7.2 FUNDAMENTAL DISCUSSION OF MOIST AIR The composition of atmospheric air is variable, particularly with regard...

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7.2 FUNDAMENTAL DISCUSSION OF MOIST AIR The composition of atmospheric air is variable, particularly with regard to amounts of water vapor and particulate matter. Before we can discuss thermodynamic properties, the substance must be precisely defined. The working substance in air conditioning problems is called moist air. Moist air is defined as a binary mixture of dry air and water vapor. Goff [2], in a final report of the Working Subcommittee, International Joint Committee on Psychrometric Data, has defined dry air as shown in Table 7.2. Although somewhat arbitrary, this composition is regarded as exact, by definition. The molecular weights for dry air and water vapor are 28.966 and 18.016, respectively. The respective gas constants can be obtained by dividing the universal gas constant, R = 1.986 Btu/lbmole- R = 1545 ft-lbf/lbmole - R (8.314 kJ/kmole - K), by the appro- priate molecular weight. dry air: R = R 28.966 water vapor: R = 0.0686 Btu/lbmR = 53.35 ft-lbf/lbm- R = 287 J/kg. K R = 0.110 Btu/lbm R = 85.78 ft-lbf/lbm. R 18.016 462 J/kg. K Moist air may contain variable amounts of water vapor from zero (dry air) to that of saturated moist air. Goff [3] has defined saturation of moist air as that condition where moist air may coexist in neutral equilibrium with associated condensed water, presenting a flat surface to it. The humidity ratio, W, is defined as the mass of water vapor per unit mass of dry air in a moist air mixture. W = Chap. 7 / Thermodynamic Properties of Moist Air Two measures of humidity relative to saturation conditions are commonly used. Degree of saturation is defined by the relation m ma = W W (7.4) 179 180 where W, is the humidity ratio at saturation for the same temperature and pressure as those of the actual state. Relative humidity is defined by the relation TABLE 7.2 Composition of Dry Air Substance Oxygen (O) Nitrogen (N) Argon (A) Carbon dioxide (CO) Molecular Weight = 32.000 28.016 39.944 44.01 = SOURCE: Reprinted by permission from Trans. ASHVE 55, 463. 1+ Xw Xw,5 1+ Mol-fraction Composition in Dry Air 0.622 W 0.622 W 0.2095 0.7809 0.0093 0.0003 1.0000 = where x is the mole fraction of the water vapor in the mixture and xs is the mole frac- tion of water vapor at saturation for the same temperature and pressure as those of the actual state. We may convert Eq. (7.5) to the forms 0.622 + W, 0.622 + W Part III / Psychrometrics (7.5) Partial Molecular Weight in Dry Air 6.704 21.878 0.371 0.013 28.966 (7.6) It is important to observe that neither nor are defined when the temperature of moist air exceeds the saturation temperature of pure water corresponding to the moist air pressure. For sea-level pressure, W, approaches infinity at 212 F (100 C). Thus for 14.696 psia (101.325 kPa), and are undefined for temperatures higher than 212 F (100 C). Three moist-air properties are associated with temperature. The dry-bulb tempera- ture, t, is the true temperature of moist air at rest. The dew-point temperature, ta, is defined as the solution t(P, W) of the equation W,(P,t) = W (7.7) In words, the dew-point temperature of a moist air state is the saturation temperature cor- responding to the humidity ratio and pressure of the state. Another way to describe the dew-point temperature is to consider a moist air mixture defined by P, W, and t. If you slowly reduce the temperature of the mixture while holding P and W constant, then the temperature at which saturation is reached is the dew-point temperature, td. The third temper 4/21 125% + rty is the thermodynamic wet- batic saturation temperature. bulb tem However, before defining this temperature it is necessary to introduce the enthalpy of the (Note that this form of the definition is particularly convenient for humidification and dehumidification processes where the flow rate of dry air remains constant as the moist air passes through the humidifier or dehumidifier.) Thus 182 or 7.3 THERMODYNAMIC WET-BULB TEMPERATURE h=h + me ho ma h = h + Wh Moist air 11, W, P, ma Adiabatic saturation temperature is that temperature at which water, by