Computational Fluid Dynamics Simulation of Flows in an Oxidation Ditch Driven by a New Surface Aerator

Abstract

In this article, we present a newly designed inverse umbrella surface aerator, and tested its performance in driving flow of an oxidation ditch. Results show that it has a better performance in driving the oxidation ditch than the original one with higher average velocity and more uniform flow field. We also present a computational fluid dynamics model for predicting the flow field in an oxidation ditch driven by a surface aerator. The improved momentum source term approach to simulate the flow field of the oxidation ditch driven by an inverse umbrella surface aerator was developed and validated through experiments. Four kinds of turbulent models were investigated with the approach, including the standard k−ɛ model, RNG k−ɛ model, realizable k−ɛ model, and Reynolds stress model, and the predicted data were compared with those calculated with the multiple rotating reference frame approach (MRF) and sliding mesh approach (SM). Results of the momentum source term approach are in good agreement with the experimental data, and its prediction accuracy is better than MRF, close to SM. It is also found that the momentum source term approach has lower computational expenses, is simpler to preprocess, and is easier to use.

Introduction

An oxidation ditch is a kind of artificial closed-loop open channel, in which the wastewater is driven by the impeller of surface mechanical aerators and mixed with activated sludge and dissolved oxygen for the biological treatment. Presently, the oxidation ditch process is one of the widely used wastewater treatment methods. However, with the presence of the bend channel, the flow patterns in the ditch are complex and the distribution of the velocity is inhomogeneous. Thus, the areas with a lower velocity may appear and the activated sludge would be induced to settle down in these regions, which will deteriorate the performance of the oxidation ditch process. As a result, it is very important to understand the flow patterns and hydrodynamic characteristics for the design and operation of oxidation ditches (Littleton and Daigger, 2001; Littleton et al., 2007).

With the help of computational fluid dynamics (CFD), the complex flow in the ditch can be investigated, thus attracting a lot of attention (Glover et al., 2006). Some mathematical models for the hydrodynamic simulation of oxidation ditches have been presented. De Clercq et al. (1999), Lesage et al. (2003), and Stamou (1997) studied simple hydrodynamic features in oxidation ditches by using a one-dimensional model. Stamou (1993) applied a two-dimensional vertical averaged k−ɛ turbulence model in an oxidation ditch simulation. Littleton and Daigger (2001) applied a three-dimensional (3D) fluid dynamic model for a rotor disc test tank and a full-scale Orbal oxidation ditch to investigate fluid flow features by imparting a fraction of the known momentum source of the fluid, calculated from the measured flow velocity in the oxidation ditch (Littleton et al., 2007). Luo et al. (2003, 2005) used a 3D standard k−ɛ turbulence model and the moving mesh approach to simulate the motion of the brush aerators and their interaction with the fluid in a small-scale ditch. Fan et al. (2010) simulated the 3D solid–liquid two-phase flow field in a laboratory-scale oxidation ditch aerated with surface aerators using the multiple rotating reference frame approach (MRF) and the standard k−ɛ turbulence model. Yang et al. (2011) predicted the flow field in a full-scale Carrousel oxidation ditch with the 3D standard k−ɛ turbulence model and a modified moving mesh approach. In previous study, we have proposed a momentum source term approach, only based on dimensions and the rotational velocity of aerators, to calculate the flow field in a full-scale Carrousel oxidation ditch (Jiang et al., 2010).

Recently, we further developed the momentum source term approach, in which the friction resistance between the fluid and the impeller has been included (Li, 2011). We applied the improved approach to calculate the flow field of the Ruston turbine stirred tank. The prediction results agreed well with the experimental data (Murthy and Joshi, 2008), and its accuracy was much better than MRF and similar to the sliding mesh approach (SM), but with higher computational speed and simpler preprocessing (Li, 2011).

In the present study, we designed a new inverse umbrella surface aerator, and tested the flow field in oxidation ditch driven by the surface aerator. We also developed a CFD model for the aerator. The improved momentum source term approach was applied to simulate the flow field in the oxidation ditch with the surface aerator, and the predictive capabilities of four turbulence models, that is, the standard k−ɛ model, RNG k−ɛ model, realizable k−ɛ model, and Reynolds stress model (RSM), were investigated. The validity of the momentum source term approach in simulating the flow field of the oxidation ditch was verified with the experimental data. The calculated flow velocity distribution and predicting power inputs were also compared with those of MRF and SM.

