Abstract

The rate of cooling of domesticated pig bones is investigated within the temperature range of 20°C-320°C. Within the afore-mentioned temperature range, it was found that different behaviors in the rate of cooling were taking place. For bones reaching a temperature within the lower temperature range of 20°C-50°C, it was found that the rate of cooling is mostly governed by the empirical Newton’s law of cooling. It is also shown that a transition is taking place somewhere within 50°C-100°C, where both the heat conduction equation and Newton’s law apply. As bones can be raised at a fairly high temperature before burning, it was found that the rate of cooling within the range 125°C-320°C is mostly behaving according to the heat conduction equation and Stefan-Boltzmann radiation law. A pulsed CO2 laser was used to heat the bones up to a given temperature and the change of temperature as a function of time was recorded by non-contact infrared thermometer during the cooling period.

© 2014 Optical Society of America

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References

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    [CrossRef] [PubMed]
  2. K.-W. Guan, Y.-Q. Jiang, C.-S. Sun, and H. Yu, “A two-layer model of laser interaction with skin: A photothermal effect analysis,” Opt. Laser Technol.43(3), 425–429 (2011).
    [CrossRef]
  3. J. H. Torres, M. Motamedi, J. A. Pearce, and A. J. Welch, “Experimental evaluation of mathematical models for predicting the thermal response of tissue to laser irradiation,” Appl. Opt.32(4), 597–606 (1993).
    [CrossRef] [PubMed]
  4. M. M. Ivanenko, S. Fahimi-Weber, T. Mitra, W. Wierich, and P. Hering, “Bone Tissue Ablation with sub- µs Pulses of a Q-switch CO2 Laser: Histological Examination of Thermal Side Effects,” Lasers Med. Sci.17(4), 258–264 (2002).
    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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  8. Y. Yener and S. Kakaç, Heat Conduction, 4th ed. (Taylor & Francis, 2008) pp. 210–212.
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    [CrossRef]
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    [CrossRef]
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  14. A. Sagi, A. Shitzer, A. Katzir, and S. Akselrod, “Heating of biological tissue by laser irradiation: theoretical model,” Opt. Eng.31(7), 1417–1424 (1992).
    [CrossRef]
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  17. M. R. Spiegel, “Mathematical Handbook of Formulas and Tables,” McGraw-Hill Inc., New York (1974).
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    [CrossRef] [PubMed]
  19. H. S. Carslaw and J. C. Jaeger, Conduction of Heat in Solids, 2nd ed. (Oxford University Press 1959) pp. 305–306.
  20. L. Lévesque, “Temperature control of water-based substances by CO2 laser for medical applications,” Appl. Opt.52(16), 3856–3863 (2013).
    [CrossRef] [PubMed]

2013 (1)

2012 (1)

M. R. Tatara, W. Krupski, B. Tymczyna, and I. Luszczewska-Sierakowsky, “Biochemical bone metabolism markers and morphometric, densitometric and biomechanical properties of femur and tibia in female and gonadectomised male Polish Landrace pigs,” J. Pre-Clin. Clin. Res.6(1), 14–19 (2012).

2011 (3)

K.-W. Guan, Y.-Q. Jiang, C.-S. Sun, and H. Yu, “A two-layer model of laser interaction with skin: A photothermal effect analysis,” Opt. Laser Technol.43(3), 425–429 (2011).
[CrossRef]

J. Lee, Y. Rabin, and O. B. Ozdoganlar, “A new thermal model for bone drilling with applications to orthopaedic surgery,” Med. Eng. Phys.33(10), 1234–1244 (2011).
[CrossRef] [PubMed]

A. Mathur and Y. K. Agrawal, “An overview of methods used for estimation of time since death,” Aust. J. Forensic Sci.43(4), 275–285 (2011).
[CrossRef]

2008 (1)

Q. Peng, A. Juzeniene, J. Chen, L.O Svaasand, T. Warloe, K-E Giercksky, and J. Moan, “Lasers in Medicine,” Rep. Prog. Phys.71, 056701 (2008).

