Analysis and interpretation of data is the crucial phase of the decision making process. The interplay between variables must be considered as to their relative significance upon the final result, and sometimes time-sensitive decisions must be made when actual events deviate from predicted information, such as Apollo 13. As the number of variables increases past say four, the traditional method of cross-plotting tends to break down, and digital/analog results cannot present a sharply defined method of analysis. A graphical system, the carpet plot, is presented in which an unlimited number of complicated relationships of variables can be evaluated

Byrne, J. M.

English

NASA Technical Reports Server

CARPET PLOT DATA FORMAT
JOSEPH M. O'BYRNE
ASSOCIATE PROFESSOR
DEPARTMENT OF MECHANICAL AND AERO-SPACE ENGINEERING
UNIVERSITY OF MASSACHUSETTS,AMHERST
Analysis and interpretation of data is the crucial phase of the decision
making process. The interplay between variables must be considered as to
their relative significance upon the final result, and sometimes time-sensitive
decisions must be made when actual events deviate from predicted information,
such as Apollo 13. As the number of variables increases past say four, the
traditional method of cross-plotting tends to breakdown, and dlgital/analog
results cannot present a sharply defined method of analysis. A graphical
system is suggested. I named it "carpet plot," (since it covers anything),
in which an unlimited number of complicated relationships of variables can
be evaluated.
Introduction
The presentation and analysis of data involving
more than two variables has traditionally been
accomplished by cross-plottlng data and results.
An engine test is typical of this scheme, in which
rpm, dynamometer load and engine temperatures are
varied.
Three graphs are required, none of which define
the interlationshlp completely, as shown in
Figures i, 2, and 3, in that one of the variables
is always missing. As more variables are brought
into the picture, such as spark timing, compression
ratio, octane number, etc., the proliferation of
graphs necessary to present the data becomes
excessive, possibly increasing exponentially.
An entirely different situation exists when a
fixed result can he achieved by a multiple combin-
ation of several variables. Taking a simple case,
the Fourier equation for one dimensional conductive
heat flow,
dt
q = -KA d""x (1)
shows that q can be adjusted to a predetermined
value by an almost infinite combination of vari-
ables. For instance, if dt is doubled for a
constant K and A, dx should be doubled. If K
changes by a factor of 1.4, A by 0.8, dx by 0.6,
then dt can change by a factor of 1.9 and still
have the same numerical value of q.
This type of problem is entirely different than
the engine test type. As the system becomes more
complex, and the number of variables increase and
commence to interact, the difficulty of analysis
increases. The carpet plot represents a new
method of data presentation to simplify analysis.
There is theoretically no limit to the number of
variables.
/
x 5
- i
ENGINE TEMPERATURE, OF
Figure i. Plot shows the effect of dynamometer
load variation on fuel consumption and
other operating parameters of an
internal-combustion engine
W
W
DYNAMOMETER LOAD, Ib
Figure 2. Plot shows the effect of engine
temperature on fuel consumption and
other operating parameters of an
internal-combustlon engine
392
https://ntrs.nasa.gov/search.jsp?R=19720010002 2020-03-17T03:49:20+00:00Z
ENGINE SPEED, rPm
Figure 3. Plot shows the effect of engine speed
on fuel consumption and other operat-
ing parameters of an internal-
combustion engine.
Example of a Carpet Plot
A simple heat transfer problem was selected
which could be analyzed mathematically and verif-
ied by laboratory equipment. Figure 4 shows an
insulated cylinder filled with m lb. of water; its
average temperature, t, is measured by six thermo-
couples wired in parallel. A helical cooling coil
is placed in the tank, through which water flows
at a mass flow rate W, regulated by an outlet
valve and measured by a rotameter. A variable
power heater is installed inside the tank, whose
energy can be controlled by a rheostat and
measured with a wattmeter.
WATTMETER
RHEOSTAT
9_
COOLING
COIL
"T-T"
T I JL W _ T 2
ROTOMETER_I 1 *------VALVE
Figure 4. Schematic diagram of equipment used
in a transient heat transfer
experiment
With no heat application to the system, water is
passed at a constant rate through the cooling coil
until thermal equilibrium is reached in the tank.
At time zero, the heater is turned on, and the
problem is to predict the transient and steady-
state values of the tank water as a function o_
time.
For a fixed value of W and M, various values of
q were selected and the tank temperature calculated
by conventional methods as a function of time, 8; a
typical parametric graph is shown in Figure 5, in
which the 8 scale is constructed by conventional
methods.
Assuming that the tank temperature is of little
interest at low values of 8, Figure 6 modifies
Figure 5 by ignoring the low 8 values, bringing the
0 values vertically into the graph, and erasing the
"old" 8 scale; each of the 8 values are labeled.
_c
z
Figure 5.
e'l e'2 e's e'4
ELAPSED,,ME,M.,.J,.s
Traditional time-temperature plots
