37
Ind. Eng. Chem. Fundam. 1082, 21, 37-43
Hydrodynamic Analysis of Horizontal Solids Transport Yang 1. Shlh, Hamld Arastoopour; Depertment of
and Sanford A. Well
Gss Enginwlng, Illinois InstirUte of
Technology, Institute of
Gas Technobgy, Chicago, Illinois 60616
Pressure drop and saltation velocity were calculated for horizontal gas-solid flow using two-dimensional steady state, continuity, and momentum equations. The onedimenslonal hydrodynamic models proposed in the literature were formulated and analyzed for two-dimensional cases. The characterlstics of the equations as an initial value problem are Imaginary. To obtain a well-posed system of equations, alternative modifications were introduced. Only two of the modified models, the pressure drop in gas phase and the relative velocity model, were well-posed. The calculated values for pressure drop before saltation compare well with Konchesky’s experimental data (1975). Uncertainties in drag correlations for dense flow regimes after saltation at present prevent an equally good mathematical description of the solids flow behavior.
Introduction The horizontal pneumatic transport of solids has attracted attention because of ita importance for coal conversion processes, fluidized catalytic cracking units, combustion of solid fuels, etc. Successful operation of the horizontal solids flow depends upon accurate estimation of the design parameters, including saltation velocity and pressure drop. A number of empirical correlations for predicting the saltation velocity have been reported, including those of Culgan (1952), Thomas (19621, Doig and Roper (1963), Zenz (1964), Rose and Duckworth (1969), Mataumoto et al. (1974), Duckworth and Kakka (1971), and Wen et al. (1979). Also, previous investigators have reported several correlations for pressure drop based on their experiments, including Vogt and White (1948), Korn (1950), Lapple (1951), Culgan (1952), Clark et al. (1952), Hinkle (1953), Mehta et al. (1957), Richardson and McLeman (1960), Curran and Gorin (1968), Rose and Duckworth (1969), Duckworth and Kakka (1971), Konchesky et al. (1975), and Yang and Keairns (1976). Recently, Leung and Jones (1978) compared eight correlations for predicting the saltation velocity based on experimental data. Arastoopour et al. (1979) also compared several correlations for predicting the saltation velocity and the pressure drop for coal and related materials. They also showed a large deviation between all the available data and the parameters predicted by empirical correlations. These empirical correlations cannot be generalized a t different operating conditions for different solid particles. To develop a more general and fundamental formula to predict the characteristics of the horizontal solid-gas flow, hydrodynamic equations based on thermodynamic and fluid mechanical principles should be used. One-dimensional hydrodynamic equations have been used successfully by Jackson (1971), Shook and Masliyah (1974), and Arastoopour and Gidaspow (1979). For horizontal gas-olids flow, gravity (one of the most important forces) is not in the direction of flow. One-dimensional models, therefore, are inadequate to describe the solids flow behavior. Thus two-dimensional mass and momentum balances should be employed. Arastoopour and Gidaspow’s (1979) one-dimensional, steady-state equations are well-posed, and there are no mathematical difficulties. For non-steady state cases, however some models are illposed as an initial value problem because of the complex valued characteristics of the equations (Lyczkowski et al., 1978). For two- and three-dimensional cases similar to the unsteady case, mathematical difficulties may exist and must be analyzed.
