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  • CADET-Match
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    • @@ -67,6 +67,7 @@

    Releases

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    • @@ -81,6 +82,7 @@

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    The following build arguments are exclusive to builds on MS windows:

    • DVCPKG_TARGET_TRIPLET: We use vcpkg to manage our dependencies. This triplet specifies which version of the dependencies should be installed. It takes the form of architecture-os-linking, so x64-windows-static for our use cases.

    +

    The following build arguments can only be used with the Clang or GCC compilers:

    +
      +

    • DENABLE_ASAN: enables the address sanitizer.

      • +

    • DENABLE_UBSAN: enables the undefined behaviour sanitizer.

      • +
    diff --git a/chore/developer_guide/developer_guide/cadet_core_architecture.html b/chore/developer_guide/developer_guide/cadet_core_architecture.html index c4da744cf..f0a97bf6b 100644 --- a/chore/developer_guide/developer_guide/cadet_core_architecture.html +++ b/chore/developer_guide/developer_guide/cadet_core_architecture.html @@ -58,12 +58,13 @@

    Navigation

  • Model Expansion
  • CADET-Core debugging
  • CADET-Core testing
    • +
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    • +
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    • @@ -82,6 +83,7 @@

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    • @@ -86,6 +87,7 @@

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    • @@ -77,6 +78,7 @@

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      @@ -125,6 +127,7 @@

      Branches

      } +

      To debug memory related issues, you can compile the code with the address sanitizer ASAN and the undefined behaviour sanitizer UBSAN by enabling the cmake arguments DENABLE_ASAN and DENABLE_UBSAN.

      diff --git a/chore/developer_guide/developer_guide/index.html b/chore/developer_guide/developer_guide/index.html index 5e792abca..89a88b96f 100644 --- a/chore/developer_guide/developer_guide/index.html +++ b/chore/developer_guide/developer_guide/index.html @@ -54,12 +54,13 @@

      Navigation

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      • +
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    • @@ -78,6 +79,7 @@

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      @@ -108,7 +110,7 @@

      Branches

      Developer Guide

      This developer guide is intended to help expand and maintain CADET-Core and is written to serve developers with diverse backgrounds.

      If you find any gaps or have suggestions for improvement, your input is valuable in refining our documentation for better comprehension by all developers. -Please feel free to open an issue on GitHub or, even better, write a draft and create a pull request.

      +Please feel free to open an issue on GitHub or, even better, write a draft and create a pull request.

      When planning to extend or contribute to CADET-Core, please adhere to our notation style and research software engineering (RSE) best practices. For detailed guidance, you can e.g. refer to the RSE guidelines. Since some developers may be new to collaborative software development using Git and GitHub, we specifically recommend consulting the Git RSE guide.

      @@ -147,6 +149,11 @@

      Branches

    • Maintenance of the tests
    +
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    • diff --git a/chore/developer_guide/developer_guide/model_expansion.html b/chore/developer_guide/developer_guide/model_expansion.html index b6afa05bb..2571bfa37 100644 --- a/chore/developer_guide/developer_guide/model_expansion.html +++ b/chore/developer_guide/developer_guide/model_expansion.html @@ -60,12 +60,13 @@

    Navigation

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    • @@ -84,6 +85,7 @@

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      diff --git a/chore/developer_guide/developer_guide/release_new_version.html b/chore/developer_guide/developer_guide/release_new_version.html new file mode 100644 index 000000000..917f038cb --- /dev/null +++ b/chore/developer_guide/developer_guide/release_new_version.html @@ -0,0 +1,192 @@ + + + + + + + + CADET-Core version release — CADET + + + + + + + + + + + + + + + + + + + +
      + + +
      +
      + + +
      + + +

      + + + You're reading the documentation for a development version. + For the latest released version, please have a look at v5.0.0. + + +

      + + +
      +

      CADET-Core version release

      +

      CADET-Core releases follow the semantic versioning system, which is documented here.

      +
      +

      Release checklist

      +
        +

      • Run the release tests:

        +
          +

        • The release tests contain extensive testing that is not included in our CI, such as EOC tests. +Running these tests might take a while and this should be done on the server.

          • +

        • Some tests are implemented in CADET-Core, and can be run with the [ReleaseCI] flag.

          • +

        • More tests are implemented in Python, the code can be found in CADET-Verification

          • +
        +

        Compare the results with the previous run. +The release process can only be continued if the results are reasonable.

        +
        • +

      • Run performance benchmarks:

        +
          +

        • If numerical algorithms were refactored or if performance-critical infrastructure was changed, you should run performance benchmarks to compare the latest release with the planned new one. +To this end, you can refer to the performance benchmark templates in CADET-Reference, e.g. the benchmark for the modified Newton method

          • +
        +
        • +

      • Create a version bump commit, which will be the target commit for the release. +The bump commit contains:

        +
          +

        • Update of the version number in the version.txt and in the zenodo.json

          • +

        • Update of the authors list if needed: CONTRIBUTING.md and zenodo.json

          • +

        • Update of the copyright (years)

          • +

        • Update of the file format if needed

          • +
        +
        • +

      • Create the release on github here.

        +
          +

        • Add the version number according to the semantic versioning system as the tag and set the master branch as target.

          • +

        • Add release notes with these categories:

          +
            +

          • Added: New features, enhancements, or functionalities introduced in this release.

            • +

          • Fixed: Bug fixes and corrections made to resolve issues from previous versions.

            • +

          • Changed: Modifications to existing features and breaking changes for major releases including changes in the interface.

            • +

          • Updated: Improvements to documentation, minor tweaks, or other updates that don’t fit into the other categories.

            • +
          +
          • +

        • Attach the (zipped) source code.

          • +
        +
        • +

      • Check success of zenodo archiving:

        +
          +

        • Upon release, Zenodo automatically archives the release, generating a version-specific DOI (Digital Object Identifier) for it and storing a copy of the source code, along with any associated files. +The concept DOI, which is also given in the repository README, does not change but represents the repository as a whole and always points to the latest version.

          • +
        +
        • +
      +
      +
      +

      Release of binaries on conda-forge

      +

      To ensure CADET-Core is accessible to a broad community, it is available as a Python package on conda-forge. +Other software, such as our Python interface, CADET-Python, and the frontend, CADET-Process, import this package.

      +
      +
      + + + +
      + +
      +
      +
      +
      +
      + ©2008-2024. + + | + Page source +
      + + + + + + \ No newline at end of file diff --git a/chore/developer_guide/developer_guide/testing.html b/chore/developer_guide/developer_guide/testing.html index 9dd77a796..dbabbe49b 100644 --- a/chore/developer_guide/developer_guide/testing.html +++ b/chore/developer_guide/developer_guide/testing.html @@ -16,7 +16,7 @@ - + @@ -59,12 +59,13 @@

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    • Maintenance of the tests
    +
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    • @@ -83,6 +84,7 @@

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    @@ -67,6 +67,7 @@

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      diff --git a/chore/developer_guide/examples/index.html b/chore/developer_guide/examples/index.html index 0360e79a1..b9e0cb2e9 100644 --- a/chore/developer_guide/examples/index.html +++ b/chore/developer_guide/examples/index.html @@ -48,7 +48,7 @@

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    @@ -67,6 +67,7 @@

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    • +
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      diff --git a/chore/developer_guide/genindex.html b/chore/developer_guide/genindex.html index 3eeccd334..d7f9a2fc0 100644 --- a/chore/developer_guide/genindex.html +++ b/chore/developer_guide/genindex.html @@ -47,7 +47,7 @@

      Navigation

    • CADET-Match
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      • -
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      • +
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    @@ -66,6 +66,7 @@

    Releases

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      diff --git a/chore/developer_guide/getting_started/build_linux.html b/chore/developer_guide/getting_started/build_linux.html index bc2f5f926..d0f71ef17 100644 --- a/chore/developer_guide/getting_started/build_linux.html +++ b/chore/developer_guide/getting_started/build_linux.html @@ -48,7 +48,7 @@

      Navigation

    • CADET-Match
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      • -
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    @@ -67,6 +67,7 @@

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    • Branches

      @@ -135,19 +136,23 @@

      LAPACK

      Build CADET

      -
        -

      • Clone the CADET source code git clone https://github.com/modsim/CADET.git CADET

        • +
          +

        • Clone the CADET source code git clone https://github.com/cadet/cadet-core.git CADET

        • Create the directories CADET/build and CADET/install

        • Open a terminal and change to CADET/build

        • If using MKL, execute export MKLROOT=/opt/intel/mkl

          • -

        • Using standard LAPACK: Execute cmake -DCMAKE_INSTALL_PREFIX="../install" ../

          -
            -

          • Using MKL (sequential): Execute cmake -DCMAKE_INSTALL_PREFIX="../install" -DBLA_VENDOR=Intel10_64lp_seq ../

            • -

          • Using MKL (parallel): Execute cmake -DCMAKE_INSTALL_PREFIX="../install" -DBLA_VENDOR=Intel10_64lp ../

            • -
          +

        • To compile: +- Using standard LAPACK: Execute cmake -DCMAKE_INSTALL_PREFIX="../install" ../

          +
            - Using MKL (sequential): Execute `cmake -DCMAKE_INSTALL_PREFIX="../install" -DBLA_VENDOR=Intel10_64lp_seq ../`
+
+  - Using MKL (parallel): Execute `cmake -DCMAKE_INSTALL_PREFIX="../install" -DBLA_VENDOR=Intel10_64lp ../`
+
          +
          • -

        • Execute make

          • -

        • Execute make install

          • +

        • To build: +- Execute make

          • +

        • To install: +- Execute make install

      diff --git a/chore/developer_guide/getting_started/build_osx.html b/chore/developer_guide/getting_started/build_osx.html index 59545298f..df29a710f 100644 --- a/chore/developer_guide/getting_started/build_osx.html +++ b/chore/developer_guide/getting_started/build_osx.html @@ -48,7 +48,7 @@

      Navigation

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      • +
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    @@ -67,6 +67,7 @@

    Releases

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    • +
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    • Branches

      @@ -132,7 +133,7 @@

      LAPACK

      Build CADET

        -

      • Clone the CADET source code git clone https://github.com/modsim/CADET.git CADET

        • +

      • Clone the CADET source code git clone https://github.com/cadet/cadet-core.git CADET

      • Create the directories CADET/build and CADET/install

        diff --git a/chore/developer_guide/getting_started/build_windows.html b/chore/developer_guide/getting_started/build_windows.html index eaf9da254..10a51fe1d 100644 --- a/chore/developer_guide/getting_started/build_windows.html +++ b/chore/developer_guide/getting_started/build_windows.html @@ -48,7 +48,7 @@

        Navigation

      • CADET-Match
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        • -
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        • +
      • Legal notice
      @@ -67,6 +67,7 @@

      Releases

    • v4.3.0
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      • +
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    Branches

    @@ -73,6 +73,7 @@

    Releases

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    • +
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    • Branches

      diff --git a/chore/developer_guide/getting_started/installation.html b/chore/developer_guide/getting_started/installation.html index 96fc927f9..61981d284 100644 --- a/chore/developer_guide/getting_started/installation.html +++ b/chore/developer_guide/getting_started/installation.html @@ -58,7 +58,7 @@

      Navigation

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      • -
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      • +
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    @@ -77,6 +77,7 @@

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      diff --git a/chore/developer_guide/getting_started/tutorials/breakthrough.html b/chore/developer_guide/getting_started/tutorials/breakthrough.html index 7e88029e1..158904fa3 100644 --- a/chore/developer_guide/getting_started/tutorials/breakthrough.html +++ b/chore/developer_guide/getting_started/tutorials/breakthrough.html @@ -59,7 +59,7 @@

      Navigation

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      • +
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    @@ -78,6 +78,7 @@

    Releases

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    • +
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      diff --git a/chore/developer_guide/index.html b/chore/developer_guide/index.html index a05cf03f6..6a9679279 100644 --- a/chore/developer_guide/index.html +++ b/chore/developer_guide/index.html @@ -50,7 +50,7 @@

      Navigation

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    @@ -69,6 +69,7 @@

    Releases

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  • Output Group
    • @@ -70,7 +71,7 @@

    Navigation

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    • -
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    • +
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    • @@ -89,6 +90,7 @@

    Releases

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    • @@ -70,7 +71,7 @@

    Navigation

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    • @@ -89,6 +90,7 @@

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    • @@ -70,7 +71,7 @@

    Navigation

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    • +
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    • @@ -89,6 +90,7 @@

