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The increasing demands made on mixing processes
mean that the efficient solution of the process
problems involved in designing stirred vessels are
continually gaining in importance. In recent years, the
dramatic increase in computing power has led to the
increased use of computational fluid dynamics (CFD)
as a design tool. Linking this closely to experimental
results from model/pilot-scale tests and knowledge
gained from process technology enables problems to
be analyzed cost-effectively in a minimum of time. An
expert system uses the results from laboratory tests,
CFD simulations and industrial experience to produce
an optimized agitator design.
PROBLEM ANALYSIS
The first step in designing mixing processes is toanalyze whether the available expertise, such as thatformalized in the EKATO expert system ANTOS, issufficient, or whether there are problems which canonly be solved by first performing tests and/or CFDsimulations, see Fig. 1.
The process parameters relevant to the problemsconcerned are required as basis:– Vessel dimensions– Product physical properties, such as density, ρ
and viscosity η– Discontinuous or continuous production mode– Process description covering formulation, chan-
ges in level and viscosity– Is only one product to be manufactured, or is a
complete range of products involved, each withvery different requirements for the agitatorsystem?
The CFD simulation process and/or the test work cover:
– Definition of the principal mixing duty– Selection of a suitable agitator system– Scale-up rule to be applied or direct CFD simula-
tion on a commercial scale
Fig. 2 shows a list of the key parameters andcharacteristics for the various mixing tasks. In indu-strial practice it is common for several mixing tasksto overlap, thus increasing the complexity of theproblems to be solved.
In the problem analysis the key parameters mustbe considered over the full time of the process.Numerous reaction stages often occur in the course ofa chemical process. The agitator system can impactdecisively on integrated process operations such asprecipitation, crystallization, filtration distillation atvarious stages of their process. When manufacturing asingle product, not only can the volume and temper-ature of the vessel contents change, but also theviscosity, and often by very large factors.
The sensitive microorganisms present in bio-reactors mean than only gentle gradients are permis-sible for the process parameters. It is therefore neces-
49
EKATO Handbook of Mixing Technology Problem analysis
Mix
ing
pro
cess
des
ign
Mixing process design
Problem analysis
Testing
Computational fluid dynamics
Expert system
Problemanalysis
Process engineeringKnowledge baseExpert system
Computationalfluid dynamics
Experiments with models/ testing
Fig. 1 Techniques when designing mixing processes
sary to distribute the substrate, oxygen and thetemperature throughout the entire contents of thevessel as rapidly and as uniformly as possible. In afermenter for producing primary metabolic substancesin fed-batch operation, for example, there is a highoxygen uptake rate, therefore accompanied by a largeamount of heat being evolved. Although the volume ofliquid is low when fermentation begins, and there isthus only a small surface area available for heat trans-fer, the heat that is developed must nevertheless beremoved. The agitator/vessel/internals system musttherefore ensure the required high rate of heat trans-fer.
50
Problem analysis EKATO
Fig. 2 Typical parameters for mixing processes
Product
characterization
Vessel
features
Mixing process
requirements
Blending Flow properties(Newtonian/non-Newtonian
Preparation orstorage vessel
Required blend time; flow velocity;special requirements such as drawingin at the surface or strongagglomeration tendencies,entrainment of air must be avoided
Suspending Particle diameter, solidsconcentration, density of liquidand solids
Correct locationof outlet
Is a uniform suspension required or isoff-bottom suspension enough; agita-tor start-up with sediment build-up
Dispersing Required stoichiometric excess of reactant gas; are solids (e.g. catalysts)present which must be held in suspension?
Type of gassing,self-gassing orseparate internal(ring sparger,gassing lance)
Course of reaction; desired masstransfer rate; gassing rate; superficial gas velocity; mass transfer coefficient kL·a;local/global power input per unitvolume
Heat transfer Specific heat capacity cp, and thermal conductivity λ ofthe product
Heat transfer areaof the vessel;temperature inheating and coo-ling circuit; exter-nal film coeffi-cient αS; materialand thickness ofvessel wall
Temperature difference at beginningand end of process; desiredheating/cooling time; local/globalheat transfer coefficients
00 0.2 0.4 0.6 0.8 1
50
100
150
0
0.4
0.8
0.2
1.2
0.6
1.0
Oxygen uptake rate OUR [mol 02/(m3 · h)]
Relative time of fermentation
Relative volume of liquid V/VmaxRelative heat generation Q/Qmax
OU
R [
mo
l 02/
(m3
· h)]
V/V
max
, Q/Q
max
Fig. 3 Agitator/relevant process parameters for a500 m3 fermenter
TESTING
Application-oriented testing
Basic testing
“Basic Testing” is carried out to determine the hydro-
dynamic properties of agitator impellers, such as the
blend time characteristics, the measurement of the
energy dissipated and of flow profiles using laser-
Doppler anemometry, plus the determination of
power numbers and radial force coefficients. The
results of these tests form the basis for agitator
designs. So-called “customer tests” are often carried
out for particular applications so that the special
process characteristics and produce properties can be
taken into account in the design work. Tests of this
type offer the following advantages:
– Rapid performance of the test, usually with the
original products and within 1 to 5 days
– Exhaustive testing of a variety of agitator
impellers at the standard test scales of 3 l
and 50 l (1 m3)
– Analysis and optimization of the mixing
process in terms of the desired process
specifications
– Determination of design-relevant physical
properties of the original product
– Assured scale-up.
51
EKATO Testing
Mix
ing
pro
cess
des
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Fig. 4 Laboratory and pilot-scale test equipment for basic testing and application-oriented tests
APPLICATION-ORIENTED TESTING
Product characterization and rheometry
Determination of the effective viscosity at the agitatorimpeller is of fundamental importance for many agi-tator designs. To design calculations at different vis-cosities illustrate the typical effect of viscosity on therequired agitator power.
In both cases a two-stage EKATO VISCOPROPagitator system with an impeller diameter d2 = 1100 mmserves as basis for the calculations.
The viscosity of Newtonian media is independentof the agitator shaft speed or shear rate. If non-New-tonian flow characters are displayed, however, therelations η= f (n) or η= f (γ· ) must be determined, sincethe scale-up process results in different shaft speeds atthe test scale and the commercial scale.
In order to avoid making numerous individualmeasurements at various agitator shaft speeds, it isoften more practical to carry out a measurement usinga rotational viscometer. With this measurementtechnique a rotating cylinder is accelerated in a shorttime from rest up to a maximum speed of rotation(maximum shear rate), shearing the liquid for a definedtime, after which the speed is reduced again to zero.During this time the viscosity and the shear stress areplotted as a function of the shear rate.
The measurement processes described below arenormally used for determining the effective viscosity atthe agitator impeller.
Viscosity measurement via the agitator impeller usingthe Rieger-Novak method [1]
The following schematic illustrates the measurementprocedure. This process can be used in the laminarflow range and for identifying the turbulent flow range(Ne = constant).
To calculate the power on a commercial scale, theoperating viscosity is read off from the agitator vis-cosity curve, as shown in Fig. 5. This is used to calculatethe Reynolds numbers, after which the correspondingpower number is read off from the power character-istic. This gives the impeller power on the commercialscale.
Measurement with rotational viscometer and deter-mination of the shear rate for the agitator shaft speedusing the Metzner-Otto method [2]
This process applies to the laminar flow range.
The scale-up process gives the agitator shaftspeed, the assumption being made in the subsequentsteps that the shear rate γ· developed by the impeller isproportional to the frequency of rotation n of theagitator. The shear rate is calculated using the Metzner-Otto constant kM/O, which is a proportionality factorspecific to the impeller concerned.
γ· full-scale = kM/O · n
52
Testing EKATO
Measurement of torque Mt at laboratory scale at various shaft speeds
Calculation of agitator power from measuredtorque and shaft speed P = 2π · n · Mt
Determination of the Reynolds number from the power characteristic Re = f(Ne)
Calculation of the associated apparent or effective viscosity ηapp
The two values (ηapp,n) are used to plot theviscosity curve for the agitator
Selection of suitable rotating body
Attemperation at 20–90°C possible
Plotting of viscosity against shear rate η= f(γ·)
Approx. fitting of measured viscosity curve todefined flow curve, e.g. Ostwald-de Waele power
law curve with consistency factor K and flowbehavior index m.
