The present study concerns the feedback stabilisation of the unstable equilibria of a two-dimensional nonlinear pool-boiling system with essentially heterogeneous temperature distributions in the fluid-heater interface. Regulation of such equilibria has great potential for application in, for instance, micro-electronics cooling. Here, as a first step, stabilisation of these equilibria is considered. To this end a control law is implemented that regulates the heat supply to the heater as a function of the Fourier-Chebyshev modes of its internal temperature distribution. These modes are intimately related to the physical eigenmodes of the system and thus admit robust and efficient regulation on the basis of the natural composition of the temperature field. Key to this modal-control strategy and its application for controller design are two equivalent and interchangeable PDE and state-space forms of the linearised pool-boiling model. Derivation of these forms is a central theme of this study. Performance of modal controllers thus designed is demonstrated and analysed by simulations of the nonlinear closed-loop system.

1. Feedback stabilisation of
non-uniform pool boiling
states
Rob van Gils1,2 Michel Speetjens2 Henk Nijmeijer1
1 Mechanical Engineering, Dynamics and Control Group
2 Mechanical Engineering, Energy Technology Group
March 31, 2010
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2. Introduction 2/12
Pool-boiling system
Heater surface submerged in pool of boiling liquid
Cooling based on boiling heat transfer
Boiling heat transfer
Cooling capacities beyond that of conventional methods
schematic pool-boiling system
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3. Introduction 2/12
Pool-boiling system
Heater surface submerged in pool of boiling liquid
Cooling based on boiling heat transfer
Boiling heat transfer
Cooling capacities beyond that of conventional methods can
Controlling the dynamics of pool-boiling systems thus
serve as basis for state-of-the-art cooling schemes
schematic pool-boiling system
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4. Presentation outline 3/12
Introduction to pool boiling
Two-dimensional (2D) pool boiling model description
Results from analysis of one-dimensional (1D) simplification
Approach
Some results of 2D analysis
Conclusions and recommendations
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5. Introduction to pool-boiling 4/12
Pool-boiling
Uniform heat supply
Non-uniform and nonlinear heat extraction
Boiling modes
Nucleate boiling: efficient schematic pool-boiling model
Transition boiling: highly unstable
Film boiling: collapse of cooling capacity
Goal:
Stabilisation of transition boiling
Global boiling curve
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6. Two-dimensional model description 5/12
Heater only modelling approach
Heat transfer is modelled by
∂T
∂t
=κ 2
T
Boundary conditions are given by
∂T Two-dimensional rectangular heater
∂ x x=0,1
=0
∂T
∂ y y=0
= − 1 (1 + u(t))
∂T
∂ y y=D
=− 2
q F (TF )
Output
z(t) = T (t, x, y)
Local boiling curve
Experiments confirm qualitative validity of model
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7. Equilibria / Linearisation of the model 6/12
Equilibria are of the form
∞
cosh(nπ y) D−y
T∞ (x, y) = Tn cos(nπ x) +
