AGITATOR DESIGN METHODS The routines used in the agitator design section encompass much of the known theory of mixing. Different correlations and methods are used for different types of agitators. All methods and procedures used in this program are available in the literature through technical articles or in standard texts on mixing. POWER CORRELATIONS The basic methods used for all agitators is based upon pump technology. The power required by an impeller is proportional to the pumping flowrate and the velocity head produced. P = f(Q*H*density) Power is also proportional to the the rotational speed to the third power and the diameter of the impeller to the fifth power. P = f(N^3*D^5*density) These relationships are usually correlated in terms of power numbers and flow numbers that are measured and are specific for different types of agitators. The correlations are in general based upon a tank that contains four baffles. The Flow number Nq is a dimensionless group that is used to quantify the flow characteristics of the agitator. It is given by the following relationship. Nq = Q/N*D^3 Values of experimentally derived flow numbers are presented in Uhl and Gray and in OldShues Fluid Mixing Technology Table 8-3. and also in the Chemineer articles referenced earlier. The power number Np is correlated against the mixer Reynolds number. These two dimensionless variables are defined as follows. Np = (1.523 * 10^13 * P) / (N^3 * D^5 * density) Nre = (10.754 * N * D^2 * density)/Viscosity In order to determine the power requirements of a specific agitator it is necessary to read the power number for that type of agitator from the Reynold's number correlations. There are power number correlations in the literature for a major types, such as Axial flow, Radial Flow, propellers and other types. Once the power number has been read from the graph the power requirement is determined by multiplying the power number by the liquid density, the RPM to the third power and the Agitator Diameter to the fifth power. The Power requirements for a given agitator are calculated from the power number by the following equation. P = Np * N^3 * D^5 * density / 1.523*10^13 The correlations for power numbers are given for different agitators in several standard texts. The graphs given here were published in Chemical Engineering Progress Sept 1950, Page 470 and Vol 2 Oct 1963. The power number of the agitators under turbulent conditions reaches a constant value that is specific for the agitator. The graphs were also developed for a certain number of blades and standard blade widths. They are also based upon a standard tank geometry with 4 baffles sized at T/12 width. The program uses a curve fit of these graphs, and covers the range from Newtonian flow at very low Nre to very high levels where the power number is constant. Additional correlations are used to adjust the power results for different number of blades, blade width, and proximity of the agitator to the bottom and top of the tank. The specific correlations for the different agitators selected are defined as follows for each type. Flat and Disk Blade Turbines This type of agitator (also known as a Rushton Type turbine) generally consists of 6 blades that are horizontal to the shaft. The Blade width is normally 0.2 of the turbine diameter. A disk agitator has a center disk that the blades are attached. The disk serves to direct the flow of the liquid (or gas) to the blades. Disk agitators have a somewhat higher power number than a flat blade turbine without a disk. AGITATOR POWER For agitator Reynold's number above 10,000 ie highly turbulent flow, the power numbers reach a constant value. The power number of the flat blade turbine is 4.0 for a BW of 1/5. A disk blade turbine has a Np of 5.0 for this case. For both the flat blade turbine: The power Number is adjusted for blade width as follows: Np = Np * BW/0.2 This relationship is based upon Nagata, where he shows that power is roughly proportional to blade width, for W/D ration of 0.1 to 0.4 (Nagata 'Mixing Principles and Applications'). The power number is adjusted for number of blades by correction for the Power number of the 6 bladed turbine by the number of blades to the 0.8 power as follows: Np = Np * Np(6blades) * (Number of Blades / 6 ) ^ 0.8 The power numbers at different Reynolds Numbers were curve fit from the following graph. ( Graphs and Specific Correlations are deleted from Demo Text) POWER NUMBER CORRELATIONS RADIAL FLOW TURBINES AGITATOR FLOW The total circulation flow that is generated from a turbine is comprised of the flow from the blades of the turbine and an induced flow. The flow from the turbine blades is called the discharge flow from the turbine. The discharge flow entrains fluid to produce the total circulation flow. It is the discharge flow numbers that have been measured and presented in the standard references (see Oldshues Fluid Mixing Technology Table 8-3 ) The discharge flow from a radial turbine is given by Q = Nq * N * D^3 Nq is a constant known as the pumping of discharge flow number, it is dependent on the type of impeller, system geometry and the Reynolds number. Under turbulent flow conditions, the radial flow discharge number is calculated by the following equation. Nq = 6.0 * B * BW * (D/T)^0.3 WHERE B = (NUMBER BLADES / 6)^ 0.7 D = IMPELLER ID inches T = tank diameter inches The Nq number for a radial flow turbine is generally about 0.8, the total induced flow rate is generally about 3 times higher, giving a flow number of around 2.75, For axial flow turbines the Nq is also about 0.77 but the induced flow is