Continuous Variation is a Known as What Type of Varitation

Pyrometry and Control

J.D. GILCHRIST B.Sc., Ph.D., A.R.C.S.T., F.I.M. , in Fuels, Furnaces and Refractories, 1977

(2) Floating or Integral Control

allows continuous variation of heat input rate between limits which may cover the whole range from "Off" to "Full On", but is more likely to cover only a part of the range. The result need not be very different from step control as the input can change to its extreme value and stay there till the sign of the deviation is reversed. The rate of change of input (speed of opening a fuel valve) may be constant or may depend on the magnitude of the deviation so allowing faster response to rapidly changing situations. Properly tuned floating control should be able to bring the fuel input rate to the ideal value for the required temperature in a few cycles of operations and maintain it even if the load fluctuates. It is particularly useful where the capacity of the process is low and it can accommodate large changes in load if these are not too fast.

Read full chapter

URL:

https://www.sciencedirect.com/science/article/pii/B9780080204307500335

Drying

J.F. RICHARDSON , ... J.R. BACKHURST , in Chemical Engineering (Fifth Edition), Volume 2, 2002

Principles of the theory

The capillary theory of drying has been proposed in order to explain the movement of moisture in the bed during surface drying. The basic importance of the pore space between granular particles was first pointed out by Slichter ⁽¹⁶⁾ in connection with the movement of moisture in soils, and this work has been modified and considerably expanded by Haines ⁽¹⁷⁾ . The principles are now outlined and applied to the problem of drying. Considering a systematic packing of uniform spherical particles, these may be arranged in six different regular ways, ranging from the most open to the closest packing. In the former, the spheres are arranged as if at the corners of a cube with each sphere touching six others. In the latter arrangement, each sphere rests in the hollow of three spheres in adjacent layers, and touches twelve other spheres. These configurations are shown in Figure 16.5. The densities of packing of the other four arrangements will lie between those illustrated.

Figure 16.5. Packing of spherical particles. (a) Cubic arrangement, one sphere touching six others. (b) Rhombohedral packing, one sphere touching twelve others, with layers arranged in rhombic formation

In each case, a regular group of spheres surrounds a space which is called a pore, and the bed is made up of a series of these elemental groupings. The pores are connected together by passages of various sizes, the smallest portions of which are known as waists. The size of a pore is defined as the diameter of the largest sphere which can be fitted into it, and the size of a waist as the diameter of the inscribed circle. The sizes of the pores and waists will differ for each form of packing, as shown in Table 16.2.

Table 16.2. Properties of packing of spheres of radius r

Packing arrangement	Pore space (per cent total volume)	Radius of pore	Radius of waist	Value of x in equation 16.20 for:
limiting suction potential of pores	entry suction potential of waists
Cubical	47.64	0.700r	0.414r	2.86	4.82
Rhombohedral	25.95	0.288r	0.155r	6.90	12.90

The continuous variation in the diameter of each passage is the essential difference between a granular packing and a series of capillary tubes. If a clean capillary of uniform diameter 2r' is placed in a liquid, the level will rise in the capillary to a height h_s given by:

(16.17) $h_s=\left(\frac{2\sigma }{r\text{'}\rho g}\right)\cos \alpha$

where:

ρ is the density of the liquid,

σ is the surface tension, and

α is the angle of contact.

A negative pressure, known as a suction potential, will exist in the liquid in the capillary. Immediately below the meniscus, the suction potential will be equivalent to the height of the liquid column h_s and, if water is used, this will have the value:

(16.18) $h_s=\frac{2\sigma }{r\text{'}\rho g}$

If equilibrium conditions exist, the suction potential h ₁ at any other level in the liquid, a distance z ₁ below the meniscus, will be given by:

(16.19) $h_s=h_1+z_1$

Similarly, if a uniform capillary is filled to a height greater than h_s , as given by equation 16.17, and its lower end is immersed, the liquid column will recede to this height.

The non-uniform passages in a porous material will also display the same characteristics as a uniform capillary, with the important difference that the rise of water in the passages will be limited by the pore size, whilst the depletion of saturated passages will be controlled by the size of the waists. The height of rise is controlled by the pore size, since this approximates to the largest section of a varying capillary, whilst the depletion of water is controlled by the narrow waists which are capable of a higher suction potential than the pores.

The theoretical suction potential of a pore or waist containing water is given by:

(16.20) $h_t=\frac{x\sigma }{r\rho g}$

where:

x is a factor depending on the type of packing, shown in Table 16.2, and

r is the radius of the spheres.

For an idealised bed of uniform rhombohedrally packed spheres of radius r, for example, the waists are of radius 0.155r, from Table 16.2, and the maximum theoretical suction potential of which such a waist is capable is:

$\frac{2\sigma }{0.155r\rho g}=\frac{12.9\sigma }{r\rho g}$

from which x = 12.9.

The maximum suction potential that can be developed by a waist is known as the entry suction potential. This is the controlling potential required to open a saturated pore protected by a meniscus in an adjoining waist and some values for x are given in Table 16.2.

