Live API reference

Base URL: https://www.engivault.com/api. These 44 numerical calculation endpoints plus the sample-statistics and unit-conversion endpoints are available without an API key during the public beta. Each entry below includes the input units, output fields, an executable example and the model assumptions.

For the numerical calculators, send a JSON object with numeric input values using POST and Content-Type: application/json. Unit labels apply individually: temperatures may be °C or K, and elastic modulus may be Pa or GPa. Select inputs use the numeric codes listed in their options.

Responses and errors

Successful calculations return HTTP 200 with {"success":true,"data":{...}}. Invalid JSON or inputs return HTTP 400 with an error message. An unknown calculation returns HTTP 404. Request bodies longer than 8192 characters return HTTP 413, except sample statistics which permits 32768 characters. Error responses always include an error field; do not rely on a success field being present in every error response.

GET /health reports service status. GET /calculators provides calculator input/output metadata, assumptions and source links. The unit converter uses string category and unit IDs; its metadata links to the dedicated /units endpoint.

Engineering unit conversion

GET /units lists category IDs, supported unit IDs, names, symbols, conversion mode, notes and category references. An empty sources array means references have not been recorded for that legacy category; it is not a completed source review.

POST /units requires exactly four fields: category, fromUnit and toUnit are case-sensitive strings from the catalog; value is a finite JSON number. Both units must belong to the selected category. Unknown fields, invalid units, incompatible quantities and numerical overflow or underflow return HTTP 400.

curl 'https://www.engivault.com/api/units' -H 'Content-Type: application/json' --data '{"category":"volume","value":1,"fromUnit":"gal_us","toUnit":"L"}'

Example response

{
  "success": true,
  "data": {
    "category": "volume",
    "fromUnit": "gal_us",
    "toUnit": "L",
    "convertedValue": 3.785411784,
    "symbol": "L"
  }
}

The response data includes category, fromUnit, toUnit, numeric convertedValue and the target symbol. The API returns the calculated number without the browser's display rounding; extra digits do not imply measurement accuracy.

Use category and unit IDs from GET /api/units. Both units must belong to the selected category.
Temperature points use scale offsets; temperature differences use factors only. Values are scale arithmetic, not a physical-state assessment.
Unit scaling preserves pressure reference and gas reference conditions. Dynamic and kinematic viscosity cannot be interconverted without density.
Sources are available for selected categories. Missing references mean the legacy category has not completed a source review. Conversion does not add measurement precision.

For example, use temperature, celsius and fahrenheit for temperature points: 10 °C becomes approximately 50 °F. Use temperature_difference, delta_celsius and delta_fahrenheit for intervals: a 10 °C rise becomes an 18 °F rise.

Open the browser unit converter

Repeated measurement statistics

POST /statistics/sample takes exactly one field, values, containing 2–1,000 finite JSON numbers in the same unit. Numeric strings and extra fields are rejected.

curl 'https://www.engivault.com/api/statistics/sample' -H 'Content-Type: application/json' --data '{"values":[10,12,14,16,18]}'

Example response

{
  "success": true,
  "data": {
    "count": 5,
    "mean": 14,
    "median": 14,
    "minimum": 10,
    "maximum": 18,
    "range": 8,
    "sampleVariance": 10.000000000000002,
    "sampleStandardDeviation": 3.1622776601683795,
    "standardError": 1.4142135623730951,
    "degreesOfFreedom": 4
  }
}

count: Number of readings
mean: Arithmetic mean (reading unit)
median: Median (reading unit)
minimum: Minimum (reading unit)
maximum: Maximum (reading unit)
range: Range (reading unit)
sampleVariance: Sample variance (reading unit²)
sampleStandardDeviation: Sample standard deviation s (reading unit)
standardError: Standard error of the mean s/√n (reading unit)
degreesOfFreedom: Sample variance degrees of freedom

Standard error requires independent observations under the same conditions for interpretation as uncertainty of the mean. It is not total measurement uncertainty. No confidence interval or outlier removal is performed.

Open the statistics tool, assumptions and references

Reynolds, Prandtl and Péclet Numbers

POST /api/thermal/transport-numbers

Calculate dimensionless transport groups and diffusivities from fluid properties, a speed and a characteristic length. Defaults are an illustrative property set, not a certified fluid state.

Open interactive calculator

Inputs, outputs, example and references

Inputs

Supply every listed field. Defaults below form the example, and are not automatically filled by the API.
JSON field	Meaning and units	Example
density	Fluid density (kg/m³)	1000
viscosity	Dynamic viscosity (Pa·s)	0.001
specificHeat	Specific heat at constant pressure (J/(kg·K))	4000
conductivity	Thermal conductivity (W/(m·K))	0.5
speed	Characteristic speed (m/s)	1
length	Characteristic length (m)	0.01

Outputs

Fields inside the response data object. Some model limits return explanatory strings; inspect the example and assumptions.
JSON field	Meaning	Unit
kinematicViscosity	Kinematic viscosity (momentum diffusivity)	m²/s
thermalDiffusivity	Thermal diffusivity	m²/s
reynoldsNumber	Reynolds number Re	Dimensionless / text
prandtlNumber	Prandtl number Pr	Dimensionless / text
pecletNumber	Thermal Péclet number Pe	Dimensionless / text

Example request

curl 'https://www.engivault.com/api/thermal/transport-numbers' \
  -H 'Content-Type: application/json' \
  --data '{"density":1000,"viscosity":0.001,"specificHeat":4000,"conductivity":0.5,"speed":1,"length":0.01}'

Example response

{
  "success": true,
  "data": {
    "kinematicViscosity": 0.000001,
    "thermalDiffusivity": 1.25e-7,
    "reynoldsNumber": 10000,
    "prandtlNumber": 8,
    "pecletNumber": 80000
  }
}

Model and limits

Calculate dimensionless transport groups and diffusivities from fluid properties, a speed and a characteristic length. Defaults are an illustrative property set, not a certified fluid state.
Supply positive scalar properties for the same fluid and evaluation state. Dynamic viscosity μ is in Pa·s, not cP or m²/s; specific heat cp is in J/(kg·K), not kJ/(kg·K). This tool does not evaluate properties from temperature or pressure.
Kinematic viscosity ν = μ/ρ and thermal diffusivity α = k/(ρ cp), both in m²/s. Prandtl number Pr = ν/α = μ cp/k compares momentum and thermal diffusion.
Reynolds number Re = UL/ν. Choose the speed and length required by your intended model, such as bulk speed and inside diameter for a circular pipe. A plate length, hydraulic diameter and computational-cell size are different choices and are not automatically interchangeable.
Thermal Péclet number Pe = UL/α = Re × Pr when the same speed and length are used. It compares advection and thermal diffusion on those scales. This is not a mass-transfer Péclet number; that uses a mass diffusivity instead.
Zero speed gives Re = Pe = 0 while the material diffusivities and Pr remain defined. No universal laminar/turbulent classification, Nusselt correlation, heat-transfer coefficient or boundary-layer thickness is inferred. Match geometry, boundary conditions, property evaluation and validity range when selecting a correlation.

References

NASA Glenn: Reynolds number and kinematic viscosity (accessed 2026-09-08)
MIT Unified Engineering: Prandtl number and thermal diffusivity, equation 17.7 (accessed 2026-09-08)
Idaho National Laboratory MOOSE: Péclet number UL/alpha (cell-length application) (accessed 2026-09-08)

Two-Input Measurement Uncertainty

POST /api/controls/measurement-uncertainty

Propagate two input standard uncertainties through addition, subtraction, multiplication or division. Standard uncertainty is expressed as a standard deviation; do not enter a tolerance bound or expanded uncertainty unchanged.

Open interactive calculator

Inputs, outputs, example and references

Inputs

Supply every listed field. Defaults below form the example, and are not automatically filled by the API.
JSON field	Meaning and units	Example
operation	Measurement equation `0`: A + B `1`: A − B `2`: A × B `3`: A / B	2
valueA	Estimate A (A unit)	10
valueB	Estimate B (B unit)	2
uncertaintyA	Standard uncertainty of A (A unit)	0.1
uncertaintyB	Standard uncertainty of B (B unit)	0.02
correlation	Correlation coefficient (−1 to 1)	0
coverageFactor	Coverage factor k	2

Outputs

Fields inside the response data object. Some model limits return explanatory strings; inspect the example and assumptions.
JSON field	Meaning	Unit
estimate	Output estimate	output unit
sensitivityA	Sensitivity to A	output unit / A unit
sensitivityB	Sensitivity to B	output unit / B unit
standardUncertainty	Combined standard uncertainty	output unit
relativeStandardUncertainty	Relative standard uncertainty	%
coverageFactor	Coverage factor k	Dimensionless / text
expandedUncertainty	Expanded uncertainty U	output unit
lowerEstimate	Estimate minus U	output unit
upperEstimate	Estimate plus U	output unit
method	Method	Dimensionless / text

Example request

curl 'https://www.engivault.com/api/controls/measurement-uncertainty' \
  -H 'Content-Type: application/json' \
  --data '{"operation":2,"valueA":10,"valueB":2,"uncertaintyA":0.1,"uncertaintyB":0.02,"correlation":0,"coverageFactor":2}'

Example response

{
  "success": true,
  "data": {
    "estimate": 20,
    "sensitivityA": 2,
    "sensitivityB": 10,
    "standardUncertainty": 0.28284271247461906,
    "coverageFactor": 2,
    "expandedUncertainty": 0.5656854249492381,
    "lowerEstimate": 19.434314575050763,
    "upperEstimate": 20.565685424949237,
    "relativeStandardUncertainty": 1.4142135623730951,
    "method": "First-order approximation — assess nonlinearity"
  }
}

Model and limits

Propagate two input standard uncertainties through addition, subtraction, multiplication or division. Standard uncertainty is expressed as a standard deviation; do not enter a tolerance bound or expanded uncertainty unchanged.
Use the same unit for A and B in addition/subtraction. For multiplication the output unit is A-unit × B-unit; for division it is A-unit / B-unit. Enter each absolute standard uncertainty in the same unit as its corresponding estimate. No unit conversion is performed.
Let cA = ∂f/∂A and cB = ∂f/∂B. Combined variance uc² = (cA uA)² + (cB uB)² + 2r cA cB uA uB, where r is the correlation coefficient. A value of zero specifies uncorrelated inputs; it is not inferred from the data.
Sensitivities are (1, 1) for a sum, (1, −1) for a difference, (B, A) for a product and (1/B, −A/B²) for a quotient. This is first-order propagation; nonlinear effects can matter for products and ratios, especially with a denominator near zero. A zero first-order result does not prove zero uncertainty for a nonlinear model. No Monte Carlo propagation, distribution or degrees-of-freedom estimate is provided.
Expanded uncertainty U = k uc. The displayed estimate ± U does not have an automatically assigned confidence level. Choosing k = 2 alone does not establish 95% coverage. Relative standard uncertainty is 100 uc/|estimate| and is undefined at a zero estimate.

References

NIST TN 1297 Appendix A: Law of Propagation of Uncertainty (accessed 2026-09-08)
NIST TN 1297 Section 6: Expanded Uncertainty (accessed 2026-09-08)

Wet Steam Mixture Properties

POST /api/thermal/wet-steam-properties

Pure water at liquid–vapor equilibrium, using absolute pressure in Pa. The range is 611.657 Pa through pressures strictly below 22064000 Pa. Quality is undefined at the critical point; numerically unresolved phase boundaries are rejected.

Open interactive calculator

Inputs, outputs, example and references

Inputs

Supply every listed field. Defaults below form the example, and are not automatically filled by the API.
JSON field	Meaning and units	Example
pressure	Absolute saturation pressure (Pa)	101325
quality	Vapor mass fraction (quality, 0 to 1) (kg vapor / kg mixture)	0.9

Outputs

Fields inside the response data object. Some model limits return explanatory strings; inspect the example and assumptions.
JSON field	Meaning	Unit
pressure	Absolute saturation pressure	Pa
temperature	Saturation temperature	K
celsius	Saturation temperature	°C
quality	Vapor mass fraction (quality)	kg vapor / kg mixture
moisture	Liquid mass fraction (moisture)	kg liquid / kg mixture
vaporVolumeFraction	Vapor volume fraction	m³ vapor / m³ mixture
specificVolume	Mixture specific volume	m³/kg
density	Mixture density	kg/m³
enthalpy	Mixture specific enthalpy	J/kg
entropy	Mixture specific entropy	J/(kg·K)
internalEnergy	Mixture specific internal energy	J/kg
liquidEnthalpy	Saturated liquid specific enthalpy	J/kg
vaporEnthalpy	Saturated vapor specific enthalpy	J/kg
latentHeat	Latent heat of vaporization	J/kg
phase	Equilibrium phase	Dimensionless / text

Example request

curl 'https://www.engivault.com/api/thermal/wet-steam-properties' \
  -H 'Content-Type: application/json' \
  --data '{"pressure":101325,"quality":0.9}'

Example response

{
  "success": true,
  "data": {
    "pressure": 101325,
    "temperature": 373.12429580541163,
    "celsius": 99.97429580541166,
    "quality": 0.9,
    "moisture": 0.09999999999999998,
    "vaporVolumeFraction": 0.9999307217457116,
    "specificVolume": 1.506163131216676,
    "density": 0.6639387057577234,
    "enthalpy": 2450053.2909022365,
    "entropy": 6750.135480966874,
    "internalEnergy": 2297441.3116317065,
    "liquidEnthalpy": 419054.4102440765,
    "vaporEnthalpy": 2675719.833197587,
    "latentHeat": 2256665.4229535107,
    "phase": "Liquid–vapor mixture"
  }
}

Model and limits

Pure water at liquid–vapor equilibrium, using absolute pressure in Pa. The range is 611.657 Pa through pressures strictly below 22064000 Pa. Quality is undefined at the critical point; numerically unresolved phase boundaries are rejected.
Saturation temperature and separate liquid/vapor properties use IAPWS SR1-86(1992). The release’s rounded lower pressure endpoint maps to the triple-point temperature. Ice and dissolved substances are excluded.
Quality x is vapor mass divided by total mixture mass. Enter a fraction, such as 0.9 for 90%. Specific volume, enthalpy and entropy are mass-weighted: y = (1 − x)y_liquid + x y_vapor. Density is 1/v, not a mass-weighted average of the two densities.
Vapor volume fraction = x v_vapor / v_mixture. This describes a saturated inventory. It is not a flowing-pipe void-fraction prediction from vapor mass flow fraction; phase slip and flow patterns are not modeled.
Mixture internal energy u = h − pv. Enthalpy and entropy retain the IAPWS reference convention. For the inverse calculation, x = (h − h_liquid)/(h_vapor − h_liquid); values outside the model’s saturated enthalpy interval are rejected rather than clamped. Near-critical inversion is sensitive as the enthalpy gap shrinks.

References

IAPWS SR1-86(1992): Saturation phase properties (accessed 2026-09-08)
DOE Thermodynamics Volume 1: quality, mixture properties and enthalpy definitions, pages 18, 34 and 49 (accessed 2026-09-08)

Steam Quality from Enthalpy

POST /api/thermal/steam-quality-enthalpy

Open interactive calculator

Inputs, outputs, example and references

Inputs

Supply every listed field. Defaults below form the example, and are not automatically filled by the API.
JSON field	Meaning and units	Example
pressure	Absolute saturation pressure (Pa)	101325
enthalpy	Mixture specific enthalpy (J/kg)	2450000

Outputs

Fields inside the response data object. Some model limits return explanatory strings; inspect the example and assumptions.
JSON field	Meaning	Unit
pressure	Absolute saturation pressure	Pa
temperature	Saturation temperature	K
celsius	Saturation temperature	°C
quality	Vapor mass fraction (quality)	kg vapor / kg mixture
moisture	Liquid mass fraction (moisture)	kg liquid / kg mixture
vaporVolumeFraction	Vapor volume fraction	m³ vapor / m³ mixture
specificVolume	Mixture specific volume	m³/kg
density	Mixture density	kg/m³
enthalpy	Mixture specific enthalpy	J/kg
entropy	Mixture specific entropy	J/(kg·K)
internalEnergy	Mixture specific internal energy	J/kg
liquidEnthalpy	Saturated liquid specific enthalpy	J/kg
vaporEnthalpy	Saturated vapor specific enthalpy	J/kg
latentHeat	Latent heat of vaporization	J/kg
phase	Equilibrium phase	Dimensionless / text

Example request

curl 'https://www.engivault.com/api/thermal/steam-quality-enthalpy' \
  -H 'Content-Type: application/json' \
  --data '{"pressure":101325,"enthalpy":2450000}'

Example response

{
  "success": true,
  "data": {
    "pressure": 101325,
    "temperature": 373.12429580541163,
    "celsius": 99.97429580541166,
    "quality": 0.8999763851115482,
    "moisture": 0.1000236148884518,
    "vaporVolumeFraction": 0.9999307035687359,
    "specificVolume": 1.5061236387349746,
    "density": 0.6639561150769278,
    "enthalpy": 2450000,
    "entropy": 6749.992657508752,
    "internalEnergy": 2297392.0223051785,
    "liquidEnthalpy": 419054.4102440765,
    "vaporEnthalpy": 2675719.833197587,
    "latentHeat": 2256665.4229535107,
    "phase": "Liquid–vapor mixture"
  }
}

Model and limits

Pure water at liquid–vapor equilibrium, using absolute pressure in Pa. The range is 611.657 Pa through pressures strictly below 22064000 Pa. Quality is undefined at the critical point; numerically unresolved phase boundaries are rejected.
Saturation temperature and separate liquid/vapor properties use IAPWS SR1-86(1992). The release’s rounded lower pressure endpoint maps to the triple-point temperature. Ice and dissolved substances are excluded.
Quality x is vapor mass divided by total mixture mass. Enter a fraction, such as 0.9 for 90%. Specific volume, enthalpy and entropy are mass-weighted: y = (1 − x)y_liquid + x y_vapor. Density is 1/v, not a mass-weighted average of the two densities.
Vapor volume fraction = x v_vapor / v_mixture. This describes a saturated inventory. It is not a flowing-pipe void-fraction prediction from vapor mass flow fraction; phase slip and flow patterns are not modeled.
Mixture internal energy u = h − pv. Enthalpy and entropy retain the IAPWS reference convention. For the inverse calculation, x = (h − h_liquid)/(h_vapor − h_liquid); values outside the model’s saturated enthalpy interval are rejected rather than clamped. Near-critical inversion is sensitive as the enthalpy gap shrinks.

