Tangency condition

Fig. 5b. Relation of T-curve to crack growth and strength (b) tangency condition for crack growth for a flaw of initial size t...

This corresponds to the tangency condition shown in Fig. 7.4-5. Note that both cleaning and processing times can be dependent on equipment size. 

For a reactor of given dimensions and a fixed value of T, that is a particular Qq line, there will be the family of Qq lines shown in Fig. 21. One of the Qq lines will be tangential to the heat loss line alternatively, one can think of a particular concentration which defines the Qg lines and varying either or the slope of the Qq line until the tangency condition is just fulfilled. This situation is shown in Fig. 21 and it corresponds to the condition where a steady reaction is Just possible any increase in the reactant concentration or the temperature of the surroundings (T ) or reduction in the rate of heat loss, will result in explosion. At the tangency point, the reactant temperature has a critical value Tc and also Qq = Qq, i.e. 

General tangency condition with inflow of autocatalyst... 

The tangency condition can only have real roots so long as (8 + 9K 1)/ 0 < 1. This provides limits on either the autocatalyst inflow concentration... 

Our general interest has been to find the conditions, in terms of the extra unfolding parameters p, q, r, etc., at which the qualitative nature of the stationary-state locus changes (e.g. the appearance or disappearance of a hysteresis loop or an isola). In some cases we have been able to make use of special techniques such as factorization or the tangency condition. Now we seek a more widely applicable approach. This will involve the stationary-state condition F = 0 and also a series of equations obtained by differentiation of this expression with respect to the variable x and the parameter tres. 

When the residence time is varied so that we approach an ignition or extinction point in the stationary-state locus, then the flow and reaction curves L and R become tangential. The condition for tangency is R = L and 8R/da = SL/da. Thus the difference between the slopes of R and L decreases to zero. From eqn (8.17) we see that the tangency condition also causes the value of the eigenvalue A to tend to zero. An alternative interpretation, in... 

To determine the response of the cubic autocatalytic system to perturbations in the vicinity of a turning point in the locus, we must return to eqn (8.6). The first two terms (not involving A a) again cancel exactly, because of the stationary-state condition. If we are also at an ignition or extinction point, the tangency condition in any of its forms discussed above ensures that the coefficient of the A a term is also zero. Thus the first non-zero term is that involving (Aa)2 ... 

The critical point can be determined by imposing the tangency conditions [17], i.e.,... 

Mathematical identification of the tangency condition is straightforward when equation (56) applies. Equality of generation and loss rates requires 3e = 9, and equality of the slopes requires 3e = 1. Hence 9—1 at the point marked criticality in Figure B.3, and 1/e is the critical value of 3. Solutions of equation (56) for various values of 3, shown schematically in Figure B.4, confirm that 3 = 1/e can properly be interpreted as the critical measure of the ratio of the generation rate to the loss rate, above which... 

FIGURE 3.14 Tangency conditions of both S and M vectors on a residue curve at a CPM pinch point, x. 

P and Q, of with SB represent potential stationary states is a stable intersection and Q an unstable intersection the tangency condition of 0tz with SB at point C represents the critical conditiOH and T the highest stable temperature attainable... 

FfgHK 10 Critkality as a tangency condition between integral curves (0,y) and a target curve TC derived from a Liapunov Junction or arbitrary inflection criterion curve (a) satisfies the criterion at all times and would be deemed stable, curve (b) represents the critical integral solution, and curve (c) violates the criterion for some period of time... 

Thus while q e M, q and p together satisfy a separate constraint relationship g (q)M p = 0. This is sometimes referred to as a co-tangency condition or the hidden constraint. The set of pairs (q,p) satisfying, simultaneously, the equations... 

Guard band A design technique for a color CRT intended to improve the purity performance of the device by making the Kghted area of the screen smaller than the theoretical tangency condition of the device geometry. 

Substitute Equation (25.6) into Equation (25.7) for each phase. This leads to the common tangency condition that the slope of the line must be the same at... 

Now consider the nearfield tangency condition and form the vertical derivative... 

If we evaluate Equation 5-112 at y = 0 and apply tangency conditions (see Equation 3-18), we obtain, instead of Equation 3-23, the equation... 

However, a similar classification of two-dimensional diffeomorphisms, or of three-dimensional fiows, is not that trivial. Let us illustrate this with an example. Consider a diffeomorphism T which has two saddle fixed points 0 and O2 with the characteristic roots )Ai) < 1 and i > 1 at (z = 1,2). Suppose that Wq and have a quadratic tangency along a heteroclinic orbit as shown in Fig. 8.3.1. The quadratic tangency condition implies that all similar diffeomorphisms form a surface of codimension-one in the space of all diffeomorphisms with a C -norm. 

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