heat transfer2010 estonia

Where innovation starts

Feedback stabilisation oftwo-dimensional non-uniformpool-boiling states

Rob van Gils1,2 Michel Speetjens2 Henk Nijmeijer1

1 Mechanical Engineering, Dynamics and Control Group2 Mechanical Engineering, Energy Technology Group

Heat Transfer 2010; July 14-16, 2010

2/11

/ dept. mechanical engineering, R.W. van Gils MSc, Heat Transfer 2010; July 14-16, 2010

Motivation

Pool-boiling systemI Heater surface submerged in pool of boiling liquidI Cooling based on boiling heat transfer

Boiling heat transferI Cooling capacities beyond that of conventional methods

schematic pool-boiling system

Controlling the dynamics of pool-boiling systems thus canserve as basis for state-of-the-art cooling schemes

3/11


Introduction to pool-boiling

Pool-boilingI Uniform heat supplyI Non-uniform heat extraction

Boiling modesI Nucleate boilingI Transition boilingI Film boiling

Goal:I Stabilisation of transition boiling

schematic pool-boiling model

Global boiling curve

4/11


Two-dimensional model description

Heater only modelling approachI Heat transfer is modelled by

∂T∂t = κ∇

2T

I Boundary conditions are given by∂T∂x

∣∣x=0,1 = 0

∂T∂y

∣∣∣y=0

= −13(1+ u(t))

∂T∂y

∣∣∣y=D

= −523

qF(TF)

I Output

z(t) = T (t, x, y)

Two-dimensional rectangular heater

Local boiling curve

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Equilibria of the model

I Temperature distribution of equilibria: T∞(x, y)I 3 = D = 0.2,52 = 2 and κ = 3D/|1−52|

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.05

0.1

0.15

0.2

0

0.5

1

1.5

2

2.5

3

x

y

Homogeneous equilibrium in transition boiling

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.05

0.1

0.15

0.2

0

0.5

1

1.5

2

2.5

3

x

y

Mode-1 heterogeneous equilibrium

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.05

0.1

0.15

0.2

0

0.5

1

1.5

2

2.5

3

x

y


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.05

0.1

0.15

0.2

0

0.5

1

1.5

2

2.5

3

x

y


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Equilibria of the model

I Temperature distribution of equilibria: T∞(x, y)I 3 = D = 0.2,52 = 2 and κ = 3D/|1−52|

I TF,∞(x) = T∞(x, y = D)

0 0.2 0.4 0.6 0.8 10

0.5

1

1.5

2

2.5

3

x

TF,∞

(x)

2nd Homogeneous eq.Mode-1 eq.Mode-2 eq.Mode-3 eq.

Fluid heater interface temperature of equilibria

6/11


Linearisation about an equilibrium

I Linearisation: T (x, y, t) = T∞(x, y)+ v(x, y, t)

∂v

∂t= κ∇2v, and

∂v∂x

∣∣x=0,1 = 0

∂v∂y

∣∣∣y=0

= −13

u(t)

∂v∂y

∣∣∣y=D

= −523γ (x)v(x, D, t)

• γ (x) = dqFdTF

∣∣∣TF=TF,∞

7/11


Control Strategy

I Spectral method: v(x, y, t) =∞∑

p=0

∞∑q=0

vpq(t)φq(θ(y)) cos(pπx)

• θ(y) = 2D y − 1

• φq(θ) = cos(q arccos(θ))

I Modal control: u(t) =∞∑

p=0

∞∑q=0

vqp(t)kqp

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Some Results (1/2)

Stabilisation of a heterogeneous equilibriumI System parameters

3 = D = 0.2, 52 = 2, κ = 3D/|1−52|

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.05

0.1

0.15

0.2

0

0.5

1

1.5

2

2.5

3

x

y

Heterogeneous equilibrium

0 0.2 0.4 0.6 0.8 10

0.5

1

1.5

2

2.5

3

xT

F,∞

(x)

