9  Occupant Thermal Comfort

Published

April 7, 2026

9.1 Lecture Overview

By the end of this module, students will be able to:

  • Recreate the basic equation for the thermal balance of a human body, which gives rise to expressions about heat dissipation to the environment
  • Understand latent heat (and its distinction from sensible heat, which is what we’ve covered so far), and Fick’s law for moisture transfer (relating it to Fourier’s law for conduction)
  • Be able to calculate shape factors of simple symmetric geometries, and understand their relevance in radiative heat transfer
  • Derive the Mean Radiant Temperature and its approximation when surfaces have similar temperature
  • Derive the concept of “Operative Temperature” by starting with convective and radiative heat transfer equations between the outer clothing and the environment
  • Be familiar with clothing insulation values (and their units of clo)
  • Calculate total conductive plus radiative sensible heat loss for a person if given their metabolic rate and clothing insulation (as well as the operative temperature)
  • Use the Predictive Mean Vote method for calculating a thermal sensation index
  • Reason about comfort levels using ASHRAE’s comfort chart relating operative temperature and humidity ratio
  • Human body thermal balance equations
  • Heat generation through metabolism
  • Heat dissipation mechanisms: conduction, convection, radiation, and evaporation
  • Sensible heat vs. latent heat
  • Fick’s law for moisture transfer
  • Shape factors for radiative heat transfer
  • Mean Radiant Temperature (MRT) and its calculation
  • Operative Temperature concept and derivation
  • Clothing insulation and the clo unit
  • Total sensible heat loss calculations
  • Predictive Mean Vote (PMV) thermal sensation index
  • ASHRAE comfort charts and their application

Understanding occupant thermal comfort is critical for designing autonomous HVAC control systems that can:

  • Maintain appropriate comfort levels for building occupants
  • Balance energy efficiency with occupant satisfaction
  • Adapt to different occupancy patterns and preferences
  • Account for seasonal variations in clothing insulation
  • Respond to both temperature and humidity conditions

This module provides the theoretical foundation for incorporating comfort models into your final project’s control strategies.

9.2 From Building Thermodynamics to Human Thermodynamics

In Lectures 5 and 6, we developed an understanding of how buildings exchange heat with the external environment. We studied conduction through walls, convection at surfaces, radiation between surfaces, and thermal network models that describe how building temperatures evolve over time. However, there’s a crucial piece missing from this picture: the people inside the buildings.

After all, an empty building does not need to be heated or cooled nearly as much as an occupied one. Buildings are not designed and built merely to shield us from the external environment—they exist to improve the quality of life (or at least the productivity, in the case of commercial buildings) of their occupants. From this perspective, maintaining a proper temperature is but one of many aspects required to achieve this greater objective. The study of the quality of the indoor environment—including acoustic, visual, and thermal aspects and their objective and subjective impact on occupants—is generally referred to as Indoor Environment Quality (IEQ), which includes sub-fields such as Indoor Air Quality (IAQ) that deal with specific aspects of the environment. That said, we will only be discussing a small sliver of the full IEQ field.

In particular, in this lecture, we shift our focus from building thermodynamics to human thermodynamics. Just as buildings must balance heat gains and losses to maintain desired indoor temperatures, our bodies must balance the heat they generate (as a byproduct of metabolizing the chemical energy we ingest in food) by dissipating excess heat to the environment to avoid overheating. This balance is the basis for human thermal comfort.

The good news is that the same fundamental heat transfer modes we studied in previous lectures—conduction, convection, and radiation—still apply. The geometry, materials, and boundary conditions are different (we’re now dealing with human skin and clothing rather than concrete walls and windows), but the underlying physics remains the same. The main new element we’ll introduce is latent heat transfer associated with moisture evaporation, since water mediates a significant portion of the heat exchange between our bodies and the environment. This mechanism, which we haven’t covered in detail before, plays a crucial role in thermal comfort.

9.2.1 Why Thermal Comfort Matters for Building Control

Understanding occupant thermal comfort is essential for designing effective autonomous building control systems. The primary goal of HVAC systems is to maintain acceptable thermal comfort for occupants while doing so as efficiently as possible. This creates an inherent tension between energy consumption and occupant satisfaction.

Consider these facts:

  • Thermal complaints dominate building issues: Studies consistently show that thermal comfort is the most common source of occupant complaints in commercial buildings, far exceeding concerns about air quality, lighting, or acoustics. In a typical office building, 30-50% of occupants may be dissatisfied with their thermal environment at any given time.

  • Impact on productivity: Research has demonstrated that thermal discomfort significantly affects cognitive performance and productivity. Studies have shown productivity decreases of 5-10% when temperatures deviate from the comfort zone by just 2-3°C. For knowledge workers in office buildings, the cost of this lost productivity can far exceed the building’s annual energy costs.

  • Energy implications are enormous: Space heating and cooling account for approximately 40% of total energy consumption in commercial buildings and over 50% in residential buildings. Even small improvements in how we manage thermal comfort—such as expanding acceptable temperature ranges by 1-2°C or implementing occupancy-based control—can yield energy savings of 10-30%.

  • Comfort drives setpoint battles: In the absence of sophisticated control, occupants often override thermostats, open windows in air-conditioned buildings, or bring in space heaters—all behaviors that waste energy and indicate failed comfort management. Understanding the physics and perception of thermal comfort allows us to design smarter control strategies that maintain satisfaction while minimizing these inefficiencies.

WarningFact or hallucination?

Find reliable sources to confirm or deny the numbers claimed in the previous bullet point list.

For autonomous building systems, this means we need more than just temperature sensors and simple on-off control. We need models that predict how occupants will perceive their thermal environment based on multiple factors: air temperature, radiant temperature from surrounding surfaces, humidity, air movement, their activity level, and their clothing. This lecture provides the theoretical foundation for building those models.

9.3 The Human Body as a Thermal System

Like the buildings we’ve been studying, the human body is a thermal system that must carefully manage heat flows to maintain appropriate temperatures. However, unlike buildings where we can tolerate a relatively wide range of indoor temperatures (say, 15-30°C), the human body must maintain a relatively constant core temperature of approximately 37°C (98.6°F) to function properly. This tight requirement must be met despite widely varying environmental conditions (outdoor temperatures from -40°C to +50°C) and activity levels (from sleeping to intense exercise).

To maintain this critical core temperature, our bodies employ several sophisticated thermoregulation mechanisms:

  • Vasodilation and vasoconstriction: The body can increase or decrease blood flow near the skin surface by dilating or constricting blood vessels. When hot, increased blood flow to the skin enhances heat dissipation to the environment. When cold, reduced skin blood flow conserves heat for the core organs.

  • Sweating: When the body needs to dissipate large amounts of heat (during exercise or in hot environments), sweat glands secrete moisture onto the skin surface. As this moisture evaporates, it carries away significant heat through the latent heat of vaporization—an extremely effective cooling mechanism.

  • Shivering: When cold, involuntary muscle contractions (shivering) generate additional metabolic heat to help maintain core temperature.

  • Behavioral adaptations: Beyond physiological responses, we instinctively adjust our behavior—adding or removing clothing, seeking shade or sun, adjusting posture to change surface area exposure, or simply moving to a more comfortable location.

However, these thermoregulation mechanisms have limited capacity. If environmental conditions or activity levels push the body beyond its ability to maintain thermal balance, serious health consequences follow:

  • Hypothermia occurs when core temperature drops below ~35°C, impairing physical and cognitive function and potentially leading to death
  • Heat exhaustion and heat stroke occur when core temperature rises above ~40°C, causing organ damage and potentially death

The fact that our thermoregulation range is relatively narrow compared to the environments we inhabit explains why humans have developed technologies—clothing, shelter, heating and cooling systems—to extend our effective operating range.

At a high level, the thermal balancing act performed by our bodies can be understood through a simple energy balance: the total rate at which our bodies produce energy (in the form of heat and mechanical work) must equal the total rate at which we dissipate heat to the environment. When these rates are in balance and our core temperature is maintained at ~37°C, we experience thermal comfort. When they’re out of balance, we feel too hot or too cold, and our bodies must deploy their thermoregulation mechanisms to restore equilibrium—or signal us to take behavioral action.

9.3.1 Metabolic Heat Generation

The energy balance we just described starts with the body’s energy production, which comes from metabolism—the process of converting chemical energy stored in food into forms the body can use. Through cellular respiration, our cells combine glucose and oxygen to produce ATP (adenosine triphosphate), the energy currency of the cell.

This energy is used for two purposes:

  1. Mechanical work (\(\dot{W}\)): Moving muscles, pumping blood, breathing, etc.
  2. Heat (\(\dot{Q}\)): All metabolic processes ultimately generate heat, either directly or as a byproduct of work (due to inefficiencies)

For most typical indoor activities, the mechanical work component is relatively small compared to the total metabolic rate. For example, when sitting at a desk, virtually all metabolic energy becomes heat. Even during walking, only about 20-25% of the metabolic energy goes into mechanical work; the rest becomes heat.

The total energy production rate is called the metabolic rate and is denoted \(\dot{M}\). In the field of thermal comfort, metabolic rate is commonly expressed using the met unit:

\[\boxed{1 \text{ met} = 58.2 \text{ W/m}^2}\]

Notice that this is expressed in units of power per unit area. The area refers to the body surface area (\(A_{sk}\), where “sk” stands for skin). This normalization by body surface area makes sense because larger people generally have higher absolute metabolic rates but similar rates per unit surface area for the same activity.

The typical body surface area for an adult is approximately 1.8 m² (though this varies with height and weight and can be estimated using the DuBois formula). Thus, 1 met corresponds to an absolute metabolic rate of:

\[\dot{M}_{absolute} = 1 \text{ met} \times 58.2 \text{ W/m}^2 \times 1.8 \text{ m}^2 \approx 105 \text{ W}\]

This value (1 met) was chosen to represent the metabolic rate of a seated, resting adult in thermal comfort—essentially the baseline human energy consumption rate.

