Simplified modeling of circulating flow of solids between a fluidized bed and a
vertical pneumatic transport tube reactor connected by orifices
1 2
Karel Svoboda, 2David Baxter, 3Francesco Miccio, Sylwester Kalisz, 1Michael PohoY
elż
1
Institute of Chem. Process Fundam., Academy of Sciences of Czech Republic, Rozvojová 135 ,
165 02 Praha 6  Suchdol 2 , Czech Republic, Tel.: + 420 220 390 241
Fax: +420  220 920 661 E-mail: [e-mail]
2
Institute for Energy, Joint Research Centre of EC, 1755 ZG Petten , The Netherlands
Tel: [phone] Fax: [phone] E-mail: [e-mail]
E-mail: [e-mail] E-mail: [e-mail]
3
Istituto di Ricerche sulla Combustione, Consiglio Nazionale delle Ricerche, Napoli, Italy
E-mail: [e-mail]
Key words: fluidized bed ; pneumatic transport ; circulation ; solids ; orifice
Abstract
Interconnected and dual fluidized bed (FB) systems with various regimes of gas-solids flow are
commonly studied with the aim to find and ensure suitable circulation of solids and control of flow
between two chambers/reactors with different reacting gases. Dual separated fluidized beds or
fluidized and transporting bed systems are convenient for cleaning of gases with regeneration of a
solid sorbent or for gasification of solid fuels and production of rich fuel gas with practically no
content of nitrogen. We concentrated our attention on development, comparison and checking of
three simplified models for solid particle velocity, pressure drop and mass flux in a pneumatic
transport reactor (riser). The simplified models have been used in a more general model for flow of
solids via orifices between a FB near the incipient fluidization and a tube with vertical pneumatic
transport (PT) of solids and circulation back to the FB. The model predictions have been compared
with experimental data and validated in a limited range of operating conditions: for lower solids
fluxes of sand particles (dp = 0.3 mm, mass fluxes below 40 kg/m2/s) at gas velocities higher than
terminal velocity UT of solids and for relatively high solid fluxes (over 150 kg/m2/s) and gas
velocities around or lower than UT. Predictions from the models we have used for estimation of
effects of particle size, gas velocity in pneumatic transport column, height of FB and temperature on
circulating mass flux of solids. Considering simplified equations for estimation of pressure drop in
the cyclone/impactor and for gas leakage from annular FB to the PT-column we tentatively
estimated their possible effects on mass flux of solids. Comparison of a theoretical forecast for
saturation carrying capacity of gas in pneumatic transport and forecast of  choking , computed by
selected Yang correlation, with the model-computed mass fluxes of solids has indicated possible
problems and instabilities of solids flow in regimes with higher mass fluxes of solids - particularly
arrangements with larger orifices, higher FB and/or lower gas velocities in riser (PT-column) and
bigger particles. For control of circulation rate of a given particulate material in the dual FB-PT
systems three main, practically important parameters should be considered: superficial gas velocity
in the riser (PT-column), height of FB and cross sectional area of orifices.
1. Introduction
The interconnected [1,2] and dual fluidized bed (FB) systems [3-5] with various regimes of gas-
solids flow have been studied with the aim to find and ensure suitable circulation of solids and
control of flow between generally two or more chambers/reactors with different reacting gases.
Dual separated FB reactor systems or FB and transport systems are convenient for cleaning of gases
with regeneration of a solid sorbent or for gasification of solid fuels (e.g. wood) and production of
high calorific value fuel gas practically free of nitrogen by means of air used for combustion and
generation of needed heat [5,6].
In the case of dual FB gasification an inert particulate material (e.g. quartz sand or alumina)
circulates between a fluidized bed, where endothermic gasification by steam takes place, and
another fast FB or transporting bed, where the char formed in gasification is burnt to produce heat
for gasification. The heat from combustion is transported generally by both the circulating sand and
by transfer through the walls separating the combustion and gasification chambers (parts). The rate
of circulation of solids for systems with two FB connected by an orifice and differing in fluidization
regime was described by Korbee et al. [2]. Experimental description and modeling of circulation of
solids in the system with one FB reactor and a pneumatic transport (PT) reactor communicating via
orifices is rare in the open literature. We have found only six partly relevant experimental studies
[7-12] describing internally circulating fluidized/moving bed either with a relatively short draft tube
[7-9] with rather a fluidization regime inside the draft tube and system [10] with a conical
distributor, an annular downwards moving bed and with a draft (riser) tube with regime of fast
fluidization and pneumatic transport. Another solid circulation system experimentally studied and
modeled [11] used downward moving annular bed and description (modeling) by means of
measured pressure gradients in the PT- column. Modeling of coal gasification in an internally
circulating fluidized bed (ICFB) with a draft tube was based [12] in fact on annular moving bed and
dense fluidized bed inside the draft (vertical transport) tube. In literature, rather typical circulating
fluidized bed systems with riser, cyclone and L-valve or fluidized bed siphon have been studied and
described [13-16]. The usual regime described and modeled in the transport reactor has been fast or
turbulent fluidization, pneumatic transport or dense fluidized bed flow being the boundary and
marginal regime only.
