equations for determining humidity from dewpoint and psychrometric data
TRANSCRIPT
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N A S A
w
n
z
VI
z
T E C H N I C A L
N O T E
E Q U A T I O N S F O R THE D E T E R M I N A T I O N
OF H U M I D I T Y F R O M D E W P O I N T
A N D P SY CH RO M ET RIC D A T A
0 Owen Parish
and
Terrill
W.
Putnam
Dryden
Fiight Research
Center
Edwards, Cali 93523
N A T I O N A L A E R O N A U T I C S A N D S PA C E A D M I N I S T R A T I O N W A S H I N G T O N , D . C. J A N U A R Y
1977
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TECH
LIBRARY
KAFB,
NM
IllillI11
111
l111
lllllllll
I1
0334324
9. Performing Organization Name and Address
NASA Dryden Flight Research Center
P . 0 . Box 273
Edwards, California 93523
.-
1. Report No.
NASA TN-D-8401
4. Title and Subtitle
10. Work Unit No.
505-11-26
11.
Contract or Grant No.
13. Type
o f
Repor t and Period Covered
3. Recipient's Catalog No.
I
. Government Accession No.
5.
R epor t Date
17. Key Words (Suggested by Author(s) )
EQUATIONS FOR THE DETERMINATION OF HUMIDITY FROM
DEWPOINT A N D PSYCHROMETRIC DATA
18. Distribution Statement
I
January
1977
19. Security Classif. (of this report)
Unclassified Unclassified
20; Security Classif. (of this page)
~
6.
Performing Organization Code
I
21.
NO.
of Pages
22 Price'
24
3.25
7. Author(s)
0 .
Owen Parish and Terrill
W .
Putnam
8. Performing Organization Report No.
H-937
12.
Sponsoring Agency Name and Address
I Technical Note
National Aeron autics an d Spa ce Administration
Washington, D . C . 20546
14.
Sponsoring Agency Code
I
15. Supplementary Notes
16. Abstract
A
general expression based on the Claperon-Clausius differential
equation that relates saturation vapor pre ssu re, absolute tempera ture,
and the latent heat of transformation was derived that exp res ses
saturat ion vapor pre ssu re as a function of absolute tempera ture. This
expression was then used to derive general expressions for vapor
pr es su re , absolute humidity, and relative humidity a s functions of
either dewpoint and ambient temperature or psychrometric parameters.
Constants for all general ex pressions w ere then evaluated to give
specific expres sion s in both the Internati onal System of Units
(SI)
and
U .S
.
Customary Units for temperatures above and below freezing.
The temperature r ang e considered for all expression's was - 5 O O C to
l o o o C - 5 8 O
F
to 2 1 2 O F) over water and
- 5 O O
C to O o C - 5 8 O F to
32O F) over ice . The genera l expre ssio ns, along with the values of the
constants that give the specific expressions, appear in a table for easy
reference.
Humidity
Vapor pressure
Saturation vapor pressure
Unclassified Unlimited
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EQUATIONS FOR THE DETERMINATION OF HUMIDITY
FROM DEWPOINT
A N D
PSYCHROMETRIC DATA
0 . Owen Parish and T err ill W . Putnam
Dryden Flight Research Center
INTRODUCTION
The measurement of humidity
is
important to many ar ea s of scien tific study .
One such area is the reduction and ana lys is of acoustic data obtained in uncontrolled
field environments-aircraft and engine noise data , for example. Acoustic data
acquired und er these circumstances are usually c orrected to a standard day with a
specified humidity and temperature. Such corrections a r e functions of the actual
ambient temperature and humidity at the time the data were taken.
There are two primary methods for determining humidity. The fir st re qu ire s
that the dewpoint and ambient temperature s be know n. The second method req ui re s
that the barometric pr es su re and the psychrom etric parameters-dry-bulb (ambient)
and wet-bulb temperatures-be known . Both methods normally re qu ire refer enc e to
tab les, the use of special slide ru le s , or the solution of sev er al complicated equa-
tio ns , all of which a re tedious and time consuming and incr ea se the probability of
er ro rs in the final res ult .
The Claperon-Clausius differential equation (re f.
1)
, which re lates saturation
vapor pressure
,
absolute temperature
,
and the latent heat of transformation
,
was
used as the star ting point in the development of the equations presented he re in ,
which give humidity as a direc t function of eith er se t of param ete rs stated above.
Humidity can be found with these equations on a computer or calculator with a
significant reduction in time and with g rea ter reliability than with the previo us
methods. Furthermore
,
the need for tables and special slide rul es
is
virtually
eliminated for most practi cal pu rp os es .
These equations a re developed and presen ted herein in the ir most general form.
The constants for the equations a re prese nted in tab ular format fo r the ambient
temperature range from - 5 O O C to looo C (-58O F to
212O
F) and in both the
International System of Units
(SI)
and
U .S
. Customary Units.
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SYMBOLS
A
vapor pres su re proportionality fac tor
a , a l , b ,
bl,
constants (table
1)
~ 1 ,
,
f Y g , k
specific heat of ice at Oo C . 3 2 O
F ) ,
J / k g
K
(Btu/lbm
OF)
i
C specific heat of water at
1 5 O
C 5 9 O F) , J / k g
K
(Btu/lbm
OF)
W
C
P V
D
e
S
specific heat of water vapor at constant pre ss ur e, J /k g K
(Btu/lbm
O F )
dewpoint temperature ,
OC
( O F )
saturation vapor pre ssu re , m b (in. Hg)
e vapor pre ssu re,
m b
(in. Hg)
V
e.
vapor pre ssu re over ice at Oo C 3 2 O F) , m b (in. Hg)
10
e
H
absolute humidity, kg/m (lbm/ft )
Li
vapor pres sure over water at Oo C
3 2 O F) , mb
(in.
