Moments equations for

In applying the resulting state space model for control system design, the order of the state space model is important. This order is directly affected by the number of ordinary differential equations (moment equations) required to describe the population balance. From the structure of the moment equations, it follows that the dynamics of m.(t) is described by the moment equations for m (t) to m. t). Because the concentration balance contains c(t)=l-k m Vt), at I east the first four moments equations are required to close off the overall model. The final number of equations is determined by the moment m (t) in the equation for the nucleation rate (usually m (t)) and the highest moment to be controlled. 

By virtue of the fixed control volume framework we first derive a generalized moment equation for (V )m- The average quantity iP)m is defined by (2.55). To find the equation that (V )m satisfies, we multiply on both sides of the Boltzmann equation (2.185) by ip and integrate over all velocities c. Thus,... 

In this analysis the moment of a force with respect to an axis, namely, torque, is important. Although the linear momentum equation can be used to solve problems involving torques, it is generally more convenient to apply the angular moment equation for this purpose. By forming the moment of the linear momentum and the resultant force associated with each particle of fluid with respect to a point in an inertial coordinate system, a moment of moment equation that relates torque and angular momentum flow can be obtained. 

Use the moment equations for constant growth rate in a MSMPR crystallizer to calculate (a) the surface-volume mean size and (6) the mass average size, (c) Compare these values with the sizes where a maximum occurs in the corresponding distribution curves. 

Next, the moment equations for the CLD of dead polymer chains is derived. The zeroth moment of the CLD for dead chain is... 

Table 2.6 Summary of moment equations for homopolymerization in a batch reactor...

It seems clear that, if we were to proceed in this fashion, developing population balances and moment equations for copolymerization would be a procedure even more tedious (as hard as it may be to imagine ) than the one used for homopolymerization. But, do not despair. A few concepts will be introduced that will allow us to translate the homopolymerization equations to copolymerization equations with minimum effort. 

In [31], Oh and Orin extend the basic method of Orin and McGhee [33] to include simple closed-chain mechanisms with m chains of N links each. The dynamic equations of motion for each chain are combined with the net face and moment equations for the reference membo and the kinematic constraint equations at the chain tips to form a large system of linear algebraic equations. The system unknowns are the joint accelerations for all the chains, the constraint fcwces applied to the reference memba, and the spatial acceleration of the reference member, lb find the Joint accelerations, this system must be solved as a whole via standard elimination techniques. Although this approach is sbmghtforward, its computational complexity of 0(m N ) is high. 

It should be noted that for termination by recombination (Fig. 10.5A), inherently an error is introduced regarding its contribution in the classical moment equations. For the individual living polymer continuity equations, it can be derived that the contribution of termination by recombination is... 

One encounters similar constraints with aggregation processes to generate closed integral moment equations, i.e., a constant aggregation rate and at most a linear growth rate. In order to demonstrate the moment equations for this case, we recall the population balance equation (3.3.5) for the constant aggregation rate, a x, x ) = a, incorporate a linear growth term X(x, t) = kx, and take moments. The result is... 

The ensuing moment equation for the compactly supported moments is identical to that for the fip a,b) moments, except for the appearance of boundary terms. These boundary terms will take on the form ... 

In the following model development, we will use the polymerization kinetics mechanism described by Eqs. (1)-(14) to derive the population balances and moment equations for homopolymerization with a catalyst containing only one site type. Catalysts containing two or more site types are handled similarly by defining a set of equations with distinct polymerization kinetic constants for each different site type. 

A similar procedure leads to the moment equations for the chain length distribution of dead polymer molecules [Eqs. (80) and (81)). 

Tab. 9.3. Moment equations for radical polymerization with transfer to polymer and random scission by H-abstraction.

Next, we vill derive the higher moment equations. For N = 1 and M = Owe obtain the set of population balance equations for the pseudo-distributions of order (1,0) as listed in Table 9.13. 

The points and weights of the Gaussian quadrature solve the moment equations for A = 0,1,..., 2 1 exactly as the pseudospqctrum of order n and the following... 

From the theory of structural frames, the basic moment equations for a rectangular firame under internal pressure loading P when the two pain of opposite sides have equal thickness and equal length, as shown in Fig. 17.9, are as follows ... 

This equation is a generalization of the macroscopic moment equation for dilute gas to dense gas of identical, rigid spherical molecules. 

With more complicated molecules, additional isotopic data are needed. For a linear molecule such as XYZ, the moments of inertia for two molecular species and the expression from Table n give two equations to be solved for the two bond lengths. Alternately, the coordinates of, for example, the X-atom zx, that is, the distance from the center of mass, can be evaluated from Kraitchman s equation (see Section VIII.D) using isotopic data from X YZ. Subsequently, this coordinate can be used in the moment-of-inertia and first-moment equations for the XYZ species. 

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