AC : A STUDENT PROJECT ON RAYLEIGH-BENARD CONVECTION

Transcription

1 AC : A STUDENT PROJECT ON RAYLEIGH-BENARD CONVECTION John Matsson, Oral Roberts University O. JOHN E. MATSSON is an Associate Professor of Mechanical Engineering and Chair of the Engineering, Physics and Physical Science Department at Oral Roberts University in Tulsa, Oklahoma. He earned M.S. and Ph.D. degrees from the Royal Institute of Technology in Stockholm, Sweden in 1988 and 1994, respectively. American Society for Engineering Education, 2008 Page

2 A Student Project on Rayleigh-Bénard Convection Abstract This paper describes a project where a group of five undergraduate engineering students in the heat transfer course designed, built and tested a portable Rayleigh-Bénard convection apparatus for visualization of the instability patterns that appear in this flow during laminar to turbulent transition. The students had to choose appropriate materials, optimize production cost, determine fabrication techniques for the apparatus, and implement this in the workshop. The students designed the apparatus using SolidWorks TM. Heating of the copper plate was provided by a heating element. The different flow patterns that appear in this flow were visualized by mixing a small amount of Afflair 100 Silver Pearl Pigment in Dow Corning 200 silicon oil. COSMOSFloWorks TM software was used to simulate the instabilities that occurred in the experiments. The constructed device is a valuable learning tool for both present and future engineering students. Introduction Rayleigh-Bénard convection and Taylor-Couette flow has been classical cases to study the process of laminar-turbulent transition. The first case occur when a layer of fluid between parallel plates is heated from below while the second flow case appear between two concentric vertical rotating cylinders. In this paper we will describe a student project on Rayleigh-Bénard convection. Many textbooks on introduction to heat transfer includes this fundamental flow case. It is always desirable to have the ability to demonstrate the different topics that are covered in a course. Therefore, it was decided to include the design, fabrication and testing of a Rayleigh- Bénard convection apparatus as a course project. Bénard 1 was the first scientist to experimentally study convection in a fluid layer with a free surface heated from below. Rayleigh 2 used linear stability analysis to theoretically explain and study the stability of the fluid motion between horizontal parallel plates with the hotter plate at the bottom. Chandrasekhar 3 completed the linear stability analysis of Rayleigh-Bénard convection and Koschmieder 4 showed the development of the research in this area during the following couple of decades. There are only a few experiments including Koschmieder and Pallas 5 and Hoard et al. 6 that have been able to produce the concentric ring cells in Rayleigh-Bénard convection that according to stability theory should appear with a rigid upper boundary condition. A group of five undergraduate senior students designed the experimental setup for flow visualizations. The only requirement for the design from the instructor was that the fluid layer would have a specified diameter and that the thickness of the layer could be varied. One student modeled the convection cell using SolidWorks TM while another student completed the bill of materials. A third student concentrated on the COSMOSFloWorks TM simulations after the model was finished in SolidWorks TM. The remaining two students were building the apparatus under Page

3 ( the guidance from the technician. All five students were working together during testing of the apparatus including flow visualizations and measurements of temperatures for determination of Rayleigh numbers. Theory The instability of the flow between two parallel plates heated from below is governed by the Rayleigh number Ra. )L c 3 Ra = g T 1 T 2 Pr (1) 2 where g is acceleration due to gravity, is the coefficient of volume expansion, T 1,T 2 are the temperatures of the hot and cold surfaces respectively, L c is the distance between the surfaces, Pr is the Prandtl number and is the kinematic viscosity of the fluid. Below the critical Ra crit =1708 for a rigid upper surface, the flow is stable but convective currents will develop above this Rayleigh number. For the case of a free upper surface, the theory predicts a lower critical Rayleigh number Ra crit =1101. The non-dimensional wave number of this instability is determined by = 2 L c (2) where is the wave length of the instability. The critical wave numbers are crit =3.12,2.68 for rigid and free surface boundary conditions, respectively. Construction The design of the experimental apparatus was developed using SolidWorks TM, see Figure 1. The design consisted of a bottom unit including a quarter inch thick copper plate and a lateral outer circular wall made of Plexiglas. Another unit including the top wall was designed to slide inside the outer unit and thereby enable the depth of the fluid layer to be varied. The top wall was cooled using water and the bottom wall was heated using a heating element from Hi-Heat Industries Inc. The diameter of the fluid layer was 28 cm and the depth of the fluid layer could be as large as 3 cm. The fluid used was Dow Corning 200 silicon oil with a viscosity of 100 cs (centistokes). The outer unit was insulated with a layer of Styrofoam. Care has to be taken when heating the copper plate so that a linear temperature profile is obtained. Page

4 Figure 1. SolidWorks TM model of the Rayleigh-Bénard convection apparatus. The apparatus was initially designed for the cooling water to enter at one location on the rim and exit on the opposite side. However, it has been shown by Koschmieder 4 that in order to get the concentric circular instability pattern in experimental Rayleigh-Benard convection, you have to introduce the cooling water at the center and let is spread out in the radial direction. Also, the circular flow patterns are easier to find if the top wall is made of sapphire instead of glass. However, due to the very high cost for a sapphire plate, it was decided to use transparent tempered glass in this student project. Figure 2 shows the appearance of the cell pattern when the top plate is in place. It can be seen that the outermost cells extend almost around the whole circumference. However, closer to the center region the cells are more irregular and ordered randomly. Several test were made by varying the flow rate of the cooling water and the thickness of the silicon fluid layer but we have so far not been successful in obtaining a concentric circular pattern throughout the whole fluid region. Such a pattern is desirable in order to make comparisons with linear stability theory that predicts the appearance of such flows, see Koschmieder 4. Page

