Reaction Kinetics and Process Intensification of Continuous-Flow Synthesis of Propylene Glycol in a Spiral Microchannel Reactor

Abstract

This study investigates the continuous-flow hydrolysis reaction of propylene oxide (PO) in a spiral microchannel reactor, integrating experiments, computational fluid dynamics (CFD) simulations, and response surface methodology (RSM). To the best of our knowledge, experimentally determined apparent Arrhenius parameters for PO hydrolysis under microscale continuous-flow conditions remain rarely reported, and afterwards they were incorporated into CFD-based numerical simulations. This combined experimental–numerical framework provides a robust methodology for quantifying and optimizing liquid-phase kinetics in microscale flow environments. Subsequently, CFD simulations were employed to examine key process parameters, including reaction system temperature, inlet flow rate, and reactor length. Finally, RSM was utilized to identify the optimal process conditions (reaction system temperature of 298.15 K, inlet flow rate of 6 × 10⁻³ m·s⁻¹, and reactor length of 4 m), achieving a predicted PO conversion rate of 81.68%. The study provides a reference for designing and optimizing spiral microchannel reactors for PO hydrolysis.

1. Introduction

Propylene glycol (PG), as a valuable chemical intermediate, is widely employed in the fabrication of polyester resins, unsaturated resins, pharmaceuticals, cosmetics, etc. [1]. Currently, its industrial production primarily relies on the direct hydration of propylene oxide (PO). However, the reaction is constrained in traditional batch reactors by mass transfer efficiency owing to poor mutual solubility between PO and water. This phenomenon results in prolonged reaction times, low selectivity, numerous side reactions, and difficulty in controlling exothermic heat release [2].

In recent years, microreactor technology has emerged as a key method for enhancing chemical processes. Microreactors typically contain channels with millimetre or smaller diameters, offering high specific surface area and high mass transfer efficiency [3,4]. Compared to traditional reactors, microreactors substantially enhance heat and mass transfer, and increase reaction rates and spatiotemporal yield. They also offer advantages such as operational safety, compact system size, and ease of scale-up [5,6,7,8,9].

The development of continuous-flow chemistry has further expanded the application potential of microreactors, enabling stable, controllable, and continuous reaction operations while maintaining efficient mass and heat transfer characteristics. The integration of continuous-flow processes with microreactors substantially reduces liquid hold-up, enhancing process safety and scalability [10,11,12,13]. Moreover, implementation of this technology within green chemistry has become a vital direction for the synthesis of high-value-added chemicals.

Currently, microreactors are widely employed in various exothermic or hazardous reactions, including chlorination, bromination, fluorination, and esterification [14,15,16]. As a representative example, the temperature rise in a microreactor during the phenol nitration is less than 5 °C, whereas that in a traditional semibatch reactor can reach up to 55 °C, with precise temperature control increasing the product yield from 55% to 75% [17]. Guerrero-Corella et al. achieved the Michael addition reaction of 2-hydroxybenzophenone imine in a continuous-flow microreactor, yielding 90% conversion within 30 min—approximately 40 times faster than in a batch reactor [18]. Li et al. reported up to 91% yield for the synthesis of 2-amino-5-chloro-4-methylbenzenesulfonic acid in a continuous-flow microreactor, outperforming the 65.7% yield achieved using conventional processes [19]. These studies demonstrate that microreactors can achieve high selectivity, high reaction rates, and process safety [20,21,22,23,24].

Beyond their synthetic advantages, microreactors are uniquely valuable for reaction kinetics research. Their precise temperature and concentration control effectively mitigates uncertainties caused by intermittent sampling errors and temperature fluctuations [25]. While microreactors have been widely applied to kinetic studies of various continuous-flow reactions—such as esterification, nitration, and oxidation—the hydrolysis of propylene oxide (PO) under microscale continuous-flow conditions remains largely unexplored. Most existing studies on PO hydrolysis rely on kinetic parameters derived from conventional batch reactors, which may not capture the reaction–transport coupling effects that become significant in microscale environments. Furthermore, previous research on the hydrolysis of PO to produce 1,2-propanediol has primarily focused on side-reaction suppression, catalyst development, and process optimization. As a result, experimental kinetic parameters specifically determined at the microscale for this reaction are extremely rare [26].