The total power generated by our bodies, which must equal the total rate at which we dissipate energy to the environment (in steady state), is:

\[\dot{M} \cdot A_{sk} = \dot{Q} + \dot{W}\]

Or, rearranging to emphasize the heat that must be dissipated:

\[\dot{Q} = \dot{M} \cdot A_{sk} - \dot{W}\]

For most indoor activities where \(\dot{W}\) is negligible, we can approximate:

\[\dot{Q} \approx \dot{M} \cdot A_{sk}\]

This heat \(\dot{Q}\) is what must be transferred to the environment through the heat dissipation mechanisms we’ll discuss shortly.

9.3.1.1 Typical Metabolic Rates

Metabolic rates vary significantly depending on the activity level. The following table, based on ASHRAE Standard 55-2013, shows typical values for common activities:

Activity Metabolic Rate (met) Metabolic Rate (W/m²) Absolute Rate (W)*
Sleeping 0.7 40 72
Reclining 0.8 46 83
Seated, quiet 1.0 58 105
Standing, relaxed 1.2 70 126
Walking, 0.9 m/s (2 mph) 2.0 115 207
Office work, typing 1.1 64 115
Cooking 1.6-2.0 95-115 171-207
House cleaning 2.0-3.4 115-200 207-360
Dancing 2.4-4.4 140-255 252-459
Heavy exercise 4.0-6.0 230-350 414-630

*Assuming body surface area \(A_{sk} = 1.8\) m²

Key observations:

  • Metabolic rate can vary by nearly an order of magnitude between sleeping and vigorous exercise
  • The 1 met baseline (seated, quiet) is close to the lowest sustainable long-term metabolic rate for awake adults
  • For HVAC control purposes, typical office occupants can generally be assumed to be at 1.0-1.2 met
  • Residential spaces may see wider variation depending on activities (cooking, cleaning, exercising)

Understanding these variations is crucial for comfort modeling: a person exercising at 4 met generates roughly 400 W of heat that must be dissipated, compared to only 100 W when seated. The same environmental conditions (air temperature, humidity) that feel comfortable for a sedentary occupant may feel oppressively hot for someone who is physically active.

9.3.2 Heat Dissipation Mechanisms

The heat generated by metabolism must be dissipated to the environment to maintain thermal balance. The human body uses four main heat transfer mechanisms, three of which we’ve already encountered in our study of building thermodynamics:

  1. Conduction: Direct heat transfer through contact (e.g., feet on floor, body on chair)
  2. Convection: Heat transfer to surrounding air via fluid motion
  3. Radiation: Electromagnetic heat exchange with surrounding surfaces
  4. Evaporation: Heat removal through moisture vaporization (new mechanism!)

For most indoor scenarios, conduction is relatively minor (typically <5% of total heat loss) unless sitting on a cold surface or submerged in water. The dominant heat transfer paths are convection, radiation, and evaporation.

The key insight for modeling human thermal comfort is that these heat transfer mechanisms can be represented as two parallel thermal network paths, analogous to the electrical circuits we used for building thermal analysis:

Path 1: Sensible Heat Transfer (\(\dot{Q}_{sen}\))

This path handles heat transfer that changes temperature (sensible heat):

\[\dot{Q}_{sen} = \dot{Q}_{conv} + \dot{Q}_{rad}\]

The thermal network for sensible heat looks like this:

Skin (T_sk)
    |
    R_cl (clothing thermal resistance)
    |
Clothing surface (T_cl)
    |
    +------ R_conv -----> Air (T_a)
    |
    +------ R_rad ------> Surrounding surfaces (T_mrt)

Heat flows from the skin surface (at temperature \(T_{sk}\)) through the clothing insulation (thermal resistance \(R_{cl}\)) to the outer clothing surface (at temperature \(T_{cl}\)). From there, it splits into two parallel paths:

  • Convective path: Heat flows to the surrounding air (at temperature \(T_a\)) through convective resistance
  • Radiative path: Heat radiates to surrounding surfaces, which we characterize using a single equivalent temperature called the Mean Radiant Temperature (\(T_{mrt}\))

Path 2: Latent Heat Transfer (\(\dot{Q}_{lat}\))

This path handles heat transfer associated with moisture evaporation (latent heat):

\[\dot{Q}_{lat} = \dot{Q}_{evap}\]

The thermal network for latent heat is similar in structure but involves moisture transfer:

Skin surface (W_sk)
    |
    R_e,cl (clothing moisture resistance)
    |
Clothing surface (W_cl)
    |
    R_e,evap (evaporative resistance)
    |
Indoor air (W_a)

Moisture evaporates from the skin surface (at vapor concentration \(W_{sk}\)), diffuses through the clothing (moisture vapor resistance \(R_{e,cl}\)), and then evaporates from the clothing surface into the surrounding air (at vapor concentration \(W_a\)). Each kilogram of water that evaporates carries away approximately 2,430 kJ of heat (at body temperature)—an extremely efficient cooling mechanism.

Putting the networks together:

These two networks operate in parallel—sensible heat and latent heat are transferred simultaneously and independently. The total heat dissipated is:

\[\dot{Q}_{total} = \dot{Q}_{sen} + \dot{Q}_{lat}\]

Which, again, must equal the heat generated by metabolism (minus any mechanical work):

\[\dot{Q}_{total} = \dot{M} \cdot A_{sk} - \dot{W}\]

Important notes:

  • The clothing layer appears in both networks, acting as thermal resistance for sensible heat and as moisture vapor resistance for latent heat. Different clothing materials have different properties for each path.

  • The Mean Radiant Temperature (\(T_{mrt}\)) is a clever simplification: rather than tracking radiation to every surface in the room (walls, ceiling, floor, windows), we define a single equivalent temperature that gives the same net radiative heat exchange.

  • The relative importance of each path depends on environmental conditions and activity level. In cool, dry environments with sedentary occupants, sensible heat dominates. In warm, humid environments or during exercise, evaporative cooling becomes critical.

We’ll develop detailed models for each of these components in the sections that follow. For now, keep this parallel network structure in mind—it’s the conceptual foundation for all quantitative thermal comfort calculations.

9.3.3 The Thermal Balance Equation

Now we can write down the complete thermal balance equation for the human body. At its core, this equation expresses conservation of energy: the rate at which energy is produced must equal the rate at which it’s dissipated (in steady state).

Starting from our earlier observation that the body produces energy at rate \(\dot{M} \cdot A_{sk}\), which is split between mechanical work \(\dot{W}\) and heat \(\dot{Q}\), we can expand the heat dissipation term into its constituent mechanisms:

\[\boxed{\dot{M} \cdot A_{sk} - \dot{W} = \dot{Q}_{conv} + \dot{Q}_{rad} + \dot{Q}_{evap} + \dot{Q}_{cond} + \dot{Q}_{res,sen} + \dot{Q}_{res,lat} + \Delta \dot{Q}_{stored}}\]

where:

  • \(\dot{M}\) = metabolic rate per unit body surface area (W/m² or met)
  • \(A_{sk}\) = body surface area (typically ~1.8 m² for adults)
  • \(\dot{W}\) = rate of mechanical work done by the body (W)
  • \(\dot{Q}_{conv}\) = convective heat loss rate through skin (W)
  • \(\dot{Q}_{rad}\) = radiative heat loss rate through skin (W)
  • \(\dot{Q}_{evap}\) = evaporative heat loss rate through skin (W)
  • \(\dot{Q}_{cond}\) = conductive heat loss rate (W, usually very small)
  • \(\dot{Q}_{res,sen}\) = sensible (temperature-based) heat loss through respiration (W)
  • \(\dot{Q}_{res,lat}\) = latent (moisture-based) heat loss through respiration (W)
  • \(\Delta \dot{Q}_{stored}\) = rate of change of heat stored in the body (W)

Typical magnitudes and simplifications:

For most thermal comfort analyses, we can make several simplifying assumptions:

  1. Steady-state assumption: For thermal comfort (as opposed to thermal stress), we assume the body is in steady state, meaning core temperature isn’t changing significantly. Therefore: \(\Delta \dot{Q}_{stored} \approx 0\)

  2. Negligible work: For typical indoor activities (sitting, standing, light office work), mechanical work is negligible compared to metabolic rate: \(\dot{W} \ll \dot{M} \cdot A_{sk}\), so \(\dot{W} \approx 0\)

  3. Small conduction: Unless sitting on a very cold or hot surface, or submerged in water, conductive heat transfer is typically <5% of total: \(\dot{Q}_{cond} \approx 0\)

  4. Small respiration losses: Respiration accounts for breathing in cool, dry air and exhaling warm, moist air. While non-negligible, it’s typically <10% of total heat loss for sedentary activities: \(\dot{Q}_{res,sen} + \dot{Q}_{res,lat} \approx 0.1 \times (\dot{Q}_{conv} + \dot{Q}_{rad} + \dot{Q}_{evap})\)

With these simplifications, the practical thermal balance equation becomes:

\[\boxed{\dot{M} \cdot A_{sk} \approx \dot{Q}_{conv} + \dot{Q}_{rad} + \dot{Q}_{evap}}\]

Or, using our earlier notation grouping sensible and latent heat:

\[\boxed{\dot{M} \cdot A_{sk} \approx \dot{Q}_{sen} + \dot{Q}_{lat} = (\dot{Q}_{conv} + \dot{Q}_{rad}) + \dot{Q}_{evap}}\]

This is the fundamental equation we’ll use for thermal comfort analysis. The left side is determined by the occupant’s activity level. The right side depends on environmental conditions (air temperature, mean radiant temperature, humidity, air velocity) and personal factors (clothing insulation). Thermal comfort occurs when this equation is balanced at the appropriate skin temperature (typically 33-34°C for comfort).

