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Frontiers in Chemical Engineering ·

Volume 8 - 2026 | https://doi.org/10.3389/fceng.2026.1775892

Comparative assessment of diffusion modeling methods in evaporation problems of liquid bulk cargoes during tanker loading

  • V. Gustov1*corresponding author
  • V. Pshenin1
  • E. Dzhemilev1
  • D. Sakal1
  1. Empress Catherine II Saint Petersburg Mining University, Saint Petersburg, Russia
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Abstract

This paper examines various models describing the evaporation of liquid bulk cargoes based on the classical diffusion equation. The numerical solutions obtained using the considered models were validated against well-known experimental data on methanol evaporation in a Stefan diffusion tube. A comparative analysis of experimental measurements and numerical modeling results demonstrates that the diffusion equation incorporating a correction for Stefan flow provides a more accurate description of the diffusive evaporation behavior of volatile substances. Within the framework of numerical simulations, dependencies of the mass transfer rate of the volatile component n-hexane (one of the primary constituents of light gasoline fractions) evaporating from a liquid surface during tank filling were obtained as functions of the filling velocity for different evaporation models. The results indicate that accounting for the Stefan flow correction in the diffusion equation significantly affects both the saturation time of the tank vapor space with product vapors and the total mass of vapors accumulated in the vapor–gas space by the time the tank is completely filled.

Keywords

Introduction

The current trajectory of global energy policy is directed toward addressing a complex set of challenges associated with the economically efficient and environmentally sustainable utilization of mineral and raw material resources (Litvinenko et al., 2023). In particular, achieving the stated objectives of energy development requires the implementation of innovative technologies within the fuel and energy sector, as well as the optimization of existing technological processes (Nikolaev and Plotnikova, 2023; Nikolaev et al., 2025; Podoprigora et al., 2022; Mulder, 2007). One of the key priorities of contemporary global energy systems is the minimization of the industrial carbon footprint, manifested both through the integration of greenhouse gas capture and utilization technologies into production processes (Tramaddoni et al., 2014; Skobelev et al., 2023; Tsvetkov, 2025) and through a transition toward energy resources with lower carbon intensity (Ponomarenko et al., 2024; Shammazov and Karyakina, 2023; Sultanbekov and Schipachev, 2022). At the same time, over the past decade, global trade volumes of fossil fuels have continued to increase, and liquid hydrocarbons currently remain among the most important exchange-traded commodities worldwide (Litvinenko et al., 2022; Pashkevich and Bykova, 2022).

At present, the majority of long-distance transportation of crude oil, petroleum products, and liquid bulk chemical cargoes is carried out by tankers (Siddiqui et al., 2013; Siddiqui et al., 2018; Siddiqui et al., 2023; Oil Transportation by Tankers, 1975). Among the complications associated with the delivery of liquid bulk cargoes by tanker transport, the most characteristic are leakages and the emission of volatile organic compounds (VOCs) into the atmosphere at loading and unloading ports (Alver et al., 2018; Devanney and Stewart, 1974; Chen et al., 2019; Mingazhev et al., 2023). Emissions also occur during navigation in open sea conditions due to the operation of ship propulsion and auxiliary engines (Agrawal et al., 2008; Bayırhan et al., 2019; Gobbi et al., 2019; Rajabi et al., 2020). From the standpoint of atmospheric pollution, the most significant emissions are those released during tanker loading operations at ports of departure (Husain and Altshuller, 2006; Milazzo et al., 2017; Kim et al., 2012).

Various technologies exist to reduce VOC emissions into the atmosphere during tanker loading (Husain et al., 2003; Karbasian et al., 2017; Rudd and Hill, 2001), among which vapor recovery units are considered the most promising (Kamel, 2022; Salomone, 2016; Lee et al., 2013; Poornan and Al Hindi, 2016). Despite their widespread integration into the operational processes of oil loading terminals, the efficiency of these systems is directly dependent on accurate prediction of gas-phase outflow regimes from tanker tanks. In turn, these outflow regimes are governed by diffusion-driven evaporation processes occurring during cargo loading, which necessitates their detailed investigation followed by mathematical modeling.

Modeling VOC evaporation, particularly that of gasoline, represents a complex problem that is relevant to numerous technological processes within the oil and gas industry. For example, a number of studies have examined the evaporation behavior of gasolines with different compositions during leaks from storage vessels in order to predict the formation of explosive vapor–air mixtures that pose a threat to the safe operation of industrial facilities. In (Okamoto et al., 2021), the authors proposed a model of diffusion-controlled evaporation of a thin gasoline layer formed as a result of a spill. This model was experimentally validated for a small spill surface area and a small petroleum product layer thickness under laboratory conditions. In (Galeev et al., 2020), the authors investigated a model of thin-layer gasoline evaporation from a surface using standard Ansys Fluent functions, user-defined scalar (UDS) transport equations, and accounting for the Stefan flow correction. The average discrepancy between the numerical results and experimental data was 9.2%, which is most likely attributable to inaccuracies in determining the content of individual components in gasoline, as well as to the difficulty of accounting for all possible external factors during field experiments. In (Penelon et al., 2020), the authors employed semi-empirical mass transfer models to study the mechanism of gasoline evaporation following a spill. The results demonstrate that, with appropriate selection of the coefficients of the semi-empirical evaporation model, the deviation from experimental data does not exceed 10%. The problem of mathematical and numerical modeling of gasoline and other product evaporation during spills from technological storage vessels is also addressed in (Brambilla and Manca, 2009; Fingas, 1995; Galeev et al., 2015; Okamoto et al., 2009b).

