Flow Characteristics of Gas and Liquid through a Cell Porous Disk

This paper describes applications of cell porous materials. The authors investigated the flow characteristics of a gas-liquid mixture in a rotating porous disk. For theoretical analyses, the gas is assumed to permeate the entire disk surface. A simple one-dimensional model illustrates that the residence time of the liquid is much greater than that of the gas. Violent interaction in small cells is likely to enhance the chemical reaction between gas and liquid. Cell porous materials might also be exploited for chemical reaction purposes.


INTRODUCTION
Applications of porous materials have been investigated in various fields.The Institute of Theoretical and Applied Mechanics investigated gas flow around and inside the cell porous rotating disk [1,2].Simple one-dimensional (1D) stationary model of the flow based on Darcy's model [3] revealed that highly permeable cell porous rotors made of metal or ceramics are promising for gas transport use [4].
This study examines the use of rotating porous materials for chemical reaction between a gas and a liquid.Chemical reactions are enhanced by increasing interface and mixing.If a liquid is introduced into a rotating porous disk, it moves to the periphery, splitting into small droplets [5].The gas and the small droplets interact violently in cell porous materials.These peculiarities may be effectively exploited in a chemical reaction between a gas and a liquid.
Theoretical analyses of gas-liquid mixture flow characteristics in the porous rotating disk are carried out using a simplified 1D stationary model.The aim of the present work is to estimate the residence times of the gas and the liquid in the disk.

Scheme of a model and assumptions
The configuration considered in this study is axisymmetric, as shown in Figure 1.In this figure, r o and D, respectively, represent the disk radius and its width.The rotating disk is sealed from the bottom faceplate.The ambient gas enters the porous disk through its upper surface.At the periphery, the gas moves in a radial direction with nonzero angular velocity.Near the rotation axis, the liquid is supplied at a certain flow rate to a small hole with radius r i .
In this model, the packed density of the porous disk material is assumed to be large.Under that assumption, it might be reasonable that the tangential velocity u is nearly equal to the rotating velocity rω.
The 1D stationary momentum equation inside the rotating porous disk [4] can be approximated as where ν is the radial velocity component, p is the pressure, ρ is the fluid density, ω is the angular velocity of disk rotation, and r is the radius in cylindrical coordinates.The last term is presumably proportional to the square of the gas velocity relative to the disk with drag force coefficient k [6].
It is known that the difference between tangential velocities of the gas and the solid can be neglected when k 1 (this condition is satisfied in our model as shown in Section 3.2) [4].
The characteristic flow velocities are smaller than the sound velocity, which allows for neglecting the gas compressibility effects.In this model, the gas velocity is known to be uniform everywhere at the upper surface of the rotating disk [4].

Research Letters in Materials Science
Within the framework of the 1D model, the continuity equation of the gas is written as On the other hand, the liquid is treated as a cloud of small droplets.The mass density of the cloud (φρ L ) is governed by the following equation:

Mathematical model for the gas
The residence time of the gas is studied in this section.When r i ≈ 0, in Figure 1, (2) can be simplified to The equation of pressure at r = r o is In reality, the sum of kinetic energy and static pressure in the left-hand side of ( 5) exceeds the static pressure of stagnating gas far from the disc p ∞ [2,4].In this model, equality in (5) is assumed; that is, pressure loss is neglected outside the disk.
By solving (1), (4), and (5), p(r) is written as The equation of pressure at r = r i can be written as follows: Substituting ( 7) into (6) gives When r i is small, the gas flow rate is represented as Using the following relation:  with (4), the residencetime of the moving gas from r to r o is written as The volume element of gas coming to the disk between r and r + dr is expressed as Therefore, the average residence time of the gas is represented as where q is the total flow rate as Equation ( 13) can be rewritten by

Mathematical model for liquid
In this section, the residence time of the liquid is considered.The liquid is assumed to be a cloud of droplets with volume fraction φ.The continuity of mass is represented as where m, ρ L , and ν L denote the feeding rate of the liquid, the density of the liquid, and the radial component of the liquid velocity, respectively.The momentum equation can be expressed by the following relationship: In ( 17), liquid motion is assumed to be affected only by the interactive force with the rotating disk because the force from the surrounding gas is small.Using the boundary condition, we obtain the following: Equation ( 19) is obtained by solving (17) which is a linear equation in ν 2 .The liquid velocity is obtained as Similarly, the residence time of the liquid is represented as It is convenient to introduce nondimensional variables: Equations ( 9), ( 15), ( 16), ( 19), and (20) can then be expressed as ( 22), ( 25), ( 26), (23), and (24), respectively:

Numerical solutions
Figure 2 shows the nondimensional relation between the drag force coefficient of gas and the gas flow rate for The flow rate is determined by K G .As shown in ( 9), the flow rate is proportional to the angular velocity ω.The residence time is directly related to 1/ω.The variation of nondimensional residence time is illustrated as a function of K G in Figure 3 for H = 0.1.
Figure 4 depicts the variation of the radial component of liquid velocity with the nondimensional radius evaluated for K L = 1000, H = 0.1, M = 10 −4 , ζ i = 0.1, and φ i = 1.
Velocity change occurs steeply in a narrow range near ζ i .Figure 5 portrays velocity variations for various values of φ i with enlarged ζ; the figure is greatly exaggerated for convenient representation.The right-hand side of (17) is positive; the liquid is accelerated there.On the contrary, if it is negative, the liquid is decelerated.Variation of the volume fraction along ζ is shown in Figure 6.
As portrayed in Figure 5, the change of initial φ i has no influence, except in a very narrow region near ζ i .
Equation ( 26) is calculated as a function of K L ; its results are shown in Figure 7.
Comparison of Figures 2 and 7 reveals that the residence time of liquid is approximately five times longer than that of gas for K L = 100 − 1000.

Experiments
Experiments were carried out using the cell porous disk of stainless steel as shown Figure 1.Its packed density was 440 kg m −3 .The outer and inner diameters were 0.15 m and 0.02 m, respectively.The width was 0.027 m.The frequency of disk rotation was stable and equal to 3000 rpm.
The gas flow rate of 0.0297 m 3 s −1 was obtained by the method of JIS (Japanese industrial standard no.B8330).Drag force coefficient of the gas and its resistance time were estimated to be K G = 300 and T G,a = 5 from ( 22) and (23), respectively.
The measurement of the residence time for liquid was conducted.The water was injected at r i = 0.01 m instantaneously and it came out at the edge of the disk after certain time delay.The residence time of liquid, T L = 200, was obtained from the observation of the time delay by high-speed video camera (FASTCAM ultima SE, Photron).The value of the drag force coefficient for the liquid was estimated to be K L = 23000 from (25).
Results of experiments revealed that K L is much larger than K G .Therefore, the ratio of T L /T G,a becomes much larger in a real rotating disk system.

CONCLUSION
Considering the application of cell porous material to a chemical reactor, the authors investigated gas-liquid mixture flow characteristics in a rotating porous disk for use in reactors.
The gas is assumed to be soaked through the entire disk surface.The theoretical results for the simplified 1D model indicate that the residence time of the gas in the disk is inversely related to the angular velocity.
On the other hand, the residence time of liquid is much longer than that of gas.The residence time of liquid injected into the disk near the center axis is also inversely related to the angular velocity, but it is almost independent of the initial volume fraction.
The residence time of the liquid is much longer than that of the gas.The relation of T L /T G 1 indicates that a rotating cell porous disk system is suitable for the reaction of a smallvolume liquid with a large-volume gas.

Figure 1 :Figure 2 :
Figure 1: Scheme showing gas and liquid flows inside the rotating porous disk.

Figure 3 :Figure 4 :
Figure 3: Relation between the drag force coefficient of gas, K G , and the mean residence time of gas, T G,a , for H = 0.1.

Figure 5 :Figure 6 :
Figure 5: Velocity variations for various values of φ i with enlarged ζ.

Figure 7 :
Figure 7: Residence time of liquid, T L , versus the drag force coefficient of water, K L , evaluated for H = 0.1, M = 10 −4 , and φ i = 1.The inlet position of liquid ζ i = 0.1.

NOMENCLATUREh:
Porous disk width (m) H: Nondimensional porous disk width k: Drag force coefficient (m −1 ) K: Nondimensional drag force coefficient m: Feeding rate of liquid (m s −1 ) M: Nondimensional feeding rate of liquid q: Gas flow rate (m 3 s −1 ) Q: Nondimensional gas flow rate r: Radial position coordinate (m) T: Nondimensional time u: Tangential velocity component (m s −1 ) v: Radial velocity component (m s −1 ) V : Nondimensional radial velocity component w: Axial component of gas velocity at the upper surface of the porous disk (m s −1 ) z: Axial coordinate GREEK SYMBOLS ζ: Nondimensional radial position coordinate ρ: Density (kg m −3 ) τ: Residence time (s) ω: Angular velocity of porous disk (rad s −1 ) φ: Volume fraction of liquid SUBSCRIPTS a: Average G: Gas i: Value at inner radius L: Liquid O: Value at outer radius