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# LMTD Calculator – Log Mean Temperature Difference

What is LMTD for a heat exchanger?Formula for LMTD – Counter flow and Parallel flowLMTD correction factorHow to calculate LMTDExample: Using the LMTD calculatorFAQs

You can use the LMTD calculator to determine the logarithmic mean temperature difference (LMTD) for a heat transfer process. When you calculate heat transfer, you must have noticed the term for temperature difference in the equation along with heat transfer coefficient, mass flow rate, and the area. The equation is generally used to estimate heat transfer through walls, shells, and also heat exchangers. The heat exchanger is a device specifically designed to take advantage of the phenomenon for either heating or cooling processes.

This device has several applications ranging from the air conditioners in your room to the vehicle you drive, all the way to massive nuclear reactors and everything in between. Since they are used in so many places, there are several types and arrangements in which an engineer designs a heat exchanger. The most common types of heat exchangers are parallel flow and counter flow. Read on to understand what is the definition of LMTD, and how to differentiate the LMTD for parallel flow and counter flow?

## What is LMTD for a heat exchanger?

Before getting into the log mean temperature difference or LMTD, let us look at the process of heat transfer in a heat exchanger. There are two types of fluid in this device, one being the hot fluid and the other being the colder one.

Let's consider a simple concentric pipe heat exchanger such that the hot fluid is moving in the inner tube and the cold liquid in the outer tube, as shown in the figure below.

Based on this arrangement, both fluids can move either in the same direction or in different directions, hence the names — parallel flow and counter flow heat exchangers. Despite the arrangement, the temperature of hot fluid would decrease and increase for the cold liquid. That said, the temperature difference varies based on the arrangement.

Parallel flow – Counter flow
Some books and references use the terms co-current and countercurrent instead of parallel and counter flow, respectively.

Definition of LMTD: The term LMTD stands for "logarithmic mean temperature difference", which is the logarithmic mean of the difference between the inlet and outlets temperatures for hot and cold fluids. At our natural log calculator you can discover why logarithms! Now let's have a look at the LMTD method. The log mean temperature difference formula is:

$\quad \Delta T_{\mathrm{lm}} = \frac{\Delta T_1 - \Delta T_2}{\ln{\frac{\Delta T_1}{\Delta T_2}}}$

where:

• $\Delta T_1$ and $\Delta T_2$Temperature differences for fluids at the inlet and outlet of the heat exchanger; and
• $\Delta T_{\mathrm{lm}}$ – Logarithmic mean temperature difference.

But why is LMTD used instead of arithmetic mean (the one we help you find with our mean calculator)? — This method is based on tracking the temperature change in fluids along the length of the heat exchanger. This temperature profile follows an exponential curve, i.e., it decreases or increases exponentially). Further, the arithmetic mean would lead to the overestimation of heat transfer. This error increases when the difference between $\Delta T_1$ and $\Delta T_2$ is large.

Similarly, the effectiveness-NTU method is also used for the design and assessments of heat exchangers. This one is preferred when the outlet temperatures of the fluids are unknown since, in these cases, the LMTD requires an iterative solution.

## Formula for LMTD – Counter flow and Parallel flow

The variation in the LMTD formula for these two types of flow depends on the inlet and outlets. The formula for the temperature difference terms $\Delta T_1$ and $\Delta T_2$ vary as per the flow. However, the convention is temperature difference at left and right sides, respectively.

LMTD for Parallel flow: Since the flow direction of both fluids are the same, the formula is

$\quad \Delta T_1 = T_{\mathrm{hi}} - T_{\mathrm{ci}}$
$\quad \Delta T_2 = T_{\mathrm{ho}} - T_{\mathrm{co}}$

where:

• $T_{\mathrm{hi}}$ – Temperature of the hot fluid at the inlet;
• $T_{\mathrm{ho}}$ – Temperature of the hot fluid at the outlet;
• $T_{\mathrm{ci}}$ – Temperature of the cold fluid at the inlet; and
• $T_{\mathrm{co}}$ – Temperature of the cold fluid at the outlet.