5 Figure 2. Photo of Rayleigh-Bénard experimental set up with a rigid boundary condition at the top wall (Ra = 2490). Figure 3. Photo of Rayleigh-Bénard experimental set up with a free upper surface at Ra = In Figure 3 it is shown the flow that occur when there is no top wall. In this case regular hexagonal cells are formed due the surface tension of the silicon oil. The flow pattern in this case is much more regular than in the case with a top lid on the fluid layer where surface tension will not affect the stability of the flow. Page

6 CosmosFloWorks Simulations A model of the apparatus was designed in SolidWorks TM and exported to CosmosFloWorks TM. The dimensions of the model used in the simulations were the same as for the experimental apparatus. Using the COSMOSFloWorks TM wizard, the SI unit system was first chosen followed by the choice of the internal analysis type option. The cold top wall was set to 20 ºC and the hot bottom wall was set to ºC. Simulations were made only for the case of a rigid boundary condition for the cold wall as CosmosFloWorks TM is not able to handle flows with a free surface boundary condition. The fluid used in the simulations was water but the Rayleigh number was the same as in the experiments. Obviously, gravity was included as a physical feature in the general setting of COSMOSFloWorks TM. The upper part of figure 4 shows the temperature field and the lower part shows the wall normal velocity component (Z-velocity) at a Rayleigh number Ra=2.8Ra crit. Regions of hot fluid are seen moving away from the hotter wall indicating the presence of the roll cell instability. Figure 4. The temperature field (above) and wall normal velocity component (below) in a cross section of the fluid layer at Ra = The hot wall is the bottom wall. Page

7 In the lower part of figure 4 are contour plots also shown of the wall normal velocity component (Z-direction). Alternating regions of low and high velocity are visible in the X-direction. The wave number was determined to be =2.66 for this Rayleigh number. The computer simulations were made as 2-D steady calculations in order to get computation time down to a reasonable level. From the experiments the wave number was determined to be =3.8 which is 43% higher than in the simulations. However, the simulations reflect the general roll cell patterns observed in the experiments. The students have learned through this process that it sometimes can be difficult to get good agreement between experimental and simulated results. Assessment There were ten students enrolled in the ME 433 Heat Transfer course. Two projects were given with five students in each project. Each student had to write an individual project report at the time of the completion of the project. The project contributed to ten percent of the final grade in the course. One of the projects has been described in this paper and the other project was completed by the junior students in the course. The student performance was assessed based on the written final project report but also on their diligence in working on the project towards its completion. A log-book was available for each project where the student filled out a detailed description of their contribution and the time spent working on the project outside of class. The students also had to write a one-page summary as an appendix to the final project report that indicated their contribution. The projects contributed to certain course outcomes and proficiencies/capacities such as intellectual creativity, critical thinking, communication skills, leadership capacity and interpersonal skills. Conclusions This paper has shown a Rayleigh-Bénard experimental project by undergraduate students. It was initiated as a project in the heat transfer course for visualization of fluid flow instabilities caused by a vertical temperature gradient and buoyancy or surface tension depending on the upper boundary condition. Improvements of the existing experimental set up would be to be able to visualize the ring cells that are the most ordered form of the instability for the rigid upper boundary condition. The cost of building the experiment described in this paper was $613. The project cost is detailed in Table 1. Page

8 Vendor Part Number Item Description Quantity Price Grainger 4CB72 Temperature Probe 1 $72.30 Hi-Heat Industries 07INT010LP Heating Element, 1 $ , 120 V, 340 W McMaster-Carr 94115K275 Neoprene O-ring 4 $14.67 McMaster-Carr 8560K381 Clear Cast Acrylic 2 $ Sheet 1.25" Thick, 12" X 12" McMaster-Carr 9821K17 Alloy 110 Copper Sheet 1 $ #8 Mirror Finish.250" Thick, 12" X 12" Robertson Glass N/A Tempered Glass 2 $86.62 Total $ Table 1. Detailed Project Cost Bibliography 1. Bénard, H. Les tourbillons cellulaires dans une nappe liquide, Rev. Gen. Sciences Pure Appl. 11, , , Rayleigh, L. On convection currents in a horizontal layer of fluid when the higher temperature is on the under side., Phil. Mag. 32, , Chandrasekhar, S. Hydrodynamic and Hydromagnetic Stability, Dover, Koschmieder, E.L. Bénard cells and Taylor vortices, Cambridge University Press, Koschmieder, E.L. and Pallas, S.G. Heat transfer through a shallow, horizontal convecting fluid layer., Int. J. Heat Mass Transfer 17, , Hoard, C.Q., Robertson, C.R., and Acrivos, A. Experiments on the cellular structure in Bénard convection., Int. J. Heat Mass Transfer 13, , Page