Splash Reducing Surfaces

Abstract

Systems and methods for reducing splashing of a liquid impinging on a surface include two sidewalls symmetrically oriented about a vertical plane. The sidewalls are joined along the vertical plane defining a back corner of a splash reducing surface. Each sidewall forms an angle less than a critical angle relative to a horizontal line extending from the sidewall to a vertical axis. The vertical axis is on the vertical plane at a specified distance from the back corner.

Description

TECHNICAL FIELD

The present disclosure relates to surfaces that reduce splashing of impinging liquids.

BACKGROUND

When a liquid drop or jet impinges on a solid surface, a splash can form. Splashes can generate smaller secondary droplets that can be ejected at sometimes high velocity. Aerosolized droplets can spread far from the location of impact ultimately coating large areas of surfaces surrounding the impact with the liquid. The formation of splashes depends on many factors including liquid properties (e.g., density, viscosity, surface tension), surface properties (e.g., surface roughness, wettability, stiffness), and impact parameters (e.g., velocity, angle, drop or jet size). Splashing occurs in many natural and industrial processes. For example, urine can splash from urinals in public restrooms, which can result in large areas of floors and walls of the public restroom being coated by urine droplets thereby leading to unpleasant and unsanitary conditions.

SUMMARY

A splash reducing surface can be designed based on an impingement angle that a fluid stream makes when impinging on the surface. For example, when an impingement angle is less than a critical angle (e.g., 40 degrees (°) or less, 30° or less), then the number of satellite droplets generated by the impacting fluid are reduced or eliminated compared with when the impingement angle is greater than the critical angle. When the fluid originates from or passes through a common reference point relative to the splash reducing surface, the surface can be designed so that fluid passing through the common reference point to the splash reducing surface will impact the splash reducing surface with an impingement angle less than the critical angle thereby reducing splashing from the surface.

For example, considering a cross-section of a splash suppressing surface, a line drawn from an origin (e.g., the fluid origination point) to the splash suppressing surface will form an angle less than the critical angle. If the line is rotated about the origin, for rotation angles that the line intersects the splash reducing surface, the line forms an angle less than the critical angle with the splash reducing surface. The splash reducing surface can be symmetrically oriented about a central line or plane. For example, a central line can extend from the origin to a back surface or corner of the splash reducing surface. The distance from the origin to the back surface defines a length scale of the splash reducing surface. Increasing the distance will increase the relative scale of the splash reducing surface, and decreasing the distance will decrease the relative scale of the splash reducing surface. A sidewall of the splash reducing surface is defined by the intersection of a line with a line segment oriented at an angle less than or equal to the critical angle as the line is rotated through a range of angles about the origin relative to the central line. For example, the line can be rotated from 0° (e.g., coincident with the central line) to an angle of 30° or more on either side of the central line.

In some examples, the splash reducing surface is a part of a urinal. The urinal can include a vertical upper portion and a curved lower portion where the lower portion forms a bowl to collect fluid from the upper portion. The vertical upper portion can have a horizontal cross-section designed based on the impingement angle and the common reference point. For example, the common reference point can be at a distance from the back surface or corner of the urinal where a user of the urinal would likely stand. In other words, and more generally, the distance from the back surface or corner of the urinal can be a minimum fluid origination distance from a splash resistant surface, formed according to the techniques described herein, to achieve effective splash reduction. For example, the minimum fluid origination distance can serve, in some implementations, as both a design parameter and an operational limitation. Dispensing of fluids at distances less than the minimum fluid origination distance, may subject the individual (in the case of a urinal) or fluid dispenser (in the case of a machine) to increased fluid splash back, while fluid dispensing at greater distances will have no increase in splash back. The vertical portion can be formed by extruding the cross section vertically. The lower portion can be formed by sweeping the cross-section through a sweep curve. Sidewalls of the lower portion can be smoothly continuous with sidewalls of the upper portion. The sweep curve can be defined based on a radius about a rotation point that is located between the fluid origination point and the back corner or surface of the urinal. The lower portion can include a drain to drain fluid collected in the lower portion. In some implementations, the fluid origination point is located above a front most portion of the lower portion relative to the back corner or surface. The urinal can be formed of porcelain or other suitable material. One or more fluid pathways can be formed in a top portion of the vertical portion. A flush fluid (e.g., water) can be dispensed through the fluid pathways to rinse or flush the urinal. The fluid pathways and drain can be configured to fluidly couple with a plumbing system of a building.

In some examples, the splash suppressing surface is based on a vertical cross-section. A fluid jet emanating from a fluid origination point or origin, will have a parabolic trajectory due to the effects of gravity. A higher fluid velocity of the jet will increase the horizontal distance that the fluid will travel. The splash suppressing surface can be formed by the locations where the surface would intersect a fluid jet at an angle less than or equal to the critical angle. For example, for a critical angle of 30°, a splash suppressing surface is generated corresponding to the locations where the fluid jet would impinge on the surface at an angle of 30° or less for a range of jet velocities. The defined surface cross section can be rotated about a vertical axis going through the origin to form a surface that meets the splash reducing criterion relative to the origin.