Materials and Methods

Experimental setup and methods

In this work, a test oxidation ditch located at Anhui Guozhen Environmental Protection Science and Technology Co., Ltd. was investigated. The ditch width was 19 m, and the water depth was 4.240 m. The total horizontal flow distance around the ditch at its centerline was ∼111 m, and the liquid volume in the oxidation ditch was about 3300 m³. The details of the oxidation ditch are provided in Figure 1a.

FIG. 1.

Detailed information of the experiment: (a) planform of the oxidation ditch; (b) measuring locations of the cross section; (c) front view of the DSC325 inverse umbrella surface aerator (unit: mm); (d) front view of the original surface aerator.

A new inverse umbrella surface aerator, DSC325, was designed from an original aerator as shown in Figure 1c and d. The impeller of the aerator has seven blades that are 1.132 m long and 0.974 m high. It has a higher height and a smaller diameter than the original aerator. The aerator was installed at the water level of the oxidation ditch as shown in Figure 1. The aerator was eccentrically installed, and the distance between the center axis of the aerator and the midline of the central wall was 400 mm. In the experiment, the actual rotational velocity N=29.01 rpm.

The measured velocity data were obtained at four test cross sections in the straight channel as shown in Figure 1a (Test locations 1–4). With the middle cross section x=0 m, Test location 1 was located at x=3.4 m, Test locations 2 and 3 were both located at x=16 m, and Test location 4 was located at x=4 m. For each test cross section, measurements were made 0.500 m away from the outside and inside wall, and in the middle of the ditch, each at three depths below the water surface (1.000, 2.100, and 4.140 m; Fig. 1b).

The flow velocity measurements were carried out with an ultrasonic Doppler instrument (model 6526, Starflow Ultrasonic Doppler Instrument with Micrologger; Unidata Pty., Ltd.). The accuracy of the instruments is 2% of measured velocity. The ultrasonic probe was fixed to a metal bar and carefully oriented perpendicular to the flow during measurement and the velocity values reported were the average of several recordings.

Impeller modeling methods

In this article, three numerical approaches of modeling impeller motion have been compared, that is, the momentum source term approach, MRF, and SM.

Momentum source term approach

In this approach, the momentum source term was added into the momentum equation to represent the effect of the impeller. It is proposed that the interaction between the plate blade and fluid includes the propelling force in the tangential direction and the friction resistance in the radial direction. From this, the forces exerted by a small area of blade dS on the fluid can be, respectively (Jiang et al., 2010):

Tangential propelling force

(1)

Radial friction resistance

(2)

where ρ is density (=998.2 kg/m³), μ is viscosity (=1.003×10⁻³ Pa·s), u is linear velocity of the area element on the blade surface dS, u_θ is fluid tangential velocity rotating with the impeller before the fluid is propelled by the impeller, u_r is radial velocity of the fluid, dS is the cross-sectional area of the interface between fluid and the impeller blade, and C_f is local resistance coefficient related to Re_d, which is calculated approximately by the following:

(3)

(4)

where Re_d=ρu_rd/μ, and d is the distance between the cell center and the rotation center.

In the momentum source term approach, the impeller power can be calculated from the integration of the momentum source in the impeller region (Xu and McGrath, 1996):

(5)

where P is the impeller power input, F_θ is the tangential momentum source term, and V is the cell volume.

As it is unnecessary to construct the impeller in the momentum source term approach, the tank geometry and mesh generations are simpler than MRF and SM. The oxidation ditch discussed in this article was discretized into 609,021 unstructured cells (96×20×96 cells in length direction×height direction×width direction in the impeller region, and about 225×21×95 cells in other region). A grid-independent study has been carried out for the momentum source term approach with the standard k−ɛ model in different grid resolutions. In the simulation, the cell width in the impeller region was specified as 0.040625 m and the maximum cell width in the other region of the ditch was 0.20 m (Fig. 2).

FIG. 2.

Computational mesh for the momentum source term approach simulation: (a) the whole oxidation ditch, (b) the free surface, and (c) the impeller region.