Our attention in this theoretical study was focused on a simplified description and modeling of
the circulating flow of solid particles from a dense fluidized bed through orifices to a pneumatic
riser, where after separation in a cyclone/impactor the particles are returned back to the fluidized
bed. Three simplified models for the description of vertical transport of solids within the riser have
been used in a more general model [17] for circulating flow of particles between a FB near incipient
fluidization and a tube (column) with PT of solids communicating via orifices. Because of lack of
relevant experimental data for validation of the models we have used our measured experimental
data and partly data from literature with dense fluidization flow in the riser [7,8]. The models have
been used further for estimating effects of gas velocity, particle size, height of fluidized bed (or
mass inventory of solids), gas temperature and orifice cross sectional area on particle velocity, mass
flux of particles and pressure drop in a PT column. We concentrated on smaller diameter and
shorter length of riser tube and rather shallow fluidized bed. Simplified empirical equations for
estimation of pressure drop for a cyclone/impactor and for gas leakage from the FB to the PT-
column enabled us tentatively evaluate their possible effects on the circulation of solids. Possible
regimes and instabilities in pneumatic/fluidized transport originating from operation and design
parameters have been estimated by means of correlation for saturation carrying capacity, dilute and
dense regions in circulating fluidized beds [18] and choking [19].
2. Model for circulating flow of FB particles through an opening to PT-column
A schematic picture of a system for circulation of solids with interconnected FB reactor, PT-
column, return impactor and L-valve is shown in Fig. 1.
Flue gas
Impactor
for circulation
of solids (sand)
Syngas
PT-column
Auxiliary gas for L-valve
Orifices
Fluidized bed of sand
height LFB
Gas
distributor
Steam (gas) for gasification
Air for combustion of char
Fig. 1 : Simplified scheme of possible arrangement for a circulating system of solids with dual
FB-PT-column system  as an example for application in gasification.
Mass flux of solids GFB from a fluidized bed through an orifice at the bottom side wall to the
atmosphere, another fluidized bed or to a pneumatic transport bed is described by equation [17]:
GFB = CD * (µu)2.35 * (Sorif/SFB)* [2 Ás (1- µmf ) "Porif ]0.5 ( 1 )
where µu is the voidage at the other side of the orifice (e.g. bottom of another FB or PT-column).
The discharge coefficient CD in eq. (1) has typically values between 0.3 and 0.6 and often is taken
to be 0.5. In fact the real value of CD depends on geometrical arrangement, orifice edge parameters
and thickness of the wall [20]. The equation holds for flow of solids through relatively smaller
orifices in comparison with FB cross sectional area (Sorif/SFB << 1).
Under steady state conditions of solids flow the mass flow ms is constant and identical in the
fluidized bed (down-flow), the flow through orifice(s) and in the pneumatic transport (up-flow):
GFB * SFB = Gorif * Sorif = GPT * SPT = ms ( 2 )
where SFB , Sorif and SPT are cross sectional areas of FB, orifice(s) and PT-column respectively.
The pressure difference between FB pressure at the orifice and outside the orifice (i.e. at the
bottom of PT-column) can be expressed as:
"Porif = "PFB - "PPT ( 3 )
where for a smaller orifice at the bottom of FB with a regime near minimum fluidization the excess
pressure "PFB can be expressed as:
"PFB = LFB*(1 - µ mf)*Ás * g H" mFB*g/ SFB ( 4 )
where LFB is distance between orifice axis and top level od a dense fluidibed bed and µ mf is voidage
of the FB at incipient fluidization.
After substitution and rearrangement:
(GFB)2 = 0.5* (µu)4.7 * (Sorif/SFB)2 * Ás * (1- µmf )* [LFB*(1 - µ mf)*Ás * g - "PPT ] ( 5 )
For the pressure drop in pneumatic transport of solids the equation with 4 terms only was
considered (pressure drop due to acceleration of gas and gas gravity are neglected):
"PPT = LPT * (1 - µ)*Ás * g + 0.5 (1 - µ)*Ás * (Up)2 + LPT * Ffw + LPT *Fpw ( 6 )
where Up is the actual particle exit velocity, LPT is the length of the vertical PT-column,
Ffw and Fpw are gas-wall and particle-wall friction force per unit volume of the column respectively.
In the case of PT-column diameters over 4 cm, gas velocities lower than 6 m/s and smooth
transport tube without bends the contribution of gas-wall friction Ffw can be neglected [21,22] and
for conditions of pneumatic transport with relatively lower concentrations of solids Fpw can be
expressed by means of a correlation [23,24]:
Fpw = 0.057* GPT * (g/D)0.5 ( 7 )
Although there is agreement with direct proportionality between Fpw and GPT, for some cases
corrections of eq. (7) by means of voidage (or particle concentration) term and particle
characteristics (dp, roughness) are discussed and suggested in literature [22].
2.1 Simplified empirical models for particle velocity and pneumatic transport
For the estimation of particle velocity and pressure drop in PT ("PPT) three different model
situations have been considered:
a) Up is computed by means of terminal velocity of particle UT :
The slip velocity (difference between gas velocity and particle velocity) is supposed to be equal
to UT. The particle velocity Up1 in this case can be written:
Up1 = Uf /µ - UT = Uf /[1  GPT/(Ás Up1)]  UT ( 8 )
Where for substitution for voidage µ the definition equation for mass flux in PT was used:
GPT = (1 - µ) Ás Up1 ( 9 )
Validity of dependence of Up1 on GPT according to eq. ( 8 ) can be supposed mainly for very low
and lower concentrations of solid particles in PT-column, where voidage µ is near to 1 (or over
0.96).
