Phosphate-Pebble Rotary Drying Process

• Phosphate-Pebble Rotary Drying Process is a convective, direct-contact thermal treatment that removes moisture from phosphate rock pebbles while ensuring energy efficiency.
• The process integrates first-principles lumped-parameter modeling with nonlinear dynamic control to optimize drying performance and maintain product quality.
• Operational guidelines emphasize precise feed rates, controlled temperature zones, and effective exhaust heat recovery to achieve uniform drying and reduce energy consumption.

The phosphate-pebble rotary drying process is a convective, direct-contact thermal treatment method used for the moisture removal of phosphate rock pebbles, integral to phosphate beneficiation and downstream handling. The process is characterized by its nonlinear multivariate dynamics, direct coupling of heat and mass transfer processes, and the need for tightly regulated controls to achieve both product quality and energy efficiency objectives.

1. Rotary Dryer Configuration and Operating Stages

A phosphate-pebble rotary dryer is a longitudinally partitioned, direct-fired drum consisting of three functionally distinct sections:

• Preheating Zone: This zone elevates the incoming pebble temperature from ambient (~25 °C) to the wet-period threshold (~100 °C) utilizing hot combustion gases (900–1 200 °C), fresh air, and wet feed (15 wt % moisture). Key state variables include solid and gas temperatures ($T_{s1}(t)$, $T_{g1}(t)$) and gas mass ($M_{g1}(t)$).
• Constant-Rate Drying (Main Drying Zone): The dominant phase for liquid water removal, this zone maintains an approximately constant evaporation rate until the surface moisture is depleted. The typical solids feed rate $F_s(t)$ lies within 2–5 kg/s, incoming combustion gas temperatures are 700–900 °C, and the principal state variables are moisture content ($M_w(t)$), bed temperature ($T_{s2}(t)$), and gas properties ($T_{g2}(t)$, $M_{g2}(t)$).
• Cooling Zone: Here, dried pebbles are cooled to sub-60 °C for safe post-processing using ambient or preheated air. State variables of concern are $T_{s3}(t)$, $T_{g3}(t)$.

Major manipulated variables include combustion fuel flow $m_f(t)$, combustion air $m_a(t)$, solids feed rate $F_s(t)$, and the exhaust fan-induced flow $m_{\text{stack}}(t)$. Disturbances comprise pebble inlet moisture $X_{\text{in}}(t)$, ambient temperature $T_{\text{amb}}(t)$, and gas inlet temperature $T_{\text{ai}}(t)$.

2. First-Principles Nonlinear Dynamic Model

The underlying model employs lumped-parameter balances, treating each zone as a perfectly mixed volume at constant pressure, with negligible radiation and axial conduction. Gas and solid phases possess uniform temperatures throughout each control volume, and the gas (flue) is ideal.

• Moisture Mass Balance:

$$\frac{dM_w}{dt} = \dot{m}_{\mathrm{in},w} - \dot{m}_{\mathrm{out},w} - r_e$$

where $M_w$ is the mass of water in the pebble bed, $\dot{m}_{\mathrm{in},w}=F_s X_{\mathrm{in}}$ and $\dot{m}_{\mathrm{out},w}=F_s X_{\mathrm{out}}$ denote water transport with inflowing and outflowing solids, and $r_e$ is the evaporation rate.

• Solid-Phase Energy Balance:

$$M_s c_{p,s} \frac{dT_s}{dt} = Q_{g\to s} - \Delta H_v r_e$$

with $M_s$ as the dry-solid mass, $c_{p,s}\approx800$ J/kg·K, $Q_{g\to s}$ the convective heat input, and $\Delta H_v\approx2.26\times10^6$ J/kg.

• Gas-Phase Energy Balance:

$$M_g c_{p,g} \frac{dT_g}{dt} = \dot{Q}_{\mathrm{in}} - Q_{g\to s} - \dot{Q}_{\mathrm{out}}$$

with $M_g$ as gas mass, $c_{p,g}\approx1\,000$ J/kg·K, $\dot{Q}_{\mathrm{in}}$ enthalpy inflow via combustion, and $\dot{Q}_{\mathrm{out}}$ exhaust losses.

• Convective Heat Transfer:

$Q_{g\to s} = U A (T_g - T_s)$

Here, $U$ is the overall heat-transfer coefficient (50–200 W/m²·K), and $A$ is the effective gas–solid interface area.

• Evaporation Kinetics:

$$r_e = k_e A_{\text{surf}} \frac{p_{\text{sat}}(T_s) - p_g}{p_{\text{atm}}}$$

with $k_e=5\times10^{-6}$–$5\times10^{-5}$ kg/m²·s the mass-transfer coefficient, $A_{\text{surf}}$ the wetted area, and $p_{\text{sat}}(T_s)$, $p_g$, $p_{\text{atm}}$ the saturation, partial vapor, and atmospheric pressures.

3. Parameter Estimation and Typical Process Values

Parameter estimation is performed via a combination of transient system tests and dedicated lab-scale calorimeter experiments:

• Step Disturbance Tests: Applied to fuel or air flows; least-squares fits are used to identify $U$ and $c_{p,s}$.
• Calorimeter Drying: Used to estimate $k_e$ and $A_{\text{surf}}$ from evaporation curves.
• Physical Measurements: Direct assessment of pebble density, porosity ($\varepsilon \approx 0.4$, $A/V \approx 10$ m²/m³), and moisture diffusion.

