In this communication we utilize an improved version for model proposed by Manning for the waste flow via an open channel Lyari. We have computed the status of discharges, storages and depth for all locations of the Lyari waste flow using this proposed model. We have found that the result obtained using this model is good agreement for the recent data sets.Open Channel, Proposed Model, Flow, Storage, Depth, Comparison

Asif Mansoor

Rana Khalid Naeem

English

Research Papers in Economics

Indus Journal of Management & Social sciences 
Vol. 1, No. 2: 187-194 (Fall   2007) 
 
Scaling Aspects of Lyari River Flow Routing 
Rana Khalid Naeem
* and Asif Mansoor
**
ABSTRACT 
In this communication we utilize an improved version for model proposed by Manning 
for the waste flow via an open channel Lyari.  We have computed the status of 
discharges, storages and depth for all locations of the Lyari waste flow using this 
proposed model. We have found that the result obtained using this model is good 
agreement for the recent data sets.  
 
JEL Classification: O21; O22; Q22; Q24; Q25; R14;R15 
 
Key words: Open Channel, Proposed Model, Flow, Storage, Depth, Comparison 
 
1. INTRODUCTION 
 
Chezy (1776) developed an equation, which computes mean velocity in an open channel 
flow and that equation is, 
V = C RS       ( 1 )  
where  R  is the hydraulic radius in meters and  is the channel slope. In equation (1), C 
is called Chezy’s coefficient and is dimensional, having dimensions (length)
S
1/2 per time. 
The Chezy’s coefficient is determined by experiments. Manning performed series of 
experiments and found that dependence on hydraulic radius is actually not as given in 
equation (1), and modified equation (1). The modified equation, called Manning 
equation, is 
V = S R
n
3 / 2 1       ( 2 )  
in which n is Manning’s resistance coefficients and its value dependents on the surface 
material of the Channel’s, wetted perimeter and is determined from experiments.  This 
formula is more accurate then Chezy and is widely being used now a days. The 
relationship between Chezy’s C and Manning’s n is easily shown to be 
                                                 
The material presented by the authors does not necessarily represent the viewpoint of the editors and the 
management of the Indus Institute of Higher Education (IIHE) as well as the authors’ institutes. 
 
IJMSS is published by Indus Institute of Higher Education (IIHE) Plot. # ST-2D, Block-17, 
Gulshan-e-Iqbal, Karachi, Pakistan. 
 
*  Prof. Dr. Rana Khalid Naeem is Professor of Mathematics at University of Karachi, Pakistan.  E-mail: 
rknwindsor@yahoo.com 
 
**  Asif Mansoor is Research Fellow at University of Karachi, Pakistan.E-mail: asifmansure@yahoo.com 
 
 
 Rana Khalid Naeem and Asif Mansoor 
 
C = 6 / 1 1
R
n
      ( 3 )  
After the appearance of the Manning work, many formulae are proposed for C by 
researchers; here we quote some of them like:  (Kutter 1869 in Yen 2004; Lacey 1929; 
Chow 1964; Chitale 1966; Diplas 1990; Strupczweski 1990; Sharma 1990; Choudhary 
1993; Naot 1996) 
 
Bazin (1897) considered Chezy C to be a function of R but not of S and presented the 
following formula for computation of C 
R
M
C
+
=
1
87       ( 4 )  
in which  M is the coefficient of roughness of the channel.  
 
Pavlovsky (1925) in  (Sharma 1990) modified C in Manning formula. The modified C is 
C =
y R
n
1
      ( 5 )  
where ) 10 . 0 ( 75 . 0 13 . 0 5 . 2 − − − = n R n y . It depends upon the roughness 
coefficient (n) and hydraulic radius (R). The formula is widely used in U.S.S.R. 
 
Kennedy (1895) gave a classic empirical equation correlating mean velocity V in a 
regime canal with vertical depth D, measured on the approximately horizontal silted bed. 
The mean velocity involves critical velocity Vo given by  
Vo = 0.55 D
 0.64      ( 6 )  
The hydraulic diagrams given by Kennedy are extensively used in India for designing 
irrigation canals. These diagrams enable to design any number of channel section for 
given slope with different bed width and depth. 
 
