difflow#
Differentiable Flowsheet Framework for Chemical Processes
A JAX-based framework for building and optimizing chemical process flowsheets with automatic differentiation.
Source code: jkitchin/differentiable-flowsheets
Documentation: https://kitchingroup.cheme.cmu.edu/differentiable-flowsheets/
Issue tracker: jkitchin/differentiable-flowsheets#issues
Features#
Fully Differentiable: All unit operations and flowsheet calculations support automatic differentiation via JAX
Sensitivity Analysis: Compute gradients of outputs with respect to any inputs, parameters, or operating conditions
Optimization Ready: Use gradient-based optimization for process design, parameter estimation, and economic optimization
Modular Design: Unit operations can be composed into complex flowsheets with recycle streams
Technoeconomic Analysis: Comprehensive TEA module with equipment costs, operating costs, and profitability metrics (NPV, IRR, MSP)
Bio Manufacturing: Specialized unit operations for biopharmaceutical processes (bioreactors, chromatography, filtration)
Gas Networks: Steady-state gas transmission networks with a topology-computed sequential decomposition and differentiable tear solving
Flowsheets Without Code: A machine-readable catalog of every unit, JSON round trip, Python code generation, a browser-based editor served on
localhost, and one-file interactive HTML for publishing a modelAgent Tools (MCP):
difflow mcpserves difflow to AI agents over the Model Context Protocol: build, solve, diagnose and converge flowsheets, and run sensitivity, optimization, uncertainty and cost studies from plain-language requests
Plugins#
Six domain plugins ship with difflow:
Plugin |
Domain |
|
|---|---|---|
|
Biomanufacturing |
|
|
Rare earth separations |
|
|
Carbon capture |
|
|
Gas transmission networks |
|
|
Electrical grids |
|
|
Petroleum refining |
⚠️ ALPHA SOFTWARE#
This project is under active development and not ready for production use. APIs may change without notice. This notice will be removed when the project reaches stable release.
It is highly recommended that you confirm the equations and physical properties used in the models you make; these were generated by Claude. We have endeavored to ensure they seem reasonable, but cannot guarantee they are accurate in all cases.
This package uses jax solvers (e.g. diffrax, optimistix, etc.) and does not rely on IPOPT or pounce, or pyomo / IDAES. It is a pure Python / jax focused package that was developed as a proof of concept.
LLM usage#
Claude Code is heavily used to generate the code, examples and tests. This has allowed the project to develop faster than it can be used, and to develop more features than are immediately needed. This may mean there are modules that do not match the performance or output of other projects. You should perform your own diligence when using the code to ensure the library does what you expect it to. Ultimately this is a proof of concept in differentiable flowsheets that wouldn’t be possible without Claude Code.
We regularly run all of the notebooks to ensure they run without errors, and review them to make sure the results make sense. We are happy to take issues and / or pull requests to fix problems. We also use Claude to review the code to look for issues.
We actually anticipate that Claude Code is used when using this library (See CLAUDE.md). The library is large enough that it would take a long time to learn all the capabilities in addition to learning the nuances of differentiable programming. This repo provides all the information Claude needs to help you translate your flowsheet ideas into differentiable programs.
Using difflow with an AI agent (MCP)#
difflow includes an MCP server, so an AI agent such as Claude Code or Claude Desktop can use it directly: find unit operations, build and solve flowsheets, explain why a solve fails and search for settings that converge it, and run sensitivity, optimization, uncertainty and capital-cost studies, all with exact derivatives through the solve.
pip install "difflow[mcp]" # 0.3.0 or later
claude mcp add difflow -- difflow mcp
Then ask in plain language, for example “build a flash drum for an equimolar water and ethanol feed at 362 K and tell me the vapor composition”, or “this flowsheet does not converge; find out why”. The server discovers operations and plugins from the installed code, so new units are available without changes to it. difflow mcp --no-exec leaves out the tools that run Python, for shared or desktop clients. Setup for Claude Desktop and other clients, the full tool list and troubleshooting are in the agent documentation.
Installation#
# From PyPI
pip install difflow
# With examples and tutorials (includes matplotlib, jupyter)
pip install "difflow[examples]"
# The MCP server for AI agents (see "Using difflow with an AI agent")
pip install "difflow[mcp]"
# Everything
pip install "difflow[all]"
For development, install from source:
git clone https://github.com/jkitchin/differentiable-flowsheets.git
cd differentiable-flowsheets
uv venv
uv pip install -e ".[dev]"
# Install everything
uv pip install -e ".[all]"
Quick Start#
import jax.numpy as jnp
import jax
from difflow import (
make_stream, get_flows,
IdealThermo, SpeciesData,
CSTR, CSTRParams,
)
# Define species
species_data = {
"A": SpeciesData("A", MW=100.0, Cp_coeffs=(75.0, 0.0, 0.0, 0.0),
Hvap_coeffs=(35000.0, 0.38, 500.0),
antoine_coeffs=(10.0, 3000.0, -50.0)),
"B": SpeciesData("B", MW=100.0, Cp_coeffs=(75.0, 0.0, 0.0, 0.0),
Hvap_coeffs=(30000.0, 0.38, 450.0),
antoine_coeffs=(10.0, 2800.0, -40.0)),
}
thermo = IdealThermo(species_data)
# Define reaction kinetics
def rate_fn(C, T, params):
k = params["A"] * jnp.exp(-params["Ea"] / (8.314 * T))
return jnp.array([k * C["A"]])
