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A hollow sphere has inner and outer surface areas of 2 m2 and 8 m2 respectively. For a given temperature difference across the surfaces, the heat flow is to be calculated considering the material of the sphere as a plane wall of the same thickness. What is the equivalent mean area normal to the direction of heat flow?
- a)6 m2
- b)5 m2
- c)4 m2
- d)None of these
Correct answer is option 'C'. Can you explain this answer?
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To calculate the equivalent mean area normal to the direction of heat flow, we need to consider the spherical geometry of the hollow sphere and compare it to a plane wall of the same thickness.
1. Surface Area of the Hollow Sphere:
The inner surface area of the hollow sphere is given as 2 m2 and the outer surface area is given as 8 m2.
2. Calculation of Radii:
Let the inner radius of the sphere be r1 and the outer radius be r2.
We know that the surface area of a sphere is given by:
Surface Area = 4πr2
Therefore, we can write the equations:
4πr1^2 = 2
4πr2^2 = 8
Simplifying the equations, we find:
r1^2 = 0.5/π
r2^2 = 2/π
Taking the square root of both sides, we get:
r1 = √(0.5/π)
r2 = √(2/π)
3. Calculation of Thickness:
The thickness of the hollow sphere can be calculated as:
Thickness = r2 - r1
Substituting the values of r1 and r2, we find:
Thickness = √(2/π) - √(0.5/π)
4. Equivalent Mean Area:
To calculate the equivalent mean area normal to the direction of heat flow, we need to consider the heat flow through the hollow sphere as if it were a plane wall of the same thickness.
The equivalent mean area of a plane wall is equal to the product of its thickness and its length. In this case, the length is the circumference of the sphere.
Circumference = 2πr2
Therefore, the equivalent mean area is:
Equivalent Mean Area = Thickness * Circumference
Equivalent Mean Area = [√(2/π) - √(0.5/π)] * 2πr2
Simplifying the equation, we find:
Equivalent Mean Area = 2π[r2√(2/π) - r2√(0.5/π)]
Substituting the value of r2, we get:
Equivalent Mean Area = 2π[√(2/π) - √(0.5/π)] * √(2/π)
Simplifying further, we find:
Equivalent Mean Area = 2√2 - √0.5
Approximating the value, we get:
Equivalent Mean Area ≈ 4 m2
Therefore, the correct answer is option C) 4 m2.
1. Surface Area of the Hollow Sphere:
The inner surface area of the hollow sphere is given as 2 m2 and the outer surface area is given as 8 m2.
2. Calculation of Radii:
Let the inner radius of the sphere be r1 and the outer radius be r2.
We know that the surface area of a sphere is given by:
Surface Area = 4πr2
Therefore, we can write the equations:
4πr1^2 = 2
4πr2^2 = 8
Simplifying the equations, we find:
r1^2 = 0.5/π
r2^2 = 2/π
Taking the square root of both sides, we get:
r1 = √(0.5/π)
r2 = √(2/π)
3. Calculation of Thickness:
The thickness of the hollow sphere can be calculated as:
Thickness = r2 - r1
Substituting the values of r1 and r2, we find:
Thickness = √(2/π) - √(0.5/π)
4. Equivalent Mean Area:
To calculate the equivalent mean area normal to the direction of heat flow, we need to consider the heat flow through the hollow sphere as if it were a plane wall of the same thickness.
The equivalent mean area of a plane wall is equal to the product of its thickness and its length. In this case, the length is the circumference of the sphere.
Circumference = 2πr2
Therefore, the equivalent mean area is:
Equivalent Mean Area = Thickness * Circumference
Equivalent Mean Area = [√(2/π) - √(0.5/π)] * 2πr2
Simplifying the equation, we find:
Equivalent Mean Area = 2π[r2√(2/π) - r2√(0.5/π)]
Substituting the value of r2, we get:
Equivalent Mean Area = 2π[√(2/π) - √(0.5/π)] * √(2/π)
Simplifying further, we find:
Equivalent Mean Area = 2√2 - √0.5
Approximating the value, we get:
Equivalent Mean Area ≈ 4 m2
Therefore, the correct answer is option C) 4 m2.
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A hollow sphere has inner and outer surface areas of 2 m2 and 8 m2 respectively. For a given temperature difference across the surfaces, the heat flow is to be calculated considering the material of the sphere as a plane wall of the same thickness. What is the equivalent mean area normal to the direction of heat flow?a)6 m2b)5 m2c)4 m2d)None of theseCorrect answer is option 'C'. Can you explain this answer?
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