Question

The forming section of a plastics plant puts out a continuous sheet of plastic that is \(1.2-\mathrm{m}\) wide and \(2-\mathrm{mm}\) thick at a rate of $15 \mathrm{~m} / \mathrm{min}$. The temperature of the plastic sheet is \(90^{\circ} \mathrm{C}\) when it is exposed to the surrounding air, and the sheet is subjected to airflow at \(30^{\circ} \mathrm{C}\) at a velocity of $3 \mathrm{~m} / \mathrm{s}$ on both sides along its surfaces normal to the direction of motion of the sheet. The width of the air cooling section is such that a fixed point on the plastic sheet passes through that section in $2 \mathrm{~s}$. Determine the rate of heat transfer from the plastic sheet to the air.

Short answer

Answer: The rate of heat transfer from the plastic sheet to the air is approximately 14.4 W.

Step-by-step solution

Find the plastic sheet's surface area.

In order to determine the heat transfer rate, we first need to find the surface area of the plastic sheet. We know that the sheet is 1.2 meters wide and 2 mm thick. The equation for the area of a rectangle is A = length * width, so we find the surface area by multiplying the given dimensions: A = 1.2 m * 0.002 m = 0.0024 m^2 Since the plastic sheet has two sides, we multiply this result by 2 to get the total surface area: Total Surface Area = 0.0024 m^2 * 2 = 0.0048 m^2

Calculate the heat transfer coefficient.

To determine the heat transfer coefficient (h), we will use Newton's law of cooling, which states: q = hA(T_hot - T_cold) where q is the heat transfer rate, A is the surface area, T_hot is the temperature of the hot substance (in this case, the plastic sheet), and T_cold is the temperature of the surroundings (in this case, the airflow temperature). We are given T_hot as 90°C and T_cold as 30°C, so we can plug them into the equation: q = h * 0.0048 m^2 * (90°C - 30°C) We now need to find the heat transfer coefficient (h). Since there's no information given about the material properties, we can assume a value for h. In general, for forced convection, h ranges between 10 and 100 W/(m^2*K). We can assume h = 50 W/(m^2*K) as a reasonable approximation.

Calculate the rate of heat transfer.

With the heat transfer coefficient found in Step 2, we can now calculate the rate of heat transfer (q) by plugging in the values into the Newton's law of cooling equation: q = 50 W/(m^2*K) * 0.0048 m^2 * (90°C - 30°C) q = 50 W/(m^2*K) * 0.0048 m^2 * 60 K q = 14.4 W So, the rate of heat transfer from the plastic sheet to the air is approximately 14.4 W.