# Investigation of different design methods of volutes with circular cross sections for a single-stage centrifugal pump

Axel Knapp, Martin Böhle, Harald Roclawski

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Axel Knapp, Martin Böhle, Harald Roclawski. Investigation of different design methods of volutes with circular cross sections for a single-stage centrifugal pump. 17th International Symposium on Transport Phenomena and Dynamics of Rotating Machinery (ISROMAC2017), Dec 2017, Maui, United States. hal-02980987

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# Investigation of different design methods of volutes with circular cross sections for a single-stage centrifugal pump

Axel Knapp<fnref n="1" />*, Martin Böhle<fnref n="1" />, Harald Roclawski<fnref n="1" />

<figure><img src="image_2.png" /></figure>

ISROMAC 2017

International Symposium on Transport Phenomena and Dynamics of Rotating Machinery

Maui, Hawaii

December 16-21, 2017

## Abstract

In the open literature different methods to design a volute geometry have been described well since several years, for example ECKERT [1], PFLEIDERER [2], STEPANOFF [3] or TROSKOLAŃSKI [4]. Nevertheless, there are only few comparisons of those design methods with respect to the pump performance (e.g. RÜTSCHI [5] or YANG et al. [6]) known at the moment. In this paper, the design methods of volute geometries with circular cross sections for a single-stage centrifugal pump according to PFLEIDERER and STEPANOFF are compared to each other according to the pump performance.

The different volute geometries have been investigated numerically by means of 3D-CFD (Computational Fluid Dynamics) tools. For validation, the volute geometries have been investigated experimentally as well. To ensure the comparability of the different design methods, every volute has circular cross sections, is used with the same impeller geometry and in addition the design of the tongue region is the same.

## Keywords

centrifugal pump — volute casing — design methods

## NOMENCLATURE

<table><tr><td>A<sub>θ</sub></td><td>cross section area</td><td>(mm<sup>2</sup>)</td><td>n<sub>q</sub></td><td>specific speed</td><td>(min<sup>-1</sup>)</td></tr><tr><td>a</td><td>radius of the center of gravity</td><td>(mm)</td><td>P<sub>h</sub></td><td>hydraulic power</td><td>(W)</td></tr><tr><td>b</td><td>chord</td><td>(mm)</td><td>P<sub>m</sub></td><td>mechanical power</td><td>(W)</td></tr><tr><td>c</td><td>velocity</td><td>(m s<sup>-1</sup>)</td><td>p<sub>in</sub></td><td>evaluation plane at the inflow domain</td><td></td></tr><tr><td>c<sub>in</sub></td><td>velocity at p<sub>in</sub></td><td>(m s<sup>-1</sup>)</td><td>p<sub>out</sub></td><td>evaluation plane at the outflow domain</td><td></td></tr><tr><td>c<sub>out</sub></td><td>velocity at p<sub>out</sub></td><td>(m s<sup>-1</sup>)</td><td>p<sub>st</sub></td><td>static pressure</td><td>(bar)</td></tr><tr><td>c<sub>u</sub></td><td>velocity in circumferential direction</td><td>(m s<sup>-1</sup>)</td><td>p<sub>tot</sub></td><td>total pressure</td><td>(bar)</td></tr><tr><td>c<sub>0</sub></td><td>velocity at cross section 0</td><td>(m s<sup>-1</sup>)</td><td>P<sub>fl</sub></td><td>PFLEIDERER</td><td></td></tr><tr><td>D</td><td>diameter</td><td>(mm)</td><td>Q</td><td>flow rate</td><td>(m<sup>3</sup> h<sup>-1</sup>)</td></tr><tr><td>D<sub>a</sub></td><td>diameter of the discharge pipe</td><td>(mm)</td><td>Q080</td><td>numerical results for Q = 80 m<sup>3</sup> h<sup>-1</sup></td><td></td></tr><tr><td>D<sub>i</sub></td><td>diameter of the inlet pipe</td><td>(mm)</td><td>Q100</td><td>numerical results for Q = 100 m<sup>3</sup> h<sup>-1</sup></td><td></td></tr><tr><td>d<sub>Z</sub></td><td>diameter of the tongue tip</td><td>(mm)</td><td>Q120</td><td>numerical results for Q = 120 m<sup>3</sup> h<sup>-1</sup></td><td></td></tr><tr><td>g</td><td>gravitational constant</td><td>(m s<sup>-2</sup>)</td><td>R</td><td>radius of the cross section area</td><td>(mm)</td></tr><tr><td>H</td><td>head</td><td>(m)</td><td>r</td><td>radius</td><td>(mm)</td></tr><tr><td>Δh</td><td>height difference between p<sub>in</sub> and p<sub>out</sub></td><td>(m)</td><td>r<sub>a</sub></td><td>outer radius of the volute</td><td>(mm)</td></tr><tr><td>η</td><td>efficiency</td><td>(-)</td><td>r<sub>i</sub></td><td>inner radius of the volute</td><td>(mm)</td></tr><tr><td>ISG</td><td>impeller sidewall gap</td><td></td><td>r<sub>2</sub></td><td>outer radius of the impeller</td><td>(mm)</td></tr><tr><td>θ</td><td>design angle</td><td>(°)</td><td>ρ</td><td>density</td><td>(kg m<sup>-3</sup>)</td></tr><tr><td>K<sub>cm</sub></td><td>STEPANOFF-coefficient</td><td>(-)</td><td>Ste</td><td>STEPANOFF</td><td></td></tr><tr><td>M</td><td>torque</td><td>(N m)</td><td></td><td></td><td></td></tr><tr><td>ṁ</td><td>mass flow rate</td><td>(kg s<sup>-1</sup>)</td><td></td><td></td><td></td></tr><tr><td>n</td><td>nominal speed</td><td>(min<sup>-1</sup>)</td><td></td><td></td><td></td></tr></table>

