Granular Matter (2016) 18:78 DOI 10.1007/s10035-016-0674-5

ORIGINAL PAPER

New one-dimensional hydrodynamics of circulating fluidized bed risers Francisco J. Collado1

Received: 22 January 2016 © Springer-Verlag Berlin Heidelberg 2016

Abstract A one-dimensional analysis of the hydrodynamics of circulating fluidized bed risers is performed following brand new mass and momentum balances for a gas–solids flow that the author has recently proposed. This analysis also includes a characteristic slip velocity between the phases derived from classical particle mechanics (the single particle equation). The predictions of the pressure drop and the solids holdup axial profiles, and the dense bed height are successfully compared with a rather limited amount of data. Keywords Circulating fluidized beds · One-dimensional gas–solid flows · Characteristic particle velocity · Void fraction · Pressure drop

Np p r Re u up u s (r, z) ut v

List of symbols

vγ

Ap CD dp d ps

z

FD, p FD FD,bed g G Gs Hd mp

B 1

Projected area of the particle Drag coefficient Particle diameter Mean surface diameter of the particles population Drag force on a single particle γ v 2 Drag force per unit volume Total drag force along the bed Gravitational constant Gas mass flux Solids mass flux Height of the dense bed Mass of the particle

Francisco J. Collado [email protected] Mechanical Engineering Department, Universidad de Zaragoza, Maria de Luna 3, 50018 Zaragoza, Spain

Number of particles per unit volume Gas pressure Radial coordinate with origin the bed vertical axis ρvd Reynolds number μ p

G Gas velocity ρ(1−ε s) Particle velocity i.e., characteristic one dimensional velocity of the population of particles Local solids velocity Terminal velocity (for the mean surface diameter d ps ) Slip velocity i.e., characteristic one dimensional slip velocity of the population of particles derived from one dimensional Newton’s laws of motion, v = up u m− ·g p ‘Friction’ velocity γ v,d ( p) Vertical coordinate (height), positive upwards and origin the gas distributor

Greek symbols εs γ ρ ρs

Cross-sectional average of solids void fraction Friction coefficient between gas and particles,  γ = γ v, d p Gas density Solids mass density

Subindex i d o R

Inlet Dense bed surface Outlet Return line

123

78

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1 Introduction Circulating fluidized bed (CFB) reactors have been used for a wide range of industrial applications, such as fluidized catalytic cracking, combustion and gasification, metallurgical processes, etc., over the past 40 years [1–3]. CFB are gas–solids reactors containing very fine particles that are fluidized at a rather high gas velocity and are blown out of the bed reactor. Individual particles are blown off the bed when the gas velocity exceeds the terminal velocity of particles u t [1]. To achieve fast fluidization, solids must be continuously fed to maintain the required solids holdup in a vertical riser. This is usually realized by capturing solids leaving at the top and returning them to the bottom of the riser through a recirculation system. Since the key features of CFB result from the gas–solids cocurrent upflow in the riser where most reactions occur [3], this paper focuses on the riser. Understanding solids axial dispersion and flow behaviour in CFB risers is key to successful design and scale-up of CFB systems. The weight and acceleration of solids govern the axial profile of the pressure drop occurring along the CFB riser, and they also determine heat and mass transfer rates and chemical reaction performances [3–5]. However, it is well known [6,7] that the flow structure in CFB risers is rather complex and highly heterogeneous both in radial and axial directions. Along the riser, the axial profiles of the concentration of solids can be divided, in general, into three zones [1,2]: a dense bed at the bottom, a splash or transition zone, and an upper dilute or transport zone. Spatially, the dense bed has a time-average solids concentration that is uniform to a certain extent. Strong back-mixing above the dense bed (in the splash zone) causes rapid decay in the concentration of solids over a few metres, followed by a dilute zone with less pronounced back-mixing occurring mainly at the furnace walls. In the radial direction, a coreannulus (wall) flow structure is typical [4,8–10] with a dilute core upward gas–solids flow and a downward dense flow of solids near the wall. Some authors have recently pointed out that [10,11] the simplest and most convenient analysis approach to evaluate the effects of solids acceleration on pressure drop, despite gas–solids flow structure complexities, would be based on one-dimensional models, where flow properties are crosssectionally averaged and change from various flow regimes along the riser [1,2,4,5,11–13]. Others refine their 1-D models [4,8–10], splitting the cross-section of the riser into a core (solids upwards) and an annulus (solids downwards), in which the annulus thickness decreases upwards. However, predictions of both 1-D models only yield some reasonable flow structures in the upper dilute zone, or freeboard [10]. These solids holdup predictions along the lean zone are usually based on the work of Kunii and

123

F. J. Collado

Levenspiel [1,4]. They propose that solids are distributed into two regions in  the vessel, a constant solids fraction εsd m3 solids/m3 in the lower dense region of height Hd and an upper lean region of height Hl in which the solids fraction εs (z) falls exponentially with height (z) from εsd towards the lowest limiting value or transport carrying capacity εs∗ , see Fig. 1. Thus, four numerical values would be needed, namely a (an empirical decay coefficient in the exponential function), εsd , εs∗ , and a solids fraction at the outlet, εso . The latter is related to the solids mass velocity G s by a solids material balance at the reactor outlet [4] G S = ρ S (u o − vo ) εso ,

(1)

where ρ S is the material solids density, u o is the outlet gas velocity and vo the outlet slip velocity. The relative velocity of the gas respect to the particles i.e., v, see Fig. 1, is defined as [1,4] v = u − u p,

(2)

where u and u p are the gas and particle velocities, respectively. This relative speed v is well known in fluidization [1] as the slip velocity. Values for all these constants can be found in the literature [1,4], although no generalised agreement has been reached. For example, a correlation for εsd proposed in [5] predicts experimental data with a relative deviation of less than 30 %. Furthermore, the reported values for the outlet slip velocity vo , see Eqs. (1, 2), differ widely [4]. Consequently, for the dense bed and the transition zone, the 1-D predictions are significantly deviated from reality [11]. For instance, cross-sectional averaged solids concentration in a riser flow has been roughly estimated from pressure drop measurements by equating the gravitational force from local solids holdup to the local axial gradient of pressure [13]. However, by neglecting the effects of solids acceleration, the converted volumetric solids holdup is much higher than the actual (measured) solids concentrations [8]. Finally, the severe difficulties current models experience in capturing the axial pressure profile, as well as the axial and radial solids profiles, are clearly recognized in [14]. The conclusion is that the current state of the art about CFB hydrodynamics is rather unsatisfactory. Given some solids properties, a superficial gas velocity and a mass solids velocity flowing along the CFB riser at a steady state, some key questions about the flow structure have not yet received a rational answer [1–14]. In particular, how the solids volumetric fraction is defined at the dense bed, in addition to the bed height; which would be a coherent function of the solids distribution along the transition and the lean zone; in which zones solids acceleration begins and finishes; and, finally,

