Hydraulics

Note: Once you have read through all the documentation, you can then see business package
Ruralwater models two kinds of rural water supplies: “gravity” and “from a borehole”, also known as “station” (short form for “pump station from borehole”).
Remark in all the images in this page, the vertical dimension has

been enhaced to make it more evident all calculations based on altimetric elevation and hydraulic energy. So, the slopes of the pathways appear very steep, much more than the reality found walking on the ground.

here is a pdf file pdf

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What the software does

The software tackles two distinct kinds of hydraulic problems:

  • the hydraulic design problem - given the required water flow, find the pipes that will provide such water at the minimum cost

  • the hydraulic check problem - given the pipes, find the water flows and perform a control to see whether the working conditions do not exceed the rated limits of the installed hydraulic hardware.

Gravity

The physical system

In a gravity water supply we have a water spring and a network of pipes, shaped as a tree, which brings the water to a number of water reservoirs at a lower elevation than the spring itself.

General concepts

The names in bold below correspond to those used by the classes of the software.

The ‘topology’ of the water supply system
A tree represents the water supply system composed of its pathways from the source to each water reservoir. A pathway represents the trench along which the pipes are laid down.
A pathway has a ‘tail end’ and a ‘head end’ and water flows from tail to head. A pathway has an altimetric profile associated with it.
Since the pathways are laid down as the branches of a tree, then each pathway may have ‘children’ pathways that branch off its head end.
The network is a tree; therefore water flows in a unique path and direction towards each reservoir.

The ‘water demand’ in the water supply system
Water abstraction takes place only at the reservoirs; indeed, we assume that the users live ‘nearby’ the reservoirs. One may imagine that the reservoirs are placed in the centre of the various village neighbourhoods (or near a school, near the market, near the location where the village houses or huts are more scattered). There are no water abstractions along the pathway. Therefore water flow along each pathway remains constant along the pathway itself.

The ‘independent parts’ in the water supply system (a.k.a. the ‘pressure zones’)
Let’s imagine that we have a spring that serves two villages located in a valley, and the second village is further downstream in the valley. In this case the topology of the water supply will be: (1) a pathway from the spring to the ‘nearest village’ and a second pathway from the ‘nearest’ to the ‘farthest village’. In this case, all the water that reaches the second village (the farthest) will flow through the reservoir of the first (the closest) village. This is a common scenario and more complex topologies could also exists, like in the images of the tutorial, where there is a Y-shaped water supply, and one of the reservoirs feeds another village, more far away. Remark: in all such cases, the water reservoir behaves like a ‘spring’ for any other pathway that abstracts water from such reservoir and transports it further away downstream. Therefore a tree of pathways may be split into one or more disjoint pressure zones.
From the point of view of hydraulic equations, every pathway whose head end is a reservoir gives rise to an independent water supply system stemming from each of its child pathways. In fact, from the point of view of solving the hydraulic equations, a reservoir is identical to the spring: it is “what provides water” into the “downstream” pipes (otherwise said: it provides water into the downstream section of the water supply). This independent system is called a pressure zone. Each pressure zone is independent of the others and their hydraulic energy equations are solved separately.
How water flow is controlled in the water supply system
Nothing controls the water flow better than the law of gravity. Since we are talking about reliable solutions and intrinsic technology, with as little maintenance as possible, we avoid using ‘gate-valves’ installed along the pathways and then partially closed to the required amount of water. Instead, the pipes have no gate-valves, the flow flows continuously along the 24 hours and fills the water reservoirs. The water flow is set by the diameters of the commercial pipes and by the continuous flow under gravity. The next section introduces the equations of flow under the pure force of gravity.

The equations that model the system

The gravity water supply is modelled by two kinds of equations:
  • the continuity of water flows.

  • the continuity of hydraulic energy.

Continuity of water flows:
For each pathway, its water flow is the sum of the water flows in its children pathways plus the water flow, if any, required at its head end. If such water flow at the head end exists, then the head end is a reservoir and each of the children pathways represents the source for a “separated” or “independent” water supply.

Continuity of hydraulic energy:
This is the law of Bernoulli. For each pathway, the hydraulic energy at its head end is equal to the energy at the tail end minus the hydraulic friction in the pathway due to its water flow and to the sequence of pipes laid down along the trench.

Example
Imagine you have the following gravity water supply:
  • there is a spring

  • from th spring the pipeline goes to a junction where the pipe branches in two

  • the two branches go to Tank1 and Tank2 respectively.

This is therefore a Y-shaped water supply, a typical configuration where one source, the spring, must be shared by two villages or by two water tanks at the opposite ends of a larger village.
The equations that describe this system are:
  • the conservation of water flows

  • the energy difference in each of the 3 branches is the hydraulic friction in the pipes

These equations are written below:

We start with the conservation of flows:

\[\begin{eqnarray} Q_{Spring\rightarrow{Tee}} = Q_{Tee\rightarrow{Tank1}} + Q_{Tee\rightarrow{Tank2}} \end{eqnarray}\]

