WATER TRANSMISSION LINES
6.5. CHOICE BETWEEN GRAVITY AND PUMPING SYSTEMS
A pumping system can be adopted in any type of topographic configuration. On the other hand, the gravity system is feasible only if the input point is at a higher elevation than all the withdrawal points. If the elevation difference between the input point and the with-drawal point is very small, the required pipe diameters will be large, and the design will not be economic in comparison with the corresponding pumping system. Thus, there exists a critical elevation difference at which both gravity and pumping systems will have the same cost. If the elevation difference is greater than this critical difference, the gravity system will have an edge over the pumping alternative. Here, a criterion for adoption of a gravity main was developed that gives an idea about the order of magnitude of the critical elevation difference (Swamee and Sharma, 2000).
6.5.1. Gravity Main Adoption Criterion
The cost of gravity main Fgconsists of the pipe cost only; that is,
Fg¼ kmLDm: (6:49)
The head loss occurring in a gravity main is expressed as
hf ¼ z0 zL H ¼ 8fLQ2
p2gD5: (6:50)
Equation (6.50) gives the diameter of the gravity main as
D¼ 8fLQ2
p2g zð 0 zL HÞ
15
: (6:51)
Equations (6.49) and (6.51) yield
Fg¼ kmL 8fLQ2 p2g zð0 zL HÞ
m5
: (6:52)
Similarly, the overall cost of the pumping main is expressed as
FP¼ kmLDmþ kTrgQh0, (6:53a) and the pumping head of the corresponding pumping main can be rewritten as
h0¼ H þ zL z0þ8fLQ2
p2gD5: (6:53b)
Using Eqs. (6.53a) and (6.53b) and eliminating h0, the optimal pipe diameter and optimal pumping main cost can be obtained similar to Eqs. (6.9) and (6.11) as
D ¼ 40 kTrfQ3 The second term on the right-hand side of Eq. (6.54b) is the cost of pumping against gravity. For the case where the elevation of entry point z0is higher than exit point zL, this term is negative. Because the negative term is not going to reduce the cost of the pumping main, it is taken as zero. Thus, Eq. (6.54b) reduces to the following form:
Fp¼ kmL 1þm optimality criteria for a gravity main to be economic is derived as
z0 zL H .L
Equation (6.56) states that for economic viability of a gravity main, the left-hand side of inequality sign should be greater than the critical value given by its right-hand side. The critical value has a direct relationship with f and km. Thus, a gravity-sustained system becomes economically viable by using smoother and cheaper pipes. As m , 2.5, the critical elevation difference has an inverse relationship with Q. Therefore, for the same topography, it is economically viable to transport a large discharge gravitationally.
6.5. CHOICE BETWEEN GRAVITY AND PUMPING SYSTEMS 129
Equation (6.56) can be written in the following form for the critical discharge Qc for which the costs of pumping main and gravity main are equal:
Qc¼ L
For a discharge greater than the critical discharge, the gravity main is economic. Thus, (6.57) also indicates that for a large discharge, a gravity main is economic.
Example 6.6. Explore the economic viability of a 10-km-long cast iron gravity main for carrying a discharge of 0.1 m3/s. The elevation difference between the input and exit points z02 zL¼ 20 m and the terminal head H ¼ 1 m. Adopt kT/km¼ 0.0185 units.
Solution. Adopt m ¼ 1.62 (for cast iron pipes), g ¼ 9.8 m/s2, and f ¼ 0.01. Consider the left-hand side (LHS) of Eq. (6.56), z02 zL 2H ¼ 19 m. On the other hand, the right-hand side (RHS) of Eq. (6.56) works out to be 11.48 m. Thus, carrying the discharge through a gravity main is economic. In this case, using Eq. (6.52), Fg¼ 1717.8km, and using Eq. (6.55), Fp ¼ 2027:0km. The critical discharge as computed by Eq. (6.57) is 0.01503 m3/s. For the critical discharge, both the pumping main and the gravity main have equal cost. Thus, LHS and RHS of Eq. (6.56) equal 19 m; and further, Eqs. (6.52) and (6.55) give Fg¼ Fp¼ 503:09km.
EXERCISES
6.1. Design a cast iron gravity main for carrying a discharge of 0.3 m3/s over a distance of 5 km. The elevation of the entry point is 180 m, whereas the elevation of the exit point is 135 m. The terminal head at the exit is 5 m.
6.2. Design a ductile iron pumping main carrying a discharge of 0.20 m3/s over a dis-tance of 8 km. The elevation of the pumping station is 120 m and that of the exit point is 150 m. The required terminal head is 5 m. Use iterative design procedure for pipe diameter calculation.
6.3. Design a ductile iron pumping main carrying a discharge of 0.35 m3/s over a dis-tance of 4 km. The elevation of the pumping station is 140 m and that of the exit point is 150 m. The required terminal head is 10 m. Estimate the pipe diameter and pumping head using the explicit design procedure.
6.4. Design a multistage cast iron pumping main for the transport of 0.4 m3/s of water from a reservoir at 150 m elevation to a water treatment plant situated at an elevation of 200 m over a distance of 100 km. The water has n ¼ 1.0 1026m2/s and r ¼ 1000 kg/m3. The pipeline has 1 ¼ 0.25 mm, m ¼ 1.6 and hb¼ 60 m. The terminal head H ¼ 10 m. The ratio kT/km¼ 0.025, and hc¼ 160 m.
6.5. Design a multistage cast iron pumping main for carrying a discharge of 0.3 m3/s from a river intake having an elevation of 100 m to a location at an elevation of 1050 m and situated at a distance of 25 km. The pipeline has 1 ¼ 0.25 mm and hb¼ 60 m. The terminal head H ¼ 4 m. The ratio kT/km¼ 0.019 units, and E/km¼ 15,500 units.
6.6. Explore the economic viability of a 20-km-long cast iron gravity main for carrying a discharge of 0.2 m3/s. The elevation difference between the input and exit points z02 zL¼ 35 m and the terminal head H ¼ 5 m. Adopt kT/km¼ 0.0185 units.
REFERENCES
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Swamee, P.K. (1996). Design of multistage pumping main. J. Transp. Eng. 122(1), 1 – 4.
Swamee, P.K. (2001). Design of high-rise pumping main. Urban Water 3(4), 317 – 321.
Swamee, P.K. and Sharma, A.K. (2000). Gravity flow water distribution network design. Journal of Water Supply: Research and Technology-AQUA, IWA. 49(4), 169 – 179.
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