The Closest Continuer Schema is Compatible With the Principle of Independence

Common Cause Analysis

Peng Wang , in Civil Aircraft Electrical Power System Safety Assessment, 2017

6.2.1 Common Mode Analysis Inputs

CMA is generally aimed at catastrophic and hazardous failure conditions.

The CMA inputs include:

•

CMA checklist

•

Independence principles and/or independence requirements identified by the PSSA (or PASA at aircraft level)

•

Some characteristics of the system, including characteristics relevant to system operation and installation. It mainly includes the following contents:

•

System design architecture and installation plan

•

Characteristics of the equipment and its components

•

Maintenance and test tasks

•

Crew procedures

•

The technical specifications of systems, equipment, software, etc.

When considering the system characteristics, it is necessary to understand some safety precautions that eliminate the common causes, including the following:

•

Differences (nonsimilarity, redundancy, etc.) and isolation

•

Test and preventive maintenance procedures

•

Design control and prevention used in the design process (quality procedures, design reviews, etc.)

•

Review of procedures and technical specifications

•

Personnel training

•

Quality control, etc.

From above, we can sort out the data sources and supporting documents of the input information, including the following:

•

CMA checklist

•

System requirements documents

•

System architecture and design description documents

•

System interface description documents

•

AFHA report and SFHA report

•

System PSSA report

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Mechanisms of Water Transport Across Cell Membranes and Epithelia

Luis Reuss , in Seldin and Giebisch's The Kidney (Fourth Edition), 2008

High P_os/P_d

If the membrane contains pores, then osmotic water flow does not occur via a diffusion-like mechanism, which obeys the independence principle. Instead, it involves some form of interaction between water molecules, either Poiseuille-like (viscous) flow in thin capillaries or single-file transport. As discussed previously, in both cases the flow of individual water molecules depends on the flow of other water molecules. It follows that the value of P_os/P_d is significantly greater than unity (the ratio is proportional to the square of the pore radius in case of viscous flow and equal to the number of water molecules contained in the case of single-file transport). However, unstirred-layer effects can cause a disproportionately large underestimation of Pd relative to P_os, with the end result of an artifactually large ratio. This error can be prevented by measuring the unstirred-layer equivalent thickness and making appropriate corrections for P_os and P_d (25). In conclusion, a correctly obtained P_os/P_d > 1 is strong evidence for water transport via pores. The value of P_os/P_d can denote either the pore radius or the number of water molecules in the pore. Resolving this issue requires additional experimental work, such as permeation studies with solutes of varying sizes to estimate the pore radius.

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Re-Entry Operations Safety

Paul D. Wilde , ... Nicholas Johnson , in Safety Design for Space Operations, 2013

Simple Object Aerodynamics

The bodies to be treated are in principle blunt and the destructive re-entry occurs at hypersonic Mach values. Therefore the Mach number independence principle applies for the drag coefficient. The drag coefficient, C _D, only depends on the geometric parameters in continuum and free molecular flow, but in the rarefied transitional flow a bridging method is used.

Table 9.3.3 summarizes for a sphere and for cylinders the drag coefficient dependence on geometry and attitude motion.

Table 9.3.3. Aerodynamic drag coefficients for spheres and cylinders in continuum and free molecular flow

Object shape motion	Continuum flow	Free molecular
Sphere, Aref = (π/4)D2	CD,CF = 0.92	CD,FM = 2.0
Cylinders, Aref = D∗L
Broadside and spinning	CD, CF = 1.22	CD,FM = 2.0
End-over-end tumbling, spinning	CD, CF = 0.519 +0.556D/L	CD,FM =1.273+D/L
Random tumbling and spinning	CD, CF = 0.720 +0.326D/L	CD,FM = 1.570 +0.785D/L
End-on and spinning	CD, CF = 1.307D/L	CD,FM = 1.570D/L

Rarefied transitional flow regime is usually covered by a bridging relation, which gives for Kn<<1 the continuum drag and for Kn >>1 the free molecular drag coefficient. If one uses the normalized drag coefficient definition as shown in the following equation the bridging relation has the following limiting values: continuum flow F_bridge = 0, and free molecular flow F_bridge = 1.

$\frac{C_D(Kn)-C_{Dcont}}{C_D-C_{Dcont}}=F_{bridge}(Kn)$

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Multiscale Numerical Simulation of Heart Electrophysiology

Andres Mena , Jose A. Bea , in Advances in Biomechanics and Tissue Regeneration, 2019

7.2.3.2.2 Goldman-Hodgkin-Katz Equation

Assuming that the cell membrane is permeable to a single ion only is not valid. However, it is assumed that when several permeable ions are present, the flux of each is independent of the others (known as the independence principle). According to this principle, and assuming: (i) the membrane is homogeneous and neutral, and (ii) the intracellular and extracellular ion concentrations are uniform and unchanging, the membrane potential is governed by the well-known Goldman-Hodgkin-Katz equation. For N monovalent positive ion species and for M monovalent negative ion species, the potential difference across the membrane is as follows

(7.24) $\begin{array}{l}\hfill E=-\frac{RT}{F}log\left(\frac{\sum _{j=1}^N{P}_j{[c_j^{+}]}_i+\sum _{j=1}^M{P}_j{[c_j^{-}]}_e}{\sum _{j=1}^N{P}_j{[c_j^{+}]}_e+\sum _{j=1}^M{P}_j{[c_k^{-}]}_i}\right),\end{array}$

where [c _j ] _i and [c _j ] _e are the intracellular and extracellular concentrations for the jth ion, P _j is the permeability of the intracellular and extracellular concentrations for the jth ion, and E is the membrane potential. The permeability for the jth ion is defined as

$\begin{array}{l}\hfill P_j=\frac{D_j{\beta }_j}{h},\end{array}$

where h is the thickness of the membrane, D _j is the diffusion coefficient, and β _j is the water partition coefficient of the membrane. Both D _j and β _j depend on the type of ion and the type of membrane.

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Chemorheology of Benzoxazine-based Resins

Sarawut Rimdusit , ... Wanchat Bangsen , in Handbook of Benzoxazine Resins, 2011

3.1 Rheological Property Measurement for Gel Point Determination

Two major types of rheological measurements can be used to determine the gel point of materials: i.e., steady shear test and oscillatory shear test [6,10 ]. The steady shear experiment is highly appealing to some researchers due to its simplicity, as the gel point is coincident with the divergence of steady shear viscosity and the appearance of the equilibrium modulus. Extrapolation of the data is needed, however, the exact gel point not being accessible by any steady mode rheometers. As a result, the oscillatory shear experiment is preferable because minimum deformation is applied to the material, particularly the delicate gel materials, at the gel point. The use of the frequency independence principle of the loss tangent to define the gel point makes the oscillatory experiment even more powerful and accurate [ 6,43–63]. The gel point can be determined from various rheological criteria including (1) the point where the storage modulus, G′, and the loss modulus, G″, of the dynamic mechanical analysis are equal or tan δ = 1 [64], (2) the maximum in the tan δ [65,66], (3) the crossing point between the tangent line of the elastic modulus curve and the baseline G′ = 0 [66], (4) the onset of decrease in the rate of growth of the viscous modulus during the curing process of the polymer [67], and (5) the point at which tan δ is independent of the frequency [68]. In this chapter, the last criterion has been used to determine the gel point of the benzoxazine-based resins.

