Stability Analysis and Maldistribution Control of

Two-Phase Flow in Parallel Evaporating Channels

TieJun Zhang∗,a,b, John T. Wena,b,c, Agung Juliusc, Yoav Pelesb, MichaelK. Jensenb

aCenter for Automation Technologies and Systems,Rensselaer Polytechnic Institute, 110 8th Street, Troy, NY 12180, USA

bDepartment of Mechanical, Aerospace & Nuclear Engineering,Rensselaer Polytechnic Institute, 110 8th Street, Troy, NY 12180, USA

cDepartment of Electrical, Computer & System Engineering,Rensselaer Polytechnic Institute, 110 8th Street, Troy, NY 12180, USA

Abstract

Parallel channel heat transfer system is an attractive means to enhanceheat transfer by increasing the total surface area in the heat exchanger. Uni-form distribution of flow in the channels is an important operation attributein order to avoid channel dryout and the subsequent hot spots and possibledevice failure. For the system operating in the single-phase regime, uniformflow distribution is a stable equilibrium. However, in the two-phase boil-ing regime, stable equilibrium bifurcates – the uniform distribution becomesunstable, and new stable or unstable non-uniform distributions, or maldis-tributions, equilibria emerge. As a result, parallel channel heat exchangerstypically operate in the single phase regime to avoid such “maldistributioninstability.” This paper presents a stability analysis and active flow andtemperature control of a parallel channel evaporator in a pumped two-phaseboiling heat transfer loop. We show that for identical channels, the system isuncontrollable with the pump alone and unobservable from the overall flowrate measurement. A conventional solution involves introducing additionalvalve-induced pressure drop, but the drawback is a resulting higher pumpingpower. We present a new, somewhat surprising, result that if the channelcharacteristics are all distinct, the system actually becomes controllable from

∗Corresponding author: TieJun Zhang, Tel.: +1-518-276-2125, Fax: +1-518-276-4897,E-mail: tjzhrpi@gmail.com

Preprint submitted to Int. J. Heat Mass Transfer August 8, 2011

the pump and observable from the total loop flow rate. For non-identicalparallel systems, we show a controller design approach and demonstrate itsefficacy through a simulation example.

Key words: Flow Instability; Flow Distribution; Multiple Evaporators;Parallel Channel; Two-Phase Flow

Nomenclature

A cross-sectional area (m2)L length (m)N channel numberP pressure (Pa)T Temperature (◦C)V volume (m3)∆P pressure drop (Pa)m mass flow rate (g/s)s pressure drop slope (Pa s/g)t time (s)

Subscripts

0 initiali i-th channelin inlete exitw wallr restrictorD demandS supply

Superscripts

i i-th channel

1. Introduction

Evaporating two-phase flow has been widely used in power generation,thermal management, chemical and other industries due to its excellent heat

2

transfer performance [1, 2, 3, 4]. For example, in next-generation solar ther-mal power plants, an array of parallel parabolic trough solar collectors areused to generate steam directly instead of using oil as the secondary heatingmedium [5, 6]. The commercialization of large-scale direct steam genera-tion technologies is hindered by the lack of understanding of two-phase flowbehavior within the absorbing pipes and the fear of possible occurrence ofcritical heat flux, nonuniform heating, instabilities and uneven flow rate dis-tribution [7, 8]. Cooling is also a critical problem for high heat-flux electronicsas heat is fast becoming the performance bottleneck [3, 4, 9]. Microchannelflow boiling is a promising approach as it utilizes both the latent heat ofevaporation and the larger area to volume ratio in microchannels to enhancethe heat transfer performance [1, 10, 11]. Much effort has been made todesign boiling microchannel heat exchangers with a large number of parallelchannels. Such a configuration is attractive as it maximizes the heat transferwith minimal pump demand [3, 4, 9, 12].

