Question

In a production facility, 3 cm thick large brass plates (k = 110 W/mC, α = 33.9 × 10-6 m2 /s) that are initially at a uniform temperature Ti = 25C are heated by passing through an oven maintained at 700C. The plates remain in the oven for a period of 10 minutes. The average heat transfer coefficient in the oven h = 80 W/m2 C. Determine the surface temperature (C) of the plates when they come out of the oven

Answer 1

Answer:

Explanation:

Given.

Thickness of brass plate $t = 3 cm$

Thermal conductivity of brass $k = 110 W/m.°C$

Density of brass $\rho = 8530 kg/m^3$

Specific heat of brass $C_p =380J/kg.°C$

Thermal diffusivity of brass $\alpha = 33.9\times 10^{-6} m^2/s$

Temperature of oven $T_{\infty} = 700°C$

The initial temperature $T_i= 25°C$

Plate remain in the oven $t =10 min$  

Heat conduction in the plate is one-dimensional since the plate is large relative to its thickness  and there is thermal symmetry about the center plane.

The thermal properties of the plate are constant.

The heat transfer coefficient is constant and uniform over the entire surface.

The Fourier number is > 0.2 so that the one-term approximate solutions (or the transient  temperature charts) are applicable (this assumption will be verified).

The Biot number for this process $Bi = \frac{hL}{k}\\\\Bi=\frac{(80 W/m^2.°C)(0.015 m)}{(110 W/m.°C)}\\=Bi =0.0109$

The constants $\lambda_1$ and $A_1$ corresponding to this Biot are, from 11-2 tables.

The interpolation method used to find the

$\lambda_1=0.1039$  and $A_1=1.0018$  

The Fourier number $\tau=\frac{\alpha t}{L^2}\\\\\tau=\frac{(33.9\times 10^{-6} m^2/s)(10 min \times 60 s/min)}{(0.015m)^2} \\\\\tau=90.4>0.2$

Therefore, the one-term approximate solution (or the transient temperature charts) is applicable.

Then the temperature at the surface of the plates becomes

$\theta(L,t)_{wall}=\frac{T(x,t)-T_{\infty}}{(T_i-T_{\infty})}\\\\\theta(L,t)_{wall}=A_1e^{-\lambda_1^2\tau}\cos(\lambda_1L/L)\\\\\theta(L,t)_{wall}=(1.0018)e^{-(0.1039^2(90.4))}\cos(0.1039)\\\\\theta(L,t)_{wall}=0.378\\\\\frac{T(L,t)-700}{25-700}=0.378\\\\T(L,t)=445°C$

Answer 2

Answer: final temperature is 40.27°C

Explanation:

Detailes calculation is shown in the image below