7.8.1 Primary Reference
- Chapter 2 of Reddy (as noted in syllabus)
7 Thermal Dynamics of Buildings: Part I
7.1 Lecture Overview
7.2 Building Thermodynamics
We are now ready to address some of the fundamental technical concepts in building science, which are needed for us to reason about (plan) the data that our autonomous technologies will be gathering (i.e., what they sense). As we saw in the previous chapter, energy use in buildings is dominated by heating and cooling services. Thus, we will begin by reviewing basic concepts of thermodynamics leading up to the establishment of simple dynamic models of the temperature inside buildings and the energy required to maintain it.
7.2.1 Heat Loss in Buildings: A High-Level View
In a very general sense, buildings lose (and gain) heat through interactions with the outdoor environment, and through internal processes that compensate for that. During a whole season (say, a full year), the energy lost through the building envelope is (again, at a high level) proportional to the leakiness of the building envelope, and to the temperature demand (i.e., the average difference between the indoor and outdoor temperatures). So, if we were to write an equation for the total (say, heating) losses \(Q\) that need to be compensated to keep the building at a given internal temperature (\(T_i\)), we can write the following high-level formula:
\[Q = \text{leakiness} \times \text{temperature-demand}\]
This leakiness is a function of the properties of the building’s envelope. A bit more specifically, it is mostly affected by the way the envelope handles heat conduction and air infiltration and is usually expressed in units of energy per time per degree of temperature difference. The temperature demand is a function of the accumulated difference between the exterior temperature \(T_o\) and the average temperature we wish to keep the indoor space \(T_i\) over the season in question. This is usually described and tabulated as Heating Degree-Days (HDD) or Cooling Degree-Days (CDD), where Degree-Days refers to the accumulation (e.g., integral) of the difference in temperature (\(T_o - T_i\)) over the season (expressed as number of days).
The leakiness in the above equation represents the rate at which heat is lost through the building enclosure in proportion to the temperature difference (i.e., it’s in units of power per degree of temperature difference). This happens through a combination of thermal conduction, radiation and convection as well as through infiltration (air exchange).
Tip
Read Appendix E of MacKay (2008) for more details on this topic. See if you can re-calculate the numbers in Figure E.12 that refer to “my house, before” and “my house, after”.
7.2.2 Heating and Cooling Degree-Days
Degree-days provide a practical way to quantify the “temperature demand” in our high-level heat loss equation. A degree-day compares the mean outdoor temperature recorded for a location to a standard base temperature, typically 65°F (18.3°C) in the United States. This base temperature represents the outdoor temperature at which, on average, buildings require neither heating nor cooling to maintain comfortable indoor conditions.
7.2.2.1 Calculating Degree-Days
The calculation is straightforward. In theory, we would just compute the integral of the difference between outdoor temperature and a base temperature reference. But since we’re using discrete measurements we perform a sum. For any given day:
Since temperature during the day changes, we need to start with an average value for the day. You may, for example, compute the mean daily temperature from the extreme values recorded: \(T_{mean} = \frac{T_{high} + T_{low}}{2}\)
then compare to the base temperature (65°F):
- If \(T_{mean} < 65°F\): Heating Degree-Days (HDD) \(= 65 - T_{mean}\)
- If \(T_{mean} > 65°F\): Cooling Degree-Days (CDD) \(= T_{mean} - 65\)
- If \(T_{mean} = 65°F\): Both HDD and CDD are zero
For example, a day with a high of 90°F and a low of 66°F has a mean of 78°F, yielding \(78 - 65 = 13\) cooling degree-days. Conversely, a day with a high of 33°F and a low of 25°F has a mean of 29°F, yielding \(65 - 29 = 36\) heating degree-days.
Finally, annual degree-days are simply the sum of daily values over a year, providing a single number that characterizes how much heating or cooling a location requires.
7.2.2.2 Typical Values Across Climates
The table below shows approximate annual HDD and CDD values for selected U.S. cities, illustrating the dramatic variation across climate zones:
| City | Climate | Annual HDD | Annual CDD |
|---|---|---|---|
| Miami, FL | Hot-humid | ~200 | ~4,400 |
| Phoenix, AZ | Hot-dry | ~1,200 | ~4,600 |
| Atlanta, GA | Mixed-humid | ~2,800 | ~1,800 |
| Pittsburgh, PA | Cold | ~5,800 | ~800 |
| Chicago, IL | Cold | ~6,200 | ~900 |
| Minneapolis, MN | Very cold | ~7,600 | ~700 |
Values are approximate based on 1981-2010 climate normals. For current data, see the NOAA Climate Prediction Center or DegreeDays.net for worldwide locations.