When a bed is composed of granular material with particles of mixed sizes, the suction potential cannot be calculated and it must be measured by methods such as those given by Haines ⁽¹⁷⁾ and Oliver and Newitt ⁽³⁾ .

Drying of a granular material according to the capillary theory

If a bed of uniform spheres, initially saturated, is to be surface dried in a current of air of constant temperature, velocity and humidity, then the rate of drying is given by:

(16.21) $\frac{\text{d}w}{\text{d}t}=k_{G}A(P_{w0}-P_w)$

where P_{w 0} is the saturation partial pressure of water vapour at the wet bulb temperature of the air, and P_w is the partial pressure of the water vapour in the air stream. This rate of drying will remain constant so long as the inner surface of the "stationary" air film remains saturated.

As evaporation proceeds, the water surface recedes into the waists between the top layer of particles, and an increasing suction potential is developed in the liquid. When the menisci on the cubical waists, that is the largest, have receded to the narrowest section, the suction potential h_s at the surface is equal to 4.82σ/rρg from Table 16.2. Further evaporation will result in h_s increasing so that the menisci on the surface cubical waists will collapse, and the larger pores below will open. As h_s steadily increases, the entry suction of progressively finer surface waists is reached, so that the menisci collapse into the adjacent pores which are thereby opened.

In considering the conditions below the surface, the suction potential h ₁ a distance z ₁ from the surface is given by:

$h_s=h_1+z_1$

(equation 16.19)

The flow of water through waists surrounding an open pore is governed by the size of the waist as follows:

(a)

If the size of the waist is such that its entry suction potential exceeds the suction potential at that level within the bed, it will remain full by the establishment of a meniscus therein, in equilibrium with the effective suction potential to which it is subjected. This waist will then protect adjoining full pores which cannot be opened until one of the waists to which it is connected collapses.

(b)

If the size of the waist is such that its entry suction potential is less than the suction potential existing at that level, it will in turn collapse and open the adjoining pore. In addition, this successive collapse of pores and waists will progressively continue so long as the pores so opened expose waists having entry suction potentials of less than the suction potentials existing at that depth.

As drying proceeds, two processes take place simultaneously:

(a)

The collapse of progressively finer surface waists, and the resulting opening of pores and waists connected to them, which they previously protected, and

(b)

The collapse of further full waists within the bed adjoining opened pores, and the consequent opening of adjacent pores.

Even though the effective suction potential at a waist or pore within the bed may be in excess of its entry or limiting suction potential, this will not necessarily collapse or open. Such a waist can only collapse if it adjoins an opened pore, and the pore in question can only open upon the collapse of an adjoining waist.

Effect of particle size. Reducing the particle size in the bed will reduce the size of the pores and the waists, and will increase the entry suction potential of the waists. This increase means that the percentage variation in suction potentials with depth is reduced, and the moisture distribution is more uniform with small particles.

As the pore sizes are reduced, the frictional forces opposing the movement of water through these pores and waists may become significant, so that equation 16.19 is more accurately represented by:

(16.22) $h_s=h_1+z_1+h_f$

where h_f , the frictional head opposing the flow over a depth z ₁ from the surface, will depend on the particle size. It has been found ⁽²⁾ that, with coarse particles when only low suction potentials are found, the gravity effect is important though h_f is small, whilst with fine particles h _ƒ becomes large.

(a)

For particles of 0.1-1 mm radius, the values of h ₁ are independent of the rate of drying, and vary appreciably with depth. Frictional forces are, therefore, negligible whilst capillary and gravitational forces are in equilibrium throughout the bed and are the controlling forces. Under such circumstances the percentage moisture loss at the critical point at which the constant rate period ends is independent of the drying rate, and varies with the depth of bed.

(b)

For particles of 0.001-0.01 mm radius, the values of h ₁ vary only slightly with rate of drying and depth, indicating that both gravitational and frictional forces are negligible whilst capillary forces are controlling. The critical point here will be independent of drying rate and depth of bed.

(c)

For particles of less than 0.001 mm (1 μm) radius, gravitational forces are negligible, whilst frictional forces are of increasing importance and capillary and frictional forces may then be controlling. In such circumstances, the percentage moisture loss at the critical point diminishes with increased rate of drying and depth of bed. With beds of very fine particles an additional factor comes into play. The very high suction potentials which are developed cause a sufficient reduction of the pressure for vaporisation of water to take place inside the bed. This internal vaporisation results in a breaking up of the continuous liquid phase and a consequent interruption in the free flow of liquid by capillary action. Hence, the rate of drying is still further reduced.

Some of the experimental data of Newitt et al. ⁽²⁾ are illustrated in Figure 16.6.