A urinal formed by a splash reducing surface based on a vertical cross-section can also include sidewalls that align with the edges of the splash reducing surface. A bottom surface can be formed such that a jet impacting the bottom wall will also impact at an angle less than or equal to the critical angle. An opening can be formed in an upper portion of the urinal between the side walls and the splash reducing surface. The bottom wall joins with the sidewalls and the splash reducing surface. A drain can be located at the lowest point in the bottom of the urinal to drain fluid collected in the urinal. Fluid pathways can be formed in the top of the splash reducing surface to allow a flush fluid to rinse the urinal.

In some implementations, an ideal origin or ideal fluid origination point is based on statistics of a human population. For example, the distance from the back corner of a splash reducing surface can be based on an average distance a person stands away from a urinal. A height of the origin can be based on an average male crotch height.

In some implementations, the specified distance is a minimum design distance that a fluid stream would originate relative to the splash reducing surface. When an actual fluid stream originates at a distance further away from the back corner of the splash reducing surface, the impingement angle will be less than the critical design angle by the nature of the design of the splash reducing surface.

The splash reducing surface can be used in myriad applications where a fluid splashes onto a surface. For example, splash reducing surfaces can be utilized for a pet bowl from which a pet (e.g., a cat or a dog) drinks water, a coffee machine, chemical processing equipment that collects fluid streams, and beverage processing equipment. Splash reducing surfaces can also be used in spray cooling and/or jet cooling of electronic or other equipment and in other applications using liquid streams or drops for transferring heat from a surface.

In an example implementation, a splash reducing surface includes two sidewalls symmetrically oriented about a vertical plane, the sidewalls being joined along the vertical plane defining a back corner of the splash reducing surface, each sidewall forming an angle less than a critical angle relative to a horizontal line extending from the curved sidewall to a vertical axis, the vertical axis being on the vertical plane at a specified distance from the back corner.

In an aspect combinable with the example implementation, the sidewalls form the angle less than the critical angle relative to the horizontal line for a range of rotation angles of the horizontal line relative to the vertical plane.

In another aspect combinable with any of the previous aspects, the two sidewalls form an upper portion having a cross section in a horizontal plane, and wherein the splash reducing surface further include a lower portion joined to the upper portion and wherein a cross-section of the lower portion is continuous with the cross-section of the upper portion translated along a curve.

In another aspect combinable with any of the previous aspects, the splash reducing surface is a urinal.

Another aspect combinable with any of the previous aspects includes a drain formed in the lower portion.

In another aspect combinable with any of the previous aspects, the sidewalls of the splash reducing surface are defined by

$r=B\exp \left(\frac{\alpha }{\Theta }\right),$

where r is a distance from the vertical axis to the sidewall, B is the specified distance from the back corner, α is an azimuthal angle of r relative to the vertical plane, and Θ=±tan θ is the tangent value of the critical impinging angle θ.

In another aspect combinable with any of the previous aspects, the critical angle is 40 degrees or less.

In another aspect combinable with any of the previous aspects, the critical angle is 30 degrees or less.

In another example implementation, a splash reducing surface includes a surface forming an angle less than a critical angle at an intersection of a parabolic line extending from a reference point and the surface, where the reference point is located at a specified distance from the splash reducing surface at a specified height above a bottom of the splash reducing surface.

In an aspect combinable with the example implementation, the back surface is defined by

$C=\frac{3}{\sqrt{8{\Theta }^2-1}}\arctan \left(\frac{4\Theta z-r}{r\sqrt{8{\Theta }^2-1}}\right)+\frac{1}{2}\ln \left(2\Theta z^2-\mathrm{rz}+\Theta r\right),$

where Θ=±tan θ is the tangent value of the critical impinging angle θ, r is a distance between the fluid origination point and the back surface, z is a vertical distance relative to the fluid origination point, and C is a constant related to the specified distance.

In another aspect combinable with any of the previous aspects, the critical angle is 40 degrees or less.

In another aspect combinable with any of the previous aspects, the critical angle is 30 degrees or less.

In another example implementation, a urinal includes a bowl including a back surface forming an angle less than a critical angle between a line extending from a fluid origination point to the back surface, the fluid origination point being a specified distance from the back surface and a specified height from a bottom of the bowl; two sidewalls, each sidewall forming a sidewall angle between a central vertical plane including the fluid origination point and the sidewall; and a front wall joining the two sidewalls and a bottom portion of the back surface, the two sidewalls and the back surface define an opening.

In an aspect combinable with the example implementation, the line is a parabolic curve.

In another aspect combinable with any of the previous aspects, the back surface is defined by

$C=\frac{3}{\sqrt{8{\Theta }^2-1}}\arctan \left(\frac{4\Theta z-r}{r\sqrt{8{\Theta }^2-1}}\right)+\frac{1}{2}\ln \left(2\Theta z^2-\mathrm{rz}+\Theta r\right),$

where and Θ=±tan θ is the tangent value of the critical impinging angle θ., r is a distance between the fluid origination point and the back surface, z is a vertical distance relative to the fluid origination point, and C is a constant related to the specified distance.

