Design And Analysis Of Pipe Systems: Hardy Cross Method

Friction Intensity Factor for Pipe Systems

The piping system for the friction intensity constraint for every pipe should always be known, and it should be free of the flow situations for the scope of flow states of intrigue. In the event that the distribution water condition is utilized to relate head loss h_L to velocity V or flow rate Q (Spiliotis and Tsakiris, 2010), the representation of friction intensity factor is denoted by f, this has a steady value on a specific pipe for completely turbulent flow, however it does not imply on the transitional or laminar flow. In the event that the Hazen Williams equation (Kumar, Narasimhan and Bhallamudi, 2010) is utilized, the friction intensity factor is CHW, which is thought to be identified for a specific pipe.

The Hardy cross method used in analyzing the pipe network system illuminate the nonlinear conditions associated with network investigation by making certain assumptions. The higher power rectification terms can be dismissed and the loop number is little for a solitary loop despite the fact that the underlying guess is weak. However, dismissing contiguous loops and considering just a single amendment condition at once can influence the arrangement and furthermore number of iteration required for joining increments as the measure of the system increments.

Altered Hardy Cross strategy can be connected to enhance merging and lessen the quantity of loops. In any case, this number can be very substantial for genuine systems. Consequently, rather than considering just a single amendment condition at once, all the adjustment conditions can be illuminated by thinking about the impact of every single contiguous circle. So joining can be accomplished in fewer loops. Additionally, a portion of the conditions engaged with pipe network investigation is nonlinear.

Design and analysis of pipe systems are imperative to undeveloped town or cities, not just in light of the fact that water is an essential monetary improvement parameter, yet in addition since water is a central factor later on of peace. 

1. To determine pipe discharges, Q
2. To determine nodal heads, H
3. To determine pipe resistance constants, R
4. To determine nodal inflows or outflows, q  

Considering a network that has different loops, at normal circumstances there will be channels regular to bordering loops with a clockwise stream in one loop showing up as unfriendly to clockwise in the other. Each loop must be distinguished and the rectifications made efficiently to each loop thus. The remedy to the streams must be made each time before proceeding onward to the following loop. For in excess of two loop in a system, the procedure turns out to be extremely intricate and computer strategies should be utilized. 

In consideration of analyzing pipe  network system, the conventionally approach is known as the Hardy Cross procedure (Huang, Vairavamoorthy and Tsegaye, 2010). This strategy is appropriate if the entire pipe sizes (lengths and breadths) are settled, and either the head losses between the outlets and inlets are known yet the flow are not, or the flow at each inflow and overflowing point are known, yet the head losses are definitely not. This last case is investigated straightaway.

The system incorporates making a guess with respect to the flow to rate in each pipe, taking consideration of making a guess to such an extent that the total flow into any crossing point approaches the total flow out of that convergence. By then the head loss in each pipe is found out, in perspective of the normal flow and the picked flow versus head loss relationship. Next, the system is checked whether the head loss around each loop is zero. Since the fundamental flow were speculated, this will undoubtedly not be the circumstance. The flow rates are then adjusted with the end goal that continuity will in any case be fulfilled at each crossing point, aside from the head loss around each loop is more similar to be zero. This strategy is repeated until the point that the progressions are attractively little. The definite procedure is according to the following

The Hardy Cross Method for Pipe Network Analysis

Divided the network into loops

  For each loops done the fallowing steps

1. Assumptions on the flow, , flow course in the pipes, direction of flow in the loop where positive will be taken to be clockwise or negative will be taken to be counterclockwise, with an application of  equation of continuity condition at every node. Evaluated pipe flows are associated with iteration until head loss in the clockwise direction is equivalent to the counterclockwise bearing in each loop.
2. The equivalent resistance K for each pipe will be required to be calculated based on the given parameters on the demand for each node, similarly pipe length and diameter, together with temperature and finally pipe material are expected to be unique if not it is assumed to be equal.
3. Calculation for each pipe. The sign from procedure 1 is retained  and computation is done for the  sum of the loops h_f
4. Computation of  h_f⁄Q for each and every pipe and the summation  for each and every loop
5. Calculation of the  correction by the fallowing formula.
6. Application of correction to Q_new= Q+ΔQ
7. Repeat procedure (3) to (6) until Δh become very small.
8. Finally solving of the total pressure at each and every node using energy method

