Solid-Mass Flow-Rate Prediction in Dense-Phase Pneumatic

Article pubs.acs.org/IECR

Solid-Mass Flow-Rate Prediction in Dense-Phase Pneumatic Conveying of Pulverized Coal by a Venturi Device Haifeng Lu, Xiaolei Guo, Yi Liu, Peng Li, and Xin Gong* Key Laboratory of Coal Gasification and Energy Chemical Engineering of Ministry of Education, Shanghai Engineering Research Center of Coal Gasification, East China University of Science and Technology, Shanghai 200237, P. R. China ABSTRACT: This paper aims at predicting the solid-mass flow rate of pulverized coal in dense-phase pneumatic conveying systems. A onedimensional model was developed based on the proper modification of a dilute-flow model by inserting a two-phase flow multiplier. The relationships between the model parameters (discharge coefficient and pressure ratio parameter) and the hydrodynamic nondimensional numbers (Reynolds and Stokes numbers) were determined, and the fitting coefficients were obtained from calibration tests by using regression analysis. Two methods attached to the model were proposed to predict the solid-mass flow rate, and both can provide satisfactory predictions with acceptable errors. These results indicated the possibility of applying a venturi device to measure the solid-mass flow rate at high-pressure and dense-phase conditions.

1. INTRODUCTION Entrained-flow pulverized coal gasification is recently undergoing rapid industrialization around the world with the characterization of large-scale, high-efficiency, and clean emissions.1,2 The most eminent environmental advantage of coal gasification lies in its inherent reaction features that produce negligible sulfur and nitrogen oxides, as well as other pollutants.3 Handling coal is a major process concern, and reliable feeding of pulverized coal into the gasifier is an important issue during the industrial process because it can affect the final product quality and the efficiency of the processes. Consequently, accuracy and repeatability of the flowrate measurements are necessary for process development and control. Dense-phase pneumatic conveying technology has been commonly applied to feed pulverized coal into the gasifier because of much smaller quantities of carrier-gas consumption and more efficient power utilization.4,5 In such a process, the conveying pressure and solid concentration are high with up to 4 MPa and 500 kg/m3, respectively.6 Because the dense-phase gas−solid two-phase flow in the pneumatic conveying pipeline is an unsteady and complex nonlinear dynamical system, it is not easy to measure the solid-mass flow rate at such highpressure and dense-phase conditions.7 In some cases, a weighing hopper is used to monitor the mass variation of bulk solids, where weighing cells are installed in the hopper and the solid-mass flow rate can be determined from the weight versus time curve. However, the weighing hopper measurement always brings errors due to friction in the fulcrum bearings, and to date, no absolute online measurement has been achieved.8 In the other cases, a mass flow meter system is used to measure the flow of bulk solids through pneumatic lines. The system is composed of two independent sensors and a single transmitter, where the velocity and © XXXX American Chemical Society

concentration sensors operate using capacitance technology. The system is expensive and complex and needs to be calibrated before use, and the measurement result is affected significantly by the powder properties, especially the water content.9 So, it seems the fundamental study of solid-mass flowrate measurement at high-pressure and dense-phase conditions is still very limited and imperfect in comparison with increasing industrial applications. In the last decades, many investigations focused on flow measurement using differential pressure devices. The most common differential pressure device is the venturi meter. Venturi has a structure of first narrowing down in diameter and then opening up back to the original pipe diameter. The smooth flow pattern in venturi reduces frictional losses, which increases the reliability of the venturi. Changes in the crosssectional area cause changes in the velocity and pressure of the flow.10 Calculation of the fluid flow rate by reading the pressure loss across a pipe restriction is perhaps the most commonly used flow measurement technique in industrial applications. Venturi is routinely applied for metering the flow of a singlephase gas or liquid,11 while successful applications have generated considerable interest in the use as a gas−solid flow meter.12 The application of venturi as a gas−solid flow meter has several advantages. Compared to the weighing method applied in blast furnace spurts coal technology, the venturi device can measure the mass flow rate timely and accurately, while the weighing method cannot provide absolute online measurement. Compared to the solid-mass flow meter applied in entrained-flow pulverized coal gasification technology, the Received: July 25, 2016 Accepted: September 12, 2016

A

DOI: 10.1021/acs.iecr.6b02845 Ind. Eng. Chem. Res. XXXX, XXX, XXX−XXX

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Industrial & Engineering Chemistry Research Table 1. Physical Properties of Pulverized Coal material pulverized coal

size distribution (μm) for 10%, 50%, 90%

mean diameter (μm)

external moisture (wt %)

particle density (kg/m3)

bulk density (kg/m3)

angle of repose (deg)

2, 20, 109

42.5

1.9

1532

419

45.7

pulverized coal is characterized by poor flowability and certain cohesive forces. The purpose of this work is to assess the possibility of using a venturi to measure the solid-mass flow rate at high-pressure and densephase conditions. The experimental program was conducted on the facility of a dense-phase pneumatic conveying system, as shown in Figure 1. The whole system consists of a gas cylinder, feeding and

