Diode laser beam absorption in laser transformation hardening of low alloy steel

Transcription

1 Diode laser beam absorption in laser transformation hardening of low alloy steel Henrikki Pantsar and Veli Kujanpää Citation: Journal of Laser Applications 16, 147 (2004); doi: / View online: View Table of Contents: Published by the Laser Institute of America Articles you may be interested in Correlation between measured voltage and observed wavelength in commercial AlGaInP laser diode J. Appl. Phys. 115, (2014); / Optimization of production rate in diode laser hardening J. Laser Appl. 20, 1 (2008); / Examination of laser hardened steel surfaces using interference microscopy J. Laser Appl. 17, 263 (2005); / Deformation behavior and microstructural changes of a hardened high carbon alloy steel in laser bending J. Laser Appl. 14, 83 (2002); / Laser surface treatment of steels with CrB 2 and Ni 2 B powders J. Laser Appl. 11, 216 (1999); /

2 JOURNAL OF LASER APPLICATIONS VOLUME 16, NUMBER 3 AUGUST 2004 Diode laser beam absorption in laser transformation hardening of low alloy steel Henrikki Pantsar a) and Veli Kujanpää Laser Processing Laboratory, Lappeenranta University of Technology, FIN Lappeenranta, Finland Received 3 June 2003; accepted for publication 4 November 2003 Defining and controlling the absorption of the laser beam is important since all of the heating energy is brought to the material through absorption. Even small variations in the absorption change the laser power needed by hundreds of W. In this study the absorption of a diode laser beam to low alloy steel has been measured by a liquid calorimeter and the surface temperature has been measured with a dual wavelength pyrometer. The varied processing parameters were the power intensity of the beam, the interaction time, and the angle between the surface and the optical axis of the laser beam. Surface temperatures during hardening varied from the A c1 temperature to the melting point. Tests were done with a3kwdiode laser with a 12 5 mm hardening optic. The absorptivity of a machined clean steel surface ranged from 46% to 72% depending on the processing parameters. Aluminum oxide blasting of the surface increased the relative amount of energy absorbed to the work piece. The coupling rates for blasted surfaces varied from 66% to 81%. Best absorptivity was achieved by applying graphite coating on the surface. Absorptivity values in excess of 85% were measured Laser Institute of America. Key words: laser transformation hardening, diode laser, absorptivity, beam coupling efficiency I. INTRODUCTION Laser surface hardening or laser transformation hardening is a technique for producing a hard, wear-resistant surface on components. The surface is heated rapidly to the phase by scanning a high intensity laser beam across the hardened area. Efficient conduction of heat to the surrounding material induces the surface to quench very quickly to temperatures below the M s temperature, forming a martensitic layer on the surface, or in high alloy steels a martensitic structure with portions of retained austenite. Due to the high cooling rates, the hardness of martensite reaches higher values than in components hardened with conventional methods. When a component is hardened with a laser, the laser beam interacts with the substrate, heating it very rapidly. The amount of heat transferred to the material is defined by the absorptivity of the material. The absorptivity is the ratio between the energy absorbed by the work piece and the laser energy on the surface. Absorptivity is also often referred to as the coupling rate and it is a dimensionless number between zero and unity. After the energy is absorbed to a very thin surface layer, it is transferred into the surrounding material by thermal conduction. The principle of laser transformation hardening and the conduction of heat are presented in Fig. 1. If the laser beam material interaction is rapid or the traverse rate is high, the absorbed energy has only little time to conduct to the surrounding material. This results in high surface temperatures. On the contrary, if the process is slow, a Electronic mail: pantsar@lut.fi the surface will not reach as high temperatures, even though the laser energy is equal in both cases. Laser transformation hardening was one of the first industrial applications of lasers. Traditionally surface hardening has been carried out by CO 2 lasers, which have provided adequate power levels for surface treatments from the early 1970s. However, absorption of the nm wavelength of a carbon dioxide laser to steel is modest. For low alloy steels the values vary from 4.5% to 11%, 1 depending on the temperature. Therefore an absorptive coating must be added to the heat-treated surfaces in order to improve the effectiveness of the process. Shorter wavelengths absorb more readily to steel. To utilize the increase in absorption Nd:yttrium aluminum garnet YAG 1064 nm and excimer lasers 100 nm 400 nm have been used in surface hardening. Recently high power diode lasers have become available on the market. This type of laser is ideal for surface treatments. The laser device can be mounted to a robot, making possible the processing of more complex three dimensional geometries. The beam shape is rectangular, which is in many cases the best shape for surface hardening and the wavelength is short enough 808 and 940 nm to absorb without an absorptive coating, although absorptive coating may be used to increase the coupling rate. Graphite is one of the most common absorptive coatings used in surface treatments. The spectral absorptivity of graphite for 650 nm wavelength is 97%. 