Drawing the water surfaces in open-channel for gradually varied flow is relatively complicated and difficult. In order to identify which type of the water surfaces among 12 water-surface styles we have to base on critical depth (yc) and normal depth (yo). In this case, to calculate the critical depth (yc) that particularly need to use the Semi empirical equations. This article generally the way to compute the critical depth; the way to compute flow in circular sewers; analyze the application of existing formulas and then offering a new equation to compute the critical depth. This new equation will help to have more accurate result. Also it is more comfortable to non-uniform flow in the circular section/ circular sewers.

1. http://www.iaeme.com/IJCIET/index.asp 438 editor@iaeme.com
International Journal of Civil Engineering and Technology (IJCIET)
Volume 10, Issue 05, May 2019, pp. 438-445, Article ID: IJCIET_10_05_047
Available online at http://www.iaeme.com/ijmet/issues.asp?JType=IJCIET&VType=10&IType=5
ISSN Print: 0976-6308 and ISSN Online: 0976-6316
© IAEME Publication
CREATING A NEW CRITICAL DEPTH
EQUATION FOR GRADUALLY VARIED FLOW
IN CIRCULAR SECTION
Le Van Nghi
Key Laboratory of River and Coastal Engineering
Vietnam Academy for Water Resources
Nguyen Minh Ngoc;
Faculty of Urban Environmental and Infrastructural Engineering
Hanoi Architectural University
ABSTRACT
Drawing the water surfaces in open-channel for gradually varied flow is relatively
complicated and difficult. In order to identify which type of the water surfaces among
12 water-surface styles we have to base on critical depth (yc) and normal depth (yo). In
this case, to calculate the critical depth (yc) that particularly need to use the Semi
empirical equations.
This article generally the way to compute the critical depth; the way to compute
flow in circular sewers; analyze the application of existing formulas and then offering
a new equation to compute the critical depth. This new equation will help to have
more accurate result. Also it is more comfortable to non-uniform flow in the circular
section/ circular sewers.
Key words: Critical depth, Non-uniform flow, circular pipes, critical flow, specific
energy.
Cite this Article: Le Van Nghi and Nguyen Minh Ngoc, Creating a New Critical
Depth Equation for Gradually Varied Flow in Circular Section, International Journal
of Civil Engineering and Technology 10(5), 2019, pp. 438-445.
http://www.iaeme.com/IJCIET/issues.asp?JType=IJCIET&VType=10&IType=5
1. INTRODUCTION
Research on water surfaces play an important role in open channel design [6]. Usually using
of the semi empirical equations to compute the critical depths in different types of channel
sections are studied and applied for open-channel design [5]. Until now, researches usually
focus on the critical depths for rectangular, trapezium and parabolic sections [1,2,3,4], there
were only few studies about critical depth of circular section whereas sewage systems are
mainly designed for circular sewer.

2. Le Van Nghi and Nguyen Minh Ngoc
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When calculating the critical depth (yc), only the rectangle channel is constructed by
theoretical analysis [6-9], while other types of channels use empirical equations or trial and
error method.
When calculating by using trial and error method, the calculator will need a lot of time, or
he has to use a sophisticated search algorithm, thus, semi-empirical formulas have been
introduced to make calculations faster but still ensure that the results are within tolerable
limits.
In application of a sewer calculation, it is indeed necessary to calculate the critical depth
for circular section, with existing empirical equations, after the calculation, there is a large
and unstable error (from 0,06% to 7,5% - Table 1).
This research is focused on analysing of the critical flow in circular sewers and proposed
quick calculation formula for the critical depth of circular section.
2. THEORY
2.1. Baseline analysis to determine the critical depth
The flow of critical state, when Froude number is equal one. It means, the specific energy has
the smallest value in a section for a given discharge [6]
+ Specific energy
2
2
Q
e y
2gA

  (2.1)
+ Take the derivative with depth:
2
2
de 1 Q
d y
dy dy 2gA
 
  
 
2 2
3
Q dA V dA
1 1
gA dy gA dy
 
   
+ If y  0 so
de
dy
  : Thus, e invert with the depth y, the horizontal axis (y = 0) makes
the horizontal asymptoic.
+ If y  + so
de
1
dy
 : Thus, e Increasing with the depth y, the Bisector makes the tangent
line.
So function e = f(y) will be minimum, along with its depth (yc). This depth (yc) has called
critical depth.
Show:
2
min c 2
c
Q
e y
2gA

