Annealing is a process by which steel is reheated and then cooled to make it less brittle. Consider the reheat stage for a 100-mm-thick stee

Question

Annealing is a process by which steel is reheated and then cooled to make it less brittle. Consider the reheat stage for a 100-mm-thick steel plate ( 7830 kg/m3, 550 J/kg·K, 48 W/m·K), which is initially at a uniform temperature of 170°C and is to be heated to a minimum temperature of 550°C. Heating is effected in a gas-fired furnace, where products of combustion at [infinity] 800°C maintain a convection coefficient of 250 W/m2·K on both surfaces of the plate. How long should the plate be left in the furnace?

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Thu Giang 4 years 2021-08-21T14:44:55+00:00 1 Answers 611 views 1

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  1. Amity
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    2021-08-21T14:46:26+00:00
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    Answer:

    The answer to the question is;

    The plate be left in the furnace for 905.69 seconds.

    Explanation:

    To solve the question, we have to check the Bi number as follows

    Bi = \frac{hL}{k} = \frac{250\frac{W}{m^{2} K} *0.05 m}{48\frac{W}{mK} } = 0.2604

    As the Bi number is > 0.1 we have to account for the variation of temperature with location in the mass.

    We perform nonlumped analysis

    The relation for heat transfer given by

    Y =  \frac{T_f-T_{inf}}{T_i- T_{inf}}

    =\frac{550-800}{170- 800} = 0.3968 = C₁ exp (ζ₁² F₀)  

    where

    C₁ and ζ₁ are coefficients of a series solution

    We therefore look for the values of C₁ and ζ₁ from Bi tables to be

    ζ₁ = 0.4801 +(0.26-0.25) (0.5218-0.4801)/(0.3-0.25) ≈ 0.4884 and

    C₁ = 0.4801 +(0.26-0.25) (1.0450 – 1.0382)/(0.3-0.25) ≈ 1.03956 and  

    This gives the relation

    0.3968 = 1.03956 exp (ζ₁² F₀)  

    or ζ₁² (\frac{\alpha t}{L^2})

    where

    α = Thermal diffusivity of solid = k/(ρ·c_p) = \frac{48}{7830*550} = 1.1146×10⁻⁵

    c_p = Specific heat capacity of solid at constant pressure = 550 J/kg·K

    ρ = Density of the solid = 7830 kg/m³

    =㏑(\frac{0.3968 }{1.03956 }) = -0.9631 from where we have

    t = \frac{0.9631 *0.05^{2} }{0.4884^2*1.11*10^{-5}}  = 905 seconds.

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