OAK RIDGE NATIONAL LABORATORY - International Nuclear ...

OAK RIDGE NATIONAL LABORATORY operated by

UNION CARBIDE CORPORATION NUCLEAR DIVISION

for the U.S. ATOMIC ENERGY COMMISSION

ORNL- TM-2802 UC-80 — Reactor Technology

COMPUTER CALCULATION OF TWO-DIMENSIONAL, SJizADXzSXAXB--HEAT TRANSFER INCONCENTRIC CYLINDERS

WITH INTERNAL HEAT GENERATION!

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G . A . Crisfy

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Submitted as cT- dissertation to the Graduate School of the University of Tennessee^ in partial .'fulfillment of the requirements for the Degree of Master of Science. '

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Printed in the United States of America. Available from Clearinghouse for Federal Scientific and Technical Information, National Bureau of Standards,

U.S. Department of Commerce, Springfield, Virginia 22151 Price: Printed Copy $3.00; Microfiche $0.65

LEGAL NOTICE

This report was prepared as an account of Government sponsored work. Neither the United States, nor the Commission, nor any person acting on behalf of the Commission: A. Makes any warranty or representation, expressed or implied, with respect to the accuracy,

completeness, or usefulness of the information contained in this report, or that the use of any information, apparatus, method, or process disclosed in this report may not Infringe privately owned rights; or

B. Assumes any l iabi l i t ies with respect to the use of, 'or for damages resulting from the use of any information, apparatus, method, or process disclosed in this report.

As used in the above, "person acting on behalf of the Commission" Includes any employee or contractor of the Commission, or employee of such contractor, to the extent that such employee or contractor of the Commission, or employee of such contractor prepares, disseminates, or provides access to, any information pursuant to his employment or contract with the Commission, or his employment with such contractor.

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Contract Wo. W-7^05-eng-26

DIRECTOR'S DIVISION

Civil Defense Research Project

COMPUTER CAICUIATION OF TWO-DIMENSIONAL, STEADY-STATE HEAT TRANSFER IN CONCENTRIC CYLINDERS

WITH INTERNAL HEAT GENERATION

G. A. Cristy

Submitted as a dissertation to the Graduate School of the University of Tennessee in partial fulfillment of the requirements for the Degree of Master of Science.

FEBRUARY 1970

OAK RIDGE NATIONAL LABORATORY Oak Ridge, Tennessee

operated by UNION CARBIDE CORPORATION

for the U.S. ATOMIC ENERGY COMMISSION

DISTRIBUTION OF THIS DOCUMENT IS UNLIMITED:

ii

ACKNOWLEDGEMENTS

The author wishes to acknowledge -with appreciation the assistance and guidance of his major professor, Dr. J. W. Prados, Associate Dean of Engineering at the University of Tennessee, without whose encouragement this would never have been accomplished. Thanks are also due the members of the review committee for their advice and encouragement.

Research performed at the Oak Ridge National Laboratory was sponsored by the U. S. Atomic Energy Commission under contract with Union Carbide Corporation.

The assistance provided by Dr. J. C. Bresee, Civil Defense Project Leader and the Union Carbide Corporation through the educational assis-tance program is gratefully acknowledged. The help provided by G. H. Llewellyn in making unpublished programs available and for consultation on the use of other heat transfer programs is deeply appre-ciated.

The thanks of the author are extended to the programming aides at the Oak Ridge National Laboratory for advice on programming details and their explanations of the idiosyncrasies of the IBM and the ORNL compilers. Thanks are also due Mrs. LaVerne Stoddard and Mrs. Susan Price for the excellent typing and to the author's family for their patience and under-standing.

• • •

111

ABSTRACT

A computer program -was developed for calculating steady state temperature distributions in concentric axisymmetric cylinders having internal heat generation. The program, written in F0RTRAN IV, takes into account heat transfer by conduction, convection and radiation. The input format is simple and easy to use. Input consists of the dimensions of the cylinders and the thermal characteristics of the materials. The program divides the cylinders into nodes, arranges the connectors and calculates the temperature distribution in two dimensions. After calculating the temperature distribution in an array of nodes, the program subdivides the nodes and calculates the temperature distribution in the more detailed nodal system. The subdivision can be continued as long as desired provided the number of nodes does not exceed seventy-five (75) in either direction.

The program provides an ability to calculate heat transfer by conduction and radiation across gas filled gaps within the cylinder, and to calculate the variation of the width of the gap due to thermal expan-sion. A program listing and instructions for assembly of input data are included.

V

TABLE OF CONTENTS

CHAPTER PAGE X. INTRODUCTION 1

Objective 1

Method* 1 Related Work by Others 6

II. DESCRIPTION OF THE PROGRAM . 11

General Procedure (Roadraap) . . U Setup of Nodal System . . . . . . . . 13 Calculation of Data on Regular Nodes. . . . . . . . . 16 Calculation of Data on Internal Special Nodes . . . . 17

Calculation of Data on Internal and External Surface Nodes 17

Calculation of Temperature Distribution . . . . . . . 17 Output 19

III. HEAT TRANSFER EQUATIONS . 21 Regular Nodes 21 Interior Special Nodes. . 25

Surface Nodes . . . . . . . . . . . . . . . 26

VT. COMPARISON WITH ONE DIMENSIONAL ANALYTICAL SOLUTION . . 35 V. USER'S GUIDE k8

Input Preparation * . Convergence Controls 1+8 Overrelaxation 1+8

vi

CHAPTER PAGE Output Options Control Cards . . . . . lj.9

VI. DISCUSSION 50

REFERENCES 53

APPENDIXES - 55

A. FORTRAN UST 56 B. DATA CARDS 78

C. INPUT FORMAT 79 D. EXPANSION OF TEMPERATURE ARRAY 91

E. OUTPUT EXAMPLES 98 F. NOMENCLATURE (WITH UNITS) 123 C-. FORTRAN NAMES AND SYMBOIS 126

vii

LIST OF TABLES

TABLE PAGE

I. One Dimensional Problem Calculations - (Heat Generated and Heat Transfer Area) * 38

II. One Dimensional Problem Calculations - (Temperatures at bl

III. One Dimensional Problem Computer Calculations Problem 100 . ij-2

IV. One Dimensional Problem Computer Calculations Problem 125 • V3 V. One Dimensional Problem Computer Calculations Problem lk-5 • kk

VI. One Dimensional Problem Computer Calculations Problem 160 . k6

VII. One Dimensional Problem Computer Calculations Problem 180 . hi

viii

LIST OF FIGURES

FIGURE PAGE

1. Fuel Holder for Coated Particle Irradiation Tests. . • . . . 2 2. Schematic of Heat Transfer Network 4

3. Schematic of Nodal Connections 5 k. Typical Cylinder lf 5. Establishing Nodal Boundaries 15 6. Gas Gap 31 7. One Dimensional Problem • • • 36 8. Data Cards 78 9. Cylinder for Example Problem 99 10. Scale Drawing of Cylinder . 120 11. Graph of T vs R. 121 12. Graph of T vs EL 122

1

CHAPTER I

INTRODUCTION

Objective

The purpose of this research is to develop a digital computer program, using the IBM 360 computer, to provide a quick and easy method of calcula-ting the steady state temperature distribution within composite cylinders having internal heat generation. The cylinders to be analyzed are made up of a number of axially symmetrical concentric cylindrical shapes which have widely different heat transfer properties. Some of the cylinders have narrow gaps which contain an inert gas. The tops and bottoms of the cylinders are usually insulated and the outer surface cooled by convection. However, it is desirable to be able to calculate any combination of methods of cooling the cylinders (i.e. conduction, convection or radiation from any or all surfaces). An example of the type of problem to be solved is to find the temperature distribution in a typical fuel holder employed in coated particle irradiation tests, as shown in Figure 1.

Method The program, written in FORTRAN IV, can be used with any IBM % 0 machine

which uses the IBM compiler and cathode ray tube plotting equipment. The program is also compatible with the local F 0 R T R A N compiler at- the Oak Ridge National Laboratory.

The program is based on the concept of lumped parameters. The method is a modification of the ones used by Dusinberre ( 5 ) and Schenck ( 1 7 ) - A

nodal network represents the physical problem. The "real" node points are centered within the lumps and temperatures at the nodes are considered

2

GRAPHITE LID

GRAPHITE CAN

FUEL PARTICLES

CENTRAL HOLE FOR MELT WIRES

CENTRAL POST

0.500 in

Fig. 1. Fuel Holder for Coated Particle Irradiation Tests

3

uniform throughout the lump. Conduction "between two lumps is assumed to be from the center of one lump through half of that lump to the interface (a zero volume lump) then through half of the second lump to its center. The network system is illustrated schematically in Figure 2 and Figure 3. The conductors represent the capability for transmitting thermal energy from one node to the next. These conductors, or "conductanceshave the units of energy flow rate per degree (watts per degree Centigrade or BTU per hour per degree Fahrenheit). Consistent sets of units, either Metric system or English system, may be used in this program. The program uses two constants which depend on the system used, the value of the Boltzmann constant and the conversion constant for absolute temperatures. Therefore, the input data must indicate which set of units is being used. This report indicates the appropriate units in each system for all dimensional quantities.

The accuracy of the method of lumped parameters depends to a large extent on taking the node size small enough so that the physical conditions approach the assumptions inherent in the method. On the other hand, the smaller the node size the longer it takes to accomplish the calculation. This requires a balance between precision and calculation time. In order to assist the user of the program in making that decision, this program sets up the problem first in the coarsest geometry possible. After making preliminary calculations to get an approximate temperature distribution, the nodes are subdivided to produce a finer geometry and a more accurate solution. This type of procedure can be continued until the geometry is as fine as desired, within the limits of the program (i.e. 75 nodes in each direction).

k

SURFACE

LEGEND

• REGULAR (REAL) NODES ( I = E V E N , J = E V E N )

• PLANE SPECIAL NODES ( I = ODD, J = EVEN)

G CYLINDRICAL SPECIAL NODES ( I = EVEN, J=ODD)

Fig. 2. Schematic of Heat Transfer Network

5

a. REGULAR NODE

C 3 ( I , J )

• (I,J)

J)

b. PLANE SPECIAL NODE

C 2 ( M ) C, (I,J) C2 ( I , J )

-A/S—© HA ( I , J )

(I,J)

c. CYLINDRICAL SPECIAL NODE d. CONVECTIVE CYLINDRICAL SURFACE NODE

LEGEND

O REGULAR NODE

• PLANE SPECIAL NODE

G CYLINDRICAL SURFACE NODE

Fig. 3. Schematic of Nodal Connections

6 )

Related Work "by Others

A generalized heat conduction code, GHT, was written by Fowler and

Volk (8) for use on the IBM 70k. It solves steady state and/or transient

heat conduction problems in three dimensional, cartesian coordinate

geometry. Material properties and film coefficients may be considered

a function of position but not of temperature nor time. Boundary tempera-

tures and heat generation may be functions of time and position. The

program is limited to 950 nodal points, including internal and boundary

points. The program requires eight magnetic tape units and two drum

storage units. All the data for each node must be calculated externally.

The program is so difficult to use that several node generator codes for

use with GHT have been written. The HEATING Program of Liguori and

Stephenson (12) has such a node generator. The nine geometries offered

by the program are:

Cylindrical Rectangular

1. R-0-Z 6. X-Y-Z

2. R-0 7. X-Y

3. R-Z 8. X-Z

4. R 9- x

5. z

HEATING reduced the labor of input considerably but the program is

still limited to the IBM 70I+ and 7090.

Brune11 (k) and Nickols wrote a modification of HEATING for the

IBM 709^ in F0RTRAN II language. This version, HEATING2, has a 750 node

capacity and can take up to ^0 materials and 100 material regions.

7

The GHT code was modified by Moore (15) to include radiation,

increasing the complexity of the code another order of magnitude.

Wilson (21,22) devised GRIN to prepare input for GHT, GHTR and updated

GHTR for use on the IBM 360. The GHT code is still an unusually difficult

one to use. It requires that the points (lumps or nodes) "be numbered in

a special way, namely all radiation points must be assigned to lowest

numbers, starting with 1. The user must specify for each node complete

data necessary to describe the thermal system. These include the initial

temperature, thermal capacity, heat generation rate, conductances and

radiances, and the data used to prescribe temperatures and heat generation

for nodes having such prescribed values.

Bagwell (l) developed the TOSS program for two or three dimensional

systems of coordinates with internal heat generation as a function of time

and space, transferring heat by conduction, radiation and forced or free

convection. It uses the first forward finite difference method of calcula

ting the heat transfer. It is limited to 800 nodes and is frequently in

danger of overloading the machine storage. It is written in machine

language and limited to the IBM 7090. Pierce (l6) modified the TOSS

program to include material properties that change with temperature, and

to solve the heat transfer equations by the Gauss-Siedell method. The

modification reduced the number of nodes to 400 but reduced computer time

per problem. Both T0SS—the Carbide version (l)--and T0SS l~the

Westinghouse version (l6)--require complicated calculations to produce

the input data. Consequently several node generators for use with TOSS

have been developed. Bagwell (2,3) developed SIFT, a simplified input

8

program, which generates a T0SS input deck, then uses a subroutine version of T0SS to solve the heat conduction equations. The SIFT program requires an IBM-7090 with 32K core storage, 4 or 5 on-line tape drives plus usual off-line equipment. The SIFT program is in FAP language. The SIFT-T0SS combination is a powerful tool for solving complicated heat transfer problems but still takes considerable time for setup.

Llewellyn (13) has recently produced a program, written in FORTRAN IV for use on the IBM 360 computer, designed to solve for temperature distri-butions in concentric multiregional irradiation assemblies. The solutions may be obtained for: (l) radial and axial temperature distribution; (2) triaxial gamma heat distribution; (3) triaxial thermal flux distribu-tion; (k) triaxial fast-flux distribution; or (5) radius. Heat may be transferred only in the radial direction in all regions except in a single outer coolant region where it is transferred in both radial and axial direction. The physical properties of materials are stored in a subroutine N0ALL in which the properties of viscosity, thermal conductivity, emissivity, thermal expansion, specific heat, and density are expressed as a function of temperature as point pairs. It provides the option of up to 100 pairs as well as the desired order of Legrangian interpolation. Only one coolant region is allowed, the outermost cylinder.

Another recent Oak Ridge National Laboratory development is Wilson's (23) HETRARZ, a F0RTRAN IV code which calculates steady state temperature distribution in RZ geometry. It takes into account heat generation, heat transfer by conduction and radiation, variation of radial gap widths with thermal expansion, and pool boiling boundary conditions.

9

The nodal array is established from coordinates in the RZ plane. Programs which have been published or are in use at other installations:

(1) ARGUS (l8) at Argonne National Laboratory (primarily for flow problems, written in FORTRAN Ii).

(2) TOODEE (llj.) at Phillips Petroleum Co. (primarily for calculation of power excursions, written in FORTRAN IV and FAP).

(3) HOT-1 (19) at Westinghouse - Bettis Lab. (for two dimensional plane and axisymmetrical steady-state heat conduction problems with diagonal boundaries and interfaces).

(4) LI0N (7) at Knolls Atomic Power Lab. (primarily to perform structural analysis of structures at elevated temperatures).

(5) CINDA-3G at TRW., A UNIVAC (third generation) version of CINDA (9), the Chrysler programs for the Aero-space industry.

(6) CINDA-3G (9) for IBM 360 at Phillips Petroleum Co. (7) RATH-MESHER (20) at Los Alamos (primarily a code for generating

nodal systems for use in other heat transfer codes, CINDA, TOSS, HGT etc.)

(8) TRUMP (6) at Lawrence Radiation Lab (Code is written in F0RTRAN II for the IBM 709^ and in F0RTRAN II for the IBM 709^ and in F0RTRAN -^00 for the CDC 3600 and CDC 6600).

Gaski and Lewis (9) have pointed out that the work in this field is either undocumented or underdocumented. Also many programs are propietary or are written for machines not widely available.

At the time this program was undertaken, none of the existing programs were particularly convenient or efficient to use for the heat transfer

10

calculations frequently required in the design and analysis of irradiation capsules, i.e. two-dimension, steady-state heat conduction in multiple axisymmetric regions with internal heat generation, containing gas gaps whose dimensions are altered by thermal expansion. The present study was undertaken in an attempt to fill the need for such a program. To a certain extent, Llewellyn's (13) and Wilson's (23) programs are parallel develop-ments to this work.

11

CHAPTER II

DESCRIPTION OF THE PROGRAM

A. General Procedure (Roadmap) The main program CYL sets the counter NC0UNT at 1 and calls subroutine

PRESET. Subroutine PRESET zeros all the major arrays to prevent data from previous programs entering the problem, reads the data cards, prints out a summary of the input data, establishes the nodal boundaries from the dimen-sions in the input data, puts one real node in each nodal cylinder, calcu-lates the heat transfer coefficients, conductances, of each real and special node, establishes and prints an initial temperature distribution from input data then returns to the main program. The main program calls subroutine STEADY. Subroutine STEADY, by a Gauss-Siedel iterative procedure calculates a new temperature distribution based on the heat balance equations of Chapter III. The iterations are repeated until the convergence criteria are met or the limit of the number of iterations is reached, whichever occurs first. When either of these limits is reached, subroutine STEADY calls subroutine PRESET. The temperature distribution of the cylinder as calculated by subroutine STEADY is stored temporarily. Subroutine PRESET then subdivides each node within each nodal cylinder into four nodes. That is, each cylinder is divided into smaller cylinders which have half the length and half the thickness of the original cylindrical node.

The stored temperature array is used to give the four smaller nodes the same temperature the one larger node had at the end of the first series of iterations. The program then returns to subroutine STEADY where the iterative calculations are again performed on the finer geometry. When the limit of iterations has been reached or the convergence criteria are

1 2

met, subroutine STEADY again calls subroutine PRESET. Subroutine PRESET

then stores the temperature pattern and subdivides the nodes again. Each

time PRESET is called the counter NG0UNT is advanced by one. The number

of nodes that are established between cylindrical nodal boundaries and

the number of nodes that are established between plane nodal boundaries

will be equal to NC0UNT unless one of the control parameters MRS'or MLS

is less than NC0UKT. If WHS is less than NC0UNT, the number of nodes

between cylindrical boundaries will be equal to HRS. If NLS is less than

NC0UNT, the number of nodes between plane boundaries will be equal to ELS.

Each time, after the subdivision of the nodes, the temperature distribution

of the previous iteration is imposed on the new, finer mesh pattern of nodes.

After each such subdivision the program returns to subroutine STEADY for

the iterative calculations. When NC0UKC reaches the value of the larger

of the control parameters (HRS, MLS), control returns to the main program.

Subroutine PICT is called. Subroutine PICT uses the cathode ray tube

plotter to draw a half sectional scale drawing of the cylinder and to

print out a few of the characteristics of the cylinder and the calculations.

Control then returns to the main program. If the convergence criteria

were met on the last series of iteration, subroutine RPLOT and ELPLOT are

called in succession. Subroutine RPLOT uses the cathode ray tube plotter

to draw a series of graphs of temperature vs. radial position at selected

values of axial position. Subroutine ELPLOT uses the cathode ray tube

plotter to draw a series of graphs of temperatures vs. axial position at

selected values of radial position. Control then returns to the main

program which is then complete. If the convergence criteria were not

1 3

met on the last series of iterations in subroutine STEADY the main program

ends without calling subroutines RPLOT and ELPIAT.

B. Setup of Nodal System

The nodal boundaries are established on the first call of PRESET.

The half section of a typical composite cylinder shown in Figure k illus-

trates the input data used to calculate the nodal boundaries. The cylin-

drical boundaries, Rl(l) and R2(l), of each region are extended the full

length of the cylinder and the plane boundaries, ELl(X) and EI2(l), of each

region are extended to the full diameter of the cylinder as shown by the

dotted lines in Figure 5- One region (other than region 1) can be sub-

divided in the radial direction by setting the control parameter MPR equal

to the number of that region. Figure 5 shows region 3 so .subdivided. Any-

one region (other than region l) can be subdivided in the axial direction

by setting the control parameter MPL to the number of that region. Figure

5 shows region 2 so subdivided. These extra subdivisions can be very help-

ful in regions of high thermal gradients.

The nodal regions are established from the surface data by arrays

assembling all the RSI and RS2 values into a singly subscripted R array

and all the ELI and EL2 values into an EL array. The values in these two

arrays are then rearranged in order of increasing size and deleting dupli-

cate values from each array. The R and EL arrays as established at this

time are copied into storage arrays RT and ET to be used in all subsequent

calls to subroutine PRESET. The R and EL arrays are then expanded to

provide a regular node between each boundary. The values in the expanded

Ik

<L

cr cc oc dc oc oc

Fig. k. Typical Cylinder

1 5

TOP OF CYLINDER

Fig. 5- Establishing Nodal Boundaries

1 6

R and EL arrays represent the locations of all the nodes. The odd numbered

values represent special node locations and the even values represent real

node locations.

Nodes are coded according to type. Regular (or real) nodes always

have even numbers for both I and J. Cylindrical special nodes, both

internal and external, have even numbers for I, odd numbers for J. Plane

special nodes have odd numbers for I, even numbers for J. Nodes having

both I and J odd are not used,

C. Calculation of Data on Regular Nodes

The R and EL arrays contain all the information needed to permit

calculation of the size, shape and volume of each regular node. Using

these calculated values, the conductances, Clf C2, C3, C4 are calculated

for each node using equations 3, 6, 10, l6, and 19 from Chapter III.

The heat generation rate for each node is calculated, using equation 21

from Chapter III. On the first call to PRESET (i.e. NC0UNT = l) the

temperature of each real node is set at the initial temperature of the

region in which it is located. On subsequent calls, the temperatures of

the real nodes are obtained from the temporary array TT which contains the

temperature distribution of the preceding series of iterations. The

assignment of temperatures to the new nodes from the old array is controlled

by indexes calculated by integer arithmetic, explained in detail in

Appendix D.

1 7

D. Calculation of Data on Internal Special Nodes

Plane special nodes have no heat transfer in the radial direction,

therefore C1 and C2 are both zero. The value of C3 will be the value of

C4 of the real node just above the C4 will be the value of C3 of the node

below.

Cylindrical special nodes have no heat transfer in the axial direction

therefore C3 and C4 are zero. The value of C^ will be the same as C2 of

the next outward real node and the value of C2 will be the same as Cx of

the next inward real node.

E. Calculation of Data on Internal and External Surface Nodes

The coefficients for heat transfer for all surface nodes, both internal

and external, are calculated. The specific coefficients depend on the type

of heat transfer, conduction, convection, conduction and radiation across a

gas gap or convection and radiation to a sink. An insulated surface node

requires that the conductance which connects that surface to the real node

for both the surface and the real node be zero.

F. Calculation of Temperature Distribution

Subroutine STEADY takes the data established in subroutine PRESET

and calculates a new temperature for each node of the array based upon the

estimated temperature of surrounding nodes and the coefficients of each node

using the equations of Chapter III. When the new temperature has been

calculated it is inserted into the array as the next "best estimated

temperature". This iteration technique is continued until the three

criteria for convergence have been met or the limit of the number of

1 8

iterations set by the input data is reached. The three criteria for conver

gence are:

(1) First Convergence Test

The largest fractional change in temperature produced by a

single calculation during one iteration, TEST, is compared with

an input control value, DTM. When TEST is less than DTM the

first convergence test has been met. Experience shows that DTM

should be of the order of 0.00001.

(2) Second Convergence Test

The sum of all the absolute values of the fractional changes

in temperature produced by the calculations of one iteration, SUM,

is compared with the input control value, TOTAL. When SUM is less

than TOTAL the second convergence test has been met. Experience

shows that TOTAL should be of the order of 0.001.

(3) Third Convergence Test

The sum of the heat transferred from all surface nodes to a

sink, CUE, is compared with the calculated value of total heat

generated within the cylinder, HO- QTEST is the fractional

variation of CUE from HG. SPEC is calculated from the input

control parameter FAC by the equation SPEC = FAC/WCgf. When

QTEST is less than SPEC the third criterion for convergence has

been met. Experience has shown that a value of 0.005 for FAC

will give results that deviate from analytical solutions by less

than 0.1% (or 1/2 degree at 500 C).

1 9

The program allows the use of an "overrelaxing" method similar to the

one suggested by Lapidus (11). This method can hasten the convergence of

the iterations by using the two overrelaxation controls TENT and TINT. The

overrelaxation causes the new value of T (l,J) calculated by the iteration

to be set at a value slightly overcorrected. That is:

T(I,J) = TNEW + TEN* (TNEW-T0LD)

Where: T0LD is the previous temperature of node (l,j)

TNEW is the calculated new temperature of node (l,j)

T(l,J) is the value entered into the matrix as the new

temperature of node (l,J)

TEN is the overrelaxation factor calculated by the equation

TEN = TENT + TINT* NC0

Values for TENT of 0.3 and for TINT of 0.1 have worked very well.

G. Output

The standard outputs always printed by the program are:

(1) Numbers of the data cards read.

(2) All input data arranged as illustrated in Appendix E.

(3) Flags to indicate the start of PRESET calculations and the start

of the iteration calculations in STEADY for each value of NC0UNT.

(b) Final assignments of nodal boundaries both radial (R) and axial

(EL).

(5) The calculated total heat generated (S Q,) total convective

transfer conductance (llh A), and an average AT between outer

surface and heat sink.

20

(6) The initial temperature distribution established in PRESET

for each value of NC0UNT.

(7) An array of NNT (l,J) for each value of NC0UNT.

(8) Values of TEST, DTM, and ITER at each tenth iteration for

•which TEST < DIM.

(9) Values of SUM, TOTAL, and ITER at each tenth iteration for

which SUM < T0TAL.

(10) Values of QTEST, SPEC, and ITER at each tenth iteration for

which QTEST < SPEC.

(11) Final temperature distribution at the end of each series of

iterations, illustrated in Appendix E.

(12) A cathode ray tube plot showing the shape and dimensions of the

cylinder, the problem number, the materials composing the various

regions, the number of iterations performed for each value of

NC0 and the values of TENT and TINT used. Shown in Appendix E,

Fig. 10.

(13) Cathode ray tube plots of temperature versus radial position and

of temperature versus axial position. Shown in Appendix E,

Fig. 11 and Fig. 12.

In addition to the standard output, optional outputs may be called

by use of the appropriate output option parameters which are explained

in the list of F0RTRAN Names and Symbols, in the comment statements

of the F0RTRAN list and in Appendix B Input Format.

21

CHAPTER III

HEAT TRANSFER EQUATIONS

In this program heat transfer is calculated "by using a heat balance

equation. The program sets up a two dimensional array for calculation of

the heat transfer through the cylinder. Each regular node represents a

hollow cylinder (except the centerline node which is a solid cylinder).

All heat transfer from a regular node is made to a special node. The

size of a regular node varies depending on the initially specified nodal

boundaries and the number of times the nodes have been subdivided. Special

nodes which are not surface nodes merely serve as connectors between two

adjacent regular nodes. A special node which is truly a surface node may

provide any of the following conditions.

(a) Insulated (no transfer)-

(b) Convection to a sink.

(c) Radiation to a sink.

(d) Radiation and convection to a sink.

(e) Conduction between two regular nodes.

(f) Conduction and radiation across a gas gap.

In every case a true surface node will be conducting heat to (or from) at

least one regular node. All sinks are assumed to be at a constant tempera-

ture no matter how much heat is transferred (i.e. infinite sinks).

A. Regular Nodes

The three dimensional cylinder is symmetrical around the centerline.

Therefore, the heat transfer problem is two dimensional. Each regular

node is assumed to be a point, but exhibiting properties (i.e. temperature,

22

volume, mass, heat generation, etc.) of the surrounding material.

The nodal volxime is assumed to be small enough to be able to

represent the heat transfer by two orthogonal vectors one in the radial

direction and one in the axial direction. Conduction then occurs to the

node (I, J) from the four special nodes (I, J+l), (I, J-l), (l+l,j) (i-l, J).

The convention is employed that heat transferred to the node is positive,

away from the node is negative.

(l) Conduction between regular node (I, j) and special node (I, J+l).

If R(j+l)/R(j) ^1.5 the difference between the logarithmic

mean and the arithmetic average is less than 1°j0* For hollow

cylinder nodes (i.e. when J > k) this ratio is always less

than 1.5 (usually less than k/j>). For the solid cylinder nodes

(i.e. J = 2) R( J+l)/R( J) = 2 and equation (l) becomes:

The equation for radial conduction in a cylinder is:

(1)

qj, = K 2jti Qm 2 [T(I, J+l) - T(I,J)] (2)

Let = 9-06V7 0 J = 2 (3)

Then « When J > 2 equation (l) becomes:

l 2 R(J+1) - R(J)

(5)

23

Let J > 2 (6)

Then equation (4) satisfies equation (5) as well as equation (2)

Conduction between regular node (I, J) and special node (I, J"-l):

QLa = k 2itZ [t(i, j:1) - t(l,j)] fbn ITjT (7)

When

When

J = 2; q3 = 0

Ml ^ 1-5 (8)

Therefore equation (7) becomes:

q2 = K 1R(J) + R(J-

fR(d) - R( J-l)~j

|R(J) + R(J-l)j

L* J Let Cs = KX R(J) + R(J-l)

2 -p(J) - R(J-l)]

t(i, j-l) - t(i, jjj

(9)

(10)

Then equation (9) becomes:

q2 = c2 [t(I,j-1) - t(i,j)J

Conduction between node (I, j) and special node (i-l, J).

The linear conduction equation is: K A AT

(11)

<13 = X (12)

a = * £r3(j+1) - r2(j-1)] (13)

|r(J+1) - R(j-l)] ' [r(J+1) + r(j-1)J A = it

2k

A = 2jt R(j) • [r(J+1) - R(J-l)] (14)

(15)

Let C3 = K 2rt r(j) [r(J+1) - R(J-l)] EL(K) - EL(I-l) (16)

Then equation (15) becomes:

q* = C3 [T(I,J-1) - T(I,J)] (17)

(4) Conduction between node (I, J) and special node (1+1, J)

(18)

let Go. - K 2jtR(f} [ R ( J + 1? 7 , N l e Z ^ ~ EL(I+1) - EL(I) (19)

then equation (l8) becomes:

q4 = C4 [t(I+1,J) - t(i, j)] (20)

(5) Rate of heat generation within node (l,j):

Q = G - p - V (21)

V = A ' Si

V = 2n r(j) [r( J+l) - r( J-l)j |EL(l+l) - EL(l-l)J

(22)

(6) At steady state the energy balance gives:

]Cq = 0

or

q i + q s + q3 + a 4 + Q = o

2 5

C i

C 3

T(l,J+l) - T(l,j)

T(l-1, j) - T(l,j)

+ C2

+ C4

T(I, J-l) - T(I, J)

T(l+1, j) - T(I,J)

+

+ Q = 0

(23)

Solve equation (23) for T(l,J):

T(I j) = Q + cit(t, j+l) + C3T(I,J-1) + C3t(l-l, J) + C4T(l+l,j)

C i + c2 + c3 + c4

(7) Translating the equations into F0RTRAN language:

VOL = PIS * R(J) * DELL * DELR

QW(I, J) = RHO(K) * Q(K) * VOL

Cj,(l,2) = 9.0647 * CON(K) * DELL

Ci(ljJ) = PI2 * CON(K) * DELL * DILR * ROM

C2(L, J) = RI2 * CON(K) * DELL * DILR * RIM

C3(I, J) = RI4 * CON(K) * R(J) * DELR * DILL

C4(l,j) = CS(I,J)

ADEN(I, j) = Cl(l,j) + Cs(l,j) + C3(I,J) + C4(l,j)

ANUM = ci(l,j) * t(I,J+1) + C3(I,J) * T(l,J-l) +

C3(l,j) * t(I-1, J) + C4(I,J) * T(I+1,J)

tnew = (qjff(i,j) + mm) / aden(i, j)

B. Interior Special Nodes

(l) Cylindrical Nodes

A cylindrical special node transfers heat only from one

regular node to another in the radial direction. The energy

"balance equation is:

(24)

(22?)

