## 2 Description of the Pump Model  
Multistage pumps are used when a high discharge pressure is required, but it is beyond the duty range of single-stage pumps. There are several factors to consider while choosing the number of stages required for the pump model. The number of stages is determined by the head requirement, the Net Positive Suction Head Required (NPSHr), the minimum flow requirement and the performance curve. The more stages a pump has, the higher the final discharge pressure. The proposed multistage pump is designed in such a manner that the number of stages can be increased or decreased, depending on the requirements. A five-stage pump is chosen here to meet the high head and flow rate requirements of several units in the petrochemical refinery. The five-stage centrifugal pump model consists of a double-suction impeller and a twin volute staggered at 180\({}^{\circ}\) in the first stage. The suction chamber has a semispherical shape, with baffle plates installed near the inlet of the double-suction impeller to reduce any possible swirling action toward the impeller inlet. The second stage consists of an impeller, diffuser and return vane. Stages three and four are the repetition of the second stage. An impeller, diffuser and outlet volute are installed in the fifth stage. The impeller is designed to attain a high head with optimum efficiency, while the diffuser is intended to maximize the pressure in the diffuser vanes and minimize the pressure gradient in the return vanes. The pump design indicating each component is shown in Figure 1. The preliminary design of the pump model was obtained from a dynamically similar existing pump provided by the manufacturer. The existing pump was a scaled-down model used for non-hazardous fluids in low flow rate applications. The proposed pump is specifically scaled-up for handling hazardous fluids at a high flow rate and high head operation. The design is extracted using pump affinity laws as shown below:  
\[\frac{Q_{1}}{Q_{2}}=\ \left(\frac{n_{1}}{n_{2}}\right);\ \frac{H_{1}}{H_{2}}=\ \left(\frac{n_{1}}{n_{2}}\right)^{2};\ \frac{P_{1}}{P_{2}}=\ \left(\frac{n_{1}}{n_{2}}\right)^{3} \tag{1}\]  
where the quantities denoted by subscripts 1 and 2 refer to the model and the prototype. The quantities \(n\), \(Q\), \(H\) and \(P\) correspond to the rotational velocity, flow rate, head and input power, respectively. The impeller blade, diffuser vanes and return vanes were designed using BladeGen software. BladeGen is a comprehensive blade design tool commonly used by turbomachin  
Figure 1: The multistage centrifugal pump design. ery designers for the design and control of a blade's properties, including but not limited to the meridional shape, blade angles, thickness and cut-offs [21]. The meridional view of the blades is shown in Figure 2. The design specification of the multistage centrifugal pump is given in Table 1.  
The head coefficient and the flow coefficient are calculated using the following relations [22]:  
\[\psi=\frac{gH}{n^{2}D^{2}} \tag{2}\]  
\[\varphi=\frac{Q}{nD^{3}} \tag{3}\]  
where \(\psi\) is the head coefficient, \(\varphi\) is the flow coefficient, \(D\) is the impeller diameter and \(g\) is the acceleration due to gravity. The pump performance is calculated theoretically using the parameters obtained from Equation (1), and the efficiency is calculated using Equation (4):  
\[\eta=\ \frac{\rho QgH}{P} \tag{4}\]  
Figure 3 shows the pump performance curve obtained by theoretical calculations. The preliminary design was tested for water, as the affinity laws were guaranteed for water or pure liquids with kinematic viscosities less than 10 cS [23]. The efficiency, dimensionless groups and head and flow coefficients were normalized using the values associated with the design point of the pump, which corresponded to the best efficiency point (BEP) of the pump. The Hmax and Hmin correspond to the head of the maximum and minimum impeller diameters, respectively. The head obtained by the actual pump operation must be between these values.