Notice how heating-dominated climates (Pittsburgh, Chicago, Minneapolis) have HDD values 5-10 times larger than their CDD values, while cooling-dominated climates (Miami, Phoenix) show the opposite pattern. This has profound implications for building design: a well-insulated envelope is critical in Minneapolis, while shading and cooling efficiency matter more in Phoenix.
7.2.2.3 Using Degree-Days for Energy Estimation
Returning to our high-level equation, we can now be more precise. If we express the leakiness as an overall heat loss coefficient \(K\) (in units of power per degree of temperature difference, such as W/°C), then the seasonal heating energy can be estimated as:
\[Q_{heating} = K \times \text{HDD} \times 24 \text{ hours/day}\]
The factor of 24 converts degree-days to degree-hours, giving us energy (since \(K\) has units of power per degree). This simple model, sometimes called the degree-day method, allows quick estimates of heating fuel requirements. For instance, if a building in Pittsburgh (\(\approx 5,800\) HDD) has the same envelope as one in Miami (\(\approx 200\) HDD), the Pittsburgh building would require roughly \(5800/200 = 29\) times more heating energy annually.
Note
The degree-day method is a useful first approximation, but it has limitations. It assumes a fixed base temperature, ignores internal heat gains from occupants and equipment, and treats the building as if it responds instantaneously to outdoor temperature changes. We will revisit these assumptions later when we discuss dynamic thermal behavior.
7.3 Fundamental Thermal Concepts
Before diving into heat transfer mechanisms, we need to establish some fundamental definitions.
7.3.1 Temperature
Temperature is a measure of the average kinetic energy of the molecules in a substance. When we heat a material, its molecules move faster (or vibrate more intensely in solids), and we perceive this as a higher temperature. Importantly, temperature is an intensive property—it does not depend on the amount of material present. A cup of boiling water and a pot of boiling water are at the same temperature, even though the pot contains more thermal energy.
7.3.1.1 Temperature Scales
Four temperature scales are commonly encountered in building science:
| Scale | Symbol | Water Freezes | Water Boils | Absolute Zero |
|---|---|---|---|---|
| Celsius | °C | 0 | 100 | -273.15 |
| Fahrenheit | °F | 32 | 212 | -459.67 |
| Kelvin | K | 273.15 | 373.15 | 0 |
| Rankine | °R | 491.67 | 671.67 | 0 |
The conversions between these scales are:
\[T_F = \frac{9}{5}T_C + 32 \qquad T_K = T_C + 273.15 \qquad T_R = T_F + 459.67\]
In building science, we typically use Celsius (or Fahrenheit in the U.S.) for everyday temperatures, but Kelvin becomes essential when dealing with radiation heat transfer, where absolute temperature matters.
7.3.1.2 Temperature vs. Heat
A critical distinction that often causes confusion:
- Temperature is a state property—it describes the current thermal condition of a substance (how “hot” it is).
- Heat is energy in transit—it refers to the transfer of thermal energy between systems due to a temperature difference.
An analogy: temperature is like the water level in a tank, while heat is like the flow of water between tanks. Water flows from higher to lower levels; heat flows from higher to lower temperatures. Two objects at the same temperature exchange no heat, regardless of how much thermal energy each contains.
This distinction matters practically: when we say a building “loses heat,” we mean thermal energy is flowing out of the building because the interior is warmer than the exterior. The rate of this heat loss depends on the temperature difference, as we will see repeatedly in the sections that follow.
7.3.2 Heat Capacity
Heat capacity describes a material’s ability to store thermal energy. The heat capacity \(C\) of an object is defined as:
\[C = cm\]
where \(c\) is the specific heat capacity of the material and \(m\) is its mass. The specific heat capacity can be understood through the relationship:
\[c = \frac{Q}{m \Delta T}\]
where \(Q\) is the heat added and \(\Delta T\) is the resulting temperature change.
7.3.2.1 Specific Heat of Building Materials
The table below lists specific heat capacities for materials commonly encountered in buildings:
| Material | Specific Heat \(c\) (J/(kg\(\cdot\)K)) | Density \(\rho\) (kg/m³) |
|---|---|---|
| Water | 4,186 | 1,000 |
| Wood (typical) | 1,200 | 400-700 |
| Gypsum board | 1,000 | 800 |
| Air (at 20°C) | 1,005 | 1.2 |
| Concrete | 880 | 2,300 |
| Brick | 840 | 1,700 |
| Glass | 840 | 2,500 |
| Stone/Marble | 880 | 2,600 |
| Steel | 420 | 7,850 |
| Aluminum | 900 | 2,700 |
Values are approximate and vary with composition and moisture content. Sources: Engineering ToolBox, GreenSpec.
Notice that water has an exceptionally high specific heat—roughly 4-5 times that of most solid building materials. This is why water-based heating and cooling systems are so effective at transporting thermal energy.