# Create CSTR
stoich = jnp.array([[-1.0], [+1.0]]) # A → B
cstr_params = CSTRParams(
V=jnp.array(1.0),
rate_fn=rate_fn,
stoich=stoich,
rate_params={"A": jnp.array(1e6), "Ea": jnp.array(50000.0)},
species_order=["A", "B"],
# Concentration basis: tau = V*rho/F. Pass eos=<cubic EOS> +
# reaction_phase instead for the real density at reactor conditions;
# with neither, the CSTR falls back to liquid water and warns.
molar_density=55500.0,
)
cstr = CSTR(cstr_params, thermo=thermo, mode="isothermal")
# Run simulation
inlet = make_stream({"A": 10.0, "B": 0.0}, T=300.0, P=101325.0)
outlet, info = cstr(inlet, T_spec=350.0)
print(f"Conversion: {info['conversion']['A']*100:.1f}%")
# Compute gradient of outlet B w.r.t. reactor volume
def outlet_B(V):
params = CSTRParams(V=V, rate_fn=rate_fn, stoich=stoich,
rate_params={"A": jnp.array(1e6), "Ea": jnp.array(50000.0)},
species_order=["A", "B"])
cstr = CSTR(params, thermo=thermo, mode="isothermal")
outlet, _ = cstr(inlet, T_spec=350.0)
return outlet["F_B"]
dFB_dV = jax.grad(outlet_B)(jnp.array(1.0))
print(f"dF_B/dV = {dFB_dV:.4f} mol/s per m³")
Unit Operations#
CSTR (Continuous Stirred Tank Reactor)#
Multiple reactions with user-defined kinetics
Isothermal, adiabatic, or specified heat duty modes
Automatic material and energy balance solving
PFR (Plug Flow Reactor)#
ODE-based design equation: dF/dV = stoich @ r
Isothermal or adiabatic operation
GasPFR variant for gas-phase reactions with:
Pressure drop (Ergun equation)
Variable volumetric flow from mole change
RK4 integration via
lax.scan(fully differentiable)
from difflow import PFR, PFRParams, GasPFR, GasPFRParams
# rate_fn, stoich and inlet are those of the Quick Start
params = {"A": jnp.array(1e6), "Ea": jnp.array(50000.0)} # rate_fn's parameters
# Liquid-phase PFR
pfr = PFR(PFRParams(V=2.0, rate_fn=rate_fn, stoich=stoich,
rate_params=params, species_order=["A", "B"]))
outlet, info = pfr(inlet, volumetric_flow=0.01, T_spec=350.0) # m³/s
# Gas-phase with pressure drop (A → 2B, mole increase)
gas_pfr = GasPFR(GasPFRParams(V=1.0, rate_fn=rate_fn, stoich=stoich,
rate_params=params, species_order=["A", "B"],
alpha=50000.0)) # Pressure drop parameter
outlet, info = gas_pfr(inlet, T_spec=500.0)
# info contains: conversion, profiles (V, F, T, P, Q), pressure_drop
Flash Separator#
TP flash (temperature and pressure specified)
Rachford-Rice equation for VLE
Ideal thermodynamics (Raoult’s law)
Liquid-Liquid Extraction (LLE)#
MultistageCascade: Counter-current or co-current mixer-settler cascade
Kremser equation for stage calculations (differentiable in n_stages)
Continuous stage relaxation for optimization
DifferentialContactor: Packed column extractor
HETP-based equilibrium model
Rate-based mass transfer model
Equilibrium Models:
Distribution coefficients (K-values) with temperature dependence
NRTL activity coefficient model
UNIQUAC activity coefficient model
from difflow import (
MultistageCascade, CascadeParams,
LLEEquilibrium, DistributionCoeffs,
)
# Define distribution coefficients for rare earth extraction
K_coeffs = DistributionCoeffs(
species=("La", "Nd", "Dy"),
K0=(0.5, 2.0, 8.0), # K at reference temperature
)
equilibrium = LLEEquilibrium(
solutes=["La", "Nd", "Dy"],
aqueous_carrier="H2O",
organic_carrier="Organic",
K_coeffs=K_coeffs,
)
cascade = MultistageCascade(CascadeParams(
n_stages=5,
equilibrium=equilibrium,
flow_config="counter_current",
))
feed_stream = make_stream({"La": 1.0, "Nd": 1.0, "Dy": 0.5, "H2O": 50.0, "Organic": 0.0},
T=298.15, P=101325.0)
solvent_stream = make_stream({"La": 0.0, "Nd": 0.0, "Dy": 0.0, "H2O": 0.0, "Organic": 50.0},
T=298.15, P=101325.0)
raffinate, extract, info = cascade(feed_stream, solvent_stream)
Utilities#
Mixer: Combine multiple streams
Splitter: Split stream by fraction
Fed-Batch Reactor#
General-purpose fed-batch (semi-batch) reactor for chemical reactions
Time-varying feed addition with configurable feed profiles
RK4 integration for batch dynamics
Supports multiple reactions with user-defined kinetics
from difflow import FedBatchReactor, FedBatchParams
# Fed-batch reactor with continuous reagent addition
def rate_fn(C, T, params):
k = params["k0"] * jnp.exp(-params["Ea"] / (8.314 * T))
return jnp.array([k * C["A"] * C["B"]])
params = FedBatchParams(
V0=jnp.array(1.0), # Initial volume (m³)
rate_fn=rate_fn,
stoich=jnp.array([[-1.0], [-1.0], [1.0]]), # A + B → C
rate_params={"k0": jnp.array(1e6), "Ea": jnp.array(50000.0)},
species_order=["A", "B", "C"],
)
reactor = FedBatchReactor(params)
# Initial charge (mol/m³) and a constant feed of B (mol/m³) at 0.001 m³/s
C0 = {"A": 1000.0, "B": 0.0, "C": 0.0}
def feed_rate(t): return jnp.array(0.001) # m³/s
final, info = reactor(C0, T0=350.0, P=101325.0, t_final=3600.0,
feed_rate_fn=feed_rate,
feed_composition={"A": 0.0, "B": 1000.0, "C": 0.0}, feed_T=350.0)
# info contains the time profiles: t, V, C, T
Distillation Columns#
ShortcutColumn: Fenske-Underwood-Gilliland method for quick design estimates
Minimum stages (Fenske equation)
Minimum reflux ratio (Underwood equations)
Actual stages for given reflux (Gilliland correlation)
DistillationColumn: Rigorous stage-by-stage calculation
MESH equations (Material, Equilibrium, Summation, Heat balance)
Supports partial/total condenser and reboiler
from difflow import ShortcutColumn, ShortcutColumnParams, IdealThermo
from difflow.database import get_species_data
names = ["benzene", "toluene", "ethylbenzene"]
aromatics = IdealThermo({s: get_species_data(s) for s in names})
params = ShortcutColumnParams(
species_order=names,
light_key="benzene",
heavy_key="toluene",
x_D_LK=0.99, # 99% benzene recovery in distillate
x_B_HK=0.99, # 99% toluene recovery in bottoms
)
column = ShortcutColumn(params, thermo=aromatics)
feed = make_stream({"benzene": 40.0, "toluene": 35.0, "ethylbenzene": 25.0},
T=370.0, P=101325.0)
distillate, bottoms, info = column(feed, R=3.0, q=1.0)
