The Pennsylvania State University. The Graduate School. College of Engineering PREDICTION OF CLEAN-BED HEAD LOSS IN CRUMB RUBBER FILTERS.

Transcription

1 The Pennsylvania State University The Graduate School College of Engineering PREDICTION OF CLEAN-BED HEAD LOSS IN CRUMB RUBBER FILTERS A Thesis in Environmental Engineering by Hao Tang 2008 Hao Tang Submitted in Partial Fulfillment of the Requirements for the Degree of Master of Science December 2008

2 The thesis of Hao Tang was reviewed and approved* by the following: Yuefeng F. Xie Professor of Environmental Engineering Thesis Co-Advisor John M. Regan Associate Professor of Environmental Engineering Thesis Co-Advisor Shirley E. Clark Assistant Professor of Environmental Engineering Peggy A. Johnson Professor of Civil Engineering Head of the Department of Civil & Environmental Engineering *Signatures are on file in the Graduate School

3 iii ABSTRACT Pilot crumb rubber filters were tested to study their clean-bed head loss under the influences of three design and operational parameters (media size, media depth, and filtration rate). Filter media compressions were observed during the filtration test. Statistic analysis of the field data was used to investigate the applicability of the Kozeny and Ergun equations in predicting the clean-bed head loss in crumb rubber filters. A statistical model was also developed to evaluate the effects of the three parameters and to better predict the clean-bed head loss. Both the Kozeny and Ergun equations were unacceptable for crumb rubber filters, especially when the compressed media depth and porosity from the filtration tests were not available. The statistical model developed based on the original media depth and porosity was statistically valid and was able to provide best-fit to the actual head loss data without using the compressed media depth and porosity obtained in filter tests. The results from this study indicated that the statistical model could be used in place of the Kozeny and Ergun equations for the prediction of clean-bed head loss in crumb rubber filters.

4 iv TABLE OF CONTENTS LIST OF FIGURES... v LIST OF TABLES... vi ACKNOWLEDGEMENTS... vii Chapter 1 INTRODUCTION... 1 Statement of Problem... 1 Objectives... 2 Thesis Organization... 3 Chapter 2 BACKGROUND... 4 Manufacture of Crumb Rubber... 4 Characterization of Filter Media... 5 Theory of Crumb Rubber Filtration... 7 Head Loss Theory Chapter 3 MATERIALS AND METHODS Filter Setup Filter Media Design and Operational Conditions The Kozeny and Ergun Equations Statistical Modeling Regression Verification Chapter 4 RESULTS AND DISCUSSION Media Sphericity Prediction by the Kozeny Equation Prediction by the Ergun Equation Development of a Statistical Model Effects of Parameters Effect of filtration rate Effect of interaction between filtration rate and media depth Effect of filter depth Effect of media size Effect of interaction between media size and filtration rate Prediction by the Statistical Model Chapter 5 CONCLUSIONS Bibliography... 46

5 v LIST OF FIGURES Figure 1-1: Overview of research objectives and scope for model development and verification Figure 2-1: Sieve results of crumb rubber media Figure 2-2: Schematic diagram of filter medium configurations Figure 2-3: Comparison of performances between a crumb rubber filter and a sand/anthracite filter treating secondary effluent at 6gpm/ft Figure 2-4: Crumb rubber filter performances under various media size conditions Figure 3-1: Experimental setup of the crumb rubber filter Figure 4-1: Prediction of clean-bed head loss by the Kozeny equation Figure 4-2: Residual analysis of the Kozeny equation using compressed media depth and porosity Figure 4-3: Prediction of clean-bed head loss by the Ergun equation Figure 4-4: Residual analysis of the Ergun equation using compressed media depth and porosity Figure 4-5: Actual head loss profiles at all filter configurations Figure 4-6: Pareto chart of the standardized effects showing the results of medium and high level factorial calculation Figure 4-7: Impact of filtration rate on filter depth for a crumb rubber filter that has media size of 1.20 mm and filter depth of 0.9 m Figure 4-8: Impact of media compression for the filter configuration of 0.66 mm crumb rubber media and 0.6 m media depth Figure 4-9: Prediction of clean-bed head loss by the statistical model Figure 4-10: Residual analysis of the statistical model using initial media depth and porosity

6 vi LIST OF TABLES Table 2-1: Physical characteristics of crumb rubber media... 7 Table 2-2: Typical properties of several filter media Table 4-1: Sphericity estimates by the Kozeny and Ergun equations for crumb rubber media Table 4-2: Parameters in the statistical model... 35

7 vii ACKNOWLEDGEMENTS I would like to thank my advisors Dr. Yuefeng Xie and Dr. John Regan, for many years of mentoring, and for helping me to become the person I was meant to be. I would also like to thank Dr. Shirley Clark for being on my committee, Mr. Joe Swanderski and his staff at University Park Wastewater Treatment Plant for providing research facilities, and my colleagues at Penn State for their friendship and encouragement. Finally, I wish to thank my parents; my wife, Wen; and my sons, Shunyu and Jinyu. Without them, I could not have accomplished this.