Against this backdrop, this study employs computational fluid dynamics (CFD) to investigate the process intensification of PO hydrolysis within a spiral microchannel reactor, including the effects of reaction system temperature, residence time, and reactor length on the PO conversion rate. The spiral microchannel structure generates a characteristic Dean vortex pattern, which enhances radial mixing and heat transfer capabilities [27,28]. Optimal process parameters were determined using orthogonal and response surface designs in RSM Finally, a kinetic model for PO hydrolysis in a spiral microchannel reactor was established through experiments and numerical simulations, providing a theoretical basis and technical support for continuous and environmentally sustainable PG production.

2. Experimental Section

2.1. Preparation of Materials

PO (AR ≥ 99.0%) was purchased from Sinopharm Group (Shanghai, China). Hydrochloric acid (HCl) (36–38%) was purchased from Shanghai Titan Technology Co., Ltd. (Shanghai, China). All reagents were directly utilized without further purification. All weighing and preparation operations were conducted at room temperature. Deionized water was prepared in house.

The key equipment included a peristaltic pump (LabN1; Baoding Shenchen Precision Pump Co. Ltd., Hebei, China), near-infrared spectrometer (Nicolet 6700; Thermo Scientific, Waltham, MA, USA), electronic balance (ME204E; Mettler Toledo, Columbus, OH, USA), water bath (HH-2L; Shanghai Yukang Educational Instrument Equipment Co., Ltd., Shanghai, China), and cooling circulation pump (ECO RE 630; LAUDA Trading Co., Ltd., Shanghai, China). Chemometric models were developed using GRAMS IQ 9.2 (Thermo Fisher Scientific, Waltham, MA, USA).

2.2. System Description

The experimental system is illustrated in Figure 1, which primarily comprised a peristaltic pump, Y-shaped tee (inner diameter = 3 mm), constant-temperature water bath, near-infrared spectrometer, and polytetrafluoroethylene single-helix microchannel reactor. The microchannel possessed outer and inner diameters of 3 and 2 mm, respectively, with the entire microchannel forming a spiral configuration with a diameter of 145 mm. Residence time in the reactor system was controlled by adjusting the length of the microchannel [29].

Prior to entering the system, the PO and water were preheated to the reaction system temperature. Subsequently, two peristaltic pumps delivered PO and water—the latter containing HCl as a catalyst—into the Y-shaped three-way mixer. After thorough mixing, the mixture flowed into the reaction zone. The entire microchannel system was fully submerged in the constant-temperature water bath. The temperature of the reaction system was controlled within ±0.1 °C by the cooling circulation pump and water bath tank. Reaction products were continuously collected at the outlet, immediately quenched in an ice bath (0–5 °C), and analyzed via infrared spectrometry. The hydrolysis reaction is described using Equation (1) [30].

C₃H₆O + H₂O → C₃H₈O₂

(1)

2.3. Analysis

No substantial reaction was observed without the catalyst (36–38 wt% HCl solution) and at a reaction system temperature of 313.15 K, even when the PO and water were thoroughly mixed, demonstrating that the reaction was a typical acid-catalyzed process. To establish a quantitative analysis model, 50 sets of mixed solutions with varying PG and PO concentrations were prepared, with each mixed solution supplemented with a fixed mass of water. Reaction systems at different conversion stages were simulated by adjusting the PG to PO ratio. The PO conversion rate in each mixed solution ranged from 40% to 89%, and the PG yield ranged from 30% to 79%. Preliminary experiments confirmed that this data range met the accuracy requirements for subsequent experimental modelling. Near-infrared spectra were collected for each mixed solution, upon which a quantitative analysis model was established for rapid assessment of the PO conversion rate during the reaction process.

The PO conversion rate (X) was calculated using Equation (2).

$$X={\displaystyle \frac{{\mathrm{c}}_{\mathrm{P}\mathrm{O}0}-c_{PO}}{{\mathrm{c}}_{\mathrm{P}\mathrm{O}0}}}\times 100\mathrm{\%},$$

(2)

where ${\mathrm{c}}_{\mathrm{P}\mathrm{O}0}$ and $c_{PO}$ represent the PO concentrations at the reactor inlet and outlet, respectively.

2.4. Reaction Kinetic Model

Elucidating the kinetics of a reaction is crucial for understanding the associated mechanisms and optimizing the process [31,32]. Previous studies have primarily focused on kinetic models of PO hydrolysis at conventional scales, while the reaction kinetics within a spiral microchannel reactor remain poorly understood. Consequently, this study investigates PO hydrolysis kinetics at the microscale.