FTMS is one of the rheometric techniques used recently by many researchers as an effective tool for the gelation studies of both chemical and physical networks [43–63]. The technique is based on the Winter and Chambon hypothesis of the fractal geometry of the gel network. It was observed experimentally that at a gel point, the stress relaxation modulus, G(t), can be represented by a power law, i.e., G(t) ∼ t ⁻ⁿ [54]. The Laplace transform of this time domain stress relaxation relation together with the Kramers-Kronig relationship of storage modulus, G′, and loss modulus, G″, results in the conclusion that at gel point, storage and loss moduli will behave with the same scaling law with frequencies, i.e., G′, G″ ∼ ω ⁿ .

Consequently, tan δ = G′/G″ = tan(nπ/2) is frequency independent at the gel point [6,36,43]. By knowing the value of tan δ at the gel point, the viscoelastic exponent, n, can be directly obtained. The values of n lie primarily in the range of 0.4-0.8 and have been reported to be dependent on the thermal history and concentration of the sol fraction [6,36,43–63]. By applying multiple waveforms, which is the superposition of mechanical waves of different frequencies, to a rheometer, i.e., a parallel plate rheometer in this study, the gel point can be determined in a single experiment. The above technique eliminates the unwanted curing time variation of the multiple-run experiments of different frequencies [36].

In this investigation, the composite waveform was obtained by the superposition of mechanical waves with five different frequencies, i.e., 1, 3, 10, 30, and 100   rad/s. The strain amplitude is constant for each frequency, i.e., 2.5%, etc. For simplicity, Figure 10 and Figure 11 exhibit the mechanical sine waves of only three different frequencies, i.e., 1, 3, and 10   rad/s and their corresponding multiple waveforms. Experimentally, a discrete Fourier transform is performed on the obtained stress data to provide the individual stress values for each frequency used. Finally, the tan δ value is determined from the known input strain and the obtained stress.

Figure 10. Mechanical sine waves with strain amplitude of 2.5%.

Figure 11. Complex waveform resulting from the superposition of mechanical waves in Figure 10.

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Real-time adjustment of zinc powder dosage based on fuzzy logic

Chunhua Yang , Bei Sun , in Modeling, Optimization, and Control of Zinc Hydrometallurgical Purification Process, 2021

5.1 Copper removal performance evaluation based on ORP

In order to precisely control the copper ion concentration to its set-point, the zinc powder dosage has to be determined accurately to account for the actual need of copper removal [1] [2]. In practice, the operators evaluate the operating condition and adjust the zinc powder dosage according to the inlet and outlet copper ion concentrations. However, with the discrete-delayed measurement of copper ion concentrations, the operation is not able to follow the variation of inlet and outlet copper ion concentrations in time, which may lead to instability of the process [3]. According to the kinetics of multielectrode reactions, ORP, which is related to the reaction rate, is a real-time external indicator of the internal reaction state. Therefore, ORP can be utilized in the monitoring and adjusting of the copper removal process [4] [5] [6].

5.1.1 Relationship between copper ion concentration and ORP

Before evaluating the copper removal performance based on ORP, the relationship between ORP and the outlet copper ion concentration has to be analyzed. The copper removal process mainly involves two oxidation–reduction reactions:

(i)

Copper cementation: $\mathrm{CuS}{\mathrm{O}}_{\mathrm{4}}+\mathrm{Zn}\to \mathrm{ZnS}{\mathrm{O}}_{\mathrm{4}}+\mathrm{Cu}$ .

(ii)

Copper comproportionation: $\mathrm{CuS}{\mathrm{O}}_{\mathrm{4}}+\mathrm{Cu}+{\mathrm{H}}_{\mathrm{2}}\mathrm{O}\to \mathrm{C}{\mathrm{u}}_{\mathrm{2}}\mathrm{O}\downarrow +{\mathrm{H}}_{\mathrm{2}}\mathrm{S}{\mathrm{O}}_{\mathrm{4}}$ .

The two reactions follow first-order chemical reaction dynamics, whose reaction rate can be expressed as

(5.1) $r_i=-k_{i}C,$

where $r_i$ is the reaction rate of the ith main reaction ( $i=1,2$ ), C is the outlet copper ion concentration, and $k_i$ is the rate constant of the ith main reaction ( $i=1,2$ ). Based on the independence principle of coexisting chemical reactions and mass balance principle, the reaction rate of copper ions can be expressed as

(5.2) $\frac{dC}{dt}=r_1+r_2=-(k_1+k_2)C.$

According to the Arrhenius equation, the activation energy $E_{a\cdot i}$ can affect the rate constant $k_i$ :

(5.3) $k_i=A_i\exp (-\frac{E_{a\cdot i}}{RT}),$

where $A_i$ and $E_{a\cdot i}$ are the preexponential factor and activation energy of the ith main reaction, respectively, R is the gas constant, and T is the absolute temperature.

From the perspective of the electrode reaction, the main electrochemical reaction on the anode is

(5.4) $\mathrm{Zn}\to \mathrm{Z}{\mathrm{n}}^{\mathrm{2}+}+\mathrm{2}{\mathrm{e}}^{-},E^0=-0.736\text{V}.$

On the cathode, the reactions include

(5.5) $\mathrm{C}{\mathrm{u}}^{\mathrm{2}+}+\mathrm{2}{\mathrm{e}}^{-}\to \mathrm{Cu},E^0=0.337\text{V},$

(5.6) $\mathrm{2C}{\mathrm{u}}^{\mathrm{2}+}+{\mathrm{H}}_{\mathrm{2}}\mathrm{O}+\mathrm{2}{\mathrm{e}}^{-}\to \mathrm{C}{\mathrm{u}}_{\mathrm{2}}\mathrm{O}+{\mathrm{H}}^{+},E^0=0.203\text{V}.$

According to the independency principle of parallel electrode reactions, these reactions share a common electrode potential. The electrode potential, also known as mixed potential, determines the reaction rate by affecting the activation energy or the electron transfer rate between oxidant and reductant [7] [8].

According to the kinetics of the electrode reaction, the actual activation energies of the anode and cathode reactions can be formulated as [9]

(5.7) $\{\begin{array}{c}{E}_{ea}=E_{ea}^0-(1-\lambda )nF(e_{mix}-e_{eq}),\hfill \\ E_{ec}=E_{ec}^0-\lambda nF(e_{mix}-e_{eq}),\hfill \end{array}$

where $E_{ea}^0$ and $E_{ec}^0$ are the standard activation energy of the anode and cathode electrode reactions, respectively, $e_{mix}$ is the mixed potential, $e_{eq}$ is the standard equilibrium potential, n is the number of electron moles, F is the Faraday constant, and λ and ( $1-\lambda$ ) are the coefficients describing the effects of electrode potential variation on the activation energy of the cathode and anode electrode reactions [9].

There are two opinions on the relationship between mixed potential and ORP. One is that ORP and mixed potential exhibit a linear relationship:

(5.8) $e_{mix}=p{e}_{\mathit{orp}}-q,$

where p and q are parameters to be identified. The identification result indicates that p is close to 1 and q is close to 0. The other opinion is that ORP equals the mixed potential. Then, by combining Eq. (5.2) and Eq. (5.8), the mathematical relationship between ORP and copper ion concentration is

(5.9) $\ln \left(\frac{C_t}{C_0}\right)=t[A_1\exp (-\frac{E_{e\cdot 1}+2\lambda F(p{e}_{\mathit{orp}}+q-e_{eq\cdot 1})}{RT})+A_2\exp (-\frac{E_{e\cdot 2}+\lambda F(p{e}_{\mathit{orp}}+q-e_{eq\cdot 2})}{RT})],$

where R, F, $e_{eq\cdot 1}$ , and $e_{eq\cdot 2}$ are constant; the values of $C_0$ and T are obtained by measurement; $E_{e\cdot 1}$ , $E_{e\cdot 2}$ , λ, $A_1$ , $A_2$ , p, and q are unknown and need to be identified by minimizing the difference between the predicted outlet copper ion concentration ( $C_{kinetic}$ ) and the measured concentration ( $C_{real}$ ):

(5.10) $\min \frac{1}{n_T}\sum _{k=1}^{n_T}{[C_{kinetic}(k)-C_{real}(k)]}^2,$

where $n_T$ is the number of identification samples.

