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3.4 Continuity
The
equation of continuity works under the assumption that the flow in will equal
the flow out. This can be useful to solve for many properties of the fluid and
its motion :
Q1 = Q2
This
can be expressed in many ways, for example: A1*v1 = A2*v2 . The equation of continuity applies to any
incompressible fluid. Since the fluid cannot be compressed, the amount of fluid
which flows into a surface must equal the amount flowing out of the surface.
Applying the Continuity Equation
You can
observe the continuity equation's effect in a garden hose. The water flows
through the hose and when it reaches the narrower nozzle, the velocity of the
water increases. Speed increases when cross-sectional area decreases, and speed
decreases when cross-sectional area increases. This is a consequence of the
continuity equation. If the flow Q is held constant, when the area A decreases,
the velocity v must increase proportionally. For example, if the nozzle of the
hose is half the area of the hose, the velocity must double to maintain the
continuous flow.
Continuity Equation
When a
fluid is in motion, it must move in such a way that mass is conserved. To see
how mass conservation places restrictions on the velocity field, consider
the steady flow of fluid through a duct (that is, the inlet
and outlet flows do not vary with time). The inflow and outflow are one-dimensional,
so that the velocity V and density \rho are constant over the area A (figure
14).
Figure 14. One-dimensional duct showing control volume.
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Now we apply the principle of mass conservation. Since there is no flow through the side walls of the duct, what mass comes in over A_1 goes out of A_2, (the flow is steady so that there is no mass accumulation). Over a short time interval\Delta t,
This is a statement of the principle of mass conservation for a steady, one-dimensional flow, with one inlet and one outlet. This equation is called the continuity equation for steady one-dimensional flow. For a steady flow through a control volume with many inlets and outlets, the net mass flow must be zero, where inflows are negative and outflows are positive.
Example : Water runs through a water main of cross-sectional area
0.4 m2 with a velocity of 6 m/s. Calculate the velocity of the
water in the pipe when the pipe tapers down to a cross-sectional area of 0.3 m2.
Answer:
Example: Water enters a typical garden hose of diameter 1.6 cm with
a velocity of 3 m/s. Calculate the exit velocity of water from the garden hose
when a nozzle of diameter 0.5 cm is attached to the end of the hose.
Answer: First, find the cross-sectional areas of the entry (A1)
and exit (A2) sides of the hose.
Next,
apply the continuity equation for fluids to solve for the water velocity as it
exits the hose (v2).
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