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Non-Newtonian Flow

CHEN 4401W

Unit Operations Laboratory

Section 003, Group 6

Section Instructor: Raul Caretta

September 17th, 2013

Planner: Alvaro de la Garza Musi

Experimenter: Chen Fang

Analyzer: Shaw Su

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Abstract The Uni-Minn Corporation was asked to design a piping system to transport a dilute guar-xanthan solution by a personal care products client. It was specified by the client that a day’s solution should be transferred from a storage tank to a target tank 2000 ft. away at a flow rate of 150 gallons/min. The specifications also included a height difference between the tanks of 65 ft. Data was obtained from the Rheological Analysis Equipment (RAE) and the Brookfield Viscometer, during to laboratory sessions at the University of Minnesota pilot plant. The rheological constants calculated after were transferred to construct the requested scale-up design. A comparative analysis between the two test methods was also done to verify the values of the behavior and consistency indexes, while also evaluating accuracy and precision. The effect of temperature on the fluid was also studied. The solution was analyzed at different Reeve’s pump settings and recycle stream valve positions. A variation from 3 to 6.5 was done in the pump, and the recycle stream valve was opened until there was almost no flow or closed completely to reach higher fluid velocities. Similarly, the polymer solution was studied using the Rotational viscometer, with RPM varying from 1 to 100. Using the RAE, flow rates in the range of 0.15 to 0.65 ± 0.06 kg/s were attained; the pressure drops recorded at four different points spanned from 300 to 3,000 ± 15 Pa. Calculations resulted in a consistency index of 0.21 Nsn/m2 and a flow behavior index of 0.65. The Brookfield viscometer produced torque measurements ranging from 10 to 50 ± 0.1 Nm, and apparent viscosities around 250 to 550 ± 15 cp. Using this data the consistency and flow behavior indexes were found to be 4.90 Nsn/m2 and 0.67 respectively. These values correspond to thinning behavior which indicates the power law fluid lies in the pseudoplastic fluid region. Tests changing the temperature of the polymer solution to 24, 25 and 27± 0.2 ºC were done using a hot water bath and the Brookfield viscometer. Viscosity was recorded and presented an exponential decreasing behavior with increasing temperature. The consistency index was calculated at changing temperatures and was found to be 3.6, 3.5, and 4.3 Nsn/m2 at 24, 25 and 27ºC respectively. In addition, the flow behavior index was calculated to be 0.72, 0.73, and 0.68 at 24, 25 and 27 º C. These values show a decreasing flow behavior index and therefore increasing pseudoplastic behavior. Density was measured to be 1090 ± 10 during both lab periods using a 1L beaker. The polymer content of the solution was obtained by drying the solution for two days in a convection oven and was calculated to be 0.43 ± 0.01wt%. The data could have been improved on accuracy and precision by having one person run and record data for the RAE and using a broader range of temperatures in the Brookfield viscometer tests.

The scale-up piping system was designed using the calculated values for the consistency and the flow behavior indexes. Two open 190,290 gallon tanks, each with a 25.3 ft diameter, a 50.6 ft height and 20in freeboard were sized to hold a day’s supply for the process. A 6-in diameter schedule 40 steel pipe was chosen for the piping in the system to minimize cost. The Reynolds number was calculated to be 1,655 for the polymer solution validating laminar flow. A Reynolds number of 197,000 was calculated for water, indicating turbulent flow. A positive displacement 4HP pump is recommended for its use in the transfer of the polymer solution from the storage tank to the receiving tank. In case of water, vacuum pump with 3-HP motor can be used. A recycle stream is also recommended to have more flexibility when it comes to fluid flow control if the displacement pump does not offer many settings. The total cost for the design essentials was calculated to be $1,710,000 not including the recycle stream, valves, fittings or temperature gages and a rotameter.

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TABLE OF CONTENTS 1. Introduction, Theory and Technical Background

1.1 Introduction 4

1.2 Theory and Technical Background 4

1.3 Design Problem Theory 7

1.4 Error Analysis 8

2. Description of Apparatus

2.1 Rheological Analysis Equipment 10

2.2 Brookfield Rotational Viscometer 12

3. Experimental Procedure

3.1 Rheological Analysis Equipment 12

3.2 Brookfield Rotational Viscometer 14

4. Results

4.1 Polymer Solution Characteristics Data 14

4.2 Rheological Analysis Equipment Data 15

4.3 Brookfield Rotational Viscometer Data 17

5. Final Data and Results

5.1 Polymer Content and Density 19

5.2 RAE Results 20

5.3 Brookfield Rotational Viscometer Results 21

5.4 Temperature Effects 22

5.5 Comparison of RAE and Rotational Viscometer results 22

6. Discussion of Results, Conclusions and Recommendations

6.1 Discussion of Results 22

6.2 Conclusions 23

6.3 Recommendations 23

7. Design Problem

7.1 Introduction 25

7.2 Design Parameters 25

7.3 Design Calculations 26

7.4 Non-Newtonian and Water Comparative 26

8. Nomenclature 27

9. References 28

10. Appendices

A. Original Data Sheets 29

B. Sample Calculations 43

C. Design Problem Calculations 49

D. Error (Uncertainty) Analysis 56

E. Data Transfer Sheet 60

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1. Introduction, Theory and Technical Background

1.1 Introduction

A personal care products client asked the Uni-Minn Development Corporation to design a piping system that will efficiently transport a guar and xanthan blend for their process. It was specified that the solution is initially stored in an open storage tank, and it is to flow at the rate of 120 gallons/minute through 2,000 feet of pipe. The target storage tank is 65 feet above the tank where the guar and xanthan solution is initially stored. Rheological constants were obtained in lab to provide an accurate design including optimal pipe diameter, and pump characteristics. The independent variables were determined to be the pump setting in the RAE, the RPM in the Brookfield viscometer, the temperature, and the concentration in both of them. The dependent variables were the flow rate, and pressure drop in the RAE. In the case of the Brookfield viscometer they were determined to be the angular velocity, the torque, and the apparent viscosity.

Two laboratory sessions were used to perform the Rheological Analysis Equipment and Brookfield Viscometer experiments. The RAE was ran at a controlled temperature of 22±1°C, pressure drop measurements were done at different points along the length of the copper piping. Data obtained regarding pressure changes and flow velocities were modeled using a power-law fluid function and rheological properties were calculated through log-log plots of shear rate vs. shear stress.

