The General Form of the Continuity Equation for Incompressible Flow I

Continuity Equation

The two independent continuity equations indicate the need for two equations of motion to predict the mass average velocity vγ, and the species velocity vAγ or the mass diffusion velocity uAγ.

From: Advances in Heat Transfer , 1998

Reservoir Simulation

John R. Fanchi , in Integrated Reservoir Asset Management, 2010

13.1.2 Incompressible Flow

The continuity equation for the flow of a fluid with density ρ, velocity $\overrightarrow{v}$ , and no source or sink terms may be written as

(13.1.12) $\frac{\partial \mathrm{\rho }}{\partial t}+\mathrm{\nabla }\cdot \left(\mathrm{\rho }\overrightarrow{v}\right)=0$

If we introduce the differential operator

(13.1.13) $\frac{D}{\mathit{Dt}}=\frac{\partial }{\partial t}+\overrightarrow{v}\cdot \mathrm{\nabla }$

into Eq. (13.1.12), the continuity equation has the form

(13.1.14) $\frac{D\mathrm{\rho }}{\mathit{Dt}}+\mathrm{\rho }\mathrm{\nabla }\cdot \overrightarrow{v}=0$

For an incompressible fluid, density is constant, and the continuity equation reduces to the following condition:

(13.1.15) $\mathrm{\nabla }\cdot \overrightarrow{v}=0$

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Fluid Flow Equations

John R. Fanchi , in Shared Earth Modeling, 2002

Incompressible Flow

The continuity equation for the flow of a fluid with density ρ, velocity $\overrightarrow{v}$ and no source or sink terms may be written as

(9.2.9) $\frac{\partial \rho }{\partial t}+\nabla \cdot \left(\rho v\right)=0$

If we introduce the differential operator

(9.2.10) $\frac{D}{Dt}=\frac{\partial }{\partial t}+\overrightarrow{\nu }\cdot \nabla$

into Equation (9.2.9), the continuity equation has the form

(9.2.11) $\frac{D\rho }{Dt}+\rho \nabla \cdot \overrightarrow{\nu }=0$

In the case of an incompressible fluid, density is constant and the continuity equation reduces to the following condition for incompressible fluid flow:

(9.2.12) $\nabla \cdot \overrightarrow{\nu }=0$

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GENERALIZED GOVERNING EQUATIONS FOR MULTIPHASE SYSTEMS: AVERAGING FORMULATIONS

Amir Faghri , Yuwen Zhang , in Transport Phenomena in Multiphase Systems, 2006

4.4.1 Continuity Equation

The continuity equation for phase k in the multifluid model is expressed by eq. (4.22). Summing the continuity equations for all Π phases together, one obtains

(4.90) $\frac{\partial }{\partial t}\left({\displaystyle \sum _{k=1}^{\Pi }{\epsilon }_k{〈{\rho }_k〉}^k}\right)+\nabla \cdot {\displaystyle \sum _{k=1}^{\Pi }{\epsilon }_k{〈{\rho }_k〉}^k{〈V_k〉}^k=}{\displaystyle \sum _{k=1}^{\Pi }{\displaystyle \sum _{j=1(j\ne k)}^{\Pi }{{{\dot{m}}^{″}}^{\prime }}_{jk}}}$

The right-hand side of equation (4.90) must be zero because the total mass of all phases produced by phase change must equal the total mass of all phases consumed by phase change. Considering this fact and eq. (4.20), the continuity equation becomes

(4.91) $\frac{\partial 〈\rho 〉}{\partial t}+\nabla \cdot {\displaystyle \sum _{k=1}^{\Pi }{\epsilon }_k{〈{\rho }_k〉}^k{〈{\text{V}}_k〉}^k=0}$

The bulk velocity of the multiphase mixture is the mass-averaged velocity of all the individual phases:

(4.92) $\tilde{\text{V}}=\frac{1}{〈\rho 〉}{\displaystyle \sum _{k=1}^{\Pi }{\epsilon }_k{〈{\rho }_k〉}^k{〈{\text{V}}_k〉}^k}$

Substituting eq. (4.92) into eq. (4.91), the final form of the continuity equation for a multiphase mixture is

(4.93) $\frac{\partial 〈\rho 〉}{\partial t}+\nabla \cdot 〈\rho 〉\tilde{\text{V}}=0$

It can be seen that eq. (4.93) has the same form as the local continuity equation (3.38), except that the volume-averaged density and velocity are used in eq. (4.93), where $〈\rho 〉={\displaystyle \sum _{k=1}^{\Pi }{\epsilon }_k}{〈{\rho }_k〉}^k$ .

