Fountain hydraulics and calculations

Fountain hydraulics and calculations

From nozzle flow and pressure to path loss and determining the working point of the pump

Hydraulic design is the relationship between the expected image of the fountain and the equipment needed to create it. The height of the fountain, the thickness of the flow, the uniformity of the nozzles and the speed of the change of effects, all depend on the behavior of the water in the system.

A water fountain may have the right equipment, but due to the wrong choice of pipe diameter, high loss of connections or mismatch between the pump and the network, it may not achieve the desired result. On the contrary, careful calculation can avoid selecting a pump larger than needed, excessive energy consumption, and difficult setups during start-up.

In this chapter, the method of calculating the fountain pressure circuit will be introduced and an educational example will be provided from the nozzle requirement to the required flow rate and head of the pump. The detailed selection of nozzles and pumps will be completed in later chapters.

20–1. Let's start the calculation from the hydraulic circuit

Before using the formulas, the path of the water must be determined:

Water source → suction path → pump → transfer path → collector → branch → nozzle → receiving basin and return path

For each section, it should be known how much water passes through it, what is the difference in height and which equipment is placed in the path.

In many projects, the display circuit and the filter circuit are not the same. If the filtration works in an independent circuit, the filter loss of that circuit should not be added to the display pump head. Also, the flow rate of the purification circuit does not add up to the flow rate of the nozzles without checking the connection method.

In the initial map, the location of the free water surface, the minimum and maximum level of exploitation, the location of the pump and the connection point of the nozzles should be specified. This map is the basis for subsequent calculations.

Figure note from the source manuscript

Figure 1-20: Schematic of display circuit with open tank, pump, collector and multiple nozzles; The independent filter circuit is displayed in a different color. Calculation reference points and flow direction are clear. Suggested subtitle: "Knowing the actual path of water is the first step in calculating flow rate and head."

20–2. Basic quantities and units

Commonly used conversions are:

$1L/s=3.6m^3/h$

$1m^3/h\approx 16.67L/min$

For water with approximate density $1000kg/m^3$:

$H_p=\frac{p}{\rho g}$

Therefore, a pressure of 1 bar is approximately equivalent to 10.2 metres of water column.

Pump head expresses the energy added per unit weight of fluid and is not simply the height of the pump installation or the height of the fountain. Head definition in KSB reference ^[117]

In calculations, gauge pressure and absolute pressure should be different from each other. This difference is especially important in the study of suction and cavitation.

20–3. Flow rate and flow velocity in the pipe

The continuity relationship for water flow in a section is:

$Q=Av$

For a filled circular tube:

$A=\frac{\pi D^2}{4}$

So:

$v=\frac{4Q}{\pi D^2}$

In these relations, $D$ must be the actual inside diameter. The nominal diameter or outer diameter of the pipe, regardless of the material, thickness and product category, is not a substitute.

With flow constant, reducing the inside diameter increases velocity and usually increases path loss. Enlarging the diameter should also be compared with cost, installation space and operating conditions.

The appropriate speed for all sectors is not a fixed value. The suction path, the main line, the nozzle branch and the return circuit have different conditions.

20–4. The relationship between fountain height and output speed

For an ideal vertical jet, neglecting air resistance and other losses, the rise height relative to the nozzle opening is obtained from the following equation:

$h_{ideal}=\frac{v_0^2}{2g}$

Therefore:

$v_0=\sqrt{2g{h}_{ideal}}$

For an ideal height of 4 meters:

$v_0=\sqrt{2\times 9.81\times 4}\approx 8.86m/s$

This result only shows the exit speed in the ideal model. It is not possible to directly select the project pump.

In practice, nozzle internal drop, flow shape, air entrainment, jet break and wind conditions affect the result. Also, this model is not sufficient to determine the behavior of foam nozzles, water curtains or multi-directional effects.

