# Derivation of the Mathematical Approach to the Radial Pump’s Meridional Channel Design Based on the Controlment of the Medial Axis

Authors: Yeming Lu; Xiaofang Wang; Rong Xie
Journal: Mathematical Problems in Engineering
 (2017)

---

## Abstract

The meridional channel is the base for designing the radial pumps, and a new design approach is proposed here. Different from the previous studies, research here tries to establish the design model simply controlled with the radial coordinate. With the combination of a series of mathematical equations, the new design approach can shape the meridional contours directly by using the initial design variables. As for the mathematical constraint in the new design approach, it was presented in two forms, and each form had its corresponding solution. For the first form (Constraint I), the midpoints of the design points on the hub and shroud contours were thought to be located on the medial axis, and the PSO algorithm was adopted to search for the suitable results. Continually, to accelerate the design process, the second form (Constraint II) to simplify the mathematical constraint was added, and the explicit mathematical expressions calculating the coordinates on the hub and shroud contours were deduced. Finally, to check out the feasibility of the design approach in engineering, it was applied to redesign some typical meridional channels proposed by previous studies, and, through comparative analysis, the effectiveness of the new approach was evaluated and demonstrated.


---

## Body

## 1. Introduction

Radial pumps mainly consist of the centrifugal pump and the mixed-flow pump [1], which are widely applied in various areas such as the chemical industry, the industrial water applications, and the nuclear industry [2]. As the heart of the power equipment in these related areas, the pumps could provide stable energy for the circulation of fluid medium. The need for designing a higher efficiency pump consuming lower energy is increasing with the advocacy for green power, nowadays [3]. Hence, an effective design method for configuring the pump with high performance would be significant [4]. To design the pump, two aspects, namely, the blade shape and the meridional channel, can be considered [5]. Though many inverse design methods about the blade shape have already been conducted by former studies [6–8], little attention has been paid to the meridional channel design. Nevertheless, the meridional shape is the foundation for the pump design, and Bing et al. [9] had proven that the meridional shape has great effects on the pump performance, so that the related design method about the meridional channel should be treated seriously.In terms of the meridional channel design approach, Casey [10] used the Bezier curves to design the hub and shroud contours of the pump, but it is hard to guarantee the good flow characteristics on the cross sections due to the nonsynchronicity design of the hub and shroud contours. Additionally, minor adjustment of the control points on the Bezier curves can lead to a wide change of the meridional shape, which is not useful for local modification and optimization, sometimes. Apart from the shortcomings in optimization, the Bezier curves may confront troubles in the preliminary design of the meridional shape; for that the control points on the Bezier curves are just given according to the expert experience without too much consideration of the related mathematical constraints in most of the conditions.Based on the ISO 2858-1975 standard, Guan [11] used many linear lines and circular arcs to design the meridional shapes of some industrial centrifugal pumps, and they are widely used in China due to the good anticavitation property and simple structure. Nevertheless, as for this traditional design method, a large amount of tests have to be done to settle down the specific values determining the arcs and lines. When faced with the new design requirement beyond the previous design cases, this traditional meridional design method cannot give the specific procedures to identify the design parameters except for the abundant tests, which is rather time-consuming. Moreover, apart from the application in the centrifugal pump design area, the traditional design method is not so popularized in the mixed-flow pump design; therefore, there is a lack of a united design approach to design the meridional shape of the centrifugal pump and the mixed-flow pump.Instead of geometrically designing the meridional channel with linear lines and circular arcs, an important medial axis transform theory was firstly proposed by Choi et al. [12], which was subsequently extended to meridional shape design by Zou et al. [13] and Wang et al. [14]. In succession, Wang et al. [15] adopted the medial axis method in his study and had designed the meridional shapes with different inlet conditions. However, in terms of the previous meridional design approach transformed with the medial axis, too much variables are adopted to design the meridional channel with the complex constrained equations, and among these design variables, the main determined variables are not so clear, which is not beneficial for further modification and optimization.Although it can be concluded that former studies have already done relevant works, there are still some challenges waiting to be solved; consequently, the new meridional channel design approach making up for the partial shortcomings of the previous design methods is proposed in this study. In comparison with the previous methods, the proposed design approach has many advantages. Firstly, it provides a united way to design the meridional channel of the centrifugal pump as well as the mixed-flow pump, which would be verified in the design cases of this study. Secondly, taking the rigid mathematical constraints guaranteeing the good hydraulic characteristics into consideration, the proposed design approach selects the radial coordinate as the single control variable, which is much more simple and manipulable. Moreover, instead of the complex implicit equations, simpler explicit expressions are in great demand in engineering, and an explicit design formula is proposed in the new design approach according to the simplified condition (Constraint II in Section 3.3). Last but not least, the proposed design approach could design the meridional shape including the hub, the medial axis, and the shroud contours directly, which would provide the skeleton and structural parameters for the inverse design of the blade, and this would be further discussed in Section 6.2. Plus, the new meridional shape design approach can also help to extend the design theory of the pump.Taking the envelope formula of the medial axis transform (MAT) theory [12–15] into consideration, research here transformed it into another simple manner controlled by the independent variable of the radial coordinate. Then based on the transformed envelope formula, the circular equation, and the cross section area equation, two kinds of mathematical constraints which had the corresponding solutions for the shape design were introduced. For the first constraint(Constraint I), the partial swarm optimization (PSO) algorithm was adopted to search for the suitable points in the design process. And for the second one(Constraint II), the deduced mathematical expressions calculating the coordinates on the hub and shroud contours were presented. Finally, the design method was applied to redesign the typical meridional shapes, and its effectiveness was verified by comparison with the previous design. The paper is organized as follows: Section 2 gives the details of the basic design theory; in particular, the simplification process for the envelope formula in MAT theory is given. Section 3 describes the mathematical constraints and gives the corresponding solution for the meridional channel design. Section 4 shows the design results of the typical structures. Section 5 analyzes the design results and explores the possibilities of the new design approach in other design conditions. Section 6 provides the conclusions and future work.


## 2. Basic Design Theories

### 2.1. Simplification of Envelope Formula in MAT Theory

The medial axis transform (MAT) method was firstly proposed by Blum [17] to recognize the biological shape, and then it was extended to various areas, for example, the finite-element meshing [18] and the pattern recognition [19]. To calculate the cross section areas and adjust the hub and shroud contours conveniently, the MAT theory was adopted here to design the meridional channel of the radial pump. The enveloping formula in the MAT method is given as [12](1)St=Mt+dt-d′tpt+1-d′t2qtΗt=Mt+dt-d′tpt-1-d′t2qt,where t denotes the location in the natural coordinate; d(t) is the radius of the inscribed circle; p(t) denotes the tangent vector, whose norm is 1; and q(t) represents the vector which is perpendicular to p(t), and its norm equals 1, too.Continually, different from the previous studies [13–15], research here would use the radial coordinate r to take the place of the natural coordinate t; thus, r becomes the independent variable in the envelope formula of the MAT theory. Comparatively, d(t), p(t), q(t), and z(t) in Figure 1 can be replaced with dr, p(r), q(r), and z(r). As a result, based on these separate changes, the details of the transformation process for envelope formula (see (1)) would be presented next.Figure 1
Sketch representation of the MAT.As shown in Figure1, the tangent of θ can be calculated as(2)tan⁡θ=drdz=1dz/dr=1z′r.Consequently,p(t) and q(t) in (1) can be expressed as(3)pt=cos⁡θsin⁡θ=z′r1+z′r211+z′r2,qt=-sin⁡θcos⁡θ=-11+z′r2z′r1+z′r2.Next,d′(t) can be deduced as(4)d′t=dddt=dddz2+dr2=dd1+z′r2dr=11+z′r2d′r.With (2)~(4), (1) can be further expressed as (5)ZsRs=zrr+dr-d′r1+z′r2z′r1+z′r211+z′r2+1-z′r2-d′r21+z′r2-11+z′r2z′r1+z′r2ZhRh=zrr+dr-d′r1+z′r2z′r1+z′r211+z′r2-1-z′r2-d′r21+z′r2-11+z′r2z′r1+z′r2.Also, (5) can be further transformed as(6)Zs=zr-drd′rz′r1+z′r2-1-z′r2-d′r21+z′r2,Rs=r-drd′r1+z′r2+z′r1-z′r2-d′r21+z′r2,Zh=zr-drd′rz′r1+z′r2+1-z′r2-d′r21+z′r2Rh=r-drd′r1+z′r2-z′r1-z′r2-d′r21+z′r2.With the combination of (6), the final simplified envelop formula from the MAT theory can be expressed as(7)z′rZs-zr+Rs-r=-d′rdr,z′rZh-zr+Rh-r=-d′rdr.From (7), it can be concluded out that the envelope formula has been successfully transformed into a simpler manner controlled by the independent variable r of the radial coordinate.


