Perform rigorous thermal rating and hydraulic pressure drop calculations for TEMA Type E industrial shell and tube heat exchangers using Kern's method. Solves true mean temperature difference FT × LMTD, shell crossflow mass velocity, clean and fouled overall heat transfer coefficient U, and shell/tube pressure drops.
1. Thermal Stream Parameters
2. Shell & Tube Bundle Geometry
3. Thermal Rating & Pressure Drops
→ [ Multi-Pass Tube Bundle (Nt Tubes, do × di, Pt Pitch) ] → [ Floating / Fixed Tubesheets ]
→ [ Tube-Side Channel Head with Pass Partitions ] ↔ [ Shell Outlet Nozzle ]
Mathematical Foundations & Kern's Shell-Side Derivations
Heat exchanger rating couples the basic heat transfer rate equation $Q = U cdot A cdot F_T cdot ext{LMTD}$ with Kern's hydrodynamic crossflow correlations:
$$ ext{LMTD} = rac{Delta T_1 - Delta T_2}{ln(Delta T_1 / Delta T_2)}$$ $$F_T = rac{sqrt{R^2 + 1} lnleft(rac{1-P}{1-PR} ight)}{(R-1) lnleft(rac{2 - P(R+1-sqrt{R^2+1})}{2 - P(R+1+sqrt{R^2+1})} ight)}$$ Prevents unfeasible temperature cross designs.
$$S_m = rac{D_s cdot (P_t - d_o) cdot B}{P_t} quad [ ext{m}^2]$$ $$G_s = rac{dot{m}_{shell}}{S_m} quad [ ext{kg}/( ext{m}^2cdot ext{s})]$$ $$Re_s = rac{G_s cdot D_e}{mu_s}$$
$$rac{1}{U_{fouled}} = rac{1}{h_o} + R_{fs} + rac{d_o ln(d_o/d_i)}{2 k_{wall}} + R_{ft} rac{d_o}{d_i} + rac{1}{h_i}rac{d_o}{d_i}$$ Evaluates true fouled margin under operating cycles.
$$Delta P_s = rac{f_s cdot G_s^2 cdot D_s cdot (N_b + 1)}{2 ho_s cdot D_e cdot (mu_s / mu_w)^{0.14}}$$ $$f_s = exp(0.576 - 0.19 ln Re_s)$$
5 Fatal Traps in Shell & Tube Exchanger Design
When high-velocity shell-side gas or vapor crosses the tube bundle, Karman vortex shedding or fluid-elastic instability can lock onto the acoustic natural frequency of the shell cavity ($f_a = c / 2 D_s$). The resulting standing acoustic wave creates deafening 130 dB sonic screams and induces violent tube bundle vibration. Tubes repeatedly hammer against baffle holes, severing tubes cleanly at baffle edges and releasing high-pressure toxic chemicals into cooling water systems. Always verify Connors instability constants and baffle spacing.
When the cold fluid outlet temperature exceeds the hot fluid outlet temperature ($T_{c,out} > T_{h,out}$), a temperature cross occurs. In a multi-pass 1-2 TEMA E shell, this causes the local temperature driving force in the cocurrent pass to reverse, heating the cold fluid backwards. The configuration factor $F_T$ drops steeply below 0.75. Novice engineers compensate by arbitrarily multiplying surface area, but the exchanger will physically never achieve the target duty. The only engineering solution is splitting the duty into two or more shells in series (TEMA F or multi-shell E).
Process fluids entering the shell inlet nozzle at high velocity carry immense kinetic energy density ($ ho v^2$). If $ ho v^2 > 2,230 ext{ kg}/( ext{m}cdot ext{s}^2)$ and no TEMA solid impingement plate or annular distribution belt is installed under the nozzle, the incoming jet strikes the top row of tubes directly. Entrained liquid droplets or particulate fines erode the 1.6 mm tube wall within months, causing catastrophic tube puncture and inter-stream cross-contamination.
Specifying segmental baffle cuts greater than 35% diameter in an attempt to lower shell pressure drop creates massive low-velocity dead zones behind the baffles and allows 40% of the shell fluid to bypass the tube matrix through the window zones without participating in crossflow heat exchange. The effective heat transfer coefficient collapses by 50%, while the dead zones accumulate stagnant sludge and severe under-deposit pitting corrosion.
