thermofluids-ai999x — offline, verified thermofluids solver with step-by-step answers

Install · Capability guide · Evaluation · AI999X on Hugging Face · Website · Support: Buy Me a Coffee

License: Apache-2.0 · Author: Ibnul Jaif Farabi (creator and engineer) · Organization: AI999X · Version 1.0.0

Field: Engineering · Tasks: Question Answering, Token Classification, Text Classification, Sentence Similarity, Feature Extraction

thermofluids-ai999x is an offline, deterministic Python solver that answers thermofluids questions with verified results from 126 certified capabilities, gives 1010 documented answers identically on Windows and Linux, and refuses rather than guesses when an answer cannot be verified.

thermofluids-ai999x is a compact, deterministic thermofluids engine for Python. It reads a question in plain English (or a typed call), picks a certified thermofluids capability, computes the answer exactly, checks it, and returns the result with its steps, units, LaTeX and assumptions — or an honest refusal that says what is missing. It runs fully offline on any CPU, needs no GPU, no API key and no internet, and is built to give zero wrong answers: when it is not sure, it refuses instead of guessing.

Highlights

• Verified, not generated. Every answer is computed by a certified thermofluids capability and checked (substituted back, dimension-checked or compared with an independent computation) before it is returned.
• Zero wrong answers by design. Seven explicit outcomes — SOLVED, NEEDS_INPUT, AMBIGUOUS, INVALID_INPUT, OUT_OF_SCOPE, UNRECOGNIZED_FORMAT, INTERNAL_ERROR — never a silent guess. Every one of its 1010 documented answers is reproduced exactly on every tested platform.
• Plain-English questions and a guaranteed typed interface. ask() reads documented question forms; solve(capability_id, **inputs) always solves a supported task with valid inputs.
• Explains itself. Steps, formulas in LaTeX, units, assumptions and the model used, ready for notebooks, tutoring apps and homework checkers.
• Small and portable. Pure Python (sympy, mpmath); identical answers on Windows and Linux (Python 3.10, 3.11, 3.12, 3.13, 3.14).
• Coverage: fluid statics and dynamics, pipe flow, boundary layers, aerodynamics and compressible flow, heat conduction, convection and radiation, fins, heat exchangers, thermodynamic cycles, psychrometrics and combustion.

Model overview

Property	Value
Package	thermofluids-ai999x 1.0.0 (import thermofluids_ai999x)
Type	Deterministic neuro-symbolic thermofluids engine (rule-based reading + certified symbolic capabilities)
Capabilities	27 certified derivations, 13 certified procedures, 86 certified relations
Knowledge pack	version 0.1.1
Golden answers	1010 documented answers, identical on every tested platform
Platforms tested	Windows (Python 3.13), Linux x86-64 (Python 3.10, 3.11, 3.12, 3.13, 3.14)
Runtime	CPU only, offline; no GPU, no network, no API key
License	Apache-2.0
Creator / engineer	Ibnul Jaif Farabi
Organization	AI999X

Quickstart

pip install thermofluids-ai999x
# or straight from this repository:
pip install https://huggingface.co/AI999X/thermofluids-ai999x/resolve/main/dist/thermofluids_ai999x-1.0.0-py3-none-any.whl

import thermofluids_ai999x as thermofluids

thermofluids.ask("Water with density 998 kg/m^3 and dynamic viscosity 0.001 Pa*s flows at an average velocity of 1.5 m/s in a pipe of diameter 0.05 m. What is the Reynolds number?").show()
# -> SOLVED — 74850 1  (reynolds_number)
thermofluids.ask("Oil of density 900 kg/m^3 and viscosity 0.1 Pa*s flows in laminar flow at a mean velocity of 0.5 m/s in a pipe of diameter 0.02 m. What is the friction factor?").show()
# -> SOLVED — 0.711111 1  (reynolds_number, laminar_friction_factor)
thermofluids.ask("An air-standard Otto cycle has a compression ratio of 8. What is its thermal efficiency?").show()
# -> SOLVED — 0.564725 1  (otto_efficiency_air)
thermofluids.ask("Dew point temperature: T = 25 \u00b0C, RH = 60%").show()
# -> SOLVED — T_dew = 289.851, p_v = 1901.53  (dew_point_temperature)
thermofluids.solve("lmtd", Th_in="150 \u00b0C", Th_out="90 \u00b0C", Tc_in="30 \u00b0C", Tc_out="70 \u00b0C", arrangement="counterflow").show()
# -> SOLVED — lmtd = 69.5212, dT1 = 80, dT2 = 60  (lmtd)
thermofluids.ask("Who created you?").message
# -> thermofluids-ai999x was created and engineered by Ibnul Jaif Farabi at AI999X.

Command line: thermofluids-ai999x "<question>", thermofluids-ai999x --capabilities. The outputs above were produced by running this exact code on thermofluids-ai999x 1.0.0.

How every answer is verified

1. Reading — the question is turned into typed quantities (value, unit, kind, role) and one asked quantity; every stated number must be used or the question is refused.
2. Planning — only certified capabilities may answer; a capability applies only under the conditions the wording states (for example a pendulum, a liquid at rest, boiling).
3. Computing — exact symbolic or high-precision numeric evaluation.
4. Checking — dimensions, substitution back into the defining equation, and agreement between independent readings; any disagreement becomes a refusal, never a number.