References

IAPWS SR1-86(1992): Saturation phase properties (accessed 2026-09-08)
DOE Thermodynamics Volume 1: quality, mixture properties and enthalpy definitions, pages 18, 34 and 49 (accessed 2026-09-08)

Water Saturation Pressure

POST /api/thermal/water-saturation-pressure

Pure water liquid–vapor equilibrium from the triple-point temperature 273.16 K to the critical point 647.096 K. Pressure is absolute, never gauge; temperature input uses kelvin.

Open interactive calculator

Inputs, outputs, example and references

Inputs

Supply every listed field. Defaults below form the example, and are not automatically filled by the API.
JSON field	Meaning and units	Example
temperature	Saturation temperature (K)	373.15

Outputs

Fields inside the response data object. Some model limits return explanatory strings; inspect the example and assumptions.
JSON field	Meaning	Unit
temperature	Saturation temperature	K
celsius	Saturation temperature	°C
pressure	Absolute saturation pressure	Pa
liquidDensity	Saturated liquid density	kg/m³
vaporDensity	Saturated vapor density	kg/m³
liquidSpecificVolume	Saturated liquid specific volume	m³/kg
vaporSpecificVolume	Saturated vapor specific volume	m³/kg
liquidEnthalpy	Saturated liquid specific enthalpy	J/kg
vaporEnthalpy	Saturated vapor specific enthalpy	J/kg
liquidEntropy	Saturated liquid specific entropy	J/(kg·K)
vaporEntropy	Saturated vapor specific entropy	J/(kg·K)
latentHeat	Latent heat of vaporization	J/kg
pressureSlope	Slope of saturation pressure with temperature	Pa/K

Example request

curl 'https://www.engivault.com/api/thermal/water-saturation-pressure' \
  -H 'Content-Type: application/json' \
  --data '{"temperature":373.15}'

Example response

{
  "success": true,
  "data": {
    "temperature": 373.15,
    "celsius": 100,
    "pressure": 101417.99381792783,
    "liquidDensity": 958.346796277354,
    "vaporDensity": 0.5980991686478763,
    "liquidSpecificVolume": 0.0010434636019908926,
    "vaporSpecificVolume": 1.671963534510007,
    "pressureSlope": 3619.2201978599674,
    "liquidEnthalpy": 419162.83483193157,
    "vaporEnthalpy": 2675760.4697580603,
    "liquidEntropy": 1307.2024431909222,
    "vaporEntropy": 7354.630112830823,
    "latentHeat": 2256597.6349261287
  }
}

Model and limits

Pure water liquid–vapor equilibrium from the triple-point temperature 273.16 K to the critical point 647.096 K. Pressure is absolute, never gauge; temperature input uses kelvin.
Uses IAPWS SR1-86(1992), equation (1), with critical temperature 647.096 K and pressure 22.064 MPa. Pressure-to-temperature conversion numerically inverts the same monotonic correlation.
The lower pressure input limit is the release table’s rounded triple-point pressure, 611.657 Pa. The tiny interval up to the equation’s approximately 611.657070 Pa endpoint maps to 273.16 K. The upper limit is 22064000 Pa.
At 101325 Pa, saturation temperature is approximately 373.1243 K (99.9743 °C). This is equilibrium for pure water, not a boiling-temperature prediction for solutions or transient heating.
At a saturation pair, pressure and temperature do not specify vapor fraction or mixture enthalpy. The two densities and reciprocal specific volumes describe separate saturated phases using IAPWS equations (2) and (3), not mixture density. Enthalpy and entropy use equations (4)–(9) and the release reference convention: liquid internal energy and entropy are zero at the triple point, subject to coefficient rounding. Latent heat is the vapor-minus-liquid enthalpy difference. This tool does not calculate steam quality, superheat or subcooled-liquid properties. At the critical endpoint the distinct liquid and vapor phases merge.

References

IAPWS SR1-86(1992): Saturation Properties, equations (1)–(9) and Table 1 (accessed 2026-09-08)

Water Saturation Temperature

POST /api/thermal/water-saturation-temperature

Pure water liquid–vapor equilibrium from the triple-point temperature 273.16 K to the critical point 647.096 K. Pressure is absolute, never gauge; temperature input uses kelvin.

Open interactive calculator

Inputs, outputs, example and references

Inputs

Supply every listed field. Defaults below form the example, and are not automatically filled by the API.
JSON field	Meaning and units	Example
pressure	Absolute saturation pressure (Pa)	101325

Outputs

Fields inside the response data object. Some model limits return explanatory strings; inspect the example and assumptions.
JSON field	Meaning	Unit
temperature	Saturation temperature	K
celsius	Saturation temperature	°C
pressure	Absolute saturation pressure	Pa
liquidDensity	Saturated liquid density	kg/m³
vaporDensity	Saturated vapor density	kg/m³
liquidSpecificVolume	Saturated liquid specific volume	m³/kg
vaporSpecificVolume	Saturated vapor specific volume	m³/kg
liquidEnthalpy	Saturated liquid specific enthalpy	J/kg
vaporEnthalpy	Saturated vapor specific enthalpy	J/kg
liquidEntropy	Saturated liquid specific entropy	J/(kg·K)
vaporEntropy	Saturated vapor specific entropy	J/(kg·K)
latentHeat	Latent heat of vaporization	J/kg
pressureSlope	Slope of saturation pressure with temperature	Pa/K

Example request

curl 'https://www.engivault.com/api/thermal/water-saturation-temperature' \
  -H 'Content-Type: application/json' \
  --data '{"pressure":101325}'

Example response

{
  "success": true,
  "data": {
    "temperature": 373.12429580541163,
    "celsius": 99.97429580541166,
    "pressure": 101325,
    "liquidDensity": 958.365236793104,
    "vaporDensity": 0.5975862349130926,
    "liquidSpecificVolume": 0.0010434435240433125,
    "vaporSpecificVolume": 1.6733986520714133,
    "pressureSlope": 3616.472097158746,
    "liquidEnthalpy": 419054.4102440765,
    "vaporEnthalpy": 2675719.833197587,
    "liquidEntropy": 1306.9121275421815,
    "vaporEntropy": 7354.93807579184,
    "latentHeat": 2256665.4229535107
  }
}

Model and limits

Pure water liquid–vapor equilibrium from the triple-point temperature 273.16 K to the critical point 647.096 K. Pressure is absolute, never gauge; temperature input uses kelvin.
Uses IAPWS SR1-86(1992), equation (1), with critical temperature 647.096 K and pressure 22.064 MPa. Pressure-to-temperature conversion numerically inverts the same monotonic correlation.
The lower pressure input limit is the release table’s rounded triple-point pressure, 611.657 Pa. The tiny interval up to the equation’s approximately 611.657070 Pa endpoint maps to 273.16 K. The upper limit is 22064000 Pa.
At 101325 Pa, saturation temperature is approximately 373.1243 K (99.9743 °C). This is equilibrium for pure water, not a boiling-temperature prediction for solutions or transient heating.
At a saturation pair, pressure and temperature do not specify vapor fraction or mixture enthalpy. The two densities and reciprocal specific volumes describe separate saturated phases using IAPWS equations (2) and (3), not mixture density. Enthalpy and entropy use equations (4)–(9) and the release reference convention: liquid internal energy and entropy are zero at the triple point, subject to coefficient rounding. Latent heat is the vapor-minus-liquid enthalpy difference. This tool does not calculate steam quality, superheat or subcooled-liquid properties. At the critical endpoint the distinct liquid and vapor phases merge.

References

IAPWS SR1-86(1992): Saturation Properties, equations (1)–(9) and Table 1 (accessed 2026-09-08)

First-Order Process Response with Delay

POST /api/controls/first-order-response

EU means engineering unit. Use consistent input/output units; gain is output-unit change divided by input-unit change. This is a stable linear first-order-plus-dead-time process initially at equilibrium, with one sustained step at time zero.

Open interactive calculator

Inputs, outputs, example and references

Inputs

Supply every listed field. Defaults below form the example, and are not automatically filled by the API.
JSON field	Meaning and units	Example
initialValue	Initial equilibrium output (output EU)	20
gain	Process gain (output EU / input EU)	2
inputStep	Input step change at time zero (input EU)	10
timeConstant	Time constant (s)	30
deadTime	Response delay (s)	5
time	Time since input step (s)	35

Outputs

Fields inside the response data object. Some model limits return explanatory strings; inspect the example and assumptions.
JSON field	Meaning	Unit
outputValue	Output at observation time	output EU
finalValue	Final equilibrium output	output EU
finalChange	Final output change	output EU
percentResponse	Completed fraction of final change	%
responseRate	Output rate (right-hand value at delay boundary)	output EU/s
timeTo90	Time from input step to 90% response	s
timeTo95	Time from input step to 95% response	s
timeTo98	Time from input step to 98% response	s
phase	Response phase	Dimensionless / text

Example request

curl 'https://www.engivault.com/api/controls/first-order-response' \
  -H 'Content-Type: application/json' \
  --data '{"initialValue":20,"gain":2,"inputStep":10,"timeConstant":30,"deadTime":5,"time":35}'

Example response

{
  "success": true,
  "data": {
    "finalChange": 20,
    "finalValue": 40,
    "outputValue": 32.64241117657115,
    "responseRate": 0.24525296078096154,
    "timeTo90": 74.07755278982138,
    "timeTo95": 94.87196820661973,
    "timeTo98": 122.36069016284438,
    "percentResponse": 63.212055882855765,
    "phase": "Exponential response (rounded at long times)"
  }
}

Model and limits

EU means engineering unit. Use consistent input/output units; gain is output-unit change divided by input-unit change. This is a stable linear first-order-plus-dead-time process initially at equilibrium, with one sustained step at time zero.
Final change = gain × input step; final output = initial output + final change. Negative gains and steps are allowed.
Before delay θ, output stays at its initial value. At and after θ: y = y_initial + final_change × [1 − exp(−(t − θ)/τ)], with positive time constant τ. At θ + τ, 63.212% of the final change is complete.
After the delay, output rate = (final_change/τ) × exp(−(t − θ)/τ). At the delay boundary the tool reports the right-hand derivative; the derivative jumps from zero for a nonzero step.
Time to fraction f is θ − τ ln(1 − f), measured from the input step. For this monotonic model, 90%, 95% and 98% correspond to remaining-error bands of 10%, 5% and 2% of the final change, not of the absolute output.
If the final change is zero, output stays constant, rate and response times are zero, and percent completion is not applicable. Exact final equilibrium is approached asymptotically for a nonzero change; displayed numbers eventually round to it.
This does not fit parameters, tune a controller or simulate a closed loop, oscillation, actuator saturation, multiple steps, nonlinear behavior or an unstable/integrating process. Use identified model parameters valid around the operating point.

References

APMonitor Process Dynamics and Control: First Order Plus Dead Time (accessed 2026-09-08)

Current-Loop Voltage Budget

POST /api/controls/current-loop-budget

Model one DC series loop with a two-wire loop-powered transmitter, resistive receiver and total outgoing-plus-return cable resistance. Use the lowest available supply and worst-case device data for your selected current.

Open interactive calculator

Inputs, outputs, example and references

Inputs

Supply every listed field. Defaults below form the example, and are not automatically filled by the API.
JSON field	Meaning and units	Example
supplyVoltage	Available supply voltage (V)	24
transmitterMinimum	Transmitter minimum operating voltage (V)	12
current	Required loop current (mA)	20
receiverResistance	Total receiver resistance (Ω)	250
wireResistance	Total outgoing and return wire resistance (Ω)	100
otherDrop	Other series voltage drops at this current (V)	1

Outputs

Fields inside the response data object. Some model limits return explanatory strings; inspect the example and assumptions.
JSON field	Meaning	Unit
budgetStatus	Voltage-budget check	Dimensionless / text
requiredSupply	Minimum calculated supply	V
voltageMargin	Supply voltage margin	V
transmitterVoltage	Voltage left for transmitter at requested current	V
receiverDrop	Receiver voltage drop	V
wireDrop	Total wire voltage drop	V
resistanceHeadroom	Additional series resistance headroom (negative means deficit)	Ω
receiverPower	Receiver resistive dissipation at requested current	W
wirePower	Wire resistive dissipation at requested current	W

Example request

curl 'https://www.engivault.com/api/controls/current-loop-budget' \
  -H 'Content-Type: application/json' \
  --data '{"supplyVoltage":24,"transmitterMinimum":12,"current":20,"receiverResistance":250,"wireResistance":100,"otherDrop":1}'

Example response

{
  "success": true,
  "data": {
    "receiverDrop": 5,
    "wireDrop": 2,
    "requiredSupply": 20,
    "voltageMargin": 4,
    "transmitterVoltage": 16,
    "resistanceHeadroom": 200,
    "receiverPower": 0.1,
    "wirePower": 0.04,
    "budgetStatus": "Positive voltage margin — other device limits still require checks"
  }
}

Model and limits

Model one DC series loop with a two-wire loop-powered transmitter, resistive receiver and total outgoing-plus-return cable resistance. Use the lowest available supply and worst-case device data for your selected current.
Convert mA to A by dividing by 1000. Receiver and cable voltage drops are I × R. Include additional series device drops separately without counting the same resistance twice.
Minimum supply = transmitter minimum + receiver drop + wire drop + other drops. Voltage margin = available supply − minimum supply. Negative margin means the requested current cannot be supported under these assumptions; this tool does not predict the actual saturated current.
Additional resistance headroom = voltage margin / I. A negative result is a deficit, not an allowable resistance. Receiver and cable dissipation each equal I²R; select physical ratings with the applicable derating.
Original example: 24 V supply, 12 V transmitter minimum, 20 mA, 250 Ω receiver, 100 Ω total wire and 1 V other drop require 20 V and leave 4 V margin. Include both cable conductors.
Check the greatest operating or alarm current you require, which may exceed 20 mA. Also check transmitter maximum voltage at low current, receiver ratings, supply tolerance/current capacity, temperature effects, isolation and grounding. A positive margin is not a complete device or hazardous-area approval.

References

NI: Current-loop system design and power supply selection (accessed 2026-09-08)

4–20 mA to Engineering Value

POST /api/controls/current-loop-scale

EU means your chosen engineering unit: use the same unit for both endpoint values and the engineering input/output. No unit conversion is performed. Current is in mA, not A.

Open interactive calculator

Inputs, outputs, example and references

Inputs

Supply every listed field. Defaults below form the example, and are not automatically filled by the API.
JSON field	Meaning and units	Example
current	Measured current (mA)	12
valueAt4	Engineering value at 4 mA (EU)	0
valueAt20	Engineering value at 20 mA (EU)	100

Outputs

Fields inside the response data object. Some model limits return explanatory strings; inspect the example and assumptions.
JSON field	Meaning	Unit
value	Engineering value, unclamped	EU
current	Current, unclamped	mA
percentSpan	Position along configured signal span	%
rangeStatus	Nominal current-range check	Dimensionless / text

Example request

curl 'https://www.engivault.com/api/controls/current-loop-scale' \
  -H 'Content-Type: application/json' \
  --data '{"current":12,"valueAt4":0,"valueAt20":100}'

Example response

{
  "success": true,
  "data": {
    "current": 12,
    "value": 50,
    "percentSpan": 50,
    "rangeStatus": "Within nominal 4–20 mA range"
  }
}

Model and limits

EU means your chosen engineering unit: use the same unit for both endpoint values and the engineering input/output. No unit conversion is performed. Current is in mA, not A.
Configure the engineering values corresponding to exactly 4 mA and 20 mA. They must differ; decreasing scales are supported. This assumes a linear transmitter output.
Forward: fraction = (current − 4)/16; value = valueAt4 + fraction × (valueAt20 − valueAt4). Inverse: fraction = (value − valueAt4)/(valueAt20 − valueAt4); current = 4 + 16 × fraction. Percent span = 100 × fraction.
Example: a −50 to 150 °C linear scale gives 50 °C at 12 mA and 75% span at 16 mA. Percent span is measured from the 4 mA endpoint, not from zero engineering value.
Results outside 4–20 mA are mathematical extrapolations, not valid process readings or feasible output commands. They are flagged and never clamped. Check device-specific alarms and configured limits; this is not a fault diagnosis or NAMUR compliance test.
No square-root extraction, sensor linearization, calibration uncertainty, loop supply/burden check, hardware communication or safety-interlock validation is included.