# info contains: N_min, R_min, N, N_feed, Q_condenser, Q_reboiler, ...
Heat Exchangers#
Heater/Cooler: Single-stream with utility (steam, cooling water)
Specified duty mode
Specified outlet temperature mode
Rating mode (given UA and utility temperature)
Constant
Cp, or athermofor a real enthalpy balance (carries latent heat)
CounterCurrentHX: Two-stream counter-current (shell-and-tube style)
CoCurrentHX: Two-stream co-current (parallel flow)
EnthalpyCounterCurrentHX: Two-stream, closed on EOS enthalpies through phase change
The constant-Cp units use the effectiveness-NTU method; all are fully differentiable
from difflow import (
Heater, HeaterParams,
CounterCurrentHX, HeatExchangerParams,
design_heat_exchanger,
)
# Streams of the Quick Start's species A and B (mol/s)
cold_feed = make_stream({"A": 10.0, "B": 5.0}, T=300.0, P=101325.0)
hot_stream = make_stream({"A": 10.0, "B": 5.0}, T=450.0, P=101325.0)
cold_stream = make_stream({"A": 10.0, "B": 5.0}, T=300.0, P=101325.0)
# Single-stream heater with steam
heater = Heater(HeaterParams(T_out=400.0, Cp=75.0))
heated_feed, info = heater(cold_feed)
# info: Q, T_in, T_out, LMTD (if utility temp specified)
# Duty from the thermo instead of a constant Cp -- required if the stream
# vaporizes, since a constant Cp carries no latent heat
heater = Heater(HeaterParams(T_out=400.0), thermo=thermo)
heated_feed, info = heater(cold_feed)
# Two-stream counter-current heat exchanger
hx = CounterCurrentHX(HeatExchangerParams(
UA=2000.0, # W/K
Cp_hot=75.0, # J/(mol·K)
Cp_cold=80.0,
))
hot_out, cold_out, info = hx(hot_stream, cold_stream)
# info: Q, effectiveness, NTU, LMTD, approach temperature
# Design: calculate required area
result = design_heat_exchanger(
Q=jnp.array(100000.0), # 100 kW
T_hot_in=jnp.array(450.0), T_hot_out=jnp.array(380.0),
T_cold_in=jnp.array(300.0), T_cold_out=jnp.array(360.0),
U=jnp.array(500.0), # W/(m²·K)
)
print(f"Required area: {result['A']:.1f} m²")
Bio Manufacturing Operations#
The difflow_bio plugin provides specialized unit operations for biopharmaceutical manufacturing:
Bioreactors#
ContinuousBioreactor: Chemostat with Monod kinetics
FedBatchBioreactor: Fed-batch with substrate feeding strategy
from difflow_bio import (
ContinuousBioreactor, BioreactorParams,
FedBatchBioreactor, FedBatchParams,
monod_kinetics,
)
# Create a continuous bioreactor (chemostat)
params = BioreactorParams(
V=1000.0, # Volume (L)
Y_xs=0.5, # Biomass yield (g/g)
kinetic_fn=monod_kinetics,
kinetic_params={"mu_max": 0.3, "K_s": 0.5}, # 1/h, g/L
alpha=0.1, # Growth-associated product yield
)
bioreactor = ContinuousBioreactor(params)
feed_stream = make_stream({"cells": 0.0, "substrate": 20.0, "product": 0.0},
T=310.0, P=101325.0)
outlet, info = bioreactor(feed_stream, D=0.1) # D: dilution rate (1/h)
Downstream Processing#
DiscStackCentrifuge: Cell removal with Stokes’ law separation
Ultrafiltration: Protein concentration via TFF
Diafiltration: Buffer exchange
ProteinAChromatography: Affinity capture for mAb purification
IonExchangeChromatography: Polishing step (bind-elute or flow-through)
SizeExclusionChromatography: Aggregate removal
from difflow_bio import (
DiscStackCentrifuge, DiscStackParams,
Ultrafiltration, UltrafiltrationParams,
ProteinAChromatography, ProteinAParams,
)
# Disc-stack centrifuge for cell removal (Sigma from the disc geometry)
centrifuge = DiscStackCentrifuge(DiscStackParams(
n_discs=100, r_outer=0.15, r_inner=0.05, # m
rpm=7000.0,
))
# Protein A capture
proa = ProteinAChromatography(ProteinAParams(
column_volume=10.0, # CV (L)
q_max=40.0, # g mAb / L resin
yield_factor=0.95,
))
# Ultrafiltration for concentration (the concentration factor is a call argument)
uf = Ultrafiltration(UltrafiltrationParams(
membrane_area=1.0, # m²
))
Gas Transmission Networks#
The difflow_gas plugin models steady-state gas transmission networks
as sequential-modular differentiable flowsheets. The sequential
decomposition of a meshed network (spanning tree, tear set, balance
schedule) is computed from the topology, so multi-loop networks need
no hand derivation:
import difflow_gas as dg
net = dg.GasNetwork(
arcs={
"p1": ("src", "a", "pipe"),
"cs1": ("a", "b", "compressor"),
"p2": ("b", "c", "pipe"),
"p3": ("b", "d", "pipe"),
"p4": ("c", "d", "pipe"), # closes a loop: the tear
},
beta={aid: dg.weymouth_beta(L, 0.6, 1e-4)
for aid, L in [("p1", 20e3), ("p2", 40e3),
("p3", 60e3), ("p4", 80e3)]},
supply_kg_s={"src": 120.0, "c": -50.0, "d": -70.0},
)
fs, dec = dg.build_network_flowsheet(net, root="src",
p_slack_pa=60e5,
ratios={"cs1": 1.3})
streams = fs.solve(tol=1e-8) # signed flows, Anderson tears
assert dg.residual_report(streams, net, dec).ok
# exact gradients through the converged tear iteration
obj = fs.make_objective_fn(
lambda s: dg.total_compressor_power_w(s, dec, net.gas_temp_k))
dW_dr = jax.grad(obj)({"cs_cs1.ratio": 1.3})
Pipes, resistors, compressor stations, open valves, control valves and
short pipes are supported; see docs/unit-operations-gas.md.
