A long 0.6 m OD ANSI 347 stainless steel (use k=15.8 W/mK) cylindrical billet at 290 degrees C is placed into cooling chamber where the temperature is 30 degrees C. If the average heat transfer coefficient is 170 W/m^2K, estimate the time in minutes required for the center line temperature to reach 232 degrees C.
To estimate the time required for the centerline temperature of the cylindrical billet to reach 232 degrees C, we can use Fourier's law of heat conduction.
Fourier's law states that the heat transfer rate (Q) through a material is proportional to the temperature difference (ΔT), the cross-sectional area (A), and the thermal conductivity of the material (k). Mathematically, it can be expressed as:
Q = k * A * (ΔT / L)
where:
Q = heat transfer rate (W)
k = thermal conductivity of the material (W/mK)
A = cross-sectional area (m^2)
ΔT = temperature difference (K)
L = length of the material (m)
We can rearrange this equation to solve for the temperature difference ΔT:
ΔT = Q * L / (k * A)
Given that the outer diameter (OD) of the cylindrical billet is 0.6 m, we can calculate the cross-sectional area A:
A = π * (OD/2)^2
Substituting the given values:
OD = 0.6 m
k = 15.8 W/mK
Q = 170 W/m^2K (average heat transfer coefficient)
L = unknown
ΔT = 290 - 30 = 260 degrees C
We need to convert the temperature difference from degrees Celsius to Kelvin by adding 273.15:
ΔT = 260 + 273.15 = 533.15 K
Now we can solve for the length of the cylinder L:
L = Q * A / (k * ΔT)
Substituting the known values:
L = 170 * π * (0.6/2)^2 / (15.8 * 533.15)
L ≈ 0.0039 m
Finally, to estimate the time required for the centerline temperature to reach 232 degrees C, we'll use the concept of thermal diffusivity (α) defined as the ratio of the thermal conductivity (k) to the product of the material density (ρ) and specific heat capacity (c). Mathematically, α = k / (ρ * c).
Given that this is a stainless steel billet, we can approximate the density of stainless steel as around 8000 kg/m^3, and the specific heat capacity at a constant pressure as approximately 500 J/(kg*K).
Using the thermal diffusivity, we can calculate the time required for the centerline temperature to reach 232 degrees C (505.15 K).
t = (L^2 / α) * ln((T_final - T_initial) / (T_final - T_center))
Where:
t = time (s)
L = length of the material (m)
α = thermal diffusivity (m^2/s)
T_final = final temperature (K)
T_initial = initial temperature (K)
T_center = centerline temperature (K)
Now, we can substitute the known values:
t = (0.0039^2 / (15.8 / (8000 * 500))) * ln((505.15 - 303.15) / (505.15 - 505.15))
t ≈ 7.09 seconds
Therefore, converting the time to minutes:
t ≈ 7.09 * 1/60 ≈ 0.12 minutes
So, it would take approximately 0.12 minutes (or 7.09 seconds) for the centerline temperature to reach 232 degrees C.