The air humidity indoors is assumed to be free floating. Five vapor fluxes then contribute to maintaining the equilibrium in a room:

These five fluxes load and unload the room air with vapor. There is even a sixth flux: vapor inflow and outflow by diffusion through the envelope. Its magnitude, however, is insignificant compared to the five fluxes mentioned, hence ignoring it does not introduce any significant error.

Assuming ideal air mixing, the balance per room is expressed by

where

Ga,in	=	all airflows entering room, including infiltration from outdoors, ventilation supply air, and infiltration from adjacent rooms, kg/s
xv,in	=	ratio of water vapor to dry air in entering air, kg/kg
Ga,out	=	all airflows leaving room, including exfiltration to outdoors, exhaust air, and exfiltration to adjacent rooms, kg/s
xv,i	=	ratio of water vapor to dry air in room, kg/kg
βj	=	surface film coefficient for diffusion at each hygroscopic surface, kg/(m2·s·Pa)
Aj	=	area of each hygroscopic surface in room, m2
pi	=	water vapor pressure in room, Pa
psat,Aj	=	vapor saturation pressure at different hygroscopic surfaces in room (fabric, furniture, and furnishings), Pa
ϕsat,Aj	=	relative humidity at different hygroscopic surfaces (fabric, furniture, and furnishings) in room on a scale from 0 to 1
βk	=	surface film coefficient for diffusion at all room surfaces where vapor condenses or evaporates, kg/(m2·s·Pa)
Aj	=	area of all room surfaces experiencing surface condensation or surface condensate drying, m2
psat,Aj	=	vapor saturation pressure for room’s various condensing and evaporating surfaces, Pa
Gv,P	=	vapor release in room, kg/s
V	=	air volume of room, m3
t	=	time, s

Solving Equation (1) requires knowledge of all in- and exfiltrating airflows between the room and all adjacent rooms and between the room and the outdoors, the ventilation supply airflow, exhaust airflow, water vapor ratio transported by all four airflows, surface temperatures and vapor balances at all hygroscopic surfaces, surface temperatures at all condensing and drying surfaces, and the surface film coefficients for diffusion at each of these surfaces (Hens 2012). Calculating requires numerical methods (preferably with dedicated software) to account for the transient nature of the variables involved (Woloszyn and Rode 2008).

In reality, because of stack effects, supply air jets, and convective plumes around people, the air in a room is never fully mixed, and vapor pressure gradients exist inside every room. Wetter, warmer air is also lighter and creates a positive temperature and humidity gradient along a room’s height, even in stagnant conditions. Despite these internal gradients, whose quantification requires computational fluid dynamics (CFD) calculations, further discussion is based on the fully mixed assumption.

When considering steady-state conditions where the room air maintains an average humidity that does not change with time (e.g., because sorption/desorption by the hygroscopic surfaces no longer intervenes and all airflows and vapor releases are time averaged), in the absence of surface condensation and surface drying, the humidity balance per room simplifies to

If the only incoming airflow is from the outdoors (e.g., infiltration, natural ventilation, air-side economizer applications), equality between supply and exhaust yields the mean indoor relative humidity:

where

ϕi	=	indoor relative humidity on scale from 0 to 1
psat,i	=	vapor saturation pressure for indoor temperature, Pa
ϕo	=	outdoor relative humidity on scale from 0 to 1
psat,o	=	vapor saturation pressure for outdoor temperature, Pa
R	=	gas constant for water vapor, equal to 462 (Pa · m3)/(kg · K)
θi	=	indoor temperature, °C
Gv,P	=	average indoor vapor release, kg/h
Va	=	outdoor air ventilation or/and infiltration flow, m3/h

Relative humidity over longer periods of time thus depends on mean outdoor temperature (through p_sat,e), mean outdoor relative humidity, mean indoor air temperature (through p_sat,i), mean indoor vapor release, and mean outdoor air ventilation and/or infiltration flows.

