Growth-Differentiation Balance Model

Similar to the CNBM, the GDBM has been regarded as a supply-side model, as it emphasizes that the levels of defenses are indirectly determined by the supply of secondary metabolites, rather than actively regulated in order to fulfill defensive demands. The GDBM is based on the premise that there is a physiological trade-off 

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Nevertheless, as discussed previously, the physical model for a crystallizer is an integro-partial differential equation. A common method for converting the population balance model to a state-space representation is the method of moments however, since the moment equations close only for a MSMPR crystallizer with growth rate no more than linearly dependent on size, the usefulness of this method is limited. The method of lines has also been used to cast the population balance in state-space form (Tsuruoka and Randolph 1987), and as mentioned in Section 9.4.1, the blackbox model used by de Wolf et al. (1989) has a state-space structure. 

An alternative scheme, proposed by Garside et al. (16,17), uses the dynamic desupersaturation data from a batch crystallization experiment. After formulating a solute mass balance, where mass deposition due to nucleation was negligible, expressions are derived to calculate g and kg in Equation 3 explicitly. Estimates of the first and second derivatives of the transient desupersaturation curve at time zero are required. The disadvantages of this scheme are that numerical differentiation of experimental data is quite inaccurate due to measurement noise, the nucleation parameters are not estimated, and the analysis is invalid if nucleation rates are significant. Other drawbacks of both methods are that they are limited to specific model formulations, i.e., growth and nucleation rate forms and crystallizer configurations. 

For effective control of crystallizers, multivariable controllers are required. In order to design such controllers, a model in state space representation is required. Therefore the population balance has to be transformed into a set of ordinary differential equations. Two transformation methods were reported in the literature. However, the first method is limited to MSNPR crystallizers with simple size dependent growth rate kinetics whereas the other method results in very high orders of the state space model which causes problems in the control system design. Therefore system identification, which can also be applied directly on experimental data without the intermediate step of calculating the kinetic parameters, is proposed. 

In this paper, three methods to transform the population balance into a set of ordinary differential equations will be discussed. Two of these methods were reported earlier in the crystallizer literature. However, these methods have limitations in their applicabilty to crystallizers with fines removal, product classification and size-dependent crystal growth, limitations in the choice of the elements of the process output vector y, t) that is used by the controller or result in high orders of the state space model which causes severe problems in the control system design. Therefore another approach is suggested. This approach is demonstrated and compared with the other methods in an example. 

This results In a set of first-order ordinary differential equations for the dynamics of the moments. However, the population balance Is still required In the model to determine the three Integrals and no state space representation can be formed. Only for simple MSMPR (Mixed Suspension Mixed Product Removal) crystallizers with simple crystal growth behaviour, the population balance Is redundant In the model. For MSMPR crystallizers, Q =0 and hp L)=l, thus ... 

The model for this crystallizer configuration has been shown to consist of the well known population balance (4), coupled with an ordinary differential equation, the concentration balance, and a set of algebraic equations for the vapour flow rate, the growth and nucleatlon kinetics (4). The population balance is a first-order hyperbolic partial differential equation ... 

Ramkrishna et al.m proposed a similar model at about the same time—this too was an unsegregated model which also divided the biomass into two compartments. They referred to the material in the two compartments as G-mass and D-mass, respectively, and suggested that these materials were formed in parallel. They also proposed that the micro-organism produced a toxic substance which inhibited its growth. They produced a set of differential equations obtained from material-balance considerations, to describe the behaviour of such a system in both batch and continuous culture. For batch culture ... 

The last term in equation 5.245 represents the dilution of active component /, by the expansion of the biomass. Esener et al.m also present a two-compartment model which takes this effect into account and they emphasise the need to devise the theory so that it can be tested by experiment. In their model they identify a K compartment of the biomass which comprised the RNA and other small cellular molecules. The other compartment contained the larger genetic material, enzymes, and structural material. The model assumes that the substrate is absorbed by the cell to produce, in the first instance, K material, and thence it is transformed into G material. Additionally, the G material can be reconverted to K material, a feature intended to account for the maintenance requirement of the micro-organism. A series of material balances for the cellular components during growth in a CSTF produced the following differential equations ... 

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... 

Clearly, no simple analytical solution of the differential equation derived from the biomass balance can be obtained. Adopting numerical methods based on this differential equation the increase in biomass in time in a batch culture can stiU be calculated. In Fig. 28 batch growth is simulated based on the model ofjassby and Platt including spectral resolution. This simulation is based on C. sorokiniana grown under a constant photon flux density of 1.5 X 10 molph m s. The biomass concentration increases in time, but the rate of increase slows down. This is more apparent in the yield of biomass on photons which is maximal after about 24 h after which it decreases again. At 24 h the biomass concentration is optimal in the sense that at this point in time the volumetric productivity, as well as the biomass yield on light, is maximal. 

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... 

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... 

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