A 3-in-thick slab is 12 in wide and 15 ft long. Thickness of the slab is reduced in three steps in a hot rolling operation. In each step, thickness is reduced by 20% and width increases by 3%. If the entry speed of the slab in the first step is 40 ft/min, and roll speed is the same for the three steps, determine (a) length and (b) exit velocity of the slab after the final reduction.

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zainsubhani zainsubhani · Answer 1 · 2020-01-23T08:18:10+01:00

Answer:

[tex]L_f=26.8108 ft[/tex]

Part B:

For Final Reduction

[tex]v_f=48.5436ft/min[/tex]

Explanation:

Part A:

At each step 0.8 (100-20)% of thickness is left

Final Thickness t_f:

[tex]t_f=(0.80)(0.80)(0.80)*3\\t_f=1.536 in[/tex]

Width increases by 0.03 in each step so (100+3)%=1.03

Final Width w_f:

[tex]w_f=(1.03)(1.03)(1.03)*12\\w_f=13.1127 in[/tex]

Conservation of volume:

[tex]t_ow_oL_o=t_fw_fL_f\\L_f=\frac{t_ow_oL_o}{t_fw_f} \\L_f=\frac{3*12*(15*12)}{1.536*13.1127}\\L_f=321.730 in\\L_f=26.8108 ft[/tex]

Part B:

[tex]t_ow_ov_o=t_fw_fv_f[/tex]

At First reduction exit Velocity:

[tex]v_f=\frac{t_ow_oL_o}{t_fw_f} \\v_f=\frac{3*12*(40)}{0.8*3*1.03*12}\\v_f=48.5439ft/min[/tex]

At 2nd Reduction:

[tex]v_f=\frac{0.8*3*1.03*12*40}{0.8^2*3*1.03^2*12}\\v_f=48.5436ft/min[/tex]

For Final Reduction:

[tex]v_f=\frac{0.8^2*3*1.03^2*12*40}{0.8^3*3*1.03^3*12}\\v_f=48.5436ft/min[/tex]