Heat Transfer Modeling
and Applications
1
A SHORT COURSE
REZA TOOSSI, PH.D, P.E.
CALIFORNIA STATE UNIVERSITY, LONG BEACH
Outline
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 Scope and Types
 Energy Equation



Formulation
Conversion Mechanisms
Dimensionless Parameters in Heat Transfer
 Modes of Heat Transfer



Conduction, Convection, and Radiation
Correlations
Combined Modes
 Heat Transfer in Multiple Phases



Conjugate Heat Transfer
Composites
Phase Change
Scales
3
 T  0 K – 1.4x1032 K
 L  1.6x10-35 m – 1.6x1026 m (15 bly)
 t  5.4x10-44 s – 2.7x1017 s
 m  10-69 kg - 1054 kg
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5
Mechanical Systems
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 Mechanically determined problems

Can be solved only by Newton’s Law of Motion and
Conservation of Mass
Examples:
 Free fall of a body; F = ma
 Dynamics of rigid bodies in absence of friction
 Flow of ideal fluids between two parallel plates
 Mechanically undetermined problems
 Additional laws are needed
Examples:
 Dynamics of rigid bodies with friction
 Dynamics of deformable body
Thermal Systems
7
 Thermodynamically determined
problems

Can be solved by the general laws of mechanics and the first
and second laws of thermodynamics.
Example:

Flow of steady 1-D isentropic and subsonic fluid through a nozzle.
mass
d(rAV) = 0
Momentum dp+ rAdV = 0
Energy
du +pd(1/r) = 0
 Thermodynamically undetermined
problems


Heat Transfer (modes of heat transfer)
Gas Dynamics (equation of state)
Knudsen Number
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 Continuum regime (Kn < 0.01)
 Slip flow regime (0.01 < Kn < 0.1)
 Transition regime (0.1 < Kn < 3)
 Free molecular flow regime (Kn > 3)
Energy Equation
9
Formulations
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 Differential
 Integral
 Integral-Differential
 Thermal Nodes
Differential-Volume
11
Fire Walking
Infinite flat plate
Solution:
Interface Temperature (no contact resistance)
Integral Volume (lumped)
12
 Heat losses under the condition of hypothermia
Take rcvV = 5x105 J/K, Sv = 400 W, and QA= Qr = 800 W; T = 10oC
Get t =3.47 hr
Surface Coating
13
 Droplet Impingement on a Hot Surface




Temperature is uniform within the droplet
(particle)
Particle temperature varies during its flight
Sensible heating until particle reaches its
melting temperature. Melting occurs at T = Tsl
Time to reach melting temperature
Integral-Differential
14
Thermal Nodes
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- Slab
- Cylinder
- Sphere
Energy Conversion Mechanisms
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Heat Source
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 Phase Change
 Chemical Reaction
 Nuclear Fission and Fusion
 Surface Friction Heating
 Viscous Heating
 Ultrasound Heating
 Microwave Heating
 Joule Heating
 Thermolectric Heating (and Cooling)
Phase Change
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Chemical Reaction
19
ar is pre-exponential factor = func (f)
Nuclear Fission and Fusion
20
Surface Friction Heating
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 Interfacial Energy Conversion of Mechanical Energy
μF is the friction coefficient
pc is the contact or joint pressure
Δui is the interface relative velocity
Viscous Heating
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 Volumetric Conversion of Mechanical Energy Due to Fluid
Viscosity
Example: Viscous heating in a ball bearing. The bearing is
0.2 mm in diameter, and ui = 1 m/s. Engine oil at STP
has a viscosity of mf = 0.366 Pa/s
Ultrasound Heating
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 Volumetric Conversion of Longitudinal Acoustic Waves to
Thermal Energy
sac acoustic absorption coefficient
= 1.4 for blood
= 14 for muscle
= 31 for skin
= 161 for bone
as
speed of sound
= 1,519 m/s in tissues
= 3,445 m/s in bones
f
μ
γ
Pr
frequency
dynamic viscosity
specific heat ratio g = cp/cv
Prandtl number
Microwave (Dielectric) Heating
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 Volumetric Conversion of Electromagnetic to Molecular
Vibration (Heat)
ee (V/m) electric field intensity
eec
e0
f (Hz)
dielectric loss factor
(relative permittivity)
permittivity of free space
oscillation frequency
Joule heating
25
 Conversion of Electrical Energy to Heat
Seebeck Thermoelectric
26
Peltier Thermoelectric Cooling and Heating
27
Thermoelectric Properties for Some Materials
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Thermoelectric Power Generation Unit
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 Direct Electrical Power Generation by