Two-phase heat exchangers pose unique challenges as they are prone tovarious flow boiling instabilities [13], such as the Ledinegg flow excursion,parallel-channel flow maldistribution, pressure-drop and density-wave flowoscillations. The Ledinegg instability arises when the flow boiling system op-erates in the two-phase negative-slope region of the demand pressure curve,where the demand pressure drop decreases with increasing mass flow rate[12, 13, 14, 15]. Slight changes in the supply pressure drop will trigger asudden flow excursion to either a subcooled or a superheated operating con-dition. Pressure-drop oscillations could occur when there exists large up-stream compressibility of the flow boiling system, pressure-drop oscillations[1, 13]. For microchannel systems, flow has been observed to redistributeamong the parallel channels in a nonuniform fashion both spatially and tem-porally. This “maldistribution instability” could cause large uncontrolledmicrochannel wall temperature difference. Different flow conditions were ex-perimentally demonstrated and found to be heavily dependent on the priorstate of the microchannels [16]. Even for conventional-scale two-phase heatexchangers, flow maldistribution also has dramatic negative consequenceson thermal and mechanical performance [17, 18]. Attributed to local flowinstabilities and maldistribution, the critical heat flux condition could beprematurely initiated before the advantage of phase change is realized [19].Local channel dryout would severely compromise heat transfer performanceand could lead to severe safety problems.

In a parallel-channel flow boiling system, both pressure drop oscillation

3

and non-uniform flow distributions take place under certain operating condi-tions. Experimental, analytical, and simulation studies have been reported toestablish stability criteria [7, 20, 21, 22, 23, 24] With the simplified parallel-pipe flow analysis by [25], a control procedure was proposed in [8] to regu-late the individual pipe flow rates with a desired pipe exit quality througha set of inlet control valves. To suppress the two-phase flow instability inmicrochannels, the methods of inlet control valve [8], inlet restrictors [26]and inlet seed bubble generators [27] were proposed. Introducing these ad-ditional control devices leads to either larger pressure loss or higher powerconsumption, and increased fabrication cost. Active two-phase flow controloffers the potential as an alternative to mitigate microchannel flow instabil-ities. In [28], we proposed a dynamic model-based pump control approachto suppress pressure-drop flow oscillations in microchannel boiling systems,where the inlet flow from the pump can be regulated to compensate for theupstream compressibility.

In this paper, we address the two-phase flow maldistribution instabilityamong multiple parallel evaporating channels. Our focus is on using thepump to regulate the channel flow rather than regulating the pressure of theindividual channels as in the past. A dual problem is to estimate the individ-ual channel flow rate based on the measurement of the total flow rate. Forheat exchangers with identical channels, the equilibria in the single-phaseregimes (subcooled liquid or superheated vapor) for a given total flow rateare unique and globally stable. In the two-phase regime, the single-phaseequilibria bifurcate to multiple equilibria, with the uniform flow equilibriumbecoming unstable while the non-uniform flow equilibria are (locally) stable.Furthermore, the hope of using active feedback control to stabilize the un-stable equilibrium is dashed as we show that the linearized system about theunstable equilibrium is neither controllable nor observable, implying that nolinear time invariant stabilizing controller (or channel flow observer) exists.We are motivated by the observation that balancing multiple parallel invertedpendulums (i.e., multiple pendulums on a common cart) is controllable fromthe common base force except when the pendulums are identical. Similarly,the system is observable from the base motion except when the pendulumsare identical. This also follows as a consequence of systems with symmetryrequiring more than one control parameter [29]. Indeed, when the channelsare non-identical, meaning that the slope of the pressure drop versus themass flow curve at the unstable equilibrium is different for each channel, thesystem becomes both controllable from the pump and observable from the

4

total mass flow rate. Once the controllability and observability properties areestablished, we develop a stabilizing feedback controller based on the modelpredictive control approach using the first-principle flow dynamics model.Simulation results have demonstrated that the open-loop unbalanced flow inparallel evaporating channels can be effectively regulated to the desired oper-ating point with balanced two-phase flow distribution, resulting in improvedtwo-phase heat transfer performance and lower channel surface temperatures.

