basic Heat Transferencia Model Library Manual comsol 4.3

VERSION 4.3b

Heat Transfer ModuleModel Library Manual

C o n t a c t I n f o r m a t i o n

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Part number: CM020802

H e a t T r a n s f e r M o d u l e M o d e l L i b r a r y M a n u a l 19982013 COMSOL

Protected by U.S. Patents 7,519,518; 7,596,474; and 7,623,991. Patents pending.

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Version: May 2013 COMSOL 4.3b

Solved with COMSOL Multiphysics 4.3b

2 0 1 3 C O M S O L

Rad i a t i v e Hea t T r a n s f e r i n a U t i l i t y Bo i l e r

Introduction

In recent years, many studies have been conducted in the field of performance oplifboprcoteimanforemcom

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timization of large power plant boilers. The main aims have been to extend the etime, increase the thermal efficiency, and reduce the pollutant emissions of the ilers. A good furnace design is the most important part in the energy conversion ocess in the boilers. A furnace is where the fuel is burnt and the chemical energy is nverted into heat to be transferred into the water walls of steam boilers. The mperatures in fuel furnaces are high enough that the radiation becomes the most portant mechanism in heat transfer. Due to complexity of the radiation mechanism d its dependence on the enclosure's geometry, there is no analytical solution except r very simple problems. This fact along with expensive experimental modeling leads searchers to develop numerical models for analyzing these enclosures. Three of the ost attractive methods, as far as accuracy and computational requirement are ncerned, are the discrete transfer, the discrete ordinates and the finite-volume ethods.

ote: Solving this model requires approximately 22 GB of memory.

odel Definition

f practical relevance is the radiative heat transfer in furnaces containing obstacles, ch as protrusions and obstructions. In some applications, the thickness of the stacles is very small as it occurs in utility boilers, where panels are often hanged in e radiation chamber.

order to reduce the mesh (and then the computational cost), these obstructions are odeled as baffles (zero thickness). This study handles zero thickness obstacles ntaining an emitting-absorbing medium.


2 | R A D I A T I V E

A three-dimensional enclosure resembling the combustion chamber of a utility boiler is modeled. The enclosure contains five baffles, as shown in Figure 1, which simulates the panels of superheaters suspended at the top of the combustion chamber.

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Tthore H E A T TR A N S F E R I N A U T I L I T Y B O I L E R 2 0 1 3 C O M S O L

igure 1: Utility Boiler with obstructions

this example, the S8 discrete ordinate method was employed for predicting the heat ux on the side walls of enclosures and incident radiation distribution within the rnace. It results in a set of 80 discrete directions to represent radiative intensity ansport.

he main assumption is using an existing uniform temperature and properties within e volume and surface zones, as proposed in Ref. 1. The temperature and emissivity

f the boundaries, including the surface of the baffles, are taken as 800 K and 0.65, spectively, except at x = 10 m and for 22 z 30 m, where the temperature was set


2 0 1 3 C O M S O L

equal to 1200 K and a black-body surface is assumed. An emitting-absorbing medium is assumed, with the following distribution of temperature and absorption coefficient:

T

Tdi(Rar

T

w

Tem

w

TABLE 1: DISTRIBUTION OF TEMPERATURE AND ADSORPTION COEFFICIENT

COORDINATE [M] ABSORPTION COEFFICIENT[1/M]

TEMPERATURE [K]

z 5 0.20 16005 < z 10 0.25 200010 < z 20 0.20 16002 3 | R A D I A T I V E H E A T TR A N S F E R I N A U T I L I T Y B O I L E R

H E R M A L A N A L Y S I S

he discrete-ordinates method (DOM) relies on the discrete representation of the rectional dependence of the radiation intensity. The radiative transfer equation TE) is solved for a set of directions which span he total solid angle range of 4 ound a point in space.

he RTE for this type of configuration can be written as:

here

I is the radiative intensity at a given position s, following direction T is the temperature

, , s are absorption, extinction, and scattering coefficients, respectively IbT is the blackbody radiative intensity1a10 and 0. ( is the cosine of the scattering angle).

he boundary intensities in the furnace walls are given physically by the effective itted intensity plus reflected incident intensities into a respective direction.

here

w is the surface emissivity, which is in the range [0, 1] d1w is the diffusive reflectivity n is the outward normal vector

0 < z 30 0.18 1200

I s Ib T I s s4------ I 0

4+=

Ibnd wIb T d------qout for all such that n 0+=



qout is the heat flux striking the wall:

The above equations can be discretized in Cartesian coordinates for monochromatic or gray radiation as:

T

Fcoeq

Fis

R

Fmco

qout n I s n j 0=

I I T I s------ I 4

+= H E A T TR A N S F E R I N A U T I L I T Y B O I L E R 2 0 1 3 C O M S O L

he Sn approximation of the RTE in the m direction can be expressed as:

or a discrete direction, i, the values of i,1, i,2, and i,3 define the direction sines of i obeying the condition i,12 + i,22 + i,321. The j index in the above uation denotes the direction of incoming radiation contributing to the direction i.

or a diffuse reflecting surface on a wall boundary, the boundary condition equation transformed as:

esults and Discussion

igure 2 and Figure 3 show the predicted incident radiation surface plots. The aximum incident radiation occurs at the level where the temperature and absorbing efficient of the medium are the highest (that is, at the boilers burner level).

i i b i 4 0

i Ii Ib T Iis4------ wjIj j i

j 1=

n

+=

Ii bnd wIb T d------ wjIjn jn j 0 for all i such that n i 0+=


2 0 1 3 C O M S O L

Fi

Fi 5 | R A D I A T I V E H E A T TR A N S F E R I N A U T I L I T Y B O I L E R

gure 2: Incident radiation.

gure 3: Incident radiation on the front of the boiler (W/m2).



Predicted outgoing heat flux is shown on and in good agreement with published data (Ref. 1).

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2 H E A T TR A N S F E R I N A U T I L I T Y B O I L E R 2 0 1 3 C O M S O L

igure 4: Outgoing heat flux on walls of the boiler (W/m2)

eference

. P.J. Coelho, J.M. Goncalves, and M.G. Carvalho, Modelling of Radiative Heat ransfer in Enclosures with Obstacles, International Journal of Heat and Mass ransfer, vol. 41, no. 45, pp. 745756, 1998.

odel Library path: Heat_Transfer_Module/Thermal_Radiation/boiler

odeling Instructions

O D E L W I Z A R D

Go to the Model Wizard window.

Click Next.


2 0 1 3 C O M S O L

3 In the Add physics tree, select Heat Transfer>Radiation>Radiation in Participating Media (rpm).

4 Click Add Selected.

By default, the Discrete ordinates method setting is S4, which corresponds to 24 discrete angular directions. To obtain the maximum resolution of 80 directions, select S8 from the list. Note, however, that this requires approximately 20 GB of memory.

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Click Next.

Find the Studies subsection. In the tree, select Preset Studies>Stationary.

Click Finish.

L O B A L D E F I N I T I O N S

rametersIn the Model Builder window, right-click Global Definitions and choose Parameters.

In the Parameters settings window, locate the Parameters section.

In the table, enter the following settings:

ecewise 1Right-click Global Definitions and choose Functions>Piecewise.

In the Piecewise settings window, locate the Function Name section.

In the Function name edit field, type T.

Locate the Definition section. In the Argument edit field, type z.

Find the Intervals subsection. In the table, enter the following settings:

ame Expression Description

hot 1200[K] Temperature, hot surface zone

low 800[K] Temperature, cool surfaces

m 0.65 Emissivity, cool surfaces

cattC 0[1/m] Scattering coefficient

tart End Function

5 1600

10 2000

0 20 1600

0 30 1200



6 Locate the Units section. In the Arguments edit field, type m.

7 In the Function edit field, type K.

Piecewise 21 Right-click Global Definitions and choose Functions>Piecewise.

2 In the Piecewise settings window, locate the Function Name section.

3 In the Function name edit field, type absC.

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Locate the Definition section. In the Argument edit field, type z.

Find the Intervals subsection. In the table, enter the following settings:

Locate the Units section. In the Arguments edit field, type m.

In the Function edit field, type 1/m.

E O M E T R Y 1

ork Plane 1In the Model Builder window, under Model 1 right-click Geometry 1 and choose Work Plane.

In the Work Plane settings window, locate the Plane Definition section.

From the Plane list, choose xz-plane.

zier Polygon 1In the Model Builder window, under Model 1>Geometry 1>Work Plane 1 right-click Plane Geometry and choose Bzier Polygon.

In the Bzier Polygon settings window, locate the Polygon Segments section.

