Lecture 17: Natural (Free) convection

1

1. Assume that gravitational field affects the flow

0v

div

gvp

vvtv

TTvtT

http://www.youtube.com/watch?v=qIlcXp-cKHg

2

20

0

00

0

0

00

00

0

0

0 1

1

pppppppppp

5. Linearization of the pressure term

6. Substituting this into Navier-Stokes equation

gvppp

vvtv

2

0

0

00

0

or vgp

vvtv

00

3. We separate the static parts and the non-static contributions:

00 ,ppp

4. Assume that 1,1

00

pp

2. Statics: 00

0 gp

3

gTvp

vvtv

0

TTvtT

0div v

Let us non-dimensionalise these equations.

7. In incompressible flow, density variations are due to temperature non-isothermalities

T 000

vgTp

vvtv

0

Finally, the governing equations for thermal convection in Boussinesq approximation

4

L -- length scale (typical size)

2L -- time-scale (also called convective time scale)

L

Lv

-- scale of velocity

2

0

2

0

L

vp -- scale of pressure

T -- scale of temperature (Θ is the typical temperature difference)

sm2

Dimensions of the viscosity and thermal conductivity coefficients

K1Dimension of the heat

expansion coefficient

Gravity acceleration kgg

dimdimdimdimdimdimdimdim ,,, nonnonnonnon TTpvpvvvtt

Non-dimensionalisation

5

PrTPr

TvtT

,1

0v

div

Non-dimensionalization of the Navier-Stokes equation:

kgTv

Lvp

Lv

vvLv

tvv

20

20

2

or

2

3

gL

GrkTGrvpvvtv

,

Heat transfer equation,

TL

TvLv

tT

2

or

Non-dimensionalization of the continuity equation is trivial:

Non-dimensionalised equations of thermal convection

6

2

3

gLGr

Non-dimensional parameters:

Pr

-- Grashof number (defines the strength of the forcing (buoyancy) term; characterises intensity of the convective flow)

-- Prandtl number (characterises the fluid properties; defines the relative importance of thermal conduction and convection as two mechanisms of heat transfer, Pr>>1: convection dominates, Pr<<1: convection can be disregarded)

kTGrvpvvtv

TPr

TvtT

1

0v

div

Czochralski process

7

The Czochralski process is a one of the methods of crystal growth from melt, used to obtain single crystals of semiconductors (e.g. silicon, germanium and gallium arsenide), metals (e.g. palladium, platinum, silver, gold), salts, and synthetic gemstones.

When the silicon is fully melted (~15000C), a small seed crystal mounted on the end of a rotating shaft is slowly lowered until it just dips below the surface of the molten silicon. The seed crystal's rod is slowly pulled upwards.

Silicon ingot

Main difficulty: strong convective flows in the melt. From one hand, convection makes the melt more uniform, which is positive. From another hand, strong flows near the crystallization plane introduce crystal defects. Convection is controlled by rotation/vibration of ingot and crucible, by magnetic field, …

Solutal convection

8

Consider incompressible isothermal fluid flow with an admixture.

Equation of mass conservation for an impurity:

CDCvtC

(this equation is valid for not very large C)

Diffusion coefficient

Variations in density can be caused by admixture: C000

gCvp

vvtv

0

CDCvtC

0v

div

Equations of solutal convection:

P.S. If flow is non-isothermal, the thermosolutal (or double-diffusive) convective flow is induced.

Lecture 1Flows driven by surface tension gradients

Spatial variations in surface tension result in added tangential stresses at an interface, giving rise to fluid motions in the underlying bulk liquids. The motion induced by tangential gradients of surface tension is termed the Marangoni effect.

Examples:

the camphor ball ‘dances’ on a water surface (http://www.youtube.com/watch?v=Pe88T45VdR8),

the calming effect of ‘oil troubled waters’ (http://www.youtube.com/watch?v=00PPPt7EJqo).

9

10

T The surface tension coefficient is function of temperature; the thermocapillary flows are induced.

C function of concentration; the difusocapillary flows.

q function of electric charge; the electrocapillary flows.

This motion is induced at an interface, where, besides the normal force, there is another force tangential to the surface, . Adding this force, we obtain the following boundary condition

i

kikiki xnn

RRpp

,2,1

2121

11

is the unit normal vector directed into medium 1.n

tf

Thermocapillary motion in a thin liquid layer

The thermocapillary motion generated in an open rectangular shallow pan with a very thin layer at the bottom.