2002 (1)

M. M. Ivanenko, S. Fahimi-Weber, T. Mitra, W. Wierich, and P. Hering, “Bone Tissue Ablation with sub- µs Pulses of a Q-switch CO2 Laser: Histological Examination of Thermal Side Effects,” Lasers Med. Sci.17(4), 258–264 (2002).
[CrossRef] [PubMed]

2001 (1)

N. M. Fried and D. Fried, “Comparison of Er:YAG and 9.6-µm TE CO2 Lasers for Ablation of Skull Tissue,” Lasers Surg. Med.28(4), 335–343 (2001).
[CrossRef] [PubMed]

1998 (1)

M. M. Ivanenko and P. Hering, “Wet bone ablation with mechanically Q-switched high-repetition-rate CO2 laser,” Appl. Phys. B67(3), 395–397 (1998).
[CrossRef]

1995 (1)

B. Anvari, T. E. Milner, B. S. Tanenbaum, S. Kimel, L. O. Svaasand, and J. S. Nelson, “Selective cooling of biological tissues: application for thermally mediated therapeutic procedures,” Phys. Med. Biol.40(2), 241–252 (1995).
[CrossRef] [PubMed]

1993 (2)

M. Forrer, M. Frenz, V. Romano, H. J. Altermatt, H. P. Weber, A. Silenok, M. Istomyn, and V. I. Konov, “Bone-Ablation Mechanism Using CO2 lasers of Different Pulse Duration and Wavelength,” Appl. Phys. B56(2), 104–112 (1993).
[CrossRef]

J. H. Torres, M. Motamedi, J. A. Pearce, and A. J. Welch, “Experimental evaluation of mathematical models for predicting the thermal response of tissue to laser irradiation,” Appl. Opt.32(4), 597–606 (1993).
[CrossRef] [PubMed]

1992 (1)

A. Sagi, A. Shitzer, A. Katzir, and S. Akselrod, “Heating of biological tissue by laser irradiation: theoretical model,” Opt. Eng.31(7), 1417–1424 (1992).
[CrossRef]

1990 (1)

A. L. McKenzie, “Physics of thermal processes in laser-tissue interaction,” Phys. Med. Biol.35(9), 1175–1210 (1990).
[CrossRef] [PubMed]

Agrawal, Y. K.

A. Mathur and Y. K. Agrawal, “An overview of methods used for estimation of time since death,” Aust. J. Forensic Sci.43(4), 275–285 (2011).
[CrossRef]

Akselrod, S.

A. Sagi, A. Shitzer, A. Katzir, and S. Akselrod, “Heating of biological tissue by laser irradiation: theoretical model,” Opt. Eng.31(7), 1417–1424 (1992).
[CrossRef]

Altermatt, H. J.

M. Forrer, M. Frenz, V. Romano, H. J. Altermatt, H. P. Weber, A. Silenok, M. Istomyn, and V. I. Konov, “Bone-Ablation Mechanism Using CO2 lasers of Different Pulse Duration and Wavelength,” Appl. Phys. B56(2), 104–112 (1993).
[CrossRef]

Anvari, B.

B. Anvari, T. E. Milner, B. S. Tanenbaum, S. Kimel, L. O. Svaasand, and J. S. Nelson, “Selective cooling of biological tissues: application for thermally mediated therapeutic procedures,” Phys. Med. Biol.40(2), 241–252 (1995).
[CrossRef] [PubMed]

Chen, J.

Q. Peng, A. Juzeniene, J. Chen, L.O Svaasand, T. Warloe, K-E Giercksky, and J. Moan, “Lasers in Medicine,” Rep. Prog. Phys.71, 056701 (2008).

Fahimi-Weber, S.

M. M. Ivanenko, S. Fahimi-Weber, T. Mitra, W. Wierich, and P. Hering, “Bone Tissue Ablation with sub- µs Pulses of a Q-switch CO2 Laser: Histological Examination of Thermal Side Effects,” Lasers Med. Sci.17(4), 258–264 (2002).
[CrossRef] [PubMed]

Forrer, M.