### 2.2. The Circular Equation

As for the enveloping circle shown in Figure1, its center point is Mzr,r, and the radius is d(r). Thus, according to the circular equation, the mathematical relationships between the coordinates of S and H can be expressed as follows:(8)Zh-zr2+Rh-r2=dr2,(9)Zs-zr2+Rs-r2=dr2.Similarly to (7), it is known that (8) and (9) from the circular equation are still controlled by the independent variable r of the radial coordinate, which would provide a united form for further simplification next.


### 2.3. The Cross Section Area Equation

As mentioned in Figure1, the meridional channel is tangent to a series of enveloping circles, whose centers are located on the medial axis. For a random enveloping circle in the channel, it has a corresponding cross section arc. After rotating the cross section arc around the axial axis, the revolutionary surface called the cross section area can be generated. The cross section area F(r) is an important variable, which can be calculated with the following equation [11]:(10)Fr=2πrcbc,where rc is center of the cross section arc and bc denotes the length of the cross section arc. They can be settled down with the following formula:(11)rc=13r+Rh+Rsbc=23dr+Zs-Zh2+Rs-Rh2,where d(r) can be got from (8) or (9).In [14, 15], the distribution of the cross section area F(r) can be taken as a previous settled variable, so that the variable F(r) can still be taken as a known variable before the meridional channel design in this study. However, in the other design conditions [13], the previous given cross section area F(r) can also be replaced with some other design variables, for instance, the volume flow rate Q, the mean meridional velocity Cm, and the blockage factor τ, but actually, the distribution of F(r) can be settled down previously with the following equation:(12)Fr=QCmτ,where Cm can be got from the simulation result or the experienced formula [14] (Cm=A+Br, where A and B are the constants determined by the inlet and outlet variables). τ is closed to 1 and can be ignored sometimes; moreover, it can also be calculated from some experienced formulas [11].


## 2.1. Simplification of Envelope Formula in MAT Theory

The medial axis transform (MAT) method was firstly proposed by Blum [17] to recognize the biological shape, and then it was extended to various areas, for example, the finite-element meshing [18] and the pattern recognition [19]. To calculate the cross section areas and adjust the hub and shroud contours conveniently, the MAT theory was adopted here to design the meridional channel of the radial pump. The enveloping formula in the MAT method is given as [12](1)St=Mt+dt-d′tpt+1-d′t2qtΗt=Mt+dt-d′tpt-1-d′t2qt,where t denotes the location in the natural coordinate; d(t) is the radius of the inscribed circle; p(t) denotes the tangent vector, whose norm is 1; and q(t) represents the vector which is perpendicular to p(t), and its norm equals 1, too.Continually, different from the previous studies [13–15], research here would use the radial coordinate r to take the place of the natural coordinate t; thus, r becomes the independent variable in the envelope formula of the MAT theory. Comparatively, d(t), p(t), q(t), and z(t) in Figure 1 can be replaced with dr, p(r), q(r), and z(r). As a result, based on these separate changes, the details of the transformation process for envelope formula (see (1)) would be presented next.Figure 1
Sketch representation of the MAT.As shown in Figure1, the tangent of θ can be calculated as(2)tan⁡θ=drdz=1dz/dr=1z′r.Consequently,p(t) and q(t) in (1) can be expressed as(3)pt=cos⁡θsin⁡θ=z′r1+z′r211+z′r2,qt=-sin⁡θcos⁡θ=-11+z′r2z′r1+z′r2.Next,d′(t) can be deduced as(4)d′t=dddt=dddz2+dr2=dd1+z′r2dr=11+z′r2d′r.With (2)~(4), (1) can be further expressed as (5)ZsRs=zrr+dr-d′r1+z′r2z′r1+z′r211+z′r2+1-z′r2-d′r21+z′r2-11+z′r2z′r1+z′r2ZhRh=zrr+dr-d′r1+z′r2z′r1+z′r211+z′r2-1-z′r2-d′r21+z′r2-11+z′r2z′r1+z′r2.Also, (5) can be further transformed as(6)Zs=zr-drd′rz′r1+z′r2-1-z′r2-d′r21+z′r2,Rs=r-drd′r1+z′r2+z′r1-z′r2-d′r21+z′r2,Zh=zr-drd′rz′r1+z′r2+1-z′r2-d′r21+z′r2Rh=r-drd′r1+z′r2-z′r1-z′r2-d′r21+z′r2.With the combination of (6), the final simplified envelop formula from the MAT theory can be expressed as(7)z′rZs-zr+Rs-r=-d′rdr,z′rZh-zr+Rh-r=-d′rdr.From (7), it can be concluded out that the envelope formula has been successfully transformed into a simpler manner controlled by the independent variable r of the radial coordinate.


## 2.2. The Circular Equation

As for the enveloping circle shown in Figure1, its center point is Mzr,r, and the radius is d(r). Thus, according to the circular equation, the mathematical relationships between the coordinates of S and H can be expressed as follows:(8)Zh-zr2+Rh-r2=dr2,(9)Zs-zr2+Rs-r2=dr2.Similarly to (7), it is known that (8) and (9) from the circular equation are still controlled by the independent variable r of the radial coordinate, which would provide a united form for further simplification next.


## 2.3. The Cross Section Area Equation

As mentioned in Figure1, the meridional channel is tangent to a series of enveloping circles, whose centers are located on the medial axis. For a random enveloping circle in the channel, it has a corresponding cross section arc. After rotating the cross section arc around the axial axis, the revolutionary surface called the cross section area can be generated. The cross section area F(r) is an important variable, which can be calculated with the following equation [11]:(10)Fr=2πrcbc,where rc is center of the cross section arc and bc denotes the length of the cross section arc. They can be settled down with the following formula:(11)rc=13r+Rh+Rsbc=23dr+Zs-Zh2+Rs-Rh2,where d(r) can be got from (8) or (9).In [14, 15], the distribution of the cross section area F(r) can be taken as a previous settled variable, so that the variable F(r) can still be taken as a known variable before the meridional channel design in this study. However, in the other design conditions [13], the previous given cross section area F(r) can also be replaced with some other design variables, for instance, the volume flow rate Q, the mean meridional velocity Cm, and the blockage factor τ, but actually, the distribution of F(r) can be settled down previously with the following equation:(12)Fr=QCmτ,where Cm can be got from the simulation result or the experienced formula [14] (Cm=A+Br, where A and B are the constants determined by the inlet and outlet variables). τ is closed to 1 and can be ignored sometimes; moreover, it can also be calculated from some experienced formulas [11].


## 3. New Meridional Channel Design Approach Based on Different Mathematical Constraints

### 3.1. The Introduction of the Mathematical Constraint

As with previous studies [13, 14], the medial axis (z(r),r) and the distribution of the cross section area F(r) are chosen to be the prescribed design variables. And these known variables would be input to calculate the coordinates determining the meridional channel based on the basic design theories above. Next, the specific details for the calculation process are then to be presented.In Figure1, it is known that the meridional channel has been designed successfully after the settlement of the coordinates Zs,Rs,Zh,Rh, and these four variables should be in accord with the MAT theory, the circular equation, and the cross area equation set up in Section 2. To eliminate the irrelevant variables d(r) and d′(r), (7) got from the MAT theory can be preliminarily transformed as(13)z′rZs-zr+Rs-r=z′rZh-zr+Rh-r.Accordingly, the circular equation can be expressed as(14)Zh-zr2+Rh-r2=Zs-zr2+Rs-r2.To representZh,Zs with other variables, (13) and (14) can be further transformed as(15)Zh=Rs-Rh+2zrz′r+Rs+Rh-2rz′r22z′rZs=Rh-Rs+2zrz′r+Rs+Rh-2rz′r22z′r.Moreover, from (8)–(11), the cross section area equation can be further deduced as (16)Fr=49πr+Rh+RsZs-zr2+Rs-r2+Zs-Zh2+Rs-Rh2.So far, based on the basic design theory in Section2, three equations (see (15) and (16)) are finally got to settle down the four variables Zs,Rs,Zh,Rh. Apparently, to close the calculation process, an extra equation is needed. Then two kinds of mathematical constraints got from different assumptions are imported to settle down the design variables.


### 3.2.Constraint I and the Solution with PSO Algorithm

#### 3.2.1. Introduction ofConstraint I

As shown in Figure1, point B is the midpoint of point SZs,Rs and point HZh,Rh. Since the distance between point B and point M is very small, the location of B can still be taken as on the prescribed medial axis. Therefore, the following mathematical constraint can be established as(17)Zs+Zh2=zRs+Rh2,where zRs+Rh/2 can be calculated and got from the prescribed distribution of the medial axis z(r) by the replacement of r with Rs+Rh/2.To get the specific values ofZs,Rs,Zh,Rh, the PSO algorithm is imported to search for the right solution.