In cold and temperate climates, the outdoor air ventilation required by building standards easily dilutes the vapor released in residences, offices, and schools. In temperate climates, heating tends to keep relative humidity levels below the upper acceptability threshold. In very cold climates, ambient temperature and outdoor relative humidity may be such that normal ventilation rates cannot raise the indoor relative humidity above 15 to 20% at room temperature. Humidification may then become necessary. In hot, humid climates, outdoor temperature and relative humidity can be so high that the combined action of ventilation, cooling, and vapor release raises the indoor relative humidity above the upper acceptability threshold. Supplemental air dehumidification then is required.

Where outdoor air infiltrates into the room and the ventilation system delivers supply air at a fixed condition, the mean relative humidity indoors becomes

where

ϕo	=	outdoor relative humidity on scale from 0 to 1
psat,o	=	vapor saturation pressure for outdoor temperature, Pa
ϕs	=	relative humidity of supply air on scale from 0 to 1
psat,s	=	vapor saturation pressure for supply air temperature, Pa
V̇a,inf	=	infiltrating volumetric outdoor airflow, m3/h
V̇a,s	=	volumetric supply airflow, m3/h

    In hygric buffering, vapor from the air is adsorbed into hygroscopic materials when the relative humidity rises, and released back into the air when the relative humidity lowers. On shorter time scales, hygric buffering by various elements such as indoor air and the building fabric (e.g., envelope, furniture, furnishings) dampens and shifts fluctuations in relative humidity and consequently in indoor vapor pressure compared to the fluctuations of the exterior vapor pressure and internal vapor release.

Figure 2 compares measured indoor vapor pressures in an office building with the exterior vapor pressure over the course of a winter month when exterior conditions changed from cold and dry to warmer and more humid (Hens 2009a). Indoor vapor pressure remained higher than the outdoors during the cold, dry spell but dropped lower than outdoors during the warmer and more humid spells. Figure 3 shows the calculated buffering effect on the daily indoor vapor pressure in a well-ventilated two-person bedroom during winter (Hens 2012a). The effects of hygric buffering in keeping the daily changes in vapor pressure much lower than if only the air acted as buffering volume are obvious.

Antretter et al. (2012) performed a series of buffering experiments in a test room, simulating 4 to 6 day periods of summer and winter conditions. For the summer simulation, conditions were kept constant at 20°C, 90% rh, and 1 air change per hour (ach) for 1 to 3 days, then altered to an air change rate of zero and brought up to 35°C in 2 h. These conditions were held for 10 h, after which the temperature was returned to 20°C in 2 h and held there for the next 3 days. For the winter simulation, conditions were kept constant at 23°C, 50% rh, and 1 ach for 1 to 3 days, followed by 3 days with two vapor release peaks and a drop in the air change rate from 1 to 0.5 ach. The reference experiment was initially done without moisture buffering, and then repeated with 27 0.3 × 0.3 m sorptive tiles in the room. Figure 4 shows how the relative humidity changed the day after the initializing period. The effect of buffering is evident.

Hygroscopic buffering can even induce dampening and phase shifting on an annual basis in buildings with massive construction, as shown by Figure 5 for the same bedroom as in Figure 3 (Hens 2012a). For the same vapor release indoors and constant ventilation rate, the monthly mean indoor vapor pressures form an inclined ellipse, with the highest value in winter, lowest in summer, and values lower in springtime than in autumn.

Estimating moisture loads requires knowledge of occupancy patterns and releases per individual, activity, plant, and water surface present. In most cases, the real occupancy is unknown, and the sources are even more difficult to quantify. Some activities in the same classification differ in the amount of vapor released (e.g., cooking, which varies with the type of meals prepared). Literature sources only give global average numbers and should therefore be used with care and understanding of the implicit assumptions. Tables 1 to 5 list example amounts by type (IEA-ECB 1990; Kumaran and Sanders 2008; Sanders 1996; TenWolde and Walker 2001).

The means in Table 6 yield, as least-square regressions,

The individual data, however, produce a different relationship with considerably more scatter:

In both equations, n is the number of family members weighted by the ratio between the hours each is at home and 24 h a day. For example, if four people occupy the home for 16 h daily, n is

Equation (7) indicates that each person who is at home 24 h/day adds 2.82 kg of water vapor to the indoor ambient. Measurements in some German apartments, however, suggested that this release rate is too high: for a family of three, the rates documented ranged from 5.6 to 7.8 kg per day (Hartmann et al. 2001), giving a value between 0.8 and 2.48 kg per person per day. Data from Estonia suggest a person present 24 h/day adds on average 1 kg to the vapor released (Kumaran and Sanders 2008).