H g )
wo
3 3
latent heat of sublimation
,
J/kg (Btu/lbm)
latent heat of evaporation, J/kg (Btu/Ibm)
Lw
m
m
d
molecular weight
molecular weight of dr y a i r , k g mol
(lbm
mol)
m
molecular weight of water vap or , kg mol
lbm
mol)
V
P
R
R
atmospheric pre ss ure
, m b
(in. Hg)
specific gas constant , J / k g K (Btu/lbm OF)
universal gas constant , J /k g mol K (Btu/lbm mol O F )
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Rd
T
T'
TO
U
V
Sub scrip ts:
i
W
X
specific gas constant for dry a i r , J / kg K (Btu/lbm OF)
specific gas constant
for
water vapor, J /k g
K
(Btu/lbm OF)
dry-bulb
or
ambient temperature,
OC
(OF) or K (OR)
wet-bulb temperature,
O C
(OF)
or
K
(OR)
reference temperature, Oo C (32O F)
rela tive humidity (decimal value)
volume, m 3 (ft
3
ratio of molecular weight of water vapor to molecular weight of
d r y a i r ,
m
/m
v d
reference value
ice
water
dummy varia ble , replaced by i
or
w
DEFINITIONS
AND
RELATIONSHIPS
Vapor Pressure
The vapor pr es su re of moist a i r , e
is
defined as the partial pre ssu re of the
V '
water vapor present in the air mass. The vapor pre ssu re is said to be with respect
to water
(or
ice) i f the air mass
is
over a plane su rface of water (or ice) at the same
temperature and pre ssu re.
Saturation Vapor Pressure
Saturation vapor pressure, e
is
defined as the vapor p res sur e that exists
when the air mass is at the tem perature at which
two
phases of water coexist in
neutral equilibrium.
S
The saturation vapor pressure is said to be with respect to water (ice) when
the a ir mass is over a plane surface of water (ice) at the same temperature and
pre ss u re . Any reduction in temperature resu lts in the formation of dew (f ro st) .
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D
ewpoint
The dewpoint
,
D
, is
the temperatu re to which moist ai r must be cooled to
become saturated at initial pr es su re and moisture content. The dewpoint is also the
temperature at which the saturation vapor pressure equals the actual vapor pres-
su re . Any furthe r cooling results i n the formation of dew
or
fr os t. When the dew-
point
is
at or below fre ezing
,
it is called th e frostpoint.
Absolute Humidity
Absolute humidity,
H
, is defined as the ma ss, inv , of the water vapor p resen t
per unit volume V of air at a given temperature and barometric pr es su re . Absolute
humidity may be expressed in equation form as follows:
H = mv/V
1)
To expre ss absolute humidity a s a function of the vapor pr es su re ev and absolute
ambient temperature T , the gen eral law for perfect gase s
is
used to give the
following equation:
e V = m R T
V
v v
where
Rv
is the gas constant for water vapor and ev is with respect to water for T
above freezing and with respect to ice for T at or below freezing. Then equation
( 2 )
is solved for m
/V
and the result is substituted into equation
1)
, as follows:
V
H
=
eV/RVT
( 3 )
To permit the use of any desired pressure and temperature units, equation
( 3 )
may be written in the following, more general, form:
H
=
kev/(T
+
d)
4)
where
k is
the product of all required conversion factors divided by
Rv
and the
value of d determines the temperature units used . Equation
(4 )
is the form used
her ein. The constants k and d a re evaluated in appendix A , and their values are
listed
in
table 1.
Since water vap or
is
not a perfect gas and
is
subject to compressibility effects,
equations
(3 )
and
(4 )
should contain a compressibility factor in the denominator.
However , sinc e the value of that factor is between
1
OOOO and 0 . 9 9 5 6 for normal
atmospheric temperature and press ure ra nges (ref. 2 ) , it may be taken as 1 . 0 0 0 0
for most purpo ses with excellent resu lts .
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Relative Humidity
The relative humidity of a i r ,
U , is
defined a s the ratio of the partial pr es su re
of the water vapor pres ent at a given temperature and barometric pr es su re , ev , to
the partial pr es su re of the water pre sent at saturation for the given temperature
and pressu re , e
.
In equation form,
S
e
e
V
S
u = -
( 5 )
which g ives re lative humidity a s a decimal valu e.
To express relative humidity as
a percentag e, the resu lt of equation ( 5 ) can be multiplied b y 100.
The techniques and procedures recommended in reference
2
req uir e ev to be
evaluated with resp ect to water fo r ambient temperatures above freezing and with
respe ct to ice for ambient tempe rature s at or below freez ing and es always to be
evaluated with respect to water in equation ( 5 ) .
EXPRESSIONS
FOR SATURATION VAPOR PRESSURE
Equation Derivation
When water
or
ice is transformed into va po r, heat must b e abso rbe d. The total
heat absorbed for a water-to-vapor transformation is called the latent heat of evap-
oration, L
latent heat of sublimation, L The ra te of change in the laten t heat of transforma-
tion with absolute temperature and at constant pressure may be written as follows:
. The heat absorbed for
an
ice-to-vapor transformation is called the
W
i
dLX
dT
= c
-
pv
where x
is
replaced by
w
for evaporation and by
i
for sublimation and
Lx is
the
latent hea t of evaporation or sublimation, c
constant pr es su re , and cx is the specif ic heat of water or ice.
is
the specific heat of water vap or at
PV
Throughout the temperature ra ng es of int er es t, th e variatio ns in specific heat
a re small enough
so
that c
normal atmospheric temperature range, equation ( 6 ) may b e integra ted a s follows:
and cx may be taken as constant. Hence, for the
PV
Lx = p p v -
- To) Lxo
7)
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where Lxo is the known la ten t heat of evaporation
or
sublimation at a reference
temperature T o.
