Minimization of Total Pumping Cost from an Aquifer to a Water Tank, Via a Pipe Network
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Minimization of Total Pumping Cost from an Aquifer to a Water Tank, Via a Pipe Network N. Nagkoulis 1
& K.L. Katsifarakis
1
Received: 22 April 2020 / Accepted: 31 August 2020/ # Springer Nature B.V. 2020
Abstract
In this paper, we have investigated minimization of total cost to pump a given flow rate from any number (n) of wells up to a water tank, under steady-state flow conditions. Regarding groundwater flow, we have considered infinite or semi-infinite aquifers, to which the method of images applies. Additional regional groundwater flow can be taken into account, too. The pipe network connecting the wells to the tank can include junctions at the locations of the wells only. Moreover, all pumps have equal efficiency. We have derived a new analytical formula, which holds at the critical points of the total cost function. Based on this formula, we derived a system of n equations and n unknowns, to calculate the well flow rate combinations which correspond to the critical points of the total cost function. The n-1 equations are 2nd degree polynomials, while the remaining one is linear, expressing the constraint that the sum of well flow rates must be equal the required total flow rate. The solution of the system can be achieved using commercial solvers. Moreover, we have concluded that there is one feasible solution that minimizes the total cost. Finally, we present a tabulation process to facilitate the use of solvers and we provide and discuss two illustrative examples. Keywords Hydraulic friction losses . Groundwater flow . Optimization . Pipe network . Pumping cost . Water resources
1 Introduction Water is directly and indirectly vital to humanity. From drinking water, sanitation and agriculture to building infrastructures, manufacturing and religion, water is shaping our everyday life. Moreover, not only historically “the availability of water … has been considered an essential part of a civilized way of life in different periods” (Vuorinen et al. 2007), but still water scarcity is one of the main problems of humanity, as its lack is considered to accelerate
* K.L. Katsifarakis [email protected]
1
Department of Civil Engineering, Aristotle University of Thessaloniki, Thessaloniki, Greece
Nagkoulis N., Katsifarakis K.L.
diseases’ spread and to cause millions of deaths every year (Montgomery and Elimelech 2007). Climate change seems to aggravate water scarcity on a global scale. In this framework, optimal management of water resources, including both the supply and the demand side of the water balance (Rijsberman 2006) is a high priority issue. For this reason, the respective bibliography is quite large. A substantial part of it is dedicated to groundwater resources. The respective problems can be broadly classified in two groups: a) maximization of pumping rate under physical or financial constraints and b) minimization of pumping cost of the total flow rate from a system of wells (probably combined with other cost items). This group includes also aquifer restoration problems, where groundwater pu
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