Principle of Continuity Principle of Erosion

Continuity Principle

Also note that the continuity principles used here are exactly the same as the principles used to derive the classical two-point flux formula.

From: Reservoir Simulations , 2020

Fluid mechanics concepts

Chris Baker , ... Mark Sterling , in Train Aerodynamics, 2019

2.4 Forces and energy in fluids

As with any mechanical system, fluid flows follow the three basic conservation laws of mass (the continuity principle); momentum (Newton's second law) and energy (the first law of thermodynamics). In words, these can be expressed as follows:

Conservation of mass – the amount of fluid entering a region of space in unit time is equal to that leaving plus that stored within the region through density changes. For incompressible flows, the latter is of course zero.

Conservation of momentum can be expressed in one of two ways, either directly as the amount of fluid momentum leaving a region of space minus the amount entering in unit time being equal to the sum of the fluid forces on the boundaries of that space or indirectly as the mass of the fluid element multiplied by the total (i.e., spatial and temporal) acceleration being equal to the sum of the forces. There are three types of forces that act on the fluid – pressure forces which act perpendicular to the faces of the fluid element; friction forces which act parallel to the faces of the element and body forces such as gravity. The latter are usually small in low-speed atmospheric flows.

Conservation of energy – the energy leaving the region of space minus the energy entering the unit of space is equal to the heat transfer minus the work done by the fluid. The types of energy within the airflows we are considering here are kinetic energy, potential energy and thermal energy. The work done by the fluid is a function of the pressures at the faces of the fluid element, which do work on adjacent fluid elements.

The forces imposed on the surfaces over which the fluid passes, in our case trains and the local trackside infrastructure, are also of the two main types outlined above – pressure forces and friction forces. This distinction will be seen to be important in what follows. However, it is often difficult to distinguish between these two types of force in any one situation, and very often it is only possible to specify the overall forces and moments on structures caused by a combination of pressure and friction effects, usually specified by the force or moment coefficients in each of the three directions as defined above.

Read full chapter

URL:

https://www.sciencedirect.com/science/article/pii/B9780128133101000022

The Hydrodynamic Transport Processes of Cohesive Sediments and Governing Equations

Emmanuel Partheniades , in Cohesive Sediments in Open Channels, 2009

4.1.1 The Development of the General Transport Equations

A theoretically rigorous study of the hydrodynamic behavior of cohesive sediments in a turbulent flow field has to start with the formulation of the fundamental transport equation for these kinds of sediments. The process is controlled by two groups of variables: (a) the relevant sediment properties and (b) the set of large- and small-scale flow parameters which fully define and control flocculation, erosion, deposition, and resuspension.

As already indicated, the properties of coarse sediments can be adequately represented by either an average grain size or by an appropriate grain size distribution. In contrast, the size and other properties of cohesive sediment aggregates depend on the stresses induced by the turbulent velocities. The latter generate the mechanism for interaggregate collision while at the same time limiting the maximum aggregate size. In addition, aggregates are subjected to a continuous process of growth and disintegration, which has to be properly introduced in the transport equations.

The first general cohesive sediment transport equation was developed by McLaughlin [80 ] and was based on the following continuity principle:

The net rate of inflow of settling units with a diameter d_ithrough a control surface enclosing a control volume plus the rate of generation of such units within the control volume is equal to the rate of concentration increase of the same settling units within the control volume.

It is more convenient to describe the elements in suspension by their settling velocities, w_s , rather than by their diameters, because the former are the relevant parameters in sedimentation. In the foregoing analysis, w_s is subdivided into n classes with an n number of equal increments differing by Δw. Thus, w _s1 refers to the class of settling velocities between 0 and Δw, w _s2 refers to velocities between Δw and 2Δw, and, in general, w_si refers to values of settling velocities between (i–1)Δw and iΔw. Particles with settling velocities w_i are called i particles, and their concentration is indicated as C_i . Denoting the total concentration by C and the average settling velocity by w_{si ,} the continuity principle leads to the following relation:

(4.1) $C(x,y,z,t)=\sum _{i=1}^n{C}_i(x,y,z,t)$

and

(4.2) $w_s(x,y,z,t)C(x,y,z,t)=\sum _{i=1}^n{w}_{si}(x,y,z,t)C_i(x,y,z,t)$

It is noted that, in general, settling velocities and concentrations are functions of space and time and that the left side of Equation 4.2 represents the total rate of settling of suspended sediment.