New one-dimensional hydrodynamics of circulating fluidized bed risers

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78

Fig. 1 Sketch of the circulating fluidized bed reactor

how the pressure drop evolves along the different regions following solids distribution and acceleration. In short, this study intends to improve the 1-D hydrodynamics knowledge of CFB risers by exploring answers to the above questions, but based on first principles. The analysis is mainly conducted on new mass and momentum balances for a vertical gas–solids flow. These balances are based on new hyperbolic conservation laws for a two-phase flow that the author has recently derived [15]. The analysis also includes a new slip velocity expression derived from classical particle mechanics. Finally, the pressure drop and holdup predictions of the model are successfully compared against only two tests from the Technical University of Hamburg [8]. However, the promising results could support new and more exhaustive comparisons of the new hydrodynamic model, which could then be useful for CFB design and scale-up. These new results would also be the first experimental confirmation of the new two-phase flow fundamentals.

is cross-sectional averaged, and all the velocities considered are upwards. Furthermore, the solids are distributed in two regions in the riser: in the first instance, a constant solids fraction εsd in the lower dense region of height Hd and an upper lean region, or freeboard, in which the solids fraction εs (z) falls with height (z) from εsd towards the outlet solids holdup εso , see Fig. 1. However, as a main difference from [1,4], the exponential decay of solids concentration is not assumed here. On the other hand, and following the new two-phase flow fundamentals, the constancy of the solids fraction in the dense bed will be slightly modified later. Finally, concerning the thermodynamic properties of the gas since the model is compared against experimental data taken at a cold model unit at TUHH [8], isothermal (ambient) flow and ideal gas are used.

2 One-dimensional hydrodynamic model 2.1 General assumptions

Classical equations [1–4] are used for gas continuity at a steady state. Furthermore, as a first approach, gas density changes along the riser are neglected,

The general assumptions are mainly based on [1,4]. Therefore, the steady one-dimensional model proposed here is the simplest, in other words, any solids void fraction value εs

G = u i ρi = uρ (1 − εs ) = constant ⇒ ui u i ρi ≈ , u= ρ (1 − εs ) (1 − εs )

2.2 Gas continuity

(3)

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F. J. Collado

where G is the gas mass velocity, u i is the gas velocity just before bed inlet or superficial velocity [1,4], and ρi is the inlet gas density. Moreover, u stands for the gas velocity at any riser height z, see Fig. 1, in which the gas voidage is (1 − εs ). At the dense bed, assuming a practically constant solids fraction εsd , and also considering that the gas density variations in the dense bed are not very high, the gas velocity along the dense zone u d would also be practically constant [12]. Then, from Eq. (3), ud ≈

ui . − (1 εsd )

(4)

Regarding gas velocity above the dense bed, along the freeboard, and as mentioned above, there is a general agreement about a strong decay of the solids fraction just from the bed height thus the gas velocity should mainly follow the decay of the solids fraction, see Eq. (3). Finally, at the riser outlet, for isothermal flow and ideal gas, the whole gas density decrease along the riser would be related to the whole riser pressure drop p = pi − po . Although, in first instance, the outlet gas velocity u o would not include gas density changes uo =

u i ρi u i ( po + p) ui , = ≈ (1 − εso ) ρo (1 − εso ) po (1 − εso )

(7)

From the last equation, the slip velocity just at the bed height vd would be v (z = Hd ) = vd = u d − u t ,

(8)

where u d is defined in Eq. (4).

2.3 New particle characteristic velocity along the riser From Eq. (2), a one dimensional upward characteristic particle velocity (of the population of particles) is directly derived

2.4 New momentum balance for vertical one-dimensional gas–solids flow Here we use a new momentum conservation equation for a two-phase global mass system following a gas phase, which the author has recently derived [15]. For a one-dimensional steady state and without mass transfer between the phases, the new global (gas–solids) momentum proposed is

(6)

Point out that, after the gas velocity has been defined by Eq. (3), u p is a direct function of the slip velocity v, which will be derived later from classical mechanics using the single particle drag equation. On the other hand, notice that the actual local solids velocity u s (r, z) may be upwards (in the core of the bed) or downwards (near the walls). In this work, the particle characteristic velocity just at the bottom of the riser has been considered as zero u p (z = 0) = 0. This could be supported by the previous assumption about the initial motion of solids [12], and the above-commented core-annulus models [4,8–12]. Since the strong back mixing of solids is produced by an upward flow in the core of the riser combined with downward falling clusters at the wall. Then, at the reactor bottom, the fluidizing gas must change the flow direction of the solids from a downward vertical flow to an upward one. Therefore, just at the bottom (z = 0) , the net

123

u p (z ≥ Hd ) = u t = u (z) − v (z) .

(5)

where pi and po are the inlet and outlet pressures, respectively. See Fig. 1.

u p (z) = u (z) − v (z) .

movement of solids could be considered horizontal, in other words, with zero vertical velocity. Moreover, following [12], the height of the solids acceleration region is assumed as the height at which solids—initially held motionless at the reactor distributor and then suddenly released in a dense bed of constant voidage—reach their terminal velocity u t under constant gas velocity, namely u d . Indeed, as we will show later, this height would physically coincide with the dense bed height Hd . Therefore, after the solids reach their terminal velocity u t , which is the velocity the particle reaches when the net force experimented by the mean diameter particle is zero [16], the net force on the particles is held equal to zero. Thus, just from the bed height, the particles would move at a constant terminal velocity until the riser outlet.

−

 τw 4π D d   dp = gρs εs +gρ (1 − εs ) + u ρs εs u p + dz Ac dz d + (9) [u (ρ (1 − εs ) u)] , dz

where the forces (per unit volume) considered over the whole system are gas pressure forces, solids and gas gravity forces, and shear forces on the boundary of the system, which are accounted for by a wall shear stress. It is noticeable that the drag force of the gas on the solids is internal to the control volume considered here. However, classical treatments [10– 12] usually separate the forces on the gas from solids forces. The only difference between Eq. (9) and classical onedimensional models is the particles acceleration term. The particles mass flux is multiplied by gas velocity, whereas, in classical models, the solids velocity multiplies this solids flux. This discrepancy is justified in [15].

