Simulation of Steam Pressurizing Tank Transients by Analog Computer
SIMULATION OF STEAM
PRESSURIZING TANK TRANSIENTS
BY ANALOG COMPUTER
Donald B. Bos ley-
Roth S. Leddick
6854
LEY
1956
THESIS
B725
Letter en cover:
SIMULATION OF STEAM PRESSURIZING
TANK TRANSIENT
COMPUTER
Donald Eritton Bosley
SIMULATION OF STEAM
PRESSURIZING TANK TRANSIENTS
BY ANALOG COMPUTER
3Y
Donald Britton Bosley
Lieutenant, United States Navy
AND
Roth Sumner Leddick
Lieutenant, United States Navy-
Submitted in partial fulfillment
of the requirements
for the degree of
MASTER OF SCIENCE
IN
MECHANICAL ENGINEERING
United States Naval Postgraduate School
Monterey, California
1956
This work is accepted as fuii'illing
tne thesis requirements for the degrees or
MASTER OF SCIENCE
IN
MECHANICAL ENGINEERING
from the
United States Naval Postgraduate School
PREFACE
"Underway on nuclear power."
i
With this simple but historically
significant message, the U.S.S. Nautilus heralded a new age in power
generation.
Powered by the first mobile nuclear reactor, she set the
stage for dramatic developments in military and industrial fields.
A
powerful vessel, but unwieldy and undesirably large for a submarine,
her size was largely determined by the size of the power plant components.
One of the more significant of these components is the pres-
surizer for the reactor primary coolant system.
This tank is extremely
large due to the conservative design necessitated oy ignorance of the
thermodynamic transient behavior in its pressure range.
The objective of this thesis is to produce design data for steam
pressurizing systems, Dy electronic analog simulation of the thermodynamic transients which occur in the pressurizer vessel.
The writers wish to thank Professor Eugene E. Drucker, and Professor Hugo U. Martinez of the U.S. Naval Postgraduate School for their
assistance and enthusiastic cooperation in the prosecution of this
project.
•
ii
TA3LE oF CONTENTS
Item
Title
Chapter I
Introduction
1
Chapter II
Development of Theoretical Equations
4
Chapter III
Application of Equations to the
Analog Computer
10
Analog Computer Solution of
Theoretical Equations
14
Experimental Results and Conclusions
19
Chapter IV
Chapter V
Page
Bibliography
Appendix I
29
Derivation of Empirical Relations for
Pressure and Compressibility Factor
30
Appendix 11
Derivation of Machine Equations
40
Appendix III
Hand Solution of Theoretical Equations
Appendix 17
Actual Tank Transient Data
iii
....
44
50
.
LIST UF ILLUSTRATlu^
Figure
Page
Boeing Electronic Analog Computer
and Sanborn xtecorder
3a
2
Equivalent System
4
3.
Block Diagram of Computer Circuit
11
4.
Hand Solution Curves
15
5.
Sanborn Recording or Test Run
16, 17
6.
K Determination for Typical Run
2U
7-
K - Surge Duration Diagram
22
8.
Pressure Prediction Curves
25
9.
Temperature- Entropy Chart
26
1.
10.
Enthalpy - Pressure
11.
Internal Energy - Pressure
12.
Constant - Specific Volume, for e computation.
13.
Compressibility Factor - Pressure Diagram
14.
Constant - Specific Volume, for Z computation.
15.
Computer Schematic Diagram
31
r
IV
33
.
.
35
37
.
.
39
43
.
TABLE OF SYMBOLS AND ABBREVIATIONS
a
potentiometer setting for analog computer
C
capacitance in micro-farads
c
mathematical constant
c
P
specific heat
E
total internal energy in B.T.U.
e
specific internal energy in B.T.U. per pound
F
temperature in degrees Fahrenheit
H
total enthalpy in B.T.U.
h
specific enthalpy in B.T.U. per pound
I.C.
initial conditions
J
energy conversion factor, equal to 778 ft lb/BTU
K
effective thermal conductance
k
mathematical constant
lb
pound
1.1
megohms resistance
m
mass
P
absolute pressure
p.p.
patch panel (analog computer)
Q
total heat energy in B.T.U.
q
specific heac energy in B.T.U. per pound
R
electrical resistance
R
temperature in degrees Rankine
T
absolute temperature
t
time in seconds
o
o
At
duration or surge in seconds
U
amplitude 01 driving function, equal to one-half the
change in spec
.
volume during a surge
V
total volume of steam in cubic feet
v
specific volume in cubic feet per pound
W
total work done on system in B.T.U.
w
specific work in B.T.U. per pound
Z
compressibility factor
ex
scaling factor for analog computer
(Jj
natural frequency in radians per second
SUBSCRIPTS
o
initial value
e
pertains to internal energy
f
final value
P
pertains to pressure
q
11
"
heat energy
s
"
"
steam
T
"
"
temperature
t
"
"
time
v
"
"
specific volume
w
"
"
work
z
"
"
compressibility factor
vi
SUPERSCRIPTS
a bar over any symbol represents the voltage equivalent
to that quantity (analog computer}
a dot over any symbol indicates the first derivative of
the quantity with respect to time
OPERATORS
d
exact differential of a quantity
1_
exact integral of a quantity (Heaviside notation)
P
Vll
CHAPTER 1
INTRODUCTION
The prototype (Lark 1) of the Submarine Thermal Reactor (STR) was
the first full scale power reactor in the world to be completed and
successl'ully operated.
