Transient response of a 2-tank flash evaporator
by Thomas William Holzberger
A thesis submitted to the' Graduate Faculty in partial fulfillment of the requirements for the degree of
DOCTOR OF PHILOSOPHY in Chemical Engineering
Montana State University
© Copyright by Thomas William Holzberger (1969)
Abstract:
To provide information for the design of flash desalination plants, the dynamics of a 2-tank flash
evaporator were studied. Using the 2-tank evaporator available in the Montana State University
Chemical Engineering Laboratory, transient data were obtained for step increases and decreases in the
temperature of the hot feed water to the first flash tank in the system. A digital computer model was
written for the 2-tank evaporator that describes the dynamic response of the system. The computer
model was rewritten to describe a 4-tank evaporator, and tested for stability to upsets in temperature of
the inlet flow streams.
The digital computer model proved to be very versatile in predicting both steady-state operating
conditions and responses to various upsets in the inlet streams. Stability of a multi-tank flash
evaporator follows from the stability of the end tanks of the system. Stable operation exists only over a
narrow range of temperature and flow rates near specified design values. Upsets in the entering stream
temperatures of greater than 5—10 degrees Centigrade cause the water levels in the evaporator tanks to
rise or fall beyond the range of stable operations. Increasing the evaporator tank size slows the rate of
response to upsets but does not change the equilibrium operating conditions. Changing the tube bundle
heat transfer rate changes the equilibrium operating conditions but does not affect stability.
Proportional-Integral control of the inlet temperature and flow rate to the first flash tank is necessary
for stable operation. If the feed brine temperature is variable, it must also be controlled. Temperature
upsets entering the system are quickly corrected by Proportional-Integral control, and do not upset the
system more than 1/2 degree Centigrade past the second tank. TRANSIENT RESPONSE OF A 2-TANK E L A S K EVAPORATOR
Kr
.■THOMAS WILLIAM HOLZBEROER,
A tliesi's, submitted to the' Graduate- Faculty in partial
fulfillment of the requirements for the degree
of
DOCTOR OF PHILOSOPHY
in
Chemical Engineering
A p p roved;
____________ ZI&pNL
Head, Major Depart%@«1f/
mU
’
U L ^
CKairman, Examining Committee
Graduate Dean
MONTANA STATE UNIVERSITY
Bozeman, Montana
' 1DeeemberV '1969
tli
' TABLE OF- CONTENTS
Page
V i t a ........... ..
. ■......... ..
Li
Table of C o n t e n t s ......... ^
List of Figures-............................................... ..
Abstract
.
Iif
.'. .
iv
. . . . .... . . -. . . . -...................... . . .yii
Introduction ..........................................................
I
Background
3
..................- . . . . ■.......... ..
Description of Equipment
Experimental Data
8
,. . . . ■. . . '. . . . . . . .■ . . . . ,.
Theoretical Development
Analog Computer Model
. . . -......... -.
• .....
. I . ’. . ■.......... .'
. . .
.........
25
.........
29
Digital Computer M o d e l .................. ..........................
1.
2-Tank. Simulation . . . . . . . . . . . . . . . . . . . .
2.
4—Tank Simulation . . . . .
3.
Stability of a Flash E v a p o r a t o r ....................
4.
Rate of Response to Upsets
Control of a Flash Evaporator
13
.
34
34
. . ..............■ ......... 49’
. . .
........................
. ............ . ■....................
63
65
.
69
Conclusion . ■. .............. . . . . •• •.................... ..
84
Suggestions for Further Work . . . . . .
86
..........................
A p p e n d i x ......... •. . . ... . ,. . -.......... ..
. .. ■.............
88
Literature Cited ........................................................ 119
iy
LIST OF FIGURES
Figure
Page
1.
R-Siage Flash. E v a p o r a t o r ......... ..
2.
Existing Instrumentation
3.
Modified Equipment D i a g r a m ......... ................. ..
4.
Data Acquisition S y s t e m .........
5.
Experimental Data, 2—Tank. Evaporator, Step Rise in T q , T^
Response Curve
. . . . .. ■........... -. .. . . . .
..
6.
7.
8.
9-
.
. . ..................
......................... ..
...
4
9
. . .
11
. . ................. ' 1 2
.. • 15
Experimental Data,.2-Tank Evaporator, Step Rise in T , AT
Response Curve
. . . . . . . . . . .
........... . . . . . . .
l6
Experimental Data, 2-Tank Evaporator, Step Rise in T , AP
Response Curve
...................... . . . . . . . . . . . .
17
Experimental Data, 2-Tank Evaporator, Step Rise in T ,
Height Response Curves
........... . . . . . . . . . . . .
18
Experimental Data, 2-Tank Evaporator, Step Drop in Tq , T^
Response Curve
. . . . .. '. .r . . . . .............. t .
19
. :.
10.
Experimental-Data, .2-Tank.Evaporator, Step Drop in Tq , AT
Response Curve
. . . .............. ............................ 20
11.
Experimental Data, .2-Tank Evaporator, Step Drop in Tq , AP
Response C u r v e .................... ................... ..
21
12.
Experimental Data, 2—Tank Evaporator, Step Drop in Tq ,
Height Response Curves
. . . , ........... '...................22
13.
Analog Simulation - Step Response D a t a ............. ..
14.
Analog Simulation -
15.
' Comparison Between Experimental Data and 2—Tank Simulation,
Step Drop in T q , T^ Response C u r v e .......................... 40
16.
Comparison Between Experimental Data and 2—Tank Simulation,
Step Drop in T q , AT Response C u r v e ........... ..
as a Function of F q
. . ,
........... ..
. .
32
33
41
T
L I S T 'OF FIGURES. CCont.}
Figure
Page
17. ■ Comparison Between Experimental Data: and P-TanK Simulation ^
Step Drop in T q , AP Response Curye
. . . . . . . . .
18.
Comparison Between. Experimental Data and 2—Tank. Simulation,
Step Drop in T q , Height Response Curves . . . . . . . . . . .
42
.43
19.
Comparison Between Experimental Data and 2—Tank Simulation,
Step Rise in T q , T^ Response Curve
. . .......................,44
20.
Comparison Between Experimental Data and.2—Tank Simulation,
Step Rise in T q , AT Response Curve
. . . . . . . . . . . . .
45
Comparison Between Experimental Data and 2-Tank Simulation,
Step Rise in T q , AP Response Curve
. . . . . . . . . . . . .
46
Comparison Between Experimental Data and 2—Tank Simulation,
Step Rise in T q , Height Response Curves . . ............ ..
47
4-Tank Simulation, Step Drop in T q from 104 — 101 ° C ,
Temperature Response Curves
............. . . . . . .
52
4-Tank Simulation, Step Drop in T from 104 — 101 .°C,
AT Response Curves
. ..........................................
53
4-Tank Simulation, Step Drop in T from 104 - 101 0C ,
AP Response C u r v e s ........... .. . ■...................... - -
54
4-Tank Simulation, Step Drop in T from 104 - 101 ° C ,
Height Response Curves
...................... . . . . . . . .
55
4-Tank Simulation, Step Drop in T^ from. 73 - 68 ° C ,
Temperature Response Curves ........... ..
57
21.
22.
23.
24.
25.
26.
27.
28.
4-Tank'Simulation, Step Drop in T^ from 73 - 68 0C ,
Height Response C u r v e s ............... .. . . ................ 58
29.
4-Tank Simulation,, Step Rise in T from 103 — 10.6 0C ,
Response of T for Evaporators wi?h. Tank Areas of
8 ft2 ,-'20. ft2^ and 50 ft2
60
yi
LIST OF FIGURES. (Cont..)
Figure
Page
4—Tank. Simulation, Step Rise in T from 103 — 10 6 0C ,
Responso of T, for Eyaporators wi£h_ TuLe Bundle Heat^-Remoyal
Rates of 20Q0J"BTU/Lr °F» .Vo'OO BTU/Lr -9F , and. 8000 BTU/hr °F .
62
31.
Frequency- Response Study
66
32.
Existing Control
33.
U - T a n k ,Simulation, P-I Control of T .and F- , 5 qC Step Rise
in Inlet Stream to T Heat Exchanger, Temperature Response
C u r v e s ......... . . ? . ■........... .. . . . . -. -......... .. .
73
U-Tank Simulation, P-I- Control of T and F , 5 °C Step Rise
in Inlet Stream to T Heat Exchanger, HeigSt Response
Curves
. , . . .
. . . "................-. - * * •
*
7U ■
U-Tank Simulation, P-I Control of T and F , Step Drop
in T from 73 - 68 0C, Temperature Response Curves'- . . . . .
75
30.
34.
35.
36.
37-
....................................
........... .. -.
.
........... 70
C
■
■
■'
.
U-Tank Simulation, P-I Control of T and F , Step Drop
in T c from 73 - 68 0C, Height Response C u r v e s ............... 76
U-Tank Simulation, P-I Control of T and F , 5 0O Step Rise
in Inlet Stream to T Heat Exchanger, Temperature Response
.
C u r v e s ........... ?
38.
U-Tank Simulation, P-I Control of T and F , F °C Step Rise
in Inlet Stream to T Heat Exchanger, HeigSt Response
Curves
....................... ....................................78
39*
U-Tank Simulation, P-I Control of T and F , Change of T q
Set Point from IOU ° - 101 0C,.Temperature0Response Curves
Uo.
Ul.
42.
.
77
80
U-Tank Simulation, P-I Control of T and F , Change of T q
Set Point from IOU 0 - 101 0C,. HeigSt Response Curves . . . .
81
U-Tank Simulation, P-I Control of T and F , Chauge of T c
Set Point from 73 0 - 76 0C , Temperature Response Curves
. .
82
U-Tank Simulation, P-I Control of T and F , -Change of Tc
Set Point from 73 0 - 76 °C, Height "^Response Curves . . . . .
83
ABSTRACT
To provide information for the design of flash desalination p l a n t s ,
the dynamics of a 2-tank flash evaporator were studied.
Using the 2-^tank
evaporator available in the Montana State University Chemical Engineering
Laboratory, transient data were'obtained...for step increases and decreases
in the temperature of the hot feed water to the first flash tank in the
system.
A digital computer model was written for the 2-tank evaporator
that describes the dynamic response of the system.
The computer model
was rewritten to describe a 4-tank evaporator, and tested for stability
to upsets in temperature of the inlet flow streams.
The digital computer model proved to be very versatile in
predicting both steady-state operating conditions and responses to
various upsets in the inlet streams.
Stability of a multi-tank flash
evaporator follows from the stability of the end tanks of the system.
Stable operation exists only over a narrow range of temperature and flow
rates near specified design values. Upsets in. the entering stream
temperatures of greater than 5—10'degrees Centigrade cause the water
levels in the evaporator tanks to rise or fall beyond the range of stable
operations.
Increasing the evaporator tank size slows the rate of response to upsets but does h o t change the equilibrium operating
conditions.
Changing the tube bundle heat ,transfer rate changes the
equilibrium operating conditions but does not affect stability.
Proportional-Integral control of the inlet temperature and flow rate to
the first flash tank is necessary for stable operation.
If the feed
brine temperature is variable, it must also be controlled.
Temperature
upsets entering the system are quickly corrected by Proportional-Integral
.control, and do not upset the system more than 1/2 degree Centigrade past
the second tank.
, INTRODUCTION
The steady growth of urban population centers in arid areas and
the fixed quantities of ground water in non-arid areas have forced man
to look to the sea to meet the ever-increasing demand for fresh waters .
P r e sently, the cheapest way to purify seawater or brackish water for
human consumption is b y flash desalination.
In this evaporative pr o c e s s ,
seawater is heated to about 250 0C and cascaded through a series of tanks,
each at a succeedirigly lower temperature and pressure.
Vapor flashes off
from the brine and is condensed, yielding pure fresh water.
Two flash
desalination plants, one in Florida and one in Cuba, are currently
producing fresh water from seawater where other supplies of fresh water
are not available.
The Office of Saline water, an agency of the
Department of the Interior, is working to develop the, technology of
flash desalination to the point that fresh water from th_e sea will be
economically attractive.
The proper design and instrumentation of a large flash
desalination plant demands a knowledge of both steady-state and
transient response data.
No transient data for a flash evaporator has
been published in the literature, and no digital computer model is
available that describes unsteady-state behavior.
This study was
undertaken to obtain transient data from the 2-stage flash- evaporator
available in the Montana State University chemical Engineering laboratory
and to describe the transient behavior of the system in a digital
computer simulation.
It is hoped, that the results of this study- may "be
useful in the design of future flash evaporators.
BACKGROUND
A schematic drawing of an N-stage flash evaporator is given in
Figure I.
Fresh seawater feed is circulated, through a tube bundle
starting with stage N .
As the feed moves toward stage I 5 it is warmed
from its entering temperature.
The warmed seawater exits from the
evaporator and enters the' steam heat exchanger, where its temperature
reaches the inlet temperature.
Seawater-enters tank I and flashes to
an equilibrium temperature less than t h e ,inlet temperature and reaches
a corresponding vapor pressure.
The water that flashes is. condensed
into drip trays and removed as product.
the incoming feed in the tube bundle.
The heat of condensation warms
Tank 2 contains water at a lower
temperature and vapor p r e s s u r e; hence water flashes across the orifice
between the two tanks.
This process continues through tank R 5 where
roughly $0% of the water, has been flashed off.
discarded.
The waste brine is
A more detailed flowsheet is available (lp)•
The first practical multi-stage flash evaporation plant constructed
in the United States was completed at San Diego i n '1962 and moved to the
Guantanamo Naval -Base in Cuba in I 96U (12,6), where it was combined with
a steam power generation plant.
Subsequently, the Bolsa Island project
was proposed in Southern California as a combined power-desalting project,
but was discontinued when cost estimates rose dramatically.
The Office
of Saline Water (OSW) has proposed other large projects, ranging from 50
to 250 million gallons product water per day.
Designs w i t h comprehensive
FRESH BRINE
STEAM
WASTE BRINE
Figure I.
N-Stage Flash Evaporator
-5- '
cost estimates and complete scale drawings of equipment and
instrumentation, were'done (18 ,1 9 ,20), but no flash, desalination plants
were constructed.
A comprehensive hybrid computer simulation (2l) was
written by Electronic A s s ociates, Inc. , to test the design submitted
b y Fluor (19) to the Office of Saline Water.
Using 12 differential
equations and 44 supporting equations to describe the conditions in
each flash tank, EAI developed a mathematical model for a 39-tank
flash evaporator.
Because no experimental data were available describing
the operation of a flash evaporator, Electronics Associates, Inc., could
not test the model for transient response to upsets.
B y estimating data
where n e e d e d , EAI did develop a startup procedure and predict .-that for
steady-state operation the specified product water rates could be met.
To supply practical experimental data for large flash desalination
p l a n t s , the OSW is presently building a flash test module with a capacity
of IT million gallons product water per day (22).,
Kogan (l4) derived equations that describe the flash evaporator
w i t h respect to greatest steam economy and maximum thermodynamic
efficiency.
Arad (4) has done a system analysis of a large flash
desalination plant in which he studied plant cost versus plant life,
performance, efficiency, and effectiveness.
Karsimhan (17 ) derived the
differential equations describing a multiple-effect concentrating
evaporator, simplified these equations, applied boundary conditions, and
solved the differential equations mathematically.
Andre and Ritter (3)
-6took experimental data on a double—effect concentrating evaporator and
wrote a digital computer model describing their system.
Andersen,
G l a s s o n , and Lees (2) compared experimental■data from a single-effect
concentrating evaporator w i t h the results of an analog computer
simulation of the system, and suggested possible control schemes.
Itahara Cll) used dynamic programming to optimize the design and
operation of both a multiple-effect concentrating evaporator and a
multi-stage multi-effect flash evaporator in the steady state.
Dynamic studies have been done concerning the behavior of
reactors (7,15), packed liquid-liquid extraction columns (5) , and
distillation columns (2^).
Several other papers, perhaps more .closely
related, have been written concerning heat exchanger' dynamics (l,8,13).
In each paper, a mathematical model was analyzed using an analog
computer, a digital computer, or a tabulated solution to a differential
equation.
When this project was begun in October 1966, no experimental data
could be found in the literature describing either the transient or
steady-state operation of a flash evaporator.
No computer simulation
of a flash evaporator was available at that time.
The EAI hybrid
computer simulation was completed in October 1967, and the results
were published in condensed form in January 1969.
