Severe asphyxia of the fetus or new born is known to be an important cause of in

Question

Severe asphyxia of the fetus or new born is known to be an important cause of injury to the development of brain. Lowering the cerebral temperature over several days after a hypoxicischemic insult can markedly protect the brain from damage. Consider a newborn head as a sphere of 10 cm in diameter with average thermal conductivity of 0.5 W/mK, metabolic heat generation and positive heat contribution from blood flow are 10,000 W/m3. The air surrounding the infant is circulating with a heat transfer coefficient of 15 W/m2K. The normal temperature of the center of the head without cooling is 37

Explanation / Answer

Under steady state

(1/r^2)*d(r^2*dT/dr)/dr + q'''/k = 0

r^2*dT/dr = -(q'''/k)*(r^3/3) + C1

T(r) = -(q'''/k)*(r^2/6) - C1/r + C2

Boundary conditions

At r = 0

dT/dr = 0 (since symmetric profile )------(1)

At r = 0

T(0) = To = 35 deg (after steady state)----(2)

From 1

C1 = 0

From 2

T0 = C2

C2 = To = 35 deg = 308 K

T(r) = -(q'''/k)*(r^2/6) + 308

At surface T(r = 5 cm)

Ts = -(10,000/0.5)*25/6*10^-4 + 308 = 299.667 K = 26.667 deg

We also know

h*As*(Ts-Tamb) = q'''*V

Tamb is air temperature

h*4*pi*r^2*(Ts-Tamb) = q'''*4/3*pi*r^3

h*(Ts-Tamb) = q'''*1/3*r

15*(299.667-Tamb) = 10000*1/3*5*10^-2

Tamb = 288.556 K = 15.556 deg

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