Developing Laminar Flow and Heat Transfer in the Entrance Region of Regular Polygonal Ducts, Int.. Optimum Separation of Asymmetrically Heated Sub-channels Forming a Bundle: Influence of
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qmax
HLW ≤ 0.57
k
L2(Tmax− T0) Pr4/99
1+ t
Dopt
−67/99
· Be47/99 (5.91)
with
Be= ∆PLµα2
forthe range
104≤ ReD h ≤ 106 106≤ ReL≤ 108 1011≤ Be ≤ 1016
• Turbulent flow and entrance lengths:
X
D 10
X T
• Turbulent flow friction factor:
f 0.046Re −1/5 D 2× 104≤ ReD ≤ 105 (see Fig 5.13) (5.68)
• Turbulent flow heat transfer:
St· Pr2/3 f
forPr≥ 0.5
NuD= hD k = 0.023Re4/5
forPr≥ 0.50
2× 104≤ ReD≤ 106
NuD = 0.023Re4/5
wheren = 0.4 forheating the fluid and n = 0.3 forcooling the fluid in the range L
D > 60 0.7 ≤ Pr ≤ 120 2500≤ ReD ≤ 1.24 × 105
NuD = 0.027Re4/5
D · Pr1/3µ
µ0
0.14
(5.78)
in the range
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0.70 ≤ Pr ≥ 16,700 ReD ≥ 104 Here
µ0= µ(T0) (T0is the wall temperature)
µ = µ(T m ) (T mis the bulk temperature)
1.07 + 900/ReD − 0.63/(1 + 10Pr) + 12.7(f/2)1/2 (Pr2/3 − 1) (5.79a)
NuD = (f/2)Re D· Pr
1.07 + 12.7(f/2)1/2 (Pr2/3 − 1) (5.79b)
where
0.5 ≤ Pr ≤ 106 4000≤ ReD ≤ 5 × 106 andf from Fig 5.13.
NuD = (f/2)(ReD− 103)Pr
1+ 12.7(f/2)1/2 (Pr2/3 − 1) (5.80)
where
0.5 ≤ Pr ≤ 106 2300≤ ReD ≤ 5 × 106 andf from Fig 5.13.
NuD = 0.0214Re0D .8− 100Pr0.4 (5.81a) where
0.5 ≤ Pr ≤ 1.5 104≤ ReD ≤ 5 × 106
NuD = 0.012Re0D .87− 280Pr0.4 (5.81b) where
1.5 ≤ Pr ≤ 500 3× 103 ≤ ReD≤ 106
NuD=
6.3 + 0.0167Re0.85
D · Pr0.93 q0 = constant (5.82)
4.8 + 0.0156Re0.85
D · Pr0.93 T0= constant (5.83) where for eqs (5.82) and (5.83),
0.004 ≤ Pr ≤ 0.1 104≤ ReD ≤ 106
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• Total heat transfer rate:
• Isothermal wall:
∆T lm= ∆Tin− ∆Tout
q = ˙mcp ∆Tin
1− e −hA w / ˙mc p
(5.87)
• Uniform heat flux:
NOMENCLATURE
Roman Letter Symbols
A cross-sectional area, m2
(a) pressure at point 1, Pa
B cross-section shape number, dimensionless
(b) pressure at point 2, Pa
C cross-section shape factor, dimensionless
Cf,x local skin friction coefficient, dimensionless
cp specific heat at constant pressure, J/kg·K
f friction factor, dimensionless
Gz Graetz number, dimensionless
h heat transfer coefficient, W/m2·K
specific bulk enthalpy, J/kg
N numberof plate surfaces in one elemental channel,
dimensionless
Nu Nusselt number, dimensionless
Nux local Nusselt number, dimensionless
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pressure difference, dimensionless
∆P pressure difference, Pa PrPrandtl number, dimensionless
Prt turbulent Prandtl number, dimensionless
p perimeter of cross section, m
Ra Rayleigh number, dimensionless
ReD Reynolds numberbased onD, dimensionless
ReD h Reynolds numbers based onDh, dimensionless
ReL Reynolds numberbased onL, dimensionless
St Stanton number, dimensionless
Tin inlet temperature, K
Tout outlet temperature, K
∆Tavg average temperature difference, K
∆T lm log-mean temperature difference, K
u longitudinal velocity, m/s
u∗ friction velocity, m/s
v transversal velocity, m/s
x∗ longitudinal position, dimensionless
x+ longitudinal position, dimensionless
X T thermal entrance length, m
yVSL viscous sublayerthickness, m
Greek Letter Symbols
α thermal diffusivity, m2/s
H thermal eddy diffusivity, m2/s
M momentum eddy diffusivity, m2/s
θ∗m bulk temperature, dimensionless
ν kinematic visocity, m2/s
τapp apparent sheer stress, Pa
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τavg averaged wall shear stress, Pa
φ fully developed temperature profile, dimensionless
Subscripts
0-x averaged longitudinally
Superscripts
fluctuating components
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Bejan, A., and Morega, A M (1994) The Optimal Spacing of a Stack of Plates Cooled by
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Fowler, A J., Ledezma, G A., and Bejan, A (1997) Optimal Geometric Arrangement of
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CHAPTER 6
Forced Convection: External Flows
YOGENDRA JOSHI
George W Woodruff School of Mechanical Engineering Georgia Institute of Technology
Atlanta, Georgia
WATARU NAKAYAMA
Therm Tech International Kanagawa, Japan
6.1 Introduction 6.2 Morphology of external flow heat transfer 6.3 Analysis of external flow heat transfer 6.4 Heat transfer from single objects in uniform flow 6.4.1 High Reynolds numberflow overa wedge 6.4.2 Similarity transformation technique for laminar boundary layer flow 6.4.3 Similarity solutions for the flat plate at uniform temperature 6.4.4 Similarity solutions for a wedge
Wedge flow limits 6.4.5 Prandtl number effect 6.4.6 Incompressible flow past a flat plate with viscous dissipation 6.4.7 Integral solutions for a flat plate boundary layer with unheated starting length Arbitrarily varying surface temperature
