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Discharge of Newtonian and Non-Newtonian Liquids from Tanks
0263–8762/06/$30.00+0.00 # 2006 Institution of Chemical Engineers Trans IChemE, Part A, December 2006 Chemical Engineering Research and Design, 84(A12): 1194– 1198
www.icheme.org/cherd doi: 10.1205/cherd.05138
SHORTER COMMUNICATION
DISCHARGE OF NEWTONIAN AND NON-NEWTONIAN LIQUIDS FROM TANKS ´ SKI and A. MARCINKOWSKI M. DZIUBIN Faculty of Process and Environmental Engineering, Lodz Technical University, Lodz, Wolczanska, Poland
R
esults of experiments concerning gravitational discharge of Newtonian and nonNewtonian shear thinning liquids from a tank have been presented. Survey has been carried out for Newtonian liquids of very high viscosity up to 26.2 Pa s and for non-Newtonian liquids with rheological parameters: k ¼ 0.11– 15.1 Pa sn and n ¼ 0.457 – 0.771. Discharge orifices had the diameters 5, 8, 12.5 and 17 mm, and the ratio of orifice length to diameter L/d was 0, 0.35, 0.5, 0.75, 1 and 3. Equations describing discharge coefficient as a function of the Reynolds number and orifice geometry have been proposed. In the case of Newtonian liquid discharge the proposed correlation equation described experimental data with mean error equal to 7.1%, and in the case of non-Newtonian liquid discharge the mean error was equal to 5.7%. Keywords: discharge from tank; Newtonian liquid; non-Newtonian liquid.
INTRODUCTION The subject of the discharge of Newtonian liquids from tanks has been discussed extensively in literature (Medaugh and Johnson, 1940; Lienhard and Lienhard, 1984), except for liquids of very high viscosity (Kiljan´ski, 1993, Dziubin´ski et al., 2003). The discharge of non-Newtonian liquids from tanks has been considered in a very few papers only (Dziubin´ski and Marcinkowski, 2003; SalasValerio and Steffe, 1990). A very narrow range of experimental data has been shown in the paper (Dziubin´ski and Marcinkowski, 2003), moreover it has been presented in conference materials only. The paper (Salas-Valerio and Steffe, 1990) concerns the discharge of non-Newtonian liquids, but the liquids were pumped through a horizontal pipe and discharged through a small orifice in the pipe. Additionaly non-Newtonian liquids used in those investigations represented weak non-Newtonian properties (0.68 , n , 0.80). A similar geometry of the discharge of oil-in-water emulsions through an orifice and Venturi meters was investigated in paper (Pal, 1993). In the case of the discharge of highly viscous liquids, to which small Reynolds numbers correspond, the discharge velocity is strongly limited by inner friction due to which, with a decreasing Reynolds number the coefficient of discharge also decreases. Assuming that flow in this case is laminar and neglecting the kinetic energy of discharged liquid
Correspondence to: Dr M. Dziubin´ski, Faculty of Process and Environmental Engineering, Lodz Technical University, 93-005 Lodz, Wolczanska 213, Poland. E-mail: [email protected]
stream, which is negligible as compared to the energy dispersed due to friction, the energy balance for discharge through a sharp-edged circular orifice is represented as H rg ¼ j
n2 r 2
(1)
where
j¼
K Re
(2)
Substituting equation (2) to equation (1) with the use of the classical Reynolds number definition and the equation that determines discharge velocity from tanks pffiffiffiffiffiffiffiffiffi n ¼ f 2gH (3) we have H rg ¼
K h pffiffiffiffiffiffiffiffiffi f 2gH 2d
After simple transformations we obtain: pffiffiffiffiffiffiffiffiffi 2gH dr Re f¼ ¼ Kf Kh
(4)
(5)
Hence,
1194
pffiffiffiffiffiffi f ¼ K 0 Re
(6)
