Consider an apple in a crate, which is stored in cold storage withair temperatur
ID: 1815656 • Letter: C
Question
Consider an apple in a crate, which is stored in cold storage withair temperature of 4oC. The convection heat transfercoefficient is approximately 2.0 W/(m2*K). Thermalenergy is uniformly generated at a total rate of 5000 J/(kg*day).The density and thermal conductivity of the apple are 850kg/m3 and 0.3 W/(m*K), respectively. The apple can bemodeled as a sphere with a diameter of 90 mm. If the temperatureanywhere inside the apple should not exceed 5o C, checkwhether the temperature of the air in the storage is coldenough.Explanation / Answer
*Cp*dT/dt = -div[-k*grad(T)] + generation dT/dt = 0 0 = k*laplacian(T) + generation 0 = k*(1/r2)*d[r2dT/dr]/dr +generation (-gen/k)*r2 = d[r2dT/dr]/dr (-gen/3k)*r3 + C1 = r2*dT/dr (-gen/3k)*r + C1/r2 = dT/dr T = (-gen/6k)*r2 - C1/r + C2 boundary conditions, -k*dT/dr = h*(T -Tair) at r = 45 mm also dT/dr = 0 ar r = 0 (symmetry condition) dT/dr = (-gen/3k)*r + C1/r2 eval at r = 0 0 = (-gen/3k)*0 + C1/02 0 =/= => C1 =0 =>T = (-gen/6k)*r2 +C2 with dT/dr = (-gen/3k)*r Use -k*dT/dr = h*(T -Tair) at r = 45 mm = Radius of sphere = R -k*(-gen/3k)*R = h*[(-gen/6k)R2 + C2 - Tair] (gen/3)*R/h + (gen/6k)*R2 + Tair = C2 Thus T = (-gen/6k)*r2 + (gen/3)*R/h +(gen/6k)*R2 + Tair T = (gen/6k)*(R2 - r2 ) + (gen/3)*R/h +Tair gen = 5000 J/(kg*day)*(850 kg/m3)*(1 day/24hr)*(1 hr/3600s) = 49.1898148 J-m-3s-1 T = ( 49.1898148 J-m-3s-1 / 6*0.3J-m-1C-1s-1)*[(0.045m)2 - r2] + (49.1898148J-m-3s-1/3)*(0.045 m/ 2.0J-m-2C-1s-1) + 4 C T = 27.3276749 C * (0.002025 - r2) +(0.368923611)C + 4C T = 27.3276749 C * (0.002025 - r2) + (4.36892361) C Let T = 5 C 5C = 27.3276749 C * (0.002025 - r2) + (4.36892361) C ((5-4.36892361)/27.3276749) = (0.002025 - r2) (0.002025 - r2) = 0.0230929412 -0.0210679412 = r2 No r exists, that when squared, equals a negative number Seems that the temperature of the air is cold enough