Poppe Method¶
Historical Background¶
The Poppe method was developed by M. Poppe and H. Rogener in 1991 as a more rigorous alternative to the Merkel method. It removes the key simplifying assumptions of the Merkel approach — specifically, the Lewis factor is not assumed equal to unity, and evaporation losses are explicitly tracked.
The Poppe method solves a system of three coupled ordinary differential equations (ODEs) simultaneously, providing more accurate predictions especially when:
- The air approaches or exceeds saturation within the tower
- The L/G ratio deviates significantly from 1.0
- High accuracy in evaporation rate prediction is required
Governing Equations¶
The Poppe method tracks three state variables as functions of water temperature:
1. Humidity Ratio \( w(T_w) \)¶
2. Air Enthalpy \( h_a(T_w) \) (via mass-enthalpy relation)¶
In practice, the air enthalpy is updated as:
3. Merkel Number (cumulative)¶
The total Merkel number (KaV/L) is obtained by integrating from \( T_{w,out} \) to \( T_{w,in} \).
Key Differences from Merkel¶
| Aspect | Merkel Method | Poppe Method |
|---|---|---|
| Lewis factor | Assumed = 1 | Calculated (Bosnjakovic) |
| Evaporation | Neglected | Explicitly tracked |
| Air saturation | Ignored | Handled (supersaturation check) |
| Number of ODEs | 1 (integral) | 3 (coupled system) |
| Integration method | Simpson's rule | Runge-Kutta 4th order |
| Accuracy | Good for standard conditions | Superior for all conditions |
| Computational cost | Low | Moderate |
| Industry standard | CTI acceptance testing | Research and advanced design |
Lewis Factor Calculation¶
The Poppe method uses the Bosnjakovic (1965) formula for the Lewis factor:
where \( w_s \) is the saturation humidity ratio at the water temperature and \( w \) is the current humidity ratio of the air.
For typical cooling tower conditions, \( Le_f \) ranges from approximately 0.9 to 1.1. The deviation from unity becomes more significant at:
- High water temperatures (large evaporation rates)
- Low humidity (large \( w_s - w \) difference)
- Near-saturation conditions
Handling Supersaturation¶
A unique feature of the Poppe method is its ability to handle supersaturated air (fog conditions) within the tower. When the calculated humidity ratio exceeds the saturation value at the local air temperature:
In this case, the governing equations switch to the supersaturated formulation, where excess moisture is assumed to exist as fine liquid droplets (fog) in the air stream. The enthalpy calculation accounts for the liquid water content.
Runge-Kutta 4th Order Integration¶
The system of ODEs is solved using the classical RK4 method:
Given the state vector \( \mathbf{y} = [w, \, h_a, \, Me]^T \) and the derivative function \( \mathbf{f}(T_w, \mathbf{y}) \):
[ \mathbf{k}_1 = \mathbf{f}(T_w, \mathbf{y}) ] [ \mathbf{k}_2 = \mathbf{f}\left(T_w + \tfrac{h}{2}, \, \mathbf{y} + \tfrac{h}{2}\mathbf{k}_1\right) ] [ \mathbf{k}_3 = \mathbf{f}\left(T_w + \tfrac{h}{2}, \, \mathbf{y} + \tfrac{h}{2}\mathbf{k}_2\right) ] [ \mathbf{k}_4 = \mathbf{f}(T_w + h, \, \mathbf{y} + h\,\mathbf{k}_3) ] [ \mathbf{y}(T_w + h) = \mathbf{y}(T_w) + \frac{h}{6}\left(\mathbf{k}_1 + 2\mathbf{k}_2 + 2\mathbf{k}_3 + \mathbf{k}_4\right) ]
where \( h = dT_w = (T_{w,in} - T_{w,out}) / N \).
The default number of steps is N = 50, which provides excellent accuracy for typical cooling tower conditions.
Implementation Algorithm¶
Input: Tw_in, Tw_out, Twb, Tdb, L/G, Patm, N_steps
Output: KaV/L, w_out, h_a_out
1. Compute inlet conditions: w_in, h_a_in from (Tdb, Twb, Patm)
2. Initialize state: y = [w_in, h_a_in, 0.0]
3. Set step size: dTw = (Tw_in - Tw_out) / N
4. For each step i = 0 to N-1:
a. Tw = Tw_out + i * dTw
b. Compute PoppeDerivatives(Tw, y, L/G, Patm):
- w_s = saturation humidity at Tw
- h'_s = saturation enthalpy at Tw
- Le_f = Lewis factor (Bosnjakovic)
- h_fg = latent heat at Tw
- Check if w > w_s(T_air): choose unsaturated or supersaturated equations
- Return [dw/dTw, dima/dTw, dMe/dTw]
c. Compute k1, k2, k3, k4 (RK4 stages)
d. Update: y = y + (dTw/6) * (k1 + 2*k2 + 2*k3 + k4)
5. KaV/L = y[2] (accumulated Merkel number)
6. w_out = y[0], h_a_out = y[1]
Comparison with Merkel Method¶
For the classic test case (Tw_in = 40 °C, Tw_out = 30 °C, Twb = 20 °C, L/G = 1.0):
| Result | Poppe | Merkel | Difference |
|---|---|---|---|
| KaV/L | 0.8144 | 0.8030 | +1.4% |
| Q rejected (kW) | 11.60 | 11.60 | 0.0% |
| Air outlet T (°C) | 32.35 | 25.28 | +7.07 K |
| Air outlet w (kg/kg) | 0.02693 | 0.02904 | -7.3% |
| Evaporation (kg/s) | 0.00393 | 0.00452 | -13.1% |
When to Use Each Method
- Merkel: Industry-standard for acceptance testing (CTI), quick parametric studies, preliminary design.
- Poppe: Detailed design, research, cases with high L/G or near-saturation air, when accurate evaporation prediction is needed.
References¶
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Poppe, M. and Rogener, H. "Berechnung von Ruckkuhlwerken." VDI-Warmeatlas, Section Mi 1-15, VDI-Verlag, Dusseldorf, 1991.
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Kloppers, J.C. and Kroger, D.G. "The Lewis factor and its influence on the performance prediction of wet-cooling towers." International Journal of Thermal Sciences, Vol. 44(9), pp. 879-884, 2005. https://doi.org/10.1016/j.ijthermalsci.2005.03.006
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Kloppers, J.C. and Kroger, D.G. "A critical investigation into the heat and mass transfer analysis of counterflow wet-cooling towers." International Journal of Heat and Mass Transfer, Vol. 48(3-4), pp. 765-777, 2005. https://doi.org/10.1016/j.ijheatmasstransfer.2004.09.004
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Bosnjakovic, F. Technische Thermodynamik. Theodor Steinkopff, Dresden, 1965.