For the derivation of the general fundamental transport equation for a three-dimensional flow field, we consider an infinitesimal control volume dx.dy.dz in a Cartesian coordinate system x, y, z, where x and y are the two horizontal directions and z is the direction of gravity. Then the transport and accumulation of sediment through the control surface and within the control volume, respectively, are evaluated. The transport of i particles takes place through the following two processes: (a) by convection due to the temporal mean flow velocity, $\overrightarrow{V}=u\hat{i}+v\hat{j}+w\hat{k}$ , where u, v, and w are the velocity components in the x, y, and z directions, respectively; and $\hat{i},\hat{j}$ , and $\hat{k}$ are the three unit vectors in the same corresponding directions; the total convective sediment transport is equal to $\overrightarrow{V}C$ ; and (b) by turbulent diffusion, which follows Fick's law and can be expressed as $-D_{ij}\text{undefined}(\partial C_i/\partial x_j)$ , where j = (1, 2, 3) and where D_ij is the turbulent diffusion coefficient of i particles in the j direction. Within the control volume there is a net rate of positive or negative concentration change by coagulation and destruction, expressed as $(\partial C_i/\partial t)dx.dy.dz.$ There can also be a possible source generating i particles per unit time and unit volume at the rate of ω_ι (x, y, z, t); that is, within the control volume, there may be a rate of i particle production equal to ω_ι.dx.dy.dz.

Applying the continuity principle to all i particles, adding the resulting equations for all i values, and introducing Equation 4.1 and Equation 4.2 into the sum and considering the diffusion coefficients independent of particle size, we obtain the following equation:

(4.3) $\begin{array}{l}\frac{\partial C}{\partial t}+\frac{\partial (w_{\text{s}}C)}{\partial z}+u\frac{\partial C}{\partial x}+v\frac{\partial C}{\partial y}+w\frac{\partial C}{\partial z}-\frac{\partial }{\partial x}\left(\sum _{i=1}^n{D}_{tx}\frac{\partial C_i}{\partial x}\right)\\ -\frac{\partial }{\partial y}\left(\sum _{i=1}^n{D}_{ty}\frac{\partial C_i}{\partial y}\right)-\frac{\partial }{\partial z}\left(\sum _{i=1}^n{D}_{tz}\frac{\partial C_i}{\partial z}\right)-\sum _{i=1}^n{\omega }_i=0\end{array}$

where D_tx , D_ty and D_tz are the turbulent diffusion coefficients in the x , y and z ,directions, respectively.

If the i particles are created solely by flocculation of smaller particles and/or by destruction of larger agglomerates, that is, if Σω_i = 0, and if the turbulent diffusion coefficients, are assumed to be independent of the agglomerate size, Equation 4.3 reduces to the following simpler form:

(4.4) $\begin{array}{l}\frac{\partial C}{\partial t}+u\frac{\partial C}{\partial x}+v\frac{\partial C}{\partial y}+w\frac{\partial C}{\partial z}+\frac{\partial (w_{s}C)}{\partial z}\\ -\frac{\partial }{\partial x}\left(D_{tx}\frac{\partial C}{\partial x}\right)-\frac{\partial }{\partial y}\left(D_{ty}\frac{\partial C}{\partial y}\right)-\frac{\partial }{\partial z}\left(D_{tz}\frac{\partial C}{\partial z}\right)=0\end{array}$

where the effect of flocculation and/or hindered settling is included implicitly in the term w_sC.

The two-dimensional near uniform flow case is of particular interest since it approximates the conditions in a wide variety of open channel flows. If x is considered to be the direction of flow, then v =w = 0 and, moreover, u is a function of z only; that is, u =f(z). Equation 4.4 then simplifies into the following one:

(4.5) $\frac{\partial C}{\partial t}+u\frac{\partial C}{\partial x}+\frac{\partial (w_{s}C)}{\partial z}-\frac{\partial }{\partial x}\left(D_{tx}\frac{\partial C}{\partial x}\right)-\frac{\partial }{\partial z}\left(D_{tz}\frac{\partial C}{\partial z}\right)=0$

If it is further assumed that only the mean sediment concentration changes spatially while the average settling velocity of the aggregates remains constant, Equation 4.5 takes the following form:

$\frac{\partial C}{\partial t}+u\frac{\partial C}{\partial x}+w_s\frac{\partial C}{\partial z}-D_{tx}\frac{{\partial }^{2}C}{\partial x^2}-\frac{\partial D_{tx}}{\partial x}\frac{\partial C}{\partial x}-D_{tz}\frac{{\partial }^{2}C}{\partial z^2}-\frac{\partial D_{tz}}{\partial z}\frac{\partial C}{\partial z}=0$

or

(4.6) $\frac{\partial C}{\partial t}+u\frac{\partial C}{\partial x}=D_{tx}\frac{{\partial }^{2}C}{\partial x^2}+\left(\frac{\partial D_{tz}}{\partial z}-w_s\right)\frac{\partial C}{\partial z}+D_{tz}\frac{{\partial }^{2}C}{\partial z^2}+\frac{\partial D_{tx}}{\partial x}\frac{\partial C}{\partial x}$

