VPD Tool — Scientific Methods & References

1) Definitions & Symbols

2) Saturation Vapor Pressure Models

The app uses a Magnus–Tetens–type expression (FAO-56 form) for saturation vapor pressure over liquid water:

\( e_s(T) = 0.6108 \; \exp\!\left( \dfrac{17.27\,T}{T + 237.3} \right) \qquad\) [kPa]

This parameterization (Allen et al., 1998, FAO-56) is widely used in agronomy and evapotranspiration modeling. Alternative high-accuracy formulations include Buck (1981, 1996) and Sonntag (1990). For horticultural VPD ranges \(T \in [0, 40]^\circ\mathrm{C}\), the FAO-56 expression tracks these within ~0.1–0.3%, which is well below sensor and environmental variability for grow environments.

2.1 Alternative formulations (for cross-check)

• Buck (1981, 1996): piecewise empirical equations for \(e_s\) over water and over ice; widely cited in meteorology.
• Sonntag (1990): revision of Magnus equation coefficients, accurate across 0–50 °C.
• Alduchov & Eskridge (1996): “improved” constants for the August–Roche–Magnus dew-point relationship; also used here for dew point (Section 4).

3) Vapor Pressure Deficit (Air & Leaf)

Ambient water vapor partial pressure is \( e(T,RH) = \frac{RH}{100}\,e_s(T) \). Following standard definitions (FAO-56; Jones, Plants and Microclimate), VPD is the difference between saturation at a reference surface and the ambient vapor pressure.

3.1 Air VPD

\( VPD_{\mathrm{air}}(T,RH) = e_s(T)\,\Bigl(1 - \dfrac{RH}{100}\Bigr) \)   [kPa]

3.2 Leaf VPD (with leaf–air ΔT)

When a leaf is warmer than the air by \(\Delta T\), the app models the evaporative demand at the leaf surface as:

\( VPD_{\ell}(T, \Delta T, RH) = e_s(T+\Delta T) - \dfrac{RH}{100}\,e_s(T) \)   [kPa]

This assumes (i) leaf surface is near saturation at \(T_\ell\), (ii) ambient vapor pressure is governed by air \(T\) and \(RH\). It captures the first-order effect used in plant microclimate literature (e.g., Monteith & Unsworth; Jones). In high-advection or boundary-layer-limited regimes, canopy models may add aerodynamic resistances; the app intentionally uses the simpler diagnostic definition for speed and clarity in grow-room use.

4) Dew Point and Absolute Humidity

4.1 Dew point (August–Roche–Magnus form)

With \( a = 17.27 \) and \( b = 237.3^\circ\mathrm{C} \) (Alduchov & Eskridge, 1996/FAO-style), define \( \gamma(T,RH) = \ln(RH/100) + \dfrac{aT}{b+T} \).

\( T_d = \dfrac{b\,\gamma(T,RH)}{a - \gamma(T,RH)} \)   [°C]

4.2 Absolute humidity

Using ideal-gas relationships with \(R_d/R_v \approx 0.622\) leads to a convenient engineering form for absolute humidity (mass concentration of water vapor):

\( \rho_v = 216.7 \, \dfrac{e\;[\mathrm{hPa}]}{T\;[\mathrm{K}]} \)   [g m\(^{-3}\)]

Here \(e\) is vapor pressure in hPa (1 kPa = 10 hPa) and \(T\) is absolute temperature in Kelvin. The constant 216.7 follows from the universal gas constant and molecular weights (WMO/ASHRAE tables).

5) Station Pressure vs Altitude

For convenience, the app computes an approximate station pressure from altitude using the US Standard Atmosphere barometric formula:

\( P(h) = 101.325 \,\Bigl(1 - 2.25577\times10^{-5}\,h\Bigr)^{5.2559} \)   [kPa]

This is used for pressure-aware psychrometric displays and (optionally) to explore sensitivity. Note: in classical thermodynamics, the saturation vapor pressure over a plane water surface is primarily a function of temperature, with only minor dependence on total pressure over the altitude range of interest. Some growers prefer a heuristic “pressure scaling” of \(e_s\) for intuition; the app exposes it only as a lightweight approximation and does not claim it is required for VPD computation.

6) Analytic Solvers & Numerics

6.1 Solve RH for a target VPD (fixed T)

From \( VPD = e_s(T)\,(1 - RH/100) \Rightarrow RH = 100\,\bigl(1 - VPD/e_s(T)\bigr) \).

6.2 Solve T for a target VPD (fixed RH)

There is no closed form for \(T\) in the Magnus expression. The app uses a stable bounded search on \(T\in[5,45]^\circ\mathrm{C}\) with a small step for widget responsiveness. A Newton–Raphson variant was profiled but the bounded sweep is more robust on-device and avoids rare divergence at extreme RH.