2 The diode laser beam is not absorbed as efficiently to graphite due to the slightly longer wavelengths. However, the absorption of a diode laser beam is expected to be almost as high as the absorption of 650 nm wavelength and applying graphite will significantly increase the coupling rate compared to oxidized X/2004/16(3)/147/7/$ Laser Institute of America

3 148 J. Laser Appl., Vol. 16, No. 3, August 2004 H. Pantsar and V. Kujanpää TABLE II. Physical properties of steel 42CrMo4. 42CrMo4 Reference Specific heat capacity, 100 C 477 J kg C 1 6 Specific heat capacity, 800 C 883 J kg C 1 6 M s temperature 360 C 7 A c1 temperature 730 C 7 A c3 temperature equilibrium 789 C 7 FIG. 1. Principle of laser transformation hardening and the conduction of heat. or clean metal surfaces. The energy coupling efficiency using an absorptive coating is dependent not only on the reflectivity of the coating but also on the heat transfer efficiency between the coating and the work piece. 3 The coating can also be evaporated from the surface, 3 or if melting occurs, the coating can blend into the melt. When hardening in an oxidizing atmosphere, surface oxidation has a great effect on the coupling rate. Oxidation is expected to increase surface absorptivity by approximately 50%. 4 The surface shape, in the form of roughness, may be such as to cause a single ray to be reflected twice or more, before leaving the surface, consequently, greatly reducing the surface reflectance. 5 For a milled surface a linear agreement between the surface roughness and absorptivity has been reported. However polished, ground, and sandblasted surfaces do not fit the relationship between the arithmetic average surface roughness and absorptivity. 1 Surface temperature affects the absorptivity, especially with CO 2 and CO laser wavelengths. The absorptivity is, however, almost constant regardless of the temperature for wavelengths from 0.5 to 1 m. Experimental results 1 and values calculated from electrical conductivity 4 show that the absorptivity is not affected by the surface temperature in this wavelength range. Since all of the heating energy is brought to the material via absorption, it is essential to better understand the factors affecting absorptivity. A lot of work has been done to establish the absorptivity for CO 2 and Nd:YAG lasers, but only little experimental data are available for diode laser hardening, especially in a normal atmosphere, where surface oxidation will be a major factor affecting absorptivity. The present work was undertaken to study the absorption of a diode laser beam in laser surface hardening, or in cases where the surface temperature varies from A c1 temperature to the melting point. Also the affecting factors, graphite coating, aluminum oxide blasting of the surface, and the angle between the laser s optical axis and the surface, were studied. II. EXPERIMENTAL PROCEDURE A. Test material The steel used in this study was a heat treatable low alloy steel EN CrMo4 in quenched and tempered condition. Composition of the steel is presented in Table I. Surfaces were machined and cleaned with acetone. The arithmetic average surface roughness, R a, was from 0.2 to 0.4 m. Some specimens were graphite coated or aluminum oxide blasted in order to increase absorptivity. The surface roughness value R a for blasted samples was between 2.0 and 2.7 m. Physical properties of steel 42CrMo4 are presented in Table II. The specific heat for the steel was calculated from the experimental data of 1% Cr Mo steel, 6 assuming a linear increase with temperature. A value at 25 C was used in calculations. B. Laser processing A high power diode laser was used for hardening. The diode laser consists of stacks of separate diodes delivering radiation at wavelengths of and nm. The power distribution between the two wavelength ranges could not be selected. For each power level half of the delivered laser power comprised of nm radiation and half of nm radiation. The maximum nominal power of the laser P N was 3 kw. Time to attain full power was 10 ms, and the time from full power to power off was 10 ms. Focal length of the optic was 500 mm and the measured spot size, calculated by second moment, was 5.3 mm in the traverse direction and 12.3 mm across the traverse direction. Figure 2 illustrates the measured profile of the beam. Beam power on the work piece, P W, was measured with a laser power and energy meter, and was 95% of the nominal laser power. The laser was mounted on a six axis industrial robot. Load bearing capacity of the robot was 125 kg. Processed work pieces were placed on a thermally insulating material, in order to avoid energy loss via conduction during processing. A 130 mm long track was hardened on the machined surface of a 150 mm long and 25 mm thick work piece. Work pieces were made from steel rods, 25 mm in diameter. The most important processing parameters in surface hardening are laser power and traverse speed or the interac- TABLE I. Composition of the 42CrMo4 used wt %. C Si Mn P S Cr Ni Mo W V Cu Al