For a near uniform flow, the diffusion coefficients can be assumed independent of the coordinate x and dependent only on z. Furthermore, as will be explained later in Chapter 6 and Chapter 7, the effect of suspended sediment on the diffusion coefficients is negligible for the range of concentrations encountered in rivers and estuaries. Only when in excess of 20,000   ppm do sediment concentrations appear to affect the diffusive and dispersive properties of the fluid system. The gradient of D_tx with respect to x is, therefore, practically zero, so that Equation 4.6 reduces to its final simplest form:

(4.7) $\frac{\partial C}{\partial t}+u\frac{\partial C}{\partial x}=D_{tz}\frac{{\partial }^{2}C}{\partial z^2}+\left(\frac{\partial D_{tz}}{\partial z}-w_s\right)\frac{\partial C}{\partial z}+D_{tx}\frac{{\partial }^{2}C}{\partial x^2}$

Equation 4.7 is similar to that developed by Dobbins [17] for the case of two-dimensional steady flow without the time derivative of C.

For the solution of Equation 4.7, appropriate initial and boundary conditions are needed. The obvious initial condition is

At t = 0, C = constant and equal to a prescribed value C_o .

For the free surface the no sediment flux condition can be expressed by

(4.8) $D_{tz}\frac{\partial C}{\partial z}=-w_{s}C\text{undefined}at\text{undefined}z=y_o$

where y_o is the depth of flow.

The most complicated and crucial boundary condition is that at the bed. Its general form is [100, 103]

(4.9) $E+(1-p_r)w_{s}C=-D_{tz}\frac{\partial C}{\partial z}\text{undefined}at\text{undefined}z=0$

where E is the rate of erosion and/or resuspension, p_r is the proportion of all near-bed settling sediment that reaches the bed coming into contact with the latter and sticking to it, and z is the distance from the bed.

Equation 4.9 states that the amount of bed sediment eroded per unit bed area and unit time plus the amount of deposited sediment that cannot reach the bed must be entrained back to the main flow by turbulence. This equation leads to the following special cases:

(a)

Net erosion without simultaneous deposition: E≠ 0 and p_r = 1;

(b)

Neither erosion nor deposition: E = 0, p_r = 0. This is a case of wash load transport (Section 7.2) and Refs. [100, 103];

(c)

Net deposition without erosion: E = 0, 0 <p_r < 1.

There was considerable speculation in earlier studies as to the actual near-bed sedimentation processes. For instance, Dobbins hypothesized that, under steady conditions, the rate of particle and/or floc pickup is equal to the rate of deposition, an assumption which can be stated by Equation 4.9 with p_r = 0 [17]. Krone claimed that during deposition there is an exchange between bed flocs and suspended flocs [62]. This hypothesis appeared to originate from the basic idea behind Einstein's bed load function for coarse sediment [25] and from some limited and fragmentary experimental evidence. The more recent and extensive experimental investigations, to be discussed in detail in Chapter 6 and Chapter 7, revealed that this is not the case and that for cohesive sediments, apart possibly from a very short transient time period, no simultaneous erosion and deposition occurs.

Equation 4.3 describes the most general three-dimensional unsteady case with sediment input from outside. Equation 4.4 is a simpler form of the previous one. Both equations are significant and have to be solved in cases in which the sediment distribution is of importance. Such is the case, for example, in environmental studies, where turbidity plays a significant role. In many cases, however, it is sufficient to know the average cross-sectional distribution of the concentration, the location of shoaling, and the rates of deposition and/or resuspension. In estuarial shoaling control, for instance, primary attention is focused on the shoaling sites and rates without much concern about the vertical concentration distribution. In such cases, moreover, we only need the average over the depth flow velocities or the total discharge over either the depth of flow per unit width for the two-dimensional flows or over the entire cross-section for the one-dimensional flows. The first apply to relatively wide estuaries and bays and the second to relatively narrow channels with width-to-depth ratios of the order of unity. The sediment transport equations in either case are developed by an over-the-depth integration of Equation 4.4 and an introduction of the overall dispersion coefficients in place of the turbulent diffusion coefficients. The analysis for both cases has been presented elsewhere [100, 103] and in abbreviated form in Chapter 11 of Ref. [110].

The two-dimensional sediment transport equation is

(4.10) $\begin{array}{l}\frac{\partial C}{\partial t}+u\frac{\partial C}{\partial x}+v\frac{\partial C}{\partial y}=\frac{\partial }{\partial x}\left(D_x\frac{\partial C}{\partial x}\right)+\frac{\partial }{\partial y}\left(D_y\frac{\partial C}{\partial y}\right)\\ +F_e(x,y,C,t)-F_d(x,y,C,t)\end{array}$

where u and v are the average over-the-depth velocities in the direction of flow, x, and in the normal to x horizontal direction, y, respectively, given by

(4.11) $u=\frac{q_x}{y_o}$

(4.12) $v=\frac{q_y}{y_o}$

where C is the average over-the-depth suspended sediment concentration; q_x and q_y are the flow rates per unit width in the x and y directions, respectively; and F_e and F_d are the sediment source and sink functions representing the resuspension and deposition rates per unit bed area. These last two functions depend on the location, as defined by the coordinates x and y, on the suspended sediment concentration, C, and on the time. D_x and D_y are the dispersion coefficients in the x and y directions, respectively.

