ABSTRACT

Mixing is an important process used in chemical industries for producing homogeneous mixtures, to ensure proper heat and mass transfers, etc. The relationship between mixing speeds and power consumption for a propeller of diameter 0.066m, a 6-blade turbine of diameter 0.0064 and, flat blade impeller of diameter 0.100m, was obtained by recording the armature current of the motor used to drive the impellers, for different agitator speeds. It was found that the shape of an impeller and its diameter determines the power consumption and mixing time for a given fluid at a given temperature. The propeller consumed the least amount of power for given angular velocities; and the flat blade consumed the most amount of power. The relationship between mixing speeds and mixing time was obtained by measuring the time for conductivity of water to attain a steady reading for 10 seconds, after 10 ml of potassium chloride solution is added, for the propeller, 6-blade turbine, and flat blade impeller. The mixing speed was slowest for the propeller for a given angular velocity, and fastest for the flat blade

impeller. The Fox and Gex equation [ ] was used to correlate the mixing data

obtained and the constant k of the equation was found to be 4805.58, 4138.03, and 2575.01 for the

propeller, 6-blade turbine, and flat blade impeller, respectively. However, plots of against

for the three impellers showed that the experimental mixing data did not show any

resemblance to the Fox and Gex relation. A recommendation was made to determine the mixing time for more agitator speeds to increase the accuracy of results to better determine if the Fox and Gex relation is an accurate description for the type of mixing in this experiment.

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APPARATUS/PROCEDURE

The following apparatus and materials were used to conduct the experiment:

o Mixing rig consisting of a mixing tank, a variable speed motor, impellers (propeller, flat

blade, and 6-blade turbine), power meter, conductivity meter and recorder, tachometer, etc [shown in figure 1]

o Potassium chloride

o Water

The following procedure was taken in conducting the experiment.

Done for EACH impeller:

Part A (power consumption)

1. Propeller being studied was attached to the shaft of the motor of mixing vessel; motor-plate was then placed on top of the glass vessel that was filled with water up to a measured point, and secured with thumbnuts to hold it in place.

2. Tachometer and speed controller were connected to mixing vessel.3. Starting at 200 rpm and increasing by 200 rpm up until 1000 rpm, the % armature current

was recorded.4. Step 3 was repeated.

Part B (mixing time)

1. A solution comprising 75g of potassium chloride and 250 ml of water was made.2. Propeller being studied was attached to the shaft of the motor of mixing vessel; motor-

plate was then placed on top of the glass vessel that was filled with water up to a measured point, and secured with thumbnuts to hold it in place.

3. Conductivity meter was attached to mixing vessel.4. Agitator was started set at 200 rpm.5. At steady state (constant conductivity reading), 10 ml of potassium chloride solution was

measured and added to the mixing vessel, simultaneously starting a stopwatch. 6. Once the conductivity reading remained constant for ten seconds, the stopwatch was

stopped and the time recorded. 7. Steps 5 and 6 were repeated.8. Steps 5 to 7 were done with agitator set at 300 rpm and 500 rpm.

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RESULTS

Table 1 shows results for procedure PART A, power consumption.

Speed (rpm)

% Armature Current

for Propeller

% Armature Current

for 6-Blade Turbine

% Armature Current

for Flat Blade

C1 C2 CAVE C1 C2 CAVE C1 C2 CAVE

200 6 6 6 6 5.5 5.75 7 6.5 6.75

400 7 7 7 7 6.5 6.75 10 10 10

600 8.5 8 8.25 8 8 8 14.5 14 14.25

800 10 10 10 10 10 10 17 17 17

1000 10.5 10 10.25 10.5 10.5 10.5 20 20 20

Table 2 shows results for procedure PART B, mixing time.

