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Humidification requirements in economizer-type HVAC systems.

INTRODUCTION

The humidification requirement of a heating, ventilation, and air-conditioning (HVAC) system is determined by the amount of outside air (OA) that the system uses and the difference between the psychrometric properties of OA versus space design.

This requirement is computed relatively easily when the amount of OA used by the HVAC system is fixed. The situation is more complicated when the amount of OA varies as a function of its psychrometric properties: this is the case when an economizer-type system is used, because it is designed with the intent to minimize energy consumption (Mumma 2005). Two types of economizer systems are in widespread use: temperature-based and enthalpy-based economizers. This research derives explicit formulae for the humidification loads that occur in non-economizer, temperature-based economizer, and enthalpy-based economizer systems.

The Properties of Moist Air

Moist air is a mixture of air (itself composed primarily of diatomic molecules, notably [N.sub.2] and [O.sub.2]) and water vapor ([H.sub.2]O). The specific heat [c.sub.V] of a diatomic gas at constant volume is (Toth 2010):

[c.sub.v.sup.dg] = 5R/[2[M.sub.n]] (1)

where R [congruent to] 8.31[JK.sup.-1][mol.sup.-1] [congruent to] 3.40 ft*lb*[R.sup.-1][mol.sup.-1] is the ideal gas constant and [M.sub.n] (#[congruent [to]0.029 kg/mol [congruent [to] 0.064 lb/mol for air) is the molar mass of the gas. For dry air, we get

[c.sub.v.sup.da] = 716 [JK.sup.-1]k[g.sup.-1] = 133 1b*f[t.sub.f]*[R.sup.-1]1[b.sub.w.sup-1] (2)

which agrees well with the observed value 718 [JK.sup.-1] [kg.sup.-1] = 133 1b*f[t.sub.f]*[R.sup.-1]*[1b.sub.w.sup.-1]

The factor of five in the numerator of Equation 1 is the number of kinetic degrees of freedom for a diatomic molecule at room temperature; in addition to the three translational degrees of freedom, diatomic molecules also have two rotational degrees of freedom. Water molecules are triatomic, with six kinetic degrees of freedom at room temperature (three translational, three rotational). Given a molar mass of [M.sub.n] = 0.018 kg/mol (0.040 lb/mol) for water vapor, we get

[c.sub.v.sup.wv] = 6R/[2[M.sub.n]] = 1.39 kJ[K.sup.-1]k[g.sup.-1] = 257 1b*f[t.sub.f]*[R.sup.-1]1[b.sub.w.sup-1] (3)

which agrees well with the observed value 1.38 k[JK.sup.-1] [kg.sup.-1] = (257 1b f[t.sub.f]*[R.sup.-1]*[1b.sub.w.sup.-1] at room temperature.

The water vapor content of moist air is usually measured using a dimensionless number, the humidity ratio w, which is the ratio of the mass of water vapor and the mass of dry air. Thus, for a given quantity of moist air characterized by w, the specific heat can be calculated as

[c.sub.v.sup.ma] = [w/(1 + w)][c.sub.v.sup.wv] + [(1/1 + w)][c.sub.v.sup.da] (4)

Given that w [much less than] 1, we have 1/(1 + w) [congruent to] 1 - w, and terms quadratic in w can be dropped, allowing us to obtain a further simplification:

[c.sub.v.sup.ma] = ([c.sub.v.sup.wa] - [c.sub.v.sup.da])w + [c.sub.v.sup.da] (5)

The specific heat at constant pressure, [c.sub.P], is connected with [c.sub.V] by:

[c.sub.P] - [c.sub.V] = R/[M.sub.n] (6)

which is known as Mayer's relation and can be derived from axiomatic thermodynamics. Thus, for dry air we get

[c.sub.p.sup.da] = [7/5][c.sub.v.sup.da] (7)

and for water vapor:

[c.sub.p.sup.wv] = 4/3 [c.sub.v.sup.wv] (8)

again in good agreement with the observed values of 1.01 and 1.84 kJ*[K.sup.-1](0.24 and 0.44 Btu*[R.sup.-1]l[b.sup.-1])], respectively. For moist air, we get

[c.sub.p.sup.ma] = w/1+w [c.sub.p.sup.wv] + 1/1+w [c.sub.p.sup.da] [congruent to] [c.sub.p.sup.da] + ([c.sub.p.sup.wv] - [c.sub.p.sup.da])w (9)

The enthalpy of moist air, [h.sup.ma], is the sum of the heat content of moist air plus the latent heat of vaporization [h.sup.vap]:

[h.sup.ma] = [c.sub.p.sup.ma]T + [h.sup.vap]w (10)

By convention, ASHRAE chooses the heat content of moist air as zero at the starting point [T.sub.0] of the temperature scale, (1) while calculating the vaporization heat [h.sub.0.sup.vap] at [T.sub.0] (ASHRAE 2005). Thus, the enthalpy is written as:

