Experimental analysis of natural convection with Joule heating by means of particle image velocimetry

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1 Experimental analysis of natural convection with Joule heating by means of particle image velocimetry Tatsuya Kawaguchi *, Takushi Saito and Isao Satoh Department of mechanical and control engineering, Tokyo institute of technology Ookayama , Meguro-ku, Tokyo , Japan * correspondent author: kawat@mep.titech.ac.jp Abstract In the present study, the natural convection in a square cavity with internal Joule heating by the electric current was experimentally investigated by means of particle image velocimetry (PIV). The mixture of glycerol and water was used as a working fluid with an electrolyte. Relation between electrolyte concentration and electric conductivity was calibrated in advance, temperature dependency of the conductivity as well. A transparent acrylic cubic cavity of 100 mm on a side was used as an enclosure of the fluid flow. A pair of opposed vertical carbon plates was used as the electrode. Electric conductivity of working fluid was set to σ = 0.16 S/m, mean electric current was I=1.5 A. Consequently the density of internal heating by electric current was 136 kw/m 3 assuming that the thermal energy supply per unit volume was spatially homogeneous. Resultant temperature difference between top and bottom in a cavity was 40 deg. Rayleigh number was varied as a parameter, the maximum Rayleigh number with internal heating was Ri = 1.38x The measured velocity vector maps showed that the magnitude of the velocity was 0.01 m/s approximately. Characteristic unstable downward flows were observed in the center of the channel. Temperature fluctuation in the cavity was measured by thermocouple and resulted that the mean temperature of the fully developed flow was 68 deg, the amplitude was 5 deg. 1. Introduction Natural convection with internal heating is the flow due to the temperature difference within the enclosure. The Rayleigh-Bénard convection is one of the fundamental flows of the chaotic and transitional fluid phenomena. In the present study, the natural convection in a square cavity with internal Joule heating by electric current was experimentally investigated by means of particle image velocimetry. In many experiments, the energy source of the internal heating is electric current for Joule heating, absorption of electromagnetic wave and such molecular or atomic reactions as chemical reactions and nucleorrhexis. For example, mantle convection is one of the well-known flows of the natural convection with internal heating (Turcotte, 2001). In the industrial application, the high temperature molten glass is manufactured by using the electrical Joule heating since the electrical conductivity of the high temperature glass is significantly high. In the present study, we investigated the fluid behavior of natural convection due to the volumetric Joule heating of liquid by using a model liquid instead of a molten glass. The liquid we used was the mixture of glycerol and water with lithium chlorine (LiCl) so that they could simulate the kinematic viscosity as well as the resistivity of model liquid with soda-lime-silica glass. The study by Stanek et al. (1969) was one of the first approaches for the modeling of the electric melting of glass in furnaces and they consider not only Grashof number but also Galileo analogy for the modeling of the convection of molten glass by electric heating. They also investigated the location and orientation of the multiple electrodes, moreover the phase of the electric current was considered. Natural convection could be scaled by Grashof number, Gr (=gβ TL 3 /ν 2 ), Prandtl number, Pr (=ν/α), and Damköhler number, Da (=ql 2 /k T) in the presence of the internal heat source. Characteristic length, L, is equal to the size of the channel. q is the input energy density per unit volume. g represents gravitational acceleration. β, ν, α, and k are the thermal expansion coefficient, - 1 -

