Liquid Mixing Fundamentals Piero M. Armenante 2008 © Special Topics - Modules in Pharmaceutical...

Liquid Mixing Fundamentals Piero M. Armenante 2008 Special Topics - Modules in Pharmaceutical Engineering ChE 702

Piero M. ArmenanteChE7022 Instructional Objectives of This Section By the end of this section you will be able to: Identify the geometric, physical and dynamic variables of importance for the analysis of mixing in a stirred tank Assess the relative importance of those variables Quantify the power dissipation, pumping effects, and blend time in a mixing vessel

Piero M. ArmenanteChE7023 Summary Basic Rheology Power Dissipation Impeller Pumping Effects Blend Time in Stirred Tanks

Basic Rheology

Piero M. ArmenanteChE7025 Basic Rheological Concepts Consider a fluid contained between two plates separated by a distance y. One plate is set in motion parallel to the other, with velocity v x. For many fluids it has been found experimentally that the force applied to the plate is directly proportional to v x and inversely proportional to y.

Piero M. ArmenanteChE7026 Basic Rheological Concepts

Piero M. ArmenanteChE7027 Newtons Law of Viscosity Mathematically: i.e.: This constitutes Newtons Law of Viscosity.

Piero M. ArmenanteChE7028 Newtons Law of Viscosity Definitions Shear Stress: Shear rate: (Dynamic) Viscosity: Kinematic Viscosity:

Piero M. ArmenanteChE7029 Newtonian Fluids Newtonian fluids are fluids having constant viscosity.

Piero M. ArmenanteChE70210 Dynamic Viscosities of Various Fluids

Piero M. ArmenanteChE70211 Dynamic Viscosities of Various Fluids

Piero M. ArmenanteChE70212 Focus of This Section Only the mixing behavior of Newtonian fluids, and, more specifically, liquids, will be examined in this section.

Piero M. ArmenanteChE70213 Schematic of a Stirred Tank

Piero M. ArmenanteChE70214 Important Variables in the Analysis of Mixing Phenomena The variables of importance in the analysis of mixing phenomena in stirred tanks can be classified as: geometric variables physical variables dynamic variables

Piero M. ArmenanteChE70215 Geometric Variables Geometric variables include the geometric characteristics of: tank (shape, sizes) shaft liquid height baffles (shape, size, position) impellers (type, dimensions, position)

Piero M. ArmenanteChE70216 Geometric Variables: Tank, Shaft, and Liquid Height Tank shape (e.g., cylindrical) Tank bottom shape (e.g., dish, flat) Internal diameter, T Internal height, H T Shaft diameter Shaft length Liquid height, H (or Z)

Piero M. ArmenanteChE70217 Geometric Variables: Baffles Number of baffles, n B Shape (e.g., rectangular) Baffle width, B Baffle height (e.g., full, half) Baffle thickness Gap between baffles and tank wall Gap between baffles and tank bottom

Piero M. ArmenanteChE70218 Geometric Variables: Impellers Number of impellers, n Impeller type (e.g., disc turbine) Diameter, D Blade angle [Pitch, p] Blade width (height), w [Blade width projected across the vertical axis, w b ]

Piero M. ArmenanteChE70219 Geometric Variables: Impellers (continued) Clearance off the tank bottom measured from the midpoint, C [Clearance off the tank bottom measured from the impeller bottom, C b ] Spacing between impellers, S Disc diameter (disc turbines) Blade thickness Hub diameter

Piero M. ArmenanteChE70220 Physical Variables Liquid density, or L Liquid rheology (e.g., newtonian, non-netwonian, shear-thinning, etc.) and corresponding parameters (e.g., power law exponent) Dynamic viscosity, [Kinematic viscosity, (= /)]

Piero M. ArmenanteChE70221 Dynamic Variables Impeller rotational (agitation) speed, N Impeller angular velocity, Impeller tip speed, v tip Torque, Power dissipation (consumption), P Impeller pumping flow, Q Gravitational acceleration, g

Piero M. ArmenanteChE70222 Relationship Between N, and V tip The agitation speed, N, must be expressed in revolutions per unit time such as: revolutions per minute (rpm) revolutions per second (rps) The tip speed, v tip, is not independent of N but it is related to N as follows (with in rad/s, N is in rps, D in m, v tip in m/s):

