Free Convection in a Water Glass

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1 Solved with COMSOL Multiphysics 4.1. Free Convection in a Water Glass Introduction This model treats free convection in a glass of water. Free convection is a phenomenon that is often disregarded in chemical equipment. Yet, in certain circumstances, it can be of great importance. This can include fermentation processes, casting, and biochemical reactors. Natural convection can also be the leading contributor to transport in small reactors. Model Definition This example considers free convection in a glass of cold water in an environment at room temperature. You model the flow using the CFD Non-Isothermal Flow physics interface. The aim of the model is to compute the flow pattern and temperature distribution. Initially, the glass and the water are both at 5 C, as if they had been taken directly from a refrigerator. The surrounding air and table are held constant at 25 C. The glass wall has a finite thickness with a specific thermal conductivity. Due to rotational symmetry, you can model the whole system in 2D, using axisymmetric geometry. The geometry and model domain are shown in Figure 1 below. Figure 1: Geometry and computational domain. FREE CONVECTION IN A WATER GLASS 1

2 Solved with COMSOL Multiphysics 4.1. COPYRIGHT 2010 COMSOL AB. The global mass and momentum balances for non-isothermal flow are coupled to an energy balance, where transport occurs through convection and conduction. In the energy balance in the wall of the glass, consider conduction as the only transport mechanism. The thermal properties for the glass wall are assumed to be of Silica Glass. BOUNDARY CONDITIONS Assuming perfect contact between the surface and the bottom of the glass, you can set the boundary condition to a temperature of 25 C. The boundary at the symmetry axis is set to the symmetry/insulating condition, and the other exterior boundaries include a convective heat flux, driven by the temperature difference between the glass and the surrounding atmosphere: q = h( T T ) Here q is the inward heat flux and h is the heat transfer film coefficient. The Heat Transfer Module comes with a library of heat transfer coefficient functions (Ref. 1) that you can access easily and use in this model. For the flow field, no-slip conditions apply on the interior boundaries (between the glass and the water) while an axisymmetry condition applies on the axis of rotation and a slip condition on the open surface. In this case, the simulation runs for a period of 5 minutes. Results The heat fluxes through the top surface, side wall and bottom of the glass are shown in Figure 2. Because of the low values of the heat transfer film coefficients, a large part of the heat is conducted to the water through the bottom boundary When the fluid is heated at the bottom of the glass, the local density decreases, thereby inducing a flow within the glass. Figure 3 shows temperature distributions for equally distanced time intervals. The buoyancy-driven flow induces recirculation zones in the glass. These recirculation zones are clearly seen in a streamline plot of the velocity field. Figure 4 shows the streamlines for the same output times as the previous figure. 2 FREE CONVECTION IN A WATER GLASS

3 Solved with COMSOL Multiphysics 4.1. Figure 2: Heat flux through the top surface (solid line), side wall (dotted line), and bottom of the glass (dash line). Figure 3: Temperature distribution at t = 72, 144, and 216 s. FREE CONVECTION IN A WATER GLASS 3

The following plot shows the temperature distribution in the glass after 5 minutes.

4 Solved with COMSOL Multiphysics 4.1. COPYRIGHT 2010 COMSOL AB. Figure 4: Velocity field after t=72, 144, and 216 s visualized with streamlines. The following plot shows the temperature distribution in the glass after 5 minutes. Figure 5: Temperature distribution after 5 minutes. Reference 1. A. Bejan, Heat Transfer, Wiley, FREE CONVECTION IN A WATER GLASS

5 Solved with COMSOL Multiphysics 4.1. Model Library path: Heat_Transfer_Module/Tutorials/cold_water_glass Modeling Instructions MODEL WIZARD 1 In the Model Builder window, click Untitled.mph (root). 2 Go to the Model Wizard window. 3 Click the 2D axisymmetric button. 4 Click Next. 5 In the Add Physics tree, select Fluid Flow>Non-Isothermal Flow>Laminar Flow (nitf). 6 Click Next. 7 In the Studies tree, select Preset Studies>Time Dependent. 8 Click Finish. GEOMETRY 1 Polygon 1 (pol1) 1 In the Model Builder window, right-click Model 1 (mod1)>geometry 1 and choose Polygon. 2 Go to the Settings window for Polygon. 3 Locate the Coordinates section. In the r edit field, type In the z edit field, type Polygon 2 (pol2) 1 In the Model Builder window, right-click Geometry 1 and choose Polygon. 2 Go to the Settings window for Polygon. 3 Locate the Coordinates section. In the r edit field, type In the z edit field, type Click the Build All button. FREE CONVECTION IN A WATER GLASS 5

