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Chemical Enginewing Science, Vol. 48. NO. 9, pp. 1619-1628, 1993. oooY_25‘W/93 $6.00 + 0.00 Printed in Great Britain. 0 1993 Axgamon Press Ltd

TURBULENT MIXING IN DILUTE POLYMER SOLUTIONS*

V. V. RANADE and R. A. MASHELKAR* Chemical Engineering Division, National Chemical Laboratory, Pune 411 008, India

(First received 12 March 1992; accepted in revised form 14 October 1992)

Abstract-Addition of polymer molecules to solvents influences the turbulence characteristics and in turn it influences the macromixing and micromixing behaviour in such flows in a profound way. The mixing lengths in pipe flows of dilute polymer solutions are known to be several times larger than those for Newtonian flows. A simple phenomenological model of polymer-turbulence interaction is developed to evaluate the reduction in friction factor and it is then used for analysing mixing in one-dimensional turbulent flows. The extent of mixing in dilute polymer solutions is then predicted quantitatively. The limited experimental data available show that the model simulates mixing in flows of Newtonian fluids as well as in mildly viscoelastic drag-reducing fluids very well.

1. INTRODUClTON

The aspects of turbulence-polymer interaction in drag reducing fluids has evoked a considerable interest over the years. A number of reviews (Virk, 1975; Astarita and Mashelkar, 1977; Berman, 1978; Kulicke et al., 1989) are available. We ourselves have studied the aspects of friction reduction in internal flows (Kelkar and Mashelkar, 1972), in external rotational flows (Mashelkar, 1973; Kale et al., 1973), influence on mass transfer (Mashelkar, 1984; Mashelkar and Soylu, 1984), etc. However, the phenomenon of macromixing and micromixing in drag-reducing fluids in pipes seems to have been poorly understood. The only data available in the literature where some interpretation of mixing in dilute polymer solutions in agitated vessels are by Quraishi et al. (1977). These authors showed, somewhat qualitatively, the effect of drag- reducing additives on mixing time.

The problem of quantitatively predicting the changes in mixing characteristics in turbulent pipe flows has remained unsolved. There is a special in- centive for studying this problem. One of the profit- able exploitation of drag-reducing additives has been in enhancing the performance of a pipeline carrying crude oil or products (Sellin et al., 1982; Schmerwitz and Reher, 1986). Indeed, such addition has been successfully demonstrated in trans-Alaska pipelines (Burger et al., 1980). In such cases, the polymer additive is added at the entrance of the pipe and it mixes as it moves down the pipe. Can we predict the length required for this mixing to be complete? There is nothing in the published literature which could answer such questions. The present study was motiv- ated by such concerns.

Before understanding the complex problem of dispersion and mixing of drag-reducing polymer molecules, it was thought desirable to understand the aspects of turbulent mixing of small molecules. Even there, we found that no guidelines were available for

‘NCL Communication number 5276. ‘Author to. whom correspondence should be addressed.

estimating the mixing length in pipe flows of drag- reducing polymers. We have, therefore, taken up this problem for analysis first, since at least a lower bound on the mixing lengths for turbulent mixing of large polymer molecules could then be obtained through such an effort.

Let us first convince ourselves that the problem is interesting enough for study and that there are funda- mentally significant differences in turbulent mixing of Newtonian and dilute polymer solutions. This seems to be the case. Flow visualisation using fast reaction can be used elegantly to investigate the mixing characteristics (Smith, 1969; Berman and Tan, 1985). Smith (1969) has reported drag-reduction data along with the mixing data (decolouration lengths) for various polymer concentrations and Reynolds number. He has observed two major differences between Newtonian flows and flows of drag-reducing fluids. Firstly, the decolouration lengths (equivalent to mixing lengths) required for the flow of water remain almost independent of Reynolds number. This is in contrast to the observation in the case of polymer solutions, where these lengths depend strongly on Reynolds number. Secondly, it was observed that when experiments were done with dilute polymer solutions, the rate of decrease in intensity of segregation reached a limiting value as the Reynolds number increased. This again is in marked contrast to the behaviour without the polymer. Therefore, there are fundamental differences that need to be resolved.

