Parameter Estimation with difflow.estimation

This notebook demonstrates the difflow.estimation module — a structured API for parameter estimation inspired by pyomo’s parmest, powered by JAX autodiff.

What You’ll Learn

1. Experiment objects — packaging experimental data

2. Estimator.fit() — fitting parameters with exact gradients

3. Confidence intervals — Fisher information CIs via jax.hessian

4. Diagnostics — R², RMSE, AIC/BIC, residual analysis

5. Summary reports — one-line result tables

Why Use difflow.estimation?

Compared to manual parameter estimation (see notebook 14), this module:

• Encapsulates the boilerplate (loss functions, scipy calls, CI computation)

• Provides JAX-compiled objectives with exact gradients automatically

• Offers a consistent Experiment data structure

• Includes diagnostics (R², AIC, BIC) and formatted summaries

import jax
import jax.numpy as jnp
import numpy as np
import matplotlib.pyplot as plt

jax.config.update("jax_enable_x64", True)

from difflow.estimation import Estimator, Experiment

print("Setup complete")

Setup complete

1. A Simple Linear Model

Start with the simplest case: fitting a line \(y = a x + b\) to noisy data.

Step 1: Define the model function

The model function must follow this convention:

def model_fn(theta_dict, experiment) -> dict[str, float]

• theta_dict: parameter names → values (e.g., {'a': 2.0, 'b': 1.0})

• experiment: an Experiment object (read inputs from experiment.inputs)

• Returns: predictions dict with keys matching experiment.observed

def linear_model(theta, exp):
    """y = a * x + b"""
    return {'y': theta['a'] * exp.inputs['x'] + theta['b']}

Step 2: Create Experiment objects

Each Experiment holds the inputs, observed outputs, and optional uncertainties.

# True parameters (unknown to the estimator)
a_true, b_true = 2.5, 0.8

# Generate noisy data
rng = np.random.default_rng(42)
x_data = np.linspace(0.5, 5.0, 12)
noise_std = 0.3

experiments = []
for x in x_data:
    y = a_true * x + b_true + noise_std * rng.standard_normal()
    experiments.append(Experiment(
        inputs={'x': float(x)},
        observed={'y': float(y)},
    ))

print(f"Created {len(experiments)} experiments")
print(f"First experiment: {experiments[0]}")
print(f"Output names: {experiments[0].output_names}")

Created 12 experiments
First experiment: Experiment(inputs={'x': 0.5}, observed={'y': 2.141415123926329}, uncertainties=None, name='', metadata={}, measured=None)
Output names: ['y']

Step 3: Create an Estimator and fit

est = Estimator(linear_model, param_names=['a', 'b'])

result = est.fit(experiments, theta_init={'a': 1.0, 'b': 0.0})

print(f"Converged: {result.converged}")
print(f"Estimated a = {result.theta_opt['a']:.4f} (true: {a_true})")
print(f"Estimated b = {result.theta_opt['b']:.4f} (true: {b_true})")
print(f"Objective (SSE) = {result.objective_value:.6f}")

Converged: True
Estimated a = 2.5328 (true: 2.5)
Estimated b = 0.6673 (true: 0.8)
Objective (SSE) = 0.883816

Step 4: Confidence intervals and diagnostics

ci = est.confidence_intervals(result, experiments)

print("95% Confidence Intervals:")
for name in ['a', 'b']:
    print(f"  {name}: [{ci.ci_lower[name]:.4f}, {ci.ci_upper[name]:.4f}]  "
          f"(std err = {ci.std_errors[name]:.4f})")

# Check if true values fall within CIs
for name, true_val in [('a', a_true), ('b', b_true)]:
    in_ci = ci.ci_lower[name] <= true_val <= ci.ci_upper[name]
    print(f"  True {name}={true_val} within CI: {in_ci}")

95% Confidence Intervals:
  a: [2.3974, 2.6682]  (std err = 0.0608)
  b: [0.2487, 1.0859]  (std err = 0.1879)
  True a=2.5 within CI: True
  True b=0.8 within CI: True

# Full summary report
print(est.summary(result, experiments))

============================================================
Parameter Estimation Summary
============================================================
Converged: True
Objective value: 0.883816
Iterations: 5

Parameters:
  Name                   Value      Std Err     CI Lower     CI Upper
  ---------------------------------------------------------------
  a                    2.53282    0.0607705      2.39742      2.66823
  b                   0.667271     0.187867     0.248678      1.08586

Diagnostics:
  R-squared:     0.994276
  Adj R-squared: 0.993004
  RMSE:          0.271388
  AIC:           -27.3010
  BIC:           -26.3311
  N obs:         12
  N params:      2
============================================================

# Visualize the fit
fig, axes = plt.subplots(1, 3, figsize=(14, 4))

# Data and fit
x_plot = np.linspace(0, 5.5, 100)
y_true = a_true * x_plot + b_true
y_est = result.theta_opt['a'] * x_plot + result.theta_opt['b']

axes[0].scatter([exp.inputs['x'] for exp in experiments],
                [exp.observed['y'] for exp in experiments],
                label='Data', alpha=0.7, s=50)
axes[0].plot(x_plot, y_true, 'g--', label=f'True (a={a_true}, b={b_true})', lw=2)
axes[0].plot(x_plot, y_est, 'r-', label=f'Fit (a={result.theta_opt["a"]:.2f}, b={result.theta_opt["b"]:.2f})', lw=2)
axes[0].set_xlabel('x')
axes[0].set_ylabel('y')
axes[0].set_title('Model Fit')
axes[0].legend()
axes[0].grid(True, alpha=0.3)

# Residuals
diag = est.diagnostics(result, experiments)
residuals = [r['y'] for r in diag.residuals]
axes[1].stem([exp.inputs['x'] for exp in experiments], residuals)
axes[1].axhline(0, color='k', linewidth=0.5)
axes[1].set_xlabel('x')
axes[1].set_ylabel('Residual')
axes[1].set_title(f'Residuals (RMSE={diag.rmse:.4f})')
axes[1].grid(True, alpha=0.3)

# Correlation matrix
im = axes[2].imshow(ci.correlation, cmap='RdBu_r', vmin=-1, vmax=1)
axes[2].set_xticks([0, 1])
axes[2].set_xticklabels(['a', 'b'])
axes[2].set_yticks([0, 1])
axes[2].set_yticklabels(['a', 'b'])
axes[2].set_title('Parameter Correlation')
for i in range(2):
    for j in range(2):
        axes[2].text(j, i, f'{float(ci.correlation[i, j]):.2f}',
                     ha='center', va='center', fontsize=14, fontweight='bold')
plt.colorbar(im, ax=axes[2])

plt.tight_layout()
plt.show()

2. Nonlinear Model: Exponential Decay

Fit \(y = A \cdot \exp(-k \cdot t)\) — a common kinetic model.

The JAX-powered Hessian gives exact second derivatives for confidence intervals, even for nonlinear models.

def decay_model(theta, exp):
    """y = A * exp(-k * t)"""
    return {'y': theta['A'] * jnp.exp(-theta['k'] * exp.inputs['t'])}

# True parameters
A_true, k_true = 5.0, 0.3

# Generate data
rng = np.random.default_rng(123)
t_data = np.linspace(0.0, 10.0, 20)
decay_experiments = []
for t in t_data:
    y = A_true * np.exp(-k_true * t) + 0.15 * rng.standard_normal()
    decay_experiments.append(Experiment(
        inputs={'t': float(t)},
        observed={'y': float(y)},
    ))

# Fit
est_decay = Estimator(
    decay_model,
    param_names=['A', 'k'],
    param_bounds={'A': (0.1, 20.0), 'k': (0.01, 5.0)},
)
result_decay = est_decay.fit(decay_experiments, theta_init={'A': 3.0, 'k': 0.1})

print(est_decay.summary(result_decay, decay_experiments))

============================================================
Parameter Estimation Summary
============================================================
Converged: True
Objective value: 0.284552
Iterations: 15

Parameters:
  Name                   Value      Std Err     CI Lower     CI Upper
  ---------------------------------------------------------------
  A                    4.96811    0.0861885      4.78703      5.14918
  k                   0.293621   0.00797516     0.276866     0.310376

Diagnostics:
  R-squared:     0.992554
  Adj R-squared: 0.991678
  RMSE:          0.119279
  AIC:           -81.0515
  BIC:           -79.0600
  N obs:         20
  N params:      2
============================================================

# Plot fit and confidence band
ci_decay = est_decay.confidence_intervals(result_decay, decay_experiments)

fig, axes = plt.subplots(1, 2, figsize=(12, 4))

t_plot = np.linspace(0, 10, 200)
y_fit = result_decay.theta_opt['A'] * np.exp(-result_decay.theta_opt['k'] * t_plot)
y_true_plot = A_true * np.exp(-k_true * t_plot)

axes[0].scatter([e.inputs['t'] for e in decay_experiments],
                [e.observed['y'] for e in decay_experiments],
                label='Data', alpha=0.7, s=40)
axes[0].plot(t_plot, y_true_plot, 'g--', label=f'True (A={A_true}, k={k_true})', lw=2)
axes[0].plot(t_plot, y_fit, 'r-', lw=2,
             label=f'Fit (A={result_decay.theta_opt["A"]:.3f}, k={result_decay.theta_opt["k"]:.4f})')
axes[0].set_xlabel('Time')
axes[0].set_ylabel('y')
axes[0].set_title('Exponential Decay Fit')
axes[0].legend()
axes[0].grid(True, alpha=0.3)

# Parity plot
diag_decay = est_decay.diagnostics(result_decay, decay_experiments)
observed = [e.observed['y'] for e in decay_experiments]
predicted = [e.observed['y'] + r['y'] for e, r in zip(decay_experiments, diag_decay.residuals)]
# Actually: predicted = observed + residual, so predicted = observed + (pred - obs) = pred
# Let's compute correctly
predicted = []
for e in decay_experiments:
    p = decay_model(result_decay.theta_opt, e)
    predicted.append(float(p['y']))

axes[1].scatter(observed, predicted, s=50, alpha=0.7)
lims = [min(min(observed), min(predicted)) - 0.5, max(max(observed), max(predicted)) + 0.5]
axes[1].plot(lims, lims, 'k--', lw=1)
axes[1].set_xlabel('Observed')
axes[1].set_ylabel('Predicted')
axes[1].set_title(f'Parity Plot (R² = {diag_decay.r_squared:.4f})')
axes[1].set_aspect('equal')
axes[1].grid(True, alpha=0.3)

plt.tight_layout()
plt.show()

3. Multi-Output Estimation

Estimate parameters from models that predict multiple outputs simultaneously. Here we fit a Michaelis-Menten model: \(v = V_{max} \cdot S / (K_m + S)\) and predict both the rate and the substrate consumption.

def mm_model(theta, exp):
    """Michaelis-Menten kinetics with two outputs."""
    S = exp.inputs['S']
    v = theta['Vmax'] * S / (theta['Km'] + S)
    # After time dt, substrate consumed
    dt = exp.inputs['dt']
    S_final = S - v * dt
    return {'rate': v, 'S_final': S_final}

# True parameters
Vmax_true, Km_true = 10.0, 2.0

# Generate data at different substrate concentrations
rng = np.random.default_rng(7)
dt = 0.1
mm_experiments = []
for S in [0.5, 1.0, 2.0, 3.0, 5.0, 8.0, 12.0, 20.0]:
    v_true = Vmax_true * S / (Km_true + S)
    S_final_true = S - v_true * dt
    mm_experiments.append(Experiment(
        inputs={'S': float(S), 'dt': dt},
        observed={
            'rate': float(v_true + 0.3 * rng.standard_normal()),
            'S_final': float(S_final_true + 0.05 * rng.standard_normal()),
        },
    ))

est_mm = Estimator(mm_model, param_names=['Vmax', 'Km'],
                    param_bounds={'Vmax': (0.1, 100.0), 'Km': (0.01, 50.0)})
result_mm = est_mm.fit(mm_experiments, {'Vmax': 5.0, 'Km': 1.0})

print(est_mm.summary(result_mm, mm_experiments))

============================================================
Parameter Estimation Summary
============================================================
Converged: True
Objective value: 0.0666396
Iterations: 11

Parameters:
  Name                   Value      Std Err     CI Lower     CI Upper
  ---------------------------------------------------------------
  Vmax                 10.0908    0.0683073      9.94429      10.2373
  Km                   2.08909    0.0487065      1.98462      2.19355

Diagnostics:
  R-squared:     0.999806
  Adj R-squared: 0.999776
  RMSE:          0.0645366
  AIC:           -83.6967
  BIC:           -82.1515
  N obs:         16
  N params:      2
============================================================

# Visualize Michaelis-Menten fit
fig, axes = plt.subplots(1, 2, figsize=(12, 4))

S_plot = np.linspace(0.1, 25, 200)
v_true_plot = Vmax_true * S_plot / (Km_true + S_plot)
v_fit_plot = result_mm.theta_opt['Vmax'] * S_plot / (result_mm.theta_opt['Km'] + S_plot)

axes[0].scatter([e.inputs['S'] for e in mm_experiments],
                [e.observed['rate'] for e in mm_experiments],
                label='Data', s=60, alpha=0.7)
axes[0].plot(S_plot, v_true_plot, 'g--', lw=2, label=f'True (Vmax={Vmax_true}, Km={Km_true})')
axes[0].plot(S_plot, v_fit_plot, 'r-', lw=2,
             label=f'Fit (Vmax={result_mm.theta_opt["Vmax"]:.2f}, Km={result_mm.theta_opt["Km"]:.2f})')
axes[0].set_xlabel('[S] (concentration)')
axes[0].set_ylabel('Rate v')
axes[0].set_title('Michaelis-Menten Fit')
axes[0].legend()
axes[0].grid(True, alpha=0.3)

# Lineweaver-Burk plot (1/v vs 1/S)
S_data = np.array([e.inputs['S'] for e in mm_experiments])
v_data = np.array([e.observed['rate'] for e in mm_experiments])
axes[1].scatter(1/S_data, 1/v_data, s=60, alpha=0.7)
S_inv = np.linspace(0.01, 1/0.4, 100)
v_inv_fit = (result_mm.theta_opt['Km'] / result_mm.theta_opt['Vmax']) * S_inv + 1/result_mm.theta_opt['Vmax']
axes[1].plot(S_inv, v_inv_fit, 'r-', lw=2)
axes[1].set_xlabel('1/[S]')
axes[1].set_ylabel('1/v')
axes[1].set_title('Lineweaver-Burk Plot')
axes[1].grid(True, alpha=0.3)

plt.tight_layout()
plt.show()

4. Weighted Least Squares

When measurements have different uncertainties, use weighted SSE. Experiments with smaller uncertainties get more weight in the fit.

# Create experiments with varying measurement uncertainty
rng = np.random.default_rng(99)
weighted_experiments = []

for x in np.linspace(0.5, 5.0, 10):
    # Uncertainty grows with x (e.g., harder to measure at extremes)
    sigma = 0.1 + 0.15 * x
    y = a_true * x + b_true + sigma * rng.standard_normal()
    weighted_experiments.append(Experiment(
        inputs={'x': float(x)},
        observed={'y': float(y)},
        uncertainties={'y': float(sigma)},
    ))

# Compare SSE vs WSSE
est_w = Estimator(linear_model, ['a', 'b'])
result_sse = est_w.fit(weighted_experiments, {'a': 1.0, 'b': 0.0}, objective='sse')
result_wsse = est_w.fit(weighted_experiments, {'a': 1.0, 'b': 0.0}, objective='wsse')

print(f"{'':>15} {'a':>10} {'b':>10}")
print(f"{'True':>15} {a_true:>10.4f} {b_true:>10.4f}")
print(f"{'SSE fit':>15} {result_sse.theta_opt['a']:>10.4f} {result_sse.theta_opt['b']:>10.4f}")
print(f"{'WSSE fit':>15} {result_wsse.theta_opt['a']:>10.4f} {result_wsse.theta_opt['b']:>10.4f}")
print()
print("WSSE gives more weight to precise measurements (low x),")
print("often recovering the true parameters more accurately.")