11

z

x

h1h2

g

h(x)

l

T1

(hot)

T2

(cold)

The difference in side wall temperatures results in a temperature gradient along the surface. For liquids,

0T

α

α1

α2

x

1hl

The flow in nearly lateral. Any flow nonuniformities at the side walls are small and the flow is essentially 2D.

Except at the side walls, which are far removed from the bulk flow, the vertical velocity component is very much smaller than the horizontal component and any effects of the free surface curvature may be neglected.

We assume that the liquid layer is thin enough to neglect inertial effects, implying Re<<1 (‘shallow water’ approximation).

Governing equations:

12

2

2

zu

xp

gzp

x-projection of the momentum equation,

z-projection,

0,0,zuv 21 RR

The continuity equation (in integral form) for the fully developed flow is

0d0

zzuxh

(1)

(2)

(3)

Boundary conditions:

13

0u1. Bottom (z=0):

2. free surface (z=h):

ikikiatm x

nnpp

Or, for our flow,

1,0,0n

dxd

zu

atmpp

Integration of (1) with the use of (4) and (5) gives

2

21

zxp

zxp

hdxd

u

(4)

(5)

(6)

Integration of (2) with the use of (6) gives

zhgpp atm

(7)

(8)

(8) specifies the relation between the pressure gradient and variation of free surface height in the x-direction as

14

dxdh

gxp

(9)

By using (7) and (9), condition (3) results in dxdh

ghdxd

32

Or, 21

21 3

hhg Here, we assumed that

α=α1 and h=h1 at x=0.

The velocity profile:

dxd

hzz

u

1

23

2

umax

dxdh

u

4max

The requirement Re<<1 may be expressed as

15

14

211max dxdhhu

Re

or

dxdh

2

21

4

The variation of α with T for liquids is close to linear. For water,

KmmN

T

15.0 Thus

with mK

dxdT

100 This would give a value 215

mmN

dxd

sm2

610 3310mkgand Hence, mmh 11

gc 2 -- capillary length

1

21

2

1

ghh

Boc

We did not consider the gravity-driven convection. The gravity between the gravity and capillary forces is defined by the Bond number:

Here,

16

For water/air interface, . And, for our configuration, 3.0BommN

731

This means that, for the configuration examined, the flows driven by the surface tension effect (Marangoni convection) dominate (in comparison with the gravity-driven convective flows).

The surface tension gradients can be important in such very thin layers of mm size or less, or in a reduced gravity environment. For example, a crystal grown from its melt under reduced gravity is governed by convection driven by thermally induced surface tension gradients rather than buoyancy forces.

17

Ludwig Prandtl (4 February 1875, Freising, Upper Bavaria – 15 August 1953, Gottingen) was a German scientist. He was a pioneer in the development of rigorous systematic mathematical analyses which he used to underlay the science of aerodynamics, which have come to form the basis of the applied science of aeronautical engineering. In the 1920s he developed the mathematical basis for the fundamental principles of subsonic aerodynamics in particular.

18

Carlo Giuseppe Matteo Marangoni (29 April 1840, Pavia Italy – 14 April 1925, Florence, Italy) was an Italian physicist.He held a position of High School Physics Teacher at the Liceo Dante (Florence) for 45 years, until retirement in 1916.He mainly studied surface phenomena in liquids and also contributed to meteorology.

Joseph Valentin Boussinesq (13 March 1842, Saint-Andre-de-Sangonis, France –Died 19 February 1929, Paris, France) was a French mathematician and physicist who made significant contributions to the theory of hydrodynamics, vibration, light, and heat.

Franz Grashof (July 11, 1826 in Dusseldorf - October 26, 1893 in Karsruhe) was a German engineer. He was a professor of Applied Mechanics at the Technische Hochschule Karlsruhe. He developed some early steam-flow formulas but made no significant contribution to free convection.

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Jan Czochralski (23 October 1885 – 22 April 1953) was a Polish chemist who invented the Czochralski process, which is used for growing single crystals and in the production of semiconductor wafers. He was educated at Charlottenburg Polytechnic in Berlin, where he specialized in metal chemistry. Czochralski discovered the Czochralski method in 1916, when he accidentally dipped his pen into a crucible of molten tin. He immediately pulled his pen out to discover that a thin thread of solidified metal was hanging from the nib. The nib was replaced by a capillary, and Czochralski verified that the crystallized metal was a single crystal. Czochralski's experiments produced single crystals a millimeter in diameter and up to 150 centimeters long. In 1950, the method was used to grow single germanium crystals, leading to its use in semiconductor production.