A low-cost experiment to understand hydraulic head losses and their effect on flowrates in a network

Abstract

Understanding how head losses affect the flow configuration in hydraulic networks is essential for analyzing both natural and human-made systems, including physiological systems, biomedical devices, and urban infrastructure. In this study, a low-cost experimental system was designed and implemented to visualize and quantify the impact of both major and minor head losses on the flow distribution within a bifurcated network, using a submersible pump, ball valves, flowrate sensors, and components introducing geometric variations. Experimental results were compared with a theoretical model based on mass conservation, mechanical energy conservation, the performance curve of the pump, and correlations obtained from bibliographic data of head loss coefficients, showing good agreement. The proposed low-cost experimental setup is presented as an effective educational tool for introducing key concepts in fluid mechanics to undergraduate students, and shows how in-classroom topics are applied to a simple but real-life hydraulic configuration.

Keywords

Low-cost experiment hydraulic network head loss impact flowrate sensors flowrate estimation

Introduction

Existing educational approaches in fluid mechanics have contributed significantly to improve conceptual understanding of fundamental topics by students. Early hydraulic design projects focused on system-level engineering calculations and infrastructure layout, but generally lacked experimental implementation and real-time validation.¹ Demonstration-based methodologies have been shown to increase students motivation and conceptual engagement, particularly for visualizing laminar and turbulent flow behavior² or the dynamics of waves.³ However, these approaches are predominantly qualitative and rarely incorporate quantitative hydraulic analysis. Real-world experimental projects have highlighted the pedagogical value of connecting theory with practice, although these activities often rely on project-specific implementations that are difficult to reproduce as standardized laboratory platforms.⁴ In contrast, classroom-scale experiments have successfully introduced concepts such as Bernoulli’s equation, Reynolds number, and pressure losses using simple apparatuses,^5–8 yet these systems generally investigate isolated hydraulic effects rather than network interactions. Recent low-cost educational devices investigating pipe-flow parameters or Pitot-tube measurements provide accessible alternatives for teaching fluid behavior, though they are typically limited to single-line configurations or localized measurements.^10–12 In this work, a low-cost experiment, easy to implement under laboratory conditions, is presented, which allows to perform a quantitative analysis of the flow through an hydraulic network, and to compare with the correponding theoretical predictions.

Thanks to the smartphone revolution and the rise of open hardware platforms, low-cost of sensors, microcontroller units and other electronic components are now accessible for educational purposes. In addition, curriculum-oriented reforms integrating sensors, data acquisition, and inductive learning have demonstrated the importance of modern instrumentation in fluid mechanics education.⁹ Reproducible low-cost engineering education platforms have demonstrated the value of combining construction methodology, instrumentation, and experimental validation. The regular incorporation of these new technologies into the teaching of fluid mechanics can help students to strengthen the acquired knowledge and to observe its practical application. The experimental setup that is introduced in this article, includes the employment of flowrate sensors and their digital instrumentation.

In particular, hydraulic head loss is a fundamental concept in fluid dynamics, as it represents the reduction of useful mechanical energy, due to friction and flow disturbances within a conduit. Although this phenomenon is routinely addressed in theoretical coursework, students often lack opportunities to observe its consequences in practical settings, particularly in the context of flow distribution in networks and branched systems. The relationship between flowrate and head loss, specifically at branches located downstream from a bifurcation, determines the behavior of flows in many human-made and natural systems. These includes: (1) biological systems, such as cardiovascular or respiratory networks^13,14; (2) urban infrastructure, such as water supply and sewerage networks¹⁵; (3) residential hydraulics¹⁶; and (4) biomedical devices, in which fluids are employed or transported.¹⁷ Despite its importance, this topic is often underrepresented in practical undergraduate courses of different professional degrees, including physics, chemistry and engineering. To address this issue, we propose a simple and accessible experiment, enhanced with basic digital instrumentation, to help students understand how head loss or pressure drop generated by a pipe component (for instance, a valve or a joint) affects the overall flow configuration in an hydraulic network. The objective is to provide a low-cost, hands-on experimental setup, as an educational tool to illustrate how flowrates in a network respond to changes in component-induced resistance.

An experimental setup was built, consisting of a simple flow circuit with one bifurcation, two valves, two flowrate sensors, and a coupling. The working fluid(water at room temperature) is taken from a reservoir with a submersible pump, circulates through the network, and is discharged back into the same reservoir. The experimental results are compared with predictions from a theoretical model, based on mass conservation, mechancial energy conservation and the performance curve of the pump. Empiric expressions to quantify head losses of the components are also considered. This hands-on approach strengthens students’ understanding of fundamental principles in fluid dynamics and demonstrates their relevance in both natural and engineered systems.

Experimental setup

Flow circuit

The proposed flow circuit is shown in Figure 1. A pump, immersed in a water reservoir, provides a constant volumetric flow $Q_{1}$ through a circular pipe of inner diameter $ϕ_{1}$ and length $L_{1}$ . This pipe is connected to a tee joint, which divides the flow into two branches, each of which has a ball valve (with opening angles $θ_{2}$ and $θ_{3}$ ) and a Hall effect flowrate sensor,¹⁸ and conducts the corresponding volumetric flowrates $Q_{2}$ and $Q_{3}$ . The inner diameters of the circular pipes, composing the two branches, are $ϕ_{2}$ and $ϕ_{3}$ , respectively, whereas their total lengths, without taking into account the elements in the branches (valve and sensor), are $L_{2}$ and $L_{3}$ , respectively. The branch through which $Q_{3}$ flows includes a coupling, formed by a sudden expansion from $ϕ_{3}$ to $ϕ_{exp}$ , and an immediate contraction, going back to $ϕ_{3}$ . Both branches discharge their flows back into the water reservoir.