The first central station nuclear power plant
in the United States will be the Pressurized Water Reactor
nearing completion at Shippingport, Pennsylvania.
(P'.VR),
The many highly
desirable aspects of the pressurized light water reactor cause it to be
one of the most promising types, to date.
A basic requirement for these light water reactors is that a very
high pressure is maintained on the primary coolant.
Pressures in the
neighborhood of 2000 psia and higher are presently in use.
High pres-
sures permit the use of sub-cooled water at high temperatures in the
reactor without danger of boiling.
When any control program other than constant average primary loop
temperature is used, a change in the volume of the primary coolant is
to be expected for a corresponding change in power.
A volumetric tran-
sient is consequently induced and a surge tank is required in the
system.
Since the primary coolant must be maintained at a high pressure
the surge vessel is also used to perform this function.
The pressurizer must be strong enough to maintain steam and water
in equilibrium at high pressures.
It must be large enough to absorb
normal and accidental surges without permitting excessive pressures
in the primary loop.
Although thermally insulated, it must have internal
heaters, capable of maintaining the water and steam at the saturation
temperature.
These heaters must have additional heating capacity cap-
able of generating steam at sufficient rate so as to prevent excessive
pressure drop in the primary coolant during negative (out) surges.
Specifically, in this thesis we will assune a primary loop working
pressure of 2000 psia, normal volume surges of 4 cubic feet and accidental
surges of 7 cubic feet as representative values.
At steady state water
and steam are maintained at saturation temperature (636 ° F) in the tank,
but only the pressure (2000 psia) is transmitted to the cooler primary
loop via a relatively long stand-pipe.
Assuming that the installed
heaters adequately compensate for negative surges, this type surge is
not considered a design limitation.
Therefore only those surges caused
by an increase in primary loop volume, or positive surges, will be
considered here.
When designing or sizing a pressurizer, limits must be imposed on
the various properties of the fluids in the tank.
The maximum allowable
pressure change must be determined from consideration of the strength
of the members involved.
The expected change in average density and
therefore volume of the primary coolant must be computed.
The duration
of the surge is determined from changes in power, either accidental or
deliberate.
The one remaining variable, the size of the tank must now
be determined on the basis of the aforementioned parameters.
The pressure change caused by a positive surge may be minimized by
introducing a portion of the surge water into the top of the tank as a
finely divided spray.
The most conservative basis for pressurizer
design is to assume that no fepray is employed, the steam is dry saturated,
2
and undergoes isentropic compression.
Should the steam not be ary, the
resulting final pressure would be lower than in the case of dry steam.
The optimum and therefore minimum size required can only be positively
determined after a complete understanding of the thermodynamic and heat
transfer mechanisms involved.
In this project, it is felt that valuable information is available,
both in the fields of design limitation and in determining the characteristics of high pressure saturated steam.
A Boeing Electronic Analog
Computer with associated 4-channel Sanborn Recorder (see Fig. 1) is
utilized to simulate the pressurizer, with the above objectives in
view.
An important assumption which is made at the outset is that the
driving function, specific volume (which is directly related to tank
level) follows a cosine curve.
Analysis of actual tank transients
indicates that this approximation is a good one for a large portion of
them.
This type of function can be easily produced on the analog com-
puter, while production of more exact functions is difficult.
In the analysis of results, every attempt is made to utilize the
principle of geometric similarity, to more nearly generalize these
results.
BOEING ELECTRONIC ANALOG COMPUTER AND SANBORN RECORD a
Figure 1
3a
CHAPTER 11
DEVELOPMENT OF THEORETICAL EQUATII
The initial step in the analog simulation
oi'
this problem consists
of a description of the system in terms of theoretical equations.
.Vhenever possible known thermodynamic relationships are used, and when
this is not possible empirical relationships are developed.
No reference
can be found in the literature treating the following development of
descriptive eolations , and it is believed that this approach may be a
new one.
As an initial simplifying assumption, an equivalent system is devised
which lends itself more readily to analysis than does the actual tank.
v
\
HEAT SINK
\V
H IM It I
PERFECT
INSULATION
W*~>
J
Equivalent System
Figure 2
The actual tank is well insulated, therefore the concept of perfect
insulation for the equivalent arrangement does not introduce significant
error.
The thermodynamic system is defined as the mass of steam in the
tank.
In the actual tank tnere is installed internally a spray and
degasifier assembly.
In this analysis of positive surges, without spray,
its only effect is to function as a heat sink.
The heat sink in the
equivalent arrangement represents the tank walls, spray and degasifier
assembly, the mass of water in the tank, and all other apparatus within
The piston represents the water surface, which
the tank insulation.
in the actual tank performs' work upon the steam during an in-surge.
Thermodynamic equilibrium is assumed at all times.