Case 1 Case 2
measured viscosity 2000 18,000η (mPa·s)
n (min–1) 50 50
ρ (kg/m3) 1000 1000
Re 504 56
Ne = f (Re) 1.4 2.1
Pimpeller (W) 1304 1957
The following approximate values are typicalMetzner-Otto constants for commonly encounteredagitator systems:
Propeller agitator kM/O = 10Flat blade disk turbine kM/O = 12Ribbon agitator kM/O = 30Anchor agitator kM/O = 25
The Metzner-Otto constant kM/O is determined forindividual agitator systems by comparing the measure-ment results from the rotational viscometer with thoseof the “agitator viscometer” by plotting correspondingvalues of (γ·, n) for the same viscosity η. The effectiveviscosity at the impeller is then determined from theviscosity curve using the calculated shear rate γ·, seeFig. 5. The agitator impeller power for the full-scalesystem can then be determined as for the Rieger-Novakprocess.
Viscosity measurement processes
With viscosity measurement processes a fundamentaldistinction must first be made between the measure-ment of absolute values and the determination ofempirical values. Various industries often use charac-teristic, media-specific scales with arbitrarily selectedunits as a measure of viscosity. Conversion of relativeviscosity data of this type into a physically defined unitof viscosity can only be performed reliably if accurateknowledge of the measurement process concernedand the specific characteristic data of the measuring
instrument are available. If the viscosity data are un-reliable with regard to the effective viscosity at theagitator impeller, the actual flow properties of themixture of substances should therefore always betested using an original sample of the actual product.EKATO offers the facilities for doing this in the cus-tomer laboratory.
Frequently used relative measurement pro-cesses, which cannot however be directly converted tothe viscosity that is effective at the impeller are forexample:– The flow cup to DIN 53211
(determination of flow time as a measure of theviscosity, mainly used for paints and varnishes).
– The Engler viscometer to DIN 51560(comparison of the outflow time of the liquidconcerned to the outflow time for the samequantity of distilled water at 20°C; mainly used inthe oil industry).
Absolute viscometers are characterized by being ableto set exact test conditions for both the shear rate γ· andthe shear stress τ. Fundamental distinction is madebetween two measurement principles: capillary androtational viscometers. Capillary viscometers aresuitable only for low-viscosity Newtonian media,whereas rotational viscometers can also be applied tonon-Newtonian liquids over a wide range of viscos-ities.
1. Capillary viscometerThe form of the outflow nozzle as a capillary tubemeans that the conditions for laminar flow accor-ding to the Hagen-Poiseuille equation areattained. The viscosity of the liquid can thereforebe determined, since the geometric measure-ments are known and the outflow time of theliquid is measured.
2. Rotational viscometerThe requirement for a defined shear gradient γ· isnearly fulfilled in rotational viscometers by anarrow coaxial annular space filled with product,in which laminar flow is created by the rotation ofthe inner cylinder. At a constant speed of rotationand sufficiently small annulus the shear rate isdefined, with the simplifying assumption that thevelocity profile across the annular gap is linear. Atorquemeter is used to measure the torque, an asthe area is known (2·π ·Ri·h), the shear stress τ.can be calculated. Pairs of measurement readings(γ·, τ) can be determined by varying the speed ofrotation and plotted as a flow curve in the normalmanner.
3. Dropping ball viscometerThis measurement principle is based on Stoke’slaw for the drag exerted by the laminar flowaround a sphere in a viscous medium. The time
53
EKATO Testing
Mix
ing
pro
cess
des
ign
Viscosity curve
Metzner-Otto:n
Haake rotationalviscometer
Viscomon agitatorviscometer
nRieger-Novak:
Power characteristic
γ = kM/O · n
γ
ηapp
η ηappη
Re = n · d22 · ρ/ηapp
P = Ne · ρ · n3 · d25
Re
Ne
n
Fig. 5 Calculating agitator power for non-Newtonianmedia
needed by the ball to fall through a specifieddistance is a measure of the viscosity of the liquidbeing tested.
4. Brookfield viscometerThis type of viscometer is used in many produc-tion plants for process monitoring and is similarin design to the rotational viscometer describedabove. The design of this instrument is such thata defined shear gap is only formed at two pointson the rotating cylinder (anchor). Measurementsmade with this instrument should therefore notbe regarded as absolute values. The viscositiesdetermined with the Brookfield viscometer arenormally considerably higher than those foundwith the rotational viscometer. Brookfield vis-cometers should therefore be classified as instru-ments determining a relative viscosity.
Blend time determination
In application tests the original product is nearlyalways used. Since these are usually not transparentmedia, blend time tests can only be carried out in rarecases (without using expensive sensor techniques).Colors or silicone dyes are metered into the product tomake the blending processes visible, see Fig. 6. As arule, the product data that are acquired, such as theviscosity, are measured and the blend time is deter-mined through empirical correlations. The reader isreferred to “Basic mixing tasks”, pages 25ff, 46, for thecorrelations used and literature sources.
Suspending
A cost-effective agitator design that fulfills suspensionrequirements requires an accurate knowledge anddemarcation of the various possible suspension con-ditions. Fig. 7 illustrates the degrees of suspension“on-bottom motion”, “off-bottom suspension” and“visually uniform suspension”.
With “on-bottom motion” although most of thesolids are in motion, large clear (i.e. particle-free) zonescan occur in the upper part of the vessel. Temporarydeposits of solids are tolerated underneath the im-peller and in the corners.
“Off-bottom suspension” defines a condition inwhich individual particles never come to rest for longerthan a second at the bottom of the vessel. Zwietering[3] was the first to work with this visual suspension cri-terion. He determined minimum shaft speeds njs (js = justsuspended) at which this condition first occurs. The off-bottom suspension point can be determined by methodssuch as the ultrasonic Doppler anemometer [4], whichdo not, however, take into account the uniformity ofdistribution in the vessel. This means that particularlywith particles with high settling velocities, “on-bottommotion” and “off-bottom suspension” are fulfilled, thereare often clear zones near to the liquid surface with noparticles in them. In these cases a further assessmentprocess is employed, the so-called “layer thicknesscriterion” [5], at which the layer of particles in suspen-sion has reached 90 % of the height of the liquid.
In contrast, “visually uniform suspensions” haveno large clear zones. In this condition, settling of thecoarser particles can still occur, if for example there isa wide range of particle sizes, even though the fineparticle fraction is already held in suspension near theliquid surface. If “visually uniform suspension” is usedas the sole criterion, the shaft speed required toachieve it can therefore be greater or less than that for“off-bottom suspension”.
For the more demanding suspension duties, e.g.continuously operated equipment with an overflow ofthe suspension, a “uniform suspension” is required. Toachieve this both the “one second criterion” at thebottom of the vessel and the “visually uniform sus-pension” criterion must be fulfilled.
The Society for Chemical and Process Engineer-ing (GVC) of the German Association of Engineers (VDI)has produced a standard test method [6] to improve thecomparison and assessment of suspension testing.Fig. 8 illustrates this method with the results of a seriesof tests carried out with EKATO ISOJET impellersarranged as single-stage and two-stage agitators. Thepercentage difference in concentration of a solidssuspension between two measurement points locatedat 0.2 h1 and 0.8 h1 has been plotted against the agitatorpower input. A lower power is required to achieve theone-second criterion with the single-stage EKATOISOJET agitator than with the two-stage agitator,because the entire power input in the former system isnear to the bottom of the vessel. Nevertheless,selecting a two-stage agitator means a uniform powerinput into the vessel can achieve a uniform suspensionwith Δcv less than one percent for a lower overall powerinput.
54
Testing EKATO
Fig. 6 Color drawn in at the surface of a high-viscosity product
Measurement methods for suspension duties
As the degree of uniformity can be determined neithervisually nor by laser-Doppler anemometry, the distri-bution of the solids concentration must be ascertainedby other measurement processes. For an overview ofthe measurement methods (sampling, pressuremeasurement, photoelectric process, density measure-ment) the reader is referred to authors such as Einenkeland Mersmann [7]. EKATO mainly uses the followingtwo methods.
Sampling at various heights
A sampling tube is inserted horizontally through thewall of the vessel or vertically downwards through theliquid surface into the suspension. About 0.5% of thesuspension are then withdrawn by a pump and trans-ferred to a graduated cylinder. The concentration ofsolids in the sample is determined either from itsdensity or from the height of the sediment once all thesolids have settled. Care must be taken that thesampling is isokinetic.
Conductivity probe [8]
The conductivity probe consists essentially of twoplatinum electrodes molded in plastic with a gapbetween them. If a non-conducting solid is in the gapbetween the electrodes, the conductivity is reducedaccording to the proportion of the solid by volume. The
instrument is calibrated at two points (in clear liquidand in the settled settlement) before the measurementis carried out.
Between these two extremities, cv = 0% and cv = max.%, it is assumed that the conductivity varieslinearly with the concentration of the solids.