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Real-Time Transient Model–Based Leak Detection

Morgan Henrie PhD, PMP, PEM , ... R. Edward Nicholas , in Pipeline Leak Detection Handbook, 2016

Continuity Equation (Conservation of Mass)

The continuity equation ( Eq. 4.1) is the statement of conservation of mass in the pipeline: mass in minus mass out equals change of mass.

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The first term in the equation, $\partial (\rho vA)/\partial x$ , is "mass flow in minus mass flow out" of a slice of the pipeline cross-section.

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The second term, $\partial (\rho A)/\partial t$ , is the rate of change of mass of a slice of pipeline cross-section.

Equation 4.1. Continuity Equation (Mass Balance)

where $\rho$ is mass density, v is the velocity, A is the pipe cross-sectional area, $x$ is the coordinate along the pipe centerline, and $t$ is time.

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27th European Symposium on Computer Aided Process Engineering

Qi Zhang , ... Ignacio E. Grossmann , in Computer Aided Chemical Engineering, 2017

3.4 Continuity Constraints

Continuity equations are required at the boundaries of each season in order to maintain mass balance and feasible transitions:

(3a) $y_{imh,0}=y_{imh,\left|{\overline{T}}_h\right|}\forall i,m\in M_i,h$

(3b) $z_{im{m}^{\prime }ht}=z_{im{m}^{\prime }h,t+\left|{\overline{T}}_h\right|}\forall i,\left(m,m^{\prime }\right)\in T{R}_i,h,-{\theta }_i^{\text{max}}+1\le t\le -1$

(3c) $y_{imh,\left|{\overline{T}}_h\right|}=y_{im,h+1,0}\forall i,m\in M_i,h\in H\backslash \left\{\left|H\right|\right\}$

(3d) $z_{im{m}^{\prime }h,t+\left|{\overline{T}}_h\right|}=z_{im{m}^{\prime },h+1,t}\forall i,\left(m,m^{\prime }\right)\in T{R}_i,h\in H\backslash \left\{\left|H\right|\right\},-{\theta }_i^{\text{max}}+1\le t\le -1$

(3e) ${\overline{Q}}_{jh}=Q_{jh,\left|{\overline{T}}_h\right|}-Q_{jh,0}\forall j,h$

(3f) $Q_{jh,0}+n_h{\overline{Q}}_{jh}=Q_{j,h+1,0}\forall j,h\in H\backslash \left\{\left|H\right|\right\}$

(3g) $Q_{j,\left|H\right|,0}+n_{\left|H\right|}{\overline{Q}}_{j,\left|H\right|}\ge Q_{j,1,0}\forall j.$

The cyclic schedules are enforced by applying Eqs. (3a)–(3b) while Eqs. (3c)–(3d) match the state in which the system is at the end of one season to the beginning of the next. Also, by adding xEqs. (3e)–(3g), we allow excess inventory $\left({\overline{Q}}_{jh}\right)$ accumulated over the course of a season to be carried over to the next.

Finally, the objective is to minimize the total annual cost, which consists of capital and operating expenses, for which piecewise-linear approximations have been incorporated.

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Numerical modeling methodologies for friction stir welding process

Rahul Jain , ... Shiv B. Singh , in Computational Methods and Production Engineering, 2017

5.5.3 Governing equation

Continuity, momentum, and energy equations are the conservation equations in Eulerian analysis as specified as follows:

Continuity equation

(5.25) $\frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}+\frac{\partial w}{\partial z}=0$

Momentum equation

(5.26) $\rho \left(u\frac{\partial u}{\partial x}+v\frac{\partial u}{\partial y}+w\frac{\partial u}{\partial z}\right)=-\frac{\partial p}{\partial x}+\frac{\partial }{\partial x}\left(\mu \frac{\partial u}{\partial x}\right)+\frac{\partial }{\partial y}\left(\mu \frac{\partial u}{\partial y}\right)+\frac{\partial }{\partial z}\left(\mu \frac{\partial u}{\partial z}\right)$

