Document KJwVXo206GORdj2gmdprLzGy6

16 CHAPTER 3 1960 Guide of tile thermometer. Arnold1 has developed a theory which makes possible the calculation of the thermodynamic wet' bulb temperature from observed data through the use of suit able corrections to be applied to the readings of the wet-bulb thermometer. However, it is frequent practice to take the ob served equilibrium readings of a wetted-wick thermometer as being essentially equal to the thermodynamic wet-bulb tem perature, if the air velocity past the wick is about 1000 fpm and if the wick is exposed to radiation exchange with sur roundings at temperatures differing from the dry-bulb tem perature by no more than about 20 F deg. USEFUL PERFECT-GAS RELATIONSHIPS In early thermodynamics, with particular reference to the properties of gases, pure and mixed, scientists expressed rela tionships between the properties of gases in formulations which they called laws. Today, with evidence from highly precise measurements and with greatly improved scientific knowledge of the details involved', these early so-called fats* have been found to be approximations. Many useful engi neering results, however, can be accomplished with.the aid of judicious approximations, if the accuracy and limitations in volved can be established. Calculations making use of socalled perfect gas relationships fall into this category. Thermodynamically considered, a perfect gas c*n be de fined as one for which the pressure, volume, and temperature ore related by the following equation of state: PV-w^T (9) where P * pressure, pounds per square foot, ahsolute. V * volume occupied, cubic feet. w 0 mass of gas, pounds. fit TM constant (which is the same for all perfect gases) <* 1644 foot pounds per (mol) (Fahrenheit degree, ab solute). m molecular weight, pounds per mol. T <= temperature, Fahrenheit, absolute. From the above relationship follow: (1) Boyle's Law, PiPi b P,Vi when T% * T* , and (10) (2) Charles' or Gay-Lussac's Law, r, Vt when Pt Ti Tt Pt (U) where subscripts 1 and 2 refer to two different states of the gas. Historically, the above laws were derived from experi mental observations by the men for whom they are named. Common diatomic and triatomic gases, such as are dealt with in air conditioning, follow perfect-gas relationships fairly well at pressures less than a few atmospheres and tem peratures which are far enough removed from the saturation temperature. A quantitative measure of departure from per fect-gas behavior may be had in magnitudes of the compressi bilityfactor, Fc, defined as PIVM C ~ (RVm)T (12) For a perfect gas, Fc is unity. Magnitudes of Fc are given in many treatises on thermodynamics.4 * The adopted definition of partial pressure means that the pressure of a gas mixture is exactly the sum of the partial pres sures of the constituents. This is a present-day formulation of a concept first given by Dalton, who indicated that each gas in a mixture occupies the total volume of the mixture just as though the other gases were not present. If this concept of di rect addition of constituent properties to obtain mixture prop erties were generally applicable, then the gas-mixture proper ties on a per-mol basis such as enthalpy, entropy, specific heat, and internal energy, all could be determined by ample sum mations of mol fractions times the property in question for each constituent of the mixture. In reality this simple rela tionship is not true, but it is approximated closely at low or moderate pressures where the gases approach perfect-gas behavior. Mechanistically considered, a perfect gas may be described as having perfectly elastic point-mass molecules with no mo lecular attraction or repulsion forces either between the mol ecules or involving the .container wails. Real gases approach these attributes more closely as the pressure diminishes and the molecules move about more freely without collision. The following gaseous-state equations are in common use (the symbols used are defined after Equation 26): Equation of State: for air for water vapor for mixture where P. V. P. V, 0 wJi.Tw Pm Vm = WmRmTm. U3H (14)t - d5)t V. V, . K. ^"P'i (16) Wm 0 tt, + tt>, Sm -- gas constant for dry air = 53.3 ft lb per Ob) {F deg abs). R, = gas constant for water vapor " 85.6 ft lb per (lb) (F deg abs). - (--B.-+ (---x--) 8, + u'./ \ip -f- tP,/ l Denotes precise Application to perfect cases 00)7. (17) (18) J Thermodynamics 17 Relative humidity: P. ,, * " Pm " V, (is): Humidity ratio: 28066P. P. P - P. (2on where 18.016 and 28.966 are the molecular weights of water and air respectively. P - P. + P. (21) />. 0 - - P; P, 0 -P n + n, n + (22) M " " 28066' u>. " 18.016 (23) In the thermodynamic properties which follow, it is con venient and conventional to express all mixture properties as per unit mass of dry air in the mixture of air and water vapor. Specific heat: 0 e,, + Wc^ (24)| where 0 0.24 and c,. - 0.44 Btu/(lb) (F deg) in the ordinary temperature range. Enthalpy: hm 0 A. + WK (25) t Entropy: + Ws, (28)* where s,-aad s, are evaluated at the respective partial pressures in the mixture. Symbols for Equations 13-26 are given below. Subscripts: a denotes dry air v denotes water vapor n denotes air-vapor mixture vs denotes saturated pure water vapor P pressure, pounds per square foot, absolute. V 0 volume, cubic feet, v 0 specific volume, cubic feet per pound. w 0 mass, pounds. R gas constant per unit mass, foot-pounds per (pound) (Fahrenheit degree, absolute). T => temperature, Fahrenheit, absolute. 4 0 relative humidity,- decimal. W humidity ratio, pounds vapor per pound dry air. n 0 mala. Cf 0 specific heat, Btu per (pound) (Fahrenheit degree). h 0 enthalpy, Btu per pound; note especially that A. is Btn per pound of dry air in the mixture, s 0 entropy, Btu per (pound) (Fahrenheit degree abso lute) ; note especially that a. is in Btu per (pound of dry air in the mixture) (Fahrenheit degree absolute). The mixture enthalpy fu , Equation 25, can be empirically t Denotes precise Application to perfect paw only. expressed, within about 0.1 percent accuracy over the range 32 to 100 F at 1 atmosphere pressure by the equation A. 0 (0.2402 + 0.44lF)t + 1061FF (27)* where t 0 dry-bulb temperature, Fahrenheit. Equation 27 states that: h, 0 0.2402 t, and A, 0 1061 + 0.441 ' (28) (29) TABULATED THERMODYNAMIC PROPERTIES OF MOIST AIR AND OF WATER AT SATURATION FOR ACCURATE CALCULATIONS Engineering designs and analyses often require accuracies which cannot be achieved with perfect-gas assumptions. This need is expeditiously met by using properties in tabular form. Properties of steam* and many gases* have been tabulated. The latest and most precise formulation of the thermody namic properties of moist air is found in the work of Goff and Gratch.7- * Much of the work baric to this material was per formed as an ASHAE cooperative research activity in the Towne Scientific School of the University of Pennsylvania. Table 2 gives complete data for moist air at a pressure of 1 standard atmosphere and temperatures from minus ;160 F to plus 200 F. In Table 2 there are 15 columns of values, each <*dtitnn being headed by a symbol; in the following subpara graphs brief explanations of these columns are given. {(F) 0 Fahrenheit temperature defined in terms of absolute temperature 7* by the relation, T - t + 459.67 (30) W, 0 humidity ratio at saturation. Saturation is the condi tion at which the vapor phase (moist air) may exist in equi librium with a condensed phase (liquid or solid) at the given temperature and pressure (standard atmospheric pressure in the case of Table 2). At given values of temperature and pres sure, the humidity ratio Vv can have any value from sero to W,. v 0 specific volume of dry air, cubic feet per pound. B v. --. p , the difference between the volume of moist air at saturation, per pound of dry air; and the specific volume of the dry air itself, cubic feet per pound of dry air. v, 0 specific volume of moist sir at saturation per pound of dry air, cubic feet per pound of dry air. A. 0 specific enthalpy of dry air, Btu per pound of dry air. The specific enthalpy of dry air has been assigned the value sero at 0 F, standard atmospheric pressure. The energy unit Btu is related to the-foot-pound by definition, as follows: 1 Btu - 778.3 ft-lb. h., * A, -- A. , the difference between the enthalpy of moist air at saturation, per pound of dry air, and the specific enthalpy of the dry air itself, Btu per pound of dry air. A, * enthalpy of moist air of saturation per pound of dry air, Btu per pound of dry air. *. 0 specific entropy of dry air, Btu per (pound) (Fahren / heit degree absolute). It will be noticed that the specific en tropy of dry air has been assigned the value sero at 0 F and standard atmospheric pressure. =, -- <, the difference between the entropy of moist sir at saturation, per pound o! dry air, and the specific entropy of the dry air itself, Btu per (pound of dry air) (Fahrenheit degree absolute). s, 0 entropy of moist air at saturation per pound of dry air, Btu per (pound of dry air) (Fahrenheit degree, absolute). A* 0 specific enthalpy of condensed water (liquid or solid) in equilibrium with saturated air at standard atmospheric pressure, Btu per pound of water. The specific enthalpy of tCsntiatW tn p. S8)