M. Forrer, M. Frenz, V. Romano, H. J. Altermatt, H. P. Weber, A. Silenok, M. Istomyn, and V. I. Konov, “Bone-Ablation Mechanism Using CO2 lasers of Different Pulse Duration and Wavelength,” Appl. Phys. B56(2), 104–112 (1993).
[CrossRef]

Frenz, M.

M. Forrer, M. Frenz, V. Romano, H. J. Altermatt, H. P. Weber, A. Silenok, M. Istomyn, and V. I. Konov, “Bone-Ablation Mechanism Using CO2 lasers of Different Pulse Duration and Wavelength,” Appl. Phys. B56(2), 104–112 (1993).
[CrossRef]

Fried, D.

N. M. Fried and D. Fried, “Comparison of Er:YAG and 9.6-µm TE CO2 Lasers for Ablation of Skull Tissue,” Lasers Surg. Med.28(4), 335–343 (2001).
[CrossRef] [PubMed]

Fried, N. M.

N. M. Fried and D. Fried, “Comparison of Er:YAG and 9.6-µm TE CO2 Lasers for Ablation of Skull Tissue,” Lasers Surg. Med.28(4), 335–343 (2001).
[CrossRef] [PubMed]

Giercksky, K-E

Q. Peng, A. Juzeniene, J. Chen, L.O Svaasand, T. Warloe, K-E Giercksky, and J. Moan, “Lasers in Medicine,” Rep. Prog. Phys.71, 056701 (2008).

Guan, K.-W.

K.-W. Guan, Y.-Q. Jiang, C.-S. Sun, and H. Yu, “A two-layer model of laser interaction with skin: A photothermal effect analysis,” Opt. Laser Technol.43(3), 425–429 (2011).
[CrossRef]

Hering, P.

M. M. Ivanenko, S. Fahimi-Weber, T. Mitra, W. Wierich, and P. Hering, “Bone Tissue Ablation with sub- µs Pulses of a Q-switch CO2 Laser: Histological Examination of Thermal Side Effects,” Lasers Med. Sci.17(4), 258–264 (2002).
[CrossRef] [PubMed]

M. M. Ivanenko and P. Hering, “Wet bone ablation with mechanically Q-switched high-repetition-rate CO2 laser,” Appl. Phys. B67(3), 395–397 (1998).
[CrossRef]

Istomyn, M.

M. Forrer, M. Frenz, V. Romano, H. J. Altermatt, H. P. Weber, A. Silenok, M. Istomyn, and V. I. Konov, “Bone-Ablation Mechanism Using CO2 lasers of Different Pulse Duration and Wavelength,” Appl. Phys. B56(2), 104–112 (1993).
[CrossRef]

Ivanenko, M. M.

M. M. Ivanenko, S. Fahimi-Weber, T. Mitra, W. Wierich, and P. Hering, “Bone Tissue Ablation with sub- µs Pulses of a Q-switch CO2 Laser: Histological Examination of Thermal Side Effects,” Lasers Med. Sci.17(4), 258–264 (2002).
[CrossRef] [PubMed]

M. M. Ivanenko and P. Hering, “Wet bone ablation with mechanically Q-switched high-repetition-rate CO2 laser,” Appl. Phys. B67(3), 395–397 (1998).
[CrossRef]

Jiang, Y.-Q.

K.-W. Guan, Y.-Q. Jiang, C.-S. Sun, and H. Yu, “A two-layer model of laser interaction with skin: A photothermal effect analysis,” Opt. Laser Technol.43(3), 425–429 (2011).
[CrossRef]

Juzeniene, A.

Q. Peng, A. Juzeniene, J. Chen, L.O Svaasand, T. Warloe, K-E Giercksky, and J. Moan, “Lasers in Medicine,” Rep. Prog. Phys.71, 056701 (2008).

Katzir, A.

A. Sagi, A. Shitzer, A. Katzir, and S. Akselrod, “Heating of biological tissue by laser irradiation: theoretical model,” Opt. Eng.31(7), 1417–1424 (1992).
[CrossRef]

Kimel, S.