2 kinematic viscosity, thermal diffusivity and thermal conductivity of liquid respectively. The dimensionless parameter, Ri, could be used that quantifies the natural convection with internal heating and hence the strength of the buoyancy force. Ri is defined as Ri 5 = gβ ql / kνα = Pr Gr Da (1) The dimensionless parameter contains the intensity of the volumetric heating instead of a temperature difference between horizontal Dirichlet boundaries. Since the parameter increases directly with the fifth power of the characteristic length, the growth of the cavity size significantly affects the internal fluid behavior of cavity. Critical Rayleigh number was Ric = 1400 given by Robert (1967). The numerical prediction and relevant experiments of the physics of a natural convection in a cavity has been investigated by Tritton and Zarraga(1967), Kulacki and Goldstein(1972), Tveitereid(1978), Gurnis and Davis(1985), Char and Chiang(1994). Liaquat and Baytas(2000) have numerically investigated the heat transfer characteristics of internally heated square cavity at high Rayleigh numbers from 10 7 to They concluded that the flow pattern with lower Rayleigh number below 10 7 showed periodic oscillation whereas the higher Rayleigh number above 10 8 caused non-periodic time-dependent behavior. Since the instability of the flow field is essentially present and the considerable range of Rayleigh number is widely varied, the verification of each numerical model is quite complicated without the detailed and precise experimental data. The objective of the present study is to analyze a higher Rayleigh number natural convection in a rectangular cavity with Joule heating by means of particle image velocimetry. 1. PIV Measurement of natural convection 1.1 Electric conductivity of model liquid. Figure 1. Temperature dependency of electric conductivity of molten glass in terms of the frequency of applied current

3 Figure 1 depicts the electrical property of the high temperature molten glass as a function of the temperature. Portable LCR analyzer was used to measure the electric conductivity. Sheathed thermocouple with data logging system was used to measure the temperature of molten glass. The frequency effect of the electrical current was also considered. A pair of inconel alloy electrode was used whose diameter, effective length and separation were 3 mm, 25 mm and 50 mm respectively. The electric conductivity of molten glass beyond 1300 K is as high as that of the semiconductor materials. The electric current through a molten glass is not due to the free electron but due to the ion in the high temperature glass. It must be noted that the electric conductivity of the solution of electrolyte such as LiCl and KCl is affected by the frequency in contrast to the frequency independent conductivity of molten glass shown. Therefore the frequency of the electric current used in the following experiment was fixed to 100 Hz. Figure 2. Electric conductivity of glycerol 80% water solution in terms of a temperature. The electric conductivity at 70 deg is eight times as large as that of 20 deg. Figure 2 depicts the relation between electric conductivity of LiCl solution and temperature. The result shows that the increase of temperature causes the higher electric conductivity. In the model experiment of molten glass, the positive relation between electric conductivity and liquid temperature is one of the significant properties for understanding of the fluid phenomena. If the local temperature was increased by any reason such as the stagnation or vortex, the electric current tend to concentrate in the high temperature liquid and consequently the local temperature was incrementally changed. This feedforward characteristic of the fluid flow complexifies the behavior of the convection in a cavity. In the following PIV measurement, the ratio between water and glycerol was varied in order to control Pr number of working fluid. In this study, two solvent were compared, the one was the glycerol 80% water solution, and the other was 50% mixture of water and pure glycerol. Pr numbers of each mixture were and 19.7 respectively

4 Figure 3. Electric conductivity of working fluid as a function of the electrolyte concentration. The conductivity is linearly proportional to the concentration of electrolyte. Relation between LiCl concentration and electric conductivity was firstly confirmed in advance. The model experiment of Joule heating of molten glass must be simulating the characteristics of electric conductivity of glass. That is the reason we employed the LiCl solution as a working fluid. Figure 3 depicts the calibration line between electric conductivity of working fluid and ion concentration of the LiCl solution at room temperature. Resultant molar electric conductivity of the working fluid we used was 0.01 Sm 2 /mol, which is the same value given by OIML (convention establishing an international organization of legal metrology) reference data for the various electrolytes. Since the ion concentration was relatively low, the conductivity of liquid was linearly proportional to the concentration of electrolyte. In the following experiment, the electric conductivity of the liquid was regulated to σ = S/m, electric current for heating was I = 1 to 2 A. The value of electric current enables to assume that the local magnetic flux density as well as the Lorentz force was negligibly small. 1.2 Experimental Setup. Figure 4 illustrates the simplified set up of the PIV measurement that consisted of a square transparent acrylic open channel with a pair of carbon plates as electrode, and a copper plate as a constant temperature ceiling. The size of the cubic cavity was 100 mm on a side. Characteristic length, L, in equation (1) was equal to the size of the channel. Copper plate with internal water channel was mounted on the top of cavity. The channel was connected to the heat exchanger with an automatic temperature control system. The degree of temperature fluctuation of the plate was less than 0.1 K during the following experiments. The other five surfaces were assumed to be adiabatic by which the heat fluxes through the surfaces were equal to zero. The K-type wire thermocouples whose diameter was 30 µm were installed for the measurement of the temperature transition within the cavity. One was located on the center position of the cavity. The others were in the vicinity of the ceiling copper plate and bottom acrylic plate. Resultant temperature difference between top and bottom surface was 20 K to 60 K depending on the Rayleigh number