2. PROPOSED MODEL 
 
In the present work we propose the following formula for computing the mean velocity 
through an open channel. The formula is: 
model C S R
n
V 2
1
3
2 1
=      (7) 
This formula is an improvement to Manning formula and Cmodel is a dimensionless 
number defined in (13). It is worth noting that in the proposed V, the Cmodel  is non-
dimensional while in the earlier works in which Chezy C is modified, are all dimensional. 
The formula is employed in computing V through the Lyari River. The computed V is 
compared with the observed values and the values obtained through Manning formula. It 
is seen that the computed values are in good agreement with the observed values. 
 
3.  FLOW ROUTING MODEL 
 
The discharge Q through cross sectional area  A is given by (Chow 1959; Ponce 1994) 
   
 
Vol. 1, No. 2: 187-194 (Fall   2007) 
 
189 Scaling Aspects of Lyari River Flow Routing 
 
AV Q =      ( 8 )  
where V  is the mean velocity of flow.  
The Lyari River can be approximated as an open channel with trapezoidal cross section. 
The mean velocity for flow through it, employing Manning formula, is given by (Souders 
2002; Vehoeven 2003) 
[ ]
3
2
2
3
2
3
2
1 z zh W n
s W h
V
+ +
=      (9) 
where W, h and z are river width, channel flow depth and side slope respectively.  
Since the cross sectional area A is given by 
Wh A =      ( 1 0 )  
The equation (8), utilizing equation (9) and equation (10), yields 
[ ]
3
2
2
3
5
3
5
1 z zh W n
s h W
Q
+ +
= manning     ( 1 1 )  
In order to obtain dimensionless expression for C we set  
) , ( W R C C =       ( 1 2 )  
The dimensional analysis indicates that C can be expressed in terms of π -term 
‘R/W’’and, therefore, we write . 
) / ( W R C C model model =       ( 1 3 )  
Many rational approximations for Cmodel are attempted by others and it is found that ratio 
of two cubics for Cmodel produces very good results. The rational approximation for Cmodel 
is 
3 2
3 2
1
1
fx ex dx
cx bx ax
Cmodel + + +
+ + +
=      ( 1 4 )  
where and   W R x / =
 a = 1.09E-3,  b = 8.29E-3,  c = 1.08E-3,  d = 8.99E-4, e = 1.17E-3,  f = 9.15E-4 
The discharge using Cmodel is   
⎥
⎥
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎢
⎢
⎣
⎡
⎥ ⎦
⎤
⎢ ⎣
⎡
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
+ +
+ + +
+ + +
=
3
2
2 z 1 zh W n
s 3
5
h 3
5
W
3 x 000915 . 2 x 00117 . x 000899 . 1
3 x 00108 . 2 x 00829 . x 00109 . 1
Qmodel
 (15) 
The balance of the surface water store, S, within two river bridges is given by 
(Kraijenhoff and Moll 1986) 
Q I
dt
dS
− =        ( 1 6 )  
in which I is the inflow 
The surface water, S, is assumed to be a linear function of outflow discharge (Hudson and  
Hazen 1964), so 
Q   S τ =         ( 1 7 )  
   Indus Journal of Management & Social Sciences  190 Rana Khalid Naeem and Asif Mansoor 
 
in which ‘τ ’ is the travel time between two bridges under consideration. If ‘L ’ is the 
distance between two bridges, then  
V
L
= τ         ( 1 8 )  
Equation (16), utilizing equation (8) and (17), yields  
[ WhV I
LW dt
dh
− =
1 ]       ( 1 9 )  
Equation (19) describes the flow in terms of rate of change of the flow depth for a given 
river section.  
 
On using an explicit forward step finite difference approximation for equation (19), we 
obtain 
[ t t t t WhV I
LW
t
h h − ]
∆
+ = +1      ( 2 0 )  
Equation (16) is solved numerically. The discretization of equation (19) on xt-plane 
(Figure 1) leads to  
2
S S
2
Q Q
2
I I t 1 t 1 t t 1 t t −
=
+
−
+ + + +      ( 2 1 )  
in which  
It = inflow at time level 1,  It+1 = inflow at time level 2,  Qt = outflow at time level 1, 
Qt+1 = outflow at time level 2, St = storage at time level 1, St+1 = storage at time level 2 
t ∆ = time interval 
 
 
   
 
 
 
 
 
 
 
 
 
 
 
Time level 2 
Time level 1 
It                        Qt
St
It+1                      Qt+1
St+1
∆t 
t-axis
x-axis
 
