Special Topics - Modules in Pharmaceutical Engineering ChE 702

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

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. Armenante ChE702

Summary Basic Rheology Power Dissipation Impeller Pumping Effects
Blend Time in Stirred Tanks Piero M. Armenante ChE702

Basic Rheology

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 vx. For many fluids it has been found experimentally that the force applied to the plate is directly proportional to vx and inversely proportional to y. Piero M. Armenante ChE702

Basic Rheological Concepts
Piero M. Armenante ChE702

Newton’s Law of Viscosity
Mathematically: i.e.: This constitutes Newton’s Law of Viscosity. Piero M. Armenante ChE702

Newton’s Law of Viscosity
Definitions Shear Stress: Shear rate: (Dynamic) Viscosity: Kinematic Viscosity: Piero M. Armenante ChE702

Newtonian Fluids Newtonian fluids are fluids having constant viscosity. Piero M. Armenante ChE702

Dynamic Viscosities of Various Fluids

Focus of This Section Only the mixing behavior of Newtonian fluids, and, more specifically, liquids, will be examined in this section. Piero M. Armenante ChE702

Schematic of a Stirred Tank

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. Armenante ChE702

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. Armenante ChE702

Geometric Variables: Tank, Shaft, and Liquid Height
Tank shape (e.g., cylindrical) Tank bottom shape (e.g., dish, flat) Internal diameter, T Internal height, HT Shaft diameter Shaft length Liquid height, H (or Z) Piero M. Armenante ChE702

Geometric Variables: Baffles
Number of baffles, nB 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. Armenante ChE702

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, wb] Piero M. Armenante ChE702

Geometric Variables: Impellers (continued)
Clearance off the tank bottom measured from the midpoint, C [Clearance off the tank bottom measured from the impeller bottom, Cb] Spacing between impellers, S Disc diameter (disc turbines) Blade thickness Hub diameter Piero M. Armenante ChE702

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. Armenante ChE702

Dynamic Variables Impeller rotational (agitation) speed, N
Impeller angular velocity,  Impeller tip speed, vtip Torque,  Power dissipation (consumption), P Impeller pumping flow, Q Gravitational acceleration, g Piero M. Armenante ChE702

Relationship Between N,  and Vtip
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, vtip, is not independent of N but it is related to N as follows (with  in rad/s, N is in rps, D in m, vtip in m/s): Piero M. Armenante ChE702

Power Dissipation in Low Viscosity Liquids in Stirred Tanks

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. Armenante ChE702

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. Armenante ChE702

Velocity Fluctuations in Turbulent Flow

Turbulent Flow In a turbulent flow, “pulsations” consisting of disorderly displacement of fluid bodies (eddies), are superimposed on an average flow. Piero M. Armenante ChE702

Isotropic Turbulence In isotropic turbulence all the fluctuation components are equal, and there is no correlation between the fluctuations in different directions Piero M. Armenante ChE702

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. Armenante ChE702

During this process smaller and smaller eddies are generated One can conceptually introduce an eddy Reynolds Number: Piero M. Armenante ChE702

As long as Reeddy>>1 no viscous dissipation will occur, and the kinetic energy will simply be transferred to smaller and smaller eddies However, at Reeddy~1 viscous forces will begin to dominate Piero M. Armenante ChE702

For Reeddy<<1, the eddy will not break up and the eddy kinetic energy will be transformed into heat by the viscous forces (energy dissipation). Piero M. Armenante ChE702

Such a transition occurs at the Kolmogoroff’s length scale, equal to: where  is the power dissipated per unit mass and  is the kinematic viscosity. k is the size of the smallest eddy in the turbulent fluid Piero M. Armenante ChE702

Energy Cascade: Summary
Big whorls have little whorls That feed on their velocity, And little whorls have lesser whorls And so on to viscosity. Lewis F. Richardson ( ) The poem summarizes Richardson's 1920 paper ‘The Supply of Energy from and to Atmospheric Eddies‘. (A play on Jonathan Swift's "Great fleas have little fleas upon their backs to bite 'em, And little fleas have lesser fleas, and so ad infinitum." (1733)) Piero M. Armenante ChE702

