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F r e e C o nv e c t i o n i n a Li gh t Bu l b
Introduction
This model simulates the nonisothermal flow of argon gas inside a light bulb. The
purpose of the model is to show the coupling between energy transport—through
conduction, radiation, and convection—and momentum transport induced by density
variations in the argon gas.
Model Definition
A light bulb contains a tungsten filament that is resistively heated when a current is
conducted through it. At temperatures around 2000 K the filament starts to emit
visible light. To prevent the tungsten wire from burning up, the bulb is filled with a
gas, usually argon. The heat generated in the filament is transported to the
surroundings through radiation, convection, and conduction. As the gas heats up,
density and pressure changes induce a flow inside the bulb.
Figure 1 shows a cross section of the axially symmetric model geometry.
Figure 1: The model geometry.
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The filament is approximated with a solid torus, an approximation that implies
neglecting any internal effects inside the filament wire.
The equations governing the nonisothermal flow in the argon gas are
u
2
 ------ +   u  u  = – p +     u +  u  T  – -------    u I + g
t
3
----- +    u  = 0
t
where  denotes the density (kg/m3), u the velocity (m/s),  the viscosity (Pa·s), p
the pressure (Pa), and g the gravity vector (m/s2). The density is given by the ideal
gas law
 = Mp
--------RT
where M denotes the molar weight (kg/mol), R the universal gas constant (J/
(mol·K)) and T the temperature (K).
The convective and conductive heat transfer inside the bulb is described by the
equation
T
C p ------- +    – kT  = – C p u  T + Q
t
where Cp denotes the heat capacity (J/(kg·K)), k is the thermal conductivity (W/
(m·K)), and Q refers to the power density (W/m3) in the filament that serves as a heat
source. The total light bulb power is 60 W.
BOUNDARY CONDITIONS
At the bulb’s inner surfaces, radiation is described by surface-to-surface radiation. This
means the mutual irradiation from the surfaces that can be seen from a particular
surface, and radiation to the surroundings are accounted for. At the outer surfaces of
the bulb, radiation is described by surface-to-ambient radiation, which means that
there is no reflected radiation from the surroundings (blackbody radiation).
The top part of the bulb where the bulb is mounted on the cap is assumed to be
insulated:
– n   – kT  = 0
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Results
The heating inside the bulb has a long and a short time scale from t0, when the light
is turned on. The shorter scale captures the heating of the filament and the gas close
to it. The following series of pictures shows the temperature distribution inside the
bulb at t2, 6, and 10 s.
Figure 2: Temperature distribution at t = 2, 6, and 10 s. The color ranges differ between
the plots.
When the temperature changes, the density of the gas changes, inducing a gas flow
inside the bulb. The following series of pictures shows the velocity field inside the bulb
after 2, 6, and 10 s.
Figure 3: Velocity field after 2, 6, and 10 s. The color ranges differ between the plots.
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On the longer time scale, the glass on the bulb’s outer side heats up. The following
plot shows the temperature distribution in the bulb after 5 minutes.
Figure 4: Temperature distribution after 5 minutes.
Figure 5 shows the temperature distribution at a point on the boundary of the bulb at
the same vertical level as the filament. This plot shows the slow heating of the bulb.
After 5 minutes, the bulb has reached a steady-state temperature of 580 K.
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Figure 5: Temperature distribution at a point on the boundary of the bulb at the same
vertical level as the filament.
Heat is transported from the boundary of the bulb through both convective heat flux
and radiation. The net radiative heat flux leaving the bulb at t300 s is plotted in
Figure 6, as function of the z-coordinate. The top boundaries of the bulb where the
bulb is mounted on the cap are excluded from this plot. The distinct bump in the curve
occurs at the same vertical level as the filament.
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Figure 6: The net radiative heat flux leaving the bulb.
Notes About the COMSOL Implementation
To set up the model, use the Heat Transfer Module’s Non-Isothermal Flow predefined
multiphysics coupling. The model uses material from the Material Library to accurately
account for temperature-dependent properties over a wide range. The model setup is
straightforward and also shows how to create your own material to treat argon as an
ideal gas. When working with surface-to-surface radiation in COMSOL Multiphysics,
you have to specify which subdomains are transparent and which are opaque.
Furthermore, to allow for surface-to-surface radiation, an opaque domain must be
adjacent to a transparent domain. To achieve this, you can set the argon gas to be
transparent and the glass and filament to be opaque. The assumption that the glass on
the bulb is opaque, might seem odd, but is in fact valid because glass is almost opaque
to heat radiation, but transparent to radiation in the visible spectrum.