5.1.2 ORP-based process evaluation

Analyzing ORP is a feasible approach to evaluate the reaction state of the copper removal process [6]. However, copper removal is a complex process with various influencing factors. Other impurities in the solution can also affect the reaction state, which introduces uncertainties in the relationship between ORP and copper removal performance. To handle these uncertainties, fuzzy logic is selected for process evaluation [10] [11] [12].

ORP is an indicator of the current state of the copper removal process, while the variation of ORP indicates the trend of the copper removal performance [13]. If we only utilize the current value of ORP in process evaluation, then the obtained evaluation result is inaccurate, which misguides the process operation. For example, if the ORP is more positive than its normal upper limit, then the process is considered to be in a serious state (extreme shortage of zinc powder) and more zinc powder is required. However, if the inlet copper ion concentration decreases sharply and the required zinc powder dosage is decreased accordingly, then the process will enter another serious state (excessive zinc powder). Therefore, both ORP and its variation trend should be utilized in the process evaluation.

In the evaluation process, ORP and its trend are classified into several fuzzy sets with different fuzzy membership functions. The kinetic model and the production limitation of the concentrations are used to determine the parameters of these membership functions. After the fuzzification of ORP and its trend, a set of fuzzy inference rules is adopted to evaluate the process performance [14].

The evaluation schedule mainly consists of four stages. The first stage is preparing the input variables. A linear regression method is applied to extract the real-time trend of ORP. The trend extraction includes the following steps.

Step 1:

Determine the initial window size of the time series as W, and set the threshold θ.

Step 2:

In the time window, build a linear model to approximate the time series of ORP:

(5.11) $\hat{x}(k)=a(k-k_0)+b,$

where k is the time step in the window, $k_0$ is the initial time step, $\hat{x}(k)$ is the approximated value, and a and b are parameters of the linear regression model, whose values are determined by minimizing the sum of least-squares:

(5.12) $S=\sum _{i=0}^W{(\hat{x}(k_i)-x(k_i))}^2,$

where $x(k_i)$ is the actual value of ORP at time step $k_i$ in the original window. The fitted parameter a represents the initial ORP trend. The singular value decomposition method is used to account for the matrix singularity.

Step 3:

Add a new data point to the window.

Step 4:

Calculate the sum of least-squares with the fitted model over the new windows. If S is less than θ, then repeat step 3. Otherwise, the new data point will be treated as the initial data of the new window, and step 2 is repeated.

In stage 2, the input variables are converted to linguistic variables, which is known as fuzzification. Three types of membership functions are used in this stage.

Type 1:

The generalized bell membership function:

${\mu }_i^j(\chi )=\frac{1}{1+|(x-{\delta }_i^j)/w_i^j{|}^{2m}},$

where ${\delta }_i^j$ and $w_i^j$ ( $i=-3,\cdots ,3$ , $j=1,2$ ) are the center and amplitude of the curve, respectively, and m is a positive constant.

Type 2:

Z-shaped membership function:

${\phi }^j(\chi )=\{\begin{array}{cc}1,\hfill & x⩽{\alpha }^j,\hfill \\ 1-2{\left(\frac{\chi -{\alpha }^j}{{\beta }^j-{\alpha }^j}\right)}^2,\hfill & {\alpha }^j⩽\chi ⩽\frac{{\alpha }^j+{\beta }^j}{2},\hfill \\ 2{\left(\frac{\chi -{\beta }^j}{{\beta }^j-{\alpha }^j}\right)}^2,\hfill & \frac{{\alpha }^j+{\beta }^j}{2}⩽\chi ⩽{\beta }^j,\hfill \\ 0,\hfill & \chi ⩾{\beta }^j,\hfill \end{array}$

where both ${\alpha }^j$ and ${\beta }^j$ are parameters deciding the slope of the curve ${\phi }^j(\chi )$ .

Type 3:

S-shaped membership function:

${\psi }^j(\chi )=\{\begin{array}{cc}0,\hfill & x⩽c^j,\hfill \\ 2{\left(\frac{\chi -c^j}{d^j-c^j}\right)}^2,\hfill & c^j⩽\chi ⩽\frac{c^j+d^j}{2},\hfill \\ 1-2{\left(\frac{\chi -d^j}{d^j-c^j}\right)}^2,\hfill & \frac{c^j+d^j}{2}⩽\chi ⩽d^j,\hfill \\ 1,\hfill & \chi ⩾d^j,\hfill \end{array}$

where both $c^j$ and $d^j$ are parameters deciding the slope of the curve ${\psi }^j(\chi )$ .

According to the reaction kinetics, the increase of ORP indicates an increasing copper ion concentration and vice versa. However, the reflection is vague. The fuzzified value should be of physical meaning and suitable for control. Therefore, the definition of those membership functions should take the limitation of the outlet copper ion concentration into account. Denote the limitation of the outlet copper ion concentration as $[L_{\mathit{MIN}},L_{\mathit{MAX}}]$ . The parameters in the ORP membership function are calculated according to the limitations by using $e_{\mathit{orp}}=g(C_{C{u}^{2+}})$ , which is the inverse function of Eq. (5.9). To make the evaluation more intuitive, the fuzzy sets are labeled by fuzzy languages indicating reaction states of the copper removal process. If the outlet copper ion concentration is at the center of production limitations and ORP is in a stable situation, then it is abbreviated as S. Similarly, if ORP is relatively low or very low, it is labeled as little low (LL) or very low (VL). If ORP is relatively high or very high, it is labeled as little high (LH) or very high (VH).

The trend of ORP indicates its underlying variation, which determines the future value of ORP. A sharp trend can force ORP to return to a stable situation from an unstable one within minutes. It can also cause the deviation of ORP from an originally stable situation within seconds. Therefore, the parameters of the ORP trend membership functions are determined according to the influence of the ORP trend on the ORP state. Denote the time threshold for ORP trend analysis as τ(min). If ORP skips or drops from one condition to a higher or lower situation during τ minutes, its trend is viewed as low positive or low negative. If ORP skips or drops from a stable situation to the highest or lowest situation, its trend is labeled as high positive or high negative. The relevant information of fuzzification of ORP and its trend are presented in Tables 5.1 and 5.2. Fig. 5.1(a and b) shows the shapes of the memberships.

Table 5.1. Parameter calculation and fuzzy language definition for ORP.

