2. Basic Data. 

Procedures for computing the effective roughness coefficient n to be used in the Manning formula for channels with different bed and bank roughnesses have been developed by Horton(3), Colebatch(4), Einstein(5), and the U. S. Army Engineer District, Los Angeles, California(6). In each case the effective n value is a function of the bed and bank roughnesses and their respective segments of the wetted perimeter or flow area. In their simplest form, the equations for effective n values
can be written as

n_eff = ΣnA / ΣA (Los Angeles District) ----------------------- (1)

n_eff = [ Σ(n^(3/2) P) / P ]^(2/3) (Horton or Einstein) -------- (2)

n_eff = [ Σ(n^(3/2) A) / A ]^(2/3) (Colebatch) ----------------- (3)

A and P are the channel flow subareas and wetted perimeter segments, respectively; n is the respective Manning roughness coefficient for each segment considered.
 
3. Study of the equations given in paragraph 2 indicates that for channels with smooth inverts and rough banks, use of the Horton-Einstein equation results in more conservative design than use of either the Colebatch or the Los Angeles District equation. Laboratory and field investigations are needed for complete evaluation of the equations. The use of the Horton-Einstein equation is suggested for design purposes pending availability of additional test data.

4. For rectangular or trapezoidal channels, equation 2 can be written in the form

n_eff = [ (n1^(3/2) P1 + 2 n2^(3/2) P2) / (P1 + 2 P2) ]^(2/3) --- (4)

where the subscripts 1 and 2 refer to the bed and bank wetted perimeters, respectively. The terms are further defined in the sketch in Hydraulic Design Chart 631-4/1.

5. Application. 

Chart 631-4 provides a rapid graphical method for determining the solution of equation 2 to obtain an effective n value for use in the design of uniform channel sections with different bed and bank roughnesses. The ordinates of the chart indicate the bed, bank, and combined effective roughness coefficients. The abscissas are values of the ratio of the bed and bank wetted perimeters. The effective
n value is determined in the following manner. The chart is entered vertically from the bottom with the given value of 2P2/P1 to its intersection with an imaginary line connecting nl and n2 . The value of n_eff at this point is read on the right side of the chart.

6. Chart 631-4/1 can be used to obtain the required wetted perimeter ratio for use with Chart 631-4. Chart 631-4/1 presents bank-bed wetted perimeter relations for trapezoidal and rectangular channel sections as functions of the bed width, flow depth, and bank slope. These charts can be used with Charts 63J_and 631-1 for the design of channels with riprapped banks.


7. References.

(1) King, H. W., Handbook of Hydraulics for the Solution of Hydraulic Problems, revised by E. F. Brater, 4th ed. McGraw-Hill Book Co., Inc. , New York, N. Y., 1954, Table 76, p 20.

(2) Chow, V. T., Open-Channel Hydraulics. McGraw-Hill Book Co., Inc., New York, N. Y., 1959, Tables 5 and 6, p 111.

(3) Horton R. E., “Separate roughness coefficients for channel bottom and sides.” Engineering News-Record, vol iii, No. 22 (30 November 1933), pp 652-653.

(4) Colebatch, G. T., “Model tests on Liawenee Canal roughness coefficients.”Transactions of the Institution, Journal of the Institution of Engineers, vol 13, No. 2, Australia (February 1941), pp 27-32.

(5) Einstein.. H. A.. “Der hvdraulische oder Profil-Radius.” Schweizerische Bauzeitung ,“vol 103, No. 8 (24 February 1934), pp 89-91.

(6) U. S . Army, Office, Chief of Engineers, Hydraulic Design of Flood Control Channels. EM 1110-2-1601 (unpublished Engineer Manual draft ).

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Using equation 4 with water depth d, for rectangular channel, 

where P1 is width of channel and d is water depth.

For trapezoidal channel, if side slope is 1:z, n_eff can be calculated as followings.

P2 = d*( 1 + z^2 )^(1/2)