Explanation - Heat Integration

I. How Composite Curves Are Constructed and Interpreted

1. Selection of streams

All hot streams are first considered together and plotted on a temperature–enthalpy ( $T-H$ ) diagram. Each stream is represented by a straight line between its supply and target temperatures, with a slope determined by its heat-capacity flow rate ( $C_{P}$ ).

At this stage, the absolute position on the enthalpy axis is arbitrary. Only the temperature span and enthalpy change of each stream matter.

2. Combining hot streams into a hot composite curve

The hot composite curve is constructed by examining the temperature intervals formed by the hot streams:

Divide the temperature range into intervals wherever a hot stream starts or ends.
In each temperature interval:
- Identify all hot streams present in that interval.
- Add their heat-capacity flow rates to obtain a combined $C_{P}$ .
Calculate the total enthalpy change across that interval using the combined heat capacity.
Plot a single line segment for that interval.

This process is repeated for all temperature intervals, producing a single continuous hot composite curve that represents the combined heat release behavior of all hot streams.

The composite curve therefore represents a virtual single hot stream whose heat capacity changes stepwise with temperature.

3. Constructing the cold composite curve

The same procedure is applied independently to all cold streams:

Plot cold streams on the same $T-H$ diagram.
Divide the temperature axis into intervals defined by cold stream temperatures.
In each interval:
- Identify all cold streams present.
- Add their heat-capacity flow rates.
- Calculate the combined enthalpy change.
Plot the resulting cold composite curve.

The cold composite curve represents the total heat absorption requirement of the process as a function of temperature.

4. Plotting both composite curves together

Both the hot composite curve and the cold composite curve are plotted on the same temperature–enthalpy diagram.

At this point:

Their horizontal (enthalpy) positions are still arbitrary
Only their shapes and temperature spans are fixed

5. Sliding the composite curves and identifying the pinch

The cold composite curve is now slid horizontally relative to the hot composite curve.

The curves are moved until they just touch at one point
This point of closest approach is called the pinch temperature

At the pinch:

The temperature driving force between the hot and cold streams is at its minimum
No further horizontal movement is possible without violating heat-transfer feasibility

6. Minimum temperature difference ( $\Delta T_{min}$ )

The composite curves are separated vertically by a minimum temperature difference to ensure feasible heat transfer.

This vertical separation corresponds to the minimum allowable approach temperature, known as the minimum temperature difference ( $\Delta T_{min}$ ).

ΔTmin represents:

Economic trade-off between heat-exchanger area and utility consumption
The fundamental constraint governing heat recovery

7. Determination of minimum utilities

Once the curves are positioned at the pinch condition:

Minimum cold utility ( $Q_{C,min}$ )

Defined as the horizontal enthalpy difference between:
- The starting point of the hot composite curve
- The starting point of the cold composite curve

This represents the minimum external cooling requirement.

Minimum hot utility ( $Q_{H,min}$ )

Defined as the horizontal enthalpy difference between:
- The end point of the hot composite curve
- The end point of the cold composite curve

This represents the minimum external heating requirement.

8. Physical meaning of composite curves

Each composite curve represents the aggregate thermal behavior of multiple process streams
Changes in slope indicate changes in the number or heat capacity of streams active in that temperature range
The pinch divides the system into two thermodynamically independent regions:
- Above the pinch: heat deficit (requires hot utility)
- Below the pinch: heat surplus (requires cold utility)

9. Key conceptual takeaways

Composite curves depend only on enthalpy changes and temperature ranges
Absolute enthalpy positioning is irrelevant until curves are aligned
Stream heat capacities are summed interval-wise, not stream-wise
Utility targets are obtained graphically, without assuming a heat-exchanger network

II. Problem Table Algorithm (PTA)

The Problem Table Algorithm is a numerical alternative to composite curves. Instead of plotting temperature–enthalpy lines, the same information is organized in a table to identify the pinch point and minimum utility requirements.

1. Selection of streams

All hot and cold streams in the process are listed with:

Supply temperature
Target temperature
Heat-capacity flow rate ( $C_{P}$ )

Each stream represents a potential source or sink of heat over a defined temperature range.

2. Apply minimum temperature difference ( $\Delta T_{min}$ )

To account for the minimum allowable driving force for heat transfer:

Hot stream temperatures are reduced by $\frac{\Delta T_{min}}{2}$
Cold stream temperatures are increased by $\frac{\Delta T_{min}}{2}$

This ensures that any heat exchange identified by the algorithm is thermodynamically feasible.

3. Create temperature intervals

Collect all shifted stream temperatures (hot and cold).
Sort them in descending order.
Form temperature intervals between each adjacent pair.

Each interval represents a temperature range where the set of active streams remains unchanged.

4. Identify active streams in each interval

For every temperature interval:

A hot stream is active if it passes through the interval while cooling.
A cold stream is active if it passes through the interval while heating.

Only streams that actually exist in that interval contribute to heat flow.