For the one-dimensional case, the sediment transport equation reads

(4.13) $\frac{\partial C}{\partial t}+V\frac{\partial C}{\partial x}=\frac{1}{A}\frac{\partial }{\partial x}\left(A{D}_x\frac{\partial C}{\partial x}\right)+F_e(x,C,t)-F_d(x,C,t)$

where A is the cross-sectional area of the channel, being in general a function of the flow direction x; V is the instantaneous average over the cross-section velocity, i.e., V =Q/A, where Q is the discharge; D_x is, as in the previous case, the dispersion coefficient in the direction of flow, x, while the sediment source and sink functions, F_e and F_d , representing the corresponding rates per unit length of the channel, depend now only on x, C, and t.

Read full chapter

URL:

https://www.sciencedirect.com/science/article/pii/B9781856175562000044

Deposition and Resuspension of Cohesive Soils

Emmanuel Partheniades , in Cohesive Sediments in Open Channels, 2009

7.3.2 Fundamental Considerations

The rate of erosion, E , expressed as the mass of sediment eroded per unit bed area and unit time, is related through the sediment continuity principle to the time rate of the increase of the suspended sediment concentration, C, by the equation [85]

(7.41) $E=h\frac{dC}{dt}$

Similarly, the time rate of change of the depth of erosion, dz/dt, is given by the following relationship in terms of the rate dC/dt, the local dry density of the bed, ρ_sb (z) at an elevation z, and the total depth of flow, h:

(7.42) $\frac{dC}{dt}=\frac{{\rho }_{sb}(z)}{h}\frac{dz}{dt}$

From these last two relations, the rate of erosion can be determined if we know the readily determinable rate of increase of the suspended sediment concentration, dC/dt, and ρ_sb (z). The determination of the latter was the subject of a separate research effort.

The hydraulic parameter controlling resuspension is expected to be the bed shear stress, τ_b , as in the cases of erosion of uniform consistency beds and in the cases of deposition. As indicated in Section 6.3, a cohesive bed possesses a certain erosive strength, τ_s , which is equal by definition to the minimum or critical bed shear stress, τ_c , for the initiation of erosion. One is reminded that the nature of the erosive strength of a cohesive bed is different from the mass shear strength of the soil and that the latter exceeds the former by two to three orders of magnitude. In the first preliminary studies on resuspension, reported and discussed in Chapter 6, C increased initially rapidly but at gradually diminishing rates reaching, under certain conditions, a constant value, C_s , at which point any further erosion stops. It can then be assumed that at that stage, the bed shear stress is equal to the erosive strength of the bed, i.e., τ_s = τ_b = τ_c . The latter can be readily determined by controlled tests under continuous varying bed shear stresses until the suspended sediment concentration reaches a nearly constant value.

The following factors may affect the erosive resistance of a cohesive bed and were investigated: (a) the conditions of bed preparation; (b) the consolidation time, that is, the time the deposited bed was left in a quiescent state before the start of erosion; and (c) the salinity of the eroding water.

Read full chapter

URL:

https://www.sciencedirect.com/science/article/pii/B978185617556200007X

Equations of Fluid in Motion

David S-K. Ting , in Basics of Engineering Turbulence, 2016

2.3.1 Conservation of Mass

In the absence of nuclear reactions and other extreme conditions such as those involving a mass traveling at the speed of light, mass can neither be destroyed nor created. This conservation of mass principle is also referred to as the continuity principle. Consider an infinitesimal control volume of dimensions dx dy dz, as shown in Fig. 2.1. At the center of the volume, point 0 with coordinates (0, 0, 0), the fluid density is ρ and the velocity is

Figure 2.1. A differential control volume in the Cartesian coordinate.

(Created by Z. Yang).

(2.9) $\overrightarrow{U}=U\hat{i}+V\hat{j}+W\hat{k}$

The pointed hats are used to denote unit vectors. Invoking Taylor series expansion about point 0 leads to terms such as

(2.10) ${\left.\rho \right)}_{x+\frac{dx}{2}}=\rho +\frac{\partial \rho }{\partial x}\frac{dx}{2}+\frac{{\partial }^2\rho }{\partial x^2}\frac{1}{2!}{\left(\frac{dx}{2}\right)}^2+\mathrm{..}\text{.}$

Neglecting the much smaller, higher order terms, we are left with

(2.11) ${\left.\rho \right)}_{x+\frac{dx}{2}}=\rho +\left(\frac{\partial \rho }{\partial x}\right)\frac{dx}{2}$

(2.12) ${\left.U\right)}_{x+\frac{dx}{2}}=U+\left(\frac{\partial U}{\partial x}\right)\frac{dx}{2}$

where ρ, U, ∂ρ/∂x, ∂U/∂x are evaluated at point 0.

