Chezy’s Resistance Coefficient in a Rectangular Channel

The Chezy’s resistance coefficient plays an important role in the calculation of the normal depth in the open channels. When using the Chezy’s relationship for the calculation of the normal depth, the main unknown parameter of the problem is the Chezy’s coefficient. There is no explicit and complete relationship for the evaluation of the Chezy’s resistance coefficient. Current relations are either implicit or do not take into account all the parameters that influence the flow, such as channel slope or kinematic viscosity. Most of them do not apply to the whole domain of turbulent flow because the kinematic viscosity is not taken into account. For these reasons, one affects arbitrarily a constant value for Chezy's resistance coefficient as a given data of the problem, in most practical applications. This arbitrary choice is not physically justified because the Chezy’s resistance coefficient must be calculated according to the parameters that influence the flow, especially the normal depth sought. The purpose of this paper is to show how to calculate the Chezy’s resistance coefficient in a rectangular channel, using the minimum of practical data. In this article, it is expressed the dimensionless Chezy's coefficient in order to give it a general validity character. The expression of this dimensionless coefficient is deduced from the comparison between the Chezy’s relationship and the general formula of the discharge valid for all geometric profiles. The detailed study of this relationship gives interesting results. It is clearly demonstrated that the dimensionless Chezy’s resistance coefficient depends on the relative roughness, the aspect ratio of the wetted area and the modified Reynolds number. This allows concluding that the obtained relationship is applicable to the entire domain of turbulent flow. The graphical representation of this relationship shows that the dimensionless Chezy coefficient increases with the decrease of the aspect ratio of the wetted area, whatever the value of the modified Reynolds number. This is reflected in the increase of the Chezy’s coefficient when the normal depth increases. In addition, the obtained curves intersect the x-axis at points corresponding to the particular case of the narrow rectangular channel, for which the aspect ratio tends to zero. This corresponds to a rectangular channel of small width and large depth. For this particular case, the relationship expresses the dimensionless Chezy coefficient is established, showing the influence of both the relative roughness and the modified Reynolds number. The aspect ratio of the wetted area has no effect.
Through a detailed practical example, it is shown how to calculate the Chezy resistance coefficient in a rectangular channel, from practical data. This calculation depends on the value of the relative normal depth in a rough rectangular channel that is easily determined using the rough model method. A cubic equation is obtained whose resolution is facilitated by the hyperbolic and trigonometric functions.