New one-dimensional hydrodynamics of circulating fluidized bed risers

Equation (9) is now simplified assuming that [4–6] gas velocity variations are much lower than particle velocity variations, gas gravity is negligible against the weight of the solids, and the whole system wall friction is small compared to the other terms, −

   dp = gρs εs + ρs d u εs u p . dz

(10)

Therefore, the gas pressure gradient (N /m 3 ) must basically overcome the gravity and acceleration of solids. Finally, integrating along the riser from the inlet i to any station at a z distance from the inlet, the profile of the pressure drop, relative to the atmospheric pressure po , p (z) = p (z) − po would be pi − p (z) = p − [ p (z) − po ] = p − p (z)  z  z    = gρs εs dz + ρs d u εs u p , 0

(11)

0

where the whole riser pressure drop p has been included. Equation (11) needs expressions for gas and particles velocities, both in function of z. Both velocities are closely related to the slip velocity here, see Eq. (6). A new function for εs (z) is also necessary since, although the solids void fraction is nearly constant along the dense bed, we will see later that few variations may be significant. 2.5 New characteristic slip velocity Here, as we have already advanced above, a one-dimensional upward characteristic slip velocity of the population of particles is derived from classical mechanics using the single particle drag equation, which will have the same expression at the dense bed as along the freeboard. However, as we will show later, the respective characteristic slip velocities will be different since, in the dense bed, the gas velocity is considered practically constant whereas, along the freeboard, such velocity decreases following the solids void fraction collapsing. 2.5.1 Single particle drag equation Following Tong [16] (page 34), we state the classical force balance for an individual particle of mass m p , with onedimensional upward velocity u p , dragged by an upward gas flow with velocity u, Eq. (3), and neglecting buoyancy effects [12]. Therefore, the particle velocity relative to the gas velocity would be the opposite of the slip velocity, see Eq. (1), namely −v = u p − u. Hence, the equation of motion is given by

Page 5 of 14

− mp

dv = dt

+

−m p g   particle gravity

=

1  g

γ v2  

⇒

particle drag

dv  2  = dt

1 − vvγ

78

dv 1

g 1 − γ v2 m pg

(12)

where the mass particle multiplied by its acceleration is equated to the particle weight plus the (quadratic) drag force on particle γ v 2 , in which γ is called the coefficient of friction. By definition [17], this drag force is parallel to the free-stream velocity. In Eq. (12), for the sake of convenience, also based on [16], the coefficient of friction γ is considered constant. Moreover, the factor mγp g , which multiplies to the square of the slip velocity in Eq. (12), is also taken as a constant. As the square root of the inverse of this factor has dimensions of velocity, here it is called ‘friction’ velocity vγ because it would be, in some manner, equivalent to the assumed constant coefficient of friction  vγ =

m pg . γ

(13)

However, strictly speaking [17], the friction coefficient γ is not a constant but a function of the different diameters of the particle population and also of the drag coefficient C D  1  γ v, d p = C D ρ A p , 2

(14)

where A p is the projected area of the particle (π d 2p /4). The drag coefficient C D is a function of the Reynolds number, which, in turn, is based on the relative speed and a characteristic dimension of the body, namely the particle diameter d p ρvd [17]. Hence, with Re = μ p and C D = C D (Re), we have   C D = C D v, d p . Therefore, substituting Eq. (14) into Eq. (13), the so-called ‘friction’ velocity would be  vγ =  =

m pg =  γ v, d p



4ρs d p g   . 3C D v, d p ρ

2m p g   C D v, d p ρ A p (15)

In the above equation, mass particle m p has been substituted by the product of solids density and particle volume, so ρs V p = ρs (π d 3p )/6, and then combined with the previous projected particle area A p . Highlight that the actual terminal velocity of a particle u t is by definition [11] the velocity reached when the net force on the particle is zero. Therefore, from Eq. (12) and having into account Eq. (15),

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 ut =

F. J. Collado

4ρs d p g   . 3C D u t , d p ρ

(16)

Notice  that,  in Eq. (15), we first calculate the drag coefficient C D v, d p entering with any value of the slip velocity v, for a specific particle diameter, and we directly get the corresponding ‘friction’ velocity vγ , which is definitely different from the inlet value of v. On the other hand, in Eq. (16), we have to enter into the drag coefficient with a u t value that must be equal to the outlet result from Eq. (16). For any particle diameter of the size distribution, this terminal velocity is found by trial and error since it is not possible to directly derive u t from the drag coefficient expression. So, Table 1 shows the calculation of the terminal velocity, Eq. (16), for the diameters of the sand particle size distribution used in the CFB tests from TUHH [8]. Note that the inlet velocity has been varied until the outlet result has coincided with such entry. On the other hand, Table 2 displays, for some particle diameters, the ‘friction’ velocities found from Eq. (15) entering with two possible values of the characteristic slip velocity. 2.5.2 Characteristic slip velocity along the dense bed Next, the integration of Eq. (12) would supply slip velocity v = v (z). However, for computing the previous pressure

Table 1 Terminal velocities, Eq. (16) d p (μm)

u t (m/s)

Reynolds

CD

65

0.27

1.16

d ps = 130

0.83

7.15

24.55 5.33

200

1.48

19.60

2.58

340

2.73

61.47

1.29

Quartz sand (φ = 0.95) [20]

Table 2 ‘Friction’ velocities, Eq. (15) Test A, z = 0

Following the basic assumptions at the dense zone, the gas velocity is practically constant and set to u d , Eq. (4). Considering vγ constant, the integration of Eq. (17), which is related to atanh (x) [16,18], is immediate and yields 

z 0

=



v

u d dv   2  ⇒ z vi 1 − vvγ     u d vγ atanh vvγ − atanh vvγi

1 dz = g

 = vγ tanh

g



zg ud + atanh u d vγ vγ

⇒ v (z)

 .

