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A solid sphere with an initial temperature Ti is immersed in a large thermal reservoir of
temperature To . The sphere reaches a steady temperature after a certain time t1 . If the radius of the
sphere is doubled, the time required to reach steady-state will be
temperature To . The sphere reaches a steady temperature after a certain time t1 . If the radius of the
sphere is doubled, the time required to reach steady-state will be
- a)t1/4
- b)t1/2
- c)2t1
- d)4t1
Correct answer is option 'C'. Can you explain this answer?
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A solid sphere with an initial temperature Ti is immersed in a large t...
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A solid sphere with an initial temperature Ti is immersed in a large t...
Solution:
Given:
Ti: Initial temperature of solid sphere
To: Temperature of thermal reservoir
t1: Time taken by sphere to reach steady-state
Let's consider the heat flow from the thermal reservoir to the solid sphere at steady-state. The heat flow is given by the formula:
Q = k * A * (T1 - T2) / d
where
Q: Heat flow
k: Thermal conductivity of the material
A: Surface area of the sphere
T1: Temperature of the thermal reservoir (To)
T2: Temperature of the sphere
d: Diameter of the sphere
As we know that the sphere reaches steady-state after time t1, so the temperature of the sphere at steady-state is:
T2 = To + (Ti - To) * exp(-t1/tau)
where
tau = (rho * V * c) / (4 * k * pi * r^2)
rho: Density of the material
V: Volume of the sphere
c: Specific heat capacity of the material
r: Radius of the sphere
Now, let's consider the case where the radius of the sphere is doubled (2r). The new volume of the sphere is:
V' = (4/3) * pi * (2r)^3 = 8 * V
The new surface area of the sphere is:
A' = 4 * pi * (2r)^2 = 16 * A
The new diameter of the sphere is:
d' = 2 * (2r) = 4r
The new tau is:
tau' = (rho * V' * c) / (4 * k * pi * (2r)^2) = tau / 2
Now, let's find the time required for the sphere to reach steady-state with radius 2r (t2):
T2' = To + (Ti - To) * exp(-t2/tau')
T2' = To + (Ti - To) * exp(-2*t1/tau)
T2' - To = (Ti - To) * exp(-2*t1/tau)
(T2' - To) / (Ti - To) = exp(-2*t1/tau)
ln((T2' - To) / (Ti - To)) = -2*t1/tau
ln((T2' - To) / (Ti - To)) = -t1/tau'
ln((T2' - To) / (Ti - To)) = -2*t1/(tau/2)
ln((T2' - To) / (Ti - To)) = -4*t1/tau
t2 = 4 * t1
Therefore, the time required for the sphere to reach steady-state with radius 2r is 4 times the time required for the sphere to reach steady-state with radius r. Hence, the correct option is (C).
Given:
Ti: Initial temperature of solid sphere
To: Temperature of thermal reservoir
t1: Time taken by sphere to reach steady-state
Let's consider the heat flow from the thermal reservoir to the solid sphere at steady-state. The heat flow is given by the formula:
Q = k * A * (T1 - T2) / d
where
Q: Heat flow
k: Thermal conductivity of the material
A: Surface area of the sphere
T1: Temperature of the thermal reservoir (To)
T2: Temperature of the sphere
d: Diameter of the sphere
As we know that the sphere reaches steady-state after time t1, so the temperature of the sphere at steady-state is:
T2 = To + (Ti - To) * exp(-t1/tau)
where
tau = (rho * V * c) / (4 * k * pi * r^2)
rho: Density of the material
V: Volume of the sphere
c: Specific heat capacity of the material
r: Radius of the sphere
Now, let's consider the case where the radius of the sphere is doubled (2r). The new volume of the sphere is:
V' = (4/3) * pi * (2r)^3 = 8 * V
The new surface area of the sphere is:
A' = 4 * pi * (2r)^2 = 16 * A
The new diameter of the sphere is:
d' = 2 * (2r) = 4r
The new tau is:
tau' = (rho * V' * c) / (4 * k * pi * (2r)^2) = tau / 2
Now, let's find the time required for the sphere to reach steady-state with radius 2r (t2):
T2' = To + (Ti - To) * exp(-t2/tau')
T2' = To + (Ti - To) * exp(-2*t1/tau)
T2' - To = (Ti - To) * exp(-2*t1/tau)
(T2' - To) / (Ti - To) = exp(-2*t1/tau)
ln((T2' - To) / (Ti - To)) = -2*t1/tau
ln((T2' - To) / (Ti - To)) = -t1/tau'
ln((T2' - To) / (Ti - To)) = -2*t1/(tau/2)
ln((T2' - To) / (Ti - To)) = -4*t1/tau
t2 = 4 * t1
Therefore, the time required for the sphere to reach steady-state with radius 2r is 4 times the time required for the sphere to reach steady-state with radius r. Hence, the correct option is (C).
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