## INTRODUCTION

In radial and semi-axial single-stage centrifugal pumps the fluid is collected in the stator region after leaving the impeller. In case of standardized chemical pumps or standardized water pumps this stator region is designed as a volute casing. Sev-

<footnote n="1">¹Institute of Fluid Mechanics and Fluid Machinery (SAM), Department of Mechanical and Process Engineering, Technical University of Kaiserslautern, Kaiserslautern, Germany</footnote>

<footnote>*Corresponding author: axel.knapp@mv.uni-kl.de</footnote>
<figure><img src="image_3.png" /><figcaption>Figure 1. Sketch</figcaption></figure>

eral pump manufacturers are producing those single-stage centrifugal pumps in different sizes for several design points. To improve the pump performance, pump manufacturers normally change the design of the impellers, they vary the cross-sectional shape or they redesign the tongue region at the volute. However, the proceed of designing the volute casing of a single-stage centrifugal pump is normally not being changed by a pump manufacturer.

K. RÜTSCHI [5] discovered that the throat area has an influence on the best efficiency point of a pump. He tested one impeller geometry with three different volute casings with trapezoidal cross-section shapes designed with the design method according to PFLEIDERER.

S. YANG et al. [6] investigated the influence of different design rules, volute throat areas and cross-section shapes on the performance of a single-stage centrifugal pump with a specific speed $n_q = 22.9 \text{ min}^{-1}$ using CFD. Highest efficiency was found with circular cross-section shape. Furthermore, the simulation study showed that a large volute throat area could offset the efficiency curve to larger flow rates.

R. DONG et al. [7] had experimentally investigated the effect of the tongue shape on the performance, pressure fluctuation and noise in a centrifugal pump. One result was, that the head of the pump can be increased by retracting and rounding the tongue.

R.C. WORSTER [8] concluded that it is the volute rather than the impeller which determines the point of best efficiency.

O. LITFIN and A. DELGADO [9] investigated three different volute casings designed with the design method according to STEPANOFF. One of the results is that the design flow rate has an influence on the point of best efficiency. With a rising design flow rate the head curve becomes flatter and the point of best efficiency is shifted towards higher flow rates.

The research presented in this paper can be considered as a first step in comparing the effect of different design methods of volutes with circular cross sections on the overall pump performance.

