Get 20M+ Full-Text Papers For Less Than $1.50/day. Start a 14-Day Trial for You or Your Team.

Learn More →

Characteristics of Laminar Flow in Pipelines of Homogenous Alum Sludge Approximated with use of the Vočadlo Model for Viscoplastic Liquids

Characteristics of Laminar Flow in Pipelines of Homogenous Alum Sludge Approximated with use of... The study presents the manners of determination of the Darcy friction factor for a homogenous hydromixture of alum sludge of varied hydration and temperature for the laminar flow zone. The rheological evaluation of the hydromixture as a viscoplastic body has been conducted with use of measurements of viscosity. The curves of flow were approximated with use of the generalized Vocadlo model. The Darcy friction factor of the pipeline was determined with use of the non-dimensional criterion (Regen) and (Re, He). INTRODUCTION The use of pipeline transport of hydromixtures as an optimum means of transport does not require a detailed justification. However, the determination of hydrotransport parameters influencing the costs of the planned venture may be questionable. The essential parameter of hydrotransport, which enables to determine the pressure loss of the transported mixture is the Darcy friction factor . The knowledge of this parameter enables eventually to conduct the analysis of the operation of the pump and pipeline system and to design it correctly. However, the regimen of flow for the planned transport has to be determined. The analysis of the diagrams of the correlation (Re, ) allows us to determine with certainty that for a given mixture, the critical value of the Reynolds number Recr (transition from laminar flow to the transition zone ­ turbulent flow) corresponds with the minimum value of the Darcy friction factor min for a specific diameter of the pipe. Further slight decrease in the value of the Darcy friction factor in the turbulent zone is however connected with the constant increase in the flow rate, which causes a significant pressure loss in the pipeline. To sum up, the final zone of laminar flow is characterised both by lower flow rates v and by lower values of the Darcy friction factor , which in turn leads to a decrease in the pressure loss in the designed pump and pipeline installation. The economic aspect of pipeline hydrotransport should also be taken into account, as it requires transporting the hydrated material in the most efficient possible way. The fact that hydrotransport is preferred at high concentrations of the solid component is reasonable from the economic point of view. The authors believe that the hydrotransport of homogenous mixtures should take place in the laminar flow zone, due to both hydraulic and economic reasons. In order to provide the proper dimensions of the pump and pipeline installation for hydrotransport it is required to conduct the measurements of the viscosity of the mixture, necessary for the correct determination of the Darcy friction factor for the adopted flow parameters. Thus, the analysed problem is the determination of the rheological parameters of the mixture, and basing on that, of the pressure loss in the pipeline. This issue is important for designing the transport systems of concentrated sediments from water treatment and wastewater treatment [2, 3, 11]. OBJECTIVE OF THE STUDY The aim of the present analysis is to present the methodology of the determination of the rheological parameters and of the Darcy friction factor for a homogenous viscoplastic hydromixture, described by the tri-parametric Vocadlo model, basing on the analysis of the methods suggested by various authors [1, 6, 13, 14]. Physical and rheological properties of the analysed mixtures The study encompasses the sediments taken from alum sludge tanks of the waterworks utility collecting surface waters and using aluminium sulphate in the coagulation process. Apart from the rheological properties, the following characteristics were measured: the temperature, hydration and density of the sludge, chemical composition of the solid component and structure of the sediments. The rheological properties of the sludge were measured with use of a rotational rheometer Rheotest 2. The measurements determined the values of shear stress on the surface of the measurement cylinder for various assumed pseudo deformation rates. As a result of the measurements series of points were obtained, determining the pseudo-curves of flow. The measurements were conducted at the temperatures of 273.45 K and 293.15 K. Constant temperature of the sediment in the rheometer was maintained with use of an ultrathermostat equipped with contact thermometers. The hydration of the sediments was determined by the mass share of water and measured as the difference between the mass of humid sediment and the mass of the sediment dried to constant weight at the temperature of 378 K. The chemical composition of the solid fraction of the sludge was determined with use of chemical analysis, and its structure through the measurement of the isotherms of sorption of CO2 and C6H6 in high-vacuum gravimetric apparatus [13]. The mathematical form of the model and the determination of values of rheological parameters were selected with use of methods of statistical analysis of flow curves. The pseudo-curves of flow