Numerical Analysis for Engineers
Numerical Solution for 1D Heat Equation
with a Moving Heat Source
Keith Brooky
May 8, 2001
Table of Contents
Section
Page
Nomenclature
Material Values Used
Introduction
Problem Description and Formulation
Numerical Solution
Results
Error Analysis
Conclusion
References
Computer Algorithm
Complete Data Output
Nomenclature3
2
3
3
4
5
6
8
11
13
14
Appendix A
Appendix B
Table 1
Symbol
Description
Units(metric)

T
u

k
A
gs
Thermal Diffusivity
Temperature
Velocity
Density
Thermal conductivity
Area
Energy Generation
M2/s
C
M/s
kg/M3
W/MC
M2
W/M2
Table 2
Values Used for Project3
Material: Low Carbon Steel

u

k
A
gs
1.35E-05 M2/s
.001M/s
7860 kg/M3
51.9 W/MC
1.57E-07 M2
10,000,000W/M2
Introduction
3
Situations with moving heat sources are very common in many manufacturing processes.
Even examples of this simple one-dimensional situation, can be found many situations,
such as thermal treatment of wire and thin bar. Typically, these processes involve
moving a wire or bar by an induction hardening coil, flame, or laser. By solving the one
dimensional heat equation with a moving heat source, one can determine the temperature
in the wire or bar during the process. This knowledge of temperatures in the material
being treated is of interest to the engineer designing the process. Maximum temperature,
heating rate, and cooling rate are all parameters that will have a great influence in the
final properties in the material being treated.
There are several different ways to solve the one dimensional heat equation with a
moving heat source.
It also can be solved by the Finite-Difference Methods for Linear
Problems, which is a numeric method to give an approximate solution. This project will
compare the results from the two solutions.
Problem Description and Formulation1
4
The first step is to solve the one-dimensional heat equation with a moving heat source for
the exact solution. This is done as follows:
”- (
u 2
)  + 1/k Gs(X) = 0
2
Eq. 1 - 1
The common solution of this equation is as follows:
 = C1 e-(u/2) X + C2 e(u/2) X
Eq. 1 - 2
In Terms of Temperature, this equation can be represented as:
-(u/2) X
T = C1 e
 C1- e-(u/2) X + C2- ; X 0
+ C2 =
Eq. 1 - 3
+ -(u/2) X
+
 C1 e
+ C2 ; X 0
To determine the constants C1 and C2, we first see that as X approaches +infinity,
Temperature approaches 0. Also, for negative values of X, the values are constant, due
to no loss in heat. In other words:
k
dT 
dT 
k
 gs
dX
dX
Eq. 1 - 4
Thus:
 (/ u k) gs ; X 0
T=
Eq. 1 - 5
 (/2 u k ) gs e-(u/) X ; X 0
This solution assumes that there is no heat loss. This can be seen as a bar with a perfect
insulation rating. This is the simplest solution to this type of heat equation, and will
allow us to compare it to a numerical solution.
5
Numerical Solution4
The method selected for this report is The Backward-Different method for solving
Parabolic Partial Differential Equations. This method involves solving PDE’s by
breaking up the desired solution span into little cells and computes slopes to approximate
the next point on the curve. The Finite-Difference Method takes the following PDE:
2
u
2  u
( x, t )  
( x,.t )  0
t
x 2
Eq. 2 - 1
From this we turn to the backward-difference quotient for an unconditionally
stable solution of the PDE. Since this is a one-dimensional solution, we can simplify the
backward-difference quotient to:
wi  wi 
 2k
h2
( w(i 1)  2wi  w(i 1) )
Eq. 2 - 2
For this physical condition, where a heat source has moved over part of a thin
insulated wire, we have to take into account the energy input. For the areas that have
seen the heat, the value of the heat input is added to the above approximation in the form
of the following:
wi  wi 
 2k
h
2
(w(i 1)  2wi  w(i 1) ) 

(u * k )
g s Eq. 2 - 3
The solution to this algorithm is done with the use of a Fortran 77 program. The
program uses the Finite Difference formula above to approximate the value of the heat
equation with a moving heat source. The first step of the computer algorithm is to find an
6
approximate condition for wo, which is the heated left boundary of the wire. Once this is
done, the above equation is used to calculate the values of “w”. The above equation is
calculated for all the values of xi, and can then be solved as a system of linear equations.
