Heat Exchanger LMTD Calculator

Hot oil cools from 120 °C to 70 °C, heating water from 20 °C to 55 °C.

In counterflow the streams enter at opposite ends, so:

• ΔT₁ = 120 − 55 = 65 °C
• ΔT₂ = 70 − 20 = 50 °C
• LMTD = (65 − 50) ÷ ln(65 ÷ 50) = 57.2 °C

In parallel flow both streams enter at the same end, so the differences are 100 °C and 15 °C, and LMTD = 44.8 °C.

Temperature difference does not fall in a straight line along the exchanger. When the two end differences are close, the plain average is near enough: 57.5 °C against 57.2 °C. When they are far apart, it is not. In the parallel case the average of 100 and 15 is 57.5 °C, which overstates the true 44.8 °C by 28%. An exchanger sized that way would be 28% too small.

Heat duty, area and LMTD are tied together: Q = U × A × LMTD.

So A = Q ÷ (U × LMTD). For a 500 kW duty and a U of 800 W/m²K:

The U value depends on the fluids, velocities and fouling, so take it from a test, a vendor or a proper thermal design. The duty comes from the flow: Q = ṁ × c_p × ΔT. Fluid speed drives U, and the Reynolds number calculator tells you whether the flow is turbulent. The guide to heat transfer covers the three modes.

Most shell-and-tube units are neither pure counterflow nor pure parallel flow. Their mean difference is F × LMTD, using the counterflow LMTD. For one shell pass and two tube passes:

• R = (T_h,in − T_h,out) ÷ (T_c,out − T_c,in)
• P = (T_c,out − T_c,in) ÷ (T_h,in − T_c,in)
• F = [S × ln((1 − P) ÷ (1 − P × R))] ÷ [(R − 1) × ln((2 − P × (R + 1 − S)) ÷ (2 − P × (R + 1 + S)))], with S = √(R² + 1)

For the oil and water case, R = 1.43, P = 0.35 and F = 0.90.

A common rule of thumb is to keep F at 0.75 or above. Below that, F falls steeply with small changes in temperature, so the design is unreliable. The usual fix is to add shells in series. Plate exchangers run close to true counterflow, and vendors publish their own factors.

A cross means the cold stream leaves hotter than the hot stream leaves. In a true counterflow exchanger this is normal. In a 1-2 shell and tube, it costs dearly.

In parallel flow a cross cannot happen, since the two streams approach the same temperature. If the hot stream ever falls below the cold stream at the same point, the entered temperatures are impossible, and the calculator tells you so.

When steam condenses at 120 °C and heats water from 20 °C to 55 °C, the hot side stays at one temperature. Enter 120 for both hot temperatures. The differences are 100 °C and 65 °C, and LMTD = 81.2 °C, whichever way the water flows. This is the one case where counterflow gives no advantage. F is 1 too, since the arrangement does not matter.

If both end differences are the same, the formula would give 0 ÷ 0, and the LMTD is simply that difference. This happens in counterflow when both streams have equal heat capacity rates: 100→50 °C against 30→80 °C gives 20 °C.

LMTD is a temperature difference, so converting it is different from converting a temperature. Multiply by 1.8 and do not add 32. A result of 57.2 °C is a difference of 102.9 °F. Or enter all four temperatures in °F, and the result comes out in °F. The thermal conductivity converter handles related units, and heat duty in kW or BTU/h converts with the power converter.