Given Below is the Continuity or Mass Conservation Equation The Navier Stokes

Introduction

Shuyu Sun , Tao Zhang , in Reservoir Simulations, 2020

1.7.1.1 Conservation of mass (continuity equation)

The principle of conservation of mass indicates that in the absence of mass sources and sinks, a region will conserve its mass on a local level. The differential form of the mass conservation or continuity equation is given by

(1.29) $\frac{\partial \rho }{\partial t}+\nabla \cdot \left[\rho \mathbf{v}\right]=0$

In the absence of any significant absolute pressure or temperature changes, it is acceptable to assume that the flow is incompressible; that is, the pressure changes do not have significant effects on density. This is almost invariably the case in liquids and is a good approximation in gases at speeds much less than that of sound.

The incompressibility condition indicates that ρ does not change with the flow. This is equivalent to saying that the continuity equation for incompressible flow is given by

(1.30) $\nabla \cdot \mathbf{v}=0$

Eq. (1.30) states that for incompressible flows the net flow across any control volume is zero, that is, "flow out"="flow in."

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Aeroacoustics of wind turbines

Dan Zhao , ... Arne Reinecke , in Wind Turbines and Aerodynamics Energy Harvesters, 2019

8.4 Governing equations

The principles of conservation of mass and momentum govern the motion of a fluid or acoustic medium such as air [82]. The compressible continuity equation can be written in tensor notation as,

(8.9) $\frac{\partial \rho }{\partial t}+\frac{\partial }{\partial x_i}\left(\rho u_i\right)=0.$

The Navier–Stokes equations for conservation of momentum in an unsteady compressible flow can be written as,

(8.10) $\frac{\partial }{\partial t}\left(\rho u_i\right)+\frac{\partial }{\partial x_j}\left(\rho u_i{u}_j+p{\delta }_{\mathit{ij}}-{\sigma }_{\mathit{ij}}\right)=0,$

where i  =   1, 2, 3…, δ _ij is the Kronecker delta and the viscous stress σ _ij is given by,

(8.11) ${\sigma }_{\mathit{ij}}=\mu \left(\left[\frac{\partial u_i}{\partial x_j}+\frac{\partial u_j}{\partial x_i}\right]-\frac{2}{3}\frac{\partial u_k}{\partial x_k}{\delta }_{\mathit{ij}}\right).$

It has been observed that while fluid compressibility effects are important for the propagation of sound, most of the dominant sources are pressure fluctuations generated near the wind turbine blade, for which the pressure field can be adequately described by incompressible flow solutions.

The Mach number is used to determine the relative importance of compressibility in a fluid flow. It is given by,

(8.12) $M=\frac{\left|u\right|}{c},$

where |  u  | is the magnitude of the flow velocity and c is the local speed of sound measured at the same point in the flow.

The Mach number is much less than unity for most HAWT blades. This has led to the development of de-coupled models, where noise sources and propagation can be solved separately. Typically, local pressure fluctuations are solved in a small region near the blade using a flow solver, and this data is used as an input to a simpler wave propagation method in a larger spatial domain that is less expensive to solve [75].

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Basic Equations of Fluid Mechanics and Thermodynamics

Aniko Toth , Elemer Bobok , in Flow and Heat Transfer in Geothermal Systems, 2017

2.2.1 The Principle of Conservation of Mass

The principle of conservation of mass states that the mass of a body is constant during its motion. This can be stated in the rate form as the rate of change with time of the mass of a body being zero. It is obvious that for a material system the above statement can be expressed mathematically.

Consider the volume $\text{V}\left(\overrightarrow{\text{r}},\text{t}\right)$ flowing with the fluid. It is constituted of the same particles of fixed identity. The volume $\text{V}\left(\overrightarrow{\text{r}},\text{t}\right)$ and its bounding surface $\text{A}\left(\overrightarrow{\text{r}},\text{t}\right)$ vary in time representing successive configurations of the same fluid particles. This is illustrated in Fig. 2.1.

Let an infinitesimal volume element be located at a point P, characterized by the position vector $\overrightarrow{\text{r}}$ , within the flowing volume under consideration. The scalar density point function $\mathrm{\rho }\left(\overrightarrow{\text{r}},\text{t}\right)$ is the sum of the infinitesimal mass elements, thus it is the volume integral of the density over the volume $\text{V}\left(\overrightarrow{\text{r}},\text{t}\right)$ . The principle of conservation of mass can be expressed as the material derivative of this volume integral:

(2.4) $\frac{\text{d}}{\text{dt}}\underset{{\text{V}}_{\text{t}}}{\int }\mathrm{\rho }\text{dV}=0$

Applying Euler's transport theorem, this expression becomes:

(2.5) $\underset{\text{V}}{\int }\frac{\partial \mathrm{\rho }}{\partial \text{t}}\text{dV}+\underset{\left(\text{A}\right)}{\int }\mathrm{\rho }\overrightarrow{\text{v}}\text{d}\overrightarrow{\text{A}}=0$

Therefore, the sum of rate of change of mass within the fixed volume V, which is an instantaneous configuration of V(t), and the mass flux $\mathrm{\rho }\overrightarrow{\text{v}}$ across the bounding surface of V is zero. The first term represents the local rate of change of mass within the fixed volume V. The surface integral represents the mass which crosses the bounding surface A. This convective mass flux equals the local rate of change of mass.

