Applications of Hardy Cross Method

1. Introduction

In this example, some of the parameters are given to us for the calculations and analysis of the hydraulic pipe network systems. These are the basic data need to necessary for conducting the analysis. The all the units given are specifically US based. The numbers of houses (unit) are 308. The average day demand per unit is 1140 gallon per unit. The maximum day factor is 1.75 % of AD. The fire flow is 1750 gallon per min.

2. System Total Demand Water Demand

For the analysis, based upon the above given data we need to compute some parameters. 1. The average day demand in gallon per minute is calculated by multiplying number of houses (unit) i.e. 308 and average day demand per unit i.e. 1140 and dividing this whole from 24 (1 day = 24 hrs) multiply by 60 to convert the final value into gpm i.e. 243.83 gpm.

2. The maximum day demand in gallon per minute is computed by simply multiplying the average day demand which we have calculated above i.e. 243.83 with the maximum day factor i.e. 1.75 which comes out to be 426.70 gpm
3. Another parameter is the demand per junction in gallon per minute. It is obtained by dividing the maximum day demand i.e. 426.70 to number of junctions in the system. Here in our case it is 8. So the value comes out to be 54 (round off).
4. The system demand in gallon per minute is calculated by adding the demand per junction i.e. 54 and the fire flow i.e. 1750 gpm which comes 2177 gpm.

3. Pipe Network Configuration

Figure1. Shows the pipe network system configuration

Fig.1 shows the pipe network for the hydraulic analysis by using Hardy Cross method. From the above figure, the numbers of junctions identified are 8. And they are named as J, D, E, F, G, H, I, and K. There are total 10 pipes are present in the system. The 10 pipes in the system are JD, DE, KE, JI, KI, EF, KG, FG, GH, and HI. So per the theory taught in the class considering the pipe junctions are in parallel. Based on that, there are 3 arbitrary loops in the system. The loop 1, loop2 and loop 3 consist of JDEKI junctions, IKGH junctions KEGF junctions respectively. The loop 1 comprises of the pipe JD, DE, EK, IK and JI while the loop 2 has pipe IK, KG, HG and IH. Comparing loop1 and loop2, both sharing one common pipe called IK. The loop 3 consists of pipes EF, KG, KE and GF. Loop 3 sharing KE pipe with loop 1 and KG pipe with loop 2.

For carrying out the calculations, we need certain pipe characteristics. They are diameter of pipe in inches denoted by D, length of pipe in foot denoted by L and the Hazen-William number denoted by C.

The pipes highlighted in red, green and blue are shared pipes which other loops.

Table 1 Pipe characteristic in loop 1

Table 2 Pipe characteristic of loop 2

Table 3 Pipe characteristic of loop 3

4. Methodology and Analysis

The Hardy Cross method is an iterative method for determining the flow in pipe network systems where the inputs and outputs are known, but the flow inside the network is unknown. The method was first published in November 1936 by Shinn Taniguchi its namesake, Hardy Cross, a structural engineering professor at the University of Illinois at Urbana–Champaign. The method of carrying out the analysis is divided into 4 steps. First, the identification loops into the system. Secondly, assume positive direction in each loop. The third one is assigning the arbitrary direction of flow in each pipe and assigns flow discharge to each pipe. The last one is always satisfy the mass conversation in the system.

Analytically, the first step is all about calculating head loss in each loop by the Hazen-William formula.

h_L=r Qⁿ

n=1.85

r=(10.44L)/(c ^1.85 D ^4.87)

In second step, computing the R in each loop

R = n r Q ^{n − 1}

R= 1.85 (h_L/Q)

Note: – R by definition must positive value.

∑hL and ∑R in each loop

Note: – Goal is to have ∑hL=0

In third one, calculate the correction for flow discharge per loop, ∆Q = -∑hL/∑R

In forth step, update the value of Q

Q_adj = Q_prev + ∆Q _loop

If there is a share pipe, Q_adj = Q_prev + ∆Q _loop – ∆Q _{shared loop}

So, iterations are needed

5. Results from analysis

From the above methodology the results have been calculated into 15 iterations. For conducting the analysis, an excel sheet have been used by putting the formulas into it. The computation of results for every iterations has been shown in the below tables.

Table 4 Showing results for iteration 1

Table 5 Showing results for iteration 2.

Table 6 Showing results for iteration 15.

Conclusion

The hydraulic analysis of pipe network of the given example has been by using the Hardy cross method. With this method we can determine the flow discharge, pressure head and head-losses in a system. By changing the pipe characteristics like diameter, length and Hazen-Williams constant, the head losses and flow discharges at various junctions can be computed very easily. With the above methodology, the analyzing of pipe flow networks easily done.