an experimental investigation of the relation between the
TRANSCRIPT
Brigham Young UniversityBYU ScholarsArchive
All Theses and Dissertations
1972-8
An Experimental Investigation of the RelationBetween the Cooling Rate and Welding Variablesin Fusion WeldingSujit BiswasBrigham Young University - Provo
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BYU ScholarsArchive CitationBiswas, Sujit, "An Experimental Investigation of the Relation Between the Cooling Rate and Welding Variables in Fusion Welding"(1972). All Theses and Dissertations. 7075.https://scholarsarchive.byu.edu/etd/7075
AN EXPERIMENTAL INVESTIGATION OF THE RELATION
BETWEEN THE COOLING RATE AND WELDING
VARIABLES IN FUSION WELDING
A Thesis
Presented to the
Department of Mechanical Engineering
Brigham Young University
In Partial Fulfillment
of the Requirements for the Degree
Master of Science
bySujit Biswas
August 1972
This thesis by Sujit Biswas is accepted in its present form
by the Department of Mechanical Engineering Science of Brigham Young
University as satisfying the thesis requirement for the degree of Master
of Science.
i / L L-Date
ii
DEDICATED
To my parents
ACKNOWLEDGEMENTS
The author of this thesis is indebted for the guidance and help
rendered by Dr. Milton G. Wille and Mr. William Hayes, both of the
Department of Mechanical Engineering and takes this opportunity to
forward many thanks for the same.
iv
TABLE OF CONTENTS
Page
ACKNOWLEDGEMENTS................................................ ivCHAPTER
I. INTRODUCTION ............................................ 1
II. EXPERIMENTAL APPARATUS .................................. 6
III. PROCEDURE.............................................. 8
IV. RESULTS AND DISCUSSION OF RESULTS........................ 11
V. CONCLUSIONS............................................ 14BIBLIOGRAPHY .................................................... 16APPENDIX
A. DIMENSIONAL ANALYSIS .................................... 18B. COMPUTER PROGRAM........................................ 22
v
CHAPTER I
INTRODUCTION
In the field of welding metallurgy, much attention is given to
the effects which the welding arc or flame will have on the structure and properties of the metals being joined. The engineer is interested in the intensity and extent of physical changes brought about by the
unavoidable "heat treatment" which accompanies the execution of a weld.In some alloys, particularly hardenable steels, the effects of
welding heat on structure are as dependent upon the rate with which
the material cools through a certain temperature range as upon the peak
temperatures. In general, with steel, the higher the peak temperatures (above the critical) and the faster the cooling rate, the greater is
the likelihood that martensite will form in the heat-affected zone.Hence, a relation between cooling rate and welding variables
and pre-heat temperature of material is very desirable in predetermining the cooling rate that will follow.
For example, in steel, it is usually of interest to ascertain the
cooling rate prevailing in the critical temperature range. Having
determined the prevailing cooling rate, it then becomes a question of whether or not this exceeds the critical cooling rate at the
temperature characteristic of the heat affected zone. If so, the pre
heat temperature may be elevated to the point where the weld cooling rate becomes less than the critical cooling rate, and martensite
1
formation is avoided.
2
The first step in investigating a relation between cooling rate and welding variables and material property was to determine the
variables which affected cooling rate. Cooling rate was determined at 1300°F since it is usually of interest to ascertain the cooling rate
prevailing at the critical temperature. It was concluded after
investigation that the cooling rate depends upon the variables which
are included in the function as follows:
CR = f(VI, s, t, w, 0p, k, p, c)
whereVI = heat input (voltage, V, times current, I),
s = welding speed,
t = thickness of plate, w = width of plate,
0p = pre-heat temperature, k = thermal conductivity of the plate metal,
p = mass density of the plate metal,
c = specific heat of the plate metal.To reduce the number of variables dimensionless products were
investigated by dimensioned analysis which is discussed in Appendix A.
There were many different independent sets of dimensionless
products that could be formed from the given set of variables.
The criterion for arranging the variables was to obtain the
maximum amount of convenience in utilizing the dimensionless variables. This was accomplished by having the significant variables each occur in
only one dimensionless term.
3
In the dimensionless matrix, the first variable is the dependent variable, the second variable is that which is easiest to
regulate experimentally, the third variable is that which is next easier to regulate experimentally, and so on.
The dimensionless products obtained were, as shown in Appendix
A, as follows:
4CRt c 4> (VIw
perst _t a ’ w )
To determine the relationship between the dimensionless products, the procedure adopted was as follows:
Let, cRtVor
VIw--- = 7T3 ^pa3
st m a * *3
and
tw
9pc
Let the relation between the dimensionless products be
. a b e d ’l * k "2 *3 \ ”5
Now, if five experiments are conducted changing the variables in
such a way as to change at least one dimensionless product, we will get
a set of five equations.
