# Consequences of the Integration of a Hyperbolic Funnel into a Showerhead for Droplets, Jet Break-Up Lengths, and Physical-Chemical Parameters

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## Abstract

**:**

## 1. Introduction

#### 1.1. Motivation

_{2}emissions [2]. Water reduction devices are commonly used to decrease domestic water consumption [3,4] and therefore contribute to decreasing in CO

_{2}emissions. Among such devices, those dedicated to shower water consumption are especially challenging as it is known that total flow influences consumer satisfaction [5]. More analytically, consumer (shower) satisfaction depends on total pressure exerted by water on consumer’s skin, within a certain limit, i.e., both high- and low-pressure cause discomforts. Consequently, just reducing the water flow in order to save water and energy will also decrease customer satisfaction. For this purpose, smaller nozzles with higher pressures are often used, which are also more prone to clogging issues involving, e.g., particle deposition and lime precipitation. In this work, we presented an alternative approach, which produces higher spray velocities at the same flow rate compared to a normal showerhead. Since such a device builds on purely geometric modifications of the showerhead and there is no change in energy input, it provides a low cost and easy to implement a solution for more sustainable, equally comfortable showers. The consequences of such modifications on the overall water characteristics and shower performance would be presented in sequence.

#### 1.2. Hyperbolic Vortices

_{1}|, |PF

_{2}| to two fixed points, F

_{1}and F

_{2}(the foci), is constant, usually denoted as 2a with a > 0. Such geometric space can be represented as

- the velocity vector field of such structures is quite particular, meaning particles immersed in vortex structures, depending on their size, will be subjected to different tangential, axial, and radial velocities. This varies (considerably) with position and time,
- when considering liquid-based hyperbolic flow structures, there is always a well-defined air-liquid internal interface, which could be eventually used to enhance gas-diffusion in the liquid,
- for liquid structures, there is also, and necessarily, a solid-liquid interface, which would contribute to enhancing shear stresses and would be partially responsible (together with viscous stresses) for the axial velocity gradient and energy losses of the tangential component of the liquid velocity.

## 2. Materials and Methods

#### 2.1. Showerheads

#### 2.2. Jet Break-Up Length, Jet Velocity, and Droplet Characteristics

^{®}. The program can differentiate between jets and droplets from one single image by using their circularity, i.e., the ratio between the smallest and the largest distance between two internal points of the projected object. Jets normally have circularity below 0.4 and droplets above this value, with perfect spheres having a circularity equal to 1. After differentiating the objects, ImageJ can also calculate the jet break-up lengths and droplet diameters by using their maximum Feret diameter [14], i.e., the maximum distance between two parallel lines enclosing the projected body. The jet length was calculated as the distance from the nozzle tip until the jet tip just after a break-up event. The liquid velocity was calculated using the movement of the jet between two consecutive break-up events. No retraction at the jet tip just after filament break-up was observed or considered.

#### 2.3. Physical and Chemical Parameters

## 3. Results and Discussion

#### 3.1. Optical Spray Analysis

^{−1}.

^{−1}(additional data for 6, 7 and 9 L⋅min

^{−1}in the Appendix A in Figure A1, Figure A2 and Figure A3, respectively). In an aqueous vortex, perturbations in the air/water surface create a significant air boundary layer that stays associated with the surface. As these perturbations move inwards, they pull the boundary layer with them, creating a force that draws the air inwards into the vortex [16,17]. Due to the resultant pressure gradient, a certain volume of air is drawn into the water and thereby increases the flow rate through the most peripheral nozzles. This would result in an increased liquid kinetic energy for these nozzles and a consequently bigger break-up length (also observable in Figure 4). Since the sub-pressure is highest in the center of the vortex, the air intake takes place primarily at the innermost nozzles (as, for example, nozzle 5, right side, in Figure 4b). In order to understand the consequences of this result for the “shower experience” [5], the obtained median velocities for the five nozzles were used to fit a proportionality constant between flow rate and velocity for each nozzle and each showerhead, using a simple least-squares linear regression model. Taking into consideration the number of nozzles present for each radius, a weighted average was taken of the ratio between these constants in order to make our results comparable to those of Okamoto et al. [5]. The results of this calculation showed that the vortex showerhead could provide the same jet velocity as a normal showerhead when using 14.4 ± 5.6% less water (p < 0.01). Or alternatively, when using the same amount of water, the velocities (see Figure 5) and jet lengths produced by a hyperbolic showerhead were higher than those produced by a normal showerhead. It has been shown that jet velocity is related to the “shower experience” [5]. Thus, according to the results of Okamoto et al. [5], a vortex showerhead could provide the same comfort level with less water. It should be pointed out, though, that the central nozzles are used as air inlets resulting in a different spatial spray distribution, which may or may not give a desirable effect.