Energy Cascade: Summary
Big whirls have little whirls, That feed on their velocity, Little whirls have smaller whirls, And so on to viscosity. Piero M. Armenante ChE702

Power Dissipation The power dissipated (or consumed) by the impeller, P, is one of the most important variables to describe the performance of an impeller in a tank P is a function of all the geometric and physical variables of the system Dimensional analysis can be used to establish a relationship between P and the independent variables Piero M. Armenante ChE702

Experimental Determination of Power Consumption
The power dissipated by various impellers under different conditions has been experimentally obtained by many investigators Power data are available in the literature (as non-dimensional Power Numbers) Piero M. Armenante ChE702

It is relatively easy to determine the cumulative overall power drawn by a mixing system (including motor, drives, seals, impellers, etc.) It is much more difficult to determine the power dissipated by the impeller alone The power dissipated by the impeller in the fluid is the only important power dissipation parameter for the mixing process Piero M. Armenante ChE702

The total power dissipation in a system is given by: If one needs to know Pimpeller, Ptotal and all other power dissipation sources must be known under the dynamic conditions in which the impeller operates This can be quite difficult Piero M. Armenante ChE702

A number of methods have been used to measure the power dissipated by impellers including: electric measurements dynamometers (coupled to the motor or the tank) strain gages and torquemeters calorimetric measurements Piero M. Armenante ChE702

Example of Strain Gage System for Power Measurement

Power Dissipation For the case in which a number of geometric variables have been defined (e.g., tank shape, tank bottom, impeller type, baffle position, etc.) the dependence between P and the other variables can be written as: Piero M. Armenante ChE702

Power Dissipation Using dimensional analysis (Buckingham pi theorem) the previous equation can be rewritten in non-dimensional terms, as: Piero M. Armenante ChE702

Power Number, NP (also referred to as Po or Ne)
The impeller Power Number, Np (also called Po, or the Newton number, Ne) is a non-dimensional variable defined as: If English units are used then: Piero M. Armenante ChE702

Power and Power Number The power consumed by an impeller and the Power Number are related to each other via the equation: where Np is a function of the impeller type and the geometric and dynamic characteristic of the system Piero M. Armenante ChE702

Impeller Reynolds Number
The impeller Reynolds number, Re, defined as: is a product of the non-dimensional analysis. Compare this Re with the Reynolds number for a pipe: Piero M. Armenante ChE702

Impeller Reynolds Number
As usual, a physical interpretation can be associated with the impeller Reynolds number, Re. Accordingly: Piero M. Armenante ChE702

Froude Number Another non-dimensional number arising from the non-dimensional analysis is the Froude number, Fr, defined as: Piero M. Armenante ChE702

Froude Number It can be shown that the Froude number has the following physical interpretation: Piero M. Armenante ChE702

Power Equation The power equation can be rewritten as: i.e.:

Geometrical Similarity
Two systems are geometrically similar if all corresponding dimensional ratios are the same in both systems D T H Cb 1.5 H 1.5 Cb 1.5 D 1.5 T Piero M. Armenante ChE702

Geometrical Similarity
For geometrically similar systems: Piero M. Armenante ChE702

Power Equation for Geometrically Similar Systems
For geometrically similar (including same type of impeller) stirred tanks and impeller all geometric ratios are the same Hence, NP does not change with scale between tanks: Piero M. Armenante ChE702

Power Equation for Baffled, Geometrically Similar Systems
When baffles are present, no vortex occurs, i.e., the gravitational forces become unimportant, and the Power Number becomes independent of Fr: Piero M. Armenante ChE702

Typical Power Curve for Impellers in Baffled Tanks

Power Curve: Laminar Flow Regime
For Re<10 the flow in a baffled tank is laminar Theoretical and experimental evidence shows that: i.e.: Piero M. Armenante ChE702