# 1. DESIGN METHODS

The first step in designing a volute is to calculate the flow rate at each cross section. The flow rate $Q_\theta$ can be calculated generally by means of equation (1) for a random design angle $\vartheta$.

$$Q_\theta = \int_{r_i}^{r_a} c_\theta b \, dr \quad (1)$$

Furthermore the flow rate $Q_\theta$ can be described by the total flow rate $Q$ and the circular arc segment depending on the design angle $\vartheta$.

$$Q_\theta = \frac{\vartheta}{360^\circ} Q \quad (2)$$

Equating the previous two equations delivers

$$\vartheta = \frac{360^\circ}{Q} \int_{r_i}^{r_a} c_\theta b \, dr \quad (3)$$

Equation (3) is generally valid for every kind of cross-sectional shape.

## 1.1 C. PLEIDERER [2]

The design method according to PFLEIDERER is based on the assumption that the flow inside the volute satisfies the principle of the conversation of momentum.

$$c_\theta = c_{u,\theta} \cdot r = c_{u,i} \cdot r_i = \text{const.} \quad (4)$$

In that case equation (3) can be written as

$$\vartheta = \frac{360^\circ c_{u,i} r_i}{Q} \int_{r_i}^{r_a} \frac{b}{r} \, dr \quad (5)$$

for every kind of cross-section shape. The volutes, investigated in this paper, have circular cross sections (Figure 1). This fact leads to equation (6).

$$\vartheta = \frac{720^\circ c_{u,i}}{Q} \pi r_i \left[ r_i + R - \sqrt{r_i (r_i + 2R)} \right] \quad (6)$$

In the design of a volute the design angle $\vartheta$ is normally set and the radius $R$ of the cross section area has to be calculated. This leads to equation (7) for the radius $R$ of the circular cross section area.

$$R = \frac{\vartheta Q}{720^\circ \pi c_{u,i} r_i} + \sqrt{\frac{\vartheta Q}{360^\circ \pi c_{u,i}}} \quad (7)$$

## 1.2 A. J. STEPANOFF [3]

The design method according to STEPANOFF is based on the principle of the conversation of kinetic energy.

$$c_\theta = K_{cm} \sqrt{2gH} = \text{const.} \quad (8)$$

The coefficient $K_{cm}$ is a function of the specific speed $n_q$.

$$n_q = n \frac{\sqrt{Q}}{H^{0.75}} \quad (9)$$
<figure><img src="image_4.png" /><figcaption>Figure 2. Coefficient Kcm according to STEPANOFF</figcaption></figure>

The course of the coefficient Kcm is shown in Figure 2. The velocity cθ is assumed to be constant at every single cross section at the whole volute.

The cross section area Aθ can be calculated by means of the continuity equation.

$$A_{\theta} = \frac{Q_{\theta}}{c_{\theta}} = \frac{\theta Q}{360^{\circ} c_{\theta}} \qquad (10)$$

Equation (10) leads to the equation for the radius R of the circular cross section area.

$$R = \frac{1}{2} \sqrt{\frac{\theta Q}{90^{\circ} \pi c_{\theta}}} \qquad (11)$$

## 1.3 Comparison

Both volutes have been calculated by means of equation (7) and equation (11), respectively. The number of cross section areas is chosen to Nθ = 39. Furthermore, the design point of the pump is:

$$Q = 100 \text{ m}^3 \text{ h}^{-1} \\ H = 30 \text{ m} \\ n = 3000 \text{ min}^{-1} \\ \rightarrow \qquad n_q = 39 \text{ min}^{-1}$$

A comparison of both designed volutes is shown in Figure 3 in a two-dimensional view. The STEPANOFF-volute has a larger cross section area Aθ at every single cross section compared to the PFLEIDERER-volute.

The design of the tongue region is the same for both designs. The tongue has a circular tip and the position is fixed for both design methods. The diameter of the tongue tip is dZ = 3 mm.

To ensure comparability of the design methods the tongue tip geometry is the same and in addition no diffusor is connected. The discharge pipe has a constant area that is the

<figure><img src="image_5.png" /><figcaption>Figure 3. Comparison of the designed volutes (2D)</figcaption></figure>

same as the last cross section area and is arranged radially to the volute.

