ub

v

d

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tand computer programming. 2010 Elsevier Ltd. All rights reserved.

1. Introduction

The critical and normal depths play a major role in the design,operation, and maintenance of open channels [1]. The criticaldepth is a quantity of fundamental importance to understandingthe flow characteristics in open channels. If the actual depth isless than the critical depth, the flow is considered supercritical,which is fast-flow and is impacted by the upstream conditions.If the actual depth is greater than the critical depth, the flowis considered subcritical, which is slow-flow and is impacted bythe downstream conditions [2,3]. The occurrence of the criticaldepth at the upstream end of a reach under supercritical flowconditions or at the downstream end of a reach under subcriticalflow conditions provides a control section for gradually varied flowcomputations [4].

The normal depth occurs in a steady uniform flow for a givenchannel geometry, slope, and roughness, and a specified value ofthe discharge. The normal depth is an important parameter for thehydraulic design of open channels and has ramifications for floodprediction. It is also important to control and make efficient use ofsuch channels.

For circular, trapezoidal, and horseshoe channels, the governingequations for the critical and normal depths are implicit andno analytical solutions exist. For these channels the criticaland normal depths are presently obtained by trial procedures,

Corresponding author.E-mail addresses: arvatan@ut.ac.ir, alireza_vatankhah@yahoo.com

(A.R. Vatankhah), seasa@ryerson.ca (S.M. Easa).

numerical and graphical methods, or explicit regression-basedequations. Specifically for the critical depth, explicit equations areavailable for only trapezoidal [47] and circular channels [4,8,9],but not for horseshoe channels. For the normal depth, explicitequations are available for only rectangular [1013] and horseshoechannels [14,15], but not for trapezoidal or circular channels.

In this paper, explicit solutions for the critical and normaldepths are presented for trapezoidal, circular, and horseshoechannels using the curve fitting method. The proposed equationseither fill the current gap or improve upon existing equations interms of accuracy and simplicity. A comparison of the accuracyof the proposed and existing solutions is also presented. Beforepresenting the derivation of these explicit equations, it is necessaryto describe the geometric properties of the channel cross sections.

2. Geometric properties

2.1. Trapezoidal channels

Considering Fig. 1 for a trapezoidal channel section, theapplicable equations are as follows

A = (b+ zy)y (1)P = b+ 2y

1+ z2 (2)

T = b+ 2zy (3)whereA is cross section area, b is bedwidth, z is horizontal distancecorresponding to 1 m vertical distance (side slope of the channel),y is flow depth, P is wetted perimeter, and T is width of the channelat the water surface.Flow Measurement and Instr

Contents lists availa

Flow Measurement a

journal homepage: www.else

Explicit solutions for critical and normalAli R. Vatankhah a,, Said M. Easa ba Department of Irrigation and Reclamation Engineering, University College of Agricultureb Department of Civil Engineering, Ryerson University, Toronto, ON, Canada M5B 2K3

a r t i c l e i n f o

Article history:Received 19 August 2010Received in revised form1 November 2010Accepted 12 December 2010

Keywords:Critical and normal depthsOpen channels flowExplicit equations

a b s t r a c t

Critical and normal depths aroperation, and maintenancecomputation can only be derobtain such equations for thtrapezoidal, circular, and horthus the use of trial procedurexplicit solutions for the criimproved explicit regressiondifferent shapes. A comparisequations are simple, have a0955-5986/$ see front matter 2010 Elsevier Ltd. All rights reserved.doi:10.1016/j.flowmeasinst.2010.12.003mentation 22 (2011) 4349

le at ScienceDirect

nd Instrumentation

ier.com/locate/flowmeasinst

epths in channels with different shapes

and Natural Resources, University of Tehran, P.O. Box 4111, Karaj, 31587-77871, Iran

e important for computing gradually varied flow profiles and for the design,of open channels. A closed-form analytical equation for the normal depthived for triangular channels. For exponential channels, it is also possible tocritical depth. This is not possible, however, for other geometries, such as

seshoe channels. In these channels, the governing equations are implicit andes, numerical methods, and graphical tools is common. Some channels haveical and normal depths, while others do not. This paper presents new and-based equations for the critical and normal depths of open channels withon of the proposed and existing equations is also presented. The proposedmaximum error of less than 1%, and are well-suited for manual calculations