(2UF)

(3F)

(6f) (10F)

(i6F)

(19f)

(24F)

2 6

Qi + Is = 0 or

Ci(l, J) [T(I,JH) - T(I,J)] + C2(l,j) [T(I,J-1) - T(X,J)] = 0

solving for T(l, j):

T(l,j) = C ^ J ) T(l,J+l) + C2(l,j) T(I,J-1) ( 2 5 )

Cx(l, J) + Ca(l,j)

Ci(l^J) is the conductance for heat transfer from the regular

node (I, J+l) to the special node (I, J); therefore,

CX{1,J) = C2(l,J+l) (26)

Similarly:

C2(l,j) = Cx(l,j-i) (27)

(2) Plane Nodes

A plane special node transfers heat from one regular node to

another in the axial direction. By analog;/- with equations (25),

(26), and (27): rp/j Tv _ Ca(l,J) T(l-1, J) + C4(X, J) T(I+1, J) J/ = (28)

Ca(l,J) + C4(l,j)

C3(I,J) = C4(l-l,j) (29)

C4(l,j) = C3 (1+1, J") (30)

C• Surface Nodes

Each surface node will "be conducting to (or from) one regular node.

The conductances Cx, C2, C3, or C4 will "be determined as for the interior

special nodes in Paragraph B above. The transfer in the other direction

is as follows:

27

(1) Insulated Nodes (NT = l)

In the PRESET calculations, the conductance for heat transfer to

(or from) an adjacent regular node to an insulated node is set equal

to zero. During the iterative calculations in subroutine STEADY, the

temperature of the insulated node is calculated by an equation which

maintains the same temperature gradient across both halves of the

regular node. For example, if the top surface is insulated, this

equation is used:

T(I,J) = T(2,J) * 2 - T(3, J) (31)

This comes from the equation:

T(1,J) - T(2,J) + [T(2,J) - T(3,J)] (31a)

(2) Convection to a sink (NT = 2)

The rate of heat transfer to the sink is:

qg = hA [T(I,J) - T G] (32)

If the surface is a plane surface:

A = 2JRR(J) |R(J+1) - R(J-1)J (Ik)

If the surface is a cylinder:

A = 2tcR( J) [EL(I+1) - EL(I-I)] (33) L. J

The energy balance equation gives:

28

For a plane surface:

C3(I,J) [T(I-1,J) - T(I,J)] + hA(l,j) [Ts - T(I,J)] = 0

T(I,J) = C3(I,J) T(I-1,J) + hA(l,j) T s

CS(I,J) + hA(l,j) (35)

or

C4(l,j) |T(I+1,J) - T(I,J) ] + hA(l,j) £RG - T(I,J)J = 0

T(I,J) = C4(l,j) T(I+1) + hA(l, J) T s

C 4 ( J ) + hA(l,J) ( 3 6 )

For a cylindrical surface:

Ca(l,j) [T(I,J-1) - T(I,J)J + hA(l,j) [Tg - T(I,J)J = 0

and

T(I,J) = C,(l,j) T(L, J-l) + hA(l,j) Tg ( j Ca(l, J) + hA(l, J)

(3) Radiation to a Sink (MT = 3)

The heat transfer equation for radiation to a sink is:

q^. = AFsFe.a (et - e%) (38)

where:

Fg = surface shape factor (input data)

F = emissivity factor (input data)

and d3 are absolute temperatures (deg K or deg R)

of the node and the sink.

o = Boltzmann Constant. The program automatically uses the

2 9

appropriate unit:

If KA = 1 BZM = cr = 1.366 x 10 - 1 2 watts/cm3 (deg K)4

If KA = 2 BZM = c j = 1.714 x 10"9 B T U P e r -hr

ft2 (deg R)4

To put the calculation in the same form as the others we note

that:

et - e% = (0? + eg) (ef - e|)

= (ef + eg) (ea + 9S) (ex - ea)

Let TRAD = (0? + eg) (6x + 0 2) (39)

and RA(I, J) = AF F o (40) S 6

also note that:

0X - 02 = Tx - T3

Equation (38) then becomes:

TRAD * RA(l,j) * |t(I,J) - Tg J (4l)

The heat balance equation is:

% " 1r = 0 (42)

If the surface is cylindrical, equation (42) becomes:

C2(L, J) * |T(I, J-l) - T(I,J)J + (TRAD) RA(L,J) * - T(L,J)J- 0 m

Solving for T-(l, J):

T(I, J) = Ca(l, J) * T(l, J-l) + TRAD * RA(I,J) * Tg ^

C2(I, J) + (TRAD) * RA(I,J)

30

If the surface is the top surface:

T(1,J) = C4(l,j) * T(2,J) + (TRAD) * RA(1,J) * T

C4(l,j) + (TRAD) * RA(1, J) ^ ^

If the surface is the "bottom surface:

T(MAXI, j) = CS(MAXI, j) * T(MAXI-1,J) + (TRAD) * RA(MAXI, J) * Tt

C3(MAXI, J) + (TRAD) * RA(MAXI, J) 0*6)

(4) Conduction and radiation across a gas gap (NT = 4 or 7),

see Figure 6.

The equation for conduction through the gas is:

q = K A. AT g c - s — m

Let C = K A 6 (48)

Then q g c = C g AT

The program provides for calculating K g from the

equation:

K = K 0O 1/2(0! + 08)

00 for noble gases:

a ~ 0.62

the input data required for this calculation are

C0NTZ = K£

TZER0 = 60

AEX = a

%

(50)

3 1

top of cylinder

Fig. 6. Gas Gap

32

0i and 03 are the absolute temperatures of the surface nodes

on each side of the gas gap. At each iteration they are

changed to the values of the temperature obtained on the

previous iteration.

6 = r a(l + ofe(T2 - T r e f) - rx(l + g^Tx - T r e f) ) (51)

where a is the coefficient of expansion of the material

adjacent to the respective surfaces, rj, and r3 are as shown

in Figure 6, and T ^ is the temperature at which and r2

were measured (usually room temperature).

The radiation equation is:

% = A FeFg a (ef - 0|) (52)

In this case, the gap is so small that we assume:

F = 1.0 s

Let RA(I, J) = A F a (53) e

and TRAD = (0X + 02) (ef + 6%) (54)

For nodes on the outer surface of the gas gap (NT ~ 7):

qr = (TRAD) RA(I,J) J T ( X , J ~ 2 ) - T(l,J)J (55)

Note: PRESET always establishes one regular node

between the two gas gap surfaces but the

calculation does not use that node, therefore,

the two surfaces are two J units apart.

For nodes on the inner surface of the gas gap (NT = 4):

qr = (TRAD) RA(l, j) |T(I,J+2) - T(l,J)J (56)

3 3

The energy balance equation is:

«c + V + «r - 0 ( 5 7 )

For the outer surface (ET = 7):

Cx(l,j) |t(I, J+l) - T(l, J)j + Cg |t(I, J-2) - T(I,J)J +

(TRAD) RA(l,j) |T(I,J-2) - T(l,j)J = 0 (58)

Solving for T(l, J):

T(l, J) = Cx(l,j) T(l,J+l) + (C ) T(l,J-2) + (TRAD) RA(l,j) T(l,J-2) . £2 1 Cj(l,j) + c + (TRAD) RA(I,J)

g (59) Similarly for the inner surface (NT =4):

T(I,J) = C2(l,j) T(l,J-l) + (C ) T(l,J+2) + (TRAD) RA(l,j) T(lj,J+2) . $ . C2(L,J) + c + (TRAD) RA(I,J)

g • (60) For a horizontal gas gap, equation (51) becomes:

a = (L - EL2)(1 + (^(Ta - T r e f) ) - (L-ELL)(l + Oi(Ts - T r e f) )

(51a)

.and the equations comparable to (59) and (60) are:

T(L, J) = C4(I,J) T(L+1, J) + (C ) T(L-2, J) + (TRAD) RA(L,J) T(L-2,J) § C4(L,J) + C + (TRAD) RA(I, J)

g (59a) T(I,J) = C3(I,J) T(I-1,J) + (C ) T(l+2, J) + (TRAD) (RA(I,J) T(l+2, j) £2 .

C3(I,J) + c + (TRAD) RA(I,J) §

Convection and radiation to a sink (NT = 6).

The energy balance gives:

% - as - Or = 0

3^

The equations of paragraphs 2 and 3 above can be combined to get:

For cylindrical surface:

T(I, J) = C2(I,J) T(I,J-1) + [hA(l,j) + (TRAD) RA(l,j)J Tg

Ca(l,J) + hA(l,j) + (TRAD) RA(I, j)

(kka)

For the top surface:

T(1,J) = C4(1,J) T(2,J) + hA(l,j) + (TRAD) RA(1,J) T , J s C4(1,J) + hA(l,j) + (TRAD) RA(l,j)

For the bottom surface:

T(MAXI,J) = Ca(MftXI,J) T(MAXI-l,j) + [hA(l,j) + (TRAD) RA(l,j)j Tg

Ca(MAXI,j) + hA(l,j) + (TRAD) RA(l, j)

(46a)

Conduction between two nodes (NT = 5)»

The surface is handled like any other special node between

two surfaces as described in Section B.

3 5

CHAPTER IV

COMPARISON WITH ANALYTICAL SOLUTION

As a check on the operation of the program, a one dimensional (radial)

problem was set up and calculated both by hand and by the computer program.

The test problem is shown in Figure 7* Equations for the analytical solu-

tion follow:

The steady state heat transfer equation is:

where G is the heat generation rate (watts/gm) and K is the thermal

conductivity (watts/cm3 deg C).

In one-dimensional (radial) cylindrical geometry this reduces to:

The solution to this differential equation is:

T = A1 + B 0nr - SjSf. (3)

Where A' and B are constants of integration. At the surface of the

cylinder, T6 can be calculated by the equation:

Q = h Ae(T6 - Ts) (if)

where: As = 2jt r6 I cm2

6

Q, = ^ q^ watts i=l

q. = j r.2 - ^ p.G. watts

36

i <L

TOP OF CYLINDER 1

1

To-«c— )

A.-2 x=3 X = 4 x=5 x=6

X- cm

lii H UJ

K UJ H

To

SPA

CE X CL <

tr e> FU

EL X

Q. < ce o FU

EL X a < CC o

ro CM — -

Z o

z o O o

z o

z o

_

o UJ a:

CD UJ CC

O UJ Q:

CD UJ CX

o UJ a:

o UJ a:

rO r4 r2 r3 r4 r5 r6 0.0 0.16 0.28 0.35 0.50 0.57 0.635

ALL DIMENSIONS IN cm

Fig. 7« One Dimensional Problem

3 7

h = 1.0 watt/cm deg C

T = 100°C s

A = 1 cm

Calculations are summarized in Table 1.

Calculated surface temperature:

Analytical: Ts = 201.9l)-6OC

Computer: T6 = 201.9^59°C

Equation (3) holds in each hollow cylinder. For the innermost subcylinder

(the solid cylinder) the Rvalue disappears. The boundary conditions for

subcylinder i are:

BC No. 1 r = r. ; T = T. 1 1

BC No. 2 r = r. : j=i-l

P. = K. /3T\ ^ = / = 1 1 W r . . ^ A(i=l) = A(i-1) ™

i.e. the flux at the inner boundary of cylinder i is equal to the total

heat flow rate crossing the boundary divided by the area of the boundary

and is also equal to minus the gradient of the temperature at the same

surface multiplied by the thermal conductivity.

Simplifying equation (5) gives:

* v K i A ( i - D . 1-1

TABLE I. One Dimensional Problem Calculations - (Heat Generated and Heat Transfer Area)

• 1 = 1 2 . 3 4 5 6 Units

r. 1 0.16 0.28 0.35 0.50 0.57 0.635 cm

r? i 0.0256 0.0784 0.1225 0.250 0.3249 0.4032 cm3

r2 i-1 0.0 0.0256 0.0784 0.1225 0.250 0.3249 cm2

- ri-l 0.0256 0.0528 0.0441 O.1275 0.0749 0.0783 cm2

Pi 0.01 2.0 5.0 2.0 5.0 2.0 gm/cm3

G. i 0.01 1.0 216.73 1.0 216.73 1.0 •watts/gram

0.000008 0.331752 150.1333 0.8011 254.9884 0.4921 •watts

0.000008 O.33176 150.4651 151.2662 406.2546 406.7467 •watts

Computer calculated (HG) = 406.746 •watts

A. l 1.0053 1.759 2.199 3.142 3.581 3.9898 cm3

3 9

Apply BC No. 1 to equation (3):

G.p.r." T. = A'. + B.J07Z r. - ^ 1 1 1 1 4K. (7)

rearranging gives:

A'. = T. 1 1 G 0 r 3

B.&n r. + i T i 1 1 TFkT 1 substituting into (3) gives;

( 7 a )

r. G.p. T = T. - B.0n-i + -r^i fr.2 - r2

1 1 r 4K. \ 1 (8)

Apply BC No. 2 to equation (8)

3T\ fi dri r(i-l)

r(i-l) - 2r /.

G .P. - Q / .

(i-1) T k 7 ~ KjA(i-l) ( 9 )

Solve for B. 1

Bi • r (i-D Gipi

2K. K. A/. t \ 1 (l-l) (10)

Noting that:

A. _ = 2it r,. -n 4 l-l (i-l) (J8 = 1)

Bi * /. _ % G. p. Q/. ..x (i-l; 1 1 _ (i-1)

2K. 1 2jt K. 1

Substituting into Equation (8) gives:

(11)

T 1 r I gjt IC G. p.r /. 1 1 (^lil 2K.

G.p. (r2 - r2) 1 i 1 J

40

Equation (12) is used to calculate all the nodal "boundary temperatures.

Substitute the known value of T6, calculate T5, then with the known value

of Tb calculate T4, etc. Details of the calculations are shown in Table II.

The test problem has been calculated by the computer program a number

of times. The temperature of the points all having the same J value were

constant within 0.01 degree by the time convergence reached acceptable

values by all three convergence tests. The values of the temperature at

the boundaries between regions agreed with the analytical solution within

0.01$ in cases where five subdivisions were made and FAC was set at 0.005«

Tables III, IV, V, VI and VII summarize a series of computations of

the one dimensional problems which were made to explore the effects of the

overrelaxing factor TEN, the advantages of continued subdivisions of the

nodes, and the variations to be expected in the convergence test functions.

Problem 100, Table III, used no overrelaxing factor (i.e. TEN = 0.0).

The iterations did not meet all the convergence tests until the fifth

division. The line LIMIT shows the number of iterations allowed for each

subdivision.

Problem 125, Table IV, used TEN - 0.25 with the same limit on the

number of iterations used in Problem 100. All the convergence criteria

were met with fewer iterations. The temperatures at the boundaries between

regions fluctuated around the analytical solution but remained within O.yjo

at all times.

Problem 145, Table V, used TEN = 0.45 but was restricted to fewer

iterations. All convergence criteria were met by the end of the second

nodal division iterations and from that time on the boundary temperatures

stayed within 0.2$ of the analytical solution.

TABLE II. One Dimensional Problem Calculations - (Temperatures at Nodal Boundaries) 1 " - — • - —• —

• 1 = 1 2 " 3 4 5 6 Units

2 r. 1

G. 1

Pi K. l

0.0256

0.01

0.01

0.002

F. * 0.00032

fa (rj/ri_i) "

Q d - 1 )

Q(i-l)/2«Ki ° GiPiri-l/2Ki * M.** 0

T. 1

l-l

655.5068

655.5071

0.0784

1.0

2.0

0.1225

2 1 6 . 7 3

5 . 0

0 . 2 5 0

1.0

2.0

0.3249

216.73

5-0

0.4032

1.0

2.0

cm

watts/gram

grams/cm3

0.35 0.02 0.35 0.02 0.35 watts/cm

0.0754 597.362 0.1821 1014.567 .1119 deg C

0.5596 0.2231 O . 3 5 6 6 0 . 1 3 1 0 0.10795 -

0.000008 O . 3 3 1 7 6 150.4651 1 5 1 . 2 6 6 2 406.2546 watts

O .OOOOO36 2.640 68.4208 1203.739 184.736 deg C

O . O 7 3 I 2123.95 0.350 6 7 7 2 . 8 1 .9283 deg C

0.0409 -473.264 2 4 . 2 7 4 -729.548 19.842 deg C

655.472 . 531.375 5 0 6 . 9 1 8 9 221.8999 201.9^6 deg C

655.5068 655.472 531-375 5 0 6 . 9 1 8 9 221.8999 deg C

42

TABLE III. One Dimensional Problem Computer Calculations Problem 100

NUMERICAL SOLUTION ANAL. I DEV. I DEV. NC0UNT 1 2 3 4 5 SOL.

ITER 200 400 800 1600 1240

To 673-71 666.39 660.84 656.88 655.66 655.507 0.023

Ti 673.69 666.33 660.80 656.87 655.65 566.507 0.022

Ts 673.52 666.18 660.07 656.82 655.62 655.472 0.022

T3 543.20 538.51 534.87 532.23 531.49 531.375 0.022

T4 518.04 513.57 510.16 507.70 507.03 506.919 0.021

TS 224.63 223.78 222,, 84 222.12 222.02 221.900 0.052

TS 204.30 203.56 202.75 202.13 202.04 201.946 0.049

QTEST 0.023 0.0158 O.OO79 0.00182 O.OOO987 SPEC 0.005 0.0025 O.OOI67 0.00125 0.001

TEST 0.000268 0.000041 0.0000107 0.0000037 0.000003 DTM 0.001 0.001 0.001 0.001 0.001

SUM 0.0069 0.00526 0.00317 0.00164 0.00137

TOTAL 0.02 0.02 0.02 0.02 0.02 TEN 0.0 0.0 0.0 0.0 0.0

LIMIT 200 4oo 800 1600 3200 1

Input Data: FAC = 0.005; DTM = 0.001; T0TAL =•• 0.02

43

TABLE III. One Dimensional Problem Computer Calculations Problem 100

NUMERICAL SOLUTION ANAL. % DEV.

NC0UNT 1 2 3 4 5 SOL. % DEV.

ITER 160 310 46o 510 500

to 654.80 657.46 656.63 656.02 655.53 655.507 0.0035

ti 6p4.76 657.48 656.63 656.01 655.53 655.507 0.0035

t3 654.70 657.44 656.60 656.OO 655.50 655.472 0.0043

t3 530.89 532.68 532.14 531.77 531.43 531.379 0.0096

t4 506.57 508.16 507.64 507.29 506.97 506.919 0.010

t5 221.47 222.19 222.10 222.04 222.00 221.90 0.045 • !

t6 201.57 202.91 202.11 202.07 202.04 201.946 0.047

Q TEST 0.0037 0.0024 0.00166 0* 00122 0.00091

SPEC 0.005 0.0025 0.00167 0.00125 0.0010

TEST O.OOOO95 0.00002 0.000010 0.000006 0.000005

DTM 0.0001 0.00005 0.000025 0.000012 0.000003

SUM 0.00208 0.00204 0.0023 0.0027 0.0028

TOTAL 0.01 0.01 0.01 0.01 0.01

TEN 0.25 0.25 0.25 0.25 0.25

LIMIT 200 400 800 1600 3200

Input Data: FAC = 0.005; DTM = 0.001; T0TAL =•• 0.02

TABLE V. One Dimensional Problem Computer Calculations Problem 145

NUMERICAL SOLUTION ANAL. °!O DEV.

NC0UNT 1 2 3 4 5 SOL.

°!O DEV.

ITER. 100 150 4oo 550 530

T0 656. 4 65^.97 656.62 655.96 655.45 655.507 0.009

Ti 635.9 654.98 656.62 655.96 655.45 655.507 0.009

T2 635.7^ 654.96 656.59 655.93 655.42 655.472 0.007

T3 518.18 531.44 531.18 531.7^ 531.38 531.375 0.0009

T4 494.58 507.06 507.68 507.23 506.93 506.919 0.002

T5 218.14 222.11 222.14 222.04 222.01 221.900 0.0045

T6 19k.66 202.15 202.15 202.07 202.04 201.946 0.0047

QTEST 0.0323 0.00203 0.00202 0.00121 0.00092

SPEC 0.005 0.0025 O.OOI67 0.00125 0.001

TEST 0.000783 0.00030 0.0000082 0.0000058 0.0000047

DTM 0.001 0.0005 0.0005 0.00025 0.000125

SUM 0.0176 0.00187 0.00291 0.00248 0.00254

TOTAL 0.02 0.02 0.02 0.02 0.02

TEN 0.45 0.45 0.45 0.45 0.45

LIMIT 100 200 400 800 1600

Input Data: FAC = 0.005; DTM = 0.001; T0TA1 =0.02

45

Problem l6o, Table VI, and Problem 185, Table VII, show that not

much improvement is gained by the larger values of TEN.

In the one dimension (radial) case, the emphasis has been on the

boundary temperatures because they are readily calculated. The fact that

the program gives very accurate temperatures early is encouraging. However,

the temperatures within the regions are not as accurate in the early stages

because of the lumping effect. Division of the nodes into 4 or 5 nodes

per node boundary will give a much more accurate picture of the temperature

within the undivided materials.

46

TABLE VI. One Dimensional Problem Computer Calculations Problem 160

NUMERICAL SOLUTION ANAL. °lo DEV. SOL. °lo DEV.

NC0UNT 1 2 3 4 5

ITER 80 160 320 390 370

To 646.5 658.85 657.68 656.70 656.08 555.507 0.088

Tl 646.2 658.87 657.69 656.71 656.08 555.507 0.088

645.89 658.86 657-67 656.59 656.06 555.472 0.090

524.77 533.66 532.86 532.23 531.81 531.375 0.082

T4 500.69 509.09 508.31 507.72 507.34 p06.919 0.082

Ts 219.81 222.46 222.29 222.17 222.12 221.900 0.098

Ts 200.08 202.43 202.28 202.18 202.14 201.946 0.094

QTEST 0.0182 0.00478 0.00331 0.0023 0.0019

SPEC 0.01 0.005 0.00333 0.0025 0.002

TEST 0.000653 0.000047 0.000017 0.00001 0.0000108

DTM 0.002 0.002 0.002 0.001 0.005

SUM 0.0131 0.0044 0.00415 0.00447 0.00453

TOTAL 0.05 0.05 0.05 0.05 0.05

TEN 0.6 0.6 0.6 0.6 0.6

LIMIT 80 160 320 640 1280

Input Data: FAC = 0.01; DTM = 0.002; T0TAL.= 0.05

47

TABLE VII. One Dimensional Problem Computer Calculations Problem 180

NUMERICAL SOLUTION ANAL. % DEV.

NC0UNT 1 2 3 4 5 SOL.

ITER 60 150 260 350

To 656.91 657.03 656.50 656.17 655.507 0.101

ti 656.91 657.08 656.60 656.22 655.507 0.108

t2 656.86 657.09 656.60 656.21 655-472 0.112

t3 532.32 532.53 532.11 531.87 531.375 0.093

t4 507.92 508.06 507.62 507.38 506.919 0.091

t5 221.84 222.17 222.09 222.04 221.90 O.O63

Te 201.89 202.19 202.11 202.06 201.946 0.056 Q.TEST 0.00054 0.0024 0.0016 0.0012

SPEC 0.005 0.0025 0.00166 0.00125 TEST 0.000036 0.000043 0.000016 0.000012

DTM 0.0005 0.00025 0.000125 0.000063 SUM 0.00077 0.00419 O.OO378 0.00406

TOTAL 0.01 0.01 0.01 0.01

TEN 0.8 0.8 0.8 0.8

LIMIT 100 200 400 800

Input Data: FAC = 0.005; DTM = 0.0005; TOTAL = 0.01

48

CHAPTER V

USER'S GUIDE

1. Input Preparation

It will be a help in preparing the input data cards if a sketch of

the cylinder is prepared first. Since the overall cylinder must be called

region 1, it must include all materials including cladding. Wo limit is

imposed on the amount or shape of any region except that it must be able

to be approximated by a cylinder (either solid or hollow) concentric with

the overall cylinder.

If one region is expected to have a much higher temperature gradient

than the rest, it should be chosen for the double subdivision.

The initial guess for the temperature of the regions is not highly

critical. However, deviations of several hundred degrees between initial

guess and final temperature can cause slow convergence.

2. Convergence Controls

FAC is the most accurate and sensitive control to use. A value of

0.005 is not unreasonable to expect. The answers should be accurate to

within less than 0.1$ (0.5 degree at 500°). To get the same accuracy,

DTM needs to be about 0.00001 and TOTAL about 0.005.

3. Overrelaxation

The optimum overrelaxing factor cannot be predicted with any accuracy.

It is suggested that the first attempt at solution be made with a low value

of TENT. If convergence is too slow then successively larger values of

TENT and TINT can be tried. It should be noted that the iterations tend

to become unstable as the value of TEN approaches 1.0.

49

Output Options The output options can be helpful in diagnosing problems that do not

converge properly. However, indiscriminate use of the options can generate

tremendous quantities of printout paper. It is recommended that normally

no output option other than 11 be used. Option 11 is not really necessary

but it does provide for a fairly detailed survey of the iteration process.

Option 6 is not an output control. It arbitrarily sets the outer cylindri-

cal surface at a temperature calculated from the total heat generated and

the sum of the convactive conductances. It is useful only on one dimen-

sions 1 problems and problems where there is little axial variation in

surface temperature.

5- Control Cards

Control cards vary so much with the installation, and with the

preferences of the user, that no guide to control cards will be attempted

here. The user should consult his computer center personnel for guidance

if he is not familiar with the system. The control cards must make

provision for use of the cathode ray tube plotting equipment if the graphs

are desired. If no plotting is desired, the cards for the subroutines

RPL0T, ELPL0T and PICT should be left out of the F0RTRAN deck and the

three cards v/hich call these subroutines should be removed from the main

program CYL.

50 >

CHAPTER VI

DISCUSSION

The program has a number of limitations which could be reduced with

some further work.

The restriction to constant physical properties could be removed

by including another subroutine, for example, one similar to Llewellyn's

(10) N0ALL, which contains data on the temperature dependence of the

physical properties of materials normally used in capsule development

work. This subroutine could be called from PRESET to adjust the

coefficients if the temperature of any of the materials varied signifi-

cantly from the previous calculation. Alternately, the input format

could be changed to allow input of data on the temperature dependence of

the physical properties and to introduce equations for calculating the

physical properties at node temperature.

Provision could be made to calculate the convective transfer from

data on the coolant flow and take into account the temperature rise in

the coolant.

The program would be more useful if provisions were made to adjust

the heat generation rates in accordance with some given (or calculated)

neutron flux distribution

Modification of the program for calculating temperature distributions

under transient conditions should be accomplished.

The overrelaxation process should be studied in more detail to attempt

to establish better guidelines for its use. Other means of speeding

convergence, such as extrapolation, could be tried.

5 1

The dimensions of the arrays could be increased to allow for more

detailed division of the nodes or to allow more regions and/or gas gaps,

within the limits of the memory of the machine.

227

REFERENCES

1. Bag-well, David, "TOSS - An IBM-7090 Code for Computing Transient or Steady State Temperature Distributions," Union Carbide Nuclear Company Oak Ridge, Tennessee, K-1494, December 1961.

2. Bagwell, David, SIFT (Simplified Input to TOSS), Union Carbide Nuclear Company, Oak Ridge, Tennessee, K-3.528, July 6, 1962.

3. Bagwell, David, Supplement to SIFT Simplified Input to TOSS, Union Carbide Nuclear Company, Oak Ridge, Tennessee, Unpublished Memorandum, March 9, 1965-

Brunell, R. P., HEATING and HEATING2, Atomics International, San Diego, California, AI-64-MEMO-177, September 1965.

5. Dusinberre, George Merrick, Heat Transfer Calculations by Finite Differences, International Textbook Company, 1961.

6. Edwards, Arthur L., TRUMP: A Computer Program for Transient and Steady State Temperature Distributions in Multi-Dimensional Systems, Lawrence Radiation Laboratory, UCRL-14754, February 24, 1966.

7. Fischer, W. W., LI0N TALES A User's Manual for the LI0N Thermal-Struc-tural Evaluation Code, Knolls Atomic Power Laboratory, Gener.-.l Electric Company, KAPL-M-b533> July 31, 1967-

8. Fowler, T. B. and E. R. Volk, Generalized Heat Conduction Code for the IBM-704 Computer, Oak Ridge National Laboratory, ORNL-2754, October 19, 1959-

9. Gaski, J. D. and D. R. Lewis, CINDA: Chrysler Improved Numerical Differencing Analyzer, Chrysler Corporation Space Division, TN-AP-66-157 April 30, 1966.

10. Katsma, K. R. and M. L. Uptmor, IBM 560/75 Version of CINDA-3G, Phillips Petroleum Company, Atomic Energy Division, Informal Memo, May 19, 1969.

11. Lapiaus, Leon, Digital Computation for Chemical Engineers, McGraw-Hill 1962.

12. Liguori, R. R. and J. W. Stephenson, The HEATING Program (Heat Engineering and Transfer in Nine Geometries), ASTRA 417-5-0; Astra, Inc., Raleigh, N. C., January 1,

13. Llewellyn, G. H., Capsule Design Program CDP-1, Unpublished Memorandum, Oak Ridge National Laboratory, Oak Ridge, Tennessee, 1969-

14. McClure, John A., T00DEE - A Two Dimensional, Time-Dependent Heat Conduction Program, National Reactor Testing Station, Idaho, Phillips Petroleum Company, April 1967.

54

15- Moore, S. E., Code GHTR, Oak Ridge National Laboratory (Unpublished).

16. Pierce, B. L., "Modified Transient and/or Steady State (TOSS) Digital Heat Transfer Code,,r Westinghouse Electric Corporation Astronuclear Laboratory, WANL-TMI-1028, April 1964, WANL-TME-1108, April 1964.

17. Schenck, Hilbert, Jr., Fortran Methods in Heat Flow, Ronald Press, 1963.

18. Schroeberle, D. F., J. Heestand, and L. B. Miller, A Method of Calculating Transient Temperature in a Multiregion, Axisymmetric Cylindrical Configuration, The ARGUS Program, 1089/RE 248, Written in FORTRAN II, Argonne National Laboratory, ANL 6654, November I9&3-

19. Smith, R. B., and J. Spanier., H0.T.-1: A Two Dimensional Steady-State Heat Conduction Program for the Philco-2000, Bettis Atomic Power Laboratory, WAPD-TM^i£5, July 1964'/

20. Thomas, Richard F, Jr., The RATH MBSHER - A 7090 Program for Preparing Input Data for Heat Conduction Codes, Los Alamos Scientific Laboratory of the University of California, Los Alamos, New Mexico, LAMS 2809^ December 1962.

21. Wilson, J. V., GRIN, A Program to Prepare Input to Codes GHT and GHTR, Oak Ridge National Laboratory, Unpublished Memorandum April 29, 1966.

22. Wilson, J. V., Code GHTR for the IBM 360, Draft of unpublished ORNL-CF-Memo, Oak Ridge National Laboratory, April 28, 1966.

23. Wilson, J. V., HETRARZ - A Code for Calculating Heat Transfer in RZ Geometry, ORNL CF 68-10-74, Oak Ridge National Laboratory, Oak Ridge, Tennessee, October 15, 1968.

APPENDICES

56

• * I - T N , G , L , F , M . PROGRAM CYL

C CYLINDRICAL H p AT TRANSFER PROGRAM , MAIN PROGRAM COMMON ^ " " 3 , 7 5 ) , C1 C c , 7 * ) ("»•», "''S C3( , ^ ^ } , C M 7 5 t 7 5 ) ,

' 3W(7? , 7 5 ) ,ADFN(75 ,7 ' ? ) t H A ( 7 5 , 7 R | ,PA(7«; ,7e; j , N N T ( 7 c f 75 ) t T ( 7 f , 75 J , ? K R ( 7 * , 7 5 ) , R ( 7 5 ) , F L ( 7«5) , ST( ??) ,TZEROC ,TPE C ( 1 c J ,CONTZP «5) , ?R1 ( 1 5 ) ,P? (15 ) , E L 1 <151 , FL2'f ! 5 ) , M A X I T P n , AFX P. 5 1 , ALP HA < 1

RFAL*» M T L ( l O ) COMMON/AB/MTL DATA 701/ NCOUNT = 1

"< CALL PRF SFT ( DTM»PAC iHG» IH , I L» IN tKA t LMT,MAXI ,MAXJ»MMXt ! N C O U N T , N G A P , N I , N J , N L S , N O P R , N R S , T F N T , T I N T , T O T A L )

C\ SUBROUTINF PRFSFT READS IN THF INPUT DATA, ESTABLISHES THC ARRAYS, C1 CALCULATFS TH c VARIOUS TRANSCFP ^ ? r " I r I = N T S » CI COP^S THF NODFS BY SHAPE AND EY HF AT TRANSFER MOD^S C1 SUBROUTINE STEADY CALCULATFS TPMPCRATURF UNDER STEADY STATE CI CONDITIONS USING RELAXATION MFTHOD C!