Find the Added segments subsection. Click the Add Linear button.

Find the Control points subsection. In row 1, set yw to 4.

In row 2, set yw to 30.


Find the Control points subsection. In row 2, set xw to 10.


tart End Function

5 0.20

10 0.25

0 20 0.20

0 30 0.18


2 0 1 3 C O M S O L

9 Find the Control points subsection. In row 2, set yw to 22.

10 Find the Added segments subsection. Click the Add Linear button.

11 Find the Control points subsection. In row 2, set xw to 8.

12 In row 2, set yw to 20.

13 Find the Added segments subsection. Click the Add Linear button.

14 Find the Control points subsection. In row 2, set xw to 10.

15 In row 2, set yw to 18.

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Find the Control points subsection. In row 2, set yw to 4.



In row 2, set yw to 0.




Find the Control points subsection. Click the Close Curve button.

ectangle 1Right-click Plane Geometry and choose Rectangle.

In the Rectangle settings window, locate the Size section.

In the Width edit field, type 4.

In the Height edit field, type 10.

Locate the Position section. In the yw edit field, type 20.

xtrude 1In the Model Builder window, right-click Geometry 1 and choose Extrude.

In the Extrude settings window, locate the Distances from Plane section.


istances (m)


10 | R A D I A T I V

M A T E R I A L S

Add a material to specify the absorption and scattering coefficients inside the boiler.

Material 11

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Distances (m)E H E A T TR A N S F E R I N A U T I L I T Y B O I L E R 2 0 1 3 C O M S O L

In the Model Builder window, under Model 1 right-click Materials and choose Material.

Right-click Material 1 and choose Rename.

Go to the Rename Material dialog box and type Chamber in the New name edit field.

Click OK.

In the Material settings window, locate the Material Contents section.


nalogously, specify the emissivity of the boiler walls using a material.

aterial 2In the Model Builder window, right-click Materials and choose Material.

Right-click Material 2 and choose Rename.

Go to the Rename Material dialog box and type Walls in the New name edit field.

Click OK.

In the Material settings window, locate the Geometric Entity Selection section.

From the Geometric entity level list, choose Boundary.

From the Selection list, choose All boundaries.

Locate the Material Contents section. In the table, enter the following settings:

A D I A T I O N I N P A R T I C I P A T I N G M E D I A

In the Radiation in Participating Media settings window, locate the Participating Media Settings section.

roperty Name Value

bsorption coefficient kappaR absC(z)

cattering coefficient sigmaS scattC

roperty Name Value

urface emissivity epsilon_rad em


2 0 1 3 C O M S O L

2 From the Pindex list, choose 1.

Radiation in Participating Media 11 In the Model Builder window, under Model 1>Radiation in Participating Media click

Radiation in Participating Media 1.

2 In the Radiation in Participating Media settings window, locate the Model Inputs section.

3 In the T edit field, type T(z).

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paque Surface 1In the Model Builder window, under Model 1>Radiation in Participating Media right-click Opaque Surface 1 and choose Rename.

Go to the Rename Opaque Surface dialog box and type Low Temp in the New name edit field.

Click OK.

In the Opaque Surface settings window, locate the Model Inputs section.

In the T edit field, type Tlow.

paque Surface 2In the Model Builder window, right-click Radiation in Participating Media and choose Opaque Surface.

Right-click Opaque Surface 2 and choose Rename.

Go to the Rename Opaque Surface dialog box and type Hot Temp in the New name edit field.

Click OK.

In the Opaque Surface settings window, locate the Model Inputs section.

In the T edit field, type Thot.

Locate the Wall Settings section. From the Wall type list, choose Black wall.

Select Boundaries 82, 84, 86, 88, 90, and 92 only.

E S H 1

apped 1In the Model Builder window, under Model 1 right-click Mesh 1 and choose More Operations>Mapped.

Select Boundary 5 only.


12 | R A D I A T I V

Free Quad 11 In the Model Builder window, right-click Mesh 1 and choose More Operations>Free

Quad.

2 Select Boundary 2 only.

3 Right-click Mesh 1 and choose Swept.

Size1 In the Model Builder window, under Model 1>Mesh 1 click Size.

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10E H E A T TR A N S F E R I N A U T I L I T Y B O I L E R 2 0 1 3 C O M S O L

In the Size settings window, locate the Element Size section.

From the Predefined list, choose Extra fine.

In the Model Builder window, right-click Mesh 1 and choose Build All.

T U D Y 1

In the Model Builder window, right-click Study 1 and choose Compute.

he default plot groups show the Incident Radiation slice plot and the Net Radiative eat Flux surface plot.

ollow instructions below to replace the slice plot for Incident Radiation by a contour lot.

E S U L T S

cident Radiation (rpm)In the Model Builder window, expand the Results>Incident Radiation (rpm) node.

Right-click Slice 1.1 and choose Delete.

Click Yes to confirm.

Right-click Incident Radiation (rpm) and choose Contour.

In the Contour settings window, locate the Levels section.

From the Entry method list, choose Levels.

In the Levels edit field, type range(0e5,1e5,21e5).

Locate the Coloring and Style section. From the Contour type list, choose Filled.

Click to expand the Quality section. From the Smoothing list, choose None.

Click the Plot button.

Rescale the zoom to get results shown in Figure 2.

Click the Zoom Extents button on the Graphics toolbar.

Switch the yz perspective to display results from Figure 3.


2 0 1 3 C O M S O L

11 Click the Go to YZ View button on the Graphics toolbar.

Net Radiative Heat Flux (rpm)Now, replace the Net Radiative Heat Flux by the Outgoing Radiative Heat Flux in the second graph.

1 In the Model Builder window, under Results right-click Net Radiative Heat Flux (rpm) and choose Rename.

2 Go to the Rename 3D Plot Group dialog box and type Outgoing Radiative Heat

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Flux (rpm) in the New name edit field.

Click OK.

utgoing Radiative Heat Flux (rpm)In the Model Builder window, expand the Outgoing Radiative Heat Flux (rpm) node.

Right-click Surface 1.1 and choose Delete.

Click Yes to confirm.

Right-click Outgoing Radiative Heat Flux (rpm) and choose Contour.

In the Contour settings window, click Replace Expression in the upper-right corner of the Expression section. From the menu, choose Radiation in Participating Media>Boundary fluxes>Outgoing radiative heat flux (rpm.qout).

Locate the Coloring and Style section. From the Contour type list, choose Filled.

Locate the Quality section. From the Smoothing list, choose None.

Go to Default 3D View to obtain Figure 4.

Click the Go to Default 3D View button on the Graphics toolbar.


14 | R A D I A T I V E H E A T TR A N S F E R I N A U T I L I T Y B O I L E R 2 0 1 3 C O M S O L


2 0 1 3 C O M S O L

Hea t Gen e r a t i o n i n a D i s c B r a k e

Introduction

This example models the heat generation and dissipation in a disc brake of an ordinary car during panic braking and the following release period. As the brakes slow the car, they transform kinetic energy into thermal energy, resulting in intense heating of the brsc

Inbrofpead

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Fi 1 | H E A T G E N E R A T I O N I N A D I S C B R A K E

ake discs. If the discs overheat, the brake pads stop working and, in a worst-case enario, can melt. Braking power starts to fade already at temperatures above 600 K.

the model, the car (1800 kg) initially travels at 25 m/s (90 km/h) when the driver akes hard for 2 s, causing the vehicles eight brake pads to slow the car down at a rate 10 m/s2. The wheels are assumed not to skid against the road surface. After this riod of time, the driver releases the brake and the car travels at 5 m/s for an ditional 8 s without any braking. The questions to analyze with the model are:

How hot do the brake discs and pads become during the braking stage?

How much do they cool down during the subsequent rest?

odel Definition

odel the brake disc as a 3D solid with shape and dimensions as in Figure 1. The disc s a radius of 0.14 m and a thickness of 0.013 m.

gure 1: Model geometry, including disc and pad.


