AN ABSTRACT OF THE THESIS OF
STEPHEN DE MONT HILL
(Name)
for the
DOCTOR OF PHILOSOPHY
(Degree)
in CHEMICAL ENGINEERING presented on
a/24.4 0471
(Date)
/
2_
Title: HEAT AND MASS TRANSFER IN KROLL PROCESS TITANIUM
SPONGE DURING SALT EVAPORATION
Abstract approved:
Redacted for Privacy
Robert V. Mlazek
Laboratory experiments were conducted to study those factors which
control the rate at which salts, such as magnesium chloride and sodium
chloride, are removed from titanium sponge. The sponge was prepared by
reducing titanium tetrachloride with either magnesium or sodiumiand the
resulting sponge salt mixtures were treated by vacuum evaporation to remove
the salts. As the temperature, bulk density, and surface to volume ratio
of samples were varied, measurements were made of the rate of salt
evaporation in a vacuum. A constant-rate and falling-rate period were
observed during the drying process. An apparatus was constructed and
measurements were made of the thermal diffusivity of titanium sponge
compacts and industrial titanium sponge. A computer program was
developed to incorporate experimentally determined data to simulate both
heat and mass transfer during the salt removal process for an industrial
sized sponge cake. The program yields transient concentration and
temperature profiles throughout the sponge cake and shows the effect
of controllable variables such as sponge geometry, reactor size and
heat transfer boundary conditions. An estimate of the mass transport
coefficient within a sponge block was made by comparing experimentally
determined and computer calculated salt concentration profiles of
partially distilled sponge compacts.
Heat and Mass Transfer in Kroll Process
Titanium Sponge During Salt Evaporation
by
Stephen De Mont Hill
A THESIS
submitted to
Oregon State University
in partial fulfillment of
the requirements for the
degree of
Doctor of Philosophy
June 1972
APPROVED:
Redacted for Privacy
Professor of Chemical Engineering
In Charge of Major
Redacted for Privacy
Head of Department of Chemical Engineering
Redacted for Privacy
Dean of Graduate School
Date Thesis is presented
.//,/..c.,( /1/
Pir)
Typed by Grace Danton for Stephen De Mont Hill
ACKNOWLEDGEMENT
The author wishes to thank all those who assisted in the progress
of this work. Special acknowledgement goes to Dr. R. V. Mrazek for his
aid and guidance. The friendly assistance of the Staff of the Chemical
Engineering Department of Oregon State University is greatly appreciated,
as is the complimentary appointment assistantship.
This research was conducted at the Albany Metallurgy Research
Center of the U. S. Bureau of Mines and was supported by Bureau funds
and service facilities. The assistance of Mr. Frank E. Block, former
Research Supervisor of the Chemical Processes Projects, is hereby
gratefully acknowledged. Special credit is also given to Herbert 0. Boren,
Engineering Technician, for his aid in preparing samples and to the
analytical services group for their continued interest and support.
The author would also like to express his gratitude to the Oregon
State University Computer Center for its partial financial support and
particularly to R. Jay Murray for his continued help in computer programming,
Finally, those who have directly participated in the final preparation
of this thesis deserve special mention--in particular Mrs. Grace Danton,
who typed the thesis, Mr. James Wilderman who prepared the final drawings, and Mr. Bob Nelson who reproduced the final form.
TABLE OF CONTENTS
Page
INTRODUCTION
1
LITERATURE REVIEW
5
THEORETICAL CONSIDERATIONS
10
EXPERIMENTAL APPARATUS AND PROCEDURES
Weight-Loss Apparatus
Sample Preparation
Thermal Conductivity
14
21
24
EXPERIMENTAL RESULTS
Evaporation Studies on Magnesium Reduced
29
Constant-Rate Period
Critical Concentration
Falling-Rate Period
Evaporation Studies on Sodium Reduced Sponge
Thermal Conductivity Results
29
33
36
Sponge
41
47
COMPUTER SIMULATION OF EVAPORATION PROCESS
51
COMPARISON BETWEEN REMOVAL OF NaC1 and MgC12
63
SUMMARY
69
BIBLIOGRAPH Y
71
APPENDICES
I. Nomenclature
II. Data
III. Computer Program
75
76
118
LIST OF FIGURES
Page
Figure
1.
Kroll Process for Titanium Sponge
2.
Heat Transfer in Differential Element
12
3.
Recording Balance Detail
15
4.
Control Thermocouple Calibration
17
5.
Weight-Loss System
19
6.
Typical Weight-Loss Results for Titanium Sponge
20
7.
Variation of Salt Evaporation Rate with
Temperature for Various Samples
23
8.
Thermal Condu ctivity Apparatus
26
9.
Typical Sine-Wave Recording
28
Variation of Surface Evaporat ion Rate with
Temperature f or Titanium Sponge Compacts
31
Variation of Surface Evaporation Rate with
Temperature for Titanium Sponge Compacts
32
12.
Effect of Sample Size and Temperature on
Critical Concentration
34
13.
Constant-Rate and Falling-Rate Periods
35
14.
Falling-Rate Period for Titanium Sponge
38
10.
2
Compact Containing MgC12
Containing MgC12
11.
Containing MgC12
Compacts Containing MgC12
Figure
Page
Smoothed Data for Evaporation from Titanium
40
Typical Weight-Loss Results for Titanium
43
Variation of Initial Rate of Evaporation
With Temperature for Titanium Sponge
45
18.
Some Falling-Rate Data for NaC1 from
Titanium Sponge Compacts
46
19.
Experimental Thermal Conductivities
49
20.
Cutaway View of Titanium
52
21.
Comparison of Experimental and Computer
Calculated Concentration Profiles
59
22.
Comparison of Experimental and Computer
Simulated Evaporation
60
23.
Computer Simulation of Industrial Sponge
Cake Evaporation
62
24.
Proposed Models for Drying of Magnesium
and Sodium Reduced Titanium Sponge.
64
25.
Results of NaC1 Partial Evaporation Runs.
66
26.
Results of MgC12 Partial Evaporation Runs.
67
15.
16.
17.
Sponge Compacts Containing MgC12
Sponge Compact Containing NaC1
Compacts Containing NaC1
Sponge Batch
HEAT AND MASS TRANSFER IN KROLL PROCESS
TITANIUM SPONGE DURING SALT EVAPORATION
INTRODUCTION
Kroll (20) published the first description of a laboratory-scale
magnesium reduction process for the preparation of titanium metal in
1940. Although sodium reduction processes for production of metal
from titanium tetrachloride are now coming into industrial use both in Great
Britain and the United States, by far the greater part of the titanium available
at the present time is produced by the Kroll magnesium reduction process.
In all essentials the industrial method differs very little from Kroll's
original laboratory process.
Figure 1 depicts the well known Kroll process and shows the
individual steps which comprise the entire operation. A mixture of rutile
and coke is first fed into a chlorination furnace where they react with
chlorine to form titanium tetrachloride which is transported, as a vapor,
to a tetrachloride purification and condensing tower. Metallic impurities
from the rutile such as silicon, iron, and vanadium are removed from
the titanium tetrachloride stream in this purification chain. Purified
titanium tetrachloride liquid is then fed into a reduction retort where it
reacts with magnesium metal to form titanium metal sponge and byproduct
magnesium chloride salt. Part of the magnesium chloride salt may be
RUTILE
;*
TITANIUM
TETRACHLORIDE COKE
Y!:
VAPOR
I
CONDENSER
CHLORINE
,....
HEAT
PUMP
t
TITANIUM
PURIFICATION
1\
f
RAW SPONGE
.
a
.
HEAT
ai,)..).
SPONGE
44
'V.
MAGNESIUM
CHLORIDE
MAGNESIUM
ELECTROLYTIC
CELLS
FIGURE I.-Kroll Process for Titanium Sponge.
3
drained from the bottom of the reduction retort as a liquid, and recycled
through the electrolytic cells where magnesium and chlorine are recovered
for recycle to the reduction furnace and chlorinator. The titanium sponge
resulting from the batch reduction step is still saturated with 30-40 weight
percent magnesium chloride, along with a small excess of reducing metal.
These must be removed from the titanium sponge to obtain a titanium product
which can be melted to form a usable ingot. The salt removal, or purifica-
tion step of the process is accomplished either by high temperature evaporation or by dilute acid leaching. Leaching has the major disadvantage that
residual hydrated magnesium chloride in the sponge decomposes during
subsequent arc-melting. The resulting volatile products of decomposition
cause ingot porosity and complicate the melting process. Removal of the
salt by high temperature evaporation yields sponge generally considered to
be of higher quality, but evaporation is a slower, more costly procedure.
In a recent Journal of Metals article entitled "Process MetallurgyRenaissance or Continued Stagnation": (38) the author writes
There exists a vast potential for work in the metals
industry, aimed at the development of transport or
systems based on models of processes, with a view
of their optimization and optimal control. The realization of this potential could provide the marked
improvement in overall performance that is badly
needed if metals are to remain competitive with the
alternatives that are becoming available.
4
This author is of the opinion that the titanium processing industry
is badly in need of optimization in order to decrease production costs,
improve quality and therefore make titanium more competitive with
alternative materials. This investigation was carried out in an attempt to
define those factors which influence and control the rate at which salt is
evaporated from titanium sponge during the purification step of the process.
If rate controlling mechanisms could be determined and rate equations
defined, it was anticipated that a model could be described and that a
computer program could be devised to predict salt concent rations remaining
inside a sponge mass as evaporation occurs. Thus the effect of controllable
parameters could be studied as operating conditions were varied. Such
a program might be employed to optimize reactor design and operating
conditions for the sponge purification step of the Kroll process. In order
to optimize the entire process there is, of course, a need for optimization
of the individual operations. This study is only a first step in studying
the factors which control the rate which salt may be removed by high
temperature evaporation, a first attempt to determine the mechanism of
sal t removal and to model the purification step of the process using
experimentally determined information.
5
LITERATURE REVIEW
The subject of simultaneous heat and mass transfer in porous
systems has long been of interest. Most experimental studies have been on
porous systems containing water or some other low boiling, high volatility
material. There has been no previous work on the vacuum evaporation of
magnesium chloride, or any other salt, from a porous metal sponge media,
but this process may be considered similar to the drying of any porous
material. A recent publication by Harmathy (14) has summarized the
present state of the art of simultaneous heat and mass transfer in porous
systems with particular reference to drying. Another recent publication by
Fulford (12) contains a very good survey of recent Soviet research on the
drying of solids.
There are numerous possible mechanisms by which moisture may
be transferred within the material, depending on the nature of the material,
the type of moisture bonding, the moisture content, the temperature and
the pressure in the pores, etc. Some of these possible mechanisms are
as follows:
(a) Liquid moisture movement due to capillary forces;
(b)
Liquid moisture diffusion due to differences in concentration;
(c) Surface diffusion in adsorbed liquid layers on the pore surfaces;
6
(d) Vapor diffusion in partially filled pores;
(e) Vapor flow due to differences in total pressure in the pores;
(f)
Liquid moisture flow due to gravitational force; (mainly in
coarse- pore materials).
Evidently there are several mechanisms of internal moisture
transfer active simultaneously, and the role of these mechanisms may
change as drying proceeds. From among the several theories so far
suggested to explain migration of moisture in porous media, three have won
general recognizition: the diffusion theory, the capillary flow theory and
the evaporation-condensation theory.
The movement of moisture by diffusion was explicitly proposed as
the principal flow mechanism by Lewis in 1921 (23). Further studies on
this mechanism of transport were made by Tuttle (39), Sherwood (32) (33)
(34) (35), Newman (28) (29) and Childs (10) (11).
The fundamentals of capillary flow theory were laid down by
Buckingham in 1907 (6). He introduced the concept of "capillary potential"
and postulated the mechanism of "unsaturated capillary flow." A later
version of this theory presents the basic assumption that the moisture flux is
proportional to the gradient of the chemical potential of the moisture. With
this and other assumptions the fundamental equation of moisture migration
turns out to be similar to Fick's second law with a concentration-dependent
7
diffusion coefficient. Miller and Miller (25) and Remson and Randolph (31)
discuss this diffusion mechanism in detail.
The evaporation-condensation theory assumes that migration of
moisture takes place entirely in the gaseous phase. Experimental studies
by Gurr et al (13) and Hutcheon (16) showed that when a solid system is sub-
jected to a temperature gradient this assumption is essentially correct,
even at relatively high pore saturation. The evaporation-condensation
mechanism was utilized by Henry (15) and Cassie et al (7) (8) (9) in
describing movement of moisture in beds of textile materials and by
Nissen et al (30) and Breyer (5) in describing the movement of several
organic liquids through beds of porous materials.
In addition to the attempts to explain internal drying phenomena in
terms of a diffusion or a capillary mechanism alone, other workers have
assumed that drying of solids can be divided into two zones: one zone in
the interior which is still wet and in which there is little moisture transfer
resistance, and the other near the drying interface, the thickness of which
increases as drying occurs, and through which moisture is transferred by
vapor diffusion providing most of the resistance to internal transfer. The
problem, therefore, reduces in this case essentially to calculating the
changing position of the submerged evaporation interface as the drying
process proceeds (2, 17, 26, 27). While these mechanisms may be
8
adequate in individual cases, their over-emphasis has led to contradictions
and difficulties in other cases, as pointed out in the standard reference
on drying, e.g. (3).
One approach to generalizing the problem has been made by Krischer
in his paper (19). He attempts to allow for two main mechanisms of
moisture transfer (capillary and diffusional). He sets up differential
heat and mass transfer equations in which it is assumed that material may
be transferred by these two mechanisms in series, in parallel, or in more
complex series and parallel combinations. This involves the use of two
coeffficients to relate the rate of moisture transfer to the diffusional
driving force and to the capillary driving force. Unfortunately, in general,
both of these quantities depend on the nature of the material, the nature of the
pore structure, the moisture content and the temperature.
Anotle r approach to generalizing the problem of internal heat and
moisture transfer during drying has been made by Luikov's work in the
Soviet Union (24). This approach has been based on application of
methods of thermodynamics of irreversible process to the case of heat and
moisture transfer in drying. Essentially the moisture transfer is split
into two parts, one due to moisture transfer driving force, which is
specially defined in an attempt to encompass most of the mechanisms of
moisture transfer and which is characterized by a moisture diffusivity
9
coefficient, and one due to temperature gradient which is characterized by
a thermo gradient coefficient. Both coefficients have to be evaluated
experimentally at present, and in general turn out to be dependent upon
moisture content and temperature as well as upon the nature of the
material.
In very crude terms, the situation can perhaps be summed up by
saying that since all these coefficients vary in a complex manner with the
dependent variables, it is convenient and appropriate at the present time
to lump all the variability into one variable coefficient until sufficient
information is available to break down the transport mechanism further.
10
THEORETICAL CONSIDERATIONS
The drying of solids is usually taken to mean the removal of a liquid
from the solid by evaporation. In the evaporation process, heat is supplied
to the material which causes a simultaneous transfer of heat and mass to
occur. The evaporated liquid is usually carried away by means of an
external drying medium circulated over the drying solid. Often this
medium consists of a dry gas which may also be heated to act as the heat
transfer medium. In the first drying period (constant-rate drying) the
rate of drying per unit surface area depends entirely on the parameters
affecting evaporation such as gas velocity, flow patterns, temperature and
moisture content of the drying gas, and evaporation occurs at the surface
of the solid. In the case of drying by vacuum evaporation, the usual
picture of drying under constant external conditions is that the rate
during the constant-rate period is dependent upon the vapor pressure of
the evaporating liquid and the surf ace area available for evaporation.
If one considers the evaporation of any material from a porous
medium, it is necessary to consider both unsteady state heat transfer and
mass transfer for the system. To determine the equation which governs
heat transfer within a differential element in the interior of a porous
material in which evaporation occurs only at the surface, one need
consider energy transferred by conduction and mass flow, as well as
11
energy which is used to change the temperature of the element. These
heat transfer mechanisms are depicted in the three dimensional rectangular
differential element shown in Figure 2. The partial differential equation
which results from a heat balance on this element for the unsteady state
process is shown in equation (1) and is commonly found in many references
on heat and mass transfer, e.g. , (Bird, Stewart and Lightfoot). (4).
(1)
(
DT
+v
3T
+v
BT
+v
aT
=k
(
a2T
+
a.21,
+ .4)
The nomenclature for the terms used in the equations throughout this thesis
is shown in the nomenclature list in the Appendix.
The equation which represents mass transfer in an interior
differential element may be written in a similar fashion.
(2)
iE + vx -@C + vy
aC
+ vz Tz- De
(
2c
+
a2c
@2c
+ 7-zz)
Equations (1) and (2) are applied, in finite-difference form, to all
interior elements as they are written. For any element which has a surface
exposed to the drying medium, an additional term to account for evaporation
(or energy used in evaporation) must be included in each of these expressions.
These terms are included as boundary conditions for the surface nodes.
(MASS FLOW)z
(MASS FLOW)x
(CONDUCTION)y
(MASS FLOW)y
(CONDUCTION )x
(CONDUCTION)z
FIGURE 2.-Heat Transfer in Differential Element.
13
It is evident that in order to simulate the process mathematically
it is necessary to determine experimentally the rate of evaporation and the
factors which affect this rate, the effective thermal conductivity of the
porous medium, and the effective mass diffusivity or mass transport
coefficient for the system.
For this process, in which the heat transfer by mass flow is
small compared to conductive and latent heats and in which diffusion and
mass flow or capillary flow are lumped together, the equations reduce to
the following form:
(3)
(4)
V2T
ac
Ke
2
V C
These are the equations which are t he basis for the mathematical model
proposed for simul ation of the process of salt removal in titanium sponge.
To use these equations for a computer simulation of the process one
must experimentally measure the rate of evaporation, the thermal
diffusivity and the mass transport coefficient and the factors which affect
these quantities.
14
EXPERIMENTAL APPARATUS AND PROCEDURES
Weight-loss Apparatus
The rate at which a substance can be evaporated from a porous
material can best be determined by a thermogravimetric technique which
allows continuous recording of sample weight and temperature. When
determining the diffusion coefficient for an evaporating material in a
porous body, it is also necessary to know the decrease in weight of the
body during the process of drying, and the concentration of the diffusing
material in each part of the specimen must be determined for partially
dried specimens. To study the factors which influence and control the
rate at which magnesium chloride salt can be removed from titanium
sponge by high temperature vacuum evaporation, it was necessary
therefore to assemble a system which would allow a continuous recording
of sample weight and temperature during the salt removal process.
This was accomplished by suspending the sample from a Cahn RH
automatic recording electrobalance having a 100 gram capacity.
Details of the recording balance assembly are shown in figure 3.