7.3.2.2 Thermal Mass in Buildings
While specific heat tells us how much energy is needed to raise a unit mass of material by one degree, what often matters in buildings is how much energy is needed to raise a unit volume. This is the volumetric heat capacity:
\[\rho c = \text{density} \times \text{specific heat} \quad \text{[J/m}^3\text{·K]}\]
Materials with high volumetric heat capacity can store significant amounts of thermal energy without large temperature swings. This property, called thermal mass, has important implications for building design:
- Temperature stabilization: High thermal mass materials (concrete, brick, stone) absorb excess heat during warm periods and release it during cool periods, moderating indoor temperature swings.
- Peak load shifting: Thermal mass can delay peak cooling loads by several hours, potentially shifting air conditioning demand to off-peak electricity rates.
- Passive solar design: In climates with significant day-night temperature differences, thermal mass can capture daytime solar gains and release them at night, reducing heating loads.
For comparison, water has a volumetric heat capacity of approximately 4,186 kJ/m³·K, while concrete is around 2,000 kJ/m³·K—meaning water stores about twice as much heat per unit volume. This is why phase-change materials and water tanks are sometimes used for thermal energy storage in buildings.
Note
Thermal mass is most beneficial in climates with significant diurnal (day-night) temperature variation. In consistently hot or consistently cold climates, the benefits are reduced, and insulation typically provides better returns.
7.4 Modes of Heat Transfer
Let’s now more formally study thermal conduction, convection and radiation.
7.4.1 Conduction
Joseph Fourier (1768 - 1830) came up with the law of conduction heat transfer, which relates the rate at which heat transfer occurs through a material, to the temperature difference it is subjected to, and the distance through which conduction is occurring. It is actually very similar to Ohm’s law relating current (the rate at which electric charge flows through a conductor, to the voltage difference and the resistance). In particular, Fourier’s law shows that:
\[\dot{Q} = -k A \frac{dT}{dx}\]
Where \(k\) is the thermal conductivity of the material, \(A\) is the area through which heat is flowing, and \(\frac{dT}{dx}\) is the temperature gradient at a specific point \(x\) in the material.
7.4.1.1 Conduction Through a Plane Wall
For steady-state conduction through a plane wall:
\[\dot{Q} = k A \frac{T_1 - T_2}{\Delta x}\]
(Assuming many things, including \(T_1 > T_2\), no heat sources within the material, constant thermal conductivity, etc.)
This can also be stated as:
\[\dot{Q} = \frac{T_1 - T_2}{\Delta x / (kA)}\]
Which is similar to how one would define Ohm’s Law (i.e., \(I = V/R\)), and shows that the term \(\Delta x / (kA)\) can be thought of as the thermal resistance \(R\) of the wall.
NoteWorked Example: Heat Loss Through a Concrete Wall
Consider a 200 mm (0.2 m) thick concrete wall with thermal conductivity \(k = 1.4\) W/m·K and area \(A = 10\) m². If the inside surface is at 20°C and the outside surface is at 5°C, what is the rate of heat loss?
\[\dot{Q} = k A \frac{T_1 - T_2}{\Delta x} = 1.4 \times 10 \times \frac{20 - 5}{0.2} = 1{,}050 \text{ W}\]
The thermal resistance of this wall is:
\[R = \frac{\Delta x}{kA} = \frac{0.2}{1.4 \times 10} = 0.0143 \text{ K/W}\]
Or, expressed per unit area (which is more commonly tabulated):
\[R'' = \frac{\Delta x}{k} = \frac{0.2}{1.4} = 0.143 \text{ m}^2\text{·K/W}\]
This area-normalized thermal resistance is what building codes typically call the R-value. Its inverse is the U-value (thermal transmittance): \(U = 1/R'' = 7.0\) W/m²·K. We will explore R-values and U-values in more detail in the next section.
7.4.2 Convection
Convection is heat transfer between a surface and a moving fluid (which, in buildings, is usually air). Unlike conduction, which occurs through stationary material, convection involves bulk fluid motion that carries thermal energy.
7.4.2.1 Newton’s Law of Cooling
The rate of convective heat transfer is described by Newton’s law of cooling:
\[\dot{Q} = h A (T_s - T_\infty)\]
where:
- \(h\) is the convection heat transfer coefficient (W/m²·K)
- \(A\) is the surface area (m²)
- \(T_s\) is the surface temperature
- \(T_\infty\) is the bulk fluid temperature far from the surface
This can be rewritten in resistance form:
\[\dot{Q} = \frac{T_s - T_\infty}{1/(hA)}\]
showing that the convective thermal resistance is \(R_{conv} = 1/(hA)\), or per unit area: \(R''_{conv} = 1/h\).