2 | H E A T G E N E

Neglecting drag and other losses outside the brakes, the brakes retardation power is given by the negative of the time derivative of the cars kinetic energy:

Here m is the cars mass, v denotes its speed, R equals the wheel radius (0.25 m), is the angular velocity, and is the angular acceleration. The acceleration is constant in this case, so (t) t.Bbthsucowog

Yb

Cth

(N

Uthfr

Tanco

Ptd

d mv2

2----------- m v

vdtd

------ m R2 t = = =R A T I O N I N A D I S C B R A K E 2 0 1 3 C O M S O L

0

y definition, the retardation power equals the negative of the work per unit time done y the friction forces on the discs at the interfaces between the pads and the discs for e eight brakes. You can calculate this work as eight times an integral over the contact rface of a single brake pad. The friction force per unit area, ff, is approximately nstant over the surface and is directed opposite the disc velocity vector, vdvde,

here e denotes a unit vector in the azimuthal (angular) direction and the magnitude f vd at the distance r from the center equals vd( r, t )(t) r. Thus, writing fdffe ives the following result for the retardation power:

ou can approximate the last integral with the pads area, A (0.0035 m2), multiplied y the distance from the center of the disc to the pads center of mass, rm (0.1143 m).

ombining the two expressions for P gives the following result for the magnitude of e friction force, ff:

ote that is negative during retardation.)nder the previously stated idealization that retardation is due entirely to friction in e brakes, the heat power generated per unit contact area at time t and the distance r om the center becomes

he disc and pad dissipate the heat produced at the boundary between the brake pad d the disc by convection and radiation. This example models the rotation as nvection in the disc. The local disc velocity vector is

P 8 ff Ad vd 8 ff t t r Ad= =

ffmR28rmA-----------------

=

q r t f f vd r t mR2

8rmA----------------- r 0 t+ = =


2 0 1 3 C O M S O L

The model also includes heat conduction in the disc and the pad through the transient heat transfer equation

where k represents the thermal conductivity (W/(mK)), Cp is the specific heat cath

Athto

Inm

Tfo

Hde

Tth

TA

P

C

k

vd t y x( , )=

Cp T t------- + kT Q CPu T= 3 | H E A T G E N E R A T I O N I N A D I S C B R A K E

pacity (J/(kgK)), and Q is the heating power per unit volume (W/m3), which in is case is set to zero.

t the boundary between the disc and the pad, the brake produces heat according to e expression for q given earlier. The heat dissipation from the disc and pad surfaces the surrounding air is described by both convection and radiation

this equation, h equals the convective film coefficient (W/(m2K)), is the aterials emissivity, and is the Stefan-Boltzmann constant (5.67108 W/(m2K4)).o calculate the convective film coefficient as a function of the vehicle speed, v, use the llowing formula (Ref. 1):

ere l is the discs diameter. The material propertiesthe thermal conductivity, k, the nsity, , the viscosity, , and the specific heat capacity, Cpare those for air.

able 1 summarizes the thermal properties, which come from (Ref. 1). You calculate e density of air at a reference temperature of 300 K using the ideal gas law.

BLE 1: MATERIAL PROPERTIES

ROPERTY DISC PAD AIR

(kg/m3) 7870 2000 1.170

p (J/(kgK)) 449 935 1100

(W/(mK)) 82 8.7 0.026

0.28 0.8 -

(Pas) - - 1.810-5

qdiss h T Tref T4 Tref4 =

h 0.037kl

------------------ Re0.8 Pr0.33 0.037kl

------------------ l v---------- 0.8 Cp

k-----------

0.33= =


4 | H E A T G E N E

Results and Discussion

The surface temperatures of the disc and the pad vary with both time and position. At the contact surface between the pad and the disc the temperature increases when the brake is engaged and then decreases again as the brake is released. You can best see these results in COMSOL Multiphysics by generating an animation. Figure 2 displays the surface temperatures just before the end of the braking. A hot spot is visible at the contact between the brake pad and disc, just at the pads edgethis is where the tetete

F(t

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mperature could become critical during braking. The figure also shows the mperature decreasing along the rotational trace after the pad. During the rest, the mperature becomes significantly lower and more uniform in the disc and the pad.

igure 2: Surface temperature of the brake disc and pad just before releasing the brake = 1.8 s).

o investigate the position of the hot spot and the time of the temperature maximum, is helpful to plot temperature versus time along the line from the center to the pads ge depicted in Figure 3. The result is displayed in Figure 4. You can see that the aximum temperature is approximately 415 K. The hot spot is positioned close to the dially outer edge of the pad. The highest temperature occurs approximately 1 s after gaging the brake.


2 0 1 3 C O M S O L

Fi

Fi(z 5 | H E A T G E N E R A T I O N I N A D I S C B R A K E

gure 3: The radial line probed in the temperature vs. time plot in Figure 4.

gure 4: Temperature profile along the line indicated in Figure 3 at the disc surface = 0.013 m) as a function of time.

r (m)t


6 | H E A T G E N E

To investigate how much of the generated heat is dissipated to the air, study the surface integrals of the produced heat and the dissipated heat. These integrals give the total heat flux (J/s) for heat production, Qprod, and heat dissipation, Qdiss, as functions of time for the brake disc. The time integrals of these two quantities, Wprod and Wdiss, give the total heat (J) produced and dissipated, respectively, in the brake disc. Figure 5 shows a plot of the total produced heat and dissipated heat versus time. You can see that 8 s after disengagement the brake has dissipated only a fraction of the produced heat. The plot indicates that the resting time must be extended significantly to dissipate al

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1mR A T I O N I N A D I S C B R A K E 2 0 1 3 C O M S O L

l the generated heat.

igure 5: Comparison of total heat produced (solid line) and dissipated (dashed).

he results of this model can help engineers investigate how much abuse, in terms of ecific braking sequences, a certain brake-disc design can tolerate before overheating. is also possible to vary the parameters affecting the heat dissipation and investigate eir influence.

eference

. J.M. Coulson and J.F. Richardson, Chemical Engineering, vol. 1, eq. 9.88; aterial properties from appendix A2.


2 0 1 3 C O M S O L

Model Library path: Heat_Transfer_Module/Thermal_Contact_and_Friction/brake_disc

Modeling Instructions

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O D E L W I Z A R D


Click Next.

In the Add physics tree, select Heat Transfer>Heat Transfer in Solids (ht).

Click Next.

Find the Studies subsection. In the tree, select Preset Studies>Time Dependent.

Click Finish.



Define the global parameters by loading the corresponding text file provided.


Click Load from File.

Browse to the models Model Library folder and double-click the file brake_disc_parameters.txt.

E O M E T R Y 1

ylinder 1In the Model Builder window, under Model 1 right-click Geometry 1 and choose Cylinder.

In the Cylinder settings window, locate the Size and Shape section.

In the Radius edit field, type 0.14.

In the Height edit field, type 0.013.

Click the Build Selected button.


8 | H E A T G E N E

Cylinder 21 In the Model Builder window, right-click Geometry 1 and choose Cylinder.

2 In the Cylinder settings window, locate the Size and Shape section.

3 In the Radius edit field, type 0.08.

4 In the Height edit field, type 0.01.

5 Locate the Position section. In the z edit field, type 0.013.

6 Click the Build Selected button.

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ork Plane 1Right-click Geometry 1 and choose Work Plane.

In the Work Plane settings window, locate the Plane Definition section.

In the z-coordinate edit field, type 0.013.

Click the Show Work Plane button.

zier Polygon 1In the Model Builder window, under Model 1>Geometry 1>Work Plane 1 right-click Plane Geometry and choose Bzier Polygon.

In the Bzier Polygon settings window, locate the Polygon Segments section.

Find the Added segments subsection. Click the Add Cubic button.

Find the Control points subsection. In row 1, set yw to 0.135.

In row 2, set xw to 0.02.

In row 2, set yw to 0.135.






Find the Control points subsection. In row 2, set xw to 0.03.




In row 4, set xw to 0.



2 0 1 3 C O M S O L

18 Find the Added segments subsection. Click the Add Cubic button.

19 Find the Control points subsection. In row 2, set xw to -0.035.

20 In row 2, set yw to 0.09.

21 In row 3, set xw to -0.03.


23 In row 4, set xw to -0.04.


25

26

27

28

29

30

31

32

33

34

35

E1

2

3

4

D



Find the Control points subsection. In row 2, set xw to -0.05.


In row 3, set xw to -0.02.


Click the Close Curve button.

To complete the pad cross section, you must make the top-left and top-right corners sharper. Do so by changing the weights of the Bzier curves.

Find the Added segments subsection. In the Added segments list, select Segment 1 (cubic).

Find the Weights subsection. In the 3 edit field, type 2.5.

Find the Added segments subsection. In the Added segments list, select Segment 4 (cubic).

Find the Weights subsection. In the 2 edit field, type 2.5.


xtrude 1In the Model Builder window, under Model 1>Geometry 1 right-click Work Plane 1 and choose Extrude.

In the Extrude settings window, locate the Distances from Plane section.



The model geometry is now complete.

istances (m)

.0065


10 | H E A T G E N

Next, define some selections of certain boundaries. You will use them when defining the settings for model couplings, boundary conditions, and so on.

