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Chapter 5: Problem 16

Molten plastic at a temperature of \(510^{\circ} \mathrm{F}\) is angered through an extruder barrel by a screw oscupying \(s\) of the tarrel's volume (Fig. \(P 5.16\) ). The extruder is \(16 \mathrm{ft}\) long and has an inner dianeter of 8 in. The barrel is connected to an adspter having a volume of \(0.48 \mathrm{ft}^{3} .\) The adapter is then connected to a die of equal volume. The plastic exiting the die is immediately rolled into sheets. The line is producing 4 -ft widths of material at a rate of \(30 \mathrm{ft} / \mathrm{min}\) and a gauge thickness of 187 mil. What is the axial velocity, \(V_{1},\) of the plastic in the barrel' Assume that the plastic density is constant as it solidifies from a liquid (in the extruder) into a solid sheet.

Short Answer

Expert verified
Axial velocity \(V_1\) can be obtained after performing the above calculations using the given parameters and continuity equation, ensuring units are kept consistent through the calculations.

Step by step solution

01

Identify given Parameters

Given parameters include:- Length of extruder barrel, \(\ L = 16 ft\)- Inner diameter of barrel, \(\ d = 8 in\) (which needs to be converted to feet)- Volume of adapter, \( = 0.48 ft^3\)- Volume of die is said to be of equal volume to that of adapter, \( = 0.48 ft^3\)- Width of sheet produced \(W = 4 ft\)- Rate at which material is produced, \( = 30 ft/min\)- Thickness of sheet \( = 187 mil\) (which needs to be converted to feet)
02

Calculate Total Volume of Plastic Produced

The volume of plastic produced per minute can be calculated using the dimensions of the produced sheet. The volume is given by the product of width, thickness, and rate of production. Now, convert given thickness from mil to feet. \(1 mil = 1.0 * 10^{-3} in\), and \(1 in = 1/12 ft\). Therefore, thickness in feet is \(187 * 10^{-3} * 1/12 ft\). Next, calculate the volume per minute, \(volume = W * thickness * rate = 4 * 187 * 10^{-3} * 1/12 * 30 ft^3/min\)
03

Calculate the Axial Velocity

The axial velocity \(V_1\) is calculated using the volume flow rate and the cross-sectional area of the extruder barrel. The cross-sectional area \(A\) can be calculated using the given diameter of the barrel. \(A = \pi * (d/2)^2\). After converting the given diameter from inches to feet, substitute in the calculation. Finally, the axial velocity \(V_1\) can be calculated using the continuity equation where \(V_1 = Volume Flow rate / A\) with the volume flow rate from step 2 and A from this step.

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