Figure 1.

Scheme of the flow circuit proposed for the experiment. Variables and parameters are explained in Section “Flow circuit”.

Equipment and cost

The components used in the actual experimental setup are shown in Figure 2. Reservoir, submersible pump, tee joint, ball valves, flowrate sensors, and coupling were placed in the flow circuit as depicted in Figure 1. Both sensors were connected with Dupont cables to a protoboard and an ESP32 module, which was in turn connected to a computer with a USB cable, as illustrated in Figure 2g. An electronic balance and a graduated cylinder were also employed during the calibration procedures of the flowrate sensors.

Figure 2.

Photographs of the main components of the experimental setup: (a) water reservoir, (b) submersible pump, (c) tee joint, (d) ball valve with goniometer, (e) flowrate sensor, (f) coupling, (g) ESP32 module with protoboard and cables, and (h) electronic balance and graduated cylinder.

The cost of the experiment, separated for each of its components and gathered for the total, is presented in Table 1. They correspond to the USD equivalents of the local costs of the components and total. The componets used in the actual experiment correspond to the combination of Common elements and Option 1 elements, while a lower cost alternative is given by substituting Option 1 elements by Option 2 elements. All the components can be obtained through e-commerce sites, such as AliexPress or Amazon, besides local distributors.

Table 1.

Cost of the elements of the experiment. Option 1 elements were employed for the actual experiment, presented in this work, whereas option 2 elements are proposed replacements to reduce the total cost. Common elements are used for both options.

Common elements
Equipment	Cost [USD]
Plastic hose (5 m)	$ 6.00
Ball valves, connectors and PTFE tape	$ 41.10
Flowrate sensor YF_S201 (2 pcs)	$ 7.00
Plastic reservoir (20 L)	$ 2.60
Plastic graduated cylinder (1 L)	$ 25.70
ESP32-DevKitC V4 with	$ 6.90
ESP32-WROOM $-$ 32 module
USB-A to Micro USB data cable (1 m)	$ 1.20
Protoboard	$ 8.70
Dupont cables (10 pieces $\times$ 10 cm)	$ 1.10
Submersible pump LittleGIANT NK-2	$ 180.00
DLT $-$ 4100 electronic balance (4 kg max.)	$ 51.40
Total (Common + Option 1)	$ 331.70
Option 2 elements
Bilge pump NMBP12-G1100 $-$ 40 $-$ 12	$ 12.10
Switched-mode power supply (12 V, 5 A)	$ 15.00
Truper 102317 BASE $-$ 3 electronic	$ 21.80
balance (3 kg max.)
Total (Common + Option 2)	$ 149.20

Theoretical model

Considering the aforementioned configuration (see fig. 1), considering a steady state system and an incompressible fluid, mass conservation dictates that:

Q_{1} = Q_{2} + Q_{3},

(1)

where the volumetric flowrate

Q_{j}

, of either the pump mainstream (

j = 1

) or the discharge branches (

j = 2

and

j = 3

), is related to the corresponding average flow speed

U_{j}

, as follows:

Q_{j} = (π / 4) ϕ_{j}^{2} U_{j} .

(2)

In turn, the application of mechanical energy conservation applied to a control volume, which connects the fluid surface at the reservoir and the outlet of the

j

th branch, yields:

\begin{aligned} \frac{P_{atm}}{ρ g} + \frac{U_{0}^{2}}{2 g} + z_{0} + h_{pump} & = \frac{P_{atm}}{ρ g} + \frac{α_{j} U_{j}^{2}}{2 g} + z_{j} + \sum_{j} h_{f, j} + \sum_{j, i} h_{K, j i}, \end{aligned}

(3)

where

P_{atm}

is the atmospheric pressure,

ρ

is the density of the fluid,

g

is the gravitational acceleration,

U_{0}

is the flow speed at the surface of the reservoir,

z_{0}

and

z_{j}

are the vertical positions of the fluid surface and the discharge of the

j

th branch, respectively, both measured with respect to the bottom of the reservoir, and

h_{pump}

is the hydraulic head supplied by the pump. The coefficient

α_{j}

is the kinetic energy correction factor of the

j

th pipe section. Its value depends on the flow regime, usually considering

α_{j} = 2

for laminar flows and

α_{j} \approx 1.05

for transitional and turbulent flows.¹⁹ The

j

index corresponds to the

j

th pipe section, whereas the

i

index corresponds to a particular element (tee joint, ball valve or coupling) within its corresponding

j

th pipe section. The last two terms on the right-hand side of eq. (3) correspond to the sum of the major head losses

h_{f, j}

, throughout the pipe sections (

j = 1, 2,

and

3

) of the streamline, and to the sum of the minor head losses

h_{K, j i}

, for all the elements (

i =

tee joint, ball valve and coupling) at the same pipe sections of the streamline, respectively. For instance, two control volumes are defined for the hydraulic network depicted in Fig. 1. Both of them start at the free surface of the water reservoir, going down towards the entrance of the pump, being propulsed by the pump to the mainstream (

j = 1

). The first turns right at the tee joint (

j = 2

i =

tee) towards the right-side branch (

j = 2

), going through a ball valve (

j = 2,

i =

ball valve) and the corresponding flowrate sensor, ending at the discharge of the right-side branch (

j = 2

) above the water surface of the reservoir. The second turns left at the tee joint (

j = 3

i =

tee) towards the left-side branch (

j = 3

), going through a ball valve (

j = 3,

i =

ball valve), the corresponding flowrate sensor and a coupling (

j = 3

i =

coupling), ending at the discharge of the left-side branch (

j = 3

) above the water surface of the reservoir.