were transformed into curves of flow with use of the known Krieger ­ Elrod ­ Maron equation [8, 9, 10]. The values of m and dm/dlogR in this equation were determined from the polynomial approximation function of pseudo-curves of flow, determined in a logarithmic system of co-ordinates (logGp, logR), with use of the classical method of least squares. Polynomial functions and the Bingham model are linear correlations with respect to their constant parameters. Due to this linearity, the application of the least squares method also leads to linear systems of normal equations for these parameters, so the solution thereof does not present any difficulties. On the other hand, the Vocadlo model is a non-linear correlation, which however can be reduced to a linear form. After the switching of the variables a linear form was obtained for new parameters, followed by the estimation of the value of one of the parameters and the application of the least squares method with respect to the linear forms of the analysed models. The limits of flow determined with use of the said models allow to test which of the measurement points fall within the zone of partial shearing of the sediment. Such points were rejected, and the described procedure was repeated until the condition of hydration of the whole sediment sample in the slit of the rheometer was fulfilled. Apart from the calculated values of the rheological parameters of the models, also the values of the correlation coefficient and standard deviation were calculated [13]. Values of the coefficient of correlation ­ R and of the standard deviation ­ S are presented in Table 1 below. Table 1. Statistical parameters of the flow curves Temperature K 293.15 273.45 Hydration % 93.60 88.05 95.05 88.00 Models Bingham R 0.9934 0.9531 0.9904 0.9400 S 1.4039 5.4154 2.1740 8.3210 R 0.9945 0.9985 0.9989 0.9982 Vocadlo S 0.1546 0.4731 0.1251 1.0634 These values enabled the evaluation of the compliance of the specified models with data obtained from the experiments. The tri-parametric model approximated the course of the flow curves of the sediments within the whole range of the shearing rate very well. Moreover, the applicability of the bi-parametric Bingham model for the description of the rheological properties of alum sludge was proven, in particular within the higher hydration range. The accurateness of this approximation decreased noticeably with the decrease in hydration. The results of the analysis of the structure of sediments are presented in Table 2. The presented results allow to classify the analysed alum sludge as a sediment with a well-developed structure. This degree of development of the sediment structure depends on the content of aluminium hydroxide and has a vital influence on the rheological properties of the sludge. Table 2. Physical properties of the sediment structure properties of the structure ultramicropores micropores mesopores (1.5÷3.0) m (3.0÷30) m 3 -1 (30÷100) m total value pore capacity, cm ·g 0.007 0.012 0.011 -1 pore surface area, m ·g 18.4 24.6 9.9 For the purpose of approximation of pseudo-curves of flow the triparametric, generalized Vocadlo model was used (1). = 0 for R 0 R = + K v 0 (1) for R > 0 : R ­ shear stress on the surface of cylinder [Pa] 0 ­ yield stress [Pa] ­ shear rate [s-1] 1 n Kv ­ consistency index for Vocadlo model [ Pa n s ] n ­ flow behaviour index [-] The physical and rheological properties of the analysed mixtures are presented in Table 3. Table 3. Physical and rheological properties of the analysed mixtures Hydration [%] Density [kg/m3] Temperature [K] K V Pa n s n o [Pa] n [-] Determination of the parameters of flow of the mixture For laminar flow of Newtonian liquids the Darcy friction factor is calculated with use of formula (2) 64 Re (2) the Reynolds number Re is calculated with use of equation (3) Re = vD (3) Analogically to the Newtonian liquid, Parzonka [12], Kempiski [6], Sozaski [13] adopt the determination of parameter for non-Newtonian liquids with use of formula (4) 64 Re gen (4) Regen is the generalized Reynolds number for the adopted rheological model. In this case, the determination of the dimensions of the installation is based on the unconditional criterion (Regen). Vocadlo and Charles [14] and Czaban [1] suggest that the determination of dimensions within the laminar zone of the flow should be based on the criterion (Re, He), thus creating a family of curves shifted parallelly depending on the value of the Hedström number He. A similar approach was suggested by Govier and Aziz [4]. Non-dimensional criterion (Regen) The application of this criterion requires the knowledge of the generalized Reynolds number Regen for the adopted rheological model describing the viscoplasticity of homogenous hydromixture. Such hydromixtures are characterised by the occurrence of the so-called yield stress 0. The omission of this value causes errors in the determination of the Reynolds number [7]. The lack of knowledge of Regen led to the abandonment of this method of determination of dimensions. Kempiski [6] presented a full generalized Reynolds number for the Vocadlo model along with a precise approach for the application thereof. The formulas quoted below enable to determine the required stress on the pipeline wall w (6), the generalized Reynolds number (5) and the Darcy friction factor (4). Generalized Reynolds number for the Vocadlo model according to Kempiski [6] p 1 = 4 1+ L 3n 0 6 vK 1+ 1 D n D 0 (5) Shear stress on the surface of the pipeline according to Kempiski [6] w = Dn (2vK )n n 1 0 - 3n + 1 3 w 1 0 3(3n + 1) w 3+ (6) Sozaski [13] presented the generalized Reynolds number Regen (7) for viscoplastic mixtures as a modification of the known Reynolds number for pseudoplastic bodies Re (8), by means of introduction of the plasticity number Lp (9), and relative error p (10). The generalized Reynolds number according to Sozaski [13] is determined with use of the formula (7) Re gen = Re 1 + 1 Lp 6 -n (1 - ) -1 (7) Re = v2-n D n 2 Kn 6+ n 8 (8) The plasticity number (9) D 0n Lp = vK Relative error (10) (9) p = p L (10) Absolute error is determined with use of formula (11) p = p p - L L (11) 1 0 6vK p 1+ = 4 1+ 1 3n D L D 0n (12) this is an approximate value of pressure loss, while p is an exact value of pressure L loss which, according to the methodology adopted by Sozaski [13] can be determined numerically, e.g. by means of halving the range with use of equation (13). p L p 1- L 3 1+ 1+ 3 n (13) 1 3n 1+ 3 n 1+ 6vK D 0n as the first approximation of the searched value. Non-dimensional criterion (Re, He) Vocadlo and Charles [14] determined the friction factor f (Re, X) in form of the formula (14) f = X Re 1 - f 16 n Re 2 n (n - 2 ) 1 X 1+ 1- 3n f Re 6 n-2 (14) f = (15) and Re is the Reynolds number as for the pseudoplastic model, determined by formula (8). The parameter X is determined by the correlation (16) 2n 2-n X = 0 2 2-n 8- n D 1 2Kv 3 + n (16) Czaban [1] suggested a solution based on the correlation (Re, He, ): 64 1 + 3n Re 1 - 3n 1 + 3n n 1 +3 n (17) He = 0 Re 2 v 2 (18) Re is, as in the solution of Vocadlo and Charles, the Reynolds number determined as in the pseudoplastic model (8), and He is the Hedström number (18), 8He Re 2 (19) while can be calculated with use of formula (19). ANALYSIS OF THE RESULTS OF THE CALCULATIONS The initial viscometric measurements enabled us to select the correct rheological model along with the determination of its parameters. This in turn allowed to conduct the calculation of the parameters of hydrotransport, in particular the Darcy friction factor , according to the methodologies discussed here above, suggested by various authors. The parameters of flow of alum sludge were calculated for an adopted pipeline diameter D = 200 mm, within the range of flow rates from 0.1 to 2.4 m/s, depending on the hydration and temperature of the mixture, basing on the adopted critical Reynolds number Recr = 2300. The values of generalized Reynolds numbers Regen and of the Darcy friction factors of the pipeline , calculated basing on the correlations presented by various authors, are presented in Tables 4, 5, 6, 7. The results of calculation presented in form of tables confirm the similarity of methodological solutions proposed by various authors and enable us to calculate the Darcy friction factor in an unequivocal manner. The methodology developed by Kempiski [6] and Sozaski [13] enables however, apart from the determination of the factor, to evaluate the nature of flow of a homogenous hydromixture (laminar or turbulent), basing on the generalized Reynolds number Regen. However, the evaluation of the regimen of flow of viscoplastic mixtures basing on the Reynolds number Re calculated as for pseudoplastic bodies may lead to significant errors [7]. Significant differences between Table 4. Hydraulic parameters of sludge mixture of the hydration of 93.6%, at temperature T = 293.15 K v [m/s] 0.1 0.2 0.4 0.6 0.8 Kempiski Regen 71.5 274 1029 2206 3762 0.8953 0.2338 0.0622 0.0290 0.0170 Sozaski Regen 74.9 283 1050 2236 3799 0.8543 0.2261 0.0609 0.0286 0.0168 Vocadlo Re 1666 3775 8554 13803 19381 0.8947 0.2319 0.0618 0.0290 0.0170 Re Czaban 0.8947 0.2319 0.0618 0.0290 0.0170 1666 3775 8554 13803 19381 Table 5. Hydraulic parameters of sludge mixture of the hydration of 88.05%, at temperature T = 293.15 K v [m/s] 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 Kempiski Regen 42.0 155 328 554 830 1150 1514 1918 2360 1.5222 0.4133 0.1953 0.1155 0.0772 0.0556 0.0423 0.0334 0.0271 Sozaski Regen 42.4 155 328 555 830 1151 1514 1918 2361 1.5084 0.4117 0.1949 0.1154 0.0771 0.0556 0.0423 0.0334 0.0271 Vocadlo Re 77.4 240 465 743 1070 1441 1854 2305 2794 1.5215 0.4132 0.1953 0.1155 0.0771 0.0556 0.0423 0.0334 0.0271 Re Czaban 1.5215 0.4132 0.1953 0.1155 0.0771 0.0556 0.0423 0.0334 0.0271 77.4 240 465 743 1070 1441 1854 2305 2794 Table 6. Hydraulic parameters of sludge mixture of the hydration of 95.05%, at temperature T = 273.45 K v [m/s] 0.2 0.4 0.6 0.8 Kempiski Regen 315 1122 2306 3800 0.2030 0.0571 0.0278 0.0168 Sozaski Regen 318 1126 2310 3803 0.2001 0.0568 0.0278 0.0169 Vocadlo Re 1175 2934 5011 7326 0.2024 0.0568 0.0277 0.0169 Re Czaban 0.2024 0.0568 0.0277 0.0169 1175 2934 5011 7326 Table 7. Hydraulic parameters of sludge mixture of the hydration of 88.00%, at temperature T = 273.45 K v [m/s] 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6 Kempiski Regen 22.3 83.4 179 305 460 643 851 1084 1340 1619 1920 2242 2585 2.875 0.7674 0.3582 0.2098 0.1390 0.0995 0.0752 0.0590 0.0477 0.0395 0.0333 0.0285 0.0248 Sozaski Regen 22.6 84.0 179 306 461 644 852 1085 1341 1620 1921 2243 2585 2.8366 0.7623 0.3568 0.2092 0.1388 0.0994 0.0751 0.0590 0.0477 0.0395 0.0333 0.0285 0.0248 Vocadlo Re 44.0 138 270 435 629 850 1096 1367 1660 1976 2313 2670 3048 2.8745 0.7672 0.3581 0.2097 0.1390 0.0996 0.0752 0.0591 0.0478 0.0395 0.0333 0.0285 0.0248 Re Czaban 2.8745 0.7672 0.3581 0.2097 0.1390 0.0996 0.0752 0.0591 0.0478 0.0395 0.0333 0.0285 0.0248 44.0 138 270 435 629 850 1096 1367 1660 1976 2313 2670 3048 Regen and Re are noticeable. These differences grow with the increase of the rheological parameter n. Figures 1. 