Cutting mixed oil in product pipelines has a great influence on the economy of the pipeline operation processes. The reasonable prediction of CDMS (the concentration distribution in the mixed segment) is important for cutting mixed segments. The classical model cannot explain the tailing phenomenon well which should not be neglected during operation processes. Based on Fick’s diffusion law, a new model for calculating the diffusion coefficient is proposed in this article, which originates from the essence of the diffusion phenomenon and considers the effects of both physical properties of oil products and the turbulence. At the same time, the dynamic fluid equilibrium model of CDMS near the pipe wall is given which has considered the adsorption effect of wall roughness. Based on these two factors, a novel numerical model for simulating the quantity of tailing oil is proposed, which is solved via the characteristic method and the finite difference method. The effects of different physical properties, as well as the adsorption, on both LFMS (the length of the front of the mixed segment) and LTMS (the length of the tail of the mixed segment), are analyzed. The comparison between the simulation results and the experimental data is utilized to validate the proposed numerical model. The simulation results show that the novel model can well describe the mixed segment tailing phenomenon and also explain the mixing essence of two miscible but dissimilar fluids in the pipeline more clearly. To sum up, this model can provide theoretical guidance for the prediction of CDMS and cutting process in practical operation processes; therefore, more economic benefit can be obtained.
Sequential transportation of multiproduct pipelines is a process by which refined oils are transported continuously with a certain sequence in the pipeline, as shown in Figure
Characterization of the mixing process at a different distance and different time (convection and diffusion) [
During the process of batch transportation, mixed oil is influenced by heat, mass, and momentum transfer concurrently [
The calculation method of the oil mixture is divided into the theoretical formula and empirical formula. The mixing quantity of the mixed segment between two batches is calculated directly by the empirical formula. Among the previous studies, the empirical formula proposed by American scholars Austin and Palfery [
Convection-diffusion equation is a theoretical formula used to solve the governing equation of the concentration curve in the mixed segment. The convection-diffusion equation is established by convection and diffusion effects which are the two most important factors affecting the mixed segment. The convection-diffusion equations are mostly 3D but can be reduced to 1D or 2D depending on geometry or as a result of simplifications. The one-dimensional equation considers that the concentration and the velocity are constant in the pipe cross section and are independent of the radial position. The two-dimensional equation can better describe the mixed phenomenon, and its calculation is relatively accurate. But it is relatively difficult to perform the calculation in a long-distance pipeline at the same time. The diffusion coefficient is very important to calculate the mixed segment in the diffusion term. Empirical formulas are often used to calculate the effective diffusion coefficients in one-dimensional equations, which are also applied together with turbulent diffusion coefficients to two-dimensional models.
It is widely believed that the effective diffusion coefficient is related to the Re number [
However, when using the empirical formula or the convection-diffusion equation to solve the mixed segment, it is the average values of the density, viscosity, and temperature of the two oils that are used to solve the model. When these empirical formulas are used to calculate the mixed segment, the sequence of transportation has no influence on the calculated volume of the mixed segment. However, it is found in the engineering practice that, with the same parameters, the mixture quantity of gasoline followed by diesel is smaller than that of diesel followed by gasoline. When diesel is followed by gasoline, diesel will adhere to the pipe wall more firmly, which leads to a harder work for gasoline to replace diesel by continual scouring. On the wall of the pipeline, the replacement process of the oil at the tail of LMS is slowing down, resulting in an increase of LTMS. The existing models cannot describe the influence of the oil sequence on mixed segment quantity and distribution nor can describe the phenomenon of mixed segment trailing. The above are the factors influencing the mixed segment in the theoretical and empirical formulas. In addition, some engineering factors may also affect the mixed oil, including mixed oil trailing, the unstable quality potential of oil source products, the variable demand of pipeline transportation changing with the market, the possession of mixed oil tank capacity, the limitation of oil mixing, and other factors. In order to improve the accuracy of mixed segment prediction, this paper describes the tailing phenomenon of the mixed segment, considers the difference in physical properties of oil products at the front and tail of the mixed segment, and takes the influence of the pipe wall adsorption on CDMS into consideration.