Continuity Formula

The sum of pipe amount of flows into and out of the respective nods equals to the amount of flow that is entering or leaving the system through each node (Cunha and Sousa, 2010).

Hence, from the statement it means that the following equation will be resulted: Q_Total = Q₁ + Q₂

Where,

Q = Total inflow, Q1 + Q2= Total outflow

Energy conservation formula

The total algebraic Summation of head loss h_f  around any closed loop is zero (Giustolisi, 2010).

Therefore, _f(loop) = 0

 Where,

Q= Actual inflow,

ΔQ= Correction

 K= Head loss coefficient,

n= Flow exponent.

Always the following formula should be used for general relationship between discharges and head-losses for each pipe in loops:

h_f = k*Qⁿ

3.0 Hazen-William equation

K =

 n = 1.87 

Assume the C for all pipes = 100

(1/K)^1.85 =  

Where,  and flow rate is in l/s and diameter is in meters 

Q_{iteration 2loop 1}=  

Q_{iteration 2loop 2}=   

The last condition gives a way to deal with determining an estimation of ?Q which will influence the estimation of the head loss wherever the loop to be zero. For the underlying couple of loops, that iteration is likely not to be correct, so the registered estimation of ?Q won’t influence the value of head loss around the loop to be definitely zero, anyway it will influence the head loss to be closer to zero when contrasted with the past loop. The estimation of ?Q would then have the capacity to be added to the main estimations of Q for each one of the pipe distribution network of loops, and iteration can be finished. This same strategy can be used for each one of the circles in the system. In case a pipe is a bit of no less than two particular loops, the change factors for each one of the loop that contain it are associated with it. 

As represented already, the figure gauge of the flow rates is totally optional, as long as movement is satisfied at each convergence. On the off chance that one makes great conjectures for these flow rates, the issue will consolidate quickly, and in case one influences poor guess, it will take more loops for the last course of action is found. Regardless, any assessments which meet the mass change model will finally provoke the same, and amendment will be done on the  last result.  

Loop	Pipe	Diameter	Length	(1/K)^1.85	Assumed Q	Q^1.85	H	H/Q	Correction	New Q
(m)	(m)	(L/s)		(m)	(m/L/s)	L/s
A	ab	0.2	300	0.00453	30	540.35	2.4499469	0.081664897	-0.536235	29.4638
	bc	0.2	300	0.00453	13.5	123.34	0.55922356	0.041423967	-0.27507998	13.2249
	cda	0.25	600	0.00306	-40	-920.05	-2.8171931	0.070429828	-0.24	-40.24
Total							0.19197736	0.193518692
B	bc	0.2	300	0.00453	-13.5	-123.34	-0.5587302	0.041387422	0.27507998	-13.225
	bef	0.2	500	0.00756	15.6	161.17	1.21796169	0.078074467	-0.26115502	15.3388
	cf	0.25	160	0.00082	-35.2	-726.28	-0.59337076	0.016857124	-0.26115502	-35.461
Total							0.06586073	0.136319013