Table 7 provides some country-related statistical information about the vapor release expected in different types of residential buildings.

Evaporation from the pool surface, wet swimmers, and the wet floor around the pool are the main sources of water vapor. The evaporation rate is calculated based on the pool surface, multiplied by a factor that accounts for the average number of pool users and the wet floor around the pool:

where

Gv,P	=	evaporation rate, kg/s
β	=	surface film coefficient for diffusion at water surface in pool, kg/(m2 · s · Pa)
f	=	factor that accounts for average number of pool users and wet floor around pool
Apool	=	water surface area, m2
psat, pool	=	vapor saturation pressure at pool’s water temperature, Pa

In cases where the airflow comes from outdoors, Equation (9) can be combined with Equation (3) to yield the mean vapor pressure in a natatorium:

Measurement of the water surface, water temperature, air temperature, relative humidity, ventilation supply flow, and number of users in a university and a recreation natatorium allowed estimating a surface film coefficient for diffusion β, with inclusion of the factor f (Table 8). For the university natatorium, the table includes all measured evaporation rates in g/(m² · h).

Using the data from the university pool in Table 8, the unknown factor f could be derived as a function of the number of pool users (Hens 2009b):

where n is the number of users per square metre of pool surface.

For zero users, f is 1. The least-square line is depicted in Figure 6.

For example, given a water surface of 50 × 20 m, a water temperature of 28°C, an air temperature of 30°C, an indoor relative humidity of 60%, and 100 pool users, evaporation amounts to 80 kg/h.

In North America and Europe, there have been many monitoring programs in residential buildings that gathered data on the indoor/outdoor vapor pressure and vapor concentration difference. The least-square linear relationships found between these data, averaged over a given time span, and the outdoor air temperature averaged over the same time span typically show a decreasing trend at higher temperatures. Figure 7 shows such a measured relationship for a region where massive construction prevails and homes tend to be naturally ventilated (Hens 2016)

There are two reasons for the decrease shown in Figure 7: (1) windows are more frequently opened in warmer weather, resulting in more outdoor air ventilation and thus lower indoor/outdoor vapor pressures and concentration differences at higher outdoor air temperatures; and (2) the massive building fabric that absorbs and releases significant amounts of vapor and consequently dampens and shifts the indoor vapor pressure compared to the outdoors even on an seasonal basis.

Ridley et al. (2008) took long-term measurements in 1065 U.K. living rooms and 916 U.K. bedrooms. The least-square regressions, based on half-hour averages, are shown in Figures 8 and 9. They are given by

Living rooms:

Bedrooms:

Figure 10 and Table 9 are based on measurements in 101 dwellings in Finland and Estonia (Kalamees et al. 2006). The data gained were approximated by three distinct linear segments with the running weekly average indoor/outdoor vapor pressure differences constant below 5°C and above 15°C outdoors and varying linearly in between. A distinction was also made between high, average, and low vapor load.

The data collected in northern Canada (Kumaran and Sanders 2008) are represented by dots in Figure 10. The average indoor/outdoor vapor pressure the dot at top left represents a four-day period where the average outdoor temperature was −18°C and is substantially higher than the Finnish and Estonian high humidity results, apparently because a boil-water safety advisory led to an extremely high vapor release. The other dot at 13°C outdoor temperature is close to the Finnish and Estonian low-humidity curve, even though the number of household members was higher than typical for southern Canada. The dwellings monitored were measured to have air change rates up to 12 ach at 75 Pa.

Measurements in the living rooms of 10 German homes are shown in Figure 11. The main variable seems to be the number of household members and their living habits, including window operation (Antretter et al. 2010).

