A gas-solid flow is characterized by the flow of gases with suspended solids. This type of flow is fundamental to many industrial processes such as pneumatic transport, particulate pollution control, combustion of pulverized coal, drying of food products, sand blasting, plasma-arc coating and fluidized bed mixing. The dynamics and thermal history of particles in gases also affect the performance of rocket motors using metallized fuels, the quality of some pharmaceutical products and the design of advanced techniques for materials processing.

Various features of gas-solid flows can be best understood by considering the motion and thermal history of particles in a gas flow field. The equation of motion for a spherical particle of diameter D accelerating in a gas stream is


where m is the particle mass, g is the acceleration due to gravity, ρg is the gas density, CD is the drag coefficient and up and ug are the particle and gas velocities respectively. Dividing through by the particle mass yields


where ρp is the material density of the particle and Rep is the particle Reynolds Number based on the relative velocity, particle diameter and gas properties. The factor CDRep/24 is the ratio of the drag coefficient to Stokes drag and is represented by the factor f. This factor is a function of the particle Reynolds number but may also depend on other parameters such as particle shape, Mach number based on the relative velocity and turbulence level (Clift et al., 1978). For small particle Reynolds number (Rep ≤ 1) the factor approaches unity which corresponds to Stokes flow. (See also Stokes Law and Spheres, Flow Around and Drag)

The coefficient of the drag term in the particle motion equation has dimensions of reciprocal time and leads to the definition of the particle velocity response time


This time is a measure of the responsiveness of a particle to a change in gas velocity. For example, the time required for a particle released from rest in a gravity-free environment to achieve 63% of the gas velocity in Stokes flow is one velocity response time. Also the distance a particle would penetrate into a stagnant gas before stopping is u0τV, where u0 is the initial particle velocity. This distance is called the stopping distance. The terminal velocity a particle would achieve falling through a stagnant gas is gτv.

Neglecting radiative heat transfer and assuming there is no change of phase of the particle material, the equation for particle temperature assuming a uniform internal temperature is


where c is the specific heat of the particle material, Nu is the Nusselt Number and Tp and Tg are the particle and gas temperatures respectively. Dividing through by the mass and specific heat yields a coefficient for the heat transfer term which is the reciprocal of the thermal response time; namely,


which is a measure of the thermal responsiveness of the particle to a change in gas temperature. At low Reynolds numbers, the Nusselt number is 2. Empirical expressions are available for higher particle Reynolds numbers and other effects (Clift et al., 1978). For particles in a gas, the thermal response time and velocity response time are of the same order of magnitude.

The response times are used in the definition of Stokes number which is the ratio of the particle response time to a time characteristic of the flow system. For example, consider the gas-solids flow through a venturi geometry shown in Figure 1 A time which would be representative of the fluid residence time in the venturi would be


where L is the throat diameter and uT is the flow velocity in the throat region. The Stokes number based on the velocity response time is:


For Stokes numbers much less than unity, the particles have sufficient time to maintain near velocity equilibrium with the gas and the gas-solids flow could be regarded as a single-phase flow with modified density. On the other hand, if the Stokes number is large compared to unity, the particle motion is unaffected by the change in gas flow velocity through the venturi.

Figure 1. 

The same definition of Stokes number applies to thermal response time as well. For low Stokes numbers, the particles maintain near thermal equilibrium with the gas. At large Stokes numbers, the particles have no time to respond to the changes in gas temperature.

A basic definition in gas solids flows is dilute and dense flows. A dilute flow is a gas-particle flow in which the particle motion is controlled by the drag and lift forces on the particle. In a dense flow, on the other hand, the particle motion is controlled primarily by particle-particle collisions. A dilute flow would correspond to the conditions where the stopping distance is less than that between particle-particle collisions. This definition is important in understanding and modeling the velocity and thermal fields in a gas-solid flow.