Heat Absorption at a Hot Junction
Rejecting the Peltier Heat at the Cold Junction



Bismuth-telluride cold p-n junction
S,p=230x10-6 V/k, S,n= -210x10-6 V/k, Je=10 A
The Peltier heat absorbed at the cold junction
 Tc = 120oC  Q = -1.73 W
 Tc = 20oC  Q = -1.29 W
 Tc = - 80oC  Q = -0.85 W
Volumetric and Surface Radiation Absorption
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Classification
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Solar Irradiation
32
Flame Irradiation
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Laser Irradiation
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Summary of Heat Source Terms
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Modes of Heat Transfer
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 Diffusion (transfer of heat within one medium or
from one medium to another medium)
 Conduction (diffusion of heat in moving or stationary
rigid bodies)
 Convection (diffusion of heat in moving deformable
bodies)
 Radiation (transfer of heat by electromagnetic waves)
Conduction
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Convection
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Heat Transfer Coefficient
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 Convective heat transfer is affected by the geometry,
surface condition, and fluid properties.
 To find h, the detailed flow field must be known.




Mass (1)
Momentum (3)
Energy (1)
Equation of State (1)
 Solve for u, v, w, T, P, and ρ
 Find qw = h A (TW -T∞) {and also tw = m du/dyw}
 Π-Buckingham Theorem (f = n – m)
 h = h (k, m, cp, r, L, e, T, gbT, hst, t)
 Nu = Nu (Gr, Pr, Ste, Fo, e/L)
Dimensionless Parameters in HT
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Dimensionless Parameters in HT
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Boundary Conditions
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1.
2.
3.
4.
5.
6.
7.
8.
Prescribed Temperature
Prescribed Heat Flux (Fourier’s Law)
Insulation
Convective (Newton’s Cooling Law)
Radiative (Stefan-Boltzmann Law)
Prescribed Heat Flux Acting at a Distance
Interface (Continuity, Conservation Law)
Moving Boundary
How to insulate a boundary?
43
 q0

Efficient electric heater
Guard Heater
When is the assumption of isothermal or insulated wall justified?
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 h  0; h  ∞

Flow through a thin heated tube

Insulated wall [
Ti  Tm (Boiling)
 k  0; k  ∞

Lumped vs. distributed (Biot number)

Insulated wall k  large (lumped)
k  small (distributed)

Bare tube, h0<< hi  (qo0)
]
Critical thickness of Insulation
45
Moving Boundary
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 Ice layer forming on surface of a pond on a
clear night with calm wind
Tsky = 0 K; T∞ = 3oC; Tℓ = 4oC; Tice = 0oC)
Forced Convection (External Flow)
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Forced Convection (Internal Flow)
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Free Convection: Flat Plates
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Free Convection: Cylinders and Spheres
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Conjugate Heat Transfer
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 Both flow field in the fluid and temperature field in
a solid must be known



Requires two physical models for fluid and the solid materials
Point-to-point connectivity at the boundary zone
Energy must be conserved at the interface.
 This requires interfacial matching of appropriate
boundary conditions at the interface
 In general energy and momentum equations are
coupled
Interfaces
52
 Solid-Solid
Pressure Contact (glue, contact resistance)
 Relative motion(grinding wheel, gears)

 Solid-Liquid

Phase Change at the interface (ablation, melting of ice)
 Solid-Gas
No phase change (fins)
 Phase Change at the interface (condensation)

 Liquid-Gas

Phase Change at the interface (pool nucleate boiling)
Examples of CHT Problems
53
 Heat transfer through composites and porous
media.
 Laminar boundary-layer flow past a plate of finite
thickness with internal heat generation.
 Steady and unsteady duct-flow in which axial wall
conduction is important.
 Liquid jet spray into a quiescent gas (with and
without impingement)
Effective Conductivity (LTE)
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• Low porosity, low conductivity beads
Local thermal equilibrium (<T>g= <T>s  <k>eff)
Composites (Layered)
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 1-D Treatment
 Layered
 Non-layered
Composites (Non-layered)
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 Solid-gas composites or porous solids (fiberglass insulation)
 Solid-solid composites (fiberglass filled with resin or thermo-
plastic polymers)
 Liquid-liquid composites (liquid emulsion)
Heat transfer through porous media
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 High porosity, open-celled metal foams