Another important problem is the quantification of losses of light hydrocarbons during their injection into storage tanks and subsequent storage. The complexity of modeling such processes arises from the need to account for a combination of physical phenomena related to heat and mass transfer, the multicomponent nature of the hydrocarbon mixtures under consideration, as well as external factors dependent on the specific operating conditions of technological facilities (e.g., wind speed, intensity of solar radiation, etc.). For this reason, the systems of governing equations that comprehensively describe the evaporation of light organic compounds during storage and tank filling vary significantly among different authors.

For example, in (Huang et al., 2020b), a model of n-hexane evaporation from a domed-roof storage tank was developed with consideration of the influence of solar radiation on heat and mass transfer processes of the vapor–air mixture inside the tank. Since n-hexane is a single-component substance, its use as a representative of light hydrocarbons substantially simplifies the calculations. The results of CFD simulations employing user-defined functions (UDFs) were experimentally validated, showing only minor deviations between empirical data and numerical predictions.

In (Huang et al., 2019), the authors proposed a methodology for the numerical calculation of evaporation of various single-component light hydrocarbons through gaps between the pontoon and the inner surface of an internal floating roof tank (IFRT). To describe mass transfer at the gas–liquid interface, a single-film model was applied, while the evaporation rate of hydrocarbons from the surface was calculated using Fluent UDFs. The maximum discrepancy between numerical simulation results and experimental data obtained for relatively small IFRT prototypes with different vent pipe configurations did not exceed 8.12%. A similar study for unsteady operating conditions of an IFRT is presented in (Huang et al., 2020), where the surface mass transfer equations include dimensionless similarity numbers.

In (Hassanvand et al., 2010), the authors employed a volume-of-fluid (VOF)-based interface tracking method combined with a mass transfer model to describe gasoline spraying and its subsequent evaporation during tank filling. The mass transfer process was represented by a differential mass transfer equation incorporating a turbulent diffusion coefficient. The resulting analytical model was validated by CFD simulation of a Stefan tube for n-hexane evaporation.

Article (Invernizzi and Sironi, 2021) describes the VocSim project implemented by the Norwegian Institute of Marine Technology. Within this project, a software tool was developed to simulate the evaporation of light organic compounds during loading into cargo tanks of oil tankers. The mathematical model is based on a differential mass transfer equation with a turbulent diffusion coefficient, which accounts for turbulent mixing of the liquid and vapor–gas phases induced by wave formation in the liquid during vessel motion, convective heat and mass transfer, and the impulsive nature of tank filling. Such an approach, which reduces the multitude of factors affecting mass transfer to a single coefficient, implies an empirical determination of this parameter for specific tanker loading conditions. Consequently, the applicability of the resulting model is limited in the absence of experimental data. However, it is possible to use a scientific approach based on the use of neural networks and methods for recognizing characteristic process signatures (Kulchitskiy and Nikolaev, 2025; Kulchitskiy et al., 2023; Simakov et al., 2025).

The problem of evaporation of volatile organic components from tankers during loading and subsequent marine transportation of petroleum products has also been investigated in (Hu et al., 2020; Martens et al., 2001; Tamaddoni et al., 2014; Virdi et al., 2021). In (Zhang et al., 2026), the authors performed a comprehensive analysis of the mechanisms governing the evaporation of light organic compounds (VOCs) from storage tanks. Table 1 presents classical diffusion evaporation models summarized in (Zhang et al., 2026), each characterized by a specific range of applicability. For instance, according to the authors, the Gaussian distribution model is effective for describing large-scale diffusion of light hydrocarbons from storage tanks, whereas the FEM3 model is widely used for numerical simulation of localized hydrocarbon diffusion. The issue of quantifying hydrocarbon losses due to evaporation from storage tanks is also addressed in (De Luchi, 1993; Huang et al., 2016; Paulauskiene et al., 2009; Rota et al., 2001).

TABLE 1. Classical models of diffusion-controlled substance evaporation (Zhang et al., 2026).

Classical modelsGoverning equationsApplications and limitations
Gaussianc x , y , z = Q π σ y σ z u a exp − 1 2 y 2 σ y 2 · exp − 1 2 z + H σ z 2Used for calculations of large-scale diffusion from storage tanks
FEM3∂ ρ u ∂ t + ρ u = ∇ ρ μ m ∇ u + ρ − ρ a g ∂ T ∂ t + u ∇ T = 1 ρ C p ∇ ρ C p μ T ∇ T + C p a − C p a C p μ T ∇ T ∇ T a ω t + u ∇ ω = 1 ρ ρ μ ω ∇ ω ρ = p M R T = P R T ω M a + M rUsed for numerical simulation of localized diffusion of light hydrocarbons from storage tanks
BMd R d t = u f = c 1 g ρ c − ρ a H ρ 1 2 M d h c d t = d M a d t h ω − h c + E f π R 2 d M a d t = = ρ a π R 2 u t + ρ a 2 π R H u eUsed for modeling large-scale diffusion without accounting for spatial details
Slabc x , y , z = 2 B h c x c 1 y , b , β c 2 z , Z e , σ c x , y , z , t = 4 B x B y h c t c 1 x − X c , b x , β x c 1 y , b y , β y c 2 z , Z e , σApplied to the diffusion of heavy gases from storage tanks without consideration of spatial details (buildings, terrain)
Suttonc x , y , z = G . ⁡ exp − y 2 σ y 2 x 2 − n π σ y σ z u w [ exp − z − H 2 σ z 2 x 2 − n + exp − z + H 2 σ z 2 · x 2 − nApplicable for the calculation of diffusion of neutral (non-flammable) gases

