Mass inside
∀ ρ d∀
the
=
(2.11)
control volume
The statement of conservation of mass for the control volume calls for the time rate of change in
mass. In mathematical notation,
Time rate of change
∂
∀ ρ d∀
in mass in the
=
(2.12)
∂t
control volume
For the reach of river, the statement of the conservation of mass becomes
∂
A ρ 2 v 2 dA 2 - A ρ1 v 1 dA 1 + ∂t ∀ρ d∀ = 0
(2.13)
2
1
It is often convenient to work with average conditions at a cross-section, so we define an average
velocity V such that
1
V=
v dA
(2.14)
A
A
The symbol v represents the local velocity whereas the velocity V is the average velocity at the
cross-section.
Because water is nearly incompressible the density ρ of the fluid is considered constant, ρ1 = ρ2 =
ρ. When the flow is steady
∂
ρ d∀ = 0
(2.15)
∂t
∀
and Equation 2.13 reduces to the statement that inflow equals outflow or
ρV2 A 2 - ρV1 A 1 = 0
That is, for steady flow of incompressible fluids
V1 A 1 = V2 A 2 = Q = VA
(2.16)
where Q is the volume flow rate or the discharge.
Equation 2.16 is the familiar form of the conservation of mass equation for steady flow in rivers. It
is applicable when the fluid density is constant, the flow is steady and there is no significant
lateral inflow or seepage.
2.5
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