D E F I N I T I O N S

Explicit 11 In the Model Builder window, under Model 1 right-click Definitions and choose

Selections>Explicit.

2

3

4

5

6

7

E1

2

3

4

5

6

7

E1

2

3

4

5

6E R A T I O N I N A D I S C B R A K E 2 0 1 3 C O M S O L

Right-click Explicit 1 and choose Rename.

Go to the Rename Explicit dialog box and type Disc faces in the New name edit field.

Click OK.

In the Explicit settings window, locate the Input Entities section.


Select Boundaries 1, 2, 46, 8, 1315, and 18 only.

xplicit 2In the Model Builder window, right-click Definitions and choose Selections>Explicit.


Go to the Rename Explicit dialog box and type Pad faces in the New name edit field.

Click OK.



Select Boundaries 9, 10, 12, 16, and 17 only.

xplicit 3In the Model Builder window, right-click Definitions and choose Selections>Explicit.


Go to the Rename Explicit dialog box and type Contact surface in the New name edit field.

Click OK.



To select the contact surface boundary, it is convenient to temporarily switch to wireframe rendering.


2 0 1 3 C O M S O L

7 Click the Wireframe Rendering button on the Graphics toolbar.


To select this boundary, you typically need to click twice before right-clicking; the first click highlights the pad's top surface.

9 Click the Wireframe Rendering button on the Graphics toolbar.

Integration 11 In the Model Builder window, right-click Definitions and choose Model

2

3

4

In1

2

3

4

In1

2

3

4

D

St1

2


Couplings>Integration.

In the Integration settings window, locate the Source Selection section.



tegration 2Right-click Definitions and choose Model Couplings>Integration.



From the Selection list, choose Disc faces.

tegration 3Right-click Definitions and choose Model Couplings>Integration.



From the Selection list, choose Pad faces.

efine a step function for use in variables.

ep 1Right-click Definitions and choose Functions>Step.

In the Step settings window, click to expand the Smoothing section.

In the Size of transition zone edit field, type 2*0.01.


12 | H E A T G E N

4 Click the Plot button.

D

Va1

2

3

4

5

N

Q

QE R A T I O N I N A D I S C B R A K E 2 0 1 3 C O M S O L

efine some variables by loading the corresponding text file provided.

riables 1Right-click Definitions and choose Variables.

In the Variables settings window, locate the Variables section.

Click Load from File.

Browse to the models Model Library folder and double-click the file brake_disc_variables.txt.

In the table, add variables for the produced heating power and dissipated heating power defined in terms of these variables and the integration operators you defined previously:


ame Expression Description

_prod intop1(q_prod) Produced heating power

_diss intop2(-q_d_disc)+intop3(-q_d_pad)

Total dissipated heating power


2 0 1 3 C O M S O L

M A T E R I A L S

Material 11 In the Model Builder window, under Model 1 right-click Materials and choose Material.

2 In the Material settings window, locate the Material Contents section.

3 In the table, enter the following settings:

M1

2

3

4

H

HIn

Tr1

2

3

4

Property Name Value

T

D

H

P

T

D

H

-

x


aterial 2In the Model Builder window, right-click Materials and choose Material.

Select Domain 3 only.

In the Material settings window, locate the Material Contents section.


E A T TR A N S F E R I N S O L I D S

eat Transfer in Solids 1 the Model Builder window, expand the Model 1>Heat Transfer in Solids node.

anslational Motion 1Right-click Heat Transfer in Solids 1 and choose Translational Motion.

Select Domains 1 and 2 only.

In the Translational Motion settings window, locate the Translational Motion section.

In the utrans table, enter the following settings:

hermal conductivity k k_disc

ensity rho rho_disc

eat capacity at constant pressure Cp C_disc

roperty Name Value

hermal conductivity k k_pad

ensity rho rho_pad

eat capacity at constant pressure Cp C_pad

y*omega x

*omega y

z


14 | H E A T G E N

Symmetry 11 In the Model Builder window, right-click Heat Transfer in Solids and choose Symmetry.


Heat Flux 11 Right-click Heat Transfer in Solids and choose Heat Flux.

2 In the Heat Flux settings window, locate the Boundary Selection section.

3 From the Selection list, choose Disc faces.

4

5

6

H1

2

3

4

5

6

B1

2

3

4

In1

2

3

S1


Locate the Heat Flux section. Click the Inward heat flux button.

In the h edit field, type h_air.

In the Text edit field, type T_air.

eat Flux 2Right-click Heat Transfer in Solids and choose Heat Flux.

In the Heat Flux settings window, locate the Boundary Selection section.


Locate the Heat Flux section. Click the Inward heat flux button.

In the h edit field, type h_air.

In the Text edit field, type T_air.

oundary Heat Source 1Right-click Heat Transfer in Solids and choose Boundary Heat Source.

In the Boundary Heat Source settings window, locate the Boundary Selection section.

From the Selection list, choose Contact surface.

Locate the Boundary Heat Source section. In the Qb edit field, type q_prod.

itial Values 1In the Model Builder window, under Model 1>Heat Transfer in Solids click Initial Values 1.

In the Initial Values settings window, locate the Initial Values section.

In the T edit field, type T_air.

urface-to-Ambient Radiation 1In the Model Builder window, right-click Heat Transfer in Solids and choose Surface-to-Ambient Radiation.

In the Surface-to-Ambient Radiation settings window, locate the Boundary Selection section.


2 0 1 3 C O M S O L

3 From the Selection list, choose Disc faces.

4 Locate the Surface-to-Ambient Radiation section. From the list, choose User defined. In the associated edit field, type e_disc.

5 In the Tamb edit field, type T_air.

Surface-to-Ambient Radiation 21 Right-click Heat Transfer in Solids and choose Surface-to-Ambient Radiation.

2 In the Surface-to-Ambient Radiation settings window, locate the Boundary Selection

3

4

5

6

G1

2

3

N

W

W 15 | H E A T G E N E R A T I O N I N A D I S C B R A K E

section.


Locate the Surface-to-Ambient Radiation section. From the list, choose User defined. In the associated edit field, type e_pad.

In the Tamb edit field, type T_air.

To compute the heat produced and the heat dissipated, integrate the corresponding heating-power variables, Q_prod and Q_diss, over time. For this purpose, define two ODEs using a Global Equations node.

In the Model Builder windows toolbar, click the Show button and select Advanced Physics Options in the menu.

lobal Equations 1Right-click Heat Transfer in Solids and choose Global>Global Equations.

In the Global Equations settings window, locate the Global Equations section.


Here, W_prodt (W_disst) is COMSOL Multiphysics syntax for the time derivative of W_prod (W_diss). The table thus defines the two uncoupled initial value problems

To obtain the first-order ODEs, take the time derivative of the integrals

ame f(u,ut,utt,t) Initial value (u_0)

Initial value (u_t0)

Description

_prod W_prodt-Q_prod 0 0 Produced heat

_diss W_disst-Q_diss 0 0 Dissipated heat

W

prod/diss t Qprod/diss t =Wprod/diss 0 0=


16 | H E A T G E N

The initial values follow from setting t0.

M E S H 1

Free Triangular 11

2

3

4

S1

2

3

SIn

D1

2

3

Wprod/diss t Qprod/diss t' t'd0

t

=E R A T I O N I N A D I S C B R A K E 2 0 1 3 C O M S O L

In the Model Builder window, under Model 1 right-click Mesh 1 and choose More Operations>Free Triangular.

Click the Transparency button on the Graphics toolbar.

Select Boundaries 4, 7, and 11 only.

Click the Transparency button on the Graphics toolbar again to return to the original state.

izeIn the Model Builder window, under Model 1>Mesh 1 click Size.

In the Size settings window, locate the Element Size section.

From the Predefined list, choose Extra fine.

wept 1 the Model Builder window, right-click Mesh 1 and choose Swept.

istribution 1In the Model Builder window, under Model 1>Mesh 1 right-click Swept 1 and choose Distribution.

In the Distribution settings window, locate the Distribution section.

In the Number of elements edit field, type 2.


2 0 1 3 C O M S O L

4 In the Model Builder window, right-click Mesh 1 and choose Build All.

The complete mesh consists of roughly 5700 elements.

S

St1

2

3

4

5

So1

2


T U D Y 1

ep 1: Time DependentIn the Model Builder window, expand the Study 1 node, then click Step 1: Time Dependent.

In the Time Dependent settings window, locate the Study Settings section.

In the Times edit field, type range(0,0.05,1) range(1.2,0.2,3) range(4,10).

In the Model Builder window, right-click Study 1 and choose Show Default Solver.

