Steady-state population balance equation

Given expressions for the crystallization kinetics and solubility of the system, the population balance (equation 2.4) can, in principle, be solved to predict the performance of both batch and of continuous crystallizers, at either steady- or unsteady-state... 

Nicmanis, N. and Hounslow, M.I., 1998. Finite-element methods for steady-state population balance equations. American Institution of Chemical Engineers Journal, 44(10), 2258-2272. 

The population balance simulator has been developed for three-dimensional porous media. It is based on the integrated experimental and theoretical studies of the Shell group (38,39,41,74,75). As described above, experiments have shown that dispersion mobility is dominated by droplet size and that droplet sizes in turn are sensitive to flow through porous media. Hence, the Shell model seeks to incorporate all mechanisms of formation, division, destruction, and transport of lamellae to obtain the steady-state distribution of droplet sizes for the dispersed phase when the various "forward and backward mechanisms become balanced. For incorporation in a reservoir simulator, the resulting equations are coupled to the flow equations found in a conventional simulator by means of the mobility in Darcy s Law. A simplified one-dimensional transient solution to the bubble population balance equations for capillary snap-off was presented and experimentally verified earlier. Patzek s chapter (Chapter 16) generalizes and extends this method to obtain the population balance averaged over the volume of mobile and stationary dispersions. The resulting equations are reduced by a series expansion to a simplified form for direct incorporation into reservoir simulators. 

The population balance equation is employed to describe the temporal and steady-state behavior of the droplet size distribution for physically equilibrated liquid-liquid dispersions undergoing breakage and/or coalescence. These analyses also permit evaluation of the various proposed coalescence and breakage functions described in Sections III,B and C. When the dispersion is spatially homogeneous it becomes convenient to describe particle interaction on a total number basis as opposed to number concentration. To be consistent with the notation employed by previous investigators, the number concentration is replaced as n i,t)d i = NA( i t)dXi, where N is the total number of particles per unit volume of the dispersion, and A(xj t) dXi is the fraction of drops in increment X, to X( + dxi- For spatially homogeneous dispersions such as in a well-mixed vessel, continuous flow of dispersions, no density changes, and isothermal conditions Eq. (102) becomes... 

Venneker et al [118] made an off-line simulation of the underlying flow and the local gas fractions and bubble size distributions for turbulent gas dispersions in a stirred vessel. The transport of bubbles throughout the vessel was estimated from a single-phase steady-state flow fleld, whereas literature kernels for coalescence and breakage were adopted to close the population balance equation predicting the gas fractions and bubble size distributions. 

Adoption of this approach to microbial process development cannot occur until methods exist for determining the influence of reactor design and operating parameters on single-cell metabolic control actions and reaction rates. If this Information is available, population balance equations and associated medium conservation equations provide the required bases for reactor analysis (1, ). For example, for a well-mixed, continuous-flow Isothermal mTcroblal reactor at steady-state, the population balance equation may be written ... 

As a test of the range of applicability of the kinetics determined in the steady-state measurements, the transient population balance equation has been solved, using the kinetics determined from steady state, to simulate the sequence of protein content frequency functions obtained in synchronous growth of this organism. The simulation results are in very good qualitative agreement with the experimental measurements of the corresponding quantities (28). 

When the crystallizer is well mixed and operated continuously at steady state, with no attrition and breakage of the crystals at a size-independent crystal growth rate, the simplest population balance equation is... 

In identifying the steady-state population balance equation for the number density function/ (x, c, t), we appeal to the general form (2.8.3) and drop the time derivative. Also we take note of the fact that drops which appear in the vessel either by entering with the feed or by breakage of larger droplets must necessarily be of age zero so that they are accounted for in the boundary condition at age zero. Thus, the population balance equation becomes... 

Our goal is to formulate the steady-state population balance equation. This equation could then be used to study the effect of mixing on the conversion in the reactor. Since the sink term is easily identified, from considerations in (hi) the population balance equation for the reactor is seen to be... 

