Matching Criterion of Submersible Mixer and Pool

Abstract

Matching problem of submersible mixer and pool has been a difficult problem. Based on in-depth study of the flow field characteristics of a submersible mixer, combined with fluid mechanics theory, the matching problem is discussed in this paper. Calculation formulas of the axial advance distance x and effective stirring radius R are obtained; meanwhile the critical dimension of matched pool is proposed: length × width × height = xv_0.3 × 2k₁Rv_0.3 × 2k₂ Rv_0.3, and calculation formula of the critical impeller diameter D_critical is obtained. They are theoretical foundation of matching criterion between pool and submersible mixer. Matching relationship of pool and submersible mixer is widely used in engineering practice with the help of experiment and CFD numerical simulation, and it can provide significant help for the selection of a submersible mixer.

1. Introduction

Fluid state stirred by a submersible mixer in the pool is very complicated. The fluid is a submerged nonfree turbulent swirling jet with a circular cross-section and a radial wall jet. Ultimately, with pool wall's interaction, impinging jet phenomenon occurs, causing fluid circulation formed in the entire pool, meeting the mixing requirements.

How can we make jet caused by the interaction of submersible mixer and pool wall achieve the expected mixing effect? The authors studied the historical development of the submersible mixer and wastewater treatment in the decades. We have researched a large number of materials, but no records are found about the matching of the submersible mixer and pool. In practical engineering, matching is often based on experience, repeating the installation test, so the whole project will cost a lot of human and material resources. In scientific research, distribution of the flow field is investigated in a fixed pool with one or more submersible mixers. However, problems on how to reasonably select the submersible mixer in a specific pool or choose a suitable pool for a submersible mixer, namely, matching problem between pool and submersible mixer, have not been solved yet.

2. Flow Field Characteristics of Submersible Mixer

Due to the interaction of the submersible mixer and the pool wall, watershed in the pool is divided into a number of regions. Among these, typical and large regions are as follows: (1) free jet region; (2) the region impacting with sidewall c; (3) the region impacting with the bottom of the pool; (4) the impact zone on the bottom of the pool; (5) the region impacting with the surface of the pool; (6) the jet region of the pool wall of the bottom; (7) the jet region of radial sidewall c; (8) the jet region of radial surface of the pool, as shown in Figure 1.

Figure 1:

Flow diagram in pool. (I) Free jet region; (II) the region impacting with side wall c; (III) the region impacting with the bottom of the pool; (IV) the region impacting with the bottom of the pool; (V) the region impacting with the surface of the pool; (VI) the region of the bottom of the pool wall jet; (VIII) the region of radial wall jetting with the side wall c; (IX) the region of radial wall jetting with the pool surface.

The flow in the pool is very complicated because of the interaction of the submersible mixer and the pool wall. Thus, there are three characteristic phenomena as follows.

(a) Entrainment Phenomenon. The fluid ejects from the impeller at the initial velocity u and forms a velocity discontinuity face with the surrounding stationary fluid. As known from turbulent mechanics theory, the velocity discontinuity face is unsteady, so fluctuation phenomenon appears and develops into vortex, thus causing turbulence. It will involve the original surrounding stationary fluid into the jet. So this is called entrainment phenomenon. With the development of turbulence, the fluid which is taken and moving with the jet together becomes increasingly great. The jet boundary gradually extends to both sides, and the quantity of flow gradually increases along the way. Mixed with the surrounding stationary fluid and the jet, the resistance of the jet flow is accordingly generated. It reduces the velocity in the jet edges; however, the same initial velocity is hard to keep. The farther downstream, the wider the jet boundary and the larger the quantity of the flow, while the velocity is smaller.

(b) Coanda Effect. Because of the asymmetry of the submersible mixer installation position in the pool, the fluid will move to the closer side when it is flowing, as it is absorbed over by the wall, which is called Coanda Effect. It makes the fluid at the bottom of the pool stirred more fully.

