Stream vector

A typical module showing the input stream vectors xijy output stream vectors yik, specified equipment parameter vector pf, unspecified equipment parameter vector qt, and the retention (dependent) variable vector r-. 

Figure 5,10 FLOWTRAN Subroutine for a Pump Block. The dummy arguments of the subroutine are (1) F and B, the feed and effluent stream vectors containing stream flow information (2) the equipment paramenter vector, R, which contains the outlet pressure and (3) the retention vector, R, which contains the flowrate, pressure increase, fluid Hp, pump efficiency, brake Hp, motor efficiency, and electrical power in kilowatts, all of which are to be printed out. From Seader, I. D., W. D. Seider, and A. C. Pauls, Flowtran Simulation An Introduction 3rd ed., CACHE, Austin, TX (1987).

Thus, defining the stream vector 5 (number of particles per square centimetre per second) by the components... 

During execution of a unit subroutine, the stream vectors and equipment parameters are accessed, from a so-called B vector in ASPEN PLUS, and changes are recorded when new... 

In the schematic of the FLASH2 subroutine in Figure 4.7, on the second line, the B vector, which contains the stream vectors and equipment parameters for all of the streams and simulation units, is referred to in the B common storage. When simulating the flash vessel, FI, the stream variables are taken from the FEED vector and two of the equipment parameters, TEMP, PRES, VFRAC, and DUTY, are taken from the subroutine inputs. As the flash equations are solved, the stream variables are stored in the VAP and LIQ vectors and the heat duty is stored as a parameter, QCALC. 

There are only one- and two-dimensional data streams (vectors and matrices) of complex data types (large integer, fix). Typical vector sizes are 10-500 elements. Arrays with two dimensions have less than 10 entries in... 

Vector (length 20) of stream composition (I = 1,N). Contribution from temperature dependence of UNIQUAC binary interaction parameters, here taken as 0. 

The matrix of measurements is rearranged into a stacked vector where each subsequent set of stream measurements follows the one above. As an example, the component flows in the Xf matrix are placed in the vector of measurements as follows ... 

To use turboexpanders for condensing streams, the rotor blades must be shaped so that their walls are parallel at every point to the vector resultant of the forces acting on suspended fog droplets (or dust particles). The suspended fog particles are thus unable to drift toward the walls. Walls would otherwise present a point of collection, interfering with performance and eroding the blades. Hundreds of turboexpanders are in successful operation involving condensing liquids. 

The directing jet is supplied at a right angle to the mam stream axis with an initial velocity of I os from the nozzle with an inner diameter (d, -) located at the distance (Iq) from the plane of main stream supply and at the distance Yq from its geometrical axis. The momentum vector component along the Y axis remains constant and equal to the initial momentum (Fig. 7.57) ... 

The state variables are the minimal set of dependent variables that are needed in order to describe fully the state of the system. The output vector represents normally a subset of the state variables or combinations of them that are measured. For example, if we consider the dynamics of a distillation column, in order to describe the condition of the column at any point in time we need to know the prevailing temperature and concentrations at each tray (the state variables). On the other hand, typically very few variables are measured, e.g., the concentration at the top and bottom of the column, the temperature in a few trays and in some occasions the concentrations at a particular tray where a side stream is taken. In other words, for this case the observation matrix C will have zeros everywhere except in very few locations where there will be 1 s indicating which state variables are being measured. 

A novel approach [98], proposed for generating starting configurations of amorphous dense polymeric systems, departs from a continuous vector field and its stream lines. The stream lines of continuous vector fields never intersect. If the backbones of linear polymer chains can be associated with such stream lines, the property of the stream lines partly alleviates the problem of excluded volume, which - due to high density and connectivity - constitutes the major barrier to an efficient packing method of dense polymeric systems. This intrinsic repulsive contact can be compared to an athermal hard-core potential. Considering stream lines immensely simplifies the problem. 

The stream lines of a plain vector field, however, do not in general have a polymer-like behavior. The problem, then, consists of generating a vector field whose stream lines have the mesoscopic properties of real polymers. 

In order to investigate a continuous vector field, one first needs to parameterize this continuum of vectors. Since one is interested in the stream lines, only unit vectors indicating the direction of propagation of the stream lines are relevant. Thus, it is sufficient to employ a set of spatially varying polar... 

The stream lines of a vector field v(x) are those trajectories where the vector v(x) is tangential to the path. In analogy to trajectories of atoms subject to the influence of a Hamiltonian, the stream lines obey an equation of motion of first order given by... 

The vector field entirely and uniquely determines the stream lines and their properties. As we focus our attention on the mesoscopic properties of stream lines, assuming that they can resemble a polymer-like amorphous packing of chain backbones, we have to consider in greater detail their intrinsic properties. As shown in the next section, Santos and Suter [98] elaborated a model for generating packing structures of Porod-Kratky chains. 

Note that the curvature is independent of the kind of parameterization s of the curve, which indicates that, in the case of a stream line, the curvature can be made to depend on the location of this point in the vector field and not on the parametrization along the stream line itself. By virtue of Eqs. (3.3), (3.4), and (3.5) and some manipulations, the curvature of a stream line at a location x can be expressed as ... 

In the case of the Porod-Kratky model, the polymer backbones have a constant curvature c0. Accounting for the polymer stiffness in generating the dense configuration of stream lines, the vector field used must have a homogeneous curvature field with a unique value cq in the entire simulation box T. In order to quantify the success in creating such a vector field, the deviation of the curvature from the ideal Porod-Kratky case, a volume integral has been used by Santos as a penalty function ... 

So far, Santos has been able to express the relation between a set of coefficients af, aj J 6 / describing a vector field and the overall curvature of the stream lines of this vector field. Based on the curvature field, they constructed the measure E of the curvature distribution in the simulation box. Provided that the homogeneous curvature field of curvature c0 is the one that minimizes E, the problem of packing has been recast as a minimization problem. However, the lack of information about the gradient of the error function to be minimized does not facilitate the search. Fortunately, appropriate computer simulation schemes for similar minimization problems have been proposed in the literature [105-109]. 

Every minimization departs from an initial estimation for the vector field. The minimizations were carried out with a starting configuration obtained by randomizing the coefficients aj and a the resulting vector field has no preferential orientation and the distribution of curvature in the simulation box exhibits a long tail mainly due to abrupt changes in the direction of the stream lines (see Fig. 3.2A). 

The final vector fields have a curvature field, the stream lines of which exhibit a fluctuating curvature around the goal curvature c0. Santos and Suter reported that they had not been able to reduce the width of the curvature distribution below the limit of 0.2c0. This lower limit is reached at the end of the cooling process for each of the goal curvatures as shown in Fig. 3.3. The final vector fields obtained have a curvature field, the stream lines of which are characterized by a fluctuating curvature around the target c0. 

Fig. 3.4. Four stream lines and their projection onto the xy, yz and xz planes, respectively, constructed by integration for a vector field of curvature c0 = 10 in a box of unit edge length...

Fig. 3.5. Curvature along the four stream lines depicted in Fig. 3.4 with target curvature c0 = 10 for the underlying vector field...

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