Prabal TalukdarPrabal Talukdar - libvolume6.xyzlibvolume6.xyz/mechanical/btech/semester6/heatandmasstransfer/free... · Prabal TalukdarPrabal Talukdar ... Buoyancy ForceBuoyancy Force

NATURAL/FREE CONVECTION

Prabal TalukdarPrabal TalukdarAssociate Professor

Department of Mechanical EngineeringDepartment of Mechanical EngineeringIIT Delhi

E-mail: [email protected]

Natural/freeNatural/free convection

P.Talukdar/Mech-IITD 2

Natural ConvectionNatural Convection


Buoyancy ForceBuoyancy ForceThe upward force exerted by a fluid on a body completely or partially immersed in it is called the buoyancy force. The magnitude of the b f i l t th i ht f th fl id di l d b th b dbuoyancy force is equal to the weight of the fluid displaced by the body

F = ρfluidgVbody

Fnet = W – Fbouyancy= ρbodygVbody - ρfluidgVbody= (ρbody - ρfluid)gVbody(ρbody ρfluid)g body

It i th b f th t k th hi fl t


It is the buoyancy force that keeps the ships afloat in water (W = Fbuoyancy ) for floating objects

Volume Expansion CoefficientVolume Expansion CoefficientThe coefficient of volume expansionis a measure of the change in volume of a substance with temperature at constant pressure

11 ⎞⎛ ∂⎞⎛ ∂

PP T1

T1

⎟⎠⎞

⎜⎝⎛∂ρ∂

ρ−=⎟

⎠⎞

⎜⎝⎛∂ν∂

ν=β

11 ρρρΔTT

1T

1−ρ−ρ

ρ−=

ΔρΔ

ρ−≈β

∞

∞ At constant P

( )−ρβ=ρ−ρ TT

For an ideal gas P = ρRT

( )∞∞ −ρβ=ρ−ρ TT


T1

gas ideal =β

Equation of MotionEquation of Motion


Newton’s 2nd lawNewton s 2 lawNewton’s 2nd law gives: x,bodyx,surfacex FFam +=⋅δ

Mass

A l ti

,y,

)1dydx(m ⋅⋅ρ=δ

Acceleration

yu

xuu

dtdy

yu

dtdx

xu

dtdua x ∂

∂ν+

∂∂

=∂∂

+∂∂

==

Forces

yy

)1.dy.dx(g)1dy(dxP)1dx(dyFx ρ−⋅⎟⎠⎞

⎜⎝⎛∂∂

−⋅⎟⎟⎠

⎞⎜⎜⎝

⎛∂τ∂

=

)1dydx(gxP

yu

)y(g)y(x

)(yy

2

2

x

⋅⋅⎟⎟⎠

⎞⎜⎜⎝

⎛ρ−

∂∂

−∂

∂μ=

ρ⎠⎝ ∂⎟

⎠⎜⎝ ∂


⎠⎝

⎟⎠⎞⎜

⎝⎛

∂∂μ=τ y

u

Momentum Equationg

xP

yu

yu

xuu 2

2ρ−

∂∂

−∂

∂μ=⎟⎟

⎠

⎞⎜⎜⎝

⎛∂∂

ν+∂∂

ρ

The x-momentum equation for the quiescent field outside the boundary layer can be found by applying the above equation as u = 0

yy ∂⎠⎝

The y-momentum equation results:

gx

P∞

∞ ρ−=∂∂

0P=

∂ )x(P)x(PP ∞==The y-momentum equation results: y∂

gx

PxP

∞∞ ρ−=

∂∂

=∂∂

)x(P)x(PP ∞

g)(y

uyu

xuu 2

2ρ−ρ+

∂

∂μ=⎟⎟

⎠

⎞⎜⎜⎝

⎛∂∂

ν+∂∂

ρ ∞

P.Talukdar/Mech-IITD 8)TT(g

yu

yu

xuu 2

2

∞−β+∂

∂ν=⎟⎟

⎠

⎞⎜⎜⎝

⎛∂∂

ν+∂∂ )TT( ∞∞ −ρβ=ρ−ρ

Grashof NumberGrashof Number

( )2

2

L2L

2

3cs

yu

Re1

ReTLTTg

yuv

xuu

∗

∗∗∞

∗

∗∗

∗

∗∗

∂

∂+

⎥⎥⎦

⎤

⎢⎢⎣

⎡

ν

−β=

∂

∂+

∂

∂

2

3cs

LL)TT(g

Gr−β

= ∞

The Grashof number Gr is a measureof the relative magnitudes of the buoyancy

2Lν


of the relative magnitudes of the buoyancy force and the opposing viscous force acting on the fluid.