Power Dissipation in Low Viscosity Liquids in Stirred Tanks

Piero M. ArmenanteChE70224 Instructional Objectives of This Section By the end of this section you will be able to: Calculate Re, Fr in stirred tanks Distinguish agitation regimes Calculate the power dissipated by an impeller from available power numbers Calculate the power dissipation as a function of operating variables

Piero M. ArmenanteChE70225 Turbulence and Mixing Turbulent flows are associated with rapid, apparently random fluctuations of all three components of the local velocity vector with time To this day turbulence is still a relatively poorly understood phenomenon Many mixing phenomena are associated with turbulence

Piero M. ArmenanteChE70226 Velocity Fluctuations in Turbulent Flow

Piero M. ArmenanteChE70227 Turbulent Flow In a turbulent flow, pulsations consisting of disorderly displacement of fluid bodies (eddies), are superimposed on an average flow.

Piero M. ArmenanteChE70228 Isotropic Turbulence In isotropic turbulence all the fluctuation components are equal, and there is no correlation between the fluctuations in different directions

Piero M. ArmenanteChE70229 Energy Cascade in Isotropic Turbulent Flow During the process of energy transfer and ultimate decay in a turbulent system the largest eddies receive fresh kinetic energy from an outside source (e.g., an impeller) and pass it on to smaller eddies that are produced as a result of the instability of the primary eddies

Piero M. ArmenanteChE70230 Energy Cascade in Isotropic Turbulent Flow During this process smaller and smaller eddies are generated One can conceptually introduce an eddy Reynolds Number:

Piero M. ArmenanteChE70231 Energy Cascade in Isotropic Turbulent Flow As long as Re eddy >>1 no viscous dissipation will occur, and the kinetic energy will simply be transferred to smaller and smaller eddies However, at Re eddy ~1 viscous forces will begin to dominate

Piero M. ArmenanteChE70232 Energy Cascade in Isotropic Turbulent Flow For Re eddy 10,000) the flow in a baffled tank is turbulent Theoretical and experimental evidence shows that N P is independent of Re: i.e.:

Piero M. ArmenanteChE70261 Power Curve: Turbulent Flow Regime In the turbulent flow region the power dissipated by an impeller is given by: where k is a proportionality constant equal to N PT, the asymptotic value of N P that depends on: type of impeller geometry ratios for the system

Piero M. ArmenanteChE70262 Power Dissipation in the Turbulent Flow Regime In turbulent regime, power dissipation is: independent of viscosity directly proportional to the density of the liquid very strongly affected by the agitation speed (PN 3 ) extremely sensitive to the impeller diameter (PD 5 )

Piero M. ArmenanteChE70263 Sensitivity of Power Dissipation In the turbulent regime P is very sensitive to N and D Examples: a 10% increase in agitation speed, N increases the power dissipated by 33% a 20% increase in N increases P by 73% a 10% increase in impeller diameter, D increases the power dissipated by 61% a 20% increase in D, increases P by 148%

Piero M. ArmenanteChE70264 Sensitivity of Power Dissipation Because of the sensitivity of the power dissipation to impeller diameter and agitation speed small adjustments to the impeller size or agitation speed can rectify situations in which an existing motor is underpowered

Piero M. ArmenanteChE70265 Power Number Curves for Various Impellers After Bates et al., Ind. Eng. Chem. Proc. Des. Devel. 1963

Piero M. ArmenanteChE70266 Power Number Curves for 45 Pitched-Blade Turbines (4-Blades) and HE-3 Impeller After K. Myers and R. J. Wilkens, Personal Communication

Piero M. ArmenanteChE70267 Equation for Power Number Curves An equation for Power Number as a function of Re has been proposed: where A, B, and C are coefficients that depend on the type of impeller. After John Smith, Unpublished Data

Piero M. ArmenanteChE70268 Coefficients in Equation for Power Number Curves

Piero M. ArmenanteChE70269 Turbulent Power Number Most low viscosity systems and industrial stirred tanks operate in the turbulent regime where N P is constant A simple and meaningful way to compare the power performance of various agitators is to compare their turbulent Power Numbers, N PT The term Power Number is often used to mean Turbulent Power Number

