# D. M. Mohan's Modified Hazen-Williams Formula for Head Loss

**Design of Pressure Pipelines:**

This article is intended to help the hydraulic design of pressure pipelines flowing under pressure wherein the hydraulic gradient over its entire length lies above the crown soffit of the pipes. The Hazen-Williams formula has been commonly in use for such designs.

**Hazen-Williams Formula:**

Where,

h = head loss (m)

D = diameter (m)

V = velocity (m/s)

C = Hazen-Williams C-factor

L = length (m)

k = 6.79 for V in m/s, D in m

The C-factor ranges between 0 to 150 depending on the material and age of the pipe.

**Limitations of Hazen-Williams Formula:**

The commonly used Hazen-Williams Formula has following inherent limitations.

The numerical constant in Hazen-Williams formula (1.318 in FPS units or 0.85 in MKS units) has been calculated for an assumed hydraulic radius of 1 foot and friction slope of 1/1000. However, the formula is used for all ranges of diameter and fiction slopes. This practice may result in an error up to +/- 30% in the evaluation of velocity and +/- 55% in the estimation of frictional resistance head loss.

The Darcy-Weisbach formula is dimensionally consistent. The Hazen-Williams coefficient C is usually considered independent of pipe diameter, velocity of flow and viscosity. However, to be dimensionally consistent and to be representative of friction conditions, it must depend on relative roughness of pipe and Reynolds's number. A comparison between estimates of Darcy-Weisbach friction factor 'f', and its equivalent value computed from Hazen-Williams C for different pipes materials brings out the error in estimation of 'f' up to +/- 45% in using Hazen-Williams formula. It has been observed that for higher "C' values (new and smooth pipes) and larger diameter, the error is less, whereas it is appreciable for lower 'C' values (old and rough pipes) and lower diameters at higher velocities.

The Hazen-Williams formula is dimensionally inconsistent, since the Hazen-Williams 'C' has dimension of L^(-0.37) T^(-1) and therefore is dependent on units employed.

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**Modified Hazen-Williams Formula:**

In *1986*, *Dr. D. M. Mohan* submitted thesis in partial fulfillment of the requirements for the degree of Doctor of Philosophy at the *Center for Environmental Science and Engineering of Indian Institute of Technology, Bombay*. In this research study, *Dr. Mohan*, proposed the Modified Hazen-Williams Formula, stated below:

where,

V = velocity of flow in m/s,

Cr = pipe roughness coefficient, ( 1 for smooth pipes; < 1 for rough pipes),

r = hydraulic radius in m,

S = friction slope,

h = friction head loss in m.

**Discussions and Advantages of Modified Hazen-Williams Formula:**

The Modified Hazen-Williams Formula has been derived from Darcy-Weisbach and Colebrook-White equations and obviates the limitations of Hazen-Williams formula.

Analysis carried out to evaluate effect of temperature (viscosity) on value of Cr reveals that the maximum variation of Cr for a temperature range of 10 Degree Celsius to 30 Degree Celsius is 4.5% for a diameter of 2000 mm at a velocity of 3 m/s. In the light of this revelation, Cr values are presented for average temperatures of 20 Degree Celsius which are okay for practical purposes.

The use of Hazen-Williams C results in under utilization of new pipe material. The extent of under-utilization varies from 13 to 40 percent for CI (metallic) pipes; 23 percent for RCC and AC pipes; and 8.4 percent for HDPE and PVC pipes.

The resulted values of head loss using Modified Hazen-Williams formula are nearer to head loss resulted using Darcy-Weisbach formula when compared to Modified Hazen-Williams formula.

**Government Bodies/ Manuals that recommend use of Modified Hazen-Williams Formula for Design of Water Networks, in India:**

**Manual on Water Supply and Treatment by Central Public Health and Environmental Engineering Organization, Ministry of Urban Development, Government of India****Guidelines for Planning and Design of Piped Irrigation Network, Central Water Commission, Government of India****State Government Water Boards and Public Health Engineering Departments (PHEDs) in India****Urban Local Bodies, Rural Water Boards and Urban local bodies/ Municipal Corporations in India.**

*©Copyright DTK Hydronet Solutions, 2019*

*(This article is purely based on the study by Dr. D. M. Mohan and we duly acknowledge the inventor's contribution, DTK does not claim any Intellectual Authority over Modified Hazen-Williams Formula discussed in the above article.)*