Design and Internal Flow Characteristic Investigation of High-Temperature H₂/Steam-Mixed Working Fluid Turbine

Abstract

In this paper, an improved RSM-CFD method is used to optimally design a mixed turbine of non-equilibrium condensing steam (NECS) and hydrogen (H₂), of which the response surface method (RSM) and computational fluid dynamics (CFD) are coupled to take into account the effects of the wet steam non-equilibrium condensation process of the multimixed working fluid. A single-stage H₂/Steam (NEC)-mixed turbine was developed based on the improved RSM-CFD, and the effect mechanism of the H₂ component ratio (ω_H2) on the flow characteristics, internal power, and isentropic efficiency within the turbine stage were investigated. The results show that the isentropic efficiency (η) increases with the increase in the hydrogen component ratio (ω_H2), since hydrogen, as a non-condensable component, can inhibit the nucleation and growth of steam, reducing the pressure pulsation on the blade surface; furthermore, it accelerates the transport and diffusion of liquid droplets, inhibits the flow separation, and reduces the flow loss in the flow channel. However, the internal power of the turbine (P) tends to decrease with increasing ω_H2, since the increase in hydrogen reduces the pressure difference on the blade and lowers the torque of the fluid acting on the blade, and at the same time, the vortex and radial flow intensify, and the enthalpy drop inside the stage decreases. On this basis, the optimum operating conditions are found where the hydrogen component ratio (volume percent) ω_H2 = 53%. Accordingly, the hydrogen component ratio should be maintained in the range of 38–68%, considering the work capacity and hydrogen yield of the mixed working fluid.

1. Introduction

Until now, the primary source of energy for industrial production has been fossil fuels such as coal, oil, and natural gas, which has resulted in major environmental degradation and the greenhouse effect. Clean and sustainable energy is a major concern because of the substantial environmental concerns caused by the overuse of fossil fuels [1]. Because it only creates water when burned, hydrogen is usually regarded as one of the most promising energy solutions for the future [2,3]. Hydrogen energy is a clean, carbon-free secondary energy source with a high energy density and a variety of conversion methods that play a critical role in the energy system. It is mostly obtained through fossil fuel reforming [4,5], biogenic hydrogen production [6], hydrolysis [7], and other techniques. These reactions, however, require a lot of energy and expensive catalysts. Active metal hydrolysis is a popular research topic and trend in hydrogen generation because of its high theoretical hydrogen production rate, low cost, and ease of operation [8,9]. Researchers are particularly interested in hydrogen synthesis via the magnesium–water reaction at high temperatures because of its advantages, such as high efficiency and high-value-added products [10,11,12,13]. Under high temperature conditions, the magnesium–water reaction releases a large amount of heat and produces a mixture of hydrogen and water vapor. In the reaction, 1 kg of magnesium reacts with 0.75 kg of water to produce 1.67 kg of magnesium oxide and 0.083 kg of hydrogen gas [14,15,16]. The temperature at which the reaction takes place and the flow rate of high-temperature steam affect the temperature of the gas mixture and the mixing ratio, which in turn affects the temperature of the turbine inlet and the hydrogen component ratio.

There are two primary areas of current research on hydrogen production from the magnesium–water reaction. The first is the use of magnesium metal as a high energy density propulsion fuel [17,18], and the second is the use of the magnesium–water reaction to make hydrogen, which is then condensed and dried for storage and usage [11,19]. The former is commonly found in small power machines and is distinguished by its small size and flexibility. It typically simply uses the thermal energy provided by the metal–water reaction, with the hydrogen produced being directly vented to generate thrust. The latter can only use the chemical energy contained in the hydrogen, and a significant portion of the heat generated during the magnesium–water reaction is squandered since it is not recovered and utilized properly. Bergthorson developed a concept that could use all of the chemical energy in metallic fuels, but only in theory [15].

In this paper, the high-temperature H₂/Steam-mixed gas produced by the magnesium–water reaction hydrogen production system under laboratory conditions is investigated and used as the working fluid in a turbine to establish a high-temperature magnesium–water hydrogen production energy recovery cycle system, as shown in Figure 1 (HT, HP, and LT denote high temperature, high pressure, and low temperature, respectively). As a result, this research constructed and optimized a high-temperature H₂/Steam (HT-H₂/Steam)-mixed working fluid turbine to recover waste heat from the HT-H₂/Steam mixture created during the high-temperature magnesium–water reaction for hydrogen production. To achieve the recycling of heat energy, the HT-H₂/Steam mixture is employed as the working fluid of the power cycle. The internal energy is turned into mechanical energy in the turbine, which is then transferred into electrical energy.

The flow of H₂/Steam gas mixtures is a multiphase multicomponent flow with thermodynamic properties different from those of either component, hence the large number of studies aimed at determining the viscosity, thermal conductivity, PVT (pressure volume temperature) properties, and other properties [20,21,22]. The mass and heat transfer and physical field parameters are also affected by the concentration, flow rate, pressure, and other factors of the mixed components [23,24]. Hydrogen as a non-condensable component affects the non-equilibrium condensation of steam, causing complex fluid excitation [25]. Because of the flow instability and flow losses induced by the mixing and transportation of components, multicomponent-mixed flows are highly unpredictable. Therefore, it is necessary to study the flow characteristics and workability of the mixed working fluid in turbine power generation. Since little attention has been paid to the study of the power generation of a H₂/Steam mixture, the work in this paper was developed concerning the study of the physical parameters and dynamic cycle characteristics of other types of mixture.