Assumption of the local thermal equilibrium is not reasonable (ks >> kg)
Betchen, L. , and Straatman A., “Interface conditions for a non-equilibrium heat transfer model for conjugate fluid /porous/solid domains,” Numerical Heat
Transfer, Part A, 49, pp. 543–565.
Coating Substrate
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Lim, J. S., Bejan, A., and Kim, J. H., “The optimal thickness of a wall with convection on one side,“ Int. J. Heat mass Transfer, Vol. 35, 1992, pp..1673-1679.
Uncoated Plate (Similarity Solution)
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Coated Plate
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Natural Convection along a Vertical Wall
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1.
Isothermal wall, Isothermal pool of fluid
(simplest case)
2.
Wall with internal heat generation,
Isothermal pool of fluid (Cooling of nuclear fuel rod by
liquid metals)
3.
Isothermal wall, Thermally stratified fluid
(heated wall bounded by the ceiling)
Conjugate boundary layer (single-pane window)
5. Channel flow (chimney)
6. Consideration of free convection (mixed
4.
convection)
Thermobuoyant Flow along a Vertical Wall
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Mixed Convection
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Lloyd, J. R., and Sparrow, E. M., “Combined forced and free convection flow on vertical surfaces, Int. J. Heat Mass Transfer, Vol. 13, 1970, pp.
434-438.
Cooling of a Nuclear Fuel Element by Upward Stream of Liquid Metals
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S. Jahangeer, M.K. Ramis, G. Jilani, « Conjugate heat transfer analysis of a heat generating vertical plate,” Int J Heat and Mass Transfer, 50 (2007) 85–93
Cooling of a Nuclear Fuel Element by Liquid Metals (Continued)
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Cooling of a Nuclear Fuel Element by Liquid Metals (Continued)
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Cooling by Free Convection (Re 0)
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Heat conduction equation for the slab
M. D. Kelleher and K. YANG,” A Steady Conjugate Heat Transfer analysis of a heat generating vertical plate,” Int. J. Heat Mass Transfer, 50
(2007) 85-93.
Analytical Solution
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 Conduction in Slab

Poisson Equation  Use Fourier sin transform
 Free Convection
 Similarity Solution  Define stream function
 Interfacial Condition
Results
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Single-Pane Windows
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R. Anderson and A. Bejan, “Heat transfer through single and double vertical walls in natural convection: theory and experiment,” Int. J. Heat Mass Transfer,
Vol. 24, 1981, pp. 1611-1620
Impermeable Wall Imbedded in a Porous Medium
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Bejan, A., and Anderson, R., “Heat transfer across a vertical impermeable partition imbedded in a porous medium,” Int. J. Heat Mass Transfer, Vol.
24, 1981, pp. 1237-1245.
Vertical Channel Flow
(Chimney)
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Anderson, R., and Bejan, A., “Heat transfer through single and double vertical walls in natural convection: theory and experiment”, Int. J. Heat
Mass Transfer, Vol. 24, 1981, pp. 1160-1245.
Boussinesq Approximation
73
What is the effect of buoyant force due to temperature variation within the system?
Solution: Vertical Channel Flow
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Moving Boundary Problems
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Solid-Liquid Transition
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Analytical Solutions in Phase Change Problems
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 Contact Melting (melting of a solid under its own weight)
Solidification (One-Region Problem)
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A solid PCM at Ti = Tm is suddenly exposed to a constant heat flux q”0. Find the
transient location of the solid-liquid interface.
Solidification (Two-Region Problem)
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Convective Effects
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Numerical Simulation in Phase Change Problems
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 Analytical  1D and some 2D conduction-controlled
 Numerical
 Strong (Classical ) numerical solution
 Velocity

u and pressure p satisfy Navier Stokes equations
pointwise in space-time.
Weak (Fixed-Grid) solution
 Enthalpy Method (Shamsunder and Sparrow, 1975)
 The Equivalent Heat Capacity Method ( Bonacina et al ., 1973)
 The Temperature-Transforming Model ( Cao and Faghri, 1990)
Enthalpy Method
82
 Two-Region Melting of a Slab