The temperature range of the reaction system was carefully selected based on laboratory conditions. Figure 2 presents the PO conversion rate over time within a temperature range of 283.15–298.15 K. Results indicated that an increase in reaction system temperature accelerated the hydrolysis rate.

Considering water as the excess reactant, with a PO-to-water molar ratio of 1:3.73, a pseudo-first-order kinetic model was employed to fit the experimental data illustrated in Figure 2. It should be noted that the present pseudo-first-order model should be understood as an apparent kinetic approximation rather than a strict intrinsic kinetic model. The PO hydrolysis rate can be expressed as Equation (3).

$$r=k\left[\mathrm{P}\mathrm{O}\right],$$

(3)

The differential form of the reaction rate equation is expressed as Equation (4). In this study, the high aspect ratio of the spiral microchannel and the Dean vortex structure effectively suppressed axial recirculation, resulting in flow behavior that closely approximated an ideal parallel flow reactor (PFR)—a prerequisite for eliminating non-ideal flow disturbances from the intrinsic kinetic measurements [33,34].

$$r=-{\displaystyle \frac{{\mathrm{d}}_{c_{PO}}}{\mathrm{d}t}}=k{c}_{PO},$$

(4)

where ${\mathrm{c}}_{\mathrm{P}\mathrm{O}}$ denotes the PO concentration, and k represents the reaction rate constant.

The PO concentration at reaction time t was calculated as Equation (5).

$$c_{PO} ={\mathrm{c}}_{{\mathrm{P}\mathrm{O}}_0}\left(1-X\right),$$

(5)

Integration of Equation (5) yielded Equation (6).

$$f\left(x\right)=\mathrm{l}\mathrm{n}(1-X)=-kt,$$

(6)

Figure 3 presents the relationship between $\mathrm{f}\left(\mathrm{x}\right)$ and $\mathrm{t}$ at different reaction system temperatures. The k value under various conditions can be obtained from the slope of each line. As shown in Figure 3, within the temperature range of 283.15–293.15 K, ln(1 − X) exhibits a linear relationship with reaction time, supporting the validity of the pseudo-first-order assumption under these conditions. In contrast, at 298.15 K, particularly at longer reaction times, a significant deviation from linearity was observed, indicating that the reaction deviates from ideal first-order kinetics at higher temperatures. This behavior suggests increased system complexity. From a reaction engineering perspective, this transition reflects a shift in the relative time scales of the reaction and mixing. At lower temperatures, the system operates in the kinetically controlled regime; at higher temperatures, the characteristic reaction time approaches the mixing time scale, leading to reaction-mass transfer coupling [35].

The linear relationship between k and 1/T is illustrated in Figure 4.

The Arrhenius equation is expressed as Equation (7).

$$\mathrm{l}\mathrm{n}k=-{\displaystyle \frac{E_{\mathrm{a}}}{RT}}+\mathrm{l}\mathrm{n}A,$$

(7)

where E_a denotes the activation energy, A represents the pre-exponential factor, R refers to the universal gas constant (8.314 J·mol⁻¹·K⁻¹⁾, and T represents the absolute temperature.

Although the data at 298.15 K show deviations from ideal first-order behavior, including them in the fit did not obscure the overall Arrhenius trend, as evidenced by the high correlation coefficient (R² = 0.9796). On the contrary, this allowed the model to provide a more representative description of the system across the entire operating range. Therefore, the obtained kinetic parameters should be interpreted as apparent kinetic constants, reflecting the combined effects of intrinsic reaction kinetic and, mass transfer phenomena. Although the data at lower temperatures better represent the intrinsic behavior, the overall model captures the system’s primary temperature dependence. These results provide a physically meaningful and practically relevant kinetic description, and the aforementioned kinetic parameters will be used in subsequent CFD simulations [36].

The parameters of the linear fitting equation indicated that Ea = 2.3731 × 10⁴ J·mol⁻¹, A = 36.89 s⁻¹, and the correlation coefficient (R²) = 0.9796. These fitting parameters were utilized to establish the kinetic model. The resulting parameters serve as apparent kinetic parameters valid only within the tested reactor configuration.

2.5. Numerical Simulations

The physical model used in the numerical simulation maintained the same microchannel geometry as in the experiment, with a diameter of 2 mm, a spiral radius of 145 mm, and a pitch of 2 mm. The structure of the microchannel is shown in Figure 5.