5 Figure 4. Experimental rig. Figure 5 shows the temporal transition of the input electric energy as well as the control voltage of energy source. The alternating current was applied for the energy source of the Joule heating. The frequency of the current was 50 Hz due to the commercial electric supply. After 2500 sec, the intensity of internal energy source was kept constant. Input electric energy for the heating was 136W under the steady flow, consequently the internal Joule heating density was q = 1.36x10 5 W/m 3. Rayleigh numbers were Ri = 8.46x10 8 and 1.38x Figure 5 : Applied voltage was controlled in order to stabilize the intensity of internal heating. Electric energy input was maintained constant after 2500 sec

6 The electric conductivity of the carbon graphite as a pair of electrode was 10 6 S/m. Since the value was sufficiently larger than that of the working fluid used in the experiment, the electric potential in the carbon electrode was assumed to be constant. The copper plate used for the cooled surface has so high electric conductivity that the electric potential in the plate could be homogeneously distributed. Therefore the plate was protected by the no conducting substance against short circuit in order to avoid the undesired potential distribution in the test section. The boundary condition of velocity on the surface was assumed to be zero. The Ar-Ion laser was used as a light source for the illumination of the tracer particles in the cavity. Output optical energy was 4 W for the continuous emission configuration. Cylinder lens in conjunction with the rectangular thin slit were used to generate the light sheet in the test section. The thickness of the light sheet was 1 mm. For the image acquisition, a high-speed CMOS camera (IDT Corp) was applied, which was synchronized by the signal generator. Acquisition interval of consecutive images was 0.01 to 1 sec depending on the maximum displacement of particle in images. The pixel number and grayscale accuracy were 1016x1016 and 10 bit respectively. The successive particle images in the illuminated slice were firstly stored into the on-board high-speed memory in the camera. Since the capacity of the internal memory of the camera was 1.3 GB, the capacity restricted the total amount of the image in one sequence. i.e., the maximum recordable duration of the image sequence was several ten second in the present configuration. Afterward, the stored image data were transferred to the host computer through the direct gigabit ether connection. A pixel size of the imaging system in physical domain was 0.1 mm by noting the magnification ratio of the receiving optics and pixel size of the camera. A telecentric imaging system, by which the magnification ratio is independent of the depth of field, was used. Spherical polystyrene particle was used as a tracer of fluid. The arithmetic mean diameter of the tracer particle was 50 µm approximately. The DFT based coarse pre-processing followed by the fine direct cross correlation computation with Gaussian sub-pixel interpolation was performed by means of a portable workstation WSP17G2-G4 with Nvidia GTX480M dual GPGPU (general-purpose computing on graphics processing units). The sizes of reference and interrogation windows for the real-domain correlational computation were 20 pixels and 24 pixels respectively. Resultant spatial resolution of the measured velocity was 2 mm due to the 50 % overlapping window methodology of PIV image processing. Figure 6. An example of the measured particle image after background subtraction