Electrical Grids and AC-OPF#
The difflow_power plugin models steady-state electrical transmission
and distribution networks, and solves the AC optimal power flow — the
nonconvex problem every wholesale market and control centre sits on
top of. Because the model is differentiable, the quantities a grid
study is actually after are derivatives rather than separately-derived
sensitivity factors: locational marginal prices, shift factors,
marginal loss factors, and the value of relaxing any binding limit.
import difflow_power as dp
net = dp.cases.case9() # WSCC 9-bus benchmark
pf = dp.solve_power_flow(net) # Newton-Raphson, implicit-diff gradients
pf.losses_mw # 4.9547 (MATPOWER: 4.9547)
opf = dp.solve_acopf(net) # interior-point AC-OPF, written in JAX
opf.cost # 5296.69 $/h (MATPOWER: 5296.69)
opf.lmp_mw # locational marginal prices, $/MWh
opf.binding() # binding limits and their shadow prices
# the multipliers ARE prices: check them against jax.grad of the optimum
max(opf.check_prices().values()) # ~1e-12 $/MWh
# and everything the classical factor tables give, as derivatives
dp.loss_sensitivity(net) # marginal loss factors
dp.ptdf(net), dp.lodf(net) # shift and outage factors
One branch model covers lines, transformers and phase shifters;
generator boxes, voltage limits and thermal ratings are carried as
inequalities by a primal-dual interior-point solver written in JAX
(no IPOPT, so the differentiability survives). Radial feeders also
solve sequentially, by the backward/forward sweep, which agrees with
Newton to 1e-12. Every benchmark result is asserted against MATPOWER’s
published answer; see docs/unit-operations-power.md.
Refinery: Crude Distillation#
difflow_refinery.CrudeDistillationUnit is a crude unit: a TBP assay
characterized into pseudo-components, a fired heater solved together with
an atmospheric column (side strippers, pumparounds, stripping steam), and
products reported as yields, API gravities and TBP ranges. Specs follow a
simulator’s degrees of freedom (product rates, pumparound duties,
overflash), and every yield or duty has an implicit-function gradient with
respect to the specs, the feed and the assay.
import difflow_refinery as dr
unit = dr.CrudeUnit(assay, column_params) # dr.Assay, dr.column.CrudeColumnParams
res = unit.solve(95_000, T=513.15, P=6e5) # bbl/d at the furnace inlet
print(res.table()) # yields, API, TBP 5/50/95
Refinery: Vacuum Distillation#
difflow_refinery.vacuum characterizes a crude assay into
pseudocomponents and runs a vacuum distillation unit on the atmospheric
residue. It covers LVGO and HVGO pumparound sections, a wash zone with an
overflash spec, a flash zone fed by the furnace, and a steam-stripped
residue. The MESH equations of every stage are solved simultaneously, and
the result carries exact implicit-function gradients, so the VGO/residue
cut point is a decision variable with a derivative. That includes the
cut’s derivative with respect to a single point of the assay’s TBP curve.
import difflow_refinery as dr
char = dr.vacuum.characterize(dr.vacuum.heavy_crude()) # 300-800 C cuts + residue lump
feed = dr.vacuum.atmospheric_residue(char, crude_rate_kg_s=100.0)
vdu = dr.VacuumColumn(dr.VacuumColumnParams(components=char.components))
overhead, lvgo, hvgo, slop, residue, info = vdu(feed)
info["properties"]["hvgo"] # rate, SG, S, N, CCR, Ni+V, TBP 5/50/95
Any output can be specified in place of the knob that controls it, for example an HVGO end point instead of the furnace temperature.
The crude unit, the vacuum column and the blend pool can share one
characterization. An Assay with a HeavyEnd is carried into the vacuum
range, closed by a residue lump, and given sulfur, nitrogen, CCR and metals
per component. The CDU runs on it, and the VDU’s components are
char.pseudo_components(), so the CDU’s "residue" outlet feeds the VDU
directly in a Flowsheet. The balance closes, and gradients cross the
connection:
char = dr.characterize(assay) # assay has heavy_end=dr.HeavyEnd()
cdu = dr.CrudeDistillationUnit(dr.CrudeDistillationUnitParams(assay=assay, column=params))
vdu = dr.VacuumColumn(dr.VacuumColumnParams(components=char.pseudo_components()))
grid = dr.BlendCharacterization.from_characterization(char) # products into the pool
See docs/unit-operations-refinery.md and
examples/36_crude_to_vacuum.ipynb.