Speed (rpm)

Mixing Time (s)

for Propller

Mixing Time(s)

for 6-Blade Turbine

Mixing Time (s)

for Flat Blade

τ1 τ2 τAVE τ1 τ2 τAVE τ1 τ2 τAVE

200 51.69 60.09 55.89 53.1 52.81 52.96 34.44 33.2 33.82

300 47.06 47.6 47.33 43.5 46.15 44.83 18.54 19.9 19.22

500 34.37 30.37 32.37 25.91 20.81 23.36 15.35 17.35 16.35

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CALCULATIONS/DISCUSSION for PART A; Power Consumption

Calculating Power Consumed

Power consumed, P = (flux constant) ωIa

For mixing apparatus, the flux constant is 0.2381

Where, ω_ angular velocity and Ia _ armature current

ω = [rpm × (2π/60)] radians per second [1]

Ia = [CAVE × 3] Amperes

Sample calculation for power consumed.

For propeller at 200 rpm:

ω = [rpm × (2π/60)]

ω = [200 × (2π/60)]

= 20.94 rad s-1

Ia = 3 × 6% = 3 × 0.06 = 0.18 A

Therefore, P = 0.2381 ωIa = 0.2381 × 20.94 × 0.18 = 0.8974 W

Table 3 shows calculated power consumption for each blade at the different agitator speeds.

Propeller 6-Blade Turbine Flat Blade

Impeller Speed (rpm) ω (rad/s) Ia (A) P (W) Ia (A) P (W) Ia (A) P (W)

200 20.94 0.18 0.8976 0.17 0.8602 0.20 1.0098

400 41.89 0.21 2.0944 0.20 2.0196 0.30 2.9921

600 62.83 0.25 3.7027 0.24 3.5905 0.43 6.3955

800 83.78 0.30 5.9841 0.30 5.9841 0.51 10.1730

1000 104.72 0.31 7.6671 0.32 7.8541 0.60 14.9603

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Microsoft Excel was used for all calculations for this lab. For the calculated values in the sample calculations, approximate previously calculated values (such as that for angular velocity) were used; while calculations in excel would have used the exact values. That is, for instance, 200 rpm = 20.94 rad/s was used in sample calculation; while excel calculations used

200 rpm = [200 × (2π/60)] = 20.943951023932 rad/s

Hence, results of sample calculations would be inaccurate by a small amount.

Graph 1 above shows the variation of power consumption with impeller speed.

The power meter was not functioning properly, especially when the impeller speeds were at 1000 rpm. The reading fluctuated continuously between 9 and 11% for propeller and 6-blade turbine, therefore an average was taken. Graph 1 and table 3 shows that the propeller of diameter 0.066 m and 6-blade turbine of diameter 0.064 m showed almost the same power consumption for the different impeller speeds. While the power consumption for the flat blade of diameter 0.100 m was significantly higher than that of the propeller and the 6-blade turbine. It is difficult to determine if the 6-blade has greater power consumption than the propeller or if the diameter of an impeller determines the power consumption at a given angular velocity. Graph 1 shows that the power consumption variation with impeller speed is one of exponential increase, for the propeller, 6-blade turbine and flat blade impeller.

Calculating Power Number and Reynolds Number

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Power Number =

Where, ρ _ liquid density = density of water = 996 kg m-3 at 30°C. [2]

N _ impeller speed

Di _ impeller diameter

For propeller, Di = 0.066 m

For 6-Blade Turbine, Di = 0.064 m

For Flat Paddle, Di = 0.100 m

Reynolds Number =

Where, µ _ liquid viscosity = viscosity of water = 798 × 10-6 N m-1 s-1 at 30°C. [3]

Sample calculation for Power Number and Reynolds Number

For Propeller at 200 rpm:

Power Number = = = 0.0784

Reynolds Number = = = 113847

Table 4 shows calculated Power number and Reynolds number for the impellers at different speeds.

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Propeller 6-Blade Turbine Flat Blade

Impeller Speed (rpm) Re No. Power No. Re No. Power No. Re No. Power No.