(c) Vortex Phenomenon. After the fluid is speeded up by the submersible mixer, it crashes onto the wall c and is divided into two parts, keeping crashing, respectively, onto wall b and wall d. It causes reflux flow in the pool, leading to an obvious whirlpool phenomenon and can be evidently observed. The jet flows made by the crashes result in the whirlpool phenomenon, which energizes the fluid in the pool. Thus, the fluid circle is fully spread. Because of this, the appearance of the whirlpool phenomenon is also a necessary condition for the proper stirring of the fluid in the pool.

It can be concluded from the above analyses that, because of the effect of the submersible mixer, the fluid in the pool changes into jet flow, which obviously moves forward axially. Therefore, the submersible mixer is usually installed on the shorter wall, referring to wall a or wall c in Figure 1.

3. Matching Criterion of Submersible Mixer and Pool

3.1. Vortex Derivations

According to jet mechanics theory, assuming the jet flow from the circular section is uniform, with the same velocity v_o, the jet boundary layer can be calculated linearly, as shown in Figure 2; the diffusion angle is 2α [1 –12]:

R = x^{'} \tan α,

(1)

where x′ is the distance of the section away from the pole; namely, x′ = x + x₀. x is the distance of the section away from the outlet of the nozzle jet.

Figure 2:

Diagram of irrotational jet with circular section.

The experimental results show that the diffusion angle of the cylindrical stream spurted by cylindrical nozzle jet is α = 14.85° [1 –12]. In the main section, relationship between the axial velocity v_m along the jet axis and radial diffusion radius R is as follows:

\frac{R}{R_{0}} = 3.283 \frac{v_{0}}{v_{m}},

(2)

where R₀ is the radius of cylindrical nozzle and v₀ is the initial velocity of fluid jetted from cylindrical nozzle [1 –12].

According to formulas (1) and (2), in the main section, axial velocity decreases with the increase in the distance away from the outlet of nozzle, that is, a relationship as follows:

\frac{v_{m}}{v_{0}} = 3.283 \frac{R_{0}}{x^{'} \tan α} = 12.38 \frac{R_{0}}{x^{'}} = 12.38 \frac{R_{0}}{x + x_{0}} .

(3)

Namely,

v_{m} = 3.283 \frac{R_{0}}{x^{'} \tan α} v_{0} = 12.38 \frac{R_{0}}{x + x_{0}} v_{0} .

(4)

The expansion angle of swirling jet is 2α. Drake and Hubbard, and Groe and Ranz, respectively, proposed an estimation formula in 1964 [1 –12]:

α^{'} = 14.8 + 14 S = 14.8 + 14 \frac{M}{F D} .

(5)

Formula (5) shows that the expansion angle of swirling jet is greater than that of irrotational round jet, Δα = 14S = 14(M/FD), which is generated due to curl [1 –12].

Loitsyanskii, Gortler, Chigier, and Beer's studies show that axial velocity attenuates at the speed of x⁻¹ and the rate of decay increases with the increasing of curl [1 –12].

According to formula (4), axial attenuation of the fluid stirred by submersible mixer is faster than that of irrotational round jet because of α′ > α and tanα′ > tanα. Derived from (4) and (5), we conclude

x = 3.283 \frac{R_{0}}{v_{m} \tan α^{'}} v_{0} - x_{0} .

(6)

That is

R = x \tan α = (3.283 \frac{R_{0}}{v_{m} \tan α^{'}} v_{0} - x_{0}) \tan α .

(7)

When v_m = 0.3 m/s, x_v0.3 is the effective axial advance distance of submersible mixer, and R_v0.3 is the effective disturbance radius of submersible mixer.

According to formula (6) and (7), the critical size of pool is

b \times a \times h = x_{v_{0.3}} \times 2 k_{1} R_{v_{0.3}} \times 2 k_{2} R_{v_{0.3}},

(8)

where a is the length of submersible mixer installation wall; b is the length of the adjacent pool wall; and h is the depth of the pool. Consider that k₁ < 1, k₂ < 1. This is because the length of the installation wall must be less than the effective disturbance radius, and then impact flow can be generated by the adjacent pool wall. The wall of the pool can play a role of boundary. In general, the pool depth will be smaller than the width of the water pool because the submersible mixer is usually installed near the bottom; namely, k₂ ≤ k₁. The size is a theoretical reference value, needing further experimental correction, where

x_{{v_{0.3}}_{}} = 3.283 \frac{R_{0}}{v_{m} \tan α} v_{0} - x_{0} < \frac{10.94 R_{0}}{\tan (14.8 + 14 (M / F D))} v_{0},

