Hydrodynamic model for the flow of granular solids in the S-valve

Powder Technology 230 (2012) 77–85

Contents lists available at SciVerse ScienceDirect

Powder Technology journal homepage: www.elsevier.com/locate/powtec

Hydrodynamic model for the flow of granular solids in the S-valve G.M. Guatemala a, F. Santoyo b, L. Virgen a, R.I. Corona c, E. Arriola c,⁎ a

Centro de Investigación y Asistencia en Tecnología y Diseño del Estado de Jalisco, A. C., Avenida Normalistas N° 800, Colonia Colinas de la Normal, CP 44270 Guadalajara, Jalisco, Mexico b Departamento de Ciencias Exactas, CUSur, Universidad de Guadalajara, Ciudad Guzmán, Jalisco, Mexico c Centro Universitario de Ciencias Exactas e Ingenierías, Universidad de Guadalajara, Calzada Marcelino García Barragán y Olímpica, Sector Reforma, CP 44420 Guadalajara, Jalisco, Mexico

a r t i c l e

i n f o

Article history: Received 13 November 2011 Received in revised form 22 June 2012 Accepted 29 June 2012 Available online 5 July 2012 Keywords: S-valve Granular solids flow Hydrodynamic modeling

a b s t r a c t This article proposes a phenomenological model to predict the discharge of granular solids (group D in the Geldart classification) through a valve known as the S-valve (or spitting valve), which controls the flow of solids with the injection of a gas. The model predicts the flow of solids as a function of the density of the solid (ρs), the friction coefficient (fs), the void fraction (ε), the gas flow (Q), the valve diameter (Dv), the pressure (P), and the spitting factor (SF). The friction coefficient, fs, and the void fraction, ε, were estimated based on the surface velocity of the aeration gas using empirical relationships that were integrated into the model. The solids used were rice, lentils, and green coffee beans. The deviation from the model was ±3% for rice and lentil grains and ± 2% for green coffee for a range of operation between 0.4 and 0.7 MPa. © 2012 Elsevier B.V. All rights reserved.

1. Introduction Granular material is one of the most common solid forms in the industrial world today. The treatment of granular materials involves 10% of the energy resources worldwide [1]. Weight estimates show that approximately 75% of raw materials and 50% of chemical industry products are manipulated in granular form [2]. Within the chemical industry, $61 billion is devoted to research on particle technology each year, and approximately 1.3% of the electric energy produced in the United States is expended in the pulverization of minerals [3]. Problems related to the poor handling of granular solids account for failures in approximately 1000 silos, containers and nozzles in North America each year [4]. In Mexico, 30% of the 5 million tons of corn stored in silos each year is lost due to poor handling of the grain [5]. Solid granular particles such as coffee, rice, mustard, salt, sand, and sugar grains will cling together due to their nature, which leaves gaps between grains that, in most cases, are filled with gas. Modeling the transport of two-phase gas and solid flows and calculating the loss in pressure is a complex task because of the different weight distributions in the mixture. These different weight distributions imply different types of flow, each with its own model [6]. Some techniques have been developed to treat the phenomenon of two-phase flows, such as empirical estimates and methodologies [4,7,8], semi-analytical estimates [9], and empirical models with reduced application ranges [10–12]. According to Zuriguel [13], the technology to manipulate and control granular materials is not yet adequately developed. Likewise, according ⁎ Corresponding author. E-mail address: [email protected] (E. Arriola). 0032-5910/$ – see front matter © 2012 Elsevier B.V. All rights reserved. doi:10.1016/j.powtec.2012.06.055

to Huang et al. [14], the study of the hydrodynamics of non-mechanical valves is still deficient given the complex relationship between the gas and the solid that occurs within the valves. Of the many studies on non-mechanical valves (which mostly refer to the use of “group B” solids in the Geldart classification scheme), the studies of Matsumoto et al. [15], Geldart and Jones [8], Yang and Knowlton [16], Huang et al. [14], Kunii and Levenspiel [17], Daous and Al-Zahrani [18], and Hua et al. [9] stand out. These works omit the study of Geldart group D granular solids that occur frequently in industry. To operate an experimental system of staged spouted beds, Arriola [19] designed and used a modified L-valve for the first time. The researcher dubbed this modified L-valve an “S-valve”—a device that has not yet been characterized and documented. In some of these experimental tests, the use of an oscillating pressure was observed to yield a more uniform flow of solids than when a stable pressure was used. Solids are continuously ejected from the system using pulses of compressed air that literally “spit” the solids (this is how the name “spitting valve” was derived). To prevent the problem of solids circulating uncontrolled toward the exterior of the pressurized system, the S-valve has an ingenious seal for these solids. The S-valve (shown in Fig. 1) is a non-mechanical valve that controls the flow of granular materials with only the injection of a gas. Non-mechanical valves include the L-valves, the J-valves [20], the W-valves [17,21], the V-valves [22], and the N-valves [23]. The transport of solids takes place in a diluted stage when the solid particles move and are distributed throughout the section of the conduit that transports them, particularly the S-valves. In the diluted stage, the particles collide with each other or against the walls of the conduit; however, because contact between particles is minimal, it can be disregarded [24]. The void fraction (the space between particles) is

