İzmir Institute of Technology
CHEMICAL ENGINEERING DEPARTMENT
2008-2009 Spring Semester
CHE 310
CHEMICAL ENGINEERING LABORATORY I
Efflux Time for a Tank
with an Exit Pipe
1
EFFLUX TIME FOR A TANK WITH AN EXIT PIPE
Objective: In this experiment one observes the rate of drainage of liquids from a
tank through exit tubes of various sizes. With this system, one analyzes the use of a
macroscopic mass balance combined, with a macroscopic mechanical energy balance.
Indirectly estimation of the loss of mechanical energy caused by viscous dissipation
within the fluid can be examined.
Principle and Theory
Consider a vertical, cylindrical tank such as that shown in Figure 7B.9 (See Ref.1)
that contains a Newtonian liquid up to the level h above the bottom of the tank. The top
of the tank as well as the bottom of the drain pipe are exposed to the atmosphere. Thus,
the liquid drains from the tank under the influence of gravity. A model of the draining
process is sought in order to predict the rate at which the tank empties.
First, one may write a macroscopic mass balance for the liquid in the system,
which is defined as the entire liquid volume from the top surface to the outlet of the pipe.
The total mass of liquid in the system is
mtot   ( R02 L  R 2 h)
(E-1)
where  is the density of the liquid, R is the radius of the tank, R0 is the pipe radius, and L
is its length. According to Equation 7.1-1 in TrPh
d
mtot     R02 v
dt
(E-2)
where v=v2 is the average velocity out of the bottom of the pipe. There is no v1 term
because there is no flow into the top of the tank.
Taking the density to be constant and substituting Equation 1 into Equation 2, one
obtains
2
dh
R 
 0  v
 R
dt
(E-3)
This equation may be integrated from the initial liquid level h1 to a subsequent level h2 to
obtain the corresponding efflux time te. That is,
te  
te
0
 R
dt    
 R0 
2

h2
h1
dh
v
(E-4)
In order to perform the desired integration, one must know the velocity v as a function of
liquid level h. That relationship comes from the macroscopic mechanical energy balance.
2
If one neglects the rate of change of the total mechanical energy in the system,
which is called the pseudo-steady-state approximation, then one can use the macroscopic
mechanical energy balance given as Equation 7.4-2 in TrPh. This equation is also called
the Bernoulli equation. In addition to the pseudo-steady-state approximation, one can
assume that there is no work term, that there is no overall pressure differential, and that
the velocity profile in the tube is essentially flat. With those assumptions Equation 7.4-2
becomes
1 2
E 
v  g  L  h    v 
 w
2
(E-5)
where g is the acceleration of gravity, w is the mass flow rate through the pipe, and Ev is
the rate of viscous dissipation in the liquid.
From the derivation of the Bernoulli equation, one can identify the source of the
viscous dissipation term. According to Equation 7.4-1, it is given by
E v      v dV
(E-6)
V
where V is the total volume of the system, and  is the shear stress tensor defined in
Section 1.2 of TrPh. For a Newtonian fluid  is directly proportional to the velocity
gradient v and the viscosity . Therefore, it is possible to apply dimensional analysis to
Equation 6 to arrive at the conclusion that
 Ev  1 2
   v ev
 w 2
(E-7)
where ev is a dimensionless function of Reynolds number and geometrical ratios and is
called the friction loss factor.
Because Equation 6 is integrated over the volume of the system, the contributions
to Ev or ev from various regions of the system are additive. For regions where most of the
viscous dissipation arises from form friction and steep velocity gradients, such as flow
through valves or around elbows, one can identify a value of ev that is relatively constant.
Some representative values for obstacles, valves, and fittings are tabulated in Table 7.5-1
of TrPh. A more complete listing is given in Perry's Handbook (Perry and Green, 1984).
For the geometry considered in this experiment, an ev value of interest with the
exit pipes is that associated with the entrance into the pipe. According to Table 7.5-1 in
  R0  2 
  . There is no exit loss because the liquid
TrPh, this value should be 0.451  
  R 
leaves the tube in a streamline manner, without any extra turbulence generated by its
contact with air. For the orifice, the value of ev is 2.71   1   2 /  2 where  is the

ratio of the area of the orifice to that of the tank.
3

For regions of straight pipe, one can replace the friction loss factor with the
Fanning friction factor f:
ev  4 f
L
D
(E-8)
where D is the diameter of the pipe, L is its length, and f is a function of Reynolds
number, Re, as shown in Chapter 6. These relations are the basis for the Bernoulli
equation given as Equation 7.5-10 in TrPh.
For the case under consideration, the Bernoulli equation may now be written as

1 2
1 
L
v  g L  h   v 2  2 f  ev,i 
2
2  R0 i

(E-9)
where the summation of friction loss factors includes all obstacles and disturbances that
the fluid encounters. All terms in Equation 9 may be viewed as energy per unit mass of
fluid flowing. The first term on the left-hand side represents the kinetic energy carried out
of the pipe by the fluid. Rearrangement yields an explicit expression for the
instantaneous velocity in the pipe




