Minimize Blending Time

Calculating the time actually needed can lead to economic and operational benefits.

By David S. Dickey, MixTech, Inc.

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Most industrial applications blend liquids for much longer than required — incurring unnecessary burdens on both energy budgets and equipment. Reducing blend time offers an opportunity for increased productivity and decreased costs. Knowing and understanding process requirements is the first step in estimating proper blend time for your application.

This article focuses on stirred-tank applications, typically involving center-mounted top-entering mixers. Most low-viscosity turbulent applications require baffles to prevent uncontrolled swirling because solid-body rotation of liquid doesn't create effective mixing. These systems have been studied well enough to provide some actual quantified guidance for predicting process blend times. Other mixers, such as angle-mounted portable mixers, when appropriately positioned quickly blend low-viscosity batches, usually in less than a minute or two [1].

Testing
Blend time generally is determined by adding a small quantity of liquid to an agitated batch of similar property liquid. Laboratory tests most often add a liquid that results in a color change, from some hue to clear, and conduct tests in transparent tanks — allowing observation of the final color change indicating a degree of complete blending, no matter the location in the tank. Because low-viscosity mixing is a turbulent process the exact location and time will vary from test to test as flow patterns fluctuate. Changing flow patterns and random velocities both are mechanisms by which rapid blending may occur in turbulent conditions.

Some more-common color change methods use either a pH indicator solution or an iodine color remover. With the pH approach, a color-to-clear indicator, such as phenolphthalene, first is set to the color form by a dilute caustic solution mixed in the test tank. Then a small quantity of a more-concentrated acid is added on the surface of the liquid. The quantity and concentration of the acid suffice to take the batch from alkaline to acidic condition, making the colored indicator disappear. Repeated tests with careful addition and timing will establish a good average for blend time at the prescribed degree of uniformity.

Blend-time test results typically are correlated as a dimensionless blend time (Θ), which is expressed as measured blend time multiplied by the mixer's rotational speed. This group is dimensionless because blend time has the units of time and rotational speed has the units of reciprocal time. The value will be the same for any unit of time, so long as both variables are expressed in that unit. Sometimes Θ is affectionately known as the Betty Crocker number because the time to uniformity effectively is related to the number of impeller revolutions (beater strokes). For turbulent mixing, Θ is a constant.

Blend time also depends upon the impeller type and the impeller-to-tank diameter ratio. (We'll address the effects of fluid properties separately.) However, Θ is independent of absolute vessel size. So for geometrically similar mixers and tanks, blending in a small or large tank requires the same number of impeller revolutions. This ability to apply laboratory measurements to process vessels is essential to the practical use of blend time correlations. Scale-up holding blend time constant is impractical as power requirements quickly become excessive.

Determining Blend Time
Correlations have been developed in a form involving Θ, rotational speed (N), impeller-to-tank diameter ratio (D/T), liquid-level-to-tank diameter ratio (Z/T), and number of similar impellers

(ni): ΘN(D/T)i(Z/T) -jnik = A (1)

If the units for time and length are consistent, A is a constant for an impeller type. Of more practical value is a rearrangement that gives actual blend time for specific types of impellers.

For a four-blade 45° pitched-blade turbine, blend time for 99% uniformity can be expressed as:

Θ99% = (6.34/N)(D/T)-2.3(Z/T)0.5 ni-0.7 (2)

For a four-blade straight-blade turbine, which creates radial flow, the expression becomes:

Θ99% = (4.80/N)(D/T)-2.3(Z/T)0.5 ni-0.6 (3)

For hydrofoil impellers, typical three-blade, narrow-blade or marine propellers, the expression is:

Θ99% = (16.4/N)(D/T)-1.7(Z/T)0.5 ni-0.8 (4)

Achieving 99% uniformity may not suffice for some applications. However, because uniformity by blending follows an exponential relationship, it's possible to adjust for other degrees of uniformity (Table 1).

While Eqs. 2–4 show that a hydrofoil impeller needs more time for blending (because of its larger constant) than pitched-blade and straight-blade turbines, it offers benefits in power and torque reductions because of its lower turbulent-flow power number (Np):

Np = 1.37 for 4-blade 45° pitched-blade turbine;
Np = 3.96 for 4-blade straight-blade turbine; and
Np = 0.31 for 3-blade hydrofoil impeller.

Example Calculation
Suppose a blending application involves adding a small quantity of an active agent to about 3,000 gal. of a water-like liquid in a 96 in.-dia. tank. Liquid level would be about 96 in. A single 30-in.-dia. pitched-blade turbine turning at 68 rpm (1.13 s-1) provides mixing. Substituting these conditions into Eq. 2 yields an 81-sec. blend time. So in less than one-and-a-half minutes, the 3,000 gal. of liquid can be blended to 99% uniformity.

Feed Rate
Figure 1: Feed rate --  Rate of addition can significantly
impact the amount of time needed to attain desired uniformity.
It's also possible to find turbulent impeller power (P) in hp. via:

P = (Np sp.gr. N3D5)/1.524 × 1013 (5)

where sp. gr. is fluid specific gravity, N is in rpm and D is in in. In this case, P is about 0.69 hp., so a 1-hp. motor would suffice.

Now as a potential process improvement, let's consider replacing the pitched-blade turbine with a hydrofoil impeller. It could be used at the same speed and sized for the same power draw as the pitched-blade turbine. Because the hydrofoil impeller has a lower power number, it must have a larger diameter — 40 in. in this case — for the same power. The hydrofoil impeller still will require 0.69 hp. but will reduce blend time to only 41 sec. Thus, with the same power and speed, which is also the same torque, the hydrofoil gives more rapid blending. Before making a switch, though, it's important to check the natural frequency of the shaft to ensure the weight of the larger impeller doesn't cause mechanical problems.

