Technical & Sanitaryware Ceramics, Darcy-Washburn Plaster Slip-Casting Kinetics
Ceramic Slip Casting Parabolic Wall-Buildup Formula
Calculate Wet Cast Green Wall Thickness L = SQRT(2*K*t) (mm), Instantaneous Buildup Rate dL/dt (mm/min), Fired Ceramic Wall Thickness (mm), and Dwell Time for 8 mm Cast (min).
Doubling plaster-mold dwell time NEVER doubles ceramic cast thickness—because consolidated clay cake chokes capillary suction as SQRT(t)!
During **Drain Slip Casting** of porcelain sanitaryware, alumina crucibles, and silicon-nitride turbine rotors, micro-porous `beta`-hemihydrate **Plaster of Paris (`CaSO_4·2H_2O`)** molds draw filtrate water out of deflocculated ceramic slip by `~0.1–0.2` MPa capillary suction, depositing a dense leather-hard **Cast Cake of Thickness `L(t)`** (`mm`) against the mold wall! Because filtrate water must percolate through the *growing* low-permeability clay/alumina cake itself (`Darcy’s Law dL/dt = K / L`), integrating yields **Adcock & McDowall’s Parabolic Root-Time Law `L(t) = SQRT(2 * K * t)`** (`mm`), where **Casting Rate Constant `K`** (`mm²/min`) lumps plaster suction pressure, slip solids loading, filtrate viscosity, and cake permeability, **Instantaneous Growth Rate `dL/dt = K / SQRT(2 * K * t)`** (`mm/min`) decays continuously, and reaching `8.0` mm wet wall takes **`t_{8mm} = 64 / (2 * K)`** (`4x` dwell time for `2x` wall thickness!)!
Verify on a Deflocculated Vitreous China Sanitaryware Slip—Casting_Rate_Constant `K` = 0.42 mm²/min (B2), Mold_Dwell_Time `t` = 48.0 min (B3), and Total_Wet-to-Fired_Linear_Shrinkage `S_{tot}` = 11.5% (B4): Wet Green Wall `L` reaches `6.35` mm (E2), Instantaneous Growth Rate slows to `0.066` mm/min (E3), Post-Kiln Fired Wall equals `5.62` mm (E4), and building a heavy `8.0` mm wet rim requires `76.2` minutes before draining excess slip (E5)!
| Ceramic Slip-Casting KPI (t=48 min, Shrinkage=11.5%: Fresh Mold K=0.42 vs Wet Mold K=0.18) | Formula | Verified Output |
|---|---|---|
| Wet Cast Green Wall L (mm) | SQRT(2 * Casting_K_mm2_min * Dwell_min) | 6.35 mm (4.16 mm thin wall @ wet mold!) |
| Instantaneous Growth Rate (mm/min) | Casting_K_mm2_min / SQRT(2 * K * t) | 0.066 mm/min (0.043 mm/min @ K=0.18) |
| Fired Sintered Wall Thickness (mm) | Wet_Green_Wall_mm * (1 - Shrink_Pct / 100) | 5.62 mm (3.68 mm slumping risk!) |
| Dwell Time for 8.0 mm Wet Wall (min) | 64 / (2 * Casting_K_mm2_min) | 76.2 min (177.8 min = ~3 hr bottleneck!) |
Darcy-Washburn Parabolic Slip-Casting Cake Buildup & Fired Shrinkage Formula
Validate `Casting_Rate_Constant K` in `[0.01, 5.0]` mm²/min, `Dwell_Time t` in `[1, 600]` min, and `Total_Linear_Shrinkage` in `[0%, 35%]`!
=LET(kc_r, B2, td_m, B3, sh_p, B4, IF(OR(kc_r < 0.01, kc_r > 5, td_m < 1, td_m > 600, sh_p < 0, sh_p > 35), "Invalid Slip Cast Input", ROUND(64 / (2 * kc_r), 1)))Water-Logged / Sodium-Silicate Blinded Plaster Mold (`K = 0.18` mm²/min) Thin-Cast Check (Set K B2 = 0.18)
Re-cast into a damp plaster mold saturated from the morning turn or glazed by sodium-silicate deflocculant (`K` drops to `B2 = 0.18` mm²/min): a `48`-minute timer produces only `4.16` mm wet wall (E2 -> `3.68` mm fired wall in E4, sagging during bisque firing!), while reaching `8.0` mm wet wall balloons E5 to **`177.8` minutes**!
=IF(ISTEXT(E5), E5, ROUND(SQRT(2 * B2 * B3), 2))Try the example
Use the original values or change them to test your understanding.