The basis of selection should be the documented relation of flow rate, pressure and water effect of the same nozzle. In the GSO collection, comet, foam, cascade, laminar and control nozzles have been introduced in different groups; It is not suitable to calculate all these groups with a simple jet model. Types of nozzles and urban fountain equipment GSO ^[118]

20–5. Correct reading of nozzle information

At least the following information must be specified for each nozzle:

• Product code and exact size;

• The height or extent of the desired effect;

• Flow rate required for that effect;

• The required pressure and its measurement reference location;

• installation conditions and possible dependence on the water level;

• Permissible operating range.

"Nozzle Inlet Pressure" is not the same quantity as "Fountain Height". If the pressure required to create the effect has been taken from the nozzle data, the height of the fountain should not be added back.

It should also be clarified whether the manufacturer's data is related to the inlet static pressure or the required total head. If the total header is declared, re-adding the input speed header will cause an error.

In the calculation sheet, it is better to write its reference location next to the pressure value; For example: "Gauge pressure at nozzle inlet connection, before nozzle body".

20–6. The equation of energy and required head of the pump

For stable and incompressible flow, between two points 1 and 2, we can write:

$H_{pump}=(z_2-z_1)+\frac{p_2-p_1}{\rho g}+\frac{{\alpha }_2{v}_2^2-{\alpha }_1{v}_1^2}{2g}+h_L$

In this regard:

• $z$: height compared to the base level;

• $p/\rho g$ : pressure head;

• $\alpha v^2/2g$: speed head with kinetic energy correction factor;

• $h_L$: the sum of the path drops between two points.

For many basic turbulent flow calculations, $\alpha$ is assumed to be approximately one; This assumption should be checked according to the circumstances. The basis of separation of pressure head, height, velocity and drops is also explained in the KSB head reference. Pump head and system head

If point 1 is the free surface of a large open tank, the gauge pressure and velocity are almost zero. If point 2 is nozzle inlet:

$H_{pump}=\Delta z+\frac{p_{nozzle}}{\rho g}+\frac{v_{inlet}^2}{2g}+h_L$

In this selection of points, the internal drop of the separate nozzle is not added; The required inlet pressure is taken from the performance characteristics of the nozzle.

The depth of placement of the pump below the water level should not be added to the height difference regardless of the reference points. The location of the pump is important for the suction condition, but the height difference in this equation is calculated from the free surface of the tank to the selected end point.

20–7. Pipe friction loss

For a full circular pipe, the Darcy-Weissbach relation is as follows:

$h_f=f_D\frac{L}{D}\frac{v^2}{2g}$

where $f_D$ is the Darcy friction coefficient, $L$ is the length and $D$ is the inner diameter of the pipe.

The friction coefficient depends on the flow condition and relative roughness. The Reynolds number is calculated from the following equation:

$Re=\frac{vD}{\nu }$

For turbulent flow in a pipe, one of the common relations to determine the friction coefficient is Colebrook's relation:

$\frac{1}{\sqrt{f_D}}=-2{\log }_{10}(\frac{\epsilon }{3.7D}+\frac{2.51}{Re\sqrt{f_D}})$

In this regard, $\epsilon$ is absolute roughness and $\nu$ is kinematic viscosity. It is usually solved iteratively.

The Darcy factor is not the same as the Fanning factor. Using the wrong factor can make the calculated head loss four times its correct value or one quarter of it. KSB head-loss reference ^[119]

20–8. Loss of connections, valves and equipment

The local loss of a component can be calculated from the following equation if it has a valid coefficient:

$h_m=K\frac{v^2}{2g}$

The reference speed must be compatible with the definition of the factor $K$. For equipment with a pressure drop curve, such as a strainer or control valve, it is better to use the data of the same product at the corresponding flow rate.

Three important points are:

1. If the speed is different in different sections, the drop of each section is calculated with the speed of the same section. 2. The loss of a joint should not be recalculated with both the $K$ factor and the equivalent length. 3. The drop of the equipment in a clean state and in a state requiring service can be different.

In the actual circuit, the tee, transformer, check valve, strainer and regulating valve should be checked according to the arrangement and flow conditions.

20–9. Calculation of multi-channel networks

In a network with parallel nozzles:

$Q_{total}=\sum Q_i$

But the drop head of all branches is not added together. Each path from the source to the nozzle must be checked separately; The path with the highest head requirement in design conditions is the critical path.