Refinery Product Blending#
difflow_refinery.BlendPool blends component streams into finished
products (gasoline, jet, ULSD, fuel oil) with the nonlinear rules
refiners use: Ethyl RT-70 octane interactions, the RVP^1.25 index (and
Raoult on the pseudocomponents as a check), Hu-Burns flash and cold-flow
indices, Refutas viscosity, and distillation and cetane index computed
from the blend’s composition. It is differentiable in the recipe and in
every component property, and it reports signed spec margins with a
smooth-violation option.
from difflow_refinery import BlendComponent, BlendPool
reformate = BlendComponent.from_properties(
"reformate", SG=0.80, RON=98.0, MON=88.0, RVP_psi=3.5, S_ppm=1.0,
olefins_vol=1.0, aromatics_vol=65.0)
alkylate = BlendComponent.from_properties(
"alkylate", SG=0.70, RON=95.0, MON=93.0, RVP_psi=4.5, S_ppm=5.0,
olefins_vol=0.5, aromatics_vol=0.5)
butane = BlendComponent.from_properties(
"butane", SG=0.58, RON=93.0, MON=90.0, RVP_psi=52.0, S_ppm=10.0,
olefins_vol=0.5, aromatics_vol=0.0)
components, recipe = [reformate, alkylate, butane], [0.45, 0.50, 0.05] # volume fractions
pool = BlendPool("gasoline") # RON, MON, RVP, S specs
res = pool(components, recipe) # properties, margins, product stream
pool.linear_blend_error(components, recipe) # what an LP's back-off must cover
pool.as_block(components) # a difflow.planning.Block
See docs/unit-operations-refinery.md and
examples/33_refinery_gasoline_blending.ipynb, which compares the
nonlinear optimum with a linear-by-volume LP plus successive back-off.
Every refinery unit (preheat train, crude and vacuum units, gas plant,
isomerization, hydrotreater, hydrocracker, FCC, reformer, alkylation and
blending) is listed, with its model and how far it has been validated, in
docs/refinery-summary.md and src/difflow_refinery/README.md.
examples/40_refinery_flowsheet.ipynb joins them into a small whole
refinery, from the crude to the product pools and the hydrogen header, and
difflow_refinery.plant (Chain, Stage, AD_MODES) composes library units
into one differentiable function across their different AD modes (the
reformer is forward-only, a default hydrotreater reverse-only).
Data Reconciliation#
Plant measurements are noisy and, taken at face value, contradict the
model. difflow.reconciliation finds the smallest statistically
weighted adjustment that satisfies the model equations, and returns
estimates sharper than the raw measurements:
from difflow.reconciliation import reconcile, global_test, measurement_test
# a splitter with three metered flows that do not close: F1 = F2 + F3
names = ["F1", "F2", "F3"]
residual_fn = lambda x, params: jnp.array([x[0] - x[1] - x[2]])
y = jnp.array([100.0, 60.0, 38.0]) # measured, kg/s
sigma = jnp.array([1.0, 1.0, 1.0])
res = reconcile(residual_fn, y, sigma, names=names)
print(res.summary()) # measured vs reconciled, with standard errors
global_test(res) # is the data set consistent with the model?
measurement_test(res) # which sensor is lying?
An entry of sigma set to inf marks a variable to be estimated
rather than reconciled, so joint parameter estimation and reconciliation
are the same solve. Whether an unknown can be recovered at all is decided
before solving, so an ill-posed problem raises a named error instead of
returning NaN. Works with any differentiable residual function; see
docs/data-reconciliation.md and
examples/28_data_reconciliation.ipynb for the gas-network case, and
examples/29_model_updating.ipynb for when to update a model parameter
rather than the data.
Delta-Base Planning#
Refinery and value-chain planning runs on linear programs whose unit
submodels are base plus delta vectors, y ~= y0 + J (u - u0). Every
commercial system builds J by perturbing a rigorous simulator one
variable at a time, which costs O(n) evaluations. A flowsheet is a pure
function with its flash and recycle solves embedded, so jax.jacobian
returns the same reduced Jacobian for a cost independent of n:
from difflow.planning import Block, Network, DeltaBasePlanner
net = Network([ngl, power], links=[("ngl.residue_F", "power.fuel_F")])
res = DeltaBasePlanner(net, prices={"ngl.NGL_C2": 9.0, "power.Power": 55.0},
specs=[("ngl.T_colfeed", "<=", 236.0)],
radius=0.3).solve()
res.plan # optimal decisions
res.delta_vectors # the J blocks actually used
res.pyomo_model # emitted for the Pyomo/IDAES ecosystem
res.plan_sensitivity(wrt="prices") # d(plan)/d(price), not just the plan
Measured on a two-plant chain, the AD gradient costs 1-2 model
evaluations from 5 to 80 decisions while central differences cost 2n.
Every LP proposal is checked against the nonlinear blocks before it is
accepted, violations are charged from the real model rather than from LP
slacks, bang-bang levers are vertex-seeded, and a linearisation that
straddles a phase boundary raises a warning instead of quietly
extrapolating a branch that no longer exists. Pooling/blending
bilinearity and the commercial trappings (assay libraries, blending
correlations, scheduling) are explicitly out of scope. See
docs/planning.md and examples/30_delta_base_planning.ipynb.
Thermodynamics#
Ideal Thermodynamics (for VLE)#
Ideal gas behavior
Antoine equation for vapor pressures
Polynomial Cp correlations
Watson correlation for heat of vaporization
SpeciesData(
name="species_name",
MW=100.0, # Molecular weight (g/mol)
Cp_coeffs=(a, b, c, d), # Cp = a + bT + cT² + dT³
Hvap_coeffs=(A, n, Tc), # Hvap = A(1 - T/Tc)^n
antoine_coeffs=(A, B, C), # log10(Psat) = A - B/(T+C)
Hf=0.0, # Heat of formation (J/mol)
)
Equations of State (for non-ideal VLE)#
Peng-Robinson: Cubic EOS for hydrocarbon and gas systems
Soave-Redlich-Kwong (SRK): Alternative cubic EOS
Fugacity coefficients for both vapor and liquid phases
Flash calculations with non-ideal K-values
Binary interaction parameters (kij) support
from difflow import PengRobinson, SRK, CriticalProperties, flash_TP_eos
# Define critical properties
props = {
"methane": CriticalProperties("methane", Tc=190.6, Pc=4.6e6, omega=0.011),
"ethane": CriticalProperties("ethane", Tc=305.3, Pc=4.87e6, omega=0.099),
}
# Create EOS
eos = PengRobinson(props)
# or: eos = SRK(props)
# Compressibility factor
z = eos.solve_Z(T=300.0, P=1e6, y=jnp.array([0.7, 0.3]), phase="vapor")
# Fugacity coefficients
phi = eos.fugacity_coefficient(T=300.0, P=1e6, y=jnp.array([0.7, 0.3]), phase="vapor")
# Flash calculation
V_frac, x, y = flash_TP_eos(eos, z=jnp.array([0.5, 0.5]), T=250.0, P=2e6)
Property Database#
Built-in database with 55+ common species including critical properties and ideal thermo data:
from difflow import (
get_species_data, get_critical_props, list_species,
get_alkanes, get_btex, get_common_solvents,
)
# Get species data for ideal thermodynamics
methanol = get_species_data("methanol")
thermo = IdealThermo({"methanol": methanol, "water": get_species_data("water")})
# Get critical properties for EOS
methane = get_critical_props("methane")
eos = PengRobinson({"methane": methane, "ethane": get_critical_props("ethane")})
# Convenience functions
alkanes = get_alkanes() # methane through n-decane
btex = get_btex() # benzene, toluene, ethylbenzene, xylenes
solvents = get_common_solvents() # water, methanol, ethanol, acetone, etc.