In another aspect combinable with any of the previous aspects, the critical angle is 40 degrees or less.

In another aspect combinable with any of the previous aspects, the critical angle is 30 degrees or less.

Another aspect combinable with any of the previous aspects includes a drain formed in the bottom portion.

In another example implementation, a method for forming a splash suppressing surface includes determining a fluid origination point a specified length from a back portion of the splash reducing surface; defining a surface that intersects a line extending from the fluid origination point at an angle less than a critical angle; and fabricating the defined surface to form the splash suppressing surface.

In an aspect combinable with the example implementation, the line is a horizontal line.

In another aspect combinable with any of the previous aspects, fabricating the defined surface comprises fabricating a porcelain casting of the defined surface.

In another aspect combinable with any of the previous aspects, the critical angle is 40 degrees or less.

In another aspect combinable with any of the previous aspects, the critical angle is 30 degrees or less.

In another aspect combinable with any of the previous aspects, the back surface is defined by

$C=\frac{3}{\sqrt{8{\Theta }^2-1}}\arctan \left(\frac{4\Theta z-r}{r\sqrt{8{\Theta }^2-1}}\right)+\frac{1}{2}\ln \left(2\Theta z^2-\mathrm{rz}+\Theta r\right),$

where and Θ=±tan θ is the tangent value of the critical impinging angle θ., r is a distance between the fluid origination point and the back surface, z is a vertical distance relative to the fluid origination point, and C is a constant related to the specified distance.

In another aspect combinable with any of the previous aspects, the curved sidewalls of the upper portion are defined by

$r=B\exp \left(\frac{\alpha }{\Theta }\right),$

where r is a distance from the fluid origination axis to the curved sidewall, B is the specified distance from the back surface, α is an azimuthal angle of r relative to the central vertical plane, and Θ=±tan θ is the tangent value of the critical impinging angle θ.

In another aspect combinable with any of the previous aspects, the surface is defined by rotating the line through a range of angles relative to a central vertical plane containing the fluid origination point.

In another aspect combinable with any of the previous aspects, the surface is a first surface, and the method further includes defining a second surface that is a mirror image of the first surface across the central vertical plane. Fabricating the surface includes fabricating the first surface and the second surface together to form a unified body.

In another aspect combinable with any of the previous aspects, the first surface and the second surface form a urinal.

Implementations of the systems, methods, and devices of this disclosure can have particular technological advantages. For example, urinals incorporating splash reducing surfaces as described herein can reduce or eliminate urine splash back in comparison to conventional urinals. The substantial suppression of urine splash back can reduce utilization of water, cleaning chemicals, and human resources required for the maintenance of restrooms as compared with conventional urinals. For example, assuming ten times the amount of water is used to clean a volume of splashed urine, a water savings of up to 10 million liters (L) of water could be saved per day across the United States. Reduced cleaning and maintenance requirements also reduce financial and environmental costs of operating public restrooms. Reducing urine splash back can also provide cleaner and healthier restrooms. The urinals described herein can improve accessibility of urinals by providing a surface that can reduce splashing for a large range of fluid origination heights meeting accessibility standards. The systems, methods, and devices of this disclosure achieve reduced splashing by manipulating the geometry of the urinal and does not require incorporating expensive or complicated surface materials or coatings. These systems, methods, and devices are therefore compatible with conventional materials and suitable for mass implementation.

The details of one or more embodiments of these systems and methods are set forth in the accompanying drawings and the description below. Other features, objects, and advantages of these systems and methods will be apparent from the description and drawings, and from the claims.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1A is a perspective view of an example urinal including splash reducing surfaces.

FIG. 1B is a perspective view of another example urinal including splash reducing surfaces.

FIG. 2A is a schematic of a coordinate system for defining splash reducing surfaces.

FIG. 2B is a plot illustrating example side profiles of splash reducing surfaces.

FIG. 2C is a plot illustrating example horizontal profiles of splash reducing surfaces.

FIG. 3A is a front view of an example urinal including splash reducing surfaces.

FIG. 3B is an angled view of the example urinal in FIG. 3A.

FIG. 3C is a back view of the example urinal in FIG. 3A.

FIG. 3D and 3E are side views of the example urinal in FIG. 3A.

FIG. 3F is a top view of the example urinal in FIG. 3A.

FIG. 4A is a top view of another example urinal including splash reducing surfaces.

FIG. 4B is an angled view of the example urinal in FIG. 4A.

FIG. 4C is a front view of the example urinal in FIG. 4A.

FIG. 4D is a back view of the example urinal in FIG. 4A.

FIG. 4E is a side view of the example urinal in FIG. 4A.

FIG. 5A is a side view schematic of an example test apparatus for measuring splashing on a urinal.

FIG. 5B is a top view schematic of the example test apparatus in FIG. 5A.

FIG. 6 is a plot showing comparisons between theoretically determined splashing and experimental measured splashing from a splash reducing surface.

Like reference symbols in the various drawings indicate like elements.