8 Chapter 1 INTRODUCTION Statement of Problem Scrap tire stockpiles cause health and environmental concerns by presenting a potential fire hazard and providing breeding ground for vectors of disease. The properties of scrap tires such as volume, flammability, toughness, durability, etc. make their storage and elimination difficult (USEPA, 1993). According to the data compiled by Rubber Manufacturers Association, 275 million tires remained in stockpiles across the United States in 2003, and approximately 290 million new scrap tires are generated each year (USEPA, 2006). Various recycling technologies have been recommended for conservation of natural resources and minimization of environmental impacts of these tires. The use of crumb rubber, a scrap-tire-derived material, as a filter medium is an innovative technology that has been investigated as a potential green engineering solution for wastewater treatment and disposal of scrap tires. As a compressible material, crumb rubber could form an ideal porosity gradient in filters because the top layer of the media was least compressed while the bottom layer was most compressed. Crumb rubber filters favored in-depth filtration and could allow longer filtration time and higher filtration rate, which could substantially increase the filtration efficiency (Graf et al., 2000; Xie et al., 2001). Their light weight, compact size and capability to remove turbidity, phytoplankton and zooplankton also allowed them to be used in ballast water treatment (Tang et al., 2006a). Since crumb rubber filtration has beneficial engineering applications, it is important to understand the effects of design and operational parameters on the filter performance. Clean-bed head loss is such an important filter performance parameter, because water head must be provided

9 2 to accommodate the increase of head loss resulting from the accumulation of particulates in the filter media (Cleasby and Logsdon, 1999). There are numerous models for the prediction of clean-bed head loss, but the Kozeny and Ergun equations are most accepted models (Trussell et al., 1999a). These two equations, however, were developed for conventional rigid granular media filters. Their applicability in the crumb rubber filter was not clear because of the crumb rubber media compression, which changed the filter configurations during the filtration process, and much higher filtration rates. Objectives and Scope The principle objective of this research was to evaluate the applicability of the Kozeny and Ergun equations for clean-bed head loss prediction in crumb rubber filters. Statistic analysis was also conducted to investigate the impacts of three design and operational parameters and to develop a statistical model for the prediction of head loss in crumb rubber filters. A general flow chart showing an overview of the research scope, with regard to model development and verification, is shown in Figure 1-1 and summarized as follows:

10 3 Figure 1-1: Overview of research objectives and scope for model development and verification. (1) Conduct filtration tests by varying media size, media depth, and filtration rates to obtain actual clean-bed head loss data under various filtration conditions. (2) Estimate the crumb rubber media sphericity based on empirical fitting of filtration test data. (3) The filtration test data and sphericity estimates are applied to the Kozeny and Ergun equations, respectively, to determine whether the two equations can be used in crumb rubber filtration. (4) Develop a statistical model based on three design and operational parameters. Thesis Organization The thesis is composed of five chapters. Chapter 1 describes the problem and objectives of the study. Chapter 2 reviews previous studies on crumb rubber filtration and head loss theories. Methodologies are described in Chapter 3. Chapter 4 elaborates on the results and discussion of the study. Conclusions are summarized in Chapter 5.

11 Chapter 2 BACKGROUND Manufacture of Crumb Rubber Crumb rubber is made by a combination or application of several size reduction technologies. Typically, the crumbing processes involve shredding the tire into progressively smaller pieces, removing the metals (belts and bead wire) using magnets, blowing away the fibers, and then grinding to further reduce the material to a given size. The manufacturing technologies can be further divided into two major categories, mechanical grinding and cryogenic reduction. Mechanical grinding is the most commonly used process. The method consists of mechanically breaking down the rubber into small particles using a variety of grinding techniques, such as cracker mills and granulators. The steel components are removed by a magnetic separator (sieve shakers and conventional separators, such as centrifugal, air classification, and density are also used). The fiber components are separated by air classifiers or other separation equipment. These systems are well established and can produce crumb rubber (varying particle size, grades, quality, etc.) at relatively low cost. The cryogenic process consists of freezing the shredded rubber at an extremely low temperature (far below the glass transition temperature of the compound). The frozen rubber compound is then easily shattered into small particles. The fiber and steel are removed in the same manner as in mechanical grinding. The advantages of this system are cleaner and faster operation resulting in the production of a fine mesh size. The most significant disadvantage is the slightly higher cost due to the added cost of cooling (e.g., liquid nitrogen).

12 5 The crumb rubber media used in this study was produced using the first process. No cooling was applied to make the rubber brittle. The tires are first developed into chips of 50 mm in size in a preliminary shredder. The tire chips then entered into a granulator, where the chips were reduced to a size to less than 10 mm at ambient temperature and most of the steel and fiber were liberated from the rubber granules. The steel then was removed magnetically and fiber was removed using shaking screens and wind sifters. The fine grinding process generally was conducted by cracker mills, which was currently the most common and productive method of producing tire rubber. The end product was usually an irregularly shaped particle with a large surface area, varying in size from 4.75 mm to 0.45 mm (Hsiung, 2003). Characterization of Filter Media Crumb rubber has been sieved for three media sizes according to the sieve analysis using American Society for Testing and Materials (ASTM) Standard Test C136-01, Sieve analysis of Fine and Coarse Aggregates (ASTM, 2001). Sieve results were plotted in Figure 2-1 for determination of media characteristics. As summarized in Table 2-1, the effective size (for which 10% of the grains are smaller by weight) was 0.66 mm, 1.20 mm, and 1.90 mm for the size range of mm, mm, and mm crumb rubber media. Their uniformity coefficients (Ratios of sieve size that will permit passage of 60% of media by weight to the sieve size that will permit passage of 10% of media by weight) were determined to be 1.39, 1.53, and 1.28, respectively (Tang et.al., 2006a). Density of crumb rubber is not correlated with media size, and the value of 1130 kg/m 3 is uniform for all sizes crumb rubber media. Porosity is defined as the ratio of volume of the void space to the bulk volume of crumb rubber media (Trussell et al., 1999). It is expressed in Eq. 1, and was determined by measurements of the dry weight of the media loaded into the filter column and the height of the media in the column.

