in this example we are given that in a venchuri-meter,
the diameter of pipe is 4 centimeter, and
that of constriction is, 3 centimeter. when
water flows, the height of water level in
the pressure tubes are 15 and 20 centimeter,
and we are required to find the discharge
rate of pipe. we just draw the situation here
we can see, in a ventchuri-meter, there are
2 pressure tubes, 1 is attached to a pipe
and, the other is connected to the constriction
which is of smaller cross sectional area.
and say in this situation, we are given that
the diameter, is 4 centimeter at the level
of pipe, and at the level of constriction
it is 3 centimeter. so obviously the flow
velocity here will increase and pressure will
drop. so water level here will be less, and
the water level, in this tube will be relatively
more. and the difference in the 2 levels here
we can see, it is given to be 5 centimeter
because, here it is 15 and 20 centimeter.
now if flow rate here is v 1, flow rate here
is v 2, we know well that, in venchuri-meter,
the flow velocity, of fluid, in pipe is given
as, this we already calculated, in the separate
section of venchuri-meter analysis and, this
v 1 can be given by the expression as, root
of 2 g h by, a-1 square by, a-2 square minus
1, where a-1 and a-2 are the cross sectional
areas of pipe and constriction. and here if,
diameters are given and area is proportional
to square of diameter, so, if we substitute
the values here we can see, this’ll be root
of, 2, g we can take as 10, height difference
between the 2 pressure tubes is 5 centimeter,
point zero 5, divided by, ratio of the 2 areas
will be proportional to the ratio of, square
of their diameters, so here we can use it
as, 4 by 3 to power 4, minus 1. on substituting
the values we are getting it as root of, we
can say, it is 20 multiplied by point zero
5 it is 1, divided by, 2 56 by 81, minus 1.
further simplifying the expression, we’ll
get it as, 9 by, 5 root 7, meters per second.
that is the speed of, liquid flow in the pipeline,
water flow in the pipeline. and if we wish
to find out the discharge rate, of pipe, discharge
rate is, the rate at which it is ejecting,
or the volume flow rate of water can be given
as, the product of area and, the flow velocity.
and, the area of the pipe can be given as,
pi r square, radius is 2 centimeter. so we
can just calculate it in centimeter per second
so it is, pi into 2 square, multiplied by
the speed of, water flow, in centimeter per
second if we take it is, 900 by, 5 root 7,
centimeter per second. so on calculation this’ll
give us, 36 hundred pi by, 5 root 7, centimeter
cube per second. that’ll be the discharge
rate and, answer to this problem. if required,
we can further numerically simplify it.