7.4.2.2 Natural vs. Forced Convection
The convection coefficient \(h\) depends strongly on whether the fluid motion is driven by external means or by buoyancy:
Natural (free) convection: Fluid motion is driven by density differences caused by temperature variations. Warm air rises, cool air sinks. This is the dominant mode for interior building surfaces in still air.
Forced convection: Fluid motion is driven by external means such as wind, fans, or pumps. This typically results in much higher heat transfer rates.
7.4.2.3 Typical Values for Buildings
| Situation | \(h\) (W/m²·K) |
|---|---|
| Still air (natural convection, vertical surface) | 5-10 |
| Still air (natural convection, horizontal surface, heat rising) | 7-12 |
| Light wind (~2 m/s) on exterior surface | 10-15 |
| Moderate wind (~5 m/s) on exterior surface | 20-30 |
| Strong wind (~10 m/s) on exterior surface | 40-60 |
Values are approximate and depend on surface geometry, orientation, and temperature difference.
Building codes typically use standardized surface resistances for interior and exterior surfaces. For example, ASHRAE specifies an interior surface resistance of approximately \(R''_{si} = 0.12\) m²·K/W (corresponding to \(h \approx 8\) W/m²·K) and an exterior surface resistance of \(R''_{se} = 0.03\) m²·K/W for winter conditions (corresponding to \(h \approx 33\) W/m²·K, assuming some wind).
7.4.3 Radiation
Radiation is heat transfer via electromagnetic waves. Unlike conduction and convection, radiation requires no physical medium—it can occur across a vacuum. All objects above absolute zero emit thermal radiation, with the intensity and spectrum depending on their temperature.
7.4.3.1 The Stefan-Boltzmann Law
The maximum rate at which a surface can emit thermal radiation is given by the Stefan-Boltzmann law:
\[\dot{Q}_{emit} = \varepsilon \sigma A T^4\]
where:
- \(\varepsilon\) is the emissivity of the surface (0 to 1, dimensionless)
- \(\sigma = 5.67 \times 10^{-8}\) W/(m\(^2\)\(\cdot\)K\(^4\)) is the Stefan-Boltzmann constant
- \(A\) is the surface area
- \(T\) is the absolute temperature in Kelvin
A surface with \(\varepsilon = 1\) is called a blackbody—it emits the maximum possible radiation for its temperature. Real surfaces have \(\varepsilon < 1\).
7.4.3.2 Emissivity of Common Building Materials
| Material | Emissivity \(\varepsilon\) |
|---|---|
| Brick, concrete, stone | 0.90-0.95 |
| Glass | 0.90-0.95 |
| Wood | 0.85-0.90 |
| White paint | 0.85-0.95 |
| Aluminum (polished) | 0.04-0.06 |
| Aluminum (anodized) | 0.70-0.80 |
| Low-e coating | 0.04-0.10 |
The low emissivity of polished metals and specialized low-e coatings is exploited in window design to reduce radiative heat transfer while maintaining visible light transmission.
7.4.3.3 Radiation Exchange Between Surfaces
When two surfaces “see” each other, they exchange radiation. The net heat transfer from surface 1 to surface 2 depends on their temperatures, emissivities, and geometry:
\[\dot{Q}_{1 \to 2} = \varepsilon_{eff} \sigma A_1 F_{1-2} (T_1^4 - T_2^4)\]
where \(F_{1-2}\) is the view factor—the fraction of radiation leaving surface 1 that reaches surface 2. View factors depend on geometry and can be complex to calculate, though tabulated values exist for common configurations.
For building applications, a useful simplification for small temperature differences is to linearize the radiation exchange:
\[\dot{Q}_{rad} \approx h_r A (T_1 - T_2)\]
where \(h_r\) is a radiative heat transfer coefficient:
\[h_r = 4 \varepsilon \sigma T_m^3\]
with \(T_m\) being the mean absolute temperature of the two surfaces. At typical building temperatures (~300 K), this gives \(h_r \approx 5\) W/m²·K for high-emissivity surfaces—comparable to natural convection.
Tip
You might want to prove to yourself that the linear approximation is appropriate (and under what conditions). Reading Chapter 2 (page 51) of Reddy et al. (2016) could help with this.
7.4.3.4 Solar Radiation
The sun is a special case of radiative heat transfer. Solar radiation incident on buildings is typically characterized by:
- Direct (beam) radiation: Arrives in a straight line from the sun
- Diffuse radiation: Scattered by the atmosphere, arrives from all directions
- Reflected radiation: Bounced off surrounding surfaces (ground, other buildings)
The total solar radiation on a surface is called irradiance and is measured in W/m². Peak values on a surface perpendicular to the sun can reach 1,000 W/m² on a clear day. The fraction of incident solar radiation that is absorbed depends on the surface’s solar absorptance \(\alpha_s\), which can differ significantly from its thermal emissivity (especially for selective surfaces like low-e coatings).
