# LMTD Calculator – Log Mean Temperature Difference

Created by Gabriela Diaz
Last updated: Sep 07, 2022

Use this LMTD calculator to find the logarithmic mean temperature difference (LMTD) for different heat exchanger configurations. The log mean temperature difference, or LMTD, represents the logarithmic average temperature difference between a heat exchanger's cold and hot fluids. With this value, we can easily determine the heat transfer rate of a given heat exchanger.

• The LMTD formula;
• How to calculate the LMTD for parallel-flow equipemnts;
• The LMTD for the counter-flow configuration; and
• How to calculate the correction factor to find the LMTD for cross-flow and tube and shell heat exchangers.

## What is the log mean temperature difference? – LMTD and heat exchangers

In the analysis and design of heat exchangers, the logarithmic mean temperature difference, or LTMD, is the temperature difference we use when calculating the heat transfer rate that takes place in a particular heat exchanger.

The general formula to determine the log mean temperature difference is:

$\small \Delta T_{\text{lm}} = \cfrac{\Delta T_{\text{1}} - \Delta T_{\text{2}}}{\text{ln} \left( \Delta T_{\text{1}} / \Delta T_{\text{2}}\right)}$

Where:

• $\Delta T_{\text{lm}}$ – Logarithmic mean temperature difference or LMTD, in temperature units;
• $\Delta T_{\text{1}}$ – Temperature difference between hot and cold fluid at one of the ends; and
• $\Delta T_{\text{2}}$ – Temperature difference between fluids at the other end.

In the following sections, we'll see how to adjust this general formula to fit the different heat exchanger configurations.

Notice we don't use the arithmetic mean temperature difference $(\Delta T_{\text{am}} =\left( \Delta T_{\text{1}} - \Delta T_{\text{2}}\right)/2)$, as it always yields a greater value than the log mean $\Delta T_{\text{lm}}$, thus we'll be overestimating the heat transfer rate.

## LMTD formula for parallel-flow

Parallel-flow heat exchangers are those in which the hot and cold fluids flow in the same direction. For this type of heat exchanger, we'll use the same LMTD formula described in the previous section.

What differentiates the LMTD formula from one to another type of heat exchangers is how we define the $\Delta T_{\text{1}}$ and $\Delta T_{\text{2}}$ temperature differences. In the case of parallel-flow heat exchangers, these temperature differences are as follows:

\small \begin{aligned} \Delta T_{\text{1}} &= T_{\text{hi}} - T_{\text{ci}}\\ \Delta T_{\text{2}} &= T_{\text{ho}} - T_{\text{co}} \end{aligned}

Where:

• $T_{\text{ci}}$ and $T_{\text{co}}$ – Inlet and outlet temperatures of the cold fluid; and
• $T_{\text{hi}}$ and $T_{\text{ho}}$ – Inlet and outlet temperatures of the hot fluid.

We can substitute these into the LMTD formula and obtain the LMTD for parallel-flow $\Delta T_{\text{lm, PF}}$:

$\small \Delta T_{\text{lm, PF}} = \cfrac{\left(T_{\text{hi}} - T_{\text{ci}} \right) - \left(T_{\text{ho}} - T_{\text{co}}\right)}{\text{ln} \left(\cfrac{T_{\text{hi}} - T_{\text{ci}}}{ T_{\text{ho}} - T_{\text{co}}}\right)}$

## LMTD formula for counter-flow heat exchangers

A commonly used configuration for heat exchangers is the counter-flow. In this case, the hot and cold fluids move in opposite directions. Usually, this configuration is preferred over the parallel-flow one as it requires smaller areas to achieve the same heat transfer rate. This means that the LMTD in counter-flow is always greater than the one on parallel flow for the same inlet and outlet conditions, $\Delta T_{\text{lm, CF}} > \Delta T_{\text{lm, PF}}$.

We are using the same initial LMTD formula, but in this case the values for the temperatures differences $\Delta T_{\text{1}}$ and $\Delta T_{\text{2}}$ are:

\small \begin{aligned} \Delta T_{\text{1}} &= T_{\text{hi}} - T_{\text{co}}\\ \Delta T_{\text{2}} &= T_{\text{ho}} - T_{\text{ci}} \end{aligned}

By substituting these in the LMTD formula, we can get the specific expression to obtain the LMTD for counter-flow heat exchangers $\Delta T_{\text{lm, CF}}$:

$\small \Delta T_{\text{lm, CF}} = \cfrac{\left(T_{\text{hi}} - T_{\text{co}} \right) - \left(T_{\text{ho}} - T_{\text{ci}}\right)}{\text{ln} \left(\cfrac{T_{\text{hi}} - T_{\text{co}}}{ T_{\text{ho}} - T_{\text{ci}}}\right)}$

## LMTD formula for shell and tube and cross-flow heat exchangers – LMTD correction factor

The general log mean temperature difference LMTD formula is easy to use for parallel-flow and counter-flow heat exchangers. However, the resulting expressions for cross-flow and multipass shell-and-tube heat exchangers are more complicated.