A
form of the differential equation that relates saturation vapor p re ssur e ,
absolute tem peratu re, and latent heat of transformation
is
1
desx LX
esx dT R ~ T '
- - - -
where e
on the ambient temperature. Th is equation is a form of the Claperon-Clausius
equation that relates pressure to temperature in a system in which two phases of a
substance a re in equilibrium.
is the saturation vapor pr es su re with respe ct to water
or
ice , depending
sx
The general expression for saturation vapor p re ss ur e over water o r ice as a
function of absolute temperature
is
obtained by substituting equation
(7 )
into equa-
tion (8) and integrating the res ul ts, a s follows:
log esx = a log T b/ T + c
(9 1
where
a
=
(epv - cx)/RV
=
[ p p v - Cx)To - Lxo / E tv In 1 0 )
c = log e
- a log To - b/To
(12)
xo
Equation
(9)
is derived
in
reference
1
for saturation vapor pr ess ure over water
with esw in cen tibars and T in kelvi ns, as follows:
log esw =
-4.9283
log T
-
2937.4/T
2 2 . 5 5 1 8
(13)
This is known a s Magnus' formula.
Equation
(9)
is the ba sis for the exp ressio ns developed herein fo r absolute
humidity
,
relative humidity, and vapor pr es su re . However
,
the form derived
herein
is
more gen eral than equation ( 9 ) . That form is obtained by taking the anti-
logarithm of equation
(9)
as follows:
ITa
e =
1 0
sx
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and then including a temperature conversion constant, d
,
to permit temperatures
to be expressed in d egre es Celsius
o r
Fahrenheit. Thus
This equation
is
the desired form for obtaining saturation vapor p re ss ur e.
The values of a , b c and d in equation
(15)
depend on the temperature and
pre ssu re units desired; on whether saturation vapor pre ssu re
is
required over
water or ice; on the refe ren ce temperature us ed; and on the sourc e used to obtain
the values of the other parameters involved in their computation.
The values of
a ,
b ,
c and d are evaluated in appendix A for the temperature ra nge from
-5OO
C
to l o o o C
(-58O
F to
212O
F) .
The resu lts, listed in table 1 , give saturation vapor
pr es su re over water and ice for both SI and U .S . Customary Units.
Comparison of Equation Values With Smithsonian
Meteorological Tables
Since equation ( 1 5 ) is used to develop the expressions used to find vapor pres-
su re and humidity, any err or in it
is
transmitted to those exp ression s. Therefore,
to test the validity
of
equation
( 1 5 ) ,
values obtained by using it with the constants
given in table
1
for SI we re compared to the corresponding values given in the
Smithsonian Meteorological Tables (SMT) (r ef . 2 ) .
The res ult s, expressed a s the
percentage of difference between the values found with equation
( 1 5 )
and those in
the Smithsonian Meteorological Ta ble s, ar e presented in figu res 1 a) and
1 b)
.
T,
O F
-80 -40 0 40 80 120 160 200 240
1 7 7 1 I
D i f f e r e n c e , I
p e r c e n t
- 1
-60
-40
-20
0
20 40
60 80
100
T,
C
a ) esw values.
Figure 1 . Difference between Celsius values give n b y equation 1 5 ) and those
gi ve n b y Smithsonian Meteorological Tabl es for esw and e
si
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T, O F
-80 -40
0
40
I
I
I
1
D i f f e r e n c e ,
p e r c e n t 0
-1
-60
-40 -20 0
T, C
b ) esi values.
Figure 1 Concluded.
The results of comparing saturation vapor p re ss ur e over wa ter , given by
-4.92830
(T 273)
=
1 0 i23.5518-2937.4/ (T+273)]
sw
appear in figure
1
a ), and the results over ice , given by
-0.32286
(T + 273)
1 . 4 8 1 6 - 2 7 0 5 . 2 /
(T+273)]
esi = 10
appear in figure
1
b)
.
Figure l( a ) shows that equation (15) gives values for saturation vapor pres -
su re over water that differ from SNIT va lues by le ss than
0 . 1
percent over the
temperature range from - 3 O O C to
70
C - 2 2 O F to 158O F) and by le ss than
0 . 8
per-
cent over the ran ge from
- 5 O O
C to l o o o
C
- 5 8 O F to
2 1 2 O
F j .
Figure
l b )
shows that equation (15) gives values over ice that differ by le ss
than
0 . 2
percent over the rang e from
-3Q0 C
to
Qo C - 2 2 O
F to
3 2 O
F) and by less
than 0 . 6 percent over the rang e from - 5 O O C to Oo C - 5 8 O F to 3 2 O F) . Since the
normal atmospheric tempe rature s lie within the ra ng e from - 4 O O C to
60
C
(-40
F
to
140
F ) , equation
(15)
can be used with the constants in table
1
to find saturation
vapo r pre ss ur e for most practical applications with excellent res ul ts , in most case s
within the err or of the temperature-measuring device us ed .
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EXPRESSIONS FOR VAPOR PRESSURE
Dewpoint Expressions
Because of the definition of dewpoint , equation (15) gives vapor press ure as a
di rect function of dewpoint
(or
frostpoint i f the ambient temperature
is
at or below
freezing)
i f
T
is
replaced by
D
.
The same constants in table
1
ar e used . With the
appropriate changes in notation, equation (15) assum es the following form:
where x is replaced by w for ambient temperatures greate r than freezin g and by i
for ambient temperatures at or le ss than freezing. The units used fo r dewpoint
determine which se t of tab le 1 constants should be used.
Psychrometric Expressions
The most common expression used to find vapor pressure as a function of
psychrometric param eters is taken from reference 2 and given he re in the functional
notation form
e
(T
,
T'
,
p) = esx(T')
-
Ap(T
-
T')
(17)
vx
where
evx(T
,
T'
,
p)
T
T'
P
esx(T'>
A
vapor pres sur e in air over water or ice, depending or, T'
dr y -bulb temperature
w et-bulb temperature
barometric press ure
saturation vapor pr es su re over water
or ice, depending on
T' (given by eq .
(15)
evaluated
or
T')
proportionality factor
The press ure s evx , p , and esx must be in the same system of units.
The proportionality factor A has the form
A =
(f + gT')
(18)
where f and g ar e cons tants. Although thi s factor has been determined empirically
9
I
I 111111111 11111
1111111
111111 Ill111 I111 I
I1
111111111
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and verified by many investigators (re fs. 2 and 3 1 , it
is
credited to Fe rre l. The
values for f and g appear in table
1;
the values for U .