n=0
cosh(nπ D)
Tn given by
∞
TF (x) := T (x, y = D, t) = Tn cos(nπ x)
n=0
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8. Equilibria / Linearisation of the model 6/12
Equilibria are approximated by
N
cosh(nπ y) D−y
T∞ (x, y) ≈ Tn cos(nπ x) +
n=0
cosh(nπ D)
Linearisation: T (x, y, t) = T∞ (x, y) + v(x, y, t)
∂v
∂ x x=0,1
=0
∂v ∂v 1
= κ 2 v, and ∂ y y=0 = − u(t)
∂t ∂v
∂y
= − 2 γ (x)v(x, D, t)
y=D
dq F
where γ (x) = dTF T =T
F F,∞
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9. Equilibria / Linearisation of the model 6/12
Homogeneous equilibria are given by
D−y
T∞ (x, y) = T0 + , Tn = 0, for n > 0
Local boiling curve
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10. Results from one-dimensional analysis 7/12
Approach:
One-dimensional analysis, only x-independent equilibria
Spatial discretisation heater domain
Modal control: stabilisation by feedback of spectral modes
Disadvantage: size of obtained ODE-system
Therefore, a method to apply modal control to the infinite dimensional
system is devised
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11. Overview of approach 8/12
Continuous System Continuous state feedback Quasi pole-placement
D 1
v(x, t) = κ 2 v(x, t)
˙
u(t) = v(x, t)g(x)dxdy, Use
with
0 0 - characteristic equation
∂v - pole trajectory plots
∂ x x=0,1 = 0 is considered, where g(x) ≈
∂v to obtain satisfactory
∂ y y=0 = − 1 u(t) N ,K closed-loop dynamics
∂v 2 γv gqp pq (θ )wC (θ ) cos( pπ x)
∂ y y=D = − F
p,q=0
Discretised System k Simulations
gqp = D qp
¯
2 Cq C p 4.5
N ,K 4
Initial profile
Equilibrium
Intermediate profiles
v= vnk pn (θ ) cos(kπ x), 3.5
3
n,k=0 Discrete state feedback
TF (x)
2.5
2
with the Chebyshev- u = Kv, 1.5
Fourier spectrum 1
is considered, where 0.5
T
v = v0,0 · · · v N ,K 0
0 0.2 0.4
x
0.6 0.8 1
K = k0,0 · · · k N ,0 k0,1 · · · k N ,K
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12. Overview of approach 8/12
Continuous System Continuous state feedback Quasi pole-placement
D 1
v(x, t) = κ 2 v(x, t)
˙
u(t) = v(x, t)g(x)dxdy, Use
with
0 0 - characteristic equation
∂v - pole trajectory plots
∂ x x=0,1 = 0 is considered, where g(x) ≈
∂v to obtain satisfactory
∂ y y=0 = − 1 u(t) N ,K closed-loop dynamics
∂v 2 γv gqp pq (θ )wC (θ ) cos( pπ x)
∂ y y=D = − F
p,q=0
Discretised System k Simulations
gqp = D qp
¯
2 Cq C p 4.5
N ,K 4
Initial profile
Equilibrium
Intermediate profiles
v= vnk pn (θ ) cos(kπ x), 3.5
3
n,k=0 Discrete state feedback
TF (x)
2.5
2
with the Chebyshev- u = Kv, 1.5
Fourier spectrum 1
is considered, where 0.5
T
v = v0,0 · · · v N ,K 0
0 0.2 0.4
x
0.6 0.8 1
K = k0,0 · · · k N ,0 k0,1 · · · k N ,K
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13. Overview of approach 8/12
Continuous System Continuous state feedback Quasi pole-placement
D 1
v(x, t) = κ 2 v(x, t)
˙
u(t) = v(x, t)g(x)dxdy, Use
with
0 0 - characteristic equation
∂v - pole trajectory plots
∂ x x=0,1 = 0 is considered, where g(x) ≈
∂v to obtain satisfactory
∂ y y=0 = − 1 u(t) N ,K closed-loop dynamics
∂v 2 γv gqp pq (θ )wC (θ ) cos( pπ x)
∂ y y=D = − F
p,q=0
Discretised System k Simulations
gqp = D qp
¯
2 Cq C p 4.5
N ,K 4
Initial profile
Equilibrium
Intermediate profiles
v= vnk pn (θ ) cos(kπ x), 3.5
3
n,k=0 Discrete state feedback
TF (x)
2.5
2
with the Chebyshev- u = Kv, 1.5
Fourier spectrum 1