CALL STFADY(NCOUNT,NOPR 1 T T M , H G , I H , I L , I N , L M T , MAXI,MAXJ,NLS» 1 N P S , T E N T , T I N T , T O T A L , K M I N , L I T , T c A C , c A C , N C O , M M X » K A , N G A P » N I » N J )

CALL P ICT(MAXI ,MAXJ ,NGAP,NOPR,NCO,MMX, T ^NT ,T INT) I F ( L I T . F Q . < M GO TO CALL R P L O T i M A X J , M A X I , K M I N , T F A C , N I , N J ) CALL F L P L O T ( M A X J , M A X I , K M I N A C , N I , N J 5

i r STOP END

57

SUBROUTINE PR ESET( DTM,FAC,HG,IH,IL,IN,KA,LMT,MAX I,MAXJ»MMXt iNCOUNT,NGAP,NI,NJ »NLS » NOPP,NRS,TENT,TINT,TOTAL) COMMON Nl ,-'?), Cl(75 ,7?) ,C2(75,75),C3(75,75) ,C4(75,75) , IQW(75 ,ADEN(75 ,75) ,HA(7 5T75) ,RA(75 75) ,NNT(75,75) ,T( 75,75) , 2KR(75,75),R(75),FL(75),ST(25),TZpRO(15),TREF(15), C ONTZ(15), 3R1(1K),R2(15) ,EL1 (15)tEL2(15),MAXIT(10),AEX(15),ALP HA<10) DIMENSICN N0(15),RH0(lr),C0N(lr), Q(10),TT(75,75),TI(10), 1NB(5P) ,NT(5>) ,NG(50),RT(cr ),PT(5r>) , 2RS1 (50) ,RS2(5 0) ,SL1 (50) ,SL2(50), H(50),FS(50),FR{50),RF(15) RFAt*8 MTLC.O) sOUTP,OPT,TI TLF(Q) COMM.ON/AB/MTL DATA TOL/.000001/,PI 2/6.?P3186/,PI4/1 2.566 37?/,BZ/1 .366E-12/ C01 c m THF FIRST T IMF PRESET IS CALLFD NCOUNT = 1 . THIS CAUSES THE PROGRAM TO CO1 START INPUT DATA C01 C n 1 - MEANING Oc INPUT NAMES r o i APX FX PONc NT A IN EQUATION FOR K (G) C01 ALPHA COFFEICIFNT OP FXPANSION C*1 C O N ( I ) CONDUCTIVITY 0*= MTL ( I ) - WATTS/CM.OEG.C OR BTU/HR/FT-DEG F CO1 CONTZ CONDUCTIVITY OF GAS GAP AT TZERO (SAME UNITS AS CON) ro - 1 OTM - DELTA T MAX - USED AS A TEST FOR CONVERGENCE C I E L I ( I ) DISTANCE BETWEEN TOP Oc CYL £ TOP OE R P G I O N U ) r r i P L 2 ( I ) DISTANCF BETWEEN TOP O c CYL £ BOTTOM O c R P G I O N ( I ) C01 CAC PACTOP USFD IN SPFCIFYING ACCURACY Oc RESULTS• (3A1 CR ( I ) E M I S S I V I T Y OP SURFACE( I ) -DIMFNSIONLESS

F S ( I ) SHAPE FACTOR OF SUP p ACE( I ) -D IMPNSIONLESS CO"! H ( I > HPAT TRANSPER COEFFICIENT OF S U R F A C E ( I ) , WATTS/SQ.CM.DEG.C OR CO1 BTU/HR/SQ FT-DEG F c KA DIMENSION INDPX CO! 1 = METRIC SYSTEM C01 2 = C NGLISH SYSTCM CA1 LMT A CONTROL PAPAMFTFP TO L I M I T THF NUMBER OP ITERATIONS I F CO" CONVERGENCE IS TOO SLOW. MAX MO OP ITPRS = LMT* (?* *NCOUNT) C01 MMX NO. OF SFPARATF REGIONS (REGION 1 INCLUDFS ALL OTHERS- OVERALL DIM) (7 r MPL NUMBFR OP A PFGION IN WHICH I T I S DESIRED TO HAVE TWICE THE REGULAR CO1 NUMBER OP NODPS BFT HFEN PLANE SURFACES CO1. MPP NUMBER OP A REGION I N WHICH I T I S ?ESIRPP TO HAVE TWICE THE REGULAR C <M NUMBER Oc NODE'S BFTWFFN CYLINDRICAL SURcACcS CO"1 M T L ( I ) MATPRIAL OP REGION I CO1 NB( I ) NUMBER OP SUREACE( I ) USFD FOR CHECKING CARD SEQUFNCE f r i NCA CARD NUMBER - HELPS USCR TO CHECK CARD SPQUENCF CO1 NG( I ) SHAPE AND LOCATION OP SURPACE CO"! NG FOR PLANE NOrE = 1 0 0 + NUMBER OP THE SURPACE f > i NG FOR CYL NODc = 2 m + NUMBER OF THF SURCACE C01 NGAP NUMBER OF GAS GAPS IN CYLINDER r o i N I INTERVAL BETWEEN PLOTS OF RADIAL- NODES (EVFN INTEGER . LE . 2*NC0UNT) C^ I NJ INTERVAL BPTWEFN PLOTS OF AXIAL NODES(EVFN I N T E G E R . L F . 2 * N C 0 U N T ) C01 NLS NUMBPR OF NODES BFTWFFN PLANE SURFACES C 01 N O ( I ) NUMBER OF REGION I - USED ONLY TO CHECK POR CARD SEQUENCE c NOPR - N r . O c THF PROBLEM C r 1 NRS NUMBER OF NODES BETWFEN CYLINDRICAL S U R E A ^ s C 01 NS NUMBER OF SURPACES (INCLUDING INTERIOR SURPACES) C n l NT( I ) TYPE OF HEAT TRANSFER AT THE SURCACE - AS FOLLOWS c ^ i 1- = NONE (SURFACE IS INSULATED) C01 2 = CONVECTION TO A SINK C01 3 = RADIATION TO A SINK C ^ 4 = CONDUCTION AND RADIATION TO GAS GAP I N POSIT IVE DIRECTION C07 5 = CONDUCTION ( INTFRIOR SURFACES) CO1 6 = CONVECTION AND RADIATION TO A SINK

58

7 = COND(JfT ION AND RADIATION TO GAS GAP IN NCGATIVF DIRECTION) OPTIONS CALL FCR PRINTOUT AS cOLLOWS R AND FL AP"AYS AS FIRST FSTABLISHCD (ONLY WHEN NCCUNT = 1) P AND PL APR AYS AFTER REARRANGEMENT (ONLY WHCN NCOUNT = 1) VOL»ROM,RIM, D ^ ,nFLLtnILR »nlLL,SIZE AMD REGION OP REG VJODFS PRELIMINARY VALUES O'F CONSTANTS cOR RCGUL AR NODES IF NCOUNT = 1 RFGION ANn Nrnt: DESIGNATIONS OF GAS GAPS CALCULATES AVERAGE SURPACF T*MP P«OM HcAT GrN L TRANSF DATA TOLD,TNEW»T(I ,J) ,TPST,SUM,CUF,IS cOR PIRST * ITERATIONS RE SETS IG TQ AT START Oc c/<CH SERIES Dc CALCULATIONS COEccICIENTS OF PLANP AND CYLINDRICAL SPCCIAL NOD=S COEEFICICNTS Oc REGULAR NOCcS TEMPPRATURp ARRAY cOR PACH Op cIP$T E THEN EVERY TENTH ITERATION KX » LX» NX Ic NCOUNT = ? KX,LX»KK,LL,NX IP NCOUNT.GT,7 T0LD,TNEW»5ASK,CELG cOR cflH GAS GAP MQDF cACH ITERATION HEAT GENERATION RATF OE MTL(I) - WATTS/GRAM OP BTU/HR/LB MASS RADIATION FACTOR CR CALCULATION Oc RADIATION ACROSS GAS GAP (IJ MINIMUM RADIUS OE MTL(I) MAXIMUM RADIUS OF MTL(I) DENSITY ge M T L ( I ) - GRAMS/CUBK CM OR L B S / r u . c T MINIMUM RADIUS Oc SUREACE(I)

TEMPERATURE Oc THF SINK TO WHICH SURPACC (1) IS TRANSFERRING HEAT EACTOR USEH TO HASTEN PELAXATIQN INITIAL (ESTIMATED) TCMPFCATURE OP REGION(I) MAX TOTAL DFVIAT ION DURING ONc ITERATION EACH TIME PRESET IS CALLED THC VALUC Op NCOUNT INCREASES BY ONE VALUES OE TEMPERATURE ARF STORED IN TEMPORARY LOCATION TT. DC 1 I = 1-75 DO 1 J = 1,7" ADFN(I,J) =0. C1(1,JI = 0. C2( I,J) = o. C (I,J> = 0. CM I, J) = 0. HA(I,J) = r, KR(I,J) = 0 N {I, J ) = 0 NNT(It J) = QW(I,J) = 0. PA(I,J) =0. CONTINUE IF(NCOUNT.CQ.1) GO TO ? DO 2 I = T,MAXI DO ? J = 11M A XJ TT(I,J) = T(I,J) GO TO "" DO I = 1,2C S T ( I ) = 0 . R (I ) =0. P (? *I ) = PC3*! I = p L (• I) = 0. EL(I) = FL(I ) = n. CONTINUE DO c I = I , " " 1

r n i CO1 OUTPUT I C^T 1 I A r r i "> I B CO1 * IC CO1 L IP C^1 c I C

COT 6 I P COT 7 IG CD 1 R IH CO1 o I J CO! IK c n I L CO1 I Z CO! 13 IN

Q U I COT RP( I ) CO! RT ( I ) C r 1 R2 { I ) rcP PHO ( I ) C0"< R ST I I ) cv P S ? ( 1 ) C O1 SL111 ) r o 1 SL? ( I ) cri ST ( 11 C^ i TENT -ro'y T I ( I ) C^ i TOTAL-fP 7 CO"i FACH T I C"1! rni VALUES COT

59

DO 5 J = 1,75 TCI,J| = 0. TTCI ,.J1 = E CONTINUE DO 7 I = 1,50 ET( I) = n. RTC 11 = FS(I) = 1. 7 FRC I» = 1. READ 2 !,TITLE,NCA RINT 300 ,TITLE PRINT 301 ,NC A READ 2 2 ,NQPR,MMX , NS* NGAP, NPS, NLS, MRR»M.PL»LMT, NI ,NJ,KA,NCA .PRINT 30! ,NCA READ 203 ,DTM,TOTAL ,FAC,TENT,TINT,NCA PRINT 3*.! ,NC A READ QUTP,OPT,IA,IB,IC ,1D,IE,IF,IG,1H,IJ,IK,ILfIZ,IN,NCA PPINT 30!,NCA DO R 1 m \,MMX READ 20,5,MTL(I) ,R1(I),P2(I ) ,EL! (11 ,FL2< I) ,RHO< I) tCON< 11 ,Q( I) , TI (Ii ! ,NO (I) , NC A, AL. PHA (I) ,N0(I),NCB INT 3 1 ,NCA PRINT 30!,NCB 8 CONTINUE IF(NGAP.EQ.*) GO TO l*1 IGM = !0 + NGAP DO o I = 1! ,IGM READ ?C6, RP(I),R!{I),R2{I),EL!(I),EL2(I),CONTZUJjTZEROCjj, 1 TP F F { I) T A EX' I »i NO' I i, NC A PPINT 30!,NCA o CONTINUE 1r DO !! I = !,NS READ ?OP,NT(I),NG< I),RS!( I ),RS2U),SIH 11 , SL2 (I)»ST (I) ,H( I), FS( I)» 1 FR(I),NB(I),NCA PRINT 3^ , NC A 1! CONTINUF NCCK = + 2*MMX •»• NS + NGAP IF(NCA.NE.NCCK) PPINT CO? C02 FND OF INPUT DATA - START 0* PRINTOUT OP INPUT DATA Cn2 PRINT 2°°,TITLE PRINT 30?,NOPR,MMX »NS,NRS,NLS,LMT IF(KA.EQ.l) PRINT 31.6 IF(KA.E0.2) PRINT 21f PRINT 303,DTM,TOTAL,FAC,TENT,TINT IF(MPR.GT.!» PRINT 3T&, MPR IF(MPL.GT.l) PRINT 30?, MPL PRINT 306 PRINT 3C7,(N0(I),MTL(I) ,RHO< I ) ,CONU),Q(I)»TI(I),ALPHA(I),1=1,MMXI PRINT ?0P PRINT 30°,(N0(I),R1(I), R2(II,ELI(I),EL2<I),1=1,MMX) IF(NGAP.EQ«P) GO TO 1? PRINT 30°,(I,Rl(Ii,R2(H,ELl(II,EL2CIit I = l! «IGM ) PRINT 3!0 PPINT 31!,(NO(I),RF(I)fTZERO(II,TREF(I)tCONTZ(I)»AEX(IltIslltIGHJ 12 PPINT *5! PRINT 3!?,(NB(I) ,NT(I) ,NG( l»,RSlCn,RS2(I)»SLl(II»SL2(II,I=l,NSI PPINT 3!3,(NB(I) ,ST(II ,H( I ) , FSU I , FR< I I , 1 =1 ,NSI PRINT 3!4, OUTP,OPTtIA,IB,iC,ID,IE,IF,IGtIH,IJ,IK,IL,IZ,IN PRINT 253

CO"1

dO

c^r> pi' pPIMTQUT IN«>UT r A T f l r o*» C c S T A e L l S M S ! Z C L O r f t T i P N Oc NO n c S ff 1

?ZM = P l I c ( K A . c C . *>» BZM = B Z * '

p c i K - ' - 1 0 , N r Q U N T

1 P ( N rOUWT• N P . ! ) GO T 0 DP ->i. I s 1 ,NS L= L + ^ C L ( L - * ) = S I M ! ) F L ( L ) = S L ? ' f } F ( L - ? ) = D S! ( I ) R ( l > = F S ? | I ) CONTINUE I P ( I A . N E . N f O U N T ) GO TO 1 F PPINT PPINT T(J,R(J) , J=! , L J PPT NT -ii PPIMT ( I t FL ( T > »1= * tL I ft

CO* PC LOOP ARRANGES THF COORDINATES O c THC SURFACES INTO TWO ARRAYS r o t a n R ftp RAY THC PLANE SURCACES

AN PL ARRAY cOF THE CYLINDRICAL S'JRCACFS CO* PC LOOP ?0 RFAPRANG-FS THP R ARRAY INTO AN INCREASING ORDER CO^ PO LOOP REARRANGES THC =L ARPAY INTO AN INCREASING ORDER

IA IS OUTPUT OPTION ! I P ( I A . E Q . l ) PROGRAM PRINTS PRELIM R AND EL ARRAYS IP IS OUTPUT OPTION ? I F d P . F Q . ? ) PROGRAM PRINTS REARRANGED PRELIM NOCc

CO^ BOUNDARIES

MJ * r JM = L - " po j = : , j m JP = J + 1 DO I® J J = J P , L I F ( A B S ( R ( J J ) - R ( J ) ) . L c » T O L ) GO TO I C ( R ( J ) . L T . R ( J J > ) Gt) TO ! c

TEMP = R ( J ) PCJ) = P ( J J > R ( J J I = TFMP GO T 0 ! e

y P R t J J 1 = TOO.+ PLOATJMJ) NJ =MJ • !

, c CCNT1MUP ?0 CONTINUE MI = r

IM = L - - 1

DO ?0 I = ! • I M IP= l + l DC *»« J = IP» L I F ( A B S t c L ( J } - P L ( I ) ) . L c . T O l ) GO TO I P C P L ( I ) . L T . P L ( J ) > GO TO 9 o TFMP = r L ( 1 1 EL ( I ) = c H J I F L ( J J = TPMP GC TO « »» CLI J) = ">00.• PLOAT(MI) MI : Ml t 1

, c CCNTINUC

' ' 0 CONTINUc

I c ( I H . N F . ' . O R .NCOUNT.GT.M GO TO "

6 1

roc COE r Qc

p p i n t PPINT J,R(J), J=!,L) PPINT u o,(J,eL(i ) , I®1 » L) " K = L- HJ L = L- MI KM = K LM = L - 1 I M I B . F Q . ?.AND.NCOUNT.FQ.- ! )

DC LOOP L<\ EXPANDS THF R PO LOOP c0 EXPANDS THF EL

PRINT ?16VKM«LM ARRAY TO GIVE ROOM t=OP REGULAR NODES ARRAY TO GIVP ROOM for REGULAR NODES

NRP = NRS + 1 NLP = NLS + i

PO *? I = !»K **> RTU)= P( II DO <a? I = 1 ,1 « ETUJ = EL( I ) IF(IB.NE.2) GO to PPINT ?1 7 PPLNT iro.U tRT( I ) ,K) PRINT 0° »(I» ET<I)11 = !*L) -XL NRP = NCOUNT + 1 NLP = NR° I*C NCOUNT. GT. NRS J Ic(NCOUNT.GT.NLS) IGM » + NGAP IX = ' DO 1= 11KM NPR = NRP - 1 IF (NGAP.c0.0) GO TO ?7 DP K = , IGM IF(ABS(PT(I )-RMK)).LF.TOL ) NPR = * ** CCNTINUC •>"* XP MRTCH-')- RT(Ii)/ FLOAT(NPR) IP (MPP.LE.l) GO TO ?p Ic( APSCR''(MPRI-RT( I)) .GT-TCLI GO TO »» XR = O.E*XR NPR s ?*NPR

CO" COCO

oe

ic

coe CO e i

u c

MPR = NO OF PPGION TO HAVC TWICC AS MANY RADIAL NODES AS THE RFST rp - > c j = 1 , n p p

R{IX) = PT(I) + XP* cLOAT(J-1) IX a IX +? CONTINUc CONTINUE MAXJ = IX P(MAXJ) « p?(! I MPL = NO Oc REGION Tp HAV* TWICC AS MANY AXIAL NODES AS THE REST

IX = 1 DO c0 1= 1»LM NPL = NL° - ? XL = (pT(I+11- FT(I II/ cLOAT(NPL) IF(MPL•LE.* J GC TO iP IF(ABS(ELI<MPL»-FT(I)).GT.TOH GO TO XL = r•e*XL NPL = * * NPL 00 « J = * , NPL EL<IX) = FTC I|+ XL* cL0AT(J-? ) IX - IX + ?. CPNTINUC

6 2

CPNTTMUF MAX I = IX F L I M A X I J = C L ? < 1 I PRINT ? ,NCOUNT PRINT &0«f(JtR(J)t J=-»,MAXJi?> PPINT ?1?fNCOUNT PRINT A r e f { i f g L ( I I * I , M A X I , ? )

rof-COA fr* CO* CO'

CO* r Of ro* CO*

CODING ALL

ffst rr7 ro"*

NOOpS N= < N= ' N= 0

N= A N= B

N N CYLINDRICAL

I N THF ARRAY PLANE MCnE I CYLINDRICAL N 0 0 c I RC6ULAR NOOF I UNUSFC NOD^S I GAS GAP NODES I

PLANE NOrF AT SUP«=ACC \

ODD CVFN EVFN ODD CVCN

J EVFN J ODD J FVFN J ODD J EVEN

NODc SURP ACc a

OC I s 1 »MAXI DO R S J = 1 ,MAXJ t ' Q W ( I , J ) ' 0 .

B P N ( I » J ) = A DO c c J = ? * M A X J t 9

QWU i J ) = 0 . c ° N ( I » J ) = 1 * 0 CPNTINUC

r«P f c I = 7 »MAXI t 1

DP 6? J = 1 » MAX J • ' O W U t J ) = 0 . f N(I,J ) - ? np 64 J = ' , M A > J t - 5

f i N i l , J ) = « 6 C CONTINUE

CALCULATE TH* C O F F c i d P N T S AND TEMPERATURE O c REGULAR NODES

NCM = NCOUNT - i NCT = NCOUNT * ? HTOT = 0 . HG = 0 . r o i = M A X I t ? NX = 0 I p ( N C OUNT.GT .1 ) KX = I T + ? ) * N C M / N C T * ? I F ( N C O U N T . G T . 2 ) KK= ( 1 + 1 J * N C M / N r T * ? I P ( T C . E Q . a . A N D . N C O U N T . E C . 1 ) PRINT 91 q I P = I+"» IM = I - -F L U ) = 0 . 5 * ( F L I I P ) + F L ( I M ) ) D F L L = F l ( I P ) - P L U M ? D I L L = 1 . /DELL CO 7 0 J = ? ,MAXJ t ? IF(NCOUNT.EQ.2) LX = ( J + ? + N X ) / * * 2 TP(NCOUNT.L e . ?1 GO TO 6 f LX » ( J + P - N X ) *NCM/NCT<"? + NX LL = ( J + 1 - N X ) * N C M / N C T * ? + NX JP = J+1 JM = J - 1 P ( J ) = 0 . c * ( P ( J P ) + PC JN11 POM = R ( J ) + R ( J P ) PIM = P ( J ) + R(JM) DFLR = R(JP)— R ( J M i DILF = . /DFLR

6 3

VCL = PI' * °(J) * rEL R * PCLL NVT(I,J) = c I F I K . M c . ? . D ° . NC 3 U N T . N E . 1 ) GO T O PPINT c - f , I t J t P O M t P ] M , D F L F , P C L L , P I L P 1 0 1 L L , V O L

r i r I S O U T P U T O P T I O N ? I c ( I C . c Q . ' i P R O G R A M P R I N T ? O U T V O L , P O M , R I M , D E L R ,

P P L L , n i L P , O I L L F O R E A C H P F G U L A R N O D E f O P A N D L A T E R P R I N T S O U T S I Z F A N O R E G I O N O c

f 0 ° E A C H r e g u l a r N O D E r ^ P V O L = V O L U M E O F R E G U L A R N O ^ F r O c C Q M s S U M O E R A D I U S O c P E G U L A 0 N O D c A N D R A D I U S C c I T S O U T E R B O U N D A R Y r O P P J M s S U M O c P A P I U S C E R E G U L A R N O D E A N D R A D I U S P c I T S I N N E R B O U N D A R Y

D E L P = D E L T A R 0 c R C G f O R D ^ L L = D E L T A L D C P P G N C D C

C 0 » 0 I L P = I N V E R S E 0 c D E L R C r P D I L L = I N V E R S C O F D E L L COR

I F ( N G A P . E O . O ) GO to fo DO f p K = 1 1 , I G M I E ( R ( J ) . L T . R - » < K ) . O R . R ( J I . G T . R 7 I K ) ) GO TO 69 I E ( E j _ ( ] ) . L P , C | _ . I ( K J . O P . E L H ) . G E . E L 5 I K ) ) GO T O

K P ( I , J - " ! ) = K K P U t J + i ) = K K R U t J ) = K N ( I , J ) = 5 KR ( I + » J I = K I c ( N r O U N T , N E . T ) GO "»0 T { I t J ) = T I ( i ) T(I+!»J) = TIC!) T< I, J-1 ) = TI C. ) T ( I , J + ? ) = T I ( t ) I E ( I E . F Q . « 5 ) PRINT 9 ? F , 1 , - j , K R n , J ) « N U f J )

h5 CONTINUE (•o n C n-p K=1 »MMX

Ip(N( It J) .GE.M GO r0 "»7. I F ( R ( J ) . L T . R l ( K ) . O R . R ( J ) . G T . R ' ( K ) ) GO T O I C ( E L ( I ) . L T . E L ! ( K ) . O R . E L d ) . G T . F L ? ( K ) ) GO T O K P ( I , J ) = K I E < N C C U N T . E O . H T ( I , J ) = T 1 ( K ) Q W ( I « J ) = Q ( K ) * P H O ( K ) * V O L C'(ItJ>= P l ' ^ * C D N t K l * R(J) * PELP * D I L L CMI, J)= C3 ( I ,J) IF(J.NE.?1 GO TO CMltJ) * CON(K)*PELL C'UtJ) =0. GO TO TC = P I ? * CON < K ) * D E L L * D I L P C M ! » J ) = CC * R O M C ? ( I , J ) = C C » R I M

" " A D E N ( I , J ) = C ^ Ic( IC .EQ. AND. NCOUNT. FQ.l ) PRINT AH , 1, J t DELE r PFL L , K" (I , J ) lE(NrrUNT.FQ.9) T ( l t J ) = TT(KX,LX) Ic ( IZ • NE • 1 ? ) GO TO

I F t N C P U N T . E Q . T ; ) P R I N T ?">*-, I , J , KX , L < » NX I c ( NCCUNT . G T . I ) * P R I N T , I , J , K X * L X , KK T L L , NX

" '? I E ( NCCUNT , L C . 2 ) GO TO " I E ( N C O U N T . L c . N R S . A N D , N r C U N T . G T . N L S )

! T ( I , J ) = * ( T T ( I t L X ) + T T C I » L L ) ) IF(NCPUNT.GT.NPS.ANP.NCOUNT.LE.NLS) !T(TtJ) = 0.c * (TT(KX,J) + TT ( KK J ) )

6h

I c (NCOUNT . L p . NP S . ANC. NC HUNT. LF.. NLS) 1 T ( I , J ) = 0 . c * ( T T ( K X , L X ) + TT(KK »LL) )

""= CPNTINUC

Pr CONTINUE r oc CO° i n IS OUTPUT OPTION A I « = U C . F Q . A ) PROGRAM PRINTS D R F L I M VALUFS r o p REGULAR NODES I c NCOUMT FO 1 CO°

I F U C . f ^ . A . O R . N C O U N T . N F . i ) G O T O «? DO P! 1= ? , M A X I » ? PRINT ?6 C

PPINT A i t , n » ? L U I , J»P ( J1 , K « U »J) »T( I , J », QW( 11 J ) » C M I » J l f C 3 ( I , J | 1. A C E N ( I » J ) tC? ( 1 , J ) ,CAt MIM = MAXI - 1 PJM = MAX J - 1

DO 1= »MI M PC q-? J= ?,MJM" I E ( N ( I , J ) « G E . 3 ) GO TO v N N T ( I t J ) = « I F ( N C O U N T . N c . ! ) GO TO P c

T ( I » J ) = T I C ! ) DO K = ?,MMX I P ( P ( J ) . L T . P i ( K ) . O P . R ( J ) . G T . R ' ( K ) 1 GO TO «A I P ( C L ( I ) . L T . F L T < K J . O R . P L ( I ) . G T . F L ? ( K » l GO TO T ( I t J ) = TT ( K )

OA CONTINUE P? I F ( N ( T , J ) . F Q . l ) G O T Q «

C M I , J ) = r . C A { I , J ) = 0 . C 1 ( I • J ) = C ? ( I , J + 1 ) C ( I » J ) = C 1 ( I , J - l ) I E ( N C O U N T . e q . 1 ) GO TO P 7

T ( T , J ) = 0 . ( T ( I , J—1) + T ( I , J + i ) » GO TO P"7

06 C" ( T , J ) = 0 . C ? ( I , J ) = 0 . r » { I , J J ft C M 1 - : y J ) CA( I , J } = C ? ( 1 + 1 , J ) I F ( NCOUNT.EQ. 1) GO TO P"7

T ( I , J ) = r . 5 * ( T U - l »J ) + T ( I + ' , j n P7 CONTINUE OP CONTINUE n i

Cl1 SUM HEAT GENERATED (HG) ri' C I ' CALCULATE SURE&CF NODES - INTERNAL AND FXTFRNAL C "

DO " , 1 0 1 = 1 , M A X ! DO ' 0 ° J= ' r MAXJ HG = HG + Q W ( I t J ) I F ( N ( 11 J ) .GE..3 ) GO TO 1 0<? H A ( I , J i = 0 . R A ( I , J ) = r . no TOP K=! ,NS I ' M ( N G ( K ) - K ) .EQ.-» 0 0 ) G O T O °P I F ( A B S < R < J ) - P S ! < K J ) . G T . T Q L ) GO T 0 1 r « I F ( E L ( I J . L T . S L T ( K ) . O R . c L ( T J . G T . S L ? ( K ) ) GO TO ! 0 ° N N T { I « J ) = NT ( K)

65

N ( I » J I = N G ( K ) L = NNT(I ,J) GO T O ( 9 1 , 9 2 , 9 3 , 9 4 , 9 8 , 9 2 , 9 7 ) L 91 CI ( I,J) =0. C2(I,J) = 0. I F ( J . N E . M A X J ) C 2 ( I , J + l ) = 0 . CI ( I , J - l ) = GO TO 1 0 *

Q ? HA( I , J ) = H( K) * P12 * R ( J ). * ( FL ( 1 + 1 ! —EL'C 1 - 1 1 ) I F ( L . E Q . 6 ) GO TO 93 HTOT = HTOT + H A ( I , J , ) GO TO 108 «3 R A (I , J ) « BZM* PI.2 * FS(I<) * R(J) *< EL (1+1 )-EL( 1-1 ) ) # FR(K| HTOT = HTOT + HA(I,J) + BZM*RAU,J) GO TO 168 CKI,JI = r. QE RA( I , J ) = BZM*PI?*RF<K)*R<J)*(EL<I+l)-EL(I-m GO .TO 108

Q7 C 2 ( 1 , J ) ' = GO TO ° 5 I F ( A B S ( E L ( I ) - S L ! ( K ) J . G T . T O L J GO TO 1 0 ? 1 F ( R ( J ) . L T . R S 1 ( K ) , 0 R . R ( J ) . G T . R S ? ( K ) ) G 0 TO 103 N ( I , J ) a NG(K) NNT ( I , J ) = NT ( K) L= N N T ( I , J ) GO TO ( l P ! , l P 2 , i r 3 , ! n ^ , i r p , i r 2 , ' ' n p ) L C 3 ( I , J ) = 0 . GO TO 1 0 5 C 3 ( I , J ) « n . C M I , J ) = 0 . I F < I . N E . l ) C M I - 1 , J ) = 0 . I E ( I . N E . M A X I ) C 3 ( I + 1 , J ) = GO TO 108 HA ( I , J ) = H ( K ) * P I 2 * R I J ) * ( R | J + 1 H I J - 1 M I F ( L . E Q . 6 ) GO TO 1^3 HTOT = HTOT + H A ( I , J ) GO TO 10P RA( I , J ) = BZM* PI 2 * «=S (K ) * F r ( K ) * ( R (J+l I - R ( J-l I j * R(J) HTOT = HTOT + HA(I ,J) + BZM*RA(I,J) GO TO 10R C M I , J ) = P . R A { I , J ) = B Z M * P I 2 * R F < K - ) * R < J ) * ( R ( J + ! ) - R ( J - l ) ) C O N T I N U E C O N T I N U E C O N T I N U E I F ( H T O T . N E . O . ) OFLTAT = HG/ HTOT PRINT 2 ? P , H G , HTOT ,DELTAT

1 00

in

102

1 p-a

1 OA. ire 108 1 n o lir

CI1 5 C11 5 C11 5 C11 5 CUE CI! 5 CUE

11! 112

I F I S OUTPUT OPTION 6 IFUF.EQ.6i PROGRAM ADJUSTS OUTER CYLINDRICAL SURFACE (NO.? 1 TO A VALUE CALCULATED BY DELTA T = HG/HTOT T(SURFACE I=T(SINKl+DELTA 1 IF(!F«NE®61 OUTER SURFACE TAKES SAME TEMP AS THE ADJACENT REGULAR NODE' I F ( I F , N F . 6 ) GO TO 112 DO 1 1 1 I = 2» MAX I , 2 T ( I t M A X J ) = S T ( 3 1 + D5LTAT C ? ( L , M A X J ) = CI ( I , MJMI CON-TINUE GO TO 11A DO 1 1 3 1= 2 , M A X I , 2 T ( I , M A X J ) = T ( I , M J M )

66

C ? U f M A X J ) = CI ( I t M J M ) 11? CONTINUE 11 * DO 115 J =» 2 , M J M , 2

T (1 , J ) = T( 2 t J) * 2 . - T ( 3 t J » K M A X I t J l = T < M l M t J ) * 2 . - T < M A X I - 2 , J i C A ( 1 , J ) = C 3 ( 2 » J I C3 ( MAX I f J ) * C M M I M f J )

