# Assignment 7

## Problem 1

### Part a

Consider radiation between two parallel plates with uniform temperature $T_g = 1500$ K and absorption coefficient $k=1$ m$^{-1}$. The walls are black at a temperature of 300 K.

Compute the exact heat flux **to** the wall for various plate spacings $H$.
Normalize the axes using $q_{wall}/\sigma(T_g^4-T_w^4)$ versus $kH$. On the
same plot, show the corresponding curve for the two flux model (see the notes).

In the derivation of the two flux model the average of $\cos(\theta)$ was used. If we let $\mu=\cos(\theta)$, then we used $\bar{\mu} = 1/2$. A better value of $\bar{\mu}$ can be found: $\bar{\mu}=0.711$. Plot the curve using this value too.

### Part b

Make another plot of $Q/\max(Q_{exact})$ versus x/H for H=11 m. Include the three cases of Part a. Use $Q=dq/dx$ instead of $Q=-dq/dx$ so that $Q$ will be positive. (That is, plot the volumetric heat loss rate instead of the standard heat gain, which is negative here since the gas temperature is greater than the wall temperature.)

## Problem 2

### Part a

Using RadLib or the data on the TNF Workshop page , compute the Planck mean absorption coefficient for stoichiometric methane-air at 2000 K. What is the corresponding optical lengthscale? Compare this to typical flame/fire scales.

### Part b

By how much does the absorption coefficient increase if the flame has a soot volume fraction of 0.1 ppmv? Use $k_{soot} = 1817f_vT$, which has units of $m^{-1}$.