The Log Mean Temperature Difference (LMTD) is a mathematical formula used in the design of heat exchangers to calculate the temperature difference between the hot and cold fluid streams at different points in the exchanger. The LMTD is used to determine the heat transfer rate, which in turn is used to design the heat exchanger size, type, and cost.

**Table of Content:**

LMTD Equation

LMTD Calculation

How to calculate when the formula does not work?

## LMTD Equation

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

Î”T_{1 }â†’ the temperature difference between hot and cold fluids at one end of the heat exchanger

Î”T_{2 }â†’ the temperature difference between hot and cold fluids at the other end of the heat exchanger

## LMTD Calculation

**Problem Statement: **Calculate the 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 ^{0}C

Hot fluid out temperature TH2 = 90 ^{0}C

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

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

**Solution**

(i) For counter current heat exchanger,

Î”T_{1} = TH1 - TC2 = 100 - 50 = 50^{0}C (At one end hot fluid enters and cold fluid exits.)

Î”T_{2} = TH2 - TC1 = 90 - 30 = 60^{0}C

(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 = 54.85^{0}C.

This can be also verified quickly in EnggCyclopedia's LMTD calculator.

(ii) For co-current heat exchanger,

Î”T_{1} = TH1 - TC1 = 100 - 30 = 70^{0}C (At first end hot and cold fluids enter the heat exchanger.)

Î”T_{2} = TH2 - TC2 = 90 - 50 = 40^{0}C

(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 = 53.61^{0}C. 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.

## How to calculate LMTD when the formula does not work?

**Problem statement: **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 â€“ (TH2-TC1 / log ( (TH1-TC1) / (TH2-TC2) ).. (for counter-current flow)

= (100-50) â€“ (80-30) / log (50/50)

= 0 /0Â â†’ 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.

Click here to thoroughly understand LMTD for heat exchanger design.