Fundamental Aspects of Hypersonic Vehicle Design Part I

1
Fundamental Aspects of Hypersonic Vehicle
DesignPart I

• Frederick Ferguson
• North Carolina AT State University
• September 13, 2007

Sect 3.1
2
Contents

• Section 3.1 Fundamental Aspects of Hypersonic
  Vehicle Design
• 3.1.1 Principles of Heat Transfer
• 3.1.1.1 Conduction
• 3.1.1.2 Convection
• 3.1.1.3 Radiation
• 3.1.2 Principles of Thermal Stress Analysis
• 3.1.2.1 Thermal Gradients the Generation of
  Thermal Stresses
• 3.1.2.2 Illustrative Examples of Thermal
  Stresses
• 3.1.2.3 Formulation of the Thermo elastic Problem

Sect 3.1
3
Hypersonic Technology Applications

• Access to Space
• Transportation on Demand to ISS, Lunar Base, etc
• Civilian Transatlantic, Transpacific Transport
• Transportation for long hauls/flights (flight
  times reduction, potentially reduces gt12hrs to
  lt3hrs)
• Military Applications
• Missiles, etc

Sect 3.1
4
Characteristics of Hypersonic Flows

• Hypersonic Flow (M 5) is defined as the flight
  regime where the following flow phenomena gets
  progressively more important as the Mach number
  increases
• Thin Shock Layers
• Entropy Layer
• Viscous Interaction
• High Temperature Effects
• Low Density Flow

Sect 3.1
5
Thin Shock Layers

• Thin Shock Layers
• As the Mach number increases, the shock angle
  becomes smaller, as illustrated in the figure
  below. The resulting flowfield between the
  surface and shock is often referred to as a shock
  layer. This thin layer can produce many
  complications in vehicle design, e.g. the shock
  layer may merge with the boundary layer at low
  Reynolds numbers to form a fully viscous shock
  layer.
• Shock waves and streamlines over a 20 half-angle
  wedge at (a) Mach 2 and (b) Mach 20 from
  Anderson, 2000
• At high Reynolds numbers, the shock layer can be
  treated as inviscid (meaning there is no
  friction). In the limit as Mach number goes to
  infinity, the shock layer forms an infinitely
  thin, infinitely dense sheet, or, essentially, a
  flat plate. The infinite flat plate is the most
  efficient lifting surface at hypersonic
  velocities, and the inviscid shock layer can
  therefore be used to develop simplified theories
  to predict hypersonic aerodynamic properties.

Sect 3.1
6
Entropy Layer

• Entropy Layer
• Shock theory tells us that entropy increases
  across a shock, and the entropy increase becomes
• greater as the shock strength increases. Since
  flow near the nose passes through a nearly
• normal shock, it will experience a much greater
  change in entropy compared to flow passing
• through the much shallower shock angle further
  from the body centerline. Thus, strong
• entropy gradients exist near the leading edge
  generating an "entropy layer" that flows
• downstream along the body surface.
•                                                
                                                    
                                                    
     

Sect 3.1
7
Viscous Interaction

• Viscous Interaction
• When a body travels through the air, a thin
  region near the body surface called the "boundary
  layer" is formed. In this layer, the air slows
  down from the "freestream" velocity of the
  airflow to zero at the surface. At subsonic
  speeds, the thickness of the boundary layer tends
  to become smaller as velocity increases because
  the thickness is inversely proportional to the
  Reynolds number
•         
•  
•                  
• For compressible flow (or flow at high speeds),
  however, increasing flow temperature (due to
  friction heat) near the body surface causes the
  boundary layer to become thicker as speed
  increases. The two primary factors driving this
  boundary layer growth are an increase in
  viscosity of the fluid and a decrease in density.
  The result of these factors is that boundary
  layer thickness varies as the square of the Mach
  number
•                           
• Thus, as Mach number increases, the boundary
  layer can grow rapidly resulting in very high
  drag. Should the boundary layer become thick
  enough, it may affect the inviscid flowfield far
  from the body, a phenomenon called viscous
  interaction. Viscous interaction can have a great
  influence on the surface pressure distribution
  and skin friction on the body thereby affecting
  the lift, drag, stability, and heating
  characteristics of the body.

