Abstract
The results of twodimensional mathematical modeling of heat and mass transfer in a highly viscous hydrocarbon liquid by inductive and radiofrequency (RF) electromagnetic (EM) heating are presented. The model takes into account the dependence of the liquid's viscosity and thermal conductivity on the temperature and also the presence of thermal convection effects. It is shown that the occurrence of volumetric heat sources inside the liquid caused by EM radiation yields an intensive deep heating as compared with inductive heating. Numerical calculations show that, in both these cases, the single vortex flow structure is formed in the whole volume of the liquid. However, RF EM heating provides a more homogeneous distribution of heat in the medium and requires threefold less power consumption in comparison with induction heating.
Keywords:
Radiofrequency electromagnetic field; Induction heating; Hydrocarbon liquid; Thermal convection; Heat transfer; Mathematical model
INTRODUCTION
The study of convective heat transfer in multiphase multicomponent media, such as natural (oil and bitumen) or manmade (sludge and petroleum products) hydrocarbon systems, are associated with a number of problems arising in the oil and gas industry. One of them is the utilization of oil sludge, formed as a result of tank and reservoir cleaning. Heating significantly reduces the viscosity of such systems and may cause convective motion of the liquid (Kovaleva et al., 2005Kovaleva, L. A., Nasyrov, N. M., Maksimochkin, V. I. and Suf'yanov, R. R., Experimental and numerical modeling of the thermal conductivity of highviscosity hydrocarbon systems. Journal of Applied Mechanics and Technical Physics, 46(6), pp. 851856 (2005)., 2010). The area covered by the motion, as well as the temperature field in this area can continuously and irregularly change. There are various ways of transfering heat energy from the source to the medium and inductive and electromagnetic heating are among these. In recent years, the use of highfrequency changing EM fields has become a widely used technology for heating complex hydrocarbon systems (Carrizales et al., 2008Carrizales, M. A., Lake, L. W. and Johns, R. T., Production improvement of heavyoil recovery by using electromagnetic heating. Colorado, USA, 2124 September, SPE 115723 (2008).; Kasevich et al., 1994Kasevich, R. S., Price, S. L., Faust, D. L. and Fontaine, M. F., Pilot testing of a radio frequency heating system for enhanced oil recovery from diatomaceous earth. 69 Annual Technicel Conference and Exhibition held in New Orleans, USA, SPE 28619, pp. 105118 (1994).; Kovaleva et al., 2008, 2009Kovaleva, L. A., Kireev, V. N. and Musin, A. A., Modeling of heat and mass transfer in a hydrocarbon fluid under inductive heating. Journal of Applied Mechanics and Technical Physics, 50(1), pp. 8085 (2009)., 2011aKovaleva, L. A., Minnigalimov, R. Z. and Zinnatullin, R. R., Destruction of waterinoil emulsions in radiofrequency and microwave electromagnetic fields. Energy and Fuels, 25(8), 37313738 (2011a)., 2011bKovaleva, L., Musin, A., Zinnatullin, R. and Akhatov, I. S., Destruction of waterinoil emulsions in electromagnetic fields. Proceedings of ASME 2011 International Mechanical Engineering Congress & Exposition, IMECE201162935, Denver, Colorado, USA, November 1117 (2011b).; Bientinesi et al., 2013Bientinesi, M., Petarca, L., Cerutti, A., Bandinelli, M., De Simoni, M., Manotti, M., Maddinelli, G., A radiofrequency microwave heating method for thermal heavy oil recovery based on a novel tightshell conceptual design. Journal of Petroleum Science and Engineering, v. 107, pp. 1830 (2013).). The urgency of this technology is connected with the fact that, for the considered media, it is important not only to reduce the viscosity by heating, but to a greater extent to destroy stable structures of highmolecular hydrocarbon compounds by the effect of electromagnetic radiation. The most effective method of applying heat and also provoking a "destructive" effect in such media is by heating with a radiofrequency electromagnetic field. The advantage of this method of heating as compared with the other types of treatment is that it involves a large volume of liquid and the speed of the process is very high.
The results of a numerical study of the processes of heat and mass transfer in a hydrocarbon liquid under the influence of the induction and RF EM heaters are given in the paper. The results are obtained by the solution of the full system of equations of thermal convection, taking into account the dependence of the thermophysical parameters on the temperature of the liquid.
PROBLEM STATEMENT
We have studied the thermal motion of a heavy hydrocarbon liquid with temperaturedependent viscosity and thermal conductivity in the space between two coaxial cylindrical tubes using various methods of heating. The reservoir model consists of two coaxial metal tubes (1 and 2 in Figure 1), the lower part of which is closed by a plug (4). The space between the tubes is filled with the hydrocarbon liquid (3).
Scheme of the experimental zone: 1) reservoir wall; 2) inner tube; 3) hydrocarbon liquid; 4) bottom of the reservoir.
The selected geometry of the physical model, from the electrodynamics point of view, is a cylindrical capacitor, which gives the possibility to use different methods of heating of the highviscosity hydrocarbon medium. Two of these methods differ not only in the way of implementation, but also in the nature of the forces leading to heating of the medium:

Induction (or contact) heating, i.e., heating of the liquid filling the annular space of the reservoir with an induction heater placed at its center. In this case heating is carried out by heat applied on the inner tube.

Heating of the liquid by exposure to radiofrequency electromagnetic field (RF EMF) is due to EMF energy dissipation in the liquid (volumetric heating). The outer tube of the reservoir and the inner tube in this case act as electrodes.
MATHEMATICAL MODEL
A mathematical model of the impact on the hydrocarbon medium includes the system of equations of mass conservation, energy and momentum. From the point of view of mathematical modeling, the physical model is a multilayer system: a reservoir wall (outer tube), an inner tube, a hydrocarbon liquid, concrete bottom of the reservoir.
The distribution of temperature in the solid parts of the model (1, 2 and 4 in Figure 1) is described by the thermal conductivity equation:
where ρ_{i}, c_{i}, k_{i} are density, heat capacity and coefficient of thermal conductivity of the ith material, respectively; q_{i} is the density of the distributed heat sources in the ith material; I = 1,2,4.
Temperature and velocity fields in the layer of liquid are described by the system of equations of thermal convection in the Boussinesq approximation (Gershuni and Zhukhovitsky, 1976Gershuni, G. Z. and Zhukovitskii, E. M., Convective Stability of Incompressible Fluids. Nauka, Moscow (1972). Translated from the Russian by Louvish D. Keter Publications, Jerusalem/Wiley, p. 330 (1976).):
Here ρ_{3},η_{3}k_{3}, k_{3} are density, coefficient of dynamic viscosity, heat capacity and thermal conductivity of a hydrocarbon liquid; v is a vector field of velocity of free convective motion of the liquid; P is pressure; T is temperature; f is the vector field of mass forces; q_{3} is the density of distributed heat sources in the liquid.
In the given mathematical model (2)(4) the following assumptions have been made:

the medium is homogeneous, isotropic and incompressible;

electrodynamic and thermal properties except for the viscosity and the thermal conductivity are constant;

in the equations of motion, the effect of terms containing the derivatives of viscosity with respect to temperature is negligible;

the temperature change caused by heat generation due to the dissipation of energy by internal friction is negligible;