(9)

R_{v_{0.3}} = x_{v_{0.3}} \tan α = (\frac{10.94 R_{0}}{\tan (14.8 + 14 (M / F D))} v_{0} - x_{0}) \times \tan (14.8 + 14 \frac{M}{F D}) = 10.94 R_{0} v_{0} .

(10)

v₀ equals the maximum axial velocity at the impeller outlet. Submersible mixer can only be applied to the pool with a size less than the critical value.

The critical diameter of the impeller can be derived according to the axial advancing distance of formula (6) if we know the length of installation wall is set to a and the length of the adjacent wall is set to b:

R_{0 b} \geq \frac{0.3 (x + x_{0}) \tan α}{3.283 v_{0}} \approx \frac{\begin{matrix} b \tan (14.8 + 14 (M / F D)) \end{matrix}}{10.94 v_{0}} .

(11)

But also according to formula (7), critical diameter of impeller can be derived from effective disturbance radius:

R_{0 a} \geq \begin{matrix} \frac{0.3 ((a / \tan α) + x_{0}) \tan α}{3.283 v_{0}} \end{matrix} = ((\frac{a}{\tan 14.8 + 14 (M / F D)} + x_{0}) \times \tan (14.8 + 14 \frac{M}{F D})) \times {(10.94 v_{0})}^{- 1} \geq \frac{a}{10.94 v_{0}} .

(12)

In order to meet formulas (11) and (12), the critical radius must satisfy the following conditions:

R_{0} \geq \max (R_{0 a}, R_{0 b}) > \frac{a}{10.94 v_{0}} .

(13)

The critical diameter of impeller is

D = 2 R_{0} = \frac{a}{5.47 v_{0}} .

(14)

The diameter of the selected impeller should be greater than the critical diameter because the increasing of effective disturbance radius will result in the impact flow generated by the interaction of fluid through the submersible mixer impeller and the wall surface, which makes the most of the pool wall.

3.2. Efficiency Calculations

Efficiency can be estimated according to external characteristic parameters of submersible mixer. Estimation formula is [13]

η = \frac{F^{\begin{smallmatrix} 3 / 2 \end{smallmatrix}}}{3.14 D M n} \times 100 % .

(15)

F is hydraulic thrust, N; M is torque, N·m; n is rotational speed, r/min; D is impeller diameter, m.

Therefore, the submersible mixer and mixing pool matched with each other can be selected according to formulas (8), (14), and (15) in practical engineering. If the impeller diameter and axial velocity at the outlet are known, the mixing range can be roughly estimated, so the critical size of the matching pool can be derived; if the pool size is known, the diameter of the submersible mixer impeller matched with the pool can be identified.

4. Experiments and CFD Verification

The widely used WJ0.75-4-210 submersible mixer was investigated deeply in this paper. As shown in Figure 3, the main parameters of the submersible mixer are impeller diameter D = 210 mm, the number of blades is 2, the rotational speed n = 1400 r/min, and the hub diameter d_h = 70 mm. It was installed at the location which is 800 mm far from the bottom of the pool (the impeller shaft is 800 mm far from the bottom of the pool) where the distance ratio of the narrow side and the other is 3: 4. The size of the pool is 4000 mm × 1000 mm × 1500 mm.

Figure 3:

Model of the submersible mixer.

To make the numerical calculation closer to the actual situation, we take the influence of motor in the fluid into account when calculating. In the modeling, the shape of the motor was simplified to a drum with the diameter D = 144 mm and length l = 300 mm. The computational domain was modeled by PRO/E software.

4.1. Simulation Setting

To simplify calculation, the stirring medium of submersible mixer is water and its density is ρ = 1 × 10³ kg/m³. The tetrahedral unstructured grids of computational domain are built by ICEM software, and local grid refinements near the impeller blades are performed. The node number of the whole computational domain is 3672548.

