Question

A man is found dead in a room at \(12^{\circ} \mathrm{C}\). The surface temperature on his waist is measured to be \(23^{\circ} \mathrm{C}\), and the heat transfer coefficient is estimated to be $9 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\(. Modeling the body as a \)28-\mathrm{cm}$ diameter, \(1.80\)-m-long cylinder, estimate how long it has been since he died. Take the properties of the body to be $k=0.62 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}\( and \)\alpha=0.15 \times 10^{-6} \mathrm{~m}^{2} / \mathrm{s}$, and assume the initial temperature of the body to be \(36^{\circ} \mathrm{C}\). Solve this problem using the analytical one-term approximation method.

Short answer

Answer: It has been approximately 6.45 hours since the man's death.

Step-by-step solution

Calculate the Biot number (Bi)

The Biot number (Bi) is a dimensionless number that represents the ratio of heat transfer resistance within the body to the heat transfer resistance between the body's surface and surrounding environment. Formula for Biot number (Bi): \(Bi = hL_c/k\) Where: \(h = 9 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\) (Heat transfer coefficient) \(L_c = (V/A_s) = (28\,\mathrm{cm})/2 = 14\,\mathrm{cm} = 0.14\) m (Characteristic length - the ratio of volume to surface area) \(k = 0.62 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}\) (Thermal conductivity of the body) Computing the Biot number: \(Bi = \frac{9 \times 0.14}{0.62} = 2.03\)

Calculate the time constant (τ)

Time constant (τ) is a parameter that characterizes the time response of a system to various stimuli. For this problem, it is given by: Formula for time constant (τ): \(τ = L_c^2/α\) Where: \(L_c = 0.14\) m (Characteristic length) \(α = 0.15 \times 10^{-6} \mathrm{~m}^{2} / \mathrm{s}\) (Thermal diffusivity of the body) Computing the time constant: \(τ = \frac{(0.14)^2}{0.15 \times 10^{-6}} = 130933.33\, \mathrm{s}\)

Calculate the Fourier number (Fo)

Use the temperature formula for a cylinder to calculate the Fourier number (Fo). Formula for Fourier number (Fo): \(Fo = τ/Bi\) Where: \(τ = 130933.33\, \mathrm{s}\) (Time constant) \(Bi = 2.03\) (Biot number) Computing the Fourier number: \(Fo = \frac{130933.33}{2.03} = 64468.47\)

Calculate the dimensionless temperature (Θ)

We'll use the dimensionless temperature formula for a cylinder in conjunction with the one-term approximation method to determine the time it has been since the man's death. Formula for dimensionless temperature (Θ): \(Θ = \frac{T - T_\infty}{T_i - T_\infty}\) Where: \(T = 23^{\circ} \mathrm{C}\) (Surface temperature on his waist) \(T_\infty = 12^{\circ} \mathrm{C}\) (Room temperature) \(T_i = 36^{\circ} \mathrm{C}\) (Initial temperature of the body) Computing dimensionless temperature: \(Θ = \frac{23 - 12}{36 - 12} = \frac{11}{24} = 0.4583\)

Estimate the time (t) using one-term approximation

Using the one-term approximation method, we can write the following equation: \(Θ = e^{-Bi \cdot t/Fo}\) Substituting the values we obtained for Θ, Bi, and Fo: \(0.4583 = e^{-2.03 \cdot t/64468.47}\) Now, solve for t: \(t = \frac{-\ln(0.4583) \times 64468.47}{2.03} = 23207.62\, \mathrm{s}\) Now, convert the time to hours: \(t = \frac{23207.62}{3600} = 6.446\, \mathrm{hours}\) The estimated time since the man's death is approximately 6.45 hours.