„ a b c d *1(1) = K *2(1) *3(1) *4(1) *5(1)
*1(2) " K *2(2) *3(2) *4(2) *5(2)
*1(3) * K *2(3) *3(3) *4(3) *5(3)
v a b c d Tl(4) = *2(4) *3(4) *4(4) *5(4)
v a b c d Tl(5) “ * *2(5) *3(5) *4(5) *5(5)
Taking Logarithm,
log *1(1) = lo& K + a l o 8 ^2(1) + b log ^3(1) + c 108 *4(1) +3(1) 4(1)
d log ir5(1)
log tt1(2) “ log K + a log *2(2) + b log *3(2) + C l0g *4(2) +
d log IT5(2)
108 *1(3) " log K + a log *2(3) + b log *3(3) + c log *4(3) +
d log 7T5(3)
l0g *1(4) = log K + a log *2(4) + b log *3(4) + c log *4(4) +
d log tr5(4)
5
108 "1(5) “ l o 8 K + a log * 2 ( 5 ) + b lo 8 *3 (5) + c l o 8 *4 (5) +
d 108 ’5(5)
The unknowns a, b, c, d, and K can be solved by Gaussian's
Elimination method. The actual testing involved seven sets of data.
To average the values of a, b, c, d and K a computer program was written which could solve all the possible combinations of 5 x 5 matrices from a set of seven equations and a subroutine which could solve the martices
by Gaussian Elimination method which is discussed in Appendix B.As discussed later on, the above developed method for
investigating a relationship between cooling rate and welding variables appears to be applicable to predict the cooling rate.
CHAPTER II
EXPERIMENTAL APPARATUS
The experimental apparatus consisted of steel plates, thermo
couples, strip chart recorders and a welding machine.
Steel Plates
The plates used in the experiments were A1S1 1020, commercially
known as mild steel. The sizes used were: 1/8" x 2" x 12",1/2" x 4" x 12", 1/8" x 10" x 12", 1/4" x 2" x 12", 1/4" x 4" x 12",1/4" x 10" x 12", 3/8" x 2" x 12", 3/8" x 4" x 12" and 3/8" x 10" x 12".
Twelve plates were used choosing at least one plate of all the above
mentioned sizes. The width of the plates were drastically changed so as to study the effect of width on the cooling rate, minimum
width being at least the width of the heat affected zone (HAZ).Maximum thickness of the plate was limited by the capacity of the welding generator.
Thermocouple
Chromel-Alumel thermocouples were used as their range extends
to about 2500°F, and thus they are appropriate to this application.The ends which were imbedded into the plate were shielded by
porcelain tubes.
Brush RecorderTwo two-channel. Brush Mask 220 Recorders, GOULD, mel?0902, were
6
7
used. The wires from the four thermocouples were connected to
these four channels. The paper speed was set at 1 mm/sec, and gain set at 1 mV/div. The gain was chosen to give maximum possible
deflections of the recording pens during welding. The first channel was connected to the thermocouple which was at 1/4" distance from the
welding axis, the second was connected to the thermocouple at 1/2"
distance, and so on in 1/4 " increments.
Welding Machine
A D.C. amp, Lincoln Arc Welding Machine, Serial No. A2170826, was used. The unit had a motor-driven, differentially compound wound D.C. generator.
Automatic Arc-WelderAn automatically controlled arc-welding research apparatus was
used for contrciling the rod feed and welding speed. The apparatus was built to automatically arc weld specimens for research work by
controlling weld velocity and weld current. Weld velocity was
controlled by a reversible variable speed series motor driving a
moving table carriage which was mounted with roller brushings on two supporting guide bars, through a screw drive. A weld specimen placed
on this table could then be moved at a constant, pre-selected velocity underneath the arc of a vertically or inclined positioned electrode.By monitoring the welding current a servo-control system automatically
produced the proper rod feed rate or rod position such as to maintain
the current constant. An error signal between a reference voltage and
a current-monitoring shunt voltage drove the system to adjust the electrode's arc length to maintain a constant current.
CHAPTER III
PROCEDURE
Thermocouples were imbedded at distances of 1/4”, 1/2", 3/4" and 1" from the weld axis, on one side of the plate. Imbedding was
done by drilling slightly over-sized holes to about half the thickness of the plate depth, and squeezing the metal around the thermocouple after inserting it in the hole.
By this arrangement only the first point gave the maximum
temperature (higher than the upper critical temperature), while the remaining three gave temperatures lower than the critical temperature.
Welding was done in the heat-treatment laboratory. The low
temperature ends of the thermocouples were kept in ice. The hot junction of the thermocouples was shielded with porcelain tubes.
The welding generator was set at 40 amps to start with and was
subsequently varied between 40 amps and 150 amps for different sets of experiments.
The welding apparatus was set at AUTO and the direction of the
table was set on either LEFT or RIGHT. The speed of the table was varied, by changing the position of the knob of the silicon-
controlled rectifier (SCR). The sensitivity of the amplifier was preadjusted such as to maintain a constant arc-length when the START button was pushed.