S
. Customary Units a r e those
obtained by F er re l, and the values for SI ar e conversions of those values (app. A) .
However, reference 2 points out that when the wet bulb is covered by a thin layer
of ice, F er re l's constants a re invalid and must b e multiplied by 0.882, which is the
ratio of the latent heat of evaporation to the latent heat of sublimation. Th ere fore ,
for
this condition the values given in table 1 for
f
and
g
should also b e multiplied
by this ratio.
A general e xpression for vapor pr es su re a s a function of psychrometric data is
obtained by substitu ting equation (15) evaluated for T' and equation (18) into equa-
tion (17) as follows:
where the values for the constants a re given in table 1. The use of over ice o r over
water values
is
determined by TI.
EXPRESSIONS FOR HUMIDITY
Once expressions for saturation vapor p res sur e and vapor pre ssu re have been
de fine d, expre ssions for absolute and relative humidity can b e obtained by subs ti-
tuting equation (15)
,
( 16 )
,
o r
(19),
evaluated for the ap pro pria te conditions, into
the definitions of abso lute humidity (eq.
(4))
or relative humidity (eq .
(5 ) ) .
Absolute Humidity as a Function of Dewpoint and
Ambient Temperature
The ge neral expre ssio n for absolute humidity a s a function of dewpoint tempera-
t u r e , D
,
and ambient temperature, T , is obtained by substituting equation ( 1 6 ) into
equation
(4 )
as follows:
H(T,
D)
=
k(T d)-1 0 [c+b (D+d)
D
d ) a
whe re the appr opriate values for the constants a re taken from table
1
for specific
ex pre ssio ns. The use of over ice or over water values is determined by T.
Absolute Humidity a s a Function of Psychrometric Param eters
The general expr ess ion for absolute humidity a s a function of dry-bulb tempera-
t u r e , T , wet-bulb temperature, T ', and ambient pr es su re , p , is obtained by substi-
tuting equation (19) into equation ( 41 , as follows:
1 0
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H(T, T' , p)
=
k ( T + d)-l[lO{C+[b/(T'+d)l}(Tr + d)a - (f gT')p(T - T')] ( 2 1 )
where the appropriate values for the constants are taken from table
1
or specific
expr essio ns. The us e of over ice or over water values is determined by T '.
Relative Humidity as a Function of Dewpoint and
Ambient Temp erature
The genera l ex pres sion for rela tive humidity a s a function of dewpoint tempera-
t u r e , D
,
and ambient temperature, T , is obtained by subs tituting equation (16) into
the numerator and equation
(15)
into the denominator of equation
(5)
as follows:
(22)
U(T, D) = 10[(c-cl)+b/ (D+d)-bl/ (T+d)
+
d)a(T + d )
al
To expr ess relativ e humidity as a perc entag e, the re su lt of equation ( 2 2 ) can be
multiplied by 1 0 0 . The appro priate values of the constants a r e taken from table
1.
The use of over ice or over water values is determined by D . The subscripted
constants differ from the corre spon ding nonsubscripted constants only when the
dewpoint temperature,
D
, is at or below fre ezin g.
When
D is
above free zing,
equation ( 2 2 ) red uces to the following exp ression:
U(T, D ) = [(D
+
d ) / ( T + d)la10b [@fd)-'- (T+d)-']
Relative Humidity as a Function of Psychrometric Param eters
The genera l expres sion for relative humidity as a function of dry-b ulb tempera-
tu re , T , wet-bulb temperature, T'
,
and ambient pre ss ur e, p
,
is obtained by substi-
tuting equation
(19)
into the numerator and equation (15) into the denominator of
equation (5) as follows:
U(T, T',p) = 10 {cl+[bl/ (T+d)]} (T d)-al [~O{~+[~/(~'+~)~'(T'd)a (f + gT')p(T - T')]
(24)
where the appropriate valu es for the constants ar e taken from table 1. Relative
humidity can b e e xpre ssed as a percentage by multiplying the r esu lt of equation
(24)
by
100.
The use of ove r ice or over water valu es
is
determined by T'. The sub-
scripted constants differ from the corresponding nonsubscripted constants only
when the wet-bulb tempera ture, T' , is at
or
below fr eez ing.
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APPLYING TABLE 1
All constants appea ring in table
1
ar e evaluated in appendix
A .
The constants
a , b , and c a re essentially for the saturation vapor pr es su re equation, which is
fundamental to all the other equations. When saturation vapor pr es sur e over water
is
specifically required, as in the denominator of the relative humidity equation,
the corresponding subscripted constants a 1 , b
1 ,
and c1 ar e used.
The constant d
is
a temperature conversion constant and depends only on the
temperature units required.
The constants f and g a r e empirical constants that give the proportionality
factor essential for vapor pr es su re a s a function of psychrometric data and must b e
multiplied by
0.882 if
the wet bulb
is
covered by a thin layer of ice.
The constant k is used in the absolute humidity equations to convert f r o m one
system of units to another and is a multiple conversion factor divided by the
specific gas constant
for
water vapor.
The use of over ice o r over water values for constants is determined by the
value of T'
o r
D in those equations in which they ap pe ar . When T'
or
D is less than
or equal to freezi ng, the over ice values ar e use d. When T' or D is greater than
freezing, the over water values a re used.
Examples using table 1 may be found in appendix
B
.
SUMMARY OF
RESULTS
General equations a re derived herein that give saturation vapor pressure ,
vapor pressure, absolute humidity, and relative humidity
as
functions
of
either
dewpoint and ambient temperature or psychrometric param eters over a plane
surface of either water or ice and for both the International System of Units (SI) and
U .S
Customary Units.
