| Reynolds Number
Why we cannot use tank GT values when sizing a static mixer.
The work of Thomas R Camp established the use of the root mean square velocity gradient G expressed in reciprocal seconds when dealing with flocculation and mixing problems as early as 1943. G is defined as:
G = (W/m )0.5 sec-1
where W = total power dissipated in the tank volume
and m = absolute viscosity of the fluid
Later, based on a study of flocculation basins in 1955 Camp introduced the term GT where T is the flocculation or mixing time in seconds. G values ranged from 25 to 74 while the dimensionless GT values found in the 21 plants studied ranged from 23,400 up to 210,000. It was assumed that all these coagulation basins were operating satisfactorily.
As static mixers came into common use in the water treatment field in the 1960's engineers started to specify G and GT values for static mixers based on tank mixer data. This was done despite the fact that:
The residence time distribution a static mixer is very narrow when compared with a tank type mixer.
The plug flow characteristic of a well designed static mixer means that all material passing through a static mixer has received the same mixing experience characterized by the GT value involved.
These differences mean that both the G and the GT values that are applied to tank mixer specifications cannot be applied to all static mixer designs. In general, much higher values for G can be tolerated in static mixers since the G values throughout the mixer structure have a narrow distribution. This, combined with the low residence time avoids floc damage. Formation of floc actually begins after the liquid and flocculating agent have left the high shear rate region in the static mixer. In general the shear rate in a static mixer can be much larger than that experienced in a mechanically stirred flocculation tank without floc damage as discussed above. Applying a mixing time T value to a full scale static mixer design based on a flocculation tank retention time can result in a static mixer design of impractical and unnecessary length.
The following equations apply to a static mixer.
G = 5939(D PQ/d2l)0.5 sec-1 , T = 0.204ld2/Q sec , GT = 1212d(D Pl/Q)0.5 - dimensionless D P [=] psi, Q [=] gpm, d [=] in, l [=] in
Typical flocculation tank values would be G = 40 sec-1, GT = 50,000 , T = 1250 sec. with a flow rate through the system of 22,588 gpm. This is to be replaced by a 48" diameter static mixer system where D P = 0.5 psi and mixer length = 15 feet. This would give values G = 980 sec-1, GT = 3,675 and T = 3.75 sec. To insist on the flocculation tank value of T = 1250 seconds would require a static mixer length of over 204 ft long!
Velocity range in Komax water treatment mixers is 1 ft/sec to 8ft/sec. Typical propeller tip speed with a mechanical mixer is over 30 ft/sec. Which one will do more floc damage?
Komax Systems static mixers are excellent for mixing polymer and other water treatment chemicals into small or large flow rates. Over the last thirty years more then 8,000 Komax static mixers have been installed in municipal and industrial water and waste water treatment plants.
ADVANTAGES OVER MECHANICAL MIXERS
Lower equipment cost.
Lower installation cost.
Lower energy requirement ( 95% less)
Longer life. (25 years +)
This portion of this document was originally produced for internal use at Komax to provide a common source for the way we make many of our calculations. It can also suggest a route to help solve unusual problems and to provide assistance to customers. It is certainly not the last word on the subject and suggestions for additions, corrections and changes are welcome.
During the last 30 years Komax has developed into an engineering, manufacturing and problem solving company whose core products are based on static or motionless mixer technology. Many patented and non patented designs have been applied world wide to a range of industrial fields including water treatment, plastics, pulp and paper, steam heating and desuperheating, reactors and the chemical industries in general. Process applications in these fields can involve mixing and separation, heating and cooling, contacting and mass transfer applications where residence times can range from a few milliseconds to hours.
Much of the success of Komax designs is based on the concept that many in-line mixing problems are best approached by paying attention to what we call the "entrance conditions" of the problem at hand. In other words, if we are to mix components A and B, how are A and B first introduced to one another? Most manufacturers have paid little attention to this consideration and indeed, the literature available on the subject is sparse. Over many years Komax has devoted considerable research and development effort on this subject, often in cooperative programs with customers. This work has produced a number of devices we call Distribution Heads or DH units.
DH units control the manner in which inlet components are first introduced to each other before entering any downstream mixing device. In general a large interfacial area is generated between components together with a small interfacial thickness. This approach can solve many mixing problems with low capital expenditure and modest energy cost. In some cases the DH unit alone can solve the entire mixing problem. In other situations the contribution of the DH unit can make a mixing task simple that would normally be considered impossible. This type of design approach is typified by the well known Komax logo, "Mixing by Design".
Komax offers test facilities to customers and clients where concepts and hardware for solutions to mixing problems can be tried on a confidential basis.
The flow of fluids through pipes and other conduits can be characterized by two flow regimes known as laminar flow and turbulent flow. Laminar flow occurs at very low fluid velocities and high fluid viscosities. Turbulent flow happens when the velocity is high and the viscosity is low. In laminar flow the fluid moves with non intercepting stream lines, but in turbulent flow many large and small vortices exist.