200 113868 0.07833 107072 0.08755 261406 0.01104

400 227737 0.02285 214144 0.02569 522811 0.00409

600 341605 0.01197 321215 0.01353 784217 0.00259

800 455473 0.00816 428287 0.00952 1045623 0.00174

1000 569342 0.00535 535359 0.00640 1307029 0.00131

Now, the relation between the power number and Reynolds number is:

Hence,

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Graph 2 showing the linear relation of: log10 [ ] versus log10 [ ] for the propeller.

Hence, for a propeller of diameter 0.066 m: a = -1.6457

And log10 K = 7.1999 therefore, K = 15845283

For a propeller:

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Graph 3 showing the linear relation of: log10 [ ] versus log10 [ ] for the 6-blade turbine.

Hence, for a 6-blade turbine of diameter 0.064 m: a = -1.6075

And log10 K = 7.0072 therefore, K = 10167168

For a 6-blade turbine:

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Graph 4 showing the linear relation of: log10 [ ] versus log10 [ ] for the flat blade impeller

Hence, for a flat blade impeller of diameter 0.064 m: a = -1.3155

And log10 K = 5.1585 therefore, K = 144046

For a flat blade impeller:

Graphs 2, 3 and 4 shows that each impeller has a linear relation for log10(power number) against log10(Reynolds number). As graph 1 had shown that there was very little difference in power consumption by the 0.066m propeller and the 0.064m 6-blade turbine, graphs 2 and 3 showed very little difference in the linear relation of log10 (power number) against log10 (Reynolds number). The gradient and intercept for these linear relations for the propeller was -1.6457 and 7.1999, respectively, while those for the 6-blade turbine are -1.6075 and 7.0072, respectively. Graph 4 shows that the gradient and intercept for the linear relationship of the flat blade impeller are -1.3155 and 5.1585. Now it was obvious that the power consumption for the flat blade was much greater than the other two impellers investigated. Hence, with

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the gradient and intercept decreasing (for a linear relation for log10 (power number) against log10(Reynolds number)) with an impeller of greater power consumption characteristic, then the 6-blade turbine of diameter 0.064m has a greater power consumption than the propeller of 0.066m for given angular

velocities. Graphs 2, 3 and 4 also show that the value of K in the relation:

decreased with increasing power consumption for impellers at given angular velocities.

CALCULATIONS/DISCUSSION for PART B; Mixing Time

Graph 5 shows the variation of mixing time with impeller speed for the propeller, 6-blade turbine and the flat blade.

Table 2 and graph 5 shows that the mixing time, at different agitator speeds, was slowest for the propeller of 0.066m and fastest for the flat blade impeller of diameter 0.100m. As the power consumption was the

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smallest for the propeller and largest for the flat blade, this suggests that the greater the power consumption, the shorter the mixing time.

Calculating the Fox and Gex Equation

The Fox and Gex equation correlates mixing time data:

Where, DT _ vessel diameter = 0.190 m

ZL _ liquid height

ZL = (height of vessel) – (height of liquid from top of vessel) = (0.55 – 0.061) m = 0.498 m.

Re _ Reynolds Number

Fr _ Fronde Number

Fr = where, g _ gravitational acceleration = 9.81 m s-2.

Sample calculation for k of the Fox and Gex equation

Now, =

For Propeller:

At propeller speed 200rpm = 20.94 rad/s = N

Fr = = = 2.95

Reynolds Number = = = 113847

Mixing time at 200rpm = 55.89 seconds

Hence,

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So, k = 4202

Similarly, k for propeller at 300 rpm and 500rpm were found to be 4990 and 5224, respectively.

Therefore k for a propeller of diameter 0.066m is [(4202 + 4990 +5224)/3] = 4805.

Table 5 shows Fe, Re at various speeds for propeller, 6-blade turbine, and flat blade and the corresponding k for these impellers.