44 A.R. Vatankhah, S.M. Easa / Flow Measurem

Notations

A Cross section areaa, b, and c Coefficientsb Bed widthD Channel diameterg Gravitational accelerationH Height of the conduitn Mannings roughness coefficientP Wetted perimeterQ DischargeR, r Circle radiussin() Longitudinal slope of the channelS0 Longitudinal slope of the channelt = R/r A characteristic parameterT Width of the channel at the water surfacey Flow depthyn Normal flow depthz Side slope of the channel Energy correction factor Unit conversion constant Water surface angle in radians = y/D Dimensionless depthc = Q 2/[gD5 cos()] Dimensionless discharge for critical

depth computations (circular cross section)h = Q 2/[gH5 cos()] Dimensionless discharge for critical

depth computations (standard horseshoe crosssection)

t = z3Q 2/[gb5 cos()] Dimensionless discharge for criti-cal depth computations (trapezoidal cross section)

c = 2 cos1(1 2cc) For critical depth computationsn = 2 cos1(1 2nc) For normal depth computationsc = nQ/(D8/3S0) Dimensionless discharge for normal

depth computations (circular cross section)h = nQ/(H8/3S0) Dimensionless discharge for normal

depth computations (standard horseshoe crosssection)

r = nQ/(b8/3S0) Dimensionless discharge for normaldepth computations (rectangular cross section)

t = nQ/(b8/3S0) Dimensionless discharge for normaldepth computations (trapezoidal cross section)

cc = yc/D Dimensionless critical depth (circular cross sec-tion)

ch = yc/H Dimensionless critical depth (standard horse-shoe cross section)

ct = zyc/b Dimensionless critical depth (trapezoidal crosssection)

nc = yn/D Dimensionless normal depth (circular crosssection)

nh = yn/H Dimensionless normal depth (standard horse-shoe cross section)

nr = yn/b Dimensionless normal depth (rectangular crosssection)

nt = yn/b Dimensionless normal depth (trapezoidal crosssection)

Subscripts

c and n Denote critical and uniform flow conditions, respec-tively

c Denotes circular cross sectionh Denotes standard horseshoe cross sectionr Denotes rectangular cross sectiont Denote trapezoidal cross section

I and II Denote Type I and Type II, respectivelyent and Instrumentation 22 (2011) 4349

Fig. 1. Cross section of a trapezoidal channel.

Fig. 2. Cross section of a partially filled pipe channel.

2.2. Circular channels

For a partially filled circular channel section, shown in Fig. 2, thegeometric elements are as follows

A = D2

8( sin ) (4)

P = 12D (5)

T = D sin 2

(6)

= 2 cos1(1 2) (7)where D is channel diameter, is water surface angle in radians,and = y/D.2.3. Standard horseshoe channels

A horseshoe cross section, shown in Fig. 3, consists of four arcsegments: a top arc (BC) with radius r , a bottom arc (AD) withradius R, and two lateral arcs (AB and DC) with the same radiusR but with different circular centers [14]. Horseshoe cross sectionscan be classified using the characteristic parameter t = R/r . Fort = 3 and t = 2, the cross sections are called standard Type I andType II horseshoe cross sections, respectively [14]. Note that whent = 1, the horseshoe cross section becomes circular.

Fig. 3 shows a general horseshoe cross section and thecorresponding geometric symbols for three ranges of water depths(y): (a) 0 y e, (b) e y r , and (c) r y 2r , where e isthe height of the bottom arc, which is given by e = 0.12917r ( =0.294515 rad) for Type I, and e = 0.17712r ( = 0.424031 rad)for Type II cross sections, respectively [15].