venturi device shows better flexibility, let alone much lower cost. The total cost of ownership of a venturi is favorable because of savings in installation, operating, and maintenance costs. The study of gas−solid flow through the venturi was faced by researchers most on an experimental basis and described with empirical approaches. For example, Lee and Crowe13 carried out an experimental investigation to determine the scaling parameters applicable to measurement of the mass flow rate of gas−particle suspensions through the venturi. Azzopardi et al.14 proposed a quasi-one-dimensional model to predict the behavior for gas−solid flow in the venturi, where both acceleration and deceleration behaviors of the gas and solid particles were described. Giddings et al.15 applied venturi as a two-mixed-phase meter to measure the flow rate of pulverized coal in coal-fired power stations. Liang et al.16 studied the pressure letdown flow characteristics through the venturi in dense-phase pneumatic conveying at high pressure. Liu et al.17 carried out an experimental study on the flow characteristics and pressure drop of a gas−coal mixture through the venturi, where the distributions of pressure, loading ratio, and gas velocity along the venturi were obtained and analyzed. Nevertheless, the majority of papers reporting investigations on gas−solid flow through the venturi are focused on dilute or low-pressure flows. It seems that there is still a lack in the literature of quantitative and systematic investigation on the venturi aiming at measurement of the solid-mass flow rate at high-pressure and dense-phase conditions. The purpose of this work is to exhibit experimental results showing the possibility of the application of the venturi to measure the solid-mass flow rate of pulverized coal in the dense-phase pneumatic conveying system. First, a one-dimensional model for solid-mass flow-rate prediction was developed with proper modifications of the dilute-flow formula. Second, a series of calibration tests were carried out to obtain the required model parameters based on regression analysis. Finally, a series of verified tests were carried out to examine the model application in high-pressure and dense-phase gas−solid flow.

Figure 1. Schematic diagram of the experimental setup: (1) gas cylinder; (2) gas distributor; (3) metal-tube rotameter; (4) feeding vessel; (5) pressure sensor; (6) solid concentration sensor; (7) solid velocity sensor; (8) venturi; (9) weighing cell; (10) receiving cell; (11) dust filter. receiving vessels, pipelines, a venturi, a dust remover, and a data acquisition system. The test rig is described in detail elsewhere.17 The venturi is inserted into the pipework in the vertical riser pipe with a given structure of inlet angle θ1 = 10°, diffuser angle θ2 = 16°, throat diameter dt = 6 mm, and throat length Lt = 260 mm. The test was carried out by maintaining a normal atmosphere of the receiving vessel and increasing the gas flow rate to increase the pressure of the feeding vessel (Pf). N2 from the gas cylinder was used as the carrier gas and divided into three parts. Two parts of the gas were introduced into the feeding vessel from the top (Qg1) and cone (Qg2), and the rest was introduced to the front end of the conveying pipe (Qg3). Pulverized coal was conveyed from the feeding vessel by the carrier gas to the receiving vessel along a 35-m-long pipeline with a 15 mm internal diameter (D). Gas and solids were separated in the receiving hopper, where the gas was discharged to the atmosphere through the filter and the solids were retained in the receiving vessel for recirculation. The gas-volume flow rates (Qg1, Qg2, and Qg3) were adjusted by three metal-tube rotameters to achieve the desired operating conditions. The pressure along the venturi was measured by several pressure sensors (P1, P2, P3, P4, and P5) located at the inlet, throat, and outlet of the venturi, as shown in Figure 2. All of the pressure values measured in this paper are gauge pressures. Capacitance sensors DK 13 and DC 13 (model 2019 system) produced by Thermo Ramsey Co, were used to measure the solid velocity (Us) and solid concentration (Cs), respectively. The solid velocity sensor DK 13 was about 600 mm away from the upstream pressure tap of the venturi

2. EXPERIMENTAL SECTION Pulverized coal was used as the experimental material in this work, whose physical properties are shown in Table 1. The particle-size distribution was measured using a Mastersizer 2000 laser granulometer with a wet dispersion unit (Malvern Instruments). The particle-size distribution of pulverized coal covered a wide range of sizes with a mean particle diameter of about 42.5 μm. The moisture content was measured by an MA150 infrared moisture analyzer (Sartorius) according to the National Standards of the People’s Republic of China.18 The detected external moisture content was very small, lower than 2%, suggesting a limited influence on the flow characteristics that can be ignored. The particle density was determined by mercury intrusion analysis.19 A powder characteristics tester (PT-X) was used to measure the bulk density and angle of repose. Correlating powder flowability and some simple physical measures, Carr20 suggested that an angle of repose below 30° indicated good flowability, 30−45° some cohesiveness, 45−55° true cohesiveness, and >55° sluggish or very high cohesiveness and very limited flowability. Under Carr’s criteria, B


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Industrial & Engineering Chemistry Research

where μg is the gas viscosity. For two-phase flow in the venturi, the mixture-to-gas pressure ratio is defined as a measure of the increase in the differential pressure owing to the presence of particles in the flow. It is considered that the magnitude of this ratio is proportional to the degree to which the particles accelerate through the venturi and to the concentration of particles in the gas. Dimensional analysis has shown that venturi pressure-flow data should be correlated in terms of dimensionless parameters including the mixture-to-gas pressure ratio, the Stokes number, and the particle-to-gas mass loading ratio:

Figure 2. Structure diagram of the venturi and located pressure taps. (P1). Weighing cells were installed in the receiving vessel to monitor the mass variation of pulverized coal. All of the signals were acquired at a frequency of 1 Hz by a computer through a data acquisition board.