Equation #1 is the well-known work equation:
w s
where
-If dv
w - work (BTU/lbj, positive when work is done on the system
P = system pressure (psia)
v = specific volume of steam (cubic feet/lb)
Considering units,
w =
-144 |P dv
-
-.1853JP dv
In Heaviside operator form,
w -
-if. 1853 P(dv)dtl- -.1853 JP (dv) dt
L
dt
dt
J
J
p
Equation #2 is the First Law of Thermodynamics:
Ae =
q + w
or
e -
q +- w + e
where
e -
specific internal energy (BTU/lb)
q =
heat flow (BTU/lb), positive when into the system
,
Equation #3 is the equation of state for a non-ideal gas:
Pv =
where
c
1
Z T
Z is defined as Pv
(dimensionless compressibility factor,)
C]T
T c
absolute temperature (° A)
l z 1545
s
13.01x144
3
.5957 lb f ft
(gaa constant)
lb.° E
iri
ill
Equation #4 relates P to e and v empirically from steam table
values (see Appendix. 1):
2
P = 8.51 e + 52,590 v
- 30,700 v
- 3161
Equation #5 relates Z to P and v empirically, in the saturated
and superheated region, 1'rom steam table values(see Appendix 1):
Z a 1.715 x 10~ 4 P - 9.47 v
Note:
2
+
6. 043
v
- .5693
Equation #5 is not required nat hematic ally, but is desirable
for computer circuit simplicity.
Equation #6 relates heat flow to temperature differences and is
developed as follows:
First, consider the sink as a system,
where
^ sink
d ^sink =
msink Csink
'^sink =
Quantity of heat energy (BTU)
m sirk =
mass ° f
Tsink s
temperature of sink (° R)
c
sink heat ca P acit y (BTU/lb° a)
sink =
sifik
(lb)
Rearranging
dT
S in k
-
=Arj
"HainkCsink
dJ 3ink
—
Integrating,
T
sink =
msink c sink
but
Q
and
^steam
+
-isink
I
.
(
T ^si.
" '^ink = Q
stea7i *
(T o)
To
sin k "
Then,
T
I'Jow,
_^
sink =
m sink c sink
+
Q
T
considering the original system (the steam):
d^
. -K (T - Tsink)
dt
K represents the effective thermal conductance of the system boundary,
and will be considered a constant in the equivalent system for any given
surge.
The units of K are (BTU/sec° Ii) •
K is ordinarily the product
hA, when each is determinable.
Substituting for T
d£ =
.
.
,
+
- K (T
Since Q =
m
da s
a
q
- T
^
dt
)
sink c sink
where m s = mass of steam (lb)
,
- K
(T - T
ms
dt
c)
-
K
q
m sir.k c sink
The problem which is now encountered is that of evaluating the
equation constants.
Since accurate design data are not available, these
constants must necessarily be evaluated in an arbitrary and approximate
manner.
For the subject tank, m
approximated as follows:
.
,
can be taken as 3000 lb.
This is
mass of metal
~
2000 lb.
mass of water
«
1000 lb.
Total
3000 lb.
The heat capacity of water at the pressure and temperature range
involved is approximately 2.0 BTU/lb° R
The heat capacity of stainless
.
steel (tank wall material) is approximately .13 BTU/lb R
.
A weighted
average heat capacity of the sink is computed as follows,
c
2
slnK =
.
.
x 1000
+
.13
x 2000
. .75 3TU/lb° R
3000
The mass of steam in the system will depend on the initial tank
level, and saturation conditions, but rarely varies greatly from an
average value of 175 lb.
Properly, the mass of steam should be computed
and Equation #6 modified for every run simulated, but for simplicity
this average value is used for all surges.
ra
In all simulated surges,
anc* m are taken as constants, while the quantity K is
s
sink» c sink»
adjusted to cause computed transient properties to coincide with experimental data.
Equation #6 with constants evaluated becomes,
da,
-
- 5.72 x 10" 3 K (T - T
)
- .445
x 10~3 K q
dt
In Heaviside operator notation, the equation takes the form,
q
"
= - I C 5 72 x 10 3k
-
?
(
t -
V
+
"
-^ * 10 3k
During all runs the assumption is made that the steam mass remains
constant throughout the surge.
Since the energy introduced as work
during a surge is on the order of 3000 3TU, the maximum mass change
would be in the neighborhood of 6 lb., if all of the energy were involved
8
in a phase change.
Therefore, the percentage mass change can never be
a significant value, especially since heat flows abundantly to the sink.
Any phase change will significantly affect K however, because of the
heat transfer mechanism involved (heat of vaporization).
The six theoretical equations as developed and in their most useful
forms appear as follows:
#1
w
- 1
p
=
j
c
L.
P
12
e
=
q * w + e
#3
T
t
P v
Z
e
- Co v
J
j
at
,/2
Ct,
(dv) dt
+ c^ v
2
-
c
"
c
#4
P
=
c
#5
Z
.
c^ P t
c„ v
#6
q
=
- 1 Pc
K (T - T
c
z .5957
c
± 8.51
c
2
P L
- c
v
g
Q
)
+
c
±1
5
9
K q "j
where,
c
c
1
2
3.07 x 10
r
3 " "
4
'
= 5.259 x 10
c
6
g
4
4
= 1.715 x 10"
= 6.048
- 9.47
cn =
9
c r = 3161
c
?
c
c
1Q3
.5693
= 5.72
x 10"10
x 10
xl = .445
c
12
= .1853
-3
CHAPTi^t III
APPLICATION OF EQUATIONS TO THE ANALOG COMPUTER
After the derivation of the six theoretical equations, a computer
circuit must be designed to accomplish their simultaneous solution.
In
the circuit design many things must be considered; such as, scaling
factors, follow-up time of multiplier components, amplifier drift, voltage magnitudes, etc.
A certain amount of latitude is allowed in circuit
component selection, but definite limits are present due to stability
considerations.
The six basic equations in computer notation appear as below:
ffl
w
1 [l.lll (.IF dv dt
dt
p 1-
c
)
!
