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3/12/15
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Introduction to Heat Transfer in Soils and Other Materials
ME 7710 Spring 2015
Instructor: Pardyjak
Surface/Skin Temperature
• Ts - The temperature at the air-soil interface. For an “ideal” surface which varies in time in response to energy fluxes at the surface – Depends on:
• Radiation Balance • Surface exchange processes (e.g. turbulence) • Vegetative cover • Thermal properties of the subsurface
– Difficult to Measure (very large temperature gradients near the surface both in the air & soil) • Extrapolate air/soil temps • Radiometer – uses
4~ sL TR εσ−↑
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Diurnal Soil & Air Temperatures
Atmosphere
Soil
Fig from Stull, 1988 An Introduction to Boundary Layer Meteorology
Surface/Skin Temperature • Diurnal Range – In dry desert ~ 40-50 oC – Surface & subsurface moisture moderate range • Increased evaporation from the surface • Increased heat capacity (c) & conductivity of the soil (k)
– Wet soils may dry changing the temperature response
– Vegetation moderates diurnal range • Intercepts incoming solar – lower surface temps during
the day • Intercepts outgoing longwave • Enhanced latent heat flux due to evapotranspiration
(ET) • Increased Turbulence
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Sub-surface Soil Temperature • Much easier to measure – thermocouple • Amplitude of the temperature fluctuations
decrease exponentially with depth • Depends on – – Latitude – Time of year – Net radiation – Soil texture (porosity) and moisture content – Ground cover – Surface weather conditions
Thermal Properties of Soil • Specific Heat – c (J kg-1 K-1) – the amount of heat absorbed by
a material to raise the temperature of a unit mass of material by 1o K
• Thermal Conductivity – k (W m-1 K-1) – material property; the ability of a material to conduct heat
• Thermal Diffusivity – αh (m2 s-1) Ratio of thermal conductivity to heat capacity
( ) 2
2
zTkcT
t ∂
∂=
∂
∂ρ
1D Thermal Conduction
2
2
zT
tT
h ∂
∂=
∂
∂α
Ck
ck
h ==ρ
α
( )zHcT
tG
∂
∂−=
∂
∂ρ z
TkHG ∂
∂−=
Soil heat capacity
Fourier’s conduc2on law T(z)
z
HG
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Solu2ons
• Analy2cal – mul2ple methods • Numerical (e.g.) – e.g. finite difference • Force Restore – 2 Layer Slab Model (See Stull Ch. 7, Blackadar, 1976)
Diurnal & Annual Soil Temperature Temporal Variability
Diurnal Wave
Annual Wave
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Soil Heat Transfer • 1D Thermal Conduction
( )
( ) ( ) ( )
( )∫
∫
∫
=
=
=
=
=
=
∂
∂+=
∂
∂==−=
⎭⎬⎫
⎩⎨⎧
∂
∂−=
∂
∂
Dz
zDG
Dz
z
Dz
z
dzCTt
HH
dzCTt
DzHzH
dzzHCT
t
0
0
0
0
T(z)
z
Reference depth
HG
Surface Heat Flux Storage
Governing Parameters
• Thermal conduc2vity -‐ k • Heat Capacity -‐ Cs • Thermal Diffusivity – α (some2me κ) • Thermal AdmiXance -‐ μ
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Governing Parameters
• Thermal conduc2vity – k (W m-1 K-1) – Def. - the ability of a material to conduct heat – Depends on: • Soil particles • Porosity • Moisture content
Governing Parameters
• Heat Capacity – Cs = ρ c (J m-3 K-1) – c specific heat of the soil (J kg-1 K-1) – Relates to the ability of a material to store heat – Def. The amount of heat (J) necessary to increase a
unit volume (m3) of a substance by 1 K. – Water (~5 J m-3 K-1) has a very high heat capacity,
air is quite low – Depends on porosity, mineral content, organic
content, air, etc.
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Governing Parameters
• Thermal Diffusivity – α = k/Cs (m2 s-1) – Controls the speed at which temperature waves
move through the soil & the depth of thermal influence of an active surface
– Water (~5 x 10^6 J m-3 K-1) has a very high heat capacity, air is quite low
Let’s Look at example Data from Sage Brush at DPG
Governing Parameters
• Thermal AdmiXance – Surface Property (not a “soil property”)
• μ = (kCs)1/2 (J m-2 s-1/2 K-1) – Def. The ability of a surface to accept or release
heat – High μ – metals – low μ – wood – High μ materials feel cooler to the touch even though they have the same surface temperature
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Thermal AdmiXance
• See MATERHORN surface temperature video • k = thermal conduc2vity ~ 0.9 W m-‐1 K-‐1
• α = thermal diffusivity ~ 0.4 x 10-‐6 m2s-‐1 • C = rho*c = heat capacity ~ 2.2 x 10^6 • μs = (kCs)1/2 thermal admiXance ~ 1407 J m^-‐2 s^-‐1/2 K^-‐1
Typical Values
From Oke, 1988
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Effect of Soil Moisture on Thermal Proper2es
From Oke, 1988
Comparison between the WRF-‐simulated and Data from a tower in Oklahoma City
D01
D03 D04
D05
Ver2cal momentum fluxes observed by crane sonics (points), and simulated by WRF (lines)
−0.5 −0.4 −0.3 −0.2 −0.1 0 0.1 0.20
10
20
30
40
50
60
70
80
90
vetical momentum flux u’w’, v’w’ [m2/s2]
Heig
h [m
]
Vertical momentum fluxes at the location of crane sonics for IOP 2 (07.02.2003 16:00UTC)
u’w’ sonicsu’w’ WRFv’w’ sonicsv’w’ WRF
0 0.05 0.1 0.15 0.2 0.250
10
20
30
40
50
60
70
80
90
vertical turbulent heat flux w’T’ [m K/s]
Heig
ht [m
]
w’T’ at the location of crane sonics for IOP 2 (07.02.2003 16:00UTC)
w’T’ sonicsw’T’ WRF
Ver2cal heat flux observed by crane sonics (points), and simulated by WRF (lines)
From Kochanski et al 2014
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Varia2on of Fluxes in Urban Areas
From Kastner-‐Klein, P., & Rotach, M. W. (2004). Mean Flow and Turbulence Characteris2cs in an Urban Roughness Sublayer. Boundary-‐Layer Meteorology, 111(1), 55–84.
Varia2on of Fluxes in Urban Areas
From Kastner-‐Klein, P., & Rotach, M. W. (2004). Mean Flow and Turbulence Characteris2cs in an Urban Roughness Sublayer. Boundary-‐Layer Meteorology, 111(1), 55–84.
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Hysteresis
Net all-wave Radiation
( )dzCTt
HS ρ∫ ∂∂
=Δ
Urban Areas
Grimmond & Oke 1999
Increasing time
More Energy is transferred to the “urban fabric” in the morning - Asymmetry
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Urban Areas
Grimmond & Oke 1999
