convection part1 external flow. introduction recall: convention is the heat transfer mode between a...
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Convection Part1
External Flow
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IntroductionRecall: Convention is the heat transfer mode between a fluid and a solid or a 2 fluids of different phases
In order to simplify the process we used Newton’s correlation
Where h is the convective heat transfer coefficient also called the film coefficient.
h is a function of: Fluid flowFluid propertiesGeometry of the solid
ThA
q
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There are four means to evaluate the heat transfer coefficient
1) Dimensional analysis2) Exact analysis of boundary layer3) Approximate integral analysis of the boundary
layer4) Analogy between energy and momentum transfer
Significant Parameters:
Nusselt Number Nu
v TTs
TTs x
y
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The heat transfer rate between the surface and the fluid is
At the surface itself
Where k is the thermal conductivity of the fluid. Therefore:
TThAq sy
0
y
sy TTy
kAq
TThATTy
kA s
y
s
0
TT
TTy
k
h
s
y
s
0
LTT
TTy
k
hLNu
s
y
s
0
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Momentum Diffusivity
Thermal Diffusivity
The ratio of the momentum diffusivity over the thermal diffusivity is a combination of fluid properties and is also thougth of as a property (Named Prandtl Number Pr).
Dependent on fluid and temperature
pC
k
k
C pPr
Prandtl Number Pr
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Dimensional Analysis of Convective Heat Transfer
Forced Convection: movement dictated by v
Variable Symbol Dimensions
Tube Diameter D L
Fluid density ρ M L-3
Fluid viscosity μ M L-1 t-1
Fluid heat capacity Cp Q M –1 T –1
Fluid thermal conductivity k Q t –1 L –1 T –1
Velocity v L t –1
Heat transfer coefficient h Q t –1 L –2 T –1
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Using the Buckingham method we group the variables in dimensionless number:
This dimensional analysis for a forced convection in a circular conduit indicates the possibility of correlating the variables as
Similarly we could have developed the Stanton number instead of the Nusselt
Dv
vkD dcba Re1
k
CCpvkD phgfe
Pr2
k
hLNuhvkD lkji 3
PrRe,1fNu
vCp
hSt
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Free Convection: movement dictated by buoyancy
Given the coefficient of thermal expansion β: T 10
TggFbuoyant 00
Variable Symbol Dimensions
Significant length D L
Fluid density ρ M L-3
Fluid viscosity μ M L-1 t-1
Fluid heat capacity Cp Q M –1 T –1
Fluid thermal conductivity k Q t –1 L –1 T –1
Fluid Coef. Therm. Exp. β T –1
Gravitational acceleration G L t –2
Temperature difference ΔT T
Heat transfer coefficient h Q t –1 L –2 T –1
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Using the Buckingham method we group the variables in dimensionless number:
Define the Grashof number as
This dimensional analysis for a forced convection in a circular conduit indicates the possibility of correlating the variables as
k
CpCpgkL edcba Pr1
Pr,2 GrfNu
TTgkL onmlk 3
k
hLNuhgkL tsrqp 4
2
23
2 gL
gkL jihgf
Gr32
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Nu vs Re
0
25
50
75
100
100 1000 10000
Re
Nu
Pr = 2
Pr = 1
Pr = 0.5
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Nu vs f ( Re,Pr)
0
25
50
75
100
10 30 50 70
Re0.5 Pr 0.33
Nu
Pr = 2
Pr = 1
Pr = 0.5
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Group Symbol Definition Interpretation
Grashof Number Gr Ratio buoyancy to viscous forces
Colburn Factor jH Dimensionless heat transfer coefficient
Nusselt Number Nu Dimensionless surface temperature gradient
Prandtl Number Pr Ratio momentum to thermal diffusivity
Reynolds Re Ratio inertia to viscous forces
Stanton Number St Modified Nusselt number
Peclet Number Pe RePr Independent heat transfer parameter
Selected Dimensionless Groups
k
C p
k
hL
Dv
32
PrSt
2
23
TgL
vCp
h
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Flat Plate in Parallel Flow
Laminar FlowTurbulent Flow
Tra
nsit
ion
Reg
ion
δ(x)
xL
Lv
L Re
xv
x Re
Properties of fluid evaluated at the film temperature Tf