When using a conductivity probe to measureabsolute values, there is a risk of measurement errorsoccurring, since the conductivity of the liquid can beinfluenced more by impurities and solubility in theliquid phase than by the difference in solids concen-tration.
Higher measurement accuracy can be attained byusing two conductivity probes with differentialcircuitry. In a uniform suspension the difference inconcentration between two different measurementpoints is by definition zero.
The difference in solids concentration Δcv shownin Fig. 8 was determined by means of conductivitymeasurements.
As an alternative to the differential conductivitymeasurement, the second probe can be used as areference probe by inserting it into a cage throughwhich a liquid that is free of solids flows. This arrange-ment has a more sensitive response than the singleprobe for small values of cv.
55
EKATO Testing
Mix
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pro
cess
des
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On-bottom suspension
Fig. 7 Degrees of suspension
Off-bottom suspension Uniform suspension
1-second
criterion fulfilled
1-second
criterion fulfilled
”visually uniform
suspension“
temporary local
deposition
Evaluation of conductivity measurements
Fig. 9 shows the results of a typical series of measure-ments in which the vertical position of the conductivityprobe in the vessel was varied. At the lowest shaftspeed (n = 348 min–1) the solids concentration in thelower part of the vessel is considerably more than theaverage value, while in the upper part a large clear zoneis still present.
From about n = 580 min–1 the solids are uniformlydistributed. In the upper part a so-called “concen-tration bulge” can be observed. As the curves indicate,no further increase in uniformity can be achieved byincreasing the shaft speed/power input.
The variance can be calculated from the meas-ured distribution of solids concentration:
σ2 = · ( –1)2
(1)cv,i
cv
n
∑i =1
1n
Where n is the number of equidistant measurementpoints. The variance (the square of the standarddeviation) is an important parameter in the design ofagitators for suspension duties and is described infurther detail on page 30.
Measurement of mass transfer
In nearly all industrial processes such as fermentation,wastewater aeration, hydrogenation, oxidation, chlorin-ation, etc. the mass transfer from the gaseous to theliquid phase is of critical importance. The key para-meters in these processes are the liquid film coeffi-cients kL and the specific interfacial area a = A/V(surface area of all bubbles/volume of liquid). NeitherkL nor a can be measured individually. What can bemeasured, however, is the mass transfer coefficientwhich is equal to the product kL·a.
Parameters interacting on the value of kL·a (→ = causes)
– High viscosity → low flow velocity around thebubbles → diminished mass transfer
– Higher proportion of dissolved salts in the liquidphase → increasing ionic concentration → re-duced coalescence tendencies → smaller bubbles→ larger specific interfacial area → better masstransfer; see Fig. 10.
– Increased power input via the agitator → greaterlocal energy dissipation → greater dispersingeffect → smaller bubbles → better mass transfer
– Increased gassing rate (below the flooding limitfor the impeller) → greater proportion of gas inthe liquid phase (hold-up) → increased masstransfer
Measurement methods for mass transfer
At EKATO two different measurement processes areused for determining the value of kL·a.
56
Testing EKATO
00 20 40 60 80
P [W]
Δ c
v [%
]
100 120 140 160 180
4
8
12
16
ISOJET 1-stageISOJET 2-stage
➀ 1-s criterion 1-st. fulfilled➁ 1-s criterion 2-st. fulfilled
Fig. 8 Determination of degrees of suspension bythe GVC standard method
0.0 0.2 0.4 0.6 0.8 1.00
5
10
15
20
Relative height of probe h/h1
c v [
%]
➀ n = 348 min-1
➁ n = 406 min-1➂ n = 464 min-1
➃ n = 522 min-1➄ n = 580 min-1
➅ n = 696 min-1
Fig. 9 Concentration profile at various shaft speeds
Ionic concentration I
k L• a
(so
luti
on
) / k
L• a
(w
ater
)
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.80
1
2
3
4
5
6
7
8
Fig. 10 kL·a as a function of ionic concentration
Chemical method
A defined quantity of sodium sulfite (Na2SO3) is addedto a continuously aerated electrolyte (salt) solutionsaturated with oxygen in the presence of a Co++ catalyst.The oxygen content in the solution is monitored bymeans of a dissolved oxygen (DO) probe installed inthe vessel. The dissolved oxygen is first consumed inoxidizing the salt to sodium sulfate (Na2SO4), a processwhich begins immediately, the measurement readingfalling to zero (cO2
= 0 ppm). The amount of oxygendispersed into the liquid phase is alone enough tooxidize the total amount of Na2SO3. Once it has allbeen oxidized to Na2SO4, the concentration of dis-solved oxygen in the liquid phase rises again to itssaturation value. The oxidation time of the Na2SO3,which determines the mass transfer, is measured; seeFig. 11.
Evaluation:
kL · a = (2)
c* [g/m3] : saturated concentrationmNa2SO3
[g] : initial mass of Na2SO3M~
[kg/kmol] : mol masstoxi [h] : oxidation timeV [m3] : liquid volume
Test conditions to be complied with
– compensation for the temperature-dependent sat-urated concentration of O2.
– correction for the salt content because it affectsthe saturation concentration of oxygen c* = f (csalt).
mNa2SO3 · 0.5 · ~MO2
~MNa2SO3 · toxi · V · c*
– the DO probe must always be in a stream of liquid[9].
– the DO probe must be checked for good workingorder (electrode wear, calibration).
Advantages of the method
– It is possible to measure comparative values ofkL·a in a noncoalescing system. (Practice-orien-ted!).
– From an ionic concentration I = 0.5 upwards thesalt concentration has virtually no influence onthe coalescence properties, see Fig. 10. If thecorrection is made for the salt concentration,several measurements can be made one after theother without needing to replace the test liquid.
– The oxidation time can be adjusted by selectingthe quantity of sulfite added so that the responsetime of the probe does not falsify the measure-ment result.
Alternative measurement process
Hydrazine can also be used for the chemical methodinstead of Na2SO3, but the latter is generally preferablesince it is safer to handle.
Dynamic method
In this method, which is easy to carry out experimen-tally, oxygen is stripped from the product system in theagitated vessel by means of nitrogen. The system isthen again saturated with oxygen by aerating it, andthe increase in the concentration of dissolved oxygenis recorded with a DO probe, see Fig. 12. The value ofkL·a can be determined from the exponential curve thatresults.
Evaluation:
kL · a = ln (3)
Advantages of the method:
– For measurements made in water there is noaddition of salt to influence the ionic concentra-tion and therefore the coalescence properties.
Disadvantages of the method:
– With high kL·a values (short saturation time) theresponse time of the DO probe falsifies themeasurement result to an increasing degree(correction factors).
– This measurement method can hardly be used onan industrial scale since enormous quantities ofnitrogen would be necessary.
c* – ca(c* – ce)1
t oxi
57
EKATO Testing
Mix
ing
pro
cess
des
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0 3 6 9 12 150
2
4
6
8
10
Oxidation time
c oxy
gen
[pp
m]
➀ Sodium sulfite added ➁ Oxidation complete
t [min]
Fig. 11 Measured oxygen concentration withNa2SO3 method for determining kL·a
Correlation of test results:
Evaluation of the volume-related mass transfer coeffi-cient kL·a results in the following form of correlationsfor scale-up to industrial dimensions:
kL · a = c1 · εc2 · vsgc3 (4)
vsg [m/s] : superficial gas velocityε [W/kg] : energy dissipation per unit massc1, c2, c3 : empirical factor and exponents
Conditions to be complied with for scale-up
– There must be geometrical symmetry with thetest equipment.
– The product and the process conditions must beretained.
Note
As the value of kL·a is dependent on many parameters,the following must be stated:
An absolute value which is stated, or a value ofkL·a determined from equation (3) applies strictlyspeaking only to the product system in which it wasmeasured. For other product systems these values ofkL·a must be converted using the usual dimensionlessmass transfer correlations (“Sherwood relations”).See “Basic mixing tasks”, page 36.
Determination of heat transfer properties
The (internal) heat transfer in agitated vessels has beeninvestigated in numerous research projects forstandard agitator impellers and a range of specialEKATO agitator systems. See “Basic mixing tasks”,page 38. The influence of product physical propertiessuch as density ρ, viscosity η, specific heat capacity cpand thermal conductivity λ together with the flowconditions in the vessel are therefore sufficiently well-known and can be applied in appropriate Nusseltequations and computer programs based on these.The testing technology involved in determining heattransfer properties in agitated vessels is thereforemostly limited to measuring physical properties as yetunknown, particularly flow properties, which can thenbe used in the calculations.