the flow is laminar.
Additional Relations
The thermal motion of the liquid occurs in the gravitational field in the presence of spatial heterogeneity of density caused by the inhomogeneity of the temperature (Gershuni and Zhukhovitsky, 1976Gershuni, G. Z. and Zhukovitskii, E. M., Convective Stability of Incompressible Fluids. Nauka, Moscow (1972). Translated from the Russian by Louvish D. Keter Publications, Jerusalem/Wiley, p. 330 (1976).). The density, temperature, and pressure of the medium are interconnected through the equation of state ρ = ρ(T,P). The form of this dependence for different media is different and depends on many factors, including the magnitude of the changes of temperature and pressure. Hydrocarbon liquid is a complex heterogeneous system. Its composition and its physicochemical properties can vary within very wide limits. In particular, it concerns the value of the boiling point of the liquid. Every component of a hydrocarbon medium has its own boiling point and, in the process of heating, their transition into the gaseous state occurs at different temperatures. Therefore, it is natural to assume that, in the process of heating, there can be simultaneously two phases in one hydrocarbon medium: gas and liquid. Moreover, the concentration of the gaseous phase rises with the temperature increase.
In the general case, for the simulation of such a system, the twophase nature of the medium should be taken into account and the transfer equations should be solved for each of the phases. However, if we assume that the gas and liquid phases behave as a single system and if we neglect the compressibility of the medium, the quasihomogeneous approximation can be applied (Kutateladze and Nakoryakov, 1984Kutateladze, S. S. and Nakoryakov, V. E., Teplomassoobmen i volny v gazozhidkostnykh sistemakh (Heat and mass transfer and waves in gasliquid systems). Novosibirsk, Nauka Publishers, p. 301 (1984).). In this case, it is possible to solve the system of equations mentioned above using effective values of some of the physical quantities: density, coefficient of dynamic viscosity, coefficient of thermal conductivity, heat capacity, and the averaged values of velocity, temperature and pressure in the medium.
The density of the mixture is assumed to be an additive quantity, while the concentration of the gas phase is associated with intensity of degassing of the hydrocarbon system, which depends on the temperature. Then the dependence of the mixture density on temperature can be represented by the following empirical relation:
Here Т is the current temperature, T_{s}=70 °C is the temperature of the beginning of oil degassing, B = 0.0542 and γ = 93 °C are the empirical coefficients of the degassing process intensity. Then, following the Boussinesq approximation, the components of the mass forces f in equation (2) under the conditions of thermal convection will be present in the following form:
where g is the acceleration due to gravity.
The viscosity of the hydrocarbon liquid is also temperature dependent: the approximate dependence of viscosity on temperature in the form of two experimentally obtained exponents has been used (Kovaleva et al., 2005Kovaleva, L. A., Nasyrov, N. M., Maksimochkin, V. I. and Suf'yanov, R. R., Experimental and numerical modeling of the thermal conductivity of highviscosity hydrocarbon systems. Journal of Applied Mechanics and Technical Physics, 46(6), pp. 851856 (2005).):
where η_{01} Pa·sec is the viscosity of the hydrocarbon liquid at the temperature of T_{01}= 35.5 °C; η_{02} Pa·sec is the viscosity of the hydrocarbon liquid at the temperature of T_{02} = 54.2 °C; γ_{1} = 0.497 K1 is the temperature coefficient within the range T_{01} < T < T_{02}; γ_{2} K^{1} is the temperature coefficient within the T > T_{02} range.
The dependence of the conductivity coefficient of the liquid on the temperature was set as linear (Kovaleva et al., 2005Kovaleva, L. A., Nasyrov, N. M., Maksimochkin, V. I. and Suf'yanov, R. R., Experimental and numerical modeling of the thermal conductivity of highviscosity hydrocarbon systems. Journal of Applied Mechanics and Technical Physics, 46(6), pp. 851856 (2005).):
Here k_{30} is the coefficient of thermal conductivity at the temperature of T = T_{0}; b=0.0095 K^{1} is an empirical constant.
The definition of the ratios for the density of distributed heat sources q in equations (1), (3) and their values depends on the type of heating:

 induction (contact) heating is modeled by placing the distributed sources of heat on the inner tube of the reservoir (2 in Figure 1) (Kovaleva et al., 2005Kovaleva, L. A., Nasyrov, N. M., Maksimochkin, V. I. and Suf'yanov, R. R., Experimental and numerical modeling of the thermal conductivity of highviscosity hydrocarbon systems. Journal of Applied Mechanics and Technical Physics, 46(6), pp. 851856 (2005).):
where N_{I} is the net power of the inductor; R_{1}, R_{2} are the inner and outer radii of the inductor tubes; Z_{2}  Z_{1} is the height of the inductor.

