# minimum fluidization velocity

The point at which These may be calculated from equation 6.32, which is identical to equation 5.84 in Chapter 5: Richardson and Zaki(11) found that ui corresponded closely to u0, the free settling velocity of a particle in an infinite medium, for work on sedimentation as discussed in Chapter 5, although ui was somewhat less than u0 in fluidisation. Experimentally, the most common method of measurement requires that pressure drop One suggestion for improvement is to insert baffles into the bed and thereby cut down the effective bubble size. $\frac{1.75}{\epsilon_{mf}^3 \phi} \left( \frac{d_p u_{mf} \rho_g}{\mu} \right)^2 + \frac{150(1-\epsilon_{mf})}{\epsilon_{mf}^3 \phi^2} \left( \frac{d_p u_{mf} \rho_g}{\mu} \right) = \frac{d_p^3 \rho_g (\rho_s - \rho_g) g}{\mu^2}$, $\frac{1.75}{\epsilon_{mf}^3 \phi} Re_{p, mf}^2 + \frac{150 (1 - \epsilon_{mf})}{\epsilon_{mf}^3 \phi^2} Re_{p, mf} = Ar$, $Ar = \frac{d_p^3 \rho_g (\rho_s - \rho_g) g}{\mu^2}$, $Re_{p,mf} = \frac{d_p u_{mf} \rho_g}{\mu}$, $K_1 = \frac{1.75}{\epsilon_{mf}^3 \phi}$, $K_2 = \frac{150(1-\epsilon_{mf})}{\epsilon_{mf}^3 \phi^2}$, $Re_{p,mf} = \left( a^2 + b Ar \right)^{1/2} - a$, $u_{mf} = \frac{Re_{p,mf} \mu}{d_p \rho_g}$, $u_{mf} = \frac{d_p^2 (\rho_s - \rho_g) g}{150 \mu} \frac{\epsilon_{mf}^3 \phi^2}{1 - \epsilon_{mf}}$, $u_{mf}^2 = \frac{d_p (\rho_s - \rho_g) g}{175 \rho_g} \epsilon_{mf}^3 \phi$. In very dilute cumene–air mixtures the kinetics are essentially first order reversible with respect to cumene with an equilibrium conversion of 94%. For inelastic fluids exhibiting power-law behaviour, the bed expansion which occurs as the velocity is increased above the minimum fluidising velocity follows a similar pattern to that obtained with a Newtonian liquid, with the exponent in equation 6.31 differing by no more than about 10 per cent. What conversion do you expect to find under these conditions? Neglecting effects due to the container wall then: where Re’c is the Reynolds number ucdρ/μ. The general form of relation between velocity and bed voidage is found to be similar for both Newtonian and inelastic power-law liquids. [1.3] into Eq. Liquid–solid fluidised systems are generally characterised by the regular expansion of the bed which takes place as the liquid velocity increases from the minimum fluidisation velocity to a value approaching the terminal falling velocity of the particles. An alternative method of calculating the value of Re’mf (and hence umf) is to substitute for Re’0 from equation 6.21 into equation 6.35, and to put the voidage e equal to its value emf at the minimum fluidising velocity. Qualitatively similar results have been reported by many workers, and in a recent review [Chhabra, 1993a,b] it has been shown that equation (5.22) correlates most of the data available for inelastic power-law fluids (0.6 ≤ n ≤ 1; (d/D)≤0.16). What would be the conversion in a downflow packed bed (Hs=100 cm and u0=20 cm/s)? Richardson, in Non-Newtonian Flow and Applied Rheology, 2008. Calculate minimum fluidization velocity for very small particles where Relation between fluid velocity (uc) and voidage (e) for the fluidisation of 6.4 mm steel spheres in water. Minimum fluidization velocity and local gas holdup are important parameters used to characterize the hydrodynamic behavior of a material inside the fluidized bed. When void fraction and sphericity are not known, values for $$a$$ and Refer to Equation 25 and Table 4 in Chapter 3 of Kunii and Levenspiel Liquid–solid fluidized systems are generally characterized by the regular expansion of the bed which takes place as the liquid velocity increases from the minimum fluidization velocity to a value approaching the terminal falling velocity of the particles. When the gas velocities increase beyond the minimum fluidization velocity, bubbles can form. The minimum fluidization velocity (Umf), defined as the superficial gas velocity at which the drag force of the upward moving gas becomes equal to the weight of the particles in the bed, is one of the most important parameters associated with a fluidized bed system. Furthermore, Re’0 is given by equation 6.21. J.F. known. Minimum fluidization velocity¶ For a bed of particles, the minimum fluidization velocity is the gas velocity at which the drag force of the upward moving gas equals the weight of the particles. Drag, particles settling; Pressure-drop: Ergun's equation; Examples on pressure drop calculations; Fluidization; Minimum fluidization velocity; Examples on fixed and fluidized beds; Filtration. (equation 6.9)Reynolds number Re’0 at terminal falling velocity is given by equation 6.21: The value of n in equation 6.31 is given by equation 6.32 for small values of d/dt as: The voidage e at a velocity of 0.25 m/s is then given by equation 6.31 as: R.P. The simulations are performed with two particle phases with diameter 153 µm and 960 µm. Specifically, it is the point at which all the particles become suspended. This approach is based on the Ergun pressure drop equation for a bed of As air flow is increased above the minimum fluidization velocity, the bed may exhibit behaviors M1, can have values ranging from very small to very large, with 1 representing perfect mixing. Determine minimum fluidization velocity from particle and gas properties. At this air velocity or flowrate all of the bed particles are completely suspended by the air stream. The approximate values for a and b give two similar functional relationships between ϕ and εmf, εmf is the bed voidage at minimum fluidization. Cs can have values between −1 and +1 with 0 being perfect mixing. By using gas (or liquid) flowing upwards through a layer of a particulate material supported on a distributor, at certain fluid upwards velocity the particles start to move. The coefficient of vertical segregation, Cs, is defined as: where xB and xT are the concentrations of the component of interest in the bottom and top halves of the bed, respectively. What would be the conversion in a larger fluidized bed pilot plant (Hs=100 cm and u0=20 cm/s) in which the estimated bubble size is 8 cm? Minimum fluidization velocity at elevated temperature in tapered fluidized bed. Typical bed expansion data for 3.57 mm glass spheres fluidised by shear-thinning polymer solutions [Srinivas and Chhabra, 1991], R.P. deductions could be made. Chapter 4: 76 Figure 4-5 Bubble fluidised bed Figure 4-6 The plot of pressure drop against gas velocity at T=300°C, Hs=0.5D of silica sand (500- 600 μm), fluidization and de-fluidization curves. At this point, bubbles of gas form and rise through the bed. Butterworth-Heinemann, 2nd edition, 1991. Figure 5.18 Minimum fluidization velocity and pressure drop are important hydrodynamic parameters in the design and scale up of fluidized bed reactors. Reynolds number < 20 and for very large particles where Reynolds number > 1000. By substituting Eq. However, if the particle density is smaller than the fluid density (as in the case of ping pong balls in water), particles float and are compressed onto the distributor placed at the top of the column. So far we have treated the powder as if it were a homogeneous fluid, though we have allowed the ‘fluid’ properties (for example, minimum fluidization velocity and bulk density) to change with particle size and density. Richardson and Zaki(11) showed that, for sedimentation or fluidisation of uniform particles: uc is the observed sedimentation velocity or the empty tube fluidisation velocity. This formula is based on the Ergun pressure drop equation The general relation between velocity and volumetric concentration or voidage is found to be similar to that between sedimentation velocity and concentration for particles in a suspension. particles. For systems where only gravity and vertical fluid drag force are acting on particles, the bed is fluidized first and the fluid supply is reduced to determine the minimum fluidization point. Beyond the minimum fluidisation velocity ( Determine the gas flow rate in the bubble phase and emulsion phase for a fluidized bed combustor under the following two operating conditions: T=1173K, P=1.1 MPa, D=1.2 m, particle loading Mp=1600 kg, dp=0.51 mm, ϕs=0.84, ρp=2422 kg/m3, u0=1.0 m/s. Determine minimum fluidization velocity using experimental coefficients from More recently, Khan and Richardson(14) have proposed the following relation to account for the effect of the walls of the vessel in fluidisation: If logarithmic co-ordinates are used to plot the voidage e of the bed against the superficial velocity uc (Figure 6.5), the resulting curve can be represented approximately by two straight lines joined by a short transitional curve. Figure 6.5. The minimum fluidization velocity is commonly measured with decreasing fluidization velocity to avoid reliance on the incipient loading. This formula is based on the Ergun pressure drop equation … Catalyst: Pc = 1.83 g/cm3, € = 0.45 Catalyst: Pc = 1.83 g/cm3, E = 0.45 Omf Gas: Pg 1 = 1x 10-3g/cm3, u = 1.7 x 10-4 g/cm.s D(Pp-pf) For laminar flow 16504 (Omp)? Values calculated from equation (5.30) and experimental values of Z differ by less than 10%. This increases the effective volume of the bed. At low velocities the voidage remains constant corresponding to that of the fixed bed, and for the fluidised state there is a linear relation between log uc and log e. The curve shown refers to the fluidisation of steel spheres in water. On plotting particle Reynolds