POWER CONSUMPTION AND MIXING EFFICIENCY
IN THE AGITATION OF LIQUIDS
INSTRUCTIONS FOR STUDENTS
1. Wear safety glasses at all times.
2. Wear jeans or slacks, a long sleeved shirt, and sturdy shoes that give good traction on possibly wet floors.
3. Guard against electrical hazards by making sure that all equipment is well grounded using three-wire plugs and other means. All instruments should be plugged into a GFCI protected power strip.
4. Guard against falls, burns, cuts, and other physical hazards.
5. THINK FIRST OF SAFETY IN ANY ACTION YOU TAKE. If not certain, ask the TA or a faculty member before you act.
6. The materials used in this experiment, namely water, corn syrup, ketchup and sodium or potassium chloride, pose no unusual safety problems. However loose clothing should not be worn near rotating equipment.
The goals in carrying out this experiment are:
o To determine the torque (and power consumption) as a function of impeller speed for water and then for corn syrup in several different vessels, to find the effect of baffles, impeller design, and tank diameter on torque and power, and to compare the results with correlations from the literature. Ketchup, which is non-Newtonian, may also be used.
o To compare the measured torque for a 20 L tank with the torque calculated from results obtained using a 5 L tank, based on the principle of similitude.
o To determine the efficiency of mixing in a baffled and unbaffled vessel filled with water, as a function of impeller speed and dimensions and baffle location, by using a tracer injection method. The resulting transient conductivity data are analyzed by a parameter estimation program that finds the least-squares values for two parameters of a model for the flow pattern in the tank.
o To measure the effect of impeller RPM, and other parameters, on the heat transfer coefficient from a heated aluminum cylinder immersed in a stirred 20 liter tank.
o To use a photosensor at the tank wall to detect the arrival of small beads immersed in the agitated tank. On the assumption that the beads do not interact, a Poisson distribution of waiting times is expected.
LACIMEHC CHEMICAL Inc. is moving into the production, in batch and semi-batch reactors, of several new products. It has become necessary to specify to vendors the size of the reactor vessels, the diameter and design of the impellers, the design of the baffles, and the power and speed of the stirrer motors. Your group, working with Dr. Ole Aginous, has been tasked to set up a pilot plant and gather data on which these specifications can be based. It will also test a novel method, based on the transient response to the injection of a tracer, to measure the efficiency of mixing.
The agitation and mixing apparatus is shown in the attached schematic diagram. The major components are as follows:
Tanks and Baffles
There is a polycarbonate tank of 6.5 inches ID and 8 inches height, with removable baffles, and volume of 4 L A third tank, 11.5 inches ID and 12 inches high, with removable baffles and a volume of about 20 L, is also available. This tank is equipped with a platinum-electrode conductivity probe located at the wall near the bottom of the tank, and a funnel located near the top of the tank and across from the conductivity probe. The funnel is used to reproducibly inject 30 ml of NaCl or KCl solution into the tank in the tracer response runs. A drain line with a stainless ball valve is also provided for the 20 liter tank.
Each tank is supplied with a baffle assembly, four stainless baffles attached to an acrylic top plate. The baffle width is 1/12 of the tank diameter, a standard ratio.
Mixer Speed and Torque Measurement
The impellers are driven by an electronically controlled DC motor. The motor shaft is hollow and can accommodate a 3/8 inch shaft (stainless or fiberglass) on which the impeller is mounted, a chuck being used to lock the impeller shaft into the motor shaft. Three 3/8 inch shafts are supplied, two of fiberglass-epoxy, one of stainless steel. The non-conductive shafts must be used when tracer is injected into the tank and the transient conductivity of the water in the tank is measured. The motor is connected to a control unit which is used to set the motor speed (in RPM), and which also measures the torque (in oz-in). The speed is adjustable in 10 RPM steps from 60 to 2200 RPM. The torque is indicated in 1 ounce inch steps from 0 to 45 oz in. Note that 1 oz in = 0.00706 N m. At any given speed the torque reading can be zeroed, using a button on the motor control box.
