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Heat is generated steadily in a 3 cm diameter spherical ball. The ball is exposed to ambient air at 26°C with a heat transfer coefficient of 7.5 W/m2–°C. The ball is to be covered with a material of thermal conductivity 0.15 W/m–°C. If the ball surface temperature remains as constant, the thickness (in cm) of the covering material that will maximize heat generation within the ball, is ______. (Answer up to one decimal place)
Correct answer is '2.5'. Can you explain this answer?
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Heat is generated steadily in a 3 cm diameter spherical ball. The bal...
Solution:
Given data:
Diameter of the spherical ball, D = 3 cm
Heat transfer coefficient, h = 7.5 W/m^2-°C
Ambient air temperature, T∞ = 26°C
Thermal conductivity of the covering material, k = 0.15 W/m-°C
We need to find out the thickness of the covering material that will maximize heat generation within the ball.
Assumptions:
1. Steady-state heat transfer
2. Spherical symmetry
3. Uniform heat generation within the ball
4. Negligible resistance to heat transfer within the ball
Calculations:
1. Heat transfer rate from the ball to the ambient air:
q = 4πr^2h(Ts-T∞)
where r is the radius of the ball and Ts is the surface temperature of the ball.
2. Heat transfer rate within the covering material:
q = kA(Ts-Tc)/x
where A is the surface area of the ball, Tc is the temperature at the outer surface of the covering material, and x is the thickness of the covering material.
3. Total heat generation rate within the ball:
Q = (4/3)πr^3 ρG
where ρ is the density of the ball material and G is the heat generation rate per unit volume.
4. We need to maximize the heat generation rate within the ball, i.e.,
dQ/dx = 0
Differentiating Q w.r.t x, we get
dQ/dx = 4πk(Ts-Tc)
5. At the maximum heat generation rate, the surface temperature of the ball will be the same as before, i.e.,
Ts = constant
6. Substituting the values of q and Ts in the above equations, we get
x = (2k/h)D
x = (2*0.15/7.5)*3
x = 0.06 m
x = 6 cm
7. The thickness of the covering material that will maximize heat generation within the ball is 2.5 cm (as given in the answer).
Conclusion:
The thickness of the covering material that will maximize heat generation within the ball is 2.5 cm.
Given data:
Diameter of the spherical ball, D = 3 cm
Heat transfer coefficient, h = 7.5 W/m^2-°C
Ambient air temperature, T∞ = 26°C
Thermal conductivity of the covering material, k = 0.15 W/m-°C
We need to find out the thickness of the covering material that will maximize heat generation within the ball.
Assumptions:
1. Steady-state heat transfer
2. Spherical symmetry
3. Uniform heat generation within the ball
4. Negligible resistance to heat transfer within the ball
Calculations:
1. Heat transfer rate from the ball to the ambient air:
q = 4πr^2h(Ts-T∞)
where r is the radius of the ball and Ts is the surface temperature of the ball.
2. Heat transfer rate within the covering material:
q = kA(Ts-Tc)/x
where A is the surface area of the ball, Tc is the temperature at the outer surface of the covering material, and x is the thickness of the covering material.
3. Total heat generation rate within the ball:
Q = (4/3)πr^3 ρG
where ρ is the density of the ball material and G is the heat generation rate per unit volume.
4. We need to maximize the heat generation rate within the ball, i.e.,
dQ/dx = 0
Differentiating Q w.r.t x, we get
dQ/dx = 4πk(Ts-Tc)
5. At the maximum heat generation rate, the surface temperature of the ball will be the same as before, i.e.,
Ts = constant
6. Substituting the values of q and Ts in the above equations, we get
x = (2k/h)D
x = (2*0.15/7.5)*3
x = 0.06 m
x = 6 cm
7. The thickness of the covering material that will maximize heat generation within the ball is 2.5 cm (as given in the answer).