Major head losses

A major head loss $h_{f, j}$ is due to the shear stress generated by the moving fluid within a straight pipe of the $j$ th section. It is described by the Darcy-Weisbach equation ^16,19, which reads:

h_{f, j} = f_{j} ({Re}_{j}, ϵ_{j}) (\frac{L_{j}}{ϕ_{j}}) \frac{U_{j}^{2}}{2 g}

(4)

where

L_{j}

and

ϕ_{j}

are the length and inner diameter of the

j

th pipe section, and

U_{j}

is the corresponding average flow speed. The friction factor

f_{j}

is a function of the internal roughness

ϵ_{j}

of the

j

th pipe section and of the Reynolds number

{Re}_{j}

, defined as:

{Re}_{j} = \frac{ρ U_{j} ϕ_{j}}{μ},

(5)

where

μ

is the dynamic viscosity of the fluid. Depending on the flow regime, which is identified by means of the value of the Reynolds number, a precise expression is employed to determine the friction factor. For laminar flows, observed when

{Re}_{j} \leq 2300

, it is computed using the theoretical Hagen-Poiseuille equation ,^16,19 which reads:

f_{j, Lam} = \frac{64}{{Re}_{j}} .

(6)

For turbulent flows, observed when

4000 < {Re}_{j}

, the friction factor is computed using the empirical Colebrook-White correlation,^16,19,given by:

\frac{1}{\sqrt{f_{j, Turb}}} = - 2 \log_{10} (\frac{ϵ_{j}}{3.7 ϕ_{j}} + \frac{2.51}{{Re}_{j} \sqrt{f_{j, Turb}}}) .

(7)

Nevertheless, in order to have a single formula for the entire range of the Reynolds number, including the transition zone

2300 < {Re}_{j} < 4000

, a weighted geometric mean can be used,²⁰ reading:

f_{j} = f_{j, Lam}^{α} f_{j, Turb}^{(1 - α)},

(8)

where

α

is the weighting exponent given by:

α = \frac{1}{1 + {({Re}_{j} / 2720)}^{9}},

(9)

which has been obtained by fitting Nikuradse’s experimental data.

Minor head losses

A minor head loss $h_{K, j i}$ occurs when the fluid moves through the $i$ th component, located at the $j$ th pipe section. Components include fittings and couplings, such as diffusers, nozzles, elbows, wye and tee joints, and other elements that are different from straight pipes, such as valves and obstructions. Each minor head loss is estimated with the following expression^16,19:

h_{K, j i} = K_{j, i} ({geometry}_{i}, U_{ref}, U_{j}) \frac{U_{ref}^{2}}{2 g},

(10)

where

K_{j, i}

is the loss coefficient of the

i

th component, being a function of the specific geometry and characteristics of the

i

th component, of the average flow speed

U_{j}

of the

j

th pipe section, and of a reference flow speed

U_{ref}

. In most cases,

U_{ref}

is equal to

U_{j}

, but when the cross-section area of the pipe changes (nozzles or diffusers) or when there is a junction or division into branches (wye or tee joints),

U_{ref}

must be specified.

For instance, for an insert (or threaded) tee joint, in a symmetric configuration for dividing a stream, $K_{j, tee}$ is given by Idelchik²¹:

K_{j, tee} (U_{1}, U_{j}) = 1 + \frac{3}{2} {(\frac{U_{j}}{U_{1}})}^{2},

(11)

where

U_{ref} = U_{1}

is the flow speed of the incoming stream, and

U_{j}

corresponds to the analyzed outgoing branch.

In turn, considering a ball valve at the $j$ th pipe section, $K_{j, bv}$ depends mainly on the angle $θ_{j}$ . Experimental data leads to relations between the loss coefficient and the angle of the ball valve,²¹ such as the following expression:

K_{j, bv} (θ_{j}) = 1.9 [\sin^{- 9} (\frac{π}{2} - θ_{j}) - 1],

(12)

which is depicted in Figure 3. The angle is

θ_{j} = 0

when the valve is fully open, whereas

θ_{j} = π / 2

when the valve is totally closed.

Figure 3.

Head loss of a ball valve $K_{j, bv}$ as a function of the opening angle $θ_{j}$ . Data points found in the literature,²¹ and fitting curve, described by eq. (12).

The loss coefficient of a flowrate sensor of paddle-wheel type,^16,19 such as the ones employed in this experiment, is usually very low, since it does not present a permanent obstruction to the flow. Consequently, for this model, we neglect its contribution to the minor head losses.

