© Fluent Inc. 8/10/2015G1 Fluids Review TRN-1998-004 Heat Transfer.
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Transcript of © Fluent Inc. 8/10/2015G1 Fluids Review TRN-1998-004 Heat Transfer.
© Fluent Inc. 04/19/23G1
Fluids ReviewTRN-1998-004
Heat Transfer
© Fluent Inc. 04/19/23G2
Fluids ReviewTRN-1998-004
Outline
Introduction Modes of heat transfer Typical design problems Coupling of fluid flow and heat transfer Conduction Convection Radiation
© Fluent Inc. 04/19/23G3
Fluids ReviewTRN-1998-004
Introduction
Heat transfer is the study of thermal energy (heat) flows Heat always flows from “hot” to “cold” Examples are ubiquitous:
heat flows in the body home heating/cooling systems refrigerators, ovens, other appliances automobiles, power plants, the sun, etc.
© Fluent Inc. 04/19/23G4
Fluids ReviewTRN-1998-004
Modes of Heat Transfer
Conduction - diffusion of heat due to temperature gradient Convection - when heat is carried away by moving fluid Radiation - emission of energy by electromagnetic waves
qconvection
qconduction
qradiation
© Fluent Inc. 04/19/23G5
Fluids ReviewTRN-1998-004
Typical Design Problems
To determine: overall heat transfer coefficient - e.g., for a car radiator highest (or lowest) temperature in a system - e.g., in a gas turbine temperature distribution (related to thermal stress) - e.g., in the walls of a
spacecraft
temperature response in time dependent heating/cooling problems - e.g., how long does it take to cool down a case of soda?
© Fluent Inc. 04/19/23G6
Fluids ReviewTRN-1998-004
Heat Transfer and Fluid Flow
As a fluid moves, it carries heat with it -- this is called convection Thus, heat transfer can be tightly coupled to the fluid flow solution Additionally:
The rate of heat transfer is a strong function of fluid velocity Fluid properties may be strong functions of temperature (e.g., air)
© Fluent Inc. 04/19/23G7
Fluids ReviewTRN-1998-004
Conduction Heat Transfer
Conduction is the transfer of heat by molecular interaction In a gas, molecular velocity depends on temperature
hot, energetic molecules collide with neighbors, increasing their speed In solids, the molecules and the lattice structure vibrate
© Fluent Inc. 04/19/23G8
Fluids ReviewTRN-1998-004
Fourier’s Law
“heat flux is proportional to temperature gradient”
where k = thermal conductivity in general, k = k(x,y,z,T,…)
y
T
x
TkTkq
A
Q
hot wall cold wall
dx
dT
1
temperature profile
x
heat conduction in a slab
units for q are W/m2
© Fluent Inc. 04/19/23G9
Fluids ReviewTRN-1998-004
Generalized Heat Diffusion Equation
If we perform a heat balance on a small volume of material…
… we get:
qTkt
Tc
2
c
k
thermal diffusivity
Theat conductionin
heat conductionoutq
heat generation
rate of changeof temperature
heat cond.in/out
heatgeneration
© Fluent Inc. 04/19/23G10
Fluids ReviewTRN-1998-004
Boundary Conditions
Heat transfer boundary conditions generally come in three types:
T = 300Kspecified temperature
Dirichlet condition
q = 20 W/m2
specified heat fluxNeumann condition
q = h(Tamb-Tbody)external heat transfercoefficientRobin condition
Tbody
© Fluent Inc. 04/19/23G11
Fluids ReviewTRN-1998-004
Conduction Example
Compute the heat transfer through the wall of a home:
shinglesk=0.15 W/m2-K
sheathingk=0.15 W/m2-K
fiberglas insulationk=0.004 W/m2-K
2x6 studk=0.15 W/m2-K
sheetrockk=0.4 W/m2-K
Tout = 20° F Tout = 68° F
Although slight, you can see the “thermal
bridging” effect through the studs
© Fluent Inc. 04/19/23G12
Fluids ReviewTRN-1998-004
Convection Heat Transfer
Convection is movement of heat with a fluid E.g., when cold air sweeps past a warm body, it draws away warm air
near the body and replaces it with cold air
often, we want to know the heat transfer coefficient, h (next page)
flow over a heated block
© Fluent Inc. 04/19/23G13
Fluids ReviewTRN-1998-004
Newton’s Law of Cooling
Tbody
T
ThTThq body )(
average heat transfer coefficient (W/m2-K)h
q
© Fluent Inc. 04/19/23G14
Fluids ReviewTRN-1998-004
Heat Transfer Coefficient
h is not a constant, but h = h(T) Three types of convection:
Natural convection fluid moves due to buoyancy
Forced convection flow is induced by external means
Boiling convection body is hot enough to boil liquid
3
1
4
1
, ThTh
consth
2Th
Typical values of h:
4 - 4,000 W/m2-K
80 - 75,000
300 - 900,000
Thot Tcold
Thot
Tcold
Tcold
Thot
© Fluent Inc. 04/19/23G15
Fluids ReviewTRN-1998-004
Looking in more detail...
