“Partial pressures” of humid air in wide pressure and temperature ranges

Abstract

“Partial pressure” in humid air is a question very much concerned by scientists and no satisfactory answer has been found to date. This paper proposes a novel method to obtain the “partial pressures” of the water vapor and dry air in humid air. The results obtained by the proposed method are quite different from that obtained by Dalton’s partial pressure law. The fundamental behaviors of water vapor and dry air are studied in depth in wide pressure and temperature ranges. Semi-permeable membrane models are proposed and applied for both saturated and unsaturated humid air. “Improvement factors” are developed to quantitatively describe the magnitude of the interaction between dissimilar molecules. One discovery is that the “partial pressure” of the water vapor in saturated humid air equals P_s, rather than (f·P_s) which was formerly believed. The other is that the interaction between dissimilar molecules may be omitted when temperature is above “cutting-off temperature” for unsaturated humid air. This paper satisfactorily answers the quest of “partial pressures” in humid air from a new perspective.

Keywords

partial pressure / Dalton’s partial pressure law / humid air / saturated / unsaturated

Introduction

Humid air is composed of dry air and water vapor, which is one kind of typical gas mixture. Humid air has been widely used in the field of refrigeration, air cooling [1] and humid air turbine (HAT) cycles [2].

The exact “partial pressures” of the dry air and the water vapor in humid air is very much concerned by scientists. In the past, researchers always directly applied Dalton’s partial pressure law to describe the “partial pressures” for gas components in humid air [3]. Such practices regarded humid air as an ideal mixture and omitted the interactions between molecules. Errors were caused in these practices because of such omissions.

This paper treats the “partial pressures” of humid air from a new perspective. The semi-permeable membrane model is constructed for humid air to detect “partial pressures”. The “partial pressure” of the water vapor in saturated humid air is discovered, and the result obtained is quite different from that obtained in the past. Such difference can be quantitatively described by “improvement factors”. The semi-permeable membrane model is also fit for unsaturated humid air. The “Improvement factors” are studied for unsaturated humid air in depth. It is discovered that “cutting-off temperature” is a crucial benchmark temperature. The numerical results of “partial pressures” obtained by the novel model proposed are depicted and discussed, which demonstrate that the real “partial pressures” deviate even more than 10% from those obtained by Dalton’s partial pressure law. The “Partial pressures” obtained by the semi-permeable membrane model and those obtained by Dalton’s partial pressure law converge with the increase of temperature.

“Partial pressures” in saturated humid air

Saturated humid air and semi-permeable membrane model

As shown in Fig. 1, the “cube” contains saturated humid air. The “triangles” represent saturated water vapor molecules while the “black dots” represent dry air molecules. The solid lines and dotted lines represent the interactions between similar and dissimilar molecules. On the top of the “cube”, there is a semi-permeable membrane which allows the water vapor molecules to pass through freely but obstructs the dry air molecules to pass through. A cylinder is installed on the semi-permeable membrane, in which there is a pressure sensor (a thin piston with a weight on it). Thus, the “partial pressure” of saturated humid air can be detected by the weight in the cylinder.

There can only be three possibilities for the pressure of the water vapor in the cylinder, that is P_{Cylinder}>P_s; ; and P_{Cylinder}=P_s, as shown in Fig. 1. Here, P_s is the saturated pressure for the existing alone saturated water vapor at temperature T.

The possibility of P_{Cylinder}>P_s cannot hold because the existing alone water vapor in the cylinder cannot exceed P_s. The possibility is also impossible, because means the pure water vapor in the cylinder is not saturated. It is supposed that some liquid water at the same temperature is added into the cylinder. The added liquid water in the cylinder would evaporate immediately at the same temperature, because the pure water vapor in the cylinder is not yet saturated. During the evaporation of the liquid water, the pressure in the cylinder must increase and exceed the “partial pressure” of the water vapor in the “cube”. The equilibrium on the two sides of the semi-permeable membrane would be broken and some water vapor in the cylinder would move into the “cube”. However, the humid air in the “cube” is already saturated and no water vapor could be taken in any more. Here appears a contradiction which is caused by the wrong assumption of . The only possibility left is P_{Cylinder}=P_s. Thus, it is discovered that when the humid air in the “cube” is saturated, the pure water vapor in the cylinder must be saturated as well.

