Population balance macroscopic

For output-input due to growth or other internal variables , we must add 

For the net generation terms we have birth, B, minus death, D. Combining all these terms we have the microscopic population balance  

This differential equation is the fundamental population balance. This equation together with mass and energy balances for a system form a dynstmic multidimensional accounting of a process where there is a change in the particle size distribution. This equation is completely general and is used when the particles are distributed along both external and internal coordinate space. External coordinate space is simply the position x, y, and z in Cartesian coordinates. Internal coordinates Xj are, for example, the shape, chemical composition, and the size of the particles. More convenient and more restrictive forms of the population balance will be subsequently developed. 

A population balance over a macroscopic region has many engineering applications. For this type of balance the general population balance developed in equation (3.7) can be simplified. Into a macroscopic volume, we can have an arbitrary number of inputs and outputs at flow rates . In addition, if we assume that the suspension is well mixed 

The terms involving the velocity of the moving surfaces are essentially equal to the sum of the input-output streams multiplied by the population contained in them  

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Ca ], can be balanced using the general macroscopic population balance equation as follows... 

This is the macroscopic population balance, which is a more useful form of population balance for describing transient and steady state particle size distributions in well-mixed vessels. This population balance in conjunction with mass and energy balances gives a complete description of particulate processes in well-mixed vessels. 

Simplify the macroscopic population balance to describe the particle size distribution in a continuous constant volume isothermal well-mixed crystallizer with mixed product removal operating at steady state. Assume the crystallizer feed streams are free of suspended particles, that the crystallizer operates with ne igible breakage, and that agglomeration and crystallization cause no change in the volume of the system. 

Macroscopic Population Balance on a Discrete Mass Basis... 

Several classification functions, C(L), are given in Figure 4.20. Here, the fraction of particles by mass reporting to the recycle stream is given as a function of particle size, L, for a screen and a cyclone. Several authors have used empirical classification functions instead of dassifier performance curves with reasonable results for the overall comminution-classification circuit control. The steady state (i.e., dmidt = 0) macroscopic population balance on a discrete mass basis over the grind-... 

The role a lomeration plays in a CSTR is explained by considering the macroscopic population balance at a steady state ... 

In this book the macroscopic population balance equation formulation is presented following the original notation and nomenclature of Luo [73] and Luo and Svendsen [74]. 

In this section the macroscopic population balance formulation of Prince and Blanch [92], Luo [73] and Luo and Svendsen [74] is outlined. In the work of Luo [73] no growth terms were considered, the balance equation thus contains a transient term, a convection term and four source terms due to binary bubble coalescence and breakage. 

This relation expresses that not all collisions lead to coalescence. The modeling of the coalescence processes thus means to find adequate physical expressions for hc d d, Y) and pc d d, Y). Kamp et al [39], among others, suggested that microscopic closures can be formulated in line with the macroscopic population balance approach, thus we may define ... 

For bubbly flows most of the early papers either adopted a macroscopic population balance approach with an inherent discrete discretization scheme as described earlier, or rather semi-empirical transport equations for the contact area and/or the particle diameter. Actually, very few consistent source term closures exist for the microscopic population balance formulation. The existing models are usually solved using discrete semi-integral techniques, as will be outlined in the next sub-section. 

In the case of droplets of volume so small that their content can be regarded as uniform, the process can be modeled by applying to each droplet a macroscopic population balance (72) in the form... 

Using the population balance approach is most readily seen by considering several examples. One could rederive all of the age distribution formalism by choosing Ci = (age) in the macroscopic population balance Eq. 12.6-2 ... 

All of the age distribution formalism could be derived by starting from the single-particle joint-PDF transport equation (12.4.1-11) or the macroscopic population balance equation (12.6.1-3). Indeed, the only property of interest is the age, so that = a, the age. Now intervals of age are also intervals of clock time, so that Q = daldt= 1 and (12.6.1-3) reduces to ... 

Equation (69) gives the macroscopic population balance for a CSTR, where the left-hand side accounts for the accumulation of particles in the reactor, the first term on the right-hand side accounts for the entry of particles into the reactor, the second for the exit of particles from the reactor, the third for the formation and loss of particles of unswollen volume v due to particle growth, the fourth for the loss of particles by coagulation with other particles and the fifth term accounts for the formation of particles of unswollen volume v by particle coagulation. In Eq. (69) n(v) and n-i (v) are the reactor and inlet number density of polymer particles, Qs [m s ]... 

Consider a continuous crystallizer of volume V, as shown in Figure 6.4.2(a). A feed stream having a particle (crystal) number density function ra/(rp) (which is also the population density function), volumetric flow rate Qf and species i mass concentration enters the crystallizer continuously. Product stream 1, having a particle (crystal) number density function n (tp), volumetric flow rate Qi and species i mass concentration Pf, leaves the crystallizer continuously. The particle (crystal) number density function n rp) in the well-mixed crystallizer is the same throughout the crystallizer. The macroscopic population balance equation for a stirred tank separator may be written using equations (6.2.60) and (6.2.61) as follows ... 

There is an additional phenomenon, namely random fluctuations in growth rates of the different crystals around a mean value. For example, one crystal will display several growth rates during its growth in the crystallizer for the time period fres however, there may be a mean, as shown in Figures 6.4.5(e) and (f). The governing macroscopic population balance equation for an MSIVIPR crystallizer, where /Zp) = 0, B = He = 0, may be written as... 

Sometimes it is convenient to define a macroscopic population balance by averaging over the physical space and over file inlet and outlet streams. Let us define... 

In the work of Fleischer et al. [24] and Hagestether et al. [30] one dimensional population balance formulations and closure laws very similar to those of Luo and Svendsen [80] were employed. In the bubbly flow simulations by Hagestether et al. [31, 32] and Wang et al. [133, 134] the macroscopic population balance formulation of Luo and Svendsen [80] was still adopted (no fluid dynamic calculations were performed) but with extended versions of the kernel functions. 

In a first modeling approach, a macroscopic population balance is formulated directly on the averaging scales in terms of number density functions [80, 102], A corresponding set of macroscopic source term closures are presented as well. Reviews of numerous fluid particle breakage and coalescence kernels on macroseopie scales can be found elsewhere [60,73,74, 122], This modeling framework resembles the mixture model concept. 

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