Flow Through a Packed Bed

The majority of gas-phase reactions are catalyzed by passing the reactant through a packed bed of catalyst panicles. 

The equation used most to calculate pressure drop in a packed porous bed is the Ergun equation  

Term 1 is dominant for laminar flow, and Term 2 is dominant for turbulent flow. 

In calculating the pressure drop using the Ergun equation, the only parameter that varies with pressure on the right-hand side of Equation (4-22) is the gas density, p. We are now going to calculate the pressure drop through the bed. 

Because the reactor is operated at steady state, the mass flow rate at any point down the reactor, in (kg/s). is equal to the entering mass flow rate. i, (i.e., equation of continuity), 

H = superficial velocity = volumetric flow rate cross-sectional area of pipe. 0/1) (m/s) p = gas density, lb /ft (kg/m ) 

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Flow distribution in a packed bed received attention after Schwartz and Smith (1953) published their paper on the subject. Their main conclusion was that the velocity profile for gases flowing through a packed bed is not flat, but has a maximum value approximately one pellet diameter from the pipe wall. This maximum velocity can be 100 % higher than the velocity at the center. To even out the velocity profile to less than 20 % deviation, more than 30 particles must fit across the pipe diameter. 

The column may be packed or it may be an open tube but in this example, a packed column will be specifically considered. The column is considered to have a length (L) and inlet and outlet pressures and inlet and outlet velocities of (Pi), (Po) (ui) and (uo), respectively. The pressure and velocity at a distance (x) from the front of the column is (Px) and (ux), respectively. According to D Arcy s equation for fluid flow through a packed bed, at any point in the column. 

A certain amount of information on particle-to-liquid and bed-to-wall heat transfer is available for single-phase fluid flow through a packed bed. It is not clear, however, to what extent this information can be applied to... 

Since the exact profile of the mobile phase flow through a packed bed is unknown, only an approximate description of the ] and broadening process can be attained. For packed column gas chronatography at low mobile phase velocities, equation (1.35) provides a reasonable description of the band broadening process [70,82,83]. 

Problems are stiff when the time constants for different phenomena have very different magnitudes. Consider flow through a packed bed reactor. The time constants for different phenomena are ... 

A reacting gas flows through a packed bed that has a Peclet number Pe =... 

A Reynolds number for flow through a packed bed can be defined as... 

If a fluid is passed upwards in laminar flow through a packed bed of solid particles the superficial velocity u is related to the pressure drop AP by equation 9.33 ... 

These methods depend on the fact that at low flowrates the flow through a packed bed is directly proportional to the pressure difference, the proportionality constant being proportional to the square of the specific surface (surface volume ratio) of the powder. From this method it is possible to obtain the diameter of the sphere with the same specific surface as the powder. The reliability of the method is dependent upon the care with which the sample of powder is packed. Further details are given in Chapter 4. 

Reproducible correlations for the heat transfer coefficient between a fluid flowing through a packed bed and the cylindrical wall of the container are very difficult to obtain. The main difficulty is that a wide range of packing conditions can occur in the vicinity of the walls. However, the results quoted by Zenz and Othmer(44) suggest that ... 

From d Arcy s Law for fluid flow through a packed bed, at the optimum mobile phase velocity the length of the column is given by,... 

In Reprint C in Chapter 7, the behavior of a tracer pulse in a stream flowing through a packed bed and exchanging heat or matter with the particles is studied. It is shown that the diffusion in the particles makes a contribution to the apparent dispersion coefficient that is proportional to v2 fi/D. The constant of proportionality has one part that is a function of the kinematic wave speed fi, but otherwise only a factor that depends on the shape of the particle (see p. 145 and in equation (42) ignore all except the last term and even the suffixes of this e, being unsuitable as special notation, will be replaced by A. e is defined in the middle of p. 143 of Chapter 7). In this equation, we should not be surprised to find a term of the same form as the Taylor dispersion coefficient, for it is diffusion across streams of different speeds that causes the dispersion in that case just as it is the diffusion into stationary particles that causes the dispersion in this.7 What is surprising is that the isothermal diffusion and reaction equation should come up, for A is defined by... 

Except in the case of reactions at high pressure, the pressure drop which must be maintained to cause flow through a packed bed of particles is usually insufficient to produce forced flow in the capillaries of the solid, and the gas flow is diverted around the exterior periphery of the pellets. Reactants then reach the interior of the porous solid by Knudsen or molecular diffusion. 

The physical understanding of gas flows through a packed bed is important for the study of the gas-solid transport system as it represents the limiting case of the gas-solid transport. A typical example is the minimum fluidization condition of gas-solid fluidization. In a gas-solid flow system, the presence of unsuspended particles is also common, and, thus, understanding the gas flow in a packed bed is essential. 

Experiments were conducted by Darcy (1856) to study viscous flows through homogeneous porous media, which resulted in the well-known Darcy law. There are various models from which Darcy s law may be derived to describe gas flow through a packed bed... 

The generalized relation for the pressure drop for flows through a packed bed was formulated by Ergun (1952). The pressure loss was considered to be caused by simultaneous kinetic and viscous energy losses. In Ergun s formulation, four factors contribute to the pressure drop. They are (1) fluid flow rate, (2) properties of the fluid (such as viscosity and density), (3) closeness (such as porosity) and orientation of packing, and (4) size, shape, and surface of the solid particles. 

Flow Through a Packed Bed of Spheres. The Ergun Equation (46), gives the pressure drop through a packed bed of spheres 2... 

In the preceeding section, we have shown how fluid flows through a packed bed may be observed at various levels, leading to completely different interpretations. Actually, when modelling a transport process, it is always necessary to consider at least two observation levels the bed scale and the particle scale. 

Equation 8.26 shows that diffusion-limited selectivity depends on the surface conditions. Therefore, when a reacting mixture flows through a packed bed reactor the local selectivity is dependent on the position. The total yield (the ratio of desired product B... 

The void fraction is the most significant factor determining the loss of pressure during fluid flow through a packed bed. At low flow rates (Re 50) the pressure drop is proportional to... 

Significant literature on the axiaj dispersion in gas and liquid phases for countercurrent-flow packed-bed columns have been reported. Trickle- and bubble-flow regimes have been considered. Unlike the holdup, there is quite a discrepancy in the results of various investigators. Almost all the RTD data are correlated by a single-parameter axial dispersion model. A summary of the reported axial dispersion studies in countercurrent flow through a packed bed is given in Table 8-1. 

In porous solids made of larger elements such as fixed packed beds, where the characteristic dimension of the packing is d (for example the diameter of a packed solid), the frequency of the velocity change is a = v/d (after each flow through an element of the packed bed, the local fluid velocity v changes its direction). Now if we use this value of a in the dispersion coefficient, we obtain the famous relation Pe = (vd)/D = 2, which gives the value of the dispersion coefficient when a fluid flows through a packed bed [4.90]. 

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