Liquid storage gauging method and apparatus

The density, volume and mass of fuel held in an aircraft tank are measured by a system of four pressure sensors interfaced with an associated microcomputer. The sensors are disposed in a predetermined array at unequal depths below the surface plane of the contained fuel. Electrical transducers associated with the pressure sensors develop electrical signals representing the associated pressures P1, P2, P3, and P4 and these signals are processed by the microcomputer which is programmed to correlate the individual pressure signals and known coordinate locations of the sensors with certain unknown parameters including the fuel density .rho., and the orientation and distance of the fuel surface plane relative to each pressure sensor. These relationships establish four simultaneous equations that are processed by the programmed microcomputer to yield simultaneous solutions for the density .rho., and for the position of the fuel surface plane. Further processing by the programmed microcomputer of the signals representing the position of the surface plane yields output data signals representing the volume of the fuel, i.e., that which fills the tank up to the now-determined plane of the fuel surface level. Thereafter the previously measured density .rho. is automatically multiplied in the microcomputer by the determined volume of fuel to yield data signals representing the fuel mass. The resulting signals for the density .rho., volume V and mass M of the fuel are displayed on readout devices responsive to output signals from the microcomputer. In an alternative embodiment, a system of three pressure sensors is employed in a similar manner to produce information signals representing volume V and mass M of the fuel when the density .rho. is either known or assumed.

BACKGROUND 
The invention generally relates to liquid gauging systems, and it more 
particularly pertains to method and apparatus for measuring the density, 
volume and/or mass of fuel contained in a partially filled tank such as an 
aircraft aviation fuel tank. 
While the invention disclosed hereinafter has general application to the 
technology of liquid gauging, the preferred embodiment of the invention 
concerns the problems and requirements that are present in measuring fuel 
contained in aircraft tanks. Aviation fuel tanks are of varied and complex 
shape and are generally mounted in the wings and fuselage with the shape 
of the tank being dictated by the profile of the aircraft structure. The 
irregular geometry of the tanks and the different attitudes that can be 
assumed by the aircraft at any given instant that the fuel level is 
monitored, are factors that contribute to the difficulty of accurately 
measuring the remaining quantity of fuel. Additionally, the aircraft tanks 
are routinely filled from a variety of aviation fuel sources, and the 
density of fuel from these different sources can and typically does vary 
widely. Since the fuel density is related to the energy content in any 
given unit volume of fuel, it is important to know not only the volume of 
the remaining fuel but also its density and hence mass or energy content, 
thus adding an additional design complication to the developement of 
suitable gauging instrumentation. 
One of the more common fuel metering systems uses a plurality of electrical 
capacitance sensors. Each such sensor comprises a cylindrical open pipe 
disposed vertically in the tank and surrounding a coaxial center 
conductor. An AC electrical signal is applied across the outer pipe and 
center conductor. The fuel fills the space between the inner wall of the 
cylindrical pipe and the center conductor to the surrounding level of fuel 
in the tank, thus providing a different (larger) dielectric constant for 
the portion of the sensor that is submerged below the fuel level compared 
to the unsubmerged portion, where the dielectric constant of air causes a 
lower effective dielectric. The plurality of sensors in a given tank are 
wired in parallel to sum the overall capacitances and this sum is then 
correlated to the total fuel content. Usually the capacitance sensors are 
tailored to a particular tank geometry by shaping the inner center 
conductor, although such shaping does not always precisely match the 
sensor output to the content of the remaining fuel. Furthermore, such 
capacitance sensors are usually not augmented with devices for measuring 
the density of the fuel and hence deviations in the mass (energy content) 
of the fuel due to variations in density are not measured. Rather, an 
average fuel density is assumed or estimated and that value is used to 
compute the mass of the fuel. 
The capacitance-type sensors are frequently impaired by corrosion, microbe 
growth and other contaminants present in the fuel which introduce 
deviations in the dielectric constant and hence in the measurement 
performed by the sensor. Electromagnetic interference may also cause the 
capacitance sensors to deviate from a true fuel content measurement. 
Different pitch and roll attitudes of the aircraft introduce variations 
between the actual remaining fuel content and the amount of fuel as 
measured by capacitance sensors and the relations between various 
attitudes and fuel measurements are not readily correlated and accounted 
for in the instrumentation. These factors, which contribute to imprecise 
fuel content measurements, are compensated for by carrying additional fuel 
in each of the tanks using worst-case assumptions, sufficient to more than 
compensate for tolerances in the fuel measurement instrumentation based on 
capacitance sensors. 
