Free Convection in a Water Glass

This example 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 — for example, in 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 at room temperature. You model the flow using the Nonisothermal Flow interface. The aim of this tutorial is to compute the flow pattern and the 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 an axisymmetric geometry. The geometry and model domain are shown in Figure 1 below.

Figure 1: Geometry and computational domain.

The global mass and momentum balances for nonisothermal flow are coupled to an energy balance, where heat transport occurs through convection and conduction.

For the energy balance in the wall of the glass, only the conduction is considered. The thermal properties for the glass wall are assumed to be of silica glass.

Boundary Conditions

Assuming perfect contact between the table surface and the bottom of the glass, you can set the boundary condition to a temperature of 25°C. At the top and outer surfaces, use a convective heat flux boundary condition driven by the temperature difference between the glass and the surrounding atmosphere:

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 application.

For the flow field, no slip conditions apply on the interior boundaries (between the glass and the water) while an axial symmetry 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 2 minutes.

Results and Discussion

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, most of the heat is conducted to the water through the bottom boundary.

Figure 2: Heat flux through the top surface (dottet line), side wall (dashed line), and bottom of the glass (solid line).

When the fluid is heated at the bottom of the glass, the local density decreases, thereby inducing a flow inside the glass. Figure 3 shows temperature distributions for 30, 60 and 81s.

Figure 3: Temperature distribution at times 30, 60, and 81 s.

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.

Figure 4: Velocity field at times 30, 60, and 81 s visualized with streamlines.

The following plot shows the temperature distribution in the glass after 2 minutes.

Figure 5: Temperature distribution at time 2 minutes.

Reference

1. A. Bejan, Heat Transfer, John Wiley & Sons, 1993.

Heat_Transfer_Module/Tutorials,_Forced_and_Natural_Convection/cold_water_glass

Modeling Instructions

From the menu, choose .

New

In the window, click

Model Wizard

In the window, click

In the tree, select .

Click .

Click

In the tree, select .

Click

Global Definitions

Parameters 1

In the window, under click .

In the window for , locate the section.

In the table, enter the following settings:


Name	Expression	Value	Description
r_top	4.5[cm]	0.045 m	Radius on the top
r_bottom	3.5[cm]	0.035 m	Radius at the bottom
Hg	10[cm]	0.1 m	Height of the glass
h_wall	0.13[cm]	0.0013 m	Thickness of the glass wall
h_bottom	0.3[cm]	0.003 m	Thickness of the bottom
Vlength	sqrt((r_top-r_bottom)^2+Hg^2)	0.1005 m	Length of the outer wall
rho0	1000[kg/m^3]	1000 kg/m³	Density reference

Geometry 1

Polygon 1 (pol1)

In the toolbar, click

In the window for , locate the section.

In the table, enter the following settings:


r (m)	z (m)
0	h_bottom
r_bottom-h_wall	h_bottom
r_top-h_wall	Hg
0	Hg

In the toolbar, click

Polygon 2 (pol2)

In the toolbar, click

In the window for , locate the section.

In the table, enter the following settings:


r (m)	z (m)
0	Hg
0	0
r_bottom	0
r_top	Hg

In the toolbar, click

Add Material

In the toolbar, click

to open the window.

Go to the window.

In the tree, select .

Click in the window toolbar.

In the tree, select .

Click in the window toolbar.

In the toolbar, click

to close the window.

Materials

Silica glass (mat1)

In the window, under click .

Select Domain 1 only.

Water, liquid (mat2)

In the window, click .

Select Domain 2 only.

In the window for , locate the section.

Click

In the dialog box, type Water in the text field.

Click .

Laminar Flow (spf)

Set the flow as incompressible to ensure mass conservation, and then lock the pressure in the closed cavity to get a well-posed model. Use the Boussinesq approximation to account for the buoyancy force.

In the window, under click .

In the window for , locate the section.

From the list, choose .

Locate the section. From the list, choose .

Select the check box.

Specify the rref vector as


r_top-h_wall	r
Hg	z

Click to expand the section. From the list, choose .

This setting gives quadratic elements for the velocity field.

Pressure Point Constraint 1

In the toolbar, click

and choose .

Select Point 6 only.

Wall 2

In the toolbar, click

and choose .

Select Boundary 5 only.

In the window for , locate the section.

From the list, choose .

Multiphysics

Nonisothermal Flow 1 (nitf1)

In the window, under click .

In the window for , locate the section.

Select the check box.

Set the ambient temperature to be used in boundary conditions of the Heat Transfer interface.

Definitions

Ambient Properties 1 (ampr1)

In the toolbar, click

and choose .

In the window for , locate the section.

In the Tamb text field, type 298.15[K].

Heat Transfer in Fluids (ht)

Initial Values 1

In the window, under click .

In the window for , locate the section.

In the T text field, type 278.15[K].

Solid 1

In the toolbar, click

and choose .

Select Domain 1 only.

Temperature 1

In the toolbar, click

and choose .

Select Boundary 2 only.

Use the ambient temperature defined before in the section of the interface.

In the window for , locate the section.

From the T0 list, choose .

Heat Flux 1

In the toolbar, click

and choose .

Select Boundary 7 only.

In the window for , locate the section.

From the list, choose .

In the L text field, type Vlength.

In the ϕ text field, type acos(Hg/Vlength).

For the external temperature and pressure, use the values defined in the section of the interface.

From the Text list, choose .

From the pA list, choose .

Heat Flux 2

In the toolbar, click

and choose .

Select Boundaries 5 and 8 only.

In the window for , locate the section.

From the list, choose .

When the Rayleigh number is less than 104 for laminar flow, the conduction becomes far more dominant than the convection. Therefore, set the heat transfer coefficient to 2 W/(m2·K) which corresponds to the thermal resistance within a small layer of air at 298.15 K that floats above the glass.

In the h text field, type 2.

For the external temperature, use the value defined before in the section of the interface.