Heat Transfer Miscellaneous
 For the threedimensional object shown in the figure below, five faces are insulated. The sixth face (PQRS), which is not insulated, interacts thermally with the ambient, with a convective heat transfer coefficient of 10W/m^{2}K. The ambient temperature is 30°C. Heat is uniformly generated inside the object at the rate of 100 W/m^{3}. Assuming the face PQRS to be at uniform temperature, its steady state temperature is

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Given data;
h_{o} = W/m^{2}K
h_{o} = 30 °C
q_{G} = 100 W/m^{3}
Volume, V = 2×1×2=4 m^{3}
Heat generated, Q = q_{G} × V = 100 × 4 = 400 W
Convection heat transfer from face PQRS,
Q = h_{o} A (T_{s} – T_{o})
400 = 10 × 2 × 2 (T_{s} – 30)
T_{s} = 40°CCorrect Option: D
Given data;
h_{o} = W/m^{2}K
h_{o} = 30 °C
q_{G} = 100 W/m^{3}
Volume, V = 2×1×2=4 m^{3}
Heat generated, Q = q_{G} × V = 100 × 4 = 400 W
Convection heat transfer from face PQRS,
Q = h_{o} A (T_{s} – T_{o})
400 = 10 × 2 × 2 (T_{s} – 30)
T_{s} = 40°C
 A stainless steel tube (k_{s} = 19 W/mK) of 2 cm ID and 5 cm OD is insulated with 3 cm thick asbestos (k_{a} = 0.2 W/mK). If the temperature difference between the inner most and outermost surfaces is 600°C, the heat transfer rate per unit length is

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Given: d_{1} = 2 cm
r_{1} = 1 cm
d_{2} = 5 cm
r_{2} = 2.5 cm
d_{3} = 5 + 2 × 3 = 11
r_{3} = 5.5 cm
Equivalent thermal resistanceR_{t} = 1 In r_{2} + 1 In r_{3} 2πk_{st} r_{1} 2πk_{a} r_{2} = 1 In 2.5 + 1 In 5.5 2π × 19 1 2π × 0.2 2.5
= 7.67 × 10^{–3} + 0.627 = 0.635
∴ Heat transfer rate per unit lengtht_{1}  t_{2} = 600 = 944.88 W/m R_{t} 0.635 Correct Option: C
Given: d_{1} = 2 cm
r_{1} = 1 cm
d_{2} = 5 cm
r_{2} = 2.5 cm
d_{3} = 5 + 2 × 3 = 11
r_{3} = 5.5 cm
Equivalent thermal resistanceR_{t} = 1 In r_{2} + 1 In r_{3} 2πk_{st} r_{1} 2πk_{a} r_{2} = 1 In 2.5 + 1 In 5.5 2π × 19 1 2π × 0.2 2.5
= 7.67 × 10^{–3} + 0.627 = 0.635
∴ Heat transfer rate per unit lengtht_{1}  t_{2} = 600 = 944.88 W/m R_{t} 0.635
 A well machined steel plate of thickness L is kept such that the wall temperature are Th and Tc as seen in the figure below. A smooth copper plate of the same thickness L is now attached to the steel plate without any gap as indicated in the figure below. The temperature at the interface is T_{t}. The temperatures of the outer walls are still the same at T_{h} and T_{c}. The heat transfer rates are q_{1} and q_{2} per unit area in the two cases respectively in the direction shown. Which of the following statements is correct?

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For caseI:Q_{1} = T_{h}  T_{c} R_{1} where R_{1} = L K_{S}A
For CaseII:Q_{2} = T_{h}  T_{c} R_{2} + R_{3} where R_{2} = L and R_{3} = L K_{s}A K_{cu}A
As R_{2} + R_{3} > R_{1} at constant temperature difference in both cases.Then q_{1} or Q_{1} > q_{2} or Q_{2} A A
For CaseII:
R_{2} > R_{3}
∵ k_{cu} > k_{s}Q_{2} = T_{h}  T_{1} R_{2} or R_{2} = T_{h}  T_{1} Q_{2} also Q_{2} = T_{i}  T_{c} R_{3} or R_{3} = T_{i}  T_{c} Q_{2} ∴ T_{h}  T_{i} > T_{i}  T_{c} Q_{2} Q_{2}
or T_{h} – T_{i} > T_{i} – T_{c}
or T_{h} + T_{c} > 2T_{i}or T_{i} < T_{h} + T_{c} 2 Correct Option: D
For caseI:Q_{1} = T_{h}  T_{c} R_{1} where R_{1} = L K_{S}A
For CaseII:Q_{2} = T_{h}  T_{c} R_{2} + R_{3} where R_{2} = L and R_{3} = L K_{s}A K_{cu}A
As R_{2} + R_{3} > R_{1} at constant temperature difference in both cases.Then q_{1} or Q_{1} > q_{2} or Q_{2} A A
For CaseII:
R_{2} > R_{3}
∵ k_{cu} > k_{s}Q_{2} = T_{h}  T_{1} R_{2} or R_{2} = T_{h}  T_{1} Q_{2} also Q_{2} = T_{i}  T_{c} R_{3} or R_{3} = T_{i}  T_{c} Q_{2} ∴ T_{h}  T_{i} > T_{i}  T_{c} Q_{2} Q_{2}
or T_{h} – T_{i} > T_{i} – T_{c}
or T_{h} + T_{c} > 2T_{i}or T_{i} < T_{h} + T_{c} 2
 In a case of one dimensional heat conduction in a medium with constant properties, T is the temperature at position x, at time t.
Then δT is proportional to δt

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One dimensional heat conduction equation is,
δ^{2}T = 1 δT δx^{2} α δt ∴ δ^{2}T ∝ δT δx^{2} δt Correct Option: D
One dimensional heat conduction equation is,
δ^{2}T = 1 δT δx^{2} α δt ∴ δ^{2}T ∝ δT δx^{2} δt
 Heat flows through a composite slab, as shown below. The depth of the slab is 1 m. The k values are in W/mK. The overall thermal resistance in K/W is

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L_{1} = 0.5 = 25 k_{1} 0.02 2L_{2} = 5 k_{2} 2L_{3} = 12.5 k_{3}
R_{total} = 25 + 3.6 = 28.6 K/WCorrect Option: C
L_{1} = 0.5 = 25 k_{1} 0.02 2L_{2} = 5 k_{2} 2L_{3} = 12.5 k_{3}
R_{total} = 25 + 3.6 = 28.6 K/W