DETAILED DESCRIPTION

A splash reducing surface can be designed based on an impingement angle that a fluid stream makes when impinging on the surface. For example, when an impingement angle is less than a critical angle (e.g., 40° or less, 30° or less), then the number of satellite droplets generated by the impacting fluid are reduced or eliminated compared with when the impingement angle is greater than the critical angle. When the fluid originates from or passes through a common reference point relative to the splash reducing surface, the surface can be designed so that fluid passing through the common reference point to the splash reducing surface will impact the splash reducing surface with an impingement angle less than the critical angle thereby reducing splashing from the surface.

For example, considering a cross-section of a splash suppressing surface, a line drawn from an origin (e.g., the fluid origination point) to the splash suppressing surface will form an angle less than the critical angle. If the line is rotated about the origin, for rotation angles that the line intersects the splash reducing surface, the line forms an angle less than the critical angle with the splash reducing surface. The splash reducing surface can be symmetrically oriented about a central line or plane. For example, a central line can extend from the origin to a back surface or corner of the splash reducing surface. The distance from the origin to the back surface defines a length scale of the splash reducing surface. Increasing the distance will increase the relative scale of the splash reducing surface, and decreasing the distance will decrease the relative scale of the splash reducing surface. A sidewall of the splash reducing surface is defined by the intersection of a line with a line segment oriented at an angle less than or equal to the critical angle as the line is rotated through a range of angles about the origin relative to the central line. For example, the line can be rotated from 0° (e.g., coincident with the central line) to an angle of 30° or more on either side of the central line.

FIG. 1A is a perspective view of an example urinal 100 including splash reducing surfaces 102 and 104. The splash reducing surfaces 102 and 104 function to reduce splashing from jets or drops of liquid impinging on the splash reducing surfaces 102 and 104. The urinal 100 can reduce splashing from jets or drops of liquid that pass through a reference axis (e.g., a fluid origination point) that is a specified distance from the back of the urinal. The splash reducing surfaces 102 and 104 are defined such that a horizontal line passing through the reference axis will form an angle with the surfaces 102 and 104 that is less than a critical angle (e.g., a splashing threshold angle). The urinal 100 includes plumbing fixtures 106 to supply water to flush the urinal 100 after use. The urinal 100 can reduce splashing from fluid jets or drops that originate at a variety of heights relative to the bottom 108 of the urinal.

FIG. 1B is a perspective view of another example urinal 150 including splash reducing surfaces 152, 154, 156, and 158. Similar to the urinal 100, the urinal 150 is designed such that a fluid jet or drop passing through a reference point will intersect the surfaces 152-158 at an angle that is less than a critical splashing angle. In particular, the urinal 150 is designed using a parabolic curve passing through a fluid origination point that is a specified distance above the bottom of the urinal 150. The urinal 150 includes plumbing fixtures 160 to supply water to the urinal 150 to flush the urinal 150 after use.

Splash generated from a jet or droplet train impinging on a flat surface is a complex phenomenon depending on many factors, including the impact speed (U), impinging angle (θ), dynamic viscosity (μ), density (ρ), diameter of the jet or droplet (D), surface tension (γ) of the liquid, ambient pressure (P), as well as the wettability and roughness of the surface and the impinging angle. Non-dimensional numbers quantify the relative importance of these factors; for example, the Reynolds

$\left(Re=\frac{\rho \mathrm{UD}}{\mu }\right),$

capillary

$\left(\mathrm{Ca}=\frac{\mu U}{\gamma }\right),$

and Weber

$\left(\mathrm{We}=\frac{\rho U^{2}D}{\gamma }\right)$

numbers. In the context of urination, many of these factors cannot be changed; however, the impinging angle can be controlled to reduce or eliminate splash.

In designing a splash reducing surface, a reduction in splash flow rate with change in the impinging angle can be determined by a theoretical model using a splash ratio, Q*, defined as splashed mass divided by total impinged water mass. Q₉₀*, is the nominal value of Q* for a given flow condition impinging perpendicular to the surface. Q₉₀* can be used to normalize the splash ratio, Q*. To better predict the normalized splash ratio, Q*/Q₉₀*, a modified Weber number, We†=W e sin² θ, can be used, where θ is the impinging angle of the droplet. The modified Weber number includes the fluid velocity normal to the surface on which the droplet or jet impacts.

Upon impact with a surface, a droplet forms lamella which moves radially outwards; the lamella experiences aerodynamic lift and subsequently splash. Assuming that as more lift is required for more ejecta, the volume of splash scales to surplus lift over the threshold. The splash fraction scales with average surplus lift expressed in the following integral:

$Q^{*}~\frac{1}{\pi }{\int }_0^{{\varphi }^{*}}{V}_l^2-V_l^{*2}d\varphi .$

where V_l is the lamellar velocity, V_l* is a threshold lamellar velocity, and ϕ* is a critical angle associated with V_l*, beyond which, no splash occurs.

