Equations for Determining Humidity from Dewpoint and Psychrometric Data

8/10/2019 Equations for Determining Humidity from Dewpoint and Psychrometric Data

1/26

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


2/26

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


3/26

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.


4/26

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 )

2


5/26

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) .

3


6/26

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 .

4


7/26

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)

5


8/26

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

6


9/26

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

7


10/26

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 .

8

H-937


11/26

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


12/26

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


13/26

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.

11


14/26

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

H-937


15/26

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:

H-937

13


16/26

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


17/26

, ,

.

.

_.

.

.

. . ..

.

.....

.

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


18/26

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

. ..

.,

.

...

H-937


19/26

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

H-937

1 7


20/26

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.

18


21/26

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

1 9


22/26

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.

20


23/26

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.

H-937

2 1


24/26

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

. . .

. .

. . _. .... .. .-

_ _

. .


25/26

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

23


26/26

AN D SPAC E A D M I N I S T R A T I O N -

W ASH I N G T O N . D . C .

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.

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SPACE ACT OF

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Equations for Determining Humidity from Dewpoint and Psychrometric Data

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