Mixed-component working fluid research for thermal power generation has recently been a prominent focus of research. More studies have been conducted on the H₂O/CO₂ mixed working fluid for the supercritical water gasification process of coal for hydrogen production and the low-boiling-point organic-mixed working fluid for the organic Rankine cycle for low-temperature heat sources for some types of mixed components. Many experiments and molecular dynamics simulation approaches have been utilized to investigate the physical property parameters of the mixed working fluid.

Peter H. Huang [26] studied the thermodynamic property parameters, including specific enthalpy, specific entropy, and Gibbs energy, of a mixture of hydrogen–water vapor and air–water vapor in a fuel cell. The effect of interaction between hydrogen molecules and air molecules was further investigated. Yuanbin Liu [22] developed a thermodynamic model of the H₂O-CO₂-H₂ mixture and a PVTs dataset of the ternary mixture under near-critical and supercritical conditions using molecular dynamics methods. The PVTs parameters of the mixture in near-critical and supercritical regions can be predicted using this model. Xueming Yang [21] analyzed the thermal conductivity of H₂O-CO₂-H₂ mixtures in the supercritical region using equilibrium molecular dynamics simulations, and established a prediction model as guidance for the determination of the thermal conductivity of the mixed working fluid. The above study has established an approximate method for calculating the physical parameters of the mixed working fluid without further investigation of the changes in their physical parameters, such as their work capacity. In addition, for the research on the dynamic circulation of the mixed working fluid, Shuo Zhang [27] conducted an experimental analysis of the two-dimensional leaf grille and applied the pressure basis solver transition SST (shear stress transport) turbulence model to numerically analyze the flow field loss distribution of the seventh-stage static blade of the supercritical H₂O/CO₂-mixed mass turbine at different impulse angles and exit Mach number after the grille. However, only the static cascade channel was investigated, and the flow inside the rotational cascade channel was not involved. Wang [28] investigated the characteristics and cycle efficiency of CO₂–H₂O binary mixtures for the solar-driven Rankine cycle, and performed a thermal analysis of the kinetic cycle with a binary mixture of H₂O/CO₂ as the working fluid. The results of the study showed that the condensation temperature increases continuously with the increase in the component ratio of water vapor, while the cycle efficiency increases. Costante M. Invernizzi [29] studied the cycle efficiency, the maximum operating pressure of hydrocarbons including eight pure hydrocarbons (linear and cyclic), and four linear hydrocarbon binary mixtures as biomass-powered organic Rankine cycle (ORC) working fluids, and found that for iso-octane-n-octane binary mixtures under the same thermodynamic conditions, the turbine output power increased significantly with the change in mixture component ratio. Kyoung Hoon Kim [30] studied the effect of turbine inlet pressure on the heat exchanger performance of ammonia water-based Rankine (AWR) and regenerative Rankine (AWRR) systems for a comparative analysis of ammonia water-based Rankine (AWR) and regenerative Rankine (AWRR) power generation cycles using an ammonia–water binary mixture as the working fluid. To demonstrate the viability of employing a CO₂ mixture as a working fluid to improve the thermodynamic cycle performance, Abubakr Ayub [31] proposed a supercritical CO₂ cycle and ORC cycle using flue gas as a waste heat source and performed a thermodynamic comparison. Abhay Patil [32] investigated the two-phase operation of a Terry turbine using a mixture of air and water as the working fluid and assessed the change in the Terry turbine’s properties as the mass fraction of gas in the mixture changed.

The practicality and superiority of mixed gas as a working fluid in thermodynamic cycles have been proven in the studies above. Due to the complex fluctuation in the physical properties of the mixture, the thermodynamic properties of the gas mixture are closely related to its composition, component ratios, temperature, and pressure, which can result in unforeseen impacts on the internal flow and efficiency of the turbine. The design of a reasonable and reliable mixed-gas turbine has become a critical point.

Currently, the design methods for turbine structures can be broadly classified into two categories: direct design using commercial software such as AxSTREAM (3.9.13), and optimization design using RSM-CFD methods. However, neither of these methods is suitable for addressing the specific problem investigated in this study. AxSTREAM (3.9.13) software simplifies the non-equilibrium condensation process of wet steam into an equilibrium condensation process. This deviation from the actual operation during the design phase can lead to unexpected energy losses and a decrease in efficiency. On the other hand, the commonly used RSM-CFD methods require presetting appropriate initial structural parameters. These methods are applicable for optimizing existing products [33] or simple flow structures [34]. However, they are insufficient for designing novel flow structures for complex working fluids, such as multicomponent non-equilibrium condensing flows. Therefore, there is currently a lack of a universal optimization design method for special turbines of complex working fluid.