# Alias support: "CO2", "MeOH", "isopropanol", "IPA" all work
co2 = get_critical_props("CO2")
# List all available species
print(list_species())
Activity Coefficient Models (for LLE)#
NRTL: Non-Random Two-Liquid model with temperature-dependent parameters
UNIQUAC: Universal Quasi-Chemical model
Technoeconomic Analysis (TEA)#
The difflow.economics module provides comprehensive technoeconomic analysis capabilities, all fully differentiable for gradient-based optimization.
Capital Costs#
Equipment cost correlations with CEPCI escalation and installation factors:
import difflow.economics as econ
import jax.numpy as jnp
# Equipment costs (2024 dollars)
reactor_cost = econ.reactor_cost(jnp.array(5.0), "cstr_jacketed") # 5 m³
hx_cost = econ.heat_exchanger_cost(jnp.array(100.0), "shell_tube_floating") # 100 m²
pump_cost = econ.pump_cost(jnp.array(10.0), "centrifugal_single") # 10 kW
# Installed cost with Lang factor
installed = econ.installed_cost(reactor_cost, lang_factor=4.74)
# Total capital investment
tci = econ.total_capital_investment(
purchased_equipment_cost=reactor_cost + hx_cost + pump_cost,
lang_factor=4.74,
working_capital_fraction=0.15,
)
Available equipment types:
Reactors: CSTR (jacketed, coil), PFR, batch
Vessels: Pressure vessels, storage tanks, flash drums
Heat Exchangers: Shell-tube, double-pipe, plate-frame, air coolers
Columns: Tray columns, packed columns
Pumps: Centrifugal, reciprocating, gear
Compressors: Centrifugal, reciprocating, screw
Separators: Mixer-settlers, centrifuges, filters, extraction columns
Utility Costs#
# Steam cost from heat duty
heating_cost = econ.steam_cost_from_duty(jnp.array(1e6), "medium_pressure") # 1 MW
# Cooling water
cooling_cost = econ.cooling_water_cost(jnp.array(500e3)) # 500 kW
# Electricity
electricity_cost = econ.electricity_cost(jnp.array(100.0)) # 100 kW → $/hour
Profitability Metrics#
All metrics are JAX-differentiable:
# Net Present Value
cash_flows = jnp.ones(20) * 500000 # $500k/year for 20 years
npv = econ.npv(cash_flows, jnp.array(0.10), jnp.array(2e6)) # 10% discount, $2M investment
# Internal Rate of Return
irr = econ.irr(cash_flows, jnp.array(2e6))
# Minimum Selling Price
msp = econ.minimum_selling_price(
total_annual_cost=jnp.array(1e6),
annual_production=jnp.array(50000.0), # kg/year
)
# Annualized cost for optimization
tac = econ.annualized_cost(
capital_cost=jnp.array(5e6),
annual_opex=jnp.array(1e6),
discount_rate=jnp.array(0.10),
plant_life=jnp.array(20.0),
)
Gradient-Based Economic Optimization#
import jax
def annual_profit(params):
V, T = params[0], params[1]
# Simulate process
outlet, info = simulate_reactor(V, T)
# Economics
capex = econ.reactor_cost(V, "cstr_jacketed")
installed = econ.installed_cost(capex)
utility_cost = econ.cooling_water_cost(jnp.abs(info["Q"]))
annual_utility = utility_cost * 8000 * 3600 # $/year
revenue = outlet["F_product"] * product_price * 8000 * 3600
crf = econ.capital_recovery_factor(jnp.array(0.10), jnp.array(20.0))
return revenue - annual_utility - installed * crf
# Optimize design for maximum profit
grad_profit = jax.grad(annual_profit)
