Numerical Simulation and Analysis of the Heat and Mass Transfer of Oil-Based Drill Cuttings in a Thermal Desorption Chamber

Abstract

To reveal the coupled heat and mass transfer process during thermal desorption of oil-based drill cuttings, a numerical model of heat and mass transfer was established, which is divided into four components = evaporation. The C language programming catch-up method was used to solve the discrete equation, and the interactive effects of the oil-based drill cuttings’ particle size, water content, oil content, content of light and heavy components in oil, heating temperature, heating time, and other operating parameters on the mass and heat transfer of the oil-based cutting particles were investigated. Results showed that the time at which the center point temperature of oil-based drill cuttings with radius of 1 mm and 5 mm reached 600 °C was 441 s and 63 s, respectively, and the temperature difference between the center and the surface increased with particle size. The desorption process of water, light components, and heavy components was not completed individually but mixed alternately. The duration of each desorption component was closely related to the temperature at the center of the oil-based drill cuttings. The smaller the particle size was, the lower the water and oil contents were, and the higher the heating temperature and time were. These conditions were beneficial to thermal desorption, mass transfer, and heat transfer.

1. Introduction

Oil-based drill cuttings are oil-bearing wastes produced by oil-based drilling fluid in the process of oil and natural gas exploitation. In China, they are listed under the HW08 waste mineral oil category in the National Hazardous Waste List because of their toxic characteristics [1,2]. They require harmless treatment and resource utilization. Thermal desorption is a harmless physical treatment method for solid–liquid separation via heating. When used to dispose oil-based drill cuttings, it has the advantages of recoverable oil resources and high efficiency [3,4]. It has been widely used in industrialization and has attracted increasing scholarly attention.

Many studies have been conducted on thermal desorption technology for oil-based drill cuttings [5,6,7]. This technology includes microwave thermal desorption [8], the thermal desorption mechanism [9], standard treatment of residue oil content [10], oil recovery and influencing factors of components [11], and oil-based drill cuttings and recycling of treated residues [12,13]. However, only a few studies have focused on thermal desorption for the heat and mass transfer of oil-based drill cuttings.

On the basis of thermogravimetric or thermal desorption experiments, the kinetic parameters and thermodynamic characteristics of different research objects can be obtained through reaction kinetics experiments and numerical simulation [14,15,16]. Wu et al. [17] and Park [18] studied the pyrolysis of oily sludge from different sources by using the mode function method and solved the kinetic parameters of the reaction process. Liu et al. [16] found that the thermal desorption process of oil-based drill cuttings at low temperatures follows nonlinear, minimum, second-order exponential dynamics (R² > 0.9). Chiang et al. [19] discovered that the pyrolysis process of oil-containing sludge in a refinery is of 2.5 reaction order and 11.06 kJ/mol activation energy. Choudhury et al. [20] studied the nonlinear pyrolysis kinetics of oily sludge in an oil refinery and showed that the pyrolysis reaction of oily sludge is suitable for the second-order reaction model. Shie et al. [21] calculated the first-order, second-order, and third-order reaction kinetic parameters and determined the optimal model via numerical simulation. Li et al. [22] found that the exponential decay model can effectively fit the dynamics, and the thermal desorption equilibrium is consistent with the Langmuir isothermal model. They also found that diesel organic matter (C₁₀–C₂₈) is mainly controlled by physical adsorption.

Previous studies have shown that the particle size [23], water content, and other parameters [24] of materials exert a considerable influence on the thermal desorption heat transfer process, which is similar to the thermal desorption heat transfer process of oil-based drill cuttings. However, oil-based drill cutting particles in hot stripping cavities involve a process of momentum, mass, and energy coupling [25,26,27], and the current understanding of the convection heat transfer characteristics remains insufficient [28,29]. Moreover, information on oil-based drill cuttings’ moisture and oil contents, physical and chemical properties (e.g., particle size, heating temperature, and heating time), and hot stripping operation parameters (e.g., heat transfer process in the interaction and the influence law) is lacking [30]. As a result, the energy consumption in the process of thermal desorption of oil-based drill cuttings is relatively high, which restricts further development of this technology.

Therefore, on the basis of experiments and numerical analysis of convection heat and mass transfer, a numerical model is established in this study. The convection heat transfer is examined under the influence of the particle size, moisture and oil contents, and composition of diesel oil in terms of heating time, heating temperature, and influence of factors on particle heat transfer and regular features. A hot stripping device is also developed to serve as a reference in designing and applying operation parameters.

2. Experimental and Numerical Simulation Methods

2.1. Experimental Instruments and Methods

Oil-based drill cuttings were obtained from oil-based drilling fluid prepared with No. 0 diesel oil, particle radius 1–5 mm, which contains (mass percentage content): SiO₂ 55%, CaO 10%, Al₂O₃ 15%, and the rest is SO₃, MgO, Fe₂O₃, K₂O, Na₂O, etc. A Netronics STA449F3 synchronous integrated thermal analyzer (DSC/DTA-TG) was used in the experiment. In accordance with the device performance and thermal desorption process characteristics, the mass of the oil-based drill cuttings was 10–15 mg, the water content was 6.94%, the oil content was 9.65% (the rest was solid), and the temperature range was controlled from 35 °C to 800 °C. Inert nitrogen gas was used, and the flow rate could be adjusted. A TG-DSC thermal analyzer was employed to measure the sample mass, weight loss rate, and the corresponding changes in heat with temperature in the thermal desorption experiment [31,32], and a set of data was recorded every 1 s.