Expand the Study 1>Solver Configurations node.

lver 1In the Model Builder window, expand the Study 1>Solver Configurations>Solver 1 node, then click Time-Dependent Solver 1.

In the Time-Dependent Solver settings window, click to expand the Absolute Tolerance section.

In the Tolerance edit field, type 1e-4.


18 | H E A T G E N

4 Locate the Time Stepping section. From the Steps taken by solver list, choose Intermediate.

This setting forces the solver to take at least one step in each specified interval.

5 In the Model Builder window, right-click Study 1 and choose Compute.

R E S U L T S

The first of the two default plots displays the surface temperature of the brake disc and pad at the end of the simulation interval. Modify this plot to show the time just before re

Te1

2

3

4

5

Tfo

11

2

3

4

5

6

7

8

9

10

11


leasing the brake.

mperature (ht)In the Model Builder window, under Results click Temperature (ht).

In the 3D Plot Group settings window, locate the Data section.

From the Time list, choose 1.8.


Click the Zoom Extents button on the Graphics toolbar.

Compare the result to the plot shown in Figure 2.

o compare the total heat produced and the heat dissipated, as done in Figure 5, llow the steps given below.

D Plot Group 3In the Model Builder window, right-click Results and choose 1D Plot Group.

In the 1D Plot Group settings window, click to expand the Title section.

From the Title type list, choose None.

Click to collapse the Title section. Locate the Plot Settings section. Select the x-axis label check box.

In the associated edit field, type Time (s).

Right-click Results>1D Plot Group 3 and choose Point Graph.

Select Point 1 only.

In the Point Graph settings window, locate the y-Axis Data section.

In the Expression edit field, type log10(W_prod+1).

Click to expand the Coloring and Style section. Find the Line style subsection. From the Color list, choose Blue.

Click to expand the Legends section. Select the Show legends check box.

From the Legends list, choose Manual.


2 0 1 3 C O M S O L


14 Right-click Results>1D Plot Group 3>Point Graph 1 and choose Duplicate.

15 In the Point Graph settings window, locate the y-Axis Data section.

16 In the Expression edit field, type log10(W_diss+1).

17

18

19

20

21

22

D1

2

3

4

5

6

Legends

log10(W_prod+1), heat produced

L

l 19 | H E A T G E N E R A T I O N I N A D I S C B R A K E

Locate the Coloring and Style section. Find the Line style subsection. From the Line list, choose Dashed.

Locate the Legends section. In the table, enter the following settings:


In the Model Builder window, right-click 1D Plot Group 3 and choose Rename.

Go to the Rename 1D Plot Group dialog box and type Dissipated and produced heat in the New name edit field.

Click OK.

Finally, follow the steps below to reproduce the plot in Figure 3.

ata SetsIn the Model Builder window, under Results right-click Data Sets and choose Cut Line 3D.

In the Cut Line 3D settings window, locate the Line Data section.

In row Point 1, set z to 0.013.

In row Point 2, set x to -0.047.

In row Point 2, set y to 0.1316.

In row Point 2, set z to 0.013.

egends

og10(W_diss+1), heat dissipated


20 | H E A T G E N

7 Click the Plot button.

8

9

10

21

2

3

4

5

6

7


In the Model Builder window, right-click Data Sets and choose More Data Sets>Parametric Extrusion 1D.

In the Parametric Extrusion 1D settings window, locate the Data section.

Click and Shift-click in the list to select all time steps from 0 through 3 s.


Right-click 2D Plot Group 4 and choose Surface.

In the Surface settings window, locate the Coloring and Style section.

From the Color table list, choose ThermalLight.

Right-click Results>2D Plot Group 4>Surface 1 and choose Height Expression.

Right-click 2D Plot Group 4 and choose Rename.

Go to the Rename 2D Plot Group dialog box and type Temperature profile vs time in the New name edit field.

Click OK.


2 0 1 3 C O M S O L

Buoyan c y F l ow i n Wa t e r

Introduction

This example studies the stationary state of free convection in a cavity filled with water and bounded by two vertical plates. To generate the buoyancy flow, the plates are heated at different temperatures, bringing the regime close to the transition between la

Alathapcosoprge

Aflo

Fi

Prco

C 1 | B U O Y A N C Y F L O W I N W A T E R

minar and turbulent.

n important step in setting up a convection model is to assess whether the flow stays minar or becomes turbulent. It is also important to approximate how fine should be e mesh needed to resolve velocity and temperature gradients. Both of these proximations rely on the velocity scale of the model. This makes the set-up of natural nvection problems challenging since the resulting velocity is part of the non-linear lution. There are however tools to approximate scales for natural convection oblems. These tools are demonstrated in this model using simple 2D and 3D ometries.

first 2D model representing a square cavity (see Figure 1) focuses on the convective w.

gure 1: Domain geometry and boundary conditions for the 2D model (square cavity).

Hot temperature (20C)

essure point

Water

Thermal insulation

Thermal insulationSquare side length (10 cm)

on the right plate

nstraint (0 Pa)

old temperature (10C)on the left plate


2 | B U O Y A N C Y

The 3D model (see Figure 2) extends the geometry to a cube. Compared to the 2D model, the front and back sides are additional boundaries that may influence the fluid behavior.

F

Bpp

M

2

Fimrired

Tsiva

Pressure pointconstraint (0 Pa) F L O W I N W A T E R 2 0 1 3 C O M S O L

igure 2: Domain geometry and boundary conditions for the 3D model (cubic cavity).

oth models calculate and compare the velocity field and the temperature field. The redefined Non-Isothermal Flow interface available in the Heat Transfer Module rovides appropriate tools to fully couple the heat transfer and the fluid dynamics.

odel Definition

D M O D E L

igure 1 illustrates the 2D model geometry. The fluid fills a square cavity with permeable walls, so the fluid flows freely within the cavity but cannot leak out. The

ght and left edges of the cavity are maintained at high and low temperatures, spectively. The upper and lower boundaries are insulated. The temperature ifferential produces the density variation that drives the buoyant flow.

he compressible Navier-Stokes equation contains a buoyancy term on the right-hand de to account for the lifting force due to thermal expansion that causes the density riations:

Cube side length (10 cm)

Water

Cold temperature (10C)on the left plate

Hot temperature (20C)on the right plate

u u p + u u T+ = 23--- u g+


2 0 1 3 C O M S O L

In this expression, the dependent variables for flow are the fluid velocity vector, u, and the pressure, p. The constant g denotes the gravitational acceleration, gives the temperature-dependent density, and is the temperature-dependent dynamic viscosity.

Because the model only contains information about the pressure gradient, it estimates the pressure field up to a constant. To define this constant, you arbitrarily fix the pressure at a point. No-slip boundary conditions apply on all boundaries. The no-slip co

A

Hsp

TteF

3

Fnosuin

M

BennuTte

TA

P

k

C

3 | B U O Y A N C Y F L O W I N W A T E R

ndition results in zero velocity at the wall but does not set any constraint on p.

t steady-state, the heat balance for a fluid reduces to the following equation:

ere T represents the temperature, k denotes the thermal conductivity, and Cp is the ecific heat capacity of the fluid.

he boundary conditions for the heat transfer interface are the fixed high and low mperatures on the vertical walls with insulation conditions elsewhere, as shown in igure 1.

D M O D E L

igure 2 shows the geometry and boundary conditions of the 3D model. The cavity is w a cube with high and low temperatures applied respectively at the right and left rfaces. The remaining boundaries (top, bottom, front and back) are thermally sulated.

O D E L A N A L Y S I S

efore starting the simulations, it is recommended to estimate the flow regime. To this d, four indicators are presented: the Reynolds, Grashof, Rayleigh, and Prandtl mbers. They are calculated using the thermophysical properties of water listed in

able 1. The thermophysical properties are given at 290 K which is in the range of the mperatures observed in the model.