The contribution and innovation of this paper are as follows: (1) A novel model is proposed to simulate CDMS, which takes into account the turbulence effect, the difference of the physical properties of the fluid, and the adsorption effect of the pipe wall roughness. The novel model holds a new diffusion coefficient. (2) Sensitivity analysis is carried out on the key factors affecting the quantity of mixed oil as well as LTMS and LFMS.
Taylor [
Here, it is noticed that
Aunicky [
Here, the definitions of the variables are the same as before.
In order to minimize the quantity of mixed oil formed in the process of refined oil batch transportation and to enhance the economy of pipeline operation, the pipeline for sequential transportation requires a turbulent flow state. The turbulent flow in a circular tube contains momentum exchange between different flow layers, which makes the velocity distribution in the center of the pipe more uniform. The laminar region is close to the wall. The center region is located where turbulence is fully developed. The buffer region is between laminar flow and fully developed turbulent flow. The two-dimensional convection-diffusion equation can be simplified differently in different regions according to different oil-mixing mechanisms. The specific division of different regions is shown in Figure
Division of the flow area in a pipeline.
During the mixing process, the flow regime has a great impact on the volume of mixed oil. More uniform velocity distribution in the central portion of the pipe is resulted from the momentum exchange between different flow layers caused by the pulsation. Because of the restriction of the pipe wall, the pulsation is completely eliminated near the pipe wall. Table
Simplified formulas of the flow regions [
Region | Distance to the pipe wall | Velocity distribution | Turbulent diffusion coefficient | Effective axial diffusion coefficient |
---|---|---|---|---|
Laminar layer |
|
|
|
|
Buffer layer |
|
|
|
|
Turbulent core area |
|
|
|
|
Tichacek et al. [
Suppose that, in the beginning, only oil A with a density of
Effects of the difference in density and viscosity of adjacent oils on the diffusion coefficient [
Effects of different properties | Density difference | Viscosity difference |
---|---|---|
Assumptions |
|
|
An empirical formula for the density or viscosity of mixed oils |
|
|
|
||
Conversion of formulas |
|
|
Because of the roughness of the pipe, the oil molecules will be absorbed in the rough inner surface of the pipe when oil flows and then form a stable adsorption layer because of the action of the molecular force and electrostatic field. A maximum concentration balanced with the component concentration will reach the adsorption layer. The equilibrium adsorption concentration equation, also known as the Langmuir adsorption isotherm [
Here,
The derivate
When the pipe wall has an adsorption effect on the liquid molecule, the variation of concentration over time is as follows:
Here,
The right-hand side term
Also, we have
Substituting this into the unsteady convection-diffusion equation, the following is obtained:
In this nonlinear partial differential equation, the coefficient of the time term is a second-order variable coefficient which is associated with state variable
Based on the above contents, a two-dimensional convection-diffusion model considering the difference of physical properties in adjacent oils can be described as follows:
Definite conditions contain initial conditions and boundary conditions. The concentration of front oil is denoted as
Initial conditions [
Boundary conditions [
Among them,
Supposing that the front and tail of the mixed oil segment between two batches are located at
Because of
The initial conditions are
At
Numerical simulation of the movement of a single oil-mixing interface in a long-distance pipeline.
In this paper, the internal node method is adopted for the solution, and the distribution relationship between nodes and control volume is shown in Figure
Internal node and control volume in a cylindrical coordinate system.
The fractional step method is applied to solve this problem [
Flow chart of the calculation.
The numerical simulation results are compared with the empirical formula under the same conditions. A 30 km pipeline was simulated with the 0# diesel as front oil and the 90# gasoline as tail oil. The temperature is 20°C, and the pipe diameter is 560 mm. The average flow velocity is 1.14 m/s. The mixed oil volume obtained from empirical formulas and numerical simulation results were compared.
As can be seen in Figure
Comparison of calculated results to those of empirical formulas.
In Table
Empirical formulas for calculating VMS.
Formula name | Expression |
---|---|
Austin [ |
|
Brige Edwin [ |
|
Fowler and Brown [ |
|
Smith and Schulze [ |
|
Taylor [ |
|
Jablonskij [ |
|
Sjenitzer [ |
|
The comparisons of LMS results obtained from numerical simulation to others such as Austin and Palfrey [
We have compared the numerical simulation results to the experimental results under the same conditions. The total length of the experimental pipeline is 200 m, made of plexiglass. The pipe diameter is DN25, and the probe position of the interface detector is shown in Figure
Distribution and marking of the probes in the experimental pipe.