The computation of all of this is done with th1e Fortran77 algorithm, and the output is
given in values of “x” with the corresponding “w”, which is the approximate temperature
of the wire at position “x”.
This type of algorithm had a compounded error, since the value of wI is dependent
on the solution of wI-1. After several iterations, it can be seen that the solution can
quickly deviate from the exact solution. By making the cells smaller and smaller, it is
possible to reduce the error of approximating the slope over larger spans. Unfortunately,
by solving these algorithms with very small step sizes, a new error caused by truncation is
created. This algorithm was used to approximate the values of w over various distances
and mesh spacing. It was found that by using 1100 individual calculations over a 60cm
wire, an accurate result was computed, without increasing the computer computation time
required for the solution.
Results
Because we are looking for a solution of an infinite wire with no energy loss due
to convection, we expect that the temperature to the left of the heat source be constant. It
7
can be seen in the following table that because the initial approximation done by the
algorithm was not exact, there is a gradual increase in temperature up to the area of the
heat source, and then it rapidly decreased once the heat source is passed. The heat that is
ahead of the source is being conducted and is the area that we will focus our attention to.
The following table details the results of both methods of solving the Heat
Equation. The first column is the distance in meters from the heat source. Positive
numbers indicate that the heat source has not reached it yet. The exact solution was
calculated by using Equation 1 – 5 on page 5. The Numeric solution was calculated using
the algorithm in appendix A:
Distance from
Numeric Exact Solution Difference
% error
Heat Source(m) Solution (oF)
(oF)
(oF)
-0.010
2622.82
2601.156
-21.664
-0.83286
0.000
2509.67
2532.52
22.85
0.902263
0.010
1210.52
1223.79
13.27
1.084336
0.021
562.619
567.98
5.361
0.943871
0.030
272.246
274.46
2.214
0.806675
0.040
131.732
132.63
0.898
0.677072
0.050
61.2195
61.55
0.3305
0.536962
0.060
29.6194
29.74
0.1206
0.405514
0.070
14.3295
14.37
0.0405
0.281837
0.080
6.65762
6.671
0.01338
0.20057
0.090
3.22013
3.223
0.00287
0.089047
0.100
1.55724
1.558
0.00076
0.04878
It can be seen that for the exact and the approximate values of the temperature in
the wire are very close over the length of the wire. If the algorithm were run with a larger
or finer mesh spacing, the values would not be as precise. . Values used in the equations
can be found in Table 2 on Page 3.
8
Numeric Solution
3000
Degrees C
2500
2000
1500
1000
500
211
196
181
166
151
136
121
106
91
76
61
46
31
16
1
0
Location
Exact Solution
3000
Degrees C
2500
2000
1500
1000
500
222
209
196
183
170
157
144
131
118
105
92
79
66
53
40
27
14
1
0
Location
The total comparison of the algorithm and exact solution can be seen in Appendix
B. This table contains 225 data points from the 1100 that were calculated. This data
table has been truncated for simplicity.
9
Error Analysis
As can be seen, the resulting errors in the temperatures of the wire are relatively
small over the 60 cm that is being examined. As previously mentioned, the error was
minimized by running the program several times with different mesh spacing and interval
10
length. With this type of algorithm, it is important to realize that large mesh spacing
generate error because the approximations between the different points of “w” are less
accurate the more they get larger. Also, if the mesh becomes too small, significance in
the small numbers is lost due to truncation of the number that the computer will do.
Typically, computers will cut off numbers after the eighth decimal place, unless it is set to
double precision, and will then extend the calculations to sixteen decimal places.
From an engineer’s point of view, the relative error for the approximation is not
very significant. Actual variations in material properties, current, heat transfer rates, and
dimensional aspects would most likely increase the error to greater levels than the
algorithm does. Performing actual measurements to confirm the true accuracy of a
physical reality to these solutions would most likely not be as accurate as these solutions
are to each other.
The location at the heat source is the location of the greatest error in the
computations. The error there is only 24.6 degrees Celsius different, which is
approximately 1.14% error. Most heat treaters would be very happy to know the
temperatures of their product during any process in their heat treating department.