Figure-1.  Discretization of Storage Equation in xt-plane   
 
 
Then Storage ( ) at any time ‘t+1’, after rearranging equation (20), is given by   1 + t S
[] [ ] t   Q Q
2
1
t I I
2
1
S S 1 t t 1 t t t 1 t ∆ + − ∆ + + = + + +    (22) 
 
 
Vol. 1, No. 2: 187-194 (Fall   2007) 
 
191 Scaling Aspects of Lyari River Flow Routing 
 
4.  COMPARISON OF COMPUTED AND OBSERVED VALUES 
 
Comparison are made between discharges (mean monthly, mean annual), mean monthly 
storage, mean monthly depth obtained from Manning and proposed Cmodel with each other 
and observation at all eleven bridges of the Lyrai River it is seen that at all locations of 
the Lyari River Cmodel produces better results than the Manning formula.  For the sake of 
convenience we can give only graphs for a few locations of the Lyari River. These are 
depicted in figures (2-5).  
 
Figure-2.  Comparison of Mean Annual Discharges 
 
6.21530
6.21535
6.21540
6.21545
6.21550
6.21555
123456789 1 0 1 1 1 2
Month
M
e
a
n
 
A
n
n
u
a
l
 
F
l
o
w
 
(
C
u
m
e
c
s
)
Observe Flow Calculated Flow Model Flow
 
 
 
 
5.56594
5.56596
5.56598
5.56600
5.56602
5.56604
5.56606
5.56608
5.56610
5.56612
5.56614
123456789 1 0 1 1 1 2
Month
M
e
a
n
 
A
n
n
u
a
l
 
F
l
o
w
 
(
C
u
m
e
c
s
)
Observe Flow Calculated Flow Model Flow
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
   Indus Journal of Management & Social Sciences  192 Rana Khalid Naeem and Asif Mansoor 
 
Figure-3.  Comparison of Mean Monthly Discharges 
 
0.9926
0.9927
0.9928
0.9929
0.9930
0.9931
0.9932
0.9933
0.9934
1 13 25 37 49 61 73 85 97 109 121 133 145 157 169 181 193 205 217
Month
M
e
a
n
 
M
o
n
t
h
l
y
 
F
l
o
w
 
(
C
u
m
e
c
s
) Observe Flow Calculated Flow Model Flow
 
5.3213
5.3214
5.3215
5.3216
5.3217
5.3218
5.3219
5.3220
5.3221
1 11 21 31 41 51 61 71 81 91 101 111 121 131 141 151 161 171 181 191 201 211 221
Month
M
e
a
n
 
M
o
n
t
h
l
y
 
F
l
o
w
 
(
C
u
m
e
c
s
)
Observe Flow Calculated Flow Model Flow
 
 
Figure-4.  Comparison of Depths at Different Locations of River Lyari 
 
1.1460
1.1480
1.1500
1.1520
1.1540
1.1560
1.1580
1 12 23 34 45 56 67 78 89 100 111 122 133 144 155 166 177 188 199 210 221
Distance (Meter)
D
e
p
t
h
 
(
M
e
t
e
r
)
Observed Depth Calculated Depth Model Depth
 
   
 
Vol. 1, No. 2: 187-194 (Fall   2007) 
 
193 Scaling Aspects of Lyari River Flow Routing 
 
0.6560
0.6580
0.6600
0.6620
0.6640
1 11 21 31 41 51 61 71 81 91 101 111 121 131 141 151 161
Distance (Meter)
D
e
p
t
h
 
(
M
e
t
e
r
)
Model Depth Calculated Depth Observed Depth
 
 
Figure 5.  Comparison of mean monthly storage between two consecutive bridges. 
8.75812
8.75814
8.75816
8.75818
8.7582
8.75822
8.75824
8.75826
8.75828
123456789 1 0 1 1 1 2
Month
M
e
a
n
 
M
o
n
t
h
l
y
 
S
t
o
r
a
g
e
Model Storage Calculated Storage Observe Storage
 
 
4. CONCLUSION 
 
In this work an expression for Cmodel  is proposed for the computation of mean flow 
velocity V through an open channel. The discharges, storages and depths are computed 
using the proposed Cmodel.  These are compared with using Manning formula and the 
observed formula, and it is seen that results obtained through proposed Cmodel are much 
closer to the observed values. This indicates that Cmodel is an improvement over the 
Manning formula. 
 
