**2. Pump Geometry**

In this study, we have chosen a standard Electrical Submersible Pump (ESP) as the research objective. The main design parameters of this ESP are as follows: design flow rate *Q*des = 36 m3/h; rotating speed *n* = 2850 r/min; total head *H* = 75 m; number of stages *N* = 7; specific speed *n*s = 175.8; design efficiency η = 72%; and maximum power *P* = 11 kW. More details about the parameters of pump geometry could be found in one of our previous paper [22].

This kind of ESP is usually manufactured with plastic materials, in order to ensure there is enough space for the unloading of casting, so the impeller was designed with five blades. In most cases, the number of diffuser vanes is selected to be relative prime number with the number impeller blades. In order to study the effects of the diffuser vanes number on the unsteady flow and pressure fluctuation, the same profile was used for the diffuser vanes and their number was adjusted to 6, 7, and 8 for three different configurations. The solid model of the impeller is shown in Figure 1, and the three configurations with different diffuser vanes number are shown in Figure 2.

**Figure 1.** Solid model of impeller with five blades.

**Figure 2.** Solid model of three configurations with different diffuser vanes number.

## **3. Mathematical Model**

The Navier–Stokes equation (N–S) and continuity equation of an incompressible fluid are [23]:

$$
\frac{\partial u\_i}{\partial t} + u\_j \frac{\partial u\_i}{\partial \mathbf{x}\_j} + \frac{1}{\rho} \frac{\partial p}{\partial \mathbf{x}} - \nu \nabla^2 u\_i = 0,\tag{1}
$$

$$\frac{\partial u\_i}{\partial \mathbf{x}\_i} = 0.\tag{2}$$

where <sup>∇</sup><sup>2</sup> is Laplace operator, *ui* is the flow velocity component in *i* direction, *i* = 1, 2, 3 respectively represent the flow velocity component in the three directions of *x*, *y*, *z*.

Carrying out partial derivative operations on the Equation (1) we get:

$$\frac{\partial^2 u\_i}{\partial t \partial \mathbf{x}\_i} + \frac{\partial u\_j}{\partial \mathbf{x}\_i} \frac{\partial u\_i}{\partial \mathbf{x}\_j} + u\_j \frac{\partial^2 u\_i}{\partial \mathbf{x}\_i \partial \mathbf{x}\_j} + \frac{1}{\rho} \frac{\partial^2 p}{\partial \mathbf{x}\_i^2} - \nu \nabla^2 \frac{\partial u\_j}{\partial \mathbf{x}\_i} = 0. \tag{3}$$

Using formula (2) into Equation (3) we get:

$$\frac{\partial^2 p}{\partial \mathbf{x}\_i^2} = \nabla^2 p = -\rho \left( \frac{\partial \mathbf{u}\_i}{\partial \mathbf{x}\_j} \frac{\partial \mathbf{u}\_j}{\partial \mathbf{x}\_i} \right). \tag{4}$$

If *ui* = *ui* + *u i* , *uj* = *uj* + *u j* , *p* = *p* + *p* , put them into Equation (4) and Equation (2) to get:

$$
\nabla^2 \overline{p} + \nabla^2 p' = -\rho \frac{\partial^2}{\partial \mathbf{x}\_i \partial \mathbf{x}\_j} \Big( \overline{u\_i u\_j} + \overline{u\_i} u'\_j + u'\_i \overline{u\_j} + u'\_i u'\_j \Big), \tag{5}
$$

$$\begin{aligned} \frac{\partial \overline{u\_i}}{\partial \mathbf{x}\_i} + \frac{\partial u\_i'}{\partial \mathbf{x}\_i} = 0, \end{aligned} \tag{6}$$

where *ui*, *uj*, *p* represent time-averaged value, *ui* , *uj* , *p* represent fluctuating value, *ui* = *ui*, *uj* = *uj*, *p* = *p*, *ui* = 0, *uj* = 0, *p* = 0.

Taking the average of Equations (5) and (6) separately we get:

$$\nabla^2 \overline{p} = -\rho \frac{\partial^2}{\partial \mathbf{x}\_i \partial \mathbf{x}\_j} (\overline{u\_i u\_j} + \overline{u\_i' u\_j'})\_\prime \tag{7}$$

$$\frac{\partial \overline{\boldsymbol{u}\_i}}{\partial \mathbf{x}\_i} = 0.\tag{8}$$

*Processes* **2019**, *7*, 354

Substituting Equation (7) from Equation (5) to obtain the expression of pressure fluctuation equation:

$$\nabla^2 p' = -\rho \left[ \frac{\partial^2 \{\overline{u\_i} u\_j' + u\_i \overline{u\_j} \}}{\partial \mathbf{x}\_i \partial \mathbf{x}\_j} + \frac{\partial^2}{\partial \mathbf{x}\_i \partial \mathbf{x}\_j} \left( u\_i' u\_j' - \overline{u\_i' u\_j'} \right) \right]. \tag{9}$$

Equation (6) minus Equation (8) we obtain the expression of pressure fluctuation continuous equation:

$$\frac{\partial u\_l'}{\partial \mathbf{x}\_l} = 0.\tag{10}$$

Using Equation (10) into Equation (9) we finally get:

$$\nabla^2 p' = -\rho \left[ 2\frac{\partial \overline{u\_i}}{\partial \mathbf{x}\_j} \frac{\partial u\_j'}{\partial \mathbf{x}\_i} + \frac{\partial^2}{\partial \mathbf{x}\_i \partial \mathbf{x}\_j} (u\_i' \, u\_j' - \overline{u\_i'} \overline{u\_j'}) \right]\_{\mathbf{y}'} \tag{11}$$

where ρ is the density of water, *p* is pressure fluctuation, *ui* is fluctuating velocity, *ui* is time-averaged velocity.

Equation (11) indicates that the pressure fluctuation is mainly caused by the fluctuation of velocity, the first term on the right-hand side of the equation represents the interaction between mean velocity and fluctuating velocity, which is the mean velocity field distortion and generally called a "quick response term" [24,25]. The time-averaged velocity gradient expresses mean shear stress term, therefore, it is also known as the "turbulence-shear". The second term on the right side of the equation is the change of pressure fluctuation caused by the nonlinear effect in the fluctuating velocity field, commonly known as "turbulence-turbulence". Equation (11) can be understood as the basic equation of pressure fluctuation, in which we can see that pressure fluctuation is equal to the comprehensive effects of the two categories of the effects in brackets of the right side: the first category is the interaction of "shear-turbulence", and the second category is the interaction of "turbulence-turbulence". At the same time, it can be seen from the Equation (11) that there is no direct relation between the viscosity of the fluid and the pressure fluctuation. The viscosity of the fluid only plays an indirect role in pressure fluctuation [25].

#### **4. Numerical Setup**

## *4.1. Computational Domain*

The solid models were created and assembled to form the computational domain in Pro/E Wildfire 5.0 software, which include the inlet section, impeller, diffuser, lateral cavity, and outlet section [26]. In order to balance the total grid number of points, calculation time, and the computer capability [27], only one stage model was chosen in this study, as shown in Figure 3.

**Figure 3.** Computational domain.
