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H5) Empirical study on terminal water velocity of drainage stack
(1)C.L. Cheng, Dr. (2) K.C. He, Mr. (3)C.L.Lin ,Ms.
[email protected] [email protected]
[email protected] National Taiwan University of Science
and Technology, Department of Architecture, 43 Keelung Road Sec.4,
Taipei, Taiwan, R.O.C. Abstract
An actual discharge of vertical drainage stack has a complex
phenomenon and may consist of triple phase flow feature with
incorporated solid, liquid and air. The guideline of National
Plumbing Code (NPC) of US was used to set the permit flow rate as
the regulation of building drainage system. Following initial work
of the HASS 203 of Japan in 1970s, the method of steady flow
condition was merged as the provision reference and evaluation
technique. According to the importance of permit flow rate
regulation, the terminal velocity in drainage stack was also seen
as one of the crucial issues in building drainage studies. Couple
theories and predictions were reported in previous researches. This
paper would also introduce a prediction method with empirical
approach by theoretical study from air pressure distribution
research. A new technology with digital high speed camera was used
to validate the prediction of terminal velocity in drainage stack
in this paper. The theoretical study reveals the practical sense
and the validation also approximately responses to the prediction
results. Keywords terminal velocity, the fluid phenomena of the
water, air-pressure, drainage stack
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List of symbols symbols content unit
Qw Water flow rate l/s Qa
Air flow rate in stack vent m3/s
R
Diameter of stack m
wV
Velocity of water flowing
m/s
aV Velocity of airflow
m/s
tV Terminal velocity of water
m/s
ww V The water resistance in the stack.
2a a
V The air action force to falling water
g Gravity acceleration m/s2
t
time interval
sec Distance m
SD The accumulation distance of falling water m 1.
Introduction
Appliance discharges to a vertical stack of drain may be
described as an unsteady or time dependent flow, and the form of
the appliance discharge flow contributes to this flow condition. An
actual discharge of vertical drainage stack has a complex
phenomenon and may consist of triple phase flow feature with
incorporated solid, liquid and air. Airflow in the drainage stack
is promoted by through-flow mixing as well as the interaction of
friction with the falling water and air. This mechanism causes the
negative pressure on the upper floors and the positive pressure on
the lower floors in the building vertical drainage system. Hunter1)
explored the flow phenomenon of drainage stack in1940s. Afterward,
Wyly2)3) & Dawson first issued the theory of the terminal
velocity at 1960s. The guideline of National Plumbing Code (NPC) of
US was used to set the permit flow rate as the regulation of
drainage system. Following initial work of the HASS 203 of Japan in
1970s, the method of steady flow condition was merged as the
provision reference and evaluation technique; hence it conducted
series researches of steady flow method with reference to building
drainage network. According to the importance of permit flow rate
regulation, the terminal velocity in drainage stack was also seen
as one of the crucial issues in these series researches. Couple
theories4)5) and predictions were reported in previous researches.
However, the validation and accuracy were still criticized so far.
This paper would also introduce a prediction method by theoretical
study from air pressure distribution research. Meanwhile, a new
technology with digital high speed camera was used to validate the
prediction of terminal velocity in drainage stack in this
paper.
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2. Technical Reviews The theory of the annulated flowing in
drainage stack was first issued by Wyly2) in 1960s. Afterward, some
researches tried to figure out the velocity of flowing water in the
stack by the experimental method and theory, however, no results
were reported in that period. In 1980s, Tukagoshi6) conducted
electricity to the salt solution in Japan, and put the sensor of
the electricity into the pipe which perpendicular to the pipes
section and divided into 1-25 points as observational points, when
salt solution flowing into the vertical stack and pass through the
sensor would evaluated the velocity and quantity of the water
flowing. In 1994, Sakaue7) in Japan continuously infused water into
vertical stack for testing the velocity of the water flowing, and
to return to original equation for evaluated the water flowing rate
in the vertical stack. However, all these researches have not
reached a clarified and validated conclusion on the terminal
velocity on drainage stack. According to the previous
researches8)9)10) on air pressure distribution, the airflow rate
(Qa) was identified as a critical parameter for a prediction model
which can express the mechanism of vertical drainage flow.