Another important feature of gas-solid flows is coupling which is the interaction between phases. If the gas affects the motion and temperature of the particles but the particles do not change the gas velocity or thermal flow fields, then flow is one-way coupled. On the other hand, if there is a mutual interaction between phases, the flow is two-way coupled. As an example of coupling, consider hot particles being transported by a cold gas in a duct as shown in Figure 2. Assuming one-way coupling, the particle temperature decreases toward the gas temperature while the gas temperature remains constant. With two-way coupling, the gas temperature increases due to heat exchange with the particles and the rate of particle temperature is reduced. Actually all flows are two-way coupled, but for low particle concentrations, the effect of the particles on the gas field may be negligible. Therefore, the assumption of one-way coupling would be justified thereby simplifying the analysis of the flow system.

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Figure 2. 

Pneumatic transport is an important example of gas-solid flows because of its wide use in industry to transport metal particles, grains, ores, cement, coal and other products not susceptible to damage by contact with the pipe walls. The main advantage of pneumatic transport is the flexibility of line location and the capability to tap the line at arbitrary locations. The gas velocity for vertical transport has to exceed the settling (terminal) velocity of the particles to maintain transport.

For horizontal pneumatic transport, various flow patterns are identified which depend on several factors such as flow velocity and particle loading. The various flow patterns are shown in Figure 3. Homogeneous flow occurs when the velocity is sufficiently high to keep the particles in suspension. Dune flow begins as the velocity is lowered and particles begin to settle out on the wall forming a dense flow region with a pattern like sand dunes. The velocity at which particles begin to settle out is called the saltation velocity. Further reduction in velocity leads to slug flow where there are alternate regions where particles fill the pipe and where they are in suspension. The flow behaves like slug flow of the gas-liquid flow regimes. Finally, with further reduction in velocity the particles become packed in the pipe and form a packed bed while the gas moves through the interstitial region between the particles. Even though the particles fill the pipe there may still be a slow motion of the particle bed.




(a) Homogeneous Flow( )





(b) Dune Flow( )





(c) Slug Flow( )





(d) Packed Bed( )


Figure 3. 

A typical variation in pressure drop with flow velocity is shown in Figure 4. At high velocities (in the homogeneous flow regime), the pressure drop varies with nearly the square of the velocity as with a single-phase flow. The pressure drop for the particle-laden flow is higher because the particles lose momentum on contact with the wall. The force applied to the fluid over a length ΔL of pipe by the drag on the particles is


where n is the number of particles per unit volume and A is the cross-sectional area of the pipe. Equating this force to the augmented pressure gradient in the pipe due to the presence of the particles yields


where is the mass of particles per unit volume or the apparent or bulk particle density. Thus one notes that the pressure loss increases with increased particle concentration and gas-particle velocity difference (more momentum loss at the wall). The increase in pressure gradient corresponds to a two-way coupling effect. There are several empirical formulations available in the literature to estimate the pressure drop due to the particulate phase.

Figure 4. 

As the velocity is reduced to the saltation velocity, the cross-sectional area for the flow is reduced because of the deposits on the wall. The effective flow velocity is increased and the pressure loss is higher. To avoid deposition, higher pressure loss and possible plugging, the pneumatic system should be designed with velocities exceeding the saltation velocity of the slug flow regime. Various empirical expressions for saltation velocity and friction factors for pressure drop due to the solid phase can be found in Klinzing (1981).

The Fluidized Bed is another important example of a gas-solids flow and is a key element in many chemical processes, particularly coal gasification, combustion and liquification (Azbel and Cheremisinoff, 1983). Fluidized beds are also used for roasting ores and for the disposal of organic, biological and toxic wastes. In essence, the fluidized bed consists of a vertical cylinder loaded with particles and supplied with a gas through a distributor plate in the bottom of the cylinder. As the gas flow is increased the bed goes through several flow regimes (Hetsroni, 1982) as shown in Figure 5. At low flow rates, there is no significant motion of the particles as the gas passes through a packed bed. With increasing gas flow a point is reached where the particles are just supported by the hydrodynamic forces which is called particle fluidization. Further increase in gas velocity leads to the formation of bubbles (region of low particle concentration) which move upward in the bed and enhance mixing. With more gas flow, the bubbles grow to fill the tube and the slug flow regime is realized and is similar to the slug flow pattern in vertical gas-liquid flows. As the gas flow rate is further increased clusters of particles move about the field in an irregular fashion. Finally, at the highest gas flow rate, clusters move up the tube and out, with some downward motion near the wall, so the particles have to be reintroduced at the bottom. This condition is called fast fluidization. Thus, the fluidized bed has a range of flow regimes from dense to dilute flows. The fluidized bed is attractive as a reactor because the gas is exposed to a large solids surface area to enhance surface reactions and heat transfer. Also the bed operates at a nearly uniform temperature which provides control of the reaction rates. Depending on the application, heat exchanger tubes may be located in the bed itself (Howard, 1983). The flow in a fluidized bed is very complex and difficult to scale up from bench scale to prototype operation.