Second iteration

Loop	Pipe	Diameter	Length	(1/K)^1.85	Q	Q^1.85	H	H/Q	Correction	New Q
(m)	(m)	(L/s)		(m)	(m/L/s)	L/s
A	ab	0.2	300	0.00453	29.463765	522.62	2.36955908	0.080422821	-3.13432	26.32944
	bc	0.2	300	0.00453	13.22492	118.73	0.53832182	0.04070511	-2.001841	11.22308
	cda	0.25	600	0.00306	-40.53624	-930.29	-2.84854798	0.070271646	-3.13432	-43.67056
Total							0.05933292	0.191399577
B	bc	0.2	300	0.00453	-13.22492	-118.73	-0.5378469	0.040669214	2.001841	-11.22307
	bef	0.2	500	0.00756	15.338845	156.21	1.18047897	0.076960095	-1.132479	14.20637
	cf	0.25	160	0.00082	-35.46116	-736.28	-0.60154076	0.016963372	-1.132479	-36.59363
Total							0.04109131	0.134592681

Third iteration

Loop	Pipe	Diameter	Length	(1/K)^1.85	Q	Q^1.85	H	H/Q	Correction	New Q
(m)	(m)	(L/s)		(m)	(m/L/s)	L/s
A	ab	0.2	300	0.00453	26.329445	424.44	1.92441096	0.07308969	-3.13432	23.19512
	bc	0.2	300	0.00453	11.223079	87.64	0.39735976	0.035405592	-2.001841	9.221238
	cda	0.25	600	0.00306	-43.67056	-1068.76	-3.27254312	0.074937062	-3.13432	-46.80488
Total							-0.9507724	0.183432345
B	bc	0.2	300	0.00453	-11.22307	-87.64	-0.3970092	0.035374372	2.001841	-9.221233
	bef	0.2	500	0.00756	14.206366	135.55	1.02435135	0.072105094	-1.132479	13.07389
	cf	0.25	160	0.00082	-36.59363	-780.37	-0.63756229	0.017422765	-1.132479	-37.72611
Total							-0.01022014	0.124902231

Sewerage systems are designed and built to give the services of collection, diverting, treatment and transfer of sewage and reuse of treated waste water. The design  of sewerage includes design of sewer lines that limit blockage and negligible disintegration of sewer channels sub-current of gravity. Pumped sewerage is debilitated as the cost of pumping sewer is high. The sewer ought to be outlined in such a way they can accomplish self-purging speed once a day with a most extreme of 3.0m/s to maintain a strategic distance from the disintegration of sewer dividers and channel.it ought to likewise approach openings (sewer vents) at the particular separations for adjusting of the sewer lines on the off chance that there are blockages

The Hazen Williams Equation for Pipe Flow

A sewer system is a system of pipes used to pass on storm spillover and additionally sanitary sewer in a city.

The design of sewer framework includes the determination of, diameter, incline slope, and invert rises for each pipe in the framework.

Free surface flow exits for the design discharge; o that is, where the flow is by the gravitational force; pumping stations and pressurized sewers ought not to be considered.

The sewers are of commercially accessible sizes.  The design distance across should be less economically accessible pipe having flow limit equivalent to or more noteworthy than the plan release and fulfilling all the suitable limitations.

Sewers must be set at a profundity with the end goal that they o won’t be vulnerable to ice, o will have the capacity to deplete storm cellars, and o will have adequate padding to forestall breakage because of ground surface stacking. o To these closures, least cover profundities must be determined.

The sewers are joined at intersections with the end goal that the crown rise of the upstream sewer is no lower that of the downstream sewer.

To avoid or lessen exorbitant affidavit of strong material in the sewers, a base passable stream speed at configuration release or at scarcely full-pipe gravity stream is determined.

To anticipate scour and other unwanted impacts of high-speed stream, a greatest passable stream speed is additionally indicated.

At any intersection or sewer vent, the downstream sewer can’t be littler than any of the upstream sewers at that intersection.