[h.sub.ASHRAE.sup.ma] = [c.sub.p.sup.ma](T - [T.sub.0]) + [h.sub.0.sup.vap]w (11)

The latent heat of vaporization for water vapor at T0 is [h.sub.0.sup.vap]] [congruent to] 2501 kJ/kg (1061 Btu/lb) _

Therefore, per unit of dry air (1 unit of dry air is contained in 1 + w units of moist air), we obtain in SI units,

[h.sub.ASHRAE.sup.ma] = (1.85w + 1.00)(T - [T.sub.0]) + 2501w kj/kg (12a)

This agrees closely with the standard ASHRAE formulation, which reads

[h.sub.ASHRAE.sup.ma] = (1.86w + 1.006)(T - [T.sub.0]) + 2501w kj/kg (13a)

Similarly, in I-P units, we obtain

[h.sub.ASHRAE.sup.ma] = (0.44w + 0.24)(T - [T.sub.0]) + 1061w Btu/1b (12b)

which is again in close agreement with the ASHRAE result:

[h.sub.ASHRAE.sup.ma] = (0.444w + 0.240)(T - [T.sub.0]) + 1061w Btu/1b (13b)

At a given atmospheric pressure, the temperature T and humidity ratio w completely determine the properties of moist air. These two properties are usually plotted in the form of a state point in a psychrometric chart, in which temperature is represented by the horizontal axis, the humidity ratio by the vertical axis, and additional plot lines indicate states of constant enthalpy, relative humidity, and wet-bulb temperature (Figure 1).

[FIGURE 1 OMITTED]

When quantities of air are mixed, the properties of the resulting mixed air can be computed by recognizing that the following quantities are additive and conserved: the amount of dry air, the amount of moisture, and the energy content (enthalpy). Therefore, given [m.sub.1], [w.sub.1], and [h.sub.1] representing the dry air mass, humidity ratio, and enthalpy of a first quantity of air, with [m.sub.2], [w.sub.2], and [h.sub.2] the same for a second quantity of air, a mixture would be represented by

m = [m.sub.1] + [m.sub.2] (14)

w = [w.sub.1][m.sub.1] + [w.sub.2][m.sub.2]/[m.sub.1] + [m.sub.2] (15)

h = [h.sub.1][m.sub.2] + [h.sub.2][m.sub.2]/[m.sub.1] + [m.sub.2] (16)

In a standard HVAC system, a fixed amount of OA is taken in, either by itself (makeup air system) or mixed with a quantity of recirculated air (mixed air system). Clearly, such a system is not always energy efficient. It may draw in more OA when necessary (e.g., during summer months), requiring excessive cooling; in winter months, it may be drawing in less air than possible, making it necessary to cool interior parts of larger buildings actively.

Economizer Systems

Issues of energy efficiency are meant to be addressed by economizer-type HVAC systems, in which the ratio of OA and recirculated air is varied in response to changing OA conditions.

Generally speaking, an economizer-type system has three operating modes:

1. When the OA is hotter than the design conditions, the amount of OA being drawn in is the minimum amount of fresh air required;

2. When the OA is colder than the design conditions but warmerlint free tissue than a predetermined minimum supply air temperature, the system operates in a 100% OA mode;

3. When the OA is colder than the minimum supply air temperature, the ratio is modulated to ensure that the supply air temperature does not fall below the permitted minimum; however, minimum fresh air requirements are still observed and take precedence (requiring perhaps that the OA be preheated in order to achieve the minimum supply air temperature).

The difference between temperature-based and enthalpybased economizer systems boils down to the definition of "hotter:" specifically, is OA considered hotter when its temperature is higher than the design conditions, or when its enthalpy is higher than the design enthalpy?

The behavior of economizer systems is illustrated in Figure 2.

[FIGURE 2 OMITTED]

The difference between the two types of economizer systems can be striking. For instance, a temperature-based economizer may be drawing in 100% very moist air that is only slightly colder than the design conditions; after being heated, this air would be significantly more moist than what is acceptable, requiring excessive dehumidification.

Insofar as humidification requirements are concerned, the difference is effectively non-existent. For a non-economizertype system, the worst-case humidification condition occurs at the bottom edge of the shaded region, when the humidity ratio of the OA is lowest, which is when the most water vapor needs to be added in order to achieve the desired design conditions. If this point falls within region III (in Figure 2), the location of the worst-case point does not change; humidification requirements may, however, increase because 100% OA is being used. More likely, however, the worst-case condition falls within region IV, where the amount of OA is being modulated. In this case, as a result of the modulation, less humidification is required than in a corresponding makeup air system; indeed, chances are that the worst-case humidification scenario coincides with the minimum supply air temperature as at lower temperatures, the amount of OA being used rapidly decreases due to modulation.