# Use gradient for optimization...
Uncertainty Propagation#
Leverage JAX’s automatic differentiation for uncertainty quantification:
from difflow import linear_propagation, monte_carlo_propagation, sensitivity_analysis
# Define a process model
def reactor_model(params):
k = params['k0'] * jnp.exp(-params['Ea'] / (8.314 * params['T']))
conversion = 1 - jnp.exp(-k * params['tau'])
return conversion
nominal = {'k0': jnp.array(1e6), 'Ea': jnp.array(50000.0),
'T': jnp.array(350.0), 'tau': jnp.array(100.0)}
uncertainties = {'k0': 1e5, 'Ea': 2000.0, 'T': 5.0, 'tau': 10.0}
# Linear (Jacobian-based) propagation - fast, first-order approximation
mean, std, info = linear_propagation(reactor_model, nominal, uncertainties)
print(f"Conversion: {mean:.3f} ± {std:.3f}")
print(f"Variance by input: {info['variance']}")
# Monte Carlo propagation - handles non-linear models
mean_mc, std_mc, info_mc = monte_carlo_propagation(
reactor_model, nominal, uncertainties, n_samples=10000
)
# Sensitivity analysis with gradient information
sens = sensitivity_analysis(reactor_model, nominal)
# Returns: gradient, elasticity (normalized sensitivity), variance contribution
Available functions:
linear_propagation(): First-order Jacobian-based uncertainty propagationmonte_carlo_propagation(): Parallel sampling using JAX vmapsensitivity_analysis(): Local gradient-based sensitivity with variance contributionssobol_indices(): Global sensitivity via Sobol samplingpropagate_covariance(): Full covariance matrix propagation for correlated inputs
Flowsheets with Recycles#
from difflow import (Flowsheet, Unit, make_stream, IdealThermo, CSTR, CSTRParams,
Mixer, Splitter, mass_action_kinetics)
from difflow.database import get_species_data
# n-hexane -> 2-methylpentane in a CSTR, 80 % of the effluent recycled
species = ["n_hexane", "2_methylpentane"]
thermo = IdealThermo({s: get_species_data(s) for s in species})
kin = mass_action_kinetics(
[{"reactants": {species[0]: 1.0}, "products": {species[1]: 1.0},
"rate_params": {"A": 0.05}}], species_order=species)
reactor = CSTR(CSTRParams(V=1.0, molar_density=10.0, **kin.params_kwargs()), thermo)
fs = Flowsheet(species_order=species)
fs.add_feed("fresh", make_stream({species[0]: 1.0, species[1]: 0.0}, T=340.0, P=101325.0))
fs.add_unit(Unit("mixer", Mixer(species, thermo), ["fresh", "recycle"], ["reactor_feed"]))
fs.add_unit(Unit("reactor", reactor, ["reactor_feed"], ["effluent"], params={"T_spec": 340.0}))
fs.add_unit(Unit("split", Splitter(species), ["effluent"], ["recycle", "product"],
params={"split_frac": 0.8}))
fs.add_recycle("recycle", "recycle") # tear the splitter outlet
streams = fs.solve(tol=1e-8)
Building Flowsheets Without Code#
A flowsheet is a graph with numbers on it. Writing that as Python is the flexible route, not the only one: difflow can also describe a model as data, edit it in a browser, write it back out as a script, and publish it as a page that needs nothing installed.
The operation catalog#
difflow.catalog answers what you can do with a unit: how many streams go in
and out, what parameters it takes, which are required, and which hold code rather
than data.
from difflow import catalog, describe_operation
spec = describe_operation("Heater")
spec.ports.inlets # ['inlet']
spec.ports.n_outlets # 1
spec.required_parameters() # [] -- every field has a default
spec.equations # LaTeX governing equations
spec.is_buildable # True: constructible from data alone
spec.to_dict() # JSON-serializable, for a UI or code generator
describe_operation("Flash").is_buildable # False
describe_operation("Flash").constructor_extras # ['thermo'] -- an object, not data
All of it is derived by introspection (parameters from dataclasses.fields,
ports from the __call__ signature), so it cannot drift from the code, and
plugin units appear with no extra work.
JSON round trip and code generation#
from difflow import serialize, codegen
serialize.save(fs, "plant.json") # flowsheet -> JSON
fs = serialize.load("plant.json") # JSON -> flowsheet
print(codegen.to_python(fs)) # flowsheet -> runnable script
Rate laws are the usual obstacle to writing a reactor down as data, because a
callable is code. mass_action_kinetics builds one from plain dictionaries
instead, so a reaction network can be stored and edited as data.
The local editor#
difflow with no arguments opens a flowsheet editor in the browser, served on
localhost with the installed package doing the solving:
difflow # an empty canvas
difflow gui plant.json # ...on a flowsheet
difflow gui plant.py # ...on one a script builds (saves plant.json beside it)
difflow gui --port 9000 --no-browser
python -m difflow.gui plant.json # where the console script is not on PATH
from difflow import gui
gui.serve(fs, path="plant.json") # on a flowsheet you already have
Units are dragged from a palette of everything registered and wired on the
canvas; feeds are filled in from the inspector. Solving shows the stream table,
the recycle solver’s own diagnostics, and derivatives: pin a lever for one
jax.jvp over every stream, or pin an output for one jax.grad over every
lever. About a third of the registered operations (34 of 87) need something no
form can supply, such as a thermo object or a rate law; their palette entries
are dimmed with the reason, and a short Python code context in the editor can
define them. It is single-user and a development tool: do not expose it to a
network.
Publishing a model#
difflow.publish turns a flowsheet into a self-contained HTML page anyone can
open with nothing installed, the form a model needs for supplementary material
or a project page.
from difflow import publish, SweepAxis
publish(
fs,
axes=[SweepAxis("reactor.V", 0.5, 5.0, n=21, label="Reactor volume", units="m³")],
outputs={"isomer_out": lambda streams: streams["product"]["F_2_methylpentane"]},
path="model.html",
)
JAX has no WebAssembly build, so the browser cannot run the real solver. Instead
the solve is pre-computed: the flowsheet is evaluated on a grid with jax.vmap,
with exact gradients from jax.grad, and both are baked into the page, which
interpolates between grid points and shows local sensitivities. It is exact at
the grid points and only as good as the grid between them, and it can vary only
what the axes name.
See docs/streams-and-flowsheets.md for the
full documentation.
