- Email: [email protected]
Simulating irradiation power density on body surface in postmortem cooling
Legal Medicine 6 (2004) 131–140 www.elsevier.com/locate/legalmed
Simulating irradiation power density on body surface in postmortem cooling Gita Mall*, Michael Hubig, Andreas Bu¨ttner, Wolfgang Eisenmenger Institute of Legal Medicine, University of Munich, Frauenlobstrasse 7a, 80337 Munich, Germany Received 23 June 2003; received in revised form 15 December 2003; accepted 24 December 2003
Abstract Irradiation poses a major problem to determining the time since death by temperature-based methods. Neither empirical nor heat-flow postmortem cooling models have so far been able to assess irradiation. Heat-flow models seem overall better suited to calculate irradiation because of their direct relation to the physics of heat transfer. An implementation of irradiation boundary conditions in heat-transfer models requires the knowledge of the irradiation power density on the body surface. The present study develops formulae and implements them in a computer program to simulate the radiation power density on a semicylindrical body surface coming from irradiation by a rectangular radiant heater nearby or from the sun. The formulae are valid for deliberate geometrical arrangements of either body and radiant heater or body and sun. In case of the radiant heater scenario shading functions for the shading of the semi-cylinder by itself and by the rear panel of the radiant heater are developed. In case of the sun scenario only the shading by the semi-cylinder is relevant. In examplary analyses of typical irradiation scenarios the power density coming from a 2000 W radiant heater nearby on the body surface amounted to a maximum of 418 W/m2, the radiation power density originating from sunlight on a clear summer afternoon in middle-Europe amounted to a maximum of 422 W/m2. q 2004 Published by Elsevier Ireland Ltd. Keywords: Irradiation; Postmortem cooling; Time since death
1. Introduction Analysing postmortem cooling is essential for determining the time since death in the early postmortem period. Two completely different approaches can be cited from the forensic literature. Most studies [i.e. 1,5– 7,14] followed the approach of mathematical or empirical modelling, which * Corresponding author. Tel.: þ49-89-5160-5111; fax: þ 49-895160-5144. E-mail address: [email protected] (G. Mall).
consists in describing experimental temperature time curves by analytical formulae. Its advantage is that it implicitly includes all factors influencing postmortem cooling. Its disadvantages are that it is only valid for cooling scenarios similar to the experimental standard conditions and that its parameters which are pure curve fitting parameters are not directly related to the physics of cooling. Only few studies [8 – 12,15] followed the approach of thermodynamical or heat flow modelling, which consists in describing postmortem cooling by the underlying physical laws. Its advantages are that it
1344-6223/$ - see front matter q 2004 Published by Elsevier Ireland Ltd. doi:10.1016/j.legalmed.2003.12.004
132
G. Mall et al. / Legal Medicine 6 (2004) 131–140
inherits the validity of the physical laws and is thus by nature more valid than empirical modelling and that it can in principle be applied to any non-standard cooling scenarios. Its disadvantage is that solving the heat transfer equation can become very difficult for complex cooling conditions. All heat-flow models therefore approximate the shape of the human body by a two-dimensional homogeneous cylinder. Since Sellier [15] and Joseph and Shickele [10] solved the heat transfer equation analytically they are restricted to very simple boundary conditions. The more recent models of the workgroup around Hiraiwa [8,9,11,12] used the finite-difference-method as a numerical procedure instead and were therefore able to assess much more complex boundary conditions especially a variable environmental temperature. A forensic pathologist in practice is often faced with irradiation cooling scenarios e.g. by a radiant heater near the cooling body or by the sun. Irradiation poses major problems to analysing postmortem cooling since neither empirical nor heat-flow models have so far implemented radiation heat gain from external sources. While empirical models will by nature be restricted to special standardized irradiation scenarios, heat-flow models and especially the numerical model of the workgroup around Hiraiwa [8,9,11,12] seem much better suited to include irradiation power in solving the heat transfer equation. The present study develops a method for simulating the radiation power density on a cylindrical body surface caused by irradiation from a radiant heater of deliberate position near the body with a given radiation power and from sunlight of deliberate direction and intensity.
2. Method Calculating the radiation power absorbed by a body exposed to irradiation first of all requires a quantitative description of the position of the body in relation to the radiation field. This is achieved by geometrical modelling of the body surface and in case of nearby located radiation sources of the radiating surfaces as well. Additionally the shading of parts of the absorbing body surface by itself and or the rear panels of radiation sources located nearby have to be quantified. The radiation power absorbed by the body
surface is calculated by so-called viewfactors and by applying Kirchhoff’s law. 2.1. Geometry 2.1.1. Body The shape of the body is approximated by a semicylinder of length L and radius D with the ground area G (Fig. 1A). The centre of the ground area G is the origin of a rectangular coordinate system whose x~ -axis points in the direction of the main cylinder axis. The semi-cylinder’s surface is divided into rectangular segments Hij by Nf þ 1ði ¼ 0; …Nf Þ endpoints on the semi-circular end of the cylinder and Nx þ 1ðj ¼ 0; …; Nx Þ endpoints along the x~ -axis. Thus a regular network on the surface of the semi-cylinder is constructed. One segment Hij has the area: AðHij Þ ¼
pDL Nf Nx
ð1Þ
~ ij is defined pointing For each segment Hij a vector h from the coordinate origin to the centre of the segment. 2.1.2. Radiator The emitting radiator R of deliberate spatial position in relation to the body is modelled as a parallel epiped with the edge lengths L1 and L2 (Fig. 1B). The parallel epiped is anchored in the coordinate system by vector b~ and spanned by the unit vectors v~1 and v~ 2 : A regular mesh of radiation emission points ~rlm is defined on the parallel epiped. In the direction of vector v~1 on the edge of length L1 there are N1 points ðl ¼ 1; …; N1 Þ defined, in the direction of vector v~ 2 on the edge of length L2 there are N2 points ðm ¼ 1; …; N2 Þ: The radiation emission points ~rlm can then be calculated: ~ þ v~1 lL1 þ v~ 2 mL2 ~rlm ¼ b N1 N2
ð2Þ
The number NR of radiation emission points is: NR ¼ N1 N2
ð3Þ
The area of the radiator’s surface Rlm belonging to the radiation emission point rlm in case the vectors v~ 1
G. Mall et al. / Legal Medicine 6 (2004) 131–140
Fig. 1. Explanatory scheme for semi-cylinder model.
