- Email: [email protected]
Dispersion in non-Newtonian laminar flow through a tube
Chemicd EngineeringScience,1976,Vol. 31, pp. 241-242. PergamonPress. Printed in Great Britain
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Dispersion in non-Newtonian laminar flow through a tube (Received 25 September 1974, accepted 3 July 1975)
In 1%5, Fan and Hwang[l] extended Taylor’s[f, 31 analysis of dispersion for Newtonian fluids to include the non-Newtonian Ostwald-de Waele (Power Law) model fluid. FrenchM has presented the numerical solution of the complete convective diffusion equation for the case of the Power Law model fluid. At present, few experimental studies are available to validate the dispersion results predicted from these analyses. In the present study, the problem of longitudinal dispersion of a solute in laminar flow of a Power Law non-Newtonian fluid has been studied experimentally in two different size capillaries. A wide range of parameters has been investigated and the range of applicability of various theoretical solutions is discussed.
values of the Power Law constants n = 0.981and m = 0.175dynes sec”/cm*. The results from the displacement experiments were plotted on probability paper as concentration versus the length of the tube. The length of the mixing zone was determined from these lines. Using the value of mixingzone length and corresponding flow time, the dispersion and diffusion coefficients were calculated. The complete tabulations of data for the 1 and 2 mm tubes can be found in[S]. The experimental data were converted into dimensionless mixing zone length L’ and dimensionless time 7. The present results were compared with French’s numerical solution together with the Taylor-Fan approximate solution in Fig. 1. Experimental results agree within +2% with the Taylor-Fan approximate solution for the r range of 0.7 I 7 5 4.0. This was to be expected as the numerical solution predicts that the Taylor-Fan approximate solution is valid for 7 20.7.
ANALYSIS OF DATA In order to use the present experimental data and the Taylor-Fan approximate solution, it is necessary to introduce the dimensionless average concentration Cm*‘, where,
I.0
cmz+=l-Cm’.
(1)
Introducing (1) into the Fan and Hwang’s[l] solution of the convective diiusion equation with a step input is Cm*+=03
[
l+erf$&j
1
Numerical
-
where the dispersion coefficient k is given by
k=R2? -. D
n2
o 2
1
mm tube
[ 2(3n + l)(Sn + 1)
o~ooolt‘,‘,‘,I’0001
If L is the length of the mixing zone in which concentration of solute changes from 10 to 90% then from (2) it follows that
‘,.,,.
‘,,‘I’,’ 001
01
‘,,,,,‘I I.0
‘,‘.AIO
T = Dt/ff2
L = 3.62d(kr)
or k = L2/13.1t.
(4)
The value of the diffusion coefficient, D can be determined from (3). EXPERIMENTAL Dispersion experiments were carried out with DuPont Pontamine Sky Blue 6Bx dye dissolved in a non-Newtonian fluid (1.0% Polyox-a polyethylene oxidd resin) by displacing it with pure 1% polyox solution in capillary tubes. The dye concentration along the tube length was determined by a Cary 14 spectrophotometer. Measurements were carried out after displacing liquid from the capillary tube by a syringe pump. Precise diameters of the capillary tubes used were 1.040and 2.106mm and the length was approx. 103cm. The concentration of dye solution used with the 1.0mm tube was 200ppm and with the 2.0 mm tube, 100ppm was used. The calibration of the spectrophotometer was based on a reading of zero absorption for the pure 1.0% Polyox solution. A linear relationship was observed between the absorbance and various concentrations. The concentration was measured at 5 cm intervals along the tube. All measurements were done at a temperature of 23 + O.S”C. RESULTSANDDISCUSSION It was found that the 1.0% Polyox solution obeyed the Power Law model over a wide range of shear rate (4-300 see-‘) with the