 RF EM heating is modeled by placing a distributed heat source in the liquid (3 in Figure 1). For a cylindrical capacitor the following equation is used (Landau and Lifshitz, 1960Landau, L. D. and Lifshitz, E. M., Electrodynamics of Continuous Media (Volume 8 of A Course of Theoretical Physics), Pergamon Press (1960).):
where N_{R} is the net power of the RF generator; r is radiusvector; R_{3} is the radius of the reservoir; Z_{3}  Z_{2} is the reservoir depth.
Boundary Conditions
Initially, the liquid is motionless and the model has a constant temperature equal to the ambient:
Adhesion conditions are set for all velocities at solid walls:
All the internal borders of the reservoir are marked by the index "G".
In accordance with the terms of the experiments, the following boundary conditions were chosen:
 in the case of induction heating, the condition of the heat exchange with the environment on the upper surface of the reservoir is given according to the law of free convection in an unbounded space. The outer wall of the reservoir and its lower base are heat insulated. The condition of heat exchange with the air in a confined space is applied on the inner surface of the inductor tube. The corresponding boundary conditions are:
Here α_{h}, α_{v} are heat transfer coefficients along the horizontal and vertical walls; T_{0} is the initial temperature of the medium and the temperature of the air surrounding the experimental unit.
 In the case of RF EM heating, the reservoir with liquid is heat insulated on all sides. The corresponding boundary conditions are:
RESULTS OF CALCULATIONS
The system of Equations (1)(4) with the closing ratios (514) was solved numerically by the method of control volume with the help of the SIMPLE algorithm (Patankar, 1980Patankar, S., Numerical Heat Transfer and Fluid Flow. Hemisphere Publishing Corporation, New York (1980).).
The numerical values of the physical and geometrical parameters of the model were taken in accordance with the parameters of laboratory facilities used in experimental research. Model properties are given in Table 1.
Values of the unknown quantities were determined by comparing the results of mathematical modeling with experimental results.
In the process of mathematical modeling, temperature fields and streamlines in a hydrocarbon liquid heated by induction and RF EM sources were obtained. Low temperature and high viscosity of the liquid at the initial time exclude the possibility of any motion of the liquid. Therefore, for a sufficiently long time after the start of the experiment, the medium is heated only by the distributed heat sources and the mechanism of molecular heat conduction. Then the zone warmed up to the temperature of fluidity is formed near the inner tube (rod) and that is sufficient for the emergence of the thermal motion of the liquid and the mechanism of convective heat transfer becomes part of the heating process. It is revealed that in both cases a single vortical flow structure is observed in the whole volume of the liquid (Figure 2). Most of the time the core of the flow is located in the upper part of the reservoir. By the end of the process it is shifted to the center of the reservoir. The asymmetry of the main flow that is observed throughout the heating process is the result of the clearly defined dependence of viscosity on temperature.
Streamlines in the liquid at the 80th minute of the experiment: a) induction heating, b) RF EM heating.
The intensity of expansion of the zone heated to the fluidity temperature where convective motion is taking place is relatively higher in the case of RF electromagnetic heating than in the case of induction heating (Figure 3). Due to a more homogeneous nature of the temperature distribution under RF EM heating, a large part of the volume of the liquid is covered with a convective flow as compared to induction heating. This in its turn significantly speeds up the process of heat transfer in the liquid.
Temperature zones in the liquid at the 80th minute of the experiment: a) induction heating, b) RF EM heating.
The main difference between RF electromagnetic and induction heating is that the electromagnetic heating covers a large volume of liquid. The distinctive feature of RF EM heating can be seen through the comparison of the dynamics of the temperature change measured with thermocouples located at different distances from the axis of the reservoir during induction (Figure 4) and RF EM heating (Figure 5).
Dynamics of the temperature distribution in the reservoir liquid during induction heating: symbols  experimental data; lines  calculated data; curves 1, 2, 3 correspond to the positions of the first, second and third thermocouples.
Dynamics of changes in temperature of the liquid in the reservoir in the process of RF EM heating: symbols  experimental data; lines  calculated data; curves 1, 2, 3 represent readings on 1, 2, 3 thermocouples respectively.
It is shown in Figure 4 that, in the process of induction heating, the temperature increases first on the surface of the inner tube (curve 1 in Fig. 4) and only after 50 minutes is a consistent rise in temperature observed on the second (curve 2) and the third (curve 3) thermocouples. In the case of RF electromagnetic heating (Fig. 5), the temperature begins to rise with varying intensity simultaneously on all thermocouples (curves 1, 2 and 3 in Fig. 5). This occurs due to the fact that the more homogeneous redistribution of the energy in the case of RF EM heating prevents the local overheating of the liquid that takes place in induction heating and a steady temperature rise is observed throughout the volume of the liquid. After reaching the temperature of fluidity, a larger part of the liquid is covered by a convection flow, thereby increasing its heating volume. Points in Figs. 4 and 5 show the temperature readings on the thermocouples obtained in the process of laboratory experimental studies.
CONCLUSION
The comparative efficiency of radiofrequency electromagnetic field heating in comparison with induction heating of a heavy hydrocarbon liquid system was shown by mathematical modeling. It was proved that, in both cases, a single vortex flow structure is formed in the whole volume of the liquid, which expands as the liquid reaches the fluidity temperature. However, RF EM heating provides a more homogeneous redistribution of heat and prevents local overheating of the liquid, which is observed in the process of induction heating. Besides, in the case of induction heating, a threefold lower power consumption is required for adequate heating; this fact indicates that RF electromagnetic heating is significantly more efficient when dealing with highviscosity hydrocarbon liquids in comparison with induction heating.
NOMENCLATURE