The paper speed and the sensitivity on the Brush Recorder was
8
9set at 1 rnm/sec and 1 mv/div respectively.
The welding generator was turned on and the START button of the welding apparatus was pushed to start welding.
After welding, the plates were allowed to cool off at the room temperature and the time-temperature history was obtained on the
Chart of the Brush-Recorder. Cooling rates were found by measuring the slope of the temperature traces on the chart (see Figure 1) at 1300°F.
The above procedure was repeated for each of the plates with different welding speeds and currents.
Twelve experiments were conducted for different values of the
welding variables. Only seven experiments gave maximum temperatures higher than the upper critical temperature.
The experimental data obtained are tabulated in Table I.
TABLE I
EXPERIMENTAL DATA
NoVoltage
VVolts
CurrentI
Amps
Length of Weld L
Inches
Thicknesst
Inches
WidthwInches
1 30 120 8 1/4 22 32 130 7.75 3/8 23 32 135 7 3/8 24 31 125 7.25 1/4 45 30 125 7.5 1 /8 106 31 150 6 1/4 107 28 85 5.6 1 /8 4
Values of thermal conductivity and specific heat were averagedout over the temperature range of each experiment.
I I I I I I I I I I I I I
I I I ------I ------- 1--------------- J I I I I I ' I--------1--------1--------1------- 1-------- 1—:----I— - J 1— ,| L
■I 1 h— I 1 1 1— H f— I 1 1 1 b—1 1 1 1 h— -- ^
I 1— I 1 1 f I h— I 1 I I— 1 h— I 1 1 1 1 1 1 1 1 1 1 1 1 1-
Channel 44------ 1------ 1------1------ 1------ L----- 1----- X------ 1------ 1------1------ 1------ 1------ 1------ 1----- 1---- 1 ____i____ L___I____ I------ 1------ 1------ l.. i ... i____t
Figure 1
10A computer program was written, to read in the different values
of specific heat and thermal conductivity, which is described later.
CHAPTER IV
RESULTS AND DISCUSSION OF RESULTS
Seven experiments which gave maximum temperature higher than
the upper critical temperature were chosen for analysis. Tangents were
drawn at 1300°F and the slope measured to find out cooling rate at 1300°F.
Conversion factors for all the dimensionless groups were
calculated. The sample calculation for the dependent dimensionless group is as follows:
, cR t**cTfl = -----a 3
° v68 (— ) x (sec3) (f t1*) x c(BTU)“ S c C
a3(ft6) 4x12 (lbm-°F)
Converting lb into lbr we have: m t
_ (°F) (sec3) (ft1*) (BTU) (ft)__(sec) (ft6) (lb ̂_ oF) (gec2)
BTU = lbf - ft
Converting BTU into lb^ - ft we have:
11
12778.17 lb, - ft * _________r_____
lbf - ft
which is dimensionless.Hence, the conversion factor is
= J x g
= 778.17 x 32.2
- 25500.64
Seven equations were obtained each having five unknowns. The
unknowns, which were the exponents of the dimensionless group and a constant quantity, were obtained by solving twenty-one (the maximum
number of 5 x 5 different matrices formed from seven equations of
five unknowns) by the Gaussian Elimination method. The value of the constant coefficient and experiments were evaluated by averaging out
the twenty-one values obtained by twenty-one 5 x 5 matrices.A computer program was written in Fortran IV programming
language to calculate all the dimensionless groups. It solved all
twenty-one matrices averaging out the values of the constant
coefficient and exponents and calculated the cooling rates for all the seven experiments using relation obtained between the cooling rate and
welding variables. It also compared the experimental cooling rates with the cooling rate obtained by the relation obtained, calculating
the deviation, percentage deviation and average percentage deviation.
The deviations were found to be within five percent. Hence, the
relationship obtained fit the data well.
The final relationship with the exponents and constant
coefficient calculated by the computer program, which is discussed
in Appendix B, is as follows:
13
CR t**c .178 x 10-15 .Vlt.2.7' per
st.0.126 'a '
( 1)*469'w' '•6pc;0.73
The experimental cooling rates and the cooling rates obtained by the relationship obtained compared as follows:
TABLE IICOMPARISON BETWEEN EXPERIMENTAL
AND CALCULATED DATA
PercentageDeviation
68 62 8.773 74.4 -1.9
112 109.1 2.584 78.4 6.634 33 2 .868 70.7 -4.018 15.9 11 .8
The accuracy for the seven sets of data was 5 percent.
CHAPTER V
CONCLUSIONS
The amount of deviation clearly indicates the success of the
relation developed. From the deviation obtained, the following conclusions can be derived.
1. The experimental relation developed in this thesis can
predict cooling rate with + 5 percent accuracy.
2. A few inconsistencies in results can be avoided if the heat input is uniform throughout the length of the weld for seven experiments.
