Application of Rheological Data for Non-Newtonian Sludges; Use of the Differential Viscosity for Mixing Simulations and System Friction Calculations Marilyn Pine 1, Kent Keeran 1, Glenn Dorsch 1 1 Vaughan Company, Inc., Montesano, WA ABSTRACT Proper sizing of mixing and pumping systems requires a good understanding of the viscosity of the unique sludge or slurry under consideration. A Newtonian fluid is one in which the viscosity does not change with shear rate; its viscosity can be described with one value. However, because of the solid components present, many biological slurries (TWAS, digested sludges, primary sludges, etc.) are non-newtonian in nature. By definition, a non-newtonian fluid is one whose viscosity is not constant and which varies with shear rate. Its viscosity can no longer be described with one value and must be described as a set of values, or as an equation which calculates viscosity as a function of shear rate. This relationship between viscosity and shear rate can be used as an input into a Computational Fluid Dynamics (CFD) simulation to predict fluid velocities and mixing effectiveness or it can be used to approximate frictional losses in piping systems. Viscosity as a function of shear rate can be derived from a fluids flow curve. The flow curve is a set of data points which plot measurements of shear stress versus shear rate. This data can be obtained from a rotational viscometer or ideally, a rheometer. The most commonly used convention for describing the viscosity of a non-newtonian fluid is with an expression for apparent viscosity ; it can be calculated by taking a shear stress value at any point on the fluid s flow curve and dividing by the corresponding shear rate at that point. This is essentially the slope of a line from any given data point back to the origin. This method is typically used since it can be calculated from a single data point and can be measured in the field at a given shear rate. Experience with simulations and actual observed results over the last 10-13 years, however, has shown that using apparent viscosity values or equations in CFD simulations of mixing tank applications can lead to non-realistic and overly conservative results which might predict the need for higher mixing power than necessary. This is especially true for fluids with higher yield stresses and higher viscosities (typically sludges with higher percent solids and/or with polymer thickening). This paper discusses an alternate method to calculate viscosity by defining an equation for the actual slope, or first derivative, of the fluid s flow curve to describe the fluid s differential viscosity at varying shear rates. Better correlation has been found between simulated results and actual observed system performance when using this method for predicting mixing system velocities. This method requires a complete set of flow curve data in order to determine the true slope, and thus viscosity, at any given shear rate. Figure A1 demonstrates flow curve data (shear stress versus shear rate measurements) for a non- Newtonian slurry. The apparent viscosity at Point A is calculated as the slope of line A1 which extends from Point A back to the origin. The differential viscosity at point A is calculated as the slope of line A2 which is tangent to the flow curve at point A. Note that both the apparent and differential viscosity values vary at different shear rates. 496
Figure A1: Comparison of Apparent Viscosity versus Differential Viscosity at Point A Understanding the rheology of each unique application is important to predict how pumping and mixing systems will work in the field and to avoid overestimating the amount of energy usage and power required. Lessons learned over the last decade have helped to refine techniques for interpreting and applying flow curve data for sizing mixing and pumping systems. Figure A2 below illustrates CFD results from a study where the actual viscosity of the thickened sludge being introduced into a mixing system was measured. The application of field data and differential viscosity provided the best representation of actual observed mixing velocities and system performance. Figure A2: Comparison of Simulation Results for Customer Mixing System This paper will discuss: The importance of accurate flow curve data over a wide range of shear rates. How apparent and differential viscosity can be calculated and an example of how the interpretation of the flow curve data can impact results, in some cases dramatically. Examples of real-life applications and physical validation. How the differential viscosity can also be applied to frictional losses in piping. KEYWORDS Non-Newtonian, differential viscosity, apparent viscosity, mixing, CFD 497