Capabilities are certified before they ship: each must reproduce hand-worked reference cases, reject a deliberately wrong variant (a control), and leave every benchmark answer unchanged.

Capabilities

27 certified derivations, 13 certified procedures, 86 certified relations. A few of them:

• Derivations: derive bernoulli horizontal; derive blasius local skin friction; derive blasius momentum thickness; derive brayton efficiency air; derive buoyant force; derive continuity speed; derive darcy weisbach head loss; derive darcy weisbach pressure drop; derive diesel efficiency air; derive dynamic pressure; derive effectiveness phase change; derive humidity ratio vapour pressure; derive hydraulic power; derive hydrostatic force plane; derive minor loss head; derive nusselt heat transfer coefficient; derive otto efficiency air; derive overall heat rate; derive overall u plane wall; derive plane wall conduction; derive pump shaft power; derive radiation net exchange; derive radiation parallel plates; derive reynolds number …
• Procedures: Churchill-Bernstein correlation for a cylinder in cross flow; Churchill-Chu correlation for natural convection on a vertical plate; Darcy friction factor from the Colebrook equation; Dew-point temperature from dry-bulb temperature and relative humidity; Efficiency of a pin fin with an insulated tip; Equivalence ratio of a fuel-air mixture; Gnielinski correlation for turbulent pipe flow; Heat exchanger rating by the effectiveness-NTU method; Humidity ratio from temperature, relative humidity and pressure; Log-mean temperature difference of a heat exchanger; Saturation pressure of water vapour; Stoichiometric air-fuel ratio of a gaseous fuel; Thermodynamic wet-bulb temperature
• Relations: Absolute pressure at a depth in a liquid; Actual outlet temperature of an adiabatic air compressor; Adiabatic flame temperature with constant product specific heat; Average Nusselt number for laminar flow over a flat plate; Average skin-friction coefficient of a laminar flat plate; Bernoulli's equation along a horizontal streamline; Biot number; Blasius friction factor for turbulent flow in a smooth pipe; Buoyant force (Archimedes' principle); Carnot efficiency of a reversible heat engine; Coefficient of performance of a Carnot heat pump; Coefficient of performance of a Carnot refrigerator; Continuity: speed after a change of pipe diameter; Critical radius of insulation on a cylinder; Darcy friction factor in laminar pipe flow; Darcy-Weisbach head loss in a pipe; Darcy-Weisbach pressure drop in a pipe; Displacement thickness of a laminar flat-plate boundary layer; Dittus-Boelter correlation, fluid being cooled; Dittus-Boelter correlation, fluid being heated; Drag force from the drag coefficient; Dynamic pressure of a moving fluid; Effectiveness of a heat exchanger with one stream changing phase; Effectiveness of a parallel-flow heat exchanger …

thermofluids_ai999x.templates() lists every question form ask() reads; the capability guide (PDF, every example executed) is included in this repository.

All 99 formulas

Show every formula, by topic

Aerodynamics

Lift coefficient of a thin symmetric airfoil

$$C_L=2\pi \alpha C\_\{L\}\; =\; 2\; \backslash pi\; \backslash alpha$$

Boundary layers

Average skin-friction coefficient of a laminar flat plate

$$\text{cfm}=\frac{1.3282293448607851}{\sqrt{\text{ReL}}}\backslash text\{cfm\}\; =\; \backslash frac\{1.3282293448607851\}\{\backslash sqrt\{\backslash text\{ReL\}\}\}$$

Displacement thickness of a laminar flat-plate boundary layer

$$d_s=\frac{1.7207876575205x}{\sqrt{\text{Rex}}}d\_\{s\}\; =\; \backslash frac\{1.7207876575205\; x\}\{\backslash sqrt\{\backslash text\{Rex\}\}\}$$

Local skin-friction coefficient of a laminar flat plate

$$c_f=\frac{0.6641146724303926}{\sqrt{\text{Rex}}}c\_\{f\}\; =\; \backslash frac\{0.6641146724303926\}\{\backslash sqrt\{\backslash text\{Rex\}\}\}$$

Momentum thickness of a laminar flat-plate boundary layer

$$t_h=\frac{0.6641146724303926x}{\sqrt{\text{Rex}}}t\_\{h\}\; =\; \backslash frac\{0.6641146724303926\; x\}\{\backslash sqrt\{\backslash text\{Rex\}\}\}$$

Combustion

Adiabatic flame temperature with constant product specific heat

$$\text{Tad}=T_0+\frac{\text{LHV}}{(1+\text{AFR})c_p}\backslash text\{Tad\}\; =\; T\_\{0\}\; +\; \backslash frac\{\backslash text\{LHV\}\}\{\backslash left(1\; +\; \backslash text\{AFR\}\backslash right)\; c\_\{p\}\}$$

Combustion / mixture strength

Equivalence ratio of a fuel-air mixture

$$\varphi =\frac{AF{R}_s}{AFR}\backslash phi\; =\; \backslash frac\{AFR\_s\}\{AFR\}$$