    Releases

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    • +
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    @@ -76,6 +76,7 @@

    Releases

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    • +
  • v5.0.1

    • Branches

    +
  • Population balance model +
  • Reaction models
    • -
  • Spatial discretization methods -
  • Flux reconstruction methods
  • Nonlinear solver for consistent initialization
    • +
  • Spatial discretization methods +
  • Output Group
  • Output Group
    • @@ -70,7 +71,7 @@

    Navigation

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      @@ -211,6 +213,11 @@

      Branches

    +
  • Population balance model +
  • Reaction models
    • -
  • Spatial discretization methods -
  • Flux reconstruction methods
  • Nonlinear solver for consistent initialization
      @@ -264,6 +261,17 @@

      Branches

    • +
  • Spatial discretization methods +
    • diff --git a/chore/developer_guide/interface/introduction.html b/chore/developer_guide/interface/introduction.html index 555131392..dc8b34cc8 100644 --- a/chore/developer_guide/interface/introduction.html +++ b/chore/developer_guide/interface/introduction.html @@ -63,7 +63,7 @@

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    • @@ -82,6 +82,7 @@

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      @@ -112,9 +113,9 @@

      Branches

      Introduction

      Global structure

      -

      The global structure (see Fig. 10) is divided into three parts: input, output, and meta. -Every valid CADET file needs an input group (see Fig. 11) which contains all relevant information for simulating a model. -It does not need an output (see Fig. 13) or meta (see Fig. 10) group, since those are created when results are written. +

      The global structure (see Fig. 11) is divided into three parts: input, output, and meta. +Every valid CADET file needs an input group (see Fig. 12) which contains all relevant information for simulating a model. +It does not need an output (see Fig. 14) or meta (see Fig. 11) group, since those are created when results are written. Whereas the output group is solely used as output and holds the results of the simulation, the meta group is used for input and output. Details such as file format version and simulator version are read from and written to the meta group.

      If not explicitly stated otherwise, all datasets are mandatory. @@ -124,25 +125,25 @@

      Global structure ../_images/file_format_structure.png
      -

      Fig. 10 Structure of the groups in the root group of the file format

      +

      Fig. 11 Structure of the groups in the root group of the file format

      ../_images/file_format_input.png
      -

      Fig. 11 High-level structure of the groups in the input part of the file format

      +

      Fig. 12 High-level structure of the groups in the input part of the file format

      ../_images/file_format_input_model_unit.png
      -

      Fig. 12 Structure of the groups in a column unit operation (/input/model group)

      +

      Fig. 13 Structure of the groups in a column unit operation (/input/model group)

      ../_images/file_format_output.png
      -

      Fig. 13 Structure of the groups in the output part of the file format

      +

      Fig. 14 Structure of the groups in the output part of the file format

      diff --git a/chore/developer_guide/interface/meta_group.html b/chore/developer_guide/interface/meta_group.html index 9087fbfbf..b504b28f6 100644 --- a/chore/developer_guide/interface/meta_group.html +++ b/chore/developer_guide/interface/meta_group.html @@ -57,7 +57,7 @@

      Navigation

    • CADET-Match
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    • Bibliography
      • -
    • Legal notice
      • +
    • Legal notice
    @@ -76,6 +76,7 @@

    Releases

  • v4.3.0
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    • +
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    • Branches

      diff --git a/chore/developer_guide/interface/output_group.html b/chore/developer_guide/interface/output_group.html index 204bb6c2a..1048cb358 100644 --- a/chore/developer_guide/interface/output_group.html +++ b/chore/developer_guide/interface/output_group.html @@ -17,7 +17,7 @@ - + @@ -64,7 +64,7 @@

      Navigation

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      • +
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    @@ -83,6 +83,7 @@

    Releases

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    • +
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  • Output Group
    • @@ -70,7 +71,7 @@

    Navigation

  • CADET-Match
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  • Bibliography
    • -
  • Legal notice
    • +
  • Legal notice
    • @@ -89,6 +90,7 @@

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      @@ -198,9 +200,144 @@

      Correlations

      Parameter-State Dependencies

      +

      Currently, the dependence of surface diffusion on the particle liquid salt component is the only available parameter-state dependence.

      Group /input/model/unit_XXX

      -

      Parameter-State Dependencies are not fully implemented yet.

      +

      PAR_SURFDIFFUSION_DEP

      +
      +

      Parameter dependence of \(\texttt{PAR_SURFDIFFUSION}\) on the particle liquid salt component (i.e. component with index 0). Valid dependencies are:

      +
        +

      • \(\texttt{NONE}\) Original parameter is used unmodified.

        • +

      • \(\texttt{LIQUID_SALT_EXPONENTIAL}\) Original parameter is modified by exponential law of liquid phase salt concentration.

        • +

      • \(\texttt{LIQUID_SALT_POWER}\) Original parameter is modified by power law of liquid phase salt concentration.

        • +

      • \(\texttt{LIQUID_SALT_COLLOIDAL_AFFINITY}\) Original parameter is modified by colloidal binding affinity based on liquid phase salt concentration.

        • +
      +

      Optional: If left out, no parameter dependence is assumed and the original surface diffusion coefficients are used unmodified.

      + + + + + + +

      Type: string

      Length: \(1 / \texttt{NPARTYPE}\)

      +
      +

      PAR_SURFDIFFUSION_EXPFACTOR

      +
      +

      Factor \(\texttt{p1}\) in exponential law particle surface diffusion relation +\(D_{s, i, m} = \tilde{D}_{s, i, m} p_{1, i, m} exp \left(p_{2, i, m} c_{0}^{p} \right)\), where \(\tilde{D}_{s, i, m}\) is the original surface diffusion coefficient. +Only required if \(\texttt{PAR_SURFDIFFUSION_DEP}\) is \(\texttt{LIQUID_SALT_EXPONENTIAL}\).

      + + + + + + + +

      Type: double

      Range: \(\geq 0\)

      Length: \(\texttt{NBOUND}\)

      +

      PAR_SURFDIFFUSION_EXPFACTOR +\(D_{s, i, m} = \tilde{D}_{s, i, m} \left[ p_{4, i, m} \left( k_{i, m} \left( c_{0}^{p} \right) \right)^{p_{5, i, m}} p_{6, i, m} exp \left( p_{7, i, m} k_{i, m} \left( c_{0}^{p} \right) \right) \right]\) +where \(\tilde{D}_{s, i, m}\) is the original surface diffusion coefficient and +\(k_{i, m} \left( c_{0}^{p} \right) = p_{1, i, m}\left( c_{0}^{p} \right)^{p_{2, i, m}} + p_{3, i, m}\). +Only required if \(\texttt{PAR_SURFDIFFUSION_DEP}\) is \(\texttt{LIQUID_SALT_COLLOIDAL_AFFINITY}\).

      + + + + + + + +

      Type: double

      Range: \(\mathbb{R}\)

      Length: \(\texttt{NBOUND}\)

      +
      +

      PAR_SURFDIFFUSION_EXPARGMULT

      +
      +

      Factor \(\texttt{p2}\) in exponential law particle surface diffusion relation +\(D_{s, i, m} = \tilde{D}_{s, i, m} p_{1, i, m} exp \left(p_{2, i, m} c_{0}^{p} \right)\) +where \(\tilde{D}_{s, i, m}\) is the original surface diffusion coefficient. Only required if \(\texttt{PAR_SURFDIFFUSION_DEP}\) is \(\texttt{LIQUID_SALT_EXPONENTIAL}\).

      + + + + + + + +

      Type: double

      Range: \(\mathbb{R}\)

      Length: \(\texttt{NBOUND}\)

      +
      +

      PAR_SURFDIFFUSION_POWFACTOR

      +
      +

      Factor \(\texttt{p1}\) in power law particle surface diffusion relation +\(D_{s, i, m} = \tilde{D}_{s, i, m} p_{1, i, m} \left( c_{0}^{p} \right)^{p_{2, i, m}}\) +where \(\tilde{D}_{s, i, m}\) is the original surface diffusion coefficient. Only required if \(\texttt{PAR_SURFDIFFUSION_DEP}\) is \(\texttt{LIQUID_SALT_POWER}\).

      + + + + + + + +

      Type: double

      Range: \(\geq 0\)

      Length: \(\texttt{NBOUND}\)

      +
      +

      PAR_SURFDIFFUSION_POWEXP

      +
      +

      Fjactor \(\texttt{p2}\) in power law particle surface diffusion relation +\(D_{s, i, m} = \tilde{D}_{s, i, m} p_{1, i, m} \left( c_{0}^{p} \right)^{p_{2, i, m}}\) +where \(\tilde{D}_{s, i, m}\) is the original surface diffusion coefficient. Only required if \(\texttt{PAR_SURFDIFFUSION_DEP}\) is \(\texttt{LIQUID_SALT_POWER}\).

      + + + + + + + +

      Type: double

      Range: \(\mathbb{R}\)

      Length: \(\texttt{NBOUND}\)

      +
      +

      PAR_SURFDIFFUSION_LOGKEQFACTOR

      +
      +

      Factor \(\texttt{p1}\) in colloidal affinity law particle surface diffusion relation +\(D_{s, i, m} = \tilde{D}_{s, i, m} \left[ p_{4, i, m} \left( k_{i, m} \left( c_{0}^{p} \right) \right)^{p_{5, i, m}} p_{6, i, m} exp \left( p_{7, i, m} k_{i, m} \left( c_{0}^{p} \right) \right) \right]\) +where \(\tilde{D}_{s, i, m}\) is the original surface diffusion coefficient and +\(k_{i, m} \left( c_{0}^{p} \right) = p_{1, i, m}\left( c_{0}^{p} \right)^{p_{2, i, m}} + p_{3, i, m}\). +Only required if \(\texttt{PAR_SURFDIFFUSION_DEP}\) is \(\texttt{LIQUID_SALT_COLLOIDAL_AFFINITY}\).

      + + + + + + + +

      Type: double

      Range: \(\mathbb{R}\)

      Length: \(\texttt{NBOUND}\)

      +
      +

      PAR_SURFDIFFUSION_LOGKEQEXP

      +
      +

      Factor \(\texttt{p2}\) in colloidal affinity law particle surface diffusion relation +\(D_{s, i, m} = \tilde{D}_{s, i, m} \left[ p_{4, i, m} \left( k_{i, m} \left( c_{0}^{p} \right) \right)^{p_{5, i, m}} p_{6, i, m} exp \left( p_{7, i, m} k_{i, m} \left( c_{0}^{p} \right) \right) \right]\) +where \(\tilde{D}_{s, i, m}\) is the original surface diffusion coefficient and +\(k_{i, m} \left( c_{0}^{p} \right) = p_{1, i, m}\left( c_{0}^{p} \right)^{p_{2, i, m}} + p_{3, i, m}\). +Only required if \(\texttt{PAR_SURFDIFFUSION_DEP}\) is \(\texttt{LIQUID_SALT_COLLOIDAL_AFFINITY}\).

      + + + + + + + +

      Type: double

      Range: \(\mathbb{R}\)

      Length: \(\texttt{NBOUND}\)

      +
      +

      PAR_SURFDIFFUSION_LOGKEQCONST

      +
      +

      Factor \(\texttt{p3}\) in colloidal affinity law particle surface diffusion relation +\(D_{s, i, m} = \tilde{D}_{s, i, m} \left[ p_{4, i, m} \left( k_{i, m} \left( c_{0}^{p} \right) \right)^{p_{5, i, m}} p_{6, i, m} exp \left( p_{7, i, m} k_{i, m} \left( c_{0}^{p} \right) \right) \right]\) +where \(\tilde{D}_{s, i, m}\) is the original surface diffusion coefficient and +\(k_{i, m} \left( c_{0}^{p} \right) = p_{1, i, m}\left( c_{0}^{p} \right)^{p_{2, i, m}} + p_{3, i, m}\). +Only required if \(\texttt{PAR_SURFDIFFUSION_DEP}\) is \(\texttt{LIQUID_SALT_COLLOIDAL_AFFINITY}\).

