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A metallic ball in shape of sphere is made up of steel (thermal conductivity of 35W/m-K) of diameter 0.38 m . The outside heat transfer coefficient is 15W/m2. It is to be insulated with film of thermal conductivity 3W/m-K. The thickness of insulation should be
- a)0.21m
- b)0.02m
- c)0.38m
- d)0.19m
Correct answer is option 'A'. Can you explain this answer?
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A metallic ball in shape of sphere is made up of steel (thermal conduc...
Thickness=0.4-0.19=0.21m
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A metallic ball in shape of sphere is made up of steel (thermal conduc...
Given:
- Diameter of the metallic ball = 0.38 m
- Thermal conductivity of steel = 35 W/m-K
- Outside heat transfer coefficient = 15 W/m2
- Thermal conductivity of insulation film = 3 W/m-K
To find:
The thickness of insulation required
Solution:
Step 1: Calculate the heat transfer through the metallic ball without insulation
The rate of heat transfer through a spherical object can be calculated using the formula:
Q = 4πkR(T1 - T2)/d
Where:
- Q is the heat transfer rate
- k is the thermal conductivity of the material
- R is the radius of the sphere
- T1 is the temperature inside the sphere
- T2 is the temperature outside the sphere
- d is the diameter of the sphere
Given:
- k (thermal conductivity of steel) = 35 W/m-K
- R (radius of the sphere) = 0.38/2 = 0.19 m
- T1 (temperature inside the sphere) = T2 (temperature outside the sphere)
- d (diameter of the sphere) = 0.38 m
Q = 4π * 35 * 0.19 * (T1 - T2)/0.38
Step 2: Calculate the heat transfer through the insulation
The rate of heat transfer through the insulation can be calculated using the formula:
Q = 2πkL(T1 - T2)/ln(r2/r1)
Where:
- Q is the heat transfer rate
- k is the thermal conductivity of the insulation material
- L is the thickness of the insulation
- T1 is the temperature inside the insulation
- T2 is the temperature outside the insulation
- r2 and r1 are the outer and inner radii of the insulation
Given:
- k (thermal conductivity of insulation) = 3 W/m-K
- L (thickness of insulation) = ?
- T1 (temperature inside the insulation) = T2 (temperature outside the insulation)
- r2 (outer radius of insulation) = 0.19 + L
- r1 (inner radius of insulation) = 0.19
Q = 2π * 3 * L * (T1 - T2)/ln((0.19 + L)/0.19)
Step 3: Equate the heat transfer through the metallic ball to the heat transfer through the insulation
Since the heat transfer through the metallic ball and the insulation should be equal, we can equate the two equations:
4π * 35 * 0.19 * (T1 - T2)/0.38 = 2π * 3 * L * (T1 - T2)/ln((0.19 + L)/0.19)
Simplifying the equation:
35 * 0.19/0.38 = L/ln((0.19 + L)/0.19)
L = (35 * 0.19/0.38) * ln((0.19 + L)/0.19)
Step 4: Solve for L
To solve the equation numerically, we can use iterative methods or a numerical solver. After solving the equation, we find that the value of L
- Diameter of the metallic ball = 0.38 m
- Thermal conductivity of steel = 35 W/m-K
- Outside heat transfer coefficient = 15 W/m2
- Thermal conductivity of insulation film = 3 W/m-K
To find:
The thickness of insulation required
Solution:
Step 1: Calculate the heat transfer through the metallic ball without insulation
The rate of heat transfer through a spherical object can be calculated using the formula:
Q = 4πkR(T1 - T2)/d
Where:
- Q is the heat transfer rate
- k is the thermal conductivity of the material
- R is the radius of the sphere
- T1 is the temperature inside the sphere
- T2 is the temperature outside the sphere
- d is the diameter of the sphere
Given:
- k (thermal conductivity of steel) = 35 W/m-K
- R (radius of the sphere) = 0.38/2 = 0.19 m
- T1 (temperature inside the sphere) = T2 (temperature outside the sphere)
- d (diameter of the sphere) = 0.38 m
Q = 4π * 35 * 0.19 * (T1 - T2)/0.38
Step 2: Calculate the heat transfer through the insulation
The rate of heat transfer through the insulation can be calculated using the formula:
Q = 2πkL(T1 - T2)/ln(r2/r1)
Where:
- Q is the heat transfer rate
- k is the thermal conductivity of the insulation material
- L is the thickness of the insulation
- T1 is the temperature inside the insulation
- T2 is the temperature outside the insulation
- r2 and r1 are the outer and inner radii of the insulation
Given:
- k (thermal conductivity of insulation) = 3 W/m-K
- L (thickness of insulation) = ?
- T1 (temperature inside the insulation) = T2 (temperature outside the insulation)
- r2 (outer radius of insulation) = 0.19 + L
- r1 (inner radius of insulation) = 0.19
Q = 2π * 3 * L * (T1 - T2)/ln((0.19 + L)/0.19)
Step 3: Equate the heat transfer through the metallic ball to the heat transfer through the insulation
Since the heat transfer through the metallic ball and the insulation should be equal, we can equate the two equations:
4π * 35 * 0.19 * (T1 - T2)/0.38 = 2π * 3 * L * (T1 - T2)/ln((0.19 + L)/0.19)
Simplifying the equation:
35 * 0.19/0.38 = L/ln((0.19 + L)/0.19)
L = (35 * 0.19/0.38) * ln((0.19 + L)/0.19)
Step 4: Solve for L
To solve the equation numerically, we can use iterative methods or a numerical solver. After solving the equation, we find that the value of L
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A metallic ball in shape of sphere is made up of steel (thermal conduc...
Correct your units like h = 15 w/m2k
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A metallic ball in shape of sphere is made up of steel (thermal conductivity of 35W/m-K) of diameter 0.38 m . The outside heat transfer coefficient is 15W/m2. It is to be insulated with film of thermal conductivity 3W/m-K. The thickness of insulation should bea)0.21mb)0.02mc)0.38md)0.19mCorrect answer is option 'A'. Can you explain this answer?
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A metallic ball in shape of sphere is made up of steel (thermal conductivity of 35W/m-K) of diameter 0.38 m . The outside heat transfer coefficient is 15W/m2. It is to be insulated with film of thermal conductivity 3W/m-K. The thickness of insulation should bea)0.21mb)0.02mc)0.38md)0.19mCorrect answer is option 'A'. Can you explain this answer? for GATE 2024 is part of GATE preparation. The Question and answers have been prepared according to the GATE exam syllabus. Information about A metallic ball in shape of sphere is made up of steel (thermal conductivity of 35W/m-K) of diameter 0.38 m . The outside heat transfer coefficient is 15W/m2. It is to be insulated with film of thermal conductivity 3W/m-K. The thickness of insulation should bea)0.21mb)0.02mc)0.38md)0.19mCorrect answer is option 'A'. Can you explain this answer? covers all topics & solutions for GATE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for A metallic ball in shape of sphere is made up of steel (thermal conductivity of 35W/m-K) of diameter 0.38 m . The outside heat transfer coefficient is 15W/m2. It is to be insulated with film of thermal conductivity 3W/m-K. The thickness of insulation should bea)0.21mb)0.02mc)0.38md)0.19mCorrect answer is option 'A'. Can you explain this answer?.
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