Rough Model Method Research Articles

Chezy's resistance coefficient in vaulted rectangular conduit

ABSTRACT Calculating resistance coefficients for flows in pipes and channels, such as the Darcy–Weisbach coefficient of friction and the Chezy or Manning coefficients, is a complex process aimed at accurately reflecting flow under load or a free surface. Generally, these coefficients are given as constants, although they can be challenging to ascertain due to implicit models expressing these coefficients. Therefore, the design of pipes and channels necessitates an explicit and comprehensible expression of the resistance coefficient, incorporating numerous flow parameters, including aspect ratio, kinematic viscosity, pipe slope, and roughness of internal walls. To achieve this outcome, we suggest employing the rough model method (RMM) giving the volume flowrate to identify the Chezy's resistance coefficient C in uniform and free surface flow for the vaulted rectangular conduit. This applies in both cases where the diameter of the geometric profile of the conduit is known or unknown.

Chezy’s Resistance Coefficient in a Round-cornered Rectangle Channel

Abstract The determination of resistance coefficients, such as Chezy’s or Manning’s coefficients, requires a great deal of sensible thought in order to express these coefficients better and more extensively in free-surface channels and aqueducts. This can be achieved if the expression of the resistance coefficient is well stated and takes into account the maximum number of parameters for governing flows in channels. However, in most practical cases, if these coefficients are not expressed by implicit models, they are generally taken as constant and arbitrary. To this end and in a rational manner, the dimensioning and design of channels requires the expression of the resistance coefficient in an easily and explicit form by adopting numerous flow parameters, namely the roughness of the walls, the aspect ratio, the slope of the channels and essentially the viscosity of the liquid. To achieve this aim, the Chezy’s resistance coefficient C is identified using the rough model method (RMM), which gives the discharge under uniform flow conditions appropriate to a round-cornered rectangle channel.

Optimisation of the choice of the sizing method for drinking water networks in the developing countries: A case of Cameroon

In developing countries (DC), particularly in Cameroon, the demand for drinking water is increasing. To date, the methods of design, sizing and extension of existing drinking water networks are done through the EPANET application which is a simulation software based on the Hardy-Cross method. The main objective of this study is to set up a local tool adapted to the context of D.C. allowing the design, sizing and extension of networks based on the rough model method (RMM). Thus, on the basis of a previously modelled existing network, a comparison was made between the Hardy-Cross method and the RMM (according to the criteria: costs, speed of execution, aptitude for automation, fidelity and precision) with the multi-criteria analysis tool PROMETHEE II. The results of this analysis show that the best adapted method for the design, sizing and extension of drinking water networks is the RMM. It appears that the setting up of water supply networks in DC must be done by the RMM for affordable costs, rapid execution, easy reproducibility and reliable precision.

Manning's roughness coefficient in a truncated triangular open-channel flow section

Abstract In uniform flows, Manning's roughness coefficient n plays an important role in the calculation of the normal depth in the open channels. When calculating uniform flows in open conduits and channels, Manning's coefficient is arbitrarily chosen. This arbitrary choice is not physically justified because the coefficient n must be determined according to the parameters that influence the flow, particularly the normal depth sought. In this paper, a new method is presented to compute Manning's coefficient in a truncated triangular channel section. In the first step, the study proposes to establish the general relationship that allows computing Manning's coefficient in this type of channel. The relationship is presented in dimensionless terms, giving it a character of general validity, including the dimensionless Manning's resistance coefficient, denoted as N. The latter is shown to depend on the aspect ratio of the wetted area, the side slope of the channel, the relative roughness and the modified Reynolds number, which is physically justified since flow resistance is closely related to flow depth. In the second step, the rough model method is proposed to express Manning's coefficient when the horizontal dimension of the channel is not a given data of the problem.

Estimation of natural stream-water flow rates in semi-arid climate with the rough model method

Estimation of natural stream-water flow rates in semi-arid climate with the rough model method

Estimation of natural stream-water flow rates in semi-arid climate with the rough model method

Estimation of natural stream-water flow rates in semi-arid climate with the rough model method

Chezy’s resistance coefficient in a horseshoe-shaped tunnel

The calculation of Chezy’s resistance coefficient (CRC) is typically not provided a priori in a design problem, and its value is often selected subjectively from the literature in most open channels and conduits for the uniform flow. The evaluation of this coefficient is crucial to channel design and for computing its normal depth. The primary purpose of this research study is to revisit the mathematical formulation for the resistance coefficient. A general explicit relation of the resistance coefficient in turbulent flow is set with different geometric profiles of conduits and channels, mainly the horseshoe-shaped tunnel using the rough model method (RMM). CRC is firmly based on the internal walls' absolute roughness of the channel, the liquid kinematic viscosity, the longitudinal slope, the discharge and the filling rate. Additionally, a simplified method is proposed to determine CRC with a restricted number of variables such as the kinematic viscosity, the absolute roughness, the slope of the conduit, and the discharge. Based on studying the variation of CRC as a function of the filling rate, another explicit expression is provided to compute this coefficient efficiently when its maximum value is reached. To demonstrate how Chezy’s resistance coefficient can be calculated in a horseshoe-shaped tunnel, some examples of calculations are proposed.