                         a          b
           True     2.5000     0.8000
        SSE fit     2.5005     0.7545
       WSSE fit     2.4827     0.7992

WSSE gives more weight to precise measurements (low x),
often recovering the true parameters more accurately.

5. CSTR Kinetic Parameter Estimation

Now let’s use the estimation module with a difflow CSTR model. We estimate the rate constant k from outlet concentration data at different flow rates.

from difflow import CSTR, CSTRParams, make_stream

# Rate function: A -> B, r = k * C_A
def rate_fn(C, T, params):
    return jnp.array([params['k'] * C['A']])

stoich = jnp.array([[-1.0], [+1.0]])

# True rate constant
k_true_cstr = 0.15

# Generate synthetic CSTR data at different flow rates
rng = np.random.default_rng(0)
flow_rates = [0.5, 1.0, 1.5, 2.0, 2.5, 3.0, 4.0, 5.0]
cstr_experiments = []

for F_A_in in flow_rates:
    cstr = CSTR(
        CSTRParams(V=1.0, rate_fn=rate_fn, stoich=stoich,
                   rate_params={'k': k_true_cstr}, species_order=['A', 'B']),
        mode='isothermal',
    )
    inlet = make_stream({'A': F_A_in, 'B': 0.0}, T=350.0, P=101325.0)
    outlet, _ = cstr(inlet, T_spec=350.0)

    # Add measurement noise
    F_A_out_noisy = float(outlet['F_A']) + 0.01 * rng.standard_normal()
    cstr_experiments.append(Experiment(
        inputs={'F_A_in': F_A_in},
        observed={'F_A_out': F_A_out_noisy},
        name=f'F_A_in={F_A_in}',
    ))

print(f"Created {len(cstr_experiments)} CSTR experiments")
for e in cstr_experiments:
    print(f"  F_A_in={e.inputs['F_A_in']:.1f}  F_A_out={e.observed['F_A_out']:.4f}")

/tmp/ipykernel_1384/1245159964.py:24: CSTRDensityWarning: CSTR: neither eos nor molar_density was given, so the concentration basis falls back to 55500 mol/m^3 -- liquid water. Density sets the volumetric flow and hence the residence time (tau = V * rho / F_total), so a wrong density is a proportionally wrong conversion: ~8x for a C3-C8 hydrocarbon liquid, ~200x for a gas. Pass molar_density=<mol/m^3>, or eos=<cubic EOS> with reaction_phase='liquid'|'vapor' for a phase-consistent basis, or an explicit volumetric_flow= to the call.
  outlet, _ = cstr(inlet, T_spec=350.0)

/tmp/ipykernel_1384/1245159964.py:24: CSTRDensityWarning: CSTR: neither eos nor molar_density was given, so the concentration basis falls back to 55500 mol/m^3 -- liquid water. Density sets the volumetric flow and hence the residence time (tau = V * rho / F_total), so a wrong density is a proportionally wrong conversion: ~8x for a C3-C8 hydrocarbon liquid, ~200x for a gas. Pass molar_density=<mol/m^3>, or eos=<cubic EOS> with reaction_phase='liquid'|'vapor' for a phase-consistent basis, or an explicit volumetric_flow= to the call.
  outlet, _ = cstr(inlet, T_spec=350.0)

/tmp/ipykernel_1384/1245159964.py:24: CSTRDensityWarning: CSTR: neither eos nor molar_density was given, so the concentration basis falls back to 55500 mol/m^3 -- liquid water. Density sets the volumetric flow and hence the residence time (tau = V * rho / F_total), so a wrong density is a proportionally wrong conversion: ~8x for a C3-C8 hydrocarbon liquid, ~200x for a gas. Pass molar_density=<mol/m^3>, or eos=<cubic EOS> with reaction_phase='liquid'|'vapor' for a phase-consistent basis, or an explicit volumetric_flow= to the call.
  outlet, _ = cstr(inlet, T_spec=350.0)

/tmp/ipykernel_1384/1245159964.py:24: CSTRDensityWarning: CSTR: neither eos nor molar_density was given, so the concentration basis falls back to 55500 mol/m^3 -- liquid water. Density sets the volumetric flow and hence the residence time (tau = V * rho / F_total), so a wrong density is a proportionally wrong conversion: ~8x for a C3-C8 hydrocarbon liquid, ~200x for a gas. Pass molar_density=<mol/m^3>, or eos=<cubic EOS> with reaction_phase='liquid'|'vapor' for a phase-consistent basis, or an explicit volumetric_flow= to the call.
  outlet, _ = cstr(inlet, T_spec=350.0)

/tmp/ipykernel_1384/1245159964.py:24: CSTRDensityWarning: CSTR: neither eos nor molar_density was given, so the concentration basis falls back to 55500 mol/m^3 -- liquid water. Density sets the volumetric flow and hence the residence time (tau = V * rho / F_total), so a wrong density is a proportionally wrong conversion: ~8x for a C3-C8 hydrocarbon liquid, ~200x for a gas. Pass molar_density=<mol/m^3>, or eos=<cubic EOS> with reaction_phase='liquid'|'vapor' for a phase-consistent basis, or an explicit volumetric_flow= to the call.
  outlet, _ = cstr(inlet, T_spec=350.0)

/tmp/ipykernel_1384/1245159964.py:24: CSTRDensityWarning: CSTR: neither eos nor molar_density was given, so the concentration basis falls back to 55500 mol/m^3 -- liquid water. Density sets the volumetric flow and hence the residence time (tau = V * rho / F_total), so a wrong density is a proportionally wrong conversion: ~8x for a C3-C8 hydrocarbon liquid, ~200x for a gas. Pass molar_density=<mol/m^3>, or eos=<cubic EOS> with reaction_phase='liquid'|'vapor' for a phase-consistent basis, or an explicit volumetric_flow= to the call.
  outlet, _ = cstr(inlet, T_spec=350.0)

/tmp/ipykernel_1384/1245159964.py:24: CSTRDensityWarning: CSTR: neither eos nor molar_density was given, so the concentration basis falls back to 55500 mol/m^3 -- liquid water. Density sets the volumetric flow and hence the residence time (tau = V * rho / F_total), so a wrong density is a proportionally wrong conversion: ~8x for a C3-C8 hydrocarbon liquid, ~200x for a gas. Pass molar_density=<mol/m^3>, or eos=<cubic EOS> with reaction_phase='liquid'|'vapor' for a phase-consistent basis, or an explicit volumetric_flow= to the call.
  outlet, _ = cstr(inlet, T_spec=350.0)

/tmp/ipykernel_1384/1245159964.py:24: CSTRDensityWarning: CSTR: neither eos nor molar_density was given, so the concentration basis falls back to 55500 mol/m^3 -- liquid water. Density sets the volumetric flow and hence the residence time (tau = V * rho / F_total), so a wrong density is a proportionally wrong conversion: ~8x for a C3-C8 hydrocarbon liquid, ~200x for a gas. Pass molar_density=<mol/m^3>, or eos=<cubic EOS> with reaction_phase='liquid'|'vapor' for a phase-consistent basis, or an explicit volumetric_flow= to the call.
  outlet, _ = cstr(inlet, T_spec=350.0)

Created 8 CSTR experiments
  F_A_in=0.5  F_A_out=0.0013
  F_A_in=1.0  F_A_out=-0.0012
  F_A_in=1.5  F_A_out=0.0067
  F_A_in=2.0  F_A_out=0.0015
  F_A_in=2.5  F_A_out=-0.0046
  F_A_in=3.0  F_A_out=0.0047
  F_A_in=4.0  F_A_out=0.0150
  F_A_in=5.0  F_A_out=0.0125

# Model function that wraps the CSTR
def cstr_model(theta, exp):
    """Predict outlet flow from CSTR."""
    cstr = CSTR(
        CSTRParams(V=1.0, rate_fn=rate_fn, stoich=stoich,
                   rate_params={'k': theta['k']}, species_order=['A', 'B']),
        mode='isothermal',
    )
    inlet = make_stream({'A': exp.inputs['F_A_in'], 'B': 0.0}, T=350.0, P=101325.0)
    outlet, _ = cstr(inlet, T_spec=350.0)
    return {'F_A_out': outlet['F_A']}

# Fit
est_cstr = Estimator(cstr_model, ['k'], param_bounds={'k': (0.001, 10.0)})
result_cstr = est_cstr.fit(cstr_experiments, theta_init={'k': 0.05})

print(est_cstr.summary(result_cstr, cstr_experiments))

/tmp/ipykernel_1384/4119124814.py:10: CSTRDensityWarning: CSTR: neither eos nor molar_density was given, so the concentration basis falls back to 55500 mol/m^3 -- liquid water. Density sets the volumetric flow and hence the residence time (tau = V * rho / F_total), so a wrong density is a proportionally wrong conversion: ~8x for a C3-C8 hydrocarbon liquid, ~200x for a gas. Pass molar_density=<mol/m^3>, or eos=<cubic EOS> with reaction_phase='liquid'|'vapor' for a phase-consistent basis, or an explicit volumetric_flow= to the call.
  outlet, _ = cstr(inlet, T_spec=350.0)
/tmp/ipykernel_1384/4119124814.py:10: CSTRDensityWarning: CSTR: neither eos nor molar_density was given, so the concentration basis falls back to 55500 mol/m^3 -- liquid water. Density sets the volumetric flow and hence the residence time (tau = V * rho / F_total), so a wrong density is a proportionally wrong conversion: ~8x for a C3-C8 hydrocarbon liquid, ~200x for a gas. Pass molar_density=<mol/m^3>, or eos=<cubic EOS> with reaction_phase='liquid'|'vapor' for a phase-consistent basis, or an explicit volumetric_flow= to the call.
  outlet, _ = cstr(inlet, T_spec=350.0)
/tmp/ipykernel_1384/4119124814.py:10: CSTRDensityWarning: CSTR: neither eos nor molar_density was given, so the concentration basis falls back to 55500 mol/m^3 -- liquid water. Density sets the volumetric flow and hence the residence time (tau = V * rho / F_total), so a wrong density is a proportionally wrong conversion: ~8x for a C3-C8 hydrocarbon liquid, ~200x for a gas. Pass molar_density=<mol/m^3>, or eos=<cubic EOS> with reaction_phase='liquid'|'vapor' for a phase-consistent basis, or an explicit volumetric_flow= to the call.
  outlet, _ = cstr(inlet, T_spec=350.0)
/tmp/ipykernel_1384/4119124814.py:10: CSTRDensityWarning: CSTR: neither eos nor molar_density was given, so the concentration basis falls back to 55500 mol/m^3 -- liquid water. Density sets the volumetric flow and hence the residence time (tau = V * rho / F_total), so a wrong density is a proportionally wrong conversion: ~8x for a C3-C8 hydrocarbon liquid, ~200x for a gas. Pass molar_density=<mol/m^3>, or eos=<cubic EOS> with reaction_phase='liquid'|'vapor' for a phase-consistent basis, or an explicit volumetric_flow= to the call.
  outlet, _ = cstr(inlet, T_spec=350.0)
/tmp/ipykernel_1384/4119124814.py:10: CSTRDensityWarning: CSTR: neither eos nor molar_density was given, so the concentration basis falls back to 55500 mol/m^3 -- liquid water. Density sets the volumetric flow and hence the residence time (tau = V * rho / F_total), so a wrong density is a proportionally wrong conversion: ~8x for a C3-C8 hydrocarbon liquid, ~200x for a gas. Pass molar_density=<mol/m^3>, or eos=<cubic EOS> with reaction_phase='liquid'|'vapor' for a phase-consistent basis, or an explicit volumetric_flow= to the call.
  outlet, _ = cstr(inlet, T_spec=350.0)

/tmp/ipykernel_1384/4119124814.py:10: CSTRDensityWarning: CSTR: neither eos nor molar_density was given, so the concentration basis falls back to 55500 mol/m^3 -- liquid water. Density sets the volumetric flow and hence the residence time (tau = V * rho / F_total), so a wrong density is a proportionally wrong conversion: ~8x for a C3-C8 hydrocarbon liquid, ~200x for a gas. Pass molar_density=<mol/m^3>, or eos=<cubic EOS> with reaction_phase='liquid'|'vapor' for a phase-consistent basis, or an explicit volumetric_flow= to the call.
  outlet, _ = cstr(inlet, T_spec=350.0)
/tmp/ipykernel_1384/4119124814.py:10: CSTRDensityWarning: CSTR: neither eos nor molar_density was given, so the concentration basis falls back to 55500 mol/m^3 -- liquid water. Density sets the volumetric flow and hence the residence time (tau = V * rho / F_total), so a wrong density is a proportionally wrong conversion: ~8x for a C3-C8 hydrocarbon liquid, ~200x for a gas. Pass molar_density=<mol/m^3>, or eos=<cubic EOS> with reaction_phase='liquid'|'vapor' for a phase-consistent basis, or an explicit volumetric_flow= to the call.
  outlet, _ = cstr(inlet, T_spec=350.0)
/tmp/ipykernel_1384/4119124814.py:10: CSTRDensityWarning: CSTR: neither eos nor molar_density was given, so the concentration basis falls back to 55500 mol/m^3 -- liquid water. Density sets the volumetric flow and hence the residence time (tau = V * rho / F_total), so a wrong density is a proportionally wrong conversion: ~8x for a C3-C8 hydrocarbon liquid, ~200x for a gas. Pass molar_density=<mol/m^3>, or eos=<cubic EOS> with reaction_phase='liquid'|'vapor' for a phase-consistent basis, or an explicit volumetric_flow= to the call.
  outlet, _ = cstr(inlet, T_spec=350.0)

/tmp/ipykernel_1384/4119124814.py:10: CSTRDensityWarning: CSTR: neither eos nor molar_density was given, so the concentration basis falls back to 55500 mol/m^3 -- liquid water. Density sets the volumetric flow and hence the residence time (tau = V * rho / F_total), so a wrong density is a proportionally wrong conversion: ~8x for a C3-C8 hydrocarbon liquid, ~200x for a gas. Pass molar_density=<mol/m^3>, or eos=<cubic EOS> with reaction_phase='liquid'|'vapor' for a phase-consistent basis, or an explicit volumetric_flow= to the call.
  outlet, _ = cstr(inlet, T_spec=350.0)

/tmp/ipykernel_1384/4119124814.py:10: CSTRDensityWarning: CSTR: neither eos nor molar_density was given, so the concentration basis falls back to 55500 mol/m^3 -- liquid water. Density sets the volumetric flow and hence the residence time (tau = V * rho / F_total), so a wrong density is a proportionally wrong conversion: ~8x for a C3-C8 hydrocarbon liquid, ~200x for a gas. Pass molar_density=<mol/m^3>, or eos=<cubic EOS> with reaction_phase='liquid'|'vapor' for a phase-consistent basis, or an explicit volumetric_flow= to the call.
  outlet, _ = cstr(inlet, T_spec=350.0)
/tmp/ipykernel_1384/4119124814.py:10: CSTRDensityWarning: CSTR: neither eos nor molar_density was given, so the concentration basis falls back to 55500 mol/m^3 -- liquid water. Density sets the volumetric flow and hence the residence time (tau = V * rho / F_total), so a wrong density is a proportionally wrong conversion: ~8x for a C3-C8 hydrocarbon liquid, ~200x for a gas. Pass molar_density=<mol/m^3>, or eos=<cubic EOS> with reaction_phase='liquid'|'vapor' for a phase-consistent basis, or an explicit volumetric_flow= to the call.
  outlet, _ = cstr(inlet, T_spec=350.0)