References

NI: Custom Scales, linear input/output scaling (accessed 2026-09-08)

Engineering Value to 4–20 mA

POST /api/controls/current-loop-output

EU means your chosen engineering unit: use the same unit for both endpoint values and the engineering input/output. No unit conversion is performed. Current is in mA, not A.

Open interactive calculator

Inputs, outputs, example and references

Inputs

Supply every listed field. Defaults below form the example, and are not automatically filled by the API.
JSON field	Meaning and units	Example
value	Engineering value (EU)	50
valueAt4	Engineering value at 4 mA (EU)	0
valueAt20	Engineering value at 20 mA (EU)	100

Outputs

Fields inside the response data object. Some model limits return explanatory strings; inspect the example and assumptions.
JSON field	Meaning	Unit
value	Engineering value, unclamped	EU
current	Current, unclamped	mA
percentSpan	Position along configured signal span	%
rangeStatus	Nominal current-range check	Dimensionless / text

Example request

curl 'https://www.engivault.com/api/controls/current-loop-output' \
  -H 'Content-Type: application/json' \
  --data '{"value":50,"valueAt4":0,"valueAt20":100}'

Example response

{
  "success": true,
  "data": {
    "current": 12,
    "value": 50,
    "percentSpan": 50,
    "rangeStatus": "Within nominal 4–20 mA range"
  }
}

Model and limits

EU means your chosen engineering unit: use the same unit for both endpoint values and the engineering input/output. No unit conversion is performed. Current is in mA, not A.
Configure the engineering values corresponding to exactly 4 mA and 20 mA. They must differ; decreasing scales are supported. This assumes a linear transmitter output.
Forward: fraction = (current − 4)/16; value = valueAt4 + fraction × (valueAt20 − valueAt4). Inverse: fraction = (value − valueAt4)/(valueAt20 − valueAt4); current = 4 + 16 × fraction. Percent span = 100 × fraction.
Example: a −50 to 150 °C linear scale gives 50 °C at 12 mA and 75% span at 16 mA. Percent span is measured from the 4 mA endpoint, not from zero engineering value.
Results outside 4–20 mA are mathematical extrapolations, not valid process readings or feasible output commands. They are flagged and never clamped. Check device-specific alarms and configured limits; this is not a fault diagnosis or NAMUR compliance test.
No square-root extraction, sensor linearization, calibration uncertainty, loop supply/burden check, hardware communication or safety-interlock validation is included.

References

NI: Custom Scales, linear input/output scaling (accessed 2026-09-08)

Beam Bending from Section Properties

POST /api/mechanics/section-beam

Uniform homogeneous linear-elastic slender beam, constant EI, small deflection and principal-axis bending. Loads act in the bending plane without torsion. I is the area second moment about that axis, not polar or mass inertia; c is the farthest material fibre from the neutral axis.

Open interactive calculator

Inputs, outputs, example and references

Inputs

Supply every listed field. Defaults below form the example, and are not automatically filled by the API.
JSON field	Meaning and units	Example
support	Support and point-load location `0`: Simply supported; point load at center `1`: Cantilever; point load at free tip	0
span	Span (m)	3
secondMoment	Second moment about the bending axis (m⁴)	0.000027786666666666686
extremeDistance	Farthest fibre from neutral axis (m)	0.1
youngModulus	Young’s modulus (Pa)	200000000000
pointLoad	Downward point load (N)	1000
lineLoad	Downward load per length over full span (N/m)	500

Outputs

Fields inside the response data object. Some model limits return explanatory strings; inspect the example and assumptions.
JSON field	Meaning	Unit
sectionModulus	Elastic section modulus	m³
rigidity	Flexural rigidity EI	N·m²
totalLoad	Total downward load	N
leftReaction	Upward left reaction	N
rightReaction	Upward right reaction	N
fixedReactionMoment	Counterclockwise fixed-end reaction moment	N·m
maximumMoment	Maximum bending moment magnitude	N·m
maximumBendingStress	Maximum bending stress magnitude	Pa
maximumShear	Maximum shear force magnitude	N
maximumDeflection	Maximum downward deflection	m
deflectionSpanRatio	Deflection / span	Dimensionless / text
criticalLocations	Critical locations	Dimensionless / text

Example request

curl 'https://www.engivault.com/api/mechanics/section-beam' \
  -H 'Content-Type: application/json' \
  --data '{"support":0,"span":3,"secondMoment":0.000027786666666666686,"extremeDistance":0.1,"youngModulus":200000000000,"pointLoad":1000,"lineLoad":500}'

Example response

{
  "success": true,
  "data": {
    "sectionModulus": 0.00027786666666666684,
    "rigidity": 5557333.333333338,
    "totalLoad": 2500,
    "leftReaction": 1250,
    "rightReaction": 1250,
    "fixedReactionMoment": 0,
    "maximumMoment": 1312.5,
    "maximumShear": 1250,
    "maximumDeflection": 0.00019610912008157373,
    "deflectionSpanRatio": 0.00006536970669385791,
    "criticalLocations": "Moment and deflection at midspan",
    "maximumBendingStress": 4723488.483685218
  }
}

Model and limits

Uniform homogeneous linear-elastic slender beam, constant EI, small deflection and principal-axis bending. Loads act in the bending plane without torsion. I is the area second moment about that axis, not polar or mass inertia; c is the farthest material fibre from the neutral axis.
S = I/c; stress magnitude = Mmax/S. Use the larger extreme distance for an asymmetric section. The tool cannot validate a supplied I and c against a real section. Shear force is reported; no shape-independent shear stress is inferred.
Simply supported: Rleft = Rright = (P + wL)/2; Mmax = PL/4 + wL²/8; δmax = PL³/(48EI) + 5wL⁴/(384EI), both at midspan.
Cantilever: Rleft = P + wL, Rright = 0; reaction moment magnitude = PL + wL²/2; δmax = PL³/(3EI) + wL⁴/(8EI). Moment is largest at the root, deflection at the tip.
Include self-weight in w if relevant. No shear deformation, torsion, yielding, fatigue, local/lateral buckling, connection or design-code assessment. Transferred geometry does not establish a material grade, load or safe capacity.

References

MIT Solid Mechanics: Deflections due to Bending, standard cases (accessed 2026-09-08)
MIT Solid Mechanics: Stresses, Beams in Bending (accessed 2026-09-08)

Circular Section Properties

POST /api/mechanics/circular-section

Concentric circular opening; zero inner diameter selects solid. A = π(Do² − Di²)/4; Ix = Iy = π(Do⁴ − Di⁴)/64. A circle has the same centroidal second moment about every in-plane diameter.

Open interactive calculator

Inputs, outputs, example and references

Inputs

Supply every listed field. Defaults below form the example, and are not automatically filled by the API.
JSON field	Meaning and units	Example
outerDiameter	Outer diameter (m)	0.05
innerDiameter	Inner diameter (0 for solid) (m)	0.03

Outputs

Fields inside the response data object. Some model limits return explanatory strings; inspect the example and assumptions.
JSON field	Meaning	Unit
area	Material area	m²
centroidX	Centroid x from left of outer bounding box	m
centroidY	Centroid y from bottom of outer bounding box	m
secondMomentX	Second moment about horizontal centroidal x-axis	m⁴
secondMomentY	Second moment about vertical centroidal y-axis	m⁴
polarAreaMoment	Polar second moment of area Ix + Iy	m⁴
sectionModulusX	Elastic section modulus about x-axis	m³
sectionModulusY	Elastic section modulus about y-axis	m³
radiusOfGyrationX	Area radius of gyration about x-axis	m
radiusOfGyrationY	Area radius of gyration about y-axis	m

Example request

curl 'https://www.engivault.com/api/mechanics/circular-section' \
  -H 'Content-Type: application/json' \
  --data '{"outerDiameter":0.05,"innerDiameter":0.03}'

Example response

{
  "success": true,
  "data": {
    "area": 0.0012566370614359175,
    "centroidX": 0.025,
    "centroidY": 0.025,
    "secondMomentX": 2.6703537555513247e-7,
    "secondMomentY": 2.6703537555513247e-7,
    "polarAreaMoment": 5.340707511102649e-7,
    "sectionModulusX": 0.000010681415022205297,
    "sectionModulusY": 0.000010681415022205297,
    "radiusOfGyrationX": 0.014577379737113252,
    "radiusOfGyrationY": 0.014577379737113252
  }
}

Model and limits

Concentric circular opening; zero inner diameter selects solid. A = π(Do² − Di²)/4; Ix = Iy = π(Do⁴ − Di⁴)/64. A circle has the same centroidal second moment about every in-plane diameter.
All dimensions in metres. Axes pass through the centroid: x horizontal, y vertical. Centroid coordinates are measured from the bottom-left of the outer bounding box. These are area properties, not mass moments of inertia.
Sx = Ix/(outer height/2), Sy = Iy/(outer width/2); rx = √(Ix/A), ry = √(Iy/A). The section modulus has dimensions of length cubed, but is not a volume of material.
Polar area moment Jp = Ix + Iy. It equals the Saint-Venant torsion constant for a circular solid/annulus, not for a rectangle. No rounded corners, eccentric void, strength rating, local buckling or design-code check is included.

References

Engineering Statics: Moments of Inertia of Common Shapes (accessed 2026-09-08)
Engineering Statics: Radius of Gyration (accessed 2026-09-08)
MIT Engineering Beam Theory, Appendix A (accessed 2026-09-08)

Rectangular Section Properties

POST /api/mechanics/rectangular-section

Concentric rectangular opening with sharp corners. Set both inner dimensions to zero for solid. A = bh − bi hi; Ix = (b h³ − bi hi³)/12; Iy = (h b³ − hi bi³)/12. Wall thickness may differ horizontally and vertically.

Open interactive calculator

Inputs, outputs, example and references

Inputs

Supply every listed field. Defaults below form the example, and are not automatically filled by the API.
JSON field	Meaning and units	Example
width	Outer width (m)	0.1
height	Outer height (m)	0.2
innerWidth	Inner width (0 for solid) (m)	0.08
innerHeight	Inner height (0 for solid) (m)	0.18

Outputs

Fields inside the response data object. Some model limits return explanatory strings; inspect the example and assumptions.
JSON field	Meaning	Unit
area	Material area	m²
centroidX	Centroid x from left of outer bounding box	m
centroidY	Centroid y from bottom of outer bounding box	m
secondMomentX	Second moment about horizontal centroidal x-axis	m⁴
secondMomentY	Second moment about vertical centroidal y-axis	m⁴
polarAreaMoment	Polar second moment of area Ix + Iy	m⁴
sectionModulusX	Elastic section modulus about x-axis	m³
sectionModulusY	Elastic section modulus about y-axis	m³
radiusOfGyrationX	Area radius of gyration about x-axis	m
radiusOfGyrationY	Area radius of gyration about y-axis	m

Example request

curl 'https://www.engivault.com/api/mechanics/rectangular-section' \
  -H 'Content-Type: application/json' \
  --data '{"width":0.1,"height":0.2,"innerWidth":0.08,"innerHeight":0.18}'

Example response

{
  "success": true,
  "data": {
    "area": 0.0056000000000000025,
    "centroidX": 0.05,
    "centroidY": 0.1,
    "secondMomentX": 0.000027786666666666686,
    "secondMomentY": 0.000008986666666666672,
    "polarAreaMoment": 0.000036773333333333356,
    "sectionModulusX": 0.00027786666666666684,
    "sectionModulusY": 0.0001797333333333334,
    "radiusOfGyrationX": 0.07044078904941911,
    "radiusOfGyrationY": 0.0400594795867583
  }
}

Model and limits

Concentric rectangular opening with sharp corners. Set both inner dimensions to zero for solid. A = bh − bi hi; Ix = (b h³ − bi hi³)/12; Iy = (h b³ − hi bi³)/12. Wall thickness may differ horizontally and vertically.
All dimensions in metres. Axes pass through the centroid: x horizontal, y vertical. Centroid coordinates are measured from the bottom-left of the outer bounding box. These are area properties, not mass moments of inertia.
Sx = Ix/(outer height/2), Sy = Iy/(outer width/2); rx = √(Ix/A), ry = √(Iy/A). The section modulus has dimensions of length cubed, but is not a volume of material.
Polar area moment Jp = Ix + Iy. It equals the Saint-Venant torsion constant for a circular solid/annulus, not for a rectangle. No rounded corners, eccentric void, strength rating, local buckling or design-code check is included.

References

Engineering Statics: Moments of Inertia of Common Shapes (accessed 2026-09-08)
Engineering Statics: Radius of Gyration (accessed 2026-09-08)
MIT Engineering Beam Theory, Appendix A (accessed 2026-09-08)

Normal Shock: Downstream State and Pressure Loss

POST /api/fluids/normal-shock

Stationary, one-dimensional normal shock; calorically perfect ideal gas, constant R and γ. Upstream Mach must exceed 1 for a shock; exactly 1 is the zero-strength limit. Inputs use the shock-fixed frame, absolute pressure and Kelvin.

Open interactive calculator

Inputs, outputs, example and references

Inputs

Supply every listed field. Defaults below form the example, and are not automatically filled by the API.
JSON field	Meaning and units	Example
absolutePressure	Upstream static absolute pressure (Pa)	101325
absoluteTemperature	Upstream static temperature (K)	288.15
specificGasConstant	Specific gas constant (J/(kg·K))	287.05
heatCapacityRatio	Constant heat-capacity ratio γ	1.4
upstreamMach	Upstream Mach (1 = zero-strength limit)	2

Outputs

Fields inside the response data object. Some model limits return explanatory strings; inspect the example and assumptions.
JSON field	Meaning	Unit
upstreamSpeed	Upstream speed	m/s
downstreamMach	Downstream Mach	Dimensionless / text
pressureRatio	Static pressure ratio p₂/p₁	Dimensionless / text
densityRatio	Density ratio ρ₂/ρ₁	Dimensionless / text
temperatureRatio	Static temperature ratio T₂/T₁	Dimensionless / text
downstreamPressure	Downstream static absolute pressure	Pa
downstreamTemperature	Downstream static temperature	K
downstreamDensity	Downstream density	kg/m³
downstreamSpeed	Downstream speed	m/s
stagnationPressureRatio	Stagnation pressure ratio p₀₂/p₀₁	Dimensionless / text
upstreamStagnationPressure	Upstream stagnation absolute pressure	Pa
downstreamStagnationPressure	Downstream stagnation absolute pressure	Pa
stagnationPressureLoss	Stagnation pressure loss	Pa
stagnationTemperature	Conserved stagnation temperature	K
entropyIncrease	Specific entropy increase	J/(kg·K)

Example request

curl 'https://www.engivault.com/api/fluids/normal-shock' \
  -H 'Content-Type: application/json' \
  --data '{"absolutePressure":101325,"absoluteTemperature":288.15,"specificGasConstant":287.05,"heatCapacityRatio":1.4,"upstreamMach":2}'

Example response

{
  "success": true,
  "data": {
    "upstreamSpeed": 680.5845737305541,
    "downstreamMach": 0.5773502691896257,
    "pressureRatio": 4.5,
    "densityRatio": 2.666666666666667,
    "temperatureRatio": 1.6874999999999998,
    "downstreamPressure": 455962.5,
    "downstreamTemperature": 486.2531249999999,
    "downstreamDensity": 3.2666993759751852,
    "downstreamSpeed": 255.21921514895777,
    "stagnationPressureRatio": 0.7208738614847455,
    "upstreamStagnationPressure": 792812.3017003256,
    "downstreamStagnationPressure": 571517.6653593228,
    "stagnationPressureLoss": 221294.6363410028,
    "stagnationTemperature": 518.67,
    "entropyIncrease": 93.94891209201757
  }
}

Model and limits

Stationary, one-dimensional normal shock; calorically perfect ideal gas, constant R and γ. Upstream Mach must exceed 1 for a shock; exactly 1 is the zero-strength limit. Inputs use the shock-fixed frame, absolute pressure and Kelvin.
With stations 1 upstream and 2 downstream: p₂/p₁ = 1 + 2γ(M₁² − 1)/(γ + 1); ρ₂/ρ₁ = (γ + 1)M₁²/[(γ − 1)M₁² + 2]; T₂/T₁ = (p₂/p₁)/(ρ₂/ρ₁).
M₂² = [(γ − 1)M₁² + 2]/[2γM₁² − (γ − 1)]. Mass conservation gives V₂/V₁ = ρ₁/ρ₂. Total temperature is conserved; total pressure falls and entropy rises.
p₀₂/p₀₁ = [(ρ₂/ρ₁)^γ/(p₂/p₁)]^[1/(γ − 1)]; Δs = −R ln(p₀₂/p₀₁). Extremely weak shocks may display zero loss at floating-point precision.
No oblique shock, boundary-layer interaction, nozzle location, moving-shock frame conversion, heat transfer, real gas or variable heat capacity. This tool does not determine whether a shock forms. Check the constant-property assumption against the calculated temperatures.