Label	Meaning	Function	Calculation of parameters
VL	Very low	φ 1(x)	${\alpha }^1=g(L_{\mathit{MIN}}),{\beta }^1={\alpha }^1-p_2{\omega }_{-1}^1$
LL	Little low	${\mu }_{-3}^1(x)$	${\sigma }_{-1}^1=g(q_1{L}_{\mathit{MIN}}+(1-q_1)L_{\mathit{MAX}}),{\omega }_{-1}^1=p_1({\sigma }_0^1-{\alpha }^1),m=2$
S	Stable	${\mu }_0^1(x)$	${\sigma }_0^1=g(q_0{L}_{\mathit{MIN}}+(1-q_0)L_{\mathit{MAX}}),{\omega }_0^1=p_0({\sigma }_1^1-{\sigma }_{-1}^1),m=2$
LH	Little high	${\mu }_1^1(x)$	${\sigma }_1^1=g(q_1{L}_{\mathit{MIN}}+(1-q_1)L_{\mathit{MAX}}),{\omega }_1^1=p_1(d^1-{\sigma }_0^1),m=2$
VH	Very high	ψ 1(x)	$d^1=g(L_{\mathit{MAX}}),c^1=d^1+p_2{\omega }_1^1$

Table 5.2. Parameter calculation and fuzzy language definition for the trend of ORP.

Grade	Meaning	Function	Calculation of parameters
HN	High negative	φ 2(x)	${\alpha }^2=({\alpha }^1-d^1)/\tau ,{\beta }^2={\alpha }^2-r_2{\omega }_{-1}^2$
LN	Low negative	${\mu }_{-1}^2(x)$	${\sigma }_{-1}^2=({\sigma }_{-1}^1-{\sigma }_1^1)/\tau ,{\omega }_{-1}^2=r_1({\sigma }_0^2-{\alpha }^2),m=2$
Z	Zero	${\mu }_0^1(x)$	${\sigma }_0^2=0,{\omega }_0^2=r_0({\sigma }_1^2-{\sigma }_{-1}^2),m=2$
LP	Low positive	${\mu }_1^1(x)$	${\sigma }_1^2=({\sigma }_1^1-{\sigma }_{-1}^1)/\tau ,{\omega }_1^2=r_1(d^2-{\sigma }_0^2),m=2$
HP	High positive	ψ 2(x)	$d^2=(d^1-{\alpha }^1)/\tau ,c^2=d^2+r_2{\omega }_1^2$

Figure 5.1. Fuzzy logic process evaluation based on ORP.

In stage 3, a set of rules is applied on the fuzzy sets obtained in stage 2 for evaluating the copper removal process. Both ORP and its trend are considered in the process evaluation. If both ORP and its trend are in an acceptable situation without fluctuation, then the copper removal process is considered in a stable state. If ORP is in an acceptable situation, however, with a sharp trend, then the process is not as stable as the value of ORP indicated. Similarly, if ORP exceeds its normal limitation, however, with a good trend leading to an acceptable situation, then the process situation is not as bad as it seems to be [15]. Therefore, according to this idea, the fuzzy inference rules for process evaluation are formulated, as shown in Fig. 5.1(c) [16]. The evaluation linguistic variables are obtained by the Max-Prod operator following the Larsen fuzzy inference method [17].

In stage 4, the linguistic variables are converted into a single numerical value, i.e., the evaluation grade. The centroid defuzzification method is applied where the crisp value of the output variable is computed by finding the center of gravity of the membership function for the fuzzy value [18] [19]. The output value is calculated as

(5.13) $y^{⁎}=\frac{{\sum }_{i=1}^n\mu (y_i)y_i}{{\sum }_{i=1}^n\mu (y_i)}.$

A more positive grade means a situation with a higher concentration and vice versa. An evaluation grade closer to zero indicates a more stable process condition. Fig. 5.1(d) shows the evaluation grade surface of the process, which displays the relationship between two inputs (ORP and its trend) and the response output (evaluation grade).

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Mechanisms of Ion Transport Across Cell Membranes and Epithelia

Luis Reuss , in Seldin and Giebisch's The Kidney (Fourth Edition), 2008

DIFFUSION AND ELECTRODIFFUSION

Diffusion and electrodiffusion are the main processes of passive solute transport across homogeneous phases (e.g., lipid membranes or aqueous pores) by independent motion of the solute molecules. Diffusion applies to uncharged particles and electrodiffusion to ions. Although diffusion does not strictly apply to ion transport, its analysis is simpler and helps in understanding electrodiffusion.

Diffusion of a solute in aqueous solution is the result of the random thermal motion of solute molecules. Disregarding convection, if there are differences in concentration between different sectors of the solution, then random solute motion will tend to make its distribution homogeneous (equilibrating transport). All solute particles move randomly at uniform average velocities, dependent on the solution temperature. Hence, more particles will tend to move from regions of high concentration to sectors of low concentration than in the opposite direction, simply because there are more particles per unit volume in the high-concentration regions. In other words, differences in concentration cause unequal unidirectional fluxes in a regime of diffusion because of differences in the number of particles flowing in each direction per unit of time, not because of different velocities of individual particles flowing in one direction or the other. In diffusion, the molecules move independently of each other and of other particles present in the solution, that is, there is no flux coupling. This is the independence principle.

Diffusion of a nonelectrolyte in solution is described by Fick's first law (25):

(5) $J_s=-D_s\frac{d{C}_s}{dx}$

where J_s is the solute flux (moles·cm²·sec⁻¹), D_s is the solute diffusion coefficient (cm²·sec⁻¹) and dC_s/dx is the concentration gradient. The negative sign denotes the direction of the flux.

Fick's second law of diffusion considers the time course of the process:

(6) $\frac{dC}{dt}=D_s\frac{d^2{C}_s}{d{x}^2}$

where dC/dt is the rate of change in solute concentration and x denotes distance.

The average time required by diffusing particles to cover a given distance is inversely proportional to the diffusion coefficient and directly proportional to the square of the traveled distance. Einstein approximated the second law of diffusion with λ = (D_st)^1/2, where λ is the traveled distance in the x-axis. The dependence of t on the square of the distance makes diffusion a very slow transport process for long distances. For a typical D_s = 10⁻⁵ cm²·sec⁻¹, it takes 1 millisecond for the solute to diffuse 1 μm, but it takes 1000 seconds (∼16.7 minutes) to diffuse 1 mm. Convective flow (see Chapter 9) is a much more effective mass transport mechanism for long distances.

Now we consider a thin lipid membrane of thickness δ _m separating two aqueous compartments (Fig. 2). The solutions on both sides are well stirred, so that the solute concentrations are homogeneous in both. Inserting the solute partition coefficient (β _s ) to denote its lipid solubility relative to its water solubility, at the steady state, the following expression is obtained for the solute flux:

FIGURE 2. Diffusion across a membrane. A: A membrane separates two aqueous solutions (1 and 2). The dots represent molecules of a solute to which the membrane is permeable; the solute concentration (Cs) is greater in solution 1. Solute molecules move randomly in each solution and collide with the membrane with a probability proportional to the concentration. Solutes collide, dissolve in and diffusing across the membrane. The unidirectional fluxes (J_1→2 and J_2→1) are proportional to the solute concentrations in sides 1 and 2, respectively; the net flux (J_net) is proportional to the concentration difference. The concentration difference does not accelerate the molecules, and hence it is not a force, although it is usually referred to as the chemical driving force. Diffusion is passive, equilibrating transport, i.e., net transport ceases when the concentrations on both sides of the membrane are equal. B: Lines denote solute concentration profiles in the solutions and the membrane depending on the partition coefficient (β_s). When β_s = 1, solute concentration in the membrane boundaries are identical to those in the adjacent solutions; concentrations in the membrane are greater or smaller than those in the adjacent solution if β_s is greater or smaller than unity, respectively.

