Modeling the Conditions for Stabilizing Aqueous Phase Evaporation in Highly Stable Water-Hydrocarbon Emulsions Under Mechanical Turbulence to Suppress Unstable Phase Transfer and Reduce Accident Risks

Abstract

Vast quantities of liquid hydrocarbon and oil-containing wastes are generated and accumulate annually. Dewatering such sludges presents a significant technological challenge due to the high content of emulsified and chemically bound water. Consequently, the development of integrated approaches, particularly thermomechanical methods, have emerged as a promising strategy. These methods aim to disrupt the emulsion stability and enhance water evaporation efficiency. This study provides a theoretical basis for stabilizing the evaporation of the aqueous phase through mechanical agitation within boiling emulsions. A quantitative mathematical model is developed to identify critical conditions that prevent explosive boiling. Under intensive mixing, water globule diameters decrease by 80–85% within the first 5 s, while their settling time exceeds the dispersion time by hundreds of times—effectively inhibiting the accumulation of a critical aqueous-phase mass. Energy analysis reveals that, at a superheat temperature of 110 °C, the maximum permissible droplet diameter is approximately 0.5 mm; at 150 °C, it must not exceed 0.25 mm to avoid explosive boiling. To ensure safe operation, mixer rotational speeds of at least 100–200 rpm are required, with higher speeds (>200 rpm) necessary near 150 °C. The mechanical agitation modes proposed herein enable controlled, non-explosive evaporation of water from complex emulsions. Collectively, these findings lay a theoretical foundation for the industrial-scale deployment of thermomechanical dewatering technologies—offering a safer, more efficient pathway for managing challenging sludge streams.

1. Introduction

Annually, substantial volumes of liquid hydrocarbons, petroleum, and petroleum-contaminated waste are produced and accumulated throughout the processes of oil production, refining, and transportation, as well as within metallurgical industries [1,2,3,4]. Dewatering hydrocarbon and oil-containing sludges presents a complex technological process aimed at reducing the water content of waste materials to enhance their recyclability and energy value, while also mitigating their adverse environmental impact [5]. These sludges contain significant amounts of bound and emulsified water, retained within the system due to the presence of surfactants and complex constituents such as resins, asphaltenes, and mineral impurities, which substantially complicate their separation processes [6,7]. Highly stable water-in-hydrocarbon emulsions are also common in oil refineries, where inadequate equipment preparation or even minor process upsets—such as transient flooding under turbulent flow—can generate persistent emulsified systems resistant to conventional breaking methods. The presence of these “contaminated” flows subsequently leads to instability in downstream separation processes [8].

Conventional dewatering methods encompass various approaches: thermal [9,10,11,12], chemical (coagulation and flocculation with demulsifiers) [13,14], physical (centrifugation [15,16], filtration [17,18], ultrasonic [19,20], microwave [21] treatment, flotation [22,23], etc.), and biological treatments [24,25] and combined methods [26]. However, many such methods rely fundamentally on gravity-driven settling, which is only moderately effective for coarse, weakly stabilized emulsions—and proves slow, inefficient, and often impractical for viscous, heavily stabilized systems. Chemical approaches are highly sensitive to demulsifier selection and emulsion composition; their performance deteriorates when the densities of the hydrocarbon and aqueous phases are closely matched—limiting applicability in waste treatment. Mechanical methods like centrifugation and filtration demand high energy inputs and capital investment, yet still fail to ensure complete emulsion breakdown in challenging cases. Thermal evaporation techniques, which rely on water phase vaporization, are restricted by issues such as foaming and unstable (potentially explosive) boiling phenomena.

These conventional approaches, when employed separately, generally fail to achieve satisfactory results in processing complex water–hydrocarbon systems, especially real-world industrial conditions. Therefore, the development of integrated processing methods that leverage the advantages of multiple technologies is increasingly critical. Among these, thermomechanical dewatering—which couples controlled heating with mechanical agitation—has shown particular promise [27,28,29,30,31,32]. This method disrupts emulsion microstructure, promotes fine dispersion of aqueous droplets, and enhances uniform vaporization of the water phase, delivering both high efficiency and deep dewatering in practice.

Prior studies have identified the initial water content as a critical factor influencing thermomechanical processing. The literature distinguishes between concentrated emulsions (>5% water) and dilute emulsions (<5% water) [33]. In concentrated systems, mechanical agitation increases bulk temperature and nucleation site density—boosting bubble formation frequency while reducing bubble size [34,35,36,37]. In contrast, dilute emulsions exhibit reduced contact with heated surfaces, lower nucleation rates, and thus require higher degrees of superheating to initiate boiling [38,39,40,41]. Moreover, as water globules shrink and surfactant concentrations rise, greater energy input is needed to overcome interfacial cohesive forces—leading to higher operational costs.

The preliminary research was conducted on investigations into the efficiency of thermomechanical dehydration of various emulsion samples collected from industrial facilities [27,28,29,30,31]. These samples included cutting fluid emulsions from metallurgical plants, pyrolysis resin emulsions, intermediate layers from crude oil processing units, mixtures of oil-containing waste generated during the cleaning and washing of petroleum product transportation and storage facilities, storage sludge, natural bitumen, and other related materials. The studies demonstrated the feasibility of employing thermomechanical dehydration to utilize highly stable emulsions, thereby obtaining products with residual water content approaching trace levels [27,32]. Additionally, the effect of heating temperature on dehydration efficiency was systematically evaluated using laboratory-scale equipment [42]. To attain trace amounts of the aqueous phase in the dehydrated products, the final heating temperature varied within the range of 110 °C to 150 °C, considering the complex nature of intermolecular interactions present in highly stable emulsions. Preliminary correlations were developed to predict the optimal dehydration temperature based on the properties of the hydrocarbon phase, including density, coking ability, pour point, and viscosity [32,42].

Despite significant progress, robust modeling and systematic evaluation of the critical parameters governing stable droplet evaporation during thermomechanical dewatering remain scarce—especially for complex, real-world emulsions. This knowledge gap impedes both theoretical validation and reliable scale-up to industrial implementation. Existing models do not sufficiently account for the complex multiphase composition of real water–hydrocarbon emulsions; the dynamic changes in emulsion structure and properties during thermomechanical treatment; coupled heat- and mass-transfer phenomena under combined boiling and agitation—including the role of hydrodynamic turbulence in promoting uniform evaporation and mitigating localized overheating; and the kinetics of water droplet dispersion, growth, and evaporation.