Figure 16.6. Rates of drying of various materials as a function of percentage saturation. A–60 µm glass spheres, bed 51 mm deep. B-23.5 µm silica flour, bed 51 mm deep. C–7.5 µm silica flour, bed 51 mm deep. D-2.5 µm silica flour, bed 65 mm deep. Subscripts: 1. Low drying rate 2. High drying rate

Freeze drying

Special considerations apply to the movement of moisture in freeze drying. Since the water is frozen, liquid flow under capillary action is impossible, and movement must be by vapour diffusion, analogous to the "second falling rate period" of the normal case. In addition, at very low pressures the mean free path of the water molecules may be comparable with the pore size of the material. In these circumstances the flow is said to be of the 'Knudsen' type, referred to in Volume 1, Section 10.1.

Read full chapter

URL:

https://www.sciencedirect.com/science/article/pii/B9780080490649500278

DESIGN OPTIMIZATION

Dominick Rosato , Donald Rosato , in Plastics Engineered Product Design, 2003

Fiber Strength Theory

The deformation and strength of filamentary structures subjected to combined loading can be theoretically predicted using experimentally-determined intrinsic stiffnesses and strength of the individual constituent layers. In order to have an integrated material and structure design, the gross properties as functions of the micromechanical parameters represent an important issue on the continuing and expanding use of RPs. It has been established, both in theory and experiment, that four principal elastic moduli and three principal strengths govern the deformation and strength of unidirectional fiber RPs. With the aid of a yield condition, the initial failure of filamentary structures can be predicted. After the initial failure, the structure may carry additional loads. An analysis of a partially failed or degraded structure can be used to predict the ultimate deformation and strength.

With an understanding of the gross behavior of a filamentary structure, a proper assessment of the mechanical and geometric properties of the constituent materials is possible. In particular, the use of fiber strength, the binding resin matrix, and the interface may be placed in a perspective based on the results of a mathematical analysis. They provide accurate guidelines for the design of RPs.

A better understanding exists of the elastic stiffness of filamentary materials than of the strengths. The generalized Hooke's law is usually accepted as the governing equation of the linear elastic deformation of RP materials. The simultaneous or sequential modes of deformation and fracture are difficult to describe from the phenomenological standpoint. In general, a strength theory on one criterion will not be sufficient to cover the entire range of failure modes of RP. In addition, fabrication variables and test methods are also known to introduce uncertainties in strength data that makes the verification of theories more difficult.

A macroscopic theory of strength is based on a phenomenological approach. No direct reference to the mode of deformation and fracture is made. Essentially, this approach employs the mathematical theories of elasticity and tries to establish a yield or failure criterion. Among the most popular strength theories are those based on maximum stress, maximum strain, and maximum work. The maximum stress theory states that, relative to the material symmetry axes x-y, failure of the RP will occur if one of three ultimate strengths is reached. There are three inequalities, as follows:

(2-24) ${\sigma }_x\le X$

(2-25) ${\sigma }_y\le Y$

(2-26) ${\sigma }_s\le S$

With negative normal stress components, compressive strengths designated by X' and Y' must be used:

(2-27) ${\sigma }_z\le X\text{'}$

(2-28) ${\sigma }_y\le Y\text{'}$

Shear strength S has no directional property and it retains the same value for both positive and negative shear stress components.

The maximum strain theory is similar to the maximum stress theory. Associated with each strain component, relative to the material symmetry axes, e_x, e_y , or e_s , there is an ultimate strain or an arbitrary proportional limit, X_e, Y_e , or S_e , respectively. The maximum strain theory can be expressed in terms of the following inequalities:

(2-29) $e_x\le X_e$

(2-30) $e_y\le Y_e$

(2-31) $e_x\le S_e$

Where e_x and y_y are negative, use the following inequalities:

(2-32) $e_x\le X{\text{'}}_e$

(2-33) $e_y\le Y{\text{'}}_e$

The maximum work theory in plane stress takes the following form:

(2-34) ${\left(\begin{array}{c}{\sigma }_x\\ X\end{array}\right)}^2-\left(\begin{array}{c}{\sigma }_{\text{z}}\\ \text{X}\end{array}\right)\left(\begin{array}{c}{\sigma }_y\\ X\end{array}\right)+{\left(\begin{array}{c}{\sigma }_y\\ Y\end{array}\right)}^2+{\left(\begin{array}{c}{\sigma }_{\text{s}}\\ \text{S}\end{array}\right)}^2=\text{1}$

If σ _x and σ _y are negative, compressive strengths X' and Y' should be used in Eq. 2-34, respectively.