Piero M. ArmenanteChE70270 Turbulent Power Numbers Turbulent Power Numbers have been obtained experimentally for many impellers Typically, N PT is measured for a standard configuration of the agitation system (H=T, D/T=1/3, C=D) Data also exist for other non-standard systems (e.g., N P as a function of C/D)

Piero M. ArmenanteChE70271 Turbulent Power Numbers for Various Impellers

Piero M. ArmenanteChE70274 Effect of D/T Ratio on Power Number for Disc Turbines

Piero M. ArmenanteChE70275 Effect of D/T Ratio on Power Number for Pitched-Blade Turbines

Piero M. ArmenanteChE70276 Effect of Impeller Clearance on Power Number for Disc Turbines

Piero M. ArmenanteChE70277 A correlation between the Power Number and the impeller clearance off the impeller bottom, C b1, is: Effect of Impeller Clearance on Power Number for Disc Turbines

Piero M. ArmenanteChE70278 Effect of C b on Power Number for Disc Turbines

Piero M. ArmenanteChE70279 Effect of C b on Power Number for Flat-Blade Turbines

Piero M. ArmenanteChE70280 Effect of Impeller Clearance on Power Number for Pitched-Blade Turbines

Piero M. ArmenanteChE70281 Effect of C b on Power Number for Pitched-Blade Turbines

Piero M. ArmenanteChE70282 Effect of C b on Power Number for HE-3 Impellers

Piero M. ArmenanteChE70283 Power Dissipation in Multiple Impeller Systems

Piero M. ArmenanteChE70284 Power Dissipation in Multiple Impeller Systems If the H/T ratio is larger than 1.2-1.5 multiple impellers are typically used The Power Number and the power drawn by two impellers mounted on the same shaft and spaced by a distance S is not usually twice that of the individual impeller For large S, N P double 2 N P single

Piero M. ArmenanteChE70285 Power Dissipation in Multiple Disc Turbine Systems

Piero M. ArmenanteChE70286 Power Dissipation in Double Disc Turbine Systems

Piero M. ArmenanteChE70291 Power Dissipation in Triple Disc Turbine Systems

Piero M. ArmenanteChE70292 Power Dissipation in Triple Disc Turbine Systems

Piero M. ArmenanteChE70293 Power Dissipation in Double Pitched-Blade Turbine Systems

Piero M. ArmenanteChE70296 Power Curves for Impellers in Baffled and Unbaffled Tanks

Piero M. ArmenanteChE70297 Power Curves for Impellers in Baffled and Unbaffled Tanks N P vs. Re plots for baffled systems show that N P reaches an asymptotic value at high Reynolds Number N P vs. Re plots for unbaffled systems show that N P keeps decreasing with Re even at high Reynolds Numbers

Piero M. ArmenanteChE70298 Power and Torque The power drawn by an impeller, P, and the torque, , required by the same impeller rotating at N are related to each other by the following equation: Remark: the same power dissipation can be achieved using a higher torque and smaller agitation speed or vice versa

Piero M. ArmenanteChE70299 Power Dissipation and Operating Cost of Mixing The power dissipated by the impeller, P, is just the energy consumed by the impeller per unit time, typically as electric energy Hence, the operating cost of the mixing operation are proportional to P:

Piero M. ArmenanteChE702100 Torque and Capital Cost The capital cost of a mixing operation is significantly dominated by the cost of the gear box The cost of the gear box is directly related to the its torque rating, typically through an power law:

Piero M. ArmenanteChE702101 Important Mixing Operating and Scale-up Parameters Traditionally, mixing processes have been scaled up and operated by maintaining constant one the following parameters: Power per unit liquid volume in the tank, P/V, or per unit liquid mass, P/V Torque per unit liquid volume in the tank, /V, or per unit liquid mass, /V

Piero M. ArmenanteChE702102 Power per Unit Volume The power dissipated by the impeller per unit liquid volume in the tank: is one of the most important mixing parameters used in scale up of mixing processes The units for P/V are W/L, kW/m 3 or hp/1000 gal

Piero M. ArmenanteChE702103 Power per Unit Mass The power dissipated by the impeller per unit liquid mass in the tank, : is an alternative to the use of P/V (since the only difference is the presence of ) is also widely used for scale-up The units for are m 2 /s 3