2. 3 and 4 present the diagrams of correlations Regen (v) and Re(v) that enable us to calculate the maximum acceptable values of flow rate vmax, that meet the conditions of laminar flow. It is clearly visible that the zone of laminar flow determined basing on the diagram of Regen(v) increases along with the decrease in hydration, thus for the hydrations 95.05% and 93.60% the acceptable actual rate vmax is, respectively, 0.60 m/s and 0.61 m/s, while for lower hydration values 88.05% and 88.00% this rate is, respectively, 1.78 m/s and 2.4 m/s. The acceptable range of flow rates increased 3­4 times in this case. Fig. 1. Correlations Regen(v) and Re(v) for sludge of the hydration of 93.60%, at temperature T = 293.15 K Fig. 2. Correlations Regen(v) and Re(v) for sludge of the hydration of 95.05%, at temperature T = 273.45 K The rates of laminar flow, determined basing on the correlation Re(v) narrow the zone of flow significantly, in particular for high hydration values, limiting the flow to the rate vmax ranging from 0.13 m/s to 0.33 m/s (Figure 1, 2). For lower hydration values the rate vmax ranges from 1.6 m/s to 2.2 m/s (Figures 3, 4). Fig. 3. Correlations Regen(v) and Re(v) for sludge of the hydration of 88.05%, at temperature T = 293.15 K Fig. 4. Correlations Regen(v) and Re(v) for sludge of the hydration of 88.00%, at temperature T = 273.45 K The approach developed by Kempiski [6], based on indirect determination of shear stress on the surface of the pipe w, enables also to determine, with use of this parameter and the value of the yield stress 0, the radius of the core of the cross-section r0 for the laminar flow of homogenous viscoplastic hydromixture. This enables us to obtain deeper knowledge about the nature of flow of the analysed hydromixture. SUMMARY AND CONCLUSIONS The description of rheological properties of homogenous viscoplastic hydromixtures should be based on tri-parametric, generalized rheological models, such as the Vocadlo model, which is simplified to simpler, bi- and uni-parametric models. The selection of the rheological model should be based on the statistical analysis of the results obtained from experimental viscometric tests. The authors suggest to apply, for the purpose of determination of dimensions of pump and pipe installations, the generalized, non-dimensional criterion (Regen). It allows not only to determine the Darcy friction factor , but also, basing on the knowledge of the generalized Reynolds number Regen, to determine the range of acceptable flow rates ensuring the laminar flow of the analysed homogenous hydromixture. The criterion based on the correlation (Re, He), which also enables the precise determination of the Darcy friction factor , does not allow for the evaluation of the regimen of flow. The evaluation of the flow rate range in laminar flow is another fundamental task of the designer. The lack of possibility to evaluate the critical Reynolds number Recr directly for the Vocadlo model allows the authors to adopt a generally applied value of Recr on the level 2300. For the analysed mixtures of alum sludge the following conclusions, resulting from theoretical considerations on the parameters on their flow, may be drawn: the Vocadlo model describes the rheological properties of mixtures correctly; for the adopted pipeline of a diameter D = 200 mm, the maximum rates vmax for laminar flow fell within the range 0.6­2.4 m/s, depending on the temperature and concentration of the mixtures; the Darcy friction factor varied within the range from 0.0168 to 2.875. depending on the flow rate and the temperature and hydration of the mixtures; the analysed theoretical methods of determination of the Darcy friction factor lead to similar obtained results of calculations and thus can be considered fully applicable in engineering calculations. The models of applied rheology were used to describe the flow characteristics of alum sludge hydromixtures. Importance of the results described here will continuously grow with the development of water treatment plants, increase of water pollution or increasing length of pipelines for the waste conveying. The issues of waste transport and utilization are closely related to environmental engineering in general [3] and, particularly, to sediment management at water treatment plants and sewage treatment plants [5, 15]. The presented work expands the area of applicable knowledge of hydrotransport methods in design of sludge management equipment. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Archives of Environmental Protection de Gruyter

Characteristics of Laminar Flow in Pipelines of Homogenous Alum Sludge Approximated with use of the Vočadlo Model for Viscoplastic Liquids

Download PDF
13 pages

Loading next page...
