Modeling of the operation of a disc pump with the wall roughness consideration

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BACKGROUND

Dynamic disk pumps, unlike their centrifugal counterparts, do not feature profiled blades within their impellers (Fig. 1). Instead, the impeller of a classic disk pump consists of a package of two or more smooth disks. These pumps are particularly suited for applications that involve pumping high-viscosity liquids [1, 2]. However, the use of disk pumps in handling low-viscosity media has also been noted, sparking interest in calculating and predicting their characteristics under such conditions [3].

Research on this subject is scarce, with some studies focusing on theoretical descriptions and their comparison with experimental results [4], while others examine these pumps through numerical hydrodynamic modeling [5]. Theoretical approaches to modeling flow with developed turbulence encounter significant limitations, which cause discrepancies in the obtained characteristics.

In studies that employ numerical modeling to explore disk pump flows, a frequent observation is the absence of experimental data for validation or significant discrepancies in such comparisons [6]. Given the extremely widespread use of CFD packages in engineering, validating the results from these models has become particularly interesting scientifically and practically.

This paper presents the results of numerical modeling and experimental research into disk pumps operating on water. A key finding has been the significant influence that flow path element roughness has on the characteristics of this type of hydraulic machine, which must be considered when formulating the CFD calculation problem.

AIM

The objective of this study was to determine the factors influencing the convergence of the main characteristics with experimental data when performing CFD calculations of a disk pump operating with low-viscosity fluids.

DESCRIPTION OF THE MATHEMATICAL MODEL

The numerical modeling method used in this study relies on solving distinct analogs of the basic hydrodynamic equations. For an incompressible fluid (ρ = const), these equations are expressed as follows [7]:

1) mass conservation equation (continuity equation):

$\frac{\partial {\tilde{u}}_j}{\partial x_j}=0$,

where ${\tilde{u}}_j$ is the average value of the fluid velocity in projection onto the j-th axis (j = 1,2,3);

2) equation for conservation of momentum (Reynolds averaging):

$\rho \left[\frac{\partial U_i}{\partial t}+U_j\frac{\partial U_i}{\partial x_j}\right]= -\frac{\partial P}{\partial x_i}+\frac{\partial }{\partial x_i}\left[{\tilde{T}}_{ij}^{\left(v\right)}-\rho u_i{u}_j\right]$,

where U and P are the average speed and pressure, respectively; ${\tilde{T}}_{ij}^{\left(v\right)}=2\mathrm{\mu }{\tilde{s}}_{ij}$ is the viscous stress tensor for an incompressible fluid;

${\tilde{s}}_{ij}= \frac{1}{2}\left[\frac{\partial {\tilde{u}}_i}{\partial x_j}+\frac{\partial {\tilde{u}}_j}{\partial x_i}\right]$ is the strain rate tensor; and $\mathrm{\rho }{u}_i{u}_j$ is the Reynolds stress.

To complete the given system of equations, the semi-empirical turbulence model k–ω SST was used, demonstrating satisfactory convergence with the experimental data when simulating dynamic pumps [8, 9, 10].

To simulate flow near solid walls, a high-Reynolds version of the turbulence model is used. This approach involves using wall functions to describe the velocity profile instead of solving the equations of momentum conservation. In the logarithmic domain, these functions take the following form:

$u^{+}=\frac{1}{K}\ln \left(\frac{E}{f}{y}^{+}\right)$,

where $y^{+}=\frac{{\text{yu}}^{\text{*}}}{\text{ν}}$ is the dimensionless distance from the wall; K and E are constants; and f is the roughness function.

In this approach, roughness is described by a function that directly affects the wall function in the logarithmic region, effectively reducing the velocity magnitude relative to the solid wall. The roughness function is described as follows:

$f=\left\{\begin{array}{c}1 \text{если} R^{+}<R_{smooth}^{+} \\ {\left[B\cdot \left(\frac{R^{+}-R_{smooth}^{+}}{R_{rough}^{+}-R_{smooth}^{+}}\right)\right]}^a \text{если} R_{smooth}^{+}<R^{+}\\ B+C{R}^{+} \text{если }{R}^{+}>R_{rough}^{+}\end{array}\right.<R_{rough}^{+}$,

where $R^{+}=\frac{r{u}^{\text{*}}}{\nu }$; r is the value of equivalent roughness; v is kinematic viscosity; u* is the characteristic velocity near the wall, determined depending on the turbulence model used; $a=\sin \left[\frac{\pi }{2}\frac{\text{log}\left(\raisebox{1ex}{$R^{+}$}\!\left/ \!\raisebox{-1ex}{$R_{smooth}^{+}$}\right.\right)}{\text{log}\left(\raisebox{1ex}{$R_{rough}^{+}$}\!\left/ \!\raisebox{-1ex}{$R_{smooth}^{+}$}\right.\right)}\right]$; $R_{smooth}^{+}$ is the R⁺ value characterizing the roughness corresponding to smooth walls; and $R_{smooth}^{+}$ is value for roughness corresponding to a rough wall.

 

Fig. 1. A disc pump.

Рис. 1. Дисковый насос.

 

The computational mesh comprises polyhedral cells in the flow core and prismatic layers near the solid walls (Fig. 2).

 

Fig. 2. The simulation mesh in the section of the flow part.

Рис. 2. Расчётная сетка в сечении проточной части.

EXPERIMENTAL STUDY

A disk pump featuring a replaceable impeller configuration was tested (the test bench is presented in Fig. 3).