133
134
G. Mall et al. / Legal Medicine 6 (2004) 131–140
~ ij with a factor 0 # a # 1 : combination of ~rlm and h
and v ~ 2 are perpendicular is: AðRlm Þ ¼
L1 L2 N1 N2
ð4Þ
~ ij of the The distance dijlm between the centre h segment Hij on the semi-cylinder’s surface and the radiation emission point ~rlm is: ~ ijj 2 ~rlm l dijlm ¼ lh
ð5Þ
2.1.3. Irradiation scenarios The present study calculates the radiation power absorbed by the semi-cylindrical surface for two typical geometrical arrangements: Scenario ‘R’: A radiator R with rectangular radiating surface is positioned besides the semicylinder H: The radiating surface is inclined towards the cylinder or x~ -axis of the coordinate system and oriented parallel towards the z~-axis of the coordinate system. (Fig. 1C) Scenario ‘S’: A point-like radiation source (the sun) is positioned very far—virtually infinitely far—from the semi-cylinder H: This means that the semicylinder is exposed to a homogenous radiation field: The rays of the radiation field can be approximated to run parallel near the body (Fig. 1D). 2.1.4. Shading of the semi-cylinder by itself The determination of the shading of the semicylinder by itself is demonstrated for scenario ‘R’ first and then transferred to scenario ‘S’. The shading function attributes to every segment ~ ij of the segment Hij on the semi-cylinder H centre h and to every radiation emission point ~rlm on the radiator’s surface a shading factor aH ijlm ; which assumes the value 1 in case the segment centre can be directly seen from the radiation emission point, or the value 0 in case the segment centre is hidden from sight to the radiation emission point by other parts of the semi-cylinder surface. If the line drawn between the radiation emission points ~rlm and the segment centre h~ ij crosses the semicylinder’s surface, the shading factor assumes the value 0, else the value 1. Each deliberate point ks on the line between the radiation emission point ~rlm and ~ ij can be described by a convex the segment centre h
~s ¼ ð1 2 aÞ~rlm þ ah~ ij
ð6Þ
Any shading conditions for deliberate spatial positions of ~rlm and h~ ij can be derived using the projection p~ onto the two-dimensional plane spanned by one of the semi-circular ends of the semi-cylinder (see Fig. 1E). The fact that the difference vector of ~rlm and h~ ij crosses the semi-cylinder’s surface is equivalent to the condition that the point ks on the line ~ ij which is nearest between the endpoints of ~rlm and h to the cylinder axis (or x-axis) has a distance from the x-axis smaller than the cylinder radius D: Using the corresponding projections this is: l~pð~sÞ 2 p~ðx 2 axisÞl ¼ l~pð~sÞ 2 0l , D
ð7Þ
~ ð~sÞ to the line To know the factor a of a vector p ~ ij Þ nearest to ~ ðh between the endpoints of p~ ð~rlm Þ to p ~ ð~sÞ has zero it is sufficient to propose that the vector p to be perpendicular to the difference vector of p~ ð~rlm Þ ~ ij ). This proposition can be formulated using ~ (h and p the scalar product: ~ ij Þl ~ ðh ~ ð~rlm Þ 2 p 0 ¼ k~pð~sÞ; p
ð8Þ
Substituting ks by the linear combination ð1 2 aÞ ~ ij and solving the equation for the convex r~lm þ ah factor a leads to:
a¼
~ ij Þl ~ ðh l~ pð~rlm Þl2 2 k~ pð~rlm Þ; p 2 ~ ij Þl ~ ðh l~pð~rlm Þ 2 p
ð9Þ
Thus all steps of the algorithm for determining the ~ ij ) of shading factor aH ijlm for a deliberate pair (~rlm ; h ~ ij radiation emission points ~rlm and segment centres h are derived and can be summarized: ~ ð~rlm Þ and p~ ðh~ ij Þ: This is done by just (1) Calculate p ~ ij : leaving out the x-coordinates of ~rlm and h (2) Calculate the factor
a¼
~ ij Þl ~ ðh l~ pð~rlm Þl2 2 k~ pð~rlm Þ; p : 2 ~ ij Þl ~ ðh l~ pð~rlm Þ 2 p
If a is smaller than or equal to 0 or equal to or larger than 1 ða 0; 1½Þ; then there is no shading ðaH ijlm ¼ 1Þ and the algorithm stops. If a ranges between 0 and 1 ða [0; 1½Þ; then the algorithm continues.
G. Mall et al. / Legal Medicine 6 (2004) 131–140
~ ij Þ and ~ ðh (3) Calculate pðxÞ ¼ ð1 2 aÞ~pð~rlm Þ þ ap see, whether: l~ pð~sÞl , D: If l~ pð~sÞl , D; there is shading and aH ijlm ¼ 0: If l~pð~sÞl . D; there is no shading and aH ijlm ¼ 1: For scenario ‘S’ the shading function is different. There is only one radiation emission point ~r: Its position in the coordinate system is calculated by reversing the unit vector ~e pointing in the direction of the rays of the homogenous radiation field of the sun and extending it by the factor U which has to be much larger than L and D (see Fig. 1F): U @ L; D
ð10Þ
2.2. Physics Thermal radiation is emitted due to an excitation of electrons to energetically higher states in the electron shell of atoms by mechanic impacts among neighbouring atoms. The electrons quickly return to their original energetically lower states and thereby emit energy in form of electromagnetic radiation. The radiation power P of a body in the vacuum according to the law of Stefan and Boltzmann depends on the emissivity 1; the Stefan-Botzmann constant s and the absolute temperature in the fourth power T 4 : P ¼ 1 sT 4
~r ¼ 2U~e
ð11Þ
The shading factor aH ij in case of scenario ‘S’ only depends on the segments Hij since there is only one radiation emission point ~r: 2.1.5. Shading of the semi-cylinder by the radiator A shading by the radiator is only relevant to scenario ‘R’ (Fig. 1G). The plane C which is the plane of the radiating surface of the radiator divides the three-dimensional space into two subspaces, the subspace Cþ in the direction of the radiation emitted by the radiator and the subspace C2 in the direction averted from the radiation. To be able to attribute the segments Hij of the semi-cylinder’s surface to either of the two subspaces Cþ or C2 the two-dimensional ~ plane C is described as the set of all vectors q perpendicular to the normal vector n~ in direction Cþ of the radiator’s surface translated by the fixed ~: vector b ~ l ¼ 0} þ b~ C ¼ {~qlk~q; n
ð12Þ
Now the following algorithm can be used to calculate the value of the shading factor aRij in the segment ~ ij of a segment Hij on the semi-cylinder’s centre h surface: ~ ij 2 b: ~ (1) Calculate h ~ nl . 0; then h ~ ij belongs to Cþ and (2) If kh~ ij 2 b; ~ ij 2 b; ~ aR ¼ 1 since there is no shading. If kh ~ l # 0; then h~ ij belongs to C2 or lies in C and n aRij ¼ 0 since there is shading.
