### The Manning Formula and the Colebrook-White Equation

**By Stephen Webster**

**Introduction**

The hydraulic capacity of drainage pipes is a complex theoretical problem because in real drains the flow is turbulent. The different layers of water flow are constantly mixing with each other creating small eddies within the flow which reduces the hydraulic capacity in complex and unpredictable ways. For this reason the formulas used by drainage design engineers are a mix of empirical and theoretical formulas.

There are two main methods in use today for estimating the capacity of drainage pipes for design purposes. These are most commonly known as the Manning Formula and the Colebrook-White Equation. Each formula has a different theoretical basis and different empirical corrections.

**Manning Formula**

In the US and many other parts of the world the Manning Formula is most commonly used for drainage pipe design. It is also included as a possible method in the European codes of practice. The Manning Formula is an entirely empirically derived formula used to calculate the average velocity and the flow of any open channel including a circular pipe not running under pressure. The Manning Formula as used for drainage pipe design is often expressed as shown below.V = Average Water Velocity (can be multiplied by flow area to calculate the flow capacity)

n = Manning Coefficient. This is an empirical roughness correction coefficient which is used to calibrate the formula to allow for the different energy losses caused by different pipe materials.

R = Hydraulic Radius. This is the area of flow divided by the length of the water-pipe interface. For circular pipes flowing full this can be taken as the pipe diameter divided by 4.

S = Hydraulic Gradient. This is simply the slope of the pipe (in m/m).

In the past one of the advantages of the Manning Formula was its simplicity. Nomographs and tabulated solutions were very useful before the proliferation of scientific calculators, particularly when designs needed to be altered onsite.

The disadvantage of the Manning formula is its lack of accuracy. The empirical formula was originally derived from a very limited dataset and does not have a strong theoretical basis. While the formula can be used to produce a good estimate of the hydraulic capacity of a circular drainage pipe for conditions similar to the original dataset, it loses accuracy the further conditions deviate from this. Generally the Manning Formula produces good results for surface water drains less than 300mm and for slimed foul drainage pipes less than around 750mm diameter. For larger diameter pipes the accuracy of the Manning Formula deteriorates and has been shown to overestimate the capacity of surface water drains in some cases.

**Colebrook-White Equation**

The Colebrook-White Equation was developed in 1939 through experiments with commercial drainage pipes with artificially roughened internal surfaces. The results were combined with the von Karman-Prandtl and Darcy-Weisbach formula to produce the design equation. Initially the equation was considered too complex for practical use but subsequent publication of design charts and tabulated values allowed the more accurate equation to be used in some standard design conditions. Nowadays programmable calculators and simple excel spreadsheet programs can be used to complete the calculations allowing designers to use the more accurate equation in all conditions.g = Gravitational Constant. Can be taken as 9.81m/s2.

D = Internal Pipe Diameter

S = Hydraulic Gradient. As in the Manning Formula this is the slope of the pipe (in m/m).

vk = Kinematic Viscosity of the water. This can be taken as 1.139mm2/s for water at around 15°C.

ks = Equivalent Sand Roughness Coefficient. This coefficient describes the internal roughness of the pipe. The value of this coefficient must be derived from hydraulic tests of the pipe materials. European standards state that values of 0.6mm and 1.5mm are used for surface water and foul water drains respectively. These conservative values include allowances for some grit in surface waters and sliming of foul water drains.

While the Colebrook-White Equation is more accurate than the Manning Formula for most design conditions, there are some cases where the Colebrook-White Equation is not well suited. These include corrugated pipes and pipes with significant sediment deposits. The complexity of the Colebrook-White Equation also means that it is only suitable for calculating the water velocity. It cannot be converted to calculate the hydraulic gradient or the pipe diameter when the velocity of the water is known. Approximations of these equations have been developed recently which are suitable for most practical design situations where the water velocity is known.

**Flow in Partially Full Pipes**

In most design standards it is accepted practice to calculate the maximum hydraulic capacity of drainage pipes when they are flowing just full. In actuality the maximum capacity of circular drainage pipes does not occur when they are running full but when the water level is at around 94 percent of the maximum height. This is because friction at the pipe-water interface slows down the water and reduces the flow. So after the 94 percent mark the ratio of flow area to length of pipe-water interface reduces the hydraulic capacity. The difference between the capacity of a circular drainage pipe flowing full and the true maximum flow capacity is around 8 percent.

The main reason this is allowed in drainage design standards is because calculating the true maximum pipe capacity is a more complex calculation and before programmable calculators were widely used it was felt that the simpler more conservative calculation would be sufficient. Nowadays any drainage design software or even a simple drainage design spreadsheet can instantly calculate the true hydraulic capacity of drainage pipes. Where these programs are used it is often justifiable to allow for the true capacity rather than a conservative estimate used only for reasons of simplicity rather than to allow for a specific practical variability.

**Conclusion**

In some cases the designer is not allowed to choose the hydraulic design methodology as it is dictated by the specification or national standards. However in most cases the designer should consider which method is more suitable to the design conditions. In some cases the two formulas are roughly equivalent, but in many cases the Colebrook-White Equation will deliver more accurate results where they are required. Similarly the designer should consider the partially full pipe condition as the full pipe condition specified in most national standards can be quite conservative, both in terms of the flow capacity and the minimum water velocity. The slightly more complex calculations can lead to significant savings where the hydraulic performance of the drainage pipes is critical.

**Stephen Webster** is a Chartered Civil Engineer based in the UK who runs a civil design consultancy and blog dealing with civil and structural engineering design. He can be reached at stephen@civilweb-spreadsheets.com.