DISCHARGE OF NEWTONIAN AND NON-NEWTONIAN LIQUIDS FROM TANKS It follows from equation (6) that discharge coefficient f is directly proportional to square root of the Reynolds number. From the review of literature presented above it follows that the subject of a discharge of Newtonian liquids with very high viscosity and non-Newtonian liquids from tanks has been described very scarcely in the literature. The aim of this study is to discuss the gravitational discharge of Newtonian and non-Newtonian liquids from tanks and to propose relations that would describe discharge coefficient f for a broad range of rheological properties of liquids, Reynolds numbers and orifice geometry. METHODS AND MATERIALS The main element of the experimental set-up was a cylindrical glass tank 0.2 m in diameter. Discharge orifices were situated in the steel bottom of the tank. They had the diameters 5, 8, 12.5 and 17 mm, and the ratio of orifice length to diameter L/d was 0, 0.35, 0.5, 0.75, 1 and 3. The orifices were equipped with electronically opened and shut valves. To determine precisely the mass of discharged liquid, laboratory scales were used. Rheological properties of the applied media were determined by a Bohlin CVO 120 rotary rheometer (UK). The Newtonian media were water, ethylene glycol and water solutions of starch syrup, while the non-Newtonian liquids were water solutions of carboxymethylcellulose CMC. Table 1 shows rheological properties of these media. Rheological parameters of the non-Newtonian liquids were determined in the shear rate range covering the experimental values. Normal stresses differences exhibited by the experimental media measured by the Bohlin CVO 120 rotational rheometer were close to zero in the examined range of shear rate. Results of the measurements and the difference of normal stresses showed that experimental media did not exhibit viscoelastic properties. Classical relations concerning a gravitational discharge of liquids from tanks determine the discharge time as
t¼
pffiffiffiffiffiffi pffiffiffiffiffiffi 2D2 ( H0 H1 ) pffiffiffiffiffi fd2 2g
(7)
1195
The Reynolds number for Newtonian liquid discharge from tanks was defined in a classical way, while in the case of non-Newtonian liquids the definition proposed in (Metzner and Reed, 1955) was applied ReMR ¼
n2n d n r k(ð3n þ 1Þ=4n)n 8n1
(9)
RESULTS AND DISCUSSION Discharge of Newtonian Liquids from Tanks Figure 1 illustrates the values of discharge coefficient f calculated from equation (8) as a function of the Reynolds number Re and orifice geometry L/d for the discharge of Newtonian liquids used in the experiments. From analysis of this diagram it follows that the discharge coefficient f grows with an increase of the Reynolds number and shows a tendency to become constant at Re . 100. For majority of the experimental data the value of f was about 0.62 in this range of the Reynolds number. It confirms results of similar studies carried out by other authors (Kiljan´ski, 1993; Troskolan´ski, 1954; Prosnak, 1970; Bolshakov, 1977; Agroskin et al., 1954). At Re , 100, experimental data show that the value of f depends on the Reynolds number and discharge orifice geometry. For Re , 10, the experimental points can be approximated by the curves that are described by the power equation:
f ¼ BReC
(10)
The constants B and C in equation (10) do not depend on the discharged liquid type and they change according to the orifice geometry assuming the values given in Table 2. As follows from Table 2, the values of coefficient C are close to 0.5 which is in agreement with the previous theoretical consideration of liquid discharge from tanks—cf. equation (6). After assuming coefficient C equal to 0.5, we obtain pffiffiffiffiffiffi f ¼ B0 Re
(11)
The discharge coefficient f was calculated from a relation resulting from equation (7): pffiffiffiffiffiffi pffiffiffiffiffiffi 2D2 ( H0 H1 ) pffiffiffiffiffi f¼ td 2 2g
(8)
Table 1. Properties of the liquids. Liquid Water Ethylene glycol Starch syrup solutions 1.6 wt% CMC 3.1 wt% CMC 3.5 wt% CMC 4.0 wt% CMC 4.5 wt% CMC 5.0 wt% CMC
Density, kg m23
Rheological properties
Shear rate, s21
998.9 1101 1270– 1391
h ¼ 1.14 mPa s h ¼ 15.9 mPa s h ¼ 0.1–26.2 Pa s
— 100–3400 4 –2800
k ¼ 0.11 Pa sn, n ¼ 0.771 k ¼ 1.45 Pa sn, n ¼ 0.606 k ¼ 4.54 Pa sn, n ¼ 0.531 k ¼ 6.53 Pa sn, n ¼ 0.512 k ¼ 10.6 Pa sn, n ¼ 0.471 k ¼ 15.1 Pa sn, n ¼ 0.457
