What is LMTD for a heat exchanger?
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 the 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. A heat exchanger is a device specifically designed to take advantage of this phenomenon for either heating or cooling processes.
This device has several applications ranging from the air conditioner 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.
Before getting into the log mean temperature difference, let us look at the process of heat transfer in a heat exchanger. There are two fluids in this device: one is the hot fluid and the other is the colder one. Consider a simple concentric pipe heat exchanger such that the hot fluid moves in the inner tube and the cold liquid in the outer tube.
Based on this arrangement, both fluids can move either in the same direction or in opposite directions — hence the names parallel flow and counter flow heat exchangers. Regardless of the arrangement, the temperature of the hot fluid decreases and the temperature of the cold fluid increases. That said, the temperature difference varies based on the arrangement.
Definition of LMTD
The term LMTD stands for "logarithmic mean temperature difference", which is the logarithmic mean of the difference between the inlet and outlet temperatures for the hot and cold fluids. Some books and references use the terms co-current and countercurrent instead of parallel and counter flow, respectively.
Formula for LMTD – Counter flow and Parallel flow
The log mean temperature difference formula is:
ΔTLM = (ΔT₁ − ΔT₂) / ln(ΔT₁ / ΔT₂)
where ΔT₁ and ΔT₂ are the temperature differences between the hot and cold fluids at each end of the heat exchanger. How you pair the temperatures depends on the flow arrangement:
Counter flow (countercurrent)
- ΔT₁ = Thot,in − Tcold,out — hot inlet meets the cold outlet
- ΔT₂ = Thot,out − Tcold,in — hot outlet meets the cold inlet
In counter flow, the two fluids move in opposite directions. This arrangement keeps the temperature difference more uniform along the exchanger and produces a higher LMTD, making it the more efficient configuration.
Parallel flow (co-current)
- ΔT₁ = Thot,in − Tcold,in — both inlets at the same end
- ΔT₂ = Thot,out − Tcold,out — both outlets at the same end
In parallel flow, both fluids enter at the same end and travel in the same direction. The temperature difference is largest at the inlet and shrinks toward the outlet, so the cold fluid can never be heated above the hot fluid's outlet temperature.
LMTD correction factor
The simple LMTD formula assumes a true counter flow or parallel flow exchanger. Real shell-and-tube and cross-flow exchangers have more complicated flow paths (multiple passes, baffles), so the effective mean temperature difference is slightly lower. To account for this we multiply by a dimensionless correction factor F (0 < F ≤ 1):
ΔTLM,corrected = F × ΔTLM
The factor F is read from standard charts based on two ratios (P and R) of the four terminal temperatures. For a pure counter flow exchanger, F = 1. Enable the advanced options in the calculator to enter your own F value.
How to calculate LMTD
- Choose the flow arrangement: counter flow or parallel flow.
- Determine the four terminal temperatures: hot fluid inlet and outlet, cold fluid inlet and outlet.
- Compute the two end temperature differences ΔT₁ and ΔT₂ using the formulas above.
- Plug them into ΔTLM = (ΔT₁ − ΔT₂) / ln(ΔT₁ / ΔT₂).
- If ΔT₁ = ΔT₂, the formula reduces to ΔTLM = ΔT₁ (the logarithmic mean equals the arithmetic mean).
- Optionally multiply by the correction factor F, and combine with U and A to find the heat duty: Q = U · A · F · ΔTLM.
Example: Using the LMTD calculator
Consider a counter flow heat exchanger with these temperatures:
- Hot fluid in: 80 °C, hot fluid out: 50 °C
- Cold fluid in: 20 °C, cold fluid out: 40 °C
The end differences are:
- ΔT₁ = 80 − 40 = 40 °C
- ΔT₂ = 50 − 20 = 30 °C
Therefore:
ΔTLM = (40 − 30) / ln(40 / 30) = 10 / 0.2877 ≈ 34.76 °C
If the overall heat transfer coefficient U = 300 W/(m²·K) and the area A = 5 m², the heat duty would be Q = 300 × 5 × 34.76 ≈ 52.1 kW. Switch the calculator to the American (imperial) system to work in °F, BTU/(h·ft²·°F), ft² and BTU/h.
FAQs
Why use a logarithmic mean instead of a simple average?
The temperature difference between the fluids changes exponentially along the exchanger, not linearly. The logarithmic mean correctly weights this exponential profile, so the LMTD is the true average driving force for heat transfer. The arithmetic average would overestimate the heat duty.
Is counter flow or parallel flow better?
Counter flow is almost always better. For the same terminal temperatures it gives a larger LMTD, which means more heat transfer for the same area — or a smaller (cheaper) exchanger for the same duty. Counter flow can also raise the cold fluid above the hot fluid's outlet temperature, which parallel flow can never do.
What happens when ΔT₁ equals ΔT₂?
The formula contains ln(ΔT₁/ΔT₂), which becomes ln(1) = 0 and would divide by zero. In this limiting case the logarithmic mean equals the value itself, so ΔTLM = ΔT₁ = ΔT₂. The calculator handles this automatically.
Why did I get an error about a temperature cross?
LMTD requires the hot fluid to be hotter than the cold fluid at both ends (both ΔT₁ and ΔT₂ must be positive). If your temperatures produce a negative or zero end difference, the configuration is physically impossible — check that the hot inlet is the highest temperature and the cold inlet is the lowest.
Does LMTD depend on the temperature unit?
LMTD is a temperature difference, so a value in kelvin equals the same value in degrees Celsius. Working in °F gives a number 1.8 times larger. The calculator reports the result in both °C (K) and °F so you can use it directly in your heat transfer equation.