Conservation of Mass in Fluid Mechanics

shayan pakravan

The mass conservation equation is one of the basic equations in fluid mechanics, which is solved in all Fluid Mechanics CFD simulations. In this article, we will examine this equation and prove this equation.

Contents

What is the Conservation of Mass in Fluid Mechanics?

The mass conservation equation, which is known as the continuity equation in fluid mechanics, states that the difference in the input and output mass to the control volume per unit time is equal to the change in the mass inside it. The control volume is a part of the space that is investigated, for example, it can be the space inside a pump. It is possible to calculate the changes in the mass of the fluid inside the pump by measuring the input and output flows from the pump.

Conservation of Mass (Continuity) Formula

In Figure 1, the control volume element is shown. A two-dimensional problem is assumed, with one unit thickness in the z-direction. The velocities u and v represent the fluid velocities in the x and y directions, respectively. The dimensions of the element are dx and dy. The density of the fluid is ρ, and ρdxdy represents the total mass of the fluid within the control volume.

Fig 1. The control volume element, flow rates, and total mass are shown in the figure

The difference in the flow rates at different levels is equal to the change in the mass of the control volume per unit of time:

$\left( \left( \rho u \right)_2 - \left( \rho u \right)_1 \right)dy+\left( \left( \rho v \right)_2 - \left( \rho v \right)_1 \right)dx = \frac{-d \rho dxdy }{dt}$

We divide the sides by dxdy:

$\frac{\left( \left( \rho u \right)_2 - \left( \rho u \right)_1 \right)}{dx}+\frac{\left( \left( \rho v \right)_2 - \left( \rho v \right)_1 \right)}{dy} = \frac{-d \rho }{dt}$

It should be noted that the dimensions of the element are fixed, and to change the mass inside the element, its density must change. Because the dimensions of the element are very small, the difference in velocities can be written as a differential. Additionally, because the velocities and density can be functions of several variables, they can be expressed as differential quantities:

$\frac{\left( \partial \rho u \right)}{\partial x}+\frac{\left( \partial \rho v \right)}{\partial y} = \frac{-\partial \rho }{\partial t}$

Using a similar method, the continuity equation for three dimensions becomes:

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

This equation can be written in the following form:

$\nabla \rho \overrightarrow{V} = \frac{-\partial \rho }{\partial t}$

$\overrightarrow{V} = (u,v,w)$

If the fluid is incompressible, the density is constant and does not change with time, then:

$\nabla \overrightarrow{V} = 0$

The continuity equation can be proved by another method. In this method, as shown in Figure 2, the flow rates are written based on the Taylor series.

Fig 2. The control volume element, flow rates based on the Taylor series, and total mass are shown in the figure

The flow rate difference is equal to the change of element mass based on time:

$\left( \rho u+ \frac{\partial \rho u}{\partial x } dx - \rho u \right)dy + \left( \rho v+ \frac{\partial \rho v}{\partial y } dy - \rho u \right)dx = \frac{-\partial \rho dxdy }{\partial t}$

By dividing the sides by dxdy:

$\frac{\left( \partial \rho u \right)}{\partial x}+\frac{\left( \partial \rho v \right)}{\partial y} = \frac{-\partial \rho }{\partial t}$

The continuity equation for the control volume that is not an element is:

$\int_{CV} \frac{\partial \rho}{\partial t} d\vartheta + \int_{CS} \rho (\overrightarrow{V}.n )dA = 0$

CV and CS stand for control volume and control surface, respectively. In cylindrical coordinates, the continuity equation is:

$\frac{\partial \rho }{\partial t} + \frac{1}{r} \frac{\partial}{\partial r} (\rho r v_r)+\frac{1}{r} \frac{\partial}{\partial \theta} (\rho v_{\theta}) + \frac{\partial}{\partial z}(\rho v_z)=0$

$\overrightarrow{V}= (v_r,v_{\theta},v_z)$

An Example Conservation of Mass Fluid Mechanics Examples

In Figure 3, V and A represent the average velocity and cross-sectional area of the inlet or outlets, respectively. Calculate the value of V₂ based on the other given parameters. Assume the fluid is incompressible.

Fig 3. A control volume with 1 inlet and 2 outlets.

Because the fluid is incompressible and the control volume is completely filled with fluid, the amount of fluid entering and exiting the system is equal:

$V_1A_1=V_2A_2+V_3A_3 \Rightarrow V_2 = \frac{V_1A_1-V_3A_3}{A_2}$

The Role of Conservation of Mass Equation (Continuity Equation) in CFD and ANSYS Fluent

Continuity and Navier-Stokes equations are the basis of Fluid Mechanics CFD simulations. Simulation algorithms are determined based on their solution. ANSYS Fluent offers different methods and algorithms to solve these equations. Users can see the answer of these equations in each iterate, and adjust their details. There are many options to adjust these equations in Fluent.

In ANSYS Fluent, we can verify the satisfaction of the law of conservation of mass by using the ‘Report’ feature. To do this, select ‘Report’, then ‘Fluxes’, followed by ‘Mass Flow Rate’. Next, choose the inlet and outlet boundaries of your model. If the net mass flow rate (the difference between inflow and outflow) is very close to zero, it indicates that the law of conservation of mass is satisfied within the computational domain.

The window related to the flux report in ANSYS Fluent allows you to display the error of the continuity equation within the desired boundaries for each iteration.

At CFDLAND, we have completed numerous Fluid Mechanics CFD simulation projects. After reviewing our work and gaining confidence in our specialists’ abilities, you can place your order on order CFD project.

Click to access the CFD Heat Transfer

Conclusion

In conclusion, the continuity equation is widely used in fluid mechanics and in Fluid Mechanics CFD simulations it is solved within each cell of the mesh. We have seen how this equation can be reached and proven by different methods and how this equation looks in cylindrical coordinates.

more information:

Energy Conservation Equation

Dimensionless Numbers in Fluid Mechanics