A ball-bearing low-profile torque table is also supplied as part of the experiment. Its use is optional, but it provides a wider range of, and more accurate, torque measurements than the motor control box. The tank being used must be mounted in the center of the table. An arm extending from the table presses on an Omega LCGC-500 load cell (0 to 500 g range), and the load cell is connected to an Omega DP-25S panel meter. The load cell/meter can be calibrated by attaching the 500 g spring scale to the innermost part of the torque table arm, taring the panel meter when no force is exerted on the arm, extending the scale until a reading of 400 g is achieved, and noting the panel meter reading (typically 190 units). The reading corresponds to xxx N m of torque.
Tracer Injection and Response Equipment
A 50 ml Erlenmeyer flask is used to pour 30 ml of a NaCl solution (about 20 g NaCl/L) into a funnel mounted on the side of the 20 L tank. A conductivity cell with (unplatinized) platinum electrodes is mounted near the bottom of the tank. The cell is connected to a conductivity meter (Cole-Parmer) which sends a signal to a Real Time Devices A/D board in a computer. The computer acquires conductivity data at a rate of 10 (or 5) points per second and plots the conductivity as a function of time, and the data are then written to a file.
Note: The conductivity meter may exhibit a somewhat nonlinear response to tracer concentration at higher tracer concentrations. This does not affect the parameter estimation program, since only small increments of tracer are added in any one run and the tracer response curve is scaled to lie in the range of 0 to 1 for each run. But it will limit the number of runs made, before the water in the tank is drained and refilled, to 10 or 20.
The program used to acquire and analyze the tracer response data is typically located in a subdirectory (e.g. C:\ZZQB45) and is called gtagit01.bas. The directory must also contain QuickBASIC Version 4.5.
Bead Statistics Measurement
A phototransistor (sensor) can be mounted on the side of the 20 L tank, with a Tensor lamp sitting on the shelf illuminating the region in front of the phototransistor. A 6 volt battery sends current through a 1 megohm resistor to the phototransistor. About 200 white beads can be added to water in the 20 L tank, and the voltage at the sensor will drop from about 5 volts to 1 or 2 volts when a bead is near the sensor. The sensor voltage is digitized by the RTD A/D board, and acquired and processed by a QuickBASIC program (bbb.bas). The program records an event when the sensor voltage drops below an arbitrary level, and calculates the time interval (dead time) between events. Finally, the program calculates the dead time distribution. In the absence of interaction between beads, a Poisson (exponential) distribution is expected.
Heat Transfer Coefficient Measurement
Also supplied with the experiment is a 2 inch diameter by 3 inch long aluminum cylinder containing a 300 watt heater connected to a Powerstat variable transformer. The cylinder also contains an LM35CAZ temperature sensor that produces a voltage directly proportional to the temperature in degrees Celsius. At a given power input to the heater, the cylinder temperature will drop as the heat transfer coefficient from the cylinder to the water increases. The temperature sensor voltage is acquired and displayed by a QuickBASIC program ttt.bas.
Torque and Power
We will base the torque and power (note that power = 2*Pi*RPS*torque) calculations on the plots in McCabe, Smith, and Harriot (Figs. 9-13 and 9-14), in which the power number Np is plotted against the Reynolds number NRe, for six-bladed turbines and three-bladed propellers, and for several shape factors related to baffle width, tank diameter, etc. For a given run the Reynolds number will be calculated from the stirring speed, impeller diameter, and liquid density and viscosity. The graph will be used to find the power number, and from the power number the torque will be determined, and compared with the observed torque. (The difference between a 3-bladed propeller and a 4-bladed propeller can be assumed negligible.)
In general the theory predicts that torque will be a linear function of impeller speed at low Reynolds numbers (laminar flow) and a quadratic function of impeller speed at high Reynolds numbers (turbulent flow).