A sample of titanium sponge containing salt is placed inside a
quartz basket which is suspended from the balance by using a quartz
fiber inside a Vycor tube. The sponge temperature was measured using
15
1---1._._._e
1--
Cahn RH
recording
balance
Water cooling
coils
Tare
.23 weight
D
To vacuum
system
D
-\
CIP:N-Inverted copper
cup
fiber
Vycor tube
Sample
.......1
0
0
0
Quartz hanger
0
0
0
Resistance heating---,,,c)
elements
o
FIGURE
o
Thermocouple
shield
3.- Recording Balance Detail
70-182
16
a shielded platinum-platinum 10%-rhodium thermocouple which was
located directly beneath the sample and which had been calibrated
previously against a thermocouple attached to the surface of a test
sample. A water cooled, inverted copper cup was placed above the
sample to condense the salt which evaporated from the sample. The
entire system was evacuated to about 104 torr using a mechanical
vacuum pump, a silicon oil diffusion pump, and a liquid nitrogen cold
trap. This vacuum was usually maintained overnight before heating the
sample. This served to remove any moisture which was absorbed on the
sample from the loading operation. The sample was heated inside the
Vycor vacuum chamber with a fluidized sand bath heater. The sand bath
was constructed of a 5-inch-diameter nickel cylinder having a conical
bottom which terminated in a nickel microfilter. The filter served as
a distribution plate for the preheated air which fluidized the sand. The
sand bath was heated with a 5-inch diameter combustion tube furnace
and the furnace was counter balanced so it could easily be repositioned
vertically. This allowed the sand bath to be heated for several hours
before it was brought up around the vacuum chamber containing the
sample. Consequently the sample attained an equilibrium temperature
in a minimum time after the experiment was started. Figure 4 shows
a temperature equilibrium curve as measured by the thermocouple
17
900
1
I
I
I
I
1
Control thermocouple
800
Sample thermocouple
700
600
0
cr 500
11-1
a_
2
400
300
200
100
0
10
FIGURE
80
70
50
60
40
TIME, minutes
4.-Typical Control Thermocouple Calibration.
20
30
71- 224
18
located directly beneath the sample and one attached to a sample. This
shows that equilibrium was established after about 10 minutes of heating.
The furnace temperature was controlled with a Wheelco precision control
unit which consisted of a Vane-type proportional milli voltmeter controller,
a magnetic amplifier and a saturable core reactor. Figure 5 shows a
schematic diagram of the complete weight-loss system. The thermocouple
output and the thermobalance control out put were recorded on a two pen
Electronik, model 194, lab recorder. Figure 6 shows a typical recorder
trace during a test. It can be seen from this trace that the equilibrium
temperature of the sample is attained in about 10 minutes, that there is a
period of heating up followed by a period of constant rate of evaporation,
and finally that there is a period during which the overall evaporation
rate falls off. These three periods are usually observed during the
drying of any porous material. In early tests, the rate of weight-loss
was measured at various salt concentrations by determining the tangents
to the weight loss curve. Later, a time derivative computer was used
with the Cahn balance which allowed both the sample weight and time
derivative, or slope to be recorded simultaneously on the two pen
recorder. The temperature of the sample was then recorded on a
separate recorder and periodically checked using a portable potentiometer.
Cahn balance
Balance
control
Liquid ---"T
nitrogen
trap
Temperature
controller
0
0
0
Vacuum
pump
0
0
0
i
Diffusion
pump
L
Sliding sandbath heater
0
0
0
0
0
.
0
0
L__1
FIGURE
5.- Weight- Loss System.
I
Two pen
recorder
12
1,000
20
900
Temperature
800
700
600
500
cc.)
rr
400 LI
300 LA
200
Sample weight
100
0
7
I
0
FIGURE
I
I
I
I
I
I
I
40
50
60
70
TIME, minutes
6.- Typical Weight- Loss Results for Titanium
10
20
30
Sponge Compact Containing MgC12.
71-159X
21
Sample Preparation
Titanium sponge produced on a large industrial scale normally has
a higher average bulk density than the material produced on a small
laboratory scale. Sponge produced in the laboratory also has a wider
variation of properties, such as bulk density, salt content, and porosity.
It was therefore necessary to modify the sponge prepared in the laboratory
to resemble more closely the sponge product produced industrially. The
sponge for this study was produced by reducing titanium tetrachlorides with
magnesium metal in a 5-inch diameter retort heated in a furnace at about
900° C. The crucible containing the dendritic sponge and byproduct
magnesium chloride salt was then inverted and again heated to about
900° C, in an inert atmosphere to drain excessive salt free from the metal
sponge. Specimens were then prepared by grinding the freshly reduced
dendritic sponge still containing 30-40 weight percent salt in a small
hammer mill and then recompacting the ground material into cylindrical
or rectangular compacts. These compacts were made by compressing
the sponge isostatically while contained in a rubber sock. By controlling
the particle size and compacting pressure, it was possible to obtain
samples which were representative of any type of industrial grade sponge.
This method of sample preparation also gave specimens which yielded
reproducible results when determining the effects of sample size,
22
temperature and density on the evaporation rates of salt from the samples.
Because of the hygroscopic nature of the raw sponge, it was necessary to
handle it in an inert atmosphere glove box throughout the sample preparation procedure.
To obtain the proper characteristics of the compacted sponge, the
effect of temperature on the rate of evaporation during the constant-rate
period for compacts of various particle size and compacting pressure
was determined. These values were compared with rates determined for
laboratory sponge samples of various grades of sponge of approximately
the same size and weight. Figure 7 shows the rate of evaporation per
gram of sample during the constant-rate period as a function of tempera-
ture for various compact conditions. The data were plotted as the
logarithm of the rate vs the inversie of the absolute temperature because
the relationship should be similar to the Langmuir evaporation equation
for vacuum evaporation. From this figure it is evident that this compacting technique yields samples which may represent any type of industrial
grade sponge. Therefore this compacting technique was used as a
method of preparing samples of various geometry and surface to volume
ratios. The compacts used for the rest of the experimental work were
all compressed to 10,000 p.s.i. This method gave specimens which
initially contained uniformly distributed salt concentrations and which
23
o Very porous sponge
A Medium sponge
o Dense sponge
Porous compact 10,000 psi
Medium compact 15,000 psi
Dense compact 30,000 psi:,
10.00
5.00
°\:
No
A
No
.10
.05
09
1.0
1.1
1.2
1.3
1.4
1.5
1.6
1,000/T, °K-'
FIGURE 7.- Variation of Salt Evaporation Rate With Tem-
perature for Various Samples of Titanium
Sponge Containing MgCl2.
69-59
24
yielded reproducible weight-loss results.
A second method used to determine the proper characteristics of the
compacts used in this study was to compare measured thermal conductivities
of specimens prepared by the compacting technique and specimens which
were machined from actual industrial titanium sponge. The technique of
measuring thermal conductivity and the results of these measurements
are discussed in detail in later sections.
Thermal Conductivity Apparatus
To simulate a drying operation, in addition to having evaporation
rate data, it is necessary to know the thermal conductivity of the heat
transfer media. In this study, it was also necessary to compare the
heat transfer characteristics of sponge compacts produced from ground
laboratory sponge and actual industrial titanium sponge. The goal was
not only to measure the thermal conductivity of dried sponge, but to
determine how the salt content affected the conductivity of the porous
sponge medium. A thermal conductivity technique was therefore selected
which would allow a continuous measurement of the conductivity to be made
as the salt was evaporated from porous sponge specimens.
The apparatus was patterned after one described by Siddles and
Danielson (36) and later by Ables, Cody and Beers (1). A schematic
25
diagram of the apparatus is shown in figure 8. It includes a cylindrical
specimen containing two butt-welded AWG-28 chromel-Alumel thermo-
couples which were peened into small holes drilled along the diameter
of the sample a known distance apart. It was thus possible to locate
the thermocouple junction at the center of the rod. The thermocouple
separation was selected so that the time required for a heat pulse to
travel between the two thermocouples could be measured conveniently.
A sinusoidally varying heat input to the sample was provided by a small
electric heater, flat in shape and in thermal contact with the sample. The
heating element was a 40-mil thick Alundum platelet, metallized on both
sides with a 1-mil layer of molybdenum. Grooves were inscribed on one
side of the metallized ceramic to increase the electrical resistance of the
molybdenum film to about 10 ohms at room temperature. The heating
element with its grooved side facing down, the platinum power tabs, and
the ceramic insulating plate were all held firmly in place against the nickel
heat sink with the tungsten spring. The platinum tabs served as current
leads to the heating element and the ceramic plate insulated it electrically
from the nickel sink, The sinusoidal power to the heater was produced
by an Exact, model 250, function generator and a DC coupled amplifier
capable of about 75 watts. The function generator was continuously
variable over a frequency range of 0.001 hertz to 10 kilohertz. The
26
2
/ Water outlet
2 Water jacket
3 Bellows
4 Ball joint to
vacuum
8
9
/0
//
/2
5 Vycor tube
6 Heating elements
7 Tungsten spring
8 Ceramic insulator
9 Specimen
/0 Thermocouples
//
Heater
/3
/2 Power tabs
/4 /3 Ceramic insulator
/4 Nickel block
/5 /5 Thin wall support
tube
/6 Baffles
/6 /7 Vacuum 0-ring
/8 Water inlet
/7
/9 Quick disconnect
/8
FIGURE
8.
Thermal Conductivity
Apparatus.
70-179x
27
sample holder was mounted in a vacuum furnace which could be evacuated
to about 10-5 torr and its temperature was regulated by a Wheelco
saturable reactor controller. Measurements could be made in the
temperature range of about 100° C to 700° C.
The two sine waves which are produced from the thermocouple
outputs are of different amplitude and offset by a lag time which is
dependent on the distance of separation of the thermocouples and the
thermal diffusivity of the material. The theory of the method is discussed
by both sets of authors, (1) (36), and will not be described in detail.
Figure 9 shows a typical trace of the output of the two thermocouples as
recorded on the two pen recorder. By measuring the ratio of the
amplitudes and the lag time of the two curves, the thermal diffusivity
of the specimen could be calculated and the change in the thermal properties
could be noted by continuously monitoring the two thermocouple outputs as
the salt was being removed.
TIME
FIGURE
9.- Typical Sine Wave Recording.
70 -224%
29
EXPERIMENTAL RESULTS
Evaporation Studies on Magnesium
luceci Sponge
From preliminary tests on actual sponge samples prepared in the
laboratory and titanium sponge compacts containing magnesium chloride,
it was apparent that both a constant-rate and a falling-rate period occurred
during the vacuum evaporation of these materials. It was also apparent that
the length of the constant-rate period was very temperature dependent
and that the fraction of salt removed during the constant-rate period was
greater at the higher temperatures. In order to determine a mathematical
expression to represent the rate of evaporation for the total drying operation, tests were conducted over a temperature range of about 600 -800° C
and the data were analyzed by considering the constant-rate period, the
critical concentration, and the falling-rate period for each run.
Constant-Rate Period
Since the initial rate data appeared to follow a Clausius type relation-
ship, the mechanism of evaporation during the constant-rate period
should be similar to Langmuir evaporation of any material. In this case,
the logarithm of the rate should be directly proportional to the inverse
of the absolute temperature and should be dependent, also on the external
surface area available for evaporation. To test this hypothesis, compacts
of similar geometry but of varying size were made, and the rates of
30
evaporation during the constant-rate period were determined over a
temperature range of about 600 - 800° C for each of four different
sized compacts. Compacts were made in the form of cylinders with a
diameter to height ratio of one and having nominal sizes of 1/4, 3/8, 5/8,
and 7/8 inch. This gave a relative weight ratio of about 1 : 45 and a
relative surf ace area ratio of about 1 : 12 between the smallest and
largest compacts. Figure 10 shows a plot of the logarithm of the
constant weight-loss rates, in mg-salt/min-gr of metal, versus the
inverse of the absolute temperature. The linear relationships for each
of the four different sized compacts all have about equal slopes which
justifies the assumption that during the cnnstant-rate period evaporation
is occurring primarily from the outer surface of the compacts. When
the rate data are converted to a mass flux based on the external surface
area of the compacts (mg-salt/min-cm2), the data from all four
different compacts fall on a single straight line. This representation is
shown in figure 11. All data are for compacts which were pressed at
10,000 psi and should be representative of commercial titanium sponge.
From this information an equation for the temperature dependency of
mass flux for evaporation during the constant-rate period was written in
the form R* = Ae-Btr, where T is the absolute temperature (°K).
The solid line shown on figure 11 represents the least squares line for
31
50.0
E
1.7
1/4 inch
3/8 inch
5/8 inch
7/8 inch
._
10.0
O
cr,
5.0
1.0
.5
09
1.0
1.1
12
1.3
1.4
1,000/T, °K-1
FIGURE I0.-Variation of Surface Evaporation
Rate With Temperature for Titanium
Sponge Compacts Containing
MgC12.
70-103
32
NE 5.00
I
I
El
E
I
1
1-
\V
0
a
A
V\
0
cy)
E 1.00
%.0
NE,
.50
cn
U)
2
H
<:(
cc
D=H
Nominal size
0
.10
1/4 inch
I
A 3/8 inch
H
a .05
2
5/8 inch
6
--ID r-i
12
V 7/8 inch
0
0.9
1.0
17
O
Relative area
I
1.1
I
I
1.2
I
1.3
1.4
1.5
1,000/T, °K-I
FIGURE II.-Variation of Surface Evaporation
Rate With Temperature for Titanium
Sponge Compacts Containing
MgCl2.
70-104
33
all data taken on different size compacts during the constant-rate period
of evaporation.
Critical Concentration
For any drying or evaporation process in which a constant-rate
and a falling-rate period are observed, the concentration at which the
mechanism changes from constant-rate to falling-rate is defined as the
critical concentration. The critical concentration may be a function of
drying temperature, density, porosity or even gas flow rate in the case
of a flowing system.
To establish the variables affecting the critical content in the
system of magnesium chloride evaporating from porous titanium sponge
compacts, curves extending to total dryness were determined for three
different size compacts at several temperatures between about 600 - 800° C.
The results of these tests are shown in figure 12. The critical content
appears to be independent of sample size but highly dependent on tempera-
ture. Figure 13 shows rate curves for three different size samples all at
the same drying temperature. The rate is plotted as a dimensionless flux
(R/R*) versus the dimensionless concentration (c/c*). Here R* and c*
are the rate during the constant-rate period and the critical concentration,
respectively.
The slight hump in all three curves near the right end was
34
1.0
1
.8
I
I
I
O 1/4 inch compact
A 3/8 inch compact
CI 5/8 inch compact
0
I
I
A
0
0
.6
0
U
,k
cn
Lu
_J
Z
c)
O
4
0
.2
2
500
z
(7)
LL.1
0
0
700
TEMPERATURE, °C
600
800
FIGURE I2.Effect of Sample Size and
Temperature on Critical Concentration.
for Titanium Sponge Compacts
Containing MgCl2.
70-105
35
1.2
I.0
.
6
ii
/
.4
lit
D =H
Nominal size
A
0
.2
0
0-p
0
LVO-C1
0
1
I
Temperature, °C
1/4 inch
3/8 inch
750
750
750
5/8 inch
i
2
0A
0.....-/CI
1
3
a1
i
4
i
5
DIMENSIONLESS CONCENTRATION, c/c*
FIGURE
13. Constant- Rate and FallingRate Period.
70 -181
6
36
observed in all runs. It was apparently due to the small amount of
magnesium present in all samples, which apparently vaporized before the
magnesium chloride. Analysis of several samples showed that based on
magnesium and chlorine analysis there was about 4 percent excess
magnesium and about 26 percent magnesium chloride in the compacts.
Falling-Rate Period
In a period of falling-rate of drying which follows a period of
constant-rate, the distribution of moisture content and temperature at
the beginning of the falling-rate period can be described by a parabolic
distribution for the one dimensional case. The same distribution should,
theref ore, be observed in the system involving magnesium chloride and
porous titanium sponge. Lui kov (24) shows that for these initial boundary
conditions at the beginning of the falling-rate period the moisture content
and evaporation rate are functions of the critical content and temperature.
Thus if the ratio of the instantaneous rate to the rate during the constantrate period is plotted versus the ratio of the instantaneous concentration
to the critical concentration, a correlation of data should result.
Experimental data of Lebedev and Lisenkov (22), as reported in Luikov
(24), show a correlation of drying data as a function of two dimensionless
37
parameters, Bim/Pn, where Bim is the Biot mass transfer number and
Pn is dependent on mass transfer potential. Since for this system
information to determine the Bim/Pn ratios was unavailable for the
various operating conditions, the dimensionless rate was simply plotted
versus the dimensionless concentration for various conditions of sample
size and drying temperature. Figure 14 shows some of the data. It is
apparent from these curves that a good correlation with respect to size
or temperature was not obtained; however, all curves have the same
general shape. For simulation purposes, it was therefore assumed
that the falling-rate drying period could be represented by a second
order polynomial fit and the scatter was due to the difficulty in measuring
instantaneous rates and critical concentrations as well as a considerable
variation in the sponge uniformity.
A least squares best fit of a second order polynomial of the
dimensionless parameters was therefore used to describe the falling-rate
period, the exponential relationship to describe the constant-rate period,
and the temperature dependency for the critical concentration. Thus
the entire evaporation process was described in equation form. The
equation to describe the entire drying operation is of the form:
38
1.0
D =H
.9
Nominal size
Temperature, °C
1/4 inch
3/8 inch
o 5/8 inch
v 1/4 inch
3/8 inch
o
1/4 inch
3/8 inch
6
5/8
inch
750
745
750
690
695
725
805
815
5
4
3
2
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
DIMENSIONLESS CONCENTRATION, c/c
14. -Falling-Rate Period for Titanium Sponge
Compacts Containg MgCl2.
FIGURE
70-174X
39
(5)
R/R* = C(c/c*) + D (c/c*) 2
or
(6)
Ae-B/T
D (c/c*)2]
The constants C and D apply only during the falling-rate period and the
expression in the brackets is set to unity during the constant-rate period.
Figure 15 shows a representation of some of the data for the
entire drying operation. This figure shows that the mass flux can be
evaluated during the constant-rate period from a knowledge of tempera-
ture alone. Knowing the critical concentration, which is also a function
of temperature, the point at which the mechanism changes to a falling-
rate can also be determined. Finally, during the falling-rate period the
rate can be expressed as a function of temperature and concentration.
The equation determined from all the experimental data for the rate of
salt removal expressed in pound of salt per hour-square foot of external
surface area for magnesium reduced titanium sponge is defined by
equation (7). The constants in this equation were determined by least-
squares fit of all the experimental data and the temperature is expressed
in degrees Kelvin.
40
5.0
1
I
1
1
r
1
r
1
1
MN
753° C
'N
717° C
\\*CONSTANT RATE
\-...
658° C
\
1.0
.5
\*
.
\-
608° C
FALLING
RATE
* 541° C
mml
1
I
I
I
I
1
f
1.0
0.8
0.6
0.4
0.2
0
DIMENSIONLESS CONCENTRATION,
c/co
FIGURE I5.-Smoothed Data for Evaporation
from Titanium Sponge Compacts
Containg MgCl2.
10-283
41
(7)
R=
1230e8' 52012 [0.2 (c/c*)
+
0.8
(c/c*) 2]
Evaporation Studies on Sodium Reduced Sponge
Titanium sponge is produced by sodium reduction of titanium
tetra-chloride by one U. S. Firm, Reactive Metals Corporation, Ashtabula,
Ohio. This process is a two-step reduction and the resulting sponge
product is a much finer grained sponge than the normal Kroll process
magnesium reduced sponge. Sodium reduced material also contains
more salt (NaC1) than the magnesium reduced material (MgC12). Because
of the finer grain of the sodium reduced sponge, the lower vapor pressure
of NaCl, and the presence of more salt, the sodium process usually
uses an aqueous leach to remove excess salt before the sponge is melted
into ingot form. However, Reactive Metals Corporation was interested
in knowing the relative ease or difficulty of removing the salt by vacuum
evaporation compared to magnesium reduced material. They were therefore willing to supply some of their product for this study. This finer
grain sponge product containing sodium chloride and a small amount of
lower chlorides of titanium was ground in a hammer mill, passed through
a 20 mesh screen and compacted in an isostatic press by the same
technique used to make homogeneous sponge compacts used for the Kroll
process sponge study. The salt from these compacts was then removed
42
by high temperature vacuum evaporation using the previously described
recording balance apparatus to determine evaporation rates for various
size samples over a temperature range of about 700 - 900° C.