References

NASA Glenn: Normal Shock Wave Equations (accessed 2026-09-08)

Gas Speed of Sound, Mach and Stagnation State

POST /api/fluids/gas-flow-state

Calorically perfect ideal gas with constant specific gas constant R and γ > 1. Enter static absolute pressure and Kelvin temperature, not stagnation conditions. Defaults approximate dry air; no atmospheric or composition lookup is performed.

Open interactive calculator

Inputs, outputs, example and references

Inputs

Supply every listed field. Defaults below form the example, and are not automatically filled by the API.
JSON field	Meaning and units	Example
absolutePressure	Static absolute pressure (Pa)	101325
absoluteTemperature	Static absolute temperature (K)	288.15
specificGasConstant	Specific gas constant (J/(kg·K))	287.05
heatCapacityRatio	Constant heat-capacity ratio γ = cp/cv	1.4
speed	Flow speed (m/s)	100

Outputs

Fields inside the response data object. Some model limits return explanatory strings; inspect the example and assumptions.
JSON field	Meaning	Unit
density	Static density	kg/m³
soundSpeed	Local speed of sound	m/s
mach	Mach number	Dimensionless / text
specificHeat	Implied constant-pressure specific heat	J/(kg·K)
temperatureRatio	Stagnation / static temperature	Dimensionless / text
pressureRatio	Stagnation / static pressure	Dimensionless / text
densityRatio	Stagnation / static density	Dimensionless / text
stagnationTemperature	Ideal stagnation temperature	K
stagnationPressure	Ideal stagnation absolute pressure	Pa
stagnationDensity	Ideal stagnation density	kg/m³
stagnationPressureRise	Ideal stagnation minus static pressure	Pa
dynamicPressure	Dynamic pressure ½ρv²	Pa

Example request

curl 'https://www.engivault.com/api/fluids/gas-flow-state' \
  -H 'Content-Type: application/json' \
  --data '{"absolutePressure":101325,"absoluteTemperature":288.15,"specificGasConstant":287.05,"heatCapacityRatio":1.4,"speed":100}'

Example response

{
  "success": true,
  "data": {
    "density": 1.2250122659906943,
    "soundSpeed": 340.29228686527705,
    "mach": 0.293865020924175,
    "specificHeat": 1004.6750000000002,
    "temperatureRatio": 1.017271330104553,
    "pressureRatio": 1.0617660071347597,
    "densityRatio": 1.0437392421406722,
    "stagnationTemperature": 293.1267337696269,
    "stagnationPressure": 107583.44067292953,
    "stagnationDensity": 1.2785933741181548,
    "stagnationPressureRise": 6258.4406729295315,
    "dynamicPressure": 6125.061329953472
  }
}

Model and limits

Calorically perfect ideal gas with constant specific gas constant R and γ > 1. Enter static absolute pressure and Kelvin temperature, not stagnation conditions. Defaults approximate dry air; no atmospheric or composition lookup is performed.
ρ = p/(RT); a = √(γRT); M = v/a. Mach depends on the local gas state, not a universal speed factor. R is per kilogram, not the universal molar gas constant.
Let F = 1 + (γ − 1)M²/2. Then T0/T = F, p0/p = F^[γ/(γ − 1)], and ρ0/ρ = F^[1/(γ − 1)]. The stagnation state represents ideal isentropic deceleration to rest.
cp = γR/(γ − 1) and cp(T0 − T) = v²/2. Dynamic pressure q = ρv²/2 is generally different from p0 − p in compressible flow; they approach one another at low Mach number.
No shocks, friction, heat exchange, nozzle solution, variable heat capacities, condensation, dissociation or real-gas effects. The ideal p0 is not the downstream reading of a supersonic Pitot probe across a shock. Check constant-property validity over the calculated temperature range.

References

NASA Glenn: Isentropic Flow Equations (accessed 2026-09-08)
NASA Glenn: Speed of Sound (accessed 2026-09-08)

Circular Shaft Torsion and Power

POST /api/mechanics/shaft-torsion

Uniform circular shaft with concentric bore, homogeneous isotropic material, constant torque and linear elastic small-strain response. Enter diameters, not radii. Inner diameter zero selects a solid shaft.

Open interactive calculator

Inputs, outputs, example and references

Inputs

Supply every listed field. Defaults below form the example, and are not automatically filled by the API.
JSON field	Meaning and units	Example
outerDiameter	Outer diameter (m)	0.05
innerDiameter	Inner diameter (0 for solid) (m)	0.03
length	Shaft length (m)	1
shearModulus	Shear modulus (Pa)	80000000000
torque	Signed torque (N·m)	500
speed	Signed rotational speed (rpm)	1500

Outputs

Fields inside the response data object. Some model limits return explanatory strings; inspect the example and assumptions.
JSON field	Meaning	Unit
polarMoment	Polar second moment of area	m⁴
torsionalStiffness	Torsional stiffness	N·m/rad
twistRadians	Signed end-to-end twist	rad
twistDegrees	Signed end-to-end twist	°
maximumShearStress	Maximum shear stress magnitude	Pa
maximumShearStrain	Maximum engineering shear strain	Dimensionless / text
angularSpeed	Signed angular speed	rad/s
power	Signed mechanical power	W
strainEnergy	Stored elastic strain energy	J

Example request

curl 'https://www.engivault.com/api/mechanics/shaft-torsion' \
  -H 'Content-Type: application/json' \
  --data '{"outerDiameter":0.05,"innerDiameter":0.03,"length":1,"shearModulus":80000000000,"torque":500,"speed":1500}'

Example response

{
  "success": true,
  "data": {
    "polarMoment": 5.340707511102649e-7,
    "torsionalStiffness": 42725.6600888212,
    "twistRadians": 0.011702569344992302,
    "twistDegrees": 0.6705078329272351,
    "maximumShearStress": 23405138.689984605,
    "maximumShearStrain": 0.0002925642336248076,
    "angularSpeed": 157.07963267948966,
    "power": 78539.81633974484,
    "strainEnergy": 2.9256423362480755
  }
}

Model and limits

Uniform circular shaft with concentric bore, homogeneous isotropic material, constant torque and linear elastic small-strain response. Enter diameters, not radii. Inner diameter zero selects a solid shaft.
J = π(Do⁴ − Di⁴)/32; τmax = |T|Do/(2J). J is a second moment of area, not a mass moment of inertia.
Twist θ = TL/(GJ), stiffness = GJ/L, γmax = τmax/G, elastic energy = Tθ/2. θ is relative end rotation, not the total rotation of the running shaft.
Angular speed ω = 2πn/60 and mechanical power P = Tω. Use the same positive axis for torque and speed. Opposite signs give negative power; reversing torque reverses twist but preserves stress magnitude and elastic energy.
Does not assess keyways, stress concentrations, yielding, combined bending, fatigue, buckling, critical speed, bearings or losses. A noncircular section cannot use this J as its torsion constant. No safe torque or equipment rating is inferred.

References

MIT OpenCourseWare, David Roylance: Shear and Torsion, equations 8 and 12–14 and strain energy (accessed 2026-09-08)

AC Power: Single and Balanced Three Phase

POST /api/electrical/ac-power

Sinusoidal steady-state AC power for a passive single-phase load or a balanced three-phase load. Enter RMS values and displacement power factor from 0 to 1.

Open interactive calculator

Inputs, outputs, example and references

Inputs

Supply every listed field. Defaults below form the example, and are not automatically filled by the API.
JSON field	Meaning and units	Example
phases	Supply configuration `1`: Single phase `3`: Balanced three phase	3
voltage	RMS voltage (line-to-line for three phase) (V)	400
current	RMS current (line current for three phase) (A)	10
powerFactor	Displacement power factor	0.8
reactiveDirection	Load behavior `0`: Lagging, inductive load `1`: Leading, capacitive load	0

Outputs

Fields inside the response data object. Some model limits return explanatory strings; inspect the example and assumptions.
JSON field	Meaning	Unit
activePower	Total active power	W
reactivePower	Total reactive power, positive inductive	var
apparentPower	Total apparent power	VA
phaseAngle	Specified phase angle, positive lagging	°
activePerPhase	Active power per phase	W

Example request

curl 'https://www.engivault.com/api/electrical/ac-power' \
  -H 'Content-Type: application/json' \
  --data '{"phases":3,"voltage":400,"current":10,"powerFactor":0.8,"reactiveDirection":0}'

Example response

{
  "success": true,
  "data": {
    "apparentPower": 6928.203230275509,
    "activePower": 5542.562584220407,
    "reactivePower": 4156.921938165305,
    "phaseAngle": 36.86989764584401,
    "activePerPhase": 1847.5208614068024
  }
}

Model and limits

Sinusoidal steady-state AC power for a passive single-phase load or a balanced three-phase load. Enter RMS values and displacement power factor from 0 to 1.
Single phase: S = VI. Balanced three phase: S = √3 VL IL, where VL is line-to-line RMS voltage and IL is line RMS current. The latter applies to balanced wye or delta loads without substituting phase values for line values.
P = S cosφ = S PF. Q = ±S√(1 − PF²); positive for lagging inductive current and negative for leading capacitive current. Specified angle φ = ±acos(PF) is the per-phase impedance angle, not the angle between a line-to-line voltage and line current. At zero current, all powers are zero; the angle only reflects the specified input.
P is in watts, Q in var and S in volt-amperes. For balanced three phase, active power per phase is P/3. Unity power factor gives Q = 0.
Does not model harmonic distortion, unbalanced phases, exported active power, transients or motor efficiency. Displacement power factor need not equal true power factor for distorted waveforms. Results are not conductor ampacity, protection or equipment-rating selections.

References

US DOE Electrical Science Volume 3, ES-09: power triangle and balanced three-phase loads (accessed 2026-09-08)

Series RLC Circuit and Resonance

POST /api/electrical/series-rlc

Sinusoidal steady-state series RLC circuit with ideal linear components and positive total series resistance. Enter RMS source voltage, frequency in Hz, inductance in henries and capacitance in farads.

Open interactive calculator

Inputs, outputs, example and references

Inputs

Supply every listed field. Defaults below form the example, and are not automatically filled by the API.
JSON field	Meaning and units	Example
voltage	Source RMS voltage (V)	120
frequency	Source frequency (Hz)	60
resistance	Total series resistance (Ω)	30
inductance	Series inductance (0 omits inductor) (H)	0.1
capacitance	Series capacitance (0 bypasses capacitor) (F)	0.0001

Outputs

Fields inside the response data object. Some model limits return explanatory strings; inspect the example and assumptions.
JSON field	Meaning	Unit
current	Series RMS current	A
impedance	Impedance magnitude	Ω
reactance	Net reactance, positive inductive	Ω
inductiveReactance	Inductive reactance	Ω
capacitiveReactance	Capacitive reactance magnitude	Ω
phaseAngle	Voltage phase relative to current	°
powerFactor	Displacement power factor	Dimensionless / text
activePower	Active power in resistance	W
reactivePower	Net reactive power	var
apparentPower	Apparent power	VA
resistorVoltage	Resistor RMS voltage	V
inductorVoltage	Inductor RMS voltage	V
capacitorVoltage	Capacitor RMS voltage	V
resonanceFrequency	Ideal LC resonance frequency	Hz

Example request

curl 'https://www.engivault.com/api/electrical/series-rlc' \
  -H 'Content-Type: application/json' \
  --data '{"voltage":120,"frequency":60,"resistance":30,"inductance":0.1,"capacitance":0.0001}'

Example response

{
  "success": true,
  "data": {
    "inductiveReactance": 37.69911184307752,
    "capacitiveReactance": 26.525823848649225,
    "reactance": 11.173287994428293,
    "impedance": 32.01315924126257,
    "current": 3.7484585353053492,
    "phaseAngle": 20.427490994154418,
    "powerFactor": 0.9371146338263373,
    "activePower": 421.52824172710575,
    "reactivePower": 156.99521475339793,
    "apparentPower": 449.81502423664193,
    "resistorVoltage": 112.45375605916048,
    "inductorVoltage": 141.31355756161489,
    "capacitorVoltage": 99.43095081147537,
    "resonanceFrequency": 50.329212104487034
  }
}

Model and limits

Sinusoidal steady-state series RLC circuit with ideal linear components and positive total series resistance. Enter RMS source voltage, frequency in Hz, inductance in henries and capacitance in farads.
ω = 2πf; XL = ωL; XC = 1/(ωC); X = XL − XC. The special input C = 0 bypasses the capacitor (XC = 0); it does not represent a physical zero-capacitance component. L = 0 omits the inductor.
Z = R + jX, |Z| = √(R² + X²), I = V/|Z|, φ = atan2(X,R), PF = R/|Z|. Positive φ means inductive impedance and lagging current; negative φ means capacitive impedance and leading current.
P = I²R; Q = I²X; S = VI. Component RMS voltages are IR, IXL and IXC. They add as phasors: V² = VR² + (VL − VC)², not as scalar magnitudes.
With both L and C present, f0 = 1/(2π√(LC)). At resonance the net reactance vanishes, but the individual L/C voltages can exceed the source voltage. Finite positive resistance is required; the ideal zero-resistance resonance singularity is excluded.
No switching transient, parasitics beyond the supplied series resistance, saturation, frequency-dependent losses, nonlinear load, harmonics or thermal rating calculation. Component and source ratings require a separate assessment.

References

MIT OpenCourseWare Physics II: AC Circuits, series RLC and average power, sections 12.3–12.4 (accessed 2026-09-08)

Liquid Water Properties by Temperature and Pressure

POST /api/fluids/water-properties

Pure liquid water properties from temperature and absolute pressure, using IAPWS SR6-08(2011) and SR1-86(1992). Supported range: 0.01–110 °C, pressure at or above saturation and no greater than 0.3 MPa (300000 Pa).

Open interactive calculator

Inputs, outputs, example and references

Inputs

Supply every listed field. Defaults below form the example, and are not automatically filled by the API.
JSON field	Meaning and units	Example
temperature	Water temperature (0.01–110 °C) (°C)	20
pressure	Absolute pressure (maximum 300000 Pa) (Pa)	100000

Outputs

Fields inside the response data object. Some model limits return explanatory strings; inspect the example and assumptions.
JSON field	Meaning	Unit
density	Density	kg/m³
viscosity	Dynamic viscosity	Pa·s
kinematicViscosity	Kinematic viscosity	m²/s
specificHeat	Specific heat at constant pressure	J/(kg·K)
conductivity	Thermal conductivity	W/(m·K)
thermalDiffusivity	Thermal diffusivity	m²/s
prandtlNumber	Prandtl number	Dimensionless / text
thermalExpansion	Volumetric thermal expansion coefficient	1/K
compressibility	Isothermal compressibility	1/Pa
soundSpeed	Speed of sound	m/s
vaporPressure	Saturation vapor pressure	Pa
specificVolume	Specific volume	m³/kg

Example request

curl 'https://www.engivault.com/api/fluids/water-properties' \
  -H 'Content-Type: application/json' \
  --data '{"temperature":20,"pressure":100000}'

Example response

{
  "success": true,
  "data": {
    "density": 998.2065464241152,
    "viscosity": 0.0010015672646030015,
    "kinematicViscosity": 0.0000010033667563000116,
    "specificHeat": 4184.1890783969275,
    "conductivity": 0.5980046517984032,
    "thermalDiffusivity": 1.4317686440030226e-7,
    "prandtlNumber": 7.007883295269969,
    "thermalExpansion": 0.00020681031228794193,
    "compressibility": 4.589143232318951e-10,
    "soundSpeed": 1482.3443444561285,
    "vaporPressure": 2339.1937366227544,
    "specificVolume": 0.0010017966758305778
  }
}

Model and limits

Pure liquid water properties from temperature and absolute pressure, using IAPWS SR6-08(2011) and SR1-86(1992). Supported range: 0.01–110 °C, pressure at or above saturation and no greater than 0.3 MPa (300000 Pa).
Calculate the 0.1 MPa reference specific volume and derivatives from SR6 equations 2–3, heat capacity from Table 2, then apply the limited-pressure corrections in Table 4. Density is the reciprocal of specific volume. Table 3 gives expansivity, compressibility and sound speed.
Dynamic viscosity and thermal conductivity use SR6 equations 7–8; section 6 permits these temperature correlations throughout the supported pressure range. Kinematic viscosity = μ/ρ, thermal diffusivity = k/(ρ cp), and Prandtl number = μ cp/k.
Saturation pressure uses SR1 equation 1. Below-saturation pressure, freezing-range temperatures and pressures above 0.3 MPa are rejected. At saturation, outputs describe the liquid branch, not a liquid-vapor mixture.
SR6 reports stable-region uncertainties of about 0.0001% for density, 0.1% for heat capacity, 0.005% for sound speed, 1% for viscosity and 1.5% for conductivity. Extra displayed digits aid reproducibility; they do not imply matching measurement accuracy.
Ordinary pure water only. Salinity, glycol, dissolved additives, steam, ice, supercooled liquid and high-pressure service require other property models. This implementation is independently verified against release tables, not certified by IAPWS.

