Resistances in Series and Parallel

5-3 Steady-state temperature profile in a composite wall with constant thermal conductivities k kg, and kc and no energy sources in the wall. The thermal circuit is shown in (b). The total resistance is the sum of the three resistances shown. 

5-4 Thermal circuit for Example 1. Steady-state conduction in a furnace wall with heat losses from the outside surface by convection (hc) and radiation (hR) to the surroundings at temperature Tsur. The thermal conductivities kD, kg, and ks are constant, and there are no sources in the wall. The heat flux q has units of W/m2. 

Tsur is 290 K. We want to estimate the temperature of the inside wall T. The wall consists of three layers deposit [kD=1.6 W/(m-K), AxD = 0.080 m], brick [kB = 1.7 W/(m-K), AxB = 0.15 m], and steel [ks = 45 W/(m-K), Axs = 0.00254 m]. The outside surface loses heat by two parallel mechanisms—convection and radiation. The convective heat-transfer coefficient hc = 5.0 W/(m2K). The radiative heat-transfer coefficient hR = 16.3 W/(m2K). The latter is calculated from 

Referring to Fig. 5-4, the steady-state heat flux q (W/m2) through the wall is 

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Example 1 Conduction with Resistances in Series and Parallel Figure 5-4 shows the thermal circuit for a furnace wall. The outside surface has a known temperature T2 = 625 K. The temperature of the surroundings... 

Example 1 Conduction with Resistances in Series and Parallel. 5-5... 

The overall resistance using resistances in series and parallel can be written as... 

Ohm s law and resistance in series and parallel circuits (V-I graphs for ohmic conductors Rj-=RNR2 stc. 

R Thermal resistance, equals x/kA, 1/UA, l/hA Ri, Ro, R l, R for thermal resistance of sections 1, 2, 3, and n of a composite body Rj for sum of individual resistances of several resistances in series or parallel R -, and for dirt or scale resistance on inner and outer surface respectively Ratio of total outside surface of finned tube to area of tube having same root diameter (s-K)/J (h- F)/Btu... 

The a.c. impedance technique [33,34] is used to study the response of the specimen electrode to perturbations in potential. Electrochemical processes occur at finite rates and may thus be out of phase with the oscillating voltage. The frequency response of the electrode may then be represented by an equivalent electrical circuit consisting of capacitances, resistances, and inductors arranged in series and parallel. A simplified circuit is shown in Fig. 16 together with a Nyquist plot which expresses the impedance of the system as a vector quantity. The pattern of such plots indicates the type and magnitude of the components in the equivalent electrical network [35]. 

The effect of the adsorbent shape on mass transfer is much more complex. There are several mass transfer resistances in series and/or parallel, each one of which may be controlling for a particular set of conditions. However, in general, mass transfer rates will be larger for particles with larger specific surface... 

Fig. 9. Simple electrical circuits illustrating the relationship between driving force, V, current flow, /, and resistance, R. (a) Simple series circuit, (b) More complex circuit with multiple resistances Ri connected in series and parallel fashion.

The net diffusivity of component A within the pores of a catalytic pellet is obtained by adding mass transfer resistances for Knudsen diffusion and ordinary molecular diffusion, where convection reduces the resistance due to ordinary molecular diffusion but Knudsen flow occurs over length scales that are much too small for convective mass transfer to be important. This addition of resistances is constructed to simulate resistances in series, not parallel. Consider the trajectory of a gas molecule that collides with the walls of a channel or other gas molecules. In the pore-size regime where Knudsen and ordinary molecular diffusion are equally probable, these collisions occur sequentially, which suggests that gas molecules encounter each of these resistances in series. Hence, for binary mixtures. 

Resistance of conductors in series and parallel n. The total resistance of any number of resistances joined in series is the sum of the separate resistances. The total resistance of conductors in parallel whose separate resistances are ri, r2, rs,..., r is given by the formula... 

Fig. 8.4 Complex plane and Bode plots for circuit consisting of solution resistance in series with parallel connection of CPE and resistance R. Parameters = 10 D, T = 20 pF cm s, ...

The r term of Eq. (5.1) may be factored into component resistances in series or parallel. For example, for transpiration, r can be factored into and r, where Va = the boundary layer or air resistance, and... 