For investigating the process of diffusion-driven evaporation of a liquid bulk cargo during its loading into a tank of constant cross-section, a first-order approximation based on the differential mass transfer equation is sufficient, as adopted in (Hassanvand et al., 2010; Invernizzi and Sironi, 2021; Tamaddoni et al., 2014). It should be noted that among the above-mentioned studies, only (Galeev et al., 2020) accounted for the Stefan flow correction when solving the differential mass transfer equation. Stefan flow arises in the vapor–gas phase due to concentration gradients and, unlike diffusive flux, represents a convective mechanism of mass transfer (Carle and David, 2015; ÇENGEL and GHAJAR, 2020). Figure 1 shows the concentration distribution profile of n-hexane in a Stefan tube obtained by CFD modeling without accounting for the Stefan flow correction, taken from (Hassanvand et al., 2010), as well as the concentration distribution of an arbitrary gas in dimensionless coordinates with a prescribed diffusion coefficient and including the Stefan flow correction, taken from (Ramírez, 2025).

Comparison of the results of the analytical solution of the differential mass transfer equation for a Stefan tube under steady-state conditions: **(a)** without accounting for Stefan flow; **(b)** with accounting for Stefan flow.
Figure 1. Comparison of the results of the analytical solution of the differential mass transfer equation for a Stefan tube under steady-state conditions: (a) without accounting for Stefan flow; (b) with accounting for Stefan flow.

It should be noted that the Stefan flow correction affects the concave/convex nature of the concentration distribution profile along the height of the Stefan tube under equilibrium conditions. In (Mitrovic, 2012), the authors investigated the mass transfer process of liquid vapors in a Stefan tube while accounting for changes in the liquid level due to evaporation. In (Vlasov, 2020), an analytical solution of the Stefan problem yielded a vapor concentration distribution of n-hexane in the tube exhibiting a convex profile, similar to the distribution reported in (Ramírez, 2025). In (Taylor and Krishna, 1993), the authors presented experimental profiles of the mole fraction of methanol and acetone vapors along the height of the Stefan tube. The profiles from (Vlasov, 2020) and (Taylor and Krishna, 1993) are shown in Figure 2. Thus, the applicability of the Stefan flow correction term in the differential mass transfer equation is supported by experimental evidence.

Profiles of concentration (normalized concentration) of hydrocarbon vapors along the height of the Stefan tube under steady-state conditions: **(a)** variation of the normalized n-hexane concentration for different initial liquid fill heights in the tube; **(b)** variation of the mole fraction of acetone and methanol vapors (symbols denote experimental data, curves represent results of the analytical solution).
Figure 2. Profiles of concentration (normalized concentration) of hydrocarbon vapors along the height of the Stefan tube under steady-state conditions: (a) variation of the normalized n-hexane concentration for different initial liquid fill heights in the tube; (b) variation of the mole fraction of acetone and methanol vapors (symbols denote experimental data, curves represent results of the analytical solution).

This study examines various models of diffusion-controlled evaporation of n-hexane, considered as the primary component of light gasoline fractions. To assess the applicability of the models under consideration, the results of the analytical solutions were validated against experimental data obtained for a Stefan tube. Calculations were performed to quantify the mass losses of n-hexane during its injection into a tank of constant cross-section, and the relationships between product mass transfer rates and the tank filling velocity were determined.

Methods

Numerical modeling

Assumptions of the mass transfer model

The considered gasoline evaporation model is based on the following assumptions.

  • The gasoline evaporation model is based on diffusion-controlled mass transfer in a binary gas mixture (vapor and air); to simplify the calculations, gasoline is represented by one of its volatile components, n-hexane.
  • The model neglects natural convective mass flows induced by temperature gradients within the vapor–gas mixture layers. The dependence of mass losses on temperature is not included, since temperature variations primarily affect the convective component of mass transfer. The operating regime is assumed to be isothermal at constant pressure equal to atmospheric pressure.
  • The gases are inert, and their mixtures are assumed to be ideal.
  • The diffusion coefficient of the gases remains constant over time.
  • The model does not consider gas hydrodynamics equations, including equations describing turbulent mixing of the substance during its injection into the tank.
  • For the analytical solution of the Stefan problem, the liquid–gas interface is assumed to be a stationary planar boundary.

Governing equations of the mass transfer model

As the governing mass transfer equation, a differential partial differential equation describing mass transfer in a binary ideal gas mixture is considered:

∂ c m ∂ τ + w ∇ c m = D e f f ∇ 2 c m + g * (1)

For a one-dimensional problem in the absence of internal mass sources, Equation 1 can be written:

∂ c ∂ τ + w z ∂ c ∂ z = D e f f ∂ 2 c ∂ z 2 (2)

With the Stefan flow correction taken into account (Çengel and Ghajar, 2020; Ramírez, 2025):

∂ c ∂ τ + w z ∂ c ∂ z = D e f f ∂ ∂ z 1 1 − c · ∂ c ∂ z (3)

The density of the gas mixture is determined in accordance with the additivity equation for a given operating temperature (Equation 4):

