Rational Psychrometric Formulae/Appendix No. 4

Rational Psychrometric Formulae by Willis H. Carrier
Appendix No. 4

APPENDIX No. 4

DERIVATION OF FORMULA [6] GIVING THE EQUATION OF THE ADIABATIC SATURATION LINE

${\displaystyle r'(W'-W)=c_{pa}(t-t')+c_{ps}W(t-t')}$

86Assuming 1 lb. of pure air having the temperature ${\displaystyle t}$ containing ${\displaystyle W}$ lb. of moisture with the corresponding dew point hand vapor pressure ${\displaystyle e_{1}}$ having a resultant adiabatic saturation temperature of ${\displaystyle t'}$, assume also a moisture increment ${\displaystyle dW}$ under adiabatic conditions resulting in a temperature increment of ${\displaystyle -dt}$. This moisture increment ${\displaystyle dW}$ is evidently evaporated at a vapor pressure ${\displaystyle e_{1}}$ corresponding to temperature ${\displaystyle t_{1}}$ and superheated to temperature ${\displaystyle t}$. The temperature of the liquid is evidently constant at temperature ${\displaystyle t'}$, from principle C. The total heat of the vapor in the increment is ${\displaystyle H_{1}dW+C_{ps}(1-t_{1})dW}$, where ${\displaystyle H_{1}}$ is the total heat of steam corresponding to temperature ${\displaystyle t_{1}}$ and vapor pressure ${\displaystyle e_{l}}$, and ${\displaystyle C_{ps}(t-t_{1})dW}$ is the heat required to superheat from saturation temperature ${\displaystyle t_{1}}$ to dry-bulb temperature ${\displaystyle t}$. The heat of the liquid evaporated, however, is ${\displaystyle q'dW}$ corresponding to temperature of saturation ${\displaystyle t'}$.

87 The total heat interchange required to evaporate ${\displaystyle dW}$ under these conditions is therefore

[40]

${\displaystyle \Sigma =\{H_{1}-q'+[C_{ps}(t-t_{1})]\}dW}$

The change in sensible heat of 1 lb. of air and ${\displaystyle W}$ lb. of water vapor due to the temperature increment ${\displaystyle -dt}$ is

[41]

${\displaystyle \Sigma =-(C_{pa}+WC_{ps})dt}$

Since the change is adiabatic these values may be related by the equation

[42]

${\displaystyle \{H_{1}-q'+[C_{ps}(t-t_{1})]\}dW-(C_{pa}+WC_{ps}dt=0}$

[43]

${\displaystyle \int _{W}^{W'}\{H_{1}-q'+[C_{ps}(t-t_{1})]\}dW=\int ^{t'}(C_{pa}+WC_{ps})dt}$

in which ${\displaystyle H_{1}}$ and [/itex]t_1[/itex] are variables corresponding to the variable ${\displaystyle W}$ while ${\displaystyle t}$ is a variable related to ${\displaystyle W}$ by the different equation. A constant corresponding to ${\displaystyle t'}$ is ${\displaystyle q'}$ while ${\displaystyle C_{ps}}$, may be taken approximately as a mean between its values at ${\displaystyle t_{1}}$ and at ${\displaystyle t'}$ and ${\displaystyle C_{ps}}$ as a mean between its values at ${\displaystyle t}$ and at ${\displaystyle t'}$. The temperature of saturation is ${\displaystyle t'}$, and ${\displaystyle W'}$ is the corresponding moisture content at saturation.

88It is not necessary, however, to solve this equation in this form as this relationship may be simpliﬁed.

[44]

${\displaystyle \{H_{1}-q'+[C_{ps}(t-t_{1})]\}dW=\{H_{1}-q'+[C_{ps}(t'-t_{1}))+(C_{ps}(t-t')]\}dW}$

It may be shown thermodynamically, assuming steam to be a perfect gas. that [45]

${\displaystyle H_{1}-q'+C_{ps}(t'-t_{1})]=H'-q'=r'}$

89This may also be demonstrated approximately for the range of temperatures under discussion by computation from the values given in the steam tables of Marks and Davis, as in Table 6:

 TABLE 6 COMPARISON OF ACTUAL VALUES OF ${\displaystyle r'}$ WITH VALUES OF ${\displaystyle r'}$ COMPUTED FROM THE TOTAL HEAT AT DIFFERENT TEMPERATURES ${\displaystyle t_{1}}$ ${\displaystyle t'=80{\text{deg., }}r'=1046.7}$ ${\displaystyle t'}$ ${\displaystyle t_{1}}$ ${\displaystyle C_{ps}}$ ${\displaystyle H_{1}}$ ${\displaystyle q'}$ ${\displaystyle r'}$(Computed) 80 70 0.44365 1090.2 48.03 1046 .70 80 60 0.44356 1085.9 48.03 1046 .74 80 50 0.44347 2081.4 48.03 1046 .78 80 40 0.44347 1076.9 48.03 1046 .5 ${\displaystyle t'=100{\text{deg., }}r'=1035.6}$ 100 90 1099.2 0.44401 67.97 1035 .67 100 80 0.44392 1094.8 67.97 1035 .7 100 70 0.44383 1090.8 67.97 1035 .64 100 60 0.44374 1085.9 67.97 1035 .67 100 50 0.44365 1081.4 67.97 1035 .81 100 40 0.44356 1076.9 67.97 1035 .54 ${\displaystyle t'=120{\text{deg., }}r'=1024.4}$ 120 110 0.44419 1108.0 87.91 1024 .4 120 100 0.44410 1103.6 87.91 1024 .57 120 90 0.44401 1099.2 87.91 1024 .52 120 80 0.44392 1094.4 87.91 1024 .64 120 70 0.44383 1090.3 87.91 1024 .58 120 60 0.44374 1085.9 87.91 1024 .50 120 50 0.44365 1081.4 87.91 1024 .54 120 40 0.44356 1076.9 87.91 1024 .47 ${\displaystyle r'=H_{1}-q'+[C_{pa}(t'-t_{1})]}$ or ${\displaystyle H'=H_{1}+[C_{ps}(t'-t_{1})]}$

Hence substituting in equation [43]

[46]

${\displaystyle r'\int _{W}^{W'}dW+C_{ps}\int _{W}^{W'}(t-t')dW=C_{pa}\int _{t'}^{t}dt+C_{ps}\int _{t'}^{t}Wdt}$

[47]

${\displaystyle r'(W'-W)+C_{pa}\int _{W}^{W'}\left[\int _{t'}^{t}dt\right]dW=C_{pa}(t-t')+C_{ps}W\int _{t'}^{t}dt+C_{ps}\int _{W}^{W'}\left[\int _{t'}^{t}dt\right]dW}$

[48]

${\displaystyle r'(W'-W)=C_{pa}(t-t')+C_{ps}W(t-t')}$

90The same result may be obtained by equating the total heat in the air in any state with its total heat when in the state of adiabatic saturation. The total heat in a mixture of 1 lb. of pure air and saturated water vapor at a temperature ${\displaystyle t'}$ calculated from a base temperature of 0 deg. fahr. and deducing the heat of the liquid, ${\displaystyle q'}$, which as we have shown is unaffected by the adiabatic change, is [49]

${\displaystyle \Sigma =C_{pa}t'+r'W'}$

91The total heat under any other adiabatic condition, where temperature is ${\displaystyle t}$ and moisture ${\displaystyle W}$, is

[50]

${\displaystyle \Sigma =C_{pa}t+[(H_{1}-q_{1})+C_{ps}(t'-t_{1})]W}$

which is substantially equivalent to

[51]

${\displaystyle \Sigma =C_{pa}t+r'W+C_{ps}(t-t')W}$

Therefore since the change is adiabatic we may equate [47] and [49].

[52]

${\displaystyle C_{pa}t+r'W+C_{ps}(t-t')W=C_{pa}t'+r'W'}$

[53]

${\displaystyle C_{pa}(t-t')+C_{ps}(t-t')W=r'(W'-W)}$

where

${\displaystyle (t-t')}$ = the true wet-bulb depression
${\displaystyle (W'-W)}$ = the moisture absorbed per lb. of pure air when it is adiabatically saturated from an initial dry-bulb temperature to and an initial moisture content ${\displaystyle W}$
${\displaystyle C_{pa}}$ = mean speciﬁc heat of air at constant pressure between temperature ${\displaystyle t}$ and ${\displaystyle t'}$
${\displaystyle C_{ps}}$ = speciﬁc heat of steam at constant pressure between ${\displaystyle t}$ and ${\displaystyle t'}$
${\displaystyle r'}$ = latent heat of evaporation at wet-bulb temperature ${\displaystyle t'}$

This is identical with equation [20] obtained by the differential method.