## Question

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

*Hot fluid in temperature TH1 = 100*

^{0}C*Hot fluid out temperature TH2 = 80*

^{0}C*Cold fluid out temperature TC2 = 50*

^{0}C*Cold fluid in temperature TC1 = 30*

^{0}C

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.

## Answer and Explanation

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 ^{0}C 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 ^{0}C. Hence the mean temperature difference should also be equal to 50 ^{0}C and clearly is not indeterminate.