Thermo-Fluids Laboratory

      Airflow over Flat Plate-Boundary Layer Numerical Simulation and Experimental Design Introduction The objective of this numerical experiment includes two aspects: 1) simulate airflow over a flat plate in COMSOL (a commercial finite element package, available on vlabs/Mechanical Engineering or General Engineering 2/COMSOL Multiphysics 5.5.); 2) use the simulation results to guide the design of an experiment, in which you will measure the thickness of the velocity boundary layer over the surface of a flat plate. Reading: Heat transfer from surface of flat plates has been investigated extensively. Thermal boundary layers for fully heated or partially heated flat plates are presented in your heat transfer textbook by Incropera, DeWitt, Bergman, and Lavine entitled “Fundamentals of Heat and Mass Transfer”, Chapter 7; Theories of velocity boundary layer over a flat surface has been elaborated in detail in Fundamentals of Fluid Mechanics textbook by Munson, Chapter 9. The Velocity Boundary Layer Consider flow over a flat plate, as shown in Figure 6.1 (Heat transfer textbook). When fluid particles contact the surface, they assume zero velocity. These particles then act to retard the motion of particles in the adjoining fluid layer, which act to retard the motion of particles in the next layer, and so on until, at a distance y=δ from the surface, the effect becomes negligible. This retardation of fluid motion is associated with shear stresses τ acting in planes that are parallel to the fluid velocity. With increasing distance y from the surface, the x velocity component of the fluid, u, must then increase until it approaches the free stream value uꝏ. The quantity δ is (0,0) 2 DOE, Fall 2020 termed the velocity boundary layer thickness, and it is typically defined as the value of y for which, u u 0.99 = ∞ (1) For laminar flow over an isothermal plate, based on the similarity solution from the heat transfer textbook, the velocity boundary layer thickness could be expressed as: 5 5 Re / x x u vx δ ∞ = = (2) For turbulent flow over an isothermal plate, to a reasonable approximation, the velocity boundary thickness may be shown as: 1/5 0.37 Rex δ x − = (3) Comparing equation (2) and (3), we see that turbulent boundary layer growth is much more rapid (δ varies as x4/5 in contrast to x1/2 for laminar flow). Transition from Laminar to Turbulent Flow Boundary layer development on a flat plate is illustrated in Figure 6.6. In many cases, laminar and turbulent flow conditions both occur, with the laminar section preceding the turbulent section. The transition from laminar to turbulent flow is ultimately due to triggering mechanisms, such as the interaction of unsteady flow structures that develop naturally within the fluid or small disturbances that exist within many typical boundary layers. It is reasonable to assume that transition begins at some 3 DOE, Fall 2020 location xc. The location is determined by the critical Reynolds number Rex,c. For flow over a flat plate, Rex,c is known to be: 5 Re , 5 10 c x c ρu x µ ∞ = = × (4) For a fixed location x, there is a critical velocity associated with the critical Reynolds number Rex,c, it can be calculated by: , , Rex c c u x µ ρ ∞ = (4’) The Thermal Boundary Layer Just as a velocity boundary layer develops when there is fluid flow over a surface, a thermal boundary layer must develop if the fluid free stream and surface temperatures differ. Consider flow over an isothermal flate plate (Figure 6.2). At the leading edge, the temperature profile is uniform, with T(y)=Tꝏ. However, fluid particles that come into contact with the plate achieve thermal equilibrium at the plate’s surface temperature. In turn, these particles exchange energy with those in the adjoining fluid layer, and temperature gradients develop in the fluid. The region of the fluid in which these temperature gradients exist is the thermal boundary layer, and its thickness δt is typically defined as the value of y for which: 0.99 s s T T T T∞ − = − (5) With increasing distance from the leading edge, the effects of heat transfer penetrate further into the free stream and the thermal boundary layer grows. For laminar flow, the ratio of the velocity to thermal boundary layer thickness is: 1/3 Pr t δ δ ≈ (6) For turbulent flow, boundary layer development is influenced strongly by random fluctuations in the fluid and not by molecular diffusion. Hence relative boundary layer growth does not depend on the value of Pr. Therefore, for turbulent flow, 4 DOE, Fall 2020 t δ δ ≈ (7) Flat Plat in parallel flow with unheated starting length When there is an unheated starting length upstream of a heat section, as shown in Figure 7.4. Velocity boundary layer growth begins at x=0, while thermal boundary layer development begins at x=ξ. Hence there is no heat transfer for 0 ≤ ≤ x ξ . Through the use of an integral boundary layer solution, you can calculate the average heat transfer coefficient: , 1 theory x theory L h h dx L = ∫ (8) Here, hx,theory can be obtained by the heat flux balance at y=0, ( ) ( ) 0, 0, x xs f y x f x s y x T q hT T k y k T h TT y ∞ = ∞ = ∂ = − =− ∂ ∂ = − − ∂ (9) On the other hand, if you know the power generated in the heating element, you could calculate the average heat transfer coefficient using: ( ) t t s Q q hT T L w = = − ∞ ⋅ (10) Here, w is the width of the plate; Qt is the power in W; qt is the power density in W/m2 . Numerical Simulation 5 DOE, Fall 2020 Figure 1 Schematic Illustration of the flat plate in parallel flow with unheated length As shown in Figure 1, we are tying to simulate the laminar flow passing through a flat plate, which is made of resin in the unheated length (1.2 inch) and copper in the heated length (4.8 inch). The width of the flat plat is 1.25 inch. From x=0 to x=ξ, the flat plate surface temperature Ts equals the air temperature Tꝏ; with x> ξ, a piece of copper has been heated with an electric heater by 4 Watts, which will raise the flat plate surface temperature. A short area has been added before the leading edge of the flat plate to avoid the effect of boundary condition brought to the edge. According to the introduction, there will be two boundary layers developed along the flat plate surface: velocity boundary layer and thermal boundary layer. A COMSOL model has been built to simulate the phenomenon. Please follow the steps below to analyze the simulation results. 1. Analyze how the velocity boundary layer thickness has been developed in the x direction. Use a 1D graph to demonstrate it (x axis is the x coordinate, y axis is the boundary layer thickness). 2. Analyze how the thermal boundary layer thickness has been developed in the x direction. Use a 1D graph to demonstrate it (x axis is the x coordinate, y axis is the boundary layer thickness). 3. Select a x coordinate, perform a parametric study of velocity. Use a 1D graph to demonstrate the correlation of velocity and boundary layer thickness (x axis is the velocity, y axis is the velocity boundary layer thickness). Use Eq. 4’ to back calculate the critical velocity to ensure that Rex is smaller than Rex,c, so the flow is laminar. 6 DOE, Fall 2020 4. Plot δ as a function of Reynolds number with a fixed x (same x you picked in question 3). Compare the results with δ from equation (2). 5. (Bonus) Plot δt as a function of Reynolds number with a fixed x (same x you picked in question 3). Compare the results with δtfrom equation (6). 6. (Bonus) Calculate the average heat transfer coefficient based on equation (10). Note: Ts has been calculated by the average of temperature across the copper surface in the Comsol model as a global variable. Design of Experiment Objective: Measure the velocity profiles in the velocity boundary layer of laminar flow over a flat plate in a wind tunnel. Recommended Devices and Instruments: A Flat Plate: the flat plate should have a specified designed leading edge for the smooth development of the velocity boundary layer. It is recommended that the flat plate being placed in a wind tunnel to provide a uniform and stable laminar flow. Funnel with pressure measurement: The funnel is connected with a Betz micro manometer, which can measure the absolute pressure of the air before existing the funnel. With this you can carefully set the speed of the flow from the funnel and calibrate the hot wire. Flow velocity measurements: Hot Wire Anemometry (HWA) of Dantec is capable to measure flow speeds in and around the velocity boundary layer. It is capable to measure velocity from a few cm/s to 200 m/s. Miniature wire probe should be used. The probe is 1.25mm long plated tungsten wire with a diameter of 5μm, which results in a short response time. The height of the HWA should be able to be varied by a small increment (depending on the thickness of the boundary layer). Figure 2 left: Betz manometer Middle: Funnel with probe for calibration Right: Flat plate Task 1: Describe how you would scope your experiment and select appropriate apparatus. Be sure to specify: the dimension of the flat plate; the air velocity range 7 DOE, Fall 2020 you would like to test; the critical velocity associated with the critical Reynolds number; the thickness of the velocity layer; what are the operating range and accuracy requirements for each apparatus. Task 2: Describe how you would calibrate the HWA. Be sure to specify: how to calibrate the air velocity of HWA by the pressure from Betz manometer; the velocity range you should calibrate. Task 3: Describe how you would set up the experiment. Be sure to specify: where should you set up the HWA (x coordinate); how to vary the height of HWA probe to measure velocity at different y (how much is your increment for each measurement). Task 4: Describe how you would postprocess your data. Be sure to specify: what theoretical calculation are you going to use as a comparison for your experimental data; how are you going to make comparison.