J
-0333 w
+
e
4.255 e
- 5.1167 v
+
4.3825 (.02 v^j
.5145 e
+
+
!f2
e*
- .0333 q
#3
T
«
3.3574(~j
ifk
P
-
lb
Z
=
#6
q
s - lf.1716 KT
3.024v
- 2.3675 (.02 v
- .1716 K T
Q
+
2
)
.000445
- 52.6833
- 2S.465
Kql
P
The bar over a quantity represents the quantity as a voltage.
The
detailed conversion of the theoretical equations to the above "machine"
equations comprises Appendix II.
Fig. 3 shows the resulting machine circuit in block diagram form.
For the detailed circuit, see Appendix II.
designed to function as follows:
10
Briefly, the circuit is
>
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f
-oItj
Q-
C^
(X
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Z <S
^
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or
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>' Uft
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r4
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cT)
H
^
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f
3
1
s
.
'
/
C_>
or
1
MM
o
L
S.T
'
UJ
h-
>
<\i
5
cvi
C*
#
QC
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- i£
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1- £ -
—
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MULTIPL
FUNCTIO
3
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r
CO
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or ^ S
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—
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uJ
,
.
In the driving function circuit a cosine function is added electrically to a constant voltage, the resultant voltage transient very
closely approximates the specific volume throughout a surge.
The
amplitude of the cosine function represents one-half the change in v
,
while the period of the function is twice the time of the surge (a t)
Within this same circuit the time derivative of specific volume, dv
dt
is also generated.
A function multiplier is used to obtain the quantity lr t while a
servo-multiplier produces the products
P dv
An integrating
and P v.
dt
circuit integrates
_
w
.
P dv
dt
as a function of time to produce specific work,
Another integrating circuit produces the time integrated specific
heat flow,
q
,
using temperature inputs.
A summing amplifier sums e
e, v, and
v^ to produce P.
,
w
,
and q
A third sums
A dividing circuit performs the division
to produce
P, v, and
P y
Z
v
e.
Another sums
to produce Z.
to produce T~.
The circuit required to solve the set of equations consists of
eleven amplifiers, four sign-changing amplifiers, two function multipliers,
one duo-channel servo-multiplier, and twenty-one 50,000 ohm potentio-
meters.
The net result is a rather complex circuit, but one which does
not seem to be subject to further simplification without sacrificing
necessary accuracy.
Since instability in this type q£ computer is a direct
function of the number of components used, circuit simplicity has been
a constant objective.
This phase, the designing and setting up of the circuit, requires
a great deal of computer experience and proficiency.
For the uninitiated,
many trial and error situations are encountered, and with a circuit of
12
this complexity the process can be very time consuming.
The following chapter will deal with the problem of producing
quantitive results, assuming the circuit has been designed, balanced
and tested.
13
•
CHAPTLR IV
ANALOG COIJPUTEil SOLUTION OF THEOKhriCAL EQUATIONS
After the analog computer circuit is designed, assembled, and rough
qualitative results obtained, the next phase consists of producing
accurate quantitative solutions.
consists of static adjustment.
The first step in accomplishing this,
With various initial values of e
,
and
v set, the e, P, Z and T computing circuits are adjusted to give steam
table values for these quantities over the entire anticipated range.
This step of the co puter set-up is very exacting and time consuming
but is a necessary step prior to adjusting for dynamic accuracy.
Prior to the dynamic adjustment, a hand solution of the system of
theoretical equations is obtained for use as an adjustment reference.
A set of initial conditions, and a driving function are arbitrarily
selected, and using the Heun Methodof numerical integration, a point-
by-point solution of the equations is calculated (see Appendix III).
The various thermodynamic quantities as calculated are plotted, the
resulting curves serving as the final reference for dynamic adjustment
(see Fig. 4)
Next, the computer is made to duplicate the hand solution.
This
is done by adjusting the w and q integrating circuits until, the w and q
curves, as viewed on a properly calibrated Sanborn Recorder, are of the
proper shape and magnitude (see Fig. 5).
These adjustments in general
are minor.
The value of careful static adjustment is now apparent since no
further adjustment of the e, P, Z and T circuits is found necessary.
1
Special case of Runge-Kutta Method; see Kopal, Numerical Analysis,
pp. 202-205, Wiley (1955).
j-4
SMM
(I
/-a4oo
-\&oo
TIME
FI6DRE
S
SANBORN RECORDING OF TEST RUi'J
Fl&URE
5"
(CO NT.}
17
The curves of these quantities are
1,01V
found to match the correspono
hand solution curves very well (see Figs. 4 and 5/.
Once the hand solution is duplicated, a reference for measuring K
is now available.
of K is .9254.
In the hand solution, the arbitrarily selected value
The corresponding setting of potentiometer a,,
J-4
is now
readily observed (usually in the neighborhood of .100, varying slightly
from day to dayj
.
iiince K and the a.,
setting are proportional, K can
be determined at any time by the simple relation,
*= KoxCajJ
It is now possible to vary K over a large range of values by merely
adjusting a single potentiometer, thus varying the integrated q curve,
and all other curves of thermodynamic properties.
By simulating the
driving function for a given surge, K can be varied until the proper
pressure curve is obtained (i.e. a pressure curve which matches the
pressure curve resulting from an actual tank transient;.
The ne^t
chapter discusses the investigation of actual surges by simulator, to
determine values of K for these surges.
types is also discussed.