(5.27) $\rho \left(u\frac{\partial v}{\partial x}+v\frac{\partial v}{\partial y}+w\frac{\partial v}{\partial z}\right)=-\frac{\partial p}{\partial y}+\frac{\partial }{\partial x}\left(\mu \frac{\partial v}{\partial x}\right)+\frac{\partial }{\partial y}\left(\mu \frac{\partial v}{\partial y}\right)+\frac{\partial }{\partial z}\left(\mu \frac{\partial v}{\partial z}\right)$

(5.28) $\rho \left(u\frac{\partial w}{\partial x}+v\frac{\partial w}{\partial y}+w\frac{\partial w}{\partial z}\right)=-\frac{\partial p}{\partial z}+\frac{\partial }{\partial x}\left(\mu \frac{\partial w}{\partial x}\right)+\frac{\partial }{\partial y}\left(\mu \frac{\partial w}{\partial y}\right)+\frac{\partial }{\partial z}\left(\mu \frac{\partial w}{\partial z}\right)$

Energy equation

(5.29) $\rho C_p\left(u\frac{\partial T}{\partial x}+v\frac{\partial T}{\partial y}+w\frac{\partial T}{\partial z}\right)=\frac{\partial }{\partial x}\left(k\frac{\partial T}{\partial x}\right)+\frac{\partial }{\partial y}\left(k\frac{\partial T}{\partial y}\right)+\frac{\partial }{\partial z}\left(k\frac{\partial T}{\partial z}\right)+S_t$

where u, v, and w are the velocity in X, Y, and Z directions, respectively, and S_t is the viscous heat dissipation. It has been reported that viscous heat dissipation is small as compared to the plastic heat dissipation (Nandan et al., 2008) and is given by Eq. (5.30).

(5.30) $S_t=\alpha \varphi$

where α is a constant reflecting the extent of mixing and it is defined as 0.05. ϕ is the viscous dissipation heat as shown in Eq. (5.31).

(5.31) $\varphi =\mu \left[2{\left(\frac{\partial u}{\partial x}\right)}^2+2{\left(\frac{\partial v}{\partial y}\right)}^2+2{\left(\frac{\partial w}{\partial z}\right)}^2+{\left(\frac{\partial v}{\partial x}+\frac{\partial u}{\partial y}\right)}^2+{\left(\frac{\partial v}{\partial z}+\frac{\partial w}{\partial y}\right)}^2+{\left(\frac{\partial w}{\partial x}+\frac{\partial u}{\partial z}\right)}^2\right]$

The coupled equations were solved by using pressure-based solver with second-order discretization of momentum and energy equation

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Numerical simulation of a wind turbine

Farschad Torabi , in Fundamentals of Wind Farm Aerodynamic Layout Design, 2022

6.1.1.1 Conservation of mass (continuity)

The continuity equation applies to all kinds of flows, including compressible, incompressible, Newtonian, and non-Newtonian flows. It says that matter is conserved in a flow, meaning that the difference between the entering and exiting flow is equal to the mass rate. With assumption of incompressible flow, the mass conservation law is expressed using

(6.2) $\frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}+\frac{\partial w}{\partial z}=0,$

in which u, v, and w represent the velocity components in the x, y, and z directions (Sayma, 2009).

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Proceedings of the 8th International Conference on Foundations of Computer-Aided Process Design

Dung A. Pham , ... Sun-Keun Lee , in Computer Aided Chemical Engineering, 2014

2.1 Continuity equation

The continuity equation is expressed as follows:

(1) $\frac{\partial \rho }{\partial t}=-\nabla \cdot \left(\rho \overrightarrow{\mu }\right)$

where ρ is the density (kg/m³), and $\overrightarrow{u}$ is the velocity vector. The continuity equation means the overall mass balance. The Hamiltonian operator (∇) is a spatial derivative vector. The independent variables of the continuity equation are t, x, y, and z. The first term of Eq. (1) is the accumulation term of the total mass within a controlled volume. The second term denotes the convection term of the total mass. If a fluid is incompressible like a liquid or gas under a mild temperature and pressure (normally an absorber with structured-packing bed is operated at about 25–100 °C, and the pressure drop per a pack unit is about 24–36Pa. The Mach number of the flow is therefore below 0.3(Durst, 2008)), the density may be constant with time and space. At this case, Eq. (1) is simplified into