(18)

In Eq. (18), vi has been substituted by u d . This is based on Eq. (6) and the assumption that the particle characteristic velocity just at the bottom is zero i.e., u p (z = 0) = 0. From Eq. (15), it is clear that vγ is not a constant but a function of v and d p . However, for the sake of convenience, it has been assumed that the drag coefficient here is not a strong function of the slip velocity in the interval [v (z = 0) , v (Hd )]. Another question is what the representative diameter of a size distribution of particles to be used in Eqs. (15, 16) should be. In this work, vγ is varied until the calculated pressure drop profile is fitted to data then, knowing the slip velocity, the characteristic particle diameter may be derived from Eq. (15).

d p (μm)

v (m/s)

Reynolds

CD

vγ (m/s)

290

u d ≈ 4.10

78.74

1.13

2.70

A, z = 0

340

u d ≈ 4.10

92.32

1.04

3.04

A, z = Hd (0.4 m)

290

vd ≈ 3.27

62.80

1.27

2.54

A, z = Hd (0.4 m)

307

vd ≈ 3.27

66.48

1.24

2.65

A, z = Hd (0.4 m)

340

vd ≈ 3.27

73.63

1.17

2.87

Test B, z = 0

290

u d ≈ 5.13

98.52

1.01

2.85

B, z = 0

340

u d ≈ 5.13

115.51

0.94

3.21

B, z = Hd (0.24 m)

290

vd ≈ 4.30

82.58

1.10

2.73

B, z = Hd (0.24 m)

300

vd ≈ 4.30

85.43

1.08

2.80

B, z = Hd (0.24 m)

340

vd ≈ 4.30

96.82

1.02

3.08

Quartz sand (φ = 0.95) [20]

123

drop profile, Eq. (11), instead of solving v in function of time t, an expression in function of z is needed. Bearing in mind that gas velocity is a space-time ratio, Eq. (12) is accordingly modified [16] where the gas velocity is taken as reference   1 udv dz  ⇒ dz = (17) dt =  2  . u g 1 − vvγ

New one-dimensional hydrodynamics of circulating fluidized bed risers

Finally, using the formula of the hyperbolic tangent of the sum of two arguments [18], the new expression of the slip velocity along the dense bed is 

  + tanh u dzgvγ  ,   v (z ≤ Hd ) = vγ 1 + vuγd tanh u dzgvγ ud vγ

(19)

2.5.3 Characteristic slip velocity along the freeboard Strictly speaking, according to the proposed model, the integration of Eq. (17) along the whole riser should be split into two parts following gas velocity changes. So, the first integration has been performed previously, see Eqs. (18, 19). Then, the second one would be from the bed surface to the reactor outlet (from z = Hd to z = H ), in which the gas velocity is now variable and equal to the constant solids terminal velocity plus the variable characteristic slip velocity, see Eq. (6). However, it can be checked that this second integration along the freeboard gives an implicit function of the slip velocity making analysis difficult. As an alternative for the zones above the dense bed, an inspection of Eq. (19) suggests using the same function for the slip velocity, but with different parameters. If z went to infinite, the slip velocity would go to vγ . Whereas, above the dense bed, if z went to infinite, the slip velocity would go to the outlet slip velocity vo , which is related to the solids mass velocity G s , see Eq. (1). In conclusion, and as a first approximation, the slip velocity function along the free board would be 





d )g + tanh (z−H u a vo  .   v (z > Hd ) = vo d )g 1 + vvdo tanh (z−H u a vo

vd vo

(20)

Indeed, Eq. (20) is the integration of Eq. (17) from z = Hd to z, assuming that the average gas velocity is u a , in other words, the arithmetic mean of the gas velocity variations between the dense bed surface and the outlet, u a = (u d + u o ) /2.

78

(20), combined with the solids terminal velocity u t , makes it possible to calculate the gas velocity profile along the freeboard



and the characteristic particle velocity along the dense bed can be performed from Eq. (6).



Page 7 of 14

(21)

This now eases the analytic integration along the transition and the lean regions of the riser. In this work, as before with vγ , vo is varied to fit the calculated pressure drop profile along the freeboard to the measured one. In the proposed model, the solids velocity holds constant after they reach their terminal velocity, see Eq. (7). Hence, Eq.

u (z ≥ Hd ) = u t + v (z > Hd ) .

(22)

These two integrations of the force balance for a particle, see Eqs. (19, 20), are clearly not strict: vγ has been assumed constant to ease integration, and the gas density changes have not been taken into account. Furthermore, Eq. (20) works with an average gas velocity above the bed, which is obviously a crude simplification, since gas velocity is a function of slip velocity. However, these simplified analytic slip velocities ease the analysis of the main hydrodynamic trends in the CFB and, as we will see later, also give acceptable predictions of the pressure drop and the solids holdup. 2.6 Solids profile along the riser 2.6.1 Solids profile in the dense bed The new hyperbolic conservation equations for a two-phase flow without mass exchange between the phases [15], on which this work is partially based, suggest that a new mass conservation of the whole mass (gas and particles) should be the so-called zero-net-flux mass condition. Its one-dimensional expression is a simple function of the variations of the concentration of solids and slip velocity along the dense bed. Taking the values just at the bed surface as a reference, d (ρs εs v) = 0 ⇒ εs (z) v (z) = εsd vd ⇒ εs (z ≤ Hd ) dz εsd vd . (23) = v (z) 2.6.2 Solids profile along the freeboard Equation (23) has been checked above the dense bed and it does not work. This could be due to the strong discontinuity in the solids concentration profile arising when the solids reach their terminal velocity, and end their acceleration, just at the dense bed surface. Using simplified gas continuity is suggested here as an alternative, Eq. (3), to arrive at the solids void fraction. The gas velocity above the bed has already been derived, Eq. (22). Thus, from Eq. (3), εs (z > Hd ) = 1 −

ui ui =1− . u (z) u t + v (z)

(24)

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78

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F. J. Collado

(a)

3 Results

11

u =3 [m/s] i εsd=0.2675

10 9

v =2.65 [m/s] t v =2.28 [m/s]

8

u =0.83 [m/s]

123

t

pd=5.16 [kPa]

7

p

slip velocity v cal. Δp meas. Δp cal. εs

H =0.40 [m]

meas. εs

d

6 5 4 3 2 1 0

0

0.5

1

1.5

2

z [m]

(b) 11

u =4 [m/s] i εsd=0.22

10

gas velocity u particle velocity u

v =2.8 [m/s] t vo=3.15 [m/s]