Simple cooling tests on a 50 l scale are suitablefor the rapid and approximate verification of designcalculations. cooling tests are preferred because theheat losses from the system are low. It is particularlyrecommended to carry out a test if the product hascomplex flow properties and when the specificationscall for a uniform temperature throughout the vessel.It can therefore be checked whether the recirculation ofthe product is sufficient or if the agitator system issuitable to prevent temperature differences in thevessel.
Frequent objectives when carrying out a coolingtest are:
– Determination of the temperature distribution inthe whole vessel,
– Investigation of boundary layers on the vesselwall that necessitate the use of scrapers,
– Comparison of the measured cooling time on thetest scale with the calculated cooling time,
– A heat balance via the cooling water, to determinethe rate of heat transfer,
– Testing to see whether flash cooling would bemore suitable for the product.
Scale-up
Calculation of the quantity of heat to be removed on anindustrial scale per unit time:
Q·
= m · cp · Δϑ/Δt (5)
Calculation of the required overall heat transfer coeffi-cient k:
k = · ln (6)
Calculation of the required product-side film coefficient
αi = (1/k – s/λwall – 1/αS)–1 (7)
ϑa – ϑS
ϑe – ϑS
m · cp
Δt · A
58
Testing EKATO
0 10 20 30 40 50 600
2
4
6
8
10
c oxy
gen
[pp
m]
Oxidation time
➀ cstart
t [min]
➁ cend ➂ csat
Fig. 12 Increase in oxygen concentration with thedynamic method for determining kL·a
Determination of the required agitator data to achievethe necessary product-side film coefficient αi (turbulentflow):
αi = C · Re2/3 · Pr1/3 · (η/ηw)0.14 · λ/d1 (8)
As a rule, the product-side film coefficient αi is thelimiting parameter, i.e. the factor that determines thecooling/heating time, as it is in turn determined by theoften high viscosity at the product side and by the ratioof volume to heat transfer surface area, which alwaysbecomes less favorable on scale-up. The utility-sidefilm coefficient αS is normally not critical, on account ofthe higher flow velocities of the cooling/heatingmedium in the vessel coil or jacket.
BASIC TESTING
Blend time determination
When blending two miscible liquids it is often im-portant to know what blend time is required to achievea desired degree of homogeneity. In the turbulent flowrange there is, within certain limits, a linear relationbetween the recirculation rate and blend time. This isdescribed in further detail on page 26. A definition ofthe degree of homogeneity can be found in theliterature [10].
Various techniques are used to determine theprimary or total circulation rates, e.g. photographingparticle movements by the light plane method mea-suring circulation times by means of virtually iner-tialess floating particles (flow followers) or measuringthe velocity field (hot-wire and vane-type currentmeters, laser Doppler imager).
Numerous physical and chemical methods areused for determining blend time [10]. A basic distinc-tion is made between those employing probes, e.g.conductivity, pH, temperature) and visual chemicalmethods (e.g. color change of chemical indicators, thecolorizing reactions, refraction of light).
Temperature equalization method
Several highly sensitive resistance thermometers areinstalled in the vessel at various levels.
A small proportion of the liquid is taken from theagitated vessel and heated sufficiently so that when itis returned to the surface of the liquid in the vessel aneasily measurable temperature increase, e.g. overall0.2°C, results, which however does not change therheological properties of the system. After this hotliquid has been added to the vessel, the probes at theirvarious levels measure the local change of tempe-rature until complete temperature equalization isestablished in the vessel, see Fig. 13. The probe at the
liquid surface is contacted by the hot liquid, as yetbarely diluted, so that initially a local temperature riseis seen. The last probe to respond is the one at thebottom of the vessel. After some time in the exampleshown about 50 seconds, a uniform temperature existsin the vessel. The plot thus shows the degree of homo-geneity as a function of blend time. After about 39seconds the temperatures at all four probes haveconverged toward the final temperature to within 10%of the difference between the initial and final tempe-ratures. By definition the degree of homogeneity isthen 90%. With this measuring technique, the tempe-rature increase due to the agitator power input must betaken into consideration (rising straight line curve).
Visual chemical method
One example of the color change method is the H2SO4-NaOH-phenolphthalein system. A defined quantity ofOH–– ions (0.1 NaOH solution) is added to the initiallyneutral liquid. These form a red complex with theindicator as shown in Fig. 14. After adding the samenumber of moles of hydrogen ions (H2SO4) at time tathe liquid is decolorized locally where the pH fallsbelow about 8.5. The blend time Θ is then completewhen decolorization is achieved. This method has thedisadvantage of sudden decolorization with no inter-mediate stages identified at a pH of 8.5. It cannot beidentified as to whether the local concentration of OH––
ions in the red or decolorized zones deviates onlyslightly, e.g. by 1% or by a factor of 1000 (pH 11.5, or5.5) from the average concentration. The assessmentof degree of homogeneity can therefore be falsified,especially in highly viscous media, by streaks of colorthat do not easily dissipate.
The decolorization reactions are less ambiguousin the iodine-starch-sodium thiosulfate system. Iodineand starch in aqueous solution form a complex with adeep violet color. If at the time ta the equivalent numberof moles of Na2S2O3 are added, the iodine is reduced to
59
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t [s]
ϑ [
°C]
0
0 0.5 0.8 0.9
10
21.4
21.5
21.6
21.7
20 30 40 50 60
ϑ ∞ (t)
ϑ ∞ - ϑ 0
ϑ ϑ 0-
ϑ 0 (t)θ 90
Fig. 13 Temperature method: blend time as functionof degree of homogeneity
iodide and the solution is decolorized. Even locally thedecolorization process does not take place suddenlybut in proportion to the remaining iodine concen-tration. The blend time θ is when complete decolori-zation is achieved.
Methods using probes and those with visualchemical reactions are used to the same extent. Theformer methods allow a larger degree of objectivity,but their results are localized because of the finitenumber of measuring points. The methods involving avisual chemical change do not disturb the natural flowpatterns and involve the entire volume of the vessel,but undesirable reactions with the agitated productmay take place.
Droplet size distribution
The primary objective of liquid-liquid dispersion is toproduce a large value of a, the interfacial area per unitvolume, between the dispersed and continuousphases. For a given dispersed phase concentration cv,d,the smaller the droplet diameter, the greater is theavailable interfacial area for mass transfer:
a = 6 · cv,d / d32 (9)
Where d32 is the Sauter diameter. The latter is calcula-ted from the droplet size distribution over the n diffe-rent size classes (i = sequential number).
d32 = (10)∑i=1
n
ni · dd,i3
∑i=1
n
ni · dd,i2
One problem in the measurement of droplet size is thepossible falsification of readings due to increasedcoalescence as soon as the droplets are no longersubject to shear, e.g. after sampling. Another source oferror may be a progressive contamination of thedispersion with surface-active substances from thesurroundings.
Measurement methods
Grossmann [13] describes the most commonly usedprocesses: photography with subsequent analysis ofthe films, the Coulter counter and photoelectric tech-niques (reflection, refraction and extinction of light). Inrecent years two methods have mainly been used atEKATO. Solidification of molten wax droplets asdescribed by Brooks [12] and analysis of the settlingrate distribution of the disperse phase as described byGrossmann [11].
In the first method molten carnauba wax isdispersed at 85°C in water. This dispersion is thenquenched with chilled water and the solidified wax“pills”" are then strained out and dried. This is followedby a gentle sifting process to break up any undesiredagglomerates.
In the second method, after a steady-state dropletsize distribution has been obtained, a maximum of50 ml of the dispersion is pumped out of the stirredvessel with a time of 10 ms into a dilution vessel. Theprobability of coalescence is drastically reduced by thedilution. Starting at time ta, several hundred drop-lets pass downwards through a vertical tube. Afterfalling through a certain distance l at a constant settlingrate wd they pass through a laser beam and trigger apulse by reflection. The distribution of the pulses withtime can be used to calculate the settling rate distri-bution:
wd,i = (11)
using the Stoke’s law equation
wd,i = (12)
the droplet size distribution can then be determinedand presented graphically. The Sauter diameter andthe interfacial area are calculated from equations (9)and (10).
High measurement accuracy results from thesettling times, which are over one half-hour fordroplets smaller than 100 μm. Calibration is not neces-sary, but the measurement apparatus can be verified,for example using glass beads with a known sizedistribution.
dd,i2 · (ρd – ρc) · g
18 · ηc
lΔti
60
Testing EKATO
Fig. 14 Color change method: preparatorycolorization; left with propeller; right withEKATO VISCOPROP
Measurement of power number and hydraulic force
coefficients
Power number Ne
Every impeller type, whether propeller, EKATO VISCO-PROP or whatever, is basically characterized in theturbulent flow range by a constant, impeller-specificpower number Ne. The mechanical drive power re-quired can only be determined when the value of Ne isknown.
