Rheologica Acta
Rheol. Acta 21, 725
-
729 (1982)
Diffusion of macromolecular solutions in the turbulent boundary layer of a cylindrical pipe. III. Development of parametric models M. Bues, C. Gebel, and H. Reitzer Institut de M6canique des Fluides, Universit6 Louis Pasteur and Ecole Nationale Supbrieure des Arts et Industries, Strasbourg Abstract: This publication completes the model of the evolution of the transversal concentration profile proposed in the last paper, with a development of parametric models concerning the length of diffusion plume and the characteristic exponent (R --- 40000). Key words: Turbulent diffusion, macromolecular solution, pipe flow, concentration profile, diffusion plume
Notation C
m
c,
-
X
y 2 H
R D a,b F,G
-
(ii) " F a r f r o m the sink": concentration wall concentration initial concentration before injection downstream distance from the slot normal distance from the wall characteristic height of diffusion, i.e. the value of 9 at which C/Cw = 0.5 characteristic exponent radius of pipe diameter of pipe constants annex functions
1. Introduction In the previous publication [1] we introduced the parameters 2 (characteristic length of diffusion plume) and n (characteristic exponent). For completely defining the model of transversal diffusion, it is necessary to develop a parametric model concerning these two parameters. Recall the earlier relationships which define 2 and H.
(i) " N e a r the sink": C / C w = exp [ - 0.693 ( y / 2 ) n] , 854
(1)
C / C w = a / e x p [ - 0.693 ( y / 2 ) n] t
+ exp
-0.693
,
(2)
l / a = 1 + e x p [ - 0 . 6 9 3 ( D / 2 ) n] . The physical analysis will be carried out only for the characteristic length of the diffusion plume 2. A n identical a p p r o a c h would have allowed us to show that the exponent n comes under the same influences as 2.
2. Physical analysis The presence o f polymer hinders the " n a t u r a l " development o f diffusion p h e n o m e n a . The magnitude o f viscoelastic forces explains this "wall adhesion". Experimentally, we have found: (i) for Ci constant: 2 increases with the abscissa x. (ii) for x constant: 2 increases as the initial concentration C i decreases.
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Rheologica Acta, Vol. 21, No. 6 (1982)
We wish to find a variable q which satisfies the two conditions. Another manner in which to express these conditions is: 2 varies in the same direction when x increases and when Ci decreases. The variable including these two effects may is the wall concentration C w. Indeed: (i) for Ci constant: Cw decreases when the abscissa x increases, (ii) for x c o n s t a n t : Cw decreases when the initial concentration Ci decreases. With the object of generalizing our model, it is more interesting to take as the variable q the reduced wall concentration C w / C i which varies as Cw. Thus it is evident that 2
=
(3)
f(Cw/Ci).
Similarly we can write:
We notice that the injection flow rate and the total flow rate enter the relationship (3) and (4) as boundary values of C w / C i [2, 3].
3. Complementary assumptions The quantitative development of models (3) and (4) requires the estimation of boundary conditions. The knowledge of 2 and n values for the abscissas x = 0 and x = L m (mixing length) is necessary. 3.1. B o u n d a r y values o f 2
For x = 0, to the right of the injector, the diffusion boundary layer is not developed. It seems reasonable to say that for ;t = 0 the reduced wall concentration takes the value 1. For x = Z m, the diffusion process takes place in the final zone [2]. We have noticed before that the diffusion boundary layer coincides with the hydrodynamic boundary layer. The latter has as thickness the radius R in cylindrical pipe flow. Consequently, it appears that the characteristic height 2, which cannot in any case be larger than the thickness of the diffusion boundary layer, will have R as a limit. A discussion of the validity of this assumption follows. The boundary conditions concerning the characteristic height )t are: for
For x = Lm, we shall assume the existence of a boundary value. Let n o be this upper limit. If we were to assume, for x = 0, that n takes the value zero, the reduced concentration at the middle of the pipe would be equal to 0.5 at the slot, eq. (1). However, this would be an incorrect assumption. It will be evident that the lower boundary of n is not zero. The boundary conditions are written: n/no= a
for
C w / C i = 1,
(6) n/n o = 1
for
Cw/Ci = Qi/Qt .
where a is defined as the minimal value of ratio n / n o . The two parameters 2 and n are function of the single variable C w / C i . It would seem that a correlation of type n / n o = h ( 2 / R ) should exist.
(4)
n = g(Cw/Ci).
2/R = 0
3.2. B o u n d a r y values o f n
4. Development of models of parameters 2 and n. Experimental results 4.1. M o d e l 2 = f ( C w / C i )
In deciding upon the mathematic form of this function, the analysis of primary results suggested that an exponential relationship would be suitable. The introduction of boundary conditions allows us to write. C w / C i = exp [In Q i / Q t ( ) t / R ) ml ] .
(7)
Figure 1 represents the variation of A / R versus ( C w / C i - Q i / Q t ) / ( l - Q i / Q t ) . The exponent m~ has been calculated by a method of least squares and is equal to 0.184. The dashed and full lines appearing in figure 1 are the model curves with flow ratios Q i / Q t = 2 • 10 -3 and 12 • 10 -3, respectively.
4.2. M o d e l n = 9 ( C w / C i )
Assume that the minimal value of ratio n / n o is zero. The analysis of primary results shows that an exponential f o r m is suitable. In this case: Im21. /
(8)
m = m l / m 2.