In the following reviews, the tensile and compressive strengths of unidirectional and laminated RPs, based on the three theories, is computed and compared with available data obtained from glass fiber-epoxy RPs. The uniaxial strength of unidirectional RPs with fiber orientation θ can be determined according to the maximum theory. Strength is determined by the magnitude of each stress component according to Eqs. 2-24, 2-25, and 2-26 or Eqs. 2-27 and 2-28. As fiber orientation varies from 0° to 90°, it is only necessary to calculate the variation of the stress components as a function of θ. This is done by using the usual transformation equations of a second rank tensor, thus:

(2-35) ${\sigma }_{\text{x}}={\sigma }_1{\text{cos}}^{\text{2}}\theta$

(2-36) ${\sigma }_x={\sigma }_1{sin}^2\theta$

(2-37) ${\sigma }_x={\sigma }_{\text{1}}\text{sin}\theta \text{cos}\theta$

where σ _x , σ _y , σ _s are the stress components relative to the material symmetry axes, i.e., σ_x is the normal stress along the fibers, σ _y , transverse to the fibers, σ_s , the shear stress; σ ₁ = uniaxial stress along to the test specimen. Angle θ is measured between the 1-axis and the fiber axis. By combining Eqs. 2-35, 2-36, and 2-37 with 2-24, 2-25, and 2-26, the uniaxial strength is determined by:

(2-38) ${\sigma }_1\le X/{cos}^2\theta$

(2-39) $\le Y/{sin}^2\theta$

(2-40) $\le S/\left(sin\theta /{cos}^2\theta \right)$

The maximum strain theory can be determined by assuming that the material is linearly elastic up to the ultimate failure. The ultimate strains in Eqs. 2-29, 2-30, and 2-31 as well as 2-32 and 2-33 can be related directly to the strengths as follows:

(2-41) $X_e=X/E_{11}$

(2-42) $Y_e=X/E_{22}$

(2-43) $S_e=S/G$

The usual stress-strain relations of orthotropic materials is:

(2-44) $e_x=\frac{1}{E_{11}}\left({\sigma }_x-v_{12}{\sigma }_y\right)$

(2-45) $e_y=\frac{1}{E_{22}}\left({\sigma }_y-v_{12}{\sigma }_x\right)$

(2-46) $e_s=\frac{1}{G}\sigma ,$

Substituting Eq. 2-35, 2-36, and 2-37 into 2-44, 2-45, and 2-46 results in,

(2-47) $e_x=\frac{1}{E_{11}}\left({cos}^2\theta -v_{12}{sin}^2\theta \right){\sigma }_1$

(2-48) $e_y=\frac{1}{E_{22}}\left({sin}^2\theta -v_{21}{cos}^2\theta \right){\sigma }_1$

(2-49) $e_s=\frac{1}{G}\left(sin\theta cos\theta \right){\sigma }_1$

Finally, substituting Eqs. 2-47, 2-48, and 2-49 and 2-41, 2-42, and 2-43 into Eqs. 2-29, 2-30, and 2-31, and after rearranging, one obtains the uniaxial strength based on the maximum theory:

(2-50) ${\sigma }_1\le X/\left({cos}^2\theta -v_{12}{sin}^2\theta \right)$

(2-51) $\le Y/\left({sin}^2\theta -v_{21}{cos}^2\theta \right)$

(2-52) $\le X/\left(sin\theta -cos\theta \right)$

The maximum work theory can be obtained directly by substituting Eq. 2-35, 2-36, and 2-37 into Eq. 2-34:

(2-53) $\frac{1}{{\sigma }_1^2}=\frac{{cos}^4\theta }{X^2}+\left(\frac{1}{S^2}-\frac{1}{X^2}\right){cos}^2\theta {sin}^2\theta +\frac{{sin}^4\theta }{Y^2}$

Determining the strength of laminated RPs is no more difficult conceptually than determining the strength of unidirectional RPs. It is only necessary to determine the stress and strain components that exist in each constituent layer. Strength theories can then be applied to ascertain which layer of the laminated composite has failed. Stress and strain data is obtained for E-glass-epoxy, and cross-ply and angle-ply RPs. Under uniaxial loading, only N, is the nonzero stress resultant and when temperature effect is neglected, the calculations become:

(2-54) $e_i=\left(A{\text{'}}_{i1}+zB{\text{'}}_{i1}\right)N$

(2-55) ${\sigma }_i^{\left(k\right)}=c_i^{\left(k\right)}j\left[A{\text{'}}_{j1}+z{B}_{j1}^i\right]N_1$

where A' and B' matrices are the in-plane and coupling matrices of a laminated anisotropic composite.

The stress and strain components can be computed from Eqs. 2-54 and 2-55. They can then be substituted into the strength theories, from which the maximum Ni, the uniaxial stress resultant can be determined. Uniaxial tensile strengths of unidirectional and laminated composites made of E-glass-epoxy systems are obtained. Also, uniaxial axial-compressive strengths are obtained. The three strength theories can be applied to the glass-epoxy RP by using the following material coefficients:

(2-56) $\begin{array}{c}{E}_{11}=7.8\times {10}^{6}psi\\ E_{22}=2.6\times {10}^{6}psi\\ V_{12}=0.25\\ G=1.25\times {10}^{6}psi\\ X=150ksi\\ X\text{'}=150ksi\\ Y=4ksi\\ Y\text{'}=20ksi\\ S=8ksi\end{array}$

The maximum stress theory is shown as solid lines in Fig. 2.28. On the right-hand side of the figure is the uniaxial strength of directional RPs with fiber orientation θ from 0° to 90°; on the left-hand side, laminated RPs with helical angle α from 0° to 90°. Both tensile and compressive loadings are shown. The tensile data are the solid circles and the compressive are squares. Tensile data are obtained from dog-bone specimens. Compressive data are from specimens with uniform rectangular cross-sections.