By using more than 1100 mesh spacings with this program, one is able to
calculate a more accurate solution to the problem. Unfortunately, by doing this the
programmer ends up creating a program that takes several minutes to determine a solution
and also generates a very large amount of data. Typically only a handful of measurements
are desired as a final result of this type of investigation, and having thousands of data
points to sort through is not productive. By using larger mesh spacing for the program,
greater error is in the computations is created and some accuracy is lost. For the average
11
application, the precision found in this report is not required and far fewer spacings can
be used to determine a more general output.
Conclusion
The use of The Finite Difference algorithm did approximate the heating curve of
wire as it approached the heat source. By using 1100 computations over the 60-cm
distance, the numeric solution was very close to the exact solution. One does not truly
need to use a mesh spacing so fine, depending on the required accuracy of what is being
12
examined. Mesh spacings of only one hundred points will give a rough approximation of
what the system will do with much finer mesh spacing.
For an engineer that is trying to determine how fine of a mesh to use, without
knowing the exact solution to the problem, it would be beneficial to run the algorithm
with finer and finer mesh spacings, until the plot of the data points smoothes out and
sharp jumps are eliminated. With finer mesh spacing, the area around the heat source
generates a spike that is a result of error caused by a courser mesh. The solution done for
this report was done with a fine enough mesh to eliminate such numerical errors.
The Finite Difference Algorithm is a powerful tool to approximate Ordinary
Differential Equations. By selecting the appropriate mesh spacing, accuracy can be
increased to the point where a useful solution can be achieved. Of course the derived
solution is the more accurate one, it is not always the easiest to formulate.
References
1.
HTTP://www.rh.edu/~ernesto/C_S2000/cht/Notes/cht10.htm
2.
HTTP://www.rh.edu/~ernesto/C_S2001/nae
3.
Materials Science and Engineering, William D. Callister, Jr., John Wiley & Son , Inc.,
1991
13
4.
Numerical Analysis, Richard L. Burden, J. Douglas Faires, 7th Edition, Brooks/Cole,
2000
5.
Fundamentals of Heat Flow In Welding, Welding Research Council, P.S. Myers, O.A.
Uyehara, G. L. Borman, July 1967.
Appendix A
C
C
C
C
C
C
C
C
C
C
Finite Difference Explicit Method for Heat Equation
Moving Planar Source of Heat
Find y(x,t) such that
dy/dt = a*d^2y/dx^2
x in [0,xl]
dy(0,t)/dx = 0
dy(xl,t)/dx = 0
Source of strength g moving from L to R with velocity u
14
parameter(nx=1101)
! Use nx = odd
C
dimension w(nx),wo(nx)
C
C
C
Input Data
eka = 51.9
rho = 7860
cp = 486
ele = .60
g = 1e7
u = 0.001
dt = 0.01
winit = 0.0
C
C
C
! W/m C
! kg/m^3
! J/kg C
! m
! W/m^2
! m/s
! s
! degree C
Preliminaries
timet = 0.4/u
alpha = 1.35e-05
