Resistance-in-series concept

A more simple approach to describe the membrane flux in case of a gel or a cake layer formation is based on the use of the resistance-in-series concept. According to this model, the osmotic pressure of the feed solution can be neglected and the permeate flux can be expressed as ... 

The resistance-in-series concept. The outward movement of a solute molecule under non-particle resuspending conditions is potentially controlled by transport resistances on both sides of the interface. Water is the continuous phase from the particle surface where the local equilibrium assumption (LEA) applies, through the bed porewater pathways and then through the interface to the water column. Derived elsewhere (9) the resistance-in-series flux algorithm is shown below. Using a concentration in solution difference it is... 

The overall mass-transfer coefficient. As shown above the resistance-in-series concept provides a means for combining transport processes on either side of the interface. Defined as the ratio of the flux to the concentration on particles, Ks, the overall sediment soluble release mass-transfer coefficient (L/t) is... 

The purpose of this section was to review the known or suspected individual processes for moving soluble chemicals near and within bed sediments. From this set those that 1) are supported by data, 2) have a sound theoretical foundation and 3) have algorithms for making computations were selected as the candidate theoretical processes. They are listed in Table I. The well known and accepted resistance-in-series concept was used to connect processes on either side of the interface and to represent the combined contributions as the overall mass-transfer coefficient. Some coefficients are normally defined using a solute concentration basis and some are defined using a particle concentration basis. The particle concentration form was chosen to express all the coefficients so that numerical values can be compared directly. 

The essence of the interface compartment concept has existed in the chemical engineering literature for decades (see Cussler, 1997 Bird et al., 2002). The idea originated and evolved through the efforts of Nernst, Whitman, and Lewis over the 20-year time-period 1904-1924. One current chemical engineering application is for gas-to-liquid mass transfer where it appears as a step in the derivation of the overall MTC, relating it to the individual phase MTCs (resistance-in-series concept). Another use is estimating concentrations at the gas-liquid interface. 

This equation expresses the two-film concept of Whitman (40), the numerator being a concentration driving force and the denominator being the sum of two mass transfer resistances in series. Analogous expressions may be written for two liquid film resistances in series. Figure 1 illustrates qualitatively the concentration profiles through two liquid films. 

Table 3.1 illustrates that the separation between the different processes is not precise, as the processes overlap. Therefore, filtration and separation models are generally applicable to mote than one process. Often several phenomena are operative simultaneously and which one dominates depends on the membrane and the solute or particle in question. Concepts such as the resistance-in-series model, the osmotic pressure model or concentration polarisation are principles which are applicable to any membrane operation. These wiU be described in the MF section. 

The usual and the most established design method of catalytic slurry reactors consists of a concept which assumes resistances in series(see Figure ).For instance,for a first order reaction this treatment given ... 

Figure 9.14 illustrates another use of the Safe Zone construct and the concept of a threshold power. In this assessment, simulations were performed for two otherwise identical 2.6-Ah 18650 cells having different anode materials one with MCMB 2528 (25 pm mean diameter, low surface area), and one with MCMB 628 (6 pm mean diameter, high surface area). The surface heat transfer coefficient was assumed to be 11 W/m -K. The FEA model for this particular set of simulations also accounted for the kinetics of separator shutdown. In this simulation, the short, rather than being modeled as a constant power source, is represented as a constant 25-mf2 resistance in series with the cell s 50-mQ equivalent series resistance. The total power dissipated in the cell due to the short (both i R heating in the short and Joule heating in the jelly roll) was calculated as 220 W initially. The total power dissipated in the short was then set to decrease with time based on a function that... 

This concept of resistances in series can be applied to various processes (see also chapter V6.2 and VI.4.4.1) and the approach is to determine the mass transfer coefficients by means of the semi-empirical relationships given in table VII. 1 (see also ref. [ 1 ]). When the resistances in the boundary layers are small compared to that of the membrane resistance the permeation rate is given by eq. VII - 49. 

Although the basic features of tiiis process are the same as those of the spherical cavity (Fickian diffusion through a variable area), we use it to cast a somewhat wider net by considering a composite cylinder made up of different materials witii different dijfusivities. This leads us to the concept of resistances in series, and the principle that flows from it, the additivity of resistances. Botii the concept and the principle are so pervasive in mass (and heat) transfer tirat tiiey deserve an early introduction. 

The above alternative scenario development for liquid-to-gas phase chemical transport has a long history of successful application in the area of chemical separation in the chemical process industries (Bird et al., 2002). It gives support to the assumptions behind the interface compartment concept. However, the resistance-in-series algorithm shown in Equation 4.16 will work correctly only if the flux equations are of the diffusive concentration-difference-type. 

When a process consists of several individual steps in series, the concept of the ratecontrolling step simply states that one of the several steps involved provides the major resistance to the overall process. In such a case, this slow step is termed the ratecontrolling step and can be considered as acting alone (Levenspiel, 1972). 

The resistances of electric resistors arranged in series may be added to obtain the overall circuit resistance. This concept also applies for a heat... 

In practice we often encounter plane walls that consist of several layers of different materials. The tbermal resistance concept can still be used to detennine the rate of steady heat transfer through such composite walls. As you may have already guessed, this is done by simply notiifg that the conduction resistance of each wall i.s IJkA connected in series, and using the electrical analogy. That is, by dividing the temperature difference between two surfaces at known temperatures by the total thermal resistance between them. 

According to Cheryan [6], neither of the two models discussed the entire pressure flux behavior observed during typical UF and MF. A better approach that describes the entire pressure flux behavior observed during typical UF or MF is the resistance in the series concept. For an ideal membrane and feed solution, flow rate through the membrane can be rewritten as follows ... 

To visualize the concept of the rate-determining step, electrical circuit analogues are useful. For a consecutive reaction, an electrical circuit with a series of two or more resistances and power can be used, and for a parallel reaction, parallel resistance and power as shown in Figure 5.10 can be used. For consecutive reactions with three reaction steps, the electric circuit has three resistors, Kj, Rj, and in series in addition, the power source (a fuel cell or a battery) has an internal resistance, R. The current (f) through the electrical circuit is given by the expression... 

Water flows (viscosity, /r = 1 X 10 Pa-S) through two channels of same length (L = 50 mm) connected in series at a flow rate of 10 " m /s. Both channels have same width, that is, 5 mm. However, height of channel-1 is 50 pm and channel-2 is 1 mm. Can you use the resistance concept to calculate the total pressure drop inside the channel Explain your answer. 

The mathematics of mass diffusion within a single phase has thus been well estab-Ushed. Diffusion between phases such as air and water was not fully understood until 1923 when Whitman proposed the two-film theory in which transfer is expressed using two mass transfer coefficients in series, one for each phase (Whitman, 1923). This concept has been rediscovered in pharmacology and probably in other areas and is now more correctly termed the two-resistance theory. ... 

The Ragone concept is based on the equivalent circuit of an electrical storage system connected to a load or a generator. The battery can be modeled by a pure power source of uniform potential Foe in series with the internal resistance, R, generating ohmic losses when the current i draws the load represented by the resistance / l so that the potential across the load is given by the basic Ohm s law ... 

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