References

IAPWS SR6-08(2011): Properties of Liquid Water, coefficients, pressure corrections and verification Table 8 (accessed 2026-09-08)
IAPWS SR1-86(1992): Saturation Properties, vapor pressure equation 1 (accessed 2026-09-08)

Water Pipe Pressure Loss with Temperature

POST /api/hydraulics/water-pipe-loss

Calculate pure-water density and viscosity from temperature and inlet pressure, then estimate straight horizontal pipe friction. Liquid-water range: 0.01–110 °C and saturation pressure through 300000 Pa absolute.

Open interactive calculator

Inputs, outputs, example and references

Inputs

Supply every listed field. Defaults below form the example, and are not automatically filled by the API.
JSON field	Meaning and units	Example
temperature	Water temperature (0.01–110 °C) (°C)	20
inletPressure	Absolute inlet pressure (maximum 300000 Pa) (Pa)	200000
flowRate	Volume flow rate (m³/s)	0.001
length	Straight horizontal pipe length (m)	20
diameter	Actual internal diameter (m)	0.025
roughness	Absolute pipe roughness (m)	0.000015

Outputs

Fields inside the response data object. Some model limits return explanatory strings; inspect the example and assumptions.
JSON field	Meaning	Unit
pressureDrop	Friction pressure loss	Pa
headLoss	Friction head loss	m
outletPressure	Estimated absolute outlet pressure	Pa
velocity	Mean velocity	m/s
reynoldsNumber	Reynolds number	Dimensionless / text
frictionFactor	Darcy friction factor	Dimensionless / text
density	Calculated inlet water density	kg/m³
viscosity	Calculated water dynamic viscosity	Pa·s
vaporPressure	Water saturation pressure	Pa
flowRegime	Flow regime	Dimensionless / text

Example request

curl 'https://www.engivault.com/api/hydraulics/water-pipe-loss' \
  -H 'Content-Type: application/json' \
  --data '{"temperature":20,"inletPressure":200000,"flowRate":0.001,"length":20,"diameter":0.025,"roughness":0.000015}'

Example response

{
  "success": true,
  "data": {
    "pressureDrop": 37945.023505827514,
    "frictionFactor": 0.0228978279432567,
    "reynoldsNumber": 50761.0191471177,
    "velocity": 2.0371832715762603,
    "outletPressure": 162054.97649417247,
    "headLoss": 3.8760895853659307,
    "density": 998.2523576546281,
    "viscosity": 0.0010015672646030015,
    "vaporPressure": 2339.1937366227544,
    "flowRegime": "Turbulent"
  }
}

Model and limits

Calculate pure-water density and viscosity from temperature and inlet pressure, then estimate straight horizontal pipe friction. Liquid-water range: 0.01–110 °C and saturation pressure through 300000 Pa absolute.
IAPWS SR6-08(2011) supplies temperature-dependent water properties with limited-pressure corrections. Properties are held at their inlet values along this constant-diameter, isothermal pipe.
Velocity = 4Q/(πD²), Re = ρvD/μ. Darcy f = 64/Re below Re 2300, otherwise the Swamee–Jain approximation. The transitional range 2300–4000 is flagged as uncertain; EPANET transitional interpolation is not implemented. Q = 0 gives zero loss.
ΔP = f(L/D)ρv²/2; head loss = ΔP/(ρg), g = 9.80665 m/s². Outlet pressure = inlet pressure − ΔP. Reject a predicted outlet pressure below saturation rather than applying a single-phase result there.
No fittings, valves, elevation change, pump, developing-flow correction, water hammer or heat transfer. Use actual internal diameter and a documented roughness; defaults are illustrative. A passing saturation check is not a cavitation assessment of other system components.

References

IAPWS SR6-08(2011): Liquid water density and viscosity with pressure corrections (accessed 2026-09-08)
IAPWS SR1-86(1992): Saturation vapor pressure (accessed 2026-09-08)
US EPA: EPANET 2.2 User Manual, Darcy–Weisbach and friction-factor methods (accessed 2026-09-08)

Liquid Control Valve Kv and Choking

POST /api/valves/liquid-sizing

For turbulent liquid flow with no attached fittings correction: ΔP = P1 − P2; FF = 0.96 − 0.28√(Pv/Pc); ΔPchoked = FL²(P1 − FF Pv). All pressures must be absolute and use the same units.

Open interactive calculator

Inputs, outputs, example and references

Inputs

Supply every listed field. Defaults below form the example, and are not automatically filled by the API.
JSON field	Meaning and units	Example
flowRate	Required upstream liquid flow (m³/s)	0.005
density	Liquid density at inlet conditions (kg/m³)	1000
inletPressure	Absolute inlet pressure (Pa)	300000
outletPressure	Absolute outlet pressure (Pa)	250000
vaporPressure	Absolute vapor pressure at inlet temperature (Pa)	3000
criticalPressure	Fluid absolute critical pressure (Pa)	22064000
recoveryFactor	Manufacturer pressure recovery factor, FL	0.9

Outputs

Fields inside the response data object. Some model limits return explanatory strings; inspect the example and assumptions.
JSON field	Meaning	Unit
kv	Required Kv using limiting pressure drop	m³/h
uncorrectedKv	Kv ignoring choking, for comparison only	m³/h
pressureDrop	Actual pressure drop	Pa
chokedPressureDrop	Calculated choking pressure drop	Pa
effectivePressureDrop	Pressure drop used for sizing	Pa
criticalRatio	Liquid critical pressure ratio factor, FF	Dimensionless / text
regime	Choking assessment	Dimensionless / text
phaseAssessment	Outlet phase screening	Dimensionless / text

Example request

curl 'https://www.engivault.com/api/valves/liquid-sizing' \
  -H 'Content-Type: application/json' \
  --data '{"flowRate":0.005,"density":1000,"inletPressure":300000,"outletPressure":250000,"vaporPressure":3000,"criticalPressure":22064000,"recoveryFactor":0.9}'

Example response

{
  "success": true,
  "data": {
    "kv": 25.455844122715714,
    "uncorrectedKv": 25.455844122715714,
    "pressureDrop": 50000,
    "chokedPressureDrop": 240675.1338284906,
    "effectivePressureDrop": 50000,
    "criticalRatio": 0.9567350500038809,
    "regime": "Below calculated choking limit; cavitation not ruled out",
    "phaseAssessment": "Outlet above vapor pressure; assess internal cavitation separately"
  }
}

Model and limits

For turbulent liquid flow with no attached fittings correction: ΔP = P1 − P2; FF = 0.96 − 0.28√(Pv/Pc); ΔPchoked = FL²(P1 − FF Pv). All pressures must be absolute and use the same units.
Use ΔPeffective = min(ΔP, ΔPchoked). Convert Q from m³/s to m³/h and ΔPeffective from Pa to bar. Kv = Q(m³/h)√[(ρ/1000)/(ΔPeffective in bar)]. Kv is a flow coefficient referenced to water and 1 bar, not the actual service flow.
Assumes turbulent service, liquid at the inlet and piping geometry factor FP = 1. No Reynolds/viscosity correction, attached reducers, staged trim, gas, steam or inlet two-phase sizing is implemented.
FL must match manufacturer data for the candidate valve, trim and travel. Defaults are teaching assumptions, not a selected valve or verified fluid-property state. Repeat at minimum, normal and maximum operating cases.
Below the choking limit does not establish cavitation-free operation. Outlet pressure below vapor pressure indicates flashing; damage and noise are not predicted. Assess pressure recovery, materials and service limits with manufacturer data.
Kv alone does not determine nominal valve size, opening, rangeability, actuator capability or control stability. The comparison ignoring choking must not replace the limiting-pressure result.

References

Bürkert: Fluid Calculator, liquid Kv equation (accessed 2026-09-08)
Emerson: Understanding Choked Flow in Fisher Valves, D104173X012, September 2017 (accessed 2026-09-08)
Emerson: Control Valve Sourcebook, liquid sizing and limiting pressure drop (accessed 2026-09-08)

Pipe Heat Transfer and Insulation

POST /api/thermal/insulated-pipe

Steady one-dimensional radial heat flow, constant properties and bulk temperatures along the length; perfect layer contact, no internal heat generation. Actual diameters are required, not nominal pipe sizes.

Open interactive calculator

Inputs, outputs, example and references

Inputs

Supply every listed field. Defaults below form the example, and are not automatically filled by the API.
JSON field	Meaning and units	Example
length	Pipe length (m)	10
innerDiameter	Actual pipe inner diameter (m)	0.020
pipeOuterDiameter	Actual pipe outer diameter (m)	0.022
pipeConductivity	Pipe thermal conductivity (W/(m·K))	380
insulationThickness	Radial insulation thickness (0 for bare pipe) (m)	0.020
insulationConductivity	Insulation thermal conductivity (W/(m·K))	0.040
innerFilm	Inside convection coefficient (W/(m²·K))	1000
outerFilm	Outside convection coefficient (W/(m²·K))	10
fluidTemperature	Bulk fluid temperature (K)	333.15
ambientTemperature	Ambient fluid temperature (K)	293.15

Outputs

Fields inside the response data object. Some model limits return explanatory strings; inspect the example and assumptions.
JSON field	Meaning	Unit
heatRate	Heat transfer, positive outward	W
heatPerLength	Heat transfer per length	W/m
thermalResistance	Total thermal resistance	K/W
innerSurface	Pipe inner surface temperature	K
pipeOuterSurface	Pipe outer surface temperature	K
outerSurface	Exposed surface temperature	K
bareHeatRate	Bare pipe heat transfer, same film coefficients	W
magnitudeReduction	Reduction in heat-transfer magnitude vs bare pipe	W
outerDiameter	Diameter including insulation	m

Example request

curl 'https://www.engivault.com/api/thermal/insulated-pipe' \
  -H 'Content-Type: application/json' \
  --data '{"length":10,"innerDiameter":0.02,"pipeOuterDiameter":0.022,"pipeConductivity":380,"insulationThickness":0.02,"insulationConductivity":0.04,"innerFilm":1000,"outerFilm":10,"fluidTemperature":333.15,"ambientTemperature":293.15}'

Example response

{
  "success": true,
  "data": {
    "heatRate": 85.98753076301945,
    "heatPerLength": 8.598753076301945,
    "thermalResistance": 0.4651837265828635,
    "innerSurface": 333.013146594348,
    "pipeOuterSurface": 333.0128033437506,
    "outerSurface": 297.5646259887745,
    "bareHeatRate": 273.4447173431028,
    "magnitudeReduction": 187.45718658008337,
    "outerDiameter": 0.062
  }
}

Model and limits

Steady one-dimensional radial heat flow, constant properties and bulk temperatures along the length; perfect layer contact, no internal heat generation. Actual diameters are required, not nominal pipe sizes.
For length L: inner-film resistance = 1/(hi π Di L); pipe resistance = ln(Do/Di)/(2π kp L); insulation resistance = ln((Do + 2t)/Do)/(2π ki L); outer-film resistance = 1/(ho π (Do + 2t) L).
Add the four resistances. Q = (Tfluid − Tambient)/Rtotal; heat per length = Q/L. Each surface temperature follows from the preceding temperature minus Q times that layer resistance. Negative Q means heat enters the pipe.
Bare comparison removes insulation and uses the bare outer surface area with the same film coefficients. Magnitude reduction = |Qbare| − |Q|; a negative result means increased heat transfer. Added insulation need not reduce heat transfer for every radius and fixed coefficient.
Defaults are a constructed teaching case, not certified copper or insulation properties. Supply conductivities at the relevant temperatures and convection coefficients for the actual flow conditions. The comparison assumes these coefficients remain unchanged.
Radiation, axial fluid cooling, fittings/supports, thermal bridges, fouling/contact resistance, moisture, condensation and transient effects are excluded. This is not an insulation product selection, minimum thickness requirement or economic optimum.

References

US DOE Fundamentals Handbook, Heat Transfer: cylindrical conduction pp. 11–17 and combined convection pp. 20–23 (accessed 2026-09-08)

Rectangular Beam Stress and Deflection

POST /api/mechanics/rectangular-beam

Analyze a solid rectangular beam under a downward point load plus a uniform full-span load. Choose simple supports with a central point load, or a left-fixed cantilever with a tip load.

Open interactive calculator

Inputs, outputs, example and references

Inputs

Supply every listed field. Defaults below form the example, and are not automatically filled by the API.
JSON field	Meaning and units	Example
support	Support and point-load location `0`: Simply supported; point load at center `1`: Cantilever; point load at free tip	0
span	Span from left support to right end (m)	3
width	Rectangle width, parallel to neutral axis (m)	0.1
depth	Rectangle depth, in bending direction (m)	0.2
youngModulus	Young’s modulus (Pa)	200000000000
pointLoad	Downward concentrated load (N)	1000
lineLoad	Downward load per length over full span (N/m)	500

Outputs

Fields inside the response data object. Some model limits return explanatory strings; inspect the example and assumptions.
JSON field	Meaning	Unit
area	Section area	m²
secondMoment	Second moment of area about bending axis	m⁴
sectionModulus	Elastic section modulus	m³
rigidity	Flexural rigidity EI	N·m²
totalLoad	Total downward load	N
leftReaction	Left support reaction, upward	N
rightReaction	Right support reaction, upward	N
fixedReactionMoment	Fixed-end external reaction, counterclockwise	N·m
maximumMoment	Maximum bending moment magnitude	N·m
maximumBendingStress	Maximum elastic bending stress magnitude	Pa
maximumShear	Maximum shear force magnitude	N
maximumShearStress	Maximum rectangular-section shear stress	Pa
maximumDeflection	Maximum downward bending deflection	m
deflectionSpanRatio	Deflection / span	1
criticalLocations	Critical locations	Dimensionless / text

Example request

curl 'https://www.engivault.com/api/mechanics/rectangular-beam' \
  -H 'Content-Type: application/json' \
  --data '{"support":0,"span":3,"width":0.1,"depth":0.2,"youngModulus":200000000000,"pointLoad":1000,"lineLoad":500}'

Example response

{
  "success": true,
  "data": {
    "area": 0.020000000000000004,
    "secondMoment": 0.00006666666666666668,
    "sectionModulus": 0.0006666666666666669,
    "rigidity": 13333333.333333336,
    "totalLoad": 2500,
    "leftReaction": 1250,
    "rightReaction": 1250,
    "fixedReactionMoment": 0,
    "maximumMoment": 1312.5,
    "maximumShear": 1250,
    "maximumDeflection": 0.00008173828125,
    "deflectionSpanRatio": 0.000027246093749999998,
    "criticalLocations": "Moment and deflection at midspan",
    "maximumBendingStress": 1968749.9999999995,
    "maximumShearStress": 93749.99999999999
  }
}

Model and limits

Analyze a solid rectangular beam under a downward point load plus a uniform full-span load. Choose simple supports with a central point load, or a left-fixed cantilever with a tip load.
Dimensions define the bending axis: I = bd³/12, S = bd²/6, A = bd. Depth is the dimension in the direction of transverse deflection.
For simple supports: each reaction = (P + qL)/2; Mmax = PL/4 + qL²/8; δmax = PL³/(48EI) + 5qL⁴/(384EI).
For a cantilever: upward reaction = P + qL; external reaction moment = PL + qL²/2; δmax = PL³/(3EI) + qL⁴/(8EI).
Bending stress magnitude = Mmax/S; maximum rectangular-section shear stress = 1.5Vmax/A. These maxima occur at different section locations and are not a combined failure criterion.
Assumes a straight, prismatic, homogeneous beam with linear elasticity, small deflections and ideal supports. Deflection includes bending only; shear deformation, torsion, axial force and support settlement are excluded.
Enter total service loads, including self-weight in the uniform load if needed. Both loads are downward and may be zero. No material strength, buckling, fatigue, connection, deflection-limit or building-code check is performed.
With the illustrative defaults: each support carries 1250 N, maximum moment is 1312.5 N·m, and maximum deflection is 0.0000817383 m. The modulus is an example, not a verified material selection.

References

MIT Solid Mechanics: beam displacement formulas, page 2 (accessed 2026-09-08)
MIT Mechanics of Materials: stresses in beams (accessed 2026-09-08)

Humid Air Properties

POST /api/hvac/humid-air

Calculate humidity ratio, dew/frost point, enthalpy, specific volume and moist-air density from temperature, relative humidity and absolute pressure.

