Wind-Chill Factor

Lawrence Davis

86 Wind-Chill Factor

A table with rows labels running from 5 to 60 mph of wind speed and columns labeled from 40 to -45 in Fahrenheit degrees of temperature. The cells of the table show the wind-chill temperature caused by the wind speed and temperature in the same row and column. Wind chill value run from 36 to -63 across the top row and 25 to -98 on the bottom row. In the center values run from 28 to -87. — The wind chill chart displays the theoretical calm air temperature which would produce the same cooling rate as the actual air temperature + wind combination. Image Credit: “Wind Chill Chart” by National Weather Service, NOAA

As stiff breeze can feel refreshing on a hot day and make a cool day feel quite cold. This phenomenon is known as the wind chill effect or just wind chill. Wind chill is a significant factor in thermal injuries such as hypothermia and frost bite, which is tissue damage caused by the formation of ice crystals within the tissue. The wind chill chart above shows what calm air temperature would be necessary to produce the same heat loss rate as the actual air temperature plus wind combination. The chart is makes this calculation specifically for the skin on a human face. The chart also shows the exposure time which is likely to result in frostbite. For example, a day with 10 °F air temperature and 10 mph wind would feel as cold as a -7 °F day without wind.

Everyday Example

The author grew up in a small remote town on the tundra in Western Alaska. School was cancelled on days when the wind chill was less than -75 °F. According to the windchill chart, school could be cancelled on a -30 °F day with 55 mph wind or on a -45 °F day with 15 mph wind. Both of these combinations were entirely possible in Bethel, Alaska and growing up we usually had one or two wind chill cancellation days per year.

Forced Convection

In the absence of wind, a layer of warm air will form next to your skin, which effectively provides an additional layer of insulation known as a boundary layer. The thermal image below color-codes warm and cool air so that we can visualize the formation of the warm boundary layer. (In the next chapter we will learn how thermal images like this one are created.)

Thermal image showing air (green) warmed by conduction from the skin then rising within the surrounding cooler and more dense air due to natural convection, which is discussed in the following section. Forced convection is the reduction of this boundary layer by fluid due to factors other than the warming of the fluid itself, such as wind. Image Credit: “Thermal Plume from human hand” by Gary Settles via Wikimedia Commons

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We can see that the skin-warmed layer is thin, but air has a very low thermal conductivity, so this layer can make and important contribution to slowing conduction. Wind tends to partially strip this insulating layer away and replace it with cooler air. The thickness of the warm boundary layer that is able to form depends on the wind speed, with higher speeds leading to thinner layers and causing greater wind chill effect. Wind chill is an example of forced convection, in which warm and cold fluid exchange places due to fluid motion caused by external factors such as blowing wind or flowing water.

Everyday Example: Hot Springs and Saunas

When submerged in fluid with a temperature higher than body temperature, such as in a hot spring or sauna, you may notice that the fluid suddenly feels hotter when you move around. Whether a fluid moves around you, or you move through the fluid, forced convection will occur in either case. When the fluid is warmer than your body then heat transfers out of the fluid into your body, leaving a slightly cooler boundary layer of fluid next to your skin. When you move this boundary layer gets left behind and replaced with new hot fluid that has not yet been cooled down by your body.

The rate of heat transfer by forced convection can be calculated using an empirical equation that looks very similar to the conduction equation:

(1) $\begin{equation*} Q/t = hA(T_2-T_1) \end{equation*}$

Once again the heat transfer rate is proportional to a difference between the object and environment temperatures. Contact surface area (A), again plays a role, in this case between the object and the fluid. Finally the convective heat transfer coefficient (h) incorporates fluid properties and accounts for the dependence of boundary layer thickness on fluid speed. The convective heat transfer coefficient is often determined experimentally. For example, the following graphs shows experimental data on the heat transfer coefficient for air at a range of wind speeds:

Heat transfer coefficient in units of (W/square meter K) on the vertical axis and velocity in m/s on the horizontal axis. The curve starts from a heat transfer coefficient value near 22 at roughly 3 m/s and curving gently upward and leveling out before reaching roughly 35 at 20 m/s. — Graph of convective heat transfer coefficient data for air. The “m2” in the units of the vertical axis means square meters.

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Reinforcement Exercises

Everyday Example: Cold Weather Survival

Cutting down on wind chill (forced convection) is an important part of the overall cold weather survival strategy. Let’s evaluate the effects of forced convection during a wilderness survival situation at 25 °F (-3.9 °C) with a 10 mph (4.5 m/s) wind. Let’s assume you are wearing thin fabrics that are fairly permeable to wind. Now we can approximate the effect of convection by assuming the fabric surface is the same as body temperature and applying the convection equation:

$\begin{equation*} Q/t = hA(T_2-T_1) \end{equation*}$

The difference between body temperature and the air temperature is 37 °C – (-3.9 °C) = 40.9 C°. The heat transfer coefficient is about $27, bold{frac{W}{m^2cdot K}}$ according to the previous graph. Using the methods in Chapter 17 we estimate the surface of the upper body to be $1,bold{m}$ . Entering these values into the convection equation:

$\begin{equation*} Q/t = (27\, \bold{\frac{W}{m^2\cdot K}})(1\,\bold{m^2})(40.9\,\bold{C^{\circ}}) \approx 1100\, \bold{W} \end{equation*}$

The convective heat loss rate in this situation would completely overwhelm the 100 W resting thermal power of a typical person and body temperature would drop quickly. To see how quickly, check out the Heat Capacity chapter. Shivering can boost the thermal power by up to 2.5 times, to rougly 250 W^[4], but even that would fall well short of balancing the convective heat loss rate. Finding shelter from the wind is an important part of the survival strategy. Carrying wind-stopping clothing is an important part of being prepared in the wilderness. Even a large plastic trash bag that could be worn over most of the body would significantly cut down on heat loss by convection.