Two-Dimensional Solids-Gas Unequal Velocity Models The horizontal solids-gas mixture system can be described by means of two-dimensional, isothermal, steadystate mass and momentum balances. Two-dimensional analysis is necessary because of segregation and settling of the particles. The effect of vertical forces on the solids cannot be neglected; thus one-dimensional hydrodynamic equations are inadequate to describe the solids flow behavior. The isothermal assumption is a good approximation for dispersed flow resulting from good heat transfer to the wall and low rates of heat generation by friction. The steady-state assumption discards any fluctuations. The following assumptions were also made in formulating the two-dimensional hydrodynamic equations: (1) There is no interphase mass or heat transfer. (2) Collisions among the particles are negligible and do not affect the fluid-particle drag forces. (3) The same kind of drag force expressions apply to both horizontal and vertical directions. (4) Both the gas and solid phases are incompressible. The continuity equations for gas and solid phases and the mixture momentum equations in the horizontal ( x ) and vertical (y) directions are as follows. Continuity Equations. gas phase
solid phase
The x direction indicates the direction of the flow and the y direction indicates flow from the top toward the bottom of the pipe. Mixture Momentum Equations. x direction dug ax: + (1-
epgug
au, a% - + ep u - + ,ax g g ay
e)p u
y direction
0198-43l3/82/1021-0037$Ol.25/00 1982 American Chemical Society
38
Ind. Eng. Chem. Fundam., Vol. 21, No. 1, 1982
and fswy are the friction forces between the gas phase and the pipe wall in the x and y directions. The solid-wall friction forces have been neglected. To determine the horizontal gas and solid velocities, u,, and us, the vertical gas and the solid velocities, u, and us, the system pressure, P, and the volume fraction, t, two more equations are needed. There are two possibilities for derivation of the equations. One approach is to calculate the momentum balance of the solid phase by using the continuum concept, which assumes the dispersed solid phase also forms continua. The second approach is to use the constitutive equations of relative motion of the two phases using the principles of nonequilibrium thermodynamics, Gidaspow (1978). Solid-Phase Momentum Balances. x direction
fgwx
PSU,
au,
aus
ax +
Paus
= -77 aY
ap
P
at
&- + Y - + fsx (1 - ax
au, au, ax + ay = PSUS
-a
ap
P at (1 - e ) ay
- + P -- + Peg + f a y aY
a
a
fax
(u, - us) - (u, - us) + (u, - us) - (u, - us) = -ax aY P S y direction
a
a
(7)
f SY
(ug- us)ax (up - us) + (u, - us) a Y (up - us) = g - PS
(5)
y direction PSU,
Fick’s law of diffusion for particles which includes an inertial correction usually deleted in writing Fick’s law. In the unsteady operation this correction allows the signals to propagate at finite rather than at infinite speeds. The use of Fick’s law to describe transport of particles and its essential equivalence to relative velocity equations is taken as a proof of the validity of the model. The two-dimensional relative velocity equations used to describe the horizontal solids transport are x direction
(6) The particle-particle interaction forces have been neglected in eq 5 and 6 because of the low solid volume fraction, and present uncertainties in the proper expression of the interaction force term. The values of the four parameters, 7, y, a,p, in eq 5 and 6 are based on the following different assumptions for the pressure drop expressions: case A: pressure drop in both solid and gas phases 77 = (y = 1; y = p = 0 case B: pressure drop in gas phase only tl=y=(y=p=o