SUMMARY OF THE INVENTION 
According to one aspect of the invention, method and apparatus are provided 
for simultaneously measuring the density, volume and mass of liquid held 
in a container, such as aviation fuel carried in an aircraft tank. The 
method comprises the steps of measuring pressures, by suitable pressure 
sensors, at four or more sensor points of known coordinate locations, such 
locations being disposed in an array at different distances beneath the 
surface plane of the liquid. These four pressure measurements, P1, P2, P3, 
and P4, are processed by a microcomputer, which is programmed to establish 
a series of simultaneous relationships, relating the value of the pressure 
at each sensor location to various unknown parameters including the 
density .rho. of the liquid and the position (orientation and distance to 
sensors) of the surface plane. Simultaneous solutions to the various 
relationships based on the pressure measurements, P1, P2, P3, and P4, are 
obtained and the microcomputer produces signals representing density 
.rho., and the orientation of the surface plane and its distance from the 
locations of the pressure measurements. Further processing of these output 
signals by the microcomputer yields the volume of the contained liquid 
(amount of liquid that fills the tank up to the now-located surface plane) 
and the mass of the liquid (determined by multiplying the measured density 
.rho. times the measured liquid volume). The resulting signals 
representing the density .rho., volume V, and mass M of the liquid are 
displayed on readout devices and/or are stored in volatile or permanent 
memory devices coupled to the output ports of the microcomputer. 
The apparatus comprises the array of pressure sensors mounted on a bracket 
that in turn is anchored inside a liquid storage tank and arranged as 
described above in the summarized method, transducer means for converting 
the sensed pressures P1, P2, P3, and P4 into representative electrical 
signals, signal input means for feeding such pressure signals into a 
programmed microcomputer that is programmed in accordance with the 
above-summarized method, and suitable readout devices for displaying the 
measured density .rho., volume V, and mass M. 
The above-summarized method and apparatus advantageously measure the volume 
V and mass M by taking into account variations in density .rho.. Other 
advantages include the adaptability of the method and apparatus to make 
measurements in tanks of varied and complex shapes and immunity to such 
environmental conditions that are common problems for liquid, and 
particularly fuel-metering systems, including corrosion, microbe growth, 
the presence of other contaminants, and electromagnetic interference. 
Another advantage of this method and aparatus, when used for measuring 
fuel reserve in aircraft tanks is an automatic accountability for various 
pitch and roll conditions of the aircraft during gauging. 
Another aspect of the invention is to provide a method and apparatus 
similar to those summarized above, but employing a minimum of three or 
more pressure sensor locations and associated sensors, and processing 
representative pressure signals P1, P2, and P3 to yield output signals of 
liquid volume V and mass M, when the density .rho. is either known or 
assumed. In this case, the required number of sensors is less because the 
density is known, thereby eliminating one variable and hence the need for 
one of the simultaneously solvable relationships.

DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENT 
With reference to FIG. 1, the preferred form of the invention is to provide 
for measuring density, volume and mass of fuel contained in an aviation 
fuel tank 11, which is typically of irregular geometry, because of being 
shaped to fit inside the profile of the aircraft wing, fuselage or other 
body section. Further complicating the gauging methodology is the fact 
that the density .rho. of the fuel 12 contained by tank 11 varies 
considerably depending upon its source, the ambient temperature and 
pressure, and other variables outside the control of the aircraft user. As 
mentioned above, variations in fuel density are manifested by different 
energy content per unit of volume of fuel and hence it is necessary to 
measure and account for the actual density .rho. in determining the 
effective quantity of the fuel reserve in tank 11. The density .rho. may 
vary by as much as 0.2 to 0.3 pounds per gallon, and a typical range is 
from approximately 6.2 pounds per gallon to 6.7 pounds per gallon. Such a 
difference in density can have a significant effect on the available 
energy content of the fuel reserve. 
