Question

5.5 You are a detective investigating a murder, and need to know when the murder occurred. There are two suspects: Howard, who was seen leaving town the previous day, and Mickey, who could have committed the crime only in the last 30 hours. The victim's body (which is of cylindrical geometry) is very long, and has a radius of 15 cm. The average thermal conductivity, density, and heat capacity of a human body are 1.5 x 10- cal/cm sec K, 1.1 g/cc, and 0.83 cal/g K, respectively, and the outside heat transfer coefficient is approx- imated to be 8 W/m² K. The room temperature has been constant at 20°C, and the temperature at the center of the body is measured to be 29°C. How long ago was the crime committed, and who is the culprit?

Answer 1

(5,5) body of long cylindrical length. given: 8 15 cm = 0,15m cal/cm seek conductivity ) = 1.5x 10 kolthermal - 0.627 J/m. Kas 1.1 g/cm3 = 1.18103 kg/m3 density (P) = heat capacity (Cp): 0.83 cal/g k. . 3452.8 II leg.k. room tump (To): 20 c = 293 h. 8 W/m²K Temp of human body : 37°C hat the the body was that Assume at CT: 2 372) a temp of initially Ti : 310 k.

x = Thermal = difficsivity 1 k cop x= 0.627 X.X. m. 3452.8 T x l.1x10 3 m Ing.K t. 1.650x10 m²/s. Biot No (Bi) Find the Bi = kw (No k Bi- 8x(0.15) 0.9569 0.627 contre line temperature dimensionless Now for the a long cylinder i given as

Ti-To Tot centre une temperature of cylinder where To= room temperature Tia initial tump of body. used in C, G, are woefficients get c,, e, at various Bi values we can from tables. have we table From 1. 1902 1.2048 0. 9 x2 al 0.95 1.207 1.2558

determine e, and c, at Bi=0,95 by interpolation ri 1-0.95 : 1.2558-21 1.2558 - 1.20 48 1-0.9 S -1.23 al 123 and 1.2071 X₂ 1-0.95 . 1-0.9 1.2071 - 1.1902 a 1,198, C, : 1.198 and substituting 0 to eqn coming back the values in, we get,

Over 29- 20 : 0.529C, exp(-8,Fo) 37-20 0.5293 1198 exp(- (1.23)* Fo ) -1.5129 Fo 0.441 e nidis en on both Taking ene -1.5129 Fo In 0.441 Fo 15129 -0.817 = Foo 0.54 Cty) T= 29c to reach taken 1. Time -73636.366 tes Fo (ro), 0.54 X10,15). 2.31 1.650 X10-7

Imin Ihr to 7 3 636.36 n 6on 60 min tje 20 hrs the crime Mickey committed