Typical operating and geometric parameters include:

• Drum diameter: 2–3 m; length: 10–15 m (volume $30–100$ m³)
• Heat-transfer $U$: 50–150 W/m²·K; $k_e$: $1 \times 10^{-5}$–$1 \times 10^{-4}$ kg/m²·s
• Specific heats: $c_{p,s}=750–900$ J/kg·K; $c_{p,g}=950–1\,050$ J/kg·K

4. Linearization and Control System Design

To facilitate control synthesis, the nonlinear ODEs are linearized around a steady-state operating point. Letting $x=[M_w, T_s, T_g]^T$ and $u=[\dot m_\mathrm{in}, \dot m_{\mathrm{air}}]^T$, linearization yields:

$\frac{d\,\delta x}{dt} = A\,\delta x + B\,\delta u$

where explicit matrix entries are derived from first derivatives at steady state.

Control Loops

A decentralized PI control architecture is implemented via direct synthesis (DS) and internal model control (IMC):

• Moisture (Loop 1): Feed rate $F_s$ as MV for $X_{\text{out}}$. The small-signal TF is $G_1(s)=K_1/(\tau_m s+1)$, yielding a PI by direct synthesis:

$$G_{c1}(s)=K_{c1}\left(1 + \frac{1}{T_{i1} s}\right)$$

with $K_{c1} = \tau_m/(K_1 \tau_c)$ and $T_{i1} = \tau_m$.

• Chamber Temperature (Loop 2): Combustion air $m_a$ controls $T_c$; a non-minimum-phase second-order plant $G_2(s)$ is controlled using IMC/DS with filter $F_2(s)=(\lambda s+1)^{-2}$, tuned for $M_s \le 2$, $PM \ge 50°$.
• Draft Pressure (Loop 3): Exhaust fan speed $m_{\text{stack}}$ as MV for pressure $P$, plant $G_3(s)$ is double-integrating with a lag/RHP zero. IMC yields a realizable controller with $F_3(s)=(\alpha s+1)^{-2}$.
• NMPC Alternative: A multioutput nonlinear MPC may be formulated:

$$\min_{\{u(k)\}} \sum_{k=0}^{N_p} \|y(k)-y_{\text{sp}}\|_Q^2 + \sum_{k=0}^{N_c-1}\|\Delta u(k)\|_R^2$$

subject to process constraints.

Setpoints and Constraints

• Moisture: $X_{\text{out,sp}}=3\%\pm0.2\%$
• Bed Temperature: $T_s = 350\pm5$°C
• Gas Temperature: $T_g = 900\pm10$°C
• Feed, fuel, and draft limited as $F_s \le 6$ kg/s, $\dot m_f \le 0.02$ kg/s, draft $\ge-3\,000$ Pa.

5. Dynamic Simulation and Performance Assessment

Simulations using the full nonlinear model (e.g., in Simulink) have been conducted under representative disturbances:

• 50 % fuel flow drop at $t=200$ s,
• Inlet-moisture increase (15 % → 23 %) at $t=500$ s,
• Draft-pressure step ($-1\,500$ Pa → $-2\,500$ Pa) at $t=800$ s,
• Chamber temperature step ($900$ → $1\,050$ °C) at $t=1\,000$ s.

Performance metrics (over 0–1\,500 s) for the main loops are summarized below:

Variable	ISE	Overshoot (%)	Final Steady Error (%)
$X_{\text{out}}(t)$	0.019	11.2	2.8
$T_c(t)$	$2.1\times10^5$	9.5	1.4
$P(t)$	$4.2\times10^8$	13.8	2.0

All closed-loop overshoots remained below 20%, with steady-state errors under 5%. Energy consumption was reduced by 12% over open-loop operation, control actions respected actuator limitations, and setpoint tracking was achieved during severe transients.

6. Practical Operation and Energy-Efficiency Recommendations

Best-practice operating guidelines for phosphate-pebble rotary dryers are as follows:

• Feed rates of 2–5 kg/s yield residence times of 5–10 min.
• Inlet gas temperatures of 850–950 °C, with velocities 1–1.5 m/s, provide efficient drying.
• Bed depth should be maintained at 0.5–0.8 m for drying uniformity.
• Exhaust temperature targets of 350–450 °C optimize trade-offs between energy efficiency and corrosion risk.
• Draft pressure should be kept in the range $-1\,500$ to $-2\,500$ Pa.

Energy efficiency enhancements include preheating inlet air via exhaust recovery to 100–150 °C, minimizing exhaust enthalpy loss ($U_e A_e < 100$ W/K), and maintaining operation in the constant-rate drying regime to utilize latent heat effectively (with efficiency exceeding 60% in this phase).

7. Summary and Recommendations

The phosphate-pebble rotary drying process comprises an integrated dynamic system, in which accurate first-principles modeling of combustion, drying, and exhaust subsystems is critical for control and optimization. Lumped-parameter mass and energy balances formalize the interconnected dynamics as nonlinear ODEs, which can be systematically linearized for classical control synthesis. Decentralized PI controllers, derived via direct synthesis and IMC, have demonstrated robust setpoint tracking ($<$15% overshoot, $<$5% steady error) and a 12% reduction in energy consumption relative to open-loop operation. Process efficiencies range from 35–60%, with highest values realized in the constant-rate drying regime. Parameter estimation rigor, accurate steady-state mapping, and robust filter tuning are identified as decisive for meeting industrial throughput, quality, and energy performance objectives. Optimal operation is achieved at inlet temperatures near 900 °C, bed depths of 0.6 m, and systematic exhaust heat recovery.