2.2. Reaction Rates of the Thermogravimetric Test and Thermal Desorption at Various Stages

2.2.1. TG/DTG Curves

The TG/DTG curves of the oil-based drill cuttings at a heating rate of 5 °C/min (Figure 1) show that the weight-loss process of the oil-based drill cuttings could be divided into three stages. The desorption stage of the light-carbon component (light component) of 0# diesel oil in the oil-based drill cuttings occurred from room temperature to 350.48 °C, and the weight loss rate was 11.35%. The desorption process of the heavy-carbon component (referred to as the heavy component) of diesel oil in the oil-based drill cuttings occurred from 350.48 °C to 559.2 °C, with a weight loss rate of 2.23% and a small amount of drilling fluid treatment agent decomposition. The decomposition of organic matter and carbonate in formation rocks occurred from 559.2 °C to 771.69 °C [33], with a weight loss rate of 6.75%. In accordance with the DSC instrument, the internal reaction heat of the water evaporation and light component desorption stage was −361.04 kJ/kg, and the internal reaction heat of the heavy component was −70.94 kJ/kg, where the minus sign indicates endothermic.

2.2.2. Kinetic Analysis

According to basic kinetic theory and the Arrhenius law, considering that the reaction mechanism function f(α) is related to the reaction rate and number of classes in which the reaction occurs, the basic kinetic equation of thermal desorption reaction is as follows:

$$\frac{da}{dt}=kf\left(\alpha \right)=A{e}^{\frac{-E}{RT}}{(1-\alpha )}^n$$

(1)

where α is the conversion rate of thermal desorption of oil-based drill cuttings (%), t is the reaction time (s), k is the rate constant (s⁻¹), A is the pre-exponential factor (s⁻¹), E is the activation energy (kJ/mol), R is the gas constant (kJ/(mol·K)), and T is the reaction temperature (K).

The thermal desorption transformation rate can be expressed by the change in the mass of oil-based drill cuttings. It is the difference between the mass of oil-based drill cuttings and the initial mass at a certain moment divided by the difference between the mass of the residue and the initial mass after the thermal desorption is complete, as shown in the Equation (2):

$$\alpha =\frac{m_t-m_0}{m_{\infty }-m_0}$$

(2)

where m₀ is the initial weight (g) of the oil-based drill cutting sample, m_t is the weight (g) of the oil-based drill cutting sample at time T in the process of thermal desorption, and m_∞ is the weight (g) of the residue after the thermal desorption reaction.

Under non-isothermal conditions, the temperature dT at a certain time is inversely proportional to heating rate β, and Equation (1) can be changed as follows:

$$\frac{da}{{(1-\alpha )}^n}=\frac{A}{\beta }{e}^{\frac{-E}{RT}}dT$$

(3)

The Coats–Redfern method combined with the KAS method was used to integrate the variables of Equation (3), that is, α between the initial reaction 0 and a certain time α and T between the initial temperature T₀ and a certain time T. The equations were obtained after taking logarithms of both sides of the equation as follows:

$$\left\{\begin{array}{c}\mathit{ln}\left[\frac{-\mathit{ln}\left(1-\alpha \right)}{T^2}\right]=\mathit{ln}\left[\frac{AR}{\beta E}\left(1-\frac{2RT}{E}\right)\right]-\frac{E}{RT}n=1\\ \mathit{ln}\left[\frac{{\left(1-\alpha \right)}^{\left(1-n\right)}}{-T^2(1-n)}\right]=\mathit{ln}\left[\frac{AR}{\beta E}\left(1-\frac{2RT}{E}\right)\right]-\frac{E}{RT}n\ne 1\end{array}\right.$$

(4)

where the R value is 8.314 × 10⁻³ kJ/(mol∙K). The oil-based rock fragment thermal desorption heating temperature is 600 °C, namely, 873.15 K, and the free energy E value is far greater than the product of R and T, i.e., E $\gg$ RT. The $\frac{2RT}{E}$ limit value approaches 0, namely, $\underset{E\gg \mathit{RT}}{\lim }\frac{2RT}{E}\to 0$, which is further simplified by Equation (5).

$$\left\{\begin{array}{c}\mathit{ln}\left[\frac{-\mathit{ln}\left(1-\alpha \right)}{T^2}\right]=\mathit{ln}\frac{\mathit{AR}}{\beta E}-\frac{E}{R}\frac{1}{T}n=1\\ \mathit{ln}\left[\frac{{\left(1-\alpha \right)}^{\left(1-n\right)}}{-T^2(1-n)}\right]=\mathit{ln}\frac{AR}{\beta E}-\frac{E}{R}\frac{1}{T}n\ne 1\end{array}\right.$$

(5)

Whether the reaction index is 1 or n ≠ 1, the reaction functions $\ln \left[\frac{-\ln \left(1-\alpha \right)}{T^2}\right]$ and $\ln \left[\frac{{\left(1-\alpha \right)}^{\left(1-n\right)}}{-T^2(1-n)}\right]$ are linearly correlated with time 1/T. The activation energy E and pre-exponential factor A of the thermal desorption of oil-based drill cuttings can be calculated with the linear fitting method according to different reaction indices.

The thermal desorption of oil-based drill cuttings approximately conforms to the first-order reaction law [34]. In accordance with TG experimental data, the fitting equation can be obtained through linear fitting of kinetics experimental curves in different reaction processes of oil-based drill cuttings’ thermal desorption. The results are shown in

Table 1. Fitting results of dynamic parameters in different thermal desorption processes of oil-based drill cuttings.

Temperature (°C)	Fitting Equation	E (kJ/mol)	A (s−1)	R2
25 °C–176.78 °C	y = −0.22877x − 9.41986	19.02	1.23 × 104	0.9880
176.78 °C–350.48 °C	y = 0.41108x − 13.00847	34.18	4.46 × 105	0.9736
350.48 °C–600 °C	y = 0.5939x − 13.5831	49.38	7.93 × 105	0.9871

.

In Table 1, y is the reaction function $\ln \left[\frac{-\ln \left(1-\alpha \right)}{T^2}\right]$, and x is the temperature function 1/T. Therefore, the reaction rate constants of the water evaporation stage, light component desorption, and heavy component desorption can be calculated.