Next comes the conservation of energy:

\[\begin{split}\begin{eqnarray} Z_{Spring} - kQ_{Spring\rightarrow{Tee}}^2(\sum_{j=1,...,segments} {\frac{L_j}{D^5_j}}) = H_{Tee} \\ H_{Tee} - kQ_{Tee\rightarrow{Tank1}}^2(\sum_{j=1,...,segments} {\frac{L_j}{D^5_j}}) = Z_{Tank1} \\ H_{Tee} - kQ_{Tee\rightarrow{Tank2}}^2(\sum_{j=1,...,segments} {\frac{L_j}{D^5_j}}) = Z_{Tank2} \end{eqnarray}\end{split}\]
Where the symbols represent:
\(Z_{Spring}\) is the elevation of the spring
\(H_{Tee}\) is the hydraulic energy at the junction of the Y-shaped pathways
\(Z_{Tank1}\) and \(Z_{Tank2}\) are the elevations of the two water tanks where the water flows to
for each segment, \(L_j\) and \(D_j\) represent xxx and \(j\) is the index over the piped segments. A piped segment is a stretch along which the same pipe is used.
Example explaining the piped segments: Imagine to have one pathway is long 1000 meters and that we use a ‘2 inches pipe rated for 60 meters of water pressure’ (short-handed as 2”PN6 pipe) for 600 meters followed by a ‘3 inches with the same pressure rating’ (short-handed 3”PN6) in the remaining 400 meters of the pathway pathlength. Then, we have two segments and the symbols in the above sum would bear the values of:

\(L_1\) would be 600 \(D_1\) would be the internal diameter of a 2 inches pipe \(L_2\) would be 600 \(D_2\) would be the internal diameter of a 2 inches pipe

In the ‘hydraulic design problem’ the unknowns are: \(Q_{Spring\rightarrow{Tee}}\) \(Q_{Tee\rightarrow{Tank1}}\) \(Q_{Tee\rightarrow{Tank2}}\) The known data are all the other symbols. Therefore we have 4 equations and 4 unknowns and the system can be solved.

Check

With reference to the above equations, the hydraulic check problem is the problem of solving the above equation, where the unknowns are the water flows.

Design

This requires much more engineering judgement than the hydraulic check.
Let’s investigate the tree of pipes. In such condition there is an interesting problem of economics. At each pathway, the hydraulic energy at its tail end and head end should remain close to the ground elevation so that the pressure along the pathway remains low and all the pipes may be selected from those commercial pipes belonging to the lowest rated pressure class. If this is the case then each pathway may be solved separately with the technique of the “simplex”. pressures.
However, real life cases may involve hilly profiles. The pathway may cross a deep valley. This is the case when the villages are scattered on the other flank of a valley from that were the water spring is located. In such cases, the values of hydraulic energy at tail and head ends of the pathways must be kept “far above” the ground elevation if a head end lies in the middle of a valley that is crossed by the water supply. In this case an interesting (and difficult) problem of optimization arises: is is preferable to burn more hydraulic energy through friction in the pipe sections with higher pressure rating or the opposite? The solution depends, case by case, from the amount of water required and from the elevation profile along the pathways. This problem of optimization cannot be solved using the “simplex” method. In fact, it is not a “convex problem” due to the discontinuities introduced by the existence of both discrete diameters and discrete pressure ratings. See :ref: algorithms_gravity_design how this translates in an algorithm.
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Borehole’s station

I link with the module gravity now i link with the class GravityDesign and even the method solve_hydraulic_design_problem()

I want to refer to the equations (system_of_equations) and to the algorithm (the_algorithm)

We now move to the second typology of rural water supply system: a small pumping station that connects a borehole, with a water reservoir which is far from the borehole and usually at higher that that of the site where the borehole is located.

We deal with two kinds of borehole stations: with and without electric power. The reason is because this software is dedicated to rural solutions and, as such, it must consider the case where electric power is not available.

Remark: with ‘not available’ here we mean an arrangement that works entirely *without electric power. With ‘electricless’ we mean a solution that is not *making use of electric power because it is preferred to have a rural plant

Where electric current is not available, it can nonetheless be produced by an electric generator set.

However, when we speak of ‘electricless’ pumping station, we mean that the presence of electric power is intentionally avoided (and so it is avoided to use an electric generator) despite it can always be generated even in remote area by a generating set. The logic is that electric power is both a difficult subject and it introduces complexity in the hardware and the need for spare parts.

The key point is that electric power might be tricky in a rural context. Knowledge about how to run the maintenance of an electric pumping station is hard and could be not available locally; perhaps the spare parts could be difficult to find on the local market. For these reasons, this software allows you to design a pumping station composed of electricity-free parts, and more precisely: a diesel engine with manual start, that drives directly a pump, with no need of electric motor in the pump (since it is driven directly by the engine through a system of belts and pulleys) nor of electric start in the diesel engine itself (which is indeed started manually with a handle keyed on the crankshaft). Readers who have experience in humanitarian water supply interventions in rural areas of the world will recognise in this ‘electricless’ station the arrangement with a Listeroid diesel engine and a monopump. We provide more information about such equipment in these pages: LINK engine and LINK pump.

  • the ‘electric’ arrangement, where we may use electric power.

IMPORTANT: this a work in progress so the sections below are now kept as stubs for future implementation.

Electric Station

Check problem

TODO

Design problem

We examine the case of a feeder. The feeder entails an interesting problem of economics.

Will it be preferable to adopt small bore feeders (and therefore limit the cost of pipes) and consequently have a high hydraulic energy to overcome (and therefore install more expensive pumps) or adopt the opposite solution (larger pipes and smaller pumps)?

Electricless Station

Check problem

TODO

Design problem

TODO