In the same manner measurements were taken using the Brookfield Viscometer to obtain different fluid or rheological properties. The parameters calculated were compared to those calculated using the RAE, to assess precision and provide a recommendation. Furthermore, temperature tests were done using the Brookfield viscometer to determine the dependence of non-Newtonian fluids with respect to temperature changes.

1.2 Theory and Technical Background

Newtonian and Non-Newtonian Fluids

A Newtonian fluid is one who follows Newton’s law of viscosity given by equation 1.2.1. The shear stress is directly proportional to the shear rate (-dv/dr) with viscosity as the proportionality constant. (1)

(1.2.1)

On the other hand, Non-Newtonian fluids do not follow Newton’s Law of viscosity, and can be divided into three categories according to their shear rate/shear stress behavior: fluids whose behavior is independent of the duration of shear, fluids whose behavior is dependent, and viscoelastic fluids which present elastic behavior. The guar and xanthan solution studied presents a time-independent non-Newtonian behavior; its characteristics correspond to a pseudoplastic fluid. Figure 1.2.1 shows a diagram of the shear stress/rate relationship of Newtonian fluids and different types of time-independent non-Newtonian fluids. (1)

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Figure 1.2.1. Shear stress vs. Shear rate graph

The modeling of the flow behavior of non-Newtonian fluids is done through a power-law equation.

(

) (1.2.2)

Equation 1.2.2 above represents the power law equation where K is the consistency

index in and n is the flow behavior index and it’s dimensionless. These two rheological constants characterize a power law fluid. The apparent viscosity is represented by the following equation.

(

)

(1.2.3)

For n<1 the apparent viscosity decreases as the shear rate increases, these are known as shear thinning fluids. Non-Newtonian solutions’ rheological characteristics are susceptible to bacterial, heat, enzyme and UV degradation.

Rheological Analysis Equipment

The RAE uses pipe flow and pressure changes to study non-Newtonian flow properties. Shear stress and shear rate are calculated using the available measurements, and logarithmic plots are used to find the rheological constants needed.

Shear stress can be calculated the following equation,

(

) (

) (1.2.4)

Where D is the pipe diameter, is the pressure drop, L is the length of the pipe, and V

is the fluid velocity. A log-log plot of shear stress versus (

) yields to a linear fit with

the slope being n the flow behavior index, and an intercept of K the consistency index.

Similarly, the shear rate at the wall of the pipe is a function of the behavior index n, the linear fluid velocity V, and the pipe diameter D as shown below.(3)

(

)

(

) (

) (1.2.5)

This relation only holds for laminar flow, therefore low fluid velocities have to be maintained. Finally using the results above the generalized viscosity coefficient is calculated according to the following relation,

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(

) (1.2.6)

Brookfield Rotational Viscometer

Another important method for measuring flow properties is by using a rotating concentric-cylinder viscometer. (2) This device is composed by a rotating cylinder, or spindle, spinning at a constant rotational speed inside another cylinder. The gap between the walls of these two cylinders is usually small and is filled with the fluid subject to analysis. A commercial device of this kind is the Brookfield viscometer.

The torque necessary to maintain a constant rotation rate is directly proportional to the resistance induced by the fluid viscosity and it’s measured using a torsion wire where the spindle is suspended from. This device provides a uniform shear rate, and a uniform corresponding shear stress.

The shear rate at the surface of the spindle for non-Newtonian fluids is as follows;

(

)

[ (

)

]

(1.2.7)

Where is the radius of the spindle, is the radius of the outer cylinder, and is the

angular velocity of the spindle. The above relationship only holds for 0.5 <

< 0.99. To

calculate the angular velocity of the spindle and the torque required the below equations are necessary,

(1.2.8)

Given that N is the revolutions per minute. And for torque in Nm,

(1.2.9)

The shear stress at the wall of the spindle is given by

(1.2.10)

Where T is the measured torque, and L is the length of the spindle. Equation 1.2.8 holds for both Newtonian and non-Newtonian fluids.

Substituting equations 1.2.7 and 1.2.10 in the power-law equation results in the following equation,

[

( (

)

)]

(1.2.11)

Which after experimental data is obtained by measuring the torque at different angular

velocities can be used to evaluate flow-property constants. A log T versus log w results

in a linear fit with the parameter n as the slope and the intercept as , the

consistency factor can be extracted easily from the intercept.

Effect of Temperature Changes

Non-Newtonian fluids apparent viscosity is very sensitive to temperature changes. As temperature increases power law fluids present a reversible softening which translate to

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a decrease in viscosity. Several experiments have been conducted using rotational viscometers and they all present an exponential decrease in viscosity with rising temperature. (7) Another correlation with temperature is the flow behavior index, as temperature increases the flow behavior index decreases, which means the degree of pseudoplasticity increases. With respect to other parameters such as shear stress, and the consistency index the temperature does not present a defined correlation, this might be due to the reversible property of non-Newtonian fluids which can lead to hysteresis and stochastic behaviors.

1.3 Design Problem Theory

The client requested a scale-up design of a piping system to transfer a guar-xanthan solution from one tank to another. The non-Newtonian fluid used in lab is identical to the one used in the client facilities, therefore the rheological constants obtained from the RAE and the Brookfield Viscometer were used for the design. Specifications for the scale up design include, sizing of the pump, providing the optimal pipe diameter needed, and correct height and diameter for both storage tanks included in the system. Lastly, differences in design between using a non-Newtonian fluid and water were provided as requested by the client.

Optimal Pipe Diameter

Cost efficiency is as important as effectiveness when designing a plant, therefore a pipe which is economically optimum as well as capable to meet the client’s requirements is needed. The following equation is therefore used to calculate the pipe diameter,

[ (

)

(

) ]

(1.3.1)

The above equation from Peters and Timmerhaus is based on economical and design optimization parameters. It includes several costs which are essential to minimize in a plant such as electricity, annual average fixed and maintenance costs, etc. The description of the symbols used is found in the nomenclature section. (8)

Reynolds Number

To validate the use of the results obtained in lab for scale up, it has to be proven that the fluid is flowing in the laminar region. The generalized Reynolds number equation can be used for the analysis,

(

) (1.3.2)

Where D is the pipe diameter, V is the linear velocity, is the fluid’s density, K and n are

flow indexes. The transition region to turbulent flow starts at , so a flow lower than that has to be attained.

In the case of water a different equation for Reynolds number is needed,

(1.3.3)

Here the laminar region ends after a Reynolds number higher than 2,000.

