Lehrstuhl Informatik V 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
Lehrstuhl Informatik V 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
Lehrstuhl Informatik V 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 · δ T 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
Lehrstuhl Informatik V A Wiremesh Model • consider rectangular plate as fine mesh of wires • compute temperature x i , j at nodes of the mesh x i,j+1 x i−1,j x i,j x i+1,j x i,j−1 h y h x 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
Lehrstuhl Informatik V 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
Lehrstuhl Informatik V 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 � � x i − 1 , j − x i , j � � • from the right: k x i + 1 , j − x i , j � � • from below: k x i , j − 1 − x i , j � � • from above: k x i , j + 1 − x i , 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
Lehrstuhl Informatik V 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 � � x i − 1 , j − x i , j � � • from the right: k x i + 1 , j − x i , j � � • from below: k x i , j − 1 − x i , j � � • from above: k x i , j + 1 − x i , j • equation for steady state: sum over all fluxes = zero: x i , j = 1 � � x i − 1 , j + x i + 1 , j + x i , j − 1 + x i , j + 1 4 for all temperatures x i , 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
Lehrstuhl Informatik V A Wiremesh Model (3) • temperature known at (part of) the boundary; for example x 0 , j = T j models a heated/cooled wall with constant temperature T j at the left boundary. • temperature flux known at (part of) the boundary; for example x i , 0 = x i , 1 models an isolated wall at the lower boundary. • heat sources: temperature given at a certain position i , j : x i , j = T s . • 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
Lehrstuhl Informatik V A Finite Volume Model • object: a rectangular metal plate (again) • model as a collection of small connected rectangular cells h y h x • 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
Lehrstuhl Informatik V 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
Lehrstuhl Informatik V 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 ( T 1 − T 0 ) between the adjacent cells • length h of the edge • e.g.: heat flow across the left edge: q ( left ) � � = k x T i , j − T i − 1 , j h y i , 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 8
Lehrstuhl Informatik V 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 ( T 1 − T 0 ) between the adjacent cells • length h of the edge • e.g.: heat flow across the left edge: q ( left ) � � = k x T i , j − T i − 1 , j h y i , j • heat flow across all edges determines change of heat energy: 1 T i , j = 2 k y h x + 2 k x h y � � � � �� k y h x T i − 1 , j + T i + 1 , j + k x h y T i , j − 1 + T i , 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
Lehrstuhl Informatik V A Finite Volume Model (3) • temperature known in boundary layer cells; for example T 0 , j = T j models a heated/cooled wall with constant temperature T j at the left boundary. • temperature flux known in boundary layer cells; for example q ( bottom ) = 0 i , 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
Lehrstuhl Informatik V A Finite Volume Model (4) • heat sources: consider additional source term F i , j due to • external heating • radiation • F i , j = f i , j h x h y ( f i , j heat flow per area) • equilibrium with source term requires q i , j + F i , j = 0: 1 T i , j = 2 k y h x + 2 k x h y � � � � � � T i − 1 , j + T i + 1 , j + k x h y T i , j − 1 + T i , j + 1 . k y h x − h x h y f i , 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
Lehrstuhl Informatik V Towards a Time Dependent Model • idea: set up an ODE for each cell • simplification: no external heat sources or drains, i.e. f i , 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
Lehrstuhl Informatik V Towards a Time Dependent Model • idea: set up an ODE for each cell • simplification: no external heat sources or drains, i.e. f i , j = 0 • change of temperature per time is proportional to heat flow into the cell (no longer 0): d k x � � dt T i , j ( t ) = T i − 1 , j ( t ) + T i + 1 , j ( t ) − 2 T i , j ( t ) h x k y � � T i , j − 1 ( t ) + T i , j + 1 ( t ) − 2 T i , j ( t ) − h y • 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
Lehrstuhl Informatik V 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
Lehrstuhl Informatik V From Discrete to Contiuous • remember the discrete model: − k x � � f i , j = 2 T i , j − T i − 1 , j − T i + 1 , j h x − k y � � 2 T i , j − T i , j − 1 − T i , j + 1 h y • assumption: heat flow accross edges is proportional to temperature difference q ( left ) � � = k x T i , j − T i − 1 , j h y i , j • in reality: heat flow proportional to temperature gradient T i , j − T i − 1 , j q ( left ) ≈ kh y i , j h x 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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