Che 303 (Transport Phenomena) : Lecture 3 - Continuity Equation

DEPARTMENT OF CHEMICAL ENGINEERING
OBAFEMI AWOLOWO UNIVERSITY, ILE IFE
CHE 303 (TRANSPORT PHENOMENA)
LECTURE 3 – CONTINUITY EQUATION

1. FLUIDS IN MOTION
We will limit our treatment to moving fluids that are incompressible (that is, whose density doesn’t change)
and ones that are at steady state. By steady state, we mean that the pressure and velocity do not change in
time in the fluid, although they may change with position.
For fluids at rest, we only needed to consider two quantities, density and pressure. If the fluid is flowing
(or moving) we need one more quantity which is the velocity of the fluid.
1.1. What do we mean by the velocity in a fluid?

The velocity in a fluid is the velocity of a “small” volume of the fluid. More specifically, it is the velocity
of a volume of the fluid as the volume approaches zero. In general the velocity in a fluid can change from
one point to another, so we can speak of the velocity at a point in the fluid. At every point in the fluid, we
can ascribe a velocity vector, which is the velocity of a small volume of the fluid at that point. The velocity
in a fluid is an example of a “vector field”. If there is a vector assigned to every point in space, this collection
of vectors is called a vector field.
We can express our velocity as a function of position and time:
𝒗 = 𝒗(𝒙, 𝒚, 𝒛, 𝒕)
So that we can have velocity components in the x, y or x – directions and these can also vary with time(or
other coordinate directions):
X – direction = 𝒗𝒙
Y – direction = 𝒗𝒚
Z – direction = 𝒗𝒛
For all practical applications however, we only consider the average velocity in the direction of interest.
1.2. Continuity Equation

In its basic form, the continuity equation is the material balance expression below:
(𝑴𝑨𝑺𝑺 𝑰𝑵𝑭𝑳𝑶𝑾) − (𝑴𝑨𝑺𝑺 𝑶𝑼𝑻𝑭𝑳𝑶𝑾) = 𝑨𝑪𝑪𝑼𝑴𝑼𝑳𝑨𝑻𝑰𝑶𝑵

Applying this to the elemental volume shown below, we can write mass balance in each direction:
In the x – direction, we have:
𝝏
𝝆𝒒𝒙 |𝒙 − 𝝆𝒒𝒙 |𝒙+∆𝒙 = 𝝏𝒕 (𝝆𝑽) (1)
Where:
𝜌 = fluid density
𝑞𝑥 |𝑥 = volumetric flow rate of the fluid in the x – direction (at position x) = (∆𝑦∆𝑧)𝑣𝑥 |𝑥
𝑞𝑥 |𝑥+∆𝑥 = volumetric flow rate of the fluid in the x – direction (at position x+∆x) = (∆𝑦∆𝑧)𝑣𝑥 |𝑥+∆𝑥
𝑣𝑥 |𝑥 = velocity of the fluid in the x – direction (at position x)
𝑣𝑥 |𝑥+∆𝑥 = velocity of the fluid in the x – direction (at position x+∆x)
𝑉 = volume of the portion of the fluid taken as a reference = ∆𝑥∆𝑦∆𝑧
Noting all these, we can rewrite (1) as:
𝝏
𝝆(∆𝒚∆𝒛)𝒗𝒙 |𝒙 − 𝝆(∆𝒚∆𝒛)𝒗𝒙 |𝒙+∆𝒙 = 𝝏𝒕 (𝝆𝑽) (2)
Taking the elemental volume V= ∆𝑥∆𝑦∆𝑧 to be constant and dividing through equation (2) by the
volume, we have:
𝝆𝒗𝒙 |𝒙 − 𝝆𝒗𝒙 |𝒙+∆𝒙 𝝏𝝆
=
∆𝒙 𝝏𝒕
Taking limits as ∆𝑥 → 0, we have
𝝏 𝝏𝝆
− 𝝏𝒙 (𝝆𝒗𝒙 ) = 𝝏𝒕
(3)
Writing similar expressions for the y – and z – directions, we have:
y = direction:
𝝏
𝝆(∆𝒙∆𝒛)𝒗𝒚 | − 𝝆(∆𝒙∆𝒛)𝒗𝒚 | = (𝝆𝑽)
𝒚 𝒚+∆𝒚 𝝏𝒕
z = direction:
𝝏
𝝆(∆𝒙∆𝒚)𝒗𝒛 |𝒛 − 𝝆(∆𝒙∆𝒚)𝒗𝒛 |𝒛+∆𝒛 = (𝝆𝑽)
𝝏𝒕
Treating the equations the same way we treated equation (2), we have:
𝝏 𝝏𝝆
− 𝝏𝒚 (𝝆𝒗𝒚 ) = 𝝏𝒕
(4)
𝝏 𝝏𝝆
− 𝝏𝒛 (𝝆𝒗𝒁 ) = 𝝏𝒕
(5)
Combining the differential equations in (3) – (5), we end up with the 3-dimensional “material balance”
expression shown below:
𝝏 𝝏 𝝏 𝝏𝝆
−[ (𝝆𝒗𝒙 ) + (𝝆𝒗𝒚 ) + (𝝆𝒗𝒁 )] = (6)
𝝏𝒙 𝝏𝒚 𝝏𝒛 𝝏𝒕
Equation (6) is known as CONTINUITY EQUATION in Cartesian coordinates.
Since velocity is a vector quantity, we can write equation (6) in vector form:
𝝏𝝆
̅) =
−𝛁 ∙ (𝝆𝒗 (7)
𝝏𝒕
𝜕𝜌
For incompressible fluids, the density is constant so that 𝜕𝑡
= 0. With this, equation (7) becomes:
̅=𝟎
𝛁∙𝒗 (8)
Our continuity equation for flow in pipes can be written simply as:
Mass flow in = Mass flow out
𝝆𝟏 𝒗𝟏 𝑨𝟏 = 𝝆𝟐 𝒗𝟐 𝑨𝟐 (9)
2
1
Where:
𝜌1 = fluid density at position 1
𝑣1 = fluid velocity at position 1
𝐴1 = Flow cross-sectional area at position 1
𝜌2 = fluid density at position 2
𝑣2 = fluid velocity at position 2
𝐴2 = Flow cross-sectional area at position 2
For incompressible fluids, the density is constant so that the continuity equation becomes:
𝑨𝟏 𝒗𝟏 = 𝑨𝟐 𝒗𝟐 (10)
Or,
𝑨𝒗 = 𝒄𝒐𝒏𝒔𝒕𝒂𝒏𝒕

Che 303 (Transport Phenomena) : Lecture 3 - Continuity Equation

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DEPARTMENT OF CHEMICAL ENGINEERING

OBAFEMI AWOLOWO UNIVERSITY, ILE IFE

CHE 303 (TRANSPORT PHENOMENA)

LECTURE 3 – CONTINUITY EQUATION

1.1. What do we mean by the velocity in a fluid?

We can express our velocity as a function of position and time:

1.2. Continuity Equation

(𝑴𝑨𝑺𝑺 𝑰𝑵𝑭𝑳𝑶𝑾) − (𝑴𝑨𝑺𝑺 𝑶𝑼𝑻𝑭𝑳𝑶𝑾) = 𝑨𝑪𝑪𝑼𝑴𝑼𝑳𝑨𝑻𝑰𝑶𝑵

Taking limits as ∆𝑥 → 0, we have

Writing similar expressions for the y – and z – directions, we have:

expression shown below:

Equation (6) is known as CONTINUITY EQUATION in Cartesian coordinates.

Mass flow in = Mass flow out

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