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

# # Homework 15

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import numpy as np
import matplotlib.pyplot as plt
get_ipython().run_line_magic('matplotlib', 'inline')

from scipy.integrate import quad


# ## Problem 1
# 
# * Use ```quad``` to solve the following integral:
# $$ I = \int_0^{\pi/2}\cos(x)dx$$
#     * See the [posted lecture notes](http://nbviewer.org/url/ignite.byu.edu/che263/lectures/quad.ipynb) for how to use ```quad```.

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# ## Problem 2
# 
# Use ```quad``` to find the numerical integral of the following function:
# 
# $$I = \int_0^{2\pi} e^{-x}\sin(3x)dx.$$ 

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# ## Problem 3
# 
# Use ```quad``` to find the numerical integral of the following function:
# $$I = \int_{x_1}^{x_2}f(x,a,b,c)dx,$$
# where,
# $$f(x,a,b,c) = ax^2 + bx + c.$$
# * That is, $f(x,a,b,c)$ is a polynomial function of $x$ with coefficients $a$, $b$, and $c$.
# 
# * Solve this for $x_1=1$, $x_2=5$, and for coefficients $a=2$, $b=3$, $c=4$. 
# * As described in the lecture notes, ```quad``` normally calls the function you pass as $f(x)$. 
#     * To get ```quad``` to call your function as $f(x,a,b,c)$ instead, you need to tell quad about the extra $a$, $b$, $c$ arguments. 
#     * ```quad``` has a parameter called ```args``` that we can set to a tuple containing our extra argments, like this: ```I,err = quad(f,x1,x2,args=(a,b,c))```.
#     * Then, when ```quad``` calls our function, it knows to call it as ```f(x,a,b,c)``` instead of ```f(x)``` like it usually does.

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# ## Problem 4
# 
# The following equation describes species heat capacities, with data following.
# 
# $$c_{p,i}(T) = R_g(a_{0,i} + a_{1,i}T + a_{2,i}T^2 + a_{3,i}T^3 + a_{4,i}T^4)$$ 
# 
# 
# | Species | $a_0$           | $a_1$             | $a_2$            | $a_3$             | $a_4$             |
# |---------|-----------------|-------------------|------------------|-------------------|-------------------|
# | CO2     | 2.356773520E+00 |   8.984596770E-03 | -7.123562690E-06 |   2.459190220E-09 |  -1.436995480E-13 |
# | H2O(g)  | 4.198640560E+00 |  -2.036434100E-03 |  6.520402110E-06 |  -5.487970620E-09 |   1.771978170E-12 |
# | O2      | 3.782456360E+00 |  -2.996734160E-03 |  9.847302010E-06 |  -9.681295090E-09 |   3.243728370E-12 |
# | N2      | 3.298677000E+00 |   1.408240400E-03 | -3.963222000E-06 |   5.641515000E-09 |  -2.444854000E-12 |
# | CH4     | 5.149876130E+00 |  -1.367097880E-02 |  4.918005990E-05 |  -4.847430260E-08 |   1.666939560E-11 |
# 
# The enthalpy of the species is given by
# 
# $$h_i(T) = h_{f,i} + \int_{T_{ref}}^{T}c_{p,i}(T)dT.$$
# * Here, $h_{f,i}$ is the heat of formation of the species at temperature $T_{ref}=298$. 
# 
# | Species | $h_{f,i}$ (J/mol)  |
# |---------|--------------------|
# | CO2     | -393509            |
# | H2O(g)  | -241818            |
# | O2      | 0                  |
# | N2      | 0                  |
# | CH4     | -74520             |
# 
# ### Part a
# * Use ```quad``` to find the enthalpy of CO$_2$ at a temperature of 1000 K.
# 
# ### Part b
# * Methane combustion follows the reaction 
# $$CH_4 + 2O_2 + 7.52 N_2 \rightarrow CO_2 + 2H_2O + 7.52N_2$$
# * The reactant mole fractions are: $x_{CH4}$= 0.095, $x_{O2}$ = 0.19, $x_{N2}$=0.715 
# * The enthalpy of the reactant mixture is $h_r = x_{CH4}h_{CH4}(T) + x_{O2}h_{O2}(T) + x_{N2}h_{N2}(T)$. 
# * Find the mixture enthalpy at T=1000 K.
# 

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