_{0}being the nozzle diameter, ρ the liquid density, Q the liquid flow rate, and γ the liquid surface tension [20]. Weber numbers smaller than one indicate the liquid break-up is happening in the so-called dripping regime. For Weber numbers between one and four, it is known as “transition regime”, as the jet in that window is not yet completely formed. Rather, a small ligament forms at the nozzle tip from which droplets break-up [21]. For Weber numbers higher than four, a jet is clearly formed at the orifice. Normally, at this level, the break-up length is around 10 times bigger than the nozzle inner diameter [13]. This regime is known as the “jetting regime”. Rayleigh [22] has thoroughly studied the physics behind these break-up mechanisms and defined that the diameter of the droplets formed from such break-up is around 1.8 times the jet diameter. The Weber numbers of the jets analyzed in this work were calculated from the flow rate according to equation 2. Their values were W

_{e}= 11.3, 15.4, 20.1, 25.5 for flowrates of 6,7,8,9 L min

^{−1}, respectively. Density and surface tension values of water at 35.8 °C were taken from the literature (ρ = 993.79 kg m

^{−1}[23] and γ = 70.27 mN m

^{−1}[24]). These calculations showed that the water was well within the jetting regime for all flowrates investigated.

#### 3.2. Chemical Parameters

_{1,2}) is the dissolved oxygen concentration at times t

_{1}and t

_{2}> t

_{1}, C

_{s}is the saturation concentration, and $\tau =\frac{1}{ka}$ is a time constant typical for the system. K is the gas transfer coefficient, and a is the diffusion area divided by the total liquid volume. This equation can be rewritten to define a relative saturation coefficient F after [25]:

_{1}and t

_{2}can also be replaced by the values at the inlet and outlet, respectively, to identify the instantaneous effect of the shower spray. F should be constant throughout the experiment, allowing us to calculate the K

_{a}coefficient, which determines the efficiency of the system. Examples of various experiments are given in Table 1. In some experiments, the aeration was faster using the vortex showerhead (experiment 1), whereas, in others, there was no measurable difference (experiment 2). Moreover, the variation of the parameter τ was of the same order between showerheads and experiments. Naturally, the observed additional mixing of air to the water by the vortex is expected to increase the amount of dissolved oxygen. On the other hand, the time for this diffusion to happen is rather small. Hence, with the measurement precision available, no statistically significant difference in aeration could be found. However, some experiments indicated better aeration of the vortex showerhead compared to the normal one.

_{2}equilibrium reaction so that CO

_{2}is expelled, comparable to stirring a glass of carbonated water,

_{2}, the changes in the redox potential plotted in Figure 10 require some more in-depth discussion. The redox potential is an electrical characteristic of a solution that shows its tendency to transfer electrons to or from a reference electrode, describing a system’s overall reducing or oxidizing capacity. In well-oxidized open waters, the redox potential is normally positive (above +300 to +500 mV), whereas, in reduced environments, it can be negative. Measuring redox potential in natural (potable) waters can yield different results depending on the method [26].

- the change of pH,
- the (missing) increase in DO.

_{0}is the standard potential at 25 °C, R is the general gas constant, T the absolute temperature in K, z the number of electrons transferred, F the Faraday constant, and A are the activities of the species involved. This allows us to derive a direct proportionality of the redox potential E and the pH, namely

## 4. Conclusions

_{2}and an increased intake of oxygen, part of which was immediately used for oxidation processes.

## Supplementary Materials

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A

**Figure A1.**Boxplot graphs of the average liquid velocities in the experiments done with 6 L⋅min

^{−1}for the regular (red boxes) and vortex (blue boxes) showerhead and nozzles 1 (outermost) to 5 (innermost). The green triangles are the calculated population mean. The horizontal black line is the expected liquid velocity calculated using continuity law [15] by taking the flow rate divided by the total nozzle surface area.

**Figure A2.**Boxplot graphs of the average liquid velocities in the experiments done with 7 L⋅min

^{−1}for the regular (red boxes) and vortex (blue boxes) showerhead and nozzles 1 (outermost) to 5 (innermost). The green triangles are the calculated population mean. The horizontal black line is the expected liquid velocity calculated using continuity law [15] by taking the flow rate divided by the total nozzle surface area.

**Figure A3.**Boxplot graphs of the average liquid velocities in the experiments done with 9 L⋅min

^{−1}for the regular (red boxes) and vortex (blue boxes) showerhead and nozzles 1 (outermost) to 5 (innermost). The green triangles are the calculated population mean. The horizontal black line is the expected liquid velocity calculated using continuity law [15] by taking the flow rate divided by the total nozzle surface area.