135
ð13Þ
The emissivity is a material property. It is the ratio of radiation emitted by a surface of a certain material PM to radiation emitted by a black body PB (which is a perfect emitter with an emissivity of 1): 1¼
PM PB
0#1#1
ð14Þ ð15Þ
Apart from the emitted radiation power the absorbed radiation power is of interest. According to Kirchhoff’s law the emissivity of a surface at a certain temperature 1ðTÞ is equal to its absorptivity gðTÞ for black body radiation at the same temperature: 1ðTÞ ¼ gðTÞ
ð16Þ
Kirchhoff’s law is commonly used in radiation analysis with acceptable results. In case of solar radiation the surface temperatures as well as the spectral distribution of the emitted wavelengths are so different from the radiation emitted by the skin that Kirchhoff’s law cannot be applied. Instead the solar absorptivity gS of the human skin has to be used [3]. 2.2.1. Radiation heat transfer between two surfaces— Scenario ‘R’ Radiation heat transfer from one surface A1 to another surface A2 strongly depends on their distance and orientation. The influence of their geometrical arrangement is described by the so-called viewfactor
136
G. Mall et al. / Legal Medicine 6 (2004) 131–140
Vf [2]:
infinitely far point (Fig. 1I):
Vf ðA1 ;A2 Þ ¼
1 ð ð cos b1 cos b2 dA1 dA2 AðA1 Þ A1 A2 pr 2
ð17Þ
dA1 and dA2 are infinitesimally small pieces of areas on A1 and A2 ;r is the distance between A1 and A2 ; b1 is the angle between the normal vector on A1 and r; b2 the angle between the normal vector on A1 and r and AðA1 Þ is the area of A1 : The formula can be transferred to scenario ‘R’ (see Fig. 1H): dA1 is Rlm the radiation emission area of point ~rlm on the radiator and dA2 is Hij the radiation absorbing segment on the semi-cylinder. Therefore the integral can be omitted: Vf ðRlm ;Hij Þ ¼
1 cos blm cos bij AðRlm ÞAðHij Þ ð18Þ ~ ij l2 AðRlm Þ pl~rlm 2 h
Since we want to know the radiation power density Qij which is independent of the absorbing surface area we have to divide the radiation power Pij by the area AðHij Þ of the segments Hij : Qij ¼
Pij AðHij Þ
ð19Þ
Since we know the radiation power P emitted by the radiator which is a technical parameter we can calculate the radiation power density Qij at the location of the semi-cylinder segment Hij by: X Vf ðRlm ;Hij ÞaH ijlm aRij P lm ð20Þ Qij ¼ 1 NR AðHij Þ with: P=NR 1 Vf aH ijlm aR ij
radiation power of any radiator segment Rlm emissivity of the skin (0.98) viewfactor self-shading factor of the semi-cylinder shading factor of the radiator.
2.2.2. Radiation heat transfer in a homogenous radiation field—Scenario ‘S’ The viewfactor calculation only depends on the orientation of the segments Hij on the semi-cylinder’s surface since the sun is represented by a virtually
Vf ðHij Þ ¼ cosbij AðHij Þ
ð21Þ
where bij denotes the angle between the normal vector n of the segments Hij and the direction of the rays. Since we know the radiation power density Q of the sun which is derived from the solar constant [4] we can calculate the radiation power density Qij on the semi-cylinder by: Qij ¼
QgS Vf ðHij ÞaHij AðHij Þ
ð22Þ
With: Q gS Vf aH ij
radiation power density of the sun absorptivity of the human skin (0.62) viewfactor self-shading factor of the semi-cylinder
3. Results Parameter values used in the presented simulations: Parameters of the semi-cylinder: length, L ¼ 1:85 m radius, D ¼ 0:35 m emissivity, 1 ¼ 0:98 solar absorbtivity, gS ¼ 0:62 number of points in x-direction, Nx ¼ 100 number of points in f-direction, Nf ¼ 50 Parameters of the radiator: anchoring vector, b ¼ (0.3 m, 0.9 m, 0.2 m) p spanning vector, v1 ¼ (2 1 m, 2 1 m, 0 m)/ 2 spanning vector, v2 ¼ (0 m, 0 m, 1 m) length, L1 ¼ 1 m length, L2 ¼ 0:75 m number of points in L1 N1 ¼ 10 number of points in L2 N2 ¼ 10 power, P ¼ 2000 W Parameters of the sun:
G. Mall et al. / Legal Medicine 6 (2004) 131–140
Actual angle of the sun above the horizon, x ¼ 408 Maximal angle above the horizon in Europe, x0 ¼ 658 Angle of sun radiation and cylinder main axis, c ¼ 458 Maximum power density of sun radiation, Qmax ¼ 970 W/m22 Actual power density of sun radiation, Q¼
Qmax sin x sin x0
For presentation reasons the coordinate system id shifted by a pure translation mapping the coordinate origin to the point ð2L=2; 2D; 0Þ in the coordinates of the original coordinate system. To distinguish the two systems the axes in Fig. 2 are marked by x; y and z instead of x˜, y˜ and z˜. In the Fig. 1A,B and F, the z-axis does not represent the third dimension but the range of the shading functions. In the Fig. 1C,D,G and H the z-axis represents the power density value. 3.1. Scenario ‘R’ Fig. 2A presents the inverted shading function 1 2 aR of the radiator on the semi-cylinder’s surface unrolled in the x – y-plane. Corresponding to the diagonal position of the radiator the part of the cylinder surface on the right side is diagonally shaded. The inverted shading function assumes the value 1 on the vertical axis in this region. The remaining cylinder surface on the left side is not shaded. The inverted shading function assumes the value 0. The edge of the shaded region assumes a cosinus shape as expected caused by the curvature of the semi-cylindrical surface. Fig. 2B demonstrates the inverted shading function 1 2 aH caused by the cylinder itself on the semicylinder’s surface unrolled in the x – y-plane. The radiator emits its radiation in the direction of the positive y-axis. Accordingly the part of the cylinder surface adjacent to the radiator is luminated ð1 2 aH ¼ 0Þ and the part averted from the radiator is shaded ð1 2 aH ¼ 1Þ: The Fig. 2C and D illustrate the power density function on the semi-cylinder surface unrolled in the x – y-plane from two different directions. The power
137
density values in W/m2 can be read from the vertical axis. Fig. 2E gives a colour-coded two dimensional plot of the power density on the unrolled semi-cylinder surface. In the red region of the surface directly adjacent to the radiating surface maximum power density values of 418 W/m2 are reached. 3.2. Scenario ‘S’ Senario ‘S’ simulates conditions that could be observed in the afternoon on a clear day in summer in middle Europe. Fig. 2F presents the inverted self-shading function 1 2 aH on the semi-cylinder’s surface unrolled in the x – y-plane in a view from the negative y-direction, which is from the side of the semi-cylinder not exposed to the sun. Accordingly, the part not exposed to the sun is shaded and the inverted shading function assumes the value 1 on the vertical axis. Fig. 2G and H illustrates the power density function on the semi-cylinder surface unrolled in x – y-direction. The power density values in W/m2 can be read from the vertical axis. In figure 16 the function is viewed from the negative y-axis, which from that side of the semi-cylinder not exposed to the sun. In figure 17 the function is viewed from the negative x-direction which is from one end of the semi-cylinder with sun radiation coming from the left. Fig. 2I gives a colour-coded two dimensional plot of the power density on the unrolled semi-cylinder’s surface. The upper part of the plot corresponds to the side of the semi-cylinder surface exposed to the sun, the lower part of the plot to the side of the semicylinder not exposed to the sun. Maximum power density values are reached near the semi-cylinders vertex in the red region with 421.7 W/m2.