 
/lp/de-gruyter/characteristics-of-laminar-flow-in-pipelines-of-homogenous-alum-sludge-w18QWNbQGm
Publisher
de Gruyter
Copyright
Copyright © 2014 by the
ISSN
2083-4810
eISSN
2083-4810
DOI
10.2478/aep-2014-0037
Publisher site
See Article on Publisher Site

Abstract

The study presents the manners of determination of the Darcy friction factor for a homogenous hydromixture of alum sludge of varied hydration and temperature for the laminar flow zone. The rheological evaluation of the hydromixture as a viscoplastic body has been conducted with use of measurements of viscosity. The curves of flow were approximated with use of the generalized Vocadlo model. The Darcy friction factor of the pipeline was determined with use of the non-dimensional criterion (Regen) and (Re, He). INTRODUCTION The use of pipeline transport of hydromixtures as an optimum means of transport does not require a detailed justification. However, the determination of hydrotransport parameters influencing the costs of the planned venture may be questionable. The essential parameter of hydrotransport, which enables to determine the pressure loss of the transported mixture is the Darcy friction factor . The knowledge of this parameter enables eventually to conduct the analysis of the operation of the pump and pipeline system and to design it correctly. However, the regimen of flow for the planned transport has to be determined. The analysis of the diagrams of the correlation (Re, ) allows us to determine with certainty that for a given mixture, the critical value of the Reynolds number Recr (transition from laminar flow to the transition zone ­ turbulent flow) corresponds with the minimum value of the Darcy friction factor min for a specific diameter of the pipe. Further slight decrease in the value of the Darcy friction factor in the turbulent zone is however connected with the constant increase in the flow rate, which causes a significant pressure loss in the pipeline. To sum up, the final zone of laminar flow is characterised both by lower flow rates v and by lower values of the Darcy friction factor , which in turn leads to a decrease in the pressure loss in the designed pump and pipeline installation. The economic aspect of pipeline hydrotransport should also be taken into account, as it requires transporting the hydrated material in the most efficient possible way. The fact that hydrotransport is preferred at high concentrations of the solid component is reasonable from the economic point of view. The authors believe that the hydrotransport of homogenous mixtures should take place in the laminar flow zone, due to both hydraulic and economic reasons. In order to provide the proper dimensions of the pump and pipeline installation for hydrotransport it is required to conduct the measurements of the viscosity of the mixture, necessary for the correct determination of the Darcy friction factor for the adopted flow parameters. Thus, the analysed problem is the determination of the rheological parameters of the mixture, and basing on that, of the pressure loss in the pipeline. This issue is important for designing the transport systems of concentrated sediments from water treatment and wastewater treatment [2, 3, 11]. OBJECTIVE OF THE STUDY The aim of the present analysis is to present the methodology of the determination of the rheological parameters and of the Darcy friction factor for a homogenous viscoplastic hydromixture, described by the tri-parametric Vocadlo model, basing on the analysis of the methods suggested by various authors [1, 6, 13, 14]. Physical and rheological properties of the analysed mixtures The study encompasses the sediments taken from alum sludge tanks of the waterworks utility collecting surface waters and using aluminium sulphate in the coagulation process. Apart from the rheological properties, the following characteristics were measured: the temperature, hydration and density of the sludge, chemical composition of the solid component and structure of the sediments. The rheological properties of the sludge were measured with use of a rotational rheometer Rheotest 2. The measurements determined the values of shear stress on the surface of the measurement cylinder for various assumed pseudo deformation rates. As a result of the measurements series of points were obtained, determining the pseudo-curves of flow. The measurements were conducted at the temperatures of 273.45 K and 293.15 K. Constant temperature of the sediment in the rheometer was maintained with use of an ultrathermostat equipped with contact thermometers. The hydration of the sediments was determined by the mass share of water and measured as the difference between the mass of humid sediment and the mass of the sediment dried to constant weight at the temperature of 378 K. The chemical composition of the solid fraction of the sludge was determined with use of chemical analysis, and its structure through the measurement of the isotherms of sorption of CO2 and C6H6 in high-vacuum gravimetric apparatus [13]. The mathematical form of the model and the determination of values of rheological parameters were selected with use of methods of statistical analysis of flow curves. The pseudo-curves of flow were transformed into