Thermal Process Engineering, TEMA Shell-and-Tube Exchangers & Temperature Cross
1-2 Shell-and-Tube LMTD Correction Factor (Ft) Formula
Calculate Heat-Capacity Ratio R, Cold-Side Thermal Effectiveness P, Exact Bowman-Mueller-Nagle 1-Shell / 2-Tube-Pass LMTD Correction Factor F_t, and Effective Mean Temperature Difference (°C).
Why U-tube 1-2 exchangers suffer a severe Ft penalty below 0.75 whenever outlet temperatures cross!
Applying pure counterflow LMTD to a TEMA 1-shell-pass, 2-tube-pass (U-tube or floating-head) heat exchanger undersizes heat-transfer area because half of the tube bundle runs in parallel co-current flow against the shell fluid! Bowman, Mueller & Nagle derived the exact analytical Configuration Correction Factor F_t from Capacity Ratio R_cap = (T_hi - T_ho) / (T_co - T_ci) and Thermal Effectiveness P_eff = (T_co - T_ci) / (T_hi - T_ci), including the smooth L’Hopital limit when R_cap = 1! Whenever process designers push Cold Outlet T_co above Hot Outlet T_ho ("temperature cross"), F_t plunges rapidly below the industry 0.75 minimum design rule—signaling that a 2-shell-pass exchanger or true counterflow PHE is required instead of adding useless extra tube surface!
Try this TEMA 1-2 Crude Preheater Sizing Check first—Hot_Inlet T_hi = 160.0 °C (B2), Hot_Outlet T_ho = 90.0 °C (B3), Cold_Inlet T_ci = 30.0 °C (B4), and Cold_Outlet T_co = 80.0 °C (B5): Capacity Ratio hits R_cap = 70 / 50 = 1.400 (E2), Effectiveness equals P_eff = 50 / 130 = 0.385 (E3), Exact Bowman 1-2 Factor F_t evaluates to 0.8640 (>= 0.75 healthy!, E4), and multiplying Pure Counterflow LMTD (69.52 °C) by F_t gives Effective Mean Temperature Difference Delta_T_eff = 60.07 °C (E5)!
| TEMA 1-2 Exchanger KPI (Thi=160 °C, Tho=90 °C, Tci=30 °C, Tco=80 °C) | Formula | Verified Output |
|---|---|---|
| Heat-Capacity Ratio R_cap | (Thi - Tho) / (Tco - Tci) | 1.400 (1.077 @ Tco = 95 °C cross) |
| Thermal Effectiveness P_eff | (Tco - Tci) / (Thi - Tci) | 0.385 (0.500 @ Tco = 95 °C cross) |
| Bowman 1-2 Correction Factor Ft | Analytical Bowman-Mueller-Nagle (incl R=1) | 0.8640 (0.7599 @ Tco = 95 °C cross!) |
| Effective Mean Temp Diff (Ft*LMTD) | Ft * Counterflow_LMTD_C | 60.07 °C (47.47 °C @ Tco = 95 °C) |
Exact Bowman-Mueller-Nagle 1-Shell / 2-Tube-Pass Ft & Effective MTD Formula
Validate T_hi > T_ho > T_ci, T_hi > T_co > T_ci, and check thermodynamic pinch feasibility inside the 1-2 logarithm arguments (rejecting pinch violations where 1 - R*P <= 0 or the denominator argument <= 1)!
=LET(th_i, B2, th_o, B3, tc_i, B4, tc_o, B5, IF(OR(th_i <= th_o, tc_o <= tc_i, th_o <= tc_i, tc_o >= th_i), "Invalid Exchanger Temps", LET(dt_1, th_i - tc_o, dt_2, th_o - tc_i, lm_c, IF(ABS(dt_1 - dt_2) < 0.000001, dt_1, (dt_1 - dt_2) / LN(dt_1 / dt_2)), rr_f, (th_i - th_o) / (tc_o - tc_i), pp_f, (tc_o - tc_i) / (th_i - tc_i), sq_r, SQRT((rr_f ^ 2) + 1), dn_a, 2 - pp_f * (rr_f + 1 - sq_r), dn_b, 2 - pp_f * (rr_f + 1 + sq_r), IF(OR(1 - rr_f * pp_f <= 0, dn_b <= 0, dn_a <= dn_b), "1-2 PINCH CROSS UNREACHABLE", LET(ft_v, IF(ABS(rr_f - 1) < 0.000001, (pp_f * SQRT(2)) / ((1 - pp_f) * LN(dn_a / dn_b)), ((sq_r / (rr_f - 1)) * LN((1 - pp_f) / (1 - rr_f * pp_f))) / LN(dn_a / dn_b)), ROUND(lm_c * ft_v, 2))))))5 °C Temperature-Cross Ft Cliff Check (Set Cold Outlet T_co = 95.0 °C > Hot Outlet 90.0 °C)
Push Cold_Outlet B5 up to 95.0 °C (a 5 °C temperature cross above Hot_Outlet 90.0 °C): Effectiveness E3 reaches 0.500, Bowman Correction Factor E4 tumbles from 0.8640 right down to the cliff edge at 0.7599, and Effective Mean Temperature Difference E5 drops from 60.07 °C to 47.47 °C!
=IF(ISTEXT(E5), E5, LET(th_i, B2, th_o, B3, tc_i, B4, tc_o, B5, rr_f, (th_i - th_o) / (tc_o - tc_i), pp_f, (tc_o - tc_i) / (th_i - tc_i), sq_r, SQRT((rr_f ^ 2) + 1), dn_a, 2 - pp_f * (rr_f + 1 - sq_r), dn_b, 2 - pp_f * (rr_f + 1 + sq_r), ft_v, IF(ABS(rr_f - 1) < 0.000001, (pp_f * SQRT(2)) / ((1 - pp_f) * LN(dn_a / dn_b)), ((sq_r / (rr_f - 1)) * LN((1 - pp_f) / (1 - rr_f * pp_f))) / LN(dn_a / dn_b)), ROUND(ft_v, 4)))Try the example
Use the original values or change them to test your understanding.