After rearrangement of eq. ( 8 ):
 (Up1)2  Up1(Uf  UT + GPT/Ás )  GPT UT/Ás = 0 ( 10 )
solution of the resulting quadratic eq. ( 10 ) is:
Up1 = 0.5 [+( Uf  UT + G/Ás ) + (( Uf  UT + GPT/Ás )2 + 4 * GPT*UT/Ás )0.5] ( 11 )
The eq. ( 6 ) for pressure drop in PT-column can be rearranged by substitution for the term (1 - µ )
from eq. ( 9 ) and considering simultaneously computation of Up = Up1 according to eq. ( 11 ):
("PPT)1 = LPT * (GPT/Up1) * g + 0.5 GPT * Up1 + LPT * 0.057* GPT * (g/D)0.5 ( 12 )
The resulting equation for computation of circulation mass flux of solids in PT-column (GPT),
applying eqs. ( 2 ), ( 5 ) and ( 12 ), has the form:
[GPT SPT ]2/(SFB)2 = 0.5 (µu)4.7 (Sorif/SFB)2 Ás (1- µmf ) [LFB*(1 - µ mf) Ás g  ("PPT)1] ( 13 )
b) Up is computed by means of a slip factor È
Such a procedure for computation and regime are probably more typical for regimes of
transition to fast or turbulent fluidization (higher values of particle concentration in PT-column).
The slip factor È and slip velocity Uslip between gas and particles are defined elsewhere [25, 26] :
È = (Uf/µ)/Up ( 14 )
Uslip = (Uf/µ - Up) = (Uf/µ) * (1  1/È) ( 15 )
For particle velocity Up2 under such conditions can be written:
Up2 = Uf/µ - Uslip = [ Uf/µ - (Uslip/UT) * UT ] = Uf/µ * 1/È ( 16 )
After substituting for µ from eq. ( 9 ) :
Up2 = [Uf (Ás Up2)/(Ás Up2 - GPT)] 1/È ( 17 )
Up2 * (Ás Up  GPT  Uf Ás /È) = 0 ( 18 )
and
Up2 = GPT/Ás + Uf /È ( 19 )
To compute the slip factor È the following correlation was suggested [25, 26]:
È = 1 + 5.6 * (g * D)0.5/Uf + 0.47 * [UT /(g*D)0.5]0.41 ( 20 )
The voidage µ in a PT-column can be expressed by means of eqs. ( 9 ) and ( 19 ):
µ = 1  GPT/(GPT + Uf Ás/È) ( 21 )
The pressure drop in the PT-column is then expressed by means of Up2(È) from eq. ( 19 ):
("PPT)2 = LPT * GPT/Up2 * g + 0.5 GPT * Up2 + LPT * 0.057* GPT * (g/D)0.5 ( 22 )
The circulating mass flux of solids in PT-column (GPT) is computed from the equation for
circulating mass flux in analogy with eq. (13).
c) Up is computed by means of correlations for relative slip velocity Uslip/UT
The correlations are valid for smaller particles (dp < 0.7 mm) and smaller column diameters
[27]. For PT-column diameters over 5 cm the forecasts have been not validated.
Uslip/UT = A1 * (1 - µ)B1 ( 23 )
A1 = 93.67/(Rep)0.994 * (dp/D)1.014*(Ás /Áf )0.706 ( 24 )
B1 = 1.075/(Rep)0.445 * (dp/D)0.476 * (Ás /Áf )0.313 ( 25 )
Where the particle Reynolds number Rep in eq. ( 24 ) and ( 25 ) is defined:
Rep = dp * UT * Áf/µf ( 26 )
The equation for the particle velocity Up3 has the form:
Up3 = [ Uf/µ - (Uslip/UT) * UT ] = Uf /[1  GPT/(Ás * Up3)] - A1*[GPT/(Ás * Up3)]B1 * UT ( 27 )
This non-linear equation for Up3 and GPT has to be solved simultaneously together with the
equation for circulating mass flux (analogy with eq. 13). It is thus necessary to solve a set of two
nonlinear equations with two unknowns, Up3 and GPT.
3. Validation of the models for circulation mass flux
The models with three different expressions for Up assume a constant value of GPT along the
height of the PT-column (at given operation conditions) and immediate acceleration of particle from
practically zero vertical velocity at the orifice level to the final velocity. Under such conditions the
particle holdup µs = (1- µ) axial profile in the column is uniform. In reality such a situation is in
vertical transport of solids rather limit. There are two possibilities of practically little variation of
axial average particle holdup: either very diluted flow of particles tending to homogeneous
pneumatic transport with relatively low values of µs (usually with low values of GPT) or, on the
other side, dense particle suspense flow characterized by relatively high and uniform particle holdup
along the height in transport column (with rather lower gas velocities and high values of solids
fluxes GPT). Typical time and radial averaged particle holdup axial profiles in a real riser are shown
in Fig. 2 . As it is obvious, for a general description of the flow of particles in the transport column
a simplified particle concentration axial profile should be considered. Problem is, that there are no
simple (e.g. linearized) and validated correlations for estimation of the axial particle concentration
profiles in risers for a broad range of operating conditions and design parameters. Bolkan et al .[28]
tried to develop simple, linear correlation for slip factor profile. But comparison of computed data
(slip factors and axial voidage) according to their model with experimental data [29] has led to
disappointment.
3.1 Validation of the models for dilute solids flow conditions
For the dilute solids flow conditions in the PT-column we validated the models by experimental
data measured in a plexi-glass apparatus [30] with annular fluidized bed and central PT-column
with orifices. The scheme of the experimental arrangement is shown in Fig. 3.
Gs = 15 kg/m2/s Gs = 38 kg/m2/s
Gs = 51 kg/m2/s Gs = 80 kg/m2/s
0,3
0,25
0,2
0,15
0,1
0,05
0
0123456
Height in riser ( m )
Fig. 2 : Dependence of axial solid particle holdup µs = (1- µ) in a riser [29]
(Hr = 5.75 m, Dr = 0.12 m) on axial distance for Uf = 3m/s and four different values
of particle mass flux Gs in the riser. Solid particles ballotini dp(average) = 0,089 mm,
Ás = 2540 kg/m3.