Piero M. ArmenanteChE702104 Substituting for P and V gives: Power per Unit Volume

Piero M. ArmenanteChE702105 The ratio of P/V at two different scales is: Power per Unit Volume at Different Scales

Piero M. ArmenanteChE702106 For geometrically similar systems: and the P/V ratio becomes: Power per Unit Volume at Different Scales

Piero M. ArmenanteChE702107 Scale-up Based on Constant Power per Unit Volume (P/V) If P/V is kept constant during scale-up of geometrically similar systems:

Piero M. ArmenanteChE702108 Tip Speed and Torque per Unit Volume For geometrically similar systems (for which D T) in fully turbulent regimes, or for the same system at different agitation speeds, if the torque per unit volume, /V, is kept constant, then:

Piero M. ArmenanteChE702109 Tip Speed and Torque per Unit Volume Simplifying: i.e., keeping constant the tip speed is equivalent to keeping /V constant, provided that the geometry of the systems is similar and the flow is fully turbulent

Piero M. ArmenanteChE702110 Typical P/V Values for Common Mixing Processes

Piero M. ArmenanteChE702111 Typical Tip Speed and P/V for Various Mixing Equipment After Arthur Etchells, Unpublished Data

Piero M. ArmenanteChE702112 Additional Power Sources in Stirred Tanks In the vast majority of cases mechanical power input in stirred tanks is provided by impellers Additional mechanical power sources can also be present, and their contribution should be incorporated in power calculations

Piero M. ArmenanteChE702113 Additional Power Sources in Stirred Tanks Mechanical power can be supplied to stirred tanks via three primary different sources, i.e.: mechanical agitation (e.g., impellers) power delivered by the expansion of a compressed gas (e.g., gas dispersers, diffusers) power delivered by the kinetic energy of a liquid (e.g., jets)

Piero M. ArmenanteChE702114 Additional Power Sources in Stirred Tanks Important Remark: some mechanical power sources, e.g., sparging a gas, typically reduces the mechanical power input by the impeller (e.g., a gassed impeller)

Piero M. ArmenanteChE702115 Power Input by Gas Sparging The mechanical power input contribution of a gas sparged inside a liquid is:

Piero M. ArmenanteChE702116 Power Input of a Liquid Jet The mechanical power input contribution of a liquid jet injected inside a liquid is:

Piero M. ArmenanteChE702117 Total Mechanical Power Input The total mechanical power input to a liquid in a stirred vessel is: In the presence of a sparged gas:

Impeller Pumping Effects

Piero M. ArmenanteChE702119 Instructional Objectives of This Section By the end of this section you will be able to: Distinguish the flow patterns generated by different impellers under different operating conditions Calculate the impeller discharge flow from available flow numbers

Piero M. ArmenanteChE702120 Impeller Pumping Action Both radial and axial impellers exert a pumping action within the tank The mixer can then be regarded as a caseless pump Different types of impellers produce different pumping actions resulting in the establishment of fluid flow circulation patterns inside the tank

Piero M. ArmenanteChE702121 Vortices Generated by Impeller Blades Both radial and axial impellers produce strong vortices behind them These vortices are primarily responsible for a number of mixing phenomena, including bubble and droplet breakup, rapid mixing of homogeneous fluids, and power dissipation

Piero M. ArmenanteChE702122 Vortices Generated by a Disc Turbine Blade Disc Ulbrecht and Patterson, Mixing of Liquids by Mechanical Agitation, 1985.

Piero M. ArmenanteChE702123 Vortices Generated by a Disc Turbine A balanced vortex pair develops behind a Rushton turbine blade, conveying away turbulent energy Source: John Smith, Mixing XX

Piero M. ArmenanteChE702124 Vortices Generated by a Pitched Blade Turbine Ulbrecht and Patterson, Mixing of Liquids by Mechanical Agitation, 1985.