In this paper, an improved RSM-CFD method is used to optimally design the turbine flow structure of a H₂/Steam (NEC) turbine, of which the AxSTREAM(3.9.13) determines the initial structural parameters of a H₂/Steam (EC) turbine, and the improved RSM-CFD are coupled to consider the effects of the wet steam non-equilibrium condensation process of the multimixed working fluid. The effect mechanism of the H₂ component ratio (ω_H2) on the flow characteristics, internal power, and isentropic efficiency within the turbine stage were investigated.

2. Materials and Methods

2.1. Experimental Process

Using an average particle size of 0.5 mm for magnesium particles, at 400 °C, and a superheated water vapor reaction producing a hydrogen–water vapor gas mixture, the temperature can reach 450 °C. The purity of the magnesium particles is higher than 99.8 wt%. The autoclave is used as a vessel for the magnesium metal hydrolysis reaction; the material is 310S stainless steel, the internal volume is 2 L, and the maximum allowable pressure is 20 Mpa. The experimental equipment is shown in Figure 2. The parameters of the gas mixture are shown in the

Table 1. Parameters of the H₂/Steam mixture.

Parameters	Units	Values
temperature	°C	450–550
pressure	MPa	0.1–0.2
hydrogen volume fraction	-	10–100%
mass flow rate	Kg/h	0–30

.

2.2. H₂/Steam Flow Control Model

According to the improved RSM-CFD method, the H₂/Steam-mixed flow needs to be numerically solved using CFD, the function expressions of the turbine structure parameters (L, B, γ, t_b), and the internal efficiency (P), as well as the isentropic efficiency (η) which needs to be fitted with RSM. The core of improved RSM-CFD is to establish the flow control model of the H₂/Steam-mixed flow field, where the effects of the steam non-equilibrium condensation process are considered.

The flow control equations developed in this study are based on the following assumptions and models:

(a)

The ideal mixing state assumption is applied, meaning that there is no variation or segregation of components within the H₂/Steam mixture. Each component is assumed to be uniformly distributed throughout the entire system.

(b)

The assumption of neglecting velocity slip between phases is made, implying that there is no relative slip in velocity between the gas phase and liquid phase. These phases are considered to move together as a single entity.

(c)

The Young model is adopted, assuming that the droplets generated from NEC has a spherical geometry. This model disregards any interactions between the droplets and the surrounding environment, as well as the influence of diffusion phenomena.

(d)

The ICCT model (improved classical nucleation theory for binary gas mixtures) is employed. It assumes that the two-component gas consists of two different monomeric gases and that there is no chemical reaction between them. Furthermore, the condensation process of the two-component gas is considered to be homogeneous nucleation, wherein the nuclei form spontaneously in the gas phase rather than on solid surfaces or impurities.

(e)

The SST turbulence model is used, which assumes that the fluid is a homogeneous, incompressible Newtonian fluid. It further assumes that the turbulent motion within the fluid is isotropic and constant.

The flow of the H₂/Steam mixture is a multiphase multicomponent flow. Assuming an ideal mixing state, the properties of the H₂/Steam mixture can be determined directly using the properties of each component and its component ratio. Neglecting the velocity slip between phases, the two-phase flow-governing equations of the mixture can be expressed as follows:

$$\frac{\partial {\rho }_m}{\partial \mathrm{t}}+\nabla \left({\rho }_m\overrightarrow{v_m}\right)=0$$

(1)

$$\frac{\partial }{\partial t}\left({\rho }_m\overrightarrow{v_m}\right)+\nabla \cdot \left({\rho }_m\overrightarrow{v_m}\overrightarrow{v_m}\right)=-\nabla \cdot p+\nabla \cdot \left[{\mu }_m\left(\nabla \overrightarrow{v_m}+\nabla \overrightarrow{v_m^T}\right)\right]+{\rho }_m\overrightarrow{g}+\overrightarrow{F}$$

(2)

$$\frac{\partial }{\partial t}{{\displaystyle \sum }}_{k=1}^n\left({\alpha }_k{\rho }_k{E}_k\right)+\nabla \cdot {{\displaystyle \sum }}_{k=1}^n\left({\alpha }_k\overrightarrow{v_k}\left({\rho }_k{E}_k+p\right)\right)=\nabla \cdot \left(k_{eff}\nabla T\right)+S_E$$

(3)

where μ_m is the mixed viscosity; $\overrightarrow{F}$ is the body force; n is the number of phases and n = 2; α_k is the volume fraction of the k phase; and S_E indicates the energy source term.

The physical properties of a multicomponent mixture are determined by parameters such as the properties, temperature, pressure, and component ratio of each component. The ideal mixture model is considered for the simplification of the multicomponent mixture flow model by neglecting the interactions between the components. The physical parameters of the mixture are given by the mass-weighted average and volume-weighted average [35,36]:

$${\chi }_m=\alpha {\chi }_1+\left(1-\alpha \right){\chi }_2$$

(4)

$${\varsigma }_m=\beta {\varsigma }_1+\left(1-\beta \right){\varsigma }_2$$

(5)

where χ_m indicates the value of density, molecular weight, viscosity, and thermal conductivity; ζ_m indicates the value of velocity, enthalpy, entropy, and energy; α is the volume fraction of the mixture; and β is the mass fraction of the mixture.