To measure density along with volume and mass, an array 14 of pressure 
sensors is mounted within tank 11 near the bottom so as to be submerged 
below the fuel surface as long as a minimum reserve quantity of fuel 
remains in tank 11. Array 14 incorporates a minimum of four separate 
pressure sensors shown more clearly in FIG. 2 as sensors S1, S2, S3 and S4 
which are mounted on a triangular shaped plate 20 which in turn is secured 
to an interior wall of tank 11 by a bracket 22 so that the sensors are 
fixed relative to the walls of tank 11 and hence relative to the tank 
geometry. The importance of fixing the location of array 14 relative to 
the tank geometry will be explained below. Sensors S1 through S4 are 
actually sensing tubes ported at a lower, sensing end so as to communicate 
thereat with fuel 12 with the resulting pressure of the fuel at that 
location being in turn transmitted by air pressure through lengths of 
flexible tubing 24, one per sensor, to a bank of pressure-to-electric 
signal transducers 26 individually designated TS1, TS2, TS3, and TS4, 
respectively. As described more fully herein, transducers 26 compare the 
liquid pressure of the fuel at the ported ends of sensors S1 through S4 
with ambient pressure and produce corresponding electrical analog signals 
represented as P1a, P2a, P3a, and P4 a. The pressure signals are adjusted 
for temperature variations by proper coeeficients calculated by the 
microcomputer 30 using temperature measurements available from temperature 
sensors mounted with transducers 26. 
Sensors S1, S2, S3, and S4 are preferably of a differential, oscillating 
quartz type in which the fluid pressure present at the open-ended sensing 
ports S1 through S4 of tubing 24 is applied to a mechanical device that 
responds to a pressure differential to develop a proportional force that 
in turn is exerted against a quartz crystal force sensor. The quartz 
crystal force sensor, which is also in communication with ambient 
pressure, oscillates at a frequency that varies in a predictable manner in 
reaction to the applied differential force and variations in the frequency 
of oscillations are calibrated to a high degree of pressure-measuring 
accuracy. This combination of the sensor and transducer is known per se 
and is described in Measurements and Data, Issue No. 56, Vol. 10, No. 2, 
March/April 1976. A device for measuring the pressure in this manner can 
be calibrated to achieve a measurement tolerance of one-tenth of a percent 
of the actual pressure. While other pressure sensors may be used, the 
relatively high accuracy of this type sensor-transducer combination makes 
it preferred for use in the apparatus disclosed herein. 
A set of analog-to-digital converters 28 convert the analog signals 
corresponding to the sensed pressures into digital form with the resulting 
digitized signals being represented as P1d, P2d, P3d, and P4d. Such 
signals provide the variable inputs to a programmed microcomputer 30. 
While a number of commercially available devices are suitable for use as 
microcomputer 30, as shown in more detail in FIG. 4 this particular 
embodiment incorporates a microprocessor 30a manufactured and available 
from California Computer Systems or S.D. Computer Products as their Model 
280 microprocessor, cooperating sets of input/output (I/O) ports 30b and 
30c, a 64K byte random-access memory (RAM) 30d, and dual disc drives 30e 
having a capacity of 500K bytes per disc, all linked by data and control 
bus 30f. 
In accordance with the principles of the invention, microcomputer 30 
cooperates with the array 14 of pressure sensors to provide a sufficient 
number of pressure readings and hence digitized pressure signals P1d 
through P4d so that a series of simultaneously solvable equations is 
established within the microcomputer to thereby allow for simultaneous 
determination of density (.rho.), volume (V) and mass (M) of the fuel. The 
magnitude of one or more of these parameters is then indicated on readout 
devices 32. The plurality of pressure measurements is processed in the set 
of simultaneously solvable equations to first determine density .rho.; 
then further processing yields the orientation of the surface plane 16 of 
fuel 12 and the distance of such plane from array 14; form the now 
established position of plane 16, the volume V of fuel 12 that fills tank 
11 up to the surface plane is computed and; finally, the mass M of fuel 12 
is computed from the product of the predetermined density .rho. and volume 
V. 
Devices 32 may be conventional LED displays and if desired a recording 
device (not shown) may be connected in parallel with devices 32 to make a 
record of variations in .rho., V, and M as a functon of flight time, to be 
used in post-flight analysis. 
Now, with reference to FIG. 3, array 14 of sensors S1 through S4 is 
disposed in tank 11 so that the various sensors are at nonequal but 
unknown depths D1, D2, D3, and D4 beneath the surface plane 16 of fuel 12. 
