# Sample Problem – Logarithmic Mean Temperature Difference (LMTD) calculation

### Sample Problem Statement

Calculate log mean temperature difference for the following shell and tube heat exchanger for two possible arrangements (i) counter-current and (ii) co-current flow patterns.Hot fluid in temperature TH1 = 100 0C Hot fluid out temperature TH2 = 90 0C Cold fluid out temperature TC2 = 50 0C Cold fluid in temperature TC1 = 30 0C

### Solution

As per the definition and equation for Log Mean Temperature Difference (LMTD), ΔT1 → the temperature difference between hot and cold fluids at one end of the heat exchanger
ΔT2 → the temperature difference between hot and cold fluids at the other end of the heat exchanger (i) For counter current heat exchanger,
ΔT1 = TH1 - TC2 = 100 - 50 = 500C (At one end hot fluid enters and cold fluid exits.)
ΔT2 = TH2 - TC1 = 90 - 30 = 600C

(At the other end cold fluid enters and hot fluid exits.)

by definition given above, LMTD for counter current flow = (60-50) / ln(60/50) = 10 / 0.1823 = 54.850C.
This can be also verified quickly in EnggCyclopedia's LMTD calculator. (ii)
For co-current heat exchanger,
ΔT1 = TH1 - TC1 = 100 - 30 = 700C (At first end hot and cold fluids enter the heat exchanger.)
ΔT2 = TH2 - TC2 = 90 - 50 = 400C

(At the other end hot and cold fluids exit the heat exchanger.)
by definition given above, LMTD for counter current flow = (70-40) / ln(70/40) = 30 / 0.5596 = 53.610C. This can be also verified quickly in EnggCyclopedia's LMTD calculator.

### Remarks

It can be readily noticed that for co-current heat exchanger the logarithmic mean temperature difference is lower compared to counter-current heat exchanger. This suggests that for same fluids, to achieve the same heat transfer, a co-current heat exchanger always requires more heat transfer area than the counter-current type exchanger. In this example, a simple heat exchanger with single shell pass and single tube pass is considered. For multiple shell and tube passes, the LMTD calculated has to be multiplied by a correction factor to account for geometric changes. EnggCyclopedia's LMTD correction factor calculator can be readily used for this correction factor calculation.