Scientific Computing I Module 6: Heat Transfer Discrete and - - PowerPoint PPT Presentation

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Scientific Computing I

Module 6: Heat Transfer – Discrete and Contiuous Models

Miriam Mehl based on Slides by Michael Bader (Winter 09/10)

Winter 2011/2012

Miriam Mehl based on Slides by Michael Bader (Winter 09/10): Scientific Computing I Module 6: Heat Transfer – Discrete and Contiuous Models, Winter 2011/2012 1

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Part I: Discrete Models

Motivation: Heat Transfer Wiremesh Model A Finite Volume Model Time Dependent Model

Miriam Mehl based on Slides by Michael Bader (Winter 09/10): Scientific Computing I Module 6: Heat Transfer – Discrete and Contiuous Models, Winter 2011/2012 2

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Motivation: Heat Transfer

• objective: compute the temperature distribution of some object
• under certain prerequisites:
• temperature at object boundaries given
• heat sources
• material parameters
• observation from physical experiments:

q ≈ k · δT heat flow proportional to temperature differences

Miriam Mehl based on Slides by Michael Bader (Winter 09/10): Scientific Computing I Module 6: Heat Transfer – Discrete and Contiuous Models, Winter 2011/2012 3

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A Wiremesh Model

• consider rectangular plate as fine mesh of wires
• compute temperature xi,j at nodes of the mesh

xi,j xi−1,j xi+1,j xi,j+1 xi,j−1 hx hy

Miriam Mehl based on Slides by Michael Bader (Winter 09/10): Scientific Computing I Module 6: Heat Transfer – Discrete and Contiuous Models, Winter 2011/2012 4

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A Wiremesh Model (2)

• model assumption: temperatures in equilibrium at every mesh

node

• What is the equilibrium? → steady state

Miriam Mehl based on Slides by Michael Bader (Winter 09/10): Scientific Computing I Module 6: Heat Transfer – Discrete and Contiuous Models, Winter 2011/2012 5

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A Wiremesh Model (2)

• model assumption: temperatures in equilibrium at every mesh

node

• What is the equilibrium? → steady state
• incoming temperature fluxes at point i,j via the four wires:
• from the left: k
• xi−1,j − xi,j
• from the right: k
• xi+1,j − xi,j
• from below: k
• xi,j−1 − xi,j
• from above: k
• xi,j+1 − xi,j
• Miriam Mehl based on Slides by Michael Bader (Winter 09/10): Scientific Computing I

Module 6: Heat Transfer – Discrete and Contiuous Models, Winter 2011/2012 5

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A Wiremesh Model (2)

• model assumption: temperatures in equilibrium at every mesh

node

• What is the equilibrium? → steady state
• incoming temperature fluxes at point i,j via the four wires:
• from the left: k
• xi−1,j − xi,j
• from the right: k
• xi+1,j − xi,j
• from below: k
• xi,j−1 − xi,j
• from above: k
• xi,j+1 − xi,j
• equation for steady state: sum over all fluxes = zero:

xi,j = 1 4

• xi−1,j + xi+1,j + xi,j−1 + xi,j+1
• for all temperatures xi,j.

Miriam Mehl based on Slides by Michael Bader (Winter 09/10): Scientific Computing I Module 6: Heat Transfer – Discrete and Contiuous Models, Winter 2011/2012 5

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A Wiremesh Model (3)

• temperature known at (part of) the boundary; for example

x0,j = Tj models a heated/cooled wall with constant temperature Tj at the left boundary.

• temperature flux known at (part of) the boundary; for example

xi,0 = xi,1 models an isolated wall at the lower boundary.

• heat sources: temperature given at a certain position i, j:

xi,j = Ts.