5. Calculate net heat flow in each interval

Within each temperature interval:

Sum the heat-capacity flow rates of all hot streams:

$\sum_{}^{}C_{P,hot}$

Sum the heat-capacity flow rates of all cold streams:

$\sum_{}^{}C_{P,cold}$

Compute the net heat-capacity flow rate:

$\Delta C_{P}=\sum_{}^{}C_{P,hot}-\sum_{}^{}C_{P,cold}$

Multiply by the interval temperature width $\Delta T$ to obtain the net heat flow:

$\Delta H=\Delta C_{P}\cdot \Delta T$

A positive value indicates heat surplus, while a negative value indicates heat deficit.

6. Heat cascade (enthalpy accumulation)

Starting from the highest temperature interval:

Assume an initial heat flow of zero.
Add the net heat flow of each interval sequentially as temperature decreases.
Track the cumulative heat flow through all intervals.

This step simulates how heat is cascaded down the temperature levels of the process.

7. Identify minimum hot utility

If any cumulative heat flow becomes negative, this indicates a heat deficit.

The most negative value represents the minimum hot utility requirement.
Add this amount uniformly to all cascade values to make the minimum zero.

This ensures that the process never violates energy feasibility.

8. Identify minimum cold utility

After adjusting the cascade:

The final cumulative heat flow at the lowest temperature interval represents the minimum cold utility requirement.

9. Locate the pinch temperature

The temperature interval where the adjusted cumulative heat flow equals zero corresponds to the pinch point.

This is the temperature level where:

No net heat can be transferred across
The system separates into two thermodynamically independent regions

10. Physical meaning of the PTA

The problem table represents the same information as composite curves, but in numerical form
The heat cascade is equivalent to sliding composite curves until they just touch
The pinch temperature from PTA is identical to the pinch identified graphically

11. Key design implications

Once the pinch is known:

No heat should be transferred across the pinch
No cold utility above the pinch
No hot utility below the pinch

Violating these rules increases utility consumption beyond the minimum.

III. Grand Composite Curve (GCC)

The Grand Composite Curve represents the net heat surplus or deficit of the entire process as a function of temperature. Unlike composite curves, it does not show individual hot or cold streams, but instead shows where utilities are required and where excess heat is available.

1. Starting point: Problem Table Algorithm

The Grand Composite Curve is constructed directly from the Problem Table Algorithm.

From the PTA, you already have:

Shifted temperature intervals
Net heat flow ( $\Delta H$ ) in each interval
The heat cascade and pinch point

The GCC is essentially a graphical representation of the heat cascade.

2. Apply minimum temperature difference ( $\Delta T_{min}$ )

As in the PTA:

Hot streams are shifted downward by $\frac{\Delta T_{min}}{2}$
Cold streams are shifted upward by $\frac{\Delta T_{min}}{2}$

All temperatures used for the GCC are shifted temperatures.

3. Define temperature intervals

Using the shifted temperatures:

Sort all temperatures in descending order
Form temperature intervals between adjacent values

These intervals are identical to those used in the Problem Table Algorithm.

4. Determine net heat flow per interval

For each temperature interval:

$\Delta H=\left(\sum_{}^{}C_{P,hot}-\sum_{}^{}C_{P,cold} \right)\Delta T$

This represents whether the process has:

A heat surplus (positive)
A heat deficit (negative)

5. Perform the heat cascade

Starting from the highest temperature interval:

Cascade the interval heat flows downward
Adjust the cascade so that the minimum cumulative value is zero

This adjustment introduces the minimum hot utility and ensures feasibility.

The pinch temperature occurs where the cumulative heat flow equals zero.

6. Construct the Grand Composite Curve

The GCC is plotted using the adjusted heat cascade data:

Axes

X-axis: Net cumulative heat flow (enthalpy)
Y-axis: Shifted temperature

Plotting method

Start at the pinch temperature with zero enthalpy
Move upward in temperature by adding cumulative heat surplus
Move downward in temperature by adding cumulative heat deficit
Connect the points interval by interval

This produces a single step-like curve representing the process heat balance.

7. Interpretation of the Grand Composite Curve

Heat deficit regions

Portions of the curve that lie to the left of the vertical axis
Indicate external hot utility requirements
The maximum leftward extent equals minimum hot utility

Heat surplus regions

Portions of the curve that lie to the right of the vertical axis
Indicate excess heat available for recovery or export
The maximum rightward extent equals minimum cold utility

8. Pinch on the GCC

The pinch appears as:

The point where the GCC touches the vertical axis
Zero net heat flow
Separation between heat-deficit and heat-surplus regions

This is the same pinch identified using composite curves and PTA.

9. Utility placement using the GCC

The GCC clearly shows:

Where hot utilities should be introduced
- At temperature levels where the curve lies to the left
Where cold utilities should be placed
- At temperature levels where the curve lies to the right

This makes the GCC especially powerful for:

Utility level selection (steam levels, refrigeration levels)
Cogeneration and heat-pump studies
Site-wide energy integration

10. Physical meaning of the GCC

The GCC represents the net heat balance of the process
It shows how much heat is available or required at each temperature level
It is independent of individual stream identities
The curve shape reflects the structure of the process heat demand

11. Relationship to Composite Curves and PTA

Method	Representation	Primary Use
Composite Curves	Individual hot & cold heat availability	Targeting
Problem Table Algorithm	Numerical heat balance	Pinch & utilities
Grand Composite Curve	Net heat surplus/deficit	Utility integration

All three methods are thermodynamically equivalent and identify the same pinch point and minimum utilities.