2.3. Establishment and Solution of the Heat and Mass Transfer Model

The thermal desorption process is prone to polymerization and cyclization reactions that produce polycyclic aromatic compounds due to the high temperature [35], but little difference exists between the recovered oil and the base oil after thermal desorption. Thus, the removal of base oil and pollutants through the thermal desorption of oil-based drill cuttings mainly involves physical reaction processes, such as distillation and desorption, and a few chemical reaction processes. According to the reaction characteristics of oil-based drill cutting thermal desorption combined with the kinetic reaction process, the thermal desorption process can be simplified into four parallel reaction stages: water evaporation, light component desorption, heavy component desorption (ignoring the decomposition and desorption of a small amount of the treating agent at high temperatures), and solid residue heating [36]. As shown in Figure 2, water evaporates as water vapor, the diesel component undergoes desorption as oil vapor and non-condensing gas, and the residue does not undergo any chemical reaction and only heats up.

In Figure 2, ρ_ODC, ρ_w, ρ_gw, ρ₁, ρ₂, ρ_g1, ρ_g2, and ρ_R are the density of oil-based drill cuttings, water, water vapor, light component, heavy component, light component steam, heavy component steam, and solid residue (g/cm³), respectively; meanwhile, k_w, k₁, and k₂ are the reaction rate constants of water vapor, light component steam, and heavy component steam (s⁻¹), respectively.

The following assumptions are made for the convective heat and mass transfer process of oil-based drill cuttings. (1) The desorption components and temperature in oil-based drill cuttings are evenly distributed before the reaction begins. (2) The oil-based drill cutting particles are assumed to be spherical, and the heat and mass transfer of the particles are one-dimensional along the diameter of the circle during heating, that is, the sum of the temperature components in the other directions (except in the r vector radius direction) is 0, and the states and changes in the points on the circumference of the same diameter of the circle are the same. (3) After the thermal desorption gas is produced, it is immediately released outward from the particle and separated without any other reaction. (4) The volume of the ball remains constant during the reaction. (5) The specific heat of each component is constant, and the density and thermal conductivity change with the reaction process.

2.3.1. Establishment of Basic Equations of Energy and Mass in Heat Transfer

(1)

Energy equation

In accordance with the energy differential equation and diffusion coefficient formula, the expression of the relationship among density, specific heat capacity at constant pressure, and thermal conductivity coefficient in the capacity differential equation can be obtained as follows:

$$\rho C\frac{\partial T}{\partial t}=\lambda \frac{1}{r^2}\frac{\partial }{\partial r}\left(r^2\frac{\partial T}{\partial r}\right)+S$$

(6)

where S is internal heat (kJ); r is the vector radius (position of the sphere reaction layer, the wall thickness is not taken into account); λ is thermal conductivity (W/(m·°C)); ρ is density (g/cm³); C is specific heat capacity (J/(kg·°C)); t is time, and the initial moment of time indicated is from 0.

In accordance with the conservation of energy, the product of the density and specific heat capacity of oil-based drill cuttings at constant pressure is equal to the product of the density and specific heat capacity of each component in oil-based drill cuttings as follows:

$${\rho }_{ODC}{C}_{ODC}={\rho }_w{C}_w+{\rho }_1{C}_1+{\rho }_2{C}_2+{\rho }_s{C}_s$$

(7)

where ${{\rho }_{ODC},\rho }_w,{\rho }_1,{\rho }_2, \mathrm{and} {\rho }_s$ are the density of oil-based drill cuttings, water, light component, heavy component, and solid residue (g/cm³), respectively; $C_{ODC},C_w,C_1{{,C}_2, \mathrm{and} C}_s$ are the specific heat of oil-based drill cuttings, water, light component, heavy component, and solid residue (kJ/(kg·°C)), respectively.

After substituting Equation (7) into Equation (6), the energy differential equation of thermal desorption containing the relationship between various components in oil-based drill cutting particles can be obtained as follows:

$$\frac{\partial T}{\partial t}({\rho }_w{C}_w+{\rho }_1{C}_1+{\rho }_2{C}_2+{\rho }_s{C}_s)=\lambda \frac{1}{r^2}\frac{\partial }{\partial r}\left(r^2\frac{\partial T}{\partial r}\right)+S.$$

(8)

Endogenous heat S is understandable for oil-based drill cutting particles in the reaction heat and external heat comprehensive reaction, namely, the internal reaction heat minus all kinds of gas migration and the diffusion of grain loss of heat. In accordance with the reaction, the hot stripping product production of the heat and light component for vapor oil steam, and the heavy oil production in the process of heat generation, the sum of the heat flux of the thermal desorption product per unit time is the cumulative sum of all the heat carried in the desorption process of water vapor, light component oil vapor, and heavy component oil vapor. Therefore, the internal heat can be expressed as

$$S=\frac{\partial \rho }{\partial t}△H-\frac{Q}{{4\pi r}^2}{C}_g\frac{\partial T}{\partial r}=\left(\frac{\partial {\rho }_{gw}}{\partial t}△h_w+\frac{\partial {\rho }_{g1}}{\partial t}△h_1+\frac{\partial {\rho }_{g2}}{\partial t}△h_2\right)-\left({{\int }_0^{r}4\pi r}^2\left(\frac{\partial {\rho }_{gw}}{\partial t}{C}_{gw}+\frac{\partial {\rho }_{g\mathrm{1,2}}}{\partial t}{C}_{g\mathrm{1,2}}\right)d_r\right)$$

(9)

where ∂_ρ/∂_t is the reaction formation rate of the product; △H is the internal reaction heat (kJ/kg); Q is the mass flow of the gas product (kg/s); Q/(4πr²) is the mass flux of the gas product per unit time (kg/(m²·s)); C_g is the specific heat capacity of the gas product (J/(kg·°C)); ∂_T/∂_r is the temperature change rate of the vector radius of oil-based debris particles; ρ_g1,2 is the density of the sum of the steam of light and heavy components (g/cm³); △h_w, △h₁, and △h₂ are the heat generated in the steam reaction stage of water vapor, light component, and heavy component, respectively (kJ/kg); and C_gw, C_g1, C_g2, and C_g1,2 are the constant pressure specific heat capacity of steam, light components, heavy components, and mixed oil steam, respectively (J/(kg·°C)).

