Question

A 200 mm thick slab is rolled using 500 mm diameter rolls under cold rolling and hot rolling conditions, separately. The coefficient of friction is 0.04 in cold rolling and 0.4 in hot rolling. The ratio of maximum possible thickness reduction in cold rolling to that in hot rolling is ................... (round off to 2 decimal places).

Hint:

The maximum draft formula \( \Delta h_{max} = \mu^2 R \) is a fundamental concept in rolling. Notice how sensitive the draft is to the coefficient of friction (a squared relationship). This is why hot rolling, with its much higher friction coefficient, allows for significantly larger reductions per pass compared to cold rolling.

Answer 1

Step 1: Understanding the Concept:
In rolling, the maximum possible reduction in thickness in a single pass, known as the maximum draft, is limited by the ability of the friction between the rolls and the slab to pull the material into the roll gap. This limit depends on the coefficient of friction and the roll radius.
Step 2: Key Formula or Approach:
The maximum possible draft (\(\Delta h_{max}\)) is given by the formula: \[ \Delta h_{max} = \mu^2 R \] where \(\mu\) is the coefficient of friction and \(R\) is the radius of the rolls. We need to calculate this for both cold and hot rolling conditions and then find their ratio.
Step 3: Detailed Calculation:
Given values:
- Roll diameter = 500 mm, so roll radius \(R = 250\) mm.
- Coefficient of friction for cold rolling, \(\mu_{cold} = 0.04\).
- Coefficient of friction for hot rolling, \(\mu_{hot} = 0.4\).
1. Calculate maximum thickness reduction for cold rolling:
\[ (\Delta h_{max})_{cold} = (\mu_{cold})^2 R = (0.04)^2 \times 250 \] \[ (\Delta h_{max})_{cold} = 0.0016 \times 250 = 0.4 \text{ mm} \] 2. Calculate maximum thickness reduction for hot rolling: \[ (\Delta h_{max})_{hot} = (\mu_{hot})^2 R = (0.4)^2 \times 250 \] \[ (\Delta h_{max})_{hot} = 0.16 \times 250 = 40 \text{ mm} \] 3. Calculate the ratio: \[ \text{Ratio} = \frac{(\Delta h_{max})_{cold}}{(\Delta h_{max})_{hot}} = \frac{0.4}{40} = 0.01 \] Step 4: Final Answer:
The ratio is 0.01.