/tmp/ipykernel_1384/4119124814.py:10: CSTRDensityWarning: CSTR: neither eos nor molar_density was given, so the concentration basis falls back to 55500 mol/m^3 -- liquid water. Density sets the volumetric flow and hence the residence time (tau = V * rho / F_total), so a wrong density is a proportionally wrong conversion: ~8x for a C3-C8 hydrocarbon liquid, ~200x for a gas. Pass molar_density=<mol/m^3>, or eos=<cubic EOS> with reaction_phase='liquid'|'vapor' for a phase-consistent basis, or an explicit volumetric_flow= to the call.
  outlet, _ = cstr(inlet, T_spec=350.0)
/tmp/ipykernel_1384/4119124814.py:10: CSTRDensityWarning: CSTR: neither eos nor molar_density was given, so the concentration basis falls back to 55500 mol/m^3 -- liquid water. Density sets the volumetric flow and hence the residence time (tau = V * rho / F_total), so a wrong density is a proportionally wrong conversion: ~8x for a C3-C8 hydrocarbon liquid, ~200x for a gas. Pass molar_density=<mol/m^3>, or eos=<cubic EOS> with reaction_phase='liquid'|'vapor' for a phase-consistent basis, or an explicit volumetric_flow= to the call.
  outlet, _ = cstr(inlet, T_spec=350.0)

/tmp/ipykernel_1384/4119124814.py:10: CSTRDensityWarning: CSTR: neither eos nor molar_density was given, so the concentration basis falls back to 55500 mol/m^3 -- liquid water. Density sets the volumetric flow and hence the residence time (tau = V * rho / F_total), so a wrong density is a proportionally wrong conversion: ~8x for a C3-C8 hydrocarbon liquid, ~200x for a gas. Pass molar_density=<mol/m^3>, or eos=<cubic EOS> with reaction_phase='liquid'|'vapor' for a phase-consistent basis, or an explicit volumetric_flow= to the call.
  outlet, _ = cstr(inlet, T_spec=350.0)
/tmp/ipykernel_1384/4119124814.py:10: CSTRDensityWarning: CSTR: neither eos nor molar_density was given, so the concentration basis falls back to 55500 mol/m^3 -- liquid water. Density sets the volumetric flow and hence the residence time (tau = V * rho / F_total), so a wrong density is a proportionally wrong conversion: ~8x for a C3-C8 hydrocarbon liquid, ~200x for a gas. Pass molar_density=<mol/m^3>, or eos=<cubic EOS> with reaction_phase='liquid'|'vapor' for a phase-consistent basis, or an explicit volumetric_flow= to the call.
  outlet, _ = cstr(inlet, T_spec=350.0)

/tmp/ipykernel_1384/4119124814.py:10: CSTRDensityWarning: CSTR: neither eos nor molar_density was given, so the concentration basis falls back to 55500 mol/m^3 -- liquid water. Density sets the volumetric flow and hence the residence time (tau = V * rho / F_total), so a wrong density is a proportionally wrong conversion: ~8x for a C3-C8 hydrocarbon liquid, ~200x for a gas. Pass molar_density=<mol/m^3>, or eos=<cubic EOS> with reaction_phase='liquid'|'vapor' for a phase-consistent basis, or an explicit volumetric_flow= to the call.
  outlet, _ = cstr(inlet, T_spec=350.0)

/tmp/ipykernel_1384/4119124814.py:10: CSTRDensityWarning: CSTR: neither eos nor molar_density was given, so the concentration basis falls back to 55500 mol/m^3 -- liquid water. Density sets the volumetric flow and hence the residence time (tau = V * rho / F_total), so a wrong density is a proportionally wrong conversion: ~8x for a C3-C8 hydrocarbon liquid, ~200x for a gas. Pass molar_density=<mol/m^3>, or eos=<cubic EOS> with reaction_phase='liquid'|'vapor' for a phase-consistent basis, or an explicit volumetric_flow= to the call.
  outlet, _ = cstr(inlet, T_spec=350.0)

/tmp/ipykernel_1384/4119124814.py:10: CSTRDensityWarning: CSTR: neither eos nor molar_density was given, so the concentration basis falls back to 55500 mol/m^3 -- liquid water. Density sets the volumetric flow and hence the residence time (tau = V * rho / F_total), so a wrong density is a proportionally wrong conversion: ~8x for a C3-C8 hydrocarbon liquid, ~200x for a gas. Pass molar_density=<mol/m^3>, or eos=<cubic EOS> with reaction_phase='liquid'|'vapor' for a phase-consistent basis, or an explicit volumetric_flow= to the call.
  outlet, _ = cstr(inlet, T_spec=350.0)

/tmp/ipykernel_1384/4119124814.py:10: CSTRDensityWarning: CSTR: neither eos nor molar_density was given, so the concentration basis falls back to 55500 mol/m^3 -- liquid water. Density sets the volumetric flow and hence the residence time (tau = V * rho / F_total), so a wrong density is a proportionally wrong conversion: ~8x for a C3-C8 hydrocarbon liquid, ~200x for a gas. Pass molar_density=<mol/m^3>, or eos=<cubic EOS> with reaction_phase='liquid'|'vapor' for a phase-consistent basis, or an explicit volumetric_flow= to the call.
  outlet, _ = cstr(inlet, T_spec=350.0)

/tmp/ipykernel_1384/4119124814.py:10: CSTRDensityWarning: CSTR: neither eos nor molar_density was given, so the concentration basis falls back to 55500 mol/m^3 -- liquid water. Density sets the volumetric flow and hence the residence time (tau = V * rho / F_total), so a wrong density is a proportionally wrong conversion: ~8x for a C3-C8 hydrocarbon liquid, ~200x for a gas. Pass molar_density=<mol/m^3>, or eos=<cubic EOS> with reaction_phase='liquid'|'vapor' for a phase-consistent basis, or an explicit volumetric_flow= to the call.
  outlet, _ = cstr(inlet, T_spec=350.0)

/tmp/ipykernel_1384/4119124814.py:10: CSTRDensityWarning: CSTR: neither eos nor molar_density was given, so the concentration basis falls back to 55500 mol/m^3 -- liquid water. Density sets the volumetric flow and hence the residence time (tau = V * rho / F_total), so a wrong density is a proportionally wrong conversion: ~8x for a C3-C8 hydrocarbon liquid, ~200x for a gas. Pass molar_density=<mol/m^3>, or eos=<cubic EOS> with reaction_phase='liquid'|'vapor' for a phase-consistent basis, or an explicit volumetric_flow= to the call.
  outlet, _ = cstr(inlet, T_spec=350.0)

/tmp/ipykernel_1384/4119124814.py:10: CSTRDensityWarning: CSTR: neither eos nor molar_density was given, so the concentration basis falls back to 55500 mol/m^3 -- liquid water. Density sets the volumetric flow and hence the residence time (tau = V * rho / F_total), so a wrong density is a proportionally wrong conversion: ~8x for a C3-C8 hydrocarbon liquid, ~200x for a gas. Pass molar_density=<mol/m^3>, or eos=<cubic EOS> with reaction_phase='liquid'|'vapor' for a phase-consistent basis, or an explicit volumetric_flow= to the call.
  outlet, _ = cstr(inlet, T_spec=350.0)

/tmp/ipykernel_1384/4119124814.py:10: CSTRDensityWarning: CSTR: neither eos nor molar_density was given, so the concentration basis falls back to 55500 mol/m^3 -- liquid water. Density sets the volumetric flow and hence the residence time (tau = V * rho / F_total), so a wrong density is a proportionally wrong conversion: ~8x for a C3-C8 hydrocarbon liquid, ~200x for a gas. Pass molar_density=<mol/m^3>, or eos=<cubic EOS> with reaction_phase='liquid'|'vapor' for a phase-consistent basis, or an explicit volumetric_flow= to the call.
  outlet, _ = cstr(inlet, T_spec=350.0)

/tmp/ipykernel_1384/4119124814.py:10: CSTRDensityWarning: CSTR: neither eos nor molar_density was given, so the concentration basis falls back to 55500 mol/m^3 -- liquid water. Density sets the volumetric flow and hence the residence time (tau = V * rho / F_total), so a wrong density is a proportionally wrong conversion: ~8x for a C3-C8 hydrocarbon liquid, ~200x for a gas. Pass molar_density=<mol/m^3>, or eos=<cubic EOS> with reaction_phase='liquid'|'vapor' for a phase-consistent basis, or an explicit volumetric_flow= to the call.
  outlet, _ = cstr(inlet, T_spec=350.0)

/tmp/ipykernel_1384/4119124814.py:10: CSTRDensityWarning: CSTR: neither eos nor molar_density was given, so the concentration basis falls back to 55500 mol/m^3 -- liquid water. Density sets the volumetric flow and hence the residence time (tau = V * rho / F_total), so a wrong density is a proportionally wrong conversion: ~8x for a C3-C8 hydrocarbon liquid, ~200x for a gas. Pass molar_density=<mol/m^3>, or eos=<cubic EOS> with reaction_phase='liquid'|'vapor' for a phase-consistent basis, or an explicit volumetric_flow= to the call.
  outlet, _ = cstr(inlet, T_spec=350.0)

/tmp/ipykernel_1384/4119124814.py:10: CSTRDensityWarning: CSTR: neither eos nor molar_density was given, so the concentration basis falls back to 55500 mol/m^3 -- liquid water. Density sets the volumetric flow and hence the residence time (tau = V * rho / F_total), so a wrong density is a proportionally wrong conversion: ~8x for a C3-C8 hydrocarbon liquid, ~200x for a gas. Pass molar_density=<mol/m^3>, or eos=<cubic EOS> with reaction_phase='liquid'|'vapor' for a phase-consistent basis, or an explicit volumetric_flow= to the call.
  outlet, _ = cstr(inlet, T_spec=350.0)

/tmp/ipykernel_1384/4119124814.py:10: CSTRDensityWarning: CSTR: neither eos nor molar_density was given, so the concentration basis falls back to 55500 mol/m^3 -- liquid water. Density sets the volumetric flow and hence the residence time (tau = V * rho / F_total), so a wrong density is a proportionally wrong conversion: ~8x for a C3-C8 hydrocarbon liquid, ~200x for a gas. Pass molar_density=<mol/m^3>, or eos=<cubic EOS> with reaction_phase='liquid'|'vapor' for a phase-consistent basis, or an explicit volumetric_flow= to the call.
  outlet, _ = cstr(inlet, T_spec=350.0)

/tmp/ipykernel_1384/4119124814.py:10: CSTRDensityWarning: CSTR: neither eos nor molar_density was given, so the concentration basis falls back to 55500 mol/m^3 -- liquid water. Density sets the volumetric flow and hence the residence time (tau = V * rho / F_total), so a wrong density is a proportionally wrong conversion: ~8x for a C3-C8 hydrocarbon liquid, ~200x for a gas. Pass molar_density=<mol/m^3>, or eos=<cubic EOS> with reaction_phase='liquid'|'vapor' for a phase-consistent basis, or an explicit volumetric_flow= to the call.
  outlet, _ = cstr(inlet, T_spec=350.0)

/tmp/ipykernel_1384/4119124814.py:10: CSTRDensityWarning: CSTR: neither eos nor molar_density was given, so the concentration basis falls back to 55500 mol/m^3 -- liquid water. Density sets the volumetric flow and hence the residence time (tau = V * rho / F_total), so a wrong density is a proportionally wrong conversion: ~8x for a C3-C8 hydrocarbon liquid, ~200x for a gas. Pass molar_density=<mol/m^3>, or eos=<cubic EOS> with reaction_phase='liquid'|'vapor' for a phase-consistent basis, or an explicit volumetric_flow= to the call.
  outlet, _ = cstr(inlet, T_spec=350.0)

/tmp/ipykernel_1384/4119124814.py:10: CSTRDensityWarning: CSTR: neither eos nor molar_density was given, so the concentration basis falls back to 55500 mol/m^3 -- liquid water. Density sets the volumetric flow and hence the residence time (tau = V * rho / F_total), so a wrong density is a proportionally wrong conversion: ~8x for a C3-C8 hydrocarbon liquid, ~200x for a gas. Pass molar_density=<mol/m^3>, or eos=<cubic EOS> with reaction_phase='liquid'|'vapor' for a phase-consistent basis, or an explicit volumetric_flow= to the call.
  outlet, _ = cstr(inlet, T_spec=350.0)

/tmp/ipykernel_1384/4119124814.py:10: CSTRDensityWarning: CSTR: neither eos nor molar_density was given, so the concentration basis falls back to 55500 mol/m^3 -- liquid water. Density sets the volumetric flow and hence the residence time (tau = V * rho / F_total), so a wrong density is a proportionally wrong conversion: ~8x for a C3-C8 hydrocarbon liquid, ~200x for a gas. Pass molar_density=<mol/m^3>, or eos=<cubic EOS> with reaction_phase='liquid'|'vapor' for a phase-consistent basis, or an explicit volumetric_flow= to the call.
  outlet, _ = cstr(inlet, T_spec=350.0)

/tmp/ipykernel_1384/4119124814.py:10: CSTRDensityWarning: CSTR: neither eos nor molar_density was given, so the concentration basis falls back to 55500 mol/m^3 -- liquid water. Density sets the volumetric flow and hence the residence time (tau = V * rho / F_total), so a wrong density is a proportionally wrong conversion: ~8x for a C3-C8 hydrocarbon liquid, ~200x for a gas. Pass molar_density=<mol/m^3>, or eos=<cubic EOS> with reaction_phase='liquid'|'vapor' for a phase-consistent basis, or an explicit volumetric_flow= to the call.
  outlet, _ = cstr(inlet, T_spec=350.0)

/tmp/ipykernel_1384/4119124814.py:10: CSTRDensityWarning: CSTR: neither eos nor molar_density was given, so the concentration basis falls back to 55500 mol/m^3 -- liquid water. Density sets the volumetric flow and hence the residence time (tau = V * rho / F_total), so a wrong density is a proportionally wrong conversion: ~8x for a C3-C8 hydrocarbon liquid, ~200x for a gas. Pass molar_density=<mol/m^3>, or eos=<cubic EOS> with reaction_phase='liquid'|'vapor' for a phase-consistent basis, or an explicit volumetric_flow= to the call.
  outlet, _ = cstr(inlet, T_spec=350.0)

/tmp/ipykernel_1384/4119124814.py:10: CSTRDensityWarning: CSTR: neither eos nor molar_density was given, so the concentration basis falls back to 55500 mol/m^3 -- liquid water. Density sets the volumetric flow and hence the residence time (tau = V * rho / F_total), so a wrong density is a proportionally wrong conversion: ~8x for a C3-C8 hydrocarbon liquid, ~200x for a gas. Pass molar_density=<mol/m^3>, or eos=<cubic EOS> with reaction_phase='liquid'|'vapor' for a phase-consistent basis, or an explicit volumetric_flow= to the call.
  outlet, _ = cstr(inlet, T_spec=350.0)

/tmp/ipykernel_1384/4119124814.py:10: CSTRDensityWarning: CSTR: neither eos nor molar_density was given, so the concentration basis falls back to 55500 mol/m^3 -- liquid water. Density sets the volumetric flow and hence the residence time (tau = V * rho / F_total), so a wrong density is a proportionally wrong conversion: ~8x for a C3-C8 hydrocarbon liquid, ~200x for a gas. Pass molar_density=<mol/m^3>, or eos=<cubic EOS> with reaction_phase='liquid'|'vapor' for a phase-consistent basis, or an explicit volumetric_flow= to the call.
  outlet, _ = cstr(inlet, T_spec=350.0)

