1. Introduction
With the development of the MHD generator, domestic and foreign scholars have found that the detonation of high-energy explosives in the combustion chamber can form plasma, and the gas has good conductivity. In the 1960s, an MHD company [1] in the United States developed a magnetic fluid generator: when the external magnetic field is 2.8T, adding some ionizing seeds to the C4 type composite, the maximum current of the device can reach 260kA. Choudhary [2] studied the conductivity of the detonation products of RDX/TNT explosive, and its maximum conductivity is 2500S/m. Beijing Institute of Technology [3] measured the maximum conductivity of TNT as1800S/m. It can be seen that the powder gas in the high temperature environment will ionize and form plasma.
The flow field of the high speed conducting gas will form the Lorenz force in the magnetic field, and the force field can change the flow state of the gas, suppress turbulence, and thus affect the heat transfer characteristics [4-5]. Bityurin’s [6] experimental results show that the hypersonic flow plasma can reduce the heat flux at the stagnation point in magnetic field, and a stable magnetic field will suppress the turbulent convective heat transfer ability and reduce the convective heat transfer capacity of conductive fluid. Dietiker [7] used the modified algebraic turbulence model to simulate plasma flow around a bluff body under the action of a magnetic field, and the calculated results show that the surface friction coefficient and heat transfer capacity can be reduced. Huang [8] provided a numerical simulation of the effect of low temperature plasma on heat transfer of aero engine tail nozzle. The results show that plasma under the control of a magnetic field can suppress the increase in temperature of the nozzle wall.
The working environment of weapon tubes is extremely bad; high temperature propellant gas lifts the wall temperature of the tube within a short time, breaking the thermal balance of the tube wall and leading to thermal softening of the surface [9]. At the same time, the mechanical friction between the projectile and the inner wall of the tube is also aggravated by the thermal softening of the metal, and the ablation of the tube produced by the high temperature propellant gas has become an important factor to reduce the service life of the weapon [10-11]. According to the analysis of the magnetic field on the heat transfer characteristics of conductive gas, a method to reduce the ablation of high temperature gas to the tube by using the magnetic field controlled plasma is presented, which aims at improving the heat resistance of the tube and prolonging the service life.
2. Turbulent Dissipation Model of Conducting Gas in Cylinder
Turbulence is a fluid flow state. When the flow rate is lower, the fluid stratifies flow, known as laminar flow. When the flow rate increases, the flow gradually forms in disorder, resulting in irregular turbulent flow. The momentum between the micro fluid mass and heat transfer rate are much higher than those of laminar flow [12]. Therefore, turbulence is an important factor to enhance the heat transfer capacity. In this paper, the turbulent dissipation model of conducting gas in a cylinder is constructed by using the method of magnetic fluid description. The effects of different magnetic fields on the flow and heat transfer characteristics of high temperature conducting gas are studied.
2.1. Physical Model
As shown in Figure 1, the gun tube is simplified into a cylindrical structure, and the conductive gas at the entrance of the muzzle velocity is U. The uniform magnetic field B is applied along the y and z directions of the cylinder.
Figure 1
Figure 1.
Diagram of physicalmodel
According to the research object of this paper, the following basic assumptions are put forward:
$\cdot$ The flow of conductive gas in the cylinder is in the continuous medium region, and it can be studied by using the magnetic fluid dynamics method.
$\cdot$ The conductivity of the conducting gas in the cylinder is assumed to be constant.
$\cdot$ The high temperature conducting gas produced by combustion is in the local thermal equilibrium state, which satisfies the ideal gas state equation.
$\cdot$ The chemical reaction in the process of plasma flow is not considered.
2.2. Governing Equation
In the electromagnetic field, the Lorenz force in the movement of the conductive gas can be expressed as follows:
In this formula: B is the magnetic induction intensityand J is the induced current density.To meet Ohm’s law:
In the formula, the conductivity is σ, the electric field intensity is E, and the velocity of flow is U. It is known that the induction current density should be calculated firstly.
According to the Maxwell equations:
μm is the magnetic permeability, and Q is the joule heat. When the magnetic induction intensity of B is known, the Lorenz force and Joule heat can be calculated.
Magnetic fluid dynamics are based on non-conducting fluid mechanics [13]. Add the Lorenz force and joule heat to the momentum and energy conservation equation, and make corresponding amendments to the equations of fluid mechanics. The mass conservation equation does not involve the effects of force. Therefore, the mass conservation equation of MHD is the same as that of traditional fluid mechanics.