The depths D1 through D4 are oriented along vectors that are assumed to be 
normal to the orientation of surface plane 16. However, the precise 
orientation of surface plane 16 is an unknown variable and is 
simultaneously determined along with the distances of depths D1-D4 at the 
time that the gauging system measures the variables of density, volume and 
mass. 
More particularly, to determine the orientation of surface plane 16 
relative to array 14, the following relationships are used. First, the 
scalar (without direction) distances or depths D1 through D4 of each of 
the four sensors relative to the surface plane 16 of the fuel can be 
represented by the relationships 
EQU Di=Pi/.rho., 
where i=1, 2, 3, and 4. Since the orientation of surface plane 16 is also 
unknown at this phase of the measurement process, the vector directions of 
such distances Di are unknown; however, it is known that the scalar value 
of each of these distances defines an imaginery spherical surface shown by 
the dotted lines 34 in FIG. 3 and that collectively these four spherical 
surfaces must contact the common surface plane 16 at different tangential 
contact points indicated at 36. Secondly, the array 14 of sensors S1 
through S4 is arranged so that nominally none of the distances or depths, 
D1, D2, and D3, and D4, are equal. By making the depths of the various 
sensors unequal, a set of simultaneous relationships may be established, 
as set forth below, and in which the solution for density .rho. depends on 
the differences between the various pressure readings at S1, S2, S3, and 
S4. 
DETERMINING DENSITY FROM PRESSURES AT FOUR KNOWN POINTS IN FUEL TANK WHEN 
THE DENSITY .rho.IS UNKNOWN 
Equation of liquid plane: 
EQU n.multidot.r=d 
n=unit vector normal to plane 
r=point on plane 
d=perpendicular distance (minimum distance) of plane from coordinates 
system origin, 
where pressure sensor locations=r.sub.i, pressure readings=Pi, and 
density=.rho.. 
Then: 
##EQU1## 
or: 
##EQU2## 
The relative sensor locations are defined as: 
r.sub.1 1. choose as origin (0, 0, 0) 
r.sub.2 2. choose on x-axis (x.sub.2, 0, 0) 
r.sub.3 3. choose on z=0 plane (x.sub.3, y.sub.3, 0) 
r.sub.4 4. choose outside of xy, yz, and zx planes (x.sub.4, y.sub.4, 
z.sub.4) 
and then system of equations is: 
##EQU3## 
where n=(n.sub.1, n.sub.2, n.sub.3) and d are unknown. 
Define the 4.times.4 matrix Ma as: 
##EQU4## 
Note that Det(Ma)=x.sub.2 y.sub.3 z.sub.4, and that the inverse of Ma is 
Ma.sup.-1, or 
##EQU5## 
Explicitly the inverse of matrix Ma can be written as: 
##EQU6## 
and then: 
##EQU7## 
Note that n depends on the differences of pressures. 
From the relationship: 
EQU n.sub.1.sup.2 +n.sub.2.sup.2 n.sub.3.sup.2 =1, 
the density .rho. can be determined as functions of pressure differences: 
EQU .sup..delta..sub.2 =P .sub.2 -P .sub.1 
EQU .sup..delta..sub.3 =P .sub.3 -P .sub.1 
EQU .sup..delta..sub.4 =P .sub.4 -P .sub.1, 
and 
##EQU8## 
The value of the density .rho. is thus measured as a function of 
differences in the pressures P1, P2, P3, and P4 and of the coordinate 
locations of the sensors within tank 11. The processing for density .rho. 
is performed in microcomputer 30 (FIG. 1) in accordance with the flow 
charts of FIG. 7, described more fully below. Upon completing the signal 
processing for density .rho., microcomputer 30, as shown in FIG. 1, 
produces an output signal that causes the computed density to be displayed 
on readout devices 32 and, if desired, suitable recording device may be 
used as mentioned above to provide a permanent or semipermanent record of 
variations in density .rho.. Additionally, the computed density value is 
made available within microcomputer 30 for determining the mass of fuel 12 
after the pressure measurement signals are processed to locate the 
position of the surface plane 16 of fuel 12 as discussed below. 
The apparatus and process described above for determining the density .rho. 
can be compared with a hydrometer. One of the characteristics of a 
hydrometer is that it will automatically assume an orientation along the 
gravitational force and hence normal to the surface plane of the measured 
liquid. In contrast, the apparatus and method described above yield a 
measurement of density .rho. without producing the hydrometer-like 
indication of the gravity direction and the corresponding orientation of 
the surface plane 16. In the above apparatus and method, the orientation 
of plane 16 must be separately determined in order to find the volume V of 
the contained fuel 12, and then from the product of the volume V and 
density .rho., a measure of the mass M of the fuel. 