In this paper, we develop a simple phenomenological model to predict the extent of mixing in dilute polymer solutions and show that the experimental observations can be successfully simulated through such a model. This paper is organised as follows: Section 2, by examining the major characteristics of turbulence, develops an appropriate mathematical model for turbulent mixing. Section 3 proposes a simple model for polymer-turbulence interactions and deduces a simple expression for predicting the reduction of friction factors. Section 4 describes the simulation of mixing in turbulent pipe flows with and

1619

1620 V. V. RANADE and R. A. MASHELKAR

without polymers, and provides comparison with the available experimental data.

2. MODELLING OF TURBULENT MIXING Hiby (1968) and Smith (1969) conducted mixing

experiments in turbulent pipe flows in the presence of polymers by using a fast reaction between acid and base. In this section, we attempt to develop a mathematical model for interpreting the results of such mixing studies in a fundamental manner.

2.1. Time scales of mixing

Mixing between two miscible fluids proceeds through the following steps (Ranade and Bourne, 1991):

Step 1: convection by mean velocity Step 2: turbulent dispersion by large eddies Step 3: reduction of segregation length scale Step 4: laminar stretching of small eddies Step 5: molecular diffusion (and chemical reaction,

if possible).

It is necessary to estimate the time scales involved in these steps and compare these with the final mixing time (may be from the experiments) to identify the most significant events, which control the mixing steps. In this section, our aim is to construct a mathematical model for the description of mixing and instantaneous chemical reaction (acid-base neutralisation) in turbulent pipe flows. For simplicity, we use one-dimensional flow approximation with no radial concentration gradients. This simplification implies that we can disregard the first two steps of convection by mean velocity and turbulent dispersion from the micromixing process. The final mixing time (t,J in such flows is the ratio of the decolouration length and mean velocity of fluid.

Consider the third step of reduction of segregation length scale. The characteristic time constant (tMS), for the reduction in segregation scale (inertial-convective mixing) is identified as (Corrsin, 1964; Baldyga, 1989)

tMS = 2(L,2/&)“3 (I)

where LS is segregation length scale and E is the rate of dissipation of turbulent energy.

Considering the fourth step, the characteristic time for the engulfment (tE) can be estimated as (Baldyga and Bourne, 1984)

t, = 17.25(v/e)“’ (2)

where v is kinematic viscosity. Considering the fifth step, the diffusion time scale

(t,J can be written as (Baldyga and Bourne, 1984)

t,, > (V/E)“2 arcsinh (0.1s~) (3)

where SC is Schmidt number defined as v/D, with D as diffusivity.

It can be shown that for systems with SC < 4000, we have t,, > t, > t,,. When tmix is of the order of tDS, i.e. when only one engulfment is sufficient for com-

plete neutralisation, the mixing will be diffusion-controlled. Such situations will occur when the amount of acid (or base) used is far in excess of the amount of base (or acid). Decolouration length experiments reported by Bourne and Tovstiga (1988) fall in this regime. When many engulfments are necessary for the completion of reaction, but tmix is less than t,,. the mixing will be mainly controlled by the engulfment process. When tmix is greater than t,,, neither diffusion nor engulfment can significantly affect mixing. When stoichiometric acid-excess ratios are not very high, the time required for the complete neutralisation will be of the order of tMS. Therefore, to predict mixing under such situations, we need to concentrate mainly on step 3 and to some extent also on step 4.

Recently, Baldyga (1989) analysed the contribu- tions of the last three steps in the mixing process. He showed that the overall time constant of mixing is not a constant in reality; however, it reaches a constant asymptotic value equal to t,, at longer times. The mixing time constant can be obtained from the pro- files of intensity of segregation (I,) as

tM= - Z,/(dZ,/dt). (4)

If we neglect the contribution of viscous-diffusive part of concentration variance in the overall concentration variance, then the overall intensity of segregation can be written as

I, = I,, + I, (5)

where I,, and I, are the inertial-convective (step 3) and viscous-convective (step 4) parts of the overall segregation. Following Baldyga (1989) the decay equations for I,, and I, can be written as

dZ,ddt = - Z&t,, (6)

dZ,jdt = Z&t,, - I&. (7)

Using eqs (4)-(7), we obtain

t It - 1 + (M - l)/[M(e”-‘)‘/‘~~- l)]. M MS- (8)

Here M is the ratio oft,, and t,. It can be readily seen that the overall mixing time constant t,,, reduces to tMS at large values of (M - l)t/tMS (* 1) and to t, at very small (4 1) values of (M - l)t/tMS- Thus, the engulfment process (step 4) wiH be important only if the decolouration lengths are very small. We will use this mixing time expression to model the mixing in pipeline flows.