Because of sewage treatment pool's undulating free surface, the rigid-lid assumption simulation is used. Assume that the free surface normal velocity is zero and the characteristic quantities to the gradient is zero, the location of the free surface does not change with time any longer; thus meshing and calculations can be carried out. This method is always applied into the free surface without much fluctuation:

\begin{matrix} {v |}_{z = 1} = 0, \\ {\frac{\partial q}{\partial y} |}_{z = 1} = 0 . \end{matrix}

(16)

All physical walls are set as no-sliding wall boundary conditions; giving mixing blades and the shaft a corresponding rotational speed, the interface of internal and external subdomains is set as “interface” boundary condition to ensure the mutual coupling in the calculation [13 –20].

The numerical simulations are performed through FLUENT 6.3 software and the finite-volume method is used to discretize the governing equations. The SIMPLEC algorithm is adopted and the spatial derivatives are computed using a second-order upwind scheme [13 –20]. The widely used RNG k~ε turbulence model is selected, and the convergence criterion is that all residuals are less than 10⁻⁵.

4.2. Analysis of the External Characteristics

Table 1 shows a comparison between the numerical simulation of the submersible mixer's external characteristics and experimental date. The results of the numerical simulation of the water thrust, torque, and efficiency of the submersible mixer are significantly higher than the experimental values, which shows the relative error is large.

Table 1:

External characteristics of the submersible mixer.

	Water thrust/N	Torque/(N·m)	Efficiency/%	Average velocity/(m/s)

Numerical simulation	315	10.28	60.08	0.35
Experiment (clear water)	255	10	44.11	—
Relative error (%)	23.5	2.5	36.2	—

Combined with the previous theoretical deduction, we make a matching test of WJ0.75-4-210 submersible mixer and the pool. Substituting the impeller diameter = 210 mm, the water thrust, and the torque which comes from the experiment into formula (5), we can obtain

α = 14.8 + 14 S = 14.8 + 14 \frac{M}{F D} = {16.5}^{°} .

(17)

By formula (9), we can get effective axial advance distance of submersible mixer:

x_{v_{0.3}} = \frac{10.94 R_{0}}{\tan (14.8 + 14 (M / F D))} v_{0} = 11.3 m .

(18)

By formula (10), we can get the radius of the radial disturbance:

R_{v_{0.3}} = 10.94 R_{0} v_{0} = 3.4 m .

(19)

Therefore, we can know WJ0.75-4-210 submersible mixer can be used to stir in the sewage treatment pool whose size is smaller than 11300 mm × 5000 mm × 5000 mm:

D_{critic} = 2 \frac{a}{10.94 ν_{0}} = 0.08 m .

(20)

So D = 210 mm > D_critic = 80 mm.

From the above, the pool whose size is 4000 mm × 1000 mm × 1500 mm can be stirred by the submersible mixer whose diameter is over 80 mm.

The average velocity of the fluid in the pool is 0.35 m/s, which fully meets the velocity requirements of fluid in the wastewater treatment pools. From the analysis above, WJ0.75-4-210 submersible mixer can be used in the pool whose size is 4000 mm × 1000 mm × 1500 mm.

5. Result

Through the theoretical analysis, significant conclusions can be reached: the axial advance distance is x < 3.283(R₀/v_mtanα)v₀ − x₀, effective stirring radius is R = xtanα < (3.283(R₀/v_mtanα)v₀ − x₀) tanα, and meanwhile the critical dimension of matching pool is proposed: length × width × height = x_v0.3 × 2k₁R_v0.3 × 2k₂R_v0.3, and calculation formula of the critical diameter of impeller is D = a/5.47v₀. This is the theoretical foundation of matching between pool and submersible mixer. At the same time, conclusions can be verified through experiment and CFD analysis.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Footnotes

Acknowledgments

This work was supported by the National Natural Science Foundation (Grant no. 51109093), the National Science and Technology Support Program (Grants nos. 2011BAF14B01 and JHB2012-38), and Senior Talent Start-Up Funds of Jiangsu University (Grant no. 13JDG029).

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