Stream lines

When this equation holds, each stream match (exchanger) must provide that one of the two streams involved reaches its target temperature. Such a network is called acycHc. In an acycHc network, it is not possible to trace a closed path along stream lines from exchanger to exchanger and return to the starting point without retracing some of the path. 

The resulting eontours of the stream lines, a elose-up of a veetor plot near the top eover plate and the profile of radial veloeity at the inner edge of the eatalyst bed (Profile C, Figure 10-15) all show us that these last-named proposed ehanges would have many benefits. The main ones are ... 

The tensile stresses acting in the direction of converging stream lines can ellipsoidally deform the big particles, but not so much as to form fine fibrils from small particles (region B). The matrix are also elongated in the converging section. As they pass the die exit (region C), recoil of the matrix occurs to release the stored energy... 

The simplified flow diagram of continuous filtration with stream lines of material balance is shown in Figure 9.3. Mass in is equal to mass out. 

Figure 2.40 Zigzag micro mixer with concentration field (left) and flow stream lines (right) obtained from a CFD simulation for a Reynolds number of 38. In [135] a sawtooth geometry of larger amplitude was considered and distinctive recirculation zones were found only at Reynolds numbers larger than 80.

Dean, W. R., The stream-line motion of fluid in a curved pipe, Philos. Mag. 5 (1928) 673-695. 

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. 

The most austere representation of a polymer backbone considers continuous space curves with a persistence in their tangent direction. The Porod-Kratky model [99,100] for a chain molecule incorporates the concept of constant curvature c0 everywhere on the chain skeleton c0 being dependent on the chemical structure of the polymer. It is frequently referred to as the wormlike chain, and detailed studies of this model have already appeared in the literature [101-103], In his model, Santos accounts for the polymer-like behavior of stream lines by enforcing this property of constant curvature. 

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 ... 

Equation (3.6) defines a scalar field, the value of which at any point in the simulation box is the curvature of the stream line passing through this point and can also be reformulated to avoid the use of the parameter s ... 

This curvature field is always positive and has no singularity for continuous differentiable fields and [Pg.62]

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...

Fig. 3.6. a End-to-end distance of the four stream lines depicted in Fig. 3.4 as a function of the contour length of the stream lines, b Ratio of the end-to-end distance to the square of the radius of gyration as a function of the contour length of the stream lines. The Porod-Kratky value of 6 is achieved only for a contour length of the same order as the box edge length... 

Santos also scrutinized the polymer-like behavior of the constructed stream lines. Two characteristics were analyzed the second moment of the end-to-end distance distribution, (r2)0, which characterizes the spatial configurations of chain molecules, and the radius of gyration, Rg, that indicates how the... 

We have not mentioned here the crucial inverse mapping of the realistic polymer structure onto the stream line. For polymers without side groups, such as polyethylene or bisphenol-A-polycarbonate, the following strategy has successfully been used [98] an energy minimization of the internal energy contributions was carried out simultaneously with a minimization of the distances of all atoms to the stream line (to this end, the sum of the squared dis-... 

Fig. 3.7. A united-atom PE chain constituted of 100 united atoms mapped around a stream line. The bond angles as well as the bond lengths have been kept fixed. The torsion angles have been subject to a minimization considering a three-fold potential accounting for the simplified chemical structure of the chain. An additional proximity function has been used to force the chain to follow the trajectory of the stream line [114]...

Stream-line antimicrobial therapy based on clinical judgment, patient response, and microbiological data... 

It is convenient to represent a heat exchanger network as a grid see Figure 3.24. The process streams are drawn as horizontal lines, with the stream numbers shown in square boxes. Hot streams are drawn at the top of the grid, and flow from left to right. The cold streams are drawn at the bottom, and flow from right to left. The stream heat capacities CP are shown in a column at the end of the stream lines. 

The stream flow-rates and compositions can be shown on the diagram adjacent to the stream lines, when only a small amount of information is to be shown, or tabulated separately. 

The data on the flow-rate of each individual component, on the total stream flow-rate, and the percentage composition, can be shown on the flow-sheet in various ways. The simplest method, suitable for simple processes with few equipment pieces, is to tabulate the data in blocks alongside the process stream lines, as shown in Figure 4.1. Only a limited amount of information can be shown in this way, and it is difficult to make neat alterations or to add additional data. 

A better method for the presentation of data on flow-sheets is shown in Figure 4.2. In this method each stream line is numbered and the data tabulated at the bottom of the sheet. Alterations and additions can be easily made. This is the method generally used by professional design offices. A typical commercial flow-sheet is shown in Figure 4.3. Guide rules for the layout of this type of flow-sheet presentation are given in Section 4.2.5. 

The stream line numbers should follow consecutively from left to right of the layout, as far as is practicable so that when reading the flow-sheet it is easy to locate a particular line and the associated column containing the data. 

The new organization, photographs, and chapter headings will stream-line the connection with a range of general chemistry textbooks and courses. For the general reader, however, our intention is still to demonstrate the wide scope and significance of chemistry and the ever-present connection of the discipline to our daily lives. 

In a laminar flow reactor the concentration varies both radially and axially and depends on the specific rate. Along a stream line, the material balance is... 

What is of interest is the mean value over the cross section. In problem P4.08.01 it is shown that the. mean value is related to the stream line values by... 

---