What happens when it’s not balanced?

  • If \(\dot{M} \cdot A_{sk} > \dot{Q}_{sen} + \dot{Q}_{lat}\): The body is generating more heat than it can dissipate. Core temperature rises, triggering thermoregulation (sweating, vasodilation, behavioral changes). The occupant feels too warm.

  • If \(\dot{M} \cdot A_{sk} < \dot{Q}_{sen} + \dot{Q}_{lat}\): The environment is extracting more heat than the body is generating. Core temperature falls, triggering thermoregulation (shivering, vasoconstriction, behavioral changes). The occupant feels too cold.

The goal of HVAC control, from a thermal comfort perspective, is to maintain environmental conditions such that this balance can be achieved at comfortable skin and core temperatures.

A note on what’s to come:

This equation introduces several concepts we haven’t fully developed yet—particularly latent heat and evaporative cooling. Before we can make this equation quantitative and calculate actual comfort levels, we need to take a detour to understand these mechanisms in detail. That’s what we’ll do in the next section.

9.4 Sensible Heat vs. Latent Heat

Up to this point in the course, we’ve focused exclusively on sensible heat transfer—heat transfer that you can “sense” with a thermometer because it changes the temperature of materials. Conduction through walls, convection to air, and radiation between surfaces all fall into this category. However, the thermal balance equation we just derived includes an evaporative heat loss term (\(\dot{Q}_{evap}\)) that represents a fundamentally different type of heat transfer: latent heat.

Sensible heat is heat transfer associated with a change in temperature:

\[Q_{sensible} = m \cdot c \cdot \Delta T\]

When you add sensible heat to a substance, its temperature rises. When you remove it, temperature falls. This is what we’ve been studying throughout the course.

Latent heat is heat transfer associated with a phase change at constant temperature:

\[Q_{latent} = m \cdot h_{fg}\]

where \(h_{fg}\) is the latent heat of vaporization (or fusion). When water evaporates, it absorbs a large amount of energy to break the molecular bonds that hold it in liquid form, but the temperature doesn’t change during this process—the water and water vapor both remain at the same temperature during evaporation. This absorbed energy is the latent heat.

Why evaporation is such an effective cooling mechanism:

The latent heat of vaporization for water is remarkably high: approximately 2,430 kJ/kg at human body temperature (~35°C). To put this in perspective:

  • Evaporating 1 kg of water absorbs 2,430 kJ of heat
  • Cooling 1 kg of water by 100°C (from boiling to near-freezing) releases only \(1 \times 4.18 \times 100 = 418\) kJ
  • Evaporation is nearly 6 times more effective per kilogram than sensible cooling over a 100°C range

For the human body, this means:

  • Evaporating just 0.1 kg (100 mL) of sweat removes 243 kJ of heat
  • For a sedentary person generating ~100 W of metabolic heat, this much evaporation provides 40 minutes of cooling
  • During heavy exercise (~400 W), the body can produce 1-2 liters of sweat per hour, providing cooling power of 650-1,300 W

This is why sweating is the body’s primary defense against overheating during exercise or in hot environments.

Why moisture migration matters beyond thermal comfort:

While our focus is on thermal comfort, moisture transfer through building materials and clothing has several other important implications:

  1. Structural integrity: Moisture accumulation in building envelopes can lead to mold growth, wood rot, corrosion of metal components, and freeze-thaw damage. Understanding moisture migration paths is critical for building durability.

  2. Insulation effectiveness: Many insulation materials (fiberglass, mineral wool) lose significant thermal resistance when wet. Water has much higher thermal conductivity (~0.6 W/m·K) than air (~0.026 W/m·K), so moisture in insulation creates thermal bridges.

  3. Clothing comfort and performance: Moisture trapped in clothing feels clammy and reduces the effectiveness of clothing insulation. High-performance athletic fabrics are designed with specific moisture vapor permeability to allow sweat to escape while blocking wind and rain.

  4. Indoor air quality: Excess indoor humidity promotes mold growth and dust mite proliferation, both of which affect occupant health. Too little humidity causes dry skin, irritated respiratory passages, and static electricity.

Understanding the parallel between heat transfer (governed by Fourier’s law) and moisture transfer (governed by Fick’s law) allows us to apply similar analytical techniques to both problems. Let’s develop that analogy now.

9.4.1 Latent Heat Transfer: Evaporation and Moisture

The human body has two primary pathways for evaporative heat loss: skin evaporation and respiratory evaporation. Both involve water changing phase from liquid to vapor, each time carrying away the latent heat of vaporization.

The latent heat of vaporization (\(h_{fg}\)):

At human body temperature (~35°C / 95°F), the latent heat of vaporization of water is:

\[h_{fg} \approx 2{,}430 \text{ kJ/kg} = 2{,}430{,}000 \text{ J/kg}\]

This value varies slightly with temperature (it’s 2,257 kJ/kg at 100°C and 2,501 kJ/kg at 0°C), but 2,430 kJ/kg is appropriate for skin temperature calculations.

Skin moisture evaporation:

The skin continuously loses water through two distinct mechanisms:

  1. Insensible perspiration: This is passive water diffusion through the skin that occurs even when we’re not consciously sweating. The epidermis (outer skin layer) is not perfectly waterproof, so water vapor continually diffuses from the moist inner layers through the skin to the drier environment. This accounts for roughly 20-30 W of continuous cooling in typical indoor conditions (about 20-30% of basal metabolic rate).

  2. Sensible perspiration (sweating): When the body needs additional cooling beyond what insensible perspiration provides, eccrine sweat glands actively secrete liquid water onto the skin surface. This sweat then evaporates, providing substantial cooling. The rate is controlled by the thermoregulatory system:

    • Minimal: 0 (no active sweating in cool conditions)
    • Light sweating: 100-200 W of cooling
    • Moderate sweating: 200-400 W
    • Heavy sweating: 400-600 W
    • Maximum sustainable: ~600-800 W (corresponding to 0.9-1.2 L/hour sweat production)

The total evaporative heat loss from skin is:

\[\dot{Q}_{evap,skin} = \dot{m}_{evap} \cdot h_{fg}\]

where \(\dot{m}_{evap}\) is the rate of water evaporation (kg/s).

Respiratory moisture evaporation:

Each breath involves:

  • Inhaling relatively cool, dry ambient air
  • Warming it to body temperature in the airways (sensible heat loss)
  • Saturating it with water vapor in the lungs (latent heat loss)
  • Exhaling this warm, moist air

The respiratory latent heat loss can be estimated as:

\[\dot{Q}_{res,lat} = \dot{m}_{air} \cdot (W_{exhaled} - W_{inhaled}) \cdot h_{fg}\]

where \(W\) represents the humidity ratio (kg water vapor / kg dry air) and \(\dot{m}_{air}\) is the breathing rate.

For typical sedentary conditions:

  • Respiratory latent heat loss: ~10-15 W
  • Respiratory sensible heat loss: ~5-8 W
  • Combined: ~15-23 W (about 15-20% of basal metabolic rate)

During heavy exercise with increased breathing rate, respiratory heat loss increases but remains a smaller fraction of total heat loss compared to skin evaporation.

Total evaporative cooling:

Combining skin and respiratory evaporation:

\[\dot{Q}_{lat} = \dot{Q}_{evap,skin} + \dot{Q}_{res,lat}\]

The body can modulate only the sweating component; insensible perspiration and respiratory evaporation are relatively constant for given environmental conditions.

Environmental factors affecting evaporation:

The effectiveness of evaporative cooling depends critically on the moisture gradient between the skin/clothing surface and the ambient air:

  • Low humidity: Large moisture gradient → rapid evaporation → effective cooling
  • High humidity: Small moisture gradient → slow evaporation → reduced cooling

This is why 35°C (95°F) with low humidity can feel more comfortable than 30°C (86°F) with high humidity—evaporative cooling is much more effective in the dry conditions.

In the extreme case of 100% relative humidity at skin temperature, evaporation effectively stops, and the body loses its most powerful cooling mechanism. This is why very humid conditions combined with high temperatures can be deadly—the body cannot dissipate metabolic heat and core temperature rises uncontrollably.

9.4.2 Fick’s Law for Moisture Transfer

Just as Fourier’s law governs the diffusion of heat through materials, Fick’s law governs the diffusion of mass (in our case, water vapor) through materials. The mathematical form is strikingly similar, which allows us to use the same analytical framework we developed for thermal networks.

Fick’s First Law states that the mass flux (mass transfer rate per unit area) is proportional to the concentration gradient:

\[\boxed{\dot{m}'' = -D \frac{dC}{dx}}\]

where:

  • \(\dot{m}''\) = mass flux (kg/s·m²) - mass transfer rate per unit area
  • \(D\) = diffusion coefficient (m²/s) - analogous to thermal diffusivity \(\alpha = k/(\rho c)\)
  • \(C\) = concentration (kg/m³) - mass of water vapor per unit volume
  • \(dC/dx\) = concentration gradient (kg/m⁴)

The negative sign indicates that mass flows from high concentration to low concentration, just as heat flows from high temperature to low temperature.

For a finite layer of thickness \(L\) with area \(A\), integrating Fick’s law gives:

\[\boxed{\dot{m} = -D \cdot A \frac{\Delta C}{L} = \frac{A \cdot \Delta C}{R_m}}\]

where \(R_m = L/D\) is the moisture diffusion resistance (analogous to thermal resistance \(R = L/k\)).