115 CONTINUE DO 116 I 21 M A X ! t 2 T ( I t l ) = T d , 2 » * 2 . - T ( 1 , 3 1 C 2 { 1 1 2 J = 0 .

116 CONTINUE C I 2 C I 2 1J I S OUTPUT OPTION o I F ( I J . E Q . 9 ) PROGRAM PRINTS COEFFICIENTS OF C12 SPECIAL NODES BOTH PLANE AND CYL C12

DO 121 I = 2 iMAXI»? I F U J . E Q . 9 ) PRINT 267,NCOUNT DO, J = i , M A X J , 2 A D E N ( I , J ) = C K I > J ) + C 2 d i J ) I F d J . N E . 9 ) GO TO 120 PPINT A I D , l , J , N N T ( I , J ) ,N t C 3 U * J ) » C A ( I , J | ,

1 H A ( I , J J , R A ( I , J ) • A 0 E N ( I T J > t T'( I , J ) 120 CONTINUE 12 1 CONTINUE

DC 123 I = I ,MAXI»2 I F d J . E Q . 9 ) PRINT 268,NCOUNT 00 122 J = 2 t M A X J , 2 ADEN( I » J ) = C 1 1 J ) + C ^ ( I » J 1 I F ( I J . N E « 9 ) GO TO ^22 PRINT A l O f I , J ,NNT ( I , J } ,N ,C3 , C M I , J ) ,

I H M I t J ) » R M 1 f J ) f A D E N ( I , J ) « T ( I i J ) 12? CONTINUE ^ CONTINUE CI. 3

C I 3 IK IS OUTPUT OPTION I D I F d K . E Q . 1 Q ) PROGRAM PRINTS COEFFICIENTS C13 OF ALL REGULAR NODES C i i

DC 1 2 f 1= 21 MAX I I F ( I K . E Q . I O ) PRINT 2 6 6 , NCOUNT DO 1 2 * J = ? , M A X J , 2 ADENt I } J ) = C K I t J I + C ? ( I t J ) + C 3 ( I , J ) + C A ( I > J ) I E U K . N E . 1 0 ) GO TO 12A PRINT A l 4 t 1 , E L ( 1 ) » J , R t J ) , K R ( I , J ) t T ( I T J ) t Q W ( I r J ) t C H I , J ) , C 3 ( I , J ) ,

l A D E N d t JJ • C ? ( I , J » , C A d t J ) I 2 A CONTINUE 12 5 CONTINUE

PFJ = MAX J/3 IF<MFJ<<5.EQ.MAXJ) GO TO 135 NPAGES= MFJ + 1 GO TO 136

135 NPAGES = MFJ 136 LOW * I

NUP' = 5 DO 13P K = 1,NPAGES PRINT A 5 ! PRINT AA5» NCCUNT I F ( N U P . G E . M A X J ) NUP = MAXJ PRINT A36 DO 137 I = l f M A X l

137 PRINT A 3 7 , ,T 11 , J > , J=LOW t NUPJ LOW = LOW + 5-

67

1 ?8 NUP = NUP + 5 PRINT A*6,NCOUNT DO 1 I = 1 >MAXI PRINT * * 7 t ( N N T ( I , J ) » l t M A X J )

139 CONTINUF 201 F 0 R M A T ( X , 9 A « , 3 X , I M 2 ° 2 F O R M A T ( X , I 7 , 1 1 I 4 - , 2 A X , I A ) 20? FORMAT ( 8 X » 5 F 8 . 0 , 2 B X , I M 2OA FORMAT(2A8,1 3 I A , 8 X , I M 2D5 F O R M A T ( A 8 , 8 F P . P , 2 I A / 8 X f E l 6 . 8 , A 8 X , 2 I A ) 206 F O R M A T ( X 1 P 7 . 0 , 8 F 8 . 0 , 2 I ^ ) 208 F O R M A T ( 2 I A , 8 F 8 . 0 , 2 I A » 217 FORMATdHI F^OUTPUT OPTION T , ir*X ,A1 HP RELI MI NAR Y ASSIGNMENT OF

1 NODE BOUNDARIES / ) 212 FORMAT (1 H I , 2 O X , 3 5 H F I N A L ASSIGNMENT OF NODE BOUNDARIES

1 6 X , 7 H N C O U N T = , I A / / ) 21? F 0 R M A T ( 1 H 1 , ! 0 X , 1 5 H 0 U T PUT OPTION 2 , 1 OX,33HREARRANGED PRELIM NODE BO

1UNDAPIES / ) 2 1 * FORMAT (2PHPN0 OF- RADIAL GROUPS, I A , AX , 1 8HN0 OF AXIAL GROUPS,14) 217 PORMAT{"! H I ,1 r X , l 5H0UTPUT OPTION 2 ,1 OX, 21HNODE GROUP BOUNDARIES/) 218 F 0 P M A T ( 1 H 1 , 1 0 X , 1 SHOUT PUT OPTION NO 3 , ^OX,

11PHSIZE DATA ON NODES / ) FCRMAT{ 1H1 . ^ ^ X , 28HSTART OF PRESET CALCULATIONS / /

1 1 H 0 , 3 0 X , 7 H N C 0 U N T = , I A ) 2?0 F0RMAT( !H1 ,1PX,1QHHEAT GENERATED(HG) = ,1 P E 1 6 . 8

11CHTP ANS COPFF. (HTOT)= , El 6 . 8 / I H D , l r x , 1 9H DELTA T (SURFACE) = , 2 E 1 6 . P )

221 FOP MATCH 0 , A F 1 0 . 0 / X , F 9 , * p 1 n E 2 ^ . 8 ) 22A F O R M A T ( 3 H 0 I = » I 3 , 5 X , ? H J = , I 3 , * X , 2 H K = , 1 3 , A X ,

1 6 H E L ( I ) = , P 1 0 . 5 , 3 X , 7 H E L 1 ( K ) = , F 1 0 . 5 , ? X , 7 H E L 2 ( K ) = , F 1 0 . 5 ) 22 * FOP MAT??HOI = , 1 3 , 5 X , 2 H J = , I ? , * X , 8 H K R ( I , J ) = , I * , A X , 7 H N ( I , J ) = , I A ) 226 FORMAT { 3 H n T ( , I ? , 1 H , t - I 2 « 7 H ) = TT( , 12 , 1 H , , I 2 ,1H I , 3HNX = , I A ) ?27 F O R M A T ( ? H O T ( , I 2 , 1 H , , I ? , 1 2 H ) = 0 . 5 * ( T T ( , 1 2 , 1 H , , 1 2 , 7 H I + T T ( ,

I I 2 , 1 H , , I 2 , 2 H ) ) , 3 H N X = , I M 25? FORMAT < l H n , 2 r X , 2 Q H E N D OF PRINTOUT OF INPUT DATA ) 265 FORMAT(1 H I , 1 0 X , 1 8 H 0 U T P U T OPTION NO * , 1 0 X ,

1A1HPRFLIMINARY COEFFICIENTS OF REGULAR NODES/ / ) 266 F0PMAT(1H1,1 6H0UTPUT OPTION 1 ^ , 3 D X ,

1 29HC0EFF1CIENTS OF REGULAR N O D E S , A X , 7 H N C 0 U N 7 = , 1 4 / / ) 2 6 7 FORMAT(1H1,?X,15H0UTPUT OPTION ° , 1 0 X t

1 *1HCOEF P IC IENTS OF CYLINDRICAL SPECIAL NODES , 1 A X , 7 H NC OUNT = , I A / 2 1 H 0 , ? X , 1 H I , 3 X , 1 H J , X»3HNNT,2X ,1HN,8X» 2 H C 1 , 1 2 X , 2 H C 2 , 1 2 X , 2 H C 2 , 1 2 X , 3 2 H C A , 1 2 X , 2 H H A , 1 2 X , 2 H R A , 1 2 X , A H A D F N , 1 2 X , 1 H T / )

268 FORMAT(1 H I , 2 X , 1 5 H 0 U T P U T OPTION 9 , I P X , 1 35HCOEFFICIENTS OF PLANE SPECIAL NODES , A X , 7 H N C 0 U N T - » I A / 2 1 H 0 , 2 X , 1 H I , 3 X , 1 H J , X , 3 H N N T , 2 X , 1 H N , 8 X , 2 H C 1 , 1 ? X , 2 H C 2 , 1 2 X , 2 H C 3 , 1 2 X , 3 2 H C A , 1 2 X , 2 H H A , 1 2 X , 2 H R A , 1 2 X , A H A D E N , 1 2 X , 1 H T / )

2oo E 0 R M A T ( ! H l r ? C X , 9 A 8 / / ) ? 0 0 FORMAT(20X,9A8/ /1HO?5X»25HNUMBERS OF THE CARDS READ) 30 1 FORMAT! I I P ) 3D2 FORMAT ( ! 7 H P PROBLEM NUMBER=I 6 , 1 2X , 1 8HNUMBER OF REGIONS=, I 2 , 1 2 X ,

121H NUMBER OF S U R F A C E S = , I 6 / / 3 3 H 0 REG.NODES BETWEFN CYL SURFACES^, 2 1 6 , 6 X ,3?HREG NODES BETWEEN. PLANE SURFACES=, IA / 3 2 1 H ^ L I M I T OF ITERATI0N5 = , I 6 / )

303 FORMAT(18H0 DELTA T M A X ( D T M ) , F 1 0 . A , A X , 1 A H M I N S U M ( T O T A L ) , 2 X , F 1 0 . A , 1 6 X , * H F A C = , F 1 0 . 5 / 6 H 0 T E N T = , F 1 0 . 5 , 6 X , 5 H T I N T = , F i r . 5 / / / )

30A F O R M A T ( 1 H O , 2 P X , 6 H R E G I O N , I * , 2 X , 56HWILL HAVE TWICE THE NUMBER OF tNODES BETWEEN CYL SURFACES / / }

305 F O R M A T ( 1 H 0 » 2 0 X , 6 H R E G I 0 N , I A , 2 X , 58HWILL HAVE TWICE THE NUMBER OF 1 NODES BETWEEN PLANE SURFACES / / )

306 FORMAT<7H R E G I O N , 7 X , 8 H M A T E R I A L , 8 X , 7 H D E N S I T Y , 8 X , 1 2 H C 0 N D U C T I V I T Y , 3 X , 113HHEAT GEN R A T E , A X , 9 H I N I T .TEMP , 1 0 X , 5 H A L P H A / I

68

307 F 0 R M A T ( 4 X » I 2 , 8 X , A 8 , 0 P 4 F 1 5 . 5 , E 2 0 . 8 / ) 308 F0RMAT(7H0REGI0N,7X,8HMIf: RAD. , 7X, 8HMAX RAD.,7X,6HMIN L . , 9 X ,

16HMAX L . / ) 3 ro FORMAT(4X , 12 • 4F1 5 • 5/ ) 3 1 0 FORMAT(7H0GAP NO* 6X,2HFR,8X,5HTZER0,7X,AHTREF,8X,5HC0NTZ

1 ,10X,3HAEX/ /> ?11 F O R M A T ( X , I 5 » 4 X , F 7 . 3 , 3 X , F R . 2 , 3 X , F 8 . 2 , 3 X , F 1 2 . 6 , 3 X , F 1 2 . 6 / ) 312 FORMAT <12H0SURFACE,N0=I2,6H TYPE=I2,7H SHAPE=IA,?^ RS1=F8.5*

15H RS2=F8.5 ,5H SL1=F8 .5 ,5H S 1 2 = F 8 . 5 , ) 3^? FORMAT (1 2H^ SURFAC t: N 0 = I 2 , l^HSINK TEMP=,F8 .2 ,2X ,3H H=F8.5,4t f FS =

1FP.5 ,AH FR=F8.51 31A FORMAT < l H 0 , 2 A 8 ? 1 9 1 2 i 315 FORMAT i T.HC, 2PX *29HEND OF PRINTOUT OF INPUT DATA ) 316 FORMAT(1HO,10X,19H METRIC SYSTEM USED ) 317 FORMAT(1H0,10X,19HENGLISH SYSTEM USED ) AOR FORMAT (1H , 1 OX,2HJ = , 1 2 , 8 X , 2 H R = , F ! 0 . 5 ) 409 FORMAT <1H , 1 OX, 2HI = , I 2,7X. ,3MEL=, F10 . 5) - 1 0 F0RMAT(M4,BE14.5. ) A13 FORMAT(3H I = , 12 ,2X ,2HJ = , 1 2 ,2X,8HDELTA R = , F i n . 6 , 2 * X , 8HDELTA L=,

l F l o .6 ,12X »6HREG10N 114} A14 FORMAT(4HOEL( , I 2 , 2 H I = , 1 P E 1 6 . 8 , A X , 2 H R ( , 1 2 , 2 H > = , E 1 6 . 8 , 4 X , 6 H R E G I O N »

l I A t A X , 2 H T = , E 2 0 . 1 O / 2 n x , 3 H Q W = , ^ 1 6 . 8 , 3 X , 3 H C l = , E 1 6 . 8 , 3 X # 3 H C 3 = , E 1 6 . 8 / 21PX»5HADEN=,E16 .8 ,3X ,3HC2= ,E16 .8 ,3X ,3HC4= t E16 .8 )

A16 FORMAT<3H0I=, 12, 2,* , 2HJ= , 12 ,6X ,4HR0M=, 1PE14.6,3X»4HR IM= ,E1 4 . 6 , 1 2X,5HDELR=,E14.6,2X*5HDELL = » El 4 . 6 / 1 6 X • 5 H D I L R = , E l 4 . 6 , 2 X , 5 H D I L L = , 2 E 1 A . 6 1 3 X , 4 H V 0 L = , E 1 4 . 6 )

A?6 FORMA.T ( 5 ( 8H 1 J K ,7X , 1HT „6X , 1H*J / J 437 FORMATC HO, I 2 ,13 , 1 2 ,1 P01 A„ 6 , 1 H * , 4 < 2 1 3 , 1 2 , D14.6 , 1H*) ) A45 FORMAT (1H , 3 ^ X , 3 9 H I N I T I A L TEMP DISTRIBUTION WITH NCOUNT = , I 4 / J A46 FORMAT(1 H I , 3 0 X , 8 H N N T ( I , J ) , 1 0 X , 8 H N C 0 U N T = , 1 4 1 4A7 FORMAT( lH0 ,30(12 ,2H- ) ) 451 FORMAT(1 H I I A61 FORMAT(X ,2H I= , I 4»2X ,2HJ= , I 4 , 2X ,2HK= , I 4 ,4X ,7HNC0UNT= ,14J 475 FORMAT( 1 HO,2OX,34HINC0RRECT NUMBERING OF INPUT CARDS J

RFTURN END

6 9

SUQPCUTINC STFADY(NCOUNT tNOPP t D T M , H G , I H , ! L , I N t L ^ T , MAXI ,MAXJt 1 NLS»NCS»T«=NT , T T N T i T O T A L t K M I N , L I T f T P a C . c A C , N C Q » M M X , K A t M G f t P f N I , MJ )

PFAL*'- M T L ( l n ) rCMMON/AB/MTL COMMON N C c , 7 s ) t C 1 P " f 7 K ) 1 C ? ( 7 * f * » * ) t C : ? ( 7 * t - ' B ) T C M 7 5 t 7 5 ) t

1 GM ,*7 C ) ,ADFN<7= ,75 ) , H A ( 7 C , 7 C ) t R A ( 7 r j f N N T ( ->c f -tc; , f T ( 7 5 f 7 ^ ) t ? K P ( 7 c : , _ r c ) tP ( 7 = ) f f l < 7 k ) , S T ( ? c ) f T Z ^ O P c ) , T R F F ( ' S) tCONTZl 1 , " ^ P l C f ) , P ? ("! c ) n e ) » F L 2 ( 1 P J , M A X I T ( 7 0 ) , A<=x (1 K ) , ALPHA (1 0}

DIMFNSICN GPWH •?) ,GPW? f e l TATA T K / ? " 7 ' a . 1 ^ / » T 0 L / . r r , r r r ' , / t P I ? / 6 . ' ' p 3 1 * * / ,TR 0 . / TA = TK I F ( K A . F Q . ? ) TA = TR I F ( NCOUNT. FQ.1, ) GO TO ?

' «"ALL P P ^ S F T ( C T M , F A C » H G , I H , I L , I N , K A , L M T , M A X I , M A X J t M M X , 1NCOUNT,NGAP,NI ,NJ tNLS»NOPB*NRS,T*NT,T INT ,TOTAL)

' I c ( I H . F Q . 8 ) IG = ~> NCO = NCOUNT LIM = LMT * ? * * N C O SPFC = c AC /FLOAT(^eOUNT) PPINT , NCOUNT NCOUNT = NCOUNT + 1 j t f p = r

DC ? I = GPW ( I ) = 0 (MAXJ) GPW ( I ) = r .

? CONTINUC

TFN = TFNT + T I N T * * l _OAT(NCO» MJM = M A X J - 1 MIM = MAXI - •» MTJ = ( M JM I T O I c ( MT J * 1 " . F Q • M J M ) GO TO ° NPAGC S = MTJ + i GC TO 1 0

e NPAGFS = MTJ T LCW = ?

NUP = 1 0 T M I N = 1 0 0 0 . TMAX = r . NT SJ = o n c k = 0 c u f = r . T ST = 0 . s u m = 0 . ITFR = IT R + I p ( I G «FQ» 7 ) P R I N T A « , I T « , t c n

DO c 0 J = 1 , MAX J re i = i , m ax i I F ( N( I , J ) • CQ . M GO TO IF (KR ( I , J ) . L T . - M > GO TQ n I F ( N ( I , J ) . c Q . f ) T ( I , J ) = 1 1 J - 1 ) + T { I t J + 1 ) ) I F ( N ( I , J ) . = 0 . 1 ) T ( I , J ) = 0 . ? * ( T ( I + 1 t J J + T ( I - i , J ) )

" 7 I F ( N ( I , J ) . F Q . C ) GO TO TCLD = T ( I , J ) I c ( J . F Q . ' " ) GO Y n I r ( J . « = Q . M A X J ) GO T 0 ">0 T F ( I . F O . " ! ) GO t o ? i I P < I . F C . M A V I ) GO TO ?? AMUM = n | i , j j * T U , j t M +<" " M I , J ) * T { I , J-"1 ) + C M I , J , j ) +

1 c l ( I , J ) * T ( I + 1 t J ) I S -

r

TO

I F ( N ( I , J ) . G T . l O O ) IS = N ( I t J I - 1 0 0 I F ( I S . G T . l ^ r ) I S = I S - 10P GO TO ??

?0 ANUM « C 2 ( I » J ) * T U , M J M ) I S = 3 GO TO 25

21 ANUM = C M I t J ) * T ( 2 , J ) 1 F ( N N T ( I , J ) . E Q . l ) T ( I « J ) = 2 . * T ( 2 , J ) - T ( 3 , J ) I S = 1 GO TO 25

2? £NUM = C ? ( I , J ) * T ( M I M , J ) I F ( N N T ( I , J) . E Q . l ) T ( I , J ) = 2 . * T ( M I M , J ) - T ( M A X I - ' ? t J ) I S = ?

2 K M = NNT ( I , J ) GO TO ( A o , 3 P f 3 1 , 3 2 , 3 5 , 3 1 , ? : ? ) M rooc

COOK CALCULAT IOM BRANCHES DEPENDING UPON TYPE OF HEAT TRANSFFR AT THE NODE f o r c M = i INSULATED SURFACE - N3 CALCULATION GO TO 49 C 0 0 c M = ? CONVECTION TO A SINK GO TO 30 C 005 M= i RADIATION TO A S I N K GO TO 3 1 Cope M = A COND TO A GAS GAP I N THE POSIT IVE OIRFCTION GO TO 32 COO* M= c CONDUCTING NODE GO TO 35 COO!? M= 6 CONVECTION AND RADIATION TO A SINK GO TO 3 1 QCC* M = 7 COND TO A GAS GAP I N THE NEGATIVE DIRECTION GO TO 33 COO*

TNEW = ( H A ( I , J ) * S T ( I S ) + ANUMJ / ( H A ( I , J ) + A D E N ( I , J ) ) GO TO A t TA1 = TA + TOLD TA? = TA + S T ( I S ) TP AD = ( T A ' < + T A 2 ) * { T - A 1 * T A ' ! + T A 2 * T A 2 ) ANUM = ANUM + TR AO*R A ( 1 , J ) *ST ( IS ) A D F N ( I , J ) = A D F N ( I , J J + T P A P * R A ( I , J ) I F ( M . N E . 6 ) GO TO 3F ANUM = ANUM + H A < I , J ) * S T ( I S I A D E N ( I » J ) = ADEN( I , J ) + H M I , J ) GO TO 3?

32 K = K R ( I i J ) KA = K R ( I , J - 1 ) K^ = K R ( I » J + 3 ) GASK « C O N T Z ( K ) * ( ( T K + r . F * ( T ( I , J ) + T ( I , J + 2 ) ) ) / T Z F R O ( K ) > * * A E X ( K ) DFLGA = P { J + ? ) * C l . + A L P H A ( K 7 ) * ( T ( I , J + 2 ) - T R E F ( K ) l ) TFLGB = P ( J ) * ( 1 . + A L P H A ( K A ) * ( T ( I f J ) - T R E * « K ) J ) DELG = DELGA - DELGB I F ( D F L G . G T . T O L ) Q G = G A S K * P I 2 * R ( J ) * ( E L ( I + 1 ) - E L ( I - 1 ) ) / D E L G TA1 = TA + TOLD TA2 = TA + T ( I , J + 2 ) TRAD = ( T A i + T A 2 ) * ( T A I * T A 1 + T A 2 * T A ? l TNEW = (ANUM + ( T R A D * R A ( I , J ) + Q G ) * T ( I , J + 2 J ) / ( A D E N ! I , J ) + QG +

1TRAD*RA( I , J ) ) I F ( D E L G . L T . G P W l ( K - l O ) ) G P W M K - 1 0 ) = DELG I F ( D F L G . G T . G P W 2 ( K - 1 0 ) ) G P W 2 ( K - ! 0 ) = DELG I F ( I N . N F . 1 3 ) GO TO 4 6 NCK = NCK + 2 PRINT A 5 5 , 1 , J , K ,TOLD, TNEW, GASK, DELG PRINT t j , K , T A 1 , T A 2 , T R A D GO TO A t K = KR ( I , J ) KA = KR ( I , J - 3 ) K 7 = K R ( I , J + l I GASA' = (TA + • 5 * ( T O L D + T ( I , J - 2 ) ) ) / TZERO(K) GASK = C O N T Z ( K ) * G A S A * # A E X ( K ) DELGA = R ( J ) + R ( J ) * A L P H A ( K 7 > * ( T O L D - T R E F ( K ) )

7 1

DELGB « R ( J - 2 ) + R ( J - 2 I * A L P H A < K 4 ) * | T ( I , J - 2 ) - T R E F < K J ) DELG = DELGA - DELGB IF(DELG.GT.TOL) Q G = G A S K * P I 2 * R < J ) * | E L C I + l l - E L < I - 1 ) I / D E L < 5 TA! - TA + TOLD TA2 = TA + T ( I » J — 2 ) TRAD = < T A ! + T A 2 ) * d A l * T A l + T A 2 * T A 2 ) TNEW=(ANUM+( TRAD*RAd » J l + Q G I * T ( I , J -2 ) I / (ADEN(1«J)+QG+TRAO#RA( I« J ) i I F ( D E L G . L T . G P W K K - I P ) ) GPWl (K- l f t ) « OELG IF(DELG.GT.GPW2(K-10J) GPW2CK-10) . DELG I F d N . N E . 1 3 ) GO TO NCK = NCK + 2 PRINT * 5 5 »I ,J,K,TOLD,TNEW,GASK,DELG PRINT A ? A , I , J , K , i ; A l f T A ? , T P A D GO TO

35 I F ( A D E N d , J ) . E Q . P . ) G O TO TNEW = (QW(I ,Jl+ANUM)/ADEN d , J )

A6 DIF«= = TNEW - TOLD ADIF = ABS(DIFF/TNEW) T ( I , J ) = TNEW + TEN * DIFF I F ( T d t J ) . L T . T M I N ) T M I N = T t I , J ) SUM '= SUM + ADIF I F ( A D I F . G T . T E S T ) TEST =ADIF I F ( M . E 0 . A . 0 R . M . E Q . 7 ) GO TO 48 I F ( H A d , J ) . N E . O . O R . R A d , J ) . N E . P )

! CUE = CUF + H A d , J ) * ( T d , J ) - S T d S ) ) • TRAP*RAI 11 J ) * ( T ! I » J I - S T I I S M I F I T I I , J ) .LE.TMAX) GO TO AP TMAX = T ( I , J ) 11 = I JJ=J GO TO

47 T d , 1 ) = 2 . * T ( I , 2 I - T d , 3 ) A8 I F ( I G . N F . 7 ) GO TO A9

NCK = NCK + 1 IF (NCK/5P *5P•ECU NCK) PRINT »ITERfTFN PRINT * 3 A , I , J , K R d , J ) , T O L D , T N E W , T d , J ) ,TEST, SUM,CUF, IS

AO CONTINUE EC CONTINUE

QTEST = ABS(CUE/HG-! • ) I F ( I T E R . G T . 2 ) IG = 0

C n r o IL IS OUTPUT OPTION V I F I L . E Q . l ! PROGRAM PRINTSTEMPERATURE COOo ARRAY FOR THE FIRST «=IVE ITERATIONS ANO COO° EVERY TENTH ITERATION. THIS TEMPERATURE C^ro ARRAY CONTAINS TEMPERATURES FOR ONLY C009 THE REGULAR NODES PIUS THE OUTER CYLINDRICAL COO° NODES COPo

I F ( I T E R . L E . 5 . A N D . I L . E Q . i l I GO TO 55 I F d T F R / 1 0*1 O.NE. ITER) GO TO 10 I F ( I L . N E . l l ) GO TO 62

5? DO 60 K= ItNPAGES PPINT A51 PRINT * 3 5 , ITER,TEST,SUM,CUE,NCO ,DTM,TOTAL,H6iTEN,QTEST,SPEC,

I TMAX, P( J J ) , E L d I ) IF(NUP.GE.MAXJ) NUP « MAXJ PRINT 4.36 00 57 I = 2 . MAXI.2

E7 PRINT * 3 7 , ( I , J 9 KR(I , J) ,T d , J I , J= LOW, NUPt 2 ) LOW = LOW + i n

60 NUP = NUP + IP PRINT P.RINT A35 , ITER,TEST,SUM,CUE,NCO ,DTM,TOTAL,HG»TFN,QTEST,SPEC,

1TMAX,P(JJ ) ,EL d 1 1

72

DO 61 1 = 2 , MAXI , 2 61 PRINT I . MAX J t T<I ,MAXJ» 6? I F U T F P . G E . L I M ) GG TO <?98

IFCTCST.GT.DTM) GO TCI 6? PRINT A57 ,TEST,DTM, ITER NTEST a NTEST + 1

6? IPCSUM.GT.TOTAL) GO TO PP1NT tSUMfTOTAL»ITER NTFST a NTEST + 1

fA 1FI0TPST.GE.SPEC) GO TO 6E PPINT A«f ,QTFST , SPEC»ITER NTEST = NTCST • 1

h* I F { N T C S T . L T . 3 ) QO TO 10 DTM = DTM/2 .

«9P MPj n MAXJ/c

IF (MPJ*c .EQ.MAXJI GO TO 1 0 E

NPAGFS= M eJ + 1 GO TO

* OE NPAGFS a M.PJ ' O f LOW = 1

NUP '« K

DC 'OP K = ' ,NPAGFS PRINT A H PPT NT u e , NCCt ITER PPI NT a ? « , S U M , T O T A L , I T E R PRINT AC-»,TEST,OTM,ITEP PRINT A f A f 0 T 5 S T , S P c C , I T E P 1CCNUP.GC .MAXJ) NUP = MAXJ PP 1 NT a " i f . TO T T j = 1 tMAXI

!<*•» PRINT CI » J » K R C I , J ) , T ( I» J ) , J = LOW,NUP) LOW s LOW + F

*0P NUP = NUP • c

I F C N G A P . P Q . f . ) GO TO PPINT A?? DC I = 1»NGAP IGM » I • \ f PPINT A?3, IGM,GPW1CI) ,GPW2CI)

1 O c CCNTINUF M6X|T(NC0) » ITER IFCNC0UNT.LE.NRS1 GO TO 1 IFCNCOUNT.LE.NLSI GO TO 1 KMAX * T M A X / i n . • 1 . KMIN a T M I N / 1 0 . TFAC ® KMAX - KMIN L I T a L IM - ITER 1 F C I T E R . L T . L I M I PRINT A « 0 , T M A X , T " T N , K M I N t T F A C , L I T I F C I T E P . G E . L I M ) PRINT A52,1TFR RFTURN

A?? FOPHATfIHl,30X»28HCALCULATED WIDTH OF GAS GAPS//5X,7HGAS GAP,6Xt 1«HMIN W10TH,6X,<?HMAX WIDTH/ ) A?' FORMATClH^^Xt^^XfPin.S.KX.Fl^.S/)

a?? F 0 P M A T U H l , 3 0 X r 3 3 H * * * * * START OF CALCULATIONS * * * * * / t l H 0 » 3 E X , 7 H N C 0 U N T = , I A )

A"»? FORMATC1H1,2f*X ,17HCHECK CALC. ITER=, 15,AX »4HTEN=,F1 n . f c / / A X , l H I , 3 X , 11HJ,2X,eHR5G10N,12X»AHT0L0,13X.AHTNEWf16Xt6HTCI»J)»12X,AHTEST,9Xt ? 3 H S U M , 9 X , 3 H C U E , 9 X , 2 H I S / )

A3A FORMATC1H , 3 1 4 , 6 X , I P 3 E 2 0 . 8 , 3 C P F 1 1 . 6 , 2 X ) , 1 4 ) A?*? FORMAT<6H I T E R ® , 1 5 , A X , 5 H T F S T = , E l 6 . 8 , A X , 6 H S U M » , F 1 6 S 8 , 6 X , 4 H C U E = ,

lEJ6 .8 ,AX,7HNC0UNT = , I 4 , / 1 5 X f E H D T M = , E 1 6 . 3 , 4 X , 6 H T 0 T A L « = , E 1 6 . 8 » 6 X , 2«HHG a , F 1 6 . 8 , 7 X , 4 H T E N « , F P . 4 / 6 6 X , 6 H Q T E S T » , E l 6 . 8 , 6 X , 5 H S P E C » , E 1 6 . 8 / 31?X,5HTMAX = » F l n . 5 , 4 X t 9 H W l T H P. 0 F , F 1 0 . 6 , 4 X , 8 H A N D L OF , F ! 0 . 6 / / )

73

A36 F0RMAT(5(8H I J K,7X ,1HT , 6 X , 1 H * ) / ) W F 0 R M A T U H ' M 2 , I 3 , , I 2 , l P D l A . 6 f l H * , A ( 2 l 3 , I 2 , D l A . 6 , ' ! H * > ) A30 F O R M A T C 1 H 0 , A X , 2 H I e , I 3 » A X » 2 H J = , I 3 , A X , 2 H T « , l P D ? 0 . 1 0 ) A * 0 FORMAT(1HI,2OX,24HPROBLEM TFRMINATED AFTER,16,3X,10HITERAT!0NS /

H H P , 2 ^ X , 3 3 H M I N . N 0 OF NODES BETWEEN SURFACES , IA) FORMAT (1H , 3r iX »39H FINAL TEMP DISTRIBUTION WITH NCOUNT = , I A ,

n 0 X , 5 H I T E R = , I 6 / l A50 FORMATUHl, 3HX, 38HALL CONDITIONS MET, END 0 * CALCULATION / /

11CX ,5HTMAX=,F10, 5 ,AX, ?HTMI N=, .F10 .5 ,^X,5HKMIN=, ' I5 ,AX,5HT C AC = , F 1 0 . 2 , 2 A X , A H L I T = , I 6 / )