with parametric heat loads
MI
W _ e4
el
Figure 6. First step in a carpet plot technique
The graph, up to this time, has been for a
constant W I and M]. The carpet plot proceeds in
Figure 7, In whic_ for the same values oF q and 8,
a different coolant flow rate W 2 is desired. This
effect is introduced by moving some convenient
physical distance (3 inches, 6 inches, etc.),
constructing a new time scale _, on the traditional
basis and plotting tank temperature t, as a functior
of the same q. - q., m., and a constant W^, on the
II I! I II I II
old t scale. (T_e new W 2 plot will probably
not be in a horizontal line with the W I graph). A
_igniflcant feature of the carpet plot-_s _h_t the
q93
result, t, for all the variables uses only one
scale.
For the W parametric graph, the same procedure2
is used as previously done; bring 0 into the graph,
ignore small values of 8, erase the e scale, and
label the vertical 8 lines; this step is shown in
Figure 8.
Attention is drawn to the fact that at say the
left hand bottom corner of the W 1 plane of Figure
8, a point exists for a WI, qA' and 8. value, M
being a constant. Likewise the left _and bottom
corner of the W 2 plane has the same q4 and 81, M at
the same constant. Thus, q4 and e. values are
common values, and any line connecting them will
have the same q4 and 81 . Figure 9 expands this
concept to account for more points.
M I
W i
qSW I
4
el
el el e3 e4
Figure 7. Extension of Fig. 6 for a different
mass flow rate
The carpet plot takes _n a 3-dimensional appear-
ance; however, it is plotted on a flat sheet of
paper and is not 3-D. It also differs from an
X-Y-Z coordinate system, in that no axis is fore-
shortened. The stories of the box represent var-
ious q values, the ends represent various W's, and
vertical slices are various values of e.
Only two values of W are shown; in practice, at
least more than three should be computed in order
to establish the correct trends; the author uses
four values of W as a minimum to assure that no
inflections, etc., occur.
Assuming that the W I and W 2 planes correctly
identify the tank temperature, recall that their
physical separation (say 3 inches) was completely
arbitrary. A plane half way between W and W 2
would logically represent conditions o3 W 1 1/2;
give your children the game of drawing in this
interpolated plane, as shown in Figure i0, whose
appearance is similar to the existing computed
planes; its numerical accuracy is only a function
of the artist's ability. In a similar fashion,
moving 6 inches to the right of plane W. would
logically represent a plane of W 3 conditions which
can be drawn in by eye. If the interconnecting
lines of common values between W. and W^ are
correct, and the performance of _he system is
assumed to be logical at higher values of W2, then
theoretically there is no limit to the extent of
extrapolation and interpolation as shown in
Figure ii.
M I
w,
81 e I
Figure 8. Common property representation for
a fixed mass
MI M I
w_ w,z
el ei
Figure 9. Extension of Fig. 8
MI MI M I M I
Wl Wl.5 W2 w2.5
q_r qlw_ i ql_ ql/r _ ,
/ i _ / I ;/r0 . //IN .(..-"
I /"1 q,/I i -, %_ _ I f -lr-o--r-i_"l- -T', ,..i_ i_1
w / _ _ I t,-"/ I _3_ i IV.-F _I , , I .., _
el " el " el--_- -- -- --_e( 02
Figure i0. Technique of extrapolation and
interpolation
394
M I
Wl W z
ql
i
4 4
_ e_
W I W,
M 2
i
• t/J_l "*_J----'-_ 04
el e I
Figure Ii. Introduction of a new variable
into the carpet plot
Tothis point, thecarpetplot techniquehas
heldthemassof tankwater,MI, as a constant.
Let us extend the previous concepts in order to
change M 1 to M 2 for the same values of e, q, and W.
The vertical location of the new M_ plane is fixed
by the temperature scale, but the _orizontal loc-
ation is arbitrary, as was the selection of the W 2
plane location. Basically, the problem is only
to select a separation distance which will not
cause overlapping of the planes• By repeating
the calculations, for the same q's, W's and e's,
using a new value of M2, Figure ii shows the
results Only one interconnecting line between M.
and M 2 is shown for clarity to permit interpolatlon
and extrapolation of the M values in a fashion
similar to the W values. The first objectives,
presentation of coherent data for more than two
variables, and logical interpolation and extra-
polation of data, have been established. The
variables were changed in a nested DO loop and the
results computed on a CDC 3600 machine•
The next phase of utilization of a carpet plot
is shown in Figure 12, which quickly shows that a
tank temperature T will result when the system is
operated with heatXrate q = q4' W = W I, e = e3 at
M = M.. Note that temperature at x can also be
1 =
reached at W = W2, e e., q = q_, an entirely
different set of operating conditions. The carpet
plot shows an almost infinite number of combin-
ations. More frequently is that an operating T is
required, and what combinations of variables wi_l
permit this T condition? Entering the carpet
plot at the Tx value, all possible combinations can
be determinedXrapidly and easily•
In addition to presenting every correct solution,
the carpet plot will indicate every incorrect com-
bination of variables. For instance, if temper-
ature T is. deslred, it is perfectly obvious that
x