We begin with (for a six-bladed turbine)
u = viscosity = 0.9 cP = 0.0009 Pa-s (water)
Da = impeller diameter = 4.0 inches = 0.1016 m n = speed = 5.00 s-1 (300 RPM = 5 RPS)
D = tank diameter = 0.298 m
d = density = 1000 kg/m3
gc = 1
baffle width = 1 1/8 inch = 0.0286 m
We find the Reynolds number to be
NRe = Da2 n d u-1 = 0.10162 5.00 1000/0.0009 = 57,347
From the graph in McCabe, Smith, and Harriot for a six-bladed turbine in a baffled tank we find the power number
Np = P gc n-3 Da-5 d-1 = 6
and from Np = 6 we have the power P as
P = Np gc-1 n3 Da5 d = 2 Pi n T
Solving for the torque T we have
T = Np gc-1 n2 Da5 d (2Pi)-1
T = 6 1 5.002 0.10165 1000 (2 3.14159)-1 = 0.258 N m
T = 0.258/0.00706 = 36 oz in
We begin with a calculation for Karo corn syrup in a baffled 4 liter tank and using a six-bladed turbine 3 inches in diameter.
u = viscosity = 2500 cp = 2.5 Pa-s (water)
Da = impeller diameter = 2.5 inches = 0.0635 m
n = speed = 1000 RPM = 16.67 s-1
Dt = tank diameter = 6 inches = 0.152 m
d = density = 1300 kg/m3
gc = 1
We find the Reynolds number to be
NRe = Da2 n d u-1 = 52.2
so that we are in the laminar range where torque should vary linearly with RPM.
From the graph of power number vs. Reynolds number in McCabe, Smith, and Harriot we find, for 4 baffles,
Np = P gc n-3 Da-5 d-1 = 4.3
and, from the equation for Np, we have the power P as
P = Np gc-1 n3 Da5 d = 2 Pi n T
Solving for the torque T we have
T = Np gc-1 n2 Da5 d (2 Pi)-1
Note that for constant Np, as occurs at high Reynolds numbers, the torque should vary as the square of the impeller speed n.
T = 4.3 1-1 16.673 0.06355 1000 (2 3.14159)-1 = 0.1964 N m
Since 1 pound force = 4.448 N (0.2248 N/lbf) we have the conversion factor 4.448 N/lbf or 0.278 N/oz (or 3.597 oz/N). Thus
T = 0.1964 N m (oz/0.278 N)(39.37 in/m) = 27.82 oz in
Note that, for a Reynolds number above about 1000, the power number is essentially constant for a baffled tank, and that this leads to a torque proportional to the RPM squared. For laminar flow the torque will vary linearly with RPM.
In general, the more vigorous the agitation the more rapidly a tracer introduced into the vessel will mix with the contents and approach a uniform concentration. But the shape of the concentration vs. time curves can be quite complex, and the mixing process can be chaotic and not reproducible over short times. Thus it is difficult to look at a tracer response curve and produce an objective number related to the rapidity of mixing and thus to the intensity of agitation.
In order to provide an objective measure of mixing time, we use a mathematical model of the flow pattern in the vessel. This model is shown schematically in Figure 1. It consists of six pools (well-mixed volumes), three (of equal volume) representing a core region in the tank in which the flow is downward, and three (of equal volume) representing a shell region in which the flow is upward. The model is thus designed to represent a baffled tank in which a propeller, driving downward, is located near the middle of the tank. Whether it will represent other situations, for example a tank with a turbine impeller, or with a propeller driving upward, or a tank without baffles, is open to question. The experiments will provide at least a partial answer to this question.
The volume of each core tank is b1 V/3, where V is the volume of the whole tank. The volume of each shell tank is (1 - b1)V/3. Thus b1 is the fraction of the tank volume V in the core region. The second parameter is b2, the volumetric flow rate (in L/min) downward in the core (inner) section and upward in the shell (outer) section.
