LMTD Correction Factors

When the exchanger is not true counter-flow: use ΔT_lm,true = F·ΔT_lm,counterflow

coredesignheat transfer

⏱️ About 16 min

Real exchangers often have multiple passes or crossflow. The temperature field is more complex, but we can still use LMTD with one extra factor.

💡

The big idea: For non-ideal flow arrangements (multi-pass shell-and-tube, crossflow), a correction factor F (≤ 1) adjusts the counter-flow LMTD to match the true mean driving force.

🎯 By the end, you'll be able to

Explain when an LMTD correction factor is needed
Define the dimensionless ratios P and R used with F-charts
Compute heat duty using Q = U·A·F·LMTD_cf

📎 Helpful to know first

Heat Exchanger Types & the LMTD Method

Why correction factors exist

The counter-flow LMTD formula assumes a simple 1D temperature field with two streams exchanging heat along a single direction. In many practical exchangers, one or both streams make multiple passes, or the streams cross each other (crossflow).

To preserve the convenience of the LMTD method, we compute an equivalent counter-flow LMTD and then apply a correction factor F obtained from standard charts.

🔑 Correction-factor equation

The true log-mean temperature difference is modeled as:

ΔT_lm,true = F · ΔT_{lm,counterflow}, with 0 < F ≤ 1.

\[ \Delta T_{\mathrm{lm,true}} = F\,\Delta T_{\mathrm{lm,cf}} \]

LMTD correction-factor relationship for multi-pass / crossflow exchangers.

Temperature ratios used on F charts

For many shell-and-tube correction charts, you compute two dimensionless temperature ratios:

P = (T_c,out − T_c,in)/(T_h,in − T_c,in) and R = (T_h,in − T_h,out)/(T_c,out − T_c,in).

You then read F from a chart based on the exchanger configuration (e.g., 1–2 shell-and-tube: one shell pass, two tube passes).

\[ P = \frac{T_{c,out}-T_{c,in}}{T_{h,in}-T_{c,in}},\qquad R = \frac{T_{h,in}-T_{h,out}}{T_{c,out}-T_{c,in}} \]

Dimensionless ratios used to determine F from standard charts.

✨ Interpretation of F

F measures how much the true mean driving force is reduced relative to ideal counter-flow. Values near 1 mean the exchanger behaves close to counter-flow; smaller values indicate a less favorable temperature field.

\[ Q = U\,A\,F\,\Delta T_{\mathrm{lm,cf}} \]

Heat duty using counter-flow LMTD with a correction factor.

📝 Worked example: A 1–2 shell-and-tube exchanger is approximated using a correction factor F = 0.85. The counter-flow LMTD based on the terminal temperatures is 35.0 K. Given U = 850 W/(m²·K) and A = 12.0 m², compute the heat rate Q.

Compute corrected mean temperature difference: ΔT_lm,true = F·LMTD_cf = 0.85 × 35.0 = 29.75 K.
Compute Q = U·A·ΔT_lm,true = 850 × 12.0 × 29.75 W.
First multiply: 850 × 12.0 = 10,200 W/K.
Then Q = 10,200 × 29.75 = 303,450 W = 3.03×10^5 W.

✓ 3.03×10^5 W

✏️ Practice: A crossflow exchanger is modeled with F = 0.78. The counter-flow LMTD is 42.0 K. Given U = 60.0 W/(m²·K) and A = 25.0 m², compute Q.

Solution

Corrected mean ΔT: ΔT_lm,true = F·LMTD_cf = 0.78 × 42.0 = 32.76 K.
Compute UA = 60.0 × 25.0 = 1500 W/K.
Heat rate: Q = UA·ΔT_lm,true = 1500 × 32.76 = 49,140 W.

Check your understanding

1. The correction factor F is introduced primarily because:

F accounts for the fact that real arrangements do not maintain the same temperature difference pattern as ideal counter-flow.

2. Which statement about F is generally true?

F is a reduction factor relative to counter-flow LMTD, so it lies between 0 and 1 in normal operation.

3. If F decreases while U and A stay constant, the predicted heat rate Q:

Q is proportional to F in Q = U·A·F·LMTD_cf.

✅ Key takeaways

Many practical exchangers are not true counter-flow (multi-pass or crossflow).
Use ΔT_lm,true = F·ΔT_lm,counterflow with F read from charts using P and R.
Heat duty becomes Q = U·A·F·LMTD_cf.

➡️ LMTD works nicely when outlet temperatures are known. Next, you’ll learn the ε–NTU method, which is often easier when outlet temperatures are unknown (especially in phase-change service).

Want to test yourself on this? Try the Chemical Aptitude test →