Question  set 156

Heat Generation Through Sphere

1.     Consider heat conduction through a solid sphere of radius R. There are certain assumptions
(i) Unsteady state conditions
(ii) One-dimensional radial conduction
(iii) Constant thermal conductivity
Identify the correct statements
a) i and iii
b) ii and iii
c) i, ii and iii
d) i and ii
Answer: b
Explanation: Statement 1 should be steady state condition.

2.     An 8 cm diameter orange, approximately spherical in shape, undergoes ripening process and generates 18000 k J/m³ hr of energy. If external surface of the orange is at 6.5 degree Celsius, find out the temperature at the center of the orange. Take thermal conductivity = 0.8 k J/ m hr degree for the orange material
a) 13.5 degree Celsius
b) 12.5 degree Celsius
c) 11.5 degree Celsius
d) 10.5 degree Celsius
Answer: b
Explanation: q _g = 5000 W/m³, k = 0.222 W/m K and t = t _W + q _g R ²/6K = 12.5 degree Celsius.

3.     Consider the above problem, calculate the heat flow from the outer surface of the orange
a) 4.82 k J/hr
b) 5.82 k J/hr
c) 6.82 k J/hr
d) 7.82 k J/hr
Answer: a
Explanation: Q = 4/3 (π R ³ q _g) = 1.34 J/s.

4.     What is the heat flow for steady state conduction for sphere?
a) 4 Q _R + Q _G = Q _{R + d R}
b) 3 Q _R + Q _G = Q _{R + d R}
c) 2 Q _R + Q _G = Q _{R + d R}
d) Q _R + Q _G = Q _{R + d R}
Answer: d
Explanation: Q _R + Q _G = Q _R + d (Q _R) d R/d R.
Where, Q _R = Heat conducted in at radius R
Q _G = Heat conducted in the element
Q _{R + d R} = Heat conducted out at radius R + d R.

5.     The general solution for temperature distribution in case of solid sphere is
a) t = t _W + q _g (R ₂ – r ₂)/4 k
b) t = t _W + q _g (R ₂ – r ₂)/8 k
c) t = t _W + q _g (R ₂ – r ₂)/6 k
d) t = t _W + q _g (R ₂ – r ₂)/2 k
Answer: c
Explanation: The temperature distribution is parabolic.

6.     A solid sphere of 8 cm radius has a uniform heat generation 0f 4000000 W/m³. The outside surface is exposed to a fluid at 150 degree Celsius with convective heat transfer coefficient of 750 W/m² K. If thermal conductivity of the solid material is 30 W/m K, determine maximum temperature
a) 444.45 degree Celsius
b) 434.45 degree Celsius
c) 424.45 degree Celsius
d) 414.45 degree Celsius
Answer: b
Explanation: q _g (4 π R³/3) = h 4 π R² (t _W – t _a), t _w = 292.22 degree Celsius
T _MAX = t _w + q _g R ²/6 k.

7.     Consider the above problem, find the temperature at 5 cm radius
a) 348.9 degree Celsius
b) 358.9 degree Celsius
c) 368.9 degree Celsius
d) 378.9 degree Celsius
Answer: d
Explanation: t – t _w/t _MAX – t _w = 1 – (r/R) ^½.

8.     Identify the correct boundary condition for a hollow sphere with inside surface insulated
a) At r = r ₁, the conduction region is perfectly insulated
b) At r = r ₁, the conduction region is partially insulated
c) Heat flow is infinity
d) Heat flow is negative
Answer: a
Explanation: In this range, the conduction region must be perfectly insulated.

9.     A hollow sphere (k = 30 W/m K) of inner radius 6 cm and outside radius 8 cm has a heat generation rate of 4000000 W/m³. The inside surface is insulated and heat is removed by convection over the outside surface by a fluid at 100 degree Celsius with surface conductance 300 W/m² K. Make calculations for the temperature at the outside surfaces of the sphere
a) 105.6 degree Celsius
b) 205.6 degree Celsius
c) 305.6 degree Celsius
d) 405.6 degree Celsius
Answer: c
Explanation: q _g 4 π (R ³ – r ³)/3 = h ₀ 4 π r ² (t ₂ – t _a).

10. Consider the above problem, also calculate the temperature at the inside surfaces of the sphere
a) 138.3 degree Celsius
b) 327.8 degree Celsius
c) 254.7 degree Celsius
d) 984.9 degree Celsius
Answer: b
Explanation: t = t ₂ + q _g (R ² – r ²)/6 k – q _g r ³ (1/r – 1/R)/3 k.