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Home , Natural convection, Nusselt number, Rayleigh number, Reynolds number

AME 60634 Int. Trans. Free : Overview

• Free Convection (or ) – motion induced by forces – buoyancy forces arise when there are gradients in a fluid and a body force proportional to density arises

• Density Gradient – due to gradient thermally driven flow • Body Force – gravity (function of mass)

Basic Principle: heavy fluid falls and light fluid rises creating vortices

D. B. Go AME 60634 Int. Heat Trans. Free Convection: Overview • Internal vs. External – free convection can be generated in a duct or enclosure (internal) – along a free surface (external) – in both internal & external cases, free convection can interact with (mixed convection)

• Free Boundary Flow – occurs in an extensive quiescent fluid (i.e., infinite, motionless fluid) • no forced convection

– induces a on a heated or cooled surface (Ts ≠ T∞) – can generate plumes and buoyant jets

D. B. Go AME 60634 Int. Heat Trans. Free Convection: Dimensionless Parameters Pertinent Dimensionless Parameters Grashoff Number 3 gβ(Ts − T∞)L buoyancy force As Grashoff # increases: buoyancy Gr = ~ L ν 2 viscous force overcomes friction and induces flow µ ν = ≡ kinematic of fluid ρ € L ≡ characteristic length g ≡ gravitational constant € 1 #∂ρ & -1 β ≡ coefficient (fluid property) β = − % ( [K ] € ρ $∂T 'p € Recall VL inertial force € Re = ~ Gr is analogous to Re L ν viscous force

Rayleigh Number 3 gβ(Ts − T∞)L comparable to Peclet number (Pe) for Ra = Gr Pr = L L να forced convection

D. B. Go

€ AME 60634 Int. Heat Trans. Free Convection: Mixed Convection

Mixed Convection • a condition where both free and forced convection effects are comparable • free convection can assist, oppose, or act orthogonally to forced convection

When is mixed convection significant? • an assessment of scales is conducted through a comparison of free and forced non-dimensional parameters " >>1 free convection dominates NuL = f (GrL ,Pr) buoyancy force Gr $ L Nu f Re ,Gr ,Pr ~ 2 #O (1) mixed convection condition L = ( L L ) inertial force ReL $ % <<1 forced convection dominates NuL = f (ReL ,Pr) € Affect on € • forced and free convection DO NOT combine linearly € € # n 3 assisting/opposing n n n = + è assisting/transverse NuL = NuL, forced ± NuL, free → $ 7 - è opposing % n = 2 or 4 transverse D. B. Go

€ AME 60634 Int. Heat Trans. Free Convection: Boundary Layers • Transition to – similar to forced flows, free convection flows can transition from a laminar state to a turbulent state – hydrodynamic instabilities – in forced convection, the transition is dependant on inertial and viscous forces – in free convection, amplification of disturbances depends on relative magnitude of buoyancy and viscous forces

Critical

9 Rax,c ≈10 vertical flat plate

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D. B. Go AME 60634 Int. Heat Trans. Free Convection: External Flow • Only empirical relations exist for most “real” geometries

hL n NuL = = CRa k L • Immersed Vertical Plate – average Nusselt numbers

9 – Churchill & Chu correlation: (RaL<10 ) 1 4 0.670RaL NuL = 0.68 + 4 " 9 % 9 $1 + 0.492 Pr 16' # ( ) &

– Churchill & Chu correlation: all flow regimes

( +2 € 1 * 6 - 0.387RaL NuL = *0.825 + 8 - " 9 % 27 * $1 + 0.492 Pr 16 ' - )* # ( ) & ,-

D. B. Go € AME 60634 Int. Heat Trans. Free Convection: External Flow • Inclined & Horizontal Plates – buoyancy force is not parallel to plate • horizontal: perpendicular buoyancy force • inclined: angled buoyancy force

– flow and heat transfer depend on • thermal condition of plate: heated or cooled • orientation: facing upward or downward

cooled (T

cold fluid jets cold fluid jets

heated (T >T ) heated (Ts>T∞) s ∞ horizontal facing heated (Ts>T∞) horizontal inclined facing up down

warm fluid jets warm fluid jets D. B. Go AME 60634 Int. Heat Trans. Free Convection: External Flow Horizontal Plates Correlation: heated/horizontal/facing up AND cooled/horizontal/facing down