18
Correlation of K with surge
CHAPTER V
EXPERIMENTAL RESULTS AND CONCLUSIONS
Appendix IV is a compilation of data obtained from the actual surge
bank.
Eighteen positive surges are represented, spray being utilized
in only one.
It is evident that possession of many more runs would be
desirable, but the present number must sui'fice siace ...ore could not be
obtained.
Fortunately these eighteen surges represent a
1'air
cross-
section of positive surges encountered.
The procedure for simulating a given surge is outlined in Chapter
IV ana briefly is as i'ollowo:
A proper driving function (specific volume J is set up on the comput-
er, providing for the generation of a cosine curve of proper amplitude
and period.
Initial values of
and a surge is generated.
v, e, T, and a trial value of K are set,
K is adjusted for several trials until the
generated property curves match the corresponding actual tank curves
(see Fig. 6 for Sanborn recording of this procedure).
Each of the eighteen surges is simulated in this manner, and the
proper value of Redetermined (see Table 1).
ing K v/ith various quantities (a v, av
,
v
o
a fair correlation with surge duration
av_
kany attempts at correlat,
etc.) were fruitless, but
AXj
{/ntj
is possible (see Fig. 7).
It is likely, in fact, that K is a function of several or all of the
variables considerea, but the effect of each seems small in comparison
to the effect of At.
3y simple curve fitting, a correlating equation
is derived,
K
= 3-75
*
25
t - 17
19
-bOO
o
P RESSl/RE
IN
RUN it 2' 129
PKifci
y(
1
'
4
1
1
c
K DETERMINATION FOR TIPICjAL
20
dm
i
Nothing i3 claimed for this equation except that it describes a K which
aids in predicting pressure rises on the safe side (i.e. in most casus
the predicted pressure rise will be somewhat larger than the actual
pressure rise).
Column 4 of Table 1 lists the K for each surge as
predicted by formula.
In general, it can be observed that K does not vary greatly for
any surge of over 30 seconds duration, and that it becomes nearly constant for surges longer than 90 seconds,
it is also true that pressure
rise is not sensitive to moderate changes in K.
Table
1
The last column of
lists errors encountered when a constant value of 3-75, the
limiting value of K, is used in the computer.
The mechanism of heat transfer for the actual tank can only be
surmised.
er,
However, it is logical to expect greater turbulence for short-
faster surges, resulting in somewhat larger values for K.
This
could be due to a larger heat transfer surface between the steam and
water in the tank.
The effectiveness of spray in increasing K is readily
observed in run 1506, Table 1.
Whereas a K of approximately 5-0 could
have been expected without spray, the actual K observed is 13.8
From the general characteristics of the
K vs.
.
Surge Duration curve
(Fig. 7;, it appears that K approaches a limiting value of about 3.75
for surges of long duration.
For geometrically similar tanks utilizing
the sane pressure range, the same limiting value of K could reasonably
be expected.
Realizing that a correlation exists between K and the driving
function, it is now possible to predict pressure rises for all conceivable driving functions.
This is done in the following manner:
21
— B
I
as
m
:
1
i
^
,
SUM NO.
b
3U
K
(exp.;
K
(for
.
(sec;
2045
75
4.31
4.17
7
2302
120
4.12
4.00
- 4
1630
120
3.96
4.00
-12
2123
100
4.26
4.05
-
1653
95
4.26
4.06
18
2229
90
^•07
4.07
-22
2023
65
2.73
4.27
-34
1556
65
4.54
4.27
12
2010
60
3.15
4.32
-21
1101
27.5
6.30
6.15
30
2240
45
6.25
4.65
12
2255
32.5
5.55
5.35
18
2248
35
3.90
5.15
7
1911
37.5
4.36
5.00
66
2113
37.5
6.50
5.00
60
2039
45
5.3^
4.65
66
2247
45
6.02
4.65
90
1506
(spray)
45
13.3
4.65
200
i_,rror
5
in predicting oressure rise (in psia) when using a constant
value of K,( the limiting value of 3.75;, rather than the exper-
imental or formula values.
23
.
A driving function is
e:
tablished, the proper value of K is set,
and the resulting pressure rise recorded.
In an attempt to use the
'inciple of geometric similarity, diraensionless parameters are used
when possible.
a v
v
,
o
aj_
.
f
Accordingly,
a
family of cui'ves is obtained relal
and at (see Fig. 8 ) .
?n
o
since the actual tan ; surges vary
1
the predictions
ax-e
fro:j
27.5 secor.ds to 120 seconds,
only valid within this range.
that safe extrapolations may be made.
However it appears
An attempt to justify liraitin
values of the prediction curves is based on the following reasoning:
Initially the system consists of dry saturated steam.
As a surge
progresses work is done on the system, and heat will flow out provided
a temperature difference exists.
Examination of Fig. 9 shows that the
isentropic compression process prescribes the practical upper limit of
over-pressure.
It is believed
that a process following the saturation
line prescribes the lower limit of over-pressure for this tank,
oince
the surge introduces colder water into the tank, a potential sink exists
which could conceivably cause the process to enter the wet steam region.
however, a careful study of actual surges indicates that the stea.i
remains superheated throughout an in-surge.
Very long surges appear to
rallel closely the saturated line with slight superheat.
In addition,
all surges appear to be initially isentropic in nature
Based on the aoove reasoning ana actual tesc results, it is believed
t
the prediction curves approach finite values as 1 1 approaches in-
(for any given A_v ) is that which
v
o
?
would be obtained in a process following the saturation line.