case C: partial pressure drop in solid and gas phases l=y=(y=p=l Parameters y and j3 are similar to the displacement factor, B, presented in Soo’s (1980) transient one-dimensional momentum equation for multiphase flow. The displacement factor is dependent upon the flow configuration, which affects the diffusion of a phase. Relative Velocity Model. The relative velocity equations which are the momentum balances between the phases were first developed by Gidaspow (1978) for the transient one-dimensional two-phase flow using the methods of nonequilibrium thermodynamics. Briefly, the procedure consists of assuming the existence of an entropy production function which in addition to being a function of internal energy, volume etc. is also a function of relative velocity rather than of individual phase velocities, to satisfy Galileo’s relativity principle. Some of the two-phase flow equations do not satisfy this principle, although frequently numerical errors encountered are small. After assuming the existence of said entropy function, an energy balance was constructed; then an expression for an entropy production was developed using the usual mathematical manipulation in transport phenomena. Following Onsager’s derivation of multicomponent diffusion equations, this entropy production was minimized. The results for the two-dimensional case are eq 7 and 8. The one-dimensional derivation of the equations (Gidaspow, 1978) and ita twoand three-dimensional extension (Arastoopour, 1978) was restricted to the case of isothermal operation with no phase changes. It can be easily extended. Gidaspow (1978) has shown that relative velocity equations give the general
The relative velocity eq 7 and 8, or the solid-phase momentum eq 5 and 6, with mixture momentum and continuity equations are sets of six simultaneousfirsborder differential equations that are used to solve for u,, us, u,, us, P, and 6. The friction force between the gas phase and the wall can be expressed by means of the usual Fanning’s equation
where f,,the friction factor, is a function of the Reynolds number and the relative roughness of the pipe. Because the pipe is assumed to be smooth in these calculations, f , is a function of the gas Reynolds number only. For the lower Reynolds number, the friction factor can be obtained from the empirical Blasius formula fg
=Oa316 for Re,
Re,’J4
< 100000
For higher gas Reynolds numbers, the friction factor can be obtained from the following expression 1 -= 2 log (Re, (fJ1J2) - 0.8
(fp2
The drag force expressions used in this study are the same as those used by Arastoopour and Gidaspow (1979). Thus
where 24 Reax 24 C D , = -(1 Rea,
+ 0.15Re,x0.68’);
Resx
+ 0.15Re,0.687);
Re,
< 1000 (16)
CDsx= 0.44; Re, > 1000
(17)
C D =~0.44; ~ Resy > 1000
(18)
CD, = -(1
and
1000 (15)
Ind. Eng. Chem. Fundam., Vol. 21, No. 1, 1982
Analysis of the The above seta of six nonlinear differential equations which consist of momentum and mass balances could be written in the matrix form and be shown in general as
au
au
A-+B-=C ax ay where A and B are (6 X 6) the coefficients matrix, C is the nonhomogeneous column matrix, and U is the column vector of the six dependent variables. The characteristics of eq 21 could be derived from IAA + BI = 0 (22) where A's are the characteristics. Two common imaginary characteristics, ki,are derived from the analysis of all of the above different seta of equations. If the characteristics of the equations are complex, it is impossible to find a numerically stable finite difference scheme to solve the equations as an initial value problem. Real characteristics are a necessary condition for such a well-posed set of equations. Therefore the sets of governing equations must be modified if it is to be solved as an initial value problem. Modification of the Equations Because vertical pressure drop before saltation of the solids is not significant, the dP/ay terms are dropped out from both the mixture and solid-phase momentum equations in the y direction. The relative velocity equations remain unchanged. Only the pressure drop in the gasphase model and the relative velocity model have real characteristics under this modification. Other cases have imaginary characteristics, and therefore are ill-posed sets of equations. The real characteristic curves for the system of equations including the relative velocity equations are A1 = A2 = 0
(23)
For the case of the equations with solid phase momentum balance and pressure drop in the gas phase only, the characteristics are A1 = A2 = 0 UL7
A2
= --
and (24) Other modifications could also be made to obtain well-posed seta of equations, such as neglecting the rotational effect of the flow of gas and solid. Shih (1980)
3Q
