modeling the heat equation in 3d for grilling a steak
TRANSCRIPT
CE291F Gaetano Andreisek · Michael Nava
MODELING THE HEAT EQUATION IN 3D FOR
GRILLING A STEAK
¡ Use Partial Differential Equations to model the heating properties of a 3D Object in Cartesian Coordinates
¡ Specifically, we wish to model the cooking process of
a piece of steak in three dimensions
OBJECTIVE
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¡ Precise modeling of temperature propagation in a three dimensional object allows us to
§ make predictions § optimize the cooking process to minimize time or energy
consumption § use differential flatness techniques to minimize waste of food
products
MOTIVATION
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Rate of internal energy accumulation =
Flow of energy into the system -
Flow of energy out of the system +
Rate of energy “generation” of some dif ferent source
cp – Specific Heat Capacity [J/(m3K)] k – Thermal Conductivity [W/(m K)] ρ – Material Density [kg/m3]
DERIVATION OF 3D HEAT EQUATION IN CARTESIAN COORDINATES
∂∂x
k ∂u∂x
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∂∂y
k ∂u∂y
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∂∂z
k ∂u∂z
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&'+ u = cpρ
dudt
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¡ Plugging into PDE:
¡ Solution:
SEPARATION OF VARIABLES (SOV)
u x, y, z, t( ) =Φ x, y, z( ) ⋅T t( )
∂2Φ∂x2
+∂2Φ∂y2
+∂2Φ∂z2
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'(T t( ) =
cpρkΦ x, y, z( ) ⋅
T t( )dt
⇔∂2Φ∂x2
+∂2Φ∂y2
+∂2Φ∂z2
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'(1Φ=cpρk⋅T 'T
⇔X"X+Y "Y+Z"Z=cpρk⋅T 'T= −λ 2
X x( ) = A ⋅sin µx( )+B ⋅cos µx( )Y y( ) =C ⋅sin νy( )+D ⋅cos νy( )Z z( ) = E ⋅sin γz( )+F ⋅cos γz( )
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T t( ) =G ⋅e−λ2
kcpρ
t
with :λ 2 = µ 2 +ν 2 +γ 2 5/53
¡ Initial condition (t = 0): ¡ Dirichlet BCs: ¡ Neumann BCs:
¡ Inhomogeneous BCs:
EXAMPLES FOR IC AND BCS
u x, y, z, 0( ) = f x, y, z( )
u 0, y, z, t( ) = u x 0 , y, z, t( ) = 0u x, 0, z, t( ) = u x, y 0 , z, t( ) = 0u x, y, 0, t( ) = u x, y, x 0 t( ) = 0
!
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ux 0, y, z, t( ) = ux x 0 , y, z, t( ) = 0uy x, 0, z, t( ) = uy x, y 0 , z, t( ) = 0uz x, y, 0, t( ) = uz x, y, x 0 t( ) = 0
!
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u 0, y, z, t( ) = ux0u x, 0, z, t( ) = uy0u x, y, 0, t( ) = uz0
!
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u x0, y, z, t( ) = ux1u x, y0, z, t( ) = uy1u x, y, z0, t( ) = uz1
!
"#
$#
D = x, y, z( ) : 0 ≤ x ≤ x0, 0 ≤ y ≤ y0, 0 ≤ z ≤ z0{ }
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¡ Using Sturm-Liouville Method
SOLUTION TO DIRICHLET BCS
u x, y, z, t( ) = Amnpp=1
∞
∑n=1
∞
∑ sin mπ xx0
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'(sin
nπ yy0
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'(sin
pπ zz0
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'(e
−π 2kcpρ
mx02 +
ny02 +
pz02
#
$%%
&
'((t
m=1
∞
∑
Amnp =8
x0y0z0f x, y, z( )sin mπ x
x0
!
"#
$
%&sin
nπ yy0
!
"#
$
%&sin
pπ zz0
!
"#
$
%&
'
()
*
+,
D∫∫∫
Amnp =32 ⋅u0
π 3 ⋅m ⋅n ⋅ pFor : f x, y, z( ) = u0 1−
zz0
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¡ Based on replacing the dif ferentials by algebraic equations (derivatives by differences)
¡ Discrete corollaries:
NUMERICAL APPROACH – FINITE DIFFERENCE METHOD
df x( )dx
≅f x +Δx( )− f x( )
Δx
dudx i+1
2
≅ui+1 −uiΔx
d 2udx2 i
≅
dudx i+1
2
−dudx i−1
2
Δx=
ui+1 −uiΔx
−ui −ui−1Δx
Δx=ui+1 − 2ui +ui−1
Δx( )2
dudx i−1
2
≅ui −ui−1Δx
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¡ For each time step, we represent temperature at § Each corner (8)
PRELIMINARY GEOMETRY - CUBE
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¡ For each time step, we represent temperature at § Each corner (8)
PRELIMINARY GEOMETRY - CUBE
6
2 1
3
4
78
5
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¡ For each time step, we represent temperature at § Each corner (8) § The edges (12)
PRELIMINARY GEOMETRY - CUBE
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¡ For each time step, we represent temperature at § Each corner (8) § The edges (12) § External areas (6) § Internal volume
PRELIMINARY GEOMETRY - CUBE
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MATLAB/Video…
NUMERICAL RESULTS
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¡ Direct analytical solution in MATLAB computationally prohibitive ¡ PDE Toolbox in MATLAB only capable of modeling one and two
dimensional partial dif ferential equations ¡ Finite Element Software like Comsol very expensive ($1,700 for
student version) ¡ Open source modeling software (FEAP) has steep learning curve ¡ Material parameters (Density, Heat Conductivity) and object
dimension change over time
DIFFICULTIES ENCOUNTERED
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¡ Continue to improve scalability and extensibility of GUI § Increase number of input parameters § Implement user-selected views
¡ Develop differential flatness techniques for higher dimensions ¡ Derivation of analytical solution for nonzero BCs and
comparing results to Finite Difference Method ¡ Testing and verification of results based on data gathered
from CE271 project
FUTURE WORK
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¡ Questions?
Disclaimer: No animal was harmed in the making of this project.
THANK YOU.
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