dx = ele/float(nx-1)
dtmax = (dx**2)/(6.0*alpha)
if(dt.gt.dtmax) dt=dtmax
C1 = dt*alpha/dx**2
nt = int(timet/dt) + 1
C
C
C
Initial Condition
50
do 50 i=1,nx
wo(i) = winit
continue
75
t = 0.0
xs = 0.0
write(6,*) t,xs
x = 0.0
do 75 i=1,nx
write(6,*) x,wo(i)
x = x + dx
continue
100
do 200 j=1,nt
t = t + dt
xs = xs + u*dt
write(6,*) t,xs
x = 0.0
do 100 i=2,nx-1
xm = x
x = x + dx
w(i) = wo(i) + C1*(wo(i-1) - 2.0*wo(i) + wo(i+1))
if(xm.le.xs.and.xs.le.x) then
w(i) = w(i) + (dt/(dx*rho*cp))*g
endif
continue
C
c
c
C
c
C
C
C
Boundary conditions
w(1) = w(2)
w(nx) = w(nx-1)
C
C
C
Reset values
15
200
do 150 i=1,nx
wo(i) = w(i)
continue
x = 0.0
do 175 i=1,nx
write(6,*) x,w(i)
x = x + dx
continue
write(66,*) t,w(1)
continue
225
x = 0.0
do 225 i=1,nx
write(67,*) x,w(i)
x = x + dx
continue
150
c
175
C
C
stop
end
Appendix B
Location
Position to Heat
0.377998
0.378543
0.379089
0.379634
0.38018
0.380725
0.381271
-0.022002
-0.021457
-0.020911
-0.020366
-0.01982
-0.019275
-0.018729
Numeric
Solution
2622.78
2622.78
2622.78
2622.79
2622.79
2622.79
2622.8
16
Exact Solution
difference
% error
2601.156
2601.156
2601.156
2601.156
2601.156
2601.156
2601.156
-21.624
-21.624
-21.624
-21.634
-21.634
-21.634
-21.644
-0.831323
-0.831323
-0.831323
-0.831707
-0.831707
-0.831707
-0.832092
0.381816
0.382361
0.382907
0.383452
0.383998
0.384543
0.385089
0.385634
0.38618
0.386725
0.38727
0.387816
0.388361
0.388907
0.389452
0.389998
0.390543
0.391088
0.391634
0.392179
0.392725
0.39327
0.393816
0.394361
0.394907
0.395452
0.395997
0.396543
0.397088
0.397634
0.398179
0.398725
0.39927
0.399816
0.400361
0.400906
0.401452
0.401997
0.402543
0.403088
0.403634
0.404179
0.404725
0.40527
0.405815
0.406361
0.406906
0.407452
0.407997
-0.018184
-0.017639
-0.017093
-0.016548
-0.016002
-0.015457
-0.014911
-0.014366
-0.01382
-0.013275
-0.01273
-0.012184
-0.011639
-0.011093
-0.010548
-0.010002
-0.009457
-0.008912
-0.008366
-0.007821
-0.007275
-0.00673
-0.006184
-0.005639
-0.005093
-0.004548
-0.004003
-0.003457
-0.002912
-0.002366
-0.001821
-0.001275
-0.00073
-0.000184
0.000361
0.000906
0.001452
0.001997
0.002543
0.003088
0.003634
0.004179
0.004725
0.00527
0.005815
0.006361
0.006906
0.007452
0.007997
2622.8
2622.8
2622.81
2622.81
2622.81
2622.82
2622.82
2622.83
2622.83
2622.83
2622.83
2622.84
2622.83
2622.83
2622.82
2622.82
2622.82
2622.83
2622.86
2622.93
2623.03
2623.18
2623.37
2623.57
2623.69
2623.64
2623.28
2622.45
2621.08
2619.18
2616.94
2614.76
2613.27
2613.23
2509.67
2409.11
2312.11
2219.03
2129.99
2044.91
1963.59
1885.79
1811.26
1739.76
1671.11
1605.16
1541.8
1480.91
1422.4
17
2601.156
2601.156
2601.156
2601.156
2601.156
2601.156
2601.156
2601.156
2601.156
2601.156
2601.156
2601.156
2601.156
2601.156
2601.156
2601.156
2601.156
2601.156
2601.16
2601.16
2601.16
2601.16
2601.16
2601.16
2601.16
2601.16
2601.16
2601.16
2601.16
2601.16
2601.16
2601.16
2601.16
2601.16
2532.52
2432.32
2335.91
2243.48
2154.56
2069.31
1987.29
1908.66
1833
1760.48
1690.82
1623.8
1559.55
1497.74
1438.48
-21.644
-21.644
-21.654
-21.654
-21.654
