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Dr. Primal Fernandoprimal@eng.fsu.eduPh: (850) 410-6323

Chapter 19 – Forced convection

Thermal-Fluids II

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Physical mechanism of convection

• Convection and conduction both require presence of a material medium unlike radiation.

• Convection requires the presence of fluid motion.

(no fluid motion: conduction)

(no fluid motion: conduction)

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Convection heat transfer coefficientExperience shows that the convection heat transfer strongly depends on fluid properties µ, k, ρ and cp.

Despite the complexity, the rate of heat transfer is observed proportional to temperature difference and expressed by Newton’s Law of cooling,

)m/W()TT(hq 2sconv ∞−= )W()TT(hAQ sconv ∞−=

No-temperature-jump-condition

)m/W(yTkqq 2

0yfluidconconv

=∂∂

−==

)C.m/W(TT

yTk

h 2

s

0yfluid

°−

∂∂

−

=∞

= )C.m/W(dxhL1h 2

L

0x °= ∫

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Nusselt number (Nu)

Convection involves when fluid layer is in motion and conduction involves when fluid layer is motionless.

)m/W(Thq 2conv ∆= )m/W(

LTkq 2

cond∆

=

number)(Nusselt ratio Nuk

hL

LTk

Thqq

cond

conv ==

==∆∆

Nu represents the convection relative to the conduction in the same fluid layer. Larger the Nu more effective the convection. Nu=1 represents pure conduction across the fluid layer.

Dimensionless temperature gradient at the surface

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Thermal boundary layer

(fluid hotter than the surface)

(surface hotter than fluid)

( )( ) 99.0

TTTT

s

s =−−

∞

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Prandtl number (Pr)

kc

Pr pµαυ===

heat ofy diffusivitMolecular momentum ofy diffusivitMolecular

(ratio of momentum and thermal diffusivities)

Pr number describes relative thickness of velocity and thermal boundary layer

Pr << 1: heat diffuses very quickly relative to momentum, thermal boundary layer thicker compared to velocity boundary layer

Pr >> 1: heat diffuses very slowly relative to momentum, thermal boundary layer thinner compared to velocity boundary layer

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Parallel flow over a flat plate I

0.60PrNu :Laminar x >== 3/15.0x

x PrRe332.0kxh

75

x

10Re105 60;Pr0.6

Nu :Turbulent

≤≤×≤≤

== 3/18.0x

x PrRe0296.0kxh

(local Nu number)

(average Nu)

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Parallel flow over a flat plate II

Heat transfer coefficient and Nu number for entire plate

(average Nu number)

(applicable for liquid metal)

(local Nu number)(all fluids; all Pr numbers

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Flat plate with unheated starting lengthKays and Crawford -1994

Thomas-1977

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Uniform heat flux

Surface temperature at any point can be calculated

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Problem solving

Step 1: Calculate the Re number and compare with the critical Re number 5105×=crRe

Step 2: Select suitable correlation according to flow region (laminar/turbulent) and given condition (ex. Pr range) and calculate the Nu

Step 3: Calculate the heat transfer coefficientkhx

=Nu

Step 4: Apply

)m/W()TT(hq 2sconv ∞−= )W()TT(hAQ sconv ∞−=or

(properties are evaluated at film temperature)2

TTT s

f∞+

=

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Flow across cylindersFlow across spheres and cylinders generally involve flow separation

Nu high at stagnation point

Nu decreases due to thickening of laminar boundary layer

Nu increases due to good mixing (wake)

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Flow over spheres

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General correlation for flow over a cylinder

C, m and n are experimentally obtained.

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Pipe flow: Mean velocity , average speed through the cross section

cA

cm dAu(r,x)AVmc

∫== ρρ rate, flow mass

Conservation of mass:

mV ity,mean veloc

u(r,x) profile,velocity

∫∫∫

===R

022

A

c

cA

m u(r,x)rdrR2

R

rdr2u(r,x)

A

dAu(r,x)

V cc

ρπ

πρ

ρ

ρ

ity,mean veloc

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The entrance region

Pipe Entrance

v vv

Inviscid region

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Critical Reynolds number

µρ

υDVDV mm ===

forceviscousforceInertialRe

PA4D c

h =

flowturbulent 10000Reflow altransition4000Re2300

flowlaminar 2300Re

>≤≤

<

Under most practical conditions

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Temperature profile

Note: mean temperature change along the tube

cm

pm

pmpfluid VdATcmTcTcmE ∫∫ ===

ρδ

Energy transported by fluid

∫∫∫

===R

02

mp2

m

R

0p

p

mp

m rdr)x,r(V)x,r(TRV2

c)R(V

)rdr2V(Tc

cm

mTcT

πρ

πρδ

2TT

T e,mi,mb

+=

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Thermal entry region

Fluid at uniform temperature enter into pipe

Fluid particles that are in contact with surface will have surface temperature

unchangedTTTT

profileetemperaturessDimentionlms

s =−−

=

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Local heat transfer coefficient

)x(fTT

)r/T(TTTT

r s

Rr

Rrms

s ≠−

∂∂−=

−−

∂∂

∞

=

=

ms

Rrx

Rrmsxs TT

)r/T(kh

rTk)TT(hq

−

∂∂=→

∂∂

=−= =

=

(independent of x)

Surface heat flux

Therefore convection heat transfer coefficient remain constant(we mention before friction factor constant for fully develop region)

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Thermal entry length

DRe05.0L arminla,h ≈

arminla,harminla,t LPrDPrRe05.0L =≈

If Pr > 1, fluid will hydrodynamically fully develop first.

If Pr ≈ 1, fluid will both will coincide

(Kays and Crawford)

D10LL lturbulent,hturbulent,t ≈≈

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Nu number variation along a tube for turbulent flow for both uniform surface temperature/heat flux condition

Nu number reach constant less than 10D

Nu much higher at the entrance region

In fully develop region, both uniform surface temperature/heat flux Nu numbers are equal

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General thermal analysis

)TT(hq msxs −=

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General thermal analysis: constant surface heat flux

)TT(hq msxs −=

)TT(cmAqQ iepss −==

p

ssie cm

AqTT

+=

hq

TT)TT(hq smsmss

+=→−=

(in fully develop region: h constant and Ts -Tm= constant

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Energy interaction for differential control volume in a tube

)pdx(qdTcm smp =

ttanconscmpq

dxdT

p

sm ==

hq

TT sms

+=dx

dTdx

dT sm =

Fully develop region dimensionless temperature profile is remain unchanged

Fully develop flow, constant heat flux: temperature gradient is independent of x

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Constant surface temperature (Ts=constant)

avemssaves )TT(hAThAQ −== ∆

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Fully developed laminar flow in a circular tube

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Fully developed laminar flow in noncircular tubes

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Developing laminar flow in the entrance region

Constant temperature, circular tube

When surface and fluid temperature difference is big

Isothermal parallel plate

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Turbulent flows I

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Turbulent flows II

( )14.0

s

4.025.14 PrRe10526.4Nu

×= −

µµ (2300 < Re < 6000)

5.15 < Pr < 5.30

Fernando et al. (2008)For small channels