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Stainless steel sphere [k = 16 W/m . OC ] having a diameter of 4 cm is exposed to a convection environment at 20OC, h = 15 W/m2. OC. Heat is generated uniformly in the sphere at the rate of 1.0 MW/m3. Calculate the steady state temperature for the center of the sphere.?
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Stainless steel sphere [k = 16 W/m . OC ] having a diameter of 4 cm is...
Problem Statement: Calculate the steady state temperature for the center of a stainless steel sphere having a diameter of 4 cm, exposed to a convection environment at 20OC with a heat generation rate of 1.0 MW/m3.
Solution:
Given:
Diameter of the sphere, D = 4 cm
Radius of the sphere, r = 2 cm
Thermal conductivity of the sphere, k = 16 W/m . OC
Convection environment temperature, T∞ = 20OC
Heat transfer coefficient, h = 15 W/m2. OC
Rate of heat generation, q'' = 1.0 MW/m3
Assumptions:
The thermal conductivity of the sphere is independent of temperature.
Heat generation is uniform throughout the sphere.
Steady-state conditions prevail.
Negligible radiation heat transfer.
Formulae:
Steady-state heat transfer rate, q = 4πkr2(Tc - T∞)
Rate of heat generation, q'' = Q/V
where Q is the heat generated and V is the volume of the sphere.
Volume of the sphere, V = 4/3 πr3
Critical radius, rC = k/h
Calculation:
Volume of the sphere, V = 4/3 πr3 = 4/3 π(2 cm)3 = 33.51 cm3 = 33.51 x 10^-6 m3
Rate of heat generation, q'' = Q/V = 1.0 MW/m3
Steady-state heat transfer rate, q = q''V = 1.0 x 10^6 W/m3 x 33.51 x 10^-6 m3 = 33.51 W
Critical radius, rC = k/h = 16 W/m . OC / 15 W/m2 . OC = 1.07 m
Since the radius of the sphere, r (2 cm) is less than the critical radius, rC (1.07 m), the heat transfer is dominated by conduction.
Therefore, the steady-state temperature distribution in the sphere is given by:
q/4πk(Tc - T∞)/r = -d/dr(r2 dT/dr)
Integrating and applying boundary conditions, we get:
Tc = T∞ + q''r2/6k = 20OC + 1.0 x 10^6 W/m3 x (2 cm)2 / (6 x 16 W/m . OC) = 20.44OC
Answer: Therefore, the steady-state temperature for the center of the sphere is 20.44OC.
Solution:
Given:
Diameter of the sphere, D = 4 cm
Radius of the sphere, r = 2 cm
Thermal conductivity of the sphere, k = 16 W/m . OC
Convection environment temperature, T∞ = 20OC
Heat transfer coefficient, h = 15 W/m2. OC
Rate of heat generation, q'' = 1.0 MW/m3
Assumptions:
The thermal conductivity of the sphere is independent of temperature.
Heat generation is uniform throughout the sphere.
Steady-state conditions prevail.
Negligible radiation heat transfer.
Formulae:
Steady-state heat transfer rate, q = 4πkr2(Tc - T∞)
Rate of heat generation, q'' = Q/V
where Q is the heat generated and V is the volume of the sphere.
Volume of the sphere, V = 4/3 πr3
Critical radius, rC = k/h
Calculation:
Volume of the sphere, V = 4/3 πr3 = 4/3 π(2 cm)3 = 33.51 cm3 = 33.51 x 10^-6 m3
Rate of heat generation, q'' = Q/V = 1.0 MW/m3
Steady-state heat transfer rate, q = q''V = 1.0 x 10^6 W/m3 x 33.51 x 10^-6 m3 = 33.51 W
Critical radius, rC = k/h = 16 W/m . OC / 15 W/m2 . OC = 1.07 m
Since the radius of the sphere, r (2 cm) is less than the critical radius, rC (1.07 m), the heat transfer is dominated by conduction.