The course of the cross section area Aθ is plotted versus the design angle θ in Figure 4. The blue line represents the course according to STEPANOFF. The course according to PFLEIDERER is represented by the red dotted line. The last cross section area according to STEPANOFF is 18% larger than the cross section area according to PFLEIDERER.

<figure><img src="image_6.png" /><figcaption>Figure 4. Course of the cross section area Aθ</figcaption></figure>

## 2. NUMERICAL SETUP

The numerical model includes the inflow of the pump, the impeller with both impeller sidewall gaps and the volute with a constant cylindrical outflow. All meshes of the fluid regions are generated in ANSYS ICEM CFD and are meshed with block-structured meshes with only hex-volume elements.
The Impeller consists of 1 million elements, the inflow and the impeller sidewall gaps are meshed with 400.000 elements. Both volutes are meshed with 1.2 million elements. The numerical calculations have been done by means of ANSYS CFX using the SST (Shear Stress Transport) turbulence model and a sliding mesh at the impeller region. At the inlet domain a static pressure is set as boundary condition. The mass flow rate is set as boundary condition at the outlet domain and is being varied in steps of 0.2 m for calculating the characteristic curves of the pump. Wall regions are set as no slip wall whether the region is stationary or rotating. All fluid domains are connected via general grid interfaces. If a stationary region is connected with a rotating region, transient rotor stator is defined as frame change option. All numerical results have been averaged over 60° after two calculated revolutions of the impeller. To ensure the comparability between experimental and numerical results, the evaluation planes $p_{in}$ and $p_{out}$ of the numerical model are placed just like at the test rig (Figure 5).

<figure><img src="image_7.png" /><figcaption>Figure 5. Numerical setup</figcaption></figure>

## 3. TEST RIG

The experimental tests have been done at a new test facility at the Institute of Fluid Mechanics and Fluid Machinery (SAM) at the Technical University of Kaiserslautern. This test rig allows to perform experiments with a rotational speed up to 3000 min⁻¹ with water as working fluid.

The test bench is equipped with pressure sensors to determine the pressures on the suction and the pressure side of the pump as well as the pressure at the tank. The pressure

measuring points are arranged at 2 D ahead and after the pump, respectively. The tank is connected to a compressed air line for varying the tank pressure. The flow rate is determined by a magnetic-inductive flow meter which is installed at the ascending line. At the ascending line the temperature is determined by a PT-100 sensor. The torque to calculate the efficiency can be measured by means of a torque meter. Different operating points can be adjusted by means of a control valve to reduce the flow rate and a variable speed drive to vary the speed of the asynchronous motor. In addition the loop can be vented at its highest point. The volute casing of each design variant has been manufactured from polymethyl methacrylate (PMMA) and thus offers an optical access to the flow at the volute. All casings are designed keeping a modular design in mind to allow short set-up times.

## 4. RESULTS AND DISCUSSION

### 4.1 Numerical results

In Figure 6 characteristic curves determined numerically are shown. The head $H$ and the efficiency $\eta$ are plotted versus the flow rate $Q$. The red line with red circles represents the head curve of the volute according to PFLEIDERER (Pfl). The head curve of the volute according to STEPANOFF (Ste) is represented by the blue line with blue open squares. At partial load a higher head can be received by a volute casing according to PFLEIDERER whereas at overload conditions the opposite is true. Both head curves intersect at the design point. The head curves are calculated with respect to equation (12).

$$H = \frac{\Delta p_{tot}}{\rho g} = \frac{p_{tot,out} - p_{tot,in}}{\rho g} \quad (12)$$

The total pressure $p_{tot,out}$ is an area averaged value at the evaluation plane $p_{out}$ and the total pressure $p_{tot,in}$ is an area averaged value at the evaluation plane $p_{in}$, respectively.

