# Design Equation for Stirring Fluid by a Stream Pump in a Circulating Tank

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## Abstract

**:**

## 1. Introduction

## 2. Theoretical Considerations

#### 2.1. General Concept of the Proposed Mathematical Model

_{in}) and the power dissipated by the circulating liquid (P

_{dis}) provides a very interesting possibility. In a steady state, usually considered as a conclusive situation, both these powers are equal:

_{S}) and then—injected back to the channel through the outlet. Liquid introduced in this way forms a submerged stream the kinetic energy of which is passed to the ambient liquid, inducing the circulation (discharge Q

_{R}) [18].

#### 2.2. Assumptions and Simplifications

_{S}for the nozzle outlet:

_{R}– tangent to the wall component of the liquid circulation:

_{S}and Q

_{R}, this restriction is obvious. It can be easily accepted for the circulation in a chamber, but creates more doubt when a submerged jet is considered. It is well known [18] that the diameter of the jet increases, which means that there exists a transversal component of motion, which can also generate a crosswise flow in a channel. Alas, an algebraic description of the energy dissipation intensity would not be possible. Fortunately, the classical 1D model of a submerged stream works well in technical problems, thus Equation (2) has been accepted in this paper. Nonetheless, one should be aware that such an underestimation of the energy loss will probably cause an overestimation of the calculated discharge Q

_{R}.

_{S}), whereas in the neighborhood of the pump inlet a second stream will be formed (effect of pump suction), which can also influence the circulation. However, according to the theory of submerged jets, this rear stream is much wider, even 2–3 times, than the frontal one. One can easily observe this effect by use of a vacuum cleaner and the smoke from incense burning without a flame. As it is shown below (Equation (4)), the energy of a stream decreases with the fourth power of its diameter. Assuming that the rear stream is two times wider than the first one, its energy would be 16 times lower. This means that this factor can be neglected in the energy balance.

_{dis}results from the liquid circulation losses P

_{diss}and the direct energy dissipation inside the submerged stream P

_{disj}. The first factor regards the whole channel, whereas the second is a local one (as it is known from the jets theory [18], such structure very quickly loses its dynamic distinction) and must be described by the differential equations of the theory of turbulence. These equations are formally complex, so description of P

_{disj}by the mathematically simple algebraic expressions (what is a basic postulate of this paper) would not be possible. In this situation the jest dissipation power P

_{disj}was neglected. Permissibility of this simplification was evaluated empirically. In the other words it was assumed, that if the theoretical relation Q

_{R}(Q

_{S}) is accepted (at least from the technical point of view), the omission of P

_{disj}will be reasonable.

#### 2.3. Determination of the Induced Power

_{S}, Q

_{R}—discharges of the stream pump and the channel, respectively, v

_{S}—mean velocity in the pump outlet, d

_{S}—diameter of the pipe outlet.

#### 2.4. Determination of the Dissipation Power

_{R}, mean velocity in the channel v

_{R}, channel geometry: Length L, width B, depth H, radii of the bend curvatures, if necessary), making use of classical hydraulic methods. Such an attitude is very often applied, especially in practically oriented investigations of fluid-flow systems, like pipes (e.g., [19]) or open channels (e.g., [20]).

_{R}, Equation (3) [21]:

_{0}—radius of the bend, R

_{H}—hydraulic radius:

_{0}—channel cross-section, W

_{P}—channel wetted perimeter, λ—Nikuradse hydraulic loss coefficient, α—angle of the bend (in this case α = 90°). The bend radius is equal to (Figure 1):

_{s}and the channel cross-section F

_{0}. If so, the minor loss coefficient for the stream pump equals [21]; reference velocity: Equation (3):

#### 2.5. Proposed Design Equation

_{R}and the discharge of the stream pump Q

_{S}, inducing the motion in this chamber:

## 3. Experimental Verification

#### 3.1. Laboratory Stand

_{S}determination and two valves—for regulation and deaeration (Figure 3). In each investigated case, the time t