      + + + + + + + +

      Type: double

      Range: \(\mathbb{R}\)

      Length: \(\texttt{NBOUND}\)

      +
      diff --git a/chore/developer_guide/interface/population_balance_model.html b/chore/developer_guide/interface/population_balance_model.html new file mode 100644 index 000000000..709fe481b --- /dev/null +++ b/chore/developer_guide/interface/population_balance_model.html @@ -0,0 +1,364 @@ + + + + + + + + Population balance model — CADET + + + + + + + + + + + + + + + + + + + + +
      + + +
      +
      + + +
      + + +

      + + + You're reading the documentation for a development version. + For the latest released version, please have a look at v5.0.0. + + +

      + + +
      +

      Population balance model

      +

      The PBM in CADET is implemented as part of the reaction module and can thus be used in any unit operation that includes reactions. +Typical applications consider crystallization in a Continuous stirred tank reactor model or, to model continuous processes, in a Dispersive Plug-Flow Reactor (DPFR), which is modelled by a Lumped Rate Model Without Pores.

      +

      The particle size domain (internal coordinate) is discretized by the FV method, giving us a finite set of particle sizes under consideration \(\{x_1, \dots, x_{N_x}\}\). +Every particle size considered is treated as an individual component of the unit operation and the field NCOMP of that unit operation in which the crystallization happens, must be specified accordingly as \(N_x + 2\). +The two additional components account for the solute \(c\) and solubility \(c_\text{eq}\). +That is, by setting the ``NCOMP`` field, you specify the number of FV cells for the internal coordinate.

      +

      Note that the first component must be solute \(c\) and the last component must be the solubility \(c_\text{eq}\).

      +

      Example code to configure the PBM can be found in CADET-Verification .

      +
      +

      Group /input/model/unit_XXX

      +

      NCOMP

      +
      +

      Number of components, which is given by two plus the number discrete particle sizes, which is given by the number of FV cells discretizing the internal coordinate

      + + + + + + + +

      Type: int

      Range: \(\geq 3\)

      Length: 1

      +
      +

      REACTION_MODEL

      +
      +

      The crystallization code is implemented as a reaction module, which is why crystallization needs to be specified here

      + + + + + + + +

      Type: String

      Range: \(\{ CRYSTALLIZATION \}\)

      Length: 1

      +
      +
      +
      +

      Group /input/model/unit_XXX/reaction_bulk - REACTION_MODEL = CRYSTALLIZATION - UNIT_TYPE = CSTR

      +

      The following parameters need to be specified under Group /input/model/unit_XXX/reaction/reaction_bulk/ for CSTR units, and Group /input/model/unit_XXX/reaction/reaction/ for transport units like the LRM.

      +

      CRY_BINS

      +
      +

      Coordinates of the cell faces, e.g. equidistant or logarithmic discretization of the internal coordinate \(x \in [x_c, x_\text{end}]\), including the end points.

      + + + + + + + +

      Type: double

      Range: \(\geq 1\)

      Length: \(\mathrm{N_x} + 1\)

      +
      +

      CRY_NUCLEI_MASS_DENSITY

      +
      +

      Nulcei mass density

      + + + + + + + +

      Type: double

      Range: \(\geq 0\)

      Length: 1

      +
      +

      CRY_VOL_SHAPE_FACTOR

      +
      +

      Volumetric shape factor of the particles

      + + + + + + + +

      Type: double

      Range: \(\geq 0\)

      Length: 1

      +
      +

      CRY_PRIMARY_NUCLEATION_RATE

      +
      +

      Primary nucleation rate constant \(k_p\)

      + + + + + + + +

      Type: double

      Range: \(\geq 0\)

      Length: 1

      +
      +

      CRY_SECONDARY_NUCLEATION_RATE

      +
      +

      Secondary nucleation rate \(k_b\)

      + + + + + + + +

      Type: double

      Range: \(\geq 0\)

      Length: 1

      +
      +

      CRY_GROWTH_RATE_CONSTANT

      +
      +

      Growth rate constant \(k_g\)

      + + + + + + + +

      Type: double

      Range: \(\geq 0\)

      Length: 1

      +
      +

      CRY_GROWTH_CONSTANT

      +
      +

      Growth constant \(\gamma\)

      + + + + + + + +

      Type: double

      Range: \(\geq 0\)

      Length: 1

      +
      +

      CRY_A

      +
      +

      Defines constant \(a\) used to determine the growth rate

      + + + + + + + +

      Type: double

      Range: \(\geq 0\)

      Length: 1

      +
      +

      CRY_G

      +
      +

      Defines constant \(g\) used to determine the growth rate

      + + + + + + + +

      Type: double

      Range: \(\geq 0\)

      Length: 1

      +
      +

      CRY_P

      +
      +

      Defines constant \(p\) used to determine the growth rate

      + + + + + + + +

      Type: double

      Range: \(\geq 0\)

      Length: 1

      +
      +

      CRY_GROWTH_DISPERSION_RATE

      +
      +

      Growth dispersion rate \(D_g\)

      + + + + + + + +

      Type: double

      Range: \(\geq 0\)

      Length: 1

      +
      +

      CRY_U

      +
      +

      Defines constant \(u\) used to determine the primary nucleation

      + + + + + + + +

      Type: double

      Range: \(\geq 0\)

      Length: 1

      +
      +

      CRY_B

      +
      +

      Defines constant \(b\) used to determine the secondary nucleation

      + + + + + + + +

      Type: double

      Range: \(\geq 0\)

      Length: 1

      +
      +

      CRY_K

      +
      +

      Defines constant \(k\) used to determine the secondary nucleation

      + + + + + + + +

      Type: double

      Range: \(\geq 0\)

      Length: 1

      +
      +

      CRY_GROWTH_SCHEME_ORDER

      +
      +

      Defines the growth flux FV reconstruction scheme. It can only be \(1\): upwind scheme; \(2\): HR Koren scheme; \(3\): WENO23 scheme; \(4\): WENO35 scheme. +We recommend using the HR Koren scheme, which showed to be the most performant in our benchmarks.

      + + + + + + + +

      Type: int

      Range: \([1, \dots, 4]\)

      Length: 1

      +
      +
      +
      + + + +
      + +
      +
      +
      +
      +
      + ©2008-2024. + + | + Page source +
      + + + + + + \ No newline at end of file diff --git a/chore/developer_guide/interface/reaction/index.html b/chore/developer_guide/interface/reaction/index.html index 647353fc1..20465d37d 100644 --- a/chore/developer_guide/interface/reaction/index.html +++ b/chore/developer_guide/interface/reaction/index.html @@ -17,7 +17,7 @@ - + @@ -52,14 +52,15 @@

      Navigation

    • System of unit operations
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    • Binding models
      • +
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    • Reaction models
      • -
    • Spatial discretization methods
    • Flux reconstruction methods
    • Nonlinear solver for consistent initialization
    • Return data
    • Parameter Sensitivities
    • Solver Configuration
    • Parameter Dependencies
      • +
    • Spatial discretization methods
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    • @@ -70,7 +71,7 @@

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      @@ -121,101 +123,97 @@

      Branches

      Group /input/model/unit_XXX/discretization/SPATIAL_METHOD - Details on the methods

      CADET offers two spatial discretization methods: Finite Volumes (FV) and Discontinuous Galerkin (DG). While both methods approximate the same solution to the underlying models, they may differ in terms of computational performance. -Generally, FV can be more performant for small problem sizes and solutions with steep gradients, while DG excels for large problem sizes and smooth solutions.

      -

      In the following, we give a brief introduction to the numerical theory that is most relevant for the computational performance of the methods. -Based on that theory and our experience, we give advice on which method to use in which scenario, how to identify the more performant method, and how to specify the discretization parameters. -For a comprehensive description on the FV and DG methods as they are implemented in CADET, we refer to our publications on CADET-FV and CADET-DG.

      +Generally, FV can perform better for small problem sizes and solutions with steep gradients, while DG excels for large problem sizes and smooth solutions.

      +

      In the following, we give a brief introduction to the numerical theory w.r.t. the computational performance of the methods. +Based on that theory and our experience, we give advice on which method to use in which scenario, how to identify the computationally best performing method, and how to specify discretization parameters. +For a comprehensive description of the FV and DG variants that are implemented in CADET, we refer to our publications on CADET-FV and CADET-DG.

      Discrete system size

      -

      Numerical methods discretize the continuous (here: spatial) domain of the equations into a finite set of discrete points. -Then, a system of equations is formulated for those points, and both system size and number of unknowns / degrees of freedom (DoF) are given by the number of discrete points. -This system can be linear or non-linear, depending on the method (this will become important again in the section on smooth solutions). -The wall clock time to compute the solution depends on the system size and, thus, on the number of discrete points. -Conversely, the numerical solution is more accurate with more discrete points. -Thus, we trade computation time for approximation accuracy by specifying the parameters that determine the number of discrete points. -For the FV scheme, the number of axial discrete points in the column is given by the number of volume cells NCOL. -For the DG scheme, the number of axial discrete points in the column is given by the number of polynomial interpolation nodes (= POLYDEG + 1) times the number of DG elements NELEM. -The LRMP and GRM additionally consider particle equations that are also discretized. -In the spatially discretized equations, a single particle is incorporated at each axial discrete point, which increases the total number of DOF per axial point, especially for the GRM where particles are fully resolved. -The parameters for the GRM particle discretization are given for FV in NPAR and for DG in PAR_POLYDEG and PAR_NELEM.

      +

      FV and DG discretize the continuous spatial domain of the partial differential algebraic equations (PDAE) into a finite set of discrete points. +Then, a system of (semi-discrete) equations is formulated on those points, resulting in a system of ordinary differential algebraic equations (ODAE). +The system size and number of unknowns, are also referred to as degrees of freedom (DoF), is given by the number of (spatial) discrete points (times the number of components for multi-component systems).

      +

      The numerical solution becomes more accurate with more discrete points used, but the compute time required to solve the equations also increases. +That is, we trade compute time for approximation accuracy.

      +

      For the FV method, the number of axial discrete points in the column is specified by the number of volume cells NCOL. +For the DG method, the number of axial discrete points in the column is specified by the number of polynomial interpolation nodes (= POLYDEG + 1) times the number of DG elements NELEM. +The GRM additionally comprises a transport equation along the radial coordinate of the particles which is also spatially discretized and the particle discrete points are correspondingly given by NPAR for FV and for DG via PAR_POLYDEG and PAR_NELEM.

      Order of convergence

      The computational performance of a numerical method depends on its theoretical order of convergence. -The order of convergence refers to the rate at which the method’s approximation approaches the exact solution under refinement of the spatial grid.

      -

      A higher-order method can be faster than a low-order method: -Imagine a high- and a low-order method’s approximation to exhibit similarly bad approximation accuracy due to a coarse spatial resolution. -Refining the grid for both methods by the same number of discrete points improves the approximation accuracy of the higher-order method more than the other one. -Thus, the low-order method requires more DOFs and ultimately more compute time to compute a solution of the same accuracy.

      -

      The theoretical order of convergence is an asymptotic property, however. +The order of convergence refers to the rate at which the numerical approximation approaches the exact solution under refinement of the spatial grid. +Consequently, higher order methods often require less spatial discrete points to compute an approximation of the desired accuracy and can thus be computationally more efficient.

      +

      The theoretical order of convergence for the CADET-FV method is globally limited to 2. +It is locally (except column boundaries) limited to 3, with an improved order of up to 5 for smooth solutions, and can be varied by specifying the parameters described in the Flux reconstruction methods section. +For the CADET-DG method, the theoretical order of convergence is \(N_d + 1\) with \(N_d\) denoting the polynomial degree, and can thus be user-defined by specifying the field POLYDEG (and PAR_POLYDEG for the particles in the GRM).

      +

      The theoretical order of convergence is an asymptotic property, i.e. holds for infinite refinements, but can be numerivcally observed for finite refinements as well. Having the exact solution, we can compute an experimental order of convergence (EOC) via the formula

      -(25)\[\begin{aligned} +(30)\[\begin{aligned} EOC_k = \frac{log(\varepsilon_{k+1} / \varepsilon_{k}) }{log(n_{k} / n_{k+1})}, \end{aligned}\]
      -

      with \(\varepsilon_{k}\) and \(n_{k}\) denoting some error norm and the degrees of freedom of the kth approximation. +

      with \(\varepsilon_{k}\) and \(n_{k}\) denoting some error norm and the degrees of freedom of the $k$th approximation. The EOC approaches the theoretical order of convergence for \(k \rightarrow \infty\) but is typically lower for underresolved problems. -High-order methods typically suffer from start-off problems, i.e. they typically won’t exhibit their high order until the grid is fine enough and a certain accuracy is already reached. -That is, increasing the number of discrete points from, e.g., 2 to 4 typically does not improve the solution according to the theoretical order of convergence but by a much smaller EOC. -The EOC is highly problem-dependent, and it is generally unknown when a high-order method will actually be faster than a lower-order method. -Experience shows that higher-order methods work well for smooth solutions.