Combustion / stoichiometry

Stoichiometric air-fuel ratio of a gaseous fuel

$$AF{R}_s=\frac{(x+y\mathrm{/}4)(M_{O_2}+3.76M_{N_2})}{x{M}_C+y{M}_H}AFR\_s\; =\; \backslash frac\{(x\; +\; y/4)(M\_\{O\_2\}\; +\; 3.76\; M\_\{N\_2\})\}\{x\; M\_C\; +\; y\; M\_H\}$$

Compressible flow

Isentropic stagnation pressure of air from the Mach number

$$P_0=P{(1+\frac{M^2}{5})}^{\frac{7}{2}}P\_\{0\}\; =\; P\; \backslash left(1\; +\; \backslash frac\{M^\{2\}\}\{5\}\backslash right)^\{\backslash frac\{7\}\{2\}\}$$

Mach number

$$M=\frac{v}{a}M\; =\; \backslash frac\{v\}\{a\}$$

Mach number behind a normal shock in air

$$M_2=\sqrt{\frac{M_1^2+5}{7M_1^2-1}}M\_\{2\}\; =\; \backslash sqrt\{\backslash frac\{M\_\{1\}^\{2\}\; +\; 5\}\{7\; M\_\{1\}^\{2\}\; -\; 1\}\}$$

Speed of sound in an ideal gas

$$a=\sqrt{\gamma RT}a\; =\; \backslash sqrt\{\backslash gamma\; R\; T\}$$

Stagnation temperature from the Mach number

$$T_0=T(1+\frac{(\gamma -1)M^2}{2})T\_\{0\}\; =\; T\; \backslash left(1\; +\; \backslash frac\{\backslash left(\backslash gamma\; -\; 1\backslash right)\; M^\{2\}\}\{2\}\backslash right)$$

Static pressure behind a normal shock in air

$$P_2=\frac{P_1(7M_1^2-1)}{6}P\_\{2\}\; =\; \backslash frac\{P\_\{1\}\; \backslash left(7\; M\_\{1\}^\{2\}\; -\; 1\backslash right)\}\{6\}$$

Conduction

Critical radius of insulation on a cylinder

$$r=\frac{k}{h}r\; =\; \backslash frac\{k\}\{h\}$$

Fourier's law: heat conduction through a plane wall

$$q=\frac{kA(T_1-T_2)}{L}q\; =\; \backslash frac\{k\; A\; \backslash left(T\_\{1\}\; -\; T\_\{2\}\backslash right)\}\{L\}$$

Heat transfer rate from the overall heat transfer coefficient

$$q=UA(T_i-T_o)q\; =\; U\; A\; \backslash left(T\_\{i\}\; -\; T\_\{o\}\backslash right)$$

Maximum temperature of a plane wall with uniform heat generation

$$T_{\text{max}}=T_s+\frac{q{L}^2}{2k}T\_\{\backslash text\{max\}\}\; =\; T\_\{s\}\; +\; \backslash frac\{q\; L^\{2\}\}\{2\; k\}$$

Overall heat transfer coefficient of a plane wall with convection on both sides

$$U=\frac{1}{\frac{1}{h_1}+\frac{L}{k}+\frac{1}{h_2}}U\; =\; \backslash frac\{1\}\{\backslash frac\{1\}\{h\_\{1\}\}\; +\; \backslash frac\{L\}\{k\}\; +\; \backslash frac\{1\}\{h\_\{2\}\}\}$$

Radial heat conduction through a cylindrical wall

$$q=\frac{2\pi kL(T_1-T_2)}{\ln \left(\frac{r_2}{r_1}\right)}q\; =\; \backslash frac\{2\; \backslash pi\; k\; L\; \backslash left(T\_\{1\}\; -\; T\_\{2\}\backslash right)\}\{\backslash ln\backslash left(\backslash frac\{r\_\{2\}\}\{r\_\{1\}\}\backslash right)\}$$

Radial heat conduction through a spherical shell

$$q=\frac{4\pi k(T_1-T_2)}{\frac{1}{r_1}-\frac{1}{r_2}}q\; =\; \backslash frac\{4\; \backslash pi\; k\; \backslash left(T\_\{1\}\; -\; T\_\{2\}\backslash right)\}\{\backslash frac\{1\}\{r\_\{1\}\}\; -\; \backslash frac\{1\}\{r\_\{2\}\}\}$$

Thermal diffusivity

$$\alpha =\frac{k}{\rho c_p}\backslash alpha\; =\; \backslash frac\{k\}\{\backslash rho\; c\_\{p\}\}$$

Convection

Average Nusselt number for laminar flow over a flat plate

$$\mathrm{N}\mathrm{u}=0.664{\text{ReL}}^{\frac{1}{2}}{\mathrm{P}\mathrm{r}}^{\frac{1}{3}}\backslash mathrm\{Nu\}\; =\; 0.664\; \backslash text\{ReL\}^\{\backslash frac\{1\}\{2\}\}\; \backslash mathrm\{Pr\}^\{\backslash frac\{1\}\{3\}\}$$

Dittus-Boelter correlation, fluid being cooled

$$\mathrm{N}\mathrm{u}=0.023{\mathrm{R}\mathrm{e}}^{\frac{4}{5}}{\mathrm{P}\mathrm{r}}^{\frac{3}{10}}\backslash mathrm\{Nu\}\; =\; 0.023\; \backslash mathrm\{Re\}^\{\backslash frac\{4\}\{5\}\}\; \backslash mathrm\{Pr\}^\{\backslash frac\{3\}\{10\}\}$$