completed a detailed analysis of such modifications. Numerical Scheme The method of linea is used in solving the well-posed sets of equations. This involves reducing the partial differential equations to ordinary differential equations by replacing the derivatives in the y direction with finite differences. If we divide the pipe diameter into n equal intervals or (n+ 1)grid points, we have 6(n + 1)simultaneous ordinary differential equations with respect to the x coordinate for 6(n + 1)dependent variables [(n + 1)points for each of the ug,ug,us,us, e, and P variables]. With the zero pressure gradient in the y-direction modification, the ( n 1) pressure grid points are reduced to 1. The 6(n + 1) equations for 6(n + 1)variables are reduced to 5(n + 1) + 1. The six boundary conditions required in this study are assigned as at y = 0: ug = 0 or very low slip velocity
+
ug = 0 us =
0 y = Dpipe: ug = 0 or very low slip velocity
(25)
ug = 0
0 (26) Along with the above boundary condition, an appropriate set of initial conditions that depend on the experimental inlet conditions was considered. The 5(n + 1)+ 11 simultaneous f i b o r d e r ordinary differential equations can be expressed in the form of an initial value problem &/dx = f(?) us =
P(0) = P (27) The Runge-Kutta method was then used to obtain the numerical results for the above set of equations, stepping in the x direction. Detailed numerical analysis of the equations has been described by Shih (1980). Parametric Study Using the Relative Velocity Model In order to use actual numerical values and to compare the calculated results from the proposed models with the experimental data, the horizontal pneumatic transport studied here is the same as that used by Konchesky et al. (1975) in their study on pneumatic transport of crushed coal in horizontal pipelines. The test section was a 1.22-m straight horizontal pipeline of 0.2 m diameter. The experiments were made with air as the gas phase and crushed coal as the solid phase. The solid-phase density was 2240 kg/m3, and the average particle diameter was 2840 pm. The inlet pressure in the experiments was 1.089 X lo6 N/m2 and the temperature was 30 OC. In the cited experimental studies, the gas volumetric and solids mass flow rates were measured in addition to the pressure drop. To make the numerical calculations using the relative velocity model, a set of initial values for the dependent variables, ug,us,ug, us, E , and P, at each inlet grid point is needed. Because Konchesky only reported the inlet pressure and superficial gas- and solid-phase velocities, reasonable numbers for the dependent variables at each inlet grid point were assigned. Initially, zero vertical phase velocities were assigned at each inlet grid point, and a uniform profile for the gas volume fraction was chosen. In the horizontal direction parabolic velocity profiles were assigned. The gas volume fraction, the solid and gas phase velocities, and pressure drop were calculated using the relative velocity model.
40
Ind. Eng. Chem. Fundam., Vol. 21, No. 1, 1982
:I I ,
,
u..x:O61
u., x: 0 0 0 m
m
286
I
‘
4
r
I
1 I43
I
W, = 2 7 4 3 k g / m z - a
up = 5 0 29 m / s 20 9985 P o = 1089 X IO5 N/mZ p , = 224 x 10) kg/m3 d , ~ 2 8 4X to-) m 6 0
2 39
-
Lz122m
D,,pe = 0 203 m
n
V
‘0
n
.
192
0.96
X
2z
E
z
n.
n.
Bn
143 0 W
W a 3 m Lo
a 0
IO
I
1
0
3.0
50
30
70 0
I
I lC.2
9) d$, X’I
1
2iJO
30
10
I
1 3.0
X’I
70
fl/1
1
1 15.2
0.1
us,
22 m
50
2.3 m / ,
1.0000
0.9995
-
-
3 v) u)
096
0.48
83 m
Figure 1. Horizontal solid-phase velocity profiles for solid-gas flow along a horizontal pipe using the relative velocity model. W, = 2 7 4 2 kg/m2-a U, ~ 5 209 m/s € 0 = O 9985 P o = I 0 8 9 X IO’ N/m2 p r 5 2 2 4 X IO3 kg/m3 d, =284 X m
h X
N
E n.
0.24
048
I
I
I
40
60
80
I
100
I
120
--f
0.12
140 f l / s m /I
SUPERFICIAL GAS VELOCITY
w
Figure 3. The effect of initial pressure on the pressure drop for the flow of a solid-gas mixture through a horizontal pipe calculated using the relative velocity model.
0.9990
i
0
5
0.9985
5 W
I
2
\,
0.9980
0
>
\
fn a W
\
\
).
1.00
095 0.90
1
0
2
I
3
4
1.22
0 61
5
6 11
1.83 m
PIPE LENGTH
Figure 2. Gas volume fractions for solids-gas flow through a horizontal pipe using the relative velocity model.