A conductive mass flux may also occur, primarily as the result of a concentration gradient. This is the so-called ordinary diffusion. The conductive mass flux $\overrightarrow{\text{j}}$ is given by Fick's law:

(2.6) $\overrightarrow{\text{j}}=-\text{D}\text{grad}\mathrm{\rho }$

where D is the so-called diffusivity. This is one of the characteristic physical properties of the fluid, with dimensions (m²/s).

Within the volume V under consideration, there may also be mass sources or sinks; for instance, the rate of mass produced within V by chemical reactions. (Note that conductive mass flux, sources and sinks relate only to some species of mass.)

If this is the case, the mass balance equation becomes:

(2.7) $\underset{\text{V}}{\int }\frac{\partial {\mathrm{\rho }}_{\text{i}}}{\partial \text{t}}\text{dV}+\underset{\left(\text{A}\right)}{\int }\left({\mathrm{\rho }}_{\text{i}}\overrightarrow{{\text{v}}_{\text{i}}}-{\text{D}}_{\text{i}}\text{grad}{\mathrm{\rho }}_{\text{i}}\right)\text{d}\overrightarrow{\text{A}}=\underset{\text{V}}{\int }{\mathrm{\xi }}_{\text{i}}\text{dV};\text{i}=1\text{,}2\text{,}\dots \text{n}$

where ξ is the strength of the sources or sinks per unit volume. Either of these equations represents the mass balance equation for the i-th species. When all n equations are added together, one obtains:

(2.8) $\underset{\text{V}}{\int }\frac{\partial \mathrm{\rho }}{\partial \text{t}}\text{dV}+\underset{\left(\text{A}\right)}{\int }\mathrm{\rho }\overrightarrow{\text{v}}\text{d}\overrightarrow{\text{A}}=0$

Since the total mass of the body is always constant, it is necessary for the conductive fluxes (the sum of the sources and the sinks), to vanish.

The conservation of mass equation may be rewritten in differential form. The term representing the surface integral may be transformed into a volume integral by means of Gauss's divergence theorem:

(2.9) $\underset{\text{V}}{\int }\left[\frac{\partial \mathrm{\rho }}{\partial \text{t}}+\text{div}\left(\mathrm{\rho }\overrightarrow{\text{v}}\right)\right]\text{dV}=0$

Since the limit of integration is arbitrary, and ρ and $\overrightarrow{\text{v}}$ are continuous functions with continuous derivatives, the integrand must be zero. Removing the integral signs, we obtain the well-known continuity equation; the differential equation form of the law of conservation of mass:

(2.10) $\frac{\partial \mathrm{\rho }}{\partial \text{t}}+\text{div}\left(\mathrm{\rho }\overrightarrow{\text{v}}\right)=0$

This equation can be expanded both for Cartesian and cylindrical coordinates:

(2.11) $\frac{\partial \mathrm{\rho }}{\partial \text{t}}+\frac{\partial }{\partial \text{x}}\left(\mathrm{\rho }{\text{v}}_{\text{x}}\right)+\frac{\partial }{\partial \text{y}}\left(\mathrm{\rho }{\text{v}}_{\text{y}}\right)+\frac{\partial }{\partial \text{z}}\left(\mathrm{\rho }{\text{v}}_{\text{z}}\right)=0$

(2.12) $\frac{\partial \mathrm{\rho }}{\partial \text{t}}+\frac{1}{\text{r}}\frac{\partial }{\partial \text{r}}\left(\text{r}\mathrm{\rho }{\text{v}}_{\text{r}}\right)+\frac{1}{\text{r}}\frac{\partial }{\partial \mathrm{\phi }}\left(\mathrm{\rho }{\text{v}}_{\mathrm{\phi }}\right)+\frac{\partial }{\partial \text{z}}\left(\mathrm{\rho }{\text{v}}_{\text{z}}\right)=0$

The continuity equation may be transformed into:

(2.13) $\frac{\partial \mathrm{\rho }}{\partial \text{t}}+\overrightarrow{\text{v}}\text{grad}\mathrm{\rho }+\mathrm{\rho }\text{div}\overrightarrow{\text{v}}=0$

It is clear that the first and second terms represent the local and the convective terms of the material derivative of the density field. Thus we obtain:

(2.14) $\frac{\text{d}\mathrm{\rho }}{\text{dt}}+\mathrm{\rho }\text{div}\overrightarrow{\text{v}}=\text{0}$

For a fluid of constant density, this equation reduces to:

(2.15) $\text{div}\overrightarrow{\text{v}}=\text{0}$

whether the flow is steady or not, i.e., whether or not the flow is locally time-dependent.

