Heat Transport
Temperature of a wolf pup
Clemson Hydro
Goblet filled with coffee and ice water
Radiating steel bar
Thermal plume, Lake Michigan
Solar radiation
Temperature distribution
Schlieren image of natural convection.
Note laminar-turbulent transition
Multiphase flow
http://www.geog.ucsb.edu/~jeff/115a/remote_sensing/thermal/thermalirinfo.html
Clemson Hydro
Concepts
• Storage
• Source
• Flux
Clemson Hydro
Heat Storage
Latent heat of
vaporization
Internal
Energy
[E/M];
[J/kg]
Latent heat
of fusion
solid
Sensible heat
liquid
vapo r
Temperature
Clemson Hydro
Heat Storage as Sensible Heat
Internal
Energy
(J/kg)
Sensible heat
cp=specific heat capacity = Energy/(Mass Temp)
Change of sensible heat
 E 
 E = c p T
 M  
 : temperature
On a per volume basis
Cpr=Energy/(Vol Temp)
Temperature
c p rT
 E M
 M  L3 
Clemson Hydro
Specific Heat Capacity
Water:
Ice:
Air:
CO2:
kJ/(kg K)
4.18
2.11
1.00
0.84
Gold:
Hydrogen:
0.13
14 (max)
Span factor of 100
Water in upper 90%
Clemson Hydro
Heat Storage as Latent Heat
Water
Internal
Energy
(J/kg)
Melting water:
Heating, 0-100oC:
Vaporing water:
334
420
2260
kJ/kg
kJ/kg
kJ/kg
solid
liquid
vapor
Temperature
Clemson Hydro
Heat Sources
Heat generated internally
 E 
 L3T 
Friction: fluids, solids, energy dissipation
Chemical reaction: Exo, Endothermic
Radioactive: Fission, fusion
Biological: Metabolism
Dielectric: Microwaves, oven
Joule: Electrical resistance, toaster
Clemson Hydro
Types of heat transfer
Conduction: Static media, Diffusion
Convection: Fluid moving, advection
Forced;
Radiation:
Free or Natural
Emit and adsorb EM, no media
Thermal flux
E
Power
J
W
=
;
= 2
2
2
2
LT
L
ms m
Clemson Hydro
Radiation
Heat transfer flux by radiation
“Black” body absorbs all incoming
radiation and is a perfect emitter.
T1
T2
e =  T14
Stefan’s Law
q hR =  (T  T )
4
1
4
2
 = Stefan-Boltzman constant
5.67x10-8 W/(m2 K4)
 = emissivity (0-1); ~1: black; ~0: shiny
http://www.thermoworks.com/emissivity_table.html
Clemson Hydro
Thermal Conduction
dT
qhx =  K h
dx
qhx =  K hT
Fourier’s Law, energy flux
 E   E 
 L2T  =  TL L 
J
W
 E  P
Kh = 
= =
= o
o

 TL   L  ms C m C
Thermal conductivity
Clemson Hydro
Thermal conductivity
W/(moC)
Vacuum:
0.001
Air:
0.02
Water:
0.6
Ice:
2
Steel:
10
Copper:
400
Diamond: 1000
http://web.mit.edu/lienhard/www/ahttv202.pdf
Clemson Hydro
Convection
Energy flux:
A = qTc p r
L3f  E M f
= 2
2
3
LcT  M L f
LcT
E
Forced: Fluid flow driven by processes other than thermal, pump etc.
Free or Natural: Fluid flow by thermally induced density differences
Clemson Hydro
Convection
Energy flux:
A = qTc p r
L3f  E M f
= 2
2
3
LcT  M L f
LcT
E
Forced: Fluid flow driven by processes other than thermal, pump etc.
Free or Natural: Fluid flow by thermally induced density differences
Clemson Hydro
Convective heat transfer —fluidsolid
Tf
b
Ts
Kh
qh =
(Ts  T f ) = h (Ts  T f )
b
E
E
=
;
2
2
L T L T
P
W 
 E
h: 2
= 2 ; 2 
 L T L  m K 
Clemson Hydro
Convective Heat Transfer Coefficient
http://web.mit.edu/lienhard/www/ahttv202.pdf
Clemson Hydro
https://docs.google.com/viewer?url=http%3A%2F%2Fwww.bakker.org%2Fdartmouth06%2Fengs150%2F13-heat.pdf
Clemson Hydro
Governing Equation
Conservation of Heat Energy
c=
Storage
E
Lc 3
c =cprnT
c
 
=S
t
= D +A
Advective Flux
A = c p rTq
Diffusive Flux (Fick’s Law)
D =- K hT
Governing
 E M  L3f   E