Model Library path: Heat_Transfer_Module/
Electronics_and_Power_Systems/light_bulb
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Modeling Instructions
MODEL WIZARD
1 Go to the Model Wizard window.
2 Click the 2D axisymmetric button.
3 Click Next.
4 In the Add physics tree, select Heat Transfer>Conjugate Heat Transfer>Laminar Flow
(nitf).
5 Click Add Selected.
6 Click Next.
7 In the Studies tree, select Preset Studies>Time Dependent.
8 Click Finish.
GEOMETRY 1
Import 1
1 In the Model Builder window, right-click Model 1 (mod1)>Geometry 1 and choose
Import.
2 Go to the Settings window for Import.
3 Locate the Import section. Click the Browse button.
4 Browse to the model’s Model Library folder and double-click the file
light_bulb.mphbin.
5 Click the Import button.
GLOBAL DEFINITIONS
Parameters
1 In the Model Builder window, right-click Global Definitions and choose Parameters.
2 Go to the Settings window for Parameters.
3 Locate the Parameters section. In the Parameters table, enter the following settings:
NAME
EXPRESSION
DESCRIPTION
h0
5[W/(m^2*K)]
Heat transfer coefficient
Qf
60[W]
Heat source in filament
p0
50[kPa]
Initial pressure
rho_glass
2595[kg/m^3]
Density, glass
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NAME
EXPRESSION
DESCRIPTION
k_glass
1.09[W/(m*K)]
Thermal conductivity, glass
Cp_glass
750[J/(kg*K)]
Heat capacity, glass
eps_glass
0.8
Surface emissivity, glass
Mw_a
39.94[g/mol]
Molar mass, argon
MATERIALS
Glass
1 In the Model Builder window, right-click Model 1 (mod1)>Materials and choose
Material.
2 Select Domain 1 only.
3 Go to the Settings window for Material.
4 Locate the Material Contents section. In the Material contents table, enter the
following settings:
PROPERTY
NAME
VALUE
Thermal conductivity
k
k_glass
Density
rho
rho_glass
Heat capacity at constant pressure
Cp
Cp_glass
5 Right-click Material 1 and choose Rename.
6 Go to the Rename Material dialog box and type Glass in the New name edit field.
7 Click OK.
8 Right-click Materials and choose Open Material Browser.
9 Go to the Material Browser window.
10 Locate the Materials section. In the Materials tree, select Material
Library>Elements>Tungsten.
11 Right-click and choose Add Material to Model from the menu.
Tungsten [solid,Ho et al]
1 In the Model Builder window, click Tungsten [solid,Ho et al].
2 Select Domain 3 only.
To apply the surface emissivity for tungsten as a material property, you also need to
define tungsten as the material for the filament surface.
3 In the Model Builder window, right-click Materials and choose Open Material Browser.
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4 Go to the Material Browser window.
5 Locate the Materials section. In the Materials tree, select Material
Library>Elements>Tungsten.
6 Right-click and choose Add Material to Model from the menu.
Tungsten [solid,Ho et al] (2)
1 In the Model Builder window, click (2).
2 Go to the Settings window for Material.
3 Locate the Geometric Entity Selection section. From the Geometric entity level list,
select Boundary.
4 Select Boundaries 15–18 only.
You will return to the Materials branch shortly to specify argon as the material for
the interior of the bulb. However, first set up the physics to let COMSOL
Multiphysics flag what properties you need to specify manually.
C O N J U G A T E H E A T TR A N S F E R ( N I T F )
1 In the Model Builder window, click Model 1 (mod1)>Conjugate Heat Transfer (nitf).
2 Go to the Settings window for Conjugate Heat Transfer.
3 Locate the Physical Model section. From the Default model list, select Fluid.
4 Select the Surface-to-surface radiation check box.
5 Locate the Radiation Settings section. From the Surface-to-surface radiation method
list, select Direct area integration.
6 Right-click Model 1 (mod1)>Conjugate Heat Transfer (nitf) and choose Heat Transfer
in Solids.
NON-ISOTHERMAL FLOW (NITF)
Heat Transfer in Solids 1
Select Domain 1 only.
Opaque 1
Right-click Model 1 (mod1)>Non-Isothermal Flow (nitf)>Heat Transfer in Solids 1 and
choose Opaque.
Heat Transfer in Solids 2
1 In the Model Builder window, right-click Non-Isothermal Flow (nitf) and choose Heat
Transfer in Solids.
2 Select Domain 3 only.
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3 Right-click Heat Transfer in Solids 2 and choose Opaque.