BLE 1: THERMOPHYSICAL PROPERTIES FOR WATER AT 290 K

ARAMETER DESCRIPTION VALUE

Density 999 kg/m3

Dynamic viscosity 1.0810-3 Ns/m2

Thermal conductivity 0.60 W/(mK)

p Heat capacity 4.18 kJ/(kgK)

Thermal expansion coefficient 0.1710-3 K-1

Cpu T k T 0=


4 | B U O Y A N C Y

Prantdl Number

Definition The Prandtl number is the ratio of fluid viscosity to thermal diffusivity. It is defined by:

For air at 300 K, Pr0.71 (see, for example, A.4 in Ref. 1) and for water at 300 K, P

B

bre

Tstd

A

wTR

D

It

w

Aan

TA

P

PrCp

k----------= F L O W I N W A T E R 2 0 1 3 C O M S O L

r5.8 (see, for example, A.6 in Ref. 1).

oundary Layers The Prandtl number also indicates the relative thickness of the outer oundary layer, , and the thermal boundary layer, T. In the present case, it is asonable to estimate the ratio T by the relation (7.32 in Ref. 2)

(1)

he outer boundary layer is the distance from the wall to the region where the fluid abilizes. It is different from the momentum boundary layer, M, which measures the istance from the wall to the velocity peak.

pplication in This Model With the values given in Table 1, the Prandtl number for ater at 290 K, is found to be of the order 1 or 10. According to Equation 1, and should then be of same order of magnitude.

eynolds Number

efinition The Reynolds number estimates the ratio of inertial forces to viscous forces. is defined by the formula

here U denotes the typical velocity and L the typical length.

t atmospheric pressure and at 300 K, air and water have the following properties (A.4 d A.6 in Ref. 1).

BLE 2: THERMOPHYSICAL PROPERTIES OF AIR AND WATER AT 300 K AND ATMOSPHERIC PRESSURE

ARAMETER AIR WATER

1.16 kg/m3 997 kg/m3

1.8510-5 N.s.m-2 8.5510-4 N.s.m-2

T------ Pr=

Re UL------------=


2 0 1 3 C O M S O L

You can thus approximate the Reynolds number by:

In these relations, U has to be in meters per second and L in meters.

The Reynolds number can be rewritten as the velocity ratio

wbe

Fl

Gflo

Asm20

M

ev

A

R

w

G

D

de

Reair 6 104UL Rewater 106UL

Re U-------------------= 5 | B U O Y A N C Y F L O W I N W A T E R

hich compares U to L. The latter quantity is homogeneous to a velocity and can seen as the typical velocity due to viscous forces.

ow Regime The value of the Reynolds number is used to predict the flow regime. enerally, low values of Re correspond to laminar flow and high values to turbulent w, with a critical value for the transition regime that depends on the geometry.

s an indication, Reynolds experiments concerning the flow along a straight and ooth pipe showed that the transition regime in this case occurs when Re is between 00 and 104 (see chapter 1.3 in Ref. 3).

omentum Boundary Layer The momentum boundary layer thickness can be aluated, using the Reynolds number, by (5.36 in Ref. 2)

(2)

pplication in This Model The typical length L of the model is equal to 10 cm so the eynolds number is evaluated as

here U is still unknown. Estimates of this typical velocity will be provided later.

rashof Number

efinition The Grashof number gives the ratio of buoyant to viscous forces. It is fined by

L

M LRe-----------

Re 105U

Gr 2g2

-------------TL3=


6 | B U O Y A N C Y

where g is the gravity acceleration (equal to 9.81 m.s-2) and T is the typical temperature difference.

At atmospheric pressure and at 300 K, air and water have the following properties (A.4 and A.6 in Ref. 1).

Tg

Y

In

T

w

T

F

isinR

A

is

TABLE 3: THERMOPHYSICAL PROPERTIES OF AIR AND WATER AT 300 K AND ATMOSPHERIC PRESSURE

PARAMETER AIR WATER

1.16 kg/m3 997 kg/m3 F L O W I N W A T E R 2 0 1 3 C O M S O L

he value of for air was here obtained by the following formula, considering the ideal as approximation:

ou can then approximate the Grashof number by:

these relations, T is given in kelvin and L in meters.he Grashof number can also be expressed as the velocity ratio

here U0 is defined by

(3)

his quantity can be considered as the typical velocity due to buoyancy forces.

low Regime When buoyancy forces are large compared to viscous forces, the regime turbulent; otherwise it is laminar. The transition between these two regimes is dicated by the critical order of the Grashof number which is 109 (see Figure 7.7 in ef. 2).

pplication in This Model In this model, T is equal to 10 K so the Grashof number about 107 which indicates that a laminar regime is expected.

1.8510-5 N.s.m-2 8.5510-4 N.s.m-2

3.3410-3 K-1 2.7610-4 K-1

1T----=

Grair 108TL3 Grwater 4 10 9TL3

GrU0

2

L 2---------------------------=

U0 gTL=


2 0 1 3 C O M S O L

Table 4 provides the values of the quantities necessary to calculate U0. This velocity is here of order 10 mm/s.

Rayleigh Number

D

sok,

Nnu

Afo

In

U

w

Fl

tr

TABLE 4: THERMOPHYSICAL PROPERTIES OF WATER AT 290 K USED IN THE GRASHOF NUMBER

PARAMETER DESCRIPTION VALUE

g Gravitational acceleration 9.81 m/s2

Thermal expansion coefficient 0.1710-3 K-1 7 | B U O Y A N C Y F L O W I N W A T E R

efinition The Rayleigh number is another indicator of the regime. It is defined by

it is similar to the Grashof number except that it accounts for the thermal diffusivity, given by

ote: The Rayleigh number can be expressed in terms of the Prandtl and the Grashof mbers through the relation Ra = Pr Gr.

t atmospheric pressure and at 300 K, you can use the approximations of Ra below r air and water (A.4 in Ref. 1)

these relations, T is given in kelvin and L in meters.sing Equation 3, the Rayleigh number can be rewritten as the velocity ratio

here the ratio L can be seen as a typical velocity due to thermal diffusion.

ow Regime Like the Grashof number, a critical Rayleigh value indicates the ansition between laminar and turbulent flow. For vertical plates, this limit is about

Ra2gCp

k--------------------TL3

=

kCp----------=

Raair 108TL3 Rawater 2 1010TL3

RaU0

2

L L --------------------------------=


8 | B U O Y A N C Y

109 (9.23 in Ref. 1).

Typical Velocity Because the viscous forces limit the effects of buoyancy, U0 may give an overestimated typical velocity. Another approach (see 7.25 in Ref. 2) is to use U1 instead, defined by

th

o

Tanp

T

bth

A

bla

STincow

Fcr

U1L---- Ra= F L O W I N W A T E R 2 0 1 3 C O M S O L

at is

(4)

r

his should be a more accurate estimate of U because the fluids thermal diffusivity d viscosity are used in the calculations. From now on, U1 will therefore be the

referred estimate of U.

hermal Boundary Layer The Rayleigh number can be used to estimate the thermal oundary layer thickness, T. When Pr is of order 1 or greater, it is approximated by e formula (7.25c in Ref. 2)

(5)

pplication in This Model Here, Ra is of order 108. The laminar regime is confirmed ut the Rayleigh number found is near the transition zone. The thermal boundary yer thickness is then found to be of order 1 mm and U1 of order 10 mm/s.

ynthesiso prepare the simulation, it is very useful to follow the steps below that give dications of what results to expect. It is important to remember that the quantities mputed here are only order of magnitude estimates, which should not be considered

ith more than one significant digit.

irst evaluate the Grashof and Rayleigh numbers. If they are significantly below the itical order of 109, the regime is laminar. In this case, Equation 3 or Equation 4

U1k

CpL--------------- Ra=

U1U0Pr

----------=

T LRa4

------------


2 0 1 3 C O M S O L

provide estimates of the typical velocity U that you can use to validate the model after performing the simulation.

According to Equation 1, the Prandtl number determines the relative thickness of the thermal boundary layer and the outer layer. Equation 2 and Equation 5 then provide orders of magnitude of the thicknesses. When defining the mesh, refinements have to be done at the boundary layers by, for instance, inserting three to five elements across the estimated thicknesses.

Hfonoes

FTTisla 9 | B U O Y A N C Y F L O W I N W A T E R

ere, Gr and Ra are 107 and 108, respectively, and thus below the critical value of 109 r vertical plates. A laminar regime is therefore expected but because these values are t significantly below 109, convergence is not straightforward. In this regime, the

timates U0 and U1 of the typical velocity are both of the order 10 mm/s.

or water at 290 K, Pr is about 10 so and T are of same orders of magnitude. Here, is of the order 1 mm.

he Reynolds number calculated with U1 is about 103, which confirms that the model

close to the transition regime. Using U1 and Equation 2, the momentum boundary yer thickness M is found to be about 1 mm.


10 | B U O Y A N C

Results and Discussion

2 D M O D E L

Figure 3 shows the velocity distribution in the square cavity.

F

TvethY F L O W I N W A T E R 2 0 1 3 C O M S O L

igure 3: Velocity magnitude for the 2D model.

he regions with faster velocities are located at the lateral boundaries. The maximum locity is 4.20 mm/s which is in agreement with the estimated typical velocity U1 of e order 10 mm/s. According to Figure 4, the momentum boundary layer thickness


2 0 1 3 C O M S O L

is of order 1 mm, as calculated before.