Comparing the mean of the German averages with those from the U.K. in Figures 8 and 9 gives smaller differences at low outdoor temperatures, and higher differences at higher outdoor temperatures:

Temperature and relative humidity data collected in 60 homes, equally spread over three climate zones in the United States (Pacific northwest, a cold location in the northeast, and a hot and humid location in the southeast) gave the values plotted in Figure 12 as the average relation between the moving monthly mean outdoor temperature and the moving monthly mean indoor/outdoor vapor pressure difference (Arena et al. 2010).

Figure 12 clearly shows the dehumidifying effect of air conditioning. The least-square regression line for a running monthly mean outdoor temperature below 19.5°C equals

Above 19.5°C, the line becomes

Indoor/outdoor vapor concentration difference measurements in 10 homes in Madison, WI, and 10 homes in Knoxville, TN, over a one-year period gave as monthly means the values shown in Figure 13 (Antretter et al. 2010).

The data are consistent with what Figure 12 shows: vapor pressure deficits only occur during the summer months in the Madison climate, and between April and November in the warmer, more humid Knoxville climate. Both situations are the result of dehumidification by air conditioning.

Measurements taken in 71 homes in Rhode Island are shown in Figure 14, giving the distribution of the measured indoor/outdoor vapor pressure difference at 0°C outdoors (Francisco and Rose 2010).

The mean was 396 Pa, close to what Equation (15) suggests, whereas the 95th percentile value equaled 772 Pa. This is higher than the data measured (Hens 2016) in the temperate climate of northwestern Europe, where for larger homes the 95th percentile on a weekly basis was

whereas for small homes the relation was

To minimize risk, the 95th percentile is often used for design purposes.

These results are based primarily on homes using natural ventilation. Many codes in the United States require homes to add mechanical ventilation to compensate for tighter envelopes. This significant difference must be addressed in evaluating residences with mechanical ventilation.

Measurements in 30 homes by Francisco et al. (2009) found that unvented gas fireplaces added on average about 100 Pa to the indoor/outdoor vapor pressure difference.

Vapor pressure difference measurements in natatoriums are few; however, studies have been conducted in some countries. Figure 15 gives weekly mean indoor/outdoor vapor pressure differences in relation to the weekly mean outdoor air temperature, measured in 20 natatoriums in a temperate region (Hens 2016).

There is a considerable amount of scatter. Least-square regressions for the mean and 95th percentile are

Mean

95th percentile

The data clearly show that, although not highly correlated with the outdoor temperature, the vapor pressure differences in natatoriums are consistently high. These buildings therefore require specific measures to prevent moisture accumulation in the envelope, which can lead to degradation (Figure 16).

The few measuring campaigns undertaken in student residences and schools in a temperate-climate region gave a very diffuse picture, with a large spread in weekly mean indoor/outdoor vapor pressure differences; see Figure 17 (Hens 2005) and Table 10 (Hens et al. 2007).

The mechanical system can affect the air pressure differences among building zones, and between building zones and the outdoors. HVAC operation influences the indoor relative humidity and, consequently, indoor/outdoor vapor pressure differences. To minimize humidity concerns,

In temperate climates, ground pipes (ventilation air ducts buried in the soil near the building) are sometimes used to moderate the condition of incoming ventilation air. Rain and groundwater must be prevented from entering these pipes: water penetration turns the pipes into air humidifiers, with very negative consequences for the indoor air quality. In dry pipes, relative humidity during the warm season should not exceed the mold threshold: too high a relative humidity in the pipes can load the supply air with mold spores, which then germinate in the supply filter. These risks are high enough that ground pipes are not recommended as a safe choice (Hens 2012b).

If there is no watertight layer or capillary break inserted above grade, stone-based building fabrics may suffer from rising damp and salt transport. Building parts, wetted by rising damp, act as permanent moisture sources affecting indoor vapor release and the indoor/outdoor vapor pressure difference.

Even when raintightness is guaranteed, excessive indoor vapor pressures can induce mold and condensation on interior surfaces and favor interstitial condensation in assemblies. Chapters 25 and 27 deal in more detail with moisture transport in the building envelope.