4. Discussion As already mentioned in detail in the introduction the assessment of irradiation is a major problem in the determination of the time since death by methods based on postmortem cooling. Neither empirical nor heat-flow models of postmortem cooling have so far been able to simulate irradiation. Heat-flow models are in general better suited for irradiation analyses
138
G. Mall et al. / Legal Medicine 6 (2004) 131–140
Fig. 2. Shading functions and radiation power in the scenarios ‘R’ and ‘S’.
because of their direct relation to the physics of heat transfer. Implementation of irradiation effects in heat-flow models requires the knowledge of the irradiation
power density on the body surface. The present paper develops formulae for calculating the required irradiation power density and implements them in computer simulations performed in the graphic
G. Mall et al. / Legal Medicine 6 (2004) 131–140
language IDL. Since this quantity strongly depends on the geometrical arrangement of the body and of the irradiating source, special attention had to be paid to the geometrical modelling of the body and of the radiating sources. Going back to an approximation of the shape of the human body which has already proven well suited for radiation and semi-empirical convection analyses [13], the human body was modelled by a semi-cylinder of definite length (here: 1.85 m) and diameter (here: 0.7 m). The surface of the semicylinder was subdivided into many (here: 5000) rectangular segments. The simulation was restricted to two irradiation sources relevant in medico-legal practice: a radiant heater and the sun. The radiant heater was modelled as a rectangular surface (here: 1 m £ 0.75 m) with a network of (here: 100) points emitting radiation. In the examplary analysis the radiator was positioned 0.2 m above the ground and 458 inclined towards the cylinder main axis. The region of the radiant heater nearest to the semicylinder almost touched it. The sun was modelled as a radiating point virtually infinitely far from the body resulting in a homogeneous radiation field with parallel rays, the direction of the rays depending on the position of the sun in the sky. In the examplary analysis the sun was positioned 408 above the horizon and 458 inclined towards the cylinder axis as in the afternoon on a clear summer day in middle Europe. The irradiation power density strongly depends on the geometrical position of the emitting surface on the radiant heater or the direction of the sun’s rays towards the absorbing surface segments on the semi-cylinder. The ratio of radiation power emitted to radiation power absorbed was calculated by viewfactors [2]. The viewfactor calculation is aggravated by the fact that parts of the semi-cylindrical body surface are shaded from the radiation by the cylinder itself or in case of the radiant heater by the radiant heater aswell. We were able to develop a general shading function for the self-shading of the body and for the shading of the radiant heater valid for deliberate geometrical arrangements of body and radiant heater. In case of irradiation by the sun only the self shading function of the body had to be used. The presented formulae and computer simulations provide the radiation power density on
139
a semi-cylindrical surface coming from nearby (radiant heater) or far (sun) irradiating sources. They can be applied to deliberate geometrical arrangements of the semi-cylinder and the radiant heater or of the semi-cylinder and the position of the sun in the sky. The radiation power density is an essential boundary condition in heat-flow models of postmortem cooling. The formulae and the model presented could be applied to the model of Hiraiwa and his workgroup [8,9,11,12] or directly to a semi-cylindrical threedimensional model. They are of general use for estimating the effect of sun irradiation. Even the attenuation of sunlight by glass windows can be further calculated based on the technical parameters of the window. In sum the formulae developed and implemented in a computer program for the first time enable direct simulation of irradiation in heat-flow modelling.
References [1] Al-Alousi L. A study of the shape of the post-mortem cooling curve in 117 forensic cases. Forensic Sci Int 2002;125: 237 –44. [2] Baer HD, Stephan K. Wa¨rme- und Stoffu¨bertragung, 2nd ed. Berlin: Springer; 1996. p. 586–7. [3] Cengel YA. Heat transfer—a practical approach. Boston: McGraw-Hill; 1997. p. 513 –8. [4] Gerthsen C, Vogel H. Physik. Berlin: Springer; 1993. p. 555. [5] Green MA, Wright JC. Postmortem interval estimation from body temperature data only. Forensic Sci Int 1985;28: 35– 46. [6] Henßge C. Die Pra¨zision von Todeszeitscha¨tzungen durch die mathematische Beschreibung der rektalen Leichenabku¨hlung. Z Rechtsmed 1979;83:49–67. [7] Henßge C, Madea B. Methoden zur Bestimmung der Todeszeit an Leichen. Lu¨beck: Schmidt-Ro¨mhild; 1988. p. 133–201. [8] Hiraiwa K, Kudo T, Kuroda F, Ohno Y, Sebetan IM, Oshida S. Estimation of postmortem interval from rectal temperature by use of computer—relationship between rectal and skin cooling curves. Med Sci Law 1981;21:4–9. [9] Hiraiwa K, Ohno Y, Kuroda F, Sebetan IM, Oshida S. Estimation of postmortem interval from rectal temperature by use of computer. Med Sci Law 1980;20:115 –25. [10] Joseph AEA, Schickele E. A general method for assessing factors controlling postmortem cooling. J Forensic Sci 1970; 15(3):364–91. [11] Kuroda F, Hiraiwa K, Oshida S, Akaishi S. Estimation of postmortem interval from rectal temperature by use of
140
G. Mall et al. / Legal Medicine 6 (2004) 131–140
computer(III)—thermal conductivity of the skin. Med Sci Law 1982;22:285–9. [12] Kuroda F, Hiraiwa K, Oshida S, Akaishi S. Estimation of postmortem interval from rectal temperature (IV)—thermal diffusivity of rabbit. Med Sci Law 1983;23:125–30. [13] Mall G, Hubig M, Beier G, Bu¨ttner A, Eisenmenger W. Supravital energy production in the early postmortemn
phase—estimate based on heat loss due to radiation and natural convection. Legal Med 2002;4:71– 8. [14] Marshall TK, Hoare FE. Estimating the time of death. J Forensic Sci 1962;7:56–81. see also pages 189–210, 211–221. [15] Sellier K. Determination of the time of death by extrapolation of the temperature decrease curve. Acta Med Leg Soc 1958; 11:179–302.