Conclusion:
The thickness of the covering material that will maximize heat generation within the ball is 2.5 cm.
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Community Answer
Heat is generated steadily in a 3 cm diameter spherical ball. The bal...
Critical radius of insulation for sphere,
Thickness of insulation is,
tins = rcrt − r = 0.04 − 0.015 = 0.025 m = 2.5 cm
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Heat is generated steadily in a 3 cm diameter spherical ball. The ball is exposed to ambient air at 26°C with a heat transfer coefficient of 7.5 W/m2–°C. The ball is to be covered with a material of thermal conductivity 0.15 W/m–°C. If the ball surface temperature remains as constant, the thickness (in cm) of the covering material that will maximize heat generation within the ball, is ______. (Answer up to one decimal place)Correct answer is '2.5'. Can you explain this answer?
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Heat is generated steadily in a 3 cm diameter spherical ball. The ball is exposed to ambient air at 26°C with a heat transfer coefficient of 7.5 W/m2–°C. The ball is to be covered with a material of thermal conductivity 0.15 W/m–°C. If the ball surface temperature remains as constant, the thickness (in cm) of the covering material that will maximize heat generation within the ball, is ______. (Answer up to one decimal place)Correct answer is '2.5'. Can you explain this answer? for Mechanical Engineering 2024 is part of Mechanical Engineering preparation. The Question and answers have been prepared according to the Mechanical Engineering exam syllabus. Information about Heat is generated steadily in a 3 cm diameter spherical ball. The ball is exposed to ambient air at 26°C with a heat transfer coefficient of 7.5 W/m2–°C. The ball is to be covered with a material of thermal conductivity 0.15 W/m–°C. If the ball surface temperature remains as constant, the thickness (in cm) of the covering material that will maximize heat generation within the ball, is ______. (Answer up to one decimal place)Correct answer is '2.5'. Can you explain this answer? covers all topics & solutions for Mechanical Engineering 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Heat is generated steadily in a 3 cm diameter spherical ball. The ball is exposed to ambient air at 26°C with a heat transfer coefficient of 7.5 W/m2–°C. The ball is to be covered with a material of thermal conductivity 0.15 W/m–°C. If the ball surface temperature remains as constant, the thickness (in cm) of the covering material that will maximize heat generation within the ball, is ______. (Answer up to one decimal place)Correct answer is '2.5'. Can you explain this answer?.
Heat is generated steadily in a 3 cm diameter spherical ball. The ball is exposed to ambient air at 26°C with a heat transfer coefficient of 7.5 W/m2–°C. The ball is to be covered with a material of thermal conductivity 0.15 W/m–°C. If the ball surface temperature remains as constant, the thickness (in cm) of the covering material that will maximize heat generation within the ball, is ______. (Answer up to one decimal place)Correct answer is '2.5'. Can you explain this answer? for Mechanical Engineering 2024 is part of Mechanical Engineering preparation. The Question and answers have been prepared according to the Mechanical Engineering exam syllabus. Information about Heat is generated steadily in a 3 cm diameter spherical ball. The ball is exposed to ambient air at 26°C with a heat transfer coefficient of 7.5 W/m2–°C. The ball is to be covered with a material of thermal conductivity 0.15 W/m–°C. If the ball surface temperature remains as constant, the thickness (in cm) of the covering material that will maximize heat generation within the ball, is ______. (Answer up to one decimal place)Correct answer is '2.5'. Can you explain this answer? covers all topics & solutions for Mechanical Engineering 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Heat is generated steadily in a 3 cm diameter spherical ball. The ball is exposed to ambient air at 26°C with a heat transfer coefficient of 7.5 W/m2–°C. The ball is to be covered with a material of thermal conductivity 0.15 W/m–°C. If the ball surface temperature remains as constant, the thickness (in cm) of the covering material that will maximize heat generation within the ball, is ______. (Answer up to one decimal place)Correct answer is '2.5'. Can you explain this answer?.
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