(7) $J_s=-\frac{D_s{\beta }_s}{{\mathrm{\delta }}_m}\text{Δ}{C}_s$

where ΔC_S is the solute concentration difference between the two solutions. Defining the solute permeability (P_s ) as P_s = D_sβ_s/δ_m, Eq. 7 reduces to:

The diffusive permeability coefficient of the membrane relates the flux to the driving force and denotes the ease with which the membrane permits mass transfer of a particular species. Its units are centimeters per tenth of a second, that is, those of velocity. In the simple case of a nonelectrolyte, under isothermal and isobaric conditions, the permeability (P) of solute s is given by a rearrangement of Eq. 8: P_s = –J_S/ΔC_S. This is the phenomenological, experimentally determined permeability, calculated by dividing the steady-state solute flux by the difference between the solute concentrations of well-stirred bathing solutions. The other definition of diffusive permeability is mechanistic and considers the factors involved in solubility-diffusion, D_S, β _S, and δ _m, as described above.

The preceding discussion considers the specific case of solubility-diffusion, but the phenomenological definition of permeability can be applied in principle to any transport mechanism. Of course, its interpretation varies. An important case is that of the permeation of certain hydrophilic nonelectrolytes through aqueous pores in the membrane. If the lipid bilayer is impermeable to the solute, then diffusive transport is entirely via the pores. The permeable area of this membrane (S_p ) is only a fraction of the total membrane area, given (for 1 cm² of membrane) by S_p = nπr ², where n is the density of homogeneous pores of radius r. The partition coefficient is unity (the solute is dissolved in water both inside and outside of the pore) and is hence eliminated from the equation:

(9) $P_s=n\pi r^2{D}_s/L$

where L is the pore length (about equivalent to mem brane thickness).

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Kinetic modeling of the competitive-consecutive reaction system

Chunhua Yang , Bei Sun , in Modeling, Optimization, and Control of Zinc Hydrometallurgical Purification Process, 2021

3.3.1 Model structure determination

In the improved copper removal process, the copper cementation and cuprous oxide precipitation reactions take place parallelly. Copper ions are consumed in both reactions and one of the products of the former reaction becomes a reactant of the latter reaction. These two reactions form a "competitive-consecutive" reaction system (CCRS):

(3.23) $A+B\stackrel{k_1}{\to }C\downarrow +D,$

(3.24) $A+C+E\stackrel{k_2}{\to }R\downarrow +H,$

where $k_1$ with $k_2$ are the rate constants of the two reactions. In the context of copper removal, the symbols A, B, C, D, E, R, and H stand for $\mathrm{CuS}{\mathrm{O}}_{\mathrm{4}}$ , Zn, Cu, $\mathrm{ZnS}{\mathrm{O}}_{\mathrm{4}}$ , ${\mathrm{H}}_{\mathrm{2}}\mathrm{O}$ , $\mathrm{C}{\mathrm{u}}_{\mathrm{2}}\mathrm{O}$ , and ${\mathrm{H}}_{\mathrm{2}}\mathrm{S}{\mathrm{O}}_{\mathrm{4}}$ , respectively.

These two reactions follow the first-order reaction kinetics. Their reaction rates can be expressed as

(3.25) $r_i=-k_i{c}_{C{u}^{2+}},$

where $r_i$ and $k_i$ are the reaction rate and rate constant and $i=1,2$ . According to the results in Sections 3.2.2 and 3.2.3,

(3.26) $k_1=\frac{6K_{\mathrm{D}}(g_{\mathrm{Zn}}+{\gamma }_{\mathrm{Zn}}{g}_{\mathrm{under}})\begin{array}{c}\hfill {\sum }_{i=1}^n{\alpha }_i{d}_i^2\hfill \end{array}}{V_{\rho }\begin{array}{c}\hfill {\sum }_{i=1}^n{\alpha }_i{d}_i^3\hfill \end{array}},$

(3.27) $k_2=4\pi R_{\mathrm{S}}^2{V}^{-1}[\frac{1}{K_{\mathrm{G}}}+\frac{(\eta -1)R_{\mathrm{S}}}{D_e}+\frac{{\eta }^2}{k_{surf}}].$

However, the reaction mechanism of a single reaction cannot describe the overall reaction system. It is required to analyze the influence of these two reactions on the copper removal performance. It can be observed from the two reactions that the two reactions compete with each other as they both use Cu²⁺ as reactant. In addition, reaction (3.5) generates reactant for reaction (3.6) . Therefore, these two reactions hold a "consecutive reaction" relationship. According to the independence principle of chemical reactions, if the orders of two reactions are equal, then the consumption rate of common reactant (Cu ²⁺) equals the sum of each reaction. However, in consecutive reactions, the consumption rate of the related material (Cu) relies not only on the reaction order, but also on the reaction constant. When the reaction orders are the same, however their reaction constant are different in order of magnitude, then the consumption rate of the related material relies on the reaction with the smaller reaction constant. When the reaction orders are the same and the reaction constants are similar, then the consumption rate of the related material equals the reaction rate of consumption minus the reaction rate of production. To sum up, the reaction rates of each reactant/product of the CCRS are:

$r_{C{u}^{2+}}=-r_1-r_2,r_{Zn}=-r_1,r_{Z{n}^{2+}}=r_1,r_{H_{2}O}=-r_2,r_{C{u}_{2}O}=r_2,r_{H^{+}}=r_2,$

$r_{Cu}=\{\begin{array}{cc}{r}_2,\hfill & k_1>>k_2,\hfill \\ -r_1,\hfill & k_1<<k_2,\hfill \\ -r_1+r_2,\hfill & \text{else},\hfill \end{array}$

where $r_{C{u}^{2+}}$ , $r_{Zn}$ , $r_{Z{n}^{2+}}$ , $r_{H_{2}O}$ , $r_{Zn}$ , $r_{C{u}_{2}O}$ , $r_{H^{+}}$ , and $r_{Cu}$ are the reaction rates of $C{u}^{2+}$ , Zn, $Z{n}^{2+}$ , $H_{2}O$ , Zn, $C{u}_{2}O$ , $H^{+}$ , and Cu, respectively.

Outlet copper ion concentrations of each reactor are the KPIs of the copper removal process. Considering a single reactor, the reaction rate of Cu²⁺ can be derived according to the above result:

(3.28) $$\begin{gathered} r_{\mathrm{C}{\mathrm{u}}^{\mathrm{2}+}}=-\{\frac{6K_{\mathrm{D}}(g_{\mathrm{Zn}}+{\gamma }_{\mathrm{Zn}}{g}_{\mathrm{under}})\begin{array}{c}\hfill {\sum }_{i=1}^n{\alpha }_i{d}_i^2\hfill \end{array}}{\rho \begin{array}{c}\hfill {\sum }_{i=1}^n{\alpha }_i{d}_i^3\hfill \end{array}} \\ +4\pi R_{\mathrm{S}}^2[\frac{1}{K_{\mathrm{G}}}+\frac{(\eta -1)R_{\mathrm{S}}}{D_e}+\frac{{\eta }^2}{k_{surf}}]\}, \end{gathered}$$

where $r_{\mathrm{C}{\mathrm{u}}^{\mathrm{2}+}}$ is the consumption rate of copper ion concentration.