3. Better accuracy would be obtained if the arc length of each experiment was measured and the heat input was multiplied by the arc efficiency which varies between about 65 percent for an arc length
of 1/4" to 80 percent for an arc length of 1/8".
14
BIBLIOGRAPHY
BIBLIOGRAPHY
1. Adams, Clyde M., Jr. "Cooling Rates and Peak Temperatures inFusion Welding," Welding Journal, Supplement, Vol. 37, 1958, p. 210S-215S.
2. Datsko, J. Material Properties and Manufacturing Processes.John Wiley and Sons, New York.
3. Langhaar, Henry L. Dimensionless Analysis and Theory of Models.John Wiley and Sons, New York.
4. Pennington, Ralph H. Introductory Computer Methods and NumericalAnalysis. The MacMillan Company, New York.
APPENDICES
APPENDIX A
DIMENSIONAL ANALYSIS
The cooling rate depends upon the following nine variables:
heat input, speed, thickness, width, pre-heat temperature, thermal conductivity, mass density and specific heat of the specimen.
The basis of dimensional analysis as a formal procedure is the
Buchingham ir-theorem, which states that a complete physical equation such as
\
• £< V V ••• V
may be expressed in the form of a number of it terms, each it term
representing a product of powers of some of the Q's, which in terms of
the primary dimensions, form a dimensionless group. Thus, the above equation may be expressed as
W1 " ^ * 2’ *3* •** "n-k^
where each 17 “ Q2 ••• Qn wlth the resulting product being dimensionless when each Q is expressed in terms of the primary dimensions. The primary dimensions in this case are mass (m), time (T), length (L), and temperature (0).
n ■ number of variables 9
k - number of primary dimensions 4
18
19(n - k) =5, tt terms are the greatest number of Independent tt's which will represent the physical equation.
The arrangement of the variables with their dimensions, as discussed earlier, is given dimensional matrix form as follows:
lCR
2VI
3s
4t
50p
6w
7k
8P
9c
M 0 1 0 0 0 0 1 1 0
L 0 2 1 1 0 1 1 -3 2
T -1 -3 -1 0 0 0 -3 0 -2
e 1 0 0 0 1 0 -1 0 -1
A set of homogeneous linear algebraic equations whose
coefficients were the number in the rows of the matrix was written
R2 + R? + Rg - 0
2R- + R_ + R. + R- + R, - 3R„ + 2Rn - 01 3 4 6 7 8 9(a)
- R1 - 3R2 - Rg - 3R? - 2Rg - 0
Rx + R5 - R? - R9 " 0
Any values could be assigned to R^, R2> Rg, R^ and R,. and
Equation (a) could be solved for Rg, R7, Rg and R^. The solution was accomplished readily by the elementary elimination procedure. The
R6 = AR1 + *2 + R3 " R4 + 2R5
R? - - 3RX ~ 3R2 - Rg - 2R5
result is
20
RS " 3R1 + 2R2 + R3 + ^ 5
R9 " 4R1 + 3R2 + R3 + 3R5
(b)
Values assigned were ■ 1, R2 ■ Rg ** “ Rg " 0 andEquation (b) yielded Rg * A, R7 “ -3, Rg » 3, Rg ■ A. Similarily, for R2 " 1» R^ *» Rg = R^ = Rg ■ 0, Equation (b) yielded Rg “ 1,
R? - -3, Rg - 2, R9 - 3, for Rg ** 1, Rg - R2 - R4 - Rg - 0, Rg - 1,
R7 " ”1* Rg “ 1 , Rg = 1, fo r R^ ■ 1 , Rx = R2 - Rg ■ R5 - 0 , Rft - - 1 ,
R? - 0 , Rg - 0 , Rg - 0 and fo r Rg - 0 , Rg - R2 - Rg - R4 - 0 , Rfi - 2 ,
R? * - 2 , Rg « 2 and R0 = 3 .
The solution could be neatly arranged in the matrix form shownbelow.
CRR1
W CM
> PCSsR3
tr a
0pR5
wR6
kR7
PR8
cR
*1 1 0 0 0 0 A -3 3 A
n 2 0 1 0 0 0 1 -3 2 3
*3 0 0 1 0 0 1 -1 1 1
\ 0 0 0 1 0 -1 0 0 0
n5 0 0 0 0 1 2 -2 2 3
Accordingly, the following complete set of dimensionless
products were obtained:
TTg - CR w1* k"3 p 3cu
ir̂ * VI w k 3p2c3
TTg » s w k~^ p cir. - tw ^A
21a 2 . - 2 2 3tj s 6p w k p c
Substituting a = — we get:
ACR w cT1 = “ 1 ---a
VIwT 2 3 --------2 3pa
swr = — 3 a
i.A w
« 2 op w c
Now
s2 2 2 2 TT- s w a*6 = ~ " 2 X a 2 = ft*5 a 9p w c Qpc
Hg clearly involved less variables than n,. and hence was replaced for
VThe final dimensionless products could be represented as:
CRt4ca
.VIw st t̂ s . ̂ 3 * a ’ w ’ 0pc'
APPENDIX B
A computer program which will evaluate all possible 5 x 5
matrices from a set of seven equations and solve them by a subroutine. The subroutine solves all the 5 x 5 matrices by Gaussian Elimination
method.