SIX-POOL FLOW PATTERN MODEL USED
IN ESTIMATING PARAMETERS b1 AND b2
┌───────────┐ b2 ┌───────────┐ INJECTION
│ X1 ├<───────────┤ X6 │ <=
│ V b1/3 │ │ V(1-b1)/3 │
│ V = TOTAL VOLUME ^
│ X2 │ │ X5 │
│ V b1/3 │ │ V(1-b1)/3 │
V │ CONDUCTIVITY
┌─────┴─────┐ ┌─────┴─────┐ MEASUREMENT
│ X3 │ b2 │ X4 │ =>
│ V b1/3 ├───────────>┤ V(1-b1)/3 │
DOWN FLOW UP FLOW
V = tank volume (L)
b1 = fraction of volume in core
b2 = volumetric flow rate (L/min)
Xk = tracer concentration in kth pool
Figure 1. Schematic diagram of the six-pool flow pattern model used to find the least-squares values for parameters b1 and b2.
The model corresponds to six ordinary differential equations for the tracer concentrations X1, X2, ..., X6. Tracer injection takes place in tank 6, and the conductivity probe is located in tank 4. A finite-difference method, involving making small changes in each parameter and solving the resulting normal equations for the parameter increments, is used to fit 300 data points. The first 100 conductivity values acquired during a run (normally taken at a rate of 10 per second) are used to calculate an average initial conductivity, and the last 100 points, taken after the conductivity has become constant, are used to establish the final average value. The data for the middle 300 points are scaled to produce a response that varies from 0 at time zero to 1 at the final time of 30 seconds, and the nonlinear regression program finds a least-squares fit to these data. The speed and certainty of convergence are controlled by a user-supplied step-size parameter beta. When beta = 0.1 convergence normally occurs in about 20 to 30 iterations; larger beta (e.g. 0.3 or 0.5) produces faster convergence, but may at times lead to a failure to converge. If convergence fails, the program allows the fitting to be repeated using a smaller value for beta.
When convergence has occurred, the best parameter values are shown on the screen, as well as a plot of the data and of the best-fit curve. These data, and the best-fit curve, are also written to a file (the user selects the file name) for later plotting by spreadsheet, Excel or the equivalent. Note that an objective number for mixing time (in seconds) is V/b2, where
V = 18 L is the volume in liters of liquid in the large tank and b2 is the circulation rate in L/s.
It is not necessarily true that the model will fit the response data well, especially since only two parameters are determined. But it will be of interest to see if the model, based on a downward driving propeller, fits responses for that case better than when an upward driving propeller or turbine is used.
Given a tank, impeller (turbine or propeller), baffles (or no baffles) and liquid (water or, for the 4 liter tank, corn syrup) we will vary the shaft speed and measure the torque as a function of RPM. (The measured torque should be corrected for the torque due to friction in the motor bearings.) This will be done using one of the three available tanks. From these runs we will calculate Np and NRe, and check Np vs. the value taken from the plot, or compare the measured torque with the calculated torque. We will also determine the effect of baffles on the torque and on mixing time as determined by the tracer injection method.
Note that a very large number of runs is possible if all combinations of tank size, liquid, baffle presence, impeller design, and impeller speed are explored. Thus judgement is needed to select runs that have meaning. For example it is useful to check the dependence of torque on RPM (linear or quadratic), and to see how impeller speed and baffle presence affect mixing time.
We suggest below some possible runs.
Torque and power runs
1. Using the 4-inch 3-bladed propeller on the stainless steel shaft, raise the impeller until it is just above the liquid surface but, for safety, still within the tank. With the motor stopped, zero the torque reading using the button on the control box. Then record the torque at intervals of 100 RPM from 100 RPM to 2000 RPM. Plot the torque (due to bearing friction) vs. RPM for later use in correcting torque readings.
2. Fill the 20 L tank with water and remove the baffles. Mount the downward driving 4-inch 3-bladed propeller on the mixer shaft and move the impeller to a location in the center of the tank, and half way between the tank bottom and the water surface. Increase the stirring speed slowly until a vortex forms that pulls air into the tank. This is the maximum speed for this configuration. Set the speed at zero and zero the torque reading (which may not be zero). Then run at 10 speeds evenly spaced between 0 and the maximum speed, and record the torque reading at each speed. Repeat using other impellers. Convert the torque reading in oz-in to torque (in N m), after correcting the torque for bearing friction. N.B. 1 oz-in = 0.00706 N m.
3. Repeat the procedure above with baffles inserted in the tank, again using water. Baffles will allow higher impeller speeds and markedly affect the torque at a given RPM.
