For sand-casting a steel rectangular plate with dimensions 80mm x 120mm x 20mm, a cylindrical riser has to be designed. The height of the riser is equal to its diameter. The total solidification time for the casting is 2 minutes. In Chvorinov’s law for the estimation of the total solidification time, exponent is to be taken as 2. For a solidification time of 3 minutes in the riser, the diameter (in mm) of the riser is __________ (correct to two decimal places).Correct answer is '51.87'. Can you explain this answer?

Question

For sand-casting a steel rectangular plate with dimensions 80mm x 120mm x 20mm, a cylindrical riser has to be designed. The height of the riser is equal to its diameter. The total solidification time for the casting is 2 minutes. In Chvorinov&rsquo;s law for the estimation of the total solidification time, exponent is to be taken as 2. For a solidification time of 3 minutes in the riser, the diameter (in mm) of the riser is __________ (correct to two decimal places).Correct answer is '51.87'. Can you explain this answer?

Answer

's rule, the constant 'n' is assumed to be 2.5. The pouring temperature of the molten steel is 1500°C, and the mold temperature is 300°C. The specific heat of steel is 0.5 J/g°C, and the latent heat of fusion is 270 J/g. The density of steel is 7.8 g/cm³.

To design the riser, we need to calculate the volume of the rectangular plate and the volume of the riser. The volume of the plate is:

V_plate = 80mm x 120mm x 20mm = 192,000 mm³ = 192 cm³

The volume of the riser should be equal to the volume of the solidified metal that will shrink as it cools and solidifies. This is calculated using Chvorinov's rule:

t = (C / A)^n

where t is the solidification time, C is a constant, A is the surface area of the casting, and n is a constant. Solving for C:

C = t x A^(1/n)

The surface area of the rectangular plate is:

A_plate = 2(80mm x 120mm) + 2(80mm x 20mm) + 2(120mm x 20mm) = 32,400 mm² = 32.4 cm²

Substituting the given values:

C = 2 min x (32.4 cm²)^(1/2.5) = 1.96 cm³/min

The volume of the riser is:

V_riser = V_plate / (1 - C / (pi x D^2 x H))

where D is the diameter of the riser and H is its height. We are given that H = D, so:

V_riser = V_plate / (1 - C / (pi x D^3))

Solving for D using the given values:

V_riser = 192 cm³ / (1 - 1.96 cm³/min / (pi x D^3)) = 266.5 cm³

D^3 = 1.96 cm³/min / (pi x (1 - 192 cm³ / 266.5 cm³)) = 2.84 cm³/min

D = (2.84 cm³/min)^(1/3) = 1.34 cm

So the diameter and height of the riser should both be 1.34 cm.

Finally, we can calculate the total amount of heat that needs to be removed from the casting during solidification. The specific heat of steel is 0.5 J/g°C, so the amount of heat needed to cool the casting from 1500°C to 300°C is:

Q_sensible = (80mm x 120mm x 20mm x 7.8 g/cm³) x (1500°C - 300°C) x 0.5 J/g°C = 4,147,200 J

The latent heat of fusion is 270 J/g, so the amount of heat needed to solidify the casting is:

Q_latent = (80mm x 120mm x 20mm x 7.8 g/cm³) x 270 J/g = 4,665,600 J

The total amount of heat that needs to be removed is the

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