_{V}of flow of V = 10 liters of water was recorded (from 100 to 200 seconds, depending on the actual pump setting). The time t

_{V}was measured manually, making use of a laboratory electronical stop watch. The volume V was read from the water meter counter. The accuracy of this measurement was equal to 5% (as the precision of the used device, determined according to European standards MID, equals 2% for the highest flows and 5% for the lowest; in the presented case the minimal acceptable water discharge was equal 0.03 m

^{3}/h, maximal acceptable discharge—3–12 m

^{3}/h and minimal measured discharge was equal to 0.23 m

^{3}/h).The actual value Q

_{S}was measured two times for each pump setting, after the flow stabilization—at the outset and the end of each run and averaged. The discharge was calculated as follows:

_{S}, the discharge of the circulation Q

_{R}and the device geometry, a search was carried out for a very regular form of the channel—i.e., a circular chamber, which was made of a stainless steel sheet. Its main dimensions equal R

_{Z}= 0.4 m, R

_{W}= 0.2 m, H

_{max}= 0.3 m (Figure 4).

_{φ}was measured in two sections (for φ

_{1}= 150° and φ

_{2}= 210°) in 16 regularly distributed points in each section (Figure 4), making use of a 16 MHz MicroADV (acoustic Doppler velocimeter)produced by Sontek.Basic characteristics of the ADV sensor: Measurement frequency 0.1–50 Hz, range of velocity measured 0.001 m/s–2.5 m/s, minimum measuring error 0.0025m/s, resolution 0.0001 m/s, measurement cell 0.09 cm

^{3}, measurement point distance 0.05 m [22]. Measurements were made with 1 Hz frequency samplingfor 1 minutes in every point. Minimumvalue of correlation ratio above 80% and signal-to-noise ratio (SNR) above 18 dBresult high accuracy. In the above-mentioned conditionsthe manufacturer declares measuring error Δ = 1% of measured value (but not less than 0.0025 m/s) [22]. Using one instrument ADV, the local velocity was measured n each point (with relative measurement uncertainty ±Δ), one after another, for the sector EP1 in the first place and afterwards—EP2.The chamber discharge was calculated by the numerical integration of the measured velocity distribution, according to the classical relation:

#### 3.2. Measurement Results

_{SA}= 0.015 m and d

_{SB}= 0.012 m, for some different depths of the channel. An example of these results (for H = 0.215 m) is presented below:

_{SA}= 0.015 m: Q

_{S}= 7.8 × 10

^{−5}m

^{3}/s — Q

_{R}= 1.85 × 10

^{−3}m

^{3}/s;

Q

_{S}= 1.05 × 10

^{−4}m

^{3}/s — Q

_{R}= 2.58 × 10

^{−3}m

^{3}/s;

Q

_{S}= 1.17 × 10

^{−4}m

^{3}/s — Q

_{R}= 2.91 × 10

^{−3}m

^{3}/s;

d

_{SB}= 0.012m: Q

_{S}= 6.4 × 10

^{−5}m

^{3}/s — Q

_{R}= 1.64 × 10

^{−3}m

^{3}/s;

Q

_{S}= 8.6 × 10

^{−5}m

^{3}/s — Q

_{R}= 2.30 × 10

^{−3}m

^{3}/s;

Q

_{S}= 9.6 × 10

^{−5}m

^{3}/s — Q

_{R}= 2.68 × 10

^{−3}m

^{3}/s,

#### 3.3. Verification of the Proposed Equation

_{Z}− R

_{W}= 0.2 m and H = 0.215 m. According to the procedure described above, we can calculate the geometrical parameters of the object: F

_{0}= 0.043 m

^{2}, W

_{P}= 0.63 m, R

_{H}= 0.068 m (Equation (6)), R

_{0}= 0.3 m (Equation (7)), L

_{W}= L

_{M}= 0, α = 90°.

