Analysis of effective solid stresses in a conical spouted bed

Analysis of effective solid stresses in a conical spouted bed

A. L. T. CHARBEL¹, G. MASSARANI¹ and M. L. PASSOS^{2 *}

¹Programa de Engenharia Química, COPPE, Universidade Federal do Rio de Janeiro,

C. P. 68502, 21945-970, Rio de Janeiro - RJ, Brazil.

²Departamento de Engenharia Química, Universidade Federal de Minas Gerais,

Rua Espírito Santo 35, 6º andar, 30.160-030, Belo Horizonte - MG, Brazil.

E-mail: [e-mail]

(Received: July 26, 1999; Accepted: September 19, 1999)

Abstract - Some applications of spouted beds have been limited by problems with spout stability. In order to overcome these limitations, research should be concentrated on describing the mechanism by which the spout is developed. This work presents a theoretical and experimental study to describe the effective solid stress distribution in the annular region of a conical spouted bed. From experimental observation, the failure state of the bed of particles at the outset of spouting is identified and related to fluid-particle properties and column dimensions. Linear and angular momentum balances for the solid phase are then formulated as a function of the bed failure state. The set of equations obtained is solved using numerical methods, and results regarding stress distribution, stability and spout shape are presented and discussed.

Keywords: types of failure, solid stress distribution, pressure drop, spout shape.

INTRODUCTION

Spouted beds are a fluid-solid contact technique applied to coarse particles which are difficult to fluidize. The spouting regime is characterized by three different bed regions: the spout, the annulus and the fountain. In the spout, particles are carried upward by the inlet jet fluid. In the fountain, these particles change direction, falling back onto the top of the annulus region. In the annulus region, they move downward and inward in countercurrent to the fluid flow, until re-entraining into the spout. This cyclic solids motion, unique to this technique, results in effective contact between particles and fluid. Note that particles move in the spout as they do in pneumatic transport, while in the annulus region the motion is slow with particles sliding into one another, similar to the solids flow which occurs in the discharge of silos.

The stability of the spouting regime depends on particle and fluid properties as well as the configuration and dimensions of the column (Mathur and Epstein, 1974). Since their discovery in 1955, spouted beds have shown to have potential applications in the drying of grains, seeds and suspensions, granulation, mechanical extraction, chemical reaction, etc. However, in order to apply this technique in industrial processes, problems with spout stability must be overcome. Mathur and Epstein (1974) pointed out three distinct mechanisms which cause spout termination in beds with H > H_max: (i) fluidization of the annulus top; (ii) choking of the spout and (iii) growth of instability. To identify the first and second mechanisms in a conventional spouted bed (conical-cylindrical column design), Littman and Morgan (1988) proposed the following dimensionless parameter:

(1)

This A parameter represents the ratio of the minimum inlet energy required to sustain the spout to the minimum frictional energy lost across the spout at H = H_max. Passos et al. (1993) extended this definition of A to a two-dimensional spouted bed. Their results show that different ranges of A can distinguish well the three spout termination mechanisms. Moreover, these authors have characterized the growth of instability as a transitional mechanism between the fluidization of the top of the annulus region and the choking of the spout.

In previous work, Passos (1991) demonstrated that the occurrence of these spout termination mechanisms is directly related to how the spout is formed. As a consequence, spout stability as well as the behavior of fluid flow and solids motion depends on the type of bed failure during spout formation. Working on a half-conical spouted bed with four different types of particles spouted by ambient air, Oliveira (1997) identified three distinct spout formation mechanisms as a function of the range of A. Her experimental findings are summarized in Figure 1. As can be seen, a cavity develops just above the inlet nozzle when fluid is injected into the column. With a further increase in the fluid flow rate, an arc of particles is formed at the top of this cavity and is compressed by the inlet fluid jet until maximum compaction is reached. Up to this point, the solid matrix (i.e., the solid skeleton) behaves as an elastic material. Only at this maximum point, which corresponds to the maximum of spout pressure drop versus fluid flow rate curve, bed failure starts and the solid matrix deforms plastically. Depending on the range of A, the spout is formed differently.

Figure 1:
Diagram of the three spout formation mechanisms observed by Oliveira (1997).