curves of flow with use of the known Krieger ­ Elrod ­ Maron equation [8, 9, 10]. The values of m and dm/dlogR in this equation were determined from the polynomial approximation function of pseudo-curves of flow, determined in a logarithmic system of co-ordinates (logGp, logR), with use of the classical method of least squares. Polynomial functions and the Bingham model are linear correlations with respect to their constant parameters. Due to this linearity, the application of the least squares method also leads to linear systems of normal equations for these parameters, so the solution thereof does not present any difficulties. On the other hand, the Vocadlo model is a non-linear correlation, which however can be reduced to a linear form. After the switching of the variables a linear form was obtained for new parameters, followed by the estimation of the value of one of the parameters and the application of the least squares method with respect to the linear forms of the analysed models. The limits of flow determined with use of the said models allow to test which of the measurement points fall within the zone of partial shearing of the sediment. Such points were rejected, and the described procedure was repeated until the condition of hydration of the whole sediment sample in the slit of the rheometer was fulfilled. Apart from the calculated values of the rheological parameters of the models, also the values of the correlation coefficient and standard deviation were calculated [13]. Values of the coefficient of correlation ­ R and of the standard deviation ­ S are presented in Table 1 below. Table 1. Statistical parameters of the flow curves Temperature K 293.15 273.45 Hydration % 93.60 88.05 95.05 88.00 Models Bingham R 0.9934 0.9531 0.9904 0.9400 S 1.4039 5.4154 2.1740 8.3210 R 0.9945 0.9985 0.9989 0.9982 Vocadlo S 0.1546 0.4731 0.1251 1.0634 These values enabled the evaluation of the compliance of the specified models with data obtained from the experiments. The tri-parametric model approximated the course of the flow curves of the sediments within the whole range of the shearing rate very well. Moreover, the applicability of the bi-parametric Bingham model for the description of the rheological properties of alum sludge was proven, in particular within the higher hydration range. The accurateness of this approximation decreased noticeably with the decrease in hydration. The results of the analysis of the structure of sediments are presented in Table 2. The presented results allow to classify the analysed alum sludge as a sediment with a well-developed structure. This degree of development of the sediment structure depends on the content of aluminium hydroxide and has a vital influence on the rheological properties of the sludge. Table 2. Physical properties of the sediment structure properties of the structure ultramicropores micropores mesopores (1.5÷3.0) m (3.0÷30) m 3 -1 (30÷100) m total value pore capacity, cm ·g 0.007 0.012 0.011 -1 pore surface area, m ·g 18.4 24.6 9.9 For the purpose of approximation of pseudo-curves of flow the triparametric, generalized Vocadlo model was used (1). = 0 for R 0 R = + K v 0 (1) for R > 0 : R ­ shear stress on the surface of cylinder [Pa] 0 ­ yield stress [Pa] ­ shear rate [s-1] 1 n Kv ­ consistency index for Vocadlo model [ Pa n s ] n ­ flow behaviour index [-] The physical and rheological properties of the analysed mixtures are presented in Table 3. Table 3. Physical and rheological properties of the analysed mixtures Hydration [%] Density [kg/m3] Temperature [K] K V Pa n s n o [Pa] n [-] Determination of the parameters of flow of the mixture For laminar flow of Newtonian liquids the Darcy friction factor is calculated with use of formula (2) 64 Re (2) the Reynolds number Re is calculated with use of equation (3) Re = vD (3) Analogically to the Newtonian liquid, Parzonka [12], Kempiski [6], Sozaski [13] adopt the determination of parameter for non-Newtonian liquids with use of formula (4) 64 Re gen (4) Regen is the generalized Reynolds number for the adopted rheological model. In this case, the determination of the dimensions of the installation is based on the unconditional criterion (Regen). Vocadlo and Charles [14] and Czaban [1] suggest that the determination of dimensions within the laminar zone of the flow should be based on the criterion (Re, He), thus creating a family of curves shifted parallelly depending on the value of the Hedström number He. A similar approach was suggested by Govier and Aziz [4]. Non-dimensional criterion (Regen) The application of this criterion requires the knowledge of the generalized Reynolds number Regen for the adopted rheological model describing the viscoplasticity of homogenous hydromixture. Such hydromixtures are characterised by the occurrence of the so-called yield stress 0. The omission of this value causes errors in the determination of the Reynolds number [7]. The lack of knowledge of Regen led to the abandonment of this method of determination of dimensions. Kempiski [6] presented a full generalized Reynolds number for the Vocadlo model along with a precise approach for the application thereof. The formulas quoted below enable to determine the required stress on the pipeline wall