Figure 5. 

Another important industrial example of gas-solid flows is the removal of particulates from exhaust gases for pollution control. The two most common devices for removal of particulates from a gas stream are the cyclone separator and the electrostatic precipitator. The cyclone separator is simply a device which separates the solids from the gas by centrifugal acceleration. Standard designs are available in engineering handbooks. The performance of the cyclone separator is quantified by the cut size, which is the particle size above which all the particles are collected and below which are carried out by the exhaust gases. The cyclone separator is widely used because it is inexpensive and robust. (See Cyclones.)

The electrostatic precipitator operates by charging the particles and applying a Coulomb force to move the particles toward the collecting surface. It is capable of removing smaller particles from a gas stream than the cyclone separator and operates at a higher efficiency. It is commonly used to remove flyash from coal-fired power plants. The particulate-laden flow passes through an array of vertically suspended metal plates as shown in Figure 6. The cross-sectional configuration is shown in the same figure. The ribs protruding into the flow provide mechanical rigidity for the large plates. High voltage wires produce a corona at the wires and an electric field between the wires and the walls. For a negatively charged wire, electrons generated by the corona travel along the electric lines of force and accumulate on the particles yielding negatively charged particles. The resulting Coulomb force on the particle moves the particle toward the wall (collection surface).

Figure 6. 

Periodically the walls of the precipitator are “rapped” and the particles fall into a collection bin below the precipitator (Oglesby and Nichols, 1978).

The drift velocity toward the wall is given by


where E is the electric field intensity and q/m is the charge to mass ratio on the particle. For very small particles the drag factor f must account for noncontinuum effects because the particle size may be comparable to the mean free path of the gas. A higher drift velocity implies an improved collection efficiency (particles collected/particles entering). The modified Deutsch-Anderson equation for efficiency


where A is the plate surface area, u is the gas velocity through the precipitator and k is an empirical constant, is often used to predict efficiency but cannot be used with confidence to extrapolate performance predictions under different operating conditions. The effect of the charged particles on the electric field distribution and the influence of turbulence on particle dispersion currently preclude the capability of making an accurate predictions of collection efficiency. (See also Electrostatic Separation.)

The particle-laden jet is another important example of a gas-solid flow. Gas-particle jets are fundamental to the operation of a furnace burning pulverized coal as well as sand blasting equipment and plasma-arc coating devices. A jet conveying pulverized coal issuing into a corner-fired furnace is shown schematically in Figure 7. Intense radiative heat transfer to the entering coal particles quickly causes devolatization where the volatiles in the coal (methane, hydrogen, etc.) are driven off as gaseous fuel to support combustion in the furnace. The remaining char particles burn more slowly as they pass through the furnace and react with the water vapor and other compounds. The mixing and dispersion of the particles by the jet is important to the efficient operation of the system (Smoot and Smith, 1985).

Figure 7. 

The production of powdered foods and other products through spray drying is a further example of gas-solid flows. Some examples of spray dried products include powdered milk, laundry detergent and pharmaceutical powders. Spray drying is particularly attractive for drying products which are heat sensitive. Slurry droplets are sprayed into a chamber, dried by hot gases and the resulting particles are collected as products. A typical counterflow spray dryer is shown in Figure 8. Here the droplets are sprayed downward from the top of the chamber as the hot gases are introduced from below. The gases are introduced with a tangential velocity component which produces a swirling motion as they pass upward and out through the top. The dried particles fall through the port at the bottom where they are further treated or packaged as the final product.

