Passive Solar Design for New Mexico

Class IV

Review of previous classes:

• The Three Pillars of Passive Solar: Gain, Mass, and Insulation
• Guidelines
• Basic energy concepts: BTUs, solar constant, light transfer, heat transfer
• Heat Load Calculations

Thermal Mass:

• density (r), specific heat (c), conductivity (k)
• (Conventional) Heat Capacity (C): Q_stored = V C DT_{_material}, V = volume, C = heat capacity, DT = change in temperature of the material.
• Diurnal Heat Capacity (C_d) (See the article by Doug Balcomb called "Thermal Storage in Passive Buildings" in the Advanced Passive Solar Design Book - this appears to have appeared in Supplement One of an ASHRAE publication titled "Passive Solar Heating Analysis", published in November 1987): The formula involving diurnal heat capacity is Q_{stored, returned after 6pm} = A C_d DT. Note that the material's area is used here, not volume, in contrast to the conventional heat capacity. Thickness is still important, but is taken into account implicitly - the values for C_d must either be for a specified thickness, or for infinite thickness, in which case the material actually used must be thick enough to approximate the behavior of infinite thickness. This is possible of course because we are talking about the harmonic response, not a "dc" (static) response, wherein heat goes in and out on a daily cycle, so that it doesn't penetrate too deeply. As the general passive guidelines suggest, four inches is roughly adequate for materials with a high thermal mass index, and less for materials with low indices. Also note that the temperature swing in this case is for the air in the room. This allows one to add up a total diurnal heat capacity for many materials acting simultaneously in the same room.  In fact, once added up, the temperature swing can be predicted using DT_swing = .61 Q_solar/C_{d_total}, where Q_solar is the daily solar heat gain, and  C_{d_total} is the total diurnal heat capacity of all the material surfaces in the room. If the room temperature swing was not incorporated, for example, if you use the conventional head capacity, you'd have to determine in some way the swing of each material separately. To summarize, the use of area instead of volume, and room temperature swing instead of material temperature swing, allow the diurnal heat capacity approach to implicitly embody an enormous amount of information about the transfer of energy from the air to the material, and the propagation of heat in the material for a 24 hour cycle, making diurnal heat capacity easy to it use but extremely powerful for architectural design applications. 

Example: 1000 ft² house: Assume 120 ft² south glazing (12% of floor area); Assume thermal mass of concrete, with an area of six times south glazing = 720 ft², thickness of four inches. Volume of mass is therefore 720/3 = 240 ft³.  Heat storage accomplished with DT = 10 ^oF is Q_stored = V C DT = 240 x 30 x 10 = 72,000 Btu = .72 therm. This is a little low to provide adequate heat, but not awful. How about with thermal mass area of nine times south glazing with DT = 15 ^oF? Area is 1080 ft², Volume is 1080/3 = 360 ft³. Q_stored = V C DT = 360 x 30 x 15 = 162,000 Btu = 1.62 therm. This is more than enough (thus we have verified the 6-9 times mass area guideline). What if  diurnal heat capacity used instead? Q_stored = A C_d DT = 720 x 10.7 x 10 = 77,040 Btu = .77 therm. This is very close to the result of .72 therm calculated above using a 4 inch thickness! For the larger surface area and temperature swing,  Q_stored = A C_d DT = 1080 x 10.7 x 15 = 173,340 = 1.73 therm. Again, this is close to the corresponding value of 1.62 therm calculated above. Thus we have verified the four inch mass rule using the diurnal heat capacity concept. In fact, note from above that the diurnal heat capacity of concrete is 10.71 per square foot, which is also exactly one third of the heat capacity of concrete per cubic foot, 30. This makes sense, because a slab consisting of the first four inches of a cubic foot contains exactly one third of the mass of the cubic foot.   

• Thermal Wave Propagation: Here we will check the diurnal heat capacity concept from another perspective - by examining the dynamics of the thermal wave into concrete over a six hour time scale. Standard heat transport theory yields that:

Q = rate of heat flux into or out of an infinitely thick wall

= A k (T_outside - T_{inside-infinity}) /( p D t) ^1/2_, where D =k/(rc) (diffusion coefficient)

Q_stored = energy stored in an infinitely thick wall

= (2/p^1/2) A (rck)^1/2 (T_outside - T_{inside-infinity}) (t) ^1/2

Note that the thermal mass index directly controls the amount stored.

Example: Concrete, rck = 16.2, Assume (T_outside - T_{inside-infinity}) = 10 ^oF,

Q_stored = 45.4 A (t) ^1/2; 

The value of 111.2 Btu/ft² stored after 6 hours is almost exactly what the diurnal heat capacity would predict to be stored with a temperature change of 10 ^oF. Thus the value reported for the diurnal heat capacity looks quite reasonable.

Homework:

1. Calculate how many Btus and then therms of energy are provided by solar gain over a clear six hour period through a 200 square foot, southfacing window. Assume total transmission and misalignment losses of 20% through the window.
2. Calculate how many therms are lost through the window over a 24 hour period, if the average temperature inside and outside are 65 ^oF and 30 ^oF, respectively. Assume the window has a U value of .4 Btu/(hr ft^{2 o}F).
3. Compare the two answers above: What can you conclude?