Experimental results are compared with the numerical simulation results. (a) Interface distribution in the mixed segment. (b) Re of the experimental result is 33245.7. (c) Re of the numerical simulation result is 33245.7.
In Figure
In the process of calculating the mixed segment, the influence of various parameters on the mixed segment is shown in Figure
Influence factor.
The comparison between the adsorption effect and the nonadsorption effect is shown in Figure
CDMS (a) when considering the adsorption effect and (b) without considering the adsorption effect.
It can be seen from the figure that when considering the adsorption effect, it has little influence on CDMS at the center of the pipe, but it has a significant influence on the oil substitution near the pipe wall. When the adsorption effect is taken into account, the oil concentration gradient at the pipe wall is larger than that without adsorption effect. The adsorption effect results in front oil adhering to the pipe wall, which slows down the oil replacement.
The influence of adsorption parameters
Effects of adsorption parameters
Essentially, the diffusion is caused by a concentration gradient, and the diffusion rate is proportional to the concentration gradient of the material. The diffusion coefficient is related to the density, viscosity, and concentration gradient of the oils. The larger the concentration gradient is at a certain position in MS, the larger the corresponding diffusion coefficient value will be, which is in accordance with the nature of the diffusion phenomenon and has a clearer physical significance.
VMS is related to the value of the diffusion coefficient. The model in this paper can reflect the physical properties of oil and the influence of the concentration gradient on the diffusion coefficient and further reflect the influence of the oil transportation sequence on VMS. The influence of oil sequence exchange on MS is shown in Figure
Influence of the oil sequence on VMS.
It can be seen from Figure
Comparison of the adsorption effect and the diffusion coefficient.
In this paper, the segment with 95–99.9% of front oil concentration is used as LFMS and the segment with 0.1–5% is used as LTMS. Figure
Effects of parameters
LMS will change when the sequence of oil transportation is changed. As can be seen from Figure
Effects of oil properties on LTMS.
Compared to previous studies, a new method for calculating CDMS is proposed via analyzing the tailing phenomenon of the mixed oil segment in situ, where a new model is proposed to calculate the axial diffusion coefficient by considering the difference in physical properties and turbulence effect. Meanwhile, the pipe wall roughness is involved to process the pipe wall adsorption effect on oil molecules. Based on the new method, LFMS, LTMS, and CDMS are related to the adsorption effect and to the value of the diffusion coefficient, which is a function of oil density, viscosity, and concentration gradient. The conclusions are summarized as follows: The larger the concentration gradient is in a certain part of the mixed segment, the larger the diffusion coefficient value will be. The sequence of oil product transportation shows an effect on LFMS and LTMS and shows a greater effect on LFMS. LTMS is always greater than LFMS, no matter what sequence is used. When diesel is followed by gasoline, LMS is larger than that of gasoline followed by diesel, and LFMS and LTMS are also larger. When diesel is followed by gasoline, the difference between LFMS and LTMS is rather small and is also smaller than those of gasoline followed by diesel. When the adsorption effect on CDMS is simulated, the mixed segment increases with the increasing The simulation results show that the novel model can well describe the mixed segment tailing phenomenon of LTMS and also explain the mixing essence of two miscible but dissimilar fluids in pipelines more clearly. To sum up, this model can provide theoretical guidance for the prediction of CDMS and cutting process in practical operation processes; therefore, more economic benefit can be obtained.
Miscible but dissimilar fluids
Mixing of miscible but dissimilar fluids
The mixed segment
The length of the mixed segment
The volume of mixed oil
The length of the tail of the mixed segment
The length of the front of the mixed segment
The concentration distribution in the mixed segment
The difference between LTMS and LFMS.
The datasets generated and/or analyzed during the current study are available from the corresponding author on reasonable request.
The authors declare that there are no conflicts of interest regarding the publication of this paper.
This work was funded by the China Postdoctoral Science Foundation (2019M653481) and by the National Natural Science Foundation of China (51674212).