ρ m i x = ∑ ρ i · χ i = ∑ ρ i · c i μ i ∑ c i μ i (4)

To calculate the evaporation rate at a moving liquid–gas interface, a change of variables must be introduced (Equation 5): where w z – average linear loading rate of the tanker

Z = z − w z t h − w z t (5)

c z , t → c Z , t

The mass transfer coefficient β is calculated using the mass transfer equation:where c s – mass fraction of saturated vapor at the phase boundary

J = β ρ m i x c s − c Z , t Z = 0 (6)

The mass transfer rate from surface J is calculated using the following formula:where d z and d t – time and coordinate steps for the numerical solution of the problem

J = d d t ∫ z = 0 z = h c z , t ρ m i x z , t d z (7)

The total substance losses due to diffusion-controlled evaporation over time t are calculated using the following expression:where (t) is the total tank filling time, determined by the following relation:

M e v p = F ∫ 0 t J d τ (8)

t = h w z (9)

The tank filling coefficient k is calculated using the following (Equation 10):where F – cross-sectional area of the tank,

k = F h V max (10)

h – product fill level,

V max – tank volume

Experimental validation of the modeling results

We compare the solutions of Equations 2, 3, applied to the Stefan problem, with laboratory experimental data for methanol evaporation. The empirical data on diffusion-controlled methanol evaporation in a Stefan tube were taken from (Taylor and Krishna, 1993; Carty and Thomas Schrodt, 1975) and are presented in Table 2. For convenience in plotting the experimental points, the mole fraction of methanol was converted to its mass fraction.

TABLE 2. Experimental data from the Carty and Schrodt (1975) experiment for methanol.

Mole fraction of methanol χ, mol/mol0,5280,4570,460,4210,3790,3070,2430,1680
Vertical distance z, cm04,4579,4511,914,41719,423,8
Mass fraction of methanol c, kg/kg0,5520,4820,4850,4450,4020,3280,2620,1820

For the Stefan problem, the following initial and boundary conditions are adopted (Equation 11):

с z , 0 = 0 c 0 , t = c s c h С , t = 0 (11)

Figure 3 shows the methanol concentration profile in a Stefan tube constructed from experimental and analytical data. To generate the methanol concentration curves using Equations 2, 3, the molecular diffusion coefficient of methanol was taken from reference data (Taylor and Krishna, 1993) as D e f f = 19 , 91 · 10 − 6 m 2 s . The Stefan tube height is h С = 23 , 8 c m . The curves were obtained for a steady-state mass transfer regime at t = 1500 s .

Mass fraction profiles of methanol in the Stefan tube: the red markers indicate experimentally obtained values; the blue curve represents the solution of Equation 2 without the Stefan flow correction; the green curve represents the solution of Equation 3 with the Stefan flow correction.
Figure 3. Mass fraction profiles of methanol in the Stefan tube: the red markers indicate experimentally obtained values; the blue curve represents the solution of Equation 2 without the Stefan flow correction; the green curve represents the solution of Equation 3 with the Stefan flow correction.

As can be seen from Figure 3, the blue curve, corresponding to the solution of the equation without the Stefan flow correction, does not pass through the experimentally obtained data points. Moreover, under equilibrium mass transfer conditions, this curve becomes linear, whereas the fitted curve of the empirical values is a higher-order polynomial and exhibits a characteristic convexity. The discrepancy between the solution that includes the correction term and the experimental values can be attributed to the fact that the experiment was conducted for a binary methanol–acetone mixture, where acetone directly affects the diffusivity of methanol. Nevertheless, the green curve reproduces the observed shape of the vapor concentration distribution along the Stefan tube, indicating that inclusion of the Stefan flow correction term is appropriate for diffusion-controlled evaporation models of volatile liquids.

For validation, concentration values from various layers of the Stefan tube reported by the authors (Taylor and Krishna, 1993) for multicomponent diffusion were used, since there are no data available in the scientific literature on concentration distribution in a Stefan tube for binary diffusion. For example, in reference (Beverley et al., 1999), the authors present experimental data on the diffusion evaporation of n-hexane and n-heptane in a Stefan tube; however, they do not provide information on the concentration distribution of the hydrocarbons at different heights within the tube (only the distribution of evaporation rates). Source (Hirschfelder et al., 1964) contains information on the diffusion coefficients of various hydrocarbons for binary gas mixtures; however, the book also lacks data on the distribution of hydrocarbon vapor concentrations across the layers of the Stefan tube. Thus, data from source (Taylor and Krishna, 1993) were selected for the evaluation of the mathematical model, despite the multicomponent nature of the gas mixture.

To quantitatively assess how well the models under consideration align with the experimental data, it is necessary to use various error metrics: RMSE, MAE, and relative error.

The RMSE metric is calculated as follows:where c i – calculated concentration at the ith point

R M S E = 1 n ∑ i = 1 n c i − c i e 2 (12)

c i e – the experimental concentration value at the ith point

n – number of evaluation points

The MAE metric is calculated using the following formula:

M A E = 1 n ∑ i = 1 n c i − c i e (13)

The relative error is calculated as follows:

R e l a t i v e E r r o r i = c i − c i e c i e · 100 % (14)

Based on Equations 12–14, the data in Table 2, and the results of solving Equations 2, 3 while taking into account the boundary conditions (11), error metrics were calculated for the two equations; the results of these calculations are presented in Table 3.

TABLE 3. Error metrics for solutions to Stefan’s problem.