INPUT DETAILS
Card 1Columns Information Format Value
1-10 Cooling Rate, Exp. #1 F10.5 68°F11-20 Cooling Rate, Exp. #2 F10.5 7321-30 Cooling Rate, Exp. #3 F10.5 11231-40 Cooling Rate, Exp. #4 F10.5 8441-50 Cooling Rate, Exp. #5 F10.5 3451-60 Cooling Rate, Exp. #6 F10.5 6861-70 Cooling Rate, Exp. #7 F10.5 1871-80 Thickness of plate, F10.5 .25 inch
Exp. #1
Card 2
1-1011-2021-3031-4041-50
51-60
61-70
Thickness Exp. #2
of Plate, F10.5 .375 inchThickness
Exp. nof plate. F10.5 .375
Thickness Exp. #4
of plate, F10.5 .25Thickness
Exp. #5of plate. F10.5 .125
Thickness Exp. //6
of plate. F10.5 .25
Thickness Exp. #7
of plate. F10.5 .125Sp. heat of metal
the plate F10.5 0.15 BTU/lbiSp. heat of metal
the plate F10.5 0.16
22
71-80
23Card 3Columns Information Format Value
1-10 Sp. heat of the plate metal
F10.5 0.1511-20 Sp. heat of the plate
metalF10.5 0.15
21-30 Sp. heat of the plate metal
F10.5 0.17
31-40 Sp. heat of the plate metal
F10.5 0.1441-50 Sp. heat of the plate
metalF10.5 0.14
51-60 Sp. heat of the plate metal
F10.5 0.327 ft /hro61-70 Diffusivity of the
plate metalF10.5 0.28 ft /hr
71-80 Diffusivity of the plate metal
F10.5 0.327
Card 41-10 Diffusivity of the
plate metalF10.5 0.313
11-20 Diffusivity of the plate metal
F10.5 0.25221-30 Diffusivity of the
plate metalF10.5 0.35
31-40 Diffusivity of the plate metal
F10.5 0.36441-50 Heat input, Exp. #1 F10.5 3600. Watt51-60 Heat input, Exp. #2 F10.5 4160.61-70 Heat input, Exp. i f3 F10.5 4320.71-80 Heat input, Exp. #4 F10.5 3875.
Card 5
1-10 Heat input, Exp. #5 F10.5 3750.11-20 Heat input, Exp. i f6 F10.5 4650.21-30 Heat input, Exp. ifl F10.5 2380.31-40 Mass density of the
plate metalF10.5 490 lb /ft:IQ
41-50 Mass density of the plate metal
F10.5 49051-60 Mass density of the
plate metalF10.5 490
61-70 Mass density of the F10.5 490plate metal
Mass density of the F10.5 490plate metal
71-80
24
Card 6Columns Information Format Value
1-10 Mass density of the plate metal
F10.5 490
11-20 Mass density of the plate metal
F10.5 490
21-30 Welding speed F10.5 47.14 ft/hr31-40 Welding speed F10.5 62.8341-50 Welding speed F10.5 72.4151-60 Welding speed F10.5 72.5061-70 Welding speed F10.5 56.2571-80 Welding speed F10.5 54.54
Card 7 1-10 Welding :speed F10.5 50.90
11-20 Width of plate. Exp. //I F10.5 2 inches21-30 Width of plate, Exp. #2 F10.5 2 inches31-40 Width of plate. Exp. #3 F10.5 2 inches41-50 Width of plate. Exp. #4 F10.5 4 inches51-60 Width of plate, Exp. #5 F10.5 10 inches61-70 Width of plate, Exp. #6 F10.5 10 inches71-80 Width of plate. Exp. #7 F10.5 10 inches
Card 8
1-10 Pre heat temperature F10.5 75°F11-20 Pre heat temperature F10.5 75°F21-30 Pre heat temperature F10.5 75°F31-40 Pre heat temperature F10.5 75oF41-50 Pre heat temperature F10.5 75°F51-60 Pre heat temperature F10.5 75°F61-70 Pre heat temperature F10.5 75°F
Output25
Exponents * a * 2.7, b = 0.126, c - 0.469, d - 0.73 Constant, k = .178 x 10
Average deviation = 4.9 percent
S P p E N M X# f t c
c029
T h e s i sDIMENSION Cr (7) ,T<7) , C(7), AL<7) ,VI (7) ,RH0(7>, S f 7 i,W(7i7TMP(7) ,B(9> • #.P(9#9)#T:t(9>»S5(9)#BB(9),A(9,9>,X(9),AB<9>9),ACf9*9),D'(9,9),E(9»9 •)«F(9.9),R(9*9)*H(9»9)»0(9»9)»R(9*9)»qUM(9)»R1(9*9)»P2(9.9)>P3(9.9 *),P4<9>9)7R5<9.9>,R6(9#9),R7(9.9l,P8(9>9).R9(9#9>,R10r9,9).Rll<9.9 *>7 AVE(9), v(9t 9» »CRF.(9) .DIFC9) ,PDIF(9) .AK1<9>,AK2(9>,AKR(Q)Re a d(5*DnRfT.n* a l,vi ,r h o,s ,w,thp N = 51 FoRMAT(8Fi0.5)D 0 1 5 e l = l , 7
130 A«l(I)=(C{I)*5*378416500.)/(Al(I)**3)0021=1,72 B( I Js A L O G iatCRtI> * < T ( I>**4>*AK1(I))0 0 1 5 1 1 = 1 , 7151 AK2( I )=18r444898./(AL(I)«*3)0031=1,73 P(I»1)=1.0041=1/7