Figure 2.28. Maximum stress theory

Figure 2.29 shows the comparison between the maximum strain theory and the same experimental data shown in Fig. 2.28. The formats are similar. Fig. 2.28 shows a comparison between the maximum work theory of the same experimental data as shown in Figs 2.29 and 2.30.

Figure 2.29. Maximum strain theory

Figure 2.30. Maximum work theory

Based on a Tsai review, it shows that the maximum work theory is more accurate than the maximum stress and strain theories. The maximum work theory encompasses the following additional features.

1.

There is a continuous variation, rather than segmented variation, of the strength as a function of either the fiber orientation θ or helical angle α.

2.

There is a continuous decrease as the angles θ and α deviate from 0°. There is no rise in axial strength, as indicated by the maximum stress and strain theories.

3.

The uniaxial strength is plotted on a logarithmic scale and an error of a factor of 2 exists in the strength prediction of the maximum stress and strain theories in the range of 30°.

4.

A fundamental difference between the maximum work and the other theories lies in the question of interaction among the failure modes. The maximum stress and strain theories assume that there is no interaction among the three failure modes (axial, transverse, and shear failures).

5.

In the limit, when:

(2-57) $X=Y=\sqrt{3S}$

which corresponds to isotropic materials, equation 2-53 becomes,

(2-58) ${\sigma }_1=X$

This means that the axial strength is invariant. If equation 2-57 is substituted into equation 2-38, 2-39, and 2-40 thus:

(2-59) $\begin{array}{c}{\sigma }_1\le X/{cos}^2\theta \\ \le X/{sin}^2\theta \\ \le X/\sqrt{3sin\theta cos\theta }\end{array}$

The angular dependence does not vanish, that should not be the case for an isotropic material.

Read full chapter

URL:

https://www.sciencedirect.com/science/article/pii/B978185617416950003X

Energy Storage

Ziyad Salameh , in Renewable Energy System Design, 2014

Transmission system

The transmission system must allow a continuous variation in ratio and must operate efficiently under widely varying load conditions. The main types of transmissions are mechanical, electrical, and hydrostatic. The electrical transmission is the most common one. It uses motors and generators that can be placed within the vacuum housing. The only problem with this is the need of some sort of liquid.

The motor, which can be either brushless synchronous or induction, is fed from the mains through an inverter and a rectifier. The inverter gives a variable frequency input to the motor when the flywheel has to be charged. During discharge, the flywheel operates as the prime mover and the motor becomes a generator, its output being rectified and fed to a fixed-frequency inverter. In place of AC machines fed by inverters, it is possible to use DC machines controlled by choppers. The disadvantages of this solution are the greater size and cost of them; their somewhat lower efficiency; some speed limitations; and, above all, the inability to place the flywheel motor/generator directly within the vacuum housing due to the brushes.

Read full chapter

URL:

https://www.sciencedirect.com/science/article/pii/B9780123749918000040

Space Transformation Concept: Controlling the Path of Electromagnetic Waves

Shah Nawaz Burokur , ... Paul-Henri Tichit , in Transformation Optics-based Antennas, 2016

3.2.2 3D design, implementation and full-wave simulations

The calculated isotropic material parameters present a continuous variation of permittivity along the z-axis (E-field direction). This is not easy to achieve in practical implementations and, in general, it is necessary to carry out a discrete variation. A method that consists of discretizing the permittivity profile into several zones of different permittivity values is adopted. This discretization secures the characteristics of the ε _zz response of the lens.

According to effective medium theory, if the operating wavelength is large enough with respect to the size, the composite material can be considered isotropic and homogeneous. A discrete lens model composed of 92 unit cells is proposed. The lens is designed such that the respective permittivity of each cell is considered to be constant across the cell and is equal to the average permittivity within the cell. As illustrated in Figure 3.5, the permittivity ε_zz values range from 1.5 to 2.6 in the discrete approximation of the conformal lens. The two edges with permittivity values below 1.5 are suppressed from the continuous lens such that the permittivity ranges between 1.5 and 2.6.

Figure 3.5. Parameter reduction for the discrete model of the in-phase emission restoring lens

As only ε _zz varies from 1.5 to 2.6 in the lens design, we are able to consider using it over a broad frequency range. The lens is thus realized from non-resonant cells. Air holes in a dielectric host medium of relative permittivity ε _h  =   2.8 is, therefore, considered as for the practical implementation, we will consider a dielectric photopolymer with ε _r  =   2.8 in a polyjet 3D printing fabrication facility. Suppose that two materials are mixed together, the effective parameter can be calculated by:

[3.4] ${\epsilon }_{\mathit{eff}}={\epsilon }_a{f}_a+{\epsilon }_h{f}_h$

where ε _a  =   1 and f _a and f _h  =   1   − f _a are the volume fraction of the air holes and the host material, respectively. By judiciously adjusting the volume fraction of the air holes through the geometrical dimensions p and r _a, the effective permittivity of the cell can then be engineered as shown in Figure 3.6. Values ranging from close to 1 to 2.8 can, thus, be obtained for ε _zz over a wide frequency-band.