4. Repeat using Karo corn syrup in the 5 L tank. The maximum speed may be limited by the power available for large impellers.
5. Repeat steps 1 and 2 using the 5 L tank filled with water. Use various impellers as time permits, and investigate the effect of baffles.
6. Optionally, fill the 5 L tank with ketchup and stir using the small 3-bladed propeller. Determine torque as a function of RPM. Note movement or lack of movement of the surface of the ketchup, as a basis for commenting on Newtonian or non-Newtonian behavior.
Consider a small tank and a larger tank that are geometrically similar, and operating with possibly different liquids. If both tanks operate at the same Reynolds number the principle of similitude asserts that (under certain assumptions) all other dimensionless numbers will also be the same, including of course the power number.
Stir the 5 L tank (using the small 3-bladed propeller and water) at say 1000 RPM, measure the torque and calculate the Reynolds and power numbers. Fill the 20 L tank with water, use the 4-inch 3-bladed propeller, and stir at an RPM such that the Reynolds number is equal to that for the 5 L tank. The power numbers should be the same. From the large tank power number and RPM calculate the torque, and compare it to the measured torque.
Impulse Response Runs
1. Mount the 4-inch diameter 3-bladed down-driving propeller on a fiberglass shaft and insert in the middle of the 20 L tank filled to about 2 cm below the funnel inlet with tap water. Connect the conductivity probe to the conductivity meter, connect the meter to the terminal board, and load (from WINDOWS go to the DOS prompt, connect to C:\ZZQB45 and type QB GTAGIT01) and start the QuickBASIC program gtagit01.bas (located, for example, in C:\ZZQB45, which also contains the QuickBASIC 4.5 system) for agitation data acquisition. Insert the baffle assembly in the tank, with the conductivity probe midway between baffles. Set the speed at 60 RPM. Rapidly pour 30 ml of 20 g/L NaCl or KCl solution into the funnel mounted at the top of the tank, at the same time starting (hit Alt-R first and then hit ENTER) the data acquisition program. Make 10 to 15 additional injections at increasing stirring speeds. Make sure to use a different file name for each run. For these runs record the values of parameters b1 and b2. The tank will now contain a dilute solution of tracer, and can be drained and refilled.
2. Repeat the runs of step 5 with baffles removed from the tank.
3. If time permits, repeat some of the runs of steps 5 and 6 using the large turbine or the smaller three-bladed propeller.
Bead statistics runs
Fill the 20 L tank with water, add 100 to 200 white beads, mount the phototransistor on the side of the tank, connect the negative pole of the 6 volt battery to the A/D board pin xxx, connect the positive pole of the battery to the 1 meg resistor, and connect the green lead from the sensor to pin xxx of the terminal board. Set the RPM at about 200 (a level at which all beads are fully suspended). Mount the Tensor lamp on the shelf and adjust so that the region in front of the sensor is illuminated. Start program bbb.bas, and make 5 runs (taking about 4 minutes). The program will produce a semilog plot of the dead time distribution, and also write the distribution data to a file for plotting by Excel or other spreadsheet. Make a second set of 5 runs at about 400 to 500 RPM, and copy the output file to a diskette for later plotting.
Heat transfer coefficient runs
Insert the aluminum cylinder assembly through the 2-inch hole in the acrylic baffle plate of the 20 L tank. Connect the power cord from the cylinder to the Powerstat transformer, make sure the transformer is off, and set the transformer dial to 50. Connect the temperature sensor plug to the corresponding plug to the terminal board, and load and start program gtheat01.bas.
Fill the 20 L tank with water and insert the large propeller shaft in the motor. At 0 RPM allow the temperature to stabilize and record it. (A dial thermometer can also be used to check the water temperature, noting any disagreement between the two measurements of water temperature.)
Turn on the Powerstat. At 60, 120, 180, 240, 300, 500, 800 and 1200 RPM, allow the temperature to stabilize and record the heater and the water temperature. (Because the heater is supplying about 150 W, the water temperature will rise steadily but slowly.