A complete copy of
the EAI simulation was not obtained until September 1969.
—7—
Transient.experimental data were taken on the flash evaporator
in the Montana State University chemical engineering laboratory from
September 1968 through February 19 69 •
To explain the data, a
simulation of the 2-tank flash evaporator was needed.
First an analog
computer simulation was used, and when this proved inadequate a digital
computer simulation was written.
The subsequent examination of the
complete EAI report showed that the approach of each of the two
projects was considerably different.
made later.
A detailed comparison will be
DESCRIPTION OF EQUIPMENT
The 2— stage flash evaporator available in the Montana State
University chemical engineering laboratory had been used previously to
correlate 2-phase f l o w through an orifice (lO).
Original instrumenta­
tion consisted of 10 thermocouples used to measure various temperatures,
2 temperature recorder-controllers, a temperature recorder, a thermopile,
and 2 U-tube manometers.
(See Figure 2.)
To better monitor the evaporator and produce controlled upse t s ,
several changes in instrumentation were made.
A Masoneilan Little Scotty
1/ 2" pneumatic control valve was placed in the inlet flow line and
connected to a Foxboro proportional flow controller to regulate the
inlet flow r a t e .
Two Magnetrol level transducers were purchased, one
for each tank, to indicate the liquid levels.
A pressure transducer
was obtained to monitor the pressure in tank I, and a differential
pressure transducer was ordered to monitor the pressure drop between
Thermocouple wells 2, 3, a nd.A A 1 (Figure 2) were used to
tanks.
monitor respectively the inlet temperature, the temperature in tank I,
and the temperature drop between t a n k s .
.Modifications to the equipment included:
I.
'
Closing the valve to the product-water pump so that no air
could leak back into the system through the pump packing.
Product water condensed as b e f o r e , maintaining the heat
b a l a n c e ,•and dripped back into the tank in small quantities
that did not alter the material balance.
3
6
9
4
Flow
I
Mano- i
DP Manometer
Figure 2.
Existing Instrumentation
-IQ2.
Placing vapor "bleed lines to a vacuum pump so that any air
leaking into the system Was removed.
3.
Inserting a bypass around the pneumatic, steam-supply valve to
the steam heat exchanger so that better step changes in the
inlet temperature could be m a d e .
b.
Permanently placing a 0.875" diameter orifice in place between
tanks I and 2.
(Bee Figure 3.)
5.
Using pure 'water (no salt added).
6.
Recycling all water in a closed system so that a constant mass
balance was maintained.
A data acquisition system, developed b y the Electronics Research
Laboratory (ERL) at M S U (Figure -4)
was used to collect the raw data.
Ten input channels were monitored each 20 seconds b y the digital
voltmeter.
Output b y the teletype was b o t h a data listing and a
punched paper tape.
The punched paper tapes from each run were
converted to data card, decks to enable analysis on M S U 1s SDS Sigma-7
computer. . The reduced data were listed and plotted using the computer
lineprinter.
L
H
I
Transducer
Figure 3.
Modified Equipment Diagram
Thermopile
'D' Dry cell and typical
transducer circuit
Hewlett Packard 341*0 A
Digital Voltmeter
<30000
MSU ERL Model 862A
Analog Scanner
O
MSU ERL Model 682A
Digital Translator
C=D
i
20-second timer
Teletype Corporation Model 33
Teletype
Figure U.
Data Acquisition System
EXPERIMENTAL DATA
The data inputs read by the data acquisition system are listed
below:
I.
2.
3.
4.
56.
7-
temperature of feed brine entering tank I
To
temperature in tank I
Tl
AT"
temperature drop from tank I to tank 2
vapor pressure in tank I
P1
AP
pressure drop from tank I to tank 2
height of water in tank I
H1
height of water in tank 2
H2
(See Appendix I for a complete definition of variable names.)
The 3
additional channels of the data acquisition system were used to"monitor
the source voltage of the
1D 11 dry cell..
The scan rate of I scan every 20 seconds gave 3 data points for
each variable each minute.
The digital voltmeter read to a maximum
value of 99-99 millivolts, With an accuracy of + .01 mv.
For T q and T^,
with typical readings of 5.00 my, measurement error was + .01 mv.per
5.00 mv. or
0.h% of calibration.
For AT, with a typical reading of 1.00
mv.,, measurement error was + .01 mv. per 1.00 mv. or
P ^ , A P , H^, and
2% of calibration.
were measured above 10.00 m v . , and hence were
accurate to the accuracy of the calibration.
T
o
and T n were calibrated between an ice bath and a boiling-water
l
bath.
Since all runs were'near the boiling p o i n t , the calibrations
were felt to be good within l/2°G. .AT was calibrated between 2 boiling
water baths and an ice water—boiling water bath, a difference of 95 0C .
The total mv. output at a difference of 95 °C matched that given in the
thermocouple tables for an iron-constantan thermocouple.
Ho w e v e r ,
since a small temperature difference was measured .(3-4 °C) compared to
the calibration, the calibration accuracy was unknown.
.during every run because of drift.
was calibrated
and Hg were stable to drift but
were dependent upon the water density, which was temperature dependent.
Since temperature increase runs and temperature decrease runs covered
different temperature r a n g e s , one calibration of H^ and Hg was run for
temperature increase runs , and one calibration was run for temperature
decrease runs.
The source
'D' cell was replaced periodically to insure
source stability.
Startup and lineout of the evaporator took from 2-3 h o u r s , while
each run lasted from 35v^0 m i nutes.
Step increases and decreases in
T
were run because they were the easiest to produce a n d , therefore,
c
the most reproducible and accurate.
Eight temperature increase runs
and 4 temperature decrease runs were chosen out of many of each type
as reproducible runs at one set of operating conditions.
The runs of
each type were averaged arithmetically for initial steady-state v a l u e ,
maximum value from the .step u p s e t , and final line-out value to give a
composite run for a step rise in T q and a step drop in T q . (Figures 5-12.)
Temperature, Degrees
0 0 0 0 (30 0 0 0 0 0 0 O
Composite curve
OO
Data - 2/13/69
Time, Minutes
Figure 5*
Experimental Data, 2-Tank Evaporator, Step Rise in T
T^ Response Curve
-91-
AT, Degrees
4.1 -I
2.9
I
15
20
25
T i m e , Minutes
Figure ‘6.
Experimental Data, 2-Tank Evaporator, Step Rise in T
AT Response Curve
5- 2 — ,
Composite curve
O O
b.J -
Data - 2/13/69
so
Inches
K
1
H
„ 4.2 -
-Q
I
HDOOO
3.7
1
10
I
0
5
"T
15
T
20
5
1
30
T i m e , Minutes
Figure
J.
Experimental Data, 2-Tank Evaporator, Step Rise in T
AP Response Curve
O
I
35
Composite curve
O O
Data - 2 /13/69
Height, Inches Water
n
O O O O Q
- ) O O O O Q
Time, Minutes
Figure 8.
Experimental Data, 2-Tank Evaporator, Step Rise in T
Height Response Curves
®
DOOOO
Composite curve
Data - 2/21/69
%
99 -
QOQOOOn
0 0 0 0 0 00 6-0 0 0 0 0 0 00 00
-
T i m e , Minutes
Figure 9-
Experimental Data, 2-Tank Evaporator, Step Drop in T q
T^ Response Curve '
Composite curve
AT, Degrees
O O
Data - 2/21/69
o^ooOnOo^oo Qp n 00 0 0 no on
Time, Minutes
Figure 10.
Experimental Data, 2-Tank Evaporator, Step Drop in T
AT Response Curve
^e e o -O G
Inches Hg
3.5 *
3.0 -
%
2.5 -
Composite curve
Data - 2/21/69
T i m e , Minutes
Figure 11.
Experimental Data, 2-Tank Evaporator, Step Drop in T q
AP Response Curve
Composite curve
Data - 2/31/69
H e i g h t , Inches Water
) O Q O O
OOOOOOOOOOOCOOO
1IQaooopo
T i m e , Minutes
Figure 12.
Experimental Data, 2-Tank Evaporator, Step Drop in T q
Height Response Curves
—23"^
The solid lines are composite runs ; the points, are actual points from
one run.
A rise in T q produces a corresponding rise in T^ and upsets
in AT and A P .
A rise in T^ increases the vapor pressure in .tank, I,
forcing more water out of tank I into tank 2 , thus raising the level in
tank 2.
AT and AP rise sharply- .as■hotter water enters tank I, and then
fall to new equilibrium values as the hotter water enters tank 2 .
A drop in T q produces a corresponding drop in T^ and upsets in AT and
AP.
A drop in T^ decreases the pressure in tank I, forcing less water
out of tank I into tank 2,, thus dropping the level in tank 2.
AT and
AP drop sharply- as cooler water enters tank I, and then rise to new
equilibrium values as the cooler water enters tank. 2.
rise, the flow rate is 9^-50 #/hi.
For a temperature
For a temperature drop, the flow rate
is 9300 #/hr.
Upon examination of the composite curves, several peculiarities
of .the system were noticed.
Response of the system to a step increase
in inlet temperature is slower than the response o f the system to a step
decrease in inlet temperature.
Furthermore, although the AP curves
return nearly to the initial v a l u e s , the AT curves do n o t .
Nor do the
AT curves follow similar "but inverted paths as do the AP and T^ curves.
And, the displacement of the T^ curve from the T q curve is not constant,
but varies from run to run and within any one run itself.
In spite of the seemingly inconsistent features of these runs, the
r
-24-
curves resulting from step input temperature changes were reproducible.
The composite runs, agree very well w i t h at least one run for each
temperature u p s e t .
T
THEORETICAL DEVELOPMENT
In order to explain the transient behavior of the 2-tank flash
evaporator, the theoretical relationships between the variables were
developed.
Differential equations were written describing the process.
About stage I (see Appendix I for a complete definition of terms u s e d ) :
Heat balance:
Input [F0Cp (T0-Ir e f ) + FoCp (t2-tr e f )] 40
Output [F1Cp (T1-Ir e f ) f FoCp (P1-Pr e f )] 46
t 1+ t ?
Accumulation P ^ C p ^ l ^ r e f } + pV iv I11I + W lCp (~
' tr e f ) l6 + A6
-‘OLVp'h-hef1 + pVitA
+ wI0Pt-iT a " Vef11I9
0r- FcCptTo-Vef> + V p tV - tPef1 " Fl0p tTT tr e f 1
dV
= PpC V1 — ^ + PpC T
-FoCpttl-tref1 hL p I de
L p
dv
’ 11IpV1 d0
dpV l W1
dt
I .
H1V1 -- — + -A
i i de
2
Vde
dV
- PtC
t
jL p ref d0
2,
(l)
-26Mass Balance:
Input
F qAQ
Output F^AQ
Accumulation (V1Pl + ^ 1Pt
) |0 + AG
I
0r- fO
- Fi =
- (Y1P 7 + V 1
IkL
I
} |0
(2 )
+ Ti V I 1
’
Energy balance on vapor s p a c e :
Input
F C (T - T n ) AG
op
o
I
(UA) 1 CT1 -
Output
Accumulation
Or,
AG
Xm p V n 0 + AG
1I t I 1
F_C_(T_ - T n ) - (UA)1 CT1
o p- o
I'
1t Ip V
1 16
tI ^ 2 ,
, ■
.
dvn
'dp v
I
I
+ Xm v
T1 I d G
xT 1pV 1 dG
'• (3)
Volume in stage I:
V n + v n = Constant
I
I
IV1
dG
=
Iv1
dG
The flow of water between flash tanks ,of ,a flash evaporator through a
sharp-edged orifice was correlated b y R. C . Huntsinger (10) as:-
-27-
& 0-AP)g_^/2 1.826 ,
.7434
C-AP)'
37.83C1/2
PX
QC-AP
'
Or, with. D = .875 in.
.174 .2566
Q = 0.19207
Where,
.1564
O-AP)
.5694
#/min. .
(5)
p = # /ft.- hr.
p =
#/ft^
X = ft - Ih/lh
m
AP = Ihf/ft2
Q = flow, #/min.
Vapor pressure equation:
Los10P = 7.9668 - i f f i i i
* = °C
'
(6)
P = mmHg
For stage 2, the following equations were developed as those for
stage I:
Heat b a l a n c e , stage 2:
+ FoSCtrt^)
d.Tp
PyC
TT-
L p 2 d0
v
-^Vp
+ %v^2 a r -
" ^ ^ r e f )
av
+ P,C_1L — -
L p 2 d0
W
dt
_ p^C t
dVp
— -
L -p ref d9
dt
+ 2- C p C a T + a r >
"
dy
+ p- .H-
VgVg ae
C7)
-2,8Mass b a l a n c e , stage 2:
f I-fO
=&
cV
l
(8).
* t2 V !
C.
Energy b a l a n c e , vapor space, stage 2:
.-^Pi
f I0P c t X-1S 1
- (UA)2 CT2 -
2
1
■ AT 2T 2 d6
(9)
+ Xm
P
T 2 "T 2 d0
V o l u m e , stage 2:
^ 2
de '
-dY2
de
These equations define the 2-tank flash evaporator.
(10)
M A L O Q COMPUTER MODEL
The first attempt to solve the system of differential equations
and auxiliary equations describing the 2-stage flash evaporator was
made using the TR-48 analog computer.-
The following simplifications
were m a d e :
1.
Temperature drops t^-t^ and t^-t^ in the tube bundle were
given fixed values because of the limiting calculation
capabilities of the analog computer.
2.
The vapor pressure of water as f (T0F ) ’was linearized as .
P = 1.36*T-212.-6 cm Hg.
3.
The vapor density of water as f (Tq F) was linearized as
P.V. = .0006*T-. 0902 #/ft2
3-°F.
O
O
#/ft3 and p^ = 0.032 #/ft3 *
4.
For average v a l u e s ,
= 60.0
5*
A heat capacity for water of I BTU/#-°F was used.
The area of tank I was measured at 8.88 ft
2
was measured at 6.57 ft .
2
, and the area of tank 2
The total rectangular volume of each tank
was measured, and the tube bundle volume was subtracted from each,
giving volumes I and 2 equal to 24.9 ft
3
3
a n d .18.9 ft , respectively.
Neglecting tube bundle holdup w^ and vapor density and volume
t e r m s , changing the volume terms to height times area terms and ex­
pressing the heights in inches, and letting
gives:
,
'
■
= 0°F in Equation I
' ,
-30-
d.0
IP
—
'P-Yt 1A 1
» 0C V V
t2 i
dli
^
(111
Tlie total differential of Equation 2 is:
dV1
dp
'"T1
dpL
F o ” F i = pL d e ~ + v i d e ~
+ Ti da
+ pT1 de”
dV1
The sum of the last three terms is less than I , while p^
&Q~
greater than 100, allowing the last three terms to "be neglected.
Thus,
Equation 2 "becomes:
dV%
ae
^
(12 )
(Fo - h 1
Or,
dh
^
ae
= inches'
(13)
Similarly for stage 2, neglecting the tube bunlle holdup
density and volume t e r m s , and letting .t
19
ae
\ p A h ^f I t I
L 2 2
= 0.0 in Equation T gi v e s :
Pt t Pa p
“ F o^T 2- t 3+ t 2 ^ “
and vapor
12~
dh
d F *}
(IU)
And, for Equation 8, by expanding to the total differential and
neglecting the last 3 terms, the resulting mass balance on stage 2 is:
dim
ae
hg = inches
(15)
Auxiliary Equation 5 reduces t o :
F 1 = 2433.859 (-AP) *5^9ll1 #/hr.
(16 )
■
’
•
■
'
!•
-31The pressure drop "between t a n k s , using a linearized vapor pressure for
w a t e r , is:
(-AP) = .1867 Ch1-L2J + !.SS(T1-T2 )
(17)
This system of equations w a s ■amplitude scaled and time scaled and
wired onto the TR-48 analog computer.
Using this simple system,
k6 of
the 48 available amplifiers were u s e d , as well as all multiplier units
and one VDFQ unit to generate F 1 = f (&P).
This simplified system
simulation behaved only qualitatively like the experimental data for
H 1 and Hg corresponding to a step change in T q (Figure 13).
The '
simulation showed such great stability that values of H 1 and Hg
returned to original leve l s .
Values of Hg as a function of F q (Figure
lU) lined out to steady-state levels .from a picked starting value of
12 inches of water qualitatively the same as real evaporator data
would.