The dynamics of the copper removal reactor can be described using the classical CSTR model [17]. Then according to the above kinetic study and mass balance, the dynamics of the copper removal reactors follow

(3.29) $V{\dot{C}}_{\mathrm{C}{\mathrm{u}}^{\mathrm{2}+},\mathrm{1}}=Q{C}_{\mathrm{C}{\mathrm{u}}^{\mathrm{2}+},\mathrm{1}}^0-(1+Q_{\mathrm{uf}})Q{C}_{\mathrm{C}{\mathrm{u}}^{\mathrm{2}+},\mathrm{1}}-V{r}_{\mathrm{C}{\mathrm{u}}^{\mathrm{2}+}},$

(3.30) $V{\dot{C}}_{\mathrm{C}{\mathrm{u}}^{\mathrm{2}+},\mathrm{2}}=(1+Q_{\mathrm{uf}})Q{C}_{\mathrm{C}{\mathrm{u}}^{\mathrm{2}+},\mathrm{2}}^0-(1+Q_{\mathrm{uf}})Q{C}_{\mathrm{C}{\mathrm{u}}^{\mathrm{2}+},\mathrm{2}}-V{r}_{\mathrm{C}{\mathrm{u}}^{\mathrm{2}+}},$

where ${\dot{C}}_{\mathrm{C}{\mathrm{u}}^{\mathrm{2}+},\mathrm{i}},i=1,2$ , is the reaction rate of copper ions in the ith reactor, $C_{\mathrm{C}{\mathrm{u}}^{\mathrm{2}+},\mathrm{i}}^0,i=1,2$ , is the inlet copper ion concentration of the ith reactor, and Q and $Q_{\mathrm{uf}}$ are the flow rates of zinc sulfate solution and the returned underflow, respectively. The details of model parameters are summarized in Table 3.1.

Table 3.1. Parameters in the kinetic model.

Parameter	Physical meaning
V	Volume of reactor
Q	Flow rate of leaching solution
Q uf	Flow rate of recycled underflow
${\dot{C}}_{\mathrm{C}{\mathrm{u}}^{\mathrm{2}+},\mathrm{i}}$	Reaction rate of copper ions in the ith reactor
$C_{\mathrm{C}{\mathrm{u}}^{\mathrm{2}+},\mathrm{i}}^0$	The inlet copper ion concentration of the ith reactor
$C_{\mathrm{C}{\mathrm{u}}^{\mathrm{2}+},\mathrm{i}}$	The copper ion concentration in the ith reactor
$r_{\mathrm{C}{\mathrm{u}}^{\mathrm{2}+}}$	Consumption rate of copper ions
r 1	Reaction rate of copper cementation
r 2	Reaction rate of cuprous oxide precipitation
r 2,1	Reaction rate of external diffusion in cuprous oxide precipitation
r 2,2	Reaction rate of internal diffusion in cuprous oxide precipitation
r 2,3	Reaction rate of surface chemical reaction in cuprous oxide precipitation
$C_{\mathrm{C}{\mathrm{u}}^{\mathrm{2}+},\mathrm{g}}$	The copper ion concentration in the bulk phase
$C_{\mathrm{C}{\mathrm{u}}^{\mathrm{2}+},\mathrm{s}}$	The copper ion concentration at the outer surface of the copper particles
$C_{\mathrm{C}{\mathrm{u}}^{\mathrm{2}+},\mathrm{c}}$	The copper ion concentration on the reaction surface
R S	Particle radius
R C	Distance from the core to the reaction surface
k G	Mass transfer coefficient of external diffusion of reaction (3.6)
D e	Effective diffusivity of internal diffusion of reaction (3.6)
k surf	Surface chemical reaction rate constant of reaction (3.6)
K D	Mass transfer coefficient of reaction (3.5)
S A	Surface area of reaction (3.5)
ρ	Density of zinc powder particles (3.5)
g Zn	Weight of added zinc powder
g under	Weight of recycled solids
γ Zn	Weight fraction of zinc powder in the recycled solids

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Spatiotemporal Drought Analysis and Modeling

Zekâi Şen , in Applied Drought Modeling, Prediction, and Mitigation, 2015

5.5.2 Theoretical Treatment

In the most general case, none of the sites have equal PoPs, which implies that the RF is heterogeneous. Let the wet and dry spell probabilities in a subarea be denoted by p_j and q_j , (j  =   1, 2, …, n), respectively,where n is the total number of sites and p_j   + q_j   =   1. The areal coverage of probability, P(A  = i|n), including i dry sites, can be evaluated through enumeration technique by the application of the mutual exclusiveness and exhaustiveness rules of probability theory. For the necessary derivations the conceptual model can be visualized as in Fig. 5.14, where there are n mutually exclusive sites represented by in-square boxes.

Fig. 5.14. Probability derivation conceptual model.

If out of n sites, i sites are wet and the remaining (n  − i) sites are dry, then the whole spectrum of possible cases become to have two mutually exclusive groups. The first group includes i cases with n possibility in each site, and the second (n  − i) sites again each with n possibilities. Because the wetness on each site is independent from the others, then the probability of wet spell occurrence collectively can be found from the multiplication rule of independent events as p ₁.p ₂.p ₃…p_i for one pattern. In the case of i wet spell sites, there are i(i  −   1)(n  −   2)…1   = i!, mutually exclusive patterns each with probability of wetness. Mutually exclusive events imply summation in the probability theory; therefore, the successive summation terms express all the possible, collective, and exhaustive joint probabilities at i sites. The second part on the right-hand side of Fig. 5.14 corresponds to the joint probability of the remaining (n  − i) sites each belonging to respective pattern of joint wet occurrence at i precedent patterns. Again, the independence principle of the probability theory provides the multiplication of the remaining dry occurrence sites. These explanations can be translated into a mathematical formulation as in Eq. (5.52).

(5.72) $P\left(A=i|n\right)=\frac{1}{i!}\underset{i-\mathrm{fold}}{\underbrace{{\displaystyle \sum _{j_1=1}^n{\displaystyle \sum _{\begin{array}{l}{j}_2=1\\ j_2\ne j_1\end{array}}^n\cdots {\displaystyle \sum _{\begin{array}{l}{j}_i=1\\ j_i\ne j_k\\ k=1,2,\dots ,i-1\end{array}}^n{p}_{j_1}{p}_{j_2}\cdots p_{j_i}}}}}}\underset{\left(n-i\right)-\mathrm{fold}}{\underbrace{{\displaystyle \prod _{\begin{array}{l}m=1\\ m\ne j_l\\ l=1,2,\dots ,i\end{array}}^n{q}_m}}}$

where the i summation terms in the first horizontal big bracket includes all the possible combinations of i precipitation occurrences at n sites, whereas the second horizontal big bracket implies the multiplication term corresponding to possible dry combinations.

For heterogeneous wet spells the term in the first part in Eq. (5.72) simplifies to n(n  −   1)…(n  − i  +   1)pⁱ and the next part multiplication yields to qⁿ⁻ⁱ ; hence, it reduces similar to Eq. (5.3),

(5.73) $P\left(A=i\right)=\left(\begin{array}{l}n\\ i\end{array}\right)p^i{q}^{n-i}$

This is the well-known binomial distribution with two-stage Bernoulli trials (Feller, 1967).

