Criticality equation

Since the general concept in CEC is to use packing materials with a beads size as small as possible, the viscosity of the liquid used for slurring the beads is critical. Equation (2) rearranged to... 

The Integrated Equation of State 11 yields very good representations of (p, P, T) data and provides a clear view of the evolution of the sub-critical equation of state to the power law for the critical isotherm. 

Combining these equations gives the critical equation ... 

The conclusion that one may draw from the real-energy treatment [29] is that, in the neighborhood of a resonance whose exacf energy is E we have a pair of critical equations, both of which exhibit the significance of (T o/Eo)-... 

It is useful to regard Equation (29) as a critical equation (the characteristic equation when p is zero) for a new reactor whose characteristic operators G P are related to those of the reactor in question by ... 

The multiplication constant and perturbation. The new eigenvalue quantity, here called the multiplication constant k, is often called the effective multiplication constant and is called the static criticality factor, (7, by Weinberg and Wigner. It is the largest real eigenvalue of the criticality equation... 

The critical equation arises from more than one eigenvalue problem. One is the natural time-dependent or dynamic problem of the form... 

In changing the neutron balance equation, we transform the adjoint equation. For example, if we have transformed the critical equation by means of an operator, B, say, we compare... 

To overcome the difficulty with the critical equation, write an equation of the form... 

The derivation of pseudo-critical values for multicomponent mixtures at moderate and high temperatures has been discussed in an earlier article [7], This same technique can be extended to apply to light gases at low temperatures when quantum effects are important [17, 18]. The form of the pseudo-critical equations under these conditions can be deduced from the t5T>e of equations which result when potential functions conforming to (8) are substituted into (6) and (7). 

In the case of some measurements, it is possible to interpret the data in several different ways. For instance, measurements of migration area may be interpreted by age-diffusion, age, one group, or multigroup theory. Numerical values of M as computed by the different methods will vary coMiderably, although each will form a con-slstent set of reactor parameters in its own critical equation, hi addition, there appears to be agreement in values of M as measured by subcritical and by critical. methods. Thus, it is necessary to make any comparison between critical and subcritical experiments with a common theoretical interpretation. 

It is of considerable interest to determine to what extent rather simple results may be used for future analyses. The criticality equation... 

Using a one-group critical equation (with constant reflector savings assumed) migration areas of 44.6 cm in radial (M ) and 46.0 cm in axial direction (h were determined. Theoretical value for migration area for this lattice is 42.29 cm using Mercury code in which Deutsch s Method is used. ... 

The critical equation for an aqueous homogeneous / thermal reactor may be written as... 

Essentially, the thermodynamic modeling consists in a set of equations that relate process parameters through mass and heat balances and liquid-vapor equilibrium equations. Below are the critical equations ... 

The critical equation for the bare homogeneous reactor with mono-energetic fission neutrons may be developed from the solutions (6.72). By the usual definition of criticality, we require that the number of neutrons introduced into the system be equal to the number produced by fissions in the subsequent generation. Thus the source function (6.55) may be used to establish the criticality condition. The lethargy integral of (6.55) yields... 

We return now to the development of the critical equation which was previously recognized as the relation (6.94). This expression may be expanded by introducing the solution for p u) from (6.100), using (6.61), and from (6.108). It is easily shown that the critical equation is... 

The critical-mass calculation for any one of these reactors entails a trial-and-error procedure. One can either assume a size and compute the critical concentration, or vice versa. We will do the latter. The next step then is to evaluate the macroscopic cross sections of the reactor medium for the entire lethargy range, i.e., Sa(i ), /( )i ( )> well as the thermal-group cross sections at the operating temperature of the system. These data may now be applied directly to the computation of the various parameters which appear in the critical equation, i.e., nh, Pth, 17) /) and L. In the first trial calculation it is convenient to assume a value for the fast effect (perhaps c = 1) and check it later. All the necessary data having been collected, B may be computed from the appropri-... 

The substitution of these results into the criticality equation (7.324) gives the half-width a of the infinite-slab reactor which will be critical with the composition (7.325). This solution is found to be... 

If we apply instead the no-return current condition (7.264), the criticality equation is given by (7.265). For B/Xt = 0.3781, we obtain Ba = 1.343. The corresponding solution in mean free paths is... 

A convenient measure of merit of a reflector is the so-called reflector savings. The reflector savings gives the decrease in the critical core size of a bare reactor which could be realized by the addition of a particular reflector. To illustrate this idea consider, for example, an infinite-slab reactor of half-width a. In the one-velocity approximation, the criticality equation is given by... 

If we assume that Dc Dr, then the application of the first term in the above expression to the criticality equation (8.11) yields... 

An accurate calculation for the reflector savings based on the solution of the criticality equation for the one-velocity model will yield a curve of the shape shown in Fig. 8.3. The approximations (8.20) and (8.22) apply for the extreme values of reflector thickness. 

Some of the functions on the left side of (8.120) have been tabulated. It will be convenient, therefore, to rewrite the criticality equation in t rms of the tabulated functions (see also Fig. 8.14) ... 

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