an_model.md

CAST Toolbox - Analytical Models

The analytical model toolbox within the CAST provides the solution for $L_{max}$. The toolbox currently include 6 different models; each of them varying with the other with respect to model conditions, dimensionality, input quantities, and also orientation. Analytical model toolbox is the recommended step after the use of Data Toolbox in the CAST work flow.

Analytical models in CAST

Included models are (see individual model page for details):

1. Ham et al. (2004)

   $$ L_{max} = \frac{W_r^2}{4\pi\alpha_{Th}}\bigg(\frac{\gamma C_D^\circ}{C_A^\circ}\bigg)^2 $$

2. Liedl et al. (2005)

   $$ L_{max}\frac{4M^2}{\pi^2 \alpha_{Tv}}\ln\bigg( \frac{4}{\pi} \frac{\gamma C_{ED}+ C_{EA}}{C_{EA}}\bigg) $$

3. Chu et al. (2005)

   $$ L_{max} = \frac{\pi}{16}\frac{W^2}{\alpha_{Th}} \Bigg(\frac{\gamma C_D^\circ }{C_A^\circ -\epsilon}\Bigg)^2 $$

4. Liedl et al. (2011)

   $$ \text{erf}\Bigg(\frac{W}{\sqrt{4\alpha_{Th}L_{max}}}\Bigg),\text{e}^{-\alpha_{Tv}L_{max} \big(\frac{\pi}{2M}\big)^2} =\frac{\pi}{4}\frac{\gamma C_{Thres}+C_{A}^\circ}{\gamma C_{D}^\circ+C_{A}^\circ} $$

5. Karanovic et al. (2007) - BIOSCREEN-AT

   $$ c(x,y,z, t) = C_0 \frac{x}{8\sqrt{\pi D_x'}}\exp(-\gamma t) \times \int\limits_0^t\frac{1}{\xi^{3/2}}\exp\Bigg{(\gamma-\lambda_{EFF})-\xi \frac{(x-v'\xi)^2}{4D_x'\xi}\Bigg} \times $$

$$\times \Bigg[\text{erfc}\Bigg{\frac{y-\frac{w}{2}}{2\sqrt{D_y'\xi}} \Bigg}-\text{erfc}\Bigg{\frac{y+\frac{w}{2}}{2\sqrt{D_y'\xi}} \Bigg}\Bigg]\times$$

$$\times \Bigg[\text{erfc}\Bigg{\frac{z-H}{2\sqrt{D_z'\xi}}\Bigg} \text{erfc}\Bigg{\frac{z+H}{2\sqrt{D_z'\xi}} \Bigg}\Bigg]\text{d}\xi$$

Computing interfaces in CAST

CAST provide the following two computing interfaces for each analytical and empirical models:

1. Single computing interface
2. Multiple computing interface

Single computing interface

The single computing interface is a quick computing mode in which a set of model data can be inserted in data input box to obtain $L_{max}$. The interface output (see screenshot {numref}scm below) is actual $L_{max}$ and the plot in which the user input result is compared against the field plume length. The output graph has limited interactive options- such as zooming, panning, and obtaining figure as a png graphics.

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Single computing interface

The single computing interface also provide user interactivity with computation. Interface provide sliders for changing parameter values, which leads to computation and visualization of $L_{max}$.

Multiple computing interface

The Multiple computing interface provides a possibility to create simulation scenarios and compute $L_{max}$ for all scenarios at once (see screenshot {numref}mcm below) . Scenarios can be directly put input in the table or can be uploaded to the interface. For uploading the sample input template (a .csv) file has to be downloaded. After creating data in the .csv file, it can be uploaded to the interface for the $L_{max}$ computation.

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Multiple computing interface

The computed $L_{max}$ is graphically displayed in the interactive figure. The obtained results can then be compared with the user selected data from the database.

The workflow with the computing interfaces can be to begin with either single or multiple interface based on the user requirements.