Convective Heat Transfer: Buildings in the Cold

Engineering Toolbox has an equation (here) for calculating a convective heat transfer coefficient:
   

where v is the wind speed in m/s and h_c is in W/(m² K).  When I first found this equation I thought that I had a way to calculate the h_actual/h_NW ratio more accurately than in my earlier post about "wind chill" on buildings.  As I reviewed my previous post however, I realize that this is awkward to compare.  The ratio that I would obtain using the above equation is h_actual/h_NW = 2.833 (this is comparing convective heat transfer with 15 mph wind versus no wind).  To see that something isn't quite copacetic, observe that if we use the equation used there, namely,

       T_WC = T_S – (h_actual / h_NW)(T_S – T_actual)),

with 2.833 for the ratio and 88 °F as the skin temperature, we obtain

    T_WC = 88 – (2.833)(88 – (–31)) = –249 °F,

the implausibility of which suggestion I am satisfied. Radiative heat loss may be complicating the situation. But whatever the reason, the wind chill equation makes no promises to us about its applicability to anything other than humans.

The current day wind chill index is a bit of a backward way of looking at things. Notice that it is called an index. It is not a physical quantity. It is a measure of a phenomenon. The wind doesn't make the temperature outside colder.  It does make bodies which are warmer than the air loose heat faster. But be cautious with relative terms. Something does not really loose heat "faster".  Rather, it may lose heat "faster than something else". It is more helpful (to the understanding) to say that the absence of wind results in a building loosing heat less quickly than a wind swept building.

I say this because of the way one sets up the steady state heat equations. We set the temperature of the outside of the outer air film to be equal to the ambient outdoor temperature. And we assume a small air film with R value something like 0.15 so that the outside surface of the building is not much different in temperature than the ambient air temperature. However, this neglects the affects of sunlight. In the absence of wind, sunlight has the most beneficial effect to the exterior of a building (in winter). By heating up the exterior surface of the building, the thermal gradient across the envelope is less—there is less temperature difference from inside to outside—and therefore the rate of heat loss through the envelope is reduced. The benefit of sunlight is kept in check by the convection on the outside of the building. As the exterior of the building gets warmer, the temperature difference between it and the air increases, which increases the rate of heat loss from the surface of the building.  If there is significant wind, it will mean a larger convective heat transfer coefficient (relative to no wind) and therefore raise the rate at which exterior surface loses heat, which in time reduces the temperature difference between the air and the exterior surface.

Hence, in modeling a steady state temperature loss, the coldest the outside temperature of the envelope should be modeled at is the ambient outside temperature; this is the worst case scenario. To see that this is true, notice that at this point there is no temperature difference between the ambient and the exterior surface. Considering the convective heat transfer equation, this means there is no heat transfer. Thus,

       q = h_c·A·(T_S - T_air) = h_c·A·0 = 0,

by which observation, with some thought, you might realize that this means the outside surface is necessarily warmer than the ambient air temperature, by at least some amount. Otherwise your building would stop losing heat, which would be very efficient indeed if it was possible!

If you want to model the benefits of reduced wind conditions and sunlight, you might model the exterior temperature as being warmer than ambient.  In short, wind lessens the benefits of sunlight to the thermal performance of a building envelope, but it does not make the exterior colder than ambient.

What I am saying is that the reason why heating requirements may be higher in windier places is because the less windy places are receiving (and retaining) a greater amount of benefit from sunlight and therefore have lesser heating requirements.

If Not "Wind Chill", Then What?

If the wind chill index is not totally real (though it is fine for its intended purpose), what should we be considering? The physical quantity that really matters is called heat flux. Heat is the transfer of thermal energy (1, p. 53). Of course, when a transfer is happening, it is happening at some rate.  So you can speak of loosing or gaining heat at a rate specified in Watts, kilowatts, Btu/h, etc. Heat flux is the rate of thermal energy gain/loss per unit time and area.  Possible units of measure include W/m² and Btu/(h ft²).

Solar radiation is often expressed in these units.  Note however, that to model the effect of the sunlight effectively requires you to determine the fractional portion of the incident radiation which is actually converted into thermal energy. You also need to scale the result by the cosine of the incidence angle (measured perpendicular to the surface normal) since a non-direct (non-normal) angle "spreads out" a given amount of radiation over a broader area.

One of the merits of working with heat flux, is that you can compare a tendency to heat loss between objects of varying surface area. A really large building might be losing heat faster than a small building even though it is better insulated. If you wanted to compare their tendency to loose heat on a kind of level playing field, you would divide the rate of heat loss by surface area.  Your result would be heat flux—actually, average heat flux.

