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

# # Homework 11

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


# ### Problem 0 (not graded)
# 
# Review the notes on arrays. Memorize the key functions. Practice.

# ### Problem 1
# 
# * Make an array of $x$ from 0 to 5 with 1000 points.
# * create the following arrays
#     * $2x$
#     * $x^2 -3x + 4$
#     * $\sin(5x)$
# * Plot each array versus $x$ on a single plot.

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# ### Problem 2
# 
# * Ideal gas law
# $$\rho = \frac{MP}{RT}$$
# * M=29 kg/kmol
# * R=8314 J/kmol*K
# * P=101325 Pa
# * **Find $\rho$ for $T=300,400,\ldots1000$ K**
#     * Store $\rho$ and $T$ as arrays
#     * Plot $\rho$ versus $T$

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# ### Problem 3
# 
# * Redo Problem 2 by writing a function for $\rho(T)$ as you would have in a previous lecture.
#     * Call the function with an array of $T$ and store the result as an array of $\rho$.
#     * Plot $\rho$ versus $T$.

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# ### Problem 4
# 
# We have an array of times $t$, and we want the array of differences $dt$ between times.
# * How big is $t$?
# * How big will $dt$ be?
# * Create $dt$ using a loop.
# * Create $dt$ using slice-indexing.
# 
# <img src=http://ignite.byu.edu/che263/lectures/L13_f1.png width=300>

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t = np.array([ 
    0.01136646,  0.01655847,  0.02548247,  0.07698612,  0.17964401,
    0.19813924,  0.22419994,  0.28622096,  0.28884997,  0.3627257 ,
    0.42555072,  0.44965767,  0.47257237,  0.59335946,  0.73842352,
    0.85900369,  0.88851226,  0.89153229,  0.89355469,  0.97705246,
    ])


# ### Problem 5
# 
# * You have an array of $x$ points $x=0.0, 0.1, \ldots, 50$.
# * Create $x$
# * Create $y=\sin(x)$
# * At interior points, use a central difference approximation to find the derivative 
# $$\left(\frac{dy}{dx}\right)_i \approx \frac{y_{i+1}-y_{i-1}}{2\Delta x},$$
# where $\Delta x = x_1-x_0$.
# * At edge point use a one-sided difference approximation,
# $$\left(\frac{dy}{dx}\right)_0 \approx \frac{y_{1}-y_{0}}{\Delta x},$$
# $$\left(\frac{dy}{dx}\right)_{n-1} \approx \frac{y_{n-1}-y_{n-2}}{\Delta x}.$$
# * Plot $x$ versus $y$, and $x$ versus $y^{\prime}$, and $x$ versus $\cos(x)$.

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# ### Problem 6
# * Solve the rate equation below to $t=5$ s using the Explicit Euler method $y_{i+1} = y_i + \Delta tf(y_i,t_i).$
# $$\frac{dy}{dt} = -ky,$$
# $$y_0=1$$,
# $$k=2.$$
# 
# * Use $\Delta t = 0.2$.
# * Store $t$ and $y$ in arrays.
# * Plot $y$ versus $t$, and $y_0e^{-kt}$ versus $t$ on the same plot.

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# ### Problem 7
# * The Haaland equation relates the friction factor $f$ in turbulent pipe flow to the Reynolds number $Re$.
# $$\frac{1}{\sqrt{f}} = -1.8\log_{10}\left[\left(\frac{\epsilon/D}{3.7}\right)^{1.11}+ \frac{6.9}{Re}\right].$$
# 
# * Write a function to compute $f(Re, \epsilon/D)$.
# * Create an array of 100 points that is uniformly spaced on a log scale, for $1000\le Re\le 1\times 10^8$
# * Plot $f$ versus $\log_{10}(Re)$. 
# * Use $\epsilon/D=0.001$.

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