However, time response of the simulation was 2 to 3 times
slower than the experimental data.
■
Because the TR-48 analog computer
was used to capacity for this simplified model, no changes were made
in the model and analog computer work was dropped in favor of a digital
simulation.
-32-
205 “
CO
<u
(D
bD
<D
Q
0
200
~
td
0
1
BH
195 -
H e i g h t , Inches Water
Rose
13 -
10 -
T -r
T i m e , Hours
Figure 13•
Analog
Simulation - Step Response Data
-33-
Inches Water
F 0= 5 0 0 ( W h r
W
CM
Time, Hours
Figure l4.
Analog Simulation -
As A Function of F q
DIGITAL COMPUTER MODEL
2-Tank Simulation
A search for a general-purpose computer program that would solve
a system of non-linear differential equations using M S U l's SDS Sigma—7
digital computer showed that no such program was available.
Therefore,
a simulation computer program was written to solve, the specific set of
equations describing the •2-tank, flash evaporator.
The integration
technique chosen was the Runge-Kutta method as described b y Kreyszig(l6)
To enable the simulation to be run on a computer other t h a n -the SDS
Sigm a - 7 , the simulation was written in Fortran I V - H .
Results of the
simulation were listed and printed using the computer lineprinter.
The 2-tank simulation is given in Appendix B - I ; the plot program.islisted in Appendix B-2.
Use of the digital computer provided a simulation that described
the 2-tank flash evaporator m u c h better than the analog computer
simulation did.
The greatest advantage of the digital computer was
that it gave greater accuracy and flexibility in the selection of
■
auxiliary equations.
'
For flow from tank I to tank 2, Equation 5 was
used in its complete form having q, p ,, and
X as functions of temperature
Equation '6, the vapor pressure of water as a function of temperature,
was written into the simulation exactly, without linearization.
Work with the system of differential equations using M S U *s SDS
-35Sigma-7 digital computer sIiowed that a further simplification in the
system of differential,equations could "be made.
dh^
eliminate — — in Equation Il yields:
a ir
2
(18)
frOtiO-1I1 - rOftU t2 11
=
Using Equation 15 to eliminate
dT
Using Equation 13 to
12
ah2
,
in Equation 14 gives:
(F1 (T1-T0 ) - F_(t —t )}
2
ov 2 3;
(19)
P Aghg ^ l v I
Equations 18 and 19 were further modified "by defining (t^-t^) = TDROPl
and (tg-t^) = T D R 0 P 2 , where TDROPl and TDROP2 were functions of the
temperature difference "between the tube "bundle and the temperature of
the condensing vapor inside each tank:
dT„
a r = V r r frOtiO-R 1 - rOfmffiopi'11
(20)
Jj J_ J-
dT2
12
a T = v r r frIfV
■ Il C C
Equations
R
1 - ro (TDB0p211
(21)
20, 21, 13, and 15 comprise the system of differential
equations describing the 2-tank flash evaporator.
The performance of the evaporator was found to be highly sensitive
to the amount of heat removed by the tube bundle in each tank.
The heat
removed was expressed as a temperature rise in the cooling water flowing
through each tank.
The limiting heat transfer mode in the tube bundle
in each tank was found to be the heat transfer rate from each tube to
-36the cooling water flowing through the tuhe.
If Q. is the heat removed
b y the tube b u n d l e , M Is the flow' rate per unit time through the tube
bundle, UA is the overall heat transfer coefficient inside tube area of
the tube b u n d l e ,
ATrise
is the log mean temperature driving force, and
the Increase in the temperature of the cooling w a t e r , then the
following equation h o l d s :
Q = (UA) ATn = MG
AT .
Im
p
rise
Or,
'
(UA) AT,
rise
MG
If the temperature difference is small, a log mean temperature
difference approximately equals an arithmetic temperature difference.
For a tube of temperature'T^ filled with brine of entering temperature
T c ,'the temperature rise of the brine i s :
AT
(ml
rise .
-tC1 + (TS-1C - fflW
1
MG
P
Or,
(UA)
MG
rise
For
UA
I +
2MC
+ (T
s
- T )
c
= I BTU/#-°F, the temperature rise Is:
ATrise = 2M + (UA) + ^ s
~
The change in the temperature of the cooling ,water flowing through the
tube bundle in each tank is t h e n :
—37—
™ BC» 2 C y
ZCUAl
- a, * (DA)
■ H f 2 - Tc )
(22)
2
. 2 (UA)1
TDROPl (T15T2 ) = m + (UA^
* (T1 - T c - TDROPg(Tg))
(23)
For the 2-tank evaporator, the tuhe. bundle In tank I had an inner—
2
tube heat transfer area of 37*8 ft
•2
inside area of 26.9 ft .
, and the tank 2 tube bundle had an
The best fit of the experimental data by the
2-tank simulation was found for U 1 = 7-5 B T U /hr-ft2—°F and Ug = 90
2
BTU/hr-ft -°F.
The flow rate through the tube bundle was slightly over
9000 # / h r , which gave a tube Reynolds number of 4000 to 9000 and a
2
calculated heat transfer coefficient of approximately 200 B TU/hr-ft -0F.
The value of Ug appears reasonable because the evaporator had been run '
extensively under corrosive conditions.
The value O f U 1 is very low,
and can only be explained b y assuming that either extreme scale existed
in tube bundle I or that the difference.measured between T and T 5 was
.. 1 ■ ■
’ .
. ..
°
■1
in error.,
H o w e v e r , there are no inconsistencies•in the data; there is
less than I0C difference between T q and T1 .in/every run.
The development of the 2-tank digital simulation did show why the
laboratory evaporator behaved as it did..
The amount of heat removed b y
the tube bundle in each stage accounted for several important effects,.
The amount of heat removed governed the temperature drop .from stage to
stage.
The variation in the heat-removal rate caused the varying
-38displacement between the T q and
curves, and explains the differences
between the AT curves for a temperature rise in T q and a temperature
drop in T .
In a temperature step rise upset, hotter water entering
tank 2 increases the temperature driving force for heat transfer in the
tank 2 tube bundle.
Thus, the water in tube bundle 2 reaches a higher
temperature and warmer water enters the tank I tube bundle.
In tank I,
the temperature driving force between the tube bundle and the flashing
vapor is less, enabling T^ to reach an equilibrium temperature nearer
to T q than it had reached before the upset.
Under this same upset,
AT increases to a higher value after the upset without dropping back to
its initial level because of the increased heat transfer in the second
tube bundle.
For a temperature drop, AT tends to return closer to its
initial value because the upset decreases the temperature driving force
between the tube bundle and the flashing vapor in tank 2.
Less heat is
removed b y the tube bundle in tank 2, which raises T^ slightly from
what it would have been if the heat transfer rate- to the tube bundle
had remained constant. T^ undergoes little change because of its l ow
heat transfer coefficient, so AT increases to only about half way
between the peak upset value and initial value.
The varying heat-removal rate also gave heights of water in the
two tanks that lined out to new levels slowly, without returning to
the original height levels as the analog simulation did.
The rate of
flow of water through the evaporator and the temperature drop from
-39tank to tank set the pressure differential between ta n k s .
in tank 2,
the amount of heat lost to the' surroundings- was found to be negligible
compared to that removed by. the tube bundle.
In tank I, the heat lost
to the surroundings- m a y have had some effect because
-was so small.
The inlet temperature of the cooling water was found to be one of the
important variables regulating the performance of the entire system.
A comparison between the experimental data and the ,computer
simulation is shown in Figures 15-22., For both temperature rise and
temperature drop curves, T^ curves agree well.
The AT curves, which
were the least reliable experimentally, are off b y 0.3 to 0.5 0C , but
do agree well in shape.. The AP curves agree well for initial and final
v a l u e s , but disagree in the total deviation at the height of the upset.
Since there was some overshoot (up to" 2 0C ) in. the. feed temperature
T
in each run in a n attempt to make a sharp step'change ,■ this
difference does not seem unreasonable.
The height curves agree fairly
well in each case. All- steps ,show excellent response rates to those
rates observed experiment a l l y .
For a temperature drop., the simulation
flow rate of 9120 #/hr is 1.9% less than the experimental flow rate of
9300 #/hr.
For a temperature rise, the simulation flow rate of
9600 #/hr is 1.6% higher than the experimental flow rate of .9^50 #/hr.
A cooling water inlet temperature of 77 0C was used for the temperature
drop r u n ; one.of
Jd °C ,for the temperature rise run. •
101 -I
Experimental Data
-Otp
Simulation
Time, Minutes
Figure 15.
Comparison Between Experimental Data and 2-Tank Simulation,
Step Drop in T q , T j Response Curve
W
I
I
\
-----
Experimental Data
-----
Simulation
\
\
\
4.0
\
AT, Degrees
o
3.5
Jr-
H
I
3.0
2.5
0
T
—!
-----------1
5
10
15
I
20
I
25
I
30
Time, Minutes
Figure l6.
Comparison Between Experimental Data and 2-Tank Simulation,
Step Drop in T _ , AT Response Curve
35
A P , Inches
3.5 -
3.0 “
Experimental Data
Simulation
2.5 “
T i m e , Minutes
Figure 17.
Comparison Between Experimental Data and 2-Tank Simulation
Step Drop in T q5 AP Response Curv=.
19
Experimental Data
-----
Simulation
-Er
H e i g h t , Inches Water
\
I;
0
I
i
I
5
10
15
I
20
I
25
I
30
Time, Minutes
Figure 18.
Comparison Between Experimental Data and 2-Tank Simulation,
Step Drop in T q , Height Response Curves
I
35
101
Experimental Data
Simulation
Time, Minutes
Figure 19.
Comparison Between Experimental Data and 2-Tank Simulation,
Step Rise in T ^ , T^ Response Curve
It.I -
AT, Degrees
u 3.7
Experimental Data
Simulation
3.3 —
Time, Minutes
Figure 20.
Comparison Between Experimental Data and 2-Tank Simulation
Step Rise in T , AT Response Curve
5-2 -I
Experimental Data
A P , Inches Hg
Simulation
h.2 ~
Time, Minutes
Figure 21.
Comparison Between Experimental Data and 2-Tank Simulation,
Step Rise in T^. AP Response Curve
Experimental Data
H e i g h t , Inches Water
Simulation
Time, Minutes
Figure 22.
Comparison Between Experimental Data and 2-Tank Simulation,
Step Rise in T ^ , Height Response Curves
-48The 2-tank simulation developed in this paper is quite different
from the simulation developed hjr Electronics Associates, Inc.
The EAI
simulation uses 12 differential equations and 44 supporting equations
that require extensive design and experimental data. ' Every- possible
effect upon the operation of the system is represented.
Hot only are
heat and mass balances written for the brine, but a mass balance for
the salt concentration, the effect of boiling point rise, and the
changes in vapor pressure from the finite rate of air leaking ..into the
system are also included.
Even the pressure drop of the vapor as it
moves through the demisters to the tube bundle is represented.
If the
water level in any tank drops below the orifice opening between tanks ,
the correlation for flow between tanks changes.from 2—phase liquidvapor flow.to include vapor flow also.
The complexity of the model
gives prohibitively long execution times, for a digital computer, forcing
the solution for the system of equations to be calculated using the
hybrid computer.
The 2-tank digital simulation represents the differential mass
and energy balance for the brine in relation to the heat removal rate
of the tube b u n d l e .
The effects of salt concentration and design
features other than orifice size, tank size, and tube bundle heatremoval rate are neglected.
With these simplifications, the model still
represents accurately the response of the laboratory evaporator to
upsets in inlet temperature.
The less complicated model permits
the solution to the model to.he obtained using the digital computer.
' Although the digital simulation is m u c h less involved than the
hybrid simulation, the hybrid simulation is much faster in execution
t h a n 'the digital simulation.
Sixty- minutes of simulated time for a
39 tank evaporator takes 20 minutes of execution time using the hybrid
computer.
Sixty minutes of simulated time for a 2-tank, evaporator
takes 9-78 minutes of execution time using.the digital computer, and
extrapolates’ to 3 hours of execution time for a 39-tank digital
simulation run of 60 minutes.
However, much pertinent data.can be
obtained for a large flash evaporator b y examining only a few tanks.
T h u s , the digital model may be used to simulate less than the total
number of tanks in the system, and execution times will be less than
3 hours.
4-Tank Simulation
The 2-tank flash evaporator available in the MSU chemical
engineering laboratory was unique for several reasons.
First, the
tube bundles in each tank were different from each other, and hence
made each tank behave differently.
Second, the laboratory evaporator
consisted of only 2 tanks, and could not .show experimentally if an
evaporator with more than. 2 tank's would behave similarly.
water was recirculated in a closed system.
2 fell, giving an artificial stability.
Third, the
If tank I rose, then tank
To overcome these limitations
— ^O-
and describe more fully the dynamics of the"2-tank flash evaporator, a
simulation of a 4-tank flash, evaporator- teas teritten.
The general differential equations of a 4-tank flash, evaporator
were derived like those for the 2—tank evaporator and simplified to
the following f o r m s : -
I
;12
d0
.
F o (TBROPl)}
124)
F o (TDR0P2)}
(25)
h r tii2 (T2-T3 ) - r o Cn)R0P3l}
Ii O j
(26)
PLA lhl
dT,
dF
55S
trI tlI-1 = 1 -
dT
^
d0
dT
4
12___, (F^(T -Tlt)
P l A 4H]
d0
ahi
d0
.
,
(f - f J
(28)
12
,
(Fn-Fn )
(2?)
(Fn-Fn)
(30)
(F0- F ll)
(31)
p La 2
dh
b
d0
d0
(2?)
PLA 1
dh2
d0
dhu
12
f o (t d r o p 4 ) }
12
PLA 3
'
12
PLA 4
These equations were programmed using Fortran I V - H , and are given as
the complete program in Appendix B-3.
A program, was also written
-51(Appendix B-4) to plot the simulation results.
The physical parameters of the 4—tank evaporator were chosen to
be similar to those of the 2-tank laboratory evaporator.
standardized at 8 ft
6.57 ft
Tank area was
for each of the 4 tanks- compared to 8.88 ft
for the 2-tank evaporator.
and
The maximum allowable height of
water in each tank was- set at 20 i n ches, the same as in the 2-tank
evaporator.
The tube bundle cooling capacity was set at 2000 BTU/hr— °F
compared to 2440 BTU/hr— °F for tank 2 and 28.5 BTU/hr— °F for tank I of
the 2-tank evaporator.
The diameter of the orifice between each of the
4 tanks was set at 0.875 inches.
Using the 4-tank simulation, open loop response studies were run
of step changes in T q and T ^ .
shown in Figures 23-26.
The results of a 3 0C step drop in T q are
The temperature curves, AT curves, and AF
curves show that the temperature upset tends to decrease in size as it
progresses through the system.
The upset moves more rapidly through
the tanks than the rate of flow of water from tank to tank would indicate
the upset reaches tank 4 within 3 minutes after being initiated into
tank I, while it would take at least 8 minutes for the water in the first
3 tanks to be totally exchanged by the hotter water n o w entering tank I.
The height curves tend to line out to new equilibrium values.
an exponential-decay-shaped path, while
conditions.
and
declines and finally levels out.
follows
react to prevalent
The size of the upset
105 -I
100 -
Time, Minutes
Figure 23.
It-Tank Simulation, Step Drop in T q from 104
Temperature Response Curves
101
C
3.8
AT, Degrees
3.3 -
.AT2-3
2.8
-
ATI-2
Time, Minutes
Figure 24.
4-Tank Simulation, Step Drop in T q from 104 - 101 0C,
AT Response Curves
A p , Inches
AP3-U
API-2
Time, Minutes
Figure 25.
U-Tank Simulation, Step Drop in T^ from IOU - 101 °C
in AP Response Curves
H e i g h t , Inches Water
19 -|
I
Vl
Vl
I
I
10
15
20
25
T i m e , Minutes
Figure 26.
I-Tank Simulation, Step Drop in T
Height Response Curves
from 104 - 101 °C,
I
35
-56-
was. s m a l l , a 3 °C cha n g e , yet the system nearly went dry in tank 4.
A
larger upset would have exceeded the limits of stability of the system,
and tank 4 would have gone dry.
A 3 0C step rise in T q shows a similar
progression of the upset through the t a n k s , and corresponding changes
in heights.
is more stable to a step rise in T q w i t h these initial
heights in each tank, and tends to stabilize as long as the upset is
not too large.