2. Parallel Channel Flow Stability

2.1. Stability Analysis of Parallel Channels with Inlet Pressure Control

Consider N parallel boiling channels with the same inlet and exit man-ifolds, as depicted in Figure 1. The ith channel has length L and cross-sectional area Ai.

Figure 1: Schematic of parallel boiling channels

The overall mass flow rate through the boiling channels is m =∑N

i=1 mi,where mi is the mass flow rate in the ith channel. Applying a momentumbalance, we obtain [30]

L

Ai

dmi

dt= ∆PS −∆P i

D = Pin − Pe −∆P iD(mi) (1)

where ∆Ps, the supply pressure drop is the difference between the inlet pres-sure, Pin (controlled by the pump), and the exit pressure, Pe, ∆P i

D is thedemand pressure drop of the i-th channel, with a typical shape as in Fig-ure 2. We assume that exit pressure Pe and temperature are maintainedat a constant value by an external temperature controller immersed in thedownstream reservoir.

5

0 50 100 150 200 2500

250

500

750

1000

1250

1500

Mass flux G (kg/m2−s)

Pre

ssur

e dr

op Δ

PD (

Pa)

OFISuperheated

Subcooled

Two−Phase

Figure 2: Typical two-phase flow characteristics of a boiling channel under constant heatflux (OFI: onset of flow instability [15]; G = m/A: mass flux)

If Pin is a constant, Pin = P ∗in, then the equilibria for channel i are givenby the intersection between the horizontal line, ∆P ∗s = P ∗in − Pe, and thedemand pressure drop curve in Figure 2:

∆P ∗s = P ∗in − Pe = ∆P iD(m∗i ). (2)

For a given pressure drop, there may be up to three possible equilibria withflow rates corresponding to the subcooled, two-phase and superheated flows.For high and low pressure drops, there is only one equilibrium, correspondingto the subcooled and superheated flows, respectively.

With Pin as the control variable, we can analyze the stability of (1). Foropen loop system, i.e., Pin is a constant, the stability of the equilibria isdetermined by the linearized system

L

Ai

dδmi

dt= −d∆P i

D(m∗i )

dmi

δmi. (3)

Clearly, the equilibrium is stable if and only if the slope of the demand curveis positive. In the two-phase region, the slope of pressure-drop versus flowcurve is negative, resulting in what is known as the Ledinegg Instability

6

[13, 15]. It should be noted that the pressure-drop slope depends on manyflow system parameters (i.e. heat flux, inlet subcooling) as discussed in [15];in this paper we focus on the mass flow effect.

A common solution to the two-phase instability problem is to add extrapressure drop through valving at each channel inlet. This increases the powerrequirement on the pump and adds implementation complexity. The problemis even more acute for microchannel evaporators where there may be morethan 100 channels. If the pump alone is used to regulate the channel flows,then the linearized channel dynamics become coupled:

L

Ai

dδmi

dt=

N∑i=1

d(Pin(m∗))

dmδmi −

d∆P iD(m∗i )

dmi

δmi. (4)

Writing the equation in the vector form, we have

dδmv

dt=AD

TDsinL

δmv −AS

Lδmv (5)

where

mv =

m1...mN

, DT=

1...1

, A = diag{A1, . . . , AN},

sin =d(Pin(m∗))

dm, S = diag{s1, . . . , sN}, si :=

d∆P iD(m∗i )

dmi

. (6)

As shown in Appendix A, the linearized system is stabilizable by choosingan appropriate sin if and only if

D⊥TSD

⊥> 0 (7)

where > 0 means that the matrix is positive definite, and D⊥ ∈ RN×N−1

is the orthogonal complement of D, i.e., the rank N − 1 matrix such that

DD⊥

= 0. The matrix D⊥

is non-unique. In particular, we may choose it sothat the columns are orthonormal, i.e.,

D⊥TD⊥

= IN−1. (8)

For N = 1, D⊥ is a null matrix, so (7) is satisfied. For N = 2, the conditionbecomes s1 + s2 > 0 (same as [20] where the condition was shown for Ai =

7

A). For N = 3, using the principal minor test for positive definiteness, thecondition becomes, s1 + s2 > 0 and s1s2 + s2s3 + s1s3 > 0. Conditions forother N ’s may be similarly calculated.