Open interactive calculator

Inputs, outputs, example and references

Inputs

Supply every listed field. Defaults below form the example, and are not automatically filled by the API.
JSON field	Meaning and units	Example
dryBulb	Dry-bulb temperature (°C)	25
relativeHumidity	Relative humidity (%)	50
pressure	Total absolute pressure (Pa)	101325

Outputs

Fields inside the response data object. Some model limits return explanatory strings; inspect the example and assumptions.
JSON field	Meaning	Unit
humidityRatio	Humidity ratio	kg water/kg dry air
specificHumidity	Specific humidity	kg water/kg moist air
dewFrostPoint	Dew point over water / frost point over ice	°C
enthalpy	Enthalpy per mass of dry air	J/kg dry air
specificVolume	Volume per mass of dry air	m³/kg dry air
density	Total moist-air density	kg/m³
vaporPressure	Water-vapor partial pressure	Pa
saturationVaporPressure	Saturation vapor pressure	Pa
saturationBasis	Relative humidity reference phase	Dimensionless / text

Example request

curl 'https://www.engivault.com/api/hvac/humid-air' \
  -H 'Content-Type: application/json' \
  --data '{"dryBulb":25,"relativeHumidity":50,"pressure":101325}'

Example response

{
  "success": true,
  "data": {
    "saturationVaporPressure": 3169.2164701436277,
    "vaporPressure": 1584.6082350718139,
    "humidityRatio": 0.009881043690749623,
    "specificHumidity": 0.009784363962945561,
    "enthalpy": 50321.958802184665,
    "specificVolume": 0.8580432638526019,
    "density": 1.17695818641638,
    "dewFrostPoint": 13.863973269509785,
    "saturationBasis": "Liquid water"
  }
}

Model and limits

Calculate humidity ratio, dew/frost point, enthalpy, specific volume and moist-air density from temperature, relative humidity and absolute pressure.
Supported inputs: −50 to 80 °C, 0–100% RH and 20–120 kPa absolute. Total pressure must exceed saturation vapor pressure at the dry-bulb temperature.
ASHRAE saturation equations use ice at or below 0.01 °C and liquid water above it. RH is relative to that phase; subfreezing RH reported relative to liquid water must be converted before use.
Water-vapor pressure is RH/100 times saturation pressure. W = 0.621945 pv/(p − pv); specific humidity = W/(1 + W).
Enthalpy h = 1000[1.006T + W(2501 + 1.86T)] J/kg dry air, with T in °C and the conventional dry-air/liquid-water zero at 0 °C. It is not enthalpy per kg of moist air.
v = 287.042(T + 273.15)(1 + 1.607858W)/p; total density = (1 + W)/v. Assumes ideal moist air with no liquid droplets or real-gas corrections.
Dew/frost point inverts the saturation equation. Dry air has no finite dew point; results below −100 °C are reported outside the model range. These labels are returned as strings in the API.
Adapted coefficients from MIT-licensed PsychroLib; the numerical inversion, exact zero-humidity behavior and restricted operating domain differ from that library.

References

PsychroLib: ASHRAE-based psychrometric equations and documentation (accessed 2026-09-08)
PsychroLib MIT license and adaptation attribution (accessed 2026-09-08)

Air Heating, Cooling and Condensate

POST /api/hvac/air-temperature-process

Uniformly heat or cool an ideal moist-air stream at constant pressure to a specified outlet temperature. If cooling reaches saturation, remove liquid water and return saturated outlet air. No bypass or added moisture is modeled.

Open interactive calculator

Inputs, outputs, example and references

Inputs

Supply every listed field. Defaults below form the example, and are not automatically filled by the API.
JSON field	Meaning and units	Example
dryAirFlow	Dry-air mass flow (kg dry air/s)	1
inletTemperature	Inlet dry-bulb temperature (°C)	30
inletRelativeHumidity	Inlet relative humidity (%)	50
outletTemperature	Target outlet temperature (0–80 °C) (°C)	10
pressure	Common absolute pressure (Pa)	101325

Outputs

Fields inside the response data object. Some model limits return explanatory strings; inspect the example and assumptions.
JSON field	Meaning	Unit
heatRemoved	Heat removed, negative for heat added	W
airEnthalpyReduction	Air-stream enthalpy reduction	W
condensateEnthalpyFlow	Energy leaving in drained water	W
condensateFlow	Condensate mass flow	kg/s
condensateHourly	Hourly condensate mass	kg/h
inletHumidityRatio	Inlet humidity ratio	kg water/kg dry air
outletHumidityRatio	Outlet humidity ratio	kg water/kg dry air
outletRelativeHumidity	Outlet relative humidity	%
inletEnthalpy	Inlet enthalpy	J/kg dry air
outletEnthalpy	Outlet enthalpy	J/kg dry air
inletVolumeFlow	Inlet volume flow	m³/s
outletVolumeFlow	Outlet volume flow	m³/s
process	Ideal process	Dimensionless / text

Example request

curl 'https://www.engivault.com/api/hvac/air-temperature-process' \
  -H 'Content-Type: application/json' \
  --data '{"dryAirFlow":1,"inletTemperature":30,"inletRelativeHumidity":50,"outletTemperature":10,"pressure":101325}'

Example response

{
  "success": true,
  "data": {
    "heatRemoved": 34689.07477920524,
    "airEnthalpyReduction": 34926.84586387177,
    "condensateEnthalpyFlow": 237.77108466652973,
    "condensateFlow": 0.005680150135368603,
    "condensateHourly": 20.44854048732697,
    "inletHumidityRatio": 0.01331020383863019,
    "outletHumidityRatio": 0.0076300537032615865,
    "outletRelativeHumidity": 100,
    "inletEnthalpy": 64211.52917460966,
    "outletEnthalpy": 29284.68331073789,
    "inletVolumeFlow": 0.8771677404250905,
    "outletVolumeFlow": 0.8119717645777006,
    "process": "Cooling with liquid-water removal"
  }
}

Model and limits

Uniformly heat or cool an ideal moist-air stream at constant pressure to a specified outlet temperature. If cooling reaches saturation, remove liquid water and return saturated outlet air. No bypass or added moisture is modeled.
Dry-air mass flow must be positive. Inlet: −50 to 80 °C and 0–100% RH. Outlet: 0–80 °C, excluding ice/frost processing. Pressure: 20–120 kPa absolute and above saturation pressure at both temperatures.
Wout = min(Win, Wsat(Tout)). Without condensation, W is unchanged and RH changes with temperature. With condensation, outlet RH is 100%. Condensate flow = mda(Win − Wout).
Heat removed = mda(hin − hout) − mcond hw,out. Here h is in J/kg dry air, and liquid-water enthalpy is approximated as hw,out = 4186 Tout J/kg relative to liquid water at 0 °C. Condensate leaves at outlet temperature.
Positive heatRemoved means cooling; negative means heating. It is thermal transfer, not electrical input, compressor power or a selected equipment capacity. No arbitrary sensible/latent split is imposed.
Convert measured inlet volume flow to dry-air mass flow with mda = Qin/vin using the inlet specific volume. Moist-air mass flow is a different basis. The output shows both inlet and outlet volume flows.
No coil bypass factor, apparatus dew point, fan heat, heat leakage, pressure drop, frost, re-evaporation, humidity control or real-gas correction. A real coil can produce an unsaturated outlet; this ideal saturation model does not predict its detailed performance.

References

ASHRAE Fundamentals 2025, Chapter 1: sensible heating/cooling and cooling with dehumidification (accessed 2026-09-08)
PsychroLib moist-air enthalpy, saturation and volume relationships (accessed 2026-09-08)
PsychroLib MIT license and adaptation attribution (accessed 2026-09-08)

Adiabatic Mixing of Two Humid Airstreams

POST /api/hvac/air-mixing

Mix two ideal moist-air streams adiabatically at a common pressure, with no work, heat transfer, added water or condensation. Inputs are dry-air mass flows, not total moist-air mass flows or volume flows.

Open interactive calculator

Inputs, outputs, example and references

Inputs

Supply every listed field. Defaults below form the example, and are not automatically filled by the API.
JSON field	Meaning and units	Example
dryAirFlow1	Stream 1 dry-air mass flow (kg dry air/s)	1
dryBulb1	Stream 1 dry-bulb temperature (°C)	10
relativeHumidity1	Stream 1 relative humidity (%)	60
dryAirFlow2	Stream 2 dry-air mass flow (kg dry air/s)	3
dryBulb2	Stream 2 dry-bulb temperature (°C)	25
relativeHumidity2	Stream 2 relative humidity (%)	50
pressure	Common absolute pressure (Pa)	101325

Outputs

Fields inside the response data object. Some model limits return explanatory strings; inspect the example and assumptions.
JSON field	Meaning	Unit
dryBulb	Mixed dry-bulb temperature	°C
relativeHumidity	Mixed relative humidity	%
totalDryAirFlow	Total dry-air mass flow	kg dry air/s
stream1DryAirFraction	Stream 1 dry-air mass fraction	Dimensionless / text
moistAirFlow	Total moist-air mass flow	kg/s
vaporMassFlow	Water-vapor mass flow	kg water/s
volumeFlow	Mixed volume flow	m³/s
enthalpyFlow	Mixed enthalpy flow (0°C reference)	W
humidityRatio	Humidity ratio	kg water/kg dry air
specificHumidity	Specific humidity	kg water/kg moist air
dewFrostPoint	Dew point over water / frost point over ice	°C
enthalpy	Enthalpy per mass of dry air	J/kg dry air
specificVolume	Volume per mass of dry air	m³/kg dry air
density	Total moist-air density	kg/m³
vaporPressure	Water-vapor partial pressure	Pa
saturationVaporPressure	Saturation pressure at dry bulb	Pa
saturationBasis	RH reference phase at dry bulb	Dimensionless / text

Example request

curl 'https://www.engivault.com/api/hvac/air-mixing' \
  -H 'Content-Type: application/json' \
  --data '{"dryAirFlow1":1,"dryBulb1":10,"relativeHumidity1":60,"dryAirFlow2":3,"dryBulb2":25,"relativeHumidity2":50,"pressure":101325}'

Example response

{
  "success": true,
  "data": {
    "saturationVaporPressure": 2530.3811724783395,
    "vaporPressure": 1373.9981351015892,
    "humidityRatio": 0.008549701895843288,
    "specificHumidity": 0.00847722415640181,
    "enthalpy": 43126.08973599568,
    "specificVolume": 0.8455441914255236,
    "density": 1.1927817754805987,
    "dewFrostPoint": 11.688048771883729,
    "saturationBasis": "Liquid water",
    "totalDryAirFlow": 4,
    "stream1DryAirFraction": 0.25,
    "dryBulb": 21.27726123925074,
    "relativeHumidity": 54.30004578147607,
    "moistAirFlow": 4.0341988075833735,
    "vaporMassFlow": 0.03419880758337315,
    "volumeFlow": 3.3821767657020945,
    "enthalpyFlow": 172504.35894398272
  }
}

Model and limits

Mix two ideal moist-air streams adiabatically at a common pressure, with no work, heat transfer, added water or condensation. Inputs are dry-air mass flows, not total moist-air mass flows or volume flows.
Each inlet uses −50 to 80 °C, 0–100% RH and 20–120 kPa absolute, with pressure above saturation pressure at dry bulb. Both inlet states must be valid even if one flow is zero. Flows must be nonnegative and their sum positive.
Using dry-air flows m1 and m2, W = (m1W1 + m2W2)/(m1 + m2) and h = (m1h1 + m2h2)/(m1 + m2). Dry-air mass, water-vapor mass and enthalpy are conserved.
For h in J/kg dry air, T = (h − 2501000W)/(1006 + 1860W) °C. Relative humidity follows from vapor pressure at this mixed temperature. Do not average inlet RH or use volume-flow fractions directly.
Convert an inlet volume flow Q to dry-air flow with m = Q/v, using that inlet's specific volume in m³/kg dry air. Mixed volume flow is the total dry-air flow times mixed specific volume; total moist-air flow includes vapor mass.
RH uses ice saturation at or below 0.01 °C and liquid water above it. A supersaturated vapor-only prediction is rejected: condensation or ice and their energy effects require a separate equilibrium model.
Enthalpy flow is referenced to the model's 0 °C datum and is not a heating or cooling load. Pressure losses, fans, stratification and incomplete mixing are excluded.

References

ASHRAE Fundamentals 2025, Chapter 1: Adiabatic Mixing of Two Moist Airstreams (accessed 2026-09-08)
PsychroLib ideal moist-air property relationships (accessed 2026-09-08)
PsychroLib MIT license and adaptation attribution (accessed 2026-09-08)

Humid Air from Dry and Wet Bulb

POST /api/hvac/humid-air-wet-bulb

Calculate an ideal moist-air state from dry bulb, thermodynamic wet bulb and absolute pressure, using ASHRAE relationships adapted from MIT-licensed PsychroLib.

Open interactive calculator

Inputs, outputs, example and references

Inputs

Supply every listed field. Defaults below form the example, and are not automatically filled by the API.
JSON field	Meaning and units	Example
dryBulb	Dry-bulb temperature (°C)	30
wetBulb	Thermodynamic wet-bulb temperature (°C)	25
pressure	Total absolute pressure (Pa)	95461

Outputs

Fields inside the response data object. Some model limits return explanatory strings; inspect the example and assumptions.
JSON field	Meaning	Unit
relativeHumidity	Relative humidity at dry bulb	%
humidityRatio	Humidity ratio	kg water/kg dry air
specificHumidity	Specific humidity	kg water/kg moist air
dewFrostPoint	Dew point over water / frost point over ice	°C
enthalpy	Enthalpy per mass of dry air	J/kg dry air
specificVolume	Volume per mass of dry air	m³/kg dry air
density	Total moist-air density	kg/m³
vaporPressure	Water-vapor partial pressure	Pa
saturationVaporPressure	Saturation pressure at dry bulb	Pa
saturationBasis	RH reference phase at dry bulb	Dimensionless / text
wetBulbDepression	Dry minus wet bulb	°C
wetBulbBasis	Wet-bulb energy-balance phase	Dimensionless / text

Example request

curl 'https://www.engivault.com/api/hvac/humid-air-wet-bulb' \
  -H 'Content-Type: application/json' \
  --data '{"dryBulb":30,"wetBulb":25,"pressure":95461}'

Example response

{
  "success": true,
  "data": {
    "saturationVaporPressure": 4246.030243592604,
    "vaporPressure": 2862.2930147265274,
    "humidityRatio": 0.01922476983752272,
    "specificHumidity": 0.01886214935748411,
    "enthalpy": 79333.8915205781,
    "specificVolume": 0.9397191727619008,
    "density": 1.084605698574766,
    "dewFrostPoint": 23.30236309478508,
    "saturationBasis": "Liquid water",
    "relativeHumidity": 67.41103691020146,
    "wetBulbDepression": 5,
    "wetBulbBasis": "Liquid water"
  }
}

Model and limits

Calculate an ideal moist-air state from dry bulb, thermodynamic wet bulb and absolute pressure, using ASHRAE relationships adapted from MIT-licensed PsychroLib.
Dry bulb: −50 to 80 °C. Wet bulb: −50 °C through the dry-bulb temperature. Pressure: 20–120 kPa absolute, above saturation pressure at dry bulb. Pairs implying negative moisture content are rejected rather than clamped to a small positive value.
Let Ws = 0.621945 ps(Tw)/(p − ps(Tw)). For Tw ≥ 0 °C, W = [(2501 − 2.326Tw)Ws − 1.006(Td − Tw)] / [2501 + 1.86Td − 4.186Tw].
For Tw < 0 °C, use the ice balance: W = [(2830 − 0.24Tw)Ws − 1.006(Td − Tw)] / [2830 + 1.86Td − 2.1Tw]. The wet-bulb energy-balance branch changes at 0 °C; the saturation-pressure function changes at the 0.01 °C triple point.
pv = pW/(0.621945 + W); RH = 100pv/ps(Td). RH uses the saturation reference phase at dry bulb, which can differ from the wet-bulb balance phase. Enthalpy and specific volume are per kg of dry air; specific humidity and total density include water vapor.
Thermodynamic wet bulb is an adiabatic-saturation property. A physical wet-bulb thermometer can be affected by ventilation, radiation, wick condition and ice formation; this tool does not correct an instrument reading. No fog, liquid droplets, supersaturation or real-gas correction is included.
For a state given by relative humidity instead, use the Humid Air Properties calculator. Neither tool solves a full HVAC process or selects equipment.

References

PsychroLib: humidity ratio from wet bulb, ASHRAE equations 33 and 35 (accessed 2026-09-08)
PsychroLib MIT license and adaptation attribution (accessed 2026-09-08)

Composite Wall Heat Transfer

POST /api/thermal/composite-wall

Model up to three solid layers between two fluids. Number layers from side 1 toward side 2; set unused layer thicknesses to zero and retain positive conductivity inputs.