Figure 1shows the horizontal solid-phase velocity profiles at different distances along the pipe. The second-zone horizontal solid velocity, u%,was decelerated because the solid particles that fall into this zone have less momentum than the solids that fall out of this zone. The central-zone solid-phase horizontal velocity, uB8, does not change significantly. The fourth-zone solid-phase horizontal velocity, uB1, was accelerated because of lower particle concentration in this zone. At the top and bottom boundaries the solids are flowing at a low horizontal velocity. The solid velocity at the bottom layer should decrease along the pipe, but because the solid wall friction is neglected in the model, it remains constant. This unrealistic constant velocity prediction does not change the solid flow patterns in the system. The calculations showed similar gas-phase velocity behavior in the horizontal line. Figure 2 shows the variations of the gas volume fraction (4,along a horizontal pipe at a solid mass flow rate of 27.42 kg/mz-s, a superficial gas velocity of 50.29 m/s, and the initial gas-phase volume fraction of 0.9985. The subscripb
of t indicate the position along the vertical direction of the pipe: 1is the top and 5 is the bottom of the pipe. At the top boundary, t is gradually increased by the gradual settling of the solid particles. For the same reasons, the gas volume fraction at the second zone, t2,also increases along the pipe; however, the increasing rate is lower than that at the top boundary. The central-zone gas volume fraction, t3, does not change significantly. This could be because of the mimimum rate of solid accumulation along the pipe in this zone (annular flow of solids at the center). t4 increases rapidly; this phenomena could result either from particles being rapidly carried away by the highly accelerated horizontal gas velocity or from the fast settling of the particles from this zone ratio into the bottom boundary zone. Because of saltation of the particles, t5 at the bottom boundary decreases rapidly along the pipe. Figure 3 shows the effect of initial pressure on the pressure drop and saltation velocity in the 1.22-m test section of the pipe using the relative velocity model. The gas-phase densities at different initial system pressures were calculated using the ideal gas equation of state. The system temperature was 30 OC. Higher initial pressure, Po, implies a higher pressure drop and lower saltation velocity. A t higher initial pressure, the solid particles are accelerated more because of a higher gas density, which results in a higher drag force exerted by the gas on the solids. The higher solid particle velocity causes less settling of the solid particles; therefore, lower saltation velocity is predicted. A higher drag force exerted by the gas on the solids at high pressure generates higher values for pressure drop. Zenz’s (1960) interpretation of the pressure drop behavior in the saltation region was used to construct calculated pressure drop vs. superficial gas velocity curves. The effect of solid particle density on the pressure drop and saltation velocity is shown in Figure 4. As the physics of the problem would suggest, higher particle density results in more settling of the particles in the horizontal pipe, which in turn requires more gas to sustain the particles in the pipe. Also, higher pressure drops are predicted at higher solid particle densities.
Ind. Eng. Chem. Fundam., Vol. 21, No. 1, 1982
41
N I
. .0
K
X
N
E
3.83
e - eo
3.35
- 70
2.87
-60
z
287
239
-
-
- 239
60
W,:2742 kg/mZ-$ = O 99862 P o ~ 1 0 8 8X 10’N/m7 d , : 2 8 4 x 10‘” ( 0
.
50
-
40
-
I91
L = l2 2 m DPIP. = 0.203 m
0
B,,,
P
0 O W
a 3 Iv)n
2
2.39
- 50
1.91
- 40
1.43
- 3.0
0.96
- PO -
-
K W
191
-
- 143
40
a
I I
co r0.9990 143
- 30
-
096
~
70
a0
2133 24%
90
loa
120
140
3048 3292
3658
4267
io0
165
ff/r
80
5029 m/r
100
24.38
30.48
SUPERFICIAL GAS VELOCITY
Figure 4. The effect of solid particle density on the pressure drop for a flow of a gasaolida mixture through a 1.22-m section of horizontal pipe calculated using the relative velocity model.