LMTD for Counter flow: Here, the flow direction of both fluids is different. In this case, the inlet of hot fluid and outlet of cold fluid is on the same side (left) of the heat exchanger. Therefore, the formula is:

$\quad \Delta T_1 = T_{\mathrm{hi}} - T_{\mathrm{co}}$
$\quad \Delta T_2 = T_{\mathrm{ho}} - T_{\mathrm{ci}}$

You can use the above equations in the log mean temperature difference formula to obtain the value for the respective flow arrangement.

Counter flow vs Parallel flow
Assuming the same set of inlet and outlet temperatures, the LMTD value for counter flow would be greater than the parallel one. Therefore, it would have lesser surface area for the same amount of heat transfer.

## LMTD correction factor

In addition to the parallel and counter flow heat exchangers, there are much more complex versions where the fluids pass multiple times, i.e., multi-pass or flowing perpendicular to each other, i.e., cross flow. The shell and tube heat exchanger diagram can be seen below, along with different types of heat exchangers.

To calculate LMTD for the said configurations, a term called correction factor, $F$, is introduced in the equation below:

$\quad \Delta T_{\mathrm{LMTD}} = F \ \Delta T_{\mathrm{lm,\ CF}}$

where:

• $\Delta T_{\mathrm{LMTD}}$ – LMTD value for the cross flow or shell and tube heat exchangers; and
• $\Delta T_{\mathrm{lm,\ CF}}$ – LMTD value for counter flow arrangement.

Therefore, the steps in the case of these advanced heat exchangers are:

• Calculate the LMTD considering the counter flow arrangement; and
• Apply the correction factor.

You can obtain correction factor from the charts based on two parameters, $P$ and $R$. Such that:

$\quad P = \frac{T_{\mathrm{s2}} - T_{\mathrm{s1}}}{T_{\mathrm{c1}} - T_{\mathrm{s1}}}$
$\quad R = \frac{T_{\mathrm{c1}} - T_{\mathrm{c2}}}{T_{\mathrm{s2}} - T_{\mathrm{s1}}}$

where:

• $T_{\mathrm{s1}}$ and $T_{\mathrm{s2}}$Inlet and outlet temperature for shell side of the heat exchanger; and
• $T_{\mathrm{c1}}$ and $T_{\mathrm{c2}}$Inlet and outlet temperature for tube side of the heat exchanger.

The values of $P$ and $R$ are used along with the configuration of the heat exchangers, e.g., fluid through shell passes once, but 2 times for the fluid in the tube. Then, you should refer to the chart for the above arrangement and use the $P$ and $R$ values to obtain the correction factor. Some of the charts can be seen below.

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

## How to calculate LMTD

To calculate LMTD:

1. Select the type of heat exchangers from the list to begin.

2. Input the inlet and outlet temperatures for hot fluid.

3. Enter the inlet and outlet temperatures for cold fluid.

4. The calculator will use the LMTD formula to return the temperature difference.

In the case of crossflow or shell and tube heat exchanger design, follow the further steps, in addition to the ones mentioned above:

5. Enter the inlet and outlet temperatures for the shell side of the heat exchanger.

6. Fill in the inlet and outlet temperatures for the tube side of the heat exchanger.

7. The LMTD calculator will return the constants $P$ and $R$ values, which you can use to find the correction factor for your desired configuration.

8. Enter the correction factor.

9. The calculator will return the corrected LMTD value for the chosen heat exchanger type.

## Example: Using the LMTD calculator

Determine the log mean temperature difference for a shell and tube heat exchanger design, having the following temperatures.

• Inlet temperatures of hot and cold fluids $80\ \mathrm{\degree C}$ and $20\ \mathrm{\degree C}$, respectively.
• Outlet temperatures of hot and cold fluids as $40\ \mathrm{\degree C}$ and $50\ \mathrm{\degree C}$, respectively. Take the number of shell and tube passes as $2$ and $4$, respectively.

