## Handling units with dimensionless equations

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As we have seen, handling units with third party functions is fragile, and often requires additional code to wrap the function to handle the units. An alternative approach that avoids the wrapping is to rescale the equations so they are dimensionless. Then, we should be able to use all the standard external functions without modification. We obtain the final solutions by rescaling back to the answers we want.

Before doing the examples, let us consider how the quantities package handles dimensionless numbers.

import quantities as u

a = 5 * u.m
L = 10 * u.m # characteristic length

print a/L
print type(a/L)

0.5 dimensionless
<class 'quantities.quantity.Quantity'>


As you can see, the dimensionless number is scaled properly, and is listed as dimensionless. The result is still an instance of a quantities object though. That is not likely to be a problem.

Now, we consider using fsolve with dimensionless equations. Our goal is to solve $$C_A = C_{A0} \exp(-k t)$$ for the time required to reach a desired $$C_A$$. We let $$X = Ca / Ca0$$ and $$\tau = t * k$$, which leads to $$X = \exp{-\tau}$$ in dimensionless terms.

import quantities as u
import numpy as np
from scipy.optimize import fsolve

CA0 = 1 * u.mol / u.L
CA = 0.01 * u.mol / u.L  # desired exit concentration
k = 1.0 / u.s

# we need new dimensionless variables
# let X = Ca / Ca0
# so, Ca = Ca0 * X

# let tau = t * k
# so t = tau / k

X = CA / CA0 # desired exit dimensionless concentration

def func(tau):
return X - np.exp(-tau)

tauguess = 2

print func(tauguess) # confirm we have a dimensionless function

tau_sol, = fsolve(func, tauguess)
t = tau_sol / k
print t

-0.125335283237 dimensionless
4.60517018599 s


Now consider the ODE $$\frac{dCa}{dt} = -k Ca$$. We let $$X = Ca/Ca0$$, so $$Ca0 dX = dCa$$. Let $$\tau = t * k$$ which in this case is dimensionless. That means $$d\tau = k dt$$. Substitution of these new variables leads to:

$$Ca0*k \frac{dX}{d\tau} = -k Ca0 X$$

or equivalently: $$\frac{dX}{d\tau} = -X$$

import quantities as u

k = 0.23 / u.s
Ca0 = 1 * u.mol / u.L

# Let X = Ca/Ca0  -> Ca = Ca0 * X  dCa = dX/Ca0
# let tau = t * k -> dt = 1/k dtau

def dXdtau(X, tau):
return -X

import numpy as np
from scipy.integrate import odeint

tspan = np.linspace(0, 5) * u.s
tauspan = tspan * k

X0 = 1
X_sol = odeint(dXdtau, X0, tauspan)

print 'Ca at t = {0} = {1}'.format(tspan[-1], X_sol.flatten()[-1] * Ca0)

Ca at t = 5.0 s = 0.316636777351 mol/L


That is pretty much it. Using dimensionless quantities simplifies the need to write wrapper code, although it does increase the effort to rederive your equations (with corresponding increased opportunities to make mistakes). Using units to confirm your dimensionless derivation reduces those opportunities.

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## Units in ODEs

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We reconsider a simple ODE but this time with units. We will use the quantities package again.

Here is the ODE, $$\frac{dCa}{dt} = -k Ca$$ with $$C_A(0) = 1.0$$ mol/L and $$k = 0.23$$ 1/s. Compute the concentration after 5 s.

import quantities as u

k = 0.23 / u.s
Ca0 = 1 * u.mol / u.L

return -k * Ca

import numpy as np
from scipy.integrate import odeint

tspan = np.linspace(0, 5) * u.s

print sol[-1]

[ 0.31663678]


No surprise, the units are lost. Now we start wrapping odeint. We wrap everything, and then test two examples including a single ODE, and a coupled set of ODEs with mixed units.

import quantities as u
import matplotlib.pyplot as plt

import numpy as np
from scipy.integrate import odeint as _odeint

def odeint(func, y0, t, args=(),
Dfun=None, col_deriv=0, full_output=0,
ml=None, mu=None, rtol=None, atol=None,
tcrit=None, h0=0.0, hmax=0.0, hmin=0.0,
ixpr=0, mxstep=0, mxhnil=0, mxordn=12,
mxords=5, printmessg=0):

def wrapped_func(Y0, T, *args):
# put units on T if they are on the original t
# check for units so we don't put them on twice
if not hasattr(T, 'units') and hasattr(t, 'units'):
T = T * t.units
# now for the dependent variable units. Y0 may be a scalar or
# a list or an array. we want to check each element of y0 for
# units, and add them to the corresponding element of Y0 if we
# need to.
try:
uY0 = [x for x in Y0] # a list copy of contents of Y0
# this works if y0 is an iterable, eg. a list or array
for i, yi in enumerate(y0):
if not hasattr(uY0[i],'units') and hasattr(yi, 'units'):