Natural Convection

Wind will not penetrate a well-sealed window, so it seems like forced convection should not be a significant factor in the heat loss through a window. However, a single pane of glass does a poor job of preventing conduction, so significant thermal energy still crosses the barrier. That thermal energy will warm up a boundary layer on the outside of the window, which could then be stripped away by wind, so in fact wind chill may affect the rate of heat loss through the window. Double pane windows reduce conduction by using a layer of trapped air between two panes of glass.

A diagram showing two glass panes separated by a gap filled with a gas. Arrows indicate that conduction occurs across the glass panes from right to left. Arrows indicate that a convection cell has formed in the gap, with gas rising on the right and falling on the left. — The gap within a double pane window is too small for efficient convection cells to form. Filling the gap with a less conductive, more viscous (or slow-moving) and low pressure gas further reduces both conduction and convection through the gas. A mixture of krypton and argon gases is often used as a compromise between thermal performance and cost. Image Credit: “Gas Filled Windows” by John A Dutton e-science Institute, Penn State College of Earth and Mineral Sciences

We know that conduction is reduced across a double pane window because the thermal conductivity of air is exceptionally low, but the air gap in a double pane windows is typically only about 2 cm. Considering that the rate of heat transfer by conduction decreases as the thickness of the air layer increases, why don’t double pane windows have a much larger gap? Why not minimize conduction by making the gap almost the full thickness of the wall? Natural convection is the answer to that question. Natural convection is transfer of heat due to fluid movement caused by thermal expansion of the fluid itself, rather than by external factors such as wind. For example, you warm the air next to your skin and that air expands. After expanding, that same mass of air now has a larger volume, thus by definition it has a lower density. Being less dense than the surrounding cooler air, the warm air will float upward, as you can see in the previous thermal image of a person’s hand. (To remind yourself why the warmer and less dense air will rise, see the earlier chapter on buoyant forces).

As the warmed air rises away from the skin it is replaced by cool air moving in from each side of the warm surface, which is then warmed before rising, creating a cyclic flow pattern known as a convection cell. Overall the convection cells transport thermal energy away from the skin (or any warm object).

A cross section of a room is shown. There is a gravity furnace at the left side. The hot air from the furnace is rising up and is shown with the help of upward-pointing arrows along the left wall that are labeled hot air rises. The arrows then become horizontal and pass just under the ceiling to the right wall. The arrows then curve downward, become blue, and pass down the right wall and are labeled air cooled by room sinks. Finally, the blue arrows curve and pass along the floor to return to the gravity furnace. — A convection cell powered by thermal energy input from a furnace transfers thermal energy throughout a room. Unfortunately most of the warm air is the top half of the room, not near the bottom half where people usually hang out. Image Credit:OpenStax, College Physics

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Using double pane windows with a large air gap would allow for large convection cells to form and those cells would efficiently transfer thermal energy across the gap. Keeping the gaps size small prevents large convection cells from forming.

Everyday Examples: Down, double-pane windows, fiberglass and fur

Most insulation strategies create a fiber matrix that traps small pockets of air to exploit its low thermal conductivity while preventing large convection cells from forming. Down, fiberglass batting, and fur are examples of this strategy. The fibers of these materials have conductivity greater than air, so trapping more air with fewer fibers provides the lowest overall average conductivity. In fact, some animals, such as polar bears, have hollow fur to increase the ratio of air to matrix material.^[6] The best insulation strategies combine an air trapping matrix to minimize conduction and natural convection with a wind-stopping outer coating to prevent forced convection. In wet climates the wind-stopping layer should also be water proof to prevent water from filling the air pockets created by the underlying fiber matrix.

Wind Power

Convection cells drive heat transfer in a wide variety of systems on many scales. In fact the wind that serves to drive forced convection from your body is actually caused by natural convection cells. Such cells can form due to differential heating and cooling of the local geography. For example, the air above the ocean may remain cooler while the air above the coast warms rapidly during the day, resulting in an on-shore breeze (sea breeze) during the day. The convection cell reverses at night, creating an offshore breeze.

Reinforcement Exercises

Draw the convection cells near a coastline both during the day and at night in order to show how on-shore and off-shore breezes are created by differential heating of ocean and coastline. Indicate the relative temperature of the water, land and air during day and night.

Wind is also produced by global scale convection cells. The following graphic shows the global-scale convection cells that drive the winds at various latitudes. Notice that the spinning of the Earth combined with the inertia of the air mass creates a Coriolis Effect, which causes the wind direction to curve away from the direction indicated at the bottom of the convection cell. You may notice that the latitudes of the great deserts and forests of the world match up with the boundaries between the cells. We will learn why in the next chapter.

A diagram of the Earth shows convection cells forming in the atmosphere near the Earth's surface. Arrows indicate that air rises near the equator, and 60 degree latitudes and falls near 30 degree latitude and at the poles. — Global convection cells and associated winds. Image Credit: Earth Global Circulation-en by Kaidor [CC BY-SA 3.0 (https://creativecommons.org/licenses/by-sa/3.0)], via Wikimedia Commons

The predictability and stability of large scale convection cells allows for the growing implementation of wind turbine power plants. As our predictive skills continue to improve, along with turbine efficiency and energy storage technology, wind power is becoming a viable option for some communities.