78

G.M. Guatemala et al. / Powder Technology 230 (2012) 77–85

Fig. 1. S-valve [19] operated by a solenoid valve.

high (between 0.90 and 0.99 [25,26]), the ratio of solids to air is low, and the solids and the transporting fluid (air) are mixed within the same pipe and behave as “a single fluid,” all of which helps to achieve an average density [27]. In the modeling of granular solids in the diluted stage, the particles interact only by so-called short-range forces (mechanical contact). Long-range forces, such as electrostatic forces, are not considered in these systems. The dynamics of group-D granular materials is therefore governed by Newton's laws of motion [28]. This article presents a mathematical model that permits the prediction of the discharge of group-D granular solids through the S-valve. The model is based on a balance of forces and theoretical considerations specific to the two-phase flow. 2. Materials and methods 2.1. Solids used An experimental program was carried out with three types of grains in different sizes and shapes: rice, coffee, and lentils. The grains studied belong to the “Geldart group D” type [8]. The experimental design was based on multiple factors with different measurement levels. The “discharge” variable was left as a dependent variable, and air flow (Q), working pressure (P), valve diameter (Dv), and particle diameter (dp) were used as predicting variables (Table 1). The experiments were executed in triplicate. The data were analyzed with Statgraphics Centurion XV (StatPoint, Inc., 2005). The relevant properties of the solids used to design the S-valve were geometric particle diameter (dp), density (ρp), sphericity (ϕs), and void fraction (ε) under the assumption of a fixed bed (Table 2). The characteristic dimensions (length, width, and height) of the grains were measured with a micrometer; the density was determined by the method

Variable

Measurement levels

Units

Q

0.07 0.08 0.10 7 × 105 8 × 105 0.04 0.02 0.00280 0.00378 0.00646

m3/s

Dv dp

2.2. Equipment and procedure The system in which the experiments were performed (Fig. 2) consists of a solids feeder built from stainless steel with a coneshaped base and an output facing a transparent acrylic tube 0.04 m in diameter. The acrylic tube is connected directly to the horizontal tube that makes up the S-valve, which was also built with a transparent acrylic tube with a diameter D1 (Fig. 3). The S-valve works with compressed air as a pulsing fluid; the pulses were regulated with the use of a solenoid valve. The complete system also includes a compressed-air supply and a water trap from which two lines lead. The first line is connected to a pressure regulator, followed by a two-way ON/OFF solenoid valve, which in turn is connected to a time relay (with an adjustment accuracy of ±5%), a flow meter that operates on a 0 to 0.15 m 3/s scale, and a nozzle connected to the S-valve. The second line, with its own pressure regulator and flow meter, provides air to the solids feeder tank to provide the option of operating the system at a pressure greater than atmospheric pressure. The feeding process requires the elimination of rods and/or large solids (trash) that could get stuck in the system, create an added resistance, and prevent the solids from flowing freely. The air filters and the water trap need to be checked and cleaned frequently. The precise flow of solids is determined by using a chronometer and a scale. All of the experiments were carried out at room temperature and in triplicate. 2.3. Development of the model Fig. 3 presents a schematic diagram of the S-valve showing the four sections of the valve as used in this research; Di and Li represent the inner diameter and length, respectively, of each section i.

Table 1 Experiment design variables.

P

suggested by Shoemaker et al. [29] and by Baryeh and Mangope [30]. The sphericity was measured using the method suggested by Baryeh and Mangope [30].

Table 2 Characteristic dimensions of the solids used. Pa Particle m m

dp

ϕs

0.00280 0.00378 0.00646

ε 3

(kg/m )

(m) Rice Lentil Coffee

ρp

0.76 0.83 0.78

1299 1283 1192

0.35 0.33 0.43

G.M. Guatemala et al. / Powder Technology 230 (2012) 77–85

79

Fig. 2. Diagram of the equipment used in the study.

2.3.1. Equilibrium of forces for the horizontal section The conditions needed to achieve the equilibrium of forces in one of the horizontal sections (Fig. 4) that is bounded by the diameter of the conduit (Di) and the length of the section (Δx) are shown in the equations below:

One of the assumptions in the model is the elimination of the averaged stress among particles σ x , when the flow in the S-valve is considered to be in the diluted stage with void fraction values greater than 0.90. Eq. (4) can therefore be simplified to



PA
xþΔx þ σ x A
xþΔx −PAjx −σ x Ajx þ τx S ¼ 0

ð1Þ

dP 4 þ τ ¼0 dx Di x




π 2 π 2 
D þ τx πDi Δx ¼ 0 P
xþΔx þ σ x
xþΔx Di − P
x þ σ x
x 4 4 i

ð2Þ

where P is the operating pressure, σx is the average friction between individual particles, τx is the friction force caused by contact between the walls of the conduit and the movement of the fluid (which is, in turn, based on the velocity of the gas), A is the cross sectional area and Di is the diameter of the conduit. Dividing Eq. (2) by the control volume π4 D2i Δx and reordering the terms, we obtain:

ðP þ σ x Þ
xþΔx −ðP þ σ x Þ
x Δx

þ

4τx ¼0 Di

ð3Þ

Finally, when Δx → 0, we have: dP dσ x 4 þ þ τx ¼ 0 dx dx Di

ð4Þ

Fig. 3. Sketch of the S-valve.

