In a counterflow heat exchanger, the Logarithmic Mean Temperature Difference (LMTD) is defined as

• A. LMTD = (ΔT1 − ΔT2)/ln(ΔT1/ΔT2)
• B. LMTD = ΔT1 − ΔT2
• C. LMTD = ln(ΔT1/ΔT2)
• D. LMTD = ΔT1/ΔT2

Answer: (A)

The quantity being tested is how the driving temperature difference is averaged in a counterflow heat exchanger. Because the temperature difference between the two fluids changes along the length, we use a logarithmic mean to get an effective driving force for heat transfer. In a counterflow arrangement, the two end differences are set as ΔT1 (at one end) and ΔT2 (at the other end). The proper definition of the logarithmic mean temperature difference is the difference of those end differences divided by the natural log of their ratio: (ΔT1 − ΔT2) / ln(ΔT1/ΔT2). This form emerges when you integrate the heat transfer equation with a constant overall heat transfer coefficient and account for how temperature differences evolve along the length; it yields a single representative driving force for the whole exchanger, Q = U A ΔT_lm.

If the end differences were the same, the expression naturally reduces to that common temperature difference, as a limiting case.

The other expressions don’t capture the correct averaging. A linear subtraction of the end differences ignores the exponential variation of ΔT along the length. A logarithm of the ratio alone isn’t a temperature difference, and a simple ratio of the end differences isn’t a mean temperature difference either.