difflow

Differentiable Flowsheet Framework for Chemical Processes

A JAX-based framework for building and optimizing chemical process flowsheets with automatic differentiation.

• Source code: https://github.com/jkitchin/differentiable-flowsheets
• Documentation: https://kitchingroup.cheme.cmu.edu/differentiable-flowsheets/
• Issue tracker: https://github.com/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 model
• Agent Tools (MCP): difflow mcp serves 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
	difflow_bio	Biomanufacturing
	difflow_ree	Rare earth separations
	difflow_cc	Carbon capture
	difflow_gas	Gas transmission networks
	difflow_power	Electrical grids
	difflow_refinery	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]"     # 0.4.0 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.

Agent skills, commands and hooks

The repository also ships Agent Skills for any compatible agent (Claude Code, Codex, OpenCode, Gemini CLI, Cursor, Copilot), plus Claude Code slash commands, a session hook and the MCP server config, in the layout knowhow and npx skills install from:

Item	Name	What it does
skill	difflow	Build, solve, differentiate and debug flowsheets with difflow
skill	difflow-add-unit	Add a unit operation to the difflow codebase by its conventions
skill	difflow-gradient-check	Check gradients against finite differences; chase NaNs
command	test, test-unit, lint, example, optimize	Claude Code slash commands for working on difflow
hook	difflow-session-start	Session-start status: environment, difflow import, JAX backend
mcp	difflow, difflow-no-exec	The difflow mcp server, with or without the Python tools

uvx knowhow add jkitchin/differentiable-flowsheets --list
uvx knowhow add jkitchin/differentiable-flowsheets -s difflow -g    # for every project
uvx knowhow add jkitchin/differentiable-flowsheets --mcp difflow -a claude-code
# or: npx skills add jkitchin/differentiable-flowsheets --skill difflow

They live in skills/, commands/, hooks/ and mcp/; check them with uvx knowhow validate ..

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 a thermo for 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 propagation
• monte_carlo_propagation(): Parallel sampling using JAX vmap
• sensitivity_analysis(): Local gradient-based sensitivity with variance contributions
• sobol_indices(): Global sensitivity via Sobol sampling
• propagate_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
00_cstr_pfr_basics.ipynb	CSTR and PFR basics: conventional vs difflow
01_cstr_flash_recycle.ipynb	Complete flowsheet with CSTR, flash, and recycle
02_cstr_sensitivity.ipynb	Sensitivity analysis for CSTR parameters
03_optimization.ipynb	Gradient-based optimization problems
04_rare_earth_extraction.ipynb	Rare earth recovery using LLE
05_technoeconomic_analysis.ipynb	Comprehensive TEA with profit optimization
06_uncertainty_propagation.ipynb	Uncertainty propagation and sensitivity analysis
07_heat_exchangers.ipynb	Heat exchanger design, rating, and optimization
47_evaporators.ipynb	Single/multiple-effect evaporators, MVR, optimal number of effects
64_serialization.ipynb	Save/load flowsheets as JSON, refs=/extras=, export as Python
10_dynamic_modeling.ipynb	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
01_jax_fundamentals.ipynb	grad, jit, vmap, pytrees, jacfwd/jacrev, VJP/JVP, HVP
02_inverse_hessian_vector_products.ipynb	IHVP, conjugate gradient, Newton-CG optimization
02_optimization.ipynb	Gradient descent, Newton, Adam, constrained optimization
03_differential_equations.ipynb	ODE solvers, parameter estimation, neural ODEs
04_custom_derivatives.ipynb	custom_vjp, custom_jvp, stop_gradient
05_machine_learning.ipynb	Neural networks from scratch, training loops
06_gotchas.ipynb	Common JAX pitfalls and how to avoid them

Key Design Decisions

1. Streams as Dicts: Simple {"F_A": ..., "F_B": ..., "T": ..., "P": ...} format that's a JAX pytree by default

2. Property Database Available: Built-in database with 55+ species, or define custom species data

3. Function-Based Kinetics: Maximum flexibility via rate_fn(C, T, params) → rates

4. Unrolled Iteration: Fixed-point solvers use lax.scan for automatic differentiability

5. Continuous 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