#### 3.2.2. Applying PSO into the Calculation Process

(
1) Establishment of the Objective Function. On the basis of the four established equations (see (15)~(17)), the values of Zs,Rs,Zh,Rh are then to be identified. In the calculation process, (16) and (17) can be expressed with Y1,Y2 as follows:(18)Y1=Fr-49πr+Rh+RsZs-zr2+Rs-r2+Zs-Zh2+Rs-Rh2=0Y2=Zs+Zh2-zRs+Rh2=0.To connect the two equations in (18), an extra function Y(Zh,Rh,Zs,Rs) is defined, and it is utilized to unite the two equations together as (19)YZh,Rh,Zs,Rs=-Y1+Y2=0.Up to now, for the suitable design variablesZs,Rs,Zh,Rh satisfying (16) and (17), they would also make the calculating result of YZh,Rh,Zs,Rs in (19) equal to 0. Nevertheless, the maximum value of YZh,Rh,Zs,Rs is just 0; namely, the corresponding variables Zs,Rs,Zh,Rh determining the maximum value will also make (16) and (17) true.To find out the suitable variablesZh,Rh,Zs,Rs reaching the maximum value of YZh,Rh,Zs,Rs and making (16) and (17) true, the PSO algorithm is adopted. Plus, if the constraints of (15) are also considered during the searching process, the calculated results would make (15)~(17) true; consequently, the meridional channel can be settled down with these calculated results. Therefore, the objective function can be established:(20)maxYZh,Rh,Zs,Rs=-Y1+Y2st.Zh=Rs-Rh+2zrz′r+Rs+Rh-2rz′r2z′rZs=Rh-Rs+2zrz′r+Rs+Rh-2rz′r2z′rhmin≤Rh≤hmaxsmin≤Rs≤smaxz≤Zs+Zh2,where the variables r,z(r),z′(r) are from the prescribed variables; they are settled as the description of Fig S. 1 in APPENDIX A in Supplementary Material available online at https://doi.org/10.1155/2017/7027016; hmin,hmax,smin,smax represent the search range of PSO in finding the suitable Rh,Rs, and to save the computing resource and shorten the search time (the search range can also be defined as a larger section, but it is time-consuming), they can be settled as(21)hmin=r-b12hmax=rsmin=rsmax=r+b12,where b1 is the expected inlet width, which can be calculated from the given cross section area of the inlet.(
2) Solve the Objective Function with PSO Algorithm. PSO is firstly proposed by Kennedy and Eberhart [20] in 1995 according to the research on bird predation. As for the predation process, the easiest way to find food is to find out the partner who is nearest to the food. Inspired by this biological behavior, PSO is established and applied to solve the optimization problems. In the search space, the potential solutions called particles are assigned with a randomized velocity, a recorded coordinate, and a calculated best solution (fitness). The velocity determining the direction and distance of the particles’ movement can be adjusted dynamically based on the moving experience. Each particle keeps record of the best solution called Pbest and the related coordinates. At the same time, of all the particles so far in the population, the best solution called Gbest in the global version is recorded, too. When the particles are updated at a time, Pbest and Gbest are also updated and reserved based on the newly calculated coordinates and fitness. This searching process is shown in Figure 2. For the movement of each particle, its velocity components mainly consist of the previous motion inertia, the social velocity, and the cognition velocity. The social velocity denotes the effects from the social members, which can be calculated from the distance between the local particle and the global best solution Gbest. Similarly, the cognition velocity denotes the judgement of the particle itself in the searching process, and it can be calculated from the distance between the local particle and the best solution Pbest. The calculations of the motion inertia, the social velocity, and the cognition velocity are all effectively conducted with (23) using the related real time-update parameters.Figure 2
Movement of particles in the searching process.Assuming in the searching space ofD dimensions, the population X=X1,X2,…,Xn consisted of n particles. As for the ith particle in the space it can be represented as Xi=xi1,xi2,…,xiDT, which is on behalf of the potential solution. With the application of the objective function, the fitness value at location Xi can be calculated. In correspondence to Xi, the search velocity is defined as Vi=Vi1,Vi2,…,ViDT. Additionally, the coordinate of the corresponding Pbest is Pi=Pi1,Pi2,…,PiDT, and Gbest is Pg=Pg1,Pg2,…,PgDT. Similarly to the genetic algorithm, the PSO algorithm would find the optimal solution in the design space through a series of iterations. Thus, the k+1th step calculation is established on the calculation of the kth step, and the particles in Figure 2 update their positions and velocities with the following formulas:(22)Xk+1=Xk+vk+1(23)vk+1=vmovk+vsock+vcogk=ωkvik+c1m1Pik-Xik+c2m2Pgk-Xik,where vmovk,vsock,vcogk represent the motion inertia, the social velocity, and the cognition velocity, respectively; k is the iterative number, whose range is settled as [1,200] in this study; c1,c2 are the accelerate factors, which vary the contributions of personal best and local best attractors, and their values are all taken as 1.49445 here; and m1,m2 are the random numbers ranging from 0 to 1. In order to prevent the blind search, the generating velocities are locked in a certain range [-vmax,vmax], and vmax is approximately taken as 10% of the search scope in the design space. ω(k) in (22) denotes the kth iterative weight factor and can be settled down with the following equation:(24)ωk=ωs-ωs-ωdkTmax,where ωs denotes the initial inertia weight whose recommended value is 0.9; ωs is the inertia weight of the maximum iteration whose recommended value is 0.4; and Tmax is the number of the maximum iterations, as it is taken as 200 here.To find out the maximum value of the objective function in (20), the calculation process of PSO algorithm is as follows:(a)
Initialize the velocity and the location of the particles. Calculate the fitness of each particle by using the objective function, and find outPbest and Gbest.(b)
Continue to move and search for the optimal solution in the search area. For the particles,
(1)
calculate the new tentative velocity with (23);(2)
calculate the new location with (22);(3)
calculate the new fitness of each particle, and then, accordingly, update thePbest.(c)
UpdateGbest by the judgement of the newly calculated Pbest.(d)
If the stopping criteria are reached, outputPbest; otherwise, go to the second step.To make a much clearer description of the design process, the flow chart of the design approach with PSO is given in Figure3 along with the solution ofConstraint II. To verify the effectiveness of the PSO algorithm, the test functions in APPENDIX B are executed with the code here, and the results are listed in Fig S. 1 of APPENDIX B.Figure 3
The sketch presentation of the new approach in the design cases.


### 3.3.Constraint II and the Deduced Solution

#### 3.3.1. Introduction ofConstraint II

To accelerate the design process ofConstraint I and get the explicit design expression for the meridional channel in engineering, a simpler assumption indicating that the chord length |SH| equals the enveloping circle’s diameter in Figure 1 is proposed here. Therefore, the envelope circle’s center M becomes the midpoint of |SH|. This simplifying assumption can be briefly expressed with the following mathematical equations:(25)zr≈12Zs+Zh(26)r≈12Rs+Rh.With this assumption, the expressions calculating the coordinates of the meridional channel would be derivated.


#### 3.3.2. Deviation of the Explicit Expressions

From (13), the following equation could be got as(27)Zh-Zs=Rs-Rhz′r.And then with (25), (26), and (27), the cross section area equation of (16) can be transformed as(28)Fr=2πr1+1z′r2Rs-Rh.Continually, with the combination of (15) and (26)~(28), the deduced explicit expressions used to settle down the design variables are as follows:(29)Rh=r-Fr4πr1+1/z′r2Rs=r+Fr4πr1+1/z′r2Zh=Rs-Rh+2zrz′r+Rs+Rh-2rz′r22z′rZs=Rh-Rs+2zrz′r+Rs+Rh-2rz′r22z′r,where r,z(r),z′r are the prescribed variables, and their settlement is shown in APPENDIX A; F(r) can be given previously or calculated by using (12) from the other related variables.