evaporating into air, can bring the air to saturation adiabatically at the same temperature. It is unneces- sary to inject practical details into a discussion of adiabatic saturation, since the results the process are given by definition. However, to help us understand it better, we may con- sider how such a process could be approached. Figure 7.1 schematically shows a device which may provide adiabatic saturation. The chamber could be indefinitely long and perfectly insulated. The total quantity of water present could be large compared to that added to the air in a given length of time. We may assume no temperature gradients within the water body. Regardless of the initial tem- perature of the water, we would expect that after sufficient time the water would assume a constant temperature. This limiting temperature of water should be less than the enter- ing air dry-bulb temperature, but greater than the entering air dew-point temperature. In order to maintain steady conditions within the chamber, makeup water at tem- perature is supplied at a rate m,, which is equal to the rate at which vapor is added to the moist air. Thus, if the mass flow rate of dry air passing through the chamber is m, the rate of makeup water is given by m = m(W,2 - W) (7.11) The chamber is adiabatic, and no work is done by or on the apparatus. Therefore, the steady-flow energy equation for the process is mh1 + mWhw1 + M h,2 = mh2 + m W.2 hw, 2 Perfectly insulated (7.10) Water at 2 Saturated moist air 12, W52, P, ma my, makeup water at 12 Figure 7.1 Schematic of an adiabatic saturation device. (7.12) Part III / Psychrometrics By applying Eq. (7.10) in order to write the energy balance in terms of the moist-air enthalpy, substituting Eq. (7.11) for the rate of makeup water and noting that saturation conditions exist at the exit, the steady-flow energy equation can be written as h + (W,2W )h2 = h,2 (7.13) Since the leaving air is saturated, and since we assume a constant pressure P, the quanti- ties W,2, hs, 2, and h2, are sole functions of temperature t. We may then deduce that t is a function of h, W, and P, or that t is a function of state 1. Therefore, t is a thermody- namic property of state 1. We call this property the thermodynamic wet-bulb tempera- ture t*. Denoting all properties evaluated at t* with the superscript *, Eq. (7.13) becomes h+ (W* - W)h = h (7.14) Thus, for given values of h, W, and P (given moist air state), the thermodynamic wet-bulb temperature t* is that value of temperature which satisfies Eq. (7.14). There are numerous practical problems where the concept of thermodynamic wet-bulb tempera- ture is useful. 7.4 THE TABLES FOR MOIST AIR By applying fundamental procedures of statistical mechanics, Goff and Gratch [4] calcu- lated accurate thermodynamic properties of moist air for standard sea-level pressure. More recently, new formulations were developed by Hyland and Wexler [5]. Tables A.4E and A.4SI, extracted from the ASHRAE Handbook, 1993 Fundamentals Volume [6], pre- sents properties for dry air and saturated moist air based on the new formulations. In the following list, brief explanations of the data in Tables A.4E and A.4SI are shown. W, = humidity ratio of saturated air, mass of water vapor per mass dry air. v = specific volume of dry air under 14.696 psia (101.325 kPa) pressure, ft/lbm (m/kg). v = volume of saturated air, ft/lbm dry air (m/kg dry air). Vas = V - V, ft/lbm dry air (m/kg dry air). ha = specific enthalpy of dry air, Btu/lbm, (kJ/kg dry air). Zero enthalpy for dry air is taken at 0 F (0 C). h = enthalpy of saturated air, Btu/lbm of dry air (kJ/kg dry air). has = hs - h, Btu/lbm of dry air (kJ/kg dry air). Sa = specific entropy of dry air, Btu/lbm R (kJ/kg-K). Zero entropy for dry air is taken at 0 F (0 C). s = entropy of saturated air, Btu/lbm of dry air. R (kJ/kg dry air K). Sas= S - Sa, Btu/lbm of dry air R (kJ/kg dry air K). Calculations for volume, enthalpy, and entropy of unsaturated moist air states are closely given by the relations Chap. 7 / Thermodynamic Properties of Moist Air 6/21 V = V + V as h = h + phas S = Sa+ S as 125% + (7.15) (7.16) (7.17) 183 ( 184 EXAMPLE 7.2 Moist air exists at 66 C dry-bulb temperature and 30 percent degree of