(2)

Quality equation

As shown in Figure 2, the oil-based cutting water product, lightweight components, and heavy components of the steam generation and the reduced water, light components, and heavy components are equal in quality. As the hot stripping reaction continued, the steam products increased, whereas the component was reduced. The relationship of the change over time is in the following order (minus the reaction rate in the opposite direction on both sides of the equal sign):

$$\frac{\partial {\rho }_{gw}}{\partial t}=-\frac{\partial {\rho }_w}{\partial t}=k_w{\rho }_w$$

(10)

$$\frac{\partial {\rho }_{g1}}{\partial t}=-\frac{\partial {\rho }_1}{\partial t}=k_1{\rho }_1$$

(11)

$$\frac{\partial {\rho }_{g2}}{\partial t}=-\frac{\partial {\rho }_2}{\partial t}=k_2{\rho }_2$$

(12)

$$\frac{\partial {\rho }_{\mathrm{1,2}}}{\partial t}=\frac{\partial {\rho }_{g1}}{\partial t}+\frac{\partial {\rho }_{g2}}{\partial t}=k_1{\rho }_1+k_2{\rho }_2$$

(13)

The reaction rate constants of water evaporation, light component desorption, and heavy component desorption are obtained from Table 1.

(3)

Selection of initial boundary conditions

In the beginning of the thermal desorption of oil-based drill cutting particles, the internal temperature distribution is uniform, that is, the same as the ambient temperature, and the particle surface temperature is equal to the ambient temperature (set as 25 °C) as follows:

$${\left.T\right|}_{t=0}=25°\mathrm{C}$$

(14)

The temperature gradient at the inner center of the oil-based drill cuttings during thermal desorption is 0, that is,

$${\left.\frac{\partial T}{\partial r}\right|}_{r=0}=0$$

(15)

In other words, the inner side of the particle surface facilitates heat conduction, and the outer side facilitates heat convection. When the thermal desorption reaction starts to occur, the boundary condition is that the heat absorbed by the inner side of the particle is similar to the heat released by the outer side for thermal convection, as follows:

$${\left.-{\lambda }_{ODC}\frac{\partial T}{\partial r}\right|}_{r=R}=h\left(T_f-T_s\right)t>0,$$

(16)

where T_f is the temperature of the gas around the particle (°C); T_s is the temperature of the particle surface (°C); h is the convective heat transfer coefficient between the gas and particle surface (W/(m²·°C)); and R is the particle radius (m).

2.3.2. Parameters in the Energy and Mass Equations of Heat Transfer

(1)

Physical parameters

To simplify the model calculation, the main parameters were set according to the average value or constant. Nitrogen was assumed to enter the cavity, and the temperature was assumed to reach the temperature set by the cavity within a very short time. Then, nitrogen conducted heat transfer with the oil-based drill cutting particles through convection. When the temperature was 25 °C, the density of nitrogen was 1.138 kg/m³, and the thermal conductivity was 0.0242 w/(m∙K). The kinematic viscosity was 15.753 × 10⁻⁶ m²/s, and the Prandtl number was 0.736.

The base oil in the oil-based drilling fluid was 0# diesel, and the thermal desorption products were mainly composed of a small amount of non-condensable gas and diesel steam, including light components (boiling point ≤350.48 °C) and heavy components (boiling point >350.48 °C), which were determined by referring to the study of [36]. The physical composition of the oil-based drill cuttings and thermal desorption products is shown in

Table 2. Physical parameters of the oil-based drill cuttings and thermal desorption products.

Oil-Based Cutting Component	Water	Light Component	Heavy Component	Drill Cuttings
Initial density g∙cm−3	1	0.82	0.85	2.18
Coefficient of thermal conductivity w.(m∙°C)−1	0.68	0.11	0.12	0.89
Specific heat capacity kJ∙(kg∙°C)−1	4.18	2.6	2.69	0.9
Thermal desorption product composition	Water vapor	Light component vapor	Heavy component vapor
Specific heat capacity kJ∙(kg∙°C)−1	2.1	2.49	2.58

.

The particle size range of oil-based drill cuttings varies. When the particle size is too small, the temperature difference within particles is not obvious; when the particle size is too large, the calculation time is too long, and the mass and energy equations need to be iterated in convective heat transfer. Before entering the thermal desorption unit, oil-based drill cuttings are sifted through a vibrating screen to remove particles with a radius greater than 5 mm. To improve the calculation efficiency and in agreement with actual working conditions, the radius was set as 1–5 mm in this study.

(2)

Calculation parameter setting

a.

Initial density, thermal conductivity, and specific heat capacity of thermal desorption steam at constant pressure

Each component is the product of its initial density, and the weighted average calculation method is adopted. The thermal conductivity of oil-based drill cutting particles at any time is the sum of the product of the component content and its initial thermal conductivity. The specific heat capacity at constant pressure of thermal desorption steam at any time is the sum of the product of the content of each component at that time and the initial specific heat capacity of the component. The sum of each group at any time is 1.

b.