9

s

Δp (kPa) velocity (m/s) ε *10

Only two tests [8] reported the simultaneous measurements of the axial pressure profiles and the cross-sectional averaged solids concentration distributions in the bottom and splash zones of a cold model CFB unit. The riser has a diameter of 0.4 m and a height of 15.6 m, where the externally circulated solids were returned at a height of 1 m. In these experiments, ambient temperature air, with atmospheric pressure at the outlet, transports quartz sand   with d p5 = 65 µm, d p50 = 200 µm, d p95 = 340 µm a mean surface diameter d ps = 130 µm. Sand particle density is assumed equal to ρs = 2600 kg/m3 [19] here. The superficial velocities and inlet solids mass velocity for the two experiments (both with a riser pressure drop of around 10 kPa) were u i = 3 m/s, G s = 9.3 kg/m2 s and u i = 4 m/s, G s = 23 kg/m2 s for tests A and B, respectively. The measured maximum solids porosities for tests A and B were εsd = 0.2675 and εsd = 0.22, respectively. This research riser [8] is equipped with many pressure drop taps to measure the axial pressure profile. The deviation between the sum of the individual differential pressure measurements and the measured overall pressure drop was found to be less than 5 %. The local solids concentrations were measured by two different techniques, namely γ -ray absorption and fibre-optical reflection probes, resulting in good agreement each other. The authors [8] calculated the reported cross-sectional averaged solids concentrations through a relationship of radial voidage profiles. Figure 2a, b (for tests A and B, respectively) show the reported pressure drop profiles (square symbols) and the cross-sectional averaged solids concentration (circle symbols). Finally, as we have already commented, Table 1 shows the calculated terminal velocities for each particle diameter of the size distribution used [8]. These terminal velocities have been calculated through Eq. (16) by trial and error, using the most accurate correlation of the drag coefficient for isometric particles (RMS deviation ≈ 3 %) proposed in [20] for non-spherical particles. Particle sphericity has been set to φ = 0.95, which fits, with the same correlation, the terminal velocity value (0.93 m/s) reported in [19] for other quartz sand size distribution with d ps = 140 µm. In this study, after several preliminary runs of the model, the terminal velocity of the mean surface diameter (d ps = 130 µm), which is u t = 0.83 m/s in Table 1, is considered the representative terminal velocity of the solids size distribution.

gas velocity u particle velocity u

o

s

Δp (kPa) velocity (m/s) ε *10

3.1 Experimental data from the TUHH Circulating Fluidized Bed [8]

u =0.83 [m/s]

8

t

pd=6.31 [kPa]

7

p

slip velocity v cal. Δp meas. Δp cal. εs

H =0.24 [m]

meas. εs

d

6 5 4 3 2 1 0

0

0.5

1

1.5

2

z [m]

Fig. 2 a Pressure drop and solids holdup profiles in the TUH-CFB for u i = 3 m/s and G s = 9.3 kg/s m2 . b Pressure drop and solids holdup profiles in the TUH-CFB for u i = 4 m/s and G s = 23 kg/s m2

The corresponding ‘friction’ velocities vγ for several particle diameters have also been calculated, see Table 2, using the same drag coefficient correlation [20] and the same sphericity as before. For convenience, only the results for the larger diameters are shown. So, for tests A and B, we have entered into Eq. (15) with the two extreme values of the slip velocity v along the dense bed, namely the initial slip velocity (z = 0), which is the initial gas velocity u d , Eq. (4), and the slip velocity just at the bed height 3 = 4.1 (m/s) vd (z = Hd ), Eq. (8). Therefore, u d ≈ 1−0.2675 4 and u d ≈ 1−0.22 = 5.13 (m/s) for tests A and B, respectively whereas vd = 4.1−0.83 = 3.27 (m/s) for test A, and vd = 5.13−0.83 = 4.30 (m/s) for test B. The resulting Eq. (15) outputs are shown in Table 2.

New one-dimensional hydrodynamics of circulating fluidized bed risers

3.2 Results for the dense bed 3.2.1 Simultaneous fitting of pressure drop and solids holdup profiles along the dense bed The pressure drop profile along the dense bed is calculated through Eq. (11) including the former assumptions about the constancy of u d along the dense bed, Eq. (4), and the zero initial solids velocity, u p (z = 0) = 0, p (z ≤ Hd ) = p − gρs  z × εs (z) dz − ρs u d εs (z) u p (z) .

Page 9 of 14

particle velocity reaches its representative terminal value, that is u t = 0.83 m/s, which is that of the mean surface particle (d ps = 130  see Table 1. Then u p (z = Hd ) =  µm), u t = u d − v z, vγ ⇒ vd = u d − u t . Thus, we could say  that we have arrived to the dense bed surface when v z, vγ = vd = u d − u t . To verify this result, an analytic expression of the bed height is derived using the first part of Eq. (18), Hd =

(25)

0

Note that in Eq. (25) the measured whole riser pressure drop p must be an input. Moreover, the maximum solids void fraction εsd , at the dense bed surface, has been assumed known and equal to the measured one. This issue will be addressed later. So, known εsd , u d is calculated from Eq. (4) and vd from Eq. (8) using the above commented representative terminal velocity of the size particle distribution i.e., that of the mean surface diameter, see Table 1. Therefore, to obtain the pressure drop profile along the dense bed, Eq. (25), we need the solids void fraction profile εs (z), see Eq. (23), and the characteristic particle velocity function u p (z) i.e., Eq. (7). However, both are direct function of the characteristic slip velocity, which, in turn, is now function only of the so-called ‘friction’ velocity vγ , see Eq. (19). Consequently, we have varied vγ until the calculated pressure drop profiles fit the measured ones (see Fig. 2a, b). Point out that, logically, that same value of vγ has been simultaneously used to draw the solids void fraction profile. Integration in Eq. (25) is solved by numerical integration. The matlab files used are deposited as electronic supplementary material. The pressure drop and solids holdup profiles measured in tests A and B [8] are also included in these files. We have found that the model supplies acceptable fits of the pressure drop, as well as solids holdup, along the dense bed for vγ ≈ 2.65 m/s and vγ ≈ 2.8 m/s for tests A and B, respectively. See Fig. 2a, b. For both tests, these ‘friction’ velocities values would correspond with d p = 307 µm for test A and d p = 300 µm for test B, entering with the characteristic slip velocity value at the bed surface namely, vd , see the corresponding cursive rows in Table 2. However, about the calculated solids concentration profile, it could be clearly argued that this would be due merely to the fact that εsd is a data entry and εs (z) along the dense bed only experiments low variations around it. 3.2.2 Prediction of the dense bed height This study also found that the height of bed Hd can be approximated for both tests as the location where the characteristic