Power Curve: Laminar Flow Regime
In the laminar flow region the power dissipated by an impeller is given by: where k” is a proportionality constant that depends on: type of impeller geometry ratios for the system Piero M. Armenante ChE702

Power Dissipation in the Laminar Flow Regime
In the laminar regime power dissipation is: independent of the density of the liquid directly proportional to the viscosity strongly affected by the agitation speed (PN2) strongly affected by the impeller diameter (PD3) Piero M. Armenante ChE702

Power Curve: Transitional Flow Regime
For ~10<Re<~10,000 the flow regime cannot be well characterized as either fully laminar or fully turbulent Depending on the type of impeller NP may decrease with Re or decrease and then increase with Re before entering the turbulent flow regime Piero M. Armenante ChE702

Power Curve: Turbulent Flow Regime
At high Reynolds numbers (Re>10,000) the flow in a baffled tank is turbulent Theoretical and experimental evidence shows that NP is independent of Re: i.e.: Piero M. Armenante ChE702

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 NPT, the asymptotic value of NP that depends on: type of impeller geometry ratios for the system Piero M. Armenante ChE702

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 (PN3) extremely sensitive to the impeller diameter (PD5) Piero M. Armenante ChE702

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. Armenante ChE702

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. Armenante ChE702

Power Number Curves for Various Impellers
After Bates et al., Ind. Eng. Chem. Proc. Des. Devel. 1963 Piero M. Armenante ChE702

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. Armenante ChE702

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. Armenante ChE702

Coefficients in Equation for Power Number Curves

Turbulent Power Number
Most low viscosity systems and industrial stirred tanks operate in the turbulent regime where NP is constant A simple and meaningful way to compare the power performance of various agitators is to compare their turbulent Power Numbers, NPT The term “Power Number” is often used to mean “Turbulent Power Number” Piero M. Armenante ChE702

Turbulent Power Numbers
Turbulent Power Numbers have been obtained experimentally for many impellers Typically, NPT 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., NP as a function of C/D) Piero M. Armenante ChE702

Turbulent Power Numbers for Various Impellers

Effect of D/T Ratio on Power Number for Disc Turbines

Effect of D/T Ratio on Power Number for Pitched-Blade Turbines

Effect of Impeller Clearance on Power Number for Disc Turbines

Effect of Impeller Clearance on Power Number for Disc Turbines
A correlation between the Power Number and the impeller clearance off the impeller bottom, Cb1, is: Piero M. Armenante ChE702

Effect of Cb on Power Number for Disc Turbines

Effect of Cb on Power Number for Flat-Blade Turbines

Effect of Impeller Clearance on Power Number for Pitched-Blade Turbines

Effect of Cb on Power Number for Pitched-Blade Turbines

Effect of Cb on Power Number for HE-3 Impellers

Power Dissipation in Multiple Impeller Systems

Power Dissipation in Multiple Impeller Systems
If the H/T ratio is larger than 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, NP double  2 NP single Piero M. Armenante ChE702

Power Dissipation in Multiple Disc Turbine Systems

Power Dissipation in Double Disc Turbine Systems

Power Dissipation in Triple Disc Turbine Systems

Power Dissipation in Double Pitched-Blade Turbine Systems

Power Curves for Impellers in Baffled and Unbaffled Tanks

Power Curves for Impellers in Baffled and Unbaffled Tanks
NP vs. Re plots for baffled systems show that NP reaches an asymptotic value at high Reynolds Number NP vs. Re plots for unbaffled systems show that NP keeps decreasing with Re even at high Reynolds Numbers Piero M. Armenante ChE702

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. Armenante ChE702

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. Armenante ChE702

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. Armenante ChE702

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. Armenante ChE702

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/m3 or hp/1000 gal Piero M. Armenante ChE702

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 m2/s3 Piero M. Armenante ChE702

Power per Unit Volume Substituting for P and V gives:

Power per Unit Volume at Different Scales
The ratio of P/V at two different scales is: Piero M. Armenante ChE702

Power per Unit Volume at Different Scales
For geometrically similar systems: and the P/V ratio becomes: Piero M. Armenante ChE702

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. Armenante ChE702

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. Armenante ChE702

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. Armenante ChE702

Typical P/V Values for Common Mixing Processes

Typical Tip Speed and P/V for Various Mixing Equipment
After Arthur Etchells, Unpublished Data Piero M. Armenante ChE702

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. Armenante ChE702

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. Armenante ChE702

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. Armenante ChE702

Power Input by Gas Sparging
The mechanical power input contribution of a gas sparged inside a liquid is: Piero M. Armenante ChE702

Power Input of a Liquid Jet
The mechanical power input contribution of a liquid jet injected inside a liquid is: Piero M. Armenante ChE702

Total Mechanical Power Input
The total mechanical power input to a liquid in a stirred vessel is: In the presence of a sparged gas: Piero M. Armenante ChE702

Impeller Pumping Effects

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. Armenante ChE702

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. Armenante ChE702

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. Armenante ChE702

Vortices Generated by a Disc Turbine
Blade Disc Ulbrecht and Patterson, Mixing of Liquids by Mechanical Agitation, Piero M. Armenante ChE702

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. Armenante ChE702

Vortices Generated by a Pitched Blade Turbine
Ulbrecht and Patterson, Mixing of Liquids by Mechanical Agitation, Piero M. Armenante ChE702

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. Armenante ChE702

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. Armenante ChE702

Flow Pattern for a Typical Axial Impeller

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. Armenante ChE702

Flow Pattern for a Typical Radial Impeller (high C/T)

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. Armenante ChE702

Flow Pattern for a Typical Radial Impeller (low C/T)

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<C/T<0.2  either flow pattern can exist For C/T <0.16  “single-eight” flow pattern Piero M. Armenante ChE702

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<C/T<0.25  either flow pattern can exist For C/T <0.20  “single-eight” flow pattern Piero M. Armenante ChE702

Velocity Flow Field in a Stirred Tank

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. Armenante ChE702

Laser-Doppler Velocimetry (LDV) System

Impeller Discharge Flow
The pumping action of an impeller results in a discharge flow rate out of the impeller region, Qout, balanced by an incoming flow toward the impeller (inflow rate= Qin). Since mass is conserved it must be that: where Q is the discharge flow rate Piero M. Armenante ChE702

Cylindrical envelope to determine flow out of impeller region Piero M. Armenante ChE702

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. Armenante ChE702

Flow Number (or Pumping Number) NQ
In order to make the impeller discharge flow rate non-dimensional one can define the Flow Number, or Pumping Number, NQ: Piero M. Armenante ChE702

Turbulent Flow Numbers
The Flow Number is to the discharge flow rate what the Power Number is to power Turbulent Flow Numbers, NQT (or simply NQ) have been obtained experimentally for many impellers Typically, NQ is measured for a “standard” configuration of the agitation system (H=T, D/T=1/3, C=D) Piero M. Armenante ChE702

Flow Numbers for Various Impellers in Baffled Tanks

Flow Numbers for 45° Pitched-Blade Turbines (4 Blades)

Flow Numbers for HE-3 Impellers

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. Armenante ChE702

Relationship Between Power and Flow
For a fixed impeller geometry it is: i.e.: Piero M. Armenante ChE702

NQ/NP for Various Impellers in Baffled Tanks

Variation of Flow and Power Dissipation
Most impellers have flow numbers in the relatively narrow range of (typically ), 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. Armenante ChE702

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 NQ/NP ratios is low Radial impellers generate significant turbulence and produce high shear Piero M. Armenante ChE702

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 NQ/NP ratios will be high Axial impellers generate less turbulence and shear Piero M. Armenante ChE702