Reaction kinetic models provide a means to understand the mechanism of product synthesis, thereby enabling process intensification. The reaction kinetics used in the CFD model were not directly adopted from literature-reported batch reactor data, but were experimentally determined under the present microscale continuous-flow conditions. This treatment improves the reliability of the simulation and better reflects the actual reaction behavior in the microchannel reactor.

2.5.1. Mathematical Model

To balance accuracy and computational efficiency, the liquid flowing within the microchannels was assumed to be an incompressible fluid. A microchannel reactor model was constructed using commercial CFD software (COMSOL Multiphysics 6.2). The mass, momentum, and energy transfer behaviours during the reaction were systematically characterized by coupling multiple physical fields, including laminar flow, chemistry, dilute mass transfer, and fluid heat transfer. Within the computational domain, the governing equations comprised the continuity, momentum, energy, and mass transport equations. The mass conservation equation for the mixture is given by Equation (8) [37].

$${\displaystyle \frac{\partial \rho }{\partial t}}+\nabla ·\left(\mathsf{\rho }\overrightarrow{\nu }\right)={\mathrm{S}}_{\mathrm{m}},$$

(8)

where $\rho$ denotes the density of the reactant, $S_m$ refers to the additional mass of the continuous phase, and v represents the flow rate.

According to the Navier–Stokes equations for laminar flow, the momentum conservation formula is expressed as Equation (9) [38].

$${\displaystyle \frac{\partial }{\partial t}}\left(\rho \overrightarrow{\nu }\right)+𝜵·\left(\rho \overrightarrow{\nu }\overrightarrow{\nu }\right)=-𝜵P+𝜵·\left(\stackrel{̿}{\tau }\right)+\overrightarrow{F},$$

(9)

where P refers to the static pressure; $\mathsf{\rho }$ denotes the fluid density; F indicates the external force; and τ denotes the stress tensor, which is defined using Equation (10) [39].

$$\stackrel{̿}{\tau }=\mu \left[\left(𝜵 \overrightarrow{\nu }+𝜵 {\overrightarrow{\nu }}^T\right)-{\displaystyle \frac{2}{3}}𝜵\overrightarrow{\nu }I\right],$$

(10)

where μ denotes the molecular viscosity, and I represents the unit tensor.

The energy equation is expressed as Equation (11) [40].

$${\displaystyle \frac{\partial }{\partial t}}\left(\rho E\right)+\nabla \cdot \left[\overrightarrow{\nu }\left(\rho E+P\right)\right]=\nabla \cdot \left[k\nabla T-{\sum }_j{h}_{\mathrm{j}}{\overrightarrow{J}}_{\mathrm{j}}+\left(\stackrel{̿}{\tau }\cdot \overrightarrow{\nu }\right)\right]+S_{h },$$

(11)

where k denotes the thermal conductivity, and Sh represents the volumetric heat source and reaction heat release term.

The transport equations for each component are given by Equation (12).

$${\displaystyle \frac{\partial }{\partial t}}\left(\rho {\overrightarrow{V}}_{\mathrm{i}}\right)+\nabla \cdot \left(\rho \overrightarrow{\nu }{\mathrm{Y}}_{\mathrm{i}}\right)=-\nabla {\overrightarrow{J}}_{\mathrm{i}}+R_{\mathrm{i}}+S_{\mathrm{i}},$$

(12)

where $Y_i$ represents the mass fraction of substance i, $R_i$ denotes the net production rate of substance i, $S_i$ indicates the external source term for substance i, and $J_i$ signifies the diffusion flux of substance i.

The physical parameters of the chemical species in this study are listed in

Table 1. Thermophysical properties of all species involved in the reaction.

Physical Parameter	C3H6O	H2O	C3H8O2	Reaction Mixture
Molecular weight (g·mol−1)	58.095	18	76.095	-
Diffusion coefficient (m2·s−1)	1 × 10−9	1 × 10−9	1 × 10−9	-
Density (kg·m−3)	830	1000	1040	912.59
Heat Capacity (J·mol−1·K−1)	146.54	75.36	-	-
Thermal conductivity (W·m−1·K−1)	-	-	-	0.559
Reference temperature (K)	293	293	293	293

(original values in [41]). To ensure reproducibility, all constants, boundary conditions, and source term symbols in the numerical simulations were consistent with the definitions in Equations (8)–(12). Table 1 Thermophysical properties of all species involved in the reaction.