78

     vd ud 1 vγ u d atanh − atanh . g vγ vγ

(26)

Equation (26) is now transformed dueto natural logarithms  resulting from the atanh (x) = 21 ln 1+x function [18], 1−x since the actual arguments of the inverse hyperbolic tangents in this equation are greater than 1, but mathematically these arguments should be ≤ |1| [18].       vγ + u d vγ + vd 1  − ln  Hd = vγ u d ln 2g vγ − vd vγ − u d ⎧

⎫ vγ +vd ⎪ ⎪ ⎨ ⎬ vγ −vd 1

vγ u d ln = ⎪ 2g ⎩ (vγ +u d ) ⎪ ⎭ (vγ −u d )

(27)

Again, εsd is assumed known, thus the values of u d and vd are the same as before. The values of vγ used for both tests are the same as that we have found fitting the respective pressure drop and solids holdup profiles. See Fig. 2a, b. Therefore, the calculated values of Hd are direct function of the chosen vγ . In other words, when we vary the ‘friction’ velocity vγ trying to fit the pressure drop profile, we are simultaneously fitting not only the solids holdup profile but also the dense bed height. Finally, from Eq. (27), the calculated Hd for test A is 0.40 m, which may be compared with that reported in [8], namely around 0.39–0.40 m. Whereas for test B, the calculated Hd is 0.24 m and the measured one is around 0.22–0.26 m [8]. See Fig. 2a, b, respectively. In [8], it is suggested that the height of the bottom zone may be defined as the point where the solids concentration starts to decrease. Besides, this bed height location compares well with that proposed in [9], that is where the measured pressure profile starts to deviate from the straight line. 3.3 Results for the freeboard As already pointed out, see Eq. (8), a key point of this onedimensional model is the assumption that after reaching its terminal value, the characteristic particle velocity does not vary until the riser outlet. This is because, by definition of terminal velocity [16], the net force on the particle will be zero thereafter. From Eq. (9), the gas velocity will then follow

123

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F. J. Collado

Eq. (22), whereas the slip velocity v (z) will now follow Eq. (20).

again. Logically, the lower the inlet gas velocity, the greater the length of the splash zone.

3.3.1 Simultaneous fitting of pressure drop and solids holdup profiles along the freeboard

3.3.2 Outlet characteristic slip velocity vo and its relation with G s

The pressure drop profile above the dense bed is again calculated through Eq. (11). At the dense bed surface, when the solids reach their terminal velocity, there is strong discontinuity. The gas velocity is not a constant now, see Eq. (22), but the solids velocity is (u t ). Furthermore, the solids void fraction goes down as the gas velocity decreases, see Eq. (24),

In defining Eq. (20), it was already suggested the theoretical meaning of vo namely, when z went to infinite, the characteristic slip velocity would go to this outlet value. Thus, to indirectly check the validity of the above fittings, we are going to compare them with some possible actual value of vo now derived from the actual solids mass flux G s of the tests reported in [8]. So, at the outlet, Eq. (2) becomes

 z εs dz p (z ≥ Hd ) = ( pd − po ) − gρs H d  

vo = u o − u t =

solids gravity

− u t ρs [u (z) εs (z) − u d εsd ] .

 

(28)

gas decceleration

As before, assuming that εsd is a data entry to the model, we should now find the value vo for the characteristic slip velocity along the freeboard, see Eq. (20), that best fits the pressure drop above the dense bed and, at the same time, also fits the solids concentration decay. First, the approximated average of the gas velocity u a = (u d + u o ) /2, on which Eq. (20) is based, must be calculated. However, here, as an acceptable simplification, the outlet gas velocity is approximated to the inlet superficial velocity i.e., u o ≈ u i , see Eq. (3). It is noticeable that the first term, the gravity of solids, contributes to the pressure drop, but the second term is a deceleration, thus reducing the pressure drop. As before for the dense bed, we have varied vo until the calculated pressure drop and solids holdup profiles fit the measured ones (see Fig. 2a, b). In Eq. (28), integration is again solved by numerical method, see matlab files in electronic supplementary material. The model has been found to supply acceptable fits for the pressure drop and solids void fraction profiles above the dense bed for vo ≈ 2.28 m/s and vo ≈ 3.15 m/s for tests A and B, respectively. Although, for test A, notice that the solids void fraction fitting along the freeboard is less accurate, see Fig. 2a. Furthermore, the deceleration term in Eq. (28), see red dashed lines in Fig. 2a, b, has been checked and found not to act immediately after the dense bed. However, the gravity term in Eq. (28), black dashed lines in Fig. 2a, b, is present from the bed surface. For test A, the deceleration would take place around 1 m afterwards from the bed surface, whereas for test B this distance is about 0.50 m. This could be related to the length of the splash zone, in which particles that have escaped from the dense bed go down and must be accelerated

123

u i ( po + p) − ut , (1 − εso ) po

(29)

in which Eq. (3), namely the gas continuity at the outlet velocity, has been included. Then, the outlet solids fraction εso is derived from the solids material balance at the riser outlet, Eq. (1), and substituted in Eq. (29) εso = G S / (ρ S u t ) ⇒

vo =

u i ( po + p) − ut . (1 − G S / (ρ S u t )) po (30)

For test A, εso = 9.3/ (2600 × 0.83) = 0.0043. For a total pressure drop of 10 kPa and assuming that the outlet pressure is the standard atmospheric pressure (101.325 kPa), the actual outlet gas velocity would be u o = (3 × 111.325)/ ((1−0.0043) · 101.325) = 3.31 (m/s) . Finally, the slip outlet velocity would be vo = 3.31−0.83 = 2.48 (m/s). If the gas density changes were neglected, u o ≈ 3/(1−0.0043) = 3.01 (m/s), the outlet slip velocity would be vo ≈ 3.01 − 0.83 = 2.18 (m/s). Remember that the fitted value for vo has been 2.28 (m/s). Whereas for test B, εso = 23/ (2600 · 0.83) = 0.0107. For a pressure drop of 10 kPa, the outlet gas velocity would be u o = (4 · 111.325) / ((1−0.0107) × 101.325) = 4.35 (m/s). Finally, the slip outlet velocity would be vo = 4.35 − 0.83 = 3.52 (m/s). Again, neglecting gas density changes, u o ≈ 4/ (1 − 0.0107) = 4.04 (m/s), and the outlet slip velocity would be vo ≈ 4.04 − 0.83 = 3.21 (m/s) . Comment that, for this test, the above fitted value for vo was 3.15 (m/s). 3.4 Maximum solids void fraction εsd The maximum solids void fraction εsd has been assumed known in all the above developments. Thus, an extra equation to calculate it was needed to complete the one-dimensional