The external, annular cylinder has ID = 12.4 cm and the PT-column has ID = 4.4 cm and height
2 m. The column has four orifices (Dor = 1 cm) in the lower part in fluidized bed. Particle feed ring
served for acceleration of particles and for hindering of free flow of particles from annulus to the
column under conditions of no gas flow. Free area of orifices (above the ring) has shape of segment
of the circle with height 3 mm  as shown in Fig. 4. The area of the segment is 0.198 cm2 and total
free area of the 4 orifices is 0.792 cm2. For the sake of modeling of the circulating flow of particles
we supposed that the particles can flow only through those free segments in orifices. The superficial
gas velocity in the PT-column was varied in a range 2  3 m/s. In the model we supposed that the
gas bypassing from annular space to the PT-column can be neglected. Solid particles of quartz sand,
density 2550 kg/m3 with size 0.2  0.4 mm have been used. Effect of fluidized bed height (distance
between free segment of orifice and top level of the dense FB) was studied in a range 3.5  20 cm.
The experimental results for dependence of the mass flux of particles GPT on superficial gas
velocity Uf in the column are compared with model predictions in Fig. 5. In the models the voidage
µmf of FB under conditions near the incipient fluidization of the sand particles was taken to be 0.53.
The µu value (voidage in the PT-column at the orifice level) was assumed to be the same as at any
higher level in the column. The particle terminal velocities UT have been taken from internet official
and recommended methods for their computation [31] and for temperature of air 20 and 800 oC they
are collected in Table I. The height of the PT-column was considered in models to be 2.1 m,
because the impactor caused an additional small elevation of the pressure drop. Because of
computational problems in the case of the model with power relation for relative slip velocity
Uslip/UT only limited predictions for higher gas velocities Uf are illustrated in the Fig. 5. At lower
gas velocities the iterative procedure was unstable. The discharge orifice coefficient CD was chosen
0.25 because of relatively small free segments of circles in orifices for flow of particles.
The literature data for CD stated by Chin et al. [20] suggest for ratio Dorif/dp(aver) H" 10 values of CD
less than 0.3 (in our case it is relevant for height of free segment of the orifice equal to 3 mm).
3
3
Particle concentr. (m/m )
Impactor
Collecting
cylinder for
PT-column
solids mass
flow
measurement
Fluidized bed of
sand, height LFB
Orifices
Gas distributor
Air for fluidization
for the FB
in annulus
Air for vertical
transport
Fig. 3: Simplified scheme of the experimental arrangement for measurement of circulating flow of
sand particles in a dual FB-PT-column system.
Table I : Terminal velocities of sand particles [31] at temperatures 20 oC and 800 oC
Particle size dp Terminal velocity UT at 20 oC Terminal velocity UT at 800 oC
( mm ) ( m/s ) ( m/s )
0.1 0.557 0.317
0.15 0.969 0.6675
0.2 1.393 1.086
0.25 1.815 1.60
0.3 2.22 2.019
0.4 3.016 2.975
0.5 3.76 3.958
0.6 4.47 4.937
The trend of experimental data is slightly different from theoretical prediction of model based on
assumption that Uslip = UT, but we have to take into account that about 50 % of the sand particles
has in fact diameter lower than 0.3 mm and therefore particularly at lower gas velocities (near the
terminal particle velocity for average particle size) lower diameter and higher model predictions of
GPT can be expected.
PT-column
Orifice " 10 mm
Gas distributor
Free segment for particle flow
Particle feed ring
with suction 3
effect of gas
7
Particle feed ring
Air for vertical
transport
Fig. 4 : Scheme of arrangement of the lower part of the PT-column with orifices, acceleration ring
and position of free segment of the orifice circle
Exper. data Uslip=UT Slip factor Power rel.
30
25
20
15
10
5
0
1 1,5 2 2,5 3 3,5 4 4,5
Uf (m/s)
Fig. 5 : Comparison of model prediction of GPT with experimental data measured with quartz sand,
dp(aver) = 0.3 mm, height of fluidized bed = 11 cm, 4 orifices with free segments given in
Fig. 4
The experimental results for dependence of the mass flux of particles GPT on height of the
fluidized bed LFB in annulus at given superficial gas velocity (Uf = 2.74 m/s) in the PT-column are
compared with model predictions in Fig. 6. Again we have used the same values µmf = 0.53 and CD
= 0.25 as in the foregoing case. As follows from the figure, the trend of model predictions is very
2
PT
G
(kg/m/s)
similar to trend of experimental data dependence on LFB. However, the model utilizing
approximation of slip velocity by means of UT is closer to measured experimental data.
The models consider uniform voidage along the height of the PT-column. Such assumption is
strictly valid only for values of GPT sufficiently lower than the saturation carrying capacity Gs* [18].
At velocities lower than Gs* there is no  denser bed (layer) in the bottom of the riser (PT-column).
But even at values lower than Gs* the values of particle holdup are higher at the bottom
of the riser than at the top.
Uslip = UT Slip factor Exper. data
40
30
20
10
0
0 0,05 0,1 0,15 0,2 0,25
HFB ( m )
Fig. 6 : Comparison of model predictions with experimental data measured with quartz sand
dp(aver) = 0.3 mm, Uf = 2,74 m, 4 orifices with free segments given in Fig. 4
Comparison of fluxes for conditions of saturation carrying capacity Gs* with computed values of
GPT (model approximating Uslip with UT), experimental values and comparison of values of solids
holdup (1-µ) from the model solution with values from correlations of Bai and Kato [18] for solids
holdup and the bottom (µsb) and top (µst) of the riser are given in the Table II for a range of Uf
between 2.2 and 3.4 m/s.