B. Anvari, T. E. Milner, B. S. Tanenbaum, S. Kimel, L. O. Svaasand, and J. S. Nelson, “Selective cooling of biological tissues: application for thermally mediated therapeutic procedures,” Phys. Med. Biol.40(2), 241–252 (1995).
[CrossRef] [PubMed]

Konov, V. I.

M. Forrer, M. Frenz, V. Romano, H. J. Altermatt, H. P. Weber, A. Silenok, M. Istomyn, and V. I. Konov, “Bone-Ablation Mechanism Using CO2 lasers of Different Pulse Duration and Wavelength,” Appl. Phys. B56(2), 104–112 (1993).
[CrossRef]

Krupski, W.

M. R. Tatara, W. Krupski, B. Tymczyna, and I. Luszczewska-Sierakowsky, “Biochemical bone metabolism markers and morphometric, densitometric and biomechanical properties of femur and tibia in female and gonadectomised male Polish Landrace pigs,” J. Pre-Clin. Clin. Res.6(1), 14–19 (2012).

Lee, J.

J. Lee, Y. Rabin, and O. B. Ozdoganlar, “A new thermal model for bone drilling with applications to orthopaedic surgery,” Med. Eng. Phys.33(10), 1234–1244 (2011).
[CrossRef] [PubMed]

Lévesque, L.

Luszczewska-Sierakowsky, I.

M. R. Tatara, W. Krupski, B. Tymczyna, and I. Luszczewska-Sierakowsky, “Biochemical bone metabolism markers and morphometric, densitometric and biomechanical properties of femur and tibia in female and gonadectomised male Polish Landrace pigs,” J. Pre-Clin. Clin. Res.6(1), 14–19 (2012).

Mathur, A.

A. Mathur and Y. K. Agrawal, “An overview of methods used for estimation of time since death,” Aust. J. Forensic Sci.43(4), 275–285 (2011).
[CrossRef]

McKenzie, A. L.

A. L. McKenzie, “Physics of thermal processes in laser-tissue interaction,” Phys. Med. Biol.35(9), 1175–1210 (1990).
[CrossRef] [PubMed]

Milner, T. E.

B. Anvari, T. E. Milner, B. S. Tanenbaum, S. Kimel, L. O. Svaasand, and J. S. Nelson, “Selective cooling of biological tissues: application for thermally mediated therapeutic procedures,” Phys. Med. Biol.40(2), 241–252 (1995).
[CrossRef] [PubMed]

Mitra, T.

M. M. Ivanenko, S. Fahimi-Weber, T. Mitra, W. Wierich, and P. Hering, “Bone Tissue Ablation with sub- µs Pulses of a Q-switch CO2 Laser: Histological Examination of Thermal Side Effects,” Lasers Med. Sci.17(4), 258–264 (2002).
[CrossRef] [PubMed]

Moan, J.

Q. Peng, A. Juzeniene, J. Chen, L.O Svaasand, T. Warloe, K-E Giercksky, and J. Moan, “Lasers in Medicine,” Rep. Prog. Phys.71, 056701 (2008).

Motamedi, M.

Nelson, J. S.

B. Anvari, T. E. Milner, B. S. Tanenbaum, S. Kimel, L. O. Svaasand, and J. S. Nelson, “Selective cooling of biological tissues: application for thermally mediated therapeutic procedures,” Phys. Med. Biol.40(2), 241–252 (1995).
[CrossRef] [PubMed]

Ozdoganlar, O. B.

J. Lee, Y. Rabin, and O. B. Ozdoganlar, “A new thermal model for bone drilling with applications to orthopaedic surgery,” Med. Eng. Phys.33(10), 1234–1244 (2011).
[CrossRef] [PubMed]

Pearce, J. A.

Peng, Q.

Q. Peng, A. Juzeniene, J. Chen, L.O Svaasand, T. Warloe, K-E Giercksky, and J. Moan, “Lasers in Medicine,” Rep. Prog. Phys.71, 056701 (2008).

Rabin, Y.