The expression for saturation vapor pressure is fundamental to all other
expressions,
so
the values given by it for SI units over water and ice were
compared with values fo r corresponding conditions g iven by the Smithsonian
Meteorological Table s. The comparison showed differences of less than
0 . 1
percent
for the temperature range from - 3 O O C to 70 C
- 2 2 O
F to
1 5 8 O
F ) over water and
differences
of
less than
0.8
percent for the temp erature rang e from
-5OO
C
to
l o o o
C
- 5 8 O F to 2 1 2 O F )
.
For
over ice, the differences were less than 0 . 2 percent for
temperatures f r o m
-30
C to Oo C
- 2 2 O
F to 3 2 O
F )
and less than 0 . 6 percent for the
temperature range from - 5 O O
C
to Oo C - 5 8 O F to
3 2 O
F) .
Dryde n Flight Research Center
National Aero nauti cs and Spa ce Admin istra tion
June
8 ,
1976 Edwards California
1 2
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APPENDIX
A .
-EVALUATION OF TABLE 1 CONSTANTS
The values of the constants a , b
,
and c for use with
SI
were found fir st and
converted to their equivalents in U
. S .
Customary Units.
The va lues of
f
and g for
U .S
.
Customary Units were taken from references 2 and
3
and converted to their
equivalents in
SI.
The general expression for saturation vapor pr es su re derived in the text
(eq
. ( 1 4 ) )
is repeated he re fo r reference:
where a , b
,
and c a re as defined by equations
(10)
to (121, below:
b
=
[(c
PV
-
cx)To
-
Lxo]/(RV
In
1 0 )
c = log e - a log To - b/To
xo
and T is in absolute temperature units. The units used he re for T ar e kel vin s,
with the reference temperature, To, taken as
273
kelvins.
The values of all other parame ters used to evaluate a , b , and c may vary
considerably, depending on the sou rce they a re taken from.
The value s used
her ein , which a re listed in table
2 ,
were taken from reference 1 except as noted.
They were used because of the close agreement of thei r re su lts with the va lues in
the Smithsonian Meteorological Tables over the temperature range from
- 5 O O
C
to
l o o o C - 5 8 O
F to
2 1 2 O F )
and not because they w ere assumed to be the best va lues
available.
Evaluating a , b
,
and c Over Water
The following evaluation of the constan ts a , b
,
and c for equation (14)
is
the
same as that in referen ce 1 except tha t esw is in millibars instead of centibars.
For any specific gas of molecular weight
m
and gas constant R , it follows from
the definition of the univ ersa l ga s constant
R*
and the gen eral law fo r gases that
R*
=
m R ( 25 )
Therefore, the specific gas constants of water vapor and d ry a ir a re related as
follows:
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APPENDIX A
- Continued
R * = m R =mdRd
v v
V '
Solving equation (26) for R
1
R v = (m d m v )
R
d = F R d
(26)
where
E =
m /m
= 1 8 . 0 1 6 / 2 8 . 9 7 = 0 . 6 2 2
v d
The SI values of a , b
,
and c can be found by substitutin g equation ( 2 7 ) into equa-
tions ( 1 0 ) and ( 1 1 ) and taking the proper values from table 2 , as follows:
=
0 . 6 2 2 ( 1 9 1 1
-
4 1 8 5 ) / 2 8 7 . 0
=
- 4 . 9 2 8 3
= E [('pv
-
' w ) ~ o -
wo]/(
R d In 1 0 )
=
0 . 6 2 2 [ ( 1 9 1 1 - 4 1 8 5 1 2 7 3
-
2 . 5
X
1 o 6 ] / ~ 2 8 7 . 0
In
1 0 )
=
- 2 9 3 7 . 4
c
=
log e
-
a log To
-
b /T
wo
= log
6
l l 4 . 9 2 8 3 log 2 7 3 + 2 9 3 7 . 3 7 / 2 7 3
= 2 3 . 5 5 1 8
These co nstants permit saturation vapo r pr es su re to be found in millibars by
using equation ( 1 4 ) , but only i f T is in kelvins.
To
permit T in degrees Celsius,
the approximate temperature conversion
T(K) = T(OC)
+
2 7 3
(28)
was found to give excellent re su lts . The refore , by defining a temperature conver-
sion constant d su ch that d equals zero for T in kelvins and 2 7 3 for T in degrees
Ce lsius , equation (14) may be written in the following, more gene ral form:
1 4
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, ,
.
.
_.
.
.
. . ..
.
.....
.
APPENDIX A - Continued
without affecting the va lues of a ,
b
, and c found above. The value s for SI thu s
found appear in table 1 in the column for over w ater.
The U .S
.
Customary Unit equiva lents of a , b
,
c
,
and d a r e found as follows.
To permit T in de gre es Fah ren he it, the approximate conversion
T
(K) = 5 / 9
[T (OF)
+ 4 5 9 . 4 1
( 3 0 )
was found to give excellent re su lt s. If d is made equal to
4 5 9 . 4
for T in degre es
Fahrenheit and equation
( 3 0 )
is applied to equation
( 1 4 ) ,
the form given by equa-
tion
( 2 9 )
remains valid and the values
of
b and c change a s follows:
New b =
( 9 / 5 )
(Old b) =
( 9 / 5 ) ( - 2 9 3 7 . 4 )
=
- 5 2 8 7 . 3 2
New c
=
(Old c)
+
a log
( 5 / 9 )
=
2 3 . 5 5 1 8
-
4 . 9 2 8 3 ( 1 0 g
5
-
log
9 )
=
2 4 . 8 0 9 8 6
However, esx is still in millibars. To convert the millibar value to inches of
mer cu ry , the res ult of equation
( 2 9 )
can be multiplied by
0 . 0 2 9 5 3
by adding the
logarithm of
0 . 0 2 9 5 3
to the valu e of c jus t found.
c
= 2 4 . 8 0 9 8 6 +
log
0 . 0 2 9 5 3 = 2 3 . 2 8 0 1
These values
of
b
,
c , and d , along with a (which remains unchanged), appear in
table 1 for U
. S
. Customary Units over w ate r.