Piero M. ArmenanteChE702125 Vortices Generated by a Pitched Blade Turbine The single line vortices from pitched blade or hydrofoil impellers are less intense that those generated by the flat blade of a Rushton turbine Source: John Smith, Mixing XX

Piero M. ArmenanteChE702126 Flow Pattern for Axial Impellers in Baffled Tanks Axial impellers tend to pump downward or upward, depending on the direction of rotation Downward pumping impellers produce an axial (or angled) main flow that: impinges on the tank bottom first moves upwards near the tank wall converges radially inwards, and then returns to the impeller to feed it

Piero M. ArmenanteChE702127 Flow Pattern for a Typical Axial Impeller

Piero M. ArmenanteChE702128 Flow Pattern for Radial Impellers in Baffled Tanks Radial impellers pump the liquid radially, forming a radial jet If C/T is sufficiently high, as the liquid jet impinges on the tank wall it splits upwards and downwards Both upward and downward flows move vertically first, converge radially inwards, and then return to the impeller to feed it (double- eight flow pattern)

Piero M. ArmenanteChE702129 Flow Pattern for a Typical Radial Impeller (high C/T)

Piero M. ArmenanteChE702130 Flow Pattern for Radial Impellers in Baffled Tanks If C/T is low, the liquid jet impinging on the tank wall only forms an upward flow that first moves vertically near the wall, then converges radially inwards, and returns to the impeller to feed it (single- eight flow pattern) In the single-eight regime the lower circulation patter is suppressed because of the proximity with the tank bottom

Piero M. ArmenanteChE702131 Flow Pattern for a Typical Radial Impeller (low C/T)

Piero M. ArmenanteChE702132 Impeller Clearance and Flow Pattern Change with Disc Turbines For disk turbines a flow transition from double-eight to single- eight regimes occurs when the C/T ratio drops below a specific value: For C/T >0.2 double-eight flow pattern For 0.16

Piero M. ArmenanteChE702133 Impeller Clearance and Flow Pattern Change with Flat-Blade Turbines Also for flat-blade turbines the flow pattern changes from double-eight to single-eight regimes as C/T varies. For C/T >0.25 double-eight flow pattern For 0.20

Piero M. ArmenanteChE702134 Velocity Flow Field in a Stirred Tank

Piero M. ArmenanteChE702135 Experimental Velocity Measurement Local velocity measurements inside a stirred tank are generally difficult Techniques include: laser-Doppler velocimetry (LDV) hot-wire anemometry whole flow visualization

Piero M. ArmenanteChE702136 Laser-Doppler Velocimetry (LDV) System

Piero M. ArmenanteChE702137 Impeller Discharge Flow The pumping action of an impeller results in a discharge flow rate out of the impeller region, Q out, balanced by an incoming flow toward the impeller (inflow rate= Q in ). Since mass is conserved it must be that: where Q is the discharge flow rate

Piero M. ArmenanteChE702138 Impeller Discharge Flow Cylindrical envelope to determine flow out of impeller region

Piero M. ArmenanteChE702139 Impeller Discharge Flow The impeller discharge flow rate, Q, can be obtained by summing up the outflow contributions from all the surfaces of the cylinder enveloping the impeller:

Piero M. ArmenanteChE702140 Impeller Discharge Flow

Piero M. ArmenanteChE702141 Flow Number (or Pumping Number) N Q In order to make the impeller discharge flow rate non-dimensional one can define the Flow Number, or Pumping Number, N Q :

Piero M. ArmenanteChE702142 Turbulent Flow Numbers The Flow Number is to the discharge flow rate what the Power Number is to power Turbulent Flow Numbers, N QT (or simply N Q ) have been obtained experimentally for many impellers Typically, N Q is measured for a standard configuration of the agitation system (H=T, D/T=1/3, C=D)

Piero M. ArmenanteChE702143 Flow Numbers for Various Impellers in Baffled Tanks

Piero M. ArmenanteChE702144 Flow Numbers for 45 Pitched- Blade Turbines (4 Blades) After K. Myers and R. J. Wilkens, Personal Communication

Piero M. ArmenanteChE702145 Flow Numbers for HE-3 Impellers After K. Myers and R. J. Wilkens, Personal Communication

Piero M. ArmenanteChE702146 Relationship Between Power and Flow In a number of industrial cases it may be advantageous to use impellers that produce significant circulation within the tank, but consume little power. To determine the optimal impeller design and operation the following ratio: should be maximized.