ð5Þ

According to Levenspiel [21], within a conduit, the shear stress, τx, can be related to the loss of friction; i.e., ðforcetransmittedtothewallsÞ ¼ ðlossoffrictionenergybythefluidÞ Based on the Fanning friction factor, we obtain:

τx ¼

  f f ρu20

ave

2g c

ð6Þ

where gc is a conversion factor. The friction factor, ff, depends on the velocity and the properties of the fluid (density and viscosity), as well as on the size of the tube [25,27]. If we substitute Eq (6) into Eq. (5), we have   2 dP 2 f f ρu0 ave þ ¼0 dx Di g c

Fig. 4. Sketch showing the equilibrium of forces in the horizontal sections.

ð7Þ

80

G.M. Guatemala et al. / Powder Technology 230 (2012) 77–85

The second term of Eq. (7) represents the losses in friction caused by the circulating fluid. Because the fluid consists of two phases, two types of friction energy losses occur: one friction energy loss is caused by the friction of the solids with the walls of the tube, and the other is caused by the friction of the fluid (air) with the walls of the tube. Taking these considerations into account, we have 2

" # 2ρp ð1−εÞu2p 2ρg εu2g fs þ f dx ¼ 0 ∫ dP þ ∫ Di g c Di g c g P 0

3

;

for 3  10 < Re < 1  10

5

ð13Þ

The Reynolds number can be calculated using Eq. (14): ð8Þ

where ε is the void fraction, ρs is the solid density, ρg is the gas density, fg is the factor for friction between the air and the conduit through which it circulates and fs is the factor for friction between the solid and the conduit through which it circulates. Empirical expressions were used in this work to estimate the values of the friction coefficients. Integrating between the appropriate limits, we have the following equation for the first section of the valve:

Re ¼

Dug ρg μ

ð14Þ

where μ is the dynamic viscosity. 2.4. Integration of the equilibrium of forces of each section Presupposing that the void fraction and the change in surface velocity of both the gas and the particles can be disregarded along the length of the entire valve, we have the following expression for the four sections of the S-valve:

L1

ð9Þ

1

2.3.2. Equilibrium of forces for the oblique section To characterize the difference in pressure between the two points in the sketch shown in Fig. 5, we use a modified Bernoulli equation [17]. In this section, the change in kinetic energy for the gas is small and can be ignored. Under these conditions, the difference in pressure, ΔP, is given by Eq. (10) below [17]: P 4 −P 3 ¼

ρave gL3 sin α up Gp þ þ ∑pf gc gc

ð10Þ

where ρave is the density of the mixture that consists of the solids and the gas, Σpf is the sum of the frictional losses between the gas-to-solid mix and the walls of the conduit, up is the solid superficial velocity, Gp is the solid mass flux, g is the acceleration due gravity, gc is a conversion factor and L3 is the length of the conduit. Eq. (10) can also be written as: h P 4 −P 3 ¼

i ρp ð1−εÞ þ ρg ε gL3 sin α gc

up Gp þ þ ∑pf gc

2f g ρg εu2g L3 2f s ρs ð1−ε Þu2s L3 þ D3 g c D3 g c

P 1 −P 5 ¼

4 4 2ρp ð1−εÞu2p X 2ρg εu2g X Li Li ρp ð1−εÞgL3 sinα fs þ fg þ gc gc D g D c i¼1 i i¼1 i ð15Þ ρp ð1−ε ÞgL3 sinα ρg εgL3 sinα up Gp þ þ þ gc gc gc

Geldart [8] proposed Eqs. (16) and (17) to obtain the velocity of the particles and the velocity of the gas, respectively: up ¼

4Gs πD2 ρp ð1−εÞ

ð16Þ

ug ¼

4Q πD2 ρg ε

ð17Þ

where Gs is the flow of solids and Q is the corrected flow of the air that enters the valve. Substituting Eqs. (16) and (17) into Eq. (15) and reordering the elements, we obtain: P 1 −P 5 −2ρg εf g

" 2 # 2 2 2 ug1 L1 ug2 L2 ug3 L3 ug4 L4 þ þ þ D1 D2 D3 D4

h i − ρp ð1−εÞ þ ρg ε gL3 sin α ¼ 2ρp ð1−ε Þf s

ð12Þ

"

2

2

16 Gs L1 16 Gs L2 þ ρ2p ð1−εÞ2 π2 D51 ρ2p ð1−εÞ2 π2 D52 ! 4Gs # "2 # 2 2 ρs ð1−ε ÞπD23 16Gs L3 16Gs L4 þ 2 þ þ2 ð 1−ε Þ ð18Þ πD23 ρp ð1−ε Þ2 π 2 D53 ρ2p ð1−ε Þ2 π 2 D54