Figure 3.6. Effective permittivity of a cubic cell composed of air hole in a dielectric host medium. A parametric analysis is performed to extract the effective permittivity value according to the radius r_a of the air hole with respect to a cubic cell of period p (p changes according to the regions of the discrete lens)

Full-wave simulations using commercial electromagnetic solver HFSS from Ansys are performed to verify the functionality of the lens numerically. A microstrip patch antenna array composed of eight linear radiating elements conformed on a cylindrical surface is considered as the excitation source for the in-phase emission restoring lens as shown in Figure 3.7(a) and the schematic design of the lens is presented in Figure 3.7(b). The lens comprises of five regions with a total of 92 cells where ε _zz varies from 1.5 to 2.6 and the dimensions are set as: r  =   27   cm and d  =   2   cm. Region 2 and region 3 share similar characteristics with respectively region 4 and region 5. Region 1 is discretized into 13 rows of 4 cells with p  =   5   mm, regions 2 and 4 are each discretized into 6 rows of 3 cells with p  =   6.7   mm and regions 3 and 5 are decomposed each into 1 row of 2 cells with p  =   10   mm.

Figure 3.7. a) Schematic design of the conformal patch array source; b) perspective view of the discrete lens

The simulated 2D far-field radiation diagrams are presented in Figure 3.8. The planar array presents a directive radiation with a clear narrow main lobe, while the conformal array presents a wider main beam with a lower magnitude. The lens antenna system is able to increase the magnitude of the first lobe and decreases the level of the second lobe. The conformal array without lens has a directivity of 12.3   dB, while the conformal array in presence of lens has a directivity of 15.7   dB, which is better than the directivity of the planar array alone (15.3   dB). Such results confirm the fact that the lens is able to create a constructive interference between the emissions of the radiating elements of the conformal array and, hence, produce an overall in-phase emission. However, though the effect of restoring in-phase emission is clear on the main lobe of the radiation pattern, sidelobes level tends to be high in presence of the lens. This is due to the fact that the anisotropy in permittivity has been ignored, as discussed previously.

Figure 3.8. Simulated radiation patterns in the focusing plane (x-y plane); a) 8   GHz; b) 10   GHz; c) 12   GHz. The conformal array presents a distorted diagram with a lower radiation level than the planar array. The lens above the conformal array allows restoring the in-phase emission to create a radiation pattern with a clear directive main beam, similar to a planar array

Read full chapter

URL:

https://www.sciencedirect.com/science/article/pii/B9781785481970500032

Navigation Receiver

Rajat Acharya , in Understanding Satellite Navigation, 2014

5.2.3.3 Analog to digital converter

The signal obtained until now is in analog form with continuous variation over time. It has a bandwidth referred to as the precorrelation bandwidth. The analog values of the signal are available at the input of the analog to digital converter (ADC), which converts this continuous signal into a discrete form. The analog values are first selected at definite discrete intervals and rejected at all intermediate times. This process is called sampling; the selected values form the samples of the analog signal. These discrete sample values are first characterized and then their levels are converted to corresponding digital numbers through binary coding. The process of sampling and quantization is illustrated in Figure 5.6.

FIGURE 5.6. Sampling and quantization.

Read full chapter

URL:

https://www.sciencedirect.com/science/article/pii/B9780127999494000051

Electrochemical Measurement Methods and Characterization on the Cell Level

Têko W. Napporn , ... Viktor Hacker , in Fuel Cells and Hydrogen, 2018

9.6 Chronoamperometry and Chronopotentiometry

With the two fundamental parameters E and I , the continuous variation of the electrode potential during a CV experiment makes the interpretation of I  = f(t) responses more complex. To simplify and make the analysis much more efficient, it is recommended that one fix one of these two quantities; namely the potential E, or the intensity of the current I. This leads to two new techniques; chronoamperometry and chronopotentiometry. Chronoamperometry consists of holding the potential of the working electrode constant and measuring the variation of the current over time: I  = f(t). In the second case, maintaining the current and monitoring the spontaneous variation of the potential of the working electrode is called chronopotentiometry: E  = f(t). The chronoamperometry finds its application in the case of metal deposits, electrolysis, and the study of the stability over time of electrochemical systems. For chronopotentiometry, it is a matter of studying the capacity of a system to provide an accurate current for a given cell voltage. For fuel cell testing, both techniques can be applied. In the galvanostatic mode (current steps), the anodic and the cathodic potential, thus the overall cell voltage, are measured by chronopotentiometry: the potentials of the two half-cells evolve over time at fixed I. In the potentiostatic mode, the overall cell voltage is fixed and the steady-state current is recorded. Conclusively, the employment of these two techniques during the half-cell experiments is an intelligent and excellent strategy for anticipating, interrogating, and solving what could happen within a real fuel cell system. The desired potential to be applied during chronoamperometry should be a result of preliminary CV experiments. Rigorously performing chronopotentiometry tests supposes that one presumably knows the range of currents that the system is able to deliver by the oxidation and reduction at both working and counter electrodes. Indeed, forcing the electrodes to deliver an unsuitable current (because it's out of the range) can kill the system by provoking several cascades of undesired and parasitic processes.