w (6), the generalized Reynolds number (5) and the Darcy friction factor (4). Generalized Reynolds number for the Vocadlo model according to Kempiski [6] p 1 = 4 1+ L 3n 0 6 vK 1+ 1 D n D 0 (5) Shear stress on the surface of the pipeline according to Kempiski [6] w = Dn (2vK )n n 1 0 - 3n + 1 3 w 1 0 3(3n + 1) w 3+ (6) Sozaski [13] presented the generalized Reynolds number Regen (7) for viscoplastic mixtures as a modification of the known Reynolds number for pseudoplastic bodies Re (8), by means of introduction of the plasticity number Lp (9), and relative error p (10). The generalized Reynolds number according to Sozaski [13] is determined with use of the formula (7) Re gen = Re 1 + 1 Lp 6 -n (1 - ) -1 (7) Re = v2-n D n 2 Kn 6+ n 8 (8) The plasticity number (9) D 0n Lp = vK Relative error (10) (9) p = p L (10) Absolute error is determined with use of formula (11) p = p p - L L (11) 1 0 6vK p 1+ = 4 1+ 1 3n D L D 0n (12) this is an approximate value of pressure loss, while p is an exact value of pressure L loss which, according to the methodology adopted by Sozaski [13] can be determined numerically, e.g. by means of halving the range with use of equation (13). p L p 1- L 3 1+ 1+ 3 n (13) 1 3n 1+ 3 n 1+ 6vK D 0n as the first approximation of the searched value. Non-dimensional criterion (Re, He) Vocadlo and Charles [14] determined the friction factor f (Re, X) in form of the formula (14) f = X Re 1 - f 16 n Re 2 n (n - 2 ) 1 X 1+ 1- 3n f Re 6 n-2 (14) f = (15) and Re is the Reynolds number as for the pseudoplastic model, determined by formula (8). The parameter X is determined by the correlation (16) 2n 2-n X = 0 2 2-n 8- n D 1 2Kv 3 + n (16) Czaban [1] suggested a solution based on the correlation (Re, He, ): 64 1 + 3n Re 1 - 3n 1 + 3n n 1 +3 n (17) He = 0 Re 2 v 2 (18) Re is, as in the solution of Vocadlo and Charles, the Reynolds number determined as in the pseudoplastic model (8), and He is the Hedström number (18), 8He Re 2 (19) while can be calculated with use of formula (19). ANALYSIS OF THE RESULTS OF THE CALCULATIONS The initial viscometric measurements enabled us to select the correct rheological model along with the determination of its parameters. This in turn allowed to conduct the calculation of the parameters of hydrotransport, in particular the Darcy friction factor , according to the methodologies discussed here above, suggested by various authors. The parameters of flow of alum sludge were calculated for an adopted pipeline diameter D = 200 mm, within the range of flow rates from 0.1 to 2.4 m/s, depending on the hydration and temperature of the mixture, basing on the adopted critical Reynolds number Recr = 2300. The values of generalized Reynolds numbers Regen and of the Darcy friction factors of the pipeline , calculated basing on the correlations presented by various authors, are presented in Tables 4, 5, 6, 7. The results of calculation presented in form of tables confirm the similarity of methodological solutions proposed by various authors and enable us to calculate the Darcy friction factor in an unequivocal manner. The methodology developed by Kempiski [6] and Sozaski [13] enables however, apart from the determination of the factor, to evaluate the nature of flow of a homogenous hydromixture (laminar or turbulent), basing on the generalized Reynolds number Regen. However, the evaluation of the regimen of flow of viscoplastic mixtures basing on the Reynolds number Re calculated as for pseudoplastic bodies may lead to significant errors [7]. Significant differences between Table 4. Hydraulic parameters of sludge mixture of the hydration of 93.6%, at temperature T = 293.15 K v [m/s] 0.1 0.2 0.4 0.6 0.8 Kempiski Regen 71.5 274 1029 2206 3762 0.8953 0.2338 0.0622 0.0290 0.0170 Sozaski Regen 74.9 283 1050 2236 3799 0.8543 0.2261 0.0609 0.0286 0.0168 Vocadlo Re 1666 3775 8554 13803 19381 0.8947 0.2319 0.0618 0.0290 0.0170 Re Czaban 0.8947 0.2319 0.0618 0.0290 0.0170 1666 3775 8554 13803 19381 Table 5. Hydraulic parameters of sludge mixture of the hydration of 88.05%, at temperature T = 293.15 K v [m/s] 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 Kempiski Regen 42.0 155 328 554 830 1150 1514 1918 2360 1.5222 0.4133 0.1953 0.1155 0.0772 0.0556 0.0423 0.0334 0.0271 Sozaski Regen 42.4 155 328 555 830 1151 1514 1918 2361 1.5084 0.4117 0.1949 0.1154 0.0771 0.0556 0.0423 0.0334 0.0271 Vocadlo Re 77.4 240 465 743 1070 1441 1854 2305 2794 1.5215 0.4132 0.1953 0.1155 0.0771 0.0556 0.0423 0.0334 0.0271 Re Czaban 1.5215 0.4132 0.1953 0.1155 0.0771 0.0556 0.0423 0.0334 0.0271 77.4 240 465 743 1070 1441 1854 2305 2794 Table 6. Hydraulic parameters of sludge mixture of the hydration of 95.05%, at temperature T = 273.45 K v [m/s] 0.2 0.4 0.6 0.8 Kempiski Regen 315 1122 2306 3800 0.2030 0.0571 0.0278 0.0168 Sozaski Regen 318 1126 2310 3803 0.2001 0.0568 0.0278 0.0169 Vocadlo Re 1175 2934 5011 7326 0.2024 0.0568 0.0277 0.0169 Re Czaban 0.2024 0.0568 0.0277 0.0169 1175 2934 5011 7326 Table 7. Hydraulic parameters of sludge mixture of the hydration of 88.00%, at temperature T = 273.45 K v [m/s] 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6 Kempiski Regen 22.3 83.4 179 305 460 643 851 1084 1340 1619 1920 2242 2585 2.875 0.7674 0.3582 0.2098 0.1390 0.0995 0.0752 0.0590 0.0477 0.0395 0.0333 0.0285 0.0248 Sozaski Regen 22.6 84.0 179 306 461 644 852 1085 1341 1620 1921 2243 2585 2.8366 0.7623 0.3568 0.2092 0.1388 0.0994 0.0751 0.0590 0.0477 0.0395 0.0333 0.0285 0.0248 Vocadlo Re 44.0 138 270 435 629 850 1096 1367 1660 1976 2313 2670 3048 2.8745 0.7672 0.3581 0.2097 0.1390 0.0996 0.0752 0.0591 0.0478 0.0395 0.0333 0.0285 0.0248 Re Czaban 2.8745 0.7672 0.3581 0.2097 0.1390 0.0996 0.0752 0.0591 0.0478 0.0395 0.0333 0.0285 0.0248 44.0 138 270 435 629 850 1096 1367 1660 1976 2313 2670 3048 Regen and Re are noticeable. These differences grow with the increase of the rheological parameter n. Figures 1. 