The report should describe concisely the goals of the work, what was done, and the results and conclusions. The raw data should appear in an Appendix, and plots of torque and power vs. stirring speed should be presented and discussed in the body of the report, as well as plots of conductivity vs. time for the tracer injection runs and bead events vs dead time for the bead statistics runs and heater temperature vs RPM for the heat transfer coefficient runs. The effect of RPM, viscosity, baffles, and impeller design on torque and mixing time should be discussed. The power consumption should be compared with correlations from the literature (McCabe, Smith and Harriot).
Specifically, does the torque as a function of stirring speed follow the prediction of theory? And, how close is the observed torque (or power) to that predicted by the correlation?
1. McCabe, W.L., J.C. Smith and P. Harriot, "Unit Operations of Chemical Engineering," Fourth Edition, McGraw-Hill, New York, 1985.
2. Uhl, V.W. and J.B. Gray, "Mixing Theory and Practice, Volume 1," Academic Press, New York, 1966.
1. Emptying the 20 L tank is done using a valve-equipped drain connection, to avoid lifting the heavy water-filled tank.
2. The viscosity of Karo corn syrup can be found from the manufacturer (it is reported as 2200 to 3000 cP), or measured using a viscometer (preferable). The viscosity can be calculated from the rate of fall of a stainless sphere in a test tube filled with syrup.8. A 1/8" 440 stainless ball falls 5 inches in about 15 seconds in Karo.
3. At high Reynolds numbers viscosity does not have a major effect on torque. But impeller diameter is a very important variable, and its effect should be determined experimentally. That is, two geometrically similar impellers of significantly different size could be used.
4. In the tracer runs the presence of a baffle is expected to disrupt the rotary (vortex) flow, move tracer to the bottom of the tank rapidly, and lead to more rapid mixing. When a vortex is present the tracer response may show several pronounced peaks, as a slug of tracer moves around the tank without rapid mixing.
5. It is useful to distinguish between agitation and mixing (see McCabe et al.). Mixing occurs only when more than one substance is present.
6. If a down-driving propeller is used, tracer injected at the top outside will move in, then down, then out to the bottom-located conductivity cell, taking a relatively long time. Up-driving will carry tracer more rapidly to the cell. Using a turbine will probably give an upper and a lower torus of rotating fluid, with tracer transferring rather rapidly between tori.
7. Stirring water gives time variations of torque due to strongly turbulent flow, stirring corn syrup gives lesser variations.
8. In geometrically similar systems all dimensions (for example impeller diameter, tank diameter and height, baffle number and width, etc.) of one system are the same multiple of the corresponding dimension in the similar system. If, for two geometrically similar systems, the Reynolds numbers are equal, then all other dimensionless numbers are equal. This applies in particular to the dimensionless power number. If we set up two geometrically similar systems and set the RPMs so that the Reynolds numbers are equal, then we can calculate from the power number of the second system the torque, based on the measured power number for the first system. Measurements of torque and power for the second system can then be compared to the predictions.
9. Typical data (20 L tank, 4 inch propeller, water,
baffled) taken using a torque table:
At 280 RPM we have 134.4 (16/453.6) 8 = 37.93 oz-in,
and 37.93 oz-in (0.00706 N m/oz-in) = 0.268 N m
10. With respect to scale up, the basic principle is as follows: If two tank/impeller systems are geometrically similar, and operate at the same Reynolds number, then all other dimensionless numbers will be equal, including the power number. Assume we set up two geometrically similar tank/impeller systems, operate one with water at a given stirring speed, and measure the Reynolds and power numbers. We can then calculate the stirrer speed needed for the second tank, operating with a different fluid, so the it has the same Reynolds number as the first tank. Since the power numbers will now be the same, we can predict the torque (and thus power) needed for the second tank. (Note: It is assumed that air ingesting vortices do not occur in either tank.) So the power needed for the second (usually larger) tank can be predicted from runs made with a smaller tank, avoiding costly experiments.
11. 1 (oz in)*(2.54 cm/in)*(lbf/16 oz)*(m/100 cm)*
(0.2248 N/lbf) = 0.00706 N m