Accordingly, this study aims to identify and comparatively assess the key parameters governing the stabilization of aqueous-phase evaporation in thermomechanical dewatering, and develop a comprehensive mathematical model that accurately describes this coupled process—providing a foundation for predictive design and optimization of industrial-scale units.

2. Materials and Methods

2.1. Simulated Unit

A thermomechanical dewatering unit (Figure 1), assembled by the efforts of our scientific team, was selected as the object for modeling. This device consists of a body (1), a flat cover (2), and a flange (3). The unit body (1) is a cylindrical vessel fabricated from Grade 3 steel, with an internal diameter of 0.16 m, a height of 0.4 m, and a wall thickness of 3 mm. The vessel is operated at 50% fill volume. A paronite gasket (4) is installed between the cover and the flange to ensure leak-tight sealing. The integrated piping system fulfills the following functions:

• Pipe 5: drainage of the dehydrated product from the bottom of the unit.
• Pipe 6: measurement of the temperature of the vapor of the distillate hydrocarbon fraction and water, monitored via thermocouple 16.
• Pipe 9: measurement of the temperature of the bottom liquid, monitored via thermocouple 13.
• Pipe 7: transportation of the feed emulsion into the unit for processing.

The vapor mixture of hydrocarbon distillate and water exits through Pipe 6, where it is condensed in cooler 14 and subsequently collected as liquid condensate in receiver 15. The unit is heated by an electric heating element (8), which raises the emulsion temperature to the required process conditions. For mixing, a turbine-type stirrer is installed, driven by an electric motor 10, which includes a device for regulating and measuring the stirrer speed. As is well established, when using viscous liquids as feedstock—particularly those contaminated with mechanical impurities—mixing devices operate predominantly in a transient mode at moderate rotational speeds (100–400 rpm). The selected turbine mixer satisfies these criteria by generating radial flow under heating conditions for highly stable emulsions. This design facilitates enhanced circulation and ensures high-quality mixing of the liquid. The stirrer has a diameter of 0.08 m. All critical process parameters—including internal temperature (measured by thermocouples 13 and 16) and stirrer speed—are supervised and regulated via the central control unit (12).

2.2. Modeling Process Stabilization Conditions

To quantify the key parameters governing thermomechanical effects on the emulsion, numerical analysis was performed based on the experimental setup shown in Figure 1 and fundamental knowledge on fluid properties [43,44], mixing and evaporation processes [45,46,47,48,49,50,51,52,53,54,55,56,57,58]. The modeling was carried out using MathCad 15.0 (PTC Inc., Boston, MA, USA). As a representative case, a highly stable water–hydrocarbon emulsion—specifically, bitumen-based—was selected for modeling, reflecting real-world sludge complexity.

Two principal criteria were formulated to define stabilization conditions during the thermomechanical dewatering of such systems:

• Prevention of aqueous-phase accumulation at the heating surface:
  The dispersion time of water globules (i.e., the time required for mechanical breakup and homogenization) must be substantially shorter than their gravitational settling time for a given droplet diameter. This ensures that droplets remain finely distributed and do not coalesce or pool near the heater—thereby avoiding localized overheating and explosive boiling (see Figure 2).
• Ensuring stable, non-explosive boiling:
  The energy accumulated in the superheated liquid should not exceed the energy required to transfer the entire emulsion volume (see Figure 3).

The calculation algorithms based on these approaches are illustrated in Figure 2 and Figure 3.

2.2.1. Comparison of Water Globule Dispersion Time with Settling Time

The principal quantitative criterion adopted in the model development is the characteristic diameter of dispersed water droplets within the water-in-oil (W/O) emulsion. Under superheated conditions (≥100 °C), intense evaporation of water from the dispersed phase leads to progressive coalescence and growth of droplets, ultimately reaching dimensions on the order of several millimeters. Based on experimental observations and thermodynamic considerations, a representative droplet diameter of 3 mm was selected for subsequent modeling and parametric analysis. The basic properties of emulsions were calculated [43,44,51].

Hydrocarbon phase density (${\mathsf{\rho }}_{\mathrm{h}}$) varies with temperature according to a linear correction:

$${\mathsf{\rho }}_{\mathrm{h}}(\mathrm{t})={\mathsf{\rho }}_{\mathrm{h}}^{20}-{\mathrm{a}}_{\mathrm{h}}·(\mathrm{t}-20)$$

(1)

where ${\mathsf{\rho }}_{\mathrm{h}}^{20}$ is the density of the hydrocarbon phase of the emulsion at 20 °C, kg/m³; ${\mathrm{a}}_{\mathrm{h}}$—empirical temperature coefficient; t—the process temperature, °C.

The emulsion density (${\mathsf{\rho }}_{\mathrm{e}}$) is then calculated as a volume-weighted average:

$${\mathsf{\rho }}_{\mathrm{e}}={\mathsf{\rho }}_{\mathrm{w}}·\mathsf{\xi }+{\mathsf{\rho }}_{\mathrm{h}}·(1-\mathsf{\xi })$$

(2)

where ${\mathsf{\rho }}_{\mathrm{w}}$ denotes the density of the aqueous phase, kg/m³; ξ—volumetric water fraction (dimensionless).

The dynamic viscosity of the hydrocarbon phase (${\mathsf{\mu }}_{\mathrm{h}}$) can be derived from the kinematic viscosity (${\mathsf{\nu }}_{\mathrm{h}}$):

$${\mathsf{\mu }}_{\mathrm{h}}={\mathsf{\nu }}_{\mathrm{h}}·{\mathsf{\rho }}_{\mathrm{h}}$$

(3)

where ${\mathsf{\nu }}_{\mathrm{h}}$—kinematic viscosity of emulsion, m²/s; ${\mathsf{\rho }}_{\mathrm{h}}$—density of the hydrocarbon phase of the emulsion, kg/m³.

