Series and parallel model

Mathai and Singh have estimated the permeability coefficient P, using the formula P = kD where k is the partition coefficient and D is the diffusivity. They have used both parallel and series models to calculate P. The experimental values are always greater than measured values. The poor agreement between the experimental and calculated values is attributed to the polar-polar interaction between the epoxy group and nitrile group. 

Usually, the parallel and series models are considered to form the upper and lower bounds respectively, and the experimental data fall within these bounds We suggest that all models which express the dependence of E vs. v can be written as... 

Figure 2.11. Models for two-phase polymer systems. Elements in (a) and (b) form the basic parallel and series models. Combinations shown in (c) and (d) represent two other possible models. Note the similarity to the spring and dashpot models often invoked in explaining homopolymer behavior. (After Takayanagi et ai, 1963.) Part (a) represents an isostrain model, (b) represents an isostress model, and (c) and (d) illustrate combinations of these limiting cases.

These two simple models have been used to predict different thermal and mechanical properties of a two-state material such as thermal conductivity and elastic modulus, where they are also called parallel and series models respectively. These two models are further able to define the upper and lower bounds for the effective properties of the mixture [1], as follows ... 

If the system under examination is nonlinear, the information in the parallel and series models is not equal. Series values cannot be calculated from parallel values and vice versa, and the actual measuring circuit determines the variables and the model chosen. 

Figure 2.1 Effect of polyurethane (PU)/polycaprolactone flbn structure on the modulus of 65 layer and blend films in comparison with values predicted from the parallel and series models. From J. Du, S.R. Armstrong, E. Baer, Co-extruded multilayer shape memory materials comparing layered and blend architectures. Polymer (United Kingdom) 54 (20) (2013) 5399-5407.

Hirsch [29] proposed a model considering the stress transfer between fibre and matrix. The model is a combination of parallel and series model and can be expressed as ... 

An agreement with experimental results was obtained by taking into account the increased effective fraction of the filler, Veff, due to the glassy interphase of the bound epoxide layer and assuming a co-continuous morphology of the epoxy-silica hybrid network. Mechanical properties in dependence on the phase continuity are treated by parallel and series models for bicontinuous morphology and discontinuous phases, respectively. The equivalent box model (EBM) developed by Takayanagi (13) (eqs 2-5) and Davies model (14) (eq. 6) were used to compare the experimental data with the theory (9). 

The parallel and series models represent the limits for phase separated blends ... 

A model referred to as the equivalent box model (EBM) has shown promise in the ability to predict the modulus behavior over the entire composition. This model has similarities to earlier models by Takayangi et al. [6,7]. This model is a combination of the parallel and series models and has been developed by Kolarik [8-10]. This model will be described in detail because of its versatility for phase separated blends. This mechanical model is illustrated in Fig. 6.5. The modulus is calculated from... 

For phase separated blends, permeability versus composition can be more complex at intermediate compositions. However, at the extremes of composition, where component 1 is entirely the continuous phase or the dispersed phase, the parallel and series models can be employed. 

The Lichtenecker and Rother (1931) generalization for dielectric permittivity (see Eq. 11.5) fills the space between the extreme boundaries of parallel and series model. For a porous rock results by variation of the parameter a ... 

In this chapter, we develop some guidelines regarding choice of reactor and operating conditions for reaction networks of the types introduced in Chapter 5. These involve features of reversible, parallel, and series reactions. We first consider these features separately in turn, and then in some combinations. The necessary aspects of reaction kinetics for these systems are developed in Chapter 5, together with stoichiometric analysis and variables, such as yield and fractional yield or selectivity, describing product distribution. We continue to consider only ideal reactor models and homogeneous or pseudohomogeneous systems. 

Fig. 5.5 Parallel and series resistance models of a lossy capacitor.

Fig. 12.18. Comparison of the optimized reduced amounts that should be dosed and the corresponding internal compositions for a fixed-bed reactor (discrete dosing, top) and a membrane reactor (continuous dosing, bottom). A triangular network of parallel and series reactions was analyzed using an adapted plug-flow reactor model, Eq. 48. One stage (left) and 10 stages connected in series (right) were considered. All reaction orders were assumed to be 1, except for those with respect to the dosed component in the consecutive and parallel reactions (which were assumed to be 2) [66].

This is the simplest way of applying the spring and dashpot model, but there are others of increasing complexity. For example, the Maxwell model considers the spring and dashpot to be in series, while the so-called standard linear solid has both parallel and series arrangements. While all of these approaches are mathematically useful, they do not have an underlying physical basis in reality there are no springs and no dashpots. However,... 