4 J. Laser Appl., Vol. 16, No. 3, August 2004 H. Pantsar and V. Kujanpää 149 TABLE III. Physical properties of water and monoethylene glycol. Reference Water Density, 25 C 997 kg m 3 8 Specific heat, 25 C 4180 J kg C 1 8 Boiling temperature 100 C 8 Monoethylene glycol Density, 40 C 1101 kg m 3 9 Specific heat, 40 C 2474 J kg C 1 9 Boiling temperature 198 C 8 FIG. 2. Measured beam profile of the hardening optic used. tion time. The traverse speed was varied so that for each power level tests were made from surface temperatures below A c1 temperature to the melting temperature. The angle between the laser s optical axis and the surface was set to 85 to reduce the amount of power reflected back to the laser. Test series with 2280 W laser power on the work piece were also performed with aluminum oxide blasted and graphite-coated surfaces. Experiments with different angles between the laser s optical axis and the surface were also made. The angles were 85, 80, 75, 70, 65, 60, and 55. Laser power was adjusted in order to maintain the same power intensity with all angles. C. Measuring equipment Surface temperature was measured during hardening with a dual wavelength pyrometer. Operable temperature range was from 490 to 1600 C. Measured wavelengths were 1300 and 1700 nm. The pyrometer was set to measure the heating and cooling curves of a stationary spot from the center line of the hardened track. The energy absorbed by the work piece was measured by using a liquid calorimeter, built especially for the tests. It comprised a thermally insulated container, thermometer, precision scale, water and monoethylene glycol as the liquids, and a mixer. The design of the calorimeter is illustrated in Fig. 3. The thermistor-operated thermometer was calibrated and traceable to National Institute of Standards and Technology, and had an absolute accuracy of C and a resolution of C. The accuracy of the precision scale was 0.5 g. The physical data of water and glycol are presented in Table III. D. Operation of the calorimeter A schematic diagram of the operation of the liquid calorimeter is illustrated in Fig. 4. The work piece is hardened between times t 0 and t 2. During this time the work piece absorbs the energy of the laser beam and is heated to its maximum temperature. The work piece is then placed in the liquid calorimeter at time t 3, and the absorbed energy is FIG. 3. Schematic picture of the calorimeter. FIG. 4. Schematic diagram of calorimeter operation.

5 150 J. Laser Appl., Vol. 16, No. 3, August 2004 H. Pantsar and V. Kujanpää transferred to the calorimeter liquid. Energy transfer is complete when the temperature of the work piece equals the temperature of the liquid, at time t 5. In the following formula the temperature rise of the liquid, the work piece, and calorimeter are multiplied by their weights and specific heat capacities in order to establish the number of Joules needed for the thermal exchange. Some energy is also needed to raise the temperature of the thermistor, mixer, and the inside of the container. For this calibration the mass L is used. It is the amount of liquid that possesses the same heat capacity as the parts of the calorimeter that are heated together with the liquid. The heating of the calorimeter liquid is measured by a thermistor-operated thermometer and the measured absorbed energy E AM J) can be calculated from an energy balance equation E AM T L1 T L0 c L m L L T L1 T W0 c W m W, 1 where T L1 is the measured maximum temperature of the liquid C ; T L0 is the temperature of the liquid at t 0 ( C); T W0 is the temperature of the work piece at t 0 ( C); m L and m W are the weights of the liquid and work piece, respectively kg ; c L and c W are specific heats of liquid and work piece, respectively J kg C 1 ; and L is the liquid value kg. The liquid value for measurements using water was 12.2 and 20.7 g for monoethylene glycol, respectively. Thermal losses are experienced during the measurement. First, a minimal fraction of the absorbed energy delivered to the calorimeter is lost because of the imperfect insulation of the calorimeter. Therefore the measured peak temperature of the liquid is lower than the theoretical peak temperature of the liquid. The cooling of the liquid was measured, when cooler air