Mold Growth. Mold growth begins on interior opaque surfaces where the surface temperature and relative humidity pass the mold threshold for a long enough period of time. For example, if a material has absorbed an amount of water from a leak, drying may temporarily sustain a surface relative humidity high enough to be conductive for mold. The IEA Annex 14 reports (IEA-EBC 1990a, 1990b) contain two design formulas for estimating mold risk. The one calculates the four-week threshold, above which the likelihood that mold will develop is close to 100%. The other evaluates the surface relative humidity needed for mold development at shorter time intervals. Combining the two yields

where

ϕcrit	=	relative humidity at surface, on scale from 0 to 100
θsi	=	indoor surface temperature, °C (5°C ≤ θsi ≤ 25°C)
T	=	time span, days

The threshold for a one-day period touches 99%.

Since completion of Annex 14, two more comprehensive models have been published: Sedlbauer (2001) (Figure 18), and Viitanen and Ojanen (2007). More recent work, however, confirms that predictions with any model give varying results for growth probability and mold density (Vereecken and Roels 2012).

Surface Condensation. Each time the indoor surface temperature somewhere on the envelope drops below the dew point of the indoor air, condensate forms. Glazed surfaces are especially prone to surface condensation. Hygroscopic materials (e.g., wood, masonry, gypsum, some plasters) may buffer the local surface wetness, preventing visible condensate but increasing the moisture content in the material. When short periods of condensation or high humidity alternate with periods of drying, there tend to be no negative consequences. Long-lasting surface condensation or long periods of high humidity, however, degrade timber and other materials, and condensate deposited on aluminum frames may wet the window reveals. Thermal bridges can experience prolonged periods of condensation that cause indoor-side finish damage, such as wallpaper glue giving away.

Interstitial Condensation. Water vapor condensation within envelope parts is called interstitial condensation, and is caused by temperature and vapor pressure gradients pointing in the same direction. The driving forces are diffusion driven by vapor pressure differences, moist air leakage (in- or exfiltration), or absorption of wind-driven rain. In wood and wood-based materials, interstitial condensation may lift the surface relative humidity above the mold threshold and, worse, the moisture content above the rot threshold. When the relative humidity adjacent to nonporous, nonabsorbing, or capillary wet layers in an assembly becomes 100%, condensation with droplet formation occurs, followed by runoff and sometimes dripping, which is especially annoying for the occupants.

In temperate and mixed climates, neither humidification nor dehumidification is explicitly required except when necessary for functional reasons, such as in museums and operating rooms. Some depressurization during winter is advisable to minimize interstitial condensation risk, such as using demand-controlled exhaust ventilation or balanced ventilation with heat recovery and a correctly chosen pressure balance between supply and exhaust. The envelope must be so airtight that depressurization will not result in ventilation airflow beyond that needed for health reasons. High air permeance (low airtightness) has additional drawbacks: sound insulation degrades, drafts can occur, heat loss increases, and indoor surface temperatures might lower to a point where mold develops or surface condensation appears. Ground floors above crawlspaces must be airtight enough to prevent soil moisture, crawlspace odors, and radon from infiltrating into the conditioned space.

Acceptable envelope performance requires that the indoor surface temperature be high enough to prevent mold from occurring at the expected average vapor release indoors, the environmental conditions outdoors, the desired thermal comfort settings and the mean ventilation rate required for good IAQ and human health. Ventilation is also used to avoid acute surface condensation, which occurs each time the indoor dew-point temperature exceeds the envelope’s indoor surface temperature somewhere (a common site is glazed surfaces). For a detailed analysis of mold and surface condensation, see Hens (1999), Sedlbauer (2001), Vereecken and Roels (2012), and Viitanen and Ojanen (2007).

With well-insulating, double- and triple-glazed, gas-filled, low-e windows in new construction and retrofits, water can also condense on the outdoor surface during cold, clear-sky nights, somewhat obscuring visibility.

In hot and humid climates, to avoid mold growth and surface condensation, the indoor relative humidity should be kept below 60% through dehumidification. Brief periods of elevated relative humidity, however, do not necessarily result in mold germination. Keeping the building under positive air pressure is required to minimize interstitial condensation in wall cavities.

