CHINA PARTICUOLOGY Vol. 3, No. 5, 279-285, 2005

A CORRELATION EQUATION FOR CALCULATING INCLINED JET PENETRATION LENGTH IN A GAS-SOLID FLUIDIZED BED

Ruoyu Hong1,*, Haibing Li3, Jianmin Ding4 and Hongzhong Li2 1Department of Chemistry and Chemical Engineering, Soochow University, Suzhou 215006, P. R. China

2Institute of Process Engineering, Chinese Academy of Sciences, Beijing 100080, P. R. China 3Laboratory of Solid Waste Energy, Guangzhou Institute of Energy Conversion,

Chinese Academy of Sciences, Guangzhou 510640, P. R. China 4IBM, HYDA/050-3 C202, 3605 Highway 52 North, Rochester, MN 55901, USA

*Author to whom correspondence should be addressed. E-mail: rhong@suda.edu.cn

Abstract Numerical simulation of gas-solid flow in a two-dimensional fluidized bed with an inclined jet was per-formed. The numerical model is based on the two-fluid model of gas and solids phase in which the solids constitutive equations are based on the kinetic theory of granular flow. The improved ICE algorithm, which can be used for both low and high-velocity fluid flow, were used to solve the model equations. The mechanism of jet formation was analyzed using both numerical simulations and experiments. The emergence and movement of gas bubbles were captured numerically and experimentally. The influences of jet velocity, nozzle diameter, nozzle inclination and jet position on jet penetration length were obtained. A semi-empirical expression was derived and the parameters were correlated from experimental data. The correlation equation, which can be easily used to obtain the inclined jet penetration length, was compared with our experimental data and published correlation equations. Keywords fluidized bed, jet, penetration length, two-fluid model

1. Introduction The ash-agglomerating fluidized-bed coal gasification

technology is being developed for utilizing pulverized coal in an efficient and environmentally acceptable manner. For this purpose, many problems related to multiphase hydro-dynamics should be resolved. The inclined jet will be used in the coal gasifier to reduce slag formation on the V-shaped gas distributor. Among all the important hydro-dynamic phenomena of the fluidized-bed gasifier, the jet penetration length is the crux (Hong et al., 1996; Hong & Li, 1996; Hong & Li, 1997; Hong et al., 2003; Hong et al., 2005; Blake et al., 1990; Merry, 1971). While our previous atten-tion was focused on vertical jet (Hong et al., 1996), double jets (Hong et al., 2003) and downward jet (Hong et al., 2005), inclined jet penetration will be scrutinized in the present investigation since the inclined jet has not been as much investigated as the other jets.

Our previous two-fluid model (Hong et al., 2003; 2005), which has fewer model parameters, is used here in simu-lating the inclined jet in a gasifier. Because the jet velocity is high, the model equations are solved by the improved ICE (implicit continuum Eulerian) method at instantaneous time steps. The motions of gas and solids were demon-strated from the simulations. The influence of jet velocity, nozzle diameter, inclination angle, and nozzle location on the inclined jet penetration length was analyzed. Based on numerical simulation, a semi-empirical expression was derived. The parameters of the expression were obtained by correlating experimental data and illustrated with measured data under various conditions.

2. Experimental 2.1 Experimental apparatus and bed materials

Our previous experimental apparatus (Hong et al., 1996) with only a vertical jet was modified to include an inclined jet, as shown in Fig. 1. The thickness of the two-dimensional fluidized bed is 25 mm and its width 314 mm. The width of the central jet tube is 20 mm. In the separating zone, the angle between the tube surface and the vertical direction is 9 degree. At the bottom of the fluidized bed, there is a 45 degree angled distributor with an aperture density of 1%.

Fig. 1 Flow diagram of the 2-D experimental setup. (1 rotameters; 2

V-shaped gas distributor; 3 2-D fluidized bed; 4 inclined gas jet nozzle; 5 separation column; 6 video camera; 7 gas com-pressor; 8 gas filter; 9 buffer tank)

Millet, silica sand 1# or silica sand 2# were used as flu-idization material respectively in the experiments. Their physical properties are shown in Table 1.

