In a counter flow heat exchanger, if the hot and cold fluids enter at T₁ and T₂, respectively, whereas cold fluid leaves at T₃ and hot fluid leaves at T₄, then LMTD is given by: 

Concept:

For counterflow heat exchanger:

Log mean temperature difference is given by:

\(LMTD = \frac{{{\rm{Δ }}{T_i}\; - \;{\rm{Δ }}{T_e}}}{{\ln \left( {\frac{{{\rm{Δ }}Ti}}{{{\rm{Δ }}Te}}} \right)}}\)

Where ΔTi = Thi – Tce

ΔTe = The - Tci

Thi & Tci = Temperature at which hot or and cold water enters respectively

The & Tce = Temperature at which hot air & cold water exits respectively

In the case of the balanced counter flow heat exchanger, 

ṁhch = ṁccc = C

ṁh and ṁc are mass flow rates of hot and cold fluid, and ch and cc specific heat capacity of hot and cold fluid.

LMTD for counter-flow heat exchanger, ΔTi = ΔTe

Explanation:

Given:

Tci = T₂, Tce = T4, Thi = T1, The = T4

ΔTi = T1 – T3

ΔTe = T4 – T2

\(LMTD = \frac{{{\rm{Δ }}{T_i}\; - \;{\rm{Δ }}{T_e}}}{{\ln \left( {\frac{{{\rm{Δ }}Ti}}{{{\rm{Δ }}Te}}} \right)}}\)

\(LMTD = \rm \frac{(T_1-T_3)-(T_4-T_2)}{In\frac{(T_1-T_3)}{(T_4-T_2)}}\)