Δpmix /Δpg = f (St , Z)

(5)

where the Stokes number, St = ρpdp Ug/18μgL, is a measure of the degree of particle acceleration in a venturi and the loading ratio, Z = Ms/Mg, is a measure of the concentration of particles in the gas. The Stokes number is the ratio of the time required for particle acceleration (aerodynamic response time) to the time available for particle acceleration (gas residence time). In this work, considering the first narrowing and then expanding structure of the venturi, the Stokes number is modified by deriving a characteristic length term that provides a better approximation of the gas residence time for the venturi. As suggested by Shaffer and Bajura,23 the gas residence time for the venturi should be redefined as the sum of two parts, one for the variable area section and the other for the throat section. Thus, the characteristic length is derived as 2

3. MODEL DEVELOPMENT Model researches have been done on empirical or semiempirical correlations of the mass flow-rate measurement in the venturi, such as single- and dilute-phase flow.21 However, there is still a lack in the literature for predicting the solid-mass flow rate at high-pressure and dense-phase conditions. Derivation of the mass flow rate from measurement of the differential pressure in the venturi is commonly used for singlephase flow in a wide range of industrial applications. The differential pressure between the inlet and throat of the venturi is primarily due to gas acceleration for single-phase gas flow, and the pressure-flow relationship can be accurately predicted by the Bernoulli equation with correction for viscous effects (balance between the kinetic and static energy of a vena fluid). Generally, the energy equation for steady-state conditions and uniform distribution may be written as follows: ρg Δpg = pg,i − pg,t = (Ug,t 2 − Ug,i 2) (1) 2 where Pg,i and Pg,t and also Ug,i and Ug,t are pressures and gas velocities at the inlet and throat of the venturi. The relationship between the differential pressure and mass flow rate through a venturi can be expressed as 1 − β 4 ⎛ Mg ⎞ Δpg = ⎜ ⎟ 2ρg ⎝ CdA t ⎠

⎛ β2 ⎞ L=⎜ ⎟2Lc + β 2L t ⎝ 1 + β2 ⎠

where Lc and Lt are the lengths of the venturi contraction and throat sections, respectively. Then, the updated formula of the Stokes number specifically for a venturi gives ⎫ ⎧ ⎤⎪ ⎡⎛ β 2 ⎞ ⎪ 2 ⎥⎬ ⎢⎜ St = ρp d p2Ug /⎨ 18 μ 2 L L + β ⎟ c t g 2 ⎪ ⎥⎦⎪ ⎢⎣⎝ 1 + β ⎠ ⎭ ⎩

2

(7)

24

According to the well-known approach, there is a linear relationship between the mixture-to-gas pressure ratio and the loading ratio, which can be expressed as

(2)

where Mg is the gas-mass flow rate, Cd is the discharge coefficient, At is the throat area of the venturi, ρg is the gas density, and β is the throat diameter ratio, which is the ratio of the throat diameter dt to the inlet diameter D. In eq 2, the discharge coefficient is introduced to take into account the vena contract cross section, the additional frictional losses, and viscosity and turbulence effects. It is considered that the discharge coefficient is influenced by the Reynolds number and throat diameter ratio. For a given venturi, the throat diameter ratio is fixed (β = 0.4 in this work) and only the effect of the Reynolds number needs to be considered. As reported by Monni et al.,22 the dependence of the discharge coefficient on the Reynolds number can be approximated by an exponential law of the type with coefficients a and b obtained from the experimental data: Cd = a × Reb

(6)

Δpmix /Δpg = mZ + 1.0

(8)

Equation 8 is routinely applied for metering the low-pressure and dilute flow. For the high-pressure and dense-phase flow, the suggestion is to modify eq 8 by inserting a two-phase flow multiplier k, taking into account the effect of gas pressure and loading ratio. The updated expression is therefore as follows: (Δpmix /Δpg )k = mZ + 1.0

(9)

where the two-phase flow multiplier k forms 1 +Zρg/ρp. In order to highlight the importance of this flow multiplier, comparisons between two extreme conditions (dilute vs dense flow) are carried out in Table 2. Data of dilute flow is from the Table 2. Comparison of a Two-Phase Flow Multiplier between Dilute- and Dense-Flow Conditions

(3)

The Re number is evaluated as Re =

ρg UgD μg

(4) C

case

Z (kg/kg)

Pf (kPa)

ρg (kg/m3)

ρp (kg/m3)

k

dilute flow dense flow

0.5 50

3 3000

1.2 36.7

700 1400

1.00 2.31


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Industrial & Engineering Chemistry Research paper by Giddings et al.,8 where the gas density is orders of magnitude smaller than the solid density and the loading ratio is relatively small. In this case, the flow multiplier is in close proximity to 1 and can be ignored accordingly. Data of dense flow is from the paper by Guo et al.5 Because the gas density increases quickly with the pressure and the loading ratio is as high as 50, the flow multiplier k gives a value of 2.29, much higher than 1. As a result, for high-pressure and dense-phase conditions, the flow multiplier becomes important and much attention should be paid to it. The line slope, m, indicates the sensitivity of the venturi differential pressure to the solid flow. Associated with it, m is influenced by the venturi structure, powder properties, operation conditions, etc. According to Liu et al.,17 it can be expressed as a function of the modified Stokes number defined in eq 7: m = c × St d

(10)

where c and d are coefficients obtained from the experimental data. On the basis of the analysis above, the model for solid-mass flow-rate prediction in dense-phase pneumatic conveying can finally be obtained and expressed as Ms =

ΔPmix /ΔPg − 1 m − (ρg /ρp )(ΔPmix /ΔPg)

Figure 3. Algorithm for obtaining a solid-mass flow rate.