To bag filter

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Table 1 Physical properties of bed materials

Materials Millet Silica sand 1# Silica sand 2#

pd /(mm) 1.43 2.25 1.43

bρ /(kg⋅m-3) 878 808 817

sρ /(kg⋅m-3) 1 042 1 354 1 582

Ut/(m⋅s-1) 7.6 11.7 8.2

Umf/(m⋅s-1) 0.52 0.64 0.45

The incipient fluidization velocity (Umf) in Table 1 was obtained from measuring the bed pressure drop (△p) as a function of fluidization velocity (Uf) when △p became in-dependent of Uf.

2.2 Experimental operation range The central jet velocity (Uj) should be higher than the

particle terminal velocity (Ut), but not too high to cause channeling in the bed. The gas velocity above the gas distributor should be high enough to ensure the incipient fluidization of particles. The gas flow rate of the central jet and the two V-shaped distributors in the present experi-ments are given in Table 2.

Table 2 Experimental parameters for measuring Lj (Umf=1.5 m⋅s-1, h0=270 mm)

Parameters Millet Silica sand 1# Silica sand 2#Nozzle diameter/mm 5, 7, 8, 10 5, 7, 8, 10 5, 8, 10

Inclination angle 10°, 0°, −10° Jet position (m) 0.043, 0.103, 0.163, 0.223

Q1/(m3⋅h-1) 11 13 10.5 Q2=Q21+Q22/(m3⋅h-1) 10–24 12–24 12–24

Q3/(m3⋅h-1) 7.9–21.9 7.4–17.4 7.9–17.9 Uj/(m⋅s-1) 26.2–189.7 26.2–218.0 35.0–196.7

No. of frames 312 288 180

2.3 Experimental measurement To observe the development of the jet clearly and

measure the jet penetration length accurately, the devel-opment of a jet was recorded with a video camera in the experiments. Under the same experimental condition, the jet was recorded five times. The video was then replayed on a TV frame by frame. Thus the jet penetration length was obtained from the five values recorded.

3. Two-Fluid Model 3.1 Governing equations

The equations of the two-fluid model (Hong et al., 1996; Hong & Li, 1997; Hong et al., 2003; 2005; Ding & Gi-daspow, 1990; Gidaspow, 1994), describing gas-solid macroscopic flow in fluidized beds, are given as follows: Continuity equations for phase k (k=g or s),

( )'

' 0kk kU

tρ

ρ∂

+∇ ⋅ =∂

, (1)

where 'k k kρ ρ α= and 1k

kα =∑ .

Momentum equations for phase k (k=g or s; l =g or s; l ≠k),

( ) ( ) ( )' ' 'gk k k k k k l k k kU U U p U U g

tρ ρ ε β τ ρ∂

+∇ ⋅ = − ∇ + − +∇ ⋅ +∂

.

(2) The derivation of the two-phase model equations, the

determination of the model parameters and the solution of these equations are given in detail by Hong & Li, (1996). It was also found that these equations could be simplified to those of Davidson's (Hong et al., 1996). The model equa-tions were solved previously by the improved IPSA method (Hong et al., 1996). A computer program with this model has been used to calculate the vertical jet penetration length (Hong et al., 1996). In the present study, the pro-gram was modified using the improved ICE method to simulate the inclined jet penetration in a fluidized bed.

3.2 Constitutional equations Gas-phase stress tensor

g g g g2 Sτ α μ= , (3)

where

( )g g g g1 12 3

TS U U U⎡ ⎤= ∇ + ∇ − ∇ ⋅⎢ ⎥⎣ ⎦

I , (4)

Solid-phase stress tensor

s s s s[ ] 2ss s sp U I Sτ ε ξ α μ= − + ∇ ⋅ + , (5) where

( )s s s s1 12 3

TS U U U I⎡ ⎤= ∇ + ∇ − ∇ ⋅⎢ ⎥⎣ ⎦

, (6)

I is the unit matrix. Drag constant between the two phases

The drag model of Syamlal et al. (1993) was employed,

s s g g sD 2

r s

(1 )34

U UC

V d

ε ε ρβ

− −= , (7)

where 2

rD 0.63 4.8 VC

Re⎛ ⎞

= +⎜ ⎟⎜ ⎟⎝ ⎠

was recommended by

Dalla Valle (1948) and Vr was devised by Garside and Al-Dibouni (1977).

