 # How to calculate LMTD when the formula fails?

## Question - How do you calculate LMTD with the formula does not work?

Calculate log mean temperature difference for the following shell and tube heat exchanger with counter-current flow.

Hot fluid in temperature TH1 = 100 0C
Hot fluid out temperature TH2 = 80 0C
Cold fluid out temperature TC2 = 50 0C
Cold fluid in temperature TC1 = 30 0C

As per the definition and equation for Log Mean Temperature Difference (LMTD),

LMTD = (TH1-TC2 – (TH2-TC1 / log ( (TH1-TC1) / (TH2-TC2) ).. (for counter-current flow)

= (100-50) – (80-30) / log (50/50)

= 0 /0 → Indeterminate!!?

When you are working on online engineering degrees this is sometimes the kind of question you may have. Here is the solution. Plainly using the LMTD formula is clearly misleading in this case. It can be observed that the temperature difference between hot and old media is 50 0C at both ends of the exchanger. Following figure for temperature profile of the counter-current flow, suggests that the temperature difference is always constant and equal to 50 0C. Hence the mean temperature difference should also be equal to 50 0C and clearly is not indeterminate.