AE/ME 201-- Procedure -- Lab: Tank Draining Instrumentation and ...

AE/ME 201 – Spring 2005
the tank.
7. ρ - density
With 3 locations and 7 unknowns at each, there are
a total of 21 unknowns initially.
Using a definition of total pressure from the
Bernoulli equation (not used yet), one definition
links four of the quantities for each location. The
three state equations are:
P T1 = P s1 + ρ 1V1
2
2
+ ρ 1 gh 1 (3)
Fluid Mechanics
Figure 3: Schematic of tank.
Although, this experiment is intended to introduce
you to some different instrumentation techniques
and demonstrate experimental methods for obtaining
the constant coefficient in a simple first order
differential equation, a brief introduction to fluid
mechanics is included to help to understand what
you’re measuring and why.
Measurements of the liquid height versus time for
the tank draining will provide you with information
on the exit flow fixture (i.e. for your experiment,
this includes the section of pipe and the valve). For
example, the valve at the bottom of the tank provides
some resistance to flow and controls the rate
of draining and the rate of change of height versus
time. Also, the height in the tank sets the water
source pressure that drives the flow through the
valve. Therefore, the measurement of height versus
time provides an indication of both flow rate and
supply pressure.
There are three locations in question as shown in
Figure 3. At each location, seven quantities are used
to describe the state of the water or conditions:
1. P s - Static pressure
2. P T - Total pressure
3. V - flow velocity
4. h - fluid height
5. ṁ - mass flow
6. A - area
P T2 = P s2 + ρ 2V2
2
2
+ ρ 2 gh 2 (4)
P T3 = P s3 + ρ 3V3
2
+ ρ 3 gh 3 (5)
2
Conservation of mass applies between pairs of locations.
The two conservation of mass equations are:
and
ṁ 2 = ṁ 3 (6)
dh 1
ρ 1 A 1 = −ṁ 3 (7)
dt
Additional definitions are used to link some of the
quantities. The velocity at location 1 is the free
surface motion:
V 1 = dh 1
(8)
dt
Mass flow rates are defined from surface fluxes as:
and
ṁ 2 = ρ 2 A 2 V 2 (9)
ṁ 3 = ρ 3 A 3 V 3 (10)
The loss coefficient, K or K 23 , for the exit plumbing
(including the test section) is
K = K 23 = 2(P T 2
− P T3 )
ρ 2 V 2
2
This adds an unknown and an equation.
(11)
More equations are obtained by adopting some assumptions.
1. The total pressure loss between stations 1 and
2 is very small
2. density is constant and known - this is three
equations
3. The static pressure at station 1 is the local atmospheric
pressure
4. The static pressure at station 3 is the local atmospheric
pressure
2