ð11Þ

According to Matousek [31], the total frictional losses are due to the friction between the particle layers and the walls of the tube and the viscous friction between the fluid and the walls of the tube, ∑pf ¼

−0:25

f g ¼ 0:0791Re

2

2ρg εug dp 2ρp ð1−εÞup þ fs þ f ¼0 dx Di g c Di g c g

P2

In Eq. (12), ug is the superficial gas velocity, fg is the factor of gas friction related to the Reynolds number, Re. The term fg can be calculated using Eq. (13) proposed by Kunii and Levenspiel [17]:

Solving for Gs, we obtain an expression for the prediction of the flow of solids, Gs ¼ ½κ ðβ−δ−γ Þ

1=2

where κ≡

32f s

h

ρp ð1−εÞπ2 L1 D51

Þ þ DL25 þ DL35 þ DL45 þ ð2f1−ε D4 2

3

4

s

i

3

β≡ðP 1 −P 5 Þ

u2g1 L1 u2g2 L2 u2g3 L3 u2g4 L4 þ þ þ D1 D2 D3 D4  h i γ≡− ρp ð1−εÞ þ ρp gL3 sinα δ≡−2ρg εf g

Fig. 5. Diagram of the tilted section of the S-valve.

! ð19Þ

G.M. Guatemala et al. / Powder Technology 230 (2012) 77–85

81

The S-valve works cyclically with a pulsed air flow; the pulses are induced by a two-way solenoid valve of the open–close type and a time relay. During the On interval (tON), the particles descend the vertical tube as in a mobile bed, whereas the particles become fluid within the horizontal tube and move toward the solid seal to finally exit the S-valve. In the Off interval (tOFF), aeration is stopped to turn the horizontal tube particles fluid and, consequently, seal the solids. The solids accumulate in a packed bed up to the peak of the ascending tube from the solid seal, thereby creating a seal that effectively prevents the passage of solids. The model includes two significant factors:

3. Results and discussion

Gs;actual ¼ ðSF Þ Gs

ε ¼ a⋅ expð−bus Þ þ εmin

ð24Þ

f s ¼ k⋅ expð−d us Þ þ f s;min

ð25Þ

ð20Þ

where SF is the cyclic spitting factor—one of the characteristics of the S-valve that depends on the time relay selected for the ON–OFF cycle. 2.5. Definition of spitting factor, SF The ON–OFF cyclic system for pulsing air operates under conditions of square waves (Fig. 6). The function defined here as SF takes on the following values, depending on the case [32]:  SF ¼

0; 1;

t OFF t ON

ð21Þ

Integrating for the entire cycle yields the following: t

SF ¼ ¼

t

∫0OFF SF dt þ ∫0ON SF dt t

∫0CICLO dt

t ON t ON þ t OFF

t

¼

t

∫0OFF ð0Þdt þ ∫0ON ð1Þdt t

∫0CYCLE dt

¼

t ON t CYCLE ð22Þ

Substituting Eqs. (19) and (22) for Eq. (20) and reordering the terms, we obtain the final expression of the mathematical model used to predict the flow of solids, in kg/s, in the S-valve based on the air flow that enters the valve, Q, in m 3/s:

Gs;actual

2 31=2 4 32 f g 2 P Li −ΔP− Q −ρ gL sinα ave 3 6 7 D5 π2 ρg ε π pffiffiffiffiffi 6 7 i¼1 i ρp ð1−ε ÞðSF Þ6 ¼ 7 4 4 5 4 P Li 1 2f s ð1−ε Þ D5 þ D4 i¼1

i

3

ð23Þ The spitting factor, SF, corresponds to the percentage of time the valve was kept open (ON); for the model we present in this article, for example, the valve was synchronized for 90% in the OFF interval and 10% in the ON interval.

Determining the void fraction, ε, and the friction coefficient, fs, in the diluted state is a complex task because many empirical expressions exist for this estimate with limited application ranges [25,26,33]. To estimate these parameters and use them in Eq. (23), we developed an empirical expression in this work based on the surface velocity of the air flow and used the Gauss–Newton method for the corresponding optimization. For a given valve and particle diameter, values for fs and ε were calculated to adjust the parameters of the proposed exponential models in Eqs. (24) and (25).