Initial Values 1
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 T edit field, type 25[degC].
Fluid 1
1 In the Model Builder window, click Fluid 1.
2 Go to the Settings window for Fluid.
3 Locate the Model Inputs section. From the p list, select Pressure (nitf/fluid1).
4 In the pref edit field, type p0.
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. In the Materials tree, select Material
Library>Elements>Argon.
4 From the Phase list, select gas.
5 Right-click and choose Add Material to Model from the menu.
Argon [gas]
1 In the Model Builder window, click Argon [gas].
2 Select Domain 2 only.
As you can see, COMSOL Multiphysics warns about required properties that have
not been defined yet. Define these as follows.
3 Go to the Settings window for Material.
4 Locate the Material Contents section. In the Material contents table, enter the
following settings:
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PROPERTY
NAME
VALUE
Ratio of specific heats
gamma
1.6
Density
rho
nitf.pA*Mw_a
/(R_const*T)
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NON-ISOTHERMAL FLOW (NITF)
Heat Source 1
1 In the Model Builder window, right-click Model 1 (mod1)>Non-Isothermal Flow (nitf)
and choose the domain setting Heat Transfer>Heat Source.
2 Select Domain 3 only.
3 Go to the Settings window for Heat Source.
4 Locate the Heat Source section. Click the Total power button.
5 In the Ptot edit field, type Qf.
Surface-to-Surface Radiation 1
1 In the Model Builder window, right-click Non-Isothermal Flow (nitf) and choose
Surface-to-Surface Radiation>Surface-to-Surface Radiation.
2 Select Boundaries 8, 9, 12–14, 19, and 21–24 only.
3 Go to the Settings window for Surface-to-Surface Radiation.
4 Locate the Radiation Settings section. In the Tamb edit field, type T.
5 Locate the Surface Emissivity section. From the  list, select User defined. In the
associated edit field, type eps_glass.
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 Select Boundaries 11 and 25 only.
3 Go to the Settings window for Convective Cooling.
4 Locate the Heat Flux section. In the h edit field, type h0.
5 In the Text edit field, type 25[degC].
Surface-to-Ambient Radiation 1
1 In the Model Builder window, right-click Non-Isothermal Flow (nitf) and choose the
boundary condition Heat Transfer>Surface-to-Ambient Radiation.
2 Select Boundaries 11 and 25 only.
3 Go to the Settings window for Surface-to-Ambient Radiation.
4 Locate the Surface-to-Ambient Radiation section. In the Tamb edit field, type
25[degC].
5 From the  list, select User defined. In the associated edit field, type eps_glass.
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Surface-to-Surface Radiation 2
1 In the Model Builder window, right-click Non-Isothermal Flow (nitf) and choose
Surface-to-Surface Radiation>Surface-to-Surface Radiation.
2 Select Boundaries 15–18 only.
3 Go to the Settings window for Surface-to-Surface Radiation.
4 Locate the Radiation Settings section. In the Tamb edit field, type T.
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
-nitf.rho*g_const
z
MESH 1
1 In the Model Builder window, right-click Model 1 (mod1)>Mesh 1 and choose Build All.
2 Either right-click and choose Build All, click the Build All button in the Settings
window toolbar, or press F8 to generate a physics-controlled mesh of default size.
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To check that this mesh resolution is sufficient, you can, if you like, later come back
and change the Element size setting to Fine, build the mesh, and re-solve the model.
You will then see that the solution is nearly the same as that for the mesh shown above.
STUDY 1
Step 1: Time Dependent
1 In the Model Builder window, expand the Study 1 node, then click 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(0,0.1,1)
range(1.5,0.5,20) range(21,3,300).
4 Select the Relative tolerance check box.
5 In the associated edit field, type 0.001.
6 In the Model Builder window, right-click Study 1 and choose Show Default Solver.
7 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 Dependent Variables 1.
2 Go to the Settings window for Dependent Variables.
3 Locate the Scaling section. From the Method list, select Manual.
For a nonlinear time-dependent problem such as this one, a judicious manual scaling
of the variables can improve convergence during solving.
4 In the Model Builder window, expand the Dependent Variables 1 node, then click
mod1_u.
5 Go to the Settings window for Field.
6 Locate the Scaling section. From the Method list, select Manual.
7 In the Scale edit field, type 0.1.
8 In the Model Builder window, click Dependent Variables 1>mod1_J.
9 Go to the Settings window for Field.
10 Locate the Scaling section. From the Method list, select Manual.
11 In the Scale edit field, type 1e5.
12 In the Model Builder window, click Dependent Variables 1>mod1_T.
13 Go to the Settings window for Field.
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14 Locate the Scaling section. From the Method list, select Manual.