The conservation of mass for the control volume dx dy dz depicted in Fig. 2.1 implies that the rate of mass entering the control volume minus that exiting the control volume is equal to the rate of change of mass of the control volume (element). In other words

(2.13) ${\dot{m}}_{\text{in}}-{\dot{m}}_{\text{out}}=\frac{\partial }{\partial t}{m}_{\text{element}}$

The mass flux through each of the six surfaces of the control volume shown in Fig. 2.1 can be described as

(2.14) ${\int }_{\text{CS}}\rho \overrightarrow{U}\cdot d\overrightarrow{A}$

Here, subscript "CS" signifies the control surface, and A (or A with an arrow head) denotes the surface tensor.

For the left (−x) surface, we have

(2.15) $-\left(\rho -\frac{\partial \rho }{\partial x}\frac{dx}{2}\right)\left(U-\frac{\partial U}{\partial x}\frac{dx}{2}\right)dydz=-\rho Udydz+\frac{1}{2}U\frac{\partial \rho }{\partial x}dxdydz+\frac{1}{2}\rho \frac{\partial U}{\partial x}dxdydz-\frac{1}{4}\frac{\partial \rho }{\partial x}\frac{\partial U}{\partial x}{\left(dx\right)}^{2}dydz$

Dropping the much smaller, higher (4th) order term leaves us with

(2.16) $-\left(\rho -\frac{\partial \rho }{\partial x}\frac{dx}{2}\right)\left(U-\frac{\partial U}{\partial x}\frac{dx}{2}\right)dydz=-\rho Udydz+\frac{1}{2}\left(U\frac{\partial \rho }{\partial x}+\rho \frac{\partial U}{\partial x}\right)dxdydz$

Similarly, for the right (+x) surface, we have

(2.17) $\left(\rho +\frac{\partial \rho }{\partial x}\frac{dx}{2}\right)\left(U+\frac{\partial U}{\partial x}\frac{dx}{2}\right)dydz=\rho Udydz+\frac{1}{2}\left(U\frac{\partial \rho }{\partial x}+\rho \frac{\partial U}{\partial x}\right)dxdydz$

The expression for the bottom (−y) surface is

(2.18) $-\left(\rho -\frac{\partial \rho }{\partial y}\frac{dy}{2}\right)\left(V-\frac{\partial V}{\partial y}\frac{dy}{2}\right)dxdz=-\rho Vdxdz+\frac{1}{2}\left(V\frac{\partial \rho }{\partial y}+\rho \frac{\partial V}{\partial y}\right)dxdydz$

The flux entering the top (+y) surface can be described by

(2.19) $\left(\rho +\frac{\partial \rho }{\partial y}\frac{dy}{2}\right)\left(V+\frac{\partial V}{\partial y}\frac{dy}{2}\right)dxdz=\rho Vdxdz+\frac{1}{2}\left(V\frac{\partial \rho }{\partial y}+\rho \frac{\partial V}{\partial y}\right)dxdydz$

And for the back (−z) surface

(2.20) $-\left(\rho -\frac{\partial \rho }{\partial z}\frac{dz}{2}\right)\left(W-\frac{\partial W}{\partial z}\frac{dz}{2}\right)dxdy=-\rho Wdxdy+\frac{1}{2}\left(W\frac{\partial \rho }{\partial z}+\rho \frac{\partial W}{\partial z}\right)dxdydz$

Similarly, for the front (+z) surface

(2.21) $\left(\rho +\frac{\partial \rho }{\partial z}\frac{dz}{2}\right)\left(W+\frac{\partial W}{\partial z}\frac{dz}{2}\right)dxdy=\rho Wdxdy+\frac{1}{2}\left(W\frac{\partial \rho }{\partial z}+\rho \frac{\partial W}{\partial z}\right)dxdydz$

Summing all the terms associated with the six surfaces gives

(2.22) $-\left[\frac{\partial \rho U}{\partial x}+\frac{\partial \rho V}{\partial y}+\frac{\partial \rho W}{\partial z}\right]dxdydz=\frac{\partial \rho }{\partial t}dxdydz$

We can bring the right-hand term to the left and get

(2.23) $\frac{\partial \rho }{\partial t}+\frac{\partial \rho U}{\partial x}+\frac{\partial \rho V}{\partial y}+\frac{\partial \rho W}{\partial z}=0$

The first term shows that local changes to mass (per unit volume) can occur when there is a change in the fluid density or when the fluid is compressible. The remaining three terms signify mass changes (per unit volume) resulting from convection.

