Optimization of atomization performance of external mixing air atomizing nozzle based on response surface agent model

2.1. Nozzle modeling

Atomizing nozzles are widely used in engines, and the atomization quality directly affects the fuel combustion rate, which in turn affects the power and economy of the engine [17]. The externally mixed air atomizing nozzle studied in this paper is a multi-channel nozzle including a gas channel and a fuel channel, and the structure of the nozzle is shown in Fig. 1. The whole nozzle structure mainly consists of a nozzle, cyclone, and nozzle shell. Among the above parts that make up the nozzle, the structure of the liquid cyclone and the nozzle is relatively complex and is the key part that affects the atomization effect of the nozzle. The liquid cyclone is shown in Fig. 1(b), and three pairs of centre-symmetric cyclone grooves are opened at the top of the right end of the liquid cyclone. The three-dimensional angle between the slotting direction and the axis of the cyclone is the inclination angle of the liquid cyclone slots $\alpha$. Inside the cyclone is a gas channel. The length $L$ of the flat section with diameter $d$ is called the length of the flat section at the gas outlet. The nozzle is shown in Fig. 1(c), and the mixing chamber outlet diameter $D$ is the outlet size at the right end of the nozzle.

Fig. 1Geometric modeling of externally mixed air atomizing nozzle: a) sectional view of nozzle shell; b) cyclone structure; c) sectional view of nozzle

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b)

c)

2.2. Controlled equations

According to the actual nozzle structure parameters, the fluid domain model is established. The overall structure of the external mixed air atomizing nozzle is selected as the physical model for numerical simulation. In addition, to explore the external atomization characteristics of the nozzle. A cylindrical fluid computational domain with a diameter of 12 and a length of 10 is created at the nozzle outlet of the nozzle as shown in Fig. 2.

Fluid simulation software is used in this paper. The flow process in the internal and external fields of the external mixed air atomizing nozzle is simulated based on the VOF method in numerical calculations. HIRT et al. [18] were the first to propose the VOF method, verified through simulation and experiment to be suitable for simulation calculations of incompressible, unmixed fluids. Based on the VOF method simulation, the volume fraction $\alpha$ is defined to characterize the percentage of different fluids within the computational grid. The less-dense gas phase is usually defined as the first phase and the liquid phase as the second phase. Where $\alpha$ is equal to 0, it means that the computational domain grid is all gas. Its value equal to 1 means that the computational domain grid is all liquid. A value equal to between 0 and 1 means that the fluid distribution inside the grid is a gas-liquid mixture. The VOF model control equations are as follows:

Continuity equations:

where $\rho$ is the fluid density, $\mu$ is the time-averaged velocity.

Momentum equation:

where $\mu$ is the coefficient of dynamic viscosity, and $p$ is the fluid pressure.

Realizable $k$-$\epsilon$ is used as the turbulence model and the $k$ and $\epsilon$ equations are:

where $k$ is the turbulent kinetic energy, $\epsilon$ is the turbulent dissipation rate, $G_k$ is the generation term of turbulent kinetic energy $k$ due to the mean velocity, ${\sigma }_k$, ${\sigma }_{\epsilon }$ are the turbulent Prandtl numbers for $k$, and $\epsilon$, respectively, $u_t$ is the turbulent viscosity, $u_t=\rho C_u\frac{k^2}{\epsilon }$, and $C_u$ is a constant:

where $E$ is the average strain rate, $v$ is the kinematic viscosity, and $C_1$, $C_2$ are constants.

2.3. Mesh partitioning and its mesh-independent validation

In this paper, the ICEM module is utilized to mesh the nozzle fluid domain. Due to the complexity of the internal structure of the nozzle, a non-structural object mesh is used, and the model meshing and computational domain are shown in Fig. 2. In order to ensure that the calculation results are independent of the number of meshes, numerical simulations are performed for different numbers of meshes of the nozzle, as shown in Table 1. It can be seen that 1.5 million mesh can better simulate the atomization characteristics of the nozzle. In comparison, the error of atomization cone angle is not more than 3.55 %. Increasing the number of meshes can reduce the computational speed, so a 1.5 million mesh model is used for simulation and analysis.