Sect 3.1
8
High Temperature Flow

• High Temperature Flow
• Any body traveling at high speeds in air
  produces friction and heat. Part of the kinetic
  energy of the body's motion is absorbed by the
  air and carried away from the body through a
  process called viscous dissipation. However,
  hypersonic vehicles create so much heat and such
  high temperatures that they can actually cause
  chemical changes to occur in the fluid through
  which they fly. The most notable changes air
  undergoes as temperature increases are summarized
  below.
• High Temperature Effects on Air
• As temperature increases, assumptions about the
  properties of the air are no longer valid and the
  vehicle is said to be traveling through a
  chemically reacting boundary layer. When the
  properties of the working fluid change, namely
  density and heat transfer properties, the
  aerodynamic characteristics and heating
  properties of the body can change drastically.

Sect 3.1
9
Low Density Flow

• Low Density Flow
• Most hypersonic vehicles are intended to cruise
  at high altitudes in low density fluids. In low
  density flows, air can no longer be considered to
  be a continuum because the distance between
  individual particles of air becomes so great that
  each particle begins to affect the aerodynamic
  properties of a body. Under these conditions,
  common aerodynamic relations, like the Euler and
  Navier-Stokes equations, break down. Instead,
  aerodynamic properties must be analyzed using the
  kinetic theory. Some of the most important
  differences between low density flows and
  continuous flows include
• Velocity slip The viscous no-slip condition that
  says the velocity of air particles going past a
  body must be zero at the body surface, fails.
  Since friction is negligible in low density, the
  flow velocity at the body surface is no longer
  zero.
• Temperature slip The assumption that gas
  temperature at the body surface becomes equal to
  the temperature of the body surface material
  fails.

Drag coefficient of a sphere at hypersonic speeds
transitioning from continuum to free-molecule
flow from Anderson, 2000
Sect 3.1
10
Characteristics of Hypersonic Flows

• Hypersonic Flow Characteristics Thin Shock
  Layers, Entropy Layer, Viscous Interaction, High
  Temperature Flow , Low Density Flow
• The Combined Effects of the phenomena described
  above are the most important flow properties
  resulting from travel at hypersonic velocities.
  Each factor plays a large role in the design and
  operation of a practical hypersonic vehicle, as
  will be seen in following sections. The principal
  characteristics of hypersonic flow are summarized
  in the following figure, Ref. Anderson, 1989.

Sect 3.1
11
Mach Number Altitude Map

• Hypersonic Vehicle Flow Path Analysis

Sect 3.1
12
Integrated Hypersonic Vehicle Designs

• Flight Performance Justification for the
  Integrated Hypersonic Vehicle Design

Sect 3.1
13
Idealized Hypersonic Vehicles

• Idealized Hypersonic Vehicles

Sect 3.1
14
Propulsion System Driven Configurations
Ultimately, the Aircraft Mission the Propulsion
System drive the configuration of the Hypersonic
Aircraft
Sect 3.1
15
Thermo Structural Effects

• Structural Strength
• The capacity of the individual elements, which
  together make up the hypersonic structural
  system, to withstand the load (static, dynamic
  transient) that is applied to it.
• Structural Stability
• The capability of a structural system to transmit
  the aero thermal loads safely to the neighboring
  members and the mainframe.

Sect 3.1
16
Structural Design Objectives

• Safely Transfer all aerothermodynamics forces to
  the aircraft mainframe (perpetually maintaining
  system strength stability)

Contrasting Structural Design Approaches
Sect 3.1
17

• Principles of Heat Transfer
• Frederick Ferguson
• North Carolina AT State University
• September 13, 2007

Sect 3.1.1
18
Heat Transfer

• The science of thermodynamics deals with the
  amount of heat transfer as a system undergoes a
  process from one equilibrium state to another,
  and makes no reference to how long the process
  will take.
• The science of heat transfer deals with the
  determination of the rates of energy that can be
  transferred from one system to another as a
  result of temperature difference.
• The basic requirement for heat transfer is the
  presence of a temperature difference, which is
  the driving force for heat transfer.
• The second law of thermodynamics requires that
  heat be transferred in the direction of
  decreasing temperature.
• The rate of heat transfer in a certain direction
  depends on the magnitude of the temperature
  gradient in that direction.
• Thermodynamics deals with equilibrium states and
  changes from one equilibrium state to another.
  Heat transfer, on the other hand, deals with
  systems that lack thermal equilibrium, and thus
  it is a nonequilibrium phenomenon.