Application to human thermal comfort:

For moisture transfer from skin through clothing to the environment, we need to track the moisture concentration (or more commonly, the water vapor pressure or humidity ratio) at each layer:

  1. At the skin surface (\(C_{sk}\) or \(W_{sk}\)): Nearly saturated due to sweat and insensible perspiration
  2. At the clothing surface (\(C_{cl}\) or \(W_{cl}\)): Lower than skin but higher than ambient
  3. In the ambient air (\(C_a\) or \(W_a\)): Determined by indoor humidity conditions

The moisture flow rate through the clothing is:

\[\dot{m}_{vapor} = \frac{A_{cl} (W_{sk} - W_{cl})}{R_{e,cl}}\]

where:

  • \(W\) = humidity ratio (kg water vapor / kg dry air) - more common than concentration in HVAC
  • \(R_{e,cl}\) = clothing moisture vapor resistance (m²·Pa/W or \(\text{clo}\) equivalent for moisture)
  • \(A_{cl}\) = clothed surface area

Then from the clothing surface to ambient air:

\[\dot{m}_{evap} = h_e \cdot A_{cl} (W_{cl} - W_a)\]

where \(h_e\) is the evaporative heat transfer coefficient, analogous to the convective heat transfer coefficient \(h_c\).

Converting mass flow to heat flow:

Once we know the evaporation rate \(\dot{m}_{evap}\), we convert it to latent heat loss:

\[\dot{Q}_{lat} = \dot{m}_{evap} \cdot h_{fg}\]

This is how moisture diffusion (governed by Fick’s law) connects to the thermal balance equation (energy conservation).

Practical implications:

The moisture vapor resistance of clothing varies dramatically by material:

  • Vapor-permeable fabrics (cotton, wool, technical athletic wear): Low \(R_{e,cl}\), allows moisture to escape
  • Vapor-impermeable layers (rubber, plastic, waxed fabrics): High \(R_{e,cl}\), traps moisture

When moisture cannot escape through clothing, it accumulates as liquid on the skin surface. This feels uncomfortable (clammy) and reduces the effective thermal insulation of the clothing (wet clothing conducts heat much better than dry clothing).

The parallel structure between Fourier’s law and Fick’s law means we can draw thermal network diagrams for moisture transfer that look exactly like those for heat transfer, with concentrations (or vapor pressures) playing the role of temperatures, and moisture resistances playing the role of thermal resistances.

The parallel between heat diffusion and mass diffusion is useful. This table shows the complete analogy:

Aspect Heat Transfer (Fourier) Mass Transfer (Fick)
Driving potential Temperature difference \(\Delta T\) Concentration difference \(\Delta C\) (or vapor pressure \(\Delta p_v\))
Flux equation \(\dot{q}'' = -k \frac{dT}{dx}\) \(\dot{m}'' = -D \frac{dC}{dx}\)
Material property Thermal conductivity \(k\) (W/m·K) Diffusion coefficient \(D\) (m²/s)
Integrated form \(\dot{Q} = \frac{k \cdot A}{L} \Delta T\) \(\dot{m} = \frac{D \cdot A}{L} \Delta C\)
Resistance \(R_{thermal} = \frac{L}{k \cdot A}\) (K/W) \(R_{mass} = \frac{L}{D \cdot A}\) (kg/m³ per kg/s)
Conductance \(U \cdot A = \frac{k \cdot A}{L}\) (W/K) \(G_m \cdot A = \frac{D \cdot A}{L}\) (kg/s per kg/m³)
Series resistances \(R_{total} = R_1 + R_2 + R_3\) \(R_{m,total} = R_{m,1} + R_{m,2} + R_{m,3}\)
Parallel resistances \(\frac{1}{R_{total}} = \frac{1}{R_1} + \frac{1}{R_2}\) \(\frac{1}{R_{m,total}} = \frac{1}{R_{m,1}} + \frac{1}{R_{m,2}}\)

Network diagram analogy:

For heat transfer through clothing:

T_sk ----[R_cl]---- T_cl ----[R_conv]---- T_a

For moisture transfer through clothing:

W_sk ----[R_e,cl]---- W_cl ----[R_e,evap]---- W_a

Both use exactly the same circuit analysis techniques!

Key insight: Because the mathematics is identical, every analytical tool we developed for thermal networks (series/parallel combination, time constants for transient problems, etc.) applies equally to moisture transfer networks. This is why understanding building thermodynamics prepares us well for understanding moisture-related comfort issues.

Practical example: Clothing as resistance

A cotton t-shirt has:

  • Thermal resistance: \(R_{cl} \approx 0.09\) m²·K/W (about 0.6 clo)
  • Moisture vapor resistance: \(R_{e,cl} \approx 0.01\) m²·Pa/W

The ratio of these resistances determines whether the clothing “breathes” well. Athletic fabrics are engineered to have low moisture resistance while maintaining thermal insulation—allowing sweat to escape while blocking wind.

A rubber raincoat has:

  • Thermal resistance: \(R_{cl} \approx 0.08\) m²·K/W (similar thermal insulation)
  • Moisture vapor resistance: \(R_{e,cl} \approx 1000\) m²·Pa/W (essentially impermeable!)

This is why raincoats feel clammy during exercise—moisture cannot escape, so sweat accumulates on the skin.

9.5 Radiative Heat Transfer and Shape Factors

In Lecture 5, we introduced radiative heat transfer and the Stefan-Boltzmann law, which describes the total radiation emitted by a surface:

\[\dot{Q}_{rad,emit} = \epsilon \sigma A T^4\]

We also discussed radiation exchange between two surfaces. However, we largely sidestepped a critical geometric question: of all the radiation leaving one surface, what fraction actually strikes another surface?

For the human body in an enclosed room, this question becomes essential. A person radiates heat in all directions, but the room has multiple surfaces at different temperatures—walls, ceiling, floor, windows. Each surface intercepts a different fraction of the person’s radiation, and each radiates back at a rate determined by its own temperature. To calculate the net radiative heat exchange accurately, we need to account for these geometric relationships.

This is where shape factors (also called view factors or configuration factors) come in. They quantify the geometric relationship between surfaces and allow us to calculate radiative exchange in complex environments without resorting to Monte Carlo ray tracing or other computationally expensive methods.

Understanding shape factors is crucial for thermal comfort because:

  1. Radiative asymmetry is a major source of local discomfort (e.g., sitting near a cold window in winter)
  2. Mean Radiant Temperature (MRT), a key parameter in thermal comfort models, is calculated using shape factors
  3. Radiant heating/cooling systems (floor heating, radiant panels) rely on shape factor calculations for design
  4. Even when air temperature is perfectly controlled, poor radiant conditions can make occupants uncomfortable

In this section, we’ll develop the concept of shape factors and show how they enable us to calculate the Mean Radiant Temperature—a single number that characterizes the radiative environment experienced by a person.

9.5.1 What Are Shape Factors?

The shape factor (or view factor) \(F_{1-2}\) is defined as the fraction of radiation leaving surface 1 that directly strikes surface 2.

Mathematically, if surface 1 emits (or reflects) radiation uniformly in all directions (a “diffuse” surface), then:

\[F_{1-2} = \frac{\text{Radiation leaving surface 1 that strikes surface 2}}{\text{Total radiation leaving surface 1}}\]

Shape factors are purely geometric properties—they depend only on the size, shape, and relative position of the surfaces, not on their temperatures or surface properties (emissivity).

Key properties of shape factors:

  1. Range: \(0 \leq F_{1-2} \leq 1\)

    • \(F_{1-2} = 0\) means surface 1 “sees” none of surface 2 (no line of sight)
    • \(F_{1-2} = 1\) means surface 1 “sees” only surface 2 (surface 2 completely surrounds surface 1)

  2. Summation rule: For a surface in an enclosure, the sum of view factors to all surfaces (including itself, if concave) equals 1: \[\sum_{j=1}^{n} F_{i-j} = 1\]

    This simply states that all radiation leaving surface \(i\) must go somewhere within the enclosure.

  3. Reciprocity relation: The radiation exchange between two surfaces is symmetric when accounting for their areas: \[\boxed{A_1 F_{1-2} = A_2 F_{2-1}}\]

    This powerful relation means if we know \(F_{1-2}\) and the areas, we can immediately calculate \(F_{2-1}\).

Why shape factors matter for radiation calculations:

The net radiative heat transfer between two diffuse, gray surfaces at temperatures \(T_1\) and \(T_2\) can be written as:

\[\dot{Q}_{1 \rightarrow 2} = A_1 F_{1-2} \sigma (T_1^4 - T_2^4)\]

(This is a simplified form assuming high emissivity; the full form includes emissivity factors.)

Without the shape factor \(F_{1-2}\), we would vastly overestimate radiative exchange because we’d be assuming all radiation from surface 1 reaches surface 2, which is rarely true in real geometries.

For human thermal comfort:

A person standing in a room needs to exchange radiation with all surrounding surfaces—four walls, ceiling, and floor. Each surface has:

  • A different temperature (window might be 10°C, radiator wall might be 25°C, other walls at 20°C)
  • A different shape factor from the person (the floor might have \(F_{person-floor} = 0.3\), while a distant wall might have \(F_{person-wall} = 0.15\))

The challenge is to calculate the total net radiation from the person to this complex environment. Shape factors are the tool that makes this tractable.

9.5.2 Calculating Shape Factors for Simple Geometries

Exact analytical formulas for shape factors exist only for relatively simple geometries. For complex situations (like a person in a real room), numerical methods or empirical correlations are used. However, understanding a few key cases builds intuition.

