Rhaologiea Acta

Rheol. Acta 21,720 - 724 (1982)

Diffusion of macromolecular solutions in the turbulent boundary layer of a cylindrical pipe. II. Model of evolution of transversal concentration profile C. Gebel, M. Bues, and H. Reitzer Institut de M~canique des Fluides, Universit~ Louis Pasteur and Ecole Nationale Sup~rieure des Arts et Industries, Strasbourg This paper presents a model for the evolution of a transversal concentration profile of a macromolecular solution (PEO) injected into a cylindrical pipe at turbulent flow conditions (R ~ 40000). This model, based on the diffusion of a scalar quantity emitted by two diametrically opposed point sinks, proves to be in good agreement with the experimental data.

Abstract:

Turbulent diffusion, macromolecular solution, pipe flow, concentration profile

K e y words:

Notation C

x

y 2 n

R D a,b

-

Lm

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 g at which C / C w = 0.5 characteristic exponent radius of pipe diameter of pipe constants mixing length, i.e. the value of x at which C w / C i = Qi/QT

Qi

Qr f,g no

-

flow rate of injection flow rate annex functions maximal value of n

I. Introduction

This publication is an extension of earlier work concerning the evolution of concentration profiles of macromolecules in a turbulent boundary layer of a cylindrical pipe flow [ 1 - 3 ] . A description of the experimental apparatus and analysis techniques can be found in the above-mentioned references. In twodimensional flow, the evolution of transversal concentration profiles C / C w is in good 853

agreement with theoretical profiles of the Morkovin type [7]. It would be interesting to consider a possible analogy between the profiles for twodimensional flow and the profiles for axisymmetric flow in a cylindrical pipe according to some hypotheses. It is obvious that this analogy is justified if it is considered f r o m the point of view of a diffused scalar quantity. 2. Starting point

Consider a cylindrical pipe of diameter D with the axis O x corresponding to the main flow direction. Suppose that the diffusion of a scalar quantity, emitted by two diametrically opposite point sinks, moves symmetrically between two near planes, in accordance with figure 1. Here the thin line represents the contribution of one point sink and the thicker line the simultaneous mean contribution of the two sinks. An assumption has been made equating the axial plane of the axisymmetric flow with an axial plane of twodimensional flow. However, the diffusion and the velocity fields cannot be dissociated and consequently this analogy is only justified near the wall. In the central zone of the axisymmetric flow, the homogeneity of the velocity field is not the same as in a twodimensional flow. Now this analogy will be shown to cover the whole plane of axial symmetry.

Gebel et al., Diffusion of macromolecular solutions in the turbulent boundary layer of a cylindrical pipe. II The diffusion field may be divided into two domains, both function of the abscissa x and we shall study the evolution of the concentration profile in the region 0 ~< y ~< R. (i) "Near the sink": In this domain the polymer concentration in the middle of the pipe, C(D/2), is zero. The transversal profile between the wall (y = 0) and the middle of the pipe (y = R) is restricted to the profile achieved with one sink. The contribution of the second sink does not interfere in the region 0 ~< y ~< R. By the mere fact of symmetry, an identical profile will be found in the region R ~< y ~< D (see fig. 2). (ii) "Far from the sink": In this domain, the polymer concentration in the middle of the pipe is not zero. The second sink influences the region 0 ~< y ~< R. Then it is possible to schematize the diffusion phenomena at two sinks as shown in figure 3.

721

where C1/Cwl and C2/Cw2 are the contributions of the sinks 1 and 2 respectively. For reasons of symmetry, a is equal to b.

3. Proposed m o d e l (i) "Near the sink": The proposed model is written I~y Morkovin's classical relation C/Cw = exp [ - 0.693 (y/ 2) n] .

(2)

(ii) "Far from the sink": The relation (1) represents the model. Now the expression for a will be determined. To this end, a will be assumed to be constant whatever the ordinate y may be for a given abscissa x. The experimental results justify this assumption [8]. For the general expression, we can write: c(y) - -

-

+ c2ty)l/Cw

[el(y)

(3)

Cw

with X I" Z

Cw

Fig. 1. Diffusion in pipe flow with two point sinks

=

c~(o)

+

c2(o).

Using eqs. (1) and (3) and introducing the identity, Cz(y) = CI(D - y), we find:

Y~or

C(y)

_ a CI(Y)

x=xo

0 t

~ C(y)

~

+

C1(D - Y) l J'

(4)

l / a = 1 + CI(D) Cwl

Fig. 2. Concentration profile "near the sink" By introducing Morkovin's model for C1/Cwl, the relation (4) can be rewritten as: ,

......

C(y)

_ a

Iexp [0"693 ( 2 ) n l

, ex,L-o.69'

(5)

Fig. 3. Concentration profile "far from the sink" With the assumption that the resultant C/Cw, corresponding to the concentration profile for the case of two simultaneous sinks, is a linear combination, then: C / C w = aCl/Cwl + bC2/Cw2

(1)

l / a = 1 + exp

-0.693

Remarks: First, it is possible to determine experimentally only C/Cw and y for a given abscissa. Indeed, the characteristic value of the diffusion plume ,t represents the distance from the wall at which the concentration drops to

722

Rheologica Acta, Vol. 21, No. 6 (1982)

half its original value for the case of one wall injection. Secondly, eq. (5) may not reduce to a simple form. A classical method of least squares is not sufficient to determine ;t and n.

using the reduced concentration in the middle of the pipe. Eq. (5) can be rewritten as:

C ( R ) _ 2 exp { - 0.693 ( D / 2 ~ ) n} Cw

4. Data reduction. Experimental results

1 + e x p { - 0 . 6 9 3 (D/~) n}

This relationship can be transformed and rewritten by an equation of type:

The study of concentration profiles has been made in four transversal sections, located at 5.7, 12.3, 16.5 and 23.5 diameters (D = 50 mm) downstream of the injection point. The polymer solution was injected by an annular slot.

f ( 2 , n) = G(2, n)

(6)

with

4.1. M e t h o d o f data reduction f(;t, n) = exp{ For using eq. (5), the 2 and n variations should be known in any diffusion zone [3]. An iterative method of data reduction has been carried out. It consists of

G(~.,n) -

1 2

0.693 ( D / 2 2 ) n } ,

-

C(R) {1 + e x p { - 0 . 6 9 3 (D/~.)n}}. Cw

v 3

3 ,<

v A~

x=81Emm 2

•

v

x =1175

mm

1 INITIAL

CONCENTRATION

vv.~~v

0 0.0

INITIAL

"-~v

Ci = 0 wpprn

I

I

0.2

0.4

I 0.6

CONCENTRATION Ci :

0

wpprn

I A~ N ~ * 0.8

1.0

C/Cw

Fig. 4. Profile of reduced concentration C/C w (water injection). Injection flow rate: • 8.33, V 16.67 cm3/s. Theoretical profile; eq. (2)

0,0

O.2

O.4

0.6

O.8

C/Cw

Fig. 5. Profile of reduced concentration C/C w (water injection). Injection flow rate: • 8.33, [] 1].11, • 13.89, V 16.67, • 19.44, © 22.22cm3/s. Theoretical profile; eq. (5)

1.0

Gebel et al., Diffusion of macromolecular solutions in the turbulent boundary layer of a cylindrical pipe. II The solution of this problem consists of finding the C(R) most adequate couple which satisfies eq. (6), - C~ being known. By fixing an arbitrary value of n, the corresponding value of 2 is calculated by determining the intersection of the curves f ( 2 , n) and G(2, n). Then the couple (2, n) is put in eq. (5) and it is thus possible to calculate the theoretical profile and to compare it with the experimental profile. The minimization of the deviation sum between the calculated and experimental values indicates whether the values chosen were a good estimate or whether an adjustment of the exponent n is necessary.

4.2. E x p e r i m e n t a l results

The analysis of figures 4, 5, 6 and 7 shows a good agreement between the calculated curves (solid lines) and the experimental points. To produce clear v v~ ~v

0v

~

v

D

723

figures, some experimental data have been plotted as a function of abscissa (D - y ) / 2 corresponding to profiles which would be achieved in the region R < y < D. The group of experimental data can be represented by one curve for a given section and initial concentration. Yet, for the study of parameter 2 and n variations, each manipulation will be taken into consideration and each of them will bring a pair of values (4, n).

5. Conclusions It has been shown experimentally that the analogy between the axial plane of axisymmetric flow and the axial plane of twodimensional flow (assumption verified near the wall) can be extended to cover the whole pipe in this particular case. In the same way, the assumption to consider a as a constant, whatever the ordinate y, is justified by our experimental results. Thus we have established a model for the evolution of a transversal concentration profile on the basis of the superposition of two Morkovin's models. In a following publication, we will work out models for the characteristic length of the diffusion plume ,I. and characteristic exponent n.

3

v /9

x : Ei15 mm

\

>,

v

v

v o

D

•

~

x-.B

v~

,7

2

v

v

v

D u

.A

1

INITIAL CONCENTRATION Ci = 200

|

0.0

0.2

\ \

INITIALcI _CONCENTR/~IIUN400 p;:H

~ ~ v ~ ~

wppm

f

OA

,

C/Cw

0.6

,

0.8

•

1.0

Fig. 6. Profile of reduced concentration C / C w (polymer injection). Injection flow rate: • 8.33, [] 11.11, • 13.89, V 16.67, • 19.44, © 22.22 cm3/s. - - Theoretical profile: eq. (5)

0

I

0.0

0.2

I

Q4

I

C/Cw

0.6

I

~

[]

0,8

Fig. 7. Profile of reduced concentration C/C w (polymer injection). Injection flow rate: • 8.33, [] 11.11, • 13.89, V 16.67, • 19.44, © 22.22 cm3/s. - - Theoretical profile; eq. (5)

1.0

724

References 1. Bhowmick, S. K. et al., Rheol. Acta 14, 1026-1031 (1975). 2. Gebel, C. et al., Rheol. Acta 17, 172- 175 (1978). 3. Reitzer, H. et al., Rheol. Acta 20, 35-43 (1981). 4. Poreh, M., K. S. Hsu, J. Hydronautics 6, 1, 27-33 (1972). 5. Latto, B., O. K. F. E1 Riedy, J. Hydronautics 10, 4, 135- 139 (1976). 6. Collins, D. J., A Turbulent Boundary Layer with Slot Injection of Drag Reducing Polymer. Thesis Georgia Institute of Technology (1973). 7. Morkovin, M. V., Intern. J. Heat and Mass Transfer 8, 129 (1965). 8. Bues, M., Diffusion de fluide non-newtonian en 6coulement interne. Propositions de modules d'~volution des

Rheologica Acta, Vol. 21, No. 6 (1982) concentrations. Th~se de Docteur Ing6nieur, ULP, Strasbourg (1981). (Received December 21, 1981) Authors' addresses: Dr. C. Gebel, Dr. H. Reitzer Institut de Mbcanique des Fluides Universit~ Louis Pasteur 2, rue Boussingault F-67083 Strasbourg Cedex Dr. M. Bues Ecole Nationale Sup6rieure des Arts et Industries 24 Bd. de la Victoire F-67000 Strasbourg