The temperature dependence of ν_h is approximated by the empirical correlation:

$${\mathsf{\nu }}_{\mathrm{h}}(\mathrm{t})={\mathsf{\nu }}_{\mathrm{h},100°\mathrm{C}}·{({\displaystyle \frac{100}{t}})}^{1.55}$$

(4)

where t is in °C and ν_h,_100°C denotes the kinematic viscosity at 100 °C.

The dynamic viscosity of the aqueous phase of the emulsion ${\mathsf{\mu }}_{\mathrm{w}}$ depends on the temperature as follows [48,49]:

$${\mathsf{\mu }}_{\mathrm{w}}={\displaystyle \frac{1.78·{10}^{-6}}{1+0.0337·t+0.000221·t^2}}·{\mathsf{\rho }}_{\mathrm{w}}$$

(5)

where ${\mathsf{\rho }}_{\mathrm{w}}$—water density, kg/m³; t—process temperature, °C.

Inverse water-in-oil emulsions, as typical dispersed systems, exhibit a marked increase in effective viscosity with rising water content, peaking near 50–60 vol.% and often undergoing phase inversion. For water-in-oil emulsions (ξ < 0.5), the effective dynamic viscosity (μ_e) is estimated using the Monson correlation [48,49]:

$${\mathsf{\mu }}_{\mathrm{e}}={\mathsf{\mu }}_{\mathrm{h}}·(1+2.5·\mathsf{\xi }+2.19·{\mathsf{\xi }}^2+27.45·{\mathsf{\xi }}^3)$$

(6)

where ξ—relative water content in the emulsion.

Conversely, for oil-in-water emulsions, the following empirical relation is applied:

$${\mathsf{\mu }}_{\mathrm{e}}={\mathsf{\mu }}_{\mathrm{w}}·{10}^{(3.2·(1-\mathsf{\xi }))}$$

(7)

Given that phase separation efficiency is markedly higher in the W/O regime—where the continuous hydrocarbon phase facilitates gravitational settling and minimizes interfacial instability—the dewatering process is intentionally operated below the phase-inversion threshold (ξ ≤ 0.5). Consequently, all subsequent calculations assume an inverted emulsion structure governed by Equation (6).

The first stage of the modeling focuses on establishing a quantitative relationship between droplet diameter, mixing time, and stirrer frequency. This relationship is derived by combining the expressions for the mixing time constant, Kτ, given in Equations (8) and (9), while accounting for the dependence of dispersed droplet size on dispersion time and agitation intensity.

To establish the functional dependence of water droplet diameter on agitation intensity and mixing time, the dimensionless mixing time constant Kτ is analyzed through two equivalent representations [47,55,56,58]:

Geometric–hydrodynamic form (empirical correlation):

$${\mathrm{K}}_{\mathsf{\tau }}={\mathrm{C}}_1·{\mathrm{R}\mathrm{e}}_{\mathrm{c}}^{\mathrm{a}}·{\mathrm{F}\mathrm{r}}^{\mathrm{b}}·{({\displaystyle \frac{{\mathrm{d}}_{\mathrm{m}}}{\mathrm{D}}})}^{\mathrm{e}}·{({\displaystyle \frac{{\mathrm{H}}_0}{\mathrm{D}}})}^{\mathrm{f}}$$

(8)

where ${\mathrm{C}}_1$—parameter depending on the type of mixing device; ${\mathrm{d}}_{\mathrm{m}}$—stirrer diameter, m; $\mathrm{D}$—apparatus diameter, m; ${\mathrm{H}}_0$—the initial liquid height, m; ${\mathrm{R}\mathrm{e}}_{\mathrm{c}}$—centrifugal Reynolds number; $\mathrm{F}\mathrm{r}$—Froude number; a, b, e, f—determined experimentally for specific impeller types and flow regimes.

Alternatively, Kτ can be expressed in terms of operational parameters:

$${\mathrm{K}}_{\mathsf{\tau }}={\displaystyle \frac{\mathsf{\tau }·\mathrm{n}·{\mathrm{d}}_{\mathrm{m}}^3}{\mathrm{V}}}$$

(9)

where $\tau$—mixing time, sec; $\mathrm{n}$—the impeller rotational speed, rps; ${\mathrm{d}}_{\mathrm{m}}$—stirrer diameter, m; $\mathrm{V}$—working volume, m³.

To establish a physically consistent scaling framework for the thermomechanical dewatering process, Equations (8) and (9)—representing, respectively, the dimensionless hydrodynamic correlation and the operational definition of the mixing time constant—are equated. This allows the explicit derivation of the working volume V of the mixing apparatus as a function of process parameters, which subsequently serves as the foundation for formulating the dependence of dispersed water droplet diameter on mixing time and impeller rotational frequency.

The centrifugal Reynolds number, characterizing the hydrodynamic regime under forced convection, is calculated as:

$${\mathrm{R}\mathrm{e}}_{\mathrm{c}}={\displaystyle \frac{{\mathsf{\rho }}_{\mathrm{e}}·\mathrm{n}·{\mathrm{d}}_{\mathrm{m}}^2}{{\mathsf{\mu }}_{\mathrm{e}}}}$$

(10)

where ${\mathsf{\rho }}_{\mathrm{e}}$—emulsion density, kg/m³; $\mathrm{n}$—the impeller rotational speed, rps; ${\mathrm{d}}_{\mathrm{m}}$—stirrer diameter, m; ${\mathsf{\mu }}_{\mathrm{e}}$—dynamic viscosity of emulsion, Pa·s.

Hydrodynamically, the system operates across distinct flow regimes, delineated by critical Reynolds number thresholds:

• ${\mathrm{R}\mathrm{e}}_{\mathrm{c}}$ ≤ 80: laminar regime,
• 80 < ${\mathrm{R}\mathrm{e}}_{\mathrm{c}}$ < 1000: transitional regime, characterized by predominantly laminar bulk flow with localized turbulence generated in the high-shear region near the impeller blades,
• ${\mathrm{R}\mathrm{e}}_{\mathrm{c}}$ ≥ 1000: fully turbulent regime.

