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Why an LMTD Correction Factor?
The Log Mean Temperature Difference (LMTD) formula is derived assuming pure counter-flow or pure parallel-flow between two fluid streams. Real shell-and-tube exchangers — with multiple tube passes and a single shell pass — experience a mix of counter-flow and co-current flow depending on which tube pass the fluid is in. The result is a mean temperature difference lower than the pure counter-flow LMTD.
The LMTD correction factor F (also called the F-factor or Ft) accounts for this deviation:
$$ Q = U A F \cdot \Delta T_{lm,cf} $$where ΔTlm,cf is the LMTD calculated as if the exchanger were in pure counter-flow. F ≤ 1 always; F = 1 for true counter-flow (1-1 exchangers). A low F (say F < 0.75) signals an inefficient arrangement — the exchanger needs more shells or a different configuration.
LMTD in Pure Counter-Flow
For a 1-1 counter-flow exchanger with hot fluid entering at Th1 and leaving at Th2, cold fluid entering at Tc1 and leaving at Tc2:
$$ \Delta T_1 = T_{h1} - T_{c2}, \quad \Delta T_2 = T_{h2} - T_{c1} $$ $$ \Delta T_{lm} = \frac{\Delta T_1 - \Delta T_2}{\ln(\Delta T_1 / \Delta T_2)} $$This value is always used as the reference baseline when computing F for multi-pass configurations.
Dimensionless Groups: P and R
The F-factor is a function of two dimensionless temperature ratios:
$$ P = \frac{T_{c2} - T_{c1}}{T_{h1} - T_{c1}} \quad \text{(cold-side thermal effectiveness)} $$ $$ R = \frac{T_{h1} - T_{h2}}{T_{c2} - T_{c1}} = \frac{C_c}{C_h} \quad \text{(heat capacity rate ratio)} $$P ∈ [0, 1] represents how much of the maximum possible temperature change the cold stream achieves. R is the ratio of cold-side to hot-side capacity rates. Note: when R = 1, the capacity rates are equal and the streams have equal temperature changes.
F-Factor Formula: 1 Shell Pass, 2N Tube Passes (TEMA E)
For the standard TEMA E shell (1 shell pass, 2 or more even number of tube passes):
$$ F = \frac{\sqrt{R^2+1}}{(R-1)} \cdot \frac{\ln\!\left[\dfrac{1-P}{1-PR}\right]} {\ln\!\left[\dfrac{2/P - 1 - R + \sqrt{R^2+1}}{2/P - 1 - R - \sqrt{R^2+1}}\right]} \qquad (R \neq 1) $$For R = 1 (equal capacity rates), a limiting form applies:
$$ F = \frac{\sqrt{2} \cdot P}{(1-P)} \cdot \frac{1}{\ln\!\left[\dfrac{2/P - 1 + \sqrt{2}/P \cdot \sqrt{1-P}}{2/P - 1 - \sqrt{2}/P \cdot \sqrt{1-P}}\right]} $$These formulas are used by process simulation tools (Aspen, HYSYS, UniSim) and by hand calculation per TEMA and Kern's Process Heat Transfer.
F-Factor Charts (P-R Diagram)
The traditional design approach uses F vs. P charts with R as a family parameter. Key observations:
- Steep drop region — near the maximum P for each R curve, F plunges toward 0. This "nose" corresponds to the temperature cross limit. Operating in this region is strongly inadvisable because small errors in T measurement cause massive F uncertainty and therefore massive area uncertainty.
- F > 0.80 rule of thumb — Most process design guidelines (Kern, HEDH) require F ≥ 0.80 for acceptable exchanger design. Below 0.80, add a shell in series.
- Temperature cross — When Tc2 > Th2, a temperature cross exists. A single TEMA E shell cannot achieve this; multiple shells in series are required.
Temperature Cross: The Critical Constraint
A temperature cross occurs when the cold-stream outlet exceeds the hot-stream outlet:
$$ T_{c2} > T_{h2} \quad \Rightarrow \quad \text{temperature cross} $$This is physically impossible in a single-shell 1-2 exchanger because part of the tube bundle operates in co-current (parallel) flow. In co-current flow, the outlet temperatures cannot cross. The maximum cold-outlet temperature in a 1-2 exchanger is bounded by the hot-inlet temperature, not the hot-outlet temperature. To achieve a temperature cross, you must use multiple shells in series.
Minimum Shells Required
The minimum number of shells in series (Ns) for a temperature cross is:
$$ N_s \geq \frac{\ln\!\left[\dfrac{1-P}{1-PR}\right]}{\ln\!\left[\dfrac{2/P_1 - 1 - R + \sqrt{R^2+1}}{2/P_1 - 1 - R - \sqrt{R^2+1}}\right]} $$where P1 is the per-shell effectiveness. In practice, engineers round up to the nearest integer and verify F ≥ 0.80 for each shell.
Worked Example: Single Shell — Does It Work?
Given: Hot process gas: Th1 = 200°C in, Th2 = 120°C out. Cold boiler feedwater: Tc1 = 40°C in, Tc2 = 110°C out. 1-2 TEMA E exchanger proposed. Determine F.
