13.3 ASHRAE Standard 55: Thermal Environmental Conditions, PMV/PPD & Adaptive Comfort Models

13.3 ASHRAE Standard 55: Thermal Environmental Conditions, PMV/PPD & Adaptive Comfort Models

Thermal comfort is defined by ASHRAE Standard 55 (Thermal Environmental Conditions for Human Occupancy) as "that condition of mind that expresses satisfaction with the thermal environment and is assessed by subjective evaluation." Because human thermal sensation varies based on physiological, metabolic, and psychological differences, environmental control systems are engineered to satisfy a target percentage of occupants (typically $\ge 80%$ satisfied, with $\le 10%$ dissatisfied due to general body thermal sensation and $\le 10%$ dissatisfied due to local thermal discomfort).

---

1. The Six Primary Factors of Thermal Comfort

Human energy balance is dictated by four environmental parameters and two personal parameters:

+---------------------------------------------------------------------------------------------------------+
| THE SIX CORE THERMAL COMFORT VARIABLES                                                                  |
+---------------------------------------------------------------------------------------------------------+
| ENVIRONMENTAL FACTORS:                                                                                  |
|   1. Air Temperature (T_a): Dry-bulb temperature of air surrounding the occupant.                       |
|   2. Mean Radiant Temperature (T_mrt): Uniform surface temperature of an imaginary black enclosure     |
|      exchanging the same radiant heat as the actual non-uniform environment.                            |
|   3. Relative Air Velocity (v_r): Air speed relative to occupant motion (affects convective/evaporative).|
|   4. Humidity / Vapor Pressure (p_a): Moisture content of ambient air affecting skin sweat evaporation. |
|                                                                                                         |
| PERSONAL FACTORS:                                                                                       |
|   5. Metabolic Rate (M): Heat production rate generated by human physical activity.                      |
|      1 met = 50 kcal/(h·m2) = 58.2 W/m2 = 18.4 Btu/(h·ft2). (Seated resting = 1.0 met; office = 1.2 met)|
|   6. Clothing Insulation (I_cl): Thermal resistance of garment ensembles.                               |
|      1 clo = 0.155 m2·K/W = 0.88 h·ft2·°F/Btu. (Summer shorts/polo = 0.5 clo; Winter suit = 1.0 clo)   |
+---------------------------------------------------------------------------------------------------------+

---

2. Operative Temperature ($T_o$) & Mean Radiant Temperature ($T_{\text{mrt}}$)

The human body exchanges heat simultaneously via convection with surrounding air ($T_a$) and radiation with surrounding envelope surfaces ($T_{\text{mrt}}$). The Operative Temperature ($T_o$) combines these into a single uniform temperature:

$T_o = \frac{h_c T_a + h_r T_{\text{mrt}}}{h_c + h_r}$

Where:

• $h_c = \text{Convective heat transfer coefficient } (\text{W}/(\text{m}^2\cdot\text{K}) \text{ or } \text{Btu}/(\text{h}\cdot\text{ft}^2\cdot^\circ\text{F}))$
• $h_r = \text{Radiative heat transfer coefficient } (\approx 4.5 - 5.0\text{ W}/(\text{m}^2\cdot\text{K}) \approx 0.8\text{ Btu}/(\text{h}\cdot\text{ft}^2\cdot^\circ\text{F}))$

At standard indoor air speeds ($v \le 40\text{ fpm}$ or $0.2\text{ m/s}$), $h_c \approx h_r$, simplifying the operative temperature to the direct arithmetic mean:

$T_o \approx 0.5 T_a + 0.5 T_{\text{mrt}} = \frac{T_a + T_{\text{mrt}}}{2}$

At elevated air velocities ($v > 40\text{ fpm}$): $T_o = A T_a + (1 - A) T_{\text{mrt}}$ Where $A = 0.5$ for $v < 0.2\text{ m/s}$ ($40\text{ fpm}$), $A = 0.6$ for $0.2 \le v < 0.6\text{ m/s}$, and $A = 0.7$ for $0.6 \le v \le 1.0\text{ m/s}$.