In the previous equations, the independent term represents the minimum value that the respective variable can take. When we linearize the previous equations, we obtain the following expressions: lnðε−ε min Þ ¼ lnðaÞ−bus

ð26Þ

  ln f s−f s; min; ¼ lnðkÞ−d us

ð27Þ

For fs, the maximum and minimum values were estimated based on Eq. (28), which was proposed by Yang [33]. For ε, 0.9 and 0.99 were taken as the lower and upper limits, respectively, which were, in turn, estimated using the empirical expression proposed by Klinzing [25] for horizontal pneumatic transport. " #−1:15 1−ε ð1−εÞus pffiffiffiffiffiffiffiffi f s ¼ 0:0293 3 ε gDv

ð28Þ

For experiments at low pressures (lower than 4 × 10 5 Pa), the parameters that were obtained showed little relation to the experimental results (r 2 b 0.85 and root-mean-square error (RMSE) > 0.045). We show the optimized values of ε for a pressure of 0.7 MPa and with a flow of 300 L/min (Table 3) as an example of the results obtained with the method previously described. Table 4 shows the optimized values of fs for a pressure of 0.7 MPa and a flow of 300 L/min, as well as the values obtained by Yang [33] using Eq. (28). The values thus obtained are within the same order of magnitude and the same conditions, all of which allows us to assert that the correlation obtained is acceptable and can be used in Eq. (23) to estimate the flow of solids from the S-valve. Fig. 7 shows the variation of the friction factor for solids, fs, based on the air flow, Q, and the pressure, P, for different particle diameters, dp, using an S-valve with a diameter of 0.05 m. The influence of the particle size is shown in this graph: for the coffee case, with a particle diameter, dp, greater than that of rice and lentil seeds (independent of their sphericity), the friction factor is greater because of the interactions and collisions of the grain with the duct walls. Our recommendation in this case is to use a relation Di ≥ 2.5 dp to prevent possible blockages in the pipe. Fig. 8 shows the dependence of the void fraction, ε, on the air flow, Q, and the pressure, P, for different particle diameters, dp, using a

Table 3 Values of ε. Valve diameter (m)

Fig. 6. Square wave function.

Particle

0.0127

0.0254

0.0508

Rice Lentil Coffee

0.926 0.9296 –

0.9894 0.9899 –

0.9521 0.9614 0.9726

82

G.M. Guatemala et al. / Powder Technology 230 (2012) 77–85

Table 4 Values of fs. Valve diameter (m)

Valve diameter (m) Yang [33]

Particle

0.0127

0.0254

0.0508

0.0127

0.0254

0.0508

Rice Lentil Coffee

0.0002 0.0001 –

0.0158 0.0010 –

0.0151 0.0188 0.0202

0.0002 0.0002 –

0.0019 0.0020 –

0.0127 0.0127 0.0130

valve 0.05 m in diameter, which once again points to the importance of particle size. Using the empirical correlations obtained to estimate the values of the void fraction ε (Eq. (24)) and the friction factors fg (Eq. (13)) and fs (Eq. (25)) within the model to predict the flow of solids in the S-valve (Eq. (23)), one can analyze the behavior of the solid discharge based on the gas velocity for the different solids featured in our study. Fig. 9 presents the adjustment of the model using the experimental data with lentil particles as the solid and a 2-inch S-valve (0.04 m interior diameter); an adjustment of 99.8% was obtained. Eq. (23), however, does not adjust to velocities lower than the minimum fluidization velocity. The minimum fluidization velocity is the air velocity necessary to overcome the frictional forces and thus begin the movement of the particles. Knowledge of this velocity is of the utmost importance because such knowledge permits the definition of criteria for the system operation. For large particles, the minimum fluidization velocity can be obtained with Levenspiel's expression [21]: 2

umf ¼

  dp ρp −ρg g 24:5ρg

ð30Þ

In Table 5, we show the values of the experimental minimum velocities of fluidization and those calculated with Eq. (30). Based on experimental observations, the stable flow for the particles studied is presented at a minimum horizontal transport velocity that is 1.5 times the minimum fluidization velocity, umf, as an operation criterion. Under this limit, the system becomes unstable and the model lacks accuracy in its predictions. In Fig. 10, the adjustment for the proposed mathematical model for rice grains for discharges between 0.10 kg/s and 0.35 kg/s and for velocities between 0.5 m/s and 2.5 m/s is 97%. Deviations occur in this model at velocities less than 1.5 times the minimal fluidization velocity and at velocities greater than 4.5 times the minimum fluidization velocity.

Fig. 7. Variation in the solids friction factor based on the air flow and operating pressure for an S-valve with a diameter of 0.050 m.

Fig. 8. Variation in the void fraction based on the air flow and operating pressure for an S-valve with a diameter of 0.050 m.

In Fig. 11, the adjustment interval to the proposed mathematical model for coffee grains for discharges between 0.03 kg/s and 0.55 kg/ s and at velocities of 2.22 m/s and 3.20 m/s is approximately 99%. In every case, the RMSE between the adjustment to the observed data and that predicted by the model is less than 10%, as shown in Fig. 12. This result confirms the validity of the proposed model. Lentil particles presented the most flows under the same system operating conditions. The behavior of the lentils confirms that spherical particles tend to exhibit a smoother flow than particles that are highly irregular [34]. The results obtained confirm that the average friction between individual particles, σ x , is negligible or can be disregarded for diluted flows: at high flow velocities of 4.30 umf for rice, 3.20 umf for lentils, and 2.12 umf for coffee, the theoretical model for solid discharge adjusts the experimental data by 97%. However, if the aeration velocity decreases (45% for rice, 70% for coffee, and 34% for lentils), an accumulation of solids occurs in the lower part of the tube (dunes), and air flows mainly through the top (with the flow of diluted solids). In this case, the assumption that σ x is negligible is no longer valid. The model in this proposed study was tested with the experimental data we reported previously [32] and with the experimental data from this study; the results are shown in Fig. 13. The levels of prediction and the errors between the predicted and observed values are acceptable (less than 15%). The S-valve is a design with very specific geometric and operational characteristics. Operation of the S-valve requires the collection

Fig. 9. Output of solids (lentil particles) based on the gas velocity.