Simulating irradiation power density on body surface in postmortem cooling Gita Mall*, Michael Hubig, Andreas Bu¨ttner, Wolfgang Eisenmenger Institute of Legal Medicine, University of Munich, Frauenlobstrasse 7a, 80337 Munich, Germany Received 23 June 2003; received in revised form 15 December 2003; accepted 24 December 2003
Abstract Irradiation poses a major problem to determining the time since death by temperature-based methods. Neither empirical nor heat-flow postmortem cooling models have so far been able to assess irradiation. Heat-flow models seem overall better suited to calculate irradiation because of their direct relation to the physics of heat transfer. An implementation of irradiation boundary conditions in heat-transfer models requires the knowledge of the irradiation power density on the body surface. The present study develops formulae and implements them in a computer program to simulate the radiation power density on a semicylindrical body surface coming from irradiation by a rectangular radiant heater nearby or from the sun. The formulae are valid for deliberate geometrical arrangements of either body and radiant heater or body and sun. In case of the radiant heater scenario shading functions for the shading of the semi-cylinder by itself and by the rear panel of the radiant heater are developed. In case of the sun scenario only the shading by the semi-cylinder is relevant. In examplary analyses of typical irradiation scenarios the power density coming from a 2000 W radiant heater nearby on the body surface amounted to a maximum of 418 W/m2, the radiation power density originating from sunlight on a clear summer afternoon in middle-Europe amounted to a maximum of 422 W/m2. q 2004 Published by Elsevier Ireland Ltd. Keywords: Irradiation; Postmortem cooling; Time since death
1. Introduction Analysing postmortem cooling is essential for determining the time since death in the early postmortem period. Two completely different approaches can be cited from the forensic literature. Most studies [i.e. 1,5– 7,14] followed the approach of mathematical or empirical modelling, which * Corresponding author. Tel.: þ49-89-5160-5111; fax: þ 49-895160-5144. E-mail address: [email protected] (G. Mall).
consists in describing experimental temperature time curves by analytical formulae. Its advantage is that it implicitly includes all factors influencing postmortem cooling. Its disadvantages are that it is only valid for cooling scenarios similar to the experimental standard conditions and that its parameters which are pure curve fitting parameters are not directly related to the physics of cooling. Only few studies [8 – 12,15] followed the approach of thermodynamical or heat flow modelling, which consists in describing postmortem cooling by the underlying physical laws. Its advantages are that it
1344-6223/$ - see front matter q 2004 Published by Elsevier Ireland Ltd. doi:10.1016/j.legalmed.2003.12.004
132
G. Mall et al. / Legal Medicine 6 (2004) 131–140
inherits the validity of the physical laws and is thus by nature more valid than empirical modelling and that it can in principle be applied to any non-standard cooling scenarios. Its disadvantage is that solving the heat transfer equation can become very difficult for complex cooling conditions. All heat-flow models therefore approximate the shape of the human body by a two-dimensional homogeneous cylinder. Since Sellier [15] and Joseph and Shickele [10] solved the heat transfer equation analytically they are restricted to very simple boundary conditions. The more recent models of the workgroup around Hiraiwa [8,9,11,12] used the finite-difference-method as a numerical procedure instead and were therefore able to assess much more complex boundary conditions especially a variable environmental temperature. A forensic pathologist in practice is often faced with irradiation cooling scenarios e.g. by a radiant heater near the cooling body or by the sun. Irradiation poses major problems to analysing postmortem cooling since neither empirical nor heat-flow models have so far implemented radiation heat gain from external sources. While empirical models will by nature be restricted to special standardized irradiation scenarios, heat-flow models and especially the numerical model of the workgroup around Hiraiwa [8,9,11,12] seem much better suited to include irradiation power in solving the heat transfer equation. The present study develops a method for simulating the radiation power density on a cylindrical body surface caused by irradiation from a radiant heater of deliberate position near the body with a given radiation power and from sunlight of deliberate direction and intensity.
2. Method Calculating the radiation power absorbed by a body exposed to irradiation first of all requires a quantitative description of the position of the body in relation to the radiation field. This is achieved by geometrical modelling of the body surface and in case of nearby located radiation sources of the radiating surfaces as well. Additionally the shading of parts of the absorbing body surface by itself and or the rear panels of radiation sources located nearby have to be quantified. The radiation power absorbed by the body
surface is calculated by so-called viewfactors and by applying Kirchhoff’s law. 2.1. Geometry 2.1.1. Body The shape of the body is approximated by a semicylinder of length L and radius D with the ground area G (Fig. 1A). The centre of the ground area G is the origin of a rectangular coordinate system whose x~ -axis points in the direction of the main cylinder axis. The semi-cylinder’s surface is divided into rectangular segments Hij by Nf þ 1ði ¼ 0; …Nf Þ endpoints on the semi-circular end of the cylinder and Nx þ 1ðj ¼ 0; …; Nx Þ endpoints along the x~ -axis. Thus a regular network on the surface of the semi-cylinder is constructed. One segment Hij has the area: AðHij Þ ¼
pDL Nf Nx
ð1Þ
~ ij is defined pointing For each segment Hij a vector h from the coordinate origin to the centre of the segment. 2.1.2. Radiator The emitting radiator R of deliberate spatial position in relation to the body is modelled as a parallel epiped with the edge lengths L1 and L2 (Fig. 1B). The parallel epiped is anchored in the coordinate system by vector b~ and spanned by the unit vectors v~1 and v~ 2 : A regular mesh of radiation emission points ~rlm is defined on the parallel epiped. In the direction of vector v~1 on the edge of length L1 there are N1 points ðl ¼ 1; …; N1 Þ defined, in the direction of vector v~ 2 on the edge of length L2 there are N2 points ðm ¼ 1; …; N2 Þ: The radiation emission points ~rlm can then be calculated: ~ þ v~1 lL1 þ v~ 2 mL2 ~rlm ¼ b N1 N2
ð2Þ
The number NR of radiation emission points is: NR ¼ N1 N2
ð3Þ
The area of the radiator’s surface Rlm belonging to the radiation emission point rlm in case the vectors v~ 1
G. Mall et al. / Legal Medicine 6 (2004) 131–140
Fig. 1. Explanatory scheme for semi-cylinder model.