J. Lee, Y. Rabin, and O. B. Ozdoganlar, “A new thermal model for bone drilling with applications to orthopaedic surgery,” Med. Eng. Phys.33(10), 1234–1244 (2011).
[CrossRef] [PubMed]

Romano, V.

M. Forrer, M. Frenz, V. Romano, H. J. Altermatt, H. P. Weber, A. Silenok, M. Istomyn, and V. I. Konov, “Bone-Ablation Mechanism Using CO2 lasers of Different Pulse Duration and Wavelength,” Appl. Phys. B56(2), 104–112 (1993).
[CrossRef]

Sagi, A.

A. Sagi, A. Shitzer, A. Katzir, and S. Akselrod, “Heating of biological tissue by laser irradiation: theoretical model,” Opt. Eng.31(7), 1417–1424 (1992).
[CrossRef]

Shitzer, A.

A. Sagi, A. Shitzer, A. Katzir, and S. Akselrod, “Heating of biological tissue by laser irradiation: theoretical model,” Opt. Eng.31(7), 1417–1424 (1992).
[CrossRef]

Silenok, A.

M. Forrer, M. Frenz, V. Romano, H. J. Altermatt, H. P. Weber, A. Silenok, M. Istomyn, and V. I. Konov, “Bone-Ablation Mechanism Using CO2 lasers of Different Pulse Duration and Wavelength,” Appl. Phys. B56(2), 104–112 (1993).
[CrossRef]

Sun, C.-S.

K.-W. Guan, Y.-Q. Jiang, C.-S. Sun, and H. Yu, “A two-layer model of laser interaction with skin: A photothermal effect analysis,” Opt. Laser Technol.43(3), 425–429 (2011).
[CrossRef]

Svaasand, L. O.

B. Anvari, T. E. Milner, B. S. Tanenbaum, S. Kimel, L. O. Svaasand, and J. S. Nelson, “Selective cooling of biological tissues: application for thermally mediated therapeutic procedures,” Phys. Med. Biol.40(2), 241–252 (1995).
[CrossRef] [PubMed]

Svaasand, L.O

Q. Peng, A. Juzeniene, J. Chen, L.O Svaasand, T. Warloe, K-E Giercksky, and J. Moan, “Lasers in Medicine,” Rep. Prog. Phys.71, 056701 (2008).

Tanenbaum, B. S.

B. Anvari, T. E. Milner, B. S. Tanenbaum, S. Kimel, L. O. Svaasand, and J. S. Nelson, “Selective cooling of biological tissues: application for thermally mediated therapeutic procedures,” Phys. Med. Biol.40(2), 241–252 (1995).
[CrossRef] [PubMed]

Tatara, M. R.

M. R. Tatara, W. Krupski, B. Tymczyna, and I. Luszczewska-Sierakowsky, “Biochemical bone metabolism markers and morphometric, densitometric and biomechanical properties of femur and tibia in female and gonadectomised male Polish Landrace pigs,” J. Pre-Clin. Clin. Res.6(1), 14–19 (2012).

Torres, J. H.

Tymczyna, B.

M. R. Tatara, W. Krupski, B. Tymczyna, and I. Luszczewska-Sierakowsky, “Biochemical bone metabolism markers and morphometric, densitometric and biomechanical properties of femur and tibia in female and gonadectomised male Polish Landrace pigs,” J. Pre-Clin. Clin. Res.6(1), 14–19 (2012).

Warloe, T.

Q. Peng, A. Juzeniene, J. Chen, L.O Svaasand, T. Warloe, K-E Giercksky, and J. Moan, “Lasers in Medicine,” Rep. Prog. Phys.71, 056701 (2008).

Weber, H. P.

M. Forrer, M. Frenz, V. Romano, H. J. Altermatt, H. P. Weber, A. Silenok, M. Istomyn, and V. I. Konov, “Bone-Ablation Mechanism Using CO2 lasers of Different Pulse Duration and Wavelength,” Appl. Phys. B56(2), 104–112 (1993).
[CrossRef]

Welch, A. J.

Wierich, W.