less giving a total flow Number of around 1.73. (Oldshue Mixing Tech p 174) The flow profile produced by a radial flow impeller is horizontal toward the vessel walls, and it splits with somewhat more than 50 percent flowing upward. The flow pattern is determined by the vessel geometry and depth of the agitator. In order to be consistent with the bulk of the literature, the radial discharge flow numbers are used in determining the circulation rates and for correlations of blend time. The vertical flow velocity from a radial turbine in the tank is calculated from the circulation flow generated from the equation. This flow is divided by 2 to account for the split flow at the walls with a radial turbine, and by the cross sectional area of the tank to get the vertical velocity. Flow = Nq * D^3 * N SHEAR Radial flow turbines are used for applications where shear is require to achieve the process result. Shear rate is calculated for the flat blade and disk turbines, but is not done for the axial blade or other types. The shear rate produced by these other agitators is insignificant. The flat blade turbines force the gas or fluid to be dispersed horizontally into the high velocity regions associated with the edge of the turbine blade. The axial flow turbines pump the fluid vertically though the inpeller and have very low shear rates. Shear rate calculation is important to judge the effectiveness of gas dispersion and other applications such as dispersing a second immisible liquid phase, such as acid or caustic, where the size of the droplets formed can affect reaction rates. The correlation used for maximum shear were taken from Bowen's article in Chemical Engineering June 9 1986. The correlation is based upon methods for predicting the velocity profiles for the turbines. The maximum velocity at the centerline of a radial flow turbine is given by the equation. This correlation is based upon measurements of the resultant flow generated by a turbine from the data of Cooper and Wolf.(Canadian Journal Echem. Eng. Vol 46, 1968 p 94). Vmax = 4.9 * N * D * (D/T)^0.3 This velocity is about 50 % greater than the radial discharge velocity. If the velocity profile is determined across the blade width of the impeller, the shear rate (dv/dz) is given by the slope of the curve. Shear rate is given in units of reciprocal seconds. The maximum shear rate is found on the sides of the profile where the slope is steepest. The maximum velocity decreases rapidly over the blade width, moving away from the centerline of the blade, and levels off beyond the blade. The average shear rate across the velocity profile is found by taking the difference between the centerline velocity and a point where the velocity of the fluid has leveled out, and dividing this by the distance between the measurements. The leveling off point is assumed to be a one blade width past the centerline. The velocity at one blade width from the centerline is about 15% of the centerline velocity. Consequently, the velocity difference is 0.85 * the maximum velocity. (dv/dz)ave = 0.85 * Vmax / W (dv/dz)ave = (0.85 * 4.9) * ND(D/T)^0.3/(W/D) * D where W = Blade width This equation shows that for geometrically similar impellers at the same D/T ration the average shear varies only with speed, and is independent of diameter. The maximum shear is 2.3 times the average shear. This relationship can also be shown in terms of the agitator RPM and Diameter to Tank ratio and Blade Width as follows. Shear(max) = 9.7 * N/60 * (D/T)^0.3 / (W/D) One implication of these equations is that the shear rate can be increased and the power reduced by reducing the width of the blade below the industry standard of 0.2. The flow is also reduced however by reducing the blade width and care must be taken to maintain pump circulation. Blade widths down to 0.1 times the impeller diameter can be used to save power on applications where a high shear rate is critical to the process. PROXIMITY FACTORS The location of the agitator in the tank affects the power consumption and circulation rates. If a radial flow turbine is located close to the bottom of a tank the power consumption is reduced because the circulation flow to the Bottom impeller eye is restricted by the tank geometry. An axial flow impeller will have an increase in power consumption when the impeller is closer to the bottom because the resistence to flow increases the head generated by the impeller for the same flow rate. Graphs for the proximity factors for radial and axial flow impellers are presented in OldShues text Figures 3-20,through 3-22. This program uses similar relationships, with different correlations for each type and location of agitator. The power requirement without the proximity factor is also generated by the program. For multiple impeller designs the average proximity factor for the impellers is used. AXIAL FLOW IMPELLERS POWER NUMBER The power number of Axial turbines are also correlated against the mixing Reynold's numbers. These correlations were curve fit for the 6 blade 45 degree pitch turbines with a W/D of 1/8, as was done for the Radial flow turbine. The specific correlations used are as follows as a function of Nre. The program adjusts for the actual number of blades on the turbine by the same equation described earlier for flat blade turbines. In the viscous region at low Reynold's Numbers the power number is approximately: Np := (50.0 / Nre)* Sqrt(Wd/0.125); In the turbulent range the power number becomes constant and given by: Np := 1.4* ( Wd/0.125); Note Wd = (W/D) above FLOW The correlation used for the flow number Nq of a pitched