/tmp/ipykernel_1384/4119124814.py:10: CSTRDensityWarning: CSTR: neither eos nor molar_density was given, so the concentration basis falls back to 55500 mol/m^3 -- liquid water. Density sets the volumetric flow and hence the residence time (tau = V * rho / F_total), so a wrong density is a proportionally wrong conversion: ~8x for a C3-C8 hydrocarbon liquid, ~200x for a gas. Pass molar_density=<mol/m^3>, or eos=<cubic EOS> with reaction_phase='liquid'|'vapor' for a phase-consistent basis, or an explicit volumetric_flow= to the call.
  outlet, _ = cstr(inlet, T_spec=350.0)

/tmp/ipykernel_1384/4119124814.py:10: CSTRDensityWarning: CSTR: neither eos nor molar_density was given, so the concentration basis falls back to 55500 mol/m^3 -- liquid water. Density sets the volumetric flow and hence the residence time (tau = V * rho / F_total), so a wrong density is a proportionally wrong conversion: ~8x for a C3-C8 hydrocarbon liquid, ~200x for a gas. Pass molar_density=<mol/m^3>, or eos=<cubic EOS> with reaction_phase='liquid'|'vapor' for a phase-consistent basis, or an explicit volumetric_flow= to the call.
  outlet, _ = cstr(inlet, T_spec=350.0)

/tmp/ipykernel_1384/4119124814.py:10: CSTRDensityWarning: CSTR: neither eos nor molar_density was given, so the concentration basis falls back to 55500 mol/m^3 -- liquid water. Density sets the volumetric flow and hence the residence time (tau = V * rho / F_total), so a wrong density is a proportionally wrong conversion: ~8x for a C3-C8 hydrocarbon liquid, ~200x for a gas. Pass molar_density=<mol/m^3>, or eos=<cubic EOS> with reaction_phase='liquid'|'vapor' for a phase-consistent basis, or an explicit volumetric_flow= to the call.
  outlet, _ = cstr(inlet, T_spec=350.0)

/tmp/ipykernel_1384/4119124814.py:10: CSTRDensityWarning: CSTR: neither eos nor molar_density was given, so the concentration basis falls back to 55500 mol/m^3 -- liquid water. Density sets the volumetric flow and hence the residence time (tau = V * rho / F_total), so a wrong density is a proportionally wrong conversion: ~8x for a C3-C8 hydrocarbon liquid, ~200x for a gas. Pass molar_density=<mol/m^3>, or eos=<cubic EOS> with reaction_phase='liquid'|'vapor' for a phase-consistent basis, or an explicit volumetric_flow= to the call.
  outlet, _ = cstr(inlet, T_spec=350.0)

/tmp/ipykernel_1384/4119124814.py:10: CSTRDensityWarning: CSTR: neither eos nor molar_density was given, so the concentration basis falls back to 55500 mol/m^3 -- liquid water. Density sets the volumetric flow and hence the residence time (tau = V * rho / F_total), so a wrong density is a proportionally wrong conversion: ~8x for a C3-C8 hydrocarbon liquid, ~200x for a gas. Pass molar_density=<mol/m^3>, or eos=<cubic EOS> with reaction_phase='liquid'|'vapor' for a phase-consistent basis, or an explicit volumetric_flow= to the call.
  outlet, _ = cstr(inlet, T_spec=350.0)

============================================================
Parameter Estimation Summary
============================================================
Converged: True
Objective value: 0.000138284
Iterations: 3

Parameters:
  Name                   Value      Std Err     CI Lower     CI Upper
  ---------------------------------------------------------------
  k                  0.0314402   0.00766161    0.0133234    0.0495571

Diagnostics:
  R-squared:     0.557231
  Adj R-squared: 0.483436
  RMSE:          0.00415759
  AIC:           -85.7251
  BIC:           -85.6457
  N obs:         8
  N params:      1
============================================================

# Visualize CSTR fit
fig, axes = plt.subplots(1, 2, figsize=(12, 4))

# Outlet flow vs inlet flow
F_in_plot = np.linspace(0.3, 6.0, 50)
F_out_true = []
F_out_fit = []
for F in F_in_plot:
    for k_val, out_list in [(k_true_cstr, F_out_true), (result_cstr.theta_opt['k'], F_out_fit)]:
        cstr = CSTR(
            CSTRParams(V=1.0, rate_fn=rate_fn, stoich=stoich,
                       rate_params={'k': k_val}, species_order=['A', 'B']),
            mode='isothermal',
        )
        inlet = make_stream({'A': float(F), 'B': 0.0}, T=350.0, P=101325.0)
        outlet, _ = cstr(inlet, T_spec=350.0)
        out_list.append(float(outlet['F_A']))

axes[0].scatter([e.inputs['F_A_in'] for e in cstr_experiments],
                [e.observed['F_A_out'] for e in cstr_experiments],
                label='Data', s=60, alpha=0.8, zorder=5)
axes[0].plot(F_in_plot, F_out_true, 'g--', lw=2, label=f'True k={k_true_cstr}')
axes[0].plot(F_in_plot, F_out_fit, 'r-', lw=2,
             label=f'Fit k={result_cstr.theta_opt["k"]:.4f}')
axes[0].set_xlabel('F_A_in (mol/s)')
axes[0].set_ylabel('F_A_out (mol/s)')
axes[0].set_title('CSTR Outlet vs Inlet')
axes[0].legend()
axes[0].grid(True, alpha=0.3)

# Residuals
diag_cstr = est_cstr.diagnostics(result_cstr, cstr_experiments)
resid = [r['F_A_out'] for r in diag_cstr.residuals]
axes[1].stem([e.inputs['F_A_in'] for e in cstr_experiments], resid)
axes[1].axhline(0, color='k', linewidth=0.5)
axes[1].set_xlabel('F_A_in (mol/s)')
axes[1].set_ylabel('Residual (mol/s)')
axes[1].set_title(f'Residuals (R² = {diag_cstr.r_squared:.6f})')
axes[1].grid(True, alpha=0.3)

plt.tight_layout()
plt.show()

/tmp/ipykernel_1384/2274204586.py:16: CSTRDensityWarning: CSTR: neither eos nor molar_density was given, so the concentration basis falls back to 55500 mol/m^3 -- liquid water. Density sets the volumetric flow and hence the residence time (tau = V * rho / F_total), so a wrong density is a proportionally wrong conversion: ~8x for a C3-C8 hydrocarbon liquid, ~200x for a gas. Pass molar_density=<mol/m^3>, or eos=<cubic EOS> with reaction_phase='liquid'|'vapor' for a phase-consistent basis, or an explicit volumetric_flow= to the call.
  outlet, _ = cstr(inlet, T_spec=350.0)

/tmp/ipykernel_1384/2274204586.py:16: CSTRDensityWarning: CSTR: neither eos nor molar_density was given, so the concentration basis falls back to 55500 mol/m^3 -- liquid water. Density sets the volumetric flow and hence the residence time (tau = V * rho / F_total), so a wrong density is a proportionally wrong conversion: ~8x for a C3-C8 hydrocarbon liquid, ~200x for a gas. Pass molar_density=<mol/m^3>, or eos=<cubic EOS> with reaction_phase='liquid'|'vapor' for a phase-consistent basis, or an explicit volumetric_flow= to the call.
  outlet, _ = cstr(inlet, T_spec=350.0)

/tmp/ipykernel_1384/2274204586.py:16: CSTRDensityWarning: CSTR: neither eos nor molar_density was given, so the concentration basis falls back to 55500 mol/m^3 -- liquid water. Density sets the volumetric flow and hence the residence time (tau = V * rho / F_total), so a wrong density is a proportionally wrong conversion: ~8x for a C3-C8 hydrocarbon liquid, ~200x for a gas. Pass molar_density=<mol/m^3>, or eos=<cubic EOS> with reaction_phase='liquid'|'vapor' for a phase-consistent basis, or an explicit volumetric_flow= to the call.
  outlet, _ = cstr(inlet, T_spec=350.0)

/tmp/ipykernel_1384/2274204586.py:16: CSTRDensityWarning: CSTR: neither eos nor molar_density was given, so the concentration basis falls back to 55500 mol/m^3 -- liquid water. Density sets the volumetric flow and hence the residence time (tau = V * rho / F_total), so a wrong density is a proportionally wrong conversion: ~8x for a C3-C8 hydrocarbon liquid, ~200x for a gas. Pass molar_density=<mol/m^3>, or eos=<cubic EOS> with reaction_phase='liquid'|'vapor' for a phase-consistent basis, or an explicit volumetric_flow= to the call.
  outlet, _ = cstr(inlet, T_spec=350.0)

/tmp/ipykernel_1384/2274204586.py:16: CSTRDensityWarning: CSTR: neither eos nor molar_density was given, so the concentration basis falls back to 55500 mol/m^3 -- liquid water. Density sets the volumetric flow and hence the residence time (tau = V * rho / F_total), so a wrong density is a proportionally wrong conversion: ~8x for a C3-C8 hydrocarbon liquid, ~200x for a gas. Pass molar_density=<mol/m^3>, or eos=<cubic EOS> with reaction_phase='liquid'|'vapor' for a phase-consistent basis, or an explicit volumetric_flow= to the call.
  outlet, _ = cstr(inlet, T_spec=350.0)

/tmp/ipykernel_1384/2274204586.py:16: CSTRDensityWarning: CSTR: neither eos nor molar_density was given, so the concentration basis falls back to 55500 mol/m^3 -- liquid water. Density sets the volumetric flow and hence the residence time (tau = V * rho / F_total), so a wrong density is a proportionally wrong conversion: ~8x for a C3-C8 hydrocarbon liquid, ~200x for a gas. Pass molar_density=<mol/m^3>, or eos=<cubic EOS> with reaction_phase='liquid'|'vapor' for a phase-consistent basis, or an explicit volumetric_flow= to the call.
  outlet, _ = cstr(inlet, T_spec=350.0)

/tmp/ipykernel_1384/2274204586.py:16: CSTRDensityWarning: CSTR: neither eos nor molar_density was given, so the concentration basis falls back to 55500 mol/m^3 -- liquid water. Density sets the volumetric flow and hence the residence time (tau = V * rho / F_total), so a wrong density is a proportionally wrong conversion: ~8x for a C3-C8 hydrocarbon liquid, ~200x for a gas. Pass molar_density=<mol/m^3>, or eos=<cubic EOS> with reaction_phase='liquid'|'vapor' for a phase-consistent basis, or an explicit volumetric_flow= to the call.
  outlet, _ = cstr(inlet, T_spec=350.0)

/tmp/ipykernel_1384/2274204586.py:16: CSTRDensityWarning: CSTR: neither eos nor molar_density was given, so the concentration basis falls back to 55500 mol/m^3 -- liquid water. Density sets the volumetric flow and hence the residence time (tau = V * rho / F_total), so a wrong density is a proportionally wrong conversion: ~8x for a C3-C8 hydrocarbon liquid, ~200x for a gas. Pass molar_density=<mol/m^3>, or eos=<cubic EOS> with reaction_phase='liquid'|'vapor' for a phase-consistent basis, or an explicit volumetric_flow= to the call.
  outlet, _ = cstr(inlet, T_spec=350.0)

/tmp/ipykernel_1384/2274204586.py:16: CSTRDensityWarning: CSTR: neither eos nor molar_density was given, so the concentration basis falls back to 55500 mol/m^3 -- liquid water. Density sets the volumetric flow and hence the residence time (tau = V * rho / F_total), so a wrong density is a proportionally wrong conversion: ~8x for a C3-C8 hydrocarbon liquid, ~200x for a gas. Pass molar_density=<mol/m^3>, or eos=<cubic EOS> with reaction_phase='liquid'|'vapor' for a phase-consistent basis, or an explicit volumetric_flow= to the call.
  outlet, _ = cstr(inlet, T_spec=350.0)

/tmp/ipykernel_1384/2274204586.py:16: CSTRDensityWarning: CSTR: neither eos nor molar_density was given, so the concentration basis falls back to 55500 mol/m^3 -- liquid water. Density sets the volumetric flow and hence the residence time (tau = V * rho / F_total), so a wrong density is a proportionally wrong conversion: ~8x for a C3-C8 hydrocarbon liquid, ~200x for a gas. Pass molar_density=<mol/m^3>, or eos=<cubic EOS> with reaction_phase='liquid'|'vapor' for a phase-consistent basis, or an explicit volumetric_flow= to the call.
  outlet, _ = cstr(inlet, T_spec=350.0)

/tmp/ipykernel_1384/2274204586.py:16: CSTRDensityWarning: CSTR: neither eos nor molar_density was given, so the concentration basis falls back to 55500 mol/m^3 -- liquid water. Density sets the volumetric flow and hence the residence time (tau = V * rho / F_total), so a wrong density is a proportionally wrong conversion: ~8x for a C3-C8 hydrocarbon liquid, ~200x for a gas. Pass molar_density=<mol/m^3>, or eos=<cubic EOS> with reaction_phase='liquid'|'vapor' for a phase-consistent basis, or an explicit volumetric_flow= to the call.
  outlet, _ = cstr(inlet, T_spec=350.0)

/tmp/ipykernel_1384/2274204586.py:16: CSTRDensityWarning: CSTR: neither eos nor molar_density was given, so the concentration basis falls back to 55500 mol/m^3 -- liquid water. Density sets the volumetric flow and hence the residence time (tau = V * rho / F_total), so a wrong density is a proportionally wrong conversion: ~8x for a C3-C8 hydrocarbon liquid, ~200x for a gas. Pass molar_density=<mol/m^3>, or eos=<cubic EOS> with reaction_phase='liquid'|'vapor' for a phase-consistent basis, or an explicit volumetric_flow= to the call.
  outlet, _ = cstr(inlet, T_spec=350.0)

/tmp/ipykernel_1384/2274204586.py:16: CSTRDensityWarning: CSTR: neither eos nor molar_density was given, so the concentration basis falls back to 55500 mol/m^3 -- liquid water. Density sets the volumetric flow and hence the residence time (tau = V * rho / F_total), so a wrong density is a proportionally wrong conversion: ~8x for a C3-C8 hydrocarbon liquid, ~200x for a gas. Pass molar_density=<mol/m^3>, or eos=<cubic EOS> with reaction_phase='liquid'|'vapor' for a phase-consistent basis, or an explicit volumetric_flow= to the call.
  outlet, _ = cstr(inlet, T_spec=350.0)

/tmp/ipykernel_1384/2274204586.py:16: CSTRDensityWarning: CSTR: neither eos nor molar_density was given, so the concentration basis falls back to 55500 mol/m^3 -- liquid water. Density sets the volumetric flow and hence the residence time (tau = V * rho / F_total), so a wrong density is a proportionally wrong conversion: ~8x for a C3-C8 hydrocarbon liquid, ~200x for a gas. Pass molar_density=<mol/m^3>, or eos=<cubic EOS> with reaction_phase='liquid'|'vapor' for a phase-consistent basis, or an explicit volumetric_flow= to the call.
  outlet, _ = cstr(inlet, T_spec=350.0)

/tmp/ipykernel_1384/2274204586.py:16: CSTRDensityWarning: CSTR: neither eos nor molar_density was given, so the concentration basis falls back to 55500 mol/m^3 -- liquid water. Density sets the volumetric flow and hence the residence time (tau = V * rho / F_total), so a wrong density is a proportionally wrong conversion: ~8x for a C3-C8 hydrocarbon liquid, ~200x for a gas. Pass molar_density=<mol/m^3>, or eos=<cubic EOS> with reaction_phase='liquid'|'vapor' for a phase-consistent basis, or an explicit volumetric_flow= to the call.
  outlet, _ = cstr(inlet, T_spec=350.0)