The required formulae for computing the geometric elementsof standard horseshoe cross sections (wetted perimeter and flowarea) for the three zones of flow depth were presented in [15].These formulae are presented in Table 1. In addition, formulae forcomputing the width of the channel at the water surface, which

are needed for developing the critical depth equation, are derivedin this study and are presented in the table.

ht = F(ct) = 1+ 2ct (9)

where t and ct are dimensionless variables with t = z3Q 2/[gb5 cos()] and ct = zyc/b, and the subscripts c and t denotecritical flow conditions and the trapezoidal section, respectively.

Using the curve fittingmethod, Swamee [4] obtained an explicitsolution for Eq. (9). The maximum error of that solution is lessthan 2.2% in the practical range of 0 ct 3. Vatankhah andKouchakzadeh [5] improved Swamees solution by developing asimilar equation that has a maximum error less than 0.28% inthe practical range of 0 ct 15. Using the fixed point itera-tion method, Wang [6] obtained a very accurate but complicatedsolution for the critical depth of trapezoidal sections. Subse-quently, Swamee and Rathie [7] obtained analytical solutions ofthe critical depth for trapezoidal sections in the form of converginginfinite series, based on Lagranges inversion theorem. When the

dimensionless normal depth, 0 ct 3, is given byct = 1/3t (1+ 1.15240.347t )0.339

(proposed-trapezoidal channels). (12)The maximum error involved in Eq. (12) is less than 0.06% in

the practical range of 0 ct 3. A summary of the existingexplicit solutions of the critical depth for trapezoidal channels andtheir maximum errors is shown in Table 2. As noted, the proposedsolution is preferable to other solutions in terms of both accuracyand simplicity.

3.2. Circular channels

Substituting for A and T from Eqs. (4) and (6) into Eq. (8) yieldsthe following dimensionless form

3slope of the channel, and g is gravitational acceleration. Most ofthe channels have slopes smaller than 1/100, thus it is reasonableto assume cos() 1 for these channels. Using Eq. (8), explicitequations for the critical depth are derived next for trapezoidal,circular, and horseshoe channels.

3.1. Trapezoidal channels

Substituting for A and T from Eqs. (1) and (3) into Eq. (8) yieldsthe following dimensionless form

3ct(1+ ct)3

ct = 1/3t (1+ abt )c (10)where a, b, and c are coefficients. To determine these coefficients,the percentage error (PE) of the dimensionless critical depth, ct , isexpressed as follows

PE =ct F(ct)1/3(1+ aF(ct)b)c

ct

100 (11)

in which the dimensionless function F(ct) is determined usingthe geometry of the trapezoidal cross section according to Eq. (9).Then, the sum of the squares of the PE values is minimizedas an objective function using the Solver toolbox of MicrosoftExcel. The resulting explicit equation in the practical range of theA.R. Vatankhah, S.M. Easa / Flow Measurem

Fig. 3. Horseshoe cross section and its geometric symbols for three zon

Table 1Formulae for computing geometric elements for three zones of flow depth of a horses

Zones of flow depth

0 y e or 0 e y H2 or 0 = cos1

t2t

= sin1

t2t

A = 14 t2H2

12 sin(2)

A = 14 t2H2

C 12 s

P = tH P = tH(2 )T = tH sin() T = H[1 t + t cos()]

Note: C = 2 + 1 sin(2) cos(2),H = 2r = height of the tunnel and = y/H .

3. Computation of critical depth (governing equation)

The critical flow condition in an open channel is described bythe following relationship [16]

Q 2

g cos()TA3

= 1 (8)where is the energy correction factor, sin() is the longitudinalnumber of terms in the proposed series is limited to 45 for prac-tical cases, the maximum relative error involved in the solution isent and Instrumentation 22 (2011) 4349 45

es of flow depth: (a) 0 y e; (b) e y r; and (c) r y 2r = H .

oe cross section.

