Numerical Modelling in Geosciencesgeo.geoscienze.unipd.it/sites/default/files/Lecture11.pdf ·...
Transcript of Numerical Modelling in Geosciencesgeo.geoscienze.unipd.it/sites/default/files/Lecture11.pdf ·...
Numerical Modelling in Geosciences
Lecture 11 Conservation of heat
Heat conservation equation
Amount of heat required for ΔT (K ) :ΔQ =mcPΔT (Joule)
Such amount of heat is given by the balance of all heat sources and sinks :ΔQ = ΔQint +ΔQA −ΔQB +ΔQC −ΔQD +ΔQE −ΔQF
Ex : ΔQA = qxAΔyΔzΔt
ΔQint = H is the rate of heat generated or consumed by different processes per unit volume (W m−3 )
Equating the above right hand sides and dividing by V = ΔxΔyΔz and Δt :
ρcPDTDt
= −∇⋅q +H
Describes the balance of heat in a continuum and relates temperature changes due to internal heat generation, as well as with advective and conductive heat trasport. Heat is the exchange of thermal energy among parts of the system that are at different temperatures. Temperature is an indicator of the thermal energy content (potential) of a system.
Heat generation and consumption H = Hr +Hr +Hr +Hr (W m−3 )
Hr = radioactive heat production, due to decay of radioactive elements Granite→ 2×10−6 W m−3
Basalt→ 2×10−7 W m−3
Peridotite→ 2×10−8 W m−3
Hs = shear heat production, due to dissipation of mechanical energy during irreversible non− elastic deformation Hs = $σ ij $εij where ij denotes summation 2D example : Hs = $σ xx $εxx + $σ yy $εyy + 2 $σ xy $εxy = 2 $σ xx $εxx + 2 $σ xy $εxy
Ha = adiabatic heat production / consumption, due to adiabatic heating / cooling during
increase / decrease of pressure⇒ Ha = TαDPDt
HL = latent heat production / consumption, due to phase transformations in rocks subjected to changes in pressure and temperatures. During melting HL < 0, During crystallization HL > 0,
Heat conservation equation
ρcPDTDt
= −∇⋅q +H
Fourier 's law : qi = −k∂T∂xi or q = −k∇T
ρcP∂T∂t
+v ⋅∇T
%
&'
(
)*=∇⋅ k∇T( )+H
ρcP∂T∂t
+ vx∂T∂x
+ vy∂T∂y
+ vz∂T∂z
%
&'
(
)*=
∂∂x
k ∂T∂x
%
&'
(
)*+
∂∂y
k ∂T∂y
%
&'
(
)*+
∂∂z
k ∂T∂z
%
&'
(
)*+H
If k = const, no advection and H = 0∂T∂t
=kρcP
ΔT ⇒ ∂T∂t
=κΔT (describes conduction of heat,κ is thermal diffusivitym2s−1)
If DTDt
= 0⇒ ∂T∂t
+v ⋅∇T = 0 (temperature change at Eulerian points due to advection)
If DTDt
=∂T∂t
= 0⇒ −∇⋅ q +H = 0 (used to compute steady− state geotherms→ find analytical solution)
Describes the balance of heat in a continuum and relates temperature changes due to internal heat generation, as well as with advective and conductive heat trasport.
Heat diffusion
tdiff =L2
κ
L = width of the region where the heat is generated
Heat diffusion is described by the Fourier’s law, a constitutive relation stating that the flow of thermal energy along a given direction depends on the temperature gradient and thermal conductivity. Basically, thermal energy flows in order to elimate differences in potentials (temperature) and achieve equilibrium. Diffusion (or conduction) of heat is due to propagation of kinetic energy among microscopic particles, without macroscopic displacement. Characteristic timescale for diffusion of heat depends on the square of the width of the region where heat is produced and is inversely proportional to the material diffusivity
Thermal conductivity It is the property of a material to conduct heat (W/m/°K) Low k à thermal insulation High k à heat sink Heat transport in non-metals is by way of elastic vibrations of the lattice (phonons) k in reality is a tensor à kij, because propagation of elastic vibrational waves depend on crystal structure and are limited by defects Even if isotropic à k=f(P,T,C)
Diopside thermal conductivity
Heat advection Heat advection occurs when there is macroscopic displacement of matter, which exchange places with other parcels of matter at a different temperature, so that internal energy is carried by the flow of matter. In planetary bodies heat advection is due to internal temperature gradients generating buoyancy differences (through volume expansion/contraction). This is better known as thermal convection, occurring when the Rayleigh number:
Ra = gαρ0ΔTD3
µκ> RaC (10
3 −104 )
On Earth, Ra =107
Homework
Read chapter 9 of textbook: Gerya, T. Introduction to numerical geodynamic modelling. Cambridge University Press, 345 pp. (2010)
Solving heat equation: conduction Constant k :
ρcP∂T∂t
= kΔT +H ⇒∂T∂t
=kρcP
ΔT + HρcP
2DExplicit formulation : Δt < Δx2
3κ !!!