Fi 11 | B U O Y A N C Y F L O W I N W A T E R

gure 4: Velocity profile at the left boundary.


12 | B U O Y A N C

Figure 5 shows the temperature field (surface) and velocity field (arrows) of the 2D model.

F

AbvaacY F L O W I N W A T E R 2 0 1 3 C O M S O L

igure 5: Temperature field (surface plot) and velocity (arrows) for the 2D model.

large convective cell occupies the whole square. The fluid flow follows the oundaries. As seen in Figure 3, it is faster at the vertical plates where the highest riations of temperature are located. The thermal boundary layer is of the order 1 mm cording to Figure 6, which is in agreement with the estimate provided by


2 0 1 3 C O M S O L

Equation 5. The outer layer is slightly thicker than the boundary layer.


gure 6: Temperature profile at the left boundary.


14 | B U O Y A N C

Figure 7 shows the correlation between density and temperature.

F

AatTY F L O W I N W A T E R 2 0 1 3 C O M S O L

igure 7: Water density for the 2D model.

s expected, has the same profile as the temperature field. High variations are located the boundaries, especially at lateral walls, and are responsible for the free convection. he variations are smoother at the center.


2 0 1 3 C O M S O L

3 D M O D E L

Figure 8 illustrates the velocity plot parallel to the heated plates.


gure 8: Velocity magnitude field for the 3D model, slices parallel to the heated plates.


16 | B U O Y A N C

A second velocity magnitude field is shown in Figure 9. The plot is close to what was obtained in 2D in Figure 3.

FplY F L O W I N W A T E R 2 0 1 3 C O M S O L

igure 9: Velocity magnitude field for the 3D model, slices perpendicular to the heated ates.


2 0 1 3 C O M S O L

In Figure 10, velocity arrows are plotted on temperature surface at the middle vertical plane parallel to the plates.

Fia

Nthtris 17 | B U O Y A N C Y F L O W I N W A T E R

gure 10: Temperature (surface plot) and velocity (arrows) fields in the cubic cavity, for temperature difference of 10 K between the vertical plates.

ew small convective cells appear on the vertical planes perpendicular to the plates at e four corners. They are more visible at lower Gr values, that is, far from the ansition regime. In Figure 11, the temperature difference between the vertical plates reduced to 1 K and 0.1 K to decrease the Grashof number to 105 and 106.


18 | B U O Y A N C

Observe the bigger cells at the four corners of the plane.

FwplY F L O W I N W A T E R 2 0 1 3 C O M S O L

igure 11: Temperature (surface plot) and velocity (arrows) fields in the cubic cavity, ith, for a temperature difference of 1 K (top) and 0.1 K (bottom) between the vertical ates.


2 0 1 3 C O M S O L

Notes About the COMSOL Implementation

The material properties for water are available in the Material Library. The density and dynamic viscosity are functions of the temperature.

At high Gr values, using a good initial condition becomes important in order to achieve convergence. Moreover, a well-tuned mesh is needed to capture the solution, especially the temperature and velocity changes near the walls. Use the Stationary study steps continuation option with T as the continuation parameter to get a solver seitnudodi

TtelaItP

R

1.H

2.

3.U

MTu

M

M

1 19 | B U O Y A N C Y F L O W I N W A T E R

quence that uses previous solutions to estimate the initial condition. For this model, is appropriate to ramp up T from 102 K to 101 K, which corresponds to a Grashof mber range of 104107. At Gr104, the model is easy to solve. The regime is minated by conduction and viscous effects. At Gr107, the model becomes more

fficult to solve. The regime is greatly influenced by convection and buoyancy.

o get a well-tuned mesh when Gr reaches 107, the element size near the prescribed mperature boundaries has to be smaller than the momentum and thermal boundary yer thicknesses, which are of order 1 mm according to Equation 2 and Equation 5. is recommended to have three to five elements across the layers when using 1 elements (the default setting for fluid flows).

eferences

F.P. Incropera, D.P. DeWitt, T.L. Bergman, and A.S. Lavine, Fundamentals of eat and Mass Transfer, 6th edition, John Wiley & Sons, 2006.

A. Bejan, Heat Transfer, John Wiley & Sons, 1985.

P. A. Davidson, Turbulence: An Introduction for Scientists and Engineers, Oxford niversity Press, 2004.

odel Library path: Heat_Transfer_Module/torial_Models,_Forced_and_Natural_Convection/buoyancy_water

odeling Instructions

O D E L W I Z A R D



20 | B U O Y A N C

2 Click the 2D button.

3 Click Next.

4 In the Add physics tree, select Fluid Flow>Non-Isothermal Flow>Laminar Flow (nitf).

5 Click Add Selected.

6 Click Next.

7 Find the Studies subsection. In the tree, select Preset Studies>Stationary.

8 Click Finish.

G

Pa1

2Y F L O W I N W A T E R 2 0 1 3 C O M S O L





2 0 1 3 C O M S O L


G

Sq1

2

3

4

Name Expression Description

L 10[cm] Square side length

DeltaT 10[K] Temperature difference

Tc 283.15[K] Low temperature

Th Tc+DeltaT High temperature

rho 999[kg/m^3] Density

m

k

C

b

U

U

P

G

R

R

R

e

e 21 | B U O Y A N C Y F L O W I N W A T E R

The Grashof and Rayleigh numbers should be less than 109, indicating that a laminar regime is expected.

E O M E T R Y 1

uare 1In the Model Builder window, under Model 1 right-click Geometry 1 and choose Square.

In the Square settings window, locate the Size section.

In the Side length edit field, type L.

Click the Build All button.

u 1.08e-3[N*s/m^2] Dynamic viscosity

0.60[W/(m*K)] Thermal conductivity

p 4.18[kJ/(kg*K)] Heat capacity

eta 0.17e-3[1/K] Coefficient of thermal expansion

0 sqrt(g_const*beta*DeltaT*L)

Typical velocity due to buoyancy

1 U0/sqrt(Pr) Typical velocity estimation

r mu*Cp/k Prandtl number

r (U0*rho*L/mu)^2 Grashof number

a Pr*Gr Rayleigh number

e0 rho*U0*L/mu Reynolds number approximation with U0

e1 rho*U1*L/mu Reynolds number approximation with U1

ps_t L/(Ra)^0.25 Thermal boundary layer thickness

ps_m L/sqrt(Re1) Momentum boundary layer thickness


22 | B U O Y A N C

M A T E R I A L S

Material Browser1 In the Model Builder window, under Model 1 right-click Materials and choose Open

Material Browser.

2 In the Material Browser settings window, In the tree, select Built-In>Water, liquid.

3 In the Material_browser window, click Add Material to Model.

N

Dva

In1

2

3

4

Vo1

2

3

4

P1

0

-Y F L O W I N W A T E R 2 0 1 3 C O M S O L

O N - I S O T H E R M A L F L O W

efine the initial temperature as the mean value between the high and low temperature lues.

itial Values 1In the Model Builder window, under Model 1>Non-Isothermal Flow click Initial Values 1.


In the T edit field, type (Tc+Th)/2.

In the p edit field, type rho*g_const*(L-y).

This setting gives an initial pressure field consistent with the volume force. The initial pressure field must be also consistent with the pressure constraint (see the section Pressure Point Constraint below).

lume Force 1In the Model Builder window, right-click Non-Isothermal Flow and choose the domain setting Laminar Flow>Volume Force.


In the Volume Force settings window, locate the Volume Force section.

In the F table, enter the following settings:

In this expression for the buoyancy flow, g_const is a predefined constant corresponding to the gravity acceleration.

ressure Point Constraint 1Right-click Non-Isothermal Flow and choose Points>Pressure Point Constraint.

x

nitf.rho*g_const y


2 0 1 3 C O M S O L

2 Select Point 2 only.

Fixing the pressure at an arbitrary point is necessary to define a well-posed model.

Temperature 11 Right-click Non-Isothermal Flow and choose the boundary condition Heat

Transfer>Temperature.


3 In the Temperature settings window, locate the Temperature section.

4

Te1

2

3

4

Ncr

M

1

2

3

4

S

Bhitenu

St1

2

3


In the T0 edit field, type Tc.

mperature 2Right-click Non-Isothermal Flow and choose the boundary condition Heat Transfer>Temperature.


In the Temperature settings window, locate the Temperature section.

In the T0 edit field, type Th.

ow modify the default mesh size settings to ensure that the mesh satisfies the iterion discussed in the Introduction section.

E S H 1

In the Model Builder window, under Model 1 click Mesh 1.

In the Mesh settings window, locate the Mesh Settings section.

From the Element size list, choose Extra fine.