If you want to calculate heat flux (φ), divide area out of the convective heat transfer equation:
   

(For the convective heat transfer equation, see ASHRAE Fundamentals chapter 4 or [1].)

But we have now touched on an important issue in energy efficiency: bigger isn't always better. If you can reduce your volume, you reduce the amount of stuff to heat.  (This is most pertinent in spaces with redundant air space, where the hot air rises to the top and is useful only for increasing the rate of heat loss through your roof—and when I say useful, I mean not useful.)  If you reduce your surface area, you reduce the area through which you lose heat. If you don't need big, go small. After you decide the space you really need, configure the space to reduce exterior surface area, subject of course to meeting the functional requirements of the space.

An interesting case of balancing surface area with other considerations is in solar energy greenhouses (SEGs).  A long building a few meters wide with a broad glazed wall facing south (up here in Canada) allows the building to maximize the amount of sunlight captured and stored in the north thermal mass wall. A square footprint would reduce the surface area for a given volume, but it would result in redundant air space. (Plants at the south wall would be too far from the north wall to benefit from it and the roof would need to be higher to allow the sunlight to reach the north wall.)  For more information on these greenhouses, see [2] and [3].

Sources:
1. World of Energy, Chapter 4: Transfer of Thermal Energy, http://www.physics.ohio-state.edu/p670/textbook/Chap_4.pdf
2. Bomford, M., Solar greenhouses, Chinese-style, http://energyfarms.wordpress.com/2010/04/05/solar-greenhouses-chinese-style/, 2010
3. Love, M., The solar solution, http://www.greenhousecanada.com/content/view/1562/38/
4. Wind Energy Institute of Canada, Wind Chill Temperature Index, http://www.weican.ca/links/011101-ec-windchill-index.php
5. Engineering Toolbox, Convective Heat Transfer, http://www.engineeringtoolbox.com/convective-heat-transfer-d_430.html
6. Irvine, D., Convective Heat Transfer on a Building Envelope (Wind Chill?),  http://darrenirvine.blogspot.com/2012/12/convective-heat-transfer-on-building.html, 2012

Convective Heat Transfer on a Building Envelope (Wind Chill?)

I'd like to make a brief mathematical investigation into the concept of a wind chill factor and how one particular scheme common in popular culture (and even affecting some in industry) does not relate well to modelling of a building envelope.  Wind chill is the temperature which a human being perceives accounting for the effects of wind and temperature together.  The preceding sentence is awkward because it is trying to say too much at once, so let me try again:

A wind chill index (expressed in units equivalent(ish) to temperature measurements) is answering a question:  What temperature T_WC (with wind speed = 0) will be perceived by a human being as equivalent to temperature T_actual (with wind speed = V_actual)?  (We are talking here more particularly about the exposed skin of a human being.  Also, it isn't purely a matter of perception.)

Human beings, not buildings.  Consider the following differences:

• buildings (in cold climates) have a much lower surface temperature than humans
• the amount (and temperature) of moisture on their surfaces is generally different and has a different internal transport mechanism (brick has basically no moisture and that moisture is already cold, human skin normally has moisture—perspiration—which evaporates more rapidly in windy conditions than calm conditions causing heat loss, because evaporation requires energy)
• unlike human beings, buildings do not have psychological comfort issues affecting their perception of temperature (and, admittedly, cannot reasonably be considered to perceive anything)

The equation used today for wind chill index has some theoretical basis but in the end is empirically derived.  That means they did experiments and found something that "works".  It is a very pragmatic way to deal with complex issues, especially ones that include psychological/cognitive components.  (I'm not "disrespecting" empirical equations, they're useful.)

So, where do I start? I would hard pressed with my current knowledge/understanding to deal with the evaporative cooling effects on human skin as compared with buildings, though such investigation might be meritorious. I will not attempt to compare the psychology of buildings with that of humans—I hope you understand. I will however seek to show how the concept of an "equivalent temperature in the absence of wind" yields different results depending on the surface temperature of the material in question. As such, although we will not compare human skin with face brick, we will compare warm Surface X with cool Surface X.

Convective Heat Transfer

The convective heat transfer equation is simple enough:

                q = h_c A_S (T_S – T_air)

where q is the heat energy (per unit time), h_c is the heat transfer coefficient, A_S is the surface area, T_S is the temperature of the surface and T_air is the ambient air temperature.
The heat transfer coefficient varies with wind speed.  Suppose we want to find an air temperature T_WC (no wind, h_NW) which causes the same rate of heat loss as T_actual (wind speed = W_actual, h_actual).  In that case, we need to relate:

                h_NW A_S (T_S – T_WC) = q = h_actual A_S (T_S – T_actual),

which simplifies to

               T_WC = T_S – (h_actual / h_NW)(T_S – T_actual).