number against bed voidage using logarithmic scales, good straight lines were obtained over the range of conditions for which the bed was fluidised. ε M is the porosity of the bed at minimum fluidization By comparing the pressure drop given by the Ergun equation to the pressure drop for minimum fluidization, you can calculate the superficial velocity necessary for fluidization. Determine the rate constant k‴ for this reaction. The actual static pressure difference between bed bottom and top is the sum of pressure drop and the pressure difference in a static condition, i.e.. Data: umf=3.2 cm/s, ε0≅εmf=0.5, DAB=3×10−5 cm2/s, fw=0.33. A coefficient of radial segregation, Crs, can also be defined in a similar way by considering concentrations in the inner and outer cylindrical volumes. Firstly, particles of constant density were used (samples A, B, and . For a free-standing bed there will exist a point, known as the minimum or incipient fluidisation point, whereby the bed's mass is suspended directly by the flow of the fluid stream. Once that occurs we have reached onset of fluidization and the fluid velocity at this point is called minimum fluidization velocity (umf). The following equation for fluidisation was presented: The difference is likely to be attributed to the fact that d/dt was very small in the sedimentation experiments. Find X, if we double the bed cross-sectional area? This suggests that the modified Archimedes number (equation 5.12) takes account of power-law shear-thinning behaviour. By setting this equation equal to the gravitational force of the particle bed, we can determine the minimum fluidization velocity (um). For spherical particles emf is usually 0.4 – 0.45. shows a typical bed expansion behaviour for 3.57 mm glass spheres fluidized by aqueous carboxymethyl cellulose solutions (n=0.84 and n=0.9), and the behaviour is seen to conform to the form of equation (5.29). Comparison of values of the index n calculated from equation 6.37 with experimental data. particles. [1]. For fluidization of uniform spheres by Newtonian liquids, equation (5.29), introduced earlier to represent hindered settling data, is equally applicable: where V With Eq. The minimum fluidization velocities, calculated with the modified Wen and Yu (1966) equation, are also added to Table 2. At minimum fluidization, pressure drop across bed is balanced by effective weight of the particle. The minimum fluidization velocity was measured experimentally by plotting the pressure drop against the superficial fluid velocity. Substitution into equation 6.37 then gives: equation 6.38 which applies to low values of d/dt is plotted in Figure 6.6, together with experimental points from the literature, annotated according to the d/dt range which is applicable(14). On the other hand, much larger values of Z have been reported for fluidisation with visco-elastic polymer solutions [Briend et al., 1984; Srinivas and Chhabra, 1991], but no systematic study has been made to predict the value of Z for visco-elastic liquids. Minimum Fluidization Velocity This equation can be used to calculate the minimum fluidization velocity umf if the void fraction emf at incipient fluidization is known. Chemical Engineering and Processing: Process Intensification 2008 , 47 (12) , 2391-2394. equation 6.31 is similar to equation 5.71 for a sedimenting suspension. We used the Ergun equationon the packed bed pageto describe the drag exerted on a particle bed by the fluid flow. Minimum Fluidization Velocity, V mf. Figure 6.6. In most cases the flow direction is bottom to top, where without fluid flow the bed particles settle on the bottom gauze or distributor plate. As discussed in Chapter 3 of the Kunnii and Levenspiel book, the minimum fluidization velocity ($$u_{mf}$$) can be calculated from the equation shown below. For large Reynolds number where Re = 1001. The fluidization parameters found for the pure components are listed in table 2. There is some evidence, however, that with viscoelastic polymer solutions the exponent may be considerably higher. For emf = 0.43, the calculated values of n are virtually unchanged over the range 10 < Ga < 105. approach can be used when bed void fraction and particle sphericity are not This state of incipient fluidization can be described by an equation giving the pressure drop in a gas flowing through … For a bed of particles, the minimum fluidization velocity is the gas velocity at We wish to determine a minimum flow rate allowing the particles to become fluidized, the minimum fluidization velocity (um), and the flow rate which carries the first particle out of the chamber, the terminal velocity (ut). The superficial gas velocity at which the bed of powder is just fluidized, is