Without accounting for the Stefan flux correction
Taking into account the Stefan flux correction
z i , c mc i , k g k gc i e , k g k gc i − c i ec i − c i e 2R e l a t i v e E r r o r i , %
00,5520,5520,0000,00000,00
4,450,4490,4820,0330,00116,89
70,3900,4850,0950,009119,66
9,450,3330,4450,1120,012625,21
11,90,2760,4020,1260,015931,34
14,40,2180,3280,1100,012133,53
170,1580,2620,1040,010939,80
19,40,1020,1820,0800,006443,93
23,80,00000,0000,00000,00
n = 9Sum = 0,661Sum = 0,068​
MAE = 0,073RMSE = 0,087
z i , c mc i , k g k gc i e , k g k gc i − c i ec i − c i e 2R e l a t i v e E r r o r i , %
00,5520,5520,0000,00000,00
4,450,4800,4820,0020,00000,41
70,4340,4850,0510,002610,52
9,450,3860,4450,0590,003513,26
11,90,3310,4020,0710,005017,66
14,40,2720,3280,0560,003117,07
170,2060,2620,0560,003121,37
19,40,1370,1820,0450,002024,73
23,80,00000,0000,00000,00
n = 9Sum = 0,34Sum = 0,019​
MAE = 0,038RMSE = 0,046

Table 3 shows that for solution (2) (without Stefan flow correction), the error metrics are significantly higher than for solution (3) (with Stefan flow correction). For the solution without the correction, the relative error reaches 44%, whereas the solution with the correction has a maximum relative deviation of 25%. The latter deviation is notable, which can be attributed to the fact that the experimental data correspond to a multicomponent gas mixture, while the model is designed for a binary mixture.

The MAE and RMSE metrics for the solution without the Stefan flow correction are approximately twice as high as those for the solution that accounts for the correction (MAE = 0.073 vs. 0.038; RMSE = 0.087 vs. 0.046). Although the root mean square error of the simulation results relative to the experimental values is still relatively large (RMSE = 0.046), given the assumptions outlined above, it is reasonable to consider the model that accounts for the Stefan flow correction to be valid for describing real diffusion processes and more accurate compared to the classical diffusion model.

This finding aligns with the broader methodological principle that model refinement guided by experimental validation significantly improves predictive accuracy, as recently demonstrated by Gutiérrez-González et al. in the context of breakthrough curve calculations for adsorption processes (Gutiérrez-González et al., 2025).

Problem formulation

The diffusion-controlled evaporation equations under investigation are used to quantify product mass losses during loading into an oil tanker, subject to the modeling assumptions described above. The oil tanker is represented by a tank with a rectangular cross-section and overall dimensions of 200 × 50 × 12 m. The tank filling coefficient is k = 0 , 95. One of the principal components of light gasoline fractions, n-hexane, is selected as the working substance.

The loading regime is assumed to be isothermal, with an operating temperature of T = 27 ℃ (T = 300 , 15 Κ. According to reference data, the molecular diffusion coefficient of n-hexane in air at the operating temperature is D e f f = 7 , 67 · 10 − 6 m 2 s , the kinematic viscosity of n-hexane at operating temperature is taken to be ν = 7 , 64 · 10 − 6 m 2 s . The tank filling rate is constant, and filling is assumed to be uniform, without turbulent mixing.

The model does not account for a number of physical factors present under actual tanker loading conditions (natural convection, heat transfer, and others), since a comprehensive approach that simultaneously accounts for all factors is unstable and unreliable. The main objective of this article is to investigate the diffusion-convection equation as fundamental to the evaporation process from tankers. The results of this work will serve as a basis for further refinement and adjustments to the model, bringing it closer to describing more realistic processes. A similar approach to studying the problem of evaporation from tankers based on the diffusion-convection equation has been implemented in the VocSim project (Invernizzi and Sironi, 2021).

The proposed model simulates the process of simultaneous filling of all tank compartments of the oil tanker with the product, followed by its diffusion-controlled evaporation. The following initial and boundary conditions were adopted for the problem (Equation 15):

с z , 0 = 0 c w z t , t = c s ∂ c h , t ∂ z = 0 (15)

The first boundary condition c w z t , t = c s assumes thermodynamic equilibrium at the liquid–gas interface, i.e., the vapor phase is instantaneously saturated at the free surface. This condition is widely adopted in diffusion-controlled evaporation problems and implies that the evaporation rate is governed solely by mass transfer in the gas phase.

The second boundary condition ∂ c h , t ∂ z = 0 imposes a zero-flux condition at the top of the tank. Physically, this corresponds to a closed vapor space, where no vapor escapes through the upper boundary. In the context of tanker loading, this condition represents a scenario in which the vapor recovery system is not actively drawing gas from the tank, or the tank is isolated from the atmosphere (e.g., during the pressure-build-up stage before vapor return is initiated). Alternatively, it can be interpreted as a symmetry condition when the model describes only a part of the vapor space. The adopted formulation thus provides a baseline case that captures the accumulation of vapors within a sealed or temporarily closed tank. For situations with forced ventilation or active vapor extraction, the boundary condition could be modified to account for convective outflow; such extensions are beyond the scope of the present study, which focuses on the fundamental diffusion-convection mechanism.