Case 1: Small surface to a large enclosure

If a small surface (area \(A_1\)) is completely surrounded by a much larger surface (area \(A_2 \gg A_1\)), then:

\[F_{1-2} \approx 1 \quad \text{(small surface sees only the enclosure)}\]

Using reciprocity: \(A_1 F_{1-2} = A_2 F_{2-1}\)

\[F_{2-1} = \frac{A_1}{A_2} F_{1-2} \approx \frac{A_1}{A_2} \approx 0 \quad \text{(large surface barely sees the small one)}\]

Application to human comfort: A person (surface area ~1.8 m²) in a typical room (surface area ~100 m²) can be approximated this way. The person “sees” essentially the entire room (\(F_{person-room} \approx 1\)), while any individual room surface barely “sees” the person (\(F_{surface-person} \approx 0.02\)).

Case 2: Person standing in center of a rectangular room

For a standing person (modeled as a vertical cylinder) in the center of a room with floor, ceiling, and four walls, empirical studies give approximate shape factors:

Surface Shape Factor \(F_{person-surface}\) Typical Value
Floor \(\frac{A_{floor}}{A_{total}}\) weighted by projection 0.25-0.30
Ceiling \(\frac{A_{ceiling}}{A_{total}}\) weighted by projection 0.05-0.10
Walls (total) Remaining fraction 0.60-0.70

The floor has a larger shape factor than the ceiling because the person is closer to the floor and has better “view” of it. For a standing person in a standard 2.7m ceiling room, the floor is much closer than the ceiling.

Case 3: Person sitting near a window

When a person sits close to a window or wall (distance < 1m), that surface dominates the radiative exchange. Approximate shape factors might be:

Surface Shape Factor Notes
Nearby window 0.30-0.40 High due to proximity
Floor 0.20-0.25 Reduced compared to center position
Opposite wall 0.10-0.15 Partially blocked by person’s position
Other surfaces 0.25-0.40 Remainder

This is why sitting near a cold window feels so uncomfortable—the high shape factor to the cold surface dominates radiative heat loss.

Analytical formula: Perpendicular rectangles

For completeness, here’s one exact analytical case—two perpendicular rectangles sharing a common edge (like a person standing next to a wall):

For rectangles with dimensions such that they share an edge of length \(l\), and have heights/widths \(h_1\) and \(h_2\):

\[F_{1-2} = \frac{1}{\pi h_1} \left[\text{arctan}\left(\frac{h_2}{l}\right) - \text{complicated integral terms}\right]\]

In practice, we use tabulated values or software tools rather than evaluating these integrals by hand.

Can you spot any errors in this calculation?

Problem: A person (modeled as a 1.7m tall, 0.4m diameter cylinder, total surface area \(A_p \approx 1.8\) m²) stands in the center of a 4m × 5m × 2.7m room. Estimate the shape factors from the person to each surface.

Solution:

First, calculate room surface areas:

  • Floor: \(A_{floor} = 4 \times 5 = 20\) m²
  • Ceiling: \(A_{ceiling} = 4 \times 5 = 20\) m²
  • Wall 1 (4m wide): \(A_{w1} = 4 \times 2.7 = 10.8\) m²
  • Wall 2 (5m wide): \(A_{w2} = 5 \times 2.7 = 13.5\) m²
  • Wall 3 (4m wide): \(A_{w3} = 10.8\) m²
  • Wall 4 (5m wide): \(A_{w4} = 13.5\) m²
  • Total room surface area: \(A_{total} = 88.6\) m²

Step 1: Use empirical correlations for a standing person

For a person centered in a room:

  • \(F_{p-floor} \approx 0.28\) (from empirical data for cylinder to floor)
  • \(F_{p-ceiling} \approx 0.08\) (ceiling is far from person at 1.7m height)
  • \(F_{p-walls} = 1 - F_{p-floor} - F_{p-ceiling} = 1 - 0.28 - 0.08 = 0.64\)

Step 2: Distribute wall shape factor by area

Assume the four walls are “seen” in proportion to their areas:

\[F_{p-w1} = F_{p-walls} \times \frac{A_{w1}}{A_{w1} + A_{w2} + A_{w3} + A_{w4}} = 0.64 \times \frac{10.8}{48.6} = 0.142\]

\[F_{p-w2} = 0.64 \times \frac{13.5}{48.6} = 0.178\]

Similarly: \(F_{p-w3} = 0.142\) and \(F_{p-w4} = 0.178\)

Step 3: Verify summation rule

\[F_{p-floor} + F_{p-ceiling} + F_{p-w1} + F_{p-w2} + F_{p-w3} + F_{p-w4} = 0.28 + 0.08 + 0.142 + 0.178 + 0.142 + 0.178 = 1.00 \quad \checkmark\]

Step 4: Use reciprocity to find shape factors from room surfaces to person

\[F_{floor-p} = \frac{A_p}{A_{floor}} F_{p-floor} = \frac{1.8}{20} \times 0.28 = 0.025\]

This confirms our earlier observation: room surfaces barely “see” the person (only 2.5% of floor radiation hits the person).

9.5.3 Application to Human Thermal Comfort

With shape factors in hand, we can now calculate the total radiative heat exchange between a person and their surrounding environment. This is critical because radiation typically accounts for 40-50% of total heat loss in comfortable indoor conditions (the rest being convection and evaporation).

Net radiative heat loss from a person:

The net radiative heat exchange between a person and surface \(i\) is:

\[\dot{Q}_{rad,i} = A_p F_{p-i} \epsilon_p \sigma (T_p^4 - T_i^4)\]

where:

  • \(A_p\) = person’s surface area (~1.8 m² for an adult, but this is actually the clothed area \(A_{cl}\))
  • \(F_{p-i}\) = shape factor from person to surface \(i\)
  • \(\epsilon_p\) = effective emissivity of clothing/skin (~0.95 for typical fabrics and skin)
  • \(\sigma\) = Stefan-Boltzmann constant (\(5.67 \times 10^{-8}\) W/m²·K⁴)
  • \(T_p\) = surface temperature of clothing (typically 28-34°C depending on clothing and conditions)
  • \(T_i\) = temperature of surface \(i\)

The total radiative heat loss is the sum over all surfaces:

\[\boxed{\dot{Q}_{rad,total} = A_p \epsilon_p \sigma \sum_{i=1}^{n} F_{p-i} (T_p^4 - T_i^4)}\]

Key insights from this equation:

  1. Shape factors weight the contribution of each surface: A cold window with \(F_{p-window} = 0.35\) has much more impact than a cold wall with \(F_{p-wall} = 0.10\), even if both are at the same temperature.

  2. Temperature differences are raised to the fourth power: This means cold surfaces have a disproportionate effect. A window at 5°C has much more impact than predicted by linear temperature differences.

  3. Position matters. The view factor (or shape factor) is actually different depending on where the person is located. Imagine it this way: though it is true that we radiate heat to everything outside of us, what matters is their relative proportion in our field of view: we don’t radiate as much to stars or deep space, compared to objects that are nearer.

Why “air temperature” isn’t enough:

Consider two scenarios:

  • Scenario A: \(T_a = 22°C\), all surfaces at 22°C
  • Scenario B: \(T_a = 22°C\), window at 8°C, opposite wall at 28°C (average still 22°C)

The air temperature is identical, but occupant comfort is very different! In Scenario B, a person near the window will feel cold due to high radiative loss to the cold surface, while a person near the opposite wall will feel uncomfortably warm.

This is why ASHRAE Standard 55 includes limits on radiant temperature asymmetry:

  • Warm ceiling: Max 5°C warmer than other surfaces
  • Cool wall (window): Max 10°C cooler than other surfaces
  • Cool ceiling: Max 14°C cooler than other surfaces

Exceeding these limits causes local discomfort even if mean radiant temperature and air temperature are in the comfort zone.

Radiant heating/cooling systems:

Understanding shape factors is crucial for designing radiant systems:

  • Radiant floor heating: High shape factor to floor (0.25-0.30) means floor temperature only needs to be 24-28°C to provide significant heating, allowing lower air temperatures and better comfort.

  • Radiant ceiling panels: Lower shape factor to ceiling (0.05-0.10) means panels must be at higher/lower temperatures to achieve the same effect, with risk of violating asymmetry limits.

  • Radiant wall panels: Moderate shape factors (0.15-0.20) offer a compromise, especially effective when positioned near occupant zones (near desks, seating areas).

Quantitative example: The cold window problem

Let’s calculate exactly how much extra heat loss a person experiences when sitting near a cold window compared to sitting in the room center, even though the air temperature is identical.

Setup:

  • Room: 4m × 5m × 2.7m, air temperature \(T_a = 22°C\)
  • Person: \(A_{cl} = 1.8\) m², clothing surface temperature \(T_{cl} = 30°C\), emissivity \(\epsilon = 0.95\)
  • Window: 2m × 1.5m = 3 m², inner surface temperature \(T_{window} = 8°C\) (cold winter day)
  • Other surfaces: All at \(T_i \approx 21°C\) (slightly below air temperature)

Position 1: Person in center of room

Shape factors (from earlier example):

  • \(F_{p-window} \approx 0.10\) (small fraction of view)
  • \(F_{p-other} \approx 0.90\) (rest of room)

Radiative heat loss to window: \[\dot{Q}_{rad,window} = 1.8 \times 0.10 \times 0.95 \times 5.67 \times 10^{-8} \times [(30+273)^4 - (8+273)^4]\] \[= 1.8 \times 0.10 \times 0.95 \times 5.67 \times 10^{-8} \times [8.468 \times 10^9 - 6.238 \times 10^9]\] \[= 1.8 \times 0.10 \times 0.95 \times 5.67 \times 10^{-8} \times 2.230 \times 10^9 = 21.6 \text{ W}\]

Radiative heat loss to other surfaces (at 21°C): \[\dot{Q}_{rad,other} = 1.8 \times 0.90 \times 0.95 \times 5.67 \times 10^{-8} \times [(303)^4 - (294)^4]\] \[= 1.8 \times 0.90 \times 0.95 \times 5.67 \times 10^{-8} \times 7.51 \times 10^8 = 65.4 \text{ W}\]

Total radiative loss (center position): 21.6 + 65.4 = 87.0 W

Position 2: Person sitting next to window

Shape factors change dramatically: - \(F_{p-window} \approx 0.35\) (window dominates view) - \(F_{p-other} \approx 0.65\) (reduced view of other surfaces)

Radiative heat loss to window: \[\dot{Q}_{rad,window} = 1.8 \times 0.35 \times 0.95 \times 5.67 \times 10^{-8} \times 2.230 \times 10^9 = 75.6 \text{ W}\]

Radiative heat loss to other surfaces: \[\dot{Q}_{rad,other} = 1.8 \times 0.65 \times 0.95 \times 5.67 \times 10^{-8} \times 7.51 \times 10^8 = 47.2 \text{ W}\]

Total radiative loss (window position): 75.6 + 47.2 = 122.8 W

Comparison:

Position Radiative Loss Difference
Center of room 87.0 W Baseline
Next to window 122.8 W +35.8 W (41% increase!)