Unlike the coarser magnesium reduced sponge compacts, the
sodium reduced compacts showed a very short, or no constant-rate
period over the entire temperature range. Most of the evaporatinn occurred
during a falling-rate period and it appeared that diffusion was the controlling
mechanism for the entire evaporation process. Figure 16 shows a
typical temperature and sample weight vs time curve for sodium reduced
sponge. In contrast to a typical weight-loss curve for a compact containing
magnesium chloride as shown in figure 6, the sodium reduced material
shows essentially a completely falling-rate curve for the entire evapora-
tion process. Since all of these curves showed a very short or no constant-
rate period, the initial-rate, rather than the rate during the constant-rate
period was plotted vs the inverse of the absolute temperature for three
different size samples at four different temperatures. The surface area
used to reduce the
initial flux rate was the external surface area of the
rectangular compacts determined by measuring the samples with Vernier
calipers prior to loading into the weight-loss apparatus. A measurement
of the external surface area after evaporation was not possible because
the compact usually fell apart when it was removed from the apparatus.
43
Poo
15
14
900
Temperature
13
800
12
700
600 °
10
LAI
9
500
8
400
7
300
Sample weight
200
100
0
3
0
10
100
90
80
70
60
50
TIME, minutes
FIGURE 16.- Typical Weight -Loss Results for Titanium
Sponge Compact Containg NaCI.
20
30
40
110
71-160X
44
Figure 17 shows the results of the initial rate based on the external
surface area of the compact vs the inverse of the absolute temperature.
The least squares best fit line through these data points is shown on the
graph. The slope of this line agrees well with the slope of the vapor
pressure curve for pure sodium chloride. This indicates that the temperature effect corresponds to the vapor pressure effect for the initial rate
of evaporation.
A good correlation of all data was obtained when the dimension-
less rate was plotted vs the dimensionless average salt concentration
based on the sample weight. Figure 18 shows this correlation for some of
the sodium chloride data. The triangles represent the lowest temperature
run and the squares the highest temperature run. Data from all of the
sodium chloride runs fall between these two extremes but not necessarily
in order of temperature or sample size. In this representation of the
falling rate period, the dimensionless rate is a ratio of the instantaneous
rate to the initial rate, and the dimensionless concentration is a ratio of
the instantaneous average concentration to the initial concentration based
on the sample weight. The somewhat better correlation of the sodium
chloride falling-rate data over the magnesium chloride falling-rate data
is probably due to the concentration used as a basis. The concentration
ratio used for the magnesium chloride data was based on the critical
45
I0
5
0.80
0.84
0.88
0.92
0.96
I,000/T, °K-I
FIGURE I7.-Variation of Initial Rate of
Evaporation With Temperature for
Titanium Sponge Compacts
Containing NaCI.
71-82
46
I.2
A 810° C
o 870° C
915° C
1.0
Least squares
cc
W.
U)
.8
0
p0
.6
w
z
0 .4
z
o
.2
or
I
12
1.0
0.8
0.6
0.4
DIMENSIONLESS CONCENTRATION, c/c;
FIGURE I8.-Some Falling-Rate Data for NaCI
0
0.2
from Titanium Sponge Compacts.
71-81
47
concentration which varied from run to run and was temperature dependent,
and the concentration used for the sodium chloride data was the average
initial concentration which was quite uniform for all samples tested. The
salt concentration in the sodium reduced compacts was more uniform
because of the finer grain of the sodium reduced sponge.
An equation of the same form used for the magnesium chloride
data was used to represent the rate of sodium chloride removal expressed
in pounds of salt per hour-square foot of external surface area.
The
equation is:
(8)
R = 988,000e
14,780/T
[0.5 (c/c*)
+
0.5
(c/c*)
2]
Thermal Conductivity Results
To test the reliability of the thermal diffusivity apparatus, thermal
conductivities of specimens of 304 stainless steel and pure titanium metal
were measured over a temperature range of about 100 to 600°C. These
specimens were 3/16 inch in diameter and about 1 1/2 inches long with
thermocouples located about 3/16 inch apart. Initially it was necessary
to determine the optimum frequency of the sine wave pulse and the best
range of input power to the heater. At frequencies higher than about 1.5 x
10-2 cycles per second the response time of the thermocouples was found
to affect the measured diffusivity of the specimens. An oscillation period
of about 100 seconds was chosen as a good operating frequency, eliminating
48
thermocouple response errors. It was also found that peak power level
inf luenced the measured diffusivity; however, the effect was found to be
small over a fairly wide range (4-10 watts).
With these optimum operating conditions having been established,
the thermal conductivities for the stainless steel and titanium specimens
were determined. Figure 19 shows a plot of the variation of the measured
thermal conductivity with temperature. Solid lines represent data reported
in the literature (18, 37). Data points are those from this experimental
investigation. The difference is probably due to purity of titanium.
Agreement with the literature values was well within acceptable limits
for this study.
Figure 19 also shows measured values for a specimen which was
machined from a piece of dense commercial Kroll process sponge and for
one made from a laboratory prepared sponge compact which had been
treated thermally to remove all magnesium chloride. Measured values
for the sponge compact and the specimen of commercial sponge are in
fair agreement . However, values for sponge are considerably lower than
those shown for dense titanium metal. This, of course, is to be expected
because of the porosity and lower bulk density of the sponge and compact.
These values for conductivity of salt-free sponge are the only
measurements which could be obtained. In attempting to measure the effect
49
13
12
-0 304 Stainless steel
o Commercial titanium metal
6 Titanium sponge compact
Industrial dense sponge
0
II
I0
0
0
9
0
0
8
3
2
--6
6 ""
i
A
A
"*"
A
0
"Z
I
I
I
200 300 400 500 600 700 800 900
I
I
TEMPERATURE, °K
FIGURE 19.- Experimental Thermal
Conductivities.
70-180
50
that salt has on the conductivity, the sponge-salt compacts were found to
be so fragile that it was impossible to drill holes for the thermocouples,
and when the thermocouple were attached to the surface of the specimens,
the volatile magnesium chloride from the sample reacted with the molybdenum
on the heater and destroyed its resistance characteristics. Because of the
extreme difficulty encountered in making these measurements, it was decided
to use an average value of thermal conductivity based on the measurements of
dry sponge and compacts. Once the computer simulation was operable, the
effect of thermal conductivity could be determined by changing the value
of the constant and observing its effect on the computer calculated tempera-
ture and salt concentration profiles.
51
COMPUTER SIMULATION OF EVAPORATION PROCESS
A computer program was developed to calculate the transient
temperature and salt concentration profiles through a sponge mass by
making a series of heat and mass balances on a system which is divided
into nodes. The computer program is a modification of a generalized
heat transfer program (21). The original program was used to solve the
three dimensional partial differential heat conduction equation by using
a finite difference technique to approximate the time differential equation.
This program was modified to include evaporation in the surface nodes
and diffusion in internal nodes and the mass transfer differential equation
was added to determine transient concentration as well as transient
temperature profiles. The program was used to simulate the vacuum
evaporation drying of magnesium chloride from titanium sponge compacts
for small scale runs.
The complete description of a drying problem includes the
specification of the relative position of the nodes and the description of
each node. Figure 20 shows a cutaway view of a typical batch of
titanium sponge divided into nodes. The node description includes both
physical and thermal properties as well as initial salt concentrations and
definition of both thermal and mass transfer boundary conditions on each
face. The program can handle boundary conditions of the following nature:
.,.
0
111111MIIMMININI
SPONGE
RECTANGULAR
NODES
FIGURE 20.-Cutaway View of Titanium
Sponge Batch.
53
(1) Convection to or from a constant ambient temperature.
(2) Radiation to or from a constant ambient temperature.
(3)
Surface thermal resistance.
(4) Internode radiation exchange.
(5) Evaporation from surf ace.
The numerical formul ation of the differential equations which
describe the drying process, equations (3) and (4), are carried out
simultaneously by a two step procedure. First the energy and mass
transfer which occur in a time increment is determined by a finite
difference relationship and then the energy and mass transfer which occur
due to evaporation are calculated using the equations from the experimental
data. Details of the energy balance calculations are discussed in the report
of the original heat transfer program (21). The original program was
modified to account for a temperature varying evaporation rate, and the
solution to the mass transfer equation was programmed using the same
calculation technique used in the original program.
In brief, the basic operation and input information required by
the program are as follows:
(1) The geometry of the system is described and entered using
node linkage cards. The input of problem description data is processed
such that diagnostic checks are made to insure that the node linkage is
complete and consistent.
54
(2) Each node is completely defined by physical dimensions,
thermal conductivity, weight density, specific heat, initial salt concentra-
tion, initial node temperatures, and heat transfer boundary conditions
such as radiation temperatures, convective heat transfer coefficients, etc.
(3) Evaporation rate equations are defined from the experimental
data.
(4) The rate of evaporation in each node is calculated from the
equations and the initial node temperatures, physical description and
concentrations.
(5) The new concentration for each node may be calculated from
the rate of evaporation and the time interval for each step.
(6) The concentration gradient which is created by evaporation
from the external nodes is then used as the basis of determining the
amount of liquid movement between nodes. By considering the concentra-
tion gradient, the area common to adjacent nodes, the inverse of the
diffusion distance and the mass transfer coefficient, the new concentrations
are calculated according to Fick's Second Law.
(7)
If the critical concentration is temperature dependent, it can
also be calculated and compared with the new calculated concentration.
If the ratio of c/c* is greater than unity, R = R* and is a function of
temperature only. If c/c* is less than unity R = R* f(c/c*).
55
(8) The energy required to remove the salt can be calculated
from the rate of evaporation or the change in concentration, and thus
another energy balance yields the new temperatures for each node.
(9) After the new temperatures and concentrations for a time
interval have been computed the procedure can be repeated for the next
time interval, and thus the transient temperature and concentration
profiles can be calculated for the nodal system described.
In order to determine the mass transfer coefficient to use for
the computer simulation of the evaporation process, a series of experiments
was run in which samples of titanium sponge compacts initially containing
about 40-weight percent salt were partially distilled under carefully
controlled conditions. When the recording balance showed that the
average salt content of the samples was down to about 10 percent, the sample
was quenched and divided into octants and each octant divided into 8
nodes. Then the average composition for the different positions within a
rectangular block were determined analytically using an atomic absorption
analytic technique. Thes e average experimentally determined salt concentrations were compared with values calculated using the computer simula-
tion of the run for various values of the mass transfer coefficient.
Table 1 shows all of the experimentally determined magnesium concentra-
tions for each node in the eight different octants of the rectangular block
shown in figure 21. These analysis are shown as percent magnesium. The
TABLE 1. - Comparison of experimental and computer
simulated concentration profiles
Node
No.
1
2
3
4
5
6
7
8
1
2.19
2.23
2.65
2.50
2.65
3.11
3.34
4.08
2
3
1.79
2.25
2.93
2.75
2.62
1.83
2.72
2.70
2.96
2.60
2.91
3.07
3.69
3.05
3.80
4.00
4
1.58
2.05
1.80
2.83
2.15
3.44
2.98
3.07
5
6
7
1.87
2.50
2.33
2.51
2.40
2.58
2.72
2.79
2.09
2.01
2.17
2.81
2.69
2.54
3.64
4.40
2.55
2.29
3.12
3.10
2.14
2.98
2.73
3.60
Avg. %
8
2.43
2.48
2.68
2.40
1.90
3.22
3.01
4.25
Mg
2.04
2.31
2.54
2.73
2.89
3.03
3.16
3.76
Experimental
# -salt
100# - sponge
8.7
10.1
11.4
12.1
12.4
13.6
14.3
17.0
Computer calculated # - salt
100# - sponge
Diffusion
Nondiffusion
model
model
De3.1x10-4 cm2/sec
10.3
10.7
10.8
11.3
11.0
11.5
11.6
12.1
7.7
8.3
9.2
9.9
14.4
15.1
16.1
16.8
57
table also shows the average composition for each nodal position and
the calculated magnesium chloride composition for each of the eight nodal
positions. An example of the profile comparison is shown in figure 21.
Experimentally measured and computer calculated salt concentrations are
shown as a function of position for the eight nodal positions of a rectangular
block of partially distilled sponge, shown in figure 21. The closed circles
are the concentrations which were determined experimentally, the open
circles are the computer calculated values assuming evaporation from all
nodes, with no diffusion, and the open triangles are the values calculated
using the diffusion model with a transfer coefficient of 62 which corresponds
to an effective mass diffusivity of 3.1 x 10-4 cm 2/sec for the magnesium
chloride-titanium sponge system at 840°C.
In order to test the reliability of the computer simulation for the
total evaporation process, rectangular sponge specimens were made by
the same compacting technique previously described. These compacts
were about 1" x 1" x 2" and weighed about 60 grams. The total drying
curves were obtained by using the recording balance and the temperature
indicated by the thermocouple directly beneath the sample. The average
salt concentration was then determined from the weight-loss curve and
plotted versus time. The computer simulation of the salt evaporation
from the rectangular block was determined by considering one octant
58
into eight nodes as shown in figure 21. An example of a typical simulated
run is shown in figure 22 along with the experimental result of the
simulated run. The form of the input data and the computer output for a
simulated run are shown in the appendix along with a copy of the computer
program.
A present limitation of the computer program is the requirement
that the ambient radiation temperature is not time dependent. Therefore
the process vessel was assumed to be at a constant temperature at the
start of the run. The transient concentration and temperature profiles
calculated for these conditions for nodes one and eight are shown as
solid lines on figure 22. The average concentration determined from
experimental weight-loss data is shown as a dotted line. There is a slight
lag time in the experimental curve due to the time required for the sample
to heat up. Except for the difference in heat-up time between the experi-
mental run and the computer simulation run, there is good agreement
between the two. This difference could be minimized if the computer
program were modified to include use of time varyin g ambient tempera-
tures during the heat-up period. For larger simulation runs, however,
the heat-up period would be a smaller fraction of the total time required
for complete evaporation. Larger scale simulation runs have been made
on
the computer and the results look reasonable, but because of the limited
5
0.24
.22
.20
.18
.16
a
A
.14
A
0
.10
.08
0
°A
0
0
0
0
Experimental
o Computer no diffusion
A Computer diffusion mode
5 6
3 4
2
NODE POSITION
FIGURE 21. -Comparison of Experimental and
Computer Concentration Profiles for
71 -87
Magnesium Reduced Titanium
Sponge Compact.
0.4
2,000
1,800
0
E
1,600
Temperature
0
i
/
\
1,400
Experimental
Temperature
calculated
%II%
1
A Node
1
1
1
CC
1,200 1..
cl
1
CC
u..I
1
O.
1
I
1,000 2
Li.1
1
o
\
_
%
\
800
\
0
iii
0.2
o
ILI
I
oNode 8 calculated
%
1-1-
600
400
I
0.4
I
I
0.6
I
I
0.8
I
I
1.0
I
it
1.2
I
I
1.4
i
16
TIME, hours
FIGURE 22. -Comparison of Experimental and Computer Simulated Evaporation.
from a .Rectangular Titanium Sponge Compact Containing MgCl2.
7'0-ttli
61
capacity of the recording electrobalance, the reliability cannot be checked
experimentally.
Figure 23 shows the results of a computer simulation of the evapora-
tion drying of a large rectangular that is representative of an industrial size
operation. The salt concentration and temperature for an inside and outside
node are plotted versus time. From this simulation it is apparent that
a large temperature gradient develops during the heat-up period and as
a result a large concentration gradient is also observed. This simulation
is of course for an idealized rectangular sponge cake and assumes constant
physical properties, such as density, conductivity, heat capacity and
initial salt concentration throughout the cake. An actual industrial sponge
cake would have some variation in these properties. This simulation
technique does, however, allow one to observe the changes in the drying
curves as certain variables are changed and could hopefully be used as a
guide to optimize both design and operation of this type of system.
1,800
0.4
,60.0
Temperature outside corner
E
1,400
Temperature inside corner
0
1,200 Loi-
1,000
20 ft
2 ft
I. 4 f t
Li;
F-
800 LI
2
600 ILLj
Salt concentration inside
Salt concentration outside
400
200
0
40
50
60
70
80
90
100
TIME, hours
FIGURE 23.-Computer Simulation of Industrial
Sponge Cake Evaporation.
70-235X
10
20
30
63
COMPARISON BETWEEN NaC1 and MgC12 REMOVAL FROM SPONGE
The model used for the computer simulation to describe the
vacuum evaporation drying of Kroll process sponge containing MgC12
was a porous media containing a liquid which moves by either capillary
action or concentration gradient diffusion and evaporates from the external
surf ace of the porous compact. Sodium reduced sponge containing NaC1
did not appear to be dried by this same mechanism, and the overall drying
process and concentration profiles could not be simulated using the model
that was used to simulate the vacuum evaporation of MgC12 from Kroll
process sponge. It appeared that a shrinking core model would be required
to describe the NaC1 evaporation process. Figure 24 depicts the proposed
model of drying for both materials. The representation shows spherical
porous particles of sponge containing salt and the proposed concentration
profiles for each of the materials at three periods during the drying
process. The magnesium reduced sponge appears to have liquid MgC12
moving to the surface at a fast enough rate to allow evaporation to occur
at a constant-rate for an extended period of time. Finally, when the
falling-rate period begins, the outer surface of the sponge is dry and the
salt concentration inside is considerably lower than the initial salt concentration. In contrast, the movement of sodium chloride appears to be
64
Sodium reduced
Mognesium reduced
Initial
Cj
Constant rate
Falling rate ci
4-- Falling rate
Radial position
FIGURE 24.-Proposed Models for Drying of
Magnesium and Sodium Reduced
Titanium Sponge.
71-2 36
65
exceptionally slow and the surface dries very early in the process but the
inside core remains at essentially the same concentration as initially
present. A s the core shrinks, evaporation occurs from the surface of the
shrinking core and the overall rate falls off.
In order to test this hypothesis, several cube-shaped compacts of
both materials were prepared and evaporation runs were carried out to
several degrees of completeness. The amount of evaporation which had
occurred was determined by suspending the sample from the recording
balance and when the desired amount of salt had been removed, the sample
was quenched. Samples were then taken to represent the inside and
outside salt concentrations. All sodium reduced sponge samples were
heated to about 870° ± 10°C, and all magnesium-reduced sponge samples
were heated to about 780° ± 10°C. Both salts have a vapor pressure of about
1 torr at these temperatures. Figures 25 and 26 show the average salt
concentrations determined experimentally for the outside and inside positions
of the samples at various points during the total evaporation process.
The observed concentration profile for the compacts containing sodium
chloride agree quite well with the depicted concentration profiles shown
in figure 24. The large concentration gradients observed for the samples
containing NaC1 indicate very little liquid movement during the evaporation
process. On the other hand the small concentration gradients observed for
the compacts containing magnesium chloride are an indication that liquid
66
60
50
40
0
a30
O
20
I0
O Inside
o Outside
I0
0.8
0.6
0.4
0.2
FRACTION OF SALT REMAINING
FIGURE 25.- Results of NaCI Partial
Evaporation Runs, 870 ±-10° C.
71-234
67
40
35
30
25
a)
20
2
I5
10
o Inside
A Outside
0.2
0.4
0.6
0.8
I0
FRACTION OF SALT REMAINING
FIGURE 26.-Results of MgCl2 Partial
Evaporation Runs, 780 ± 10° C
71 -2 35
68
diffusion is quite apparent for these compacts.
It was apparent from these tests that in order to simulate the salt
removal for the sodium chloride-titanium sponge system, it would be
necessary to base the model on a shrinking core rather than a liquid
diffusion porous model. At present, no attempt has been made to modify
the existing computer program to account for the shrinking core.