M. M. Ivanenko, S. Fahimi-Weber, T. Mitra, W. Wierich, and P. Hering, “Bone Tissue Ablation with sub- µs Pulses of a Q-switch CO2 Laser: Histological Examination of Thermal Side Effects,” Lasers Med. Sci.17(4), 258–264 (2002).
[CrossRef] [PubMed]

Yu, H.

K.-W. Guan, Y.-Q. Jiang, C.-S. Sun, and H. Yu, “A two-layer model of laser interaction with skin: A photothermal effect analysis,” Opt. Laser Technol.43(3), 425–429 (2011).
[CrossRef]

Appl. Opt. (2)

Appl. Phys. B (2)

M. M. Ivanenko and P. Hering, “Wet bone ablation with mechanically Q-switched high-repetition-rate CO2 laser,” Appl. Phys. B67(3), 395–397 (1998).
[CrossRef]

M. Forrer, M. Frenz, V. Romano, H. J. Altermatt, H. P. Weber, A. Silenok, M. Istomyn, and V. I. Konov, “Bone-Ablation Mechanism Using CO2 lasers of Different Pulse Duration and Wavelength,” Appl. Phys. B56(2), 104–112 (1993).
[CrossRef]

Aust. J. Forensic Sci. (1)

A. Mathur and Y. K. Agrawal, “An overview of methods used for estimation of time since death,” Aust. J. Forensic Sci.43(4), 275–285 (2011).
[CrossRef]

J. Pre-Clin. Clin. Res. (1)

M. R. Tatara, W. Krupski, B. Tymczyna, and I. Luszczewska-Sierakowsky, “Biochemical bone metabolism markers and morphometric, densitometric and biomechanical properties of femur and tibia in female and gonadectomised male Polish Landrace pigs,” J. Pre-Clin. Clin. Res.6(1), 14–19 (2012).

Lasers Med. Sci. (1)

M. M. Ivanenko, S. Fahimi-Weber, T. Mitra, W. Wierich, and P. Hering, “Bone Tissue Ablation with sub- µs Pulses of a Q-switch CO2 Laser: Histological Examination of Thermal Side Effects,” Lasers Med. Sci.17(4), 258–264 (2002).
[CrossRef] [PubMed]

Lasers Surg. Med. (1)

N. M. Fried and D. Fried, “Comparison of Er:YAG and 9.6-µm TE CO2 Lasers for Ablation of Skull Tissue,” Lasers Surg. Med.28(4), 335–343 (2001).
[CrossRef] [PubMed]

Med. Eng. Phys. (1)

J. Lee, Y. Rabin, and O. B. Ozdoganlar, “A new thermal model for bone drilling with applications to orthopaedic surgery,” Med. Eng. Phys.33(10), 1234–1244 (2011).
[CrossRef] [PubMed]

Opt. Eng. (1)

A. Sagi, A. Shitzer, A. Katzir, and S. Akselrod, “Heating of biological tissue by laser irradiation: theoretical model,” Opt. Eng.31(7), 1417–1424 (1992).
[CrossRef]

Opt. Laser Technol. (1)

K.-W. Guan, Y.-Q. Jiang, C.-S. Sun, and H. Yu, “A two-layer model of laser interaction with skin: A photothermal effect analysis,” Opt. Laser Technol.43(3), 425–429 (2011).
[CrossRef]

Phys. Med. Biol. (2)

A. L. McKenzie, “Physics of thermal processes in laser-tissue interaction,” Phys. Med. Biol.35(9), 1175–1210 (1990).
[CrossRef] [PubMed]

B. Anvari, T. E. Milner, B. S. Tanenbaum, S. Kimel, L. O. Svaasand, and J. S. Nelson, “Selective cooling of biological tissues: application for thermally mediated therapeutic procedures,” Phys. Med. Biol.40(2), 241–252 (1995).
[CrossRef] [PubMed]

Rep. Prog. Phys. (1)

Q. Peng, A. Juzeniene, J. Chen, L.O Svaasand, T. Warloe, K-E Giercksky, and J. Moan, “Lasers in Medicine,” Rep. Prog. Phys.71, 056701 (2008).