The integral evaluates to the following expression:

$\frac{Q^{*}}{Q_{90}^{*}}=\frac{1}{\pi }\frac{\sin \theta -v^{*10/3}}{1-v^{*10/3}}+\frac{{\varphi }^{*}\left({\cos }^2\theta \left(2+\sin 2{\varphi }^{*}-3{\beta }^{\mathrm{†4}/3}\right)+\frac{2\sqrt{3}}{3{\varphi }^{*}}\sin 2{\mathrm{\theta \beta }}^{\mathrm{†2}/3}\sin {\varphi }^{*}\right)}{3{\mathrm{\pi \beta }}^{4/3}\left(1-v^{*10/3}\right)},$

where

${\varphi }^{*}=\Re \left[{\cos }^{-1}\left(\sqrt{3}{\beta }^{2/3}\frac{v^{*5/3}-{\sin }^{5/3}\theta }{2\cos \theta }\right)\right],\beta =1.1\sqrt{\mathrm{We}},{\beta }^{†}=\beta \sin \theta ,{\beta }^{*}=v^{*},\mathrm{and}{v}^{*}=V^{*}/V_0$

is a non-dimensional critical velocity. V₀ is the incoming droplet velocity. V* and W e* represent critical values for a normal impact. Choosing V*=0.89, corresponding to the criterion √{square root over (Ca)}=0.35, gives the prediction of the normalized splash ratio, Q*/Q₉₀*.

Based on this model, the relative splashed volume has an almost invariant inflection point independent of We and Re under conditions of human urination. The model predicts that at θ=30°, the splash is reduced by about 95% from maximum splash. This implies the possibility of designing a urinal geometry so that the impinging angle is always at or below a critical angle, {circumflex over (θ)} (e.g., 30° or less, 40° or less, 50° or less, up to 60°), consequently, reducing or eliminating splash independently of factors dependent on the user of the urinal.

FIG. 2A is a schematic of a three-dimensional (3D) coordinate system 200 for defining splash reducing surfaces. The x-axis 202, y-axis 204, and z-axis 206 emanate from an origin point 208 (e.g., a fluid origination point). The x-axis 202 and y-axis 204 define a horizontal plane 214 (e.g., a top plane). The z-axis 206 extends downward from the origin point 208. The z-axis and the x-axis 202 define a central vertical plane. Vertical plane 210 can be offset from the x-axis 202 by an angle (α) 212 and extend a radial distance (r) 218 from the z-axis 206. The origin point 208 (and top plane 214) are defined at a height 216 above a horizontal surface (e.g., a floor). A fluid drop, jet, or stream that passes through the origin point 208 impinges a surface at the distance r 220 from the origin point at an impingement angle (θ) 220.

For a splash reducing surface (e.g., a urinal), the 3D fluid stream impinges the surface at or under a critical angle (i.e., θ≤{circumflex over (θ)}). This complex 3D problem can be simplified by instead considering a two-dimensional (2D) projection of the fluid stream impinging the surface at the critical angle, e.g., θ={circumflex over (θ)}=30°. Isogonal surface profiles can be created to meet this condition. Isogonal surface profiles are, for example, shapes which ensure a constant impinging angle.

The 3D problem can be described using a cylindrical coordinate system (r, α, z) as shown in FIG. 2A. Two different 2D projections can be defined: a side view (e.g., an r-z plane, plane 210) where the stream is a family of parabolic curves varying over the course of urination as bladder pressure drops, and a top view (e.g., an x-y plane, plane 214) where the stream forms rays depending on a user's aim. Satisfying θ={circumflex over (θ)} leads to solving for isogonal curves in each plane to intersect the urine trajectory at a constant angle. Splash reducing surfaces generated using the 2D projections have impinging angles at or below the critical angle across their entire area.

FIG. 2B shows variations of isogonal curves for different values of {circumflex over (θ)}. The side view projection results in a curve given by:

$C=\frac{3}{\sqrt{8{\Theta }^2-1}}\mathrm{arc}\tan \left(\frac{4\Theta z-r}{r\sqrt{8{\Theta }^2-1}}\right)+\frac{1}{2}\ln \left(2\Theta z^2-\mathrm{rz}+\Theta r^2\right),$

where Θ=±tan {circumflex over (θ)}, and C is a constant that is determined by an auxiliary or boundary condition associated with the installation of the urinal. Curves 232, 234, and 236 correspond to critical angles of 30°, 45°, and 60°, respectively, as shown in legend 250 in FIG. 2C. The curves 232, 234, and 236 have been normalized by a characteristic length L (e.g., a distance from the fluid origination point to the splash reducing surface). Revolving the curves 232, 234, and 236 about the z-axis defines a 3D internal surface of a urinal design. For example, the curves 232, 234, and 236 can be rotated through a range of angles a 212 to define the splash reducing surface. The parabolas are self-similar as are the isogonal curves 232, 234, and 236 and can be scaled as needed.

In some implementations, this design process can be used to design a hostile surface. For example, a hostile surface can increase splash back to deter urination in public areas (e.g., walls outside a pub) by maximizing impinging angle.

FIG. 2C shows the isogonal curve problem in the horizontal plane 214 (e.g., a top view). The isogonal curves 262, 264, and 266 produce logarithmic spirals

$r=B\exp \left(\frac{\alpha }{\Theta }\right),$

where B is a constant that controls the scale of the curve. The curves 262, 264, and 266 have been normalized by the characteristic length L. Curves 262, 264, and 266 correspond to critical angles of 30°, 45°, and 60°, respectively, as shown in legend 250. The curves 262, 264, and 266 intersect the rays 268 at the critical angles. A splash reducing surface can be defined by extruding the curves 262, 264, and 266 along a defined trajectory.