Therefore, the airflow performance in vertical drainage stack is
the dominated issue and it needs to be solved. Hence while air flow
rate is dominant in the vertical drainage stack, it plays a
critical role in the subsequent operation of vertical drainage
stack where the mechanism may be assumed to be a quasi-fan machine,
thus the laws of fan can be introduced to link with the vertical
drainage flow. The laws of fan can be expressed by the hydraulic
parameters such as air density, pressure, velocity, gravity,
resistance coefficient, lift, and et al. Practically, the operation
energy for airflow within fan is mainly from electric power, thus
potential energy of height is the dominating power for conducting
the airflow in vertical drainage stack. This antithesis mechanism
can be expressed as quasi-fan theory, namely the initial model of
vertical drainage flow was conducted from the laws of fan machine
alike. The mechanism of flow within vertical drainage is now
schematically understood. Air pressure in vertical drainage stack
is caused by series interactions between downstream water and
through-flow air in vertical pipe. Fig.1 illustrates the image of
flow state and the modified interaction, thus it conducts the main
parameters with air pressure, airflow rate, and resistance
coefficients, and they are the essential factors for prediction
model of air pressure distribution in vertical drainage stack. A
prediction model about the air pressure distribution, which
occurred in the drainage stack by high-rise experiment tower (108m)
and middle-high experiment tower (30m), was developed in Japan from
1990, then considerable progress has been made in predicting the
air pressure distribution within vertical drainage stack.
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A ZORE
B ZONE
C ZONE
D ZONE
QaAir flow
Frictional
Qa
Qa
Water inlet from lateral drain
Interaction
Qa
Qa Water mixes with air
Interaction
Gravity
Qa
Main horizontal drain Hydraulic jump
G
FL
Negative Pressure Positive Pressure
Peak negative pressure
Positive Pressure
Negative Pressure
G
Air pressure
Discharge height
Figure 1 Mechanism of vertical drainage feature and inverted
model
According to the mechanism and feature of vertical drainage flow
from the theoretical reviews, the profile of drainage stack was
divided into four zones, and each zone is individually modeled due
to the corresponding characteristics. Meanwhile, the air pressure
distribution, which reveals the time average air pressure data with
steady flow condition, does not involve the instantaneous air
pressure fluctuation in vertical drainage flow 3. Theory and
empirical observation The phenomenon of drainage vertical stack can
be divided by four zones (A, B, C, D) to express their individual
characters which were mentioned in previous researches. According
to the feature character observation, B zone is the most complex
area which is the acceleration area in both water flowing and
airflow in the stack. Fig. 2 shows the image of complex phenomenon
that water flows into the vertical stack from the branch pipe and
the interaction of water and air.
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Figure 2 The simulate diagram of the water flowing into the
stage The velocity of the flowing water in the stack is mainly
dominated by the three interaction balance including gravity force
(g) and friction drag of the pipe inside and the air interaction
toward the falling water. As the falling water in the stack becomes
the terminal velocity, that means the interaction inside the stack
reaches the condition of balance and the forces actions are totally
equal and neutralized. Therefore, the balance condition can be
expressed as the following equation (1) which means the forces of
friction drag of the pipe inside and the air interaction toward the
falling water are equal to the force of gravity.
2a
g aww VV += ..(1)
The velocity of the water flowing vary as speed increasing or
acceleration when water flowing into the vertical stack from the
sideling stack. Thus, the velocity can be expressed as equation (2)
(3).
2Vaaww
w VVgdt
d+= ..(2)
=t
o
Vdt
dV ww ..(3)
According to the equation (2)(3), the velocity of the water
Velocity in stack flowing between 0 sec and t sec can be gained
from integration function of equation (4).
( ) ( )t
tVtVw
aw +
+=
1g 2a ..(4)
g
2a aV
wQ
ww V
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Therefore, the water falling distance from branch pipe which
accumulated between 0.001 sec and t sec can also be added up by the
equation (5).
(t)tt
0.001t=
= wVSD ..(5)
Because there are 1000 specific gravity difference between water
and airthe action of the air toward water could be ignore
temporary. Thus, the velocity of falling water can be expressed as
equation (6).
ttV
ww +
=
1g
..(6)
According to the equation (6), the gravity and water resistance
would be constant theoretically. The falling water will soon reach
a constant velocity as time passes. Therefore, as the time is
setting as infinite number the approaching constant water velocity
can be seen as the terminal velocity. The function can be expressed
as the following equation (7).
wtVt
g= ..(7)
The continuous flowing phenomenon of B zone is the most complex
area in the drainage stack. When the water flows into the stack,
the initial falling velocity is zero theoretically and the velocity
is accumulated and increasing. Meanwhile, the constant air flow
rate would be instantly accumulated by the flowing section
extremely shrinking. Thus, the air velocity would be speedy than
water velocity at this zone. According to the interaction between
air and water, the falling water velocity is increasing and air
velocity is decreasing. The physical phenomenon in this area causes
the increasing of negative pressure. When the velocity of water and
air reach to the equal point, the increasing negative pressure
tendency will stop and change to decline. This critical point also
expresses the maximum negative pressure point. When the water flow
rate and air flow rate is constant, then the air velocity in the
stack can be calculated by the equation (8).
ww
w
w
w
a
w
aa QVA
V
VQA
QAA
QV
=
=
=
Qa ..(8)
According to the equation (8), the velocity equal point of water
and air happens in the maximum negative pressure area. Then, the
equation (8) can be substituted by the following equation (9) at
this point.