Pdrive = (13)
(ηdrive = efficiency of drive)
The power number of an impeller varies with theviscosity of the product and the geometric arrange-ment of the agitator-vessel system (e.g. with/withoutbaffles).
The procedure for determining the value of Ne fora system is as follows:
A strain gauge is attached to the agitator shaft.This gives a stress signal proportional to the torqueacting on the shaft.
The output signals are transferred via a telemetrysystem from the agitator shaft to an amplifier, areconverted from analogue to digital and are thenprocessed by a PC program which calculates theaverage values. The measuring system is calibrated byproducing defined torques by means of weights. Asystem for measuring the shaft speed is also neces-sary.
Determination of power number:
– The power can be calculated from the measuredtorque and shaft speedP = 2π · n · Mt
– The power number is then given by the powerequationNe = P / (ρ · d2
5 · n3)
Hydraulic force coefficients
The system used for determining the power number isalso used to measure the hydraulic force coefficients.The only difference in the test rig is in the type andorientation of the strain gauge on the agitator shaft. Tomeasure the radial force coefficient CR the strain gaugeis attached so that it determines bending moments,after which calibration is also carried out.
Ne · ρ · d25 · n3
ηdrive
Determination of CR:
– Measurement of the bending moments Mb– Calculation of the radial force FR = Mb / lW
where lW = shaft length
CR = FR / (ρ · d24 · n2).
CR is not a constant, but depends on the ratio ofthe operating shaft speed to the critical shaft speed.When the critical shaft speed is approached, CR reachesits maximum value. See “Forces on impeller andagitator shaft”, page 95.
Note:
In addition to the determination of Ne and CR, the testrig illustrated in Fig. 15 can also be used universally, forexample for measuring forces on impeller blades orbaffles.
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M
PCData acquisition
and evaluation
motor with friction gearstrain gaugeimpeller (here VISCOPROP)agitated vesselamplifier for instrument signalsdata acquisition board for PCshaft telemetry system for contact-free transmission of up to 4 strain gauge signalssimultaneouslytachogeneratorstrain gauge/telemetry signal cable (4-channel)agitator shaftstirred product
Fig. 15 Test rig for determining CR and Ne
When measuring Ne there are only relativelysmall fluctuations in the measurement readings. WithCR determination, however, a defined division of theplotted reading into time-dependent sectors (blocks) isnecessary, in order to avoid exaggerated readings.See the plots in Figs. 16 and 17 and the “MechanicalDesign” chapter, page 95.
Flow velocity
The measurement of flow velocity is of fundamentalimportance in ensuring that the specified circulationrates or blend times are achieved. The primarypumping rate q (and hence the associated pumpingnumber Q) of an impeller, for example, is determinedby defining measuring points for the local velocity wiat the discharge side of the impeller over the length ofthe radius. An annular area Aann, i is associated witheach value of velocity wi. The pumping number isdefined by
Q = q / (n · d23) (14)
where q is the circulation rate, given by
q = (wi · Aann, i) (15)∑i
Vane-type current meters are used for field measure-ments in pure water, see Fig. 18.
In contaminated media, such as wastewater,where it is impossible to use mechanical instruments,inductive flow-measuring devices can still be relied on.The light section method allows the flow in vessels tobe visualized directly. In this process, silver flakes areadded to the stirred medium. These have negligibleinertia and generally follow the streamlines of theliquid. A vertical “light sectional plane” is produced inthe stirred vessel by means of mercury dischargelamps and narrow apertures. Only those silver particlesthat are actually in the plane are visible by reflection atany given time. Photographs of the light sectionalplanes produced in this way for commonly used agi-tator systems are shown on pages 77 and 80.
Particle image velocimetry (PIV) can be applied todetermine the velocity fields in the agitator vessel. Thistechnique uses the principle of tracer particles, ex-tremely fine solids being uniformly dispersed in theflowing liquid. Their movement can then be followedby taking “snapshots” of the reflected light with acamera, allowing individual particles to be tracked andtheir velocities thus determined.
To determine local three-dimensional flow velo-cities and turbulent fluctuating velocities accurately,and hence the local energy dissipation rates, laserDoppler anemometry (LDA) is used. With this method,illustrated schematically in Fig. 19, two laser beamswhich cross each other form an interference grid con-sisting of light and dark stripes. Silvery particles addedto the liquid as tracers act as reflectors in the light grid.
62
Testing EKATO
1. 2. 3. 4. 5. 6.Duration of measurement [blocks]
Am
plit
ud
e [V
]
0 V
Fig. 16 Plot of CR
Duration of measurement [blocks]
Rea
din
g
Am
plit
ud
e [V
]
0 V
Fig. 17 Plot of Ne
Fig. 18 Vane-type current meter
Ph
oto
: O
tt C
o., K
em
pte
n
The number of light reflections per unit of time is ameasure of the speed of the particle concerned and theliquid that surrounds it. As this method works with neg-ligible inertia, even the smallest changes in velocity canbe detected, a vital prerequisite when measuring theturbulent fluctuating velocities w’. The semi-empiricalformula give by Brodkey is then used to calculate thelocal energy dissipation densities from the turbulentfluctuating velocities:
εloc = C · (16)
The results of the tests give valuable information, suchas the optimum feed point for substances to be dis-persed. It also makes it possible to evaluate which typeof impeller enables the most uniform global energyinput to be achieved or which displays optimumsuspension properties, see Fig. 20.
Author:Max Niefenthaler
w’3
d2
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EKATO Testing
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He-Ne laserLDA optical system withbeam splitter and Bragg cellsfocusing lensBragg cell generator
receiving lensphotomultiplierhigh voltage powerunitLDA counterx-y traversing table
M
Fig. 19 Laser Doppler anemometry (LDA) test rig
-0.25
vess
el h
eig
ht
z/h
1
0.25
radius 2r/d16-bladed flat blade turbined2/d1 = 0.33
radius 2r/d13-bladed propellerd2/d1 = 0.5
radius 2r/d12-stage INTERMIGd2/d1 = 0.6
0.5 1.0 0 0.5 1.0 0 0.5 1.0
0
0.25
0.5
-0.25
0
0.25
0.5
0
0
0.2
0.4
0.25
0.5
0.75
vu
Fig. 20 Radial/axial vector diagrams for various impeller types and arrangements (Re >104)
64
COMPUTATIONAL FLUID DYNAMICS
Preprocessing modeling of the agitated vessel
Numerical calculation
Postprocessing and applicability
The pace of development in the field of computer hard-
ware and software has made computational fluid
dynamics (CFD) an important mixing technology
design tool for the numerical simulation of flow in
agitated vessels. This new technique uses interdisci-
plinary knowledge from the fields of numerical ana-
lysis and process engineering plus physical models
and modern information technology methods.
The physical and mathematical models must be
validated by tests on a pilot scale or by practical ex-
perience on an industrial scale. CFD makes it possible
to create a direct model of the mixing process on an
industrial scale, so that it can also contribute to
solving scale-up problems.
CFD also makes it possible to investigate more
accurately those processes taking place in the agitated
vessel that are difficult to study with conventional
testing methods or can only be assessed globally. It
also means that much time-consuming and cost-inten-
sive experimental work on a laboratory or pilot scale
is now superfluous.
An important specialty in the CFD simulation ofmixing processes is in the analysis of stationary androtating components. Other characteristics are thethree-dimensional flows, often in the turbulent regime.The tasks involved are therefore highly complex, andsolving them by means of CFD simulation can usuallybe broken down into three steps: preprocessing, nu-merical calculation and postprocessing, see Fig. 23.
The numerical solution of systems of differentialequations is based on a discrete description in whichthe variables of state are only defined in certaindiscrete points in space at certain points in time.
65
EKATO Computational fluid dynamics
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Fig. 21 Local specific energy distribution with theEKATO ISOJET
Mixing technology
Numerical
analysis
Scientific
models
Information
technology
methods
Validation
Computational
fluid
dynamics
Fig. 22 Knowledge base for CFD
Preprocessing
• Definition of geometry
• Generation of mesh
Numerical calculation
Numerical solver
Postprocessing
Graphical evaluation
Result mesh-dependent?
Calculationconvergent?
Fig. 23 Steps in a CFD simulation
Preprocessing – modeling the agitated vessel
In the preprocessing stage the mixing problem isconverted into a mathematical model that can beprocessed by the computer. The first step is to definethe area to be analyzed. It is then divided into discretecells (volume elements) by means of a numericalmesh, see Fig. 24. The validity and accuracy of thesolution ultimately arrived at with the computationalprocess is highly dependent on the quality of thegeometry modeling and mesh generation. Theproperties of the fluid and the other design constraintsare then specified.
