Thermofluid 1

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Thermofluid 1

• Introduction to Transport Phenomena Conduction
  Heat Transfer
• Prof. Dongsik Kim

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1. Introduction

• Transport phenomena
• Thermodynamics
• Concerns initial and final states (not the
  process)
• 1st law of thermodynamics ? energy conservation
• 2nd law ? direction of energy transport
• Transport phenomena
• Fluid mechanics ? rate of momentum transport
• Momentum in transit due to velocity gradient
• Heat transfer? rate of thermal energy transport
• Energy in transit due to temperature gradient

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Symbols and units

• Thermal energy EJ
• Heat transfer rate qWJ/s
• Heat transfer rate per unit length q/LW/m
• Heat transfer rate per unit length q/A W/m2
• Temperature ToC or K T(oC)T(K)273.15
• Note When oC or K unit is in the denominator,
  unit change doesn't affect the numerical value,
  e.g., specific heat Cp 1 J/kg.oC1 J/kg.K,
  thermal conductivity 1 W/m.oC1 W/m.K

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Mechanisms of transport

• Diffusion
• Carriers molecules, atoms (crystal lattices),
  electrons - translation, vibration, and rotation
  in a random fashion.
• Mass, momentum, and thermal energy (heat or
  microscopically the kinetic energy of the
  particles) diffusion from the region of high
  potential to the region of low potential until an
  equilibrium state is reached.
• Fluids- molecular random motion
• Solids - lattice vibration and motion of free
  electrons

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Transport by diffusion
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Convection

• Transport with "bulk" or "macroscopic" flow
• Transport is not determined only by the random
  particle motion (diffusion) but also by the bulk
  fluid motion.
• Transport by bulk flow is termed as advection or
  convection in a narrow meaning
• Convection Conduction Bulk energy transport
  (advection)
• (ex) HT between a solid object and fluid or HT in
  a moving fluid

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Convective transport
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Radiation

• Thermal energy transport in the form of EM
  (electromagnetic) wave is called thermal
  radiation. Note that mass can not be transported
  by radiation.
• Thermal energy is emitted by EM wave (light) due
  to the changes in electron configurations of
  matter. The EM wave emission is always present
  for surfaces at temperatures greater than
  absolute 0 K.

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Review

• (Q1) Are the three modes diffusion, convection,
  and radiation independent of each other? If not,
  how are they related?
• (Q2) Show examples of heat, mass, and momentum
  transfer. Identify the mechanisms.

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Methodology

• Conservation principle Physical model

• Review (Q3) What is the conservation principle
  for a control surface (infinitesimally thin
  control volume)?

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Microscopic view

• Characteristic length scales
• MFP (mean free path) average distance a particle
  travels without a collision
• Energy carriers
• Phonon quantum of lattice vibration energy
• Electron
• Photon quantum of EM energy

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Understanding diffusion by kinetic theory
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Comparison with macroscopic observation

• Thermal conductivity
• In general, ksolgtkliqgtkgas (Fig. 2.4)
• Solid
• k(k by electron motion) (k by lattice
  vibration)
• Good electrical conductors are good thermal
  conductors as well (Wiedemann-Franz law)
• Well order materials (crystals) are good thermal
  conductors in comparison with amorphous materials
• Gas
• In general, kgas ? as T ? because v? as T ?
• Insensitive to P and r. (n ? as P, r ? but l ? as
  P, r ?)

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2. Heat transfer fundamentals

• Fourier's law and thermal conductivity
•  
• HT rate a vector from high T to low T
• Direction of HT perpendicular to isothermal
  surfaces
• kk(T, P)
• Often kk(T) (Incropera Figs. 2.5 - 2.7)
• In general, thermal conductivity depends on the
  direction. If the material is isotropic, k is
  independent of direction and a constant.
  Otherwise, second-order tensor kij
• How to measure thermal conductivity

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Thermal conductivity
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Fouriers law

• HT (area) ? (Temperature difference) /
  (distance)
• HT rate is proportional to the temperature
  gradient
• Minus sign (-) denotes the direction of heat
  flow.
• Ficks law and bianary diffusion coefficient
• Shear-law and viscosity

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Newtons law of cooling

• HT with "bulk" or "macroscopic" motion of fluid
  (HT between a solid object and fluid or HT in a
  moving fluid)
• Convection Conduction Bulk energy transport
  (advection)
• Convection Conduction on the wall (no slip BC).
• Classification
• Forced convection external means
• Natural or free convection self-induced flow by
  buoyancy 
• HT (area) ? (Temperature difference) ?
  (convection heat transfer coefficient)
• Convection heat transfer coefficient h fn (fluid
  properties, flow speed, geometry, etc.)