In comparison w i t h the 2-tank model, the 4-tank model reacts less
to a change in T q .
In the 4-tank model, tanks 2 and 3 are not directly
coupled on both ends as tanks I and 2 are in the 2-tank model.
Upsets
decrease in intensity while passing through tanks 2 and 3, while the
conditions within the tanks are not greatly affected.
coupled to tank 2, tank I reacts less to the upset.
weaker upset and reacts less.
B y being
Tank 4 receives a
The upset in T q is spread throughout more
tanks, upsetting any one tank less.
A.step drop in the cooling water or fresh brine temperature T^
from 73 to 68 °C is shown in Figures 27—28.
Cooler water in the tube
bundle increases the rate of heat transfer from.the vapor in each tank
to the tube b u n d l e , lowering'the temperature in each tank.
Because the
upset begins with the end tank, tank 4, the effect is strongest there.
The drop in the tank temperatures decreases until, in tank I, there is
little change in T^.
rises because more heat is removed there by
105 I
T
T
100
0
I
Temperature, Degrees
o
T
2
95 "I
i
Vl
90 "I
I
T
85
i
0
5
r
10
i
15
i- - - - - - - - r
20
25
H ---------- 1
30
35
Time, Minutes
Figure 27.
U-Tank Simulation, Step Drop in T
Temperature Response Curves
from 73 - 68 °C,
k
H e i g h t , Inches Water
19
Ik -
9 -
4
O
I
I
9
10
I
15
I
20
I
------------- 1
I
25
30
T i m e , Minutes
Figure 28.
4-Tank Simulation, Step Drop in T
Height Response Curves
from 73 - 68°C,
35
-59the tube bundle, g i v i n g 'a larger AT and AP between tanks. 3 and
k, -while
the outlet flow is constant.' H 1 drops because AT and Ap increase •
between tanks I and 2 , causing more water to flow out of tank I than
enters at the fixed f l o w rate F .■ Heights of water in tanks 2 and 3
°'
remain almost constant.
For a step rise in T^ from 73 to 78 0C 3 a
corresponding upset is produced.
Tanks-I and 4, the end tanks, are
less stable than interior.tanks in the. system.
Use of the
?
tank simulation not only gave a means to test the
stability of a flash evaporator to upsets ,in feedstream temperatures 3
but it also gave a w a y to test changes in equipment design as well.
The stability of the 4—tank model as a function of tank size was tested
using a step rise in T q from 103 to 10 6 q C .
for evaporators with tank areas of 8 ft
Values of T 1 were observed
, 20 ft
, and 50 ft
(Figure 29)
The steady-state temperatures in each evaporator were equal, and the
equilibrium heights of water in each evaporator were equal.
Increased
tank area slowed the rate of response to an upset, but it did not alter
steadyustate conditions.
The simulation run with 8 ft
2 1/2 times faster to a step upset in
20 ft
tanks reacted
T q than a simulation run with
t a n k s ; it reacted about 6 times faster than a simulation run
with 50 ft
2
tanks.
The. rate of response to a temperature upset is
related to the time constant p = (Tank Volume)/(inlet Flow Rate) and is
described in a later section.
Increased tank size is useful to regulate
minor cyclic fluctuations in T q , but it will not change the total effect
103
8 ft 2
102
-
20 ft
101 Degrees
50 ft
I
cr\
0
1
100 -
99 —
I
0
5
I
10
15
20
25
I
30
T i m e , Minutes
Figure 29.
4-Tank Simulation, Step Rise in T from 103 to 106°C,
Response of T, for Evaporators wiSh Tank Areas of 8 ft^,
20 ft2 , and 50 ft 2
H
35
-6lof a step change in T q *
The e f f e c t .of tube bundle heat^-removal capacity- upon evaporator
performance was- tested using a step rise in T q from 103 to 106 0C .
For
heat removal rates 2000 BTUVhr- 0F , '4000 BTU/hr-°F» and 8000 BTUVhr- 0F,
the responses of T^ to the step rise in T q are given in Figure 30.. Increased heat removal gives different steady— state operating
temperatures.
A larger heat-removal rate gives a greater AT between
tanks and a lower equilibrium temperature in each tank.
The rate of
response of the 4-tank model to an upset in T q is faster for the runs
having a higher heat-removal r a t e .
T^ reaches its n e w equilibrium
value within 12 minutes after the upset when the heat removal rate is
8000 BTU/hr-°F.
When the heat removal rate is 2000 BTU/hr— ° F , a new
equilibrium value of T-^ is reached in about 15 m inutes.
EFo significant
differences in the stability of the 4—tank model were observed as func­
tions of the heat removal rate of the tube bundle.
■The 4-tank flash evaporator model also enabled steady-state runs
to be made at different flow rates.
Flow between tanks is a function
of both the orifice size and the pressure drop across the orifice.
The simplest way to regulate flow rate through the evaporator is to
change the size of the orifice between each tank.
For an evaporator
w i t h fixed orifices, changes in flow must be accomplished by changing
the pressure drop across, each orifice.'
This may be done by either
103 -H
2000 BTU/hr°F
101 H
T i m e , Minutes
Figure 30.
U-Tank Simulation, Step Rise in T from 103 to 106°C,
Response of T. for Evaporators wi?h Tuhe Bundle Heat-Removal
Rates of 2000 BTUZhr0F, UOOO BTUZhr0F, and 8000 BTUZhr0F
-63- .
changing the temperature range oyer which the evaporator operates or "by
increasing the heat removal rate .in. the" tube bundle.
increasing the
temperature in the evaporator increases: the pressure drop from tank to
tank because the vapor pressure of water increases w i t h increasing
temperature faster than the temperature itself increases.
pressure drop between tanks gives more flow.
A higher
Decreasing the temperature
in the evaporator lowers the pressure drop from tank to tank and
decreases the flow rate through t h e 'evaporator.
and only small changes in flow result.
This effect is small,
For an orifice size between
tanks of 0.875 inches and a heat removal rate in each tank of 2000.
B T U /hr- 0F , a rise in T q from 103 to 1 0 6 0G changes the average flow
rate through the evaporator from 152 #/min to 153 #/min.,
A much
more satisfactory way to change the flow rate through the evaporator is
to change the tube bundle heat—removal rate in each tank.
For a heat-
removal rate of 2000 BTU/hr-°F, the average flow through the evaporator
is 1^5 #/min; for 4000 BTU/hr-°F, the average flow is 156 #/min; for
8000 B T U /hr-°F, the average flow is 166 #/min.
An increased heat-
removal rate may be obtained by decreasing T^ or by increasing the (UA)
heat transfer term for each tube b u n d l e .
Use of the 4-tank simulation
has shown that changes in flow rate through the evaporator cannot be
made without changing the temperatures within the system.
Stability of a_ Fl a s h Evaporator
The temperature upsets discussed for the 2-tank and 4—tank ■
-64evaporators "were chosen to show h ow the' evaporator would react to a
given change.
Each, upset was small, 3 - 5 0C, so that lineout values
for the variables could be obtained.
Eor upsets larger than 3 — 5 ° C ,
the evaporator might or might not have reached equilibrium before a
tank either pumped itself dry or filled to capacity.
In a mathematical
sense, for tanks of infinite size, the flash evaporator is everywhere
stable.
In actual equipment, finite tank size imposes narrow limits of
stability.
Increased tank depth increases the stability of a flash
evaporator to larger upsets because of the larger head differences
between tanks that are p o s s i b le.
Tanks of increased area slow down
the rate of response to u p s e t s , but offer no lnore stability than smaller
tanks with equal depths.
Stability depends not only upon the size of the upset but also
upon the frequency w i t h which it is applied.
A system may contain . .
resonant frequencies, frequencies at which upsets magnify themselves
in intensity.
A step upset is composed of all frequencies superimposed
upon one another.
The general stability of the' 2- and 4—tank models
suggests that there are no resonant frequencies for this system.
To .
confirm that stability existed at all frequencies of forced upsets, a
frequency response study of the flash evaporator to upsets in T q was
run.
Teasdale (23) gives a method for finding frequency response data
from step response data.
Using the response of
to a step change in
T q on the 4-tank evaporator, a frequency response curve was plotted for
—6^—
a step rise in T q and a step drop in Tq . These data..w ere .©om^aredto-data,
obtained from actual response of the model.to sinusoidal changes in T q
at varying frequencies.
(See Figure 31.)
A different curve was '
obtained for a step rise in T q than for a step drop in T q due to the
slower rate of reaction of the evaporator to a step rise in Tq .
The
simulation oscillation curve is a combination of the step increase and
step decrease functions , and lies between them.
The frequency response
study shows that the response of the system to a forcing input tempera­
ture upset decreases as the frequency increases, and contains no
resonant frequencies.
The simulation phase lag curve tends to level
out at h i g h frequencies, and the amplitude ratio curve slope approaches
2 at high frequencies, indicating that at high frequencies the system
order approaches 2 .
Rate of Response to Upsets
The rate of response of the flash evaporator to step changes can
be estimated using the idea of a time constant.
For a first order
system undergoing change to a n ew equilibrium state following a step
upset, 63 .2% of the change occurs in the time interval equal to I time
constant, 86% of the change in the time interval equal to 2 time
constants, and 95% of the' change in the time interval equal to 3 time
constants.
For the 2-tank evaporator model, the following calculations
were made for a step upset in T q .
-66-
1.0 -OTemperature Drop
Amplitude Ratio
Temperature Rise
—
Simulation Oscillation
Calculation from Inlet
Temperature Step Change
Cycles per minute
.Ol-J
Phase Log, Degrees
Temperature Drop
-
100
-
Temperature Rise
Figure 31.
Frequency Response Study
-67Temperature Rise T = Tolume/flow = 4.05. minutes
2-tank m o d e l , response of- T .
I * T = 4.25 minutes
2 * t = 8.05 minutes,
3 * 7 = 11.85 minutes,
T = . 4.03 minutes
T = 3.95 minutes
' 2-tank m o d e l , response of
1 * T = l4.30 minutes
2 * T ,= 26.15 minutes,
. 3 * 7 = 35.65 minutes,
T = 13.07 minutes
T = 11.88 minutes
Temperature Drop —
T - volume/flow = 2.48 minutes
2-tank model, response of T^
I * T =
2 * 7 =
3 * 7 =
2.05 minutes
4.40 minutes.
7-35 minutes,
T =
7 =
2.20 minutes
2.45 minutes
2-tank model, response of
1 * r = 6.50 minutes
2 * 7 = 17.15 minutes,
3 * T = 29.20 minutes,
t =
T=
8.57 minutes
9*73 minutes
The 7 calculated from volume/flow agrees w i t h the t for the
response of T^,.since the rate of temperature rise in tank I follows
almost exactly the model of a first-order mixing process'. ■The time
constant for the rate of response of
3 times greater than 7 for T .
the rate of response of H
to a step upset in T
is about
Due to the interaction of the system,
to a step input' change is about 3 times
slower than the rate of response o f -T^ to the change.
-68For the 4-tank evaporator, the rate of response to temperature
upsets is similar to the 2—t a nk evaporator.
Values of t - Tol/Flotr
agree generally within 15$; of those taken for the temperature in tank I
at 63.2%, 86%, and 95% of.the total temperature change from the up s e t .
The rate of response of the height of water in tank i is 2 to 2 1/2
times slower.than the rate of response of the temperature.
The response
rate does not depend upon the rate of heat'removal "by the tube bundle.
CONTROL OF A FLASH EVAPORATOR
The changes in xater' heights and temperatures that resulted when
load upsets were applied to the 4-tank simulation pointed out the need
for stabilizing' control action.
Because upsets die out as they pass
from tank to tank, the stability of the inner tanks of a flash
evaporator results from controlling the conditions within the end tanks..
Stability of the end tanks follows from control of T , F , T , and
the flow rate from the final tank.
In a report to the Office of Saline
Water (19), Fluor Corporation specifies the necessary control'for a
flash evaporator as shown in Figure 32.
The proper inlet temperature
is maintained by control of the steam pressure to the steam heat
exchanger.
The cooling, water temperature is maintained b y using a
liquid-liquid heat exchanger and b y mixing warm waste brine with the
cool fresh brine ‘
.from the ocean.
The level in tank I is maintained b y
control of F q using a pneumatic flow controller.
The level in the last
tank i s 'maintained b y controlling the f l o w of waste brine from the
system.
Thermocouples are used to indicate the temperature in each tank
and sight glasses are inserted in each tank to indicate liquid level.
Tb examine the response of a flash evaporator under corrective
controller action, the 4-tank simulation was modified to Include
proportional-integral (P-l) control.
added to control either T
regulate Fq .
o
A temperature controller was
or T , and a flow controller w a s added to
c
The height of water in t a n k 4 was held constant at
Fresh
Brine
Sight Glasses
Waste
Steam
Figure 32.
Existing Control
-71'
10 inches to approximate the action of an exit flour controller.
Control of T q or
depends- upon control of a process; heat
exchanger subject to variations in flow rate and inlet temperature.
Kammen and Koppel (13) report in a dynamic study of a heat exchanger
that t h ey experienced a lineout time of approximately 40 seconds for
heat exchanger response to a step upset in flow r a t e .
Aikman (l)
reported good success in controlling the exit temperature from a steam
heat exchanger subject to varying flow rates and inlet temperatures.
He gave response times for the heat exchanger of less than one m inute-.
To approximate the worst response of a heat exchanger to a change
in operating conditions, a 1-minute pure lag, was added to the
temperature controller.
Temperature controller constants were chosen
using the continuous cycling method as described b y Harriott (9).
K
max
=1
0CZ0C and an ultimate period P of I minute were obtained for
.■
the temperature controller.
K
c
A
= 0.4$ * K
max
The suggested controller settings were
= 0 . 4 5 0C/°C and T
r
= PZl.2 = 0.834 minutes.
For the
flow controller, control constants were chosen arbitrarily to give a
minimum amount of cycling and maintain F
at a constant va l u e . The
°
H gOZmin
proportional band for the flow controller was picked at K c = 6 ^nCft.~g ~ Q >
and the reset time was picked to be
= 1.5 min.
This combination of
control constants gave adequate control of H^ to upsets in T q and T c
without excessive cycling of Fq .
-72-.
Using the 4-tank simulation xith: T q and .Fq under P-I control, a
5 0C step rise in the Inlet' temperature, to. the' steam, heat exchanger
■was introduced into the system.
(See Figures 33—34.}
upset in-.-tahk I lined out to +1°C within 10 minutes.
A large initial
In tank 2, the
temperature .pulsed about 2 °C and then stabilized within 5 minutes.
Tank 3 had a small upset of less than I ° C , and tank 4 recorded a
negligible change.
All heights remained stable, w i t h
greatest oscillation.
to drift.
T
c
showing the
There was no tendency for the level in any tank
was held constant at 73 ° C .
A step decrease in T
c
from 73 to 68 °C with P-I control of T
and F q is shown in Figures 35-36.
from the T q upset run.
o
Control settings were left unchanged
Because a change in T q has no effect upon the
inlet temperature T q , the temperature curves of Figure 35 are identical
to those shown in Figure 29, which had no inlet temperature control.
The
curve oscillates within I °C of its set point, and
at its set point.
tanks.
H 0 and H
is steady
rise to n e w levels in their respective
Decreasing T q increases the AT and AP between tanks and raises
the levels of water in tanks 2 and 3.
A 5 0C step rise in T , with T q
under P-I control, is shown in Figures 37-38.
The I minute.'lag was
retained w i t h the T q temperature controller., and control constants of
K
c
= 0.45 0CZ0C and T
was left unchanged.
r
= 0.834 minutes were used.
The f l o w controller
Upsets in the tank temperatures have magnitudes of
less than I 0C,, and die out within 8 minutes.
The heights of water are
Temperature, Degrees
HO-,
95-
10
I
15
I
20
25
Time, Minutes
Figure 33.
^-Tank Simulation, P-I Control of T q and. F q , 5 0C Step Rise
in Inlet Stream to T q Heat Exchanger, Temperature Response Curves
Height, Inches Water
T i m e , Minutes
Figure 3^.
U-Tank Simulation, P-I Control of T and
in Inlet Stream to T^ Heat Exchanger, Heigr
5 0C Step Rise
Response Curves
T
0
102
T
I
to
IU
<u
M
T
(U
A
2
0)
0
td
92
i
-
Vl
I
CD
I
EH
T
87 "
O
I
5
I
10
I
15
I
20
I
25
I
30
Time, Minutes
Figure 35.