Evaluating a , b
,
and c Over Ice
To find a , b , and c over ice for
S I ,
equations
(10)
to
( 1 2 )
ar e used with the
applicable values from table
2 ,
as follows:
(pv
-
ci / R v )
( 1 9 1 1
-
2 0 6 0 ) / 4 6 1 . 5
- 0 . 3 2 2 8 6
1 5
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APPENDIX A
-
Continued
b =
[(c PV
-
ci)To
-
Lio]/ Rv In 1 0 )
= [ (1911 - 2 0 6 0 ) 2 7 3 - 2 . 8 3 4
X lo6]/
( 4 6 1 . 5 In 1 0 )
=
- 2 7 0 5 . 2 1
c = log e. - a log To - b/To
10
= log 6 . 1 0 7 + 0 . 3 2 2 8 6 log 2 7 3 + 2 7 0 5 . 2 1 / 2 7 3
=
1 1 . 4 8 1 6
These values give e in millibars for T in kelv ins. To permit T in deg rees
si
Celsius , let d equal 2 7 3 , which is equivalent to using equation ( 2 8 ) .
The
U . S
.
Customary Unit equivalents of a ,
b ,
c
,
and d for over ice a r e found
in the same manner a s for over water.
New b = ( 9 / 5 ) (Old b ) = ( 9 / 5 ) ( 2 7 0 5 . 2 1 )
= - 4 8 6 9 . 3 8
New c
=
(Old c + a log ( 5 / 9 ) + log ( 0 . 0 2 9 5 3 )
=
1 1 . 4 8 1 6
-
0 . 3 2 2 8 6
log
. (5/9)
+ log
( 0 . 0 2 9 5 3 )
=
1 0 . 0 3 4 3
Then , for T in deg rees Fahrenheit , if a value of d equal to 4 5 9 . 4 , an unchanged
value of a , and these values of a , b , and c a re substituted into equation ( 2 9 ) , the
resulting value of e is in inches of me rcury.
si
The values of the constants a l , b l , and cI in table
1
ar e the same as the
corresponding nonsubscripted constants for over water. They ar e used in the
rela tive humidity equations only to distinguish constants in the denominator from
the constants in the numerator.
Evaluating
f
and g
The constants f and g appear in equations expressing vapor pre ss ur e a s a
function of psychrometric param eters. Equation ( 1 7 ) , the general vapor pre ssu re
equation, is repeated below for reference:
(T , T ' , p ) = esx(T')
-
Ap(T
-
T')
( 1 7 )
vx
1 6
. ..
.,
.
...
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APPENDIX
A -
Continued
where the proportionality factor
A
has the form
A = (f + gT')
(18)
The expression for A , credited to Fe rre l, is given
in
references
2
and 3 for T'
in de gre es Fahrenheit as follows:
1571
= 0.000367 1 +
The values
of f
and
g
can be obtained simply by ca rryi ng out the operations
indicated by equation (18); thus,
f
equals 3.595 X
and g equals 2.336
X
To find the Celsius equivalents of
f
and g , the complete second term of
equation
(17) is
used
,
with
A
replaced by equation (18).
The term then becomes
(f + gT')p(T - T') . Since equation ( 1 7 ) is independent of temperature units, this
term
is
as well. Th ere fore, Fahrenheit temperatures may be assumed.
Then, to
convert equation (31) to an expre ssion for Celsius temperatu res, equations (28) and
(30) a re combined to g ive the following conversion
T (OF) = 9/5[T(OC) + 321
( 32)
and this expression
is
substituted into the term
(f +
gT')p(T
- T') .
This gives
[(9/5)f + (9/5I2gT' +
(9/5)
(32)g]p(T - T') . Substituting the Fahrenheit values for
f and
g
into this ex pression gives (6.60 X l o W 4+ 7.57 X 10-7T')p(T - T') .
Comparing this exp ression with the term
f
+ gT')p(T - T') , the Celsius equivalent
of
f is 6 . 6 0 X
l o F 4 and the equivalent of
g is
7 . 5 7
X
independent of the vapor p re ssu re being measured over water or ic e.
Therefore,
in either system of units
,
f and g have the same value s over both water and ic e.
The valu es of f and g ar e
Evaluating k
The constant k appe ars only in exp ressions involving absolute humidity.
Equation (31 , which defines absolute humidity as a function of vapo r p re ss ur e, is
repeated her e fo r reference:
H
= eV/RVT (3)
where T
is
in kelvin s. To convert this to an expression for T in degrees Celsius,
equation (28) is app lied , a s follows:
(33)
The units chosen for H
in
SI ar e kilograms p er cubic meter, and since
e
is
in
millibars, H must be converted to newtons pe r s qu are meter by multiplying
V
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APPENDIX
A
- Concluded
equation (33) by 100 N / m z mb, giving the form
where k equals
100/Rv
and d equals
273.
Substituting the value for
R v
from table
2
gives k equal to 100/461.5, which equals
0 . 2 1 6 6 8
k g
OC/mb
m over both ice an d
water .
3
To find the
U .S
. Customary Unit equiva lent for k
,
equation
(30)
is applied to
equation (3), which gives
H
equ al to (9/5)ev/[Rv (T +
459.4)
. Temperature is
now in degre es Fah renh eit. For e
in inch es of mercu ry , this expression must be
multiplied by 211.405359 N lbm m/kg(in. Hg) ft to convert H to pounds mass pe r
cubic foot. Then equation (35) may b e written i n the form of equation 34) for
U
. S . Customary Units, where k equals (211.405359) (9/5)/461.5 or
3
0.82455
lbm OF/ (in. Hg
ft
)
over both ice and w at er . When the wet bulb
is
covered
by a thin layer of ice , the values for f and g in table 1 must be m ultiplied by 0.882,
which is the ratio of the latent hea t of evaporation to the latent heat of sublimation.
1
3
The constants for table
1
a r e now complete.
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APPENDIX
B .--EXAMPLES U S I N G TABLE 1
Determining the Proper Equation
Table
1
is used to find the proper equation to solve for a desire d quant ity. The
equation that should b e used to find a quantity should b e chosen according to the
type of data available. The forms containing T , T
,
and p a r e used for psychrom-
etric data, and the forms containing T and D ar e used for dewpoint data.