      -

      The theoretical order of convergence for the CADET-FV scheme is fixed at 2. -For the CADET-DG scheme, it is \(N_d + 1\) with \(N_d\) denoting the polynomial degree, and can thus be user-defined by specifying the field POLYDEG (and PAR_POLYDEG for the GRM). -As a convergence order of \(\gt 6\) is hardly realized within the approximation error of engineering tolerance (due to start-off problems), we recommend a maximum polynomial order of 5. -As the FV scheme oftentimes yields an EOC of around 2.5 and is computationally more enhanced (less arithmetic operations per DOF and customized factorization) than the DG code, we recommend a polynomial degree of at least 3 to top this.

      +High-order methods typically do typically not exhibit their theoretical order on very coarse grids. +That is, increasing the number of discrete points from, e.g., 2 to 4 does often not improve the solution according to the theoretical order of convergence but by a much smaller EOC.

      +

      For smooth solutions, we typically observe an EOC of around 2.5 for the default CADET-FV method and around \(N_d\) for the CADET-DG method. +To our experience, DG with \(N_d>6\) does usually not realize an EOC of \(>6\) for approximation errors within engineering tolerances, i.e. higher rates are achieved only for excessively small error tolerances that are not relevant for most applications. +We thus recommend to choose \(3 \leq N_d \leq 5\) for the DG method.

      +

      One could still think that the higher the order of the method, the better the performance, but that is unfortunately not generally true, as the numerical solver performance can strongly depend on the smoothness of the approximated solution.

      Smooth solutions

      -

      Smoothness in functions is characterized by the absence of sudden changes, reflected in the continuity and differentiability of the function and its derivatives. +

      The smoothness of a function indicates the absence of sudden changes, reflected in the continuity and differentiability of the function and its derivatives. In numerical simulation, smoothness indicators are often based on derivatives or moments of the solution. -That is, strong gradients and high frequencies are used to identify non-smooth parts of the solution. +That is, the occurrence of strong gradients and high frequencies are used to identify non-smooth parts of the solution. Godunov’s order barrier theorem shows why the concept of smoothness plays a crucial role in the deployment of numerical methods. It states that linear high-order methods that are monotonous are at most first-order accurate. Linear higher-order (\(\gt 1\)) methods thus suffer from artificial oscillations at non-smooth parts of the solution, specifically at discontinuities and strong gradients. -Some higher-order methods, such as CADET-FV (2nd order), contain a non-linear mechanism to suppress these oscillations. -The non-linear WENO mechanism employed in CADET-FV can be fine-tuned via the fields specified here Flux reconstruction methods. -Unfortunately, non-linear higher-order methods (order \(\geq 3\)) are either not applicable (e.g., undefined boundary treatment) or have other shortcomings, such as more highly problem-dependent parameters.

      -

      CADET-DG is a linear high-order method (arbitrary order) and thus exhibits oscillatory behaviour at strong gradients, which increases the approximation error and results in a smaller EOC for lower resolutions. -As strong gradients are a local phenomena that can be captured by employing more discrete points, DG becomes more performant again for higher resolutions. +Some higher-order methods, such as the FV variants implemented in CADET, contain a non-linear mechanism to suppress these oscillations. +The non-linear WENO mechanism employed in CADET-FV can be fine-tuned via the parameters specified in the Flux reconstruction methods section. +Unfortunately, non-linear higher-order methods (here \(\geq 3\)) are either not applicable (e.g., undefined boundary treatment) or have other shortcomings, such as parameters whose optimal values can be highly proplem-specific.

      +

      The DG variant that is implemented in CADET is a linear high-order method (arbitrary order) and can thus exhibit oscillatory behaviour at strong gradients, which increases the approximation error and results in a smaller EOC for lower resolutions. +Since strong gradients are local phenomena which can be captured by employing more discrete points, DG performs better again for higher spatial resolutions. This, however, might happen after the engineering error tolerance is by far surpassed. -Hence, CADET-FV as a stabilized lower-order method can be more performant, depending on the setting. -The DG scheme reduces its oscillatory behaviour by adding artificial numerical dispersion at element interfaces. -Thus, the use of a lower polynomial degree and more elements is recommended for rather non-smooth problems.

      +Hence, CADET-FV as a stabilized lower-order method can perform better, depending on the smoothness of the approximated solution. +The DG method reduces its oscillatory behaviour by adding artificial numerical dispersion at element interfaces. +Thus, the use of a lower polynomial degree and more elements is recommended for non-smooth solutions.

      In Chromatography, mathematical discontinuities never happen, as there are always some dispersive effects in reality. Chromatography models, however, allow for discontinuities if dispersion parameters are set to zero. -Moreover, steep and self-sharpening concentration fronts might appear due to binding. -Binding models that might cause self-sharpening concentration fronts are often associated with competitive Langmuir type isotherms for components with differently strong binding properties. -Nonetheless, a lot of chromatography settings yield rather smooth concentration profiles, for which DG is the better choice in terms of computational performance.

      +Moreover, steep and self-sharpening concentration fronts might appear due to competitive adsorption. +Examples that can cause self-sharpening concentration fronts are often associated with competitive Langmuir type isotherms for components with differently strong binding properties. +Nonetheless, many chromatography settings yield rather smooth concentration profiles, for which DG is the better choice in terms of computational performance.

      Recommendations on the choice of spatial discretization methods

      We recommend the FV method for

      • Small problem sizes, e.g., low spatial resolution with the LRM

        • -

      • Problems with strong gradients, e.g., no or low dispersion and bindings that create sharp fronts

        • -

      • Bindings that mathematically require positive values or exhibit strange behaviour with negative concentration values

        • +

      • Problems with strong gradients, e.g., no or low dispersion and binding model parameters that create sharp fronts

        • +

      • Binding models that mathematically require positive values or exhibit strange behaviour with negative concentration values

      We recommend the DG method for

        -

      • Large problem sizes, e.g., high resolutions and more complex models (i.e. the LRMP and specifically the GRM)

        • -

      • Smooth problems, e.g., sufficient dispersion

        • +

      • Large problem sizes, e.g., high spatial resolutions and more complex models (i.e. the LRMP and specifically the GRM)

        • +

      • Smooth problems, i.e., sufficient band broadening

      Recommendations on DG discretization parameters

      • Employ an axial polynomial degree between 3 and 5

        • -

      • Select a lower axial polynomial degree for non smooth tendency and employ more elements instead. Converse choice for smooth problems

        • -

      • Adjust the DG particle polynomial degree to control approximation accuracy; leave the number of elements at one. Make exceptions if very steep gradients occur inside the particles or when specific parts of the particle domain are more interesting (here, you can resolve more interesting parts by a user-defined spacing of multiple elements)

        • -

      • The field EXACT_INTEGRATION specifies the DG polynomial integration method. The default value of 0 (collocation DG) is expected to be slightly more performant in most settings

        • +

      • Select a lower axial polynomial degree for approximating functions that tend to be less smooth and employ more elements instead. Converse choice for smooth problems

        • +

      • Adjust the DG particle polynomial degree to control approximation accuracy; leave the number of elements at one. Make exceptions if very steep gradients occur inside the particles or when specific regions of the particle domain are more interesting (the spatial resolution of certain regions can be refined by a user-defined spacing of multiple elements)

        • +

      • The field EXACT_INTEGRATION specifies the DG polynomial integration method. The default value of $0$ (collocation DG) is expected to be slightly more performant in most settings

      Refinement strategy

      A common problem in numerical simulation is that the number of discrete points required to yield an accurate approximation within a specific tolerance is unknown. We thus recommend determining the approximation error via comparison with a refined reference approximation. -Both the theoretical order of convergence and the EOC can be used to estimate the required number of discrete points.

      +Both the theoretical order of convergence and the EOC can be used to estimate the required number of discrete points. +An EOC that is significantly lower than the theoretical order indicates that the problem is numerically underresolved.

      Note on DG solution vector

      diff --git a/chore/developer_guide/interface/system.html b/chore/developer_guide/interface/system.html index 3fab25bf9..1734a45cb 100644 --- a/chore/developer_guide/interface/system.html +++ b/chore/developer_guide/interface/system.html @@ -52,14 +52,15 @@

      Navigation

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      @@ -240,7 +242,7 @@

      Group /input/model/unit_XXX/discretization - UNIT_TYPE - RADIAL_GENERAL_RATE -

      Parameters specified under Flux reconstruction methods can also be ignored.

      +

      Parameters specified under flux_restruction_methods can also be ignored.

      diff --git a/chore/developer_guide/license.html b/chore/developer_guide/license.html index c656d934b..41526480f 100644 --- a/chore/developer_guide/license.html +++ b/chore/developer_guide/license.html @@ -16,7 +16,7 @@ - + @@ -50,7 +50,7 @@

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      @@ -120,7 +122,7 @@

      Branches

      No interaction between the components is considered when the model has multiple components. One of the limitation of this isotherm is the first order Jacobian \(\left(\frac{dq^*}{dc_p}\right)\) tends to infinity as \(c_{p} \rightarrow 0\) for \(n>1\). To address this issue an approximation of isotherm is considered near the origin. -This approximation matches the isotherm in such a way that \(q=0\) at \(c_p=0\) and also matches the first derivative of the istotherm at \(c_p = \epsilon\), where \(\epsilon\) is a very small number, for example \(1e-14\). +This approximation matches the isotherm in such a way that \(q=0\) at \(c_p=0\) and also matches the first derivative of the istotherm at \(c_p = \varepsilon\), where \(\varepsilon\) is a very small number, for example \(1e-14\). The form of approximation and its derivative is given below for \(c_p < \varepsilon\) and \(n>1\):

      \[ \begin{align}\begin{aligned}\begin{aligned} @@ -137,7 +139,7 @@

      Branches

      \[\begin{aligned} \alpha_2 = \frac{1-n}{n}k_f \varepsilon^{\frac{1-2 n}{n}} \end{aligned}\]
      -

      This approximation can be used for any pore phase cocentration \(c_p < \epsilon\) given \(n>1\). +

      This approximation can be used for any pore phase cocentration \(c_p < \varepsilon\) given \(n>1\). For the case, when \(n \le 1\) no special treament near the origin is required. For more information on model parameters required to define in CADET file format, see Freundlich LDF.

      diff --git a/chore/developer_guide/modelling/binding/generalized_ion_exchange.html b/chore/developer_guide/modelling/binding/generalized_ion_exchange.html index a8b3c0887..943c103c7 100644 --- a/chore/developer_guide/modelling/binding/generalized_ion_exchange.html +++ b/chore/developer_guide/modelling/binding/generalized_ion_exchange.html @@ -56,6 +56,7 @@

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      diff --git a/chore/developer_guide/modelling/crystallization.html b/chore/developer_guide/modelling/crystallization.html new file mode 100644 index 000000000..0ea37d902 --- /dev/null +++ b/chore/developer_guide/modelling/crystallization.html @@ -0,0 +1,249 @@ + + + + + + + + Population Balance Models — CADET + + + + + + + + + + + + + + + + + + + + +
      + + +
      +
      + + +
      + + +

      + + + You're reading the documentation for a development version. + For the latest released version, please have a look at v5.0.0. + + +

      + + +
      +

      Population Balance Models

      +

      The population balance model (PBM) is given by a particle-number continuity equation which describes the evolution of the number density \(n\) of the particles over time \(t\) and with respect to a so-called internal coordinate \(x\) and external coordinate \(z\). +The internal coordinate can be chosen as any characteristic property of the particles such as the particle size or volume. +The external coordinate can be a characteristic dimension of the reactor itself, including its axial length.

      +

      In CADET, we chose particle size and axial coordinate as the internal and external coodinate respectively. +The PBM in CADET is implemented so that it can be used in any unit operation that supports reactions. +Typical applications consider crystallization in a CSTR or, to model continuous processes, in a Dispersive Plug-Flow Reactor (DPFR), which is described by the LRM without solid phase.

      +

      In the following, we give a brief overview on the modelling equations, for more information on the PBM as implemented in CADET, please refer to [24].