4-Tank Simulation, P-I Control of T and F q , Step Drop
in T c from 73 - 68 ° C , Temperature Response Curves
I
35
4
~9l~
H e i g h t , Inches Water
T i n e , Minutes
Figure 36,
U-Tank Simulation, P-I Control of T q and F q , Step Drop
in T from 73 - 68 ° C , Height Response Curves
c
105 -I
T
0
100
u
T
M
0)
0)
U
bO
V 95 O
T
<L>
3
S
%
I
2
3
—i
—3
I
90
Eh
85 _|---------- 1
0
5
I
I
10
15
I
20
i
25
I
30
I
35
Time, Minutes
Figure 37.
I-Tank Simulation, P-I Control of T^ and F q , 5 0C Step Rise
in Inlet Stream to T Heat Exchanger, Temperature Response Curves
H 3'
Ih
H
H
9
O
-Si-
Height, Inches Water
19
4
— p-
— f—
— J
I
I
5
10
15
20
25
I
----------- 1
30
35
T i n e , Minutes
Figure 38.
U-Tank Simulation, P-I Control of T and F , 5 0C Ster Rise
in Inlet Stream to T^ Heat Exchanger, Height Response Curves
-79very stab l e , varying less than I inch in each tank.
There is no tendency
for any of the water levels to drift to new levels.
The stability of the flash evaporator under P-I control was also
examined by changing the T q set point from 104 0C to 101 ° C . (See
Figures 39-^0.)
A much more severe response of the system was observed
than in the previous upsets under controller action, especially for the
heights of water in each tank.
Lowering the operating temperature
3 0C dropped the AT between tanks and lowered
to 13 inches.
and
from l4.5 inches
cycled between 11 1/2 inches and 8 1/2 inches before
stabilizing back toward the set point value of 10 inches.
K
and T
By increasing
for the flow controller, fluctuation in H 1 is reduced while
the variation in F q is increased.
dropped to new steady-state levels.
The temperatures in each tank
T^ reached its new value within
10 m i n u t e s , and T^ leveled out in a little more than
20 minutes.
A change in the T^ set point from 73 0C to 76 0C is shown in
Figures Ul-U2.
Changes in tank temperatures are small, with T^ changing
about I 0C and T^ changing less than I/ 0C .
New steady—state values
were reached within 15 minutes after the set point change.
There was
little change in any of the heights of water in any of the tanks.
Temperature, Degrees
105
85
I
0
5
I
10
I
15
I
20
H"
25
~T~
30
I
35
Time, Minutes
Figure 39.
U-Tank Simulation, P-I Control of T and F q , Change of T
from IOU °C to 101 ° C , Temperature Response Curves
Set Point
14 -
H
H
2
3
9 -18-
H e i g h t , Inches Water
19 I
I
O
5
I
10
I
15
r~
20
i
25
H
30
I
35
Time, Minutes
Figure lO.
I-Tank Simulation, P-I Control of T and F fi, Change of T
from 104 °C to 101 ° C , Height Response Curves
Set Point
105
■T o
Temperature, Degrees
o
T
100-
I
■T
2
95i
OD
ro
I
I
0
10
5
~T~
I
I
I
15
20
25
30
I
35
Time, Minutes
Figure
hi.
I-Tank Simulation, P-I Control of T„ and F , Change of T
from 73 0C to To ° C , Temperature Response Curves
Set Point
e i g h t , Inches Water
19
H
2
lU
K
H
H
9
3
i
It
i
CO
Co
I
"
I
I
5
10
r
15
I
20
I
25
I
30
i
35
T i m e , Minutes
Figure
h2.
It-Tank Simulation, P-I Control of T and F , Change of T
from 73 0C to 76 ° C , Height Response Curves
Set Point
CONCLUSION
Th.e development of a simplified mathematical model for the'
2-tank flash evaporator has defined the'relationship Between the
system v a r iables.and system stability.
The interaction of temperature,
p r e s s u r e , water height, and heat removal rate works to give the unique
behavior of the flash evaporator.
to the 1— tank
The extension of the 2—tank model
model has helped clarify the relationship of stability
to evaporator design.
Use of.the 4-tank model showed that P-I control
of the inlet temperature, inlet flow 1r a t e , cool brine temperature, and
exit flow rate was sufficient to control a multi-stage evaporator.
The 4-tank simulation may be used to simulate a 4—tank section of
any flash evaporator.
Because of the simple design of the model, a
minimum of data are needed to adapt to any particular situation.
Depending upon the temperature r a n g e , correlations for the physical
properties of the brine (viscosity, density, enthalpy of vaporization,
etc.) must be changed.
The orifice equation may be changed to fit
orifices of different sizes or may be substituted for b y another
equation describing another form of restricted flow, such as. flow
through a weir.
Tank size and tube bundle heat-transfer rates are
readily adaptable to any particular situation.
m a y be easily added or removed.
Various control schemes
Simulation runs taking less than 10
minutes of computer time will predict evaporator performance over a
-, 85-
real time period of 70- minutes-.
It. i s Iioped that the,, results of this
study m a y he used to.aid in the' design.of future flash evaporators.
SUGGESTIONS FOB FURTHER WORK
1.
Rewr i t e .tRe digital simulation so that it applys to an N-stage
evaporator.
2.
Add an exit f l o w controller to the lahoratoiy 2 -stage.evaporator
and test the control scheme proposed in this paper.
3.
Modify the 4—tank, simulation "by including actual heat transfer
correlations for the tube bund l e .
4.
Expand the laboratory evaporator to 3 or 4 tanks and compare its
operation w i t h the computer model.
5.
Correlate the submerged flow through a variable-area weir.
6.
Include the dynamic response of a steam heat exchanger in the
4—tank simulation for the examination of P-I control.
7.
Compare the 4-tank model with transient data from the San Diego
test module when it becomes available.
APPENDIX
APPENDIX.
A.
Definition of Terms.-.
B.
I.
Computer Listing for 2—Tank-Evaporator Model
2.
Computer Listing for ,2-Tank Plot Model.
3.
Computer Listing for 4—Tank Evaporator Model
4.
Computer Listing for 4—Tank Plot Model.
C.
Sample Experimental Data Listing.
DEFIIXTXON OF TERMS
A1
Mater' surface area .of tank X.'
A2
Mater surface area.of tank 2.
C
P
Eeat capacity- of x a t e r , X BTU/#— 0F,
ES
Temperature drop from,tank X to t a n k '2, 2-tank evaporator.
Temperature drop from tank X to tank 2, 4 - t a n k .evaporator.
** 1-2
Temperature drop from tank 2 to tank 3, 4-tank evaporator.
AT2-3
Temperature drop from tank 3 to tank 4, 4-tank evaporator.
AT3-4
C-AP)
Pressure drop across the orifice ."between 2 adjacent tanks.
^ 1-2
Pressure drop across the orifice "between tank X and tank 2,
4-tank evaporator
AP2-3
'
Pressure drop across the orifice between tank 2 and tank 3,
4-tank evaporator.
^3-4
Pressure drop across the orifice between tank 3 and tank 4,
4—tank evaporator.
F
FXow rate- of water (brine) into tank X.
O
F1
FXow rate of water from tank X into tank 2, 2-tank modei.
Fl 5 Fl —2
FXow rate of water from tank X into tank 2, 4—tank m o d e X .■
F 2 ’ F2-3
k
H
H1
H2
Flow rate of water from tank 3 into tank 4, 4-tank model.
I
sC
on
on
f4
Flow rate of water from tank 2 into tank 3, 4—tank model.
JpXow rate of water out of tank 4, 4-tank model.
Proportionality constant between force and mass.
Enthalpy of water vap o r .
Height of water in tank I.
Height of water in tank 2.
Heigkt of water in tank. 3, ^-tank. model.
Height' of water in tank 4,- 4—tank.model.
Enthalpy of vaporization of water.
Yapor pressure of water as a function of temperature.
Yapor pressure of water in tank I.
F l o w rate of water through an orifice as a function of
AT, AP,. and temperature.
Density of water (liquid).
Density of water (vapor).
System time constant.
Reference temperature, chosen arbitrarily.
Temperature of
cooling water in tube bundle
leaving tank
I
Temperature of
cooling water in tube bundle
leaving tank
2
Temperature of
cooling water in tube bundle
leaving tank
3
Temperature of
cooling water (fresh brine) entering the
tube bundle in tank n.
Temperature of feed water entering tank I.
Temperature of water in tank I.
Temperature of water in tank 2.
Temperature of water in tank 3, 4-tank model.
Temperature of water in tank 4, 4—tank model.
(UA )1
Combined area—heat transfer coefficient term for tube
bundle I.
(UA )2
Combined area—heat transfer coefficient term for tube
bundle 2 .
Liquid volume in tank n.
Yap or volume in. tank n.
Liquid holdup in tube bundle in any tank.
Differential.increment of time.
GomputeE listing for 2-tank Evaporator Model
IJ O B
1 9 0 6 9 ,SOLZBEEffEJ?, 35
IL I M I T S ( T I M E , 9 ) , (P4ffE5,20 0)
IAEEJffE E : 1 0 6 , ( D E V I C E , M T A 81)
IM E S S A G E ' M O U N T S C R A T C H T A P E O N D R I V E I A N D L E A V E ■M O U N T E D F O R N E X T JOB.
[FORTRAN
' ”
' .
- D I M E N S I O N AJ?EAJ( 9 , 7 0 1 ) ,J( 20) ^
■
__ _ _
D A T A A l i A 2 / 8 . 8 8 , 6 . 57/
'
P O N E ( A ) = ( 2 9 . 9 2 1 / 7 6 0 . ) * 1 0 i * * ( 7 . 9 6 6 8 - 1 6 6 8 . 21/( 2 2 8 + A ) ) __ '
V ' '^-DPINH G( A tB , C tD ) = ( ( C-D) /13 . & + P 0 N E ( A ) - P 0 N E ( B ) )
" R H O L( A ) - 0 . 6 29 3 717 3_E+0 2 - ( 0 . 1 7 5 4 8 5 5 E - 0 1 ) * A - ( . 13 5 5 1 52 S E - 03 ) * A * * 2 I
.
O E A B ( A ) = EEEJLE*EJffAfA*(( I . 8 * ( A + 2 7 3 i ) ) * * 4 - ( l . 8 * 2 9 8 . ) * * 4 ) / 6 0 . j I ' &
TDROPl(B) =ALPHA* (B-TCOOL) _
l:
JBEOPl(A,B)=BEJA*(A-JffOOL-JBEOP2(B))
^
T 1 D 0 T ( A , B ,C)=( 12. / (RHOL( A) *A1*C) )* ( F Z E R O * (T Z ERO -'^LJ- F Z E R O * T D R O P l ( A ,
c
1B)-SIDEA1*QRAD(A))
T l D O T (A , B ^ C iD ) = ( 1 2 . / ( R H O L ( B ) *A2 *B ) * ( F L O W (A ,B,ff,B)*(A-B) - F Z E R O * T D R O
1P ! ( B ) - S I D E ~ A 1 * Q R A D ( B ) ) )
-Hl-DO T(.A^,JL» ff,B) = (12. /J^fiHO L( A ) *A I ) ) * (F ZERO - F L O W (A ,B yC ,D ) )
H 1 D 0 T ( A ,B ,ffTB) = ( 1 2 , / ('EEOL (B ) *A 2 )) * ( F L O W (A ,B,ff,B) -FZERO )
S I D E A l '= '( 66*_82 + 4Jl*66 + 2 2 * 4 l ) / 1 4 4 _ ; '
SIDEAI = (4100 + 2 2 0 0 + 4 1 * 2 2 ) / 1 4 4 . '
ALPEA=. 23
. .. . '
.
0
BETA='. 0 3
TCOOL=IQ.
.
-'
,
' ■
'
: ' .. . \
' E E A B d O 5-, 5) -T l Z E R O ,TlZERO ,HlZERO ,HlZERO
<_ _ _ _ _ _
5 F O R M A T (4 E 1 0 .2)
\
R E A D ( 1 0 5 , 1 0 ) F Z E R O , D T H E T A iT S T E P , E P S I L N ,TZERO
\...
10 F O R M A T ( 5 E 1 0 . 1 )
■-
SIGMA = 1 . 7 3 5 - 9 - - - - - - - - - - A R R A Y (1,1) = T Z E R O
ARRAY(2,1) = TlZERO
A R R A Y O iI) = T 2 Z E R 0
A R R A Y ( ^ iI) = T 1 Z E R 0 - T 2 Z E R 0
AJ?5Ay(5,l) = -Po e e ( T i z e r o )
A R R A Y ( 6 , IJ =- D PI R H G( TIZ E R O ,T2 Z E R 0 ,H l Z E R O ,H2ZER0 )
A R R A Y (7,1) = H l Z E R O
_ _ A R R A Y ( Z iI) = H 2 Z E R 0 .
Ai?i?A7(9,1) = F L 0 W ( T 1 Z E R 0 , T 2 Z E R 0 , HlZERO ,H2ZER0)
21 TIME = 0.0
'
R E W I N D 106
W R I T E ( I O S iIlO)
.....
110. F O R M A T ( 'I T - Z E R O
T-ONE
T-TWO
DT
:;l D P
.^ J E R Q U E __ I H - T W O . . _ E L O W
TIME
W R I T E ( 1 0 8 , 1 2 0 ) ■( A R R A Y ( 1 , 1 ) , 1 = 1 , 0 ) , T I M E ,FZERO
DO 150 J = I , 70 0
.....
.. U
IF(I.GT.SOO) TZERO=TSTEP
Xl = T l D O T ( T I Z E R O , T 2 Z E R O . H l Z E R O ) * D T H E T A
.
Jl = T 2 D 0 T ( T 1 Z E R 0 ,T 2 Z E R O , H l Z E R O ,H 2 Z E R O )* D T H E T A
Zl = H 1 D 0 T ( T 1 Z E R 0 ,T 2 Z E R O , H l Z E R O , H 2 Z E R O )* D T H E T A
Wl = H 2 D O T ( T l Z E R O ,T 2 Z E R O , H l Z E R O , H 2 Z E R 0 ) * D T H E T A '
TXl = T I Z E R O + X l /2
... .
.... ...
J J l = T2ZERO+Y1/2
H Z l = H1ZERO+Z1/.2 ■
....
H W l = H 2 Z E R O + W 1 /2
X2 = T1D0 T ( JZl., J J l ,.SZI ) * D T H E T A __
J2 = T2DO T ( TXl ■,T Y l ,H Z I , H W l ) * D T H E T A
Z 2 = H1D0T(TX1,,TJ1;HZ1,HW1)*DTHETA
W 2 = 525(9 J( J J l ,J J I ,HZ I , H W l ) * D T H E T A
TX2 = Jl Z E R 0 + X 2 / 2
__
.
...
.
JJ2 = T2ZERO+Y2/2
H Z 2 = H l Z E R O __+_Z.2 /.2_..____ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _
5 5 2 = H2ZERO + 5 2 / 2
'
- ■
J 3 = T1D0T(TX2,TY2,HZ2)*DTHETA
J3 = T 2 D O T ( T X 2 , T Y 2 , H Z 2 , H W 2 ) * D T H E T A
Z S = H l D O T (T X 2 ,T Y 2 , H Z 2 ,H W 2 ) * D T H E T A
W3 = H 2 D 0 T ( T X 2 ,TY2 ,HZ 2,HW2) *1)TIiETA
TX3 = T l Z E R O + X3
TX3 = T 2 Z E R 0 ■+ 13
_2Z3
HW3
. Z4 =
14 =
..24=
WH- =
= glZ&RO + Z3
=
. .
= (Zl + 2*Z2 + 2.*mj.Jj:l/6,.
—
;
'
H 2 Z E R O + ^3
T1D0T(TX3,TY3,HZ3)*J),THE.TA
T2DOT(TX3,TY3,HZ3,HW3)*DTHETA
H l D O T i T X Z ,TY3 iHZ3 iHW3)*DTHETA.
H 2 D O T ( T X 3 ,TY 3 ,HZ 3 ,H W 3 ) * D T H E T A
_
Tl = T 1 Z E R O + D E L T A X
DELTAY = (
* Y2 + 2 *J3 ±X4 )„/6 . .