Fig. 25 shows some of the mesh models that canbe used to take the stationary and rotating componentsof the agitated vessel of the agitator system intoaccount. A distinction is made between steady-stateand transient methods.
Steady-state mesh
Implicit impeller modeling
In the simplest steady-state case the agitator impellersare modeled implicitly. One technique is to model theimpeller as values of velocity, while the other is tomodel it as a source of momentum. If the impeller isreplaced by averaged velocity vectors, data deter-mined experimentally must be available and inaccu-rate results may arise on scale-up to other flow con-ditions. The alternative is to model the impeller as asource of momentum, but in this case the flow patternin the vicinity of the impeller can only be predictedinaccurately.
Explicit impeller modeling
Explicit impeller modeling represents a more accuratemethod of analysis. The “rotating frame” technique isused to solve the problem of the fluid moving in therotating reference system, i.e. the observer rotates withthe system, that is, a system particularly suitable foragitated vessels with no baffles. In the “snapshot”approach, a “frozen” picture of the flow for a particularposition of the impeller is analyzed. The tangentialvelocities before and behind the “stationary” impellerblades are defined. The flow between the impellerblades can be predicted, but the method is not suitablefor any desired type of impeller. With the “multiplereference frame” model, several reference frames canbe defined for various parts of the mesh. In a two-stageagitator system with different types of impeller, forexample, two internal rotating reference frames maybe selected. The zone near to the vessel wall andbaffles can be analyzed via an external stationaryreference frame and complex impeller geometries andcombinations of different impeller types can also bemodeled. With systems having a weak interactionbetween the impeller and the baffles it is easier topredict the flow patterns.
66
Computational fluid dynamics EKATO
Fig. 24 Mesh structure of an agitated vesselequipped with an EKATO ISOJET
Implicit impeller modeling
Explicit impeller modeling
Impeller asvelocity vectors
Rotatingreference frame
Sliding mesh
Snapshottechnique
Multiple reference frame
Impeller assource of momentum
Ste
ady
stat
eT
ran
sien
t
Fig. 25 Agitated vessel mesh models
Sliding meshes
The transient method makes use of a “sliding mesh”to enable a time-dependent analysis of the impeller/baffleinteraction to be made. A mesh around the impellerrotates at shaft speed, while the external mesh for thebaffles and the vessel itself is fixed in the referenceframe. The systems of equations are linked at thesliding interface. This enables time-dependent pheno-mena to be investigated, and the flow in the vicinity ofthe impeller can be calculated more accurately. Thismethod is, however, demanding in terms of computertime, many iterations being required until a periodicbalance is attained. The actual time-dependency of themixing process can differ from its calculated value.
Properties of the fluid
Turbulence models
The analysis and CFD simulation of turbulence is oneof the most difficult tasks in fluid mechanics. In orderto include the smallest eddy that arises in the directCFD simulation, the balance equations must be solvedwith a mesh that is sufficiently fine. At large Reynoldsnumbers an immense memory and computing powerwould be required. For this reason various turbulencemodels are used as illustrated in Fig. 26. The modelin most widespread use in the so-called standard k-ε model: described by Launder and Spalding in 1974[15], based on an equation for turbulent eddy viscosity
μT = ρ · cμ · k2/ε (17)
where
k = turbulent kinetic energy = 0.5 (u–,2 + v–,2 + w––,2) (18)
ε = energy dissipation densitycμ = 0.09 (empirical constant).
Logarithmic wall functions are usually integrated foranalysis of the flow profile in the vicinity of the wall.
Multiphase flow
In liquid-liquid CFD simulations the two phases aretreated as a continuum in the Euler/Euler frameworkand the balance equations are calculated for eachphase. In addition, linked models can be used, such asdepict the influence of liquid flow on the movement ofthe solids, and vice versa. In the Lagrange frameworkeach particle is tracked individually, so that the comput-ing power required for a high particle content is con-siderably greater. Models for particle-particle inter-action can be integrated in the system of equations.
Physical chemical properties
The relevant parameters such as temperature, en-thalpy and heat of reaction are integrated in the energybalance for the calculation of heat transfer processes.
Special algorithms can be used for free surfaces(vortex formation), and models for non-Newtonianfluids can be integrated in the CFD simulation program.
Numerical calculation
The flow of the fluid can be calculated numerically onthe basis of the fundamental equations of fluid me-chanics. The equations for the conservation of massand momentum (Navier-Stokes equations) and theenergy balance form a system of five non-linear partialdifferential equations.
67
EKATO Computational fluid dynamics
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Turbulence simulation
Direct CFD Turbulence model
Vortex viscosity models Stress transport models
One-equation modelsAlgebraic models Two-equation models
Low Re models Two-layer modelStandard k-ε model with wall function k-ε renormalization group method
Finely structured model Statistical turbulence models
Fig. 26 Turbulence model [16]
φ(x,y,z)
φz –∂φz
∂z δz12
φz +∂φz
∂z δz12
φx +∂φx
∂x δx12
φx –∂φx
∂x δx12
φy –∂φy
∂y δy12
φy +∂φy
∂y δy12
xy
z
Fig. 27 Fluid cellwith balance
The finite volume method [17] is the numericalmethod that is mainly used for mixing processes:
1. Integration of the conservation equations over allcontrol volumes, see Fig. 27.
2. Discretization of the integrated equations3. Iterative calculation of the algebraic equation
system
The conservation equations are first integratedover all the control volumes.
The conservation equation of a variable Φ, forexample the velocity component u within a control-lable volume can be expressed as follows:
Rate of change of = Difference between inflowΦ in the control and outflow of Φ at the volume with time control volume through
convection+ Difference between inflow
and outflow of Φ at thecontrol volume throughdiffusion
+ Increase or reduction in Φthough source/sink withinthe control volume
Finite volume methods therefore make direct useof the conservation of the differential equations. Theformulation is made directly in the physical space, thusmaking the programming easier for complicatedgeometries. The unknown variables such as pressureor velocity in the control volume are represented bymeans of approximation functions dependent on theneighboring cells. The convection, diffusion or sourceterms of the integrated equations are replaced bydiscrete approximation terms and thus transferred to asystem of algebraic equations.
If the numerical solution parameters are in-correctly selected, non-physical oscillations can occuror the calculation can become unstable. The results arethen unusable [18, 19]. A solution which satisfies theaccuracy requirements is usable and is described asbeing convergent. The equation systems are solved inseveral iteration loops until the desired convergence isachieved. To obtain a solution independent of themesh, the computation is repeated with a largernumber of mesh nodes until a sufficiently close agree-ment is attained.
Postprocessing and applicability
The evaluation and representation of the computeddata is an important part of CFD. With the visualizationtools commercially available, the data output can be asprofiles, contours, streamlines or sectional planes, seeFig. 28. Depending upon the required results, additio-nal subprograms for visualization for the computationscan be integrated.
As a rule of thumb, it can be said that applicationsin single-phase flows in the fully turbulent range and inthe purely laminar range are well-suited to analysis byCFD, and reliable and rapid results can be expected fromthe computation. In contrast, problems in the transitionrange and in multiphase flow are more difficult to solve.
1. Blending: With blending duties the velocity fields,blend times and power numbers can be deter-mined. The CFD simulation results for globalmixing are normally reliable. In mixing processesthat are largely characterized by diffusion on amicro-scale, the results of the CFD simulation candeviate significantly from those obtained inpractice. In the laminar flow range inaccuracies inthe discretization process can lower concen-tration peaks. The mixing efficiency is over-estimated as a consequence of the so-called“numerical diffusion”. This source of error can beminimized by using a finer mesh or selectinganother discretization process.
2. Suspension: The CFD output can be in the form ofvelocity vectors and concentration contours inorder to evaluate the results of the calculation fora suspension reactor. Scale-up usually requiresextensive testing. Determination of the off-bottom suspension point is only feasible withadditional programs with the current status ofstandard software.
68
Computational fluid dynamics EKATO
Fig. 28 EKATO ISOJET flow pattern
3. Dispersion: CFD makes it possible to visualize thek and ε gradients for analyzing dispersion appli-cations. The relative distribution of local energydissipation can normally be predicted with someaccuracy, whereas the absolute values for localenergy input must be reviewed with a critical eye.
4. Heat transfer: CFD makes it possible to visualizetemperature gradients and local heat transfercoefficients for analyzing heat transfer appli-cations, see Fig. 29. Heating and cooling timescan also be determined. For heat transfer withlaminar flow the mesh structure selected at theheat transfer surface must be sufficiently fine sothat the boundary layers can be described insufficient detail [20].