Mg (11)

4. RESULTS AND DISCUSSION 4.1. Pressure Distribution and Pressure Tap Selection. In order to reliably describe the pressure loss of the venturi, pressure sensors should be mounted on measuring points with lower pressure gradient and fluctuation. In this work, pressure sensors were installed along the venturi, as shown in Figure 2: one at the inlet, three at the throat, and one at the outlet. As discussed above, the pressure drop between the inlet and throat was used for calculation of the solid-mass flow rate. So, selection of a proper pressure tap for pressure measurement at the throat is an essential part in this work. First, the characteristics of pressure distribution along the venturi were experimentally analyzed. Pressures in the direction of flow were determined by means of pressure sensors (P1, P2, P3, P4, and P5), and the distribution characteristics of the static pressure directly describe the flow characteristics. Figure 4 shows the change trend of pressure along the venturi at conveying condition of a solid-mass flow rate of around 756 kg/h and a gas velocity of around 3.8 m/s. Overall, there is a

It should be noted that the gas-mass flow rate presented in eq 11 is not as easy as that obtained from the single-phase flow. Usually, for single gas flow, the gas-mass flow rate throughout the venturi equals that introduced into the system according to material balance and can be expressed as Mg = (Q g1 + Q g2 + Q g3)ρg

(12)

However, for gas−solid two-phase flow, the gas-mass flow rate through the venturi should be calculated by deducting the portion that stayed in the feeding vessel from the introduced gas. It is considered that, during conveying, pulverized coal is discharged from the feeding vessel and there should be some gas used to fill the space of the departed solids to maintain the pressure of the feeding vessel. Consequently, the gas-mass flow rate in the pipeline or through the venturi is Mg = (Q g1 + Q g1 + Q g3 − Ms /ρp )ρg

(13)

Considering that the gas-mass flow rate Mg is a function of the solid-mass flow rate Ms, as shown in eq 13, an iterative solution procedure was proposed to determine both the gas and solid-mass flow rates. The iteration algorithm is shown in Figure 3. The geometries values (β and At) and physical properties (ρp) are described in the Experimental Section. The conveying parameters Qg1, Qg2, Qg3, pi, and pt are measured values. The fitting coefficients a, b, c, and d can be obtained from the calibration tests. The program is initiated after guessing a value for the solid-mass flow rate Ms′, the gas-mass flow rate through the venturi Mg is determined from eq 13, the gas pressure drop ΔPg is calculated by eq 2, and the target value of the solid-mass flow rate Ms is obtained from eq 11. These two values, Ms′ and Ms, will compare with each other. If the difference is larger than a predetermined value ξ, another Ms′ is assumed, and the process is repeated until a satisfactory result is obtained.

Figure 4. Pressure distribution along the venturi. D


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Industrial & Engineering Chemistry Research significant decrease in the static pressure when a dense gas− coal mixture flows through the venturi. In the convergent section, a remarkable increase in the gas velocity is observed because of the decrease of the cross-sectional area. Particles will be accelerated significantly by the gas drag force because the particle size associated with the low inertia is relatively small. The energy for this acceleration is always accompanied by a high energy loss and is obtained from the gas static pressure, leading to a sharp decrease in the pressure. As confirmed in Figure 4, there is about a 150 kPa decrease in the static pressure when gas−solid flows through the convergent section, giving a pressure gradient of about 1.2 kPa/mm. In the throat section, variation of the gas velocity becomes less significant because the cross-sectional area remains constant. However, particles will continue to be accelerated by the gas drag force until they catch up with the gas velocity. Apparently, the difference of the gas− solid velocity will decrease gradually and finally reach an equilibrium state in the throat section if it is long enough. The pressure drop of the venturi throat section is about 100 kPa, giving a pressure gradient of about 0.5 kPa/mm. In the diffuser section, the gas velocity will decrease quickly because an increase of the cross-sectional area, while particles can somewhat keep their velocities because of the inertia. The pressure gradient in the diffuser section is about 0.7 kPa/mm, falling between the convergent and throat sections. A pressure loss, instead of pressure recovery, is found in the diffuser section, as shown in Figure 4. For single- or dilute-phase flow, the increasing recovery in the static pressure is always the result of gas deceleration as a consequence of the increasing crosssectional area. On the contrary, for dense-phase flow, the significant growth in the number of high-velocity particles indicates the increase of collisions between particles and particles, particles and wall, etc., which finally leads to a remarkable pressure loss. Consistent with a report by Farbar et al.,24 there will be a transition from pressure recovery to pressure loss when the loading ratio is high enough, where the interactions between particles and gas−solids are highfrequency and intensive. Our previous study17 also confirmed this finding and indicated that the pressure recovery, in fact, moved downward of the venturi exit for dense-phase flow. The discussion above indicates that the pressure drop between the inlet and throat is primarily due to gas acceleration for single-phase gas flow and should be related to particle acceleration simultaneously for gas−solid two-phase flow. For single-phase gas flow, the gas acceleration process is mostly accomplished in the convergent section, and thus the pressure drop can be easily measured by mounting two pressure taps at the inlet and throat, respectively. However, for gas−solid twophase flow, because the particles passing through a venturi take a finite time to accelerate to the gas velocity, the pressure drop along the throat must develop over some distance. Therefore, the pressure tap at the throat should be carefully selected in order to obtain a credible pressure value. Figure 5 reports the rate of pressure change along the venturi, which is the ratio of the pressure decrease (between the venturi inlet and the corresponding location) to the distance. The average value was calculated first, and the gradient value was then obtained to have the stepped curve smoothing, considering the continuous change of the pressure along the venturi. Both values are plotted as a function of the distance, but we will only focus on the change trend of the gradient value, which is considered to provide a better description of the practical situation. As can be seen from the picture, the rate of pressure change decreases

Figure 5. Rate of the pressure change along the venturi.