6.4.8 Two-dimensional nonsimilarflows 6.4.9 Smith–Spalding integral method 6.4.10 Axisymmetric nonsimilar flows 6.4.11 Heat transfer in a turbulent boundary layer Axisymmetric flows
Analogy solutions 6.4.12 Algebraic turbulence models 6.4.13 Near-wall region in turbulent flow 6.4.14 Analogy solutions forboundary layerflow Mixed boundary conditions
Three-layermodel fora “physical situation”
Flat plate with an unheated starting length in turbulent flow Arbitrarily varying heat flux
Turbulent Prandtl number 6.4.15 Surface roughness effect 6.4.16 Some empirical transport correlations Cylinderin crossflow
Flow over an isothermal sphere
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6.5 Heat transfer from arrays of objects 6.5.1 Crossflow across tube banks 6.5.2 Flat plates
Stack of parallel plates Offset strips
6.6 Heat transfer from objects on a substrate 6.6.1 Flush-mounted heat sources 6.6.2 Two-dimensional block array 6.6.3 Isolated blocks
6.6.4 Block arrays 6.6.5 Plate fin heat sinks 6.6.6 Pin fin heat sinks 6.7 Turbulent jets
6.7.1 Thermal transport in jet impingement 6.7.2 Submerged jets
Average Nusselt numberforsingle jets Average Nusselt number for an array of jets Free surface jets
6.8 Summary of heat transfer correlations Nomenclature
References
6.1 INTRODUCTION
This chapter is concerned with the characterization of heat transfer and flow under forced convection, where the fluid movement past a heated object is induced by an ex-ternal agent such as a fan, blower, or pump The set of governing equations presented
in Chapter 1 is nonlinear in general, due to the momentum advection terms, vari-able thermophysical properties (e.g., with temperature) and nonuniform volumetric heat generation Solution methodologies for the governing equations are based on the nondimensional groups discussed in Section 6.3 Solutions can be obtained through analytical means only fora limited numberof cases Otherwise, experimental ornu-merical solution procedures must be employed
6.2 MORPHOLOGY OF EXTERNAL FLOW HEAT TRANSFER
Various cases arise from the geometry of a heated object and the constraint imposed
on the fluid flow Figure 6.1 shows the general configuration in which it is assumed that the body is being cooled by the flow The heated object is an arbitrary shape enclosed in a rectangular envelope The dimensions of the envelope areL, the length
in the streamwise direction,W, the length in the cross-stream direction (the width),
andH, the height Generally, the fluid flow is constrained by the presence of bounding
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Figure 6.1 Heated object in a flow overa bounding surface
surfaces The bounding surface may be a solid wall or an interface with a fluid of a different kind, forinstance, a liquid–vaporinterface
The distances x signifies the extent of the bounding surface in the streamwise direction, ands zis the distance to the bounding surface When s z H or L and
s z s x, the flow around the object is uniform and free of the effect of the bounding surface Otherwise, the object is within a boundary layer developing on a larger object In laboratory experiments and many types of industrial equipment, one often finds a situation where the object is placed in a duct When the duct cross section has dimensions comparable to the object size, the flow has a velocity distribution defined
by the duct walls and the object Hence, the foregoing relations betweenH, L, sx, and
szcan be put into more precise forms involving the velocity and viscosity of the fluid
as well In an extreme case, the object is in contact with the bounding surface; that is,
s z= 0 In such cases the flow and temperature fields are generally defined by both the
bounding surface and the object Only in cases where the object dimensions are much smaller than those of the bounding surface is the external flow defined primarily by the bounding surface
Several external flow configurations are illustrated in Fig 6.2 The symbols used
to define the dimensions are conventionally related to the flow direction For the flat
plate in Fig 6.2a, is the plate length in the streamwise direction, w is the length
(width) in the cross-stream direction, andt is the plate thickness The cylinderin Fig.
6.2b has length and diameter d Forthe rectangularblock of Fig 6.2c, is oriented in
the streamwise direction,h is the height, and w is the width Sometimes, these letters can be used as subscripts to a common symbol for the block The sphere (Fig 6.2d) is
defined by only one dimension, that is, the diameterd Although an infinite number
of configurations can be conceived from the combination of external flow and object geometry, only a limited number of cases have been the subject of theoretical studies
as well as practical applications The most common are two-dimensional objects in uniform flow, which are used in basic research and teaching