_
dty.
rhe limiting value of a_t
P
24
TEMPERATURE -ENTROPY
CHART (APPROXIMATE)
FIGURE 9
26
For very short surges, the Limiting condition appears to be
of isentropic compression, thus establishin
i
a
for any given Av
On the actual installation, surges of less than 10
v
seconds duration <.re unlikely uuc to primary loo.) circulation ti.ie
.
considerations.
Thus the extrapolated curves for values of a t from 10
seconds to 30 seconds would seem reasonable.
Using the prediction
curves (Fig. S) , Table 2 shows a comparison of predicted and actual
pressure rises.
It is believed that basic objectives of this project have been
realized; however many avenues for further study appear to be open.
Although a procedure for rational analysis has been indicated, it appears
that a study of the effect of spray, and of negative surges, by these
methods would be of value.
In the field of heat transfer, a great deal remains to be done.
The combining of all heat transfer mechanisms into a single coefficient
is perhaps an over-simplification, although effective in this analysis.
It is known that several thermal paths exist, and a more complete
analysis could perhaps isolate the effects of each.
Heat can flow from
steam to metal, from steam to \vater (directly or by condensation;, and
from metal to water.
A comprehensive simulator study coula provide
knowledge in this relatively unexplored field.
27
TA3L J 2
RUN NO.
ACTUAL
(psia;
(;>sia)
I
PREDICTION
__(j2sia_i
2302
179
174
5
2045
317
313
4
2123
310
300
10
1556
236
229
7
2229
298
303
1506
302
239
13
2023
214
242
-28
2010
212
232
-20
2243
257
261
- 4
2240
130
110
20
2247
380
314
66
2113
150
187
-37
1101
139
112
27
2255
179
152
27
1911
420
417
3
1658
335
310
25
1630
329
324
5
2039
362
341
21
28
-
5
31BL10JRAPHY
1.
Kiefer, P. J.,
Kinney, G.F.,
Stuart, ...C.
2.
Keenan, J.H., and
eyes, F.G.
Wiley, 1936
3.
Jakob, U.
and
Hawkins, G.A.
ELEMENTS OF HEAT rRANSFER AND INSULATION,
2nd Edition, .Viley, 1952
4.
Uc Adams,
HEAT rRANS ISS10N, 3rd Edition,
McGraw-Hill, 1954
5.
.heeler, R.C.H.
,
,'I.H.
PRINCIPLES
2nd Edition, .Viley, 1954
:
1C
-
LERMODYNAlttCS,
PROPERTIES OF STE
THEORY OF THE ELECTRONIC ANALOG
COtlPUTER, Donner Scientific Co., 1955
2V
APPENDIX 1
DERIVATION cF EMPIRICAL RELATIONS FOR PRESSURE
AND COMPRESSIBILITY FACTOR
Since e can be computed from the First Law,
w + e
is available as a driving function, it is convenient to express
v
and
q +
e
function of
P as a
and
e
v
.
This can be done by constructing a
empirical relationship from steam table values, which will oe accurate
within certain limits.
The pressure range which is considered signifi-
cant for this problem is 1700 - 2700 psia, and no accuracy is claimed
beyond these limits.
Since
cannot be taken directly from the st-eam tables in the
e
superheat region, a plot of enthalpy vs. pressure is made initially,
showing lines of constant temperature and constant superheat (see Fig.
10).
Next, at various temperatures, various values of
are listed.
e
h
=
-
For each of these values
144
e is
h
,
P
,
and v
computed from the relation
From this taole, a new chart is
P v (see Table 3)«
J
constructed relating
e
,
P
,
v
and T (see Fig. 11).
as a basis, initially an expression for
e = f (P,v)
Using this chart
is obtained in
the following manner:
The average slope of constant
e
=
v (see
.1176 P +
c-i
1
.
v lines is
.1176.
For any one line,
The constant c-y is determined as a function of
Table 4 and Fig. 12;,
expression for internal energy,
c-^
e
-
1562 v + 533.
» .1176 P +
The resulting
1562 v f
538
when
,
tested at the extreme limits is found to jive considerable error.
Although undesirable from the standpoint of computer circuitry,
30
a..
Wtoi
U
5
))
3
6
u
j
TABLii 3
3
v (ft /lb)
P v
e
1900
191 V.
.2553
.2296
.5540
.1992
8u.5
76.6
71.8
70.3
1105.6
1091.1
iu74-o
1070.3
1185.7
1167.0
1145.6
1130.6
18U0
1900
2000
2059.7
.24U7
.2168
.1936
.1798
30.,
76.4
71.7
68.7
1105.4
1090.6
1073.9
1061.3
1167.0
1146.3
2000
2100
2200
2203.2
.2053
.1546
.1633
.1616
76.2
71.8
06.4
66.2
1090.8
167^.5
1054.6
1052.3
1167.7
1147.3
1123.8
1104.4
'2100
2200
2300
2365.4
.1962
.1763
.1575
.1442
76.3
72.2
67.3
1091.4
1075.6
63.
1041.4
1150.0
1123.2
1099
1087.7
2300
2400
2500
2531.8
.1702
.1526
.1342
.1277
72.5
67.5
62.2
59.7
1077.5
1060.4
1132.3
1107.1
1072.5
1067.2
2500
26U0
2750
2708.1
.1484
.1319
.1137
.1115
68.7
63.5
36.5
56.0
1063
690
1156.6
1137.3
1114.3
2500
2600
2700
.1594
.1447
.1299
73.3
69.3
64.8
1082.8
1067.5
1049.5
700
1160.6
1142.5
2600
2700
.1549
.1415
74.6
70.7
1036.
r
(° f)
630
(sat.