Mass conservation equation:
Momentum conservation equation:
In the formula, P is the pressure, fv is other body force in addition to the magnetic force, τ is the shear stress tensor, and J×B is the Lorenz force.
Energy conservation equation:
In (6), εf is the gas energy, the right end of the first term is the pressure work, the second is the volume force work, and the third is the viscous force work. Because of the magnetic fluid effect, joule heat is added to the ordinary fluid energy equation.
To simulate the effect of magnetic field on the turbulent flow, we need to consider the influence of the magnetic field in the turbulent model. This paper adopts a realizable k-ε turbulence model.
Turbulent kinetic energy equation:
In the type: k and ε respectively represent the turbulent kinetic energy and dissipation rate; $\mu $is the mixture viscosity; ${{\mu }_{i}}$is the turbulent viscosity; ${{G}_{k}}$ is the turbulent kinetic energy caused by the velocity gradient; ${{\sigma }_{k}}$ is the turbulent prandtl coefficient based on k, take σk; $\varepsilon _{em}^{k}$ is the response to electromagnetic effects of turbulent kinetic energy, $\varepsilon _{em}^{k}={{C}_{1}}\frac{\sigma }{\rho }{{B}^{2}}k$, take C1 = 0.5.
Turbulent energy dissipation rate equation:
σε is the turbulent Prandtl coefficient. Take σε=1.3,C2 =1.5,C3 = 1.9, $\varepsilon _{em}^{k}={{C}_{1}}\frac{\sigma }{\rho }{{B}^{2}}k$, and C4=1.
2.3. Boundary Conditions
Set the initial velocity of the conducting gas at the inlet of the cylinder, U=1000m/s, T=3000K, and the conductivity of $\sigma $=500S/m. The wall material is made of non-magnetic steel, and the permeability is 1.26 ×10-6H/m. The outlet is pressure exit condition,P=101325Pa,and the applied magnetic field is a uniform magnetic field. The amount of heat transfer when the gas passes through the wall is derived from the one-dimensional heat transfer formula:
In the formula, ${{\lambda }_{w}}$ is wall thermal conductivity; ${{T}_{0}}$ is the normal temperature 283.15K; and $\delta $ is the wall thickness, take $\delta =40\text{mm}$.
3. Simulation Analysis of Flow and Heat Transfer Characteristics of Conducting Gas
3.1. Model Verification
According to the MHD model mentioned above, the FLUENT software is developed by UDF. The Lorentz force and Joule heat are addedinto the momentum equation and the energy equation in the form of the source term.
Figure 2 shows the velocity distribution on the center line of the cylinder when the magnetic field of different intensity is applied along the z axis. It can be seen from Figure 2 that the velocity at the outlet of the cylinder tends to decrease as the magnetic field increases. This is because the charged particles in the conductive gas are rotated by the Lorentz force, and the applied magnetic field has a certain blocking effect on the flow of the conductive gas. The Lorentz force on the conductive gas increases, so the velocity of flow decreases more. The calculated results are consistent with the trend of velocity distribution on the center line of plasma jet under the action of magnetic field in reference [14], which verifies the reliability of the model.
Figure 2
Figure 2.
Velocity curve of center line under different magnetic field intensity
3.2. Effect of Vertical Magnetic Field on the Flow Characteristics of Conductive Gas
Because of the conductivity of plasma, when it moves in a vertical magnetic field, it will cut the magnetic line to form inductive current J. The interaction of inductive current and magnetic field produces a Lorentz force in the opposite direction of motion in the plasma [15]. Figure 3 shows a schematic diagram of the plasma force. The direction of the induced current is perpendicular to the direction of the magnetic field and velocity.
Figure 3
Figure 3.
Force diagram of plasma under vertical magnetic field
Figure 4 shows the vector distribution of the induced current at the exit section after a 0.5T vertical magnetic field is applied along the z axis. According to the left-handed rule, the inductive current is generated in the vertical magnetic field and the closed loop is formed through the conducting wall. Therefore, at the same cross section, the induced current and Lorentz force along the y axis are larger, andthe distribution has the characteristics of anisotropy.
Figure 4
Figure 4.