Optimization Strategies to Maximize Pumping Efficiency
To maximize pumping efficiency (i.e., maximize the Q/P ratio): choose impellers with high NQ/NP ratios if capital cost must be minimized select impellers with the same vtip (=ND) but lower D, since this decreases N and hence the torque  (=P/2N) [recall that the cost of the gear box is proportional to the torque] Piero M. Armenante ChE702

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. Armenante ChE702

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/2N). Piero M. Armenante ChE702

The preceding analysis is valid if NP and NQ are constant. This is correct if the flow is fully turbulent. Changing the D/T ratio usually has little influence on NP and NQ provided that it is not too small or large (0.25< D/T <0.7) [Too large a D/T ratio chokes the recirculation flow]. Piero M. Armenante ChE702

Circulation Time One can define the circulation time, tcirc, as:
tcirc is a measure of how long it takes the impeller to pump the same volume of liquid as that contained in the tank (V=Qtcirc) Piero M. Armenante ChE702

Circulation Time The circulation time, tcirc, is directly related to how long it takes: a small, neutrally buoyant tracer particle to pass consecutively through the same region (e.g., the impeller region) a tracer to produce two consecutive concentration peaks in the region where the detector is The blend time is typically a multiple of the circulation time Piero M. Armenante ChE702

Blend Time in Stirred Tanks

By the end of this section you will be able to: Describe the concepts of blend time and degree of uniformity and how the can be determined in the lab Calculate the blend time for any desired degree of uniformity in a mixing tank Determine the blend time as a function of geometry and operating parameters Piero M. Armenante ChE702

Blend Time (Mixing Time)
If a miscible tracer is added to a homogenous liquid in an agitated tank the local concentration (measured with a detector) fluctuates with time The amplitude of the concentration fluctuations will decrease with time Eventually the tracer concentration will become completely uniform in the tank Piero M. Armenante ChE702

Blend Time Blend Time (also referred to as “Mixing Time”) is the time it takes the tracer-liquid system to reach a desired (and pre-defined) level of uniformity Piero M. Armenante ChE702

Blend Time Facts Blend time and the achievement of a homogeneous state can be critical in some operations (e.g., fast chemical reactions) In any real mixing tank, blend time is never zero Homogeneous phases do not mix instantaneously! Piero M. Armenante ChE702

Experimental Determination of Blend Time
Detection of tracer can be accomplished with a variety of techniques including: acid-base indicators (e.g., pH meters) ion-specific electrodes electric conductivity meters thermometers refractometers (for refractive index) light adsorption meters Piero M. Armenante ChE702

A tracer is typically added to the tank (typically at the surface) The concentration of the tracer is determined at one or more locations in the tank as a function of time Piero M. Armenante ChE702

Tracer Sensor Piero M. Armenante ChE702

Concentration Fluctuations at Sensor and Experimental Blend Time
CFinal t Piero M. Armenante ChE702

CFinal t t90% Piero M. Armenante ChE702

CFinal t t95% Piero M. Armenante ChE702

Equations for the Determination of Blend Time
Here, two approaches/equations for the determination of the blend time will be presented: Approach 1: Fasano, Bakker, and Penney’s approach Approach 2: Grenville’s approach Piero M. Armenante ChE702

Approach 1 Piero M. Armenante ChE702

Blend Time and Non-Uniformity
The level of non-uniformity (or unmixedeness) X is defined as: where Co and CFinal are the initial and final tracer concentrations in the liquid Before the tracer addition (t=0) C=Co and X=1; for t, C=CFinal and X=0 Piero M. Armenante ChE702

Non-Uniformity vs. Time

Mixing Rate Constant, k For “long” enough times the value of X(t) oscillates while decaying exponentially These damped oscillations are enveloped between an upper and lower decaying exponential curves (X=e-kt and X=-e-kt) The parameter k is called the mixing rate constant (in min-1) Piero M. Armenante ChE702