640
(sat.)
650
P
1156.1
1167.7
1145.8
1141.1
1700
1121.
(sol. )
660
(sat.
670
1118
.
.
(sat.)
650
(sat.
(psia)
h (3TU/lb)
32
(aru/ib)
101;
1037
102
.
.
.
1043.6
1016.6
1011.2
1071.8
ui
(C
v3
i
u.
TaJlK k
V
e
P
.1176 P
.12
1021.0
2620
308.5
712.5
.13
1030.4
2510
295.5
735.0
.14
1038.2
2409
284.0
754.2
.15
1045.4
2312
272. G
773.4
.16
1051.5
2217
261.0
790.5
.17
1056.8
2140
252.0
804.3
.18
1062 .0
2057
242.0
820.0
.19
1066.5
1982
233.5
333.0
.20
1070.5
1915
225.5
845.0
34
C
14
I
—— — — — — —— — —
!
...
!
|
!
;
-j—
I
.
f—
i
4-
!
!
;
I
i
!
;
.
additional modification is necessary, making
To accomolish this, the expression
points
C-j/
,
(
-
P
S.51
-
P
similar lines.
8.51 e
52,590 v
+
,
2
-
18
= 2.943 v
+
c lc,
- .318
for Z as a function of P and
-
-
30,7^0 v
v (see Fig. 13)
v lines is 1.715 x 10""
1.715 x 10~ 4 P
c. d
v
15
o
*
dio v
/
,
l.
3161
f (p, v) follows
=.
P is plotted from the steam tables,
First Z vs.
showing lines of constant
v
c. ,.
:
The calculation of an empirical relationship,
-
e
v )
Solving for these constants and substituting,
the final equation for P is
Z
,
P & 1900, 2100, 2300) three simultaneous equations in cjc
and c,r, result.
constant
v
g( P,
=
Substituting actual steam table values at three
is set up.
c-^y
+
e
4
,
.
The average slope of the
.
the equation for any one line being
is determined as a function of
c-ja
(see Table
v is,
Z
5
and Fig. 14)
= 1.715 x 10
The expression
.
P
2.943 v
+
.318
Once again a
v^ term is found necessary" for desired accuracy, and
using the same procedure as before the resulting expression for Z is:
Z
=
4
1.715 x 10" P
-
3b
9.47 v
2
+
6.048 v
-
.5693
TA3LE
5
1.715 x 10
-4
P
'18
.12
.4640
2613
.4475
.0165
.13
.4845
2509
.430u
.0545
.14
.5030
2410
.4130
.0900
.15
.5225
2312
.3960
.1265
.16
.5380
2211
.3790
.1590
.17
.5510
2140
.3670
.1840
.18
.5640
2060
.3530
.2110
.19
.5770
1985
.3405
.2365
.20
.5890
1915
.3280
.2610
.21
.6005
1850
.3170
.2835
38
APPENDIX 11
DERIVATION OF MACHINE EQUATIONS
Scaling factors are assigned as follows:
max
_
cK
60
3000
50
V
.01
:nax
50
raax
*p
=
T«aax
T
o<
e
-
Z
-
max
—
1^00
W
max
^raax
1
=
s
vv
'
'max
%L
50
max
£0
=
max
;j
50
%iax
oT
T
,02
50
w
c*\
3o
50
^uax
o<
30
50
e :oax
e
CX.
1500
r
=
1
=
1
1
t
Conversion of Theoretical Equations to Machine Equations
#1
w
=
- 1
c
P
W
x
- 1
p
W
s
-
1
12
f
'(f)
dt
.1351 (.1 P dv) dt] 10 c* p c*v c* t
dt
J
o/
° C.<*t
L
1.111
^
'
(.1
F
dv \ dtl
dt /
40
J
#2
w
e
-
q
e
=
2a q
+
.0333
q
=
¥
e
(
^
w
e
cx-e
+ cxT
c*
+
w
-0333
v
P
#3
+
.01 )T
.02/
o^
n
o^
.01 P V
ci^z^r 1.02 z
.01 F v 1
.02 L
#4
s
P
c
-
P
2
-<* v c<
e
= o<_ c
-
c,
50
c^y^v
+
C*r
<*,
#5
+ c^ v
c^ v
-
e
J
P
= 4.255
e
Z
=
c6 P
+-
c^c^p
o<
c
P
?
-»-
v
-
c
7
c
.5145 P
=
-
#6
q
= -
—x
- 50 c o^v c* v (.02 v J
fi
rr
-
v
lfc i0 K (T - T )+
lphoKoCp
;
c
(rng
41
z
2.3675 (.02 v
uK
qj
_+.
c, l k
<*t
*q° <t
<
3
V2
8
cX v
+ 3.024
-
2
o<
Z
.02 v^)
4.3825 (.02 v )- 52.6833
+
- 5.1167 v
(
)
-
cs.
o<7.
- 28.465
q
=
- 1 T.1716
p
-
K T
L
.1716 K T n
-1
.0)0445 K q
+
Calculated potentiometer settings, and components seiecied:
a
l
=
D.
