Prabal TalukdarPrabal Talukdar -...
Transcript of Prabal TalukdarPrabal Talukdar -...
NATURAL/FREE CONVECTION
Prabal TalukdarPrabal TalukdarAssociate Professor
Department of Mechanical EngineeringDepartment of Mechanical EngineeringIIT Delhi
E-mail: [email protected]
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
P.Talukdar/Mech-IITD 4
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
P.Talukdar/Mech-IITD 5
T1
gas ideal =β
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⎠⎝ ∂⎟
⎠⎜⎝ ∂
P.Talukdar/Mech-IITD 7
⎠⎝
⎟⎠⎞⎜
⎝⎛
∂∂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
P.Talukdar/Mech-IITD 9
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
P.Talukdar/Mech-IITD 10
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
P.Talukdar/Mech-IITD 11
value of the constant C is normally less than 1.
• 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
P.Talukdar/Mech-IITD 13
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
P.Talukdar/Mech-IITD 14
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
P.Talukdar/Mech-IITD 16
(except near the edges), impeding heat transfer
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?
P.Talukdar/Mech-IITD 18
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
P.Talukdar/Mech-IITD 20
, 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
P.Talukdar/Mech-IITD 21
(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
P.Talukdar/Mech-IITD 22
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
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.
P.Talukdar/Mech-IITD 24
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 <<
P.Talukdar/Mech-IITD 25
(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
P.Talukdar/Mech-IITD 26
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 <<
P.Talukdar/Mech-IITD 27
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
P.Talukdar/Mech-IITD 28
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
P.Talukdar/Mech-IITD 29
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 /
P.Talukdar/Mech-IITD 30
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
P.Talukdar/Mech-IITD 31