surrounds the calorimeter and then an exponential equation was established that describes cooling. The established exponential curve is set so that it passes through the measured temperature T L1. Following the curve backwards through a time t 5 t 4, a point is found that gives the temperature to which the liquid s temperature would rise if no thermal loss occurred. This temperature is the theoretical peak temperature T L2. The time t 4 is established such that A 3 and A 4 are equal Fig. 4. This is because a mean value is needed to approximate the temperature of the liquid. The following equation gives the temperature difference between the measured value T L1 and calculated value T L2 : T L1 T 0 T L2 T L1 exp k 1 t 5 t 4 T L1 T 0, where t 5 t 4 is the calculated time during which energy loss occurs and T 0 is the temperature of the ambient air, in which the experiments are carried out. The calorimeter was calibrated and a numeric value for the rate constant k 1 was obtained by regression analysis; k for glycol and k for water. The thermal energy loss during measurement E 1 can then be calculated from an energy balance equation 2 A small part of the absorbed energy is also lost to the surrounding air during hardening and while placing the work piece into the calorimeter. The temperature of the work piece decreases during that time from T W2, which is the theoretical peak temperature of the work piece, to T W1. T W1 describes how much the temperature of the work piece was at the time it was placed into the calorimeter. When the measured absorbed energy E AM and the thermal loss energy E 1 are known Eqs. 1 and 3, the temperature to which these energies together raise the temperature of the work piece can be calculated. This temperature is T W1 T W1 E AM E 1 T c W m W0. 4 W The cooling of a heated work piece was measured prior to these trials, so that an exponential equation that describes the cooling of a heated work piece in air could be established. The fact that the hardened part of the work piece is very hot and the other side is near room temperature is not considered an evenly heated sample is measured. For this reason in calculations the thermal conductivity was considered infinite during hardening and thus the time for thermal energy to conduct uniformly throughout the work piece was considered zero. The decrease in temperature from T W2 to T W1 occurs during the time t 3 t 1. This time is calculated by setting the time t 1 such that the area A 1 equals the area A 2 Fig. 4. During experiments work pieces were moved into the calorimeter in less than 20 s. Again following the exponential curve backwards from T W1, we find the theoretical maximum temperature to which the work piece would rise if no energy losses occurred (T W2 ). The equation that gives the temperature loss is T W1 T 0 T W2 T W1 exp k 2 t 3 t 1 T W1 T 0, 5 where k 2 is the rate constant. By measuring the cooling of a heated specimen in air a value of k was obtained by regression analysis. Again the lost temperature is converted to an energy value by E 2 T W2 T W1 c W m W. 6 Finally, the measured energy E AM is added to the two thermal loss energies, E 1 and E 2, to establish the amount of absorbed energy E A before thermal losses E A E AM E 1 E 2. 7 T 0, T L0, and T W0 are presented as equal in Fig. 4. The presented methodology of calculating energies and losses is valid even if these variables are not equal. III. RESULTS A. Absorptivity of machined surfaces The absorption of all work pieces was measured and calculated using the presented equipment. Surface temperatures were measured for all experiments, except for the work E 1 T L2 T L1 c L m L L T L2 T L1 c W m W. 3 pieces with absorptive coating. In those cases the pyrometer

6 J. Laser Appl., Vol. 16, No. 3, August 2004 H. Pantsar and V. Kujanpää 151 FIG. 5. Absorptivity as a function of traverse rate v and interaction time t i for different processing parameters. The laser energies E L shown are equal for all tests between the lines drawn. FIG. 6. Diode laser beam absorption: aluminum oxide blasted, graphite coated, and machined surfaces. Laser power on the work piece is 2280 W. failed to measure the temperature of the steel surface, but instead it measured the temperature of the absorptive coating. The measured absorbed energy E A was from 46.0% to 71.8% of the laser energy. With equal laser energy per distance, the best absorption was achieved with shorter interaction times and increased laser power. Figure 5 illustrates the absorbed energy plotted against traverse speed and interaction time. Laser energies E L calculated from the power on the work piece P W are also presented. The angle between the beam and the absorbing surface did not have a significant effect with the angles tested. The absorption varied in these experiments from 56.1% to 59.1%. C. Thermal cycles and absorption Thermal cycles from the surfaces during processing were measured for all specimens with a dual wavelength pyrometer. Figure 