To save pumping power, operators often throttle cooling water flows through condenser tubes. When cooling water tube-side velocity drops below 1.0 m/s (3.3 ft/s), silt, micro-algae, and zebra mussels settle out of suspension onto the inside tube surfaces. The resulting biofouling film increases thermal resistance tenfold and creates differential aeration cells that cause rapid localized microbiologically influenced corrosion (MIC) through-wall perforations.
Step-by-Step Worked Engineering Example
Application: Hydrocarbon Condensate Cooler (TEMA Type AES 1-2 Exchanger).
- Thermal Duty: $Q = 1,850 ext{ kW}$, Hot kerosene stream cooled from $145^circ ext{C}$ to $85^circ ext{C}$ in shell.
- Cooling Water: Heated from $30^circ ext{C}$ to $65^circ ext{C}$ in 2-pass tube bundle.
- Exchanger Specs: Shell ID $D_s = 610 ext{ mm}$, $N_t = 368$ tubes, OD $d_o = 19.05 ext{ mm}$, ID $d_i = 15.75 ext{ mm}$, Length $L = 4.88 ext{ m}$.
- Layout: $30^circ$ triangular pitch $P_t = 23.81 ext{ mm}$, Baffle spacing $B = 250 ext{ mm}$.
Step 1: Log Mean Temperature Difference & FT Factor:
$$Delta T_1 = T_{h,in} - T_{c,out} = 145 - 65 = 80^circ ext{C}$$ $$Delta T_2 = T_{h,out} - T_{c,in} = 85 - 30 = 55^circ ext{C}$$ $$ ext{LMTD} = rac{80 - 55}{ln(80 / 55)} = rac{25}{0.3747} = 66.72^circ ext{C}$$ $$P = rac{t_2 - t_1}{T_1 - t_1} = rac{65 - 30}{145 - 30} = rac{35}{115} = 0.3043$$ $$R = rac{T_1 - T_2}{t_2 - t_1} = rac{145 - 85}{65 - 30} = rac{60}{35} = 1.7143$$ $$ ext{Per 1-2 TEMA formula: } F_T = 0.942 implies Delta T_{eff} = 0.942 imes 66.72 = 62.85^circ ext{C}$$Step 2: Installed Outside Heat Transfer Area:
$$A_{ext} = N_t cdot pi cdot d_o cdot L = 368 imes pi imes 0.01905 imes 4.88 = 107.51 ext{ m}^2 quad (1,157 ext{ ft}^2)$$ $$U_{required} = rac{Q}{A cdot Delta T_{eff}} = rac{1,850,000 ext{ W}}{107.51 ext{ m}^2 imes 62.85 ext{ K}} = 273.8 ext{ W}/( ext{m}^2cdot ext{K})$$Step 3: Shell-Side Hydraulic Crossflow & Kern's Velocity:
$$S_m = rac{D_s cdot (P_t - d_o) cdot B}{P_t} = rac{0.610 imes (0.02381 - 0.01905) imes 0.250}{0.02381} = rac{0.610 imes 0.00476 imes 0.250}{0.02381} = 0.0305 ext{ m}^2$$ $$ ext{Hot kerosene mass flow: } dot{m}_h = rac{Q}{c_p Delta T_h} = rac{1850}{2.15 imes 60} = 14.34 ext{ kg/s}$$ $$G_s = rac{14.34 ext{ kg/s}}{0.0305 ext{ m}^2} = 470.2 ext{ kg}/( ext{m}^2cdot ext{s}) implies v_s = rac{470.2}{790} = 0.595 ext{ m/s}$$Step 4: Heat Transfer Coefficients & Fouled Overdesign:
$$ ext{Shell-side film: } h_o = 680 ext{ W}/( ext{m}^2cdot ext{K}), quad ext{Tube-side film: } h_i = 3,450 ext{ W}/( ext{m}^2cdot ext{K})$$ $$rac{1}{U_f} = rac{1}{680} + 0.00035 + rac{0.01905 ln(19.05/15.75)}{2 imes 16} + 0.00020left(rac{19.05}{15.75} ight) + rac{1}{3450}left(rac{19.05}{15.75} ight)$$ $$rac{1}{U_f} = 0.001471 + 0.000350 + 0.000113 + 0.000242 + 0.000350 = 0.002526 implies U_f = 395.9 ext{ W}/( ext{m}^2cdot ext{K})$$ $$mathbf{ ext{Overdesign Margin} = rac{395.9 - 273.8}{273.8} imes 100% = +44.6% implies ext{ extbf{Safe Margin for Long Production Cycles}}}$$