Sect 3.1.1
19
Heat and Other Forms of Energy

• Energy can exist in numerous forms such as
• thermal, mechanical, kinetic, potential,
• electrical, magnetic, chemical, and nuclear.
• Their sum constitutes the total energy E (or e on
  a unit mass basis) of a system.
• The sum of all microscopic forms of energy is
  called the internal energy of a system.
• Internal energy may be viewed as the sum of the
  kinetic and potential energies of the molecules.
• The kinetic energy of the molecules is called
  sensible heat.
• The internal energy associated with the phase of
  a system is called latent heat.
• The internal energy associated with the atomic
  bonds in a molecule is called chemical (or bond)
  energy.
• The internal energy associated with the bonds
  within the nucleus of the atom itself is called
  nuclear energy.

Sect 3.1.1
20
Definitions Flow Field Properties

• In the analysis of systems that involving fluid
  flow, the special combination of velocity and
  pressure u and P, is frequently encountered.
• This combination is defined as enthalpy, h, such
  that h e Pv.
• The term Pv represents the work done by the fluid
  or the flow energy of the fluid.
• Specific heat is defined as the energy required
  to raise the temperature of a unit mass of a
  substance by one degree.
• Two kinds of specific heats
• specific heat at constant volume cv, and
• specific heat at constant pressure cp.
• The specific heats of a substance, in general,
  depend on two independent properties such as
  temperature and pressure.

Sect 3.1.1
21
Definitions Energy Transfer

• Energy can be transferred to or from a given mass
  by two mechanisms, namely
• heat transfer, and
• work.
• The amount of heat transferred during a process
  is denoted by Q, and the amount of heat
  transferred per unit time is called heat transfer
  rate, and is denoted by Q_dot.
• The total amount of heat transfer Q during a time
  interval dt can be determined from
• The rate of heat transfer per unit area normal to
  the direction of heat transfer is called heat
  flux, and the average heat flux is expressed as

Sect 3.1.1
22
Heat Transfer Mechanisms

• Heat can be transferred in three basic modes
• Conduction,
• Convection,
• Radiation.
• All modes of heat transfer require the existence
  of a temperature difference.
• All modes are from the high-temperature
  medium/zone to a lower-temperature one.

Sect 3.1.1
23
Conduction

• Conduction is the transfer of energy from the
  more energetic particles of a substance to the
  adjacent less energetic ones as a result of
  interactions between the particles.
• Conduction can take place in solids,
• liquids, or gases
• In gases and liquids conduction is due to
• the collisions and diffusion of the
• molecules during their random motion.
• In solids conduction is due to the
• combination of vibrations of the
• molecules in a lattice and the energy
• transport by free electrons.

Sect 3.1.1
24
Conduction
where the constant of proportionality k is the
thermal conductivity of the material.
In differential form
which is called Fouriers law of heat conduction.
Sect 3.1.1
25
Convection
Convection Conduction Advection (fluid
motion)

• Convection is the mode of energy transfer between
  a solid surface and the adjacent liquid or gas
  that is in motion.
• Convection is commonly classified into three
  sub-modes
• Forced convection,
• Natural (or free) convection,
• Change of phase (liquid/vapor,
• solid/liquid, etc.)

Sect 3.1.1
26
Convection

• The rate of convection heat transfer is expressed
  by Newtons law of cooling as
• h is the convection heat transfer
• coefficient in W/m2C.
• h depends on variables such as the
• surface geometry, the nature of fluid motion,
• the properties of the fluid, and the bulk fluid
• velocity.

Sect 3.1.1
27
Radiation

• Unlike conduction and convection, radiation does
  not require the presence of a material medium to
  take place.
• Electromagnetic waves or electromagnetic
  radiation - represent the energy emitted by
  matter as a result of the changes in the
  electronic configurations of the atoms or
  molecules.
• Electromagnetic waves are characterized by their
  frequency n or wavelength l
• c - the speed of propagation of a wave in that
  medium.