In the COMSOL Multiphysics 6.2 environment, the reaction kinetics are implemented through the “Transport of Diluted Species” interface coupled with the “Reactions” node.

To replicate the experimental results, the computational domain was preheated to the set temperature at the initial time. Based on the pressure-solving method, the outlet position was set as a constant-pressure boundary, with Poutlet = 1 bar. The reactor wall was treated as a constant-temperature thermal boundary, with its temperature set to match the specified wall temperature. Heat exchange between the fluid and wall was naturally reflected through the energy equation. The inlet feed was specified by volume fraction/mass fraction and the volumetric flow conditions, with the feed molar ratio of PO to H₂O maintained within the range specified in

Table 2. Key parameters in the numerical model.

Parameters	Unit	Value
Inlet temperature	K	283.15, 288.15, 293.15, 298.15
Reactor wall temperature	K	283.15, 288.15, 293.15, 298.15
Export pressure	bar	1
Molar concentration ratio of PO and H2O at inlet	-	1:3.73
Reactor length	m	1, 2, 3, 4, 5
Inlet flow rate (mixed)	cm·s−1	0.3, 0.6, 0.9, 1.2, 1.5

. All other boundary conditions employed standard settings that were consistent with the experimental results, including no slip and adiabatic/isothermal conditions. Table 2 summarizes the key operating and geometric parameters for all cases, including the inlet temperature, reactor wall temperature, export pressure, PO to H₂O molar ratio, reactor length, and inlet flow rate (after the two reactants are mixed). To maintain consistency with the experiments, units and numerical values were retained as originally specified without conversion or relabeling of parameters.

To simplify the calculation, the following assumptions were made for the microchannel reactor:

First, the flow inside the spiral microchannel reactor was assumed to be steady, incompressible.

Second, only the main hydrolysis reaction was considered, while side reactions were neglected for model simplification.

Third, the non-uniform flow is assumed to be a single-phase flow to simplify the simulation.

2.5.2. Model Validation

A grid-independent analysis was conducted to verify the numerical accuracy of the model. The molar concentration of PG along the reactor centerline was selected as the criterion for comparative analysis. As shown in Figure 6, the change in PG concentration was negligible when the number of grid cells increased from 9.37 million to 16.67 million, indicating that an increase in grid density did not affect the computational results. Consequently, a mesh size of approximately 9.37 million elements was selected to balance computational complexity and accuracy. The corresponding mesh settings in COMSOL are listed in

Table 3. COMSOL mesh parameters.

Maximum	Minimum	Curvature Factor	Confined Space	Maximum Unit Growth Rate
0.605	0.25	0.2	1	1.05

.

The numerical simulation outcomes were subsequently validated against the experimental results, as shown in Figure 7. At different operating temperatures, the simulated PO conversion rates were consistent with the experimental results, with a maximum root mean square error of approximately 1.2. Agreement between the experimental results and numerical simulation outcomes validated the rationality of the governing equations, boundary conditions, and material property settings, providing a reliable numerical foundation for subsequent mechanism analysis and parameter optimization [42].

3. Results and Discussion

3.1. Effect of the Reaction System Temperature

The effect of the reaction system temperature on the hydrolysis reaction and PO conversion rate was investigated within a temperature range of 283.15–298.15 K, in increments of 5 K. As shown in Figure 8, an increase in the reaction system temperature substantially promoted the PO conversion rate, which was attributed to the increased temperature accelerating the thermal motion of reactant molecules, thereby increasing the frequency of effective collisions and accelerating the reaction process [32]. Specifically, the PO conversion rate rose from 49.73% to 68.06% when the reaction system temperature increased from 283.15 K to 298.15 K, representing an overall increase of approximately 18.33 percentage points. Moreover, at a constant reaction system temperature, the hydrolysis rate gradually decreased as the liquid flowed through the microchannel, indicating that the reaction system tended toward stability after a certain flow distance, with the mass transfer and mixing processes reaching a plateau. This finding was consistent with the reported results by Shuai Guo et al. in their study on the mixed-acid nitration of trifluoromethylbenzene (TFB) in a microreactor, where an increased reaction temperature accelerated TFB mass transfer and reaction rates, thereby enhancing the conversion rate [43].