Impeller Speed, N (rad/s) 20.94 31.42 52.36

Propeller τ (s) 55.89 47.33 32.37

Propeller: Fr 2.9512 6.6401 18.4447

Propeller: Re 113868 170802 284671

Propeller k 4202.97 4990.01 5223.74 4805.58

6-Blade Turbine mixing time τ (s) 52.96 44.83 23.36

6-Blade: Fr 2.8617 6.4389 17.8858

6-Blade: Re 107072 160608 267679

6-Blade k 3961.89 4701.74 3750.46 4138.03

Flat Blade mixing time τ (s) 33.82 19.22 16.35

Flat Blade: Fr 4.4714 10.0608 27.9466

Flat Blade: Re 261406 392109 653514

Flat Blade k 2725.66 2171.67 2827.70 2575.01

Table 5 shows:

The calculated k for the propeller of diameter 0.066 m is 4805.58

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The calculated k for the 6-blade turbine of diameter 0.064 m is 4138.03

The calculated k for the flat blade impeller of diameter 0.100 m is 2575.01

Now,

This is of the form of a straight line of slope = 1/6.

Graph 6 shows a plot of against for the propeller of diameter 0.066 m.

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Graph 7 shows a plot of against for the 6-blade turbine of diameter 0.064 m.

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Graph 8 shows a plot of against for the flat blade impeller of diameter 0.100 m.

Now, according to the Fox and Gex equation, the log10 Nτ against log10 [Fr/Re] should result in a straight-line relation of slope 1/6. However, graphs 6, 7 and 8 did not show a straight-line relation of slope 1/6. The graphs 7 and 8 were more of a parabola shape. Linear relations were still found – but none of slope 1/6. Also, no published studies with regards to correlating mixing time data using the Fox and Gex equation for different impellers was found, to compare with this experiment results. The results of graphs 6,7 and 8 shows that either experiment results were inaccurate and most likely insufficient, or the mixing process in this experiment does not follow the Fox and Gex equation of correlating mixing data. The most likely case was the experiment results being inaccurate and insufficient- as graphs 6, 7 and 8 did not even have a common shape. It was difficult to record the time for no change in conductivity reading after 10 seconds, especially for the flat blade impeller, which had the fastest mixing time of the 3 impellers investigated. A better analysis of results could be made if mixing time for a number of agitator speeds is recorded. This would improve the accuracy of the results which would allow the deduction of whether or not the mixing for this experiment follows the Fox and Gex equation.

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CONCLUSION

The relationship between mixing speeds and power consumption for a propeller of diameter 0.066m, a 6-blade turbine of diameter 0.0064 and, flat blade impeller of diameter was obtained by recording the armature current of the motor used to drive the impellers for different agitator speeds. The shape of an impeller and its diameter determines the power consumption and mixing time for a given fluid at a given temperature. The propeller consumed the least amount of power for given angular velocities; and the flat blade consumed the most amount of power. The following relations were obtained:

For propeller:

For 6-blade turbine:

For a flat blade impeller:

The relationship between mixing speeds and mixing time was obtained by measuring the time for conductivity of water to attain a steady reading for 10 seconds, after 10 ml of potassium chloride solution is added, for the propeller, 6-blade turbine, and flat blade impeller. The mixing speed was slowest for the propeller for a given angular velocity, and fastest for the flat blade impeller. The Fox and Gex equation was used to correlate the mixing data obtained and the constant k of the equation was found to be 4805.58, 4138.03, and 2575.01 for the propeller, 6-blade turbine, and flat blade impeller, respectively.

However, plots of against for the three impellers showed that the experimental

mixing data did not show any resemblance to the Fox and Gex relation. A recommendation was made to determine the mixing time for more agitator speeds to increase the accuracy of results to better determine if the Fox and Gex relation is an accurate description for the type of mixing in this experiment.

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RFERENCES

[1] http://en.wikipedia.org/wiki/Revolutions_per_minute

[2] http://en.wikipedia.org/wiki/Properties_of_water

[3] http://www.thermexcel.com/english/tables/eau_atm.htm

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