13 Eq. 1 6 Where = the porosity of crumb rubber media. = the dry weight of the media loaded into the filter column. = the density of crumb rubber = the cross section area of the filter column = the height of the dry media in the column The packed porosity was 0.62, 0.58, and 0.54 for the size range of mm, mm, and mm crumb rubber media, respectively. Figure 2-1: Sieve results of crumb rubber media (Regenerated from Hsiung, 2003)

14 7 Table 2-1: Physical characteristics of crumb rubber media (Source: Tang et al., 2006a) Theory of Crumb Rubber Filtration The crumb rubber filter resembles an ideal filter bed configuration consisting of the largest size media on the top, medium size in the middle, and the smallest size at the bottom, as shown in Figure 2-2a (Xie et al., 2001). For conventional media filters, however, the fine grains tend to settle on the top of the filter while the coarse grains accumulate at the bottom during stratification after filter backwash (Figure 2-2b). Such media size distribution of conventional media filters will easily clog the filter bed surface, which causes a high head loss. Compressibility of the crumb rubber media results in a favorably porosity gradient (Figure 2-2c). Compression increases with pressure throughout the filter columns. Consequently, the crumb rubber filter bed has the smallest pore size at the bottom and the largest pore size on the top. As a result, an ideal porosity profile can be formed in filter bed and the filtration efficiency can be increased. The crumb rubber media results in lower head loss and higher filtration rate and water production. According to a comparative study of a sand/anthracite filter and a crumb rubber filter in tertiary treatment (Hsiung, 2003), at the filtration rate of 6 gpm/ft 2, both filters performed similarly on turbidity removal. However, the crumb rubber filter could continuously run for 50 hours while the sand/anthracite filter could only run 3 hours before the filter bed was clogged (Figure 2-3). For the crumb rubber filtration, therefore, backwash

15 8 frequency and head loss development can be reduced while filter run time and water production rate can be increased. Figure 2-2: Schematic diagram of filter medium configurations. (Source: Xie et al., 2001)

16 9 Figure 2-3: Comparison of performances between a crumb rubber filter and a sand/anthracite filter treating secondary effluent at 6gpm/ft 2 : (a) Turbidity removal; (b) Head loss development. (Source: Hsiung, 2003) Besides compressibility, the crumb rubber has several other physical properties that offer advantages over conventional media. Typical properties of conventional filter media and crumb rubber media are shown in Table 2-2. The porosity of crumb rubber is higher than other media, which will capture more solids and increase the filter run time. The mechanisms of crumb rubber filtration are interception, adsorption, coagulation and sedimentation inside the filter bed. In addition, small hairs that protrude from sides of crumb rubber particle further help capture solids in voids (Xie et al., 2000).

17 10 Table 2-2: Typical properties of several filter media. Property Garnet Illmenite Sand Anthracite GAC Crumb rubber Effective size (mm) Uniformity coefficient Density (g/ml) Porosity (%) N/A N/A Hardness (Moh) Low Very low Sphericity was a shape factor of filter media. According to the literature related with filtration theories, the grain shape was usually described by either shape factor (ξ ) or sphericity ( ψ ). The relationship between them was shown in Eq. 2: surface area of equivalent volume sphere ψ = Eq. 2 actual surface area of grain where ψ = sphericity, dimentionless ξ = shape factor, dimentionless 6 ξ = Eq. 3 ψ The literature values for sphericity of common filter materials were calculated from head loss experiments and therefore, many of the sphericity values are empirical fitting parameters for head loss rather than true independent measurements of shape (Crittenden et al., 2005). In this study, the sphericity estimates of crumb rubber media were obtained from empirical fitting of head loss data based on conventional equations. The Density of crumb rubber is 1130 kg/m 3 (Tang, et.al., 2006a), which is much lower than that of conventional media (e.g., 2650 kg/m 3 for sand). This can substantially reduce the

18 11 backwash water flow rates required for filter backwash. Previous research (Hsiung, 2003) has demonstrated that a backwash flow rate of 21.4 gpm/ft 2 was required to achieve 30% bed expansion for a 3-feet-deep sand/anthracite filter while a same-depth crumb rubber filter only required a backwash flow rate of 10.5 gpm/ft 2 to achieve the same degree of bed expansion. In addition to its use in tertiary wastewater treatment, research on crumb rubber filtration for ballast water treatment indicated that crumb rubber filters could remove up to 70% of phytoplankton and 45% of zooplankton in addition to the removal of particles (Figure 2-4). Compared with conventional granular media filters and screen filters, crumb rubber filters required less backwash, and developed lower head loss (Tang, 2006a). When statistical approaches were applied to develop empirical models, including a head loss model that partially resembles the Kozeny equation, to evaluate the influences of design and operational factors, the results indicated that the behaviors of the crumb rubber filtration for ballast water treatment could not be described by the theories and models for conventional granular media filtration without modification (Tang, 2006b). Considering the much lighter weight of a crumb rubber filter compared with conventional filters, it was suggested as a competitive solution in ballast water treatment.