Convective heat transfer coefficient

The rated nitrogen inflow is 60 m³/h, and the average diameter of the thermal desorption chamber is about 350–700 mm, while the relative velocity of nitrogen in the cylinder direction of the thermal desorption chamber is 2 m/s. The convective heat transfer coefficient between nitrogen and oil-based rock particles can be determined according to the Ranz empirical formula as follows [37]:

$$Nu=2\frac{hr}{{\lambda }_N}=2+0.6{Re}^{\frac{1}{2}}{Pr}^{\frac{1}{3}}$$

(17)

After changes, h can be obtained as

$$h=\frac{{\lambda }_N}{r}(1+0.3{Re}^{\frac{1}{2}}{Pr}^{\frac{1}{3}})$$

(18)

where N_u is the Nusselt number; r is the particle radius (m); h is the heat transfer coefficient (W/(m²∙°C)); λ_N is the thermal conductivity of nitrogen W/(m∙K); Re is the Reynolds number of nitrogen flowing through the surface of oil-based drill cutting particles; and P_r is the Prandtl number of nitrogen.

2.3.3. Energy and Mass Equations Solved Discretely

(1)

Dispersion of equations

To realize the solution of the model equation, the time coordinate and space region need to be discretized. From the initial time, a time step is divided into ∆t,which is constituted as follows:

$$t_k=k∆t\left(k=\mathrm{0,1},\mathrm{2,3}\cdots \right)$$

(19)

where k = 0 is the starting time.

The outer node method was adopted in this study to divide the oil-based drill cutting particles into N pieces evenly along their radius (Figure 3), with a total of N + 1 nodes. Each node was marked as 0 − N, and the length of each node (the distance between two nodes) was δ_r.

(2)

Mass equation dispersion

At any moment P, the reaction rate and actual density of each component can be calculated based on the mass equation. According to the continuity of the reaction, the density on the right side of the equation is the density of the substance at the previous moment. Therefore, the density of each component at time P is the density change value of the initial density minus a certain time ∆t, and the density of water, light component, heavy component, and thermal desorption steam at time P is provided, respectively, as follows:

$${\rho }_{wP}={\rho }_{wP}^0-k_w{\rho }_{wP}^0∆t$$

(20)

$${\rho }_{1P}={\rho }_{1P}^0-k_1{\rho }_{1P}^0∆t$$

(21)

$${\rho }_{2P}={\rho }_{2P}^0-k_2{\rho }_{2P}^0∆t$$

(22)

$${\rho }_{g\mathrm{1,2}P}={\rho }_{\mathrm{1,2}P}^0-(k_1{\rho }_{1P}^0+k_2{\rho }_{2P}^0)∆t$$

(23)

In the formulas, the superscript 0 refers to the corresponding value at the last moment, and the subscript P refers to the node at P.

(3)

Energy equation dispersion

After combining Equations (8) and (9), the energy equation changes into four expressions as follows:

$$\begin{gathered} r^2\frac{\partial T}{\partial t}\left({\rho }_w{C}_w+{\rho }_1{C}_1+{\rho }_2{C}_2+{\rho }_s{C}_s\right)=\lambda \frac{\partial }{\partial r}\left(r^2\frac{\partial T}{\partial r}\right)+s{r}^2=\lambda \frac{\partial }{\partial r}\left(r^2\frac{\partial T}{\partial r}\right)+\frac{\partial \rho }{\partial t}△H{r}^2-\frac{Q}{{4\pi r}^2}{C}_g\frac{\partial T}{\partial r}{r}^2 \\ =\lambda \frac{\partial }{\partial r}\left(r^2\frac{\partial T}{\partial r}\right)+\frac{\partial \rho }{\partial t}△H{r}^2-(\frac{\partial {\rho }_{gw}}{\partial t}{C}_{gw}+\frac{\partial {\rho }_{gd}}{\partial t}{C}_{gd})\frac{\partial T}{\partial r}\frac{r^3}{3} \end{gathered}$$

(24)

The Taylor series method is adopted to expand the four terms of Equation (24) at node P, and the expressions of the four terms at point P are obtained as

$${\left[r^2\frac{\partial T}{\partial t}\left({\rho }_w{C}_w+{\rho }_1{C}_1+{\rho }_2{C}_2+{\rho }_s{C}_s\right)\right]}_P=\left({\rho }_{wP}{C}_{wP}+{\rho }_{1P}{C}_{1P}+{\rho }_{2P}{C}_{2P}+{\rho }_{sP}{C}_{sP}\right)r_P^2\frac{T_P-T_P^0}{∆t}$$

(25)

$$\begin{gathered} {\left[\frac{\partial }{\partial r}\left(r^2\frac{\partial T}{\partial r}\right)\right]}_P=\frac{1}{∆r}\left[{\lambda }_y{r}_y^2{\left(\frac{\partial T}{\partial r}\right)}_y-{\lambda }_x{r}_x^2{\left(\frac{\partial T}{\partial r}\right)}_x\right]=\frac{1}{∆r}\left({\lambda }_y{r}_y^2\frac{T_Y-T_P}{∆r}-{\lambda }_x{r}_x^2\frac{T_P-T_X}{∆r}\right) \\ =\frac{{\lambda }_x{r}_x^2}{{∆r}^2}{T}_X-\frac{{\lambda }_x{r}_x^2+{\lambda }_y{r}_y^2}{{∆r}^2}{T}_P+\frac{{\lambda }_y{r}_y^2}{{∆r}^2}{T}_Y \end{gathered}$$

(26)

$${\left[\left(\frac{\partial \rho }{\partial t}△H\right)r^2\right]}_P={\left(\frac{\partial \rho }{\partial t}△H\right)}_P^0{r}_P^2=\left[{\left(\frac{\partial {\rho }_{gw}}{\partial t}△h_w\right)}_P+{\left(\frac{\partial {\rho }_{g1}}{\partial t}△h_1\right)}_P+{\left(\frac{\partial {\rho }_{g2}}{\partial t}△h_2\right)}_P\right]r_P^2$$

