Solving Heat Exchanger Network Synthesis Problems by Mathematical Programming Methods

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Solving Heat Exchanger Network Synthesis Problems
by Mathematical Programming Methods
Kaj-Mikael Björk Process Design Laboratory Åbo
Akademi University Biskopsgatan 8, 20500 TURKU
Finland
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Background

• Some models with simultaneous optimization of
  investment and running costs
• Models with a continuous tempertaure
  representation
• Synheat model by Yee and Grossmann
• Models with a discrete temperature representation
• w1 and w2 in Westerlund et al.

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Sequential Optimisation

• Mathematical programming used
• Moderate computational effort
• Trade-off not handled simultaneously

STEPWISE PROCEDURE
min Area s.t. min no. of units
s.t. min util. costs
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Simultaneous Synheat ModelStagewise
superstructure model

• Trade-offs handled simultaneously
• The best solution can be found
• Assumes
• counter-current HEX
• constant Fcps
• constant Us

The Superstructure an Example
C1
H1
C2
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Synheat Model Limitations

• More computational effort needed
• Non-convex obj. fcn.
• Global optimum not guaranteed
• Assuming isothermal mixing

Isothermal Mixing
T1
T3
T2
T1 T2 T3
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Improvements of Synheat

• Modifications to ensure convergence to the global
  optimum for both with and without the isothermal
  mixing assumption
• earlier work by Zamora B B approach for
  Synheat with no stream splits
• A new convexification technique used
• a set of convex approximate subproblems to be
  solved
• convexifying posynomials (xr1yr2zr3 ... )

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Convexification technique an Example

• Convexifying a bilinearity xy
• Exponential transformation
• Need to include the relationship between the
  variables due to other constraints

X ln(x),Y ln(y), approximated by a stepwise
linear function
xy is replaced by Exp(XY) X ln(x) Y ln(y)
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Results Continuous T

• Assuming isothermal mixing
• quite time consuming
• solved 2 hot 4 cold
• Without isothermal mixing assumption
• time consuming due to more nonlinear constraints
• solved 2 hot 2 cold

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Future work Continuous T

• Speeding up convergence
• modified global optimization approach ??
• using more suitable MINLP-solver (Dicopt?)
• Testing non-global optimization approaches and
  comparing them
• Testing stocastic optimization methods

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Discrete Temp. representation

• Advantages
• Fcp calculations done a priori
• Area calculations done a priori
• Power law area costs only nonlinear functions
  remaining
• Small number of binary variables

• Disadvantages
• Increased size of problem
• No good superstructure
• Every thermo-dynamically possible structure
• Only stream splitting

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HRS-systems in paper machines
 

• Dry air into the dryer, process-water and the
  machine hall need to be heated
• Humid air from the dryer can be used as a heat
  source
• Normally all these demands and supplies are
  combined in the heat exchanger network

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Pressurized HRS-tower by Valmet
 

• Combined air-air and air-water heat exchanger
• Typical series decoupling

New model needed
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Hybrid model

• Combined discrete model (w2) with Synheat
  superstructure
• Not straight-forward since Synheat superstructure
  associated with continuous temperatures
• Need of addressing more binary variables
• Only nonlinear function still power law area cost
  functions

Large-sized and time consuming model, but gives
most promising solutions
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Global optimization procedure

• Replacing power law area costs with a stepwise
  linear fcn.

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Conclusion Discrete T

• Hybrid model needed due to the non-linear area
  and heat content calculations for HRS-systems in
  paper-machines
• Solved to the global optimum with a
  special-purpose global optimization procedure
• The Hybrid model is time consuming but finds the
  most promising solutions due to a good
  superstructure