T2 = T 2 Z E R O + D E L T A Y
D E L T A H = ( Z 1 + 2 * Z 2 +.2 *2.3 + Z 4 ) / 6 .
Hl = H l Z E R O +DEL T A H
DELH2_ = ( W.i + 2 * F/2 + 2 *K3±K4J./6 . .. . .
H2 = H 2 Z E R O + D E L H 2
• H - I + I ..
-- —
.
*1 A R R A Y ( I 9J) = T Z E R O
. A R R A Y ( 2 , J ) = Tl
' A R R A Y ( S 9J) = T2
A R R A Y ( H 9J) = T I r T 2 . ... ..
..
A R R A Y ( B 9J) = P O E E ( T l ) .
A R R A Y i S ,J) = D P I E H G i T l 9T 2 9H 1 9H2)
A R R A Y ( I 9J) = Hl
E2.
_
. ..
____ ____ _
. ...
__
__________________ _________ _
__
.
..
..
__________ ______
. __
.
_ _ _ _ _ ....
.. . _ _ _ _ _
.. . . . . . . .
.....
.. .
.
''
'<Q
.f
__
___ .
........
...
.
A R R A Y ( V 9J) = F L O W i T l 9T 2 , H I 9H2)
T l Z E R O = TX —
...—
—
-- - "
_ _ __
.. .
T 2 Z E R O = T2
H l Z E R O = Hl
H 2 Z E R O "= H2
_ _ _ T I M E = DTHEJTA*!
______
D T 1 = T D R 0 P 1 ( T 1 9T 2 )
DT2=TDR0P2(T2)
W R I T E ( 1 Q Q ,123) (A R R A Y (K.,J) 9 K = I ,3) ,TIME ,FZERO
12 0 F0RMAT^('3(FQ . 2 , 3 X j ,FQ . 3 ,3X ,FQ I3~,2X ,FQ . 3 ,3X ,FQ . 2 ? 3 X ,F Q~:2 ,HX , F l . I ,HX
I ,Fb . 2 , H X 9F l . I )
__ _ _ _ _ _ _
___
I F { E l . G T . 2 0 . . 0 R . E 1 . L T A .01))
GO TO 190
CONTINUE
W R I T E ( 1 0 8 , 2 0 0 ) D T E E T A , E P S I L N , A L P H A , B E T A ,TCOOL
F O R M A T i / / ' D T H E T A = ' , F S . 2 , 5 1 , ' E P S I L O N = ' , F A . I , S I ,'A L P H A = ' ,F A . 2,S I , ' B
I E T A = ' , F A . 2 , S X , ' T C O O L = ' ,FA.Q)
.......
J
DO 2 0 4 « 7 =2 1 0 , 6 90 , 4
W R I T E (106 ,202 ) ( A R R A I U iJ) , J = I , 9)
202 F O R M A T ( 9 ( F 8 . 3 ) )
204 C O N T I N U E
E N D F I L E 106
R E W I N D 106
205 C O N T I N U E
210 C A L L E X I T
END
F U N C T I O N F L O W i A , B ,C,D)
DATA Al,A 2 ,CONST/8.88,6.5 7 , 0 . 1 9 2 0 6 9 2 1 5 2 + 2 /
PONE(A) = ( 2 9 . 9 2 1 / % 6 0 .
9668-1668.21/(228+4)) _
RHOL(A)= 0 . 6 2 9 3 7 1 7 3 5 + 0 2 - ( 0 . I 7 5 4 8 5 5 2 - 0 1 ) * A - ( . 1 3 5 5 I 5 2 8 5 - 0 3 ) * A * * 2
|
E N T H A L ( A ) = ( . 1 0 740 6 3Z? + 4 - ( . 9 33 6 1 1 3 5 0 ) * A - ( . 1 045 313 2 5 - 0 2 ) * A * * 2 ) * 77 8_.
D P C M H G ( A , B , C , D ) = ( ( C - D ) / 1 8 . 6 + P O N E ( A ) - P O N E i B ) ) * 2 . SA
'1
D P = D P C M H G i A , B ,C,D)
I F (D P .G T . ( 0 . 0 ) ) GO TO 10
F L O W = 25.
GO TO 20
10 F L O W = C O N S t * ( V I S ( A ) * * .I l A ) * ( R H O L ( A ) * * . 2 5 6 6 ) * ( ( D P C M H G ( A , B ,C , D ) * 2 7 . 8A
1 5 ) * * , 5 6 9 A ) / ( E N T H A L ( A ) * * .1S6A)
20 R E T U R N
..
,,
END
. I
150
190
200
y r s ( 4 ) ___ _
.
I /
^
______
7 5 5 = ( 0 . 9 3 5 3 5 6 6 6 - ( 0 . 1 0 1 3 0 3 7 8 2 - 0 1 ) * A + ( 0 . 3 6 1 6 1 5 5 ) *A**2)
IF. ( 7 5 5 . 5 5 . ( 0 . 2 4 ) ). G O _ T O I S .
. . .
GO TO 20
. 15 .755 = 0 . 2 4
. . .
20
GO TO 80
IF ( 7 5 5 . 5 5 . ( 0 . 4 ) )
GO TO 3 0
' "25 7 5 5 = 0,4.0_ _
..
50"55' 25 .
—
— -- — -
^
.j
- "
'
30
VIS = V I S * 2 .42
RETURN
END
IL O A D _ _ _
\RUN
\DATA
96.5
148 .
.
.. ...
93.0
12.0
. .0 5 _ _ _ _ _ 95.0
12.0
0.1 _ _
97.
\EOD
\m~
________________
.
J
i
• )
Gomputer Listing for
2-tank
Plot Program.
19 0 6 9 ,E O L Z B E R G E R ,3 4
JOB
.L T M f m (TTM2,_12 ) ,(P4G$S,.15p)______ .
_
.. .. ...........
A S S I G B F '.1Q& , ( D E V I C E ,MTA 8 1 )
■
F O R T R A N . .. .
__ _ _ _ .
..
_ _ ._
C
T H I S P R O G R A M R E Q U I R E S TWO H E A D I N G CARDS.
. D I M E N S I O N ARRA.IS3., 15.0_) , T T Z E U 2 O I iL A B E L („10
12(12)
'
. ___
)., T I M E (.2 I X , N A M E A L l 8 ) ,N A M E
I N T E G E R S T A R / ’*
»/
D A T A LABEL./. I T -I N L i ’ T O N E I., ’ TTWO ' ,.'TT
' , ' P O N E r , 'DP.. ' ,.'HONE' ,IHTWO '
I ,'F L O W ' ,'
■ '/
.REWIND 106 . - - - _
--- - .. . . . . . . . . __ _ _ _
r
REA D { 105,1)- N A M E l
.
I F O R M A T i l Q A A X _ _ __ _ _ _ _ _ _ _ _ _ _ _ ' _ _ _ _ _ _ _ _ ___
;1
R E A D (I 05., 2) N A M E 2
2 F O R M A T ( 1 2AAJ _ _ _
....
___ .
.......... .....
R E A D ( 1 0 5 , 4 ) N S K I P iD T H E T A
... . 4 F O R M A T {12 , F l O . 3)_ _ _ _
.. _ _ _
. .
... .
D O 15
T = I , 13
, _ 1 5. TIMEi 11=11-1) * D X H E T A * N S K I P * 1,0
__
. _ _ _ _ _ _ _ _ _ _ __ _ _ _
3 D O 5 «7=1,150
T E A T ( 1 0 6 , 10_ ,ZZT=.ll,J?Eff = l l ) ( A R R A Y(I i«7)•>T = 1,9)
10 F O R M A T ( 9 ( T 8 . 3))
.N C O U N T „= J .
IFiNCOUNT .GT. 1 2 0 ) NC0UNT=120
j
..
J
... 5. CONTINUHL ..
11 C O N T I N U E
12 TQ_200 T=l,9
— --- — ~ . .. .
.—
..
XMAX=I.E-IO
X M I N = I .E + 1 0
DO 20 J = I iN C O U N T _
. .... 7—
...
_______________
:—
--- - -
_______ _
I F U R E A Y i I ,Jl . G T . X M A X ) X M A X = A R R A X ( I tJ)
I F U R R A H I ,J) .LI. X M IN) X M I N = A R R A X ( I iJ)
GO TO ( 3 1 , 3 1 , 3 1 , 5 0 , 5 6 , 6 0 , 7 0 , 7 0 , 8 0 ) ,J
3 1 'N X M A X = X M A X + I
N X M I N .= X M I N
I F ( ( N X M A X - N X M I N ) . G T . 6 .) GO TO 34
20
.. W T *
= W4Z---6
___
._
DO 33 J = I iN C O U N T
A R R A X ( I iJ) = ( A R R A X ( I iJ ) - N X M I N ) *6.
33 C O N T I N U E
XMAX = NXMAX
__ _ _ _ _
DELTA = I
— , NAXI-X „= 6 „ — -- - - — . , „
_
GO TO 90
34 I F ( ( N X M A X - N X M tN) .-GT .9) GO TO 2 00
NXMIN = NXMAX - 9
DO 35 J = I iN C O U N T 1
*
A R R A X ( I iJ) = ( A R R A X ( I iJ ) - N X M I N )* 3 6 ./9.
35 C O N T I N U E
XMAX = NXMAX
DELTA = I
_
, _ __
N A X I X = 4*
GO TO 9 0.
50 I N T = X M A X * 5 .+1.
X M A X = I N T * . 2.
I N T -= X M I N * 5.
X M I N ■ =_ I N T * (.2)
I F ((X M A X - X M I N ) . G T .1.2) GO TO 53 '
1 DIFF = I . 2 - (XMAX-XMIN)
X M A X = X M A X + D I F F /2.
XMIN = XMAX-1.2
DO 52 J = I iN C O U N T
A R R A X ( I iJ), = . ( A R R A X S I iJ ) ~ X M I N ) * 3 0 .
52 C O N T I N U E
D E L T A = 0.2
NAXIX = 6
GO TO 90
I
4>
Y5
53 J F ( ( X M A X - X M I N ).G T .I . 8) GO T O 200
X M I N r X M A X - 1 .8
D O 54 J = I tN C O U E T
5 4 „ A R E A I ( I tJ ) _= (A R E A J (J tJ )- X M I N )* 3 6 . / 1 .8
D E L T A = O. 2
M Z T Z = „4 .... . . .
GO TO 90
56 T U J M Z . . G r . , 3 . 1 1 _ S O
2 00 _ _ _ _ _
I F ( X M I N . L T . 24) GO T O 200
XMAX. = 31_ _
.
.
XMIN = 2 4
. W 5 8 J = I tNCOUNT.. . _ _ _ .
A M A J( T 1=ZJ = ( A M A J ( T , e 7 ) - Z M T A O * 3 6 . / 7 .
. 58 C O N T I N U E ..._ _ _ _ _ _
. ______
__
DELTA = 1
. - NAXIX = 5 _ _ _ _ _ _
______
OO T O 90
60 T M _=_ZMAZ* 1 0_,+l.
.
X M A X = I N T * ( .I )
. ...
.. I N T - X M I S * . 5 ^ _______ _______ i ...
.....
_____
. 63
X M I N = I N T * ( .2)
DIFF = I.2-(XMAX-XMIN).
I F (D I F F .L T . ( 0 . 0 ) ) GO TO 65
XMIN = Z M A Z - I . 2 „
DO 63 J = I tN C O U N T
A R R AIL I,, J ) = .L A R R A Z ( T tJ)- X M I N
CONTINUE
D E L T A = 0.2
...
.... .
MZTZ = 6
GO TO 90
65 D I F F = 1 . 8 -
.
67
... ..
)* 3 0 .
(XMAX-XMIN)
I F ( D I F F . L T . 0.0) OO TO 200
XMIN- = X M A X - 1.8
DO '6 7 J = I ,N C O U N T — •• ■'
A M A Z ( T 1J) = (Ai?i?AI ( I tJ ) - X M l N ) *36
CONTINUE
DELTA = 0 . 2
..-'T
. / 1". 8
"
NAXIX = 4
GO TO 90
70 X M A X = I b
XMIN= 9
'
D O 75 J = I iN C O U N T
" " A R R A X l T iJ ) = IA R R A H I,
75 C O N T I N U E
DELTA=I
NAXIX=Q
_
ffO TO '90
- .
'
"
'
""
J)-9 .T*6T~‘ ' .
'
_
__
80 T M iA M Z = X M ^ Z * . 1 + 1
DO
”>
__
' ' "
'
"
,
'
—IOQ1
X M A X = I N T M AX* 1 0 ‘
' “
IF( ( X M A X - X M I N ) .GT. 60.) GO TO 200
X M I n = X M A X - 6 0. "
......... .
DO 85 J = I iN C O U N T
85 A R R A J ( I l J ) = ( A R R A Y (I IJ )- X M I N ) * ( .6)
D E L T A = 10.
~ N A X I X =“ 6
90 S M A L L = 36.
N R E P T = -I
DO 150 Z Z Z = I , 35
NREPT = NREPT + I
BIG = SMALL
SMALL = BIG - I
'
M
'
92 T = I ,120
L I N E (J) = B L A N K
CONTINUE
DO 95 J = I iN C O U N T
I F ( B I G . GE . A R R A K I ,J) .AND_. (S M A L L .L T .A R R A Y ( I ,J) ) )
95 C O N T I N U E
*
KNT = KKK
I F (K N T .G T . I ) GO TO 100
WRITE ( 1 08 , 9 6 ) X M A X ,( L INE(J), T = l , 1 2 0 )
96 F O R M A T ( ’I »,4 Z , T 6 . 2 , ’^ ,I 2 O A l )
GO TO 150
10 0,} I F (K N T .N E . 18 ) GO TO H O
JJ = I
92
LINE(J)=STAR
' ... .
,
'
■
...
HO
112
115
GO TO 112
J J = 10
I F (N R E P T .N E .N A X I X) GO TO 120
NREPT = O
XMAX = XMAX-DELTA
WRITE, ( 1 0 8 , 1 1 5 ) L A B E L ( J J ) , X M A X A L I N E ( J ) ,
F O R M A T ( A 4 , 1 X , F 6 . 2, * *' , 1 2 0 A 1 )
f O 150
_ 1 2 0 _ M I T £ (,1_0_8.,_125 )__LAg E L _(J_J)^jLIN
125 F O R M A T ( A 4 , 7 Z , , 1 2 0 A 1 )
150 C O N T I N U E
W R I T E ( 1 0 8 , 1 7 0 ) N A M E l ,NAME2
170 F O R M A T ( I l X , , 1 8 A 4 , 1 2 A 4 )
(108,153 )
153 F O R M A T
M
U
^
J
J = I ,120)
12Oj_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _
_ _____
2 *')
WRITE( 1 0 8 , 1 8 0
F O R M A T ( 2 X , 13(
CONTINUE
R E W I N D 106
CALL EXI T
END
) (TIME(KI)., Z J = I , 13)
6 Z , J 4 . O ) )•"'
IJ O A J
IZJZ
IJ A J A
__
. . . ..
6/1/6 9
Z J Z I, T W O - T A N K
5 DEGREES CENTIGRADE.
4 0.05
|ZOJ
|ZJZ
.
_
EVAPORATOR.
-X0f
180
200
_ _ _ _ _ _ _
ST E P DROP I N~ TZERO F R O M
97 JO 9
Obmputer Listing for 4-tank Evaporator Model*
12 370 , H O L Z B E R G E R , 5 0
\M E S S A G E
M O U N T S C R A T C H TAPE ON D R I V E
{ A S S I G N F ^ l O S ,(,DEVICE ,MTAQ1)
IL I M I T S ( T I M E , 1 0 ) , ( P A G E S ,125)
IFORTRAN' "
' '"
"
D D
IJ O B
I
AND L E A V E ON FOR
"
N E X T JOB.
* .... ' "
..