To determine the orientation of surface plane 16 as a prerequisite to 
measuring the volume V, microcomputer 30 of FIG. 1 includes a routine for 
locating the position of plane 16 (orientation and distance from array 14) 
as a function of the measured pressures P1, P2, P3, and P4 and the 
coordinate locations x,y, and z of each of the respective sensors S1, S2, 
S3, and S4. In particular, each of the components of vector n are 
determined from the above matrix Ma as a function of the now computed 
density .rho. and four pressure values P1, P2, P3, and P4. The vector n 
equals n.sub.1 x+n.sub.2 y+n.sub.3 z, where x y and z are unit vectors, 
thereby defining the orientation of the surface plane 16 relative to the 
reference coordinate system x, y, and z within which tank 11 and the 
sensors of array 14 are located and the walls of tank 11 are defined. 
Similarly, the distance d of the surface plane 16 from the array 14 is 
represented by the fourth relationship of the above matrix Ma. The 
quantities n and d thereby define the position of plane 16 within the 
established coordinate system x, y, and z. 
To determine volume V of fuel 12 from the now established position of 
surface plane 16, it is necessary to relate plane 16 to the geometry of 
tank 11. In other words, it is necessary to determine how much volume of 
tank 11 is filled by fuel 12 up to the now known position of plane 16. Two 
methods (and the corresponding apparatus) are disclosed herein for making 
this volume determination. A first approach is based on the formulation of 
a mathematical relationship between the position of surface plane 16 and 
the position of the bottom and sidewalls of tank 11 relative to the 
established coordinate system x, y, and z. This initial scheme represents 
a suitable approach when the tank containing the fuel or other liquid has 
a regular geometry that is easily defined by simple mathematical 
statements. For example, a regular polygonal shaped (rectangular walls at 
right angles) tank is adapted to use of this first approach. The second 
technique for relating surface plane 16 to the partially filled volume of 
tank 11 has more general application and is preferred for a tank 11 having 
irregular geometries such as typically found in the case of aviation fuel 
tanks, including wing and fuselage tanks. This second approach is based on 
an empirically derived correlation between various positions of surface 
plane 16 and the corresponding partial volume of tank 11 bounded on the 
top by such surface plane. Predetermined empirical relationships between 
the surface plane and the tank volume are stored in a memory, such as the 
dual disc drives 30e of microcomputer 30 (FIG. 4) and once the position of 
surface plane 16 is computed by microcomputer 30, the correlative volume V 
is retrieved from memory. 
Considering now in greater detail the first of these two approaches, a 
regular polygon-shaped tank having rectangular top, bottom, and sides all 
at right angles, exhibits a geometry from which a partial volume can be 
computed mathematically from the points at which the surface plane 16 
intercepts the vertical corners of the tank. It will be observed that 
surface plane 16 will define four intercepts (points along the vertical 
edges of the polygonal tank) and that all four such intercepts lie in the 
common surface plane. from these intercepts, the volume is computed by a 
program stored in microcomputer 30 which assumes that the unknown tank 
volume is formed by a plurality of interfitting, pyramidal sections as 
shown in FIG. 6 of the drawings and for which the volume of each pyramidal 
section is represented by the formula: 
EQU V.sub.i =1/3A.sub.i h.sub.i 
where i=1,2,3-n number of pyramidal sections, A.sub.i is equal to the area 
of the pyramid base of each corresponding section, and h.sub.i is equal to 
the height of the corresponding pyramidal section. The sum of all of the 
section volumes: 
EQU V=V.sub.1 +V.sub.2 +V.sub.3 -V.sub.n 
is equal to the volume contained within the tank below the surface plane. 
The areas A.sub.i are predetermined from the bottom wall area of the 
regular polygonal tank and stored as fixed data in the microcomputer and 
the heights h.sub.i are the distances along the vertical edges of the tank 
corners measured between the bottom of the tank and the intercept points 
with the precomputed position of the surface plane. This volume 
computation procedure is described more fully below in connection with the 
alternative embodiment shown in FIGS. 5 and 6. 