Dittus-Boelter correlation, fluid being heated

$$\mathrm{N}\mathrm{u}=0.023{\mathrm{R}\mathrm{e}}^{\frac{4}{5}}{\mathrm{P}\mathrm{r}}^{\frac{2}{5}}\backslash mathrm\{Nu\}\; =\; 0.023\; \backslash mathrm\{Re\}^\{\backslash frac\{4\}\{5\}\}\; \backslash mathrm\{Pr\}^\{\backslash frac\{2\}\{5\}\}$$

Heat transfer coefficient from the Nusselt number

$$h=\frac{\mathrm{N}\mathrm{u}k}{L_c}h\; =\; \backslash frac\{\backslash mathrm\{Nu\}\; k\}\{L\_\{c\}\}$$

Local Nusselt number for laminar flow over a flat plate

$$\text{Nux}=0.332{\text{Rex}}^{\frac{1}{2}}{\mathrm{P}\mathrm{r}}^{\frac{1}{3}}\backslash text\{Nux\}\; =\; 0.332\; \backslash text\{Rex\}^\{\backslash frac\{1\}\{2\}\}\; \backslash mathrm\{Pr\}^\{\backslash frac\{1\}\{3\}\}$$

Newton's law of cooling (convection heat transfer)

$$q=hA(T_s-\text{Tinf})q\; =\; h\; A\; \backslash left(T\_\{s\}\; -\; \backslash text\{Tinf\}\backslash right)$$

Prandtl number

$$\mathrm{P}\mathrm{r}=\frac{c_p\mu }{k}\backslash mathrm\{Pr\}\; =\; \backslash frac\{c\_\{p\}\; \backslash mu\}\{k\}$$

Convection / external flow

Churchill-Bernstein correlation for a cylinder in cross flow

$$Nu=0.3+\frac{0.62R{e}^{1\mathrm{/}2}P{r}^{1\mathrm{/}3}}{[1+(0.4\mathrm{/}Pr{)}^{2\mathrm{/}3}{]}^{1\mathrm{/}4}}{[1+{\left(\frac{Re}{282000}\right)}^{5\mathrm{/}8}]}^{4\mathrm{/}5}Nu\; =\; 0.3\; +\; \backslash frac\{0.62\backslash ,Re^\{1/2\}Pr^\{1/3\}\}\{[1\; +\; (0.4/Pr)^\{2/3\}]^\{1/4\}\}\backslash left[1\; +\; \backslash left(\backslash frac\{Re\}\{282000\}\backslash right)^\{5/8\}\backslash right]^\{4/5\}$$

Convection / internal flow

Gnielinski correlation for turbulent pipe flow

$$Nu=\frac{(f\mathrm{/}8)(Re-1000)Pr}{1+12.7\sqrt{f\mathrm{/}8}(P{r}^{2\mathrm{/}3}-1)}Nu\; =\; \backslash frac\{(f/8)(Re\; -\; 1000)Pr\}\{1\; +\; 12.7\backslash sqrt\{f/8\}\backslash ,(Pr^\{2/3\}\; -\; 1)\}$$

Convection / natural convection

Churchill-Chu correlation for natural convection on a vertical plate

$$Nu={(0.825+\frac{0.387R{a}^{1\mathrm{/}6}}{[1+(0.492\mathrm{/}Pr{)}^{9\mathrm{/}16}{]}^{8\mathrm{/}27}})}^{2}Nu\; =\; \backslash left(0.825\; +\; \backslash frac\{0.387\backslash ,Ra^\{1/6\}\}\{[1\; +\; (0.492/Pr)^\{9/16\}]^\{8/27\}\}\backslash right)^2$$

External flow

Drag force from the drag coefficient

$$F=\frac{C_d\rho v^{2}A}{2}F\; =\; \backslash frac\{C\_\{d\}\; \backslash rho\; v^\{2\}\; A\}\{2\}$$

Lift force from the lift coefficient

$$F=\frac{C_L\rho v^{2}A}{2}F\; =\; \backslash frac\{C\_\{L\}\; \backslash rho\; v^\{2\}\; A\}\{2\}$$

Stokes drag on a small sphere

$$F=3\pi \mu DvF\; =\; 3\; \backslash pi\; \backslash mu\; D\; v$$

Terminal settling velocity of a small sphere (Stokes)

$$v=\frac{g{D}^2({\rho }_p-\rho )}{18\mu }v\; =\; \backslash frac\{g\; D^\{2\}\; \backslash left(\backslash rho\_\{p\}\; -\; \backslash rho\backslash right)\}\{18\; \backslash mu\}$$

Fins

Heat transfer rate of a very long fin

$$q=\sqrt{hPk{A}_c}(T_b-\text{Tinf})q\; =\; \backslash sqrt\{h\; P\; k\; A\_\{c\}\}\; \backslash left(T\_\{b\}\; -\; \backslash text\{Tinf\}\backslash right)$$

Fins / extended surfaces

Efficiency of a pin fin with an insulated tip

$${\eta }_f=\frac{\tanh (mL)}{mL},m=\sqrt{\frac{4h}{kD}}\backslash eta\_f\; =\; \backslash frac\{\backslash tanh(mL)\}\{mL\},\backslash quad\; m\; =\; \backslash sqrt\{\backslash frac\{4h\}\{kD\}\}$$