(27)

$${\left[\left(-\frac{Q}{{4\pi r}^2}{C}_g\frac{\partial T}{\partial r}\right)r^2\right]}_P=\left[{\left(\frac{\partial {\rho }_{gw}}{\partial t}\right)}_P{C}_{gw}+\left({\left(\frac{\partial {\rho }_{g1}}{\partial t}\right)}_P+{\left(\frac{\partial {\rho }_{g2}}{\partial t}\right)}_P\right)C_{g\mathrm{1,2}}\right]\frac{T_y-T_x}{∆r}\frac{r_P^3}{3}$$

(28)

Temperature T is assumed to present a linear distribution between X and P and between P and Y. The grid diagram shows that the temperature calculation method of x and y is as follows:

$$T_x=T_X+\frac{T_P-T_X}{{\left(\delta r\right)}_x}·\frac{∆r}{2}=\frac{T_P+T_X}{2}$$

(29)

$$T_y=T_P+\frac{T_Y-T_P}{{\left(\delta r\right)}_y}·\frac{∆r}{2}=\frac{T_P+T_Y}{2}$$

(30)

Equations (29) and (30) are substituted into Equation (28) as follows:

$${\left[\left(-\frac{Q}{{4\pi r}^2}{C}_g\frac{\partial T}{\partial r}\right)r^2\right]}_P=\frac{r_P^3}{6∆r}\left[{\left(\frac{\partial {\rho }_{gw}}{\partial t}\right)}_P{C}_{gw}+\left({\left(\frac{\partial {\rho }_{g1}}{\partial t}\right)}_P+{\left(\frac{\partial {\rho }_{g2}}{\partial t}\right)}_P\right)C_{g\mathrm{1,2}}\right]\left(T_X{-T}_Y\right)$$

(31)

By substituting Equation (24) to Formula Equation (27) into Equation (23), to facilitate the calculation of temperature at P time, the temperatures at points X, P, and Y (i.e., T_X, T_P, and T_Y) are sorted into one term to obtain the energy equation at point P. The relationship among point P, the temperature at X and Y, the starting time, and the radius at point P is as follows:

$$\begin{gathered} \left\{\left[\left({\rho }_{wP}{C}_{wP}+{\rho }_{1P}{C}_{1P}+{\rho }_{2P}{C}_{2P}+{\rho }_{sP}{C}_{sP}\right)\frac{r_P^2}{∆t}\right]+\frac{{\lambda }_x{r}_x^2+{\lambda }_y{r}_y^2}{{∆r}^2}\right\}{T}_P \\ =\left\{\frac{{\lambda }_x{r}_x^2}{{∆r}^2}+\frac{r_P^3}{6∆r}\left[{\left(\frac{\partial {\rho }_{gw}}{\partial t}\right)}_P{C}_{gw}+\left({\left(\frac{\partial {\rho }_{g1}}{\partial t}\right)}_P+{\left(\frac{\partial {\rho }_{g2}}{\partial t}\right)}_P\right)C_{g\mathrm{1,2}}\right]\right\}{T}_X\hspace{1em}\hspace{1em}\hspace{1em} \\ +\left\{\frac{{\lambda }_y{r}_y^2}{{∆r}^2}-\frac{r_P^3}{6∆r}\left[{\left(\frac{\partial {\rho }_{gw}}{\partial t}\right)}_P{C}_{gw}+\left({\left(\frac{\partial {\rho }_{g1}}{\partial t}\right)}_P+{\left(\frac{\partial {\rho }_{g2}}{\partial t}\right)}_P\right)C_{g\mathrm{1,2}}\right]\right\}{T}_Y \\ +\left[\left({\rho }_{wP}{C}_{wP}+{\rho }_{1P}{C}_{1P}+{\rho }_{2P}{C}_{2P}+{\rho }_{sP}{C}_{sP}\right)\frac{r_P^2}{∆t}\right]T_P^0+\left[{\left(\frac{\partial {\rho }_{gw}}{\partial t}△h_w\right)}_P+{\left(\frac{\partial {\rho }_{g1}}{\partial t}△h_1\right)}_P+{\left(\frac{\partial {\rho }_{g2}}{\partial t}△h_2\right)}_P\right]r_P^2 \end{gathered}$$

(32)

where λ_x and λ_y are the thermal conductivity (W/(m·K)) of the boundary surface on the left and right sides of the volume controlled by node P, respectively, and r_x and r_y are the radius of the boundary surface on the left and right sides of point P (mm), respectively.

(4)

Boundary condition processing

By ignoring the change in heat generated at the interface, for the discrete control unit, the heat absorbed by the particle radius interface on the left is equal to the heat transferred into and released by the gas on the right interface. The sum of the heat absorbed and released is 0. The left is heat conduction, and the right is heat convection.

$${\lambda }_{s}A\frac{T_X-T_P}{\delta r}+hA\left(T_f-T_P\right)=0$$

(33)

where λ_s is the thermal conductivity coefficient of the surface of oil-based drill cuttings calculated according to the thermal conductivity coefficient of the thermal desorption residue due to the completion of the instantaneous desorption reaction (W/(m·K)). A is the interface area (m²) of the control unit.

After the equation is sorted out, the boundary discrete equation of temperature at node P is obtained as

$$T_P=\frac{{\lambda }_s}{{\lambda }_s+∆rh}{T}_f+\frac{∆rh}{{\lambda }_s+∆rh}{T}_X$$

(34)

2.3.4. Energy and Mass Equations Solved by C Programming Language

Computer C language was used for programming, and the idea of the catch-up method was adopted for solving [36]. The initial temperature was assumed to be $T_P^0\left[i\right]$, and the initial parameters (e.g., coefficient of thermal conductivity and specific heat capacity) were set to calculate the rate of the reaction, the density of each component, and the correction coefficient of thermal conductivity. The method of generating the new parameters used in the energy equation was established, and the new temperature $T_P\left[i\right]$, contrast $T_P^0\left[i\right]$, and $T_P\left[i\right]$ error precision were higher than the error precision. $T_P^0\left[i\right]=T_P\left[i\right]$ to correct the thermal conductivity again and perform the next calculation.