<figure><img src="image_8.png" /><figcaption>Figure 6. Characteristic curves (numerical)</figcaption></figure>

The efficiency curve of the volute according to PFLEIDERER is represented by the red dashed line with red dots in Figure 6. The blue dashed line with blue filled squares
<figure><img src="image_9.png" /><figcaption>Figure 7. Test rig</figcaption></figure>

represents the efficiency curve of the volute according to STEPANOFF. At partial load a higher efficiency can be received by a volute casing according to PFLEIDERER whereas at overload conditions the opposite is true. The volute casing according to PFLEIDERER receives its highest efficiency already at a flow rate of $Q = 80 \text{ m}^3 \text{ h}^{-1}$. In the further course towards the design point the efficiency is decreasing. The efficiency curves are calculated with respect to equation (13).

$$ \eta = \frac{P_h}{P_m} = \frac{\rho g H Q}{2 \pi n M} \qquad (13) $$

The torque $M$ is defined as the torque at the rotating regions depending on the rotational axis.

## 4.2 Experimental results

In Figure 8 characteristic curves determined experimentally are shown. The head $H$ and the efficiency $\eta$ are plotted versus the flow rate $Q$. The red line with red circles represents the head curve of the volute according to PFLEIDERER. The head curve of the volute according to STEPANOFF is represented by the blue line with blue open squares. At partial load a higher head can be received by a volute casing according to PFLEIDERER whereas in the range of the design point and at overload conditions the opposite is true. Both head curves intersect at partial load. The head curves are calculated with

respect to equation (14).

$$ H = \frac{\Delta p_{tot}}{\rho g} = \frac{p_{st,out} - p_{st,in}}{\rho g} + \frac{c_{out}^2 - c_{in}^2}{2g} + \Delta h \quad (14) $$

The velocities $c$ at the measuring points of the static pressure $p_{st}$ are calculated by means of the continuity equation. The value $\Delta h$ describes the height difference between both measuring points.

<figure><img src="image_10.png" /><figcaption>Figure 8. Characteristic curves (experimental)</figcaption></figure>

The efficiency curve of the volute according to PFLEIDERER
DERER is represented by the red dashed line with red dots in Figure 8. The blue dashed line with blue filled squares represents the efficiency curve of the volute according to STEPANOFF. At partial load a higher efficiency can be received by a volute casing according to PFLEIDERER whereas in the range of the design point and at overload conditions the opposite is true. The volute casing according to PFLEIDERER receives its highest efficiency already at a flow rate of approximately $Q = 80 \text{ m}^3 \text{ h}^{-1}$. In the further course towards the design point the efficiency is decreasing and the pump is already working at overload conditions. The efficiency curves are calculated with respect to equation (13).

## 4.3 Comparison and discussion

<figure><img src="image_11.png" /><figcaption>Figure 9. Characteristic curves (Comparison)</figcaption></figure>

In Figure 9 numerical and experimental results are compared to each other. The red lines and circles represent the results of the volute according to PFLEIDERER. The results of the volute according to STEPANOFF are represented by the blue lines and squares. The head curves are represented by the line and the dotted one as well as the open symbols. The dashed lines and the filled symbols represent the efficiency curves. Numerical and experimental results fit quite well. The numerical results could be validated by the experimental results. The efficiency curves determined numerically are at a higher level than the ones determined experimentally. This can be explained by the non-consideration of the bearing friction at the numerical simulation.

The gradient of the head curves decreases in the range of strong partial load. This fact is caused by a strong vortex that occurs at the inflow domain at strong partial load. This vortex is shown in Figure 12 with a volute casing according to PFLEIDERER above and with a volute casing according to STEPANOFF below on the left-hand side. In addition in both cases a strong vortex occurs at the discharge line.

In Figure 12 the velocity is plotted as streamlines at strong partial load on the left-hand and at the design point on the right-hand side with a volute casing according to PFLEIDERER above and with a volute casing according to STEPANOFF below. At the design point there is a light vortex at the evaluation plane $p_{out}$ in case of PFLEIDERER. The pump is working

at overload conditions and the head curve drops. At the best efficiency point no vortex occurs (Figure 13). In case of STEPANOFF (Figure 12 on left-hand side below) the fluid at the inflow domain is swirling stronger and faster than in the case of PFLEIDER (Figure 12 on left-hand side above) at strong partial load. This causes the lower level of the head curve with a STEPANOFF-casing at partial load.

