#!/usr/bin/env python
# coding: utf-8

# # Homework 20

# ### Problem 1
# 
# Solve the following ODE $$\frac{dy}{dt} = ye^{-t}$$ to $t=6$.  Plot the resulting function $y(t)$. The initial condition is $y_0=1$. 

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# ### Problem 2
# 
# Solve the following rate equation for a coal particle in a furnace.
# $$ \frac{dT}{dt} = \left(\frac{hA}{m c_p}(T_f - T) + \frac{\sigma A}{m c_p}(T_f^4-T^4)\right).$$
# 
# The initial particle temperature is $T_0=500$ K.
# 
# Use D=100 $\mu$m, tend = 0.05 s.
# 
# The following data is given:
# 
# | Variable | Value  | Units        |
# |----------|--------|-------       |
# | $\rho_p$ | 1000   | kg/m$^3$     |
# | $c_p$    | 1380   | J/kg$\cdot$K |  
# | $k$      | 0.1    | W/m$\cdot$K  |  
# | $Nu$     | 2      | --           |  
# | $\sigma$ | 5.67E-8| W/m$^2$K$^4$ |
# | $T_f$    | 1500   | K            |
# 
# Also, the area, mass and Nusselt number $Nu$ are given, respectively, by:
# $$ A = \pi D^2, $$
# $$ m = \frac{\pi}{6}D^3\rho_p,$$
# $$ Nu = \frac{hD}{k}.$$
# 
# Plot the particle temperature as a function of time. Label and format your plot (include units).

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# ### Problem 3
# 
# We are performing a chemical reaction as follows.
# 
# $$\mbox{Rxn 1: }\phantom{xxx}  A+B\rightarrow C $$
# $$\mbox{Rxn 2: }\phantom{xxx}  B+C\rightarrow D  $$
# 
# Here, symbols $A$, $B$, $C$, $D$ denote species concentrations in mol/L. The initial concentrations are $A_0=1$, $B_0=1$, $C_0=0$, $D_0=0$. Also, $k_1=1\,L/mol*s$, and $k_2=1.5\,L/mol*s$. The concentrations obey the following rate equations:
# $$ \frac{dA}{dt} = -k_1AB,$$
# $$ \frac{dB}{dt} = -k_1AB - k_2BC,$$
# $$ \frac{dC}{dt} = k_1AB - k_2BC,$$
# $$ \frac{dD}{dt} = k_2BC.$$
# 
# Solve for the concentrations of $A$, $B$, $C$, and $D$ as functions of time to $t=3$ s.
# 
# Also, once the species concentrations are found, compute the selectivity defined as $S=C/(C+D)$ as a function of time. (S is initially undefined (since C and D are zero), but you can set it equal to 1 at t=0 since C forms before D.
# 
# Plot the concentrations of A, B, C, and D, and also selectivity S, as functions of time on the same plot. Label the axes as "time (s)" and “concentration (mol/L)”. Include a legend so we can see what the curves correspond to.
# 
# What can you do to maximize your selectivity? Consider $k_1$, $k_2$ and initial concentrations.

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