I often refer to dimensional analysis as "lost art" - because it is usually not heavily emphasized in engineering education today. However, for well over 100 years its has provided simply a wealth of practical design correlations that are still relied upon in virtually all aspects of chemical engineering, ranging from classes of problems dealing with heat and mass transfer, reaction kinetics, momentum exchanges in flow dynamics. Much of sedimentation theory, and indeed the basis for more sophisticated analyses beyond this book, are based on relating dimensionless groups that have been correlated with experimental observation. The following discussion will walk you through the approach of dimensional analysis as applied to the general theory of sedimentation, providing us with some expressions that will give us more than a head start in analyzing specific separation problems in water treatment.
To accomplish this, let's take a few steps back and start by examining the forces acting on a single particle settling through a continuous fluid medium. These forces are gravity, G, buoyant or Archimedes forces, A, centrifugal field, C, and an electrical field, Q. The system diagram for defining this is shown in Figure 13. Geometrically summing all the forces, we obtain:
If force P is greater than zero, the particle will be in motion relative to the continuous phase at a certain velocity, w. At the beginning of the particle's motion, a resistance force develops in the continuous phase, R, directed at the opposite side of the particle motion. At low particle velocity (relative to the continuous phase), fluid layers running against the particle are moved apart smoothly in front of it and then come together smoothly behind the particle (Figure 14). The fluid layer does not intermix (a system analogous to laminar fluid flow in smoothly bent pipes). The particles of fluid nearest the solid surface will take the same time to pass the body as those at some distance away.
The eddies are entrained by the flow and, at a certain distance from the particle, they disappear while being replaced by new eddies. Due to eddy formation and their breaking away from the particle, a low-pressure zone forms at the front of the particle. Hence, as described earlier, a pressure gradient is formed between the front and rear of the particle. This gradient is responsible primarily for the resistance to particle motion in the medium. The amount of this resistance depends on the energy expended toward eddy formation: the more intensive this formation, the greater the energy consumption and, hence, the greater the resistance force. The inertia forces generated by eddies play an important role. They are characterized by the mass and velocity of the fluid relative to the particle.
The total resistance is the sum of friction and eddy resistances. Both factors act simultaneously, but their contribution in the total resistance depends on the conditions of the flow in the vicinity of the particle. Hence, for the most general case the resistance force is a function of velocity, w, density, p, viscosity, p, the linear size of a particle, t, and its shape, ijr. Thus,
Assuming this relationship as an exponential complex, we obtain
where A' is dimensionless coefficient that includes the shape factor, ijr Noting the dimensions of all parameters appearing in this expression:
where L, M and T are the principal unit measures-length, mass and time. Expressing this in terms of its dimensions:
For the dimensions on the LHS of this expression to satisfy the RHS, the exponents on the principal units of measure must be equal. Thus, we have the following system of three equations (corresponding to the number of values with independent dimensions):
Because this system of equations cannot be solved (there are fewer equations than variables), we express all exponents in terms of a:
We may now write:
Equation 46 is a general expression that may be applied to the treatment of experimental data to evaluate exponent a. This, however, is a cumbersome approach that can be avoided by rewriting the equation in dimensionless form. Equation 42 shows that there are n = 5 dimensional values, and the number of values with independent measures is m = 3 (m, kg, sec.). Hence, the number of dimensionless groups according to the K-theorem is tu = 5 - 3 = 2. As the particle moves through the fluid, one of the dimensionless complexes is obviously the Reynolds number: Re = w ip/fi. Thus, we may write:
As one of two possible dimensionless numbers is now known, the second one can be obtained by dividing both sides of the equation through by the remaining values:
The result is a modified Euler number. You can prove to yourself that the pressure drop over the particle can be obtained by accounting for the projected area of the particle through particle size, i, in the denominator. Thus, by application of dimensional analysis to the force balance expression, a relationship between the dimensionless complexes of the Euler and Reynolds numbers, we obtain:
Coefficient A' and exponent a must be evaluated experimentally. Experiments have shown that A' and a are themselves functions of the Reynolds number. Equation 47 shows that the resistance force increases with increasing velocity. If the force field (e.g., gravity) has the same potential at all points, a dynamic equilibrium between forces P and R develops shortly after the particle motion begins. As described earlier, at some distance from its start the particle falls at a constant velocity. If the acting force depends on the particle location in space, in a centrifugal field, for example, it will move with uniformly variable speed until it travels outside of the boundary of the field's action or runs into an obstacle such as a vessel wall. We now shall define the relationship between particle motion and the acting factors. Moving under the action of force P, with acceleration ag at infinitesimal distance 6C, the particle with mass m,, performs work m^ 6G. This work is spent on overcoming the resistance force and the displacement of fluid mass, m,., in a volume equal to the volume of the particle, V, at the same distance, but in opposite direction, and with the same acceleration, ag:
Dividing through by 6G and expressing the masses of the particle and medium in terms of their volumes and densities, we obtain
Consider again the simple motion of a sphere. In this case, the equivalent diameter of a sphere, deq, is equal to its geometric diameter, d. Equating the above expressions and replacing 5 by d (and denoting the Euler umber, Eu, by Y), we obtain an expression for the resistance force:
In sedimentation, the Eule number is often referred to as the resistance number. Multiplying and dividing the RHS of equation 52 by u/8, we obtain
where CD = (8/it)Y = drag coefficient
F = 7td2/4 = the cross-sectional area of the spherical particle
Equation 54 is Newton's resistance law. Substituting Equation 53 into the definition for CD, we obtain
The Reynolds number for a sphere is
Substituting the resistance force into equation 51 and expressing F and V in terms of d, the basic equation of sedimentation theory is obtained:
where pf = pp - pc is the effective density of the system. In separation calculations for heterogeneous systems, the important parameter is settling velocity:
Application of the above formulas is difficult because the drag coefficient, CD, is a function of velocity and particle geometry. A generalized calculation procedure for settling under the influence of gravity was developed many years ago. The method, however, also may be applied to settling due to the influence of any force field, provided the relationship between particle acceleration and the coordinates of field is defined. The procedure is based on expressing the basic equation of sedimentation (equation 58) in terms of a relationship of criteria. For this, both sides of equation 58 are 2 divided by v2 , and the RHS multiplied and divided by the acceleration due to gravity, g:
The LHS of this expression contains a dimensionless group known as the Galileo number, defined as Ga = gd3/v2.
Multiplying by a simplex composed of densities results in the Archimedes number:
And introducing the ratio of accelerations, K^ = ag/g, where Ks indicates the relative strength of acceleration, ag, with respect to the gravitational acceleration g. This is known as the separation number. The LHS of equation 60 contains a Reynolds number group raised to the second power and the drag coefficient. Hence, the equation may be written entirely in terms of dimensionless numbers:
The Archimedes number contains parameters that characterize the properties of the heterogeneous system and the criterion establishing the type of settling. The criterion of separation essentially establishes the separating capacity of a sedimentation machine. The product of these criteria is:
This product contains information on the properties of the suspension and characterizes the settling process as a whole. Substituting equation 62 into 63 gives
This expression represents the first form of the general dimensionless equation of sedimentation theory. As the desired value is the velocity of the particle, equation 64 is solved for the Reynolds number:
To determine the size of a particle having a velocity, w, in the gravitational field, both sides of equation 63 are multiplied by the complex Re/ArKs:
The dimensionless complex Re3/Ar is expressed simply as: C = Re3/Ar = (w3/gv)(pf/pc)
Denoting S2 = C/Ks, then equation 62 may be rewritten as
This is the second form of the dimensionless equation for sedimentation. The Reynolds number also may be calculated from this equation:
As with S[, the Reynolds number is the dependent variable and S2 is the determining one. For settling under the influence of gravity, we note that ag = g, Ks = 1 and, hence, Si = Ar and S2 = C- Therefore, the general dimensionless equations for sedimentation are applicable in any force field. They need be transformed only into the appropriate dimensionless groups describing the type of force field influencing the process. Again, for gravity settling, Ar = %CDRe2, and C = 4Re/3CD. The dimensionless numbers of sedimentation, S, and S2, as well as CD and T are all functions of Re. The parameter T must be determined experimentally. Equation 53 can be written as a straight line when expressed in terms of its logarithms:
Coefficient A' and exponent a can be evaluated readily from data on Re and T. The dimensionless groups are presented on a single plot in Figure 15. The plot of the function CD = f,(Re) is constructed from three separate sections. These sections of the curve correspond to the three regimes of flow. The laminar regime is expressed by a section of straight line having a slope (3 = 135° with respect to the x-axis. This section corresponds to the critical Reynolds number, Re'cr < 0.2. This means that the exponent a in equation 53 is equal to 1. At this a value, the continuous-phase density term, pc, in equation 46 vanishes.