/tmp/ipykernel_1384/2274204586.py:16: CSTRDensityWarning: CSTR: neither eos nor molar_density was given, so the concentration basis falls back to 55500 mol/m^3 -- liquid water. Density sets the volumetric flow and hence the residence time (tau = V * rho / F_total), so a wrong density is a proportionally wrong conversion: ~8x for a C3-C8 hydrocarbon liquid, ~200x for a gas. Pass molar_density=<mol/m^3>, or eos=<cubic EOS> with reaction_phase='liquid'|'vapor' for a phase-consistent basis, or an explicit volumetric_flow= to the call.
  outlet, _ = cstr(inlet, T_spec=350.0)

/tmp/ipykernel_1384/2274204586.py:16: CSTRDensityWarning: CSTR: neither eos nor molar_density was given, so the concentration basis falls back to 55500 mol/m^3 -- liquid water. Density sets the volumetric flow and hence the residence time (tau = V * rho / F_total), so a wrong density is a proportionally wrong conversion: ~8x for a C3-C8 hydrocarbon liquid, ~200x for a gas. Pass molar_density=<mol/m^3>, or eos=<cubic EOS> with reaction_phase='liquid'|'vapor' for a phase-consistent basis, or an explicit volumetric_flow= to the call.
  outlet, _ = cstr(inlet, T_spec=350.0)

/tmp/ipykernel_1384/2274204586.py:16: CSTRDensityWarning: CSTR: neither eos nor molar_density was given, so the concentration basis falls back to 55500 mol/m^3 -- liquid water. Density sets the volumetric flow and hence the residence time (tau = V * rho / F_total), so a wrong density is a proportionally wrong conversion: ~8x for a C3-C8 hydrocarbon liquid, ~200x for a gas. Pass molar_density=<mol/m^3>, or eos=<cubic EOS> with reaction_phase='liquid'|'vapor' for a phase-consistent basis, or an explicit volumetric_flow= to the call.
  outlet, _ = cstr(inlet, T_spec=350.0)

/tmp/ipykernel_1384/2274204586.py:16: CSTRDensityWarning: CSTR: neither eos nor molar_density was given, so the concentration basis falls back to 55500 mol/m^3 -- liquid water. Density sets the volumetric flow and hence the residence time (tau = V * rho / F_total), so a wrong density is a proportionally wrong conversion: ~8x for a C3-C8 hydrocarbon liquid, ~200x for a gas. Pass molar_density=<mol/m^3>, or eos=<cubic EOS> with reaction_phase='liquid'|'vapor' for a phase-consistent basis, or an explicit volumetric_flow= to the call.
  outlet, _ = cstr(inlet, T_spec=350.0)

/tmp/ipykernel_1384/2274204586.py:16: CSTRDensityWarning: CSTR: neither eos nor molar_density was given, so the concentration basis falls back to 55500 mol/m^3 -- liquid water. Density sets the volumetric flow and hence the residence time (tau = V * rho / F_total), so a wrong density is a proportionally wrong conversion: ~8x for a C3-C8 hydrocarbon liquid, ~200x for a gas. Pass molar_density=<mol/m^3>, or eos=<cubic EOS> with reaction_phase='liquid'|'vapor' for a phase-consistent basis, or an explicit volumetric_flow= to the call.
  outlet, _ = cstr(inlet, T_spec=350.0)

/tmp/ipykernel_1384/2274204586.py:16: CSTRDensityWarning: CSTR: neither eos nor molar_density was given, so the concentration basis falls back to 55500 mol/m^3 -- liquid water. Density sets the volumetric flow and hence the residence time (tau = V * rho / F_total), so a wrong density is a proportionally wrong conversion: ~8x for a C3-C8 hydrocarbon liquid, ~200x for a gas. Pass molar_density=<mol/m^3>, or eos=<cubic EOS> with reaction_phase='liquid'|'vapor' for a phase-consistent basis, or an explicit volumetric_flow= to the call.
  outlet, _ = cstr(inlet, T_spec=350.0)

/tmp/ipykernel_1384/2274204586.py:16: CSTRDensityWarning: CSTR: neither eos nor molar_density was given, so the concentration basis falls back to 55500 mol/m^3 -- liquid water. Density sets the volumetric flow and hence the residence time (tau = V * rho / F_total), so a wrong density is a proportionally wrong conversion: ~8x for a C3-C8 hydrocarbon liquid, ~200x for a gas. Pass molar_density=<mol/m^3>, or eos=<cubic EOS> with reaction_phase='liquid'|'vapor' for a phase-consistent basis, or an explicit volumetric_flow= to the call.
  outlet, _ = cstr(inlet, T_spec=350.0)

/tmp/ipykernel_1384/2274204586.py:16: CSTRDensityWarning: CSTR: neither eos nor molar_density was given, so the concentration basis falls back to 55500 mol/m^3 -- liquid water. Density sets the volumetric flow and hence the residence time (tau = V * rho / F_total), so a wrong density is a proportionally wrong conversion: ~8x for a C3-C8 hydrocarbon liquid, ~200x for a gas. Pass molar_density=<mol/m^3>, or eos=<cubic EOS> with reaction_phase='liquid'|'vapor' for a phase-consistent basis, or an explicit volumetric_flow= to the call.
  outlet, _ = cstr(inlet, T_spec=350.0)

/tmp/ipykernel_1384/2274204586.py:16: CSTRDensityWarning: CSTR: neither eos nor molar_density was given, so the concentration basis falls back to 55500 mol/m^3 -- liquid water. Density sets the volumetric flow and hence the residence time (tau = V * rho / F_total), so a wrong density is a proportionally wrong conversion: ~8x for a C3-C8 hydrocarbon liquid, ~200x for a gas. Pass molar_density=<mol/m^3>, or eos=<cubic EOS> with reaction_phase='liquid'|'vapor' for a phase-consistent basis, or an explicit volumetric_flow= to the call.
  outlet, _ = cstr(inlet, T_spec=350.0)

/tmp/ipykernel_1384/2274204586.py:16: CSTRDensityWarning: CSTR: neither eos nor molar_density was given, so the concentration basis falls back to 55500 mol/m^3 -- liquid water. Density sets the volumetric flow and hence the residence time (tau = V * rho / F_total), so a wrong density is a proportionally wrong conversion: ~8x for a C3-C8 hydrocarbon liquid, ~200x for a gas. Pass molar_density=<mol/m^3>, or eos=<cubic EOS> with reaction_phase='liquid'|'vapor' for a phase-consistent basis, or an explicit volumetric_flow= to the call.
  outlet, _ = cstr(inlet, T_spec=350.0)

/tmp/ipykernel_1384/2274204586.py:16: CSTRDensityWarning: CSTR: neither eos nor molar_density was given, so the concentration basis falls back to 55500 mol/m^3 -- liquid water. Density sets the volumetric flow and hence the residence time (tau = V * rho / F_total), so a wrong density is a proportionally wrong conversion: ~8x for a C3-C8 hydrocarbon liquid, ~200x for a gas. Pass molar_density=<mol/m^3>, or eos=<cubic EOS> with reaction_phase='liquid'|'vapor' for a phase-consistent basis, or an explicit volumetric_flow= to the call.
  outlet, _ = cstr(inlet, T_spec=350.0)

/tmp/ipykernel_1384/2274204586.py:16: CSTRDensityWarning: CSTR: neither eos nor molar_density was given, so the concentration basis falls back to 55500 mol/m^3 -- liquid water. Density sets the volumetric flow and hence the residence time (tau = V * rho / F_total), so a wrong density is a proportionally wrong conversion: ~8x for a C3-C8 hydrocarbon liquid, ~200x for a gas. Pass molar_density=<mol/m^3>, or eos=<cubic EOS> with reaction_phase='liquid'|'vapor' for a phase-consistent basis, or an explicit volumetric_flow= to the call.
  outlet, _ = cstr(inlet, T_spec=350.0)

/tmp/ipykernel_1384/2274204586.py:16: CSTRDensityWarning: CSTR: neither eos nor molar_density was given, so the concentration basis falls back to 55500 mol/m^3 -- liquid water. Density sets the volumetric flow and hence the residence time (tau = V * rho / F_total), so a wrong density is a proportionally wrong conversion: ~8x for a C3-C8 hydrocarbon liquid, ~200x for a gas. Pass molar_density=<mol/m^3>, or eos=<cubic EOS> with reaction_phase='liquid'|'vapor' for a phase-consistent basis, or an explicit volumetric_flow= to the call.
  outlet, _ = cstr(inlet, T_spec=350.0)

/tmp/ipykernel_1384/2274204586.py:16: CSTRDensityWarning: CSTR: neither eos nor molar_density was given, so the concentration basis falls back to 55500 mol/m^3 -- liquid water. Density sets the volumetric flow and hence the residence time (tau = V * rho / F_total), so a wrong density is a proportionally wrong conversion: ~8x for a C3-C8 hydrocarbon liquid, ~200x for a gas. Pass molar_density=<mol/m^3>, or eos=<cubic EOS> with reaction_phase='liquid'|'vapor' for a phase-consistent basis, or an explicit volumetric_flow= to the call.
  outlet, _ = cstr(inlet, T_spec=350.0)

/tmp/ipykernel_1384/2274204586.py:16: CSTRDensityWarning: CSTR: neither eos nor molar_density was given, so the concentration basis falls back to 55500 mol/m^3 -- liquid water. Density sets the volumetric flow and hence the residence time (tau = V * rho / F_total), so a wrong density is a proportionally wrong conversion: ~8x for a C3-C8 hydrocarbon liquid, ~200x for a gas. Pass molar_density=<mol/m^3>, or eos=<cubic EOS> with reaction_phase='liquid'|'vapor' for a phase-consistent basis, or an explicit volumetric_flow= to the call.
  outlet, _ = cstr(inlet, T_spec=350.0)

/tmp/ipykernel_1384/2274204586.py:16: CSTRDensityWarning: CSTR: neither eos nor molar_density was given, so the concentration basis falls back to 55500 mol/m^3 -- liquid water. Density sets the volumetric flow and hence the residence time (tau = V * rho / F_total), so a wrong density is a proportionally wrong conversion: ~8x for a C3-C8 hydrocarbon liquid, ~200x for a gas. Pass molar_density=<mol/m^3>, or eos=<cubic EOS> with reaction_phase='liquid'|'vapor' for a phase-consistent basis, or an explicit volumetric_flow= to the call.
  outlet, _ = cstr(inlet, T_spec=350.0)

/tmp/ipykernel_1384/2274204586.py:16: CSTRDensityWarning: CSTR: neither eos nor molar_density was given, so the concentration basis falls back to 55500 mol/m^3 -- liquid water. Density sets the volumetric flow and hence the residence time (tau = V * rho / F_total), so a wrong density is a proportionally wrong conversion: ~8x for a C3-C8 hydrocarbon liquid, ~200x for a gas. Pass molar_density=<mol/m^3>, or eos=<cubic EOS> with reaction_phase='liquid'|'vapor' for a phase-consistent basis, or an explicit volumetric_flow= to the call.
  outlet, _ = cstr(inlet, T_spec=350.0)

/tmp/ipykernel_1384/2274204586.py:16: CSTRDensityWarning: CSTR: neither eos nor molar_density was given, so the concentration basis falls back to 55500 mol/m^3 -- liquid water. Density sets the volumetric flow and hence the residence time (tau = V * rho / F_total), so a wrong density is a proportionally wrong conversion: ~8x for a C3-C8 hydrocarbon liquid, ~200x for a gas. Pass molar_density=<mol/m^3>, or eos=<cubic EOS> with reaction_phase='liquid'|'vapor' for a phase-consistent basis, or an explicit volumetric_flow= to the call.
  outlet, _ = cstr(inlet, T_spec=350.0)

/tmp/ipykernel_1384/2274204586.py:16: CSTRDensityWarning: CSTR: neither eos nor molar_density was given, so the concentration basis falls back to 55500 mol/m^3 -- liquid water. Density sets the volumetric flow and hence the residence time (tau = V * rho / F_total), so a wrong density is a proportionally wrong conversion: ~8x for a C3-C8 hydrocarbon liquid, ~200x for a gas. Pass molar_density=<mol/m^3>, or eos=<cubic EOS> with reaction_phase='liquid'|'vapor' for a phase-consistent basis, or an explicit volumetric_flow= to the call.
  outlet, _ = cstr(inlet, T_spec=350.0)

/tmp/ipykernel_1384/2274204586.py:16: CSTRDensityWarning: CSTR: neither eos nor molar_density was given, so the concentration basis falls back to 55500 mol/m^3 -- liquid water. Density sets the volumetric flow and hence the residence time (tau = V * rho / F_total), so a wrong density is a proportionally wrong conversion: ~8x for a C3-C8 hydrocarbon liquid, ~200x for a gas. Pass molar_density=<mol/m^3>, or eos=<cubic EOS> with reaction_phase='liquid'|'vapor' for a phase-consistent basis, or an explicit volumetric_flow= to the call.
  outlet, _ = cstr(inlet, T_spec=350.0)

/tmp/ipykernel_1384/2274204586.py:16: CSTRDensityWarning: CSTR: neither eos nor molar_density was given, so the concentration basis falls back to 55500 mol/m^3 -- liquid water. Density sets the volumetric flow and hence the residence time (tau = V * rho / F_total), so a wrong density is a proportionally wrong conversion: ~8x for a C3-C8 hydrocarbon liquid, ~200x for a gas. Pass molar_density=<mol/m^3>, or eos=<cubic EOS> with reaction_phase='liquid'|'vapor' for a phase-consistent basis, or an explicit volumetric_flow= to the call.
  outlet, _ = cstr(inlet, T_spec=350.0)

/tmp/ipykernel_1384/2274204586.py:16: CSTRDensityWarning: CSTR: neither eos nor molar_density was given, so the concentration basis falls back to 55500 mol/m^3 -- liquid water. Density sets the volumetric flow and hence the residence time (tau = V * rho / F_total), so a wrong density is a proportionally wrong conversion: ~8x for a C3-C8 hydrocarbon liquid, ~200x for a gas. Pass molar_density=<mol/m^3>, or eos=<cubic EOS> with reaction_phase='liquid'|'vapor' for a phase-consistent basis, or an explicit volumetric_flow= to the call.
  outlet, _ = cstr(inlet, T_spec=350.0)

/tmp/ipykernel_1384/2274204586.py:16: CSTRDensityWarning: CSTR: neither eos nor molar_density was given, so the concentration basis falls back to 55500 mol/m^3 -- liquid water. Density sets the volumetric flow and hence the residence time (tau = V * rho / F_total), so a wrong density is a proportionally wrong conversion: ~8x for a C3-C8 hydrocarbon liquid, ~200x for a gas. Pass molar_density=<mol/m^3>, or eos=<cubic EOS> with reaction_phase='liquid'|'vapor' for a phase-consistent basis, or an explicit volumetric_flow= to the call.
  outlet, _ = cstr(inlet, T_spec=350.0)

/tmp/ipykernel_1384/2274204586.py:16: CSTRDensityWarning: CSTR: neither eos nor molar_density was given, so the concentration basis falls back to 55500 mol/m^3 -- liquid water. Density sets the volumetric flow and hence the residence time (tau = V * rho / F_total), so a wrong density is a proportionally wrong conversion: ~8x for a C3-C8 hydrocarbon liquid, ~200x for a gas. Pass molar_density=<mol/m^3>, or eos=<cubic EOS> with reaction_phase='liquid'|'vapor' for a phase-consistent basis, or an explicit volumetric_flow= to the call.
  outlet, _ = cstr(inlet, T_spec=350.0)

