The density of a gas¶

The density of a gas from its pressure, temperature and molar mass, by the ideal gas law as a density, rho = p M / (R T).

R is lusine.constants', a constant of nature. Source: Cengel & Boles, Thermodynamics, ch. 3. Checked against a reference in numpy, which has its own tests.

Source

Cengel & Boles, Thermodynamics, ch. 3 (the ideal gas equation of state).

Proof

reference, limit, invariant.

gas_density.trait

namespace lusine.properties;

float<kg/m3> gas_density(float<Pa> pressure, float<K> temperature, float<kg/mol> molar_mass)
{
  return pressure * molar_mass / (lusine.constants.R * temperature);
}

The ideal gas law as a density, rho = p M / (R T). R is lusine.constants.R, N_A k exactly: the kernel names the constant's row and declares no number.

In [1]:

import os, sys
root = os.getcwd()
while not os.path.isfile(os.path.join(root, 'docsite.json')):
    root = os.path.dirname(root)
sys.path.insert(0, os.path.join(root, 'tools'))
import numpy as np
import matplotlib.pyplot as plt
from lusine import trait_run
EPS = float(np.finfo(np.float32).eps)

import os, sys root = os.getcwd() while not os.path.isfile(os.path.join(root, 'docsite.json')): root = os.path.dirname(root) sys.path.insert(0, os.path.join(root, 'tools')) import numpy as np import matplotlib.pyplot as plt from lusine import trait_run EPS = float(np.finfo(np.float32).eps)

The kernel¶

The law as the library states it, and as this page runs it.

In [2]:

print(trait_run.source('properties/gas_density.trait'))

print(trait_run.source('properties/gas_density.trait'))

// lusine.properties.gas_density: the density of a gas from its pressure, temperature and molar mass, by the ideal gas
// law as a density, rho = p M / (R T). R is lusine.constants', a constant of nature.
// Source: Cengel & Boles, Thermodynamics, ch. 3.
// Checked against a reference in numpy, which has its own tests.

namespace lusine.properties;

float<kg/m3> gas_density(float<Pa> pressure, float<K> temperature, float<kg/mol> molar_mass)
{
  return pressure * molar_mass / (lusine.constants.R * temperature);
}

The program¶

Each section runs the kernel over a lattice of items, one a case: the host sets every parameter, and main() binds the kernel over the items.

In [3]:

print(trait_run.law_program('properties/gas_density.trait', 'gas_density'))

print(trait_run.law_program('properties/gas_density.trait', 'gas_density'))

namespace probe;

lusine.version version = { { number = 1 } };

struct item {
  host float<Pa> pressure;
  host float<K> temperature;
  host float<kg/mol> molar_mass;
  float<kg/m3> result_0;
};
lattice<item, 1> items(spacing = 1 m, boundary = wall);

void main() { items.result_0 = lusine.properties.gas_density(items.pressure, items.temperature, items.molar_mass); }

Reference: air at sea level¶

The International Standard Atmosphere's air at 288.15 K and 101,325 Pa, of dry air's 28.9645 g/mol, is 1.225 kg/m³.

In [4]:

rho = trait_run.law('properties/gas_density.trait', 'gas_density', pressure=101325.0, temperature=288.15,
                    molar_mass=0.0289645)[0]
print(f'{rho:.4f} kg/m3')
assert abs(rho - 1.225) < 0.001

rho = trait_run.law('properties/gas_density.trait', 'gas_density', pressure=101325.0, temperature=288.15, molar_mass=0.0289645)[0] print(f'{rho:.4f} kg/m3') assert abs(rho - 1.225) < 0.001

1.2250 kg/m3

Limit: one gas alone¶

Nitrogen alone at 300 K and 10⁵ Pa is p M / (R T) with nitrogen's 28.0134 g/mol, R = N_A k, within binary32's rounding.

In [5]:

R = 6.02214076e23 * 1.380649e-23
rho = trait_run.law('properties/gas_density.trait', 'gas_density', pressure=1e5, temperature=300.0,
                    molar_mass=0.0280134)[0]
assert abs(rho / (1e5 * 0.0280134 / (R * 300.0)) - 1) < 4 * EPS
print(rho, 'kg/m3')

R = 6.02214076e23 * 1.380649e-23 rho = trait_run.law('properties/gas_density.trait', 'gas_density', pressure=1e5, temperature=300.0, molar_mass=0.0280134)[0] assert abs(rho / (1e5 * 0.0280134 / (R * 300.0)) - 1) < 4 * EPS print(rho, 'kg/m3')

1.1230791807174683 kg/m3

Invariant: 1/T at fixed pressure, p at fixed temperature¶

Charles's law and Boyle's, in the density: rho T holds across temperatures, and twice the pressure is twice the density, bit for bit.

In [6]:

T = np.linspace(230, 330, 11)
rho = trait_run.law('properties/gas_density.trait', 'gas_density', pressure=1e5, temperature=T, molar_mass=0.0289645)
assert np.allclose(rho * np.float32(T), rho[0] * np.float32(T[0]), rtol=4 * EPS)
pair = trait_run.law('properties/gas_density.trait', 'gas_density', pressure=[1e5, 2e5], temperature=300.0,
                     molar_mass=0.0289645)
assert pair[1] == 2 * pair[0]
fig, ax = plt.subplots(figsize=(7, 3))
ax.plot(T, rho, 'o-')
ax.set_xlabel('T (K)')
ax.set_ylabel('density (kg/m3)')
plt.show()

T = np.linspace(230, 330, 11) rho = trait_run.law('properties/gas_density.trait', 'gas_density', pressure=1e5, temperature=T, molar_mass=0.0289645) assert np.allclose(rho * np.float32(T), rho[0] * np.float32(T[0]), rtol=4 * EPS) pair = trait_run.law('properties/gas_density.trait', 'gas_density', pressure=[1e5, 2e5], temperature=300.0, molar_mass=0.0289645) assert pair[1] == 2 * pair[0] fig, ax = plt.subplots(figsize=(7, 3)) ax.plot(T, rho, 'o-') ax.set_xlabel('T (K)') ax.set_ylabel('density (kg/m3)') plt.show()