saturation. Pressure is 101.325 kPa. Determine (a) the enthalpy, kJ/kg,, and (b) the specific volume, m/kg.. Solution: (a) By Table A.4SI. W, = 0.21848 kg/kg. By Eq. (7.16), h = 66.455+ (0.30) (572.116) = 238.09 kJ/kg (b) By Eq. (7.15), v=0.9608+ (0.30) (0.3350) = 1.061 m/kg. and thus from Eqs. (7.5) and (7.26) (7.25) The perfect-gas approximation for the relative humidity is obtained by observing that for a perfect gas R.T V= P-P W = P Pus (7.27) Equation (7.14) may be altered through use of the perfect-gas relations. By Eqs. (7.14), (7.23), and (7.24) (W*- W)h = c(t t*) (7.28) By Eqs. (7.14) and (7.21) ASa, mix Wh-pa (t-t*) hg - hi (7.29) The total entropy of a moist air mixture, S, which is at a pressure and temperature P and T, can be written as AS, T.P T AS.,T.P = m (p. In R In- To S = Sa, 0 + AS, T, P + ASa, mix + Sw,0 + ASw, T.P + ASw, mix (7.30) The quantities Sa,o and Swo are the entropy values at the reference conditions for the dry air and water vapor, respectively. The terms AS, T, P and AS, T, P are the entropy changes that result in going from the reference states to the state P, T and the terms AS, mix and AS, mix are the mixing entropies for the dry air and water vapor. These latter two mixing terms account for the fact that it is the partial pressures, P and P, not the total pressure, P, that are needed in evaluating the entropies of the components. For the case of perfect gases with constant specific heats and with Po, To as the ref- erence state Part III/ Psychrometrics = : -m. (R.In ) (7.26) m(cnfixum = mcpw In (7.31) (7.32) (7.33) 7.2 Calculate values of humidity ratio, enthalpy, and volume for saturated air at 14.696 psia pressure, using perfect-gas relations and Table A.1, for tem- peratures of (a) 70 F, and (b) -20 F. Compare your results with those shown in Table A.4. 7.3 The atmosphere within a room is at 70 F dry-bulb temperature, 50 percent degree of saturation, and 14.696 psia pressure. The inside surface tempera- ture of the windows is 40 F. Will moisture con- dense out of the air upon the window glass? 7.4 Assume that the dimensions of the room of Prob. 7.3 are 30 ft by 15 ft by 8 ft high. Calculate the num- ber of pounds of water vapor in the room. 7.2 FUNDAMENTAL DISCUSSION OF MOIST AIR The composition of atmospheric air is variable, particularly with regard to amounts of water vapor and particulate matter. Before we can discuss thermodynamic properties, the substance must be precisely defined. The working substance in air conditioning problems is called moist air. Moist air is defined as a binary mixture of dry air and water vapor. Goff [2], in a final report of the Working Subcommittee, International Joint Committee on Psychrometric Data, has defined dry air as shown in Table 7.2. Although somewhat arbitrary, this composition is regarded as exact, by definition. The molecular weights for dry air and water vapor are 28.966 and 18.016, respectively. The respective gas constants can be obtained by dividing the universal gas constant, R = 1.986 Btu/lbmole- R = 1545 ft-lbf/lbmole - R (8.314 kJ/kmole - K), by the appro- priate molecular weight. dry air: R = R 28.966 water vapor: R = 0.0686 Btu/lbmR = 53.35 ft-lbf/lbm- R = 287 J/kg. K R = 0.110 Btu/lbm R = 85.78 ft-lbf/lbm. R 18.016 462 J/kg. K Moist air may contain variable amounts of water vapor from zero (dry air) to that of saturated moist air. Goff [3] has defined saturation of moist air as that condition where moist air may coexist in neutral equilibrium with associated condensed water, presenting a flat surface to it. The humidity ratio, W, is defined as the mass of water vapor per unit mass of dry air in a moist air mixture. W = Chap. 7 / Thermodynamic Properties of Moist Air Two measures of humidity relative to saturation conditions are commonly used. Degree of saturation is defined by the relation m ma = W W (7.4) 179 180 where W, is the humidity ratio at saturation for the same temperature and pressure as those of the actual state. Relative humidity is defined by the relation TABLE 7.2 Composition of Dry Air Substance Oxygen (O) Nitrogen (N) Argon (A) Carbon dioxide (CO) Molecular Weight = 32.000 28.016 39.944 44.01 = SOURCE: Reprinted by permission from Trans. ASHVE 55, 463. 