Modeling scale formation in flat-sheet membrane modules during water desalination, A.I.Ch.E Journal

Year : 2013

Author : Kostoglou M., Karabelas A.J

Modeling the operation of spiral-wound membrane modules is essential for their successful design and optimization. Such models must include the main types of membrane fouling, degrading desalination plant performance, including scaling due to sparingly soluble salts. Unfortunately, the complexity of underlying physicochemical processes and the coexistence of several spatial and temporal scales render intractable modeling of membrane scaling based on first principles. Therefore, a suitable (albeit simplified) framework is developed for incorporating scaling dynamics into a fluid flow model formulated at an intermediate (i.e., mesoscopic) length scale of membrane operation. The general mesoscopic approach involves integration of spatially distributed submodels, thereby allowing predictions at the large (entire membrane sheet) scale; these submodels comprise constitutive laws and kinetic rate expressions derived at fine scales. A submodel for the effect of pre-existing bulk particles on scale formation is developed herein. Several numerical results are presented to exemplify the potential of the proposed framework.

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Call Stack
#	Time	Memory	Function	Location
1	1.3268	27948568	shutdown_action_hook( )	...\load.php:0
2	1.3268	27948568	do_action( $hook_name = 'shutdown' )	...\load.php:1124
3	1.3268	27948944	WP_Hook->do_action( $args = [0 => ''] )	...\plugin.php:517
4	1.3268	27948944	WP_Hook->apply_filters( $value = '', $args = [0 => ''] )	...\class-wp-hook.php:332
5	1.3268	27950104	wp_ob_end_flush_all( '' )	...\class-wp-hook.php:308
6	1.3268	27950104	ob_end_flush( )	...\functions.php:5279
7	1.3268	28048440	autoptimizeMain->end_buffering( $content = '<!doctype html>\r\n<html lang="en-US">\r\n<head>\r\n\t<meta charset="UTF-8">\r\n\t<meta name="viewport" content="width=device-width, initial-scale=1.0">\r\n\t<meta http-equiv="X-UA-Compatible" content="IE=edge">\r\n\t<link rel="profile" href="https://gmpg.org/xfn/11">\r\n\t<meta name=\'robots\' content=\'index, follow, max-image-preview:large, max-snippet:-1, max-video-preview:-1\' />\n<link rel="alternate" hreflang="el" href="https://nrre.cperi.certh.gr/el/modeling-scale-formation-in-flat-sheet-membrane-mo', 9 )	...\functions.php:5279
8	1.3274	28051408	autoptimizeScripts->read( $options = ['aggregate' => 'on', 'defer_not_aggregate' => 1, 'defer_inline' => 1, 'justhead' => '', 'forcehead' => '', 'trycatch' => '', 'js_exclude' => 'wp-includes/js/dist/, wp-includes/js/tinymce/, js/jquery/jquery.js', 'cdn_url' => '', 'include_inline' => '', 'minify_excluded' => 'on'] )	...\autoptimizeMain.php:591
9	1.3275	28051408	apply_filters( $hook_name = 'autoptimize_filter_js_dontmove', $value = [0 => 'document.write', 1 => 'html5.js', 2 => 'show_ads.js', 3 => 'google_ad', 4 => 'histats.com/js', 5 => 'statcounter.com/counter/counter.js', 6 => 'ws.amazon.com/widgets', 7 => 'media.fastclick.net', 8 => '/ads/', 9 => 'comment-form-quicktags/quicktags.php', 10 => 'edToolbar', 11 => 'intensedebate.com', 12 => 'scripts.chitika.net/', 13 => '_gaq.push', 14 => 'jotform.com/', 15 => 'admin-bar.min.js', 16 => 'GoogleAnalyticsObject', 17 => 'plupload.full.min.js', 18 => 'syntaxhighlighter', 19 => 'adsbygoogle', 20 => 'gist.github.com', 21 => '_stq', 22 => 'nonce', 23 => 'post_id', 24 => 'data-noptimize', 25 => 'data-cfasync', 26 => 'data-pagespeed-no-defer', 27 => 'logHuman', 28 => 'amp-mobile-version-switcher', 29 => 'data-rocketlazyloadscript', 30 => 'rocket-browser-checker-js-after'] )	...\autoptimizeScripts.php:286
10	1.3275	28051816	WP_Hook->apply_filters( $value = [0 => 'document.write', 1 => 'html5.js', 2 => 'show_ads.js', 3 => 'google_ad', 4 => 'histats.com/js', 5 => 'statcounter.com/counter/counter.js', 6 => 'ws.amazon.com/widgets', 7 => 'media.fastclick.net', 8 => '/ads/', 9 => 'comment-form-quicktags/quicktags.php', 10 => 'edToolbar', 11 => 'intensedebate.com', 12 => 'scripts.chitika.net/', 13 => '_gaq.push', 14 => 'jotform.com/', 15 => 'admin-bar.min.js', 16 => 'GoogleAnalyticsObject', 17 => 'plupload.full.min.js', 18 => 'syntaxhighlighter', 19 => 'adsbygoogle', 20 => 'gist.github.com', 21 => '_stq', 22 => 'nonce', 23 => 'post_id', 24 => 'data-noptimize', 25 => 'data-cfasync', 26 => 'data-pagespeed-no-defer', 27 => 'logHuman', 28 => 'amp-mobile-version-switcher', 29 => 'data-rocketlazyloadscript', 30 => 'rocket-browser-checker-js-after'], $args = [0 => [0 => 'document.write', 1 => 'html5.js', 2 => 'show_ads.js', 3 => 'google_ad', 4 => 'histats.com/js', 5 => 'statcounter.com/counter/counter.js', 6 => 'ws.amazon.com/widgets', 7 => 'media.fastclick.net', 8 => '/ads/', 9 => 'comment-form-quicktags/quicktags.php', 10 => 'edToolbar', 11 => 'intensedebate.com', 12 => 'scripts.chitika.net/', 13 => '_gaq.push', 14 => 'jotform.com/', 15 => 'admin-bar.min.js', 16 => 'GoogleAnalyticsObject', 17 => 'plupload.full.min.js', 18 => 'syntaxhighlighter', 19 => 'adsbygoogle', 20 => 'gist.github.com', 21 => '_stq', 22 => 'nonce', 23 => 'post_id', 24 => 'data-noptimize', 25 => 'data-cfasync', 26 => 'data-pagespeed-no-defer', 27 => 'logHuman', 28 => 'amp-mobile-version-switcher', 29 => 'data-rocketlazyloadscript', 30 => 'rocket-browser-checker-js-after']] )	...\plugin.php:205