133
134
G. Mall et al. / Legal Medicine 6 (2004) 131–140
~ ij with a factor 0 # a # 1 : combination of ~rlm and h
and v ~ 2 are perpendicular is: AðRlm Þ ¼
L1 L2 N1 N2
ð4Þ
~ ij of the The distance dijlm between the centre h segment Hij on the semi-cylinder’s surface and the radiation emission point ~rlm is: ~ ijj 2 ~rlm l dijlm ¼ lh
ð5Þ
2.1.3. Irradiation scenarios The present study calculates the radiation power absorbed by the semi-cylindrical surface for two typical geometrical arrangements: Scenario ‘R’: A radiator R with rectangular radiating surface is positioned besides the semicylinder H: The radiating surface is inclined towards the cylinder or x~ -axis of the coordinate system and oriented parallel towards the z~-axis of the coordinate system. (Fig. 1C) Scenario ‘S’: A point-like radiation source (the sun) is positioned very far—virtually infinitely far—from the semi-cylinder H: This means that the semicylinder is exposed to a homogenous radiation field: The rays of the radiation field can be approximated to run parallel near the body (Fig. 1D). 2.1.4. Shading of the semi-cylinder by itself The determination of the shading of the semicylinder by itself is demonstrated for scenario ‘R’ first and then transferred to scenario ‘S’. The shading function attributes to every segment ~ ij of the segment Hij on the semi-cylinder H centre h and to every radiation emission point ~rlm on the radiator’s surface a shading factor aH ijlm ; which assumes the value 1 in case the segment centre can be directly seen from the radiation emission point, or the value 0 in case the segment centre is hidden from sight to the radiation emission point by other parts of the semi-cylinder surface. If the line drawn between the radiation emission points ~rlm and the segment centre h~ ij crosses the semicylinder’s surface, the shading factor assumes the value 0, else the value 1. Each deliberate point ks on the line between the radiation emission point ~rlm and ~ ij can be described by a convex the segment centre h
~s ¼ ð1 2 aÞ~rlm þ ah~ ij
ð6Þ
Any shading conditions for deliberate spatial positions of ~rlm and h~ ij can be derived using the projection p~ onto the two-dimensional plane spanned by one of the semi-circular ends of the semi-cylinder (see Fig. 1E). The fact that the difference vector of ~rlm and h~ ij crosses the semi-cylinder’s surface is equivalent to the condition that the point ks on the line ~ ij which is nearest between the endpoints of ~rlm and h to the cylinder axis (or x-axis) has a distance from the x-axis smaller than the cylinder radius D: Using the corresponding projections this is: l~pð~sÞ 2 p~ðx 2 axisÞl ¼ l~pð~sÞ 2 0l , D
ð7Þ
~ ð~sÞ to the line To know the factor a of a vector p ~ ij Þ nearest to ~ ðh between the endpoints of p~ ð~rlm Þ to p ~ ð~sÞ has zero it is sufficient to propose that the vector p to be perpendicular to the difference vector of p~ ð~rlm Þ ~ ij ). This proposition can be formulated using ~ (h and p the scalar product: ~ ij Þl ~ ðh ~ ð~rlm Þ 2 p 0 ¼ k~pð~sÞ; p
ð8Þ
Substituting ks by the linear combination ð1 2 aÞ ~ ij and solving the equation for the convex r~lm þ ah factor a leads to:
a¼
~ ij Þl ~ ðh l~ pð~rlm Þl2 2 k~ pð~rlm Þ; p 2 ~ ij Þl ~ ðh l~pð~rlm Þ 2 p
ð9Þ
Thus all steps of the algorithm for determining the ~ ij ) of shading factor aH ijlm for a deliberate pair (~rlm ; h ~ ij radiation emission points ~rlm and segment centres h are derived and can be summarized: ~ ð~rlm Þ and p~ ðh~ ij Þ: This is done by just (1) Calculate p ~ ij : leaving out the x-coordinates of ~rlm and h (2) Calculate the factor
a¼
~ ij Þl ~ ðh l~ pð~rlm Þl2 2 k~ pð~rlm Þ; p : 2 ~ ij Þl ~ ðh l~ pð~rlm Þ 2 p
If a is smaller than or equal to 0 or equal to or larger than 1 ða 0; 1½Þ; then there is no shading ðaH ijlm ¼ 1Þ and the algorithm stops. If a ranges between 0 and 1 ða [0; 1½Þ; then the algorithm continues.
G. Mall et al. / Legal Medicine 6 (2004) 131–140
~ ij Þ and ~ ðh (3) Calculate pðxÞ ¼ ð1 2 aÞ~pð~rlm Þ þ ap see, whether: l~ pð~sÞl , D: If l~ pð~sÞl , D; there is shading and aH ijlm ¼ 0: If l~pð~sÞl . D; there is no shading and aH ijlm ¼ 1: For scenario ‘S’ the shading function is different. There is only one radiation emission point ~r: Its position in the coordinate system is calculated by reversing the unit vector ~e pointing in the direction of the rays of the homogenous radiation field of the sun and extending it by the factor U which has to be much larger than L and D (see Fig. 1F): U @ L; D
ð10Þ
2.2. Physics Thermal radiation is emitted due to an excitation of electrons to energetically higher states in the electron shell of atoms by mechanic impacts among neighbouring atoms. The electrons quickly return to their original energetically lower states and thereby emit energy in form of electromagnetic radiation. The radiation power P of a body in the vacuum according to the law of Stefan and Boltzmann depends on the emissivity 1; the Stefan-Botzmann constant s and the absolute temperature in the fourth power T 4 : P ¼ 1 sT 4
~r ¼ 2U~e
ð11Þ
The shading factor aH ij in case of scenario ‘S’ only depends on the segments Hij since there is only one radiation emission point ~r: 2.1.5. Shading of the semi-cylinder by the radiator A shading by the radiator is only relevant to scenario ‘R’ (Fig. 1G). The plane C which is the plane of the radiating surface of the radiator divides the three-dimensional space into two subspaces, the subspace Cþ in the direction of the radiation emitted by the radiator and the subspace C2 in the direction averted from the radiation. To be able to attribute the segments Hij of the semi-cylinder’s surface to either of the two subspaces Cþ or C2 the two-dimensional ~ plane C is described as the set of all vectors q perpendicular to the normal vector n~ in direction Cþ of the radiator’s surface translated by the fixed ~: vector b ~ l ¼ 0} þ b~ C ¼ {~qlk~q; n
ð12Þ
Now the following algorithm can be used to calculate the value of the shading factor aRij in the segment ~ ij of a segment Hij on the semi-cylinder’s centre h surface: ~ ij 2 b: ~ (1) Calculate h ~ nl . 0; then h ~ ij belongs to Cþ and (2) If kh~ ij 2 b; ~ ij 2 b; ~ aR ¼ 1 since there is no shading. If kh ~ l # 0; then h~ ij belongs to C2 or lies in C and n aRij ¼ 0 since there is shading.