In accordance with [46], power consumption in the transitional regime is conservatively estimated using the laminar-flow correlation, with ${\mathrm{R}\mathrm{e}}_{\mathrm{c}}$ fixed at 80, except for screw-type impellers, where distinct hydrodynamic behavior necessitates separate treatment.

Experimental evaluation of ${\mathrm{R}\mathrm{e}}_{\mathrm{c}}$ over the operational speed range confirms that the process resides entirely within the transitional regime:

• at n = 20 rpm (0.333 rps), ${\mathrm{R}\mathrm{e}}_{\mathrm{c}}$ ≈ 31.86,
• at n = 200 rpm (3.33 rps), ${\mathrm{R}\mathrm{e}}_{\mathrm{c}}$ ≈ 318.66,
• at n = 400 rpm (6.67 rps), ${\mathrm{R}\mathrm{e}}_{\mathrm{c}}$ ≈ 637.33.

Consequently, following [47], the exponent a in Equation (8) is assigned a value of 3 for both laminar and transitional regimes, in contrast to the turbulent regime, for which a = 0.

The modified Froude number, accounting for buoyancy effects arising from the density difference between dispersed and continuous phases, is defined as:

$$\mathrm{F}\mathrm{r}={\displaystyle \frac{{\mathsf{\rho }}_{\mathrm{e}}·{\mathrm{n}}^2·{\mathrm{d}}_{\mathrm{m}}^2}{({\mathsf{\rho }}_{\mathrm{w}}-{\mathsf{\rho }}_{\mathrm{h}})·{\mathrm{H}}_0·\mathrm{g}}}$$

(11)

where ${\mathsf{\rho }}_{\mathrm{e}}$—emulsion density, kg/m³; $\mathrm{n}$—the impeller rotational speed, rps; ${\mathrm{d}}_{\mathrm{m}}$—stirrer diameter, m; ${\mathsf{\rho }}_{\mathrm{w}}$—density of the aqueous phase, kg/m³; ${\mathsf{\rho }}_{\mathrm{h}}$—density of the hydrocarbon phase, kg/m³; ${\mathrm{H}}_0$—liquid fill height, m; $\mathrm{g}$—standard gravitational acceleration, 9.80665 m/s².

All input parameters used in the model—including physical properties, geometric dimensions, and operational ranges—are summarized in

Table 1. Initial data for modeling.

Symbol	Parameter	Unit	Value
$\mathrm{d}$	Characteristic diameter of water droplets in the emulsion prior to mechanical agitation, under superheated conditions	m	3 × 10−3
${\mathsf{\rho }}_h$	Density of the hydrocarbon continuous phase at 20 °C	kg/m3	941
${\mathsf{\rho }}_w$	Density of the aqueous dispersed phase at 100 °C	kg/m3	954
${\mathrm{a}}_h$	Empirical linear temperature coefficient for hydrocarbon density		0.581
${\mathsf{\nu }}_h$	Kinematic viscosity of the hydrocarbon phase at 100 °C	m2/s	25.4 × 10−6
$D$	Internal diameter of the mixing vessel (apparatus)	m	0.16
${\mathrm{d}}_{\mathrm{m}}$	Impeller (turbine) diameter	m	0.08
$z_c$	Number of impeller disks	-	6
$h_c$	Axial height of the impeller hub/assembly	m	0.2 × $d_m$
$b_m$	Width of the mixer’s central elements	m	0.25 × $d_m$
$\mathrm{b}$	Coefficient for the turbine mixer	-	0.03
${\mathrm{C}}_1$	Coefficient depending on the type of mixing device	-	83
${\mathrm{H}}_0$	Height of the emulsion layer	m	0.2
${\mathrm{i}}_{\mathrm{t}}$	Specific enthalpy of water	kJ/kg
	- at 100.5 °C		422
	- at 105 °C		440
	- at 110 °C		461
	- at 115 °C		482.7
	- at 120 °C		504.1
	- at 130 °C		546.8
	- at 140 °C		589.5
	- at 150 °C		632.7
${\mathrm{i}}_{100}$	Specific enthalpy of water at 100 °C	kJ/kg	419
$h$	Height of emulsion rise	m	0.4
$\xi$	Volumetric water content in the emulsion	Volume fractions	0.3

.

The coefficients e and f in Equation (8) depend on the ratio ${\displaystyle \frac{d_m^3}{D^{2.5}·H_0^{0.5}}}$. For the laboratory setup, this ratio equals 0.265, which exceeds the threshold value of 0.08. According to [47], this implies e = 3 and f = −0.5.

Using Equations (10) and (11), Expressions (8) and (9) can be transformed into Equation (12).

$${\displaystyle \frac{\tau ·\mathrm{n}·{\mathrm{d}}_{\mathrm{m}}^3}{V}}=C_1·({{\displaystyle \frac{{\mathsf{\rho }}_{\mathrm{e}}·\mathrm{n}·{\mathrm{d}}_{\mathrm{m}}^2}{{\mu }_e}})}^3·{({\displaystyle \frac{{\rho }_e·{\mathrm{n}}^2·{\mathrm{d}}_{\mathrm{m}}^2}{({\rho }_w-{\rho }_h)·H_0·\mathrm{g}}})}^{-0.3}·{({\displaystyle \frac{{\mathrm{d}}_{\mathrm{m}}}{\mathrm{D}}})}^3·{({\displaystyle \frac{{\mathrm{H}}_0}{\mathrm{D}}})}^{-0.5}$$

(12)

Rearranging Equation (12) yields the expression for the apparatus volume:

$$V={\displaystyle \frac{\tau ·\mathrm{n}·{\mathrm{d}}_{\mathrm{m}}^3}{C_1·{(\frac{{\mathsf{\rho }}_{\mathrm{e}}·\mathrm{n}·{\mathrm{d}}_{\mathrm{m}}^2}{{\mu }_e})}^3·{(\frac{{\rho }_e·{\mathrm{n}}^2·{\mathrm{d}}_{\mathrm{m}}^2}{({\rho }_w-{\rho }_h)·H_0·\mathrm{g}})}^{-0.3}·{(\frac{{\mathrm{d}}_{\mathrm{m}}}{\mathrm{D}})}^3·{(\frac{{\mathrm{H}}_0}{\mathrm{D}})}^{-0.5}}}$$

(13)

This resulting formula will be used subsequently to derive the dependence of droplet diameter on mixing time and impeller rotational frequency.

