Complete question is;
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 plate (ρ = 7830 kg/m³, Cp = 550 J/Kg.K, k = 48 W/m K). The plate initially is at 200 °C and is to be heated to a minimum temperature of 550 °C. Heating is effected in a gas-fired furnace where the products of combustion at T∞ = 800 °C maintain a convection heat transfer coefficient of h = 250 W/m.K on both surfaces of the plate. How long should the plate be left in the furnace?
Answer:
860 seconds
Step-by-step explanation:
We are given;
Initial Temperature; T_i = 200 °C
Minimum Temperature; T_o = 550 °C
T∞ = 800 °C
convection coefficient; h = 250 W/m².K
ρ = 7830 kg/m³
Cp = 550 J/kg K
k = 48 W/m K
Plate thickness = 100mm
Thus,L = 100/2 = 50mm = 0.05 m
Let's find the biot number from the formula;
Bi = hL/K
Bi = (250 × 0.05)/48
Bi = 0.2604
Now, lowest temperature in the slab is given as;
θ_o = (T_o - T∞)/(T_i - T∞)
θ_o = (550 - 800)/(200 - 800)
θ_o = 0.4167
Now, from online tables calculation, we can find the root of the biot number.
Thus, root of the biot number Bi = 0.2604 is;
ζ1 = 0.488 rad
Also, C1 is gotten as 1.0396
Now,formula for thermal diffusivity is;
α = k/ρc
α = 48/(7830 × 550)
α = 1.115 × 10^(-5) m²/s
Also, from online tables, f(ζ1) = 0.401
Thus, we can find the time the plate should the plate be left in the furnace from;
-(ζ1)²(αt/L²) = In 0.401
-(ζ1)²(αt/L²) = -0.9138
t = (-0.9138 × 0.05²)/-(0.488² × 1.115 × 10^(-5))
t ≈ 860 s