AQQ
V waa+
= ..(9)
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Fig. 3 shows the calculation model of air and water velocity in
stack and the accumulated distance of the falling water. According
to this calculation model, the maximum negative pressure point can
be predicted and the terminal velocity of falling water in stack
can also be expressed.
Figure 3 The diagram of the air in the pipe and the distance of
water flowing. (Limit: the initial velocity of water
flowing=0.001m/s =2.0l/s =40.0l/s )
Consequently, the negative pressure occurring point depends on
the air flow rate in stack. Fig. 4 and 5 shows the validation of
measured data and the calculation results by the above model.
6
8
6
4
432
2
10 5
10
7
10
20
30
40
The distance of Peak negative pressure occurred
Time (sec)
Flowing distance (m)
=
The cumulate distance of water flowing
Velocity of airflow (Va)
Velocity of water flowing (Vw)
Velocity (m/s)
The time of Peak negative pressure occurred
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Figure 4 The relation between the distance of peak negative
pressure occurred and the quantity of the air flow. ( 2.0(l/s))
Figure 5 The relation between the distance of Peak negative
pressure occurred and the quantity of the air flow. ( 1.0, 2.0,
3.0, 4.0 (l/s))
Distance (m)
Airflow (l/s)
Possible range
Measured value
Distance
(m)
Calculatedvalue
Distance to central
Airflow (l/s)
Distance
(m)
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The distance of maximum negative pressure from charge floor was
surveyed in previous researches which were measured in a 100 m
experiment tower in Japan. The validation is proved to be
approximately matched the calculation model. Accordingly, the water
resistance coefficient can be calculated by this validation
process. When the distance of maximum negative pressure is catch,
the time is the crucial parameter for the calculation of water
resistance coefficient by equation (5). Then the air resistance
coefficient also can be calculated by equation (1). The results of
the calculation are shown in Table 1. Table 1 The resistance
coefficient of water and air in stack
Water flow rate Qw w
a
1.0 l/s 3.10 0.03454
2.0 l/s 2.20 0.01727
3.0 l/s 1.85 0.01232
4.0 l/s 1.50 0.00746
Owing to the limitation of experiment device, the measurement
points are set in interval of 3 meter, thus the data can not be
precisely fit to the verification result. However, the results
approximately response to the theory and satisfy the validation.
According to the water resistance coefficient, the terminal
velocity can be predicted by equation (6) and the results are shown
in Table 2. Table 2 The terminal velocity of water flowing in stack
Water flow rate Qw
Previous researches
Wyly type Dowson typeThis study (Cheng)
Previous study (Kurabuch)
1.0 l/s 1.60 2.06 3.161 5.48
2.0 l/s 2.11 2.06 5.353 7.90
3.0 l/s 2.48 3.20 5.297 8.79
4.0 l/s 2.79 3.60 6.533 9.30
Regarding the researches of terminal velocity, Dawson (US) used
the manning equation to calculate the terminal velocity by the
following equation (10).
4.0
218.5
=
RQ
V wt ..(10)
Meanwhile, Wyly (US) reported the falling water in stack might
be annual membrane flow. Thus he amended the coefficient and
proposed the calculation equation for the terminal velocity as the
following equation (11).
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4.0
202.4
=
RQV wt ..(11)
Asano (Japan) followed the theory of annual membrane flow and
proposed more precise prediction model. However, the results were
similar to Wylys calculation. In 1990s, Kurabuchi (Japan) used the
theory of air pressure gradient, proposed a new approach to predict
the terminal velocity. However, the prediction results were almost
twice or triple of the previous researches as shown in Table 3.
Meanwhile, the results were not yet be verified by experiment.
Afterward, the researches of terminal velocity stopped and no
reports were presented so far. 4. Validation device and the process
The theory of this paper was developed almost the same age as
Kurabuch in Japan. Following the development of observation
technology, this research tries to use the digital high speed
camera to validate the prediction model for terminal velocity.
Table 2 is the specification of this digital high speed camera
which is used to observe the falling water velocity in stack. Fig.
6 is the picture of experiment in observation place which shows the
circumstance and condition of the operation.