Expression (14) shows the size of a water droplet as it breaks up during mixing [45]:

$$d_k=3.5·{({\displaystyle \frac{\mathsf{\sigma }}{{\mathsf{\rho }}_{\mathrm{e}}}})}^{0.6}·{({\displaystyle \frac{\mathrm{N}}{{\mathsf{\rho }}_{\mathrm{e}}·\mathrm{V}}})}^{-0.4}$$

(14)

where σ—surface tension, J/m²; ${\mathsf{\rho }}_{\mathrm{e}}$—emulsion density, kg/m³; N—energy input during mixing, J; V—volume of the apparatus, m³.

However, the absolute values of the droplet diameter predicted by Equation (14) were significantly larger than those observed experimentally by Braginski et al. [45]. This substantial discrepancy between theory and experiment may be attributed to the fact that the velocity pulsation amplitude is a stochastic variable, which—to a first approximation—follows a normal (Gaussian) distribution. If the droplet size is assumed to be governed by the maximum velocity pulsations, the empirical coefficient in Equation (14) must be adjusted from 3.5 to 0.83 based on data presented in [45], yielding:

$$d_k=0.83·{({\displaystyle \frac{\mathsf{\sigma }}{{\mathsf{\rho }}_{\mathrm{e}}}})}^{0.6}·{({\displaystyle \frac{\mathrm{N}}{{\mathsf{\rho }}_{\mathrm{e}}·\mathrm{V}}})}^{-0.4}$$

(15)

This expression can be reformulated as a functional dependence $d_k=f(\tau ,n)$, where τ is mixing time and n is impeller rotational speed.

The energy input during emulsion mixing, N, is given by [56,57]:

$$N=K_n·{\rho }_e·d_m^5·n^3$$

(16)

where $K_n$—power number (also referred to as the Euler number); ${\rho }_e$—emulsion density, kg/m³; ${\mathrm{d}}_{\mathrm{m}}$—stirrer diameter, m; n—impeller rotational speed, rps.

The power number K_n characterizes the ratio of pressure forces to inertial forces acting on a fluid element and depends on the flow regime (laminar or turbulent). For laminar flow with a turbine impeller, it is expressed as [46,55,56,58]:

$$K_n=z_c·{\displaystyle \frac{{\mathsf{\pi }}^2}{6}}·{\mathsf{\lambda }}_{\mathrm{c}}·{\displaystyle \frac{h_c}{d_m}}·(1-{\left({\displaystyle \frac{\frac{d_m}{2}-b_m}{\frac{d_m}{2}}}\right)}^3)·{Re}_c^{-1}$$

(17)

where $z_c$—number of central blades (or plates) on the impeller; π is a mathematical constant equal to 3.14159; $h_c$—height of the central blades, m; $b_m$—width of the central blades, m; ${\mathsf{\lambda }}_{\mathrm{c}}$—drag coefficient of the central blades; ${\mathrm{R}\mathrm{e}}_{\mathrm{c}}$—Reynolds number.

The blade drag coefficient λc is estimated by:

$${\mathsf{\lambda }}_{\mathrm{c}}=-21.501·\mathrm{l}\mathrm{n}(1.055·\left({\displaystyle \frac{\mathrm{D}-d_m}{d_m}}\right))+23.426·({\displaystyle \frac{\mathrm{D}-d_m}{d_m}})$$

(18)

where $\mathrm{D}$—diameter of the device, m; $d_m$—stirrer diameter, m.

The power number $K_n$ may be estimated using:

$$K_n=C·z_m·\zeta ·K_1(\psi 1)$$

(19)

where C = 3.87 for an unbaffled vessel; $z_m=1$—number of impellers mounted on the shaft; $\zeta =8.4$—turbine mixer drag coefficient; $K_1(\psi 1)$—dimensionless power factor, linking hydrodynamic performance to the circumferential flow characteristics.

The wall friction (or vessel-body resistance) coefficient (${\mathsf{\lambda }}_k$) is determined based on:

$${\mathsf{\lambda }}_k={\displaystyle \frac{\frac{D}{d_m}}{20.35·\frac{D}{d_m}-19.1}}$$

(20)

The filling height parameter of the apparatus ($\gamma$) is defined as:

$$\gamma =8·{\displaystyle \frac{H_0}{D}}+1$$

(21)

To evaluate the power factor, the peripheral velocity profile parameter ψ2 must be determined:

$${\psi }_2=-s_1-s_2·\psi 1$$

(22)

where $s_1=0.5$ and $s_2=1.25$ are empirical constants dependent on impeller type and the diameter ratio ${\displaystyle \frac{D}{d_m}}$.

The power factor itself is approximated by a quadratic form:

$$K_1(\psi 1)=0.1{·\psi 1}^2+0.222·\psi 1·{\psi }_2+0.125·{{\psi }_2}^2$$

(23)

The dimensionless mean peripheral velocity of the mixed fluid, $v_{sr}$, is calculated as:

$$v_{sr}={\displaystyle \frac{1+0.4·\psi 1+0.5·{\psi }_2+2·(1+\psi 1+{\psi }_2)·\mathrm{l}\mathrm{n}(\frac{D}{d_m})}{2·\frac{D}{d_m}}}$$

(24)

The dimensionless, radially averaged circumferential (tangential) velocity at radius r is given by:

$$v(r)={\displaystyle \frac{(1+\psi 1+{\psi }_2)·d_m}{2·r}}$$

(25)

The torque generated by the rotating impeller blades ($M_{kr}$) is:

$$M_{kr}=z_c·\zeta ·K_1(\psi 1)$$

(26)

where $z_c$—number of impellers on the shaft; $\zeta$—drag coefficient of a turbine mixer; $K_1(\psi 1)$—power factor.