The discovery of the fact that the “partial pressure” of the water vapor is equal to P_s is quite meaningful. In the past, most researchers believed that the partial pressure of the water vapor in saturated humid air is equal to f·P_s [4], which is based on Dalton’s partial pressure law. As will be discussed in Section 2.2, f is always bigger than 1.0000. The real “partial pressure” of the water vapor equals P_s, but is always smaller than the formerly believed f·P_s value.

Absolute humidity of saturated humid air

Absolute humidity is the mol number of water vapor in 1 mol of saturated humid air. Saturated humid air is a special condition of humid air. The gas phase water and the liquid phase water are at equilibrium. Thus, the fugacity of gas phase water vapor should equal the fugacity of liquid phase water:

F_w^V=F_w^L.

The gas phase water fugacity and liquid phase water fugacity can always be established [5].

The gas phase water fugacity is

F_w^V={\varphi }_w{y}_{w}P.

The liquid phase water fugacity is

F_w^L={\phi }_w^V{P}_s\exp \left[{\displaystyle \frac{\mathrm{1}}{RT}}{\displaystyle {\int }_{P_s}^P{v}_w^{L}dP}\right].

When gas phase and liquid phase equilibrium is established at pressure P and temperature T, its absolute humidity can be derived as

y_w={\displaystyle \frac{P_s}{P}}\cdot {\displaystyle \frac{{\phi }_w^V\exp \left({\displaystyle \frac{\mathrm{1}}{RT}}{\displaystyle {\int }_{P_s}^P{v}_w^{L}dP}\right)}{{\varphi }_w}},

where {\phi }_w^V is the fugacity coefficient of pure steam at temperature T, P_s is the saturated pressure of pure steam at temperture T, and v_w^L is the specific volume of pure liquid water at different temperatures. Using the properties of pure liquid water, the polynomial expression of {\phi }_w^V, P_s and v_w^L can be solved smoothly [6]. y_w is the mol number percentage of the water vapor, {\varphi }_w is the fugacity coefficient, and P is the pressure of the system. {\varphi }_w can be determined by the Redlich-Kwong model. This model regards humid air as a pseudo-pure gas and does not differentiate the “partial pressures” of different gas components. Such pseudo-pure gas has its own equation of state [7,8].

The expression

{\displaystyle \frac{{\phi }_w^V\exp \left[{\displaystyle \frac{\mathrm{1}}{RT}}{\displaystyle {\int }_{P_s}^P{v}_w^{L}dP}\right]}{{\varphi }_w}}

is defined as the “Enhancement Factor” (which is denoted as “f”, with the subscript s representing saturated state). Therefore, the following equation is obtained:

n_s^{mixture}=f_s\cdot {\displaystyle \frac{P_s}{P}}.

.

The results of the “Enhancement Factors” can be derived at the total pressures of 0.5, 1, 2 and 5 MPa, as illustrated in Fig. 2.

Mol number of existing alone saturated water vapor

In Section 2.2, the mol number of the water vapor in 1 mol of saturated humid air has been derived. Such results consider both the interaction between similar molecules and dissimilar molecules. In this section, the mol number of the existing alone saturated humid air in the same volume will be carefully studied. The compressibility factors for pure water vapor [6] and pure dry air [9] are well equipped. The compressibility factors may be applied to the existing alone water vapor and the existing alone dry air as

P_{s}V=n_s^{alone}{z}_{s}RT

P_{air}V=n_{air}^{alone}{z}_{air}RT

When the two equations are compared, they yield

{\displaystyle \frac{P_s}{P_{air}}}={\displaystyle \frac{n_s^{alone}}{n_{air}^{alone}}}\cdot {\displaystyle \frac{z_s}{z_{air}}}.

It is easy to understand that when the compressibility factors are considered, the partial pressures ratio is no longer equal to the simple mol number ratio, which is the assumption of Dalton’s partial pressure law. The partial pressure ratio should also be adjusted by the ratio of two compressibility factors.

For 1 mol of saturated humid air, when P and T are given, and after rearrangement, it can be obtained that

n_s^{alone}={\displaystyle \frac{P_s{z}_{air}}{(P-P_s)z_s+P_s{z}_{air}}}.

The introduction of the compressibility fully considers the interaction between similar molecules and omits the interactions between dissimilar molecules [10].

