Metalcasting & Steel Foundry Engineering, Caine (1948) AFS Hyperbolic Riser Feeding Law
Caine Sand-Casting Riser Freezing-Ratio & Yield Formula
Calculate Actual Riser-to-Casting Modulus Freezing Ratio x = Mr/Mc, Volume Ratio y = Vr/Vc, Caine Hyperbolic Minimum Soundness Ratio xmin, and Mold Casting Yield (%).
Passing Chvorinov’s modulus rule (Mr > Mc) is NOT enough—if riser liquid volume cannot feed 3.5% steel solidification shrinkage, Caine’s hyperbola exposes a pipe cavity!
Foundry engineers often size a cylindrical sand-casting feeder riser (`Diameter D_r` cm, `Height H_r` cm -> Volume `V_r = (PI()/4) * D_r^2 * H_r`, Non-Feeding Cooling Area `A_r = PI() * D_r * H_r + (PI()/4) * D_r^2`, Modulus `M_r = V_r / A_r`) solely for `M_r >= 1.2 * M_c` (`Casting Modulus M_c = V_c / A_c`)—forgetting **Volumetric Solidification Contraction `b`** (`~0.035` = `3.5%` for carbon steel, `~0.06` for aluminum)! **J. B. Caine (`1948`, `AFS Transactions`)** unified thermal freezing rate AND liquid mass feeding into his celebrated **Hyperbolic Soundness Boundary** between **Actual Freezing Ratio `x = M_r / M_c`** and **Riser-to-Casting Volume Ratio `y = V_r / V_c`**: **`x_{min} = (a / (y - b)) + c`** (`where a = 0.10, c = 1.00` for un-sleeved steel sand molds; `x >= x_{min}` guarantees X-ray soundness, whereas `x < x_{min}` draws primary shrinkage pipe straight down into the casting!), traded directly against **Pouring Metal Yield `Yield% = (V_c / (V_c + V_r)) * 100%`**!
Verify on a `4,200` cm³ (`32.9` kg) WCB Cast-Steel Valve Body—Casting_Volume `V_c` = 4,200 cm³ (B2), Casting_Surface_Area `A_c` = 1,680 cm² (`M_c = 2.50` cm, B3), Cylindrical_Riser_Diameter `D_r` = 14.0 cm (B4), Riser_Height `H_r` = 18.0 cm (B5 -> `V_r = 2,770.9` cm³, `M_r = 2.930` cm), Steel_Shrinkage_Fraction `b` = 0.035 (B6), and Caine_Constant `a` = 0.10 (B7): Actual Freezing Ratio `x` equals `1.172` (E2) at Volume Ratio `y = 0.660` (E3), clearing **Caine Minimum `x_{min} = 1.160`** (`1.172 >= 1.160` X-ray sound! E4) at `60.3%` Casting Yield (E5)!
| AFS Caine Riser Soundness KPI (Steel Vc=4,200 cm³, Ac=1,680 cm² -> Mc=2.50 cm, b=0.035) | Formula | Verified Output |
|---|---|---|
| Actual Freezing Ratio x (Mr / Mc) | (Vr / (PI()*Dr*Hr + PI()/4*Dr^2)) / (Vc / Ac) | 1.172 (0.793 freezes EARLY @ Dr=9.5 cm!) |
| Riser Volume Ratio y (Vr / Vc) | ((PI()/4) * Dr^2 * Hr) / Casting_Vc_cm3 | 0.660 (0.203 starved feed metal) |
| Caine Min Soundness Ratio xmin | (Caine_a / (Volume_Ratio_y - Shrink_b)) + 1.0 | 1.160 PASS (x=1.172>=1.160 vs 1.597 FAIL!) |
| Net Mold Casting Yield (%) | (Vc / (Vc + Vr)) * 100 | 60.3% sound (83.2% SCRAPPED BY PIPE!) |
J. B. Caine (1948) AFS Hyperbolic Freezing-vs-Volume Soundness Formula
Validate `Casting_Volume V_c > 10` cm³, `Casting_Area A_c > 10` cm², `Riser_Diameter D_r > 1` cm, `Riser_Height H_r > 1` cm, `Shrinkage b` in `[0.005, 0.10]`, positive volume excess `y > b + 0.02`, and `Caine a` in `[0.01, 0.50]`!
=LET(vc_c, B2, ac_c, B3, dr_c, B4, hr_c, B5, sb_f, B6, ca_k, B7, vr_c, (PI() / 4) * (dr_c ^ 2) * hr_c, yv_r, vr_c / vc_c, IF(OR(vc_c <= 10, ac_c <= 10, dr_c <= 1, hr_c <= 1, sb_f < 0.005, sb_f > 0.1, yv_r <= sb_f + 0.02, ca_k < 0.01, ca_k > 0.5), "Invalid Caine Riser Input", ROUND((vc_c / (vc_c + vr_c)) * 100, 1)))Undersized Greedy-Yield Riser (`D_r = 9.5` cm, `H_r = 12` cm) Centerline Shrinkage Cavity Check
Shrink the open sand riser to `B4 = 9.5` cm x `B5 = 12.0` cm chasing `83.2%` foundry yield (E5): Volume Ratio `y` E3 drops to `0.203`, pushing Caine Required `x_{min}` E4 UP to **`1.597`** while Actual `x` E2 falls to **`0.793`** (`0.793 << 1.597`!)—guaranteeing a deep shrinkage cavity under radiographic inspection (unless insulated with an exothermic sleeve where `c ~= 0.70`)!
=IF(ISTEXT(E5), E5, LET(vr_c, (PI() / 4) * (B4 ^ 2) * B5, yv_r, vr_c / B2, ROUND((B7 / (yv_r - B6)) + 1, 3)))Try the example
Use the original values or change them to test your understanding.