uY0[i] = uY0[i] * yi.units

except TypeError:
# we have a scalar
if not hasattr(Y0, 'units') and hasattr(y0, 'units'):
uY0 = Y0 * y0.units

val = func(uY0, t, *args)

try:
return np.array([float(x) for x in val])
except TypeError:
return float(val)

if full_output:
y, infodict = _odeint(wrapped_func, y0, t, args,
Dfun, col_deriv, full_output,
ml, mu, rtol, atol,
tcrit, h0, hmax, hmin,
ixpr, mxstep, mxhnil, mxordn,
mxords, printmessg)
else:
y = _odeint(wrapped_func, y0, t, args,
Dfun, col_deriv, full_output,
ml, mu, rtol, atol,
tcrit, h0, hmax, hmin,
ixpr, mxstep, mxhnil, mxordn,
mxords, printmessg)

# now we need to put units onto the solution units should be the
# same as y0. We cannot put mixed units in an array, so, we return a list
m,n = y.shape # y is an ndarray, so it has a shape
if n > 1: # more than one equation, we need a list
uY = [0 for yi in range(n)]

for i, yi in enumerate(y0):
if not hasattr(uY[i],'units') and hasattr(yi, 'units'):
uY[i] = y[:,i] * yi.units
else:
uY[i] = y[:,i]

else:
uY = y * y0.units

y = uY

if full_output:
return y, infodict
else:
return y

##################################################################
# test a single ODE
k = 0.23 / u.s
Ca0 = 1 * u.mol / u.L

return -k * Ca

tspan = np.linspace(0, 5) * u.s

print sol[-1]

plt.plot(tspan, sol)
plt.xlabel('Time ({0})'.format(tspan.dimensionality.latex))
plt.ylabel('$C_A$ ({0})'.format(sol.dimensionality.latex))
plt.savefig('images/ode-units-ca.png')

##################################################################
# test coupled ODEs
lbmol = 453.59237*u.mol

kprime = 0.0266 * lbmol / u.hr / u.lb
Fa0 = 1.08 * lbmol / u.hr
alpha = 0.0166 / u.lb
epsilon = -0.15

def dFdW(F, W, alpha0):
X, y = F
dXdW = kprime / Fa0 * (1.0 - X)/(1.0 + epsilon * X) * y
dydW = - alpha0 * (1.0 + epsilon * X) / (2.0 * y)
return [dXdW, dydW]

X0 = 0.0 * u.dimensionless
y0 = 1.0

# initial conditions
F0 = [X0, y0] # one without units, one with units, both are dimensionless

wspan = np.linspace(0,60) * u.lb

sol = odeint(dFdW, F0, wspan, args=(alpha,))
X, y = sol

print 'Test 2'
print X[-1]
print y[-1]

plt.figure()
plt.plot(wspan, X, wspan, y)
plt.legend(['X','$P/P_0$'])
plt.xlabel('Catalyst weight ({0})'.format(wspan.dimensionality.latex))
plt.savefig('images/ode-coupled-units-pdrpo.png')

[ 0.31663678] mol/L
Test 2
0.665569578156 dimensionless
0.263300470681  That is not too bad. This is another example of a function you would want to save in a module for reuse. There is one bad feature of the wrapped odeint function, and that is that it changes the solution for coupled ODEs from an ndarray to a list. That is necessary because you apparently cannot have mixed units in an ndarray. It is fine, however, to have a list of mixed units. This is not a huge problem, but it changes the syntax for plotting results for the wrapped odeint function compared to the unwrapped function without units.

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## Handling units with the quantities module

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The quantities module (https://pypi.python.org/pypi/quantities) is another option for handling units in python. We are going to try the previous example. It does not work, because scipy.optimize.fsolve is not designed to work with units.

import quantities as u
import numpy as np

from scipy.optimize import fsolve
CA0 = 1 * u.mol / u.L
CA = 0.01 * u.mol / u.L
k = 1.0 / u.s

def func(t):
return CA - CA0 * np.exp(-k * t)

tguess = 4 * u.s

print func(tguess)

print fsolve(func, tguess)

>>> >>> >>> >>> >>> >>> >>> ... ... >>> >>> >>> -0.00831563888873 mol/L
>>> Traceback (most recent call last):
File "<stdin>", line 1, in <module>
File "c:\Python27\lib\site-packages\scipy\optimize\minpack.py", line 115, in fsolve
_check_func('fsolve', 'func', func, x0, args, n, (n,))
File "c:\Python27\lib\site-packages\scipy\optimize\minpack.py", line 13, in _check_func
res = atleast_1d(thefunc(*((x0[:numinputs],) + args)))
File "<stdin>", line 2, in func
File "c:\Python27\lib\site-packages\quantities-0.10.1-py2.7.egg\quantities\quantity.py", line 231, in __array_prepare__
res._dimensionality = p_dict[uf](*objs)
File "c:\Python27\lib\site-packages\quantities-0.10.1-py2.7.egg\quantities\dimensionality.py", line 347, in _d_dimensionless
raise ValueError("quantity must be dimensionless")
ValueError: quantity must be dimensionless