D I M E N S I O N A R R A Y ( 2 0 , 7 0 1 ) , 2 ,(20)
D A T A A l , A 2 , A 3 , A 4 / 8 . ,8. ,8. ,8.7
P O N E { A ) = ( 2 9 . 9 2 1 / 7 6 0 . ) * 1 0 . * * ( 7 . 9 6 6 8 - 1 6 6 8 . 2 1 / ( 2 2 8 +A))
. . D P I N H G ( A , B , C , D ) = i i C - D ) / I Z .Q + P O N E { A ) - P O N E { B ) )
R H O L { A ) = Q .6 2 9 3 7 1 7 3 # t 0 2 - ( 0 . 1 7 5 4 8 5 5 5 - 0 1 ) * A - ( . 1 3 5 5 1 5 2 8 5 - 0 3 ) * A * * 2
‘ TDR O P4 ("P) = ( 2 . * B P L S I * 5 P L A I / ( 2 . * 5 Z 5 5 O * 6 0 . +55 LIli *BD LAI) ) * i D -T COOL)
J D R O P S iC,D) = i2. *BD L H 2 * B D L A 2 /(2 . *F Z E R O * Q Q .+ B D L H 2 * B D L A 2 ) ) * i C - T COO L - T
IDROP^iD))"
...... .
T D R O P 2 i B , C , D ) = i 2 . * B D L H S * B D L A 3 / ( 2 . * F Z E R O * S Q .+ B D L H S * B D L A S ) ) * i B - T C O O L
' i-TDRO'P^i D) - T D R O P S iC, D ) )
"
"
' ......
T D R 0 P H A , B , 0 , D ) = i2 . * B D L H ^ * B D L A ^ / ( 2 . * 5 2 5 5 0 * 6 0 . + 5 5 5 5 4 * S 5 L A 4 ) )*i A - T C O
' I O L - T D R O P i i D ) - T D R O P S iC,D ) - T b R d P 2 i B ^C,D)')
T l D O T i A , B ,C,D,E) =(12. / iRHO Li A) *A1*E) ) * i F Z E R O * i T Z E R O - A ) -F Z E R O * T D R O P
>501-
cy<^ D i o
T H I S P R O G R A M S I M U L A T E S ~A F O U R - T A N X E V A P O R A T O R .
T I N I S T H E F E E D T E M P E R A T U R E TO THE H E A T E X C H A N G E R .
T Z E R O I S T H E F E E D 'T E M P E R A T UR E ^ I N T O ' T A N K " l . . . .
" - H l (A.5,5,5))
T 2 D O T i A , B , C , D , E , F ) = i l 2 t/ i R H 0 L i B ) * A 2 * F ) ) * i F L O W iA ,B ,E , F ) * iA - B ) -FZERO
l*r&ROP2(g,C,D))
" '
" "
'
'
~
~
T S D O T i B , C , D , F , G ) = i 1 2 . /iRHOLiC)* A S * G ) ) * i F L 0 W i B , C , F ,G)*iB-C)-FZERO*T
"
' 1 D % 0 P 3 ( C , D ) ) ...............
"
'
..............
T ^ D O T i C , D , G , H ) = i l 2 . / i R H O L i D ) * A ^ * H ) ) * i F L 0 W i C , D ,G ,H )* i C - D ) - F Z E R O *TDR
' 1 0 5 4 (5) )
---—
.
—
■—
. . - ... . - -
H l D O T i A , B ,E,F) = il2";ViRHOL{A.)* A l ) ) * i F Z E R O - F L O W i A , B ,E,F))
H2D0~Ti'A ,B', C , E , F , G ) = 12 . / iRHO LiB) * A 2 ) * ( F LOWi A , B , E , F ) - F L O W i B , C,F , G'))
H S D O T i B ,C,D,F, G,H) = 1 2 . / i RHO Li C) * A J J * iF L O W iB , C ,F , G ) -F L O W iC ,D , G ,H ) )
:
|
H ' i D 0 T ( C , D , G iH) = ±2. / UlH O L ( D ) *A>+)*{FLOW{ C , D , 0 , H ) -FZERO)
R E W I N D 106
TCDO&=73.
1 0'
"12
"
--- ----
"
'
R E A D ( 1 0 5 , 1 0 ) T Z E R O , T l Z E R O ,T 2 Z E R 0 ,T SZ ERO ,T ZERO
F O R M A T ' ( 5 F 1 0. i ) "
'
' ..
'
R E A D ( 1 0 5 , 1 2 ) H l Z E R O , H 2 Z E R O , H S Z E R O iH ^ Z E R O
HO R H A T C^F IQ'. 2 ) . . .
"" ' "
'
R E A D ( 1 0 5 , 1 2 ) F Z E R O , D T H E T A iT S T E P ,TIN
R E A D i l U , 14) 'B D L Al ,BDLA2 ,BDLA.S , B D L A a ^
F O R M A T (A F l O . I)
"
‘
14
' - R Z M D ( 1 0 5 , 1 4 ) ' ' D D D R 1 V D D D R 2 ,B D D R 3 ,BDDE4' . . . :'
. .. .
R E A D i 1 0 5 , 1 8 ) P R O P T , P R O P H iS E T T , S E T H , R E S E T T , R E S E T H
18 f O m ^ C G f l O ^ T . . . . .
'
....
'
TIME = 0.0
S u m e r t = X t c o o l S t T n ')* r e s e t t / C p r 6 p t * d t h e t '
a ')
S U M E R H = ( F Z E R O - I U . ) * R E S E T H / (P R O P H * D T H E T A )
DO 30 J = I ,20 _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _
________
■“SOt;
30 J ( J ) = T C o o i
DO 210 L A P S = I , 2
W R I T E i l O Q ,110)
H O F O R M A T i 'I T - Z E R O
- IDJl' . . . D J 2
* '
1 12
T-ONE
DTQ
T-TWO
' DJ4*
T-THREE
H-ONE
'
2IME'//)
T O R M A T ( 5 ( FQ . 2 , Q X ) , 4 ( F6 . 2,5 Z) , IX ,FQ ‘2 ,6 J, F5 . *2 * 3 j,
D O 1 5 0 J = I ,700
__
E R R O R H = S E T H - H l Z ERO
S UMER H = S U M E R F+ F R R O R H
F Z F R O = 14 5 . + P R O P H kX E R R O R H + S U M E R H * D T H E T A / R E S E T H )
I F i F Z E R O . L T .120.) F Z E R O = 1 2 0 .
I F i F Z E R O .GT .'IQO .) F Z F R O = ISO
• '
ERRORT=SETT-Ti I )
~S U M E R T = S U M E R T + E R R O R T
T C O O L =T I N + P R O P T * (E R R O R T + S U M E R T * D T H E T A / R E S E T T )
D O 114 J J = I ,19"
F8 . 2 )
114 J(JJ)=J(JJtl)
T (20)=TCOOL
I F i L A P S . E Q . I)
GO
J O 120
____
__
T-FOUR
-TWO
'"
TT
i!/=J+700
GO TO 125
120
125
.
N=I
_
I F ( N . G T .250) T I N
=TSTEP
S1=T1D0T(T1ZER0_,T2ZER0_,.T3ZER0 ,THZERO ,H1ZER0)*DTHE_TA - T 1 = T 2 D 0 T ( T 1 Z E R 0 ,T 2 Z E R O , T Z Z E R O , T H Z E R O „H l Z E R O ,E 2 Z E R O )* DT H E T A
UA = T3DOT(T2.ZERg , T Z Z E R O ,THZERO ,H2 ZERO , HZZERO ) * D T H E T A
Vl = T H D O T i T Z Z E R O ,T H Z E R O ,H Z Z E R O ,H H Z E R O )* D T H E T A
--- - :m = B l D O M TI ZERO ,T2.ZER.0_, BI Z E R O „ H 2 Z E R O } *D T H E T A
...
Xl = H 2 D O T i Tl Z E R O ,T 2 Z E R O ,T Z Z E R O ,H l Z E R O ,H 2 Z E R O ,H Z Z E R O )* D T H E T A
- I l = B Z D O T i T 2 ZERO-,TZ ZERO__, T H Z E R O ,H 2 Z E R O ,H Z Z E R O ,HHZERO ) *DT H E T A
' Zl = H H D O T i T Z Z E R O ,T H Z E R O ,H Z Z E R O ,H H Z E R O )*D T H E T A
TSl = TlZERQjJS1 12
_ _ _ :_ _ _ _ .
__
T T l = T 2 Z E R O H T 1 /2
- TU 1=13 ZERO*U,l./,2- - - _ _ _ _ _ _ _ _ _ _ __ _ _ _ __ _ _ _ _ _ : _ _ _
„ _
TVl=THZERO+Vl/2
H W I = H l Z E R O + M l /.2 _ _ _ _
........
HXl=H2ZERO+Xl/2
'
HXl = H Z Z E R O + Yl./2 . . . . . .
. ........
HZl=HHZERO+Zl/2 '
■
S.2 = T 1 D 0 M L T S 1 >111,» TR3., TJX, H W I ) * D T H E T A . _ _ _ _
. .. __ _ _
T 2 = T 2 D O T i T S l , T T l , T U I , T V l iH W l ,H X l )* D T H E T A
U 2 = T Z D O T i l T l , TUl , T V l ,HXl , H J l ) * D T H E T A
V2=THDOTiTUl,TVl,HYl,MZl)*DTHETA
W2 = H1D0TXTR1-,TT1,.HW1,MXX)*D THETA
X 2 = H 2 D O T i T S l ,T T l , T U I , H W l , H X l I H Y l )* D T H E T A
Y 2 =H ZDOTX T T l , TU UT.V.1,HXl.,.HXl.,M,ZlX*DTHETA ........
Z 2 = H H D O T i T U l ,T V l , H Y I , HZ I ) * D T H E T A
T S 2 =Tl Z E R O + S 2 /2.. _ _
......
T T 2 = T 2 Z E R O + T 2 /2
TU2=TZZERO+U2/2
T V 2 = T H Z E R O + V2/2
________ ■
______
______
H W 2 = H l Z E R O + W2/2
"''
H X 2 = H 2 Z E R O + X 2 /2
HY2=HZZERO+Y2/2
'
H Z 2 = HH Z E R O + Z 2 /2
SZ=TlDOTiTS2,TT2,TU2,XV2,HW2)*PTHETA_ _
.
____
I.
,'o
I
!—
T 3 = T 2 D 0 T ( T S 2 , T T 2 ,T U 2 , T V 2 , H W 2 , H X 2 )* D T H E T A
U 3 = T 3 D O T ( T T 2 , T U 2 ,T V 2 ,H X 2 ,H Y 2 )*D T H E TA
V 3 = T A D 0 T ( T U 2 , T V 2 , H Y 2 , 5 2 2 )* D T H E T A
W 3 = H 1 D 0 T ( T S 2 ,Fff2 , 5 W 2 , H X 2 )* D T H E T A
X 3 = H 2 D O T (.T S 2 ,T T 2 , T U 2 iH W 2 . , J X 2 , H Y 2 ) * D T H E T A
J 3 - H 3 D 0 T ( ffff2 , T U 2 , T V 2 ,5 Z 2 , 5 7 2 , 5 2 2 )* D T H E T A
2 3 =5450ff( T U 2 , T 7 2 , 5 7 2 , 5 2 2 )*D T H E T A
T S 3 - T 1 Z E R 0 -VS3
T T 3 = T 2ZE H O + T 3
T U 3 = T 3ZE R O + U 3
Tys = T 1^ZERO + V3
H W 3=H l Z E R O +W3
HX3=H2ZERO+X3
H Y S = H S Z E R O + Y3
523=542550+23
S A = T l D O T i T S 3 , ffff3,ff53,ff73, 5 5 3 ) *5ff55ffA
TA = T2DOT(.TS3 ,TT3 , T U 3 , T V 3 , H W 3 , H X 3 ) * D T H E T A
U A = T S D O T (TT 3 ,ff53,ff73 , H X S ,5 7 3 )* D T H E T A
V A = T A D O T i T U 3 , T V S ,H Y 3 , 5 2 3 )* D T H E T A
W A = H l D O T i T S S ,ffff3 , 5 5 3 , 5 X 3 )* D T H E T A
X A = H 2 D O T i T S 3 , ffff3 , ff53, 5 5 3 , 5 7 3 , 5 7 3 )* D T H E T A
Y A = H S D O T i T T S , T U S ,T V S , 5 7 3 , 5 7 3 , 5 2 3 ) *5ff5£ffA
2 4 = H A D O T i T U S ,ff7 3 , 5 7 3 , 5 2 3 )* D T H E T A
D E L T A S = i S l + 2 * S 2 + 2 * S 3 + S A ) /6.
D E L T A T = (ffl+2*ff2 + 2*ff3 + ff4)/6 .
D E L T A U = (51 + 2 * 5 2 + 2 * 5 3 + 5 4 ) / 6 .
D E L T A V = (71 + 2 *72 + 2 * 7 3 + 7 4 ) / 6 .
D E L T A W = (51 + 2 * 5 2 + 2 * 5 3 + 5 4 ) / 6 .
DELTAX=(71+2*72+2*73+74)/6.,
D E L T A Y = i 71 + 2 * 7 2 + 2 * 7 3 + 7 4 ) / 6 .
DELTAZ=(21+2*22+2*23+24)/6.
TO H E = T ! Z E R O + D E L T A S
T TWO=T2ZER0+DELTAT
TTHREE=TSZERO+DELTAU
TFOUR=TAZERO+DELTAV
Hl=HlZERO+DELTAW
' J
H2=H2ZER0+DELTAX
:
I
j
'_ _ _ _ _ _ _ _ _ _ _ _ _
,
!
i
I
I
_
. .
1
EZ=HZZEr o +BELTA l
E h = H A Z E R O + B E L T AZ
J=I
ARRAY{I , J ) = T Z E R O _ _ _ _ _ __
A R R A Y ( 2 ,J)=TONE
A R R A Y i Z , J ) = T TWO
ARRAYiH,Ji=TTEREE
A R R A Y i 5 , J ) =TFOUR
_ A R R A Y j Z ,J) = H l _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _
A R R A Y j O ,J)=E2
A R R A Y i Z ,J)=EZ
A R R A Y i 9,J )=HH
A R R A Y i 10 ,J)=F_L0WiT0NE , T W O ,Hl ,H2 )
A R R A Y j 1 1 , J ) = F L O W j T T W O ,T T H R E E ,H2,H Z )
A R R A Y i l 2 , J ) = F L O W i T T H R E E ,T F O U R , H Z , H H )
A R R A Y j 13, J") = B P I N H G i T O N E , T TWO ,H1,H2 )
IH , J ) = B P I N H G l T T W O , T T H R E E ,H2 ,HZ)
A R R A Y ( I S 9J ) = B P I N H G i T T H R E E ,TFO U R , H Z , RH)
A ^ A Y ( 16 ,J ^ A ^ A Y ( 2_,JL) - A ^ Y ( 3 .J)
A R R A Y i I V 9J ) = A R R A Y i 3 ,J ) - A R R A Y iH ,J)
A R R A Y j I S 9J-) =AR R A Y j H ,Jl) -A R R A Y j S 9J)
A M A Y ( I g 9J) =F Z57?0
. _-.
ARRAYj2Q,J)=TIN
T1ZER0=T0NE
T 2 Z E R O = T T W 0,_ _ _ _ _ _ _ _ _ _ _ _ _
TZZERO=TTHREE
T h z e r o =TFOUR
.. ....
HIZERO=Hi
H2ZERO=H2
,
..
H Z Z E R o =HZ
HAZERp = H H - - - - --- —__
TIME=B THETA*jN ~ 1 )
F Y l = T B R O P l ( T O N E ,T T W O 3 T T H R E E ,T F O U R )
B T 2 = T B R 0 P 2 ( T T W O , T T H R E E ,T F O U R )
B T Z = T B R O P Z j T T H R E E ,TFOUR)
BTH=TBROPHjTFOUR)
INBEXjzJ
_ _ _ _ _ _ _ _ __
^O-T-
\
W R I T E M Z *112) ( A R R A Y ( K tJ) 3K = l,5'),DTliD T 2 , D T 3 , D T l\, ( A R R A I ( K iJ) ,K = S ,7
I ) .T I M E
.
1
I F ( H I . G T . 2 0 . O R . H I . L T . (.01)) GO TO 160
.150 C O N T I N U E
. __ _ _ _ _
____ ___________
.
j
160 W E I T E ( 1 0 8 ,162)
16 2 F O R M A T (_' X H ^ Z H R X E _ _ IUlF D R H ..L^FMOWI- 2 ^ F L Q W 2 - 3. „ F L O W S tA
DPl-2
I
DP2-3
PP3-4
DTl-2
DT2-3
DT3-H
TIME
FZE
2 R O\/ /.)_ _ _ _ _ _ _ _ _ _
_ .... ..