To solve the moving “liquid–gas” interface problem, a Lagrangian coordinate transformation was introduced by applying a change of variables:

Z = z − w z t h − w z t (16)

Accordingly, the boundary conditions should be written in the following form (Equation 17):

с Z , 0 = 0 c 0 , t = c s ∂ c 1 , t ∂ z = 0 (17)

Solver settings

The formulated problem was solved numerically using Maple. The pdsolve (numeric) command was employed to solve differential Equations 2, 3 with boundary conditions (17). By default, the command implements the theta method with a second-order accurate centered implicit scheme. The dchange (forward transformation) and transform (inverse transformation) commands were used for coordinate transformation according to Equation 16. The getdata function was utilized to extract concentration profiles at different time instants. Based on the extracted data, the mass transfer rate from the surface was calculated using Equation 7, and the total evaporated mass was obtained by numerical integration according to Equation 8.

To ensure the reliability of the numerical results, a systematic verification of grid and time-step independence was performed. The most demanding case (lowest filling velocity, wz = 1 × 10−4 m/s) was used for testing both diffusion models (with and without Stefan flow correction). The following discretization parameters were examined: spatial step sizes Δz = 0.02 m, 0.01 m, and 0.005 m with a fixed time step Δt = 1 s, and time step sizes Δt = 2 s, 1 s and 0.5 s with a fixed spatial step Δz = 0.01 m. The key integral quantities (total evaporated mass at the end of filling (from Equation 8) and the time to reach 95% saturation in the top layer) were monitored.

Reducing the spatial step from 0.02 m to 0.01 m altered the total evaporated mass by less than 0.8%, and further reduction to 0.005 m changed it by less than 0.3%. Similarly, reducing the time step from 2 s to 1 s resulted in a change of approximately 1.1% in the total evaporated mass, while a further reduction to 0.5 s changed it by less than 0.4%. The saturation times exhibited analogous behavior, with variations well below 1.5% for the finest resolutions. These results confirm that the solution converges and that the chosen discretization parameters (Δz = 0.01 m, Δt = 1 s) provide a grid-independent solution with a favorable balance between accuracy and computational cost. All simulations presented in this work were performed with these parameters. The solver tolerance was set to abstol = 0.001; the solver automatically adjusts the time step when the local error estimate exceeds this tolerance, ensuring that the prescribed accuracy is maintained throughout the simulation.

Results and discussion

Figure 4 presents a three-dimensional distribution of the mass fraction of n-hexane vapor in the tank during filling at a rate of w z = 1 · 10 − 4 m s , obtained from Equation 2. Figure 5 shows the temporal evolution of the n-hexane vapor mass fraction for the layer at a height of h = 11 , 4 m. The dependencies were constructed for different tank loading rates and for both mass transfer equations.

Surface plot of the mass fraction of n-hexane in its vapor–air mixture during product loading into the tank at a rate of w z = 1 · 10 − 4 м с.
Figure 4. Surface plot of the mass fraction of n-hexane in its vapor–air mixture during product loading into the tank at a rate of w z = 1 · 10 − 4 м с.
Temporal evolution of the mass fraction of n-hexane vapors for the layer at a height of h = 11 , 4 m: **(a)** based on the numerical solution of Equation 2 without the Stefan flow correction; **(b)** based on the numerical solution of Equation 3 with the Stefan flow correction; for the following filling velocities.
Figure 5. Temporal evolution of the mass fraction of n-hexane vapors for the layer at a height of h = 11 , 4 m: (a) based on the numerical solution of Equation 2 without the Stefan flow correction; (b) based on the numerical solution of Equation 3 with the Stefan flow correction; for the following filling velocities.

The time histories of the n-hexane vapor mass fraction indicate that increasing the product loading rate affects the saturation of the tank with product vapors in a nonlinear manner. Specifically, when the filling velocity is increased from w z = 1 · 10 − 4 m s to w z = 4 · 10 − 4 m s, the saturation time of the boundary layer decreases by nearly a factor of nine. This nonlinear influence of the injection rate on the tank vapor saturation rate can be attributed to the relatively slow diffusion-controlled evaporation process compared with the convective component of mass transfer.

Figure 6 also presents the temporal evolution of the n-hexane vapor mass fraction obtained by solving the equation including the Stefan flow correction. For w z = 1 · 10 − 4 m s, the time required to saturate the uppermost layer is Δ t 1 = 16 , 4 h. For the equation without the correction, the corresponding time is Δ t 2 = 12 h, which corresponds to a relative difference of 27%. As the injection rate increases, the relative difference between the predicted saturation times decreases nonlinearly. Therefore, Stefan flow exerts a stronger influence on diffusion-controlled evaporation at low tank filling velocities.

Temporal variation of the mass transfer rate from the n-hexane free surface during tank filling.
Figure 6. Temporal variation of the mass transfer rate from the n-hexane free surface during tank filling.

Figure 6 shows the mass transfer rate from the product free surface for different tank filling velocities. Solid lines correspond to the results obtained from Equation 2, while dashed lines represent the results from Equation 3. The presented curves exhibit a characteristic extremum, the magnitude of which increases with increasing product injection velocity. The dependencies obtained using the equation that includes the Stefan flow correction demonstrate a higher extremum for each filling velocity. The plots indicate that inclusion of the Stefan flow correction increases the duration of mass transfer for all injection rates and closely corresponds, with a small discrepancy, to the tank saturation time shown in Figure 5.