Note that a particular choice of sin that will guarantee that the linearizedsystem is stable under (7) is sin = −∞ (i.e., constant inlet flow rate). From

(30) of Appendix A, a = −sinN2 + DSDT

will be +∞, and a − fS1fT is

therefore always positive. Our stability criterion (7) is consistent with thecriterion reported in [21], which is a special case of our result.

2.2. Stability Analysis of Parallel Channels with Upstream Compressibility

To include the effect of flow compressibility, we include in the model anupstream surge tank [28], which is motivated by the compressible two-phaseflow in a drum-boiler [2, 31], as shown in Figure 3. The governing equationsbecome:

dPindt

=P 2in

ρlP0V0

(min −

N∑i=1

mi

)(9)

dmi

dt=AiL

(Pin − Pe −∆P i

D(mi))

(10)

where min is now the input variable (which is regulated by the pump). Fora given m∗in, the equilibrium state (P ∗in, m

∗1, . . . , m

∗N) is given by

N∑i=1

m∗i = m∗in (11)

∆P iD(m∗i ) = ∆P j

D(m∗j), for all i, j ∈ [1, N ] (12)

P ∗in = Pe + ∆P iD(m∗i ), for all i ∈ [1, N ]. (13)

The equilibrium condition may be viewed in terms of the flow rate in eachdemand curve corresponding to the same channel pressure drop as illustratedin Figure 4.

The linearized system about the equilibrium isdδPin

dtdδm1

dt...

dδmN

dt

=

0 −a . . . −ab1 −b1s1 . . . 0...

. . . 0bN 0 . . . −bNsN

︸ ︷︷ ︸

A

δPinδm1

δm2...

δmN

+

a0...0

︸ ︷︷ ︸B

δmin (14)

8

Figure 3: Schematic of multiple parallel boiling channels with a upstream surge tank

0 5 10 15 20 25 30 35 40 45 500

200

400

600

800

1000

1200

1400

1600

1800

2000

Mass flowrate mdi (g/s)

Pre

ssur

e dr

op Δ

P (

Pa)

(25.7,915.6) (26.3,915.6)

(42.7,1204)(9.3,1204)

Channel 1: D=16 mmChannel 2: D

r=17 mm

Figure 4: Two-phase flow characteristics of two non-identical channels

9

where

a =P ∗in

2

ρ`P0V0, bi =

AiL. (15)

As shown in Appendix B, the condition for stability is the same as (7) withthe additional condition

DSDT

=N∑i=1

si > 0. (16)

This is because the upstream surge tank regulates Pin and plays a role similarto the direct manipulation of Pin as a function of m in (5), but this also incursadditional dynamics.

For identical channels, the balanced flow condition, m∗i = m∗/N , is alwaysat equilibrium, but it is unstable (si are all negative) in the two-phase regime,as shown in the bifurcation diagram Figure 5 for the two-channel case. Fornon-identical channels, the bifurcation diagram as illustrated in Figure 6 ismore complex, but the balanced flow case in the two-phase regime is alsounstable.

0 10 20 30 40 50 600

10

20

30

40

50

60

Cha

nnel

1 fl

owra

te m

d1 (g/

s)

Channel 2 flowrate md2 (g/s)

Figure 5: Bifurcation curves of individual flow distribution m∗1 versus m∗

2 in two identicalparallel channels (blue: stable; red: unstable)

10

0 10 20 30 40 50 600

10

20

30

40

50

60

Cha

nnel

1 fl

owra

te m

d1 (g/

s)

Channel 2 flowrate md2 (g/s)

Figure 6: Bifurcation curves of individual flow distribution m∗1 versus m∗

2 in two non-identical parallel channels (blue: stable; red: unstable)

2.3. Controllability and Observability

To explore the possibility of two-phase flow maldistribution control, thecontrollability and observability analysis [32] is essential. In the context ofparallel-channel two-phase flow system in Figure 3, the controllability meansthe possibility of using the pumped inlet total flow, min, to regulate the in-dividual channel flow, mi, to any desired state, i.e., a state such that thechannel exit flow is of two phase (liquid vapor mixture) for better heat trans-fer performance; the observability means the possibility of using the totalflow measurement, m, to estimate the individual channel flow, mi.

The controllability for the linearized system (the ability of using the in-put to arbitrarily place the closed loop poles) corresponds to the full rankcondition of the controllability matrix [32]:

C =[B AB . . . ANB

](17)

With elementary row and column operations (which do not affect the rank),

11

C becomes

C1 =

1 0 0 . . . 00 1 A1s1 . . . (A1s1)

N−1

......

... . . ....

0 1 ANsN . . . (ANsN)N−1

. (18)

The lower portion of the matrix is a Vandermonde matrix [33]. Hence thismatrix loses rank if and only if

Aisi = Ajsj, for some i 6= j. (19)

or equivalently

d∆P iD(G∗i )

dGi

=d∆P j

D(G∗j)

dGj

, Gi =mi

Ai, i 6= j. (20)

This means if any of the two channels are identical in terms of channelgeometry and pressure demand curve at the equilibrium, then the systemis uncontrollable. For uniformly distributed flows in the two-phase regime,which we have seen to be unstable, the hope for feedback stabilization is lostif there is any channel symmetry.

The dual property of controllability is observability [32] – the ability toreconstruct the individual flow δmi from the total flow δm based on thelinearized model. Supposed that, m =

∑Ni=1 mi, is the measured output:

δm =[

0 1 . . . 1]︸ ︷︷ ︸

C

δPinδm1

...δmN

. (21)

A necessary and sufficient condition for observability is the full-ranknessof the observability matrix:

O =

CCACA2

...CAN

. (22)

12

Using elementary row and column operations, O may be manipulated to

O1 =

0 1 . . . 1∑Ni=1 bi b1s1 . . . bNsN∑Ni=1 b

2i si (b1s1)

2 . . . (bNsN)2

......

......∑N

i=1 bi(bisi)N−1 (b1s1)

N . . . (bNsN)N

. (23)

We can further simplify this matrix as follows. Define a polynomial of degreeN :

K(λ) ,N∏i=1

(λ− sibi) , λN +KN−1λN−1 + · · ·+K0.

We then multiply the ith row by Ki−1, i = 1, . . . , N , and add all of them tothe last row. The resulting matrix is

O2 =

0 1 . . . 1∑Ni=1 bi b1s1 . . . bNsN∑Ni=1 b

2i si (b1s1)

2 . . . (bNsN)2

......

......

−K0

∑Ni=1

1si

0 . . . 0

. (24)

Note that K0 =∏

i(−sibi). As in the controllability case, this matrixconsists of an N × N Vandermonde submatrix. The matrix loses rank ifand only if bisi = bjsj (or, equivalently, Aisi = Ajsj) for some i 6= j or

K0

∑Ni=1

1si

= 0. If all channels operate in the same regime (liquid, two-phase, or vapor), then all si’s are nonzero and have the same sign, and thusthe additional condition is always satisfied.

When any two channels, i and j, are identical, the reachable subspace, R,is the span (all linear combinations) of δPin, δmi + δmj, and all other δmk,k 6= i, j. The orthogonal complement of R, R⊥, is the span of δmi − δmj.Similarly, we have the unobservable subspace N = R⊥ and N⊥ = R. Rep-resenting the system using the decomposition of the state space, R ⊕ R⊥,partitions the system into a controllable and observable subsystem and anuncontrollable/unobservable subsystems. The uncontrollable/unobservablesubsystem is unstable in the two-phase region (where the slope of the pres-sure demand curve is negative). Therefore, the system is unstabilizable andundetectable.

13

When two channels are almost identical, the system will be “close” touncontrollability and unobservability, in the sense that the controllabilitymatrix C and observability matrix O (or the controllability and observabilitygrammians) are almost singular. This would result in large controller andobserver gains which are highly undesirable due to input saturation and thelack of robustness to measurement delay and modeling error. The differencebetween the channels is characterized by the slopes of the channel demandpressure curve Eq.(20), which is a function of system pressure, mass flux,inlet subcooling, heat flux, hydraulic diameter, channel length, the type ofworking fluid, inlet restrictor, and channel surface roughness [15]. Theywill need to be designed to achieve desired channel flow distribution andcontrollability/observability characteristics.

3. Flow Stabilization in Non-Identical Channels

When the flow operates in the two-phase regime, we have seen that thebalanced flow equilibrium is unstable. Furthermore, if any two channels areidentical, the linearized system about this equilibrium is neither controllablenor observable. In this section, we assume that the channel characteristics areall distinct, and we will design an active flow feedback controller to stabilizethe unstable, but desirable, equilibrium.

The two-phase flow maldistribution has significant thermal consequences,which can be characterized by the wall energy balance equations. For individ-ual evaporating channel, one has the evaporator wall temperature dynamics

dT iwdt

=q − qirCpwMw

, qir = αirSh(Tiw − T ir) (25)

where q is the imposed heat load, qir is the actual heat transferred to the fluid(i.e., taking into account heat storage in wall), T iw is the ith evaporator walltemperature, T ir is the fluid saturation temperature, and the heat transfercoefficient αir can be calculated based on Kandlikar’s two-phase flow heattransfer and friction correlations [34]. When the heat load is very high,the channel exit flow becomes vapor only; hence, the local heat transferperformance deteriorate significantly, and the channel exit wall temperaturebecomes very high. In the thermal model (25) and subsequent study, onlythe exit wall temperature transient is indicated.For the simulation study, we consider the following evaporator parameters:

14

Channel length, L = 0.4 mCartridge heater diameter, Dc = 0.0127 mChannel inner diameter, Di = 0.0206 mHydraulic diameter, D = 0.0162 mCross-sectional area, A = 2.0577× 10−4 m2

Heated surface area, Sh = 0.0160 m2

Wall thermal inertia, CpwMw = 115.2 J/KThe working fluid is assumed to be the refrigerant R-134a. The cartridge

heaters are used to represent the uniform heat load generated from electron-ics. These heaters, immersed in the refrigerant, emulate the evaporators ina two-phase cooling system. The operating conditions are selected below:Imposed heat load, q = 1500 WInlet flow enthalpy, hin = 55.1486 J/kgExit flow pressure, Pe = 2× 105 PaInitial system pressure, P0 = 2× 105 PaCompressible gas volume, V0 = 0.7× 10−3 m3

Figure 7 illustrates the active flow regulation about an unstable equilib-rium in a two-channel system. In this study, a standard linear-quadratic-regulator (LQR) controller [32] is used, with a hard constraint imposed onthe input flow control, min ∈[0,165] g/s. Model predictive control (MPC)may be applied to more effectively deal with the control constraints by solv-ing a constrained optimization problem while maintaining the closed-loopsystem stability [35], but will not be pursued in this paper.