• task: solve system of linear equations

Miriam Mehl based on Slides by Michael Bader (Winter 09/10): Scientific Computing I Module 6: Heat Transfer – Discrete and Contiuous Models, Winter 2011/2012 6

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A Finite Volume Model

• object: a rectangular metal plate (again)
• model as a collection of small connected rectangular cells

hx hy

• examine the heat flow across the cell edges

Miriam Mehl based on Slides by Michael Bader (Winter 09/10): Scientific Computing I Module 6: Heat Transfer – Discrete and Contiuous Models, Winter 2011/2012 7

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A Finite Volume Model (2)

• model assumption: temperatures in equilibrium in every grid cell
• What is the equilibrium? → steady state

Miriam Mehl based on Slides by Michael Bader (Winter 09/10): Scientific Computing I Module 6: Heat Transfer – Discrete and Contiuous Models, Winter 2011/2012 8

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A Finite Volume Model (2)

• model assumption: temperatures in equilibrium in every grid cell
• What is the equilibrium? → steady state
• heat flow across a given edge is proportional to
• temperature difference (T1 − T0) between the adjacent cells
• length h of the edge
• e.g.: heat flow across the left edge:

q(left)

i,j

= kx

• Ti,j − Ti−1,j
• hy

Miriam Mehl based on Slides by Michael Bader (Winter 09/10): Scientific Computing I Module 6: Heat Transfer – Discrete and Contiuous Models, Winter 2011/2012 8

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A Finite Volume Model (2)

• model assumption: temperatures in equilibrium in every grid cell
• What is the equilibrium? → steady state
• heat flow across a given edge is proportional to
• temperature difference (T1 − T0) between the adjacent cells
• length h of the edge
• e.g.: heat flow across the left edge:

q(left)

i,j

= kx

• Ti,j − Ti−1,j
• hy
• heat flow across all edges determines change of heat energy:

Ti,j = 1 2kyhx + 2kxhy

• kyhx
• Ti−1,j + Ti+1,j
• + kxhy
• Ti,j−1 + Ti,j+1
• .

Miriam Mehl based on Slides by Michael Bader (Winter 09/10): Scientific Computing I Module 6: Heat Transfer – Discrete and Contiuous Models, Winter 2011/2012 8

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A Finite Volume Model (3)

• temperature known in boundary layer cells; for example

T0,j = Tj models a heated/cooled wall with constant temperature Tj at the left boundary.

• temperature flux known in boundary layer cells; for example

q(bottom)

i,0

= 0 models an isolated wall at the lower boundary.

Miriam Mehl based on Slides by Michael Bader (Winter 09/10): Scientific Computing I Module 6: Heat Transfer – Discrete and Contiuous Models, Winter 2011/2012 9

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A Finite Volume Model (4)

• heat sources: consider additional source term Fi,j due to
• external heating
• radiation
• Fi,j = fi,jhxhy (fi,j heat flow per area)
• equilibrium with source term requires qi,j + Fi,j = 0:

Ti,j = 1 2kyhx + 2kxhy

• kyhx
• Ti−1,j + Ti+1,j
• + kxhy
• Ti,j−1 + Ti,j+1
• − hxhyfi,j
• .
• again, system of linear equations

Miriam Mehl based on Slides by Michael Bader (Winter 09/10): Scientific Computing I Module 6: Heat Transfer – Discrete and Contiuous Models, Winter 2011/2012 10

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Towards a Time Dependent Model

• idea: set up an ODE for each cell
• simplification: no external heat sources or drains, i.e. fi,j = 0
• change of temperature per time is proportional to heat flow into

the cell (no longer 0):

Miriam Mehl based on Slides by Michael Bader (Winter 09/10): Scientific Computing I Module 6: Heat Transfer – Discrete and Contiuous Models, Winter 2011/2012 11

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Towards a Time Dependent Model

• idea: set up an ODE for each cell
• simplification: no external heat sources or drains, i.e. fi,j = 0
• change of temperature per time is proportional to heat flow into

the cell (no longer 0): d dt Ti,j(t) = kx hx

• Ti−1,j(t) + Ti+1,j(t) − 2Ti,j(t)
• −

ky hy

• Ti,j−1(t) + Ti,j+1(t) − 2Ti,j(t)
• solve a system of ODEs

Miriam Mehl based on Slides by Michael Bader (Winter 09/10): Scientific Computing I Module 6: Heat Transfer – Discrete and Contiuous Models, Winter 2011/2012 11