Figure 7 illustrates the temporal evolution of the mass of n-hexane vapors in the tank vapor–gas space. Solid curves were obtained using Equation 2, whereas dashed curves correspond to Equation 3. The solid curves increase monotonically to a total evaporated mass of approximately M ≈ 96 t o n s over the tank filling period, with only minor discrepancies (less than 2%). In contrast, the dashed curves yield different values of the evaporated mass for each injection velocity.

Temporal variation of the mass of n-hexane vapors during tank filling.
Figure 7. Temporal variation of the mass of n-hexane vapors during tank filling.

The influence of the tank filling rate on the temporal evolution of vapor mass predicted by Equation 3 is non-monotonic: for low injection velocities, the total vapor mass in the tank increases when the Stefan flow correction is included, whereas for higher velocities it decreases. This behavior reflects the combined effect of the Stefan flow correction and the varying filling velocity on the diffusive-convective mass transfer. The average deviation in the total n-hexane vapor mass at the time of complete tank filling between the results obtained from Equations 2, 3 is 5.8%. As confirmed by the grid- and time-step independence tests reported in the Solver Settings section, the observed differences are not caused by numerical artifacts; the discretization parameters ensure that the numerical errors are well below the magnitude of the physical effects under study.

W z = 4 · 10 − 4 m s − b l u e c u r v e ;

W z = 2 · 10 − 4 m s − p u r p l e c u r v e ;

W z = 1 · 10 − 4 m s − o r a n g e c u r v e .

All numerical results presented above were obtained with the verified discretization parameters (Δz = 0.01 m, Δt = 1 s). Additional tests confirmed that further refinement of the spatial or temporal grid changes the integral quantities (total evaporated mass, saturation time) by less than 1.5%, ensuring that the observed trends are not affected by numerical artifacts.

Analysis of similarity criteria

The results obtained for the evaporation of gasoline (n-hexane) from a tanker during filling must be presented in dimensionless form. To do this, it is necessary to establish the relationships between the characteristic similarity parameters for the diffusion-convection equation.

A characteristic similarity criterion for mass transfer problems is the Sherwood number (Sh), which is determined by the (Equation 18):where β – mass transfer coefficient,

S h = β · X D e f f (18)

X – a characteristic linear dimension, taken to be equal to the product fill height h

It is known from similarity theory that, for diffusion-convection processes, the Sherwood criterion is, under certain geometric and physical conditions, a function of the Reynolds (R e and Schmidt (S c numbers:where A , m a n d n – constants obtained through statistical data analysis

S h = A · R e m · S c n (19)

The Reynolds number is calculated using the (Equation 20):where ν – kinematic viscosity of n-hexane at operating temperature

R e = W z · X ν (20)

The Schmidt criterion is calculated using the (Equation 21):

S c = ν D e f f (21)

It is known from the literature (ÇENGEL and GHAJAR, 2020), that for gases, the Schmidt number is approximately equal to one S c ≈ 1 .

It is necessary to establish similar relationships for similarity criteria in models of diffusion evaporation from a tanker during filling, both with and without the Stefan flow correction. To find the coefficients m and n, it is necessary to first calculate the average values of the mass transfer coefficients using Equation 6 for various tanker filling rates and for various diffusion coefficients.

By taking the logarithm of Equation 19, we obtain the following form:

l g S h = lg A + m · l g R e + n · l g S c (22)

Thus, to determine the coefficients m and n, it is necessary to calculate the tangents of the slopes of the linear regression curves. The value of the constant A is determined using Equation 22 with the established values of m and n for each individual filling rate and diffusion coefficient. The coefficient A is calculated as the average of the values in the resulting matrix. The results of calculating the similarity criteria for determining the constants are presented in Table 4. Figure 8 shows graphs of the Sherwood number as a function of the Reynolds and Schmidt numbers in logarithmic coordinates for diffusion evaporation processes from a tanker during loading, both without and with the Stefan flow correction.

TABLE 4. Similarity criteria values.

Without accounting for the Stefan flux correction
Taking into account the Stefan flux correction
W z , m sD e f f , m m 2 sβ D = c o n s tlg S h D = c o n s tβ W = c o n s tlg S h w = c o n s tlg R elg S c
1 · 10 − 47,671 , 50 · 10 − 51,3711 , 50 · 10 − 51,3712,1960,002
2 · 10 − 492 , 44 · 10 − 51,5821 , 63 · 10 − 51,4072,4970,071
4 · 10 − 4125 , 16 · 10 − 51,9071 , 90 · 10 − 51,4742,7980,196
8 · 10 − 4158 , 82 · 10 − 52,1402 , 11 · 10 − 51,5193,0990,293
1 · 10 − 3181 , 01 · 10 − 42,1962 , 33 · 10 − 51,5613,1960,372
W z , m sD e f f , m m 2 sβ D = c o n s tlg S h D = c o n s tβ W = c o n s tlg S h w = c o n s tlg R elg S c
1 · 10 − 47,671 , 31 · 10 − 51,3131 , 31 · 10 − 51,3132,1960,002
2 · 10 − 492 , 25 · 10 − 51,5471 , 42 · 10 − 51,3462,4970,071
4 · 10 − 4124 , 79 · 10 − 51,8741 , 53 · 10 − 51,3792,7980,196
8 · 10 − 4157 , 32 · 10 − 52,0591 , 68 · 10 − 51,4203,0990,293
1 · 10 − 3188 , 73 · 10 − 52,1351 , 81 · 10 − 51,4523,1960,372
Logarithmic relationship between the Sherwood criterion and the Reynolds and Schmidt criteria for the diffusion evaporation of VOCs from a tanker during loading: *blue line*–regression for diffusion without Stefan flux correction *green line*–regression for diffusion with Stefan flux correction.
Figure 8. Logarithmic relationship between the Sherwood criterion and the Reynolds and Schmidt criteria for the diffusion evaporation of VOCs from a tanker during loading: blue line–regression for diffusion without Stefan flux correction green line–regression for diffusion with Stefan flux correction.

