Engineering task and calculation objective
The ZELL module calculates the steady-state temperature distribution in a shell and tube heat exchanger using the cell method and determines from it the effectively acting temperature difference (CLMTD, corrected logarithmic mean temperature difference). To do this, the apparatus is divided into cells according to its flow configuration — number of tube-side and shell-side passes, number of baffles, cross-counterflow or cross-cocurrent flow; for each cell, the heat balance is solved with the proportionate heat transfer product k·A.
Compared with the classic approach of applying a global correction factor F to the logarithmic mean temperature difference of pure counterflow, the cell model determines the driving temperature difference much more accurately — particularly for few baffles, multi-pass configurations and partial cross-mixing. In addition, local effects become visible that an integral calculation conceals: maximum and minimum temperatures on the tube and shell sides, as well as possible temperature inversions in individual cells, where the temperature difference locally changes sign.
This calculation is needed whenever a heat exchanger rating with a given overall heat transfer coefficient and given heat transfer area is to show which outlet temperatures the apparatus actually achieves — for example when evaluating existing equipment, when reconfiguring flow arrangements, or when the temperature approach is tight.
Calculation workflow
- Specify the operating data of both sides: Inlet temperatures, mass flows and specific heat capacities of the tube side and shell side are entered; from these the program forms the heat capacity flows W = m·cp of both sides.
- Define the flow configuration and cell division: The number of tube-side and shell-side passes, the number of baffles, the tube rows per cell, and the flow arrangement (cross-counterflow/cross-cocurrent, countercurrent or cocurrent tube flow) determine how the apparatus is divided into cells; the degree of cross-mixing is chosen between unmixed and totally mixed.
- Distribute the transfer capability: From the actual overall heat transfer coefficient and the actual heat transfer area, the heat transfer product k·A is formed and distributed evenly over the cells (k·A per cell).
- Solve the cell balances: For each cell, the heat balance is solved using the cell efficiencies of both sides; the outlet temperatures of one cell are the inlet temperatures of the downstream cells, until the entire temperature field converges.
- Evaluate the results: The output comprises the outlet temperatures of both sides, the logarithmic mean temperature difference of counterflow, the correction factor and the effectively acting temperature difference, as well as extreme temperatures and the maximum temperature inversion in a cell.
Input quantities
| Quantity | Symbol | Unit |
|---|---|---|
| Shell-side inlet temperature | ta1 | °C |
| Tube-side inlet temperature | ti1 | °C |
| Shell-side outlet temperature | ta2 | °C |
| Tube-side outlet temperature | ti2 | °C |
| LMTD countercurrent flow | ΔTGeg | K (diff) |
| FN factor | FN | - |
| CLMTD corrected log. mean temperature difference | ΔTm | K (diff) |
| Shell-side mass flow | ma | kg/s |
| Tube-side mass flow | mi | kg/s |
| Number of tube-side passes | Nt | - |
| Number of shell-side passes | Ns | - |
| Number of baffles | NU | - |
| Specific heat capacity tube-side | cp,i | J/(kg·K) |
| Specific heat capacity shell-side | cp,a | J/(kg·K) |
| Heat capacity flow shell-side; Ww,a = ma∙cp,a | Ww,a | W/K |
| Heat capacity flow tube-side; Ww,i = mi∙cp,i | Ww,i | W/K |
| Actual overall heat transfer coefficient | k | W/(m²·K) |
| Actual heat transfer area | A | m² |
| Number of cells | Zellen | - |
| 0 = unmixed; 0.5 = mixed | (0..0.5) | - |
| Product k∙A | k*A | W/K |
| Product k∙A per cell | Zelle | W/K |
| Tube-side cell efficiency | innen | - |
| Shell-side cell efficiency | außen | - |
Worked example
For a heat exchanger, determine the shell-side outlet temperature and the logarithmic mean temperature difference of pure counterflow from the energy balance — a worked example of an LMTD calculation. Hot water flows on the tube side, cooling water on the shell side. (The refinement to the effectively acting temperature difference via the correction factor or cell model is handled by the module.)
Given values
| Tube-side inlet temperature | 90 °C |
| Tube-side outlet temperature | 50 °C |
| Tube-side mass flow | 2.0 kg/s |
| Shell-side inlet temperature | 20 °C |
| Shell-side mass flow | 3.0 kg/s |
| Specific heat capacity of both sides | 4,190 J/(kg·K) |
Solution
Heat duty from the tube-side balance
Tube-side heat capacity flow: Ww,i = mi · cp,i = 2.0 · 4,190 = 8,380 W/K
Q = Ww,i · (90 − 50) = 8,380 · 40 = 335.2 kW
Shell-side outlet temperature
Shell-side heat capacity flow: Ww,a = 3.0 · 4,190 = 12,570 W/K
ϑa,out = 20 + Q / Ww,a = 20 + 335,200 / 12,570 = 46.7 °C
Logarithmic mean temperature difference in counterflow
Δϑ₁ = 90 − 46.7 = 43.3 K (hot end), Δϑ₂ = 50 − 20 = 30.0 K (cold end)
Δϑlog = (Δϑ₁ − Δϑ₂) / ln(Δϑ₁/Δϑ₂) = (43.3 − 30.0) / ln(43.3/30.0) = 36.3 K
The effectively acting temperature difference of the apparatus is Δϑreal = F · Δϑlog with F ≤ 1; the module determines the correction factor from the actual flow configuration via the cell model.
Result
| Transferred heat duty Q | 335.2 kW |
| Shell-side outlet temperature | 46.7 °C |
| LMTD in counterflow | 36.3 K |
All values are illustrative. The applicable standard and project-specific boundary conditions remain authoritative.
Frequently asked questions
When is the LMTD with an F correction factor no longer sufficient?
The classic F-factor charts assume ideal conditions: many baffles, complete cross-mixing per baffle compartment and constant fluid properties. For apparatuses with few baffles, several shell-side passes or unusual configurations, the actual driving temperature difference deviates noticeably. The cell method reproduces the real flow arrangement and is then the more accurate tool — it also reveals when the apparatus is operated in a region of steeply falling F-factor, where small data deviations cause large changes in duty.
What does a temperature inversion in a cell mean?
In unfavorable configurations, the temperature of the stream being heated can locally rise above that of the stream being cooled — the driving difference changes sign in that cell and heat is transferred back. The apparatus destroys part of its transfer capability there. The module reports the maximum temperature inversion; if it occurs, the configuration (e.g. number of passes or flow arrangement) should be reconsidered.
What role does the cross-mixing setting play?
The parameter describes how strongly the shell-side stream mixes transverse to the main flow direction: 0 means unmixed partial streams, 0.5 total mixing. Real apparatuses lie in between — baffle clearances and bypass streams mix partially. The choice noticeably affects the calculated correction factor; when in doubt, both limiting cases should be calculated and the spread assessed.
Is the cell method a design or a rating method?
ZELL works as a rating calculation: the actual overall heat transfer coefficient and the actual area are given, and the resulting outlet temperatures and the effectively acting temperature difference are calculated. For design, one works iteratively — area or configuration are varied until the required outlet temperatures are reached.