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Segmented Interval Method

Overview

The segmented interval method is the core computational approach used in this MSHX implementation. It rigorously handles multi-stream heat exchange, non-constant properties, and phase change by dividing each stream's enthalpy range into discrete intervals and performing thermodynamic flashes at each point (Kamath et al., 2012).

Motivation

Classical LMTD and e-NTU methods assume:

  • Constant \(U\) and \(c_p\) along the exchanger
  • Single-phase operation (no phase change)
  • Only 2 streams

Real industrial MSHXs violate all three assumptions. The segmented interval method removes these limitations.

Algorithm

Step 1: Build H-T Curves

For each stream \(i\), the enthalpy range from inlet to outlet is divided into \(N_{\text{seg}}\) equal intervals. At each enthalpy point, a pressure-enthalpy (PH) flash determines the temperature:

\[ T_i^{(k)} = f_{\text{PH}}(P_i,\; H_i^{(k)}) \qquad k = 0, 1, \ldots, N_{\text{seg}} \]

where:

\[ H_i^{(k)} = H_{i,\text{max}} + k \cdot \frac{H_{i,\text{min}} - H_{i,\text{max}}}{N_{\text{seg}}} \]

The cumulative heat duty at each point is:

\[ Q_i^{(k)} = \dot{m}_i \cdot |H_i^{(k)} - H_{i,\text{max}}| \quad [\text{kW}] \]

Phase Change Handling

When a stream undergoes phase change, the PH flash returns the saturation temperature for the two-phase region. This creates a flat segment in the H-T curve where temperature remains constant while enthalpy changes — correctly capturing latent heat effects.

H-T curve construction
Figure 1. Construction of H-T curves for a hot stream with phase change. The flat region corresponds to condensation at constant temperature.

Step 2: Build Composite Curves

Individual stream H-T curves are combined into hot and cold composite curves. For multiple streams on the same side:

  1. Collect all unique temperature breakpoints from all stream curves
  2. At each temperature \(T\), interpolate the cumulative \(Q\) from each stream's H-T curve
  3. Sum the contributions: \(Q_{\text{composite}}(T) = \sum_i Q_i(T)\)

This process is detailed in Composite Curves.

Step 3: Zone-by-Zone UA Calculation

The heat transfer area is divided into \(N_{\text{zones}}\) zones of equal fractional heat duty. In each zone:

  1. Sample the hot composite temperature at fraction \(f\)
  2. Sample the cold composite temperature at the corresponding fraction (accounting for flow direction)
  3. Compute the temperature difference \(\Delta T\)
  4. Calculate the local LMTD and zone UA contribution

For counterflow, both composites are sampled at the same fraction \(f\):

[ T_{\text{hot}}^{(j)} = T_{\text{hot,composite}}(f_j \cdot Q_{\text{hot,total}}) ] [ T_{\text{cold}}^{(j)} = T_{\text{cold,composite}}(f_j \cdot Q_{\text{cold,total}}) ]

For co-current, the cold composite is sampled in reverse:

\[ T_{\text{cold}}^{(j)} = T_{\text{cold,composite}}((1 - f_j) \cdot Q_{\text{cold,total}}) \]

Step 4: MITA Determination

The minimum internal temperature approach is found by scanning all zone boundaries:

\[ \text{MITA} = \min_{j=0}^{N_{\text{zones}}} \left[ T_{\text{hot}}^{(j)} - T_{\text{cold}}^{(j)} \right] \]

Convergence

The segmented interval method is embedded within a Q-bisection loop that iterates on the total heat duty:

graph TD
    A[Set Q_lo = 0, Q_hi = Q_max] --> B[Q_trial = midpoint]
    B --> C[Distribute Q among streams]
    C --> D[PH flash → outlet T for each stream]
    D --> E[Build H-T curves]
    E --> F[Build composite curves]
    F --> G[Calculate UA and MITA]
    G --> H{Converged?}
    H -->|No| I[Bisect: adjust Q_lo or Q_hi]
    I --> B
    H -->|Yes| J[Done]

Convergence criteria:

Mode Criterion Tolerance
UA \(\|UA_{\text{calc}} - UA_{\text{spec}}\| / UA_{\text{spec}} < 0.001\) 0.1%
MITA \(\|\text{MITA}_{\text{calc}} - \text{MITA}_{\text{spec}}\| < 0.01\) K 0.01 K
Q bounds \(\|Q_{\text{hi}} - Q_{\text{lo}}\| / Q_{\text{hi}} < 10^{-5}\) 0.001%

Maximum iterations: 300.

Number of Segments

The NumberOfSegments parameter (default: 25) controls the resolution of the H-T curves. More segments provide:

  • Better resolution near phase boundaries
  • More accurate MITA determination near pinch points
  • Smoother composite curves

Recommended Values

  • 10–20: Quick estimates, single-phase systems
  • 25–50: General use (default)
  • 50–100: Systems with phase change or tight pinch points