In the H₂/Steam mixture, due to the low condensation and solubility of the H₂ component, the reaction of hydrogen with the droplets of water can be neglected. Therefore, the saturation temperature of the H₂/Steam mixture is:

$$T_{sat}=f\left(p_s\right)$$

(6)

$$p_s=p{\alpha }_s$$

(7)

where p_s and α_s denote the partial pressure and volume fraction of the steam component.

The application of classical nucleation theory (CNT) to calculate the nucleation rate of H₂/Steam-mixed masses does not yield continuously varying results.

The nucleation rate of H₂/Steam mixtures is calculated in this study using the ICCT model developed by Lamanna [37]. This model is a modification of the classical nucleation theory (CNT) for two-component mixtures that consist of only one condensable gas. It addresses the issue of discontinuous variation in the calculation results of CNT:

$$J_{ICCT,L}=\epsilon \frac{1}{S}\frac{{\rho }_v^2}{{\rho }_l}\sqrt{\frac{2\sigma }{\pi m^3}}\exp \left(-\frac{\Delta G^{*}}{k{T}_v}\right)\exp \left(\theta \right)$$

(8)

where ε is the correction factor obtained by Lamanna from the experiment, which takes the value of 0.01; m is the mass of a vapor molecule; S is the gas phase supersaturation; σ is the droplet surface tension; ΔG* is the critical free energy required to form a stable molecular cluster; and θ is the surface tension, which is expressed as follows:

$$\theta =\frac{a_0\sigma }{k{T}_v}$$

(9)

where $k$ is the Boltzmann constant and $a_0$ is the surface area of a gas phase molecule.

The droplet growth model uses the Young model. The droplet growth rate calculation method applicable to different condensation regions is determined using mass conservation at the intersection of the continuous and transition regions. The Young model is widely used in the non-equilibrium condensation of high-speed steam:

$$\frac{\mathrm{d}r}{\mathrm{d}t}=\frac{{\lambda }_v\left(1-\frac{r_{\mathrm{c}}}{r}\right)\left(T_{\mathrm{sat}}-T\right)}{{\rho }_l{h}_{lv}\left(\frac{2r^2}{1+2\beta l}+1.89(1-\phi )\frac{l}{P{r}_v}\right)}$$

(10)

where λ_v is the thermal conductivity of the vapor; T_sat is the saturation temperature; ρ_l is the liquid phase density; h_lv is the latent heat of vaporization; r_c is the critical radius of the droplet; l is the mean free path of vapor molecules; and (1 − φ) is semi-empirical correction formula that acts as a regulating factor, with the following expression:

$$\phi =\frac{R{T}_{\mathrm{sat}}}{h_{lv}}\left[\xi -0.5-\left(\frac{\gamma +1}{2\gamma }\right)\left(\frac{c_p{T}_{\mathrm{sat}}}{h_{lv}}\right)\left(\frac{2-q_{\mathrm{c}}}{2q_{\mathrm{c}}}\right)\right]$$

(11)

where R is the gas constant; ξ is the experimental adjustment factor; γ is the ratio of specific heat; and q_c is the specific heat at constant pressure.

The flow inside the impeller machinery has a complex three-dimensional irregular flow, so the internal flow is a typical turbulence phenomenon. The SST k-ω turbulence model is chosen to simulate the inverse pressure gradient and pressure fluctuation at the wall more accurately. The equations of the SST turbulence model are described as:

$$\frac{\partial }{\partial t}(\rho k)+\frac{\partial }{\partial x_i}\left(\rho k{u}_i\right)=\frac{\partial }{\partial x_j}\left[\left(\mu +\frac{{\mu }_t}{{\tau }_k}\right)\frac{\partial k}{\partial x_j}\right]+G_k-Y_k+S_k$$

(12)

$$\frac{\partial }{\partial t}(\rho \omega )+\frac{\partial }{\partial x_j}\left(\rho \omega u_j\right)=\frac{\partial }{\partial x_j}\left[\left(\mu +\frac{{\mu }_t}{{\tau }_{\omega }}\right)\frac{\partial \omega }{\partial x_j}\right]+G_{\omega }-Y_{\omega }+D_{\omega }+S_{\omega }$$

(13)

where k is the turbulent kinetic energy; ω is the turbulent kinetic energy dissipation rate; μ is the dynamic viscosity of fluids; and μ_t is the turbulent viscosity.

The turbine performance is analyzed in terms of both internal power and isentropic efficiency. The internal power of the stage can be calculated from the effective specific enthalpy drop of the stage and the steam flow rate, which is calculated as follows:

$$P_i=\frac{D\Delta h_i}{3600}=G\Delta h_i$$

(14)

where G is the inlet steam flow rate of the stage, kg/s, and $\Delta h_i$ is the effective specific enthalpy drop, kJ/kg.