2.2. Mathematical model Our primary aim here is to develop a model to

simulate the mixing experiments of Smith (1969) in turbulent flows of dilute polymer solutions. Smith used 0.02 m diameter pipe for the test section (schematically shown in Fig. 1). This experimental apparatus ensures fast radial mixing of two streams compared to the process of micromixing. If we assume that complete radial mixing is present, then we can formulate the model for small-scale mixing using the characteristic mixing time from eq. (8).

Turbulent mixing in dilute polymer solutions 1621 --_ M ACID --_

Fig. 1. Schematic diagram of the set up for mixing experiments (Smith, 1969).

The complete radial mixing of two streams will result in a mixture with segregation length scales of the order of integral length scales of turbulence. This scale will be gradually reduced by the action of turbulence till it reaches the Kolmogorof’s scale. Mixing below this Kolmogorof’s scale will be controlled by laminar stretching and diffusion processes, which will be fast compared to the scale reduction process. The reduction of segregation scale will increase the inter- facial area between the injected acid and base streams. The reaction will take place at this interface to form mixed zones (due to engulfments) around the segre- gated fluid lumps. These mixed zones will also contain only either acid or base depending upon the respective concentrations of the injected streams. However, these will be at different concentration levels than those at the injected point. The concentration of the base in such a mixed zone will only depend on the ratio of the acid to base concentration in the injected fluid. As the fluid moves downstream it will attain a final concentration depending on the stoichiometry of the injected streams. This physical picture can be mathematically modelled by using a framework similar to that developed by Ranade and Bourne (1991). Baldyga and Rohani (1987) have also developed a similar framework for analysing micromixing.

For the sake of simplicity, we will divide the fluid lumps into three subgroups according to concentrations, one for the base-rich lumps, another for the acid-rich lumps and the remaining for intermediate concentrations. Thus, the mixing because of scale reduction process (followed by rapid engulfment) can be written as

j-1 dej/dt = Eejej+ I + 2Eei+ 1 x e, - Ee, 5 ei

1 +j,j-1

(9)

dejC,j/dt = Ee,ej+ 1&j+ 1

f 2Eej_ lJglei(C,i + Cmj+ 1) 1

- EejC, 5 ei + ejR,,,j +j,j- I

(10)

where N is the number of subgroups, e, is the volume fraction of the subgroup j, C,,,, is the concentration of mth species in jth subgroup, R,j is the reaction rate of mth species in jth subgroup and E is the reciprocal of the characteristic mixing time t,. Large-scale mixing and transport of these small coherent fluid packets along the pipe length can be simulated by using the usual convective transport equation, where one as- sumes complete radial mixing and neglects the contribution of axial dispersion. This will allow us to write the time t in eqs (9) and (10) in terms of the distance travelled along the pipe (z) and the mean velocity (U). Making eqs (9) and (10) dimensionless, the pipe length required to complete the neutralisation reaction (decolouration length) can be written as

UD =.f(EDIiJ, x, eiO). (11)

Thus, the dimensionless decolouration length (L/D)

depends on the non-dimensional number (ED/U), acid-excess ratio(x) and the initial volume fractions of acid and base streams (ejO). Once the value of ED/U is estimated, predictions of the decolouration lengths become possible.

2.3. Estimation of characteristic mixing time, t, Equations(l), (2) and (8) can be used to estimate the

characteristic mixing time, t,. The segregation length scale of mixing, L, can be related to the turbulent kinetic energy and turbulent energy dissipation rate (Spalding, 1971). Therefore, the value of the character-


istic time constant for the inertial-convective mixing, t MS, can be estimated as

tM, - (k/4 (1-T

where k is turbulent kinetic energy. The turbulent kinetic energy and the dissipation rates can be related to mean flow characteristics as

k = (312)(f/2)~2 (13)

E = 2jU’/D (14)

where f is the friction factor, D is the pipe diameter and U is the mean velocity.