G.M. Guatemala et al. / Powder Technology 230 (2012) 77–85

83

Table 5 Minimum transport and fluidization velocity in m/s.

Minimum horizontal transport velocity (experimental) Minimum fluidization velocity (Eq. (30))

Rice

Coffee

Lentil

0.69 0.60

1.41 0.95

0.93 1.50

of basic important data to understand and operate the device. The particles are assumed to begin to move within the valve based on the minimum fluidization velocity [17]. In Fig. 14, we can observe the value of the air surface velocity under which the particles begin to move. In every case, we can observe the tendency to form a plateau that indicates the limit for the solid discharge set by the diameter of the S-valve. The maximum solid discharge values obtained with an S-valve of 0.044 m diameter were 0.557 kg/s for rice, 0.548 kg/s for lentils, and 0.451 kg/s for coffee.

4. Conclusions A mathematical model was developed to predict the discharge of solids in an S-valve by means of hydrodynamic analysis (Eq. (23)). When developing this model, we considered the spitting factor, SF, which corresponds to the percentage of time that the valve was kept open (ON). In this model, both the values of the solid friction factor and the void friction factor were adjusted based on the experimental data. The adjusted parameters were the starting point from which empirical equations (Eqs. (24) and (25)) that relate the void fraction, ε,

Fig. 10. Solids discharge (rice) based on the gas velocity.

Fig. 11. Solids discharge (coffee) based on the gas velocity.

Fig. 12. Values observed compared to values predicted by the model.

and the solid friction factor, fs, to the surface aeration velocity, uo, at fixed particle and valve diameters were developed. We were able to identify the range that corresponds to the surface velocity that permits the S-valve to operate in a stable or balanced state based on the experimental data for the minimum horizontal transport velocity and the estimate of the minimum fluidization velocity for the different solids used (Eq. (30)). The flow changes the operating system when the conduit diameter is increased and/or when the particle diameter is greater. We found it advisable to work with an operation criterion where the surface air velocity was kept within the range of 1.5 umf b uo b 4.5 umf. The proposed model allows for the prediction of the operation of the S-valve for different flow situations for Geldart group D solids. We obtained predictions for different conditions and deviations of ±3% for lentil and rice and ± 2% for coffee within a range of operation between 0.4 and 0.7 MPa. These results demonstrate the diversity of operations for the S-valve. The model that was developed was also used to verify the data presented by Martínez [32], which resulted in a deviation of less than 10% between the observed data and the predicted data. This project used two S-valve prototypes. Currently, an S-valve is operating in a pilot project for drying and toasting coffee grains at the State Research and Assistance in Technology and Design Center of Jalisco, A.C. (Centro de Investigación y Asistencia en Tecnología y Diseño del Estado de Jalisco, A. C., CIATEJ). Nomenclature A Cross sectional area (m 2) a Coefficient defined by Eq. (24) (dimensionless) b Coefficient defined by Eq. (24) (s/m) d Coefficient defined by Eq. (25) (s/m) dp Geometric particle diameter (m) Di ith-section diameter (m) DV Valve diameter (m) ff Fanning friction factor (dimensionless) fg Friction of air on horizontal wall (dimensionless) fs Friction of particle on horizontal wall (dimensionless) fs,min Minimum friction of particle on horizontal wall (dimensionless) 2 g Acceleration of gravity (9.8   m/s ) kgm gc Conversion factor 1 N s2 Mass flux of particles (kg/m 2·s) Gp Gs Mass flow rate of solids (kg/s) Gs,actual Actual mass flow rate of solids defined by Eq. (20) (kg/s) Δh Change in height (m) k Coefficient defined by Eq. (25) (dimensionless) Li ith-section length (m) P Pressure (Pa)

84

G.M. Guatemala et al. / Powder Technology 230 (2012) 77–85

Fig. 13. Values observed by Martínez [32] and Santoyo et al. [11] compared to values predicted by the model.

Fig. 14. Discharge of solids based on the air surface velocity.

