What is the steepest surface that can hold snow?

Question

I know it depends on the type of snow (dry or wet) and the rougness of the surface. I'm looking for an approximate rule of thumb answer. Assuming a reasonably smooth surface, at what angle it's likely to be free from snow?

Google results only discuss snow and slopes in context of skiing and roofs not caving in. Obviously, a snow safe roof (60+ degrees) will still accumulate a few inches of snow.

For example, imagine a sandwitch board sign. What's the minimum "uprightness" for it to not get obscured by snow and stay readable throughout the year?

Answer 1

What's the minimum "uprightness" for it to not get obscured by snow and stay readable throughout the year?

The answer to this problem is not in the angle at all. No amount of math will give you an angle at which you can reasonably presume your surface will remain clear. Snow can (and does) accumulate on vertical and even negatively sloped surfaces. Instead of thinking about the angle, you should be thinking about the materials and conditions involved.

There are lots of factors that determine whether snow will stick, but a few that stand out most are:

1. Temperature differentials between a surface and the atmosphere:

   This is particularly noticeable on surfaces of enclosures such as vehicles or buildings where the internal temperature causes the surface to be out of sync with the environment. You might notice snow collecting on the vertical glass surface of car side windows. This will be more pronounced in cars that were parked full of warm air when it started snowing. The warm glass would have melted the initial snow flakes. The wet surface will eventually freeze giving a textured surface for new snow flakes to sit on instead of sliding off.

   The same is true for objects with thermal mass that retain heat or cold after a sudden change in air temperature.

2. Surface properties such as roughness.

   If even a few snow flakes find purchase on a surface, more will find a way to build on that and obscure your surface. This is far more important than angle.

3. Atmospheric conditions such as humidity and the nature of the snow flakes formed.

To effectively keep a surface clear it should be ① smooth, ② quickly transfer heat to adapt to the current air temperature, and ③ be sheltered from being in contact with falling snow in the first place. Even highway signs with are smooth, vertical and thin metal are only so effective at staying clear and in some regions will have eaves protecting them from some snowfall.

Answer 2

An object on an angular surface will fall off, when forces exerted along the slope (gravitational, tangent to the surface: gt) are greater then those pressing the object to the slope (gravitational, perpendicular to the slope, gp) or working in the opposite direction on the slope (friction, $fr$).

As an example, $10\ \mathrm{cm}$ of snow, on a 0.5-by-1-meter surface:

• $0.1 \times 0.5 \times 1 = 0.05\ \mathrm{m^3}$ of snow, weighing approximately $0.05 \ \mathrm{m^3}\times 300\ \mathrm{\dfrac{kg}{m^3}}$ (density of snow = $300\ \mathrm{\dfrac{kg}{m^3}}$) = $15\ \mathrm{kg}$.
• Gravitational force exerted = $9.81\ \mathrm{\dfrac{m}{s^2}} \times 15\ \mathrm{kg} = 147\ \mathrm{N}$.
• Frictional force $fr = 147\ \mathrm{N}\times 0.53$ (as a lower bound guess, see link) = $77.91\ \mathrm{N}$.

The snow will fall off if $gp > fr$, and this occurs at an angle of $\arcsin{\dfrac{fr}{gp}} = 32^{\circ}$.

Recalculating for the high bound friction coefficient ($1.76$), results in the snow not even falling off if the surface were vertical.

I think you already pointed out the biggest problem: any estimate is heavily dependent on the contact surface, type of snow and other conditions (temperature, wind, etc.). I haven't done a thorough looking around, but from the friction coefficients listed in this one article, I can only guess that you'll have a wide variety of ranges, making a rule of thumb difficult.

Answer 3

The angle of 32° for the lower bound angle given in the answer by @Pytrik reminded me of the "typical" angle of repose for most materials, particularly soil and rock.

The angle of repose is defined as,

The steepest angle of descent or dip relative to the horizontal plane to which a material can be piled without slumping.

Angle of repose experiments with snow: role of grain shape and cohesion was published Cambridge University Press on 27 May 2020.

The shape of the snow particles, their cohesion and temperature and whether or not the snow had become sintered all factor into angle of repose for snow for a given set of conditions.

In the experiments, we observed a significant influence of particle shape and cohesion, with both factors increasing the angle of repose. The influence of shape was revealed by examining different snow types and spherical particles, and the cohesion, assumed to be caused by sintering, was examined by varying the temperature. Complex-shaped snow particles clearly formed larger angles than the spherical particles. More specifically, the smaller the shape parameter, the larger was the angle of repose. The influence of cohesion was negligible at temperatures below −22°C, but increased as the temperature increased to −2°C.

It was observed the angle of repose increased the more shape of a snow particle deviated from a sphere. Basically, the more angular the snow particle the higher its angle of repose.

When it comes to the effects of sintering, the angle of repose ...

increased at higher temperatures, between −15 and −5, and even more between −5 and −2°C.

Often, sintering is discussed in close connection to friction, which constitutes another temperature-dependent process for ice. However, its role in the angle of repose is not clear. While it is the key parameter in Mohr–Coulomb theory, which is often used for a theoretical analysis of the angle of repose (MCGlinchey, Reference McGlinchey2005), other studies conclude that friction plays a sub-dominant role behind particle shape and cohesion, and can therefore be neglected (Nowak and others, Reference Nowak, Samadani and Kudrolli2005).

The picture below is taken from the article. Particle with a smoother shape have a lower angle of repose; the bottom set of blue lines with an angle of repose around 20°. The more angular the shape of the particles, the higher the angle of repose; the yellow and green lines with an angle of repose around 30° to 40°, depending on temperature. In both of these situations, the angle of repose increase with temperature.

The fact that snow can deposit on a vertical surface would indicate the snow particles would be angular and would have a degree of sintering.

Answer 4

This question has too many variables to be answered mathematically.

In a practical sense, avoid snow on your near-vertical surfaces by gently heating them.

Example: Modern LED traffic lights can be obscured by snow on the lenses because they run at lower power than the old-school incandescent lights, and have much lower losses to heat. Therefore new LEDs are colder and less-able to soften and shed driven snow than a slightly warm filament-based lamp.