### Log Mean Temperature Difference (LMTD) Method with Examples

Logarithmic Mean temperature Difference is used to measure the Heat transfer rate in double pipe heat exchanger. The rate of heat transfer is got from the formula -

Q = U.A.LMTD .

U= Heat Transfer Co-efficient
A= Heat Transfer Surface Area
LMTD=  Mean Temperature Difference

LMTD is the logarithmic temperature difference between the hot and cold fluids passing through the each ends of heat exchanger pipes.

#### Formula for LMTD calculation -

LMTD or Logarithmic Mean Temperature Difference  is calculated by keeping some factors in mind

• The type of the heat exchanger
• temperature of the fluids at the four end positions
These are -
• Inlet temperature of the hot fluid
• Inlet temperature of the cold fluid
• Outlet temperature of the cold fluid
• Outlet temperature of the hot fluid
LMTD =  dT1 - dT2 /ln (dT1/dT2)

For parallel flow -

dT1 = Temperature of the hot fluid entering - Temperature of the cold fluid entering
dT2 = Temperature of the hot fluid exit - Temperature of the cold fluid exit

For counter flow -

dT1 = Temperature of the hot fluid entering - Temperature of the cold fluid exiting
dT2 = Temperature of the hot fluid exit - Temperature of the cold fluid entering

Mathematical Example :

1. For a Parallel Flow Heat Exchanger,  Hot fluid entering at temperature 100 degree Celsius and exiting at 90 degree Celsius . Cold fluid Entering at 30 degree Celsius and exiting at 50 degree Celsius. Find the LMTD .

For Parallel or co - current flow ,

According to the above mentioned formulas -

LMTD = ((100 - 30)-(90-50)) / ln (100-30/90-50) = 53.6 degree Celsius .

2. For a Counter Flow Heat Exchanger for same data calculate the LMTD .

For Counter Current HE ,

LMTD = ((100-50)-(90-30))/ln(100-50/90-30) = 54.85 degree Celsius .

So Heat transfer rate for Counter Flow is higher .