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Pump

Pump sizing starts with a mechanical energy balance including potential energy, pressure drop, kinetic energy, and the summation of the friction losses. It is given by the following equation,

-

∑ (1.3.4)

Here is the velocity correction factor, approximated to be 0.5 for laminar flow in Newtonian fluids. In non-Newtonian fluids is given by,

(1.3.5)

The pressure drop due to friction (Δp) in the mechanical energy balance is found using

the following equation,

(1.3.6)

This equation applies to laminar flow and requires the fanning friction factor .

The total frictional loss in the pipe (∑F) is calculated adding all the expansion losses (hex), contraction losses (Kc), and losses in fittings and valves (Kf). It is given by the following relation,

∑ (

)

(1.3.7)

Solving for the shaft work in the mechanical energy balance, and adding the average pump efficiency of 75% (4) the work required by the pump is calculated using the relation below,

(1.3.8)

Finally, the work required can be translated to Break horse power with the following equation,

(1.3.9)

From here the electric power input can be determined with the next relation,

(1.3.10)

1.4 Error Analysis

Error and uncertainty on experimental results are of extreme importance to validate the data obtained. These uncertainties are either due to instrument or measurement error. Propagation of error is done in all subsequent calculations. In addition and subtraction operations error propagates according to the formula:

√

(1.4.1)

Where ep is propagated absolute error and e1 through en are absolute errors of the measurements subtracted or added. Error of measurements in multiplication and division propagates like:

√

(1.4.2)

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Where %ep is the percentage error propagated and e1 through en are the percentage errors of the different measurements multiplied or divided.

Error associated with logarithms is determined using the following formula:

(

) (1.4.3)

To calculate the error associated to the mean of a value pool, the Student’s t test must be applied to a 95% confidence interval. The first standard deviation Sp is computed as,

√( )

(1.4.4)

where xi is the value of the measurement, xavg is the average of the experimental values, and n is the number of measurements made. The desired value for t to approach a confidence interval is determined from student’s t table. Using this value, the error limit Δ can be determined for the sample mean value, as follows.

(1.4.5)

Hence the final result accounting for errors is expressed in the form xavg ± Δ and it is based on the degrees of freedom and a particular confidence level.

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2. Description of Apparatus

In this experiment, two sets of equipment, a Rheological Analysis Equipment (RAE) and a Brookfield Rotational Viscometer were used to evaluate rheological properties of a solution containing a blend of 97% guar and 3% xanthan. Results obtained with RAE were compared with Brookfield Viscometer to see if the information could be obtained with a much smaller and simpler instrument.

2.1 Rheological Analysis Equipment

A process flow diagram for the rheological analysis equipment was shown as follows.

T

P1 P2

P3P4

MOYNO Pump

Bypass line

Recycle line

2.9m

Dra

in

Figure 2.1.1 Process flow diagram of the rheological analysis equipment

Bypass line and recycle line

Two valves were used in the operation of the RAE to control flow in this experiment, one on the bypass line and one on the recycle line. For a certain setting of the Reeves Drive pump speed, opening the bypass line valve more and the flow rate to the weight tank would be slower. The maximum flow rate at a certain pump speed would be obtained when the bypass line valve was closed.

The recycle line was used to recycle the polymer fluid in the weight tank back to the mixer tank. The recycle line valve was closed during a certain run to measure flow rate, and opened at the end of a run to recycle the fluid in the weight tank back to the mixer tank.

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Moyno Pumps and Reeves Drive

A Moyno pump, which is a type of positive displacement pump, was used to pump the polymer solution from the mixing tank to the weight tank. As the rotor turns inside the stator, cavities are formed which progress toward the discharge end of the pump carrying the material being handed (2). This leads to low levels of shearing applied to the pumped fluid hence have application of viscous or shear-sensitive materials. The volumetric flow rate is thus proportional to the rotation rate. The assumption that all pumps could have their flow rates adjusted by a valve attached to their outlet therefore could not be valid for this type of pump since such a valve will have no effect on the flow rate and closing the valve would cause high pressure. However, a bypass pipe that allows a certain amount of fluid to return to the inlet could be used to control flow rate for this kind of pump.

A Robbins & Myers, INC motor was used to drive the pump. The motor used is 5hp, 220/400 volts, serial B6932 KU. A Reeves Drive was used to connect the motor to the pump. The Reeves Drives used is 5hp, serial 126759, with output speeds 195-1170 RPM, gear ratio of 2.76-1.

Weight tank

The flow rate in this experiment was measured using the weight tank-stop watch method. An electronic scale is placed under the weight tank to measure weight of fluid as it accumulates over certain duration of time. A recycle line connects the weight tank to the mixer tank.

Pressure taps

There were four pressure taps, each connected to a manometer. Plastic tubes connect the nominal 1-in. pipe at the tap position to a manometer board. The distances between taps 1-2 and 3-4 are both equal to 2.9m. As a certain Reeves Drive pump speed was set and the system came to steady state, the fluid in the nominal 1-in. pipe would go to a certain constant level in the manometer corresponding to the relative pressure at that tap position. The working fluid was thus the same as the polymer solution being studied. Pressure was measured three times to reduce the random errors.

Mixer Tank and temperature monitor

The solution was contained in a big 50-gal mixer tank with a paratrol mixer which has a setting between 0 and 100 in the tank so the solution is well-mixed. The temperature inside the tank would increase a little due to the mixing. And the temperature indicator on the side of the tank was used to monitor the temperature inside the tank to keep it

within a 1 range.

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2.2 Brookfield Rotational Viscometer

Figure 2.2.1 Brookfield UL Adapter(2)

A model DV-I+ Brookfield viscometer was also used. The viscometer rotates a spindle immersed in a fluid through a calibrated spring at a certain rotational speed, and measures the torque needed to overcome the viscous resistance of the fluid by the spring deflection. The DV-I model has a RPM range of 0.0-100RPM. The %max rotation was shown. For a certain measurement, the rotational speed was set, and the viscometer measures torque in % and viscosity in cp. The spring torque for this model is 7187.0 dyne-cm.

Two types of spindles could be used to attach to the spindle coupling nut thread for measurement. A cylindrical spindle specified by sp01 or a UL Adapter specified by sp02. UL Adapter is to measure viscosities below the normal range of the instrument, and to provide greater sensitivity. The smaller space between the spindle and UL Adapter ensures a greater sensitivity. The spindle dimensions are below: For spindle #1, radius: 0.9421cm, length 7.493cm, actual length 6.510cm, for the UL adapter radius of spindle #2: 1.2555cm, radius of UL adapter: 1.3785cm.