FIGS. 3A-3F illustrate an example urinal 300 incorporating splash reducing surfaces. The urinal 300 includes sidewalls 302 and 304 that meet at a back corner 312 of the urinal 300. The inner surfaces of the sidewalls 302 and 304 are defined based on isogonal curves (e.g., curves 262, 264, or 266) in a horizontal plane. The sidewalls 302 and 304 can be mirror images of each other across the vertical plane 322.

An upper portion 306 of the urinal 300 is defined by extruding the isogonal curve in a vertical direction. A lower portion 308 of the urinal 300 is defined by sweeping the isogonal curve along a curved trajectory. The lower portion 308 is continuous with the upper portion 306. The lower portion 308 forms a bowl to collect fluid from the upper portion 306. A drain 310 is formed in the bottom of the lower portion 308 to enable the collected fluid to be drained (e.g., into a sewer system). The lower portion 308 also includes a rim 316. The sidewalls 302 and 304 form a unitary body 314.

The sidewalls 302 and 304 are symmetrically oriented about a vertical plane 322. The sidewalls 302 and 304 are joined along the vertical plane 322 at the back corner 312. Each sidewall 302 and 304 forms an angle 324 less than a critical angle relative to a horizontal line 326 that extends from the sidewall 302 or 304 to a vertical axis 318. The vertical axis 318 is defined on the vertical plane 322 at a specified distance 320 from the back corner 312. In some implementations, the distance 320 is less than or equal to a distance from the back corner 312 to the front of the rim 316. In some implementations, the distance 320 is greater than the distance from the back corner 312 to the front of the rim 316.

The sidewalls 302 and 304 form the angle 324 that is less than or equal to the critical angle relative to the horizontal line 326 for a range of rotation angles of the horizontal line relative to the vertical plane 322. The sidewalls 302 and 304 can be defined by

$r=B\exp \left(\frac{\alpha }{\Theta }\right),$

where r is a distance from the vertical axis to the sidewall, B is the specified distance from the back corner, α is an azimuthal angle of r relative to the vertical plane, and Θ=±tan θ is the tangent value of the critical impinging angle θ. The critical impinging angle can be 30° or less, 40° or less, 50° or less, up to 60°.

FIGS. 4A-4F illustrate another example urinal 400 that includes splash reducing surfaces. The urinal 400 includes a back surface 402, sidewalls 404 and 406, and a front wall 408. The sidewalls 404 and 406 and the front wall 408 define an opening 410. The opening 410 is surrounded by a rim 418. The sidewalls 404 and 406 form an angle 401 relative to each other. Alternately, angle 401 is a double angle of the angle of a sidewall 404 or 406 relative to a central vertical plane 403 that vertically bisects the urinal 400. The front wall 408, sidewalls 404 and 406, and back surface 402 form a bowl 416 to collect fluid. A drain can be formed in the bowl to enable the fluid to drain out of the urinal 400.

The back surface can be defined based on the vertical 2D projection discussed above. For example, the back surface 402 forms an angle 426 less than or equal to a critical angle between a line 428 extending from a fluid origination point 430 to the back surface 402. The fluid origination point 430 is a specified distance 424 from the back surface 402 and a specified height 422 from a bottom 414 of the bowl 416. The fluid origination point resides on the vertical plane 403. The line 428 can be a parabolic curve or a family of parabolic curves (e.g., curves 238). The urinal 400 can be mounted on a wall at different heights from the floor to accommodate users with different heights.

The back surface can be defined by

$C=\frac{3}{\sqrt{8{\Theta }^2-1}}\mathrm{arc}\tan \left(\frac{4\Theta z-r}{r\sqrt{8{\Theta }^2-1}}\right)+\frac{1}{2}\ln \left(2\Theta z^2-\mathrm{rz}+\Theta r\right),$

where and Θ=±tan θ is the tangent value of the critical impinging angle θ., r is a distance between the fluid origination point and the back surface, z is a vertical distance relative to the fluid origination point, and C is a constant related to the specified distance. The critical angle can be 30° or less, 40° or less, 50° or less, up to 60°.

An example method for forming splash reducing surfaces (e.g., surfaces in urinals 300 or 400) includes determining a fluid origination point a specified length from a back portion of the splash reducing surface; defining a surface that intersects a line extending from the fluid origination point at an angle less than a critical angle; and fabricating the defined surface to form the splash suppressing surface. For example, the surface can be fabricated from a porcelain casting. Alternatively, the surface can be fabricated by machining the defined surface out of a substate material using computer numeric controlled (CNC) machines. In some implementations, a mold for the porcelain casting is made from a CNC machined surface. The surfaces can be fabricated to form a unitary body such as a urinal.

To define the urinal surfaces (e.g., splash reducing surfaces), a urine trajectory can be projected onto either the top-view x-y plane or the side-view r-z plane (see FIG. 2A). Satisfying θ={circumflex over (θ)} leads to solving for isogonal curves that intersect the urine trajectory at a constant angle.