with the distance first and then inclines to a stable value of about 0.75 kPa/mm; two tangents of the curve cut at a point located around 220 mm. On the one hand, in order for the velocity profile to fully develop and the pressure drop to be predictable, the pressure tap at the throat should not be located too close to the convergent section. On the other hand, if the pressure tap at the throat is located too far from the convergent section, extra pressure loss will contribute to the pressure drop considering that the pressure gradient here is very considerable, as reported above. Therefore, it is considered that a suitable pressure tap at the throat should be P3 (coordinate value = 220 mm ≈ 36.7dt), which can balance the requirement of reliable pressure measurement and low pressure loss. A further consideration is that, for the relatively fine particles, the pressure tap at the throat should somewhat move upward, approaching the convergent section, because these particles are easily accelerated and reach the equilibrium state. On the contrary, for the relatively coarse particles, the pressure tap should somewhat move downward far from the convergent section to allow particles have enough length and residence time to be accelerated. 4.2. Determination of Model Parameters. In order to apply the model developed, it is necessary first to determine the discharge coefficient Cd defined in eq 2 and the line slope m defined in eq 9. A series of calibration tests were carried out to obtain these parameters. First, the single-phase instrument behavior has been analyzed in order to obtain the discharge coefficient Cd by using the experimental values of mass flow rate and pressure. The discharge coefficient was obtained by the substitution of each corresponding operating parameter such as Mg, ρg, and ΔPg and structural parameters of the venturi such as At and β into eq 2. Figure 6 shows the relationship between the discharge coefficient and the Reynolds number. The behavior in Figure 6 seems to correspond to an asymptotic decrease. For small Reynolds numbers, a progressive decrease between the discharge coefficient and Reynolds number is found. For large Reynolds numbers, the variation in the discharge coefficient is less pronounced and finally turns to a relatively stable value. As shown in Figure 6, for large Reynolds numbers Re ≥ 4 × 104, the discharge coefficient tends to a stable value, Cd = 0.86 ± 0.04. Because of the smooth gradual transition down to the throat diameter and back to the full pipe diameter, the friction loss in a venturi is not so significant.25 This leads to a high value of the venturi discharge coefficient, much larger than that of the orifice meter discharge coefficient, which E


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Figure 7 plots the pressure ratio parameter as a function of the modified Stokes number. The general relationship between the pressure ratio parameter and modified Stokes number is that sensitivity decreases with increasing modified Stokes number. The Stokes number is a measure of the degree to which a specific particle can be accelerated by a given gas flow in a given venturi. For low-Stokes-number conditions, particle inertia is relatively low, so particles undergo a large acceleration in the venturi and have a large influence on the gas flow, resulting in a high value of the pressure ratio parameter. For high-Stokes numbers, particle inertia is relatively high, so particles undergo less acceleration in the venturi and have less effect on the gas flow, resulting in a low value of the pressure ratio parameter. The coefficients c and d in eq 10 can be obtained based on regression analysis, whose values are c = 0.87 and d = −1.84. 4.3. Model Validation and Analysis. When eqs 2, 3, 10, and 11 are combined and the iteration algorithm shown in Figure 3 is adopted, it is possible to evaluate the solid-mass flow rate given the values of pressures at the inlet p1 and at the throat p3 of the venturi, of the gas-mass flow rate through the venturi Mg, of the throat area of the venturi At, of the throat diameter ratio β, of the gas density ρg, of the particle density ρp, and of the fitting coefficients a, b, c, and d. A series of verified tests were carried out to examine the model application in high-pressure and dense-phase pneumatic conveying of pulverized coal. The verified tests included 14 cases operated at different conditions where the operation parameters were provided in Table 3. Us and Cs are the solid velocity and solid concentration measured by capacitance sensors DK 13 and DC 13, respectively. Ms,load is the experimental solid-mass flow rate determined from the weight versus time curve provided by the weighing cells. ms, t +Δt − ms, t Ms,load = (15) Δt

Figure 6. Venturi discharge coefficient as a function of the Reynolds number.

usually falls between 0.58 and 0.65. Both the tendency of the curve and the final stable value agree well with Monni et al’s report22 to some degree. Because data in Figure 6 tend to polarize into two groups, a simplified piecewise function was proposed, accounting for the varying tendency of the curve. On the basis of a regression procedure of the experimental data, it gives Cd = 38.82Re−0.38 7000 < Re < 4 × 104 Cd = 0.86

Re ≥ 4 × 104

(14)

Second, data from the gas−solid flow have been analyzed to evaluate the line slope m in eq 9. The solid-mass flow rate was determined by weighing cells. The gas-mass flow rate through the venturi was determined from eq 13 and further used to calculate the gas pressure drop by eq 2. m was then obtained by the substitution of experimental pressures, the calculated gas pressure drop and loading ratio, and the gas and particle densities into eq 9. The term of m is referred to as the pressure ratio parameter by Lee and Crowe,13 whose value varies from 1 to 1.6 in their report, and is referred to as the sensitivity by Shaffer and Bajura,23 whose value varies from 0 to 1 in their report. The different values of m in researchers’ reports can be mostly attributed to the different ranges of the mixture-to-gas pressure ratio and loading ratio. It can also be found in Figure 7 that the values of m in this work are much larger than those reported by Lee and Crowe,13 Shaffer and Bajura.23 Because of its high value, the term of m in this work is therefore named as the pressure ratio parameter, as suggested by Lee and Crowe.13

where ms,t and ms,t+Δt are the solid mass inside the receiving vessel at time t and t + Δt, where Δt is time interval. As can be seen clearly, the conveying system was conducted at the conditions of the pressure of the feeding vessel ranging from 300 to 1000 kPa and of the solid concentration ranging from 200 to 400 kg/m3, typical of high-pressure and dense-phase pneumatic conveying. Figure 8 shows a comparison of the solid-mass flow rate between model predictions and experimental values. It can be seen that the model has a prediction of generally less than 20% relative error. It can also be found that the model deviation is more at higher pressure. For example, the error is −22.78% with a feeding pressure of 981 kPa. It is therefore considered that the pressure probably has a more than linear effect. Consequently, the two-phase flow multiplier k should be further modified as 1 + Z(ρg/ρp)λ, where λ is the pressure factor. Analysis of the experimental data and model predictions indicates that in order to have the smallest deviation, piecewise function is the better choice, which separates out the pressure effects and thus improves the prediction accuracy. The modified model with a pressure factor λ = 1.2 for Pf < 500 kPa and λ = 0.93 for Pf > 500 kPa will make all of the errors less than ±20%, as shown in Figure 8. In addition, the definition in eq 15 indicates that the weighing method cannot provide an absolute online measurement of the solid-mass flow rate, while for many applications, real-time and continuous measurement is required for timely characterization of the particle flow