Fig. 1. Plot of dimensionless mixing zone length vs dimensionless time for Dye-Polyox system. Experimental measurements with 1 mm tube are shown in Fig. 2 for various values of Npe and 7.Results of this study are compared with the numerical solution as well as the Taylor-Fan approximate solution. For higher value of 7, i.e. 7 > 0.7 at which the effect of radial diffusion becomes very important, and the curve approaches an error function. This is evident from linearity of the curve on the probability coordinates at 7 = 3.2702.The effect of axial and radial diffusion is negligible compared to that of pure convection for shorter times of flow. This is clear from the curvature of the plot at 7 = 0.01021.Also, for short times, the experimental results agree more closely with the numerical solution than with the approximate solution. Results obtained from the 2 mm tube are plotted in Fig. 3. As mentioned earlier, as time increases, for high Npe the plot of concentration distribution tends toward an error function and the plots become straight lines on probability paper. This is clear from Fig. 3 for 7 = 3.3066.For short times, curve results as can be seen for 7 = 0@05512and 7 = 0.33066.The results of the present study are thus in good agreement with those predicted from theory. Ananthkrishnan et a/.[61 concluded that the effect of axial diffusion becomes significantfor Npe < 500.However, this effect is important only for very short times of flow at Npe > 100.From Fig. 3 it may be noted that the experimental results at Npe = 194and
241
Shorter Communications
242
----- Taylor-
9994 99.9
CONCLUSIONS
The following conclusions are made as a result of the analysis of dispersion measurements. (1) Wide ranges of dimensionless tune (r = [email protected]) and Peclet number Npe = 200-475 x 101) were investigated. The experimental results showed agreement within +5% of the numerical solution of the convective diiusion equation as well ai with Taylor-Fan approximate solution for r >0*7 and hi values of Npe. (2) At high values of Npe (Npe > 59038)for a fixed r = 0.01, it was found that there is no effect of increasing Npe on the concentration distribution of solute. (3) The average value of D for the Dye -1% Polyox system was found to be 3.07x lo-’ cm2/sec over the concentration range 0 I C c O-2g/l and 5.09x lo-’ cm’lsec for 0 I; C 5 O-1gll. Thus the diffusion coefficient was concentration dependent decreasing with increasing concentration. (4) Tube size did not atTectthe results of the experiment in the range investigated.
Fm
-b&lWkZd A,0
998 995 99
T* 32702
Thls wak, n = 0,981
t mm
tube
A@=653
96
Acknowledgement-This work has been partly supported by the Research Allocations Committee and the Bureau of Engineering Research, The University of New Mexico.
0
05
I.0
I.5
S. N. SHAH K. E. COX
2.0
X%
-----
Fig. 2. Plots of concentration vs X’/T for various values of r and Npe (1 mm tube). 99.99
1
99.95 99.9 99-6
-
Taylor-Fan tb-rwicul
A.O.P
TMs work,
2mm
tubi
99.5 99 98 +d
s
g5 90
5
!!
80 60 70 50
f 0
Departmentof Chemical and Nuclear Engineering Universityof New Mexico Albuquerque NM 87131, U.S.A.
_
Npe-166215
40 30 20 IO 5
NOTATION Cm average concentration C’o initial concentration dimensionless average concentration, Cm+ = Cm /Co ;;I l-Cm+ D molecular diffusion coefficient k Taylor’s dispersion coefficient L axial length of the tube in which dispersion effects are appreciable L’ dimensionless mixing zone length Power Law fluid model parameters ?L Peclet number, dimensionless, Npe = 2R. vx/D R radius of the tube t time i average time, i = L I Vx VX mean axial velocity X axial coordinate xl convected axial coordinate X dimensionless axial coordinate, X’ = .rD/2R’vx
Greek symbols T dimensionless time, T = D/R’ REFERENCES 0.01
’
0
’
’
’
’
04
02
0.3
0.4
’
’
05
06
@7
0.6
0.9
X%
Fig. 3. Concentration vs X’/T plots for various values of 7 and Npe (2 mm tube). I = 3.3066show close agreement with the Taylor-Fan approximate solution. This indicates that the effect of axial diifusion is negliibly small, which is in agreement with the conclusion stated above.
[l] FanL.T.andHwangW.S.,Proc.Roy.Soc.l%5A28J576. [2] Taylor G. I., Proc. Roy. Sot. 1953A219 186. [3] Taylor G. I., Proc. Roy. Sot. 1954A225 473. [4] French C. hf., Ph.D. Dissertation, Kansas State University, Manhattan, Kansas Ml. [S] Shah S. N., Ph.D. Dissertation, University of New Mexico, Albuquerque, New Mexico 1974. [6] Anantbakrishnan V., GillW. N. and Bardohn A. J., A.1.Ch.E.I. 1%5 11 1063.