The isentropic efficiency of the turbine is often influenced by the inlet boundary conditions and the thermodynamic parameters of the working fluid. The isentropic efficiency of a turbine is defined as the ratio of the actual inlet- and outlet-specific enthalpy drop of the turbine to the ideal inlet- and outlet-specific enthalpy drop of the turbine, shown as follows:

$$\eta =\frac{h_1-h_2}{h_1-h_{2s}}$$

(15)

where h₁ is the average enthalpy at the turbine inlet, kJ/kg; h₂ is the average enthalpy at the actual outlet of the turbine, kJ/kg; and h_2s is the average enthalpy at the ideal outlet of the turbine, kJ/kg.

The enthalpy of the H₂/Steam mixture can be defined as a function of temperature, pressure, and component ratio. Due to the complexity of the flow, temperature and pressure are often related to the coordinate position in the flow field, that is, T = T(x,y,z) and p = p(x,y,z). The enthalpy of the mixture at any position (x,y,z) in the flow field can be calculated using the following equation:

$$h\left(x,y,z\right)={{\displaystyle \sum }}_i{\omega }_i{{\displaystyle \int }}_{T_{ref}}^{T\left(x,y,z\right)}{C}_{p,i}\left(x,y,z\right)dT\left(x,y,z\right)$$

(16)

where ${{\displaystyle \int }}_{T_{ref}}^T{C}_{p,i}dT$ denotes the enthalpy of component i and C_p,i denotes the constant pressure-specific heat of component i. It is a function of temperature and generally obtained by fitting a polynomial to the temperature.

The solution of the above equations can only be obtained with numerical simulation of the H₂/Steam mixture in the turbine flow field with the computational fluid dynamics software ANSYS CFX (2022 R1). The next section will use the numerical simulation method to build a numerical analysis model and solve it.

2.3. Turbine Optimization Design Method Based on Improved RSM-CFD

Four variables were identified based on the sensitivity analysis of the variables. The climbing direction and step length were determined according to the regression equation to achieve a fast approximation to the optimal region and to obtain the center point of the central combination design test. Furthermore, the geometric design constraint of the leaf grille required that the flow deviation should not exceed 3% of the original mass flow, that is:

$$0.97m_0\le m\le 1.03m_0$$

(17)

where m₀ is the mass flow rate at the outlet of the original turbine and m is the mass flow rate of the optimized turbine. The results of the steepest ascent design are shown in

Table 2. Experimental design and results of the steepest ascent design.

Run	Length/mm	Width/mm	Stagger Angle/°	Relative Pitch	Mass Flow Rate/g/s
1	26	22	77	0.6	8.333
2	26.5	22.5	78	0.7	8.458
3	27	23	79	0.8	8.581
4	27.5	23.5	79	0.8	8.65
5	28	24	79	0.8	8.723

.

As can be seen from Table 2, when the range of values exceeds the experimental group 3, the outlet flow will exceed the design-allowable range, so the experimental group 3 is used as the maximum value of the optimized parameters, which is 27 mm in leaf height, 23 mm in leaf width, 79° in stagger angle, and 0.8 mm in relative pitch. Therefore, the optimized parameters range from 23 to 27 mm blade height, 18 to 23 mm blade width, 77 to 79° stagger angle, and 0.6 to 0.8 relative pitch, respectively. The range of stagger angle is generally chosen between ±1° in optimization [38,39]. Such a range of parameters can reduce the number of experimental groups and obtain better optimization results. Narrow size optimization ranges ensure that the design results do not exceed the flow constraints. The isentropic efficiency of the optimized turbine structure has improved by 10% over the original turbine, from 59% to 69.9%, reaching the average level of single-stage turbine efficiency.

Once the H₂/Steam (EC) flow is modeled, the H₂/Steam (NEC) turbine can then be optimized using a design based on improved RSM-CFD. The flowchart is shown in Figure 3 and follows as below:

Step1:

Preliminarily design a H₂/Steam (EC) turbine based on AxSTREAM (3.9.13).

Step1.1:

Give boundary conditions (V, T, ω_H2) according to experiments in laboratory.

Step1.2:

Design turbine and determine initial structural parameters based on H₂/Steam (EC) model.

Step2:

Configure the set of turbine structure parameters (L_i, B_i, γ_i, t_bi) based on RSM.

Step2.1:

Set the characteristic parameters of the H₂/Steam (EC) turbine rotor blade structure as the length (L), width (B), stagger angle (γ), and relative pitch (t_b = t/b) according to the sensitivity analysis.

Step2.2:

Determine the variation range of sensitive variables based on steepest climb design.

Step2.3:

Set up the experiment set based on RSM: within the range of parameters (L, B, γ, t_b) determined by steepest ascent design, a four-factor, three-level experimental design was performed using RSM to obtain 29 sets of experimental parameters (L_i, B_i, γ_i, t_bi) (i = 1, 2, … n), which are shown in Appendix A.

Step3:

Numerically solve the turbine isentropic efficiency (η_i) of 29 sets of experimental parameters (L_i, B_i, γ_i, t_bi) based on CFD.

Step3.1:

Establish 29 sets of finite element models using commercial software ANSYS(2022 R1). according to the experimental parameters (L_i, B_i, γ_i, t_bi), of which the geometric models are established in BladeGen and the topology and mesh are generated in TurboGrid.

Step3.2:

Numerically solve 29 sets of turbine isentropic efficiency (Equations (2)–(15)), of which the enthalpy (Equation (16)) can be obtained by numerically solving the NEC steam (Equations (8)–(11)) using ANSYS CFX(2022 R1).

