Effectiveness-NTU Method¶

Overview¶

The e-NTU (effectiveness – Number of Transfer Units) method provides a direct solution for heat exchanger outlet temperatures without iteration, given the inlet conditions and UA product (Kays & London, 1984). It is used as the reference method for validating the MSHX 2-stream UA mode.

Definitions¶

Number of Transfer Units (NTU)¶

\[ \text{NTU} = \frac{UA}{C_{\min}} \]

where \(C_{\min} = \min(C_h, C_c)\) and \(C = \dot{m} \cdot c_p\) is the heat capacity rate (W/K).

Heat Capacity Rate Ratio¶

\[ C_r = \frac{C_{\min}}{C_{\max}} \qquad 0 \le C_r \le 1 \]

Effectiveness¶

\[ \varepsilon = \frac{Q}{Q_{\max}} = \frac{Q}{C_{\min} \cdot (T_{h,\text{in}} - T_{c,\text{in}})} \]

Analytical Formulas¶

Counterflow¶

\[ \varepsilon = \begin{cases} \dfrac{1 - \exp\!\bigl[-\text{NTU}(1 - C_r)\bigr]}{1 - C_r \cdot \exp\!\bigl[-\text{NTU}(1 - C_r)\bigr]} & C_r < 1 \\[10pt] \dfrac{\text{NTU}}{1 + \text{NTU}} & C_r = 1 \end{cases} \]

Co-current (Parallel Flow)¶

\[ \varepsilon = \frac{1 - \exp\!\bigl[-\text{NTU}(1 + C_r)\bigr]}{1 + C_r} \]

Reference

These relations are derived in Chapter 3 of Kays & London (1984) and Chapter 11 of Incropera et al. (2007). They assume constant \(U\) and \(c_p\).

Outlet Temperature Calculation¶

Once \(\varepsilon\) is known:

\[ Q = \varepsilon \cdot C_{\min} \cdot (T_{h,\text{in}} - T_{c,\text{in}}) \]

\[ T_{h,\text{out}} = T_{h,\text{in}} - \frac{Q}{C_h} \]

\[ T_{c,\text{out}} = T_{c,\text{in}} + \frac{Q}{C_c} \]

Validation Role¶

The e-NTU method serves as the analytical reference for validating the MSHX segmented interval solver in 2-stream mode:

Scenario	MSHX Solver	e-NTU Reference
Counterflow, balanced (\(C_r = 1\))	Q-bisection with PH flash	\(\varepsilon = \text{NTU}/(1+\text{NTU})\)
Counterflow, unbalanced (\(C_r < 1\))	Q-bisection with PH flash	Full counterflow formula
Co-current	Q-bisection with PH flash	Co-current formula

Validation Results

All 2-stream test cases show agreement within 2% on outlet temperatures and 3% on UA between the MSHX segmented solver and the e-NTU analytical solution. See 2-Stream Validation.

Limitations for Multi-Stream¶

The e-NTU method is inherently limited to 2-stream configurations. For 3+ stream exchangers, there is no closed-form e-NTU expression. The segmented interval method with composite curves provides the rigorous multi-stream solution.