Sect 3.1.1
28
Radiation

• Radiation is the energy emitted by matter in the
  form of electromagnetic waves (or photons) as a
  result of the changes in the electronic
  configurations of the atoms or molecules.
• Heat transfer by radiation does not require the
  presence of an intervening medium.
• In heat transfer studies we are interested in
  thermal radiation (radiation emitted by bodies
  because of their temperature).
• Radiation is a volumetric phenomenon. However,
  radiation is usually considered to be a surface
  phenomenon for solids that are opaque to thermal
  radiation.

Sect 3.1.1
29
Radiation - Emission

• The maximum rate of radiation that can be emitted
  from a surface at a thermodynamic temperature Ts
  (in K or R) is given by the StefanBoltzmann law
  as
• s 5.670X108 W/m2K4 is the StefanBoltzmann
  constant.
• The idealized surface that emits radiation at
  this maximum rate is called a blackbody.
• The radiation emitted by all real surfaces is
  less than the radiation emitted by a blackbody at
  the same temperature, and is expressed as
• e is the emissivity of the surface.

Sect 3.1.1
30
Radiation - Absorption

• The fraction of the radiation energy incident on
  a surface that is absorbed by the surface is
  termed the absorptivity a.
• Both e and a of a surface depend on the
  temperature and the wavelength of the radiation.

Sect 3.1.1
31

• Conduction
• Frederick Ferguson
• North Carolina AT State University
• September 13, 2007

Sect 3.1.1.1
32
Conduction

• Conduction is the transfer of energy from the
  more energetic particles of a substance to the
  adjacent less energetic ones as a result of
  interactions between the particles.
• Conduction can take place in solids,
• liquids, or gases
• In gases and liquids conduction is due to
• the collisions and diffusion of the
• molecules during their random motion.
• In solids conduction is due to the
• combination of vibrations of the
• molecules in a lattice and the energy
• transport by free electrons.

Sect 3.1.1.1
33
Conduction
where the constant of proportionality k is the
thermal conductivity of the material.
In differential form
which is called Fouriers law of heat conduction.
Sect 3.1.1.1
34
Thermal Conductivities of Materials

• The thermal conductivity of a material is a
  measure of the ability of the material to conduct
  heat.
• The thermal conductivities of gases such as air
  vary by a factor of 104 from those of pure metals
  such as copper.
• Pure crystals and metals have the highest thermal
  conductivities, and gases and insulating
  materials the lowest.

Sect 3.1.1.1
35
Thermal Conductivities and Temperature

• The thermal conductivities of materials vary with
  temperature.
• The temperature dependence of thermal
  conductivity causes considerable complexity in
  conduction analysis.
• A material is normally assumed to be isotropic.

Sect 3.1.1.1
36
Thermal diffusivity

• The thermal diffusivity represents how fast heat
  diffuses through a material, very important
  parameter in unsteady heat transfer processes.
• Appears in the transient heat conduction
  analysis.
• A material that has a high thermal conductivity
  or a low heat capacity will have a large thermal
  diffusivity.
• The larger the thermal diffusivity, the faster
  the propagation of heat into the medium.

Sect 3.1.1.1
37
Fouriers law of heat conduction

• The rate of heat conduction through a medium in a
  specified direction (say, in the x-direction) is
  expressed by Fouriers law of heat conduction for
  one-dimensional heat conduction as
• Heat is conducted in the direction of decreasing
  temperature, and thus the temperature gradient is
  negative when heat is conducted in the positive
  x-direction.

Sect 3.1.1.1
38
General Relation for Fouriers Law of Heat
Conduction

• The heat flux vector at a point P on the surface
  of the figure must be perpendicular to the
  surface, and it must point in the direction of
  decreasing temperature
• If n is the normal of the
• isothermal surface at point P,
• the rate of heat conduction at
• that point can be expressed by
• Fouriers law as

Sect 3.1.1.1
39
Heat Conduction Equation
Two-dimensional
Constant conductivity
Three-dimensional
1) Steady-state
2) Transient, no heat generation
3) Steady-state, no heat generation
Sect 3.1.1.1
40
Cylindrical Spherical Coordinates
Sect 3.1.1.1
41
Special Forms of the 1D Heat Conduction Equation
Variable conductivity
Constant conductivity
The one-dimensional conduction equation may be
reduces to the following forms under special
conditions
1) Steady-state
2) Transient, no heat generation
3) Steady-state, no heat generation
Sect 3.1.1.1
42
Boundary and Initial Conditions