In the case of the flow of dilute polymer solutions, the friction factors are reduced in comparison to that in the flow of solvent. This means that proportionally the energy dissipation rates reduce too. However, the turbulence kinetic energy in the core region will not be affected by the presence of polymers and it will still be proportional to the corresponding friction factor of Newtonian fluid at the same Reynolds number. This suggests the following relation for the mixing time constant:

[MS = ~(fllfp)DlU (15)

wheref, is the friction factor for the dilute polymer solution flow and LX is a proportionality constant.

3. DEVELOPMENT OF A MODEL TO PREDICT/p

3.1. Effect of polymer on turbulence characteristics Many excellent studies are available on poly-

mer-turbulence interactions (Virk, 1975; Durst and Rastogi, 1977; de Gennes, 1986). Most of the modelling efforts, however, are related to the near-wall region of the flow (Armstrong and Jhon, 1984; Ryskin, 1987). Polymer interactions with turbulence without any wall effects have not yet been modelled quantitatively. In this section, we concentrate on the polymer-turbulence interaction without any wall effects.

de Gennes (1986) proposed a scaling theory for such interactions. In fully developed turbulence, there exists a wide spectrum of eddy lengths. Polymer molecules will interact with the entire spectrum of these turbulent eddies. Although any polymer molecule physically may belong simultaneously to several eddies (since large eddy will contain smaller eddies within it), it can be assumed that the behaviour of polymer molecules will be dominantly affected by eddies of a particular scale. Polymer molecules in a turbulent eddy will be subjected to stretching because of ftuctuations and elongational flow field associated with the eddy. However, if the mean life time of the eddy is large compared to the relaxation time of the polymer molecule, the average elongation of the polymer molecule will be negligible. As the length scale of an eddy decreases, the mean life time of the eddy decreases. At a certain length scale, a stage will come such that the eddy mean life will be equal to the relaxation time of the polymer molecule. de Gennes

(1986) has proposed an equation to estimate this critical length scale using Zimm relaxation time (0):

r1 = (s(F)“2 (16)

where I, is an upper limit of length scales of polymer- affected eddies. In eddies, which are smaller than the critical length scale (I,), the polymer molecule will be significantly elongated compared to its state at rest. The energy required for this elongation of polymer chains will be extracted from the turbulent stresses of the eddy. If we continue to consider the interaction of the polymer with smaller and smaller eddies, a stage will come such that, below a certain size of the eddy (say 12), it will not be able to sustain a significant elongation of the polymer molecule. de Gennes has estimated this lower limit by equating the turbulent kinetic energy with the energy required for elongation. However, he has postulated a power law relationship between the polymer elongation and spatial scales. This makes the quantitative application of his model difficult in turbulent flows. We propose here an altern- ative hypothesis for predicting this limit. The smallest eddies (of Kolomogorof scale) can be assumed to prevent any significant orientation and elongation of the molecules and, therefore, their dynamics will not be affected by the presence of polymer. Thus, we are mainly concerned with the eddies lying in the range I, to I,, for which the effective turbulent stress will be reduced due to the elongation of the polymer molecule.

We propose here a slightly different hypothesis than the one proposed by de Gennes (1986) to estimate the lower limit on the turbulent length scales (I,), which will be affected by the presence of a polymer. The upper limit on these length scales (Zr) can be obtained from eq. (16) according to the proposal made by de Gennes (1986). We propose that the onset of drag reduction will be mainly dictated by the relaxation time of the molecule and the extent of the drag reduction will be dictated by the relaxation time of the polymer solution. The solution relaxation time can be considered as an effective value of molecular relaxation time, which depends on shear rate as well as polymer concentration. Therefore, the upper limit on the scale of the affected eddies will be the same as given by eq. (16) if we replace the Zimm relaxation time (0) by the solution relaxation time (A).