1 4 4 7 NuL = 0.54RaL →10 ≤ RaL ≤10 1 3 7 11 NuL = 0.15RaL →10 ≤ RaL ≤10

Correlation: heated/horizontal/facing down AND cooled/horizontal/ € facing up

1 4 5 10 NuL = 0.27RaL →10 ≤ RaL ≤10 note: smaller than above correlation

A € What is L in the Rayleigh number? L = s Use a characteristic length: P

D. B. Go € AME 60634 Int. Heat Trans. Free Convection: External Flow Inclined Plates Correlation: heated/bottom face only AND cooled/top face only • use immersed vertical plate correlations (Churchill & Chu) but replace g in the Rayleigh number by gcosθ

3 gcosθβ(Ts − T∞)L  RaL = → 0 ≤ θ ≤ 60 να

cooled (TsT∞) €

top face top face bottom face bottom face

Correlation: heated/top face only AND cooled/bottom face only flow is three-dimensional and no correlations exist

D. B. Go AME 60634 Int. Heat Trans. Free Convection: External Flow

• Long Horizontal Cylinder

heated (Ts>T∞)

– Rayleigh number based on diameter of isothermal cylinder – Churchill and Chu correlation: average

( +2 1 * 6 - 0.387RaD 12 NuD = *0.60 + 8 - ⇒ RaD <10 " 9 % 27 * $1 + 0.559 Pr 16 ' - )* # ( ) & ,- D. B. Go

€ AME 60634 Int. Heat Trans. Free Convection: External Flow

• Sphere – Rayleigh number based on diameter of isothermal sphere – Churchill: average Nusselt number

1 4 0.589RaD 11 NuD = 2 + 4 ⇒ RaD <10 ; Pr ≥ 0.7 " 9 % 9 $1 + 0.469 Pr 16' # ( ) &

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D. B. Go AME 60634 Int. Heat Trans. Free Convection: Internal Flow

• Parallel Plates – similar to internal forced convection, buoyancy-induced boundary layers develop on both plates – boundary layers can merge to generate fully developed flow conditions in sufficiently long (L) and spaced (S) channels è otherwise the channel behaves as two isolated plates – channels often considered vertical (θ = 0°) or inclined • inclined flows more complex and often result in three-dimensional flow structures

vertical plates fully developed criteria: * RaS S L or RaS S L <10 isolated plate criteria: * RaS S L or RaS S L >100 € D. B. Go

€ AME 60634 Int. Heat Trans. Free Convection: Internal Flow • Parallel Plates – Vertical Channels – both semi-analytical and empirical correlations exist – symmetry/asymmetry of boundary conditions play a factor • equal or unequal uniform surface or uniform heat fluxes for both plates

Isothermal −1 2 $ ' * - boundary C1 C2 q A S C1 C2 NuS = & ) = , 2 + 1 2 / T T k Ts,1 = Ts,2 576 2.87 % s − ∞ ( +, (RaS S L) (RaS S L) ./

Ts,1, q s" ", 2 = 0 144 2.87 3 gβ(T1 − T2 )S Ra = S να € € Isoflux (Uniform Heat Flux) −1 2 boundary C C % ( + . 1 2 € q s"" S - C1 C2 0 q "" = q" " NuS,L = ' * = + s,1 s,2 48 2.51 T − T k - Ra* S L * 2 5 0 & s,L ∞ ) S RaS S L , ( ) / qs" ", 1 ,qs" ", 2 = 0 24 2.51 gβq# # S 4 Ra* = s S kνα € D. B. Go € €

€ AME 60634 Int. Heat Trans. Free Convection: Internal Flow

Only empirical relations exist for most “real” geometries

Enclosures – Rectangular Cavities • characterized by opposing walls of different temperatures with the the remaining walls insulated • fluid motion characterized by induced vortices/rotating flow

horizontal cavity: τ = 0°,180° vertical cavity: τ = 90°

3 Rayleigh number a function of gβ(T1 − T2 )L Ra = → T > T distance between heated walls: L να 1 2

Heat transfer across the cavity: q" " = h(T1 − T2) D. B. Go €

€ AME 60634 Int. Heat Trans. Free Convection: Internal Flow Enclosures – Rectangular Cavities (Horizontal) • heating from below: τ = 0° • critical Rayleigh number exists below which the buoyancy forces cannot overcome the viscous forces