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To whom correspondence should be addressed
ACKNOWLEDGMENT
This research is supported by the Ministry of Education and Science of the Russian Federation (government contract Nº.3.1251.2014/К), the RFBR grant Nº140197005 and the Grant of President of the Russian Federation Nº MK9398.2016.1.
REFERENCES
 Bientinesi, M., Petarca, L., Cerutti, A., Bandinelli, M., De Simoni, M., Manotti, M., Maddinelli, G., A radiofrequency microwave heating method for thermal heavy oil recovery based on a novel tightshell conceptual design. Journal of Petroleum Science and Engineering, v. 107, pp. 1830 (2013).
 Carrizales, M. A., Lake, L. W. and Johns, R. T., Production improvement of heavyoil recovery by using electromagnetic heating. Colorado, USA, 2124 September, SPE 115723 (2008).
 Gershuni, G. Z. and Zhukovitskii, E. M., Convective Stability of Incompressible Fluids. Nauka, Moscow (1972). Translated from the Russian by Louvish D. Keter Publications, Jerusalem/Wiley, p. 330 (1976).
 Kasevich, R. S., Price, S. L., Faust, D. L. and Fontaine, M. F., Pilot testing of a radio frequency heating system for enhanced oil recovery from diatomaceous earth. 69 Annual Technicel Conference and Exhibition held in New Orleans, USA, SPE 28619, pp. 105118 (1994).
 Kovaleva, L. A., Kireev, V. N. and Musin, A. A., Modeling of heat and mass transfer in a hydrocarbon fluid under inductive heating. Journal of Applied Mechanics and Technical Physics, 50(1), pp. 8085 (2009).
 Kovaleva, L. A., Minnigalimov, R. Z. and Zinnatullin, R. R., Destruction of waterinoil emulsions in radiofrequency and microwave electromagnetic fields. Energy and Fuels, 25(8), 37313738 (2011a).
 Kovaleva, L. A., Minnigalimov, R. Z. and Zinnatullin, R. R., Development of electromagnetic technology for recycling oil slimes. Oil Industry, 9, pp. 7579 (2009).
 Kovaleva, L., Musin, A., Zinnatullin, R. and Akhatov, I. S., Destruction of waterinoil emulsions in electromagnetic fields. Proceedings of ASME 2011 International Mechanical Engineering Congress & Exposition, IMECE201162935, Denver, Colorado, USA, November 1117 (2011b).
 Kovaleva, L. A., Nasyrov, N. M., Maksimochkin, V. I. and Suf'yanov, R. R., Experimental and numerical modeling of the thermal conductivity of highviscosity hydrocarbon systems. Journal of Applied Mechanics and Technical Physics, 46(6), pp. 851856 (2005).
 Kutateladze, S. S. and Nakoryakov, V. E., Teplomassoobmen i volny v gazozhidkostnykh sistemakh (Heat and mass transfer and waves in gasliquid systems). Novosibirsk, Nauka Publishers, p. 301 (1984).
 Landau, L. D. and Lifshitz, E. M., Electrodynamics of Continuous Media (Volume 8 of A Course of Theoretical Physics), Pergamon Press (1960).
 Patankar, S., Numerical Heat Transfer and Fluid Flow. Hemisphere Publishing Corporation, New York (1980).
Publication Dates

Publication in this collection
JanMar 2016
History

Received
10 Oct 2014 
Reviewed
12 Feb 2015 
Accepted
06 Mar 2015