R.L. Snyder and K.T. Paw U

Copyright Regents of the University of California

Created January 6, 2002

Copyright
2002 Regents of the University of
California

These
teaching notes discuss psychrometric relationships that are important
for
understanding humidity and its measurement.
The first section discusses concepts related to measuring
dry-bulb
temperature. Then concepts pertaining
to thermodynamic wet-bulb temperature are presented.
Finally, some ideas on measuring the wet bulb temperature are
discussed.

The
heat transfer to and from a dry-bulb thermometer depends on the
long-wave radiation
balance and sensible heat transfer.
Assuming an emissivity *e* = 1.0, the net long-wave
radiation can be expressed in terms of a resistance to radiation
transfer (*r _{R}*) in s m

_{}
(1)

*T _{s}* is the shield temperature
(K) and

_{}
(2)

*H* is the sensible heat to and
from the thermometer and *T* is the air
temperature within the shield. When in
equilibrium, *R _{n}* =

_{}
(3)

_{}
(4)

The
goal is to have *T _{t}* be very
close to

(1)
*r _{H}* <<<

(2)
*T _{s}* very close to

In
an adiabatic system, the sum of latent and sensible heat remains
constant. The initial, energy state is
specified by
the temperature (*T*), vapor pressure (*e*),
and total pressure (*p*).
If liquid water is present and *e*
is smaller than *e _{s}*(

_{}
kg
m^{-3}
(5)

where,
*M _{v}* and

_{}
kJ
m^{-3}
(6)

where
*l* is the latent heat of
vaporization (kJ kg^{-1}). The
corresponding sensible heat used for vaporizing the water is

_{}
kJ
m^{-3}
(7)

Equating
the two expressions and solving for *e*,
we get

_{}
kPa
(8)

where
_{} is the psychrometric
constant. Actually, *g* is not a constant but a function of
barometric
pressure (*p*), which depends on
elevation and passing weather systems and on the latent heat of
vaporization (*l*), which is a weak function of temperature. If *p*
= 101.3 kPa, then *g* = 0.066 at 0^{o}C
and *g* = 0.067 at 20^{o}C.
A good formula to estimate *p* is

_{}
kPa
(9)

To
estimate *l,* use

_{}
J
kg^{-1}
(10)

Note that vapor pressure can also be determined
from *e _{s}*(

_{}
kPa
(11)

Substituting
into the psychrometric equation (Eq. 8), we get

_{}
kPa
(12)

Substituting
the slope of the saturation vapor pressure curve at air temperature (*D*) for *D*’, and solving for *T'*
provides an equation to estimate the wet-bulb temperature.

_{}
^{o}C
(13)

The
error in using *D* rather than *D*’ is small when *T* *T'*
is small, but the error increases with greater wet-bulb depression.

The
measured wet-bulb temperature (*T _{w}*)
is an estimate of the thermodynamic wet-bulb temperature (

_{}
kJ
m^{-2}s^{-1}
(14)

where *r _{HR}*
is the parallel resistance to convective and radiation heat transfer. The rate of latent heat loss from the
wet-bulb is

_{}
kJ m^{-2}s^{-1}
(15)

Recall that the psychrometric constant is

_{} so _{} and
_{}
(16)

By substitution, we get

_{}
(17)

In equilibrium, *lE* = *R _{n}*
+

_{}
(18)

then

_{}
(19)

and

_{}
(20)

Thus, the wet-bulb temperature equals the
thermodynamic
wet-bulb temperature only when *r _{v }*=

_{}
(21)

where _{}. Based on the
concepts of forced convection, the resistance to vapor transfer equals
93% of
the resistance to convective heat transfer (*r _{v}*
= 0.93

_{}
(22)

Rearranging terms, we get

_{}
(23)

and

_{}
(24)

For example, if *r _{R}*=210
s m