The resisting torque due to wall friction is:

$$M_{kor}={\displaystyle \frac{{v_{sr}}^{1.75}·{(\frac{D}{d_m})}^{2.75}·\pi ·{\mathsf{\lambda }}_k·\gamma }{2.2·{Re}_c^{0.25}}}$$

(27)

where $v_{sr}$—dimensionless mean peripheral velocity; π is a mathematical constant equal to 3.14159; ${\mathsf{\lambda }}_k$—wall (vessel-body) resistance coefficient; ${\mathrm{R}\mathrm{e}}_{\mathrm{c}}$—Reynolds number; $\gamma$—fill-height parameter.

Compliance with the equation of equality of moments allows us to determine the peripheral velocity profile parameter (ψ1):

$$M_{kr}-M_{kor}=0$$

(28)

Substituting Equations (12), (13), and (16) into Equation (15) yields the explicit form of the droplet diameter:

$$d_k=0.83·{({\displaystyle \frac{\mathsf{\sigma }}{{\rho }_e}})}^{0.6}·{({\displaystyle \frac{K_n·{\rho }_e·d_m^5·n^3}{{\rho }_e·\frac{\tau ·\mathrm{n}·{\mathrm{d}}_{\mathrm{m}}^3}{C_1·({\frac{{\mathsf{\rho }}_{\mathrm{e}}·\mathrm{n}·{\mathrm{d}}_{\mathrm{m}}^2}{{\mu }_e})}^3·{(\frac{{\rho }_e·{\mathrm{n}}^2·{\mathrm{d}}_{\mathrm{m}}^2}{({\rho }_w-{\rho }_h)·H_0·\mathrm{g}})}^{-0.3}·{(\frac{{\mathrm{d}}_{\mathrm{m}}}{\mathrm{D}})}^3·{(\frac{{\mathrm{H}}_0}{\mathrm{D}})}^{-0.5}}}})}^{-0.4}$$

(29)

By differentiating Equation (29) with respect to the dispersion time, the equation of dependence, showing the change in the diameter of a water globule over time is obtained.

$$d_k={\displaystyle \frac{d(d_k)}{d\tau }}={\displaystyle \frac{K·0.4}{{\tau }^{0.6}·n^{1.76}}}$$

(30)

where the constant K is given by:

$$K=0.332·{({\displaystyle \frac{\mathsf{\sigma }}{{\rho }_e}})}^{0.6}·{({\displaystyle \frac{K_n·{\rho }_e·d_m^5}{{\rho }_e·\frac{\tau ·{\mathrm{d}}_{\mathrm{m}}^3}{C_1·({\frac{{\mathsf{\rho }}_{\mathrm{e}}·{\mathrm{d}}_{\mathrm{m}}^2}{{\mu }_e})}^3·{(\frac{{\rho }_e·{\mathrm{d}}_{\mathrm{m}}^2}{({\rho }_w-{\rho }_h)·H_0·\mathrm{g}})}^{-0.3}·{(\frac{{\mathrm{d}}_{\mathrm{m}}}{\mathrm{D}})}^3·{(\frac{{\mathrm{H}}_0}{\mathrm{D}})}^{-0.5}}}})}^{-0.4}$$

Thus, Equation (30) provides the required functional dependence of droplet diameter on dispersion (mixing) time and impeller rotational frequency—valid for conditions near the boiling point of the dispersed phase.

The next step in the model is to relate droplet size to sedimentation behavior. As the droplets break down during mixing, both their settling velocity and settling time decrease accordingly.

The terminal settling velocity of a droplet is estimated using a modified Stokes-type expression accounting for internal circulation [45]:

$$w={d_{po}}^2·{\displaystyle \frac{({\rho }_w-{\rho }_h)}{18·{\mu }_h}}·{\displaystyle \frac{3·({\mu }_w+{\mu }_h)}{2·{\mu }_w+{3·\mu }_h}}$$

(31)

where $d_{po}$—surface-volume diameter of globules, m; ${\rho }_w$—water phase density, kg/m³; ${\rho }_h$—density of the hydrocarbon phase, kg/m³; ${\mu }_h$—dynamic viscosity of the hydrocarbon phase of the emulsion, Pa·s; ${\mu }_w$—dynamic viscosity of the aqueous phase of the emulsion, Pa·s.

The surface–volume diameter is related to the energy input by:

$$d_{po}=0.13·{({\displaystyle \frac{\mathsf{\sigma }}{{\mathsf{\rho }}_{\mathrm{e}}}})}^{0.6}·{({\displaystyle \frac{\mathrm{N}}{{\mathsf{\rho }}_{\mathrm{e}}·\mathrm{V}}})}^{-0.4}$$

(32)

where $\mathsf{\sigma }$—surface tension, J/m²; ${\mathsf{\rho }}_{\mathrm{e}}$—emulsion density, kg/m³; $\mathrm{N}$—energy spent on mixing the emulsion, J; $\mathrm{V}$—volume of the device, m³.

Comparing (32) with (15) and noting that 0.13/0.83 ≈ 0.157, and assuming proportionality between $d_{po}$ and the characteristic droplet diameter dk, one obtains:

$$w={{0.025·d}_k}^2·{\displaystyle \frac{({\rho }_w-{\rho }_h)}{18·{\mu }_h}}·{\displaystyle \frac{3·({\mu }_w+{\mu }_h)}{2·{\mu }_w+{3·\mu }_h}}$$

(33)

Finally, the droplet settling time to descend the fill height ${\mathsf{H}}_0$ is [49]:

$$\tau ={\displaystyle \frac{{\mathsf{H}}_0}{w}}$$

(34)

where ${\mathrm{H}}_0$—filling height of the apparatus, m; $w$—droplet settling velocity, m/s.

The presented formulas are used to calculate the values of the settling speed and time from the diameter of the globules at different rotation frequencies.

2.2.2. Comparison of the Energy Accumulated by the Superheated Liquid with the Energy Required to Transfer the Volume of the Emulsion

Another step of the modeling is to determine the boiling energy of a water globule within the emulsion volume, depending on its diameter.

Specifically, the energy required for a water droplet to transition from the saturated liquid state at 100 °C to a superheated state—up to a critical temperature at which explosive vaporization (“blow-up”) occurs without conventional phase separation—is evaluated. Based on experimental observations [32], this critical transition temperature ranges from 100.5 °C to 150 °C.