Fluid dynamics

Bernoulli's equation along a horizontal streamline

$$P_2=P_1+\frac{\rho (v_1^2-v_2^2)}{2}P\_\{2\}\; =\; P\_\{1\}\; +\; \backslash frac\{\backslash rho\; \backslash left(v\_\{1\}^\{2\}\; -\; v\_\{2\}^\{2\}\backslash right)\}\{2\}$$

Continuity: speed after a change of pipe diameter

$$v_2=v_1{\left(\frac{D_1}{D_2}\right)}^{2}v\_\{2\}\; =\; v\_\{1\}\; \backslash left(\backslash frac\{D\_\{1\}\}\{D\_\{2\}\}\backslash right)^\{2\}$$

Dynamic pressure of a moving fluid

$$q=\frac{\rho v^2}{2}q\; =\; \backslash frac\{\backslash rho\; v^\{2\}\}\{2\}$$

Speed of efflux from a hole (Torricelli's law)

$$v=\sqrt{2gh}v\; =\; \backslash sqrt\{2\; g\; h\}$$

Fluid statics

Absolute pressure at a depth in a liquid

$$P=P_0+\rho ghP\; =\; P\_\{0\}\; +\; \backslash rho\; g\; h$$

Buoyant force (Archimedes' principle)

$$F=\rho gVF\; =\; \backslash rho\; g\; V$$

Gauge pressure at a depth in a liquid

$$P=\rho ghP\; =\; \backslash rho\; g\; h$$

Hydrostatic force on a submerged plane surface

$$F=\rho g{h}_{c}AF\; =\; \backslash rho\; g\; h\_\{c\}\; A$$

Gas processes

Actual outlet temperature of an adiabatic air compressor

$$T_2=T_1(1+\frac{{\left(\frac{P_2}{P_1}\right)}^{\frac{2}{7}}-1}{\eta })T\_\{2\}\; =\; T\_\{1\}\; \backslash left(1\; +\; \backslash frac\{\backslash left(\backslash frac\{P\_\{2\}\}\{P\_\{1\}\}\backslash right)^\{\backslash frac\{2\}\{7\}\}\; -\; 1\}\{\backslash eta\}\backslash right)$$

Entropy change of an incompressible substance

$$d_S=mc\ln \left(\frac{T_2}{T_1}\right)d\_\{S\}\; =\; m\; c\; \backslash ln\backslash left(\backslash frac\{T\_\{2\}\}\{T\_\{1\}\}\backslash right)$$

Ideal gas law

$$PV=nRTP\; V\; =\; n\; R\; T$$

Specific entropy change of an ideal gas

$$d_s=c_p\ln \left(\frac{T_2}{T_1}\right)-R\ln \left(\frac{P_2}{P_1}\right)d\_\{s\}\; =\; c\_\{p\}\; \backslash ln\backslash left(\backslash frac\{T\_\{2\}\}\{T\_\{1\}\}\backslash right)\; -\; R\; \backslash ln\backslash left(\backslash frac\{P\_\{2\}\}\{P\_\{1\}\}\backslash right)$$

Temperature after an isentropic pressure change of air

$$T_2=T_1{\left(\frac{P_2}{P_1}\right)}^{\frac{2}{7}}T\_\{2\}\; =\; T\_\{1\}\; \backslash left(\backslash frac\{P\_\{2\}\}\{P\_\{1\}\}\backslash right)^\{\backslash frac\{2\}\{7\}\}$$

Temperature after an isentropic volume change of air

$$T_2=T_1{\left(\frac{V_1}{V_2}\right)}^{\frac{2}{5}}T\_\{2\}\; =\; T\_\{1\}\; \backslash left(\backslash frac\{V\_\{1\}\}\{V\_\{2\}\}\backslash right)^\{\backslash frac\{2\}\{5\}\}$$

Work done by an ideal gas in a reversible isothermal process

$$W=nRT\ln \left(\frac{V_2}{V_1}\right)W\; =\; n\; R\; T\; \backslash ln\backslash left(\backslash frac\{V\_\{2\}\}\{V\_\{1\}\}\backslash right)$$

Heat exchangers / LMTD method

Log-mean temperature difference of a heat exchanger

$$\mathrm{\Delta }{T}_{lm}=\frac{\mathrm{\Delta }{T}_1-\mathrm{\Delta }{T}_2}{\ln (\mathrm{\Delta }{T}_1\mathrm{/}\mathrm{\Delta }{T}_2)}\backslash Delta\; T\_\{lm\}\; =\; \backslash frac\{\backslash Delta\; T\_1\; -\; \backslash Delta\; T\_2\}\{\backslash ln(\backslash Delta\; T\_1/\backslash Delta\; T\_2)\}$$

Heat exchangers / effectiveness-NTU

Heat exchanger rating by the effectiveness-NTU method

$$Q=\epsilon (NTU,C_r)C_{min}(T_{h,i}-T_{c,i})Q\; =\; \backslash varepsilon(NTU,\; C\_r)\backslash ,C\_\{min\}(T\_\{h,i\}\; -\; T\_\{c,i\})$$