TipLearn-by-Doing Activity: Comparing Heat Transfer Modes
Consider a single-pane glass window with the following properties:
- Glass thickness: 6 mm
- Glass thermal conductivity: \(k = 1.0\) W/m·K
- Indoor air temperature: 20°C
- Outdoor air temperature: 0°C
- Interior surface convection coefficient: \(h_{in} = 8\) W/m²·K
- Exterior surface convection coefficient: \(h_{out} = 25\) W/m²·K
- Glass emissivity: \(\varepsilon = 0.9\)
Tasks:
Calculate the thermal resistance per unit area for:
- Interior convection: \(R_{conv,in} = 1/h_{in}\)
- Conduction through glass: \(R_{glass} = \Delta x / k\)
- Exterior convection: \(R_{conv,out} = 1/h_{out}\)
Calculate the total R-value and U-value for the window (neglecting radiation for now).
What is the heat loss per square meter?
If the interior glass surface is at approximately 10°C, estimate the radiative heat transfer from the glass to a room at 20°C using \(h_r \approx 5\) W/m²·K. How does this compare to the convective heat transfer on the interior surface?
Expected insights: You should find that the glass itself contributes very little resistance (R = 0.006 m²·K/W). The surface film resistances dominate, and radiation contributes significantly to the interior surface heat transfer—which is why low-e coatings are so effective at improving window performance.
7.5 Thermal Properties of Building Materials
7.5.1 Thermal Conductivity
Thermal conductivity (\(k\) or \(\lambda\)) measures how readily a material conducts heat. It is defined as the rate of heat transfer through a unit thickness of material per unit area per unit temperature difference, with units of W/m·K (or equivalently, W/m·°C).
Materials with low thermal conductivity are good insulators; materials with high thermal conductivity are good conductors. The table below shows representative values spanning several orders of magnitude:
| Material | Thermal Conductivity \(k\) (W/(m\(\cdot\)K)) | Category |
|---|---|---|
| Still air | 0.026 | Gas |
| Expanded polystyrene (EPS) | 0.035-0.040 | Insulation |
| Fiberglass batt | 0.040-0.045 | Insulation |
| Mineral wool | 0.035-0.045 | Insulation |
| Polyurethane foam | 0.020-0.028 | Insulation |
| Wood (softwood) | 0.12-0.15 | Structural |
| Gypsum board | 0.16-0.25 | Finish |
| Brick | 0.6-1.0 | Masonry |
| Glass | 0.8-1.0 | Glazing |
| Concrete (dense) | 1.4-2.0 | Structural |
| Stone (granite) | 2.5-3.5 | Masonry |
| Steel | 45-50 | Metal |
| Aluminum | 200-220 | Metal |
| Copper | 380-400 | Metal |
Values are approximate and vary with density, moisture content, and temperature. Sources: Engineering ToolBox, GreenSpec.
Notice that still air has extremely low thermal conductivity—this is why most insulation materials work by trapping small pockets of air (fiberglass batts, foam cells). The challenge is preventing air movement, which would introduce convective heat transfer.
7.5.1.1 Factors Affecting Thermal Conductivity
- Density: For many materials (especially insulation), lower density means more trapped air and lower conductivity—up to a point where the material becomes too sparse to suppress convection.
- Moisture content: Water has a thermal conductivity of about 0.6 W/m·K, much higher than air. Wet insulation performs poorly.
- Temperature: Conductivity generally increases with temperature, though the effect is small for typical building temperature ranges.
7.5.2 Thermal Resistance (R-value)
The R-value (thermal resistance) quantifies how well a material or assembly resists heat flow. For a single homogeneous layer, as we’ve seen before:
\[R'' = \frac{\Delta x}{k}\]
where \(\Delta x\) is the thickness and \(k\) is the thermal conductivity. Higher R-values mean better insulation.
7.5.2.1 Units and Conversion
R-value units differ between SI and Imperial systems, which causes frequent confusion:
| System | R-value Units | Symbol |
|---|---|---|
| SI (Metric) | m²·K/W | RSI |
| Imperial (I-P) | ft²·°F·h/BTU | R |
Tip
I may not be consistent in using \(R''\) to refer to the per-unit-area version of the thermal resistance throughout this material. So instead of relying on the variable name to distinguish between them, just make sure you look at the units that are being used.
The conversion factor is:
\[R_{Imperial} = 5.678 \times R_{SI}\]
For example, an R-19 fiberglass batt (Imperial) has RSI = 19/5.678 ≈ 3.35 m²·K/W.
Warning
When reading R-values, always check the units! An “R-19” wall in the U.S. is very different from “R-19” in a metric country. U.S. building codes use Imperial R-values; Canadian and European codes use RSI.
