Question

Heat is generated in a long \(0.3-\mathrm{cm}\)-diameter cylindrical electric heater at a rate of \(150 \mathrm{~W} / \mathrm{m}^{3}\). The heat flux at the surface of the heater in steady operation is (a) \(42.7 \mathrm{~W} / \mathrm{cm}^{2}\) (b) \(159 \mathrm{~W} / \mathrm{cm}^{2}\) (c) \(150 \mathrm{~W} / \mathrm{cm}^{2}\) (d) \(10.6 \mathrm{~W} / \mathrm{cm}^{2}\) (e) \(11.3 \mathrm{~W} / \mathrm{cm}^{2}\)

Short answer

a) 1.0 W/cm² b) 0.1 W/cm² c) 0.5 W/cm² d) 0.01 W/cm² e) 2 W/cm²

Step-by-step solution

Determine the radius and volume of the cylindrical heater

First, we need to determine the radius of the cylinder from the given diameter, which is \(0.3\) cm. Then, we will use the formula for the volume of a cylinder, which is \(V=\pi r^2h\). Since we are looking for a \(1 \mathrm{m}\) length of the rod, we take \(h = 100 \mathrm{cm}\). In our case, \(r = 0.15 \mathrm{cm}\), so we can find the volume as follows: $$ V=\pi (0.15)^2(100) $$

Calculate total heat generated in the cylinder

Now, let's use the given heat generation rate of \(150 \mathrm{W/m^3}\) to find the total heat generated in the \(1 \mathrm{m}\) length of the cylinder. To do so, just multiply the heat generation rate by the volume that we calculated in Step 1: $$ Q = (150) \times V $$

Determine the surface area of the cylindrical heater

In order to calculate the heat flux at the surface of the heater, we need to find its surface area. The formula for the surface area of a cylinder is \(A = 2\pi rh\). Given that \(r = 0.15 \mathrm{cm}\) and \(h = 100 \mathrm{cm}\), the surface area can be calculated as: $$ A = 2\pi (0.15)(100) $$

Calculate the heat flux at the surface of the cylindrical heater

Finally, with the total heat generated and the surface area of the cylinder, we can find the heat flux by dividing the heat generated by the surface area. This will give us the heat flux in \(W/cm^2\): $$ q = \frac{Q}{A} $$

Compare the calculated heat flux with the given options

Now, we need to compare our calculated heat flux with the given options (a) through (e) to see which one matches the closest. The correct option will be the one that best represents the heat flux at the surface of the cylindrical heater. Based on these steps, you should be able to find the correct answer to this problem and determine the heat flux at the surface of the cylindrical heater in steady operation.