(2) $\nabla \cdot \overrightarrow{u}=0$

In the porous medium with the porosity ε, the incompressible gas-liquid system is expressed as:

(3) $\begin{array}{c}\epsilon \frac{\partial (1-{\alpha }_L){\rho }_G}{\partial t}=-\epsilon \nabla \cdot ((1-{\alpha }_L){\rho }_G{\overrightarrow{\mu }}_G)+\epsilon r_{GL}\\ \epsilon \frac{\partial {\alpha }_L{\rho }_L}{\partial t}=-\epsilon \nabla \cdot ({\alpha }_L{\rho }_L{\overrightarrow{\mu }}_L)-\epsilon r_{GL}\end{array}$

where α_L is the liquid hold-up which changes with time and space, and r_GL (kg/m³/h) is the total mass transfer rate per unit volume from gas to liquid.

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Laws of Motion

Peter R.N. Childs FIMechE, FRSA, FHEA, mem.ASME, MIED, BSc(hons), DPhil, CEng , in Rotating Flow, 2011

2.3 Continuity Equation

The continuity equation is an expression representing the idea that matter is conserved in a flow. Per unit volume, the sum of all masses flowing in and out per unit time must be equal to the change of mass due to change in density per unit time. This can be expressed in the form of an equation by

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

The mass flux through each face of an arbitrary fluid element is illustrated in Figure 2.8.

Figure 2.8. Mass flux through an infinitesimal fluid element control volume.

Summing the inflows and outflows and equating them to the change in mass gives

(2.70) $\begin{array}{l}\left[\rho u_x-\frac{\delta x}{2}\frac{\partial \left(\rho u_x\right)}{\partial x}\right]\delta y\delta z-\left[\rho u_x+\frac{\delta x}{2}\frac{\partial \left(\rho u_x\right)}{\partial x}\right]\delta y\delta z+\left[\rho u_y-\frac{\delta y}{2}\frac{\partial \left(\rho u_y\right)}{\partial y}\right]\delta x\delta y\\ -\left[\rho u_y+\frac{\delta y}{2}\frac{\partial \left(\rho u_y\right)}{\partial y}\right]\delta x\delta y+\left[\rho u_z-\frac{\delta z}{2}\frac{\partial \left(\rho u_z\right)}{\partial z}\right]\delta x\delta y-\left[\rho u_z+\frac{\delta z}{2}\frac{\partial \left(\rho u_z\right)}{\partial z}\right]\delta x\delta y\\ =\frac{\partial }{\partial t}\left(\rho \delta x\delta y\delta z\right)\end{array}$

Collecting terms, dividing through by δxδyδy and converting this to a partial differential equation gives

(2.71) $\frac{\partial \left(\rho u_x\right)}{\partial x}+\frac{\partial \left(\rho u_y\right)}{\partial y}+\frac{\partial \left(\rho u_z\right)}{\partial z}=-\frac{\partial \rho }{\partial t}$

This can be restated in the form

(2.72) $\frac{\partial \rho }{\partial t}+u_x\frac{\partial \rho }{\partial x}+u_y\frac{\partial \rho }{\partial y}+u_z\frac{\partial \rho }{\partial z}+\rho \left(\frac{\partial u_x}{\partial x}+\frac{\partial u_y}{\partial y}+\frac{\partial u_z}{\partial z}\right)=0$

Using the substantive derivative, Equation 2.72 is given by

(2.73) $\frac{D\rho }{Dt}+\rho \left(\frac{\partial u_x}{\partial x}+\frac{\partial u_y}{\partial y}+\frac{\partial u_z}{\partial z}\right)=0$

or in vector form, using the del operator, the continuity equation for unsteady flows is given by

(2.74) $\frac{D\rho }{Dt}+\rho \nabla \cdot \mathbf{u}=0$

The continuity equation applies to all fluids, compressible and incompressible flow, Newtonian and non-Newtonian fluids. It expresses the law of conservation of mass at each point in a fluid and must therefore be satisfied at every point in a flow field.

For incompressible flow the density of a fluid particle does not change as it travels, so

(2.75) $\frac{D\rho }{Dt}=\frac{\partial \rho }{\partial t}+u_x\frac{\partial \rho }{\partial x}+u_y\frac{\partial \rho }{\partial y}+u_z\frac{\partial \rho }{\partial z}=0$

Note that the definition of incompressible flow is less restrictive than that of constant density. Constant density would require each term in Equation 2.75 to be zero. Incompressible flows that have density gradients are sometimes referred to as stratified flows, and examples of these are found in atmospheric and oceanic flows.