The principle of conservation of mass may be formulated in either integral or differential form. Both forms express the same physical principle. When applying the integral form, it is important to remember that the enclosing surface A must be closed; it encloses a finite volume V of the space through which the fluid flows. Some parts of the boundary surface may consist of real material boundaries, such as pipe walls. Since solid walls are impervious, the normal component of the velocity at the wall must be zero, i.e., here $\overrightarrow{\text{v}}\text{d}\overrightarrow{\text{A}}=0$ . Thus a solid wall is always a stream surface. Sometimes the control volume V includes an immersed body which interrupts the continuity of the fluid as shown in Fig. 2.2. In this case, the control volume is a multiply-connected continuous region, and thus the continuity equation cannot be written in the form of Eq. (2.5). The discontinuity within the fluid mass must be excluded by introducing an additional control surface (A₂) around its boundary, while it is also necessary to introduce a cut (A₃), which makes the volume V into a single connected region bounded by a single closed surface (A   +   A₂  +   A₃). In this manner, formulas written in integral form may be rendered applicable to flows involving discontinuities. In general, the integral form of a balance equation describes the relationship between certain quantities within a finite volume and across its bounding surface.

Differential balance equations, on the other hand, express the relationships for the derivatives of these quantities at a given point of the fluid. Most of the problems treated in this book make use of differential equations, but there are many cases in which application of the integral form is the more suitable.

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Transport Processes in Geothermal Reservoirs

Aniko Toth , Elemer Bobok , in Flow and Heat Transfer in Geothermal Systems, 2017

3.4 The Principle of Conservation of Mass

The principle of conservation of mass states that the mass of a body is constant during its motion. This can be stated in the rate form, as the time rate of change of the mass of a body is zero. It is obvious that this statement must be expressed mathematically for a material system.

Consider the volume $\text{V}\left(\overrightarrow{\text{r}},\text{t}\right)$ filled by the complex continuum, bounded by the closed surface (A). The mass of an infinitesimal volume element:

(3.22) $\text{dm}=\mathrm{\varphi }{\mathrm{\rho }}_{\text{F}}\text{dV}+\left(1-\mathrm{\varphi }\right){\mathrm{\rho }}_{\text{S}}\text{dV}$

The total mass of the volume V can be expressed as:

(3.23) $\text{M}=\underset{\text{V}}{\int }\mathrm{\varphi }{\mathrm{\rho }}_{\text{F}}\text{dV}+\underset{\text{V}}{\int }\left(1-\mathrm{\varphi }\right){\mathrm{\rho }}_{\text{S}}\text{dV}$

Note, we must take two material derivatives for the two volume integrals, expressing the conservation of mass:

(3.24) $\frac{\text{d}}{{\text{dt}}_{\text{F}}}\underset{\text{v}}{\int }{\mathrm{\rho }}_{\text{F}}\mathrm{\varphi }\text{dV}+\frac{\text{d}}{{\text{dt}}_{\text{S}}}\underset{\text{v}}{\int }\left(1-\mathrm{\varphi }\right){\mathrm{\rho }}_{\text{S}}\text{dV}=0$

Applying the transport theorem, the material derivative can be replaced by a local and convective term:

(3.25) $\underset{\text{v}}{\int }\frac{\partial \left(\mathrm{\varphi }{\mathrm{\rho }}_{\text{F}}\right)}{\partial \text{t}}\text{dV}+\underset{\text{v}}{\int }\frac{\partial \left[\left(1-\mathrm{\varphi }\right){\mathrm{\rho }}_{\text{S}}\right]}{\partial \text{t}}\text{dV}+\underset{\left(\text{A}\right)}{\int }\mathrm{\varphi }{\mathrm{\rho }}_{\text{F}}{\overrightarrow{\text{v}}}_{\text{F}}\text{d}\overrightarrow{\text{A}}+\underset{\left(\text{A}\right)}{\int }\left(1-\mathrm{\varphi }\right){\mathrm{\rho }}_{\text{S}}{\overrightarrow{\text{v}}}_{\text{S}}\text{d}\overrightarrow{\text{A}}=0$

The mass balance equation may be written for the fluid and solid phase separately:

(3.26) $\underset{\text{v}}{\int }\frac{\partial \left(\mathrm{\varphi }{\mathrm{\rho }}_{\text{F}}\right)}{\partial \text{t}}\text{dV}+\underset{\left(\text{A}\right)}{\int }\mathrm{\varphi }{\mathrm{\rho }}_{\text{F}}{\overrightarrow{\text{v}}}_{\text{F}}\text{d}\overrightarrow{\text{A}}=\underset{\text{v}}{\int }{\mathrm{\xi }}_{\text{F}}\text{dV}$

(3.27) $\underset{\text{v}}{\int }\frac{\partial \left[\left(1-\mathrm{\varphi }\right){\mathrm{\rho }}_{\text{S}}\right]}{\partial \text{t}}\text{dV}+\underset{\left(\text{A}\right)}{\int }\left(1-\mathrm{\varphi }\right){\mathrm{\rho }}_{\text{S}}{\overrightarrow{\text{v}}}_{\text{S}}\text{d}\overrightarrow{\text{A}}=\underset{\text{v}}{\int }{\mathrm{\xi }}_{\text{S}}\text{dV}$

where ξ_F and ξ_S is the rate at which mass of the fluid is produced within the unit volume of the system by chemical reactions. Similarly ξ_S is the same, referring to the solid phase. It is obvious that:

(3.28) $\underset{\text{v}}{\int }{\mathrm{\xi }}_{\text{F}}\text{dV}+\underset{\text{v}}{\int }{\mathrm{\xi }}_{\text{S}}\text{dV}=0$

that is, the whole mass of the system is constant. The integral form of the mass balance equation may be written in differential form applying the divergence theorem:

(3.29) $\frac{\partial \left(\mathrm{\varphi }{\mathrm{\rho }}_{\text{F}}\right)}{\partial \text{t}}+\text{div}\left(\mathrm{\varphi }{\mathrm{\rho }}_{\text{F}}{\overrightarrow{\text{v}}}_{\text{F}}\right)={\mathrm{\xi }}_{\text{F}}$

(3.30) $\frac{\partial \left[\left(1-\mathrm{\varphi }\right){\mathrm{\rho }}_{\text{S}}\right]}{\partial \text{t}}+\text{div}\left[\left(1-\mathrm{\varphi }\right){\mathrm{\rho }}_{\text{S}}{\overrightarrow{\text{v}}}_{\text{S}}\right]={\mathrm{\xi }}_{\text{S}}$

If the interphase mass transfer may be neglected, the equation are obtained as:

(3.31) $\frac{\partial \left(\mathrm{\varphi }{\mathrm{\rho }}_{\text{F}}\right)}{\partial \text{t}}+\text{div}\left(\mathrm{\varphi }{\mathrm{\rho }}_{\text{F}}{\overrightarrow{\text{v}}}_{\text{F}}\right)=0$

(3.32) $\frac{\partial \left[\left(1-\mathrm{\varphi }\right){\mathrm{\rho }}_{\text{S}}\right]}{\partial \text{t}}+\text{div}\left[\left(1-\mathrm{\varphi }\right){\mathrm{\rho }}_{\text{s}}{\overrightarrow{\text{v}}}_{\text{S}}\right]=0$

For a homogeneous, isotropic, and non-deformable solid matrix:

$\frac{\partial \mathrm{\varphi }}{\partial \text{t}}=0;{\overrightarrow{\text{v}}}_{\text{s}}\equiv 0,$

Thus we get:

(3.33) $\mathrm{\varphi }\frac{\partial {\mathrm{\rho }}_{\text{F}}}{\partial \text{t}}+\text{div}\left({\mathrm{\rho }}_{\text{F}}\mathrm{\varphi }{\overrightarrow{\text{v}}}_{\text{F}}\right)=0$

In this equation the product:

(3.34) $\overrightarrow{\text{q}}=\mathrm{\varphi }{\overrightarrow{\text{v}}}_{\text{F}}$

is the so-called local seepage velocity, which is a vector point function:

(3.35) $\overrightarrow{\text{q}}=\overrightarrow{\text{q}}\left(\overrightarrow{\text{r}}\text{,t}\right)$

The continuity equation for this case is obtained as:

(3.36) $\mathrm{\varphi }\frac{\partial {\mathrm{\rho }}_{\text{F}}}{\partial \text{t}}+\text{div}\left({\mathrm{\rho }}_{\text{F}}·\overrightarrow{\mathrm{\rho }}\right)=0$

For an incompressible fluid, if ρ_F  =   const, we get:

(3.37) $\text{div}\overrightarrow{\text{q}}=0$

It can be written by its orthogonal components as:

(3.38) $\frac{\partial {\text{q}}_{\text{x}}}{\partial \text{x}}+\frac{\partial {\text{q}}_{\text{y}}}{\partial \text{y}}+\frac{\partial {\text{q}}_{\text{z}}}{\partial \text{z}}=0$

At any instant, there is at every point in the flow domain a local seepage velocity vector with a definite direction. The instantaneous curves (which at any point tangent to the direction of the local seepage velocity at that point) are called seepage streamlines of the flow. The mathematical expression defining a seepage streamline is therefore:

(3.39) $\overrightarrow{\text{q}}\text{xd}\overrightarrow{\text{r}}=0$

or written by orthogonal coordinates:

(3.40) $\frac{\text{dx}}{{\text{q}}_{\text{x}}}=\frac{\text{dy}}{{\text{q}}_{\text{y}}}=\frac{\text{dz}}{{\text{q}}_{\text{z}}}$

In steady flow, i.e. one in which flow variables remain invariant with time, streamlines are constant. In unsteady flow, we can speak only of an instantaneous picture of the streamlines, as the picture varies continuously.

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Chemistry and Chemical Technology

James G. Speight PhD, DSc , in Handbook of Industrial Hydrocarbon Processes, 2011

3.1 Conservation of mass

The law of conservation of mass (principle of mass/matter conservation) is that the mass of a closed system (in the sense of a completely isolated system) remains constant over time. The mass of an isolated system cannot be changed as a result of processes acting inside the system but while mass cannot be created or destroyed, it may be rearranged in space, and changed into different types of particles. This implies that for any chemical process in a closed system, the mass of the reactants must equal the mass of the products.

The change in mass of certain kinds of open systems where atoms or massive particles are not allowed to escape, but other types of energy (such as light or heat) were allowed to enter or escape, went unnoticed during the nineteenth century, because the mass-change associated with addition or loss of the fractional amounts of heat and light associated with chemical reactions was very small.

Mass is also not generally conserved in open systems (even if only open to heat and work), when various forms of energy are allowed into, or out of, the system (see, for example, bond energy). Mass conservation for closed systems continues to be true exactly. The mass-energy equivalence theorem states that mass conservation is equivalent to energy conservation, which is the first law of thermodynamics. The mass-energy equivalence formula requires closed systems, since if energy is allowed to escape a system, mass will escape also.