(9)
n/no= exp(lnOi/Qt(n \no~
Equating (7) and (8) gives
C w / C i = 1,
(5) 2/R = 1
for
Cw/C i = Qi/Qt .
n/n o= ()t/R) m
with
Bues et al., Diffusion o f macromolecular solutions in the turbulent boundary layer of a cylindrical pipe. III
727
loI
~
0.5
•
\.
.
0.0 0.00
o.o5 (Cw/Ci
-
Qi/rat)/(1
" il -
rai/rat)
Fig. 1. Variation of characteristic height o f diffusion/tube radius versus ( C w / C i = Q i / Q t ) / ( 1 - Qi/Qt). Initial concentration: [] 0, v 100, o 200, • 400, • 800, • 1200wppm. - - Q i / Q t = 2 . 1 0 -3 , Q i / Q t = 1 2 . 1 0 - 3 ; eq. (7)
10 °
lO-1
a
I0 -2
i
i
i
i
i
i
I
IO-T
I
i
I
i
i
i
i
i
i
i0 o
h/R
Fig. 2. Variation of characteristic exponent n versus characteristic height of diffusion/tube radius. Initial concentration: [] 0, A 100, © 200, • 400, • 800, • 1200 wppm; - eq. (9)
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Rheologica Acta, Vol. 21, No. 6 (1982)
Figure 2 represents this relation with m = 0.488 and n o = 2.36. The use of the relationship (8) shows a good agreement between the two methods for determining n 0. A variation of less than 4 per cent has been obtained.
Consequently, the maximal difference between the theoretical and calculated profiles is less than 3 per cent. This shows that the assumption was justified. For defining model (9), we have supposed that the minimum value of n / n o is zero. This assumption is not correct because C ( R ) / C w to the right of the injector is not zero but equal to 0.5. For A / R approximately equal to 0.7, C ( R ) / C w is below 0.5. The form of (2) makes us sure of the existence of no discontinuity. Consequently, we can use the Rolle's theorem which shows that the derivative of C / C w per 2 / R is zero for a certain value. So we obtain the condition:
5. Discussion of the assumptions One way of checking the assumption about the maximal values of 2 is to calculate the concentration profile corresponding to x = L m . Theoretically, a flat profile must be obtained. In practice, we have computed C ( R ) / C w and have found a value of 0.972.
0.4981nR/R - 1 = 0 ,
i.e.)t/R = 0.13.
For values of 2 / R above 0.13, model (9) may be utilized. For these values we have chosen a linear variation for n / n o . The correlation has given:
25
n / n o = 1.32 ; t / R + 0.20.
(10)
The minimum value of n / n o is 0.20 giving n = 0.46. Experimentally figure 3 allows to verify the validity of this value.
20
6. Final model and conclusions 15
10
The investigation of one model that can be applied near and far from the sink has suggested an assumption to generalize the model for two sinks. It has been shown that this model is acceptable independently of the position of the profile with regard to the injector. In conclusion, this study has shown that the evolution of the transversal concentration profile is governed by one equation of the type:
~~" Oo
0
0.0
+exo(0693 -- t /1
,
,
0.2
0.4
[]
~A o o
0.6
~ •
o
~ 0.8
I
l/a=
1 + exp
-0.693
1.0
C/Cw
Fig. 3. Profile of reduced concentration C/C~. V Ci = 400 wppm, x = 285 mrn, Qi 16.67 cm3/s, • Ci = 800wpprn, x = 615 mm, Qi = 11.11 cm3/s, • Qi = 13.89 crn3/s, [] Ci = 1200wppm, x = 615mm, Qi = 16.67cm3/s, • x = 825 ram, Qi = 16.67 cm3/s =
A physical approach has shown that a correlation exists between the two parameters 2 and n. This point of view has not previously been mentioned. Independent of this result, each one of these parameters is influenced by the wall polymer concentration. Moreover, the limits of 2 and n variations have been
Bues et al., Diffusion of macromolecular solutions in the turbulent boundary layer of a cylindrical pipe. III found by physical considerations corresponding to the studied flow, as well as by the verification of the postulated assumptions.
References 1. Gebel, C., M. Bues, H. Reitzer, Rheol. Acta 21, 725 - 729 (1982). 2. Bues, M., Diffusion de fluide non-newtonian en 6coulement interne. Propositions de mod61es d'6volution des concentrations. Th~se de Docteur Ing6nieur, ULP, Strasbourg (1981). 3. Poreh, M., J. E. Cermak, Intern. J. Heat and Mass Transfer 7, 1083- 1095 (1964). 4. Bhowmick, S. K. et al., Rheol. Acta 14, 1026-1031 (1975).
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5. Gebel, C. et al., Rheol. Acta 17, 172- 175 (1978). 6. Reitzer, H. et al., Rheol. Acta 20, 35-43 (1981). (Received December 21, 1981) Authors' addresses: Dr. M. Bues Ecole Nationale Supbrieure des Arts et Industries ~24 Bd. de la Victoire F-67000 Strasbourg Dr. C. Gebel, Dr. H. Reitzer Institut de M6canique des Fluides Universit~ Louis Pasteur 2, rue Boussingault F-67083 Strasbourg Cedex