“Improvement effect” and “improvement factor”

In Section 2.2, n_s^{mixture} represents the mol number of the water vapor in 1 mol of saturated humid air. In Section 2.3, n_s^{alone} represents the mol number of the existing alone saturated water vapor in the same volume.

After careful comparison of the data of n_s^{mixture} and those of n_s^{alone}, it is discovered that

n_s^{mixture}>n_s^{alone}>n_s^{Dalton\prime s},

where n_s^{Dalton\prime s} is defined as P_s/P. Such inequalities may be explained by the fact that if n_s^{Dalton\prime s} is taken as the basis thatn_s^{alone}=f_{\mathrm{1},s}{n}_s^{Dalton\prime s},in which f_{\mathrm{1},s} demonstrates the interaction between similar molecules, it yields

n_s^{mixture}=f_s{n}_s^{Dalton\prime s},

in which f_s demonstrates both the interaction between similar molecules and dissimilar molecules. Thus, it is fine to obtain

f_{2,s}={\displaystyle \frac{f_s}{f_{\mathrm{1},s}}}={\displaystyle \frac{n_s^{mixture}}{n_s^{alone}}}.

where f_{\mathrm{2},s} is defined as the “improvement factor” of the water vapor, which demonstrates the interaction between dissimilar molecules. The numerical results of f_{\mathrm{2},s} at total pressures of 0.5, 1, 2 and 5 MPa are demonstrated in Fig. 3.

The in-depth understanding of f_{\mathrm{2},s} is explained as follows. As pointed out that f_{\mathrm{2},s}>\mathrm{1.0000}, such phenomenon may be referred to as the “improvement effect”. When NI{U}_s^{alone} mol water vapor exists alone in the “cube”, it is saturated. If some dry air at the same temperature is injected into the “cube”, the water vapor will become unsaturated. The “partial pressure” of the water vapor in the humid air will become lower than P_s and the water vapor becomes superheated. In order to recover the “partial pressure” of the water vapor in the humid air back to P_s, the (f_{\mathrm{2},s}-\mathrm{1})n_s^{alone} mol water vapor must be injected into the “cube”. The “improvement factor” f_{\mathrm{2},s} quantitatively describes the influence of the “partial pressure” on the water vapor by interactions between the dry air and the water vapor in the saturated humid air. The values of f_{\mathrm{2},s} can be obtained smoothly, so the interactions between dissimilar molecules can be well described.

The gas mixture pressure relationship law has proved that the total pressure equals the summation of “partial pressures” of different gas components. The “improvement factor” of dry air f_{\mathrm{2},air} is also smoothly derived [11].

“Partial pressures” in unsaturated humid air

Unsaturated humid air and semi-permeable membrane model

Unsaturated humid air is much more complicated than saturated humid air. It has much more thermodynamic states, which are dependent not only on the temperature and pressure, but also on the relative humidity. Relative humidity is the ratio of the water vapor compared to the saturation state. It is defined as\phi ={\displaystyle \frac{n_V^{mixture}}{n_s^{mixutre}}},in which n_V^{mixture} is the mol number of the water vapor in 1 mol of unsaturated humid air while n_s^{mixutre} is the mol number of the water vapor in 1 mol of saturated humid air at the same total pressure and temperature.

The semi-permeable membrane model is also applicable to unsaturated humid air. The “cube” contains 1 mol of unsaturated humid which is composed of n_V^{mixture}(mol) of unsaturated water vapor and n_{air}^{mixture}(mol) of dry air. The unsaturated water vapor may pass through the semi-permeable membrane, which allows the water vapor molecules to pass through freely but obstructs any dry air molecule to pass through. The unsaturated water vapor may go freely into and out of the cylinder without hindrance. The “partial pressure” of the unsaturated water vapor P_V^{cylinder} can be measured by the magnitude of the weight on the piston. Another “cube” with the same volume contains pure unsaturated water vapor. The pressure of the pure unsaturated water vapor in this “cube” can be denoted as P_V^{alone}. When P_V^{cylinder} equals P_V^{alone}, the mol number of the existing alone pure water vapor is n_V^{alone}. According to Dalton’s partial pressure law, only in the case that unsaturated water vapor and dry air do not have interaction with each other, the mol number will be the same in the above two “cubes”. Since the interactions between the unsaturated water vapor and dry air are taken into account, n_V^{mixture} and n_V^{alone} may be different. Such difference can be measured by

f_{\mathrm{2},V}={\displaystyle \frac{n_V^{mixture}}{n_V^{alone}}},

which is defined as the “improvement factor” of the water vapor in the unsaturated humid air. The description of another gas component dry air is similar, whose “improvement factor” can be denoted as

f_{2,air}^{unsaturated}={\displaystyle \frac{n_{air}^{mixture}}{n_{air}^{alone}}}.