MRF

For MRF (Luo et al., 1994; Fan et al., 2010), the oxidation ditch was divided into two zones. A rotating coordinate system was adopted for the inner zone, the rotating rate of which was set equal to the impeller agitation rate, and a nonmoving coordinate system was defined for the outer zone, including walls. The axis of the rotating reference was the centerline of the aerator. The angular velocity of the impeller was set zero with respect to the rotating coordinate system.

In MRF, the presence of the impeller configuration made the mesh in the impeller region nonuniform and increased the grid number. The oxidation ditch discussed was discretized into 1,006,869 unstructured cells, composed of hexahedrons and tetrahedrons. The region outside the impeller included 264×20×94 cells in length direction×height direction×width direction. The grids have been refined near the wall and the aerators to simulate precisely the flow characteristics. The minimum cell width was 0.0525 m and the maximum was 0.21 m.

SM

SM (Luo et al., 1993) is similar to MRF; the oxidation ditch was divided into two zones—the cylindrical zone, including the impeller and the rest of the ditch. The cylindrical zone rotated with the impeller, and the cylindrical surface and the bottom wall served as the interfaces, where the two regions were implicitly coupled. The axis of the rotating reference was the centerline of the aerator.

The oxidation ditch discussed was discretized into 356,522 unstructured cells, composed of hexahedrons and tetrahedrons. The region outside the impeller included 180×20×90 cells in length direction×height direction×width direction. The cell width in the impeller region was refined to 0.105 m, while in the other region, grids were coarsened to save the computational expenses, and the maximum cell width in the other region of the ditch was 0.21 m.

In MRF and SM, the power input is usually calculated from the predicted torque (Shekhar and Jayanti, 2002):

(6)

where m is the number of the blades and M is the torque of each blade.

Control equations and numerical methods

In the present study, the continuity and momentum equations of motion for a 3D incompressible flow were solved to calculate the single-phase flow of the oxidation ditch. The standard k−ɛ model, RNG k−ɛ model, realizable k−ɛ model, and RSM are widely used in the numerical simulations (Pope, 2000). Their accuracy and computational time in the simulations of the oxidation ditch were investigated to determine a proper turbulence model. In the study, they were applied for flow field simulations with the momentum source term approach, and the simulation results were compared with the experimental data. Lou et al. (2005), Fan et al. (2010), and Yang et al. (2011) adopted the standard k−ɛ turbulence model in MRF and the moving mesh approach to simulate the flow in the ditch; since all gained good results, the standard k−ɛ turbulence model was used in the MRF and SM simulation cases.

In the simulation, the free surface of the ditch was treated as a flat, rigid lid, so a slip wall was given to the surface of the oxidation ditch. The no-slip boundary conditions and the standard wall functions based on the proposal of Launder and Spalding (1974) were given to all the solid surfaces, including the bed, the sidewall, and central wall of the ditch.

For the three simulation methods, the settings were the same except the given items. The second-order upwind discretization scheme was used for the convection term of momentum, turbulent kinetic energy, and energy dissipation rate equations, and the pressure was solved by the standard algorithm. The discretized equations were solved iteratively using the SIMPLE algorithm for pressure–velocity coupling.

In the momentum source term approach and MRF, the steady-state approaches were used. In SM, second-order implicit unsteady formulation was used for computation. The initial condition for each simulation was that of still liquid and the volume-weighted average velocity was selected as a monitor. When the volume-weighted average velocity stayed steady and the residuals of the equations being solved met the prescribed tolerance, a converged solution was considered to have been obtained. All simulations were carried out on a PC with 2.4 GHz processor speed and 4 GB memory.

Results and Discussion

Test results

Test results of velocity at Test locations 1–4 are shown in Table 1. It can be seen that the velocity distribution of the flow field driven by the surface aerator was rather inhomogeneous. The surface velocity was higher than the bottom velocity in the downstream section near from the impeller. With the distance to the aerator increasing, the surface velocity decreased and the bottom velocity rose. The average velocity at the inside location of the ditch (0.309 m/s) was lower than the average velocity at the outside location of the ditch (0.363 m/s). However, the average velocity (0.309 m/s) in the ditch with the new aerator was higher compared with the old one (DS375; 0.287 m/s), and the average velocity difference (0.054 m/s) between the inside and outside was lower compared with the old one (0.166 m/s) (Li, 2011). The newly designed aerator has better performance in driving flow in oxidation ditch than the old one (Li, 2011).

Table 1.