normally called the minimum fluidization velocity or designated by Umf. ρ ε ε ρ ρ ε µ + − + − − = (6) To derive an equation for the minimum fluidization velocity, we have to apply the following physical concept: Fluidization of the packed bed starts when the pressure drop across the packed bed is equal the weight of the bed/unit area. umf (float) â Minimum fluidization velocity [m/s]. $$b$$ from Table 4 in Chapter 3 of Kunii and Levenspiel can be used to Depending on the value of Ar, Eq. [1.1] can be expressed as: For the pressure drop, we have the following Ergun correlation: where dp, u0, ϕ and μ are volume-to-surface mean particle diameter (where diameter of each particle is expressed by the diameter of a sphere of equal volume), superficial fluid velocity (fluid volume flow rate divided by cross section when no particles exist), shape factor (surface area of a sphere of diameter dp divided by real particle surface area) and viscosity of fluid, respectively. The minimum fluidization velocity observed in the experiments is 0.04 m/s. For all practical purpose minimum fluidization velocity at which the boiler has to be operated is taken as 1.5 times that … Figure 5.18. The above equation can be written in terms of the Reynolds and Archimedes The density of glass = 2500 kg/m3, the density of water = 1000 kg/m3, and the viscosity of water = 1mNs/m2. Sheng Fang, Yanding Wei, Lei Fu, Geng Tian, Haibin Qu, Modeling of the Minimum Fluidization Velocity and the Incipient Fluidization Pressure Drop in a Conical Fluidized Bed with Negative Pressure, Applied Sciences, 10.3390/app10248764, 10, 24, (8764), (2020). The corresponding fluid velocity, known as the "minimum fluidisation velocity", {\displaystyle u_ {mf}}. In such cases the pressure drop ΔP is mathematically negative corresponding to the downward fluid flow. estimate $$u_{mf}$$. This On the assumption that equation 6.31 may be applied at the point of incipient fluidisation: For a typical value of emf of 0.4, Re’mf is given by equation 6.14. Values of the index n range from 2.4 to 4.8 and are the same for sedimentation and for fluidisation at a given value of the Galileo number Ga. minimum fluidization velocities range from 0.21 to 0.34 m/s for void fraction 0.37 to 0.44 respectively. Mathis and Watson in AIChE J., 2, 518 (1956) reported on the catalytic conversion of cumene to phenol and acetone in both fluidized and packed beds of catalyst. [1.5] can be reduced to: We use cookies to help provide and enhance our service and tailor content and ads. © Copyright 2020, Gavin Wiggins CD12.3.2B Minimum Fluidization Velocity Fluidization will be considered to begin at the gas velocity at which the weight of the solids gravitational force exerted on the particles equals the drag on the particles from the rising gas. 0 is now the superficial velocity of the liquid in the fluidized bed and V is the terminal settling velocity of a single sphere in the same liquid; Z is a constant related to the particle Archimedes number and to the particle-to-column diameter ratio by equation (5.30). For fluidisation of uniform spheres by Newtonian liquids, equation (5.21) introduced earlier to represent hindered settling data, is equally applicable: where V0 is now the superficial velocity of the liquid in the fluidised bed and V is the terminal settling velocity of a single sphere in the same liquid; Z is a constant related to the particle Archimedes number and to the particle-to-vessel diameter ratio by equation (5.22). As discussed in Chapter 3 of the Kunnii and Levenspiel book, the It should be noted that whereas, in the absence of channelling, the pressure drop across a bed of a given expansion is directly proportional to its depth, the fluidising velocity is independent of depth. The experimental work was carried out in a column made up of acrylic having 60 mm outer diameter and 2 mm wall thickness and was 1000 Qualitatively similar results have been reported by many workers, and in a recent review (Chhabra, 1993, 2006) it has been shown that equation (5.30) correlates most of the data available for inelastic power-law fluids (0.6≤n≤1(d/D)≤0.16). the minimum fluidization velocity had to be kept constant throughout all the runs so that valid . An approach was made to predict the minimum fluidization velocity for binary mixtures of spherical particles differing in size and/or density. Values calculated from equation (5.22) and experimental values of Z differ by less than 10%. Refer to Equations 18 and 19 in Chapter 3 of Kunii and Levenspiel Reached onset of fluidization and the viscosity of water = 1mNs/m2 the drop!, 2008 less than 10 % 0.04 m/s a particular fuel, particles of 4 mm are... 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