Fig. 2Calculation area and boundary mesh

2.4. Finite element modeling

The atomization characteristics of an externally mixed air atomizing nozzle are investigated based on the VOF two-phase implicit algorithm. The first phase is set to be air and the second phase is fuel. At the initial moment of numerical simulation, the volume fraction of air at the fuel inlet is equal to 0, and the volume fraction of air at the air inlet takes the value 1. The two-phase interaction model is chosen as the surface tension model. Set the surface tension between air and fuel to 0.04 N/m. The PRESTO algorithm is chosen for the discrete format of the pressure phase. The inlet boundary condition is set to pressure inlet. The fuel pressure was 0.1 MPa, the air pressure was 50 kPa, and the wall boundary conditions were adiabatic with no slip. The external flow field is set up as a pressure outlet. Its pressure is standard atmospheric pressure. According to Chen Xi et al. [19], the turbulence model is selected as the Realizable $k$-$\epsilon$ turbulence model and the algorithm is calculated using the SIMPLE algorithm for the solution. The initial time step of the calculation is 1e-8, and the maximum time step is 1e-5. The initial time step of the calculation is 1e-8, and the maximum time step is 1e-5. The inlet and outlet flow of the nozzle are monitored, and the difference between the inlet and outlet flows is less than 3 %, so it can be determined that the calculation results have converged.

3.1. Determination of key structural parameters

The significance of the effect of nozzle structural parameters on its atomization effect was investigated by one-way experimental method. According to the results of the study, it can be seen that the inclination angle of the liquid cyclone tank $\alpha$, the length of the flat section of the gas outlet $L$ and the diameter of the outlet of the mixing chamber $D$ have a significant effect on the atomization performance of the nozzle, as shown in Figs. 3, 4 and 5.

Fig. 3Analysis curve of the influence of the inclination angle of the liquid cyclone tank on atomization effect

Fig. 4Analysis curve of the influence the length of the flat section of the gas outlet on atomization effect

3.2. Orthogonal experimental design

For the externally mixed air atomizing nozzle, three main structural parameters affect the nozzle atomization characteristics and fuel combustion efficiency. These include the inclination angle of the liquid rotating tank, the length of the flat section at the gas outlet, and the diameter of the mixing chamber outlet. A “three-factor, four-level” orthogonal test design was adopted. Orthogonal tests were designed in terms of the inclination angle of the liquid rotating tank, the length of the flat section at the gas outlet, and the diameter of the mixing chamber outlet. The orthogonal test table was designed by determining the range of levels corresponding to the three structural parameters according to the actual working conditions of the nozzle. The factors and levels are shown in Table 2.

Fig. 5Analysis curve of the influence the diameter of the outlet of the mixing chamber on atomization effect

3.3. Response surface agent modeling

Cao Haifeng et al. [20] carried out an optimization study of air-conditioner acoustic enclosures based on the response surface method to improve the acoustic quality of multi-layer acoustic enclosures. Zheng Lei et al. [21] analysed the stability of super high-rise buildings based on response surface method to improve the seismic performance of super high-rise buildings. Response surface agent modeling is a type of proxy model that uses regression equations to fit a functional relationship between input and output parameters, constructing regression equations to represent complex model relationships [22]. The relation effectively replaces the theoretical mathematical model and provides a line of thought for solving the optimal value problem. The second-order response surface agent model has high accuracy and requires a minimum number of ample points of $(n+1)(n+2)/2$ with the expression [23]:

where $x_i$ is $i$th design variable, $y$ corresponds to the evaluation index, and $a_0$, $b_{ij}$, $c_{ij}$, and $d_{ij}$ are the different coefficients of the polynomials of the agent model, respectively.

3.4. Application testing of response surface agent model

To determine whether the fitted agent model can accurately represent the response relationship of the external mixing air atomizing nozzle, an accuracy test is required. $R^2$ is used as an evaluation indicator. The expression for $R^2$ is given below:

where $S_T$ is the total deviation sum of squares, $S_R$ is the regression sum of squares, and $S_E$ is the residual sum of squares.

The closer $R^2$ is to 1, the more accurate the model is. When $R^2$ is greater than 0.9, the model is considered to have the required accuracy.

Since the agent model used in this paper involves three factors and therefore three variables, the test function equation is set as a ternary function according to mathematical theory. The purpose of establishing this equation is to test the reasonableness of the orthogonal experimental design and the accuracy of the response surface agent model, and to lay the foundation for subsequent research. A three-dimensional nonlinear test function is given as follows:

Three-factor three-level, three-factor four-level and three-factor five-level orthogonal experimental designs were used to obtain different orthogonal experimental tables. Response surface agent models were built separately based on the experimental data, and the predicted values of the test agent model obtained under different levels of orthogonal experiments were compared with the original values of the test function. The common point of the established table of orthogonal experiments is chosen as the point of comparison. The points of comparison are $x=\text{5}$, $y=\text{6}$, and $z=\text{4}$. Table 3 is a table comparing the accuracy of original and predicted values for different levels of orthogonal experimental designs.