7.5.2.2 R-values Add in Series
A key property of thermal resistance: R-values are additive for layers in series. For a wall assembly with multiple layers:
\[R_{total} = R_1 + R_2 + R_3 + \cdots\]
This makes R-values convenient for calculating assembly performance. For example, a wall with:
- Interior air film: R-0.68 (ft²·°F·h/BTU)
- 1/2” gypsum board: R-0.45
- 3.5” fiberglass batt: R-13
- 1/2” plywood sheathing: R-0.62
- Exterior air film: R-0.17
has a total R-value of approximately R-15.
7.5.3 Thermal Conductance (U-value)
The U-value (thermal transmittance) is simply the inverse of R-value:
\[U = \frac{1}{R}\]
While R-value measures resistance to heat flow (higher is better), U-value measures how easily heat passes through (lower is better). U-values are typically expressed in W/m²·K (SI) or BTU/h·ft²·°F (Imperial).
7.5.3.1 Why U-values?
Building energy codes often specify U-values rather than R-values because:
U-values handle parallel paths: When heat can flow through multiple parallel paths (e.g., through insulation AND through studs), the effective U-value is the area-weighted average, while R-values cannot be directly averaged.
U-values work for complex assemblies: Windows, for example, have different R-values for the glazing, frame, and spacer. The overall window U-value accounts for all paths.
7.5.3.2 Typical U-values
| Assembly | U-value (W/m²·K) | U-value (BTU/h·ft²·°F) |
|---|---|---|
| Single-pane window | 5.5-6.0 | 0.95-1.05 |
| Double-pane window (air) | 2.7-3.0 | 0.47-0.53 |
| Double-pane, low-e, argon | 1.4-1.8 | 0.25-0.32 |
| Triple-pane, low-e, argon | 0.8-1.2 | 0.14-0.21 |
| Well-insulated wall | 0.2-0.3 | 0.035-0.053 |
| Passive House wall | <0.15 | <0.026 |
Window U-values are typically rated for the entire assembly including frame.
TipLearn-by-Doing Activity: Calculating Assembly R-values
A residential wall assembly consists of the following layers (from inside to outside):
| Layer | Thickness | Thermal Conductivity \(k\) |
|---|---|---|
| Interior air film | — | \(h = 8.3\) W/(m²·K) |
| Gypsum board | 12.5 mm | 0.16 W/m·K |
| Fiberglass batt insulation | 89 mm | 0.043 W/(m·K) |
| OSB sheathing | 11 mm | 0.13 W/(m·K) |
| Air gap (drainage cavity) | 25 mm | 0.02 W/(m·K) |
| Brick veneer | 90 mm | 0.72 W/(m·K) |
| Exterior air film | — | \(h = 34\) W/(m²·K) |
Tasks:
Calculate the R-value of each layer. For the air films, use \(R = 1/h\).
Sum all R-values to find the total assembly R-value in m²·K/W.
Convert to Imperial R-value (multiply by 5.678). How does this compare to typical code requirements?
Calculate the overall U-value (\(U_{overall}\)).
Calculate the overall U-value excluding the air films and brick veneer (\(U_{insulation}\)).
If wood studs (\(k = 0.12\) W/m·K) replace the wall assembly through its thickness (except for the brick veneer), covering 2% of the wall area, calculate the effective U-value of the whole assembly (including exterior air films and brick veneer) using the parallel path method.
Answers (check your work):
- Total R-value: approximately 3.757 m²·K/W (R-21 Imperial)
- \(U_{overall}\): approximately 0.266 W/(m²·K)
- \(U_{insulation}\): approximately 0.287 W/(m²·K)
- Effective U-value of studs + insulation in parallel (5% framing): approximately 0.316 W/(m²·K)
- With 5% framing, after adding back brick and air films: effective U-value increases to approximately 0.291 W/m²·K (a 9% degradation)
7.6 Thermal Resistance Networks
The equations for heat transfer through building assemblies can become complex when multiple layers, surfaces, and parallel paths are involved. Thermal resistance networks provide a systematic way to analyze these systems by drawing on an analogy with electrical circuits.
7.6.1 The Electrical Analogy
The mathematical similarity between heat flow and electrical current flow is striking:
| Electrical Domain | Thermal Domain |
|---|---|
| Voltage \(V\) (Volts) | Temperature \(T\) (K or °C) |
| Current \(I\) (Amps) | Heat flow rate \(\dot{Q}\) (W) |
| Electrical resistance \(R_e\) (Ohms) | Thermal resistance \(R\) (K/W) |
| Ohm’s Law: \(I = \frac{V_1 - V_2}{R_e}\) | Fourier’s Law: \(\dot{Q} = \frac{T_1 - T_2}{R}\) |
This analogy means we can:
- Draw thermal circuits just like electrical circuits
- Use the same rules for combining resistances (series and parallel)
- Apply circuit analysis techniques (Kirchhoff’s laws, node analysis)
The key insight is that temperature difference drives heat flow, just as voltage difference drives current flow. And just as current is conserved at a junction, heat flow must be conserved (energy balance).