According to the resistance model, we have to consider parallel and series removal processes. Without consideration of source terms (see later), we have the budget equation (first-order law) of ... 

Fig. 5. Parallel and series Takayanagi models. (Adapted from Ref 64.)...

Takayanagi and co-workers transformed the spring and dashpot relaxation models (Section 1.5.6) to plastic and rubber elements in an effort to better explain the mechanical behavior of poly blends (Takayanagi et ai, 1963). Some simple combinations of the Takayanagi models are shown in Figure 2.11. The plastic phase is denoted by P and the rubber phase by R, while the quantities X and (p are functions of the volume fractions of parallel and series elements, respectively. 

In conclusion, the parallel and series two-component circuits are complementary models and very different from each other. A choice of model is unavoidable when electrical data are to be analyzed and presented. The choice must be based upon a presumption on the actual physical arrangement in the biomaterial/electrode system to be modeled. 

Passive oscillator mode Impedance analysis of the forced oscillation of the quartz plate provides valuable information about the coating even if the active mode is not applicable anymore. For impedance analysis, a frequency generator is used to excite the crystal to a constraint vibration near resonance while monitoring the complex electrical impedance and admittance, respectively, dependent on the applied frequency (Figure 2B). For low load situations near resonance, an equivalent circuit with lumped elements - the so-called Butterworth—van-Dyke (BVD) circuit — can be applied to model the impedance data. The BVD circuit combines a parallel and series (motional branch) resonance circuit. The motional branch consists of an inductance Lq, a capacitance Cq, and a resistance Rq. An additional parallel capacitance Co arises primarily from the presence of the dielectric quartz material between the two surface electrodes (parallel plate capacitor) also containing parasitic contributions of the wiring and the crystal holder (Figure 2B). 

Modeling the monolithic reactor as a combination of parallel and series reactors ... 

Physical properties of blends consisting of a continnons matrix and one or more dispersed (discrete) components can be predicted by nsing adapted models proposed for particulate composite systems (216-220). Most of these models do not consider effects of the particle size, but only of volnme fractions of components in the system. Thus, the increase in particle size dne to particle coalescence is not presumed to perceptibly affect mechanical properties, except for fractnre resistance, which is controlled by particle size and properties of dispersed rnbbers. As polymer blends with three-dimensional continuity of two or more components are isotropic materials, simple parallel or series models or models for orthotropic or quasi-isotropic materials are not applicable. Physical properties of blends with partially co-continuous constituents can be calculated by means of a predictive... 

The Takayanagi models were remarkably successful in providing a simple interpretation of the d3mamic mechanical behaviour of crystalline polymers and polymer blends. The theoretical basis is contained in Equations (8.1) to (8.6) of Section 8.2, and is deficient in two respects. First, only tensile deformations are considered and shear deformations are ignored. Secondly, as emphasized in Chapter 7, Voigt and Reuss schemes (i.e. parallel and series) only provide bounds to the true behaviour. 

If the modeling tool can t represent the PCB in full complexity, some attempt should be made to perform a better estimation of the thermal conductivity in Cu spreading layers connected to critical components. The overall smeared thermal conductivity of the plate should be replaced with patches of thermal conductivity that account for heat flowing either parallel or perpendicular to the Cu conductors. Eqs. 17.8 and 17.9 can be used to estimate parallel and series thermal conductivities in Cu layers respectively. Figure 17.14 shows graphically what is meant by a parallel thermal conductivity. Here, multiple stripes of thermally conductive materials pass thermal energy from the top to bottom. Each material channel can be considered as a thermal resistor. The thermal resistance of each resistor can be calculated as a function of its... 

In the present work, a packed bed cell model is used to calculate temperature and concentration profiles in the adiabatic RFBR for exothermic catalytic reactions with interphase resistance to mass and heat transfer. In particular, differences between the RFBR and the AFBR, operated at the same space velocity, are explored with respect to uniqueness, multiplicity, and stejDility of the steady state, profile location, selectivity in parallel and series reactions, and transient behavior. 

Composite piezoelectric materials may be represented by the so-called simple series, simple parallel and the modified cubes diphasic models (Fig. 6.4). The modified cubes model was developed as a generalization of the series, parallel and cubes models. It is adapted for the representation of 0-3 composite sheet materials. Estimated values of the average longitudinal piezoelectric strain coefficient 33 and the average piezoelectric voltage coefficient 33 for the composite may be evaluated in terms of these models. References to the piezoelectric ceramic and the polymer phase will be indicated by superscripts 1 and 2 respectively. 

Figure 3 The experimental storage modulus from Figure 2, compared with the parallel and series Takayanagi models...

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