Campbell CR10 PROM Instruction Manual download pdf (Page 27)

C-1

APPENDIX C. CSAT3 MEASUREMENT THEORY

C.1 THEORY OF OPERATION

C.1.1 WIND SPEED

Each axis of the CSAT3 pulses two ultrasonic

signals in opposite directions. The time of flight

of the first signal (out) is given by:

t

d

cu

o

a

=

+

(1)

and the time of flight of the second signal (back)

is given by:

t

d

cu

b

=

-

a

(2)

where t

o

is the time of flight out along the

transducer axis, t

b

is the time of flight back, in

the opposite direction, u

a

is the windspeed

along the transducer axis, d is the distance

between the transducers, and c is the speed of

sound.

The windspeed, u

a

, along any axis can be found

by inverting the above relationships, then

subtracting Eq. (2) from (1) and solving for u

a

.

u

d

tt

a

ob

=−

















2

11

(3)

The windspeed is measured on all three non-

orthogonal axis to give u

a

, u

b

, and u

c

, where the

subscripts a, b, and c refer to the non-

orthogonal sonic axis.

The non-orthogonal windspeed components are

then transformed into orthogonal windspeed

components, u

x

, u

y

, and u

z,

with the following:

u

u

u

A

u

u

u

x

y

z

a

b

c





















=





















(4)

where A is a 3 x 3 coordinate transformation

matrix, that is unique for each CSAT3 and is

stored in ROM memory.

C.1.2 TEMPERATURE

The sonically determined speed of sound can

be found from the sum of the inverses of Eq. (1)

and (2).

c

d

tt

ob

=+

















2

11

(5)

The speed of sound in moist air is a function of

temperature and humidity and is given by:

()

c P RT RT q

dv d

2

1061

== = +γργ γ

.

(6)

where

γ

is the ratio of specific heat of moist air

at constant pressure to that at constant volume,

P is pressure,

ρ

is air density, R

d

is the gas

constant for dry air, T

v

is virtual temperature, T

is the air temperature, and q is the specific

humidity defined as the ratio of the mass of

water vapor to the total mass of air (Kaimal and

Gaynor, 1991; Wallace and Hobbs, 1977).

Note that

γ

is a function of specific humidity. It

would be convenient if the effects of humidity

could be consolidated into one term.

The specific heats for moist air at constant

pressure and volume are given by:

C

p

=+−qC q C

pw pd

()1

=+Cq

pd

(.)1 0 84 (7a)

C

v

=+−qC q C

vw vd

()1

=+Cq

vd

(.)1 0 93 (7b)

where C

p

and C

v

are the specific heats of moist

air at constant volume and pressure, C

pw

and

C

vw

is the specific heat of water vapor, and C

pd

and C

vd

is the specific heat of dry air,

respectively (Fleagle and Businger, 1980).

Substitute Eq. (7a) and (7b) into (6) and ignore

the higher order terms. This yields

cRTRT q

dds dd

2

1051== +γγ(.)

(8)

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