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Chemistry and chemical technology

James G. Speight PhD, DSc, PhD , in Handbook of Industrial Hydrocarbon Processes (Second Edition), 2020

3.1 Conservation of mass

The law of conservation of mass (principle of mass/matter conservation) states that the mass of a closed system (in the sense of a completely isolated system) remains constant over time. The mass of an isolated system cannot be changed as a result of processes acting inside the system but while mass cannot be created or destroyed, it may be rearranged in space, and changed into different types of particles. Put simply, the law states that matter cannot be created or destroyed in a chemical reaction. This implies that for any chemical process in a closed system, the mass of the reactants must equal the mass of the products.

The law implies that mass can neither be created nor destroyed, although it may be rearranged in space, or the entities associated with it may be changed in form. This is illustrated in chemical reactions example in which the mass of the chemical components before the reaction is equal to the mass of the components after the reaction. Thus, during any chemical reaction and low-energy thermodynamic processes in an isolated system, the total mass of the reactants (the starting materials) will be equal to the mass of the reaction products. For example, using the molecular proportions as the weights of the reactants and the products:

CH₄ + 2O₂ → CO₂ + 2H₂O

16   +   (2   ×   32) → 44   +   (2   ×   18)

Mass of the reactants   =   mass of the products

The change in mass of certain kinds of open systems where atoms or massive particles are not allowed to escape, but other types of energy (such as light or heat) were allowed to enter or escape, went unnoticed during the 19th century, because the mass-change associated with addition or loss of the fractional amounts of heat and light associated with chemical reactions, was very small. Mass is also not generally conserved in open systems (even if only open to heat and work), when various forms of energy are allowed into, or out of, the system. Mass conservation for closed systems continues to be true exactly. The mass-energy equivalence theorem states that mass conservation is equivalent to energy conservation, which is the first law of thermodynamics. The mass-energy equivalence formula requires closed systems, since if energy is allowed to escape a system, mass will escape also.

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Heat and mass transfers in the context of energy geostructures

Lyesse Laloui , Alessandro F. Rotta Loria , in Analysis and Design of Energy Geostructures, 2020

3.12 Mass conservation equation

3.12.1 General

The mass conservation equation expresses the principle of conservation of mass. Such an expression, in particular, establishes a relation between the kinematic characteristics of a fluid's motion and the density of the fluid. This conservation equation is also termed the continuity equation.

3.12.2 Mass conservation equation

The mass conservation equation can be derived for a representative volume in which mass flows in and out, subjected to arbitrary hydraulic conditions on its surfaces with internal volumetric mass generation ${\dot{q}}_v$ per unit time (cf. Fig. 3.19). The balance for the elementary volume, as performed for the energy conservation equation, reads

Figure 3.19. Balance of variables over the representative volume.

$\begin{array}{cc}& \left[\begin{array}{c}\text{Rate of mass entering through}\\ \text{the bounding surfaces of a volume}\end{array}\right]\text{+}\left[\begin{array}{c}\text{Rate of mass}\\ \text{generation in a volume}\end{array}\right]\end{array}=\left[\begin{array}{c}\text{Rate of mass}\\ \text{storage in a volume}\end{array}\right]$

Accordingly, the mass conservation equation reads

(3.41) $-\nabla ·\left({\rho }_f{\overline{v}}_{\mathit{rf},i}\right)+{\dot{q}}_v=\frac{\partial {\rho }_f}{\partial t}$

In many practical cases, no volumetric mass generation is considered. Often the fluid is also assumed incompressible. The hypothesis of incompressibility indicates that the density of the fluid remains constant in space and over time. Based on the above, Eq. (3.41) can be rewritten as

(3.42) $\nabla ·{\overline{v}}_{\mathit{rf},i}=0$

The above indicates that the velocity field for an incompressible fluid is a solenoidal field, that is a field in which the divergence of the considered variable is equal to zero at all points in space.

3.12.3 Laplace's equation

Recalling that Darcy's law expresses a relationship between the seepage velocity and the hydraulic gradient, Eq. (3.42) can be rewritten as

(3.43) $\nabla ·\left(k\nabla h\right)=0$

Assuming the medium to be isotropic allows the writing of the following form of Laplace's equation (e.g. for the piezometric head)

(3.44) ${\nabla }^{2}h=0$

The quantity ${\nabla }^{2}h$ is as follows in various coordinate systems.

•

Cartesian coordinates $x$ , $y$ , $z$ :

(3.45) ${\nabla }^{2}h=\frac{{\partial }^{2}h}{\partial x^2}+\frac{{\partial }^{2}h}{\partial y^2}+\frac{{\partial }^{2}h}{\partial z^2}$

•

Cylindrical coordinates $r$ , $\theta$ , $z$ :

(3.46) ${\nabla }^{2}h=\frac{{\partial }^{2}h}{\partial r^2}+\frac{1}{r}\frac{\partial h}{\partial r}+\frac{1}{r^2}\frac{{\partial }^{2}h}{\partial {\theta }^2}+\frac{{\partial }^{2}h}{\partial z^2}$

•

Spherical coordinates $r$ , $\theta$ , $\varphi$ :

(3.47) ${\nabla }^{2}h=\frac{{\partial }^{2}h}{\partial r^2}+\frac{2}{r}\frac{\partial h}{\partial r}+\frac{1}{r^2\sin \theta }\frac{\partial }{\partial \theta }\left(\sin \theta \frac{\partial h}{\partial \theta }\right)+\frac{1}{r^2{\sin }^2\theta }\frac{{\partial }^{2}h}{\partial {\varphi }^2}$

Eq. (3.44) is associated with steady-state conditions and often represents the basis for analysis and design considerations.