3.2. Effect of the Inlet Flow Rate

As shown in Figure 9 and Figure 10, the PO conversion rate remained relatively high at a low inlet flow rate (e.g., 6 × 10⁻³ m·s⁻¹), reaching nearly 80%. The PO conversion rate gradually decreased as the flow rate increased. Notably, the PO conversion rate dropped to approximately 40% at a flow rate of 1.8 × 10⁻² m·s⁻¹. These results indicated that a lower inlet flow rate promoted increased residence time for reactants, thereby enhancing reaction completion and improving the PO conversion rate. However, the PO conversion rate decreased at higher inlet flow rates, with the system gradually approaching stability. The concentration and conversion rate of the hydrolysis reaction products increased linearly with the decreasing total flow rate, indicating that the reaction was influenced by mass transfer. This finding aligned with the reported results by Wan et al. on the effects of flow rate in sulfonation reactions, thereby justifying the rationale of this study [44].

3.3. Effect of the Reactor Length

Optimizing reactor performance requires setting proper operating conditions and geometric structural parameters. Given that some PO remained unconverted at the outlet, extending the reactor length was hypothesized to improve the PG yield. For this purpose, under the conditions of a reaction system temperature of 293.15 K, flow rate of 9 × 10⁻³ m·s⁻¹, and microchannel diameter of 2 mm, the reaction behaviour was examined for reactor lengths of 1, 2, 3, 4, and 5 m. As shown in Figure 11, the PO conversion rate substantially increased from 22.62% to 71.39% with increasing reactor length, indicating that the reaction proceeded more thoroughly with longer residence times. Meanwhile, as shown in Figure 12, the molar concentration of PG gradually increased with increasing reactor length and its distribution became more uniform. When the reactor length was extended to 4–5 m, the PG concentration at the outlet surface was substantially higher than that achieved at shorter reactor lengths.

However, the rate of improvement in PO conversion tended to plateau with a further increase in reactor length, while equipment costs and fluid resistance rose, suggesting that a reactor length of 3–4 m may offer the greatest practical value. Under the investigated experimental conditions, appropriately extending the reactor length not only substantially enhanced the PO conversion rate and PG yield but also achieved a favourable balance between reaction efficiency and economic viability.

3.4. Flow Characterization and Dimensionless Numbers

To quantitatively characterize the flow regime and mixing behavior in the spiral microchannel reactor, several dimensionless numbers were calculated based on the operating conditions (Table 2) and the physical properties of the reaction mixture (Table 1).

Reynolds number (Re) is defined as Equation (13).

$$Re={\displaystyle \frac{\rho u{d}_h}{\mu }}$$

(13)

where $\rho$ = 912.6 kg·m⁻³ is the density of the mixture, u is the inlet flow velocity 6 × 10⁻³–1.8 × 10⁻² m·s⁻¹, $d_h$ = 2 mm is the hydraulic diameter, and $\mu$ = 0.001 Pa·s is the dynamic viscosity. The calculated Re ranges from 10.95 to 32.85, confirming that the flow is laminar under all experimental conditions.

Dean number (De) is defined as Equation (14)

$$De=\mathrm{R}\mathrm{e}\sqrt{d_h/D_c}$$

(14)

where $D_c$ = 145 mm is the coil diameter. De ranges from 1.29 to 3.86, indicating that curvature-induced secondary flow begins to develop in the spiral microchannel reactor. Although the Dean vortex intensity remains relatively weak compared with high-De systems, the channel curvature still promotes mild radial mixing and helps reduce concentration gradients across the channel cross-section. Therefore, the enhanced reactor performance is attributed to the combined effects of convective transport and mild curvature-induced secondary flow.

Péclet number (Pe) for mass transfer is defined as Equation (15)

$$Pe=u{d}_h/D_{\mathrm{m}}$$

(15)

where $D_{\mathrm{m}}$ = 1 × 10⁻⁹ m²·s⁻¹. Pe ranges from 1.2 × 10⁴ to 3.6 × 10⁴, indicating that axial convection dominates over diffusion. This justifies the plug-flow reactor (PFR) approximation used in the kinetic analysis (Section 2.4).