1.0 Introduction Knowing the unique rheological properties of a given sludge or slurry is important to understand when sizing pumping and mixing applications. Sludge viscosity will directly impact CFD simulations for mixing systems and system head loss calculations for pumping. Because the fluids encountered in these mixing systems tend to be non-newtonian with shear thinning behavior, regions in the tank that are moving slowly have the highest local viscosity and therefore are the hardest areas in which to generate mixing. On the other hand, once the thick fluid starts to move and the local shear rate increases, the corresponding fluid viscosity in that region drops and it is subsequently easier to mix. Because the rheological characteristics of sludges can vary dramatically depending on percent solids, chemistry, temperature, processing, polymer addition, etc., the fluids that required to be mixed can sometimes be so viscous that they are very difficult to mix or pump. Depending upon each fluid s unique flow curve, mixing thick blended sludges can be a challenge, and one must be very careful not to undersize the mixing systems. However, oversizing can lead to increased power requirements and improper pump selection. This paper summarizes lessons learned over the last 10-13 years in applying rheological data to mixing systems and their simulation to achieve representative results. 2.0 Rheology Background 2.1 Flow Curves; Newtonian and Non-Newtonian Fluids Viscosity is a measurement of a fluid s resistance to flow or movement. It is derived from a fluid s flow curve (or rheogram), which is a plot of shear stress () measurements at different shear rates () (see Figure 1). Viscosity is essentially the slope of the curve. A constant slope throughout the entire flow curve indicates a constant viscosity at all shear rates. Fluids with this behavior are classified as Newtonian fluids; examples include water, light motor oils, and sludges with very low percent total solids (TS). In CFD simulations, a Newtonian fluid s dynamic viscosity can be described by one constant value for all shear rates within the fluid domain. However, due to the presence of a higher percent solids component, many biological slurries behave as non-newtonian fluids. The shape of their flow curve is no longer a straight line; the slope of the curve changes with shear rate. In this case, viscosity is not constant and is a function of how fast the fluid is moving (shear rate). For a mixing simulation of non-newtonian fluids, the dynamic viscosity can no longer be defined by one value and it is necessary to create an expression which describes the local dynamic viscosity as a function of the local shear rate, or how fast the fluid is moving in each area of a mixing tank. In Figure 1 below, the slope of the upper non-newtonian flow curve rapidly increases at the left hand side of the plot, at low shear rates. This higher slope correlates with higher viscosities at low shear rates, as shown in Figure 2, and directly impacts the amount of energy required for mixing. Typically as the shear rate increases, the slope of the non-newtonian flow curve begins to decrease, resulting in a similar decrease in viscosity. The rate at which this viscosity decreases with shear rate will impact the ability to pump the fluid and to accommodate the associated head loss and resistance to flow in the piping system. 498
Mixing Tanks Pipe Flow Shear Stress Non-Newtonian fluid (variable slope) Newtonian fluid (constant slope) 1 10 Shear Rate 100 Figure 1: Schematic of Fluid s Flow Curve (Hilts et al., 2014) (Shear Stress vs. Shear Rate ) Figure 2: Viscosity vs. Shear Rate (Hilts et al., 2014) 2.2 Comparison of Typical Rheological Models Non-Newtonian fluids are often characterized depending on how well their flow curve data appears to fit a given equation or rheological model. Some of the more common models are shown in Figure 3. 499