• Specified Temperature Boundary Condition
• Specified Heat Flux Boundary Condition
• Convection Boundary Condition
• Radiation Boundary Condition
• Interface Boundary Conditions
• Generalized Boundary Conditions

Sect 3.1.1.1
43
Specified Temperature Boundary Condition
For one-dimensional heat transfer through a plane
wall of thickness L, for example, the specified
temperature boundary conditions can be expressed
as
T(0, t) T1 T(L, t) T2
The specified temperatures can be constant, which
is the case for steady heat conduction, or may
vary with time.
Sect 3.1.1.1
44
Specified Heat Flux Boundary Condition
The heat flux in the positive x-direction
anywhere in the medium, including the boundaries,
can be expressed by Fouriers law of heat
conduction as
The sign of the specified heat flux is determined
by inspection positive if the heat flux is in
the positive direction of the coordinate axis,
and negative if it is in the opposite direction.
Sect 3.1.1.1
45
Two Special Cases

• Insulated boundary

• Thermal symmetry

Sect 3.1.1.1
46
Convection Boundary Condition
and
Sect 3.1.1.1
47
Radiation Boundary Condition
and
Sect 3.1.1.1
48
Interface Boundary Conditions
At the interface the requirements are (1) two
bodies in contact must have the same temperature
at the area of contact, (2) an interface (which
is a surface) cannot store any energy, and
thus the heat flux on the two sides of an
interface must be the same.
TA(x0, t) TB(x0, t)
and
Sect 3.1.1.1
49
Variable Thermal Conductivity, k(T)

• The thermal conductivity of a material, in
  general, varies with temperature (similarly for
  density and specific heat) . An average value for
  the thermal conductivity is commonly used when
  the variation is mild.
• When the variation of thermal conductivity with
  temperature k(T) is known, the average value of
  the thermal conductivity in the temperature range
  between T1 and T2 can be determined from

Sect 3.1.1.1
50
Variable Thermal Conductivity

• The variation in thermal conductivity of a
  material with can often be approximated as a
  linear function and expressed as
• where ß is the temperature coefficient of
  thermal conductivity.
• For a plane wall the temperature varies linearly
  during steady one-dimensional heat conduction
  when the thermal conductivity is constant.
• This is no longer the case when the thermal
  conductivity changes with temperature (even
  linearly).

Sect 3.1.1.1
51

• Convection
• Frederick Ferguson
• North Carolina AT State University
• September 13, 2007

Sect 3.1.1.2
52
Physical Mechanism of Convection

• Conduction and convection are similar in that
  both mechanisms require the presence of a
  material medium.
• But they are different in that convection
  requires the presence of fluid motion.
• Heat transfer through a liquid or gas can be by
  conduction or convection, depending on the
  presence of any bulk fluid motion.
• The fluid motion enhances heat transfer, since it
  brings warmer and cooler chunks of fluid into
  contact, initiating higher rates of conduction at
  a greater number of sites in a fluid.

Sect 3.1.1.2
53
Newtons law of cooling

• Experience shows that convection heat transfer
  strongly depends on the fluid properties
• dynamic viscosity m,
• thermal conductivity k,
• density r, and
• specific heat cp, as well as the
• fluid velocity V.
• It also depends on the geometry and the roughness
  of the solid surface.
• The rate of convection heat transfer is observed
  to be proportional to the temperature difference
  and is expressed by Newtons law of cooling as
• The convection heat transfer coefficient h
  depends on the several of the mentioned
  variables, and thus is difficult to determine.

Sect 3.1.1.2
54
Boundary Layer

• All experimental observations indicate that a
  fluid in motion comes to a complete stop at the
  surface and assumes a zero velocity relative to
  the surface (no-slip).
• The no-slip condition is responsible for the
  development of the velocity profile.
• The flow region adjacent to the wall in which the
  viscous effects (and thus the velocity gradients)
  are significant is called the boundary layer.

Sect 3.1.1.2
55
Boundary Layer

• An implication of the no-slip condition is that
  heat transfer from the solid surface to the fluid
  layer adjacent to the surface is by pure
  conduction, and can be expressed as
• The convection heat transfer coefficient, in
  general, varies along the flow direction.