19 12 Figure 2-4: Crumb rubber filter performances under various media size conditions: (a) Phytoplankton removal; (b) Zooplankton removal. (Source: Tang et al., 2006) Head Loss Theory Several disciplines have made important contributions to the development of models characterizing flow through porous media. The current theory of creeping flow through filter media is largely based on experimental work and theoretical analysis published by Henry Darcy.

20 Darcy observed that the flow through a bed of filter sand was directly proportional to the hydraulic head acting on the sand bed and inversely proportional to its depth (Darcy 1856). The following is the equation Darcy proposed: Where Q = flow rate through the sand bed KA Q = [ H + ΔL H 0 ] Eq. 4 ΔL K = a coefficient dependent on the nature of the sand (units of velocity) A = the area of the sand bed (in plain) H = the height of surface of the influent water above the top of the sand bed H 0 = the height of the surface of the effluent water above the bottom of the sand bed Δ L = depth of the sand bed The head loss through the sand could be defined: 13 ΔH = H + ΔL Eq. 5 H 0 Darcy s equation could then be expressed in a more commonly used form: Q A ΔH = K ΔL Eq. 6 Following Darcy, developments in the prediction of head loss through porous media were advanced by civil and chemical engineers. Kozeny (1927a, b), Fair and Hatch (1933), and Carman (1937), developed a model for predicting Darcy resistance from the characteristics of the porous media. And Ergun (Ergun and Orning 1949; Ergun 1952) combined the linear and nonlinear models to produce one comprehensive model of porous media flow appropriate for a wide range of Reynolds numbers.

21 head loss: In the laminar range of flow, the Kozeny equation (Fair et. al., 1968) is used to depict the 14 h L kμ = ρg ( 1 ε ) 3 ε 2 2 a v V Eq. 7 where h = head loss in depth of bed L = depth of filter bed g = acceleration of gravity ε = porosity a = grain surface area per unit of grain volume = specific surface (Sv ) = v 6d for spheres and 6 for irregular grains ψd eq d eq = grain diameter of sphere of equal volume V = superficial velocity above the bed = flow rate/bed area (i.e., the filtration rate) μ = absolute viscosity of fluid ρ = mass density of fluid k = the dimensionless Kozeny constant commonly found close to 5 under most filtration conditions (Fair et al., 1968) The Kozeny equation is generally acceptable for most filtration calculations because the Reynolds number Re based on superficial velocity is usually less than 3 under these conditions, and Camp (1964) has reported strictly laminar flow up to Re of about 6 (Cleasby and Logsdon, 1999): Re = d eq Vρ / μ Eq. 8

22 15 It was found that the actual head loss was often greater than that predicted from the Kozeny equation, particularly when higher velocities are used in some applications or when velocities approached fluidization (as in backwashing considerations). In this case, the flow may be in the transitional flow regime, where the Kozeny equation is no longer adequate. Ergun and Orning (1949) employed the hydraulic radius approach, which Carman and Kozeny used to characterize linear losses to develop a term for kinetic losses. The results were nearly the same as Eq. 7. To create an equation for losses though porous media over a wide range of flow conditions, Ergun and Orning added the Kozeny term for linear losses. The Ergun equation is adequate for the full range of laminar, transitional, and inertial flow through packed beds (Re from 1 to 2,000). Compared with the Kozeny equation, the Ergun equation includes a second term for inertial head loss. h L 2 2 ( 1 ε ) a ( 1 ε ) μ a V = V k Eq. 9 3 ρg ε v ε v g Note that the first term of the Ergun equation is the viscous energy loss that is proportional to V. The second term is the kinetic energy loss that is proportional to V 2. Comparing the Ergun and Kozeny equations, the first term of the Ergun equation (viscous energy loss) is identical with the Kozeny equation, except for the numerical constant. The value of the constant in the second term, k 2, was originally reported to be 0.29 for solids of known specific surface (Ergun, 1952a). In a later paper, however, Ergun reported a k 2 value of 0.48 for crushed porous solids (Ergun, 1952b), a value supported by later unpublished studies at Iowa State University. The second term in the equation becomes dominant at higher flow velocities because it is a square function of V (Cleasby and Logsdon, 1999). As is evident from Eq. 7 and 9, the head loss for a clean bed depends on the flow rate, media size, porosity, sphericity, and water viscosity.

23 Chapter 3 MATERIALS AND METHODS Filter Setup The filter study was carried out in the Kappe Environmental Engineering Laboratory at the University Wastewater Treatment Plant, University Park, Pennsylvania, USA. A pilot filter (Figure 3-1) was constructed with 15.2-cm-diameter transparent PVC columns. A water pump was used to supply the influent and backwash flow from a water storage tank. An air pump was connected to the bottom of the filter for air scour before water backwashing. Filtration rate was controlled by a flow meter installed in the filter outlet pipe. Influent water was kept constant by an overflow at 3.29 meters. The head loss through the filter media was measured using the difference between the water level in the filter and the water level in the glass tube connected to the bottom of the filter. Filter Media The crumb rubber media size was determined by sieve analysis using American Society for Testing and Materials (ASTM) Standard Test C136-01, Sieve analysis of Fine and Coarse Aggregates (ASTM, 2001). As summarized in Table 2-1, the effective size (for which 10% of the grains are smaller by weight) was 0.66 mm, 1.20 mm, and 1.90 mm for the size range of mm, mm, and mm crumb rubber media. Their uniformity coefficients were determined to be 1.39, 1.53, and 1.28, respectively and the packed porosity was 0.62, 0.58, and

24 for 0.66 mm, 1.20 mm, and 1.90 mm crumb rubber media, respectively. The density of all crumb rubber media was 1130 kg/m 3. Figure 3-1: Experimental setup of the crumb rubber filter.

