Question

Consider a 150-W incandescent lamp. The filament of the lamp is $5 \mathrm{~cm}\( long and has a diameter of \)0.5 \mathrm{~mm}$. The diameter of the glass bulb of the lamp is \(8 \mathrm{~cm}\). Determine the heat flux in \(\mathrm{W} / \mathrm{m}^{2}(a)\) on the surface of the filament and \((b)\) on the surface of the glass bulb, and (c) calculate how much it will cost per year to keep that lamp on for eight hours a day every day if the unit cost of electricity is \(\$ 0.08 / \mathrm{kWh}\).

Short answer

Answer: The heat flux on the surface of the filament is \(1.909 \times 10^5\, \mathrm{W/m^2}\), and on the surface of the glass bulb, it's \(7469.27\, \mathrm{W/m^2}\).

Step-by-step solution

Find the surface area of the filament and the glass bulb

To find the heat flux on the surfaces, we first need to calculate the surface area of both the filament and the glass bulb. The filament is a cylinder with a length of \(5 \mathrm{~cm}\) and a diameter of \(0.5 \mathrm{~mm}\). The glass bulb is a sphere with a diameter of \(8 \mathrm{~cm}\). The surface area of a cylinder \(A_{cylinder}\) is given by the formula \(A_{cylinder} = 2\pi r_{cylinder} h + 2\pi r_{cylinder}^2\), and the surface area of a sphere \(A_{sphere}\) is given by the formula \(A_{sphere} = 4\pi r_{sphere}^2\).

Calculate the heat flux on the filament surface and the glass bulb surface

Next, we need to find the heat flux on the surfaces, which can be determined by dividing the power of the lamp by the surface area of the respective surfaces. The heat flux on the filament surface \(q_{filament}\) can be found using this formula: \(q_{filament} = \frac{P}{A_{cylinder}}\). Similarly, the heat flux on the glass bulb surface \(q_{glass bulb}\) can be calculated using this formula: \(q_{glass bulb} = \frac{P}{A_{sphere}}\)

Calculate the yearly cost of keeping the lamp on for eight hours a day

To calculate the yearly cost, we need to find the energy consumption per day by multiplying the power of the lamp by the hours per day it is on (\(150 \mathrm{W} \times 8 \mathrm{hrs}\)) and then convert this to kilowatt-hours by dividing by 1000. Multiply the daily energy consumption by the unit cost of electricity and the number of days in a year to get the yearly cost. Now, let's plug in the values and calculate the solutions:

Calculations

Let's calculate the surface area of the filament (cylinder) and the glass bulb (sphere). For the filament: \(r_{cylinder} = \frac{0.5\,\mathrm{mm}}{2} = 0.25 \,\mathrm{mm} = 2.5 \times 10^{-4} \mathrm{m}\) \(h = 5 \,\mathrm{cm} = 0.05\,\mathrm{m}\) \(A_{cylinder} = 2\pi (2.5 \times 10^{-4}\,\mathrm{m}) (0.05\,\mathrm{m}) + 2\pi (2.5 \times 10^{-4}\,\mathrm{m})^2 = 7.854 \times 10^{-4}\,\mathrm{m}^2\) For the glass bulb: \(r_{sphere} = \frac{8\,\mathrm{cm}}{2} = 4\,\mathrm{cm} = 0.04\,\mathrm{m}\) \(A_{sphere} = 4\pi (0.04\,\mathrm{m})^2 = 2.0106 \times 10^{-2}\,\mathrm{m}^2\)

Calculations

Calculate the heat flux on the filament surface and the glass bulb surface. For the filament: \(q_{filament} = \frac{150\,\mathrm{W}}{7.854 \times 10^{-4}\,\mathrm{m}^2} = 1.909 \times 10^5\,\mathrm{W/m^2}\) For the glass bulb: \(q_{glass bulb} = \frac{150\,\mathrm{W}}{2.0106 \times 10^{-2}\,\mathrm{m}^2} = 7469.27\,\mathrm{W/m^2}\)

Calculations

Calculate the yearly cost of keeping the lamp on for eight hours a day. Daily energy consumption: \((150\,\mathrm{W})(8\,\mathrm{hrs}) = 1200\,\mathrm{Wh} = 1.2\,\mathrm{kWh}\) Yearly energy consumption: \((1.2\,\mathrm{kWh})(365\,\mathrm{days}) = 438\,\mathrm{kWh}\) Yearly cost: \((438\,\mathrm{kWh})(\$0.08/\mathrm{kWh}) = \$35.04\) The heat flux on the surface of the filament is \(1.909 \times 10^5\, \mathrm{W/m^2}\), and on the surface of the glass bulb, it's \(7469.27\, \mathrm{W/m^2}\). The yearly cost for keeping the lamp on for eight hours a day every day is $35.04.