What does 35.8 W feel like?

To compensate for this extra radiative loss through sensible heat, we’d need to either: - Increase air temperature by approximately 4-5°C (if convective heat transfer coefficient \(h_c \approx 8\) W/m²·K) - Increase clothing insulation substantially - Move away from the window!

This is why: - Office workers near windows constantly adjust thermostats or use space heaters - Window seats on airplanes feel drafty even when cabin temperature is comfortable - Radiant floor heating systems can maintain comfort with lower air temperatures (reversing this effect)

Key insight: Shape factors translate geometric position into quantifiable thermal discomfort. A 3x increase in shape factor (0.10 → 0.35) creates a 40% increase in radiative heat loss—enough to make the difference between comfort and significant discomfort, all while the air temperature remains constant.

This example demonstrates why thermal comfort modeling must account for both convective (air temperature) and radiative (surface temperatures weighted by shape factors) effects. Air temperature alone is insufficient!

9.6 Mean Radiant Temperature (MRT)

Earlier, when discussing heat dissipation mechanisms, we introduced Mean Radiant Temperature as a “clever simplification” that replaces tracking radiation to every surface with a single equivalent temperature. Now we’ll formalize this concept and show how to calculate it.

9.6.1 Definition and Physical Meaning

The Mean Radiant Temperature (\(T_{mr}\)) is defined such that the radiative heat exchange with this uniform-temperature enclosure equals the actual radiative heat exchange with the non-uniform environment:

\[A_p \epsilon_p \sigma (T_p^4 - T_{mr}^4) = A_p \epsilon_p \sigma \sum_{i=1}^{n} F_{p-i} (T_p^4 - T_i^4)\]

Simplifying (canceling \(A_p \epsilon_p \sigma T_p^4\) from both sides):

\[T_{mr}^4 = \sum_{i=1}^{n} F_{p-i} T_i^4\]

Taking the fourth root:

\[\boxed{T_{mr} = \left[\sum_{i=1}^{n} F_{p-i} T_i^4\right]^{1/4}}\]

where:

  • \(F_{p-i}\) = shape factor from person to surface \(i\)
  • \(T_i\) = absolute temperature of surface \(i\) (in Kelvin)
  • The summation is over all \(n\) surfaces in the enclosure

Physical meaning: MRT is the uniform temperature of an imaginary “black box” enclosure that would exchange the same net radiation with the occupant as the actual room with its non-uniform surface temperatures.

Example calculation: For a person in a room with floor at 21°C (\(F_{p-floor} = 0.28\)), ceiling at 19°C (\(F_{p-ceiling} = 0.08\)), and walls at 20°C (\(F_{p-walls} = 0.64\)):

\[T_{mr} = [(0.28)(294)^4 + (0.08)(292)^4 + (0.64)(293)^4]^{1/4} = 293.0 \text{ K} = 20.0°\text{C}\]

9.6.2 Approximation for Small Temperature Differences

The fourth-power formula is exact but cumbersome for hand calculations. When all surface temperatures are within about 10-15°C of each other (typical for most comfort situations), we can use a linear approximation:

\[\boxed{T_{mr} \approx \sum_{i=1}^{n} F_{p-i} T_i}\]

This is simply a shape-factor-weighted average of surface temperatures—much easier to calculate.

When is this valid? The approximation works well when \(\frac{\Delta T}{T_{avg}} < 0.05\) (i.e., temperature variations <5% in absolute terms, or <15°C around room temperature).

Example (same room as above):

\[T_{mr} \approx (0.28)(21) + (0.08)(19) + (0.64)(20) = 20.2°\text{C}\]

Compared to exact: 20.0°C. Error: only 0.2°C—negligible for comfort calculations!

Why use the approximation?

  • Much faster for hand calculations and quick estimates
  • Sufficient accuracy for typical comfort scenarios
  • Only use exact fourth-power formula when temperatures differ significantly (e.g., person near very cold window or radiant panel)

9.7 Operative Temperature

We now have two temperatures characterizing the thermal environment: air temperature (\(T_a\)) and mean radiant temperature (\(T_{mr}\)). Operative temperature (\(T_o\)) combines these into a single metric that characterizes how “warm” or “cold” the environment feels.

Starting from sensible heat loss from the clothing surface:

\[\dot{Q}_{sen} = \dot{Q}_{conv} + \dot{Q}_{rad} = h_c A_{cl} (T_{cl} - T_a) + h_r A_{cl} (T_{cl} - T_{mr})\]

where \(h_c\) and \(h_r\) are the convective and linearized radiative heat transfer coefficients.

We can define an operative temperature \(T_o\) such that:

\[\dot{Q}_{sen} = (h_c + h_r) A_{cl} (T_{cl} - T_o)\]

Equating these and solving for \(T_o\):

\[\boxed{T_o = \frac{h_c T_a + h_r T_{mr}}{h_c + h_r}}\]

This is a weighted average of air temperature and MRT, with weights determined by the relative heat transfer coefficients.

Special case: For typical indoor conditions with low air velocity, \(h_c \approx h_r \approx 4-5\) W/m²·K, giving:

\[T_o \approx \frac{T_a + T_{mr}}{2}\]

Practical use: Two environments with the same operative temperature should feel thermally similar, even if one has warmer air/cooler surfaces and the other has cooler air/warmer surfaces. This is why radiant floor heating can feel comfortable with lower air temperatures.

9.8 Clothing Insulation

Clothing acts as thermal resistance (\(R_{cl}\)) between skin and environment, appearing in the sensible heat network we discussed earlier.

The clo unit: Clothing insulation is measured in clo units, where:

\[\boxed{1 \text{ clo} = 0.155 \text{ m}^2\cdot\text{K/W}}\]

This represents typical indoor clothing that maintains comfort at 21°C, 50% RH, with minimal air movement and sedentary activity.

Typical values:

Ensemble Insulation
Naked 0 clo
Underwear 0.1 clo
Light summer (shorts, t-shirt) 0.3-0.5 clo
Business casual (pants, shirt) 0.6-0.7 clo
Indoor winter (pants, long sleeves, sweater) 0.8-1.0 clo
Business suit 1.0-1.5 clo
Heavy winter outdoor 2.0-3.0 clo

Clothing area factor: Clothing increases effective surface area beyond nude body surface (\(A_{body} \approx 1.8\) m²):

\[A_{cl} = f_{cl} \cdot A_{body}\]

where \(f_{cl} \approx 1.0 + 0.3 \cdot I_{cl}\) (empirical correlation), so 1 clo clothing gives \(f_{cl} \approx 1.3\).

Effect on heat transfer: Clothing adds series thermal resistance in the sensible heat path while also increasing surface area and changing surface temperature. Net effect: higher insulation reduces heat loss, requiring higher operative temperatures for comfort.

9.9 Calculating Total Sensible Heat Loss

Combining all concepts, sensible heat loss from skin through clothing to environment:

\[\boxed{\dot{Q}_{sen} = \frac{A_{body} (T_{sk} - T_o)}{R_{cl} + \frac{1}{h_c + h_r}}}\]

From thermal balance: \(\dot{Q}_{sen} = \dot{M} \cdot A_{sk} - \dot{W} - \dot{Q}_{evap}\)

Worked example: Sedentary office worker

  • Metabolic rate: 1.2 met = 70 W/m², \(A_{body}\) = 1.8 m² → 126 W total
  • Clothing: 1.0 clo, \(f_{cl}\) = 1.3
  • Evaporative loss: ~25 W (insensible perspiration)
  • Required sensible loss: 126 - 25 = 101 W

For comfort with \(T_{sk} \approx 33°\text{C}\), clothing at 1.0 clo (0.155 m²·K/W), and typical \(h_c + h_r \approx 8\) W/m²·K:

\[101 = \frac{1.8(33 - T_o)}{0.155 + \frac{1}{8 \times 1.3}}\]

Solving: \(T_o \approx 22°\text{C}\) required for thermal balance and comfort.

9.10 The Predictive Mean Vote (PMV) Method

So far, we’ve developed the physics of thermal comfort: heat generation (metabolism), heat dissipation mechanisms (sensible and latent), and the environmental parameters that affect them (air temperature, MRT, humidity, air velocity). But physics alone doesn’t tell us when people feel “comfortable” versus “too warm” or “too cold.” This is where empirical thermal comfort models come in.

The Predictive Mean Vote (PMV) is the most widely used thermal comfort model, combining heat balance calculations with empirical data from human subject studies to predict thermal sensation on a standardized scale.

9.10.1 Historical Context and Development

In the 1960s-70s, Danish researcher Povl Ole Fanger conducted extensive laboratory studies where hundreds of subjects were exposed to controlled environmental conditions and asked to rate their thermal sensation. Fanger combined these empirical results with the heat balance equations we’ve been studying to develop the PMV model.