FORMATdHl) FORMAT< lHl ,2OX,AnHCALCULATI0N TERMINATED ARBITRARILY AFTER,16,

12X»1OHITERATIONS i FORMAT <1H0,1 OX, 6HQTEST=,E"! fl.P ,1PX,5HSPEC = , E 1 8 . 8 , 1 P X , 5 H I T E R = , 1 6 / 1

AS7 FORMAT(1HO,11X,5HTFST«,E18 . « » , 1 1 X , A H D T M « , E 1 8 . 8 , K X , ( > H I T E R = , I 6 / I F O R M A T U H 0 , ! 2 X , 4 H S U M = , E 1 P . 8 , 9 X , 6 H T 0 T A L = , E 1 8 . 8 , 1 0 X , 5 H I T E R = , I 6 / l FORMAT P. H , 3 I 4 , 6 X , ! P 2 E 2 0 . e , A X , 5 H G A S K = , E 1 6 . 8 , A X , ' 3 H D E L G = , E 1 6 . 8 » END

SUBROUTINE RPLOT(MAXJ,MAXI,KMIN,TFAC,NI,NJ) REAL*P MTL(IC) COMMON/AB/MTL COMMON N(75,75l, C1<75,75),C2<75,75),C3<75,75),C4(75,75) 1QW(75,75),ADEN<75,75) ,HA(75,75),RA(75,75),NNT(75,75),T(75»75), 2KR(75,75 i , R ( 7 51 ,Et 175> , S,T« 25 J ,TZEROll5 > ,TREF( 151 ,CONTZ(15» , 3R1(15),R2<15 I,ELI(15),EL2<15),MAXITUO) ,AEX<15) ,ALPHA< 10) OIMENSION BUF(4PO0) CALL CRT(BUF,4000,'CRISTY$1,01 A = J6./ R(MAXJ) ' B = •? /TFAC NIS = N!/2*2 DO 20. I = 2,MAXI,NIS CALL CRTSYM(1.,1P.5,.25,*5HRADIAL PLOT OF CYLINDER TEMP PATTERN 1 EL = ,0.,*5) CALL CRTNUM(999.,999.,0.25,EL<I),0.,•(F7.4I•) CALL CRTS YM( 8 . , 6HR/RMAX ,^.,6) CALL CRTSYM( .10,4.,.?0,4HTFMP,90.,4) TM = 1 0*KMIN Y = 1. TP = KF = TPAC * 1 . KTM = KMIN * ln DO 10 J = 1,KF CALL rRT(l.,Y,0,l) CALL CRT< 17.,YtlO,l) CALL CRTNUMl 0.5 ,Y,.l ,KTM,r. , M 13) M Y = Y + TP

10 KTM = KTM +10 AI = X =1. DO 15 J = 1,11 CALL CRT< X,1.,1) CALL CRT (X,10. ,10,1) CALL CRTNUMl X-.30,0.5, 0.10,AI ,0. »•IF4.1) • ) X = X + 1 .6

15 A1 = AI +0.1 CALL CRT(A,B,0,4) CALL CRT(1./A,l./B,r,3) Z = T(I,1) - TM CALL CRT(0.,Z,0,1) DO 30 J = 2,MAXJ Z = T(I,J ) - TM

30 CALL CRT( R(J),Z,15,1) CALL CRT(l.,l.,r»,4) CALL CRT(0.0,0.0,0,3) CALL CRT(0®,Os,0001,2)

2f CONTINUE RETURN END

75

SUBROUTINE ELPLOT< MAXJ?MAX I , K M I N , T F A C , N1, N J > REAL*8 MTL( IO) CCMMON/AB/MTL COMMON N ( 7 5 , 7 5 ) , C 1 ( 7 5 , 7 5 ) ,C2 ( 7 5 , 7 5 ) , C3 <7? , 75) , C M 7 5 , 7

1QW(75,75) ,ADFN(75. ,75) , H A ( 7 5 , 7 5 1 , R A ( 7 5 , 7 5 ) , N N T ( 7 5 , 7 5 ) , T < 7 5 , 7 5 ) , 2KP(75 , 7 5 ) , R ( 7 5 ) , E L ( 7 5 ) ,ST>( 25) ,TZERO(15) ,TREF( 15 J ,CONTZ(151, 31*1(15) ( 1 5 ) ,EL1 (15 ) ,EL2( 15) ,MAXIT.(10) * A E X ( 1 5 ) , A L P H A ( 1 * )

DIMENSION BUF(4000) CALL CRT(BUF , 4 0 0 0 , ? C R I S T Y $ ' , 0 ) A = ? 6 . / F L ( M A X I ) B = «Q /TFAC NJS = N J / 2 * 2 DO J =2,MAXJ,NJS 1 F ( K R ( 2 , J ) . G F . l l ) GO TO 2n

CALL CRTSYMd . , 1 0 . 5 , . 2 ? , A 8 H A X I A L PLOT OF CYLINDER TEMP- PATTERN 1 RADIUS O F , 0 » , 4 8 )

CALL CRTNUM(9 9 9 . , ° 9 ° . , ' , . 2 f , R ( J ) , < " , . t * ( F " ? # 4 . ) » > CALL CRTSYM(8 . ,0 . , 0 . 3 0 , 8HEL/ELMAX,0 . ,81 CALL CRTSYM( . 1 0 , 4 . , . ? 0 , A H T E M P . . 4 ) TM = i r * K M I N Y = 1 . TP = 1 0 . * B KF = TFAC + 1 . KTM = KMIN * 10 DO 10 I s ! f R F CALL C R T ( 1 . , Y , o t l ) CALL CRT( I 7 . , Y , 1 0 , 1 ) CALL CRTNUM(0.5 , Y , . l , K T M , 0 . , • ( 1 3 ) • ) Y = Y + TP

10 KTM = KTM + 10 AI = 0 . X s i . DO 15 I = 1 , 1 1 CALL CRT(X»1 . , 0 , i ) CALL CRT ( X , 1 0 . , l n , ! ) CALL C R T N U M ( X — . 3 0 , 0 . 5 , 0 . 1 0 , A I , 0 . , M F 4 . 1 ) • ) X = X + 1 . 6

"»•= A I = A l + ' « . ! CALL CRT( CALL C R T ( 1 . / A , 1 , / B , 0 , ^ ) Z = T ( 1 , J ) - TM CALL C R T ( 0 . , Z , 1 | DO 30 I = ? ,MAXI Z = T ( I , J ) - TM CALL CRT ( E H I ) , Z , 1 5 , 1 )

' 0 CONTINUE CALL C R T n . f ! . ,<>, *> CALL C R T ( O . c t n . r t 0 , ? | CALL C R T ( 0 . , 0 . , 0 0 0 ! , 2 1

?0 CONTINUF RETURN END

76

SUBROUTINE PICT(M.AXI, MAXJ, NGAP,NOPR,NCO,MMX, T E N T , f I N T ) C THIS SUBROUTINE MAKES A SCALE DRAWING OF THE CYLINDER AND LISTS C MAJOR INPUT DATA

DIMENSION BUF(5000) DIMENSION Al ( 1 5 ) A 2 ( 1 5 ) , BLI ( 1 5 ) , BL2(15 ) REAL#8 MTL( IO) COMMON/AB/MTL COMMON N ( 7 5 , 7 5 ) , C 1 (75 ,75 ) . , C2 (75 , 75 ) ,C? ( 7 * , 75 ) ,C4( 75 , 7 5 ) ,

1 QW(75,75) ,ADEN(75 ,75 ) ,HA(7 5 ,75 i , R A ( 7 5 , 7 5 ) VWNT (-75,75 I , T( 7 5 , 7 5 ) , ' 2 K R ( 7 5 , 7 5 ) , R ( 7 5 ) , E L ( 7 5 ) , S T ( 2 5 ) , T Z E R 0 ( 1 5 ) , T R E F ( 1 5 ) , C O N T Z ( 1 5 ) , 3 R l ( 1 5 ) , P 2 ( 1 5 ) , E L l ' ( 1 5 ) , E L ? ( l 5 ) , MAX I T f i r ) , AEX(15) ,ALPHA(10)

CALL CRT(BUF,5000 , •CRI STY' , 0 ) C DRAW THE CENTERLINE

CLY =f r . 5 DO I 1= 1 , 6 CALL C R T ( 1 . , C L Y , 0 . 1 ) CLY = CLY + I . P CALL' CRT ( 1 • , C LY, 1 2 , 1 ) CLY = CLY * 0 . 2 5 CALL C R T ( l . t C L Y f P f l ) CLY = CLY + P . 2 5 CALL CRT< 1 . , C L Y , 1 2 , 1 ) CLY = CLY + r . 2 5

1 CONTINUE C DRAW OUTLINE OP CYLINDER,USING VERTICAL DIMENSION TO SET SCALE

A = e . / E L ( M A X I ) IF {R(MAXJ) .GT . F L ( M A X i ) ) A = 9 . / R(MAXJ) CALL CRT<A,A,0 ,A) CALL C R T ( 1 . / A , 1 . / A , 0 , 3 1 CALL C R T ( ^ . , r . t r , l ) CALL C R T ( R ( M A X J ) , 0 . , 2 0 , 1 ) CALL C P T ( R ( M A X J ) » E L ( M A X I ) , 2 0 , 1 ) CALL C R T ( " . , F L I M A X I ) , 2 D , 1 ) DO 5 K = ? , MMX CALL C R T ( R 1 ( K ) , E L 1 ( K ) , 0 , 1 ) CALL C R T ( R 1 ( K ) , E L 2 ( K ) , 1 5 , 1 ) CALL C R T ( R 2 ( K ) , E L 2 . ( K ) , 1 5 , 1 ) CALL C R T ( R ? ( K ) , E L 1 ( K ) , 1 5 , 1 ) CALL C R T ( R 1 ( K ) , E L I ( K ) , 1 5 , 1 )

e CONTINUE IF(NGAP.EO.O) GO TO 7 IGM = ! r + NGAP DO.6 , K= 11 , IGM CALL C R T ( R 1 ( K ) , E L 1 ( K ) , 0 , 1 ) CALL C R T ( R 1 ( K ) , E L 2 ( K ) , 1 5 , 1 ) CALL C R T ( R 2 ( K ) , E L 2 ( K ) , 1 5 , 1 ) CALL C R.T ( R 2 ( K ) , E L I ( K ) , 1 5 , 1 ) CALL C R T ( R 1 ( K ) , E L 1 ( K I , 1 5 , 1 )

6 CONTINUE C RETURN SCALE AND ORIGIN BACK TO I N I T I A L VALUES

CALL C R T ( 1 . , 1 . , 0 , 4 ) CALL C R T ( 0 . , P . , 1 , 3 )

C PET NT DIMENSICNS ON DRAWING DO 10 K= I,MMX A K K ) = 1 . + A * R l ( K ) A2(K) = 1 . + A * R2CK) BJLA ( K ) - 1 . + A * ELI ( K) B.L2 (K) = 1 „ + A # EL2(K) CALL CRTNUM(AKK) , 0 . , 0 . 1 , P 1 ( K ) , 9 0 . , • ( F 8 . 4 ) • > CALL CRTNUM(A2(K) , 0 « , 0 « 1 , R 2 ( K ) , 9 0 . , , ( F 0 . A . ) » )

TT

CALL C R T N U M i r . , B L 2 < K I , 0 . 1 , E L 2 ( K ) , 0 . , « r F 8 . 4 ) » ) CALL C R T N U M ( 0 . , B L 1 ( K ) , 0 . 1 , E L I ( K ) , 0 . , 8 < F 8 . 4 I • I

C LABEL EACH REGION IF ( K..EQ.1 I GO TO X = 0 . 5 * < A 2 < M + A 1 ( K ) ) Y = 0 . 5 * ( B L 2 ( K) + B L 1 ( K ) ) CALL C R T N U M ( X , Y , 0 . 2 , K , ' o . , • ( l i | • )

10 CONTINUE IF(NGAP.EQ.O)GO TO 12

C PLOT GAS GAPS DO 1 1 K =1111GM A ] ( K ) = 1 . + A * R1(K) CALL CRTNUM(AKK) , 0 . , n . \ ,R i (K) ,90 . t • (F8 . 4 ) • ) X = A ! ( K l - 0 . 0 5 Y = 0 . 5 * A * ( E L 1 ( K ) + E L 2 t K ) ) CALL CRTSYM(X,Y,0.1P,11HGAS GAP N 0 . , 9 O . t l l ) CALL CPTNUM( 9 99.. , 9 9 9 . , 0 o t K , 9 0 . , 8 ( 1 2 1 • )

11 CONTINUE C PRINT PROBLEM CONDITION'..

X = 1 r . 5 CALL C R T < X , 0 . , 1 , 3 ) CALL CRTS YM(n . , '9 . , 0 . 4 f> , IPH PROBLEM N O t r u t i p | CALL C R T N U M ( 9 9 9 . , 9 O o . , 0 . ^ 0 , N 0 P R , 0 . , » I CALL C R T S Y M ( . 5 , 8 . , 0 . 2 5 , 6 H R E G I 0 N , 0 . , 6 ) CALL CRTSYM{2.»8.,<">.2 3,8HMATERIAL , 0 . , 8 1 CALL C R T S Y M t * . , 8 . , 0 . 2 5 , 3 H N C 0 , O . t 3 ) CALL C R T S Y M ( 5 . , f f . , 0 . 2 5 , 4 H I T F R , 0 . , 4 ) Y= 8 . 0 DO 20 I = 1 , MMX Y = Y - 0 . 5 CALL CRTNUM(1 . , Y, 0 . 2 0 , I , 0 . I I P » CALL CRTSYM(2 . ,Y ,0 .20 ,MTL< I ) , 0 . , 8 )

?0 CONTINUE Y= 8 . 0 DO 25 I = 1 , NCO Y= Y - 0 . 5 CALL CRTNUM(4. t Y , 0 . 2 , I , 0 . , M I 2 ) M CALL CRTNUM(5 .»Y,0 .2 ,MAXIT ( I ) , o . , » { I 4 j » |

' 5 CONTINUE Y= Y - 0 . 5 CALL C R T S Y M ( 4 . , Y , 0 . 2 , 5 H T E N T = , 0 . 0 , 5 ) CALL C R T N U M < ? 9 9 . , Y , 0 « 2 , T E N T , 0 . , M ' = 6 . 2 ) 1 ) Y= Y - 0 . 5 CALL C R T S Y M ( 4 . t Y , 0 . 2 , 5 H T I N T = , 0 . 0 t 5 ) CALL C R T N U M i 9 9 9 . , Y , 0 . 2 , T I N T , 0 . , M P 6 . 2 ) M CALL C R T ( 0 . 0 , 0 . 0 , 0 , 3 ! CALL CRT(o.,o.,roonf2j RETURN END

8 16 2k * 1 40 48 56 | 6h 72 76 801

FIRST CARD TITLE 1

SECOND CARD N0PR MMX NS NGAP NRS NLS MFR MPL LMT MI Nil KA 1 2

THIRD CARD DTM TOTAL FAC TENT TINT 3

F0TJPH CARD 0UTP 0PT IA IB IC ID IE IF IG IH IJ IK XL IZ IN k\

REGI0N (MATERIAL) CARDS - 8 CARDS PER REGI0N | MTL(I) R1(I) R2(I) EL1(I) EL2(L) RH0(L) C0N(L) Q ( I ) TI(I) I l j0( l ) WCA

ALFHA(I) N0(I) NCB

GAS GAP CARDS -1 CARD PER GAP h RF(L) R1(I) RS(I) EL1(I) EL2(I) C0NTZ(I) TZER0(I) TRSF(I) AEX(I) Np( I) WCA

SURFACE CARDS NT(I) BG(I) RSl(l) RS2(I) SLl(l) SL2(L) ST(L) H(I) FS(L) FR(I) ME(I) NCA

See Appendix C for F0RMAT of Input Data

Fig. 8. Data Cards

79

APPENDIX C

INPUT FORMAT

First Card Title Card jritle Cardj

Columns

2-73

7^-79

8o

Second Card

Columns

1

2 - 8

9-12

1 3 - 1 6

17-20

21-21+

25-28

Symbol

Title

Format

9 REAL*8

NCA II

Description

Blank

A title or a description of the

problem assigned by user.

Blank

The number 1

Integer Control Parameters

Symbol

N0FR

MMK

NS

NGAP

M S

Format

17

Ik

Ik

Ik

Ik

MLS Ik

Description

Blank

The number of the problem. May be

any number from 1 to 9999999

The number of regions (limited to 10)

The number of surfaces (limited to 50)

The number of gas gaps (limited to 5)

The maximum number of real nodes to be

located between cylindrical nodal

boundaries.

The maximum number of real nodes to

the located between plane nodal

boundaries.

80

Columns Symbol Format 29-32 MFR Ik

33-36 MPL IU

37-^0 LMT lb

NI

NJ IU

Description The number of a region (other than

region l) which will have twice

as many nodes in the radial

direction.

The number of a region (other than

region l) which will have twice as

many nodes in the axial direction.

A number used to limit the number

of iterations in each series if

convergence is not accomplished

before the limit is reached. Max

no. of iterations = LMT *2 **

NC0UNT.

An indicator to give instructions

on how often to plot the radial

temperature distribution.

2 = every real node

b = every other real node

6 = every third real node, etc.

Same as HI except it controls the

frequency of the plot of the axial

temperature distribution.

81

Columns Symbol Format Description ij-9-52 KA IU An index to indicate whether metric

or english system of units is being

used.

1 = metric system

2 = english system

The program selects the proper values

for the Boltzmann constant and

converts temperatures to the proper

absolute scale.

5 3 - 7 9 - Blank

80 NCA II The number 2.

„ C a r a ["Bear. C o n W M e t e r s ]

Columns Symbol Format Description

1-8 Blank

9-16 DTM F8.0 A number used to check for conver-

gence. When TEST (the largest temp-

erature change made for any one node

during one iteration) is smaller

than DTM this test for convergence

has been satisfied.

17-2^ T0TAL F8.0 An indicator used as the second test

for convergence. When SUM (the total

of all changes made during one

"82

* Columns Symbol Format

25-32 FAC F8.0

33-40 TEET F8.0

41-48 TINT F8.0

49-79 " F8.0

80 NCA II

Description iteration) is smaller than T0TAL

this test for convergence has "been

met.

A factor used in the third test for

convergence. FAC controls SPEC, the

number used in the convergence check

SPEC = FAC/NC0UNT. Convergence is

assumed to be reached when QTEST <

SPEC. QTEST is the absolute value

of the fractional difference between

CUE and HG. CUE is the sum of the

heat transferred from each surface

node during a single iteration. HG

is the sum of the heat generated

within the whole cylinder (Calculated

directly from input data).

A number used to hasten the conver-

gence by overrelaxing. It may range

from 0.0 (no overrelaxation) to 0-9-

A number used to hasten convergence

by overrelaxation. It may range

from -0.5 to +0.5.

Blank

The number 5•

83

Fourth Card [output Options'] "" L J

Columns

1

2-8

9-16

17-20

21-21*.

25-28

29-32

Symbol

0UTP

0PT

IA

IB

IC

ID

Format

REAL*8

REAL*8

Ik

IU

IU

Ik

Description

Blank

The word "0UTPUT''

The word "<

The number 1 in Column 20 will call

for the printout of the R and EL

arrays as first established (only

when NC0UNT = l).

The number 2 in Column 2k will call

for the printout of the R and EL

arrays after rearrangements (only

when KC0UHT =1).

The number 3 in Column 28 will

call for the printout of the

intermediate functions used in

calculating the transfer coeffi-

cients (VOL, ROM, RIM, DILR, DILL,

size and region of regular nodes).

The number k in column 32 will call

for the printout the preliminary

values of constants for regular

nodes if NC0UNT = 1.

84

Columns Symbol Format

53-36 IE 14

37-40 IF 14

41-44 IG 14

45-48 IH 14

Description


for the printout of the region and

node designations of gas gaps.

The number 6 in Column 40 is not

an output control. It will cause

the program to set the outer

cylindrical surface nodes to an

average value calculated from the

heat generation, the heat transfer

coefficient and the sink tempera-

ture.


for printout of the values of T0LD,

TWEW, T(l,J), TEST, SUM, CUE, and

IS for each node for the first 3

iterations when NC0UNT = 1.


reset the index 7 at the beginning

of each series of integers (i.e.

the nodal temperatures and the

convergence check values will be

printed out for the first 3 itera-

tions of each series.

85


1+9-52 IJ 14

55-56 IK 14

57-60 IL Ik

61-64 IZ Ik

6 5 - 6 8 IN 14

Description


call for the printout of the

coefficients of the plane and

spherical special nodes.

The number 10 in Columns 55-56

will call for the printout of the

coefficients of the regular nodes.

The number 11 in columns 59-60 -will

call for the printout of the temper-

ature array (real nodes only) for

each of the first 5 iteration, then

each tenth iteration.

The number 12 in Columns 65-64 will

call for the printout of the indices

used in transferring the tempera-

ture array of one series of itera-

tions to the expanded array of the

next set of iterations. (KX, LX,

NX if NC0UNT = 2 KX, LX, NX, KK,

LL if NC0UNT > 2).

The number 15 in Columns 6 7 - 7 8

will call for printout of inter-

mediate values of nodal data for

each gas gap for each iteration.

86

Columns

69-79

80

Symbol Format

II

uescription

Blank

The number 4.

Region (Materials) Cards - Two cards per region - limited to 10 regions. First Card

Columns

1

2-8

9-16

17-24

25-52

55-40

41-48

49-56

57-64

Symbol

MTL(l)

Rl(l)

R2(I)

EL1(I)

EL2(I)

EH0(l)

C0N(I)

Q(l)

Format

REAL*8

F8.0

F8.0

F8.0

F8.0

F8.0

F8.0

F8.0

Description (and dimensions)

Blank

Name (or abbreviation of the name)

of the material of region I.

Inner radius of region I (cm or ft

Outer radius of region I (cm or ft

Distance from top of whole cylin-

der to top of region I (cm or ft.).

Distance from top of whole cylin-

der to bottom of region I (cm or

ft. )•

Density of region I (gm/cm3 or

lbs/ft3).

Thermal conductivity of region I

(watts/cm deg C or (BTU per hr.)/

ft. deg p).

Heat generation rate (watts/gram

or BTU per hr/lb).

87

Columns Symbol Format Description

65-72

73-76

77-80

Tl(l)

N0(l) KCA

F8.0

14

14

Initial estimated temp, region I

('deg C or deg F).

The number assigned to the region-

Number of the card, number

serially starting with 5.

Columns Symbol Second Card

Format Description

1-8

9-24

Blank

25-72

73-76

77-80

ALPHA.(l) E16.8

NO(l)

NCB

14

14

Linear coefficient of expansion

of region I. If no a for a region

is needed in the calculation, this

may be left blank. However, the

card must be in the deck in all

cases for all regions.

Blank

Number of the region (i).

Card number.

Gas Gap Cards (if any) - one card per gap - limited to 5 gas gaps

1

Description

Blank

88


2-8 RF(I) F7.0

9-16 R1(I) F8.0

17-24 R2(I) F8.0

25-32 EL1(I) F8.0

33-40 EL2(l) F8.0

41-48 C0WTZ(l) F8.0

49-56 TZER0(l) F8.0

57-64 TREF(l) F8.0

Description

Radiation factor (effective emis-

sivity-absorptivity factor) of the

surfaces of the gas gap. If no

value is entered in this space the

program uses a value of 1.0.

Inner radius of the gas gap region

cm or ft).

Outer radius of the gas gap region

(cm or ft).

Upper limit of the gas gap region

(measured from the top of the whole

cylinder) (cm or ft).

Lower limit of the gas gap region

(measured from the top of the

whole cylinder) (cm or ft).

Tnermal conductivity of the gas in

the gap at the temperature TZER0.

(watts/cm deg C or BTU per hr/ft

deg F.)

Absolute temperature at which

C0NTZ is known (deg K or deg R).

Reference temperatures for calcu-

lating expansion (deg C.)

89


65-72 AEX(I) F8.0

73-76 N0(I) Ik

77-80 NCA Ill-

Surface Cards - one card' per surface.


1-k MT(l) Ik

Description

Exponent "a" in the conductivity

equation.

Gas gap region. Number the gaps

consecutively starting with 11.

Card Number

Number consecutively starting with

the next number after the last

region (material) card.

Description

Type of heat transfer at surface

(i) coded as follows:

1 = none (surface insulated).

2 = convection to a sink.

3 = radiation to a sink.

ij- = conduction and radiation to a

gas gap in the position direction.

5 = conduction.

6 = convection and radiation to a

sink.

7 = conduction and radiation to a

gas gap in the negative direction.

90


5-8 ItfG(l) Ik

9-16 RSl(l) F8.0

17-24 RS2(I) F8.0

25-32 SLl(l) F8.0

41-48 ST(I) F8.0

49-56 H(I) F8.0

57-64 FS(l) F8.0

6 5 - 7 2 FR(I) F8.0

73-76 NB(I) 14

77-80 NCA 14

Description

Shape and location of surface

for plane node: NG = 100 + MB.

for cyl. node: NG = 200 + KB.

Min radius of surface (cm or ft).

Max radius of surface (cm or ft),

upper limit of surface (measured

from top of whole cylinder).

Temperature of the sink associated

with surface I.

Heat transfer coefficient watts/cm3

deg C or BTU per hr/ft3 deg F.

Shape factor of the surface.

Emissivity factor.

Number of the surface numbered

serially starting with 1.

Number of the card numbered serially

•with previous cards.

91

APPENDIX D

EXPANSION OF TEMPERATURE ARRAY

In order to impose the old temperature array upon the new nodal

pattern requires a procedure for instructing the machine to compute the

proper new array. The requirement is to expand from one node per node

boundary to two nodes per node boundary, then from two to three, three

to four, etc. Furthermore, it is necessary to adjust for the cases when

either the radial node or the axial node divisions will be less than the

others. Added to this, the gas gap must be allowed to occupy one and only

one real node no matter how finely the rest is divided. This last requires

special treatment because of the change in pattern of nodes after the gap

is spanned.

When PRESET is called from STEADY, the temperature array TT is estab-

lished with the temperature distribution pattern that was calculated in

STEADY. "When NC0UNT = 2, the temperature pattern will be arranged like

this:

I/J 2 k 6 8 /

2 A B c D |

h E F G H (

6 I J K L \

8 M N 0 P V

— ^

TT(2,2)

TT(2,M TT(2,6) TT(2,8) TT(U.2)

TT(1+,10

TT(6,2)

etc.

A

B C D

E

F

I

92

The temperature pattern must "be changed to this:

I/J 2 k 6 8 10 12 Ik 16

2 A A B B c C D D J A A B B n

V c D D 1 6 E E F F G G H H ( 8 E E F F G G H H

10 I I J J K K L L

12 I I J J K K L L

Ik M M N N 0 t P P

16 M M N N 9 0 •D J. P

£—A V : ^ ^ si

T(2,2) = T(2,k) = T(4,2) = T(4,4) = A

T (2,6) = T(2,8) = T(4,6) = T(4,8) = B

T(2,10) = T(2,12) = T(4,10) = T(4,12) = C

T(6,2) = T(6,4) = T(8,2) = T(8,4) = E

T(6,6) = T(6,8) = 1(8,6) = T(8/8) = F

T(10,2) = T(l0,4) = T(12,2) = T(l2,4) = I

etc.

Let T(I,J) = TT(KX,LX)

The necessary pattern can "be reduced to this:

I , J 2 k 6 8 10 12 14 16

KX,LX 2 2 k k r O 6 8 8

93

A little mental gymnastics produces'the following relationships,

using integer arithmetic

Let NCM = NC0UNT -1

NCT = NC0UNT *2

The following equations:

KX = (1+2) * NCM/NCT * 2

LX = (J+2) * NCM/NCT * 2

will produce the desired results.

For converting from 2 to 5 nodes per nodal "boundary.

When NC0UNT = 3, the pattern for TT is:

I/J 2 k 6 8 10 12

2 k

6 8

10 12

A B C D E F ! G H I J K L

M N P r >>

Q R S T

u V

W X 1 _J •

By using equation (k) and (5) plus, these equations:

KK = (1+1) * NCM/NCT * 2

LL = (J+1) * NCM/NCT * 2

T(I,J) = 0 . 5 * |TT(KX,LX) + TT(KK,LL)J

(2)

( 3 )

(4)

(5)

(6)

(7)

(8)

9^

the counting pattern becomes:

Then the T array becomes:

I,J

2 k

6

8

10 12 Ik

16 18

8 10 12

I, J . 2 k 6 8 10 12 Ik 16 18

KX, LX 2 k k 6 8 8 10 12 12

KK,LL 2 2 k 6 6 8 10 10 12

Ik 16 18

Using equations (k), (5), (6), (7), and (8) when MC0UKT = k

(NCM = 3 NCT = 8) the counting pattern becomes:

A A+B 2 B C C+D

2 D E E+F 2 F \

A+G 2

A+H 2

B+H 2

C+I 2~

C+J 2

D+J 2

E+K 2

E+L 2

F+L 2 \

r* Lr G+H 2 H I I+J

2 J K K+L 2" L {

M KH-N 2" N ? $+P

2 P h?

MfQ, 2

M+R 2

N+R 2

0+S 2

0+T 2~

P+2 2

Q Q+R 2 R S S+2 2 T

U U+V 2 V

HI U+X 2

V+X 2

w —

w+x 2 X

I, J 12 4 6 8 10 12 Ik 16 18 20 22 2k

KX,LX 2 6 6 8 10 12 12 Ik 16 18 18

KK,LL 1 2 2 k 6 8 8 10 12 Ik Ik 16 18

95

so that this:

becomes:

2 b 6

2 A B C 1 if D E F \ S" to n H I

8

2 A A+B 2

B+C 2 C

b A+D A+E B+F C+F b 2 2 2 2

6 D+G D+H E+I F+I 6 2 2 2 2

8 G G+H 2

H+l 2 I

^ —

Using the same equations and NC0UNT = 5; NCM = b NCT = 10

2 4 6 8 10 12 14 16 18 20

KX,LX 2 b 6 8 8 10 12 lb 16 16

KK, LL 2 b b 6 8 10 12 12 lb 16

so-that this:

2 b

6 8

8

A B C D j E F G H

I J K L

M N P J L/ S

96

"becomes this: 2 4 6 8 10

2 A B B+C 2

C+D 2 D [

k E F F+G 2

G+H 2 H

6 E+I 2

F+J 2

F+K 2

G+L 2 '

H+L 2

8 I+M 2

J+N 2

J+tf 2

K+P 2

L+P 2

10 M N N+0 2

0+P 2 P

When either dimension fails to expand in going from one value of

NC0UNT to the next the same index is used for the one that does not

expand while the equation for the one that does expand is as given

above. For example:

If the radial direction is not further subdivided:

T(l, J) = 0.5 * JTT(KK,J) + TT(KK,J)J (9)

If the axial direction is not subdivided:

T(l,j) = 0.5 * [TT(I,LX) + TT(I,LL)J (10)

When a gas gap is included, a counter MX is introduced. It is set

at zero each time I is advanced in the D0 loop. NX stays at zero until

the first gas gap is encountered. When that happens, EX is increased by

two. The equations for LX and LL are changed to:

LX = (j+2-NX) * NCM/NCT * 2 + NX (ll)

LL = (J+1-NX) * NCM/NCT * 2 + NX (12)

97

For an example of how this works, assume NC0UNT = 4 (NCM = 3; NCT = 8)

2 b 6 8 10 12 lb

A B C J M N 0 < D E F K P Q R

G H I L S T L — - i

Here the gas gap is located at J = 8. After expansion of the array,

the gas gap will be at J = 10.

I 2 b 6 8 10 12 14 16 18

KX 2 b 6 6 8 10 12 14 14 KK 2 2 b 6 8 10 10 12 14

J 2 4 6 8 10 12 14 16 18 LX 2 4 6 6 8 10 12 14 14 LL 2 2 4 6 8 10 10 12 14

NX = 0 NX = 2 Then the pattern becomes:

2 b 6 8 10 12 14 16 18

2 A A+B 2

B+C 2 c ! i

J M M+N 2 2 0

4 A+D 2

A+E 2

B+F 2

i C+F i •mmi^mmm 2

J+K 2

IVH-P 2

M+Q 2

N+R 2

g+R 2

6 D+G 2

D+H 2

E+I 2

F+I 2

K+L 2

P+S 2

P+T 2

Q+U 2

R+U 2

8 G G+H 2

n+i 2 I L S S+T

2 T+U 2 U

98

APPENDIX E

OUTPUT EXAMPLES

The cylinder illustrated in Figure 9 used to demonstrate the

output. The printout of the input data is shown starting on page

100. An example of the final temperature array is shown starting on

page 103- In that array, the temperatures of all nodes are printed,

including the unused nodes (I = odd, J = odd) which are shown as 0.