Heat exchangers

Effectiveness of a heat exchanger with one stream changing phase

$$\epsilon =1-e^{-N}\backslash varepsilon\; =\; 1\; -\; e^\{-N\}$$

Effectiveness of a parallel-flow heat exchanger

$$\epsilon =\frac{1-e^{-N(1+C_r)}}{1+C_r}\backslash varepsilon\; =\; \backslash frac\{1\; -\; e^\{-N\; \backslash left(1\; +\; C\_\{r\}\backslash right)\}\}\{1\; +\; C\_\{r\}\}$$

Heat added to a flowing stream (steady-flow energy balance)

$$Q=\text{mdot}{c}_p(T_{\text{out}}-T_{\text{in}})Q\; =\; \backslash text\{mdot\}\; c\_\{p\}\; \backslash left(T\_\{\backslash text\{out\}\}\; -\; T\_\{\backslash text\{in\}\}\backslash right)$$

Heat transfer rate from the effectiveness

$$Q=\epsilon C_{\text{min}}(\text{Thi}-\text{Tci})Q\; =\; \backslash varepsilon\; C\_\{\backslash text\{min\}\}\; \backslash left(\backslash text\{Thi\}\; -\; \backslash text\{Tci\}\backslash right)$$

Number of transfer units of a heat exchanger

$$N=\frac{UA}{C_{\text{min}}}N\; =\; \backslash frac\{U\; A\}\{C\_\{\backslash text\{min\}\}\}$$

Natural convection

Grashof number

$$\mathrm{G}\mathrm{r}=\frac{g\beta (T_s-\text{Tinf})L^3}{{\nu }^2}\backslash mathrm\{Gr\}\; =\; \backslash frac\{g\; \backslash beta\; \backslash left(T\_\{s\}\; -\; \backslash text\{Tinf\}\backslash right)\; L^\{3\}\}\{\backslash nu^\{2\}\}$$

Rayleigh number

$$\mathrm{R}\mathrm{a}=\mathrm{G}\mathrm{r}\mathrm{P}\mathrm{r}\backslash mathrm\{Ra\}\; =\; \backslash mathrm\{Gr\}\; \backslash mathrm\{Pr\}$$

Open channel

Froude number of an open-channel flow

$$\mathrm{F}\mathrm{r}=\frac{v}{\sqrt{gh}}\backslash mathrm\{Fr\}\; =\; \backslash frac\{v\}\{\backslash sqrt\{g\; h\}\}$$

Pipe flow / friction factor

Darcy friction factor from the Colebrook equation

$$\frac{1}{\sqrt{f}}=-2{\log }_{10}(\frac{\epsilon \mathrm{/}D}{3.7}+\frac{2.51}{Re\sqrt{f}})\backslash frac\{1\}\{\backslash sqrt\{f\}\}\; =\; -2\backslash log\_\{10\}\backslash left(\backslash frac\{\backslash varepsilon/D\}\{3.7\}\; +\; \backslash frac\{2.51\}\{Re\backslash sqrt\{f\}\}\backslash right)$$

Pipe flow

Blasius friction factor for turbulent flow in a smooth pipe

$$f=0.3164{\mathrm{R}\mathrm{e}}^{\frac{-1}{4}}f\; =\; 0.3164\; \backslash mathrm\{Re\}^\{\backslash frac\{-1\}\{4\}\}$$

Darcy friction factor in laminar pipe flow

$$f=\frac{64}{\mathrm{R}\mathrm{e}}f\; =\; \backslash frac\{64\}\{\backslash mathrm\{Re\}\}$$

Darcy-Weisbach head loss in a pipe

$$h_L=\frac{fL{v}^2}{2gD}h\_\{L\}\; =\; \backslash frac\{f\; L\; v^\{2\}\}\{2\; g\; D\}$$

Darcy-Weisbach pressure drop in a pipe

$$d_P=\frac{fL\rho v^2}{2D}d\_\{P\}\; =\; \backslash frac\{f\; L\; \backslash rho\; v^\{2\}\}\{2\; D\}$$

Haaland explicit friction factor for turbulent pipe flow

$$f=\frac{1}{{\left(\frac{1.8\ln ({\left(\frac{\epsilon }{3.7D}\right)}^{\frac{111}{100}}+\frac{6.9}{\mathrm{R}\mathrm{e}})}{\ln \left(10\right)}\right)}^2}f\; =\; \backslash frac\{1\}\{\backslash left(\backslash frac\{1.8\; \backslash ln\backslash left(\backslash left(\backslash frac\{\backslash varepsilon\}\{3.7\; D\}\backslash right)^\{\backslash frac\{111\}\{100\}\}\; +\; \backslash frac\{6.9\}\{\backslash mathrm\{Re\}\}\backslash right)\}\{\backslash ln\backslash left(10\backslash right)\}\backslash right)^\{2\}\}$$

Hagen-Poiseuille flow rate in laminar pipe flow

$$Q=\frac{\pi D^4{d}_P}{128\mu L}Q\; =\; \backslash frac\{\backslash pi\; D^\{4\}\; d\_\{P\}\}\{128\; \backslash mu\; L\}$$

Kinematic viscosity from dynamic viscosity and density

$$\nu =\frac{\mu }{\rho }\backslash nu\; =\; \backslash frac\{\backslash mu\}\{\backslash rho\}$$

