Multiple rate vectors

Figure 4.12 (a) PFR trajectories can never intersect each other, and (b) this phenomenon would imply that multiple rate vectors exist at the intersection point. 

Although points found in the complement that satisfy Equation 8.9 are ones that are feasible, points that do not satisfy Equation 8.9 on their own may still be feasible when considered in combination with other points. That is, linear combination of multiple rate vectors that each does not satisfy Equation 8.9 may be combined to produce a resulting rate vector that does satisfy Equation 8.9. Systems that exhibit this behavior are rare, for one might already imagine the highly specific behavior required by the rate function to achieve this behavior. We should nonetheless be mindful of it in the discussions. 

For n components in the system, n CSTR equations may be written corresponding to each component. Difficulties in solution arise when nonlinear rate expressions are used, since then multiple solutions may exist for a fixed t and Cf. This is due to the fact that the rate vector is evaluated at the exit eoncentration C and not at the feed concentration Cf. 

If the value of the effluent concentration C is known, the associated rate vector r(C) may be evaluated. From liquation 4.9, it is clear that r(C) and v are scalar multiples of each other by t. Moreover, since t can only assume positive values, r(C) and v must both point in the same direction. Hence for any point C satisfying Equation 4.9, this point must exist as a CSTR solution for the feed point Cf. We therefore arrive at the following result for CSTRs Fora specified feed point and rate expression r( C), the... 

In Section 7.3.3.1, we showed the candidate AR for a DCR boundary. But a DCR might be viewed as a unique reactor type in itself, and combinations of DCRs with other reactor types could also be used to expand the region further. For example, a PFR could be operated from any point inside the candidate region of the DCR in Figure 7.26. Physically, this arrangement would represent a PFR in series with the DCR with bypass of feed. The analysis is made slightly more complex because now temperature is involved and thus each point in c-r space is associated with many rate vectors, since each point is associated with multiple temperatures. 

Natorally, the above multiplication of a stoichiometric matrix and a rate vector can be performed by suits of mathematical packages. Figure 2.1 shows the operation sequence of such automatic developing the mathematical model of the considered reaction in the Mathcad environment. 

Banked Memory. Another characteristic of many vector supercomputers is banked memory. The main memory is usually divided into a small number of electronically separate banks. A given memory bank can absorb or supply operands at a much slower rate than the rate at which the central processing unit (CPU) can produce or use data. If the data can be spread across multiple memory banks, the effective memory bandwidth, or rate at which memory can absorb or supply data, is increased. For example, if a single memory bank can supply one operand every 16 clock cycles, then 16 memory banks would enable the entire memory subsystem to deflver one operand per clock cycle, assuming that the data come sequentially from different memory banks. 

Fig. 16.4 Schematic representation of a pair of bond vectors. The relaxation rate of the multiple quantum coherence (MQ) is dependent on the angle 6 measured between the bond vectors. 

For simplicity we speak of a mechanism or a reaction, rather than a mechanism vector or reaction vector. The distinction lies in the fact that a reaction r (or mechanism) is essentially the same whether its rate of advancement is p or a, whereas pr and or are different vectors (for p a). Therefore, a reaction could properly be defined as a one-dimensional vector space which contains all the scalar multiples of a single reaction vector, but the mathematical development is simpler if a reaction is defined as a vector. This leaves open the question of when two reactions, or two mechanisms, are essentially different from a chemical viewpoint, which will be taken up... 

The maximum vector capability occurs for matrix multiplication, for which the measured time on the CRAY-1 is twenty times faster than the best hand coded routines on the CDC 7600 or IBM 360/195. The maximum rate is circa. 135 Mflops (Millions of floating-point operations per second) for matrices that have dimensions which are a multiple of 64, the vector register size. The rate of computation for matrix multiplication is shown in figure 1 as a function of matrix size. 

Figure 1. Matrix multiplication timing. Execution rate, in MFLOPS, is plotted as a function of vector length.

Finally, most doubly or triply subscripted array operations can execute as a single vector instruction on the ASC. To demonstrate the hardware capabilities of the ASC,the vector dot product matrix multiplication instruction, which utilizes one of the most powerful pieces of hardware on the ASC, is compared to similar code on an IBM 360/91 and the CDC 7600 and Cyber 174. Table IV lists the Fortran pattern, which is recognized by the ASC compiler and collapsed into a single vector dot product instruction, the basic instructions required and the hardware speeds obtained when executing the same matrix operations on all four machines. Since many vector instructions in a CP pipe produce one result every clock cycle (80 nanoseconds), ordinary vector multiplications and additions (together) execute at the rate of 24 million floating point operations per second (MFLOPS). For the vector dot product instruction however, each output value produced represents a multiplication and an addition. Thus, vector dot product on the ASC attains a speed of 48 million floating point operations per second. 

For multiple reactions, Eq. (2.1-3) is solvable for the full vector T tot if and only if the matrix n has full rank NR, i.e., if and only if the rows of v are linearly independent. If any species production rates Ri,tot are not measured, the corresponding columns of u must be suppressed when testing for solvability of Eq. (2.1-3). Gaussian elimination (see Section A.4) is convenient for doing this test and for finding a full or partial solution for the reaction rates. 

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