The sewer framework is a dendritic, or spreading, network converging the downstream way without shut loop  

Average sewer flow is calculated based on consumption and population

Average sewage flow Q =  0.8 * consumption

Q_design = 2*peak factor * Q + infiltration (10%) + storm water (100% of peak flow)

Design equation using Manning`s formula (Vongvisessomjai, Tingsanchali and Babel, 2010) for sewage flowing under gravity

V =  R^2/3 * S^1/2

Where,

V = velocity of flow in m/sec

R = hydraulic mean depth

S = slope of the sewer

n = coefficient of roughness for pipes (n = 0.013 for RCC pipes)

Cleansing velocity => for partially combined sewer = 0.7 m/sec

Maximum velocity used should not be greater than 2.4 m/sec, to avoid abrasion

Minimum sewer size to be used 225 mm to avoid chocking of sewer with bigger size objects through the man hole

Minimum cover to be used = 1 m to avoid damage by live loads on sewer

1. Determination of present population of projected area
2. Drawing of the system layout while considering the streets and road layout.
3. Identification of the sewer line and numbering of  the manhole.
4. Allocate plots to each sewer line
5. Measurement of the sewer line length as per scale of the map provided
6. Adopt the per capita sewage flow as 70% of water consumption and calculate the average sewage flow and infiltration for all the sewer line., at this point take infiltration as 10% of the average sewage flow
7. Calculation of peak sewage flow and design flow for the sewer lines (Hvitved-Jacobsen, Vollertsen, and Nielsen, 2010)
8. By the usage of back calculation determine the appropriate diagram and sewer in assumption that the sewer is fully flowing
9. In the end find the invert levels for all the sewer
10. Draw the  profile for all  sewer line

	Present  year 2018	Design period 2038
Plots	7	10
Apartments	400	600
Flats	200	400

Assume the number of plots in the area = 280

Assume the number of apartment  in the area = 3

Number of flats in the area = 3

Take the design period to be = 20 years

Present population (Pd) = 7 * 280 + 400 * 3 + 200 *3 = 3767

Annual population growth rate for Australia = 2.1%

Population density  in 2038  Pd = 3767(1 + 2.1%)²⁰  = 5709

Pd = 10*281 + 600 * 3 + 400 * 3 = 5810 

Per capita water consumption = 300 lpc + 103 plc = 403 plc

Design and Analysis of Pipe Systems in Undeveloped Towns and Cities

Average sewer flow = pd * pcwc * 0.8/1000

                             = 5810 * 403 * 0.8/1000 = 1873 m³/day

= 0.0217 m³/s

Peak factor = 4

A = (D/4)²

V = 1/N * r^2/3 * S^1/2

r = a/p = d/4

q = a*v

Q = 0.0217 m³/s

s =0.001

n = 0.015 

d^8/5 = 0.033022

d = 0.004267 M

d = 0.4 (take)

v = q/a

v = 0.0217/a

v = 1517 m/s which is greater than 0.6 m/s 

Determination of manhole distance and pipe slope

Procedure

1. Demonstrate the inverted height on profile for each pipe entering and leaving the sewer manhole at within sewer manhole wall.
2. Since the design sewer pipe has a diameter of 0.004m and is less than 1.2 m then the following will be considered, the distance between center to center is 100 m
3. a) The pipe distance will be determined by subtracting one half inside the dimension of on which the sewer manhole, for both manhole from the total distance between the centerline of both manholes.

In the event that we take the two sewer manhole are having a most extreme of 2.0 m measurement, at that point the aggregate separation distance between the two sewer manhole with respect to centerline to centerline is 100m, and the separation distance between the centerline of the sewer manhole to the inside wall of the sewer manhole is 1.0 m. 

Since the two sewer manhole are a similar width, subtract 2.0 m from the aggregate separation distance  100 – 2= 98 m, the pipe remove between sewer manhole will be 98 m however a separation distance of  100 m ought to be appeared on the profile.

1. b) To decide the pipe slope, subtract the two sewer manhole inverts and divide the difference by the pipe distance and multiply by one hundred (100) to acquire the percent review of the pipe.

If the manhole invert elevations are 52.5 m as per the contours for one manhole and 50 for the other, then the difference between the two manhole inverts will be 2.5.00 m

Take the invert difference 5 .00 m and divide it by the pipe distance (98 m). The pipe slope will be 0.025 m per hundred meters or 2.5%. Show the pipe slope on the profile. 