2. 3 and 4 present the diagrams of correlations Regen (v) and Re(v) that enable us to calculate the maximum acceptable values of flow rate vmax, that meet the conditions of laminar flow. It is clearly visible that the zone of laminar flow determined basing on the diagram of Regen(v) increases along with the decrease in hydration, thus for the hydrations 95.05% and 93.60% the acceptable actual rate vmax is, respectively, 0.60 m/s and 0.61 m/s, while for lower hydration values 88.05% and 88.00% this rate is, respectively, 1.78 m/s and 2.4 m/s. The acceptable range of flow rates increased 3­4 times in this case. Fig. 1. Correlations Regen(v) and Re(v) for sludge of the hydration of 93.60%, at temperature T = 293.15 K Fig. 2. Correlations Regen(v) and Re(v) for sludge of the hydration of 95.05%, at temperature T = 273.45 K The rates of laminar flow, determined basing on the correlation Re(v) narrow the zone of flow significantly, in particular for high hydration values, limiting the flow to the rate vmax ranging from 0.13 m/s to 0.33 m/s (Figure 1, 2). For lower hydration values the rate vmax ranges from 1.6 m/s to 2.2 m/s (Figures 3, 4). Fig. 3. Correlations Regen(v) and Re(v) for sludge of the hydration of 88.05%, at temperature T = 293.15 K Fig. 4. Correlations Regen(v) and Re(v) for sludge of the hydration of 88.00%, at temperature T = 273.45 K The approach developed by Kempiski [6], based on indirect determination of shear stress on the surface of the pipe w, enables also to determine, with use of this parameter and the value of the yield stress 0, the radius of the core of the cross-section r0 for the laminar flow of homogenous viscoplastic hydromixture. This enables us to obtain deeper knowledge about the nature of flow of the analysed hydromixture. SUMMARY AND CONCLUSIONS The description of rheological properties of homogenous viscoplastic hydromixtures should be based on tri-parametric, generalized rheological models, such as the Vocadlo model, which is simplified to simpler, bi- and uni-parametric models. The selection of the rheological model should be based on the statistical analysis of the results obtained from experimental viscometric tests. The authors suggest to apply, for the purpose of determination of dimensions of pump and pipe installations, the generalized, non-dimensional criterion (Regen). It allows not only to determine the Darcy friction factor , but also, basing on the knowledge of the generalized Reynolds number Regen, to determine the range of acceptable flow rates ensuring the laminar flow of the analysed homogenous hydromixture. The criterion based on the correlation (Re, He), which also enables the precise determination of the Darcy friction factor , does not allow for the evaluation of the regimen of flow. The evaluation of the flow rate range in laminar flow is another fundamental task of the designer. The lack of possibility to evaluate the critical Reynolds number Recr directly for the Vocadlo model allows the authors to adopt a generally applied value of Recr on the level 2300. For the analysed mixtures of alum sludge the following conclusions, resulting from theoretical considerations on the parameters on their flow, may be drawn: the Vocadlo model describes the rheological properties of mixtures correctly; for the adopted pipeline of a diameter D = 200 mm, the maximum rates vmax for laminar flow fell within the range 0.6­2.4 m/s, depending on the temperature and concentration of the mixtures; the Darcy friction factor varied within the range from 0.0168 to 2.875. depending on the flow rate and the temperature and hydration of the mixtures; the analysed theoretical methods of determination of the Darcy friction factor lead to similar obtained results of calculations and thus can be considered fully applicable in engineering calculations. The models of applied rheology were used to describe the flow characteristics of alum sludge hydromixtures. Importance of the results described here will continuously grow with the development of water treatment plants, increase of water pollution or increasing length of pipelines for the waste conveying. The issues of waste transport and utilization are closely related to environmental engineering in general [3] and, particularly, to sediment management at water treatment plants and sewage treatment plants [5, 15]. The presented work expands the area of applicable knowledge of hydrotransport methods in design of sludge management equipment.

Journal

Archives of Environmental Protection – de Gruyter

Published: Dec 23, 2014

There are no references for this article.