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**Figure 1.**Schematics of the regular (

**a**) and the vortex (

**b**) showerhead. Both are encapsulated in the same handle and nozzle plate (

**c**). The numbers in blue in subfigures (

**a**) and (

**b**) are the dimensions in mm.

**Figure 2.**Recirculating setup for measuring chemical parameters for either the normal or the vortex showerhead.

**Figure 4.**Individual jet comparison of the spray of regular (

**a**) and vortex (

**b**) showerhead at 8 L h

^{-1}flow rate, 22.7 °C temperature using a tray filter, allowing only one line of nozzles to produce an undisturbed jet path to be imaged.

**Figure 5.**Boxplot graphs of the average liquid velocities in the experiments done with 8 L⋅min

^{−1}for the regular (red boxes) and vortex (blue boxes) showerhead and nozzles 1 (outermost) to 5 (innermost). The green triangles are the calculated population mean. The horizontal black line is the expected liquid velocity calculated using continuity law [15] by taking the flow rate divided by the total nozzle surface area.

**Figure 6.**Jet lengths for the various showerheads and nozzles at 8 L min

^{−1}and 35.8 °C taken from frames right after a filament had broken off.

**Figure 7.**Minimal Feret diameter of droplets for different flow rates for regular showerhead (

**left**) and vortex showerhead (

**right**) at 35.8 °C. Normalized intensities (total surface is 1) and Gaussian distributions were drawn for the various flow rates, and the vertical lines indicate the means of their respective distributions.

**Figure 8.**DO (dissolved oxygen) content before (blue) and after (green) the vortex showerhead expressed as the logarithm of 1−C/C

_{s}, where C is the concentration in ppm, and C

_{s}is the saturation concentration determined from fitting the data to formula 3. The fits are shown in red. In this notation, 0 represents zero DO, and −2 represents a DO value of 0.99 C

_{s}.

**Figure 9.**pH against time for both showerheads before (full) and after (dot-dashed) the spray in a circulating water setup at 35.8 °C. The lower part is the difference between the two pH sensors for both showerheads, shown as a moving average over 3 minutes.

**Figure 10.**Redox potential against time for both showerheads before (full) and after (dot-dashed) the spray in a circulating water setup. The lower part is the difference between the two redox potential sensors for both showerheads.

**Table 1.**Fitting parameters of equation 1 to data of the DO (dissolved oxygen) content for two identical experiments of regular and vortex showerheads before and after the shower using a least-squares method. t

_{1}indicates the start of the shower spray. C

_{s}is the saturation concentration for oxygen found by the fit and C(t

_{1}) is the concentration at time t

_{1}. The goodness of fit is indicated with R

^{2}.

Experiment | Showerhead | Sensor | C_{s} − C(t_{1})/ppm | τ/h | C_{s}/ppm | R² |
---|---|---|---|---|---|---|

1 | Vortex | In | 3.13 | 0.21 | 6.13 | 0.9992 |

Out | 2.13 | 0.20 | 6.11 | 0.9972 | ||

Regular | In | 3.14 | 0.22 | 6.17 | 0.9998 | |

Out | 2.09 | 0.24 | 6.17 | 0.9987 | ||

2 | Vortex | In | 3.23 | 0.25 | 6.17 | 0.9999 |

Out | 2.17 | 0.26 | 6.16 | 0.9989 | ||

Regular | In | 3.21 | 0.26 | 6.14 | 0.9998 | |

Out | 2.20 | 0.25 | 6.12 | 0.9994 |

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**MDPI and ACS Style**

van de Griend, M.V.; Agostinho, L.L.F.; Fuchs, E.C.; Dyer, N.; Loiskandl, W.
Consequences of the Integration of a Hyperbolic Funnel into a Showerhead for Droplets, Jet Break-Up Lengths, and Physical-Chemical Parameters. *Water* **2019**, *11*, 2446.
https://doi.org/10.3390/w11122446

**AMA Style**

van de Griend MV, Agostinho LLF, Fuchs EC, Dyer N, Loiskandl W.
Consequences of the Integration of a Hyperbolic Funnel into a Showerhead for Droplets, Jet Break-Up Lengths, and Physical-Chemical Parameters. *Water*. 2019; 11(12):2446.
https://doi.org/10.3390/w11122446

**Chicago/Turabian Style**

van de Griend, Maarten V., Luewton L. F. Agostinho, Elmar C. Fuchs, Nigel Dyer, and Willibald Loiskandl.
2019. "Consequences of the Integration of a Hyperbolic Funnel into a Showerhead for Droplets, Jet Break-Up Lengths, and Physical-Chemical Parameters" *Water* 11, no. 12: 2446.
https://doi.org/10.3390/w11122446