Numerical Modelling in Geosciencesgeo.geoscienze.unipd.it/sites/default/files/Lecture11.pdf ·...

14
Numerical Modelling in Geosciences Lecture 11 Conservation of heat

Transcript of Numerical Modelling in Geosciencesgeo.geoscienze.unipd.it/sites/default/files/Lecture11.pdf ·...

Page 1: Numerical Modelling in Geosciencesgeo.geoscienze.unipd.it/sites/default/files/Lecture11.pdf · Numerical Modelling in Geosciences Lecture 11 Conservation of heat . Heat conservation

Numerical Modelling in Geosciences

Lecture 11 Conservation of heat

Page 2: Numerical Modelling in Geosciencesgeo.geoscienze.unipd.it/sites/default/files/Lecture11.pdf · Numerical Modelling in Geosciences Lecture 11 Conservation of heat . Heat conservation

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.

Page 3: Numerical Modelling in Geosciencesgeo.geoscienze.unipd.it/sites/default/files/Lecture11.pdf · Numerical Modelling in Geosciences Lecture 11 Conservation of heat . Heat conservation

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,

Page 4: Numerical Modelling in Geosciencesgeo.geoscienze.unipd.it/sites/default/files/Lecture11.pdf · Numerical Modelling in Geosciences Lecture 11 Conservation of heat . Heat conservation

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.

Page 5: Numerical Modelling in Geosciencesgeo.geoscienze.unipd.it/sites/default/files/Lecture11.pdf · Numerical Modelling in Geosciences Lecture 11 Conservation of heat . Heat conservation

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

Page 6: Numerical Modelling in Geosciencesgeo.geoscienze.unipd.it/sites/default/files/Lecture11.pdf · Numerical Modelling in Geosciences Lecture 11 Conservation of heat . Heat conservation

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

Page 7: Numerical Modelling in Geosciencesgeo.geoscienze.unipd.it/sites/default/files/Lecture11.pdf · Numerical Modelling in Geosciences Lecture 11 Conservation of heat . Heat conservation

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

Page 8: Numerical Modelling in Geosciencesgeo.geoscienze.unipd.it/sites/default/files/Lecture11.pdf · Numerical Modelling in Geosciences Lecture 11 Conservation of heat . Heat conservation

Homework

Read chapter 9 of textbook: Gerya, T. Introduction to numerical geodynamic modelling. Cambridge University Press, 345 pp. (2010)

Page 9: Numerical Modelling in Geosciencesgeo.geoscienze.unipd.it/sites/default/files/Lecture11.pdf · Numerical Modelling in Geosciences Lecture 11 Conservation of heat . Heat conservation

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

Page 10: Numerical Modelling in Geosciencesgeo.geoscienze.unipd.it/sites/default/files/Lecture11.pdf · Numerical Modelling in Geosciences Lecture 11 Conservation of heat . Heat conservation

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

Page 11: Numerical Modelling in Geosciencesgeo.geoscienze.unipd.it/sites/default/files/Lecture11.pdf · Numerical Modelling in Geosciences Lecture 11 Conservation of heat . Heat conservation

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

Page 12: Numerical Modelling in Geosciencesgeo.geoscienze.unipd.it/sites/default/files/Lecture11.pdf · Numerical Modelling in Geosciences Lecture 11 Conservation of heat . Heat conservation

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.

Page 13: Numerical Modelling in Geosciencesgeo.geoscienze.unipd.it/sites/default/files/Lecture11.pdf · Numerical Modelling in Geosciences Lecture 11 Conservation of heat . Heat conservation

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.

Page 14: Numerical Modelling in Geosciencesgeo.geoscienze.unipd.it/sites/default/files/Lecture11.pdf · Numerical Modelling in Geosciences Lecture 11 Conservation of heat . Heat conservation

Homework

Read chapter 10 of textbook: Gerya, T. Introduction to numerical geodynamic modelling. Cambridge University Press, 345 pp. (2010)