Minor head loss in a fitting

$$h_L=\frac{K{v}^2}{2g}h\_\{L\}\; =\; \backslash frac\{K\; v^\{2\}\}\{2\; g\}$$

Reynolds number from density, velocity, diameter and viscosity

$$\mathrm{R}\mathrm{e}=\frac{\rho vD}{\mu }\backslash mathrm\{Re\}\; =\; \backslash frac\{\backslash rho\; v\; D\}\{\backslash mu\}$$

Reynolds number from velocity, diameter and kinematic viscosity

$$\mathrm{R}\mathrm{e}=\frac{vD}{\nu }\backslash mathrm\{Re\}\; =\; \backslash frac\{v\; D\}\{\backslash nu\}$$

Swamee-Jain explicit friction factor for turbulent pipe flow

$$f=\frac{0.25}{{\left(\frac{\ln (\frac{\epsilon }{3.7D}+{\left(\frac{6.97}{\mathrm{R}\mathrm{e}}\right)}^{\frac{9}{10}})}{\ln \left(10\right)}\right)}^2}f\; =\; \backslash frac\{0.25\}\{\backslash left(\backslash frac\{\backslash ln\backslash left(\backslash frac\{\backslash varepsilon\}\{3.7\; D\}\; +\; \backslash left(\backslash frac\{6.97\}\{\backslash mathrm\{Re\}\}\backslash right)^\{\backslash frac\{9\}\{10\}\}\backslash right)\}\{\backslash ln\backslash left(10\backslash right)\}\backslash right)^\{2\}\}$$

Psychrometrics

Humidity ratio from the partial pressure of water vapour

$$W=\frac{0.621945p_v}{P-p_v}W\; =\; \backslash frac\{0.621945\; p\_\{v\}\}\{P\; -\; p\_\{v\}\}$$

Specific enthalpy of moist air

$$h=\text{cpa}(T-T_0)+W(h_g+\text{cpv}(T-T_0))h\; =\; \backslash text\{cpa\}\; \backslash left(T\; -\; T\_\{0\}\backslash right)\; +\; W\; \backslash left(h\_\{g\}\; +\; \backslash text\{cpv\}\; \backslash left(T\; -\; T\_\{0\}\backslash right)\backslash right)$$

Specific volume of moist air per kilogram of dry air

$$v=\frac{\mathrm{R}\mathrm{a}T(1+1.607858W)}{P}v\; =\; \backslash frac\{\backslash mathrm\{Ra\}\; T\; \backslash left(1\; +\; 1.607858\; W\backslash right)\}\{P\}$$

Psychrometrics / dew point

Dew-point temperature from dry-bulb temperature and relative humidity

$$p_{ws}(T_{dp})=\varphi p_{ws}(T)p\_\{ws\}(T\_\{dp\})\; =\; \backslash phi\backslash ,p\_\{ws\}(T)$$

Psychrometrics / humidity ratio

Humidity ratio from temperature, relative humidity and pressure

$$W=0.621945\frac{\varphi p_{ws}}{P-\varphi p_{ws}}W\; =\; 0.621945\backslash ,\backslash frac\{\backslash phi\; p\_\{ws\}\}\{P\; -\; \backslash phi\; p\_\{ws\}\}$$

Psychrometrics / saturation

Saturation pressure of water vapour

$$\ln p_{ws}=C_8\mathrm{/}T+C_9+C_{10}T+C_{11}{T}^2+C_{12}{T}^3+C_{13}\ln T\backslash ln\; p\_\{ws\}\; =\; C\_8/T\; +\; C\_9\; +\; C\_\{10\}T\; +\; C\_\{11\}T^2\; +\; C\_\{12\}T^3\; +\; C\_\{13\}\backslash ln\; T$$

Psychrometrics / wet bulb

Thermodynamic wet-bulb temperature

$$W=\frac{(2501-2.326t^{\ast })W_s^{\ast }-1.006(t-t^{\ast })}{2501+1.86t-4.186t^{\ast }}W\; =\; \backslash frac\{(2501\; -\; 2.326\backslash ,t^*)W\_s^*\; -\; 1.006(t\; -\; t^*)\}\{2501\; +\; 1.86\backslash ,t\; -\; 4.186\backslash ,t^*\}$$

Pumps

Hydraulic power delivered to a fluid

$$P=\rho gQHP\; =\; \backslash rho\; g\; Q\; H$$

Shaft power required by a pump

$$P=\frac{\rho gQH}{\eta }P\; =\; \backslash frac\{\backslash rho\; g\; Q\; H\}\{\backslash eta\}$$

Radiation

Emissive power of a blackbody (Stefan-Boltzmann law)

$$E=\sigma T^{4}E\; =\; \backslash sigma\; T^\{4\}$$

Linearised radiation heat transfer coefficient

$$h_r=\epsilon \sigma (T_s+\text{Tsur})(T_s^2+{\text{Tsur}}^2)h\_\{r\}\; =\; \backslash varepsilon\; \backslash sigma\; \backslash left(T\_\{s\}\; +\; \backslash text\{Tsur\}\backslash right)\; \backslash left(T\_\{s\}^\{2\}\; +\; \backslash text\{Tsur\}^\{2\}\backslash right)$$