Open interactive calculator

Inputs, outputs, example and references

Inputs

Supply every listed field. Defaults below form the example, and are not automatically filled by the API.
JSON field	Meaning and units	Example
area	Common wall area (m²)	10
temperature1	Side 1 bulk fluid temperature (K)	293.15
temperature2	Side 2 bulk fluid temperature (K)	273.15
film1	Side 1 convection coefficient (W/(m²·K))	10
film2	Side 2 convection coefficient (W/(m²·K))	25
thickness1	Layer 1 thickness (zero to omit) (m)	.1
conductivity1	Layer 1 conductivity (W/(m·K))	1
thickness2	Layer 2 thickness (zero to omit) (m)	.05
conductivity2	Layer 2 conductivity (W/(m·K))	.04
thickness3	Layer 3 thickness (zero to omit) (m)	0
conductivity3	Layer 3 conductivity (W/(m·K))	1

Outputs

Fields inside the response data object. Some model limits return explanatory strings; inspect the example and assumptions.
JSON field	Meaning	Unit
heatRate	Heat rate, side 1 toward side 2	W
heatFlux	Heat flux, side 1 toward side 2	W/m²
overallCoefficient	Overall heat-transfer coefficient	W/(m²·K)
resistancePerArea	Area-normalized thermal resistance	m²·K/W
thermalResistance	Total thermal resistance	K/W
surface1	Side 1 wall surface	K
interface12	Layer 1 / 2 interface	K
interface23	Layer 2 / 3 interface	K
surface2	Side 2 wall surface	K

Example request

curl 'https://www.engivault.com/api/thermal/composite-wall' \
  -H 'Content-Type: application/json' \
  --data '{"area":10,"temperature1":293.15,"temperature2":273.15,"film1":10,"film2":25,"thickness1":0.1,"conductivity1":1,"thickness2":0.05,"conductivity2":0.04,"thickness3":0,"conductivity3":1}'

Example response

{
  "success": true,
  "data": {
    "heatRate": 134.22818791946307,
    "heatFlux": 13.422818791946307,
    "overallCoefficient": 0.6711409395973154,
    "resistancePerArea": 1.4900000000000002,
    "thermalResistance": 0.14900000000000002,
    "surface1": 291.80771812080536,
    "interface12": 290.46543624161075,
    "interface23": 273.6869127516779,
    "surface2": 273.6869127516778
  }
}

Model and limits

Model up to three solid layers between two fluids. Number layers from side 1 toward side 2; set unused layer thicknesses to zero and retain positive conductivity inputs.
R″ = 1/h₁ + Σ(Lᵢ/kᵢ) + 1/h₂; U = 1/R″. Each term uses the same wall area.
q″ = U(T₁ − T₂); Q = Aq″. Positive heat flow is from side 1 toward side 2.
Subtract q″ times each successive resistance to obtain surface and interface temperatures. Zero-thickness layers have no temperature drop.
Assumes steady one-dimensional heat flow, constant properties and perfect layer contact, without heat generation, radiation, air gaps, thermal bridges or phase change. Supply coefficients and conductivities appropriate to the operating conditions.
Illustrative inputs are not specified material properties or regulatory U-values. At the defaults, R″ = 1.49 m²·K/W and Q = 134.2282 W.

References

US DOE Fundamentals Handbook: conduction and convection, HT-02 pages 9–10 and 20–22 (accessed 2026-09-08)

Reservoir-to-Reservoir Pump Duty

POST /api/pumps/duty

Combine reservoir elevation and pressure differences with pipe and local losses at an entered flow rate.

Open interactive calculator

Inputs, outputs, example and references

Inputs

Supply every listed field. Defaults below form the example, and are not automatically filled by the API.
JSON field	Meaning and units	Example
flowRate	Imposed liquid flow (m³/s)	0.01
length	Total straight-pipe length (m)	100
diameter	Uniform internal diameter (m)	0.1
roughness	Absolute pipe roughness (m)	0.000045
density	Liquid density (kg/m³)	998
viscosity	Dynamic viscosity (Pa·s)	0.001
elevationRise	Destination minus source surface elevation (m)	10
pressureDifference	Destination minus source surface pressure (Pa)	0
lossCoefficient	Total fitting and entry/exit loss coefficient (1)	5
pumpEfficiency	Pump efficiency, fraction (0–1)	0.7
motorEfficiency	Motor efficiency, fraction (0–1)	0.9

Outputs

Fields inside the response data object. Some model limits return explanatory strings; inspect the example and assumptions.
JSON field	Meaning	Unit
systemHead	Signed system head	m
pumpHead	Head to add by pumping	m
surplusHead	Surplus head requiring dissipation	m
frictionHead	Straight-pipe friction head	m
fittingHead	Local-loss head	m
pressureHead	Reservoir pressure head difference	m
hydraulicPower	Hydraulic power	W
shaftPower	Pump shaft power	W
electricalPower	Motor electrical input	W
velocity	Pipe velocity	m/s
reynoldsNumber	Reynolds number	1
flowRegime	Flow regime	Dimensionless / text

Example request

curl 'https://www.engivault.com/api/pumps/duty' \
  -H 'Content-Type: application/json' \
  --data '{"flowRate":0.01,"length":100,"diameter":0.1,"roughness":0.000045,"density":998,"viscosity":0.001,"elevationRise":10,"pressureDifference":0,"lossCoefficient":5,"pumpEfficiency":0.7,"motorEfficiency":0.9}'

Example response

{
  "success": true,
  "data": {
    "frictionHead": 1.619543713520909,
    "fittingHead": 0.4132754147128235,
    "pressureHead": 0,
    "systemHead": 12.032819128233731,
    "pumpHead": 12.032819128233731,
    "surplusHead": 0,
    "hydraulicPower": 1177.6564241248552,
    "shaftPower": 1682.3663201783647,
    "electricalPower": 1869.295911309294,
    "velocity": 1.2732395447351625,
    "reynoldsNumber": 127069.30656456923,
    "flowRegime": "Turbulent"
  }
}

Model and limits

Combine reservoir elevation and pressure differences with pipe and local losses at an entered flow rate.
Hsystem = Δz + Δp/(ρg) + hf + K v²/(2g). Reservoir velocities are negligible; all loss coefficients must refer to the same pipe velocity.
Pipe friction uses Darcy–Weisbach with 64/Re below Re 2300 and Swamee–Jain otherwise. The transition range is uncertain.
For positive head: Phydraulic = ρgQH, Pshaft = Phydraulic/ηpump, Pelectrical = Pshaft/ηmotor. Enter efficiencies as fractions, not percentages.
Negative system head indicates surplus driving head at the imposed flow. Pumping power is zero in that case; control or dissipation would be needed. This is not turbine-output prediction.
Use liquid properties for the operating temperature and pressure. Assumes steady incompressible single-phase flow and one uniform pipe bore; include entrance, exit, valve and fitting losses in K.
This is a duty estimate, not a pump selection, actual operating-point solution, motor nameplate rating or NPSH/cavitation check. Use manufacturer curves and an application-specific margin separately.

References

US EPA EPANET 2.2: Pipe friction and minor losses (accessed 2026-09-08)
KSB: Pump power input (accessed 2026-09-08)
KSB: Pump efficiency (accessed 2026-09-08)

Sound Intensity Level

POST /api/acoustics/sound-intensity-level

Convert positive intensity to level: L = 10 log₁₀(I/Iref).

Open interactive calculator

Inputs, outputs, example and references

Inputs

Supply every listed field. Defaults below form the example, and are not automatically filled by the API.
JSON field	Meaning and units	Example
intensity	Sound intensity (W/m²)	0.000001
referenceIntensity	Reference intensity (W/m²)	1e-12

Outputs

Fields inside the response data object. Some model limits return explanatory strings; inspect the example and assumptions.
JSON field	Meaning	Unit
intensityLevel	Intensity level relative to reference	dB

Example request

curl 'https://www.engivault.com/api/acoustics/sound-intensity-level' \
  -H 'Content-Type: application/json' \
  --data '{"intensity":0.000001,"referenceIntensity":1e-12}'

Example response

{
  "success": true,
  "data": {
    "intensityLevel": 60
  }
}

Model and limits

Convert positive intensity to level: L = 10 log₁₀(I/Iref).
Default reference: 10⁻¹² W/m². Example: I = 10⁻⁶ W/m² gives 60 dB.
Negative levels are valid below the reference. Zero intensity corresponds to negative infinity and is excluded.
This is intensity level, not sound-pressure level, A-weighted level or perceived loudness.

References

OpenStax University Physics: Sound Intensity (accessed 2026-09-08)

Spherical Sound Spreading

POST /api/acoustics/spherical-sound-spreading

Ideal full-sphere spreading: I = P/(4πr²), then L = 10 log₁₀(I/Iref).

Open interactive calculator

Inputs, outputs, example and references

Inputs

Supply every listed field. Defaults below form the example, and are not automatically filled by the API.
JSON field	Meaning and units	Example
acousticPower	Radiated acoustic power (W)	0.01
distance	Distance from point source (m)	10
referenceIntensity	Reference intensity (W/m²)	1e-12

Outputs

Fields inside the response data object. Some model limits return explanatory strings; inspect the example and assumptions.
JSON field	Meaning	Unit
intensity	Sound intensity	W/m²
intensityLevel	Intensity level relative to reference	dB

Example request

curl 'https://www.engivault.com/api/acoustics/spherical-sound-spreading' \
  -H 'Content-Type: application/json' \
  --data '{"acousticPower":0.01,"distance":10,"referenceIntensity":1e-12}'

Example response

{
  "success": true,
  "data": {
    "intensity": 0.000007957747154594767,
    "intensityLevel": 69.00790135977903
  }
}

Model and limits

Ideal full-sphere spreading: I = P/(4πr²), then L = 10 log₁₀(I/Iref).
Example: 0.01 W at 10 m gives about 7.958 × 10⁻⁶ W/m², or 69.01 dB relative to 10⁻¹² W/m².
Assumes an isotropic point source in a lossless free field. Excludes reflections, ground effects, barriers, atmospheric absorption and near-field effects.
Use radiated acoustic power, not electrical input power. This model is not a noise-exposure assessment.

References

OpenStax University Physics: Sound Intensity (accessed 2026-09-08)

Surface Convection

POST /api/thermal/surface-convection

Calculate surface convection using Q̇ = hAΔT and q″ = hΔT.

Open interactive calculator

Inputs, outputs, example and references

Inputs

Supply every listed field. Defaults below form the example, and are not automatically filled by the API.
JSON field	Meaning and units	Example
coefficient	Mean convection coefficient (W/(m²·K))	10
area	Heat transfer area (m²)	2
temperatureDifference	Surface minus bulk fluid temperature (K)	30

Outputs

Fields inside the response data object. Some model limits return explanatory strings; inspect the example and assumptions.
JSON field	Meaning	Unit
heatRate	Heat transfer rate	W
heatFlux	Heat flux	W/m²
thermalResistance	Convection resistance	K/W

Example request

curl 'https://www.engivault.com/api/thermal/surface-convection' \
  -H 'Content-Type: application/json' \
  --data '{"coefficient":10,"area":2,"temperatureDifference":30}'

Example response

{
  "success": true,
  "data": {
    "heatRate": 600,
    "heatFlux": 300,
    "thermalResistance": 0.05
  }
}

Model and limits

Calculate surface convection using Q̇ = hAΔT and q″ = hΔT.
Positive results mean heat leaves the surface; negative results mean heat enters it. R = 1/(hA).
Example: h = 10 W/(m²·K), A = 2 m² and ΔT = 30 K give 600 W, 300 W/m² and 0.05 K/W.
Supply a mean coefficient appropriate to the geometry, fluid and flow conditions. This calculator does not estimate h or model variation along the surface.
Assumes a uniform surface temperature and a defined bulk-fluid temperature. Conduction, radiation and transient temperature changes are excluded.

References

MIT Unified Engineering: Convective Heat Transfer (accessed 2026-09-08)

Thermal Radiation to Large Surroundings

POST /api/thermal/enclosure-radiation

Net thermal radiation: Q̇ = εσA(Ts⁴ − Tsur⁴). Absolute temperatures must be entered in kelvin.

Open interactive calculator

Inputs, outputs, example and references

Inputs

Supply every listed field. Defaults below form the example, and are not automatically filled by the API.
JSON field	Meaning and units	Example
emissivity	Surface emissivity (0–1)	0.8
area	Radiating surface area (m²)	1
surfaceTemperature	Surface absolute temperature (K)	400
surroundingsTemperature	Surroundings absolute temperature (K)	300

Outputs

Fields inside the response data object. Some model limits return explanatory strings; inspect the example and assumptions.
JSON field	Meaning	Unit
heatRate	Net radiation heat rate	W
heatFlux	Net radiation heat flux	W/m²

Example request

curl 'https://www.engivault.com/api/thermal/enclosure-radiation' \
  -H 'Content-Type: application/json' \
  --data '{"emissivity":0.8,"area":1,"surfaceTemperature":400,"surroundingsTemperature":300}'

Example response

{
  "success": true,
  "data": {
    "heatRate": 793.85241866,
    "heatFlux": 793.85241866
  }
}

Model and limits

Net thermal radiation: Q̇ = εσA(Ts⁴ − Tsur⁴). Absolute temperatures must be entered in kelvin.
Uses σ = 5.670374419 × 10⁻⁸ W/(m²·K⁴), rounded from the Stefan–Boltzmann constant.
Positive heat rate leaves the surface; hotter surroundings produce a negative result. Equal temperatures give zero net exchange.
Example: ε = 0.8, A = 1 m², Ts = 400 K and Tsur = 300 K give approximately 793.85 W outward.
Model: a gray surface surrounded by a much larger isothermal enclosure, with view factor one and a nonparticipating intervening medium.
Does not solve exchange between finite surfaces, solar absorption, semitransparent materials, participating gases or combined convection. Select emissivity for the actual surface condition.

References

OpenStax: Mechanisms of Heat Transfer (accessed 2026-09-08)
NIST fundamental physical constants (accessed 2026-09-08)

Sensible Heat

POST /api/thermal/sensible-heat

Q = mcΔT; heat capacity C = mc. Negative Q represents heat removed.

Open interactive calculator

Inputs, outputs, example and references

Inputs

Supply every listed field. Defaults below form the example, and are not automatically filled by the API.
JSON field	Meaning and units	Example
mass	Mass (kg)	10
specificHeat	Specific heat capacity (J/(kg·K))	1000
temperatureChange	Final minus initial temperature (K)	20

Outputs

Fields inside the response data object. Some model limits return explanatory strings; inspect the example and assumptions.
JSON field	Meaning	Unit
heat	Heat added	J
heatCapacity	Total heat capacity	J/K

Example request

curl 'https://www.engivault.com/api/thermal/sensible-heat' \
  -H 'Content-Type: application/json' \
  --data '{"mass":10,"specificHeat":1000,"temperatureChange":20}'

Example response

{
  "success": true,
  "data": {
    "heat": 200000,
    "heatCapacity": 10000
  }
}

Model and limits

Q = mcΔT; heat capacity C = mc. Negative Q represents heat removed.
Example: 10 kg, 1000 J/(kg·K), and a 20 K rise requires 200,000 J. The specific heat is illustrative.
Assumes constant specific heat and no phase change. Choose the specific heat appropriate to the process, such as constant pressure or constant volume. Excludes equipment heat capacity and environmental losses.
Source: https://openstax.org/books/university-physics-volume-2/pages/1-4-heat-transfer-specific-heat-and-calorimetry

References

OpenStax University Physics Volume 2: Sensible Heat (accessed 2026-09-08)

Free Linear Thermal Expansion

POST /api/thermal/linear-expansion

ΔL = αL₀ΔT; final length = L₀ + ΔL. Cooling or a negative expansion coefficient can produce contraction.

Open interactive calculator

Inputs, outputs, example and references

Inputs

Supply every listed field. Defaults below form the example, and are not automatically filled by the API.
JSON field	Meaning and units	Example
originalLength	Original length (m)	10
coefficient	Mean linear expansion coefficient (1/K)	0.000012
temperatureChange	Final minus initial temperature (K)	50

Outputs

Fields inside the response data object. Some model limits return explanatory strings; inspect the example and assumptions.
JSON field	Meaning	Unit
lengthChange	Length change	m
finalLength	Final length	m

Example request

curl 'https://www.engivault.com/api/thermal/linear-expansion' \
  -H 'Content-Type: application/json' \
  --data '{"originalLength":10,"coefficient":0.000012,"temperatureChange":50}'

Example response

{
  "success": true,
  "data": {
    "lengthChange": 0.006,
    "finalLength": 10.006
  }
}

Model and limits

ΔL = αL₀ΔT; final length = L₀ + ΔL. Cooling or a negative expansion coefficient can produce contraction.
Example: 10 m, α = 0.000012/K and 50 K gives 0.006 m expansion. The coefficient is illustrative.
Assumes uniform temperature and free expansion with a suitable mean coefficient over the temperature interval. It does not calculate restraint forces, thermal stress, buckling or joint capacity.
Source: https://openstax.org/books/university-physics-volume-2/pages/1-3-thermal-expansion

References

OpenStax University Physics Volume 2: Free Linear Thermal Expansion (accessed 2026-09-08)

Ideal Gas Density

POST /api/gases/ideal-gas-density

Ideal gas relation p = ρRT gives density ρ = p/(RT). Specific volume is 1/ρ.