Piero M. ArmenanteChE702147 Relationship Between Power and Flow For a fixed impeller geometry it is: i.e.:

Piero M. ArmenanteChE702148 N Q /N P for Various Impellers in Baffled Tanks

Piero M. ArmenanteChE702149 Variation of Flow and Power Dissipation Most impellers have flow numbers in the relatively narrow range of 0.4-0.9 (typically 0.5-0.8), i.e., their ability to pump is of the same order of magnitude The same impellers have power numbers ranging between 0.25 and 6, a much wider range

Piero M. ArmenanteChE702150 Variation of Flow and Power Dissipation Impellers with blades oriented parallel to the shaft produce radial flow, and have high power dissipation rates although their pumping action is significant As a consequence, their N Q /N P ratios is low Radial impellers generate significant turbulence and produce high shear

Piero M. ArmenanteChE702151 Variation of Flow and Power Dissipation Impellers with blades forming a (small) angle with the plane of rotation produce axial flow, and have relatively low power dissipation rates although their pumping action is also significant As a consequence, their N Q /N P ratios will be high Axial impellers generate less turbulence and shear

Piero M. ArmenanteChE702152 Optimization Strategies to Maximize Pumping Efficiency To maximize pumping efficiency (i.e., maximize the Q/P ratio): choose impellers with high N Q /N P ratios if capital cost must be minimized select impellers with the same v tip (=ND) but lower D, since this decreases N and hence the torque (=P/2N) [recall that the cost of the gear box is proportional to the torque]

Piero M. ArmenanteChE702153 Optimization Strategies to Maximize Pumping Efficiency If a specific flow rate Q must be achieved then, by rearranging it is: To lower P at constant Q one can lower N while increasing D. This approach decreases the operating cost ( P).

Piero M. ArmenanteChE702154 Optimization Strategies to Maximize Pumping Efficiency If a specific power input P must be maintained then, by rearranging it is: To increase Q at constant P one can lower N while increasing D. This approach increases the capital cost (proportional to the torque =P/2N).

Piero M. ArmenanteChE702155 Optimization Strategies to Maximize Pumping Efficiency The preceding analysis is valid if N P and N Q are constant. This is correct if the flow is fully turbulent. Changing the D/T ratio usually has little influence on N P and N Q provided that it is not too small or large (0.25< D/T

Piero M. ArmenanteChE702199 Blend Time at Different Reynolds Numbers f Re is the correction factor to account for Re effects when Re is below 10,000 and the fluid is not fully turbulent. Remark: f Re =1 for Re>10,000. Once t U has been calculated for Re >10,000 it is possible to obtain t U at other Reynolds Number using the diagram obtained by Norwood and Metzner (1964).

Piero M. ArmenanteChE702200 Blend Time at Different Reynolds Numbers

Piero M. ArmenanteChE702201 Effect of Viscosity Ratio on Blend Time f * is the correction factor to account for viscosity effect, when the viscosity of the added fluid is greater than that of the liquid in the tank. In order to calculate f * the viscosity ratio: must be determined first.

Piero M. ArmenanteChE702202 Effect of Viscosity Ratio on Blend Time Fasano et al., Chem. Eng., 1994

Piero M. ArmenanteChE702203 Effect of Density Difference on Blend Time f is the correction factor to account for the effect of differences in densities between the added liquid an the liquid in the tank. If the density difference is zero (standard case) f =1. In order to calculate f the Richardson Number, Ri, must be calculated first.

Piero M. ArmenanteChE702204 Effect of Density Difference on Blend Time To account for the effect of density differences the Richardson Number is introduced:

Piero M. ArmenanteChE702205 Effect of Density Difference on Blend Time Fasano et al., Chem. Eng., 1994

Piero M. ArmenanteChE702206 Blend Time and Geometric Similarity In fully turbulent, geometrically similar systems the equation below still holds: This implies that blend time experiments can be conducted in small scale equipment to determine the above constant, and that this equation can be used for scale-up purposes

Piero M. ArmenanteChE702207 Procedure to Calculate Blend Time The procedure to calculate t U is then: set the desired value of U fix D, T, H and the impeller type set N calculate k calculate the standard blend time correct this value to account for Re, viscosity and density effects

Piero M. ArmenanteChE702208 Scale-up and Blend Time Scale-up based on blend time is extremely costly to achieve since the power consumption would increase enormously. For constant N: This implies that P/V increases with the square of the scale-up factor