Finally, the loss coefficient of a coupling, formed by a sudden expansion and an immediate contraction, placed very close to each other, is obtained as 3 times the addition of the empiric relations for the expansion and the contraction, obtained from experimental data,²¹ correspondingly given by:

K_{j, exp} ({Re}_{j}, ϕ_{j}, ϕ_{exp}) = \frac{26}{{Re}_{j}^{0.9}} + {[1 - {(\frac{ϕ_{j}}{ϕ_{exp}})}^{2}]}^{2},

(13a)

K_{j, cont} ({Re}_{j}) = \frac{27}{{Re}_{j}^{0.9}} + [1 - {(\frac{ϕ_{j}}{ϕ_{\exp}})}^{2}],

(13b)

\begin{aligned} K_{j, cpl} ({Re}_{j}, ϕ_{j}, ϕ_{exp}) & = 3 [K_{j, exp} ({Re}_{j}, ϕ_{j}, ϕ_{exp}) + K_{j, cont} ({Re}_{j})] \end{aligned}

(13c)

where

K_{j, exp}

K_{j, cont}

, and

K_{j, cpl}

correspond to the expansion, contraction and coupling loss coefficients, respectively, and

ϕ_{exp}

is the expansion diameter of the coupling. The data and fitting curves of the expansion and contraction loss coefficients are depicted in Figures. 4(a) and (b), respectively. As it can be observed, the corresponding expressions describe the behavior of the coefficients in the range

1 < Re < 10^{5}

Figure 4.

Head loss of a sudden expansion $K_{j, exp}$ and a contraction $K_{j, cont}$ as functions of the Reynolds number ${Re}_{j}$ . Data points found in the literature,²¹ for different ratios of cross-section area $\in [0.1, 0.6]$ , and fitting curves, described by eqs. (13).

Therefore, the sum of major and minor head losses, for the flow circuit depicted in Fig. 1, are given by:

\begin{aligned} \sum_{j} h_{f, j} & = f_{1} ({Re}_{1}, ϵ_{1}) (\frac{L_{1}}{ϕ_{1}}) \frac{U_{1}^{2}}{2 g} + f_{j} ({Re}_{j}, ϵ_{j}) (\frac{L_{j}}{ϕ_{j}}) \frac{U_{j}^{2}}{2 g}, \end{aligned}

(14a)

\begin{aligned} \sum_{j, i} h_{K, j i} & = K_{j, tee} (U_{1}, U_{j}) \frac{U_{1}^{2}}{2 g} + [K_{j, bv} (θ_{j}) + K_{j, cpl} ({Re}_{j}, ϕ_{j}, ϕ_{exp})] \frac{U_{j}^{2}}{2 g}, \end{aligned}

(14b)

which must be considered in eq. (3). It is worthy to notice that both, major and minor head losses, tend to increase with the average flow speed

U_{j}

, of the corresponding

j

th pipe section, and, as a consequense, with the flowrate

Q_{j}

Details on essential components

Submersible pump

A submersible pump (LittleGIANT model NK $-$ 2), which has a $100 W$ ( $115 V$ at $60 Hz$ and $1.7 A$ ) electric power consumption, has been employed. The manufacturer provides general information and specifications of the pump, among which one can find some data to reconstruct the performance curve.²² This curve relates the hydraulic head $h_{pump}$ with the volumetric flowrate $Q_{1}$ provided by the pump to the mainstream. In Figure 5, the manufacturer’s data and a fitting curve are shown. This curve is given by the following expression:

h_{pump} = - A Q_{1}^{2} - B Q_{1} + C,

(15)

where the coefficients values are

A = 7.285 \times 10^{- 3} m \cdot {min}^{2} / L^{2}

B = 1.232 \times 10^{- 2} m \cdot min / L

and

C = 3.723 m

, for the employed pump.

Figure 5.

Performance curve of the LittleGIANT submersible pump model NK $-$ 2 for water. Data points provided by the manufacturer,²² showing the hydraulic head $h_{pump}$ as a function of the volumetric flowrate $Q_{1}$ , and fitting curve, described by eq. (15).

Flowrate sensors

Two low-cost flowrate sensors YF_S201, paddle-wheel type which operation is based on a rotating mechanisms that employs the Hall effect,¹⁸ were employed in this work. The application and reliability of this kind of flowrate sensors has been successfully tested for water consumption measurements,^23–25 leakage detection,^26,27 and monitoring of waterways.^28–30

Volumetric flowrate

The mass flowrate ${\dot{m}}_{j}$ is related to the volumetric flowrate $Q_{j}$ by means of the density of the fluid $ρ$ , as follows:

{\dot{m}}_{j} = ρ Q_{j} .

(16)

The density was obtained from the mass, measured with an electronic balance (Denver instrument model DLT

-

4100), which can measure up to

4100 g

, and the corresponding volume, determined by a graduated cylinder, with a capacity of

1000 mL

and an empty mass of

205 g

. Water was employed as the test fluid, which led to a density of

ρ = 988.200 \pm 5.097 kg / m^{3}

Also, the mass flowrate can be obtained from the measurement of the accumulated mass $Δ m_{j}$ in the graduated cylinder during a given timelapse $Δ t_{m}$ , reading:

{\dot{m}}_{j} = \frac{Δ m_{j}}{Δ t_{m}} .

(17)

Therefore, the volumetric flowrate is determined with the following expression:

Q_{j} = \frac{Δ m_{j}}{ρ Δ t_{m}} .

(18)

Digital signal

The YF_S201 flowrate sensor has a multi-pole magnet attached to a rotating element, a paddle-wheel that is driven by the crossing fluid. A Hall effect probe sends a step signal (low-high) according to the polarity (positive-negative) of the magnet that is close to it. The signal from the sensor is sent to a microcontroller unit ESP32, through one of its analog-to-digital converter (ADC) pins. From the step signal, we can define a pulse, each time the signal rises from low to high, and we can count the number of pulses $p_{j}$ in a time interval $Δ t_{p}$ , which allows us to determinate the frequency $ν_{j}$ , as follows:

ν_{j} = \frac{p_{j}}{Δ t_{p}} .