Here, we will first obtain the log mean temperature difference for counter flow heat exchanger and then apply the appropriate correction factor.
To calculate LMTD:

1. Select the type of heat exchangers from the list as Shell and tube/Crossflow.
2. Input the inlet and outlet temperatures for hot fluid as $80\ \mathrm{\degree C}$ and $40\ \mathrm{\degree C}$.
3. Enter the inlet and outlet temperatures for cold fluid as $20\ \mathrm{\degree C}$ and $50\ \mathrm{\degree C}$.
4. The temperature difference is obtained and used in the LMTD method:
\scriptsize \qquad \begin{align*} \Delta T_1 &= T_{\mathrm{hi}} - T_{\mathrm{co}} \\ &= 80 - 50 = 30 \ \degree\text{C} \\ \\ \Delta T_2 &= T_{\mathrm{ho}} - T_{\mathrm{ci}} \\ &= 40 - 20 = 20 \ \degree\text{C} \\ \\ \Delta T_{\mathrm{lm}} &= \frac{\Delta T_1 - \Delta T_2}{\ln{\frac{\Delta T_1}{\Delta T_2}}} \\ &= \frac{30 - 20}{\ln{\frac{30}{20}}} = 24.663 \ \degree\text{C} \end{align*}
1. Enter the inlet and outlet temperatures for the shell side of the heat exchanger as $20\ \mathrm{\degree C}$ and $50\ \mathrm{\degree C}$.
2. Fill in the inlet and outlet temperatures for the tube side of the heat exchanger as $80\ \mathrm{\degree C}$ and $40\ \mathrm{\degree C}$.
3. The values of $P$ and $R$ are:
\scriptsize \qquad \begin{align*} P &= \frac{T_{\mathrm{s2}} - T_{\mathrm{s1}}}{T_{\mathrm{c1}} - T_{\mathrm{s1}}} \\ \\ &= \frac{40 - 80}{20 - 80} = 0.6667 \end{align*}
\scriptsize \qquad \begin{align*} R &= \frac{T_{\mathrm{c1}} - T_{\mathrm{c2}}}{T_{\mathrm{s2}} - T_{\mathrm{s1}}} = \frac{20 - 50}{40 - 80} = 0.75 \end{align*}
1. Using the values of $P$ and $R$, the value of the correction factor is obtained from the chart as $0.91$.
2. The corrected log mean temperature difference for the heat exchanger is:
$\scriptsize \qquad \Delta T_{\mathrm{LMTD}} = 0.91 * 24.667 = 22.443\ \degree\text{C}$

Alternatively, you can also input the log mean temperature difference in the calculator and some inlet or outlet temperatures to find the rest of the parameters for the heat exchanger.

Our calorimetry calculator is here to help you with practical thermodynamics' problems! And don't e scared of asking for joule heating or the counterintuitive heat capacity: at Omni we even have the heat capacity calculator!

FAQs

### What is log mean temperature difference?

The log mean temperature difference is defined as the logarithmic mean of the temperature difference of the inlet and outlet of the heat exchanger.

### Why LMTD is used?

LMTD is used because the temperature profile for the change in temperature across the length of the heat exchanger increases or decreases exponentially.

### How do I calculate LMTD?

To calculate LMTD:

1. Find the temperature difference on the left side of the heat exchanger, ΔT1.

2. Obtain the temperature difference on the right side of the heat exchanger, ΔT2.

3. Subtract the temperature difference, ΔT2 from ΔT1.

4. Divide the resultant with the natural log of the ratio of temperature difference. Mathematically that's:

ΔTlm = (ΔT1 - ΔT2)/(ln (ΔT1 / ΔT2) )

### How do I find LMTD correction factor?

To obtain the correction factor:

1. Find the value of chart parameters, P and R.
2. Use the chart as per the description of heat exchangers, say two shell passes and four tubes.
3. Locate the values of P and R on the chart to find the correction factor.

### How do I find LMTD for counter flow?

To calculate LMTD:

1. Find the difference between the inlet temperature of hot fluid and outlet temperature of cold fluid, ΔT1.

2. Obtain the difference between outlet temperature of hot fluid and inlet temperature of cold fluid, ΔT2.

3. Subtract the temperature difference, ΔT2 from ΔT1.

4. Divide the resultant with the natural log of the temperature difference ratio. Mathematically, that's:

ΔTlm = (ΔT1 - ΔT2)/(ln (ΔT1 / ΔT2))