Only one
f o r m for saturation vapor pre ss ur e is given in table 1; it may be used for data of
either type .
Once the equation has been chose n, the values for its constants can be chosen
according to the temperature rang e and units of the given dat a. If T' and D are less
than or at freez ing, the section entitled Over ice is used for the forms containing
them;
otherwise the Over water section
is
use d. For the saturation vapor pr es su re
equation, the Over ice section is used for T less than
or
at freezing and the Over
water section is used for T gre ate r than freezing. The column for SI
is
used for
data in SI units, and the U .S . Customary Units column is used for data in
U .S
.
Customary Units.
Example 1
Problem:
Given an ambient pressure, p
,
of
2 9 . 7
inch es of mercury; a dry-
bulb temperature
,
T , of 75O F; and a wet-bulb temperature , T'
,
of 6 5 . 5 F
,
find
rela tive humidity
,
U .
Solution: Find the cor rec t equation in table
1
and solve it using the given data .
Relative humidity for psychrome tric data
is
require d; therefore, the correct equa-
tion form is that for U (T
,
T'
,
p)
.
The values for the constants ar e taken from the
Over water section since T' is above freezing and f r o m the column
for
U .S
.
Custom-
ar y Units since the given data a re in
U
.S
.
Customary Units. The cor rec t equation
is , therefore,
Solving this equation for the given data gives
U
equal to
0 . 6 0 3 or
6 0 . 3 percent.
Example
2
Problem: Given an ambient tem peratur e,
T
,
of
1 2 . 2 O
C and a dewpoint tempera-
ture , D
,
of - 1 0 . 6 O C , find relative humidity, U .
Solution:
Relative humidity for dewpoint data
is
required; therefore, the
cor rec t equation form
is
that for
U
(T
, D)
in table 1 . The units of the given data are
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APPENDIX
B - Concluded
in SI and
D is
below free zing; therefo re, the equation constants a re taken from the
Over ice section and the SI column in tab le
1.
The correc t equation is , therefore,
4.9283(D 273)-0 .3228610{-12 0702- 2705.21/ D+273) [2937.4/ T+273)] }
U =
(T+
273)
Solving this equation for the given data gives U equal to 0.173 or 17.3 percent.
Example 3
Problem: Given an ambient temperature
,
T , of 67.8O F and a dewpoint temper-
ature,
D
, of 63O F , find absolute humidity, H , and relative humidity, U .
Solution: For absolute humidity, the equation form is that for H (T,
D )
with the
constants taken from the Over water section since
D is
above freezing and from the
column for U .S . Customary Units since the data a re in U .S
.
Customary Units. The
correct equation for absolute humidity is, therefore,
3
H = 0.82455(T+ 459.4)-l (D + 459.4) 4.928310[23 .2801-5287.32/ D+459.4)]
Solving for the given data gives H equal to 9 . 1
X
pounds mass per cubic
foot.
For relative humidity, the equation form is that for U (T ,
D )
; however, since D
is
above freezing, the values for a , b
,
and c wil l be the same a s their corresponding
subscripted constants a l ,
b
, and el; the refo re, the reduced form of U(T,
D )
may
be us ed . That form is
1
a b [(D+d>-l- (T+ d)-l ]
U =
[ @
-
d ) / ( T d ) ]
10
The constants used ar e the same a s for
H ,
above. Th e correc t equation for relative
humidity is therefore
U = [(D
+
459.4)/(T + 459.411 -4.928310-5287 .32 (D+459.4)-1-(T+459.4)-1]
Solving for the given data gives U equal to 0.846 or
84.6
percent. A check shows
that the long form for U (T
,
D ) gives identical results.
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REFERENCES
1. Holmboe , Jorgen; Forsythe, George E . ; and Gustin,
W i l l i a m :
Dynamic Meteor-
ology. John Wiley and Sons , In c. ,
1957.
2 .
List, Robert J . Smithsonian Meteorological Tables. Sixth ed . Smithsonian
Institution Press,
1971.
3.
Marvin, C . F .: Psychrometric Tables for Obtaining the Vapor Pre ss ur e,
Relative Humidity, and Temperature of the Dew Point.
W . B . N o .
235,
U
.S
. Government Prin ting Office, 1941.
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T A B L E 1 - EQ U A TI O NS A N D C O N S T A N T S F O R F I N D IN G S A T U R A T I O N V A P O R P R E S S U R E ,
V A P O R P R E S S U R E , A B S O L U T E H U M I D I T Y , A N D R E L A T I V E H U M ID I TY
AS
F U N C T I O N S OF E I T H E R D E W P O I N T O R P S Y C H R O M E T R I C D A T A
F O R E I T H E R S I O R U
.S
C U S T O M A R Y U N I T S
(a) E q u a t i o n s
T A B L E 1 - C o n t i n u e d
03
C o n s t a n t s
O v e r w a t e r
C u s t o m a r y
U n i t s
~ . . .
. -
- 4 . 9 2 8 3 - 4 . 9 2 8 3
- 2 9 3 7 . 4 - 5 2 8 7 . 3 2
2 3 . 2 8 0 1
3 . 5 5 1 8
2 7 3 4 5 9 . 4
6 . 6 0 0
X
3 . 5 9 5 x
7 . 5 7 0
X
2 . 3 3 6 X
0 . 2 1 6 6 8 0 . 8 2 4 5 5
- 4 . 9 2 8 3 - 4 . 9 2 8 3
- 2 9 3 7 . 4 - 5 2 8 7 . 3 2
2 3 . 2 8 0 1
3 . 5 5 1 8
2
O v e r ice
U . S .
C u s t o m a r y
U n i t s
- 0 . 3 2 2 8 6
- 4 8 6 9 . 3 8
1 1 . 4 8 1 6 1 0 . 0 3 4 3
2 7 3 4 5 9 . 4
I
-~
.