..... .......................................................................................... .
Dispersion in non-Newtonian laminar flow through a tube (Received 25 September 1974, accepted 3 July 1975)
In 1%5, Fan and Hwang[l] extended Taylor’s[f, 31 analysis of dispersion for Newtonian fluids to include the non-Newtonian Ostwald-de Waele (Power Law) model fluid. FrenchM has presented the numerical solution of the complete convective diffusion equation for the case of the Power Law model fluid. At present, few experimental studies are available to validate the dispersion results predicted from these analyses. In the present study, the problem of longitudinal dispersion of a solute in laminar flow of a Power Law non-Newtonian fluid has been studied experimentally in two different size capillaries. A wide range of parameters has been investigated and the range of applicability of various theoretical solutions is discussed.
values of the Power Law constants n = 0.981and m = 0.175dynes sec”/cm*. The results from the displacement experiments were plotted on probability paper as concentration versus the length of the tube. The length of the mixing zone was determined from these lines. Using the value of mixingzone length and corresponding flow time, the dispersion and diffusion coefficients were calculated. The complete tabulations of data for the 1 and 2 mm tubes can be found in[S]. The experimental data were converted into dimensionless mixing zone length L’ and dimensionless time 7. The present results were compared with French’s numerical solution together with the Taylor-Fan approximate solution in Fig. 1. Experimental results agree within +2% with the Taylor-Fan approximate solution for the r range of 0.7 I 7 5 4.0. This was to be expected as the numerical solution predicts that the Taylor-Fan approximate solution is valid for 7 20.7.
ANALYSIS OF DATA In order to use the present experimental data and the Taylor-Fan approximate solution, it is necessary to introduce the dimensionless average concentration Cm*‘, where,
I.0
cmz+=l-Cm’.
(1)
Introducing (1) into the Fan and Hwang’s[l] solution of the convective diiusion equation with a step input is Cm*+=03
[
l+erf$&j
1
Numerical
-
where the dispersion coefficient k is given by
k=R2? -. D
n2
o 2
1
mm tube
[ 2(3n + l)(Sn + 1)
o~ooolt‘,‘,‘,I’0001
If L is the length of the mixing zone in which concentration of solute changes from 10 to 90% then from (2) it follows that
‘,.,,.
‘,,‘I’,’ 001
01
‘,,,,,‘I I.0
‘,‘.AIO
T = Dt/ff2
L = 3.62d(kr)
or k = L2/13.1t.
(4)
The value of the diffusion coefficient, D can be determined from (3). EXPERIMENTAL Dispersion experiments were carried out with DuPont Pontamine Sky Blue 6Bx dye dissolved in a non-Newtonian fluid (1.0% Polyox-a polyethylene oxidd resin) by displacing it with pure 1% polyox solution in capillary tubes. The dye concentration along the tube length was determined by a Cary 14 spectrophotometer. Measurements were carried out after displacing liquid from the capillary tube by a syringe pump. Precise diameters of the capillary tubes used were 1.040and 2.106mm and the length was approx. 103cm. The concentration of dye solution used with the 1.0mm tube was 200ppm and with the 2.0 mm tube, 100ppm was used. The calibration of the spectrophotometer was based on a reading of zero absorption for the pure 1.0% Polyox solution. A linear relationship was observed between the absorbance and various concentrations. The concentration was measured at 5 cm intervals along the tube. All measurements were done at a temperature of 23 + O.S”C. RESULTSANDDISCUSSION It was found that the 1.0% Polyox solution obeyed the Power Law model over a wide range of shear rate (4-300 see-‘) with the