“Cutting-off temperature” and “improvement factors”

When the total pressure P of humid air is given, the saturated temperature T_s of pure water vapor corresponding to P can be obtained by International Association for Properties of Water and Steam [6]. T_s is defined as “cutting-off temperature” [12].

The “cutting-off temperature” at different total pressures are listed in Table 1.

The “cutting-off temperature” is an important benchmark. As described in Ref. [12], when temperature is lower than the “cutting-off temperature”, the “improvement factor” of the water vapor can be expressed as

f_{\mathrm{2},V}={\displaystyle \frac{\mathrm{1}}{\mathrm{1}-(\mathrm{1}-\mathrm{1}/f_{\mathrm{2},s})(\mathrm{1}-n_V^{mixture})/(\mathrm{1}-n_s^{mixture})}},

where, f_{\mathrm{2},s} is the “improvement factor” of the saturated water vapor in the saturated humid air at the same temperature T and the same total pressure P. The mol number of the humid air is 1.00 mol.

For saturated humid air, Eq. (1) can be established as

(2)(f_{\mathrm{2},s}-\mathrm{1})n_s^{alone}=\alpha (\mathrm{1}-n_s^{mixture})n_s^{mixture}.

The left side represents the magnitude of the “improvement effect”, whereas the right side represents the magnitude of the interaction between dissimilar molecules. α is a constant at such given P and T. For unsaturated humid air, Eq. (1) can be established as

(f_{\mathrm{2},V}-\mathrm{1})n_V^{alone}=\alpha (\mathrm{1}-n_V^{mixture})n_V^{mixture}.

<Number>(3)</Number>

After comparing Eqs. (2) with (3), Eq. (1) can be obtained.

Similar result also can be established for the “improvement factor” of dry air:

f_{2,air}^{unsaturated}={\displaystyle \frac{\mathrm{1}}{(\mathrm{1}-n_V^{mixture}/n_s^{mixture})(\mathrm{1}-\mathrm{1}/f_{2,air}^{saturated})}}.

When the composition ratio and total pressure of humid air are given, saturated humid air can only exist at one specific temperature. Such temperature of saturated humid air may be denoted as “starting-up temperature”, because humid air with this composition ratio and total pressure cannot exist below such temperature. With the increase of temperature, the humid air becomes unsaturated.

The most typical situation is studied in depth. It is 1.00 mol humid air which is composed of 0.50 mol of water vapor and 0.50 mol of dry air. The “starting-up temperatures” for such composition ratio at different total pressures are listed in Table 2.

The values of the two “improvement factors” can be obtained from the “starting up temperature” to the “cutting-off temperature” at different total pressures, which are illustrated in Fig. 4.

It is demonstrated that the “improvement factors” move towards 1.0000 when the temperature increases to the “cutting-off temperature”. These trends are examined at P = 0.5, 1, 2, and 5 MPa. The result shows that such trends are universally true for different total pressures. According to such trends, it is reasonable to set the “improvement factors” to 1.0000 for both the water vapor and dry air when the humid air is above its “cutting-off temperature”.

“Partial pressures” illustration

The “partial pressures” of 1.00 mol of humid air which is composed of 0.5 mol of water vapor and 0.5 mol of dry air, from the “starting up temperature” to 1000 K at different total pressures of 0.5, 1, 2 and 5 MPa are demonstrated in Fig. 5.

The partial pressures obtained by Dalton’s partial pressure law for water vapor and dry air are always 0.25, 0.5, 1 and 2.5 MPa respectively for such fixed 0.5 mol of water vapor and 0.5 mol of dry air composition ratio regardless of the change of temperature. However, the exact “partial pressures” are not constant at different temperatures.