## 3.1. The Introduction of the Mathematical Constraint

As with previous studies [13, 14], the medial axis (z(r),r) and the distribution of the cross section area F(r) are chosen to be the prescribed design variables. And these known variables would be input to calculate the coordinates determining the meridional channel based on the basic design theories above. Next, the specific details for the calculation process are then to be presented.In Figure1, it is known that the meridional channel has been designed successfully after the settlement of the coordinates Zs,Rs,Zh,Rh, and these four variables should be in accord with the MAT theory, the circular equation, and the cross area equation set up in Section 2. To eliminate the irrelevant variables d(r) and d′(r), (7) got from the MAT theory can be preliminarily transformed as(13)z′rZs-zr+Rs-r=z′rZh-zr+Rh-r.Accordingly, the circular equation can be expressed as(14)Zh-zr2+Rh-r2=Zs-zr2+Rs-r2.To representZh,Zs with other variables, (13) and (14) can be further transformed as(15)Zh=Rs-Rh+2zrz′r+Rs+Rh-2rz′r22z′rZs=Rh-Rs+2zrz′r+Rs+Rh-2rz′r22z′r.Moreover, from (8)–(11), the cross section area equation can be further deduced as (16)Fr=49πr+Rh+RsZs-zr2+Rs-r2+Zs-Zh2+Rs-Rh2.So far, based on the basic design theory in Section2, three equations (see (15) and (16)) are finally got to settle down the four variables Zs,Rs,Zh,Rh. Apparently, to close the calculation process, an extra equation is needed. Then two kinds of mathematical constraints got from different assumptions are imported to settle down the design variables.


## 3.2.Constraint I and the Solution with PSO Algorithm

### 3.2.1. Introduction ofConstraint I

As shown in Figure1, point B is the midpoint of point SZs,Rs and point HZh,Rh. Since the distance between point B and point M is very small, the location of B can still be taken as on the prescribed medial axis. Therefore, the following mathematical constraint can be established as(17)Zs+Zh2=zRs+Rh2,where zRs+Rh/2 can be calculated and got from the prescribed distribution of the medial axis z(r) by the replacement of r with Rs+Rh/2.To get the specific values ofZs,Rs,Zh,Rh, the PSO algorithm is imported to search for the right solution.


### 3.2.2. Applying PSO into the Calculation Process

(
1) Establishment of the Objective Function. On the basis of the four established equations (see (15)~(17)), the values of Zs,Rs,Zh,Rh are then to be identified. In the calculation process, (16) and (17) can be expressed with Y1,Y2 as follows:(18)Y1=Fr-49πr+Rh+RsZs-zr2+Rs-r2+Zs-Zh2+Rs-Rh2=0Y2=Zs+Zh2-zRs+Rh2=0.To connect the two equations in (18), an extra function Y(Zh,Rh,Zs,Rs) is defined, and it is utilized to unite the two equations together as (19)YZh,Rh,Zs,Rs=-Y1+Y2=0.Up to now, for the suitable design variablesZs,Rs,Zh,Rh satisfying (16) and (17), they would also make the calculating result of YZh,Rh,Zs,Rs in (19) equal to 0. Nevertheless, the maximum value of YZh,Rh,Zs,Rs is just 0; namely, the corresponding variables Zs,Rs,Zh,Rh determining the maximum value will also make (16) and (17) true.To find out the suitable variablesZh,Rh,Zs,Rs reaching the maximum value of YZh,Rh,Zs,Rs and making (16) and (17) true, the PSO algorithm is adopted. Plus, if the constraints of (15) are also considered during the searching process, the calculated results would make (15)~(17) true; consequently, the meridional channel can be settled down with these calculated results. Therefore, the objective function can be established:(20)maxYZh,Rh,Zs,Rs=-Y1+Y2st.Zh=Rs-Rh+2zrz′r+Rs+Rh-2rz′r2z′rZs=Rh-Rs+2zrz′r+Rs+Rh-2rz′r2z′rhmin≤Rh≤hmaxsmin≤Rs≤smaxz≤Zs+Zh2,where the variables r,z(r),z′(r) are from the prescribed variables; they are settled as the description of Fig S. 1 in APPENDIX A in Supplementary Material available online at https://doi.org/10.1155/2017/7027016; hmin,hmax,smin,smax represent the search range of PSO in finding the suitable Rh,Rs, and to save the computing resource and shorten the search time (the search range can also be defined as a larger section, but it is time-consuming), they can be settled as(21)hmin=r-b12hmax=rsmin=rsmax=r+b12,where b1 is the expected inlet width, which can be calculated from the given cross section area of the inlet.(
2) Solve the Objective Function with PSO Algorithm. PSO is firstly proposed by Kennedy and Eberhart [20] in 1995 according to the research on bird predation. As for the predation process, the easiest way to find food is to find out the partner who is nearest to the food. Inspired by this biological behavior, PSO is established and applied to solve the optimization problems. In the search space, the potential solutions called particles are assigned with a randomized velocity, a recorded coordinate, and a calculated best solution (fitness). The velocity determining the direction and distance of the particles’ movement can be adjusted dynamically based on the moving experience. Each particle keeps record of the best solution called Pbest and the related coordinates. At the same time, of all the particles so far in the population, the best solution called Gbest in the global version is recorded, too. When the particles are updated at a time, Pbest and Gbest are also updated and reserved based on the newly calculated coordinates and fitness. This searching process is shown in Figure 2. For the movement of each particle, its velocity components mainly consist of the previous motion inertia, the social velocity, and the cognition velocity. The social velocity denotes the effects from the social members, which can be calculated from the distance between the local particle and the global best solution Gbest. Similarly, the cognition velocity denotes the judgement of the particle itself in the searching process, and it can be calculated from the distance between the local particle and the best solution Pbest. The calculations of the motion inertia, the social velocity, and the cognition velocity are all effectively conducted with (23) using the related real time-update parameters.Figure 2
Movement of particles in the searching process.Assuming in the searching space ofD dimensions, the population X=X1,X2,…,Xn consisted of n particles. As for the ith particle in the space it can be represented as Xi=xi1,xi2,…,xiDT, which is on behalf of the potential solution. With the application of the objective function, the fitness value at location Xi can be calculated. In correspondence to Xi, the search velocity is defined as Vi=Vi1,Vi2,…,ViDT. Additionally, the coordinate of the corresponding Pbest is Pi=Pi1,Pi2,…,PiDT, and Gbest is Pg=Pg1,Pg2,…,PgDT. Similarly to the genetic algorithm, the PSO algorithm would find the optimal solution in the design space through a series of iterations. Thus, the k+1th step calculation is established on the calculation of the kth step, and the particles in Figure 2 update their positions and velocities with the following formulas:(22)Xk+1=Xk+vk+1(23)vk+1=vmovk+vsock+vcogk=ωkvik+c1m1Pik-Xik+c2m2Pgk-Xik,where vmovk,vsock,vcogk represent the motion inertia, the social velocity, and the cognition velocity, respectively; k is the iterative number, whose range is settled as [1,200] in this study; c1,c2 are the accelerate factors, which vary the contributions of personal best and local best attractors, and their values are all taken as 1.49445 here; and m1,m2 are the random numbers ranging from 0 to 1. In order to prevent the blind search, the generating velocities are locked in a certain range [-vmax,vmax], and vmax is approximately taken as 10% of the search scope in the design space. ω(k) in (22) denotes the kth iterative weight factor and can be settled down with the following equation:(24)ωk=ωs-ωs-ωdkTmax,where ωs denotes the initial inertia weight whose recommended value is 0.9; ωs is the inertia weight of the maximum iteration whose recommended value is 0.4; and Tmax is the number of the maximum iterations, as it is taken as 200 here.To find out the maximum value of the objective function in (20), the calculation process of PSO algorithm is as follows:(a)
Initialize the velocity and the location of the particles. Calculate the fitness of each particle by using the objective function, and find outPbest and Gbest.(b)
Continue to move and search for the optimal solution in the search area. For the particles,
(1)
calculate the new tentative velocity with (23);(2)
calculate the new location with (22);(3)
calculate the new fitness of each particle, and then, accordingly, update thePbest.(c)
UpdateGbest by the judgement of the newly calculated Pbest.(d)
If the stopping criteria are reached, outputPbest; otherwise, go to the second step.To make a much clearer description of the design process, the flow chart of the design approach with PSO is given in Figure3 along with the solution ofConstraint II. To verify the effectiveness of the PSO algorithm, the test functions in APPENDIX B are executed with the code here, and the results are listed in Fig S. 1 of APPENDIX B.Figure 3
The sketch presentation of the new approach in the design cases.


## 3.2.1. Introduction ofConstraint I

As shown in Figure1, point B is the midpoint of point SZs,Rs and point HZh,Rh. Since the distance between point B and point M is very small, the location of B can still be taken as on the prescribed medial axis. Therefore, the following mathematical constraint can be established as(17)Zs+Zh2=zRs+Rh2,where zRs+Rh/2 can be calculated and got from the prescribed distribution of the medial axis z(r) by the replacement of r with Rs+Rh/2.To get the specific values ofZs,Rs,Zh,Rh, the PSO algorithm is imported to search for the right solution.