COMPUTATION CHEZY’S COEFFICIENT IN A SEMI-ELLIPTICAL PIPE

In the hydraulic field, the Chezy’s flow resistance coefficient in canals and pipes is often chosen arbitrarily. This value is tabulated independently of the depth of the flow or hydraulic radius and even less of the Reynolds number. This coefficient is usually influenced by a number of parameters that must be defined and considered. The objective of this study is to examine, on the one hand, the variation of the Chezy’s flow resistance coefficient for a semi-elliptical pipe under the hypothesis of an uniform flow with free surface, and to determine on the other hand, the expressions which govern them. Further, it must search expression for Chezy’s with consideration of the geometric characteristics of the pipe and hydraulic flow. This study is based on the rough model method (RMM).

Chezy’s resistance coefficient in an egg-shaped conduit

Abstract When calculating uniform flows in open conduits and channels, Chezy’s resistance coefficient is not a problem data and its value is arbitrarily chosen. Such major disadvantage is met in all the geometric profiles of conduits and channels. Knowing the value of this coefficient is essential to both the design of the channel and normal depth calculation. The main objective of our research work is to focus upon the identification of the resistance coefficient relationship. On the basis of the rough model method (RMM) for the calculation of conduits and channels, a general explicit relation of the resistance coefficient in turbulent flow is established with different geometric profiles, particularly the egg-shaped conduit. Chezy’s resistance coefficient depends strongly on the filling rate, the discharge, the longitudinal slope, the absolute roughness of the internal walls of the conduit and the kinematic viscosity of the liquid. Moreover, in this work, a simplified method is presented to determine Chezy’s resistance coefficient with a limited number of data, namely the discharge, the slope of the conduit, the absolute roughness and the kinematic viscosity. Last but not least, after studying the variation of Chezy’s resistance coefficient as a function of the filling rate, an equally explicit expression is given for the easy calculation of this coefficient when its maximum value is reached. Examples of calculation are suggested in order to show how the Chezy’s coefficient can be calculated in the egg-shaped conduit.

Chezy’s Resistance Coefficient in a Circular Conduit

In the literature, there is no explicit method for calculating the resistance coefficient of Chezy, especially for a circular conduit. Existing relationships are either implicit or do not take into account all parameters influencing the flow such as kinematic viscosity or the slope of the conduit. In many practical cases, one affects arbitrarily a constant value for Chezy’s coefficient. It is a physically unjustified approach, because Chezy’s coefficient varies with flow parameters, especially the filling rate of the conduit and the absolute roughness. In this paper, simple and explicit relationships are presented for the calculation of Chezy’s resistance coefficient in a circular conduit. These relationships have been established based on the rough model method. The Chezy’s resistance coefficient is expressed in terms of known hydraulic parameters of the flow in a referential rough model. For fast calculation of Chezy’s coefficient, the simplified method is the most appropriate since it requires only four parameters which are the discharge, the absolute roughness, the slope and the kinematic viscosity. The study also shows that the Chezy’s resistance coefficient reaches a maximum whose expression is well defined. Some examples are presented showing how to calculate Chezy's coefficient in a circular conduit with a minimum practical data.

Analytical Solution for Normal Depth Problem in a Vertical U-shaped Open Channel Using the Rough Model Method

Normal depth plays a significan t role in the design of open channels and in the analysis of the non uniform flow as well. Currently there is no analytical method for calculating the normal depth in the open channels. Current methods are either iterative or approximate and consider, unre asonably, Chezy’ coefficient or Manning’s roughness coefficient as a given data of the problem. Yet, both of these coefficients depend in particular on the normal depth sought and it is therefore unjustified to fix them beforehand. To overcome this drawbac k, the rough model method (RMM) seems to be the most appropriate tool. The RMM takes into account, in particular, the effect of the absolute roughness which is a readily measurable parameter in practice. The method is based on known referential rough model characteristics used to deduce the normal depth by means of a non dimensional correction factor.

Computation of Normal Depth in Parabolic Cross Sections Using The Rough Model Method

A new method is applied to calculate the normal depth in an open channel of parabolic cross section. This is the rough model method whose main particularity is to ignore the flow resistance coefficients, such as Chezy’s coefficient and manning’s roughness coefficient. The method is applied to a referential rough model, whose friction coefficient is constant, which explicitly express the hydraulic and geometric characteristics of the model such as aspect ratio. By means of a non-dimensional correction factor, the normal depth is explicitly deduced. The rough model method is applicable to the entire domain of turbulent flow.

Design of a Pressurized Rectangular-Shaped Conduit Using the Rough Model Method (Part 1)

The rough model method is successfully used to design a pressurized rectangular shaped conduit characterized by two linear dimensions. In this study, the focus is on the calculation of the horizontal linear dimension of the conduit. In a first step, the method is applied to a referential rough model in order to establish the relationships that govern its hydraulic characteristics. The obtained equations are of the third degree and are easily solved by trigonometric and hyperbolic functions. In a second step, these equations are used to easily deduce the linear dimension sought by introducing a non-dimensional correction factor. Practical example is taken to enable the hydraulic engineer to better understanding the advocated method and to observe the facility with which design of such a geometric profile can be performed. The calculation uses a strict minimum of data measurable in practice, in particular the absolute roughness.