Shear Stress = K n (Power Law) = A + B n (Sisko) = y + K n (Herschel-Bulkley) = y + K (Bingham Plastic) y = K (Newtonian) Shear Rate Figure 3: Common Non-Newtonian Rheological Models Depending upon the particular type of slurry or sludge, one form of equation may best fit the data for one sludge or for a given range of shear rates, but not another. Three commonly used rheological models are the Bingham Plastic, Power Law, and Herschel-Bulkley. They have the following forms: Bingham Plastic K [1] y Power Law n Herschel-Bulkley n K [2] y K [3] where: = shear stress (Pa) y = yield stress (Pa) = shear rate (s -1 ) K and n are fitting coefficients (with appropriate units) It is important to note that these models are simplifications to characterize a given fluid; it can be difficult to fit a set of measurements neatly to any one model. Additionally, one form of equation may fit one portion, or shear rate range, of data and another may best describe another range of the data. 2.3 Apparent Viscosity As discussed above, because viscosity varies for different shear rates for non-newtonian fluids, the viscosity varies with the slope of the flow curve for different shear rates. The conventional way to describe the viscous behavior is by measuring the shear stress at a given shear rate and reporting what is defined as the apparent viscosity. This is merely the measured shear stress divided by the corresponding shear rate (A) at which it was measured. A A apparent [4] A = dynamic viscosity (Pa s) = shear stress (Pa) at a given shear rate A = shear rate (s -1 ) at a given shear rate A Or, given an equation ) to describe shear stress as a function over a range of shear rates: 500
apparent evaluated at varying values of shear rate. [5] 2.4 Differential Viscosity Another method to define viscosity is the differential viscosity which is defined as the derivative of shear stress with respect to shear rate. It is found by calculating the slope of the tangent to the flow curve at a given shear rate (the first derivative of the flow curve). d diff [6] d 2.5 Graphic Comparison/Visualization of Apparent and Differential Viscosity The following figure and discussion from Rheology Modifiers Handbook: Practical Use and Applications provides an excellent description and schematic of the apparent viscosity and the differential viscosity discussed above. Figure 4: Definition of Apparent and Differential Viscosity (Braun & Mosen, 1999) The differential viscosity is equal to the slope of the shear stress versus shear rate curve at some point A (or the tangent of the angle ). The apparent viscosity is equal to the slope of a line that connects the origin with a given point A on the shear stress versus shear rate curve (or the tangent of the angle ). Of the two methods for expressing the coefficient of viscosity, the apparent viscosity is usually chosen. This is because an apparent viscosity is easily measured at one fixed shear rate while a differential viscosity requires measurements at several shear rates followed by measurement of the slope at the shear rate of interest. (Braun & Mosen, 1999) 3.0 Validation of Mixing Simulation Results with Physical Observations 3.1 Early Experiences with non-newtonian CFD Mixing Simulations One of the first experiences with a thickened blended sludge simulation arose with a mixing system in New Zealand. The use of the apparent viscosity function (derived from non-newtonian flow curve equations provided by the owner) resulted in simulations depicting very poor mixing velocities. As a result, the proposed pump size, mixing flow, and horsepower were increased to ensure that desired mixing velocities and flow patterns would match typical supplied mixing systems. However, subsequent startup results and testing indicated that these tanks actually were mixing at much higher velocities than originally simulated. A re-simulation, at a later time, with the differential viscosity method demonstrated a much more realistic assessment of the mixing system. In hindsight, it is generally agreed that these systems were most likely oversized based on the results obtained with the apparent viscosity method. More recent systems installed at the same location were designed with less power and flow. 501
3.2 Simulation of TWAS Blend Tank and Proposed Modifications The use of a differential viscosity equation for another owner helped to provide a much more realistic assessment of their system and a realistic understanding of how proposed modifications would improve the mixing. Their installed 7 m (23 ft.) diameter x 6.1 m (20 ft.) high cylindrical blend tank mixing system was not providing visible mixing at full tank level when sludge viscosity was at its highest. Once alerted to the problem, CFD modeling was used to investigate the system and to make recommendations for improvement. This was a good opportunity to evaluate the mixing system with blended sludge rheology measurements obtained from their facility. The particular fluid that was desired to be simulated was polymer thickened waste activated sludge (TWAS); this type of sludge commonly exhibits a very high yield behavior at the left side of the flow curve. These blend tank mixing systems can be very challenging and may require extra mixing energy to overcome high viscosity at low shear rates. 