The effect of the inlet gas volume fraction on the design parameters was computed. Figure 5 shows a significant effect on the initial gas volume fraction, to,on the pressure drop along the 1.22-m test section of the pipe at various superficial gas velocities, calculated using the relative velocity model. The initial system pressure was 1.089 X lo5 N/m2. A decrease in void fraction results in a decrease in the solids velocity, which, in turn, causes higher pressure drop and more settling of the particles. Thus, more gas at the higher velocities is needed to exert a sufficient drag force to sustain a dispersed suspended flow in the line. A complete parametric study using the relative velocity model has been presented by Shih (1980). Numerical Comparison of Unequal Velocity Models The calculated phase velocities, gas volume fraction, and pressure variation along the pipe were compared using the well-posed two-dimensional relative velocity and pressure drop in gas-phase-only models for the flow of air and crushed coal particles through a horizontal pipe. The horizontal pneumatic transport studied here is the same as those used by Konchesky et al. (1975). The initial pressure is 1.089 X lo5 N/m2. The initial gas and solid velocity profiles are the same as shown in the parametric studies. Figure 6 shows the calculated gas volume fraction at the bottom boundary of the pipe, t6, using the two cited models. The pressure drop in the gas-phase-only model predicts that solid particles rapidly settle down at the bottom of the pipe in a region about 0.6 m away from the entrances. The relative velocity model predicts different phenomena: the gas volume fraction decreases gradually and the packed layer of solids forms after 2.3 m of the pipe. The gradual change of the gas volume fraction from the top layer to the bottom layer predicted by the relative
120
140
36.58
42.67
160
tl/S
48.76
m h
SUPERFICIAL GAS VELOCITY
Figure 6. The effect of initial gas volume fraction on the pressure drop for the flow of a solid-gas mixture through a horizontal pipe calculated using the relative velocity model. I.oo
P j 0.95
P-
a U
0
ock
0.90
$ z
0.85
I
m
E
3
I I
0.80
U K
W. ~ 2 7 4 kg/m‘-s 2
W
Up = 5 0 29 m / s
3 9 9
0.75
-
0.70
-
(0
€ 0
!
= o 9985
P o = IO89 X I 0 5 N / m 2 p s ~ 2 2 4x 103 k9/m3 d , ~ 2 . 8 4x IO-^ m B iPRESSURE DROP IN GAS-PHASE-ONLY
MODEL
D; RELATIVE-VELOCITY MODEL
0
2
4
6
ft
0
0.6 I
1.22
1.83
m
PIPE LENGTH
Figure 6. Gas volume fractions at the bottom boundary of the pipe for solid-gas flow through a horizontal pipe calculated using two unequal velocity models.
velocity model shows a macroscopic diffusion-type vertical flow of solid particles. This diffusion mechanism of the solid particles could be mathematically obtained from the relative velocity equations. Gidaspow (1977) and Arastoopour (1980) obtained a solids diffusivity term from the continuity and relative velocity equations. Further, Arastoopour and Gidaspow (1979) pointed out that the
42
Ind. Eng. Chem. Fundam., Vol. 21, No. 1, 1982 N
'E!
,*
x
=,
"E
z
n
0
e 87
-
60
i2 7 42 kg/m2-s r o 20 99862 P O I 088 x 1 0 ~ ~ 1 ~ ' p 1 1 2 2 4 x 103 kg/m3 d , :284 x IO-^ m L:I 2 2 m
W,
2 39
-'50
1.9
- 4.0
n 0 K
n w K 3 v) v)
w
n a
00
- VALUES CALCULATED USING RELATIVE-VELOCITY MOOEL
1.4?