The model was groundbreaking because it:

  • Provided a quantitative prediction of thermal sensation (not just “comfortable” or “uncomfortable”)
  • Was based on physical principles (heat balance) validated with empirical data (human responses)
  • Could be calculated from measurable parameters (temperature, humidity, etc.) and known characteristics (clothing, activity)

PMV was adopted by:

  • ISO 7730 (International Organization for Standardization, 1984, updated 2005)
  • ASHRAE Standard 55 (Thermal Environmental Conditions for Human Occupancy, 1992 onward)
  • EN 15251 (European standard for indoor environmental criteria)

It remains the dominant model for predicting thermal comfort in mechanically conditioned buildings worldwide.

9.10.2 The PMV Equation

PMV is calculated from six input parameters:

Personal factors:

  1. Metabolic rate (\(\dot{M}\), in met or W/m²) — activity level
  2. Clothing insulation (\(I_{cl}\), in clo) — what the person is wearing

Environmental factors:

  1. Air temperature (\(T_a\), in °C)
  2. Mean radiant temperature (\(T_{mr}\), in °C)
  3. Air velocity (\(v_a\), in m/s)
  4. Relative humidity (RH, in %) or water vapor partial pressure (\(p_a\), in Pa)

The PMV equation itself is quite complex. Without going into full detail, it has the form:

\[\text{PMV} = \left[0.303 \exp(-0.036 \dot{M}) + 0.028\right] \times \text{(thermal load)}\]

where the thermal load is the difference between metabolic heat production and total heat loss (sensible + latent). The empirical factors (0.303, 0.036, 0.028) were determined from regression analysis of Fanger’s experimental data.

The output is a thermal sensation vote on a 7-point scale:

PMV Value Thermal Sensation
+3 Hot
+2 Warm
+1 Slightly warm
0 Neutral (comfortable)
−1 Slightly cool
−2 Cool
−3 Cold

Interpretation:

  • PMV = 0: Thermal neutrality — the “ideal” thermal environment where heat production equals heat loss at comfortable skin and core temperatures
  • PMV > 0: Warm sensation — body is retaining more heat than desired, triggering cooling responses (vasodilation, sweating)
  • PMV < 0: Cool sensation — body is losing more heat than desired, triggering warming responses (vasoconstriction, shivering)

9.10.3 Calculating PMV

The full PMV calculation is iterative because clothing surface temperature (\(T_{cl}\)) appears in both the equations for heat loss and in the determination of thermal load. The calculation procedure is:

  1. Assume initial \(T_{cl}\) (often start with \((T_a + T_{mr})/2\) or similar estimate)
  2. Calculate heat losses: Use \(T_{cl}\) to compute convective, radiative, and evaporative heat loss
  3. Calculate thermal load: Difference between metabolic rate and total heat loss
  4. Calculate PMV: Apply PMV formula
  5. Update \(T_{cl}\) based on energy balance
  6. Iterate until \(T_{cl}\) converges (typically 3-5 iterations)

In practice, nobody calculates PMV by hand. Instead, use:

9.10.4 Predicted Percentage Dissatisfied (PPD)

Even when the average person feels thermally neutral (PMV = 0), not everyone is satisfied. Individual differences in metabolism, clothing preferences, acclimatization, and personal preference mean that some people will always be dissatisfied.

Fanger developed the Predicted Percentage Dissatisfied (PPD) metric to quantify this:

\[\boxed{\text{PPD} = 100 - 95 \exp\left(-0.03353 \cdot \text{PMV}^4 - 0.2179 \cdot \text{PMV}^2\right)}\]

Key observations:

  1. Minimum dissatisfaction at PMV = 0: Even under “perfect” neutral conditions, PPD = 5%. This irreducible minimum reflects individual physiological and psychological differences.

  2. Symmetric around neutral: PPD increases equally whether the environment is too warm (PMV > 0) or too cool (PMV < 0).

  3. Rapid increase away from neutral: Small deviations from PMV = 0 cause significant increases in dissatisfaction.

PMV PPD (%) Interpretation
0 5 Best achievable (5% still dissatisfied)
±0.5 10 ASHRAE 55 Category I (highest quality)
±0.7 15 ASHRAE 55 Category II (normal expectation)
±1.0 26 ASHRAE 55 Category III (acceptable minimum)
±1.5 52 More than half dissatisfied
±2.0 75 Unacceptable for occupied spaces

ASHRAE Standard 55 compliance:

  • Requires PMV between −0.5 and +0.5 (PPD < 10%) for acceptable thermal comfort
  • Stricter designs may target PMV between −0.2 and +0.2 (PPD < 6%)

Let’s calculate PMV for a typical office scenario using a computational tool (since hand calculation is impractical).

Scenario:

  • Person: Office worker, sitting (1.2 met), business casual clothing (0.7 clo)

  • Environment:

    • Air temperature: \(T_a = 23°\text{C}\)
    • Mean radiant temperature: \(T_{mr} = 22°\text{C}\) (slightly cooler surfaces)
    • Air velocity: \(v_a = 0.1\) m/s (typical for no draft)
    • Relative humidity: 50%

Using the CBE Thermal Comfort Tool or pythermalcomfort: calculate the PMV and PPD.

9.10.5 Limitations of PMV

While PMV is the industry standard, it has important limitations:

  1. Steady-state assumption: PMV assumes the body is in thermal equilibrium. It doesn’t capture:
    • Transient thermal discomfort (e.g., entering a cold building from outside)
    • Time-varying conditions (temperature ramping, intermittent heating)
    • Thermal history effects
  2. Population bias: Developed primarily from studies of young, healthy European and North American subjects. May not generalize well to:
    • Different ethnic/geographic populations
    • Elderly occupants
    • Children
    • People with certain health conditions
  3. Mechanically conditioned buildings: PMV works best in HVAC-controlled environments. In naturally ventilated buildings, occupants:
    • Adapt their expectations based on outdoor conditions
    • Have higher tolerance for temperature variations
    • Adjust clothing and behavior more dynamically
  4. No adaptation or expectation: PMV doesn’t account for:
    • Psychological adaptation (getting used to conditions)
    • Seasonal acclimatization
    • Cultural/regional preferences
    • Occupant control (having a window or thermostat increases tolerance)
  5. Individual variation: The 5% minimum PPD at PMV = 0 is somewhat optimistic. In real buildings:
    • Actual dissatisfaction often exceeds predicted values
    • Local discomfort factors (drafts, radiant asymmetry, vertical temperature gradients) not fully captured
    • Metabolic rates and clothing are estimated, not measured

For naturally ventilated buildings: ASHRAE 55 includes an adaptive comfort model that better predicts comfort in these contexts, allowing wider acceptable temperature ranges based on outdoor climate.

Despite these limitations, PMV remains extremely useful for:

  • Design: Sizing HVAC systems and selecting setpoints
  • Commissioning: Verifying that buildings meet comfort targets
  • Comparison: Evaluating different design alternatives
  • Research: Standardized metric for thermal comfort studies

9.11 ASHRAE Comfort Charts

While PMV/PPD provides a quantitative prediction of thermal sensation, designers and building operators often need a quicker way to assess whether environmental conditions meet comfort requirements. ASHRAE Standard 55 provides graphical comfort zones that show acceptable combinations of operative temperature and humidity for typical office occupancies.

These comfort charts are based on the PMV model (specifically, the range where −0.5 ≤ PMV ≤ +0.5, corresponding to PPD ≤ 10%), but presented in an easily interpretable format.

9.11.1 Reading the Comfort Chart

The ASHRAE comfort chart has the following structure:

Axes:

  • X-axis: Operative temperature (\(T_o\), in °C or °F)
  • Y-axis: Humidity ratio (g water/kg dry air) or relative humidity (%)

Comfort zones (shaded regions) show acceptable combinations. Conditions inside the zone satisfy the PMV criterion (−0.5 ≤ PMV ≤ +0.5). Conditions outside are either too warm, too cool, too humid, or too dry.

Assumptions:

  • Typical office activity: 1.0-1.2 met (sedentary)
  • Low air velocity: < 0.2 m/s (still air, no draft)
  • Separate zones for different clothing levels (summer vs. winter)

Why two comfort zones?

ASHRAE 55 defines separate comfort zones for:

  1. Summer conditions (0.5 clo clothing): Light, short-sleeved clothing
  2. Winter conditions (1.0 clo clothing): Long pants, long sleeves, possibly sweater

This reflects seasonal behavioral adaptation: people dress differently in summer vs. winter based on outdoor conditions and social norms, even though indoor temperatures could theoretically be the same year-round.

9.11.2 Acceptable Ranges for Thermal Comfort

For typical office environments (1.0-1.2 met activity, air velocity < 0.2 m/s):

Summer comfort zone (0.5 clo):

  • Operative temperature: 23.5°C to 26.0°C (74°F to 79°F)
  • Humidity ratio: 0.000 to 0.012 kg water/kg dry air
  • Equivalent RH at mid-range: approximately 30-60%

Winter comfort zone (1.0 clo):

  • Operative temperature: 20.0°C to 23.5°C (68°F to 74°F)
  • Humidity ratio: 0.000 to 0.012 kg water/kg dry air
  • Equivalent RH at mid-range: approximately 30-60%

Key observations:

  1. ~3.5°C difference between summer and winter: The winter zone is shifted cooler because occupants wear more clothing. This allows energy savings—heating to 21°C in winter can be just as comfortable as cooling to 25°C in summer, if occupants dress appropriately.

  2. Overlap region (23.5°C): This temperature is acceptable year-round, regardless of season. It’s often used as a year-round setpoint in buildings where seasonal clothing variation is minimal.