6 Solved with COMSOL Multiphysics 4.1. COPYRIGHT 2010 COMSOL AB. DEFINITIONS Integration 1 (intop1) 1 In the Model Builder window, right-click Model 1 (mod1)>definitions and choose Integration. 2 Go to the Settings window for Integration. 3 Locate the Source Selection section. From the Geometric entity level list, select Boundary. 4 Click Paste Selection. 5 Go to the Paste Selection dialog box. 6 In the Selection edit field, type 7. 7 Click the OK button. Integration 2 (intop2) 1 In the Model Builder window, right-click Definitions and choose Integration. 2 Go to the Settings window for Integration. 3 Locate the Source Selection section. From the Geometric entity level list, select Boundary. 4 Select Boundary 5 only. Variables 1 1 In the Model Builder window, right-click Definitions and choose Variables. 2 Go to the Settings window for Variables. 3 Locate the Geometric Scope section. From the Geometric entity level list, select Boundary. 4 Click Paste Selection. 5 Go to the Paste Selection dialog box. 6 In the Selection edit field, type 7. 7 Click the OK button. 8 Go to the Settings window for Variables. 9 Locate the Variables section. In the Variables table, enter the following settings: NAME Vlength EXPRESSION intop1(1) 6 FREE CONVECTION IN A WATER GLASS

7 Solved with COMSOL Multiphysics 4.1. Variables 2 1 In the Model Builder window, right-click Definitions and choose Variables. 2 Go to the Settings window for Variables. 3 Locate the Geometric Scope section. From the Geometric entity level list, select Boundary. 4 Select Boundary 5 only. 5 Go to the Settings window for Variables. 6 Locate the Variables section. In the Variables table, enter the following settings: NAME Hlength EXPRESSION intop2(1) MATERIALS 1 In the Model Builder window, right-click Model 1 (mod1)>materials and choose Open Material Browser. 2 Go to the Material Browser window. 3 Locate the Materials section. Clear the selection in the Materials tree. 4 Go to the Material Browser window. 5 Locate the Materials section. In the Materials tree, select Built-In>Silica glass. 6 Right-click and choose Add Material to Model from the menu. Silica glass 1 In the Model Builder window, right-click Materials and choose Open Material Browser. 2 Go to the Material Browser window. 3 Locate the Materials section. Clear the selection in the Materials tree. 4 Go to the Material Browser window. 5 Locate the Materials section. In the Materials tree, select Built-In>Water, liquid. 6 Right-click and choose Add Material to Model from the menu. Water, liquid 1 In the Model Builder window, click Water, liquid. 2 Select Domain 2 only. NON-ISOTHERMAL FLOW (NITF) 1 In the Model Builder window, click Model 1 (mod1)>non-isothermal Flow (nitf). 2 Go to the Settings window for Non-Isothermal Flow. FREE CONVECTION IN A WATER GLASS 7

8 Solved with COMSOL Multiphysics 4.1. COPYRIGHT 2010 COMSOL AB. 3 Click to expand the Discretization section. 4 From the Discretization of fluids list, select P2 + P1. Initial Values 1 This setting gives an initial pressure field consistent with the volume force. The initial pressure field must be consistent with the constraint at the pressure (see the section Pressure Point Constraint below). 1 In the Model Builder window, click Initial Values 1. 2 Go to the Settings window for Initial Values. 3 Locate the Initial Values section. In the p edit field, type nitf.rho*g_const*(0.1-z). 4 In the T edit field, type [K]. Heat Transfer in Solids 1 1 In the Model Builder window, right-click Non-Isothermal Flow (nitf) and choose Heat Transfer in Solids. 2 Select Domain 1 only. Volume Force 1 1 In the Model Builder window, right-click Non-Isothermal Flow (nitf) and choose the domain setting Laminar Flow>Volume Force. 2 Select Domain 2 only. 3 Go to the Settings window for Volume Force. 4 Locate the Volume Force section. Specify the F vector as 0 r -g_const*nitf.rho z Wall 2 1 In the Model Builder window, right-click Non-Isothermal Flow (nitf) and choose the boundary condition Laminar Flow>Wall. 2 Select Boundary 5 only. 3 Go to the Settings window for Wall. 4 Locate the Boundary Condition section. From the Boundary condition list, select Slip. Because this is a closed cavity flow, you must lock the pressure. 8 FREE CONVECTION IN A WATER GLASS

