Hydraulic calculations for architectural fountains: nozzles, piping and water pumps.

The purpose of this article is to attempt to solve the problems faced by designers, planners and installation workers of architectural water features when they take on a new project. Between when the water is driven by the pump units and when it exits through the nozzles it passes through a series of pipes and components that cause pressure losses. It is essential to be able to calculate these losses in order to successfully tackle a project for an architectural water feature.

This article sets out a series of theoretical and practical calculations made on a spreadsheet, which is freely available, from which the optimal results necessary for resolving the hydraulic calculations of an architectural water feature can be obtained.Read More

The height of a water jet from a nozzle depends on its type (lance, cascade, geyser, foam jet, etc.), its flow and the pressure at its base. As regards pressure variations along the length of a pipe, it is known that pressure drops when elevation or height is increased with regard to an outflow point from a tank or a pump discharge. Pressure also drops in proportion to the distance traveled by the water and due to the presence of accessories: elbows, valves, etc.

Bernoulli's equation can be used to calculate pressure at any point along a pipe. If, for example, sub-index 1 in figure 3.1 is the pump's outflow point and 2 is the point at the base of a nozzle, the relationship between the elevations, velocities and energy losses can be expressed as being caused by the effects of a pipe's length and accessories:

All of the terms in the above equation are automatically expressed in m.w.c. if the elevations are in meters, the velocities in m/s, the pressures in m.w.c. and losses along straight sections and in accessories are in m.w.c. It should be mentioned that if the pressures are in Pascals or multiples of Pascals, the height value equivalent to the pressure can be obtained by multiplying by the equivalence 1 Pa ≈ 1. 02 * 10^-4 m.w.c for a water temperature of 20ºC. Examples: What water height value corresponds to a pressure of 1 KPa? Answer: 1 KPa = 1 000 Pa*1. 02 * 10^-4 = 0. 102 m.w.c. What water height value corresponds to a pressure of 1 MPa? Answer: 1 MPa = 1000 000 Pa*1. 02 * 10^-4 = 102 m.w.c.

In summary, the variables in Bernoulli's equation are as follows:

1. The elevations of the points of the pipe axes between which the calculations are made. In the example in figure 3.1, Z₁ and Z₂.
2. The mean flow velocities at the points of the pipes between which the calculations are made. In the example in figure 3.1, V₁ and V₂. If the diameter of the pipe remains the same, V₁ = V₂, if there is no flow output or input in the section that is considered.

3. Pressures expressed in meters of water column (m.w.c.) at the points of the pipes between which the calculations are made. In the example in figure 3.1, p₁ and p₂. Pressures relative to atmospheric pressures tend to be used. In other words, gage pressures expressed in m.w.c., e.g.: A pressure of 1 kg/ cm² at a point is approximately equivalent to 10 m.w.c. A pressure of 400 kPa is approximately equivalent to 400*1000 Pa*1. 02 * 10^-4 = 40 *1. 02 40. 8 m.w.c. In architectural water features, it is necessary to calculate pressures at the bases of the different nozzles installed along the length of a pipe. To do this, the following aspects first need to be determined: elevations Z₁ and Z₂., velocities V1 and V2., pressure at a start point (for example, the pump outflow) and pressure losses. A spreadsheet can be used to organize these calculations.
4. Pressure losses (energy per unit of weight of the fluid). When water flows along a pipe and through elbows and valves etc., energy losses occur due to resistance to movement. Energy losses due to water flowing through straight sections of pipe (hf) depend on the interior roughness (ε) of its material (PVC, brass, stainless steel, galvanized steel, etc.), on length (L), on interior diameter (D) and on the velocity of the water (V). In the International System of units, it is expressed in meters of water column (m.w.c.). In turn, energy losses in accessories or "local" losses (hl) depend on the type of accessory: elbows, valves, etc. In the International System of units, this is also expressed in meters of water column (m.w.c.).

Pressure losses in a straight section of pipe can be calculated using different expressions. These include the expressions of Hazen- Williams, Chezy, Manning and Darcy- Weisbach. The Darcy-Weisbach equation is used in this manual, due to its more general nature:

Where:

• f: Darcy- Weisbach friction factor. Depends on the nature and temperature of the liquid and its Reynolds number. The Reynolds number is the quotient of the product of the velocity through the interior diameter of the pipe and of the kinematic viscosity of the fluid at its flowing temperature.
• L: length of the straight section of piping. All of the straight segments forming part of the section of interest tend to be added together.
• D: interior diameter of the piping.
• V: flow velocity.

Pressure losses in each accessory of a pipe can be calculated using the following expression:

Where:

• K_accessory: coefficient depending on the type of accessory: 90º elbow, 45º elbow, valve, etc.
• V: flow velocity.

An indirect way of calculating pressure losses in accessories is by using the concept of the equivalent length of the accessories. In this case, it can be obtained using tables or the following expression: L_equivalent = K _accessory *D/ f, the lengths of straight pipe corresponding to each accessory. The sum of all the equivalent lengths and the total length of the straight sections is used as the length for calculating energy losses in a pipeline.

Performing Bernoulli equation calculations with the help of a computer

The use of different IT resources, widely available nowadays, makes it possible to perform hydraulic calculations for architectural water features in a more precise, quick and efficient manner. Computers allow you to free yourself from tedious and excessively time consuming "traditional" graph-analytical calculations, so you can concentrate on the details of the esthetics of your architectural water feature, on different alternatives for nozzle systems, on different possibilities for combining water supply networks with your groups of nozzles and on the choice of pumps, etc.

In general, spreadsheets can be used to perform calculations for architectural water features along with computer programs such as EPANET. The Blog shows how Excel can be used to perform calculations to solve the most common water feature-related problems. If you require calculations for more complex water features, you may benefit from a copy of the free EPANET software. Explanations of the software and solutions for numerous practical examples are provided in the book entitled "Hydraulics of architectural water features and hydraulic installations" by the author.

Spreadsheets.

A spreadsheet can be thought of as a digital piece of squared paper, each cell of which can contain text, numbers, calculation formulas and photos, etc. Each cell is identified by the letter of its column, followed by the number of its row. Figure 3.2 is a diagram showing an Excel spreadsheet.

The equations entered into the yellow cells are shown in the box with blue letters, superimposed over the Excel screen shot, figure 3.2. These equations are not visible in the normal spreadsheet view, as the numerical result of the calculation in each cell is displayed to the user.

A certain advantage of the use of computerized spreadsheets over manual procedures with calculators is that once the required spreadsheet has been "built", multiple variations on the calculation can be made reliably in a shorter time.

Spreadsheet applications allow different spreadsheets to be created in a single file, which can then be linked to each other. In other words, a personalized "book" of calculations can be created. Figure 3.3 shows a book of pipe calculations comprising spreadsheets for "Water properties", and "Calculating h_f”, etc.

Calculation spreadsheets are made by different companies, and some are open source. Excel spreadsheets are the tool of choice in this manual, as it is a powerful and user-friendly calculation program included in various Microsoft Office packages.

Figure 3. 4 show the Excel spreadsheets that are free to download, in which the equations required to reliably and quickly obtain the calculations for pressure loss in pipes have been programmed.

Figures 3. 5 to 3. 7 show examples of the use of an Excel book, with a practical example of a simple fountain.

Download the EXCEL Spreadsheet used in this post.