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Convection heat transfer coefficients

•  HT in liquid gt HT in gas (material property,
  kliqgtkgas, etc)
• Aluminum k202 W/m.K, water 0.6, air 0.026
• Effect of latent heat critical

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Stefan-Boltzmann law of radiation

• Energy transportation by EM (electromagnetic)
  wave
• Thermal energy is emitted by the changes in
  electron configurations of matter
• Emission is always present at temperatures
  greater than absolute 0 K
• No medium required
• Important in high-temperature HT, solar energy
  utilization, space applications, etc.
• EM wave can be visible (light) at high
  temperature
• Energy emitted by a surface per unit area
  (emissive power) T4

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Radiation heat transfer

• Blackbodymaximum value of thermal radiation at a
  given absolute temperature
• Stefan-Boltzmann constant s5.67?10-8 W/m2K4, e
  emissivity (01)
• Experimental implementation of a blackbody
  isothermal cavity (material independent)
• A small object surround by large surroundings at
  Tsur, (AltltAsur)  

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Review

• (Q1) While thermal conductivity is a material
  property given as a function of thermodynamic
  state, heat transfer coefficient h is not a
  material property. What is the relation between k
  and h?
• (Q2) What are the typical values of thermal
  conductivity for common materials?
• (Q3) When is radiation heat transfer important?
  Give a quantitative answer.
• (Q4) A woman informs her engineer husband that
  hot water will freeze faster than cold water.
  He calls this statement nonsense. She answers by
  saying that she has actually timed the freezing
  process for ice trays in the home refrigerator
  and found that hot water does indeed freeze
  faster. As a friend, you are asked to settle the
  argument. Is there any logical explanation for
  the womans observation? (J. P. Holman, Prob.
  1-35)

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Governing equation (review)
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Conduction equation

• Cartesian
• Cylindrical
• Spherical

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Initial and Boundary Conditions

• Initial condition initial temperature
  distribution
• Boundary conditions are assigned from physical
  observation of the domain boundary
• BC of the 1st kind known temperature boundary
  condition, e.g., phase-change interface,
  temperature controlled surface
• BC of the 2nd kind known heat-flux boundary
  condition, e.g., or , insulated surface
• (insulated)
• Mixed BC when a surface is exposed to air and
  neither temperature nor heat flux is known
• Phase-change interface

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Review

• (Q1) For problems involving constant thermal
  properties and temperature boundary conditions,
  according to the governing equation, the
  temperature profile is independent of thermal
  conductivity. It depends only on thermal
  diffusivity. The thermal conductivity of iron is
  about 3000 times larger than that of air but the
  thermal diffusivity of iron is approximately
  equal to that air. How come can the temperature
  profile be the same? Is this puzzle related with
  dynamic viscosity vs. (kinematic) viscosity?
• (Q2) What are the implicit assumptions the
  diffusion equation is based on?
• (Q3) (Advanced problem) In heat conduction
  equation, is the heat capacity Cp or Cv? Why? Can
  you derive the governing equation from the 1st
  law of thermodynamics, i.e.,
  ?

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3. Solutions of 1-d steady conduction equation

• Steady ?
• 1-d ?
• The governing equation thus reduces to
• If thermal conductivity is constant
• linear profile
• Note For prescribed temperatures at the two
  boundaries, T is independent of material
  properties. What is going on?

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Thermal resistance

• Analogy between heat transfer and electricity
• ?
•  
• Definition of thermal resistance, overall heat
  transfer coefficient
• Thermal resistance of a plate K/W
• Thermal resistance of a cylinder
• Thermal resistance of a sphere
• Convection thermal resistance m2.K/W
• Thermal contact resistance

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Thermal contact resistance
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Equivalent thermal circuit
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Critical thickness of insulation layer
Thickness of the insulation layer that maximizes
(minimizes) heat transfer?
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4. 1-d steady conduction II

• Variable thermal conductivity, cross-sectional
  area
• In 1-d steady state HT, heat flux is conserved.
• Direct integration of Fourier's law
• (Ex1) Thermal conductivity (a,bgt0)
• (Ex2)

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Conduction with internal heat generation
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Example

• T(x)? TsT(l)?
• Method 1
• Method 2

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Heat generation in a cylinder and a sphere

• Cylinder
• Governing equation
• BC's
• Heat flux increases linearly with r
• Temperature profile?
• Sphere of radius r0
• Similarly,

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5. Extended surfaces

• Heat transfer enhancement
• Natural convection
• Forced convection
• Extended surfaces
• Phase change
• Fins

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Fin analysis

• Deriving a governing equation
• Steady-state, constant cross-sectional area

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Solutions to fin equations
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Fin performance

• Fin resistance
• Fin effectiveness
• HT with a fin vs. HT without a fin
• Fin efficiency
• HT with a fin vs. HT with an ideal fin
• What is an ideal fin?

