Drop size turbulent pipe flow

S. Middleman, Drop size distribution produced by turbulent pipe flow of immiscible liquids through a static mixer, bid. Eng. Chem., Process. Des. Dev. 13(1), 78-83 (1974). 

This equation was found to describe collisions of droplets in turbulent pipe flow very well (K17). Coulaloglou and Tavlarides (C13) assumed that collision frequencies between drops of nonequal sizes are given by the following equation by analogy to kinetic theory of gases ... 

DROP-SIZE DISTRIBUTIONS PRODUCED BY TURBULENT PIPE FLOW OF IMMISCIBLE LIQUIDS. 

Almost all flows in chemical reactors are turbulent and traditionally turbulence is seen as random fluctuations in velocity. A better view is to recognize the structure of turbulence. The large turbulent eddies are about the size of the width of the impeller blades in a stirred tank reactor and about 1/10 of the pipe diameter in pipe flows. These large turbulent eddies have a lifetime of some tens of milliseconds. Use of averaged turbulent properties is only valid for linear processes while all nonlinear phenomena are sensitive to the details in the process. Mixing coupled with fast chemical reactions, coalescence and breakup of bubbles and drops, and nucleation in crystallization is a phenomenon that is affected by the turbulent structure. Either a resolution of the turbulent fluctuations or some measure of the distribution of the turbulent properties is required in order to obtain accurate predictions. 

Determine the type of flow that exists. Flow is laminar (also termed viscous) if the Reynolds number Re for the liquid in the pipe is less than about 2000. Turbulent flow exists if the Reynolds number is greater than about 4000. Between these values is a zone in which either condition may exist, depending on the roughness of the pipe wall, entrance conditions, and other factors. Avoid sizing a pipe for flow in this critical zone because excessive pressure drops result without a corresponding increase in the pipe discharge. 

A dispersion of one liquid in another can be obtained by passing the mixture in turbulent flow through a pipe. The largest stable drop size depends on the ratio of the disruptive forces caused by turbulent shear to the stabilizing forces of surface tension and drop viscosity. For low-viscosity drops such as benzene or water, the elfect of viscosity is negligible, and a force balance for drops smaller than the main eddies leads to... 

As already noted, Eq. (14.13) applies for drops suspended in a homogeneous and isotropic turbulent flow. The movement of a gas-liquid mixture may be considered as homogeneous and isotropic in the bulk flow, but near a wall this assumption does not hold. Besides, drops can be blown away from a liquid film at the wall, but other drops may be deposited at the wall. These phenomena result in a change of Eq. (14.13). Measurements of the average steady-state size of drops of gas-liquid mixtures in a turbulent flow in a pipe, as described in ref [3], have shown that the average drop size is described by the following relationship ... 

Consider as an example the mass exchange for water-methanol solution drops that were input into a turbulent gas flow in a pipe at the following parameter values pipe diameter d = 0.4 m pressure P = S MPa gas temperature Tg = 313 °K inhibitor flow rate q = 1 kg/thousand nm initial mass concentration of methanol in the solution xmo = 95%. Fig. 21.6 shows the dependence of the characteristic relaxation time of the system, t q, on the flow rate Q of the gas under normal conditions. The decrease of t q with the growth Q is explained by the reduction of the average drop size. All other factors being equal, the time teq grows with the increase of the pipe diameter and the pressure. Knowing Tgq, it is... 

The fundamentals are as follows. For immiscible liquids flowing in turbulent flow in a pipe of diameter, D, the dispersed phase breaks up into drops with the diameter of the maximum (or 95th percentile) size drop predicted as follows (Dp 95/D) = 4 [1/We] where We = Weber number. Since most drop size distributions are geometrically distributed and since the geometric standard deviation is about 2, the geometric mass average is about 30% of the Dp 95. Thus, the average size drop would be 300 pm if the predicted Dp, 95 = 1000 pm. 

When drops encounter shear or turbulence different from turbulent flow in a pipe, such as flow across a valve, or a rotating impeller in a pump, then (for interfacial surface tension of 30 mN/m), the following are typical drop size distributions from a reciprocating pump pumping an oil water mixture will produce... 

Given that a processing plant is a network of pipes and vessels, it is clearly important to be able to size every pump and all of the pipes. Thus techniques to calculate the pressure drop between the ends of each pipe are important. As detailed later, there are two main types of flow which will simply be called ordered streamline and chaotic turbulent for the moment. Detailed knowledge of the flow pattern is unimportant when calculating the pressure drop. Knowledge of pipe size, viscosity, density and velocity of the fluid enable the engineer to select the appropriate equation. The one equation for ordered streamline flow and the small set of equations for chaotic turbulent flow will be given later. 

Caution This equation only holds for turbulent flow when expressed in the following units. It was derived from charts used to size industrial piping systems. It is designed to provide a conservative high value for pressure drop. To verify that the flow is turbulent, refer to Sec. 48.3.3. 

Values of k and k for various polymer/tube systems are given in Table 5.10. (Values of k and ki can be determined for a given polymer solution from laboratory measurements of pressure drop in smooth tubes at two flow rates in the turbulent range.) These values can be used with the model to predict friction loss for that solution at any Reynolds number in any size pipe. If the Colebrook equation for smooth tubes is used, the appropriate generalized expression for the friction factor is... 

Figure 5.9 is useful for calculating from a known pipe size and flow rate, but it cannot be used directly to determine the flow rate for a given pressure drop, since is not known until V is determined. However, since / changes only slightly with Nr, for turbulent flow, a trial-and-error solution converges quickly, as shown in Example 5.1, page 109. 

The pumping cost goes down rapidly as the pipe size goes up. The pumping cost is proportional to the pressure drop (see Example 6.3), which for turbulent flow is proportional to the velocity to the 1.8 to 2.0 power divided by the diameter. The velocity (for constant flow rate) is proportional to the reciprocal of the square of the diameter, so the pumping cost is proportional to the reciprocal of the diameter to the 4.6 to 5 power. 

We shall demand that the final size of drops must be equal to the average size R (see (14.14)) formed in the turbulent flow inside the pipe. Using the expression Vat, = 4tiR /3 and the correlation (14.5) for lo and taking t= L/U, one gets the required distance from the DPC to the separator ... 

---