For two non-identical parallel channels, open-loop and closed-loop sim-ulation results are shown in Figure 7. The simulation is divided into threetime zones:

• Open loop for t < 1 s: With min = 52 g/s and no feedback controlbefore t = 1 s, two-phase flow maldistribution occurs (correspondingto the circles in Figure 4). In particular, channel 2 is subcooled withhigh mass flow rate, resulting in a very high wall temperature for thatchannel. At this initial maldistribution equilibrium, P = 201.2 kPa,m1 = 9.3 g/s, m2 = 42.7 g/s, T1 = 0.9 ◦C, T2 = 201.1 ◦C.

• LQR Control for t ≥ 1 s: The active feedback control (using the fullstate feedback in this case) stabilizes the flow about the desired bal-anced flow equilibrium (corresponding to the squares in Figure 4). Thehigh wall temperature of channel 2 is now reduced to the level of chan-nel 1 due to the improved heat transfer performance.

15

• Pulse exit flow disturbance at t ∈ [3, 3.05] s: Even in the presence of apulsed disturbance, the balanced flow distribution is still maintained.

At the stabilized and balanced flow equilibrium, P = 200.9 kPa, m1 = 26.3g/s, m2 = 25.7 g/s, T1 = 0.6 ◦C, T2 = 3.2 ◦C, and the eigenvalues of Afor the open-loop unstable flow system (14) are 7.2504 + 12.2950i, 7.2504−12.2950i, 21.4465. The singular values of the controllability matrix (17) are11748, 239, 3, and the singular value of the observability matrix (22) are6411, 231, 7, evidently, both of them are of full rank.

In this simulation study, the weight matrices of the LQR cost functionare chosen as Q = 0.001 · I3, R = 0.1, and the optimal gain matrix K isobtained as K = [−1.6383, 10.8404, − 46.9680].

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5195

200

205

P (

kPa)

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50

50

100

mdi (

g/s)

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5−100

100

300

Twi (

°C)

Channel 1Channel 2

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50

100

200

Time (s)

mdin

(g/

s)

Figure 7: Open-loop vs. closed-loop constrained linear quadratic regulator control re-sponses: open-loop flow maldistribution [0-1] seconds, closed-loop balanced flow distribu-tion [1-5] seconds, flow disturbance attenuation at t = 3 second (mi

d: ith channel flowrate)

We have also considered a three-channel example. The demand curves areshown in Figure 8. At t = 0, a pulse input flow disturbance is applied. TheLQR controller is able to maintain the equilibrium after a short transient, asshown in Figure 9.

16

0 5 10 15 20 25 30 35 40 45 500

400

800

1200

1600

2000

Mass flowrate mdi (g/s)

Pre

ssur

e dr

op Δ

P (

Pa)

Channel 1Channel 2Channel 3

Figure 8: Two-phase flow characteristics of three non-identical channels

0 0.5 1 1.5 2 2.5 3200.5

201

201.5

P (

kPa)

0 0.5 1 1.5 2 2.5 315202530

mdi (

g/s)

0 0.5 1 1.5 2 2.5 30

1

2

Twi (

°C)

0 0.5 1 1.5 2 2.5 350

100

150

Time (s)

mdin

(g/

s)

Figure 9: Three-parallel-channel flow maldistribution control via linear quadratic regulator(mi

d: individual channel flow rate; T iw: channel wall temperature)

17

4. Conclusions

This paper analyzes the stability, controllability, and observability ofparallel-channel two-phase flow systems. Although an inlet control valve foreach individual channel can achieve a uniform two-phase flow distribution,the drawback is that the flow system has to sustain a higher pressure lossand the instrumentation is more complex. When the upstream pump is usedas the control variable for an identical-channel flow system, it is only possi-ble for the control of pressure-drop flow oscillations but not for the controlof flow in the individual channels at the desired balanced flow equilibrium,which is open loop unstable. We demonstrate that a necessary and sufficientcondition for the balanced flow equilibrium to be controllable is that the flowcharacteristics of each channel must be distinct. Similarly, as a dual result,the channel flow rates are not observable from the total flow rate under theidentical-channel (between any two channels) case, but are observable whenthe channels characteristics are all distinct. Most microchannel heat sinksare fabricated to have multiple uniform parallel channels. This paper showsthat from the control point of view, this is actually undesirable. We arenow actively pursuing this research direction to exploit non-identical channeldesigns.