It will be appreciated that the foregoing computational method based on 
pyramidal sections is not preferred for an irregular tank geometry such as 
wing tank 11 shown in FIG. 1. For such an irregular tank shape, the 
following empirical approach to determining the partial volume of fuel 12 
is preferred. Thus, with reference to FIG. 1, tank 11 is initially 
calibrated by incrementally adding volumes of fuel 12 of known density 
(and hence mass) and taking and storing pressure measurements at S1, S2, 
S3, and S4 as each fuel volume increment is added. Also, measurements are 
made of the locations where the surface plane 16 intercepts the tank edges 
as indicators of the position of plane 16 relative to the x, y, and z 
coordinates. The resulting pressure measurements and tank intercepts are 
fed to microcomputer 30 along with data characterizing the cumulative 
amount of fuel. Since the density is known, data developed by the 
calibration processing of tank 11 measures correlative values for the 
accumulated volume V of fuel 12 and the pressures P1, P2, P3, and P4. This 
data is stored in a memory device, such as the dual disc drives 30e of 
microcomputer 30 as shown in FIG. 4 and represents the position of the 
surface plane 16 of fuel 12 as three or more coordinates in the x, y, and 
z reference system and the correlative values of both the accumulated 
volume of fuel 12 and the pressures P1, P2, P3, and P4. 
A rough calibration of tank 11 may be performed with the assumption that 
the fuel 12 will be measured only when the aircraft is in level flight or 
on the ground with tank 11 assuming a nominal or level orientation. In 
such case, the calibration process need only take into account this level 
orientation of tank 11 and ignore variations in the pitch and roll of the 
aircraft. In a more accurate and preferred embodiment, the calibration of 
tank 11 is performed not only for a nominal attitude of the aircraft (zero 
pitch), but also for variations in pitch from approximately -3 degrees 
through +8 degrees. This is accomplished by mechanically tilting the tank 
at various increments between these limits and then recording the volume 
and pressure measurements as described above. A suitable computer 
processing routine for this procedure is shown in FIG. 9 as discussed 
hereinafter. The above range of attitude variations accounts for most of 
flight and ground attitudes at which measurements of the fuel reserve are 
made. Variations in roll of the aircraft may be calibrated in a similar 
fashion. However, it is noted that gauging of the fuel tanks normally is 
not needed during the limited times that the aircraft is in a roll. 
In the presently preferred embodiment, it is assumed that the direction of 
gravity during gauging of fuel tank 11 is earthward and does not change 
when the attitude of the aircraft varies through the angles of pitch 
mentioned above. However, if desired, the embodiments of the invention 
disclosed herein can be augmented by method and apparatus for 
independently measuring the direction and magnitude of the instantaneous 
gravitational force on the aircraft and the corresponding vector quantity 
can be processed in microcomputer 30 along with the above-mentioned 
calibration data to compensate for directions and magnitudes of gravity 
other than the normal earth gravity. 
Once tank 11 has been calibrated in the foregoing manner, the volume V for 
any quantity of reserve fuel 12 will be available in microcomputer 30 by 
retrieving the volume as a function of the measured pressures P1, P2, P3, 
and P4 and the previously measured density .rho.. A signal representing 
the computed volume V in now made available for display by readout devices 
32 (FIG. 1). 
The remaining quantity to be gauged, mass M, is determined by microcomputer 
30 from the product M (mass)=.rho..times.V. The resulting signal 
representing mass M is then applied to readout devices 32 for display. 
It is observed that the accuracy of the method and apparatus is dependent 
on the degree of depth separation established for pressure sensors S1 
through S4 of array 14. The greater the relative separation of these 
sensors (along a direction generally normal to the anticipated orientation 
of the surface plane 16), the greater accuracy of the measurements. 
However, there is a constraint on the amounts of separation at which the 
sensors are deployed inasmuch as all of the of sensors must be submerged 
at all times. This condition requires a minimum reserve of fuel below 
which the gauging method does not work. Thus, array 14 is generally 
confined to the lowermost regions of tank 11 and it is only within this 
limited volume that the depth separations of sensors S1 through S4 can be 
realized. 