3.2.1 Field Measurements of TWAS The on-site consulting engineer was able to take some shear stress measurements with a rotational viscometer at varying shear rates. The results of two combined days of samples are shown in Figure 7, plotting shear stress (Pa) versus shear rate (s -1 ). Although it would have been desired to have more flow curve data at higher shear rates, the data allowed the generation of an equation to describe the flow curve at lower shear rates. This equation was then used to input a differential viscosity formula into various CFD simulation trials to evaluate ways to improve the mixing system. These measurements from an actual process sample also helped with comparison of the rheology of this TWAS sample with other sludge data and to understand the magnitude of the resistance to flow at low shear rates. Figure 7: Field Measured Rheology Data Sept 2006 Rotational Viscometer Flow Curve Data, 6-7% TWAS Sample (Bishop et al., 2007) In this example, the flow curve was chosen to be represented by the power law equation [2]: 0.1 ( ) POWER 29.1 and its corresponding first derivative has the form shown in [6]: d n 1 0. 9 ( ) diff nk 2.91 (Differential Viscosity, Pa s) d The conventional method of deriving apparent viscosity [5] yields 0. 9 ( ) app 29.1 (Apparent Viscosity, Pa s) 502
Depending upon the method used to derive an expression for viscosity, different equations were input into the CFD analysis for comparison. 3.2.2 CFD Analysis Results of Initial Mixing System Initial simulations provided a baseline for comparison with the actual observed mixing system. Prior to any modifications, the tank was not able to be mixed when full of TWAS; reported surface mixing could only be seen to levels of about 2.1 m (7 ft.). Figures 8a and 8b are contour plots of fluid velocity through a vertical cross-section of an initial mixing tank configuration with a simulated fluid depth of 6.1 m (20 ft.). Even though this was a challenging thickened sludge to mix and both simulation results demonstrated poor mixing, the differential viscosity results were more representative of actual observations. Simulation results were useful in helping the owner to understand that even though they may not have seen surface movement as the tank levels increased, the fluid was still being mixed in the lower regions of the tank. Figure 8a: Differential Viscosity Figure 8b: Apparent Viscosity Figure 8: Fluid Streamlines and Velocity Contour Plots through Cross-Sectional Planes. Baseline Trials on Initial Mixing System: 193 m 3 /hr, 11kW (850 GPM, 15HP). Simulated Fluid Depth = 6.1 m (20 ft.). 3.2.3 CFD Analysis Results after Final Changes Made to Modify System Various changes were proposed to improve mixing using the same pump and pump horsepower. Different alternatives included evaluating different numbers of nozzles, nozzle diameters, nozzle aiming, etc. Figures 9a and 9b show simulated results with the same 6.1 m fluid depth for the final modified system. Some of the proposed changes included doubling the number of nozzles in the mixing system while adjusting the nozzle exit diameter to keep the pump operating in an acceptable portion of the pump curve, as well as re-aiming nozzle angles and adding an elbow to help direct more of the flow towards the upper regions in the tank. The simulated flow and horsepower correspond to the current modified system in operation. Again, based on actual observations of the mixing system at various operating levels, the results with the differential viscosity more closely matched with actual system performance and mixing after the modifications were put into place. Surface mixing was reported at a depth of 3.7 m (12 ft.); the apparent viscosity results were not an accurate portrayal of the system in operation. 503
Figure 9a: Differential Viscosity Figure 9b: Apparent Viscosity Figure 9: Fluid Streamlines and Velocity Contour Plots through Cross-Sectional Planes. Final Modified Mixing System: 239 m 3 /hr, 11 kw (1050 GPM, 15HP). 3.2.4 CFD Analysis Results of Estimated System to Provide Full Mixing A question also arose as to how much horsepower and flow might have been required to completely mix these tanks completely full with 100% TWAS. Figures 10a &10b below show the results of simulations with increased mixing flow and power. As expected, the result with the differential viscosity shows higher mixing velocities at all levels in the tank as the flow and power are increased. Results using the apparent viscosity show that the contents of the tank are still virtually at a standstill, except for small high velocity areas directly next to the nozzle outlets. If the apparent viscosity results were relied upon, the amount of power required to fully mix these tanks would have been significantly overestimated. While the owner did not upgrade the system as shown in Figure 10, these simulations were useful in understanding how much mixing energy and power might be required for future tanks with thickened sludges having a similar rheology. Figure 10a: Differential Viscosity Figure 10b: Apparent Viscosity Figure 10: Fluid Streamlines and Velocity Contour Plots through Cross-Sectional Planes Alternate Proposed Mixing System: 427 m 3 /hr, 30 kw (1880 GPM, 40HP). 504