Figure 7. Pressure ratio parameter as a function of the modified Stokes number. F


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Industrial & Engineering Chemistry Research Table 3. Data of Experimental Pneumatic Conveying and for Model Prediction case

Qg1 (Nm3/h)

Qg2 (Nm3/h)

Qg3 (Nm3/h)

Pf (kPa)

P1 (kPa)

P3 (kPa)

Us (m/s)

Cs (kg/m3)

Ms, load (kg/h)

1 2 3 4 5 6 7 8 9 10 11 12 13 14

0.00 2.08 2.17 2.43 2.24 3.81 3.15 4.31 4.13 5.21 4.58 5.91 5.89 5.31

4.57 3.83 6.18 5.07 4.82 6.07 5.66 8.17 8.25 8.72 8.94 10.76 10.69 11.71

4.73 5.02 0.00 5.51 8.01 4.99 10.17 7.02 14.80 8.67 4.35 12.24 7.77 6.45

298.34 351.55 400.38 451.25 500.54 543.23 601.01 700.06 752.79 799.55 866.81 921.14 931.82 981.13

195.53 239.99 206.03 317.82 380.77 332.27 479.42 535.34 617.05 625.34 633.79 746.89 702.21 742.06

64.71 100.54 64.67 145.10 185.50 165.99 246.17 275.94 332.59 324.32 295.42 405.00 377.44 402.27

3.94 4.22 2.86 4.30 4.98 3.90 5.89 4.93 6.96 4.54 4.61 5.75 5.65 5.25

218.50 292.34 348.50 317.85 242.56 318.39 207.27 332.15 183.22 285.49 371.05 277.93 388.65 326.52

536.21 621.60 657.14 709.48 665.28 786.60 677.83 896.92 722.06 888.00 1008.01 964.80 1083.60 1076.40

Figure 9. Comparison of solid-mass flow rates with data obtained from the weighing cells (■), from the capacitance sensors, DK 13 and DC 13 (○), from the complete model coupled with the algorithm in Figure 3 (△), from the complete model with a pressure factor λ (▲), and from the simplified model with gas velocity represented by the solid velocity (▽).

Figure 8. Comparison between the model predictions and experimental solid-mass flow rates: (■) model without pressure factor λ; (Δ) model with pressure factor λ.

behavior. From this perspective, the research work in this paper shows big progress and is a useful supplement for the existing technology. Model predictions were also compared with the results obtained from the mass flow meter system where the solid-mass flow rate Ms,sensor was calculated by multiplying the solid velocity, solid concentration, and flow cross-sectional area: π Ms,sensor = D2UC s s (16) 4

system, it may be simplified by assuming that the gas velocity is equal to the solid velocity inside the conveying pipeline, Ug = Us. Although the solid velocity is always less than the gas velocity because of the drag forces between the gas and solid,26 for fine powders those conveyed under dense-phase and highpressure conditions, the solid velocity is, however, very close to the gas velocity.27 Geldart and Ling28 also considered that, in the case of dense-phase conveying using fine powders, the solid velocity is small and the voidage is high so that there is little error in assuming that Ug = Us. In this work, fine pulverized coal was used as the experimental material and conveyed under high-pressure and dense-phase conditions, so it is reasonable to make the assumption that “gas velocity is equal to solid velocity inside the conveying pipeline”. A comparison between these two velocities was reported in Figure 10 to assess this assumption. The gas velocity here is expressed in terms of the superficial gas velocity, which is the gas flow rate divided by the cross-sectional area of the pipeline. It can be calculated from eq 13 by substituting Ms,load, ⎛ ⎞ 1 2 M ⎟/ Ug = ⎜Q g1 + Q g1 + Q g3 − s,load πD . The solid velocity ρp ⎠ 4 ⎝ is measured by the capacitance sensor DK 13. Acceptable agreement was found in Figure 10, confirming the feasibility of the assumption proposed. We can also find that in some cases the solid velocity is higher than the gas velocity because solids

Ms,load is regarded as a reference and compared with other solid-mass flow rates. E1, E2, E2′, and E3 given in Table 5 are the percentage differences of each with Ms,load at the same conveying condition. The comparison in Table 5 and Figure 9 indicates that the model allows one to predict the solid-mass flow rate, giving errors comparable with those produced by capacitance sensors. The point is that the model prediction is based on a venturi device associated with the values of the gas flow rate and pressure, which can be measured by the gas flow meter and pressure sensor. These devices are common and much cheaper than the capacitive solid-mass flow meter. From this perspective, the research work in this paper is very meaningful and may provide an alternative for solid-mass flowrate measurement. Further consideration is that, because the model is applied in the high-pressure and dense-phase pneumatic conveying

(

G

)