3.3 Boundary conditions For the boundary conditions used in this investigation:

(1) at the inlet, all the variable are known; (2) at the exit, the flow is assumed to be fully developed; (3) at walls, the gas velocities were taken to be zero, while partial slip was used for the solids (Ding & Gidaspow, 1990) as shown below,

ss w p w| ( ) |VV L

r∂

=∂

, (8)

where Lp is the mean free path of particles (Ding & Gi-daspow, 1990).

Hong, Li, Ding & Li: A Correlation Equation for Calculating Inclined Jet Penetration Length

281

3.4 Numerical method To obtain instantaneous jet penetration details, the ICE

algorithm (Harlow & Amsden, 1975) is used. The method improves the numerical computation in such a way that, if

gUp

∂

∂, gV

p∂

∂, sU

p∂∂

, sVp

∂∂

, gep

∂

∂ and se

p∂∂

are evaluated

from the unconverged solution, then the pressure correc-tion (p') is

g

g s

g s

g s

'

se e

pe ep p

ρ ρ

ρ ρ

+

= −∂ ∂∂ ∂+

. (9)

The correction for Ug, Vg, Us and Vs can be obtained too. Details about the numerical algorithm and solution proce-dure can be found in reference (Harlow & Amsden, 1975). Numerical computations were conducted using a DEC Alpha workstation, and the numerical results in this inves-tigation were post-processed using the Tecplot (version 7.5) (Amtec Engineering Inc., 1999) on a personal computer.

4. Results and Discussion 4.1 Definition of jet penetration length (Lj)

The inclined jet penetration length (Lj) is defined as the maximum length from the farthest point of the jet to the jet nozzle when the jet is detached from the nozzle (Hong et al., 1996) (see Fig. 2). In the numerical computation, the first-order upwind difference scheme is used in discretizing the solid continuity equation to obtain the solid volume fraction. The voidage of the jet surface is defined as the voidage value of 0.8 (Hong et al., 1996), which is consis-tent with Yang's definition (Yang & Keairns, 1980).

Fig. 2 A sketch of horizontal jet penetration length (Lj).

4.2 The mechanism of the jet It is found by numerical simulation that there are two

processes taking place above the nozzle. One is the movement of particles under the drag force exerted by the gas of high velocity. This movement will result in a torch-like vacant space due to the jet. The other is the gas

and particles entrainment by the jet due to the high gas velocity and the low gas pressure at the neck of the jet. Therefore, the jet neck is compressed by the entrainment process, and to some extent, further compression leads to the detachment of the jet from the nozzle. When the gas is first introduced through the nozzle, there is no jet, and the first process predominates, leading to the formation of a jet. At this instant, the gas velocity at the base of the jet is very large, the second process predominates, and the jet be-comes unstable and detaches. Therefore, the jet emerges near the nozzle and detaches from the nozzle, alterna-tively.

4.3 Factors affecting inclined jet penetration length (Lj)

4.3.1 Influence of jet velocity (Vj) Hong et al. (1996) found that the vertical jet penetration

length (Lj) increased with increasing jet velocity (Vj) until it reached a plateau at high jet velocity. For the inclined jet, it was found by numerical simulation that the jet velocity is a major factor in determining the inclined jet penetration length, which increases obviously with increasing jet ve-locity even at relatively high jet velocity (Fig. 3).

Fig. 3 Influence of jet velocity (Vj) on dimensionless jet penetration

length (Lj/dj); bed material, millet.

4.3.2 Influence of jet diameter (dj) It is found from numerical simulation that the inclined jet

penetration length (Lj/dj) can be increased (Fig. 4) by re-ducing the cross-sectional area of the nozzle when the flow rate of the jet is fixed at Q3=15 m3⋅h-1. This is due to the increase of the jet velocity (Vj

2), leading to increase of jet momentum of unit cross-section area (ρj·Vj

2).

Fig. 4 Influence of jet diameter (dj) on dimensionless jet penetration

length (Lj/dj) ; bed material, millet.