As shown in Table 3, the agent model obtained by sampling with 3 factors and 4 levels is close to the original function in terms of accuracy and can meet the requirements of engineering applications. Therefore, the response surface agent model used in this paper can be designed as a 3-factor, 4-level orthogonal experiment.

3.5. Genetic algorithms

The genetic algorithm mainly includes the following control parameters: crossover probability $P_c$, population size $N_{pop}$, and mutation probability $P_m$ [24, 25]. Different control parameters lead to different accuracy of genetic algorithms.

Write the code in MATLAB according to the principle of genetic algorithm. Set the initial population number, crossover rate, and mutation rate, and establish the fitness function based on the agent model. According to the algorithm output fitness function value is to determine whether the genetic algorithm results meet the requirements, meet the output results, or do not meet the loop iteration.

4.1. Analysis of variance

Factors were analysed for Analysis of variance using the $F$ function as shown in Tables 6 and 7:

where $n_1$ is the degree of freedom corresponding to each factor or interaction between factors, and $n_2$ is the sum of error degrees of freedom.

Checking the table, $F_{0.05}\left(\mathrm{2,12}\right)=\text{3.89}$, when $F>F_{0.05}$, it indicates that the factor has a significant effect on the indicator. $K_i$ denotes the experimental mean at the $i$th level of a factor [27].

As shown in Table 6, the variance values $F_{\alpha }>F_{0.05}$ and $F_D>F_{0.05}$ are significant for the effects of liquid cyclone tank inclination and mixing chamber outlet diameter on the atomization cone angle. From $K_{1\alpha }>K_{4\alpha }>K_{3\alpha }>K_{2\alpha }$, $K_{4L}>K_{1L}>K_{3L}>K_{2L}$, $K_{4D}>K_{2D}>K_{3D}>K_{1D}$, the following law can be obtained. The atomization cone angle decreases and then increases with the inclination angle of the liquid rotating tank and the length of the flat section at the gas outlet. The atomization cone angle shows a tendency to increase and then decrease and then increase with the outlet diameter of the mixing chamber.

As shown in Table 7, the effect of the length of the flat section at the gas outlet and the diameter of the mixing chamber outlet on the homogeneity of the circumferential distribution of the fuel atomization is significant, as derived from the values of the variance $F_L>F_{0.05}$ and $F_D>F_{0.05}$. From $K_{3\alpha }>K_{4\alpha }>K_{2\alpha }>K_{1\alpha }$, $K_{4L}>K_{1L}>K_{3L}>K_{2L}$, $K_{1D}>K_{3D}>K_{4D}>K_{2D}$, can be obtained the law. The uniformity of circumferential distribution of atomization shows a tendency to decrease, then increase, and then decrease with the inclination angle of the liquid rotating tank and the outlet diameter of the mixing chamber. The uniformity of atomization circumferential distribution tends to decrease and then increase with the length of the flat section at the gas outlet.

4.2. Regression analysis

To reflect the law and trend of the influence of nozzle structure parameters on the atomization cone angle and fuel atomization circumferential uniformity. So, the relationship between the indicators in Table 6 and Table 7 and the factors were fitted to get the law curve. In this case, the relationship between the atomization cone angle and the mixing chamber outlet diameter was fitted using a cubic fit. The relationship between fuel atomization circumferential uniformity and the length of the flat section at the gas outlet and the diameter of the mixing chamber outlet, respectively, were fitted three times. The relationships of other structural parameters with the atomization cone angle and fuel atomization uniformity index were fitted by conformal fitting.

Fig. 10Fitted curve of atomization cone angle

a) Fitted relationship between liquid swirl tank angle and circumferential uniformity of fuel atomization

b) Fitted relationship between the length of the flat section at the gas outlet and the circumferential uniformity of fuel atomization

c) Fitted relationship between mixing chamber outlet diameter and circumferential uniformity of fuel atomization

It can be obtained from the three curves in Fig. 10. The mixing chamber outlet diameter has the most significant effect on the atomization cone angle. The second is the liquid swirl tank inclination. The least influence is the length of the flat section at the gas outlet. Among them, the atomization cone angle shows a tendency to decrease and then increase as the inclination angle of the liquid cyclone trough increases. When $\alpha$ is taken in the range of 40 to 45, the atomization cone angle has a significant tendency to decrease. As the length of the flat section at the gas outlet increases, the atomization cone angle shows a tendency to decrease and then increase. When the value of $L$ is taken in the range of 3 to 4, there is a significant tendency for the atomization cone angle to decrease. The significance of the increase in the atomization cone angle was decreasing when $L$ was taken in the range of 4 to 6. As the diameter of the mixing chamber outlet increases, the atomization cone angle shows a tendency to increase, then decrease, and then increase. When $D$ is taken in the range of 4.5 to 5, there is a significant tendency to increase the atomization cone angle.