The probability, p _A, that all the sites, hence the whole area, are covered by wet spell at any instant can be found from Eq. (5.72) as

(5.74) $p_{\mathrm{A}}={\displaystyle \prod _{i=1}^n{p}_i}$

In practical applications, a group of randomly or systematically selected k wet or dry spell occurrences is of great interest. The joint probability that any k ₁ of n available sites has significant wet spell occurrences has already been obtained for homogeneous PoPs in chapter: Temporal Drought Analysis and Modeling, Chapter 4, Section 4.4. The corresponding general expression similar to Eq. (4.18) for the areal heterogeneous probabilities can be obtained by the substitution of Eq. (5.72) instead of the binomial distribution yields

(5.75) $P\left(A=i|X_1>x_0,X_2>x_0,\dots ,X_{k_1}>x_0\right)={\left(\frac{i}{np}\right)}^{k_1}\frac{1}{i!}\underset{i-\mathrm{fold}}{\underbrace{{\displaystyle \sum _{j_1=1}^n{\displaystyle \sum _{\begin{array}{l}{j}_2=1\\ j_2\ne j_1\end{array}}^n\cdots {\displaystyle \sum _{\begin{array}{l}{j}_i=1\\ j_i\ne j_k\\ k=1,2,\dots ,i-1\end{array}}^n{p}_{j_1}{p}_{j_2}\cdots p_{j_k}}}}}}\underset{\left(n-i\right)-\mathrm{fold}}{\underbrace{{\displaystyle \prod _{\begin{array}{l}m=1\\ m\ne j_l\\ l=1,2,\dots ,i\end{array}}^n{q}_m}}}$

Similarly, the conditional ACP of wet spell, given that a group of k ₂ sites have dry spell, can be found for homogeneous PoPs as

(5.76) $P\left(A=i|X_1<x_0,X_2<x_0,\dots ,X_{k_1}<x_0\right)={\left(\frac{n-i}{n}q\right)}^{k_2}\left(\begin{array}{l}n\\ i\end{array}\right)q^i{p}^{n-i}$

and

(5.77) $P\left(A=i|X_1<x_0,X_2<x_0,\dots ,X_{k_1}<x_0\right)={\left(\frac{n-i}{n}q\right)}^{k_2}\underset{i-\mathrm{fold}}{\underbrace{\frac{1}{i!}{\displaystyle \sum _{j_1=1}^n{\displaystyle \sum _{\begin{array}{l}{j}_2=1\\ j_2\ne j_1\end{array}}^n\cdots {\displaystyle \sum _{\begin{array}{l}{j}_i=1\\ j_i\ne j_k\\ k=1,2,\dots ,i-1\end{array}}^n{p}_{j_1}{p}_{j_2}\cdots p_{j_k}}}}}}\underset{\left(n-i\right)-\mathrm{fold}}{\underbrace{{\displaystyle \prod _{\begin{array}{l}m=1\\ m\ne j_l\\ l=1,2,\dots ,i\end{array}}^n{q}_m}}}$

Finally, the conditional ACP of wet spell, given that a group of k ₁ sites has wet spell and another group of k ₂ sites has dry spell, is obtained as

(5.78) $P\left(A=i|X_1>x_0,X_2>x_0,\dots ,X_{k_1}>x_0,X_{k_1+1}<x_0,X_{k_1+2}<x_0,\dots ,X_{k_2}<x_0\right)={\left(\frac{i}{nq}\right)}^{k_1}{\left(\frac{n-i}{n}q\right)}^{k_2}\left(\begin{array}{l}n\\ i\end{array}\right)q^i{p}^{n-i}$

With its heterogeneous counterpart as

(5.79) $P\left(A=i|X_1>x_0,X_2>x_0,\dots ,X_{k_1}>x_0,X_{k_1+1}<x_0,X_{k_1+2}<x_0,\dots ,X_{k2}<x_0\right)={\left(\frac{i}{nq}\right)}^{k_1}{\left(\frac{n-i}{n}q\right)}^{k_2}\underset{i-\mathrm{fold}}{\underbrace{\frac{1}{i!}{\displaystyle \sum _{j_1=1}^n{\displaystyle \sum _{\begin{array}{l}{j}_2=1\\ j_2\ne j_1\end{array}}^n\cdots {\displaystyle \sum _{\begin{array}{l}{j}_i=1\\ j_i\ne j_k\\ k=1,2,\dots ,i-1\end{array}}^n{p}_{j_1}{p}_{j_2}\cdots p_{j_k}}}}}}\underset{\left(n-i\right)-\mathrm{fold}}{\underbrace{{\displaystyle \prod _{\begin{array}{l}m=1\\ m\ne j_l\\ l=1,2,\dots ,i\end{array}}^n{q}_m}}}$

The probability expressions in Eqs. (5.76)–(5.79) can be effectively employed to study regional wet (dry) spell spatiotemporal occurrence patterns.

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Aspects of Flow and Convective Heat Transfer Fundamentals for Compact Surfaces

John E. Hesselgreaves , ... David A. Reay , in Compact Heat Exchangers (Second Edition), 2017

5.8 Observations on Three-Dimensional Flows

It seems paradoxical that heat transfer surface designers have tried to ensure that surfaces are presented to the incident flow as close to 'normal' as possible, on the assumption that these are the optimum configurations. In reality it is probable that two-dimensional flow is regarded as 'ideal' (and also easier to analyse!). In addition, a major, if not determining, factor historically has been that manufacturing constraints have driven this practice: oblique tubes and fins are inconvenient and might take up more space (although, having said this, many applications, such as in aerospace, are now common whereby awkwardly-shaped spaces are imposed on designers, who then are forced to think in other than simple rectangular terms).

But flows 'left to themselves' will naturally become three-dimensional on encountering any obstacle or curved surface, frequently developing regular longitudinal (flow-wise) vortex patterns, arising from well-known instability theories (Taylor instability for convex surfaces, and Goertler instability for concave surfaces). Boundary layer transition is an example of another form of instability.

We now examine, briefly, some examples of 'designed-in' three-dimensional flows.

5.8.1 Oblique Flow Over Tubular Elements

To illustrate the potential advantage of three-dimensional flows, many authors have published experimental data on the flow through inclined tube bundles. Typical of the latter are the results of Jenkins and Noie-Baghban (1988), for low Reynolds numbers typical of those of interest for pin fins, although Zukauskas (1989) states that the influence of Reynolds number on the effect of yaw is small. The overall effect of yaw is illustrated in Fig. 5.26A and B for pressure drop and heat transfer respectively.

Fig. 5.26. (A) Pressure drop and (B) heat transfer corrections for inclined cylindrical arrays. β is the angle of yaw from axes of cylinders and c_β is the ratio of yawed to un-yawed performance.

(Adapted from Zukauskas, A., 1989. In: Karni, J., (Ed.), High-Performance Single-Phase Heat Exchangers. Hemisphere, New York.)

These figures show a progressive reduction in the ratio of yawed performance to un-yawed performance, but the pressure drop falls faster that the heat transfer as yaw increases. It is probable that the cause of this is that separation is delayed in the yawed situation. This has more effect on the form drag than on the heat transfer, which is dominated by the attached flow. This is reflected in a higher j/f value for yawed arrays. It gives justification for the growing number of developments of spiral-baffled shell and tube exchangers, and for the skewed corrugations in plate heat exchangers.

The heat transfer to yawed cylinder banks can be approximated by a relationship derived from the so-called independence principle, whereby the flow is assumed to be independent of the direction of the cylinder axis (see Schlichting, 1979). The Nusselt number is then given by Jenkins and Noie-Baghban (1988) as

(5.115) $N{u}_{\beta }=N{u}_{\pi /2}{sin}^n\beta \text{,}$

where n is the exponent of the Reynolds number used for the un-yawed heat transfer (Nusselt number) correlation (typically 0.5–0.8). Clearly the independence condition cannot be met at the boundaries (eg, at the end, or separating plates), so this relationship should be treated with caution.

An approximation to the effect of yaw angle on the pressure drop of banks of staggered cylinders is given by Zukauskas and Ulinskas as

(5.116) $k_{\beta }=1.245exp\left(-0.487{\beta }^{-1.733}\right)\text{,}$

where β is the angle of yaw (in radians this time), to the cylinder axis and k_β is the ratio of pressure drop to un-yawed pressure drop.