A Little Problem

This leaves me with a problem. I want to have some type of reasonable value for the convective heat transfer coefficient ratio (h_actual / h_NW).  How do I get that?  I will engage in a little pragmatism and pretend that the wind chill equation will still give me a way to get an (h_actual / h_NW) ratio. This is perhaps a dubious step, but may suffice for illustrative purposes. From the wind chill equation we get a 15 mph wind at -31°F being equivalent to no wind at -59°F. Here's what we get for the ratio for human skin (which I have to use since that's what the wind chill equation is based on):

            (h_actual / h_NW) =  (T_S – T_WC) / (T_S – T_actual)
                                   =  (88 – (-59)) / (88 – (-31))
                                   = 1.235

(I've used 88°F for human skin, but that temperature will be different depending on the ambient temperature. Oh, well, what do you do when you live in a shoe?)

Theoretically, the convective heat transfer coefficient is largely independent of temperature and area, but it will change based on the interaction of the fluid and the surface which will include things like geometry, orientation, surface roughness, moisture, etc.  Unfortunately, temperature differences may affect the actual coefficient. They will affect the result if you consider radiation losses as well, since radiation transport is proportional to absolute temperature to the fourth power.

Wind Chill Effect on Surfaces at Different Temperatures Than Human Skin?

So, let's consider a surface which is maintained at a temperature of 10°F and is surrounded by an ambient temperature of -31°F and the wind speed is 15 mph.  (This is not the temperature in the boundary layer. The point of the convective heat transfer equation is to deal with this phenomenon in a simplified way—essentially to circumvent it.)  So, what's my "equivalent temperature in the absence of wind"?

               T_WC = T_S – (h_actual / h_NW)(T_S – T_actual)
                       = 10 – 1.235 (10 – (-31))
                       = -40.6 °F

So, the rate of heat loss from Surface X under the actual conditions is the same as the rate of heat loss for Surface X with no wind and a temperature of -40.6°F.

We obtained a different value than we did for the wind chill index applicable to human skin. The reason is straightforward: the rate of heat loss is proportional to the difference between the ambient air temperature and the surface temperature. Now, if it is -31°F outside, what temperature is the outside surface of a building? Perhaps -21°F? This makes quite a difference:

               T_WC = T_S – (h_actual / h_NW)(T_S – T_actual)
                       = -21 – 1.235 (-21 – (-31))
                       = -33.4 °F

So the effect is not zero, but it isn't much to talk about. If my h ratio is actually much higher, we will get a greater difference.

Considerations on Building Outside Surface Temperatures

The thickness and properties of the building envelope materials may change the relationship between ambient air temperature and surface temperature. Such changes will change the rate of heat loss in proportion to the temperature difference. Is there a higher difference at glazing surfaces than concrete surfaces?

There's also a sticky point regarding solar heat gains. Such gains on the surface will increase the rate of heat loss at the surface. But watch out! This is mainly behaviour occurring at the surface. We're dealing with a cold climate situation. Radiant heat from the sun heats up the surface only to be removed by increased rates of convection and radiation loss. This does not help us analyze the heat losses from the interior of the building! Plain "horse sense" tells me that adding thermal energy to the exterior does not cause a net increase to the rate of heat loss from the building interior. At worst, the increased rate of convective (and radiation) loss will remove all of the radiant energy added. To deal thoroughly with this issue, we would either need to model both together (allowing them to interact numerically with each other—for transient analysis) or use a "net direct solar gain" approach that deals with the interaction of convection and radiation (gains and losses) at the surface in an integrated way. The obvious outcome is that the rate of heat loss from the interior is reduced, but determining how much could be fun times.

Note that in warm climate conditions (cooling conditions), heat on the outside surface migrates toward the interior and we must therefore consider it. It is no longer starting to go in only to double back* (as it were), but actually conducting through the envelope and so directly affecting interior conditions. (*Analogical language enjoys talking about physics.)

CAUTION: This is NOT a Final Answer

In closing, this article is not trying to present the final answer to this problem. Rather I am attempting to expose the fallacy that the wind chill index (and corresponding formula) can be naively applied to energy modelling problems in building science. Furthermore, my criticism is not directed against nor does it address considerations of the effects of wind on air leakage analysis.

Note that ASHRAE Fundamentals chapter 26 (2009) gives effective R-values which can be used. The point of these values is to bypass all of the stuff I'm writing about in this article and account for convection and radiation losses at the surface under a 15 mph wind design condition. You don't actually need all that wind chill stuff—chapter 26 R-values already give a reasonable estimate of the wind effects, i.e., R-0.17 at 15 mph on the exterior instead of R-0.68 on the interior for still air at a wall.