Fig. 1. Plot of dimensionless mixing zone length vs dimensionless time for Dye-Polyox system. Experimental measurements with 1 mm tube are shown in Fig. 2 for various values of Npe and 7.Results of this study are compared with the numerical solution as well as the Taylor-Fan approximate solution. For higher value of 7, i.e. 7 > 0.7 at which the effect of radial diffusion becomes very important, and the curve approaches an error function. This is evident from linearity of the curve on the probability coordinates at 7 = 3.2702.The effect of axial and radial diffusion is negligible compared to that of pure convection for shorter times of flow. This is clear from the curvature of the plot at 7 = 0.01021.Also, for short times, the experimental results agree more closely with the numerical solution than with the approximate solution. Results obtained from the 2 mm tube are plotted in Fig. 3. As mentioned earlier, as time increases, for high Npe the plot of concentration distribution tends toward an error function and the plots become straight lines on probability paper. This is clear from Fig. 3 for 7 = 3.3066.For short times, curve results as can be seen for 7 = 0@05512and 7 = 0.33066.The results of the present study are thus in good agreement with those predicted from theory. Ananthkrishnan et a/.[61 concluded that the effect of axial diffusion becomes significantfor Npe < 500.However, this effect is important only for very short times of flow at Npe > 100.From Fig. 3 it may be noted that the experimental results at Npe = 194and
241
Shorter Communications
242
----- Taylor-
9994 99.9
CONCLUSIONS
The following conclusions are made as a result of the analysis of dispersion measurements. (1) Wide ranges of dimensionless tune (r = [email protected]) and Peclet number Npe = 200-475 x 101) were investigated. The experimental results showed agreement within +5% of the numerical solution of the convective diiusion equation as well ai with Taylor-Fan approximate solution for r >0*7 and hi values of Npe. (2) At high values of Npe (Npe > 59038)for a fixed r = 0.01, it was found that there is no effect of increasing Npe on the concentration distribution of solute. (3) The average value of D for the Dye -1% Polyox system was found to be 3.07x lo-’ cm2/sec over the concentration range 0 I C c O-2g/l and 5.09x lo-’ cm’lsec for 0 I; C 5 O-1gll. Thus the diffusion coefficient was concentration dependent decreasing with increasing concentration. (4) Tube size did not atTectthe results of the experiment in the range investigated.
Fm
-b&lWkZd A,0
998 995 99
T* 32702
Thls wak, n = 0,981
t mm
tube
A@=653
96
Acknowledgement-This work has been partly supported by the Research Allocations Committee and the Bureau of Engineering Research, The University of New Mexico.
0
05
I.0
I.5
S. N. SHAH K. E. COX
2.0
X%
-----
Fig. 2. Plots of concentration vs X’/T for various values of r and Npe (1 mm tube). 99.99
1
99.95 99.9 99-6
-
Taylor-Fan tb-rwicul
A.O.P
TMs work,
2mm
tubi
99.5 99 98 +d
s
g5 90
5
!!
80 60 70 50
f 0
Departmentof Chemical and Nuclear Engineering Universityof New Mexico Albuquerque NM 87131, U.S.A.
_
Npe-166215
40 30 20 IO 5
NOTATION Cm average concentration C’o initial concentration dimensionless average concentration, Cm+ = Cm /Co ;;I l-Cm+ D molecular diffusion coefficient k Taylor’s dispersion coefficient L axial length of the tube in which dispersion effects are appreciable L’ dimensionless mixing zone length Power Law fluid model parameters ?L Peclet number, dimensionless, Npe = 2R. vx/D R radius of the tube t time i average time, i = L I Vx VX mean axial velocity X axial coordinate xl convected axial coordinate X dimensionless axial coordinate, X’ = .rD/2R’vx
Greek symbols T dimensionless time, T = D/R’ REFERENCES 0.01
’
0
’
’
’
’
04
02
0.3
0.4
’
’
05
06
@7
0.6
0.9
X%
Fig. 3. Concentration vs X’/T plots for various values of 7 and Npe (2 mm tube). I = 3.3066show close agreement with the Taylor-Fan approximate solution. This indicates that the effect of axial diifusion is negliibly small, which is in agreement with the conclusion stated above.
[l] FanL.T.andHwangW.S.,Proc.Roy.Soc.l%5A28J576. [2] Taylor G. I., Proc. Roy. Sot. 1953A219 186. [3] Taylor G. I., Proc. Roy. Sot. 1954A225 473. [4] French C. hf., Ph.D. Dissertation, Kansas State University, Manhattan, Kansas Ml. [S] Shah S. N., Ph.D. Dissertation, University of New Mexico, Albuquerque, New Mexico 1974. [6] Anantbakrishnan V., GillW. N. and Bardohn A. J., A.1.Ch.E.I. 1%5 11 1063.