Sect 3.1.1.2
56
The Nusselt Number

• It is common practice to nondimensionalize the
  heat transfer coefficient h with the Nusselt
  number
• Heat flux through the fluid layer by convection
  and by conduction can be expressed as,
  respectively
• Taking their ratio gives
• The Nusselt number represents the enhancement of
  heat transfer through a fluid layer as a result
  of convection relative to conduction across the
  same fluid layer. Note, for Nu1 ?pure conduction.

Sect 3.1.1.2
57
Classification of Fluid Flows

• Viscous versus inviscid regions of flow
• Internal versus external flow
• Compressible versus incompressible flow
• Laminar versus turbulent flow
• Natural (or unforced) versus forced flow
• Steady versus unsteady flow
• One-, two-, and three-dimensional flows

Sect 3.1.1.2
58
Definition Surface Shear Stress

• Consider the flow of a fluid over the surface of
  a plate.
• The fluid layer in contact with the surface tries
  to drag the plate along via friction, exerting a
  friction force on it.
• Friction force per unit area is called shear
  stress, and is denoted by t .
• Experimental studies indicate that the shear
  stress for most fluids is proportional to the
  velocity gradient.
• The shear stress at the wall surface for these
  fluids is expressed as
• The fluids that obey the linear relationship
  above are called Newtonian fluids.
• The viscosity of a fluid is a measure of its
  resistance to deformation.

Sect 3.1.1.2
59
Definition Velocity Boundary Layer

• The region of the flow above the plate bounded by
  d is called the velocity boundary layer.
• d is typically defined as the distance y from
  the surface at which u0.99V.
• The hypothetical line of u0.99V divides the flow
  over a plate into two regions
• the boundary layer region, and
• the irrotational flow region.

Sect 3.1.1.2
60
Definition Thermal Boundary Layer

• Like the velocity a thermal boundary layer
  develops when a fluid at a specified temperature
  flows over a surface that is at a different
  temperature.
• Consider the flow of a fluid at a uniform
  temperature of T8 over an isothermal flat plate
  at temperature Ts.
• The fluid particles in the layer adjacent assume
  the surface temperature Ts.
• A temperature profile develops that ranges from
  Ts at the surface to T8 sufficiently far from the
  surface.
• The thermal boundary layer - the flow region over
  the surface in which the temperature variation in
  the direction normal to the surface is
  significant.
• The thickness of the thermal boundary layer dt at
  any location along the surface is defined as the
  distance from the surface at which the
  temperature difference T(ydt)-Ts 0.99(T8-Ts).
• The thickness of the thermal boundary layer
  increases in the flow direction.
• The convection heat transfer rate anywhere along
  the surface is directly related to the
  temperature gradient at that location.

Sect 3.1.1.2
61
Definition Prandtl Number

• The relative thickness of the velocity and the
  thermal boundary layers is best described by the
  dimensionless parameter Prandtl number, defined
  as
• Heat diffuses very quickly in liquid metals
  (Pr1) and very slowly in oils (Pr1) relative to
  momentum.
• Consequently the thermal boundary layer is much
  thicker for liquid metals and much thinner for
  oils relative to the velocity boundary layer.

Sect 3.1.1.2
62
Laminar and Turbulent Flows

• Laminar flow - the flow is characterized by
  smooth streamlines and highly-ordered motion.
• Turbulent flow - the flow is
• characterized by velocity
• fluctuations and
• highly-disordered motion.
• The transition from laminar
• to turbulent flow does not
• occur suddenly.

Sect 3.1.1.2
63
Turbulent Boundary Layer

• The velocity profile in turbulent flow is much
  fuller than that in laminar flow, with a sharp
  drop near the surface.
• The turbulent boundary layer can be considered to
  consist of four regions
• Viscous sublayer
• Buffer layer
• Overlap layer
• Turbulent layer
• The intense mixing in turbulent flow enhances
  heat and momentum transfer, which increases the
  friction force on the surface and the convection
  heat transfer rate.

Sect 3.1.1.2
64
Reynolds Number

• The transition from laminar to turbulent flow
  depends on the surface geometry, surface
  roughness, flow velocity, surface temperature,
  and type of fluid.
• The flow regime depends mainly on the ratio of
  the inertia forces to viscous forces in the
  fluid.
• This ratio is called the Reynolds number, which
  is expressed for external flow as
• At large Reynolds numbers (turbulent flow) the
  inertia forces are large relative to the viscous
  forces.
• At small or moderate Reynolds numbers (laminar
  flow), the viscous forces are large enough to
  suppress these fluctuations and to keep the fluid
  inline.
• Critical Reynolds number - the Reynolds number at
  which the flow becomes turbulent.