We further propose that polymer molecules will affect the turbulence characteristics of the eddies smaller than this upper limit in such a way that their characteristic time will not be less than the solution relaxation time of polymer molecules. The logic behind this hypothesis is that, if the eddy time is less than the relaxation time, the polymer chains will elongate. This stretching will absorb some energy without affecting the length scale of the eddy. This will lead to an increase in the eddy characteristic time. If this increase makes the eddy time more than the relaxation time, the elongation of the polymer molecule will be reduced. This process will ultimately lead to eddies with characteristic time equal to the

Turbulent mixing in dilute polymer solutions 1623

relaxation time of polymer. Thus, the turbulent shear stress corresponding to the eddies of length scale equal to 1 can be related to the length scale I and solution relaxation time as

u,u, = 12/h2. (17)

However, there will be a lower limit on the length scales of eddies affected in such a way because the turbulent stresses of the eddies cannot be lower than the square of Kolomogorof’s velocity. Let us denote this lower limit by I,, which can be written as

1,/A = (v&)“*_ (18)

For eddies smaller than I,, the characteristic velocity will remain the same (equal to Kolomogorof’s velocity) and the eddy time will progressively reduce to Kolomogorof’s time scale with a decrease in the eddy size. We can reasonably assume that the polymer influence on eddies smaller that I, will be negligible. It will be instructive to examine the relation between I, and lz_ These limiting length scales are related to Kolomogorof’s scales as

1,/l, = (A/Q’2 (19)

1214 = WtA. (20)

Thus, as the Kolomogorof’s time scale, tk, decreases, the range of length scales affected by the presence of polymer widens.

In the following, we will attempt to make use of these concepts for quantitative prediction of influence of polymer molecules on energy dissipation rates in pipe flows. The energy dissipation rate can he written as (Davies, 1972)

s = <"iuj>(u/le) (21)

where < > denotes averaging over all length scales up to 1,. For flow of dilute polymer solutions, the energy dissipation rate E is lower compared to Newtonian fluids. This may be because of the decrease in turbulent stresses (<uiuj>) or increase in scale of energy- containing eddies (I,). There are no clear indications in the literature about the influence of presence of polymer molecules on the value of I,. Berman and Tan (1985) have reported the values of integral scale of turbulence for the submerged jets. They have obtained almost same values for water, 1OOppm polyacrylamide and 100 ppm polyethylene oxide solutions. Therefore, for flow of polymer solutions, the turbulent fluctuating velocity u and length scale of energy-containing eddies 1, can be assumed to be the same as those of Newtonian fluids (Davies, 1972). The presence of polymer molecules causes damping of turbulent stresses over eddy scales 1, to 12, which results in lowering of overall turbulent stresses and, thus, lower dissipation rates. Therefore, one can write

&P/s = <Wj>P/<Wj>* (22)

The turbulent stresses for the Newtonian fluid can be estimated as the average of ratio of square of length scale of eddy to the square of time scale of eddy (which

CES 48:9-F

can be expressed in terms of the rate of energy dissipation and length scale) over all length scales:

<u*u,> = (l/I,) s

1as2/312/3dI = (3/5)(~~‘~ l,““). (23) 0

Polymer molecules, however, reduce these stresses for eddies of length scales smaller than 1,. If the rate of energy dissipation is a,,, the turbulent stresses aver- aged up to Iength scale 1, for Newtonian and polymer solutions (by substituting value of A in terms of 1, and sp after averaging) can be written as

<<upj>> = (3/5)&F” 1:‘3 (24)

<< uiuj >> p = (l/3).$3 1:‘J (25)

where << >> indicates averaging up to length scale I,. Therefore, overall turbulent stress in presence of polymer can be written as

<uiuj>p = (3/5)(&:‘3r2’3) - (4/l 5)E e s/3/:/3 . (24)

Combining eqs (22), (23) and (26) one can write

Ep/& = fp/f = [l - (4/9)(li/&)“‘]3. (27)

This model satisfies all the qualitative tests discussed by de Gennes (1986) without needing the assumption about the power law between the polymer elongation and spatial scale. Using eq. (27), one can now develop an equation for friction factor in terms of mean velocity, pipe diameter and solution relaxation time. The length scale of energy-containing edddies for Newtonian fluids can be obtained using the following equation:

1, = 2/E. (28)

The fluctuating velocity can be related to the friction factor (Davies, 1972) as

u2 = (f/2)V. (29)

Combining eqs (14), (28) and (29) one can write

I, = (0/4)(f/2)“‘. (30)

Similarly, using an analogous eq. (14) for polymer solutions and using eq. (16) one can write

I, = D(2 fp)“2(uA/D)3’2. (31)

Rearranging eqs (27), (30) and (31) one can write

fplf = I/Cl + (16/9WJw)13 (32)

where A is the relaxation time of the dilute polymer solution. Thus, with the knowledge of the relaxation time of the polymer solution, one can estimate drag reduction in pipe flow using eq. (32).

Equation (32) can be written in the normalised form similar to the friction factor ratio correlations proposed by Astarita et al. (1969), Kelkar and Mashelkar (1972) and Mashelkar (1973). Equation (32) shows that, for realising the friction factor ratio of 0.6, the value of (WA/D) should be 0.1044. If we make use of this fact and write the solution relaxation time as

A = o.1044/(u/D)o., (33)

1624 V.V. RANADE and R.A. MASHELKAR

Fig. 2. Comparison of drag-reduction correlations (internal flows): ( -) predictions from present work; (- - -) cor-

relation by Kelkar and Mashelkar (1972).

“ID

Fig. 3. Comparison of drag-reduction correlations (external flows): ( -) predictions from present work; (- - -) cor-

relation by Mashelkar (1973),

eq. (32) can be written as

&!f- = l/11 + 0.1g56*c(~/~)/(~l~),.~l~3 (34)

where (U/D),,, is the value of ratio of mean velocity and pipe diameter at which friction factor ratio (Jplf) becomes 0.6. The comparison of eq. (34) with the correlation of Kelkar and Mashelkar (1972) is shown in Fig. 2. Kelkar and Mashelkar’s correlation has been shown to represent a wide range of drag-reduction data in pipe flows. The comparison of eq. (32) (modified following the same method for critical friction factor ratio of 0.7) with the correlation of Mashelkar (1973), which has been shown to represent wide range of drag-reduction data in external flows, is shown in Fig. 3. The predictions of eq. (32) are in good agreement with these two correlations except at the friction factor ratios below 0.3. This agreement is indeed highly satisfying, since eq. (32) has been de- rived theoretically and does not contain any adjustable parameters.

Since the objective of the present paper is to simulate mixing experiments of Smith (19693. a suitable correlation for the drag-reduction data is needed. Smith’s (1969) drag-reduction data in the Reynolds number range of 1 x lo4 to 3 x lo4 can be correlated

Fig. 4. Parity plot of predicted and experimental friction factors [for experimental data provided by Smith (1969) on

dilute solutions of polyethylene oxide].

by the following equation for the relaxation time for the solutions of PEO in the concentration range 25-200 ppm:

A = 1.4692 x 10-4c/(1 + 0.0217~). (35)

Figure 4 shows a comparison of the data by Smith (1969) and predictions based on eqs (32) and (35). The agreement is indeed satisfactory.

In the following section, we will make use of this drag-reduction model to develop the mixing model for the turbulent pipe flows with drag-reducing polymers. This mixing model will be validated with Smith’s (1969) data on decolouration lengths for the acid-base neutralisation reactions in pipe flow.

4. SIMULATION OF MIXING IN PIPE FLOW

In this section, we will simulate the mixing experiments reported by Smith (1969). The experimental assembly is shown schematically in Fig. 1. Smith carried out decolouration experiments in 0.02m diameter pipe (10m long) with equal flow rates of acid and base solutions. In all the cases, the strength of the solution of alkali was kept at 0.02N. Bromothymol blue indicator was used. In this section, we will examine the suitability of the model to simulate mixing in the Reynolds number range of 1 x lo4 to 3 x 104.