Acknowledgments

This work is supported in part by the Office of Naval Research (ONR)under the Multidisciplinary University Research Initiative (MURI) AwardN00014-07-1-0723 entitled “System-Level Approach for Multi-Phase, Nan-otechnology Enhanced Cooling of High-Power Microelectronic Systems,” andin part by the Center for Automation Technologies and Systems (CATS) un-der a block grant from the New York State Foundation for Science, Technol-ogy and Innovation (NYSTAR).

Appendix

A. Open Loop Stability Condition with Inlet Pressure Control

We can write the system matrix in (5) as

Ac =A

L(D

TDsin − S). (26)

18

Applying similarity transform by pre- and post-multiplying Ac by A12 and

A− 1

2 , respectively, we get

LA12AcA

− 12 = A

12 (D

TDsin − S)A

12

which is a symmetric matrix. The stability of the system is then equivalent

to A12 (D

TDsin − S)A

12 < 0. Since A

12 is symmetric (in fact, diagonal) and

invertible, this condition is further equivalent to

DTDsin − S < 0. (27)

Let D⊥ ∈ RN×N−1 be the orthogonal complement of D

T, i.e.,

DD⊥

= 0

D⊥TD⊥

> 0.

The N ×N matrix[DT

D⊥]

is therefore invertible and (27) is equivalent

to [D

D⊥T

](−DT

Dsin + S)[DT

D⊥]> 0. (28)

This may be further manipulated to

M =

[a ffT S1

]> 0 (29)

where

a = −sinN2 +DSDT, f = DSD

⊥, S1 = D

⊥TSD

⊥.

Since sin is assumed to be arbitrary, a can be made an arbitrary positivenumber. Using the principal minor criterion for positive definiteness, thematrix M is positive definite if and only if S1 > 0 and detM > 0. Thedeterminant of M may be expressed as

det(M) = det(S1)(a− fS1fT ). (30)

Since a may be made arbitrarily large, we have M > 0 if and only if S1 > 0,or

D⊥TSD

⊥> 0. (31)

19

B. Open Loop Stability Condition with Upstream Surge Tank

In this section, we will show that the system matrix, A, of the linearizedsystem (14) is identical to the stability condition (7) of the parallel-channelsystem with direct inlet pressure regulation.

Write A as

A =

[0 −aD

ADT

L−AS

L

](32)

with D, A, S and a defined as in (6) and (15). An eigenvalue λ of A satisfies[0 −aD

ADT

L−AS

L

] [zx

]= λ

[zx

](33)

where[z xT

]Tis the corresponding eigenvector. Expanding the equation,

we have

−aDx = λz (34)

ADTz − ASx = λLx. (35)

Since A is invertible, and D⊥TDT

= 0, we have from (35):

−D⊥TSx = λLD

⊥TA−1x. (36)

If we set x = D⊥y, then we have a generalized eigenvalue problem

−D⊥TSD

⊥y = λLD

⊥TA−1D⊥y. (37)

Since D⊥TA−1D⊥

is positive definite, all N eigenvalues are negative if andonly if

D⊥TSD

⊥> 0

which is the same as (7).There is one additional eigenvalue (from the compressibility of the flow).

We now multiply (35) by D to obtain

DDTz −DSx = λLDA

−1x.

20

Note that DDT

= N . Solve for z and substitute into (34), we have

λ(DS + λLDA−1

)x = −aNDx.

Choose x = DT

, we obtain a quadratic equation of λ:

LDA−1DTλ2 +DSD

Tλ+ aN

2= 0. (38)

The roots will be negative if and only if

DSDT

=N∑i=1

si > 0.

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