H2 y H or 0 = 2 cos1 (2 1)

in(2)+ 2(t1)t sin(2)

A = 14H2Ct2 + 12 ( + sin())

P = H2 (4t + )T = H sin 2

about 2.5%. In such a case, the advantage of the presented equationsover his previous regression-based equation is rather questionable.Based on the infinite series presented by Swamee and Rathie [7],Srivastava [10] found that a fitted series would be more accuratethan a truncated one.

To further develop an improved explicit solution in the currentstudy, Eq. (9) is first numerically inverted using the curve fittingmethod as followsc = h1(cc) = (c sin c)83 sin c2(13)

aocc =(1+ 0.01060.26c 0.01321.863c )10.022

. (16)

The maximum error of Eq. (16) is less than 0.25% in the practicalrange of 0 cc 0.92.

In the current study, the following regression-based equation isproposed for computing the critical depthcc = (13.62.1135c 132.1c + 1)0.1156

(proposed-circular channels). (17)The maximum error of Eq. (17) is less than 0.27% in the practicalrange of 0.01 cc 1. Table 2 presents a summary of the pro-posed and existing explicit equations for circular cross sections.Clearly, the proposed solution offers both simplicity and accuracycompared with other solutions.

3.3. Standard horseshoe channels

Substituting for A and T from Table 1 into Eq. (8) yields thefollowing dimensionless form

h = h2(ch) = A3ch

H5Tch(18)

in which h and ch are dimensionless variables with h = Q 2/

(19) and (20), are depicted graphically in Fig. 4. The actual data usedfor estimating these equations using regression are also shown. The(almost) perfect match between the actual and proposed criticaldepths is evident.

4. Computation of normal depth (governing equation)

The uniform flow condition in an open channel is described bythe following Mannings formula [17]

Q = S0

nA5/3

P2/3(21)

in which is the unit conversion constant, 1.0 (SI), 1.49 (CU), S0 isthe longitudinal slope of the channel, and n isMannings roughnesscoefficient.

4.1. Trapezoidal channels

Substituting for A and P from Eqs. (1) and (2) into Eq. (21), theMannings equation becomes

t = 5/3nt (1+ znt)5/3

(1+ 2nt1+ z2)2/3 (22)46 A.R. Vatankhah, S.M. Easa / Flow Measurem

Table 2Summary of proposed and existing explicit equations for critical depth in trapezoidal

Equation reference Proposed formulae

Trapezoidal channels

Swamee [4] ct = (0.7t + 0.7470.42t )0.476

Wang [6] ct = 12[1+

1+ 41/3t {1+ 41/3t [1+ 41/3t (1+ 41t

Vatankhah andKouchakzadeh [5]

ct = 1/3t[1+

1/3t /

20.4P]1/P

P = 2.1290.01565t

Srivastava [10] ct = 1/3t 0.336622/3t + 0.135291.075t

1ct = (2t )0.2 + 0.379130.4t + 0.0717760.6t 0.0240Proposed ct = 1/3t (1+ 1.15240.347t )0.339

Circular channels

Straub [8] cc = 1.01 D0.01 0.25cSwamee [4] cc = (1+ 0.773c )0.085Vatankhah andBijankhan [9]

cc = 0.95840.25c(1+0.01060.26c 0.01321.863c )10.022

Proposed cc = (1+ 13.62.1135c 132.1c )0.1156There are no explicit equations available in the literature for the horseshoe cross secti

where c and cc are dimensionless variables with c = Q 2/[gD5 cos()], c = 2 cos1(1 2cc), and cc = yc/D.Straub [8] proposed a semi-empirical equation for the critical

depth in circular open channels as follows

cc = 1.01 D0.01 0.25c . (14)As noted, Eq. (14) is dimensionally inhomogeneous. This equationis not very accurate and its error depends on the diameter of thechannel. For example, for D 0.25 m, the maximum error is 5.8%.

Swamee [4] numerically inverted Eq. (13) for the dimensionlesscritical depth, cc , using the curve fitting method as follows

cc = (0.773c + 1)0.085. (15)The maximum error of Eq. (15) is less than 1.46% in the practicalrange of 0.02 cc 1.