Mixing Rate Constant, k The greater the k value is:
the faster the oscillations will die out the faster blending will be the shorter the mixing time will be The extent of the damping effect will depend on the geometric (e.g., D, T, H) as well as dynamic (e.g., N) variables Piero M. Armenante ChE702

Non-Uniformity Peaks The absolute values of the height of the oscillation peaks, p(t), in the X-t curve will determine whether a required level of homogeneity has been achieved The values of p(t) can be found from: Piero M. Armenante ChE702

Non-Uniformity Peaks p(t) determines the level of non-homogeneity (non-uniformity) One can arbitrarily decide when sufficient uniformity has been achieved by selecting a small enough p(t) value (e.g., 0.05, implying that the largest fluctuation is 5% of the final X value) For t p(t) 0 Piero M. Armenante ChE702

Degree of Uniformity, U It is convenient to introduce the Degree of Uniformity, U, defined as: where U is just the complement of p (for example, if p=0.05, U=95%, implying that the liquid is 95% homogeneous). Then: Piero M. Armenante ChE702

Blend Time and Degree of Uniformity
Then: This equation relates the blend time, tU, required to achieve a desired level of U, to U and k. For example, the time required to achieve 99% homogeneity is: Piero M. Armenante ChE702

Blend Times to Achieve Various U’s
It is possible to establish a relationship (independent of k) between a two blend times to achieve two different degrees of uniformity (e.g., U’ and U): Piero M. Armenante ChE702

Example: tU for a specific U and that for 99% (t99): Example, it takes twice as long to blend to U=99.99% that to blend to 99% Piero M. Armenante ChE702

Mixing Rate Constant In order to calculate tU one needs to determine the mixing rate constant, k As usual, dimensional analysis is used: Piero M. Armenante ChE702

Mixing Rate Constant Then:
For baffled, fully turbulent systems Re and Fr have no effect Piero M. Armenante ChE702

Mixing Rate Constant Although all geometric variables could play a role only a few are important. The most important geometric variables affecting k/N are T, H, D, and the impeller type. Hence: Piero M. Armenante ChE702

Mixing Rate Constant The equation for k/N is then:
where the parameters a and b depend on the type of impeller used Piero M. Armenante ChE702

Parameters to Calculate the Mixing Rate Constant

Blend Time Equation for Fixed Geometries
For a fixed set of geometric variables i.e.: same impeller same D/T ratio same H/T ratio k/N =constant Piero M. Armenante ChE702

If k/N is constant: i.e.: (for same impeller, same D/T ratio, same H/T ratio), irrespective of scale Piero M. Armenante ChE702

Blend Time Equations for H/T=1, T/D=3

Blend Time and Impeller Speed
The higher the agitation speed is the shorter the blend time will be For geometrically similar systems this equation: does not change with scale Geometrically similar small and large vessels have the same blend time only if the agitation speed N is the same at both scales Piero M. Armenante ChE702

From: it follows that if mixing time is to remain unchanged during scale-up the agitation speed N must remain constant provided geometric similarity is maintained Piero M. Armenante ChE702

Effect of Other Factors on Blend Time
The procedure outlined before can be used to obtain the blend time tU for the case in which: the flow is turbulent the viscosities of the added liquid and the liquid in the tank are equal the densities of the added liquid and the liquid in the tank are equal Piero M. Armenante ChE702

Corrective factors can be applied to tU to account for: different flow regimes viscosity differences between the two liquids density differences between the two liquids Piero M. Armenante ChE702

The corrective factors can be applies as follows: where: f Re = corrective factor for the effect of Re f* = corrective factor to account for the effect of viscosity differences f  = corrective factor to account for the effect of density differences Piero M. Armenante ChE702

tU (Re; *=1; =0) is the “standard” tU (i.e., under fully turbulent conditions, with an added fluid having the same viscosity an density as the liquid in the tank) calculated as outline before; tU (Re; *; ) is the mixing time calculated to account for the effect of Re, viscosity and density differences. Piero M. Armenante ChE702