9 Solved with COMSOL Multiphysics 4.1. Pressure Point Constraint 1 1 In the Model Builder window, right-click Non-Isothermal Flow (nitf) and choose Points>Pressure Point Constraint. 2 Select Vertex 6 only. Temperature 1 1 In the Model Builder window, right-click Non-Isothermal Flow (nitf) and choose the boundary condition Heat Transfer>Temperature. 2 Select Boundary 2 only. 3 Go to the Settings window for Temperature. 4 Locate the Temperature section. In the T 0 edit field, type [K]. Convective Cooling 1 1 In the Model Builder window, right-click Non-Isothermal Flow (nitf) and choose the boundary condition Heat Transfer>Convective Cooling. 2 Go to the Settings window for Convective Cooling. 3 Locate the Boundaries section. Click Paste Selection. 4 Go to the Paste Selection dialog box. 5 In the Selection edit field, type 7. 6 Click the OK button. The function you loaded is valid for cases of natural convection to air for horizontal surfaces. It requires you to input the ambient air temperatures plus a length scale, L. 7 Go to the Settings window for Convective Cooling. 8 Locate the Heat Flux section. From the Heat transfer coefficient list, select External natural convection. 9 In the L edit field, type Vlength. 10 In the T ext edit field, type [K]. Convective Cooling 2 1 In the Model Builder window, right-click Non-Isothermal Flow (nitf) and choose the boundary condition Heat Transfer>Convective Cooling. 2 Select Boundary 5 only. The inclination of the wall is small enough that you can assume it to be vertical when calculating the heat transfer coefficient 3 Go to the Settings window for Convective Cooling. FREE CONVECTION IN A WATER GLASS 9

10 Solved with COMSOL Multiphysics 4.1. COPYRIGHT 2010 COMSOL AB. 4 Locate the Heat Flux section. From the Heat transfer coefficient list, select External natural convection. 5 From the [[ExtNaturalConvectionType_combobox0]] list, select Horizontal plate, upside. 6 In the L edit field, type Hlength. 7 In the T ext edit field, type [K]. MESH 1 1 In the Model Builder window, right-click Model 1 (mod1)>mesh 1 and choose Build All. 2 In the Model Builder window s View Menu, select Show More Options. Free convection is a rather slow phenomenon and therefore, run the problem for a period of 5 minutes. Furthermore, you need to specify the solver tolerances in accordance with the physics. Furthermore, make use of a small time step to visualize the intricate details of the transient flow STUDY 1 Step 1: Time Dependent 1 In the Model Builder window, click Study 1>Step 1: Time Dependent. 2 Go to the Settings window for Time Dependent. 3 Locate the Study Settings section. In the Times edit field, type range(0,0.2,5)[min]. 4 In the Model Builder window, click Step 1: Time Dependent. 5 Go to the Settings window for Time Dependent. 6 Locate the Study Settings section. Select the Relative tolerance check box. 7 In the associated edit field, type 1e-5. 8 In the Model Builder window, right-click Study 1>Solver Configurations and choose Show Default Solver. 9 Expand the Study 1>Solver Configurations node. Solver 1 1 In the Model Builder window, expand the Study 1>Solver Configurations>Solver 1 node, then click Time-Dependent Solver 1. 2 Go to the Settings window for Time-Dependent Solver. 3 Click to expand the Time Stepping section. 4 Select the Maximum step check box. 10 FREE CONVECTION IN A WATER GLASS