Therefore, the inertia forces have an insignificant influence on the sedimentation process in this regime. Theoretically, their influence is equal to zero. In contrast, the forces of viscous friction are at a maximum. Evaluating the coefficient B in equation 55 for a = 1 results in a value of 24. Hence, we have derived the expression for the drag coefficient of a sphere, CD = 24/Re.
Figure IS. Dimensionless sedimentation theory plot,
The first critical values of the dimensionless sedimentation numbers, S, and S2, are obtained by substituting for the critical Reynolds number value, Re'cr = 0.2, into the above expressions:
Substituting the expression for CD, we again obtain the settling velocity of an isolated particle in larninar flow:
Changing kinematic viscosity, v, to dynamic viscosity, the velocity of particle sedimentation in the laminar regime is:
From equations 64 and 60 at S, < 3.6, we obtair the first critical value of the particle diameter:
In applying this equation it is possible to determine the maximum size particle in larninar flow, taking into account the given conditions of sedimentation (pc, pp, ft and ag). However, this equation does not determine what the flow regime is when d > d'cr.
The turbulent regime for CD is characterized by the section of line almost parallel to the x-axis (at the Re"cr > 500). In this case, the exponent a is equal to zero. Consequently, viscosity vanishes from equation 46. This indicates that the friction forces are negligible in comparison to inertia forces. Recall that the resistance coefficient is nearly constant at a value of 0.44. Substituting for the critical Reynolds number, Re'cr > 500, into equations 65 and 68, the second critical values of the sedimentation numbers are obtained:
S"lcr a 82,500
And substituting CD = 0.44 into equation 59, w = 1.75[d (pf/pc)ag]'
At S "icr > 82,500, we obtain the second critical value of particle size:
Those particles with sizes d > d"ct at a given set of conditions (v, pc, pp, and ag) will settle only in the turbulent flow regime. For particles with sizes d'cr < d, d"cr will settle only when the flow around the object is in the transitional regime. Recall that the transitional zone occurs in the Reynolds number range of 0.2 to 500. The sedimentation numbers corresponding to this zone are: 3.6 < S, < 82,500; and 0.0022 < S2 < 1,515.
The slope of the curve in the transitional zone changes from 135 to 180°. It shows that the exponent in changes as follows: 0 £ a £ 1. This means that the friction and inertia forces are commensurable in the process of sedimentation. Several empirical formulas have been proposed for estimating the resistance coefficient in the transition zone. One such correlation is
Introducing this to equation 59 produces:
When we consider many particles settling, the density of the fluid phase effectively becomes the bulk density of the slurry, i.e., the ratio of the total mass of fluid plus solids divided by the total volume. The viscosity of the slurry is considerably higher than that of the fluid alone because of the interference of boundary layers around interacting solid particles and the increase of form drag caused by particles. The viscosity of a slurry is often a function of the rate of shear of its previous history as it affects clustering of particles, and of the shape and roughness of the particles. Each of these factors contributes to a thicker boundary layer.
Experimental measurements of viscosity almost always are recommended when dealing with slurries and extrapolations should be made with caution. Most theoretically based expressions for liquid viscosity are not appropriate for practical calculations or require actual measurements to evaluate constants. For nonclustering particles, a reasonable correlation may be based on the ratio of the effective bulk viscosity, /tB, to the viscosity of the liquid. This ratio is expressed as a function of the volume fraction of liquid x' in the slurry for a reasonable range of compositions:
A correction factor, Rc, incorporating both viscosity and density effects can be developed for a given slurry, which provides a more convenient expression based on the following equation:
where vH is the terminal velocity in hindered settling.
Measurements of the effective viscosity as a function of composition may be fitted to equation 80 or presented in graphic form as in Figure 16. The correction factor, R,. also may be determined by accounting for the volume fraction, r|v, of particles through the Andress formula:
In summarizing sedimentation principles, we note that the particle settling velocity is the principal design parameter that establishes equipment sizes and allowable loadings for separating heterogeneous systems. However, design calculations are not straightforward because prediction of the settling velocity requires knowledge of the flow regime in the vicinity of the particles. Therefore, the following generalized design method is recommended.
From known values of d, pp, pc, fi and ag, compute the first sedimentation dimensionless number (equation 63). From the plot given in Figure 17, obtain the corresponding Reynolds number, Re, and evaluate the theoretical settling velocity. If the flow regime is laminar, the settling velocity may be calculated directly from equation 74 and the regime checked by computing the Reynolds number for the flow around an individual particle. After determining w, determine the appropriate shape factor, i|r (either from literature values or measurements) and correction factor, R,.. The design settling velocity then will be:
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