When the total pressure equals 0.5, 1, 2 and 5 MPa, it is discovered that the biggest deviation of the “partial pressure” obtained by the semi-permeable membrane model from the partial pressure (which is exactly 1/2 of the total pressure in this case) obtained by Dalton’s partial pressure law always appears at the “starting-up temperature”. It is also discovered that the “partial pressure” of the water vapor is always smaller than 1/2 of the total pressure and the “partial pressure” of the dry air is always bigger than 1/2 of the total pressure. When the total pressure are 0.5, 1, 2 and 5 MPa, the biggest “partial pressure” deviations from Dalton’s partial pressure law are 2.8%, 4.6%, 6.8% and 11.2% respectively. This demonstrates that when the total pressure is higher, the deviation of the “partial pressure” from Dalton’s partial pressure law is much more manifest.

Another discovery is that the true “partial pressures” of both the water vapor and dry air converges to the partial pressure obtained by Dalton’s partial pressure law with the increase of temperature. This indicates that both gas components behave more like ideal gas components when the temperature increases.

Conclusions

This paper proposes a semi-permeable membrane model to study the “partial pressures” of humid air from a new perspective. In the past, Dalton’s partial pressure law is always applied, but it deviates from the reality owing to its oversimplification of neglecting the molecular interactions. In the novel model proposed, molecular interactions are taken into full account by means of “improvement factors”. The first discovery is that the “partial pressure” of the water vapor in saturated humid air equals P_s, rather than formally believed (f·P_s). The second discovery reveals that the interaction between dissimilar molecules may be omitted when the temperature is above the “cutting-off temperature”. To illustrate, the calculation results of the “partial pressures” are demonstrated. It is found out that the biggest difference between the ideal and real “partial pressures” is up to 11.2% when the total pressure is 5 MPa. Such large deviation is caused by the oversimplification of Dalton’s model. The model proposed in this paper fully considers the molecular interactions and obtains the real “partial pressures” in humid air.

Notations

References

[1] Melling A, Noppenberger S, Still M, Venzke H. Interpolation correlation for fluid properties of humid air in the temperature range 100ºC to 200ºC. Journal of Physical and Chemical Reference Data, 1997, 26(4): 1111-1123

[2] Kim T S, Song C H, Ro S T, Kauh S K. Influence of ambient condition on thermodynamic performance of the humid air turbine cycle. Energy, 2000, 25(4): 313-324

[3] Perry R H, Green D W. Perry’s Chemical Engineers’ Handbook (7th edition). McGraw-Hill, 1997

[4] Herrmann S, Kretzschmar H J, Gatley D P. Thermodynamic properties of real moist air, dry air, steam, water, and ice. International Journal of Heating, Ventilating, Air-Conditioning, and Refrigerating Research, 2009, 15(5): 961-986

[5] Ji X, Lu X, Yan J. Survey of experimental data and assessment of calculation methods of properties for the air-water mixture. Applied Thermal Engineering, 2003, 23(17): 2213-2228

[6] Wagner W, Pruss A. The IAPWS formulation 1995 for the thermodynamic properties of ordinary water substance for general and scientific use. Journal of Physical and Chemical Reference Data, 2002, 31(2): 387-535

[7] Ji X, Yan J. Saturated thermodynamic properties for the air-water system at elevated temperatures and pressures. Chemical Engineering Science, 2003, 58(22): 5069-5077

[8] Ji X, Yan J. Thermodynamic properties for humid gases from 298 to 573 K and up to 200 bar. Applied Thermal Engineering, 2006, 26(2-3): 251-258

[9] Lemmon E W, Jacobsen R T, Penoncello S G, Friend D G. Thermodynamic properties of air and mixtures of nitrogen, argon, and oxygen from 60 to 2000 K at pressures to 2000 MPa. Journal of Physical and Chemical Reference Data, 2000, 29(3): 1-54

[10] Cengel Y A, Boles M A. Thermodynamics: An Engineering Approach (5th edition). McGraw-Hill, 2005, 686-687

[11] Wang Z, Chen H, Weng S. New calculation method for thermodynamic properties of humid air in humid air turbine cycle—general model and solutions for saturated humid air. Energy, 2013,in press)

[12] Wang Z, Chen H, Weng S. Revised Dalton’s method for calculation of thermodynamic properties of unsaturated humid air and gas mixture after combustion in humid air turbine cycle. Energy, 2013, in press)

RIGHTS & PERMISSIONS

Higher Education Press and Springer-Verlag Berlin Heidelberg