Net radiation between two large parallel grey plates

$$q_f=\frac{\sigma (T_1^4-T_2^4)}{\frac{1}{e_1}+\frac{1}{e_2}-1}q\_\{f\}\; =\; \backslash frac\{\backslash sigma\; \backslash left(T\_\{1\}^\{4\}\; -\; T\_\{2\}^\{4\}\backslash right)\}\{\backslash frac\{1\}\{e\_\{1\}\}\; +\; \backslash frac\{1\}\{e\_\{2\}\}\; -\; 1\}$$

Net radiation exchange of a small grey surface with large surroundings

$$q=\epsilon \sigma A(T_s^4-{\text{Tsur}}^4)q\; =\; \backslash varepsilon\; \backslash sigma\; A\; \backslash left(T\_\{s\}^\{4\}\; -\; \backslash text\{Tsur\}^\{4\}\backslash right)$$

Wien's displacement law

$$\lambda =\frac{b}{T}\backslash lambda\; =\; \backslash frac\{b\}\{T\}$$

Thermodynamic cycles

Carnot efficiency of a reversible heat engine

$$\eta =1-\frac{T_c}{T_h}\backslash eta\; =\; 1\; -\; \backslash frac\{T\_\{c\}\}\{T\_\{h\}\}$$

Coefficient of performance of a Carnot heat pump

$$\text{cop}=\frac{T_h}{T_h-T_c}\backslash text\{cop\}\; =\; \backslash frac\{T\_\{h\}\}\{T\_\{h\}\; -\; T\_\{c\}\}$$

Coefficient of performance of a Carnot refrigerator

$$\text{cop}=\frac{T_c}{T_h-T_c}\backslash text\{cop\}\; =\; \backslash frac\{T\_\{c\}\}\{T\_\{h\}\; -\; T\_\{c\}\}$$

Thermal efficiency of the air-standard Brayton cycle

$$\eta =1-r_p^{\frac{-2}{7}}\backslash eta\; =\; 1\; -\; r\_\{p\}^\{\backslash frac\{-2\}\{7\}\}$$

Thermal efficiency of the air-standard Diesel cycle

$$\eta =1-\frac{r^{\frac{-2}{5}}(r_c^{\frac{7}{5}}-1)}{\frac{7(r_c-1)}{5}}\backslash eta\; =\; 1\; -\; \backslash frac\{r^\{\backslash frac\{-2\}\{5\}\}\; \backslash left(r\_\{c\}^\{\backslash frac\{7\}\{5\}\}\; -\; 1\backslash right)\}\{\backslash frac\{7\; \backslash left(r\_\{c\}\; -\; 1\backslash right)\}\{5\}\}$$

Thermal efficiency of the air-standard Otto cycle

$$\eta =1-r^{\frac{-2}{5}}\backslash eta\; =\; 1\; -\; r^\{\backslash frac\{-2\}\{5\}\}$$

Transient conduction

Biot number

$$\mathrm{B}\mathrm{i}=\frac{h{L}_c}{k}\backslash mathrm\{Bi\}\; =\; \backslash frac\{h\; L\_\{c\}\}\{k\}$$

Fourier number

$$F_o=\frac{\alpha t}{L_c^2}F\_\{o\}\; =\; \backslash frac\{\backslash alpha\; t\}\{L\_\{c\}^\{2\}\}$$

Temperature of a lumped body cooling or heating in a fluid

$$T=\text{Tinf}+(T_i-\text{Tinf})e^{\frac{-hAt}{\rho Vc}}T\; =\; \backslash text\{Tinf\}\; +\; \backslash left(T\_\{i\}\; -\; \backslash text\{Tinf\}\backslash right)\; e^\{\backslash frac\{-h\; A\; t\}\{\backslash rho\; V\; c\}\}$$

Thermal time constant of a lumped body

$$\tau =\frac{\rho Vc}{hA}\backslash tau\; =\; \backslash frac\{\backslash rho\; V\; c\}\{h\; A\}$$

Out of scope

• Open-ended explanation, essay writing or tutoring dialogue: it is a solver, not a chatbot.
• Questions outside its certified thermofluids capabilities, and any question whose stated numbers it cannot all use — it refuses with the reason instead.
• High-stakes engineering, medical or legal decisions without independent review.

Limitations

• It answers what its certified capabilities cover (fluid statics and dynamics, pipe flow, boundary layers, aerodynamics and compressible flow, heat conduction, convection and radiation, fins, heat exchangers, thermodynamic cycles, psychrometrics and combustion); anything else is refused with a reason.
• ask() reads documented question forms and common phrasings; unusual wording may be refused — use solve() for guaranteed results.
• Results are as correct as the certified relations and their stated conditions; a certified relation is a checked model, not a claim about every physical situation.

Citation

@software{thermofluids_ai999x_1_0_0,
  title  = {thermofluids-ai999x: an offline, verified thermofluids solver},
  author = {Farabi, Ibnul Jaif}, organization = {AI999X},
  year   = {2026}, version = {1.0.0},
  url    = {https://huggingface.co/AI999X/thermofluids-ai999x}
}

Contact

Questions, bug reports and requests: open a discussion on https://huggingface.co/AI999X/thermofluids-ai999x/discussions or visit https://ai999x.com.

Support the project

thermofluids-ai999x is free and open source. If it helps you, you can support further development on Buy Me a Coffee.

Created and engineered by Ibnul Jaif Farabi · AI999X · ai999x.com