Mean Radiant Temperature Calculation

For an enclosure of $N$ surrounding surfaces with surface temperatures $T_i$ and angle/view factors $F_{p-i}$ relative to the person (where $\sum F_{p-i} = 1.0$):

$T_{\text{mrt}}^4 = \sum_{i=1}^N F_{p-i} T_i^4 \quad \text{(Temperatures in Rankine or Kelvin)}$

For small temperature differences ($|T_i - T_{\text{mrt}}| < 30^\circ\text{F}$), the linearized form is accurate: $T_{\text{mrt}} \approx \sum_{i=1}^N F_{p-i} T_i$

Globe Thermometer Measurement

In field assessments, $T_{\text{mrt}}$ is measured using a standard $6\text{ inch}$ ($150\text{ mm}$) black globe thermometer:

$T_{\text{mrt}} = \left[ (T_g + 273.15)^4 + \frac{1.10 \times 10^8 \cdot v^{0.6}}{\epsilon \cdot D^{0.4}} (T_g - T_a) \right]^{1/4} - 273.15$

Where $T_g$ is globe temperature ($^\circ\text{C}$), $v$ is air speed ($\text{m/s}$), $D$ is globe diameter ($\text{m}$), and $\epsilon$ is emissivity ($\approx 0.95$).

3. Fanger's PMV and PPD Mathematical Models

Developed by P.O. Fanger, the Predicted Mean Vote (PMV) predicts the average thermal sensation vote of a large population of people on the ASHRAE 7-point thermal sensation scale, derived from human heat balance equations:

+---------------------------------------------------------------------------------------------------------+
| ASHRAE 7-POINT THERMAL SENSATION SCALE                                                                  |
+---------------------------------------------------------------------------------------------------------+
|  +3 : Hot                                                                                               |
|  +2 : Warm                                                                                              |
|  +1 : Slightly Warm                                                                                     |
|   0 : Neutral (Optimal thermal comfort)                                                                |
|  -1 : Slightly Cool                                                                                     |
|  -2 : Cool                                                                                              |
|  -3 : Cold                                                                                              |
+---------------------------------------------------------------------------------------------------------+

Standard 55 General Thermal Comfort Envelope

To comply with Standard 55 via the PMV model: $-0.5 \le \text{PMV} \le +0.5$

Predicted Percentage Dissatisfied (PPD)

The Predicted Percentage of Dissatisfied (PPD) represents the quantitative percentage of occupants who will feel thermally uncomfortable (voting $+3, +2, -2$, or $-3$). PPD is related non-linearly to PMV:

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

+---------------------------------------------------------------------------------------------------------+
| FANGER PMV VS. PPD CHARACTERISTIC CURVE                                                                 |
+---------------------------------------------------------------------------------------------------------+
|   PPD (%)                                                                                               |
|    100% |                                                               *                               |
|     80% |                                                                                               |
|     60% |                     *                                   *                                     |
|     40% |                                                                                               |
|     20% |                         *                           *                                         |
|     10% |---------------------------[======= COMFORT =======]-------------------------------------------|
|      5% |                                       *                                                       |
|      0% +-------+-------+-------+-------+-------+-------+-------+-------+                               |
|                -3      -2      -1      -0.5     0     +0.5     +1      +2      +3   (PMV)               |
+---------------------------------------------------------------------------------------------------------+

Key PMV/PPD Benchmark Values

• When $\text{PMV} = 0.0$ (thermal neutrality): $\text{PPD} = \mathbf{5%}$ (5% baseline dissatisfaction due to individual biological variation).
• When $\text{PMV} = \pm 0.5$ (Standard 55 boundary): $\text{PPD} = \mathbf{10%}$.
• When $\text{PMV} = \pm 1.0$: $\text{PPD} = \mathbf{26.1%}$.
• When $\text{PMV} = \pm 2.0$: $\text{PPD} = \mathbf{76.8%}$.

4. Local Thermal Discomfort Criteria

Even when the overall body is in thermal neutrality ($\text{PMV} = 0$), occupants may experience acute discomfort due to localized thermal asymmetry. Standard 55 establishes strict limits across four independent local discomfort mechanisms:

+---------------------------------------------------------------------------------------------------------+
| FOUR MECHANISMS OF LOCAL THERMAL DISCOMFORT                                                             |
+------------------------------------+----------------------------------+---------------------------------+
| DISCOMFORT MECHANISM               | PHYSICAL THRESHOLD LIMIT         | MAXIMUM DISSATISFACTION CRITERIA|
+------------------------------------+----------------------------------+---------------------------------+
| 1. Draft Rate (DR)                 | Air speed v <= 30-40 fpm         | DR <= 20%                       |
|                                    | (at Tu = 40%, T_a = 72°F)        |                                 |
+------------------------------------+----------------------------------+---------------------------------+
| 2. Vertical Temperature Gradient   | Delta_T (head to ankle) <= 5.4°F | Dissatisfaction <= 5%           |
|    (Head 43" vs. Ankle 4" seated)  | (3.0°C)                          |                                 |
+------------------------------------+----------------------------------+---------------------------------+
| 3. Floor Surface Temperature       | 66°F <= T_floor <= 84°F          | Dissatisfaction <= 10%          |
|                                    | (19°C to 29°C for occupants with |                                 |
|                                    |  standard footwear)              |                                 |
+------------------------------------+----------------------------------+---------------------------------+
| 4. Radiant Temperature Asymmetry   | Warm Ceiling: Delta_T_pr <= 9°F   | Warm Ceiling: <= 5%             |
|    (Plane radiant temp difference  | Cool Wall:    Delta_T_pr <= 25°F  | Cool Wall:    <= 10%            |
|     Delta_T_pr)                    | Cool Ceiling: Delta_T_pr <= 25°F  | Cool Ceiling: <= 5%             |
|                                    | Warm Wall:    Delta_T_pr <= 41°F  | Warm Wall:    <= 5%             |
+------------------------------------+----------------------------------+---------------------------------+

Mathematical Formulation for Draft Rate (DR)

$\text{DR} = \left( (34 - T_a)(v - 0.05)^{0.62} \right) \times (0.37 \cdot v \cdot Tu + 3.14) \quad [\%] \quad (\text{for } v \ge 0.05\text{ m/s})$ Where $T_a$ is air temp ($^\circ\text{C}$), $v$ is local mean air speed ($\text{m/s}$), and $Tu$ is turbulence intensity ($Tu = 100 \times \sigma_v / v [%]$).

---

5. The Adaptive Thermal Comfort Model

The Adaptive Comfort Model is based on the principle that in naturally ventilated buildings without mechanical cooling, occupants adapt their clothing, metabolic rhythm, window openings, and thermal expectations in response to prevailing outdoor weather patterns.

Applicability Constraints

• Applies ONLY to occupant-controlled naturally conditioned spaces where operable windows open directly to outdoors.
• Mechanical cooling systems must NOT be running in the space.
• Occupants must be engaged in sedentary activity ($1.0 - 1.5\text{ met}$) with free clothing adaptation ($0.5 - 1.0\text{ clo}$).
• Prevailing mean outdoor temperature must be within: $10.0^\circ\text{C} \le T_{\text{pma(out)}} \le 33.5^\circ\text{C}$ ($50^\circ\text{F} \le T_{\text{pma(out)}} \le 92.3^\circ\text{F}$).

Adaptive Comfort Equations

$\text{Optimal Operative Comfort Temperature: } T_{c} = 17.8 + 0.31 \cdot T_{\text{pma(out)}} \quad (^\circ\text{C})$

$\text{In IP Units: } T_c = 60.4 + 0.31 \cdot (T_{\text{pma(out)}} - 32) \quad (^\circ\text{F})$

Where $T_{\text{pma(out)}}$ is the prevailing mean outdoor air temperature, calculated as an exponentially weighted rolling average of the preceding 7 to 30 sequential days:

$T_{\text{pma(out)}} = (1 - \alpha) \cdot \left[ T_{e(d-1)} + \alpha T_{e(d-2)} + \alpha^2 T_{e(d-3)} + \dots \right] \quad (\text{where } \alpha \approx 0.7 - 0.8)$

Acceptability Temperature Ranges

• $80%$ Acceptability Limit (Standard Design Target): $T_o = T_c \pm 3.5^\circ\text{C} \quad (T_o = T_c \pm 6.3^\circ\text{F})$
• $90%$ Acceptability Limit (High-Performance Target): $T_o = T_c \pm 2.5^\circ\text{C} \quad (T_o = T_c \pm 4.5^\circ\text{F})$