/tmp/ipykernel_1384/2274204586.py:16: CSTRDensityWarning: CSTR: neither eos nor molar_density was given, so the concentration basis falls back to 55500 mol/m^3 -- liquid water. Density sets the volumetric flow and hence the residence time (tau = V * rho / F_total), so a wrong density is a proportionally wrong conversion: ~8x for a C3-C8 hydrocarbon liquid, ~200x for a gas. Pass molar_density=<mol/m^3>, or eos=<cubic EOS> with reaction_phase='liquid'|'vapor' for a phase-consistent basis, or an explicit volumetric_flow= to the call.
  outlet, _ = cstr(inlet, T_spec=350.0)

/tmp/ipykernel_1384/2274204586.py:16: CSTRDensityWarning: CSTR: neither eos nor molar_density was given, so the concentration basis falls back to 55500 mol/m^3 -- liquid water. Density sets the volumetric flow and hence the residence time (tau = V * rho / F_total), so a wrong density is a proportionally wrong conversion: ~8x for a C3-C8 hydrocarbon liquid, ~200x for a gas. Pass molar_density=<mol/m^3>, or eos=<cubic EOS> with reaction_phase='liquid'|'vapor' for a phase-consistent basis, or an explicit volumetric_flow= to the call.
  outlet, _ = cstr(inlet, T_spec=350.0)

/tmp/ipykernel_1384/2274204586.py:16: CSTRDensityWarning: CSTR: neither eos nor molar_density was given, so the concentration basis falls back to 55500 mol/m^3 -- liquid water. Density sets the volumetric flow and hence the residence time (tau = V * rho / F_total), so a wrong density is a proportionally wrong conversion: ~8x for a C3-C8 hydrocarbon liquid, ~200x for a gas. Pass molar_density=<mol/m^3>, or eos=<cubic EOS> with reaction_phase='liquid'|'vapor' for a phase-consistent basis, or an explicit volumetric_flow= to the call.
  outlet, _ = cstr(inlet, T_spec=350.0)

/tmp/ipykernel_1384/2274204586.py:16: CSTRDensityWarning: CSTR: neither eos nor molar_density was given, so the concentration basis falls back to 55500 mol/m^3 -- liquid water. Density sets the volumetric flow and hence the residence time (tau = V * rho / F_total), so a wrong density is a proportionally wrong conversion: ~8x for a C3-C8 hydrocarbon liquid, ~200x for a gas. Pass molar_density=<mol/m^3>, or eos=<cubic EOS> with reaction_phase='liquid'|'vapor' for a phase-consistent basis, or an explicit volumetric_flow= to the call.
  outlet, _ = cstr(inlet, T_spec=350.0)

/tmp/ipykernel_1384/2274204586.py:16: CSTRDensityWarning: CSTR: neither eos nor molar_density was given, so the concentration basis falls back to 55500 mol/m^3 -- liquid water. Density sets the volumetric flow and hence the residence time (tau = V * rho / F_total), so a wrong density is a proportionally wrong conversion: ~8x for a C3-C8 hydrocarbon liquid, ~200x for a gas. Pass molar_density=<mol/m^3>, or eos=<cubic EOS> with reaction_phase='liquid'|'vapor' for a phase-consistent basis, or an explicit volumetric_flow= to the call.
  outlet, _ = cstr(inlet, T_spec=350.0)

/tmp/ipykernel_1384/2274204586.py:16: CSTRDensityWarning: CSTR: neither eos nor molar_density was given, so the concentration basis falls back to 55500 mol/m^3 -- liquid water. Density sets the volumetric flow and hence the residence time (tau = V * rho / F_total), so a wrong density is a proportionally wrong conversion: ~8x for a C3-C8 hydrocarbon liquid, ~200x for a gas. Pass molar_density=<mol/m^3>, or eos=<cubic EOS> with reaction_phase='liquid'|'vapor' for a phase-consistent basis, or an explicit volumetric_flow= to the call.
  outlet, _ = cstr(inlet, T_spec=350.0)

/tmp/ipykernel_1384/2274204586.py:16: CSTRDensityWarning: CSTR: neither eos nor molar_density was given, so the concentration basis falls back to 55500 mol/m^3 -- liquid water. Density sets the volumetric flow and hence the residence time (tau = V * rho / F_total), so a wrong density is a proportionally wrong conversion: ~8x for a C3-C8 hydrocarbon liquid, ~200x for a gas. Pass molar_density=<mol/m^3>, or eos=<cubic EOS> with reaction_phase='liquid'|'vapor' for a phase-consistent basis, or an explicit volumetric_flow= to the call.
  outlet, _ = cstr(inlet, T_spec=350.0)

/tmp/ipykernel_1384/2274204586.py:16: CSTRDensityWarning: CSTR: neither eos nor molar_density was given, so the concentration basis falls back to 55500 mol/m^3 -- liquid water. Density sets the volumetric flow and hence the residence time (tau = V * rho / F_total), so a wrong density is a proportionally wrong conversion: ~8x for a C3-C8 hydrocarbon liquid, ~200x for a gas. Pass molar_density=<mol/m^3>, or eos=<cubic EOS> with reaction_phase='liquid'|'vapor' for a phase-consistent basis, or an explicit volumetric_flow= to the call.
  outlet, _ = cstr(inlet, T_spec=350.0)

/tmp/ipykernel_1384/2274204586.py:16: CSTRDensityWarning: CSTR: neither eos nor molar_density was given, so the concentration basis falls back to 55500 mol/m^3 -- liquid water. Density sets the volumetric flow and hence the residence time (tau = V * rho / F_total), so a wrong density is a proportionally wrong conversion: ~8x for a C3-C8 hydrocarbon liquid, ~200x for a gas. Pass molar_density=<mol/m^3>, or eos=<cubic EOS> with reaction_phase='liquid'|'vapor' for a phase-consistent basis, or an explicit volumetric_flow= to the call.
  outlet, _ = cstr(inlet, T_spec=350.0)

/tmp/ipykernel_1384/2274204586.py:16: CSTRDensityWarning: CSTR: neither eos nor molar_density was given, so the concentration basis falls back to 55500 mol/m^3 -- liquid water. Density sets the volumetric flow and hence the residence time (tau = V * rho / F_total), so a wrong density is a proportionally wrong conversion: ~8x for a C3-C8 hydrocarbon liquid, ~200x for a gas. Pass molar_density=<mol/m^3>, or eos=<cubic EOS> with reaction_phase='liquid'|'vapor' for a phase-consistent basis, or an explicit volumetric_flow= to the call.
  outlet, _ = cstr(inlet, T_spec=350.0)

/tmp/ipykernel_1384/2274204586.py:16: CSTRDensityWarning: CSTR: neither eos nor molar_density was given, so the concentration basis falls back to 55500 mol/m^3 -- liquid water. Density sets the volumetric flow and hence the residence time (tau = V * rho / F_total), so a wrong density is a proportionally wrong conversion: ~8x for a C3-C8 hydrocarbon liquid, ~200x for a gas. Pass molar_density=<mol/m^3>, or eos=<cubic EOS> with reaction_phase='liquid'|'vapor' for a phase-consistent basis, or an explicit volumetric_flow= to the call.
  outlet, _ = cstr(inlet, T_spec=350.0)

/tmp/ipykernel_1384/2274204586.py:16: CSTRDensityWarning: CSTR: neither eos nor molar_density was given, so the concentration basis falls back to 55500 mol/m^3 -- liquid water. Density sets the volumetric flow and hence the residence time (tau = V * rho / F_total), so a wrong density is a proportionally wrong conversion: ~8x for a C3-C8 hydrocarbon liquid, ~200x for a gas. Pass molar_density=<mol/m^3>, or eos=<cubic EOS> with reaction_phase='liquid'|'vapor' for a phase-consistent basis, or an explicit volumetric_flow= to the call.
  outlet, _ = cstr(inlet, T_spec=350.0)

/tmp/ipykernel_1384/2274204586.py:16: CSTRDensityWarning: CSTR: neither eos nor molar_density was given, so the concentration basis falls back to 55500 mol/m^3 -- liquid water. Density sets the volumetric flow and hence the residence time (tau = V * rho / F_total), so a wrong density is a proportionally wrong conversion: ~8x for a C3-C8 hydrocarbon liquid, ~200x for a gas. Pass molar_density=<mol/m^3>, or eos=<cubic EOS> with reaction_phase='liquid'|'vapor' for a phase-consistent basis, or an explicit volumetric_flow= to the call.
  outlet, _ = cstr(inlet, T_spec=350.0)

/tmp/ipykernel_1384/2274204586.py:16: CSTRDensityWarning: CSTR: neither eos nor molar_density was given, so the concentration basis falls back to 55500 mol/m^3 -- liquid water. Density sets the volumetric flow and hence the residence time (tau = V * rho / F_total), so a wrong density is a proportionally wrong conversion: ~8x for a C3-C8 hydrocarbon liquid, ~200x for a gas. Pass molar_density=<mol/m^3>, or eos=<cubic EOS> with reaction_phase='liquid'|'vapor' for a phase-consistent basis, or an explicit volumetric_flow= to the call.
  outlet, _ = cstr(inlet, T_spec=350.0)

/tmp/ipykernel_1384/2274204586.py:16: CSTRDensityWarning: CSTR: neither eos nor molar_density was given, so the concentration basis falls back to 55500 mol/m^3 -- liquid water. Density sets the volumetric flow and hence the residence time (tau = V * rho / F_total), so a wrong density is a proportionally wrong conversion: ~8x for a C3-C8 hydrocarbon liquid, ~200x for a gas. Pass molar_density=<mol/m^3>, or eos=<cubic EOS> with reaction_phase='liquid'|'vapor' for a phase-consistent basis, or an explicit volumetric_flow= to the call.
  outlet, _ = cstr(inlet, T_spec=350.0)

/tmp/ipykernel_1384/2274204586.py:16: CSTRDensityWarning: CSTR: neither eos nor molar_density was given, so the concentration basis falls back to 55500 mol/m^3 -- liquid water. Density sets the volumetric flow and hence the residence time (tau = V * rho / F_total), so a wrong density is a proportionally wrong conversion: ~8x for a C3-C8 hydrocarbon liquid, ~200x for a gas. Pass molar_density=<mol/m^3>, or eos=<cubic EOS> with reaction_phase='liquid'|'vapor' for a phase-consistent basis, or an explicit volumetric_flow= to the call.
  outlet, _ = cstr(inlet, T_spec=350.0)

/tmp/ipykernel_1384/2274204586.py:16: CSTRDensityWarning: CSTR: neither eos nor molar_density was given, so the concentration basis falls back to 55500 mol/m^3 -- liquid water. Density sets the volumetric flow and hence the residence time (tau = V * rho / F_total), so a wrong density is a proportionally wrong conversion: ~8x for a C3-C8 hydrocarbon liquid, ~200x for a gas. Pass molar_density=<mol/m^3>, or eos=<cubic EOS> with reaction_phase='liquid'|'vapor' for a phase-consistent basis, or an explicit volumetric_flow= to the call.
  outlet, _ = cstr(inlet, T_spec=350.0)

/tmp/ipykernel_1384/2274204586.py:16: CSTRDensityWarning: CSTR: neither eos nor molar_density was given, so the concentration basis falls back to 55500 mol/m^3 -- liquid water. Density sets the volumetric flow and hence the residence time (tau = V * rho / F_total), so a wrong density is a proportionally wrong conversion: ~8x for a C3-C8 hydrocarbon liquid, ~200x for a gas. Pass molar_density=<mol/m^3>, or eos=<cubic EOS> with reaction_phase='liquid'|'vapor' for a phase-consistent basis, or an explicit volumetric_flow= to the call.
  outlet, _ = cstr(inlet, T_spec=350.0)

/tmp/ipykernel_1384/2274204586.py:16: CSTRDensityWarning: CSTR: neither eos nor molar_density was given, so the concentration basis falls back to 55500 mol/m^3 -- liquid water. Density sets the volumetric flow and hence the residence time (tau = V * rho / F_total), so a wrong density is a proportionally wrong conversion: ~8x for a C3-C8 hydrocarbon liquid, ~200x for a gas. Pass molar_density=<mol/m^3>, or eos=<cubic EOS> with reaction_phase='liquid'|'vapor' for a phase-consistent basis, or an explicit volumetric_flow= to the call.
  outlet, _ = cstr(inlet, T_spec=350.0)

/tmp/ipykernel_1384/2274204586.py:16: CSTRDensityWarning: CSTR: neither eos nor molar_density was given, so the concentration basis falls back to 55500 mol/m^3 -- liquid water. Density sets the volumetric flow and hence the residence time (tau = V * rho / F_total), so a wrong density is a proportionally wrong conversion: ~8x for a C3-C8 hydrocarbon liquid, ~200x for a gas. Pass molar_density=<mol/m^3>, or eos=<cubic EOS> with reaction_phase='liquid'|'vapor' for a phase-consistent basis, or an explicit volumetric_flow= to the call.
  outlet, _ = cstr(inlet, T_spec=350.0)

/tmp/ipykernel_1384/2274204586.py:16: CSTRDensityWarning: CSTR: neither eos nor molar_density was given, so the concentration basis falls back to 55500 mol/m^3 -- liquid water. Density sets the volumetric flow and hence the residence time (tau = V * rho / F_total), so a wrong density is a proportionally wrong conversion: ~8x for a C3-C8 hydrocarbon liquid, ~200x for a gas. Pass molar_density=<mol/m^3>, or eos=<cubic EOS> with reaction_phase='liquid'|'vapor' for a phase-consistent basis, or an explicit volumetric_flow= to the call.
  outlet, _ = cstr(inlet, T_spec=350.0)

/tmp/ipykernel_1384/2274204586.py:16: CSTRDensityWarning: CSTR: neither eos nor molar_density was given, so the concentration basis falls back to 55500 mol/m^3 -- liquid water. Density sets the volumetric flow and hence the residence time (tau = V * rho / F_total), so a wrong density is a proportionally wrong conversion: ~8x for a C3-C8 hydrocarbon liquid, ~200x for a gas. Pass molar_density=<mol/m^3>, or eos=<cubic EOS> with reaction_phase='liquid'|'vapor' for a phase-consistent basis, or an explicit volumetric_flow= to the call.
  outlet, _ = cstr(inlet, T_spec=350.0)

/tmp/ipykernel_1384/2274204586.py:16: CSTRDensityWarning: CSTR: neither eos nor molar_density was given, so the concentration basis falls back to 55500 mol/m^3 -- liquid water. Density sets the volumetric flow and hence the residence time (tau = V * rho / F_total), so a wrong density is a proportionally wrong conversion: ~8x for a C3-C8 hydrocarbon liquid, ~200x for a gas. Pass molar_density=<mol/m^3>, or eos=<cubic EOS> with reaction_phase='liquid'|'vapor' for a phase-consistent basis, or an explicit volumetric_flow= to the call.
  outlet, _ = cstr(inlet, T_spec=350.0)

/tmp/ipykernel_1384/2274204586.py:16: CSTRDensityWarning: CSTR: neither eos nor molar_density was given, so the concentration basis falls back to 55500 mol/m^3 -- liquid water. Density sets the volumetric flow and hence the residence time (tau = V * rho / F_total), so a wrong density is a proportionally wrong conversion: ~8x for a C3-C8 hydrocarbon liquid, ~200x for a gas. Pass molar_density=<mol/m^3>, or eos=<cubic EOS> with reaction_phase='liquid'|'vapor' for a phase-consistent basis, or an explicit volumetric_flow= to the call.
  outlet, _ = cstr(inlet, T_spec=350.0)

/tmp/ipykernel_1384/2274204586.py:16: CSTRDensityWarning: CSTR: neither eos nor molar_density was given, so the concentration basis falls back to 55500 mol/m^3 -- liquid water. Density sets the volumetric flow and hence the residence time (tau = V * rho / F_total), so a wrong density is a proportionally wrong conversion: ~8x for a C3-C8 hydrocarbon liquid, ~200x for a gas. Pass molar_density=<mol/m^3>, or eos=<cubic EOS> with reaction_phase='liquid'|'vapor' for a phase-consistent basis, or an explicit volumetric_flow= to the call.
  outlet, _ = cstr(inlet, T_spec=350.0)