is considered, where 0.5
T
v = v0,0 · · · v N ,K 0
0 0.2 0.4
x
0.6 0.8 1
K = k0,0 · · · k N ,0 k0,1 · · · k N ,K
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14. Overview of approach 8/12
Continuous System Continuous state feedback Quasi pole-placement
D 1
v(x, t) = κ 2 v(x, t)
˙
u(t) = v(x, t)g(x)dxdy, Use
with
0 0 - characteristic equation
∂v - pole trajectory plots
∂ x x=0,1 = 0 is considered, where g(x) ≈
∂v to obtain satisfactory
∂ y y=0 = − 1 u(t) N ,K closed-loop dynamics
∂v 2 γv gqp pq (θ )wC (θ ) cos( pπ x)
∂ y y=D = − F
p,q=0
Discretised System k Simulations
gqp = D qp
¯
2 Cq C p 4.5
N ,K 4
Initial profile
Equilibrium
Intermediate profiles
v= vnk pn (θ ) cos(kπ x), 3.5
3
n,k=0 Discrete state feedback
TF (x)
2.5
2
with the Chebyshev- u = Kv, 1.5
Fourier spectrum 1
is considered, where 0.5
T
v = v0,0 · · · v N ,K 0
0 0.2 0.4
x
0.6 0.8 1
K = k0,0 · · · k N ,0 k0,1 · · · k N ,K
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15. Overview of approach 8/12
Continuous System Continuous state feedback Quasi pole-placement
D 1
v(x, t) = κ 2 v(x, t)
˙
u(t) = v(x, t)g(x)dxdy, Use
with
0 0 - characteristic equation
∂v - pole trajectory plots
∂ x x=0,1 = 0 is considered, where g(x) ≈
∂v to obtain satisfactory
∂ y y=0 = − 1 u(t) N ,K closed-loop dynamics
∂v 2 γv gqp pq (θ )wC (θ ) cos( pπ x)
∂ y y=D = − F
p,q=0
Discretised System k Simulations
gqp = D qp
¯
2 Cq C p 4.5
N ,K 4
Initial profile
Equilibrium
Intermediate profiles
v= vnk pn (θ ) cos(kπ x), 3.5
3
n,k=0 Discrete state feedback
TF (x)
2.5
2
with the Chebyshev- u = Kv, 1.5
Fourier spectrum 1
is considered, where 0.5
T
v = v0,0 · · · v N ,K 0
0 0.2 0.4
x
0.6 0.8 1
K = k0,0 · · · k N ,0 k0,1 · · · k N ,K
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16. Overview of approach 8/12
Continuous System Continuous state feedback Quasi pole-placement
D 1
v(x, t) = κ 2 v(x, t)
˙
u(t) = v(x, t)g(x)dxdy, Use
with
0 0 - characteristic equation
∂v - pole trajectory plots
∂ x x=0,1 = 0 is considered, where g(x) ≈
∂v to obtain satisfactory
∂ y y=0 = − 1 u(t) N ,K closed-loop dynamics
∂v 2 γv gqp pq (θ )wC (θ ) cos( pπ x)
∂ y y=D = − F
p,q=0
Discretised System k Simulations
gqp = D qp
¯
2 Cq C p 4.5
N ,K 4
Initial profile
Equilibrium
Intermediate profiles
v= vnk pn (θ ) cos(kπ x), 3.5
3
n,k=0 Discrete state feedback
TF (x)
2.5
2
with the Chebyshev- u = Kv, 1.5
Fourier spectrum 1
is considered, where 0.5
T
v = v0,0 · · · v N ,K 0
0 0.2 0.4
x
0.6 0.8 1
K = k0,0 · · · k N ,0 k0,1 · · · k N ,K
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17. Overview of approach 8/12
Continuous System Continuous state feedback Quasi pole-placement
D 1
v(x, t) = κ 2 v(x, t)
˙
u(t) = v(x, t)g(x)dxdy, Use
with
0 0 - characteristic equation
∂v - pole trajectory plots
∂ x x=0,1 = 0 is considered, where g(x) ≈
∂v to obtain satisfactory
∂ y y=0 = − 1 u(t) N ,K closed-loop dynamics
∂v 2 γv gqp pq (θ )wC (θ ) cos( pπ x)
∂ y y=D = − F
p,q=0
Discretised System k Simulations
gqp = D qp
¯
2 Cq C p 4.5
N ,K 4
Initial profile
Equilibrium
Intermediate profiles
v= vnk pn (θ ) cos(kπ x), 3.5
3
n,k=0 Discrete state feedback
TF (x)
2.5
2
with the Chebyshev- u = Kv, 1.5
Fourier spectrum 1
is considered, where 0.5
T
v = v0,0 · · · v N ,K 0
0 0.2 0.4
x
0.6 0.8 1
K = k0,0 · · · k N ,0 k0,1 · · · k N ,K