2s
f
TTT
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Forced ConvectionFlat Plate in Parallel FlowLaminar flow: Re<2 x 105
Prandtl number >0.6The local Nusselt number is
The average Nusselt number
All Prandtl number and Pe >100The local Nusselt number is
The average Nusselt number
xL
31
21
PrRe664.0 xL
k
LhNu
31
21
PrRe332.0 xx
k
xhNu
41
32
31
21
Pr/0468.01
PrRe3387.0
xNu
41
32
31
21
Pr/0468.01
PrRe6774.0
xNu
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Forced ConvectionFlat Plate in Parallel FlowTransition flow: Rec=5 x 105
60>Prandtl number >0.63 x 106 >Re > 2 x 105
The average Nusselt number
31
54
Pr871Re037.0 LL
k
LhNu
L
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Forced ConvectionFlat Plate in Parallel FlowTurbulent flow: Re>3x106
60>Prandtl number >0.6107 >Re >3 x 106
The average Nusselt number
The local Nusselt number
31
54
PrRe037.0 LNu
L
31
54
PrRe0296.0 xNu
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Cylinder in a Cross Flow
5102Re D Separation
v D
TransitionLaminar
Turbulent
D
Separation5102Re D
v
Properties of fluid evaluated at the film temperature Tf
2s
f
TTT
Dv
D Re
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Forced ConvectionCylinder in a Cross Flow
The average Nusselt number
If ReDPr>0.2
54
85
32
31
21
282000
Re1
Pr4.01
PrRe62.03.0
41
DDDNu
ReD C m
0.4-4 0.989 0.330
4-40 0.911 0.385
40-4000 0.683 0.466
4000-40,000 0.193 0.618
40,000-400,000 0.027 0.805
31
PrRemDD CNu
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Forced ConvectionVarious Object in a Cross Flow
The average Nusselt number 31
PrRemDD CNu
Geometry ReD C mSquare 5x103-105 0.246 0.588Square 5x103-105 0.102 0.675
Hexagon 5x103-1.95x104
1.95x104 -105
0.160
0.0385
0.638
0.782Hexagon 5x103-105 0.153 0.638
Vertical Plate 4x103-1.5x104 0.228 0.731
D
D
D
D
D
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Sphere in a Cross Flow
All properties of fluid evaluated at temperature , except μs at TsT
Dv
D Re
41
4.032
21
PrRe06.0Re4.02
sDDDNu
Restrictions 0.71 < Pr < 3803.5 < ReD < 7.6x104
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Bank of Tubes in a Cross Flow
V
Fluid in cross flow over tube bank
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Aligned Bank of Tubes in a Cross Flow
ST
DA1
SL
Tv,
Properties of fluid evaluated at the film temperature Tf
max
max,ReDv
D vDS
Sv
T
T
max
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Staggered Bank of Tubes in a Cross Flow
ST
A1
Tv,
Properties of fluid evaluated at the film temperature Tf
If
else
max
max,ReDv
D
vDS
Sv
T
T
max
SL
D
DSDS TD 2
vDS
Sv
D
T
2max
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Number of row (NL) greater or equal to 102000 < ReD,max < 40000Pr > 0.7
C1 in table 7.5
If number of row is smaller than 10
C2 in table 7.6
31
max,1 PrRe13.1 mDD CNu
10210
LL NDND NuCNu
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Number of row (NL) greater or equal to 201000 < ReD,max < 2x106
500 > Pr > 0.7
C in table 7.7
If number of row is smaller than 10
C2 in table 7.8
41
Pr
PrPrRe 36.0
max,
s
mDD CNu
20220
LL NDND NuCNu
All properties of fluid evaluated at the average temperatureexcept Prs at Ts
2outin TT
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In this case the temperature difference in the convective heattransfer equation is defined as the log-mean temperature difference ΔTlm
Where Ti is the temperature of the fluid entering the bankTo is the temperature of the fluid leaving the bank
And the outlet temperature can be estimated using
Where N is the total number of tube and NT the transverse number of tube. Finally the heat transfer rate per unit length is
os
is
osislm
TTTT
TTTTT
ln
CpSvN
hDN
TT
TT
TTis
os
exp
lmTDhNq '
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Packed Bed
v
Properties of fluid evaluated at the the average temperature
ε is the porosity or void fraction of the bed (0.3 to 0.5)
Valid for
gas flow
Dv
D Re
2outin TT
575.0Re06.2 DHj
4000Re90
7.0Pr
D
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Ap,T is the total area of the particles and Ab,c is the bed cross sectional area
CpvA
hA
TT
TT
bc
Tp
is
os
,
,exp
lmTp TAhq ,