Recently, Vatankhah and Bijankhan [9] used the curve fittingmethod to obtain amore accurate equation for the critical depth ofa circular channel by inverting Eq. (13) as follows

0.95840.25c[gH5 cos()], ch = yc/H,H is height of the conduit, and the sub-script h denotes the standard horseshoe cross section.ent and Instrumentation 22 (2011) 4349

nd circular channels.

Application range Maximum relative error (%)

0 ct 3 2.20/3)1/5]1/6}1/6

1/2]None 0.015

0 ct 15 0.28

0 ct 0.347 0.02190.75t ct > 0.347 0.014

0 ct 3 0.06

0.1 cc 0.85 5.8 (D 0.25 m)0.02 cc 1 1.460 cc 0.92 0.250.01 cc 1 0.27

n, and the proposed equations are presented in the paper.

As previouslymentioned, there is currently no explicit equationfor calculating the critical depth of horseshoe channels. Therefore,using the curve fitting method the following explicit equationswere developed for computing the critical depth for standardhorseshoe channels

chI = (1.710.7h + 0.0000068.877h )0.02799

(1+ 1.62.4h )0.11(proposed-standard Type I horseshoe channels) (19)

chII = (3.3712.74h + 0.00005310.954h )0.02275(1+ 1.072.15h )0.14

(proposed-standard Type II horseshoe channels) (20)where the subscripts I and II denote Type I and Type II, respectively.The maximum error of Eq. (19) is less than 0.65% in the practicalrange of 0.01 ch 0.988 and that of Eq. (20) is less than 0.55%in the practical range of 0.01 ch 0.99.

The proposed explicit equations of the dimensionless criticaldepth for trapezoidal, circular, horseshoe channels, Eqs. (12), (17),where nt = yn/b, yn is the normal depth, t = nQ/(b8/3 S0)and the subscript n denotes normal flow conditions.

AlirezaCross-Out

AlirezaReplacement Text+

A.R. Vatankhah, S.M. Easa / Flow Measurem

Fig. 4. Comparison of actual and proposed dimensionless critical depth: (a)trapezoidal channels and (b) circular and standard horseshoe channels.

Swamee and Rathie [11] proposed two infinite series solutionsthat help in evaluating the normal depth for trapezoidal crosssections.When two different solutions are given for a cross section,a range of applicability should be determined. However, the choiceof either solution for the most widely used nt range dependson the nt-value and the side slope of the channel, and this factcomplicates their applications in practice [12].

To overcome this limitation, an explicit equation is developedhere for the normal depth using the fixed point iteration scheme.Rearranging Eq. (22), then

nt = 3/5t (1+ 2nt0

1+ z2)2/5

1+ znt0(proposed-trapezoidal channels) (23)

in which nt0 is an initial guess for the normal depth which isproposed as follows

nt0 = 1+ 0.8563/5t (1+ z1.263)(1 0.0585z3/5t )3/5t + 1.945z

. (24)

The maximum error involved in Eq. (23) is less than 0.7% in thepractical range of 0 z 3 and 0 nt 1.

For rectangular channels (z = 0), Eq. (21) becomesnr = 3/5r (1+ 23/5r + 1.7126/5r )2/5

(proposed-rectangular channels) (25)

where nr = yn/b, r = nQ/(b8/3S0). The maximum errorinvolved in Eq. (25) is less than 0.08% in the practical range, 0 nr 3.

In comparison, Srivastava [10] proposed an expression for thenormal depth of rectangular channels, based on truncation of theiterative algorithm and the curve fitting method, as followsnr = 3/5r1+ 2.4040.6321r (1+ 2.0300.9363r )0.3929

2/5. (26)ent and Instrumentation 22 (2011) 4349 47

Themaximumerror involved in Eq. (26) is less than 0.06% in therange of 0 nr 100. Note that from an engineers viewpoint,it would be more useful to have a simple and reasonably accurateexpression for the practical range rather than a complicated andmore accurate expression for a much wider range. Eq. (26) is moreaccurate over a larger range, but Eq. (25) is simpler to use, lesscomputationally intensive, and reasonably accurate. It is applicablefor the entire practical range of nr .