(19)

Calibration

Since the angular velocity of the rotating element in the sensor is related to the flowrate that goes through it, then the frequency $ν_{j}$ of the digital signal is related to the volumetric flowrate $Q_{j}$ . This relation is presented explicitly in Figure 6 for the two flowrate sensors employed in this experiment. The time interval was fixed to $Δ t_{p} = 9.5 s$ and 25 measurements of $ν_{j}$ were performed, according to eq. (19), whereas 15 measurements of $Q_{j}$ were performed, according to eq. (18), for each data point in the calibration curve of the flowrate sensor at the $j$ th branch. Once the calibration curve has been obtained for both sensors, we proceed to measure $ν_{j}$ , in order to calculate $Q_{j}$ using the corresponding calibartion curve, for all the experiments. As shown in Fig. 6, both sensors present linear behaviors of the flowrate $Q_{j}$ in terms of the measured frequency $ν_{j}$ , almost throughout the entire tested frequency domain. This linear trends are described by:

Q_{j} = (a_{j} \pm σ_{a j}) ν_{j},

(20)

where

a_{j}

is the slope, of the sensor in the

j

th branch, and

σ_{a j}

is the corresponding calibration uncertainty. The nonlinear behavior of the sensors occurs for frequencies below

ν = 5 s^{- 1}

. As a consequence, flowrates below

Q = 1 L / min

can not be accurately measured with them, since their magnitudes will be always understimated.

Figure 6.

Calibration curves of the (a) YF_S201 flowrate sensor in the $j = 2$ branch and (b) YF_S201 flowrate sensor in the $j = 3$ branch. Data points of the volumetric flowrate $Q_{j}$ as a function of the signal frequency of the sensor $ν_{j}$ , and linear fittings $Q_{j} = (a_{j} \pm σ_{a j}) ν_{j}$ , where $a_{j}$ and $σ_{a j}$ are the slope and its uncertainty, respectively.

Implementation and example results

The whole flow circuit was built with the same kind of pipe, therefore one can consider a single inner diameter $ϕ_{j} = ϕ = 9.525 mm$ , for $j = 1, 2$ and $3$ . Also, a unique roughness $ϵ_{j} = ϵ = 0.4 μ m$ , measured by profilometry at the inner surface of the pipe, was also employed. The lengths of the pipes, according to the notation employed in Fig. 1, are $L_{1} = 142 cm$ , $L_{2} = 235 cm$ and $L_{3} = 145 cm$ . The expansion diameter of the coupling is given by $ϕ_{exp} / ϕ = 5 / 3$ . Additionally, the discharge branches ( $j = 2$ and $j = 3$ ) are both fixed at the vertical position slightly above the fluid surface at the reservoir, so that $z_{2} = z_{3} ≳ z_{0}$ , and the flow speed at the surface of the reservoir, far enough from the discharges, is considered to be negligible, thus $U_{0} \approx 0$ . Experiments were performed by setting the opening angles $θ_{2}$ and $θ_{3}$ of the two ball valves, to specific values, which led to particular measurements of the volumetric flowrates $Q_{2}$ and $Q_{3}$ .

The simultaneous solution of eq. (1) and eqs. (3), one for $j = 2$ and another for $j = 3$ , allows to find the values of the volumetric flowrates $Q_{1}$ , $Q_{2}$ , and $Q_{3}$ , for a given combination of configurations $(θ_{2}, θ_{3})$ of the two ball valves. Therefore, the system of nonlinear algebraic equations, that must be solved to estimate the flow configuration of the circuit depicted in Figure 1, simplifies to:

0 = Q_{1} - Q_{2} - Q_{3},

(21a)

\begin{aligned} 0 & = (\frac{f_{1} L_{1}}{ϕ} + K_{2, tee}) Q_{1}^{2} + (α_{2} + \frac{f_{2} L_{2}}{ϕ} + K_{2, bv}) Q_{2}^{2} - (\frac{π^{2} g ϕ^{4}}{8}) h_{pump}, \end{aligned}

(21b)

\begin{aligned} 0 & = (\frac{f_{1} L_{1}}{ϕ} + K_{3, tee}) Q_{1}^{2} + (α_{3} + \frac{f_{3} L_{3}}{ϕ} + K_{3, bv} + K_{3, cpl}) Q_{3}^{2} - (\frac{π^{2} g ϕ^{4}}{8}) h_{pump}, \end{aligned}

(21c)

Examples of the behavior of the system are presented in Figure 7. The angle $θ_{3}$ of the ball valve in the $j = 3$ branch has been set to four different values $θ_{3} = 0$ , $π / 6$ , $π / 3$ and $π / 2$ , whereas the value of $θ_{2}$ has been varied from 0 (completely open) to $π / 2$ (fully closed). When $θ_{2} = π / 2$ , the ball valve in the $j = 2$ branch is completely closed, therefore $K_{2, bv} \to \infty$ and $Q_{2} \to 0$ , also making $Q_{3} = Q_{1}$ . An equivalent situation occurs when $θ_{3} = π / 2$ , the valve in the $j = 3$ branch is closed, $K_{3, bv} \to \infty$ , $Q_{3} \to 0$ , and $Q_{2} = Q_{1}$ . Obviously, when $θ_{2} = θ_{3} = π / 2$ , no flow is observed and $Q_{1} = Q_{2} = Q_{3} = 0$ . Nevertheless, this all-closed valves configuration must be avoided, since the pump will work at a shut-off condition, leading either to the damage of the pump or to leakage somewhere in the pipe sections or at any component.

