Composite Curves

Overview

Composite curves are fundamental tools in heat exchanger network design and pinch analysis. They combine multiple individual stream H-T curves into aggregate hot and cold profiles, enabling graphical and numerical determination of MITA, UA, and thermodynamic feasibility (Linnhoff & Hindmarsh, 1983).

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Construction

Individual Stream Curves

Each stream is characterized by its temperature-enthalpy (T-H) curve, built via PH flashes as described in the segmented interval method. The curve goes from the highest temperature (Q = 0) to the lowest temperature (Q = Q_total).

Combining Multiple Streams

For the hot side (analogously for cold):

1. Collect temperature breakpoints from all hot stream T-H curves
2. Sort in descending order (highest T first)
3. At each temperature \(T_k\), interpolate the cumulative Q from each stream's curve
4. Sum the contributions:

\[ Q_{\text{hot,composite}}(T_k) = \sum_{i \in \text{hot}} Q_i(T_k) \]

Figure 1. Construction of the hot composite from two individual stream curves. The composite reflects the combined heat capacity of all hot streams at each temperature level.

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Interpretation

Temperature-Heat Duty Diagram

The composite curves are plotted on a T-Q diagram where:

• x-axis: Cumulative heat duty (kW)
• y-axis: Temperature (K or °C)
• Hot composite: Red curve (descending left to right)
• Cold composite: Blue curve (ascending left to right)

Figure 2. T-Q diagram showing hot composite (red), cold composite (blue), and individual stream curves (dashed). The vertical gap between curves represents the temperature driving force.

Key Points on the Diagram

Point	Meaning
Pinch point	Location of minimum temperature difference (MITA)
Hot end	Where the hot composite enters at its maximum temperature
Cold end	Where the cold composite enters at its minimum temperature
Overlap region	Range of Q where both composites exist — this is where heat is exchanged

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MITA and Pinch Point

The Minimum Internal Temperature Approach occurs at the pinch point — the location along the exchanger where the hot and cold composites are closest:

\[ \text{MITA} = \min_{f \in [0,1]} \left[ T_{\text{hot}}(f) - T_{\text{cold}}(f) \right] \]

Feasibility Check

If MITA < 0 at any point, the hot composite crosses below the cold composite. This indicates a thermodynamically infeasible design — the specified conditions cannot be achieved without violating the Second Law.

The pinch point divides the exchanger into two thermodynamically independent regions:

• Above the pinch: Heat surplus — hot streams have more heat than cold streams can absorb
• Below the pinch: Heat deficit — cold streams require more heat than hot streams provide

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Flow Direction Effects

Counterflow

In counterflow, both composites are traversed in the same direction (from high-T end to low-T end). At fraction \(f\):

• Hot composite: \(Q_h = f \cdot Q_{h,\text{total}}\)
• Cold composite: \(Q_c = f \cdot Q_{c,\text{total}}\)

This produces the maximum overlap and highest thermal efficiency.

Co-current

In co-current flow, the cold composite is reversed relative to the hot:

• Hot composite: \(Q_h = f \cdot Q_{h,\text{total}}\)
• Cold composite: \(Q_c = (1 - f) \cdot Q_{c,\text{total}}\)

This typically results in a larger MITA (less efficient) compared to counterflow.

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Implementation Details

The MSHX plugin displays composite curves in the built-in OxyPlot chart:

• Solid thick red line: Hot composite curve
• Solid thick blue line: Cold composite curve
• Dashed thin lines: Individual stream curves (color-coded by role)
• Axes: Heat Duty (kW) on x-axis, Temperature (K) on y-axis

The chart updates automatically after each successful calculation, providing immediate visual feedback on the heat exchange process and pinch point location.