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3. Experimental Procedure

3.1 Rheological Analysis Equipment

Start up

To start up, the main power switch located at the back of the panel was turned on. The tank mixer was also turned on to mix the solution. The motor was started then the Reeves Drive pump speed set to 3 for a start-up setting.

Steady state

For a certain Reeves Drive pump speed setting and bypass line valve position, whether pressure reading was stable was used to determine if steady state was reached. For a new run, when the pump speed or the bypass valve was set, it was observed that it took a few seconds for the fluid inside the manometers to reach a new level and stay stable. When the pressure reading was stable, the steady state was reached and the measurements were taken after.

Measurements

For all measurement, flow rate was controlled by both the Reeves Drive pump speed and the position of the bypass line valve. Measurements were taken after both the pump speed and the valve position were fixed and the system reached steady state. For a certain run, time and mass for the weight tank and pressure at four tags were measured. At the start of a run, initial mass on the electronic balance of the weight tank was recorded, a period of time (most frequently 60s) was recorded by a stop watch and the final mass on the balance was recorded. The recycle line valve was closed during the entire run and opened at the end of a run to recycle the fluid in the weight tank back to the mixer tank. In the meantime while flow rate was measured, pressure measurement was recorded. Both the weight tank-and-stopwatch and pressure measurements were taken 3 times for each run to get the experimental errors and reduce random errors.

In order to get a volumetric flow rate from the mass flow rate, mass of the sample was measurement along with volume of the sample using a 1000 mL graduated cylinder. 5 samples were taken to get a more accurate density measurement.

The polymer content of the solution was determined by placing a measured amount of polymer solution inside a petri dish with lid on and put in an oven. After two days, the mass of the dried polymer was measured again to get the polymer content of the material.

Run conditions and range of variables

The Reeves Drive pump was started at a setting of 3, and increased in a 0.5 interval to a setting of 6.5. In order to obtain results at very low flow rates to compare with the Brookfield Viscometer, 7 sets of flow rate were obtained at a pump setting 3 during 2nd week, from the bypass line valve widely open (where the flow was very narrow and slow), closed a little bit to the bypass valve completely closed. Lowest flow rate was obtained with Reeves Drive Pump setting at 3 and bypass valve widely open, highest flow rate was obtained with Reeves Drive Pump setting at 6.5 and bypass valve closed. Range of flow rate from 0.01 gal/min to 8.21 gal/min was obtained.

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Shutdown

Recycle line was opened to empty the remaining solution in the weight tank to the mixer tank. The level of polymer solution in tank was checked to be about 3 to 4 inches below top of barrel. The mixer was turned off. The Reeves Drive was returned to the setting of 3 and then the motor was stopped. The main power switch was finally turned off.

Precautions

The guar-xanthan solution has been determined to be nonhazardous by the Metropolitan Waste Control Commission and the University Department of Environment Health and Safety. The solution has a BOD5 (biological oxygen demand) equal to 6.51g/L and a COD(chemical oxygen demand) of 8.14g/L and can be discharged to the sewer weekly by the shop personnel because it is biodegradable.(2)

3.2 Brookfield Rotational Viscometer

Start up

First, check that the viscometer is level. The two leveling screws on the base were adjusted to keep the viscometer level if necessary. The power switch on the real panel of the viscometer was then turned on with no spindle attached. The DV-I+ then begins its Autozero. The spindle (or the UL Adapter) was then taken out from the case and screwed to the lower shaft of the viscometer and the S01 or S02 was shown on the display corresponding to the spindle attached.

Measurements

A 1000mL flask containing the polymer solution was placed under the spindle. The Brookfield viscometer was lowered so that the water line level mark on spindle was at the same height of the polymer solution level in the flask. Once the rotational speed was set, it took less than 30 seconds for the system to come to steady state and the reading to be stabilized. The stability of reading was used to confirm that the system has reached steady state. The torque in % and viscosity in cp were then displayed and recorded. The rotational speed was varied from 5 RPM to 100 RPM for the range of variables. Thermometer was used to measure the polymer solution temperature in the flask, and was determined to be 21 .

Temperature effect on the rheological properties of the fluid was studied using the Brookfield UL Adapter during the 2nd lab period. The 1000 mL flask containing the polymer fluid was immersed in a 4L water bath containing the hot water from the tap. The system was equilibrated and the temperature was measured. The same sets of measurements as above were taken at the new temperature. The temperature for the water bath and solution system decreased slowly to near the room temperature and there is enough time for measurement at a certain temperature. And during the process the same sets of measurements were taken at the intermediate temperature.

Shut down

The viscometer motor was turned off. The spindle or the UL Adapter was taken out of the polymer solution. The spindle and the UL Adapter were cleaned up. The flask was cleaned.

Precautions

Both guar and xanthan are irritants materials and gloves were worn at all times when handling the solution using the Brookfield viscometer.

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4. Results: Calculated Data Tables

4.1 Polymer Solution Characteristics Data

During the 2nd week, mass of petri dish (hold the polymer solution), mass of wet polymer solution, mass of dry polymer plus the petri dish were obtained. Mass of wet polymer and dry polymer were obtained as follows. The balance used has accuracy to the 4th digit, with the last digit fluctuating during measurement hence used for error here. This result would be used to calculate the polymer content of the solution as in the Final Results part.

Table 4.1.1 Measurement of polymer content of the solution

Mass of wet polymer solution(g)( 0.0005) 11.7500g

Mass of dry polymer (g)( 0.0007g) 0.0500g

During the 2nd week, 5 samples were taken to get measurement uncertainty, with each the mass and volume of the sample measured. Density was calculated as shown below and in the Final Data and Results section. Density measurement if necessary to get volumetric flow rate from the mass flow rate measured and hence get velocity for calculation of rheological constant using RAE.