Using the simple electricity laws of calculating the resistance of resistors in series and parallel quickly leads to the results ... 

Figure 1.17 Various configurationsof 48 two-volt lead-acid cells In series and parallel. Effect of configuration on current, voltage and total electrical resistance. Internal resistance, Rc = 0.1 n/cell. Total external electrical resistance (Rext) = 0.05 n...

The resistance to plastic flow can be schematically illustrated by dashpots with characteristic viscosities. The resistance to deformations within the elastic regions can be characterized by elastic springs and spring force constants. In real fibers, in contrast to ideal fibers, the mechanical behavior is best characterized by simultaneous elastic and plastic deformations. Materials that undergo simultaneous elastic and plastic effects are said to be viscoelastic. Several models describing viscoelasticity in terms of springs and dashpots in various series and parallel combinations have been proposed. The concepts of elasticity, plasticity, and viscoelasticity have been the subjects of several excellent reviews (21,22). 

In either equation, /c is given by Eq. (16-84) for parallel pore and surface diffusion or by Eq. (16-85) for a bidispersed particle. For nearly linear isotherms (0.7 < R < 1.5), the same linear addition of resistance can be used as a good approximation to predict the adsorption behavior of packed beds, since solutions for all mechanisms are nearly identical. With a highly favorable isotherm (R 0), however, the rate at each point is controlled by the resistance that is locally greater, and the principle of additivity of resistances breaks down. For approximate calculations with intermediate values of R, an overall transport parameter for use with the LDF approximation can be calculated from the following relationship for sohd diffusion and film resistance in series... 

Capacitors are often combined in series or parallel, with the resulting circuit capacitance calculated as depicted in Figure 4. An important relationship is the time constant of a capacitor. The time constant is based on the product of the resistance and capacitance and is known as the RC time constant. A capacitor in a dc circuit will charge or discharge 63.2 percent in one RC time constant. The time dependence of a capacitor is shown in the equations. 

To find the relation between the values of R and measured experimentally in terms of the circuit of Fig. 12.11a and the parameter values in the circuit of Fig. 12.14a, we must first convert [with the aid of Eq. (12.23)] the parameters of the circuit with parallel elements Ry and Q into the parameters of a circuit with a resistance and capacitance in series, and to the value of resistance obtained we must add R.. As a result, we have... 

The impedance data have been usually interpreted in terms of the Randles-type equivalent circuit, which consists of the parallel combination of the capacitance Zq of the ITIES and the faradaic impedances of the charge transfer reactions, with the solution resistance in series [15], cf. Fig. 6. While this is a convenient model in many cases, its limitations have to be always considered. First, it is necessary to justify the validity of the basic model assumption that the charging and faradaic currents are additive. Second, the conditions have to be analyzed, under which the measured impedance of the electrochemical cell can represent the impedance of the ITIES. 

The R-X plot shows the most variation in the subthreshold region, while the G-B plot shows the most variation above threshold. One sees from the G-B plot that the high frequency response of the diode is independent of bias (>1 MHz). To fit the data, one models each material phase or interface as a parallel R-C combination. These combinations are then added in series, and an overall series resistance and series inductance are added. For the data in Figure 10.6, three R-C elements are used. One R-C element is associated with the Schottky barrier. Another is associated with the high frequency bias-independent arc, which we believe is associated with the capacitance of the alkoxy-PPV. The thinness of the film... 

This means that the pressure drops, i.e., the flow resistances along the separate flow paths, determine the flow distribution in a fluidic network. Furthermore, such fluidic networks can be calculated analogous to electrical networks with Kirchhofif s law for parallel and series connection of electrical resistances in series ... 

Accounting for these five resistances in series—parallel, eliminating cloud and emulsion concentrations, and integrating from the bottom to the top of the... 

The bipolar pulse technique for measuring solution resistance minimizes the effects of both the series and parallel cell capacitances in a unique way. The instrumentation for this technique is illustrated in Figure 8.15. The technique consists of applying two consecutive voltage pulses of equal magnitude and pulse width but of opposite polarity to a cell and then measuring the cell current precisely at the end of the second pulse [18]. 

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