For similar nozzles at the same height, the difference in the resistance of the branches can make the flow rate and the height of the fountains different. Hydraulic balance should be predicted with appropriate network geometry and adjustment method.

In the length of the collector, the flow rate changes after each branch. The assumption that the total flow passes through the entire length of the collector is not enough for accurate calculation.

The timing of the nozzles must be obtained from the actual display program. If all nozzles are allowed to fire, reducing the design flow rate by a nominal empirical factor can disrupt the performance of the same scene.

Figure note from the source manuscript

Figure 2-20: Collector with multiple branches; Determine the flow rate of each part and critical path. Suggested caption: "Parallel branch flow accumulates; The head is obtained by examining the flow paths.

20–10. Complete example of flow and head calculation

This example is purely educational. Nozzle specifications, roughness, fittings and equipment drop are hypothetical and do not belong to the actual product or project.

We assume that a system has 12 identical and parallel nozzles that operate at the same time. The requirement of each nozzle in the selected effect:

• Flow rate: 30 liters per minute;

• Head static pressure gauge at the entrance: 8 meters of water column;

• Nozzle inlet height compared to the free surface of the tank: 1 meter.

In this example, the filter circuit is independent and its drop is not included in the display circuit.

A) Total discharge

$Q_{total}=12\times 30=360L/min$

$Q_{total}=6L/s=0.006m^3/s=21.6m^3/h$

b) Critical path specifications

The coefficients $K$ in the table are the hypothetical input data of the example. In the project, they should be calculated from the actual list of connections.

Fluid and pipe assumptions:

$\rho =1000kg/m^3$

$\nu =1.0\times 10^{-6}{m}^2/s$

$\epsilon =0.00001m$

By calculating the speed and iteratively solving Colebrook's relation:

c) Falling of pipes and connections

For example, main line friction loss:

$h_{f,main}=0.01862\times \frac{30}{0.075}\times \frac{1.358^2}{2\times 9.81}\approx 0.700m$

Path results:

To complete the example, the collector drop in the critical path is assumed to be 0.30 m and the flat drop in the calculation state is 0.40 m. These two values ​​are not included in the coefficients of the previous table.

Therefore:

$h_L=0.029+0.091+0.700+0.564+0.268+0.159+0.300+0.400$

$h_L\approx 2.511m$

It is assumed that other branches are balanced to provide equal flow and the selected path remains critical.

d) Velocity head at the nozzle inlet

The internal diameter of the pressure reference location in this example is 25 mm:

$\frac{v_{inlet}^2}{2g}=\frac{1.019^2}{2\times 9.81}\approx 0.053m$

e) required head of the pump

$H_{required}=1+8+0.053+2.511$

$H_{required}\approx 11.56m$

So the computational requirement of this situation is equal to:

$Q=21.6m^3/hH\approx 11.6m$

This result is the network requirement in the assumed state. The selected pump curve should also be checked to determine the actual operating point. At this stage, the engine model or rated power has not been selected yet.

20–11. Network curve and pump operating point

The pump does not deliver the flow rate written in the calculation sheet independently of the network. The working point is obtained by meeting the pump curve with the network demand curve. Definition of work point in reference KSB ^[120]

In a fixed arrangement, in some ranges, the network curve can be expressed approximately as follows:

$H_{system}=H_{static}+C{Q}^2$

This approximation depends on the arrangement and behavior of the circuit components being fixed. The required pressure of the nozzle also usually changes with the flow rate; Therefore, the value of 8 m for example is not a constant component for all discharges.

Different modes should be considered for the design: all nozzles active, part of nozzles active, minimum tank level, and increased flatness loss.

Also, the "Max Flow" and "Max Head" listed for the pump are usually not achieved at the same point. The pump must provide the required flow at the required head.

Figure note from the source manuscript

Figure 3-20: Schematic diagram of pump curve and two network curves, one at base condition and one with higher drop. Determine the location of intersections. Suggested caption: "Changing network resistance shifts the actual operating point of the pump." Chart titled "Educational; without assigning to the product" should be inserted.