      +
      +../_images/PBM_Part_I.png +
      +

      Fig. 8 Nucleation, growth and growth rate dispersion in PBM. Note that dispersion is used to model (random) variance in growth speed, not the reduction of particle size, which is feasible for a convection dominated process.

      +
      +
      +
      +

      Population Balance Model in a CSTR

      +

      We assume a well-mixed tank and choose the particle size \(x\in (x_c, \infty)\) as the internal coodinate, with \(x_c>0\) being the minimal particle size considered. +The corresponding PBM is given as

      +
      +\[\begin{aligned} + \frac{\partial (n V)}{\partial t} = F_{in}n_{in} - F_{out}n - V \left( \frac{\partial (v_{G}n)}{\partial x} - D_g \frac{\partial^2 n}{\partial x^2} - B_0 \delta (x-x_c) \right), +\end{aligned}\]
      +

      where \(F_{in}, F_{out}\in \mathbb{R}^+\) are the volumetric inflow and outflow rates, \(V\in\mathbb{R}^+\) is the reactor volume, +\(n(t, x)\colon [0, T_\text{end}] \times (x_c, \infty) \mapsto \mathbb{R}^+\) is the number density distribution, +\(n_{in}\in\mathbb{R}^+\) is the number density distribution of the inlet feed, \(v_{G}\in\mathbb{R}^+\) is the particle growth rate, +\(D_g\in\mathbb{R}^+\) is the growth dispersion rate.

      +

      The boundary conditions are given by the regularity boundary condition

      +
      +(27)\[\begin{aligned} + \left. \left( nv_{G} - D_g \frac{\partial n}{\partial x} \right) \right|_{x \to \infty}=0, +\end{aligned}\]
      +

      and the nucleation kinetics boundary condition

      +
      +(28)\[\begin{aligned} + \left. \left( nv_{G}-D_g \frac{\partial n}{\partial x} \right)\right|_{x=x_c} = B_0, +\end{aligned}\]
      +

      where \(B_0\in\mathbb{R}^+\) is the nucleation kinetics factor representing particle nucleations of size \(x_c\in\mathbb{R}^+\).

      +

      The model is complemented by the following mass balance equation which accounts for the mass transfer between the particle phase and the solute phase

      +
      +\[\begin{aligned} + \frac{\partial (cV)}{\partial t} = F_{in}c_{in} - F_{out}c -\rho k_v V \left( B_0x^3_c + 3\int_{x_c}^{\infty} v_{G}n\ x^2 \;\mathrm{d}x \right), +\end{aligned}\]
      +

      where \(c(t)\colon [0, T_\text{end}] \mapsto \mathbb{R}^+\) is the solute concentration in the bulk phase, \(c_{in}\in\mathbb{R}^+\) is the inlet solute mass concentration, \(\rho > 0\) is the nuclei mass density and \(k_v > 0\) is the volumetric shape factor of the particles.

      +

      Evolution of the reactor’s volume is governed by

      +
      +\[\begin{aligned} + \frac{\mathrm{d}V}{\mathrm{d}t} &= F_{\text{in}} - F_{\text{out}}. +\end{aligned}\]
      +
      +
      +

      Population Balance Model in a DPFR

      +

      The PBM can also be formulated for a DPFR to model continuous processes. +That is, we choose the axial position within a DPFR as the external coordinate \(z\in[0, L]\) and formulate the \(2D\) PBM

      +
      +\[\begin{aligned} + \frac{\partial n}{\partial t} = -v_\text{ax} \frac{\partial n}{\partial z} +D_{ax} \frac{\partial^2 n}{\partial z^2} - \frac{\partial (v_{G}n)}{\partial x} + D_g \frac{\partial^2 n}{\partial x^2}, +\end{aligned}\]
      +

      where \(n(t, x, z)\colon [0, T_\text{end}] \times (x_c, \infty) \times [0, L] \mapsto \mathbb{R}^+\) is the number density distribution, +\(v_\text{ax}\in\mathbb{R}^+\) is the axial velocity and \(D_{ax}\in\mathbb{R}^+\) is the axial dispersion coefficient.

      +

      Boundary conditions for the internal coordinate are again given by Eq. 27 and Eq. 28.

      +

      For the external coordinate \(z\), Danckwerts boundary conditions are applied:

      +
      +\[\begin{aligned} + \left. \left( n v_\text{ax}-D_{ax}\frac{\partial n}{\partial z} \right) \right|_{z=0} = v_\text{ax} n_{in,x}, \qquad \left.\frac{\partial n}{\partial z}\right|_{z=L}=0. +\end{aligned}\]
      +

      The mass balance equation for the solute \(c(t, z)\colon [0,T-\text{end}] \times [0,L] \mapsto \mathbb{R}^+\) is given by

      +
      +\[\begin{aligned} + \frac{\partial c}{\partial t} = -v_\text{ax} \frac{\partial c}{\partial z} +D_{ax} \frac{\partial^2 c}{\partial z^2} -\rho k_v \left( B_0x^3_c + 3\int_{x_c}^{\infty} v_{G}n x^2 \;\mathrm{d}x \right). +\end{aligned}\]
      +

      As for the particle phase, the solute mass concentration subjects to the Danckwerts boundary conditions

      +
      +\[\begin{aligned} + \left.\left( c v_\text{ax}-D_{ax}\frac{\partial c}{\partial z} \right) \right|_{z=0} = v_\text{ax} c_{in}, \qquad \left.\frac{\partial c}{\partial z}\right|_{z=L}=0. +\end{aligned}\]
      +
      +
      +

      Constitutive equations

      +

      Constitutive equations describe the kinetic processes in the governing equations. The relative supersaturation \(s>0\) is:

      +
      +\[\begin{aligned} + s=\frac{c-c_{eq}}{c_{eq}}, +\end{aligned}\]
      +

      where \(c_{eq}>0\) is the solute solubility in the solvent. +The nucleation kinetics can be split into primary and secondary nucleation:

      +
      +\[\begin{aligned} + B_0 = B_p + B_s, +\end{aligned}\]
      +

      Which are in turn defined by the following constitutive equations. +An empirical equation for primary nucleation is given by:

      +
      +\[\begin{aligned} + B_p=k_ps^u, +\end{aligned}\]
      +

      where \(k_p\in\mathbb{R}^+\) is the primary nucleation rate constant and \(u\in\mathbb{R}^+\) is a constant. +An empirical power-law expression is used for the secondary nucleation:

      +
      +\[\begin{aligned} + B_s=k_bs^bM^k, +\end{aligned}\]
      +

      where \(k_b\in\mathbb{R}^+\) is the secondary nucleation rate constant, \(b\in\mathbb{R}^+\) and \(k\in\mathbb{R}^+\) are system-related parameters and \(M\in\mathbb{R}^+\) is the suspension density defined as

      +
      +\[\begin{aligned} + M=k_v\rho\int_{0}^{\infty}n\ x^3\;\mathrm{d}x. +\end{aligned}\]
      +

      The following expression for the growth rate is implemented:

      +
      +\[\begin{aligned} + v_{G}=k_gs^g(a+\gamma x^p), +\end{aligned}\]
      +

      where \(k_g\in\mathbb{R}^+\) is the growth rate constant, \(\gamma\in\mathbb{R}^+\) quantifies the size dependence, and \(g, a, p\in\mathbb{R}^+\) are system-related constants.

      +

      For information on model parameters and how to specify the model interface, see Population balance model.

      +
      +
      + + + +
      + +
      +
      +
      +
      +
      + ©2008-2024. + + | + Page source +
      + + + + + + \ No newline at end of file diff --git a/chore/developer_guide/modelling/equations.html b/chore/developer_guide/modelling/equations.html index ecd5dd2d6..441dd0633 100644 --- a/chore/developer_guide/modelling/equations.html +++ b/chore/developer_guide/modelling/equations.html @@ -49,7 +49,7 @@

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    • diff --git a/chore/developer_guide/modelling/networks.html b/chore/developer_guide/modelling/networks.html index d0879bca6..ea013dfa4 100644 --- a/chore/developer_guide/modelling/networks.html +++ b/chore/developer_guide/modelling/networks.html @@ -16,7 +16,7 @@ - + @@ -55,6 +55,7 @@

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      @@ -125,13 +127,13 @@

      Branches

      The network of unit operations uses “connection”-variables \(c_{\text{con}}\) to connect the different unit operation ports with each other. The inlet port variables \(c_{\text{in},n,k}\) of unit operation \(n\) are attached to \(c_{\text{con},n}\) via

      -(22)\[\begin{aligned} +(25)\[\begin{aligned} c_{\text{in},n,k,i} &= c_{\text{con},n,k,i}, \qquad k = 1, \dots, N_{\text{port},\text{in},n},\quad i = 1, \dots, N_{\text{comp},n}. \end{aligned}\]

      While \(N_{\text{port},\text{in},n}\) denotes the number of inlet ports of unit operation \(n\), the number of outlet ports is given by \(N_{\text{port},\text{out},n}\). The connection variables \(c_{\text{con},n,k,i}\) collect all inflows of component \(i\) into port \(k\) of unit operation \(n\):

      -(23)\[\begin{aligned} +(26)\[\begin{aligned} c_{\text{con},n,k,i} &= \frac{\sum_{m=1}^{N_{\text{units}}} \sum_{\ell = 1}^{N_{\text{port},\text{out},n}} \sum_{j = 1}^{N_{\text{comp},m}} S_{(n,k,i),(m,\ell,j)} Q_{m,\ell} c_{\text{out},m,\ell,j}}{\sum_{m=1}^{N_{\text{units}}} \sum_{\ell=1}^{N_{\text{port},\text{out},m}} \hat{S}_{(n,k),(m,\ell)} Q_{m,\ell} }, \end{aligned}\]

      where \(F_{m,\ell}\) denotes the volumetric flow rate from outlet port \(\ell\) of unit operation \(m\), \(S_{(n,k,i),(m,\ell,j)} \in \{0, 1\}\) is a connection matrix indicating whether component \(i\) at outlet port \(k\) of unit operation \(n\) is connected to component \(j\) at inlet port \(\ell\) of unit operation \(m\), and \(\hat{S}_{(n,k),(m,\ell)} \in \{0, 1\}\) is another connection matrix indicating whether outlet port \(k\) of unit operation \(n\) is connected to inlet port \(\ell\) of unit operation \(m\), that is

      @@ -188,13 +190,13 @@

      Branches

      \[Q = Q_0 + Q_1(t - t_s) + Q_2(t-t_s)^2 + Q_3(t-t_s)^3,\]

      where \(t_s\) is the beginning of the time section that triggers the valve switch.

      -

      Note that the denominator in Eq. 23 must always be positive. +

      Note that the denominator in Eq. 26 must always be positive. That is, the flow rate coefficients have to be chosen such that the flow into every connected inlet port is strictly positive at all times.

      Solution of the linear systems

      -

      Each time step in the simulation requires the solution of a nonlinear system Eq. 24 (see Sec. Time stepping). -The nonlinear problem is solved by a Newton iteration, which, in turn, requires the solution of a linear system that essentially consists of the Jacobians of the unit operations and some coupling matrices from Eqs. 22 and 23.

      +

      Each time step in the simulation requires the solution of a nonlinear system Eq. 29 (see Sec. Time stepping). +The nonlinear problem is solved by a Newton iteration, which, in turn, requires the solution of a linear system that essentially consists of the Jacobians of the unit operations and some coupling matrices from Eqs. 25 and 26.