300–3000 180–2700 50 –2000 30 –1800 20 –1400 10 –800
1009 1016 1020 1023 1027 1027
Figure 1. Dependence of discharge coefficient on Reynolds number. This figure is available in colour online via www.icheme.org/cherd
Trans IChemE, Part A, Chemical Engineering Research and Design, 2006, 84(A12): 1194 –1198
´ SKI AND MARCINKOWSKI DZIUBIN
1196
Table 2. Coefficients B and C in equation (10). L/d B C
0
0.35
0.5
0.75
1
3
0.179 0.468
0.149 0.505
0.136 0.479
0.110 0.477
0.109 0.506
0.0804 0.483
Coefficient B0 depends only on the ratio of orifice length to diameter L/d. Its value is approximated by the equation B0 ¼ A1 þ A 2
A3 L d
(12)
By correlating experimental data the values of coefficients A1, A2 and A3 in equation (12) were obtained. On this basis the following correlation equation describing discharge coefficient f at the flow of Newtonian liquid from tanks through a small cylindrical orifice was proposed: "
0:333 # pffiffiffiffiffiffi L f ¼ 0:186 0:0756 Re d
(13)
Figure 2 shows a comparison of discharge coefficient f obtained experimentally and calculated from equation (13). The correlation coefficient for 365 experimental points was R 2 ¼ 0.986, while mean relative error of the description of experimental data by equation (13) was equal to 7.1%. Curves defined by equation (13) are shown in Figure 1 as solid lines. Equation (13) holds for the range: 0.005 m , d , 0.017 m; 0 , L/d , 3; 0.273 Pa s , h , 26.2 Pa s; 0.00226 , Re , 10. The generalized equation (13) proposed in this study describes discharge coefficient f in a broad range of liquid discharge velocity from tanks, liquid properties and the discharge orifice geometry. Figure 3 illustrates a comparison of our experimental data with relevant data presented elsewhere (Kiljan´ski,
Figure 3. Comparison of our experimental data and the data discussed in literature (Kiljan´ski, 1993).
1993). In the studies, the tested liquids had viscosities around 0.02, 0.15, 0.40 and 10 Pa s, the discharge orifices had diameters 2, 3 and 5 mm and L/d ratio equal to 0.5. Additionally, two orifices 3 mm in diameter were applied. They were characterized by a different geometry L/d ¼ 0 and L/d ¼ 1. The experimental data discussed by (Kiljan´ski, 1993) were obtained for the Reynolds number 0.01 , Re , 462. Points in Figure 3 represent experimental data given in (Kiljan´ski, 1993), while solid lines stand for the proposed correlation equation (13). When analysing Figure 3, it can be found that there is good agreement between the experimental data. This is also confirmed by a high correlation coefficient of the experimental data published in study (Kiljan´ski, 1993) correlated by equation (13). In the case of the discharge orifice geometry L/d ¼ 0.5, equation (13) describes very well the experimental data. The presented experimental points overlap the curve of equation (13). In this case the correlation coefficient assumed the value R 2 ¼ 0.998. Very good agreement for the geometry L/d ¼ 0.5 is the more significant as most experimental data presented in study (Kiljan´ski, 1993) referred to that geometry. In the case of the other two orifice geometries with the ratio L/d ¼ 0 and L/d ¼ 1, the discharge coefficients for the compared data are slightly lower than the values calculated from equation (13). However, it is worth emphasizing that in our experiments the range of Reynolds number was from 2.26 1023 to 4.69 104 and was much broader than that obtained in study (Kiljan´ski, 1993). Contrary to the correlation equations given in study (Kiljan´ski, 1993), equation (13) has a general character. In a broad range of changes in process parameters it makes dependent the value of coefficient f on the Reynolds number and L/d ratio. (Kiljan´ski, 1993) proposed some very simple equations that hold only for particular orifice geometries L/d.
Discharge of Non-Newtonian Liquids from Tanks Figure 2. Comparison of discharge coefficients of Newtonian liquids flowing from tanks obtained experimentally and calculated from equation (13).