## 3.2.2. Applying PSO into the Calculation Process

(
1) Establishment of the Objective Function. On the basis of the four established equations (see (15)~(17)), the values of Zs,Rs,Zh,Rh are then to be identified. In the calculation process, (16) and (17) can be expressed with Y1,Y2 as follows:(18)Y1=Fr-49πr+Rh+RsZs-zr2+Rs-r2+Zs-Zh2+Rs-Rh2=0Y2=Zs+Zh2-zRs+Rh2=0.To connect the two equations in (18), an extra function Y(Zh,Rh,Zs,Rs) is defined, and it is utilized to unite the two equations together as (19)YZh,Rh,Zs,Rs=-Y1+Y2=0.Up to now, for the suitable design variablesZs,Rs,Zh,Rh satisfying (16) and (17), they would also make the calculating result of YZh,Rh,Zs,Rs in (19) equal to 0. Nevertheless, the maximum value of YZh,Rh,Zs,Rs is just 0; namely, the corresponding variables Zs,Rs,Zh,Rh determining the maximum value will also make (16) and (17) true.To find out the suitable variablesZh,Rh,Zs,Rs reaching the maximum value of YZh,Rh,Zs,Rs and making (16) and (17) true, the PSO algorithm is adopted. Plus, if the constraints of (15) are also considered during the searching process, the calculated results would make (15)~(17) true; consequently, the meridional channel can be settled down with these calculated results. Therefore, the objective function can be established:(20)maxYZh,Rh,Zs,Rs=-Y1+Y2st.Zh=Rs-Rh+2zrz′r+Rs+Rh-2rz′r2z′rZs=Rh-Rs+2zrz′r+Rs+Rh-2rz′r2z′rhmin≤Rh≤hmaxsmin≤Rs≤smaxz≤Zs+Zh2,where the variables r,z(r),z′(r) are from the prescribed variables; they are settled as the description of Fig S. 1 in APPENDIX A in Supplementary Material available online at https://doi.org/10.1155/2017/7027016; hmin,hmax,smin,smax represent the search range of PSO in finding the suitable Rh,Rs, and to save the computing resource and shorten the search time (the search range can also be defined as a larger section, but it is time-consuming), they can be settled as(21)hmin=r-b12hmax=rsmin=rsmax=r+b12,where b1 is the expected inlet width, which can be calculated from the given cross section area of the inlet.(
2) Solve the Objective Function with PSO Algorithm. PSO is firstly proposed by Kennedy and Eberhart [20] in 1995 according to the research on bird predation. As for the predation process, the easiest way to find food is to find out the partner who is nearest to the food. Inspired by this biological behavior, PSO is established and applied to solve the optimization problems. In the search space, the potential solutions called particles are assigned with a randomized velocity, a recorded coordinate, and a calculated best solution (fitness). The velocity determining the direction and distance of the particles’ movement can be adjusted dynamically based on the moving experience. Each particle keeps record of the best solution called Pbest and the related coordinates. At the same time, of all the particles so far in the population, the best solution called Gbest in the global version is recorded, too. When the particles are updated at a time, Pbest and Gbest are also updated and reserved based on the newly calculated coordinates and fitness. This searching process is shown in Figure 2. For the movement of each particle, its velocity components mainly consist of the previous motion inertia, the social velocity, and the cognition velocity. The social velocity denotes the effects from the social members, which can be calculated from the distance between the local particle and the global best solution Gbest. Similarly, the cognition velocity denotes the judgement of the particle itself in the searching process, and it can be calculated from the distance between the local particle and the best solution Pbest. The calculations of the motion inertia, the social velocity, and the cognition velocity are all effectively conducted with (23) using the related real time-update parameters.Figure 2
Movement of particles in the searching process.Assuming in the searching space ofD dimensions, the population X=X1,X2,…,Xn consisted of n particles. As for the ith particle in the space it can be represented as Xi=xi1,xi2,…,xiDT, which is on behalf of the potential solution. With the application of the objective function, the fitness value at location Xi can be calculated. In correspondence to Xi, the search velocity is defined as Vi=Vi1,Vi2,…,ViDT. Additionally, the coordinate of the corresponding Pbest is Pi=Pi1,Pi2,…,PiDT, and Gbest is Pg=Pg1,Pg2,…,PgDT. Similarly to the genetic algorithm, the PSO algorithm would find the optimal solution in the design space through a series of iterations. Thus, the k+1th step calculation is established on the calculation of the kth step, and the particles in Figure 2 update their positions and velocities with the following formulas:(22)Xk+1=Xk+vk+1(23)vk+1=vmovk+vsock+vcogk=ωkvik+c1m1Pik-Xik+c2m2Pgk-Xik,where vmovk,vsock,vcogk represent the motion inertia, the social velocity, and the cognition velocity, respectively; k is the iterative number, whose range is settled as [1,200] in this study; c1,c2 are the accelerate factors, which vary the contributions of personal best and local best attractors, and their values are all taken as 1.49445 here; and m1,m2 are the random numbers ranging from 0 to 1. In order to prevent the blind search, the generating velocities are locked in a certain range [-vmax,vmax], and vmax is approximately taken as 10% of the search scope in the design space. ω(k) in (22) denotes the kth iterative weight factor and can be settled down with the following equation:(24)ωk=ωs-ωs-ωdkTmax,where ωs denotes the initial inertia weight whose recommended value is 0.9; ωs is the inertia weight of the maximum iteration whose recommended value is 0.4; and Tmax is the number of the maximum iterations, as it is taken as 200 here.To find out the maximum value of the objective function in (20), the calculation process of PSO algorithm is as follows:(a)
Initialize the velocity and the location of the particles. Calculate the fitness of each particle by using the objective function, and find outPbest and Gbest.(b)
Continue to move and search for the optimal solution in the search area. For the particles,
(1)
calculate the new tentative velocity with (23);(2)
calculate the new location with (22);(3)
calculate the new fitness of each particle, and then, accordingly, update thePbest.(c)
UpdateGbest by the judgement of the newly calculated Pbest.(d)
If the stopping criteria are reached, outputPbest; otherwise, go to the second step.To make a much clearer description of the design process, the flow chart of the design approach with PSO is given in Figure3 along with the solution ofConstraint II. To verify the effectiveness of the PSO algorithm, the test functions in APPENDIX B are executed with the code here, and the results are listed in Fig S. 1 of APPENDIX B.Figure 3
The sketch presentation of the new approach in the design cases.


## 3.3.Constraint II and the Deduced Solution

### 3.3.1. Introduction ofConstraint II

To accelerate the design process ofConstraint I and get the explicit design expression for the meridional channel in engineering, a simpler assumption indicating that the chord length |SH| equals the enveloping circle’s diameter in Figure 1 is proposed here. Therefore, the envelope circle’s center M becomes the midpoint of |SH|. This simplifying assumption can be briefly expressed with the following mathematical equations:(25)zr≈12Zs+Zh(26)r≈12Rs+Rh.With this assumption, the expressions calculating the coordinates of the meridional channel would be derivated.


### 3.3.2. Deviation of the Explicit Expressions

From (13), the following equation could be got as(27)Zh-Zs=Rs-Rhz′r.And then with (25), (26), and (27), the cross section area equation of (16) can be transformed as(28)Fr=2πr1+1z′r2Rs-Rh.Continually, with the combination of (15) and (26)~(28), the deduced explicit expressions used to settle down the design variables are as follows:(29)Rh=r-Fr4πr1+1/z′r2Rs=r+Fr4πr1+1/z′r2Zh=Rs-Rh+2zrz′r+Rs+Rh-2rz′r22z′rZs=Rh-Rs+2zrz′r+Rs+Rh-2rz′r22z′r,where r,z(r),z′r are the prescribed variables, and their settlement is shown in APPENDIX A; F(r) can be given previously or calculated by using (12) from the other related variables.


## 3.3.1. Introduction ofConstraint II

To accelerate the design process ofConstraint I and get the explicit design expression for the meridional channel in engineering, a simpler assumption indicating that the chord length |SH| equals the enveloping circle’s diameter in Figure 1 is proposed here. Therefore, the envelope circle’s center M becomes the midpoint of |SH|. This simplifying assumption can be briefly expressed with the following mathematical equations:(25)zr≈12Zs+Zh(26)r≈12Rs+Rh.With this assumption, the expressions calculating the coordinates of the meridional channel would be derivated.