Table II: Comparison of mass fluxes of sand (dp = 0.3 mm) for saturation carrying capacity Gs*
with experimental values (Gs)exp and mass flux values (Gs)mod computed from the model with Uslip =
UT under assumption of uniform solid holdup (µu = µs = const.) and with computed values (Gs)mod
(µu = µsb) and comparison of computed uniform µs (from the model) with (µsb) and top (µst) computed
from correlations of Bai and Kato [18]
Uf Gs* (Gs)exp (Gs)mod (Gs)mod µu=µsb µs = const. µsb [20] µst [20]
Uslip=UT Uslip=UT (model) (correl.) (correl.)
(m/s) (kg/m2/s) (kg/m2/s) (kg/m2/s) (kg/m2/s) ( - ) ( - ) ( - )
2.2 33.6 4.9 2.2 2.1 0.0234
2.4 39.5 8.8 12 8.9 0.0205 0.273 0.0484
2.6 45.8 12.5 17.5 13.3 0.0162 0.195 0.0309
2.74 50.5 15.9 19.9 15.5 0.014 0.162 0.0247
3.0 59.7 19.8 22.75 18.8 0.011 0.120 0.0177
3.4 75.3 22.6 25.1 21.7 0.0081 0.084 0.0121
2
PT
G
(kg/m/s)
As the Table suggests, fitting of experimental data by the model utilizing values µu = µsb is better.
On the other hand both values of particle holdup from correlations (µsb and µst) are higher than the
uniform value (µs = const.) computed on the basis of the model with Uslip = UT.
3.2 Validation of the models for conditions of dense suspension solids flow
We have found in literature [7, 8] only experimental data on solids circulation between a
fluidized bed and a short draft tube with orifices at the bottom. Ahn et al. [7] have measured
circulating flux of solids in the annular fluidized bed (GFB) . The relation between the flux in the
annular FB and the solids flux in the draft tube is under steady state conditions given by the
equation:
GFB * SFB = GPT * SPT or GPT = GFB * SFB / SPT ( 28 )
Where SFB and SPT is cross sectional area of the annular fluidized bed and inert part of the draft
tube (PT-column, riser) respectively. The geometrical parameters of the system were: ID and ED of
the draft tube = 0.1 m and 0.106 m respectively, ID of the annular FB = 0.3 m. GPT = GFB * 7.8764.
The four orifices in the draft tube (length = 0.9 m) had diameter 3 cm. The annular fluidized bed
(LFB H" 0.86 m) of sand particles (dp = 0.3 mm, Ás = 2620 kg/m3) was either at minimum fluidization
(Ua/Umf = 1) or at a moderately higher gas velocity (Ua/Umf = 1.4). The superficial air velocity in the
draft tube was 0.9, 1.35 and 1.5 m/s. Because of relatively large orifices and small particles of sand
the discharge coefficient CD was considered to be 0.5. The voidage µmf in the FB was considered to
be 0.5. The UT value for the sand particles was taken from internet [31] - computed by official
recommended method.
The experimental data of Ahn et al. [7] are compared with model predictions in Fig. 7. For such
conditions of dense suspension flow in the draft tube the assumption Uslip = UT is not valid and
therefore better predictions are attained by the model utilizing the slip factor.
Uslip=UT Slip factor Power rel.
Exp. Data, Ua/Umf=1 Exp.data, Ua/Umf=1.4
60
50
40
30
20
10
0
0,5 1 1,5 2 2,5
Uf (m/s)
Fig. 7 : Comparison of experimental data [7] with model predictions based on different
expressions for Uslip.
2
G
FB
(kg/m /s)
The model based on power correlation for solids holdup seems to be inconvenient for such
conditions, where the particle volumetric fraction in the draft tube is higher than 0.25. Because of
some small but significant bypassing of the gas from the annular FB to the draft tube the circulating
flow of particles is significantly higher at higher gas velocities (Ua/Umf = 1.4) in the fluidized bed 
as shown in Fig. 7.
Similarly comparison of experimental data of Kim et al. [8] gained on a circulating system
with an annular fluidized bed operating at Ua/Umf = 1 and a draft tube (length 0.90 m, ID = 0.1 m)
communicating with the FB by 4 orifices (ID = 0.03 m) confirmed superiority of model predictions
based on slip factor (Fig. 8). The authors [8] suggested on the basis of direct measurement of "Porif
and supposing for µmf = 0.5 the value 0.31 for CD to fit their measured values GFB. In our
comparison (Fig. 8) we compared model predictions for CD = 0.5 and CD = 0.31 with the measured
data for GFB. We accepted in modeling the value UT = 1.5 m/s for sand particles used by Kim et al.
[8] despite of the fact that rather values about 2.2 m/s are recommended for UT by official
computation methods [31]. The model with slip factor and CD = 0.31 fits fairly trend of dependence
of GFB on superficial velocity in the draft tube (riser) particularly when we take into account
bypassing of gas from annulus to the draft tube increasing with increasing gas velocity Uf [7, 8].
Uslip=UT, CD=0.5 Uslip=UT, CD=0.31 Slip factor, CD=0.5
Slip factor, CD=0.31 Power rel., CD=0.5 Exper.data-Kim
60
50
40
30
20
10
0
1 1,2 1,4 1,6 1,8 2
Uf (m/s)
Fig. 8: Comparison of experimental data [8] with model predictions based on different
expressions for Uslip and two different adopted values for the discharge coefficient CD.
Ua/Umf = 1 , µmf = 0.5.
Unfortunately for conditions of mass fluxes in the riser 40 < GPT < 200 kg/m2/s we have not
found relevant literature experimental data on circulating flow of solids in a dual FB-PT system for
validation of the models.
An attempt has been made further to use the models for estimation of effect of particle size,
effect of temperature on circulating flux of solids and to estimate effects of cyclone, impactor and
gas leakage (bypassing) from annular FB to the riser on solids flux in the riser.