Table 4.1.2 Measurement of density of the polymer solution

Run sample Mass of sample(kg)

Volume of sample(mL)

Density of

sample( )

Average density of polymer

( )

1 1.10 1000 1100

1088 14 2 1.08 1000 1080

3 1.08 1000 1080

4 1.08 1000 1080

5 1.10 1000 1100

4.2 Rheological Analysis Equipment Data

A three number two dash notation, as x-x-x was used for notation of all run number in the table below. During 1st week, the 1st number was 1 and 2nd week the 1st number 2. During 1st week, for a single run, three replications were made each with run number for example 1-1-1, 1-1-2, and 1-1-3 on the original data sheet with table name “S3N6 NNF Week1 Evaluate rheological properties using RAE”. This was not right since they are replications of just one run. The replications are combined as the average calculated as follows. Velocity and in SI units are calculated to be used for the shear stress shear rate plot as in the Final results section. The uncertainty was based on the individual replications of the measurements.

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Table 4.2.1 Rheological Analysis Equipment results for the 1st week

Run Flow rate (lbs/s)*10

Velocity(kg/s)*10 (1-2) (Pascal)

(3-4) (Pascal)

1-1-(1,2,3) 1.09 0.02 0.94 0.02 1513.5 21.3 1493.9 12.7

1-2-(1,2,3) 0.02 0 0.01 0 519.8 48.7 483.8 26.6

1-3-(1,2,3) 0.62 0.02 5.36 0.06 3654.6 47.6 3671.0 45.2

1-4-(1,2,3) 3.4 0.06 2.94 0.06 2762.2 35.4 2817.8 36.2

1-5-(1,2,3) 7.40 0.06 6.39 0.08 3961.9 50.2 4011.0 51.3

1-6-(1,2,3) 9.14 0.04 7.90 0.09 4426.1 56.8 4527.4 58.5

1-7-(1,2,3) 2.33 0.03 2.01 0.04 2239.2 31.3 2239.2 33.5

1-8-(1,2,3) 2.79 0.03 2.41 0.03 2406.0 32.0 2402.6 32.2

For the run number notation, the 1st number 2 indicated the 2nd week, the 2nd number 1 was used for the rheological analysis equipment. Same quantities were calculated as below.

Table 4.2.2 Rheological Analysis Equipment results for the 2nd week

Run Flow rate (lbs/s)*10

Velocity(kg/s)*10 (1-2) (Pascal)

(3-4) (Pascal)

2-1-1 5.93 0 5.04 0.07 2987.5 38.4 3094.2 39.8

2-1-2 0.40 0 0.34 0 558.4 15.5 553.0 15.2

2-1-3 1.03 0. 03 0.88 0.03 992.3 15.5 985.2 14.1

2-1-4 0.37 0 0.31 0 496.1 8.3 492.6 9.4

2-1-5 0.29 0.02 0.25 0.02 361.0 8.3 366.3 10.8

2-1-7 0.41 0.02 0.35 0.02 503.3 10.4 497.9 12.8

2-1-8 6.32 0.05 5.37 0.07 3083.5 42.5 3133.3 40.8

2-1-9 7.21 0. 11 6.12 0.11 3279.1 44.0 3353.8 44.0

2-1-10 8.03 0.07 6.82 0.09 3478.3 45.0 3581.4 46.5

2-1-11 9.09 0.05 7.72 0.09 3693.5 48.0 3844.6 50.0

2-1-12 10.11 0.04 8.59 0.10 3930.0 52.0 4143.4 53.7

2-1-13 11.33 0 9.63 0.10 4203.8 57.0 4445.7 58.5

2-1-14 12.42 0.48 10.55 0.42 4442.1 57.8 4712.4 61.9

2-1-14’ 0.98 0. 02 0.83 0.02 928.3 14.8 917.6 14.7

2-1-15 0.51 0.43 0.02 576.2 583.3 9.7

2-1-16 1.64 0.14 0.02 1298.1 20.8 1280.4 19.6

2-1-17 1.03 0 0.88 0.01 967.4 17.5 935.4 14.9

Certain parameters of the RAE would be needed to calculate the results as indicated in the final results section. The parameters noted were as follows:

Table 4.2.3 Parameter used for calculation of the Rheological Analysis Equipment

Parameter of the RAE Value

Diameter of the pipe(d)(m) 0.025m

Distance between pressure taps(L)(m) 2.9m

For the run number notation, during 1st week, only two numbers were used. For easy reference here, a first number 1 was added. So the Run number below 1-2-1 correspond to the run number 2-1 on the original data sheet of the 1st week “S3N6 NNF Week 1 Rheological Constants using Brookfield Rotational Viscometer”. The rotational speed

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was calculated, and the torque measured in % was converted to the absolute value using the full torque for this model 7187.0 dyne-cm. Since the RPM measured has no

errors associated, the calculated has no errors. The results were as follows:

4.3 Brookfield Viscometer Data

Table 4.3.1 Brookfield Viscometer results for the 1st week

Run Temp. ( ) RPM( 1%) (rad/s) ( 1%) Torque(NM)

* ( .0001)

1-2-1 16.5 5 0.5 1.5

1-2-2 16.5 10 1.0 2.8

1-2-3 16.5 20 2.1 4.8

1-2-4 16.5 50 5.2 9.2

1-2-5 16.5 100 10.5 14.4

1-2-6 16.5 30 3.1 6.3

1-2-7 16.5 60 6.3 10.3

Brookfield UL Adapter was used for the 2nd week to study the temperature effects. The same quantities were calculated as below:

Table 4.3.2 Brookfield UL Adapter results for the 2nd week

Run Temp. ( ) RPM( 1%) (rad/s) ( 1%)

Torque(NM)* ( .0001)

2-2-1 21 5 0.5 6.3

2-2-2 21 10 1.0 10.1

2-2-3 21 20 2.1 16.5

2-2-4 21 50 5.2 30.6

2-2-5 21 60 6.3 34.4

2-2-6 21 100 10.5 46.8

2-2-7 21 30 3.1 21.9

2-2-8 27 5 0.5 5.6

2-2-9 27 10 1.0 9.1

2-2-10 27 20 2.1 15.1

2-2-11 27 50 5.2 28.0

2-2-12 27 100 10.5 42.9

2-2-13 27 30 3.1 19.6

2-2-14 27 60 6.3 31.0

2-2-15 25 5 0.5 4.7

2-2-16 25 10 1.0 8.5

2-2-17 25 20 2.1 14.5

2-2-18 25 50 5.2 27.4

2-2-19 25 100 10.4 42.4

2-2-20 25 30 3.1 19.4

2-2-21 25 60 6.3 30.8

2-2-22 24 5 0.5 4.9

2-2-23 24 10 1.0 8.8

2-2-24 24 20 2.1 15.0

2-2-25 24 50 5.2 28.2

2-2-26 24 100 10.5 43.5

2-2-27 24 30 3.1 20.0

2-2-28 24 60 6.3 31.7

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Several parameters regarding the Brookfield viscometer were needed to calculate the rheological constants as in the final results section. The parameters as shown as below:

Table 4.3.3 Parameter used for calculation of the Brookfield Viscometer

Parameter of the cylindrical spindle Parameter of the UL Adapter

Radius of spindle(m)

0.00942 Radius of spindle(m) 0.0126

Length of spindle(m)

0.0749 Radius of UL Adapter(m) 0.0138

Actual length(m) 0.0651

0.911

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5. Final Data and Results The guar and xanthan polymer solution is expected to be a non-Newtonian power law fluid. It is passed through a rheological analysis equipment to evaluate the flow behavior index n, and the consistency index K. The Brookfield Rotational Viscometer is used alternatively to compare the results. 5.1 Polymer Content and Density The polymer content of the solution is given as the weight of the dry sample over the weight of the original wet sample. The density of the solution is obtained by weighing it in a 1L cup. The results are given in the table below.

Table 5.1.1 Properties of the guar-xanthan solution

Polymer content (%) Density (kg/m3)

0.43 ± 0.01 1090 ± 10

5.2 RAE Results

The polymer solution is pumped through a capillary tube of 1 inch in diameter, where the pressure drops across a 2.9 m straight section is measured by a manometer. The Reynolds number at the highest flow rate is determined to be 930 ± 40, which indicates laminar flow. This condition is required for the power law model 1.2.2 used for analysis, where the shear stress at the wall is plotted against shear rate in the logarithmic scale.

Figure 5.2.1 log-log Flow curve of the guar-xanthan solution, where the shear stress τw is given by DΔP/4L, and the shear rate 8V/D. Note the small error bars. The slope and the intercept are given as the exponent n and constant K respectively.

y = 0.2282x0.6551 R² = 0.998

21 ºC

0.1

1

10

100

1 10 100 1000shea

r st

ress

(N

/m2

)

shear rate (1/s)

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Table 5.2.1 Rheological constants from RAE analysis

n K (N•sn’/m2)

0.6551 0.2282

The errors in the above constants are from the regression analysis alone and are approximated to be less than 1%. They are therefore ignored. The temperature under which the RAE analysis was performed was 21 ºC. 5.3 Brookfield Rotational Viscometer Results A cylindrical spindle attached to an electric motor is dipped into the polymer solution where the angular velocity is controlled to shear the liquid. The torque needed to keep the RPM constant is calculated and correlated to give the rheological constants in the following figure.

Figure 5.3.1 log log plot of torque vs angular velocity, where the slope is the

exponent, and intercept the constant in the equation above.

The constant 0.0098 is used again to obtain the consistency factor K that relates shear stress to shear rate. They are given in the table below.

Table 5.3.1 Rheological constants from rotational viscometer

A n K (N•sn/m2)

0.0098 0.6778 4.9019

There are ignorable errors from regression analysis for the constants above. The temperature of the solution was 21 ºC.

y = 0.0098x0.6778 R² = 0.9992

21 ºC

0.001

0.01

0.1

1

0.1 1 10 100

Torq

ue

(Nm

)

ω

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5.4 Temperature Effects The temperature effects on the rheological constants and viscosity are evaluated. The log log plots of torque vs angular velocity are given below.

Figures 5.4.1-3 log log plot of torque vs angular velocity at their respective

temperatures, values of A and n given in the boxes

y = 0.0089x0.6834 R² = 0.9993

27 ºC 0.001

0.01

0.1

1

0.1 1 10 100

Torq

ue

(N

m)

ω

y = 0.0083x0.7294 R² = 0.9968

24 ºC

0.001

0.01

0.1

1

0.1 1 10 100

Torq

ue

(N

m)

ω

y = 0.0081x0.7315 R² = 0.9971

25 ºC 0.001

0.01

0.1

1

0.1 1 10 100

Torq

ue

(N

m)

ω

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The rheological constants at its respective temperature are given in the following table. The associated uncertainties are again negligible and from regression alone.

Table 5.4.1 Rheological constants at 24, 25, and 27 ºC

Temperature (ºC) A n K (N•sn/m2)

24 0.0083 0.7294 3.6731

25 0.0081 0.7315 3.5668

27 0.0089 0.6834 4.3930

The plot of effective viscosity vs shear rate is given below to demonstrate the effect of temperature. The uncertainties are less than 1% and negligible.

Figure 5.4.4 Plot of effective viscosity vs shear rate at their respective

temperature from the rotational viscometer

5.5 Comparison of RAE and Brookfield Rotational Viscometer results

The conditions upon which the results are compared are 100% concentration and 21 ºC, error bars are too small to show .They are discussed in the subsequent section.

Figure 5.5.1 log log plot of shear stress vs shear rate from both the RAE and the

rotational viscometer results, values of K and n are given in previous tables

0

0.5

1

1.5

2

2.5

3

0 50 100 150

μ (

Pa

s)

shear rate (1/s)

21 C

27C

25C

24C

y = 4.9242x0.6778 R² = 0.9992

y = 0.2282x0.6551 R² = 0.998 0.1

1

10

100

1000

1 10 100 1000

she

ar s

tre

ss (

N/m

2)

shear rate (1/s)

Brookfield results

RAE results

Power (Brookfieldresults)

Power (RAE results)

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6. Discussion of Results, Conclusions and Recommendations 6.1 Discussion of Results The experimental results obtained from week 1 by means of rotational viscometer have no readily available empirical correlations to deduct the rheological constants, and therefore are not used for analysis. This is because the UL adapter wasn’t used and the ratio of the radius of spindle Rb over the radius of cup Rc wasn’t less than 0.1. The RAE results from week 1 have demonstrated shear thickening effects that contradicts to what was observed in the rotational viscometer, and thus were not used as well.

Week 2 results indicate that the guar-xanthan solution has a flow index n of 0.6551 and 0.6778 from RAE, and rotational viscometer analysis. The observed effect is that as the shear rate is increased, the viscosity is decreased. This type of non-Newtonian shear thinning behavior is consistent with the pseudoplastic fluids defined in the theory section. The consistency factors K for both techniques from figures in the previous section are where they differ by more than 20 times. They are the result of a combination of experimental error and measurement error. For the most part the experimental error comes from the drifting steady state in RAE where the flow rates and temperature are difficult to control. The measurement errors are also more significant in RAE because of two separate sources. The first source is due to multiple measurements of pressure drop, mass, time. These measurements were also recorded by two separate persons, therefore making it the second source. However, the consistency factor K does not have the dominant effect on the shear stress and the viscosity compared to the flow index n, which is the exponent of the power law correlation.