Distribution of inductive current
Figure 5 shows the distribution of the turbulent kinetic energy of the conducting gas at the exit section of the cylinder without applying an external magnetic field. Figure 6 and Figure 7 show the 0.5T magnetic field applied in different directions. As shown below, without a magnetic field, the turbulent kinetic energy in the yoz section is symmetric, and it gradually increases from the inside to the outside. The external magnetic field, the charged ion of conductive gas constraints by the Lorenz force, changes from irregular thermal motion to spiral motion around the magnetic field lines. This movement can reduce the particle collision probability parallel to the field direction, thereby reducing the particle kinetic energy exchange efficiency. Its distribution has the characteristics of anisotropy in macro performance.
Figure 5
Figure 5.
Turbulent kinetic energy without magnetic field
Figure 6
Figure 6.
Turbulent kinetic energy undermagnetic field applied along the z axis
Figure 7
Figure 7.
Turbulent kinetic energy undermagnetic field applied along the y axis
Comparing Figure 6 and Figure 7, the turbulent kinetic energy along the magnetic field direction remains in a low level state at the center. The rapid decrease in velocity at the junction on the wall leads to a sudden increase in turbulence intensity near the boundary of wall. In the direction perpendicular to the magnetic field, the turbulence kinetic energy of conducting gas increases steadily as the distance to the wall decreases. Therefore, the turbulent kinetic energy along the magnetic field direction is significantly less than the turbulent kinetic energy perpendicular to the magnetic field.
Eddy viscosity reflects the internal friction between molecules. From Figure 8, where there is an absence of magnetic field, the maximum eddy viscosity is at the center of the exit section: it is 0.337Pa·s. After the external magnetic field, the charged particle spirals around the magnetic field line under the action of the Lorentz force, thereby reducing the particle collision probability parallel to the field direction. From Figure 9 and Figure 10, it can be seen that the magnetic field reduces the eddy viscosity. On both sides of the vertical magnetic field, the eddy viscosity is larger, and the maximum value is 0.259Pa·s.
Figure 8
Figure 8.
Eddy viscosity without magnetic field
Figure 9
Figure 9.
Eddy viscosity under magnetic field applied along the z axis
Figure 10
Figure 10.
Eddy viscosity under magnetic field applied along the y axis
3.3. Effect of Vertical Magnetic Field on the Heat Transfer Characteristics of Conducting Gas
Figure 11 shows the effect of different magnetic field intensities on the wall heat flux q. It can be seen that with an increase in magnetic field intensity, the heat flux density of conductive gas to the wall decreases obviously. This is because the magnetic field can suppress the turbulence intensity of the conducting gas, which results in a decrease in the turbulent kinetic energy near the wall boundary layer. The heat transfer coefficient decreases between gas and wall surface, eventually leading to an decreasein the heat flux.
Figure 11
Figure 11.
Variation of wall heat flux with magnetic
Figure 12 and Figure 13 show the temperature distribution of the inner wall of the cylinder under a magnetic field of 2T in the vertical direction of the cylinder along the z axis. It can be seen from Figure 12 that the temperature of the wall at the entrance is higher. The maximum value is 1830K. Along the flow direction of high temperature gas, the wall temperature decreases gradually, and the lowest wall temperature at the exit is 565K. The turbulent kinetic energy along the magnetic field is lower than that along the vertical magnetic field. The decrease in turbulent kinetic energy leads to the decrease in heat transfer rate of fluid micelles. Therefore, the wall temperature in the direction of the z axis at the same cross section is slightly lower than that in the direction of the y axis. Figure 13 shows the temperature variation curves of the inner surface of the cylinder in the range of 2 ~ 5m along the x axis. The wall temperature in the parallel magnetic field direction is obviously lower than that in the vertical magnetic field direction at the same cross section.
Figure 12
Figure 12.
Temperature distribution of inner wall surface of 2T magnetic field applied along z axis
Figure 13
Figure 13.
Temperature curve of inner wall
4. Experimental Study on Heat Transfer Characteristics of Conductive Gas in Vertical Magnetic Field
4.1. Establishment of Magnetic Controlled Plasma Test System
Dielectric barrier discharge (DBD) is a kind of gas discharge with insulating dielectric insertion discharge space [16]. The advantage of this kind of discharge plasma is that it can form a large volume of plasma discharge region, and the discharge phenomenon is stable and uniform. Therefore, dielectric barrier discharge is used to ionize high temperature gas to produce conducting gas.
Figure 14 is the schematic diagram of test system, this test system consists of a plasma reactor, high temperature gas generator, magnetic field generator, temperature measuring device, and test rig.
Figure 14
Figure 14.