## 3.3.2. Deviation of the Explicit Expressions

From (13), the following equation could be got as(27)Zh-Zs=Rs-Rhz′r.And then with (25), (26), and (27), the cross section area equation of (16) can be transformed as(28)Fr=2πr1+1z′r2Rs-Rh.Continually, with the combination of (15) and (26)~(28), the deduced explicit expressions used to settle down the design variables are as follows:(29)Rh=r-Fr4πr1+1/z′r2Rs=r+Fr4πr1+1/z′r2Zh=Rs-Rh+2zrz′r+Rs+Rh-2rz′r22z′rZs=Rh-Rs+2zrz′r+Rs+Rh-2rz′r22z′r,where r,z(r),z′r are the prescribed variables, and their settlement is shown in APPENDIX A; F(r) can be given previously or calculated by using (12) from the other related variables.


## 4. Design Cases to Verify the Feasibility of the New Approach

As for the established new approach in Section3, the specific design process for its application in engineering is shown in Figure 3. In the design process, controlled by the discrete medial axis from the inlet to the outlet, the related coordinates determining the meridional channel are designed with the new approach of different constraints(Constraint I or II).To explore the effectiveness of the meridional design approach proposed above, the typical structures from the previous studies are adopted here to be the design targets, and the redesign results with the new approach are evaluated by comparison of the targets. Specific details for the application of the new approach in different conditions would be discussed next.


### 4.1. Meridional Shape Design for the Centrifugal Pump

Constructed with the linear lines and circular arcs, a series of meridional channels with good performance are designed based on the ISO 2858-1975 in [11]. Those pumps’ specific speeds are ranging from 0.11 to 0.97, and they have similar structures. We take one typical structure as the design target, whose working conditions are listed as follows: the volume flow is Q=100 m3/h; the head is H=43 m; and the rotational speed is n=2900 rpm. As for the meridional shape of this typical pump, Zou et al. [13] have already extracted the medial axis (z(r),r) and the distribution of the cross section area F(r) in their study, which are shown in Figure 4.Figure 4
Initial design variables extracted from the previous design [13].At first,Constraint Isolved with PSO algorithm is applied in the design process. As for the identified medial axis, it is discretized into 24 points which are marked as P1,P2,…,P24. Based on the previous settled distribution, the related design variables z(ri),F(ri)(1≤i≤24) of the 24 points can be got through fitting and interpolation, whose final results are also given in Figure 4. And then with the application of the design flow chart shown in Figure 3, the corresponding design coordinates controlled by the 24 points are calculated from one point to another. Consequently, the evolutionary generations of the design process are shown in Figure 5(a), and it is found out that the related objective functions get close to 0 just within 200 generations, which means that the new design results are well in accord with (15)~(17), and they are just the design coordinates we are searching for. Plus, in Figure 5(b), the final design results are presented along with the previous structure constructed with linear lines and circular arcs (marked with red lines in the figure), and in the figure, it is verified that the new design approach based onConstraint I can well capture the meridional contours and fit in well with the previous design.Figure 5
Application ofConstraint I and PSO algorithm in the meridional channel design of the centrifugal pump.
(a)
The evolutionary generations in the design process
(b)
Comparison between the new design result and the previous designAfter verifying the effectiveness ofConstraint Iin the design process,Constraint IIand the deduced explicit expressions of (29) are then applied in the design process. In correspondence to the same 24 discretized control points on the medial axis above, the new design results are shown in Figure 6. In comparison with the previous design result, it is known that the new design fits in well with the previous design except for the minor error at the turning section (in the marked box of Figure 6), which would be further discussed in Section 5.1.Figure 6
The new design result with the explicit expressions fromConstraint II.To compare the aerodynamic parameters between the new design results and the target, the 3D meridional channels which are got by rotating the designed 2D profiles around the axial coordinate are simulated and analyzed. The parameters of the blade are not given in the reference, and the main focus is about the meridional shape design here, so that the simulation of the meridional channels without the installation of the blades is conducted here.With the generated configurations and the working conditions proposed above, the inner flows are then simulated with the CFX solver. For the setup of the solver, the standardk-ε turbulent model and the second-order upwind discretization are selected. Plus, to reach maximal y+ around 10 in the computational domains, the minimum height of the cells is controlled to be 0.06 mm at the walls.Correspondingly, the computational domain and the simulating results about the static pressure and the relative velocity are presented in Figure7. From the figure, it is known that there are almost no differences between the design results and the target; however, the design result withConstraint Iis much more accurate than the result withConstraint II.Figure 7
Comparison of the aerodynamic parameters between the design results and the target (without blades) for the centrifugal pump.
(a)
The bladeless computational domain with a rotational speed of 2900 rpm
(b)
Area averaged pressure distribution along the circumference
(c)
Area averaged relative velocity distribution along the circumference


### 4.2. Meridional Shape Design for the Mixed-Flow Pump

The radial pumps consist of the centrifugal pump and the mixed-flow pump. After designing the meridional shape of the centrifugal pump, the new design approach is then applied to design the meridional shape of the mixed-flow pump.The nuclear coolant pump AP1000 is a typical mixed-flow pump, and Wang [16] has designed its meridional shape with another method in his study before. This typical mixed-flow pump is then taken as the design target here, and the specific working conditions for the pump are as follows: the volume flow is Q=17880 m3/h, the head is H=111 m, and the rotational speed is n=1800 rpm. Moreover, the other related variables can be got from Wang’s study, and the initial design variables are presented in Figure 8.Figure 8
Initial design variables extracted from the previous design [16].The medial axis here is discretized into 22 points from the inlet to the outlet. After the application of the design approach based onConstraint I, the corresponding design results are got and shown in Figure 9.Figure 9
Application ofConstraint I and PSO in the design process for the mixed-flow pump.
(a)
The evolutionary generations in the design process
(b)
Comparison between the new design result and the previous designContinually, to explore the feasibility ofConstraint II, the mathematical expressions of (29) are utilized to redesign the meridional shape, whose final design result is shown in Figure 10.Figure 10
The new design result based on the explicit expressions ofConstraint II.According to the design results in Figures9(b) and 10, it is known that, similarly to the design results of the centrifugal pump, the new design approach established on these two kinds of constraints can well design the meridional contours of the mixed-flow pump in relation to the previous structureContinually, to compare the characteristics of the inner flows between the new design results and the target, the meridional channels are then configured with the corresponding designed 2D profile.Since the bladeless meridional channels have already been investigated in Figure7, the situation that the meridional channels are installed with blades should also be considered. With the blade geometry provided by Xie et al. [21], the meridional channels containing seven blades are then to be simulated and investigated with CFD. The setup of the solver is the same as the description of Section 4.1. With the boundary conditions above, the simulating results are ultimately presented in Figure 11. The results denote thatConstraint Isolved with PSO is much more accurate than the explicit formulas ofConstraint IIin the design process, which is the same as the design results of the centrifugal pump.Figure 11
Comparison of the aerodynamic parameters between the design results and the target (with blades) for the mixed-flow pump.
(a)
The computational domain with a rotational speed of 1800 rpm
(b)
Area averaged pressure distribution along the circumference
(c)
Average relative velocity distribution along the circumference


## 4.1. Meridional Shape Design for the Centrifugal Pump

Constructed with the linear lines and circular arcs, a series of meridional channels with good performance are designed based on the ISO 2858-1975 in [11]. Those pumps’ specific speeds are ranging from 0.11 to 0.97, and they have similar structures. We take one typical structure as the design target, whose working conditions are listed as follows: the volume flow is Q=100 m3/h; the head is H=43 m; and the rotational speed is n=2900 rpm. As for the meridional shape of this typical pump, Zou et al. [13] have already extracted the medial axis (z(r),r) and the distribution of the cross section area F(r) in their study, which are shown in Figure 4.Figure 4
Initial design variables extracted from the previous design [13].At first,Constraint Isolved with PSO algorithm is applied in the design process. As for the identified medial axis, it is discretized into 24 points which are marked as P1,P2,…,P24. Based on the previous settled distribution, the related design variables z(ri),F(ri)(1≤i≤24) of the 24 points can be got through fitting and interpolation, whose final results are also given in Figure 4. And then with the application of the design flow chart shown in Figure 3, the corresponding design coordinates controlled by the 24 points are calculated from one point to another. Consequently, the evolutionary generations of the design process are shown in Figure 5(a), and it is found out that the related objective functions get close to 0 just within 200 generations, which means that the new design results are well in accord with (15)~(17), and they are just the design coordinates we are searching for. Plus, in Figure 5(b), the final design results are presented along with the previous structure constructed with linear lines and circular arcs (marked with red lines in the figure), and in the figure, it is verified that the new design approach based onConstraint I can well capture the meridional contours and fit in well with the previous design.Figure 5
Application ofConstraint I and PSO algorithm in the meridional channel design of the centrifugal pump.
(a)
The evolutionary generations in the design process
(b)
Comparison between the new design result and the previous designAfter verifying the effectiveness ofConstraint Iin the design process,Constraint IIand the deduced explicit expressions of (29) are then applied in the design process. In correspondence to the same 24 discretized control points on the medial axis above, the new design results are shown in Figure 6. In comparison with the previous design result, it is known that the new design fits in well with the previous design except for the minor error at the turning section (in the marked box of Figure 6), which would be further discussed in Section 5.1.Figure 6
The new design result with the explicit expressions fromConstraint II.To compare the aerodynamic parameters between the new design results and the target, the 3D meridional channels which are got by rotating the designed 2D profiles around the axial coordinate are simulated and analyzed. The parameters of the blade are not given in the reference, and the main focus is about the meridional shape design here, so that the simulation of the meridional channels without the installation of the blades is conducted here.With the generated configurations and the working conditions proposed above, the inner flows are then simulated with the CFX solver. For the setup of the solver, the standardk-ε turbulent model and the second-order upwind discretization are selected. Plus, to reach maximal y+ around 10 in the computational domains, the minimum height of the cells is controlled to be 0.06 mm at the walls.Correspondingly, the computational domain and the simulating results about the static pressure and the relative velocity are presented in Figure7. From the figure, it is known that there are almost no differences between the design results and the target; however, the design result withConstraint Iis much more accurate than the result withConstraint II.Figure 7
Comparison of the aerodynamic parameters between the design results and the target (without blades) for the centrifugal pump.
(a)
The bladeless computational domain with a rotational speed of 2900 rpm
(b)
Area averaged pressure distribution along the circumference
(c)
Area averaged relative velocity distribution along the circumference