The rotational viscometer is also used to test the effect of temperature of the polymer solution. This is done with a hot and cold water bath to the cup in which the spindle and its UL adapter is immersed. It is observed that the as the temperature is increased, the viscosity decreases and the solution becomes thinner. The effect of hot water bath was given at 27 ºC above where the flow index n increased from 0.6778 to 0.6834, indicating a slower drop in viscosity as the shear rate is increased; the consistency factor K decreased from 4.9019 to 4.3930, reducing the overall scale of viscosity. Because the room was conditioned to 21ºC and the outside temperature was about 25ºC (78ºF), the cold water from tap had a warmer temperature and made the solution equilibrate at about 25 ºC. The flow index n increased to about 0.73 while the consistency factor K dropped to about 3.6, indicating an even smoother drop in viscosity with the scale reduced the to the minimum with respect to changes in the shear rate. Overall the pattern is within expectation only that a lower temperature wasn’t achieved to give a broader range to investigate.

The experimental errors in the measurements recorded in the rotational viscometer are less significant because of the computerized set up for inputs and outputs, as well as its delicate scale. The source of its error is predominantly from the imperfection of the instrument. There are also measurement errors in determining the polymer density as well as its content; they include uncertainties in mass, volume. Replications in these measurements have statistically replaced these measurement errors with the notably bigger random errors.

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6.2 Conclusions

The rheological constants obtained from both techniques indicate shear thinning non-Newtonian behavior in consistency with the theory. The difference in the consistency factor K is the result of equipment scale and easiness to operate. These contributed to the experimental and measurement uncertainties of the rheological properties. The precision levels of both methods are equally high under the same temperature of 21 ºC.

The effect of temperature on the viscosity and rheological constants have shown thinning of the solution with increased temperature. Due to the range of temperature tested being from 21 to 27 ºC with only 6 ºC difference, the behavior of shear stress vs shear rate are relatively close to each other. The rheological constants obtained from both techniques have less than 1% of uncertainty and thereby not included their corresponding values.

The RAE is a realistic model to perform testing for scaling up in bigger plants. It takes into account the complexities of the bigger process where many equipments, bends, valves and pumps are involved. These process components make the thermodynamic steady state have sizable fluctuations, as well as taking longer to achieve. Therefore it is necessary to use such model when the objective desires such.

The rotational viscometer is an economic and useful alternative for obtaining the generic property information prior to doing testing on a bigger model like RAE. The results have been consistently repeatable and precise. It did however produce a different correlation for the rheology of the guar-xanthan solution than that of the RAE. Knowing the limitation and the specifications of the design process are therefore the determining factor on choosing the testing method. This method would also be suitable for processes that are small and simple in scale.

6.3 Recommendations

For improvement on data accuracy and precision, it’s best to have a single person in charge of running the RAE, verifying the steady state and taking measurements. The same should be done for using the rotational viscometer as well. The third person can be in charge of data analysis, recording experimental procedures, and giving recommendations to the operations of either teammate.

To have a broader range of temperature effects on the rheology of the guar-xanthan solution, ice bath can be used on the testing with the rotational viscometer. Making sure the solution is thoroughly mixed before any testing is vital to the quality of data as well.

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7. Design Problem

7.1 Introduction

The Uni-Minn development corporation was asked to design an efficient piping system for the transport of a 1% guar and xanthan blend to future use in the client’s personal care products. The client required for the solution to be transferred through 2000 feet of piping at flow rate of 120 gallons/minute from an open storage tank to a target tank 65 feet above the initial tank. Some of the scale-up design specifications were optimal pipe diameter, pump characteristics, and required instruments, valves or fittings. A process flow diagram for the suggested piping system is shown below in figure 7.1.1.

V-1

V-2

V-3

Storage Tank 1

Storage Tank 2

T1

T2

Rotameter

Positive displacementPump

P-9

2000 feet

65

fee

t

15

fee

t

Figure 7.1.1 Process Flow Diagram for the scale-up design

7.2 Design Parameters

According to the specifications given by the client, two tanks were identically sized to store one day’s supply. To avoid overflow and spilling the storage tanks’ size was established to be 10% greater than the total volume of a day’s supply. As shown in Figure 6.1.1 the positive displacement pump was placed 15 feet below the storage tank to provide the necessary suction head to avoid cavitation. The pump is used to provide the mechanical energy required to transfer the polymer solution from one tank to the other. A recycle stream was added to direct flow back to tank 1, this was done to control flow rate and pressure. Three gate valves were placed in the design, the first one just before the pump, the second one leads to the recycle stream, and the last one before the target tank. Furthermore two temperature gages and a rotameter were placed to monitor temperature in both tanks and measure the flow rate.

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7.3 Design Calculations

Design calculations were done transferring the results from lab using the Rheological Analysis Equipment (RAE) due to the similarity of the apparatus with the design requirements. The approximations done in the scale-up design were the following, laminar flow remains therefore the superficial velocity can be approximated to be the fluid’s velocity. The temperature change is negligible through the piping system, both tanks are well mixed, and there is zero polymer degradation because the solution is run only for a day. For sakes of accuracy and precision only week 2 data was transferred to scale up.

Two identical storage tanks were sized to a 110% of the total volume of a day’s solution supply. The calculated diameter was 25.3ft, the height 50.6ft and a freeboard of 20in, equaling a total storage volume of 190,290 gallons. Local provider USA Tank provides tanks this size for a price of $830,800.(4) Using equation 1.3.1 the optimal pipe diameter was calculated to minimize the total cost; a schedule 40 steel 6in diameter pipe was decided due to commercial availability. To fulfill the 2000ft required piping, ninety eight 21ft pieces are provided by Metals Depot, the total cost of piping was calculated to be $45,667. (5) Equation 1.3.2 was used to calculate the Reynolds number and verify the flow remains in the laminar region. A NRe of 1,655 resulted, which proves the flow is still laminar. The velocity correction factor was found to be 0.306 by equation 1.3.5.