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Fluid Dynamics

NICOLAE CRACIUNOIU , BOGDAN O. CIOCIRLAN , in Mechanical Engineer's Handbook, 2001

1.13.6 EQUATION OF CONTINUITY

The equation of continuity is obtained from the principle of conservation of mass. For steady flow, the principle of conservation of mass becomes

(1.32) ${\rho }_1{A}_1{V}_1={\rho }_2{A}_2{V}_2=\text{const},$

or

(1.33) ${\gamma }_1{A}_1{V}_1={\gamma }_2{A}_2{V}_2,$

that is, the mass of fluid passing all sections in a stream of fluid per unit time is the same. If the fluid is incompressible (γ₁ = γ₂), Eq. (1.33) yields

(1.34) $Q=A_1{V}_1=A_2{V}_2=\text{const},$

where A ₁ and A ₂ are the cross-sectional areas of the stream at sections 1 and 2, respectively, and V ₁ and V ₂ are respectively the velocities of the stream at the same sections. Commonly used units of flow are cubic feet per second (cfs), gallons per minute (gpm), or million gallons per day (mgd).

For steady two-dimensional incompressible flow, the continuity equation is

(1.35) $A_{n_1}{V}_1=A_{n_2}{V}_2=A_{n_3}{V}_3=\text{const},$

where A_n terms are the areas normal to the respective velocity vectors.

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Dynamic Model Development

Mauricio Sales-Cruz , Rafiqul Gani , in Computer Aided Chemical Engineering, 2003

2 Mathematical Models

Mathematical models for a process may be derived by applying the principle of conservation of mass, energy and/or momentum on a defined boundary (representing the process) and its connections to the surroundings. A process may be divided into a number of sections where each section is defined by a boundary and connections with other sections and the surroundings. In this way, models for different sections of a process may be aggregating together into a total model for the process. In general, the model equations may be divided into three main classes of equations,

•

Balance Equations (mass, energy and/or momentum equations)

•

Constitutive Equations (equations relating intensive variables such as temperature, pressure and/or composition to constitutive variables such as enthalpies, reaction rates, heat transfers, etc.)

•

Connection and Conditional Equations (equations relating surroundings-system connections, summation of mole fractions, etc.)

The appropriate model equations for each type of model may be derived based on the specific model needs. The model needs are translated to a set of model assumptions and together, help to describe the boundary and its connections. Therefore, based on this description, different version of a model for the same process may be derived. For example, a simple process model may include only the mass balance equations and the connection/conditional equations because the energy and momentum balance effects are assumed to be negligible and the constitutive variables are assumed to be invariant with respect to composition. A more rigorous model may include the mass and energy balance equations, the connection/conditional equations as well as the constitutive equations. There could be two modes of these models (steady state or dynamic). In the steady state mode, the rate of change of accumulation is assumed to be zero (or negligible) while in the case of dynamic mode, they vary with respect to time (the independent variable). An even more rigorous model may add the distribution of the intensive variables as a function of space (in one or more dimensions).

In general, the balance equations in a process model are based on the laws of conservation and take one or more of the following forms,

(1) $0=f\left(y,z,t\right)$

(2) $dy/dt=f\left(y,z,t\right)$

(3) $\frac{\partial y}{\partial t}+\frac{\partial y}{\partial u}=f\left(y,u,z,t\right)$

In the above equations, Eqs. 1 represents a set of AEs (Algebraic Equations) and models the steady state behaviour; Eqs. 2 represents a set of ODEs (Ordinary Differential Equations) and models the dynamic behaviour when the independent variable is time; while Eq. 3 represents a set of PDEs (Partial Differential Equations) and may be used to model both steady state and dynamic behaviours, depending on the dimension of the problem and the type of the corresponding independent variables. y represents a vector of state variables, z a vector of specified variables in Eqs. 1–2 but is an independent variable in Eqs. 3, t is an independent variable, whereas u represent another independent variable (in a 2-dimensional PDE system).

The constitutive equations are usually algebraic (but could also be of the ODE and/or PDE type). In the algebraic form, they may be written as,

(4) $0=q-\text{g}\left(T,P,x,d\right)$

Where q is a constitutive variable, which may be a function of temperature (T), pressure (P), composition (x) and parameters (d). Usually, T, P and/or x are represented by y in the balance equations. These are usually explicit in nature, that is, knowing T, P, x and/or d, it is possible to compute q.

The connection and/or conditional equations (to be called henceforth connection equations) are also usually algebraic and may be represented as,

(5) $0=r-h\left(y,z,t\right)$

As Eqs. 5 implies, some connection may be explicit (if they are not functions of r) while others may be implicit (if they are functions of r).

