Psychrometric equations

The psychrometric charts are very useful for determining air-water vapor mixture properties however, for pressure conditions outside the range, it is useful to calculate the properties directly using fully detailed equations found in ASAE Standard Psychrometric Data ASAE D271. 

Properties Equation 1 Temperature or pressure range Experimental coefficients 

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Equation 15.10 is the fundamental psychrometric equation which permits wet-bulb temperatures to be calculated, as pointed out by Davies (1978) and others. Thus a psychrometric chart can be used to estimate steady-state droplet temperature by finding the wet-bulb temperature corresponding to a given ambient temperature and relative humidity. This wet-bulb temperature is the evaporating droplet temperature ... 

One usually has the initial air conditions specified in terms of the relative humidity and temperature. Drying calculations are carried out in terms of the humidity ratio. Atypical use of psychrometric equations may follow the following sequences use T to calculate P, use RH and P to calculate P, and use P and P to calculate the humidity ratio H. For more equations relating other properties of moist air, see American Society of Agricultural Engineers (ASAE) standards [25]. 

Equation TlO.l calculates the vapor pressure at drying temperature, whereas Equation T10.2 is the psychrometric equation. Equation TlO.l and Equation T10.2 are used to calculate the water activity at drying conditions (i.e., temperature T and air humidity Y). Equation T10.3 calculates the equilibrium material moisture content at drying conditions, whereas Equation T10.4 estimates the drying time constant at drying conditions. Both Equation T10.3 and Equation T10.4 are used in Equation T10.5, which calculates the required drying time. 

Rearranging equation 32 and defining the ratio h -a/iky-ci-Cf ) as r, the psychrometric ratio, give... 

Work in the area of simultaneous heat and mass transfer has centered on the solution of equations such as 1—18 for cases where the stmcture and properties of a soHd phase must also be considered, as in drying (qv) or adsorption (qv), or where a chemical reaction takes place. Drying simulation (45—47) and drying of foods (48,49) have been particularly active subjects. In the adsorption area the separation of multicomponent fluid mixtures is influenced by comparative rates of diffusion and by interface temperatures (50,51). In the area of reactor studies there has been much interest in monolithic and honeycomb catalytic reactions (52,53) (see Exhaust control, industrial). Eor these kinds of appHcations psychrometric charts for systems other than air—water would be useful. The constmction of such has been considered (54). 

Figures 12-37 to 12-39 show humidity charts for carbon tetrachloride, oenzene, and toluene. The lines on these charts have been calculated in the manner outlined for air-water vapor except for the wet-bulb-temperature lines. The determination of these hnes depends on data for the psychrometric ratio /j Z/c, as indicated by Eq. (12-22). For the charts shown, the wet-bulb-temperature hnes are based on the following equation ...

Psychrometric coefficient The coefficient in the equation for the determination of the water vapor partial pressure from the wet bulb depression. 

Comparing equations 13.8 and 13.9, it is seen that the adiabatic saturation temperature i > equal to the wet-bulb temperature when s = h/hDpA. This is the case for most water vapour systems and accurately so when Jf = 0.047. The ratio (h/hopAs) = b is sometimes known as the psychrometric ratio and, as indicated, b is approximately unity for the air-water system. For most systems involving air and an organic liquid, b = 1.3 - 2.5 and the wet-bulb temperature is higher than the adiabatic saturation temperature. This was confirmed in 1932 by SHERWOOD and COMINGS 2 who worked with water, ethanol, n-propanol, n-butanol, benzene, toluene, carbon tetrachloride, and n-propyl acetate, and found that the wet-bulb temperature was always higher than the adiabatic saturation temperature except in the case of water. 

Prior to a discussion on the impact of processing air dew point and temperature on the drying rate behavior of a product, it is necessary to consider heat and mass transfer. Water will move from the granule to air in an attempt to reach an equilibrium, or saturated condition, determined by thermodynamics, which can be read from a phase diagram or psychrometric chart. The rate at which water will move from liquid in the granule to vapor in the air increases the further away the system is from equilibrium. When the water evaporates, it requires an amount of energy, the heat of vaporization, in order to change from liquid to vapor. Because of this, we must also consider transfer of heat as well as movement of material. These concepts can be described by equations shown in Table 5. 

For practical purposes, the properties of humid air are recorded on psychrometric (or humidity) charts such as those of Figures 9.1 and 9.2, but tabulated data and equations also are available for greater accuracy. A computer version is available (Wiley Professional Software, Wiley, New York). The terminal properties of a particular adiabatic humification of air are located... 

What role does the Lewis relation (Equation 3.69) play on the psychrometric chart (Figure 2.5) ... 

Derive" the equations for saturated volume, humid heat and the adiabatic cooling lines for the psychrometric chart. 

Related Calculations. This example is adapted from Process Drying Practice by Cook and DuMont, published in 1991 by McGraw-Hill. More details are available in that source. Similar calculation for direct dryers is far more complex, involving the psychrometric relations of moist air. Trial-and-error loops are required, and manual calculation is not only time-consuming but also error-prone. A sequence of equations suitable for setting into a computer program can be found in the aforementioned Cook and DuMont. 