# RaL < RaL,c → no induced fluid motion RaL,c =1708 → $ % RaL > RaL,c → induced fluid motion

RaL< RaL,c: thermally stable – heat transfer due solely to conduction

NuL =1 €

Ra < Ra < 5×104: thermally unstable – heat transfer primarily due to induced fluid L,c L motion è advection

ۥ thermal instabilities yield regular convection pattern in the form of roll cells

1 3 0.074 NuL = 0.069RaL Pr (first approximation)

4 9 5×10€< RaL< 7×10 : thermally unstable turbulence 1 3 0.074 D. B. Go NuL = 0.069RaL Pr

€ AME 60634 Int. Heat Trans. Free Convection: Internal Flow Enclosures – Rectangular Cavities (Horizontal) • Heating from above: τ = 180° • unconditionally thermally stable – heat transfer due solely to conduction

NuL =1 Enclosures – Rectangular Cavities (Vertical) • critical Rayleigh number exists below which the buoyancy forces cannot overcome the viscous forces € # Ra < Ra → no induced fluid motion Ra =103 → $ L L,c L,c % RaL > RaL,c → induced fluid motion

Ra < Ra : thermally stable – heat transfer due solely to conduction L L,c € NuL =1

4 RaL,c < RaL< 5×10 : thermally unstable – heat transfer primarily due to induced fluid motion è advection •€ a primary flow cell is induced – secondary cells possible at large Rayleigh numbers

D. B. Go AME 60634 Int. Heat Trans. Free Convection: Internal Flow • Enclosures – Rectangular Cavities (Vertical)

– RaL,c < RaL: thermally unstable • Nusselt number a function of cavity geometry and fluid properties

1 * 2 < H L <10 - 0.28 − 4 " Pr RaL % " H % , 5 / NuL = 0.22$ ' $ ' ⇒ , Pr <10 / # 0.2 + Pr& # L & 3 10 +,10 < RaL <10 ./ * - 0.29 1< H L < 2 " Pr RaL % , −3 5 / NuL = 0.18$ ' ⇒ , 10 < Pr <10 / # 0.2 + Pr& 3 +,10 < RaL Pr (0.2 + Pr)./

−0.3 + 10 < H L < 40 . 1 4 0.012" H % - 4 0 NuL = 0.42Ra Pr $ ' ⇒ 1< Pr < 2 ×10 L # L & - 0 -10 4 < Ra <1070 € , L / + 1< H L < 40 . 1 3 - 0 NuL 0.046Ra 1 Pr 20 = L ⇒ - < < 0 6 9 ,-10 < RaL <10 /0 D. B. Go

€ AME 60634 Int. Heat Trans. Free Convection: Internal Flow • Enclosures – Annular Cavities – Concentric Cylinders • heat transfer across cavity

2πkeff q" = (Ti − To) ln(Do Di )

• keff is effective è thermal conductivity a stationary fluid € would need to transfer the same amount of heat as a moving fluid k Ra* <100 ⇒ eff =1 c k 1 k # Pr & 4 1 100 < Ra* <107 ⇒ eff = 0.386% ( Ra* 4 c k $ 0.861+ Pr' ( c )

4 ln(Do Di ) € • Critical Rayleigh number: * [ ] Rac = RaL → L ≡ (Do − Di ) 2 3# −3 −3 & L % D 5 + D 5 ( $ i o '

D. B. Go

€ AME 60634 Int. Heat Trans. Free Convection: Internal Flow • Enclosures – Annular Cavities – Concentric Spheres • heat transfer across cavity

πkeff DiDo q = (Ti − To) L

• keff is effective thermal conductivity è thermal conductivity a stationary fluid € should have to transfer the same amount of heat as a moving fluid k Ra* <100 ⇒ eff =1 s k 1 k # Pr & 4 1 100 < Ra* <104 ⇒ eff = 0.74% ( Ra* 4 s k $ 0.861+ Pr' ( s )

€ • Critical Rayleigh number: * LRaL Ras = → L ≡ Do − Di 2 7 7 5 ( ) 4# − 5 − 5 & (Do Di ) % Di + Do ( $ '

D. B. Go

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