Step 1 — Check for temperature cross:
$$ T_{c2} = 110°C < T_{h2} = 120°C \quad \Rightarrow \quad \text{No temperature cross. Single shell feasible.} $$Step 2 — Compute P and R:
$$ P = \frac{110 - 40}{200 - 40} = \frac{70}{160} = 0.4375 $$ $$ R = \frac{200 - 120}{110 - 40} = \frac{80}{70} = 1.143 $$Step 3 — Counter-flow LMTD (reference):
$$ \Delta T_1 = T_{h1} - T_{c2} = 200 - 110 = 90°C $$ $$ \Delta T_2 = T_{h2} - T_{c1} = 120 - 40 = 80°C $$ $$ \Delta T_{lm,cf} = \frac{90 - 80}{\ln(90/80)} = \frac{10}{0.1178} = 84.9°C $$Step 4 — Compute F using the 1-2 formula:
$$ \sqrt{R^2+1} = \sqrt{1.143^2+1} = \sqrt{2.307} = 1.519 $$ $$ \frac{R+1}{R-1} \text{ … using the full formula:} $$ $$ \text{Numerator: } \frac{1.519}{(1.143-1)} \ln\!\left[\frac{1-0.4375}{1-0.4375 \times 1.143}\right] = 10.62 \times \ln\!\left[\frac{0.5625}{0.5}{}\right] = 10.62 \times \ln(1.125) = 10.62 \times 0.1178 = 1.251 $$ $$ \text{Denominator: } \ln\!\left[\frac{2/0.4375 - 1 - 1.143 + 1.519}{2/0.4375 - 1 - 1.143 - 1.519}\right] = \ln\!\left[\frac{4.571 - 2.143 + 1.519}{4.571 - 2.143 - 1.519}\right] = \ln\!\left[\frac{3.947}{0.909}\right] = \ln(4.342) = 1.469 $$ $$ F = \frac{1.251}{1.469} = 0.852 $$F = 0.852 > 0.80 ✅ — the single-shell design is acceptable.
Step 5 — Effective MTD:
$$ \Delta T_{eff} = F \cdot \Delta T_{lm,cf} = 0.852 \times 84.9 = 72.3°C $$The required area would be calculated as A = Q / (U × 72.3).
Worked Example: Temperature Cross — Multiple Shells Required
Given: Hot oil: Th1 = 180°C, Th2 = 80°C. Cold solvent: Tc1 = 30°C, Tc2 = 130°C. Check feasibility of 1-2 shell.
$$ T_{c2} = 130°C > T_{h2} = 80°C \quad \Rightarrow \quad \textbf{Temperature cross — single shell impossible.} $$Compute P and R for the overall exchanger:
$$ P = \frac{130-30}{180-30} = \frac{100}{150} = 0.667, \quad R = \frac{180-80}{130-30} = \frac{100}{100} = 1.0 $$For N = 2 shells in series, per-shell effectiveness P1 satisfies:
$$ P = \frac{N \cdot P_1 - (N-1) \cdot P_1^2 \cdot R^{N-1} \cdot \ldots}{\ldots} $$In the common R = 1 case, per-shell P1 is found from:
$$ P = 1 - \left(1 - P_1\right)^N \quad \text{(for R=1 approximately)} $$ $$ 1 - P_1 = (1-0.667)^{1/2} = (0.333)^{0.5} = 0.577 $$ $$ P_1 = 1 - 0.577 = 0.423 $$Check F for each shell with P1 = 0.423, R = 1.0 → F ≈ 0.88 ✅. Two shells suffice.
Practical Rules for Multi-Shell Design
- F ≥ 0.80 — Design minimum. If F drops below this, add a shell in series.
- Temperature cross > 10°C — Nearly always requires 2+ shells. Even a small cross of 5°C typically pushes F below 0.75.
- High R (R > 4) — Low-capacity stream (gas cooling against a large liquid flow). Multi-shell arrangements help significantly.
- Shells in series vs. parallel — Series increases effective counter-flow and allows temperature crosses. Parallel splits flow and reduces pressure drop but does not improve F.
- TEMA J and H shells — For very large exchangers where pressure drop is critical, divided-flow (J) or double-flow (H) shells offer alternatives. Their F-factor charts differ from TEMA E; use dedicated charts.
F-Factor for Other Configurations
| Configuration | F Range | Notes |
|---|---|---|
| 1-1 Counter-flow | F = 1.0 always | Benchmark; used in hairpin / double-pipe exchangers |
| 1-2 TEMA E (1 shell, 2 tube passes) | 0.70 – 0.97 | Most common industrial type |
| 2-4 TEMA E (2 shells, 4 tube passes) | 0.85 – 0.99 | Handles moderate temperature crosses |
| Crossflow (both unmixed) | 0.80 – 0.95 | Air-cooled heat exchangers, fin-and-tube |
| Crossflow (one stream mixed) | 0.75 – 0.92 | Shell-side mixed, tube-side unmixed |
| Condenser / Evaporator | F = 1.0 always | Phase change → isothermal stream, no flow-arrangement penalty |
Relationship to the NTU Method
The F-factor and NTU methods are mathematically equivalent; F is simply a remapping of the NTU-effectiveness relationship back into the LMTD framework:
$$ F = \frac{NTU_{cf}}{NTU_{actual}} \cdot \frac{C_{min}}{UA} \cdot \frac{1}{\Delta T_{lm,cf}} \cdot Q $$In design practice, many engineers use NTU for rating (unknown outlet T) and LMTD+F for sizing (known all four terminal temperatures). Both paths converge to the same required area.
Design Checklist
- Calculate all four terminal temperatures from the heat balance.
- Check for temperature cross (Tc2 vs Th2).
- Compute P and R.
- Calculate counter-flow LMTD (ΔTlm,cf).
- Apply F-factor formula or read from chart.
- If F < 0.80, increase number of shells in series and repeat.
- Effective MTD = F × ΔTlm,cf; compute required area A = Q / (U × MTD).
Try the Calculator
Use the Heat Transfer Coefficient Calculator on this page to compute U, duty, and thermal performance. Combine with the F-factor from the chart above to determine the true effective MTD and required exchanger area.
Heat Exchanger Rating
Apply this methodology directly in the ChemProCal calculator.
Open Calculator →Live Shell & Tube Heat Exchanger (LMTD & Bell-Delaware) Estimator
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