 

Fig. 3. The test bench.

Рис. 3. Испытательный стенд.

 

The impeller consists of two metal disks and plastic elements produced using additive manufacturing technologies, namely a shaft bushing, fastening blockings (possibly with blades), and a front (end) groove seal bushing seal. The impeller is shown in Fig. 4.

 

Fig. 4. A collapsible impeller.

Рис. 4. Разборное рабочее колесо.

 

RESULTS OF THE CALCULATIONS AND EXPERIMENTS

 

Fig. 5. Characteristic curves of a pump with a 13 mm wide disc impeller: experimental (b2-13), calculated with roughness considered (CFD b213-R) and without consideration (CFD b213).

Рис. 5. Характеристики насоса с дисковым рабочим колесом шириной 13 мм: экспериментальная (b2-13), расчётные с учётом шероховатости (CFD b213-R) и без (CFD b213).

 

Figure 5 compares the obtained characteristics of a pump with a 13-mm-wide disk impeller with the results of numerical simulations (CFD) with and without considering surface roughness.

 

Fig. 6. Characteristic curves of a pump with a 18 mm wide disc impeller: experimental (b2-18), calculated with roughness considered (CFD b218-R) and without consideration (CFD b218).

Рис. 6. Характеристики насоса с дисковым рабочим колесом шириной 18 мм: экспериментальная (b2-18), расчётные с учётом шероховатости (CFD b218-R) и без (CFD b218).

 

The parameters established for incorporating roughness in the simulation of a 13-mm-wide impeller were also applied to an 18-mm-wide impeller. The results are presented in Fig. 6.

 

Fig. 7. Field distribution of the velocity amplitude at a supply of 15 m³/h: a) with roughness; b) without roughness.

Рис. 7. Поле распределение амплитуды скорости при подаче 15 м³/ч: a) с учётом шероховатости; b) без учёта шероховатости.

 

For a qualitative comparison of the flow dynamics, Figs. 7 and 8 presents the velocity fields in the meridional section of the flow path (including the impeller, outlet, and lateral hollows) for an 18-mm-wide impeller.

 

Fig. 8. The circumferential velocity of the fluid at the outlet of the impeller at a supply of 15 m3/h: a) with roughness; b) without roughness.

Рис. 8. Окружная скорость жидкости на выходе из рабочего колеса при подаче 15 м³/ч: a) с учётом шероховатости; b) без учёта шероховатости.

 

ANALYSIS OF THE CALCULATION RESULTS

Based on the given characteristics and value distribution patterns, the following conclusions can be drawn:

1. Numerical simulations of dynamic pumps with disk impellers with smooth surfaces for the flow path elements lead to a considerable discrepancy (up to 30% for both impellers) when calculating pressure.
2. Incorporating the influence of roughness into the numerical models significantly improves the convergence of the simulated results with the experimental data. The maximum error was 9% for an impeller width of 13 mm and 7.5% for an impeller width of 18 mm. The maximum errors were observed in the extreme right part of the pressure characteristic, which may be associated with an inaccurate estimation of roughness values.

The velocity distribution fields clearly show a decrease in fluid velocity near solid walls, which is attributed to the consideration of roughness. This results in additional acceleration in the impeller. Consequently, the average exit speed from the impeller increased by 17.7% (from 10.2 m/s to 12.4 m/s) at 15 m³/h.

ADDITIONAL INFORMATION

Authors’ contribution. V.A. Cheremushkin — numerical modeling and experimental research, preparation and writing of the text of the article; V.O. Lomakin — expert support of the experiment, collection and analysis of literary sources, preparation and writing of the article. The authors confirm that their authorship complies with the international ICMJE criteria (all authors made a significant contribution to the development of the concept, research and preparation of the article, read and approved the final version before publication).

Competing interests. The authors declare that they have no competing interests.

Funding source. This study was not supported by any external sources of funding.

ДОПОЛНИТЕЛЬНО

Вклад авторов. В.А. Черемушкин — выполнение численного моделирования и экспериментального исследования, подготовка и написание текста статьи; В.О. Ломакин — экспертное сопровождение эксперимента, сбор и анализ литературных источников, подготовка и написание статьи. Авторы подтверждают соответствие своего авторства международным критериям ICMJE (все авторы внесли существенный вклад в разработку концепции, проведение исследования и подготовку статьи, прочли и одобрили финальную версию перед публикацией).

Конфликт интересов. Авторы декларируют отсутствие явных и потенциальных конфликтов интересов, связанных с проведённым исследованием и публикацией настоящей статьи.

Источник финансирования. Авторы заявляют об отсутствии внешнего финансирования при проведении исследования.

About the authors

Viacheslav A. Cheremushkin

Bauman Moscow State Technical University

Author for correspondence.
Email: wcheremushkin@gmail.com
ORCID iD: 0009-0006-7832-3952
SPIN-code: 3708-7709

Junior Researcher

Russian Federation, 5, 2nd Baumanskaya street, 105005 Moscow

Vladimir O. Lomakin

Bauman Moscow State Technical University

Email: lomakin@bmstu.ru
ORCID iD: 0000-0002-9655-5830
SPIN-code: 3467-7126

Dr. Sci. (Engin.), Chief of the Department of Hydromechanics, Hydromachines and Hydro-Pneumoautomatics

Russian Federation, 5, 2nd Baumanskaya street, 105005 Moscow

References

Supplementary files