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LMTD (Log Mean Temperature Difference) is the effective average temperature difference between hot and cold fluids in a heat exchanger. You calculate it using the formula:
LMTD = (ΔT₁ – ΔT₂) / ln(ΔT₁ / ΔT₂)
Where ΔT₁ and ΔT₂ are the temperature differences at each end of the heat exchanger. For flow configurations other than pure counterflow, you multiply LMTD by a correction factor F to get the Corrected LMTD = F × LMTD.
LMTD stands for Log Mean Temperature Difference. It represents the logarithmic average of the temperature differences between the hot fluid and the cold fluid at both ends of a heat exchanger. Engineers use LMTD because the temperature difference between the two streams does not stay constant along the length of the exchanger it changes continuously as heat transfers from the hot side to the cold side.
A simple arithmetic average would overestimate or underestimate the driving force for heat transfer. The logarithmic mean corrects for this non-linearity, giving you a single representative temperature difference you can use directly in the fundamental heat transfer equation:
Q = U × A × LMTD
Where:
Q = heat duty (W or BTU/hr)
U = overall heat transfer coefficient (W/m²·K or BTU/hr·ft²·°F)
A = heat transfer area (m² or ft²)
LMTD = log mean temperature difference (°C or °F)
This equation is the backbone of heat exchanger design and rating. Every process engineer, chemical engineer, and HVAC engineer works with this relationship daily.
The LMTD calculation formula is:
LMTD = (ΔT₁ – ΔT₂) / ln(ΔT₁ / ΔT₂)
Where:
ΔT₁ = temperature difference between hot and cold fluid at End 1 of the exchanger
ΔT₂ = temperature difference between hot and cold fluid at End 2 of the exchanger
ln = natural logarithm
You assign ΔT₁ and ΔT₂ based on the flow configuration:
ΔT₁ = T_hot,in – T_cold,out
ΔT₂ = T_hot,out – T_cold,in
ΔT₁ = T_hot,in – T_cold,in
ΔT₂ = T_hot,out – T_cold,out
When both terminal temperature differences are equal, the logarithm becomes undefined (ln(1) = 0 in the denominator). In this special case, LMTD simply equals ΔT₁ = ΔT₂. This situation rarely occurs in practice but you must handle it in your spreadsheet or calculator to avoid a division-by-zero error.
Using this online LMTD calculator is straightforward. Enter the four terminal temperatures and select your flow configuration. The calculator instantly returns:
Inputs you need:
The calculator checks automatically whether your temperature cross condition is violated meaning it warns you if your cold fluid outlet temperature exceeds your hot fluid outlet temperature in a parallel flow arrangement, which is thermodynamically impossible.
Write down all four temperatures clearly. Label them:
Counterflow gives you the highest LMTD and the best thermal efficiency. Parallel flow gives a lower LMTD. Cross flow and multi-pass shell-and-tube arrangements fall somewhere in between and require a correction factor.
Apply the correct terminal difference definitions for your flow arrangement (shown above).
Plug ΔT₁ and ΔT₂ into the formula:
LMTD = (ΔT₁ – ΔT₂) / ln(ΔT₁ / ΔT₂)
For anything other than a pure single-pass counterflow or parallel flow exchanger, multiply your LMTD by the correction factor F:
Corrected LMTD = F × LMTD
Let's walk through a real calculation so you can see exactly how this works.
Problem: A shell-and-tube heat exchanger (1 shell pass, 2 tube passes) cools hot oil using cooling water.
ΔT₁ = T_h,in – T_c,out = 120 – 45 = 75°C
ΔT₂ = T_h,out – T_c,in = 60 – 25 = 35°C
LMTD = (75 – 35) / ln(75 / 35)
LMTD = 40 / ln(2.143)
LMTD = 40 / 0.7635
LMTD = 52.4°C
For a 1-2 shell-and-tube exchanger, calculate the P and R parameters:
R = (T_h,in – T_h,out) / (T_c,out – T_c,in) = (120 – 60) / (45 – 25) = 60 / 20 = 3.0
P = (T_c,out – T_c,in) / (T_h,in – T_c,in) = (45 – 25) / (120 – 25) = 20 / 95 = 0.211
Using the TEMA correction factor chart or formula for a 1-2 exchanger, F ≈ 0.97
Corrected LMTD = 0.97 × 52.4 = 50.8°C
This corrected LMTD is what you use in the Q = U × A × LMTD equation to size your heat exchanger area.
The LMTD correction factor F accounts for the fact that real heat exchangers rarely operate in pure counterflow or pure parallel flow. In a 1-2 shell-and-tube exchanger, one shell-side fluid flows in one direction while the tube-side fluid makes two passes once in each direction. This mixed arrangement reduces the effective temperature driving force compared to pure counterflow.
F is always between 0 and 1. The closer F is to 1, the closer the exchanger performs to pure counterflow. When F drops below 0.75, most engineers treat this as a warning sign the exchanger design becomes thermally inefficient and sensitive to small changes in operating conditions.