7 Figure 6 is the snapshot of the captured image. The consecutive images were firstly processed to subtract the common background intensity pattern including the reflection from the walls, background image due to the multiple reflections. Computation time for the 1000 images was several tens of minute including pre-processing, main PIV computation and post-processing. Figure 7 depicts the example of the instantaneous velocity map in the cross section of a cavity at t=300 s after the start of Joule heating. Rayleigh number was 1.38x At the instance, flow was not steady and the temperature variation within the cavity was insufficient. Therefore the velocity magnitude remains very low order due to the weak buoyant force. Figure 8 compares the mean velocity vector map from the captured consecutive images during 10 sec. The velocity maps show that the maximum magnitude of downward velocity at the center of the cavity was 5 mm/s, the cycle of the transition of the global flow structure was almost 100 sec, and the spatial scale of the larger vortex in the cavity was 100 mm that is almost the same as the size of cavity. Figure 7. Instantaneous velocity map at the cross section of a cavity at 300 sec after the start of internal heating. The downward plumes were occurred due to the development of the thermal boundary layer in the vicinity of the cooled ceiling (Cu plate). Figure 8. Mean velocity distribution during 10 sec at t=1000 sec (left) and t=1500 sec (right). Long term transition of the entire structure was observed, the direction of rotation was reversed

8 Figure 9. Temporal fluctuations of each velocity components at the center of the cavity measured by PIV. t=480, 1200 and 3000 sec were depicted. Rayleigh number is 8.46x10 8. Figure 9 depicts the temporal profiles of each velocity components at the center of the cavity. At the beginning of the heating process, the temperature difference between working fluid and cooling plate was small and thermal boundary layer was not developed. Therefore the buoyancy effect was small and notable convection was not appeared. After several minute, the bulk temperature was gradually increased due to the homogeneous Joule heating by electricity. The top surface, however, was cooled by the circulating water, the thin thermal boundary layer was grown and downward plumes were induced periodically

9 Figure 10. Temporal transition of velocity components at the center of the cavity in terms of Rayleigh number. The natural convection under higher Rayleigh number causes the frequent velocity modulation due to the unsteady downward plume from the cooling surface. Figure 10 compares the temporal velocity profiles under the different Rayleigh number. Although the time after the start of heating seems to be different, both profiles corresponds the stable flow, moreover the dimensionless time was almost the same between them. Under the stable state, the thermal boundary layer in the vicinity of the cooled surface was fully developed and the mean value of velocity magnitude was also stable. With lower Rayleigh number, interval of velocity fluctuation is longer and horizontal velocity component is almost zero. In contrast, the velocity fluctuation at Ri = 1.38x10 10 contains high frequency component and the magnitude of the downward plume was beyond 10 mm/s. The fluctuation interval, however, seems to be varied due to the temporally and spatially instable detachment of cooled liquid from the top surface. Conclusions Natural convection in a square cavity with internal Joule heating was experimentally investigated by means of particle image velocimetry. The flow field with the vertical temperature gradient due to the homogeneous internal heating and a low temperature ceiling is essentially unstable. With the higher Rayleigh number condition, the flow structure was governed by buoyant and viscous forces in the vicinity of the surface. The instability of the thermal boundary layer induces the characteristic downward flow in a cavity and the turbulent mixing around the center of cavity was significantly affected by the convective behavior of fluid. The temporal interval of the downward plume was varied in terms of the Rayleigh number with internal heating. The resultant magnitude of the velocity was beyond 10 mm/s in the present configuration. References Char M I, Chiang K T (1994) Stability analysis of Benard Marangoni convection in fluids with internal heat generation. J. Phys. D: Appl. Phys. 27: Gurnis M, Davis G F (1985) Numerical study of high Rayleigh number convection in a medium with depth-dependent viscosity. Geophys. J. R. astr Soc. 186: Kulacki F A, Goldstein R J (1972) Thermal convection in a horizontal fluid layer with uniform volumetric energy sources. J. Fluid Mech. 55 :

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