The boiling energy of a superheated water globule is determined by [49,52,57]:

$$E=m_w·(i_t-i_{100})={\rho }_w{\displaystyle \frac{\pi ·d^3}{6}}·(i_t-i_{100})$$

(35)

where E—energy required for heating the droplet from 100 °C to temperature t, J; $m_w$—mass of water, kg; ${\rho }_w$—water density, kg/m³; $\mathsf{\pi }$—a mathematical constant equal to 3.14159; $d$—diameter of water globules, m; $i_t$—specific enthalpy of water at temperature t, kJ/kg; $i_{100}$—specific enthalpy of water at 100 °C, kJ/kg.

The enthalpies for water vapor are presented in Table 1.

Next, the energy required to displace the overlying emulsion column during the explosive expansion of the droplet is estimated as [48]:

$$W=F·h$$

(36)

where F—net upward force required to lift the emulsion (without phase separation), N; $h$—emulsion lifting height during blow-up, m.

Hydrodynamic equilibrium occurs when forces are equal (Figure 4) [46]:

$$F+F_{a }=F_{t }+F_{m }$$

(37)

The buoyancy (Archimedes) force is calculated using formula:

$$F_{a }=\mathsf{\Delta }m·g$$

(38)

where $\mathsf{\Delta }m={\rho }_w·\pi ·d·{\displaystyle \frac{d^2}{4 }}$—weight of a droplet of water raised to the surface of the liquid, kg.

The force of gravity is calculated using the formula:

$$F_{m }= m·g$$

(39)

where $m$—the mass of the emulsion layer above the droplet, kg, while the weight of the water droplet is neglected, since it is infinitesimally small in relation to the weight of the emulsion; g is the acceleration due to gravity, 9.80665 m/s².

$$m={\rho }_e·\pi ·H_0·{\displaystyle \frac{d^2}{4 }}$$

(40)

where ${\rho }_e$—emulsion density, kg/m³; $\pi$—a mathematical constant equal to 3.14159; $H_0$—emulsion layer height, m; $d$—diameter of water globules, m.

The friction force is calculated using the formula:

$$F_{t }=3·\pi ·d·{\mu }_h·\upsilon$$

(41)

where υ—characteristic radial velocity of the droplet (assumed 0.5 m/s for estimation); $\pi$—a mathematical constant equal to 3.14159; ${\mu }_h$—dynamic viscosity of the continuous (hydrocarbon) phase, Pa·s.

Intermolecular (e.g., Van der Waals) forces are neglected, as their contribution is minor at the droplet scale considered.

Instability—and hence explosive ejection of the emulsion—occurs when the net upward driving force exceeds the resisting forces:

$$F+F_{a }>F_{t }+F_m$$

(42)

When the boiling energy exceeds the calculated blow-up energy, explosive boiling of the emulsion occurs, propelling it into the vapor phase.

This formula is universal, as it eliminates the dependence on the apparatus diameter and allows comparison of blow-up energies depending on the filling height of the apparatus. Thus, the modeling allows for setting the minimum required mixing time and intensity for stable thermomechanical dewatering without unstable phase transfer.

3. Results and Discussion

At the experimental level, the feasibility of employing a thermomechanical method for recycling hydrocarbon- and oil-containing waste in the form of highly stable emulsions was demonstrated, successfully achieving residual water content at trace levels in the dehydrated products. However, to facilitate scale-up to pilot- and industrial-scale applications, it became essential to develop a more accurate model capable of predicting the stable boiling of the aqueous phase in highly stable water-hydrocarbon emulsions [27,28,29,30,31,32,42].

A quantitative mathematical model describing the evaporation of water from a water–hydrocarbon emulsion under mechanical agitation over a boiling liquid phase has been developed to characterize the underlying thermomechanical process. The primary objectives of formulating this model are as follows:

• To establish the operational parameters of mechanical agitation sufficient to prevent overheating of the liquid phase, coalescence of dispersed water globules, and the onset of explosive (violent) boiling within the emulsion;
• To quantitatively determine the functional dependence of water globule size distribution on key system parameters—including geometrical and hydrodynamic characteristics of the apparatus, physicochemical properties of the hydrocarbon feedstock (e.g., viscosity, interfacial tension, density), intensity of mechanical mixing and duration of mixing;
• To model the kinetics of gravitational settling of water globules, thereby enabling prediction of phase separation rates under transient thermal and mechanical conditions;
• To evaluate the mechanical energy input required to sustain stable emulsion transfer and mixing during the unstable boiling regime of the superheated liquid.

3.1. Comparison of Dispersion Time with the Settling Time of Water Globules

An important indicator is the comparison of the water globule dispersion rate with its settling rate. Figure 5 shows the dependence of water globule diameter on dispersion time and the intensity of mechanical agitation. According to calculations, the highest dispersion rate occurs during the initial period of mixing.

In the first 5 s, globule size decreases by approximately 80–85%, from about 3 mm to 0.5 mm, when subjected to moderate mixing rates exceeding 100 rpm. With further mixing, the dispersion rate decreases severalfold. It should be noted that, starting from a certain dispersion time, the size of the dispersed globules is practically independent of the initial globule size. According to calculations, the minimum dispersion time required to obtain water globules with droplet diameters less than 1.5 mm is 5 s at a stable mixing speed of 20 rpm (0.33 rps), and less than 1 s at a mixing intensity of 100–200 rpm (1.66–3.33 rps). Based on modeling, to achieve water globules with a diameter less than 0.5 mm at a stirring speed of 20 rpm, process duration exceeding 37 s is required. Increasing the stirring intensity to above 100 rpm reduces this time to approximately 5 sec. Thus, during prolonged dispersion, the stirrer speed can be maintained at a moderate level. But if the process is recommended to be carried out with a sufficiently short dispersion time, it is necessary to ensure high mixing intensity.

The settling velocity of water globules was calculated for the entire range of water globule diameters (Figure 6). Figure 6b represents an enlarged detail from Figure 6a. As expected, the settling velocity exhibits a strong positive dependence on droplet diameter. Accurate determination of the settling velocity is essential for estimating the accumulation time required for a critical mass of water to collect on the heating surface. If this accumulated water layer undergoes rapid boiling, it may trigger vapor expansion sufficient to cause liquid carry-over, vessel overpressure, or even mechanical failure.