With reference to FIG. 5, an alternative embodiment is illustrated in which 
a lesser number of pressure sensors S1', S2' and S3' are deployed in an 
array 14' to measure the volume and mass of the remaining fuel 12' under 
conditions in which the density .rho. is known or assumed. In such case, a 
series of three simultaneously solvable relationships is established 
between the pressures measured at the three sensors of array 14', at 
different relative depths with respect to surface plane 16', and then 
solved to yield the orientation and distance of plane 16' with respect to 
a coordinate system of x, y, and z, in a fashion similar to the foregoing 
description of the embodiment of FIG. 1. The location coordinates for the 
sensors S1', S2', and S3' of array 14' are predetermined. The pressures 
sensed by array 14' are communicated over tubing 24' to a bank of 
pressure-to-electrical transducers 26' which produce analog pressure 
signals that are in turn converted by analog-to-digital converters 28' 
into digital pressure signals P'1d, P'2d, and P'3d as shown. Microcomputer 
30' receives the digitized pressure values and also receives a digital 
input representing the known or assumed density .rho. of fuel 12' and 
performs the following processing 
The liquid or fuel surface plane 16' is again related, using vector 
notation, to the coordinate locations and pressure readings of sensors 
S1', S2', and S3' and with the density .rho. being known, three equations 
are established, as set forth below, which when simultaneously solved 
yield the position of plane 16'. The corresponding volume of fuel 12' is 
then computed by mathematically modeling the corresponding volume as 
illustrated in FIG. 6 in which the intercepts B, C, E, and G of surface 
plane 16' with the corner edges of tank 11' define different heights of a 
series of interfitting pyramidal segments. Collectively these 
pyramid-shaped segments constitute the volume of fuel 12'. Individually 
the pyramid segments have volumes that are readily determined using the 
relationship between the volume of a pyramid, the area A of the base of 
the pyramid, and its height as shown by the following mathematical 
formulations. 
DETERMINING VOLUME AND MASS OF FUEL FROM PRESSURES MEASURED AT THREE TANK 
LOCATIONS UNDER CIRCUMSTANCES IN WHICH DENSITY IS KNOWN OR ASSUMED 
Note that the geometry of tank 11' consists of seven planes: six forming 
the permanent tank geometry and one representing the surface 16' of the 
liquid, or in this case fuel, 12'. 
The volume calculation is based on the following observation: 
EQU V=.intg.d.sup.3 .gamma.=1/3.intg..gradient...gamma.d.sup.3 
.gamma.=1/3.intg..gamma..dA 
(the latter step from the integral divergence theorem). Thus, 
EQU V=1/3.SIGMA.d.sub.i A.sub.i 
where 
A.sub.i =base of pyramid from some reference point, and 
d.sub.i =distance of base from reference point. 
Each base of a pyramid is really one of the above seven planes. Therefore, 
if the reference point is permanently chosen at the intersection of three 
of the permanent tank planes (sides), three of the pyamids are eliminated 
because the corresponding d.sub.i= 0. This leaves at most four pyramidal 
volumes to be computed. 
SPECIFICATION OF PLANES 
The equations for the characterizing planes are written in normal form as: 
EQU n.multidot..gamma.=d 
where 
.sub.n =unit vector normal to plane pointing out from liquid volume volume 
( over a vector implies a unit vector), 
.gamma.=point on plane, and 
d=distance of plane from origin of coordinate system. 
SPECIFICATION OF LINES 
A line is given by the equation: 
EQU r=r.sub.o +cs 
where 
.sub.r =variable point on line, 
.sub.r =fixed point on line, 
.sub.c =unit vector along line, i.e., direction cosines of line, and 
s=distance along line from fixed point. 
DETERMINATION OF LIQUID SURFACE PLANE 
The liquid surface plane is thus: 
EQU n.sub.4 .multidot.r=d.sub.4 
The measurements of pressure are made Pi, i=1, 2, 3. These define spherical 
distances Pi/.rho. (.rho.=density), which must be on liquid plane, i.e., 
##EQU9## 
where r.sub.c represents the points where pressure sensors are mounted. 
Then 
##EQU10## 
These three equations together with the equation: 
EQU n.sub.4.sup.2 =1 
are sufficient to determine n.sub.4 and d.sub.4. 
PYRAMID BASE AREAS 
The pyramid bases are in general N-gons where N ranges from 3 to 6. That 
is, the pyramid bases can be triangles, quadrilaterals, pentagons, or 
hexagons. These base figures are determined by their vertices, which form 
a set of points defined by the intersections of the relevant planes. Part 
of this set of points consists of the points of intersections on the 
permanent tank structure. The remainder of such set of points is derived 
from the intersections of the liquid surface plane with the tank 
structure. These points can be enumerated as follows: 
1. Seven permanent points (there are really eight but one is chosen as the 
origin and is omitted from consideration). 