69
SUMMARY
This investigation has shown that the vacuum evaporation of
magnesium chloride from porous titanium sponge, prepared by the Kroll
process, proceeds by a mechanism similar in most respects to that which
prevails in the drying of most porous solids. Evaporation proceeds
initially at a constant -rate that is dependent on the temperature and external
surface area of the titanium sponge. During this period liquid magnesium
chloride moves toward the outer surf aces of the sponge either by capillary
action or by concentration gradient diffusion. This liquid movement may
be represented by a Fick's law relationship with a mass transfer coefficient
used to represent the effective diffusivity. As the concentration of magnesium
chloride in the sponge decreases, a critical point is reached where the
controlling mechanism for evaporation changes and the overall evaporation
rate changes to a falling-rate which is dependent on both temperature and
salt concentration within the porous medium. By making simplifying
assumptions to the heat and mass transfer equations which describe this
process, a computer program was developed which simulates the entire
cycle involving the thermal evaporation of magnesium chloride from
titanium sponge in a vacuum. The program yields, at predetermined time
intervals, the temperature and concentration of salt remaining in each of
70
several uniformly shaped nodes comprising the ent ire sponge batch.
This calculational method has been applied only to relatively small
samples, as dictated by the size of laboratory apparatus.
It would be highly
desirable to apply the method to sponge cakes that vary in size through many
orders of magnitude to determine the applicability to batches approaching
the scale of operation employed industrially. Unfortunately, data with
which to compare are not available for anything approaching industrial
scale. However, even at the present state of development, and with its
inherent uncertainties, the method appears to be useful for the study of the
salt evaporation process to indicate the effects, in this unit operation, of
the design of the apparatus and the operating parameters.
The vacuum evaporation of sodium chloride in fine grain titanium
sponge prepared by a two-step sodium reduction process was also investi-
gated in the laboratory. The rate of liquid movement in this material was
found to be very slow and an entirely different model is required to describe
this drying process. In order to simulate this evaporation process, it
would require that a shrinking-core model be developed and that vapor
diffusion be considered. Experimental data on both the magnesium reduced
sponge and the sodium reduced sponge indicated that vacuum evaporation
removal of sodium chloride from sodium reduced sponge is much slower
and because of this, the present practice of using an aqueous leach rather
than vacuum evaporation is justified for sodium reduced sponge.
71
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APPENDICES
I. NOMENCLATURE
Significance
Symbol
A, B, C, D
Empirical constants in various equations.
Cp
Heat capacity.
c
Salt concentration.
co
Initial concentration of MgC12.
c*
Critical concentration of MgC12.
ci
Initial concentration of NaCl.
De
Effective mass diffusivity.
ke
Effective thermal conductivity.
Evaporation rate.
Ri
Initial rate.
R*
Rate during constant-rate period.
T
Temperature.
t
Time.
vx, vy, vz
Velocity components.
x, y, z
Space coordinates.
cte
Effective thermal diffusivity.
K
e
Effective mass transfer coefficient.
A
Heat of vaporization.
p
Density.
V2
Laplacian operator.
75
76
II. DATA
On the following pages are the data for this work. The heading on
each table indicates the conditions of that run. Table 2 shows the chemical
analysis for various types of titanium sponge and sponge compacts.
Summary of Data
Table
3-9
Constant-rate data for sponge and compacts containing
MgC12.
10-13
Constant-rate data for 10,000 psi compacts containing
MgC12.
14-20
Falling-rate data for sponge compacts containing
MgC12.
21-29
Total drying curves for sponge compacts containing
MgC12.
30-33
Thermal conductivity data.
34-47
Total drying curves for sponge compacts containing
NaCl.
77
Table 2. Chemical Analyses of Ti-Mg-MgC12
Sample
No.
Description
Percent
ME
Percent
Percent
MgC19
Ti
Sponge
1
2
3
4
5
6
Medium dense (top)
Very porous (top)
Very dense (bottom)
Very dense (Salt appear)
Medium dense (bottom)
Medium dense (center)
3.40
7.54
3.95
1.86
3.87
4.89
25.38
13.66
34.65
43.64
30.23
22.71
71.22
78.88
63.40
54.50
62.90
60.20
20,000 psi compact (end)
20,000 psi compact (center)
30,000 psi compact (center)
30,000 psi compact (center)
10,000 psi compact (end)
4.1
4.4
4.3
4.1
4.3
25.0
25.2
25.6
26.4
25.2
70.9
70.4
70.1
69.5
70.5
Compact
1
2
3
4
5
78
Table 3. Initial Rate Data, Magnesium Reduced Sponge
Very Dense Sponge, 1.288 gr. sample, % loss = 44.3
Temperature
1000/°K
°C
266
299
389
422
455
487
514
548
582
614
642
1.855
1.748
1.510
1.438
1.373
1.315
1.273
1.218
1.169
1.127
1.092
Wt. -loss Rate
mg-salt/min-gr-metal
0.016
.031
.073
.164
.345
.439
.647
1.190
1.720
2.160
4.120
Table 4. Initial Rate Data, Magnesium Reduced Spogge, Very Porous
Sponge, 1.548 gr. sample, % loss = 32.8
Temperature
-°C
367
401
431
461
499
529
593
654
1000/°K
1,562
1.488
1.420
1.362
1.295
1.246
1.154
1.078
Wt. -loss Rate
mg-salt/min-gr-metal
0.039
.193
.532
1.230
2.135
2.995
3.190
4.900
79
Table 5. Initial Rate Data, Magnesium Reduced Sponge,
Medium Dense Sponge, 2.078 gr. sample, % loss = 39.7
°C
341
371
406
435
468
500
529
563
595
640
677
704
754
Temperature
1000/°K
1.628
1.552
1.472
1.412
1.349
1.293
1.245
1.196
1.152
1.095
1.052
1.023
0.978
Wt. -loss Rate
mg-salt/min-gr-metal
0.008
.037
.120
.240
.480
.809
1.250
1.630
3.200
3.050
6.490
9.160
14.200
Table 6. Initial Rate Data, Magnesium Reduced Sponge,
30,000 psi compact, 1.804 gr. sample, % loss = 30.4
Temperature
°C
396
460
504
554
597
645
693
741
695
1000/°K
1.494
1.364
1.287
1.209
1.149
1.089
1.035
.986
1.033
Wt. - loss Rate
mg-salt/min-gr-metal
0.007
.032
.065
.145
.371
1.290
3.645
22.900
3,550
80
Table 7.
Initial Rate Data, Magnesium Reduced Sponge,
15,000 psi compact, 1.721 gr. sample, % loss = 28.9
Temperature
1000/6K
°C
440
486
532
548
590
635
672
712
749
783
1.402
1.317
1.242
1.218
1.158
1.101
1.058
1.015
.978
.946
Wt. -loss Rate
mg-salt/min-gr-metal
0.051
.165
.338
.372
.702
1.490
2.215
3.826
6.710
10.150
Table 8. Initial Rate Data, Magnesium Reduced Sponge,
5,000 psi compact, 1.518 gr. sample, % loss = 29.5
Temperature
°C
1000/°K
442
495
542
599
700
771
1,400
1,305
1.230
1.145
1.030
0.957
Wt. -loss Rate
mg-salt/min-gr-metal
0.37
0.81
1.52
2.68
6.00
12.79
81
Table 9. Initial Rate Data, Magnesium Reduced Sponge,
10,000 psi compact, 1.216 gr. sample, % loss = 29.9
Temperature
°C
1000/°K
425
456
494
529
564
629
680
1.435
1.374
1.305
1.248
1.195
1.110
1.045
Wt. -loss Rate
mg-salt/min-gr-metal
0.18
.33
.64
.81
1.16
3.10
4.80
82
Table 10. Initial Rate Data, Magnesium Reduced Compact, 1/4"
10,000 psi compact, 0.794 gr. metal, S. G. =2.38
% loss = 29.9, D =H= 0.75 cm
°C
Temperature
507
548
626
725
788
Wt. -loss Rate
1000/°K
mg-salt/min-cm2
1.282
1.218
1.112
1.002
.942
0.154
.281
.711
1.199
2.246
Table 11. Initial Rate Data, Magnesium Reduced Compact, 3/8" 10,000
psi Compact, 1,621 gr metal, S. G. = 1.83 ,%loss = 29.8,
D = H = 1.04cm
°C
Temperature
468
512
559
656
608
714
757
1000/°K
1.349
1.273
1.201
1.076
1.135
1.013
.970
Wt. -loss Rate
mg-salt/min-cm2
0.128
.227
.395
1.026
.671
1.342
2.823
83
Table 12. Initial Rate Data, Magnesium Reduced Compact, 5/8" 10,000
psi compact, 6.214 gr. metal, S. G. = 1.78, % loss = 29.6,
D = H = 1.63 cm
°C
Temperature
411
457
510
559
651
604
715
680
777
1000/°K
1.461
1.369
1.277
1.201
1.082
1.140
1.013
1.049
.952
Wt-loss Rate
mg-salt/min-cm2
0.040
.096
.193
.367
.925
.646
2 .043
1.213
3.527
Table 13. Initial Rate Data, Magnesium Reduced Compact, 7/8" 10,000
psi compact, 21.84 gr. metal, S. G. = 1.98, % loss = 29.9,
D = H = 2.40 cm
°C
500
540
598
644
696
753
Temperature
1000/°K
1.294
1.230
1.148
1.090
1.031
.974
Wt-loss Rate
mg-salt/min-cm2
0.146
.250
.529
.878
1.733
3.166
84
Table 14, Falling-Rate Data, Magnesium Reduced Compact, 3/8"
10,000 psi Compact, 745 ° C, 2.081 gr. metal, S. G. = 2.29,
%loss = 29.7
conc.
c/c*
Rate
880
849
805
730
578
443
300
168
158
139
104
5.3
5.0
4.7
4.3
3.4
2.6
1.8
1.0
.90
.76
.62
.45
.24
.12
.06
0
75
40
19
12
0
0
21.0
31.2
29.5
29.1
28.7
29.1
28.7
21.8
16.6
12.5
8.3
4.2
2.1
1.05
0
R/R*
0
.73
1.08
1.03
1.02
1.00
1.02
1.00
.76
.57
.43
.28
.14
.07
.03
0
85
Reduced Compact, 5/8" 10,000
Table 15. Falling-Rate Data, Magnesium
psi Compact, 750° C, 7.354 gr. metal, S. G. = 2.13,
% loss = 30.4
conc.
c/c*
Rate
R/R*
3,200
3,150
3,000
2,800
2,400
2,000
1,600
1,200
5.33
5.25
5.00
4.67
4.00
3.33
2.67
2.00
1.33
1.05
1.00
.80
.66
0
26
0
800
630
600
525
396
240
158
67
30
10
0
.40
.26
.11
.05
.016
0
52
49
48
49
50
50
50
50
50
38
27
16
12
5
4
2
0
0.52
1.04
.98
.96
.98
1.00
1.00
1.00
1.00
1.00
.75
.54
.32
.24
.10
.08
.04
0
86
Table 16. Falling-Rate Data, Magnesium Reduced Compact, 1/4" 10,000
psi Compact, 750°C, 0,724 gr. metal, S. G. = 2.44,
% loss = 30.6
conc.
c/c*
Rate
R/R*
320
310
300
272
232
5.33
5.17
5.00
4.53
3.87
3.20
2.53
1.87
1.20
1.00
.97
.89
.78
.53
.33
.23
.13
.05
0.
0.
192
152
112
72
60
58
53
47
32
20
14
8
3
0
0
4.2
10.1
11.9
10.9
10.9
10.1
11.3
10.9
10.9
9.9
8.6
7.0
4.5
2.6
2.0
1.0
.5
0
.39
.92
1.09
1.00
1.00
.93
1.04
1.00
1.00
.90
.79
.64
.41
.24
.18
.09
.045
0
87
Table 17. Falling-Rate Data, Magnesium Reduced Compact, 1/4" 10,000
psi Compact, 690° C, 0.782 gr. metal, S. G. = 2.21,
% loss = 29.6
conc.
c/c*
Rate
R/R*
328
320
310
280
268
1.17
1.14
1.10
1.00
.96
.91
.84
.75
.61
.43
.32
0
0
255
236
210
170
120
90
0
0
3.0
6.5
6.5
4.0
3.0
2.0
1.0
.8
.5
.25
0
.46
1.00
1.00
.61
.46
.30
.17
.12
.07
.035
0
88
Table 18. Falling-Rate Data, Magnesium Reduced Compact, 3/8" 10,000
psi Compact, 695° C, 1.950 gr. metal, S. G. = 2.18,
% loss = 30.1
conc.
c/c *
Rate
R/R*
845
800
775
1.28
1.22
1.18
1.06
1.00
.92
.86
.80
.68
0
14
23
0
700
660
600
570
530
445
382
318
228
128
40
0
.58
.49
.35
.19
.08
0
22
22
13
10
8
5.0
4.4
3.1
1.9
1.0
.5
0
.64
1.04
1.00
1.00
.60
.45
.36
.23
.20
.14
.09
.045
.023
0
89
Table 19. Falling-Rath Data, Magnesium Reduced Compact, 3/8" 10,000
psi Compact, 805° C, 2.032 gr. metal, S. G. = 1.88,
% loss = 30.6
conc.
c/c *
Rate
900
875
816
736
709
680
600
500
5.0
4,85
4.55
4.10
0
400
300
200
178
153
120
85
56
30
0
3.95
3.85
3.33
2.75
2.20
1.67
1.10
1.00
.85
.65
.45
.30
.15
0
10
20
50
60
70
70
70
72
75
70
70
46
30
20
10
5
0
R/R*
0
.14
.28
.71
.86
1.00
1.00
1.00
1.03
1.07
1.00
1.00
.66
.43
.28
.14
.07
0
90
Table 20. Falling-Rate Data, Magnesium Reduced Compact, 1/4" 10,000
psi Compact, 725° C, 0.852 gr. metal, S. G. = 1.89,
% loss = 30.0
conc.
c/c*
Rate
R/R*
362
350
330
310
290
270
250
230
1.45
0
0
20
20
1.01
1.01
1.00
.67
.43
190
150
110
70
30
5
0
1.41
1.32
1.25
1.16
1.07
1.00
.91
.75
.59
.43
.27
.17
.01
0
11.6
22.0
21.0
19.8
13.2
8.6
6.0
4.0
2.6
1.6
.8
0
.58
1.11
1.06
.30
.20
. 13
.08
.04
0
91
T able 21. Total Drying Curve, Magnesium Reduced Compact, 10,000
psi compact, 658° C, 1.277 gr. metal, S. G. = 1.89,
D=H= . 95 cm, % loss = 24.7
Time (min)
0
24
28.7
31.5
33.7
35.7
37.6
40.0
42.6
45.8
49.3
54.0
62.8
74.2
88.6
106.5
129.3
158.2
194.8
239.2
301.0
381
Loss (mg)
0
20
40
60
80
100
120
140
160
180
200
220
240
260
280
300
320
340
360
380
400
419
% remaining
100
95.4
90.6
85.9
81.1
76.3
71.5
66.8
62.0
57.2
52.5
47.7
42.9
37.9
33.4
28.6
23.8
19.0
14.3
9.5
4.7
0
92
Table 22.
Total Drying Curve, Magnesium Reduced Compact,
10,000 psi compact, 608°C, 1,121 gr. metal, S. G. = 1.85,
D=H= 0.92 cm, % loss = 24.6
Time (min)
Loss (mg)
0
0
38.5
42
44.6
47.5
51.5
57
64
73
85
101
125
169
226
296
382
434
490
560
626
700
780
846
20
40
60
80
100
120
140
160
180
200
220
240
260
280
300
310
320
330
340
350
360
367
Total sample wt. 1.488 g
% remaining
100
94.5
89.1
83.6
78.2
72.7
67.3
61.8
56.4
50.9
45.5
40 .0
34.6
29.1
23.7
18.2
15.5
12.8
10.0
7.3
4.6
1.9
0
93
Table 23. Total Drying Curve, Magnesium Reduced Compact,
10, 000 psi compact, 753° C, 1.268 gr. metal,
S. G. = 1.85, D=H= 0.96 cm, % loss = 27.0
Time (min)
0
7
14.6
17.2
19.0
22.0
24.2
26.3
28.3
30.1
31.7
33.0
34.6
36.0
37.4
39.0
39.0
40.6
42.0
43.4
44.7
45.7
47.1
48.7
50.8
53.8
55.8
58.8
64.7
80
Loss (mg)
0
10
20
30
40
60
80
100
120
140
160
180
200
220
240
260
260
280
300
320
340
360
380
400
420
440
450
460
470
471
Total sample wt 1.739 g
% remaining
100
97.8
95.7
93.6
91.5
87.2
83.0
78.7
74.5
70.2
66.0
61.7
57.5
53.2
49.0
44.7
44.7
40.5
36.3
32.0
27.8
23.5
19.3
15.0
10.8
6.5
4.4
2.3
0.2
0
94
Table 24. Total Drying Curve, Magnesium Reduced Compact,
10,000 psi compact, 706° C, 1.592 gr. metal,
S. G. = 1.84, D=H= 1.03 cm, % loss = 27.9
Time (min)
0
14.4
19.4
22.2
24.5
26.4
28.2
29.8
31.4
32.7
34.0
35.2
36.6
38.3
40.1
42.6
45.4
48.8
52.5
56.4
61.0
66.1
71.4
77.2
83.5
90.2
97.3
105.0
113.1
122.9
137
153
177
Loss (mg)
0
20
40
60
80
100
120
140
160
180
200
220
240
260
280
300
320
340
360
380
400
420
440
460
480
500
520
540
560
580
600
610
618
Total sample wt.