Figure 4 shows discharge coefficient f calculated in equation (8) as a function of generalised Metzner – Reed
Trans IChemE, Part A, Chemical Engineering Research and Design, 2006, 84(A12): 1194– 1198
DISCHARGE OF NEWTONIAN AND NON-NEWTONIAN LIQUIDS FROM TANKS
1197
equation describing discharge coefficient f was proposed: " 0:48 # L f ¼ 0:101 0:0164 (14) Re0:426 MR d
Figure 4. Discharge coefficient f versus generalized Reynolds number ReMR for selected L/d values. This figure is available in colour online via www.icheme.org/cherd
Curves described by equation (14) are shown in Figure 4 as solid lines. Figure 5 presents a comparison of discharge coefficient f obtained experimentally and calculated from equation (14). The correlation coefficient for 240 experimental points obtained in this study was R 2 ¼ 0.991, and mean relative error was equal to 5.7%. The proposed equation (14) that describes discharge coefficient f of power-law liquids from tanks in a broad range of discharge velocity changes and geometries, has not been presented in literature so far. It was developed for the range: 0.005 m , d , 0.017 m; 0 , L/d , 3; 0.457 , n , 0.606; 1.45 Pa sn , k , 15.1 Pa sn; 0.0495 , ReMR , 100. CONCLUSIONS
Reynolds number ReMR—equation (9) and orifice geometry L/d for non-Newtonian liquids with different rheological properties. As in the case of discharge of Newtonian liquids from tanks, from the analysis of this diagram it follows that irrespective of the type of discharged liquid, experimental points are arranged along the curve described by a constant L/d ratio. On the basis of analysis of Figure 4 it may be concluded that discharge coefficient f increases with an increase of the Reynolds number and shows a tendency to become constant at ReMR . 100. The mean value of discharge coefficient f in this range of the Reynolds number is 0.67. So, this is similar to the value of coefficient f in the case of Newtonian liquid discharge from tanks. For the Reynolds number ReMR , 100, the experimental data were described as in the case of the discharge of Newtonian liquids from tanks. The following correlation
(1) Extensive experimental investigations not available in literature so far, were performed and analysed. The experiments covered over 1600 experimental points referring to the discharge of Newtonian and nonNewtonian liquids from tanks. (2) Original correlation equations (13) and (14) were proposed to describe the discharge coefficient for Newtonian and non-Newtonian liquids flowing from tanks, depending on the Reynolds number and orifice geometry: (a) in the case of Newtonian liquid discharge the proposed correlation equation (13) described a broad range of our experimental data with mean error equal to 7.1%, (b) in the case of non-Newtonian liquid discharge the proposed correlation equation (14) is the first such equation in literature; it describes a wide range of experimental data with mean error equal to 5.7%. (3) Applicability of numerical simulations for the determination of Newtonian and non-Newtonian liquid discharge parameters was confirmed. NOMENCLATURE A1, A2, A3, B, B0 , C d D g H H0 H1 0 1 K, K0 k, n
Figure 5. Comparison of discharge coefficients of non-Newtonian liquids obtained experimentally and calculated from equation (14).
L R Re ReMR v h r
constants orifice diameter, m tank diameter, m acceleration due to gravity, m s22 water head, m initial water head, m final water head, m initial value final value constants rheological parameters of Ostwald-de Waele power-law model, Pa sn length of an orifice, m correlation coefficient Reynolds number Metzner–Reed Reynolds number average liquid velocity in the orifice, m s21 liquid viscosity, Pa s liquid density, kg m23
Trans IChemE, Part A, Chemical Engineering Research and Design, 2006, 84(A12): 1194 –1198
´ SKI AND MARCINKOWSKI DZIUBIN
1198 t f j
discharge time, s discharge coefficient local drag coefficient
REFERENCES Agroskin, I.I., Dmitrev, G.T. and Pikalov, F.I., 1954, Gidravlika (Moscow, Leningrad). Bolshakov, W.A., 1977, Spravotshnik po gidravlikie (Moscow). Dziubin´ski, M., Kiljan´ski, T. and Marcinkowski, A., 2003, Discharge of highly viscous liquids from tanks, Inz˙ Apar Chem (in Polish), 3s: 35-36. Dziubin´ski, M. and Marcinkowski, A., 2003, The discharge of nonNewtonian liquids from tanks, 4th European Congress of Chemical Engineering, Grenada, Paper P.5.5-014. Kiljan´ski, T., 1993, Discharge coefficient for free jets from orifices at low Reynolds number, Transactions of the ASME, 115: 778–781. Lienhard, J.H. and Lienhard, J.H., 1984, Velocity coefficient for free jets from sharp-edged orifices, Journal of Fluids Engineering, 106: 13 –17. Medaugh, F.W. and Johnson, G.D., 1940, Investigation of the discharge and coefficients of small circular orifices, Civil Engineering, 10(7): 422– 424.