-21.664
-21.664
-21.674
-21.674
-21.674
-21.674
-21.684
-21.674
-21.674
-21.664
-21.664
-21.664
-21.674
-21.7
-21.77
-21.87
-22.02
-22.21
-22.41
-22.53
-22.48
-22.12
-21.29
-19.92
-18.02
-15.78
-13.6
-12.11
-12.07
22.85
23.21
23.8
24.45
24.57
24.4
23.7
22.87
21.74
20.72
19.71
18.64
17.75
16.83
16.08
-0.832092
-0.832092
-0.832476
-0.832476
-0.832476
-0.83286
-0.83286
-0.833245
-0.833245
-0.833245
-0.833245
-0.833629
-0.833245
-0.833245
-0.83286
-0.83286
-0.83286
-0.833245
-0.834243
-0.836934
-0.840779
-0.846545
-0.85385
-0.861539
-0.866152
-0.86423
-0.85039
-0.818481
-0.765812
-0.692768
-0.606652
-0.522844
-0.465562
-0.464024
0.9022634
0.954233
1.0188749
1.0898247
1.140372
1.179137
1.1925788
1.1982228
1.1860338
1.1769517
1.1657066
1.1479246
1.1381488
1.123693
1.1178466
0.408543
0.409088
0.409634
0.410179
0.410724
0.41127
0.411815
0.412361
0.412906
0.413452
0.413997
0.414542
0.415088
0.415633
0.416179
0.416724
0.41727
0.417815
0.418361
0.418906
0.419451
0.419997
0.420542
0.421088
0.421633
0.422179
0.422724
0.42327
0.423815
0.42436
0.424906
0.425451
0.425997
0.426542
0.427088
0.427633
0.428179
0.428724
0.429269
0.429815
0.43036
0.430906
0.431451
0.431997
0.432542
0.433088
0.433633
0.434178
0.434724
0.008543
0.009088
0.009634
0.010179
0.010724
0.01127
0.011815
0.012361
0.012906
0.013452
0.013997
0.014542
0.015088
0.015633
0.016179
0.016724
0.01727
0.017815
0.018361
0.018906
0.019451
0.019997
0.020542
0.021088
0.021633
0.022179
0.022724
0.02327
0.023815
0.02436
0.024906
0.025451
0.025997
0.026542
0.027088
0.027633
0.028179
0.028724
0.029269
0.029815
0.03036
0.030906
0.031451
0.031997
0.032542
0.033088
0.033633
0.034178
0.034724
1366.2
1312.2
1260.34
1210.52
1162.68
1116.72
1072.58
1030.19
989.472
950.365
912.803
876.725
842.074
808.792
776.825
746.122
716.632
688.308
661.103
634.973
609.876
585.771
562.619
540.382
519.023
498.509
478.805
459.881
441.704
424.245
407.477
391.371
375.902
361.045
346.774
333.068
319.903
307.258
295.114
283.449
272.246
261.485
251.149
241.222
231.688
222.53
213.734
205.286
197.171
18
1381.46
1326.8
1274.21
1223.79
1175.37
1128.78
1084.12
1041.15
999.96
960.32
922.32
885.83
850.72
817.06
784.67
753.63
723.75
695.12
667.57
641.15
615.78
591.38
567.98
545.46
523.88
503.12
483.21
464.06
445.7
428.06
411.09
394.83
379.18
364.18
349.74
335.9
322.59
309.82
297.57
285.77
274.46
263.59
253.16
243.12
233.5
224.25
215.37
206.85
198.65
15.26
14.6
13.87
13.27
12.69
12.06
11.54
10.96
10.488
9.955
9.517
9.105
8.646
8.268
7.845
7.508
7.118
6.812
6.467
6.177
5.904
5.609
5.361
5.078
4.857
4.611
4.405
4.179
3.996
3.815
3.613
3.459
3.278
3.135
2.966
2.832
2.687
2.562
2.456
2.321
2.214
2.105
2.011
1.898
1.812
1.72
1.636
1.564
1.479
1.1046284
1.1003919
1.0885176
1.0843364
1.07966
1.0684101
1.0644578
1.0526821
1.048842
1.0366336
1.0318545
1.0278496
1.0163156
1.0119208
0.9997833
0.9962448
0.9834888
0.9799747
0.9687374
0.9634251
0.958784
0.9484595
0.9438713
0.9309574
0.9271207
0.9164812
0.9116119
0.9005301
0.8965672
0.8912302
0.878883
0.8760732
0.8644971