4 P< I,2)=AlP)Gl0<VI (I )*T( I >*AK2( I ) )D05I=1» 7T 3 ( I ) * T ( M / l 2 .
5 P( 1,3) = AUnGl0((S(I)*T3(I))/AL(I))0061=1,7P(I,4)=ALnCl0(T(I)/W(I))
6 AK5(I)=l,/t75,*C(I)*778.17*32.2)DoS I=1,7S5( I );S( I 1/3.600,
8 P( I,5)=ALhOl0(S5<I>*S5(I)*AK3(I))DO10I=1,NB B d ) s B ( I )
D 010 J = 1, N IS A ( I # J ) s P ( I»J>
hall elim j n f a * m ,r q ,x) noic:i=i,M
1st SUM <I>-X(T >D011I*1.MK:I>1H r (I )=R(K)D 011J = 1» N
11 AR( I»J)=R(K,J)C a l l ei.i h t n i a b .n .b b .xj n o i c n = i . M
1C1 SiR'M ! )=SUm ( I > + V( I )noi2i=i,NK s I + 2ftq( I )=p(K)nni2J=i,R
I P AC<I»JJsPfK.J)c a l l e l i m t W(A c .n .b b .x > P0172I =1.n
l c-p SUM (I j=sum(t >+x (I)nol3Isl,NK S ! + 3IF(K-7)14714.1B!
15 KrK-7 14 R a (P = P (K >
D013J*1.N 13 He I ,J)=P(K,J)
c a l l e l i m t N(0, m ,»i b .x > d o i :3 i =i ,m
1::3 suM( I )=SUm( i ) +x (I ) 00161=1.N
029
S3"J
Ka I +4IF(K*7)17"17*18
18 KaK-717 B R ( I ) s B ( K )D016Js1,N 16 E ( I # J ) = P ( K » J )
r, a l l e l i m t n ( E , m , 9 b » x >DO104I =1*N1 0 4 S u M ( I ) = S U M f l ) * X ( I ) D019I*1»NKs I +5l F ( K - 7 ) 2 0 l 2 0 , 2 l
21 KaK-720 B q ( I ) = B ( K )
D019J=1#N19 F ( I , J ) = P ( k « J >
CALL ELIMtN(F,m ,8B»X> D 0 l S 5 I = l * N
105 S U M < I ) = S U M < i ) +x (!) Do22 I =1» NKs I +6l F ( K - 7 ) 2 3 ; 2 3 » 2 4
24 KaK-7 23 Br ( I ) s B ( K)
D022J=1»N22 G ( I i J ) = P ( K » J )
Call eli m tN(G,n ,b b»x > D0lg6I=1» n
106 SuM(I) = SUm M)*XCI) D025!=1,NK a I! F ( I - 4 ) 2 6 7 2 6 * 2 7
020
N>CO
27 Ksl+l2* BR C I ) = F> C K ̂
D025J=lfM 25 H( I,J)=P(K,J)
C a l l e l i m t ^ h .kj.r b .x )D 01 7 7 I = 1» M
107 SU’U I >=SUM( I >+X( ! )Do 2 8 I=1,N Ksl+lIr(1-4)29,29*3^
30 K =K +129 Bn(I>sR<K>
D028J=1/L28 Q ( I ,J)=P(K,J)
C a l l 8 L j h t L{Q, k',R8»x )DOl'? 8 1=1,Nj
128 SuM{I)=SU m (I>+y ( 1> Dn3j 1=1,M K = I +2I F < 1-4)3?"3?,33
33 K = K -6 3? Re? ( I ) =R(L )
D03lJrl,M31 R( I , J)=P(K,.J)
c a l l e l i m t n (R,w ,b b ,x >Dnl v 9 I=1,m1.9 9 SUM( I) = SUM« T >+X( I)
D034I=1,L K = I +3!F( 1 -4 )35:35.38
3ft K=K-6 35 BR( I ) = B ( K i
029
K3VO
D034J=1»N 34 RiC I, J)«P(K*, J)
call elimtncR1,n #b b,x >n o n c i = i , N110 SUM cI )=SUM(I)+y(l>D 0 3 7 I =1# NKs I 4.4I F < 1—45 38“ 39 * 4040 KsK+l
39 KsK-738 BR (I)*B(K J
Do37J=l,N 37 R?<I,J)=P(K,J)
call eliMtN(R2;n ,b b,x ).DOlllI=1» N
111 SuH(1)=SUM(I>+XCI) D041I=1,NK = I!F(I-4>42743'44
44 K=K+1 43 KsK+1 42 Br (I>sB(K)
D041J = 1,N41 R3(I,J)*PfK,J)
Call eli mtn<R3,n »b b,x > D0U2I=1*N