11 Solved with COMSOL Multiphysics In the associated edit field, type 1. 6 In the Model Builder window, right-click Study 1 and choose Compute. RESULTS 2D Plot Group 1 1 Go to the Settings window for 2D Plot Group. 2 Locate the Data section. From the Time list, select Click the Plot button. Produce the other three snapshots in Figures 3 and 5 in the same way but with Solution at time set to 144, 216, and 300 respectively 2D Plot Group 2 1 In the Model Builder window, right-click Results and choose 2D Plot Group. 2 Go to the Settings window for 2D Plot Group. 3 Locate the Data section. From the Time list, select Right-click Results>2D Plot Group 2 and choose Streamline. 5 Go to the Settings window for Streamline. 6 In the upper-right corner of the Expression section, click Replace Expression. 7 From the menu, choose Non-Isothermal Flow (Laminar Flow)>Velocity field (u, w). 8 Click the Plot button. Produce the other three snapshots in Figure 4 in the same way but with Solution at time set to 144, and 216 respectively Derived Values 1 In the Model Builder window, right-click Results>Derived Values and choose Line Integration. 2 Go to the Settings window for Line Integration. 3 Locate the Selection section. Click Paste Selection. 4 Go to the Paste Selection dialog box. 5 In the Selection edit field, type Click the OK button. 7 Go to the Settings window for Line Integration. 8 Locate the Expression section. In the Expression edit field, type -nitf.ntflux. 9 Select the Description check box. FREE CONVECTION IN A WATER GLASS 11

12 Solved with COMSOL Multiphysics 4.1. COPYRIGHT 2010 COMSOL AB. 10 In the associated edit field, type Flux_top. 11 Click to expand the Integration Settings section. 12 Select the Compute surface integral (axial symmetry) check box. 13 Click the Evaluate button. 14 In the Model Builder window, right-click Derived Values and choose Line Integration. 15 Go to the Settings window for Line Integration. 16 Locate the Selection section. Click Paste Selection. 17 Go to the Paste Selection dialog box. 18 In the Selection edit field, type Click the OK button. 20 Go to the Settings window for Line Integration. 21 Locate the Expression section. In the Expression edit field, type -nitf.ntflux. 22 Select the Description check box. 23 In the associated edit field, type Flux_wall. 24 Locate the Integration Settings section. Select the Compute surface integral (axial symmetry) check box. 25 Click the Evaluate button. 26 In the Model Builder window, right-click Derived Values and choose Line Integration. 27 Select Boundary 2 only. 28 Go to the Settings window for Line Integration. 29 Locate the Expression section. In the Expression edit field, type -nitf.ntflux. 30 Select the Description check box. 31 In the associated edit field, type Flux_bottom. 32 Locate the Integration Settings section. Select the Compute surface integral (axial symmetry) check box. 33 Click the Evaluate button. 1D Plot Group 3 1 In the Model Builder window, right-click Results and choose 1D Plot Group. 2 Go to the Settings window for 1D Plot Group. 3 Locate the Plot Settings section. Select the Title check box. 4 In the associated edit field, type Heat Flux vs Time. 5 Select the x-axis label check box. 12 FREE CONVECTION IN A WATER GLASS

13 Solved with COMSOL Multiphysics In the associated edit field, type Time. 7 Select the y-axis label check box. 8 In the associated edit field, type Heat Flux. 9 Click to expand the Grid section. 10 Select the Manual spacing check box. 11 In the x spacing edit field, type In the y spacing edit field, type Right-click Results>1D Plot Group 3 and choose Table Plot. 14 Go to the Settings window for Table Plot. 15 Locate the Coloring and Style section. Find the Line style subsection. From the Color list, select Blue. 16 Click to expand the Legends section. 17 Select the Show legends check box. 18 In the Model Builder window, right-click 1D Plot Group 3 and choose Table Plot. 19 Go to the Settings window for Table Plot. 20 Locate the Data section. From the Table list, select Table Locate the Coloring and Style section. Find the Line style subsection. From the Line list, select Dashed. 22 From the Color list, select Blue. 23 Locate the Legends section. Select the Show legends check box. 24 In the Model Builder window, right-click 1D Plot Group 3 and choose Table Plot. 25 Go to the Settings window for Table Plot. 26 Locate the Data section. From the Table list, select Table Locate the Coloring and Style section. Find the Line style subsection. From the Line list, select Dash-dot. 28 From the Color list, select Blue. 29 Locate the Legends section. Select the Show legends check box. 30 Click the Plot button. FREE CONVECTION IN A WATER GLASS 13

Non-Isothermal Heat Exchanger

Non-Isothermal Heat Exchanger The following example builds and solves a conduction and convection heat transfer problem using the Heat Transfer interface. The example concerns a stainless-steel MEMS heat