Then the friction factor of the yawed array is

(5.117) $f_{\beta }=k_{\beta }{f}_{\pi /2}\text{.}$

5.8.2 Plate Heat Exchanger and Printed Circuit Heat Exchanger Surfaces

Plate heat exchangers are classed as compact by virtue of their low hydraulic diameter and their tortuous flow passages, giving enhanced secondary flows. These are illustrated in Fig. 5.27.

Fig. 5.27. Typical three-dimensional flow structure in a plate exchanger passage.

(From Focke, W.W., Zacharides, J., Oliver, I., 1985. The effect of the corrugation angle on the thermohydraulic performance of plate heat exchanger. Int. J. Heat Mass Transfer 28, 1469–1479.)

Depending on the corrugation angle, a significant proportion of the flow is in the form of a swirling core along the bottom of the corrugations (Heggs and Walton, 1999; Heggs et al., 1997).

The experimental correlations are similar to those of developing laminar flow, with Reynolds and Prandtl number exponents related to the repeated flow length, as described by Shah and Sekulic (2003):

(5.118) $\mathit{Nu}=0.724{\left(\frac{\beta }{30}\right)}^{0.646}R{e}^{0.583}P{r}^{1/3}\text{.}$

A more sophisticated approach is given in Chapter 7. As remarked by Shah and Sekulic, plate design is largely undertaken in a proprietary manner by the manufacturers.

Although progress will undoubtedly be made with further reductions in hydraulic diameter and perhaps with inserts, the constraint to all-primary surface will curtail their use for highly compact and weight-limited applications, especially for low pressure gas flows.

Printed circuit heat exchangers have flow passages which are nominally corrugated in two dimensions but in fact the flow is three-dimensional because secondary flows automatically develop on curved passages. Again, plate design is proprietary, but the same principles will apply: the shorter the wavelength of the corrugations the greater is the heat transfer and pressure drop.

5.8.3 The Use of Vortex Generators (vgs)

Vortex generators were originally studied in the late 1940s as a means of controlling (delaying) separation on aircraft wings and in wind tunnels. In these applications the boundary layers were relatively thick and it was found that the most effective kind were the delta-type winglet pairs, at incidence angles of between 10 and 15 degrees to the flow direction, inducing counter-rotating vortices. The vgs were typically of the height of the local boundary layer, and drew in high energy flow from outside the boundary layer. The vortices persist many tens of generator heights downstream, and the boundary layer is significantly thinned between the vortex cores in the 'common down' configuration, in which the bulk flow is 'induced' towards the surface by divergent pairs of vgs, that is, in the same way as aircraft trailing vortices. The rectangular counter-rotating vgs in an equi-spaced arrangement, as tested by Tanner et al. (1954) are shown in Fig. 5.28. An even further improvement was obtained by 'biplane' counter-rotating pairs. The features of vortex flows generated (by a triangular generator in this case) are shown in Fig. 5.29, and their effect on a turbulent boundary layer flow is clearly displayed in Fig. 5.30. These data were selected because of the fine detail of velocity distributions.

Fig. 5.28. Equi-spaced divergent vortex generator arrangement type 8: common flow down, showing vortex paths.

(From Tanner, L.H., Pearcey, H.H., Tracy, C.M., 1954. Vortex Generators; Their Design and Their Effects on Turbulent Boundary Layers. Aeronautical Research Council, F.M. 2015, Perf 1196.)

Fig. 5.29. Vortex flows from generators: (A) arrays of counter-rotating rectangular generators.

(From Fiebig., M., Valencia, A., Mitra, N.K., 1993. Wing-type vortex generators for fin- and tube-heat exchangers. Exp. Thermal Fluid Sci. 7, 287–295.); (B) Flows from a single triangular generator (From Torii et al. (1994).)

Fig. 5.30. Velocity contours at stations downstream of vortex generators showing striking reduction in thickness in a turbulent boundary layer, together with removal of low energy flow away from the wall. Data: height h  =   0.375   in., length l  =   0.375   in., lateral spacing s  =   0.75   in.

(From Tanner, L.H., Pearcey, H.H., Tracy, C.M., 1954. Vortex Generators; Their Design and Their Effects on Turbulent Boundary Layers. Aeronautical Research Council, F.M. 2015, Perf 1196.)

Low energy fluid is simultaneously removed from the surface of the outer side of the vortex cores. The pressure distribution and boundary layer structure are significantly affected, and this delays separation in the adjacent flow.

The function of delayed separation is the feature most utilised to date in heat exchangers, as exemplified by the placing of vgs near the equator of tubes in a tube-fin exchanger to delay separation on the tubes (primary surface). This will indirectly improve heat transfer by increasing the proportion of attached flow. A small element of fin effect may be present on the fin. Heat transfer will also be significantly improved by the thinning of the boundary layer downstream of the vgs, as has been shown experimentally by Fiebig (1995) and others. The generator height should obviously be similar to the thermal boundary layer thickness for full effectiveness.

5.8.4 Structured Surfaces Employing Three-Dimensional Flow

A vortex generator is a means to superimpose a three-dimensional flow onto an otherwise two-dimensional flow. An obvious progression from this or from skewed—or inclined flow and secondary surface development is to design—in a three-dimensional flow to the whole finned surface instead of treating it as an augmentation. Although, superficially, some finned surfaces on the market have employed, for example, expanded metal fins on tubes (see Fig. 5.31), the orifices have merely acted as surface interruptions: the main flow is still along each side of the fins. In Fig. 5.32 the expanded metal surface is presented as strands normal to the incident flow. In this latter case, complex fin efficiency calculations would appear to be necessary.

Fig. 5.31. Expanded metal fin on tube.

Fig. 5.32. Plane expanded metal fin.

A development by the senior author Hesselgreaves (1989) now being exploited in commercial exchangers, called the Porous Matrix Heat Exchanger, employs layered perforated plated (originally of flattened expanded metal) which can be bonded by brazing or diffusion bonding in any number of layers between separation plates. The basic concept is illustrated in Fig. 5.33, and a simplified sectional view is shown in Fig. 5.34.

Fig. 5.33. Schematic of Porous Matrix Heat Exchanger surface (two layers shown here).

Fig. 5.34. Simplified section of surface (12 layers), showing "Christmas tree" structure.

Each plate layer has its perforations offset in relation to its neighbour, and because the plates are contiguous, the flow is forced to flow in and out of adjacent downstream perforations, and because the strands are at an angle (about 60 degree, but controllable) to the incident flow, the latter is highly three-dimensional. It will be seen that the main functioning parts of the surface—the strands are tertiary, and the heat is conducted to the separating plates by the secondary 'columns' of contact between plates. The performance of a reference plate-fin surface is shown in Fig. 5.35, with estimations of j and f factors (based on an adapted Webb model) included to verify the test method, and an example of a flattened expanded metal version is given in Fig. 5.36. A further embodiment of the principle is shown in Fig. 5.37, which is analogous to the inclined tube bank geometry mentioned above. In these examples the Reynolds numbers are based on hydraulic diameter.

Fig. 5.35. Performance of reference surface. The full lines show the predicted performance, the points are the experimental values.

Fig. 5.36. Performance of expanded metal surface of six layers, with two-dimensional estimation.

Fig. 5.37. Performance of inclined-stranded surface of 10 layers (40 degree to flow direction), with two-dimensional performance estimation.

These data indicate the enhanced performance potential of three-dimensional flows, suggesting advantages in both friction factor and j factor over two-dimensional flows. As with the technology of vortex generators, it offers much potential for the next generation of compact heat exchanger surfaces.

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