Step4:

Analytically solve the optimal turbine structure parameters.

Step4.1:

Fit the turbine isentropic efficiency (η) as the function of structure parameters (L, B, γ, t_b).

Step4.2:

Calculate the optimal structural parameters (L₀, B₀, γ₀, t_b0) of the H₂/Steam (NEC) turbine by solving the maxima of the above function based on Equation (18).

$$\begin{array}{c}\frac{\partial \eta }{\partial L_0}=0,\frac{\partial \eta }{\partial B_0}=0,\frac{\partial \eta }{\partial {\gamma }_0}=0,\frac{\partial \eta }{\partial t_{b0}}=0\\ \frac{{\partial }^2\eta }{\partial L_0^2}<0,\frac{{\partial }^2\eta }{\partial B_0^2}<0,\frac{{\partial }^2\eta }{\partial {\gamma }_0^2}<0,\frac{{\partial }^2\eta }{\partial t_{b0}^2}<0\end{array}$$

(18)

ANOVA of the model and variables was performed to check the significance of the model and parameters. The blade height, blade width, stagger angle, and relative pitch of the turbine were labeled as A, B, C, and D, respectively, and the response value (R) was taken as the isentropic efficiency of the turbine. The response surface test data were analyzed using ANOVA, and the data are shown in

Table 3. ANOVA for quadratic model and variables.

Source	Sum of Squares	Degree of Freedom	Mean SQUARE	F-Value	p-Value
Model	0.0121	9	0.0013	35.43	<0.0001	significant
A	0.0005	1	0.0005	14.31	0.0013
B	0.0023	1	0.0023	61.52	<0.0001
C	0.0004	1	0.0004	9.60	0.0059
D	0.0057	1	0.0057	148.83	<0.0001
BD	0.0006	1	0.0006	15.53	0.0009
CD	0.0001	1	0.0001	1.69	0.2096
A2	0.0005	1	0.0005	13.18	0.0018
B2	0.0011	1	0.0011	28.74	<0.0001
D2	0.0007	1	0.0007	17.56	0.0005
Residual	0.0007	19	0.0000
Lack of Fit	0.0007	15	0.0000
Pure Error	0.0000	4	0.0000
Cor Total	0.0129	28

.

The model F-value of 35.43 implies the model is significant. p-values less than 0.0500 indicate model terms are significant. In this case A, B, C, D, BD, A², B², D² are significant model terms. Values greater than 0.1000 indicate the model terms are not significant. The model p < 0.0001 for this experiment indicates that the model is significant with a good fit and can be used for analysis and prediction of response values.

After multiple regression fitting, the effects of blade height, blade width, and relative pitch of the turbine on the isentropic efficiency can be expressed by the regression equation as:

$$\begin{array}{l}R=1.0266+0.111366A+0.055229B-0.02253C-5.30135D+0.048641B\ast D\\ +0.040071C\ast D-0.00216A^2-0.002041B^2+0.997151D^2\end{array}$$

(19)

where A, B, C, D denote leaf height, leaf width, mounting angle, and relative pitch, respectively.

By solving the regression equation, the conditions for the existence of R extrema of the regression model were: A = 25.7209, B = 22.9517, C = 79, D = 0.8, which means that the blade height is 25.7209 mm, the blade width is 22.9517 mm, the stagger angle is 79°, and the relative pitch is 0.8, and the isentropic efficiency under these conditions is 69.91%.

The software used in the above optimization method includes AxSTREAM (3.9.13), ANSYS CFX (2022 R1), and Design Expert (10.0), where AxSTREAM was used for the preliminary design of the turbine, ANSYS CFX was used for the numerical calculation of the internal flow field of the turbine, and Design Expert was used for the experimental design of the response surface and for solving the optimal solution.

3. Results and Discussion

The changes of the hydrogen component ratio (ω_H2) have a great impact on the NECS flow characteristics as well as the turbine stage efficiency. To illustrate the influence mechanism, H₂/Steam (NEC) was designed based on the improved RSM-CFD method when the hydrogen component ratio was assumed as 50%; then, the influence of ω_H2 to NECS was studied using 10 calculated sets where the values of ω_H2 were taken from 0.1 to 1, respectively.

3.1. Numerical Model of H₂/Steam (NEC) Turbine

According step 1, a single-stage turbine within 45 stator blades and 50 rotor blades was designed. The structural parameters of the rotor blades and their variation range of an initial H₂/Steam (EC) turbine were calculated as shown in

Table 4. The structural parameters and their variation ranges of an initial H₂/Steam (EC) turbine.

Parameters	Units	Minimum	Maximum	Mean
L	mm	23.00	27.00	25.0
B	mm	18.00	23.00	20.0
γ	°	77.00	79.00	78.0
tb	-	0.60	0.80	0.70

using the given inlet boundary conditions (V, T, ω) in

Table 5. Inlet and outlet parameters of the H₂/Steam-mixed working fluid.