2.1 Forms of Models and Solution Modes

Models represented by AE sets (1, 4–5) usually represent the total steady state model; by DAE (ODE + AE) sets (2, 4–5) usually represent the total dynamic model; while models represented by PDAE (PDE + ODE + AE) sets (2–5) usually represent models in continuous domain.

Solution of the model equations depends on the forms of the model. Models forms of the AE-type require a linear or nonlinear equations solver depending on whether the model is linear or nonlinear with respect to the unknown variables. Usually, the AE set can be ordered and decomposed into subsets of implicit and explicit algebraic equations. The explicit equations can be solved analytically and this means that some models of AE-type may be explicit and solved analytically.

Models of the DAE-type may or may not include AE sub-sets. Most process models, however, include AE subsets, which when inserted into the ODEs, yield a system with ODEs only. Models of DAE-type may be solved in the dynamic-mode and/or steady state mode (where the condition under which the accumulation term becomes zero is sought). If the AE subsets are explicit, models of DAE-type are usually solved in the ODE-dynamic mode while the DAE-dynamic mode is employed when a part of the AE subset is implicit.

Models of the PDAE-type may or may not include AE subsets and ODE subsets. The PDE set is usually discretised with respect to the independent variables to yield a set of ODEs. Thus solution of PDAEs involves a discretisation step before solution of the resulting DAEs or AEs.

Note that introducing or removing model assumptions also generate different versions of a process model.

2.2 Model Generation

Model generation here implies generation of various versions of an available (reference) model according to the modelling needs. A process separating a stream into two phases through a separator (see Figure 4) will now be used as an example to highlight the different forms and versions of models. The starting point is a simple mass balance model.

Figure 4. A two-phase separation process

M1: Simple mass balance model (steady state); Si is split factor for component i; NC is number of components; MB is mass balance; C1 is constitutive equation; C2 is connection equation	$\begin{array}{lll}0={\text{F\hspace{0.17em}z}}_{\text{i}}-{\text{V\hspace{0.17em}y}}_{\text{i}}-{\text{Lx}}_{\text{i}}\hfill & \text{i}=1,\text{NC}\hfill & \text{MB}\hfill \\ 0={\text{S}}_{\text{i}}-\left({\text{Vy}}_{\text{i}}\right)/\left({\text{Fz}}_{\text{i}}\right)\hfill & \text{i}=1,\text{NC}\hfill & \text{C}1\hfill \\ 0=1-\Sigma {\text{S}}_{\text{i}}\hfill & \text{i}=1,\text{NC}\hfill & \text{C}2\hfill \end{array}$
M2: Simple mass balance model (steady state); replace Si with equilibrium constant Ki for component i; selected C1 model for Ki assumes ideal system; pi is vapour pressure of component i	$\begin{array}{lll}0={\text{F\hspace{0.17em}z}}_{\text{i}}-{\text{V\hspace{0.17em}y}}_{\text{i}}-{\text{Lx}}_{\text{i}}\hfill & \text{i}=1,\text{NC}\hfill & \text{MB}\hfill \\ 0={\text{K}}_{\text{i}}-{\text{p}}_{\text{i}}\left(\text{T}\right)/\text{P}\hfill & \text{i}=1,\text{NC}\hfill & \text{C}1\hfill \\ 0={\text{y}}_{\text{i}}-{\text{x}}_{\text{i}}{\text{K}}_{\text{i}}\hfill & \text{i}=1,\text{NC}\hfill & \text{C}2\hfill \\ 0=1-\Sigma {\text{y}}_{\text{i}}\hfill & \text{i}=1,\text{NC}\hfill & \text{C}2\hfill \end{array}$
M3: Mass balance model (steady state); select another model for Ki for component i that does not assume ideal system; γi is activity coefficient for component i in liquid phase; $\text{φ}$ i is fugacity coefficient of component i in vapour phase	$\begin{array}{lll}0={\text{F\hspace{0.17em}z}}_{\text{i}}-{\text{V\hspace{0.17em}y}}_{\text{i}}-{\text{Lx}}_{\text{i}}\hfill & \text{i}=1,\text{NC}\hfill & \text{MB}\hfill \\ 0={\text{K}}_{\text{i}}-\left({\text{p}}_{\text{i}}\left(\text{T}\right){\text{γ}}_{\text{i}}\right)/\left(P{\text{φ}}_{\text{i}}\right)\hfill & \text{i}=1,\text{NC}\hfill & \text{C}1\hfill \\ 0={\text{γ}}_{\text{i}}-\text{f}\left(\text{T},\underset{\_}{x}\right)\hfill & \text{i}=1,\text{NC}\hfill & \text{C}1\hfill \\ 0={\text{φ}}_{\text{i}}-\text{f}\left(\text{T},\text{P},\underset{\_}{y}\right)\hfill & \text{i}=1,\text{NC}\hfill & \text{C}1\hfill \\ 0={\text{y}}_{\text{i}}-{\text{x}}_{\text{i}}{\text{K}}_{\text{i}}\hfill & \text{i}=1,\text{NC}\hfill & \text{C2}\hfill \\ 0=1-\Sigma {\text{y}}_{\text{i}}\hfill & \text{i}=1,\text{NC}\hfill & \text{C2}\hfill \end{array}$
M4: Rigorous steady state model with mass & energy balance (EB); Hk is enthalpy for stream k; different constitutive (enthalpy) models can be selected.	M3 plus $\begin{array}{cc}0=\text{Q}-\left({\text{FH}}^{\text{F}}-{\text{VH}}^{\text{V}}-{\text{LH}}^{\text{L}}\right)& \text{EB}\\ 0={\text{H}}^{\text{k}}-\text{f}\left(\text{T},\text{P},\text{composition}\right)& \text{C}1\end{array}$
M5: Rigorous dynamic model with mass & energy balance; same model as M4 except new MB & EB (only the extra equations are shown); assume negligible vapour hold-up; ni is molar liquid holdup for compound i; E is energy holdup	$\begin{array}{ll}{\text{dn}}_{\text{i}}/\text{dt}={\text{Fz}}_{\text{i}}-{\text{Vy}}_{\text{i}}-{\text{Lx}}_{\text{i}}\text{i}=1,\text{NC}\hfill & \text{MB}\hfill \\ \text{dE}/\text{dt}=\text{Q}-\left({\text{FH}}^{\text{F}}-{\text{VH}}^{\text{V}}-{\text{LH}}^{\text{L}}\right)\hfill & \text{EB}\hfill \\ 0={\text{x}}_{\text{i}}-{\text{n}}_{\text{i}}/\left(\Sigma {\text{n}}_{\text{i}}\right)\hfill & \text{C}2\hfill \\ 0=\text{L}-\text{f}\left(\underset{\_}{\text{n}},\text{A},\text{T},\text{P}\right)\hfill & \text{C}2\hfill \\ 0=\text{V}-\text{f}\left(\underset{\_}{\text{n}},\text{T},\text{P}\right)\hfill & \text{C}2\hfill \end{array}$
M6: Rigorous two-phase reactor model; add reaction terms to MB & EB and add kinetic (constitutive) model; $\underset{\_}{\text{d}}$ are kinetic parameters; ΔHR is heat of reaction; k is reference reactant	Reaction term (mass)=(Lxk)Ratei Reaction term (energy)= ΔHRL $\text{0}={\text{Rate}}_{\text{i}}-\text{f}\left(\text{T},\text{P},\underset{\_}{\text{n}},\underset{\_}{\text{d}}\right)$