REFERENCES 
 
Bazin, H.E. 1897. Étuded’une nouvelle formule pour calculer le débit des canaux 
découverts-Mémoire No. 41. Annales des ponts et chaussées . Vol 14 No. 7  
20-70. 
   Indus Journal of Management & Social Sciences  194 Rana Khalid Naeem and Asif Mansoor 
 
Yen, Ben Chie. 2004. Channel Flow Resistance: Centennials of Manning’s formula. 
USA: Water Resources Publication, LLC, Highlands Ranch Colorado, 80163-
0026, 158. 
Chezy, A. 1776. Formule pour trouver la vitesse de l’eau conduit dan une rigole donnée. 
Dossier 847 (MS 1915) of the manuscript collection of the École Nationaldes 
Ponts et Chaussées, Paris. Reproduced in: Mouret, G. (1921). Antoine Chézy: 
histoire d’une formule d’hydraulique. Annales des Ponts et Chaussées 61,   
165-269. 
Chitale, S. V. 1966. Hydraulics of Stable Channels. Central Water and Power 
Commission, Ministry of Irrigation and Power, Government of India 
Choudhry, M. H. 1993. Open Channel Flow. New Jersey: Prentice Hall: 45-87 
Chow, V.T. 1959. Open Channel Hydraulics. New York: McGraw-Hill Book Co. 
580-680 
Chow, V.T. 1964. Hand book of applied Hydrology. New York: McGraw-Hill Book Co. 
7-24. 
Diplas, P. 1990. Characteristics of Self Formed Straight Channels. Journal of Hydraulic 
Engineering, Vol. 116. No5: 271-311. 
Hudson Jr., H. E., and R. Hazen. 1964. Droughts and Low Stream Flow. In Handbook of 
Applied Hydrology, ed. Van Te Chow. New York: MaGrow-Hill Book 
Company. 
Kennedy, R.G. 1895. The Provision of Silting in Irrigation Canals.  Proc. No: 2826, 
Proc. ICE, Vol. 119, London 
Krajenhoff, D.A. and J.R. Moll. 1986. River Flow Modeling and Forecasting, D. Reidel: 
Dordrecht. 
Lacey, G. 1929. Stable Channels in Alluvium. Proceedings of the Institute of Civil 
Engineers, London, 229, 259-292. 
Naot, D., I. Nezu and H. Nakagawa. 1996. Hydrodynamic Behaviour of Partly Vegetated 
Open Channels. Journal of Hydraulic Engineering ASCE, Vol. 122. No. 11: 
625-633 
Ponce, Victor Miguel. 1994. Engineering Hydrology Principles and Practices. Prentice 
Hall: Englewood Cliff, N.J. 07632: 233-315 
Sharma, R.K. and T.K Sharma. 1990. Irrigation and Drainage. New Dehli: Khanna 
Publisher: 157-186 
Souders, D.T and C.W. Hirt. 2002. Modeling Roughness Effect in Open Channel Flows. 
Flow Science, Inc. FSI-02-TN60. 
Strupczweski, W. and J.J Napioprkwoski. 1990. Linear Flood Routing Model for Rapid 
Flow. Hydrological Sciences Journal, Vol.35 No. ½: 149-164 
Vehoeven, R., R. Banasiak and T.Okrusko. 2003. Hydraulic Modeling of River Flow. 
Proc.of the Int Wethydrol Workshop, Goniadz, Poland 
 
   
 
Vol. 1, No. 2: 187-194 (Fall   2007) 
 
195 

Characteristics of Self Formed Straight Channels.

Droughts and Low Stream Flow.

Étuded’une nouvelle formule pour calculer le débit des canaux découverts-Mémoire

Formule pour trouver la vitesse de l’eau conduit dan une rigole donnée.

Hydraulics of Stable Channels. Central Water and Power Commission, Ministry of Irrigation and Power,

Hydrodynamic Behaviour of Partly Vegetated Open Channels.

Linear Flood Routing Model for Rapid Flow. Hydrological Sciences Journal,

of the manuscript collection of the École Nationaldes Ponts et Chaussées,

Rana Khalid Naeem and Asif Mansoor Yen, Ben Chie.

Stable Channels in Alluvium.

The Provision of Silting in Irrigation Canals.

Scaling Aspects of Lyari River Flow Routing

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Scaling Aspects of Lyari River Flow Routing

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