3. Results and Discussion

3.1. Simulation Results of the Temperature Field in a Particle

In Figure 4, dr represents a certain radius inside a particle. The smaller the dr value is, the closer it is to the center point of oil-based drill cuttings.

The results showed that when the same heating time was needed to reach the set temperature, the center required 414 s, but the surface needed only 9 s. At the same time, the surface temperature was significantly higher than the internal temperature. The smaller the dr was, the lower the temperature was, indicating that a temperature difference existed between the 5 mm particle center and the surface (△T).

According to the particle collision thermal resistance formula [38], it can be observed that as oil-based drill cuttings particles move from the surface toward the center, the particle radius gradually decreases, leading to an increase in thermal resistance. According to Fourier’s law, it is known that under the same amount of heat, a higher thermal resistance results in a larger ΔT (temperature difference). As time continues to pass, the outer surface of the oil-containing drilling cuttings reaches the set temperature, no longer absorbing heat, and heat begins to transfer to the interior. Consequently, the total heat absorption decreases. Under the condition where remains constant, ΔT gradually decreases. When the oil-based drill cutting particle at position dr = 0 mm has been exposed to heat for more than 441 s, it no longer absorbs heat, and ΔT becomes 0. At this point, the temperature on the surface and at the center of the oil-based drill cutting particle becomes identical, which aligns with the results obtained from the model simulation. The temperature dropped in a gradient, decreasing with time until the internal and surface temperatures matched. Increasing the heating temperature or prolonging the heating time could reduce the temperature difference between the particle surface and center point.

3.2. Effect of Particle Size on Heating Time at the Particle Center

The calculation results of the model assignment in Figure 5 showed that at a constant heating temperature, with the increase in particle size, the heat transfer resistance inside the particle increased, and the time required for the particle center to reach the set temperature also increased. For the oil-based drill cuttings with diameters of 1, 2, 3, 4, and 5 mm, the time required to reach 600 °C was 63, 171, 234, 333, and 441 s, respectively.

The smaller the particle size was, which means the shorter the distance for heat transfer from the surface to the center, the lower the thermal resistance that needs to be overcome within the particle [39]. Under the same heating temperature conditions, this accelerates the rate of temperature increase at the particle’s center, reducing the time required for the particle’s center to reach the design temperature. Higher heating temperatures lead to a greater temperature difference on the particle’s spherical surface on both sides of Δr. This results in a stronger driving force for heat transfer and speeds up the transfer rate. Under the same particle size conditions, it shortens the time for the center point to heat up. Therefore, the heating time of the oil-based drill cuttings could be shortened by decreasing the particle size or increasing the heating temperature [40]. However, it is important to note that excessively high heating temperatures and excessively small particle sizes can both increase energy consumption. In engineering applications, it is necessary to find an optimal range that balances these factors.

3.3. Effect of Heating Temperature on Heating Time at the Particle Center

The thermal desorption temperature is the main factor that affects the thermal desorption capacity, residual oil content, and energy consumption of oil-based drill cuttings. Therefore, a simulation was carried out in this study at different thermal desorption temperatures for the oil-based drill cuttings with a particle diameter of 5 mm, water content of 6.94%, and oil content of 9.65%. Figure 6 shows that in the process of thermal desorption of the oil-based drill cuttings with the same particle size, as the heating temperature of thermal desorption increased, the heating rate of the oil-based drill cuttings increased rapidly, and the heating time decreased. When the thermal desorption temperature was 400 °C, 450 °C, 500 °C, 550 °C, and 600 °C, the time for the 5 mm particles to reach the thermal desorption temperature was 1557, 1107, 756, 558, and 450 s, respectively.

Heating temperature affects the microscale heat transfer processes within spherical particles by influencing factors like thermal diffusion, temperature gradients, the rate of temperature increase, and phase changes [41]. At higher heating temperatures, the thermal energy within the particle increases. This results in more rapid thermal diffusion within the particle [42], as the particles absorb heat more quickly and attain higher internal temperatures. Higher heating temperatures generate a steeper temperature gradient within the particle. The temperature difference between the particle’s center and surface (ΔT) is larger. This greater temperature gradient drives heat transfer from the surface to the center more efficiently. With higher heating temperatures, the rate of temperature increase at the particle’s center is faster. This means that the time required for the particle’s center to reach the desired temperature is shorter. Depending on the composition of the particle, higher temperatures induce oil and water phase changes, which affects both the internal pressure and the mass transfer processes. Under the same conditions, the higher the thermal desorption temperature was, the shorter the time of the oil-based drill cuttings in the thermal desorption chamber was, and the higher the disposal capacity of the device was. It is essential to strike a balance between achieving the desired heating rate and controlling energy consumption in practical applications.

3.4. Effect of Moisture Content on Heating Time and Temperature

The water in oil-based drill cuttings, which is mainly from formation rocks and drilling fluids, usually has less than 20% water content. As can be seen from Figure 7, when the water content increased from 5% to 20%, the heating time increased from 414 s to 696 s, the water content increased by 15%, and the heating time was extended by 68.12%. Figure 8 indicates that, as the water content increased, the heating temperature required also increased.

Because water has a high specific heat capacity, cuttings with a higher moisture content need to absorb more heat to warm up to the same temperature, and the evaporation of water absorbs a lot of heat, which may limit the temperature increase in cuttings because some of the heat is used to evaporate the water, rather than warming the cuttings themselves. The water content affected the evaporation quality and speed of water, thus affecting the internal temperature distribution during the thermal desorption of the oil-based drill cuttings. With the increase in the temperature inside the particle, the reaction of water molecules becomes more active, and the migration speed from the inside to the surface is accelerated. However, the higher the moisture content was, the longer the heating time was, and the slower the heating rate was.

