Physics

Physics · thermodynamics

Problem

d. Estimate the maximum possible value of PwP_{w} as a fraction of PP, to one significant figure. Briefly explain how you obtained this estimate. Given constants: g=9.8 N/kgG=6.67×10−11 N⋅m2/kg2k=1/4πϵ0=8.99×109 N⋅m2/C2km=μ0/4π=10−7 T⋅m/Ac=3.00×108 m/skB=1.38×10−23 J/KNA=6.02×1023( mol)−1R=NAkB=8.31 J/(mol⋅K)σ=5.67×10−8 J/(s⋅m2⋅K4)e=1.602×10−19C1eV=1.602×10−19 Jh=6.63×10−34 J⋅s=4.14×10−15eV⋅sme=9.109×10−31 kg=0.511MeV/c2(1+x)n≈1+nx for ∣x∣≪1sin⁡θ≈θ−θ3/6 for ∣θ∣≪1cos⁡θ≈1−θ2/2 for ∣θ∣≪1\begin{array}{ll} g=9.8 \mathrm{~N} / \mathrm{kg} & G=6.67 \times 10^{-11} \mathrm{~N} \cdot \mathrm{m}^{2} / \mathrm{kg}^{2} \\ k=1 / 4 \pi \epsilon_{0}=8.99 \times 10^{9} \mathrm{~N} \cdot \mathrm{m}^{2} / \mathrm{C}^{2} & k_{\mathrm{m}}=\mu_{0} / 4 \pi=10^{-7} \mathrm{~T} \cdot \mathrm{m} / \mathrm{A} \\ c=3.00 \times 10^{8} \mathrm{~m} / \mathrm{s} & k_{\mathrm{B}}=1.38 \times 10^{-23} \mathrm{~J} / \mathrm{K} \\ N_{\mathrm{A}}=6.02 \times 10^{23}(\mathrm{~mol})^{-1} & R=N_{\mathrm{A}} k_{\mathrm{B}}=8.31 \mathrm{~J} /(\mathrm{mol} \cdot \mathrm{K}) \\ \sigma=5.67 \times 10^{-8} \mathrm{~J} /\left(\mathrm{s} \cdot \mathrm{m}^{2} \cdot \mathrm{K}^{4}\right) & e=1.602 \times 10^{-19} \mathrm{C} \\ 1 \mathrm{eV}=1.602 \times 10^{-19} \mathrm{~J} & h=6.63 \times 10^{-34} \mathrm{~J} \cdot \mathrm{s}=4.14 \times 10^{-15} \mathrm{eV} \cdot \mathrm{s} \\ m_{e}=9.109 \times 10^{-31} \mathrm{~kg}=0.511 \mathrm{MeV} / \mathrm{c}^{2} & (1+x)^{n} \approx 1+n x \text { for }|x| \ll 1 \\ \sin \theta \approx \theta-\theta^{3} / 6 \text { for }|\theta| \ll 1 & \cos \theta \approx 1-\theta^{2} / 2 \text { for }|\theta| \ll 1 \end{array} Extra Supplementary Reading Materials: Green Revolution 1{ }^{1} In this problem, we will investigate a simple thermodynamic model for the conversion of solar energy into wind. Consider a planet of radius RR, and assume that it rotates so that the same side always faces the Sun. The bright side facing the Sun has a constant uniform temperature T1T_{1}, while the dark side has a constant uniform temperature T2T_{2}. The orbit radius of the planet is R0R_{0}, the Sun has temperature TsT_{s}, and the radius of the Sun is RsR_{s}. Assume that outer space has zero temperature, and treat all objects as ideal blackbodies. Context question: a. Find the solar power PP received by the bright side of the planet. (Hint: the Stefan-Boltzmann law states that the power emitted by a blackbody with area AA is σAT4\sigma A T^{4}.) Context answer: P=πσTs4R2Rs2R02P=\pi \sigma T_{s}^{4} \frac{R^{2} R_{s}^{2}}{R_{0}^{2}} Extra Supplementary Reading Materials: In order to keep both T1T_{1} and T2T_{2} constant, heat must be continually transferred from the bright side to the dark side. By viewing the two hemispheres as the two reservoirs of a reversible heat engine, work can be performed from this temperature difference, which appears in the form of wind power. For simplicity, we assume all of this power is immediately captured and stored by windmills. Context question: b. The equilibrium temperature ratio T2/T1T_{2} / T_{1} depends on the heat transfer rate between the hemispheres. Find the minimum and maximum possible values of T2/T1T_{2} / T_{1}. In each case, what is the wind power PwP_{w} produced? Context answer: The minimum value is simply zero, Pw=0P_{w}=0. The maximum value is T2/T1=1T_{2} / T_{1}=1. Pw=0P_{w}=0. Context question: c. Find the wind power PwP_{w} in terms of PP and the temperature ratio T2/T1T_{2} / T_{1}. Context answer: Pw=x3(1−x)1+x3PP_{w}=\frac{x^{3}(1-x)}{1+x^{3}} P

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