## 4.2. Meridional Shape Design for the Mixed-Flow Pump

The radial pumps consist of the centrifugal pump and the mixed-flow pump. After designing the meridional shape of the centrifugal pump, the new design approach is then applied to design the meridional shape of the mixed-flow pump.The nuclear coolant pump AP1000 is a typical mixed-flow pump, and Wang [16] has designed its meridional shape with another method in his study before. This typical mixed-flow pump is then taken as the design target here, and the specific working conditions for the pump are as follows: the volume flow is Q=17880 m3/h, the head is H=111 m, and the rotational speed is n=1800 rpm. Moreover, the other related variables can be got from Wang’s study, and the initial design variables are presented in Figure 8.Figure 8
Initial design variables extracted from the previous design [16].The medial axis here is discretized into 22 points from the inlet to the outlet. After the application of the design approach based onConstraint I, the corresponding design results are got and shown in Figure 9.Figure 9
Application ofConstraint I and PSO in the design process for the mixed-flow pump.
(a)
The evolutionary generations in the design process
(b)
Comparison between the new design result and the previous designContinually, to explore the feasibility ofConstraint II, the mathematical expressions of (29) are utilized to redesign the meridional shape, whose final design result is shown in Figure 10.Figure 10
The new design result based on the explicit expressions ofConstraint II.According to the design results in Figures9(b) and 10, it is known that, similarly to the design results of the centrifugal pump, the new design approach established on these two kinds of constraints can well design the meridional contours of the mixed-flow pump in relation to the previous structureContinually, to compare the characteristics of the inner flows between the new design results and the target, the meridional channels are then configured with the corresponding designed 2D profile.Since the bladeless meridional channels have already been investigated in Figure7, the situation that the meridional channels are installed with blades should also be considered. With the blade geometry provided by Xie et al. [21], the meridional channels containing seven blades are then to be simulated and investigated with CFD. The setup of the solver is the same as the description of Section 4.1. With the boundary conditions above, the simulating results are ultimately presented in Figure 11. The results denote thatConstraint Isolved with PSO is much more accurate than the explicit formulas ofConstraint IIin the design process, which is the same as the design results of the centrifugal pump.Figure 11
Comparison of the aerodynamic parameters between the design results and the target (with blades) for the mixed-flow pump.
(a)
The computational domain with a rotational speed of 1800 rpm
(b)
Area averaged pressure distribution along the circumference
(c)
Average relative velocity distribution along the circumference


## 5. Further Discussion

### 5.1. Comparison of the Two Constraints in the Design Approach

In order to identify the differences between the two kinds of constraints and evaluate their applicabilities, the related design results in Section4 would be further analyzed. The design accuracy and the design time are chosen to be the analysis indexes.To weigh the design accuracy quantitatively, the design error is imported and defined. As shown in Figure12, the expression evaluating the design accuracy of the new design hub or shroud is given as (30)δ¯=1n∑i=1nΔiRi,where Ri is the radial coordinate of the previous hub or shroud; Δi is the difference between the new design and the previous design along the radial coordinate; and the value of n is taken as 5 here, and, as shown in Figure 12, the calculating points are of the same interval (0.25L) along the axial coordinate.Figure 12
Definition of the design error for the design hub and shroud.After recording the design time and calculating the design errors with (30), the analysis indexes of the design results are ultimately got and shown in Table 1.Table 1
Comparison of the constraints in the new design approach.
Constraints
Items
Centrifugal Pump
Mixed-flow Pump
Constraint I
δ
¯ of the shroud
0.4%
0.16%
δ
¯ of the hub
1.5%
1.6%
Design time
1481 s with 24 control points
0.304 s
Constraint II
δ
¯ of the shroud
0.81%
0.21%
δ
¯ of the hub
2.6%
1.81%
Design time
1139 s with 22 control points
0.202 sAs for the design errorδ, the closer it is to 0, the more accurate the design result will be. In the table, the design errors are ranging from 0.16% to 1.81%, which are close to 0, so that the design result could be quantitatively demonstrated to be desirable. As for the design errors of these two constraints, the related results withConstraint I are a little lower than the results withConstraint IIin most of the conditions (0.4% < 0.81%, 1.5% < 2.6%, 0.16% < 0.21%, and 1.6% < 1.81%), so that the design approach withConstraint I is a little more accurate in designing the meridional shape.The design process is conducted on the Dell station with an Intel Core i5-6500 CPU, whose clock speed is 3.2 GHz. As for the design withConstraint I, the total design time for the meridional channel of the centrifugal pump is 1481 s; namely, the corresponding average design time for one point on the medial axis is 1481/24=61.7 s. Also, the total time for designing the mixed-flow pump is 1139 s, so that, in correspondence with the 22 discretized points on medial axis, the average time generating a pair of coordinates on the hub and shroud contour lasts 1139/22 = 51.78 s. However, in terms of the total design time withConstraint II, it is just within 1 second. By comparison of these two constraints, it can be found that the design approach withConstraint II is much quicker.


### 5.2. Application of the New Design Approach in the Other Design Conditions

Instead of the cross section area, the distribution of the mean meridional velocity is taken as the initial design variable sometimes [15]; can the new design approach here deal with this kind of problem? Also, apart from its application in radial pump, can the new design approach here be applied in the other design area, for example, the impeller? To explore the possibility of these two conditions, the fan impeller in [14] is taken as the design target, and the meridional velocity Cm(r) is given as the initial design variable, which is shown in Figure 13(a). According to the previous study, the blockage τ is ignored and taken as 1, the volume flow is 1.32 m3/s, and the rotational speed is 2900 rpm. As a result, the distribution of F(r) can be calculated with (12); then the final design results designed withConstraint I andConstraint II are presented in Figure 13(b). From the figure, it can be known that the new design approach with these two constraints can be well adaptable to the previous design. Moreover, the bladeless meridional channels are then to be simulated with the boundary conditions stated above, and from the distribution of the pressure and relative velocity in Figure 14, it can be further concluded that the proposed design approach can well design the meridional shape of the impeller by comparison with the previous design.Figure 13
The fan impeller’s meridional channel design with the new approach.
(a)
The mean meridional velocity extracted from the previous study [14]
(b)
Comparison between the new design results and the previous designFigure 14
Comparison of the aerodynamic parameters between the design results and the target (without blades) for the vane.
(a)
Area averaged pressure distribution along the circumference
(b)
Area averaged pressure distribution along the circumference


