Chemical Enginewing Science, Vol. 48. NO. 9, pp. 1619-1628, 1993. oooY_25‘W/93 $6.00 + 0.00 Printed zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA in Great Britain. 0 1993 Axgamon Press Ltd zyxwvuts 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 zyxwvutsrqponmlkjihgfedcbaZYXWVUTS 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 inter- est 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; M ashelkar and Soylu, 1984), etc. However, the phenomenon of ma- cromixing and micromixing in drag-reducing fluids in pipes seems to have been poorly understood. The only data available in the literature where some interpreta- tion 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 dis- persion 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 charac- teristics (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 mix- ing 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 segre- gation reached a limiting value as the Reynolds num- ber 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 phenomeno- logical 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