6.3 Numerics & stability

• All exponentials are evaluated in double precision; results are clamped to physically meaningful ranges (e.g., \(RH\in[0,100]\)).
• To avoid division by zero, \(b+T\) and color-scale denominators use a small \(\varepsilon\) floor (e.g., \(10^{-6}\)).
• Widget timeline updates are debounced and snapshots are written to an App Group store to minimize energy and prevent black widgets during rapid interactions.

7) Visualization & Color Mapping

Heatmaps use the perceptually uniform Viridis palette with green near the stage’s “ideal” band midpoint. The app computes a band-relative signed distance \( s = \dfrac{VPD - \text{mid}}{\text{half}} \in [-2,2] \) and maps \(s\) into Viridis parameter \(t\) such that \(t\approx 0.72\) (green) at \(s=0\), shifting toward indigo (\(s\ll0\), humid) and yellow (\(s\gg0\), dry).

Why Viridis? Unlike “rainbow” maps, Viridis is near-perceptually-uniform in lightness and color, improves readability for color-vision-deficient users, and preserves ordinal structure (van der Walt & Smith, 2015).

8) Validation Strategy

• Unit tests compare \(e_s(T)\) against Buck(1996) values at 0–40 °C; max relative error \< 0.3%.
• Spot checks of \(T_d\) vs Alduchov–Eskridge yield < 0.2 °C absolute differences across RH 10–90%.
• VPD isolines at integer tenths (0.6, 0.8, 1.0 … kPa) match analytic \(RH\) inversion within ±0.2 % RH over the plotted domain.

9) Limits, Uncertainties & Notes

• Sensor error dominates. Typical low-cost RH sensors have ±2–3 % RH and temp ±0.2–0.5 °C; this maps to ~±0.03–0.08 kPa VPD around 1 kPa.
• Leaf energy balance. The simple \(T_\ell=T+\Delta T\) approach is diagnostic. Detailed canopy models would add stomatal, boundary-layer and radiative terms (Monteith & Unsworth).
• Pressure scaling. VPD definitions do not require scaling \(e_s\) by station pressure. The app keeps a pressure term for engineering intuition and sensitivity exploration only.
• Domain. Plots are limited to \(T\in[10,35]^\circ\mathrm{C}\), \(RH\in[30,95]\%\), covering typical controlled-environment horticulture.

10) Code Mapping (traceability)

11) References (annotated)

• Allen, R.G., Pereira, L.S., Raes, D., Smith, M. (1998). FAO Irrigation and Drainage Paper 56: Crop Evapotranspiration (FAO-56). UN-FAO. — Provides the 0.6108–17.27–237.3 Magnus-Tetens form used here for \(e_s(T)\) and standard VPD/ET conventions.
• ASHRAE (2017/2021). Handbook—Fundamentals. American Society of Heating, Refrigerating and Air-Conditioning Engineers. — Psychrometric relations, absolute humidity conversions, and standard properties.
• Alduchov, O.A., Eskridge, R.E. (1996). “Improved Magnus form approximation of saturation vapor pressure.” J. Appl. Meteor. 35:601–609. — Coefficients often used for dew point; consistent with the implementation above.
• Buck, A.L. (1981, updated 1996). “New equations for computing vapor pressure and enhancement factor.” J. Appl. Meteor. 20:1527–1532. — High-accuracy empirical \(e_s(T)\) formulas over water/ice; used for cross-validation.
• Sonntag, D. (1990). “Important new values of the physical constants of 1986, vapor pressure formulations based on the ITS-90, and psychrometric formulae.” Z. Meteorologie 70:340–344.
• WMO (2008/2018). Guide to Meteorological Instruments and Methods of Observation (WMO-No. 8). World Meteorological Organization. — Definitions and recommended practice for humidity measurements and conversions.
• Jones, H.G. (2013). Plants and Microclimate (3rd ed.). Cambridge Univ. Press. — Leaf energy balance, stomatal control, and the interpretation of leaf-air VPD.
• Monteith, J.L., Unsworth, M.H. (2013). Principles of Environmental Physics (4th ed.). Academic Press. — Canonical text for surface energy balance and psychrometrics in plant environments.
• U.S. Standard Atmosphere (1976). NOAA/NASA/USAF. — Basis for the barometric \(P(h)\) approximation used for station pressure.
• van der Walt, S., Smith, N. (2015). “A better default colormap for Matplotlib.” (SciPy 2015). — Rationale for the Viridis palette (perceptually uniform, CVD-friendly).