numerical solutions of the problem following the algorithm given in refs. 4 and 5 and assuming parabolic initial velocity proflies. For Newtonian slit flow some of the results are given in Table 1, and for the pseudoplastic (n = 0.44) conical problem in Table 2.
R 0.101
0.301
0.600
12 12
1.778 1.346
8.18 5.49
47.6 20.6
12
1.791
8.25
51.8
Numerical, Ve # 0 Numerical, Ve = 0 Creep flow approx.
125 125 125
0.367 0.420 0.127
2.30 2.45 0.51
22.6 25.2 1.98
Eqn. (9)
125
0.566
3.17
33.5
Analysis of results obtained can be summarized as follows. The predictions of pressure drop given by eqns. (9) and (11) agree fairly well with the numerical results. This agreement is closer for Newtonian fluids and tapered slit geometry. The first observation can be seen as a direct result of linearization which relates to nonlinear fluids exclusively, while growth of inertial forces, slower in tapered slits than in conical ducts, accounts for the second observation. In general, the creeping flow solution, identical with the viscous terms in eqns. (9) and (ll), provides lower results for pressure drop, and this discrepancy grows together with R and the pseudoplasticity of fluids. As may be seen from Table 2, for n < 1 the effect of viscous forces can easily be outweighed by that of the inertial forces, thus accounting for the small susceptibility to nonisothermal effects in accelerated flows of pseudoplastic fluids 14951.
I (11)
(1- R)2 _- ’
Re
245
Nomenclature power-law index = (pO - p)/p V& dimensionless pressure ; drop = 1 - r/r,,, dimensionless coordinate R along the flow length of duct r0 Re Reynolds number mean inlet velocity (negative) vo = u, / V. , dimensionless velocity VR Greek symbols cone (wedge) half-angle $0 consistency index PO density P
Subscripts 0 refers to the inlet
References T. H. Forsyth, Polym.-Plast. Technol. Eng., 6 (1976) 101. W. L. Harrison, Proc. Cambridge Philos. Sot., 19 (1916) 307. S. Oka, Rheol. Acta, 12 (1973) 224. A. B. Jarzebski and W. L. Wilkinson, J. NonNewtonian Fluid Mech., in press. A. B. Jarzebski and W. L. Wilkinson, J. Non-Newtonian Fluid Mech., paper submitted for publication.