## 5.1. Comparison of the Two Constraints in the Design Approach

In order to identify the differences between the two kinds of constraints and evaluate their applicabilities, the related design results in Section4 would be further analyzed. The design accuracy and the design time are chosen to be the analysis indexes.To weigh the design accuracy quantitatively, the design error is imported and defined. As shown in Figure12, the expression evaluating the design accuracy of the new design hub or shroud is given as (30)δ¯=1n∑i=1nΔiRi,where Ri is the radial coordinate of the previous hub or shroud; Δi is the difference between the new design and the previous design along the radial coordinate; and the value of n is taken as 5 here, and, as shown in Figure 12, the calculating points are of the same interval (0.25L) along the axial coordinate.Figure 12
Definition of the design error for the design hub and shroud.After recording the design time and calculating the design errors with (30), the analysis indexes of the design results are ultimately got and shown in Table 1.Table 1
Comparison of the constraints in the new design approach.
Constraints
Items
Centrifugal Pump
Mixed-flow Pump
Constraint I
δ
¯ of the shroud
0.4%
0.16%
δ
¯ of the hub
1.5%
1.6%
Design time
1481 s with 24 control points
0.304 s
Constraint II
δ
¯ of the shroud
0.81%
0.21%
δ
¯ of the hub
2.6%
1.81%
Design time
1139 s with 22 control points
0.202 sAs for the design errorδ, the closer it is to 0, the more accurate the design result will be. In the table, the design errors are ranging from 0.16% to 1.81%, which are close to 0, so that the design result could be quantitatively demonstrated to be desirable. As for the design errors of these two constraints, the related results withConstraint I are a little lower than the results withConstraint IIin most of the conditions (0.4% < 0.81%, 1.5% < 2.6%, 0.16% < 0.21%, and 1.6% < 1.81%), so that the design approach withConstraint I is a little more accurate in designing the meridional shape.The design process is conducted on the Dell station with an Intel Core i5-6500 CPU, whose clock speed is 3.2 GHz. As for the design withConstraint I, the total design time for the meridional channel of the centrifugal pump is 1481 s; namely, the corresponding average design time for one point on the medial axis is 1481/24=61.7 s. Also, the total time for designing the mixed-flow pump is 1139 s, so that, in correspondence with the 22 discretized points on medial axis, the average time generating a pair of coordinates on the hub and shroud contour lasts 1139/22 = 51.78 s. However, in terms of the total design time withConstraint II, it is just within 1 second. By comparison of these two constraints, it can be found that the design approach withConstraint II is much quicker.


## 5.2. Application of the New Design Approach in the Other Design Conditions

Instead of the cross section area, the distribution of the mean meridional velocity is taken as the initial design variable sometimes [15]; can the new design approach here deal with this kind of problem? Also, apart from its application in radial pump, can the new design approach here be applied in the other design area, for example, the impeller? To explore the possibility of these two conditions, the fan impeller in [14] is taken as the design target, and the meridional velocity Cm(r) is given as the initial design variable, which is shown in Figure 13(a). According to the previous study, the blockage τ is ignored and taken as 1, the volume flow is 1.32 m3/s, and the rotational speed is 2900 rpm. As a result, the distribution of F(r) can be calculated with (12); then the final design results designed withConstraint I andConstraint II are presented in Figure 13(b). From the figure, it can be known that the new design approach with these two constraints can be well adaptable to the previous design. Moreover, the bladeless meridional channels are then to be simulated with the boundary conditions stated above, and from the distribution of the pressure and relative velocity in Figure 14, it can be further concluded that the proposed design approach can well design the meridional shape of the impeller by comparison with the previous design.Figure 13
The fan impeller’s meridional channel design with the new approach.
(a)
The mean meridional velocity extracted from the previous study [14]
(b)
Comparison between the new design results and the previous designFigure 14
Comparison of the aerodynamic parameters between the design results and the target (without blades) for the vane.
(a)
Area averaged pressure distribution along the circumference
(b)
Area averaged pressure distribution along the circumference


## 6. Conclusions and Future Work

### 6.1. Conclusions

The meridional channel is the foundation of the radial pump design, which was chosen from the database with lines and arcs in previous studies. To extend the design database and provide a simpler design approach in engineering, the new approach controlled by the medial axis is presented here. To verify the effectiveness of the new approach, it was applied to redesign some typical structures. The related conclusions in the study can be got as follows:(
1
) Established on the simplified envelope formula, the circle equation, the cross section area equation, and another mathematical constraint, the new design approach can shape the meridional contours directly based on the medial axis contour and other settled variables. Instead of using the natural coordinate in previous studies, the new design approach sets r from the radial coordinate as the independent variable.(
2
) Two kinds of separate solutions based on the corresponding mathematical constraints are presented in the design approach. For the first constraint(Constraint I), the partial swarm optimization (PSO) algorithm was adopted to search for the suitable results in the design process. Furthermore, the second constraint(Constraint II) is imported, and the explicit mathematical expressions calculating the design points were finally got after a series of deviations.(
3
) The effectiveness of the new approach was evaluated through designing the typical structures. When compared to the previous design, the design errors are ranging from 0.16% to 1.81%, which are close to 0, so that the feasibility of the new approach could be quantitatively demonstrated. Moreover, by comparison of the simulating aerodynamic parameters (pressure and relative velocity), the minor differences between the design results and the target can also illustrate the feasibility of the proposed design approach.(
4
) In terms of the two constraints in the design approach, they have the separate advantages: the design approach withConstraint I is a little more accurate thanConstraint IIin designing the meridional shape; nevertheless, the latter is much simpler and can save a lot of time and computing resources in designing the meridional channel.(
5
) Apart from the radial pump, the new approach presented here can also design the meridional shape of the impeller. In terms of the initial design variable with the cross section area, it can still be replaced by the mean meridional velocity in the design process.


### 6.2. Future Work

Peng et al. [7, 22] and Cao et al. [23] have already proposed the inverse blade design theory controlled by the distribution of the torque velocity, and a series of pumps working in different industrial areas have been got with the blade design theory. But there is a lack of a specific meridional channel design approach to provide the skeleton and structural parameters for the inverse design of the blade. Therefore, we would try to do some joint works in the future; namely, with the combination of the newly proposed design approach and Peng-Cao’s blade design theory, our future work would try to establish a pump inverse design platform which can design the good hydraulic meridional shape as well as the installed blade. Sketch presentation of the design process is briefly presented in Figure 15.Figure 15
Sketch presentation of the pump design with the joint design methods.In the design platform of Figure15, the meridional shape including the hub, the middle (or the medial axis), and the shroud contours is designed with the proposed method at first. Continually, the inverse design of the blade would then be conducted on the basis of the settled meridional contours.


## 6.1. Conclusions

The meridional channel is the foundation of the radial pump design, which was chosen from the database with lines and arcs in previous studies. To extend the design database and provide a simpler design approach in engineering, the new approach controlled by the medial axis is presented here. To verify the effectiveness of the new approach, it was applied to redesign some typical structures. The related conclusions in the study can be got as follows:(
1
) Established on the simplified envelope formula, the circle equation, the cross section area equation, and another mathematical constraint, the new design approach can shape the meridional contours directly based on the medial axis contour and other settled variables. Instead of using the natural coordinate in previous studies, the new design approach sets r from the radial coordinate as the independent variable.(
2
) Two kinds of separate solutions based on the corresponding mathematical constraints are presented in the design approach. For the first constraint(Constraint I), the partial swarm optimization (PSO) algorithm was adopted to search for the suitable results in the design process. Furthermore, the second constraint(Constraint II) is imported, and the explicit mathematical expressions calculating the design points were finally got after a series of deviations.(
3
) The effectiveness of the new approach was evaluated through designing the typical structures. When compared to the previous design, the design errors are ranging from 0.16% to 1.81%, which are close to 0, so that the feasibility of the new approach could be quantitatively demonstrated. Moreover, by comparison of the simulating aerodynamic parameters (pressure and relative velocity), the minor differences between the design results and the target can also illustrate the feasibility of the proposed design approach.(
4
) In terms of the two constraints in the design approach, they have the separate advantages: the design approach withConstraint I is a little more accurate thanConstraint IIin designing the meridional shape; nevertheless, the latter is much simpler and can save a lot of time and computing resources in designing the meridional channel.(
5
) Apart from the radial pump, the new approach presented here can also design the meridional shape of the impeller. In terms of the initial design variable with the cross section area, it can still be replaced by the mean meridional velocity in the design process.


## 6.2. Future Work

Peng et al. [7, 22] and Cao et al. [23] have already proposed the inverse blade design theory controlled by the distribution of the torque velocity, and a series of pumps working in different industrial areas have been got with the blade design theory. But there is a lack of a specific meridional channel design approach to provide the skeleton and structural parameters for the inverse design of the blade. Therefore, we would try to do some joint works in the future; namely, with the combination of the newly proposed design approach and Peng-Cao’s blade design theory, our future work would try to establish a pump inverse design platform which can design the good hydraulic meridional shape as well as the installed blade. Sketch presentation of the design process is briefly presented in Figure 15.Figure 15
Sketch presentation of the pump design with the joint design methods.In the design platform of Figure15, the meridional shape including the hub, the middle (or the medial axis), and the shroud contours is designed with the proposed method at first. Continually, the inverse design of the blade would then be conducted on the basis of the settled meridional contours.



---

*Source: 7027016-2017-10-19.xml*