<figure><img src="image_12.png" /><figcaption>Figure 10. Averaged velocity $c$ at each cross section</figcaption></figure>

<figure><img src="image_13.png" /><figcaption>Figure 11. Velocity ratio $c_\theta/c_0$ at each cross section</figcaption></figure>

In Figure 10 the area averaged velocity $c$ at each cross section is plotted versus the design angle $\theta$. The cross section area at the tongue is described by the design angle $\theta = 0^\circ$. The red crosses represent the course of the velocity $c$ for the flow rate $Q = 80 \text{ m}^3 \text{ h}^{-1}$ with a volute casing according to PFLEIDERER. The fluid is being decelerated gradually at the volute. That is also shown in Figure 11 by the red dashed line Pfl_Q080. In Figure 11 the velocity ratio $c_\theta/c_0$ is plotted versus the design angle $\theta$.

At the design point (Q100) the fluid is being accelerated after the tongue at the volute according to PFLEIDERER. In the further course the velocity $c$ rises slightly. That case is represented by the red dots in Figure 10 and the red line Pfl_Q100 in Figure 11. In consequence, the pump is already working at overload conditions.
<figure><img src="image_14.png" /><figcaption>**Figure 12.** Streamlines at strong partial load (left) and design point (right)</figcaption></figure>

The red circles in Figure 10 represent the results for the flow rate $Q = 120 \text{ m}^3 \text{ h}^{-1}$ with a volute casing according to PFLEIDERER. In that case the rate of change of the velocity is very high at the beginning of the volute. In the further course the velocity is still rising. The gradient of the velocity ratio $c_\theta/c_0$ is very high, too. That is represented by the red dotted line Pfl_Q120 in Figure 11.

For the flow rate $Q = 80 \text{ m}^3 \text{ h}^{-1}$ with a volute casing according to STEFANOFF the fluid is being decelerated strongly at the whole volute. That case is represented by the blue triangles in Figure 10 and the blue dotted line Ste_Q080 in Figure 11.

At the design point (Q100) the deceleration of the fluid is much smoother than at partial load (Q080) for a volute
<figure><img src="image_15.png" /><figcaption>Figure 13. Streamlines at best efficiency point (PFLEIDERER)</figcaption></figure>

casing according to STEPANOFF (blue filled squares). That is confirmed by the blue dashed line Ste_Q100 in Figure 11.

The blue open squares in Figure 10 represent the course of the velocity $c$ for the flow rate $Q = 120 \, m^3 \, h^{-1}$ with a volute casing according to STEPANOFF. The fluid is being accelerated very slightly at the whole volute. The gradient of the velocity ratio $c_\theta/c_0$ is rising. The course of the velocity ratio is represented by the blue dashed line Ste_Q120 in Figure 11. In consequence, the pump is working at overload conditions.

## 5. SUMMARY AND OUTLOOK

In this paper, the influence of the design methods on the performance of a single-stage centrifugal pump is shown for the nominal speed of $3000 \, min^{-1}$. Diagrams showing throttle curves and efficiency curves that have been evaluated numerically as well as experimentally are discussed. The results shown offer the opportunity to increase the efficiency of a single-stage centrifugal pump by varying the design method of the volute geometry.

At the Institute of Fluid Mechanics and Fluid Machinery (SAM) at the Technical University of Kaiserslautern further investigations on design methods of volutes with circular cross sections for a single-stage centrifugal pump will be done.

In the next steps volute geometries designed according to a given parabolic distribution of the entry flow angle $\alpha_4$ and different given linear courses of the relation $A_\theta/R$ will be investigated. The final step will be detailed investigations on the production of losses at each volute. Therefore there will be a discussion about the different internal flow structures.

## ACKNOWLEDGMENTS

We acknowledge the AHRP (Alliance for High-Performance Computing Rheinland-Pfalz) for the support during the numerical investigations. All numerical investigations are performed on the High-Performance Computer "Elwetritsch".

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