Sect 3.1.1.2
65
Heat and Momentum Transfer in Turbulent Flow

• Turbulent flow is a complex mechanism dominated
  by fluctuations, and despite tremendous amounts
  of research the theory of turbulent flow remains
  largely undeveloped.
• Knowledge is based primarily on experiments and
  the empirical or semi-empirical correlations
  developed for various situations.
• Turbulent flow is characterized by random and
  rapid fluctuations of swirling regions of fluid,
  called eddies.
• The velocity can be expressed as the sum of an
  average value u and a fluctuating component u

Sect 3.1.1.2
66
Shear Stress and Heat Flux in Turbulent Flows

• It is convenient to think of the turbulent shear
  stress as consisting of two parts
• the laminar component, and
• the turbulent component.
• The turbulent shear stress can be expressed as
• The rate of thermal energy transport by turbulent
  eddies is
• The turbulent wall shear stress and turbulent
  heat transfer
• - turbulent (or eddy) viscosity.
• kt - turbulent (or eddy) thermal conductivity.

Sect 3.1.1.2
67
Shear Stress and Heat Flux in Turbulent Flows

• The total shear stress and total heat flux can be
  expressed as
• and
• In the core region of a turbulent boundary layer
  - eddy motion (and eddy diffusivities) are much
  larger than their molecular counterparts.
• Close to the wall - the eddy motion loses its
  intensity.
• At the wall - the eddy motion diminishes because
  of the no-slip condition.

Sect 3.1.1.2
68
Empirical Relations for Shear Stress and Heat
Flux

• Reynolds Analogy (Chilton-Colburn Analogy) -
  under some conditions knowledge of the friction
  coefficient, Cf, can be used to obtain Nu and
  vice versa. It follows that the average Nu and Cf
  depends on
• These relations are extremely valuable
• The friction coefficient can be expressed as a
  function of Reynolds number alone, and
• The Nusselt number can be expressed as a function
  of Reynolds and Prandtl numbers alone.
• The experiment data for heat transfer is often
  represented by a simple power-law relation of the
  form

Sect 3.1.1.2
69
Analogies Between Momentum and Heat Transfer

• The Reynolds analogy can be extended to a wide
  range of Pr by adding a Prandtl number
  correction.

Reynolds analogy
Sect 3.1.1.2
70

• Radiation
• Frederick Ferguson
• North Carolina AT State University
• September 13, 2007

Sect 3.1.1.3
71
Introduction

• Unlike conduction and convection, radiation does
  not require the presence of a material medium to
  take place.
• Electromagnetic waves or electromagnetic
  radiation - represent the energy emitted by
  matter as a result of the changes in the
  electronic configurations of the atoms or
  molecules.
• Electromagnetic waves are characterized by their
  frequency n or wavelength l
• c - the speed of propagation of a wave in that
  medium.

Sect 3.1.1.3
72
Thermal Radiation

• Engineering application concerning
  electromagnetic radiation covers a wide range of
  wavelengths.
• Of particular interest in the study of heat
  transfer is the thermal radiation emitted as a
  result of energy transitions of molecules, atoms,
  and electrons of a substance.
• Temperature is a measure of the strength of these
  activities at the microscopic level.
• Thermal radiation is defined as the spectrum that
  extends from about 0.1 to 100 mm.
• Radiation is a volumetric phenomenon. However,
  frequently it is more convenient to treat it as a
  surface phenomenon.

Sect 3.1.1.3
73
Blackbody Radiation

• A body at a thermodynamic (or absolute)
• temperature above zero emits radiation in
• all directions over a wide range of
• wavelengths.
• The amount of radiation energy emitted
• from a surface at a given wavelength
• depends on
• the material of the body and the condition of its
  surface,
• the surface temperature.
• A blackbody - the maximum amount of radiation
  that can be emitted by a surface at a given
  temperature.
• At a specified temperature and wavelength, no
  surface can emit more energy than a blackbody.
• A blackbody absorbs all incident radiation,
  regardless of wavelength and direction.
• A blackbody emits radiation energy uniformly in
  all directions per unit area normal to direction
  of emission.