30

24

18

12

6

020 48 76 104 132 160

Jet velocity Vj /(m⋅s-1)

L j/d

j

Nozzle diameter dj /mm

L j/d

j

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282

When the inclined jet velocity (Vj) is maintained at the same value, by enlarging the nozzle diameter, the inclined penetration length (Lj) is also increased. 4.3.3 Influence of jet inclination angle (α)

To the author’s knowledge, no researchers studied the influence of jet inclination angle (α) on jet penetration length (Lj). The present study showed that jet inclination angle has definite influence on jet penetration, as shown in Figs. 5 and 6. In particular, when other conditions maintain the same, horizontal jet penetration length is less than that for vertical jet for jet velocities in the range, as shown in Fig. 5.

Fig. 5 Influence of jet angle (α) on dimensionless jet penetration

length (Lj/dj) (△: −10°; ○: horizontal; ●: +10°) with upward in-clining as position.

Fig. 6 Comparison of vertical with horizontal jet penetration length

(△: horizontal; ○: vertical).

Evidently the jet is pushed upward by the fluidizing gas as well as by buoyancy, as shown in Fig. 2. The force act-ing on the jet is in the same direction for the vertical jet, thus elongating the jet. At higher jet velocity, the vertical jet penetration length (Lj) increases slowly with jet velocity (Vj) (Hong et al., 1996), while for horizontal jet, its penetration length increases more notably with jet velocity. In Yang's correlation equation (Yang & Keairns, 1980), the vertical jet penetration length (Lj) is directly proportional to the 0.374-power of jet velocity (Vj), while in Merry's correlation equation, the inclined jet penetration length (Lj) to the 0.80-power of jet velocity (Vj). 4.3.4 Influence of jet position (h)

Hong et al. (1996) found that bed depth had little influ-ence on vertical jet penetration length. The present ex-

periments showed that, placing the jet nozzle at the height of 0.2h0, 0.4h0, 0.6h0, and 0.8h0 above the bed bottom re-sulted in little difference in jet length, as can be seen in Fig. 7.

Fig. 7 Influence of horizontal jet position (h) on jet penetration length

(Lj/dj) (△: Vj=102.8 m⋅s-1; ○: Vj=161.4 m⋅s-1).

4.4 Derivation of a correlation equation A semi-empirical expression for calculating the inclined

jet penetration length (Lj) is derived in this section, to ac-count for the influence on inclined jet penetration length (Lj) by jet velocity (Vj), nozzle diameter (dj), jet inclination angle (α) and jet position (h).

Consider a horizontal gas jet penetrating into a station-ary medium, as shown in Fig. 8. The horizontal gas velocity (V) of a given cross-section can be expressed as (Merry, 1971):

32

m

1.0VV

ξ= − , (10)

where Vm is the velocity on the jet axis and bx /=ξ , where b is the cross-sectional radius of the jet.

Fig. 8 A horizontal gas jet penetrating into a fluidized bed.

The following equation can be obtained by using mo-mentum balance:

2 2 2 2j j jd V b Vρ ρ= , (11)

where ρ and V are respectively the mean density and mean velocity of the cross section. In turbulent flow, the relation between mV and V is (Merry, 1971)

m

0.26VV

= . (12)

Combination of Eqs. (10), (11) and (12) yields

Vj/(m⋅s-1)

L j/c

m

L j/d

j

Jet velocity Vj /(m⋅s-1)

L j/d

j

Dimensionless height h/h0

Hong, Li, Ding & Li: A Correlation Equation for Calculating Inclined Jet Penetration Length

283

232j 2

2j j j

1 1.0dV

V A y yρ ξρ

⎛ ⎞= −⎜ ⎟

+ ⎝ ⎠, (13)

where ( )j0.26 tan 0.26b y+yA θ= = , y being the distance

from the nozzle to the cross-section. Assume constant density of the gas, and the above

equation can be rewritten as 23

j 2

j j

1.0dVA

V y yξ

⎛ ⎞= −⎜ ⎟

+ ⎝ ⎠. (14)

The entrainment velocity (Ve) at the boundary layer can be expressed as (Merry, 1971)

23je 2

j j

1.0dVA

V y yξ

⎛ ⎞= −⎜ ⎟

+ ⎝ ⎠. (15)

When 0ξ = , j jy y L+ = , and the inclined jet penetration

length (Lj) can be given as j j

j e

1L Vd A V

= ⋅ . (16)

Similar to gas penetrating into a stationary medium, when gas penetrates into a fluidized bed, the penetration length may be expressed as (Merry, 1971)

12 2

j j j2

j s b

1(1- )