/tmp/ipykernel_1384/2274204586.py:16: CSTRDensityWarning: CSTR: neither eos nor molar_density was given, so the concentration basis falls back to 55500 mol/m^3 -- liquid water. Density sets the volumetric flow and hence the residence time (tau = V * rho / F_total), so a wrong density is a proportionally wrong conversion: ~8x for a C3-C8 hydrocarbon liquid, ~200x for a gas. Pass molar_density=<mol/m^3>, or eos=<cubic EOS> with reaction_phase='liquid'|'vapor' for a phase-consistent basis, or an explicit volumetric_flow= to the call.
  outlet, _ = cstr(inlet, T_spec=350.0)

/tmp/ipykernel_1384/2274204586.py:16: CSTRDensityWarning: CSTR: neither eos nor molar_density was given, so the concentration basis falls back to 55500 mol/m^3 -- liquid water. Density sets the volumetric flow and hence the residence time (tau = V * rho / F_total), so a wrong density is a proportionally wrong conversion: ~8x for a C3-C8 hydrocarbon liquid, ~200x for a gas. Pass molar_density=<mol/m^3>, or eos=<cubic EOS> with reaction_phase='liquid'|'vapor' for a phase-consistent basis, or an explicit volumetric_flow= to the call.
  outlet, _ = cstr(inlet, T_spec=350.0)

/tmp/ipykernel_1384/2274204586.py:16: CSTRDensityWarning: CSTR: neither eos nor molar_density was given, so the concentration basis falls back to 55500 mol/m^3 -- liquid water. Density sets the volumetric flow and hence the residence time (tau = V * rho / F_total), so a wrong density is a proportionally wrong conversion: ~8x for a C3-C8 hydrocarbon liquid, ~200x for a gas. Pass molar_density=<mol/m^3>, or eos=<cubic EOS> with reaction_phase='liquid'|'vapor' for a phase-consistent basis, or an explicit volumetric_flow= to the call.
  outlet, _ = cstr(inlet, T_spec=350.0)

/tmp/ipykernel_1384/2274204586.py:16: CSTRDensityWarning: CSTR: neither eos nor molar_density was given, so the concentration basis falls back to 55500 mol/m^3 -- liquid water. Density sets the volumetric flow and hence the residence time (tau = V * rho / F_total), so a wrong density is a proportionally wrong conversion: ~8x for a C3-C8 hydrocarbon liquid, ~200x for a gas. Pass molar_density=<mol/m^3>, or eos=<cubic EOS> with reaction_phase='liquid'|'vapor' for a phase-consistent basis, or an explicit volumetric_flow= to the call.
  outlet, _ = cstr(inlet, T_spec=350.0)

/tmp/ipykernel_1384/2274204586.py:16: CSTRDensityWarning: CSTR: neither eos nor molar_density was given, so the concentration basis falls back to 55500 mol/m^3 -- liquid water. Density sets the volumetric flow and hence the residence time (tau = V * rho / F_total), so a wrong density is a proportionally wrong conversion: ~8x for a C3-C8 hydrocarbon liquid, ~200x for a gas. Pass molar_density=<mol/m^3>, or eos=<cubic EOS> with reaction_phase='liquid'|'vapor' for a phase-consistent basis, or an explicit volumetric_flow= to the call.
  outlet, _ = cstr(inlet, T_spec=350.0)

/tmp/ipykernel_1384/2274204586.py:16: CSTRDensityWarning: CSTR: neither eos nor molar_density was given, so the concentration basis falls back to 55500 mol/m^3 -- liquid water. Density sets the volumetric flow and hence the residence time (tau = V * rho / F_total), so a wrong density is a proportionally wrong conversion: ~8x for a C3-C8 hydrocarbon liquid, ~200x for a gas. Pass molar_density=<mol/m^3>, or eos=<cubic EOS> with reaction_phase='liquid'|'vapor' for a phase-consistent basis, or an explicit volumetric_flow= to the call.
  outlet, _ = cstr(inlet, T_spec=350.0)

/tmp/ipykernel_1384/2274204586.py:16: CSTRDensityWarning: CSTR: neither eos nor molar_density was given, so the concentration basis falls back to 55500 mol/m^3 -- liquid water. Density sets the volumetric flow and hence the residence time (tau = V * rho / F_total), so a wrong density is a proportionally wrong conversion: ~8x for a C3-C8 hydrocarbon liquid, ~200x for a gas. Pass molar_density=<mol/m^3>, or eos=<cubic EOS> with reaction_phase='liquid'|'vapor' for a phase-consistent basis, or an explicit volumetric_flow= to the call.
  outlet, _ = cstr(inlet, T_spec=350.0)

/tmp/ipykernel_1384/2274204586.py:16: CSTRDensityWarning: CSTR: neither eos nor molar_density was given, so the concentration basis falls back to 55500 mol/m^3 -- liquid water. Density sets the volumetric flow and hence the residence time (tau = V * rho / F_total), so a wrong density is a proportionally wrong conversion: ~8x for a C3-C8 hydrocarbon liquid, ~200x for a gas. Pass molar_density=<mol/m^3>, or eos=<cubic EOS> with reaction_phase='liquid'|'vapor' for a phase-consistent basis, or an explicit volumetric_flow= to the call.
  outlet, _ = cstr(inlet, T_spec=350.0)

/tmp/ipykernel_1384/2274204586.py:16: CSTRDensityWarning: CSTR: neither eos nor molar_density was given, so the concentration basis falls back to 55500 mol/m^3 -- liquid water. Density sets the volumetric flow and hence the residence time (tau = V * rho / F_total), so a wrong density is a proportionally wrong conversion: ~8x for a C3-C8 hydrocarbon liquid, ~200x for a gas. Pass molar_density=<mol/m^3>, or eos=<cubic EOS> with reaction_phase='liquid'|'vapor' for a phase-consistent basis, or an explicit volumetric_flow= to the call.
  outlet, _ = cstr(inlet, T_spec=350.0)

/tmp/ipykernel_1384/2274204586.py:16: CSTRDensityWarning: CSTR: neither eos nor molar_density was given, so the concentration basis falls back to 55500 mol/m^3 -- liquid water. Density sets the volumetric flow and hence the residence time (tau = V * rho / F_total), so a wrong density is a proportionally wrong conversion: ~8x for a C3-C8 hydrocarbon liquid, ~200x for a gas. Pass molar_density=<mol/m^3>, or eos=<cubic EOS> with reaction_phase='liquid'|'vapor' for a phase-consistent basis, or an explicit volumetric_flow= to the call.
  outlet, _ = cstr(inlet, T_spec=350.0)

/tmp/ipykernel_1384/2274204586.py:16: CSTRDensityWarning: CSTR: neither eos nor molar_density was given, so the concentration basis falls back to 55500 mol/m^3 -- liquid water. Density sets the volumetric flow and hence the residence time (tau = V * rho / F_total), so a wrong density is a proportionally wrong conversion: ~8x for a C3-C8 hydrocarbon liquid, ~200x for a gas. Pass molar_density=<mol/m^3>, or eos=<cubic EOS> with reaction_phase='liquid'|'vapor' for a phase-consistent basis, or an explicit volumetric_flow= to the call.
  outlet, _ = cstr(inlet, T_spec=350.0)

/tmp/ipykernel_1384/2274204586.py:16: CSTRDensityWarning: CSTR: neither eos nor molar_density was given, so the concentration basis falls back to 55500 mol/m^3 -- liquid water. Density sets the volumetric flow and hence the residence time (tau = V * rho / F_total), so a wrong density is a proportionally wrong conversion: ~8x for a C3-C8 hydrocarbon liquid, ~200x for a gas. Pass molar_density=<mol/m^3>, or eos=<cubic EOS> with reaction_phase='liquid'|'vapor' for a phase-consistent basis, or an explicit volumetric_flow= to the call.
  outlet, _ = cstr(inlet, T_spec=350.0)

/tmp/ipykernel_1384/2274204586.py:16: CSTRDensityWarning: CSTR: neither eos nor molar_density was given, so the concentration basis falls back to 55500 mol/m^3 -- liquid water. Density sets the volumetric flow and hence the residence time (tau = V * rho / F_total), so a wrong density is a proportionally wrong conversion: ~8x for a C3-C8 hydrocarbon liquid, ~200x for a gas. Pass molar_density=<mol/m^3>, or eos=<cubic EOS> with reaction_phase='liquid'|'vapor' for a phase-consistent basis, or an explicit volumetric_flow= to the call.
  outlet, _ = cstr(inlet, T_spec=350.0)

/tmp/ipykernel_1384/2274204586.py:16: CSTRDensityWarning: CSTR: neither eos nor molar_density was given, so the concentration basis falls back to 55500 mol/m^3 -- liquid water. Density sets the volumetric flow and hence the residence time (tau = V * rho / F_total), so a wrong density is a proportionally wrong conversion: ~8x for a C3-C8 hydrocarbon liquid, ~200x for a gas. Pass molar_density=<mol/m^3>, or eos=<cubic EOS> with reaction_phase='liquid'|'vapor' for a phase-consistent basis, or an explicit volumetric_flow= to the call.
  outlet, _ = cstr(inlet, T_spec=350.0)

/tmp/ipykernel_1384/2274204586.py:16: CSTRDensityWarning: CSTR: neither eos nor molar_density was given, so the concentration basis falls back to 55500 mol/m^3 -- liquid water. Density sets the volumetric flow and hence the residence time (tau = V * rho / F_total), so a wrong density is a proportionally wrong conversion: ~8x for a C3-C8 hydrocarbon liquid, ~200x for a gas. Pass molar_density=<mol/m^3>, or eos=<cubic EOS> with reaction_phase='liquid'|'vapor' for a phase-consistent basis, or an explicit volumetric_flow= to the call.
  outlet, _ = cstr(inlet, T_spec=350.0)

/tmp/ipykernel_1384/2274204586.py:16: CSTRDensityWarning: CSTR: neither eos nor molar_density was given, so the concentration basis falls back to 55500 mol/m^3 -- liquid water. Density sets the volumetric flow and hence the residence time (tau = V * rho / F_total), so a wrong density is a proportionally wrong conversion: ~8x for a C3-C8 hydrocarbon liquid, ~200x for a gas. Pass molar_density=<mol/m^3>, or eos=<cubic EOS> with reaction_phase='liquid'|'vapor' for a phase-consistent basis, or an explicit volumetric_flow= to the call.
  outlet, _ = cstr(inlet, T_spec=350.0)

/tmp/ipykernel_1384/2274204586.py:16: CSTRDensityWarning: CSTR: neither eos nor molar_density was given, so the concentration basis falls back to 55500 mol/m^3 -- liquid water. Density sets the volumetric flow and hence the residence time (tau = V * rho / F_total), so a wrong density is a proportionally wrong conversion: ~8x for a C3-C8 hydrocarbon liquid, ~200x for a gas. Pass molar_density=<mol/m^3>, or eos=<cubic EOS> with reaction_phase='liquid'|'vapor' for a phase-consistent basis, or an explicit volumetric_flow= to the call.
  outlet, _ = cstr(inlet, T_spec=350.0)

/tmp/ipykernel_1384/2274204586.py:16: CSTRDensityWarning: CSTR: neither eos nor molar_density was given, so the concentration basis falls back to 55500 mol/m^3 -- liquid water. Density sets the volumetric flow and hence the residence time (tau = V * rho / F_total), so a wrong density is a proportionally wrong conversion: ~8x for a C3-C8 hydrocarbon liquid, ~200x for a gas. Pass molar_density=<mol/m^3>, or eos=<cubic EOS> with reaction_phase='liquid'|'vapor' for a phase-consistent basis, or an explicit volumetric_flow= to the call.
  outlet, _ = cstr(inlet, T_spec=350.0)

/tmp/ipykernel_1384/2274204586.py:16: CSTRDensityWarning: CSTR: neither eos nor molar_density was given, so the concentration basis falls back to 55500 mol/m^3 -- liquid water. Density sets the volumetric flow and hence the residence time (tau = V * rho / F_total), so a wrong density is a proportionally wrong conversion: ~8x for a C3-C8 hydrocarbon liquid, ~200x for a gas. Pass molar_density=<mol/m^3>, or eos=<cubic EOS> with reaction_phase='liquid'|'vapor' for a phase-consistent basis, or an explicit volumetric_flow= to the call.
  outlet, _ = cstr(inlet, T_spec=350.0)

/tmp/ipykernel_1384/2274204586.py:16: CSTRDensityWarning: CSTR: neither eos nor molar_density was given, so the concentration basis falls back to 55500 mol/m^3 -- liquid water. Density sets the volumetric flow and hence the residence time (tau = V * rho / F_total), so a wrong density is a proportionally wrong conversion: ~8x for a C3-C8 hydrocarbon liquid, ~200x for a gas. Pass molar_density=<mol/m^3>, or eos=<cubic EOS> with reaction_phase='liquid'|'vapor' for a phase-consistent basis, or an explicit volumetric_flow= to the call.
  outlet, _ = cstr(inlet, T_spec=350.0)

/tmp/ipykernel_1384/2274204586.py:16: CSTRDensityWarning: CSTR: neither eos nor molar_density was given, so the concentration basis falls back to 55500 mol/m^3 -- liquid water. Density sets the volumetric flow and hence the residence time (tau = V * rho / F_total), so a wrong density is a proportionally wrong conversion: ~8x for a C3-C8 hydrocarbon liquid, ~200x for a gas. Pass molar_density=<mol/m^3>, or eos=<cubic EOS> with reaction_phase='liquid'|'vapor' for a phase-consistent basis, or an explicit volumetric_flow= to the call.
  outlet, _ = cstr(inlet, T_spec=350.0)

/tmp/ipykernel_1384/2274204586.py:16: CSTRDensityWarning: CSTR: neither eos nor molar_density was given, so the concentration basis falls back to 55500 mol/m^3 -- liquid water. Density sets the volumetric flow and hence the residence time (tau = V * rho / F_total), so a wrong density is a proportionally wrong conversion: ~8x for a C3-C8 hydrocarbon liquid, ~200x for a gas. Pass molar_density=<mol/m^3>, or eos=<cubic EOS> with reaction_phase='liquid'|'vapor' for a phase-consistent basis, or an explicit volumetric_flow= to the call.
  outlet, _ = cstr(inlet, T_spec=350.0)

/tmp/ipykernel_1384/2274204586.py:16: CSTRDensityWarning: CSTR: neither eos nor molar_density was given, so the concentration basis falls back to 55500 mol/m^3 -- liquid water. Density sets the volumetric flow and hence the residence time (tau = V * rho / F_total), so a wrong density is a proportionally wrong conversion: ~8x for a C3-C8 hydrocarbon liquid, ~200x for a gas. Pass molar_density=<mol/m^3>, or eos=<cubic EOS> with reaction_phase='liquid'|'vapor' for a phase-consistent basis, or an explicit volumetric_flow= to the call.
  outlet, _ = cstr(inlet, T_spec=350.0)

/tmp/ipykernel_1384/2274204586.py:16: CSTRDensityWarning: CSTR: neither eos nor molar_density was given, so the concentration basis falls back to 55500 mol/m^3 -- liquid water. Density sets the volumetric flow and hence the residence time (tau = V * rho / F_total), so a wrong density is a proportionally wrong conversion: ~8x for a C3-C8 hydrocarbon liquid, ~200x for a gas. Pass molar_density=<mol/m^3>, or eos=<cubic EOS> with reaction_phase='liquid'|'vapor' for a phase-consistent basis, or an explicit volumetric_flow= to the call.
  outlet, _ = cstr(inlet, T_spec=350.0)