Measured Velocity Data for the Oxidation Ditch with Surface Aerator

		Horizontal velocity magnitude (m/s)
Cross-section	Depth (m)	Line 1	Line 2	Line 3	Layer average	Section average
Test location 1	1	0.42	0.41	0.75	0.527	0.342
	2.1	0.35	0.16	0.49	0.35
	4.14	0.21	0.11	0.18	0.167
Test location 2	1	0.38	0.248	0.57	0.393	0.299
	2.1	0.26	0.21	0.4	0.287
	4.14	0.21	0.21	0.21	0.21
Test location 3	1	0.36	0.44	0.16	0.257	0.31
	2.1	0.37	0.43	0.31	0.37
	4.14	0.29	0.24	0.19	0.24
Test location 4	1	0.59	0.25	0.2	0.343	0.285
	2.1	0.53	0.245	0.14	0.298
	4.14	0.38	0.12	0.11	0.197
Average		0.363	0.256	0.309		0.309

Comparison of the results from various turbulence models

Figure 3 shows the comparison of the results calculated by different turbulence models with experimental data at 36 measurement points. The average errors for standard k−ɛ, RNG k−ɛ, realizable k−ɛ, and RSM models were 7.13%, 7.00%, 10.14%, and 3.42%, respectively. The points with larger deviations were mainly at Line 1 of Test location 1, at Line 2 of Test location 3, and near the bottom. Figure 4a–d depicts the velocity contours of four Test locations. It can be found that the contours on Test location 1 were dense with larger flow gradients and Test location 3 was significantly influenced by the recirculation flow due to its location in the curved portion of the ditch. Therefore, the measurement of the velocity and depth at these points was difficult to be precise, because little position changing of the meter may result in a large variety of the readings.

FIG. 3.

Comparison of the velocity simulated by standard k−ɛ, RNG k−ɛ, realizable k−ɛ, and Reynolds stress model (RSM) with the experimental data at Test locations 1 (a), 2 (b), 3 (c), and 4 (d): ▪, measured; , standard k−ɛ; – – – –, RNG k−ɛ; - - - - - -, realizable k−ɛ; — - — - —, RSM.

FIG. 4.

Velocity contours at Test locations 1 (a), 2 (b), 3 (c), and 4 (d). Unit: m/s.

Generally, within 10% lack of fit was considered as a good simulation for CFD applications and within 20% was considered acceptable (Littleton and Daigger, 2001). Hence, it could be concluded that the results calculated by standard k−ɛ, RNG k−ɛ, realizable k−ɛ, and RSM were consistent with the experimental data, so they were all acceptable, and the RSM model was the best of all.

Comparison of the results from various impeller methods

Figure 5 shows the comparison of the results calculated by different impeller models with experimental data. Different researchers reported well matched computational results with the measured data using their impeller modeling methods, but all suffered the prediction errors. Fan et al. (2010) found that the errors would become more reasonable with a higher aerator speed. Luo et al. (2005) thought that the measurement points near the eddy areas, such as near the aerators and bend channel, incurred more discrepancies. Besides, Yang et al. (2011) presented that the measurement of the bottom velocity was difficult and uncertain, and emphasized the influence of the probe position on the sampling error. It was revealed that the error on the section with more homogeneous distribution of the velocity was smaller. Our study was consistent with the above-mentioned findings, and most of the computational results agreed well with the measured data. Hence, all simulation methods employed in this article can be considered to be reliable. The average errors for momentum source term approach, MRF, and SM, were 7.13%, 14.66%, and 7.04%, respectively. It can be seen that the accuracy of momentum source term approach was better than MRF and close to SM. In the simulation of our previous momentum source term approach, however, the computational results deviated from the measured data with an average error of 9.7% (Jiang et al., 2010). Hence, compared to the existing approach, the present momentum source approach has made an improvement on the prediction accuracy.

FIG. 5.

Comparison of the velocity simulated by momentum source term approach, the multiple rotating reference frames approach (MRF), and the sliding mesh approach (SM) with the experimental data at Test locations 1 (a), 2 (b), 3 (c), and 4 (d): ▪, measured; , momentum source term approach; - - - - - -, MRF; – – – –, SM.