Fortunately, a simplified methodology eases this calculation by incorporating some charts and the LMTD correction factor $\text{F}$.

In this case, we part from the LMTD formula for counter-flow, and by multiplying it by the correction factor $\text{F}$, we get the LMTD for cross-flow and shell and tube heat exchangers:

$\small \Delta T_{\text{lm, ST}} = \text{F} \times \Delta T_{\text{lm, CF}}$

Where:

• $\Delta T_{\text{lm, ST}}$ – LMTD for shell and tube or cross-flow,
• $\text{F}$ – Correction factor; and
• $\Delta T_{\text{lm, CF}}$ – LMTD for the counter-flow case.

We get the correction factor $\text{F}$ from charts as the ones below: One-shell pass and 2, 4, 6... tube passes (top) and One pass cross-flow with one fluid mixed (bottom).

Here you can find for the two other configurations, i.e., single-pass cross-flow with unmixed fluids and two shell passes and multiples of 4 tube pass.

Notice that to enter this chart, we need the values for the parameters $\text{P}$ and $\text{R}$, these we can easily calculate as the ratio between the temperature differences of the shell and tube sides of the heat exchanger:

$\small \text{P} = \cfrac{T_{\text{s2}} - T_{\text{s1}}}{T_{\text{c1}} - T_{\text{s1}}}$
$\small \text{R} = \cfrac{T_{\text{c1}} - T_{\text{c2}}}{T_{\text{s2}} - T_{\text{s1}}}$

Where:

• $T_{\text{c1}}$ and $T_{\text{c2}}$ – Respectively, the inlet and outlet temperatures on the shell side; and
• $T_{\text{s1}}$ and $T_{\text{s2}}$ – Inlet and outlet temperatures on the tube side.

We can expect values between zero and one for $\text{P}$. In the case of $\text{R}$, it can take values between zero and infinity. When this one is equal to zero, it indicates phase change on the shell, whereas the phase change takes place on the tube side when it tends to infinity.

Once you have the values for $\text{P}$ and $\text{R}$, go to the corresponding heat exchanger chart, read the correction factor$\text{F}$, and finally plug this value into the corrected LMTD formula we mentioned at the beginning of this section.

💡 We have more heat transfer tools to help you keep expanding your knowledge in the subject! Why not take a look at the thermal resistance calculator or the heat transfer coefficient calculator.

## Using the LMTD calculator

With the help of the LMTD calculator, you can easily find the logarithmic mean temperature difference for different heat exchanger configurations. Let's see how to calculate the LMTD with this tool:

#### For parallel-flow and counter-flow heat exchangers:

1. Begin by selecting the type of heat exchanger you're interested in studying.

2. For the Hot fluid, enter its respective inlet and outlet temperatures, Thi and Tho.

3. Similarly, input the inlet and outlet temperatures associated with the Cold fluid, Tci, and Tco.

4. In the Temperature difference section, the calculator will display the temperature differences terms ΔT1 and ΔT2, and the Log mean temperature difference (LMTD).

#### For shell and tube and cross-flow configurations:

1. Select the Shell and tube/Cross flow option in the heat exchanger section.

2. Enter the input and output temperature values for the Hot fluid and Cold fluid.

3. The calculator will display the temperature differences ΔT1 and ΔT2, and the Log mean temperature difference (LMTD). These are not our final results. We still need to determine the correction factor F for cross-flow and shell and tube.

4. For this, enter the inlet and outlet temperatures for the shell side of the heat exchanger, Tc1 and Tc2.

5. Similarly, fill in the inlet and outlet temperatures for the tube side of the heat exchanger, Ts1 and Ts2.

6. With these numbers, the calculator will determine the P and R parameters that you need to obtain the correction factor F from the charts.

7. Finally, enter the Correction factor, and the calculator will display your Corrected LMTD for your problem.

🙋 Here you can find the for the one-shell pass and 2, 4, 6... tube passes and one pass cross-flow with one fluid mixed, and here the for the single-pass cross-flow with unmixed fluids and two shell passes and multiples of 4 tube pass.

Gabriela Diaz
Select the type of heat exchanger — Parallel flow, Counter flow, or Shell and Tube/Cross flow from the list below.
Heat exchanger
Counter flow
Hot fluid
Inlet temperature, Thi
°F
Outlet temperature, Tho
°F
Cold fluid
Inlet temperature, Tci
°F
Outlet temperature, Tco
°F
Temperature difference
ΔT₁ (Thi - Tco)
°F
ΔT₂ (Tho - Tci)
°F
Logarithmic mean temperature difference (LMTD)
°F
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