2g  L + h

v=
1 + 2f L  e 
i v,i 

R0

1
2
(E-10)
Before Equation 10 is substituted into Equation 4, it is necessary to assess the relative
magnitudes of the terms in the denominator. In particular, if the term containing the
friction factor is significant and f varies with Re, it is necessary to deal with the velocity
dependence of that term.
First, suppose that the liquid is in laminar flow, then
f
16 
2 R 0 v
(E-11)
as given in Equation 6.2-11 of TrPh. If one assumes in addition that the kinetic energy
and other friction terms are negligible, then
v=
gR 20  L  h
(E-12)
8L
which is the result of the Hagen-Poiseuille analysis given in Chapter 2. If this expression
for v is substituted into Equation 4, one obtains the efflux time to be
te 
8LR 2  L  h 1 
ln 

gR 40  L  h 2 
(E-13)
4
At the other extreme, if the flow is sufficiently turbulent or the pipe is sufficiently
rough that f is a constant (See Figure 6.2-2 in TrPh.), then the result of the integration is
L


  e v ,i 
2  1  2f
 R
R0
i

t e  2  

2g
 R0  




1
2
 L  h  12   L  h  12  (E-14)
1
2


where any of the three resistance terms might be dominant.
Finally, in the intermediate range of turbulent flow, for 2100<Re<105, in smooth
pipes, the friction factor may be fit by the Blasius formula, which is given in Equation
6.2-12 of TrPh as
f
0.0791
Re
1
(E-15)
4
If only the term containing f in Equation 10 is significant, then the velocity is given by
 2 L  h 4 R 50 g 4  
v

4 4
 0.0791 L  
1
7
(E-16)
In that case, Equation 4 becomes
1 

7 R  0.0791 L 4 
te 
3R 20  g 2R 5 14 
0


2


4
7
 L  h  3 7   L  h  3 7 
1
2


(E-17)
It is possible, as a liquid drains from a tank and the level h drops, that the
Reynolds number within the pipe can change from one regime to another. In that case,
the integral of Equation 4 must be evaluated in several parts, using the appropriate
expression for the velocity in each region of h.
Materials and Methods
Materials
Water and 50-weight-percent aqueous sucrose solution will be used in this
experiment.
Equipment
The apparatus for this experiment is a cylindrical tank fitted with a depth gauge,
as shown in Figure 1. The tank has a 16 cm diameter, and its total depth is 25 cm.
Various interchangeable exit pipes can be screwed into the bottom of the tank. The
5
available exit pipes are listed in Table 1. Also available are a blank plug and a simple
plug containing an orifice with diameter 0.4775 cm.
Table 1. Dimensions of the Available Exit Pipes
Length, cm
Inside Diameter, mm
60
4
60
6
60
7
30
7
15
7
7,5
7
Additional equipment includes weighing containers, platform scales, a stopwatch,
a thermometer, and a graduated cylinder. In addition to water, there is available a 50weight-percent aqueous sucrose solution to be used as a working fluid. Properties of
aqueous sucrose solutions can be found in any handbooks. (Chemical Engineers’
Handbooks or the others).
Figure -1. Sketch of the draining-tank apparatus with an exit pipe in place
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Experimental Procedure
1. Calibrate the depth gauge on the tank as follows:
a. Insert the blank plug in the bottom of the tank.
b. Fill the tank by adding successively known volumes of water and noting the
change in level on the gauge as each portion is added. (This calibration could also be
done by adding known weights of water.)
2. Determine efflux times for draining the tank and instantaneous flow rates as follows:
a. Attach one of the exit pipes or the orifice to the bottom of the empty tank, and fill
the tank and pipe with water or sucrose solution. During the filling, seal off the end
of the tube with a fingertip or a stopper.
b. Starting with a full tank, allow the liquid to start flowing from the pipe.
c. Record the times at which the liquid level in the depth gauge passes
predetermined locations.
d. Measure the total time that it takes for the liquid level to travel from the top to the
bottom of the depth gauge.
3. Make the above measurements with all pipes and with the simple orifice with water.
Record the temperature of the liquid in each case.
4. Repeat the measurements for several pipes with the sucrose solution as time permits.
Start with the shorter tubes, and then test the longer ones if there is time.
5. During the laboratory session perform preliminary analysis of the data. In particular,
estimate initial and final Reynolds numbers for each run. Try to schedule experiments
so that data are taken for both laminar and turbulent flow.
6. When changing tubes, take care not to bend the tubes out of vertical alignment. Also,
the O-ring seals need not be tightened very hard.
7. Return sucrose solution to its storage reservoir after use. It contains preservative to
prevent spoilage. Do not taste.
8. Avoid entraining air bubbles in the sucrose solution by pouring it down the sides of
the tank and of the storage reservoir.
9. Wash all equipment well after use with the sucrose solution, especially the depth
gauge and the drain valve just below the gauge. Wash the tubes well before placing
them in the soaking trays after use.
10. Try to avoid vortex formation in the tank during draining. Swirling flow can be
prevented by placing a circle of polyethylene sheet or another floating material on the
surface of the liquid.
11. Check and record the dimensions of the apparatus.
7
Data Analysis
Calculate the experimental exit velocity v as a function of h for each run. Use
these values to calculate the Reynolds number at the beginning and end of each run. Use
the appropriate formula from the Theory section to calculate te, and compare these values
with the experimental values.
Plot efflux time te values from all water runs versus (L/R0) on a single graph as
individual data points. Show on the same graph, as lines, the two theoretical relationships
given by Equations 13 and 17.
In a table compare the experimental efflux times for the sucrose solution
compared with the theoretical values calculated with the appropriate model, depending on
the Reynolds number. In the table show also L/R0 and the average value of Re for each
run.
By reference to the theoretical models, investigate the expected accuracy of the te
predictions and compare the uncertainty in those values with the deviation from
experimental values. As part of this exercise calculate the relative magnitudes of the
contributions to the viscous dissipation term for the various experimental conditions
studied.
Acknowledgment: Author thanks Dr. A. Sofuoğlu for her helps in preparing this
manual.
References
Bird, R.B., W.E. Stewart, and E.N. Lightfoot, Transport Phenomena, 2nd ed. John Wiley
and Sons, New York (2002).
Perry, R.H., and D. Green (eds.), Perry's Chemical Engineers' Handbook, 6th ed.,
McGraw-Hill Book Co., New York, 1984.
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