      These linear systems are either solved in parallel or sequentially. The parallel method first solves each unit operation (in parallel) to compute the solution at its outlet. Using these values, the inlets are adjusted and the unit operations are solved again. This is iterated until the system is fully solved.

      diff --git a/chore/developer_guide/modelling/reaction/index.html b/chore/developer_guide/modelling/reaction/index.html index 274a96bc6..9b26de2b6 100644 --- a/chore/developer_guide/modelling/reaction/index.html +++ b/chore/developer_guide/modelling/reaction/index.html @@ -52,6 +52,7 @@

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      @@ -119,7 +121,7 @@

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      However, due to packing irregularity and inhomogeneous flow at the inlet (i.e., frits), this assumption may be a crude approximation. This model can be improved by introducing a radial coordinate \(\rho \in [0, R]\), where \(R\) is the column radius, in the interstitial volume Eq. 1:

      -(16)\[\begin{split} \varepsilon_c \frac{\partial c^\ell_i}{\partial t} = &-\varepsilon_c u \frac{\partial c^\ell_i}{\partial z} + \varepsilon_c D_{\text{ax},i} \frac{\partial^2 c^\ell_i}{\partial z^2} + \frac{1}{\rho} \frac{\partial}{\partial \rho} \left( \rho D_{\text{rad},i} \frac{\partial}{\partial \rho} \left( \varepsilon_c c^\ell_i \right) \right) \\ +(19)\[\begin{split} \varepsilon_c \frac{\partial c^\ell_i}{\partial t} = &-\varepsilon_c u \frac{\partial c^\ell_i}{\partial z} + \varepsilon_c D_{\text{ax},i} \frac{\partial^2 c^\ell_i}{\partial z^2} + \frac{1}{\rho} \frac{\partial}{\partial \rho} \left( \rho D_{\text{rad},i} \frac{\partial}{\partial \rho} \left( \varepsilon_c c^\ell_i \right) \right) \\ &- \left(1 - \varepsilon_c\right) \sum_j d_j \frac{ 3 k_{f,j,i} }{r_{p,j}} \left[ c^\ell_i - c^p_{j,i}(\cdot, \cdot, \cdot, r_{p,j}) \right] + \varepsilon_c f_{\text{react},i}^\ell\left(c^\ell\right).\end{split}\]

      Here,

      @@ -135,26 +137,26 @@

      Branches

      For simplicity, it is assumed that the parameters are piecewise constant, that is, the range \([0, R]\) is divided into disjoint zones in which all parameters are constant. These zones are used for radial discretization and can be supplied to the simulator. Continuous dependence of the parameters can be realized by piecewise constant approximation.

      -

      The Danckwerts boundary conditions at the column in- and outlet, Eq. 12 and 11, are modified to account for the radial coordinate:

      +

      The Danckwerts boundary conditions at the column in- and outlet, Eq. 3 and 2, are modified to account for the radial coordinate:

      -(17)\[\begin{aligned} +(20)\[\begin{aligned} u(\rho) c_{\text{in},i}(t,\rho) &= u(\rho) c^\ell_i(t,0,\rho) - D_{\text{ax},i}(\rho) \frac{\partial c^\ell_i}{\partial z}(t, 0, \rho) & \forall t > 0, \rho \in (0,R), \end{aligned}\]
      -(18)\[\begin{aligned} +(21)\[\begin{aligned} \frac{\partial c^\ell_i}{\partial z}(t, L, \rho) &= 0 & \forall t > 0, \rho \in (0,R). \end{aligned}\]

      Conditions for the radial direction are added:

      -(19)\[\begin{split}\begin{aligned} +(22)\[\begin{split}\begin{aligned} \frac{\partial{c^\ell_i}}{\partial \rho}(\cdot, \cdot, 0) &= 0, \\ \end{aligned}\end{split}\]
      -(20)\[ \begin{aligned} +(23)\[ \begin{aligned} \frac{\partial{c^\ell_i}}{\partial \rho}(\cdot, \cdot, R) &= 0. \end{aligned}\]
      -

      While the inner condition Eq.19 represents symmetry at the column center, the outer condition Eq. 20 is a no-flux condition.

      -

      Using the inlet boundary condition Eq. 17, each radial zone is equipped with its own inlet and outlet port. +

      While the inner condition Eq.22 represents symmetry at the column center, the outer condition Eq. 23 is a no-flux condition.

      +

      Using the inlet boundary condition Eq. 20, each radial zone is equipped with its own inlet and outlet port. That is, this unit operation has as many inlet and outlet ports as it has radial zones (parameter NRAD in the discretization group). This allows each radial zone to have its own inlet profile, which enables modeling of flow distribution in the frits by sending the feed through varying hold-up volumes before injecting it into a radial zone.

      diff --git a/chore/developer_guide/modelling/unit_operations/cstr.html b/chore/developer_guide/modelling/unit_operations/cstr.html index 3b7c47adf..32b3b4a93 100644 --- a/chore/developer_guide/modelling/unit_operations/cstr.html +++ b/chore/developer_guide/modelling/unit_operations/cstr.html @@ -59,6 +59,7 @@

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      @@ -195,18 +197,18 @@

      Branches

      \frac{\partial c^\ell_i}{\partial t} = -u \frac{\partial c^\ell_i}{\partial z} + D_{\text{ax},i} \frac{\partial^2 c^\ell_i}{\partial z^2} &- \frac{1}{\beta_c} \sum_j d_j \frac{3}{r_{p,j}} k_{f,j,i} \left[ c^\ell_i - c^p_{j,i}(\cdot, \cdot, r_{p,j}) \right] \\ &+ f_{\text{react},i}^\ell\left(c^\ell\right). \end{aligned}\end{split}\] -

      Here, \(c^\ell_i\colon \left[0, T_{\text{end}}\right] \times [0, L] \rightarrow \mathbb{R}^{\geq 0}\) denotes the concentration in the interstitial column volume, \(c^p_{j,i}\colon \left[0, T_{\text{end}}\right] \times [0, L] \times [r_{c,j}, r_{p,j}] \rightarrow \mathbb{R}^{\geq 0}\) the liquid phase concentration in the beads, \(k_{f,j,i}\) the film diffusion coefficient, \(D_{\text{ax},i}\) the dispersion coefficient, \(u\) the interstitial velocity, \(d_j\) the volume fraction of particle type \(j\), and \(\beta_c = \varepsilon_c / (1 - \varepsilon_c)\) the column phase ratio, where \(\varepsilon_c\) is the column porosity (ratio of interstitial volume to total column volume). +

      Here, \(c^\ell_i\colon \left[0, T_{\text{end}}\right] \times [0, L] \rightarrow \mathbb{R}^{\geq 0}\) denotes the concentration in the interstitial column volume, \(c^p_{j,i}\colon \left[0, T_{\text{end}}\right] \times [0, L] \times [r_{c,j}, r_{p,j}] \rightarrow \mathbb{R}^{\geq 0}\) the liquid phase concentration in the beads, \(k_{f,j,i}\geq 0\) the film diffusion coefficient, \(D_{\text{ax},i}\geq 0\) the dispersion coefficient, \(u\in\mathbb{R}\) the interstitial velocity, \(d_j > 0\) the volume fraction of particle type \(j\), and \(\frac{1}{\beta_c} = (1 - \varepsilon_c) / \varepsilon_c\) the column phase ratio, where \(\varepsilon_c\in (0,1]\) is the column porosity (ratio of interstitial volume to total column volume). If reactions are considered, the term \(f_{\text{react},i}^\ell\left(c^\ell\right)\) represents the net change of concentration \(c_i\) due to reactions involving component \(i\).

      Danckwerts boundary conditions [8] are applied to inlet and outlet of the column:

      -(11)\[\begin{aligned} +(2)\[\begin{aligned} u c_{\text{in},i}(t) &= u c^\ell_i(t,0) - D_{\text{ax},i} \frac{\partial c^\ell_i}{\partial z}(t, 0) & \forall t > 0, \end{aligned}\]
      -(12)\[\begin{aligned} +(3)\[\begin{aligned} \frac{\partial c^\ell_i}{\partial z}(t, L) &= 0 & \forall t > 0. \end{aligned}\]
      -

      Note that the outlet boundary condition Eq. 11 is also known as “do nothing” or natural outflow condition.

      +

      Note that the outlet boundary condition Eq. 2 is also known as “do nothing” or natural outflow condition.

      In the liquid phase of the porous beads (see Fig. 2) the mass balance is given by

      (4)\[\begin{split}\begin{aligned} @@ -215,7 +217,7 @@

      Branches

      &+ \underbrace{\frac{1 - \varepsilon_{p,j}}{F_{\text{acc},j,i} \varepsilon_{p,j}} D_{s,j,i} \left[\frac{\partial^2}{\partial r^2} + \frac{2}{r} \frac{\partial }{\partial r} \right] \sum_{m_{j,i}} c^s_{j,i,m_{j,i}} }_{\text{Surface diffusion}} \\ &+ f_{\text{react},j,i}^p\left( c_j^p, c_j^s \right) + \frac{1 - \varepsilon_{p,j}}{F_{\text{acc},j,i} \varepsilon_{p,j}} f_{\text{react},j,i}^s\left( c_j^p, c_j^s \right), \end{aligned}\end{split}\]
      -

      where \(c^s_{j,i,m_{j,i}}\colon \left[0, T_{\text{end}}\right] \times [0,L] \times [r_{c,j}, r_{p,j}] \rightarrow \mathbb{R}^{\geq 0}\) denotes the solid phase concentration of the \(i\)th component’s \(m_{j,i}\)th bound state in the beads of \(j\)th type, \(D_{p,j,i}\) the effective diffusion coefficient in the beads, \(D_{s,j,i}\) the surface diffusion coefficient, \(F_{\text{acc},j,i} \in [0,1]\) the pore accessibility factor, and \(\varepsilon_{p,j}\) the particle porosity (ratio of pore volume to total bead volume). +

      where \(c^s_{j,i,m_{j,i}}\colon \left[0, T_{\text{end}}\right] \times [0,L] \times [r_{c,j}, r_{p,j}] \rightarrow \mathbb{R}^{\geq 0}\) denotes the solid phase concentration of the \(i\)th component’s \(m_{j,i}\)th bound state in the beads of \(j\)th type, \(D_{p,j,i}>0\) the effective diffusion coefficient in the beads, \(D_{s,j,i}\geq 0\) the surface diffusion coefficient, \(F_{\text{acc},j,i}\geq 0 \in [0,1]\) the pore accessibility factor, and \(\varepsilon_{p,j}\in (0,1]\) the particle porosity (ratio of pore volume to total bead volume). The inner bead radius \(r_{c,j} \in [0, r_{p,j})\) is assumed to be \(0\) by default, but can be positive in order to account for core-shell particles that have an impermeable core. Reaction terms in liquid and solid phase are collected in \(f_{\text{react},j,i}^p( c_j^p, c_j^s)\) and \(f_{\text{react},j,i}^s(c_j^p, c_j^s)\), respectively.

      The GRM is used with both quasi-stationary (Eq. 5) and dynamic (Eq. 6) binding models.

      @@ -347,7 +349,7 @@

      Branches

      The radial flow GRM describes transport of solute molecules through the interstitial column volume by radial convective flow, band broadening caused by radial dispersion, mass transfer resistance through a stagnant film around the beads, pore (and surface) diffusion in the porous beads [5, 6, 7], and adsorption to the inner bead surfaces.

      The main assumptions are:

        -

      • The shells of the column are homogenous in terms of interstitial volume, fluid flow, and distribution of components. +

      • The cylindrical shells of the column are homogenous in terms of interstitial volume, fluid flow, and distribution of components. Thus, only one spatial coordinate in radial direction \(\rho\) is needed and axial transport is neglected in the column bulk volume.

      • The bead radii \(r_{p}\) are much smaller than the column radius \(\mathrm{P}-\mathrm{P}_c\), with \(\mathrm{P}\) and \(\mathrm{P}_c\) being the inner and outer column radius respectively, and the column length \(L\). Therefore, the beads can be seen as continuously distributed inside the column (i.e., at each point there is interstitial and bead volume).

        • @@ -355,23 +357,23 @@

        Branches

        That is, the volumetric flow rate at the inner and outer column radius are the same.