References pf Q Re S SF tCYCLE tOFF tON umf up us ug u0 x Δx

Frictional losses (dimensionless) Air flow (m 3/s) Reynolds number (dimensionless) Parallel area to flow (m 2) Spitting factor defined by Eq. (20) (dimensionless) Cycle time in timer relay (s) OFF interval time in timer relay (s) ON interval time in timer relay (s) Superficial gas velocity at minimum fluidizing conditions (m/s) Particle velocity (m/s) Superficial particle velocity (m/s) Air velocity (m/s) Superficial air velocity (m/s) Horizontal position (m) Change in horizontal position (m)

Greek letters α Slope angle in S-valve (°) β Coefficient defined by Eq. (19) δ Coefficient defined by Eq. (19) ε Void fraction (dimensionless) μ Dynamic viscosity (kg/m·s) εmin Minimum void fraction (dimensionless) γ Coefficient defined by Eq. (19) ρg Air density (kg/m 3) ρp Particle density (kg/m 3) σx Averaged friction between individual particles (N/m 2) τx Shear stress on horizontal wall (N/m 2) φs Sphericity (dimensionless)

Subscripts ave Weighted mean

Acknowledgments The authors wish to thank the University of Guadalajara, the State Research and Assistance in Technology and Design Center of Jalisco, A.C. (Centro de Investigación y Asistencia en Tecnología y Diseño del Estado de Jalisco, A. C.), and the State Council on Science and Technology for the State of Jalisco (Consejo Estatal de Ciencia y Tecnología del Estado de Jalisco), Project COECYTJAL-UDG-PS-2009-564, for the support they provided to this project.

[1] D.J. Sables, Poudres et Grains, Eyrolles Sciences, Paris, France, 1997. [2] R.M. Nedderman, Statics and Kinematics of Granular Materials, Cambridge University Press, Cambridge, 1992. [3] B.J. Ennis, J. Green, R. Davies, The legacy of neglect in the U.S. Chemical Engineering Progress 90 (1994) 32–43. [4] T.M. Knowlton, I. Hirsan, L-valves characterized for solids flow, Hydrocarbon Process 3 (1978) 149–156. [5] G. Joseph, B. Mena, E. Moreno, E. Sansores, O.R. Walton, Granular flow in a solar hexagonal silo, In: in: A. Aït-Kadi, J.M. Dealy, D.F. James, M.C. Williams (Eds.), Proceedings XIIth International Congress on Rheology, 1996, pp. 795–796, Québec (Québec, Canada). [6] T.E. Torres, M.Y. Retirado, Mathematical modeling of pneumatic transportation of laterite mineral in dense phase, Mining Geology 23 (2007) 1–31. [7] K. Terasaka, S. Akashi, H. Tsuge, Development of a J-shaped pneumatic valve to control the solid particle circulation rate in a circulating fluidized bed, Powder Technology 126 (2002) 13–21. [8] D. Geldart, P. Jones, The behavior of L-valves with granular powders, Powder Technology 67 (1991) 163–174. [9] T. Hua, L. Hongzhong, L. Xuesong, Z. Qiayu, Hydrodynamic modeling of the L-valve, Powder Technology 129 (2002) 8–14. [10] G.M. Guatemala, Tostado de café (Coffea arabica) utilizando un sistema de lechos fuente en multietapa. Tesis de Doctorado, Universidad de Guadalajara, 2007. [11] F. Santoyo, G. Guatemala, J. Bon, I. Orozco, F. García, E. Arriola, Modelo estadístico que describe el funcionamiento de una válvula de reciente creación para el control del flujo de sólidos granulares, Revista Alimentaria 8 (2008) 395–398. [12] F. Santoyo, G. Guatemala, I. Orozco, E. Arriola, La válvula-S: modelado del flujo de sólidos granulares utilizando números adimensionales (The S-valve: modeling granular solids flow using dimensionless numbers), Revista Mexicana de Ingeniería Química 9 (2) (2010) 179–188. [13] D.I.Z. Ballaz, Flujos y atascos en medios granulares. Memorias, Dpto. de Física y matemática aplicada. Facultad de Ciencias, Universidad de Navarra, España, 2005. [14] W. Huang, G. Xu, H. Li, A. Kang, Mathematical model for the L-valve and its application in matter transfer control, Journal of Huazhong University of Science and Technology 23 (1995) 100–104. [15] S. Matsumoto, M. Kikuta, S.J. Maeda, Effect of particle size on the minimum transport velocity for horizontal pneumatic conveying of solids, Journal of Chemical Engineering of Japan 10 (1977) 273–279. [16] W.C. Yang, T.M. Knowlton, L-valve equations, Powder Technology 77 (1993) 49–54. [17] D. Kunii, O. Levenspiel, Circulating fluidized-bed reactor, Chemical Engineering Science 52 (1997) 2471–2482. [18] M.A. Daous, A.A. Al-Zahrani, Modeling solids and gas flow through an L-valve, Powder Technology 9 (1998) 86–89. [19] E. Arriola, Residence Time Distribution of Solids in Staged Spouted Beds, Ph.D. Dissertation. Oregon State University, U.S.A., 1997. [20] J.M. Aragón, “La Válvula J,” Computlense University of MadridAvailable at: http://www.ucm.es/info/otri/complutecno/fichas/tec_jmaragon2.htm(Last accessed: 10 January 2010). [21] O. Levenspiel, Flujo de Fluidos e Intercambio de Calor, Editorial Reverte, España, 1993. [22] L.S. Leung, O. Chong, Operation of V-valves for gas-solid flow, Powder Technology 49 (1987) 271–276. [23] T. Hirama, H. Takeuchi, M. Horio, Nitric oxide emission from circulating fluidized bed coal combustion, In: Proceedings of the 9th International Fluidized Bed Combustion Conference, Boston, 1986, pp. 898–910. [24] J.M. Meléndez, B. Gutiérrez, Guía para el Laboratorio de Fenómenos de Transporte I, Departamento de Termodinámica y Fenómenos de Transferencia, Universidad Simón Bolívar, Valle de Sartenejal, 2004. [25] G.E. Klinzing, Gas–solid pneumatic transfer, In: in: G.E. Klinzing (Ed.), Gas Solid Transport, Chemical Engineering Series, McGraw-Hill, U.S.A., 1981, pp. 81–107. [26] W.C. Yang, Handbook of Fluidization and Fluid Particle System, Marcel Dekker, Inc., New York, 2003.