Figure 10 is an enlargement of the 35 frame which is produced by

subroutine PICT. The 0RNL computer center has facilities for printing

these 35 Him frames on 10" x 14" paper. Figure 11 is an enlargement

of a single one of the 35 mm frames produced by subroutine RPL0T

showing the temperature vs. radius plot at a fixed point in the axial

direction. Figure 12 is an enlargement of a single one of the 35 mm

frames produced by subroutine ELPL0T showing the temperature vs.

axial position at a fixed value of radius.

Cylinder for Example Problem

100

OUTPUT OF INPUT DATA

GRAPHITE p UEL ELEMENT FOR THESIS PROBLEM

NUMBERS OF THE CARDS READ

0 A c f 1 Q O

1 1 1 ?

1 1 6 1 "7 •I Q

1 C

?0

101

GR£0HITF. PUFL FLFMFNT FO" THcSIS P"OBLEM

PROFLcM NUMBCP= 701

PCG.NOPPS PCTCFM Cyl SUOFACFS= LIMIT ne iTro£TTCMS= '00

NUMRPD O«= PRIONS' ? NUMBFR QP SURFACCS= H RCG MODPS *PTWCCN PLANF SU«PACPS= 3

M TRir SYSTPM USSD ncLTfi T r>TM) 0.000"' MIN SUM(TrTAL> trWT= o.-OOOC T1NT= 0.1 C 000

O.O'OO 0.02000

TGIPN ' WILL HfiVP TWICP THF NUMBFB 0= NOCFS BCTWFEN CYL SURPACFS

OPGIOM ?. WILL HftVP Twice THF NUM?CR 0>= NOCFS BFTWFFN PLAN<= SUR«=ACCS

orGirM

G6P NO

K'ftTPPItL G°A PH CUPL

SPAC p

MIN RAC.

r -J-tp-JC 0.0 c c r ^ a r

PPi«JSITY 7.00000 c.00000 r nfir

KAX PAD. O.^OO c. -p&ir

0.1CBO0 r .

.000 TZFRO

27?.15

t r p c

' 0 . 0 0

CONDUCTIVITY 0.-"=000 0.70000 r

MIN L. 0 .0

.

O.POOOO " . r

CONTE

HPAT GpN RATP INIT.TFMP ALPHA .00000 00.00000 O.lSfcOOocF-OS

21 . ""OOO 500.00000 0.0 r , f i i rr '> enr.nr>orr. n.o

MAX L. '. OOO 2. 0*000

">.5 000

AFX

102

SURFACE,N0= 1 TYPE = 1 SHAPE= i n RSI = r . r R S2 = r . 6 3 5 P r S U = r . r SL2 = (7.0

SURFACE,N0= ? TYPE = 1 SHAPE= 102 RSt <= 0 . 0 RS2=- 0 . 6 3 5 0 0 SL1 = 2 . 5 4 0 0 0 SL2= 2 . 5 4 0 0 0

SURFACE,N0= 3 TYPE = 2 SHAPE= 203 RSI = 0 . 6 3 = 0 0 RS2 = 0 . 6 3 5 1 0 SL1 = r. 0 SL2 = "2 .540PP

SURFACE,N'0 = c TYPE = c SHAPE= \OA RSI = r . r RS2 = r . i ^ e s n SL1 = r. 8nrrr> SL2= 0.80000 SURFACE,N0= c TYPE = c SHAPF= 105 RSI = 0 . 0 RS2 = 0 . 1 5 8 8 0 SL1 = ?. 0&000 SL2= 2 . 0 4 0 0 0

SURFACE,N0= * TYPF= c SHAPF = 20*; RSI = o . i see r RS2 = r . 1 5 P ? r SL1 = f pprnp SL2 = 2 . 0 4 0 0 0

SURF ACE,N0 = 7 TYPE = c SHAPE= I f 7 RSI = r , 2 7 p 7 0 PS2 = r , 3 5 4trv SL1 = 0 . 6 0 0 0 0 SL2= 0 . 6 0 0 0 0

SUR*=ACP,NO = f> TYPE = r SH APE = i OP RSI = RS2= 0 . 3 5 4 4 0 sua = 2 . 34000 . SL2= 2 . 3 4 0 0 0

SURFACE,N0= o TYPE = e SHAPE= 2 r o RSI = r . ? 7 P 7 n RS2 = r , ? 7 « 7 r SL1 = r. 6 r r « n SL2 = 2 . 3 4 0 0 0

SURCACE,N0=1 r TYPF = c SH APE = 21 r RSI = r . ,3 RS2= 0 . 3 5 4 & 0 SL1 = 0 . 60000 SL2= 2 . 3 4 0 0 0

SURC ACc,NO='' ^ TYPE = u SHAPF= 211 RSI = 0 . 5 0000 R S? = 0 . 5 0 0 0 0 SL1 = 0 . 0 SL2 = 2 . 5 4 0 0 0

SURFAC c , N 0 = 1 2 TYPE = 7 SHAPE= 212 RSI = o . 5 2f«rn PS2 = r , 5 2 r p p SL l = r . P 51.2 = 2 . 5 4 0 0 C

SURCACF N0 = 1 SINK TFMP = 0 . 0 H= 0 . 0 FS = 0 . 0 FR = 0 . 0

SURFACE N0 = 2SINK TEMP- 0 . 0 H= 0 . 0 FS= 0 . 0 FR= 0 . 0

SURFACE N0= ""SINK T E N ^ i r r . p r H= r.imrn FS = r . r FR = r . r

SURFACE N0= ^SlNK TE MP= 0 . 0 H= 0 . 0 FS= 0 . 0 FR=. 0 . 0

SURPACF N0= 5SINK TFMP= 0 . 0 H" 0 . 0 FS= 0 . 0 FR= 0 . 0

SURFACE N0= 6SINK TEMP = n . n H= r . r ES = r . r FR= r . r

SURFACF N0= 7SINK TEMP= 0 . 0 H= 0 . 0 FS= 0 . 0 FR= 0 . 0

SURFACE N0= PSINK TEMP= 0 . 0 H= 0 . 0 FS= 0 . 0 FR= 0 . 0

SURPPCE N0 = °S INK TFMP= 0 . 0 H= r . r FS = r . r f r = r . r

SURFACE N0= 10SINK TFMP=: 0 . 0 H= 0 . 0 c s = 0 . 0 FR= 0 . 0

SURFACE N0=11SINK TEMP= 0 . 0 H= 0 . 0 FS= 0 . 0 FR= 0 . 0

SURFACE N0 = ,! ?S INK TEMP= r.r H= r . r FS= r . r ER= r . P

OUTPUT OPTIONS 0 0 0 0 0 0 0 0 0 0 0 0 0

END OF PRINTOUT OF INPUT DATA

103

FINAL TEMP DISTRIBUTION WITH NCOUNT = 3 ITER = 1370

SUM= 0.? 5614 E24F-0? T0TAL= 0 .99999979E-02 ITER = 1370 TEST = f .92r78tl7E-•r»5 DTM« p .49099989P- ITER* 137P QTEST= O.t 573620E-•02 SPFC=» 0 .66666640E-•02 ITER= 1370

I J K T * I J K T * I J K T » I J K T * I j K T •

1 1 i * n.P a 1 ? P 5.424475F r 2* 1 3 n P." * 1 4 0 5.419956F P2* • 1 5 P P." •

2 1 0 F.48C'°23E 02* 9 ' 1 5.483125E 02* 2 3 0 5.480276P 02* •> U 1 5.4-786 50F 02* 2 5 0 5.473«>36F 02* •a • * ^ 0 0.0 * 3 2 0 5.541 694F 02* 3 3 0 0.0 * 3 4 0 5.5372B8E 02* 3 5 0 P." * u 1 p P2* u 2 1 .K.6PP3?PE P2* 4 3 r 5 • 5 °7 53 4F u 4 1 • 5.595952E P2* 4 i f> 5.591289E 02* e 0 0.0 * c ? 0 5.71 6! 3PF 02* K 3 0 0.0 * « 4 0 5.712632F 02* 5 5 0 0.0 *

6 1 0 E.833 59°F 02* 6 2 1 5.831 e51 F 02* 6 3 0 5.83026°F 02* 6 & 1 5.82932°F P2* 6 5 p 5.82624PF P2* 7 1 r P.r * 7 2 P 5.e82?14E P2* 7 3 p «.P * 7 4 c 5.985E3"'E P2* 7 5 0 0.0 *

P I 0 6 . 004 70oe 02* n 2 i 6.007612E 02* 8 3 0 6.010054F 02* 8 4 1 6.011609F 02* 8 5 0 6.015413E 02* o 0 0.0 * a 2 0 6.031033F 02* a 3 0 0.0 * o 4 c 6.03641bF P2« 9 5 p P.P *

• i r* r?* 1" 2 1 6.P5465PF P 2* IP 3 o 6.P58816F n2* IP 4 1 6.P61245F 02* 10 c 0 6.067532F 02* t « i 0 0.0 * 11 2 0 6.07 5°0tF 02* 11 3 0 0.0 * 11 4 0 6.083970F 02* 11 5 0 0.0 *

I' i 0 6.091 C79E 02* 12 2 1 6.097173F 02* ' 2 3 n 6.1P3 2P1F f-t 12 4 1 6.1P6709E P2* 12 5 p 6.116P9]F P2» 1? t p r.r * 1 3 2 " 6.11 £P! 2E P2* i 3 3 P P." * 13 & 0 6.126255F 02* 13 5 0 0.0 • 1 u 0 6.124612F 02* 1 4 2 1 6.132861 F 02* 14 3 0 6.141067F 02* 14 4 1 6.145818F 02* 14 5 0 6.159131F 02* 1 = 1 0 0.0 * IS 2 0 6.146PArF p?* 15 3 n P.P * 15 4 0 6.16P793E P2* 15 5 p P.o *

16 1 p 6.148713E .02* 1 6 2 1 6.15 °2'°F P2* 16 P 6.1 69702E 02* 16 4 1 t .175776E 02* 16 5 0 6.193933E 02* 17 0 0.0 * 17 2 0 6.16644PF 02* •17 3 0 0.0 * 17 4 0 6.184355E 02* 17 5 0 0.0 *

18 0 6.1 61 02* 1,8 2 1 6.173687F P2* 18 3 P 6.185P84E «2*-ie 4 1 6.192952F P2* 18 5 r 6.215081E P2* 1° I n P.P * 19 2 P 6.173962F 02* 3 0 0.0 * 19 4 0 6.193220F 02* 19 5 0 0.0 *

to 1 0 02* 20 2 ? 6.72776E 02* 20 3 0 6.731°21F 02* 20 4 3 6.734355F 02* 20 5 0 6.741096E 02* 21 1 0 P.P * 21 2 O 6 • 916 °4t F P2* 21 3 n P.P * 21 & 0 6.921663F P2* 21 5 p *

22 1 r> 7.10/1.292E C2* 22 2 3 7.10611?F 02* •2i 3 0 7.107O10E 02* 22 4 3 7.108960E 02* 22 5 0 7. Ml 907 E 02$ ?3 •l 0 0.0 * ?3 2 0 7.14 0090F 02* 3 0 0.0 * 23 b 0 7.1A3354E 02* 23 5 0 0.0 *

?4 1 P .7.171 7!oe 02* 24 2 3 7.17408"'F P2* 24 3 p 7.1764P4E «2* 24 4 .".177776E P2* 24 5 p 7.181589E P2*

10U

1' I 0 0 . 0 7 0 • ' . 0 0 « 1 ( F o ? » ?•> 0 0 . 0 * 4 0 7 . 0 1 1 ° 7 0 E 0 7 * 75 R 0 0 . 0 *

i 0 T . O C ' P ? " 0 ' * • > * ? 1 T . O O f M o * 02 * 76 0 7 .00 O I 53 C « : 0 7 * 2t 4 1 7."1!7<><5F P ? * 76 P 7 . P I B A 3 6 F P 2 *

fl r p . r * 27 •> p <• . CP KlVf F P3 * 77 p • 77 4 P 6 . 9 9 0 3 7 5 6 f ? » 27 C 0 0 . 0 *

?p 1 0 0 7 * •> a •> 0 ' + ?«> 3 0 6 . ° 6 T 7 0 P F 0 2 * 78 4 1 6 . 9 6 U P 6 7 F 0 2 * ' B 5 0 fc.O7207eF 0 2 *

•>c 0 0 . 0 * 7 O ? 0 < , . °3 = 710F 0? * 2 ° 0 0 . 0 * Z9 4. p 6 . 9 3 7 7 4 1 F P Z * 2Q E 0 O . P *

ir ^ r f . c r r i t i c p r?* i n ? 1 p?» ir p 6 . 0 0 5 6 T F r>2» 30 U 1 t . ° 0 5 6 4 0 F 0 2 * 3 0 5 0 0 2 *

11 0 0 . 0 • •» 1 W 0 fliep o ? * 31 » 0 0 . 0 a 31 4 0 6 . e 7 3 e « 2 F 0 2 * 3 1 5 0 0 . 0 *

V) i 0 0 ' * ? 1 r ? » p 6.a"53"?7CE " 2 * 3Z 4 1 6 . 8 5 2 7 3 4 P P ? * 32 p 6.<?4Q!?PF n 2 *

•n p p . p * 1 ? 7 p f-.?t-rjff F r i> * 33 •a n 0.0 * 33 4 0 6 . 9 3 6 6 P 7 F 0 2 * 3 3 e 0 0 . 0 *

1L 9 0 0?» 3 4 7 t 0 7 * ?4 •a 0 fc.SZZ4«0c 0 2 * 34 1 6 . 8 Z 0 6 6 7 E 0 2 * 3 4 5 0 6 . P I 5 3 I f F 0 2 *

31 % 0 0 . 0 * fj ? rt t . 7 ? ° 3 F i ' » » 3 "5 p p . n * IC 4 c 6 . 8 1 2 1 i 1 ? P 2 * 35 p P . P »

'f- » r r ? » It 7 t 6 .P^OZC-F 0 0 ? * 36 1 6 . 9 0 3 6 7 4 F 0 2 * 3fc «> 0 6 . 7 0 7 5 8 3 F 0 2 *

* 0 0 . 0 * '.' 7 7 0 t .PO ' ' 167F o ? * 37 0 0 . 0 * 37 4 0 6 . 7 = 5 1 ? 6 E 0 7 * 3 7 5 u 0 . 0 *

105

«= 1NAL TEMP DISTRIBUTION WITH NCOUNT « 3 ITERn 1370 SUM= 0.«= -0? TOTAL= 0 .nQqoqaTCf-02 ITER= 1370 TCST = r.o?n7«ti7E- DTM= 0 .iOooaqppF-04 ITER- 1370 QTEST= SPEC = r .66666640E-12 ITER" 1?70

1 J K T * I J K T * I J K T « I J K T * I J K T *

1 6 r ?.M152fcF 1 7 o o.o * 1 8 0 5.400356E 02* 1 9 0 0.0 * I •

10 0 5.388755E 02* 2 6 i •\47Q27PF 02* ? 7 0 02* S 1 5.4 5°109R 0?* 2 9 0 5.452966F 02* 2 10 1 5.447424E 02* n 6 0 5.«='P962E fj« 3 7 n r.e * 8 o "5 • 5l78ncE 12* •> 9 0 o.n * 3 10 0 5.50604PE 02* 4 6 5.5P7fe7fF 0!>* & 7 o 5»580°06E 0'* c. 8 1 5.57652SE 02* 4 9 0 5.570291E 02* 4 10 1 5.564680F 02* 5 6 0 E.705757E 02* 5 7 0 0.0 * E 8 0 5.69c994E 02* 5 9 0 0.0 * 5 id 0 5.685068E 02* e 6 1 e.p2?ee7E r?* 7 n 5.P1P767F 09* 6 8 1 5.815466F •»2* 6 9 c 5.81022OE. 02* 6 irt 1 5.805513E 02* 7 6 r> e.oc5fic70E 02* 7 7 0 0.0 * 7 P 0 5.9O5420E 02* T 9 0 0.0 * 7 10 0 5.998762E 02* e 6 l. 6.01R39PC 02* 8 7 0 6.022654F 0?* a 8 1 6.025444E 02* 8 a 0 6.028354F 02* 8 10 1 6.031001E o'2* o 6 0 fc.^^'iAl BP r?* o 7 C * a 8 o 6.055173E r>2* 9 9 0 0.0 * 9 10 n 6.063318E 02* 10 6 1 6.0724?9E c* 1 0 7 0 6.079983F 02* 10 8 i 6. 0849! OF 02* 10 9 0 6.090549E 02* 10 10 1 6.095662F 02* 11 6 0 6•0°7Q17E o;'* 41 7 0 0.0 * 11 8 0 6.113730F 02* 11 9 0 0.0 * U 10 0 6.127117E 02* 12 t- ^ 6.1 o 2* 12 7 o 6.134998F 02 * 12 8 \ 6.142=%BE 0 2* 12 9 c 6.150972E 0 2* 12 10 l 6.15 8579E 02* 13 6 0 6.14J460E 02* I 3 7 0 0.0 * 1 3 8 0 6.t702Q*F 02* 13 9 0 0.0 * 13 70 0 6•18 8943E 02* I* f. 1 6.16°51 °E 02* 1 4 7 0 6.18667=F 02* 14 8 1 6.197861E 02* 14 Q 0 6.2P9126E 02* 14 10 1 6.219326E 02* le 6 r fc.l 8870fcE P2* 15 7 n o.o * 15 8 o 6.224578E 12* 15 9 p C.O * 15 10 0 6.249001 E 02* 16 6 6.208086F 02* 16 7 0 fr.23*250F 02* 16 e 1 6.251313E 02* 16 9 0 6.265686E 02* 16 10 1 6.278694E 02* 17 6 0 6.221001E 02* 1 7 7 0 0.0 * 17 8 0 6.278600? 12* 17 9 0 0.0 * 17 10 0 6.308613E 02* IP 6 1 6.23392'F 02* 18 7 0 6.277617E 02* 1 8 8 1 6.316191E 02* 18 9 c 6.323127E 02* 18 10 1 6.338550E 02* 19 6 0 6.234177E 02* 1 ° 7 0 0.0 * 19 8 0 6.33°668F 02* 1 9 o 0 0.0 * 19 id 0 6.3T000QE 02* 20 6 3 6.746372E 02* 20 7 0 6.75c601 F 02* 20 8 1 6.755 67op 02* 20 9 0 6.757783E 02* 20 in 1 6.759697E 02* 21 6 0 6. 93r3<«E 02* 21 7 o 1.0 * 21 8 0 6.937048P 02* 21 9 0 0.0 * 21 10 0 6.940286E 02* 22 6 o 7.11«233F 02* 22 7 0 7.11P372F 02* 22 8 1 7.I1842PE 02* 22 9 0 7.119712C 02* 22 10 1 7.120884E 02* 23 6 0 7.149404F 02* 23 7 0 0. * 23 8 n 7.154243F " 2* 23 9 p 0.0 * 23 in 0 7.156794E 02* 2* 6 3 7.1846O0E 02* 24 7 0 7.19001FF 02* 24 8 7.190C71E 02* 24 9 0 7.191453E 02* 24 10 1 _ 7.192732E 02*

106

t- 0 7.0??P">= 0?* 7 0 0.0 * 2? P 0 7.0%9"eF o?* 25 Q 0 0.0 * 25 10 0 7. 0797446 02* •>t * i •».P2,625E r?* 26 7 r> P i f»2* ?«• o p 7.046912F r>2+ 26 "JO 1 7.052444E 02* •31 f- n (..OCOrqTTp 02* 77 7 0 0.0 * ?7 P 0 7.01'A*;PF o?* ?7 9 0 0.0 * 27 10 0 7.0210576 02* •>9 (• f .e7A?|97F 0?« 7 0 R 28 9 0 «.=86036F 02* 28 10 \ 6.9B9692E 02* •jo t (« f .OWK" 7 ft * P n 6.94&P6BF 29 9 p C.P * 29 p 6.948503E ir e ? A.t>0=6?7F 0'* 30 7 Q 4.c0e?5eF o?* '0 P < 6.90£0<S0F 02* 30 9 0 6.9067976 02* 30 10 1 6.907327E 02* •ai 6 0 6.870Q£fe 0?* 31 7 0 0.0 * P 0 o?* o 0 0.0 * 31 p 6.857954F 02* 3? 6 i r?* 3? 7 A r5>* 32 n 1 6.8367*1 F n j* 22 © c 6. 83P5a8F r 2* 32 i n 1 6.825054F P2*

6 0 0?* 7 0 0.0 * P 0 6.8171.31F 07" 9 0 0.0 * 33 10 0 6.803572E 02* 1C t 1 03* 7 0 0?* P 1 £.7Q7«7F 02* It. Q c 6.78VM9F r?* 36 10 1 6.782097E 0 2 *

r t.prynr?c r 2 * 1 K 7 n r.r * 8 r 6.7<J763«>F 35 a p c.o • 35 10 0 6.77J638E 02* If- fc i. 0?* 7 0 0?» ?6 a 1 6.777'KtF 3«- s 0 6.769050F 07* 36 70 1 6.7611966 02* 17 * Q ft.7«w "!•»«(: 0?* 7 0 0.0 * 3T e 0 r 2* 37 9 p r.o * 37 IP "0 6.75C754E 02*

107

PINiU TEMP PI STRI BUT ION HITH NCOUNT = 3 ITER = 1?70 SUM= 5.cc614524=-02 TOTAL= 0.ooq9oo7OF-P9 ITER= 137" TCST= 0.c207n 617C. -05 DTM= 0 I4coaoapp c-04 ITFR = 1370 QTFST= r ,f657?62'>F--no SPEC = r .66666 64TF- ITEP= 1370

I J K T * 1 J K T * I J K T * 1 J K T * I J K T *

1 1 1 0 0.0 1 1? 0 5.3t4C?7F 0'* 1 13 0 0.0 * 1 14 0 5.364143F 07* 1 15' 0 0.0 *

7 1 1 0 5.440''OF 02* 9 12 1 5.433 41 PF 0?* 2 13 0 12* 2 14 1 5.422456E 02* 2 1$ 0 5.419783E 02* V V r r.r * 3 i ? o e.40iR55F p?* a 1 3 n r.r * ^ 14 r 5.40O693E 02* 3 15 0 0.0 *

4 11 0 5.5e71 OOP 0 2 * • 4 1? 1 5.5f03?7F 0?* 4 13 0 5.54I560F 02* 4 14 1 5.538958F 02* 4 15 0 5.536179F 02* e 11 0 0.0 a 5 12 0 S.fT"1 H3F 0?* 5 13 r * 5 14 0 5.65932°E P2* 5 15 0 r.r *

(• 11 r r2* 6 1? 1 5.7°1921F P2* 6 13 f« 5.78?Bft7F 02* f 14 1 5.77O707E 02* 6 IS 0 5.776672E 02* 7 11 0 0.0 * 7 1? 0 6.0006? 5F 0?* 7 13 0 0.0 * 1 14 0 S.978462E 02* 7 1 5 0 0.0 *

P 11 c 6.033301 F 0?* 8 12 1 6."3543?F r2* 0 13 r 6 .O3%ooi F 0 14 2 6.036465F 0 7 * 8 1 B n 6.034981F r>2*

o 11 r o.r * a i ? p 6.7110£F P9* a 1 3 0 0.0 * 0 14 0 6.077590F 09* 0 1 5 0 0.0 *

'0 11 0 6.i 01555E 0?'* 10 i? I 6.106 C75F 0'* 10 11 0 6.114«=6?c 0 2 * 10 14 2 6.1185°1E 0 2 * 10 1 5 0 6.119072E 02* 11 11 0 0.0 * 11 6.14rgO{,F T7* 11 13 n r ,r> * 11 14 r 6.154141F r 2 * 11 15 p P.P *

I? 1 1 r 6.1 6",P39E 12 i 2 1 6.174P-J4F 09* 1 9 1 3 0 6.1 R456'F 07* 17 14 2 6.189705E 0 2 * 1 2 1 c 0 6.190901E 02* 1* •r 0 0.0 * i'? 1 2 0 6.?06<=5I F 09« 1 n 1 3 0 0.0 * 13 1 4 0 6.222375F 02* 13 15 0 0.0 *

14 r 6.22°607F r 14 12 1 6.2? or,8?F ra* 14 13 r 6.949G41F n?* 1 4 14 2 6 »255r7£F P2* 14 15 o 6.256482F 02* i c r r.r * 15 1 ? 0 6.?6Cog-aF 0?* 15 13 0 0.0 * 1 5 14 0 6.2861?9F 0 2 * 15 15 0 0.0 *

ie 0 6.2°0751c 02* 16 i ? 1 6,300P°6F 0?* 16 1 3 0 6.31154PC 0?* 16 14 2 6.317183F 02* 16 15 0 6.318582E P2* 17 ii r p.n * 1 7 1 2 f> 6.3?1F r2# 17 13 o * 17 14 r 6.3473"FF 02* 17 1; r . ".r *

IP 11 0 6.350547? 0'* 18 12 1 •siti eop 02* 1 B 1 •a 0 6.37H?60F 07* 1 P U 7 6 . 3774=1 E 0 2 * IP 15 0 6.37R652E 02* 1° ii 0 0.0 ' * 1 ° 1 9 0 fc.-aoi gi-ap 0 2 * 1 O 1 0 0 0.0 * 1 e 14 0 6.4070=9= 0?* 19 15 r r.r *

•>(• 11 r f .763276F r?* 2" ' 2 1 6 .76^5 f l7 f ro-tt ?« 13 P 6.771-J77F n?* ?r 2 6.773°26F r 2 * 2 r 15 p 6.772339F op* 21 T 0 0.0 * ?1 1? 0 f. .04.50 0 2 * ' 1 0 0.0, * 71 14 0 6.952139E 02* 21 1 5 0 0.0 *

2? ii 0 7.'<2">2iflE 0?* ?' 12 1 7.1?«7*F 02* 22 13 0 7.12(i69pF 09* 92 14 2 7.13r364E p ? * 22 1 5 p 7.127937E P2* ' 2 ii r r.r * 93 i ? r -».i6i 4n F P9* 2> 13 r r.r- * 2' 14 r 7.16672F r 2 * ?-» 1 5 p 0.0 *

T" 0 7 . 1 of 176E 0 2 * 94 1? 1 7.197444F 0 2 * 24 13 0 7.200P03E 0 2 * 24 14 2 7.202595E 0 2 * 24 15 0 7.200225F 02*

108

25 11 0 0 . 0 * 25 1 2 0 7 . 08°414P 02* 25 13 0 r . o * 25 14 n 7.097510E 0 2 * 25 15 P P.P *

26 11 o 7.0=80°6F r ?* ?6 l 2 1 7 . r 6 3 3 H F 02* 26 1 3 0 7.O60053P 02* 26 14 2 7.072104F 02* 26 15 0 7.070969E 02*

Zi 1 ' 0 0 . 0 * 27 12 0 7 . 0 * 1 077F 02* Zf 13 0 0 . 0 * 27 14 0 7.039973F 02* 27 15 0 0 . 0 *

?P 11 0 6 .994465 c 0?* 2 ° 12 i 6.99P872E 02* 2P 13 n 7.nOA.-»46F " 2 * 2P 14 2 7.P07878E 0 2 * 28 15 p 7.PP69P9E P2*

2 ° 11 r o.o * 2" 12 0 6.954P17F 02* 29 13 0 0 . 0 * 2° 14 0 6.962600E 0 2 * 29 15 0 0 . 0 *

30 11 0 6 . 9 0 ° U C E 02* 30 12 1 6 . °10786F 02* 30 13 0 6 . 91505°F 02* 30 14 2 6.917339E 02* 30 15 0 6.916072E 02*

31 11 0 0 . 0 » 31 12 p 6.BAeo75F 0?* ?1 13 n n.n * 31 14 0 6.B24846E 0 2 * 31 15 p P.P *

32 11 r t . e i t r ^ f e 02 * 32 12 6.807791 F 02* 32 1 3 0 6.7O3806E 02* 32 14 1" 6.789629E 0 2 * 32 15 0 6.785122F 02*

« 11 0 0 . 0 * 33 12 0 6 . 7 8 4 8 7 ' F 02* 33 13 0 0 . 0 * 33 14 0 6.767246E 02* 33 15 0 0 . 0 *

•94 11 « 6.771 6 02F 02* 34 12 1 6 .76196rF 0 2 * 34 13 o 6 .748806F 02* 34 14 1 6.7448S5F 0 2 * 34 15 n 6.74P645F P2*

11 p. P . " * 35 12 0 6.751?16F 02* 35 13 0. 0 . 0 * 3* 14 0 6.734287E 0 2 * 35 15 0 0 . 0 *

36 11 0 6.750400F 02* 36 12 1 6.740483F 02* 36 13 0 6 .727554F 02* 36 14 1 6.723701E 02* 36 15 0 6 . 7195 83 E 02 *

•57 11 o o .o * 37 12 0 6.729751 F P2* 37 13 0 r . o * 37 14 0 6.713115E 0 2 * 37 15 0 P . i *

109

PINAL TEMP PI STR18UTION WITH NCOUNT = 3 ITER" 13*0

sUM= r . . " 6 1 l 5 2 l « : - r ? TOTAL" o.ooo<je«*7oF_«2 ITER" 13™

TFST = 0 . ° 2 07P617F-•05 0TM= 0 ITER* J 3 7 0

QTEST" 0 . 66.57^6 20F-•0' SPEC" 0 .66666 t 'OE-•02 ITER® 1370

1 J K T * I J K T * I J K T * I J K T * I J K T *

1 16 0 5.358?77E 0?* 1 17 0 0 .0 * 1 1 P 0 5.3536108 02* 1 19 0 0 . 0 I 20 0 5 .34814°E 02*

? 16 1 5 .1THBPF 0?* 2 17 0 P?* ? 18 J 5.6117-JilF " 7 * 2 19 r 5.408884E P2* 2 20 1 5.106111E 02* •3 16 o 5.175315F 02* '.7 0 0 .0 * 3 18 0 5.4697?1F 02* 3 19 0 0 . 0 * 3 20 0 • 5.464001E 02*

A 16 1 5.533177E' 0?* 17 0 5.530B O6F 0?* 4 18 1 5.52"'79IF 02* 19 0 5.524811F 02 * 4 20 1 5 .52190°E e 16 r 5 .6535°6E 0?* 5 17 p o.P * 5 18 0 5.6475°3E " 2 * cs 19 p P.P * 5 20 0 5.641326E 02 *

6 16 5.77">7i IE 02* 6 17 0 5 .77051 5F 0?* 6 18 1 5.7673°5E 02* 6 19 0 5.764033E 02* 6 20 1 5.760751E 02*

7 16 0 J.07£|>AiF 0?* 7 1 7 0 0 . 0 * 7 18 0 5.°6P1"MF 02* 7 19 0 0 . 0 * 7 20 0 5.958403E 02*

e •<6 2 6.033512E 02 * 8 17 o 6.P3«: i !*F 02* 8 1 8 2 6.02*7658 " 2 * 8 19 p 6.P21365E 0 2 * 8 20 2 6.016P86E 02*

16 0 6.07651EE 02* 9 17 0 0 . 0 * a 1 8 0 (..0703-»4F 02* Q 19 0 0 . 0 * 9 20 0 6.059192E 02*

to 16 2 6 .11°56 1 E 02* 1.0 17 0 6.1167E0F 02* 10 18 1 6 .1U00OF 0»* 10 19 0 6.108096E ,P2* IP 20 2 6.102317F 0 2 *

16 P 6.1 5K 81 8E r 2* 1 i 17 0 P.P * 11 18 0 6.15H52E 12* 11 19 p 0 . 0 * M 20 0 6.138447E 02*

12 16 2 6.19?0P7E 02* 12 1 7 0 6 .1 f l ° i 6PF 02* 17 18 2 4.186976E 02* 12 19 0 6.180686E 02* 12 20 2 6.174592E 02*

13 16 0 6.2?1C71F 0?* 13 17 0 0 . 0 * 13 18 0 1 6.21°P10F n '* 13 1<> p P.O * 13 20 0 6.207244E 0 2*

t A 16 2 6 .957P7IE r ? * M 1? p 6.255256F 14 18 7 i .252712? 09 * 11 19 p 6.246218E 0 2 * 14 20 2 6 . 23ot?27E 02*

16 0 6.288518? 02* 1 5 17 0 0.0 * 15 ie 0 6.283630F 0?* 15 19 0 0.0 * 15 20 0 6.270623E 02 *