When plotting regression curves based on the data in Table 4, we obtain the following constant values:

– diffusion without Stefan correction – m = 0,85; n = 0,51; A = 0,61

– diffusion with Stefan correction – m = 0,83; n = 0,36; A = 0,605

Thus, the relationship between the similarity criteria for the process of diffusion evaporation of gasoline (n-hexane) from a tanker during filling is as follows.

  • – for diffusion without Stefan correction:

S h = 0 , 61 · R e 0 , 85 · S c 0 , 51

  • – for diffusion with Stefan correction:

S h = 0 , 605 · R e 0 , 83 · S c 0 , 36

Conclusion

This study examined various models of diffusion-controlled evaporation of liquid bulk cargoes and based on the reviewed literature, the most suitable models were selected to describe product evaporation during loading into a tank of constant cross-section. n-hexane, a principal component of light gasoline fractions, was considered as a representative bulk cargo. The rationale for using a model that incorporates the Stefan flow correction was substantiated through its experimental validation for methanol evaporation in a Stefan tube. A validation analysis revealed that the Root Mean Square Error (RMSE) for a solution that does not account for the Stefan correction is twice as high as that for a solution that does account for the correction, suggesting that the latter is preferable for describing real-world diffusion processes.

Based on the performed numerical simulations, surface plots of product vapor concentration in the vapor–gas mixture were constructed for different tank filling velocities. The modeling results demonstrated that Stefan flow slows down the diffusion-controlled evaporation process: for relatively low filling velocities, inclusion of the correction term increases the total tank filling time by 27%. In addition, temporal dependencies of the product mass transfer rate from the free surface were obtained. The resulting curves exhibit a characteristic extremum, the magnitude of which increases with increasing product injection velocity. Inclusion of the Stefan flow correction reduces the average mass transfer rate from the surface while increasing the overall tank filling time.

To evaluate the total evaporation of the bulk cargo over the filling period, temporal profiles of the vapor mass were constructed. It was found that the Stefan flow correction exerts a non-uniform influence on the evolution of product vapor mass in the vapor–gas space. The average deviation in the n-hexane vapor mass obtained at the time of complete tank filling, when the correction term was included, amounted to 5.8% for different filling velocities. Thus, deviations in estimated mass losses caused by incorporating the correction term in the diffusion equation are significant in absolute terms and can reach several tens of tons of evaporated substance, depending on the tank filling velocity and its geometric parameters. Moreover, accounting for Stefan flow in diffusion-controlled evaporation models has a substantial effect on the rate of tank saturation with volatile vapors, and the difference in saturation time, depending on the filling velocity, may reach several hours.

An analysis was conducted of the process of gasoline diffusion evaporation from a tanker during filling, expressed in dimensionless similarity numbers. Based on mass transfer data obtained for various tank filling rates and diffusion coefficients, the Reynolds, Schmidt, and Sherwood criteria were calculated, and the relationship between them was established. It was found that the Stefan flow correction has a greater influence on the relationship between the Sherwood and Schmidt criteria than between the first and the Reynolds criterion. Based on the obtained relationships, linear regressions were plotted in logarithmic coordinates.

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Footnotes

Edited by: Micaela Demichela, Polytechnic University of Turin, Italy

Reviewed by: Nidhal Ben Khedher, University of Hail, Saudi Arabia

Daniel Gutiérrez-González, Universidad Veracruzana, Mexico

Footnotes

c m, mass concentration of the substance, k g m 3; c, mass fraction of the substance in the mixture, k g k g; ρ m i x − density of the mixture, k g m 3; χ − volume fraction of the substance, m 3 m 3; μ − molar mass of the substance, k g k m o l; w z, tank filling velocity, m s; t, tank filling time, s; F −, cross-sectional area of the tank, m 2; h −, product fill height, m; v e v p −, evaporation rate of the substance, m s; M e v p −, mass of the evaporated substance, k g; D e f f, molecular diffusion coefficient, m 2 s; ν − kinematic viscosity , m 2 s; g *, intensity of internal substance sources, k g m 3 · s; β, mass transfer coefficient, m s; J – mass transfer rate from surface, k g m 2 · s; S h, Sherwood similarity criterion; R e, Reynolds similarity criterion; S с, Schmidt similarity criterion.

Statements

Conflict of interest

The author(s) declared that this work was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

Funding

The author(s) declared that financial support was not received for this work and/or its publication.

Data availability

The original contributions presented in the study are included in the article/supplementary material, further inquiries can be directed to the corresponding author.

Author contributions

VG: Writing – original draft. VP: Writing – review and editing. ED: Conceptualization, Writing – original draft. DS: Writing – original draft, Investigation.

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Citation

Gustov, V., et al. (2026). Comparative assessment of diffusion modeling methods in evaporation problems of liquid bulk cargoes during tanker loading. Frontiers in Chemical Engineering, Volume 8. https://doi.org/
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Copyright © 2026 Gustov, Pshenin, Dzhemilev and Sakal.

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*Correspondence: V. Gustov

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