You calculate F using two dimensionless parameters:
R = (hot fluid temperature change) / (cold fluid temperature change)
R = (T_h,in – T_h,out) / (T_c,out – T_c,in)
P = (cold fluid temperature change) / (maximum possible temperature difference)
P = (T_c,out – T_c,in) / (T_h,in – T_c,in)
You then read F from a correction factor chart (published in TEMA standards and most heat transfer textbooks) or calculate it using the analytical formula specific to your exchanger configuration (1-2, 2-4, cross-flow with both fluids mixed, cross-flow with one fluid mixed, etc.).
This corrected LMTD calculator handles the F calculation automatically once you select your exchanger type.
Counterflow (counter-current) is the most thermally efficient flow arrangement. The hot fluid and cold fluid flow in opposite directions. This maximizes the temperature difference at both ends and gives the highest possible LMTD for a given set of inlet and outlet temperatures.
For counterflow:
ΔT₁ = T_h,in – T_c,out
ΔT₂ = T_h,out – T_c,in
Counterflow also allows temperature cross the cold fluid outlet can actually exceed the hot fluid outlet temperature, which is impossible in parallel flow.
Cross flow heat exchangers common in air coolers, automotive radiators, and HVAC coils have the two fluids flowing perpendicular to each other. You still calculate the raw LMTD on a counterflow basis, then apply the cross-flow correction factor F.
For cross-flow exchangers, F depends on whether each fluid is mixed or unmixed across its cross-section. An unmixed fluid (like air flowing through discrete tube rows) has a different F than a mixed fluid (like a single-pass shell-side fluid). The correction factor for cross-flow configurations typically ranges from 0.80 to 0.97 for well-designed exchangers.
Plate heat exchangers are almost always configured in counterflow arrangement between the plates. This means F = 1.0 for a single-pass plate heat exchanger in pure counterflow, and you use the raw LMTD directly.
Multi-pass plate heat exchangers require correction factors, but these are usually provided by the equipment manufacturer rather than calculated from standard TEMA charts. When you use this LMTD calculator for a plate heat exchanger, select "counterflow" for a standard single-pass gasketed plate heat exchanger.
Phase change processes condensation and evaporation occur at constant temperature on the phase-change side. This dramatically simplifies the LMTD calculation.
For a steam condenser:
ΔT₁ = T_sat – T_c,out
ΔT₂ = T_sat – T_c,in
Since one fluid is at constant temperature, the correction factor F = 1.0 regardless of the number of tube passes. The LMTD formula applies directly without any correction.
Example for a steam condenser at 100°C steam, cooling water 25°C to 40°C:
ΔT₁ = 100 – 40 = 60°C
ΔT₂ = 100 – 25 = 75°C
LMTD = (60 – 75) / ln(60/75) = –15 / ln(0.8) = –15 / (–0.2231) = 67.2°C
Note that the sign is positive because you take the absolute value of the numerator when ΔT₁ < ΔT₂.
Many engineers prefer to work in Excel for documentation and record-keeping. Here is how to set up an LMTD calculator in Excel:
B2: T_h,in (enter your value)
B3: T_h,out
B4: T_c,in
B5: T_c,out
B7: =B2-B5 (this gives ΔT₁ for counterflow)
B8: =B3-B4 (this gives ΔT₂ for counterflow)
B10: =IF(B7=B8, B7, (B7-B8)/LN(B7/B8))
(this is your LMTD with the special case handled)
The IF statement handles the case where ΔT₁ = ΔT₂ and avoids the #DIV/0! error. Add a separate cell for F and a final cell for Corrected LMTD = F × LMTD.
For the correction factor, you can build a lookup table based on R and P values, or use the explicit analytical formulas from TEMA for each exchanger configuration. Engineering references like Kern's Process Heat Transfer or the HEDH (Heat Exchanger Design Handbook) publish these formulas in closed-form equations suitable for Excel implementation.
Always be consistent. If ΔT₁ uses the hot fluid inlet, then ΔT₂ must use the hot fluid outlet. Swapping ends gives you the same LMTD mathematically, but it causes confusion when pairing with the correction factor calculation.
Many engineers calculate LMTD correctly but forget to apply F for multi-pass shell-and-tube exchangers. Using uncorrected LMTD in Q = U × A × LMTD will overestimate your available driving force and under-size your exchanger.
Always identify your actual flow arrangement before calculating. A counterflow exchanger has a higher LMTD than the same exchanger in parallel flow. Mixing these up leads to significant sizing errors.
If your corrected LMTD correction factor F falls below 0.75 or if your P and R values fall outside the chart boundaries, you have a temperature cross situation that a single shell cannot handle. You need to add more shells in series.
For condensers and evaporators, the phase-change fluid is always at saturation temperature. F = 1.0 always applies, and you do not use the standard two-temperature-change approach for that fluid.