5. Special problems, such as the excessive wear ofimpellers due to erosion and cavitation, and theimpact on chemical reactions, can also be in-vestigated using CFD. Special models for concen-tration and temperature fluctuations must beintegrated in the computations for chemical reac-tions that take place rapidly.
Fig. 30 illustrates the applicability of commer-cially available CFD programs as tools for analyzingmixing problems. Application-oriented impeller development using
CFD
The practical benefits of CFD are illustrated by the fol-lowing case study:
In hydrometallurgical treatment processes or influe-gas desulfurization systems the agitators usuallywork in abrasive suspensions. In order to keep theongoing abrasive wear within reasonable limits, eitherthe agitator impeller tip speed must be limited orexpensive measures must be taken to combat wear
69
EKATO Computational fluid dynamics
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Mixing problem
Flow pattern - 1 4 1standard geometries
Flow pattern – no standard geometries 3 4 2with vessel internals
Turbulent shear – local 3 4 –energy dissipation
Blending 2 4 3
Heat transfer 1 4 2
Chemical reactions influenced 4 4 4by agitator micromixing
Suspension – solids 3 4 –concentration profile
Suspensions – off-bottom 4 5 –suspension point
Dispersion gas-liquid 3 5 –
Dispersion liquid-liquid 4 5 4
Dispersion solids-liquid 3 5 4
Heat transfer 3 4 –
Chemical reactions 4 4 –influenced by agitation
The scale of 1 to 5 indicates increasing degree of difficulty,requiring increased special CFD and mixing technology exper-tise for problem-solving.1 : Accurate prediction without validation2 : Reliable scale-up with validation at pilot scale3 : Scale-up usually requires extensive experimental work
plus program extensions4 : Scale-up only possible with program extensions and/or
appropriate supplementary programs and extensiveexperimental work
5 : Very difficult – requires accurate detailed knowledge– : Not applicable
Fig. 30 Estimated scope of applicability forcommercially available CFD programs asanalytical tools for mixing problems. (Status: 1/2000)
Fig. 29 Thermal transfer CFD simulation withEKATO PARAVISC
ϑ - ϑ∞ϑ∞ - ϑο
relativetemperaturedistribution
0.2
0.1
0.0
-0.1
-0.2
Two
-ph
ase
Sin
gle
-ph
ase
Turb
ule
nt
Tran
siti
on
Lam
inar
(special materials of construction, plating, etc.; seepage 129). In both cases this results in expensiveagitators. In general the erosive wear is not distributeduniformly over the surface of the impeller blades, butis concentrated at locations on the blade where theflow dynamics are particularly severe. Fig. 31 shows asexample a photograph of a propeller blade from a fluegas desulfurization system. Clearly identifiable wearsymptoms at the suction side directly behind the tip ofthe blade are evident.
EKATO carried out a study using CFD to gain adeeper insight into the dynamic flow mechanisms thatcontribute to the erosion of the impeller. The exampleshown above formed the basis of this study. The firststep is to create a numerical model of the agitatedvessel and the propeller agitator on a pilot scale. Usingthe k-ε turbulence model and taking into account theboundary layers, the single-phase flow around thepropeller blade, in particular that at the blade tip, iscalculated. Figs. 32 and 33 show the results: a pro-nounced formation of vortices at the blade tip.
In the next step the interaction of the abrasiveparticles with the propeller blade are integrated in themodel by calculating the (Lagrange) particle trajec-tories, turbulence being taken into account with astochastic model of turbulent eddy life. At a certainpoint in time a defined number of particles are releasedabove the agitator blade, and their point of impact andvelocities together with the angle and frequency ofimpact on the agitator blade are tracked. The resultsare shown in Fig. 34: the relative erosive attack takinginto account the particle diameter, velocity of impact,solids concentration and frequency of impact. Theresults show a high rate of erosion on the suction sidebehind the blade tip.
Validation
The simulated single-phase flow pattern can be valida-ted with the known power numbers and pumping num-
70
Computational fluid dynamics EKATO
Fig. 31 Wear at the outer edge of an industrial-scalepropeller blade
Fig. 32 Local formation of vortices at the blade tip
Fig. 33 Formation of vortices at the blade tip,velocities in an upward axial direction
Fig. 34 Relative erosive attack simulated on a pilotscale
1.0
0.5
0.0
-0.5
-1.0
Axial velocitycomponent [m/s]
2.4
0.9
-0.6
-2.0
-3.5
Axial velocitycomponent [m/s]
very heavy
heavy
moderate
low
no attack
Erosive attack
bers. The review and adaptation of the erosive attackdetermined by computation was carried out with com-parative experimental tests on a 2 m3 pilot scale. In asuspension with 40% by weight of solids the abrasionof layers of colored paint over time was observed.Fig. 35 shows the results: scouring away of the paintlayers on the suction side of a propeller blade directlybehind its tip. Qualitatively, the erosion pattern cor-responds to that experienced on an industrial scale andthe local values predicted by CFD.
Resulting development objectives
As shown, CFD can be used to analyze the interactionbetween the vortex formation at the impeller blade andsolid particles, which cannot be measured experi-mentally. In order to reduce the erosive local inter-actions of the particles with the blade which weredetermined, it is necessary to create a flow pattern withchanged movements of the particles. One measure tomollify the backflow of the vortices at the blade tipconsists of diverting the blade vortices by a wide outerblade specifically curved upwards. The EKATO ISOJETshown in Fig. 36 with its geometry optimized in thisway can be seen to have a lower and more uniformerosion pattern. Other possibilities of minimizing the
interaction between the particles and the blade are tochange the blade angle and the specific geometry ofthe blade outside edge. The impeller on the left inFig. 36, the EKATO VISCOPROP, displays an erosionpattern which is more uniform than that of the pro-peller and which has been moved to another location.
The above example clearly shows the benefits ofCFD in analyzing local phenomena that are not directlyaccessible with instrument sensors and the possi-bilities that therefore arise to develop agitator systemsthat are specifically designed to be more erosion-resistant.
Process design
A further application of CFD in a mixing technology isillustrated by the complex absorption process in a fluegas desulfurization system [21].
The special feature of the “jet bubble reactor”shown in Fig. 37 is that several physical and chemicalprocess operations take place simultaneously in onevessel. In contrast to a conventional spray tower, theflue gas is blown directly into the liquid through severalthousand tubular lances. The zone of bubbles which is
71
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Fig. 35 Abrasion test at pilot scale
Fig. 36 More uniform erosion pattern on the EKATO VISCOPROP (left) and the EKATO ISOJET (right)
flue gas
desulfurizedflue gas
gypsumslurry
limestoneslurry
air
Fig. 37 Absorber in a flue gas desulfurizationsystem
thus created results in the immediate absorption andchemical reaction of the SO2 with the limestone slurryfed to the vessel. At the same time, the calcium sulfitewhich results from the above reaction is oxidized inseveral stages to gypsum by the air that is also blowninto the process. The gypsum is drawn off as asuspension, and the treated flue gas collects over theliquid surface and leaves the absorber through theoutlet flues.
The process has been chosen as an example toillustrate the following requirements for the agitators:the agitator system has the task of accelerating thecombined physical and chemical processes of dis-solving the limestone, SO2 absorption and neutrali-zation through a short mixing time. A large interfacialsurface is produced between the air introduced to thevessel and the liquid phase, enabling high mass trans-fer rates and hence a fast sulfite oxidation process totake place. It is a vital requirement of the agitatorsystem that the gypsum that is formed is maintained insuspension, to avoid a buildup of sediment.
Numerical calculation
CFD offers the possibility of analyzing part of theprocess which is highly significant for the reliability ofthe entire absorber in detail, i.e. the suspension oflimestone and gypsum. To do this, a 3-dimensional cellmodel of the absorber sump on both a pilot scale andan industrial scale was created. To compare theirsuspending efficacy, various agitator impellers weremodeled explicitly and integrated into the cell modelwhen required. In order to simulate the suspendingprocess with sufficient accuracy, it was necessary tocreate a numerical mesh with 90,000 cells, for whichthe mass and momentum balances must be solvediteratively. Interactions between the solid particles andthe liquid are described with Euler’s two-phase model[22] as a simplification, yet with sufficient accuracy. Thecomputing effort required to cope with problems ofthis type can be carried out, within a reasonable timeand at an economically justifiable expense. This ispossible today with the CFD software (including thespecial auxiliary programs required) and the com-puting capacities now available. Fig. 38 illustrates thecalculated distributions of the solids concentrations inthe absorber sump using an EKATO VISCOPROP and apitched blade turbine, the propeller type usually usedfor this application.