7 illustrates thermal cycles, processing parameters, and absorptivities for some samples. From the thermal cycles it could be noted that both the dwell time and the peak temperature affect the coupling rate. With the same laser energy, the best absorption was achieved with high power levels and traverse rates. The combination of high power and high traverse rate also resulted as higher cooling rates. Further, if all results for machined surfaces are plotted against surface peak temperature, a significant increase in absorption with temperature could be discovered, as illustrated in Fig. 8. B. Absorptivity of graphite coated and aluminum blasted surfaces and tests with argon shield gas Best absorption was achieved with graphite coating on the surface. The amount of absorbed energy varied from 85.7% to 86.0%. For the tested laser power the traverse rate did not have any effect on the absorption. The greatest differences in absorption compared to metal surfaces were found with high traverse rates, since the absorptivity remains constant with the traverse rate, but for noncoated metal surfaces the absorptivity decreases with the traverse rate. The surface temperature for coated surfaces could not be measured, since the optical pyrometer measured the temperature of the coating, not the metal surface beneath. Absorptivities for graphite coated and aluminum blasted surfaces are illustrated in Fig. 6. Aluminum oxide blasted surfaces absorbed energy more readily than machined surfaces. Absorptivity varied from 66.0% to 80.6%, depending on the traverse rate. Compared to machined surfaces, the absorptivity increased 33% at 1572 mm/min and 59% at 2460 mm/min traverse rate. For the same traverse speeds surface temperatures increased from 1298 to 1413 C and from 904 to 1188 C, respectively. FIG. 7. Thermal cycles during hardening. The numbers of the tests are shown in the picture near the peak temperature of each curve. Processing parameters and measured absorptivities are presented in the table.

7 152 J. Laser Appl., Vol. 16, No. 3, August 2004 H. Pantsar and V. Kujanpää FIG. 8. Absorption of machined test pieces as a function of surface peak temperature T p. IV. DISCUSSION The results clearly show that the surface temperature has the greatest effect on absorptivity in diode laser hardening of heat treatable steels, Fig. 8. At wavelengths near 1 m, however, the absorptivity of the clean metal surface does not increase with temperature. 1,4 It is therefore reasonable to believe that the coupling rate increase with temperature is caused by surface oxidation. A thicker oxide layer enables the laser radiation to absorb more effectively. The variation of absorption for a given temperature in Fig. 8 was due to different processing parameters. The rate of oxide layer growth increases with temperature. Long interaction times allow thicker layers to be formed on the surface. This effect is clearly seen in Figs. 7 and 8. Best absorption is obtained with higher surface peak temperatures. For equal surface peak temperature, the absorption is more effective when the interaction time is longer as the formed oxide layer is thicker Fig. 8. Figure 5 presents the same phenomenon from a different point of view; for equal laser energies, considerably higher absorption is attained when the traverse rate is faster and with more power, resulting as higher peak temperatures. Thus the oxide growth rate is higher, inducing a better coupling rate. Laser power has a more important effect on the absorption than the interaction time. For example, increasing the laser energy E L from 87 J/mm (P W 1170 W, v 804 mm/min) to 109 J/mm by lengthening the interaction time increases the absorptivity by 6.3%. Increasing the laser energy an equal amount by raising the laser power increases absorptivity by 16.3%. This is due to the difference in surface temperature. Lengthening the interaction time, or selecting a lower traverse rate, allows more heat to conduct away from the surface. The added energy does not raise the surface temperature much, but results as deeper penetration, or speaking of laser transformation hardening, a thicker hardened layer. Increasing laser power on the other hand raises the peak surface temperature remarkably. In this case the longer dwell time raised the surface temperature from 763 to 985 C, while by raising the laser power a peak temperature of 1147 C could be reached. Only approximately 15% of the laser energy reflected from graphite coated surfaces Fig. 6. The presented tests show that the traverse rate does not affect the absorptivity, since the absorption is not based on surface oxidation. However, if the graphite would evaporate from the surface due to a higher energy density or a longer dwell time, there would probably be a decrease in the absorptivity. Blasting the surface with aluminum oxide increased absorptivity. The increase is more considerable in low temperatures, when the surface geometry is the dominating factor. At higher