  (2.2)
From (2.2), yc only dependents on Q, A. When it Q up, yc increase and vice versa [10].
e0 emin
yc
y
e=f(y)
Fig 1: Specific energy graph

3. Creating a New Critical Depth Equation for Gradually Varied Flow in Circular Section
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2.2. Different methods for computing the Critical depth
a. Trial and error method
+ assuming a value of y, compute e from (2.2).
+ From (2.2), when two sides of the equation is equal, so y = yc.
b. Graphical method [10, 6]
+ From (2.2), we make a table different values of e and h
+ Making a curve: e = f(y)
+ From this curve, at position e = emin, y = yc.
c. Algebraic method [6]
+ when y = yc so e = emin, it be extremum of a functional at critical depth
de
0
dy

2
3
c
Q dA
0 1
gA dy

  
So: c
dA
T
dy

2
c3
c
Q
0 1 T
gA

  
32
c
c
AQ
g T

  (2.3)
In wich: Ac : area with critical depth (m2
)
Tc: Top width in critical depth (m)
Q: Discharge (m3
/s)
Compute yc for all typle of channel
+ compute :
2
Q
g

+ assuming y, compute A, T, after computing
3
A
T
+ From (2.3) comparing this value, if (2.3) is correct, when y = yc.
For quick calculations and reuse, we can make a table or drawing curve, at this value point
3
A
T
=
2
Q
g

, we find yc.
d
y
T

Fig 2. Circular section

4. Le Van Nghi and Nguyen Minh Ngoc
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3. METHOD FOR CRITICAL DEPTH OF CIRCULAR SECTION
3.1. Basic equation [6]
Solution:
+ computing:
y
a
d
 (2.3)
+ computing angle  (rad): cos(/2) = 1 – 2a (2.4)
+ computing
2
d
A ( sin )
8
    (2.5)
+ computing T: T 2 h(h d) d.sin( )
2

   (2.6)
From (2.3), leads to
32
2
d
( sin )
8Q
g d.sin
2
 
   
   

(2.7)
3.2. Trial and error method
From (2.7) trying up k following equation:
 
1
5 32
c 2 2
c c
2
16Q sin d sin
g 2
  
        
  
(2.8)
derive a formula: c
c
d
y 1 cos
2 2
  
    
  
(2.9)
3.3. Graphical method [10]
Making function relation a ~
32
c
d
( sin )
8
y ( )
d.sin
2
 
   
  

Then, computing
2
c
Q
y ( )
g

  , after find a from table [apendix - 10], since (2.3), we find yc.
Or we can use Fig 4.1 and Fig 4.2 of book “open-channel Hydraulic” – Van te chow [6], for
calculating critical depth (yc).
3.4. Semi empirical equations method
Equation 1 [5]:
0,252
c 0,26
1,01 Q
y
d g
  
   
  
(2.10)
Equation 2 [5]:
0,52
c 0,3
Q
y 0,53
d
 (2.11)

5. Creating a New Critical Depth Equation for Gradually Varied Flow in Circular Section
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Equation 3 [11]:
0,0853 15
c 6
g D
y D 0,77 1
Q

 
  
 
(2.12)
Appraisal: Using Trial and error method from (2.8 và 2.9), computing yc leads to the most
accurate value, but it's time consuming and complicated if programming. Using table, Graph
or empirical equations still have deviation and it requires available of provided table that
make complexity of the calculation. The semi empirical equations still produce certain
deviation. Therefore, the author focus on analyzing correlation to identify the semi empirical
equation that provide the most accurate result.
3.5. Research method and proposed formula
From (2.8)
 
32
5
3
( sin )Q
gd 8 sin
2
  
 

Let:
 
2
n 5
3
m
3
Q
h (2.13)
gd
( sin )
h (2.14)
8 sin
2


  


Relationship between ~ hm and proposed formula
Scope of the study: a = 0.5  0.9 (This can match the fact of design sewer) [12,13,14]
In circular section, with value a, we can compute , hm and make table 1.
Table 1. Calculated parameters
a  hm
Ln(100hm)
0.5 3.14 0.060 1.798
0.6 3.545 0.122 2.499
0.7 3.966 0.221 3.097
0.8 4.43 0.383 3.644
0.9 4.998 0.689 4.233
Because hm is very small, so Ln(hm) < 0 and we multiply hm by 100, Ln(hm) > 0, and then,
making relation curve ~ln(100hm)