New one-dimensional hydrodynamics of circulating fluidized bed risers

model. The analysis should be limited to the dense bed, in which εsd is defined. The classical particle force balance, Eq. (12), establishes a relation between solids gravity and solids acceleration on the one hand, and drag forces on the other. As the solids gravitational and acceleration forces are balanced with the pressure gradient, see Eq. (11), clearly, the extra equation would be the well-known classical momentum equation of gas phase. This equation basically balances the pressure gradient with the drag force, in addition to other minor gas forces, such as shearing, acceleration and gravitational ones [10,11]. This would be the new extra equation suggested, in which previous minor gas forces are neglected. 3.4.1 Analytic pressure drop along the dense bed We start with the pressure drop equation, Eq. (25), but now along the whole dense bed. Then z is substituted by Hd , and Eq. (23), the solids void fraction profile, is inserted,

Page 11 of 14

pbed

  sinh vHγ dugd  = ρs εsd u d vd ln ⎣  ⎩ u d /vγ ⎫ ⎤   ⎬ Hd g ⎦ + cosh + ut . ⎭ vγ u d ⎧ ⎨

⎡

(34)

3.4.2 Analytic drag along the dense bed The total drag along the dense bed would be based on the drag force on a particle FD, p (z) = γ v 2 , Eq.(12). First, the drag force per unit volume of bed FD N/m3 is worked out multiplying FD, p by the number of particles per unit volume N p . N p is related to the solids void fraction and the particle volume, see Sect. 2.5.1. Therefore, FD (z) = FD, p (z) N p (z) =

εs (z) 1 C D ρ A p v2 p 2

 V  FD, p

 pi − pd = pbed = gρs   = gρs εsd vd 0



Hd

εs (z) dz + ρs u d εsd u t  

 Acceleration (solids)

0

Gravity (solids)

Hd

dz + ρs u d εsd u t . v (z)

(31)

Note that the particles reach the representative terminal velocity u t at z = Hd . The previous integral, in the bed gravity term, can be analytically solved through the slip velocity, Eq. (18). The properties of hyperbolic functions [18] have also been taken into account, 

Hd 0

dz = v (z)



Hd

z

  vγ tanh + atanh vuγd    zg ud ud Hd ln sinh + atanh = 0 g vγ u d vγ   Hd g ud sinh + atanh vγ u d vγ ud   ln = . g sinh atanh vuγd 0



zg vγ u d

(32)

After applying the formula of the hyperbolic sine of the sum of two arguments [18], the integral in Eq. (31) is 

Hd 0

  ⎡ ⎤ Hd g   u d ⎣ sinh vγ u d dz Hd g ⎦   + cosh = ln . v (z) g vγ u d u d /vγ (33)

In conclusion, the pressure drop along the dense bed would be

78

=

v 3 C D ρv 2 εs (z) = ρs g 4d p vγ

Np

2

εs (z) .

(35)

  The previous product 4d3 p C D ρ has been substituted through Eq. (15), which defines the ‘friction’ velocity vγ . Finally, integration  along  the dense bed height is the total drag force FD,bed N/m2 . As before, the slip velocity v, Eq. (18), is needed because the solids void fraction profile is also a function of v, Eq. (23). 

FD,bed

 Hd ρs g 2 ρs g = v εs (z) dz = 2 εsd vd vdz vγ2 vγ 0 0     Hd ρs g zg ud dz. = εsd vd tanh + atanh vγ v u v γ d γ 0 (36) Hd

Also as before, the integration is immediate using the properties of hyperbolic functions [18], in particular the formula of the hyperbolic cosine of the sum of two arguments,   cosh vHγdugd + atanh vuγd   FD,bed = ρs εsd u d vd ln cosh atanh vuγd      ud Hd g + = ρs εsd u d vd ln cosh vγ u d vγ   Hd g . (37) × sinh vγ u d 3.4.3 Momentum equation of the gas phase Finally, the momentum equation of the gas phase, and its integration along the bed height, would be

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F. J. Collado

Table 3 Comparison of bed pressure drop (Pa) and bed drag (Pa) for TUHH tests (Ref. [8])   Gravity ρs εsd g H d Gravity Eq. (39) Acceleration Eq. (39)

pbed Eq. (39)

FD,bed Eq. (39)

pbed data

Test A (εsd = 0.2675) u d ≈ 4.1 m/s

2729.14

u d = 4.29 (m/s)a

2478.70

2366.78

4845.47

4595.14

5000.00

2529.66

2476.46

5006.12

5045.16

5000.00

1240.44

2435.51

3675.95

3455.81

3600.0

1252.48

2516.22

3768.71

3703.9

3600.0

Test B (εsd = 0.22) u d ≈ 5.13 m/s

1346.71

u d = 5.30 (m/s)a a Affected

−

by ( pi / pd )

dp ≈ FD (z) ⇒ pbed ≈ FD,bed . dz

(38)

Next, substituting the terms in Eq. (38) by the above expressions found for the bed pressure drop, Eq. (34), and the drag in the bed, Eq. (37), the new equation would be   ⎤   sinh uHddvgγ g H d ⎦  + cosh = FD,bed ⇒ vd ln ⎣  u d vγ u d /vγ      Hd g ud + u t = vd ln cosh + u d vγ vγ ⎛ ⎞⎤ Hd g × sinh ⎝   ⎠⎦ . (39) u d vuγd ⎡

pbed

Equation (39) can be checked for both tests. So that, we need the approximated values calculated for u d in Sect. 3.1, which are based on εsd , and also neglecting gas density variations; the characteristic slip velocity at the bed surface vd , Eq. (8); the calculated height bed Hd , see Fig. 2a, b; and, finally, the respective fitted values of the ‘friction’ velocities vγ , see Fig. 2a, b. Therefore, in Table 3, the relative differences between the calculated bed pressure drop and the bed drag are 5.2 and 6 % for tests A and B, respectively. In Table 3, the Eq. (39) is also checked with gas velocity u d affected by gas density changes, as it actually happens. For isothermal flow and ideal gas, the approximated gas velocity must be multiplied by the pressure ratio between the inlet and the bed surface, see Eq. (3). Hence, this more accurate gas velocity is slightly more than the approximated velocity. The remaining parameters are not modified with the logical exception of the slip velocity (vd = u d − u t ). Now the differences between the calculated pressure drop and the drag are 0.8 % for test A and 1.7 % for test B. Finally, for the sake of comparison, the measured bed pressure drop is also included in Table 3. For the closer gas velocity, their differences from the calculated bed pressure drop are 0.1 per cent and 4.48 % for tests A and B, respectively.