A higher moisture content speeds up heat transfer and reduces the temperature gradient within the particles [42]. Appropriately prolonging the heating time can promote the desorption of water in oil-based drill cuttings, but it can increase heat loss. In production, the moisture content must be reduced by mixing oil-based drill cuttings with a low moisture content and drying in order to shorten the heating time of thermal desorption, to improve the heat transfer efficiency, and to reduce the energy consumption.

3.5. Influence of Oil Content on Heating Time and Heating Temperature

The oil in oil-based drill cuttings mainly comes from the oil-base drilling fluid adhered to the drill cuttings. In the case of fast drilling, the contact time is short, and the amount of oil immersion is small. In the case of slow drilling or tank cleaning, the oil-based drill cuttings adhere and immerse more, and the oil content can reach 30%.

The simulated particle temperature curve under different oil-bearing rates for the oil-based drill cuttings with a particle diameter of 5 mm and oil-bearing water of 6.94% is shown in Figure 9. When the oil content was 10%, 15%, 20%, 25%, and 30%, the heating time required was 486, 639, 747, 882, and 1017 s, respectively, indicating that the oil content had a significant effect on heat transfer. Because oil thermal desorption requires a higher temperature, the higher the oil content, the longer the heating process. As shown in Figure 10, when the oil content increased, the heating temperature required to reduce the mass of the light and heavy components also increased. At the same oil content, the weight loss percentage of the light component was higher than that of the heavy component.

The change in oil content and composition in oil-based drill cuttings will have a complex effect on the microscopic process of heat and mass transfer, involving many factors such as heat capacity, thermal conductivity, phase transformation, mass transport, porosity, microstructure, thermal expansion, and stress [43]. The heat capacity of oil is usually low, and the thermal conductivity is poor, which means that the increase in oil content will lead to a reduction in the overall heat capacity of cuttings and the reduction in thermal conductivity, and the higher the oil content, the slower the heat conduction rate. Changes in the composition of the oil also affect the mass transport within the cuttings, and the speed at which the oil moves and diffuses depends on its concentration gradient, temperature, and the chemical properties of the oil. High oil content and specific oil compositions may affect these transfer rates. The higher the content of light oil components, the faster the mass transfer speed and the shorter the heating time required. According to the analysis of Figure 8, heating time and heating temperature had a significant impact on the thermal desorption rate of the heavy components, and oil content had a great impact on the heating time and heating temperature setting of the oil-based drill cuttings. With the same water and oil contents, the higher the content of heavy components in the diesel oil used by oil-based drill cuttings, the longer the heating time or the higher the heating temperature during thermal desorption.

3.6. Influence of the Change in Light Component Content of Diesel Oil in Oil-Based Drill Cuttings on the Change in Water Quality

As indicated in Figure 11, under the condition that other conditions remain unchanged, the water quality curve is basically the same as that when the content of the light components changes. The calculation results of model assignment showed that the lower the content of light components in the same heating time after 200 s, the higher the residual mass of water, indicating that the evaporation rate of water increased with the increase in light oil components. The duration of water mass reduction decreased with the increase in the content of light oil components because under the same oil content, the light component increased, the heavy component decreased, and the heavy component had a more obvious extension of water evaporation time. Combining Figure 8 and Figure 9 showed that the quality of water, light components, heavy components to reduce in the process of heating temperature, and heating time had an obvious overlap. This result suggests that in oil-based drill cuttings’ single particle convection heat transfer process, the stripping of water, light components, and heavy components is not completed separately, one by one, nor in turn; the processes overlap each other instead. Water evaporation occurs when the material enters the thermal desorption chamber, and it indicates that water evaporation and oil desorption make up an alternate and continuous process. In the thermal desorption of oil-based drill cuttings, water is greatly affected by the oil content and component distribution.

The thermal desorption of individual components within oil-based drill cuttings is influenced by various factors such as adsorption, temperature, pressure, and diffusion [44,45]. Competitive adsorption of both oil and water takes place on the particle surface, and this adsorption process is dynamic, varying with external conditions and component content. As temperature increases, the evaporation rate of water may rise, concurrently facilitating the migration and diffusion of some oil components. Conversely, the evaporation of oil can promote the evaporation of certain water fractions. These competitive adsorption, evaporation, and diffusion processes lead to alterations in the distribution of water and oil components within the particles and on their surfaces [46]. These processes persist until the thermal desorption reaction concludes. Temperature exerts the most significant influence on this reaction process. By controlling two or more temperature zones with varying temperatures, it is possible to modify the competitive adsorption, evaporation, and diffusion processes and the pressure conditions. This allows for the concentration of each component within specific temperature ranges, ultimately reducing energy consumption, improving component separation, and obtaining more valuable oil resources at reduced costs.

4. Conclusions

In the process of particle convection heat transfer, oil content (including oil components), heating temperature, water content, heating time, and particle size exerted significant effects on the heat transfer process, The desorption of water and light and heavy components in the oil-based drill cuttings did not proceed completely one by one and instead involved a simultaneous reaction process overlapping each other. The larger the particle size of the oil-based drill cuttings was, the more difficult it was to distinguish the desorption process of the three components. Therefore, this process should be fully considered when designing the structures of thermal desorption devices for oil-based drill cuttings.

Based on the thermal desorption process of oil-based drill cuttings, which can be divided into oil evaporation of water vapor and water evaporation of lightweight and heavy components, a mathematical model of heat and mass transfer was established. The effects of the oil-based drill cuttings’ characteristics (particle size, water content, oil content, and oil composition) and heating time on heat transfer were described.

However, in industry, the composition of oil is extremely complex; there are a small amount of chemical reaction processes and heat changes such as thermal cracking and polycondensation, and there may be pores in the particles. The size of pores has no effect on the diffusion of gas and heat transfer, nor on the change in thermal conductivity in the drying process of particles or on the influence of a particle’s internal structure, shape, and deformation on heat and mass transfer, which has not been well considered in the above model, and further research is needed.