Assume densities of the liquid and
solid phase are equal.
Discretization
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Algorithm (explicit scheme)
84
1.
Choose ∆t and ∆x to meet Neumann’s stability criterion
2.
Determine the initial enthalpy at every node hjo (j = 1)
Calculate the enthalpy after the first time step at nodes (j = 2 ,..., N -1)
by using equation (1).
Determine the temperature after the first time step at node (j = 1 ,..., N)
by using equations (2) and (3).
Find a control volume in which the enthalpy falls between 0 and hsl ,
and determine the location of the solid-liquid interface by using
equation (4).
Solve the phase-change problem at the next time step with the same
procedure.
3.
4.
5.
6.
Algorithm (implicit scheme)
85
 Unconditionally stable but is more complex because
two unknown variables enthalpy and temperature
are involved. [See Alexiades , A ., and Solomon , A . D ., 1993 , Mathematical Modeling of Melting and
Freezing Processes , Hemisphere , Washington , DC .]
 Transform the energy equation into a nonlinear
equation with a single variable h. [See Cao , Y ., and Faghri , A ., 1989 , " A
Numerical Analysis of Stefan Problem of Generalized Multi-Dimensional Phase-Change Structures Using the Enthalpy
Transforming Model ," International Journal of Heat and Mass Transfer , Vol . 32 , pp . 1289-1298.]
Equivalent Heat Capacity Method
86

3-D Conduction controlled melting/solidification
Heat capacity during the phase change is infinite.
 Assume Cp and k change linearly from liquid to solid

Advantage: Simplicity
 Disadvantage: Unstable if right choices for ∆x, ∆t, and ∆T are not made.

Temperature-Transforming Model
87
 Combination of the two methods [Cao , Y ., and Faghri , A ., 1990a , " A Numerical Analysis of
Phase Change Problem including Natural Convection ," ASME Journal of Heat Transfer, Vol . 112 , pp . 812-815.]
 Use finite volume approach by Patankar to solve the diffusion equation.
Enthalpy Method: Solidification in Square Section
88
 Melting/solidification with natural
convection

Assumptions




Newtonian incompressible fluid with constant properties,
except the density that is evaluated s linear function of
temperature (Bousinessqe approximation)
Effective conductivity in the mushy zone
Isotropic
Heat transfer by conduction, convection and phase change
 Algorithm

Finite volume method with body-fitted coordinates with a
SIMPLER scheme to join pressure and velocities
CARLOS HERNÁN SALINAS LIRA1, SOLIDIFICATION IN SQUARE SECTION, Theoria, Vol. 10: 47-56, 2001.
Governing Equations
89
Numerical Method
90
Finite volume method
 Staggered grid
 Poisson equation with the stretching
function
 SIMPLER algorithm (Patankar, 1980)



Central differencing scheme
Crank-Nicholson procedure for the convective terms
Gauss-Seidel method with successive
iterations
 Convergence with local deviation of
0.006% for u, v, p, and T.

Results
91
Porous Media: Averaging Techniques for Multiphase Transport
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 Eulerian Averaging
 Averaged over space, time, or both within the domain of integration

Based on time-space description of physical phenomena
Consistent with the c.v. analysis used to develop governing equations.
Eulerian time-averaging

Eulerian volume-averaging



Phase-averages:


Intrinsic phase average
Extrinsic phase average
 Lagrangian Averaging
 Follow a particle and average its properties during the flight
 Molecular Statistical Averaging
 Boltzmann statistical distribution rather than individual particle is
the independent variable.
Porous Media : One-Region Melting
93
Governing Equations:
Jany , P ., and Bejan , 1988 , " Scaling Theory of Melting with Natural Convection in an Enclosure ," International Journal of Heat
and Mass Transfer , Vol . 31 , pp . 1221-1235.
Solution: Porous Media : One-Region Melting
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Summary
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 Formulation
 Sources
 Modes
 Conduction, Convection, and Radiation
 Mixed Modes
 Phase Change
 Conjugate Heat
 Applications
Thank you…
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