If we assume that in region IV the amount of OA is modulated such that the temperature of mixed air remains constant at the minimum supply air temperature, the percentage of OA as a function of its state can be computed. First, given Equation 13, the temperature of a quantity of moist air can be computed if its enthalpy and humidity ratio are known:

T* = h - [h.sub.0.sup.vap]w/[c.sub.p.sup.wv]w + [c.sub.p.sup.da] (17)

where the notation T* = T - [T.sub.0] is used to indicate temperatures measured in a nonabsolute temperature scale.

Mixing a quantity of OA (with dry air mass [m.sub.OA], humidity ratio [w.sub.OA], and enthalpy [h.sub.OA]) and recirculated air (with dry air mass [m.sub.RA], humidity ratio [w.sub.RA], and enthalpy [h.sub.RA]), we obtain mixed air, in accordance with Equations 14-16, with mixed-air temperature as follows:

[T*.sub.MA] = [h.sub.OA][m.sub.OA] + [h.sub.RA][m.sub.RA] - [h.sub.0.sup.vap]([w.sub.OA][m.sub.OA] + [w.sub.RA] [m.sub.RA])/[c.sub.p.sup.wv]([w.sub.OA][m.sub.OA] + [w.sub.RA][m.sub.RA]) + [c.sub.p.sup.da]([m.sub.OA] + [m.sub.RA]) (18)

This equation can be solved for [m.sub.OA], yielding

[m.sub.OA] = [([c.sub.p.sup.wv][T*.sub.MA] + [h.sub.0.sup.vap])[w.sub.RA] + [c.sub.p.sup.da][T*.sub.MA] - [h.sub.RA]/([c.sub.p.sup.wv][T*.sub.MA] + [h.sub.0.sup.vap])[w.sub.OA] + [c.sub.p.sup.da][T*.sub.MA] - [h.sub.OA]] (19)

Alternatively, one can express this result in terms of the OA temperature and recirculated air temperature using Equation 13:

[m.sub.OA] = ([c.sub.p.sup.wv][w.sub.RA] + [c.sub.p.sup.da])([T*.sub.RA] - [T*.sub.MA])/([c.sub.p.sup.wv][w.sub.OA] + [c.sub.p.sup.da])([T*.sub.OA] - [T*.sub.MA])[m.sub.RA] (20)

The humidity ratio of mixed air, then, can be calculated as:

[w.sub.MA] = [[w.sub.OA][m.sub.OA] + [w.sub.RA][m.sub.RA]]/[[m.sub.OA] + [w.sub.RA]] = [([c.sub.p.sup.wv][w.sub.RA] + [c.sub.p.sup.da])([T*.sub.RA] - [T*.sub.MA]) + ([c.sub.p.sup.wv] [w.sub.OA] + [c.sub.p.sup.da])([T*.sub.OA] - [T*.sub.MA])[w.sub.RA]]/[([c.sub.p.sup.wv][w.sub.RA] + [c.sub.p.sup.da])([T*.sub.RA] - [T*.sub.MA]) + ([c.sub.p.sup.wv] [w.sub.OA] + [c.sub.p.sup.da])([T*.sub.OA] - [T*.sub.MA])] (21)

This formula can be used directly to assign mixed-air humidity ratios to all state points in a psychrometric chart; in particular, to the area representing measured weather conditions at a particular location, as indicated schematically in Figures 1 and 2. The actual humidification load is proportional to the difference between [w.sub.MA] and the humidity ratio of the desired design conditions; the lower [w.sub.MA] is, the higher the humidification load. Finding the maximum humidification load for an economizer system, therefore, amounts to finding the OA state point for which [w.sub.MA] is minimal.

Minimum Outside Air Requirement

In a typical HVAC system, a minimum amount of OA must always be used in order to satisfy fresh air requirements. Even when no such minimum exists, the amount of OA can never be negative, something that Equation 21 does not guarantee.

Given a minimum fresh air ratio expressed in the form D = [m.sub.OA] / [m.sub.RA], the OA requirement can be formulated as follows:

[m.sub.OA] = -[([c.sub.p.sup.wv][w.sub.RA] + [c.sub.p.sup.da])([T*.sub.RA] - [T*.sub.MA])]/([c.sub.p.sup.wv][w.sub.OA] + [c.sub.p.sup.da])([T*.sub.OA] - [T*.sub.MA])][m.sub.RA]

if - [[([c.sub.p.sup.wv][w.sub.RA] + [c.sub.p.sup.da])([T*.sub.RA] - [T*.sub.MA])]/[([c.sub.p.sup.wv][w.sub.OA] + [c.sub.p.sup.da])([T*.sub.OA] - [T*.sub.MA])]] > [alpha] (22)