Table 3 Specification of the digital high speed camera
Video camera
Auto Exposure ControlColor or monochrome 2,100 pictures per
second full resolution Software: Acquisition, Analytical playback,
Measurements,
Image processing and File management 256mg RAMfor files
memory
Diaphragm
Adjustable diaphragm. Resolution of the screen
MEMORY
ELECTRICAL
COMPUTER LINKS
VIDEO CAMERA DIAPHRAGM
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Fig.6 the experimental device The pictures of the transparent
pipe at each floor of the experimental tower were taken by digital
high speed camera. Water discharges are all from 12th floor with
the water flow rate of 1.0 l/s, 2.0 l/s, 3.0 l/s, 4.0 l/s,
meanwhile each floor divides into 3 layers so that each floor can
be taken video with three times. This research totally got 132
video data from the observation of 33 layers. The experimental
device includes digital high speed camera, two lamps, notebook for
recording the data and high resolution image screen. Fig. 7 and 8
show the devices of this observation. And the observation results
are shown in Fig. 9~12.
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Fig.7 The plan of experimental device
Fig.8 The elevation of experimental device
9F
lamp
lamp
high speed camera
1
2
3
building
lamp
high speed camera
lamp
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Fig. 9 Water velocity and air pressure in
stack (4.0 l/s)
Fig. 10 Water velocity and air pressure in
stack (3.0 l/s)
Fig. 10 Water velocity and air pressure in stack (2.0 l/s)
Fig. 10 Water velocity and air pressure in stack
(1.0 l/s)
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5. Conclusion According to the importance of permit flow rate
regulation, the terminal velocity in drainage stack was also seen
as one of the crucial issues in building drainage researches. This
paper introduces a prediction method with empirical approach for
terminal velocity by theoretical study from air pressure
distribution research. A new technology with digital high speed
camera was used to validate the prediction of terminal velocity in
drainage stack in this report. The theoretical study reveals the
practical sense and the validation also approximately responses to
the prediction results. However, the observation technology still
remains some difficulties and the more precise verifications are
needed to be conducted in the near future. 6. Reference
1) Roy.B.HunterBMS 79 Water Distributing System for Building,
1941
2) R.S.Wyly and H.N.Eaton;Capacities of Plumbing Stack in
Building, BMS Repoet,1321952
3) R.S. Wyly and H.N. Eaton : Capacities of Stacks in Sanitary
Drainage System for Building, N.B.S. Monograph 31, 1961
4) B.J.PinkA Study of Water Flow in Vertical Drainage Stacks by
Means of a Probe,CIB-W62 Semminar,1973
5) Yoshiharu AsanoThe basic research of the specific of the
velocity in vertical stack---terminal velocity, the report of the
architectural institute of Japan, 278(1979)
6) TukagoshiThe experimental research method of the specific of
the vertical stackTransactions of the Society of Heating,
Air-Conditioning and Sanitary Engineers of Japan (1977)
7) Sakaue: The analysis of the variation of the pressure in
vertical stack, Transactions of the Society of Heating,
Air-Conditioning and Sanitary Engineers of Japan (1979)
8) Cheng, C. L., Kamata, M., Kurabuchi, T., Sakaue, K., Tanaka,
T., A Prediction Method of Air Pressure Distribution of Drainage
Stack System in Case of Single-Point Steady Discharge, Journal of
Archit. Plann. Environ. Eng., No.481, pp83-91. (1996).
9) Cheng, C. L., Kamata, M., Kurabuchi, T., Sakaue, K., Study on
Pressure Distribution of Drainage Stack System in High-Rise
Apartment Houses, Journal of Graduate School and Faculty of
Engineering the University of Tokyo (B), Vol. XLIII, No.4,
pp467-489. (1996, EI)
10) C.L. Cheng, Lu, W. H., Shen, M.D., An Empirical Approach:
Prediction Method of Air Pressure Distribution on Building Vertical
Drainage Stack, Journal of the Chinese Institute of Engineers, Vol
28.(2004)
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7. Presentation of Author
Cheng-Li Cheng is the Professor at National Taiwan University of
Science and Technology, Department of Architecture. He is a
researcher and published widely on a range of water supply and
drainage in building. He has published extensively on a range of
sustainable issues, including the water and energy conservation for
green building. Currently he also acts as referee of Taiwan Green
Building Evaluation Committee and Nation Building Code Review
Committee. Kuen-Chi He is the Ph.D student at National Taiwan
University of Science and Technology, Department of
Architecture.
Chia-Li Lin is the master student at National Taiwan University
of Science and Technology, Department of Architecture.