      Consider a hollow (double walled) column with inner column diameter \(\mathrm{P}_c>0\) and outer diameter \(\mathrm{P}>\mathrm{P}_c\), filled with spherical beads of (possibly) multiple types with radius \(r_{p,j} \ll L\) (see Fig. 2), where \(j\) is the particle type index. The mass balance in the interstitial column volume is described by

      -
      -(10)\[\begin{split}\begin{aligned} +
      +(10)\[\begin{split}\begin{aligned} \frac{\partial c^\ell_i}{\partial t} = -\frac{u}{\rho} \frac{\partial c^\ell_i}{\partial \rho} + D_{\text{rad},i} \frac{1}{\rho} \frac{\partial}{\partial \rho} \left(\rho \frac{\partial c^\ell_i}{\partial \rho} \right) &- \frac{1}{\beta_c} \sum_j d_j \frac{3}{r_{p,j}} k_{f,j,i} \left[ c^\ell_i - c^p_{j,i}(\cdot, \cdot, r_{p,j}) \right] \\ &+ f_{\text{react},i}^\ell\left(c^\ell\right). \end{aligned}\end{split}\]
      -

      Here, \(c^\ell_i\colon \left[0, T_{\text{end}}\right] \times [\mathrm{P}_c, \mathrm{P}] \rightarrow \mathbb{R}^{\geq 0}\) denotes the concentration in the interstitial column volume, \(c^p_{j,i}\colon \left[0, T_{\text{end}}\right] \times [P_c, P] \times [r_{c,j}, r_{p,j}] \rightarrow \mathbb{R}^{\geq 0}\) the liquid phase concentration in the beads, \(k_{f,j,i}\) the film diffusion coefficient, \(D_{\text{rad},i}\) the dispersion coefficient, \(u\) the interstitial velocity, \(d_j\) the volume fraction of particle type \(j\), and \(\beta_c = \varepsilon_c / (1 - \varepsilon_c)\) the column phase ratio, where \(\varepsilon_c\) is the column porosity (ratio of interstitial volume to total column volume). +

      Here, \(c^\ell_i\colon \left[0, T_{\text{end}}\right] \times [\mathrm{P}_c, \mathrm{P}] \rightarrow \mathbb{R}^{\geq 0}\) denotes the concentration in the interstitial column volume, \(c^p_{j,i}\colon \left[0, T_{\text{end}}\right] \times [P_c, P] \times [r_{c,j}, r_{p,j}] \rightarrow \mathbb{R}^{\geq 0}\) the liquid phase concentration in the beads, \(k_{f,j,i}\geq 0\) the film diffusion coefficient, \(D_{\text{rad},i}\geq 0\) the dispersion coefficient, \(u>0\) the interstitial velocity, \(d_j>0\) the volume fraction of particle type \(j\), and \(\beta_c = \varepsilon_c / (1 - \varepsilon_c)\) the column phase ratio, where \(\varepsilon_c\in(0,1)\) is the column porosity (ratio of interstitial volume to total column volume). If reactions are considered, the term \(f_{\text{react},i}^\ell\left(c^\ell\right)\) represents the net change of concentration \(c_i\) due to reactions involving component \(i\).

      Danckwerts boundary conditions [8] are applied to inlet and outlet of the column:

      -
      -(11)\[\begin{aligned} +
      +(11)\[\begin{aligned} u c_{\text{in},i}(t) &= u c^\ell_i(t,0) - D_{\text{rad},i} \frac{\partial c^\ell_i}{\partial \rho}(t, 0) & \forall t > 0, \end{aligned}\]
      -
      -(12)\[\begin{aligned} +
      +(12)\[\begin{aligned} \frac{\partial c^\ell_i}{\partial \rho}(t, \mathrm{P}) &= 0 & \forall t > 0. \end{aligned}\]
      -

      Note that the outlet boundary condition Eq. 11 is also known as “do nothing” or natural outflow condition.

      +

      Note that the outlet boundary condition Eq. 11 is also known as “do nothing” or natural outflow condition.

      The complementing mass transport and binding equations for the liquid and solid phases of the porous beads are described by the same equations as for the axial GRM.

      For information on model parameters see Radial Flow Models in addition to General Rate Model.

      diff --git a/chore/developer_guide/modelling/unit_operations/index.html b/chore/developer_guide/modelling/unit_operations/index.html index 70ff6821d..da783c3ac 100644 --- a/chore/developer_guide/modelling/unit_operations/index.html +++ b/chore/developer_guide/modelling/unit_operations/index.html @@ -59,6 +59,7 @@

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      @@ -178,6 +180,13 @@

      Branches

      ×

      ×

      +

      Population Balance Models

      +

      ×

      +

      ×

      +

      ×

      +

      ×

      +

      ×

      +

      Moreover, the pseudo unit operations Inlet, and Outlet act as sources and sinks for the system. diff --git a/chore/developer_guide/modelling/unit_operations/inlet.html b/chore/developer_guide/modelling/unit_operations/inlet.html index 88811f481..940a9cc9b 100644 --- a/chore/developer_guide/modelling/unit_operations/inlet.html +++ b/chore/developer_guide/modelling/unit_operations/inlet.html @@ -59,6 +59,7 @@

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      diff --git a/chore/developer_guide/modelling/unit_operations/lumped_rate_model_with_pores.html b/chore/developer_guide/modelling/unit_operations/lumped_rate_model_with_pores.html index 7088b97a0..dbc3c8ded 100644 --- a/chore/developer_guide/modelling/unit_operations/lumped_rate_model_with_pores.html +++ b/chore/developer_guide/modelling/unit_operations/lumped_rate_model_with_pores.html @@ -59,6 +59,7 @@

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      @@ -117,12 +119,12 @@

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      The lumped rate model with pores [3, 4] deviates from the general rate model (see Section General rate model (GRM)) by neglecting pore diffusion. The particle phase \(c^p_j\) is still there, but no mass transfer happens except for binding and film diffusion. Hence, the model equations are given by

      -
      -\[\begin{aligned} +
      +(16)\[\begin{aligned} \frac{\partial c^\ell_i}{\partial t} &= -u \frac{\partial c^\ell_i}{\partial z} + D_{\text{ax},i} \frac{\partial^2 c^\ell_i}{\partial z^2} - \frac{1}{\beta_c} \sum_{j} d_j \frac{3}{r_{p,j}} k_{f,j,i}\left[ c^\ell_i - c^p_{j,i} \right] + f_{\text{react},i}^\ell\left(c^\ell\right), \end{aligned}\]
      -(15)\[\begin{split}\begin{aligned} +(17)\[\begin{split}\begin{aligned} \frac{\partial c^p_{j,i}}{\partial t} + \frac{1 - \varepsilon_{p,j}}{F_{\text{acc},j,i} \varepsilon_{p,j}} \frac{\partial}{\partial t} \sum_{m_{j,i}} c^s_{j,i,m_{j,i}} &= \frac{3}{F_{\text{acc},j,i} \varepsilon_{p,j} r_{p,j}}k_{f,j,i}\left[ c^\ell_i - c^p_{j,i} \right] \\ &+ f_{\text{react},j,i}^p\left( c_j^p, c_j^s \right) + \frac{1 - \varepsilon_{p,j}}{F_{\text{acc},j,i} \varepsilon_{p,j}} f_{\text{react},j,i}^s\left( c_j^p, c_j^s \right) \end{aligned}\end{split}\]
      @@ -161,11 +163,11 @@

      Radial flow LRMP\(\mathrm{P}_c>0\) and outer diameter \(\mathrm{P}>\mathrm{P}_c\), filled with spherical beads of (possibly) multiple types with radius \(r_{p,j} \ll L\) (see Fig. 2), where \(j\) is the particle type index. The mass balance in the interstitial column volume is described by

      -
      -\[\begin{aligned} +
      +(18)\[\begin{aligned} \frac{\partial c^\ell_i}{\partial t} &= -\frac{u}{\rho} \frac{\partial c^\ell_i}{\partial \rho} + D_{\text{rad},i} \frac{1}{\rho} \frac{\partial}{\partial \rho} \left( \rho \frac{\partial c^\ell_i}{\partial \rho} \right) - \frac{1}{\beta_c} \sum_{j} d_j \frac{3}{r_{p,j}} k_{f,j,i}\left[ c^\ell_i - c^p_{j,i} \right] + f_{\text{react},i}^\ell\left(c^\ell\right), \end{aligned}\]
      -

      The equations are complemented by Eq. ModelParticleLRMP and the Danckwerts boundary conditions [8]

      +

      The equations are complemented by Eq. 17 and the Danckwerts boundary conditions [8]

      \[\begin{split}\begin{aligned} u c_{\text{in},i}(t) &= u c^\ell_i(t,0) - D_{\text{rad},i} \frac{\partial c^\ell_i}{\partial \rho}(t, 0) & \forall t > 0,\\ diff --git a/chore/developer_guide/modelling/unit_operations/lumped_rate_model_without_pores.html b/chore/developer_guide/modelling/unit_operations/lumped_rate_model_without_pores.html index 4f8a775cc..c685c9e84 100644 --- a/chore/developer_guide/modelling/unit_operations/lumped_rate_model_without_pores.html +++ b/chore/developer_guide/modelling/unit_operations/lumped_rate_model_without_pores.html @@ -59,6 +59,7 @@

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      @@ -122,8 +124,8 @@

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      \end{aligned}\]

      The phase ratio is denoted by \(\beta_t = \varepsilon_t / (1 - \varepsilon_t)\) accordingly. The model equations are given by

      -
      -\[\begin{aligned} +
      +(14)\[\begin{aligned} \frac{\partial c^\ell_i}{\partial t} + \frac{1}{\beta_t} \frac{\partial}{\partial t} \sum_{m_i} c^s_{i,m_i} &= -u \frac{\partial c^\ell_i}{\partial z} + D_{\text{ax},i} \frac{\partial^2 c^\ell_i}{\partial z^2} + f_{\text{react},i}^\ell\left( c^\ell, c^s \right) + \frac{1}{\beta_t} f_{\text{react},i}^s\left( c^\ell, c^s \right), \end{aligned}\]

      where \(\beta_t = \varepsilon_t / (1 - \varepsilon_t)\) denotes the (total) phase ratio. @@ -160,8 +162,8 @@

      Radial flow LRM\(\mathrm{P}_c>0\) and outer diameter \(\mathrm{P}>\mathrm{P}_c\), filled with spherical beads. The mass balance in the interstitial column volume is described by

      -
      -(14)\[\begin{aligned} +
      +(15)\[\begin{aligned} \frac{\partial c^\ell_i}{\partial t} + \frac{1}{\beta_t} \frac{\partial}{\partial t} \sum_{m_i} c^s_{i,m_i} &= -\frac{u}{\rho} \frac{\partial c^\ell_i}{\partial \rho} + D_{\text{rad},i} \frac{1}{\rho} \frac{\partial}{\partial \rho} \left( \rho \frac{\partial c^\ell_i}{\partial \rho} \right) + f_{\text{react},i}^\ell\left( c^\ell, c^s \right) + \frac{1}{\beta_t} f_{\text{react},i}^s\left( c^\ell, c^s \right), \end{aligned}\]

      The equations are complemented by Danckwerts boundary conditions [8]

      diff --git a/chore/developer_guide/modelling/unit_operations/multi_channel_transport_model.html b/chore/developer_guide/modelling/unit_operations/multi_channel_transport_model.html index 40491dec2..ff538ac7d 100644 --- a/chore/developer_guide/modelling/unit_operations/multi_channel_transport_model.html +++ b/chore/developer_guide/modelling/unit_operations/multi_channel_transport_model.html @@ -59,6 +59,7 @@

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      The same model equations arise in describing other biological and technical processes outside of the field of plant sciences, where solutes are transported and exchanged between spatially separated compartments, for example liquid-liquid chromatography (LLC). Here, components in a mixture are separated based on their interactions with two immiscible phases of a biphasic solvent system [10]. The MCT model represents these phases by channels with respective transport and exchange properties. While the current implementation only covers linear driving forces for the exchange processes, the reaction module in CADET allows to add non-linear driving forces for the exchange processes and other chemical reactions in the channels.

      The MCT model equations are given for all channels \(l \in \{1, \dots, N_k\}\) and components \(i \in \{1, \dots, N_c\}\) by

      -(21)\[\begin{aligned} +(24)\[\begin{aligned} \frac{\partial c_{i,l}^\ell}{\partial t} = - u_l \frac{\partial c_{i,l}^\ell}{\partial z} diff --git a/chore/developer_guide/modelling/unit_operations/outlet.html b/chore/developer_guide/modelling/unit_operations/outlet.html index a01e3391e..2e1bc4e2b 100644 --- a/chore/developer_guide/modelling/unit_operations/outlet.html +++ b/chore/developer_guide/modelling/unit_operations/outlet.html @@ -58,6 +58,7 @@

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      + + +
      +
      + + +
      + + +

      + + + You're reading the documentation for a development version. + For the latest released version, please have a look at v5.0.0. + + +

      + + +

      General CADET-Core:

      +
        +

      • Leweke, S.; von Lieres, E.: Chromatography Analysis and Design Toolkit (CADET) <https://doi.org/10.1016/j.compchemeng.2018.02.025>_, Computers and Chemical Engineering 113 (2018), 274?294.

        • +

      • von Lieres, E.; Andersson, J.: A fast and accurate solver for the general rate model of column liquid chromatography <https://doi.org/10.1016/j.compchemeng.2010.03.008>_, Computers and Chemical Engineering 34,8 (2010), 1180?1191.

