Acceleration length

The pressure drop inside the draft tube is more complicated because it involves acceleration of solid particles from essentially zero vertical velocity. However, the model for calculating the pressure drop in vertical pneumatic conveying lines suggested by Yang (1977) can be applied. The acceleration length can be calculated from numerical integration of the following equation. 

Numerically integrate Eq. (11) to obtain the particle acceleration length and Eq. (15) to obtain the pressure drop across the draft tube, AP2 3. 

Yang, W. C., and Keaims, D. L., Estimating the Acceleration Pressure Drop and the Particle Acceleration Length in Vertical and Horizontal Pneumatic Transport Lines, Pneumotransport 3, pp. D7-89 (1976a)... 

Pressure Drop and Acceleration Length in Developing Regions... 

The acceleration length and the pressure loss in that length can also be estimated by using empirical correlations. Rose and Duckworth (1969) reported that... 

Example 11.2 Use Eq. (11.17) to derive a general expression for the acceleration length for dilute gas-solid pipe flows. Assume that the Stokes drag coefficient can be used. The friction coefficient of particles at the wall can be estimated by [Konno and Saito, 1969]... 

According to Newton s second law of motion, force is proportional to the product of mass and acceleration (length/time ). Natural force units are, therefore, kg-m/s (SI), g-cm/s (CGS), and lbm-ft/s (American engineering). To avoid having to carry around these complex units in all calculations involving forces, derived force units have been defined in each system. In the metric systems, the derived force units (the newton in SI, the dyne in the CGS system) are defined to equal the natural units ... 

Generation rate of species j in a flow reactor, moles //time Gravitational acceleration, length/time ... 

London dispersion force (dimensionless) colloidal force, (mass length time-2) gravitational acceleration (length time-2) effective medium conductivity (length3) conductivity for a pore of radius r (length3) conductance distribution generic function (time-1) separation distance (length)... 

Note that each term of equation (1.6.18) has the units of length times acceleration (length /time ). The quantities are usually designated as heads, namely heads due to velocity, v )/2(v) due to elevation, gZ] due to pressure,... 

Huang W, Zhu J. An experimental investigation on solid acceleration length in the riser of a long circulating fluidized bed. Chin J Chem Eng 9 70-76, 2001. 

Yang WC, Keairns DL. Estimating the acceleration pressure drop and the particle acceleration length in vertical and horizontal pneumatic transport lines. Proceedings of the Pneumotransport 3, Bedford (England) BHRA Fluid Engineering, 1977, pp D7-89-D7-98. 

The remaining term in the total pressure drop relation is the acceleration term. This contribution is essentially the similar static and frictional losses experienced by the solids,but acting over the acceleration length where the particle is being accelerated. The acceleration length may be found by the solution of Eq. 4-5, noting that dL = Up dt, to give... 

The acceleration length of a vertical transport system using Eq.4-11 is presented in Example 4-2. 

Example 4-2 Consider the vertical transport of a Pittsburgh seam coal at a rate of 0.063 kg/sec in a 0.00107-m-ID pipe. The average particle size is 237 [xm and its density is 1282 kg/m. Determine the acceleration length for the coal to reach its steady-state value. The gas velocity is 10.6 m/sec. 

Substituting the appropriate values in the expression for acceleration length, the value of Z/accel is calculated to be 0.567 m. 

A study by Shimizu, Echigo, and Hasegawa (1978), showing an entry length for vertical flow of a gas-solid system to be monotonically increasing with a modified Reynolds number, is related to this acceleration length. These investigators define their modified Reynolds number as... 

For the conditions in Prob. 4-1, determine the acceleration lengths considering the two transport gas velocities. Assume the particle velocity is introduced into the system at 0.305 m/sec. 4-3 Develop a generalized numerical procedure for solution of the implicit Yang expressions in determining the pressure losses in vertical gas-solid transfer. 

As the particle traverses a bend its velocity changes. Haag (1967) and Kovacs (1967) have analyzed the behavior of particles as they slow down in bends. The overall pressure loss in a bend may be made up of this deceleration of the particle in the bend plus the acceleration length term. Consider the horizontal bend shown in Fig. 5-4. The angle 0 varies from 0 to 90° for a right-angle turn in the horizontal plane. A single particle momentum balance on this system can be written as... 

For the same conditions as stated in Prob. 54, analyze the acceleration length using Yang s analysis. Compare with the results of Prob. 54. 

The acceleration lengths for systems involving electrostatics can now be calculated by using the accelerated flow case. 

These values relate to steady flow conditions in pipelines remote from the point at which the material is fed into the pipeline, bends in the pipeline and other possible flow disturbances. At the point at which the material is fed into the pipeline, the material will essentially have zero velocity. The material will then be accelerated by the conveying air to its slip velocity value. This process will require a pipeline length of many metres and this distance is referred to as the acceleration length . The actual distance will depend once again on particle size, shape and density. The process was illustrated earlier in Figure 4.29 in relation to the pressure drop across a bend. 

The model which has been presented appears to give a much more realistic response in terms of acceleration length and its independence from particle size effects, than previous models which have been used for the purpose. 

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