Therefore, the steady-state temperature distribution in the sphere is given by:
q/4πk(Tc - T∞)/r = -d/dr(r2 dT/dr)
Integrating and applying boundary conditions, we get:
Tc = T∞ + q''r2/6k = 20OC + 1.0 x 10^6 W/m3 x (2 cm)2 / (6 x 16 W/m . OC) = 20.44OC
Answer: Therefore, the steady-state temperature for the center of the sphere is 20.44OC.
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Stainless steel sphere [k = 16 W/m . OC ] having a diameter of 4 cm is...
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Stainless steel sphere [k = 16 W/m . OC ] having a diameter of 4 cm is exposed to a convection environment at 20OC, h = 15 W/m2. OC. Heat is generated uniformly in the sphere at the rate of 1.0 MW/m3. Calculate the steady state temperature for the center of the sphere.?
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Stainless steel sphere [k = 16 W/m . OC ] having a diameter of 4 cm is exposed to a convection environment at 20OC, h = 15 W/m2. OC. Heat is generated uniformly in the sphere at the rate of 1.0 MW/m3. Calculate the steady state temperature for the center of the sphere.? for GATE 2024 is part of GATE preparation. The Question and answers have been prepared according to the GATE exam syllabus. Information about Stainless steel sphere [k = 16 W/m . OC ] having a diameter of 4 cm is exposed to a convection environment at 20OC, h = 15 W/m2. OC. Heat is generated uniformly in the sphere at the rate of 1.0 MW/m3. Calculate the steady state temperature for the center of the sphere.? covers all topics & solutions for GATE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Stainless steel sphere [k = 16 W/m . OC ] having a diameter of 4 cm is exposed to a convection environment at 20OC, h = 15 W/m2. OC. Heat is generated uniformly in the sphere at the rate of 1.0 MW/m3. Calculate the steady state temperature for the center of the sphere.?.
Stainless steel sphere [k = 16 W/m . OC ] having a diameter of 4 cm is exposed to a convection environment at 20OC, h = 15 W/m2. OC. Heat is generated uniformly in the sphere at the rate of 1.0 MW/m3. Calculate the steady state temperature for the center of the sphere.? for GATE 2024 is part of GATE preparation. The Question and answers have been prepared according to the GATE exam syllabus. Information about Stainless steel sphere [k = 16 W/m . OC ] having a diameter of 4 cm is exposed to a convection environment at 20OC, h = 15 W/m2. OC. Heat is generated uniformly in the sphere at the rate of 1.0 MW/m3. Calculate the steady state temperature for the center of the sphere.? covers all topics & solutions for GATE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Stainless steel sphere [k = 16 W/m . OC ] having a diameter of 4 cm is exposed to a convection environment at 20OC, h = 15 W/m2. OC. Heat is generated uniformly in the sphere at the rate of 1.0 MW/m3. Calculate the steady state temperature for the center of the sphere.?.
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Stainless steel sphere [k = 16 W/m . OC ] having a diameter of 4 cm is exposed to a convection environment at 20OC, h = 15 W/m2. OC. Heat is generated uniformly in the sphere at the rate of 1.0 MW/m3. Calculate the steady state temperature for the center of the sphere.?, a detailed solution for Stainless steel sphere [k = 16 W/m . OC ] having a diameter of 4 cm is exposed to a convection environment at 20OC, h = 15 W/m2. OC. Heat is generated uniformly in the sphere at the rate of 1.0 MW/m3. Calculate the steady state temperature for the center of the sphere.? has been provided alongside types of Stainless steel sphere [k = 16 W/m . OC ] having a diameter of 4 cm is exposed to a convection environment at 20OC, h = 15 W/m2. OC. Heat is generated uniformly in the sphere at the rate of 1.0 MW/m3. Calculate the steady state temperature for the center of the sphere.? theory, EduRev gives you an
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