Pioneer work by Osbourne Reynolds established a way to define these two regimes using a dimensionless number we now call the Reynolds number Re where:
Re = 3157QS/m d
Q = flow rate in US gpm
S = specific gravity = 1 for water at 60o F
m = viscosity in centipoise = 1.0 cP for water at 60o F
d = inside pipe diameter in inches
Reynolds conducted his experiments using internally polished glass tubes about one to two inches in diameter. Dye was injected upstream to allow the flow to be visually studied. Extreme care was taken to minimize any vibration.
The Reynolds number value for transition from laminar to turbulent flow is usually accepted as being about 2,000. It has been demonstrated that below 2,000 we always have laminar flow and pressure drop in the piping is proportional to the flow rate Q. It is possible if great care is used in the piping construction and the dye injection system to get laminar flow at a Reynolds number as high as 50,000. However, as a practical matter when ordinary commercial pipes are used, by the time we reach a Reynolds number of 4,000 we usually have turbulent flow and the pressure drop is proportional to Q1.8 to Q1.9. The regime between 2,000 and 4,000 is conveniently called the transition region. By the time we reach Re = 10,000 the exponent of Q is almost exactly two.
Just about any structure installed in a pipe will reduce the Reynolds number required for the onset of turbulence, and the actual value of Reynolds number needed will depend on the particular structure used.
As a practical matter, when dealing with low viscosity fluids such as water, laminar flow conditions are rare. Conversely, when handling a material where the viscosity is more than about 100 Cp., the flow is usually laminar.
The Reynolds number can be regarded as a way of recognizing where viscous forces control the flow condition and where inertial forces dominate. Under laminar flow conditions the viscous forces dominate and the pressure drop is proportional to the flow rate Q and the viscosity but is independent of the specific gravity. Under turbulent flow conditions the effect of viscosity variations on pressure drop are small and the pressure drop is proportional to the specific gravity S and Q2.
When making pressure drop calculations in the transition region be sure to check to see if the flow type has shifted from one type of flow regime to the other.
Use caution when attempting to extend the Reynolds number concept beyond application to circular conduits. For example it is often tempting to use the concept of the "equivalent hydraulic diameter" dH where
dH = 4 x conduit area/wetted perimeter
In the case of a circular pipe:
dH = 4 x (p d2/4)/p d = d the inside pipe diameter.
If now for example we calculate the value of dH for a narrow slit andthen use this value to calculate a Reynolds number one can get incorrect and sometimes quite bizarre results.
The Reynolds number concept is meaningless when the conduit carries
a non mixed combination of different fluids having different physical properties.
The existence of turbulence in a pipe does not necessarily mean the existence of significant or adequate mixing. Case histories show this is so. A number of years ago a municipality contact Komax Systems that had a mixing problem. They wanted to mix two water steams, one form a reservoir the other from a well. Each water steam had a different water hardness and they needed them blended. They were pumping into a 42" pipeline. The Reynolds number was well over one million. Down steam at the 500 foot area sample ports on each side of the pipeline showed reservoir water on one side and well water on the other side. The municipal engineers were puzzles. Komax theory is that there is high turbulence in large diameter pipelines but it is occurring in striated bands. There is not enough radial force to move the water molecules from wall to wall of the pipeline. Now if you cannot mix water and water at a high Reynolds number in an open pipeline think how different it is to mix water treatment additives with out a static mixer. Komax solved the blending problem with a 42" two element low pressure drop static mixer. .
"A raw randomness ungoverned by words or math, an unordered whirlwind of particles as inexpressible to engineers as angels dancing on the head of a pin."
Swirling eddies variegated into smaller sub eddies, and so on, down to individual molecules."
That is close to what is happening inside a Komax Triple Action Static Mixer It is the only static mixer on today's market that produce's three mixing actions simultaneously.
Clockwise and counter clockwise rotating vortices on each side of each mixing element. Resulting in smaller and smaller diameter rotating eddies.
Two by two stream division.
Right side to left side stream impingement at a 90 degree intersection at the end of each mixing element.
HOW DO WE PROVE THE MIXERS PERFORMANCE?
Computer modeling of mixing? We think not. It is subjective not objective.
Komax can provide in field mixing test reports on small and very large static mixers proving 98% distribution.
A trace element is injected into the mixer at three flow rates and multiple samples are taken at varying points across the pipeline or channel. Sample containers are identified to their sample point. The sample containers are sent to an independent laboratory for analyses of the trace element distribution.
Very difficult mixing application solved by a Komax Triple Action static mixer.
Mixing chlorine dioxide bleach solution into 12% consistency paper pulp stock. No other static mixer on the market can mix 10% consistency let alone 12% consistency paper pulp stock. Dozens and dozens of major paper mills in North America have installed Komax static mixers replacing high horse power mechanical mixers. If the Komax static mixer can mix chemicals into 12 % consistency paper pulp stock just think how well they mix in water treatment applications.