We develop a formulation to compute the maximum humidiﬁcation load for economizer-type HVAC systems. The resulting formulae can be used, together with ASHRAE-provided climate data, to ﬁnd the maximum humidiﬁcation load in both temperature-based and enthalpy-based economizer systems. 1 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 outside air versus space design. This requirement is computed relatively easily when the amount of outside air used by the HVAC system is ﬁxed. The situation is more complicated when the amount of outside air varies as a function of its psychrometric properties: this is the case when an “economizer” type system is used [1], which is designed with the intent to minimize energy consumption. Two types of economizer systems are in widespread use: temperature-based and enthalpy-based economizers. In this paper, we derive explicit formulas for the humidiﬁcation loads that occur in non-economizer, temperature-based economizer, and enthalpy-based economizer systems. 2 The properties of moist air Moist air is a mixture of air (itself composed primarily of diatomic molecules, notably N 2and O 2) and water vapor (H 2O). The speciﬁc heat cVof a diatomic gas at constant volume is [2] cdg V=5R 2Mn, (1) where R/similarequal8.31 JK−1mol−1/similarequal3.40 ft·lb·R−1mol−1is the ideal gas constant and Mn(/similarequal0.029 kg/mol /similarequal 0.064 lb/mol for air) is the molar mass of the gas. For air, we get cda V= 716 JK−1kg−1= 133 lb ·ftf·R−1lb−1 w, (2) which agrees well with the observed value (718 JK−1kg−1= 133 lb f·R−1lb−1 w). The factor of 5 in the numerator of Eq. (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 6 kinetic degrees of freedom at room temperature (three translational, three rotational). Given a molar mass of Mn= 0.018 kg/mol for water vapor, we get cwv V=6R 2Mn/similarequal1.39 kJK−1kg−1= 257 lb ·ftf·R−1lb−1 w, (3) ∗http://www.vttoth.com/ 1 which agrees well with the observed value of 1.38 kJK−1kg−1(257 lb ·ftf·R−1lb−1 w) 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 cmoist air V =w 1 +wcwv V+1 1 +wcda V. (4) Given that w/lessmuch1, we have 1 /(1 +w)/similarequal1−w, and terms quadratic in wcan be dropped, allowing us to obtain a further simpliﬁcation: cmoist air V /similarequal(cwv V−cdryair V)w+cda V. (5) The speciﬁc heat at constant pressure, cP, is connected with cVby cP−cV=R Mn, (6) which is known as Mayer’s relation and can be derived from axiomatic thermodynamics. Thus, for dry air we get cda P=7 5cda V, (7) and for water vapor, cwv P=4 3cwv V, (8) again in good agreement with the observed values of 1.01 and 1.84 kJK−1kg−1(0.24 and 0.44 Btu ·R−1lb−1), respectively. For moist air, we get cmoist air P =w 1 +wcwv P+1 1 +wcda P/similarequalcda P+ (cwv P−cda P)w. (9) The enthalpy of moist air, hmoist airis the sum of the heat content of moist air plus the latent heat of vaporization hvap: hmoist air=cmoist air P T+hvapw. (10) By convention, ASHRAE chooses the heat content of moist air zero at the starting point T0of the temperature scale1, while calculating the vaporization heat hvap 0atT0[3]. Thus, the enthalpy is written as: hmoist air ASHRAE =cmoist air P (T−T0) +hvap 0w. (11) The latent heat of vaporization for water vapor at T0ishvap 0/similarequal2501 kJ /kg (1061 Btu /lb). Per unit of dry air (1 unit of dry air is contained in (1 + w) units of moist air), therefore, we obtain, in SI units, hmoist air ASHRAE /similarequal(1.85w+ 1.00)(T−T0) + 2501 wkJ/kg. (12a) 1A 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 hmoist airwith respect to w, given that neither Tnorhvapdepend onw: ∂hmoist air ∂w=T∂cmoist air P ∂w+hvap. Clearly, changing the starting point of the temperature scale (from 0 K to, say, 273.15 K) 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 0◦C 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◦C matches the heat content of a similar quantity of water vapor at −125.41◦C. 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. 2 Figure 1: Schematic of a psychrometric chart, with one constant relative humidity (RH) and one constant enthalpy ( h) line shown. Gray area schematically illustrates the recorded range of outside air conditions at a given location. This agrees closely with the standard ASHRAE formulation, which reads hmoist air ASHRAE-standard = (1.86w+ 1.006)( T−T0) + 2501 wkJ/kg. (13a) Similarly, in conventional units, we obtain hmoist air ASHRAE /similarequal(0.44w+ 0.24)(T−T0) + 1061 wBtu/lb, (12b) which is again in agreement (with diﬀerences being due to rounding) with the ASHRAE result: hmoist air ASHRAE-standard = (0.444w+ 0.240)( T−T0) + 1061 wkJ/kg. (13b) At a given atmospheric pressure, the temperature Tand humidity ratio wcompletely 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 on the vertical axis, and additional plotlines indicate states of constant enthalpy, relative humidity, and wet bulb temperature (see Figure 1). 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 m1,w1,h1representing the dry air mass, humidity ratio, and enthalpy of a ﬁrst quantity of air, with m2,w2andh2the same for a second quantity of air, a mixture would be represented by m=m1+m2, (14) w=w1m1+w2m2 m1+m2, (15) h=h1m1+h2m2 m1+m2. (16) 3 Figure 2: Economizer systems. Thick lines correspond to the design condition temperature and enthalpy, and the minimum supply air temperature. Temperature and enthalpy based economizers behave identically in regions I, III and IV; their behavior is diﬀerent in regions IIa and IIb and IIc. Adapted from [1]. In a standard HVAC system, a ﬁxed amount of outside air 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 outside air 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. 