3.3 Example Mixing Simulation Results for an Anaerobic Digester From about 2006 forward, the goal has been to get the best understanding of each unique sludge to be mixed, to obtain any actual shear stress versus shear rate measurements, and to develop an expression for differential viscosity for input into mixing simulations. Continued feedback on mixing test results, videos, and field observations have helped to confirm that simulated results based on the differential viscosity method better correlate with actual performance. In a typical anaerobic digester, the range of TS and viscosity of digested sludges is typically less than those seen with thickened WAS sludges as discussed in Examples 3.1 and 3.2. Figure 11 plots a couple of representative power law flow curve equations which were chosen to qualitatively illustrate the difference in mixing simulation results for a digester application. 12 10 Shear Stress vs. Shear Rate Sample 4% Digested Sludge Flow Curve Sample 2-3% Digested Sludge Flow Curve Shear Stress (Pa) 8 6 4 2 = 5 0.15 = 1.7 0.34 0 0 10 20 30 40 50 Shear Rate (sec -1 ) Figure 11: Sample Digested Sludge Power Law Flow Curves (2-3% and 4% DS) The difference in simulation results when comparing the differential and apparent viscosity methods is not as pronounced for lighter sludges and fluids with lower TS as shown below for the chosen 2-3% digested sludge in Figure 11. (Power law equation where = 1.7 0.34 ). The tank diameter chosen for this particular simulation was 33.5 m (110 ft.). Figure 12: Velocity Contour Plots and Streamlines for Anaerobic Digester Mixing System, Estimated 2-3% Digested Sludge. (Images courtesy of the City of Calgary) Left (12a): Differential Viscosity, Volume Average Velocity = 0.20 m/s Right (12b): Apparent Viscosity, Volume Average Velocity = 0.15 m/s 505
However, because average shear rates in these mixing simulations are typically very low, on the order of 1 s -1 or less, a change in yield stress has a marked difference on viscosity at low shear rates and the difference between the two viscosity methods becomes more pronounced. Figure 13 illustrates how overall mixing velocities within the tank have slowed when a sludge with the sample equation for 4% digested sludge ( = 5 0.15 ) is considered, as well as how the difference in results between the two viscosity methods is now more significant. Figure 13: Velocity Contour Plots and Streamlines for Anaerobic Digester Mixing System, Estimated 4% Digested Sludge. (Images courtesy of the City of Calgary) Left (13a): Differential Viscosity, Volume Average Velocity = 0.17 m/s Right (13b): Apparent Viscosity, Volume Average Velocity = 0.03 m/s Again, the different results shown in Figures 13a and 13b were derived from the same power law flow curve equation for 4% digested sludge. The results only differ depending upon which method was used to derive an expression for viscosity as a function of shear rate from the original power law flow curve. Based on experience with numerous digester mixing systems, the results simulated with an apparent viscosity derived from the 4% DS flow curve equation is not an accurate portrayal of how these systems actually perform or mix. 4.0 Obtaining Good Rheology Data for Calculation of Differential Viscosity 4.1 Non-Newtonian fluids cannot be described by one viscosity value Most wastewater sludges with TS values equal to or greater than 2-3% typically behave as non- Newtonian fluids; their viscosity cannot be described by one value. Because their viscosity varies with shear rate, the calculation of viscosity first requires a set of shear stress versus shear rate measurements. These values must be obtained from an instrument that is able to measure shear stress for a corresponding shear rate; equipment that assumes the fluid is Newtonian and only provides one viscosity value with no reference shear rate may not adequately describe the fluid s rheology. Shear stress versus shear rate measurements may be obtained from a rotational viscometer (example data shown earlier in Figure 7) or more ideally, a rheometer which can obtain more detailed shear stress data over a wide range of shear rates. Figure 14 below demonstrates rheometer data which is a good representation of an ideal set of measurements for a TWAS sample. From these data points, one can roughly estimate the magnitude of resistance at low shear rates, evaluate the rate of curvature at higher rates, as well have enough data points to aid in deriving a flow curve expression. 506