Before obtaining numerical answers, we must derive equations for the functional relationships expressed in Table 3.2.1, which are given in Table 3.2.2. Equation 3.2.27 is the psychrometric relation, derived by Bird et al. (3.15). This relation... 

More recently, Henry and Epstein (H3) reported data on psychrometric ratios for cylinders in cross-flow and spheres. Their experimental results, which covered the Lewis number range of 3.7 to 7.2, were identical for spheres and cylinders. Furthermore, their results could best be represented by an equation similar to that of Bedingfield and Drew (Bl) as follows ... 

Figure 1 compares the experimental data of various investigators with Eqs. (11) and (12). Equation (11) compares more favorably with the experimental results at lower values of Schmidt to Prandtl number ratios, whereas Eq. (12) compares more favorably at higher values. It is evident that further work is needed to derive a theoretical relationship which encompasses the entire range of the experimental results. Furthermore, practically no data exist for the psychrometric ratio at high temperatures and high humidities. 

Equation (23) contains G, S, and as parameters. G kg Pit,/hTR) is a function of Tr and the psychrometric ratio. The psychrometric ratio can be calculated and is fairly constant. Therefore G is a function of Tr only (Pr, being the vapor pressure at Tr). < is a function of Tr and S is a function of Tr and Tw. In Figs. 3 and 4, moisture concentration and time, in dimensionless coordinates, have been plotted with Tr and Tr — Tyr as parameters. Given Tr, Tyy, and BhTRAo/ KWy, one can determine the drying schedule by using these plots. A /W is the area of heat transfer per pound of water at the critical water content. 

There are no dots over the extensive variables in this equation because the basis of calculation is an amount, not a flow rale.) The enthalpy table for the process is shown below. Since (1) the enthalpies (/ ,) of the humid air streams are obtained from the psychrometric chart in Btu/lbn, dry air, and (2) the mass units of m, and must cancel when the two are multiplied in the energy balance, the tabulated values of m, for these streams must be in ibn, < ry air. 

Suppose now that the temperature Ti and absolute humidity m of the inlet air are specified, so that the state of the inlet air is fixed on the psychrometric chart. If we specify in addition the outlet air temperature T](< Ti), then m e/riia may be calculated from Equation 8.4-9, and it may in turn be used to calculate the absolute humidity of the outlet air. (m e +... 

For practical purposes, the properties of humid air are recorded on psychrometric (or humidity) charts such as those of Figures 9.1 and 9.2, but tabulated data and equations also are available for greater accuracy. A computer version is available (Wiley Professional Software, Wiley, New York). The terminal properties of a particular adiabatic humification of air are located on the same saturation line, one of those sloping upwards to the left on the charts. For example, all of these points are on the same saturation line (7)//) = (250,0.008), (170, 0.026) and (100,0.043) the saturation enthalpy is 72 Btu/lb dry, but the individual enthalpies are less by the amounts 2.5, 1.2, and 0, respectively. 

However, for nearly all other vapor-gas systems, particularly for organic solvents, P < 1, and hence T b > T. This is illustrated in Fig. 12-5. For these systems the psychrometric ratio may be obtained by determining h/ky from heat- and mass-transfer analogies such as the Chilton-Colburn analogy. The basic form of the equation is... 

Software is available that will perform calculations of humidity parameters for any point value, and for plotting psychrometric charts. Moreover, British Standard BS 1339 Part 2 (2006) provides functions as macros which can be embedded into any Excel-compatible spreadsheet. Users can therefore generate their own tables for any desired combination of parameters as well as perform point calculations. Hence, the need for published lookup tables has been eliminated. However, this software, like the previous lookup tables, is only valid for the air-water system. For other vapor-gas systems, the equations given in previous sections must be used. 

Use method (iv) to find p and Y. Make an initial estimate of say, using a psychrometric chart. Calculate Yas from Eq. (12-6). Find p from Table 12-1 and from Antoine equation (12-5). Repeat until iteration converges (e.g., using spreadsheet). 

The American Society of Agricultural Engineers (ASAE) http //www.asae.org. Psychrometric data in chart and equation form in both SI and English units. Charts for temperature ranges of -35 to 600°F in uses units and -10 to 120°C in SI units. Equations and calculation procedures. Air-water system and Grosvenor (temperature-humidity) charts only. 

From Sonntag equation (12-4),p, = 19,948 Pa difference from Antoine is less than 0.5 percent. Relative humidity = 100 x 3915/19,948= 19.6 percent. From a psychrometric chart, e.g.. Fig. 12-1, a humidity of 0.025 kg/kg at T = 60°C lies very close to the adiabatic saturation line for 35°C. Hence a good first estimate for Tas and T b will be 35°C. Refining the estimate of Twb by using the psychrometer equation and iterating gives... 

From the psychrometric chart, a humidity of 0.0526 kg/kg at F = 70°C falls just below the adiabatic saturation line for 45°C. Estimate Fas and F b as 45°C. Refining the estimate of F b by using the psychrometer equation and iterating gives... 

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