15 In the Scale edit field, type 1e3.
16 In the Model Builder window, click Dependent Variables 1>mod1_p.
17 Go to the Settings window for Field.
18 Locate the Scaling section. From the Method list, select Manual.
19 In the Scale edit field, type 1e4.
20 In the Model Builder window, click Time-Dependent Solver 1.
21 Go to the Settings window for Time-Dependent Solver.
22 Click to expand the Time Stepping section.
23 From the Method list, select Generalized alpha.
24 From the Steps taken by solver list, select Manual.
25 In the Time step edit field, type 0.02*if(t<10,1,10).
26 Click to expand the Advanced section.
27 From the Error estimation list, select Exclude algebraic.
28 In the Model Builder window, right-click Study 1 and choose Compute.
RESULTS
Velocity, 3D (nitf)
The first default plot shows the velocity magnitude in a 3D plot, obtained by a
revolution of the 2D axisymmetric data set, at the end of the simulation interval. Now
proceed to reproduce the velocity field plots in Figure 3.
Because the velocity magnitude is a quadratic expression in the basic velocity variables
it looks less smooth than the temperature plot. You can easily remedy the situation by
adjusting the Quality settings.
1 In the Model Builder window, expand the Velocity, 3D (nitf) node, then click Surface 1.
2 Go to the Settings window for Surface.
3 Click to expand the Quality section.
4 From the Resolution list, select Fine.
5 Click the Plot button.
6 In the Model Builder window, click Velocity, 3D (nitf).
7 Go to the Settings window for 3D Plot Group.
8 Locate the Data section. From the Time list, select 2.
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9 Click the Plot button.
10 From the Time list, select 6.
11 Click the Plot button.
12 From the Time list, select 10.
13 Click the Plot button.
Temperature, 3D (nitf)
1 In the Model Builder window, right-click Results>Temperature, 3D (nitf) and choose
Plot.
2 Click the Zoom Extents button on the Graphics toolbar.
The second default 3D plot shows the temperature at the end of the simulation
interval (Figure 4).
Next, look at the temperature field at different times. Compare the resulting series
of plots with those in Figure 2.
3 In the Model Builder window, click Temperature, 3D (nitf).
4 Go to the Settings window for 3D Plot Group.
5 Locate the Data section. From the Time list, select 2.
6 Click the Plot button.
7 From the Time list, select 6.
8 Click the Plot button.
9 From the Time list, select 10.
10 Click the Plot button.
To visualize the heating of the bulbs surface with time by plotting the temperature
at a point at the same vertical level as the filament, follow the steps below.
1D Plot Group 3
1 In the Model Builder window, right-click Results and choose 1D Plot Group.
2 Right-click Results>1D Plot Group 3 and choose Point Graph.
3 Select Vertex 24 only.
4 Click the Plot button.
Finally, study the radiative heat flux from the bulb. First plot the radiative heat flux
versus the vertical coordinate, z.
1D Plot Group 4
1 In the Model Builder window, right-click Results and choose 1D Plot Group.
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2 Go to the Settings window for 1D Plot Group.
3 Locate the Data section. From the Time selection list, select Last.
4 Right-click Results>1D Plot Group 4 and choose Line Graph.
5 Select Boundaries 11 and 25 only.
6 Go to the Settings window for Line Graph.
7 In the upper-right corner of the y-Axis Data section, click Replace Expression.
8 From the menu, choose Non-Isothermal Flow (Heat Transfer)>Radiative heat flux
(nitf.rflux).
9 In the upper-right corner of the x-Axis Data section, click Replace Expression.
10 From the menu, choose Geometry and Mesh>Coordinate>z-coordinate (z).
11 Locate the x-Axis Data section. From the Unit list, select cm.
12 Click the Plot button.
You can readily compute the total radiative heat flux from the bulb at steady state as
follows.
Derived Values
1 In the Model Builder window, right-click Results>Derived Values and choose
Integration>Line Integration.
2 Go to the Settings window for Line Integration.
3 Locate the Data section. From the Time selection list, select Last.
4 Select Boundaries 11 and 25 only.
5 In the upper-right corner of the Expression section, click Replace Expression.
6 From the menu, choose Non-Isothermal Flow (Heat Transfer)>Radiative heat flux
(nitf.rflux).
7 Click to expand the Integration Settings section.
8 Select the Compute surface integral check box.
9 Click the Evaluate button.
The result should be close to 42 W.
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