135
ð13Þ
The emissivity is a material property. It is the ratio of radiation emitted by a surface of a certain material PM to radiation emitted by a black body PB (which is a perfect emitter with an emissivity of 1): 1¼
PM PB
0#1#1
ð14Þ ð15Þ
Apart from the emitted radiation power the absorbed radiation power is of interest. According to Kirchhoff’s law the emissivity of a surface at a certain temperature 1ðTÞ is equal to its absorptivity gðTÞ for black body radiation at the same temperature: 1ðTÞ ¼ gðTÞ
ð16Þ
Kirchhoff’s law is commonly used in radiation analysis with acceptable results. In case of solar radiation the surface temperatures as well as the spectral distribution of the emitted wavelengths are so different from the radiation emitted by the skin that Kirchhoff’s law cannot be applied. Instead the solar absorptivity gS of the human skin has to be used [3]. 2.2.1. Radiation heat transfer between two surfaces— Scenario ‘R’ Radiation heat transfer from one surface A1 to another surface A2 strongly depends on their distance and orientation. The influence of their geometrical arrangement is described by the so-called viewfactor
136
G. Mall et al. / Legal Medicine 6 (2004) 131–140
Vf [2]:
infinitely far point (Fig. 1I):
Vf ðA1 ;A2 Þ ¼
1 ð ð cos b1 cos b2 dA1 dA2 AðA1 Þ A1 A2 pr 2
ð17Þ
dA1 and dA2 are infinitesimally small pieces of areas on A1 and A2 ;r is the distance between A1 and A2 ; b1 is the angle between the normal vector on A1 and r; b2 the angle between the normal vector on A1 and r and AðA1 Þ is the area of A1 : The formula can be transferred to scenario ‘R’ (see Fig. 1H): dA1 is Rlm the radiation emission area of point ~rlm on the radiator and dA2 is Hij the radiation absorbing segment on the semi-cylinder. Therefore the integral can be omitted: Vf ðRlm ;Hij Þ ¼
1 cos blm cos bij AðRlm ÞAðHij Þ ð18Þ ~ ij l2 AðRlm Þ pl~rlm 2 h
Since we want to know the radiation power density Qij which is independent of the absorbing surface area we have to divide the radiation power Pij by the area AðHij Þ of the segments Hij : Qij ¼
Pij AðHij Þ
ð19Þ
Since we know the radiation power P emitted by the radiator which is a technical parameter we can calculate the radiation power density Qij at the location of the semi-cylinder segment Hij by: X Vf ðRlm ;Hij ÞaH ijlm aRij P lm ð20Þ Qij ¼ 1 NR AðHij Þ with: P=NR 1 Vf aH ijlm aR ij
radiation power of any radiator segment Rlm emissivity of the skin (0.98) viewfactor self-shading factor of the semi-cylinder shading factor of the radiator.
2.2.2. Radiation heat transfer in a homogenous radiation field—Scenario ‘S’ The viewfactor calculation only depends on the orientation of the segments Hij on the semi-cylinder’s surface since the sun is represented by a virtually
Vf ðHij Þ ¼ cosbij AðHij Þ
ð21Þ
where bij denotes the angle between the normal vector n of the segments Hij and the direction of the rays. Since we know the radiation power density Q of the sun which is derived from the solar constant [4] we can calculate the radiation power density Qij on the semi-cylinder by: Qij ¼
QgS Vf ðHij ÞaHij AðHij Þ
ð22Þ
With: Q gS Vf aH ij
radiation power density of the sun absorptivity of the human skin (0.62) viewfactor self-shading factor of the semi-cylinder
3. Results Parameter values used in the presented simulations: Parameters of the semi-cylinder: length, L ¼ 1:85 m radius, D ¼ 0:35 m emissivity, 1 ¼ 0:98 solar absorbtivity, gS ¼ 0:62 number of points in x-direction, Nx ¼ 100 number of points in f-direction, Nf ¼ 50 Parameters of the radiator: anchoring vector, b ¼ (0.3 m, 0.9 m, 0.2 m) p spanning vector, v1 ¼ (2 1 m, 2 1 m, 0 m)/ 2 spanning vector, v2 ¼ (0 m, 0 m, 1 m) length, L1 ¼ 1 m length, L2 ¼ 0:75 m number of points in L1 N1 ¼ 10 number of points in L2 N2 ¼ 10 power, P ¼ 2000 W Parameters of the sun:
G. Mall et al. / Legal Medicine 6 (2004) 131–140
Actual angle of the sun above the horizon, x ¼ 408 Maximal angle above the horizon in Europe, x0 ¼ 658 Angle of sun radiation and cylinder main axis, c ¼ 458 Maximum power density of sun radiation, Qmax ¼ 970 W/m22 Actual power density of sun radiation, Q¼
Qmax sin x sin x0
For presentation reasons the coordinate system id shifted by a pure translation mapping the coordinate origin to the point ð2L=2; 2D; 0Þ in the coordinates of the original coordinate system. To distinguish the two systems the axes in Fig. 2 are marked by x; y and z instead of x˜, y˜ and z˜. In the Fig. 1A,B and F, the z-axis does not represent the third dimension but the range of the shading functions. In the Fig. 1C,D,G and H the z-axis represents the power density value. 3.1. Scenario ‘R’ Fig. 2A presents the inverted shading function 1 2 aR of the radiator on the semi-cylinder’s surface unrolled in the x – y-plane. Corresponding to the diagonal position of the radiator the part of the cylinder surface on the right side is diagonally shaded. The inverted shading function assumes the value 1 on the vertical axis in this region. The remaining cylinder surface on the left side is not shaded. The inverted shading function assumes the value 0. The edge of the shaded region assumes a cosinus shape as expected caused by the curvature of the semi-cylindrical surface. Fig. 2B demonstrates the inverted shading function 1 2 aH caused by the cylinder itself on the semicylinder’s surface unrolled in the x – y-plane. The radiator emits its radiation in the direction of the positive y-axis. Accordingly the part of the cylinder surface adjacent to the radiator is luminated ð1 2 aH ¼ 0Þ and the part averted from the radiator is shaded ð1 2 aH ¼ 1Þ: The Fig. 2C and D illustrate the power density function on the semi-cylinder surface unrolled in the x – y-plane from two different directions. The power
137
density values in W/m2 can be read from the vertical axis. Fig. 2E gives a colour-coded two dimensional plot of the power density on the unrolled semi-cylinder surface. In the red region of the surface directly adjacent to the radiating surface maximum power density values of 418 W/m2 are reached. 3.2. Scenario ‘S’ Senario ‘S’ simulates conditions that could be observed in the afternoon on a clear day in summer in middle Europe. Fig. 2F presents the inverted self-shading function 1 2 aH on the semi-cylinder’s surface unrolled in the x – y-plane in a view from the negative y-direction, which is from the side of the semi-cylinder not exposed to the sun. Accordingly, the part not exposed to the sun is shaded and the inverted shading function assumes the value 1 on the vertical axis. Fig. 2G and H illustrates the power density function on the semi-cylinder surface unrolled in x – y-direction. The power density values in W/m2 can be read from the vertical axis. In figure 16 the function is viewed from the negative y-axis, which from that side of the semi-cylinder not exposed to the sun. In figure 17 the function is viewed from the negative x-direction which is from one end of the semi-cylinder with sun radiation coming from the left. Fig. 2I gives a colour-coded two dimensional plot of the power density on the unrolled semi-cylinder’s surface. The upper part of the plot corresponds to the side of the semi-cylinder surface exposed to the sun, the lower part of the plot to the side of the semicylinder not exposed to the sun. Maximum power density values are reached near the semi-cylinders vertex in the red region with 421.7 W/m2.