F. depei ident
a
8
=
.4255
a
=
.4383
15
a2
11
n
aQ
=
.6864
ai6 =
.6864
a
11
n
a
=
.5117
a
=
.5000
i8 =
.0089
3
iu
=
.1111
a ii
.6333
a
r
.3333
a
12 *
.3U24
a
a6
=
.3357
a
i3 =
.5145
a
7
=
.3333
a
14 =
K
a
R
=
1 M
c
fl =
,!/£
m 10 Li
c
f43-
2/£
He - i0 M
C f8 =
,1/Af
1 m
C flO«
,l/*f
Roo z .1 M
Cf23-
.ou5/f
a
4
a
5
R, r
45
a
f
=
Rf30 s lu M
R 6o ..ill
M
R5
a
R^j_
= 10 M
.1
R 68 - .1 M
42
!7
19
20
21
=
L
=
45
.2368
=
.ii82
Q:
R?IM
-io u„ cos
urfc
€>
R-IOM
-AVvWv
io^= ioU.sin m^
*
R = /M
lou.u) sin cut:
-www-
-.ll
R<MM
-10 u. cos
wt
/{P\U.COS U>t
jH^>
R f -IM
VmEAN
R '' M
R = IM
-WWW
-
-10
RfiOM
VOLTS
R„-2M
-WWW-
_
MOTOR
/
R*MM
-I
-^f-it^i
1T1
.02V
M,
R=IM
-vwvvw-
.02 V
* 1 ^ 5£
R k -IM
-.ipjr
at
.01
PV
j-www-
m
— —
i
R = IM
10
R-Hfc-
-wvww
vvvv
M
^H§>-€
"L^ur
R-c=IOM
-'VWWVV
—
- Q*v
V.IM
a
|
—
R.'iwi
VW\
i
1
R/ IM
vVVy
-.IM
-2
K'lM
-WyWVW
R..IM
^H-<S>
PV
•»
—
R f =IOM
1
-vwww
—
•»
R=IM
—
-www
H
+2
Sz
+2
02TZ
Cf42'2/*
YVVWWRn'.IM
-(6?)
-.02T2
M,
C^-.oojy,
01
R'lM
-^wwwv
R-IM
R-IM
hVWMWK
-.02 V
-
-vwim
-VWWWV——*..
<
'
R-IM
-T
vvvvvv 1
-K
3—6
R-IM
jyWVvV
RM M
AWAV
R,,«ioM
-vAVW
"
•*
-K'
.
r
^)
R«a-.lM
VVWV—
*
M,
,
Mj
S,
COMPUTER
SCHEMATIC
FIGURt
15"
M>
DIAGRAM
Mx
FUNCTION
MULTIPLIERS
SERVO- MULTIPLIER KMPLIFIER
,
S, ,Sj
,54
SIGN
CHANGERS
APPEI'JDU 111
HAND SOLUTIOM OF THEORETICAL EQUATIONS
The first step in obtaining a hand solution consists of combining
the six theoretical equations to form two dependent differential
equations
Substituting from equation #2 into equation #4, and then into
equation
one obtains,
/'l
;
,
$
Since
\j
then
-^
gl
v^ e |
=
-clA
[
=
U
+
-
-U Co S/ilwt
u lU)S(;i
«
u,
+
„-
Co$
cot
atW
Substituting from equations
tt
v +
| v *. fVjdy
^ w .^ &
+
./ + e„ -
&
and //5 into equation ff3 , the
,74
resulting expression for T becomes,
it is now convenient to assi-r: values to the driving function,
initial conditions, and to evaluate the constants:
u
Q
OJ
=
.019 ft 3 /lb
c
2
=
8.51
C
=
.0418 rad/sec
C3
-
3.U7 x llA
c
c
+
=
5.259 * L°
1066 3TU/lb
cr
=
3161
.5957
c
=
1.714 x 10~ 4
v
mean =
eo
cx
=
'
l72 rt
ib
'
<
6
44
U
Y
8
as
6.U48
s 9.47
=
.5b93
c 12 ^
.1853
c
y
"I
Evaluating,
i*
=
/.253 *l0*%iy\ .o4/8 "t [j2 +"U7 - 36«0 V
+ £>i6G v*
-fr
6 9*]
Equation #6 is,
^
r
-C W KT + C, KT - C„K^
Arbitrarily assigning a value to K
,
evaluating constants, and
substituting the already derived expression for T into equation //6,
K
c
c
T
^
a<
.9254
=
lu
5 ' 72
X i0 3
±i
=
4.4i>
x 10~ 4
10940
°a
=
z
-5/.8v fa
-
~
=
-fxa
k_$__
\A
r
-36jo v/ fb!8ov +$>94]
W + 540 V - 3/0
i/
2-
_<?-
d- 4-/2mo q tS.770
t 30+ j
The two resulting dependent first order differential equations
bee one
where
/C,
^
/C2
,
/f3
(see Table C)
/^ are tabulated values of
.
45
V
,
^he driving function
TABLr, 6
Time
•
H
U Q COS cut
V
*L
K
.0190
.1910
-9.9o
230.5
395.7
5
.U186
,ly06
-9.88
231. U
395.5
2.6l x 10"^
10
.0174
.1893
-V.S2
232.0
395.1
5.10
15
.ol>4
.1873
-9.71
235. o
394.3
7.33
20
.0127
.1847
-9. 58
239.0
393.1
9.33
25
.0095
.1815
-9.41
242.5
391.8
10.87
30
.