% remaining
100
96.7
93.5
90.2
87.0
83.8
80.5
77.3
74.1
70.8
67.6
64.4
61.1
57.9
54.6
51.4
48.2
44.9
41.67
38.5
35.2
32.0
28.8
25.5
22.3
19.0
15.8
12.6
9.3
6.1
2.9
1.2
0
95
Table 25. Total Drying Curve, Magnesium Reduced Compact,
10,000 psi compact, 541° C, 1.189 gr. metal,
S. G. = 1.85, D=H= 0.94 cm, % loss = 26.4
Time (min)
0
31
36
39
43
48
54
62
71
83
97
115
140
170
207
257
329
410
503
606
700
790
902
1,020
1,236
1,451
1,670
1,890
2,100
2,311
2,522
2,736
3,050
3,086
Total sample wt
Loss (mg)
0
16
30
40
50
60
70
80
90
100
110
120
130
140
150
160
170
180
190
200
210
220
230
240
260
280
300
320
340
360
380
400
420
428
1.617 g
% remaining
100
96.2
92.9
90.6
88.3
85.9
83.6
81.3
78.9
76.6
74.2
71.9
69.6
67.2
64.9
62.6
60.2
57.9
55.6
53.2
50.9
48.5
46.2
43.9
39.2
34.5
29.9
25.2
20.5
15.8
11.2
6.5
1.8
0
96
Table 26. Total Drying Curve, Magnesium Reduced Compact, 10,000 psi
compact, 576° C, 1.281 gr. metal, S. G = 1.89, D°H=0.95 cm,
%loss = 26.4
Time (min)
0
31
34
36
38
41
45
49
53
57
61
65
71
78
86
105
118
131
147
185
236
292
360
442
548
657
795
958
1,154
1,416
1,630
1,745
1,960
Loss (mg)
0
21
30
40
50
60
70
80
90
100
110
120
130
140
150
170
180
190
200
220
240
260
280
300
320
340
360
380
400
420
440
450
461
Total sample wt. - 1.742 g
% remaining
100
95.4
93.5
91.3
89.2
87.0
84.8
82.6
80.5
78.3
76.1
74.0
71.8
69.6
67.5
63.1
61.0
58.8
56.6
52.3
47.9
43.6
39.3
34.9
30.6
26.2
21.9
17.6
13.2
8.9
4.6
2.4
0
97
T able 27. Total Drying Curve, Magnesium Reduced Compact, 10,000 psi
compact, 794° C, 1.525 gr. metal, S. G. = 1.85, D =H= 1.015 cm,
% loss = 25.0
Time (min)
0
2.7
5.4
7.6
9.6
11.2
12.6
13.9
15.2
16.5
17.8
19.0
20.0
20.9
21.5
22.1
22.7
23.3
23.9
24.5
25.2
26.0
27.2
29.0
31.0
33.3
40.0
Loss (mg)
0
10
30
50
70
90
110
130
150
170
190
210
230
250
270
290
310
330
350
370
390
410
430
450
470
490
510
Total sample wt. - 2.035 g
% remaining
100
98.0
94.1
90.2
86.3
82.3
78.4
74.5
70.6
66.7
62.7
58.8
54.9
51.0
47.1
43.1
39.2
35.3
31.4
27.5
23.5
19.6
15.7
11.8
7.8
3.9
0
98
Table 28. Total Drying Curve, Magnesium Reduced Compact, 10,000 psi
compact, 702° C, 1.401 gr. metal, S. G. = 1.91, D=H=0.99 cm,
% loss = 26.6
Time (min)
0
20
25
30
35
40
45
50
55
60
70
80
90
100
110
120
130
140
150
160
170
180
198
Loss (mg)
0
20
63
125
196
249
283
304
323
340
368
390
410
426
440
455
466
476
484
489
494
500
509
Total sample wt. - 1.910 g
% remaining
100
96.1
87.6
75.5
61.6
51.2
44.5
40.4
36.7
33.3
27.8
23.5
19.6
16.5
13.7
10.8
8.6
6.7
5.1
4.1
3.1
2.0
0
99
Table 29. Total Drying Curve, Magnesium Reduced Compact, 10,000 psi
compact, 806°C, 9.135 gr metal, S. G. = 1.86, D=H= 1.84
cm, % loss = 27.1
Time (min)
0
10
20
30
40
50
60
70
80
90
100
110
Loss (mg)
0
180
1,260
2,410
2,860
3,060
3,160
3,220
3,280
3,320
3,340
3,360
% remaining
1.0
94.0
62.0
28.2
14.8
8.9
5.9
4.1
2.3
1.1
0.5
0
Table 30 - Thermal Diffusivity Data, 304 Stainless Steel
Temperature
°C
°K
260
405
190
332
451
533
678
463
605
724
t
amplitude
Lag time
(sec)
a
thermal
diffusivity ft2/hr
1.28
1.23
1.27
1.22
1.20
4.6
4.9
4.6
4.9
5.1
0.169
.194
.170
.192
.210
q
ratio
k
thermal
conductivity Btu/hr-ft -°F
9.9
11.4
10.0
11.3
12.3
101
Table 31. Thermal Diffusivity Data, Titanium Ingot
Temperature
6C
°K
305
578
187
460
440
559
713
832
a
thermal
Diffusivity ft2/hr
0.241
.209
.189
.263
k
thermal conductivity
Btu /Hr -Ft - °F
9.45
8.61
8.09
9.79
Table 32. Thermal Diffusivity Data, Dense sponge (from ORE -MET)
Temperature
°C
170
210
298
406
492
428
194
°K
443
483
571
679
765
701
467
q
amplitude
ratio
1.42
1.37
1.43
1.46
1.48
1.49
1.43
t
Lag time
(sec)
4.2
4.2
4.2
4.7
5.0
4.9
4.5
a
thermal
diffusivity ft2/hr
0.133
.146
.130
.109
.100
.101
.121
thermal
conductivity Btu/hr -ft - ° F
2.31
2.53
2.40
2.10
1.94
1.92
2.27
Table 33. Thermal diffusivity data, 10,000 psi Titanium Compact with Salt, Pre-distilled
Temperature
amplitude
t
Lag time
-
q
ratio
(sec)
°C
°K
151
268
424
541
652
767
-
121
168
223
321
394
-
-
-
-
243
516
631
846
689
379
494
443
358
573
416
443
496
594
716
-
-
-
a
thermal
diffusivity ft2/hr
thermal
conductivity Btu/Hr-Ft-°F
0.094
.111
.128
.131
1.48
1.84
2.20
2.72
-
.086
.092
.098
.116
.146
1.34
1.46
1.59
1.95
2.59
-
.106
.118
.159
.130
1.73
-
-
2.01
2.94
2.33
104
Table 34. Sodium Reduced Sponge Data 10,000 psi compact, 853°C,
Rectangular Block (2.5 x 2.2 x 2.2 cm)
Time
min
0
5
10
15
20
25
30
40
50
60
70
80
90
100
110
120
140
160
180
200
210
Sample
weight, gr
28.2
28.0
26.1
24.1
22.2
20.6
19.4
17.3
15.4
13.9
12.6
11.5
10.6
9.8
9.1
8.5
7.5
6.7
6.2
5.9
5.8
Rate
mg/min-cm2
-
12.50
11.16
8.44
6.94
6.00
5.20
4.34
3.76
3.06
2.69
2.17
1.91
1.64
1.35
1.04
0.62
0.34
0
R/Ri
-
1.0
.892
.675
.555
.480
.416
.347
.300
.244
.215
.
173
. 152
.131
. 108
.063
.050
.003
c/ci
-
1.0
.896
.808
.743
.628
.524
.442
.371
.311
.262
.218
.180
.147
.092
.055
.021
.005
105
Table 35. Sodium Reduced Sponge Data 10,000 psi compact, 841°C,
Rectangular Block (2.2 x 2.2 x 1.3 cm)
Time,
Sample
min
weight, gr
0
5
14.7
14.0
12.5
11.1
10.1
10
15
20
30
40
50
60
70
80
90
100
120
140
160
180
8.4
7.2
6.4
5.6
4.9
4.4
4.0
3.7
3.3
3.2
3.1
3.1
Rate
mg/min-cm2
R/Ri
c/c.1
-
-
13.4
10.2
6.4
4.9
3.9
3.1
2.5
2.1
1.000
.760
.479
.364
.291
.233
184
. 155
. 134
.
1.8
1.4
.7
.3
0
-
.090
.053
.017
No further weight-loss
1.000
.875
.662
.512
.412
.312
.225
.162
.112
.075
.025
.012
106
Table 36. Sodium Reduced Sponge Data, 10,000 psi compact, 843° C,
Rectangular Block (2.3 x 2.3 x 0.9 cm)
Time,
min
0
5
10
15
20
30
40
50
60
80
100
120
140
Sample
weight, g
11.85
11.85
10.35
8.85
7.85
6.30
5.22
4.37
3.74
2.91
2.53
2.45
2.45
Rate
mg/min-cm2
-
15.5
9.9
7.8
6.2
4.8
3.2
2.2
.9
. 15
0
R/Ri
-
-
1.000
.
c/ci
638
.503
.400
.309
.206
. 141
.058
.009
-
No further weight-loss
1.000
.810
.683
.487
.350
.242
.163
.058
.009
_
107
Table 37. Sodium Reduced Sponge Data, 10,000 psi compact, 816° C,
Rectangular Block (2.4 x 2.4 x 1.0 cm)
Time,
min
0
5
10
15
20
25
30
35
40
50
60
80
100
140
180
220
260
300
310
Sample
weight, g
13.65
13.25
12.52
11.77
11.13
10.14
9.75
9.15
8.65
7.94
7.35
6.34
5.45
4.27
3.54
3.06
2.90
2.85
2.85
Rate,
mg/min-cm2
-
6.58
4.73
4.01
2.94
2.49
2.13
1.76
1.10
.68
.49
.10
-
R/Ri
c/c
-
-
-
-
1.000
.720
.610
.447
.379
.324
.267
.
167
.
075
.104
.015
-
No further weight-loss
1.000
.913
.840
.
738
.652
.506
.377
.205
.100
.030
.007
-
108
Table 38. Sodium Reduced Sponge Data, 10,000 psi compact, 818° C,
Rectangular Block (2.4 x 2.3 x 1.0 cm)
Time
min,
0
5
10
15
20
25
30
35
40
50
60
70
80
90
100
120
140
160
180
200
Sample
weight, g
13.30
12,25
11.00
10.05
9.25
8.64
8.12
7.62
7.25
6.60
6.00
5.52
5.04
4.62
4.20
3.60
3.15
2.82
2.70
2.70
Rate,
mg/min-cm2
-
14.53
10.97
R/R
c/c
-
-
1.000
.755
.679
.472
.382
.323
.290
.259
.216
.184
.164
.147
.141
.126
.092
.062
.036
9.88
6.86
5.55
4.70
4.22
3.77
3.15
2.68
2.39
2.15
2.05
1.84
1.34
.88
.52
No further weight-loss
1.000
.869
.769
.685
.621
.567
.515
.476
.408
.345
.295
.244
.201
. 156
.094
.046
.012
109
Table 39. Sodium Reduced Sponge Data, 10,000 psi compact, 804° C,
Rectangular Block (2.3 x 2.3 x 1.0 cm)
Time,
Sample
min .
weight, g
0
5
10
15
13.25
10.85
8.85
7.50
6.55
5.80
5.20
4.73
4.31
3.63
3.15
2.87
2.85
2.85
20
25
30
35
40
50
60
70
80
90
Rate,
mg/min-cm2
-
25.27
15.79
11.23
9.19
7.54
5.43
4.35
4.04
2.99
1.98
.70
R/R.
c/c
-
-
1
1.000
.624
.444
.363
.298
.215
.172
.
159
.118
.078
.027
No further weight -loss
1.000
.750
.581
.462
.369
.294
.235
.182
.097
.037
.025
-
110
Table 40. Sodium Reduced Sponge Data, 10,000 psi compact, 816° C,
Rectangular Block (2.5 x 2.5 x 1.0 cm)
Time,
min
0
5
10
15
20
25
30
35
40
50
60
70
80
90
100
120
140
160
180
200
Sample
weight, g
13.40
12.45
11.15
10.10
9.28
8.60
8.05
7.58
7.12
6.42
5.92
5.45
5.00
4.60
4.33
3.80
3.42
3.20
3.05
3.05
Rate,
memin-cm2
-
12.5212.45
9.50
7.92
6.45
5.87
4.95
4.26
3.69
2.68
2.36
2.11
1.84
1.47
1.08
.81
.51
R/R.
-
1.000
.
857
.655
.545
.444
.404
.341
.294
.255
.185
.163
. 146
.127
.102
.075
.056
.035
-
No further weight-loss
c/c
-
1.000
.862
.750
.663
.591
.532
.482
.433
.359
.305
.255
.207
.165
.
136
.080
.039
.016
111
Table 41. Sodium Reduced Sponge Data, 10,000 psi compact, 870° C,
Rectangular Block (2.35 x 2.30 x 1.15 cm)
Time,
min
0
5
10
15
20
25
30
35
40
50
60
70
80
90
100
120
Sample,
weight, g
14.65
13.15
11.00
9.83
8.47
7.55
6.80
6.15
5.70
4.78
4.10
3.60
3.27
3.07
3.05
3.05
Rate,
memin-cm2
-
19.38
17.22
12.57
9.69
8.05
6.64
5.71
4.78
3.50
2.71
2.03
1.35
.58
-
R/R.
c/c.
-
-
1.000
.889
.649
.500
.415
.343
.295
.247
.181
.140
.105
.070
.030
-
No further weight -loss
1.000
.787
.671
.537
.445
.371
.307
.262
.171
.042
.032
.022
.002
112
T able 42. Sodium Reduced Sponge Data, 10,000 psi compact, 915° C,
Rectangular Block (2. 2 x 2. 2 x 1. 0 cm)
Time,
min
0
5
10
15
20
25
30
35
40
50
60
70
Sample
weight, g
12.55
9.85
7.65
6.23
5.20
4.45
3.87 '
3.45
3.23
2.80
2.75
2.75
Rate
memin-cm2
31.83
18.03
12.88
9.54
7.33
6.09
4.85
3.29
1.46
RIB
1
1.000
.567
.405
.300
.231
.192
.153
.104
.046
No further weight-loss
c/c.
1
1.000
.690
.490
.345
.239
.158
.099
.068
.007
113
Table 43. Sodium Reduced Sponge Data, 10,000 psi compact, 916° C,
Rectangular Block (2.4 x 2.4 x 0.92 cm)
Time,
min
0
5
10
15
20
25
30
35
40
50
60
70
80
Sample
weight, g
12.30
9.95
7.72
6.40
5.48
4.75
4.17
3.70
3.30
2.80
2.52
2,48
2.48
Rate
mg/min-cm2
-
28.76
15.28
11.37
8.14
6.30
5.28
4.27
2.91
1.99
.77
-
R/R
c/c
-
-
1.000
.531
.395
.283
.219
.184
.149
.
101
.069
.027
-
No further weight -loss
1.000
.702
.525
.402
.304
.226
.163
. 110
.043
.005
-
114
Table 44. Sodium Reduced Sponge Data, 10,000 psi compact, 770° C,
Rectangular Block (1.3 x 1.2 x 1.1 cm)
Time,
min
0
10
20
30
40
50
60
70
80
90
100
120
140
160
180
200
220
240
260
280
300
Sample
weight, g
3.35
3.06
2.74
2.48
2.27
2.09
1.93
1.79
1.68
1.57
1.48
1.32
1.19
1.08
1.00
.93
. 87
. 83
.80
.77
.77
Rate
c/c
memin-cm2
-
4.29
2.93
2.69
2.25
1.93
1.76
1.56
1.27
1.10
.90
.
81
.67
.53
.42
.35
. 29
. 23
. 17
-
1.000
.684
.628
.525
.450
.412
.364
.297
.257
.233
. 189
.
158
. 125
. 100
.083
.
067
. 054
.041
.019
.08
No further weight-loss
1.000
.860
.746
.655
.576
.506
.445
.397
.349
.310
.240
. 183
.135
.100
.070
.
043
. 026
.013
115
Table 45. Sodium Reduced Sponge Data , 10,000 psi compact, 845° C,
Rectangular Block (1.2 x 1.1 x 1.2 cm
Time,
Sample
min
weight, g
0
5
3.54
3.30
2.88
2.53
2.25
2.04
1.87
1.73
1.63
1.43
1.29
1.16
1.07
.99
.94
.90
.88
10
15
20
25
30
35
40
50
60
70
80
90
100
110
120
130
. 88
Rat e
mg/min-cm2
-
15.63
13.82
11.23
8.36
6.91
5.61
4.79
3.86
3.29
2.36
1,89
1.48
1.11
.76
.43
-
R/R
i
-
1.000
.885
.719
.535
.442
.359
.307
.247
.211
. 151
.121
.095
.071
.049
.028
-
No further weight-loss
c/c
-
1.000
.827
.682
.566
.479
.409
.351
.310
.227
. 169
.
116
.079
.045
.025
.008
-
i
116
Table 46. Sodium Reduced Sponge Data, 10,000 psi compact, 838° C,
Rectangular Block (1.4 x 1.05 x 1.1 cm)
Time
min
0
5
10
15
20
25
30
35
40
50
60
70
80
90
100
110
Sample,
weight, g
3.59
3.32
2.78
2.38
2.10
1.90
1.72
1.58
1.46
1.28
1.14
1.04
.96
.92
.88
.88
Rate
mg/min-cm 2
-
R/11
-
13.74
11.18
7.75
6.24
4.85
1.000
1.85
1.38
1.04
.728
.480
.248
. 135
3.88
3.25
2.69
. 814
.564
.454
.354
.282
.237
.196
.
101
.076
.053
.035
.018
No further weight-loss
c/ei
-
1.000
.779
.615
.500
.418
.344
.287
.238
.164
. 106
.066
.033
.016
-
117
Table 47. Sodium Reduced Sponge Data, 10,000 psi compact, 915° C,
Rectangular Block (1.1 x 1.4 x 1.2 cm
)
Time,
min
0
5
10
15
20
25
30
35
40
45
50
55
Sample
weight, g
3.68
2.88
2.28
1.64
1.36
1.16
1.02
.96
.92
.92
.92
. 92
Rate
mg/min-cm2
-
27.45
12.91
7.98
5.55
3.66
2.06
1.26(
.52.
-
R/11
-
1.000
.471
.291
.202
.
133
. 075
.
046
.019
No further weight-loss
c/c.
1.000
.694
.368
.224
. 123
.051
.020
.008
118
HI. COMPUTER PROGRAM
Table 48 shows the input parameters and their identification
index numbers which are used in the present evaporation drying computer
program. All index numbers in the data groups 1 through 7 use the same
format which was used in the generalized heat transfer program
HE ATTRAN. Groups 8 and 9 and index number 99, in group one, have
been added to account for evaporation and diffusion of a volatile material.
A complete description of each node in the array is described by input
data cards using a technique which is normally called "groups of pairs. "
A complete description of this data input technique is given by Pagnani in
the sample problem section of his report (21). The same method of data
input is used in the modified program. The form of the computer input
data and typical calculated results along with a complete listing of the
present computer program as shown in the following pages.
119
Table 48. Input Parameters for Generalized Evaporation Drying Problem
Data
group
1
Index
No.
Property
Y
Z
K
G
4
5
2
7-12
13-18
h
Ta
3
19-24
25-30
F
4
31-36
S
5
37-42
F'
7
8
9
Convection coefficient
Convection ambient
°F
btu/hr-ft2-°F
lb/ft3
btu/lb
C1, C2, C3,
C4
Ml, M2
99
btu/hr-ft2-°F
'F
K
A
B
Physical dimensions
Thermal conductivity
watts
C p btu/lb-°F
oF
T
co lb -s alt/lb -metal
X
Remarks
btu/hr-ft-°F
T
43
44
45
46
47
48
Inches
Inches
Inches
X
1
2
3
6
Units
Symbol
H eat generation
Radiation factor
Radiation ambient
Surf. conductance
Radiation factor
Density
Specific heat
(initial value)
Initial salt concentration
Heat of vaporization
Diffusion coefficient
Constants in critical
concentration equation
Constants in falling-rate
equation (M,(c/c*)+M2(c/c*)2)
Constants in. constant rate
Equation R* = Ae -B/T
NOTE: For Groups 2 through 5, the index numbers for convection,
radiation, etc. follow a +x, -x, +y, -y, +z -z pattern. As an
example, if a convection coefficient in the -,y direction is specified
its index number is 9 and its corresponding ambient temperature
index number is 15.
iwr
SAVE FOR HILL
THERMAL PROGRAM NATA
TITANIUM SPONGE TEST (CONC PROFILE RP-8)
ONE OCTANT OF RECTANGULAR BAR 24 NODES
EVVORATION OF MGCL2 FROM EXTERIOR SURFACES
DIFFUSION COEFICIENT = 4.0E04
RADIATE ALL SIDES 1670 F
THERMAL CONDUCTIVITY = 1.5 BTU /HR FT F
DENSITY = 110 LB/FT3
MEAT CAPACITY = 0.1 BTY/LB F
INITAL SALT CONTENT = 0.4 LB SALT/LB SPONGE
NODE DIMENSIONS X=0.22, Y=0.23, 2=0.48
1.0000E-01
1
24
7.8664E-12
-8.4381E-08
3.1001E-01
2.7239E-35
1
?