        • +
      +

      CADET-Core numerics:

      +
        +

      • Breuer, J. M.; Leweke, S.; Schm?lder, J.; Gassner, G.; von Lieres, E.: Spatial discontinuous Galerkin spectral element method for a family of chromatography models in CADET <https://doi.org/10.1016/j.compchemeng.2023.108340>_, Computers and Chemical Engineering 177 (2023), 108340.

        • +

      • Leweke, S.; von Lieres, E.: Fast arbitrary order moments and arbitrary precision solution of the general rate model of column liquid chromatography with linear isotherm <http://dx.doi.org/10.1016/j.compchemeng.2015.09.009>_, Computers and Chemical Engineering 84 (2016), 350?362.

        • +

      • P?ttmann, A.; Schnittert, S.; Naumann, U.; von Lieres, E.: Fast and accurate parameter sensitivities for the general rate model of column liquid chromatography <http://dx.doi.org/10.1016/j.compchemeng.2013.04.021>_, Computers and Chemical Engineering 56 (2013), 46?57.

        • +
      +

      Selected applications and use-cases of CADET-Core:

      +
        +

      • Heymann, W.; Glaser, J.; Schlegel, F.; Johnson, W.; Rolandi, P.; von Lieres, E.: Advanced score system and automated search strategies for parameter estimation in mechanistic chromatography modeling <https://doi.org/10.1016/j.chroma.2021.462693>_, Journal of Chromatography A 1661 (2022): 462693.

        • +

      • He, Q.-L.; Leweke, S.; von Lieres, E.: Efficient numerical simulation of simulated moving bed chromatography with a single-column solver <http://doi.org/10.1016/j.compchemeng.2017.12.022>_, Computers and Chemical Engineering 111 (2018), 183?198.

        • +

      • Freier, L.; von Lieres, E.: Robust multi-objective global optimization of stochastic processes with a case study in gradient elution chromatography <http://doi.org/10.1002/biot.201700257>_, Biotechnology Journal 13,1 (2018), 1700257.

        • +

      • Freier, L.; von Lieres, E.: [Multi-objective global optimization (MOGO): Algorithm and case study in gradient elution chromatography <http://dx.doi.org/10.1002/biot.201600613>_, Biotechnology Journal 12,7 (2017), 1600613.

        • +

      • Diedrich, J.; Heymann, W.; Leweke, S.; Kunert, C.; Johnson, W.; Hunt, S.; Todd, B.; von Lieres, E.: Multi-state steric mass-action model and case study on complex high loading behavior of mAb on ion exchange tentacle resin <https://doi.org/10.1016/j.chroma.2017.09.039>_, Journal of Chromatography A 1525 (2017), 60?70.

        • +

      • P?ttmann, A.; Schnittert, S.; Leweke, S.; von Lieres, E.: Utilizing algorithmic differentiation to efficiently compute chromatograms and parameter sensitivities <https://doi.org/10.1016/j.ces.2015.08.050>_, Chemical Engineering Science, 139 (2016), 152?162.

        • +
      + + + +
      + +
      +
      +
      +
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      + ©2008-2024. + + | + Page source +
      + + + + + + \ No newline at end of file diff --git a/chore/developer_guide/requirements.html b/chore/developer_guide/requirements.html index b6f8305d6..5422390d2 100644 --- a/chore/developer_guide/requirements.html +++ b/chore/developer_guide/requirements.html @@ -48,7 +48,7 @@

      Navigation

    • CADET-Match
    • License
    • Bibliography
      • -
    • Legal notice
      • +
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    @@ -67,6 +67,7 @@

    Releases

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  • v5.0.1

    • Branches

      diff --git a/chore/developer_guide/search.html b/chore/developer_guide/search.html index 3ced3b945..911ee9cb6 100644 --- a/chore/developer_guide/search.html +++ b/chore/developer_guide/search.html @@ -54,7 +54,7 @@

      Navigation

    • CADET-Match
    • License
    • Bibliography
      • -
    • Legal notice
      • +
    • Legal notice
    @@ -63,6 +63,7 @@

    Releases

  • v4.3.0
  • v4.4.0
  • v5.0.0
    • +
  • v5.0.1

    • Branches

      diff --git a/chore/developer_guide/searchindex.js b/chore/developer_guide/searchindex.js index fbb7f5619..048150cff 100644 --- a/chore/developer_guide/searchindex.js +++ b/chore/developer_guide/searchindex.js @@ -1 +1 @@ -Search.setIndex({"alltitles": {"/output/coordinates/unit_XXX": [[48, "output-coordinates-unit-xxx"]], "0. Preliminary Steps": [[3, "preliminary-steps"]], "1. Setting Up the Model": [[3, "setting-up-the-model"], [18, "setting-up-the-model"]], "2. Setting up Time Sections and Connections": [[3, "setting-up-time-sections-and-connections"]], "3. Setting Up the Simulator and Running the Simulation": [[3, "setting-up-the-simulator-and-running-the-simulation"], [18, "setting-up-the-simulator-and-running-the-simulation"]], "4. 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a/chore/developer_guide/simulation/index.html +++ b/chore/developer_guide/simulation/index.html @@ -17,7 +17,7 @@ - + @@ -67,7 +67,7 @@

      Navigation

    • CADET-Match
    • License
    • Bibliography
      • -
    • Legal notice
      • +
    • Legal notice
    @@ -86,6 +86,7 @@

    Releases

  • v4.3.0
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  • v5.0.0
    • +
  • v5.0.1

    • Branches

      @@ -115,16 +116,16 @@

      Branches

      Simulation

      This chapter gives an overview of the simulation process and the different steps involved.

      -

      CADET uses a backward-differentiation-formula (BDF) time discretization as implemented by the IDAS solver from SUNDIALS [24]. +

      CADET uses a backward-differentiation-formula (BDF) time discretization as implemented by the IDAS solver from SUNDIALS [25]. Each time step requires the solution of a nonlinear algebraic system of equations is performed by a Newton method. Since chromatographic systems can exhibit strong nonlinearity and stiff systems, the Jacobian of the equation system is always updated (i.e., an “exact” Newton method is used).

      ../_images/time_integration.png
      -

      Fig. 8 General time integration procedure

      +

      Fig. 9 General time integration procedure

      -

      The general control flow of a simulation is shown in Fig. 8. +

      The general control flow of a simulation is shown in Fig. 9. Some aspects of the tasks involved are discussed below.

      Time sections and transitions

      @@ -134,14 +135,14 @@

      Branches

      t_0 < t_1 < \dots < t_i < t_{i+1} < \dots < t_{\text{end}}. \end{aligned}\]

      Time sections are used for various purposes, for example, for defining piecewise polynomials used as inlet profiles or external function, for changing operating conditions such as flow rates, or for changing the connectivity in a network of unit operations. -A sequence of (one or more) time sections with smooth transitions is called a time slice (see Fig. 9).

      +A sequence of (one or more) time sections with smooth transitions is called a time slice (see Fig. 10).

      A transition from one section to the next can either be smooth or discontinuous. On smooth transitions, the time integrator can just step over the section transition. Contrary to smooth transitions, a discontinuous transition requires some extra work in resetting the time integrator and finding consistent initial conditions for the next time slice.

      ../_images/sections.png
      -

      Fig. 9 Time sections, time slices, and transitions between them

      +

      Fig. 10 Time sections, time slices, and transitions between them

      @@ -206,7 +207,7 @@

      Branches

      The IDAS time integrator adaptively changes the step size \({\Delta t_\tau = t_\tau-t_{\tau-1}}\) and order \(1 \le q_{\tau} \le 5\) in each time step and automatically determines the corresponding coefficients \(\alpha_{\tau,i}\). This ansatz is inserted into the DAE and the resulting nonlinear algebraic equation system is solved for \(y_{\tau} = y\left( t_\tau \right)\) using a Newton method.

      -(24)\[\begin{aligned} +(29)\[\begin{aligned} H(y_\tau) = F\left(t_\tau,y_\tau,\frac{1}{\Delta t_\tau} \,\sum_{i=0}^{q_\tau}{\alpha_{\tau,i} \, y_{\tau-i}}, p\right) = 0 \end{aligned}\]

      This requires (possibly many) solutions of linear equation systems involving the Jacobian of \(H\) given by

      @@ -222,11 +223,11 @@

      Branches

      Adaptivity

      The BDF order \(q_{\tau}\) and size of the time step \(t_\tau\) are chosen such that the error bounds are satisfied with minimal work. -Equation 24 is solved with a Newton iteration using very few iterations (MAX_NEWTON_ITER from Table Group /solver/time_integrator) since the starting point is assumed to be close to the solution as the time steps are small. +Equation 29 is solved with a Newton iteration using very few iterations (MAX_NEWTON_ITER from Table Group /solver/time_integrator) since the starting point is assumed to be close to the solution as the time steps are small. If the solution of the Newton iteration does not pass the convergence test, the step size \(t_\tau\) is reduced and the Newton iteration is tried again. This may happen at most MAX_CONVTEST_FAIL times, otherwise time integration is aborted with failure. Having a solution of the Newton iteration at hand, a local truncation error test is performed which controls the error of the BDF method. -If this test fails, step size \(t_\tau\) and order \(q_{\tau}\) are adapted and the process restarts with solving the nonlinear system Eq. 24. +If this test fails, step size \(t_\tau\) and order \(q_{\tau}\) are adapted and the process restarts with solving the nonlinear system Eq. 29. The error test may be failed at most MAX_ERRTEST_FAIL times before time integration is aborted with failure.

      Error bounds for the local truncation error test are specified by an absolute tolerance (ABSTOL) and a relative tolerance (RELTOL). Note that the relative tolerance only works for non-zero values, whereas zero values are accounted for by the absolute tolerance. @@ -255,15 +256,15 @@

      Branches

      &= \frac{\partial F}{\partial y}(t, y, \dot{y}, p) s + \frac{\partial F}{\partial \dot{y}}(t, y, \dot{y}, p) \dot{s} + \frac{\partial F}{\partial p}(t, y, \dot{y}, p). \end{aligned}\end{split}\]

      These linear DAEs depend on the solution \(y\), \(\dot{y}\) of the original DAE system. -Consequently, the sensitivity systems are solved together with the original DAE system in a staggered approach [25].

      -

      After the nonlinear Eq. 24 has been successfully solved using Newton iteration (i.e., it has passed the convergence test), each sensitivity is solved using the same Newton iteration. +Consequently, the sensitivity systems are solved together with the original DAE system in a staggered approach [26].

      +

      After the nonlinear Eq. 29 has been successfully solved using Newton iteration (i.e., it has passed the convergence test), each sensitivity is solved using the same Newton iteration. If direct linear solvers were used, this iteration would convergence with a single iteration as there is no nonlinearity. The Newton method for the sensitivities performs at most MAX_NEWTON_ITER_SENS (see Table Group /solver/time_integrator) iterations. Assuming the convergence test has passed for each sensitivity, the local truncation error test is performed for the full set of variables (i.e., original system and sensitivities). The sensitivities can be excluded from the local truncation error test by setting ERRORTEST_SENS appropriately. Note that time integration step size is always affected by sensitivities due to possible convergence test failures.

      While the Jacobians \(\partial F / \partial y\) and \(\partial F / \partial \dot{y}\) can be computed either analytically or via AD, the last term \(\partial F / \partial p\) is always computed by AD. -In fact, the terms \(\partial F / \partial y\) and \(\partial F / \partial p\) can be computed by one AD-enabled evaluation of \(F\) using the vector mode with appropriate seed vectors [26].

      +In fact, the terms \(\partial F / \partial y\) and \(\partial F / \partial p\) can be computed by one AD-enabled evaluation of \(F\) using the vector mode with appropriate seed vectors [27].

      A parameter sensitivity is specified by identifying the involved parameters, which can be more than one (see below). Parameters are identified by name and multiple indices, which may not all be used (see Tab. Group /input/sensitivity/param_XXX). Indices that are not used for identification (e.g., flow rate is independent of bound phase or component) are set to \(-1\). All other indices are zero-based.

      Note that the sensitivity systems need to be consistently initialized, too. diff --git a/chore/developer_guide/sitemap.xml b/chore/developer_guide/sitemap.xml index 4538b8b50..0a3283052 100644 --- a/chore/developer_guide/sitemap.xml +++ b/chore/developer_guide/sitemap.xml @@ -1,2 +1,2 @@ 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