Other (6)

Y. Yener and S. Kakaç, Heat Conduction, 4th ed. (Taylor & Francis, 2008) pp. 210–212.

M. N. Özişik, Heat Conduction (John Wiley & Sons, 1980) p.276.

Foundation, www.itis.ethz.ch/itis-for-health/tissue-properties/database/density

P. Wallentén, “Heat flows in a Full Scale Room Exposed to Natural Climate,” Lund Institute of Technology, Dep. Of Building Science, ISSN 1103–4467, ISRN LUTADL/TABK −3051-SE (1998).

M. R. Spiegel, “Mathematical Handbook of Formulas and Tables,” McGraw-Hill Inc., New York (1974).

H. S. Carslaw and J. C. Jaeger, Conduction of Heat in Solids, 2nd ed. (Oxford University Press 1959) pp. 305–306.

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Figures (4)

Fig. 1
Fig. 1

Experimental set-up used to heat the domesticated pig bone samples.

Fig. 2
Fig. 2

Bone sample and laser beam cylindrical symmetry a) sample with finite dimensions b) sample modelled as semi-infinite air-bone media.

Fig. 3
Fig. 3

Temperature T of a bone sample as a function of time for steps A, B and C. In step A, the CO2 laser is delivering pulses at a modulation frequency of 5 kHz for a period of 20 seconds. In step B, the CO2 laser is delivering pulses at a modulation frequency of 20 kHz for a period of 120 seconds. In step C, the laser power is turned off and the sample is cooling off.

Fig. 4
Fig. 4

Temperature of bone samples as a function of time measured by the IR thermometer during the cooling off process (Step C). The + symbols are the experimental data points, the dashed line is the prediction from Eq. (10) (Stefan-Boltzmann radiation loss), the black solid line is the prediction from Eq. (7) (Newton’s law of cooling) and the grey line shows the prediction from the heat conduction equation Eq. (19) as formulated by Eqs. (15) to 18. a) Bone heated at moderate temperature near 60°C b) Graph shown in a) within the first 20s of cooling c) Bone heated to temperature near 130°C d) Graph shown in c) within the first 15s of cooling e) Bone heated to a temperature of 185°C f) Graph shown in e) within the first 20s of cooling g) Bone heated to a temperature above 250°C h) Graph shown in g) within the first 20s of cooling.

Tables (1)

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Table 1 Data used in predictions from Eqs. (7), (10) and (19) in Fig. 4. In these theoretical predictions we used ρ = 2500 kg/m3, τ = 12 µm, c = 1300 J/°C, k = 0.32 W/m°K, To = 21°C, σ = 5.67 x 10−8W/m2°K and ε = 1.

Equations (19)

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q conv '' =h(T T o )
q r '' =εσ( T 4 T o 4 ),
q cond. '' =kT,
I(z)= I o exp(αz),
Q=ρVcT,
dT dt +( h ρτc )T=( h ρτc ) T o .
T= T o +( T 1 T o ) e βt ,
dT dt = εσ ρτc ( T 4 T o 4 ).
1 4 T o 3 ln[ T T o T+ T o ] 1 2 T o 3 Arctan( T T o )+C= k 1 t,
k 1 t= 1 4 T o 3 { ln[ (T T o )( T i + T o ) (T+ T o )( T i T o ) ] } 1 2 T o 3 Arctan( T T o )+ 1 2 T o 3 Arctan( T i T o ).
ln[[ (T T o )( T i + T o ) (T+ T o )( T i T o ) ]]ln[ 1 T o T 1+ T o T ]2x 2 3 x 3 +... for x<1 .
q cond. =k dTs dt ,
ρVc A d T s dt k ( T s T o ) τ
d T s dt + 1 β ' T s T o β ' ,
2 T z 2 = 1 K T t
k T z | z=0 =h(T T o ) | z=0
T= T o for z > >τ
T = T i   at t= 0
T( z,t )= T i ( T i T o ) [ erfc( z 2 Kt ) exp( H 2 Kt+Hz )erfc( H Kt + z 2 Kt ) ],

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