Resistances in series

Consider heat transfer through the inner pipe of a double-pipe heat exchanger. Let s take a slice out of the heat exchanger and examine the temperature profile along a radius to the pipe at one particular axial position. 

One can identify three regions in which the temperature varies sharply. These correspond to the three resistances to heat transfer the pipe wall itself, the fluid film inside the pipe and the fluid film outside the pipe. These resistances act in series. The total resistance is related to the resistance of each film by  

That resistances add in this way is a very important concept, so I m going to take the time to prove it to you again. This will also give us an opportunity to review the definitions of the quantities. 

Consider the temperature profile along some line drawn perpendicular to the pipe wall through the center of the pipe. These temperature variations drive heat transfer down the temperature hill. The rate dq of heat transfered through some small element of the inner surface of the pipe having area dAj is 

At steady-state, this same rate of heat transfer must also occur through the pipe wall itself  

Conduction with Resistances in Series A steady-state temperature profile in a planar composite wall, with three constant thermal conductivities and no source terms, is shown in Fig. 5-3a. The corresponding thermal circuit is given in Fig. 5-3b. The rate of heat transfer through each of the layers is the same. The total resistance is the sum of the individual resistances shown in Fig. 5-3b  

Additional resistances in the series may occur at the surfaces of the solid if they are in contact with a fluid. The rate of convective heat transfer, between a surface of area A and a fluid, is represented by Newton s law of cooling as 

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Expressions similar to equations 6 and 7 may be derived in terms of an overall Hquid-phase driving force. Equation 7 represents an addition of the resistances to mass transfer in the gas and Hquid films. The analogy of this process to the flow of electrical current through two resistances in series has been analyzed (25). 

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... 

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... 

Pitting or local thinning of the measured element effectively puts a high resistance in series with the rest of the element and thus gives a highly inflated corrosion rate. 

The electrical characteristics of the cell and electrode will comprise both capacitative and resistive components, but for simplicity the former may be neglected and the system can be represented by resistances in series (Fig. 19.36 > and c). The resistance simulates the effective series resistance of the auxiliary electrode A.E. and cell solution, whilst the potential developed across by the flow of current between the working electrode W.E. and A.E. simulates the controlled potential W.E. with respect to R.E. 

The model postulates two significant resistances in series diffusion through the growing shell ( R.jyp) and polymerization at the catalyst surface (R(jat catalytic reaction resistance,... 

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. 

First, consider the transepidermal route. The fractional area of this route is virtually 1.0, meaning the route constitutes the bulk of the area available for transport. Molecules passing through this route encounter the stratum corneum and then the viable tissues located above the capillary bed. As a practical matter, the total stratum corneum is considered a singular diffusional resistance. Because the histologically definable layers of the viable tissues are also physicochemically indistinct, the set of strata represented by viable epidermis and dermis is handled comparably and treated as a second diffusional resistance in series. 

The paracellular pathway consists of the tight junction (TJ) in series with the tortuous lateral space (LS) (Figs. 6 and 7), i.e., mass transfer resistances in series ... 

The rectangular isotherm has received special attention. For this, many of the constant patterns are developed fully at the bed inlet, as shown for external mass transfer [Klotz, Chem. Revs., 39, 241 (1946)], pore diffusion [Vermeulen, Adv. Chem. Eng., 2,147 (1958) Hall et al., Ind. Eng. Chem. Fundam., 5,212 (1966)], pore diffusion with film resistance in series [Weber and Chakraborti, AIChE J. 20, 228 (1974)], the... 

The overall diffusivity, D, is obtained from Z)AB and DK by means of the conventional expression for resistances in series ... 

Figure 9.7 shows concentration profiles schematically for A and B according to the two-film model. Initially, we ignore the presence of the gas film and consider material balances for A and B across a thin strip of width dx in the liquid film at a distance x from the gas-liquid interface. (Since the gas-film mass transfer is in series with combined diffusion and reaction in the liquid film, its effect can be added as a resistance in series.)... 

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 ... 

Since the matrix, interface and boundary layer are visualized as offering resistance in succession to mass transport (14), the resultant resistance is obtained by combining the individual resistance in series. On this basis, the expression for the overall rate thus becomes ... 

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

Here we have two resistances in series, of the gas film and of the liquid film. Thus, as shown in Fig. 23.1, the rate of transfer of A from gas to liquid is given by the rate expressions, for the gas film... 

Figure 1.4 gives an example of the adsorption of a compound to suspended sediment, modeled as two resistances in series. At first, the compound is dissolved in water. For successful adsorption, the compound must be transported to the sorption sites on the surface of the sediment. The inverse of this transport rate can also be considered as a resistance to transport, Ri. Then, the compound, upon reaching the surface of the suspended sediment, must find a sorption site. This second rate parameter is more related to surface chemistry than to diffusive transport and is considered a second resistance, R2, that acts in series to the first resistance. The second resistance cannot... 

Using resistance-in-series model (Bosanquet formula), the diffusivity in pore... 

The values of k and x are usually known, while the values of and can be estimated, as stated later. It should be noted that, as in the case of electrical resistances in series, the overall resistance for heat transfer is often... 

Colloidal platinum may be prepared very easily by arcing under distilled water between platinum electrodes (Fig. 24). The current is taken from an ordinary 110-volt lighting circuit through an ammeter and an adjustable resistance in series or simply through a known... 

And the overall resistance will be the sum of three resistances in series, the bubbles, the solid, and the reaction as ... 

The reciprocals in Eq. (8.14) may be interpreted as resistances to heat transfer, and so it appears that thermal resistances in series are additive. 

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