The experiment of Kumar et al (2000) consists of continuously feeding the polydisperse suspension through a vertical column in the well-mixed state and allowing the relative motion of particles to exit at an outlet located at a suitable distance from the point of entry. The relative motion of particles will have established a steady state, spatially uniform distribution of particles with an exit number density that can be measured by a device such as a Coulter counter. The population density, / (z, v) in vertical coordinate z and particle size described by volume v, satisfies the population balance equation... 

Solution of the particle concentration profile in the particle concentration boundary layer from in the feed suspension liquid to the concentration on top of the cake (and equal to the concentration in the cake) requires consideration of the particle transport equation in the boundary layer. We will proceed as follows. We will first identify the basic governing differentied equations and appropriate boundary conditions (Davis and Sherwood, 1990) and then identify the required equations for an integral model and list the desired solutions from Romero and Davis (1988). However, we will first simplify the population balance equation (6.2.51c) for particles under conditions of steady state 8n rp)/dt = O), no birth and death processes (B = 0 = De), no particle growth (lf = 0) and no particle velocity due to external forces Up = 0), namely... 

The process inputs are defined as the heat input, the product flow rate and the fines flow rate. The steady state operating point is Pj =120 kW, Q =.215 1/s and Q =.8 1/s. The process outputs are defined as the thlrd moment m (t), the (mass based) mean crystal size L Q(tK relative volume of crystals vr (t) in the size range (r.-lO m. In determining the responses of the nonlinear model the method of lines is chosen to transform the partial differential equation in a set of (nonlinear) ordinary differential equations. The time responses are then obtained by using a standard numerical integration technique for sets of coupled ordinary differential equations. It was found that discretization of the population balance with 1001 grid points in the size range 0. to 5 10 m results in very accurate solutions of the crystallizer model. 

Population Balances. Three different models based on two approximations regarding the mode of breakage and two approximations regarding the size dependence of growth rate have been examined. The differential equations for modeling the size distribution are based on a population balance on aggregates of size L which, for a CSTR at steady state, mean residence time x, and with no particles in the feed, reduces to... 

The dispersed model after (16) with a modified solution technique was used to calculate the kinetic parameters from the experimental data in the dispersed zone. At steady state the population balance with no particle aggregation or breakage is given by the following differential equation... 

Analysis of the PSD Data. From the population balance for a CMSMPR crystallizer operated under steady-state condition, the population density n for size-independent crystal growth is given by Equation (1), where n , G, 6 and 1 are nuclei density, growth rate, residence time of reactants and particle size, respectively. 

A number of empirical size-dependent growth expressions have been developed. Of these, the ASL model given in Eq. (11.2-28) is the most commonly used. Substituting this equation into the differential population balance given by Eq. (11.2-31), the steady-state population density function can be derived as... 

A key feature of this system, and of most chemical systems that exhibit oscillations, is autocatalysis, which means that the rate of growth of a species, whether animal or chemical, increases with the population or concentration of that species. Even autocatalytic systems can reach a steady state in which the net rate of increase of all relevant species is zero— for example, the rate of reproduction of rabbits is exactly balanced by that species consumption by lynxes, and lynxes die at the same rate that baby lynxes are born. Mathematically, we find such a state by setting all the time derivatives equal to zero and solving the resulting algebraic equations for the populations. As we shall see later, a steady state is not necessarily stable-, that is, the small perturbations or fluctuations that always exist in a real system may grow, causing the system to evolve away from the steady state. 

Generally, the distribution of droplet sizes in flow can be obtained as a solution of the generalized Smoluchowski (balance population) equation describing the competition between the droplet breakup and coalescence. Various approximate approaches to the solution of the equation with various expressions for breakup and coalescence frequencies have been used in the hteratnre (101,105-115). For rather long mixing in batch mixers, achievement of a steady state in the droplet size distribution is assumed. For mixing in extruders, development of the droplet... 

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