FD : T3n = T3
0 +kΔtρcP
T50 − 2T3
0 +T10
Δx2+T40 − 2T3
0 +T20
Δy2%
&'
(
)*+
HρcP
Δt
2DImplicit formulation :
FD : T3n
Δt−
kρcP
T5n − 2T3
n +T1n
Δx2+T4
n − 2T3n +T2
n
Δy2%
&'
(
)*=
T30
Δt+HρcP
No limitation for Δt
Solving heat equation: conduction
General solution for variable grid and kImplicit formulation :
ρcP∂T∂t
= −∇⋅q +H
ρcP∂T∂t
=∇⋅ k∇T( )+H
2D : ρcP∂T∂t
= −∂qx∂x
−∂qy∂y
+H =∂∂x
k ∂T∂x
%
&'
(
)*+
∂∂y
k ∂T∂y
%
&'
(
)*+H
FD : ρ3cP3T3
t+Δt
Δt+ 2 qxA − qxB
Δx1 +Δx2+ 2
qyD − qyCΔy1 +Δy2
= H3 + ρ3cP3T3
t
Δt
Solving heat equation: boundary conditions
Constant temperature :T1 = cnst
No heat flux (insulating or symmetric boundary) :
qx = −k∂T∂x
= 0
T1 −T2 = 0
Constant heat flux :
qx = −k∂T∂x
= cnst
kAT1 −T2Δx
= cnst
Solving heat equation: advection
ρCP
DTDt
= −∇⋅q +H
Changes in temperature for the Eulerian nodes :ΔTi, j = Ti, j
t+Δt −Ti, jt
are interpolated to get the marker temperature change ΔTm :Tm
t+Δt = Tmt +ΔTm
In order to avoid numerical diffusion during advection (see Lecture 10), we can use the Lagrangian formulation of the heat equation and advect temperature with the marker-in-cell-technique. We must interpolate only temperature changes from the Eulerian nodes to the markers to minimize numerical diffusion during such interpolation.
This method, however, while preventing numerical diffusion, does not damp out small (subgrid) scale temperature differences between adjacent markers. In case of strong mixing due to thermal convection, numerical oscillations of the thermal field are produced. These oscillations do not damp out with time as would be the case if physical diffusion was active.
Solving heat equation: advection
1) Changes in temperature for the Eulerian nodes are decomposed :ΔTi, j = ΔTi, j
subgrid +ΔTi, jremaining
2) Calculate subgrid ΔT for markers :
ΔTmsubgrid = Tm(nodes )
t −Tmt( ) 1− exp −d Δt
Δtdiff
#
$%%
&
'((
)
*++
,
-..
Δtdiff = cPmρm
km 2 /Δx2 + 2 /Δy2( )
(local heat diffusion timescale for a given cell)
0 ≤ d ≤1 (dimensionless numerical diffusion coefficient)Tm(nodes )
t ,cPm ,ρm,km are interpolated from Ti, jt ,cPi, j,ρi, j,ki, j
3) Interpolate ΔTmsubgrid to Eulerian nodes to get ΔTi, j
subgrid
4) Compute ΔTi, jremaining = ΔTi, j −ΔTi, j
subgrid
5) Interpolate ΔTi, jremaining to markers to get ΔTm
remaining
6) Finally, compute new marker temperature : Tm(corrected )t = Tm
t + ΔTmsubgrid +ΔTm
remaining
In order to avoid numerical oscillations during advection, we must to introduce a consistent subgrid diffusion operation. We use part of the grid temperature change to apply subgrid temperature diffusion and thus remove non-physical subgrid oscillations.
Homework
Read chapter 10 of textbook: Gerya, T. Introduction to numerical geodynamic modelling. Cambridge University Press, 345 pp. (2010)