Question

The convection coefficient for flow over a solid sphere may be determined by submerging the sphere, which is initially at 25 °C, into the flow, which is at 75 °C, and measuring its surface temperature at some time during the transient heating process. If the sphere has a diameter of 0.1 m, a thermal conductivity of 24 W/(m·K), and a thermal diffusivity of 2.0 m2/s, at what time will a surface temperature of 60 ºC be recorded if the convection coefficient is 300 W/(m2·K)?

Answer 1

To determine the time at which a surface temperature of 60 ºC will be recorded on the solid sphere, we can use the equation for transient heat conduction. Solving for t, we find that at approximately 0.007 seconds, a surface temperature of 60 ºC will be reached.

Step-by-step explanation:

To determine the time at which a surface temperature of 60 ºC will be recorded on the solid sphere, we can use the equation for transient heat conduction:

θ = θ∞ + (θ0 - θ∞) * exp(-Bi * Fo)

Where:

• θ is the dimensionless temperature difference
• θ∞ is the dimensionless initial temperature difference
• θ0 is the dimensionless temperature difference at time t
• Bi is the Biot number (h*L/k)
• Fo is the Fourier number (α*t/r^2)
• h is the convection coefficient
• L is the characteristic length (diameter of the sphere)
• k is the thermal conductivity
• α is the thermal diffusivity
• t is the time
• r is the radius of the sphere

Plugging in the values, we have:

60 = 50 + (25 - 50) * exp(-300 * (2.0 * t / (0.1/2)^2))

Solving for t, we find that at t ≈ 0.007 seconds, a surface temperature of 60 ºC will be reached.