=

3
2 
2 
 M  L f LcT   LcT 
  ( DT )    vT 
Dh =- DhT
Dispersive Flux
Source
c cp r nT
=
t
t
3
E M Lf 
M  L3f L3c
D=
S = Sh
  (  K h  Dh  T )    qc p rT 
S=
cp r nT
t
T
=S
t
K h  Dhyd
cp r n
Sh
cp r n
= Sh
Clemson Hydro
Simulation of Heat Transport
  ( DT )    vT 
T
=S
t
T = N1
Dirichlet, Specify T
n  qh = 0
Neumann, Insulating, no flux
n  qh = N 2
Specified flux
n  qh = h(Text  T )
Convective cooling
n  qh =  (Text4  T 4 )
Surface to external radiation
n  qh =
Kh
(Text  T )
b
Cauchy-type boundary. Thin, insulating layer
Clemson Hydro
Parameters
cp heat capacity
r density
Kh thermal conductivity
q fluid flux
n
porosity
Dhyd hydraulic dispersion
Dthermal
T
  ( DT )    vT 
=S
t
3
3 2
M

L
Kh
E
f
f Lc
=
= thermal diffusivity =
cp r n
LcT E M f L3f
Dh ,disp =
Dhyd
cp r n
= thermal dispersion
 E M  L3f L3c    
S=
= 3
= 
3 
cp r n  TLc E M L f   T 
Sh
Clemson Hydro
Coupled effects
•
•
•
•
•
•
Chemical reaction rate
Phase change
Material properties
Thermal expansion
Solid-Fluid in porous media
Biological growth
Clemson Hydro
Temperature and Chemical Rxn
• Reaction Rate Constant Function
• Arrhenius eq
k = Ae

EA
RT
EA: activation energy for reaction
R: gas constant
T: temperature
A: Max rate term
Clemson Hydro
Including phase change
  (  K h  Dh  T )    qc p rT 
Internal
Energy
(J/m3)
solid
liquid
cp r nT
t
= Sh
vapor
Temperature
Area under curve =
latent heat
Cp(E/(kg3))
Temperature (
Clemson Hydro
Fluid properties
Viscosity
Density
Thermal
Conductivity
Cp(E/(kg3))
Temperature (
Clemson Hydro
Thermal expansion
Volumetric
Linear
Clemson Hydro
Solid Phase in Porous Media
Heat loss by fluid
dEs , f
dt
dEs , f
Vc dt
Tf
Ts
Heat balance between fluid and solid
= Qh ,out = qh A = Ah (T f  Ts )
=
Qh ,out
Vc
= qh
A A
= h (T f  Ts )
Vc Vc
dEs , f
=
dEs , s
dt
dt
d Es , f
d Es , s
=
dt Vc
dt Vc
Es , f
Es , s
 Es , f M f L3f  
= c p f r f nT f


3
3
M

L
L

f
c
 f
 Es , s M s L3s  
= c p s r s  n  1 Ts


3
3
 M s Ls Lc 
Clemson Hydro
Clemson Hydro
Material Properties
https://docs.google.com/viewer?url=http%3A%2F%2Fwww.bakker.org%2Fdartmouth06%2Fengs150%2F13-heat.pdf
Clemson Hydro
Heat Transfer
by Radiation and Convection
Clemson Hydro
Clemson Hydro
Conceptual Model
Radiation and convection
Radiation and convection
Clemson Hydro
Clemson Hydro
Clemson Hydro
Installation of a U-tube in a
borehole
Two pipes from a U-tube
heat exchanger in a borehole
U-tube heat exchanger
Array of U-tube heat exchangers
Clemson Hydro
Clemson Hydro
Free Convection
Rayleigh-Bernard Convection Cells
Clemson Hydro
Convection in the Earth and Atm
Clemson Hydro
Experiment
cooling at the top
Simulated lake
http://www.sciencedirect.com/science/article/pii/S0894177707001409
Clemson Hydro
Rayleigh number
 g Td 3c p r 2
Ra =
Kh
 : thermal expansion
g: gravity
T : temperature difference
Onset of natural
convection depends on Ra
For a layer with a free top
surface: Racrit=1100
d : layer thickness
c p : heat capacity
r : fluid density
 : dynamic viscosity
K h : thermal conductivity
Clemson Hydro
Coefficient of Thermal Expansion
1 L
L =
Lo T
1  V 
v = 

Vo  T  P
1  r 
v =  
r o  T  P
Clemson Hydro
Effect of varying temperature
on mixing in surface water
Clemson Hydro