112 s u m< i> = s u m(!)*x (I)Dn45I =1, NKs I <f 2J F (1-4)46747»4B 48 KsK + 5
47 KsK-5
020
OJo
tfFTC
46 B r (I )s D (K )D 0 4 5 J = 1 , N
45 P l l ( I ,J)=P<K, J>call elimtn<rii.n,bb,x> Dnii3i=i,M
H 3 sum(i> =sum(i> + x(nnn49i=i,N K r I ?I F ( I-4)c>i;t>?,53
53 K5k-2 5? K-K-551 0n( I )sB(K)D 049J =1»N 49 R4( 1 ,J)sP(K,J)
c a l l e l i m t N(R4.m »b b #x >001141=1,M
1 1 4 s u M< i > =s u m e t ) + y ( ! )Dn54I=l,N K: 1+4IF c 1-4)55:56.57
57 Ks K- 2 56 Ks K- 5 55 BR( I )=R(K)
D054J=1,N54 R«5( I , J)=P(K, J)call eliktn<R5.n,bb,x>
D0115 I = 1» M1 1 5 SljM { I ) = SUM{ I )+y< I )
D058I=1,NK * MIF < 1-4)59:60*60
6iS KaK*l
029
5 9 B B ( I ) * B ( K i DO50J«1 , N
58 R 6 ( I , J > s P ( K ; J )c a l l e l i m t N ( R 6 7 n , b b , x )0 0 1 1 6 1 = 1 , M
1 16 S u M( I >=SUM<I >+VU> D 0 H 7 1 = 1 , N
1 1 7 SlJM { I ) = S U m ( I ) + X ( I ) I)06l I s l , NK e II F < 1 - 4 ) 6 2 7 6 3 , 6 3
63 K*K*162 Br ( I ) s B ( K j
D0 6 l J a l , N 61 R 7 ( I , J ) * P f « ; J )c a l l e l i m t n < R 7 7 n » b b , x >
D 0 6 4 I s 1 , N K s I!F(1-3)65766,67
67 KaK+l 66 KsK-f 1 65 Br (I)=B(K >
D 0 6 4 J s l , N64 R 8 ( I , J ) s P( K' . J )
Call EL!MtN(R8, N, »BB»X) D 0 1 1 8 I = 1 , N
1 1 8 S t j H ( I ) = S U M ( l ) + X ( I ) D 0 6 9 I = 1 , NK b I+5I F ( 1 - 3 ) 7 0 7 7 1 , 7 2
72 K3 K+1 71 KsK-7
029
Loho
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'39 ( I >=R(K)D Q6 9 J s l # N 39 ( I , J>=P(K". J )call f.l i m tN{R9,n ,br,x >Q 0 H 9 I = 1, MStlM( I > = SUM f T )•*■¥( ! )00731*1.NKaI+3I F (I-3)74“7r ,76 KsK-7K sK + 1«R(I)sB(K)D073J=1.N 31 i’ ( I » J ) = P ( K » J 1call e l i m tN(Ri p .n ,b b,x >nni2L'i = i>Msum cI)=s u m«t>+v (Mn o i 2 i i = i . MA V F ( I ) = S U M ( I )/?1i AvE(1)=1C.**AVF(1)WRITE <6,5p) ( A V F ( I), 1*1,5) F0RMAT(1H?',5E1*.7>Oo22CI=1,7Ycl,2)= <VI<I>*T<I>*AK2<I)>002211=1,7Y ( I»3 ) = ((S(I)»T3(I)) / A L( I ))002221=1,7Y ( I , 4 ) = ( T < I ) / W( I > )002:31=1,7Y ( 1 , 5 ) = ( S 5 ( I ) * S 5 < I ) » A K * 5 ( I ) >002101=1,7Y( I , 1 )=AVF<1 ) * ? Y( 1 , 2 ) * * A VE ( 2 ) ) * ( Y( I » 3 ) * * A v F ( 3 ) ) * ( Y U , 4 ) # * a\ / E ( 4 ) ) * (
LOLo
*Y<I,5>**AvE<5)>002041=1,7204 CRE(!)=Y(T,i)*fAL(I>**3)/{(T(I)#*4)*C{I)*AK1(I>) WRlTE(6,2ci5><CP(I),1=1,7)WRITE(6,2^9) CORE(I),1=1,7)
205 FoRMAT(1H3,7F.1*.7)002061=1,7D l F ( I ) s C R c I ) - C R E < ! >
206 P0lF(I>=<nlF<I)/CR(I)>*100.WRITE(6,2S7)(DTFM)»1=1,7)WRlTE(6,2S7)(PniF(1),1=1,7)
207 FnnMAT(lH0,7El5.7)SuHPs0.002081=1,7
208 SuMP = SUMP'+PnIFM)SuMp=SUMP/7.WRITE(6,209>SUMP