cannot be used under conditions of any q, W and
_ In engineering work, the input to a data
collecting or computing system is some desired
performance which may change daily due to updating
of information from other groups; the carpet plot
permits a rapid analysis of operating conditions
to achieve the result.
Another feature of the carpet plot permits an
evaluation of the effect of the change in a vari-
able upon the result. The differential equation
method could be used, or the uncertainty method
(1), but these techniques are extremely tedious
as the number of variables increases; Figure 13
demonstrates this capability. Varying only one
property, holding all others constant, the effect
upon the result is shown schematically, in that
dt 2 and dt._ are about the same small magnitude,
dt I is somewhat larger, and dt^_ far overshadows
the others. The significance of this is immedl-
ately and clearly evident to a manager planning
his work priorities, to define which variable
should receive the major attention. In this case,
it backs the previous invalid M 2 inoperative con-
ditions for T , and shows that major design
efforts for t_e system are necessary to prevent
it from accidentiy shifting from M I to M 0.
ql MI
1 W8 ql
.... el -- e4 _ eS --e_'_
@I @!
M2 ql Mt q
4 @S Ib4
el °1
Figure 12. Method of determining all possible
solutions
qz _ Mz
/ I w'q' / I
T_T..--_--. -- - _ _
MZq!
01 @1
Figure 13. Relative effect upon a result
with changes of a variable
395
Students at the University of M_ssach_setts
proposed a detective game, in which they placed
some water in the tank, then sealed it; the
amount of water was not known by the writer; they
set in values of power and flow rate. Access to
the writer was permitted to the wattmeter, rota-
meter, and temperature recorder, and they posed
the question: "Before the experiment ends, tell
us how much water is in the tank." The approp-
riate heat loads and flow rates were rapidly
sketched into the basic carpet plot values, and
the interpolated tank water weight was established
in minutes. The significance of this experiment
is that the value of unmeasured and/or unknown
quantities can be inferred by use of the carpet plot.
The utility of a carpet plot is unlimited. For
instance, how much will the price of an automobile
change as the number of coats of paint change, the
color and weight of the dash padding, the quality
of the spark plugs, the number of instruments,
radiator size, etc. Which of the many variables
is most significant?
This technique was applied to the analysis of
variables associated with high performance insul-
ation to be used on upcoming space programs such
as nuclear stages in space. Only the thermal
aspect was considered, but factors such as manu-
facturing, fabrication, cost, etc., could easily
be brought into the picture. The George C. Marshall
Space Flight Center has evaluated the performance
of the Mars nuclear vehicle for the variables shown
in Table I.
A part of this analysis is shown.ln Figure 14 to
introduce the reader to the upcoming coordinate
system. Considering only the variables of mission
time, density, thickness, the weight penalty is
presented in a traditional fashion; note that the
remaining variables do not appear, and their
effects are unknown; the information is incomplete.
Figure 15, with time scales varying, presents
an almost complete version of a typical carpet
plot, with a few of the variables omitted due to
reproduction difficulties. Typical design
parameters are immediately evident, such as no
matter which of the assumed conduutivities is used,
large diameter tanks will have a lower weight
penalty. Insulation thickness is more critical
than density, and design effort in this area
should be emphasized.
In conclusion, I pose the following suppose
experiment to the reader, assumed to be the
manager of a complex nuclear system operating in
space: generate a carpet plot and ask your ten
year old child some numerous questions, such as:
I have designed a system for a predicted thermal
conductivity, tank diameter, percent LH^ slush,
z
density, weight penalty, temperature levels, and
manufacturing cost. Due to installation problems,
the thickness, (layer density) changes from this
to that. What will he the effect upon mission
time? (My daughter loves to solve the 100 or so
combinations of results with a school ruler.)
"A picture is worth a thousand words" applies
to this technique.
1.
2.
References
Theories of En_ineerinK Experimentation, 2nd
Edition, Hilbert Schenk, Jr.
"Air Breathing Engines," Short Course on Air-
Breathing Engine Technology, 8-12 June, 1970,
Marshall Space Flight Center, Alabama.
396
_)AAX
WP-
PH V
_AX) 2 hf 9
S"
K A T
VARIABLES
I BOUNDARY TEMPERATURES
2 INSULATION THICKNESS
3 iNSULATION DENSITY
4 MISSION TIME
5 WEIGHT PENALTY
6 THERMAL CONDUCTIVITY
7 VENT PRESSURE
8 PERCENT SLUSH
s LID RATtO
IO DESIGN BASIS
tt VEHICLE DIAMETER
Table i.
Figure 14.
Properties considered in the study
.I
I
.s
_ s
4
o-
0.10 FT
_H V I
Z J
• pt,,) h_g
XAT j
' p-4
i f
, i p =
,i _,x ,to
/ // p.I
"r_E. oA'cs
The effect of several variables
upon the performance of a nuclear-
powered space vehicle
<
_z
+
m_
_z
zo
Figure 15. Extension of Fig. 14 to account for
additional variables


Carpet plot data format

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