3.5. Multivariate Analysis

RSM is an optimization technique that integrates experimental design and mathematical modelling, enabling systematic analysis of multiple factors and their interactions while reducing the number of experiments required. Establishing a quadratic polynomial regression equation between experimental factors and response values enables the prediction of system responses and determination of optimal process parameters. A typical quadratic polynomial regression equation is represented by Equation (16).

$$Y ={\beta }_0+{\sum }_{i =1}^k{\beta }_i{\chi }_i+{\sum }_{i =1}^k{\beta }_{ii}{\chi }_i^2+{\sum }_{i < j}{\beta }_{ij}{\chi }_i{\chi }_j+e,$$

(16)

where $Y$ represents the predicted response value, ${\beta }_0$ denotes the constant term, ${\chi }_{\mathrm{i}}$ and ${\mathrm{\chi }}_{\mathrm{j}}$ refer to coded values of the dependent variables, ${\beta }_i$ indicates the linear coefficient, ${\beta }_{\mathrm{ii}}$ and ${\beta }_{\mathrm{ij}}$ represent coefficients of quadratic terms, $\mathrm{k}$ denotes the number of factors, and $\mathrm{e}$ represents the random error.

Given that the PO conversion rate in the spiral microchannel reactor was primarily influenced by the reaction system temperature, reactor length, and inlet flow rate, a Box–Behnken design (BBD) was implemented within RSM using Design-Expert 13 to systematically evaluate these factors and their interactions. Reaction system temperature, reactor length, and inlet flow rate were selected as independent variables in three-factor, three-level response surface experiments. The ranges of the three independent variables in the BBD were selected by integrating the single-factor results (Section 3.1, Section 3.2 and Section 3.3), equipment operating limitations, and literature-reported ranges of response surface optimization parameters for similar reaction systems in microreactors [45,46], as shown in

Table 4. Influencing factors and levels in continuous-flow experiments.

Independent Variable	Level
	−1	0	1
Reaction system temperature (A)	288.15	293.15	298.15
Reactor length (B)	2	4	6
Inlet flow velocity (C)	6 × 10−3	1.2 × 10−2	1.8 × 10−2

. In total, 17 experimental points were obtained (12 factorial points and five central points), with the experimental results presented in

Table 5. RSM experiment results.

Run	Factor A	Factor B	Factor C	PO Conversion Rate (%)
	Reaction System Temperature (K)	Reactor Length (m)	Inlet Flow Rate (m·s−1)
1	293.15	4	0.012	52.918
2	293.15	4	0.012	54.698
3	298.15	2	0.012	36.057
4	293.15	4	0.012	53.325
5	298.15	6	0.012	73.427
6	288.15	6	0.012	61.439
7	288.15	4	0.006	71.846
8	298.15	4	0.006	82.808
9	288.15	2	0.012	27.497
10	298.15	4	0.018	44.683
11	293.15	6	0.006	89.386
12	293.15	4	0.012	53.918
13	293.15	2	0.018	22.446
14	293.15	6	0.018	52.908
15	293.15	4	0.012	54.931
16	288.15	4	0.018	34.651
17	293.15	2	0.006	52.7

.

Quadratic polynomial regression fitting was performed on factors A–C and the response values obtained via numerical simulations, yielding the following predictive model for the PO conversion rate during continuous flow [Equation (17)].

PO conversion rate = 53.96 + 5.19A + 17.31B − 17.76C + 0.8570AB − 0.2325AC − 1.56BC − 0.1080A² − 4.25B² + 4.65C²,

(17)

Analysis of variance (ANOVA) was conducted to assess model significance and goodness-of-fit. As shown in

Table 6. ANOVA and significance test results for the regression equation describing continuous-flow PG synthesis.

Source	Sum of Squares	df	Mean Square	F-Value	p-Value
Model	5305.82	9	589.54	286.79	<0.0001
A	215.72	1	215.72	104.94	<0.0001
B	2396.4	1	2396.4	1165.75	<0.0001
C	2522.35	1	2522.35	1227.02	<0.0001
AB	2.94	1	2.94	1.43	0.2708
AC	0.2162	1	0.2162	0.1052	0.7552
BC	9.68	1	9.68	4.71	0.0666
A2	0.0491	1	0.0491	0.0239	0.8815
B2	75.87	1	75.87	36.91	0.0005
C2	90.92	1	90.92	44.23	0.0003
Residual	14.39	7	2.06
Lack of Fit	11.41	3	3.8	5.11	0.0745
Pure Error	2.98	4	0.7446
Cor. Total	5320.21	16
Standard Deviation	1.43		R2	0.9973
Mean	54.10		Adjusted R2	0.9938
Predicted R2	0.9648		Adequate Precision	63.7734

, p < 0.0001 for the overall model indicated that the model was highly significant [47], while p = 0.0745 for the nonsignificant term (greater than the threshold of 0.05) suggested that the residuals satisfied the assumption of randomness and that the overall model fit the data well.