DO 170 J = I iI N D E X
N=J.
. ______ ____ __ . .
.
___
I F i L A P S . E Q . 2) N = J + 7 0 0
_ _ -TIME= (.Nr.I )*D.THETA - - - - - - - - - - ... _ _ _ _
___
____ _
WRITEi 1 0 8 , 1 6 4 ) ( A R R A I ( I iJ) , 1 = 8 , 1 8 ) , T I M E ,A R R A Y (19 ,J")
.16.4 F O R M A T i S (FI ,2., 3.ZJ ,.6 CF8„.,3,2Z) , F 8 . 2 , 2 X iF R . 2 )
170 C O N T I N U E
__
1 0 8 , 1 8 0 ) P R O P T , P R O P H , S E T T x S E T H , R E S E T T iR E S E T H
180 F O R M A T i / ' P R O P - T = ' ,F5.2,5X, 'PROP-H=' ,F5.2,5X, 'S E T - T = ' ,F5.. 1 , 5 X , 'SET
IjJI=' ,FR . 1 ,5 X.,A R E S E T ^ T^l.,.F 5 . I ,RX., 'RESET-H=J. ,P.5,1,/) . .... , .
"
I F i L A P S . E Q . 2) GO TO 205
o
DO 195 J = 2 2 0 ,INDEX.,P.....
..
_ T I
% P P T P ( 1 0 6 , 1 9 0 ) ( A R R A K I , J ) ,1=1,9)
I. '
“ ^
190 F O R M A T ( 9 F 8 . 3)
I
195 C O N T I N U E _ _ _ _ _ _ _ _ _ _
._ _ _ _ _ _ _ _ _ :_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ __ _ _ _ _ _ A
E N D F I L E 106.
j
■■DO 198 .J=220 , I N D E X ,A
W R I T E Cl 0 6 ,190 Y ( A R R A I i T iJ) , 1 = 1 0 , 1 8 )
.. ""
" ‘'
198 C O N T I N U E
ENDFILE' 1 0 6 - - - - - - - - - - - - - - - - - - - - - -----D O 2 0 4 J= 220 , I N D E X ,1I
W R I T E ( 10 6 , 20 0 ) ' (ARRAYi I, J) ,1 = 19,20)
200 F 0 R M A T ( 2 F 8 .3)
~
204 C O N T I N U E
.
/"
'
.
E N D F I L E 106
GO TO' 210
20 5 IF. (I N D E X .GT .A 84) I N D E X = A Q ^ t
''"-DO 2 06 J = A , I N D E X , A
W R I T E ( 1 0 6 ,190) ( A R R A K I ,J) ,1 = 1,91
O
206
207
CONTINUE
E N D F I L E 106
D O 2 0 7 «7= 4 , I N D E X i 4
WBITEi 1 0 6 , 1 9 0 M A M A
CONTINUE
E N D F I L E 106
DO 2 09 j = 4 , I N D E X ^
Y ( J 1J) ,1 = 1 0 , 1 8 )
M J J J d O e ,200) ( A M A Y ( J 5J) ,J = 1 9 ,20 )
209
210
250
-1Q8.
CONTINUE~
E N D F I L E 106
CONTINUE
I
R E W I N D 106
CALL EXIT
END
F U N C T I O N F L O W iA , B iC iD)
P O N E ( A ) = ( 2 9 . 9 2 1 / 7 6 0 . ) * 1 0 . * * ( 7 . 9 6 6 8 - 1 6 6 8 . 2 1 / ( 2 2 8 +A ))
R H O L i A ) = 0 . 6 2 9 3 717 30 + 0 2 - ( 0 . 1 7 5 4 8 5 5 0 - 0 1 )* A - (. 1 3 5 5 1 5 2 8 0 - 0 3 ) * A * * 2 '
ENTHALiA)= (.10740630+4-(.933611300 )*A-(.104531320-02)*A**2)*778.
D P C M H G i A , B , C , D ) = i i C - D ) / 1 3 . 6 + P 0 N E i A ) - P O N E i B ) ) * 2 . 54
C O N S T = 0.19206921.5 J + 2
I
D P = D P C M H G i A iB 1lC ,D)
I F i D P .G T . ( 0 . 0 ) ) OO J O 10
FLOW = 2 5 .
■GO TO 20
F L 0 W = C 0 N S T * (YJ<S,( A ) * * . 1 7 4 ) * i R H O L i A ) * * . 2 5 6 6 ) * ( i D P CMHGiA , S , 0 , 0 ) *2 7 . 84 ■
10
20
1 5 )„* * , 56 9 4 ) / (M Y 5 A O ( A ) * * . 1 5 6 4 ) ..
RETURN
END
FUNCTION
_____ _
Y J S ( A ) ■' "
Y J S = ( 0 . 9 3 5 3 5 6 6 6 - ( 0 . 1 0 1 3 0 3 7 8 J - 0 I )* A + ( 0 . 3 6 1 6 1 5 5 ) * A**2)
J J ( Y J S . O J . ( 0 . 2 4 ) ) OO JO 15
GO J O 20
__
15 Y J S = 0 . 2 4
O O JO 30
.
20 J J ( Y J S . O J . (0.4) ) OO TO 25
O O J O 30
25 Y J S = O . 4 0__
_
_ _ _ _ _ _ _ _ _ _ _ _ ;_ _ _ _ _ _ _ _ _ _ _ _ _ _ _
Computer Listing for 4 - t a n k Plot Program.
YJOB
1 2 Z 1 Q , H O L Z B E R G E R ,55
IA S S I G E F : 1 0 5 ,'CD'SVI C E ^ M T A 8 1 ) .
IL I M I T S ( T I M E , 0 ) , { P A G E S ,200) ■
\F ORTRAtl
'
""
.. .
"
C
■ T H I S PROGRAM_ R E Q U I R E S TWO H E A D I H G CARDS.
D I M E N S I O N A R R A H 2 0 , 1 3 0 ) , L I N E ( l 20), L A B E L (21), T I M E
'
_
(28), N A M E l (18),
NAM
i
I
I
122(12)
' , 'T-H
','H-I ' , ' H - 2 1 ,' 5- 3 »
,'DT12 ' , '552 3' , '5
I,- '5-4 ' , '51-2 ' , ' 5 2 - 3 ' , ' 5 3 - 4 ' , ' 5 51 2 ' , '552 3' , '5534'
25.34' , 'F-IN' , 'T-IN' ,'
'/
R E W I N D 106
R E A D ( 1 0 5 , 1 )"N A M E l
'
'
1 505MA5(18A4)
READ (105,2) N A M E 2
2 FORMATS1 2 A 4 )
R E A D { 1 0 5 , 4 ) NSKiP,DTHETA
4 FORMA5(52,510.3)
D O 15
5=1,28
15 T I M E (J) = (J-I) *D T H E TA * N S K I P * 10.
DO 200 L A P S = I ,2
3 DO 5 5 = 1 , 1 3 0
R E A D ( 106 , 1 0 , 5 5 5 = 1 1 , 5 5 5 = 11) ( A R R A H I ,J) ,5 = 1,9)
10 F O R M A T ( 9 5 8 . 3 )
NCOUNT = J
..
I F i N C O U N T . GT_. 12 0) N C 0 U N T = 1 2 0
5 CONTINUE
11 DO 12 J = I ,130
R E A D ( 1 0 6 , 1 0 , 5 5 5 = 1 3 , 5 5 5 = 1 3 ) ( A 5 5 A J ( J 1J ) , 5 = 1 0 , 1 8 )
12 C O N T I N U E
-6QI-
INTEGER B L A N K f r
'j~
INTEGER STAR/'*
'/
D A T A L A B E L / vT-O '
T O N E ' T T W O ' ,'T - 3
|
13
6
14
16
20
BO 14 J = I ,130
R E A D d O S , 6 ,S TO
FORMAT(2FQ.3)
CO R T I R U E
DO 2 0 0 1 = 1 , 2 0
= IS
,ERR = IS
’'
ZzWAZ=I . S - I O
___
X M I R = I .S + 10
D O 2 0 J = I , RCQURJT _
J S ( A R R A Y (.I ,J) . G T .
)
A R R A Y {19
'
_
34
,ARRAY(20,J)
"
X M A X ) X M A X = A R R A Y i I ,J)
I F i A R R A Y i I ,J) . L T . X M I R ) X M I R = A R R A Y i I ,J)
GO
(31,31,31,31,31,7 0,70,70,70,80,80,80,60,60,60,5
'.,31 ^ R X M A X = Z M A Z T T ~
33
,J)
'™ ‘
RXMIR=XMIR
I F ( iR X M A X - R X M IR) . GT .6 . )' GO Z C 34
RXMIR = S Z M A Z - 6
DO 33 J = I , R C O U R T .
A R R A Y i I , ! ) = iA R R A Y i I , J ) - R X M I R ) * S
CORTIRUE
ZMAZ - R X M A X
...
DELTA = I
RAXIX = 6
■ r"
GO TO 90
I F i i R X M A X - R X M I R ) . G T .12) GO TO 200
R X M I R .=. R X M AX. -.12... . . .
S C 35 J = I ,RC O U R T
A S S A Z(X,.J.)_ h _(..ASSAJ.X.Z.,.J.) - R J M I R )* 3_
CORTIRUE
__
35
_ Z M A Z = S Z M A Z . ..
____ ______
DELTA = 2
S A Z Z Z =..6_ _ _ _ _ _ _ _ . _ .
GO TO 90
- 5,0 .JSZ = ZMAZ*5.+1_._ _ _ _ ' _ _ _ _ _ _
X M A X = I R T * .2
IRT = ZMZS*5.
X M I R = Z S Z * (.2)
0 ,‘5 0 , 5 0 , 8 0 , 3 1 )
•
______
■
zs( (Z M A Z - Z M Z J O ^ G 2 L A ^ 2 l _ . _ g p _ _ m 5 3 . . . . .
" *
T
.1.
DIFF
XMAX
52
= I .2-(XMAX-XMIF)
= X M A X + D IFF./2 i___
_
Z M T W = X M A X - 1 .2
DO 5 2 J = I , N C O U N T
.. . .
A W W A T ( T 9T) = ( A W W A T ( T , c 7 ) - Z M T W ) * 3 0.
CONTINUE
______
DELTA = 0.2
. W A Z T Z , = .6______ ________
53
54
GO TO 90
I F ( ( X M A X - X M I N ) .G T . 1 . 8 )
G O TO 2 0 0
ZMTW=ZMAZ-1.8
DO 54 J = I iN C O U N T
A W W A T ( T 9T ) = (A W W A T ( T 9T ) - Z M T W ) * 3 6 . / I .8
D E L T A = O,2
WAZTZ
=
______________
CO TO 90
60
67
_
....
_ .
.
=
I.2 - (ZMAZ-ZMTW)
{DIF F .L T . ( 0 . 0_) ) CO T O
65
ZMTW = ZMAZ-I.2
D O 6 3 J = I iN C O U N T
_____
A W W A T ( T 9T) = ( A W W A T ( T , T ) - Z M T W ) * 3 0 .
CONTINUE .
________ ____________
DELTA = 0.2
WAZTZ = 6
. ____
. ...
C O T O 90
D T F F = 1.8 - {XMAX-XMIN)
I F {DIFF.LT. 0 . 0 ) . C O T O 2 0 0
Z M T W = Z M A Z - 1.8
D O 67 J = I , N C O U N T
A W W A T ( T 9T ) = (A W W A T ( T 9T ) - X M I N X * 3 6 . / 1 .*8
CONTINUE
DELTA = 0.2
WAZTZ
= .4
.
-
-
-
--
%LTT
IF
65
-
INT = Z M A Z * I 0.+1.
X M A X = I N T * (.1)
.' . .
.
.
I N T = X M I N * 5.
X M I N = I N T * { , 2)
'____________ , .
DTWF
63
.
4
GO TO 90
7 0 EXMAX=XMAX+2_ _
XMAX=NXMAX
Si x m i n =X m
a x
-e
IF (XMIN.LT'.NXMIN) GO 'TO 74
DO 72 J=I,NCOUNT
72 AJf7?AJ(J,J") = (A57?AJ(T,J)-/IZ^J^)*6.
DELTA =I.
_
NAXIX=G
GO TO 90
74 ZAfAZ= 1 9
..ZAfjTAT = I
DO 75 J=I,NCOUNT
ARRAY(I,J) = (ARRAY(I,J)-I)*2
75 CONTINUE
DELTA = I
■___
NAXIX = 2
........ .
OO TO 90
80 INTMAX=XMAX* .1+1
XMAX=INTMAX*10.
JD ( (XMAX-XS1IN) .GT.60 . ) GO JO 86
X M I n = X M A X - S 0.
DO 85 J=I,NCOUNT
85 ARRAY (.I,J) = (ARRAY {I ,J)-XMIN) *( .S)
DELTA = 10.
NAXIX = 6 .....
OO TO 9 0
86 ZAfJAT =ZAfAZil20..______ _______
DO 87 J=I,NCOUNT
87 Ai?i?AJ( J ,J)=( ARRAY (I ,J) -XMIN)* (0.3 )
DELTA=20.
NAXIX=S
90 SMALL = 36.
AZDDPJ _= -I
DO 150 Z Z Z = I , 35
./ NREPT = NR-EPT + I
BrG-^-SMTALL
•---
*
SMALL
DO 92
= BIG - I
J = I ,120
L.
B & w
92
. .. . .
. _____
.
..
CONTINUE
D O 95 J = I s N C O U N T
I F (BIG. G E .A R R A U I , J~) IAND^. ( S M A L L .L T .A R R A Y (I ,J)))
95 C O N T I N U E - ■ ,
KNT' = K K K
I F (K N T .G T . I ) GO TO 10 0
: ~~
r a r r D ( 1 0 8 , 9 6 ) XMAX,.( L I N E .(J)., J = I , 120) .
96 F O R M A T ( ’l ’ , 4 Z , F 6 . 2 , , 1 2 0 A 1 ) ■
...
LINE{J)=STAR
'
"
I
OO TO 1 50
100
I F ( K N T . N E . 18)
- JJ = I
GO
TO
HO
OO T O 1 1 2
H O J J = 21
112 I F ( N R E P T . N E . N A X I X )
N R E P T J= O . _
_
115
'
120
______ _ _ _ _
.. . . . .
GO TO
.
XMAX- =' X M A X - D E L T A
W R I T E (.108,115 L L A B E L ( JJ) , X M A X , ( L I N E ( J ) , J = 1 , 1 2 0.)
____
F O R M A T ( A 4 , 1 Z - , T 6 .2,
,120A1)
GO T O 1 5 0
. . . . . . ...
.
„
.....
.
W R I T E ( 1 0 8 , 1 2 5 ) L A B E L ( J J ) A L I N E ( J ) i J= l, 120)
120
1.2 5 f 0 # A T _(A 4
; 120A.1). . . . . . . . .
150 C O N T I N U E
W R I T E (.108,170). N A M E L, N A M E H . . . . . .
170 F O i ? M A T ( H Z , l^ ' , 1 8 A 4 , 1 2 A 4 ) ;
^ T T F ( 1 0 8 , 1 5 3 ) ..
153 F O R M A T ( I I X i '
!
f
... _
..
..
..
Ji.
~g;'
_ _ _ l;'
___
.. . . _ .
_____ ________
2*')
160
I F (L A P S . E Q . 2.)... GQ,..TO 16 5
W R I T E (1 0 8 , 1 8 0 ) (TIME(KI)
165
180
200
W R I T E ( 1 0 8 , 1 8 0 ) .,(-TI M E (K I ) ,Z T = 1 3,2
F O R M A T ( 2 Z , 1 3 " ( 6 Z , F 4 . 0) )
"
CONTINUE
00. T O .2.0 0
..._ _
.
...
,' F T = I ,13) .
5)
......
"
" -
^
R E W I N D 106
CALL EX I T
END._
„ .
\LOAD
\DATA
7/21/69 '
R U N 3,
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LITERATURE CITED
1.
A i k m a n , A. R,, nFreqnencyvResponse- Analysis and Controllability of
a Chemical Rlantir, 'Trans. A.Sl-M-E.. , .Yol '76, M o t 125^, p. 1313-20.
2.
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MONTANA STATE UNIVERSITY
LIBRARIES
762 10005639 7
D378
H748
cop. 2
Holzberger, Thomas W.
Transient response of
a 2-tank flash evapo­
rator
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