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Figure 3 Relationship between  ~ hm
After, correlation analysis, finding function with multi-parameters
Suggested equation for calculating angle (k) at critical depth (applied conditions a = 0.5 
0.9)
n0.191.ln(100h ) 0.191
c n2.209e 2.209(100h )   (rad) (2.15)
Calculation method
+ Computing hn by equation (2.13), from (2.15) to find c
+ Computing yc by equation (2.9)
Table 2. Applying empirical equations to compute yc (m) with different Q, d
Discharge Diameter Computing critical depth Recommended
Q d yc (m)
angle 
(rad)
yc (m)
(m3
/s) (m)
Eq 2.8
and 2.9
Eq 2.10 Eq 2.11 Eq 2.12 Eq 2.15 Eq 2.9
0.3 0.5 0.375 0.374 0.349 0.373 4.212 0.378
0.3 0.6 0.357 0.357 0.330 0.356 3.539 0.359
0.4 0.5 0.427 0.432 0.405 0.427 4.702 0.426
0.4 0.6 0.414 0.412 0.384 0.412 3.950 0.418
0.4 0.7 0.396 0.396 0.366 0.395 3.410 0.397
0.5 0.6 0.463 0.461 0.431 0.460 4.302 0.464
0.5 0.7 0.444 0.443 0.411 0.442 3.713 0.449
0.5 0.8 0.427 0.428 0.395 0.426 3.268 0.425
0.6 0.6 0.503 0.505 0.474 0.501 4.612 0.501
0.6 0.7 0.488 0.485 0.452 0.485 3.981 0.493
0.6 0.8 0.469 0.468 0.434 0.468 3.504 0.472
0.6 0.9 0.453 0.454 0.419 0.453 3.131 0.448
0.6 1.0 0.438 0.442 0.406 0.440 2.832 0.423
0.8 0.7 0.562 0.560 0.525 0.559 4.443 0.562
0.8 0.8 0.545 0.541 0.505 0.542 3.911 0.550
0.8 0.9 0.526 0.525 0.487 0.525 3.495 0.529
0.8 1.0 0.509 0.510 0.472 0.510 3.161 0.505
1.0 0.8 0.609 0.605 0.567 0.605 4.259 0.612
y = 2.209e0.191x
R² = 0.9986
2.5
3
3.5
4
4.5
5
5.5
1 2 2 3 3 4 4 5

ln(100hm)
Compution

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1.0 0.9 0.591 0.587 0.547 0.588 3.806 0.597
1.0 1.0 0.573 0.571 0.530 0.571 3.442 0.575
1.1 0.8 0.637 0.634 0.595 0.634 4.417 0.638
1.1 1.0 0.602 0.599 0.557 0.600 3.569 0.606
1.2 0.8 0.663 0.663 0.623 0.660 4.566 0.661
1.2 1.0 0.630 0.625 0.583 0.627 3.690 0.635
Figure 4. Graph yc of all equations from table 2
4. CONCLUSIONS AND SUGGESTIONS
When calculating, drawing the water surface for unsteady flow in the sewerage, the Trial and
error method produces the most accurate results, but If we use the experimental formulas [5,
11] then the error of the calculation is unstable, the error reaches up to 7.5%. Meanwhile,
based on analyzes of theoretical critical flow in circular section, suggested the formula for
calculating the critical depth (yc) by (2.15) and apply the computational equation (2.15) for
different flow cases, comparing it with the Trial and error method results, this error is the
smallest (from 0.002% to 0.579%).
So the author propose the formula (2.15) to make it possible for quickly calculate the
critical depth, which in turn serves to calculate and draw water surface in circular sewers
more comfortable and more quickly.
NOTATION
The units shown below are SI (international system of units).
A: Flow area of water, m2
.
d : Inside diameter of circular section, m.
E : Specific energy, m.
g : Acceleration of gravity = 9.8066 m/s2
.
Q : discharge, m3
/s.
V : Velocity, m/s.
Y : Water depth, m.
Yc : Critical depth, m.
θ : Angle of water in cross-section, radians.
0.300
0.350
0.400
0.450
0.500
0.550
0.600
0.650
0.700
0 5 10 15 20 25 30
yc (m)
location
Eq 2.9
Eq 2.10

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θc : Angle at critical depth, radians.
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