2.2.1. Derive the expression of · OlympiadBench #1342

Physics · thermodynamics

Problem

2.2.1. Derive the expression of Γ\Gamma Given constants: CHANGE OF AIR TEMPERATURE WITH ALTITUDE, ATMOSPHERIC STABILITY AND AIR POLLUTION Vertical motion of air governs many atmospheric processes, such as the formation of clouds and precipitation and the dispersal of air pollutants. If the atmosphere is stable, vertical motion is restricted and air pollutants tend to be accumulated around the emission site rather than dispersed and diluted. Meanwhile, in an unstable atmosphere, vertical motion of air encourages the vertical dispersal of air pollutants. Therefore, the pollutants' concentrations depend not only on the strength of emission sources but also on the stability of the atmosphere. We shall determine the atmospheric stability by using the concept of air parcel in meteorology and compare the temperature of the air parcel rising or sinking adiabatically in the atmosphere to that of the surrounding air. We will see that in many cases an air parcel containing air pollutants and rising from the ground will come to rest at a certain altitude, called a mixing height. The greater the mixing height, the lower the air pollutant concentration. We will evaluate the mixing height and the concentration of carbon monoxide emitted by motorbikes in the Hanoi metropolitan area for a morning rush hour scenario, in which the vertical mixing is restricted due to a temperature inversion (air temperature increases with altitude) at elevations above 119 m119 \mathrm{~m}. Let us consider the air as an ideal diatomic gas, with molar mass μ=29 g/mol\mu=29 \mathrm{~g} / \mathrm{mol}. Quasi equilibrium adiabatic transformation obey the equation pVγ=p V^{\gamma}= const, where γ=cpcV\gamma=\frac{c_{p}}{c_{V}} is the ratio between isobaric and isochoric heat capacities of the gas. The student may use the following data if necessary: The universal gas constant is R=8.31 J/(R=8.31 \mathrm{~J} /( mol.K). The atmospheric pressure on ground is p0=101.3kPap_{0}=101.3 \mathrm{kPa} The acceleration due to gravity is constant, g=9.81 m/s2g=9.81 \mathrm{~m} / \mathrm{s}^{2} The molar isobaric heat capacity is cp=72Rc_{p}=\frac{7}{2} R for air. The molar isochoric heat capacity is cV=52Rc_{V}=\frac{5}{2} R for air. Mathematical hints a. ∫dxA+Bx=1B∫d(A+Bx)A+Bx=1Bln⁡(A+Bx)\int \frac{d x}{A+B x}=\frac{1}{B} \int \frac{d(A+B x)}{A+B x}=\frac{1}{B} \ln (A+B x) b. The solution of the differential equation dxdt+Ax=B\frac{d x}{d t}+A x=B \quad (with A\quad A and BB constant) is x(t)=x1(t)+BAx(t)=x_{1}(t)+\frac{B}{A} where x1(t)x_{1}(t) is the solution of the differential equation dxdt+Ax=0\frac{d x}{d t}+A x=0. c. lim⁡x→∞(1+1x)x=e\lim _{x \rightarrow \infty}\left(1+\frac{1}{x}\right)^{x}=e 1. Change of pressure with altitude. Context question: 1.1. Assume that the temperature of the atmosphere is uniform and equal to T0T_{0}. Write down the expression giving the atmospheric pressure pp as a function of the altitude zz. Context answer: p(z)=p(0)e−μgRT0zp(z)=p(0) e^{-\frac{\mu g}{R T_{0}} z} Extra Supplementary Reading Materials: 1.2. Assume that the temperature of the atmosphere varies with the altitude according to the relation T(z)=T(0)−ΛzT(z)=T(0)-\Lambda z where Λ\Lambda is a constant, called the temperature lapse rate of the atmosphere (the vertical gradient of temperature is −Λ-\Lambda ). Context question: 1.2.1. Write down the expression giving the atmospheric pressure pp as a function of the altitude ZZ. Context answer: p(z)=p(0)(1−ΛzT(0))μgRΛp(z)=p(0)\left(1-\frac{\Lambda z}{T(0)}\right)^{\frac{\mu g}{R \Lambda}} Context question: 1.2.2. A process called free convection occurs when the air density increases with altitude. At which values of Λ\Lambda does the free convection occur? Context answer: 0.0340.034 Extra Supplementary Reading Materials: 2. Change of the temperature of an air parcel in vertical motion Consider an air parcel moving upward and downward in the atmosphere. An air parcel is a body of air of sufficient dimension, several meters across, to be treated as an independent thermodynamical entity, yet small enough for its temperature to be considered uniform. The vertical motion of an air parcel can be treated as a quasi adiabatic process, i.e. the exchange of heat with the surrounding air is negligible. If the air parcel rises in the atmosphere, it expands and cools. Conversely, if it moves downward, the increasing outside pressure will compress the air inside the parcel and its temperature will increase. As the size of the parcel is not large, the atmospheric pressure at different points on the parcel boundary can be considered to have the same value p(z)p(z), with zz - the altitude of the parcel center. The temperature in the parcel is uniform and equals to Tparcel (z)T_{\text {parcel }}(z), which is generally different from the temperature of the surrounding air T(z)T(z). In parts 2.1 and 2.2, we do not make any assumption about the form of T(z)T(z). Context question: 2.1. The change of the parcel temperature Tparcel T_{\text {parcel }} with altitude is defined by dTparcel dz=−G\frac{d T_{\text {parcel }}}{d z}=-G. Derive the expression of G(T,Tparcel )G\left(T, T_{\text {parcel }}\right). Context answer: γ−1γμgRTparcel T\frac{\gamma-1}{\gamma} \frac{\mu g}{R} \frac{T_{\text {parcel }}}{T} Extra Supplementary Reading Materials: 2.2. Consider a special atmospheric condition in which at any altitude zz the temperature TT of the atmosphere equals to that of the parcel Tparcel ,T(z)=Tparcel (z)T_{\text {parcel }}, T(z)=T_{\text {parcel }}(z). We use Γ\Gamma to denote the value of GG when T=Tparcel T=T_{\text {parcel }}, that is Γ=−dTparcel dz\Gamma=-\frac{d T_{\text {parcel }}}{d z} (with T=Tparcel ).Γ\left.T=T_{\text {parcel }}\right) . \Gamma is called dry adiabatic lapse rate.

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