Validation
To validate the CFD calculations it is necessary to carryout a calibration procedure using experimental resultsfrom pilot-scale tests. It is appropriate to do this insteps. The process is at first only investigated partially,both in the experiment and in the CFD calculation, bydispensing with an expensive and time-consuminggeometric CFD simulation of the gas lances. The gasmixing times measured in the experiment and the axialand radial velocity profiles prove to be in close agree-ment with the calculated values. Fig. 39 shows acomparison of the axial velocities in the particularlycritical edge zone of the absorber sump. Afterdemonstrating the validity of the basic model used, thiscan be refined further, for example by integrating thegas lances as additional resistances to flow in themodel.
Scale-up
The numerical scale-up to industrial dimensions wascarried out with the procedure described above and thegas lances integrated into the system of equations.This made it possible to carry out an adequate andreliable CFD simulation of the process for the industrialscale and the uncertainty that arises with conventionaldesign methods in specifying the scale-up criterion tobe applied was eliminated.
The CFD simulation of the flue gas absorberprocess predicted optimum operating conditions if a
72
Computational fluid dynamics EKATO
Fig. 38 Comparison of the solids distribution byan EKATO VISCOPROP and a pitched bladeturbine
14.3
14.1
13.9
13.7
13.5
cV [%]
4-bladed EKATO VISCOPROP agitator system with a5 m diameter and 36° angle of attack were installed.Focusing on the suspension results, a conventionalpitched blade turbine with the same power input givesan increased proportion of the solids around thebottom of the sump. In contrast, the EKATO VISCO-PROP produces a uniform distribution of the gypsumwithout any build-up over the entire vessel. This ex-ample shows that a CFD analysis of flow conditions ina stirred vessel, even with difficult geometries, can helpto find an optimized agitator system for the specifiedprocess duty.
Author:Ursula Marte
73
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0 15 30 45 60 75 900.0
0.1
0.2
0.3
0.4
measured
calculated
Segment angle γ [°]
Axi
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[m
/s]
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Fig. 39 Comparison of the measured and calculatedaxial velocities
74
EXPERT SYSTEM
The knowledge-based system ANTOS has been deve-
loped at EKATO from the specialized knowledge of
mixing technology that has been accumulated over
many years. The resulting, formally structured expert
knowledge of an agitator manufacturer has a variety
of uses, and offers the customer a task-oriented agita-
tor system that is optimized for cost-effectiveness.
Knowledge from a vast wealth of experimental work,
CFD simulations and basic mechanical approaches,
using techniques such as finite element analysis (FEA)
form the basis of the system.
This is backed up by a variety of experience from
commercial practice, and includes industry- and
process-specific knowledge, engineering and design
expertise, etc. A specially customized user interface
assists the EKATO engineer in applying this readily
available knowledge in his or her daily work for the
optimized selection and design of customer-specific,
cost-effective agitator systems and a fast turnaround
with enquiries and quotations. The design system
illustrated in Fig. 40 relies on databases covering
mixing processes, customer trials, engineering
drawings, etc. This accumulation of data and infor-
mation can be considered as the raw materials, which
can only be put into practical effect with experience
and due interpretation of their knowledge content.
The design system illustrated in Fig. 40 relies on data-bases covering mixing processes, customer trials,engineering drawings, etc. This accumulation of dataand information can be considered as the raw mate-rials, which can only be put into practical effect withexperience and due interpretation of their knowledgecontent. On the one hand this means knowledge ofwhat constitutes an agitator, which physical laws areapplicable, which industrial standards must be com-plied with and which processes take place in the mix-tures of substances. On the other hand, achievingexpert status means the possession of almost a sixthsense, i.e. intuitive knowledge that cannot be baseddirectly on the reasoning processes of physics or
75
EKATO Expert system
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Des
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syst
emK
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Diagnosis
Archiving Compliance with standards
Preparing quotation
Process and mechanicaldesign, detailengineering
Process expertise
Mechanical designknowledge base
Detail engineeringknowledge base
Process designknowledge base
Mixing processesMechanicaltests
Detaildrawings
Finite elementanalysis
Customertests
Standardcomponents
Computational fluid dynamics and virtual impellers
Experience, communications, interpretation
Fig. 40 Architecture of the design system
Agitatorcomponent
Impeller
Propeller
Pitched blade turbine
Flat blade disk turbine
Anchor
Single speed
Two speed
Solid shaft
Hollow shaft
Variable speed
Mechanical seal
Lip seal
Shaft
Motor
Yoke
Shaft seal
Fig. 41 Object-oriented database
mathematics, but is plausible and derives from longyears of professional involvement with the subject.Both forms of knowledge are programmed in anobject-oriented structure.
Both hierarchies have been set up for theseobjects. In the first hierarchy, shown in Fig. 41, thecomponents are described as parts of larger units. Inthe second hierarchy the components themselves arespecified. Every object is associated with parameters,which in the process of configuring the system must becalculated, and the calculation method specified. In theempirical knowledge base the interdependence of thecomponents is filed mainly in the form of rules.
When a customer enquiry is processed, the com-puter first analyzes the problem in terms of processdesign and provides the operator with suggestions forthe basic design of the agitator: number and type ofimpellers, shaft speed, drive power rating and otherfundamental factors.
The detailed specification of the components,including their dimensions and materials of construc-tion, is carried out in the subsequent configurationprocess. It is here that the empirical knowledge base isput to work to find the correct alternative from possibleoptions. Under certain circumstances individual com-ponents or the entire arrangement must be determinedagain from scratch. For example, it can be found fromthe database that a certain drive power rating is notavailable as a standard option. If a drive with anothershaft speed or power rating to the one specified issought, ANTOS carries out all the required calculationsautomatically and adapts the design to suit the newconditions.
The ANTOS knowledge-based system results inan optimized design for practical operation, becausethe available engineering and practical knowledge isutilized systematically as a key enterprise resource.
Author:Ursula Marte
References
Testing
[1] Rieger; Novak: Chem. Eng. Science, 1974 Vol. 29[2] Metzner; Otto: AIChE (1957),1[3] Zwietering: Chem. Eng. SCI 8, 244, 1958[4] Buurman; Resoort; Plaschkes: Proc. 5th Conf.
Mixing, Würzburg, 1985, S. 15[5] Weisman; Efferding: A./Ch. E. Y. 6, 419, 1960[6] Einenkel; Mersmann: Verfahrenstechnik 11, 90,
1977[7] Franz; Schneider: (internal test report, 1988),
Vergleich Sondenmethode und Absaugmethodezur Bestimmung der Feststoffkonzentration beimSuspendieren von Glaskugeln
[8] Kraume,M.; Zehner,P.: Standardisierung von Ver-suchsbedingungen für Rührwerke; CITEAH (1999)
[9] Oxi-Fibel: Einführung in die Gelöstsauerstoff-Meßtechnik, Firmenschrift WTW, Wissenschaft-lich-Technische Werkstätten (1987)
[10] Hiby: Definition und Messung der Mischgüte inflüssigen Gemischen: Chem.-Ing.-Tech. 51 (1979),p. 704/9
[11] Grossmann: Dissertation, TU München, 1982[12] Brooks: Trans. I. Chem. Eng., 57, 210, 1979[13] Franz: Forschungsbericht EKATO/BMFT, No. PBE
03 8660[14] Schneider; Franz; Vettermann: Chemical Enginee-
ring and Technology, June 1990
DIN 1342 Pt 2 Viskosität, Newtonsche Flüssig-keiten, February 1986, Beuth-Verlag
Computational fluid dynamics
[15] Launder; Spalding: “The numerical computationof turbulent flows” Comp. Meth. in AppliedMechanics and Eng., Vol. 3, pp. 269-289
[16] Griebel: Numerische Simulation in der Strö-mungsmechanik, Vieweg, Braunschweig, Wies-baden, 1995
[17] Versteeg, H.K.; Malalasekera, W.: An introductionto computational fluid dynamics, The finitevolume method, Longman 1995
[18] Oertel; Laurien: Numerische Strömungsmecha-nik, Berlin, Heidelberg, New York, Springer 1995
[19] Ferzinger; Peric: Computational Methods for FluidDynamics, Springer Berlin, Heidelberg 1996
[20] Patankar: Numerical Heat Transfer and Fluid Flow,McGraw-Hill, New York,1980
[21] U. Marte; R. Krebs: M. Brenner: “NumerischeSimulation in der Rühr- und Mischtechnik”; CIT(70) 4/98, S. 346
[22] CFX International: CFX4.1 Flow Solver UserGuide, Harwell, Oxforshire, UK, 1995
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