[m.sub.OA] = [alpha][m.sub.RA]

if - [[([c.sub.p.sup.wv][w.sub.RA] + [c.sub.p.sup.da])([T*.sub.RA] - [T*.sub.MA])]/[([c.sub.p.sup.wv][w.sub.OA] + [c.sub.p.sup.da])([T*.sub.OA] - [T*.sub.MA])] [less than or equal to] [alpha]

Consequently, the humidity ratio of mixed air reads:

[w.sub.MA] = [[w.sub.OA][m.sub.OA] + [w.sub.RA][m.sub.RA]]/[[m.sub.OA] + [w.sub.RA]]

if - [[([c.sub.p.sup.wv][w.sub.RA] + [c.sub.p.sup.da])([T*.sub.RA] - [T*.sub.MA])]/[([c.sub.p.sup.wv][w.sub.OA] + [c.sub.p.sup.da])([T*.sub.OA] - [T*.sub.MA])] > x (23)

[w.sub.MA] = [[alpha][w.sub.OA] + [w.sub.RA]]/[1+[alpha]]

if - [[([c.sub.p.sup.wv][w.sub.RA] + [c.sub.p.sup.da])([T*.sub.RA] - [T*.sub.MA])]/[([c.sub.p.sup.wv][w.sub.OA] + ([c.sub.p.sup.da])[T*.sub.OA]) - [T*.sub.MA]] [less than or equal to] [alpha]

with [m.sub.OA] given by Equation 20.

CONCLUSION

In most design scenarios, all of the variables in Equation 23, notably, the range of OA state points to be considered (characterized by [T*.sub.OA] and [w.sub.OA]), the recirculated air state point ([T*.sub.RA], [w.sub.RA], often coinciding with design conditions), the (minimum) supply duct temperature ([T*.sub.MA]) and the minimum fresh air requirement ([alpha]), are known, and the expression can be easily and repeatedly evaluated, and the minimal value of [w.sub.MA] can be found rapidly.

Although the present focus is humidification, it is worth noting that the same approach can be used to determine the maximum heating, cooling, and dehumidification loads of a system as well. In a non-economizer system, these correspond to the maximum and minimum enthalpy, and maximum humidity ratio within the range of OA state points that occur at the design location. For economizer systems, the load is varied by the modulation effected by the system.

Lastly, we note that although data sets published by ASHRAE provide per-location state point frequency information (i.e., a frequency function in the form of f (T, w) or equivalent, assigning frequencies of occurrence to each grid point in a psychrometric chart at some grid resolution), the data are not broken down by the time-of-day [tau]. If a designer has at his disposal a three-dimensional data set available that provides frequency of occurrence of specific state points at specific times of day (i.e., a frequency function f (T, w, [tau])), this could be used to determine system loads as a function of building occupancy. This information could also be used to improve the accuracy of analyzing operational costs.

(1.) A note about a subtle trap that one can fall into at this point: we are no longer free to choose an arbitrary starting point for our temperature scale! Consider the partial derivative of [h.sup.ma] with respect to w, given that neither T nor [h.sup.vap] depend on w:

[delta][h.sup.ma]/[delta]w = [[T[delta][c.sub.P.sup.ma]]/[[delta]w + vap]]] + [h.sup.vap]

Clearly, changing the starting point of the temperature scale (from absolute zero to, say, the freezing point of water) changes the right-hand side of this equation in a way that is not accounted for by a change of dimensions. In other words, at this point, the choice of temperature scale is no longer a matter of convention; the temperature scale must be an absolute scale, otherwise the results are incorrect. Specifically, if we naively assume that we can set the enthalpy scale such that the enthalpy of moist air at the freezing point of water is its vaporization heat only, we make the invalid assumption that the heat content of air and water vapor are the same at a temperature other than absolute zero; this, of course, is wrong, as (assuming both are ideal gases) the heat content of a quantity of air at 0[degrees]C (32[degrees]F) matches the heat content of a similar quantity of water vapor at -125.41[degrees]C(-193.74[degrees]F). This problem does not affect calculations in which the humidity content of air remains constant (i.e., heating and cooling) but it can impact calculations that involve humidification or dehumidification.

REFERENCES

ASHRAE. 2005. ASHRAE Handbook--Fundamentals. Atlanta: ASHRAE.

Mumma, S.A. 2005. Role of Economizers in DOAS: Part 1. ASHRAE IAQ Applications: 6(4).

Toth, Viktor T. 2010. The virial theorem and planetary atmospheres. Quarterly Journal of the Hungarian Meteorological Service (HMS) 114(3):229-234.

Viktor T. Toth is retired and lives in Ottawa, ON, Canada.
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Title Annotation:DA-13-022
Author:Toth, Viktor T.
Publication:ASHRAE Transactions
Article Type:Report
Geographic Code:1CANA
Date:Jan 1, 2013
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