0 EXPERIMENTAL DATA OF KONCHESKY E T AL
- 3.0 80
24.38
I00
120
140
30.48
36.58
42.67
160
it/,
48.76 m/s
SUPERFICIAL GAS VELOCITY
Figure 7. A comparison of experimental data of Konchesky et al. (1975) with a numerical solution for a solids-gas flow through a horizontal pipe using the relative velocity model.
gradual change in the properties of the system using the relative velocity model indicates a distributed interaction between the phases throughout the conveying line of pneumatic transport. The relative velocity model provides a more realistic prediction for the horizontal solid-gas flow patterns than the pressure drop in gas-phase-only model because it incorporates more interaction forces and diffusion-type flow. The relative velocity model predicts a higher pressure drop than the pressure drop in the gas-phase-only model. This phenomenon also indicates that the relative velocity model took into account more interaction forces between the phases. The solid-phase velocity predicted by the pressure drop in the gas-phase-only model is lower than the relative velocity model prediction because more particles settle in a short zone. Comparison with Experiment In order to compare the calculated results with the cited Konchesky experiment, the same initial conditions and the same type of solid particles, gas, and horizontal pipe were used. The solid mass flow rate, W,, the superficial gas velocity, U,,the initial system pressure, and the pressure drop along the 1.22-m test section were taken from the experiment. In the numerical calculations, the phasevelocity distributions and the value and distribution of the gas volume fraction of the inlet points are needed. Because this information is not measured and reported in the experiments, reasonable valuee and distributions for void fraction and their velocities were chosen and have been presented in the parametric studies. In Konchesky's experiment, the saltation velocity was estimated as U, = 32.92 m/s for the flow with a mass flow rate of W, = 27.42 kg/m2-s and an inlet pressure of 1.089
X lo5 N/m2. The appropriate value of the initial gas volume fraction, eo = 0.99862, was chosen to agree with this saltation velocity. Figure 7 shows numerical calculations of pressure drop vs. superficial gas velocity using the relative velocity model and the comparison with the experimental data. The calculated values agree well with the experimental data before saltation. Nevertheless, the model is unable to generate the sharp increase in pressure drop found experimentally. This behavior could be caused by the improper drag force expression for dense solid transport that occurs after the saltation phenomena. Neglect of the friction forces between the solid particles and the pipe wall is another possible cause for the low pressure drop values calculated after the saltation in the pipe. In the dilute-phase region, the pressure drop in the gas-phase-only model predicts lower pressure drops than the Konchesky's experimental data because it accounts for less interphase interaction. This also could be caused by solids-wall friction. The model is unable to predict the increase of the pressure drop after saltation of the particles. The inconsistency of predicted saltation velocity and pressure drop with the experimental results implies that the pressure drop in the gas-phase-only model is unable to predict the physical behavior of the horizontal solid transports. This inconsistency might result from the fact that the pressure drop calculated in this model is from the gas phase only, and the dominant solids-gas interaction forces after saltation are not properly considered.