temperature, surface oxidation is the main affecting factor and the influence of the surface geometry and roughness diminishes. Depending on the processing parameters, a 33% 59% rise could be observed. The surface condition was equal with other experiments before blasting. The surface condition does not have an effect on the characteristic absorptivity of the material, but the variations are induced by the incident angle and the way the beam reflects from the surface. The probable reason for the considerable increase in the coupling rate, therefore, is the shape of the surface, allowing the incident beam to reflect more than once before leaving the surface. If the beam reflects only once from the surface, as would be the case for polished surfaces, the absorbed energy E A is defined by E A E L A, 8 where A is the absorptivity as a dimensionless number between zero and unity, and E L is the laser energy on the work piece. If the light is reflected from a point on the surface in a manner that the reflected light hits the surface again once, the absorbed energy will be E A2 E L A E L AR, 9 where reflectivity R 1 A. For multiple reflections we get a series E An E L AR n n 1 Calculating from the equations above we find that in a case of low absorptivity the second and the subsequent reflections would have a relatively larger effect on the total absorptivity than in a case of high absorptivity or more heavily oxidated surface. The angle of incidence did not have a significant effect on absorptivity with the angles tested. V. CONCLUSIONS Surface hardening with a diode laser is an effective process, which makes it possible to process materials without an absorptive coating. Even in the least absorbed experiments the absorptivity exceeded 45%. In these experiments surface oxidation was exiguous. Judging by the results and comparing the effect of surface roughness to previously published data, 1 the absorptivity of a polished, nonoxidized steel surface is expected to be approximately 35% 40%. Published experimental data 1 show that the absorption of a Nd:YAG laser beam to a polished, nonoxidized low alloy steel surface ranges from 28% to 31%. In equal conditions the absorption ofaco 2 laser beam is only 4.5% 11%. The oxidation state of the surface was found to have the greatest effect on the coupling rate in processing of ma-

8 J. Laser Appl., Vol. 16, No. 3, August 2004 H. Pantsar and V. Kujanpää 153 chined surfaces without an absorptive coating. The thickness of the formed oxide layer depends on the dwell time, or the traverse speed, and the surface temperature, the latter having a greater increasing effect. The highest beam coupling rate can be obtained with temperatures close to the melting temperature and with relatively low traverse speeds. The highest measured absorptivity to a machined surface was 71.8%. The incident angle did not present a significant effect on the coupling rate for the angles tested. The surface roughness did have a remarkable effect on the absorptivity of the surface. The surface condition does not have an effect on the characteristic absorptivity of the material itself, but the variations are induced by the incident angle and the way the beam reflects from the surface. The roughness and geometry of the surface are altered by aluminum oxide blasting. This allows the beam to reflect more than once before leaving the surface. Thus, the effect of surface roughness cannot be explained only by evaluating the arithmetic surface roughness, but also the geometry of the surface. Although the use of an absorptive coating is not necessary in diode laser transformation hardening, the use of such coating increases the efficiency of the process. Absorptivity values in excess of 85% were observed. ACKNOWLEDGMENTS This study was done as a part of the project Intelligent Laser Surface Engineering, funded by the Academy of Finland for the years The authors acknowledge the assistance of Saara Kouvo during experimental trials and Dr. John C. Ion during preparation of the manuscript. 1 G. Stern, Absorptivity of cw CO 2, CO and YAG-laser beams by different metallic alloys, Proceedings of the 3rd European Conference on Laser Treatment of Materials, Erlangen, Germany, September 1990, pp M. A. Bramson, Infrared Radiation Plenum, New York, 1968, p J. F. Ready, Industrial Applications of Lasers, 2nd ed. Academic, New York, 1997, p F. Dausinger and J. Shen, Energy Coupling Efficiency in Laser Surface Treatment, ISIJ Int. 33, W. M. Steen and J. Mazumder, Mathematical Modelling of Laser/ Material Interactions, Final Report, European Office of Aerospace Research and Development, London, Smithells Metals Reference Book, edited by E. A. Brandes and G. B. Brook Butterworth-Heinemann, London, 1992, pp Atlas zur Wärmebehandlung der Stähle Verlag Stahleisen M.B.H., 1961, p Y. A. Cengel, Introduction to Thermodynamics and Heat Transfer McGraw Hill, New York, 1997, p E. R. G. Eckert, Analysis of Heat and Mass Transfer McGraw Hill, New York, 1972, p. 779.