2
FB
G
(kg/m /s)
4. Effects of particle size, temperature, gas bypassing and particle separation
devices on circulating flow of solids
4.1 Dependence of circulating mass flux on sand particle diameter
For chosen standard conditions given in Table III and the three simplified models for solid
particle velocity and circulating mass flux of solids we compared the theoretical forecasts for effects
of particle size on the mass flux GPT. The results are given in Fig. 9 . The basic model considering
Uslip = Uf/µ  Up = UT requires the superficial gas velocity Uf in the riser (PT-column) to be higher
than the terminal velocity of particles (otherwise the sand particles would not be lifted). The other
two models are able to solve the mass flux of particles even for conditions where UT > Uf .
Table III: Standard conditions used for computations of circulation mass flux in riser
Sand density: 2550 kg/m3 DPT = 0.1 m Uf air velocity in PT: 2.5 m/s
Sand particle size: 0.1  0.6 mm LPT = 2.5 m No cyclone
Temperature : 20 oC LFB = 0.5 m No impactor
Air density (20 oC): 1.2 kg/m3 SFB/SPT = 4 No gas leakage from FB to PT
Air viscosity (20 oC): 1.8 E-05 Pa*s Sorif/SFB = 0.02 No correction of FB inventory on
inventory in PT-column
Uslip = UT Power cor. with Slip factor
140
120
100
80
60
40
20
0 0,1 0,2 0,3 0,4 0,5 0,6 0,7
dp ( mm )
Fig. 9: Dependence of mass flux of sand particles on particle size dp for the three simplified
models for particle and slip velocity (Uf = 2.5 m/s, standard conditions).
As it is obvious the model predictions are rather different and for validation of the models
reliable experimental data from a dual FB-PT system are needed.
2
G
PT
(kg/m /s)
4.2 Effect of temperature on circulating mass flux of solids
The temperature is supposed to affect terminal velocity of particles (Table I), gas (air) density Áf
and gas (air) viscosity µf only. Simplified equations have been used to estimate the temperature
dependencies of air density and viscosity:
Áf(T) = 1.2 * 293.15/(T) ( 29 )
µf(T) = 0.000018 * (T/293.15)0.66 ( 30 )
where T is the absolute temperature in K .
Comparison of computed circulating mass fluxes of solids for selected model situations (Uslip =
UT and Up computed by means of a slip factor È ) for temperature 20 oC and 800 oC and standard
conditions (Table III) is shown in Fig. 10 . As it can be seen from the figure, the effect of
temperature on GPT is very small for the case of Up computed by means of slip factor. For the
particle velocity computed on the basis of simple relation (Usl = UT) increasing temperatures cause
an increasing solids flux  mainly due to the lower particle terminal velocity at higher temperatures.
Uslip = UT: 20 oC Uslip = UT: 800 oC
with Slip factor: 20 oC with Slip factor: 800 oC
140
120
100
80
60
40
0 0,1 0,2 0,3 0,4 0,5
dp ( mm )
Fig. 10 : Effect of temperature on dependence of circulating flux of solids (sand particles) on
particle size for two simplified models for particle velocity and slip velocity
(Uf = 2.5 m/s, standard conditions).
Again, validation of the theoretical predictions on particle size and temperature is needed. On the
other hand from both models it seems that the effect of temperature on circulating mass flux of
particles is rather smaller (under comparable other conditions).
2
G
PT
( kg/m /s )
4.3 Estimation of effects of pressure drop in cyclone, impactor and gas bypassing
All the above presented figures for mass flux of solids illustrated an idealized situation with only
a PT-column, without any cyclone/impactor and operation conditions without leakage of gas (air)
from the FB to the riser (PT-column). In reality, of course, the cyclone and impactor will exhibit a
significant pressure drop and will affect (reduce) the circulating mass flux of solids. On the other
hand gas leakage from FB to PT-column will cause two effects: elevation of mass fluxes of solids
(due to higher resulting gas velocity in PT-column) and slightly higher pressure drop in
cyclone/impactor.
The pressure drop in the cyclone is either independent on solids flow or can even moderately
decrease [32] with increasing solids flow. The pressure drop in plate type impactors generally
increases with mass flux GPT [33]. In literature usually two correlations are used to estimate the
pressure drop in cyclone "Pcycl [4,34]:
Rhodes and Geldart [34] supposed:
"Pcycl ( in Pa) = 33.33 * (Uf)2 ( 31 )
Kaiser et al [4] have taken the effect of temperature on gas density into consideration:
"Pcycl = 30 * Áf(T) * (Uf)2 ( 32 )
For the pressure drop in a plate impactor separator [33] we supposed a correlation:
"Pimpact = k1 * (Uf)k2 * (1 + k3*GPT) ( 33 )
where the relevant constants were assumed:
k1 = 10 , k2 = 2 and k3 = 0.1 ( 34 )
In the simplified modeling of circulation of solids in FB-PT system we have supposed that there
is negligible leakage of fluidizing gas from the annular FB to the riser (PT-column). In reality,
according to conditions, some part of the fluidizing gas will enter the riser together with solids and
will cause slightly higher gas velocity in vertical transport. To estimate such  gas leakage or  gas
bypassing effect on mass flux of solids in the riser we adopted the following assumptions: gas
leakage (bypassing) from the annular FB to rise is approximately directly proportional to GPT (flux
of solids). Volumetric gas flow rate bypassing from FB to PT-column (Vf )bypas (m3/s) is
proportional to (Uf )FB and dependent on (LFB)0.5 and (Sorif/SFB)0.5 . At (Sorif/SFB)o = 0.01 and (Lo)FB
= 0.25 m, the gas leakage is supposed to be about 7 % of volumetric flow rate in fluidized bed [30]
(Vf)FB. An empirical equation for computation of gas leakage from the fluidized bed to the transport
column can be assumed:
(Vf)bypas = 0.07 * (Vf)FB * [LFB/(Lo)FB]0.5 * [(Sorif/SFB)/ (Sorif/SFB)o]0.5 ( 35 )
where (Vf)FB , the volumetric flow rate of gas through the fluidized bed can be expressed:
(Vf)FB = Umf(dp, Ás , Áf ,µf ) * SFB * C1 ( 36 )
C1 should be such that [(Vf)FB - (Vf)bypas ] > (Vf)mf = Umf(dp, Ás , Áf , µf) * SFB ( 37 )
or
C1 > 1 + (Vf)bypas / (Vf)mf ( 38 )
The magnitude of C1 for usual systems should be between 1.1  1.4 to ensure that the volumetric
gas flow rate after subtraction of bypassing gas will be higher than (Vf)mf .