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Fins of nonuniform cross-sectional area
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Review

• (Q1) In order to use the simple solution for an
  infinitely long fin, how long should a fin be?
• (Q2) In the above analysis, it is assumed that
  the temperature is uniform over the
  cross-sectional area of a fin. Discuss the
  validity of the assumption.

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6. Introduction to transient heat transfer problem

• Governing equation
• Thermal conductivity vs. thermal diffusivity
• Dimensionless parameters
• Biot number
• conduction thermal resistance vs. convection
  thermal resistance
• Fourier number
• Dimensionless time
• Rate of heat conduction vs. rate of heat storage
  for t (thermal diffusivity heat conduction /
  heat storage)

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Dimensionless form of governing equation
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Lumped heat capacitance method

• Neglecting spatial distribution of temperature
  T(t)
• Validity of the lumped heat capacitance analysis
  small Bi

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Review

• (Q1) In the above cooling problem, the lumped
  body experiences a negligibly small temperature
  change after sufficient time has elapsed. What is
  the characteristic time (time constant)?
• (Q2) What is the physical meaning of very
  large/small Fourier number?

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7. Solutions of 1-d transient heat conduction
equation

• ltInitial temperature Ti, suddenly exposed to
  ambient temperature T? with convection heat
  transfer coefficient hgt

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Analytical solutions

• Slab of thickness 2L  
• Cylinder of radius r0
• Sphere of radius r0

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Semi-infinite wall

• Similarity solution
• Similarity variable
• Temperature profiles at different moments are
  similar by scaling the position by some function
  of time f(t)
• PDE in x and t ? transform ? ODE in h
• Initial temperature Ti, constant temperature Ts
  at the boundary x0
• Initial temperature Ti, constant heat flux at
  the boundary x0

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Temperature profiles
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Thermal penetration depth

• Exact solution
• Scaling of governing equation

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Other related topics

• Analogy with Stokes problem of momentum
  diffusion
• Sudden thermal contact of two infinite bars at
  different temperatures (TAi, TBi)

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Review

• (Q1) A metal surface feels colder than a wood
  surface when touched, though they are at the same
  temperature. Why?
• (Q2) (Advanced topic) Can you define the speed of
  thermal wave using the concept of thermal
  penetration depth? How is the thermal wave
  different from the shock wave?

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8. Analytical methods for multidimensional heat
transfer problems

• Separation of variables
• Most common technique
• Trial solution
• Some math
• Eigenvalue problem
• Strum-Liouville theorm
• Details (Incropera pp. 163-167)

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• Are these steps crystal clear?
• Need more details?
• H.S. Carslaw and J.C. Jaeger, 1996, Conduction of
  Heat in Solids, Oxford
• M. Necati Ozisik Heat Conduction John Wiley
  Sons

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Integral method

• Write integral form of the governing equation
• Construct an approximate solution satisfying the
  boundary conditions
• Substitute the approximate solution
• An example fin problem

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Other analytical methods (advanced)

• Greens function
• Superposition using diffusion delta
• Integral transform
• Use of Laplace transform, Fourier transform

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9. Numerical methods for multidimensional heat
transfer problems

• Finite difference schemes

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Finite difference solutions

• Matrix inversion method

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• Gauss-Seidel Iteration

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Numerical experiment

• Java Applets for Engineering Education
• http//www.engapplets.vt.edu/
• A set of NSF-funded applets (primarily fluids and
  heat transfer) for web-based tutoring of simple
  Heat Transfer problems
• Try web surfing for heat transfer
• http//www.engr.colostate.edu/allan/heat_trans/pa
  ge4/page4f.html
• http//www.oilsurvey.com/EngToolkit/Process/heatma
  sstransfer.htm
• FEHT (Finite Element Heat Transfer)
• S.A. Kleins FEM package for HT education