In beds with A < 0.014, the compacted arc of particles at the top of the cavity horizontally compresses the pores of the solid matrix element in its vicinity. Then particles slip around the cavity rearranging into a new structure with the solid matrix element extending vertically. This causes the compression and deformation of another group of particles at the base of the column. Due to these plastic deformations, particles entrain into the cavity and circulate inside it. By increasing the fluid flow rate, the cavity is enlarged first radially and then vertically, until reaching the free bed surface. At this point, the true spout is formed. In beds with 0.014 £ A £ 0.02, the cavity is less circular and there is no particle circulation within. Above this cavity, the solid matrix is under tension and the spout reaches the bed surface abruptly and intermittently. The spout oscillates around its axial axis, even at high fluid flow rates. In beds with A > 0.02, few particles oscillate inside the circular cavity developed at the column base. This cavity grows axially, compressing the central region of the bed. The free surface of the bed deforms to attain a parabolic shape, as shown in Figure 1. The spout develops abruptly, but at V/V_ms > 1.2 it remains stable and centralized in the axial column axis.

As shown by Oliveira (1997), for each range of A identified in Figure 1, the spout shape as well as the solid and fluid flow behavior is distinct. Several works in the literature are aimed at modeling the fluid flow in spouted beds but only a few analyze how the spout is formed and how the mechanism of bed failure affects fluid flow, spout shape and its stability.

Lefroy and Davidson (1969) were the first to use the failure criterion of soil mechanics to solve the momentum balance for the solid matrix at the spout and annulus interface. However, they did not take into account the vertical shear stress acting on this interface. Based on the behavior of the porous material in a hopper, Bridgwater and Mathur (1972) developed a stress analysis for a solid matrix element in the annulus region of a conventional spouted bed far from the base of the column or from the top of the free surface. Assuming the spout interface to be a rough wall, these authors showed that the effective weight of this element (i.e., the actual weight minus the upward force due to fluid flow) is supported by the vertical shear stresses at the spout interface and at the vessel wall. Neglecting the effect of bed height on the stresses, they obtained an expression for the spout diameter which was corroborated by the empirical correlation reported by McNab (1972). By extending this model, McNab and Bridgwater (1974) studied the effect of radius upon normal stresses. Their conclusions demonstrate that (i) at every point in the annulus region of a spouted bed, material failure is about to occur in some direction, although movement itself may not have begun yet; (ii) a point r = r₀ in the transversal annular area, at which the shear stress becomes zero and the normal radial and axial stresses are the principal ones acting on the element of the solid matrix, must exist. Thus, the effective weight of the material between r_s£ r £ r₀ is supported only by the vertical shear stress at the spout interface. The vertical shear stress at the vessel wall supports the material in the annular region from r₀ to r_w; (iii) the normal radial stress at the spout wall is a function of the fluid flowing from spout to annulus and of the spout particles pounding the interface. Neither factor can be neglected in the location of the spout interface. Using the method of differential slices and the distribution factors proposed by Walker (see Walters, 1973 a), McNab and Bridgwater (1979) presented an analytical model for the vertical stress distribution in the cylindrical annular region of the conventional spouted bed. However, in all these studies, it is assumed that the annulus region is in the active state of failure, the spout radius is constant and the vessel wall is always vertical.

Walters (1973a, 1973b) analyzed the failure state of the stresses acting on porous material in silos and hoppers. As suggested, the active state, usually obtained when the material is poured into the vessel, is characterized by the vertical compression of the solid matrix element on the symmetry axis of the column. The effective stresses on this axis are the principal ones and the major stress, s₁, is in the vertical direction. For any isotropic material, the solid matrix element at this center responds to this compression extending horizontally. On the other hand, the passive state is obtained when the porous material is discharged through the outlet nozzle. This state is characterized by the horizontal compression of the solid matrix element on the symmetry axis of the column. For any isotropic material, the major principal stress, s₁, is in the horizontal direction and an element of the solid matrix at this center is deformed, extending vertically. The presence of the vessel walls affects the direction of the principal axis of both the effective stress and the material strain tensors. In an isotropic medium, this wall effect can be quantified by the change in the angle of friction of the porous material which, at the center of the silo, is angle of internal friction of the material, f, and, at the column wall, is the angle of friction between the material and the vessel wall, f_w.