The first-order (A, B, and C) and second-order (A², B², and C²) terms exhibited extremely significant effects (p < 0.01), while the interaction term BC approached significance (p = 0.0666). The F-values indicated that factors A–C influenced the PO conversion rate in the order of inlet flow rate (C) > reactor length (B) > reaction system temperature (A).

The model validation results are presented in Table 6. The values of the coefficient of determination (R2), adjusted coefficient of determination (adjusted R2), and predicted coefficient of determination (Predicted R²) were 0.9973, 0.9938, and 0.9648 (i.e., close to 1). Furthermore, the difference between the adjusted and predicted R2 values was only 0.029, indicating that the model did not exhibit overfitting. The model’s accuracy reached 63.7734 (substantially exceeding the threshold of 4), indicating exceptionally high predictive precision and stability. The normal probability plot of the residuals and scatterplot of predicted vs. actual PO conversion rate (Figure 13) demonstrated strong linear relationships, further validating the validity and reliability of the model [45].

To thoroughly analyze the interactions among factors A–C, three-dimensional response surface curves and two-dimensional contour plots were constructed for interaction terms AB, AC, and BC, as shown in Figure 14.

The curvature of the 3D response surface and the shape of the 2D contour lines visually represent the interaction strength between factors. A more curved response surface or more elliptical (irregular) contour lines indicate a more pronounced interaction, while a flatter (near-planar) response surface or more rounded contour lines suggest a weaker interaction.

Figure 14a,b illustrate the effects of reaction system temperature and reactor length on the PO conversion rate. Reaction system temperature and reactor length substantially enhanced the PO conversion rate, exhibiting a positive synergistic interaction. The influence of reactor length became more pronounced at lower reaction system temperatures, whereas the curve gradually flattened as the reaction system temperature increased, indicating that the effect of reaction system temperature intensified while that of reactor length diminished.

Figure 14c,d illustrate the interaction between reaction system temperature and inlet flow rate. Overall, both parameters positively influenced the PO conversion rate, although reaction system temperature had a slightly stronger effect than inlet flow rate. In the low-temperature region, variations in inlet flow rate substantially impacted the PO conversion rate, whereas the reaction system temperature became the dominant factor in the high-temperature region.

Figure 14e,f illustrate that the interaction between reactor length and inlet flow rate was the closest to being significant among the investigated interactions. The surface rose sharply in the high-reactor-length, high-inlet-flow-rate region, indicating that both factors determined the residence time within the spiral microchannel reactor. Short reactor lengths or high inlet flow rates resulted in insufficient reaction time, leading to a substantial decrease in the PO conversion rate. Conversely, extending the reactor length or reducing the inlet flow rate effectively increased the residence time, promoting reaction completion.

The regression model indicated that the optimal process conditions for the continuous-flow microreactor were a reaction system temperature of 298.15 K, inlet flow rate of 6 × 10⁻³ m·s⁻¹, and reactor length of 4 m. Under these conditions, the predicted PO conversion rate reached 81.68%, substantially outperforming other combinations.

4. Conclusions

This study proposes and validates a green, efficient route for continuous-flow PO hydrolysis in a spiral microchannel reactor. This study experimentally determined apparent Arrhenius parameters for PO hydrolysis under microscale continuous-flow conditions and incorporated them into CFD modelling. Subsequently, a CFD numerical model was constructed to systematically evaluate the influence of factors such as reaction system temperature, inlet flow velocity, and reactor length on the PO conversion rate. Results indicated that increasing the reaction system temperature substantially enhanced the PO conversion rate, while the hydrolysis rate decreased after a certain flow distance, indicating that the system gradually approached stability. Reducing the inlet flow rate improved PO conversion efficiency Extending the reactor length increased the PO conversion rate and PG yield by prolonging the residence time for reactants, although diminishing returns were observed. The quadratic response relationships and interactions among these factors were determined via BBD-based RSM, yielding optimal process conditions of 298.15 K, 0.006 m·s⁻¹ and 4 m, which achieved a predicted PO conversion rate of 81.68%. This continuous-flow approach highlights the intrinsic advantages of spiral microchannels—enhanced radial mixing, rapid heat removal, and inherent operational safety—providing a robust basis for quantitative analysis of liquid-phase reactions in microreactors. Future work will focus on refining kinetic mechanisms and advancing monitoring and control strategies for improved reaction stability inside microscale flow environments.