5
6
10
6
7
1 M 0
NODE REL LOCATIONS
9
4 11
0
8
3
0
0 13
0 17
0 21
2
6
10
14
5
3
6
0
4.8677E - 11,____1
2
2.3201E-11
2.2033F-31
1.5000E 10
- 1.5333F 34
3
4
99
1
1
1
1
1
1.2300E 01
DATA CARDS GROUP 3
29
1
1.6700E 13
23
1
5.0000E-11
2
1.6703F 33 28
5.0001F-31 22
2
4
1.6700F 13 26
4
5.0000E-31 20
DATA CARDS GROUP 6
1
1.1010E 12 43
1
1.0001E-11 44
DATA CARDS GROUP 7
1
45
7.000IE 31
DATA CARDS GROUP 8
o
1
4.0100C-1i
DATA CARDS GROUP 9
0
1
1.8100E-33
0
1
1.4701F-13
i
0
1.0901E-11
0
1
7.5001E-34
1
1
1.010IF-13
0
1
7.200IF-34
0
1
3.5000E-14
SE 10
0
1
1
1
1
1
1
14
18
22
1
8
0
0
5
9
12
0
0
4
0 14
0 18
0 22
15 13 20
0
2
0
11
15
17 24
0
0 16 1C
0
23
0
23 21
1
1
6
6
1
1
1
1
1 24
1
24
2
0
DATA CARDS
(Rad+ation factorc)-
13 13
7___13
19 19
13__19 19
(Density)
(Heat capacity)
1 24
(Initial temperature)
1-24
(Initial salt content)
1
2
1
(Diffusion areas)
7
7
8
13
14
19
20
12
13
18
19
2A
6
0
15
4
12
16
20
24
hermai ccgductivity)"
tdimeos
= Ae -8")
7
0
1
12 10
0
5
0 21
16 14 21 0
3
0
0
0 13 7
20
0
0 17 11 0
24 22
(y dimension)
7
7
0
0
8
iAtiMORSiOn)-
13 18
13 18
9
19
8
TRANSIENT IND
1 12
1 12
1
6 0-19
GE CO
24
?4
24
24
24
1
4
3
7
0.01 MRS
2
0
1.0003E-02
1.0000E-11
DATA CARDS GROUP 1
CONTROL CARD
3
11
Pam
.8000 -333.0000
.2000
PRINT OUT NODES
24
21
1
0
0 19 0
14 16 23 0 5 0
0 15 9 0
2? 20
NODE LINKING IS CONSISTANT
CASE
1
TRANSIENT RUN 0.1 HRS /
13
17
21
MT11
-60.0000
5
3
10
0
0
2
0
16
20
0
6
24
11
4
0
17 15 22
0
0
0 14
6
21 19
0 23
0 18 12
0
9
0
7
0
0
121
TYPICAL CALCULATED OUTPUT FORMAT
CONSECUTIVE NUMBERS ARE NODE NUMBERS
PRINT OUT FOR SPECIFIED TIME
TEMP - TEMPERATURE °F
CONC =SALT CONCENTRATION lb-salt/lb-metal
CRITICAL CONCENTRATION
C*
3.0162E-01
5.7495E702
3.2761E-01
5.7516E-02
9
3.3318E -01
9
5.7527E-02
10
10
3.3454E-01
5.7532E-02
1.6422E 03
3.0190E-01
5.7444E-02
14
14
14
1.6416E 03
3.2676E-01
5.7522E-02
15
15
15
1.6418E 03
3.3219E-01
5.7538E-02
18
1.6417E 03
3.3413E-01
5.7546E-02
19
19
19
1.6420E 03
3.2200E-01
5.7493E-02
20
20
20
1.6419E 02
3.4803E-01
5.7519E-02
23
23
27
1.6417F 03
3.5520E-01
5.7543E-02
24
1.6417E 03
3.5533E-01
5.7544E-02
2S
25
25
OE 00
OE OS
OE OS
5
1.6418E 03
2.9946E-01
8
7
6
3.1397E-11
5.7538E-02
11
11
11
1.6418E 03
3.3497E-01
5.7534F-02
12
12
12
1.6418E 03
3.3511F-01
5.7535C -02
13
1.6417E 01
3.3355E-01
5.7543E-02
17
1.6417E 03
3.3399E-01
5.7545E-02
18
C'
16
16
16
TEMP
CONC
C'
21
21
21
1.6418F 03
3.5347E-91
5.7534E-12
22
22
1.6418E 03
3.5478E-01
5.7540E-02
TEMP
CONC
1
1
C'
1
TEMP
CONC
6
6
TEMP
CONC
11
11
C'
11
TEMP
CONC
C'
16
TEMP
CONC
21
CONC
C'
6
TEMP
CONIC
C'
3E40
CONC
Co
C'
TEMP
CONE
6
16
16
21
21
1
1
C'
1
TEMP
CONC
6
6
TEMP
CONC
C'
11
C'
5
11
11
7
17
17
22
8
13
.17
18
24
24
PoUTINE PRINTOUT AT TIME EQUALS 1.0000E-02 NRS
1.6418E 03
4
1.6418E 03
3
1.6419E 03
2
1.64?QE 03
2.9896E-01
4
2.9735E-01
3
2
2.9078E-01
2.6220E-01
5.7527E-02
4
5.7524E-02
3
5.7515E-02
2
5.7486E-02
5
5
5.7527E...02
8
7
1.6420E 03
2.8364E-01
5.7491E-02
8
8
1.6419E 03
3.1262E-01
5.7509E-02
9
9
9
1.6419E 03
3.1923E-01
5.7518E-02
10
10
10
1.6418E 03
3.2081E-01
5.7521E-02
1.6418E 03
3.2130E-01
5.7522E-02
12
12
12
1.6418E 07
3.2145E-01
5.7522E-02
13
13
13
1.6422E 03
2.8382E-01
5.7440E-02
14
14
14
1.6419E 03
3.1157E-01
5.7516E-02
15
15
15
1.6418E 03
3.1809E-01
5.7529E-02
1.6418E 03
3.1968E-01
5.7532E-02
17
17
1.6418E 03
3.2016E-01
5.7533E -02
1.6418E 03
3.2032E-01
5.7534E-02
19
19
19
1.6420E 03
3.0442E-01
5.7489E-02
20
20
17
18
18
18
1.6419E 02
3.3341E-01
5.7512E-02
1.6418E 01
3.4000E-01
5.7523E-02
22
22
22
1.6418E 03
3.4157E-01
5.7527E-02
23
23
23
1.6418E 03
3.4204E-01
5.7528E-02
24
1.6418E 03
3.4219E-01
5.7528E-02
25
1.6415E 03
2.9962E-01
5.7528E-02
7
7
24
24
ROUTINE PRINTOUT AT TIME EQUALS 9.0000E-02 NRS
1.6418E 03
4
1.6418E 03
3
1.6419E 03
2
1.6420E 03
2.8482E-01
4
2.8295E-01
3
2
2.7527E-01
2.4410E-01
5.7525E-02
4
5.7523E-02
3
2
5.7514E-02
5.7486E-02
1.6418F 03
2.8554E-91
5.7526E-02
7
1.6418E 03
3.0760E-01
5.752CE-02
12
12
12
7
7
1.6420E 03
2.6583E-01
5.7491E-02
1.6418E 03.
3.0777E-01
5.7520E-02
8
8
A
13
13
13
1.6419E 03
2.9744E-01
5.7508E-02
9
9
1.6419E 03
3.0519E-01
5.7516E-02
1.6422E 03
2.6594E-01
5.7440E-02
14
14
14
1.6419E 03
2.9630E-01
5.7515E-02
9
20
25
25
OE 00
OE 00
DE 00
5
1.6418E 03
2.8537E-01
5.7526E-02
10
10
10
1.6419E 02
3.0706E-01
5.7519E-02
15
15
15
1.6418E 03
3.0395E-01
5.7527E-02
5
5
633 FORTRAN
VERSION 2.1
06/17/71
0918
122
PROGRAM HEATTRAN
INTEGER TITLE,DESCR
COMMON II, 12, 131 I4, 15, 16
DIMENSION XX1(350), YY2(350), ZZ3(350)
DIMENSION CIJ(6),ISCRIP(6)
EQUIVALENCE ( I1, ISCRIP(1))
DIMENSION PSUMH(350) ,CONC(350),AREA(350),CSTAR(350), AMASS(350)
*,EXPO(350),ITRANS(350)
OIMENSION X01(350),X02(350),X03(350),X04(350),X05(350),X06(350),
1X07(350),X08(350),X09(350),X10(350),X11(350),X12(350)1X13(350)/,
2X14(350),X15(350),X16(350),X17(350),X18(350),X19(350)
COMMON
3T(350),TX(350),CAP(350),TDOT(350)0(350),R(350),SAS1(350),
4SAS2(350),SAS3(350),SAS4(350),SAS5(350),SAS6(350),C1(350)J
5C2(350),C3(350),C4(350),C5(350),C6(350),A(350),IXP(350),
6IXM(350),IYP(350),IYM(350),IZP(350),IZM(350)
OIMENSION TITLE(200), ISTART(10), IEND(10), DESCR(18),B0i(350)
EOUIVALENCE (T,X01),(TX,X02),(CAP,X03),(TDOT,X04),(0,X05),(R,X06)
1(SAS10(07),(SAS2,X08),(SAS3,X09),(SAS4,X10),(SAS50(11),(SAS60(12)
2,(C1,X13),(C2,X14),(C3,X15),(C40(16),(C5017),(C60(18),(A019)
COMMON/DATA/ IMAX
OATA(IMAX=350)
*****LUN 5,IS INPUT UNIT
C*****LUN 9 IS OUTPUT STORAGE UNIT
C*****LUN 1 IS ON LINE OUTPUT UNIT
C***** LON 11 CAN BE USED TO READ IN TEMPERATURES COMPUTED
(6E12.5 FORMAT)
C***** IN A EARLIER RUN.
106 IZERO = 19*IMAX
00 601 I = 1,IZERO
601 X01(I)=0.
DO 602 I=1,IMAX
EXPO(I) = 0.0
CONC(I) = 0.0
AREA(I) = 0.0
CSTAR(I) = 0.7
ITRANS(I) = 0.0
AMASS(I) = 0.0
XX1(I) = 0.0
YY2(/) = 0.0
ZZ3(I) =0.0
602 90X(I)=0.
WRITE (114000)
4000 FORMAT (1H ,22HBEGIN INPUT PROCESSING)
IMAX IS MAXIUM NUMBER OF NODES?. CURRENTLY SET AT 350
C
C******* LL IS COUNTER FOR NUMBER OF CASES PROCESSED - -EXITS AT LL = LIO
LL=1
(5,1000)( TITLE(I) ,I=1,200)
TITLE IS A REQUIRED 10 CARD DISCRIPTIVE ARRAY
1000 FORMAT (20A4)
READ (5,1001)LIM,II,SSTEST,BB
C*****LIM IS NUMBER OF INPUT CASES
II IS NUMBER OF NODES
C
***** SSTEST- VALUE OF STEADY - STATE TEST VALUE EPS -- CONVERGENCE TEMP
C***** LIMIT.
Cu" * 9B IS RELAXATION ACCELERATION PARAMETER USED TO *OVER...RELAX*
C***** STEADY STATE ITERATIVE SOLUTIONS.
1001 rORMAT (215,2X,E8.4,2X,E8.4)
WRITE(01,9999)LIM,II,SSTEST,BB
9999 FORMATtifi '2110,2E16.4)
A1, A2, A3, A4, ARE USED TO DETERMINE CONCDNTRATION, C* (LEAST
C
RE
C
VERSION 2.1
0S1 FCPTRAN
HEATTRAN
06/17/71
0918
123
C4***SlUA4FS COFFICNTS).
C***CM1 AND ( "M2 APF USED TA DETERMINE RATE AFTER C* HAS BEEN REACHED
44" DIFFK IS DIFFUSION COFFFICENT.
RFAD(5, 3099) Al, A2, A3, Al., CM1, CM2, HCST, DIFFK
3099 FORMAT(4F12.0, 2F6.0, 2F10.0)
RFA0(5,1002)IPRTipIPRT2,IPRT3,IPRT4,IPRT5,IPRT6
C*****IPRT1 TO I °RT6 - -- SELECTED NODES FOR ON LINE PRINTING
1012 FORMAT
_ (615)
.
.EAO(5,1003)( IXP(I),IXM(I),IYP(I),IYM(I),IZP(I),
IZM(I) '1=1,11)
1
,IZM- ARE NODE LINKINGS FOR NODE(I)
C*****IXP,IXm,
+/X,Y,7 DIRECTIONS
1013 FORMAT (2413)
ARITF(9,4044)
40104 FORMAT (1H ,21HT4ERMAL PROGRAM NATA )
4RITE(9,4002)((TITLE(I)),I=1,200)
4012 FORMAT (1X,20A4)
ARITF(9,4003)LIM,II,SSTEST
4003 FORMAT (1H ,215,2X,E11.4,45X112HCONTROL CARD)
4RITE(9,879) Al, A2, A3, A4, CM1, CM2, HCST, DIFFK
9ORMAT(1X, 4E14.4, 4F10.4)
4RIT:(9,4004)IPRT1,IPRT2,IPRT3,IPRT4,IPRT5,IPRT6
4004 FORMAT (1H ,615,15X,15HPRINt OUT NODES/1H ,18HNODE REL LOCATIONS)
'4RITE(9,4005)( I,IXP(I),IXM(I),IYP(I),IYM(I),
IZP(I),IZM(I) ,I=1,II)
1
4015 FORMAT(1M 2X,13,2X,613,4X,13,2X,613,4X,13,2X,613,4X,13,2X,613)
TF1=0
;TART OF CHECK FOR NODE LINKAGE CONSISTANCY
IF IEI= MINUS, PROGRAM STOPS
10 021 I=1,II
IF (IXP(I)) 002,003,001
C"*"IF1 IS EPROP.INDICATOR.
001 N=IXP(I)
IF (I.IXM(N))002,003,002
002 IFRROR=1
SO TO 018
003 TF ( IZM(I)) 915,096,004
y14 N=IXM(I)
005,906,005
005 TFRROR=2
GO TO 018
006 IF (IYP(I)) 018,009,007
017 N=IY°(I)
IF (I-IYM(N)) 005 0099008
___
098 rE-RPOR=3
10 TO 018
019 IF (IYM(I)) 011,012,010
019 N=IYM(I)
IF (I-IYP(N)) 011,012,011
011 I7Rr)OP=4
10 Tn 118'
012 IF (IZP(I)) 014,015,013
017 J=I7P(I)
IF (I-17M(9)) 014,015,014
014 IrRPOP=c
1,1 TO 018
(I7v(I))017,321,016.
01(- '4=IZM(I)
Ir (I-17P(N)) 017,021,017
017 IFRPOR=.S
)S3 FOPT2AN
C
VERSION 2.1
HEATTRAN
06/17/71
0918
124
018 IF (IE1) 995,019,020
I JUMP TO STATEMENT 995 STOPS PROGRAM --- MALFUNCTION SOME PLACE
019 IE1=1
IF.0 =1
411E
?0
4097
021
022
looe
022
ARITE (1,4006)
NRITE (9,4006)
FORMAT (1H ,70HNOOE LINKING IS NOT CONSISTANT/1H ,9X,
28HPRORLEM WILL NOT BE EXECUTED)
ARTIE (1,4007) I,N
FORMAT (1H ,I3,3X,13)
;0 TO ( 003,006,019,012,015,021),IEPROR
CONTINUF
IF (TE1) 995,022,023
NRITE(9,4008)
FORMAT (1H 126HNOOE LINKING IS CONSISTANT)
IE1=0
_
C"'"lEGIN NEXT CASE
(OESCR (I), I=1,18)
IF(E0F(5)) CALL EXIT
C4****IF FIRST 4 COLUMNS ON DESCRIPTION CARD CONTAINS
1)1 27:A0(5,1'100)
MOANEW PROBLEM DATA SET IS READ.(JUMP TO 106)
C
IF(DESCR(1). E0.53535353B)
'TRITE (1,4009)
GO TO 106
fojq !1RMAT (1X
g15HBEGIN NEXT CASE)
C*****7.0 OF CHECK FOR NODE LINKAGE CONSISTANCY
IIATA=0
?EA0(5,1005)1C1,102,1C311C4,1C5,1C6,1C7 ,IC8, IC9
FORMAT (915)
C*44****IC1 TO IC9 ARE INDICATORS SPECIFYING NUMBER OF DATA CARDS
C"4"" IN GROUPS .1 TO 9. IC8 AND IC9 HAVE BEEN ADDED TO CONTROL INPUT
C
FOR CO1lCFNTRATION AND AREA
100
-?E A0(5,1006)TOTIME,PRTIME,STIME
.1.744""TOTImE, PPTIME,.§fIME-ChOD IS REQUIRED FOR ALL R6NS.----C***** IF TOTIME=0, STEADY STATE ASSUMED.
C
TIE IS TIME OFTWEEN PRINTOUTS.
Cm" STIME IS INITIAL TIME, USUALLY ZERO.
109E FORMAT (2X,E5.4,2X,E8.4,2X,E8..4)
ARTIF(9,401))LLO(DESCR(I)),I=1,18)
4110 FORMAT (1H ,4HCASFOX,T1/1H ,18A4)
4RITE(1,4011)ICi,IC2,IC3,IC4,IC5,IC6,IC7,
IC8, IC9,
1
TOTIME,PRTIME,STIME
4011 =nRMAT (1H ,9I5,19X,10HDATA CARDS/1H ,E11.4,2X,E11.4,2X,E11.4,
14X,17HTRANSIENT IND)
Vs,X.7.1
4RITE(9,4012),ICX
4012 FORMAT (1H ,16HDATA CARDS GPOUP,I2)
TC1=1.
GIVES GENERAL PHYSICAL DIMENSIONS- -INDEX
C*****INDEX 4 FIVES THERMAL CONDUCTIVITY.
INDEX 5 GIVES HEAT GENERATION
10 107 K=1,IC1
2c.119(5,1007)XIN,INOEX,INTO(ISTART(I),IEN1(I)Y,I=1,INT)
1017 =nRMAT (F8.4,/.12,20I3)
4P.ITE(q,4013)XIN,INDEX,INTIMSTART(I),IEND(I)),
C
1
I=1,INT)
4117 7ORMAT (1H 'Eli. 4'92X 12, 2X, I292X, 10 (1X92I3) )
IP (IWIFX) 103,1)3,102
(IN1EX-5) 104,104,1113
111? IF (INO7Y.GF.98) 1112,1C3
117 IOATI=1
;0 TI 107
111? ?FAO (6,1007) OZZ
112 T'
0S3 FORTRAN
VERSION 2.1
HEATTRAN
06/17/71
0918
125
ARITE (9,6000) 3ZZ
6000 FORMAT(1X,:29.4)
)O 1115 J =1,INT
1115
104
105
107
C
ISTA=ISTART(J)
IENDA = IEND(J)
10 1115 I=ISTA,IENDA
EXPO(I1 = XIN
10X(I)=9ZZ
INDEX = 5
10 105 J=1,INT
IN1=ISTART(J)+IMAX*(INDEX-1)
TN2=IENO(J)+IMAX4(INDEX-1)
10 105 I=IN1,IN2
X01(I) = XIN
CONTINUE
CONVERT INCHES TO FEET FOR )(01 TO X03
C'""CONVERT WATTS TO BTUS
10 123 I=1,II
XX1(I) = X01(I)
YY2(I) = X02(I)
Z73(I) = X03(I)
X0i(I)=X01(I)p0.08333
X02(I)=X02(I)40.08333
X03(I)=X03(I)p0.08333
IF (10X(I).EQ.0.) x05(1)=x05(1),3.413
IF (x01ci»109,109,110
109 IERROR=1
SO TO 116
110 IF (X02(I)) 111,111,112
111 IERROR=2
GO TO 116
112 IF (X03(I))113,113,114
113 IERROR=3
114
115
116
117
GO TO 116
IC (X04(I)) 115,115,123
IERROR=4
IF (IE1) 995,117,118
IF1 =1
IG0=1
AnTE (1,4018)
WRITE (9,4018)
4018 FORMAT (1H 131HGROUP ONE INPUT IS NOT COMPLETE/1H ,9X,
28HPROPLEM WILL NOT BE EXECUTED)
1
118 ARITE (1,4020) I,IERROR
-WYCORMAT (1H ,2X,13-,-2 ,I2)
TO (110,112,114,123),IERROR
123 CONTINUE
IE1=0
IF (IC5) 995,124,127
124 IF (IC3) 995,125,126
---tr,,m41TK-FriqTyPn--0-FR1111-010N fkANsFER.