Just as there is a viscous boundary layer in the velocity distribution, there is also a thermal boundary layer
t
wT
UT ,
y
)( yT
velocity boundarylayer edge
thermal boundarylayer edge
© Fluent Inc. 04/19/23G16
Fluids ReviewTRN-1998-004
Nusselt Number
Equate the heat conducted from the wall to the same heat transfer in convective terms:
Define dimensionless quantities:
Then rearrange to get:
)(
TThy
Tk wf
L
yy
TT
TTT
w
w
Nu
f
w
w
k
hL
Ly
TTTT
Nusselt number
“dimensionless heat
transfer coefficient”
conductivityof the fluid
© Fluent Inc. 04/19/23G17
Fluids ReviewTRN-1998-004
Energy Equation
Generalize the heat conduction equation to include effect of fluid motion:
Assumes incompressible fluid, no shear heating, constant properties, negligible changes in kinetic and potential energy
Can now solve for temperature distribution in boundary layer Then calculate h using Fourier’s law:
qTkTt
Tc
2u
0
yww y
T
TT
k
TT
qh
From calculatedtemperaturedistribution
© Fluent Inc. 04/19/23G18
Fluids ReviewTRN-1998-004
Correlations for Heat Transfer Coefficient
As an alternative, can use correlations to obtain h E.g., heat transfer from a flat plate in laminar flow:
where the Prandtl number is defined as:
Typical values are: Pr = 0.01 for liquid metals Pr = 0.7 for most gases Pr = 6 for water at room temperature
333.05.0 PrRe332.0Nu xx
k
cPr
ydiffusivit thermal
ydiffusivit momentum
© Fluent Inc. 04/19/23G19
Fluids ReviewTRN-1998-004
Convection Examples
Developing flow in a pipe (constant wall temperature)
T wT T wT T wT
TwT
x
bulk fluid temperature
heat flux from wall
TwT
© Fluent Inc. 04/19/23G20
Fluids ReviewTRN-1998-004
Convection Examples
Natural convection (from a heated vertical plate)
u
TTw
gravity
As the fluid is warmed by the plate, its density decreases and a buoyant force arises which induces flow in the vertical direction. The force is equal to:
,T
)(T
g)(
The dimensionless group that governs natural convection is the Rayleigh number:
3
RaTLg
© Fluent Inc. 04/19/23G21
Fluids ReviewTRN-1998-004
Radiation Heat Transfer
Thermal radiation is emission of energy as electromagnetic waves Intensity depends on body temperature and surface characteristics Important mode of heat transfer at high temperatures Can also be important in natural convection problems Examples:
toaster, grill, broiler fireplace sunshine
© Fluent Inc. 04/19/23G22
Fluids ReviewTRN-1998-004
Surface Characteristics
1
q W/m2 (incident energy flux) q (reflected)
q (transmitted)
q (absorbed)
absorptance
reflectance
transmittance
translucent slab
© Fluent Inc. 04/19/23G23
Fluids ReviewTRN-1998-004
Black Body Radiation
A “black body”: is a model of a perfect radiator absorbs all energy that reaches it; reflects nothing therefore = 1, = = 0
The energy emitted by a black body is the theoretical maximum:
This is Stefan-Boltzmann law; is the Stefan-Boltzmann constant (5.6697e-8 W/m2-K4)
4Tq
© Fluent Inc. 04/19/23G24
Fluids ReviewTRN-1998-004
“Real” Bodies
Real bodies will emit less radiation than a black body:
Example: radiation from a small body to its surroundings both the body and its surroundings emit thermal radiation the net heat transfer will be from the hotter to the colder
4Tq emissivity (between 0 and 1)
)( 44 TTAQ wnet
T
q
wTA
wqnetQ
© Fluent Inc. 04/19/23G25
Fluids ReviewTRN-1998-004
When is radiation important?
Radiation exchange is significant in high temperature problems: e.g., combustion
Radiation properties can be strong functions of chemical composition, especially CO2, H2O
Radiation heat exchange is difficult solve (except for simple configurations) — we must rely on computational methods
© Fluent Inc. 04/19/23G26
Fluids ReviewTRN-1998-004
Heat Transfer — Summary
Heat transfer is the study of thermal energy (heat) flows: conduction convection radiation
The fluid flow and heat transfer problems can be tightly coupled through the convection term in the energy equation when properties (, ) are dependent on temperature
While analytical solutions exist for some simple problems, we must rely on computational methods to solve most industrially relevant applications Can I go back to
sleep now?