We used the equations (31), (33), (34) and (35) to estimate such effects of cyclone, impactor and
gas leakage. The other conditions have been considered  standard (i.e. Uf = 2.5 m/s, LFB = 0.5 m,
temperature 20 oC etc.  Table III). At higher temperatures the incipient fluidization velocity will
be lower and consequently the gas bypassing at 800 oC will be less important than at temperature 20
o
C.
Comparison of computed mass fluxes of solids in a riser without any other effects, and mass
fluxes considering the effect of impactor and the most realistic case with cyclone, impactor and gas
leakage effects included are compared in Fig. 11. Increasing temperatures cause a slight increase of
circulating mass fluxes of solids (Fig. 10) in the basic case (without other effects), lower pressure
drop in a cyclone (eq. 32) and smaller gas leakage from annular FB to riser.
Normal-without impactor with impactor only gas leakage+cyclone+impactor
140
120
100
80
60
40
0 0,05 0,1 0,15 0,2 0,25 0,3 0,35
dp ( mm )
Fig. 11 : Comparison of effects of impactor and overall effects of cyclone, impactor and gas
leakage from FB to PT-column on circulating mass flux GPT . Model with Uslip = UT ,
Uf = 2.5 m/s and standard conditions (Table III)
5. Regime constraints for pneumatic transport of solids
Stability of vertical pneumatic transport and limited pressure fluctuations are prerequisite for
successful application of the dual FB-PT system. At a given gas flow rate in a PT-column there is a
characteristic maximum smooth solids flow rate possible (according to concepts called saturation
carrying capacity of gas [18 ] or  classical choking [19]) where the suspension flow of solids starts
to collapse and the flow of solid particles starts to change into unstable slug flow with significant
pressure fluctuations and time instabilities (fluctuations) of solids flow [19]. Despite of some
controversies about the concept of  choking in the literature and its relation with geometrical
parameters, arrangement, particle properties and gas blower characteristics, there is an accepted
2
PT
G
(kg/m /s)
Yang correlation [19] for estimating classical choking in pneumatic transport and circulating
fluidized beds of smaller diameters (less than 0.15 m):
Uchok (Rep )0.06 /[32 (g dp)0.5] = [Gchok /(Uchok Áf)]0.28 ( 39 )
The predicted chocking conditions (Gchok) on the basis of eq. ( 39 ) are compared with
predictions of saturation carrying capacity of gas Gs* (correlation of Bai and Kato [18 ] ):
Gs* = (0.125 µf / dp) Fr1.85 Ar0.63 (Áf /(Ás - Áf))0.44 ( 40 )
and computed mass flux data for two different heights of fluidized beds (LFB = 0.5 m and 0.25 m)
and two different Sorif/SFB ratios (0.02 and 0.01) in Fig. 12. As it is obvious, higher fluidized beds
together with larger orifices and lower gas velocities can lead to instabilities, choking and formation
of denser fluidized bed at the bottom in the riser (PT-column). This is the significant practical
constraint for the regimes in a dual FB-PT circulating system [18,19].
G(chok)=Yang correl. G(PT)1 standard
G(PT)2 lower parameters Gs*
140
120
100
80
Conditions
60 for choking
40
20
0
11,522,533,54
Uf ( m/s )
Fig. 12 : Comparison of two computed mass fluxes with forecast of chocking from the Yang
correlation (39) and forecast of saturation carrying capacity of gas Gs* (40) for dp = 0.2 mm.
GPT1 corresponds to model computation with Usl = UT and standard conditions (LFB = 0.5 m,
Sorif/SFB = 0.02) , GPT2 corresponds to model computation with Usl = UT and lower parameters
(LFB = 0.25 m, Sorif/SFB = 0.01).
6. Conclusions
Three simplified models for slip and solid particle velocity in fast fluidization/pneumatic
transport have been considered, allowing solution of the basic equation for circulating flow of solids
through orifices between FB and riser (PT-column). The effects of circulating mass flux of smaller
particles on the flux of bigger particles have not been taken into consideration in modeling.
*
2
G
chok
, G
PT
or G
s
(kg/m /s)
However, according to theoretical models described in literature [35, 36] it seems, that the terminal
velocity and particle velocity of bigger particles will be changed and the bigger particles will be
increasingly  pushed by increasing mass flux of smaller particles.
The models are able to predict qualitatively effects of operating conditions and geometrical
factors on circulating flow of solids or mass flux of solids. The absolute values of mass fluxes are
unsure, because of uncertainty in the constants CD, µmf and in term Fpw in eqs. ( 1 ), ( 4 ), ( 6 ) and
in the exponent for the term with µu in eq. ( 1 ). Those constants are generally dependent on design
(geometrical) parameters of the system(s), orifice characteristics and on operating conditions. It
means that for a given solids circulating system and geometrical design the relevant constants have
to be found on a basis of experimental measurements of mass fluxes.