1ft 16 2 6.319C63E 02* 16 17 0 6 . 3 1 7 2 2 i F 02 * 16 18 2 * . 3 U 5 C 6 E 0 2* 16 19 0 6.307866E 0 2 * 16 20 2 6.3P1326E 0 2 *

17 I f n 6.3A9888E 02* 1 7 17 p P.P * ' 7 18 0 ».344?50P 0?* 1 7 1° 0 0.0 * 17 25 0 6.330750E 02*

6 2 6.?79enE 02* 1 8 1 0 6.376P65F 02* 1 8 18 2 6.373065F 0?* 18 19 0 6.366597E 02* 18 20 2 6.360190E 02*

1° 16 0 6.109036E 0'* 19 17 r * 19 IP r> 2* 1° 19 0 0.0 * 19 20 0 6.388633E C2*

?o 16 2 6„770 801.E r ?* 2P 17 r 6. ' '6521 FF f 2* 20 18 2 ' i , 7 5 9 768F 02* 20 19 0 6.7E0322E 02* 20 20 2 6.711084E 02*

21 16 0 6.018181E 02* 21 1 7 0 0.0 * 21 18 0 6.9362°6E 02* 21 19 0 0.0 * 21 20 0 6.916787E 02*

22 16 '•> 7.126570C r 2* 22 1 7 p 7.11911PF 0 2 * 22 18 2 • \112861E 0 2* 2? 19 p 7.102581E 02* 22 2P 2 7.092515E 02 *

23 1 6 r "7.161736F P2* 23 17 (N n , 0 * 23 18 0 "T.1AO077F 02* 23 19 0 0 . 0 * 23 20 0 7.128774E 02*

SL. 1 6 2 7,19"T°20E >02* 17 0 7.191541F 0'* 2 1 IP 2 7.18 K 31 OF 0?* 21 19 0 7.175073F 02* 24 20 2 7.165054F 02*

110

2? 16 r r?* 25 1 7 ft o.p * 25 1p 0 n 25 19 C 0.0 * 25 20 0 7.O65630E 02* 9f 1 f 2 7.060(178* 02* ?6 1 7 0 7.064666F 09* 26 18 2 7.0505O?F 02* ?6 19 0 7.050496F 02* 26 20 2 7.041 597E 02* ?7 '6 0 7.037o?0E 02* ?7 1 7 0 0.0 * 27 IP 0 7.027905F 02* 27 19 0 0.0 * 27 2" p 7.P10264E 02% 2P 16 2 7.005070? 02* 28 ! 7 0 7.rrirAcp P2* 2* 18 2 6.°°6 233F. "2* 2 P. 19 e 6.987495E 0 2* 28 20 2 6.978058E 02* 70 16 0 6.°60405E 02* 99 17 0 0.0 * 29 18 0 6.9=0°64E 0?* 2° 19 0 0.0 * 29 20 0 6.O34580E .02* 30 16 7 (t.014g4£F 02* 30 1 7 0 6.°1022?F 02* 30 18 2 6.°0«71FF 02* 30 lo 0 6.897878F P2# 30 20 2 6.890220E 02* 31 16 O 6.P17722E r2* 'I 1 7 0 0.0 * 31 18 n 6.808328F P9* 31 19 0 0.0 * 31 20 0 6.796743E 02* 32 16 1 6.780725? 02* 32 17 0 6.77eoipf 02* 32 IB 1 6.7712&5P 02* 3? 19 0 6.766140E t, 2* 32 20 1 6.761157E 02* ?? 16 0 6.758611E 02* 33 17 0 0.0 * 33 18 0 6.740502? j'a 19 p o.r * 33 20 0 6.739912E 02* 1L. 16 1 6.736511E P?* ?4 17 0 6.73?P8?F P9* 34 18 1 6.727783E p?* 34 1 9 0 6.7231 qi F 02* 20 I 6.718696E 02* 16 0 6.7260?8P 02* 35 17 0 0.0 * 35 18 0 6.717A3oc 0?* 35 19 0 0.0 * 35 20 0 6.708513E 02* '6 1 6.715569E 02* 36 17 0 6.711282F 02* 36 IP 1 (t.7P7iroE " ? * 36 19 c 6.702678E 02* 36 20 1 6.698352E 02*

37 If p 6.70C11OE o?* 37 17 p O.P * 37 18 P r>2* 37 19 0 0.0 * 37 20 0 6.688191F 02*

Ill

F I N A L T P M P D I S T R I B U T I O N W I T H N C Q U N T = 3 I T > = R = 1 3 7 0

S U M = l . e f 6 1 4 5 2 4 < = - r ? T O T A L = r . o q g o o o T o p - o ? I T E R - 1 * 7 0

T E S T = C . O 2 Q 7 P 6 1 7 F - 0 5 D T M = r , 4 C Q O O ' 3 B p C _ r 4 I T F R = ! 3 7 r

0 T E S T = 0 . 6 6 5 7 3 6 9 Q F - 0 2 S P E C = 0 . 6 6 6 6 6 6 4 0 F - 0 9 I T E R = 1 3 7 0

I J K T * I J K T * 1 J K T • <« I J K T * I J K T *

1 r 0 0 . 0 * 1 2 9 0 0 2 * 1 9 3 r r . r * 1 2 * 0 5 . 3 3 6 6 7 ? E r z * 1 2 5 r P . f > *

•> 2 1 r K . 4 r ? i 7 6 F r ? * 2 9 9 1 5 . m 3 1 S F P 2 * 2 3 r 5 • 3 9 " 7 2 ° 2 F 0 2 * ? 2 4 1 5 . 3 9 4 3 4 3 F 0 2 * 2 2 5 0 5 . 3 9 1 2 2 8 F 0 2 *

a ? 1 0 0 . 0 « 3 2 2 0 . 5 . 4 5 P 0 6 4 F 0 9 * 3 ? 3 0 0 . 0 * 3 2 4 0 5 . 4 5 1 9 4 3 F 0 2 * 3 2 5 0 0 . 0 *

4 2 1 0 5 . 5 i p ? 3 3 F 0 2 * 9 9 1 . 5 . 5 1 5 P " F r ? * 4 2 3 n 5 . 5 1 2 " 6 6 4 F n 9 * 2 4 1 5 . 5 f 9 5 6 R " E r 2 * 4 2 5 p 5 . 5 P 6 2 9 9 E P 2 *

c 2 1 r> r . r * 5 2 ? r 5 . 6 3 4 P 1 0 F r j * c ' 2 3 0 0 . 0 * c 2 4 0 5 . 6 2 8 0 ° 6 E 0 2 * 5 2 5 0 0 . 0 *

6 9 1 0 E . 7 5 7 2 A 1 E 0 2 * 6 2 2 '1 5 . 7 C ? P 1 P F 0 2 * 6 2 3 0 5 . 7 5 0 1 7 P ? 0 2 * 6 2 4 1 S . 7 4 6 6 2 8 F 0 2 * 6 2 5 0 5 . 7 4 2 P ° 6 E 0 2 *

7 2 1 0 0 . 0 * 7 9 ? n 5 . 9 4 = = 6 4 F " 9 * •7 2 3 O r . r * 2 4 r 5 . 9 2 9 7 2 7 E r 2 * 7 2 5 0 O . r *

8 2 1 r 6 . r r 9 7 2 6 E P 9 * P 9 9 2 t . r n C ? ? F 0 ? * P ? 3 0 5 . 0 0 2 2 5 1 c 0 2 * 8 2 4 2 5 . 9 8 3 1 6 2 F 0 2 * 8 2 * 5 0 5 . 9 7 2 0 4 6 E 0 2 *

o 2 ' 0 0 . 0 * 9 2 2 0 6 . 0 4 3 0 7 4 E 0 2 * 0 9 3 0 0 . 0 * 0 2 4 0 6 . 0 2 2 0 5 6 E 0 2 * 9 2 5 0 0 . 0 *

' 0 2 1 r 6 . P 9 3 3 8 6 F 1 7 * 1 0 2 2 2 6 . * " R 4 £ 4 1 F C2* 1 r 2 3 0 6 . 1 7 2 6 0 3 F r 2 * i n 2 4 2 6 . 0 6 0 9 7 Q F r 2 * 1 1 2 5 r 6 . P 4 5 8 9 8 E P 2 *

1 1 9 ' r n . r * 1 1 2 9 0 6 . T 1 0 0 7 1 F 0 2 * 1 1 2 3 0 0 . 0 * 1 1 2 4 0 6 . 0 9 5 1 0 5 E 0 2 * 1 1 2 5 0 0 . 0 *

1 ? 2 1 0 6 . 1 6 4 P 5 6 F 0 2 * 1 2 2 2 2 6 . 1 5 5 3 2 C F 0 2 * 1 2 2 3 0 6 . 1 4 2 1 5 3 ' = 0 2 * I ' 2 4 2 6 . 1 2 9 2 4 6 E 0 2 * 1 2 2 5 0 6 . 1 1 2 6 6 6 F 0 2 *

1 ? 9 . 1 r P . ' " * 1 3 ? 9 r 6 . 1 8 7 E 4 f c F P 2 * i 3 2 3 r r . n * 1 3 2 4 r 6 . 1 6 P 8 7 2 E P 2 * 1 3 2 5 0 P . P . *

2 1 r 6 . 2 2 ° 7 f 3 F 0 2 * 1 4 2 2 2 6 . 2 1 o 7 o 0 F 0 ? * 1 4 2 3 0 6 . 9 0 6 0 1 8 P 0 9 * 1 4 2 4 2 6 . 1 9 2 E 1 5 F 0 2 * 1 4 2 5 0 6 . 1 7 5 2 2 ° E 0 2 *

I f 2 1 0 0 . 0 * ' 1 5 2 2 0 6 . 2 5 0 1 7 3 F 0 2 * 1 5 ? 3 0 0 . 0 * 1 5 2 4 0 6 . 2 2 2 5 2 4 E 0 2 * 1 5 2 5 0 0 . 1 *

1 6 2 1 r 6 . 2 ° P 8 4 P e 1 6 9 9 2 6 . ? p r « = 7 0 F P 2 * 1 6 2 3 r 6 . 2 6 6 4 2 P E 0 2 * 1 6 2 4 2 6 . 2 5 2 5 5 1 F T 2 * 1 6 7 5 n 6 . 2 ^ 4 8 3 9 E P 2 *

1 7 2 i 0 0 . 0 * 1 7 9 ? 0 6 . 3 0 ° 6 Q C F 0 ? * 1 7 2 " » 0 0 . 0 * i -7 2 4 0 6 . 2 P I 3 4 8 F 0 2 * 1 7 2 5 0 0 . 0 *

I P 0 6 . 3 4 < J 3 0 Q E 0 2 * 1 8 2 2 2 6 . 3 3 0 8 2 8 E 0 2 * 1 P 2 3 0 6 . 3 2 ^ 3 5 3 = 0 2 * I P 2 4 2 6 . 3 1 0 1 5 0 f 0 2 * 1 8 2 5 r 6 . 2 9 2 1 1 2 E 0 ? *

1 o •51 r r . r * 1 °

9 ? n 6 . 3 6 6 C 9 1 F < • 9 * 1 0 2 3 0 r . r * 1 0 2 4 0 6 . 3 3 7 9 1 7 F r 2 * 1 9 • 2 5 r P . P *

2 1 0 6 . 7 Z 7 q o 2 F 0 2 * 9 C ?? ? 6 . 7 1 f 0 " F 0 2 * 2 0 2 3 0 6 . 6 9 8 ° 0 6 c 0 2 * 2 0 2 4 2 6 . 6 8 1 P P 7 E 0 2 * 2 0 2 5 ' 0 6 . 6 6 1 6 6 3 E 0 2 *

2 1 2 1 0 0 . 0 * ? 1 9 9 0 6 . 8 P 9 0 1 . 7 F 0 2 * 2 1 2 3 0 0 . 0 * 2 1 2 4 r 6 . 8 5 5 9 1 P E r 2 * 2 1 2 5 ' r P . P *

' 2 ? i r 7 . r i R « H E r ? * 2 2 2 ? 2 7 . r A 4 g r f p r 2 * 2 2 2 3 n 7 . 1 4 7 2 1 4 F r 7* 2 2 2 4 2 7 . P 2 9 9 5 8 F r 2 * 2 2 2 5 ' 0 7 . 0 0 8 8 » 4 E 0 2 *

9 3 2 i 0 0 . 0 * 9 3 2 ? 0 7 . 1 0 1 1 0 1 F 0 9 * 2 3 9 3 0 0 . 0 * ? 3 2 4 0 7 . 0 6 6 Z 8 4 F 0 2 * 2 3 2 5 0 • 0 . 0 *

7L 2 " 1 0 7 . 1 5 1 0 C 6 C 0 ? * ? 9 2 7 . 7 3 7 4 1 O F 0 9 * 2 4 2 ? 0 7 . n O O 6 1 c 1 7 * 2 4 2 4 2 7 . 1 P 2 6 3 9 F r 2 * ? 4 r 7 . P P 1 5 9 2 F n ? *

112

? e r A • r ft *•» c 7 7 * i / i c l t ' r r c 0 0 . 0 ft 7 ^ ? 4 0 7 . C 0 6 6 P 7 P 0 ? * 2 5 2 C 0 0 . 0 * 9 1 0 7 . Q 7 0 7 f i C C 0 " f t 7 fc ?? 7 7 . 0 ' 1 1 * 0 C 7 A 7 7 0 C I J C A < p c 0 ' f t ? ' 7 4 2 t . < ? 8 3 4 0 6 F 0 2 * 2 6 2 5 0 fe.OAibp&c 0 2 *

? * 0 0 . 0 ft 7 7 7 7 r A A C 7 7 7 7 f* r . r ft 2 7 2 4 r 6 . 9 S 3 r P 8 F r 2 f t 2 7 2 5 r O . P * r t-.Of-f- » S= r 7 P 7 7 7 A . C ^ i 7 7 7 P 0 7 ft ? 3 0 f . O l P H f C 0 7 ft 7 P 7 4 2 f . 9 2 ? I ' O P F 0 2 * 2 8 2 5 0 6 . 9 0 3 2 1 5 F 0 2 *

7 C 7 ' 0 0 . 0 ft 2 ° 7 7 0 f - . O l 1 £ 1 P F 0 " - f t 7 C 7 7 0 0.0 ft ? c 2 4 0 6 . 6 9 1 5 C Q F 0 2 * 2 9 2 5 0 0 . 0 *

• 3 0 7 1 r r H i 1 " 7 7 7 ft. O ^ f i O i C r 7 f t 7 f i 7 7 r i t . n c u s r c r i r 2 4 2 6 . 8 4 0 5 n o p r 2 * 3 0 2 5 p 6 . 8 2 3 1 7 4 F 0 2 *

7 1 7 1 n f.r » •3 1 7 - > 0 l . 7 O 7 0 i . £ F 0 7 f t 7 1 7 7 0 0 . 0 ft 7 1 2 4 0 0 2 * ? 5 0 0 . 0 ft

3 ? 0 6 . 7 C E 7 0 i C O ' f t i 2 7 9 1 fc.7=0547F 0 7 ft 7 7 7 7 0 0 2 * 2 4 1 6 . 7 3 < > f 3 4 p 0 2 * 3 2 2 5 0 6 . 7 3 3 9 1 1 F 0 2 *

7 7 ? ' r r . r . ft •3 7 7 7 r « , . 7 ? c o r i : F r i « 1 1 7 7 r p . " ft 7 7 2 4 o t . T l = 5 1 7 P P 2 * 7 3 2 5 p 0 . 0 *

7 £ n . 7 1 7 a 7 i c 0 ? * 1 <1 7 7 1 6 . " T 0 ° 2 7 7 F Q 7 f t 1 1 ?"» 0 0 ? f t 3 ^ 2 4 1 6 . 6 0 9 = 7 1 p 0 2 * 1 4 2 5 0 6 . 6 0 4 4 7 1 - E 0 2 *

7 R 7 1 0 0 . 0 ft 7 5 7 7 0 h . f - O C 7 P C C 0 ? f t 7 7 0 - 0 . 0 ft I t 2 4 0 6 . 5 ® 9 7 3 ° F 0 2 * 7 c 2 C 0 0 - P *

7 A ? ! r ( . f m - i i t c • > 6 7 7 1 6 . 6 t > e 2 c o F r j f t 3 6 2 - » r 6 . 6 8 4 " 5 7 7 C r ? f t 2 4 1 (•. . 6 7 9 9 = 4 p r g f t 3 6 2 5 0 6 . 6 7 5 0 H 5 F 0 2 *

3 7 7 1 Q 0 . 0 * 7 7 7 7 0 t . * 7 C 7 1 i P Q 7 f t 7 7 7 7 0 0 . 0 ft 3 7 2 4 0 6 . 6 7 0 1 6 8 = 0 2 * 3 7 2 5 0 0 . 0 £

113

c T NflL TcfP f IST9I9UTICN WITH NCO|iNT = 3 SUM= TQTiL= O.CC9CC-7CT_O"> TKCT= r.c-ifipflip-pc PTM = " iCCQCOflflC-t

IT?P= 13-"" 1TEP = 1370 1TCH= I?""*

T J K QTCST= 0.t*':""t''0,:-0?

T ft 1 J K SPPr= o.*666* Ai.oc-O'' * I J K

ITERs !370 I J K T * I J K

1 ?i r r • » * 1 0 « . P * 1 70 0 0?ft 79 0 0.0 * \ '0 0 5.268664F 07 ft If- 1 O ' f t •5 7 7 0 0?ft 7 7 0 1 C . 7 5 7 7 ' ' 7 p 0 7 7 0 0 336c 01F 07* 2 30 1

C.321E4PF 02* *3 76 0 C fm •a "7 r» rt^r ft 7 7 P p e . 4 P o o 7 p c P 7 f t 7 29 p

r.r * •» 7 p P 377361E 0 2 *

i 2 * i = , 4 Q i 7 C C F <•?<• 7 7 r> 5 . 4 7 C 7 g Q C 07 ft 4 7 0 1 e . 4 i , r 4 7 c r 0-* 4 7° 0 e . t 4 f P 7 C F 02ft 4 30 1 C . 4 3 7 1 0 7 F 02* C ? A 0 0'* c 0 7 0 0.0 * c 7 P 0 =.=O0740c 02* c 2 C 0 0.0 * e 30 0 • 546e3'3 F 02* A 7 A i c . " ! 5 e / » " " p ' ? f t 6 •5 7 p r 7 f t A 7 P * R . A Q A T A r c "7ft A ? c 0 = . A 7 7 4 7 1 F r 7 ft 6 7 P 1 • 5 . 6 c < ? 8 7 7 F n 2 *

7 9 t r f i t 7 n O 0.0 ft 7 ?P 0 C t p 7 1 C O'ft 7° 0 0.0 * 7 30 0 827844F 02ft P i 5•°4P606c 0 "J* Q n 0 e . op&c O ^ f t P ? P 7 " 00 f- P 9 C 0 0?» 9 ?0 1 8?F87op 02* 0 r c . 0 0 1 j c p p Oft O r . r ft c 7 0 n e m Q 7 P 7 t t c 7 7 f t e 7 O p r . r * 0 7 P 0 •5.8P4277F 1 2 *

ir ?f- •j 6 . 0 1 4 1 1 I P 03* ' 0 0 e.CO/LTQI P 0 7 f t 1 0 ? P 1 07* 1 0 7 0 0 c K 7 F 02* 10 30 7 6.°12676E 02* 2* 0 6.04=91 pp o - > * 1 1 0 0.0 * 11 7 V* 0 c . c o r 7 r 11. 79 0 0.0 ft 30 0 ?.°4irA7F

i? 2 6 i ( j . r m n j c r \r 7 7 r 6 . P 4 7 A 7 1 . F f i t 1 7 7 0 T 7 f t 17 ?o r C . C Q 7 7 1 7 C r ? * 1 7 ?n 1 5 . C A 0 4 - 7 C F f > 2 f t

^ i 7 ' 1 0 6 . 1 0 0 2 O ^ * . 0 7 7 0 0.0 * 1 7 7 P 0 6.04P,C0C 07ft 7 7 2® 0 0.0 * 1 3 30 0 5 .c97e.R4F 02* 1 4 7 A 1 6.1?P81*F 0?* » 4 7 7 0 6 . 1 0 7 1 7 C C 0'ft 1 4 ? C *» A . o"*73 °7c O'ft 1 4 2® 0 6.050P1°F 07* 1 4 31 7 6.r?£64FF r 2 *

If ? 6 r 6 . 1 6 P l 6 ? c r 7 f t 1 c 7 7 p r . r * 1 = 9 0 r A . 7 " C 7 P 7 r « ? * 1 c 7 C P ft 5 3 r p " . r 5 3 1 7 C F 02ft ••6 21 7 A . 1 O l 0"# 1 6 7 7 0 6 . 1 t r 07* 1 6 0'* 1 A 29 0 <.1066e?c 02ft 16 30 1 6.08072PF 02* 1 7 2 A 0 f- <!.'?f 80' c 07* 7 "7 7 7 0 0.0 ft 1 7 ? P 0 '1. 1 6 1 c 0 f P 0'* * 7 0 0 p . p ft 7 7 3r p 6 . 1 0 7 ? 9 3 F P2* 1 0 26 i 6 . 2 5 i r p r p r ? f t 1 P 7 7 r 6 . ?'r 5 "F P 7 f t 7 B ? P ^ A . I q o P ^ A C 1. 0 29 r 6 . 1 6 1 r P A C T2* ! e 30 1 6.17447PF 02* 1 C 7 6 0 6 .7R1 33s c Or* 1 Q 7 7 O 0.0 ft 7 O 7 P 0 £ . 7 1 E f c 4 C = 07* 1. 0 7 c 0 0.0 * 19 • » o 0 6.1 60601 P 02*

2 * 1 6 • 6 1 ° C 4 P F 0'* ?0 0 6 .eB1 73er 02* 2 0 7g " I 7 A . K 4 = 6 4 7 C P ? f t 29 r 6 . 5 1 4 1 0 A C r 2 * 7 D ? r 1 6.18422<=F rz*

21 26 r 6 . 7 o i r ? * ? ! 7 7 p p . p * ' I O ^ f t 2' 2o r p . n » ? 1 30 0 6.65279®F 02* 7 ? 2 6 i 6.°6447?c 07* 77 7 7 0 6.O74KO0F 0?* 72 ? P 1 6.PP70?1F 07ft 7 7 2 C 0 6.8E337PC 02ft 72 30 7 6.P213e2F 02*

0 7.000OA?C 0?* '3 7 7 0 0.0 ft 7 f > r . A , O J 7 4 ' P F •> ? « 7 7 29 c r . r ft 23 3r p 6.857634F r ? *

* ? 4 ' 6 1 r ?* 21 p A . C 0 7 7 C - 7 p f 7 f t 71 2P 7 { . Q S O I O Q F 07* 7 4 2 = 0 02* 7 4 30 1 6.PM38Q9P 02*

llU

IC a ft.otafoc 0?* 9 c 97 0 0 . 0 * 25 2? 0 6 , <5ft047BC 09* 9C 0 0 . 0 * 2 = ' 0 0 6 .806057F 02*

"» f t ?6 i ft. o j» 999 c r ' 6 99 n ft ,pp99pi r ' f t 90 i A .A47ft9ftF "94 9ft 29 P e . e i E ' o o F r 9* 2ft 9f> 1 6.7848?QE 0 9*

27 2* o ft,hoi 7 r 9* 9 7 99 0 0 . 0 * 97 2° 0 6 099CC 02* 97 2° 0 0 . 0 * 27 90 0 6 .757527F 02*

9P 26 1 02* 9« 97 0 ft . P ' f *=52F ?P 7P i 6 .7010P4C 0 2* ?P 29 0 6 .759P20f 02* 28 30 1 6 . 7 3 0227F 02*

•>c 26 r* 6.&24410E r ? * 29 97 o 11 .o * 9 O 9P r ft .T^ftCOCf i ? * 90 2 = n o.O * 2 ° 3P P ft.6°P279F P2*

i r 26 i 6.7<iffc = « 07* 30 97 0 6 02* 30 2e ft 0?* 30 JO 0 6 . 69385 5F 02* 30 ? 0 l 6.6ft 6745 E 02*

3? 26 0 6 . 749 566 «= 09* 31 99 0 0 . 0 * 91 9 P 0 ft , f tpq091c 0 ? * 91 2° 0 0 . 0 * 31 30 n 6.63"55"3F P2*

32 1 6 . 7 1 3 1 B I p r ? * 32 97 n 6 , f p P 9 7op r 9* 99 9 P i ft ,ft64Iic4n 9 9* 99 29 r fc.639?90f A 2 * 32 j n 1 6.614?ft3E P2*

3 ' ' 6 0 0?* 9 9 97 0 0 . 0 * 9 9 2" 0 ft ,ft409CCF 02* 99 2° 0 0 . 0 * ? 3 30 0 6.5°984OF 02*

9 4 26 1 6 . f 7 = t i e F 09* ?4 9 7 0 6 m f . r iqq i F 02* 34 28 1 ft ,ft31 P77F 02* 34 2° 0 6.6076"'3E r 34 9 A 1 6 .584744F r 2*

9C 26 o 6.66635P «= o ? * 35 99 n f .r * 9S 2? r» ft o 2* « 29 r r * •=>5 3P P 6 .57f t f75F 02*

36 26 i 6 .6 C - M 0CF 02* ?6 99 0 6 , t 3 = 4 4 4 F 3ft 9 P i ft .6150=4F 02* ?ft 2° 0 6 .59141 IF 02* 36 30 6.56C023F 02*

97 ?ft 0 ft.ft47fl£Cp 02* 37 27 0 0 . 0 * 97 28 0 6 .f t06 64Pc 07* 7 ' 29 r r . C * 97 9n r 6 .561172F 07*

115

FINAL TFMP DISTRIBUTION WITH MCOUNT <= 3 1T>=P = l ' 7 o

SUM= 0 . 5 c 6 1 4 5 7 4 p _ 0 9 TOTAL= 0 .oooco<i->ep_oi ITFR = 137"

TCST= C.°9CT«f-i7F-o<5 DTM= 0 .4000Q090P- 0<> ITPR= 19 7 0

OTFST* r . 6 6 513620 F - i 9 SPEC = r* .66666 *4<-c-" 9 ITFR = 1 ? 9 r

I J K T * 1 J K T * I J K T * I J K T * I J K T *

1 ?1 0 0 . 0 * 1 32 0 353 F 07* 1 3 1 0 0 . 0 * 1 3"- 0 3 .446147C 07* 1 91? 0 0 . 0 *

9 31 11 C . 3 0 4 0 ° 0 F 02* 2 3211 4.301o^OE 02* 2 3311 ? .409011 r 09* 9 3 4 1 3 . 4 6 e r 9 f . p r j * 9 IC r 3 .455fei 5F 0 2 *

3 1 <• r . r * 3 39 11 4 . t ? 2 2 e P F 03* 3 33 r . r * 3 34 r 3 . 4 0 1 p M F 09* 1 ?5 0 0 . 0 *

4 3 1 1 1 5.414<3=4C 02* 3211 4 . 4 7 1 6 0 2 F 02* 4 9 3 1 1 7 .591307P 0 2 * 4 24 1 3 . 5147 lop 02* 4 3? 0 3 . 5 0 0 8 4 0 P 02*

c 0 0 . 0 * 5 3211 4 . 5 4 0772F 02* c 3? 0 r . . r * c: 3 4 r 3 . 5 5 7 0 P 1 F r ?* 5 35 r r . r *

f 1111 c . f ? o e 7 i P 6 T i l 4.627CC6P r ? * A 191 1 3 .6*.60*? c r 9 a f- 3 4 1 3 .60 1 252E 07* 6 35 0 07*

7 3* 0 0 . 0 * t i j 11 4 . £ 0 9 7 0 4 F 07* 7 ?•» 0 0 . 0 * -7 3 4 0 9 . 6 5 9 2 4 0 F 0 2 * 7 9 c 0 0 . 0 *

8 H i t 5 . n 9 4 4 4 l F 07* P 4.7=Of 4 ' F r 9* O 111 1 1 . 6 8 ^ ? ° " «9J» P 34 1 3 . 6 6 P 9 , D C ' •2* P 35 r 3 . 6 5 3 2 1 " r 9*

c 3 1 r r . r * O 3211 4 .77P7ferF r ? * O i i r r . r ft O 3 4 0 3.fc78870F 0 2 * 0 35 0 0 . 0 *

10 3 1 1 1 c .PO0784R 07* * 0 3211 4 .7C70? f P 0 2 * 10 3 3 1 1 9.7QC107P 09* 10 34 1 3 .688P33F 0 2 * 9C 0 3 . 6 7 2 8 5 6 F 07*

l ' 3 1 0 0 . 0 tt 11 32 1 1 4 . P 1 7 1 5 T 11 33 P r . r * 1 1 34 n 3 , joocp- jp r 9 * 11 95 r r . r *

3 1 1 1 5.C471.3 6F 3 2 1 ' 1? 191 1 3 .7957BCC 0 ? * 1? 34 1 3.70<? ,.C3F 02* 19 35 0 3 . 6 e 2 ° 1 " ' F 02*

13 3 ; 0 0 . 0 * 1 3 a ? 1 * 4 ® o c 5 c i o p 0 ' * 19 33 0 0 . 0 * 13 34 0 3 . 7 J C 4 6 0 F 02* 13 i e 0 0 . 0 *

U 51M •S.OO'O^F r 1 4 32 31 4.P7477CC p ? * 14 3311 I . 74«79(«P r 9* u 34 1 3 .72070CF r ? * 14 35 r 3 . 7 1 3 3 1 3 F r 2 *

1 = •ai r * 1 e 1911 4.P«J.?711 C 0 2 * 1 e 3 9 0 0 . 0 * 1 B 9 4 0 3 , 7409?5F 02* 15 35 0 0 . 0 *

' 6 01 ' 1 6 . 0 E 7 K I c c 0 ' * ' 6 ' J M 4 . 0 1 I^O^F 09* 1 fc 3 1 1 1 3.7«,-»c?o = 0 7 * 34 1 3.7506AOp 02* 16 35 0 3 .711Q40F o?»