The three curves in Fig. 11 can be shown. The mixing chamber outlet diameter has the most significant effect on the circumferential uniformity of fuel atomization. The second is the length of the flat section at the gas outlet. The least significant effect is the inclination angle of the liquid rotating tank. Among them, the fuel atomization circumferential uniformity index shows a tendency to remain constant and then increase and then decrease as the inclination angle of the liquid cyclone tank increases. When the value of $\alpha$ is taken in the range of 45 to 50, the index of circumferential uniformity of fuel atomization has a tendency to increase significantly. As the length of the flat section at the gas outlet increases, the fuel atomization circumferential uniformity index shows a tendency to decrease and then increase. When the value of $L$ is taken in the range of 5 to 6, the index of fuel atomization circumferential uniformity has a tendency to increase significantly. As the diameter of the mixing chamber outlet increases, the fuel atomization circumferential uniformity index shows a tendency to decrease, then increase, and then decrease. When the value of $D$ is taken in the range of 3.5 to 4, the fuel atomization circumferential uniformity index has a tendency to decrease significantly.

Fig. 11Fitted curve of fuel atomization circumferential distribution uniformity

a) Fitted relationship between liquid swirl tank angle and circumferential uniformity of fuel atomization

b) Fitted relationship between the length of the flat section at the gas outlet and the circumferential uniformity of fuel atomization

c) Fitted relationship between mixing chamber outlet diameter and circumferential uniformity of fuel atomization

4.3. Modeling response surface agent

Based on the principles of response surface agent modeling and orthogonal experimental tables, a response surface function is fitted with the expression:

The regression coefficient $R^2=\text{0.92}$ for the equation of atomization cone angle $y_{\theta }$ and $R^2=\text{0.93}$ for the equation of flow coefficient $y_{\sigma }$ were calculated. This indicates that the equations are fitted with high accuracy. Thus, it can be fitted with this response surface model to simplify the computational process. In this paper, the maximum atomization cone angle and the minimum fuel atomization circumferential uniformity index are taken as the optimum atomization effect. The mathematical model is:

where $y_{\theta }$ is the atomization cone angle, and $y_{\sigma }$ is the fuel atomization circumferential distribution uniformity index. where $x_1$, $x_2$, $x_3$ are the liquid cyclone tank inclination, the length of the flat section at the gas outlet, and the mixing chamber outlet diameter, respectively.

In this paper, the ideal point method of multi-objective planning is used. Since the atomization cone angle $\theta$ has a greater influence on the atomization performance than the fuel atomization circumferential uniformity index $\sigma$, the weights of 0.7 and 0.3 are used as the comprehensive indexes, respectively. The atomization cone angle $\theta$ and the oil atomization circumferential uniformity index $\sigma$ are normalized with the objective function:

4.4. Optimization calculation and result analysis

After obtaining the response relationship equation, the mathematical model was written in MATLAB. According to the control parameters required by the genetic algorithm, the number of iterations is set to 1000, the population size is 200, the crossover probability is 0.6 and the variance probability is 0.05. Then, the optimization is carried out by the optimization process of the genetic algorithm. The iteration curve is shown in Fig. 12, and the optimized nozzle parameters are shown in Table 8.

Fig. 12Genetic algorithm to optimize iteration curve

The parameters obtained after optimization with the genetic algorithm are used to build the nozzle simulation model and simulation analysis, and the simulation results obtained are shown in Table 9.

As shown in Fig. 13, the atomization cone angle increases by 23.07 % relative to the original model, and the fuel atomization circumferential uniformity index decreases by 45.06 % relative to the original model. The optimized nozzle injects fuel more evenly and improves its atomization quality. It achieves the purpose of improving the fuel combustion rate and reducing the pollution of exhaust gas emissions to the environment.

Fig. 13Comparison chart of atomization effect before and after optimization

a) Original model

b) Optimization model