3.1. Solution of a Mathematical Model of Hydraulic Performance Based on MATLAB
A multivariate regression model was constructed to solve the optimization results of the orthogonal test of all parameters of the guide vane based on MATLAB (R2010a, MathWorks, Natick, MA, USA, 2010), and the mathematical relationship between all parameters of the guide vane and the hydraulic performance was predicted. The main processes are summarized as follows:
(1) The relationships between different geometrical parameter combinations of guide vane and efficiency and head of the reactor coolant pump were sorted and recorded as txt files, respectively. The test serial number was retained in the first column and the last seven columns were the six geometric parameter values and efficiency index or head index of different guide vane schemes, respectively. Each column was further separated by a comma and preserved as the efficiency.txt and the head.txt, and then they were deposited into the MATLAB workbench for easy access.
(2) The following structure is used to analyze the efficiency relationship due to a similar process for solving multiple linear regression models of head and efficiency. For the purpose of content simplification, the order for solving the head regression model was not repeated hereof. The following command lines were entered in the window of the command line of the work interface (content after % was defined as the interpretation of the current step) after the MATLAB was opened. The definition of the analysis data is completed as follows:
Load (‘efficiency.txt’);% read the work table file of efficiency.txt;
a = load (‘efficiency.txt’);% assign the value of efficiency.txt to the contents a matrix;
x1=a (:, 2);% assigns the value of the second column of a matrix to x1;
x2=a (:, 3);% assigns the value of the third column of a matrix to x2;
x3=a (:, 4);% assigns the value of the fourth column of a matrix to x3;
x4=a (:, 5);% assigns the value of the Fifth column of a matrix to x1;
x5=a (:, 6);% assigns the value of the sixth column of a matrix to x4;
x6=a (:, 7);% assigns the value of the seventh column of a matrix to x5;
y=a (:, 8);% assigns the value of the eighth column of a matrix to x6;
X = [ones (length (y), 1), x1, x2, x3, x4, x5, x6];
The parentheses were merged by %d into the new matrix X, in which the command of ones (length (y), 1) was designed to create a column matrix that is equal to the y row number with the value of 1;
(3) Furthermore, the regression equation model as defined by the above data block
(y,X) was constructed based on the least-squares estimation method. Its regression coefficient
bi (i = 0,1,2,3,4,5,6), regression coefficient interval
bint, residual
r, confidence interval
rint, and regression model test coefficient
stats were output. Correlation coefficient
r2, significance test statistic value
F and probability
p corresponding to
F value were inc1uded. The command line was defined as follows:
The data block (Y, X) is applied with regression analysis by % d, and then b, bint, r, rint, stats, and related data were output.
The values of the regression coefficient for
b0, b1, b2, b3, b4, b5, and
b6, as well as the regression coefficient interval
bint, were output after the completion of the command line. The output values of the regression model test coefficient
stats were further read, including the correlation coefficient
r2 = 0.8659, the significance test statistic
F = 9.4896, and the corresponding significance level
p = 0.008. The test values of the independent and dependent variables
t, as well as the significant level
p, are indicated in
Table 3. Moderate correlation due to correlation coefficient (r
2 = 0.8891 < 0.9) and significant level distribution test
p = 0.008 < 0.05, significance difference distribution test
t was further performed on the coefficients, where the significant level and efficiency of x6 is less than 0.001, this is considered negligible, i.e., no significant impact on the efficiency is posted by
x6. Therefore, the re-regression analysis was performed for the remaining parameters after
x6 was removed from the regression model above. A new regression model along with its coefficients
b0,
b1,
b2,
b3,
b4,
b5 and regression coefficient interval
bint were obtained after repeating the data definition and command input process of model construction above. See
Table 4.
The output value in the regression model test coefficient
stats was successfully read, including the correlation coefficient
r2 = 0.9159, the significant test statistics
F = 3.4491, and the significant level of
p = 0.053 corresponding to value
F. It was indicated that the correlation coefficient increased to a strong correlation with a revised significance level (
p > 0.05). Therefore, the regression model shows a good significance and a good reference for predicting pump rated efficiency of different geometrical parameters of guide vanes. Furthermore, a regression model formula for efficiency can be obtained:
Since the difference test met the requirements, the fitting degree of the regression model was further analyzed by drawing the curve between the fitting value and the actual value as well as the residual confidence interval distribution graph. It can be seen from
Figure 3 that the fitted value curve and the actual value curve have a high degree of coincidence in most of the intervals. The deviation of the two curves was very small, and only a few parameter combination points have large deviations. The fitting degree between the two curves was satisfied within the allowable error range for the orthogonal experimental model. It can be seen from
Figure 4 that all residual values are within the upper and lower limits of the confidence interval, indicating that the regression model is normal.