Boundary conditions at the injection point can be specified from the knowledge of flow rates of acid and base streams. For equal flow rates of these streams, volume fraction of the first subgroup (base-rich) and third subgroup (acid-rich) can be specified as 0.5. Model equations (9) and (10) were solved using fourth-order Range-Kutta method. The quality of mixing at any location along the pipe length can be estimated in terms of the concentration segregation or the extent of neutralisation. In the present paper, the pipe length required for the complete neutralisation of the base was considered as the decolouration length.

The dimensionless decolouration length (L/D) is a function of the dimensionless time constant for inertial+onvective mixing (tMSU/D), ratio of time constants for inertial-convective mixing and viscous- convective mixing (M), and acid-excess ratio, x. It is interesting to note that the predicted decolouration


lengths for lower values of acid-excess ratio are in- sensitive to the value of M. This is consistent with the qualitative discussion on the time scales in Section 2.1. When the acid-excess ratios are very high (> lOO), dimensionless decolouration lengths will be less than 2 to 4 and the predictions will be sensitive to the value of M. Bourne and Tovstiga (1988) report decolouration lengths at very high acid-excess ratios (z 100). Here they found that the decolouration lengths increase with the viscosity of the fluid flowing at same velocity_ Increase in viscosity will reduce the Reynolds number and, therefore, the value of M, since M varies as Re3/*. This decrease in the value of M will increase the predicted decolouration lengths from our model, which is consistent with the experimental observations. However, here our interest is mainly in simulating the mixing experiments at small acid-excess ratios lying in the range of l-2. Experiments with stoichiometric quantities of acid and base can be interpreted as scalar mixing experiments (Toor, 1975). In the following, we discuss the decolouration length predictions, which do not depend on the value of M or, in other words, on the value of fluid viscosity at the same velocity.

Before presenting the mixing results in the presence of polymers, it is worthwhile to discuss some aspects of the turbulent mixing in Newtonian fluids. Pub- lished experimental data of mixing length are plotted in Fig. 5 as a function of Reynolds number. It can be seen that the decolouration length or mixing length for pipe flow is almost independent of the Reynolds number. This means that a possible increase in mixing length because of an increase in mean velocity is compensated for by the decrease in mixing time because of additional turbulence. Because different

arrangements (multiple jet, grid, etc.) by different authors have been used for conducting the mixing experiments, there is no quantitative agreement between the data published by different investigators.

The dimensionless characteristic time for inertial- convective mixing (t&Y/D) is proportional to the ratio of friction factor without and with polymer (f/f..). Therefore, for aqueous solutions of acid and base without any polymer, this dimensionless characteristic time is constant and equal to a in eq. (15). This constant CI was found to be equal to 14.5 by fitting the predicted decolouration length with the experimental data of Smith (1969). This proportionality constant will not be a function of Reynolds number or the acid- excess ratio. The predicted decolouration lengths for different acid-excess ratios are shown in Fig. 6. It can be seen that decolouration lengths decrease with an increase in acid-excess ratios. The polymer concentration might affect this proportionality constant. This is because we have assumed that turbulent fluctuations in the core of the pipe for polymer solutions will be the same as that without the polymer. The recent work of Berman and Tan (1985) shows that the effect of polymer on fluctuations is complex and can, in prin- ciple, result in increased or decreased fluctuations. Berman and Tan’s turbulent intensity data for the dilute solutions of polyethylene oxide show only mar- ginal differences from those for water. Since detailed information about how the turbulence intensity varies with polymer concentration is not available, we use the assumption that the fluctuations in the core remain approximately the same with and without polymer.

Before presenting the actual comparison of model predictions with the experimental data, it would be

Fig. 5. Experimental data on mixing lengths in Newtonian fluids

Symbol Acid-excess

ratio, x Pipe

diameter (cm) Remarks Reference

A - 0 1.5

6 27.4 1.0

15.0 Scalar mixing 2.0 -

0.31, 4.0 0.68 Multijet

Evans (1967) Smith (1969)

Toor Pohorecki and Singh and Baldyga (1973) (1983)


Fig. 6. Predicted decolouration lengths (influence of x).