Alternatively, we see that the net rate of mass flux through the control surface is

(2.24) $\left(\frac{\partial \rho U}{\partial x}+\frac{\partial \rho V}{\partial y}+\frac{\partial \rho W}{\partial z}\right)dxdydz$

and the rate of change of mass inside the control volume is

(2.25) $\frac{\partial \rho }{\partial t}dxdydz$

therefore, the net rate of change of mass is

(2.26) $\frac{\partial \rho }{\partial t}dxdydz+\left(\frac{\partial \rho U}{\partial x}+\frac{\partial \rho V}{\partial y}+\frac{\partial \rho W}{\partial z}\right)dxdydz=0$

Dividing both sides by the volume dx dy dz gives Eq. 2.23, which is the mass conservation or continuity equation with a general expression

(2.27) $\frac{\partial \rho }{\partial t}+\nabla \cdot \left(\rho \overrightarrow{U}\right)=0$

In the special case of incompressible fluid or constant-density flow, the above continuity equation is reduced to

(2.28) $\nabla ·U=0$

This expression states that the total convection of mass into the control volume minus that convected out of the control volume is zero for a constant-density flow.

Read full chapter

URL:

https://www.sciencedirect.com/science/article/pii/B9780128039700000027

Unsteady open channel flows: 1. Basic equations

Hubert Chanson , in Hydraulics of Open Channel Flow (Second Edition), 2004

Continuity equation

Considering the control volume defined by the cross-sections 1 and 2 (Fig. 16.2), located between x = x ₁ and x = x ₂, between the times t = t ₁ and t = t ₂ , the continuity principle states that the net mass flux into the control volume equals the net mass increase of the control volume between the times t ₁ and t ₂. It yields:

Fig. 16.2. Definition sketch: (a) side view, (b) top view and (c) cross-section.

(16.1) ${\displaystyle {\int }_{t_1}^{t_2}(\rho V_1{A}_1-}\rho V_2{A}_2)\text{d}t+{\displaystyle {\int }_{x_1}^{x_2}({(\rho A)}_{t_2}}-{(\rho A)}_{t_1})\text{d}x=0$

where ρ is the fluid density, V is the flow velocity, A is the flow cross-sectional area, the subscripts 1 and 2 refer to the upstream and downstream cross-section, respectively (Fig. 16.2), and the subscripts t ₁ and t ₂ refer to the instants t = t ₁ and t = t ₂, respectively.

Defining Q = VA, and dividing equation (16.1) by the density ρ, the continuity equation becomes:

(16.2) ${\displaystyle {\int }_{t_1}^{t_2}(Q_1-}{Q}_2)\text{d}t+{\displaystyle {\int }_{x_1}^{x_2}(A_{t_2}}-A_{t_1})\text{d}x=0$

Read full chapter

URL:

https://www.sciencedirect.com/science/article/pii/B9780750659789500234

24th European Symposium on Computer Aided Process Engineering

Rita Ribeiro , ... Maria do Céu Almeida , in Computer Aided Chemical Engineering, 2014

2.2 Process model

The nonlinear dynamic model developed in MATLAB^®/Simulink comprises three modules, representing the main attributes of the treatment plant: hydrodynamic behaviour, biological treatment and suspended solids separation (Ribeiro, 2011 ). The hydrodynamic model is based on the continuity principle and considers the flow characteristics of the dischargers, integrating simulation of internal flows with impact on the flow of treated effluent.

The biological treatment module consists in a simplified description of the activated sludge process based on the ASM1, a mechanistic, nonlinear model developed by the International Water Association (Henze et al., 2000). In this case study, only the processes associated to the removal of carbonaceous material (i.e., heterotrophic grow, heterotrophic decay, hydrolysis) were considered. In addition, three modifications were made in the formulation. The first concerns the integration of inert particulate products resulting from the biomass decay in the inert particulate organic material, which becomes non-conservative unlike originally established in ASM1. The second change is the division of the variable Xs in particulate and solubilised fractions, attempting to reproduce the inflow of slowly biodegradable material with different characteristics to the treatment system. Finally, the effect of temperature on the kinetics was considered in the model implementation through the Arrhenius equation.

The suspended solids separation model comprises a function which reflects the efficiency of particulate material removal in the secondary settler, assumed equivalent to the organic particulate material separation efficiency. The application of this kind of model (point settler model) is admissible only in systems with high separation efficiency with mixture effect in the clarified liquid (Vanrolleghem et al., 2003), as it is the case.