0.860838
0.8480586
0.8431081
0.8329458
0.8269318
0.825352
0.8121916
0.8066749
0.7985887
0.7943593
0.7806844
0.7760171
0.7670011
0.759623
0.7561035
0.7445255
0.435269
0.435815
0.43636
0.436906
0.437451
0.437997
0.438542
0.439087
0.439633
0.440178
0.440724
0.441269
0.441815
0.44236
0.442905
0.443451
0.443996
0.444542
0.445087
0.445633
0.446178
0.446724
0.447269
0.447814
0.44836
0.448905
0.449451
0.449996
0.450542
0.451087
0.451633
0.452178
0.452723
0.453269
0.453814
0.45436
0.454905
0.455451
0.455996
0.456542
0.457087
0.457632
0.458178
0.458723
0.459269
0.459814
0.46036
0.460905
0.461451
0.035269
0.035815
0.03636
0.036906
0.037451
0.037997
0.038542
0.039087
0.039633
0.040178
0.040724
0.041269
0.041815
0.04236
0.042905
0.043451
0.043996
0.044542
0.045087
0.045633
0.046178
0.046724
0.047269
0.047814
0.04836
0.048905
0.049451
0.049996
0.050542
0.051087
0.051633
0.052178
0.052723
0.053269
0.053814
0.05436
0.054905
0.055451
0.055996
0.056542
0.057087
0.057632
0.058178
0.058723
0.059269
0.059814
0.06036
0.060905
0.061451
189.378
181.892
174.702
167.797
161.164
154.794
148.675
142.798
137.154
131.732
126.525
121.524
116.72
112.107
107.675
103.4188
99.3307
95.4043
91.633
88.0108
84.5318
81.1903
77.9808
74.8982
71.9375
69.0938
66.3625
63.7391
61.2195
58.7994
56.475
54.2424
52.0981
50.0386
48.0605
46.1605
44.3357
42.583
40.8996
39.2827
37.7298
36.2382
34.8055
33.4296
32.108
30.8386
29.6194
28.4484
27.3237
19
190.79
183.23
175.98
169.01
162.32
155.89
149.72
143.79
138.09
132.63
127.37
122.33
117.48
112.84
108.37
104.08
99.96
96
92.2
88.54
85.04
81.67
78.44
75.33
72.35
69.49
66.73
64.09
61.55
59.12
56.77
54.53
52.37
50.29
48.3
46.39
44.55
42.79
41.09
39.47
37.9
36.4
34.96
33.58
32.25
30.97
29.74
28.57
27.43
1.412
1.338
1.278
1.213
1.156
1.096
1.045
0.992
0.936
0.898
0.845
0.806
0.76
0.733
0.695
0.6612
0.6293
0.5957
0.567
0.5292
0.5082
0.4797
0.4592
0.4318
0.4125
0.3962
0.3675
0.3509
0.3305
0.3206
0.295
0.2876
0.2719
0.2514
0.2395
0.2295
0.2143
0.207
0.1904
0.1873
0.1702
0.1618
0.1545
0.1504
0.142
0.1314
0.1206
0.1216
0.1063
0.7400807
0.7302298
0.7262189
0.717709
0.7121735
0.7030598
0.6979695
0.689895
0.6778188
0.6770716
0.6634215
0.6588735
0.6469186
0.6495923
0.6413214
0.6352806
0.6295518
0.6205208
0.6149675
0.597696
0.5976011
0.5873638
0.5854156
0.5732112
0.5701451
0.570154
0.5507268
0.5475113
0.5369618
0.5422869
0.5196407
0.5274161
0.5191904
0.4999006
0.4958592
0.4947187
0.4810325
0.4837579
0.4633731
0.4745376
0.4490765
0.4445055
0.4419336
0.4478856
0.4403101
0.4242816
0.4055145
0.4256213
0.3875319
0.461996
0.462541
0.463087
0.463632
0.464178
0.464723
0.465269
0.465814
0.466359
0.466905
0.46745
0.467996
0.468541
0.469087
0.469632
0.470178
0.470723
0.471268
0.471814
0.472359
0.472905
0.47345
0.473996
0.474541
0.475087
0.475632
0.476177
0.476723
0.477268
0.477814
0.478359
0.478905
0.47945
0.479996
0.480541
0.481086
0.481632
0.482177
0.482723
0.483268
0.483814
0.484359
0.484905
0.48545
0.485995
0.486541
0.487086
0.487632
0.488177
0.061996
0.062541
0.063087
0.063632
0.064178