7.6.2 Resistances in Series and Parallel
7.6.2.1 Series Resistances (Layers)
When heat flows through multiple layers in sequence (as through a wall), the resistances add:
\[R_{total} = R_1 + R_2 + R_3 + \cdots\]
The same heat flow \(\dot{Q}\) passes through each layer, but the temperature drops across each layer according to its resistance. This is analogous to resistors in series in an electrical circuit.
For a wall with interior air film, multiple material layers, and exterior air film:
\[R_{total} = R_{si} + \sum_i \frac{\Delta x_i}{k_i} + R_{se}\]
where \(R_{si}\) and \(R_{se}\) are the surface resistances (accounting for convection and radiation at the surfaces).
7.6.2.2 Parallel Resistances (Alternative Paths)
When heat can flow through different paths simultaneously (e.g., through studs AND insulation in a framed wall), the resistances combine in parallel:
\[\frac{1}{R_{total}} = \frac{1}{R_1} + \frac{1}{R_2} + \cdots\]
Or equivalently, using U-values (which are more convenient for parallel paths):
\[U_{total} = \frac{A_1}{A_{total}} U_1 + \frac{A_2}{A_{total}} U_2 + \cdots\]
This is an area-weighted average of the U-values, which explains why thermal bridging through studs, lintels, and other structural elements can significantly degrade the overall performance of an otherwise well-insulated wall.
7.6.3 Representing Wall Assemblies
A typical wall assembly can be represented as a series of thermal resistances from the indoor air to the outdoor air:
T_indoor T_outdoor
│ │
├──[R_si]──┬──[R_gypsum]──┬──[R_insul]──┬──[R_sheath]──┬──[R_se]──┤
│ │ │ │ │ │
T_1 T_2 T_3 T_4Where:
- \(R_{si}\) = interior surface resistance (~0.12 m²·K/W for still air)
- \(R_{gypsum}\) = gypsum board resistance = \(\Delta x / k\)
- \(R_{insul}\) = insulation resistance
- \(R_{sheath}\) = sheathing resistance
- \(R_{se}\) = exterior surface resistance (~0.03-0.06 m²·K/W depending on wind)
7.6.3.1 Worked Example: Simple Wall Assembly
Consider a wall with:
- Interior surface film: \(R_{si} = 0.12\) m²·K/W
- 12.5 mm gypsum board (\(k = 0.16\) W/m·K): \(R = 0.0125/0.16 = 0.078\) m²·K/W
- 90 mm fiberglass (\(k = 0.04\) W/m·K): \(R = 0.09/0.04 = 2.25\) m²·K/W
- 12 mm plywood (\(k = 0.13\) W/m·K): \(R = 0.012/0.13 = 0.092\) m²·K/W
- Exterior surface film: \(R_{se} = 0.04\) m²·K/W
Total R-value: \(R_{total} = 0.12 + 0.078 + 2.25 + 0.092 + 0.04 = 2.58\) m²·K/W
Overall U-value: \(U = 1/2.58 = 0.39\) W/m²·K
If the indoor temperature is 20°C and outdoor is 0°C, the heat loss per square meter is:
\[\dot{q} = U \times \Delta T = 0.39 \times 20 = 7.8 \text{ W/m}^2\]
7.6.3.2 Accounting for Thermal Bridges
In real framed walls, wood or steel studs create thermal bridges—paths of higher conductivity that bypass the insulation. A \(38 \times 89\) mm (\(2 \times 4\)) wood stud has:
\[R_{stud} = 0.089 / 0.12 = 0.74 \text{ m}^2\text{·K/W}\]
Compare this to the R-2.25 of the fiberglass it displaces! If studs occupy 15% of the wall area, the effective U-value is:
\[U_{eff} = 0.85 \times U_{insulated} + 0.15 \times U_{stud}\]
This “parallel path” calculation typically reduces the effective R-value by 10-25% compared to the nominal insulation R-value alone.