Figure 7.

Volumetric flowrates $Q_{1}$ , $Q_{2}$ and $Q_{3}$ (blue, red and yellow, respectively), as functions of the angle $θ_{2}$ of the ball valve at branch 2, for different angles of the ball valve at branch 3: (a) $θ_{3} = 0$ completely open, (b) $θ_{3} = π / 6$ , (c) $θ_{3} = π / 3$ , and (d) $θ_{3} = π / 2$ totally closed. Points represent experimental results, whereas continuous curves were obtained from the solution of eqs. (1) and (3). The uncertainty $σ_{θ} = 5 π / 180$ of $θ_{3}$ was applied on the theoretical solution, and its effect is depicted as the color areas above and below the continuous curves (only noticeable for intermediate values of $θ_{3}$ , since for $θ_{3} = 0$ and $θ_{3} = π / 2$ the effect of $σ_{θ}$ is negligible).

As it can be observed in Figure 2(d), a goniometer (made of acrylic) was incorporated to the handle of each ball valve. The resolution of the goniometer is $10 π / 180 rad$ , leading to an uncertainty of $σ_{θ} = 5 π / 180 rad$ . This uncertainty could be easily incorporated to the experimental values of $θ_{2}$ depicted in Figure 7, since $σ_{θ}$ is shown as an horizontal errobar for each measured value of $θ_{2}$ . Nevertheless, the inclusion of the uncertainty for $θ_{3}$ is not straightforward for this representation, since each subfigure (a–d) corresponds to a single value of $θ_{3}$ . Therefore, the effect of $σ_{θ}$ has been applied to the theoretical solution. Computing a theoretical solution with $θ_{3} \pm σ_{θ}$ , yields an uncertainty band around the $G_{j}$ curve (for $j = 1, 2, 3$ ) obtained with the measured value of $θ_{3}$ . It is also important to mention that, in Fig. 7, the uncertainty bands are only noticeable for intermediate values of $θ_{3}$ , since for the extreme values $θ_{3} = 0$ and $θ_{3} = π / 2$ , the effect of $σ_{θ}$ is negligible. For the experimental values of $Q_{2}$ and $Q_{3}$ , the vertical error bars correspond to the uncertainties that were calculated using the corresponding calibration curves, shown in Figure 6.

In Figure 7, for each subfigure corresponding to a fixed angle $θ_{3}$ , an increase of $θ_{2}$ leads to a growth of the head loss at the $j = 2$ branch, which affects the volumetric flowrate of both branches $j = 2$ and $j = 3$ , thus $Q_{2}$ drops while $Q_{3}$ rises. As well, the mainstream flowrate $Q_{1}$ is monotonically reduced, whereas the hydraulic head of the pump $h_{pump}$ monotonically increases. Experimental results are compared with the numerical solution of eqs. (21), for the volumetric flowrates $Q_{2}$ and $Q_{3}$ . Theoretical curves present the same trends of experimental datapoints, and a relatively good quantitative agreement is observed, mainly for $Q_{2}$ . Closing the ball valve in the $j = 2$ branch (increasing $θ_{2}$ ), leads to an enlargement of the corresponding minor head loss. Despite the reduction of the flowrate $Q_{2}$ that this provokes, an overall augmentation of the hydraulic resistance of the $j = 2$ branch occurs. Since both branches discharge into the water reservoir at the same pressure, the head loss from the entrance of the tee joint to the discharge at the reservoir must be identical for each branch. The consequence is an increase of the flowrate $Q_{3}$ , in order to make the head loss at the $j = 3$ branch attain the same growth that the head loss of the $j = 2$ branch has undergone. Nevertheless, due to the intensification of the total head loss, the hydraulic head provided by the pump $h_{pump}$ also rises, driving to a decrease of the total flowrate $Q_{1}$ .

For a fixed angle $θ_{2}$ , an increase of $θ_{3}$ (from Figure 7a to Figure 7d) leads to a growth of the head loss at the $j = 3$ branch, which affects the volumetric flowrate of both branches $j = 2$ and $j = 3$ , thus $Q_{2}$ rises while $Q_{3}$ drops. Closing the ball valve in the $j = 3$ branch (increasing $θ_{3}$ ), leads to an equivalent analysis of the variation of the head losses and flowrates, as the one previously described for the closure of the ball valve with $θ_{2}$ .

Statistical metrics, for the example results presented in Figure 7, are given in Table 2. For instance, the Root-Mean-Square Error (RMSE) is smaller than $0.51 L / min$ for $Q_{2}$ and $0.88 L / min$ for $Q_{3}$ , for all the presented values of $θ_{3}$ . As well, the Mean Absolute Percentage Deviation (MAPD) is always shorter than 14% for $Q_{2}$ . For $Q_{3}$ , the MAPD is around 137%, for the case of $θ_{3} = π / 3$ , which may be misleading, because the values of $Q_{3}$ are smaller than those of the other examples with $θ_{3} < π / 3$ . In this case, experimental values are below $Q_{3} = 0.57 L / min$ , while the model prediction does not surpass $Q_{3} = 1.15 L / min$ , yielding large values of MAPD. The $Q_{3}$ curve becomes closer to zero, dramatically, as $θ_{3}$ is increased from $π / 6$ to $π / 3$ , increasing the errors sharply, since the experimental values decrease faster than the model preditions.