G.M. Guatemala et al. / Powder Technology 230 (2012) 77–85 [27] D. Kunii, O. Levenspiel, Fluidization Engineering, 2nd ed. Butterworth Heinemann, U.S.A., 1991. [28] P. Thorsten, S. Thomas, Computational Granular Dynamics, Springer, Germany, 2005. [29] D. Shoemaker, J. Garland, W. Nibler, Experiments in Physical Chemistry, McGrawHill, U.S.A., 1989. [30] E.A. Baryeh, B.K. Mangope, Some physical properties of QP-38 variety pigeon pea, Journal of Food Engineering 56 (2002) 59–65. [31] V. Matousek, Distribution and friction of particles in pipeline flow of sand–water mixtures, In: in: A. Levy, H. Kalman (Eds.), Handbook of Powder Technology, Volume 10: Handbook of Conveying and Handling of Particulate Solids, Elsevier, The Netherlands, 2002, pp. 465–471. [32] O.E. Martínez, Caracterización de una nueva válvula de control de flujo para sólidos, la válvula-S. Tesis de maestría. Universidad de Guadalajara, 2006. [33] W.C. Yang, A correlation for solid friction factor in vertical pneumatic conveying lines, AIChE 24 (1978) 548. [34] P.K. Mehta, P.J.M. Monteiro, Concrete: microstructure, properties and materials, 3rd ed. McGraw-Hill, Westerville, 1993.

Guadalupe M. Guatemala-Morales, Birth Date October 29, 1963; Postdoctoral in Fluidization Engineering, Universidad de Guadalajara (UDG); Ph.D., ChE (UDG), 2007. She is actually working as Researcher and Professor at a federal research center (CIATEJ). His non-academic positions include positions as Junior Technician in a Research Center, Chemical Technician and Chemist, at the Department of new products development in CIBA-GEIGY MEXICANA. She is specialized in Fluidization and Food Engineering, developing products from agave (beverages and syrups), and processes for fruits, vegetables, functional foods, grains and seeds (industrial application of spouted beds). http://www.ciatej.net.mx/index.php/investigacion/tecnologiaalimentaria/investigadores-de-tecnologia-alimentaria/ guadalupe-maria-guatemala

Felipe Santoyo-Telles, Birth Date October 29, 1963; PhD in Science and Technology from the Center of Research, Technology and Design of the State of Jalisco (CIATEJ). He currently teaches at Universidad de Guadalajara. His fields of interest are Mathematical Modeling and Inferential Statistics.

85 Luis Virgen-Navarro, currently a PhD student at the Centro de Investigación y Asistencia en Tecnología y Diseño del Estado de Jalisco (CIATEJ). Birth Date February 25, 1983. M.Sc. in Biotechnological Processes (2011) and B.Sc. in Chemical Engineering (2007) both from Universidad de Guadalajara.

Rosa Isela Corona-González, Associate Professor, Chemical Engineering Department, Universidad de Guadalajara, Mexico (2000-). Birth Date October 7, 1968; Ph.D., ChE Universidad de Guadalajara (UDG), 2005; her fields of specialization are Food Engineering, and Production of metabolites of industrial interest from cells. She currently teaches courses on Chemistry, Biochemistry and Industrial Microbiology.

Enrique Arriola-Guevara, Senior Professor, Chemical Engineering Department, Universidad de Guadalajara, Mexico (1998 -). Birth Date November 3, 1943; Ph.D., ChE Oregon State University (OSU), 1997; long experience in different academic positions at several universities. His non-academic positions include Production Engineer in a Sulfuric Acid Plant, Construction manager in a Chemical Engineering Lab Construction Program, and Consultant to numerous firms, agencies and individuals. His fields of specialization are Fluidization Engineering, Transport Phenomena, Thermodynamics, and Food Engineering, developing a novel multi-stage spouted bed system for processing grains and seeds. His publications include 3 books and several papers in Refereed Technical Journals.