From the above models, it can be noted that in the life of a process, since the components in the process are unlikely to be changed, the C1 (constitutive) models would not change but the MB, EB and C2 models may change as one moves from on stage another in the life of a process. On the other hand, in the life of a process model, the EB, MB models are unlikely to be changed but the C1 and C2 models may be changed. For example, Model M5 may be further simplified by assuming rate of change of energy holdup (this will convert the EB to an algebraic equation) or the vapour holdup may not be assumed to be negligible (this will make the model more complex as now the holdup n_i will be sum of liquid and vapour holdups). Alternatively, Models M3, M4, and M5 may be simplified by selecting simpler C1 models for the liquid and vapour phase fugacities. Note that the nonlinearity of EB and MB equations becomes clear by inserting the corresponding C1 equations into them. Note also that models Ml and M2 are linear because the corresponding constitutive variables are independent of composition. Models consisting of C1 and C2 are commonly used for equilibrium saturation point calculations (does not need MB and EB) that enable the generation of phase diagrams. In a similar fashion, kinetic models represented by C1 and their corresponding C2 equations generate reaction yield diagrams (attainable regions).

In the next section, describes how the generated models can be very quickly imported into a computer-aided system and after analysis solved with the appropriate solver.

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Leak Detection Systems

Yong Bai , Qiang Bai , in Subsea Pipeline Integrity and Risk Management, 2014

Mass Balance with Line Pack Compensation

Mass balance with line pack compensation (MBLPC) relies on the principle of conservation of mass. For a normal pipeline, the flow entering and leaving the pipe can be metered. The mass of fluid in the pipe section can be calculated from the pipe dimensions and measurement of state variables of the fluid such as pressure and temperature.

The sensors required for this technique can be categorized as flow rate, pressure, and temperature of the production fluid. Flow meters are required at all inlets and outlets of the pipeline. In addition, some systems use density meters in their monitoring. [5].

If the difference of mass between upstream and downstream are larger than an established tolerance, a leak alarm is generated. This method allows the detection of a leak that does not necessarily generate a high rate of change in pressure or flow. The methods can be based on flow rate difference only, which would generate a mass or volume balance scheme or on mass balance compensated by pressure or temperature changes and inventory fluctuations in a pipeline.

The sensitivity of the mass balance technique depends on the accuracy of the estimate of pipe contents. The accuracy can be increased by considering the flow into and out of the pipe section over a long enough time period, when the mass that has flowed in and out of the pipeline is very large compared to the mass resident in the pipe. Over a long enough time period, detection is limited only by the accuracy of the flow instrumentation. Generally, the method can detect small leaks over a long time, assuming steady state line conditions. A wide range of flow variations can be accommodated without masking the leak detection process, and pipeline transients are normally filtered out by long-term averaging.

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https://www.sciencedirect.com/science/article/pii/B9780123944320000068

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Source: https://www.sciencedirect.com/topics/engineering/principle-of-conservation-of-mass

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