Temperature distribution on the fluid heater interface

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Some Results (1/2)


3 = D = 0.2, 52 = 2, κ = 3D/|1−52|

I Controller parameters:

k0,0 = −2, k1,0 = −3

−20 −15 −10 −5 0−3

−2

−1

0

1

2

3

Re(λ)

Im(λ

)

λ1 to λ13

Dominant closed-loop poles

8/11


Some Results (1/2)


3 = D = 0.2, 52 = 2, κ = 3D/|1−52|

I Controller parameters:

k0,0 = −2, k1,0 = −3

0 2 4 6 8 10−8

−6

−4

−2

0

2

4

time (nondimensional)

u(t

)

Input as function of time

0 0.2 0.4 0.6 0.8 10

0.5

1

1.5

2

2.5

3

3.5

4

4.5

Initial profileEquilibriumIntermediate profiles

x

TF(x

)

Evolution of the fluid-heater interface temperature

9/11


Some Results (2/2)

Stabilisation of a homogeneous equilibriumI System parameters

3 = D = 0.8, 52 = 2, κ = 3D/|1−52|

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0

0.5

1

1.5

2

2.5

3

x

y

Homogeneous equilibrium

0 0.2 0.4 0.6 0.8 10

0.5

1

1.5

2

2.5

xT

F,∞

(x)

Temperature distribution on the fluid heater interface

9/11


Some Results (2/2)


3 = D = 0.8, 52 = 2, κ = 3D/|1−52|

I Results from 1D analysis can be used:

k0,0 = −30, k1,0 = −10, k2,0 = 6.6

−25 −20 −15 −10 −5 0−8

−6

−4

−2

0

2

4

6

8

Re(λ)

Im(λ

)

λ1 to λ5

Dominant closed-loop poles

9/11


Some Results (2/2)


3 = D = 0.8, 52 = 2, κ = 3D/|1−52|


k0,0 = −30, k1,0 = −10, k2,0 = 6.6

0 0.5 1 1.5 2−100

−80

−60

−40

−20

0

20

40


u(t

)


0 0.5 10

1

2

3

4

5

Initial profileEquilibriumIntermediate profiles

x

TF(x

)

Evolution of the fluid-heater interface temperature

9/11


Some Results (2/2)


3 = D = 0.8, 52 = 2, κ = 3D/|1−52|


k0,0 = −30, k1,0 = −10, k2,0 = 6.6

0 0.5 1 1.5 2−100

−80

−60

−40

−20

0

20

40


u(t

)


0 0.5 1 1.5 20

0.5

1

1.5

2

2.5

3

3.5

4

4.5


∫T

F(x

,t)d

x

Mean fluid-heater interface temperature as function of time

10/11


Conclusions and future work

ConclusionsI Application of modal control to infinite dimensional systemI Stabilisation of non-uniform transition states

• uniform heat supply• modal control

I 1D results can be used for 2D case

Future workI Implement observerI Scale up to 3DI Verify simulation results with experiments

11/11


Thank you for your attention!

References:

I General pool boiling:• V. Dhir, Boiling heat transfer, Annu Rev Fluid Mech, 30(1), pp. 365 – 401, 1998

I Pool boiling model:• M. Speetjens et al., Steady-state solutions in a nonlinear pool boiling model, Comm

Nonl Sc Numer Sim, 13(8), pp. 1475 – 1494, 2008

I Control strategy:• R. van Gils et al., Feedback stabilisation of a pool boiling system, Int J Heat Mass

Transfer, 53(11), pp. 2393 – 2403, 2010

I Chebyshev-Fourier-cosine expansion/Crank-Nicholson time scheme:• C. Canuto et al., Spectral Methods in Fluid Dynamics, Springer, New York, 1987

I Separation of variables/Picard iteration scheme:• E. Kreyszig, Advanced Engineering Mathematics, Wiley, New York, 1999

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