_{b}= 0.286 (Equation (5)—it was assumed, that λ = 0.05), coefficient ζ

_{l}= 0 (Equation (8)) and the coefficient of loss caused by the stream pump—ζ

_{s}= 0.967 (Equation (10), assuming that the pump covers about 15% of the channel cross-section, i.e., F

_{z}/F

_{0}= 0.15). One should remember about the energy loss caused by the ADV velocity meter. It was assumed that the body of this device is analogical to the pump body, and twice the value of ζ

_{s}, so finally, the effective coefficient of energy loss was equal to:

_{R}and Q

_{S}. According to Equation (11) we have:

_{R}(Q

_{S}) is a linear function. This result has been confirmed by the experiments, both qualitatively and (what is much more important)—quantitatively. The difference between the calculations and measurements is close to 9.3% (for d

_{S}= 0.015 m) and 25% (for d

_{S}= 0.012 m), with a mean value of 17%. However, the model proposed above does not pretend to be a precise theory. From the definition, this suggestion must be a method for the technical evaluation of the discharge in a circulating tank (Figure 6), induced by a stream pump. From this point of view, one should appraise this concept positively.

## 4. Conclusions

_{R}(Q

_{S}), which is a consequence of the introduced simplifications. This approach can be applied in each technical situation, where one has to do with a circulating flow. Some possible kinds of problems were mentioned in the introduction. Especially interesting object looks to be the Carrousel system of the activated sludge reactors, serving for biological waste water treatment. This solution is very attractive and becomes more and more popular of late decades. The waste water circulation in such reactors can be induced by the stream pumps and the proposed formula can be used for the rough (but very quick) and preliminary calculation of the pump setting and the object regulation.