T U D Y 1

ecause the Grashof number is near the critical value of around 109, the model is ghly nonlinear. To achieve convergence, use continuation to ramp up the mperature difference value from 0.01 K to 10 K, which corresponds to a Grashof mber from 104 to 107.

ep 1: StationaryIn the Model Builder window, under Study 1 click Step 1: Stationary.

In the Stationary settings window, locate the Study Extensions section.

Select the Continuation check box.

Click Add.


24 | B U O Y A N C


6 In the Model Builder windows toolbar, click the Show button and select Advanced Physics Options in the menu.

The pseudo stepping time option is generally useful to help the convergence of a stpF

N

1

2

3

S

In

R

VeTh

TeTfi

1

2

3

4

5

Continuation parameter Parameter value list

DeltaT 1e-2 1e-1 1 10Y F L O W I N W A T E R 2 0 1 3 C O M S O L

ationary flow model. However, a continuation approach is already used here. In this recise model, disabling the pseudo time stepping option improves the convergence. ollow the instructions below to do so.

O N - I S O T H E R M A L F L O W

In the Model Builder window, under Model 1 click Non-Isothermal Flow.

In the Non-Isothermal Flow settings window, click to expand the Advanced Settings section.

Find the Pseudo time stepping subsection. Clear the Use pseudo time stepping for stationary equation form check box.

T U D Y 1

the Model Builder window, right-click Study 1 and choose Compute.

E S U L T S

locity (nitf)he first default plot group shows the velocity magnitude as in Figure 3. Notice the igh velocities near the lateral walls due to buoyancy effects.

mperature (nitf)he second default plot shows the temperature distribution. Add arrows of the velocity eld to see the correlations between velocity and temperature, as in Figure 5.

In the Model Builder window, under Results right-click Temperature (nitf) and choose Plot.

Right-click Results>Temperature (nitf) and choose Arrow Surface.

In the Arrow Surface settings window, locate the Coloring and Style section.

From the Color list, choose Black.


To reproduce the plot of Figure 7, follow the steps below.


2 0 1 3 C O M S O L

2D Plot Group 31 In the Model Builder window, right-click Results and choose 2D Plot Group.

2 In the 2D Plot Group settings window, locate the Data section.

3 From the Data set list, choose Solution 1.

4 From the Parameter value (DeltaT) list, choose 10.

5 Right-click Results>2D Plot Group 3 and choose Surface.

6 In the Surface settings window, click Replace Expression in the upper-right corner of

7

8

9

10

Inbo

D1

2

3

4

5

6

7

11

2

3

4

5


the Expression section. From the menu, choose Non-Isothermal Flow (Laminar Flow)>Density (nitf.rho).


Go to the Rename 2D Plot Group dialog box and type Density in the New name edit field.

Click OK.

Right-click 2D Plot Group 3 and choose Plot.

the following steps, the temperature and velocity profiles are plotted near the left undary in order to estimate the boundary layer thicknesses of the solution.

ata SetsIn the Model Builder window, under Results right-click Data Sets and choose Cut Line 2D.

In the Cut Line 2D settings window, locate the Line Data section.

In row Point 1, set x to 0.

In row Point 1, set y to 5[cm].

In row Point 2, set x to 1[cm].

In row Point 2, set y to 5[cm].




From the Data set list, choose Cut Line 2D 1.

From the Parameter selection (DeltaT) list, choose Last.

Right-click Results>1D Plot Group 4 and choose Line Graph.

Right-click Results>1D Plot Group 4>Line Graph 1 and choose Plot.


26 | B U O Y A N C

7 Right-click 1D Plot Group 4 and choose Rename.

8 Go to the Rename 1D Plot Group dialog box and type Temperature at boundary layer in the New name edit field.

9 Click OK.

The thermal boundary layer is around 3 mm.

1D Plot Group 51 Right-click Results and choose 1D Plot Group.

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3

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5

6

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10

N

R

In

M

1

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6Y F L O W I N W A T E R 2 0 1 3 C O M S O L


From the Data set list, choose Cut Line 2D 1.

From the Parameter selection (DeltaT) list, choose Last.

Right-click Results>1D Plot Group 5 and choose Line Graph.

In the Line Graph settings window, click Replace Expression in the upper-right corner of the y-Axis Data section. From the menu, choose Non-Isothermal Flow (Laminar Flow)>Velocity magnitude (nitf.U).



Go to the Rename 1D Plot Group dialog box and type Velocity at boundary layer in the New name edit field.

Click OK.

The momentum boundary layer is around 1 mm and the outer layer between 5 mm and 10 mm.

ow create the 3D version of the model.

O O T

the Model Builder window, right-click the root node and choose Add Model.

O D E L W I Z A R D


Click Next.

In the Add physics tree, select Recently Used>Non-Isothermal Flow (nitf).

Click Add Selected.

Click Next.

Find the Studies subsection. In the tree, select Preset Studies for Selected Physics>Stationary.


2 0 1 3 C O M S O L

7 Click Finish.

G E O M E T R Y 2

Block 11 In the Model Builder window, under Model 2 right-click Geometry 2 and choose Block.

2 In the Block settings window, locate the Size and Shape section.

3 In the Width edit field, type L.

4

5

6

M

M1

2

3

N

In1

2

3

4

Vo1

2


In the Depth edit field, type L/2.

In the Height edit field, type L.


A T E R I A L S

aterial BrowserIn the Model Builder window, under Model 2 right-click Materials and choose Open Material Browser.

In the Material Browser settings window, In the tree, select Built-In>Water, liquid.

In the Material_browser window, click Add Material to Model.

O N - I S O T H E R M A L F L O W 2

itial Values 1In the Model Builder window, under Model 2>Non-Isothermal Flow 2 click Initial Values 1.


In the T2 edit field, type (Tc+Th)/2.

In the p2 edit field, type rho*g_const*(L-z).

lume Force 1In the Model Builder window, right-click Non-Isothermal Flow 2 and choose the domain setting Laminar Flow>Volume Force.


In the Volume Force settings window, locate the Volume Force section.


28 | B U O Y A N C

4 In the F table, enter the following settings:

Pressure Point Constraint 11 Right-click Non-Isothermal Flow 2 and choose Points>Pressure Point Constraint.

2

Te1

2

3

4

Te1

2

3

4

S1

2

S1

2

M

Tac

0 x

0 y

-nitf2.rho*g_const zY F L O W I N W A T E R 2 0 1 3 C O M S O L

Select Point 4 only.

mperature 1Right-click Non-Isothermal Flow 2 and choose the boundary condition Heat Transfer>Temperature.



In the T0 edit field, type Tc.

mperature 2Right-click Non-Isothermal Flow 2 and choose the boundary condition Heat Transfer>Temperature.



In the T0 edit field, type Th.

ymmetry, Heat 1Right-click Non-Isothermal Flow 2 and choose the boundary condition Heat Transfer>Symmetry, Heat.


ymmetry, Flow 2Right-click Non-Isothermal Flow 2 and choose the boundary condition Laminar Flow>Symmetry, Flow.


E S H 2

o obtain reliable results in a reasonable computational time, create a structured mesh cording to the steps below.


2 0 1 3 C O M S O L

Mapped 11 In the Model Builder window, under Model 2 right-click Mesh 2 and choose More

Operations>Mapped.


Distribution 11 Right-click Model 2>Mesh 2>Mapped 1 and choose Distribution.

2 Select Edges 1, 3, 5, and 9 only.

3

4

5

6

7

8

9

Tan

N 29 | B U O Y A N C Y F L O W I N W A T E R

In the Distribution settings window, locate the Distribution section.

From the Distribution properties list, choose Predefined distribution type.


In the Element ratio edit field, type 3.

From the Distribution method list, choose Arithmetic sequence.

Select the Symmetric distribution check box.


he front face mesh has smaller elements near the edges because large variations in velocity d temperature are expected there.

ow extend the front mesh to the remaining structure.


30 | B U O Y A N C

Swept 1In the Model Builder window, right-click Mesh 2 and choose Swept.

Distribution 11 In the Model Builder window, under Model 2>Mesh 2 right-click Swept 1 and choose

Distribution.

2 In the Distribution settings window, locate the Distribution section.

3 From the Distribution properties list, choose Predefined distribution type.

4

5

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7

Tm

B1

2

3

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5

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7

S

S1

2

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5

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DY F L O W I N W A T E R 2 0 1 3 C O M S O L


In the Element ratio edit field, type 3.

From the Distribution method list, choose Arithmetic sequence.

Select the Reverse direction check box.

o r

basic Heat Transferencia Model Library Manual comsol 4.3

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