Aluminium wires, 3 mm in diameter, are produced by...

Aluminium wires, 3 mm in diameter, are produced by extrusion. The wires leave the extruder at an average temperature of $350^{\circ} \mathrm{C}$ and at a linear rate of $10 \mathrm{~m} / \mathrm{min}$. Before leaving the extrusion room, the wires are cooled to an average temperature of $50^{\circ} \mathrm{C}$ by transferring heat to the surrounding air at $25^{\circ} \mathrm{C}$ with a heat transfer coefficient of $50 \mathrm{~W} / \mathrm{m}^2 \cdot \mathrm{~K}$. Calculate the necessary length of the wire cooling section in the extrusion room.

Key Concepts

Lumped Capacitance Analysis

This concept involves simplifying the transient heat transfer analysis by assuming that the object’s temperature is uniform throughout its volume. When the Biot number is small, the resistance to internal conduction is negligible compared to the convective resistance at the surface, allowing the object’s temperature to be treated as a single value that decreases exponentially over time. This method is crucial for calculating the cooling time required for the extruded product, and thereby the necessary cooling length based on its linear speed.

Convective Heat Transfer Coefficient

The convective heat transfer coefficient, often denoted as h, quantifies the rate of heat transfer per unit area between the surface of the wire and the surrounding air. It reflects the efficiency of the cooling process provided by convection. In designing cooling systems, this coefficient is combined with the surface area to determine the overall rate of heat loss, which is essential for calculating the time needed for the wire to cool from its initial to its final temperature.

Cooling Section Design and Time-Length Relationship

This concept connects the time required for the wire to cool to the physical length of the cooling section in the extrusion room. By determining the cooling time using methods like lumped capacitance analysis, and knowing the linear speed at which the wire travels, one can calculate the necessary length of the cooling zone. This integration of time-dependent cooling behavior with the production process’s speed is fundamental in process design for extrusion operations.

Transcript

00:01 All right everyone, so what we have is long aluminum wires with the diameter of 5 millimeters and a density of 2 ,702 kilograms per cubic meter and a cp value of 0 .896 kilojoules per kilogram degrees celsius.

00:15 That's extruded at a temperature of 250 degrees c and cool to 50 degrees celsius in air.

00:21 That's at 25 degrees c.

00:23 Or as to solve for is the heat transfer from the wire if we know the velocity of the wire is 8 meters per minute.

00:30 So what we're going to do is we're going to write down our assumptions.

00:33 We're going to assume that things are operating at steady -state conditions, so conditions or properties are not changing with time.

00:39 And we're also going to assume that all of our thermal properties are constant as well, so there's no change in them as conditions are changing.

00:52 So what we can do next is just write down the properties of our aluminum that we were given in the problem statement.

00:57 So we know the density of aluminum is going to be 2 ,702 kilograms, cubic meter and we're also given that the cp value of aluminum is 0 .896 kilojoules per kilogram degrees celsius so what we can do first is calculate the mass flow rate of the extruded wire through the air so what we can do is we can say that the mdot wire will be equal to row times the volume flow rate and that's going to be equal to row times pi zero squared so the area times volume.

01:38 So this v is actually velocity, this v right here, since we pulled the area out.

01:43 So what this allows us to do is just use the quantities that are given to us in the problem to go ahead and solve for the mass flow rate over a wire...

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