Dehumidification. Air-handling units that mix outdoor air with return air to provide constant airflows at variable supply temperatures do not provide continuous dehumidification. At higher supply air temperatures, the cooling coil does not remove moisture from the ventilation air, which can lead to elevated indoor relative humidity and a high potential for microbial growth. This becomes even more pronounced in energy-efficient buildings with reduced sensible cooling loads and long off times for the cooling coil, which significantly reduces the effectiveness of moisture removal (Rudd 2010; Rudd et al. 2005). For variable-supply-temperature systems, the ventilation air portion can be preconditioned using a dedicated outdoor air system (DOAS). These systems dehumidify only the ventilation air, and cooling loads are handled by a separate air-handling unit, so a DOAS is a good choice for providing continuously dehumidified ventilation air. If heat recovery is economically feasible without compromising pressurization, a DOAS with heat recovery is an energy-efficient alternative if the supply and return fans maintain correct pressurization, and the relative humidity of air passing through the supply filters remains low enough to avoid mold growth there. Because schools, office buildings, hospitals, and dwellings require continuous ventilation based on the number of occupants, demand-controlled DOAS-based outdoor ventilation air dehumidification can be achieved at a constant cooling-coil temperature. Reheat of the ventilation air may be required to avoid overcooling the supply air if ventilation demands are high. The resulting system provides more stable dehumidification and improved relative humidity control.

Alternatives for residential construction include local stand-alone dehumidifiers, with or without central system mixing; continuous exhaust/supply indoor air dehumidification using either a central HVAC system with subcooling followed by reheat, or ducted dehumidifiers; or ducted direct-expansion (DX) condenser-regenerated desiccant dehumidifiers (Harriman and Lstiburek 2009; Rudd 2013).

Pressurization. Dehumidification reduces the indoor vapor pressure below the outdoor vapor pressure level. Cooling does the same for temperature. As a result, the vapor pressure and temperature gradients cause moisture migration by diffusion toward the interior. If the envelope is not airtight, building depressurization and related infiltration increase this moisture flow through the exterior wall and connected partition wall cavities. This may raise the relative humidity at or near surfaces in these cavities to levels that allow mold, or even condensation. Pressurization reverses the airflow, allowing dehumidified indoor air to exfiltrate and neutralizing inward vapor diffusion. In any case, the envelope should be constructed as airtight as possible so that pressurization does not require unnecessarily high supply ventilation.

In cold climates, the techniques for avoiding mold and surface condensation are the same as in temperate climates. During cold seasons, the indoor relative humidity can drop well below 30%. Maintaining healthy levels of relative humidity then requires humidification, which results in an increased indoor/outdoor vapor pressure difference for a temperature gradient pointing to the outdoors and vapor diffusion toward the exterior.

Humidification. At part load, on/off air-handling units that heat and humidify a mixture of outdoor and return air operate so intermittently that indoor relative humidity levels may drop substantially during off periods. Moisture buffering by the building fabric, furniture, and furnishings can reduce that variation in relative humidity (Simonson et al. 2004a, 2004b). A better choice in larger buildings is a DOAS that restricts humidification to the ventilation air, handling heating loads by a separate system. Because the ventilation required mainly depends on the number of occupants, the space can be humidified by using steam injection modulated according to demand, or by preheating the ventilation air followed by adiabatic humidification and reheat to the indoor temperature set point. The result is more stable humidification and better relative humidity control. Alternatives in homes are stand-alone humidifiers, continuous-supply air humidification at constant dew-point temperature by the central HVAC system, etc.

Pressurization/Depressurization. Pressurizing buildings in cold climates leads to air exfiltration, which provides an extra vehicle for vapor egress. Together with diffusion, this could lead to severe interstitial condensation in envelope assemblies, especially when the indoor vapor retarder is ineffective and the air barrier is not continuous. Depressurizing a building, which leads to infiltration, is as effective way to avoid interstitial condensation problems, as long as the envelope is sufficiently airtight that depressurization does not require ventilation rates beyond those needed by the occupant load. If the envelope is too air permeable, not only will large exhaust rates be required but all drawbacks mentioned in the section on Temperate and Mixed Climates will apply but be more severe.