Blend Time at Different Reynolds Numbers
fRe is the correction factor to account for Re effects when Re is below 10,000 and the fluid is not fully turbulent. Remark: fRe =1 for Re>10,000. Once tU has been calculated for Re >10,000 it is possible to obtain tU at other Reynolds Number using the diagram obtained by Norwood and Metzner (1964). Piero M. Armenante ChE702

Blend Time at Different Reynolds Numbers

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. Armenante ChE702

Effect of Viscosity Ratio on Blend Time
Fasano et al., Chem. Eng., 1994 Piero M. Armenante ChE702

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. Armenante ChE702

To account for the effect of density differences the Richardson Number is introduced: Piero M. Armenante ChE702

Fasano et al., Chem. Eng., 1994 Piero M. Armenante ChE702

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. Armenante ChE702

Procedure to Calculate Blend Time
The procedure to calculate tU 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. Armenante ChE702

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. Armenante ChE702

If P/V is kept constant during scale-up: Piero M. Armenante ChE702

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. Armenante ChE702

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. Armenante ChE702

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 tank’s contents typically takes much longer Piero M. Armenante ChE702

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. Armenante ChE702

From the definition of circulation time (tcirc=V/Q) and Flow Number, NQ: it follows that: Piero M. Armenante ChE702

The relationship between tU and tcirc can be obtained recalling that: Then: Piero M. Armenante ChE702

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 tU/tcirc is constant Piero M. Armenante ChE702

Example: Disk turbine in a standard mixing system NQ=0.8 D/T=D/H=1/3 If U=99% (=47)  t99=5.3tcirc If U=95% (=0.6547=30.5)  t95=3.5tcirc Piero M. Armenante ChE702

The previous results confirm that the blend time is typically a multiple of the circulation time tU/tcircis typically in the range 3-6 for t95/tcirc and 4-9 for t99/tcirc These results validate the empirical mixing rule of thumb stating that “Blend Time”  4  “Circulation Time” Piero M. Armenante ChE702

Approach 2 Piero M. Armenante ChE702

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. Armenante ChE702

Recalling that it is possible to calculate tU’ knowing tU: Piero M. Armenante ChE702

If U=95% is the reference degree of homogeneity: For example: Piero M. Armenante ChE702

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. Armenante ChE702

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

Impeller Efficiency in Turbulent Regime: Approach 2
From: and: it is: Piero M. Armenante ChE702

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 tU at the same power input/mass) shorter tU 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. Armenante ChE702

Blend Time Equation in Transitional Regime: Approach 2
For mixing in the transitional regime (~200<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. Armenante ChE702

Blend Time Equation in Transitional Regime: Approach 2
For mixing in transitional regime (~200<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. Armenante ChE702

Impeller Efficiency in Transitional Regime: Approach 2
From: and: it is: Piero M. Armenante ChE702

Impeller Efficiency in Transitional Regime: Approach 2
From the previous equation for the transitional regime, it is: all impellers of the same diameter are equally energy efficient (i.e., achieve the same tU at the same power input/mass) shorter tU are achieved with larger impellers at the same power input/mass blend time is proportional to viscosity and inversely proportional to density when scaling at constant power input/mass and similar geometry blend time decreases with the scale factor raised to 2/3 (however do not forget that that Re increases with scale, and larger system may no longer be in the transitional regime) Piero M. Armenante ChE702

Conclusions: Power, Flow and Blend Times in Mixing Tanks
Under turbulent conditions, the power dissipated by an impeller depends on: agitation speed (PN3) impeller size (PD5) type of impeller (PNP) density of the fluid (Pρ) Axial impellers and radial impellers generate different circulation patterns In general, axial impellers generate more flow per unit of power dissipated than radial impellers Blend time is inversely proportional to the impeller agitation speed and is not generally significantly affected by scale Piero M. Armenante ChE702

Special Topics - Modules in Pharmaceutical Engineering ChE 702

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Special Topics - Modules in Pharmaceutical Engineering ChE 702

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