Engineering & Fluid Mechanics
Kern Shell-Side Nusselt & Square-Pitch Equivalent Diameter Formula
Perform preliminary shell-and-tube exchanger sizing using Donald Q. Kern’s square-pitch hydraulic diameter De = 4*(Pt² - pi*do²/4)/(pi*do) and turbulent shell correlation Nu_s = 0.36 * Re_s^0.55 * Pr^(1/3) * (mu/mu_w)^0.14.
Fast preliminary shell-and-tube sizing via Kern’s 1950 Process Heat Transfer correlation!
Before detailed Bell-Delaware mechanical drawings exist, process engineers size shell-and-tube exchangers using Donald Q. Kern’s classic method (1950, Process Heat Transfer, McGraw-Hill) for 25%-cut segmental baffles over 2,000 <= Re_s <= 1,000,000! For removable-bundle mechanical cleaning on a 90° square tube layout of pitch Pt and tube outer diameter do, the free-flow equivalent diameter is De = 4 * (Pt^2 - pi * do^2 / 4) / (pi * do)!
Using De in the shell Reynolds and Nusselt numbers yields Kern’s shell-side heat-transfer correlation: Nu_s = (h_s * De / k) = 0.36 * Re_s^0.55 * Pr^(1/3) * (mu_b / mu_w)^0.14! For 1.00-in OD tubes (do = 0.0254 m) on 1.25-in square pitch (Pt = 0.03175 m), De = 25.13 mm; at Re_s = 18,500, Pr = 5.2, and mu_b/mu_w = 1.22, Nu_s = 142.56 (rising to 192.71 when Re_s increases to 32,000)!
| Shell-Side Operating Case (1" OD on 1.25" Sq Pitch) | Square-Pitch De (mm) | Kern Shell Nusselt Nu_s | Flow Regime Check |
|---|---|---|---|
| Re_s = 18,500, Pr = 5.2, mu_b/mu_w = 1.22 | 25.13 mm | 142.56 | Standard Kern Turbulent Window |
| Higher Velocity Re_s = 32,000 | 25.13 mm | 192.71 | Re_s^0.55 Scaling (+35.2%) |
| Pitch Pt = 0.024 m <= Tube OD 0.0254 m | Touching/Overlapping Tubes! | "Invalid Tube Pitch or Kern Inputs" | Pt > do Geometric Guard |
Kern Shell-Side Nusselt Correlation Nu_s = 0.36 * Re_s^0.55 * Pr^(1/3) * (mu_b/mu_w)^0.14
Evaluates Kern’s 25%-baffle-cut shell Nusselt number across 2,000 <= Re_s <= 1,000,000 while verifying Pt > do.
=IF(OR(B2 <= B3, B3 <= 0, B4 < 2000, B5 <= 0, B6 <= 0), "Invalid Tube Pitch or Kern Inputs", ROUND(0.36 * (B4 ^ 0.55) * (B5 ^ (1 / 3)) * (B6 ^ 0.14), 2))90° Square-Pitch Shell Equivalent Diameter De (mm) = 4*(Pt² - pi*do²/4)/(pi*do) * 1000
Computes the square-pitch hydraulic/equivalent diameter in millimeters from pitch Pt (B2, m) and outer tube diameter do (B3, m).
=IF(OR(B2 <= B3, B3 <= 0, B4 < 2000, B5 <= 0, B6 <= 0), "Invalid Tube Pitch or Kern Inputs", ROUND((4 * ((B2 ^ 2) - PI() * (B3 ^ 2) / 4)) / (PI() * B3) * 1000, 2))Try the example
Use the original values or change them to test your understanding.