Figure 8. 

Many factors have to be considered in the design of the spray dryer. The thermal coupling cools the drying gases as they pass upward through the particles thereby reducing the drying effectiveness. The gas temperatures cannot be too high to prevent burning the powder or detracting from the taste of a food product. Also the particles have to be sufficiently dry so as not to accumulate on the walls and lead to the danger of fire in the dryer.

Considerable progress has been made in the last ten years in developing numerical models for gas-solid flows (Crowe, 1991). The availability of large memory machines and high speed processors have enabled model developers to simulate sufficient detail of gas-solid flows to produce numerical models adequate for many industrial needs. Two approaches have emerged: the trajectory and two-fluid approach. In the trajectory approach, the particle field is generated by solving for the particle trajectories and the velocity and thermal history along the trajectories in the gas flow field. The local heat and momentum transfer to the gas are used to update the gas field calculations and thereby include two-way coupling effects. This method is especially attractive for dilute flows. The two-fluid approach is to treat the particulate phase as a second fluid with an effective viscosity, thermal conductivity and diffusion coefficients. This approach has found application for dense flows such as fluidized beds. The continued development of numerical models will lead to ever improving simulations to complement design and support the operation of gas-solid systems.


Azbel, D. S. and Cheremisinoff, N. P. (1983) Fluid Mechanics and Unit Operations, Ann Arbor Science Publishers.

Clift, R., Grace, J. R. and Weber, M. E. (1978) Bubbles, Drops and Panicles, Academic Press.

Crowe, C. T. (1991) The State of the Art in the Development of Numerical Models for Dispersed Phase Plows, Proc. Intl. Conf. on Multiphase Flows—’91 Tsukuba, Vol. 3, pp. 49-60.

Hetsroni, G. (ed.) (1982) Handbook of Multiphase Systems, Hemisphere Publishing Corp. DOI: 10.1016/0301-9322(83)90066-6

Howard, J. R. (ed.) (1983) Fluidized Beds Combustion and Applications, Applied Science Publishers.

Klinzing, G. E. (1981) Gas-Solid Transport, McGraw-Hill.

Masters, K. (1985) Spray Drying Handbook. G. Goodwin, Ltd, London. DOI: 10.1016/0144-8617(91)90050-M

Oglesby, Jr., S. and Nichols, G. B. (1978) Electrostatic Precipitation, Marcel-Dekker, inc.

Smoot, L. D. and Smith, P. J. (1985) Coal Combustion and Gasification, Plenum Press, NY. DOI: 10.1016/0009-2509(86)85066-7

Les références

  1. Azbel, D. S. and Cheremisinoff, N. P. (1983) Fluid Mechanics and Unit Operations, Ann Arbor Science Publishers.
  2. Clift, R., Grace, J. R. and Weber, M. E. (1978) Bubbles, Drops and Panicles, Academic Press.
  3. Crowe, C. T. (1991) The State of the Art in the Development of Numerical Models for Dispersed Phase Plows, Proc. Intl. Conf. on Multiphase Flows—’91 Tsukuba, Vol. 3, pp. 49-60.
  4. Hetsroni, G. (ed.) (1982) Handbook of Multiphase Systems, Hemisphere Publishing Corp. DOI: 10.1016/0301-9322(83)90066-6
  5. Howard, J. R. (ed.) (1983) Fluidized Beds Combustion and Applications, Applied Science Publishers.
  6. Klinzing, G. E. (1981) Gas-Solid Transport, McGraw-Hill.
  7. Masters, K. (1985) Spray Drying Handbook. G. Goodwin, Ltd, London. DOI: 10.1016/0144-8617(91)90050-M
  8. Oglesby, Jr., S. and Nichols, G. B. (1978) Electrostatic Precipitation, Marcel-Dekker, inc.
  9. Smoot, L. D. and Smith, P. J. (1985) Coal Combustion and Gasification, Plenum Press, NY. DOI: 10.1016/0009-2509(86)85066-7
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