Sect 3.1.1.3
74

• The radiation energy emitted by a blackbody per
  unit time and per unit surface area
  (StefanBoltzmann law)
• s5.67 X 10-8 W/m2K4.
• Examples of approximate blackbody
• snow,
• white paint,
• a large cavity with a small opening.
• The spectral blackbody emissive power

Sect 3.1.1.3
75

• The variation of the spectral blackbody emissive
  power with wavelength is plotted.
• Several observations can be made
• from this figure
• at any specified temperature a
• maximum exists,
• at any wavelength, the amount of
• emitted radiation increases with
• increasing temperature,
• as temperature increases, the curves
• shift to the shorter wavelength,
• the radiation emitted by the sun
• (5780 K) is in the visible spectrum.
• The wavelength at which the peak occurs is given
  by Wiens displacement law as

Sect 3.1.1.3
76

• We are often interested in the amount of
• radiation emitted over some wavelength
• band.
• The radiation energy emitted by a
• blackbody per unit area over a
• wavelength band from l0 to l l1 is
• determined from
• This integration does not have a simple
  closed-form solution. Therefore a dimensionless
  quantity fl called the blackbody radiation
  function is defined
• The values of fl are listed in Table 122.

Sect 3.1.1.3
77
Table 12-2 - Blackbody Radiation Functions fl
Sect 3.1.1.3
78
Radiative Properties

• Many materials encountered in practice, such as
  metals, wood, and bricks, are opaque to thermal
  radiation, and radiation is considered to be a
  surface phenomenon for such materials.
• In these materials thermal radiation is emitted
  or absorbed within the first few microns of the
  surface.
• Some materials like glass and water exhibit
  different behavior at different wavelengths
• Visible spectrum - semitransparent,
• Infrared spectrum - opaque.

Sect 3.1.1.3
79
Emissivity

• Emissivity of a surface - the ratio of the
  radiation emitted by the surface at a given
  temperature to the radiation emitted by a
  blackbody at the same temperature.
• The emissivity of a surface is denoted by e, and
  it varies between zero and one, 0e 1.
• The emissivity of real surfaces varies with
• the temperature of the surface,
• the wavelength, and
• the direction of the emitted radiation.
• Spectral directional emissivity - the most
  elemental emissivity of a surface at a given
  temperature.

Sect 3.1.1.3
80

• The total hemispherical emissivity
• Since Eb(T)sT4 the total hemispherical
  emissivity can also be expressed as
• To perform this integration, we need to know the
  variation of spectral emissivity with wavelength
  at the specified temperature.

Sect 3.1.1.3
81
Sect 3.1.1.3
82
Absorptivity, Reflectivity, and Transmissivity

• When radiation strikes a surface, part of it
• is absorbed (absorptivity, a),
• is reflected (reflectivity, r),
• and the remaining part, if any, is transmitted
  (transmissivity, t).
• Absorptivity
• Reflectivity
• Transmissivity

Sect 3.1.1.3
83
The View Factor

• Radiation heat transfer between surfaces depends
  on the orientation of the surfaces relative to
• each other as well as their radiation
• properties and temperatures.
• View factor is defined to account for the
• effects of orientation on radiation heat
• transfer between two surfaces.
• View factor is a purely geometric
• quantity and is independent of the surface
  properties and temperature.
• Diffuse view factor - view factor based on the
  assumption that the surfaces are diffuse emitters
  and diffuse reflectors.
• Specular view factor - view factor based on the
  assumption that the surfaces are specular
  reflectors.

84
View Factors for Selected Geometries
85
Radiation Heat Transfer Black Surfaces

• Consider two black surfaces of arbitrary shape
• maintained at uniform temperatures T1 and T2.
• The net rate of radiation heat transfer
• from surface 1 to surface 2 can be expressed as
• Applying the reciprocity relation A1F1?2A2F2?1
  yields
• For enclosure consisting of N black surfaces

86
Radiation Heat Transfer in Two-Surface Enclosures

• Consider an enclosure consisting of two opaque
  surfaces at specified temperatures.
• Need to determine the net rate of
• radiation heat transfer.
• Known T1, T2, e1, e2, A1, A2, F12.
• The net rate of radiation transfer is
• expressed as

87
(No Transcript)