Rheologica Acta

Rheol Acta 24:312-316 (]985)

Diffusion of macromolecular solutions in the turbulent boundary layer of a cylindrical pipe. IV. Visualization and model of changes occurring at the diffusion boundary layer M. Bues, H. Reitzer, and O. Scrivener l~cole Nationale Sup6rieure des Arts et Industries de Strasbourg, and Institut de M6canique des Fluides de Strasbourg Abstract." A model is presented describing the changes that occur in the diffusion

boundary layer upon injection of a macromolecular solution (PEO) into a cylindrical pipe under turbulent flow conditions (Re ~_ 40,000). A shape parameter was introduced to describe the shape of the turbulent plume. The value of this parameter was found to be the same for water and various dilute PEO solutions. The proposed model gives a good approximation at low homogeneous concentrations. Key words: Turbulent diffusion, macromolecular solution, diffusion boundary

layer, diffusion plume

Notation

x y R C C~ -

downstream distance from the slot normal distance from the wall radius of the pipe concentration wall concentration Qi - flow rate injection Qt - flow rate Cj = Ci* Qi/Qt - equivalent homogeneous polymer concentration L t / . - characteristic length of the diffusion plume 2 -characteristic height of the diffusion plume, i.e. the value ofy at which C / C w = 0.5 6 - thickness of the diffusion boundary layer x 0 - characteristic distance from the slot, i.e. the value of x at which 6/R = I/2 A + - shape parameter of the diffusion boundary layer 6 + - 6/R ~ nondimensionalized x + - x / L t / J variables

corresponds to the distance between the injector and the point at which the diffusion layer reaches the pipe axis. This length also corresponds to the characteristic length Ltf between the transition and the final zones [2]. In previous papers [3, 4] 2, the distance from the wall at which the p o l y m e r concentration falls to half its value at the wall, C ( y ) / C w = 0.5 (fig. 1), was used to model the changes in the diffusion plume. Although this p a r a m e t e r is rather arbitrary, it was introduced to set up a parametric model of the diffusion of macromolecular solutions. The present p a p e r considers the dependence o f the mean thickness 6 of the diffusion b o u n d a r y layer on the distance from the injector.

2. Experimental methods 1. Introduction Injection of a secondary fluid into a p r i m a r y fluid flow leads to a diffusion b o u n d a r y layer of the injected fluid, which grows from the injection point and divides the flow into two distinct regions: on the one hand a parietal zone completely filled with injected fluid mixed with p r i m a r y fluid and on the other hand a zone o f p r i m a r y fluid b o u n d e d by the diffusion layer. F o r a circular p i p e Bhowmick et al. [1] introduced the concept of a "cone o f p r i m a r y fluid", the length o f which 41

2.1 E x p e r i m e n t a l device

The experimental device is shown in figure 2. The p r i m a r y fluid was supplied to the 50 m m diameter pipe through a regulated constant head pressure tank (1) and an inlet regulated tank (2). The total length of the pipe (3) was 10 m. The injection system (5) was positioned inside the pipe and sufficiently far from the inlet to ensure a fully developed turbulent flow. The tank (4) downstream provided a constant pressure at t h e outlet. Regulated air pressure

Bues et al., Diffusion of macromolecular solutions in turbulent boundary layer. IV

primary

313

~R

flow

Fig. 1. Changes of diffusion plume, g-average thickness of the diffusion boundary layer, 2-distance, y, from the wall at which C(y)/Cw= 0.5; L t f - characteristic length of the diffusion plume

/

0 0

LTF

LF

X

injection

r 9 r-"--1

Fig. 2. Experimental set-up; components described in text

,I d i

I~

,! d2 I-"

I

,~

d3

I~-

q

d4

r,

o

I

Fig. 3. Analysis of the visualization pictures

was used (6) into measured rotameter

to force the polymer solution from the tank the pipe. Its flow rate was continuously and controlled by means of a calibrated (7).

2.2 Visualization of the diffusion boundary layer Walters [5, 6] and Wu [7] showed that when molecular diffusion can be disregarded a dye (fluorescein in the present investigation) may be added to the polymer solution as a tracer. A thin cross-section o f the pipe was illuminated through a slit to visualize the diffusion layer. A mirror (8) oriented at 45 ° in the tank (2)

allowed the diffusion layer to be observed through a plane window (9). Movie pictures were taken of the diffusion layer to study its evolution down the pipe. It is apparent from figure 3, which shows a typical sequence of successive pictures, that the boundary between the two regions is unstable. This also confirms the existence of a secondary vortex superimposed on the main flow, as previously reported [8]. A mean value d of the diameters ds was calculated from:

n

Z d,.

i=1

The optimum number of pictures to be processed was determined by plotting d against n. By measuring the size on the picture of a pointer located at the measuring point a magnification factor was determined and the mean value of the diffusion thickness, 3, computed. The characteristic length LiT was determined either by visual observation or by plotting 8 against the distance x from the injector. In several cases accurate measurement was difficult due to the instability of the boundary between the diffusion plume and the pri-

314

Rheologica Acta, Vol. 24, No. 3 (1985)

Fig. 4. Visualization of the diffusion boundary layer by dye tracer mary flow (fig. 4). The results were then compared with the length calculated by using the model of concentration changes proposed in previous [2, 9]. The latter was generally adopted for data processing.

Introducing these conditions into eq. (3) yields a relationship which is independent of a and c: c=--

1

and a = 1/(exp b - 1).