4.2. Circular channels

Substituting for A and P from Eqs. (4) and (5) into Eq. (21) thefollowing equation is obtained

c = h3(nc) = (n sin n)5/3

213/3 2/3n(27)

in which c = nQ/(D8/3S0), n = 2 cos1(1 2nc) andnc = yn/D.

Using the curve fitting method, the following regression-basedequation is proposed for computing the normal depth for circularchannels,

nc = 1.025(0.551.1c 14.554.136c +0.4645)

c

(proposed-circular channels). (28)

The maximum error involved in Eq. (28) is less than 0.35%in the practical range (0.005 nc 0.82). There is a rationalefor selecting the upper value of the preceding practical range.Channels with a closing top-width can be designated as channelsof the second kind [16]. In these channels, the discharge is nota single-value function of the normal depth beyond a certainrange of the flow depth. In this range, a small disturbance in thewater surface may cause it to seek alternate normal depths, thuscontributing to the instability of the water surface. In practice, itis usual to restrict the flow depth to be below this range to avoidthis double normal depth phenomenon. For circular channels, thisinstability range occurs when the flow depth is greater than 0.82of the channel height. Thus, in practice it is sufficient to restrict theflow depth to be less than this ratio.

4.3. Standard horseshoe channels

Substituting for A and P from Table 1 into Eq. (21) yields thefollowing dimensionless form

h = h4(nh) = A5/3nh

H8/3P2/3nh(29)

in which h = nQ/(H8/3S0), nh = yn/H,H is height of theconduit, and the subscript h denotes a standard horseshoe crosssection. It is important to note that for nh > 0.82, the discharge isnot a single-valued function of the normal depth for the standardType I and Type II horseshoe cross sections. Thus, in practice itis sufficient to restrict the depth of flow to be less than 0.82 ofthe channel height in both types. In addition, it is not necessaryto compute the normal depth for very small water depth [13].Therefore, the minimum value of the dimensionless normal depthis suggested as nh = 0.05.

Liu et al. [15] developed general formulae for the directcomputation of the normal depth for all types of horseshoe crosssections (for three zones) as follows,

2 (0.00216t20.0126t+0.4806)nh = (0.057t 0.3738t + 1.3849)h[for Q Q (e)] (30)

u0t

3of Eqs. (33) and (34) work very well over the entire practical rangeof depth (0.05 nh 0.82) with a maximum percentage errorless than 0.63%. Table 3, presents a summary of the proposed andexisting explicit equations for rectangular and Standard Horseshoecross sections. Clearly, the proposed solution offers both simplicityand accuracy compared with other solutions.

The proposed explicit equations of the dimensionless normaldepth for trapezoidal, circular, and horseshoe channels, Eqs. (23),(28), (33) and (34), are depicted graphically in Fig. 5, alongwith theactual data. Again, the proposed critical depth equations (almost)perfectly match actual data.

5. Conclusions

The critical and normal depths are important elements inthe design, operation, and maintenance of open channels. Thecalculation of these elements is traditionally performed using trialprocedures, numerical/graphical methods, or explicit regression-based equations. Explicit solutions for the critical depth are

Fig. 5. Comparison of actual and proposed dimensionless normal depth: (a)trapezoidal channels and (b) circular and standard horseshoe channels.

equations are available for only rectangular and horseshoechannels, but not for trapezoidal or circular channels.

This paper has presented explicit solutions of these elementsfor three types of channels: trapezoidal, circular, and horseshoe.For the critical depth, new explicit equations for horseshoechannels and improved explicit equations for trapezoidal andcircular channels are presented. For the normal depth, new explicitequations for circular and trapezoidal channels and improvedsolutions for horseshoe channels are presented.