Our function works fine with units, but fsolve does not pass numbers with units back to the function, so this function fails because the exponential function gets an argument with dimensions in it. We can create a new function that solves this problem. We need to “wrap” the function we want to solve to make sure that it uses units, but returns a float number. Then, we put the units back onto the final solved value. Here is how we do that.

import quantities as u
import numpy as np

from scipy.optimize import fsolve as _fsolve

CA0 = 1 * u.mol / u.L
CA = 0.01 * u.mol / u.L
k = 1.0 / u.s

def func(t):
return CA - CA0 * np.exp(-k * t)

def fsolve(func, t0):
'wrapped fsolve command to work with units'
tU = t0 / float(t0)  # units on initial guess, normalized
def wrapped_func(t):
't will be unitless, so we add unit to it. t * tU has units.'
return float(func(t * tU))

sol, = _fsolve(wrapped_func, t0)
return sol * tU

tguess = 4 * u.s

print fsolve(func, tguess)

4.60517018599 s


It is a little tedious to do this, but we might only have to do it once if we store the new fsolve command in a module. You might notice the wrapped function we wrote above only works for one dimensional problems. If there are multiple dimensions, we have to be a little more careful. In the next example, we expand the wrapped function definition to do both one and multidimensional problems. It appears we cannot use numpy.array element-wise multiplication because you cannot mix units in an array. We will use lists instead. When the problem is one-dimensional, the function will take a scalar, but when it is multidimensional it will take a list or array. We will use try/except blocks to handle these two cases. We will assume multidimensional cases, and if that raises an exception because the argument is not a list, we assume it is scalar. Here is the more robust code example.

import quantities as u
import numpy as np

from scipy.optimize import fsolve as _fsolve

def fsolve(func, t0):
'''wrapped fsolve command to work with units. We get the units on
the function argument, then wrap the function so we can add units
to the argument and return floats. Finally we call the original
fsolve from scipy. Note: this does not support all of the options
to fsolve.'''

try:
tU = [t / float(t) for t in t0]  # units on initial guess, normalized
except TypeError:
tU = t0 / float(t0)

def wrapped_func(t):
't will be unitless, so we add unit to it. t * tU has units.'
try:
T = [x1 * x2 for x1,x2 in zip(t, tU)]
except TypeError:
T = t * tU

try:
return [float(x) for x in func(T)]
except TypeError:
return float(func(T))

sol = _fsolve(wrapped_func, t0)
try:
return [x1 * x2 for x1,x2 in zip(sol, tU)]
except TypeError:
return sol * tU

### Problem 1
CA0 = 1 * u.mol / u.L
CA = 0.01 * u.mol / u.L
k = 1.0 / u.s

def func(t):
return CA - CA0 * np.exp(-k * t)

tguess = 4 * u.s
sol1, = fsolve(func, tguess)
print 'sol1 = ',sol1

### Problem 2
def func2(X):
a,b = X
return [a**2 - 4*u.kg**2,
b**2 - 25*u.J**2]

Xguess = [2.2*u.kg, 5.2*u.J]
s2a, s2b = fsolve(func2, Xguess)
print 's2a = {0}\ns2b = {1}'.format(s2a, s2b)

sol1 =  4.60517018599 s
s2a = 2.0 kg
s2b = 5.0 J


That is pretty good. There is still room for improvement in the wrapped function, as it does not support all of the options that scipy.optimize.fsolve supports. Here is a draft of a function that does that. We have to return different numbers of arguments depending on the value of full_output. This function works, but I have not fully tested all the options. Here are three examples that work, including one with an argument.