123

4 Discussion and conclusions A one-dimensional analysis of the riser of a circulating fluidized bed has been performed through new hyperbolic mass and momentum balances recently derived by the author [15], in addition to new expressions of a characteristic slip velocity between the phases. This representative slip velocity has been derived from classical single particle drag equation, assuming two drag coefficients, practically constant, one for the dense bed and another for the freeboard. These drag coefficients are included in a so-called ‘friction’ velocity, which is finally a fitting parameter. As we are treating with population of particles, we would also need a representative diameter of the particle size distribution. In this work, instead of searching for a specific particle diameter, the ‘friction’ velocity, which also includes the particle diameter, has been varied until the calculated pressure drop profile fits the measured one. As a first approximation to the problem, gas density variations have been neglected. Classic studies about 1-D hydrodynamics for vertical riser flow, see, for example, [10,21,22], are based on RichardsonZaki (R-Z) equation [23], which is constituted purely based on the modifications of the drag force on a single particle in the flow. The R-Z equation tries to adjust the single particle equation, when a large amount of particles exist around in fluidized bed, including the solids void fraction. However, the R-Z drag force would need some representative particle diameter (working with a population of particles) and the R-Z index, which is an empirical exponent. Instead, the model presented here starts off on classical dynamics for a single particle in the gas flow and derives the characteristic slip velocity without introducing any empirical constant. Remark that the so-called ‘friction’ velocity vγ , Eqs. (13) and (15), is merely the grouping of several intervening variables. Another key point of this simple model, suggested in [12], is the assumption that the solids accelerate from a zero vertical velocity at the bottom of the riser to their terminal velocity at the dense bed surface. At this location, the solids accelera-

New one-dimensional hydrodynamics of circulating fluidized bed risers

tion finishes and the particles move at this constant terminal velocity until the reactor outlet. The terminal velocity representative of the quartz sand particle size distribution, used in the two tests checked [8], has resulted in that of the mean   surface diameter d ps = 130 µm . This would literally mean that, at the bed surface, mp

  du p = 0 = −m p g + γ u t , d p u 2t . dt

(40)

Clearly, the question is how it is possible the simultaneous validity of Eqs. (12) and (40). A possible answer would be, see Tong [16] page 5, that inertial frames are not unique, and it is possible to map one inertial frame to another through some transformations. In particular, boosts: x = x + V * t, for constant velocity V. So, the particle velocity relative to the gas velocity at the end of the dense bed would be −v (z) = u p (z) − u d . Thus, the physical fact that the gas velocity   constant

is constant at the end of the bed allows particles to reach their terminal velocity. About the predictions of the axial profiles of the pressure drop and the solids holdup along the riser can be considered acceptable compared with only the two complete tests reported in [8] although the calculated solids holdup profile along the freeboard for test A is less accurate than test B. These predictions have been performed assuming that the maximum solids void fraction is known. As we have commented before, for the dense bed, these fittings are found to vary the ‘friction’ velocity vγ , which includes all the information about the drag coefficient along this region. The values found correspond to the greater diameters of the particle size distribution, see cursive rows in Table 2, which would be in agreement with the well-known segregation effects of larger particles at the riser bottom [7]. Besides, the simultaneous prediction of the dense bed height, at least for the two tests analysed, is acceptable provided the maximum solids void fraction is known. Along the freeboard, the pressure drop and solids holdup profiles have been fitted varying vo in Eq. (20). It has been checked that the assumed actual outlet characteristic slip velocity is close to the fitted vo . Furthermore, this actual outlet slip velocity is strongly related to the outlet solids fraction εso , which, in turn, is derived from the solids material balance at the riser outlet, see Sect. 3.3.2. Notice that a solids material balance using the vertical characteristic particle velocity u p has only physical sense just at the outlet of the riser, where it has been assumed that the actual local solids velocity distribution is fully upwards. As pointed out in [8], steady state solids mass continuity below the entry of the solids return line (z < z R ) implies that the total upward flow in the core zone must be equal to the total downward flux in the annulus. Thus, the actual

Page 13 of 14

solids mass flux in this zone needs to be zero  r u s (r, z) εs (r, z) 2πr dr = 0, G S (z < z R ) = ρs

78

(41)

0

were r is the radial coordinate from the duct axis, and z the upward axial coordinate along the riser with origin at the gas distributor [8]; ρs is the solids density, u s (r, z) is the local solids velocity and εs (r, z) is the local solids void fraction (or voidage) in the riser. On the other hand, in the splash zone just above the bed surface, the model has shown that the deceleration term in the pressure gradient does not enter into action immediately. There would be a “delay” height above the surface, in which the solids gravity would be the only component of the pressure. This could be related to the solids that, after just being blown out of the bed, fall and must be accelerated again. This could counteract gas deceleration. Finally, an extra equation is performed to calculate the maximum solids holdup, namely the classical momentum equation of gas phase. This gas equation basically balances the gas pressure drop gradient with the drag force per unit volume. Hence, the bed pressure drop should be equal to the bed drag. The closure of the integration along the bed of the momentum balance of the gas phase, using approximated gas velocity, is around 5–6 % for the two tests checked. When the gas velocity is refined to include gas density variations, the closure improves to 0.8 and 1.7 % for tests A and B, respectively. In conclusion, if the model proposed was exhaustively confirmed and refined against much more experimental data, it could exhibit some clear improvement compared to traditional Kunii-Levenspiel models in particular, reducing to a minimum the need of empirical constants. Given the scarce experimental comparison, with only two tests checked, and the simplifying assumptions made, these results should be taken with extreme care. However, acceptable predictions could support the need for exhaustive checking against far more data. Furthermore, to notice that the model proposed only consider the hydrodynamics without mass transfer between the phases. Thus, the model is not prepared yet for reactive systems with interfacial mass transfer see, for example [24], although clearly these systems also need some hydrodynamic model [25]. Compliance with ethical standards Conflict of interest The author declares that he has no conflict of interest.

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