blade turbine as a function of the Reynold's number is based upon the Chemineer Article reference earlier. The correlation is based upon 4 blades at a blade width of 0.2. The value for the Nq is adjusted by the program for the actual number of blades by the correlation given in Nagata on page 138. Nq = Nq * (Wd/0.2) Nq = Nq * ((Number Blades)/4)^0.7 BLEND TIME The Correlations used to predict Blend Time for axial flow agitators are based upon the correlation presented by Fenic and Fondy (1966 AIChE Atl. City). The correlation gives blend times as a function of Nre. Data taken from Uhl and Gray Vol 1 p 219 were used to modify this correlation for different impellers. The minimum blend time presented by the program will be at least 5 batch turnovers of the vessel by the liquid pumping rate. The correlations for blend time are only approximate and can be used for comparisons. They are no substitute for pilot plant data for critical applications. RETREAT BLADE AGITATORS Retreat blade agitators are similar to flat blade turbinesand propellers except that the blades curve backward. This type of agitator is popular with Pfaudler for use in glass lined reactors. The correlations are based upon retreat blade agitators with the use of two finger baffles. If Nre <= 10 then Np := 50.0 / Nre; If (Nre > 10 ) and (Nre <= 100.0 ) then Np := 21.832446 * Pow(Nre,-0.66690987); If (Nre > 100 ) and (Nre <= 400.0) then Np := 2.818063 * Pow(Nre,-0.20327403); If (Nre > 400 ) and (Nre <= 10000. ) then Np := 1.0399062 * Pow(Nre,-0.0559205); If Nre > 10000. then Np:= 0.55 ; FLOW The correlations for flow were taken from Nagata Page 138 Nq := 0.29 ; { with two baffles } CORRECTIONS FOR ONE BAFFLE If (Nre > 400) and (Baffle = 1.0) Np := 0.77 * Np; { for one baffle} Nq := 0.23; PROPELLER AGITATORS POWER The following correlations were used for the power number for propellers with a pitch of 1.5 In the viscous range the power number is given by Np = 45 / Nre In the turbulent range the Power Number becomes constant. Np = 0.5 The power number is adjusted for different blade pitch by the ratio Np = Np * (pitch/1.5)^ 1.5 FLOW The discharge flow number is calculated as follows as a function of pitch. Nq = 0.55 * pitch ANCHOR AGITATORS The correlations for the anchor agitator are for a pure U anchor with no cross bars. The cross bars should be estimated separately pitched turbine to estimate the effect on power of the cross bars. POWER The power number correlations are curve fit from basic data present in Oldshue text on Fiqure 3-15 Page 61. The correlation is based upon a Height to diameter ratio of 1 for the agitator. The diameter of the impeller is selected by the program at 0.95 * the tank diameter. This can be over ridden by the user from the command line. Anchors are usually only used for high viscosity applications to maximize the heat transfer coefficient at the walls. For blending applications in high viscosity service the selection of a spiral agitator is generally far superior. In the viscous range the power number is approximately. NP = 280 / Nre; In the turbulent range the power number becomes constant at Np = 0.4; BLEND TIME Blend time for anchor agitators was taken from Nagata's correlation Fiq 4.24 Time = (5/3) * 60000 / N ( Nagata used 3 turnovers I prefer to use 5) The values for circulation rate and velocity were backed out of Nagata blend time correlation assuming 5 batch turnovers for uniformity, instead of 3. I am not aware of published data on Nq values for Anchor agitators. The shear rate of the agitator was calculated by the following equation. DvDs := N * Pi * D / ((T - D) * 60); HELIX AGITATORS Helix agitators are describe very throughly in Nagata's work. The correlations used are as follows for double helix agitators. The Power requirements for a single helix with a center screw is approximately the same but somewhat lower. The correlations assume that the Diameter of the Helix is equal to 0.95 the tank diameter. The blade pitch is assumed equal to the diameter of the impeller. The Blade width is assumed equal to 0.1 * impeller diameter. POWER The following correlation is used for the power number as a function of the Reynold's Number at low Nre. Np := ( Nre * 245.663404)^ -0.8524365114); The program adjusts for different screw pitch, diameter and blade widths by the following relationship. Np = 78.0 / (Nre * A * B) Where A = SQRT ( screw pitch) B = SQRT((T - D)/D)) BLEND TIME The blend time is calculated by the relationship Time =(5/3) * 500 * 0.1 * 60 * ScrewPitch / N based upon 5 batch turnovers instead of 3. The circulation rate is given by Qcirc := (5 * A/time)*60; {for Neutonian fluids} Vel := Qcirc/Csa; The maximum shear rate is given by DvDs := rpm * Pi * D / ((T - D) * 60); based upon 5 batch turnovers instead of 3. The maximum shear rate is given by the same equation used above for Anchor Agitators. AGITATOR TORQUE The agitator torque is calculated in Inch Pounds and used in the shaft sizing programs. The following equation is used. TORQUE := Bhp * 6.3025E4 / Rpm ; Where Bhp is sum of all impellers MISCELLANEOUS RELATIONSHIPS The Prandtl number is calculated by Npr := Mfcp * Mfvis * 2.42/mftc; MfCp = Mix Phase Heat Capacity MfVis = Mix Phase Viscosity MfTc = Mix Phase Thermal Conductivity The Impeller tip speed is calculated by TS = (D/12) * Rpm * Pi The relationship called mixing intensity is used by Chemineer to correlate blending relationship. It is simply the vertical velocity of the tank contents divided by 6. The velocity is the impeller pumping rate divided by the CSA of the tank. MI = Vel/6.0 ; { Mixing intensity }