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Subroutine MAP_UTIL_INDHEAT

  1. Provenance of code.
  2. Purpose of code.
  3. Specification.
  4. Description of subroutine's operation.
  5. References.
  6. Parameter descriptions.
  7. Error indicators.
  8. Accuracy estimate.
  9. Any additional information.
  10. Example of code
  11. Auxiliary subroutines required.
  12. Keywords.
  13. Download source code.
  14. Links.

Provenance of Source Code

D. Gaude-Fugarolas,
Phase Transformations Group,
Department of Materials Science and Metallurgy,
University of Cambridge,
Cambridge CB2 3QZ, U.K.

E-mail: dg241@cam.ac.uk
 
 

Added to MAP:  2006.

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Purpose

This program is a simplified model of the  induction heating and water spray quenching of a plate of steel. The heat flow equation had been integrated analytically and this program evaluates the solved equations for different heat inputs.

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Specification

 
Language:  FORTRAN 
Product form:  Source code.
 
Complete program.

       INTEGER I, J, I2, IC, IMAX, JMAX
      PARAMETER (IMAX = 200 )
      PARAMETER (JMAX = 6 )
      DOUBLE PRECISION  X, KTH, T0, Q, TACT(0:(JMAX-1)),
     *       EVALT, R, QRK, ECOOL, TH2O, H,
     *       THI(0:(JMAX-1)),RO, CP, TMAX
      REAL TIC(0:(JMAX-1)), INCT, TIME
 

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Description

This program is a simplified model of the  induction heating and water spray quenching of a plate of steel of normalised thickness (-1<x<1). The heat flow equation had been integrated analytically and this program evaluates the solved equations for different heat inputs. These solutions contain functions erfc(x) that were implemented following [2]. Physical and thermal properties of the steel are taken to be independent of temperature. The thickness of the plate, properties of steel and thermal variables of the process (density, specific heat, thermal conductivity, severity  of the quench, initial and target temperatures, cooling water temperature) are already included in the program and the only input needed during the program run is the heat flux at the surface. The heat flux during induction heating is considered constant.

It is  also assumed that the heating stage of the process is temperature controlled, and  that the heat contribution is maintained until some desired temperature is reached. At that point the coil stops heating the material and the cooling stage begins. As the temperature is,  in practice,  measured  using a pyrometer, only the surface temperature is monitored, and a gradient between inner and outer temperature can exist. The monitoring system is expected to take periodic  measurements of surface temperature. The time step used when calculating the temperature evolution can then be made coincident with the sampling time of the control system. Finally, it is considered in this model that no time  elapses before cooling starts. As cooling starts immediately after heating, the temperature in the plate is not uniform. Naturally, a colder region inside the plate cannot start cooling before the hotter surface has reached its temperature. To allow for this, different time scales are used as function of depth into the plate. The material at a given depth x does not begin to cool (time 'stopped') until the temperature at a point x+dx is lower than at x.

The output of the program is the evolution of temperature as a function of depth with time, during the induction heating and cooling of the plate.

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References

  1. Carslaw, J.W. and Jaeger, J.C. Conduction of Heat in Solids. Oxford University Press, Oxford, 1959.
  2. Press, W.H.; Flannery, S.A.; Teukolsky, S.A. and Vetterling, W.T. Numerical Recipes. Cambridge University Press, Cambridge, 1986.
  3. Poirier, D.R. and Geiger, G.H. Transport Phenomena in Materials Processing. The Minerals, Metals & Materials Society, Warrendale, 1994.
  4. Zill, D.G.; Cullen, M.R. Differential Equations with Boundary-Value Problems. Brooks/Cole Publishing Company, Pacific Grove, 1997.
  5. Gaude-Fugarolas, D. Modelling of transformations during induction hardening and tempering (CPGS thesis). Cambridge University, Cambridge, 2000.
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Parameters

Input parameters

Q- double precision
            Q is the heat flux introduced during induction heating (in W cm ^-2).

Output parameters

TACT(I2) - double precision array of dimension JMAX
            TACT is an array containing the result temperature at position I2 in the plate at the time of the present iteration.
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Error Indicators

None.

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Accuracy

 

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Further Comments

None.