Table 2.

Root-mean-square error (RMSE) and mean absolute percentage deviation (MAPD) of the model predictions with respect to the experimental results presented in Figure 7.

	RMSE $[L / min]$		MAPD [%]
$θ_{3}$	$Q_{2}$	$Q_{3}$	$Q_{2}$	$Q_{3}$
0	0.4044	0.2852	10.9337	3.6186
$π / 6$	0.2967	0.8747	13.9745	16.0053
$π / 3$	0.3955	0.5747	10.4784	137.2220
$π / 2$	0.5021	0.0000	11.3968	N/A

Discussion

As it can be observed in Figure 1, both branches discharge at a position sligthly above the surface of the water reservoir. Thus, the head loss measured from the entrance of the tee joint to the discharge (at atmospheric pressure) must be equal for both branches. This statement can be analyzed with equations as follows. Taking eq. (2) to write eq. (11) in terms of $Q_{j}$ , then injecting the expresion of $K_{j, tee}$ with $j = 2$ into eqs. (21b) and with $j = 3$ in eq. (21c), and combining the equations to eliminate $h_{pump}$ and $Q_{1}$ , yields the flowrate ratio:

\frac{Q_{2}}{Q_{3}} = {[\frac{3 / 2 + α_{3} + (f_{3} L_{3} / ϕ) + K_{3, bv} + K_{3, cpl}}{3 / 2 + α_{2} + (f_{2} L_{2} / ϕ) + K_{2, bv}}]}^{1 / 2},

(22)

which, even though

f_{2}

is a funcion of

{Re}_{2}

, which depends on

Q_{2}

, and

f_{3}

and

K_{3, cpl}

are functions of of

{Re}_{3}

, which depends on

Q_{3}

, helps us to observe how the distribution of flowrates in the branches only depends on the hydraulic resistance configuration at both branches. Accordingly, any variation in branch resistance, e.g. due to a valve opening angle adjustment, either

θ_{2}

θ_{3}

, modifies the way the total flowrate

Q_{1}

splits into the branches as

Q_{1} = Q_{2} + Q_{3}

Now, following almost the same math procedure as before, but instead of eliminating $h_{pump}$ and $Q_{1}$ , one solves for $Q_{2}$ and $Q_{3}$ and substitute them into eq. (21a), leads us to write:

\begin{aligned} Q_{1} & = {[\frac{(π^{2} g ϕ^{4} / 8) h_{pump} - [(f_{1} L_{1} / ϕ) + 1] Q_{1}^{2}}{(3 / 2 + α_{2} + (f_{2} L_{2} / ϕ) + K_{2, bv})}]}^{1 / 2} \\ + {[\frac{(π^{2} g ϕ^{4} / 8) h_{pump} - [(f_{1} L_{1} / ϕ) + 1] Q_{1}^{2}}{(3 / 2 + α_{3} + (f_{3} L_{3} / ϕ) + K_{3, bv} + K_{3, cpl})}]}^{1 / 2}, \end{aligned}

(23)

where we can establish that the hydraulic head provided by the pump

h_{pump}

and the total flowrate delivered by the pump

Q_{1}

must adjust their values to the configuration of hydraulic resistances of all the network, but they must also obey the performance curve of the pump, given by eq. (15). Accordingly, the total flowrate

Q_{1}

and the hydraulic head

h_{pump}

, are completely constrained by the pump-hydraulic network interaction.

Despite the fact that theoretical results follow similar trends to those of the experimental measurements, there are notorious differences among them. An important source of errors comes from the employment of correlations, that were developed from the data available in the bibliography for the loss coefficients of the components that generate minor head losses.^16,19,21 Particularly, we have to mention the cases of the tee joint and the ball valve. The correlation employed for the coefficient $K_{j, tee}$ , given by eq. (11), does not take into account the geometry and dimensions of the tee joint. In turn, eq. (12), which corresponds to the correlation for $K_{j, bv}$ used to describe the operation of the ball valve , does not consider the flow speed, or equivalently the Reynolds number, as a factor. To diminish the corresponding errors, both components should have been characterized experimentally, in order to have more precise correlations that account for all the relevant parameters.³¹ Nevertheless, our goal was not to characterize the behavior of the components that cause minor losses, such as a tee joint, ball valves, and a coupling. As it can be read in the literature,^32–34 the characterization of flow control devices, such as ball valves, requires a particular experimental setup, including a pressure control device and two calibrated pressure sensors. It would be an interesting educational experiment, but it would be developed as a separated work, that could be complementary to the presented study.

Additional sources of error may arise due to conditions not considered in this work, such as curved pipe paths, which undoubtedly contribute to head loss, and uneven vertical positions of branch discharges above the surface of the water reservoir. Although we attempted to minimize the effects of these situations, they certainly contribute to the discrepancies observed between the experimental results and the theoretical predictions.

Finally, we have to mention that we have performed the calibration of the flowrate sensors, since there is a lack of information, which should have been provided by the manufacturer. The advantage of finding low-cost components comes with a negative consequence, since the corresponding datasheets may not be accurate or include a complete description of the functioning of the devices. Nevertheless, we believe that the exercise of calibrating sensors may present benefits for the students, since they represent a crucial step to obtain rigorous experimental data.