Parameters	Units	Values
inlet velocity V0	m/s	40
inlet total temperature T0	°C	450
outlet static pressure P2	MPa	0.1
hydrogen component ratio (ωH2)	-	50%
rotating speed	rpm	7500

, which were generated from the magnesium–water reaction hydrogen production system. Then, the H₂/Steam (NEC) turbine could be designed according to steps 2–3, for which the optimized structural parameters (L₀, B₀, γ₀, t_b0) of the rotor blades are show in

Table 6. The optimal structural parameters of the H₂/Steam (NEC) turbine.

Parameters	Units	Stator	Rotor
L0	mm	25.00	26.00
B0	mm	20.00	23.00
γ0	°	34.00	79.00
tb0	-	0.7	0.8
N	-	45	50

. The number of stator and rotor blades was obtained from the preliminary design based on the parameters of mass flow rate and temperature. The variation of the static and rotor blades has a large impact on the flow rate, so the number of blades was not changed to reduce the impact of the optimized design on the flow rate.

The NECS calculation domains were simplified to one stator blade and one rotor blade passage using the periodic boundary condition as shown in Figure 4. The transient rotor stator frame change/mixing model was used in the stator/rotor interface when there was unequal pitch between the stator blade and the rotor blade.

The SST turbulence model was applied since the simulation of NECS required a higher calculation reliability in the near wall area, and the grid with y⁺ = 5 was proved to be feasible for the simulation, applying the SST turbulence model of NECS in a 3D domain [40]. Based on this, four mesh cases with the same y⁺ (y⁺ = 5) but different elements (as given in

Table 7. Four different mesh cases.

Mesh Case ID	Number of Elements, Millions
Case 1	0.1
Case 2	0.3
Case 3	0.5
Case 4	0.8
Case 5	1

) were used to study the independence of the grid as given in the simulation results. The outlet velocity of the stator and rotor blades was simulated with five different mesh cases as shown in Figure 5. It was found that, when the number of meshes exceeds 800,000 (Case4), the increment of velocity became very small, therefore, the Case4 mesh was used to perform the simulations in this paper (Figure 6).

3.2. Effect of the H₂ Component Ratio on Flow Characteristics

Hydrogen as a non-condensable component can greatly impact the non-equilibrium condensation. To clarify the influence mechanisms, a comparison and analysis of the distribution of droplet mass fraction, enthalpy, pressure, and the streamlines generated from different ω_H2 are demonstrated.

The droplet mass fraction distribution of different hydrogen component ratios at 10% relative blade height section (10%L) are shown in Figure 7; it can be found that, when ω_H2 is less than 50%, significant condensation occurs. As the hydrogen component ratio increases, the steam condensation decreases, and the droplets are carried by a large amount of hydrogen gas wrapping and moving along the streamline direction. The hydrogen component occupies a certain partial pressure, which inhibits the condensation of steam. When the hydrogen component ratio increases from 10% to 40%, the maximum droplet mass fraction decreases by 7.96%. In addition, the increase in the hydrogen component ratio increases the transport capacity of the mixed working fluid to the droplets, which makes a banded distribution of the droplets along the flow direction, reducing the condensation of the droplets on the surface of the rotor blade and homogenizing the distribution of the droplets. Figure 8 shows the static enthalpy distribution for different hydrogen component ratios at different blade height cross-sections. The sudden increased location of static enthalpy indicates that steam condensation occurs with latent heat releasing. The steam condensation zone moves from the trailing edge of the suction surface to the pressure surface of the rotor blade as the hydrogen component ratio rises.

The rotor blade is the main component of energy transfer between the steam and the rotor, and the pressure distribution on the surface of the rotor blade directly affects the machine’s performance. Figure 9 shows the pressure distribution curve at 10% blade height on the rotor blade surface under different hydrogen component ratios. The negative pressure zone on the rotor blade surface increases with the increase in the hydrogen component ratio. The fluid in the negative pressure zone is unable to push the rotor to work, which reduces the turbine performance. It can be seen from Figure 9 that, in the process of increasing the hydrogen composition ratio from 50% to 90%, the negative pressure region increases from 14.8%Nc (normalized axial chord length) to 17.1%Nc. The main reason is that the mixture of H₂/Steam is more susceptible to flow disturbance than the pure steam with higher instability in the rotating domain, which is determined by the physical properties of hydrogen. The pressure difference between the rotor blade’s pressure surface and the suction surface decreases as the hydrogen component ratio rises. As a result, the torque of the fluid acting on the rotor blade also decreases, which is the primary factor contributing to the decline in turbine power.

Figure 10 shows the limiting streamlines of the rotor blade pressure surface. With the increase in the hydrogen composition ratio, the flow instability in the flow field increases. The backflow phenomenon of the fluid occurs at the blade root and 70% blade height of the pressure surface of the rotor blade, forming two vortices. The intensity of the vortices continues to increase with the increase in the hydrogen composition ratio, aggravating radial flow in the passage.

3.3. Effect of the H₂ Component Ratio on Turbine Efficiency

To study the effect of the component ratio of the mixed working fluid on the turbine efficiency, the same inlet and outlet boundary conditions are chosen. The inlet boundary is set to 40 m/s velocity and the outlet boundary has an average static pressure of 0.1 MPa. Ten groups of the mixed working fluid with different component ratios (ω_H2 = 10%, 20%, 100%) were set and numerical calculations were performed using CFX with the calculation results in

Table 8. Enthalpy drop, internal power, and isentropic efficiency at ω_H2.