Article


by the basic equation Mg = ρgUgπD2/4. With the assumption Ug = Us, model parameters Cd and m calculated from eqs 3 and 10, and the gas pressure drop ΔPg calculated from eq 2, the solidmass flow rate Ms can be calculated from eq 11 directly instead of the trial-and-error method. As shown in Table 5 and Figure 9, we can find that the simplified model is a bit less precise than the complete model. In fact, the solid-mass flow rates predicted by the simplified model are relatively smaller at high pressures. This is probably attributed to gas-phase compressibility and expansion accounting for volumetric changes due to pipeline pressure changes. In this work, the gas velocity used for mass flow-rate calculation is 600 mm upstream measured by the solid velocity sensor. This approximation will result in a smaller gas velocity considering gas-phase expansion along the pipeline, so as the gas- and solid-mass flow rates. However, despite its simplicity, the simplified model provides us with satisfactory results for all of the experiments verified, producing errors that are generally lower than ±20%. This simplification is especially important for the model application in some industries where the gas flow rate through the pipeline may not be as readily available. In that case, the pressure of the feeding vessel is usually maintained by balancing the entered and escaped gases. The entered gas is controlled by the gas flow meter, while the escaped gas is controlled by a pressure relief valve and does not have an observed value. Without the value of the escaped gas, it is not easy to get the gas mass flow rate in the pipeline as well as that through the venturi. For such conditions, the simplified model may be a good choice for solid-mass flow-rate measurement. In spite of the solid velocity requirement, the simplified model can save the investment and operation charge of a solid concentration sensor compared to the traditional solid-mass flow meter and thus reduce the cost of measurement. Nevertheless, in the detail of the model application, it appears that the model is dependent on some semiempirical coefficients, including the discharge coefficient Cd and the pressure ratio parameter m. These coefficients can be obtained by adopting a regression procedure using the single-gas flow or the gas−solid flow data at all conditions during the calibration tests. When applied to the entrained-flow pulverized coal gasification process, the model parameter should be corrected at the specific condition to minimize the measuring error. This procedure is the most appropriate if power function relationships between the discharge coefficient and Reynolds number and between the pressure ratio parameter and modified Stokes number are assumed. Two methods coupled with the model were proposed to get the solid-mass flow rate in this work. One is by an iterative solution procedure (complete model), and the other is on the basis of the equivalence between the gas velocity and solid velocity in the venturi upstream (simplified model). Both models are available to obtain the solid-mass flow rate, producing errors that are generally lower than ±20%. It appears that the model is sufficient to obtain a proper solid-mass flow rate evaluation, which will provide a suitable alternative method for the solid-mass flow-rate measurement in the high-pressure and dense-phase gas−solid flow.

Figure 10. Comparison between the gas and solid velocities.

occupying the area should be deduced. Some of these specific conditions were provided in Table 4, where the solid velocity Table 4. Velocity Comparisons at Some Specific Conditions case

solid velocity (m/s)

superficial gas velocity (m/s)

voidage

real gas velocity (m/s)

5 8 10 12

4.98 4.93 4.54 5.75

4.40 4.19 4.23 4.78

0.86 0.81 0.80 0.83

5.10 5.16 5.30 5.77

falls between the superficial gas velocity and real gas velocity. In any case, the acceptable errors suggest that it is reasonable to assume solid and gas velocities close to each other for fine pulverized coal conveyed under high-pressure and dense-phase conditions. On the basis of this assumption, a simplified model can be established where the gas velocity at P1 can be considered to be equal to the solid velocity measured by the capacitance sensor DK 13, which is 600 mm far from the upstream pressure tap of the venturi. The algorithm for the simplified model is shown in Figure 11. The difference from that reported in Figure 3 is that the gas-mass flow rate through the venturi Mg can be obtained

5. CONCLUSION The one-dimensional model derived in this study may be used to predict the solid-mass flow rate of pulverized coal in the dense-phase pneumatic conveying system. The model was developed by modifying the traditional single-phase or diluteflow model with insertion of a two-phase flow multiplier. The

Figure 11. Algorithm for the simplified model. H


Article

Industrial & Engineering Chemistry Research Table 5. Comparison between the Experimental and Predicted Solid-Mass Flow Rates case

Ms,load (kg/h)

Ms,sensor (kg/h)

E1 (%)

Ms, venturi (g) (kg/h)

E2 (%)

Ms,venturi′(g) (kg/h)

E2′ (%)

Ms,venturi(s) (kg/h)

E3 (%)

1 2 3 4 5 6 7 8 9 10 11 12 13 14

536.21 621.60 657.14 709.48 665.28 786.60 677.83 896.92 722.06 888.00 1,008.01 964.80 1083.60 1076.40

547.92 784.31 632.83 869.31 767.74 790.47 776.58 1040.56 811.20 823.56 1086.75 1015.74 1396.74 1089.56

2.18 26.18 −3.70 22.53 15.40 0.49 14.57 16.02 12.35 −7.26 7.81 5.28 28.90 1.22

676.58 640.60 741.02 686.67 697.25 629.98 720.67 761.14 730.68 803.96 1008.54 795.60 807.29 831.21

26.18 3.06 12.76 −3.21 4.81 −19.91 6.32 −15.14 1.19 −9.46 −0.05 −17.54 −25.50 −22.78

608.12 581.61 639.26 617.27 627.57 666.85 778.78 831.84 778.78 880.02 1207.41 859.07 885.62 924.38

13.41 −6.43 −2.72 −13.00 −5.67 −15.22 14.90 −7.26 7.86 −0.90 19.78 −10.96 −18.27 −14.12

683.50 645.07 760.71 687.05 687.75 643.58 698.70 744.84 699.75 795.72 905.07 771.06 765.93 781.94