45 Example of Stepped Flow Data Site J: 4.6% TWAS (GBT) Shear Stress (Pa) 40 35 30 25 20 15 10 5 Stepped Flow Data - Step 2 0 0 10 20 30 40 50 60 70 80 90 100 Shear Rate (s -1 ) Figure 14: Sample Flow Curve Data obtained from Rotational Rheometer (Hilts et al.,2014) 4.2 Assume that each sludge is unique, because it probably is In general, the yield stress and viscosity of biological slurries tend to increase with higher percent solids. All sludges are unique and their differences can become increasingly more pronounced with the type of slurry, percent solids, processing, etc. Polymer thickened sludges may tend to have the most unpredictable and variable rheologies, in terms of both yield stress behavior and resistance to flow at higher shear rates. This is especially true as the percent solids increases for > 5% thickened blended sludges (Hilts et al., 2014). Whenever possible, it is best to generate a flow curve from a representative sample of the sludge under study or consideration. Because the viscosity will vary with temperature, one should also try to test at a representative process temperature as well. Flow curves contain a wealth of information. Shear stress measurements at very low shear rates (for example, 1 s -1 or less) are very useful for evaluating a sludge s behavior when mixing; measurements at higher shear rates (up to 100 s -1 ) will help one to understand how the same sludge might react in a piping system at higher velocities. Additionally, each flow curve will tend to have a different rate of increase at higher shear rates. If a power law equation has been used to describe the data, higher exponent values will result in flow curves corresponding to fluids which do not decrease in viscosity as much at higher shear rates. This will be indicative of a fluid which may be harder to pump with associated higher friction losses. 4.3 Derive representative equation to use over shear rate range under consideration In some cases, especially as the percent solids increases or with polymer thickening, one may not be able to come up with one-size-fits-all equation that fits the whole range of data. It may be preferable to have a couple of different flow curve equations depending upon the region of shear rates that one is interested in. For example, a Bingham Plastic (BP) straight line equation is often a very good fit for data at higher shear rates, however it typically will not correctly define the viscous behavior at low shear rates; a power law or other equation might be a better choice. Figure 15 demonstrates how a BP equation fit to the sample data shown previously in Figure 14 may be a good approximation at higher shear rates but not for the lower shear rates associated with mixing. Depending upon the magnitude of a sludge s yield stress and slope of the equation, the BP equation combined with the apparent viscosity method may be overly conservative for mixing simulations. Also, one should try to keep all raw data for future reference; do not try to oversimplify the data. Once an equation has been fit to data and the original underlying data is gone, it is impossible to know how well the final flow curve equation described the original data. 507
Shear Stress (Pa) 45 40 35 30 25 20 15 Example of Flow Curve Data Site J: 4.6% TWAS (GBT) = 29.5 + 0.15 10 Stepped Flow Test Data Points 5 Example Bingham Plastic eqn to fit data 0 0 10 20 30 40 50 60 70 80 90 100 Shear Rate (s -1 ) Shear Stress (Pa) 45 40 35 30 25 20 15 = 17 0.2 Example of Flow Curve Data Site J: 4.6% TWAS (GBT) 10 Stepped Flow Test Data Points 5 Example Power Law eqn to fit data 0 0 10 20 30 40 50 60 70 80 90 100 Shear Rate (s -1 ) Figure 15: Flow curve data from Figure 14; Data fit with BP and Power Law equations. 