Dimensionless variables of the governing equations (Eqs. (8)and (21)) are very powerful tools for developing general regressionequations without the need for using actual data. The explicit48 A.R. Vatankhah, S.M. Easa / Flow Measurem

Table 3Summary of proposed and existing explicit equations for the normal depth in rectang

Equation reference Proposed formulae

Rectangular channels

Srivastava [10] nr = 3/5r1+ 2.4040.6321r (1+ 2.0300.9363r )

Proposed nr = 3/5r (1+ 23/5r + 1.7126/5r )2/5

Standard horseshoe channels

nh = (0.057t2 0.3738t + 1.3849)(0.00216t2hLiu et al. [15] (Types I and II) nh = (0.1997t2 + 1.124t 0.9243)1.2h + (

0.5401t + 1.8479)0.6h + (0.002t2 0.0166nh = (0.0111t2 + 0.0501t + 2.8232)1.2h + 0.2063t 0.3739)0.6h + (0.0001t2 0.006

Proposed (Type I) nhI = 0.5(1.7220.892h 17.64.2h +0.3956)

h

Proposed (Type II) nhII = 0.75(1.51.14h 182.466.9h +0.4414)

h

There are no proper explicit equations available in the literature for circular and trape

nh = (0.1997t2 + 1.124t 0.9243)1.2h+ (0.0956t2 0.5401t + 1.8479)0.6h+ (0.002t2 0.0166t + 0.0524)[for Q (e) Q Q (0.5H)] (31)

nh = (0.0111t2 + 0.0501t + 2.8232)1.2h+ (0.0381t2 0.2063t 0.3739)0.6h+ (0.0001t2 0.0063t + 0.3789)[for Q (0.5H) Q ]. (32)

To apply these formulae, the limits of the dischargeQ should bedetermined. Then, the equation corresponding to the flow range isused. In practice, however, it is preferable to have a single normaldepth equation for the entire range of flow (0.05 nh 0.82).Therefore, the following regression-based equations were devel-oped for computing the normal depth of horseshoe channels,

nhI = 0.5(1.7220.892h 17.64.2h +0.3956)

h(proposed-standard Type I horseshoe channels) (33)

nhII = 0.75(1.51.14h 182.466.9h +0.4414)

h(proposed-standard Type II horseshoe channels) (34)

where the subscripts I and II denote Type I and Type II, respectively.The proposed approximations of the dimensionless normal depthavailable in the literature for trapezoidal and circular channels,but not for horseshoe channels. For the normal depth, explicitent and Instrumentation 22 (2011) 4349

lar and standard horseshoe channels.

Application range Maximum relative error (%)

0.39292/5 0 nr 100 0.06

0 nr 3 0.08

0.0126t+0.4806) 0 < Q Q (e) and 0.05 nh 0.10.0956t2

+ 0.0524)Q (e) Q Q (0.5H) 0.35

(0.0381t2

t + 0.3789)Q (0.5H) Q and nh 0.82 1.35

0.05 nh 0.82 0.630.05 nh 0.82 0.63

zoidal cross sections, and the proposed equations are presented in the paper.equations were developed using regression analysis based on thepractical ranges of the corresponding dimensionless variables.

A.R. Vatankhah, S.M. Easa / Flow Measurem

The functional form of the regression equation is determinedby evaluating different mathematical functions and selectingthe one that minimizes the maximum relative error. This erroralso depends on the number of data points selected within theranges of the dimensionless variables. In addition, visualizationof the response surface of the governing equation was found togreatly help in selecting the appropriate mathematical functionsfor evaluation. These factors and the skills of the analyst couldresult in different regression equations for the same dimensionlessvariables.

The maximum relative error of the proposed regressionequations is less than 1%. The proposed equations exhibit bothsimplicity and accuracy. It is hoped that the efficient computationaltools presented in this paper will be useful in the design andanalysis of open channels.

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Explicit solutions for critical and normal depths in channels with different shapesIntroductionGeometric propertiesTrapezoidal channelsCircular channelsStandard horseshoe channels

Computation of critical depth (governing equation)Trapezoidal channelsCircular channelsStandard horseshoe channels

Computation of normal depth (governing equation)Trapezoidal channelsCircular channelsStandard horseshoe channels

ConclusionsReferences