3. Diffusion boundary layer model

By substituting A for exp b we obtain The axial symmetry of the flow in a pipe makes it possible to model changes of the diffusion layer in a meridian plane. For the "cone" model this change is linear and in the coordinate system of figure 5 can be described as: x

-

Ltf

b

(1)

R '

b=0

A 6/R - 1

A- 1

and

x=Ltf,

6=R.

X0

(2)

However, the experimental results showed that the shape of the plume was better described by an exponential model such as x _ a {exp [b am] + c} Ltf

(3)

(4)

with 0 < 6 / R < 1. Taking into account that 6 / R is 1/2 at the position x = x0 we obtain Ltf=

where the boundary conditions are respectively: x=0,

x LtU

A-1

1.

A 1/2 +

(5)

A 1/2 - - 1

Making the substitutions L t f = A+ ' Xo

__6 = 6+ ' R

x+_

x LiT'

eq. (5) can be rewritten as A = (A + -

1) 2 ,

and eq. (4) then becomes

with the same boundary conditions (2).

x+ _ (A + - 1) 26+- I a +' (A + - 2) '

__R~ 0 0

where A + is a shape parameter describing the change in shape of the diffusion boundary layer.

.

4. Experimental results

LTF

X

Fig. 5. Linear increase in the thickness of the diffusion boundary layer

As for the characteristic length of the previously defined diffusion plume 2 [2, 9], figure 6 shows that the development of the diffusion boundary layer is hindered by the presence of the polymer near the wall.

Bues et al., Diffusion of macromolecular solutions in turbulent boundary layer. IV

315

0.8

0,6

Q4

0.2

I

(3.0

I

0

[

I

10

20

I

I

I

I

30

X (era)

I

I

40

Fig. 6. Dependence of the normalized diffusion boundary layer thickness 6 / R on the distance from the injection point x • water injection, © Polyox WSR 301 injection

50

0,8

0.6 ,.w,

0.4

0.2

0.0

I

I

I

0.2

I

I

0.4

I

I

0.6

I

I

0.8

I

Fig. 7. Dependence of the normalized diffusion layer thickness 6 / R on the normalized distance from the injection point x / L t f . • - Water (Q = 30 l/h), [] - Polyox 200 ppm (Q = 30 l/h), v - Polyox 200 ppm (Q = 60t/h), o - Polyox 400 ppm (Q = 60 l/h), • - Polyox 400 ppm (Q = 30 l/h)

1

X/LTF

100

75

w

5O REYNOLDS'NUMBER:

•

Re = 4 0 0 0 0

•

Re = 5 5 0 0 0

25

I

I

I

0.5

1 (QilQt).lO0

1.5

I 2

Fig. 8. Variation of the characteristic length Ltf with the flow rate ratio Q i / Q t (water injection)

316 Given that most of the results with polymer wall injection show a thickening of the viscous sublayer our results (fig. 6) and the 4-1ayer-model proposed by Porch et al. [10] suggest that the initial zone, in which the diffusion boundary layer is overrun by the viscous sublayer, is larger for polymer than for water injection. Nevertheless, even for high homogeneous concentration Cj, the length of this zone remains small in comparison to the other characteristic length of the flow, and particularly to Ltf. The non-dimensional plot of ~/R against x/Ltf also shows that the data for both water and polymer injection fit one and the same curve. As a consequence, a single value A ÷ = 4.2 (fig. 7) was found for all the data recorded under our experimental conditions. For Newtonian fluids this confirms that the disturbance of the wall area resulting from fluid injection can be disregarded in our experiments as the relative injection rate remained small (Qi/Qt ~ 0.01) [1]. The value of the Newtonian diffusion length (Ltf) was constant and equal to 74 cm (fig. 8). When polymer solutions are injected, the value of the shape factor A ÷ remained the same as for water injection indicating that the shape o f the diffusion layer was not affected by the polymer. It should be noted that the homogeneous polymer concentration Cj was less than 5 ppm in all our experiments and that this concentration is close to the critical concentration, defined in our previous papers [2, 9], below which the non-Newtonian fluid injected seems to behave in a Newtonian manner. In conclusion, the model suggested provides a good estimation of the behaviour of the diffusion boundary layer at low homogeneous concentration. However, the

Rheologica Acta, Vol. 24, No. 3 (1985) single value for the shape parameter A + found here is probably not valid at higher homogeneous concentrations. The increase of the diffusion length observed with polymer injection is mainly due to an increase in diffusion in the initial zone, the length of which can be estimated to be a few pipe diameters. This correlates well with the linearity of the boundary layer equations if the turbulent diffusivity is supposed to change only slightly outside the viscous sublayer,

References

1. 2. 3. 4. 5. 6. 7. 8. 9. 10.

Bhowmick SK et al. (1975) Rheol Acta 14:1026-1031 Reitzer H et al. (1981) Rheol Acta 20:35-43 Gebel C et al. (1982) Rheol Acta 21:720-724 Bues Met al. (1982) Rheol Acta 21:725-729 Walters RR et al. (1971) Report n B/94000/I.C.R. Walters RR et al. (1974) Report n B/94300/I.C.R. Wu J (1972) J Hydronautics 6:46 Wetzel JM et al. (1970) Univ. of Minnesota, Report n 114 Bues M (1981) Th~se Docteur Ing~nieur, Strasbourg Poreh M e t al. (1964) I J Heat and Mass Trans 7:10831095 (Received May 30, 1984)

Authors' addresses: Dr. M. Bues t~cole Nationale des Arts et Industries 24, Boulevard de la Victoire F-67084 Strasbourg Cedex Dr. H. Reitzer, O. Scrivener Institut de M~canique des Fluides - ERA CNRS 0594 2, Rue Boussingault F-67083 Strasbourg Cedex