For an incompressible flow, with Dρ/Dt   =   0, then from Equation 2.73

(2.76) $\rho \left(\frac{\partial u_x}{\partial x}+\frac{\partial u_y}{\partial y}+\frac{\partial u_z}{\partial z}\right)=0$

Dividing through by the density gives

(2.77) $\frac{\partial u_x}{\partial x}+\frac{\partial u_y}{\partial y}+\frac{\partial u_z}{\partial z}=0$

or in vector notation

(2.78) $\nabla \cdot \mathbf{u}=0$

This equation is valid whether the velocity is time dependent or not.

In cylindrical coordinates, the continuity equation is given by

(2.79) $\frac{\partial \rho }{\partial t}+u_r\frac{\partial \rho }{\partial r}+\frac{u_{\phi }}{r}\frac{\partial \rho }{\partial \phi }+u_z\frac{\partial \rho }{\partial z}+\rho \left(\frac{1}{r}\frac{\partial \left(r{u}_r\right)}{\partial r}+\frac{1}{r}\frac{\partial u_{\phi }}{\partial \phi }+\frac{\partial u_z}{\partial z}\right)=0$

For incompressible flow, Equation 2.79 reduces to

(2.80) $\frac{1}{r}\frac{\partial }{\partial r}\left(r{u}_r\right)+\frac{1}{r}\frac{\partial u_{\phi }}{\partial \phi }+\frac{\partial u_z}{\partial z}=\frac{\partial u_r}{\partial r}+\frac{u_r}{r}+\frac{1}{r}\frac{\partial u_{\phi }}{\partial \phi }+\frac{\partial u_z}{\partial z}=0$

If the flow is turbulent, it is necessary to account for both the velocity and pressure by an average and fluctuating component. Replacing u by $\overline{\mathit{u}}+\mathit{u}\text{'}$ where the primes represent the fluctuating components, and assuming the average values of the derivatives of u′ are zero, the continuity equation using vector notation becomes

(2.81) $\nabla \cdot \overline{\mathit{u}}=0$

In component form the continuity equation for an axisymmetric flow can be stated as

(2.82) $\frac{\partial {\overline{u}}_r}{\partial r}+\frac{{\overline{u}}_r}{r}+\frac{\partial {\overline{u}}_z}{\partial z}=0$

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Conservation of Mass: The Continuity Equation

Bastian E. Rapp , in Microfluidics: Modelling, Mechanics and Mathematics, 2017

10.5 Derivation using Gauss's Theorem

The continuity equation can also be derived using Gauss's theorem (see section 7.2.1). In this case, we use $\rho \overrightarrow{v})$ (which is the directional mass flow in vector form) as the field in Eq. 7.10. We then have to consider the total mass transported across the boundaries of our control volume. Considering Fig. 7.2a as an arbitrary control volume, the mass flow across the boundary can be calculated as

(Eq. 10.18) $-{\displaystyle \int \overrightarrow{\nabla }\cdot \left(\rho \overrightarrow{v}\right)dV}$

The integral is assumed to be negative, as the flow is directed out of the control volume. The change in mass over time in the control volume is given by Eq. 10.12. Using Eq. 10.18 this results in

$\begin{array}{l}-{\displaystyle \int \overrightarrow{\nabla }\cdot \left(\rho \overrightarrow{v}\right)dV}=-{\displaystyle \int \frac{\partial \rho }{\partial t}dV}\hfill \\ \frac{\partial \rho }{\partial t}+\overrightarrow{\nabla }\cdot \left(\rho \overrightarrow{v}\right)=0\hfill \end{array}$

which is identical to Eq. 10.16. In simple terms, the continuity equation describes that if mass is transported into the system via its boundaries, the density of the system must increase in order to store this mass. On the other hand, if mass is transported out of the system via its boundaries, the density of the system must decrease. If the liquid is incompressible, the density of the system cannot change. Therefore, if liquid is flowing into the system across one side of the boundary, the same amount of mass must be flowing out of the system across a different side of the boundary. If the total boundary surface of the system is constant, Eq. 10.8 allows simplifying this statement to "The sum of all influx and outflux into and out of the system must sum up to zero."

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