/tmp/ipykernel_1384/2274204586.py:16: CSTRDensityWarning: CSTR: neither eos nor molar_density was given, so the concentration basis falls back to 55500 mol/m^3 -- liquid water. Density sets the volumetric flow and hence the residence time (tau = V * rho / F_total), so a wrong density is a proportionally wrong conversion: ~8x for a C3-C8 hydrocarbon liquid, ~200x for a gas. Pass molar_density=<mol/m^3>, or eos=<cubic EOS> with reaction_phase='liquid'|'vapor' for a phase-consistent basis, or an explicit volumetric_flow= to the call.
  outlet, _ = cstr(inlet, T_spec=350.0)

/tmp/ipykernel_1384/2274204586.py:16: CSTRDensityWarning: CSTR: neither eos nor molar_density was given, so the concentration basis falls back to 55500 mol/m^3 -- liquid water. Density sets the volumetric flow and hence the residence time (tau = V * rho / F_total), so a wrong density is a proportionally wrong conversion: ~8x for a C3-C8 hydrocarbon liquid, ~200x for a gas. Pass molar_density=<mol/m^3>, or eos=<cubic EOS> with reaction_phase='liquid'|'vapor' for a phase-consistent basis, or an explicit volumetric_flow= to the call.
  outlet, _ = cstr(inlet, T_spec=350.0)

/tmp/ipykernel_1384/2274204586.py:16: CSTRDensityWarning: CSTR: neither eos nor molar_density was given, so the concentration basis falls back to 55500 mol/m^3 -- liquid water. Density sets the volumetric flow and hence the residence time (tau = V * rho / F_total), so a wrong density is a proportionally wrong conversion: ~8x for a C3-C8 hydrocarbon liquid, ~200x for a gas. Pass molar_density=<mol/m^3>, or eos=<cubic EOS> with reaction_phase='liquid'|'vapor' for a phase-consistent basis, or an explicit volumetric_flow= to the call.
  outlet, _ = cstr(inlet, T_spec=350.0)

/tmp/ipykernel_1384/2274204586.py:16: CSTRDensityWarning: CSTR: neither eos nor molar_density was given, so the concentration basis falls back to 55500 mol/m^3 -- liquid water. Density sets the volumetric flow and hence the residence time (tau = V * rho / F_total), so a wrong density is a proportionally wrong conversion: ~8x for a C3-C8 hydrocarbon liquid, ~200x for a gas. Pass molar_density=<mol/m^3>, or eos=<cubic EOS> with reaction_phase='liquid'|'vapor' for a phase-consistent basis, or an explicit volumetric_flow= to the call.
  outlet, _ = cstr(inlet, T_spec=350.0)

/tmp/ipykernel_1384/2274204586.py:16: CSTRDensityWarning: CSTR: neither eos nor molar_density was given, so the concentration basis falls back to 55500 mol/m^3 -- liquid water. Density sets the volumetric flow and hence the residence time (tau = V * rho / F_total), so a wrong density is a proportionally wrong conversion: ~8x for a C3-C8 hydrocarbon liquid, ~200x for a gas. Pass molar_density=<mol/m^3>, or eos=<cubic EOS> with reaction_phase='liquid'|'vapor' for a phase-consistent basis, or an explicit volumetric_flow= to the call.
  outlet, _ = cstr(inlet, T_spec=350.0)

/tmp/ipykernel_1384/2274204586.py:16: CSTRDensityWarning: CSTR: neither eos nor molar_density was given, so the concentration basis falls back to 55500 mol/m^3 -- liquid water. Density sets the volumetric flow and hence the residence time (tau = V * rho / F_total), so a wrong density is a proportionally wrong conversion: ~8x for a C3-C8 hydrocarbon liquid, ~200x for a gas. Pass molar_density=<mol/m^3>, or eos=<cubic EOS> with reaction_phase='liquid'|'vapor' for a phase-consistent basis, or an explicit volumetric_flow= to the call.
  outlet, _ = cstr(inlet, T_spec=350.0)

/tmp/ipykernel_1384/2274204586.py:16: CSTRDensityWarning: CSTR: neither eos nor molar_density was given, so the concentration basis falls back to 55500 mol/m^3 -- liquid water. Density sets the volumetric flow and hence the residence time (tau = V * rho / F_total), so a wrong density is a proportionally wrong conversion: ~8x for a C3-C8 hydrocarbon liquid, ~200x for a gas. Pass molar_density=<mol/m^3>, or eos=<cubic EOS> with reaction_phase='liquid'|'vapor' for a phase-consistent basis, or an explicit volumetric_flow= to the call.
  outlet, _ = cstr(inlet, T_spec=350.0)

/tmp/ipykernel_1384/2274204586.py:16: CSTRDensityWarning: CSTR: neither eos nor molar_density was given, so the concentration basis falls back to 55500 mol/m^3 -- liquid water. Density sets the volumetric flow and hence the residence time (tau = V * rho / F_total), so a wrong density is a proportionally wrong conversion: ~8x for a C3-C8 hydrocarbon liquid, ~200x for a gas. Pass molar_density=<mol/m^3>, or eos=<cubic EOS> with reaction_phase='liquid'|'vapor' for a phase-consistent basis, or an explicit volumetric_flow= to the call.
  outlet, _ = cstr(inlet, T_spec=350.0)

/tmp/ipykernel_1384/2274204586.py:16: CSTRDensityWarning: CSTR: neither eos nor molar_density was given, so the concentration basis falls back to 55500 mol/m^3 -- liquid water. Density sets the volumetric flow and hence the residence time (tau = V * rho / F_total), so a wrong density is a proportionally wrong conversion: ~8x for a C3-C8 hydrocarbon liquid, ~200x for a gas. Pass molar_density=<mol/m^3>, or eos=<cubic EOS> with reaction_phase='liquid'|'vapor' for a phase-consistent basis, or an explicit volumetric_flow= to the call.
  outlet, _ = cstr(inlet, T_spec=350.0)

/tmp/ipykernel_1384/2274204586.py:16: CSTRDensityWarning: CSTR: neither eos nor molar_density was given, so the concentration basis falls back to 55500 mol/m^3 -- liquid water. Density sets the volumetric flow and hence the residence time (tau = V * rho / F_total), so a wrong density is a proportionally wrong conversion: ~8x for a C3-C8 hydrocarbon liquid, ~200x for a gas. Pass molar_density=<mol/m^3>, or eos=<cubic EOS> with reaction_phase='liquid'|'vapor' for a phase-consistent basis, or an explicit volumetric_flow= to the call.
  outlet, _ = cstr(inlet, T_spec=350.0)

/tmp/ipykernel_1384/2274204586.py:16: CSTRDensityWarning: CSTR: neither eos nor molar_density was given, so the concentration basis falls back to 55500 mol/m^3 -- liquid water. Density sets the volumetric flow and hence the residence time (tau = V * rho / F_total), so a wrong density is a proportionally wrong conversion: ~8x for a C3-C8 hydrocarbon liquid, ~200x for a gas. Pass molar_density=<mol/m^3>, or eos=<cubic EOS> with reaction_phase='liquid'|'vapor' for a phase-consistent basis, or an explicit volumetric_flow= to the call.
  outlet, _ = cstr(inlet, T_spec=350.0)

/tmp/ipykernel_1384/2274204586.py:16: CSTRDensityWarning: CSTR: neither eos nor molar_density was given, so the concentration basis falls back to 55500 mol/m^3 -- liquid water. Density sets the volumetric flow and hence the residence time (tau = V * rho / F_total), so a wrong density is a proportionally wrong conversion: ~8x for a C3-C8 hydrocarbon liquid, ~200x for a gas. Pass molar_density=<mol/m^3>, or eos=<cubic EOS> with reaction_phase='liquid'|'vapor' for a phase-consistent basis, or an explicit volumetric_flow= to the call.
  outlet, _ = cstr(inlet, T_spec=350.0)

/tmp/ipykernel_1384/4119124814.py:10: CSTRDensityWarning: CSTR: neither eos nor molar_density was given, so the concentration basis falls back to 55500 mol/m^3 -- liquid water. Density sets the volumetric flow and hence the residence time (tau = V * rho / F_total), so a wrong density is a proportionally wrong conversion: ~8x for a C3-C8 hydrocarbon liquid, ~200x for a gas. Pass molar_density=<mol/m^3>, or eos=<cubic EOS> with reaction_phase='liquid'|'vapor' for a phase-consistent basis, or an explicit volumetric_flow= to the call.
  outlet, _ = cstr(inlet, T_spec=350.0)

/tmp/ipykernel_1384/4119124814.py:10: CSTRDensityWarning: CSTR: neither eos nor molar_density was given, so the concentration basis falls back to 55500 mol/m^3 -- liquid water. Density sets the volumetric flow and hence the residence time (tau = V * rho / F_total), so a wrong density is a proportionally wrong conversion: ~8x for a C3-C8 hydrocarbon liquid, ~200x for a gas. Pass molar_density=<mol/m^3>, or eos=<cubic EOS> with reaction_phase='liquid'|'vapor' for a phase-consistent basis, or an explicit volumetric_flow= to the call.
  outlet, _ = cstr(inlet, T_spec=350.0)

/tmp/ipykernel_1384/4119124814.py:10: CSTRDensityWarning: CSTR: neither eos nor molar_density was given, so the concentration basis falls back to 55500 mol/m^3 -- liquid water. Density sets the volumetric flow and hence the residence time (tau = V * rho / F_total), so a wrong density is a proportionally wrong conversion: ~8x for a C3-C8 hydrocarbon liquid, ~200x for a gas. Pass molar_density=<mol/m^3>, or eos=<cubic EOS> with reaction_phase='liquid'|'vapor' for a phase-consistent basis, or an explicit volumetric_flow= to the call.
  outlet, _ = cstr(inlet, T_spec=350.0)

/tmp/ipykernel_1384/4119124814.py:10: CSTRDensityWarning: CSTR: neither eos nor molar_density was given, so the concentration basis falls back to 55500 mol/m^3 -- liquid water. Density sets the volumetric flow and hence the residence time (tau = V * rho / F_total), so a wrong density is a proportionally wrong conversion: ~8x for a C3-C8 hydrocarbon liquid, ~200x for a gas. Pass molar_density=<mol/m^3>, or eos=<cubic EOS> with reaction_phase='liquid'|'vapor' for a phase-consistent basis, or an explicit volumetric_flow= to the call.
  outlet, _ = cstr(inlet, T_spec=350.0)

/tmp/ipykernel_1384/4119124814.py:10: CSTRDensityWarning: CSTR: neither eos nor molar_density was given, so the concentration basis falls back to 55500 mol/m^3 -- liquid water. Density sets the volumetric flow and hence the residence time (tau = V * rho / F_total), so a wrong density is a proportionally wrong conversion: ~8x for a C3-C8 hydrocarbon liquid, ~200x for a gas. Pass molar_density=<mol/m^3>, or eos=<cubic EOS> with reaction_phase='liquid'|'vapor' for a phase-consistent basis, or an explicit volumetric_flow= to the call.
  outlet, _ = cstr(inlet, T_spec=350.0)

/tmp/ipykernel_1384/4119124814.py:10: CSTRDensityWarning: CSTR: neither eos nor molar_density was given, so the concentration basis falls back to 55500 mol/m^3 -- liquid water. Density sets the volumetric flow and hence the residence time (tau = V * rho / F_total), so a wrong density is a proportionally wrong conversion: ~8x for a C3-C8 hydrocarbon liquid, ~200x for a gas. Pass molar_density=<mol/m^3>, or eos=<cubic EOS> with reaction_phase='liquid'|'vapor' for a phase-consistent basis, or an explicit volumetric_flow= to the call.
  outlet, _ = cstr(inlet, T_spec=350.0)

/tmp/ipykernel_1384/4119124814.py:10: CSTRDensityWarning: CSTR: neither eos nor molar_density was given, so the concentration basis falls back to 55500 mol/m^3 -- liquid water. Density sets the volumetric flow and hence the residence time (tau = V * rho / F_total), so a wrong density is a proportionally wrong conversion: ~8x for a C3-C8 hydrocarbon liquid, ~200x for a gas. Pass molar_density=<mol/m^3>, or eos=<cubic EOS> with reaction_phase='liquid'|'vapor' for a phase-consistent basis, or an explicit volumetric_flow= to the call.
  outlet, _ = cstr(inlet, T_spec=350.0)

/tmp/ipykernel_1384/4119124814.py:10: CSTRDensityWarning: CSTR: neither eos nor molar_density was given, so the concentration basis falls back to 55500 mol/m^3 -- liquid water. Density sets the volumetric flow and hence the residence time (tau = V * rho / F_total), so a wrong density is a proportionally wrong conversion: ~8x for a C3-C8 hydrocarbon liquid, ~200x for a gas. Pass molar_density=<mol/m^3>, or eos=<cubic EOS> with reaction_phase='liquid'|'vapor' for a phase-consistent basis, or an explicit volumetric_flow= to the call.
  outlet, _ = cstr(inlet, T_spec=350.0)

6. Comparing Objectives: SSE vs NLL

The negative_log_likelihood objective assumes Gaussian errors and accounts for different measurement uncertainties per data point.

# Create experiments with known heteroscedastic noise
rng = np.random.default_rng(77)
nll_experiments = []

for t in np.linspace(0.5, 8.0, 15):
    sigma = 0.05 + 0.03 * t  # Noise grows with t
    y = A_true * np.exp(-k_true * t) + sigma * rng.standard_normal()
    nll_experiments.append(Experiment(
        inputs={'t': float(t)},
        observed={'y': float(y)},
        uncertainties={'y': float(sigma)},
    ))

est_nll = Estimator(decay_model, ['A', 'k'],
                     param_bounds={'A': (0.1, 20.0), 'k': (0.01, 5.0)})

result_sse_nll = est_nll.fit(nll_experiments, {'A': 3.0, 'k': 0.1}, objective='sse')
result_nll = est_nll.fit(nll_experiments, {'A': 3.0, 'k': 0.1}, objective='nll')

print(f"{'':>15} {'A':>10} {'k':>10}")
print(f"{'True':>15} {A_true:>10.4f} {k_true:>10.4f}")
print(f"{'SSE fit':>15} {result_sse_nll.theta_opt['A']:>10.4f} {result_sse_nll.theta_opt['k']:>10.4f}")
print(f"{'NLL fit':>15} {result_nll.theta_opt['A']:>10.4f} {result_nll.theta_opt['k']:>10.4f}")
print()
print("NLL properly weights each observation by its known uncertainty,")
print("giving more influence to precise measurements.")

                         A          k
           True     5.0000     0.3000
        SSE fit     5.0832     0.3077
        NLL fit     5.0494     0.3035

NLL properly weights each observation by its known uncertainty,
giving more influence to precise measurements.

Summary

Key API

from difflow.estimation import Estimator, Experiment

# 1. Package data as Experiments
exp = Experiment(inputs={...}, observed={...}, uncertainties={...})

# 2. Define model: theta_dict, experiment -> predictions dict
def model(theta, exp): return {'y': theta['a'] * exp.inputs['x']}

# 3. Create estimator and fit
est = Estimator(model, param_names=['a'], param_bounds={'a': (0, 10)})
result = est.fit(experiments, theta_init={'a': 1.0})

# 4. Analyze
ci = est.confidence_intervals(result, experiments)
diag = est.diagnostics(result, experiments)
print(est.summary(result, experiments))

Objective functions

Objective	When to use
'sse'	Default, equal-weight measurements
'wsse'	Measurements with known, different uncertainties
'nll'	Gaussian errors with known sigma per observation

Next steps

See notebook 23 for bootstrap uncertainty quantification and cross-validation.