1+ Xw Xw,5 1+ Mol-fraction Composition in Dry Air 0.622 W 0.622 W 0.2095 0.7809 0.0093 0.0003 1.0000 = where x is the mole fraction of the water vapor in the mixture and xs is the mole frac- tion of water vapor at saturation for the same temperature and pressure as those of the actual state. We may convert Eq. (7.5) to the forms 0.622 + W, 0.622 + W Part III / Psychrometrics (7.5) Partial Molecular Weight in Dry Air 6.704 21.878 0.371 0.013 28.966 (7.6) It is important to observe that neither nor are defined when the temperature of moist air exceeds the saturation temperature of pure water corresponding to the moist air pressure. For sea-level pressure, W, approaches infinity at 212 F (100 C). Thus for 14.696 psia (101.325 kPa), and are undefined for temperatures higher than 212 F (100 C). Three moist-air properties are associated with temperature. The dry-bulb tempera- ture, t, is the true temperature of moist air at rest. The dew-point temperature, ta, is defined as the solution t(P, W) of the equation W,(P,t) = W (7.7) In words, the dew-point temperature of a moist air state is the saturation temperature cor- responding to the humidity ratio and pressure of the state. Another way to describe the dew-point temperature is to consider a moist air mixture defined by P, W, and t. If you slowly reduce the temperature of the mixture while holding P and W constant, then the temperature at which saturation is reached is the dew-point temperature, td. The third temper 4/21 125% + rty is the thermodynamic wet- batic saturation temperature. bulb tem However, before defining this temperature it is necessary to introduce the enthalpy of the (Note that this form of the definition is particularly convenient for humidification and dehumidification processes where the flow rate of dry air remains constant as the moist air passes through the humidifier or dehumidifier.) Thus 182 or 7.3 THERMODYNAMIC WET-BULB TEMPERATURE h=h + me ho ma h = h + Wh Moist air 11, W, P, ma Adiabatic saturation temperature is that temperature at which water, by evaporating into air, can bring the air to saturation adiabatically at the same temperature. It is unneces- sary to inject practical details into a discussion of adiabatic saturation, since the results the process are given by definition. However, to help us understand it better, we may con- sider how such a process could be approached. Figure 7.1 schematically shows a device which may provide adiabatic saturation. The chamber could be indefinitely long and perfectly insulated. The total quantity of water present could be large compared to that added to the air in a given length of time. We may assume no temperature gradients within the water body. Regardless of the initial tem- perature of the water, we would expect that after sufficient time the water would assume a constant temperature. This limiting temperature of water should be less than the enter- ing air dry-bulb temperature, but greater than the entering air dew-point temperature. In order to maintain steady conditions within the chamber, makeup water at tem- perature is supplied at a rate m,, which is equal to the rate at which vapor is added to the moist air. Thus, if the mass flow rate of dry air passing through the chamber is m, the rate of makeup water is given by m = m(W,2 - W) (7.11) The chamber is adiabatic, and no work is done by or on the apparatus. Therefore, the steady-flow energy equation for the process is mh1 + mWhw1 + M h,2 = mh2 + m W.2 hw, 2 Perfectly insulated (7.10) Water at 2 Saturated moist air 12, W52, P, ma my, makeup water at 12 Figure 7.1 Schematic of an adiabatic saturation device. (7.12) Part III / Psychrometrics By applying Eq. (7.10) in order to write the energy balance in terms of the moist-air enthalpy, substituting Eq. (7.11) for the rate of makeup water and noting that saturation conditions exist at the exit, the steady-flow energy equation can be written as h + (W,2W )h2 = h,2 (7.13) Since the leaving air is saturated, and since we assume a constant pressure P, the quanti- ties W,2, hs, 2, and h2, are sole functions of temperature t. We may then deduce that t is a function of h, W, and P, or that t is a function of state 1. Therefore, t is a thermody- namic property of state 1. We call this property the thermodynamic wet-bulb tempera- ture t*. Denoting all properties evaluated at t* with the superscript *, Eq. (7.13) becomes h+ (W* - W)h = h (7.14) Thus, for given values of h, W, and P (given moist air state), the thermodynamic wet-bulb temperature t* is that value of temperature which satisfies Eq. (7.14). There are numerous practical problems where the concept of thermodynamic wet-bulb tempera- ture is useful. 7.4 THE TABLES FOR MOIST AIR By applying fundamental procedures of statistical mechanics, Goff and Gratch [4] calcu- lated accurate thermodynamic properties of moist air for standard sea-level pressure. More recently, new formulations were developed by Hyland and Wexler [5]. Tables A.4E and A.4SI, extracted from the ASHRAE Handbook, 1993 Fundamentals Volume [6], pre- sents properties for dry air and saturated moist air based on the new formulations. In the following list, brief explanations of the data in Tables A.4E and A.4SI are shown. W, = humidity ratio of saturated air, mass of water vapor per mass dry air. v = specific volume of dry air under 14.696 psia (101.325 kPa) pressure, ft/lbm (m/kg). v = volume of saturated air, ft/lbm dry air (m/kg dry air). Vas = V - V, ft/lbm dry air (m/kg dry air). ha = specific enthalpy of dry air, Btu/lbm, (kJ/kg dry air). Zero enthalpy for dry air is taken at 0 F (0 C). h = enthalpy of saturated air, Btu/lbm of dry air (kJ/kg dry air). has = hs - h, Btu/lbm of dry air (kJ/kg dry air). Sa = specific entropy of dry air, Btu/lbm R (kJ/kg-K). Zero entropy for dry air is taken at 0 F (0 C). s = entropy of saturated air, Btu/lbm of dry air. R (kJ/kg dry air K). Sas= S - Sa, Btu/lbm of dry air R (kJ/kg dry air K). Calculations for volume, enthalpy, and entropy of unsaturated moist air states are closely given by the relations Chap. 7 / Thermodynamic Properties of Moist Air 6/21 V = V + V as h = h + phas S = Sa+ S as 125% + (7.15) (7.16) (7.17) 183 ( 184 EXAMPLE 7.2 Moist air exists at 66 C dry-bulb temperature and 30 percent degree of saturation. Pressure is 101.325 kPa. Determine (a) the enthalpy, kJ/kg,, and (b) the specific volume, m/kg.. Solution: (a) By Table A.4SI. W, = 0.21848 kg/kg. By Eq. (7.16), h = 66.455+ (0.30) (572.116) = 238.09 kJ/kg (b) By Eq. (7.15), v=0.9608+ (0.30) (0.3350) = 1.061 m/kg. and thus from Eqs. (7.5) and (7.26) (7.25) The perfect-gas approximation for the relative humidity is obtained by observing that for a perfect gas R.T V= P-P W = P Pus (7.27) Equation (7.14) may be altered through use of the perfect-gas relations. By Eqs. (7.14), (7.23), and (7.24) (W*- W)h = c(t t*) (7.28) By Eqs. (7.14) and (7.21) ASa, mix Wh-pa (t-t*) hg - hi (7.29) The total entropy of a moist air mixture, S, which is at a pressure and temperature P and T, can be written as AS, T.P T AS.,T.P = m (p. In R In- To S = Sa, 0 + AS, T, P + ASa, mix + Sw,0 + ASw, T.P + ASw, mix (7.30) The quantities Sa,o and Swo are the entropy values at the reference conditions for the dry air and water vapor, respectively. The terms AS, T, P and AS, T, P are the entropy changes that result in going from the reference states to the state P, T and the terms AS, mix and AS, mix are the mixing entropies for the dry air and water vapor. These latter two mixing terms account for the fact that it is the partial pressures, P and P, not the total pressure, P, that are needed in evaluating the entropies of the components. For the case of perfect gases with constant specific heats and with Po, To as the ref- erence state Part III/ Psychrometrics = : -m. (R.In ) (7.26) m(cnfixum = mcpw In (7.31) (7.32) (7.33) 7.2 Calculate values of humidity ratio, enthalpy, and volume for saturated air at 14.696 psia pressure, using perfect-gas relations and Table A.1, for tem- peratures of (a) 70 F, and (b) -20 F. Compare your results with those shown in Table A.4. 7.3 The atmosphere within a room is at 70 F dry-bulb temperature, 50 percent degree of saturation, and 14.696 psia pressure. The inside surface tempera- ture of the windows is 40 F. Will moisture con- dense out of the air upon the window glass? 7.4 Assume that the dimensions of the room of Prob. 7.3 are 30 ft by 15 ft by 8 ft high. Calculate the num- ber of pounds of water vapor in the room.