The work shaft necessary for the pump was calculated using the mechanical energy balance described by equation 1.3.4. Friction losses through the pipe due to flow were added up with losses due to valves and fittings, and due to sudden contraction of the fluid. These calculations resulted in a required shaft work of 72.32 lbf ft/lbm, approximating a pump efficiency of 75% (1) along with a calculated mass flow rate of 22.67lb/s the brake horse power of the pump was found to be 4HP with an electrical need of 4KW. Grainger provides a positive displacement gear pump for the price of $3,572 which operates at 250RPM. (6) The total initial investment for the essentials of the scale up design, not including recycle stream, valves, and a rotameter is of $1,710,000.

Similar calculations were done for water to study the difference in behavior with the guar-xanthan solution in the scale-up design. For water, the Reynolds number was found to be 197,446 which implied turbulent flow. The pressure losses in the system translated to a shaft work of 67.9 lbf ft/lbm needed for the pump in this case. Using 75% efficiency and a mass flow rate of 16.67lbm/sec a pump of 3HP would be recommended.

Refer to Appendix C for sample calculations.

7.4 Non-Newtonian and water comparative

The comparison between water and the guar-xanthan solution presented significant differences regarding the pressure losses and the flow behavior. First off, the Reynolds number was much greater than that of the non-Newtonian fluid, the water itself presenting turbulent flow. This turbulence refutes the approximation that the superficial velocity is equal to the velocity of the fluid. Degradation is not an issue for water and therefore turbulent flow is allowed. Also, a lower pressure drop was calculated using water. This is due to the large difference between real and apparent viscosities of the two solutions.

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8. Nomenclature Symbols listed in the order in which they appear. : Shear stress

: Shear rate

Viscosity Consistency index

Flow behavior index Apparent Viscosity

Pipe diameter Pressure drop

Length of pipe Pi

Fluid linear velocity Generalized viscosity coefficient

Radius of spindle Radius of outer cylinder Angular velocity

Revolutions per minute Torque

Length of spindle Optimal pipe diameter

: Cost of electricity Mass flow rate

Working hours a year Annual fixed charges

Purchase cost of piping per feet of pipe Constant for material determined for steel pipes

Pump efficiency Density

Ratio of total cost of fittings and insulation bought per feet of pipe Reynolds number

Velocity correction factor Shaft work

Velocity Height ∑ Friction forces Compressor efficiency

Generic loss due to friction Real shaft work

Measured error Standard deviation

Value of measurement Average of experimental values

Error limit

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9. References

1. Geankoplis, Christie, “Transport Processes and Speration Process Principles,”

4th ed., Prentice Hall, Upper Saddle River, NJ (2003).

2. University of Minnesota, CHEN 4401W “Non-Newtonian Flow References”

3. University Of Minnesota, CHEN 4401W “RAE Instruction Manual”

4. USA Tanks, “USA Storage Tank Sales”

http://usatanksales.com/industrialwater.html, and phone quote (accessed Sept.15 2013)

5. Metal depot, “The Metal depot online”, https://www.metalsdepot.com/ (accessed

Sept. 15 2013)

6. Grainger, “Positive displacement gear pump”

http://www.grainger.com/Grainger/ecatalog/N-1z0dvzg (accessed Sept. 15 2013)

7. Peters, M.S. and Timmerhous, K., “Plant Design and Economics for Chemical

Engineers,” 3rd ed. McGraw-Hill Companies, NY.

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D. Error Analysis

D.1 Sources of errors

All measurements were associated with errors. For all measurements, the true value was unknown, so the true error was not known. Calibration was not performed in this experiment and the systematic errors were not considered (calibration for the Brookfield Viscometer might be needed but not a consideration for this lab). The most important sources of error were from the random errors caused by time-varying phenomena in the instrument, the surroundings and users, and the quantity being measured. The instrument sources of errors were as follows:

Sources of errors, the errors in physical quantities measured in this lab included the error in time, weight tank, pressure, Brookfield Viscometer, density and polymer content, each discussed as follows.

The flow rate was measured using the stopwatch-weight tank method. Due to the inaccuracy of human operating with stopwatch, the error in time measurement was 1s. Due to the fluctuation of electronic scale during measurement, the error in weight measurement was determined to be 0.4lbs. These two errors were accounted for by the replications.

Pressure had an error of 0.5mm the fluid in the RAE. This error was also accounted for by making replications.

The RPM had an error of 1%. The error on the reading torque was .1%, and the error on viscosity varied with the value of the viscosity. The bigger the viscosity value, the bigger the error.

Error in weight measurement was the fluctuation at the last digit of the scale: 0.0005g, and was used to determine the error on polymer content. Error in density was accounted for by replications.

D.2 Experimental uncertainties estimates with replications

In this experiment, most random errors, including error in flow rate, pressure, and density were reduced by replication.

When multiple replications were obtained, the standard deviation was calculated.

Equation

√ was used in combination with student’s T table to get an estimate on the

true error.

For the density measurement, five values obtained were 1100, 1080, 1080, 1080, 1100.

√

For 4 degrees of freedom with 95% confidence level, t=2.776

Use equation

√ to get

√

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So the density is 1088 14

Mass of wet polymer solution was measured directly as 11.75 0.0005g. Mass of dry

polymer was (4.7600 0.0005)-(4.7100 0.0005)g=0.0500g √

%polymer=

√(

)

For run 2-1-1 as in the original data sheet, in the 3 replications of time weight

measurement, flow rate were determined to be

,

,

and

, so the flow rate was lbs./s, which equals to

0.269 0kg/s.

V=

√

So V=

For run 2-1-1, three replications of pressure measurements were the same, so (1-2) =

(650 0)-(370 ) =280 √ mm polymer fluid

So )=2987.5 pascal

Use the same method to get )= 3094.2 39.8 Pascal

All error for measurements on torque was determined to be .1%, which was

D..3 Propagation of error (calculated according to sample calculation)

For run 2-1-1

So

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Shear rate

Shear rate

(

)

(

)

The correlation between the shear stress and the shear rate are from plot of logarithmic scale where the trend line is fitted to a power law behavior. The exponent n and the constant K are obtained this way such that the uncertainties are from regression alone. The regression analysis for the constants K and n are so small that are ignored as discussed in the final results section. The above equation reduce to

For run 2-2-3 of the Brookfield viscometer, the rpm has an error of 1% of the reading

The shear rate

(

)

For the shear stress,

This error was ignored to the 1st digit after the digital point.

For the Reynolds number using run 2-1-14,

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Using the formula when , b=constant

√

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