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Abstract
This paper proposes a development to reformulate the fundamental equations of primary drying into a linear-based representation to support control design. The model is expressed by transfer functions with variable gains that incorporate analytical relations derived from a phenomenological model. The time constants denote the dynamics of the heating/cooling and vacuum systems. Compared to fundamental equations, this approach simplifies the development and implementation of well-established control algorithms. The quality of the proposed representation is illustrated in the design of a control system using two different strategies: model predictive control and a feedback controller with proportional-integral action. The simulation case study allows assessing the performance of both designs with regard to cycle time reduction and robustness under parametric disturbances. Results evidence that the proposed model is detailed enough to provide accuracy in a simplified way, facilitating the implementation of in-line control strategies, which in turn enable significant reductions of drying time.
| Nomenclature | ||
| Ap | = | inner cross-sectional area of the vial (m2) |
| Av | = | outer cross-sectional area of the vial (m2) |
| c1, c2, c3 | = | empirical coefficients for the calculation of |
| Cp | = | specific heat capacity of ice (J) |
| FT | = | transfer function of the equivalent dynamics of the heating system |
| FP | = | transfer function of the equivalent dynamics of the vacuum system |
| h | = | sublimation front position (m) |
| = | front position rate (m s–1) | |
| L | = | total thickness of the frozen layer (m) |
| = | prediction horizon | |
| = | control horizon | |
| = | heat of sublimation of ice (J kg–1) | |
| Jw | = | heat flux to the product (J m–2 s–1) |
| k1, k2, k3 | = | empirical coefficients for the calculation of Qv |
| k4, k5, k6 | = | empirical coefficients for the calculation of Rp |
| K | = | gains of the LPV model |
| Kc | = | proportional gain of the PI |
| m | = | total mass of the product (kg) |
| Nw | = | sublimation flux of water vapor (kg m–2 s–1) |
| P | = | chamber pressure (Pa) |
| Pw | = | temperature-dependent water vapor saturation pressure (Pa) |
| Qv | = | overall heat transfer coefficient (J m–2 s–1 K–1) |
| Rp | = | dried layer resistance (m s–1) |
| TB | = | temperature of the product at the vial bottom (K) |
| Tc | = | critical temperature of the product (K) |
| Ti | = | temperature of the product at the interface (K) |
| Ts | = | temperature of the shelf (K) |
| u | = | system input |
| Wu | = | input increment weight |
| Wy | = | set-point tracking weight |
| y | = | system output |
| Greek symbols | ||
| λ | = | thermal conductivity (J m–1 s–1 K–1) |
| ρ | = | density (kg m–3) |
| τs | = | time constant of heating system (s) |
| τp | = | time constant of vacuum system(s) |
| τi | = | integral time constant of the PI (s) |
| Superscripts and subscripts | ||
| = | estimated value | |
| a | = | approximate |
| d | = | dried layer |
| f | = | frozen layer |
| min | = | minimum value |
| max | = | maximum value |
| r | = | reference |
Disclosure statement
The authors report no conflicts of interest. The authors alone are responsible for the content and writing of the paper.