  3. Humidity limits: The upper humidity limit (~0.012 kg/kg ≈ 60-70% RH) is set primarily for:

    • Microbial growth prevention
    • Prevention of surface condensation
    • Comfort (high humidity impairs evaporative cooling)

    The lower limit is less strict, but very low humidity (<20% RH) can cause:

    • Dry skin and mucous membranes
    • Increased static electricity
    • Increased respiratory irritation

  4. Air velocity matters: The standard zones assume low air movement. Higher air velocity (from fans, natural ventilation) extends the upper temperature limit by enhancing convective and evaporative cooling:

    • Elevated air speed provision: ASHRAE 55 allows temperatures up to 3°C higher if air velocity increases to 0.8-1.2 m/s (with occupant control)

9.11.3 Using Comfort Charts for HVAC Control

Comfort charts inform HVAC control strategies in several ways:

1. Setpoint selection: Instead of a single fixed setpoint (e.g., “maintain 22°C”), the comfort zone suggests:

  • Heating setpoint: Lower bound of winter zone (~20°C)
  • Cooling setpoint: Upper bound of summer zone (~26°C)
  • Deadband: Region between heating and cooling where no active conditioning occurs

This approach reduces energy consumption compared to tight temperature control around a single setpoint.

2. Seasonal setpoint adjustment: Some strategies adjust setpoints seasonally:

  • Winter (Nov-Mar): Heat to 20-21°C, cool above 24°C
  • Summer (May-Sep): Heat below 22°C, cool to 24-25°C
  • Shoulder seasons: Wider deadband

This acknowledges that occupants naturally adjust clothing seasonally.

3. Humidity control: While temperature is the primary control variable, the charts show that humidity matters:

  • Dehumidification: When RH > 60-65%, remove moisture even if temperature is acceptable
  • Humidification: Optional, but beneficial if RH < 30% (especially in winter in cold climates)

4. Free cooling / economizer operation: The comfort zone width allows “free cooling” (using outdoor air when conditions permit) over a wider range of outdoor temperatures, reducing mechanical cooling energy.

9.12 Putting It All Together

This lecture has covered a lot of ground, moving from basic physiology to quantitative comfort prediction. Let’s synthesize the key concepts and their implications for autonomous building control.

9.12.1 From Physics to Comfort Prediction

The chain of reasoning we’ve developed follows this logic:

1. Human body as thermal system:

  • Core temperature must be maintained at ~37°C for proper physiological function
  • Heat is continuously generated through metabolism (quantified in met units)
  • Limited thermoregulation mechanisms (sweating, shivering, vasoconstriction/dilation) have finite capacity

2. Heat dissipation pathways:

  • Sensible heat (convection + radiation): Accounts for ~60-75% of heat loss in typical conditions
  • Latent heat (evaporation): Accounts for ~25-40%, becomes dominant during exercise or in warm environments
  • These operate through parallel thermal/moisture networks with clothing as series resistance

3. Environmental characterization: Four environmental parameters determine heat exchange rates:

  • Air temperature (\(T_a\)): Drives convective heat transfer
  • Mean Radiant Temperature (\(T_{mr}\)): Drives radiative heat transfer (shape-factor-weighted average of surface temperatures)
  • Humidity (RH or \(W\)): Determines evaporative cooling potential
  • Air velocity (\(v_a\)): Enhances convective and evaporative heat transfer

4. Personal factors: Two personal characteristics modulate heat exchange:

  • Metabolic rate (met): Determines heat generation based on activity level
  • Clothing insulation (clo): Provides thermal and moisture resistance

5. Thermal balance and comfort:

  • Thermal neutrality: Occurs when heat generation equals heat dissipation at comfortable skin temperature (~33°C)
  • PMV model: Quantifies deviation from neutrality as thermal sensation on −3 to +3 scale
  • PPD metric: Translates PMV into percentage of dissatisfied occupants (minimum 5% at PMV = 0)
  • ASHRAE comfort zones: Define acceptable operative temperature and humidity ranges (PMV between −0.5 and +0.5)

This framework allows us to predict whether a given combination of environmental conditions, personal factors, and clothing will feel comfortable—enabling proactive HVAC control rather than reactive adjustment to complaints.

9.12.2 Implications for Building Control

Understanding occupant thermal comfort fundamentally changes how we think about building environmental control:

1. Multi-dimensional control problem:

Air temperature alone is insufficient. Effective control must address:

  • Temperature: Both air (\(T_a\)) and radiant (\(T_{mr}\))—a person near a cold window can feel uncomfortable even if air temperature is perfect
  • Humidity: Especially important for warm conditions and active occupants (affects evaporative cooling)
  • Air movement: Can extend upper temperature limits by 2-3°C if occupant-controlled

This has practical implications:

  • Thermostats should measure or estimate operative temperature (\((T_a + T_{mr})/2\)), not just air temperature
  • Humidity sensors should complement temperature sensors, especially in cooling mode
  • Radiant system design (floor heating, ceiling panels) must account for shape factors

2. Individual differences and flexibility:

The 5% minimum PPD at PMV = 0 understates real-world variation. In practice:

  • 10-30% dissatisfaction is common even in “well-controlled” buildings
  • Sources: individual metabolic differences, clothing preferences, acclimatization, personal thermal history, psychological factors

Control strategies must acknowledge this:

  • Personal environmental control: Operable windows, desk fans, task lighting with heat generation
  • Spatial diversity: Provide zones at different temperatures, allowing occupants to self-select
  • Temporal flexibility: Allow temperature to drift within comfort zone rather than maintaining tight setpoint
  • Feedback mechanisms: Monitor complaints, occupant behavior (opening windows, using space heaters) to adapt control

3. Seasonal adaptation for energy savings:

Supporting seasonal clothing changes (0.5 clo in summer → 1.0 clo in winter) allows:

  • Winter: Heat to 20-21°C instead of 23°C (10-15% heating energy savings)
  • Summer: Cool to 24-25°C instead of 23°C (15-25% cooling energy savings)
  • Deadband: Wider range between heating and cooling setpoints reduces equipment cycling and energy use

Implementation challenges:

  • Requires occupant education and buy-in
  • Dress codes may constrain clothing adaptation
  • Gradual seasonal transitions (1°C every 2-3 weeks) needed to allow acclimatization
  • May conflict with expectations that “air conditioning = constant temperature”

4. Integration with occupancy information:

Thermal comfort control can be dramatically improved by knowing:

  • When spaces are occupied (no need to condition unoccupied spaces to full comfort)
  • Who is present (typical vs. atypical metabolic rates, clothing preferences)
  • What activities are occurring (meeting = sedentary 1.2 met, gym = exercise 4+ met)

This links directly to occupancy sensing and modeling (Lecture 8), creating opportunities for:

  • Demand-controlled conditioning (precool before occupation, setback when vacant)
  • Activity-based control (lower setpoints for gyms, higher for sedentary spaces)
  • Personalized comfort profiles (learned preferences per occupant or group)

9.12.3 Local Discomfort Factors

While PMV/PPD provides a good overall assessment of thermal comfort, several local discomfort factors can cause complaints even when whole-body thermal balance is neutral:

1. Drafts (air velocity variability):

  • Perception of draft depends on temperature, velocity, and turbulence
  • Higher turbulence intensity increases draft perception
  • Particularly noticeable on the back of neck, ankles
  • ASHRAE 55 limits: Peak local air velocity < 0.25 m/s in winter, < 0.3 m/s in summer

2. Vertical air temperature gradients:

  • Head-to-ankle temperature difference causes discomfort
  • Common in poorly designed systems with stratification
  • ASHRAE 55 limits: < 3°C difference between 0.1m and 1.1m height (seated), or 0.1m and 1.7m (standing)

3. Warm or cool floors:

  • Direct conduction from feet makes floor temperature particularly noticeable
  • ASHRAE 55 limits: 19-29°C floor temperature
  • Radiant floor heating typically operates at 24-28°C (warm but not uncomfortable)

4. Radiant asymmetry:

  • Large temperature differences between opposing surfaces (e.g., cold window vs. warm opposite wall) create directional discomfort
  • Our earlier calculations showed 35.8 W extra heat loss near a cold window—equivalent to 4-5°C air temperature reduction
  • ASHRAE 55 limits: Cool wall (window) < 10°C cooler than average, warm ceiling < 5°C warmer

Practical considerations:

  • PMV assumes uniform environment—actual buildings have spatial variations
  • Good design minimizes these local factors through proper system selection, placement, and controls
  • When local discomfort exists, occupants may report being “too cold” or “too hot” even if PMV predicts neutral

9.13 Additional Resources

9.13.1 Primary Reference

  • Reddy, T. Agami. Heating and Cooling of Buildings: Principles and Practice of Energy Efficient Design. 3rd edition. CRC Press, 2016. Chapter 3: Elements of Heat Transfer for Buildings

9.13.2 Supplementary Reading

  • ASHRAE Standard 55-2020: Thermal Environmental Conditions for Human Occupancy
  • Fanger, P. O. Thermal Comfort: Analysis and Applications in Environmental Engineering. McGraw-Hill, 1970. (Classic reference, somewhat dated but foundational)
  • Parson, K. C. Human Thermal Environments: The Effects of Hot, Moderate, and Cold Environments on Human Health, Comfort, and Performance. 3rd edition. CRC Press, 2014.

9.13.3 Online Resources

9.13.3.1 Thermal Comfort Tools

  • CBE Thermal Comfort Tool: https://comfort.cbe.berkeley.edu/ (Interactive PMV calculator and ASHRAE comfort chart visualization)
  • ASHRAE Standard 55 Interactive Comfort Tool

9.13.3.2 PMV/PPD Calculations

9.13.3.3 ASHRAE Resources

  • ASHRAE Handbook—Fundamentals, Chapter 9: Thermal Comfort
  • ASHRAE Standard 55 documentation and user’s manual