Feed Conditions
Not all blending jobs involve adding a small quantity of liquid to a well-mixed batch. You must consider both the rate and quantity of addition in real process applications.

Figure 1 illustrates three potential feed rates. Let's look at their implications on blend time. Say a slow feed rate requires 5 min.; so uniformity can't be achieved until after 5 min. — probably more like 5 min. plus 81 sec. for our example. At an intermediate feed rate that's much less than the estimated blend time, 81 sec. may be appropriate. A high feed rate, sufficient to influence the blending flow pattern, may lead to a slightly reduced or at least different blend time.

Feed Location
Figure 2: Feed location -- Adding liquid just above the
tip of down-pumping axial-flow impeller generally
provides best results.
The quantity of addition will have effects similar to those for feed rate. If putting in that amount of liquid will take longer than estimated blend time, add addition time to estimated blend time. When feeding in a large volume, use the final liquid level to estimate blend time following the addition.

Sometimes location of the feed is more important than its rate or quantity. Published data typically come from experiments using surface feed (Figure 2). Because most tanks have bottom valves a feed location at the bottom of the tank also is possible. However, neither of these locations provides any special advantage, as both rely on adequate liquid motion between feed location and impeller region to initiate intense blending. At least part of the blend time is required to move the feed to the impeller region.

The most vigorous mixing occurs in the immediate region of the impeller. Local power-per-volume dissipation can exceed average dissipation by an order of magnitude. For fast chemical reactions, especially those with competing or consecutive reactions that produce alternative byproducts, feed near the impeller can be critical. The rate of initial blending may determine the quality and quantity of desired product. The ideal location for the feed is just above the tip of a down-pumping axial-flow impeller. At that point the feed is drawn quickly though the most-intense mixing, speeding both initial dispersion and final blend time. Velocity of flow from the dip pipe must be fast enough to prevent back-mixing at the tip and at a velocity comparable to local velocities so as not to adversely influence the flow created by the impeller.

While a wall feed location may appear to offer similar benefits to the dip-pipe location near the impeller, flow between the tank wall and impeller may follow many paths. Experience shows that wall feed causes more problems than it potentially could solve.

Physical Property Effects
Density and viscosity also can significantly impact blending and blend time. The most important influence of density appears in impeller power. For turbulent mixing impeller power is proportional to liquid density. Density differences between bulk liquid and the addition also may affect blend time. Viscosity differences can have an even bigger effect, both on bulk blending and on blend time.

The primary effect of fluid properties on both blend time and power are reflected in the impeller Reynolds number (NRe), which is expressed in terms of impeller diameter, rotational speed, density (ρ) and viscosity (μ): NRe = (D2Nρ)/μ (6)

Applying a coefficient of 10.7 makes the expression dimensionless for typical engineering units of in. for impeller diameter, rpm for rotational speed, specific gravity for density and cP. for viscosity. As viscosity increases or size, as indicated by impeller diameter, decreases, NRe becomes smaller.

NRegreater than 20,000 is characteristic of turbulent conditions — in which flow fluctuates in both magnitude and direction about a mean. NRedoesn't describe the intensity of mixing, only the type of flow conditions. At very lowNRe, less than 10, flow becomes laminar and velocities follow streamlines. In the transition region between turbulent and laminar conditions, fluid motion may be less turbulent away from the impeller.

With less turbulence mixing becomes slower, as both average and random velocities decrease. This slower blending correlates to larger values for Θ. For convenience, this effect can be handled as a correction factor for different impeller types (Figure 3). The factors show that blend times may increase by orders of magnitude as viscosity rises and NRe becomes smaller. Other correction factors must be applied to power calculations in the transition and laminar range.

Other effects of density and viscosity occur when one of the blended liquids has a distinctly different property. A significant density difference, especially when blending begins with stratified layers in a tank, may extend blend times by as much as a factor of five to eight. Avoid starting a blend with stratified layers if possible. Blending of different viscosity liquids is a very real and often difficult problem. Some common "kitchen" examples are mixing corn syrup or ketchup in water. Corn syrup is a viscous liquid. Ketchup not only is viscous but also has a yield stress. Figure 4 shows some correction factors for the effects of viscosity ratio on blend time at different bulk NRe. At high NRe even a large viscosity ratio between the feed and bulk has only a minor impact. Effects of different relative quantities of liquids with different properties and effects of nonNewtonian viscosities lengthen required processing times.

Enhance Performance
Estimation of blend time isn't a precise science — blending of individual batches may vary as a function both of turbulence variations and operating procedures. Thus, designing for a calculated blend time of a few minutes is probably inappropriate. However, if an estimated blend time is less than 5 min., then mixing for an hour or even one-half hour may be unnecessary. Some blending applications require other processes to take place, thus justifying longer mixing times. Understanding some major factors influencing blend time, from the inverse relationship with respect to rotational speed of the mixer to the effects of viscosity and viscosity difference, will help improve process results and efficiency.


David S. Dickey is senior consultant at MixTech, Inc., Dayton, Ohio. E-mail him at d.dickey@mixtech.com.

Reference:
1. Dickey, D. S., and L. B. Fenley, "Make Your Portable Mixer Work for You," Chemical Processing, p. 34, March 2007 (www.ChemicalProcessing.com/articles/2007/040.html).
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