Respuesta :
Answer:
The time for the surface temperature to reach 25 ◦C is 3.24 hr
Explanation:
Assumptions
1. 1-D heat conduction
2. constant thermal properties
3. the heat transfer coefficient is constant & uniform over the surface.
Properties
k = 0.79 W/m · ◦C
α = 5.94 × 10−7 m^2/s
ρ = 1600 kg/m^3
Cp = 0.84 kJ/kg · ◦C
Step 1: Check the Biot number
[tex]Bi = \frac{hr_0}{k}[/tex]
[tex]= \frac{(14)(0.15 m)}{0.79} = 2.658 [/tex]
As Bi > 0.1 hence lumped system analysis will not considered.
Check the Fourier number.
From Table 18-1
[tex]\lambda_1 = 1.7240, A_1 = 1.3915[/tex]
From Table 18-2, we can find
[tex]J_0 = 0.3841[/tex]
The Fourier number is determined as
[tex]\frac{T (r_0, t) - T_{\infty}}{(Ti -T\infty)} = A_1 e^{−\lambda^2 \tau} J_0(\lambda_1 r/r_0)[/tex]
[tex]\frac{25 - 28}{14 - 28} = (1.3915)e^{−(1.7240)^2 \tau} (0.3841 ·1)[/tex]
solving τ
[tex]\tau = 0.3082[/tex]
Since τ > 0.2, the one term approximation or the Heisler charts can be used.
The time for the surface temperature to reach 25 ◦C is
[tex]t = \frac{\tau r^2}{\alpha}[/tex]
[tex]= \frac{(0.3082)(0.15 m)^2}{(5.94 \times 10^{−7})}[/tex]
= 11674.24 s = 3.24 hours