4.4 Differential Viscosity is First Derivative of Flow Curve While a value for apparent viscosity can be reported with one shear stress measurement at a particular shear rate, the calculation of differential viscosity requires a series of measurements such that the rate of change in the relevant shear rate range can be determined. The use of a yield stress term (such as in the Bingham Plastic [1] or Herschel-Bulkley [3] equations) may make it difficult to calculate the initial slope of the data and to accurately model the fluid s resistance to movement at very low shear rates. Ideally, the original flow curve data points can be used instead to derive an equation which passes through zero and does not contain a yield stress term, such as with a Power Law equation. Sometimes it can be challenging to find an expression which fits the data and also which can be differentiated. As an example, one form of flow curve equation which may be used and which often helps to fit non-newtonian sludge data for a wide range of shear rates is in the form of the Sisko equation as shown on Figure 3: n ( ) A B where n1 d ( ) diff An B [7, 8] d 5.0 Application of Flow Curve Measurements and Differential Viscosity to System Friction Loss Calculations System calculations, even for a system assumed flowing with water or a very light sludge can be very challenging. Depending on the piping material, age of piping, fittings and their placement in proximity to each other, different reported loss coefficients, etc., there are a number of sources of variation which affect friction loss calculations in a piping system. For sludges with higher TS, it is generally assumed that viscosity will start to play a larger role in head loss due to friction. If a strict multiplier (based solely upon percent solids) is applied for head loss calculations, its use may lead to overly conservative estimates because it does not consider the decrease in viscosity at higher pipe velocities with a non-newtonian fluid. Also, just as with mixing simulations, as one starts to consider sludges with higher TS and/or the effect of polymers for thickening, the point when the viscosity will start to have a significant impact on friction loss calculations may also be when the actual sludge rheology is unpredictable and historical measurements from other sludges are less reliable. One way in which the differential viscosity approach has been used is by applying a given sludge s flow curve equation, when available, to estimate head loss due to 508
the friction via the steps below. Using the differential viscosity rather than apparent viscosity may give a slightly lower estimate for head loss. 1. Calculate average velocity V in each piping segment or component based on pipe diameter D and pumping flow rate Q 2. Estimate shear rate (s -1 ) in each segment of piping Although the velocity profile across the interior cross-section of a pipe varies with distance from the wall, a simplified method of estimating the shear rate at the wall for a pipe of a given diameter is based on average pipe velocity: 8 V Shear Rate N [9] D Note that [9] calculates average pipe shear rate N based on an assumed velocity distribution for laminar flow and a Newtonian fluid. The actual wall shear rate for pipe flow is higher for non-newtonian fluids but also very difficult to measure and quantify. The Rabinowitsch-Mooney equation adds a correction factor for non-newtonian fluids with assumed laminar flow (Nguyen et al., 2012). For a power law fluid: 8 V 3n 1 Shear Rate RM D 4n [10] where n = the exponent in the power law equation = K n. Corrected shear rates will be higher than for those using [9] and will increase as the exponent in the power law equation becomes smaller, indicative of more shear-thinning behavior and reduced viscosities at higher flows. Estimating shear rate with [9] should result in a lower estimate for shear rate and a higher limit for predicted head loss. 3. Calculate dynamic viscosity at the shear rate using differential viscosity function 4. Convert dynamic viscosity to kinematic viscosity 5. Calculate dimensionless Reynolds number Re from kinematic viscosity, pipe velocity, and diameter VD Re [11] 6. Calculate head loss in pipe segment due to friction losses The Darcy-Weisbach equation is commonly used to calculate head loss h f due to friction in piping with a length, or equivalent length, L based upon piping diameter D, velocity V, and acceleration constant g (Cameron, 2002). 2 L V h f [12] f DARCY D 2g Equation [12] is used in conjunction with a friction factor f which varies with the type of flow (laminar or turbulent) and the relative roughness of the pipe. For assumed laminar flow (about Re <2000), the roughness of the pipe s interior surface should not affect the friction factor; the friction factor f is estimated directly from the value of Reynolds number. Laminar flow: f 64 / Re [13] DARCY 509