2. Twelve variable points formed by intersections of the permanent tank 
edges with the liquid plane surface. 
In general it is necessary to determine: 
1. Whether these points indeed belong to or are on the volume of interest 
(many may not). 
2. Those points that determine a specific pyramid base. 
CALCULATION OF BASE AREAS 
Once a set of points has been determined to define a base area of a 
pyramid, it is necessary to calculate this base area. To do this, it is 
important to make sure the points are ordered correctly. The reason for 
this is that the area is computed by summing the area of the triangles 
that comprise the base. Let the set of points P.sub.i determine a base. 
Then define a centroid as: 
##EQU11## 
Next determine a set of vectors relative to this centroid; these will all 
lie in the base plane. 
EQU q.sub.i =p.sub.i -p.sub.s 
For convenience, we also define q.sub.i. The unit vectors q.sub.i are on a 
unit circle in the base plane with P.sub.s as the center: 
##EQU12## 
These are then ordered so that they are sequenced in order around the 
circle. If the sequence is defined by I.sub.i, I.sub.z, -In, the total 
area of the base is just 
##EQU13## 
VOLUME AND MASS 
Once the appropriate areas have been found, the volume is given by 
EQU V=1/3.SIGMA.d.sub.i A.sub.i, 
and the mass is then the product of the known density .rho. and the 
measured volume or: 
EQU M=.rho.V. 
A suitable processing program for effecting the above-described operations 
in microcomputer 30' is shown in the flow diagram of FIG. 8, and is 
described below. As is the case with the preferred embodiment, the 
foregoing computational method is not applicable to irregularly shaped 
tanks. It will be appeciated that the empirical approach described earlier 
for determining incremental volume/pressure relationships for tanks of 
this type can be used as well with the alternative embodiment shown in 
FIGS. 5 and 6. 
FLOW DIAGRAMS 
With reference to FIG. 7 a flow diagram illustrates the sequence of data 
storage and signal processing steps performed by the programmed 
microcomputer 30 in accordance with the above-described embodiment of 
FIGS. 1 through 4. While it is believed that the above description of this 
embodiment, including the stated mathematical relationships, is adequate 
to enable a person of ordinary skill in this art to prepare a program for 
operating microcomputer 30 to produce the resulting signal quantities for 
density .rho., volume V, and mass M, the flow diagrams of FIG. 7 is set 
forth here to assist in such programming. The program depicted has three 
principal sections including a temperature compensating loop 50, a 
pressure measurement read and store loop 52 and a calculation loop 54 
which cofunction to carry out the processing of the four pressure 
measurements to yield first density .rho., the volume V in conjunction 
with stored tank geometry valves 56, and then mass M. 
Similarly, the flow diagram of FIG. 8 is presented to assist in the 
programming of microcomputer 30' as used in the embodiment shown in FIGS. 
5 and 6. The FIG. 8 flow diagram is basically the same as in FIG. 7 except 
that only three pressures are inputted and processed, and the density 
.rho. is read from input data rather than being calculated. 
FIG. 9 shows a suitable processing routine for use in the preferred 
embodiment of FIGS. 1-4, for initially calibrating a tank of irregular 
geometry prior to use for measuring unknown density .rho., volume V and 
mass M. The routine consists of a sequence of tests, each test involving 
an incrementing fuel loop 70, pressure/temperature measurement and surface 
plane computation loop 72, and a tank data storing loop 74. Common to each 
test, is a conditions determining and store loop 76 that registers 
conditions of ambient temperature, fuel temperature, test parameter 
including tank pitch .phi. and roll (if desired) and other conditions that 
may influence calibration. As each increment of fuel is added pursuant to 
loop 70, the resulting pressure/density ratio for each pressure transducer 
is computed and stored along with the fuel volume in a non-volatile memory 
such as ROM, from which the volume data can be retrieved later during 
measurements of unknown fuel volume. 
While only particular embodiments have been disclosed herein, it will be 
readily apparent to persons skilled in the art that numerous changes and 
modifiations can be made thereto including the use of equivalent means, 
devices, and method steps without departing from the spirit of the 
invention.