Chvorinov Sand-Casting Modulus & Riser Sizing Formula

If a sand-casting riser freezes BEFORE the heavy boss it feeds, shrinkage cavities form straight inside the casting!

Molten gray iron, steel, and aluminum alloys shrink 3%–7% volumetrically upon freezing! Nicolas Chvorinov proved via transient 1D mold heat-conduction theory that a casting of volume V_c (cm³) and heat-dissipating mold surface area A_c (cm²) solidifies in time t_s = B * ((V_c / A_c)^n) minutes—governed by its Geometric Cooling Modulus M_c = V_c / A_c (cm), Mold Constant B (min/cm², typically ~1.8–2.4 in insulating silica green sand vs ~0.6–0.9 against iron chills), and exponent n ~= 2.0! To guarantee Directional Solidification where liquid feed metal flows continuously into the shrinking casting until the very last core freezes, foundry engineers size a sacrificial Riser to stay molten ~44% longer by setting Riser Modulus M_r = k_safe * M_c (k_safe = 1.20 -> 1.20^2 = 1.44x freeze time)! For a sand-enclosed blind cylindrical riser of optimal height H_r = D_r, modulus M_r = (PI()*D_r^3/4) / (1.5*PI()*D_r^2) = D_r / 6 yields Minimum Riser Diameter D_r = 6 * M_r cm!

Try this 3,600 cm³ Ductile-Iron Valve Body Sand Casting first—Casting_Volume V_c = 3,600.0 cm³ (B2), Cooling_Surface_Area A_c = 1,500.0 cm² (B3 -> Casting Modulus M_c = 2.40 cm in E2), Green_Sand_Mold_Constant B = 2.10 min/cm² (B4), Chvorinov_Exponent n = 2.00 (B5), and Riser_Modulus_Factor k_safe = 1.20 (B6): Casting Solidification Time hits 2.10 * (2.40^2) = 12.10 minutes (E3), Required H=D Blind Riser Diameter equals 6 * (1.20 * 2.40) = 17.28 cm (E4), and Cylindrical Riser Volume equals 4,052.5 cm³ (E5)! In a shared Arte Sheets foundry gating workbook, a `suggester` can test inserting cast-iron chills (`B = 0.85` min/cm²) in Suggest Mode while asking a hint-first AI prompt ("Why does adding an exothermic sleeve cut required riser volume in half without violating Chvorinov’s directional freeze rule?") before the `owner` releases the pattern.

Chvorinov Cooling Modulus & H=D Cylindrical Riser Sizing Formula

Validate Casting_Volume > 0, Surface_Area > 0, Mold_Constant B > 0, Chvorinov_Exponent n in [1.5, 2.5], and Riser_Modulus_Safety_Factor k_safe >= 1.05!

=LET(vc_c, B2, ac_s, B3, bm_c, B4, ex_n, B5, ks_f, B6, IF(OR(vc_c <= 0, ac_s <= 0, bm_c <= 0, ex_n < 1.5, ex_n > 2.5, ks_f < 1.05), "Invalid Foundry Input", LET(mc_m, vc_c / ac_s, dr_c, 6 * ks_f * mc_m, ROUND((PI() / 4) * (dr_c ^ 3), 1))))

Metal Chill Rapid Directional Freeze Check (Set Mold Constant B = 0.85 min/cm²)

Place external cast-iron chills against the heavy boss (lowering effective Mold_Constant B4 from 2.10 to 0.85 min/cm²): Local Boss Solidification Time E3 drops from 12.10 minutes down to 4.90 minutes—eliminating isolated hot-spot shrinkage porosity!

=IF(ISTEXT(E5), E5, ROUND(B4 * ((B2 / B3) ^ B5), 2))

Further reading

• Arte Sheets Lesson — Sand-Casting Core Buoyancy & Top Chaplet Sizing
• Arte Sheets Lesson — High-Pressure Die Casting Gate Velocity & Fill Time