LimitsLimits

For a vertical plate

Gr < 109 Laminar > 109 Turbulent

Forced convection dominates

Free convection dominatesFree convection dominates


Nu for Free ConvectionNu for Free Convection

nL

nL

c CRaPr)Gr(Ck

hLNu ===

( )Pr

LTTgPrGrRa 2

3cs

LLν

−β== ∞

Values of n and C depend on geometry

ν

g yof the surface and flow regime

The value of n is usually ¼ for laminar flow and 1/3 for turbulent flow. The value of the constant C is normally less than 1


value of the constant C is normally less than 1.

Vertical Plate RelationsVertical Plate Relations


• Constant heat flux condition for Vertical• Constant heat flux condition for Vertical surface: Same relation as constant temperature case LqhL

.temperature case

)TT(kLq

khLNu

2/L

s

∞−==

• Vertical cylinders: 4/1

L35D ≥y4/1

LGr


Inclined Hot Plate• Inclined hot plate that makes

an angle from the vertical in a cooler environment.

• The net force F = g(ρ∞‐ ρ ) (the difference between the(the difference between the buoyancy and gravity) acting on a unit volume of the fluid in the boundary layer is always in the vertical directionThe reason for this curious behavior for the upper surface is that the forceThe reason for this curious behavior for the upper surface is that the force component Fy initiates upward motion in addition to the parallel motion along the plate, and thus the boundary layer breaks up and forms plumes, as shown in the figure


In the case of a cold plate in a warmer environment ??

Use vertical plate equations for the upper surface of a cold plate and the lower surface of a hot plate L.

Inclined Plate

Replace g by g cosθ for Ra < 109

and θ < 60°

P.Talukdar/Mech-IITD

Horizontal SurfaceFor a hot surface in a cooler environment, the net force actsenvironment, the net force acts upward, forcing the heated fluid to rise.

If the hot surface is facing upward, the heated fluid rises freely, inducing strong natural convection currents and thus effective heat transfer.

B t if th h t f i f iBut if the hot surface is facing downward, the plate will block the heated fluid that tends to rise (except near the edges) impeding


(except near the edges), impeding heat transfer

Identify the conditions: Ts < / > T∞

Ts < T∞ Ts < T∞


Ts > T∞

Ts > T∞

Horizontal sphere and cylinderHorizontal sphere and cylinder

The local Nusselt number is highest at the bottom, and lowest at the top of the cylinder when the boundary layer flow remains laminarremains laminar

What will happen for a cold cylinder?


E i i lEmpirical Correlations


Natural Convection inside Enclosure

Vertical enclosureHorizontal enclosure

( ) PrLTTgPrGrRa 2

3c21

LLν−β

==

Initially, the heat transfer is by pure conduction and Nu = 1. When Ra > 1708, the buoyant force overcomes the fluid resistance and initiates natural convection currents, which are observed to be in the form of hexagonal


, gcells called Bénard cells. For Ra > 3 x 105, the cells break down and the fluid motion becomes turbulent

Rayleigh Bénard ConvectionRayleigh Bénard Convectionhttp://www.youtube.com/watch?v=xb_pHQzEFJg


(a) Formation of evolutional hexagonal structures for Ra = 2x105, Pr = 2.0. Temperature and vector-field. (b) Comparison of temperature field between numerical solution (lines) and results (dots) in the middle plane. for Ra = 2.5x105 Pr = 1.5N.M. Evstigneev , N.A. Magnitskii , S.V. Sidorov, Nonlinear dynamics of laminar-turbulent transition in three dimensional


g , g , , yRayleigh?Benard convection, Communications in Nonlinear Science and Numerical Simulation Volume 15, Issue 10 2010 2851 –2859http://dx.doi.org/10.1016/j.cnsns.2009.10.022

Convection cell in Silicone OilConvection cell in Silicone Oil


Effective Thermal ConductivityEffective Thermal Conductivity

TT

c

21s21s L

TTkNuA)TT(hAQ −=−=&

c

21scond L

TTkAQ −=&

The fluid in an enclosure behaves like a fluid whose thermal conductivity is kNu as a result of convection currents.