Metzner, A.B. and Reed, J.C., 1955, Flow of non-Newtonian fluids— correlation of the laminar, transition and turbulent flow regions, AIChE J, 1: 434 –440. Pal, R., 1993, Flow of oil-in-water emulsions through orifice and venturi meters, Ind Eng Chem Res, 32(6): 1212 –1217. Prosnak, W.J., 1970, Fluid Mechanics (PWN, Warsaw, Poland). Salas-Valerio, W. and Steffe, J., 1990, Orifice discharge coefficients for power-law fluids, J Food Proc Eng, 12: 89 –98. Troskolan´ski, A.T., 1954, Technical Hydromechanics Vol. 2 (PWT, Warsaw, Poland).
ACKNOWLEDGEMENT This study was financed by the Polish State Committee for Scientific Research within grant no. 4 T09C 019 23. The manuscript was received 27 June 2005 and accepted for publication after revision 28 July 2006.
Trans IChemE, Part A, Chemical Engineering Research and Design, 2006, 84(A12): 1194– 1198
www.icheme.org/cherd doi: 10.1205/cherd.05138
SHORTER COMMUNICATION
DISCHARGE OF NEWTONIAN AND NON-NEWTONIAN LIQUIDS FROM TANKS ´ SKI and A. MARCINKOWSKI M. DZIUBIN Faculty of Process and Environmental Engineering, Lodz Technical University, Lodz, Wolczanska, Poland
R
esults of experiments concerning gravitational discharge of Newtonian and nonNewtonian shear thinning liquids from a tank have been presented. Survey has been carried out for Newtonian liquids of very high viscosity up to 26.2 Pa s and for non-Newtonian liquids with rheological parameters: k ¼ 0.11– 15.1 Pa sn and n ¼ 0.457 – 0.771. Discharge orifices had the diameters 5, 8, 12.5 and 17 mm, and the ratio of orifice length to diameter L/d was 0, 0.35, 0.5, 0.75, 1 and 3. Equations describing discharge coefficient as a function of the Reynolds number and orifice geometry have been proposed. In the case of Newtonian liquid discharge the proposed correlation equation described experimental data with mean error equal to 7.1%, and in the case of non-Newtonian liquid discharge the mean error was equal to 5.7%. Keywords: discharge from tank; Newtonian liquid; non-Newtonian liquid.
INTRODUCTION The subject of the discharge of Newtonian liquids from tanks has been discussed extensively in literature (Medaugh and Johnson, 1940; Lienhard and Lienhard, 1984), except for liquids of very high viscosity (Kiljan´ski, 1993, Dziubin´ski et al., 2003). The discharge of non-Newtonian liquids from tanks has been considered in a very few papers only (Dziubin´ski and Marcinkowski, 2003; SalasValerio and Steffe, 1990). A very narrow range of experimental data has been shown in the paper (Dziubin´ski and Marcinkowski, 2003), moreover it has been presented in conference materials only. The paper (Salas-Valerio and Steffe, 1990) concerns the discharge of non-Newtonian liquids, but the liquids were pumped through a horizontal pipe and discharged through a small orifice in the pipe. Additionaly non-Newtonian liquids used in those investigations represented weak non-Newtonian properties (0.68 , n , 0.80). A similar geometry of the discharge of oil-in-water emulsions through an orifice and Venturi meters was investigated in paper (Pal, 1993). In the case of the discharge of highly viscous liquids, to which small Reynolds numbers correspond, the discharge velocity is strongly limited by inner friction due to which, with a decreasing Reynolds number the coefficient of discharge also decreases. Assuming that flow in this case is laminar and neglecting the kinetic energy of discharged liquid
Correspondence to: Dr M. Dziubin´ski, Faculty of Process and Environmental Engineering, Lodz Technical University, 93-005 Lodz, Wolczanska 213, Poland. E-mail: [email protected]
stream, which is negligible as compared to the energy dispersed due to friction, the energy balance for discharge through a sharp-edged circular orifice is represented as H rg ¼ j
n2 r 2
(1)
where
j¼
K Re
(2)
Substituting equation (2) to equation (1) with the use of the classical Reynolds number definition and the equation that determines discharge velocity from tanks pffiffiffiffiffiffiffiffiffi n ¼ f 2gH (3) we have H rg ¼
K h pffiffiffiffiffiffiffiffiffi f 2gH 2d
After simple transformations we obtain: pffiffiffiffiffiffiffiffiffi 2gH dr Re f¼ ¼ Kf Kh
(4)
(5)
Hence,
1194
pffiffiffiffiffiffi f ¼ K 0 Re