Read full chapter

URL:

https://www.sciencedirect.com/science/article/pii/B9780128114599000098

Scattering and Recoiling Spectrometry

J.Wayne Rabalais , in Encyclopedia of Physical Science and Technology (Third Edition), 2003

II.A TOF-SARS System

An instrument for structural studies by ion scattering and recoiling should be capable of continuous variation of the scattering Θ, beam incident α, and crystal azimuthal δ angles (see Fig. 1), generation of a pulsed kiloelectron-volt primary ion beam of low fluence, efficient detection of both ions and neutrals, in situ low-energy electron diffraction (LEED), and operation in an ultrahigh vacuum (< 10⁻¹⁰ Torr) environment. Figures 2 and 3 provide schematics of a TOF-SARS system and the pulsed ion beam with associated electronics. The primary ion beam is a 1 to 5   keV rare gas ion source which has a narrow energy spread, is mass-selected, pulsed at 10 to 40   kHz, with pulse widths of 20 to 50   ns for an average ion current density of <1   nA/cm², and has low angular divergence. The detector is a channel electron multiplier or channel plate which is sensitive to both ions and fast neutrals. The sample is mounted on a precision manipulator, and the angles α and δ are computer controlled by means of stepping motors. Scattered and recoiled particles are velocity analyzed by measuring their flight times from the sample to detector, a distance of 1   m. An electrostatic deflector plate near the flight path allows deflection of ions for collection of TOF spectra of neutrals compared to that of ions plus neutrals. Standard timing electronics are used for data collection. The trigger output of a pulse generator, delayed by the time necessary for the pulsed beam to travel from the pulse plate to the sample, starts a time-to-amplitude converter (TAC). The TAC is stopped from the signal output of a particle reaching the detector. The output of the TAC yields a histogram of the distribution of particle flight times. The data are collected into a multichannel pulse height analyzer and stored in a computer. TOF spectra can be collected with a dose of <10⁻³ ions per surface atom, making the technique relatively nondestructive. The TOF-SARS instrument also has ports that contain standard surface analysis techniques such as LEED, Auger electron spectroscopy (AES), and X-ray photoelectron spectroscopy (XPS).

FIGURE 2. Spectrometer system designed for TOF-SARS, electrostatic analysis of scattered and recoiled ions, and conventional surface analysis techniques such as AES, XPS, and LEED. A, pulsed ion beam; B, turbomolecular pump; C, ion pump; D, sample manipulator; E, detector precision rotary motion feedthrough; F, X-ray source; G, electron gun; H, 180° electrostatic hemispherical analyzer; I, sorption pumps; J, sputter ion gun; K, viewport or reverse view LEED optics; L, titanium sublimation pump; M, cryopump.

FIGURE 3. Schematic of pulsed ion beam line and associated electronics. A, ion gun; B, Wien filter; C, Einzel lens; D, pulsing plates; E, pulsing aperture; F, deflector plates; G, sample; H, channel electron multiplier with energy prefilter grid; I, electrostatic deflector.

Read full chapter

URL:

https://www.sciencedirect.com/science/article/pii/B0122274105006761

Chain Polymerization II

John M.G. Cowie , in Comprehensive Polymer Science and Supplements, 1989

22.3 Complex Stoichiometry

22.3.1 Binary Complexes

The values of x and y in equation (8) can be established spectroscopically using the 'continuous variation method' proposed by Job. ¹⁴ Both UV/visible spectroscopy and ¹H NMR data are amenable to analysis by this technique, and can be illustrated with the following examples.

When the donor–acceptor pair 2,3-dihydro-1,4-dioxin (1,4-dioxene) and MAH are dissolved in chloroform, an absorption band appears with a maximum at λ   =   366   nm, which is characteristic of the D–A complex, as shown in Figure 3. It can be seen from the figure that this band displays a maximum absorption at a 1:1 mole ratio of the donor and acceptor, when determined as a function of the mixture composition. This is interpreted as the stoichiometry of the D–A complex.

Figure 3. Analysis of the charge transfer complex formed between MAH and 2,3-dihydro-1,4-dioxin in chloroform solution, using the continuous variation method. Curves have been determined at wavelengths of 366, 400 and 430   nm (reproduced by permission of Huthig and Wepf from S. Iwatsuki and Y. Yamashita, Makromol. Chem., 1965, 89, 205)

This method has been extended by Sahai et al. ¹⁵ to encompass ¹H NMR data. They have shown that, for a 1:n complex, a plot of $\text{Δ}{\delta }_{\text{obs}}^{\text{A}}{\left[\text{A}\right]}_0$ against X _D will give a maximum at X _D  = n/(n  +   1). Here X _D is the mole fraction of the donor, [A]₀ is the initial concentration of the acceptor and $\text{Δ}{\delta }_{\text{obs}}^{\text{A}}$ is the observed chemical shift in the NMR spectrum of the acceptor protons in a complexing medium.

