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A composite hollow sphere with steady internal heating is made of 2layers of materials of equal thickness with thermal conductivities inthe ratio of 1 : 2 for inner to outer layers. Ratio of inside to outside diameter is 0.8. What is ratio of temperature drop across the inner andouter layers?
- a)0.4
- b)1.6
- c)2 ln (0.8)
- d)2.5
Correct answer is option 'D'. Can you explain this answer?
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Given:
- Composite hollow sphere with two layers of equal thickness
- Thermal conductivities ratio of inner to outer layers is 1:2
- Ratio of inside to outside diameter is 0.8
To find:
Ratio of temperature drop across the inner and outer layers
Solution:
Step 1: Understanding the problem
- The composite hollow sphere has two layers of different materials.
- The inner layer has a higher thermal conductivity compared to the outer layer.
- The sphere is subjected to steady internal heating.
- We need to find the ratio of temperature drop across the inner and outer layers.
Step 2: Assumptions
- Assuming the heat flow is radial and there is no heat loss to the surroundings.
- Assuming the thermal conductivities of the layers remain constant with temperature.
Step 3: Analysis
- Let the inner diameter of the sphere be D and the outer diameter be D'
- The ratio of inside to outside diameter is given as 0.8, so we have:
D/D' = 0.8
- Let the temperature at the inner surface of the inner layer be T1 and the temperature at the outer surface of the outer layer be T2.
- The temperature drops across the inner and outer layers can be represented as:
ΔT1 = T1 - T2 (temperature drop across the inner layer)
ΔT2 = T2 - T∞ (temperature drop across the outer layer)
Step 4: Applying Heat Conduction Equation
- The heat conduction equation for steady-state radial heat flow can be expressed as:
Q = ((4πk1D'T1) - (4πk2DT2)) / ln(D'/D)
- As the sphere is subjected to steady internal heating, the heat flow rate (Q) is constant.
- Since the thickness of the layers is the same, we can cancel out the terms involving the thickness.
Step 5: Simplification
- Rearranging the equation, we get:
T1 - T2 = (k2/k1) * (T2 - T∞)
- Substituting the given ratio of thermal conductivities (k2/k1 = 1/2), we have:
T1 - T2 = (1/2) * (T2 - T∞)
Step 6: Finding the Ratio of Temperature Drops
- Dividing both sides of the equation by (T2 - T∞), we get:
(T1 - T2) / (T2 - T∞) = 1/2
- Rearranging the equation, we have:
(T1 - T2) = (1/2) * (T2 - T∞)
- Comparing the equation with the ratio of temperature drops (ΔT1 / ΔT2), we can say:
ΔT1 / ΔT2 = 1/2
Step 7: Answer
The ratio of temperature drop across the inner and outer layers is 1:2.5, which corresponds to option 'D'.
- Composite hollow sphere with two layers of equal thickness
- Thermal conductivities ratio of inner to outer layers is 1:2
- Ratio of inside to outside diameter is 0.8
To find:
Ratio of temperature drop across the inner and outer layers
Solution:
Step 1: Understanding the problem
- The composite hollow sphere has two layers of different materials.
- The inner layer has a higher thermal conductivity compared to the outer layer.
- The sphere is subjected to steady internal heating.
- We need to find the ratio of temperature drop across the inner and outer layers.
Step 2: Assumptions
- Assuming the heat flow is radial and there is no heat loss to the surroundings.
- Assuming the thermal conductivities of the layers remain constant with temperature.
Step 3: Analysis
- Let the inner diameter of the sphere be D and the outer diameter be D'
- The ratio of inside to outside diameter is given as 0.8, so we have:
D/D' = 0.8
- Let the temperature at the inner surface of the inner layer be T1 and the temperature at the outer surface of the outer layer be T2.
- The temperature drops across the inner and outer layers can be represented as:
ΔT1 = T1 - T2 (temperature drop across the inner layer)
ΔT2 = T2 - T∞ (temperature drop across the outer layer)
Step 4: Applying Heat Conduction Equation
- The heat conduction equation for steady-state radial heat flow can be expressed as:
Q = ((4πk1D'T1) - (4πk2DT2)) / ln(D'/D)
- As the sphere is subjected to steady internal heating, the heat flow rate (Q) is constant.
- Since the thickness of the layers is the same, we can cancel out the terms involving the thickness.
Step 5: Simplification
- Rearranging the equation, we get:
T1 - T2 = (k2/k1) * (T2 - T∞)
- Substituting the given ratio of thermal conductivities (k2/k1 = 1/2), we have:
T1 - T2 = (1/2) * (T2 - T∞)
Step 6: Finding the Ratio of Temperature Drops
- Dividing both sides of the equation by (T2 - T∞), we get:
(T1 - T2) / (T2 - T∞) = 1/2
- Rearranging the equation, we have:
(T1 - T2) = (1/2) * (T2 - T∞)
- Comparing the equation with the ratio of temperature drops (ΔT1 / ΔT2), we can say:
ΔT1 / ΔT2 = 1/2
Step 7: Answer
The ratio of temperature drop across the inner and outer layers is 1:2.5, which corresponds to option 'D'.
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A composite hollow sphere with steady internal heating is made of 2lay...
Equate heat conduction equation of sphere for (R1, R2) and (R2,R3) where R1,R2,R3 are respective radius.
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A composite hollow sphere with steady internal heating is made of 2layers of materials of equal thickness with thermal conductivities inthe ratio of 1 : 2 for inner to outer layers. Ratio of inside to outside diameter is 0.8. What is ratio of temperature drop across the inner andouter layers?a)0.4b)1.6c)2 ln (0.8)d)2.5Correct answer is option 'D'. Can you explain this answer?
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