25 18 Design and Operational Conditions Factorial design was used in this study. The approach reduced the experimental burden while was effective in seeking high quality results to analyze the effects of factors and interactions. Three levels of crumb rubber media sizes and media depths and 19 filtration rates were investigated. The filter was loaded with 0.66 mm, 1.20 mm, and 1.90 mm crumb rubber media at each time. For each crumb rubber medium, the filter was loaded to a depth of 0.6 m, 0.9 m, and 1.2 m. Before each filter run, the filter was backwashed by air scour and then water flow at the rate of 29.3 m 3 /m 2 h. For each filter configuration, the filter was operated at 19 filtration rates from 0 to 73.3 m 3 /m 2 h. Head loss was measured after the media depth was stable. Porosity was determined by the measurement of the dry weight of the media initially loaded to the filter columns and the media depth (Eq. 1). The Kozeny and Ergun Equations In the laminar range of flow, the Kozeny equation (Fair et al., 1968) can be used to depict the head loss (Eq. 7), which is proportional to filtration rate (V), media depth (L), porosity term ( ( 1 ε ) 3 ε 2 ), and media size term ( 6 ( ψd eq 2 ) ). The equation is generally acceptable for the flow that has the Reynolds number (Re) less than 6 (Cleasby and Logsdon, 1999). The Ergun equation (Eq. 9) has an additional term, which is proportional to V 2 in addition to the Kozeny equation to depict the inert head loss. The equation is adequate for the full range of laminar, transitional, and inertial flow. The first term of the Ergun equation is identical to the Kozeny equation except for the numerical constant (Cleasby and Logsdon, 1999). The constant of the second term (k 2 ) was reported to be 0.48 for crushed porous solids (Ergun, 1952).

26 The sphericity for crumb rubber media is the only unknown parameter in the two equations. Their values were determined by empirical fitting of data from the filtration tests. 19 Statistical Modeling A total of 171 filtration data sets were collected (3 media sizes 3 media depths 19 runs). Based on statistical requirements, 70% of data sets were chosen to initiate the regression and the remaining 30% were used to validate the corresponding regression results. Regressions were performed using a least squares criterion and assuming that the residuals were normally distributed and independent, and had constant variance. These assumptions were checked after the model had been fitted. Sigma Plot 10.0 (SPSS, Inc., Chicago, IL) was used to initiate the fitting process for media sphericity and to conduct the statistical multiple nonlinear regression for developing a statistical model. For the regression process for media sphericity, each data set consisted of values of an actual head loss, its corresponding filtration rate, compressed media depth at this filtration rate, porosity at the compressed media depth and media size. Data except actual head loss were used to calculate head losses based on the Kozeny or Ergun equation. Regressions were then performed by varying the sphericity values in the two equations to obtain the best fit of the estimated head loss data to the actual head loss data. For the regression process for a statistical model, each data set consisted of values of an actual head loss, its corresponding filtration rate, media size, and initial media depth. Data except actual head loss were used to calculate estimated head losses based on a multiplicative power-law relationship. Then, regressions were performed by varying the model coefficients to obtain the best fit between the calculated head loss data and the actual head loss data.

27 20 Regression Verification For each attempted fit of sphericity to actual head loss data, the hypotheses tested were that the derived sphericity of the equation was significantly different from zero. Expressed mathematically, the hypothesis was as follow s : follows: : 0 : 0 Similarly, the hypotheses tested for each coefficient of the statistical model were as For numerical constant K: For exponent of filtration rate a: For exponent of media depth b: For exponent of media size c: : 0 : 0 : 0 : 0 : 0 : 0 : 0 : 0 By examining the p-value of each coefficient, the null hypothesis can be rejected and the alternative one can be concluded if the p-value was very small (<0.05 in this study). Analysis of Means (ANOM) was performed using two-sample t-test for some cases where comparison of model coefficients was necessary. For example, when comparing the

28 21 sphericity estimates from the Kozeny and Ergun equations, the hypothesis tested was that the derived sphericity estimates from the Kozeny equation was significantly different from the one from the Ergun equation. Expressed mathematically, the hypothesis was as follows: : : If the variances were not quite different, pooled variance t-test was applied by using the following equations to compute the t-statistic (Ott and Longnecker, 2000): Eq Eq. 11 With degrees of freedom equal to.. 2 If the variances were quite different, separate variance t-test was applied by using the following equations to compute the t-statistic (Ott and Longnecker, 2000): Eq. 12 With degrees of freedom equal to.. where By comparing the computed t-statistic with the corresponding one in the t-table, the null hypothesis can be rejected and the alternative one can be concluded if /, or /, where α=0.05 in this study. Coefficient of determination (R 2 ) is another criterion to verify these regressions. It represents the proportion of the total variability in the dependent variables that the regression equation accounts for (Burton and Pitt, 2001), as expressed in the following equation:

29 22 R 1 ss SS Eq. 13 Where SS err is the sum of squared errors, and SS tot is the total sum of squares. An R 2 of 1.0 indicates that the equation accounts for all the variability of dependent variables, and it is generally good and indicates a strong relation if R 2 is large. But it does not guarantee a useful equation, since high R 2 can occur with insignificant equation coefficient if only a few data observations are available. Regressions in this study were validated based on the following steps: (1) examining the regression R 2 and verification R 2 ; (2) examining the residuals of the resultant regressions statistically and graphically. In addition, the standard error of each estimate, which was computed from the variance of the predicted values, was given as a measure of the model variability. Graphical analyses of regressions were performed by examining the following requirements according to Draper and Smith, 1981: (1) residuals are independent; (2) residuals have zero mean; (3) residuals have constant variance; and (4) residuals follow a normal distribution. The purpose was to determine if the assumption of the regression error being independent and normally distributed was valid. Verification of the assumption of independent residuals was performed by graphically plotting the residuals against its variables (filtration rate, media depth, and media size) and observed values. For this assumption to be valid, the residuals should be randomly distributed in these four plots around a zero line. Verification of the assumption of normal distribution of residuals was performed using a normal probability plot. Residuals that fall along a straight line in the plot and fall into the 95% confidence interval are considered to be normally distributed. If residuals fail to pass any one of these tests, then the regression is not valid for the data. Analysis of Variances (ANOVA) was performed to check constant variances for some instances of the models if they passed the independence and normality tests. The hypothesis

30 23 tested was that the variances were significantly different from each other. The mathematical expression was as follows: : 0 : 0 The Levene s test was performed with the assistance of MINITAB 15 (The Minitab Inc). If the p-value was small (<0.1 in this case), the null hypothesis can be rejected and the alternative one can be concluded that the variances were not equal.

31 Chapter 4 RESULTS AND DISCUSSION Media Sphericity Sphericity, a physical parameter that defines the roundness of the media shape, has to be provided if the Kozeny and Ergun equations are used. For crumb rubber media, their values were estimated by empirical fitting of the actual head loss data to the predicted head loss data computed by the two equations using the compressed media depth and porosity. Two initial assumptions were made before the regressions: (1) Sphericity values for different media sizes were the same; and (2) Sphericity values at different operating conditions were the same. Based on that, regressions were first performed using 70% of all data sets without differentiating media sizes. However, the residuals of both equations could not pass the normality test, and therefore made the regressions invalid, indicating that the assumptions could be partially wrong. The first assumption was then modified as: Sphericity values for different media sizes were different. With the modified assumptions, regressions were individually performed using 70% of data sets of each media size, and the results were summarized in Table 4-1. The Kozeny equation gave a sphericity of 0.67, 0.64, and 0.62 while the Ergun equation gave a sphericity of 0.71, 0.75, and 0.79 for the 0.66 mm, 1.20 mm, and 1.90 mm crumb rubber media, respectively. All the p-values were less than , showing these coefficients were significant. Two-sample t-test indicated that a decreasing trend did not exist for the estimates from the Kozeny equation while there was an increasing trend for the estimates from the Ergun equation. In addition, it was statistically proved that the Ergun equations gave a higher sphericity estimate than the Kozeny equation. The 95% confidence intervals gave a range of variability for those estimates. It has to be pointed out

32 25 that the regressions for 0.66 mm and 1.90 mm media failed the normality test, showing that the second assumption could be wrong for the two sizes. That is, for the 1.20 mm crumb rubber media, it was acceptable to assume sphericity did not change due to compression, but for the other two sizes, this assumption might be invalid. But these estimates still could be applied in the evaluation process of two equations, since they were able to provide the highest regression R 2 and verification R 2, which addressed the most variability of dependent variables.

33 Table 4-1: Sphericity estimates by the Kozeny and Ergun equations for crumb rubber media

34 Prediction by the Kozeny Equation Figure 4-1 compared the actual head loss with the predicted head loss by the Kozeny equation using the original and compressed media depth and porosity, respectively, to evaluate the applicability of the equation in crumb rubber filters. The 45-degree-line in the figure depicted the hypothetic head loss estimates that were precisely equal to the actual values. It was found in Figure 4-1a that the R 2 value was only 0.87 and the Kozeny equation under-estimated head loss, especially at high filtration rates at each filter medium configuration. This implied that the Kozeny equation has limitation for crumb rubber filters if no compressed media depth and porosity were available to compensate the media compression. The under-estimation was likely due to the decreased porosity resulted from the decreased media depth. When the Kozeny equation was examined by using the compressed media depth and packed porosity, it was found in Figure 4-1b that the Kozeny equation could give better predicted values using the appropriate sphericity estimates. Regression of predicted and actual head loss data by the 45-degree-line gave a R 2 of 0.97, which indicated that predicted data by the Kozeny equation was close to the actual head loss data, and the equation could be used for crumb rubber filters. However, it was also noticed in the figure that some predicted data at high actual head loss values were under-estimated, which indicate that the equation had limitation when the filtration rate was high at each filter medium configuration.