+---------------------------------------------------------------------------------------------------------+
| ADAPTIVE COMFORT CHART (ASHRAE 55)                                                                      |
+---------------------------------------------------------------------------------------------------------+
|   Indoor Operative Temp (°C)                                                                            |
|     30 |                                               / -------- 80% Upper (Tc + 3.5°C)                |
|     28 |                                       / ----- / -------- 90% Upper (Tc + 2.5°C)                |
|     26 |                               / ----- / -----/                                                 |
|     24 |                       / ----- / -----/ -----/   <--- Optimum Centerline (Tc = 17.8 + 0.31*T_out)|
|     22 |               / ----- / -----/ -----/                                                          |
|     20 |       / ----- / -----/ -----/ -------- 90% Lower (Tc - 2.5°C)                                  |
|     18 | ----- / -----/ -----/ ---------------- 80% Lower (Tc - 3.5°C)                                  |
|        +-------+-------+-------+-------+-------+-------+-------+                                        |
|               10      14      18      22      26      30      34  Prevailing Mean Outdoor Temp (°C)     |
+---------------------------------------------------------------------------------------------------------+

6. Worked Engineering Calculation: Operative Temperature & Radiant Asymmetry

Problem Statement

An occupant sits in an office near an uninsulated single-pane exterior glass window on a cold winter day. The ambient indoor air temperature is $T_a = 72.0^\circ\text{F}$, and the room air velocity is $v = 20\text{ fpm}$ ($0.10\text{ m/s}$). The interior surface temperatures and corresponding view factors relative to the seated occupant are:

• Window (Wall 1): $T_1 = 45.0^\circ\text{F}$, $F_{p-1} = 0.20$
• Interior Walls (Walls 2, 3, 4): $T_2 = 70.0^\circ\text{F}$, $F_{p-2} = 0.40$
• Ceiling: $T_3 = 74.0^\circ\text{F}$, $F_{p-3} = 0.20$
• Floor: $T_4 = 68.0^\circ\text{F}$, $F_{p-4} = 0.20$

Calculate:

1. The Mean Radiant Temperature ($T_{\text{mrt}}$) using the linearized enclosure method.
2. The Operative Temperature ($T_o$) experienced by the occupant.
3. The radiant asymmetry $\Delta T_{pr}$ across the occupant and verify compliance with ASHRAE 55 local discomfort limits.

Step-by-Step Solution

1. Mean Radiant Temperature ($T_{\text{mrt}}$):

• Apply the linearized view factor formula: $T_{\text{mrt}} = \sum_{i=1}^4 F_{p-i} T_i = (0.20 \times 45) + (0.40 \times 70) + (0.20 \times 74) + (0.20 \times 68)$ $T_{\text{mrt}} = 9.0 + 28.0 + 14.8 + 13.6 = \mathbf{65.4^\circ\text{F}}$

2. Operative Temperature ($T_o$):

• Because room air velocity $v = 20\text{ fpm} \le 40\text{ fpm}$ ($0.2\text{ m/s}$), the convective and radiative weighting factor is $A = 0.5$: $T_o = \frac{T_a + T_{\text{mrt}}}{2} = \frac{72.0^\circ\text{F} + 65.4^\circ\text{F}}{2} = \mathbf{68.7^\circ\text{F}}$
• Engineering Observation: Even though the thermostat reads $72^\circ\text{F}$, the occupant perceives an operative thermal sensation of $68.7^\circ\text{F}$ due to cold radiant heat loss to the window.

3. Radiant Asymmetry Verification:

• The radiant temperature of the cool window wall is $45^\circ\text{F}$, while the opposing warm interior wall is $70^\circ\text{F}$.
• Plane radiant temperature difference: $\Delta T_{pr} = 70.0^\circ\text{F} - 45.0^\circ\text{F} = 25.0^\circ\text{F}$.
• Under Standard 55 local discomfort rules, the allowable limit for a cool wall is $\Delta T_{pr} \le 25.0^\circ\text{F}$ (corresponding to $\le 10%$ dissatisfied). The window meets the threshold at the exact boundary, but double-pane low-e glass is recommended to prevent localized draft and radiant chilling.

---

7. NCEES Reference Handbook Navigation Strategies

• Operative Temperature Formula: Search "Operative Temperature" to find $T_o = \frac{h_c T_a + h_r T_{\text{mrt}}}{h_c + h_r} \approx \frac{T_a + T_{\text{mrt}}}{2}$.
• PMV/PPD Equations: Search "Predicted Mean Vote" or "PPD" in the HVAC section to access Fanger's exponential equation and the $5%$ baseline dissatisfaction rule.
• Adaptive Comfort: Search "Adaptive Comfort" or "Prevailing Mean" to look up $T_c = 17.8 + 0.31 T_{\text{pma(out)}}$.