Piero M. ArmenanteChE702209 Scale-up and Blend Time If P/V is kept constant during scale- up:

Piero M. ArmenanteChE702210 Scale-up and Blend Time Recall that: If P/V is kept constant during scale-up of geometrically similar systems: the blend time increases with the (linear) scale factor raised to the 2/3 power

Piero M. ArmenanteChE702211 Blend Time and Other Time Scales It is always important to make sure that blend time is much shorter than the other time scales that may be important to the process If blend time is longer than other critical time scales (e.g., reaction time) mixing could become the limiting step, often inadvertently

Piero M. ArmenanteChE702212 Blend Time in Small Tanks and Large Tanks Blend time is typically short in small laboratory tanks, but much longer in larger tanks Processes that are not affected by blend time at small scales (since mixing is fast) could be limited by poor mixing at larger scales since blending the tanks contents typically takes much longer

Piero M. ArmenanteChE702213 Blend Time and Circulation Time An empirical mixing rule of thumb states that: where the proportionality constant (4) is often reported to be between 3 and 5 or even outside this range

Piero M. ArmenanteChE702214 From the definition of circulation time (t circ =V/Q) and Flow Number, N Q : it follows that: Blend Time and Circulation Time

Piero M. ArmenanteChE702215 The relationship between t U and t circ can be obtained recalling that: Then: Blend Time and Circulation Time

Piero M. ArmenanteChE702216 Finally: For a given system, and for a pre- assigned level of uniformity, U, all the factors on the right-hand side are fixed. Hence t U /t circ is constant Blend Time and Circulation Time

Piero M. ArmenanteChE702217 Example: Disk turbine in a standard mixing system N Q =0.8 D/T=D/H=1/3 If U=99% (=47) t 99 =5.3t circ If U=95% (=0.6547=30.5) t 95 =3.5t circ Blend Time and Circulation Time

Piero M. ArmenanteChE702218 Blend Time and Circulation Time

Piero M. ArmenanteChE702219 Blend Time and Circulation Time

Piero M. ArmenanteChE702220 The previous results confirm that the blend time is typically a multiple of the circulation time t U /t circ is typically in the range 3-6 for t 95 /t circ and 4-9 for t 99 /t circ These results validate the empirical mixing rule of thumb stating that Blend Time 4 Circulation Time Blend Time and Circulation Time

Piero M. ArmenanteChE702221 Equations for the Determination of Blend Time Approach 2

Piero M. ArmenanteChE702222 Blend Time Equation in Turbulent Regime: Approach 2 For mixing in the turbulent regime (Re>~10,000) Grenville (1992) found: 0.33 < D/T < 0.50 C/T = 0.33 0.50 < H/T 1 Po = Impeller Power Number

Piero M. ArmenanteChE702223 Blend Times to Achieve Various Us Recalling that it is possible to calculate t U knowing t U :

Piero M. ArmenanteChE702224 Blend Times to Achieve Various Us If U=95% is the reference degree of homogeneity: For example:

Piero M. ArmenanteChE702225 Blend Time Equation in Turbulent Regime: Approach 2 For mixing in turbulent regime (Re>~10,000) the Grenville equation becomes: 0.33 < D/T < 0.50 C/T = 0.33 0.50 < H/T 1 Po = Impeller Power Number

Piero M. ArmenanteChE702226 Blend Time for Turbulent Regime with H/T=1, T/D=3: Approach 2

Piero M. ArmenanteChE702227 Impeller Efficiency in Turbulent Regime: Approach 2 From: and: it is:

Piero M. ArmenanteChE702228 Impeller Efficiency in Turbulent Regime: Approach 2 From the previous equation for turbulent regime, it is: all impellers of the same diameter are equally energy efficient (i.e., achieve the same t U at the same power input/mass) shorter t U are achieved with larger impellers at the same power input/mass blend time is independent of fluid properties when scaling at constant power input/mass and similar geometry blend time increases with the scale factor raised to 2/3

Piero M. ArmenanteChE702229 Blend Time Equation in Transitional Regime: Approach 2 For mixing in the transitional regime (~200

Liquid Mixing Fundamentals Piero M. Armenante 2008 © Special Topics - Modules in Pharmaceutical...

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