Read full chapter

URL:

https://www.sciencedirect.com/science/article/pii/B9780444634566501162

Model-free Control

Luc Jaulin , in Mobile Robotics, 2015

Abstract

When we implement a controller for a robot perform the initial tests we rarely succeed on the first try which leads us to the problem of debugging. It might be that the compass is subject to electromagnetic disturbances, that it is placed upside-down, that there is a unit conversion problem in the sensors, that the motors are saturated or that there is a sign problem in the equations of the controller. The problem of debugging is a complex one it is wise to respect the continuity principle : each step in the construction of the robot must be of reasonable size has to be validated before pursuing construction. Thus for a robot, it is desirable to implement a simple intuitive controller that is easy to debug before setting up a more advanced one. This principle cannot always be applied. However, if we have a good a priori understanding of the control law to apply, then such a continuity principle can be followed. Among mobile robots for which a pragmatic controller can be imagined, we can distinguish at least two subclasses:

–

vehicle-robots: these are systems built by man to be controlled by man such as the bicycle, sailboat, car, etc. We will try to copy the control law used by humans and transform it into an algorithm;

–

biomimetic robots: these robots are inspired by the movement of human beings. We have been able to observe them for long periods and deduce the strategy developed by nature to design its control law. This is the biomimetic approach (see, for example, [BOY 06]). We do not include walking robots in this category because, even though we all know how to walk, it is near to impossible to know which control law we use for it. Thus, designing a control law for walking robots [CHE 07] cannot be done without a complete mechanical modeling of walking and without using any theoretical automatic control methods such as those evoked in Chapter 2.

Read full chapter

URL:

https://www.sciencedirect.com/science/article/pii/B9781785480485500036

THREE-DIMENSIONAL NUMERICAL MODELS FOR PERIODICALLY FULLY-DEVELOPED HEAT AND FLUID FLOWS WITHIN POROUS MEDIA

A. NAKAYAMA , F. KUWAHARA , in Transport Phenomena in Porous Media III, 2005

7.2.3 Method of computation

The governing equations (7.6) and (7.7), subject to the foregoing boundary and compatibility conditions (7.8)–(7.10), were numerically solved using the SIMPLE algorithm proposed by Patankar and Spalding (1972). The pressure-velocity coupling based on the SIMPLE algorithm was adopted to correct both the pressure and velocity fields simultaneously. The calculation starts by solving the three momentum equations and subsequently the estimated velocity field is corrected by solving the pressure correction equation reformulated from the discretized continuity and momentum equations such that the velocity field fulfils the continuity principle. As the w, v and w velocity fields are established, the remaining scalar transport equations, if any, are solved.

Convergence was measured in terms of the maximum change in each variable during an iteration. The maximum change allowed for the convergence check was set to 10⁻⁵, as the variables were normalized by appropriate references. The hybrid scheme was adopted for the advection terms. Further details on this numerical procedure can be found in Patankar (1980) and Nakayama (1995).

For the three-dimensional laminar flow cases, all computations have been carried out for a one structural unit H × H × H using non-uniform grid arrangements with 45 × 45 × 45, after comparing the results against those obtained with 61 × 61 × 61 for some selected cases, and confirming that the results are independent of the grid system. All computations were performed using the computer system at Shizuoka University Computer Center.

Read full chapter

URL:

https://www.sciencedirect.com/science/article/pii/B9780080444901500119

Melt Processes

Lorraine F. Francis , in Materials Processing, 2016

Questions

1.

Describe the differences in the melt structures of polymers, metals, and ceramic glasses?

2.

Why are polycrystalline ceramics, like alumina, not processed from the melt?

3.

What are the mechanical and energetic definitions of surface tension? Why is surface tension not a term used with solids?

4.

Does wetting behavior of a liquid on a solid depend on the surface tension of the liquid?

5.

What is the difference between Newtonian and non-Newtonian rheological behavior?

6.

Compare typical viscosities for metal melts, polymer melts, and ceramic glass melts. How are these typical values related to the melt structures?

7.

Explain the temperature dependence of viscosity.

8.

Explain why polymer melts tend to be shear thinning.

9.

What are the three main principles governing liquid flow?

10.

Give two examples from the chapter where mass conservation (continuity principle) is used.

11.

Sketch the velocity gradients that arise from drag flow and pressure-driven flow for (a) a Newtonian liquid and (b) a shear thinning liquid.

12.

What is the heat equation? Define thermal diffusivity.

13.

What conditions lead to "Newtonian cooling"? Would you expect a ceramic glass sheet after exiting a float glass operation to cool in this way?

14.

Under what conditions is the plot in Figure 3.24 used to predict cooling time?

15.

List three forms of shape casting of metals with advantages and disadvantages.

16.

Why is the sprue in a sand mold tapered?

17.

What factors influence solidification time in sand casting?

18.

Describe the origin of dendrites in alloys using a phase diagram. In the process define constitutional undercooling.

19.

List the similarities between die casting and injection molding.

20.

What is the sequence of events in a float glass process. How is thickness controlled?

21.

What are the advantages of the fusion downdraw process?

22.

Make a sketch of a single screw extruder and label all the parts.

23.

What are the main operating parameters in a single screw extruder? What is the operating point?

24.