Table 2 lists the predictions of impeller power inputs of different approaches. It shows that the power prediction of the momentum source term approach is similar to that of MRF and SM, which indicates that the power can be predicted by the momentum source term approach especially with the RSM turbulent model. With the same turbulent model, the predicted power input for DSC 325 is 86.14 kW and DS 375 is 89.97 kW. As the average velocity in the oxidation ditch driven by DSC is 0.309, which is more than DS 375, the results show that the DSC 325 has better driving efficiency than DS 375.

Table 2.

Predicted Impeller Power Inputs

	Momentum source term approach with standard k−ɛ	Momentum source term approach with RSM	MRF	SM
DSC 325 (kW)	86.14	81.98	89.45	82.09
DS 375 (kW)	89.97

RSM, Reynolds stress model; SM, sliding mesh; MRF, multiple rotating reference frames.

In the momentum source term approach, following Equations (1) and (2), the momentum source was calculated and introduced to the cells of the impeller region. This operation was simpler than MRF and SM without the complicated settings of boundary conditions.

In MRF and SM, substantial efforts should be made to construct the complicated impeller of the surface aerator, and owing to the presence of the impeller, the grid in the moving region was nonuniform. Therefore, it was rather convenient to generate the high-quality grid for momentum source term approach.

In the simulations, both the accuracy and economy of the approach are important. Table 3 shows the computational time for three approaches. In four turbulence models of momentum source term approach simulations, standard k−ɛ model took the least time and RSM needed the most time, but still saved 26.7% time compared with MRF. For the calculation of the flow field in the oxidation ditch, usually costs huge computational expenses, the computational time is one of the most important factors for application. As a result from the point of economy, operations, and grid generation, the momentum source term approach outperforms MRF and SM.

Table 3.

Required Computational Time

	Momentum source term approach with standard k−ɛ	Momentum source term approach with RSM	MRF	SM
Computational time (h)	26	44	61	180

As the model can predict the velocity distribution of an oxidation ditch with a surface aerator, we can compare the performance of the design with a different surface aerator and different dimension of the oxidation ditch more precisely. Thus, the model provides us a better tool to improve the design of the surface aerator and oxidation ditch.

Velocity profiles

Figure 6 represents the velocity vectors of the liquid surface and A-A cross section. From the left side of illustration, it can be seen that the higher velocity flow was at the surface near impellers, and decayed with increasing distance to the impellers. In the upper straight channel (upstream areas of the aerator), a circulation loop C1 was generated near the central wall, and a similar phenomenon was observed in the measurement. The right illustrations in Fig. 6 visualize the velocity profiles at the A-A section (Fig. 1a). Two circulation loops C2 appeared on both sides of the aerator. However, the velocity profiles and the shapes of two circulation loops were quite different, which gave an indication that the aerator exerted different effects on its two sides because of the centrifugal effect. With the circulation loops, the fluid at the top with higher velocity flowed into the bottom, which made the velocity near the bottom even larger than that in the middle of the ditch. From Fig. 6, it can be seen that all results predicted by the momentum source term approach, MRF, and SM captured the flow characteristics, which further indicates the consistency of the simulations.

FIG. 6.

Velocity profiles in the oxidation ditch simulated by (a) the momentum source term approach, (b) MRF, and (c) SM (left: velocity profiles at the surface; right: velocity profiles in A-A section).

Figures 4 and 7 show the velocity contours of four Test locations (in the positive direction of x coordinate) and for a distance from the bottom of the ditch of 0.10 m (bottom). The fluid near the impeller region was driven by the blades of the surface aerator to form a higher velocity at the surface. When the flow passed the curved portion of the ditch, due to the centrifugal effect of the bend channel, the flow with higher velocity was thrown to the outside of the ditch, spiraled down to the bottom of the ditch, and then upward on the surface of the ditch, which generated the transverse spiral flow in the cross section of the ditch (Li and Xu, 2000). Because of the higher velocity near the outside wall, the profiles on the horizontal section were inhomogeneous and the lower areas were mainly located on both sides of the central wall (Fig. 7). When the flow arrived at Test location 1, the velocity at the surface still came to 0.7 m/s. With the inertia effect, the recirculation flow extended to the straight channel, but seldom influenced the distributions of the velocity in the lower portion of the section, where lower velocity dominated (Fig. 4a). From Fig. 4b and c, it can be seen that the profiles on Test locations 2 and 3 were significantly affected by the transverse spiral flow. As the flow crashed against the central wall at the corner, the velocity direction changed, local velocity gradients were large, and the velocity contours were dense.