For the projection onto the r-z plane, assuming the initial direction of a urine jet is horizontal, the projectile motion of the fluid can be described by a family of parabolas: z=g(r)=kr², where k∈(0, ∞) is a parameter that determines the shape of the trajectory (e.g,. curves 238 in FIG. 2B). The limit of k→0 corresponds to the initial stage of urination, where bladder pressure and, consequently, jet velocity are high and the stream is almost horizontal. While k→∞ indicates a trajectory of a droplet train at the end of the urination, when the droplets fall almost vertically. The splash reducing surface is a curve or curves z=f(r) that intersects z=g(r) by {circumflex over (θ)} for arbitrary k, which can be modeled by the ordinary differential equation:

$\Theta =\frac{2f\left(r\right)-{\mathrm{rf}}^{\prime }\left(r\right)}{r+2f\left(r\right)f^{\prime }\left(r\right)},$

where Θ=±tan {circumflex over (θ)}. The solution of this equation is an implicit function given by

$C=\frac{3}{\sqrt{8{\Theta }^2-1}}\mathrm{arc}\tan \left(\frac{4\Theta z-r}{r\sqrt{8{\Theta }^2-1}}\right)+\frac{1}{2}\ln \left(2\Theta z^2-\mathrm{rz}+\Theta r\right).$

In the x-y plane projection, the stream forms a family of rays starting from the origin O, pointing in arbitrary angles α. To design an isogonal curve intersecting the rays at a constant angle, the following differential equation is established in polar coordinates:

$\frac{\mathrm{dr}}{d\alpha }=\frac{r}{\Theta },$

and the solution is the classic logarithmic spiral

$r=B\exp \left(\frac{\alpha }{\Theta }\right).$

Impinging angle maps can be generated by modeling a 3D surface composed of triangular facets. The urine streams can be simulated by parabolas initiated at the urethral outlet and ending at the center of each facet. The parabolas start horizontally and the angle between the tangent of the parabola and the normal vector of the surface can be measured. This can be repeated for all facets making up the inside surface of the urinal to generate an angle map. The angle map can be used to verify that the impinging angles are less than or equal to the desired critical angle to reduce splashing.

FIG. 5A-5B show a schematic of an example test apparatus 500 for measuring splashing on a urinal 502. FIG. 5A is a side view and FIG. 5B is a top view schematic of the test apparatus 500. The test apparatus 500 includes a urinal 502 mounted on a support structure 504. The urinal 502 can be, for example, any one of the urinals 100, 150, 300, or 400. A nozzle 506 can be positioned relative to the urinal 502 at a desired height and distance from the urinal 502. The nozzle 506 is connected to a fluid supply, and flow of the fluid supply is controlled with a valve 508 (e.g., a needle valve) and a flow meter 510. A wire frame 512 can be constructed surrounding the urinal 502 to support collection of splashing droplets from the urinal 502 (e.g., using paper towels or other absorbent media).

The urinal 502 can be installed on the support structure 504 according to applicable standards (e.g., ASME standards) to simulate installation and use in a public restroom. The nozzle 506 can be positioned horizontally at various inseam heights. Alternately, the nozzle 506 can be placed at a downward angle of 15° to direct the simulated urine flow into the urinal 502. The nozzle 506 can be made from a plastic material (e.g., using 3D printing) to represent the anatomy of a urethra.

Splashed droplets can be measured by paper towels hung on the wire frame 512 using clips, for example. The paper towel can be weighed before and after the testing. Relative humidity can be controlled to reduce evaporation from the paper towels during testing. For example, the relative humidity can be controlled to be approximately 65%.

Splatter from the urinal 502 can also be visualized to qualitatively compare the effectiveness of the splash reducing surfaces. For example, a nominal volume of water of 1.0 L can be dispensed through the nozzle 506. The splash back can be caught on a large piece of paper placed on the floor surrounding the urinal 502 and then left to dry.

The dry paper can be imaged to quantify the amount of splashed droplets. The images can be post processed by thresholding RGB values of the splatter patterns and separating them from the background with pseudo-color. Sessile droplets of known volume can be placed on the paper and imaged to provide a reference scale.

Flow rates in the tests can encompass what could be reasonably expected from the typical male human population, with a Q_max=1.9 L/min, representing the 95th percentile peak flow rate for men under 50 with an average voided volume; Q_median=0.7 L/min representing a median average flow rate in a similar case, and a Q_min=0.4 L/min representing a lower bound well below the 5th percentile average flow rate and associated with the lowest value measured by the flow rate meter 510. The high flow rate represents a condition that although high can still be reasonably encountered by a public urinal. The low flow rate can be set as low as possible rather than at a fixed percentile. As the simulated urination progresses, the flow rate can drop steadily; thus, although a high flow rate is rarely encountered, a low flow rate is always encountered. The low flow rate can explore one extreme and impinges on different regions of the urinal 502.