20–12. Hydraulic power, axis power and electric power

Power transferred to water:

$P_h=\rho gQH$

For example this chapter:

$P_h=1000\times 9.81\times 0.006\times 11.564\approx 681W$

If the assumed efficiency of the pump at this point is 60%:

$P_{shaft}=\frac{P_h}{{\eta }_p}=\frac{681}{0.60}\approx 1135W$

With an assumed motor efficiency of 88% and drive 96%:

$P_{electrical}=\frac{P_h}{{\eta }_p{\eta }_m{\eta }_d}\approx 1.34kW$

These efficiencies are merely educational assumptions. Actual values ​​should be derived from equipment specifications and operating point.

The rated power of the motor should not be selected just from rounding the above result. The required power in the entire permissible working range, installation conditions, control method and manufacturer's limitations should also be checked.

20–13. Control of suction and cavitation conditions

Providing adequate flow and head does not complete the selection without checking suction.

The two main quantities are:

• NPSH_A: value available from the system;

• NPSH_R: required pump value according to the manufacturer's data and test criteria.

For an open tank and a small water surface velocity:

$NPS{H}_A=\frac{p_{atm,abs}}{\rho g}+(z_{surface}-z_{ref})-h_{L,suction}-\frac{p_v}{\rho g}$

In this regard, atmospheric pressure and vapor pressure are absolute, and the reference level must be in accordance with the definition of the pump manufacturer.

The height of the place above the sea level, water temperature, the lowering of the tank level and the increase of the suction drop have an effect on the result. The comparison should be made with the appropriate margin determined for the application; A fixed distance cannot be prescribed for all pumps. NPSH concept and criteria in KSB reference ^[121]

NPSH control is not a substitute for checking for adequate submersion, eddy, and air entrainment. In the example of this chapter, because the pump and location conditions are not known, NPSH numerical control is not done and the final choice is open.

20–14. Changes in rotational speed, system configuration and transient flow

For a pump with a constant geometry, at the corresponding points and assuming the efficiency to be approximately constant, the similarity relations are as follows:

$\frac{Q_2}{Q_1}\approx \frac{n_2}{n_1}$

$\frac{H_2}{H_1}\approx (\frac{n_2}{n_1}{)}^2$

$\frac{P_2}{P_1}\approx (\frac{n_2}{n_1}{)}^3$

These relationships express the displacement of the corresponding points of the pump curve. To find the actual function after the round change, the new curve should be intersected with the network curve. Pump analogy rules, KSB ^[122]

Therefore, reducing the circumference to 80% does not necessarily increase the actual height of the fountain to 64%; Level differences, drops and nozzle behavior are also involved.

In a show fountain, rapid closing of valves and sudden changes in flow can create transient pressures. The steady-state calculations of this chapter are not sufficient to fully evaluate these modes. Valve operation time, pump response, path length and the need for ram impact analysis should be considered separately.

20–15. Return circuit, water storage and calculation documents

The water exiting the nozzle must be collected and returned to the circuit. The design of the return path depends on the levels, slope, collection geometry and flow conditions. The gravity return pipe should not be automatically calculated like a full pipe under high pressure.

At the start and stop of the show, some of the water may move in the pipes, basins and return lines. The volume of the tank should cover these changes, the minimum working level, the return water after stopping and the necessary free space. The pump flow alone does not determine the volume of the tank.

Circulation flow rate is also different from fresh water consumption. Compensatory water should be calculated on the basis of evaporation, splash discharge, operational discharges and other losses.

Figure note from the source manuscript

Figure 4-20: Two schematic cross-sections of the system on and off, showing the tank level and the water in the paths. Suggested subtitle: "The storage volume should cover the change in water distribution during startup and shutdown."

The final calculation book should include circuit map, nozzles data, internal diameter of pipes, path drop table, operating conditions, pump and network curves, suction check and power calculations. Assumptions and unverified data should also be exposed so that they can be replaced with factual information before implementation.

In Chapter 21, fountain nozzles and their selection principles are reviewed; From the structure of the jet and the shape of the spray to reading the performance table, depending on the water level and selecting the nozzle according to the wind, height and type of project.