Returning to Figure 1, one can see that, at least for A < 0.014, the mechanism of spout formation cannot be characterized only by the active state of bed failure. In the vicinity of the cavity, the solid matrix element, compressed horizontally, presents plastic behavior in the passive state of stress. Near the vessel wall, vertical compression causes the solid matrix element to be in the active state and the particles to slip toward the spout annulus interface.

The aim of this work is to investigate effective solid stress distribution in the annulus region of a conical spouted bed with A < 0.014, taking into account different mechanisms of bed failure at the spout-annulus interface and at the vessel wall. The approach used for modeling stress distribution is similar to that proposed by Walters (1973b), since waves of instability should occur in the bed of particles during spout formation. Experiments carried out in a half-conical spouted bed of polypropylene particles with A < 0.014 provide data for solving the model. Results obtained are analyzed in order to propose a methodology for predicting spout shape.

THEORETICAL MODEL

Initial Assumptions

Consider a horizontal slide of the solid matrix element in the annulus region of a conical spouted bed with A < 0.014, as shown in Figure 2. At minimum spouting conditions, the bed of particles is in failure state. Although groups of particles can slip over one another, the cyclic particle circulation is not yet developed. The solid matrix is isotropic and the effective solid stress variation is reduced to a plane of stresses. As a result, the Mohr-Coulomb failure criterion (Terzaghi, 1943) can be used to describe the relationship between stresses applied to the solid matrix. Moreover, the normal and shear stresses at the column wall, s_w and t_w, are distinct from those acting on the spout wall interface, s_s and t_s. Unlike in silos, the principal axis of the effective solid stress tensor coincides with the r, z coordinate axis at r = r₀. As shown in Figure 2, shear stresses become zero at r = r₀. This distance, r₀, can vary with the axial z distance.

Figure 2:
Stresses on a solid matrix slide element.

Due to different bed failure mechanisms in spouted beds with A < 0.014, the annulus region can be divided into two subregions, s and w, as shown in Figure 2. The s subregion extends from r_sms to r₀ and is characterized by the passive state of stress at the spout wall interface. The w subregion extends from r₀ to r_w and is characterized by the active state of stress at the vessel wall boundary. Shear stress applied to the solid matrix element is supposed to vary linearly with radial distance, r, in each subregion. This supposition holds for deep beds and is an approximation in shallow spouted beds (Walters, 1973b; Spink and Nedderman, 1978).

Moreover, in each subregion, it is assumed that normal radial stress, s_r (= s_rr), is constant along the cross-sectional area of the solid matrix element shown in Figure 2. Discontinuity of s_r can occur at the boundary layer between these two subregions due to the change in bed failure mechanisms. Since s_r|s can be different from s_r|w, another additional condition must be imposed at this boundary (r = r₀) to solve the momentum equation for the solid matrix element. As mentioned earlier, the free annulus surface in spouted beds with A < 0.014 is not deformed during spout formation; therefore the angular momentum due to forces applied on the solid matrix element boundary should be zero or

(2)

Based on these initial assumptions and using the analysis proposed by Walters (1973b) and by Spink and Nedderman (1978), the normal vertical stress, s_z(r), acting on the solid matrix element can be represented by its mean value, defined as

• subregion

  w in the active state of stress failure

(3)

where

obtained from the Mohr diagram circle presented in Figure 3 (for details see Charbel, 1999);

• subregion

  s in the passive state of stress failure

(4)

where

obtained by the Mohr diagram circle presented in Figure 4 (for details see Charbel, 1999)

In Equations (3) and (4), it is important to note that:

(a) Y_w and Y_s functions depend on r₀; a_w or a_s and f_w or f_s. The angle of friction between the material and the spout wall interface is assumed to be equal to the angle of internal friction of the material, i.e., f_s = f;

(b) h_w and h_s angles exist because both walls, the vessel and the spout interface are supposed to be at angles of a_w and a_s from the vertical principal axis of stresses at r = r₀. These angles are obtained from the Mohr diagram shown in Figures 4 and 5 and are expressed as

Based on Figures 3 and 4 and using trigonometry, the following equations are obtained:

(7)

By combining Equations (3) to (10), and t_s can be expressed as functions of , as shown in Table 1.