(6)
DISCHARGE OF NEWTONIAN AND NON-NEWTONIAN LIQUIDS FROM TANKS It follows from equation (6) that discharge coefficient f is directly proportional to square root of the Reynolds number. From the review of literature presented above it follows that the subject of a discharge of Newtonian liquids with very high viscosity and non-Newtonian liquids from tanks has been described very scarcely in the literature. The aim of this study is to discuss the gravitational discharge of Newtonian and non-Newtonian liquids from tanks and to propose relations that would describe discharge coefficient f for a broad range of rheological properties of liquids, Reynolds numbers and orifice geometry. METHODS AND MATERIALS The main element of the experimental set-up was a cylindrical glass tank 0.2 m in diameter. Discharge orifices were situated in the steel bottom of the tank. They had the diameters 5, 8, 12.5 and 17 mm, and the ratio of orifice length to diameter L/d was 0, 0.35, 0.5, 0.75, 1 and 3. The orifices were equipped with electronically opened and shut valves. To determine precisely the mass of discharged liquid, laboratory scales were used. Rheological properties of the applied media were determined by a Bohlin CVO 120 rotary rheometer (UK). The Newtonian media were water, ethylene glycol and water solutions of starch syrup, while the non-Newtonian liquids were water solutions of carboxymethylcellulose CMC. Table 1 shows rheological properties of these media. Rheological parameters of the non-Newtonian liquids were determined in the shear rate range covering the experimental values. Normal stresses differences exhibited by the experimental media measured by the Bohlin CVO 120 rotational rheometer were close to zero in the examined range of shear rate. Results of the measurements and the difference of normal stresses showed that experimental media did not exhibit viscoelastic properties. Classical relations concerning a gravitational discharge of liquids from tanks determine the discharge time as
t¼
pffiffiffiffiffiffi pffiffiffiffiffiffi 2D2 ( H0 H1 ) pffiffiffiffiffi fd2 2g
(7)
1195
The Reynolds number for Newtonian liquid discharge from tanks was defined in a classical way, while in the case of non-Newtonian liquids the definition proposed in (Metzner and Reed, 1955) was applied ReMR ¼
n2n d n r k(ð3n þ 1Þ=4n)n 8n1
(9)
RESULTS AND DISCUSSION Discharge of Newtonian Liquids from Tanks Figure 1 illustrates the values of discharge coefficient f calculated from equation (8) as a function of the Reynolds number Re and orifice geometry L/d for the discharge of Newtonian liquids used in the experiments. From analysis of this diagram it follows that the discharge coefficient f grows with an increase of the Reynolds number and shows a tendency to become constant at Re . 100. For majority of the experimental data the value of f was about 0.62 in this range of the Reynolds number. It confirms results of similar studies carried out by other authors (Kiljan´ski, 1993; Troskolan´ski, 1954; Prosnak, 1970; Bolshakov, 1977; Agroskin et al., 1954). At Re , 100, experimental data show that the value of f depends on the Reynolds number and discharge orifice geometry. For Re , 10, the experimental points can be approximated by the curves that are described by the power equation:
f ¼ BReC
(10)
The constants B and C in equation (10) do not depend on the discharged liquid type and they change according to the orifice geometry assuming the values given in Table 2. As follows from Table 2, the values of coefficient C are close to 0.5 which is in agreement with the previous theoretical consideration of liquid discharge from tanks—cf. equation (6). After assuming coefficient C equal to 0.5, we obtain pffiffiffiffiffiffi f ¼ B0 Re
(11)
The discharge coefficient f was calculated from a relation resulting from equation (7): pffiffiffiffiffiffi pffiffiffiffiffiffi 2D2 ( H0 H1 ) pffiffiffiffiffi f¼ td 2 2g
(8)
Table 1. Properties of the liquids. Liquid Water Ethylene glycol Starch syrup solutions 1.6 wt% CMC 3.1 wt% CMC 3.5 wt% CMC 4.0 wt% CMC 4.5 wt% CMC 5.0 wt% CMC
Density, kg m23
Rheological properties
Shear rate, s21
998.9 1101 1270– 1391
h ¼ 1.14 mPa s h ¼ 15.9 mPa s h ¼ 0.1–26.2 Pa s