Educational implementation

The low-cost components and the simple flow network proposed for this experiment makes it particularly useful for educational purposes in undergraduate courses, within the curricula of physics, chemistry and engineering degrees. By allowing the student to adjust head losses and measure the resulting flowrates, this hands-on experiment provides a useful platform for teaching and learning fundamental fluid mechanics principles, such as mass and mechanical energy conservation. This approach encourages active learning³⁵ and the use of digital instrumentation,³⁶ providing a clear example of how theoretical fluid mechanics are applied on a practical situation.^37,38

The experiment proposed in this paper allows to study and analize the flow configuration in a simple flow network, with a bifurcation. During the laboratory activity, the professor should show the components of the flow circuit and let students deduce their function. A quick glimpse of the flow bifurcation at the tee joint must lead to a deduction of the mass conservation principle. Following the path of a volume element departing from the resevoir, driven by the pump through the mainstream and the tee-joint, going through one of the branches and its components, and being discharged at the surface of the resevoir, helps to infer the mechanical energy conservation principle. Emphasis must be done on the role of ball valves and flowrate sensors, giving a brief explanation on the physical mechanisms that enable their operation. As well, the professor must describe the performance curve of the pump, focusing on the adjustment of the hydraulic head that it can provide for a given flowrate, and vice versa. After the laboratory activity, the experimental results should be compared to the theoretical curves, deduced from mass conservation and mechanical energy conservation. The theoretical results can be easily obtained by means of software with a solver for nonlinear algebraic system of equations, which can be considered as a take-home project for students. These actions will guide students towards an understanding of the effect of hydraulic head loss and its effect on the flow configuration. For instance, they will deduce how the distribution of flowrates in the branches only depends on the hydraulic resistance configuration at both branches, whereas the total flowrate at the mainstream must obey the performance curve of the pump, as a consequence of the pump-system interaction.

A student questionnaire has been developed to support the educational objectives of the proposed experiment, which is presented as supplementary material. Its main purpose is to evaluate students’ understanding of key concepts explored in the experiment, such as mass and mechanical energy conservation, major and minor head losses, and energy injection by a pump. The questionnaire also encourages critical thinking about the experimental procedure, potential sources of error, and improvements that can be done. It also serves as an assessment tool for the active learning approach, enabling educators to evaluate students’ engagement, comprehension, and the effectiveness of the experiment.

Conclusions

Most existing educational systems focus on isolated concepts, qualitative demonstrations, or simplified flow configurations. Several studies provide low-cost or experimentally accessible platforms; however, few integrate controllable hydraulic resistances, quantitative measurements, and theoretical-experimental comparisons within an hydraulic network. In particular, current educational setups rarely allow students to investigate how competing head losses influence flow redistribution between parallel branches, despite this being a fundamental concept in hydraulic systems analysis. This limitation highlights an important gap in fluid mechanics education, particularly in laboratory environments intended to connect theoretical modeling with measurable hydraulic behavior. The present work addresses this issue through a modular, low-cost experimental setup that integrates a bifurcated pipe network, adjustable hydraulic resistances, digital flow monitoring, and predictive hydraulic modeling. Unlike previous educational systems, the proposed implementation enables students to quantitatively analyze the interaction between major and minor losses, pump-system response, and branch-dependent flow distribution within a reproducible and digitally instrumented laboratory framework.

Its application is suitable for undergraduate stdents from different disciplines, including physics, chemistry, and engineering.

The experiment provides fluid mechanics professors with a tool to introduce and explore:

the mass conservation principle,

the mechanical energy conservation principle,

the performance curve of a pump, and

the effect of hydraulic head loss on the flow configuration of a network.

Additionally, a theoretical model has been developed, using generic correlations to calculate head losses, to estimate the mainstream and branches flowrates, for a given combination of opening angles of the two ball valves, each located at one of the two pipes outgoing from the bifurcation. This theoretical framework, despite its simplicity, provides students with trend curves to compare with the experimental results, and observe the effect of the network parameters and the pump-system interaction over the flowrates and the hydraulic head provided by the pump. Nevertheless, in order to reduce errors, particular behavior curves for each component of the network must be obtained instead of using the generic correlations. As well, it is advisable to perform a calibration procedure, whenever low-cost sensors are employed.

Supplemental Material

sj-pdf-1-ijj-10.1177_03064190261463316 - Supplemental material for A low-cost experiment to understand hydraulic head losses and their effect on flowrates in a network

Supplemental material, sj-pdf-1-ijj-10.1177_03064190261463316 for A low-cost experiment to understand hydraulic head losses and their effect on flowrates in a network by René Ledesma-Alonso and Benito Juárez-García in International Journal of Mechanical Engineering Education

Footnotes

Acknowledgments

We thank Gustavo Armendariz Peña for his support in taking the photographs of the equipment. We acknowledge the Laboratorio Nacional de Soluciones Biomiméticas para Diagnóstico y Terapia (LaNSBioDyT) of the Facultad de Ciencias UNAM, for providing access to its facilities and the technical support from its staff.

ORCID iDs

René Ledesma-Alonso

Benito Juárez-García

Funding

The authors disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported by the Programa de Apoyo a Proyectos de Investigación e Innovación Tecnológica (PAPIIT) of the Universidad Nacional Autónoma de México (UNAM), through the project grant IA105124.

Declaration of conflicting interests

The authors declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Data availability statement

Data sharing not applicable to this article as no datasets were generated or analyzed during the current study.

Supplemental material

Supplemental material for this article is available online.

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Supplementary Material

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