chap16sympy.ipynb

Kernel: Python 3

Modeling and Simulation in Python

SymPy code for Chapter 16

Copyright 2017 Allen Downey

License: Creative Commons Attribution 4.0 International

Mixing liquids

We can figure out the final temperature of a mixture by setting the total heat flow to zero and then solving for $T$.

from sympy import *

init_printing() 

C1, C2, T1, T2, T = symbols('C1 C2 T1 T2 T')

eq = Eq(C1 * (T - T1) + C2 * (T - T2), 0)
eq

$\displaystyle C_{1} \left(T - T_{1}\right) + C_{2} \left(T - T_{2}\right) = 0$

solve(eq, T)

$\displaystyle \left[ \frac{C_{1} T_{1} + C_{2} T_{2}}{C_{1} + C_{2}}\right]$

Analysis

We can use SymPy to solve the cooling differential equation.

T_init, T_env, r, t = symbols('T_init T_env r t')
T = Function('T')

eqn = Eq(diff(T(t), t), -r * (T(t) - T_env))
eqn

$\displaystyle \frac{d}{d t} T{\left(t \right)} = - r \left(- T_{env} + T{\left(t \right)}\right)$

Here's the general solution:

solution_eq = dsolve(eqn)
solution_eq

$\displaystyle T{\left(t \right)} = C_{1} e^{- r t} + T_{env}$

general = solution_eq.rhs
general

$\displaystyle C_{1} e^{- r t} + T_{env}$

We can use the initial condition to solve for $C_1$. First we evaluate the general solution at $t=0$

at0 = general.subs(t, 0)
at0

$\displaystyle C_{1} + T_{env}$

Now we set $T(0) = T_{init}$ and solve for $C_1$

solutions = solve(Eq(at0, T_init), C1)
value_of_C1 = solutions[0]
value_of_C1

$\displaystyle - T_{env} + T_{init}$

Then we plug the result into the general solution to get the particular solution:

particular = general.subs(C1, value_of_C1)
particular

$\displaystyle T_{env} + \left(- T_{env} + T_{init}\right) e^{- r t}$

We use a similar process to estimate $r$ based on the observation $T(t_{end}) = T_{end}$

t_end, T_end = symbols('t_end T_end')

Here's the particular solution evaluated at $t_{end}$

at_end = particular.subs(t, t_end)
at_end

$\displaystyle T_{env} + \left(- T_{env} + T_{init}\right) e^{- r t_{end}}$

Now we set $T(t_{end}) = T_{end}$ and solve for $r$

solutions = solve(Eq(at_end, T_end), r)
value_of_r = solutions[0]
value_of_r

$\displaystyle \frac{\log{\left(\frac{- T_{env} + T_{init}}{T_{end} - T_{env}} \right)}}{t_{end}}$

We can use evalf to plug in numbers for the symbols. The result is a SymPy float, which we have to convert to a Python float.

subs = dict(t_end=30, T_end=70, T_init=90, T_env=22)
r_coffee2 = value_of_r.evalf(subs=subs)
type(r_coffee2)

sympy.core.numbers.Float

r_coffee2 = float(r_coffee2)
r_coffee2

$\displaystyle 0.011610223142273859$