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18. Some Results (1/2) 9/12
Stabilisation of a heterogeneous equilibrium
System parameters
= D = 0.2, 2 = 2, D = κ|1 − 2|
3
4 2.5
3 2
T∞ (x, y)
TF,∞ (x)
2
1.5
1
1
0
0.2
1 0.5
0.1
0.5
0
y 0 0
x
0 0.2 0.4 0.6 0.8 1
x
Heterogeneous equilibrium Temperature distribution on the fluid heater interface
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19. Some Results (1/2) 9/12
Stabilisation of a heterogeneous equilibrium
System parameters
= D = 0.2, 2 = 2, D = κ|1 − 2|
Controller parameters:
k0,0 = −2, k1,0 = −3
λ1 to λ13
3
2
1
Im(λ)
0
−1
−2
−3
−20 −15 −10 −5 0
Re(λ)
Dominant closed-loop poles
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20. Some Results (1/2) 9/12
Stabilisation of a heterogeneous equilibrium
System parameters
= D = 0.2, 2 = 2, D = κ|1 − 2|
Controller parameters: 4.5
Initial profile
4 Equilibrium
k0,0 = −2, k1,0 = −3 Intermediate profiles
3.5
3
4
TF (x)
2.5
2
2
0
1.5
u(t)
−2
−4 1
−6 0.5
−8
0 2 4 6 8 10
0
0 0.2 0.4 0.6 0.8 1
time (nondimensional) x
Input as function of time Evolution of the fluid-heater interface temperature
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21. Some Results (2/2) 10/12
Stabilisation of a homogeneous equilibrium
System parameters
= D = 0.8, 2 = 2, D = κ|1 − 2|
2.5
2.5
2
T∞ (x, y)
2
TF,∞ (x)
1.5
1.5
1
1
0 0.5
0.2
0.4 0
0.6 0.4 0.2
0.8 0.6 0
1 0 0.2 0.4 0.6 0.8 1
y x x
Homogeneous equilibrium Temperature distribution on the fluid heater interface
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22. Some Results (2/2) 10/12
Stabilisation of a homogeneous equilibrium
System parameters
= D = 0.8, 2 = 2, D = κ|1 − 2|
Results from 1D analysis can be used:
k0,0 = −30, k1,0 = −10, k2,0 = 6.6
λ1 to λ5
8
6
4
2
Im(λ)
0
−2
−4
−6
−8
−25 −20 −15 −10 −5 0
Re(λ)
Dominant closed-loop poles
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23. Some Results (2/2) 10/12
Stabilisation of a homogeneous equilibrium
System parameters
= D = 0.8, 2 = 2, D = κ|1 − 2|
Results from 1D analysis can be used:
k0,0 = −30, k1,0 = −10, k2,0 = 6.6
40
5
Initial profile
20
4 Equilibrium
0 Intermediate profiles
3
TF (x)
−20
u(t)
−40
2
−60
−80
1
−100
0 0.5 1 1.5 2 0
0 0.5 1
time (nondimensional) x
Input as function of time Evolution of the fluid-heater interface temperature
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24. Some Results (2/2) 10/12
Stabilisation of a homogeneous equilibrium
System parameters
= D = 0.8, 2 = 2, D = κ|1 − 2|
Results from 1D analysis can be used:
k0,0 = −30, k1,0 = −10, k2,0 = 6.6
40 3
20 2.5
2
0
0 TF (x)dx
1.5
−20
u(t)
1
−40
0.5
1
−60
0
−80 −0.5
−100 −1
0 0.5 1 1.5 2 0 0.5 1 1.5 2
time (nondimensional) time (nondimensional)
Input as function of time Mean fluid-heater interface temperature as function of time
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25. Conclusions and future work 11/12
Conclusions
Application of modal control to infinite dimensional system
Stabilisation of non-uniform transition states
• uniform heat supply
• modal control
1D results can be used for 2D case
Future work
Implement observer
Scale up to 3D
Verify simulation results with experiments
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26. Questions? 12/12
Thank you for your attention!
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