Fig. 7. Predicted decolouration lengths (influence of VA/D’ at x = 1.5).

worthwhile to understand the trends and the sensitiv- ity of the predictions with the friction factor ratio. The friction factor ratio can be obtained from eq. (32). To understand the influence of solution relaxation time on friction factor ratio at constant Reynolds number, eq. (32) can be written as

fp/f= I/[1 + (16/9) Re (vA/D~)]~ (36)

where Re is Reynolds number and (vA/Dz) is the ratio of solution relaxation time and a characteristic momentum diffusion time. Figure 7 shows the pre-

dicted ratio of decolouration lengths with and without polymer at different values of dimensionless number (vA/D2) for acid-excess ratio of 1.5. As drag reduction increases, the decolouration length increases. It would be useful to know the upper limit on the decolouration lengths for the polymer solutions. The maximum drag-reduction asymptote discussed by Virk (1975) can be used to estimate the maximum possible friction factor ratio (flf) for a given Reynolds number as

fl& = 0.136Re0.33. (37)

The decolouration lengths for these maximum drag- reduction conditions are also shown in Fig. 7. This provides an upper bound on the mixing lengths in the case of drag-reducing polymer solutions.

For simulating the mixing experiments, we require an estimation of the friction factor ratio to predict the decolouration lengths in the presence of polymer. Equations (32) and (35) were used for this purpose. Figure 8 shows the comparison of the predicted and experimental decolouration lengths. It can be seen that the model simulates the experimental data reasonably well at two different polymer concentrations for the acid-excess ratio of 1.5. It should be noted that the model correctly predicts the increase in dimensionless decolouration lengths with an increasing Reynolds number, which is a special feature for polymer solutions. Model predictions for the flows without polymer are almost independent of the Reynolds number. The results of Brodkey (1975), Toor and Singh (1973), Pohorecki and Baldyga (1983) also show similar results with respect to Reynolds number. The direct comparison is, however, not possible because of the higher acid-excess ratios and specific geometries used in the work by these authors.

1x,* ziw 3.10

REYllQLDS NUMBER) R,

Fig. 8. Comparison of predicted and experimental data (influence of polymer concentration).


1,s 1s k ke 4 1, 11 12

Lmix

-L

M N

R,j

Fig. 9. Comparison of predicted and experimental data (influence of acid-excess ratio).

The experimental data at other acid-excess ratios and Reynolds numbers can also be examined under conditions when polymer concentration is kept constant. Smith (1969) has presented the decolouration length data for different acid-excess ratios for polymer concentration of 100 ppm. A comparison of the model prediction with his data for water and 100 ppm solution in Fig. 9 at various acid-excess ratio was made to see as to whether the model can predict the trends. It can be seen that the model simulates the influence of acid-excess ratio on decolouration length values reasonably well again.

5. CONCLUSIONS

A predictive equation without any adjustable constants has been developed for the estimation of the friction factor ratio with and without polymer. This seems to agree rather well with the empirical equations developed in the literature. A simple one-dimensional mixing model has been developed for simulating mixing in one-dimensional flows. The mixing model reasonably simulates the available experimental data on mixing lengths in turbulent flows of drag-reducing polymer solutions with the help of only one adjustable parameter.

NOTATION

fP

IE

polymer concentration concentration of component m in jth group pipe diameter volume fraction of jth group initial volume fraction of jth group reciprocal of t, friction factor for pipe flow friction factor for pipe flow for polymer solutions engulfment part of I,

t, tk

t hf

Lnix trlfs

U

inertial-convective part of I, intensity of segregation turbulent kinetic energy, wavenumber wavenumber corresponding to 1, length scale of energy containing eddies Kolomogorof’s length scale upper limit of polymer affected length scales lower limit of polymer affected length scales length required for complete mixing/de- colourisation segregation length scale ratio of t,, and t, number of subgroups rate of reaction of mth species in jth sub-

group Reynolds number Schmidt number time characteristic time for diffusion characteristic time for engulfment Kolomogorov’s time scale characteristic time for overall mixing mixing time (decolouration time) characteristic time for scale reduction process turbulence intensity turbulent stress mean velocity acid-excess ratio coordinate parallel to the length of the pipe

Greek letters

bl proportionality constant in eq. (15) E turbulent energy dissipation rate

8, turbulent energy dissipation rate for polymer solutions

9 Zimm relaxation time A solution relaxation time v momentum diffusivity

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