L Vd A V

ρε ρ

⎡ ⎤= ⎢ ⎥

⎢ ⎥⎣ ⎦, (17)

where Vb is velocity at the jet boundary. When Vb is fixed, the inclined jet penetration length can then be calculated. In a fluidized bed, there are two forces exerting on the particle phase: drag force exerted by the horizontal jet:

22 s

d g g s d1 ( )2 4

dF V V Cρ

π= − , (18)

and gravitational force: 3

g s s6F d gρπ

= . (19)

For a particle at the boundary of the jet (Merry, 1971), g s s

d g d

4tan3

F gdF C V

ρφ

ρ= = 2

b

. (20)

Therefore, the gas velocity at the jet boundary, Vb, is de-termined as

12

s sb

d g

43 tan

gdVC

ρφ ρ

⎛ ⎞= ⋅⎜ ⎟⎜ ⎟⎝ ⎠

. (21)

Substitution of Eq. (21) into Eq. (17) yields 1

2 2j j j gd

2j s s s

3 tan4 (1- )

L VCd A gd

ρ ρφε ρ ρ

⎡ ⎤= ⋅ ⋅⎢ ⎥⎢ ⎥⎣ ⎦

. (22)

For a spherical particle with diameter (ds) at high Rey-nolds number flow, the equation for steady motion of the particle is assumed to be

23 2 ss s d g g s

d 1 ( )6 d 2 4

dVd V C V Vρ ρλ

ππ= − . (23)

If rλ is defined as the distance through which the particle

travels until the relative velocity g s( )V V− falls to 1/e of its

initial value, then rλ can be calculated as

s sr

g

2.2 dρλρ

= . (24)

As a matter of fact, rλ characterizes the momentum trans-fer between the gas and the particle phase (Merry, 1971). Taking account of this momentum transfer, Eq. (22) should be multiplied by r j( / )aLλ .

Considering the influences of the nozzle inclination an-gle (α) and the nozzle position (h) on the inclined jet pene-tration length (Lj), and assuming the jet half angle constant, Eq. (22) is modified to be

1 32 542j j j g s

j s s s j 0(1- ) 2

a aa aaL V d hCd gd d h

ρ ρα

ε ρ ρ⎡ ⎤ ⎛ ⎞⎛ ⎞ ⎛ ⎞π⎛ ⎞= +⎜ ⎟⎢ ⎥ ⎜ ⎟ ⎜ ⎟⎜ ⎟⎜ ⎟ ⎝ ⎠⎢ ⎥ ⎝ ⎠⎝ ⎠ ⎝ ⎠⎣ ⎦

. (25)

4.5 Correlating the experimental data By inserting experimental data into the parameters of

Eq. (25), it can be expressed as the following empirical expression for jet penetration:

j

j

3.80Ld

+ =

0.327 0.0401.974 0.0280.1482j j g6 s

s s s j 0

1.64 10(1- ) 2

V d hgd d h

ρ ρ παε ρ ρ

−⎡ ⎤ ⎛ ⎞⎛ ⎞ ⎛ ⎞⎛ ⎞× +⎜ ⎟⎢ ⎥ ⎜ ⎟ ⎜ ⎟⎜ ⎟⎜ ⎟ ⎝ ⎠⎢ ⎥ ⎝ ⎠⎝ ⎠ ⎝ ⎠⎣ ⎦

.

(26) Comparison of above equation with experimental data is

illustrated in Fig. 9, showing a maximum difference of less than 25%.

Fig. 9 Comparison of experimental data for jet penetration length with

Eq. (26).

4.6 Comparison of simulated data with ex-perimental data

For millet particles, Vf=1.5 m⋅s-1, dj=10 mm, h=0.223 m, and α =10°, comparison of the jet penetration length pre-dicted by numerical simulation with experimental meas-urements is shown in Fig. 10. The two curves are very close.

40

35

30

25

15

10

0 5 15 25 35 40

Lj/dj predicted by Eq. (26)

20

5

010 20 30

L j/d

j mea

sure

d ex

perim

enta

l

CHINA PARTICUOLOGY Vol. 3, No.5, 2005

284

Fig. 10 Comparison of simulated results with experimental data

(△: Experimental; ○: Numerical).