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Example

1. Program text
Complete program.

2. Program data

    50

3. Program results

ITERATION     TIME     T@0.0L  T@0.2L  T@0.4L  T@0.6L  T@0.8L  T@1.0L     (^oC)
      0       0.00 s      25.0    25.0    25.0    25.0    25.0    25.0
      1       0.10 s      25.0    25.0    25.0    25.0    25.1    25.1
      2       0.20 s      25.1    25.2    25.4    25.7    26.1    26.6
      3       0.30 s      26.6    27.1    28.2    29.8    32.0    34.6
      4       0.40 s      34.6    36.3    39.5    44.3    50.7    58.8
      5       0.50 s      58.8    63.6    73.1    87.5   106.6   130.5
      6       0.60 s     130.5   139.0   156.1   181.6   215.6   258.2
      7       0.70 s     258.2   272.7   301.7   345.2   403.3   475.8
      8       0.80 s     475.8   499.8   547.7   619.7   715.6   835.5
      9       0.90 s     835.5   872.4   946.2  1057.0  1204.6  1389.1
 
 
The limit temperature  1200.0 C has been achieved
 
 
ITERATION     TIME     T@0.0L  T@0.2L  T@0.4L  T@0.6L  T@0.8L  T@1.0L     (^oC)
      9       0.90 s     835.5   872.4   946.2  1057.0  1204.6  1389.1
     10       1.00 s     835.5   872.4   946.2  1057.0  1204.6   689.6
     11       1.10 s     835.5   872.4   946.2  1057.0   982.4   560.2
     12       1.20 s     835.5   872.4   946.2  1009.8   829.1   488.3
     13       1.30 s     835.5   872.4   946.2   916.5   733.4   440.1
     14       1.40 s     835.5   872.4   939.8   838.8   665.8   404.4
     15       1.50 s     835.5   872.4   903.2   776.2   614.1   376.0
     16       1.60 s     835.5   872.4   856.3   724.1   572.0   352.3
     17       1.70 s     835.5   871.9   809.7   679.0   536.1   331.3
     18       1.80 s     835.5   861.3   765.3   638.9   504.3   312.3
     19       1.90 s     835.5   837.1   723.4   602.4   475.4   294.4
     20       2.00 s     835.5   804.7   683.9   568.7   448.7   277.5
     21       2.10 s     835.5   768.4   646.7   537.3   423.7   261.1
     22       2.20 s     831.1   730.9   611.5   507.8   400.1   245.2
     23       2.30 s     814.8   693.6   578.4   480.0   377.5   229.7
     24       2.40 s     788.2   657.3   547.0   453.7   356.0   214.4
     25       2.50 s     755.5   622.4   517.4   428.7   335.4   199.5
     26       2.60 s     720.3   589.1   489.3   404.9   315.5   184.9
     27       2.70 s     684.5   557.5   462.8   382.2   296.4   170.5
     28       2.80 s     649.2   527.5   437.5   360.5   278.0   156.4
     29       2.90 s     615.1   499.1   413.6   339.7   260.2   142.5
     30       3.00 s     582.4   472.2   390.8   319.9   243.0   128.9
     31       3.10 s     551.3   446.7   369.1   300.9   226.3   115.5
     32       3.20 s     521.8   422.5   348.4   282.6   210.2   102.4
     33       3.30 s     493.8   399.5   328.6   265.0   194.6    89.6
     34       3.40 s     467.2   377.7   309.7   248.1   179.5    77.0
     35       3.50 s     442.1   356.9   291.6   231.9   164.9    64.6
     36       3.60 s     418.2   337.1   274.3   216.2   150.7    52.5
     37       3.70 s     395.6   318.2   257.7   201.1   136.9    40.6
     38       3.80 s     374.1   300.2   241.8   186.6   123.5    29.0
     39       3.90 s     353.6   283.0   226.5   172.5   110.5    17.6
 
 
The limit temperature    25.0 C has been achieved
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Auxiliary Routines

None.

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Keywords

induction hardening analytical heat

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Download

Download source code

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MAP originated from a joint project of the National Physical Laboratory and the University of Cambridge.

MAP originated from a joint project of the National Physical Laboratory and the University of Cambridge.

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