35 28 Figure 4-1: Prediction of clean-bed head loss by the Kozeny equation: (a) Prediction using the original media depth and porosity; (b) Prediction using the compressed media depth and porosity. Figure 4-2 shows the residuals of the Kozeny equation against three variables and actual head loss. It was found that the residuals were independent against filter depth and media size

36 29 because they were almost centered at the zero line (Figure 4-2 b and c), but they were dependent on filtration rate and actual head loss, because as filtration rate or actual head loss increased, residuals went from positive to negative values (Figure 4-2 a and d). In addition to relatively lower R 2 shown in Figure 4-1, analysis of residuals is against the criterion of independent residuals as a good model. Therefore, the Kozeny equation had limitations and the derived sphericity estimates could not describe the real shape of grains. Figure 4-2: Residual analysis of the Kozeny equation using compressed media depth and porosity Prediction by the Ergun Equation To examine the performance of the Ergun equation in the changing environment of configurations in crumb rubber filters, the original media depth and packed porosity were used to

37 30 calculate the head loss, which were then compared with actual head loss as was shown in Figure 4-3a. It was found that the Ergun equation gave a R 2 of 0.95 as well as under-estimation of head loss when the actual head losses were higher than 100 cm at high filtration rates. However, the under-estimation was not as great as that of the Kozeny equation using the same data sets of original media depth and porosity, and the R 2 value was also higher than that of the Kozeny equation. This suggested that although the inert head loss included by the Ergun equation could adjust the predicted data at high filtration rates, it was still not sufficient to address the head loss increase caused by the decrease of porosity due to media compression. Same with the testing procedures for the Kozeny equation, the data sets of compressed media depth and porosity from field study were applied to reflect the effect of media compression. The predicted head loss data were calculated by the Ergun equation, and were compared with the actual head loss data. As shown in Figure 4-3b, it was found that the R 2 value was 0.99, which indicated that the Ergun equation might be suitable for crumb rubber filters. Comparing the R 2 value with that of the Kozeny equation, the Ergun equation had higher R 2 and showed better fit to the actual head loss. In addition, the Ergun equation did not under-estimate data at high filtration rates in the way the Kozeny equation did. This was because the inert head loss shown as the second term in the Ergun equation was able to adjust the prediction of head loss development at high filtration rates by portraying a linear relationship not between h and V, but between h and V 2.

38 31 Figure 4-3: Prediction of clean-bed head loss by the Ergun equation: (a) Prediction using the original media depth and porosity; (b) Prediction using the compressed media depth and porosity. Figure 4-4 shows the residuals of the Ergun equation against three variables and actual head loss. It was found that the residuals were independent against filter depth and media size

39 32 because they were centered at the zero line (Figure 4-4 b and c). For the filtration rate, however, the equation only displayed good residuals distribution when the filtration rate was high, indicating that the residuals were conditionally independent on filtration rate (Figure 4-4 a). And they were dependent on actual head loss, because when actual head loss was high, residuals went negatively higher (Figure 4-4 d). Although the R 2 of the Ergun equation was relatively higher as shown in Figure 4-3, analysis of residuals is against the criterion of independent residuals as a good model. Therefore, the Ergun equation also had limitations, although not as severe as the Kozeny equation. Figure 4-4: Residual analysis of the Ergun equation using compressed media depth and porosity

40 33 Development of a Statistical Model Both the Kozeny and Ergun equations had deficiencies for the prediction of clean-bed head loss in crumb rubber filters, and they may provided better fit when the data of compressed media depth and porosity were used to reflect the media compression. Their predictions using the data of original media depth and porosity, however, could not provide the same degree of comparison at all to the actual head loss data. The benefits of developing a statistical model for crumb rubber media are obvious: (1) No filtration tests need to be conducted to obtain the compression situations on media depth and porosity; (2) No hassle to derive a sphericity estimate since this value varies to the filtration conditions; and (3) the conventional equations are not good models for crumb rubber filtration. The statistical model development used a multiplicative power-law relationship, which resembled the Kozeny equation. The model, expressed in Eq. 14, consisted of the three factors examined in this study as independent variables: filtration rate, media depth, and media size. h ) a b c = K ( V ) ( L) ( d eq Eq. 14 Where K = dimensionless constant for the statistic model a = exponent of filtration rate b = exponent of media depth c = exponent of media size Exponents of each factor and the numerical constant were obtained via regression using 70% of data sets. The regression results were summarized in Table 4-2. The initial assumption of the regression was that the model coefficients were the same for all media sizes. However, the resultant group of coefficients failed the normality test, indicating a probably wrong assumption.

41 The assumption was then modified as: the model coefficients were different among media sizes. The model then had a form as follows: 34 h ) a b = K ( V ) ( L Eq. 15 Regressions were then performed individually using 70% of data sets for each media size, and all the three groups of coefficients passed the normality tests. The p-values were less than , indicating that the coefficients were significant. Standard errors and 95% confidence intervals gave the ranges of the variability. Regression R 2 and verification R 2 of the model were higher than 0.996, which demonstrated a potential to be a good model.

42 Table 4-2: Parameters in the statistical model

43 Effect of Parameters Three design and operational parameters (Media size, media depth, and filtration rate) were investigated for their effects. Figure 4-5 shows the actual head loss profile at all filter configurations. Factorial calculations were conducted by using these data to explore the effects of factors and possible interactions. Figure 4-5: Actual head loss profiles at all filter configurations.

44 37 Although three-level factorial design was conducted in this experiment, only the results of medium and high level factorial calculation were shown in Figure 4-6, because the results would be more meaningful since crumb rubber filters were usually designed to operate at medium and high levels of filtration rates. Four of seven effects were judged significant by Lenth s method (Lenth, 1989) with the significance level of The threshold value was denoted by the vertical line, and factors filtration rate, media depth, media size, interaction between filtration rate and media depth, and interaction between filtration rate and media size were determined to have significant effects. The interaction between filtration rate and media depth could be explained by the compression as the filtration rate increases, which correspondingly decreases media depth and therefore confounds head loss. The interaction between filtration rate and media size, although very small, was likely due to the change of sphericity during compression, because the assumption of equal sphericity at different filtration conditions were proved to be wrong for some media sizes. It is obvious that the Kozeny and Ergun equations could not be able to address these distinctive effects of crumb rubber media. The statistical model developed by the multiplicative power-law relationship could be used to analyze these effects on clean-bed head loss in crumb rubber filters. Interaction between filter depth and media size, and interaction between all three factors were statistically insignificant, therefore, they were not included in the following discussion.

45 38 Figure 4-6: Pareto chart of the standardized effects showing the results of medium and high level factorial calculation (Response is observed head loss in centimeters. Significance level = 0.05) Effect of filtration rate The exponents of filtration rate, shown as 1.55, 1.51, and 1.75 for 0.66 mm, 1.20 mm, and 1.90 mm crumb rubber media in the statistical model were all higher than 1 found in the Kozeny equation. This could explain the under-estimation of the Kozeny equation because the filtration rate in crumb rubber filters had a larger impact than that in other conventional granular media filters. This caused the actual head loss to increase faster when the filtration rate was high. The faster increase was due to the rapid development of inert head loss that could not be addressed by a linear relationship between head loss and filtration rate.

46 39 Effect of interaction between filtration rate and media depth The larger exponent of filtration rate was also believed to be caused by the interaction between filtration rate and media depth due to compression of the filter media, which was shown in Figure 4-7. The filter depth was decreased as the filtration rate increased. It was found that the crumb rubber filter that had media size of 1.2 mm and filter depth of 0.9 m was compressed by 9% as the filtration rate increased from 0 to 73 m 3 /m 2 h. However, for conventional granular filters, the filter depth and filtration rate were independent parameters. In crumb rubber filters, the phenomenon could not be addressed by the conventional head loss models Filter depth (cm) Filtration rate (m 3 /hour.m 2 ) Figure 4-7: Impact of filtration rate on filter depth for a crumb rubber filter that has media size of 1.20 mm and filter depth of 0.9 m.

47 40 Effect of media depth The exponents of media depth, shown as 1.35, 0.97, and 1.21 for 0.66 mm, 1.20 mm, and 1.90 mm crumb rubber media in the statistical model showed the influence of media compression on clean-bed head loss, because the exponent was found to be 1 in both the Kozeny and Ergun equations. The effect of media compression in crumb rubber filters was complicated according to the head loss theories. The compression decreased the media depth, which could decrease head loss, but it also decreased the porosity at the same time, which could increase head loss. Figure 4-8 showed the media depth and porosity terms change at one filter configuration. It was found that for a crumb rubber filter loaded with 0.66 mm crumb rubber media to a depth of 0.6 m, as the filtration rate gradually increased from 0 to 73.3 m 3 /m 2 h, the media depth was steadily decreased by 13.6%. However, the decrease in media depth did not cause a corresponding decrease in its exponent, which was used to be 1 as shown in the Kozeny and Ergun equations. In fact, the (1 ε ) exponent of media depth was increased to 1.35 because the porosity term 3 ε 2 was increased (1 ε ) by 69.5% due to the compression. The porosity term 3 ε in the second part of the Ergun (1 ε ) equation did not increase as much as the term 3 ε 2 in Kozeny equation, which only increased by 46.4%. But it still played an important role when the filtration rate was high and the inert head loss was dominant. Therefore, the exponent of media depth implied the overall influence of media depth decrease and porosity terms increase in crumb rubber filters.

48 41 Figure 4-8: Impact of media compression for the filter configuration of 0.66 mm crumb rubber media and 0.6 m media depth. Effect of media size The exponents of media size, shown as in the statistical model, which did not differentiate media sizes, was between -2 in the Kozeny equation and -1 in the second term of Ergun equation, indicating that the Ergun equation might be able to address the effect of this factor while the Kozeny equation could not.

49 42 Effect of interaction between media size and filtration rate Filtration rate affected crumb rubber media by changing their sphericity. The assumption of equal sphericity at one filter configuration was unacceptable for some media sizes when their sphericity estimates were verified. None of conventional equations could address this problem yet, since they both assumed that sphericity did not change. However, it was not necessarily the case for crumb rubber media. Compression could change the shape of the media, especially when the filters were operated at higher filtration rates, which correspondingly exerted higher pressure to change the media shape. Prediction by the Statistical Model Because the statistical model was developed using actual head loss, initial media depth, filtration rate, and media size, it had included the effect of media compression that changes the filter configurations. Also, it addressed the problem encountered by the Kozeny and Ergun equations, that is test must be conducted at all filtration rates to obtain the data of compressed media depth and porosity. Figure 4-9 showed the comparison of actual head loss and predicted head loss data by the statistical model at all filter configurations. The statistical model did not under-estimate head loss at high filtration rates in the way the Kozeny and Ergun equation did. The R 2 value was as high as