Engineering task and calculation objective
The FN module calculates the true (corrected) logarithmic mean temperature difference for different heat exchanger configurations. From the four process temperatures – inlet and outlet temperature on the outside and inlet and outlet temperature on the inside – the logarithmic mean temperature difference for pure counterflow is formed first and then multiplied by the configuration-dependent correction factor F.
The correction factor accounts for the fact that real equipment – shell-and-tube heat exchangers with multiple tube-side passes, cross-flow units or plate heat exchangers with mixed arrangements – does not operate in ideal counterflow. The configuration selected via the code number determines the underlying F function; the relationships go back to the classic work of Bowman, Mueller and Nagle and are documented in the VDI Heat Atlas (section on the mean temperature difference).
The true logarithmic mean temperature difference is needed in every thermal design based on the LMTD method: the heat transfer area follows from A = Q̇ / (k · Δϑm), and a Δϑm set too optimistically leads directly to an undersized heating surface. A worked example of how to calculate the LMTD correction factor is given below.
Calculation workflow
- Select the configuration: The code number of the heat exchanger configuration defines the flow arrangement – e.g. pure counterflow or cocurrent flow, a shell-and-tube unit with one shell-side and two tube-side passes (1-2 arrangement), multi-pass arrangements or cross-flow variants.
- Enter the temperatures: The inlet and outlet temperature on the outside and the inlet and outlet temperature on the inside are entered. The temperature pairs must come from a consistent heat balance.
- Form the counterflow LMTD: From the temperature differences at both ends of the exchanger, the logarithmic mean temperature difference for ideal counterflow dϑcounter is calculated.
- Determine the correction factor: From the dimensionless parameters of the temperature changes (heat capacity rate ratio R and thermal effectiveness P), the configuration-dependent correction factor F ≤ 1 is determined.
- Output the true temperature difference: The true logarithmic mean temperature difference results as the product of the correction factor and the counterflow LMTD; it enters directly into the heating surface calculation A = Q̇/(k·Δϑm).
Input quantities
| Quantity | Symbol | Unit |
|---|---|---|
| Code number for exchanger type | Wärmeübertragers | - |
| Inlet temperature | ϑei ϑea | °C |
| Outlet tempersture | ϑai ϑaa | °C |
| Inlet temperature | ϑei ϑea | °C |
| Outlet tempersture | ϑai ϑaa | °C |
| Logarithmic temperature difference (counterflow) dϑcounter | Gegenstrom | K (diff) |
| Correction factor | FN | - |
| Real logarithmic temperature difference | dϑm | K (diff) |
| Number of serial exchangers | Z | - |
| Number of tube-side passes NT | Durchgänge | - |
| Number of shell-side passes NS | Durchgänge | - |
| Number of baffles per exchanger NB | Umlenkbleche | - |
| Mass flow | mi ma | kg/s |
| Mass flow | mi ma | kg/s |
| Specific heat capacity | cpi cpa | J/(kg·K) |
| Specific heat capacity | cpi cpa | J/(kg·K) |
| Overall heat transfer coefficient | k | W/(m²·K) |
| Heat transfer area | A | m² |
| Ausführung | Ausführung | - |
Worked example
A shell-and-tube heat exchanger in a 1-2 arrangement (one shell-side, two tube-side passes) cools a process medium on the outside from 90 °C to 50 °C; on the inside, cooling water is heated from 20 °C to 45 °C. Find the counterflow LMTD, the correction factor F and the true logarithmic mean temperature difference.
Given values
| Inlet temperature outside T1 | 90 °C |
| Outlet temperature outside T2 | 50 °C |
| Inlet temperature inside t1 | 20 °C |
| Outlet temperature inside t2 | 45 °C |
| Configuration | 1 shell-side, 2 tube-side passes |
Solution
Counterflow LMTD
Temperature differences at the exchanger ends (counterflow):
Δϑ1 = T1 − t2 = 90 − 45 = 45 K
Δϑ2 = T2 − t1 = 50 − 20 = 30 K
Δϑcounter = (Δϑ1 − Δϑ2) / ln(Δϑ1/Δϑ2) = (45 − 30) / ln(45/30) = 15 / 0.4055 = 37.0 K
Dimensionless parameters
R = (T1 − T2) / (t2 − t1) = 40 / 25 = 1.60
P = (t2 − t1) / (T1 − t1) = 25 / 70 = 0.357
Correction factor F for the 1-2 arrangement
Using the Bowman relation for one shell-side and two tube-side passes:
F = [√(R²+1)/(R−1)] · ln[(1−P)/(1−P·R)] / ln[(2 − P·(R+1−√(R²+1))) / (2 − P·(R+1+√(R²+1)))]
With √(R²+1) = 1.887 it follows:
Numerator: (1.887/0.60) · ln(0.643/0.429) = 3.145 · 0.4055 = 1.275
Denominator: ln(1.745/0.398) = 1.479
F = 1.275 / 1.479 = 0.862
True logarithmic mean temperature difference
Δϑm = F · Δϑcounter = 0.862 · 37.0 K = 31.9 K
Compared with pure counterflow, the 1-2 arrangement thus provides about 14% less driving temperature difference – the heating surface must be sized correspondingly larger.
Result
| Logarithmic mean temperature difference (counterflow) | 37.0 K |
| Correction factor F | 0.862 |
| True logarithmic mean temperature difference | 31.9 K |
All values are illustrative. The applicable standard and project-specific boundary conditions remain authoritative.
Frequently asked questions
Why is the counterflow LMTD always used as the basis?
For given temperatures, counterflow provides the largest possible mean temperature difference and therefore serves as the reference. Every other flow arrangement achieves only a fraction of it, described by the correction factor F ≤ 1. This keeps the method uniform for all configurations.
Below which correction factor is an arrangement considered unsuitable?
As a rule of thumb, F should not fall below about 0.75–0.8. Smaller values mean the exchanger operates in a region where the F curves drop steeply – small deviations in the operating temperatures then lead to large errors in the calculated area. Remedies are more shell-side passes in series or a true counterflow arrangement.
Does the LMTD method also apply with phase change?
With isothermal condensation or evaporation on one side, the flow arrangement is irrelevant, the correction factor becomes F = 1 and the simple LMTD applies exactly. With gliding condensation/evaporation temperatures (mixtures, superimposed sensible heat), however, the LMTD method is only applicable section by section.
What should be done if the temperature profiles cross?
A temperature cross (outlet temperature of the cold side above the outlet temperature of the hot side) can only be achieved with arrangements close to counterflow. In a 1-2 arrangement it leads to very small or undefined F values; then several units must be connected in series or the configuration must be changed.