IF...
C
IRR IS CONTROL vARIARLf.
IRR=-1 --- NO RADIATION EFFECTS
GOTH RADIATION LOSS TO ENVIRONMENT AND INTERNOD
IRR=0
c*****
IRR=1 --- RADIATION LOSS TO ENVIRONMENT ONLY.
125 IRR=-1
C****IF
C
A. P.
126 IRR=9
GI TO 128
127 IRR=1
VERSION 2.1
0S3 FORTRAN
HEATTRAN
06/17/71
0918
128 IF (IC2) 995,164,129
129 ICX=2
WRITE(9,4012)ICX
00 135 K=1,IC2
READ(5,1007)XIN,INDEX,INT,((ISTART(I),IEND(I)),I=1,INT)
WRITE(9, 4013) XIN, INDEX,INT,((ISTART(I),IEND(I)),
I=1,INT)
IF (INDEX -6) 131,131,130
130 IF (INDEX -18) 132,132,131
131 IDATA=1
GO TO 135
1
132 10 133 J=1,INT
Cm"START PROCESSING GROUP 2 DATA CARDS.
INDEX 7 -12 ARE CONVECTION
C4**4. COEFFICIENTS FOR EACH NODE FACE. INDEX 13-18 IS THE LOCAL
TEMPERATURE ASSOCIATED WITH COEFFICIENTS.
C
IN1=ISTART(J)+IMAX*(INDEX..1)
IN2=/END(J)+IMAX*(INDEX1)
00 133 I=IN1,IN2
133 X01(I)=XIN
135 CONTINUE
00 156 I=1,II
C*****START ERROR CHECK FOR GROUP 2 DATA POINTS
IF (X07(I)) 137,138,136
136 IF (X13(I)) 138,137,138
137 IFAROR=7
GO TO 153
138 IF (X08(I)) 140,141,139
139 IF (X14(I)) 141,140,141
140 tERROR=8
GO TO 153
141 IF (X09(I)) 143,144,142
142 IF (X15(I)) 144,143,144
143 IERROR=9
GO TO 153
144 IF (X10(I)) 146,147,145
145 IF (X16(I)) 147,146,147
146 IERROR=10
GO TO 153
147 IF (X11(I)) 149,150,148
148 IF (X17(I)) 150,149,150
149 !EPROR=ii
GO TO 153
150 IF (X12(2)) 152,156,151
151 IF (X18(I)) 156,152,156
152 IERROR=12
153 IF (tE1) 995,154,155
15
IG0=1
4R/T (1,4019)
'WRITE (9,4019)
4019 FORMAT (1H ,29HAM8 TEMP AND H NOT CONSISTANT/1H ,9X,
28HPROBLEM WILL NOT BE EXECUTED)
1
-1.55- 4P/fit (1,4020) I,/ERROR
IERROR=IERROR-6
C***4*SND tRRoR CHECK FOR GROUP 2 DATA POINTS
GO TO (138,141,144,147,150,156),IERROR
156 CONTINUE
IE1=0
IF MVO 160,1 8,158
C
CONVERT DEGF TO DEG RANK/NE
126
0S3 FORTRAN
VERSION 2.1
HEATTRAN
06/17/71
0918
127
158 00 159 1=1,II
X13(I)=X13(I)+460.
X14(I)=X14(I)+460.
X15(1)=X15(I)+460.
X16(I)=X16(I)+460.
X17(1)=X17(I)+460.
X18(I)=X18(I)+460.
159 CONTINUE.
C"*"
FORM AREA TIMES CONVECTION COEFICIENT (NET SURFACE TRANSFER)
160 DO 161 1=1,11
V1=X07(I)*X02(1)*X03(I)
V2=X08(I)*X02(I)*X03(I)
V3=X09(I)4X01(I)4X03(I)
V4=X10(I)*X01(I)*X03(I)
V5=X11(I)*X01(I)*X02(I)
V6=X12(I)4X01(I)*X02(I)
C*"4"6" X19 IS EQUIVALENCED TO A
WHERE A IS LUMPED CONVECTION
C
PARAMETER.
X19(1)=V1+V24.V3+V44.V5+V6
SUMH=X13(I)*V1+X14(I)*V2+X15(I)4V34-X16(I)*V44.X17(I)*V54.X18(I)*V6
IF (B0X(I).EQ.0.) X05(I)=X05(I)+SUMH
PSUMH(I) = SUMPS
161 CONTINUE
00 162 J=1,12
KK=J*IMAX
00 162 1=1,11
K =KK +I
162 X06(K)=0.
164 IF (IC3) 995,196,165
165 ICX=3
WR1TE(9,4012)ICX
DO 171 K=1,IC3
READ (5,1007)XIN,INDEX,INTO(ISTART(I),IENO(I)),I=1,INT)
WRITE(9,4013)XIN,INDEX,INTO(ISTART(I),IEND(I)),
I1,INT)
1-
166
167
168
169
171
IF (INDEX-18) 167,167,166
IF (INDEX-30) 168,168,167
IDATA=1
GO TO 171
00 169 J=1,INT
/N1=ISTART(J)+IMAX*(INDEX-19)
IN2=IEN0(J)+IMAX*(INDEX:19)
DO 169 I=/N1,IN2
X07(I)=XIN
CONTINUE
00 192 I=1,II
TF-11(47/I))
172 IF (X13(I))
173 IERR0R=19
173,04,172
174,173,174
GO TO 189
174 IF (X08(I)) 176,177,175
175 IF (X14(1)) 177,176,117
TERROR=20
GO TO 189
377 ZF (1119(I)) 17-9,180908
178 IF (X15(/)) 180,179,180
179 IERROR=21
GO TO 189
TF TX-ITTIT) 182-0.63,161
181 IF (X16(I)) 183,182,183
,176
VERSION 2.1
OS3 FORTRAN
HEATTRAN
06/17/71
0918
128
182 IERROR=22
GO TO 189
183 IF (X11(I)) 185,186,184
184 IF (X17(I)) 186g185,186
185 IERROR=23
GO TO 189
186 IF (X12(I)) 188,192,187
187 IF (X18(I)) 192,188,192
188 IERROR=24
189 IF (IE1) 995,190,191
190 IE1=1
IG0=1
WRITE (1,4021)
WRITE (9,4021)
4021 FORMAT (1H ,29HAMB TEMP AND F NOT CONSISTANT/1H ,9X,
28HPROBLEM WILL NOT BE EXECUTED)
1
191 WRITE (1,4020) IgIERROR
IERROR=IERROR-18
GO TO (174,177,180,183,186,192) ,IERROR
192 CONTINUE
IE1=0
DO 193 I=1,II
S1=X07(I)*X02(I)*X03(I)
S2=X08(I)*X02(I)*X03(I)
S3=X09(I)*X01(I)*X03(I)
S4=X10(I)*X01(I)*X03(I)
S5=X11(I)*X01(I)*X02(I)
S6=X12(I)*X01(I)*X02(I)
X06(I)=.1712E8*($14.S24.$3+S44.S54S6)
W1=S14(X13(.I)+460.)**4
42=S2*(X14(I)+460.)"4
W3=S3*(X15(I)4.460.)**4
W4=S4*(X16(I)+460.)"4
W5=S5*(X17(I)+460.)"4
W6=S64fX18(I)+460.)**4
C
C
0 IS EQUIVALENCE() TO 1(05
NOTE
C*****
C
IF (BOX (I).t4.0.) D(I)=X05(I)+.1712E8*(W1+W24.W3+W44.W5+W6)
.1712E8'
IF (BOX(I).NE.0.) PSUMH(I) = PSUMH(I)
*(414-W2+$04444+05+146)
193 CONTINUE
DO 194 J=1,12
KK=J4IMAX
-0-0-194 1=ton
K=KK+/
194 X06(K)=0.
196.1F (IC4) 995,233,197
197 ICX=4
CALCULATE HEAT LOST BY CONDUCTION. INDEX NUMBERS 30-36 ARE
C'-"
DUNDUCT/ON TERMS
SURFA
WRITE(9,4012)ICX'
4D
READ(5,1007)X/N,INDEX,INTO(ISTART(I),IEND(I)),I=1,INT)
WRITE(1,41113)XrN,INDEX1INT,MSTART(I),IEND(I)),
I=1,INT)
1
C""
ITRUEX=3-01
198 IF (INDEX-36) 200,200,199
OS3 FORT/AN
VERSION 2.1
HEATTRAN
06/17/71
0918
199 IDATA=1
SO TO 203
200 00 201 J=1,INT
IN1=ISTART(J)+IMAX*(INDEX31)
IN2=IEND(J)+IMAX(INDEX31)
00 201 I=IN1,IN2
201 X07(I)=XIN
203 CONTINUE
00 231 I=1,II
IF (X07(I)) 206,208,205
205 IF (IXP(I)) 206,206,207
206 IERROR=31
GO TO 228
207 4=IXP(I)
X08(N)=X07(I)
208 IF (X08(I)) 210,212,209
209 IF (IXM(I)) 210,210,211
210 IERROR=32
GO TO 228
211 4=IXM(I)
X07(N)=X08(I)
212 IF (X09(I)) 214,216,213
213 IF (IYP(I)) 214,214,215
214 IERROR=33
GO TO 228
215 N=IYP(I)
Xi0(N)=X09(/)
216 IF (X10(I)) 218,220,217
217 IF (IYM(I)) 218,218,219
218 IERROR=34
GO TO 228
219 N=IYM(I)
X09(N)=X10(I)
220 IF (X11(I))222,224,221
221 1F 1/215(14) 222,222,223
222 IERROR=35
Go TO 228
223 N=IIP(I)
X12(N)=X11(I)
224 IF (X12(I)) 226,231,225
223 IF (f2M(/l) 226,226,227
226 IERROR=36
"-0 TO -228
227 N=IZM(I)
Xt1(N)=X12(I)
GO TO 231
----2215TFTIED 595,229,230
229 1E1=1
IG0=1
'WRITE (1,4022)
WRITE (9,4022)
4022 FORMAT (1H ,50HSURFACE RESISTANCE INPUT NOT CONSISTANT WITH NODE
7HLINKING/1H ,9X,28HPROBLEM WILL NOT BE EXECUTED)
230 WRITE (1,4020) I,IERROR
TEFFOR=TERROR-30
GO TO (208,212,216,220,224,231),TERROR
231 CONTINUE
IF (IE1) 995,233,232
212 Tr1=-0-
GO TO 277
129
0S3 FORTRAN
VERSION 2.1
HEATTRAN
213 00 276 I=1,II
AX=X02(I)*X03(I)
4Y=X01(I)*X03(I)
AZ=X01(I)*X02(I)
3X=X01(I)/(2.*AX*X04(I))
1Y=X02(1) /(2.*AY'X04(I))
OZ=X03(I)/(2.*AZ4X04(I))
IF (IXP(I)) 239,239,234
234 N= IXP(I)
235
236
237
238
239
240
241
242
243
0=X02(N)*X03(N)
1=X0t(N)/(2."-T*A04(N))
IF (X07(I)) 238,238,235
IF (AX-..0) 236,237,237
n=Ax
9=B+1./(X07(I)*0)
C1(I)=1./(4X+4)
GO TO 240
C1(I)=0.
IF (TXM(I))246,246,241
4=IXM(I)
D=X02(N)*X03(N)
4 = X01(N)/ (2.*D'X04(N))
IF (X08(I)) 245,245,242
IF (AX-0) 243,244,244
0=AX
244 0=4+1./(X08(I)4113)
245 C2(I)=1./(BX+B)
GO TO 247
246 C2(I)=0.
247 IF ( IYP(I)) 253 253,248
248 N=IYP(I)
0=X01(N)*X03(N)
3=X02(N)/(2.204X04(N))
IF (X09(I)) 252,252,249
249 IF (AY-.0) 250,251,251
250 D =AY
251 1=B+1./(X09(I)*D)
252 03(I)=1./(BY+B)
GO TO 254
253 C3(I)=0.
254 IF (IYM(I)) 260,260,255
255 N=IYM(I)
0=X01(N)4)(03(N)
3=X02(N)/(2.-w(1 *X04(N))
IF (X10(I)) 259,259,256
256 IF (AY-D) 257,258,258
257 n=Av
258 3=8 +1./(X10(I)4T)
259 C4(I)=1./(4Y+(3)
-GO TO 261
260 C4(I)=0.
261 IF (IZP(I)) 267,267,262
262 1=1-7151-TY
2E3
264
265
266
017X01(N)X02(N)
4=X03-(N)/(2.'0*X04(N))
IF (X11(I)) 266,266,263
IF (AZ -I1) 264,265,265
0=AZ
13=4+1./(Xli(1)*0)
15(I)=1./(BZ+13)
.
06/17/71
0918
130
0S3 FORTRAN
VERSION 2.1
HEATTRAN
06/17/71
0918
GO TO 268
267 15(1)=0.
268 IF (IZM(I)) 274,274,269
269 1=IZM(I)
3=X01(N)*X02(N)
3=X03(N)/(2.'04)(04(N))
IF (X12(I)) 273,273,270
270 IF (AZ-0) 271,272,272
271 3=AZ
272 3=84.1./(X12(I)*0)
273 C6(I)=1./(BZ+9)_
GO TO 275
274 C6(I)=0.
275 A(I)=X19(I)+C1(I)+C2(I)+C3(I)+C4(I)+C5(I)+C6(I)
276 CONTINUE
277 30 278 J=1,6
KK=J*IHAX
10 278 I=1,II
K=KK+I
278 X06(K)=0.
00 280 I=1,II
X04(I)=0.
280 CONTINUE
IF (IC5) 995,330,281
281 ICX=5
INDEX NUMBERS 3642
RADIATION BETWEEN NODES.
C"*" IC5
ARE RADIATION FACTORS (INTER NODE).
C * * * **
WRITE(9.4012)ICX
00 287 K=1.IC5
READ(5.1007)XIN,INDEX.INTO(ISTART(I),IEND(I)),I=1,INT)
WRITE(9.4013)XIN,INDEX.INTO(ISTART(I),IEND(I)),
1
IF (INDEX -36) 283,283,282
282 IF (INDEX-"42) 284.284.283
I=1.INT)
-213 IDATA=1
GO TO 287
284 00 285 J=1,INT
IN1=ISTART(J)+IMAX*(INOEX-37)
IN2=IEND(J)+IHAX*(INDEX-37)
00 285 I=IN1,IN2
285 X07(/)=XIN
287 CONTINUE
nO 314 1=1,11
IF (X07(I)) 289,291,288
288 IF (IXP(/)) 289,289,290
289 IERROR=37
G0-TO 311
290 N= IXP(I)
X08(N)=X07(I)*X02(I)*X03(I)/(X02(N)*X03(N))
291.IF (M(I)) 293,295,292
292 IF ( IXM(I)) 293,293,294
293 IERROR=38
SO TO Tii
294 N=IXM(I)
X17(N)=X08(/)*X02(I)4X03(/)/(X02(N)*X03(N))
295 IF (X09(I)) 297,299,296
296 IF (IYP(I)) 297,297,298
297 IERROR=39
SO TO 3/1
298 N=IYP(I)
131
0S3 FORTRAN
VERSION 2.1
HEATTRAN
06/17/71
0918
X10(4)=X09(I)*X11(I)*X03(I)/(X01(N)*X03(N))
299 IF (X10(I)) 301,303,300
300 IF (IYM(I)) 301,301,302
301 IFRROR=40
GO To 311
302 A=IYM(I)
X09(N)=X10(I)*X01(I)*X03(I)/(X01(N)*X03(N))
303 IF (X11(I)) 305,307,304
304 IF (IZP(I)) 315,305,306
305 IERROR=41
GO TO 311
306 q=IZP(I)
X12(N)=X11(I)*X01(I)*X02(I)/(X01(N)*X02(N))
307 IF (X12(I)) 309,314,308
308 IF (UMW) 309,309,310
309 /FRROR=42
GO TO 311
310 4=I7M(I)
X11(N)=X12(I)*X01(I)*X02(I)/(X01(N)3X02(N))
GO TO 314
311 IF (IE1) 995,312,313
312 IE1=1
IG0=1
WRITE (1,4023)
WRITE (9,4023)
4023 FORMAT (1H ,46HINTER NODE RADIATION NOT CONSISTANT WITH NODE
7HLINKING/1H ,9X,28HPROBLEM WILL NOT BE EXECUTED)
1
313 WRITE (1,4020) I,IERROR
IERROR=IERROR-42
GO TO (291,295,299,303,307,314),IERROR
314 CONTINUE
IF (IE1) 995,316,315
315 IE1=0
GO TO 330
316 90 329 1 =1,II
IF (X07(I)) 318,318,317
317 SASi(I)=.1712E-8*X07(I)*X02(I)*X03(I)
A(I)=A(I)-C1(I)
C1(I)=0.
318 IF (X08(I)) 320,320,319
319 SAS2(I)=.1712E-8*X08(I)*X02(I)*X03(I)
A(I)=A(I)-C2(I)
C2(I)=0.
320 IF (X09(I)) 322,322,321
321 SAS3(1)=.1712E8*X09(I)*X01(I)*X03(I)
A(I)=A(I)C3(I)
03(I)=0.
322 IF (X10(I)) 324,324,323
323 SAS4(I)=.1712E8*X10(I)*X01(I)*X03(I)
A(I)=A(I)-C4(I)
C4(I)=0.
324 IF (X11(I))326,326,325
325 SAS5(I)=.1712E-8*X11(I)*X01(I)*X02(I)
A(I)=A(I)-05(I)
C5(I)=0.
326 IF (X12(I)) 328,328,327
327 SAS6(I)=.1712E8*X12(I)4X01(I)*X02(I)
A(I) =A(I) -C6(I)
16111=0.
328 R(I)=X06(I)+SAS1(I)+SAS2(I)+SAS3(I)+SAS4(I)+SAS5(I)+SAS6(I)
132
0S3 FORTRAN
VERSION 2.1
HEATTRAN
06/17/71
0918
329 CONTINUE
330 00 331 I=1,II
X04(I)=X01(I)*X02(I)*X03(I)
X01(I)=0.
X02(I)=0.
X93(I)=0.