Schematic diagram of test system
The plasma reactor [17] adopts a coaxial tube structure, and the insulating medium is a quartz tube with the size of Ф30×1000mm. The internal electrode is fixed on the center of the reactor with tungsten wire as the high pressure electrode. The external electrode is made of dense steel wire mesh tightly surrounding the outer wall of the dielectric tube and used as the grounding electrode. When the high frequency power supply is used, the velocity and direction of electron motion change due to the periodic variation of applied voltage, so the number of collisions between electrons and gas atoms is greatly increased. The ionization ability of electrons is also greatly improved. Therefore, we use a 20000Hz high frequency power source as a plasma generator. The schematic diagram of dielectric barrier plasma reactor is shown in Figure 15.
Figure 15
Figure 15.
Schematic diagram of dielectric barrier plasma reactor
The high temperature gas generator can provide heat input condition for the plasma reactor. It is used to simulate high temperature gas produced by combustion. Using a WD-100 type electromagnet of Beijing Bolanton electromagnetic company to apply a uniform vertical magnetic field on the outer wall of the quartz tube, when the current is 7.5A and the distance between the poles is 40mm, the intensity of the central magnetic field in the channel is 0.5T. The physical diagram of the test system is shown in Figure 16.
Figure 16
Figure 16.
Physical diagram of test system
4.2. Analysis of Results
In order to ensure the accuracy of temperature measurement, a non-contact infrared thermal imager is used to measure the temperature of the outer wall of quartz tube. Itobtains the temperature distribution of the object surface by receiving the infrared radiation of the measured object. Figure 17 shows a real-time display of the outer wall temperature of the quartz tube measured by the infrared thermal imager, in which the blue box is the temperature analysis area. The thermal imager automatically records and calculates the maximum temperature and average temperature in the area.
Figure 17
Figure 17.
Temperature distribution of quartz tube
Figure 18 shows the maximum temperature variation curve of the outer wall of the quartz tube in the working system of 100s. From the graph, we can see that after adding a magnetic field, the temperature of the outer wall is slightly lower than that without a magnetic field. The wall temperature in the parallel direction of the magnetic field is 92.82℃, while that in the vertical direction of the magnetic field is higher; the maximum value is 105.32℃. This is consistent with the temperature distribution of the simulation results. The wall temperature in the vertical magnetic field is higher than that in the parallel direction.
Figure 18
Figure 18.
Temperature curve of quartz tube
In order to analyze the effect of magnetic field intensity on the heat transfer characteristics of plasma, the working current of the electromagnet was adjusted to 0 ~ 8A respectively, and the corresponding central magnetic field intensity of the electromagnet was 0 ~ 0.5T respectively. Table 1 shows the wall temperature of the corundum tube measured by a thermocouple at different magnetic field intensities.
Table 1. Temperature of tube wall at different magnetic field intensity
| Time | Temperature/℃ | ||
|---|---|---|---|
| B=0T | B=0.25T | B=0.5T | |
| 0s | 30.0 | 30.0 | 30.0 |
| 20s | 48.3 | 46.9 | 43.9 |
| 40s | 67.0 | 63.5 | 58.4 |
| 60s | 80.5 | 75.6 | 73.6 |
| 80s | 95.4 | 91.5 | 86.9 |
It can be seen from Table 1 that the wall temperature decreases with an increase in magnetic field intensity, which indicates that the magnetic field perpendicular to the flow direction of the conductive gas can change its flow field structure and effectively reduce its heat transfer ability.
5. Conclusions
In this paper, the turbulent dissipation model of conductive gas in the cylinder is constructed, and the effect of vertical magnetic field on the heat transfer characteristics of the conducting gas in the cylinder is analyzed. The results of the model are verified by experiments. The conclusions are as follows:
$\cdot$ The magnetic field vertical to the flow direction can effectively reduce the turbulent kinetic energy of the conductive gas and weaken the heat transfer capacity of the conductive gas. The flow distribution has anisotropic characteristics.
$\cdot$ When the vertical magnetic field is applied, the radial diffusion of the charged particles can be restricted, and the heat transfer of the conductive gas to the wall of the cylinder is reduced.
$\cdot$ The heat transfer characteristic of the high temperature conductive gas is related to the direction of application of the magnetic field, and the wall temperature in the vertical magnetic field is higher than that in the parallel direction.
Acknowledgements
This work is supported by the National Basic Scientific Research Project of China.
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