27.47 3.78 15.76 −3.16 3.38 −18.18 3.08 −16.96 3.09 −10.39 −10.21 −20.08 −29.32 −27.36

k = two-phase flow multiplier L = characteristic length of the venturi, mm Lc = length of the venturi contraction section, mm Lt = length of the throat contraction section, mm m = pressure ratio parameter defined in eq 8 ms,t = solid mass inside the receiving vessel at time t, kg ms,t+Δt = solid mass inside the receiving vessel at time t + Δt, kg Mg = gas-mass flow rate through the venturi, kg/h Ms = solid-mass flow rate, kg/h Ms,load = solid-mass flow rate obtained from the data of weighing cells, kg/h Ms,sensor = solid-mass flow rate obtained from capacitance sensors, kg/h Ms,venturi(g) = solid-mass flow rate obtained from the complete model, kg/h Ms,venturi′(g) = solid-mass flow rate obtained from the complete model with pressure factor, kg/h Ms,venturi(s) = solid-mass flow rate obtained from the simplified model, kg/h P1 = pressure at the venturi inlet, kPa P2 = pressure at the venturi throat, kPa P3 = pressure at the venturi throat, kPa P4 = pressure at the venturi throat, kPa P5 = pressure at the venturi outlet, kPa Pf = pressure of the feeding vessel, kPa Pg,i = gas pressure at the inlet of the venturi, kPa Pg,t = gas pressure at the throat of the venturi, kPa Qg1 = gas-volume flow rate introduced into the feeding vessel from the top, Nm3/h Qg2 = gas-volume flow rate introduced into the feeding vessel from the cone, Nm3/h Qg3 = gas-volume flow rate introduced at the front end of the conveying pipe, Nm3/h Re = Reynolds number St = Stokes number Ug = superficial gas velocity through the venturi, m/s Ug,i = superficial gas velocity at the inlet of the venturi, m/s Ug,t = superficial gas velocity at the throat of the venturi, m/s Ug,t = solid velocity measured by the capacitance sensor DK 13, m/s Z = loading ratio (mass flow rate of a solid/mass flow rate of a gas)

modified model, taking into account the effect of the gas pressure and loading ratio, can be applied in the gas−solid flow at high-pressure and dense-phase conditions. Calibration tests were carried out in order to determine the discharge coefficient and pressure ratio parameter in the model. These two parameters could be obtained by using the single-gas and gas−solid flow data, respectively. Power function relationships between the discharge coefficient and Reynolds number and between the pressure ratio parameter and modified Stokes number were defined, and the fitting coefficients were obtained based on the regression procedure. Two methods attached to the model were further prospered to calculate the solid-mass flow rate. One is the complete model coupled with an iteration algorithm, and the other is the simplified model with the assumption of the equivalence between the gas and solid velocities in the venturi upstream. Both models were examined by the verified tests and proven to be available to predict the solid-mass flow rate with acceptable errors. The complete model gives advantages of lower investment, while the simplified model may be favored in industrial applications in which it is difficult to obtain the gas flow rate in the pipeline.

■

AUTHOR INFORMATION

Corresponding Author

*Tel: +86 21 6425 2521. Fax: +86 21 6425 1312. E-mail: [email protected] (X.G.). Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This work was supported by the National Natural Science Foundation of China (21206041) and the Fundamental Research Funds for the Central Universities.

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NOMENCLATURE a = fitting coefficient defined in eq 3 At = throat area of the venturi, m2 b = fitting coefficient defined in eq 3 c = fitting coefficient defined in eq 10 Cd = discharge coefficient defined in eq 2 d = fitting coefficient defined in eq 10 D = diameter of the conveying pipe, mm dp = mean diameter of pulverized coal, μm dt = throat diameter of venturi, mm

Greek Symbols

ΔPg = gas pressure drop through the venturi, kPa I


Article

Industrial & Engineering Chemistry Research ΔPmix = pressure drop of a gas−solid mixture through the venturi, kPa Δt = time interval, h β = throat diameter ratio (throat diameter dt/inlet diameter D) θ1 = inlet angle of the venturi, deg θ2 = diffuser angle of the venturi, deg λ = pressure factor μg = gas viscosity, Pa s ρg = gas density, kg/m3 ρp = particle density, kg/m3

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(17) Liu, K.; Lu, H.; Guo, X.; Sun, X.; Tao, S.; Gong, X. Experimental study on flow characteristics and pressure drop of gas-coal mixture through venturi. Powder Technol. 2014, 268, 401−411. (18) GB/T 211-1996, Determination of total moisture in coal, Beijing, 1996. (19) GB/T 217-1996, Determination of true relative density of coal, Beijing, 1996. (20) Carr, R. L. Evaluating flow properties of solids. Chem. Eng. 1965, 72, 163−167. (21) Viswanathan, S. Development of a pressure drop model for a variable throat venturi scrubber. Chem. Eng. J. 1998, 71, 153−160. (22) Monni, G.; De Salve, M.; Panella, B. Two-phase flow measurements at high void fraction by a Venturi meter. Prog. Nucl. Energy 2014, 77, 167−175. (23) Shaffer, F. D.; Bajura, R. A. Analysis of venturi performance for gas-particle flows. J. Fluids Eng. 1990, 112, 121−127. (24) Farbar, L. The venturi as a meter for gas−solids mixtures. Trans. ASME 1953, 75, 943−951. (25) Hollingshead, C. L.; Johnson, M. C.; Barfuss, S. L.; Spall, R. E. Discharge coefficient performance of Venturi, standard concentric orifice plate, V-cone and wedge flow meters at low Reynolds numbers. J. Pet. Sci. Eng. 2011, 78, 559−566. (26) Agarwal, A.T. Theory and Design of Dilute Phase Pneumatic Conveying Systems. Powder handing. process. 2005, 17, 18−23. (27) Kus, F. T.; Duchesne, M. A.; Champagne, S.; Hughes, R. W.; Lu, D. Y.; Macchi, A.; Mehrani, P. Pressurized pneumatic conveying of pulverized fuels for entrained flow gasification. Powder Technol. 2016, 287, 403−411. (28) Geldart, D.; Ling, S. J. Dense phase conveying of fine coal at high total pressures. Powder Technol. 1990, 62, 243−252.

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Solid-Mass Flow-Rate Prediction in Dense-Phase Pneumatic

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