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Call Stack

Time

Memory

Function

Location

1.3268

27948568

shutdown_action_hook( )

...\load.php:0

1.3268

27948568

do_action( $hook_name = 'shutdown' )

...\load.php:1124

1.3268

27948944

WP_Hook->do_action( $args = [0 => ''] )

...\plugin.php:517

1.3268

27948944

WP_Hook->apply_filters( $value = '', $args = [0 => ''] )

...\class-wp-hook.php:332

1.3268

27950104

wp_ob_end_flush_all( '' )

...\class-wp-hook.php:308

1.3268

27950104

ob_end_flush( )

...\functions.php:5279

1.3268

28048440

autoptimizeMain->end_buffering( $content = '<!doctype html>\r\n<html lang="en-US">\r\n<head>\r\n\t<meta charset="UTF-8">\r\n\t<meta name="viewport" content="width=device-width, initial-scale=1.0">\r\n\t<meta http-equiv="X-UA-Compatible" content="IE=edge">\r\n\t<link rel="profile" href="https://gmpg.org/xfn/11">\r\n\t<meta name=\'robots\' content=\'index, follow, max-image-preview:large, max-snippet:-1, max-video-preview:-1\' />\n<link rel="alternate" hreflang="el" href="https://nrre.cperi.certh.gr/el/modeling-scale-formation-in-flat-sheet-membrane-mo', 9 )