The models for description of circulating mass flux of particles in dual FB-PT systems have
been validated by comparison with experimental data only in a limited range of parameters 
mainly superficial gas velocities and mass fluxes in a vertical transport column. At conditions of
relatively small values of GPT (approx. below 40 kg/m2/s), relatively lower values of solid holdup
(1-µ), narrow size fraction of solid particles and superficial gas velocities higher than terminal
velocity of particles (UT) the model with a simple relation Uslip = UT is able to fit the experimental
data with reasonable precision. For smaller orifices the discharge coefficient CD is commonly lower
than 0.3.
At conditions of relatively high values of GPT (approx. over 200 kg/m2/s), relatively higher
values of solid holdup in the riser, broader particle size fractions and superficial gas velocities lower
than UT it seems that the model utilizing slip factor ( È = (Uf/µ)/Up ) with the correlation for È
according to eq. ( 20 ) is applicable for prediction of mass fluxes GPT. The convenient values of CD
for larger orifices are about 0.5. Unfortunately for conditions of mass fluxes in the riser 40 kg/m2/s
< GPT < 200 kg/m2/s we have not found relevant literature experimental data on circulating flow of
solids in a dual FB-PT system with annular FB and riser with orifices for validation of the models.
From theoretical predictions of the models and limited validation by experimental data we can
conclude that for a given size of particles the circulating mass flux of solids in PT-column increases
with increasing superficial gas velocity in PT-column, height of FB, cross sectional area of orifices,
with decreasing height of PT-column and only slightly with increasing temperature (for the same
superficial gas velocity). The combined effects of cyclone, impactor and gas leakage from annular
FB to riser (vertical PT-column) usually cause decreased mass flux of solids. The changes in solids
circulating rates due to such effects are less pronounced at higher temperature.
For control of circulation rate of a given particulate material in a dual FB-PT system three
practically important parameters should be considered: superficial gas velocity in PT-column, cross
sectional area (or diameter) of orifices and height of annular FB. Increase of all three mentioned
parameters causes higher circulation rate (mass flux) of solids.
However, avoiding the choking regime and avoiding operation at GPT higher than saturation
carrying capacity of the gas (instabilities and unacceptable pressure fluctuations in solids flow in
PT-column) call for operation conditions with lower fluidized bed height LFB, smaller orifices and
higher gas flow rate in the PT-column. It means that in reality the circulating flow of solid particles
should be optimized for the given particulate material and in the frame of acceptable/desired
operating conditions.
Acknowledgements:
This research was carried out within the European Commission´s Research and Development
Programme and was supported by EU RFCS grant RFCR-CT-2007-00005
List of symbols
Ar = dp3Áf g (Ás - Áf )/µf2 Archimedes (dimensionless) number ( - )
CD discharge coefficient ( - )
dp solid particle diameter ( m )
dp(aver) average solid particle diameter ( m )
Dorif diameter of orifice (m)
DPT diameter of PT-column ( m )
Dr riser diameter ( m )
Fr = Uf /(g dp)0.5 Froude (dimensionless) number ( - )
g acceleration due to gravity ( m/s2 )
G mass flux of solid particles ( kg/m2/s )
Gchok mass flux of solids corresponding to choking conditions ( kg/m2/s )
GPT mass flux of solid particles in riser or PT-column ( kg/m2/s )
Gs flux of solids ( kg/m2/s )
Gs* flux of solids corresponding to saturation carrying capacity ( kg/m2/s )
Hr height of the riser ( m )
LFB height of fluidized bed ( m )
LPT length of PT-column ( m )
mFB mass of solid particles in fluidized bed ( kg )
ms mass flow of particles in eq. 2 ( kg/s )
"P pressure drop ( Pa )
Rep Reynolds (dimensionless) number for given particles ( - )
SFB cross sectional area of fluidized bed reactor ( m2 )
Sorif cross sectional area of orifice(s) ( m2 )
SPT cross sectional area of PT-column ( m2 )
T absolute temperature ( K )
Uf superficial gas velocity ( m/s )
Ua superficial gas velocity ( m/s )
Up particle velocity ( m/s )
Up1 particle velocity for the modelwith Usl = UT ( m/s )
Up2 particle velocity for the model with slip factor È ( m/s )
Up3 particle velocity for the model with relative slip velocity ( m/s )
Uslip slip velocity (for an solid particle) = Uf  Up ( m/s )
Uslip (for assemblage of flowing solid particles) = Uf /µ  Up ( m/s )
UT terminal velocity of solid particle ( m/s )
Uchok velocity of gas corresponding to choking ( m/s )
Vf volumetric flow rate of gas ( m3/s )
Greek symbols
µ voidage = volume occupied by gas/total volume of the gas-solid system ( - )
µmf voidage at incipient fluidization ( - )
µs = (1 - µ) , volumetric concentration of solid particles ( - )
µsb volumetric concentration of solid particles at the bottom of the riser tube ( - )
µst volumetric concentration of solid particles at the top of the riser tube ( - )
µu voidage at the bottom of PT-column ( - )
È slip factor
µf viscosity of gas ( Pa * s)
Áf gas density ( kg/m3 )
Ás density of solid particles ( kg/m3 )
Äcirc time needed for one circulation of solid particle in dual FB reactor - PT-column system ( s )
Subscripts and short form of expression
bypas relevant for gas bypassing from annular fluidized bed to the riser (PT-column)
exp relevant to experimental data
FB fluidized bed
mf relevant to conditions at incipient (minimum) fluidization
mod computed from model
orif relevant to orifice(s)
PT pneumatic transport
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