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Review:Convection
CH EN 3453 – Heat Transfer
Reminders…
• Final exam Wednesday Dec. 17, 8-10 am– This room– Two parts
• 10 multiple choice-type review questions• 3 radiation calculation problems
– You can bring three 8.5 x 11 sheets of notes (both sides)
– Don’t forget your calculator, scratch paper, pencil
• Check one last time that scores are correct– Scores have been updated
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CHAPTER 6:Introduction to Convection
• Boundary layers– Velocity– Thermal
• Boundary layer equations
Three Boundary Layers
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Laminar and Turbulent Flow
Laminar and Turbulent Flow
Figure 6.8 Variation of velocity boundary layer thickness δ and the local heat transfer coefficient h for flow over an isothermal flat plate.
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Six Steps to Solving…1. Determine flow geometry
2. Determine appropriate fluid temperature (e.g., Tfilm) and evaluate fluid properties
3. Consider fluid B (Applies only to mass transfer problems.)
4. Calculate Reynolds number to determine if laminar or turbulent flow
5. Decide whether a local or average coefficient is required
6. Select appropriate correlation
Dimensionless Groups
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CHAPTER 7:External Flow
• Geometries– Flat plate– Cylinder in cross flow– Sphere– Banks of tubes
• Heat transfer– Constant surface temperature– Constant heat flux
Flat Plate• Critical Reynolds number: 5 x 105
• Local Nusselt number (use Tfilm):– Laminar
– Turbulent
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Flat Plate• Average Nusselt number
– Laminar
– Turbulent (tripped at leading edge)
– Transition
• Unheated starting length
Flow Correlation Summary
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Cylinder in Cross Flow
• Average Nusselt number
Flow over a Sphere
• Average Nusselt number
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Flow Across Banks of Tubes
Bakes of Tubes (cont.)
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CHAPTER 8:Internal Flow
• Basics– Velocity profiles– Mean velocity, mean temperature
• Geometries– Circular tubes– Non-circular tubes– Concentric annulus
• Heat transfer– Constant surface temperature– Constant heat flux
Internal Flow
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Internal Flow Calculations
• Reynolds number (critical = 2300)
• Hydraulic diameter for non-circular tubes
• For uniform surface temperature
Laminar Flow in a Circular Tube
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Laminar Flow: Non-Circular Tube
Turbulent Flow• Circular or noncircular tube with small
temperature diffs (Dittus-Boelter)
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CHAPTER 9:Free (Natural) Convection
• Fundamentals– Buoyancy– Boundary layer development– Transition from laminar to turbulent at RaL = 109
• Geometries– Vertical surface– Horizontal plates– Long horizontal cylinder– Spheres– Vertical parallel plate channels– Enclosures/cavities
Types of Free Convection
Plume Buoyant Jet Boundary Layer
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Empirical Correlations• Similar to approach used previously for
forced convection– Correlation for Nusselt number Nu– Instead of Re, we use Ra– Careful on which expression to use
• Fluid properties determined at film temp.
• Thermal expansion coefficient β– Gases:– Liquids: Look up in appendix
Tfilm = Ts + T∞( ) / 2
Flow over Vertical Plate
• Rayleigh Number:
• Laminar Flow (RaL < 109):
• All Conditions:
(9.25)
(9.27)
(9.26)
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CHAPTER 11:Heat Exchangers
• Basics– Overall heat transfer coefficient– Phase change heat transfer one one side
• Heat exchanger configurations– Parallel flow– Counterflow
• Multiple shells• Multiple tube passes
– Cross-flow• Analysis approaches
– Log-mean temperature difference (design)– Effectiveness-NTU (performance and design)– Modified LMTD (design - fancy configurations)
Overall Heat Transfer Coefficient
• Contributing factors – Convection between the two fluids and solid– Conduction of the solid separator– Potential use of fins in one or both sides– Time-dependent surface fouling
• General expression (c and h = cold and hot)
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Log-Mean Temperature Difference
Cocurrent flow (parallel flow) Countercurrent flow
q =UAΔTlm
Special Operating Conditions
"heat capacity rate"
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Effectiveness-NTU Approach• Effectiveness:
• NTU (Number of Transfer Units):
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Parallel-Flow Heat Exchanger
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Counterflow Heat Exchanger
Shell-and-Tube Heat Exchanger(One Shell)
NOTE!! You cannot use LMTD for this
type of heat exchanger!
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Shell-and-Tube Heat Exchanger(Multiple Shells)
NOTE!! You cannot use LMTD for this type
of heat exchanger!