It has been observed that both α-MS and isoprene (IP) form donor–acceptor complexes with DCB, but their characteristic absorption bands cannot be resolved by UV spectroscopy. The Sahai method has proved successful, however, and the chemical shift data from ¹H NMR are shown in Figure 4. These curves provide evidence for the formation of 1:1 complexes in each case.

Figure 4. Stoichiometric determination of the donor–acceptor complex by the NMR continuous variation method. X _donor  =   mole fraction of donor molecules in donor–DCB mixtures; ●, α-MS–DCB; ○, IP–DCB

It was shown in Section 22.2.2 that binary complexes can form between a Lewis acid and a class B monomer by coordination of the carbonyl or cyano group to the central metal atom in the Lewis acid; see, for example, structures ( 7 ) and ( 8 ).

7

These complexes are often sufficiently stable to allow them to be isolated, usually as crystalline complexes which are very hygroscopic ^16,17 and also polymerize readily. The large shifts in the IR bands associated with the stretching of the cyano or carbonyl groups substantiate the existence of these complexes. ^18,19

¹H and ¹³C NMR have also provided useful evidence for 1:1 complexation. ^20–22 These complexes are not always 1:1 and 1:2 have also been identified.

22.3.2 Ternary Complexes

When SnCl₄ is added to methyl methacrylate (MMA), 1:1 and 1:2 complexes can form, as shown in equations (12) and (13).

(12)

(13)

These tin complexes are either trigonal, bipyramidal or octahedral. The 1:2 SnCl₄(MMA)₂ appears to be octahedral, whereas, in the 1:1 complex, two Raman lines at 355 and 396   cm⁻¹ indicate the bipyramidal form; Zn^II, B^III and Al^III complexes tend to be tetrahedral. The stoichiometry was established using gravimetric and elemental analysis. ^23,24

While the formation of binary complexes between a Lewis acid and a class B monomer is firmly established, ternary complexation between the binary system and a donor monomer is more tenuous. The strength of interaction is probably much weaker but, by careful selection of solvent, temperature and measuring conditions, some evidence for the existence of ternary complexes can be obtained.

In the SnCl₄–MMA system, a 1:2 'binary' complex can be formed (equation 13). When a donor such as S is added to this system, a 1:1 interaction between MMA and S can be identified ^21,25 using the continuous variation method based on ¹H NMR data, as shown in Figure 5.

Figure 5. Continuous variation plots of chemical shifts for the SnCl₄(MMA)₂–S system in n-hexane at 253   K; ○, methoxy protons; ●, α-methyl protons (reproduced from ref. 21 with permission from Wiley)

Similarly, when the 1:2 ZnCl₂(AN)₂ complex (AN   =   acrylonitrile) is mixed with S, a charge transfer band appears in the UV spectrum at 320–350   nm. Application of the Job method indicated a 1:1:1 complex between ZnCl₂, AN and S, ²⁶ and this type of complex formation was confirmed in the ZnCl₂–(MMA)₂–S system, and ZnCl₂–(AN)₂–divinyl ether. ²⁷

Furukawa et al. ²⁸ examined some EtAlCl₂–acrylic monomer–butadiene systems. Addition of butadiene to the EtAlCl₂/MMA complex led to the formation of a charge transfer complex with a characteristic absorption at 340   nm, which was shown to be a 1:1:1 complex. It was also proposed that, when a Lewis acid complexes with an acrylic monomer, the polarizability of the monomer is increased. NMR examination of the complexed monomer indicated an apparent delocalization of the π electrons, which was believed to be the driving force necessary for the formation of the ternary complex with a donor molecule. ²⁰

Read full chapter

URL:

https://www.sciencedirect.com/science/article/pii/B9780080967011001373

Shape Constraints in Deformable Models

Lawrence H. Staib , ... Amit Chakraborty , in Handbook of Medical Imaging, 2000

2.4 Overview

We categorize segmentation problems into two types: the delineation of structure of known shape whose form is a continuous variation around a mean shape and the finding of more complex structure that varies significantly between individuals. In the first case, we have developed an integrated method using parametric shape models applied to structures such as the caudate in the subcortex of the brain and the left ventricle of the heart. In the second, we have developed a coupled level-set approach, applied primarily to the cortex of the brain. These methods have been tested on a variety of synthetic and real images.

Read full chapter

URL:

https://www.sciencedirect.com/science/article/pii/B9780120777907500138

oxleyfellnisomer.blogspot.com

Source: https://www.sciencedirect.com/topics/engineering/continuous-variation

Share this post