— 100–3400 4 –2800
k ¼ 0.11 Pa sn, n ¼ 0.771 k ¼ 1.45 Pa sn, n ¼ 0.606 k ¼ 4.54 Pa sn, n ¼ 0.531 k ¼ 6.53 Pa sn, n ¼ 0.512 k ¼ 10.6 Pa sn, n ¼ 0.471 k ¼ 15.1 Pa sn, n ¼ 0.457
300–3000 180–2700 50 –2000 30 –1800 20 –1400 10 –800
1009 1016 1020 1023 1027 1027
Figure 1. Dependence of discharge coefficient on Reynolds number. This figure is available in colour online via www.icheme.org/cherd
Trans IChemE, Part A, Chemical Engineering Research and Design, 2006, 84(A12): 1194 –1198
´ SKI AND MARCINKOWSKI DZIUBIN
1196
Table 2. Coefficients B and C in equation (10). L/d B C
0
0.35
0.5
0.75
1
3
0.179 0.468
0.149 0.505
0.136 0.479
0.110 0.477
0.109 0.506
0.0804 0.483
Coefficient B0 depends only on the ratio of orifice length to diameter L/d. Its value is approximated by the equation B0 ¼ A1 þ A 2
A3 L d
(12)
By correlating experimental data the values of coefficients A1, A2 and A3 in equation (12) were obtained. On this basis the following correlation equation describing discharge coefficient f at the flow of Newtonian liquid from tanks through a small cylindrical orifice was proposed: "
0:333 # pffiffiffiffiffiffi L f ¼ 0:186 0:0756 Re d
(13)
Figure 2 shows a comparison of discharge coefficient f obtained experimentally and calculated from equation (13). The correlation coefficient for 365 experimental points was R 2 ¼ 0.986, while mean relative error of the description of experimental data by equation (13) was equal to 7.1%. Curves defined by equation (13) are shown in Figure 1 as solid lines. Equation (13) holds for the range: 0.005 m , d , 0.017 m; 0 , L/d , 3; 0.273 Pa s , h , 26.2 Pa s; 0.00226 , Re , 10. The generalized equation (13) proposed in this study describes discharge coefficient f in a broad range of liquid discharge velocity from tanks, liquid properties and the discharge orifice geometry. Figure 3 illustrates a comparison of our experimental data with relevant data presented elsewhere (Kiljan´ski,
Figure 3. Comparison of our experimental data and the data discussed in literature (Kiljan´ski, 1993).
1993). In the studies, the tested liquids had viscosities around 0.02, 0.15, 0.40 and 10 Pa s, the discharge orifices had diameters 2, 3 and 5 mm and L/d ratio equal to 0.5. Additionally, two orifices 3 mm in diameter were applied. They were characterized by a different geometry L/d ¼ 0 and L/d ¼ 1. The experimental data discussed by (Kiljan´ski, 1993) were obtained for the Reynolds number 0.01 , Re , 462. Points in Figure 3 represent experimental data given in (Kiljan´ski, 1993), while solid lines stand for the proposed correlation equation (13). When analysing Figure 3, it can be found that there is good agreement between the experimental data. This is also confirmed by a high correlation coefficient of the experimental data published in study (Kiljan´ski, 1993) correlated by equation (13). In the case of the discharge orifice geometry L/d ¼ 0.5, equation (13) describes very well the experimental data. The presented experimental points overlap the curve of equation (13). In this case the correlation coefficient assumed the value R 2 ¼ 0.998. Very good agreement for the geometry L/d ¼ 0.5 is the more significant as most experimental data presented in study (Kiljan´ski, 1993) referred to that geometry. In the case of the other two orifice geometries with the ratio L/d ¼ 0 and L/d ¼ 1, the discharge coefficients for the compared data are slightly lower than the values calculated from equation (13). However, it is worth emphasizing that in our experiments the range of Reynolds number was from 2.26 1023 to 4.69 104 and was much broader than that obtained in study (Kiljan´ski, 1993). Contrary to the correlation equations given in study (Kiljan´ski, 1993), equation (13) has a general character. In a broad range of changes in process parameters it makes dependent the value of coefficient f on the Reynolds number and L/d ratio. (Kiljan´ski, 1993) proposed some very simple equations that hold only for particular orifice geometries L/d.
Discharge of Non-Newtonian Liquids from Tanks Figure 2. Comparison of discharge coefficients of Newtonian liquids flowing from tanks obtained experimentally and calculated from equation (13).