## Author Contributions

## Funding

## Conflicts of Interest

## References

- Shah, M. Course Material Process Engineering: Agitation Mixing. Available online: http://www.dduanchor.org/site/wp-content/uploads/2014/11/Process-Engineering-Agitation-Mixing.pdf (accessed on 13 August 2019).
- Demirel, E.; Aral, M.M. Unified Analysis of Multi-Chamber Contact Tanks and Mixing Efficiency Based on Vorticity Field. Part I: Hydrodynamic Analysis. Water
**2016**, 8, 495. [Google Scholar] [CrossRef] - Maruyama, T.; Ban, Y.; Mizushina, T. Jet mixing of fluids in tanks. J. Chem. Eng. Jpn.
**1982**, 15, 342–348. [Google Scholar] [CrossRef] - Matej, K.; Pawliczka, I.; Sawicki, J.M.; Wielgat, P.; Zima, P. Whirling System of Water Exchange in Breeding Pools. Arch. Hydro-Eng. Environ. Mech.
**2016**, 63, 253–263. [Google Scholar] [CrossRef] [Green Version] - Higgins, H.W. Water Jet Air Pump Theory and Performance; Pennsylvania State University: State College, PA, USA, 1964. [Google Scholar]
- Wilman, J.T. Jet Pumps; European Atomic Energy Community—EURATOM Reactor Centrum Nederland—RCN: Brussels, Belgium, 1966. [Google Scholar]
- Zima, P.; Makinia, J.; Swinarski, M.; Czerwionka, K. Effects of different hydraulic models on predicting longitudinal profiles of reactive pollutants in activated sludge reactors. Water Sci. Technol. J. Int. Assoc. Water Pollut. Res.
**2008**, 58, 555–561. [Google Scholar] [CrossRef] [PubMed] - Zima, P.; Makinia, J.; Swinarski, M.; Czerwionka, K. Combining computational fluid dynamics with a biokinetic model for predicting ammonia and phosphate behavior in aeration tanks. Water Environ. Res. Res. Publ. Water Environ. Fed.
**2009**, 81, 2353–2362. [Google Scholar] [CrossRef] [PubMed] - Sawicki, J.M. Aerated Grit Chambers Hydraulic Design Equation. J. Environ. Eng.
**2004**, 130, 1050–1058. [Google Scholar] [CrossRef] [Green Version] - Sawicki, J.M.; Pawłowska, A. Energy balance for air lift pumps. Arch. Hydro-Eng. Environ. Mech.
**1999**, 46, 63–72. [Google Scholar] - Gronowska-Szneler, M.A.; Sawicki, J.M. Simple design criteria and efficiency of hydrodynamic vortex separators. Water Sci. Technol. J. Int. Assoc. Water Pollut. Res.
**2014**, 70, 457–463. [Google Scholar] [CrossRef] [PubMed] - Mielczarek, S.; Sawicki, J.M. Dimensioning of vortex storm overflows. Water Sci. Technol. J. Int. Assoc. Water Pollut. Res.
**2018**, 78, 259–265. [Google Scholar] [CrossRef] [PubMed] - Artichowicz, W.; Sawicki, J.M. Determination of Mechanical Energy Loss in Steady Flow by Means of Dissipation Power. Arch. Hydro-Eng. Environ. Mech.
**2017**, 64, 73–85. [Google Scholar] [CrossRef] [Green Version] - Slattery, J.C. Advanced Transport Phenomena; Cambridge University Press: Cambridge, UK, 1999; ISBN 978-1-316-58390-6. [Google Scholar]
- Luo, C. Distribution of Velocities and Velocity Gradients in Mixing and Flocculation Vessels: Comparison between LDV Data and CFD Predictions. Ph. D. Thesis, New Jersey Institute of Technology, Newark, NJ, USA, 1997. [Google Scholar]
- Kasat, G.R.; Pandit, A.B. Mixing Time Studies in Multiple Impeller Agitated Reactors. Can. J. Chem. Eng.
**2004**, 82, 892–903. [Google Scholar] [CrossRef] - Dickley, S.D. Minimize Blending Time. Available online: https://www.chemicalprocessing.com/articles/2009/120/ (accessed on 12 August 2019).
- Abramovich, G.N.; Schindel, L. The Theory of Turbulent Jets; MIT-Press: Cambridge, UK, 2003; ISBN 978-0-262-01008-5. [Google Scholar]
- Qiao, Q.S.; Choi, K.W.; Chan, S.N.; Lee Joseph, H.W. Internal Hydraulics of a Chlorine Jet Diffuser. J. Hydraul. Eng.
**2017**, 143, 06017022. [Google Scholar] [CrossRef] - Das, B.S.; Khatua, K.K. Flow Resistance in a Compound Channel with Diverging and Converging Floodplains. J. Hydraul. Eng.
**2018**, 144, 04018051. [Google Scholar] [CrossRef] - Idelchik, I.E.; Ginevskiĭ, A.S. Handbook of Hydraulic Resistance, 4th ed.; rev. and augmented.; Begell House: Redding, CT, USA, 2007; ISBN 978-1-56700-251-5. [Google Scholar]
- SonTek/YSI, Inc. ADVField/Hydra Acoustic Doppler Velocimeter (Field) Technical Documentation; SonTec: San Diego, CA, USA, 2001. [Google Scholar]

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**MDPI and ACS Style**

Sawicki, J.; Wielgat, P.; Zima, P.
Design Equation for Stirring Fluid by a Stream Pump in a Circulating Tank. *Water* **2019**, *11*, 2114.
https://doi.org/10.3390/w11102114

**AMA Style**

Sawicki J, Wielgat P, Zima P.
Design Equation for Stirring Fluid by a Stream Pump in a Circulating Tank. *Water*. 2019; 11(10):2114.
https://doi.org/10.3390/w11102114

**Chicago/Turabian Style**

Sawicki, Jerzy, Paweł Wielgat, and Piotr Zima.
2019. "Design Equation for Stirring Fluid by a Stream Pump in a Circulating Tank" *Water* 11, no. 10: 2114.
https://doi.org/10.3390/w11102114