Examples#
Jupyter notebooks are in the examples/ directory:
Notebook |
Description |
|---|---|
|
CSTR and PFR basics: conventional vs difflow |
|
Complete flowsheet with CSTR, flash, and recycle |
|
Sensitivity analysis for CSTR parameters |
|
Gradient-based optimization problems |
|
Rare earth recovery using LLE |
|
Comprehensive TEA with profit optimization |
|
Uncertainty propagation and sensitivity analysis |
|
Heat exchanger design, rating, and optimization |
|
Dynamic simulation, DAE systems, diffrax backend |
# Launch Jupyter to explore examples
jupyter notebook examples/
Tutorials#
The tutorials/ directory contains comprehensive JAX tutorials for differentiable programming:
Notebook |
Topics |
|---|---|
|
grad, jit, vmap, pytrees, jacfwd/jacrev, VJP/JVP, HVP |
|
IHVP, conjugate gradient, Newton-CG optimization |
|
Gradient descent, Newton, Adam, constrained optimization |
|
ODE solvers, parameter estimation, neural ODEs |
|
custom_vjp, custom_jvp, stop_gradient |
|
Neural networks from scratch, training loops |
|
Common JAX pitfalls and how to avoid them |
Key Design Decisions#
Streams as Dicts: Simple
{"F_A": ..., "F_B": ..., "T": ..., "P": ...}format that’s a JAX pytree by defaultProperty Database Available: Built-in database with 55+ species, or define custom species data
Function-Based Kinetics: Maximum flexibility via
rate_fn(C, T, params) → ratesUnrolled Iteration: Fixed-point solvers use
lax.scanfor automatic differentiabilityContinuous Relaxation: Discrete parameters (like n_stages) can be relaxed to continuous values for optimization
Dynamic Modeling#
The difflow.dynamic module provides a unified framework for transient simulation of process units:
Basic ODE Integration#
from difflow.dynamic import integrate
import jax.numpy as jnp
# Define any ODE system
def harmonic_oscillator(t, y):
x, v = y[0], y[1]
return jnp.array([v, -x]) # dx/dt = v, dv/dt = -x
result = integrate(
harmonic_oscillator,
y0=jnp.array([1.0, 0.0]),
t_span=(0.0, 10.0),
method="RK4", # or "RK45", "Euler"
)
print(f"Final state: {result.y_final}")
print(f"Trajectory shape: {result.trajectory.y.shape}")
Dynamic Unit Operations#
from difflow.dynamic import DynamicCSTR, integrate_unit
from difflow.streams import make_stream
# Define reaction kinetics
def rate_fn(C, T, params):
k = params["k"] * jnp.exp(-params["Ea"] / (8.314 * T))
return jnp.array([k * C["A"]])
# Create dynamic CSTR
cstr = DynamicCSTR(
volume=1.0,
rate_fn=rate_fn,
stoich=jnp.array([[-1], [1]]), # A -> B
species_order=["A", "B"],
rate_params={"k": 1e6, "Ea": 50000.0},
)
# Simulate startup from empty
inlet = make_stream({"A": 1.0, "B": 0.0}, T=350.0, P=101325.0)
result = integrate_unit(
cstr,
inputs={"inlet": inlet},
t_span=(0.0, 1000.0),
method="RK4",
)
Dynamic Flowsheets#
Connect multiple dynamic units for multi-unit transient simulation:
from difflow.dynamic import DynamicFlowsheet, DynamicCSTR, DynamicTank
def rate_fn(C, T, params): # A -> B, first order
k = params["k0"] * jnp.exp(-params["Ea"] / (8.314 * T))
return jnp.array([k * C["A"]])
stoich = jnp.array([[-1.0], [1.0]])
cstr = DynamicCSTR(volume=1.0, rate_fn=rate_fn, stoich=stoich, species_order=["A", "B"],
rate_params={"k0": 1e6, "Ea": 50000.0}, name="reactor")
tank = DynamicTank(max_volume=10.0, species_order=["A", "B"], name="storage")
inlet_stream = make_stream({"A": 1.0, "B": 0.0}, T=350.0, P=101325.0)
# Build flowsheet
fs = DynamicFlowsheet(species_order=["A", "B"])
fs.add_feed("feed", inlet_stream)
fs.add_unit(cstr, inlet_names=["feed"], outlet_names=["reactor_out"])
fs.add_unit(tank, inlet_names=["reactor_out"], outlet_names=["product"])
# Simulate entire flowsheet
result = fs.simulate(t_span=(0.0, 1000.0), method="RK4", n_steps=500)
DAE (Differential-Algebraic Equations)#
For systems with algebraic constraints (e.g., VLE equilibrium):
from difflow.dynamic import DynamicFlashDrum, integrate_dae
# Flash drum with VLE equilibrium constraint
flash = DynamicFlashDrum(
volume=1.0,
species_order=["A", "B"],
K_func=lambda T: jnp.array([2.0, 0.5]), # K = y/x for (A, B)
)
feed = make_stream({"A": 0.5, "B": 0.5}, T=350.0, P=101325.0)
result = integrate_dae(
flash,
inputs={"inlet": feed},
t_span=(0.0, 100.0),
method="RK4",
)
# result.x_final: differential states (moles)
# result.z_final: algebraic states (vapor fraction)
Diffrax Backend (Advanced Solvers)#
For stiff systems or when adaptive step control is needed:
pip install diffrax # Optional dependency
from difflow.dynamic import integrate
# A stiff linear system, dy/dt = -1000 (y - cos t)
stiff_ode = lambda t, y: -1000.0 * (y - jnp.cos(t))
y0 = jnp.array([0.0])
t_span = (0.0, 1.0)
# Use diffrax solvers via method string
result = integrate(
stiff_ode, y0, t_span,
method="diffrax:kvaerno5", # Implicit solver for stiff systems
rtol=1e-6, atol=1e-8,
)
# Available solvers: dopri5, tsit5, dopri8, kvaerno3/4/5, euler, heun
# Default: tsit5 (recommended for most problems)
See docs/dynamic-modeling.md for complete documentation.
Limitations#
Rigorous distillation column convergence can be sensitive to initial guesses
Gradient explosion possible with many iterations (use damping)
EOS flash limited to two-phase VLE (no three-phase VLLE yet)
Future Work#
Three-phase (VLLE) flash calculations
Extended bio operations (viral inactivation, sterile filtration)
GPU acceleration for large flowsheets
Integration with experiment databases (e.g., Cantera)
Citation#
If you use difflow in your work, please cite it. Machine-readable metadata is in
CITATION.cff,
and GitHub’s “Cite this repository” button will render BibTeX or APA from it.
License#
MIT