...\functions.php:5279

1.3274

28051408

autoptimizeScripts->read( $options = ['aggregate' => 'on', 'defer_not_aggregate' => 1, 'defer_inline' => 1, 'justhead' => '', 'forcehead' => '', 'trycatch' => '', 'js_exclude' => 'wp-includes/js/dist/, wp-includes/js/tinymce/, js/jquery/jquery.js', 'cdn_url' => '', 'include_inline' => '', 'minify_excluded' => 'on'] )

...\autoptimizeMain.php:591

1.3275

28051408

apply_filters( $hook_name = 'autoptimize_filter_js_dontmove', $value = [0 => 'document.write', 1 => 'html5.js', 2 => 'show_ads.js', 3 => 'google_ad', 4 => 'histats.com/js', 5 => 'statcounter.com/counter/counter.js', 6 => 'ws.amazon.com/widgets', 7 => 'media.fastclick.net', 8 => '/ads/', 9 => 'comment-form-quicktags/quicktags.php', 10 => 'edToolbar', 11 => 'intensedebate.com', 12 => 'scripts.chitika.net/', 13 => '_gaq.push', 14 => 'jotform.com/', 15 => 'admin-bar.min.js', 16 => 'GoogleAnalyticsObject', 17 => 'plupload.full.min.js', 18 => 'syntaxhighlighter', 19 => 'adsbygoogle', 20 => 'gist.github.com', 21 => '_stq', 22 => 'nonce', 23 => 'post_id', 24 => 'data-noptimize', 25 => 'data-cfasync', 26 => 'data-pagespeed-no-defer', 27 => 'logHuman', 28 => 'amp-mobile-version-switcher', 29 => 'data-rocketlazyloadscript', 30 => 'rocket-browser-checker-js-after'] )

...\autoptimizeScripts.php:286

1.3275

28051816

WP_Hook->apply_filters( $value = [0 => 'document.write', 1 => 'html5.js', 2 => 'show_ads.js', 3 => 'google_ad', 4 => 'histats.com/js', 5 => 'statcounter.com/counter/counter.js', 6 => 'ws.amazon.com/widgets', 7 => 'media.fastclick.net', 8 => '/ads/', 9 => 'comment-form-quicktags/quicktags.php', 10 => 'edToolbar', 11 => 'intensedebate.com', 12 => 'scripts.chitika.net/', 13 => '_gaq.push', 14 => 'jotform.com/', 15 => 'admin-bar.min.js', 16 => 'GoogleAnalyticsObject', 17 => 'plupload.full.min.js', 18 => 'syntaxhighlighter', 19 => 'adsbygoogle', 20 => 'gist.github.com', 21 => '_stq', 22 => 'nonce', 23 => 'post_id', 24 => 'data-noptimize', 25 => 'data-cfasync', 26 => 'data-pagespeed-no-defer', 27 => 'logHuman', 28 => 'amp-mobile-version-switcher', 29 => 'data-rocketlazyloadscript', 30 => 'rocket-browser-checker-js-after'], $args = [0 => [0 => 'document.write', 1 => 'html5.js', 2 => 'show_ads.js', 3 => 'google_ad', 4 => 'histats.com/js', 5 => 'statcounter.com/counter/counter.js', 6 => 'ws.amazon.com/widgets', 7 => 'media.fastclick.net', 8 => '/ads/', 9 => 'comment-form-quicktags/quicktags.php', 10 => 'edToolbar', 11 => 'intensedebate.com', 12 => 'scripts.chitika.net/', 13 => '_gaq.push', 14 => 'jotform.com/', 15 => 'admin-bar.min.js', 16 => 'GoogleAnalyticsObject', 17 => 'plupload.full.min.js', 18 => 'syntaxhighlighter', 19 => 'adsbygoogle', 20 => 'gist.github.com', 21 => '_stq', 22 => 'nonce', 23 => 'post_id', 24 => 'data-noptimize', 25 => 'data-cfasync', 26 => 'data-pagespeed-no-defer', 27 => 'logHuman', 28 => 'amp-mobile-version-switcher', 29 => 'data-rocketlazyloadscript', 30 => 'rocket-browser-checker-js-after']] )

...\plugin.php:205