209 F0RMAT(1Hpi,f15.7)STOPEND
* Y ( 1 , 5 ) * * A v E { 3 ) )002041=1,7204 CrE<I)=Y(T.i)*fAL(I)*»3)/(CT(I)**4)#C(I)*AKl<n) WRlTE(6,2n5)<CR(I),1=1,7)Wr ITE(6,2S3)(CRE(I>»1*1,7)
205 F0 RMAT( l Hpj ; 7 El 5 . 7 )0 0 2 0 6 1 = 1 , 7D I F ( I )=CRf I J - CREM )
206 P n l F ( I ) = < n l F ( n / C R ( I ) ) * 1 0 0 .WRl TE( 6 , 2« i 7 ) ( D T F U ) , 1 = 1 , 7 )W R I T E ( 6 , 2 S 7 ) ( P n i F U ) , 1 = 1 , 7 )
207 F o R MA T ( l H f l , 7 E l 5 . 7 )SlJMP = 0 .0 0 2 0 8 1 = 1 , 7
208 Sl)MP = SUMP*PnI FM >SUMP=SUMP/7,WRI TE( 6 , 209) SUMP
239 F 0 R M A T ( l H n , F l 5 . 7 )STOPEND
#F TC
«FTC
029
029SuBROUTI Nr f LIMIN(AA,N,BR,X)DIMENSION AA(9'.9>,BB(9),A<9,9>,Y(9>,X<9).Tn<9> \'N = N + 1 002:21=1,N A(I,NN)sBR(T )Do2C0J=l,N
??Q A( I , J)=AAf I". J>K*l*
1 CONTINUE 00211=1.N
21 In( I > = I ? CONTINUE
Xk =K + 1 !S = K I T = KR=APS(A(K,K))D03I=K,N 003J=K,NIF ( AOS ( A (T ,.l) )t B13»3»31
31 IR*I IT = JPsAPS(A(I,J ) )
3 CONTINUEI F ( IS-K)4',4; 41
41 no4?J=K,NM Cs A ( IS,J)A (IS,J)=A(K.J)
42 A{K,J)=C4 CONTINUE
I F ( IT-K)5;«5.51 51 I n=In «k >
LoUi
I D ( K ) # I D ( f t )I D < I T ) = IC D o 5 2 I a l » N C = A < I , IT)A ( I , I T ) a A( I " . K>
52 A M , K ) = C5 c o n t i n u eI F ( A ( K » K ) J 6 , 1 0 P » 66 c o n t i n u e
D 0 7 J = K K , N nA ( K , J ) = A ( K » J ) / A ( K , K >D 0 7 I = K K , N _W s A ( I , K ) * A ( K > J )A { I ,J ) = A ( T *J > * W! F < A B S ( A < i,J> N . 0 0 0 1 * A B $ ( W > > 7 1 , 7 , 7
71 A ( I » J )=0.7 C O N T I N U E
K s KK! F < K - N > 2 , 8 1 . 1 0 2
81 I F < A( N, N > ) 8 , 1 B ? , 8 S C O N T I N U E
Y ( N ) s A ( N » n N ) / A ( N » N >N m = N - 1 0 0 9 1 = 1 , NH Kr N- I k k = k + iY ( K ) = A ( K , n N)D 0 9 J = K K , NY ( K> = Y ( K ) . A ( t t # J > * Y < J)
9 C O N T I N U E O O lg I =1, N D O l B J = 1 , N Co
IF< I P ( J ) - T > 1 0 . 1 0 1 , 1 0101 X< I > = Y( J>
13 CONTINUE SFTUPN
102 W R l T E ( 6 , l ? i rfturn
1? FoPMAKl Hf i # ' NO UNIQUE SOLUTION*) End
u>•̂i
AN EXPERIMENTAL INVESTIGATION OF THE RELATION
BETWEEN THE COOLING RATE AND WELDING
VARIABLES IN FUSION WELDING
Sujit Biswas
Department of Mechanical Engineering
M.S. Degree, August 1972
ABSTRACT
The primary object of this research was to be able to predict the cooling rate of a given steel specimen when it is arc welded.
The cooling rate of a specimen whose temperature rises above the upper critical temperature can be predicted from the relationship arrived at.
COMMITTEE APPROVAL: