# Surface Water Drainage - Grate Inlet Capcity

The situation is a grated drain across a driveway entrance (so a thin long drain) if I know the roughness of the upstream surface (ie mannings roughness), the slope of the surface upstream (which is the same as the slope of the grate) and the effective area of the grate (ie clear opening area of the grate) how do I calculate my approach flow vs my captured flow (in the grated drain)?

Ignore the capacity of the channel of the grate I already know how to calculate that.

• Hopefully a mechanical or civil engineer can answer this. This kind of thing does not lend itself to theoretical study or physical reasoning - it's answered by applying procedures in standards and engineering specifications which have grown up from experimental and usage data over long periods of time. Unfortunately, there isn't a mech or civil eng stack exchange yet :(. I've added the "everyday-life" tag to your question but I think fluid dynamics is the only one which fits. – Selene Routley Oct 4 '13 at 8:05
• It's funny you say that, I am a practicing civil engineer in Australia and we don't have flexible and powerful tools to get the capacity of every grate type. I might approach some manufacturers and see if they can provide some guidance. – tfitzhardinge Oct 5 '13 at 3:14

I'd like to "answer" your question (not really an answer, but it's a bit long for a comment) by saying a little about why maybe physicists are not quite the right people to ask - yet! The reason is quite fascinating, so maybe you can draw some entertainment from this site if not an answer.

The answer is simple: nonlinearity. Wild nonlinearity. So far untamable nonlinearity!

The theoretical framework for thinking about your problem is the Navier-Stokes equation(s) (see Wiki page with this name) which I'm guessing you would have seen at some stage in your civil engineering degree. Not that I'm saying you should remember them to be a good civil engineer, as should become clear below.

These equations (actually the vector form is one equation, but often people split them up into several, especially when one gets down to the details of the force terms) are the essence of simplicity: at an abstract level, written out they are:

$$\rho(\vec{r})\left(\partial_t \vec{v}(\vec{r}) + \vec{v}(\vec{r})\cdot\nabla \vec{v}(\vec{r})\right) = \vec{F}(\vec{r})$$

where $\vec{v}$ is the velocity field (fluid velocity as a function of position $\vec{r}$, $\rho(\vec{r})$ the density field and $\vec{F}(\vec{r})$ the force per unit volume field. $\vec{F}(\vec{r})$ is made up of terms like the gradient of the local pressure field, the viscous drag field and any body forces (like Lorentz (electromagnetic) force in a flowing molten conductor subject to an electromagnetic field). So the terms on the left are:

$$\mathrm{mass}\times(\mathrm{apparent\,acceleration} + \mathrm{convection\,acceleration}) = \mathrm{force}$$

where the apparent acceleration is what you would infer by seeing that the velocity field changes with time and the convection acceleration is the correction one makes on understanding that the velocity field is not the velocity of individual particles, it is the field of velocities that those particles at particular points in space. So a particle at position $\vec{r}$ at time $t$ will be at position $\vec{r} + \vec{v}\Delta t$, where the velocity field is $\vec{v}(\vec{r})+\vec{v}(\vec{r})\cdot\nabla \vec{v}(\vec{r})\Delta t$ a time $\Delta t$ later. So when we track individual particles rather than velocity fields, the two terms add to give the total acceleration. The complicated looking $\vec{v}(\vec{r})\cdot\nabla \vec{v}(\vec{r})$ is nothing more than a correction for our natural, human information culling tendency to think of velocity fields rather than tracking individual particle histories - a tendeny which is borne from the real need not to be overwhelmed by our senses. So the fearsome Navier-Stokes equation is nothing more than Newton's second law $F = m a$!! Can anything be simpler than this? So you'd think that we know all about the behaviours of such an in-principle simple equation right?

Wrong! The stunning truth is that, to quote the Wiki page:

The Navier–Stokes equations are also of great interest in a purely mathematical sense. Somewhat surprisingly, given their wide range of practical uses, mathematicians have not yet proven that, in three dimensions, solutions always exist (existence), or that if they do exist, then they do not contain any singularity (smoothness). These are called the Navier–Stokes existence and smoothness problems. The Clay Mathematics Institute has called this one of the seven most important open problems in mathematics and has offered a US$1,000,000 prize for a solution or a counter-example. The culprit in all of this is the nonlinear term$ \vec{v}(\vec{r})\cdot\nabla \vec{v}(\vec{r})\$ whereby the velocity field feeds back to influence itself.

I love this / these equation (s) because they are a bit of a leveller insofar that teach us physicists a bit of humility when we're getting too full of ourselves thinking our craft and theories are oh so very clever! Especially for the kind of theorist who thinks him or herself as above experimentalists and technicians. We really do not have much of a deep theoretical understanding of so simple an equation. We have a strong hunch that these nonlinear equations would tell us about turbulence and vortices shedding off chimneys in strong winds if we could solve them properly, and with good reason- there is no evidence that there is anything other than Newton's laws at work in fluids - but we really don't know.

Much of the rest of physics (quantum physics, electromagnetism) is either linear, or, if nonlinear, concerned with ordinary differential equations. The former - linearity - is tamed by the huge mathematical knowledge afforded us by fields such as functional analysis and linear algebra. The later - ODE linearity - is essentially tamed by the Picard-Lindlöf theorem, a theorem that applies an extremely powerful and general result (contraction mapping fixed point theorem) to tell us, in theory, most of what there is to know about nonlinear ODEs. But even so, there are some wild creatures to wrestle with (Chaos).

Nonlinear partial differential equations are largely uncharted lands for us. Another field where nonlinearities are important is General Relativity. This is essentially the theory wherein (to quote John Archibald Wheeler) "Mass tells space-time how to curve, and space-time tells mass how to move." We deal with "curved" space so, intuitively, it's not surprising that the description of such a manifold is nonlinear. Only in the last thirty years have we made any real progress in bringing the Einstein field equations to life in computer simulations and the most formidable achievements in numerical relativity have been in THIS (i.e. the twenty-first) century. I understand that simple numerical problems like binary black holes have only achieved decent numerical stability in the last decade or so.

Fluid dynamics is much harder than general relativity: the nonlinearities are just as fierce but one is dealing with much more complicated systems: in GR we can learn much from the simulation of much simpler systems such as two black holes. There is an interest in GR hydrodynamics, though, to understand how things like galaxies clump. So maybe in the future the two fields will teach one another a great deal.

But for now, as far as fluid dynamics is concerned, experiment is queen. This kind of thing you're asking does not lend itself to theoretical study or physical reasoning - it's answered by applying procedures in standards and engineering specifications which have grown up from experimental and usage data over long periods of time. There are also human factors to be considered: what do people expect of a drain? Is it OK if it floods once every two, five, ten years?

So if I had a problem like this, I would rightly go to a civil engineer. My expertise as a physicist will help me understand the implications of what he or she tells me, but, for the reasons my little essay hopefully make clear, you should pay the most heed to advice from engineers, not physicists!

• In civil engineering no one has published first principles of grate capture. The US Federal Highway Administration has been the closest to this by publishing their HEC22. It provides capacity to 7 different grate types. This is somewhat limiting because there are 1000s of different grate sizes and clear opening ratios. It can be found at the link fhwa.dot.gov/engineering/hydraulics/pubs/10009/index.cfm. Some manufactures have conducted lab testing on their grate products and you can get a L/s/m rate from them. Unfortunately this is only a few and it would be great (cont next post) – tfitzhardinge Oct 17 '13 at 6:08
• to have first principle equations that you physicians and mathematicians develop. Engineers use applied mathematics and rely on you guys to develop the research. In Australia also we are currently developing our Australian Rainfall and Runoff manual for the next edition and I am hoping that there is more literature in this about this issue. – tfitzhardinge Oct 17 '13 at 6:21