3 Economizer systems Issues of energy eﬃciency are meant to be addressed by “economizer” type HVAC systems, in which the ratio of outside air and recirculated air is varied in response to changing outside air conditions. Generally speaking, an economizer type system has three operating modes: 1.When the outside air is hotter than the design conditions, the amount of outside air being drawn in is the minimum amount of fresh air required; 2.When the outside air is colder than the design conditions but warmer than a predetermined minimum supply air temperature, the system operates in a 100% outside air mode; 3.When the outside air 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 outside air be preheated in order to achieve the minimum supply air temperature.) The diﬀerence between temperature and enthalpy based economizer systems boils down to the deﬁnition of “hotter”: speciﬁcally, is outside air considered hotter if 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. 4 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- economizer type system, the worst-case humidiﬁcation condition occurs at the bottom edge of the shaded region, when the humidity ratio of the outside air 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% outside air is being used. More likely, however, the worst-case condition falls within region IV, where the amount of outside air 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 left of this temperature, the amount of outside air being used rapidly decreases due to modulation. If we assume that in region IV, the amount of outside air is modulated such that the temperature of mixed air remains constant at the minimum supply air temperature, we can compute the percentage of outside air as a function of its state. First, given (13), we can compute the temperature of a quantity of moist air if its enthalpy and humidity ratio are known: T⋆=h−hvap 0w cwv Pw+cda P, (17) where we use the notation T⋆=T−T0to indicate temperatures measured in a non-absolute temperature scale. Mixing a quantity of outside air (with dry air mass mOA, humidity ratio wOA, and enthalpy hOA) and recirculated air (with dry air mass mRA, humidity ratio wRA, and enthalpy hRA), we obtain mixed air, in accordance with Eqs. (14–16), with mixed air temperature T⋆ MA=hOAmOA+hRAmRA−hvap 0(wOAmOA+wRAmRA) cwv P(wOAmOA+wRAwRA) +cda P(mOA+mRA). (18) This equation can be solved for mOA, yielding mOA=−(cwv PT⋆ MA+hvap 0)wRA+cda PT⋆ MA−hRA (cwv PT⋆ MA+hvap 0)wOA+cda PT⋆ MA−hOAmRA. (19) Alternatively, we can express this result in terms of the outside air temperature T⋆ OAand recirculated air temperature T⋆ RAusing (13): mOA=−(cwv PwRA+cda P)(T⋆ RA−T⋆ MA) (cwv PwOA+cda P)(T⋆ OA−T⋆ MA)mRA. (20) The humidity ratio of mixed air, then, can be calculated as wMA=wOAmOA+wRAmRA mOA+mRA(21) =(cwv PwRA+cda P)(T⋆ RA−T⋆ MA)wOA+ (cwv PwOA+cda P)(T⋆ OA−T⋆ MA)wRA (cwv PwRA+cda P)(T⋆ RA−T⋆ MA) + (cwv PwOA+cda P)(T⋆ OA−T⋆ MA). 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 indi- cated schematically in Figs. 1 and 2. The actual humidiﬁcation load is proportional to the diﬀerence between wMAand the humidity ratio of the desired design conditions; the lower wMAis, the higher the humidiﬁcation load. Finding the maximum humidiﬁcation load for an economizer system, therefore, amounts to ﬁnding the outside air state point for which wMAis minimal. 5 4 Minimum outside air requirement In a typical HVAC system, a minimum amount of outside air must always be used in order to satisfy fresh air requirements. Even when no such minimum exists, the amount of outside air can never be negative, something that our Eq. (21) does not guarantee. Given a minimum fresh air ratio expressed in the form α=mOA/mRA, the minimum fresh air requirement can be formulated as follows: mOA=8 >>>>< >>>>:−(cwv PwRA+cda P)(T⋆ RA−T⋆ MA) (cwv PwOA+cda P)(T⋆ OA−T⋆ MA)mRA if−(cwv PwRA+cda P)(T⋆ RA−T⋆ MA) (cwv PwOA+cda P)(T⋆ OA−T⋆ MA)> α, αmRA if−(cwv PwRA+cda P)(T⋆ RA−T⋆ MA) (cwv PwOA+cda P)(T⋆ OA−T⋆ MA)≤α.(22) Consequently, the humidity ratio of mixed air reads: wMA=8 >>>>< >>>>:wOAmOA+wRAmRA mOA+mRAif−(cwv PwRA+cda P)(T⋆ RA−T⋆ MA) (cwv PwOA+cda P)(T⋆ OA−T⋆ MA)> α, αwOA+wRA 1 +αif−(cwv PwRA+cda P)(T⋆ RA−T⋆ MA) (cwv PwOA+cda P)(T⋆ OA−T⋆ MA)≤α,(23) with mOAgiven by (20). 5 Conclusions In most design scenarios, all of the variables in Eq. (23): notably, the range of outside air state points to be considered (characterized by T⋆ OAandwOA), the recirculated air state point ( T⋆ RA,wRA, often coinciding with design conditions), the (minimum) supply duct temperature ( T⋆ MA) and the minimum fresh air requirement (α), are known, and the expression can be easily and repeatedly evaluated, and the minimal value of wMA can be found rapidly. Although our present focus is humidiﬁcation, we note that the same approach can be used to determine the maximum heating, cooling, and dehumidiﬁcation load 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 outside air 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 τ. 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, τ )), 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.