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Part II: A Continuous Model – The Heat Equation

From Discrete to Contiuous Variants of the Heat Equation Boundary and Initial Conditions

Miriam Mehl based on Slides by Michael Bader (Winter 09/10): Scientific Computing I Module 6: Heat Transfer – Discrete and Contiuous Models, Winter 2011/2012 12

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From Discrete to Contiuous

• remember the discrete model:

fi,j = −kx hx

• 2Ti,j − Ti−1,j − Ti+1,j
• −ky

hy

• 2Ti,j − Ti,j−1 − Ti,j+1
• assumption: heat flow accross edges is proportional to

temperature difference q(left)

i,j

= kx

• Ti,j − Ti−1,j
• hy
• in reality: heat flow proportional to temperature gradient

q(left)

i,j

≈ khy Ti,j − Ti−1,j hx

Miriam Mehl based on Slides by Michael Bader (Winter 09/10): Scientific Computing I Module 6: Heat Transfer – Discrete and Contiuous Models, Winter 2011/2012 13

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From Discrete to Contiuous (2)

• replace kx by k/hx, ky by k/hy, and get:

fi,j = − k h2

x

• 2Ti,j − Ti−1,j − Ti+1,j
• − k

h2

y

• 2Ti,j − Ti,j−1 − Ti,j+1
• consider arbitrarily small cells: hx, hy → 0:

fi,j = −k ∂2T ∂x2

• i,j

− k ∂2T ∂y2

• i,j
• leads to a partial differential equation (PDE):

−k ∂2T(x, y) ∂x2 + ∂2T(x, y) ∂y2

• = f(x, y)

Miriam Mehl based on Slides by Michael Bader (Winter 09/10): Scientific Computing I Module 6: Heat Transfer – Discrete and Contiuous Models, Winter 2011/2012 14

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From Discrete to Contiuous (3)

• assumption: T is not constant over time
• change of heat energy (per time) is a result of
• transfer of heat energy across the volume’s surface D’s

surface k ∂2T(x, y) ∂x2 + ∂2T(x, y) ∂y2

• heat sources and drains (external influences)

f(x, y)

• This results in the partial differential equation (PDE)

Tt = k∆T + f, where ∆T = ∂2T

∂x2 + ∂2T ∂y2 .

Miriam Mehl based on Slides by Michael Bader (Winter 09/10): Scientific Computing I Module 6: Heat Transfer – Discrete and Contiuous Models, Winter 2011/2012 15

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Variants of the Heat Equation

• open question: how does the parameter k relate to the physical

reality? k = ¯ k ρc , where c is the specific heat capacity, ρ the density, and ¯ k the heat conductivity of the considered material.

• more common scaling of the source term f:

f = 1 ρc q.

Miriam Mehl based on Slides by Michael Bader (Winter 09/10): Scientific Computing I Module 6: Heat Transfer – Discrete and Contiuous Models, Winter 2011/2012 16

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Variants of the Heat Equation (2)

Different scenarios:

• vanishing external influence, q = 0:

Tt = k∆T alternative notation ∂T ∂t = k · ∂2T ∂x2 + ∂2T ∂y2 + ∂2T ∂z2

• equilibrium solution, Tt = 0:

0 = k∆T + q ρc − → −∆T = q ρck “Poisson’s Equation”

Miriam Mehl based on Slides by Michael Bader (Winter 09/10): Scientific Computing I Module 6: Heat Transfer – Discrete and Contiuous Models, Winter 2011/2012 17

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Boundary Conditions

Dirichlet boundary conditions:

• fix T on (part of) the boundary

T(x, y, z) = ϕ(x, y, z) Neumann boundary conditions:

• fix T’s normal derivative on (part of) the boundary:

∂T ∂n (x, y, z) = ϕ(x, y, z)

• special case: insulation

∂T ∂n (x, y, z) = 0

Miriam Mehl based on Slides by Michael Bader (Winter 09/10): Scientific Computing I Module 6: Heat Transfer – Discrete and Contiuous Models, Winter 2011/2012 18