17 31 r * 1 7 •3? 11 •\,C91 33CF 1 7 33 r r . r * 1 T 34 r 3 . 7 6 1 1 *-9F r ? * 17 9C r r . r« *

IP 3 1 1 1 e . i i r p ' ^ F 0?# 1 R 3211 4 . o 5 0 0 ? ' F 09* 1 P 3311 1.7PQ949F 07* ? 0 34 1 3 .771 685F 0 2 * i 8 3 5 0 3 .75470O F 02*

1° 11 0 0 . 0 * 1 Q 3711 « . 0 7 ? B 0 2 F 03* TO 33 0 0 . 0 * IC 94 0 r . 7 8 2 7 1 4 F 0 ? * 1 0 35 0 0 . 0 *

?r 3! 11 * .457P7PF 3 9 1 1 c . i o c r i o p r 9 * ?" 991 1 3 .0399c ?e r ? * ? r 3^ 1 3 . ° 1 2 6 6 6 E r g* 2 " 9 c r 3 . P ° 4 1 r 4 c r 2 *

' 1 3 1 0 0 . 0 ft •>1 1' 09 A 91 19 0 0 . 0 ft n 94 0 ^• .003 e 72F 07* 2i 35 0 0 . 0 *

3 1 j i 6 .70177* c 0"* 7 2 7 211 <5.4S4?i4r 0 9 * ?9 9 9 11 4.1159Q9C 0 ? * •>? 94 1 4 .0°44O0F 02* 97 9 K r 4 . r 7 4 4 7 o p r i *

' 3 91 0 0 . 0 ft 71 •a 9 1 * c ^ 0 3 1 ccp 07 » 19 Q * 9 9 ; 4 n t . l 9 P l o e c r 9* 91 9C ^ r . r *

74 • 3 1 1 1 • ' i * . e ? i H ? F r l i v 94 1 4 # * o a - r r r -I 94 > 4 . * ^ } f*r?r 94 1 r 0 4 . » 4 ; * f e p 0 ? *

116

2 e 3 1 0 0 . 0 ft 25 " ' 1 1 5 ,z.q?7£OF 0 ' * 25 3 ' 0 0 . 0 ft 25 34 0 4 . 14849£F 02* 25 35 0 0 . 0 ft

26 3? 11 r j * 96 ! c r ? 4 ?6 '311 4 . U 5 7 7 1 c 26 34 1 4 .1 452 6°F 02ft 26 35 0 4.125627E 02*

27 31 r o . r * 77 3211 c 02* 27 33 0 0 . 0 * 2T 34 0 4.140962E 02* 27 35 0 0 . 0 .* 28 3111 6« 70"11 °0 P 0 ' * 28 3211 e , A»oei7F 0") * 2" 3311 4.1=674?B 02* 2P 34 1 4.136663F 09ft 28 35 0 4 . 11731 oe 02*

2° 3 1 r O-.n * ? Q 5 • 1 67 r F 2q 3? p r.r * 2° 34 0 '4.13Zr61E 09 ft 29 35 0 n.O *

if 3111 fc.f3oqicF 0?* 30 3^11 0?* 30 3311 4.1 4705 Pc 0?ft 30 34 1 4.127A79F 02* 30 35 0 4.108462F 02*

3 1 31 0 0 . 0 * •51 ' '211 c .37°006F 02* 3 1 33 0 0 . 0 * 71 34 0 4.123552F 02* 31 35 0 0 . 0 *

•»? 3i 11 ^ .c f ioomF r 7 * ^ 2 •>21'. •5 .3645 97F 02* 39 331 1 4'. 14P177P 32 34 1 4.12C9F2F r 2» 32 35 r 4.10220OF 0 2 *

3 7 •>1 0 0 . 0 ft •a 7 c . ' '56086F 09* 31 77 0 0 . 0 * 37 34 0 4.119045F 02* 33 35 0 0 . 0 *

3 1 1 1 BCC] q£C 02* 3 4 3211 c 02* 34 3 3 1 1 4.136091 c 02* 34 34 1 4.117151F 02* 34 35 0 4.0«e557F 0 2 *

?' r r.r ft ' 5 3711 e .3431°1F 35 0 . 1 ft 7 F 34 0 4.116167F r 2* 35 35 0 " . 0 *

7* 3 ' 11 6.5*3639? 0""» 36 32 1 1 c .318801F 02* 36 3 , 1 1 4 .134 006c 02* 36 34 1 4.11E18RF 09ft 36 35 0 4.0°6677F 02*

" 0 0 . 0 ft 37 •*? *1 c .314.^1 op 02* 3 7 33 0 0 . 0 ft 37 34 0 4 .1 !42"°F 02* 37 ?5 0 0 . 0 *

117

Sl)M= PINAL TCMP TISTPieUTION WITH NCOUNT = 3

0.rc6i4524F-0'> TOTAL = o.ooooooTPF-O' l t c p = i n o

ITFPC- I I-»O

TFST= 0 . ° 7C7PA17C-QK OT«= 0 . i a c c . c o f ITFS« 1.370

QTFST= c77f.2PF-.ru SPFC = f . 6 6 t * * t 4 P F - ' ? IT5R s 137"

I J K T * I J K T ft I J K T ft I J K 7 *

1 0 2 .4?0264F 0 ?* » 7 7 0 0 . 0 * 1 ?P 0 7 . 7 0 £ 7 £ t r o»* t 1 0 0 0 . 0 *

2 1 0 ' * 2 77 0 1 ,£7017£F 02* 7 7 p 1 1.A1A77CC 0 ' *

i> 3 0 0 2.4C2P4PF r ? *

7 r 2 r 7* 7 77 o p . p * 7 IP p 7 . i i B w r r < " ? * 3 9 C P . " *

4 ' 6 1 7 . F 0 ' * 4 11 0 7 .^7777PP 0 ' * 4 70 1 7.£AQ47*C Q7ft 4 i c 0 3 P90P 0 ' *

C 0 0?ft e 77 0 0 . 0 ft s 7P 0 * . e 0 ' 0 O T P 07* K 3 0 r P . P *

6 ' 6 1 3. eT?Ofc7E r ?ft 77 p 1 , f STfctf-P r m ( 7P 1 3 .R477P1 P P 7ft t. 3° p 3 .529646c 0 ' *

7 0 O'ft 7 7 7 0 0 . 0 ft 7 70 0 7 . t o o j i c e 0">* 7 3 c 0 0 . 0 ft

P 36 1 0?ft n 77 0 02* p IP * 3 .6 ' , "4o*P " 2 * P IC f 3 . 5 ° 4 r ?7c P 7ft

c 36 r 2 . f 4 7 7 ? o E r j » c 37 r o . r ft a 70 r 3 . 6 t P " " e r "7ft O 35 0 0 . 0 ft

' 0 36 1 3 . 6 c 7 4 7 j e 07ft 1 0 77 0 7 P 0?* 10 70 1 •».627=4*P 0?ft ' 0 3 0 0 3 .6 I30^5P 02*

1 1 36 0 2 .667334P 07ft 1 1 77 0 o . r ft 3P r» 7.63777=P " 2 * IC r P.C *

1 . 2 3* 1 P 7ft » 2 77 o 3 .6* ' P<?5C r ? * ' 2 »p 1 , . 6470>7C 02* ' 7 3 0 0 2 . 6 3 ? ? 0 , P

13 36 0 3.6PT31CC 07ft 1 3 •»7 0 0 . 0 * * ? ip 0 3.6 ,!6<40' !C 02* 1 7 i® 0 0 . 0 »

26 1 , . 6 a 7 4 C 0 F 07* 1 4 77 0 3 ,(• PI P « F P 7ft 14 IP « 17* ' 4 3° r 7 , t s ? r P P P

1 = 26 p 7.-TP7BOCC r 7ft 1 K 77 r O.P * 1 c 39 0 3 .6 7 f p 7 6 c 0 ' * « r IC 0 0 . 0 ft

16 1 7 .7177PP? 0?* ' 6 77 0 y . ' 0 2 0 6 P F 0?* 1 * 70 1 3 . 6 ?6P B , C 07ft 7 0 0 3.6T200OP 0?*

1 T 36 0 2 .7?8P57P P7ft 1 7 7 7 r o . r . * t 7 3P p » 7 t* P.P. • ' P 1 7 # 77 p7 779 r j * 1 9 77 0 3 .77?*.7"»F 02* ' fl 70 1 ••,T0704fP 02* 1° 7 c 0 3 . 6 9 2 0 e 0 F 02*

IC 76 0 7 . 7 4 n t ? ! P 02* 1 0 17 0 0 . 0 * >0 7 O 0 07* 1 O IC 0 0 . 0 •

?r i f 1 7.P7617CC r ? * 2 7 r 3 . P5° i?3F r 7 * 2 " •»8 1 7.B/,»<t77C 2<* 3° r 3 . < , 2 t ° , , P

7 * « r 3.«6565OP 07 ft 77 0 0 . 0 * 71 0 3 . « 3 0 o , e r 07* 2 ' 1 9 0 0 . 0 •

2? 26 1 4 . 0 e 5 1 5 4 C 0?* 77 17 0 4 . 0 , ' - o C 4 F o»* 22 id 1 4.0 ' .o243E 02* 7 7 7C 0 4 . 0 0 2 f ^ ' P 07*

r 4 . P P P * ? " ' 3 17 n P .P * ? ' 1Q 0 4.P5541CP ' 3 3 0 p p.f" *

7 4 . 7 f 1 4 . 1 7710«'c 07* 74 l-» 0 4 . 1 0 7 C60F Q7ft 24 70 4 . 0 8 e ' l 0 P c 02* 24 IC 0 4 . 0 f r P e 6 0 r 07ft

* I J K

118

9 = I t 0 Q9ft 7 5 7 7 0 0.0 • Id 0 t ,077«.*7F 02* 2? 3 9 p r .n •

36 i ' - . i ' ^ t s e e r 9* 7fc A r?* 76 IP 1 4 .r71P"7C p 2* 26 3° r 4.r53e3«E 07*

77 "»6 0 4.10767" 07* ?7 77 0 0.0 * IB 0 4.06*041 c 2 7 39 0 0.0 •

S" 36 1 4.0ao673P 0?* 1» 77 0 4.c*0e64C 0'* 3* 1 4.r«,7Poi c "2* 2° 3 9 r 4 . 0 4 6 1 6 7 f r 2*

*6 * 4.pQ£«pe n * c o 7 7 c n.r * 7° IP p 4.05arr4F A?» 20 3 9 0 0.0 *

•>o * 4 . 0 ° 0 i 0?" 7 0 17 0 a . o ' ? ? 1 * f 0'* 3 0 ' e i 4.054O44C 02* 3 0 3 9 0 4 . 0 3 8 0 7 1 f 0 2 *

n 'A 0 l . q o t c o " : 07* 7 1 77 0 0.0 ft 71 i t p 4 .rei tc) e A7ft 71 3 9 0 r .p *

77 « i . r o i ' r c t p?t 7 7 77 p 07* 79 i a i 4.04O99<F 0 7 * 77 3 ° 0 4 . 0 3 2 3 < , c = 09*

11 ">6 0 0 '* 7 7 1 7 0 0 . 0 ft 11 70 0 4.047««.C 07* 11 3 ° 0 0 . 0 • 1 i / O f f f t f p?» 7 4 7 7 ft 14 ?P 1 A.PASPOAC **2* 3 9 r 4.r»2°n67F r ?*

i r 7*. r t / i c i f f e 7 f 7 7 0 0.0 * 10 0 4.0«c0?1c 02* 3 ' 3 9 0 0 . 0 *

7* if 1 4 . 0 , o « 0 " 0?" U 77 0 4.0*1196 F 0'* 7p 1 02* 36 3 ° 0 4.027346F 0 » *

77 r t / f O f B C 7 7 77 A r . r * 17 10 A t .P411PAC "7* 3"» 3 ° r r .0 •

119

C»S G»p HIN WIDTH i ' o . o ' i p o ' j

CALCULATED HIOTH Oc GAS GAPb MAX WIOTH O.OIO"^

120

PROBLEM NO 2 0 1

REGION MATERIAL NCO 1 2 3

GRAPH FUEL SPACE

& 8

1 2 3

ITER UOO 800 1370

TENT= 0.40 TINT= 0.10

! IS I d d d «s

Fig. 10. Scale Drawing of Cylinder

121

RADIAL PLOT OF CYLINDER TEMP PATTERN AT EL = 1.4200

s.o 0,1 a* a* • .« m m u u m IS

R/RMflX

Fig. 11. Graph of T vs R

122

AXIAL PLOT OF CYLINDER TEMP PATTERN AT RADIUS OF 0.2976

0.0 0.1 b.j 0.* o.« o.* «.s a ? m «.• IS

EL/ELMfiX

Fig. 12. Graph of T Vs EL

123

APPENDIX F

NOMENCLATURE (WITH UNITS)

SYMBOL MEANING A Area of a heat transfer surface

A'

B

C

EL

F

G

K

First constant of integration in conduction equation (Chap. IV)

Second constant of integration in conduction equation (Chap. IV)

Conductance

Z coordinate of a point within the cylinder - measured from the top

Factor in radiation equation (Fe - ei factor) (F = emissivity factor; Fg = shape

Heat generated within a node or a cylinder

Thermal Conductivity

METRIC UNITS cm3

none

none

watts deg C

cm

none

watts gram

watts en? deg C

cm

L Overall length of the cylinder cm

Q Overall heat flow rate watts

R Radial coordinate of a node cm

T Temperature of a node or surface deg C

a Exponent in gas conduction equation none

h Heat transfer coefficient watts cm? deg C

ENGLISH UNITS

ft2

none

none BTU/hr deg F

ft

none

BTU/hr lb mass

BTU/hr ft2 deg F

ft

ft

BTU/hr

ft

deg F

none

BTU/hr ft2 deg F

22k

SYMBOL MEANING

Length of a regular node q Heat flow rate in a subcylinder

or a node x Distance along a heat flow path

METRIC UNITS

cm

watts cm

ENGLISH UNITS

ft

BTU/hr ft

Greek Letters

a

6

P

0

V2

Linear coefficient of expansion Width of a gas gap Density of a material

Boltzmann constant

Absolute temperature

(deg C)"1

cm grams cm3

watts cur3 (deg K)4

deg K LaPlacian Operator:

dr \ / dz in two dimensional cylindrical geometry none

(deg F)"1

ft lbs mass

ft3

BTU/hr ft2 (deg R)4

deg R

none

g

Subscripts

Conduction Emissivity Gas

g.c. Conductivity across a gas gap h Convection

125

SYMBOL MEANING

i Subcylinder number (consecutive from £ outward)

r Badiation

ref Reference

s Shape

s Sink

0 Value of temperature where K is measured (or known) s

126

APPENDIX G

F0RTRAN NAMES AM) SYMBOLS

NAME

A

A1(K)i

A2(K))

ADEN(l, J)

AD IF

AEX(l)

AI

ALPHA(K)

ANUM

MEANING AND USE

A factor used in the plotting subroutines to scale the

abscissa.

/ Functions used in plotting subroutine.

PICT to convert dimensions of the cylinder into an

argument for subroutine CRTNUM.

The denominator of the term in the conduction equation

used to calculate T(l, J). It is calculated in subroutine

PRESET and used in subroutine STEADY. ADEN (l,j) = Cx(l, j)

+ C2(I, J) + C3(l, J) + C4(I, J) (watts/deg C or BTU per

hr/deg F.

Absolute value of the relative difference between TNEW

and TOLD of a node. It is used to find the maximum devia-

tion during a single iteration (deg C or F).

The exponent "a" in the equation for the conductivity of

gas in a gas gap (Chapter III, equation 50).

A function used in dimensioning the abscissa in subroutine

RPLOT and ELPLOT.

The coefficient of thermal expansion of region K.

The numerator of the term in the conduction equation used

to calculate T(I, J). It is calculated and used in subroutine

STEADY. It is used as part of the equation for calculating

the convection and radiation case. ANUM multiplies the

127

NAME MEANING AND USE

conductances of the node "by the temperature of the

adjacent node and sums the products.

B A factor used in the plotting subroutines to scale the

ordinate.

BL1| {Functions used in the plotting subroutines to convert

BL2J (dimensions of the cylinder into an argument for CRTNUM.

BUF The name of the buffer area for the plotting subroutines.

BZ The Boltzmann constant in metric units (watts/cm2 (deg K)4)

BZM The Boltzmann constant in the units selected for the

problem (-watts/cm8(deg)4 or BTU per hr/ft2(deg R)4).

Cl(l, J) Conductance of node (I, j) in the positive radial direction

(watts/deg C or BTU per hr/deg F)<

C2(l, J) Conductance of node (I, J) in the negative radial direction

(watts/deg C or BTU per hr/deg F).

C3(l, J) Conductance of node (I, J) in the positive axial direction


0^(1, J) Conductance of node (I, J) in the negative axial direction


CC A temporary number used in PRESET in calculating the

conductances.

CLY A number used in PICT in drawing the centerline.

C0N(l) Conductivity of material (i) Input data.

(watts/cm deg C or BTU per hr/ft deg F).

128

NAME

C0NTZ(I)

CRT

CRTSYM

CRTNUM

CUE

CYL

DELG

DELGA

DELGB

DELL

DELR

MEANING AMD USE

Thermal conductivity of the gas in gas gap(l) at temperature

0o (watts/cm deg C or BTU per hr/ft deg F).

A subroutine for drawing graphs on the cathode ray tube

plotter.

A subroutine for drawing symbols on cathode ray tube plots.

A subroutine for printing numbers on cathode ray tube plots.

A variable which is used to sum the heat transferred by

convection and radiation from each surface node. It is

compared with the total heat generated within the cylinder

as the third convergence check (watts or BTU/hr.).

The name uf the main program.

The calculated width of the gas gap (cm or ft).

The radius of the outer surface of the gas gap calculated

by the expansion of the cylinder at the calculated tempera-

ture of the surface (cm or ft).

The radius of the inner surface of the gas gap calculated

by the expansion of the cylinder at the calculated tempera-

ture of the surface (cm or ft).

Delta H. The length of a node. Calculated in subroutine

PRESET as a preliminary step in calculating the volume and

the conductances of the regular nodes (cm or ft).

Delta r. The radial width of a node. Calculated in sub-

routine PRESET as a preliminary step in calculating the

volume and the conductances of the regular nodes (cm or ft).

129

NAME MEANING AND USE DELTAT Delta T. The temperature drop between the outer surface

and the sink calculated from the total heat generated

within the cylinder and the heat transfer coefficient of

the surface (degrees C or F).

DIFF The algebraic difference between the new and old values of

the temperature of the node (I, j). It is used in calcula-

ting ADIF and in overrelaxing the temperature.

DILL The inverse of DELL.

DILR The inverse of DELR.

DTM An input parameter used to check convergence in the first

convergence test. When TEST is less than DTM convergence

is assumed, (dimensionless)

EL(l) The axial location of nodes (I, J) measured from the top of

the cylinder (cm or ft).

ELl(l) Distance from the top of the cylinder and the top of region

(I) (cm or ft).

EL2(l) Distance from the top of the cylinder and the bottom of

region (i) (cm or ft).

ELPL^T The name of the subroutine that plots temperature versus

axial position, with radial location as a parameter.

ET(l) The axial location of the extended nodal boundaries, meas-

ured from top of the composite cylinder (cm or ft).

FAC An input parameter which controls the value of SPEC which

is used as the third check for conversion. SPEC « FAC/KC0

(dimensionless)

130

NAME

FR(I)

FS(l)

GASA

GASK

GPW1(I)

GPW2(I)

H(I)

HA(I, J)

HG

HT0T

MEANING AND USE

A fraction which represents the effective ear] snivity-

absorptivity relationship between a radiative surface

and its associated sink. The program uses the value 1.0

if no value is entered in the input data, (dimensionless)

A fraction which represents the geometrical view factor

in the equation for heat transfer by radiation between a

surface node and its associated sink, (dimensionless).

The ratio of the average value of .the temperature of the

gas gap to T. Used in subroutine STEADY in calculating

Kg. (dimensionless)

Thermal conductivity of the gas in the gas gap at the

average temperature of the gas gap. Calculated in sub-

routine STEADY, (watts/cm deg C or BTU per hr/ft deg F)

Minimum calculated width of gas gap (i). (cm or ft)

Maximum calculated width of gas gap (l). (cm or ft)

Convective heat transfer coefficient (h) of surface (i).

Input data. (watts/cm2 deg C or BTU per hr/ft3 deg F)

Convective heat conductance (hA) of a surface node,

(watts/deg C or BTU per hr/deg F)

Total heat generation rate within the cylinder, (watts or

BTU/hr)

Total convective heat conductance (E h A) of a surface,

(watts/deg C or BTU per hr/deg F)

General index.

131


IA Output option No. 1. Calls for a printout of the R and

EL arrays as first established.

IB Output option No. 2. Calls for printout of R and EL

arrays after rearrangment.

IC Output option No. 3. Calls for printout of V0L, R0M,

RIM, DELR, DILR, DILL and the size and region of regular

nodes.

ID Output option No. b. Calls for printout of the coefficients

of regular nodes before adjustment for insulated or gas

gap surfaces.

IE Output option No. 5- Calls for printout of the region and

node designation of gas gaps.

IF Output option No. 6. It is a control index rather than an

output option. It causes the surface nodes to be set at

the average temperature calculated from the heat genera-

tion rate and the convective heat conductance at the

beginning of each series of iterations.

IG Output option No. 7. Calls for printout of T0LD, TNEW,

T(I,J), TEST, SUM, CUE, is for each of the first three

iterations of the first series of iterations.

IGM An index to set limit of D0 loops for gas gap input and

calculations.

IH Output option No. 8. Resets output option 7 for all

series of iterations after the first.

132

NAME MEANING AND USE II An index used during subroutine STEADY for printing the

maximum temperature during a single iteration. It

represents the I subscript of TMAX.

IJ Output option No- 9 . Calls for printout of the coefficients

of plane and cylindrical special nodes.

IK Output option No. 10. Calls for printout of the coeffi-

cients of the regular nodes.

IL Output option No. 11. Calls for printout of the tempera-

ture array of regular nodes for each of the first five

then every tenth iteration.

IM 1-1. An index used during subroutine PRESET to set the

limit for a D0 loop used to rearrange the nodal boundaries.

IN Output option No. 13. Calls for the printout of gas gap

data for each gas gap node during iterations.

IP 1+1. An index used in subroutine PRESET to control a

D0 loop for the rearranging of the nodal boundaries.

IS An index to indicate the number of the surface to -which

a particular sink is related.

ITER An iteration counter.

IX A counter used in the expansion of the R and EL arrays

during the process of subdividing the regular nodes.

IZ Output option No. 12. Calls for printout of the subscripts

established to relate the temperature distribution at the

end of one iteration series with the nodes at the beginning

133


of the next. For NC0UNT = 2 the values of KX, LX and NX

are printed. For NC0UNT greater than 2 the values of KX,

LX, KK, LL, and NX are printed.

J A general index.

JJ An index used during subroutine STEADY to printout the

maximum temperature during a single iteration. It

represents the J subscript of TMAX.

JM J-l. An index used during subroutine PRESET in the

expansion of the nodal boundaries to subdivide the

regular nodes.

JP J+1. An index used during subroutine PRESET in the

expansion of the nodal boundaries to subdivide the

regular nodrs.

K A general index. Used primarily in indicating regions.

KA An input index used to indicate -which system of units are

being used.

For metric system KA = 1.

For English system KA = 2.

KF An index used in the plotting subroutines to establish

the upper limit of the graph.

KK An index used in subroutine PRESET as a subscript in the

temporary temperature array TT. The value of KK is computed

from the value of I of the new (subdivided) node (I, j).

309


KK An index used in subroutine STEADY to reverse the order of

calculation (i.e. the iteration proceeds from the outer

surface to the centerline).

KM K-l. An index used in subroutine PRESET in expanding

the arrays to subdivide the real nodes.

KMAX An index calculated in subroutine STEADY to set the upper

limit of the temperature graphs plotted in subroutines

RPL0T and ELPL0T.

EMIN An index calculated in subroutine STEADY to set the lower

limit of the temperature graphs plotted in subroutine RPL0T

and ELPL0T.

KR(l, J) The number of the region (or material) associated with the

real node (I, J).

KTM An index used in printing the temperature (every 10 degrees)

on the ordinate of the plots made by subroutines RPL0T and

ELPL0T.

KX An index used in subroutine PRESET as a subscript in the

temperature array TT. It is calculated from the value of I

of the new (subdivided) node.

L A general index. Used also as a counting device in

establishing the boundaries of the regular nodes. Used

as an index in a computed G0 T0 statement.

LIT A number calculated in subroutine STEADY to indicate whether

convergence was reached. If LIT = 0, convergence was not

reached.

135

NAME

LL

LM

LMT

LOW

LX

M

MAXI

MAXIT(l)

MAXJ

MI

MIM

MJ

MEANING AND USE

An index used in subroutine PRESET as a subscript in the

temporary temperature array TT. The value of LL is calcu-

lated from the value of J of the new (subdivided) node

(I, J).

L-l. An index used in subroutine PRESET in expanding the

arrays to subdivide the real nodes.

An input parameter which limits the number of iteration

per cycle if convergence is not reached or is too slow.

An index used in the printout routines to control the

first value of each printout array.

An index used in subroutine PRESET as a subscript in the

temporary temperature array TT. The value of LX is calcu-

lated from the value of J of the new (subdivided) node

(I, J)-

A general index. Used in a computed G0 T0 statement.

The largest I value in the nodal array.

Largest value of ITER calculated in iteration series (l).

The largest J value in the nodal array.

An index used to count the number of duplications of EL

values of the original EL array as they are eliminated.

MAXI-1.

An index used to count the number of duplications of R

values of the original R array as they are eliminated.

MJM MAXJ-1.

136

MEANING AND USE An input parameter which indicates the number of materials regions which make up the composite cylinder. An input parameter which indicates which region (if any) is to be divided into twice as many regular nodes in the axial direction as the other regions. Used for regions with high temperature gradients. An input parameter which indicates which region (if any) is to be divided into twice as many regular nodes in the radial direction as the other regions. Used for regions with high temperature gradients. The name of the material of region (i). Input data. An index used to indicate the type of each node and the location of each surface node. 1 - plane node; 2 - cylindrical node; 3 - regular node. - unused; 5 - gas gap nodes

101 - plane node at surface 1; 20k cylindrical node at surface k, etc. The number of surface (i). Used primarily for card checking. Card number (in the 77 - 80 field). Same as NCA. A number used to check the number of cards read during

input.

137

MEANING AND USE A counter used in subroutine STEADY to count the number of lines of data printed to limit the number of lines printed per page. NC0UNT = 1.

A counter used to indicate the number of times the subroutine PRESET has been called. NC0UNT * 2. Same as NCO.

Shape and location of surface. Plane surfaces have a number equal to 100 + NB(l). Cylindrical surfaces have a number equal to 200 + NB(l). The number of gas gaps in the cylinder.

The number of nodes desired in the interval between plots of radial nodes. Must be an even integer less than or equal to 2 * NC0. Input data.

The control parameter which actually controls frequency of the radial temperature plots.

NIS = (NI/2) * 2 Input parameter which indicates the number of nodes desired between axial temperature plots. Must be less than or equal to 2 * NC0.

Actual number which controls the frequency of axial temp-erature plots. NJX = (NJ/2) * 2.

138 i

NAME MEANING AND USE NLP A counter used in subroutine PRESET to control the number

of nodes between nodal boundaries in the axial direction. NLS An input parameter which indicates the maximum number of

subdivisions desired in the axial direction.. NNT(I, J) A number which indicates the type of heat transfer which

exists at node (I,J). 1 = none (insulated surface node) 2 = convection to a sink

3 - radiation to a sink ^ = conduction and radiation to a gas gap in the

positive direction 5 = conduction 6 = convection and radiation to a sink 7 = conduction and radiation to a gas gap in the

negative direction N0(l) Number of a region or a gas gap. NOPR Number of the problem.

NPAGES Number of pages needed to print entire temperature array. NPL A counter used in subroutine PRESET to control the number

of nodes between nodal boundaries in the axial direction. NPR A counter used in subroutine PRESET to control the number

of nodes between nodal boundaries in the radial direction.

NRS An input parameter which indicates the maximum number of subdivisions desired in the radial direction.

J39

NAME MEANING AND USE NS An input number which indicates the total number of

surfaces, internal and external. NT(I) An input index which indicates the type of heat transfer

at surface (i). 1 = no transfer (insulated surface)

2 = convection to a sink 5 = radiation to a sink

k = conduction and radiation across a gas gap in the positive direction

5 = conduction 6 = convection and radiation to a sink 7 = conduction and radiation across a gas gap

NTEST A counter used in subroutine STEADY to indicate whether the iteration passes all three convergence tests.

NUP An index used in the printout routines to control the last value of each printout array.

NX An index used in subroutine PRESET to cause the regular

gas gap node to be omitted from the temperature array as the new nodes are being assigned a temperature.

jfcPT The name for the word "option". 0UTP The name for the word "output". PI2 2 * jr Value is established by a data statement. Plii- k * it Value is established by a data statement.

1^0

NAME PICT

PRESET

Q(I)

QG

QTEST

QW(l, J)

R(J) Ri(l)

R2(I)

MEANING AND USE The name of the subroutine that plots a half section scale drawing of the cylinder and prints several general items of information about the problem. The name of a subroutine which reads input data, prints out a summary of the input data, calculates the heat trans-fer coefficients for each node of the entire nodal system, subdivides the real nodes to the desired extent, establishes a temperature pattern for the divided nodal system that approximates the previous pattern.

Input value for the rate of heat generation within region (i). (watts/gram or BTQ per hr/lb mass) Conductance across a gas gap. (watts/deg C or BTU per hr/deg F)

A value used to test for convergence. QTEST is the absolute value of the deviation from 1.0 of the ratio CUE/HG. (dimensionless) The rate of heat generation within node (I, j). (watts or BTU/hr) Radius of nodes (I, J) (cm or ft). Minimum radius of region or gas gap (i). Input data (cm or ft).

Maximum radius of region or gas gap (i). Input data (cm or ft).

ll+l


RA(l, J) Radiative conductance of surface node (I, J"), ["watts/ 1 L

(deg C)4 or BTU per hr/(deg F)4)

RF(l) Fraction which represents the effective emissivity-

absorptivity relationship between the two surfaces of

gas gap (i). Input data, (dimensionless)

RH0(l) The input value of density of region (i) (grams/cm3 or

lbs mass/ft3)

RIM The sum of the radius of a regular node and its inner

boundary. Used in subroutine PRESET in calculating

nodal coefficients (cm or ft).

R0M The sum of the radius of a regular node and its outer

boundary. Used in subroutine PRESET in calculating nodal

coefficients (cm or ft).

RPL0T A subroutine which uses the cathode ray tube plotter

to make a graph of the temperature as a function of

radius using the position on the axis as a parameter.

RSl(l) An input number representing the minimum radius of

surface (i) (cm or ft).

RS2(l) An input number representing the maximum, radius of


RT(l) Radial location of the extended surfaces which serve as

nodal divisions (cm or ft).

SLl(l) Input data giving location of the uppermost position of


Ik2

NAME

SL2(I)

SPEC

ST(I)

STEADY

SUM

convergence. When SUM is less than T0TAL the convergence test has been met. (dimensioniess)

T(l, J) The temperature of the node (I, J). (degrees C or degrees F) TA The absolute temperature, depending upon the system of

units being used. (275*15 deg K or U60 deg R)

TA1 Absolute temperature of T0LD. (deg K or R)

TA2 Absolute temperature of 3T(l). (deg K or R)

TEMP A temporary value of R(j) or EL(l) used in rearranging

the nodal boundary arrays. TEN A fraction used to over correct the temperature. T(l, J)

is set at the value of TNEW + TEN * DIFF at each node of the iteration. If TNEW is less than T0LD, T(l,j) will be less than TNEW. If TNEW is larger than T0LD, T(l, J)

MEANING AND USE Input data giving location of the lowest portion of surface (I) (cm or ft). A number used in performing the third test for convergence. SPEC = FAC/NC0. When Q^EST is less than SPEC the convergence test is met. (dimensionless) Input data giving temperature of the sink associated with surface (I), (degrees C or degrees F) The subroutine that performs the iterative calculations. The sum of the absolute values of all the relative differences between old and new values of T(l, J) over an entire iteration. Used as the second test for

Ik3

MEANING AND USE •will be larger than TNEW. (dimensionless) Input parameters which control the value of TEN. TEN = TENT + TINT * NC0

The difference between the upper limit and lower limits of the temperature plots in subroutines RPL0T and ELPL0T. It is calculated in subroutine STEADY. Initial or estimated temperature of region I. (deg C or deg F)

The name of the array which contains the title of the problem being solved. Absolute temperature of zero degrees C (273«15°K). The value of the lower limit of the temperature plots and is equal to the temperature divisible by 10° just below the minimum temperature of the array, (deg C or deg F) Maximum temperature in any one iteration, (deg C or deg F) Minimum temperature in any one iteration, (deg C or deg F) The temperature calculated for a node during the iterative calculations, (deg C or deg F)

An arbitrary number used for checking small values. If the value is less than T0L, it is considered to be zero. The value of the temperature of a node before the iterative calculation is performed, (deg C or deg F)

Ikk

NAME T0TAL

TP

TR TRAD

TREF(I)

TT(I, J)

TZER0(I)

V0L X

XL

XR

MEANING AND USE An input parameter -which controls the second test for convergence; -when SUM is less than T0TAL, the second test for convergence has "been met. (dimensionless) A factor used in subroutines RPL0T and ELPL0T to space the temperature lines on the ordinate axis, (in) Absolute temperature of zero degrees F. (k60uR)

(ex + ea) * (ex * ex + q2 * e3) or (ef - / (Ti - T8) Reference temperature for calculating the expansion of the two surfaces of the gas gap. (deg C or deg F) The temporary array which holds temperature distribution while the calculations of the new network is being per-formed . The absolute temperature at which the value of K is g measured (or known), (deg K or deg R) Volume of a real node, (cm3 or ft3) A number used in the plotting subroutines to control the position of points along the X axis. The length of an individual node. Used in subroutine PRESET in establishing the nodal network, (cm or ft) The width of an individual node, used in subroutine PRESET in establishing the nodal network, (cm or ft) A number used in the plotting subroutine to control the position of points along the Y axis.

l k5

NAME MEANING AND USE Z A number used in converting T(l, J) into a number which

can "be plotted jz = T(l, j) - TmJ .

OAK RIDGE NATIONAL LABORATORY - International Nuclear ...

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