What are the advantages and uses of twin screw extrusion?

25.

List two defects in extrusion and their origins.

26.

What is the sequence of step in injection molding? Which steps is likely the longest?

27.

How are runners designed?

28.

Define the process window in injection molding using a diagram of injection pressure versus holding pressure.

29.

Compare blow molding of polymers and glasses.

30.

In fused deposition modeling, what factors control the width of the deposited strand?

31.

What are the advantages and disadvantages of FDM compared to injection molding?

Read full chapter

URL:

https://www.sciencedirect.com/science/article/pii/B9780123851321000033

Exergy analyses of renewable energy systems

Ibrahim Dincer , Marc A. Rosen , in Exergy (Third Edition), 2021

11.5.1 Wind energy systems

As a meteorological variable, wind energy refers to the energy content of wind. In electricity generation, wind plays the same role as water does for a hydraulic generation. Wind variables are important in such applications. Wind velocity deviation and changeability depend on time and location. Understanding such characteristics is the subject of wind velocity modeling. Determining the atmospheric boundary layer and modeling is a special consideration in wind power research. Much research has been carried out on these subjects. For instance, Petersen et al. [17] considered wind power meteorology and sought relationships between meteorology and wind power. During the preparation of the Denmark Wind Atlas, detailed research was performed on wind energy as a meteorological energy source [18].

Meteorological variables such as temperature, pressure, and moisture play important roles in the occurrence of wind. Generally, in wind engineering, moisture changeability is negligible and the air is assumed to be dry.

Wind as a meteorological variable can be described as a motion of air masses on a large scale with potential and kinetic energies. The pressure forces and hence velocities lead to kinetic energy. In wind engineering applications horizontal winds are important because they cover great areas.

The dynamic behavior of the atmosphere generates spatio-temporal variations in such parameters as pressure, temperature, density, and moisture. These parameters can be described by expressions based on continuity principles, the first law of thermodynamics, Newton's law, and the state law of gases. Mass, energy, and momentum conservation equations for air in three dimensions yield balance equations for the atmosphere. Wind occurs due to different cooling and heating phenomena within the lower atmosphere and over the earth's surface. Meteorological systems move from one place to another by generating different wind velocities.

With the growing significance of environmental problems, clean energy generation has become increasingly important. Wind energy is clean, but it usually does not persist continually for long periods at a given location. Fossil fuels often must supplement wind energy systems. Many technical and scientific studies have addressed this challenge with wind energy [19, 20].

During the last couple of decades, wind energy applications have developed and been extended to industrial use in some European countries including Germany, Denmark, and Spain. Successes in wind energy generation have encouraged other countries to consider wind energy as a component of their electricity generation systems. The clean, renewable, and in some instances economic features of wind energy have drawn attention from political and business circles and individuals. Development in wind turbine technology has also led to increased usage. Wind turbine rotor efficiency increased from 35% to 40% during the early 1980s, and to 48% by the mid-1990s. Moreover, the technical availability of such systems has increased to 98% (e.g., [21]). In 2019, total operational wind power capacity worldwide was approximately 650,000   MW.

Koroneos et al. [22] applied exergy analysis to renewable energy sources including wind power. This represents the first paper in the literature about wind turbine exergy analysis. However, in this paper, only the electricity generation of wind turbines is considered and the exergy efficiency of wind turbines for wind speeds above 9   m/s is treated as zero. They only considered the exergy of the wind turbine, depending on electricity generation with no entropy generation analysis. Later, numerous researchers have been carried out an exergy analysis of wind energy and considered wind power for air compression systems operating over specified pressure differences and estimated the system exergy efficiency. As mentioned before, they wanted to estimate exergy components and to show pressure differences and realized this situation by considering two different systems, a wind turbine, and an air compressor, as a united system.

Dincer and Rosen [23] investigated the thermodynamic aspects of renewables for sustainable development. They explain relations between exergy and sustainable development. The wind speed thermodynamic characteristics are given by Goff et al. [24], with the intent of using the cooling capacity of wind as a renewable energy source (i.e., using the wind chill effect for a heat pump system).

Although turbine technology for wind energy is advancing rapidly, there is a need to assess accurately the behavior of wind scientifically. Some of the thermodynamic characteristics of wind energy are not yet clearly understood. The capacity factor of a wind turbine sometimes is described as the efficiency of a wind energy turbine. But there are difficulties associated with this definition. The efficiency of a wind turbine can be considered as the ratio of the electricity generated to the wind potential within the area swept by the wind turbine. In this definition, only the kinetic energy component of wind is considered. Other components and properties of wind, such as temperature differences and pressure effects, are neglected.

Read full chapter

URL:

https://www.sciencedirect.com/science/article/pii/B9780128243725000117

hoeywifted.blogspot.com

Source: https://www.sciencedirect.com/topics/engineering/continuity-principle

Share this post