Conclusion The modified two-dimensional steady-state relative velocity model and pressure drop in gas phase-only model provide well-posed initial value problems. A numerical scheme for solving these sets of six simultaneous differential equations was developed using the method of lines. The calculated values are able to explain the physical behavior of the steady-state two-phase transport systems. For the horizontal dilute gas, coal flow, the relative velocity model predicts the design parameters and compares reasonably well with Konchesky's data before saltation. The deviation from the Konchesky's values for pressure drop is about 5% which is much better than the best available correlation prediction in the literature which is about 20% (see Arastoopour et al., 1979). In general for a dilute gas-solids flow systems, the modified models have successfully predicted the pressure drop and saltation velocity in the pressure range of 1 X lo5 to 20 X lo5 N/m2, the gas velocity of 30-50 m/s, and solids density of 1 x 103kg/m2 to 4 X lo3 kg/m2. Outside the above range of variables, the models have not yet been tested. Since the models are derived based on the fundamentals of the fluid mechanics and the thermodynamics, they should be applicable in the entire range of a steadystate solids gas flow and furthermore, they should be valid for any two-dimensional steady-state two-phase flow system, by using proper interface forces. For example, for the case of dense gas solids transport, further study is needed to develop an appropriate expression for particle-particle stress and interaction to refine the models. Nomenclature B = displacement factor CDSx = drag coefficient for solid phase in x direction CD, = drag coefficient for solid phase in y direction D = diameter of the conveying pipe dp = diameter of solid particles f = gas-wall friction factor { = gas-wall friction force in x direction = gas-wall friction force in y direction
gy
Ind. Eng. Chem. Fundam. 1982, 21, 43-46
f, = drag force per unit volume of solid particles in x direction f = drag force per unit volume of solid particles in y direction
g“= gravity acceleration
P = pressure = initial pressure Re, = gas-phase Reynolds number Re, = solid-phase Reynolds number in x direction Re, = solid-phase Reynolds number in y direction u = gas-phase horizontal velocity dg= superficial gas velocity u, = solid-phase horizontal velocity ug = gas-phase vertical velocity u, = solid-phase vertical velocity W , = solid-phase mass flow rate x = direction of the flow; horizontal direction y = direction from the top toward bottom of the pipe; vertical direction Y = column vector of dependent variables at each grid point Yo = initial value of the column vector Y Greek Letters t = volume fraction of gas phase to = initial volume fraction of gas phase a = parameter in solid-phase momentum equations /3 = parameter in solid-phase momentum equations y = parameter in solid-phase momentum equations 9 = parameter in solid-phase momentum equations pg = viscosity of the gas phase p , = density of the gas phase ps = density of the solid phase X = characteristics of the system of equations Subscripts 1 = index number for position y = 0 4 2 , ..., n - 1) = index number for position 0 < y < Dpi, n = index number for position Y = Dpi, Literature Cited
PO
43
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Received for review September 8, 1980 Accepted October 14,1981
Rates of Gas Absorption into Molten Salts Elzo Sada,” Shlgeo Katoh, Hidehumi Yoshil, and Kiyoshi Yasuda Chsmicai Engineering hpartment, Kyoto UnIverMy, Kyoto 606, Japan
Rates of physical absorption of carbon dioxide and oxygen into molten salts were measured in a stirred vessel with known gas-liquid interfacial area. The liquid-phase mass transfer coefficient for physical absorption into molten salts was correlated by a dimensionless equation in a manner similar to that used for aqueous solutions. The data on carbon dioxide absorption into a eutectic mixture of LiCl (58 mol %)-KC1 (42 mol %) containing 0.06-0.8 M NaOH were analyzed on the basis of the film theory for mass transfer with chemical reaction in the aqueous phase.
Introduction Molten salts have recently been used as reaction media and heat transfer fluids because of their stability at high temperatures, catalytic effects on some reactions, and excellent heat transfer characteristics. Although physical properties of molten salts are close to those of aqueous solutions, little is known about mass transfer rates and reaction mechanisms in gas-molten salt systems. The present work was undertaken to study the rates of gas absorption into molten salts with and without chemical reaction. Rates of the physical absorption of carbon dioxide and oxygen into a eutectic mixture of LiCl (58 mol %)-KC1 01 96-43 7318217 Q27-0043$07.2510
(42 mol %) and molten NaN03 were measured with the use of a stirred vessel with known gas-liquid interfacial area. To obtain bases for correlation, data were obtained in the same apparatus for the physical absorption of carbon dioxide into pure water and glycerol solutions. Carbon dioxide was chemically absorbed into the LiC1-KC1 eutectic containing 0.06-0.8 M NaOH to study the mechanism of gas-molten salt reaction.
Experimental Section Figure 1 shows the stirred vessel with four vertical baffles. The vessel, made of Pyrex glass, was 8 cm in inside diameter and contained 300 cm3 of liquid sample. It was 0 1982 American Chemical Society