Figure 4 shows discharge coefficient f calculated in equation (8) as a function of generalised Metzner – Reed
Trans IChemE, Part A, Chemical Engineering Research and Design, 2006, 84(A12): 1194– 1198
DISCHARGE OF NEWTONIAN AND NON-NEWTONIAN LIQUIDS FROM TANKS
1197
equation describing discharge coefficient f was proposed: " 0:48 # L f ¼ 0:101 0:0164 (14) Re0:426 MR d
Figure 4. Discharge coefficient f versus generalized Reynolds number ReMR for selected L/d values. This figure is available in colour online via www.icheme.org/cherd
Curves described by equation (14) are shown in Figure 4 as solid lines. Figure 5 presents a comparison of discharge coefficient f obtained experimentally and calculated from equation (14). The correlation coefficient for 240 experimental points obtained in this study was R 2 ¼ 0.991, and mean relative error was equal to 5.7%. The proposed equation (14) that describes discharge coefficient f of power-law liquids from tanks in a broad range of discharge velocity changes and geometries, has not been presented in literature so far. It was developed for the range: 0.005 m , d , 0.017 m; 0 , L/d , 3; 0.457 , n , 0.606; 1.45 Pa sn , k , 15.1 Pa sn; 0.0495 , ReMR , 100. CONCLUSIONS
Reynolds number ReMR—equation (9) and orifice geometry L/d for non-Newtonian liquids with different rheological properties. As in the case of discharge of Newtonian liquids from tanks, from the analysis of this diagram it follows that irrespective of the type of discharged liquid, experimental points are arranged along the curve described by a constant L/d ratio. On the basis of analysis of Figure 4 it may be concluded that discharge coefficient f increases with an increase of the Reynolds number and shows a tendency to become constant at ReMR . 100. The mean value of discharge coefficient f in this range of the Reynolds number is 0.67. So, this is similar to the value of coefficient f in the case of Newtonian liquid discharge from tanks. For the Reynolds number ReMR , 100, the experimental data were described as in the case of the discharge of Newtonian liquids from tanks. The following correlation
(1) Extensive experimental investigations not available in literature so far, were performed and analysed. The experiments covered over 1600 experimental points referring to the discharge of Newtonian and nonNewtonian liquids from tanks. (2) Original correlation equations (13) and (14) were proposed to describe the discharge coefficient for Newtonian and non-Newtonian liquids flowing from tanks, depending on the Reynolds number and orifice geometry: (a) in the case of Newtonian liquid discharge the proposed correlation equation (13) described a broad range of our experimental data with mean error equal to 7.1%, (b) in the case of non-Newtonian liquid discharge the proposed correlation equation (14) is the first such equation in literature; it describes a wide range of experimental data with mean error equal to 5.7%. (3) Applicability of numerical simulations for the determination of Newtonian and non-Newtonian liquid discharge parameters was confirmed. NOMENCLATURE A1, A2, A3, B, B0 , C d D g H H0 H1 0 1 K, K0 k, n
Figure 5. Comparison of discharge coefficients of non-Newtonian liquids obtained experimentally and calculated from equation (14).
L R Re ReMR v h r
constants orifice diameter, m tank diameter, m acceleration due to gravity, m s22 water head, m initial water head, m final water head, m initial value final value constants rheological parameters of Ostwald-de Waele power-law model, Pa sn length of an orifice, m correlation coefficient Reynolds number Metzner–Reed Reynolds number average liquid velocity in the orifice, m s21 liquid viscosity, Pa s liquid density, kg m23
Trans IChemE, Part A, Chemical Engineering Research and Design, 2006, 84(A12): 1194 –1198
´ SKI AND MARCINKOWSKI DZIUBIN
1198 t f j
discharge time, s discharge coefficient local drag coefficient
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Metzner, A.B. and Reed, J.C., 1955, Flow of non-Newtonian fluids— correlation of the laminar, transition and turbulent flow regions, AIChE J, 1: 434 –440. Pal, R., 1993, Flow of oil-in-water emulsions through orifice and venturi meters, Ind Eng Chem Res, 32(6): 1212 –1217. Prosnak, W.J., 1970, Fluid Mechanics (PWN, Warsaw, Poland). Salas-Valerio, W. and Steffe, J., 1990, Orifice discharge coefficients for power-law fluids, J Food Proc Eng, 12: 89 –98. Troskolan´ski, A.T., 1954, Technical Hydromechanics Vol. 2 (PWT, Warsaw, Poland).
ACKNOWLEDGEMENT This study was financed by the Polish State Committee for Scientific Research within grant no. 4 T09C 019 23. The manuscript was received 27 June 2005 and accepted for publication after revision 28 July 2006.
Trans IChemE, Part A, Chemical Engineering Research and Design, 2006, 84(A12): 1194– 1198