ωH2/%	ΔH/(j/kg)	P/W	η/%	ωH2/%	ΔH/(j/kg)	P/W	η/%
10	6859	3244	50.1	60	3914	1472	82.7
20	6266	2818	53.3	70	3230	1231	82.7
30	5659	2432	57.5	80	2516	1032	87.5
40	5092	2080	62.2	90	1757	856.1	91.6
50	4547	1757	69.9	100	632	657.8	95.1

.

The enthalpy drop and internal power of the stage decreased with the increasing hydrogen component ratio as shown in Table 8 at the same boundary conditions. This indicates that hydrogen is much less capable of driving the rotor to work than steam. The reason is that the molar mass of hydrogen molecules is much smaller than that of water vapor molecules, and the mass flow rate of the mixture decreases with the increase in the hydrogen component ratio under the same velocity inlet boundary conditions. In addition, the kinetic energy of hydrogen molecules with the same velocity is much smaller than that of water vapor molecules. This also corresponds to the pressure distribution curve shown in Figure 9, where the pressure difference between the pressure surface and the suction surface decreases as the hydrogen component ratio increases, and the workability decreases.

As shown in Figure 11, the enthalpy at the static blade inlet increases with the increase in the hydrogen volume fraction, showing an exponential growth trend. The enthalpy at the outlet of the rotor blade also shows a consistent change. The large increase in enthalpy makes the turbine deviate from the design operating conditions, which is not conducive to the stable operation of the turbine. When ω_H2 < 70%, the trend of inlet enthalpy increases slowly with the volume fraction of hydrogen; however, the inlet enthalpy increases sharply when ω_H2 > 70%. Therefore, to ensure the turbine operates under better working conditions, it is necessary to ensure that the hydrogen component ratio is maintained below 70%.

The graph of turbine internal power and isentropic efficiency with hydrogen component ratio (Figure 12) shows that the turbine internal power decreases with the increasing hydrogen component ratio, but the isentropic efficiency tends to increase at the same inlet volume flow rate and temperature. As the hydrogen component ratio increases, the inlet mass flow rate and enthalpy drop decrease (see Table 6) and the vortex and radial flow within the stage increase (Figure 10), the power of the turbine decreases significantly. The main reason for the increase in isentropic efficiency is the decrease in energy loss in the turbine passage as the hydrogen component ratio increases. According to Figure 12, when the volume fraction of hydrogen reaches 52.6%, the internal power and isentropic efficiency curves intersect. The intersection point indicates that the turbine operation under this mixing ratio condition considers the requirements of internal power and isentropic efficiency, which is the optimum value of hydrogen (OVH). Further considering the limitation of the actual hydrogen production and the requirement of hydrogen production, the optimal range of the hydrogen component ratio (ORH) is OVH ± 15% (38–68%), and the turbine is recommended to operate in this hydrogen component ratio range.

4. Conclusions

To achieve energy recovery in the process of high-temperature magnesium–water hydrogen production, an improved RSM-CFD method was proposed to design a single-stage turbine suitable for high-temperature H₂/Steam-mixed working fluid. The numerical analysis of the flow characteristics and working ability of the high-temperature H₂/Steam-mixed working fluid was completed. The influences of the composition ratio on the flow characteristics, power, and isentropic efficiency were studied under the condition of constant inlet velocity. Furthermore, the effects of hydrogen components on the flow characteristics and condensation of the mixed working fluid were analyzed. Based on the assumptions made on the modeling in Section 2, the following conclusions were obtained:

(1)

At constant velocity inlet conditions, the turbine internal power tends to decrease with the increase in the hydrogen component ratio due to the decrease in mass flow rate and enthalpy drop, while the isentropic efficiency tends to increase due to the decrease in flow loss. When the component ratio of hydrogen reached 53%, the turbine operated best with better isentropic efficiency and power. Considering the work capacity of the mixed mass and hydrogen production, the optimal range of the mixed working fluid component ratio was 38–68%.

(2)

Hydrogen has a greater effect on the condensation of steam as a non-condensable component. When the component ratio of hydrogen exceeded 50%, the nucleation and droplet growth processes of steam condensation were significantly inhibited and the droplet mass fraction decreased greatly. The hydrogen promoted the transport and diffusion of the droplets and reduced the collection of the droplets at the trailing edge of the rotor blade, which suppressed the chattering of the rotor blade due to the impact effect of large droplets.

(3)

The mixed mass decreases the fluid load with the increase in the hydrogen component ratio. The pressure difference between the pressure side and the suction side of rotor blade decreased, and the torque on the blade decreased, which caused the power in the turbine to decrease.

(4)

The instability of the mixed working fluid flow increased with the increase in the hydrogen component ratio. When the hydrogen component ratio increased, the vortex at pressure surface of the rotor blade increased, and the radial flow increased. Compared to steam, the flow of hydrogen in the rotating domain was more unstable and more susceptible to droplets.