Figure 3:
Stresses in active state for a solid matrix element at the vessel wall in subregion w.

Figure 4:
Stresses in passive state for a solid matrix element at the spout interface in subregion s.

Figure 5:
Diagram of the half-conical column used in experiments.

1a - Relationship between 1b - Relationship between 2a - Relationship between 2b - Relationship between 3a - Relationship between 3b - Relationship between

1a - Relationship between	1b - Relationship between
2a - Relationship between	2b - Relationship between
3a - Relationship between	3b - Relationship between

Table 1: Relationship between shear and normal stresses applied on the boundary wall of the solid matrix slide element and the mean vertical normal stress applied on this element.

Momentum Balance Equations

Under the initial considerations, the momentum balance can be developed in the z axial direction for the solid matrix slide element in both w and s subregions as

• subregion

  w, r

  ₀

  ⁽⁺⁾£ r £ r

  _w

• subregion

  s, r

  _sms£ r £ r

  ₀

  ^(-)

where dP_a/dz represents the drag force exerted by the fluid on the solid matrix element (due to fluid flow through the porous element) in the w subregion and in the s subregion; s_w and t_w are expressed by Equations (15) and (13b), respectively, and s_s and t_s are expressed by Equations (16) and (14b), respectively.

Since particles are not in continuous motion at the minimum spouting point, Equations (17) and (18) represent a quasi-static equilibrium condition. Therefore, the effective weight of the solid matrix element must be sustained by the vertical forces resulting from the effective stress distribution on the boundary of this element.

To solve Equations (17) and (18), together with Equations (2), (13b), (14b), (15) and (16) and the boundary condition at z = H, the following variables must be known: (i) particle and fluid properties, (ii) conical column design parameters and (iii) P_a(r, z) and r₀(z) or r_sms(z) functions. The first two sets of variables are determined by selecting the solid-fluid system and the conical column. Note that particle characterization should include the angles of friction, f and f_w. The third set is concerned with the fluid dynamic variables which should be measured or predicted. To date, there is no work in the literature that can describe the r₀(z) function in a spouted bed with A < 0.014. Thus, before applying the model to predict spout diameter, it is necessary to determine and analyze how r₀ varies with z when two different types of bed failure occur during spout formation. Experimental work has been developed to provide P_a(r, z) and r_sms(z) functions in order to solve the model.

EXPERIMENTAL WORK

Experiments were planned in accordance with the following objectives: to characterize the solid-fluid system by its physical properties and its angles of internal friction and wall friction; to corroborate the experimental findings reported by Oliveira (1997) regarding the spout formation mechanism in beds of particles with A < 0.014; to measure the pressure drop in the annulus region as a function of r, z, V/V_ms and H and to determine the spout radius under minimum spouting conditions as a function of z and H. Polypropylene particles were chosen as the solid phase and ambient air as the fluid phase.

These experiments were carried out in the half-conical column shown in Figure 5. This column, built of galvanized iron, has a transparent acrylic front wall which allows visual observation of the bed of particles during spout formation and permits direct measurement of spout diameter from z = 0 to z = H. For measurement of the spout and annulus pressure drop along r and z, taps connected to a set of water manometers were inserted into the front column wall, as shown in the diagram in Figure 5. To register the air flow rate, a calibrated Venturi tube was installed in the inlet air line that connected the column with the blower (ERBELE, Model: 2915 rpm, 220 V and 4.5 HP).

Air-particle properties are displayed in Table 2. Particle diameter, d_p, and density, r_s, were measured by pycnometric methods. Particle sphericity, j, and bed porosity at minimum fluidization, e_mf (= e_a), were determined from pressure drop measurements for a loosely packed bed of these particles fluidized by air (Massarani, 1997). The angle of internal friction of the air-polypropylene particle material, f, and the angle of friction between this material and the vessel wall, f_w, were measured using the direct shear box equipment and the procedure adopted in soil mechanics (see Charbel, 1999). The A value was calculated from Equation (1) using empirical correlations reported in the literature (Massarani, 1997) for determining U_T and U_mf.