FIG. 7.

Velocity contours at the bottom of the ditch. Unit: m/s.

In the bend channel flow, flow separation may occur at the convex bank of the bend channel exit (Zhang and Lui, 1993). The calculated results indicated that there existed flow separation at the exit of the second bend channel far from the surface aerator downstream, which formed the circulation loop C1 and extended with a long distance. This circulation region resulted in the lower velocity areas near the central wall in the upper channel of the ditch. In Fig. 4d, the velocity on the outside portion of Test location 4 exceeded 0.6 m/s, because the regions with higher velocity moved outward with the bend channel effect (Zhang and Lui, 1993).

It was generally found that the most notable feature of the flow pattern was nonuniform, and a spiral flow formed at the curved portion of the ditch. Similar results have been repeatedly reported in other studies (Luo et al., 1993; Littleton et al., 2007; Fan et al., 2010; Yang et al., 2011). On the other hand, both from computational results and measured data, the velocity values less than 0.25 m/s were found in many parts of the ditch (such as the regions near the central wall). It has been said that a velocity of more than 0.25 m/s on the average is necessary for the original oxidation ditch process (Grady et al., 1999). The slow flow will bring accumulation of the activated sludge and interfere with treatment. Therefore, some endeavor should be made to prevent the appearance of such lower velocity areas, such as increasing the number of the mechanical propelling equipment, modifying the configuration of the ditch, et al. With the flow pattern simulation, the velocity distributions in the ditch of various designs can be understood to improve the designs and operations of the oxidation ditch process (Yang et al., 2011).

Conclusions

In the present article, we present a newly designed inverse umbrella surface aerator, and tested its performance in driving flow of oxidation ditch. The test shows that the newly designed aerator has better performance in driving flow in oxidation ditch than the old one. The CFD approach based on the predictive momentum source (Jiang et al., 2010; Li, 2011) was applied to predict the flow field of an oxidation ditch. The predictive capabilities of the standard k−ɛ, RNG k−ɛ, realizable k−ɛ, and RSM for the oxidation ditch simulations were investigated. The predictions of momentum source term approach were compared with those of MRF and SM. All CFD results were validated through our experiments. It can be concluded that

• (1) We presented a newly designed surface aerator and the flow field of the oxidation ditch driven by the aeration through experiment and numerical simulation. The results show that it has better performance in driving the oxidation ditch than DS375.

• (2) The improved momentum source term approach was applied to simulate the flow field of the oxidation ditch driven by the inverse umbrella surface aerator, and the results predicted by standard k−ɛ, RNG k−ɛ, realizable k−ɛ, and RSM were acceptable compared with the experimental data. In the simulations, the standard k−ɛ model cost the least computational expenses and RSM cost the most, but with the highest accuracy.

• (3) The prediction accuracy of momentum source term approach is better than our previous momentum source term and MRF approach, and close to SM approach. Moreover, momentum source term approach took less time and generated grids more simply, thus easier to use than MRF and SM.

Acknowledgments

This work was supported by the Ministry of Science and Technology of the People's Republic of China (grant no. 2009BAC19B02). The authors would like to thank Anhui Guozhen Environmental Protection Science and Technology Co., Ltd. for their help in experiments.

Notation

C_f=local resistance coefficient

d=distance between the cell center and the rotation center, m

DS375=inverse umbrella surface aerator with an impeller diameter of 3.75 m (old)

DSC325=inverse umbrella surface aerator with an impeller diameter of 3.25 m (newly designed)

f=friction force between the impeller blade and fluid, N

F=propelling force from the impeller blade to fluid, N

F_θ=tangential momentum source derived from the propelling action of blade, N/m³

m=impeller blade number

M=torque, N·m

N=impeller rotational speed, rpm

P=impeller power input, W

r=radial coordinate, m

Re_d=Reynolds number of the boundary layer on the blade surface

S=impeller blade area, m²

u=line velocity of the impeller blade, m/s

u_r=fluid radial velocity, m/s

u_θ=fluid tangential velocity rotating with the impeller before propelled by impeller, m/s

V=cell volume, m³

x=axis

y=axis

z=axis

Greek Letters

θ=tangential coordinate, rad

μ=viscosity, pa·s

ρ=fluid density, kg/m³