# Nutrient stock solutions

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How stock composition, compatibility, dose volume and reservoir volume define nutrient additions without inventing ion-specific EC values.

In brief

How stock composition, compatibility, dose volume and reservoir volume define nutrient additions without inventing ion-specific EC values.

## Principle

Each stock solution is defined by the concentration of every relevant chemical species, not by a marketing label. The controller calculates the amount added to the reservoir from calibrated dose volume and stock composition.

 STK-1 Δ c i = S i j v j V R \Delta c_i=\frac{S_{ij}v_j}{V_R} Δ c i ​ = V R ​ S ij ​ v j ​ ​ Increase in reservoir concentration of ion i from stock j, where Sij is stock concentration, vj is dose volume and VR is reservoir volume.

**Explanation**The concentration increase depends on stock strength and dose volume, then is diluted by the reservoir volume.

 STK-2 c i , k + 1 = c i , k + ∑ j Δ c i − u i , k − l i , k c_{i,k+1}=c_{i,k}+\sum_j\Delta c_i-u_{i,k}-l_{i,k} c i , k + 1 ​ = c i , k ​ + j ∑ ​ Δ c i ​ − u i , k ​ − l i , k ​ Ion inventory update including additions, plant uptake ui and other losses li.

**Explanation**The next ion inventory includes new additions and subtracts plant uptake and losses.

## Stock compatibility

Concentrated calcium stocks are separated from concentrated phosphate or sulphate stocks unless compatibility has been demonstrated, because precipitation can remove nutrients and obstruct tubing. Stock identity, concentration, solvent, preparation date, lot and storage conditions are recorded.

## Recipe solving

When several stocks contribute to several ions, the system solves a constrained non-negative dosing problem rather than assigning one pump to one sensory attribute.

 STK-3 min ⁡ v ∥ S v V R − Δ c t a r g e t ∥ 2 s . t . 0 ≤ v ≤ v max ⁡ \min_{\mathbf v}\left\lVert \frac{S\mathbf v}{V_R}-\Delta\mathbf c_{\mathrm{target}}\right\rVert^2\quad\mathrm{s.t.}\quad0\leq\mathbf v\leq\mathbf v_{\max} v min ​

 ​ V R ​ S v ​ − Δ c target ​ ​ 2 s.t. 0 ≤ v ≤ v m a x ​ Constrained stock-volume selection for a target ion-change vector. **Explanation**The controller converts each candidate stock volume into a reservoir concentration change, then selects non-negative volumes that approach the target without exceeding dose limits.

 **Control rule**

EC is used as a bulk consistency and safety check. It cannot verify the individual ion vector S·v.

## Ion coupling

Stock design accounts for the fact that fertilizer salts introduce coupled ions and that precise individual-ion control requires more information than bulk EC. [R34] [R35] [R36]

## References

- [R34] Cho, Woo-Jae; Gang, Min-Seok; Kim, Dong-Wook; Kim, JooShin; Jung, Dae-Hyun; Kim, Hak-Jin (2023). Decision-tree-based ion-specific dosing algorithm for enhancing closed hydroponic efficiency and reducing carbon emissions. *Frontiers in Plant Science, 14, 1301490*. https://doi.org/10.3389/fpls.2023.1301490
- [R35] Bamsey, Matthew; Graham, Thomas; Thompson, Cody; Berinstain, Alain; Scott, Alan; Dixon, Michael (2012). Ion-Specific Nutrient Management in Closed Systems: The Necessity for Ion-Selective Sensors in Terrestrial and Space-Based Agriculture and Water Management Systems. *Sensors, 12, 13349–13392*. https://doi.org/10.3390/s121013349
- [R36] Miller, Alexander; Adhikari, Ranjeeta; Nemali, Krishna (2020). Recycling Nutrient Solution Can Reduce Growth Due to Nutrient Deficiencies in Hydroponic Production. *Frontiers in Plant Science, 11, 607643*. https://doi.org/10.3389/fpls.2020.607643