Properties dp (mm) j (-) rs (kg/m3) eo (-) emf (-) f ( º ) fw ( º ) fr ( º ) A (-) Values 2.96 0.67 960 0.38 0.41 37.8 12.2(1) 10.5(2) 28 0.0029

Properties	dp (mm)	j (-)	rs (kg/m3)	eo (-)	emf (-)	f ( º )	fw ( º )	fr ( º )	A (-)
Values	2.96	0.67	960	0.38	0.41	37.8	12.2(1) 10.5(2)	28	0.0029

Table 2: Properties of the polypropylene particles used in this work.

(1) - galvanized iron column wall;

(2) - acrylic front column wall.

Working with three different bed heights (H = 0.237, 0.337 and 0.377 m), experiments were performed by varying the air flow rate from 0 to 2.0 V_ms. The spout and annulus pressure drop data were recorded for each increase in the air flow rate. Spout diameters in the range of 1.2 £ V/V_ms£ 2.0 were measured along the z axis. A replica of each experiment was used to determine the experimental error associated with the measurements. Based on an analysis of these data, the following conclusions can be drawn:

(i) the spout is formed by the same mechanism as that reported by Oliveira (1997) for beds of particles with A < 0.014. As expected, the a _s angle varies with the z distance, as shown in the diagram in Figure 4;

(ii) DP_ms and V_ms increase as H increases, and these data can be predicted by the fluid flow model proposed by Costa Jr. (1999), as shown by Charbel (1999). Since this model was developed for full conical spouted beds, the effect of the acrylic front wall on minimum spouting conditions was neglected in the range of experiments developed here;

(iii) in Figure 6, one can see that P_a(r, z) decreases as z increases but remains almost the same for 1.0 £ V/V_ms£ 2.0. Since this is the behavior reported in the literature (Mathur and Esptein, 1974), the effect of the acrylic front wall on the annular pressure drop does not seem to be significant in the experiments developed here;

(iv) spout diameter increases as V/V_ms increases (as expected), and its profile at minimum spouting was obtained from the following correlation proposed by McNab (1972):

(19)

Figure 6:
Gauge pressure drop as a function of z: (a) in spout region; (b) at the spout-annulus interface;(c) near the vessel wall in beds of polypropylene with H = 0.377 m.

As noted in Figure 7, the values of d_s(z) at V/V_ms = 1.2, 1.4, 1.6 and 1.8 converge for the same d_sms(z), within the range of the experimental d_s error. Since Equation (19) is valid for full spout beds, the acrylic front column wall does not seem to interfere with spout stability.

Figure 7:
Spout diameter at minimum spouting calculated by Equation (19) for each V/V_ms value used in experiments: (a) H = 0.287 m; (b) H = 0.377 m.

Although data obtained for a half column are questionable in terms of their dynamic similarity with those obtained for a full column, the results suggest that the acrylic front wall’s effect on fluid dynamics has been reduced within the range of experimental variables used here. Therefore, the spout radius and annular pressure drop data are fitted with the best polynomial of z and used in Equations (17) and (18) to predict the solid stress distribution in the annulus region of conical spouted beds with A < 0.014.

RESULTS AND DISCUSSION

Using the experimental fitted curves for P_a(z) and r_sms(z), the model equations are solved by the 4^th order Runge-Kutta algorithm and/or the Simpson numerical integration method. To avoid any problem with numerical convergence, a specific subroutine was inserted into the main program. This estimates, step by step, the r₀(z) function by fitting the r₀ data generated in each iteration with the best polynomial curve. With this subroutine, both methods yield the same solution for and r₀ within an error associated with r₀£ 1´ 10^-6 m.

As suggested in Figure 8, r₀(z/H) does not vary with bed height, and its value is well described by the r₀* = r₀/(r_w - d_b/2) vs. z/H curve (see Figura 8c).

