Instrumentation & Measurement Magazine 23-3 - 11

as a flow standard. Three
existing ports on the BBB
were adapted to contain
two microwave antennas,
an acoustic speaker, and an
acoustic microphone (one
antenna and the speaker
share the same port) [3].

Volume Determined
by Microwave
Resonances
With a test gas-such as argon or nitrogen-filling the
BBB, a simplified equation
for a microwave resonance
frequency fmicro is given by:
	

a=

Fig. 2. Temperature differences as a function of time as measured on the BBB after filling to 7 MPa. The temperatures
are read from four thermistors placed as shown. Uniform temperature is reached after about 2 days (credit: NIST's Fluid
Metrology Group).

ξ micro  c0
 
2π fmicro  ng



  	(1)



where a is the radius of the BBB inner volume, c0 is the speed
of light in vacuum (which was defined exactly in 1983), ng is
the refractive index of the gas filling the BBB, and ξmicro is an
exactly calculable number related to the BBB's geometry. The
ratio in parentheses is the speed of microwaves in the gas. The
volume needed is Vmicro = 4 π a3. Since the volume is a weak func3
tion of temperature T (due to thermal expansion) and internal
pressure p, measurements of fmicro were made at a range of filling pressures from zero to 7 MPa and over a small range of
temperatures (determined by thermistors placed on the outer
surface of the BBB). The refractive index ng at the microwave
frequencies used has been inferred from tabulated values of
the dielectric constants of the test gas as a function of temperature and pressure. As a result, the inner volume of the BBB is
known as a function of temperature and pressure to about two
parts in 104, estimated at 95% confidence.

Mass of Fill Gas Determined by Acoustic
Resonances
The speed of sound w in the gas is measured by acoustic resonances [3], [4]. To take the simplest case of the gas being in
equilibrium at known pressure p and temperature T, the measured speed of sound is given by:

	

w=

(

facoust 6π 2 Vmicro

ξacoust

)

1
3

,	(2)

where facoust is the frequency of the measured acoustic resonance and ξacoust is an exactly known parameter (but different
from ξmicro) [3]. Once w is known in m/s, the density ρ in kg/
m3 for a particular gas as a function of p and w can be inferred
from standard tables so that, finally, the mass of gas in the inner
volume Macoust as determined from acoustic and microwave
resonances is given by:
May 2020	

	

Macoust = Vmicro  ρ ( p , w ) .	(3)

For the simplified equations it is assumed that the test
gas within the BBB is at a uniform temperature, determined
by thermistors placed on the outer wall. This is not usually
the case. As Fig. 2 demonstrates, filling the sphere from 1 atmosphere to 70 atmospheres by passing the gas through a
tube changes the temperature of the inflowing gas. The temperature distribution within the gas then stratifies due to
convection, with temperature increasing from bottom to top.
The approach to equilibrium is slow as inferred from the
thermistors. However, inferring the temperature from the
acoustic modes is much faster because of an inherent averaging effect [3], [4].
In [4], Pope et al. take the remaining step of calibrating
secondary flow standards using this system, including determining the time derivative of Macoust, which is "dynamic" flow.
Gas in the BBB is made to flow through the secondary standard
under calibration. The authors also describe many cross checks
with other techniques to support uncertainty claims. One of
these tests is a calibration of secondary flow standards using
the BBB as the primary standard, against results obtained using NIST's current primary standard [2]. The authors show that
dynamic flow can be conveniently monitored by measuring a
particular acoustic resonance as a function of time. To do this,
however, the resonance frequency must be acquired quickly; a
novel positive-feedback circuit has been developed to speed up
this measurement. Future steps are also presented in [4].
So, it seems possible to replace a classic flow system based
on pVTt (pressure, volume, temperature, and time) with a
system based on pVwt, where the temperature T of the gas is
replaced by the speed of sound w measured within the gas using acoustic resonances, and the volume V is determined by
microwaves resonances.

Using Light to Measure Pressure
The SI unit of pressure has the special name "pascal," symbol Pa. In terms of the base units of the SI, Pa = kg m−1 s−2. The

IEEE Instrumentation & Measurement Magazine	11



Instrumentation & Measurement Magazine 23-3

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