Engineering

"To perform numerical integration using Simpson's rule, we'll need to set up our code. We'll define a function for the integrand and then use Simpson's rule to approximate the integral. Let's do that in Python:

import numpy as np

# Constants

T_infinity = 305.15  # Initial skin temperature in Kelvin

Q_prime = 2100  # Heat flux in W/m^2

k = 0.3  # Thermal conductivity in W/m.K

alpha = 1.513e-7  # Thermal diffusivity in m^2/s

r = 0.529e-3  # Maximum burn depth in meters

# Function to integrate

def integrand(a):

    return np.exp(-a**2 * alpha * 3 / r**2) / a

# Simpson's rule for numerical integration

def simpsons_rule(func, a, b, n):

    h = (b - a) / n

    x = np.linspace(a, b, n+1)

    y = func(x)

    return h / 3 * (y[0] + 4 * np.sum(y[1:n:2]) + 2 * np.sum(y[2:n-1:2]) + y[n])

# Approximating the integral

integral_value = simpsons_rule(integrand, 0.0001, 1000, 100000)

# Calculating temperature at maximum burn depth

T_max_burn_depth = T_infinity + (Q_prime / (4 * np.pi * k)) * integral_value

print(""Temperature at maximum burn depth:"", T_max_burn_depth, ""K"")"