Table having assumption of values  

From manhole	To manhole	Length	Area increment	Coefficient C	Reduction in Area	Cumulative reduction in area	Rainfall intensity	Q	Ground surface upper end	Lower end	invert   upper end	lower end		slope of sewer (%)
1	2	100	1.5	0.7	1.05	1.05	<10	2275.5	10	9.8	8.6	8.3		2.5
2	3	100	1	0.7	0.7	1.75	<10	1517	9.8	9	8.4	7.5		8.5
3	4	100	0.9	0.7	0.63	2.38	10	1365.3	9	9	7.4	7.2		1.7
Total		300
4	5	100	0.6	0.9	0.54	0.54	12	910.2	9	9.2	7	6.9		2.5
5	6	100	0.8	0.9	0.72	1.26	12	1213.6	9.2	9	6.8	6.7		3.2
6	13	50	0.4	0.4	0.16	1.42	13	606.8	9	9	6.2	6.1		1.3
Total		250
7	8	100	1.5	0.4	0.6	0.6	<10	2275.5	9.3	9.1	7.9	7.7		3.3
8	4	100	0.8	0.7	0.56	1.16	<10	1213.6	9.1	9	7.7	7.3		2
9	10	100	1.5	0.4	0.6	1.76	<10	2275.5	9.2	8.8	7.9	7.6		1.1
Total		300
10	5	100	0.9	0.9	0.81	0.81	<10	1365.3	8.8	9.2	7.4	7.2		4
11	12	100	1.5	0.1	0.15	0.96	<10	2275.5	9.8	8.7	7.6	7.2		3.3
12	6	100	0.8	0.4	0.32	1.28	<10	1213.6	8.7	9	7.1	6.8		1.4
Total		300

sketch of a portion of sanitary sewer line profile

1^st manhole 2^nd manhole

pipe size = 0.004 m

2.5 slope = 2.5% size of the man hole extreems = 2.0

98 m

100 m

elevation difference = 60 = 55 = 5

Conclusion

The  pipe discharges, Q was determined from the use of Hardy cross method when the value of head loss tend to be zero, as 23.19512 l/s, 9.221238 l/s and -46.80488 l.s for loop A and for loop B, the flow rate is -9.221233 l/s, 13.07389 l/s and -37.72611 l/s.

The head loss at every nodal was determine from the first iteration and second iteration and the summation of head loss was  both 0.05933292 for loop A and 0.04109131 for loop B. 

References

Cunha, M.D.C. and Sousa, J.J.D.O., 2010. Robust design of water distribution networks for a proactive risk management. Journal of Water Resources Planning and Management, 136(2), pp.227-236.

Giustolisi, O., 2010. Considering actual pipe connections in water distribution network analysis. Journal of Hydraulic Engineering, 136(11), pp.889-900.

Huang, D., Vairavamoorthy, K. and Tsegaye, S., 2010. Flexible design of urban water distribution networks. In World Environmental and Water Resources Congress 2010: Challenges of Change (pp. 4225-4236).

Hvitved-Jacobsen, T., Vollertsen, J. and Nielsen, A.H., 2010. Urban and highway stormwater pollution: Concepts and engineering. CRC press..

Kumar, S.M., Narasimhan, S. and Bhallamudi, S.M., 2010. Parameter estimation in water distribution networks. Water resources management, 24(6), pp.1251-1272.

Spiliotis, M. and Tsakiris, G., 2010. Water distribution system analysis: Newton-Raphson method revisited. Journal of Hydraulic Engineering, 137(8), pp.852-855.

Vongvisessomjai, N., Tingsanchali, T. and Babel, M.S., 2010. Non-deposition design criteria for sewers with part-full flow. Urban Water Journal, 7(1), pp.61-77.