Open interactive calculator

Inputs, outputs, example and references

Inputs

Supply every listed field. Defaults below form the example, and are not automatically filled by the API.
JSON field	Meaning and units	Example
absolutePressure	Absolute pressure (Pa)	101325
absoluteTemperature	Absolute temperature (K)	300
specificGasConstant	Specific gas constant (J/(kg·K))	287

Outputs

Fields inside the response data object. Some model limits return explanatory strings; inspect the example and assumptions.
JSON field	Meaning	Unit
density	Density	kg/m³
specificVolume	Specific volume	m³/kg

Example request

curl 'https://www.engivault.com/api/gases/ideal-gas-density' \
  -H 'Content-Type: application/json' \
  --data '{"absolutePressure":101325,"absoluteTemperature":300,"specificGasConstant":287}'

Example response

{
  "success": true,
  "data": {
    "density": 1.1768292682926829,
    "specificVolume": 0.8497409326424871
  }
}

Model and limits

Ideal gas relation p = ρRT gives density ρ = p/(RT). Specific volume is 1/ρ.
Pressure must be absolute and temperature must be in kelvins. R is the mass-based gas constant for the gas or mixture, not the universal molar constant.
Example: 101325 Pa, 300 K and R = 287 J/(kg·K) gives approximately 1.17683 kg/m³. The example R is a rounded dry-air assumption.
This is an ideal-gas estimate. Dense gases, conditions near condensation or the critical point, and mixtures with changing composition require appropriate real-fluid property data.
Source: https://www1.grc.nasa.gov/beginners-guide-to-aeronautics/equation-of-state/

References

NASA Glenn: Equation of State (accessed 2026-09-08)

Axial Bar Stress and Extension

POST /api/mechanics/axial-bar

For a uniform bar under centered axial load, stress σ = F/A, strain ε = σ/E, and length change δ = εL.

Open interactive calculator

Inputs, outputs, example and references

Inputs

Supply every listed field. Defaults below form the example, and are not automatically filled by the API.
JSON field	Meaning and units	Example
force	Axial force (tension positive) (N)	10000
area	Cross-sectional area (m²)	0.001
length	Original bar length (m)	2
youngModulus	Young’s modulus (Pa)	200000000000

Outputs

Fields inside the response data object. Some model limits return explanatory strings; inspect the example and assumptions.
JSON field	Meaning	Unit
stress	Axial stress	Pa
strain	Axial strain	m/m
extension	Length change	m

Example request

curl 'https://www.engivault.com/api/mechanics/axial-bar' \
  -H 'Content-Type: application/json' \
  --data '{"force":10000,"area":0.001,"length":2,"youngModulus":200000000000}'

Example response

{
  "success": true,
  "data": {
    "stress": 10000000,
    "strain": 0.00005,
    "extension": 0.0001
  }
}

Model and limits

For a uniform bar under centered axial load, stress σ = F/A, strain ε = σ/E, and length change δ = εL.
Example: 10,000 N, 0.001 m², 2 m and E = 200 GPa gives 10 MPa stress, 0.00005 strain and 0.0001 m extension. The modulus is illustrative; use material- and temperature-appropriate data.
Negative force represents compression and produces negative strain and shortening in this model.
Valid only for small linear-elastic deformation. Does not check yield, buckling, fatigue, connections, stress concentrations or eccentric loading. Compression results do not establish column stability.
Source: https://openstax.org/books/university-physics-volume-1/pages/12-3-stress-strain-and-elastic-modulus

References

OpenStax University Physics Volume 1 §12.3 (accessed 2026-09-08)

DC Resistor Current and Power

POST /api/electrical/dc-resistor

For an ideal ohmic resistor, I = V/R and P = VI = V²/R.

Open interactive calculator

Inputs, outputs, example and references

Inputs

Supply every listed field. Defaults below form the example, and are not automatically filled by the API.
JSON field	Meaning and units	Example
voltage	Voltage across resistor (V)	12
resistance	Resistance (Ω)	100

Outputs

Fields inside the response data object. Some model limits return explanatory strings; inspect the example and assumptions.
JSON field	Meaning	Unit
current	Current	A
power	Dissipated power	W

Example request

curl 'https://www.engivault.com/api/electrical/dc-resistor' \
  -H 'Content-Type: application/json' \
  --data '{"voltage":12,"resistance":100}'

Example response

{
  "success": true,
  "data": {
    "current": 0.12,
    "power": 1.44
  }
}

Model and limits

For an ideal ohmic resistor, I = V/R and P = VI = V²/R.
Example: 12 V across 100 Ω gives 0.12 A and 1.44 W. Reversing voltage reverses current but leaves dissipated power positive.
Assumes constant DC voltage and resistance. Does not model resistance changing with temperature, reactive AC loads, source impedance or nonlinear devices.
Calculated dissipation is not a selected component rating; consult the component’s thermal derating and operating limits.
Source: https://openstax.org/books/university-physics-volume-2/pages/9-5-electrical-energy-and-power

References

OpenStax University Physics Volume 2 §9.5 (accessed 2026-09-08)

Plane Wall Heat Conduction

POST /api/thermal/wall-conduction

For steady one-dimensional conduction, heat rate = kAΔT/L and wall resistance = L/(kA).

Open interactive calculator

Inputs, outputs, example and references

Inputs

Supply every listed field. Defaults below form the example, and are not automatically filled by the API.
JSON field	Meaning and units	Example
conductivity	Thermal conductivity (W/(m·K))	0.04
area	Area normal to heat flow (m²)	10
thickness	Wall thickness (m)	0.1
temperatureDifference	Surface 1 minus surface 2 temperature (K)	20

Outputs

Fields inside the response data object. Some model limits return explanatory strings; inspect the example and assumptions.
JSON field	Meaning	Unit
heatRate	Heat rate from surface 1 to 2	W
heatFlux	Heat flux from surface 1 to 2	W/m²
thermalResistance	Wall thermal resistance	K/W

Example request

curl 'https://www.engivault.com/api/thermal/wall-conduction' \
  -H 'Content-Type: application/json' \
  --data '{"conductivity":0.04,"area":10,"thickness":0.1,"temperatureDifference":20}'

Example response

{
  "success": true,
  "data": {
    "heatRate": 80,
    "heatFlux": 8,
    "thermalResistance": 0.25
  }
}

Model and limits

For steady one-dimensional conduction, heat rate = kAΔT/L and wall resistance = L/(kA).
Example: k = 0.04 W/(m·K), A = 10 m², L = 0.1 m and ΔT = 20 K gives 80 W, 8 W/m² and 0.25 K/W. Conductivity here is illustrative; use a value appropriate to the material and operating temperature.
The temperature difference is between wall surfaces, not surrounding air. A negative difference reverses the heat-flow direction.
Assumes constant conductivity, no internal heat generation and negligible edge effects. Excludes surface convection, radiation, contact resistance and thermal bridges.
Source: https://openstax.org/books/university-physics-volume-2/pages/1-6-mechanisms-of-heat-transfer

References

OpenStax University Physics Volume 2 §1.6 (accessed 2026-09-08)

Hydrostatic Pressure

POST /api/hydraulics/hydrostatic-pressure

For a fluid at rest with constant density and gravity: Δp = ρgh.

Open interactive calculator

Inputs, outputs, example and references

Inputs

Supply every listed field. Defaults below form the example, and are not automatically filled by the API.
JSON field	Meaning and units	Example
density	Fluid density (kg/m³)	1000
depth	Depth below surface (m)	10
surfacePressure	Absolute pressure at surface (Pa)	101325
gravity	Gravitational acceleration (m/s²)	9.80665

Outputs

Fields inside the response data object. Some model limits return explanatory strings; inspect the example and assumptions.
JSON field	Meaning	Unit
pressureIncrease	Pressure increase below surface	Pa
absolutePressure	Absolute pressure at depth	Pa

Example request

curl 'https://www.engivault.com/api/hydraulics/hydrostatic-pressure' \
  -H 'Content-Type: application/json' \
  --data '{"density":1000,"depth":10,"surfacePressure":101325,"gravity":9.80665}'

Example response

{
  "success": true,
  "data": {
    "pressureIncrease": 98066.5,
    "absolutePressure": 199391.5
  }
}

Model and limits

For a fluid at rest with constant density and gravity: Δp = ρgh.
Absolute pressure at depth is surface absolute pressure + Δp. The increase is gauge pressure only when the reference is the surface pressure.
Example: 1000 kg/m³ at 10 m and g = 9.80665 m/s² gives 98,066.5 Pa increase; with 101,325 Pa at the surface, absolute pressure is 199,391.5 Pa.
Not applicable to a flowing system or a deep gas column with appreciably varying density.
Source: OpenStax, College Physics 2e §11.4 — https://openstax.org/books/college-physics-2e/pages/11-4-variation-of-pressure-with-depth-in-a-fluid

References

OpenStax College Physics 2e §11.4 (accessed 2026-09-08)

Dynamic Pressure

POST /api/hydraulics/dynamic-pressure

Dynamic pressure q = ½ρv² is kinetic energy per unit volume.

Open interactive calculator

Inputs, outputs, example and references

Inputs

Supply every listed field. Defaults below form the example, and are not automatically filled by the API.
JSON field	Meaning and units	Example
density	Fluid density (kg/m³)	1.225
velocity	Flow speed (m/s)	20

Outputs

Fields inside the response data object. Some model limits return explanatory strings; inspect the example and assumptions.
JSON field	Meaning	Unit
dynamicPressure	Dynamic pressure	Pa

Example request

curl 'https://www.engivault.com/api/hydraulics/dynamic-pressure' \
  -H 'Content-Type: application/json' \
  --data '{"density":1.225,"velocity":20}'

Example response

{
  "success": true,
  "data": {
    "dynamicPressure": 245.00000000000003
  }
}

Model and limits

Dynamic pressure q = ½ρv² is kinetic energy per unit volume.
Example: density 1.225 kg/m³ and speed 20 m/s gives 245 Pa. The example density is an input assumption, not an atmospheric property lookup.
Equating q with stagnation pressure minus static pressure requires the incompressible, lossless Bernoulli assumptions. Compressible Pitot measurements need a different relation.
Source: NASA Glenn, Dynamic Pressure — https://www.grc.nasa.gov/www/BGH/dynpress.html

References

NASA Glenn: Dynamic Pressure (accessed 2026-09-08)

Pressure Drop Calculator

POST /api/hydraulics/pressure-drop

Calculates pressure drop using the Darcy-Weisbach equation

Open interactive calculator

Inputs, outputs, example and references

Inputs

Supply every listed field. Defaults below form the example, and are not automatically filled by the API.
JSON field	Meaning and units	Example
Q	Flow Rate (m³/s)	0.1
L	Pipe Length (m)	100
D	Pipe Diameter (m)	0.1
roughness	Pipe Roughness (m)	0.000045
density	Fluid Density (kg/m³)	998
viscosity	Dynamic Viscosity (Pa·s)	0.001

Outputs

Fields inside the response data object. Some model limits return explanatory strings; inspect the example and assumptions.
JSON field	Meaning	Unit
pressureDrop	Pressure Drop	Pa
frictionFactor	Friction Factor	Dimensionless / text
reynoldsNumber	Reynolds Number	Dimensionless / text
velocity	Mean Velocity	m/s

Example request

curl 'https://www.engivault.com/api/hydraulics/pressure-drop' \
  -H 'Content-Type: application/json' \
  --data '{"Q":0.1,"L":100,"D":0.1,"roughness":0.000045,"density":998,"viscosity":0.001}'

Example response

{
  "success": true,
  "data": {
    "pressureDrop": 1361742.2071419475,
    "frictionFactor": 0.016833488077100833,
    "reynoldsNumber": 1270693.0656456924,
    "velocity": 12.732395447351626
  }
}

Model and limits

Calculates pressure drop using the Darcy-Weisbach equation
Uses Darcy friction factor: 64/Re below Re 2300, otherwise the Swamee–Jain approximation. The 2300–4000 transition range is uncertain; this implementation does not reproduce EPANET’s transitional interpolation.
Assumes steady, fully developed, single-phase incompressible flow in a full circular pipe. Input dynamic viscosity, density and roughness for the operating conditions.
Returns straight-pipe friction loss only. Fittings, valves, elevation changes and acceleration are excluded. The friction factor is Darcy, not Fanning.

References

US EPA EPANET 2.2: Pipe head loss and friction factors (accessed 2026-09-08)

NPSH Calculator

POST /api/pumps/npsh

Reservoir model: NPSHA = (surface absolute pressure − vapor pressure)/(density × g) + liquid elevation − suction losses; negligible reservoir velocity

Open interactive calculator

Inputs, outputs, example and references

Inputs

Supply every listed field. Defaults below form the example, and are not automatically filled by the API.
JSON field	Meaning and units	Example
elevation	Liquid Surface Above Pump (m)	2
atmosphericPressure	Reservoir Surface Absolute Pressure (Pa absolute)	101325
vaporPressure	Vapor Pressure (Pa)	2300
density	Liquid Density (kg/m³)	998
npshRequired	Required NPSH (manufacturer curve) (m)	3
frictionLosses	Total Suction Losses (pipe, fittings and entry) (m)	2.1

Outputs

Fields inside the response data object. Some model limits return explanatory strings; inspect the example and assumptions.
JSON field	Meaning	Unit
npshAvailable	NPSH Available	m
npshRequired	NPSH Required	m
npshMargin	NPSH Margin	m

Example request

curl 'https://www.engivault.com/api/pumps/npsh' \
  -H 'Content-Type: application/json' \
  --data '{"elevation":2,"atmosphericPressure":101325,"vaporPressure":2300,"density":998,"npshRequired":3,"frictionLosses":2.1}'

Example response

{
  "success": true,
  "data": {
    "npshAvailable": 10.017975750514966,
    "npshRequired": 3,
    "npshMargin": 7.017975750514966
  }
}

Model and limits

Reservoir model: NPSHA = (surface absolute pressure − vapor pressure)/(density × g) + liquid elevation − suction losses; negligible reservoir velocity
Uses the required NPSH entered from the manufacturer curve; it cannot be inferred from suction conditions
Returns the arithmetic margin NPSHA minus the supplied NPSHR. A positive margin alone does not establish cavitation-free operation or an adequate application-specific margin.
Check the pump curve at the actual flow and speed and confirm its cavitation criterion. NPSH3 refers to a 3% head-drop criterion, not the onset of cavitation. Include suction entry, valve and fitting losses.

References

KSB Centrifugal Pump Lexicon: NPSH and cavitation criteria (accessed 2026-09-08)

Pipe Sizing Calculator

POST /api/hydraulics/pipe-sizing

Finds a minimum internal bore satisfying the supplied mean-velocity and straight-pipe pressure-gradient limits.

Open interactive calculator

Inputs, outputs, example and references

Inputs

Supply every listed field. Defaults below form the example, and are not automatically filled by the API.
JSON field	Meaning and units	Example
flowRate	Flow Rate (m³/s)	0.05
targetVelocity	Target Velocity (m/s)	2.0
maxPressureDrop	Max Pressure Drop (Pa/m)	100
density	Liquid Density (kg/m³)	998
viscosity	Dynamic Viscosity (Pa·s)	0.001
roughness	Pipe Roughness (m)	0.000045

Outputs

Fields inside the response data object. Some model limits return explanatory strings; inspect the example and assumptions.
JSON field	Meaning	Unit
recommendedDiameter	Recommended Diameter	m
actualVelocity	Actual Velocity	m/s
actualPressureDrop	Actual Pressure Drop	Pa/m

Example request

curl 'https://www.engivault.com/api/hydraulics/pipe-sizing' \
  -H 'Content-Type: application/json' \
  --data '{"flowRate":0.05,"targetVelocity":2,"maxPressureDrop":100,"density":998,"viscosity":0.001,"roughness":0.000045}'

Example response

{
  "success": true,
  "data": {
    "recommendedDiameter": 0.2014587462045113,
    "actualVelocity": 1.5685843203294405,
    "actualPressureDrop": 99.99999999999991
  }
}

Model and limits

Finds a minimum internal bore satisfying the supplied mean-velocity and straight-pipe pressure-gradient limits.
Considers velocity constraints and pressure drop limits
Uses the pressure-drop calculator’s Darcy friction model and limitations. Transition-region results are uncertain. It does not select wall thickness, pressure class or a pipe-network design.
Returns minimum internal diameter; choose a commercial size with an equal or larger bore

References

US EPA EPANET 2.2: Pipe head loss and friction factors (accessed 2026-09-08)

Resource and material data

GET /resources lists calculators, articles, references and materials. Query parameters: q and type (calculator, article, reference or material). Search matches every query word across titles, categories, descriptions and keywords, regardless of case or word order.

GET /materials?reviewed=true lists records with at least one reviewed property. Omit reviewed to include all records; q matches words in names or categories, and category filters an exact category name, regardless of case.

Each material property contains id, label, unit, value, valueRange, valueKind, reviewStatus and source. Reviewed scalars have numeric value and null valueRange. Source-reported ranges have null value and valueRange containing min and max in the stated unit. Unreviewed scalars and ranges are null.

valueKind distinguishes a recorded minimum or typical value; null means that basis is not recorded. Sources include URL, title, conditions and access date. A partially reviewed material still has unreviewed fields. These endpoints do not apply temperature or pressure corrections or certify design suitability.

Accounts and integrations

Project storage requires a verified account and is private to that account. Calculation history is saved in this browser. Paid subscriptions, legacy API-key management, the standalone Railway API and desktop installers are not activated in this web release. Other integration guides describe the supplied standalone API and SDK source.

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