# Effect of Impeller Trimming on the Energy Efficiency of the Counter-Rotating Pumping Stage
## 2 Materials and Methods
### Mathematical Modeling  
Analytical studies were carried out under the same conditions for trimming \(t\) in a range from \(t_{0}\) [mm] to \(t_{1}\) [mm] with a certain step \(\Delta t=(t_{1}-t_{0})/n_{t}\) [mm], where \(n_{t}\)--the total number of trimming modes. These values allowed obtaining the pressure and energy characteristics of the counter-rotating stage with an arbitrary value of the impeller trimming. For this purpose, the quadratic approximation was applied:  
\[\left\{ \begin{matrix}H(t)\\ N(t)\\ \eta(t)\end{matrix} \right\}=\left[ \begin{matrix}K_{11}&K_{12}&K_{13}\\ K_{21}&K_{22}&K_{23}\\ K_{31}&K_{32}&K_{33}\end{matrix} \right]\left\{ \begin{matrix}1\\ t\\ t^{2}\end{matrix} \right\}, \tag{4}\]  
where \(t\)--trimming, mm; \(H\)--pressure head, m; \(N\)--power consumption, kW; \(\eta\)--energy efficiency, \(\%\); \([K]\)--compliance matrix. Figure 1: Flow kinematics in a centrifugal counter-rotating stage. The physical meaning of the coefficients of the matrix \([K]\) follow:. coefficients \(K_{11}=H(0)\), \(K_{21}=N(0)\), and \(K_{31}=\eta(0)\) are pressure head, power consumption, and energy efficiency for an impeller without trimming. They equal the initial values of the curves \(H(t)\), \(N(t)\), and \(\eta(t)\), respectively. Parameters \(K_{12}=dH(0)/dt\), \(K_{22}=dN(0)/dt\), and \(K_{32}=d\eta(0)/dt\) are the 1 st time derivatives of these values for an impeller without trimming. They reflect the initial slope angle of the curves \(H(t)\), \(N(t)\), and \(\eta(t)\). If these parameters are positive, the curve initially ascends and if they are negative it descends. Finally, parameters \(K_{13}=0.5\,d^{2}H(0)/dt^{2}\), \(K_{23}=0.5\,d^{2}N(0)/dt^{2}\), and \(K_{33}=0.5\,d^{2}\eta(0)/dt^{2}\) equal half of the 2 nd time derivatives of the pressure head, power consumption, and energy efficiency, respectively. They reflect the curvature of the curves \(H(t)\), \(N(t)\), and \(\eta(t)\). Moreover, if these parameters are positive, the curvature is concave and if negative it is convex. The unknown elements of the matrix \([K]\) can be evaluated based on the following matrix equation:  
\[[T] \begin{pmatrix}K_{11}\\ K_{12}\\ K_{13}\end{pmatrix} = \begin{pmatrix}H_{0}\\ \dots\\ H_{1}\end{pmatrix} ;\;[T] \begin{pmatrix}K_{21}\\ K_{22}\\ K_{23}\end{pmatrix} = \begin{pmatrix}N_{0}\\ \dots\\ N_{1}\end{pmatrix} ;\;[T] \begin{pmatrix}K_{31}\\ K_{32}\\ K_{33}\end{pmatrix} = \begin{pmatrix}\eta_{0}\\ \dots\\ \eta_{1}\end{pmatrix}  \tag{5}\]  
where \(H_{0}\),..., \(H_{1}\)--a range of pressure heads for various trimmings \(t\); [1]--\(3\times 3\) matrix of trimming coefficients:  
\[[T]= \begin{bmatrix}1&t_{0}&t_{0}^{2}\\ \dots&\dots&\dots\\ 1&t_{1}&t_{1}^{2}\end{bmatrix} . \tag{6}\]  
Since the total number of experimental points can be more than the total number of evaluated parameters, the regression analysis for an overdetermined system (5) should be applied. As a result, the following linear regression dependencies allow for estimating of the unknown elements of the matrix \([K]\):  
\[ \begin{cases}K_{11}\\ K_{12}\\ K_{13}\end{cases} =[\Theta] \begin{pmatrix}H_{0}\\ \dots\\ H_{1}\end{pmatrix} ;\; \begin{pmatrix}K_{21}\\ K_{22}\\ K_{23}\end{pmatrix} =[\Theta] \begin{pmatrix}N_{0}\\ \dots\\ N_{1}\end{pmatrix} ;\; \begin{pmatrix}K_{31}\\ K_{32}\\ K_{33}\end{pmatrix} =[\Theta] \begin{pmatrix}\eta_{0}\\ \dots\\ \eta_{1}\end{pmatrix} , \tag{7}\]  
where \([\Theta]\), the Moore-Penrose pseudoinverse matrix [24] is:  
\[[\Theta]=\Big{(}[T]^{T}[T]\Big{)}^{-1}[T]^{T}. \tag{8}\]  
Formula (4) allows determining of the extremum values of pressure head, power consumption, and energy efficiency according to the following expressions:  
\[\frac{d}{dt} \begin{cases}H(t)\\ N(t)\\ \eta(t)\end{cases} = \begin{cases}K_{12}+2K_{13}t^{*}\\ K_{12}+2K_{13}t^{***}\\ K_{12}+2K_{13}t^{***}\end{cases}