## 1.13 Zone Humidity Balance

### 1.13.1 Zone Humidity Balance

Given a zone with various flows, mj, with humidities wj, entering the zone, and with a scheduled source of water vapor,msched, a water mass balance on the zone gives:

Equation 212

which can be written as:

Equation 213

where,

M = mass of dry air in the zone; lbm of dry air.

dw/dt = the rate of change of humidity ratio in zone.

mj = air flow rate from source

wj = humidity ratio of air coming from source j; lbm H2O/lbm dry air.

w = humidity ratio of air in zone; lbm H2O/lbm dry air.

msched = scheduled rate of moisture addition to zone; lbm H2O/unit time.

Using the air perfect gas equation the last term in Equation 213 can be written

so that Equation 213 becomes

Equation 214

where T is the air temperature in absolute degrees.

This equation is solved using a forward difference rather than a backward or central difference since a forward difference uncouples the moisture balance equations of each of the zones. Integrating from time t to time t+δt, where δt is the time step, using a forward difference, gives:

Equation 215

Notice that all of the values on the right hand side of Equation 215 are determined atT(t+δt) term which represents the zone air temperature at the end of the integration period. T(t+δt) is known from the zone energy sensible energy balance at time t (see Section 2.3). The term

is assumed to be negligible and not included in the CSE code.

### 1.13.2 Stability of Solution

The time series solution of Equation 215 will become unstable unless the second term is positive. That is, stability requires

Equation 216

Solving for δt stability requires

Equation 217

Since the zone air changes per unit time is

then the stability requirement can be written in terms of air changes as:

Equation 218

If the solution is unstable at the given δt, the zone air mass M(t) can be temporarily boosted up such that:

This will lead to a higher latent capacity for the zone air, introducing some error in the zone humidity prediction. This will also lead to a zone latent heat imbalance unless this artificial increase in zone air is accounted for.

### 1.13.3 Hygric Inertia of Zone

The absorption/desorption of moisture in the zone is accounted for using the hygric inertial model of Vereecken et al. whereby a multiplier X is added to the M(t) term of Equation 10 and Equation 11. An appropriate value of X can be measured for the complete zone and all of its furnishings by using the protocol given by [Vereecken E, Roels S, Janssen H, 2011. In situ determination of the moisture buffer potential of room enclosures, Journal of Building Physics, 34(3): 223-246.]