Powder Metallurgy, Battery Cathode Calendering & Pharma Bulk Handling, Kawakita Model
Kawakita Powder Volume Reduction & Cohesiveness Formula
Calculate Kawakita Engineering Volume Strain C (%), Compressed Bed Volume V (mL), Reciprocal Yield/Cohesion Pressure 1/b (MPa), and Asymptotic Packing Realization (%).
Where Heckel shines at high punch pressures (>50 MPa), Kawakita’s hyperbolic law captures low-pressure fluffy powder rearrangement and soft granule crushing!
Fluffy spray-dried pharmaceutical granules, `NMC811` battery cathode powders, and cosmetic compacts undergo `30%–50%` volume collapse at low pressures (`1–40` MPa) where Heckel plots curve non-linearly! K. Kawakita & K.-H. Lüdde derived the exact hyperbolic **Volume-Reduction Equation** from Initial Loose Volume `V_0` (`mL`): **`C = (V_0 - V) / V_0 = (a * b * P) / (1 + b * P)`** (linearized as `P / C = P / a + 1 / (a * b)`!), governed by TWO physical constants: **Asymptotic Maximum Volume Reduction `a`** (`0 < a < 1`, equal to Initial Interparticle Bed Porosity / Carr’s Compressibility Fraction as `P -> infinity`) and **Plasticity/Rearrangement Coefficient `b`** (`MPa⁻¹`), whose reciprocal **`P_k = 1 / b`** (`MPa`) defines the pressure needed to achieve **exact `50%` of maximum possible volume reduction (`C = a / 2`)**!
Verify on a Spray-Dried Agglomerated Granule Bed—Initial_Loose_Fill_Volume `V_0` = 120.0 mL (B2), Asymptotic_Compressibility `a` = 0.38 (`38%` max volume reduction, B3), Kawakita_Coefficient `b` = 0.22 MPa⁻¹ (B4 -> Half-Compaction Pressure `P_k = 1 / b = 4.55` MPa in E4), and Pre-Compression_Pressure `P` = 25.0 MPa (B5): Engineering Volume Reduction `C` reaches `32.15%` (E2), Compacted Bed Volume shrinks to `81.42` mL (E3), realizing `84.62%` of ultimate densification `a` (E5)!
| Kawakita Powder Compaction KPI (V0=120 mL, a=0.38, P=25 MPa: Soft b=0.22 vs Hard b=0.06) | Formula | Verified Output |
|---|---|---|
| Volume Reduction Degree C (%) | ((a * b * P) / (1 + b * P)) * 100 | 32.15% (22.80% @ stiff granule b=0.06) |
| Compressed Powder Volume V (mL) | Loose_V0_mL * (1 - C_fraction) | 81.42 mL (92.64 mL @ b=0.06) |
| Half-Reduction Pressure Pk = 1/b (MPa) | 1 / Kawakita_b_inv_MPa | 4.55 MPa (16.67 MPa hard ceramic agglomerate!) |
| Realized Fraction of Max a (%) | ((b * P) / (1 + b * P)) * 100 | 84.62% (60.00% @ b=0.06) |
Kawakita-Lüdde Hyperbolic Powder Volume Strain & Pk = 1/b Formula
Validate `Loose_Volume V_0 > 1` mL, `Max_Compressibility a` in `(0.05, 0.85)`, `Kawakita b > 0.005` MPa⁻¹, and `Applied_Pressure P > 0` MPa!
=LET(vz_m, B2, am_x, B3, bk_i, B4, pa_m, B5, IF(OR(vz_m <= 1, am_x <= 0.05, am_x >= 0.85, bk_i <= 0.005, pa_m <= 0), "Invalid Kawakita Input", ROUND(((bk_i * pa_m) / (1 + bk_i * pa_m)) * 100, 2)))Hard Calcined Ceramic Granules (b = 0.06 MPa⁻¹ -> Pk = 16.67 MPa) Springback Check (Set b B4 = 0.06)
Over-calcine spray-dried ceramic granules so granule crush strength quadruples (`b` drops to `B4 = 0.06` MPa⁻¹, `P_k = 1 / b = 16.67` MPa in E4): at `25` MPa, Volume Reduction E2 stalls at `22.80%` (`92.64` mL in E3, only `60.00%` of `a` in E5)—leaving intact "blackberry" granule boundaries that survive sintering as strength-limiting macro-pores!
=IF(ISTEXT(E5), E5, LET(cf_r, (B3 * B4 * B5) / (1 + B4 * B5), ROUND(B2 * (1 - cf_r), 2)))Try the example
Use the original values or change them to test your understanding.