Compute the log mean temperature difference (LMTD) and heat duty Q for a two-stream heat exchanger from the four terminal temperatures. Choose counter-current or co-current (parallel) flow, then enter the overall heat-transfer coefficient U and area A to get the transferred power.
The log mean temperature difference (LMTD) is the correct average driving force for heat transfer in an exchanger where the temperature difference between the two streams changes along the length of the equipment. Because that difference is not constant, you cannot simply use the arithmetic average — the LMTD weights the ends logarithmically. Combined with the rate equation Q = U·A·LMTD, it is the foundation of heat-exchanger sizing and rating for shell-and-tube, double-pipe, and plate exchangers.
The log mean temperature difference is:
LMTD = (ΔT₁ − ΔT₂) / ln(ΔT₁/ΔT₂)
where ΔT₁ and ΔT₂ are the temperature differences between the hot and cold streams at the two ends of the exchanger. When ΔT₁ equals ΔT₂ the formula is indeterminate (0/0) and the LMTD simply equals that common value — this calculator handles that limit directly. The LMTD is always less than or equal to the arithmetic mean of the two end differences, which is why using the arithmetic average over-predicts the duty.
The terminal differences depend on the flow arrangement. In counter-current (the streams flow in opposite directions): ΔT₁ = T_hi − T_co and ΔT₂ = T_ho − T_ci. In co-current / parallel flow (same direction): ΔT₁ = T_hi − T_ci and ΔT₂ = T_ho − T_co.
For the same four temperatures, counter-current flow produces a larger LMTD and therefore needs less area for the same duty. It also allows the cold outlet to exceed the hot outlet — a temperature cross — which co-current flow physically cannot achieve. Counter-current is the default choice for thermal efficiency.
A temperature cross occurs when the cold stream is heated above the hot-stream outlet temperature. In co-current flow this is impossible because both streams approach a common intermediate temperature, so one of the terminal ΔT values becomes zero or negative and no valid LMTD exists — this calculator warns when that happens. Counter-current flow can handle a cross, but in a single shell-pass / multi-tube-pass shell-and-tube exchanger a cross drives the LMTD correction factor F down sharply, so designers split the duty across multiple shells in series to keep F above about 0.8.
The exchanger rate equation is Q = U·A·LMTD, where U is the overall heat-transfer coefficient (W/m²·K), A the heat-transfer area (m²), and LMTD the driving force (K). Given U, A, and the temperatures you get the duty Q in watts; rearranged as A = Q/(U·LMTD) it gives the area required to deliver a target duty. For multi-pass and cross-flow geometries the true mean difference is F·LMTD, where the correction factor F (≤ 1) accounts for the deviation from pure counter-current flow and is read from standard F-charts.
Because the temperature difference between the streams changes exponentially, not linearly, along the exchanger. The log mean is the exact average that satisfies the differential heat-balance, while the arithmetic mean over-estimates the driving force (and so under-sizes the area). The two agree closely only when ΔT₁ and ΔT₂ are within about a factor of two of each other.
The LMTD formula becomes 0/0 and is mathematically indeterminate, but the limit is well defined: when the two end differences are equal, the LMTD equals that common value. This calculator detects when |ΔT₁ − ΔT₂| is below 1e-6 and returns ΔT₁ directly, avoiding a divide-by-zero.
Counter-current is almost always better. For the same terminal temperatures it gives a larger LMTD, so it needs less area for the same duty, and it can achieve a closer temperature approach and even a temperature cross. Co-current flow is occasionally chosen to limit a wall temperature or to keep a temperature-sensitive fluid from overheating, but it is thermally less efficient.
A temperature cross is when the cold outlet temperature exceeds the hot outlet temperature. It is impossible in co-current flow and produces an invalid LMTD there. In multi-pass shell-and-tube exchangers a cross drives the LMTD correction factor F well below 1, hurting performance, so designers add shells in series to keep F above roughly 0.8.
No — it computes the true counter-current or co-current LMTD only. For 1-2, 2-4, or cross-flow exchangers the effective mean difference is F·LMTD, where F (≤ 1) comes from standard correction-factor charts based on the temperature-ratio parameters P and R. Multiply the duty or area by F when sizing multi-pass equipment.