Problem $1$: Radiative Heat Transfer $(25$ Points$)$

Your goal in this project is to estimate the increase in earth's surface temperature $T$ due to changes in emissivity to the greenhouse gas $($GHG$)$ layer. This is given by the relationship:

$T=T_p\frac{F}{4S_{\text{a}\text{v}\text{e}}}$

Where:

Current temperature of earth $(T_p)=288K$

Absorptivity of earth $()=0\mathrm{.}7$

Incoming solar radiation $(S_{\text{a}\text{v}\text{e}})=342\frac{W}{m^2}($Surface averaged$)$

$F$ is the radiative forcing or the increase in net radiative flux on the earth's surface due to the changes in emissivity of the GHG layer. This is an unknown!

You will need to determine this analytically $($hand calculations$).$

$=$ Stefan$-$Boltzmann constant $5\mathrm{.}67{10}^{-8}\frac{W}{m^2}-K^4$

Assuming the Earth and GHG as plane parallel media and using the expression:

$\frac{q_{1-2}}{A}=\frac{(T_1^4-T_2^4)}{(\frac{1}{}lo{n}_1)+(\frac{1}{}lo{n}_2)-1}$

Problem $1$: Radiative Heat Transfer $(25$ Points$)$

Your goal in this project is to estimate the increase in earth's surface temperature $T$ due to changes in emissivity to the greenhouse gas $($GHG$)$ layer. This is given by the relationship:

$T=T_p\frac{F}{4prop{S}_{\text{a}\text{v}\text{e}}}$

Where:

Current temperature of earth $(T_p)=288K$

Absorptivity of earth $()=0\mathrm{.}7$

Incoming solar radiation $(S_{\text{a}\text{v}\text{c}})=342\frac{W}{m^2}($Surface averaged$)$

$F$ is the radiative forcing or the increase in net radiative flux on the earth's surface due to the changes in emissivity of the GHG layer. This is an unknown! You will need to determine this analytically $($hand calculations$).$

$=$ Stefan$-$Boltzmann constant $5\mathrm{.}67{10}^{-8}\frac{W}{m^2}-K^4$

Assuming the Earth and GHG as plane parallel media and using the expression:

$\frac{q_{1-2}}{A}=\frac{(T_1^4-T_2^4)}{(\frac{1}{}lo{n}_1)+(\frac{1}{}lo{n}_2)-1}$

Use the following charts for estimating emissivity's of $H_{2}O$ and $C{O}_2$ :

Your goal is to estimate: $F$ and $T$ due to the $C{O}_2$ concentrations becoming: $2-$times and $4-$times present day amounts of $355$ ppm

You may make the following assumptions:

The concentration of $C{O}_2$ is uniform within this layer.

Name $q,$

Student ID $q,$

$q,$

Present day amounts $($pressure path$-$length$)$ of gases are: P ${?}_{1120.s}=23\mathrm{.}8$ bar$-$m

a$.$ Estimate the emissivity of $H_{2}O$ based on the information using the figure provided above $(3$ points$)$

b$.$ What are the pressure path lengths of $C{O}_2($in bar$-$m$)$ if it is uniformly distributed in a layer of $10$ km thickness at concentrations of: $355$ ppm $($current concentrations$),710$ ppm $(2$ times present day concentrations$)$ and $1420$ ppm $(4$ times present day concentrations$)?(3$ points$)$

c$.$ Using your answers to part $($b$),$ estimate the emissivities of $C{O}_2$ at concentrations corresponding to: $1-$time, $2-$times and $4-$times present day amounts using the figure provided above $(3$ points$)$

d$.$ "Correct" for the total emissivities of the GHG accounting for any overlap between the CO $2$ and H $2$ O bands $(3$ points$)$