import quantities as u
import numpy as np

from scipy.optimize import fsolve as _fsolve

def fsolve(func, t0, args=(),
fprime=None, full_output=0, col_deriv=0,
xtol=1.49012e-08, maxfev=0, band=None,
epsfcn=0.0, factor=100, diag=None):
'''wrapped fsolve command to work with units. We get the units on
the function argument, then wrap the function so we can add units
to the argument and return floats. Finally we call the original
fsolve from scipy. '''

try:
tU = [t / float(t) for t in t0]  # units on initial guess, normalized
except TypeError:
tU = t0 / float(t0)

def wrapped_func(t, *args):
't will be unitless, so we add unit to it. t * tU has units.'
try:
T = [x1 * x2 for x1,x2 in zip(t, tU)]
except TypeError:
T = t * tU

try:
return [float(x) for x in func(T, *args)]
except TypeError:
return float(func(T))

sol = _fsolve(wrapped_func, t0, args,
fprime, full_output, col_deriv,
xtol, maxfev, band,
epsfcn, factor, diag)

if full_output:
x, infodict, ier, mesg = sol
try:
x = [x1 * x2 for x1,x2 in zip(x, tU)]
except TypeError:
x = x * tU
return x, infodict, ier, mesg
else:
try:
x = [x1 * x2 for x1,x2 in zip(sol, tU)]
except TypeError:
x = sol * tU
return x

### Problem 1
CA0 = 1 * u.mol / u.L
CA = 0.01 * u.mol / u.L
k = 1.0 / u.s

def func(t):
return CA - CA0 * np.exp(-k * t)

tguess = 4 * u.s
sol1, = fsolve(func, tguess)
print 'sol1 = ',sol1

### Problem 2
def func2(X):
a,b = X
return [a**2 - 4*u.kg**2,
b**2 - 25*u.J**2]

Xguess = [2.2*u.kg, 5.2*u.J]
sol, infodict, ier, mesg = fsolve(func2, Xguess, full_output=1)
s2a, s2b = sol
print 's2a = {0}\ns2b = {1}'.format(s2a, s2b)

### Problem 3 - with an arg
def func3(a, arg):
return a**2 - 4*u.kg**2 + arg**2

Xguess = 1.5 * u.kg
arg = 0.0* u.kg

sol3, = fsolve(func3, Xguess, args=(arg,))
print'sol3 = ', sol3

sol1 =  4.60517018599 s
s2a = 2.0 kg
s2b = 5.0 J
sol3 =  2.0 kg


The only downside I can see in the quantities module is that it only handle temperature differences, and not absolute temperatures. If you only use absolute temperatures, this would not be a problem I think. But, if you have mixed temperature scales, the quantities module does not convert them on an absolute scale.

import quantities as u

T = 20 * u.degC

print T.rescale(u.K)
print T.rescale(u.degF)

20.0 K
36.0 degF


Nevertheless, this module seems pretty promising, and there are a lot more features than shown here. Some documentation can be found at http://pythonhosted.org/quantities/.

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## Using units in python

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I think an essential feature in an engineering computational environment is properly handling units and unit conversions. Mathcad supports that pretty well. I wrote a package for doing it in Matlab. Today I am going to explore units in python. Here are some of the packages that I have found which support units to some extent

The last one looks most promising.

import numpy as np
from scimath.units.volume import liter
from scimath.units.substance import mol

q = np.array([1, 2, 3]) * mol
print q

P = q / liter
print P

[1.0*mol 2.0*mol 3.0*mol]
[1000.0*m**-3*mol 2000.0*m**-3*mol 3000.0*m**-3*mol]


That doesn't look too bad. It is a little clunky to have to import every unit, and it is clear the package is saving everything in SI units by default. Let us try to solve an equation.

Find the time that solves this equation.

$$0.01 = C_{A0} e^{-kt}$$

First we solve without units. That way we know the answer.

import numpy as np
from scipy.optimize import fsolve

CA0 = 1.0  # mol/L
CA = 0.01  # mol/L
k = 1.0    # 1/s

def func(t):
z = CA - CA0 * np.exp(-k*t)
return z

t0 = 2.3

t, = fsolve(func, t0)
print 't = {0:1.2f} seconds'.format(t)

t = 4.61 seconds


Now, with units. I note here that I tried the obvious thing of just importing the units, and adding them on, but the package is unable to work with floats that have units. For some functions, there must be an ndarray with units which is practically what the UnitScalar code below does.

import numpy as np
from scipy.optimize import fsolve
from scimath.units.volume import liter
from scimath.units.substance import mol
from scimath.units.time import second
from scimath.units.api import has_units, UnitScalar

CA0 = UnitScalar(1.0, units = mol / liter)
CA = UnitScalar(0.01, units = mol / liter)
k = UnitScalar(1.0, units = 1 / second)

@has_units(inputs="t::units=s",
outputs="result::units=mol/liter")
def func(t):
z = CA - CA0 * float(np.exp(-k*t))
return z

t0 = UnitScalar(2.3, units = second)

t, = fsolve(func, t0)
print 't = {0:1.2f} seconds'.format(t)
print type(t)

t = 4.61 seconds
<type 'numpy.float64'>


This is some heavy syntax that in the end does not preserve the units. In my Matlab package, we had to “wrap” many functions like fsolve so they would preserve units. Clearly this package will need that as well. Overall, in its current implementation this package does not do what I would expect all the time.1