7.6.4 Representing Window Assemblies
Windows are more complex because they involve:
- Conduction through the glass
- Convection on both surfaces AND in any air gaps
- Radiation exchange between glass panes and across air gaps
A double-pane window can be represented as:
T_indoor T_outdoor
│ │
├──[R_si]──┬──[R_glass]──┬──[R_gap]──┬──[R_glass]──┬──[R_se]──┤
│ │ │ │ │ │Where \(R_{gap}\) includes both convection and radiation across the air (or gas) space. This is why:
- Argon or krypton fill improves performance (lower gas conductivity, suppressed convection)
- Low-e coatings dramatically reduce the radiative component of \(R_{gap}\)
- Wider gaps help—up to about 12-15 mm, beyond which convection currents begin to degrade performance
7.6.5 Introduction to Thermal Capacitance
So far, our thermal networks have been purely resistive—they describe steady-state heat flow where temperatures don’t change with time. But buildings have thermal mass, and temperatures do change: outdoor conditions vary, HVAC systems cycle, internal loads fluctuate.
To model dynamic behavior, we add thermal capacitance to our networks:
| Electrical Domain | Thermal Domain |
|---|---|
| Capacitance \(C_e\) (Farads) | Thermal capacitance \(C\) (J/K) |
| Charge storage: \(Q = C_e V\) | Heat storage: \(Q = C \cdot T\) |
| Current to capacitor: \(I = C_e \frac{dV}{dt}\) | Heat to thermal mass: \(\dot{Q} = C \frac{dT}{dt}\) |
The thermal capacitance of a building element is:
\[C = m \cdot c = \rho \cdot V \cdot c\]
where \(m\) is mass, \(c\) is specific heat, \(\rho\) is density, and \(V\) is volume.
7.6.5.1 The RC Circuit Analogy
A simple thermal mass (like a room) connected to the outdoor environment through a resistive envelope can be modeled as an RC circuit:
T_outdoor T_indoor
│ │
├────────[R_envelope]──────────┼────┐
│ │ │
C ===
│ │
GND GNDThe temperature response of this system follows the familiar first-order differential equation:
\[C \frac{dT_{indoor}}{dt} = \frac{T_{outdoor} - T_{indoor}}{R_{envelope}} + \dot{Q}_{internal}\]
This equation has a time constant \(\tau = RC\) that characterizes how quickly the building responds to changes. We will explore this in detail in Lecture 6, deriving thermal network models from first principles and understanding what these time constants mean for building control.
7.7 Where do we go from here?
Throughout this lecture, we have largely treated heat transfer as a steady-state phenomenon. Our equations assume that temperatures remain constant over time, that heat flow rates are unchanging, and that the only role of time is in the cumulative accounting of degree-days over a season.
But is this realistic? Consider the following questions:
How long does it take for a building to respond to a sudden change in outdoor temperature? If the temperature drops from 10°C to 0°C overnight, does the indoor temperature immediately begin falling at a new rate, or is there some delay?
When you turn on a heater, how long until the room warms up? Our steady-state analysis cannot answer this—it tells us only where the temperature will eventually settle, not how quickly it gets there.
Why do some buildings “coast” through cold snaps better than others? A massive stone building and a lightweight prefab building might have identical R-values, yet one stays comfortable much longer when the heating fails.
How should an HVAC controller anticipate occupancy? If people arrive at 8 AM, should the system start heating at 6 AM? 7 AM? This depends on the building’s dynamic response—something steady-state analysis cannot capture.
Can thermal mass help reduce peak cooling loads? We mentioned that thermal mass can shift loads to off-peak hours. But by how much? And under what conditions does this strategy actually help?
These questions all require understanding how building temperatures evolve over time. The missing ingredient is the time constant—a characteristic that emerges when we combine thermal resistance with thermal capacitance.
7.7.1 Preview of Part II
In Lecture 6, we will extend these steady-state concepts to dynamic thermal behavior by:
- Deriving thermal network models (RC circuits) from first principles using the heat equation
- Solving simple 1R1C and 2R2C networks to find building time constants
- Understanding what time constants physically mean and why they matter for control
- Writing the state-space representation of thermal network models
This dynamic perspective will be essential for the control and optimization problems you will tackle in the final project.
7.8 Additional Resources
7.8.2 Supplementary Reading
- Appendix E of MacKay (2008) - “Sustainable Energy Without the Hot Air”
7.8.3 Online Resources
7.8.3.1 Degree-Day Data
- U.S. Energy Information Administration - Degree Days Explained - Clear explanation of HDD and CDD with regional U.S. data
- NOAA Climate Prediction Center - Degree Days Statistics - Official U.S. degree-day data and forecasts
- DegreeDays.net - Free worldwide heating and cooling degree-day data
- National Weather Service - Heating and Cooling Degree Days - Definitions and calculation methods
7.8.3.2 Material Properties and Calculators
- Engineering ToolBox - Thermal Conductivity of Materials - Comprehensive database of thermal properties
- GreenSpec - Insulation Materials - Building-focused thermal properties
- Building America Solution Center - Thermal Mass - Guide to thermal mass in buildings
- UpCodes - Building code requirements including insulation R-values by climate zone