4. Discussion As already mentioned in detail in the introduction the assessment of irradiation is a major problem in the determination of the time since death by methods based on postmortem cooling. Neither empirical nor heat-flow models of postmortem cooling have so far been able to simulate irradiation. Heat-flow models are in general better suited for irradiation analyses
138
G. Mall et al. / Legal Medicine 6 (2004) 131–140
Fig. 2. Shading functions and radiation power in the scenarios ‘R’ and ‘S’.
because of their direct relation to the physics of heat transfer. Implementation of irradiation effects in heat-flow models requires the knowledge of the irradiation
power density on the body surface. The present paper develops formulae for calculating the required irradiation power density and implements them in computer simulations performed in the graphic
G. Mall et al. / Legal Medicine 6 (2004) 131–140
language IDL. Since this quantity strongly depends on the geometrical arrangement of the body and of the irradiating source, special attention had to be paid to the geometrical modelling of the body and of the radiating sources. Going back to an approximation of the shape of the human body which has already proven well suited for radiation and semi-empirical convection analyses [13], the human body was modelled by a semi-cylinder of definite length (here: 1.85 m) and diameter (here: 0.7 m). The surface of the semicylinder was subdivided into many (here: 5000) rectangular segments. The simulation was restricted to two irradiation sources relevant in medico-legal practice: a radiant heater and the sun. The radiant heater was modelled as a rectangular surface (here: 1 m £ 0.75 m) with a network of (here: 100) points emitting radiation. In the examplary analysis the radiator was positioned 0.2 m above the ground and 458 inclined towards the cylinder main axis. The region of the radiant heater nearest to the semicylinder almost touched it. The sun was modelled as a radiating point virtually infinitely far from the body resulting in a homogeneous radiation field with parallel rays, the direction of the rays depending on the position of the sun in the sky. In the examplary analysis the sun was positioned 408 above the horizon and 458 inclined towards the cylinder axis as in the afternoon on a clear summer day in middle Europe. The irradiation power density strongly depends on the geometrical position of the emitting surface on the radiant heater or the direction of the sun’s rays towards the absorbing surface segments on the semi-cylinder. The ratio of radiation power emitted to radiation power absorbed was calculated by viewfactors [2]. The viewfactor calculation is aggravated by the fact that parts of the semi-cylindrical body surface are shaded from the radiation by the cylinder itself or in case of the radiant heater by the radiant heater aswell. We were able to develop a general shading function for the self-shading of the body and for the shading of the radiant heater valid for deliberate geometrical arrangements of body and radiant heater. In case of irradiation by the sun only the self shading function of the body had to be used. The presented formulae and computer simulations provide the radiation power density on
139
a semi-cylindrical surface coming from nearby (radiant heater) or far (sun) irradiating sources. They can be applied to deliberate geometrical arrangements of the semi-cylinder and the radiant heater or of the semi-cylinder and the position of the sun in the sky. The radiation power density is an essential boundary condition in heat-flow models of postmortem cooling. The formulae and the model presented could be applied to the model of Hiraiwa and his workgroup [8,9,11,12] or directly to a semi-cylindrical threedimensional model. They are of general use for estimating the effect of sun irradiation. Even the attenuation of sunlight by glass windows can be further calculated based on the technical parameters of the window. In sum the formulae developed and implemented in a computer program for the first time enable direct simulation of irradiation in heat-flow modelling.
References [1] Al-Alousi L. A study of the shape of the post-mortem cooling curve in 117 forensic cases. Forensic Sci Int 2002;125: 237 –44. [2] Baer HD, Stephan K. Wa¨rme- und Stoffu¨bertragung, 2nd ed. Berlin: Springer; 1996. p. 586–7. [3] Cengel YA. Heat transfer—a practical approach. Boston: McGraw-Hill; 1997. p. 513 –8. [4] Gerthsen C, Vogel H. Physik. Berlin: Springer; 1993. p. 555. [5] Green MA, Wright JC. Postmortem interval estimation from body temperature data only. Forensic Sci Int 1985;28: 35– 46. [6] Henßge C. Die Pra¨zision von Todeszeitscha¨tzungen durch die mathematische Beschreibung der rektalen Leichenabku¨hlung. Z Rechtsmed 1979;83:49–67. [7] Henßge C, Madea B. Methoden zur Bestimmung der Todeszeit an Leichen. Lu¨beck: Schmidt-Ro¨mhild; 1988. p. 133–201. [8] Hiraiwa K, Kudo T, Kuroda F, Ohno Y, Sebetan IM, Oshida S. Estimation of postmortem interval from rectal temperature by use of computer—relationship between rectal and skin cooling curves. Med Sci Law 1981;21:4–9. [9] Hiraiwa K, Ohno Y, Kuroda F, Sebetan IM, Oshida S. Estimation of postmortem interval from rectal temperature by use of computer. Med Sci Law 1980;20:115 –25. [10] Joseph AEA, Schickele E. A general method for assessing factors controlling postmortem cooling. J Forensic Sci 1970; 15(3):364–91. [11] Kuroda F, Hiraiwa K, Oshida S, Akaishi S. Estimation of postmortem interval from rectal temperature by use of
140
G. Mall et al. / Legal Medicine 6 (2004) 131–140
computer(III)—thermal conductivity of the skin. Med Sci Law 1982;22:285–9. [12] Kuroda F, Hiraiwa K, Oshida S, Akaishi S. Estimation of postmortem interval from rectal temperature (IV)—thermal diffusivity of rabbit. Med Sci Law 1983;23:125–30. [13] Mall G, Hubig M, Beier G, Bu¨ttner A, Eisenmenger W. Supravital energy production in the early postmortemn
phase—estimate based on heat loss due to radiation and natural convection. Legal Med 2002;4:71– 8. [14] Marshall TK, Hoare FE. Estimating the time of death. J Forensic Sci 1962;7:56–81. see also pages 189–210, 211–221. [15] Sellier K. Determination of the time of death by extrapolation of the temperature decrease curve. Acta Med Leg Soc 1958; 11:179–302.