4.7 Comparison with Merry’s correlation Substituting the experimental conditions of jet velocity,

nozzle diameter and particle diameter, etc. into Eq. (26) and Merry's correlation equation (Blake et al., 1990) re-spectively, the horizontal jet penetration lengths at various conditions can be obtained, as shown in Fig. 11. Compared to Fig. 9, which compares Eq. (26) with measured data, it can be concluded that our empirical equation yields better agreement with experimental data than Merry's. The ad-vantage of our correlation is that it can be used for either horizontal or inclined jets.

Fig. 11 Comparison of Merry's correlation for horizontal jet penetra-

tion length with Eq. (26).

5. Conclusions The following conclusions can be drawn from the pre-

sent investigations: (1) From numerical simulations for inclined jets in a 2-D

fluidized bed using a two-fluid model, it was found that the inclined jet penetration length (Lj) was affected by gas density ( gρ ), bed structure (ε), particle properties

( sρ , ds), jet characteristics ( jρ , Vj), jet diameter (dj)

and jet inclination angle (α), and little by jet position (h).

(2) The inclined jet penetration lengths (Lj) under various conditions were obtained from numerical simulations

based on a two-fluid model. The predicted results are in accordance with the experimental data.

(3) A semi-empirical expression was obtained to deter-mine the inclined jet penetration length. The equation is in good agreement with experimental data.

Acknowledgment The project was supported by the National Natural Science

Foundation of China (NNSFC, No. 20476065), the Scientific Re-search Foundation for Returned Overseas Chinese Scholars of State Education Ministry (SRF for ROCS, SEM) and the Multi-Phase Reaction Laboratory of the Chinese Academy of Sciences (No. 2003-5).

Nomenclature ka gas or solid volume fraction

db bubble size, m dj nozzle diameter, m dp particle diameter, m D bed width, m

eg,es error of gas or solid phase continuity equation, kg⋅m-3⋅s-1

g gravitational acceleration, m⋅s-2 Hb the fluidized bed height, m h0 jet position, m K turbulent fluctuation kinetic energy, m2⋅m-2

Lj jet penetration length, m M gas axial momentum, kg⋅m-1⋅s-2 T temperature, K p gas phase pressure, Pa ps solids phase pressure, Pa p' pressure correction, Pa Q1 gas flow rate of separation column, m3⋅s-1 Q2 gas flow rate of V-shaped gas distributor, m3⋅s-1 Q3 gas flow rate of inclined jet, m3⋅s-1 Rep particle Reynolds number t time, s

kS deformation rate tensor, s-1

Ub bubble ascending velocity, m⋅s-1 Uj central jet velocity, m⋅s-1

kU =(Uk, Vk), velocity vector of phase k, m⋅s-1 Uf superficial gas velocity in bed, m⋅s-1 Umf incipient fluidization velocity, m⋅s-1 Vj inclined jet velocity, m⋅s-1 Vf bed superficial gas velocity, m⋅s-1 Vm maximum gas velocity of cross-section as defined in

Fig. 6, m⋅s-1 yj length defined in Fig. 6, m Greek letters α jet inclination angle β gas-solids drag coefficient, kg⋅m-3⋅s-1 ε dissipation rate of turbulent kinetic energy, m2⋅m-3 μeg, μes gas or solids effective viscosity, kg⋅m-1⋅s-1 μg gas viscosity, kg⋅m-1⋅s-1 ρg gas density, kg⋅m-3 ρs solids density, kg⋅m-3

40

35

30

25

15

10

0 5 15 25 35 40

Lj/dj predicted by Eq. (26)

20

5

0 10 20 30

L j/d

j pre

dict

ed b

y M

erry

’s c

orre

latio

n

L j/d

j

Jet velocity Vj /(m⋅s-1)

Hong, Li, Ding & Li: A Correlation Equation for Calculating Inclined Jet Penetration Length

285

ρj gas density at nozzle, kg⋅m-3 φ sphericity of particle, τc cohesive force, Pa

kτ stress tensor of phase k, Pa

gε gas volume fraction

sε solid volume fraction

rλ distance defined in Eq. (24), m

Subscripts g gas phase j jet p particle s solids phase Operators ∇ gradient ⋅∇ divergent

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Manuscript received November 19, 2004 and accepted May 19, 2005.