Therefore, the quantity kNu is calledthe effective thermal conductivity of the enclosure.


keff = kNu

Horizontal Rectangular EnclosureHorizontal Rectangular Enclosure

Hot plate at the top : Nu = ?Hot plate at the bottom: Significant convective current occurs when Ra > 1708

4/1R1950N 54 104R10For horizontal enclosures that contain air, Jakob recommends

can also be used for other gases

4/1LRa195.0Nu =

3/1LRa068.0Nu =

5L

4 10x4Ra10 <<7

L5 10Ra10x4 <<

gwith 0.5 < Pr < 2.

Using water, silicone oil, and mercury in their experiments, Globe and Dropkin 074.03/1

L PrRa0690Nu = 9L

5 10x7Ra10x3 <<


(1959) obtained correlation for horizontal enclosures heated from below,

L PrRa069.0Nu = L 10x7Ra10x3 <<

Inclined Rectangular EnclosureInclined Rectangular EnclosureExample:

Ai b t t i li d ll l• Air spaces between two inclined parallel plates

• flat-plate solar collectors (between the glass cover and the absorber plate) andg p )• the double-pane skylights on inclined roofs

For large aspect ratios (H/L > 12) thisFor large aspect ratios (H/L > 12), this equation correlates experimental data extremely well for tilt angles up to 70°,

++ ⎤⎡⎞⎛⎤⎡ 3/161

for Ra <105 0 < θ < 70° and H/L ≥ 12

++

⎥⎦

⎤⎢⎣

⎡−

θ+⎟

⎟⎠

⎞⎜⎜⎝

⎛

θθ

−⎥⎦

⎤⎢⎣

⎡θ

−+= 118

)cosRa(cosRa

)8.1(sin17081cosRa

1708144.11Nu3/1

L

L

6.1

L


for RaL <105, 0 < θ < 70 , and H/L ≥ 12. Any quantity in [ ]+ should be set equal to zero if it is negative. This is to ensure that Nu = 1 for RaL cosθ < 1708

Vertical Rectangular EnclosureVertical Rectangular EnclosureSmall aspect ratio

Pr2L/H1 <<

A dtl b29.0L )Ra

Pr2.0Pr(18.0Nu+

=3

L 10RaPr2.0

Pr>

+

Any prandtl number

4/128.0L )

LH()Ra

Pr2.0Pr(22.0Nu −

+=

10L/H2 <<

1010R <

Any prandtl number

Large aspect ratio

10L 10Ra <

3.0H −⎞⎛

40L/H10 <<410x2Pr1 <<012.04/1

L LHPrRa42.0Nu ⎟⎠⎞

⎜⎝⎛= 10x2Pr1 <<

7L

4 10Ra10 <<

40L/H1 <<


3/1LRa46.0Nu =

40L/H1 <<20Pr1 <<

9L

6 10Ra10 <<

Concentric Cylinders and Spheres

Raithby and Hollands (1975):

L (D D )/2Lc = (Do - Di)/2.

for 0.70 ≤ Pr ≤ 6000 and 102 ≤ FcylRaL ≤ 107.

For FcylRaL ≤ 100, natural convection currents


cyl Lare negligible and thus keff = k.

Combined Natural Convection d R di iand Radiation

Natural convection heat transfer coefficients are typically veryNatural convection heat transfer coefficients are typically verylow compared to those for forced convection. Therefore, radiation is usually disregarded in forced convection problems, but it must be considered in natural convection problems that involve a gas. p g

This is especially the case for surfaces with high emissivities. For example, about half of the heat transfer through the air space of a double pane window is by radiation


Combined Free and Forced Convection

• Natural convection is negligible when Gr/Re2 < 0.1g g• forced convection is negligible when Gr/Re2 > 10, and• neither is negligible when 0.1 < Gr/Re2 < 10.g g /


A review of experimental data suggests a correlation of the formA review of experimental data suggests a correlation of the form

Nu combined = (Nunforced ± Nun

natural)1/n

Determined from pure forced andThe value of the exponent n varies between 3 and 4 depending on the Determined from pure forced and

natural convection correlationsbetween 3 and 4, depending on the geometry involved. It is observed that n = 3 correlates experimental data for vertical surfaces well. Larger values of n are better suited for horizontal surfaces


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