Cross-Flow Heat Exchanger(Unmixed-Unmixed)
NOTE!! You cannot use LMTD for this type
of heat exchanger!
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REVIEW: Radiation
Radiation Spectrum
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Intensity vs. Wavelength and Direction
The Solid Angle
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Solid Angle Geometry
+
θ2
A2
ω =A2 cosθ2
r2 A2 cosθ2
r
Projected Area
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Radiation Heat Transfer• Energy transfer between two elements A1
and A2
q1− j = I × A1 cosθ1 ×ω j−1
= I × A1A2 cosθ1 cosθ2r2
From Example 12.1...
ω 3−1 = ω 4−1 =A3r2
=10−3m2
0.5m( )2= 4.00 ×10−4 sr
ω2−1 =A2 cosθ2
r2=10−3m2 × cos30°
0.5m( )2= 3.46 ×10−3sr
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Blackbody• Hypothetical perfect radiative surface
• Absorbs all incident radiation, regardless of wavelength and direction
• Emits maximum theoretical energy
• Diffuse emitter – Radiation emitted evenly in all directions
The Planck Distribution• Emissive power of a blackbody depends
on temperature and wavelength
• Planck figured out this relation
• Plot of E vs. λ looks like this:
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NOTES:
• Total power increases with temperature
• At any given wavelength the magnitude of emitted radiation increases with temperature
• Wavelength of radiation decreases with temperature
• Sun is approximated by blackbody at 5800 K
• At T < 800 K, most radiation in infrared
Wien’s Displacement Law• For a given temperature, spectral emission
goes through a maximum at a given wavelength.
• Wien figured this one out:
• This maximum is indicated by the dashed line in Figure 12.12
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Stefan-Boltzmann Law• If one were to integrate any of the curves
shown in Figure 12.12 over the entire range of wavelengths, one would get the total emissive power for a blackbody:
• The Stefan-Boltzmann constant σ is:
Eb =C1
λ5 exp C2 / λT( ) −1⎡⎣ ⎤⎦dλ
0
∞
∫= σT 4
Text
σ = 5.670 × 10–8 W/m2·K4
Band Emission• Amount of total emitted radiation depends on
range of wavelengths of emission
• Effective emissivity determined by integrating over wavelengths
• Table 12.1, column “F” provides fraction of total integrated area to a given wavelength
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Example: 12.29• The spectral, hemispherical emissivity of
tungsten may be approximated by the distribution given below. What is the total hemispherical emissivity when the filament temperature is 2900 K.
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Radiation Transfer Types• Emission (E)
– Associated with energy transfer due to surface temperature
• Irradiation (G)– Radiation incident onto a surface– Irradiation can have three fates:
• Absorption by the surface(α = absorptivity = fraction of G absorbed)
• Reflection by the surface(ρ = reflectivity = fraction of
• Transmission through the material(τ = transmissivity = fraction transmitted)
Irradiation onto a Surface• Irradiation can have three fates:
– Absorption by the surface (α = absorptivity = fraction of G absorbed)
– Reflection by the surface (ρ = reflectivity = fraction of G reflected)
– Transmission through the material (τ = transmissivity = fraction of G transmitted)
• Sum of α + ρ + τ = 1
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Radiosity (J)• Total radiation leaving a surface.• Sum of emission plus reflected portion of
irradiation.
View Factors• Fraction of radiation from surface i that is
captured by surface j
• Summation rule:
• Reciprocity:
Fij = 1j=1
N
∑
AiFij = AjFji
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Spaceresistance
Surfaceresistance
Review: Radiation between Surfaces
Review: Two-Surface Enclosure
Spaceresistance
Surfaceresistance
Surfaceresistance
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Radiation Shield
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Reradiating Surface
“Direct Method” for Solving Networks
• Useful for systems with >2 surfaces• Balance radiant energy around each surface
node i :
• Solve systemof equations
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Multimode Heat Transfer
Radiation with Participating Media(Gaseous Emission and Absorption)
• Gas radiation– Nonpolar gases (O2, N2) neither emit nor
absorb radiation– Polar gases (CO2, H2O, hydrocarbons) do
• In most cases, contribution of gas to radiation can be safely neglected
• Exception:
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Emissivity of Water Vapor
Emissivity of Carbon Dioxide
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Pressure Correction
H2O
CO2
H2O + CO2 Correctionεg = εw + εc − Δε
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Gas Radiation - Geometries