Moreover, for 0 £ z/H £ 0.2, the w subregion corresponds to a thin layer on the vessel wall. As noted by the visual observation of bed behavior during spout formation, there are no dead zones at 0 £ z/H £ 0.2 and particles, except those located on the vessel wall, slip towards the spout. As pointed out by Walters (1973b), when h_w < 0 (as obtained here), there must be a thin particle layer on the vessel wall which behaves as an elastic material. In this layer, increases, attaining high values. This is in accordance with the simulated data for in Figure 9. Note in the diagram presented in Figure 3 that this thin layer was compressed during spout formation; as a consequence, higher values of are also expected to occur at minimum spouting conditions. Thus, the model can predict such a effect.

Figure 9:
Stress distribution in the annular region of conical spouted beds of polypropylene particles: (a) dimensionless normal stresses, s*; (b) axial component of the dimenionless shear and normal stresses at the vessel wall, t_w* + s_w* tana_w, and at the spout wall, t_s* + s_s* tana_s.

It is important to mention that when the solid stresses are expressed in dimensionless form,

they are independent of bed height and follow the same distribution as that shown in Figure 9.

As z is increased (z/H > 0.2), the w subregion tends to extend in the r direction. At z/H = 0.5, r₀*(z/H) changes its curvature followed by an increase in and . For z/H > 0.5, surpasses (see Figure 9a) and the axial component of stresses at the spout wall (t_s* + s_s* tana_s) becomes responsible for sustaining about 40 % of the apparent bed weight. To maintain stability under these conditions, spout diameter should decrease. As seen in Figure 7, at z = 0.6 H, d_sms attains its minimum value and a_s its maximum. Thus, the model coherently describes the trends observed during the experiments. Current work is attempting to extend the r₀*(z/H) function obtained here to spouted beds of other particles with A < 0.014. This will allow application of the model for predicting spout diameter in such beds.

CONCLUSION

It is clear that type of bed failure during spout formation affects spout shape and fluid dynamics. This is confirmed by a comparison of data obtained in this work with those presented by McNab and Bridgwater (1979), where spout diameter is assumed constant and the whole bed is supposed to be in the active failure state. The approach proposed in this work to model and analyze the solids stress distribution on a bed of particles under minimum spouting conditions is more general, including different types of bed failure and variation in spout diameter along the bed height.

The model developed can effectively describe the experimental trends observed during spouting formation. A function to predict the r₀ location in spouted beds of polypropylene particles with A < 0.014 has been proposed. Future work will extend this function to other particles in order to obtain a more general equation for r₀ and to apply the model for predicting spout diameter to these spouted beds.

NOMENCLATURE

A dimensionless parameter defined by Equation (1)

d_b diameter of the conical base of the column, m

d_I inlet nozzle diameter, m

d_p mean particle diameter (based on the sphere with the same particle volume), m

g gravity acceleration, m/s²

H bed height, m

H_max maximum spoutable bed height above which the spout terminates, m

P pressure, Pa

P_a pressure in the annulus region, Pa

r radial coordinate, m

r₀ radius where the shear stress equals zero, m

r_s spout radius (= d_s/2), m; r_sms - spout radius at minimum spouting (= d_sms/2), m

r_w vessel wall radius, m

U_mf minimum fluidization superficial fluid velocity, m/s

U_T terminal particle velocity, m/s

V fluid flow rate, m³/s; V_ms - fluid flow rate at minimum spouting, m³/s

z axial coordinate, m

a_s; a_w half included angle of spout and of the column, respectively, radians or degrees

DP_ms spout pressure drop at minimum spouting, Pa

e_o bed porosity under static conditions

e_a mean bed porosity in the annulus region, assuming it is equal to e_mf

e_mf bed porosity under the minimum fluidization condition

f angle of internal friction of the material, radians or degrees

f_r repose angle of the material, radians or degrees

f_s angle of friction between the material and the spout wall interface, radians or degrees

f_w angle of friction between the material and the vessel wall, radians or degrees

h_s; h_w angles defined in Equations (6) and (5), radians or degrees

j particle sphericity

m fluid viscosity, kg/ms

r_f fluid density, kg/m³

r_s solid density, kg/m³

s normal stress, Pa

t shear stress, Pa

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^*

To whom correspondence should be addressed

Publication Dates

• Publication in this collection
  01 Mar 2000
• Date of issue
  Dec 1999

History

• Accepted
  19 Sept 1999
• Received
  26 July 1999