Head loss resulting from friction and fluid viscosity can be applied to sections of straight lengths of piping, fittings, and valves which have an equivalent length of piping. Reducers, enlargers, inlets, nozzles exits, and pipe exits which contribute to overall system head loss due to changes in velocity head are generally treated separately and not adjusted based on friction factor. 7. Sum the total losses for all segments of piping and plot for a range of different system flow rates to generate system curves. 8. Compare system curves for both assumed laminar and turbulent flows for chosen pump and application. For higher values of Reynolds number, the value of the friction factor calculated with the assumption of laminar flow gets smaller, and the friction factor for turbulent flow and the increasing effects of pipe roughness start to dominate. If the water/turbulent flow head losses begin to dominate at higher values of flow, those values are used. Where the two methods intersect the pump performance curve can serve as estimated upper and lower limits to bracket the potential head loss and flow of the final system ranging from the sludge curve estimate to the water curve estimate. This range can be compared with system requirements and capability of chosen pump. For non-newtonian sludges with assumed laminar flow, the shape of the predicted system curve will roughly correlate with the shape of its flow curve; it will depend upon the magnitude of yield stress behavior and the exponent controlling how the viscosity decreases at higher rates. Often an initial jump in predicted head loss will be apparent at very low GPM; this jump roughly corresponds to the initial yield stress of the non-newtonian fluid providing more resistance at low flow and shear rates. As viscosity drops due to non-newtonian behavior at higher shear rates, the head loss due to friction begins to drop as well. Eventually the flow transitions to turbulent flow and the system curve merges with the water curve at higher pumping flows. Many digested sludges tend to have system curves closer to the water estimate, even for higher 4-5% solids, due to the reduced viscosity at higher flows; this has been confirmed with actual pump pressure readings. Conversely, the combination of high yield stress behavior along with a high power law exponent n, as often seen with polymer thickened WAS sludges, can result in a fluid that is very difficult to get moving and whose friction losses do not decrease at higher shear rates, leading to very high friction losses in a piping system. 6.0 Conclusions and Summary This paper summarizes lessons learned over the last 10-13 years; these lessons have been applied to sizing and analysis of mixing systems, pump sizing, and hydraulic calculations in viscous slurries. Mixing these fluids can be very challenging, (for example, thickened sludge blend tanks), so it is critical to properly evaluate their expected performance. The use of an inaccurate flow curve equation or viscosity model can potentially lead to oversizing or undersizing of mixing systems and decreased confidence about the expected capability of a mixing system. Both the choice of the flow curve equation as well as the method for calculating viscosity can have a significant impact on predicted results. The choice of viscosity equation may not be as significant for some applications which have fluids that are more Newtonian in behavior or are simulated at higher shear rates. However, the derivative-based differential viscosity method may give markedly different results when considering fluids or sludges with a high yield point behavior in applications that deal with low shear rates, such as mixing systems. Based on simulation results, actual field observations, and pump pressure measurements, applying the 510
differential viscosity appears to provide much better correlation between simulated results and actual system performance. ACKNOWLEDGEMENTS The authors would like to thank the various contacts and owners throughout the years who have helped to collect sludge viscosity measurements and to provide feedback on installed mixing systems and pump performance. REFERENCES Bishop, R.; McCormick, J.; Peterson, J.; Dorsch, G.; Pine, M. (2007) Blending Raw Sludges with Pump Mix Systems, presented at 2007 PNCWA Annual Conference in Vancouver, WA. Braun, D.; Rosen, M. (1999) Rheology Modifiers Handbook: Practical Use and Applications, William Andrew Publishing, 1 st edition, Figure 1.2. Cameron Hydraulic Data: A handy reference on the subject of hydraulics and steam (2002), edited by C. C. Heald, 19 th edition, published by Flowserve Corporation. Hilts, B.; Pine, M.; Dorsch, G. (2014) Characterizing Polymer Thickened Biosolids for Pumping Applications, presented and published at WEF Residuals and Biosolids Conference 2014 in Austin, TX. Nguyen, Q. and Nguyen, N.; (2012) Incompressible Non-Newtonian Fluid Flows, Continuum Mechanics - Progress in Fundamentals and Engineering Applications, Dr. Yong Gan (Ed.), ISBN: 978-953-51-0447-6, InTech 511