According to the above operation, the corresponding command line was input to solve the mathematical regression model of the head and the geometric parameters of the different guide. The final output is as follows: the regression coefficients obtained are
b0 = 123.176,
b1 = −0.073,
b2 = −0.3085,
b3 = 0.221,
b4 = −0.0189,
b5 = −0.783,
b6 = 0.0185,
r2 = 0.9131, significant test statistics
F = 19.2651, and the probability
p = 0.064 corresponding to
F value. It was indicated that the correlation coefficient (
r2 > 0.9) shows a strong correlation with a significant level (
p > 0.05). Therefore, the regression model shows a positive significance. The regression coefficient range is shown in
Table 5.
The regression model formula for the head of the reactor coolant pump is as follows:
It can be seen from the comparison between the fitting value and the actual value curve in
Figure 5 that the coincidence degree between the fitting value curve (red line) and the actual value curve (blue line) is very high. The deviation value between the two is less than 1%, by which a very good fitting degree of the regression model was inculcated. It can also be seen from
Figure 6 that the residual values are all within the confidence interval, indicating a normal regression model.
According to the above formula, the relationship between the geometric parameters of the guide vane and the idling speed and idling flow under the idling operation condition was derived. The established mathematical model of the idling speed is shown in Formula (19):
3.3. Verification of Mathematical Models
Based on the full characteristic curve of the pump combined with the test piping setup, the following factors are defined in the mathematical model of the test system: the water tank part pressure is represented by
P0; the pipe outlet to the model pump inlet section is represented by
L1; the model pump outlet to the inlet of the water tank is represented by
L2. The
Q–H curve,
Q–P curve, and the
Q–η curve are fitted as polynomial equations, which are set as the system variables of the experimental mathematical model. The equation of fluid unsteady flow in the pressurized pipeline is combined with the principle of control process and rigid theory Equation (21) to establish the dynamic mathematical models (Equation (22)). As shown in Equation (22), the transient process of hydraulic change of the pump could be better reflected by this model when the pump was changed in a quasi-steady state under different working conditions. The model pump real-time flow
Q and head
H are defined in real-time by the input
flow-head curve signal
H(Q), while the input torque of the motor
Md is defined in real-time by the input flow-power curve signal
P(Q). The resistance moment Mf is defined by both the input flow-torque curve signal
P(Q) and the flow-efficiency curve signal slave
η (Q). The mathematical model of pump speed is concluded in Formula (23):
where,
P1 and
P2 are defined as the inlet pressure and outlet pressure of the model pump respectively,
Pa;
P0 is defined as the liquid level pressure of the tank,
Pa;
H0 is defined as the level of the water tank,
m;
L1 and
L2 are defined as length of inlet pipe and length of outlet pipe, respectively, m;
S1 and
S2 are defined as cross-sectional areas of the inlet pipe and outlet pipe, respectively, m
2.
d1 and
d2 are defined as pipe diameters of inlet pipe and outlet pipe, respectively, m;
λ1 and
λ2 are defined as the resistance coefficients of the inlet pipe and outlet pipe, respectively;
ρ is defined as the fluid density, 1000 kg/m
3;
g is defined as the acceleration of gravity;
Q is defined as the real-time flow rate of the model pump, m
3/h;
H is defined as the real-time head of the model pump, m;
CF is defined as the valve resistance coefficient;
n is defined as the pump speed, r/min;
J is defined as the moment of inertia of rotor parts, kg·m
2;
Md is defined as the input torque of the motor, kg·m;
Mf is defined as the resistance torque of the motor, kg·m;
t is defined as the simulation process time, s.
In Formulas (22) and (23), the inertia of the experimental unit is indicated by
J = 931 kg·m
2, while the pipe diameter and length are indicated by
d1 =
d2 = 0.76 m,
L1 = 20 m and
L2 = 30 m, respectively. The water tank liquid level pressure was indicated by
P0 = 1
atm and the water tank level is indicated by
H0 = 0.8 m. The starting speed of the motor is set as 1480 r/min and the mathematical model based on MATLAB was applied for the data output. The coasting output value of
dQ/dt and
dn/dt was obtained as 60 s while the flow rate and the speed curve of the corresponding time point were achieved by the integral operation. The comparison between the test output coasting speed changes and the calculation results of the mathematical model coasting speed are indicated in
Figure 9. It was shown that the trend of the rotational speed curve is similar to that of the 25 s while the error in the
t = 10 s is approximately 2.7%, which is in the acceptable range. Therefore, the coasting rotational speed mathematical model is reliable for predicting the rotational speed of the coasting transition process.