331 CONTINUE
IF (IC6) 995,347,332
IC6 -- NODE HEAT CAPACITIES
C
INDEX 43 = NODE DENSITY
C
cm.*
INDEX 44 = NODE SPECIFIC HEAT
332 ICX=6
WRITE(9,4012)ICX
00 338 K=1,IC6
READ(5,1007)XIN,INDEX,INTO(ISTART(I),IEND(I)),I=1,INT)
WRITE(9,4013)XINIINDEX,INTO(ISTART(I),IEND(I)),I=1,INT)
IF (INDEX -42) 334,334,333
333 IF (INDEX-44) 335,335,334
334 IDATA=1
GO TO 338
335 00 336 J=1,INT
IN1=ISTART(J)+IMAX*(INDEX-43)
IN2=IEND(J)+IMAX*(INDEX...43)
00 336 I=IN1,IN2
336 )(01(I) = XIN
338 CONTINUE
DO 345 I=1,II
IF (X01(I)) 339,339,340
339 IERROR=43
GO TO 342
340 IF (X02(I)) 341,341,345
341 IERROR=44
342 IF (IE1) 995,343,344
343 IE1=1
T-60=f
WRITE (1,4024)
WAITE-094024Y
4024 FORMAT (1H ,31HGROUP SIX INPUT IS NOT COMPLETE/1H ,9X,
28HPROBLEM WILL NOT BE EXECUTED)
1
344 WRITE (1,4020) I,IERROR
IF (rERRCR-43) 9450409345
345 CONTINUE
TE1=0
CALCULATE NODE HEAT CAPACITY
C
DO 346 I=1,II
X04(I)
AMASS (I) = X01(I)
= KMAS-S1TI*X02(I)
x0i(I)=0.
X0-2(/)=0.
X04(1)=0.
346 CONTINUE
347 IF (IC7) 995,354 3 8
WRITE(9,4012)IC*
'MY -353 X=
TC7
READ(5,1007)XIN,INDEX,INT,(USTART(I)gIEND(I)),I=1gINT)
WRITE(9,4013)XIN,INDEXpINT,(USTART(I),IEND(I)),
i
1
349 IDATA=1
349,35o 349
I=1,INT)
133
0S3 FORTRAN
VERSION 2.1
HEATTRAN
06/17/71
0918
134
GO TO 353
350 30 351 J=1,INT
IN1=ISTART(J)
IN2=IENO(J)
READ IN INITIAL STEADY STATE TEMPERATURES
C
DO 351 I=IN1,IN2
351 T(I)=XIN
353 CONTINUE
GO TO 355
354 READ(11,4045)((T(I)),I=1,II)
4045 FORMAT (6E12.5)
WRITE(9,4014)
4014 FORMAT (1H ,27HINITIAL TEMP READ FROM TAPE)
355 IF (IRR) 358,356,356
356 00 357 I=1,II
357 T(I) = T(I) + 460.
358 ICX = 8
WRITE(9, 4012) ICX
30 661 K = 1, IC8
READ(5,1007) XIN, INDEX, INT ,((ISTART(I),IEND(I)),I= 1,INT)
WRITE(9,4013)XIN,INDEX,INT,(USTART(I),IEND(I)),
1
00 661 J = 1, INT
IN1=ISTART(J)
IN2=IEND(J)
RO 661 I = IN1, IN2
651 CONC(I) = XIN
ICX = 9
WRITE(9, 4012) ICX
10 662 K =1,IC9
I=1,INT)
READ(51007)XINIINDEX,INT,MSTART(I),IEND(I)),I=1,INT)
WRITE(9,4013)XIN,INDEXIINT,C(ISTART(I),IEND(I)),
1
662
650
666
691
I=1,INT)
DO 662 J = 1, INT
IN1=ISTART(J)
IN2=IEND(J)
DO 662 I = IN1, IN2
AREA(I) = XIN
DO 666 I = 1,11
IF(BOX(I) .NE. 0.0) 650,666
TS0 = T(I)*T(I)
CSTAR(I) = Al + A2*T(I) + A3*TSQ + A4*TSQ*T(I)
RSV = (B0X(I)*EXP(EXP0(I)/T(I)))*3.413*HCST*AREA(I)
Q(I) = PSUMH(I) + RSV
IF(CONC(I) /CSTAR(I).LT. 1.0) ITRANS(I) =
CONTINUE
= 1,II)
WRITE(1,691) (I,CONC(I), CSTAR(I), AREA(I)
FORMATUX, 3(I3,3E12.3,1X) )
IF 110ATA) 995,360,359
359. WRITE (1,4016)
WRITE (9,4016)
4016 FORMAT (1H ,48HALL DATA CARDS NOT CONSISTANT WITH IC INDICATORS/
1H OX,28HPROBLEM WILL NOT BE EXECUTED)
1
360
361
362
363
IG0=1
rr-tmoI 995,361,999.
IF (TOTIME) 995,400,362
IF (IC6) 995,363,500
WRITE (1,4017)
WRITS (94017)
4017 FORMAT (1H ,48HTRANSIENT INDICATED DATA CARDS GROUP SIX MISSING)
VERSION 2.1
0S3 FORTRAN
HEATTRAN
06/17/71
135
0918
GO TO 999
C*****STEAOY STATE DROGRAM
400 L=0
LP=0
JUMP=0
3UM3=0.
SUM4=0.
STIME=0.
4RITE (1,4025)
4025 FORMAT (1H ,20H9EGIN S S ITERATION )
WRITE (1,4026) IPRT1,IPRT2,IPRT3FIPRT4,IPRT5IIPRT6
4026 FORMAT (1H plOHITERATIONS,6(2X0HNODE NO ,13,3X) ,7H MAX DT)
WRITE (1,4027) L,T(IPRT1),T(IPRT2),T(IPRT3),T(IPRT4),T(IPRT5),
1T(IPRT6),STIME
4027 FORMAT (1H ,2X,15,1X,6E16.6,3X,E10.4)
401 00 408 J=1,8
00 407 I=1,II
SUM = 0.0
.
/1=IXP(I)
I2=IXM(I)
I3=IYP(I)
I4=IYM(I)
I5=IZP(I)
I6=IZM(I)
IF ( C1(I) oNE. 0) 8010, 8011
8010 SUM = SUM + C1(I)* T(I1)
8012, 8013
0
)
8011 IF ( C2(I) .NE.
T(I2)
8012 SUM = SUM + C2(I)
NE. 0)
8014, 8015
8013 IF (C3 (I)
8014 SUM = SUM + C3(I)4 T(I3)
8016, 8017
NE.
0)
C4(I)
8015 IF
8016 SUM = SUM + C4(I) * T(I4)
8018, 8019
8017 IF ( C5(I) oNE. 0)
8018 SUM = SUM + C5 (I) * T(I5)
-811-9 tr (t611) .NE. 0) 8020, 8021
T(I6)
8020 SUM = SUM + C6(I)
8021 SUM- = SUM + Q(I)
(
IF (IRR) 406,403,402
402 IF (SAS1 (I) .NE. 0.0) 8110, 8111
8110 SUM = SUM + SAS1 (I) * T(11) **4
NE. 0.0) 8112, 8113
tr ISAS21I)
8112 SUM = SUM + SAS2(I)* T(I2) **4
NE. 0.0) 8114, 8115
rr ( sAmt)
8113
8114 SUM = SUM + SAS3 (I) * T(I3) **4
8115 IF ( SAS4 (I).NE. 0.0) 8116, 8111
**4
8116 SUM = SUM +SAS4(I) * T(I4)
SA(I) NE. 0.0) 8118, 810
1117
**4
8118 SUM = SUM + SAS5(I)* T(I5)
8119 IF(SAS6(1) .NE. 0.0) 8120, 403
SAS6(I)* T(I6) **4
8120.SUM = SUM +
403 02=R(I)*TII)**3
03=2.2'02
/F (131A1I)) 414,404,405
404 T(I)=SUM/(A(I)+02)
= PSUMH(T) 4.(BOX(I)
IT (BOX(T).NE.O.)
*3.413
GO TO 407
EXP(EXPO(T)/T1I)))
405 TXX4=(SUMA(I)*T(I))/R(/)
5014 = TIOW
= ABS(TXX4)**0.25
---
VERSION 2.1
0S3 FORTRAN
HEATTRAN
T(I) = SIGN(T(I),XX4)
IF (90X(I).NE.0.)
*
3.413
06/17/71
0918
= PSUMH(I) +180X(I) 4 EXP(EXPO(I) /T(I)))
SO TO 407
406 T(I)=SUM/A(I)
IF (BOX(I).NE.O.) Q(I) = PSUMH(I) +(EM(I) 4 EXP(EXPO(I)/T(I)))
*3.413
407 CONTINUE
IF (JUMP) 995,408,414
408 CONTINUE
*
9=0.
00 412 I=1III
0T=T(I)TX(I)
TX(I)=T(I)
IF (0T) 409,410,410
409 DT = -DT
410 IF'(0T-41) 412,411,411
411 9=0T
412 CONTINUE
JUMP=1
IF (BSSTEST) 413,413,401
413 WRITE (1,4028)
4028 FORMAT (IN ,23HS S SOLUTION INDICATED )
WRITE (9,4029)
,20X,22HSTEADY STATE INDICATED /,15X,
4029 FORMAT (1H
33HS S NODE TEMPERATURES DEGREES r )
IOUT=1
GO TO 980
414 JUMP=0
L=L+1
LP=LP+1
00 415 I=1,II
T(I)=T(/)+984(T(/)..TX(I))
415 CONTINUE
IF (05-25) 419,418,995
418 LP=0
419 WRITE (1,4027) L,T(IPRT1),T(IRRT2),T(IPRT3),T(IPRT4),T(IPRT5),
1T(IPRT6),13
C
GO TO 401
TRANSIENT PROGRAM
500 SPRT =O.
L=0
JUAP=0
DO 506 /=1,II
0EN=A(I)
IF (IRR) 503,501,501
51/ 02=W(1)*1(1)43
03=2.2402
IF (03 -DEN) 502,502,505
502.0EN=DEN+02
503 X02(/)=CAP(I)/DEN
IF (X02(1))995,504,506
504 WRITE (1,4034) I
4034 FORMAT (1H ,22HTIME STEP ZERO
X0211)=1.
GO TO 506
505 JUMP=1
506 CONTINUE
TT (JUMP) g95,5Z7,510507 DEL=i.
NODE 913)
136
VERSION 2.1
0S3 FORTRAN
HEATTRAN
06/17/71
137
0918
00 509 I=1,II
IF (X02(2) -DEL) 508,508009
508 DEL=X02(I)
4=/
509 CONTINUE
WRITE (1,4032) DELO
4032 FORMAT (1H ,22HINITIAL TIME STEP IS ,E11.4,16H HRS AS LIMITED,
9H BY NODE IN)
1
4=0
SO TO 511
510 WRITE (1,4033)
4033 FORMAT (1H ,32HINITIAL TIME STEP NOT CALCULATED/
1H ,9X,33HRADIATION DOMINANT FOR SOME NODES)
1
511 WRITE (1,4035)
4035 FORMAT (1H ,15HBEGIN TRANSIENT)
4RITE (1,4036) IPRT1,IPRT2,IPRT3,IPRT4IIPRT5,IPRT6
4036 FORMAT (1H ,1OHLIM. NODE ,6(2X,8HNODE NO ,13,3X),2X,4HTIME)
4RITE (9,4037) STINE
4037 FORMAT(IH ,20X, *TRANSIENT NODE TEMPERATURES (DEGREES F), *
*t CONCENTRATION, AND CRITICAL CONCENTRATION*/
,E10.4,4M HRS)
111 ,20X,21HINITIAL TIME EQUALS
1
IOUT=3
GO TO 980
512 9=1.
0 0 525 I=1,II
SUM = 0.0
/1=IXP(I)
I2=IXM(I)
I3=IYP(I)
I4=IYM(I)
I5=IZP(I)
I6=I2M(I)
IF ( C1(I) .NE. 0)18010,18011
18010 SUM = SUM + Cl(I) T(I1)
10011 tE ( C2(I) .NE. 0
)18012,18013
T(I2)
18012 SUM = SUM + C2(I)
18013 IF (C3 (I)
NE. 0) 18014,18015
18014 SUM = SUM + C3(I) T(I3)
NE.
0) 18016,18017
18015 IF ( C4(I)
T(14)
18016 SUM = SUM + C4(I)
18017 IF.
-05(/) .NE. 0) 18018,18419
SUM = SUM + C5 (I)
T(I5)
IF (06(/) .NE, 0)18020018021
SUM = SUM + C6(I)
T(I6)
SUM = SUM + Q(I)
IF (IRR) 5249515,514
514 i7 (SASS (I) .NE. 0.0)18110,1811i
18110 SUM = SUM + SAS1 (I)
T(I1) *4
18111 IF (SAS2(I)
NE. 0.0) 181120.8113
18112.SUM = SUM + SAS2(I)' T(I2) *4
18113
IF
SAS1(I)
NE. 0.0)18114,18115
18114 SUM = SUM + SAS3 (I)
T(I3) *4
18018
18019
18020
18021
.
(
(SAS4 (1).NE. 0A)18116,16117
18116 SUM = SUM +SAS4(I) ' T(I4) 4
SASS(I) .NE. 0.0)18118;18114
18118 SUM = SUM + SAS5(I) T(I5)
*4
18119 IF(SAS6(I) .NE. 0.0)18120, 515
18120 SUM = SUM +
SAS6(I) T(I6) 4'4
18i17
515 137=R(ITTTI)**303=2.2'02
,
0S3 FORTRAN
C
516
517
518
519
520
521
522
523
VERSION 2.1
HEATTRAN
06/17/71
0918
DEN=A(I)+02
TOOT(I)=(SUM-DEN*T(I))/CAP(I)
IF (03-A(I)) 516,516,517
CAP(I) IS EQUIVALENCE° TO X03 AN IS LUMPED HEAT CAPCITY.
X92(I)=CAP(I)/DEN
GO TO 520
TXX4=(SUm-A(I)*T(I))/R(I)
9TSS=TXX4,*0.25-T(I)
IF (OTSS) 518,521,518
IF (TOOT(I)) 519,521,519
X02(I)=OTSS/TIOT(I)
IF (X02(I)) 521,521,522
X02(I)=1.
TOOT(I)=0.
IF (X02(I)-9) 523,523,525
3=X02(I)
SO TO 525
524 TOOT(I)=(SUM-A(I),T(I))/CAP(I)
C.******4 TOOT IS TIME DERIVATIVE OF TEMPERATURE ---DTEMP/DTIME.
525 CONTINUE
IF (IRR) 527,526,526
526 DEL=O
527 SPRT=SPRT+DEL
STIME=STIME+DEL
IF (STIME- TOTIME) 528,529,529
528 IF(SPRT-PRTIME)531,530,530
529 IOUT=5
OT=DEL+TOTIME-STImE
3TIME=TOTIMF
GO TO 532
530 IOUT=4
DT=OEL+PRTIME-SPRT
STIME=ST/ME+DT-DEL
SPRT=0.
SO TO 532
531 IOUT=3
1T=DEL
532 00 533 I=1,II
T(I)=T(I)+DT*TDOT(I)
IF(BOX(I) .NE. 0.0) 658, 533
658 RSV = (90X(I)*EXP(EXPO(I)/T(I))).3.413
IF(ITRANS(I).EQ. 1) 653, 654
654 TSQ = T(I) *T(I)
CSTAR (I) = Al + A2Y-T(I) + A3*TSQ + AletTSQ*T(I)
CRATIO = CONC(I) / CSTAR(I)
trICRATIO.LT.1.0) 655, 657
655 ITRANS(I) = 1
GO To 656
653.CRATIO =CONC(I)/CSTAR(I)
M2*CRATIOnRATTO)
657 11= IXP(I)
12 = TXM (I)
13 = IXP(I)
t4 = IY-01(1)
15 = IZP(I)
16 = IZM(I)
XIX]. = YY2(I)PZZ3(I)
XIYi = MU)* ZZ3(I)
XIZI = XX1(I)PYY2(I)
138
Q53 FORTRAN
CIF***
C
C * ***
IIX1
3IY1
3171
SUMK
VERSION 2.1
=
=
=
=
HEATTRAN
06/17/71
0918
139
XX1(I) '0.5
YY2(I)*0.5
ZZ3(I)
0.5
0.0
10 804 JII = 1,6
ASSUMES CONTACT AREAS ARE EQUAL.
IC8 = ISCRIP(JII)
IF(ICB.LE. 0) GO TO 804
CIJ(JTI) = CONC(I)
CONC(ICO)
PACE = 0.0
DEES= 0.0
JO TO (801, 801, 802, 802, 803, 803), JII
801 'LEES = OIX1 + XX1(IC8)40.5
PACE = XIX1
GO TO 800
802 °FES = OIYi + YY2(IC6) '0.5
FACE = XIY1
GO TO 800
803 GEES = DIZ1 + ZZ3(IC8) '0.5
FACE = XIZ1
800 SUMK = SUMK + CIJ(JII)
FACE/DEES
804 CONTINUE
SUMK = SUMK 'DT*DIFFK
CONC(I) = CONC(I)
RSV'AREA(I)*DT/AMASS(I) +SUMK
D(I) = RSV*AREA(I)*HCST + PSUMH(I)
533 CONTINUE
IF (IOUT..4) 539,537,538
537 WRITE (1,4042)
4042 FORMAT (14 ,164ROUTINE PRINTOUT)
WRITE (9,4039) STINE
4039 FORMAT (14 ,20X,33H ROUTINE PRINTOUT AT TIME EQUALS ,E10.4,4H HRS)
GO TO-160
538 WRITE (1,4041)
4041 FORMAT (1H #144FINAL PRINTOUT)
WRITE (9,4040) STINE
4040 rORMAT (14 /14 ,20X,28HFINAL PRINTOUT TIME EQUALS ,E10.4,4H HRS)
GO TO 980
539 WRITE (1,4027) M,T(IPRT1),T(IPRT2),T(IPRT3),T(IPRT4),T(IPRT5),
1T(IPRT6),STIME
GO TO 512
980 fF (IRR) 983,981,981
981 90 982 1=1,11
T(2)=T(I)-460.
982 CONTINUE
983 2=-4
984 1=1+5
41=I
4?=I+1
43=1+2
44=1+3
45=1+4
46 = I + 5
IDENTF = 4HTEMP
WRITE(9,4043) IDENTF,
.
1
41,T(41)02,T(42),43,7(43)04,7(44),M5 T(M5),46,T(M6)
-41141 FORHAT(2X41A4,6(3X,I3,1,X,E11.4))-
693 FORMAT(/ )
OS3 FCRTRAN
1
VERSION 2.1
HEATTRAN
06/17/71
wRITE(9,4043) IDENTF,
M1,CONC(M1), M2, CONC(M2), M3
), M5, CONC(M5)06, CONC(M6)
14 0,
0918
CONC(M3), H4,CONC(M!,
'DENTE = 4HC*
WRITF(9,4041) IDENTF,
M1, CSTAR(M1), M2, CSTAR(M2), M3, CSTAR(M3)
1
CSTAR(M6)
T44, CSTAR(M4) , M5, CSTAR(M5)
WRITE(9, 693)
IF (II-M5) 985,985,984
985 IF(IRR) 990,988,988
988 )O 989 I=1,II
T(I)=T(I)+460.
989 CONTINUE
990 WRITE (1,4027) L,T(IPRT1),T(IPRT2),T(IPRT3),T(IPRT4),T(IPRT5),
1T(IPRT6),STIME
GO TO (991,991,512,512,991,991),IOUT
991 IF (LL-LIM) 992,999,995
992 LL=LL+1
DO 660 I = 1, IZERO
660 X01(I) = 0.0
999 END FILE 9
GO TO 101
995 WRITE (1,4015)
WRITE (9,4015)
4015 FORMAT (1M ,27HPROGRAM STOP MAL OPERATION )
END FILE 9
CALL EXIT
END
NO ERRORS FOR HEATTRAN
P 32270
RUN
C 36056
0 00001