To verify the theoretical model and find the critical angle {circumflex over (θ)} relevant to human urination, the splash generated by a jet of water from an anatomically accurate nozzle above an inclined glass plate was measured. The angle of inclination of the plate θ was varied from 20° to 90° relative to the impinging jet. The nozzle was setup vertically for the critical angle tests. The vertical stream impinges on the angled glass plate within a bucket. Flow rates were adjusted prior to testing. Paper towels were mounted on wooden rods placed around the bucket to absorb any splash. The mass of the paper towels were measured before and after testing.

In conducting critical impinging angle testing, the jet travelled a vertical distance prior to impacting the glass plate, with the low height being equivalent to the distance from 5th percentile crotch height to the lip of a standard urinal, the medium being equivalent to the distance from 50th percentile crotch height to lip height, and the high height condition being from 95th percentile crotch height to the ground.

FIG. 6 is a plot 600 showing comparisons between theoretically determined splashing and experimental measured splashing from a splash reducing surface. The experimentally measured normalized splash ratio (Q*/Q₉₀*) produces good agreement with the theoretical model. This can be visualized as the splash (e.g., number of satellite droplets) formed upon impact decreasing with the impinging angle. The normalized splash ration is nearly zero at and below an impinging angle of θ=30°. The observation of this angle provides a design criterion to meet in the development of a splashless urinals or other splash reducing surfaces. As the impinging angle increases above 30°, the normalized splash ratio also increases. Impinging angles of 40°, 50°, and 60° still see splash reduction compared to higher impingement angles. A desired amount of splash reduction can be chosen to fit a particular application by selecting an appropriate impingement angle for the surface design. For example, in some implementations, some splashing may be tolerated and a critical angle for the splash reducing surface can be selected as 20° or more, 30° or more, 40° or more, 50° or more, or up to 60°.

Various modifications and additions can be made to the exemplary embodiments discussed without departing from the spirit and scope of the present disclosure. For example, while the embodiments described above refer to particular features, the scope of this disclosure also includes embodiments having different combinations of features and embodiments that do not include all of the described features. Accordingly, the scope of the present disclosure is intended to embrace all such alternatives, modifications, and variations as fall within the scope of the claims, together with all equivalents thereof, and other embodiments are within the scope of the following claims.

Claims

1. A splash reducing surface comprising: two sidewalls symmetrically oriented about a vertical plane, the sidewalls being joined along the vertical plane defining a back corner of the splash reducing surface, each sidewall forming an angle less than a critical angle relative to a horizontal line extending from the sidewall to a vertical axis, the vertical axis being on the vertical plane at a specified distance from the back corner.

2. The splash reducing surface of claim 1, wherein the sidewalls form the angle less than the critical angle relative to the horizontal line for a range of rotation angles of the horizontal line relative to the vertical plane.

3. The splash reducing surface of claim 1, wherein the two sidewalls form an upper portion having a cross section in a horizontal plane, and wherein the splash reducing surface further comprises a lower portion joined to the upper portion and wherein a cross-section of the lower portion is continuous with the cross-section of the upper portion translated along a curve.

4. The splash reducing surface of claim 3, wherein the splash reducing surface is a urinal.

5. The splash reducing surface of claim 4, further comprising: a drain formed in the lower portion.

6. The splash reducing surface of claim 1, wherein the sidewalls of the splash reducing surface are defined by

7. The splash reducing surface of claim 1, wherein the critical angle is 40 degrees or less.

8. The splash reducing surface of claim 1, wherein the critical angle is 30 degrees or less.

9. A urinal comprising: a bowl comprising: a back surface forming an angle less than a critical angle between a line extending from a fluid origination point to the back surface, the fluid origination point being a specified distance from the back surface and a specified height from a bottom of the bowl;two sidewalls, each sidewall forming a sidewall angle between a central vertical plane including the fluid origination point and the sidewall; anda front wall joining the two sidewalls and a bottom portion of the back surface, wherein the two sidewalls and the back surface define an opening.

10. The urinal of claim 9, wherein the line is a parabolic curve.

11. The urinal of claim 9, wherein the back surface is defined by

12. The urinal of claim 9, wherein the critical angle is 30 degrees or less.

13. A method for forming a splash suppressing surface, the method comprising: determining a fluid origination point a specified length from a back portion of the splash reducing surface;defining a surface that intersects a line extending from the fluid origination point, the line intersecting the surface at an angle less than a critical angle; andfabricating the defined surface to form the splash suppressing surface.

14. The method of claim 13, wherein fabricating the defined surface comprises fabricating a porcelain casting of the defined surface.

15. The method of claim 13, wherein the critical angle is 30 degrees or less.

16. The method of claim 13, wherein the back surface is defined by

17. The method of claim 13, wherein the surface is defined by

18. The method of claim 13 wherein the surface is defined by rotating the line through a range of angles relative to a central vertical plane containing the fluid origination point.

19. The method of claim 18, wherein the surface is a first surface, and the method further comprises: defining a second surface that is a mirror image of the first surface across the central vertical plane,wherein fabricating the surface comprises fabricating the first surface and the second surface together to form a unified body.

20. The method of claim 19, wherein the first surface and the second surface form a urinal.