0.064723
0.065269
0.065814
0.066359
0.066905
0.06745
0.067996
0.068541
0.069087
0.069632
0.070178
0.070723
0.071268
0.071814
0.072359
0.072905
0.07345
0.073996
0.074541
0.075087
0.075632
0.076177
0.076723
0.077268
0.077814
0.078359
0.078905
0.07945
0.079996
0.080541
0.081086
0.081632
0.082177
0.082723
0.083268
0.083814
0.084359
0.084905
0.08545
0.085995
0.086541
0.087086
0.087632
0.088177
26.2435
25.2059
24.2094
23.2523
22.333
21.45
20.6019
19.7874
19.0051
18.2537
17.532
16.8388
16.173
15.5336
14.9194
14.3295
13.7629
13.2187
12.6961
12.1941
11.7119
11.2488
10.804
10.37681
9.96649
9.5724
9.19388
8.83033
8.48116
8.14578
7.82367
7.51429
7.21714
6.93174
6.65762
6.39434
6.14147
5.8986
5.66533
5.44128
5.22609
5.01941
4.8209
4.63024
4.44712
4.27124
4.10231
3.94006
3.78423
20
26.35
25.31
24.3
23.34
22.42
21.53
20.68
19.86
19.07
18.32
17.59
16.89
16.23
15.58
14.97
14.37
13.8
13.26
12.73
12.23
11.74
11.28
10.83
10.4
9.991
9.596
9.216
8.851
8.501
8.164
7.841
7.53
7.232
6.945
6.671
6.407
6.153
5.909
5.675
5.451
5.234
5.027
4.828
4.637
4.454
4.277
4.108
3.945
3.789
0.1065
0.1041
0.0906
0.0877
0.087
0.08
0.0781
0.0726
0.0649
0.0663
0.058
0.0512
0.057
0.0464
0.0506
0.0405
0.0371
0.0413
0.0339
0.0359
0.0281
0.0312
0.026
0.02319
0.02451
0.0236
0.02212
0.02067
0.01984
0.01822
0.01733
0.01571
0.01486
0.01326
0.01338
0.01266
0.01153
0.0104
0.00967
0.00972
0.00791
0.00759
0.0071
0.00676
0.00688
0.00576
0.00569
0.00494
0.00477
0.4041746
0.4112999
0.3728395
0.3757498
0.3880464
0.3715745
0.3776596
0.3655589
0.3403251
0.3618996
0.3297328
0.303138
0.3512015
0.2978177
0.3380094
0.2818372
0.2688406
0.311463
0.2663001
0.2935405
0.2393526
0.2765957
0.2400739
0.2229808
0.2453208
0.2459358
0.2400174
0.2335329
0.2333843
0.2231749
0.2210177
0.2086321
0.2054757
0.1909287
0.2005696
0.1975964
0.1873883
0.1760027
0.1703965
0.1783159
0.1511272
0.1509847
0.1470588
0.1457839
0.1544679
0.1346738
0.1385102
0.1252218
0.1258907
0.488723
0.489268
0.489814
0.490359
0.490904
0.49145
0.491995
0.492541
0.493086
0.493632
0.494177
0.494722
0.495268
0.495813
0.496359
0.496904
0.49745
0.497995
0.498541
0.499086
0.499631
0.500177
0.088723
0.089268
0.089814
0.090359
0.090904
0.09145
0.091995
0.092541
0.093086
0.093632
0.094177
0.094722
0.095268
0.095813
0.096359
0.096904
0.09745
0.097995
0.098541
0.099086
0.099631
0.100177
3.63456
3.49081
3.35274
3.22013
3.09276
2.97043
2.85294
2.74009
2.6317
2.52761
2.42762
2.33159
2.23936
2.15078
2.0657
1.98398
1.90549
1.83011
1.75771
1.68818
1.62139
1.55724
21
3.639
3.495
3.356
3.223
3.096
2.973
2.856
2.742
2.634
2.529
2.429
2.333
2.241
2.152
2.067
1.985
1.906
1.831
1.758
1.689
1.622
1.558
0.00444
0.00419
0.00326
0.00287
0.00324
0.00257
0.00306
0.00191
0.0023
0.00139
0.00138
0.00141
0.00164
0.00122
0.0013
0.00102
0.00051
0.00089
0.00029
0.00082
0.00061
0.00076
0.1220115
0.1198856
0.0971395
0.0890475
0.1046512
0.0864447
0.1071429
0.0696572
0.0873197
0.0549624
0.0568135
0.0604372
0.0731816
0.0566914
0.0628931
0.0513854
0.0267576
0.0486073
0.016496
0.0485494
0.0376079
0.0487805