- 0 . 3 2 2 8 6
- 2 7 0 5 . 2 1
6 . 6 0 0
X
1 0 f 4 3 . 5 9 5 x
7 . 5 7 0 X
1 0 8
2 . 3 3 6 X
0 . 2 1 6 6 8 0 . 8 2 4 5 5
- 4 . 9 2 8 3 - 4 . 9 2 8 3
- 2 9 3 7 . 4 - 5 2 8 7 . 3 2
2 3 . 5 5 1 8 2 3 . 2 8 0 1
'U se O v e r w a t e r s e c t i o n f o r f o r m s c o n t a i n i n g T' or D above f r e e z i n g .
Use
O v e r
ice section
f o r f o r m s c o n t a i n i n g
T' or
D a t
o r
b e l o w f r e e z i n g .
' C o n s t a n t s m u s t be m u l t i p l i e d by 0 . 8 8 2 i f w e t bulb is covered by t h i n
layer of ice.
22
. . .
. .
. . _. .... .. .-
_ _
. .
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TABLE
1
-Concluded
(c) Definitions
D
e
dewpoint
i f
above freezing , frostpoint
if
at
or
below freezing,
OC
( O F )
saturation vapor pr ess ure , mb (i n. Hg
e
H
p
T
T' wet-bulb temperature,
OC
( O F )
U
vapor pres su re, mb (in. Hg)
absolute humidity, kg/m3 (lbm/ft )
ambient pres sur e, mb (in . Hg)
ambient
or
dry-bulb temperature, OC (OF)
V
3
relative humidity, decimal value; may be expressed as a percentage
by multiplying by 100
TABLE
2
.-PARAMETERS AND VALUES USED FOR EVALUATING
TABLE 1 CONSTANTS FOR SI
Specific heat of ic e at
Oo C
(32O F ) , ci, J/ kg K
. . . . . . . . .
Specific heat of water a t 1 5 O
C
1 5 O F ) , c w , J / kg
K . . . . . . .
2 0 6 0
4185
Specific heat of water vapor at constant pre ss ur e,
. . . . . . . . . . . . . . . . . . . . . . .
J/kgK 1911
PV'
. . . . . . . . .
apor pr es su re of ice at Oo C (32O F ) , e io, mb
6.107
6.11
Lio, J / k g . . . . . . . . . . . . . . . . . . . . . . . . . 2.834 X l o 6
Lwo, J /kg . . . . . . . . . . . . . . . . . . . . . . . . 2 . 5 X l o 6
28.97
Vapor pr es su re of water at Oo
C
(32O F ) , ew o, mb
. . . . . . .
Latent heat of subl imat ion at Oo C (32O F) ,
Latent hea t of evaporatio n at O o
C
(32O F)
,
Molecular weight of dr y ai r , m d , k g m o l . . . . . . . . . . . .
Molecular weight of wate r vap or , m v, k g mol . . . . . . . . . 18.016
Specific gas constant for dry a i r , R d, J /k g
K
Specific gas constant for water vapor , Rv , J /k g K . . . . . . .
Universal gas constant, R , J/ kg mol
K
. . . . . . . . . . . .
8313.6
. . . . . . . . .
287.0
461.5
v a l u e taken from reference 2 .
NASA-Langley, 1977 H -937
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AN D SPAC E A D M I N I S T R A T I O N -
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20546 P O S T A G E A N D F E E S P A I D
N A T I O N A L A E R O NA U T IC S A N D
S P A C E A D M IN I S T R A T I O N
451
O F F I C I A L B U S I N ES S
PENALTY FOR PRIVATE USE 3 0 0 SPECIAL FOURTH CLASS..RATE
B O O K
025 0 0 1 C1
U
H 7 7 0 1 C 7 SF0903DS
DEPT
OF THP
IF F O R C E
AF W E A P O N S
L R B O R A T O R Y
ATTW :
T X C H N I C P L L I B R A R Y
( S U L )
K I R T L A W D A F E Y Y
57117
If Undeliverable Section
158
Postal
Mnniinl)
Do Not
Return
J3R
-
T he aeronautical and space activities
of
the United States shall be
conducted so us
t o
contribute . . . t o the expansion of hum an Lnowl-
edge of phenomena in the atmosphere and space. Th e Administrutio n
shall provide for the widest practicable and appropriate dissemination
of info rma tion concerning its activities and the results thereof.
-NATIONAL
AERONAUTICS
ND
SPACE ACT OF
1958
NASA SCIENTIFIC AND TECHNICAL PUBLICATIONS
TECHNICAL REPORTS: Scientific and ,
technical information considered important,
complete, and a lasting contribution to existing
knowledge.
TECHNICAL NOTES: Information less broad
in scope but nevertheless of imp ortan ce as a
contribution to existing knowledge.
TECHNICAL M EMORANDUMS:
Information receiving limited distribution
because of preliminary data, security classifica-
tion, or othe r reasons.
Also
includes conference
proceedings with either limited
or
unlimited
distribution.
CONTRACTOR REPORTS: Scientif ic and
technical information generated under a NA SA
contract
or
grant and considered an important
contribution to existing knowledge.
TECHN ICAL TRANSLATIONS: Information
published in a foreign language considered
to merit NASA distribution in English.
SPECIAL PUBLICATIONS : Information
derived from
or
of value t o NA SA activities.
Publications include final reports of major
projects, monographs, data compilations,
handbooks, sourcebooks, and special
bibliographies.
TECHNOLOGY UTILIZATION
PUBLICATIONS: Information on technology
used by NASA that may be of particular
interest in commercial and other-non-aerospace
applications. Publications include Tech Briefs,
Technology Utilization Reports and
Technology Surveys.
D e t a i l s o n t h e a v a i l a b i l i t y
o f t h es e p u b l i c a ti o n s m a y b e o b t a i n e d f r o m :
SCIENTIFIC AND TECHNICAL INFORMATION OFFICE
N A T I O N A L A E R O N A U T I C S A N D SPA CE A D M I N I S T R A T I O N