Given the water content in the emulsions studied, the settling time was determined based on the time it took for a 0.2 m-high emulsion layer to break down. The calculated settling time is shown in Figure 7. Figure 7b represents an enlarged detail from Figure 7a.

By comparing the dispersion time (i.e., the time needed to reduce droplet size via mechanical agitation) with the settling time, one can identify the minimum impeller speed required to maintain emulsion stability: mechanical agitation must be sufficiently intense to ensure that droplet breakup occurs faster than gravitational settling.

A comparison of the results presented in Figure 5 and Figure 7 and summarized in Figure 8 (Figure 8b represents an enlarged detail from Figure 8a) shows that the dispersion time is hundreds of times shorter than the settling time. For instance:

• At 20 rpm (Re = 31.86), the droplet diameter decreases to 1 mm within 12 s of mixing.
• At 45 rpm (Re = 71,69), the droplet diameter decreases to 1 mm within 4 s of mixing.
• At 200 rpm (Re = 318.66), the same initial droplet is reduced to 1 mm in just 1 s. In contrast, the gravitational settling time for a 3 mm water droplet is approximately 170 s.

A comparison of the data presented in Figure 8 reveals that as the degree of dispersion increases—corresponding to a reduction in water globule size of 1–2 mm—the settling time at stirring intensities exceeding 45 rpm becomes 200 to 540 times longer than the dispersion time. Under conditions of higher system dispersion, where particle sizes fall below 0.5 mm, the dispersion time is 500 times (or more) shorter than the settling time.

Based on this mathematical modeling, it is recommended—for reliable operation in pilot-plant experiments—to maintain the impeller speed within 100–200 rpm, corresponding to the transitional flow regime (Re = 159.33–318.66). This range ensures that droplet breakup outpaces settling, thereby preventing water accumulation on the heating surface.

3.2. Comparison of the Energy Accumulated by the Superheated Liquid with the Energy Required to Transfer the Volume of the Emulsion

Considering that mechanical agitation of the emulsion allows for averaging the temperature in the apparatus between the heating surface and the bulk of the emulsion, and thus reducing the superheating temperature of the emulsion layer on the heating surface, a second criterion was proposed for determining the minimum intensity of mechanical agitation on the emulsion, preventing explosive boiling and unstable liquid transfer.

The accumulated thermal energy in a superheated droplet is quantified by the enthalpy difference between the liquid at its saturation temperature (100 °C at atmospheric pressure) and at the local superheated temperature t. The temperature of the superheated liquid varied from 100.5 to 150 °C. The boundaries of this range were chosen based on experimental data.

Using Equations (35)–(42), the mechanical work required to lift the emulsion column over a droplet by a height of 0.2 m (i.e., the nominal fill height) was calculated to be 2.4 mJ. In the calculations, conditions were deliberately chosen that placed maximum demands on the intensity of mechanical agitation in order to stabilize the boiling process.

Figure 9 presents the boiling energy accumulated in a single water droplet as a function of its diameter, for H₀ = 0.2 m. From this curve, the minimum permissible droplet diameter—defined as the largest size for which the accumulated superheat energy remains below the displacement energy (2.4 mJ)—is found to be:

• 1.1 mm at t = 100.5 °C,
• 0.5 mm at t = 110 °C,
• 0.25 mm at t = 150 °C.

Assuming the water layer on the heating surface does not exceed one droplet diameter in thickness (i.e., thin-film evaporation), stable boiling is achievable when droplets are reduced to or below these critical sizes.

Referring to Figure 5:

• At 100 rpm, a 3 mm droplet reaches 1.1 mm in <1 s and 0.5 mm in <4 s, satisfying the stability criterion even at 110 °C.
• To accommodate the upper limit of observed surface temperatures (150 °C), droplets must be reduced to ≤0.25 mm, which according to the model requires >200 rpm (see Figure 5).

This dispersion time, obtained using the second criterion, is significantly lower than the settling time of water globules of a given size on the heating surface, allowing for water evaporation from the water-in-oil emulsion without the unstable mechanical transfer of liquid. Therefore, to obtain reliable results, it is also recommended to maintain the stirrer speed at least 100–200 rpm.

Since experiments have shown that heating temperatures, depending on the initial water-in-oil emulsion, can reach 150 °C [32], measures must be taken to prevent the formation of water droplets larger than 0.25 mm in diameter, for example, by using more intensive dispersion (over 200 rpm).

4. Conclusions

The possibility of stabilizing the evaporation of the aqueous phase of an emulsion by mechanical agitation on a boiling liquid was theoretically proven and demonstrated, and a mathematical model of the evaporation of the aqueous phase from an emulsion was developed. Two criteria were proposed for determining the optimal conditions for stabilizing the evaporation of the aqueous phase.

The first criterion is based on comparing the dispersion time with the settling time of the water globules. Analysis of the model revealed that effective prevention of explosive boiling and unstable liquid transfer is achieved by ensuring intensive dispersion of the water globules. Calculations showed that during the first 5 s of mixing, the globule diameter decreases by 80–85%, from about 3 mm to 0.5 mm, when subjected to moderate mixing rates exceeding 100 rpm. It was also demonstrated that the settling time of the globules is hundreds of times longer than their dispersion time.

The second criterion is based on comparing the energy accumulated by the superheated liquid with the energy required to transfer the emulsion volume. An analysis of the energy criteria for boiling revealed that at a superheat temperature of 110 °C, the permissible globule diameter, which eliminates the risk of liquid overshoot, should not exceed 0.5 mm, and at 150 °C–0.25 mm. To ensure stable and safe operation and maintain the system in a finely dispersed state, it is recommended to maintain at least a transition zone of mixing with a Reynolds number of 159.33–318.66 (stirrer speed of at least 100–200 rpm). At heating temperatures close to 150 °C, even more intensive mixing (over 200 rpm, Reynolds number greater than 318.66) is required to prevent the formation of critical globule sizes.

Therefore, adherence to the specified mechanical agitation conditions ensures reliable suppression of the risk of explosive boiling and allows for the stable evaporation of water from the emulsion without unstable liquid boiling-up phenomena.