HEAT TRANSFER: CONDUCTION
AND CONVECTION
Dr. B. Jayaraman
FDP on CFD
Lecture 5: 7
th
June 2016,
OUTLINE
•
Introduction
•
Conduction
•
Convection
Forced Convection
Free Convection
•
Other Models
MODES OF HEAT
TRANSFER
THERMAL
CHARACTERISTICS
Temperature Distributions
Amount of Heat Lost or gained
Thermal Gradients
Thermal Fluxes
Thermal Analysis
Thermal Analysis
+
Stress Analysis
PHENOMENA (MODELS)
•
Conduction
•
Convection
Forced Convection
Natural convection
Boiling (Multiphase)
•
Radiation
•
Species diffusion and Combustion
•
Conjugate heat transfer
•
Periodic heat transfer
•
Viscous Dissipation
BOUNDARY CONDITIONS
•
Heat Flux
•
Temperature
•
Convection
•
Radiation
•
Mixed – Combination of Convection and
Radiation boundary conditions
PROPERTIES
•
Fluid properties such as heat capacity, conductivity and viscosity
can be defined as:
Constant
Temperature-dependent
Composition-dependent
Computed by kinetic theory
Computed by user-defined functions
• Density can be treated as
Constant (with optional Boussinesq modeling)
Temperature-dependent
Computed as ideal gas law
Composition-dependent
User defined functions
OUTLINE
•
Introduction
•
Conduction
•
Convection
Forced Convection
Free Convection
•
Other Models
CONDUCTION
Presented on 4/6/2015 at SoME, SASTRA University, Thanjavur
9
Energy
conducted
into CV
Energy
conducted
out of CV
=
+
Heat
Generated
within CV
+
Rate of
change of
Internal
Energy in CV
2-DIM. PROBLEM
Physical domain of Slab with square cross section
2-dim, Steady state with heat generation
2-DIM,STEADY STATE
Presented on 4/6/2015 at SoME, SASTRA University, Thanjavur
11
Non-dim Boundary Conditions
At X = 0
Computational Domain of Slab with square cross section
L
Geometrically and thermally symmetric
2-DIM,UNSTEADY STATE
SQUARE PLATE
Non-dim Boundary Conditions
At X = 1
At X = 0
Computational Domain of Slab with square cross section
L
Infinitely long plate, initially at T
i
Then suddenly maintained at T
o
OUTLINE
•
Introduction
•
Conduction
•
Convection
Forced Convection
Free Convection
•
Other Models
•
Conclusion
13 of 41
CONVECTION
•
Diffusion Transport + Advection Transport
•
Advecting variable(velocity)
driver
•
Advected variable(temperature)
passenger
FORCED CONVECTION
•
Diffusion Transport
Random Molecular Motion
Molecular heat-flux includes only conduction
heat transfer
Molecular momentum-flux includes both
pressure and viscous forces
(fluid statics/dynamics)
15
FORCED CONVECTION
•
Advective Transport
Bulk or macroscopic motion of the fluid
ENERGY BALANCE
Presented on 4/6/2015 at SoME, SASTRA University, Thanjavur
17
Rate of
Energy flow
into CV
Rate of
Energy flow
out of CV
=
+
Net
Viscous
Work done
on CV
+
Rate of
accumulation
of Energy in
CV
Unsteady
term
Conduction
term
=
+ Convection
VISCOUS WORK
ENERGY EQUATION
19
S
E
= PE +
q
Q
If work done by surface stresses are included
ENTHALPY EQUATION
•
An alternative form of the energy equation is the
total enthalpy equation.
Specific enthalpy h = i + p/
ρ
Total enthalpy h
0
= h + ½ (u
2
+v
2
+w
2
) = E + p/
ρ
TRANSPORT EQUATIONS
CONSERVATION
EQUATION
23
solved by finite volume based CFD programs to calculate the flow pattern
and associated scalar fields.
1-dim, STEADY
CONVECTION-DIFFUSION
U
o
Slug Flow through a long channel of length L
EXACT SOLUTION
•
For small U
o
Pe
0 and large diffusivity,
the solution is T= x (T linear in x )
•
For large U
o
Pe>>0
Φ grows slowly with x and them suddenly rises to Φ
L
over a short
distance close to x=L
Presented on 4/6/2015 at SoME, SASTRA University, Thanjavur
25
1
)
exp(
1
)
exp(
)
(
Pe
Pex
x
NUMERICAL SOLUTION
In terms of Central difference Scheme (CDS)
2
•
Solution shows wiggles, a computational instability
•
oscillation which disappears on fine grids as Pe
c
becomes less than 2
UPWIND SCHEME
Presented on 4/6/2015 at SoME, SASTRA University, Thanjavur
27
2
Solution is stable
Known as First Order Upwind scheme
For large Pe
c
diffusion term is overestimated
OTHER SCHEMES
•
Quest for an optimal (stable as well as accurate)
advection discretization procedure continues
•
Other promising alternatives are higher order
schemes such as Second Order Upwind and QUICK
•
QUICK has been found better as compared to the
FLOW IN PARALLEL
PLATES
Presented on 4/6/2015 at SoME, SASTRA University, Thanjavur
29
Assumptions:
1. Constant heat flux
2. Thermally fully developed flow
3. Steady Flow
m
At the plate surface
h
c
Convective heat transfer Coefficient
Presented on 4/6/2015 at SoME, SASTRA University, Thanjavur
31
Energy balance on CV
dx
x
T
c
m
x
q
m
2
Solving
D
h
=2R
CIRCULAR POISEUILLE
FLOW
OUTLINE
•
Introduction
•
Conduction
•
Convection
Forced Convection
Free Convection
•
Other Models
NATURAL CONVECTION
07/06/2016
35
Hot Surface Upward
Cold Surface Downward
NATURAL CONVECTION
•
The fluid motion is induced by the heat transfer
•
Density and temperature are related; hotter gases
rise…
•
Thermal expansion coefficient is a characteristic
property of fluids
•
The momentum equation must be written in
FREE CONVECTION
Presented on 4/6/2015 at SoME, SASTRA University, Thanjavur
37
Flow over a heated vertical flat plate
Boundary conditions:
Energy equation
Momentum equation
Continuity equation
FREE CONVECTION
•
The momentum and energy equations are coupled by
via the temperature
Typically called Boussinesq fluids…
•
Boussinesq Model: Model treats density as a
constant value in all solved equations, except for the
buoyancy term in the momentum equation
•
the Boussinesq approximation is valid when
(T − T
0
) << 1
OUTLINE
•
Introduction
•
Conduction
•
Convection
Forced Convection
Free Convection
•
Other Models
•
Conclusion
39 of 41
Conjugate Heat Transfer
•
Ability to compute conduction of heat through solids,
coupled with convective heat transfer in fluid
•
In 2D Cartesian coordinates:
W
= wall
•
Properties varies with location and Temperature
Periodic Heat Transfer
•
Used when flow and heat transfer patterns are repeated
Compact heat exchangers
Flow across tube banks
•
Outflow at one periodic boundary is inflow at the other
•
Geometry and boundary conditions repeat in streamwise
direction
Presented on 4/6/2015 at SoME, SASTRA University, Thanjavur