Computation of Normal Depth in Horseshoe Shaped Tunnel Using the Rough Model Method

Normal depth plays a significant role in the design of open channels and in the analysis of the non-uniform flow as well. Currently, there is no analytical method for calculation of the normal depth in open channels, including the horseshoe profile. Current methods are either iterative or approximate. They also consider, unreasonably, Chezy’s coefficient or Manning’s roughness coefficient as a given data of the problem, despite the fact that these coefficients depend on the normal depth sought. In this study, a new analytical method is presented for calculating the normal depth in a horseshoe shaped tunnel. The method takes into account, in particular, the effect of the absolute roughness which is a readily measurable parameter in practice. In a first step, the method is applied to a referential rough model in order to establish the relationships that govern its hydraulic characteristics. In a second step, these equations are used to easily deduce the required normal depth by introducing a non-dimensional correction factor. A practical example is considered to better explain the advocated method and to appreciate its simplicity and efficiency.

Design of a Pressurized Trapezoidal Shaped Conduit Using the Rough Model Method (Part 2)

Through the Rough Model Method (RMM) an explicit calculation is proposed to solve the complex problem of the design of a pressurized trapezoidal shaped conduit characterized by three linear dimensions. Firstly, the method is applied to a rough model of the same shape in order to establish the equations governing its geometric and hydraulic characteristics. These equations are then secondly used to easily deduce the required linear dimensions of the current conduit by introducing a non-dimensional correction factor. The application of the obtained relationships requires only a strict minimum of data. All relationships provided in this article are applicable to the entire domain of the turbulent flow. Practical examples are taken to enable the hydraulic engineer to better understanding the advocated method and to observe the facility with which design of such a geometric profile can be performed.

Computation of Normal Depth in Trapezoidal Open Channel Using the Rough Model Method

The recurring problem of calculating the normal depth in a trapezoidal open channel is easily solved by the rough model method. The Darcy-Weisbach relationship is applied to a referential rough model whose friction factor is arbitrarily chosen. This leads to establish the non-dimensional normal depth relationship in the rough model. Through a non-dimensional correction factor of linear dimension, the aspect ratio and therefore normal depth in the studied channel is deduced. Keywords: Rough model method, Trapezoidal channel, Normal depth, Turbulent flow, Discharge, Energy slope.

Design of a Pressurized Circular Pipe with Benches Using the Rough Model Method (RMM)

The rough model method (RMM) is explained through its application to the design of a pressurized circular shaped conduit with benches, widely used in practice. The three basic equations of turbulent flow are firstly applied to define explicitly the geometric elements of a referential rough model characterized by an arbitrary assigned relative roughness value. The required linear dimensions of the studied conduit are then easily deduced by multiplying the homologues linear dimensions of the rough model by a non-dimensional correction factor. Friction factor is not indispensable when applying the RMM, unlike current design methods. Resulting RMM equations are not only explicit but are also valid in the entire domain of turbulent flow.

Design of a Pressurized Trapezoidal Shaped Conduit Using the Rough Model Method (Part 1)

Through the Rough Model Method (RMM) an explicit calculation is proposed to solve the complex problem of the design of a pressurized trapezoidal shaped conduit characterized by three linear dimensions. Firstly, the method is applied to a rough model of the same shape in order to establish the equations governing its geometric and hydraulic characteristics. These equations are then secondly used to easily deduce the required linear dimensions of the current conduit by introducing a non-dimensional correction factor. The application of the obtained relationships requires only a strict minimum of data. All relationships provided in this article are applicable to the entire domain of the turbulent flow. Practical examples are taken to enable the hydraulic engineer to better understanding the advocated method and to observe the facility with which design of such a geometric profile can be performed.

The Rough Model Method (RMM) Application to The Computation of Normal Depth in Circular Conduit

A new method is presented to compute the normal depth in circular conduit. This is the rough model method (RMM). It states that the linear dimension of a conduit or channel is equal to the linear dimension of a referential rough model corrected by the effect of a non-dimensional correction factor. The method is based on the Colebrook-White and Darcy-Weisbach relationships, applicable to the entire domain of turbulent flow. From the relationship governing the flow in the rough model, the normal depth in a circular conduit is explicitly deduced.

Design of Pressurized Vaulted Rectangular Conduits Using the Rough Model Method

A new theory is presented to design pressurized conduits, particularly vaulted rectangular conduit. This theory is based on a referential rough conduit model characterized by an arbitrarily assigned relative roughness taken in the fully turbulence flow regime. Thus, the geometric elements of the chosen model are well defined. These allow a direct determination of the required geometric elements of the conduit by the use of a non-dimensional correction factor. A practical example is presented to explain the procedure of calculation. Keywords: Rough Model Method, Pressurized conduit, Vaulted rectangular conduit, Turbulent flow, Discharge, Energy slope.