A mathematical study of the evolution of fouling and operating parameters throughout membrane sheets comprising spiral wound modules, Chemical Engineering Journal

Year : 2012

Author : Kostoglou M. and Karabelas A.J.

Local experimental data on key process parameters,throughoutmembrane sheets of spiral-woundmembrane (SWM) elements, are unavailable and very difﬁcult to obtain; therefore, theoretical models are essential to better understand the evolution of fouling, which is helpful for improving design and operation of these modules. The mathematical model, formulated in a fundamental manner herein, takes into account the spatial ﬂow ﬁeld non-uniformities inside narrow channels with spacers (simulating SWM elements) and their interaction with the evolving deposits. A realistic meso-scale transient model, typical of the early stages of organic membrane fouling, is employed. Appropriate non-dimensionalization of model equations allows to treat separately the two (practically signiﬁcant) cases of constant inlet pressure and constant average permeate ﬂux. Several simpliﬁed cases are examined, for speciﬁc dimensionless parameter values, and asymptotic solutions are derived. The results of this analysis have guided the development of very efﬁcient numerical techniques, most useful for future extensions towards the development of a comprehensive modeling tool, appropriate for module design calculations and for assessing detailed experimental data.

Kostoglou-Karabelas-a-mathematical-study-of-the-evolution-of-fouling-1 Download

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Call Stack

Time

Memory

Function

Location

1.4448

28301312

shutdown_action_hook( )

...\load.php:0

1.4448

28301312

do_action( $hook_name = 'shutdown' )

...\load.php:1124

1.4448

28301688

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

...\plugin.php:517

1.4448

28301688

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

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

1.4448

28302848

wp_ob_end_flush_all( '' )

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

1.4448

28302848

ob_end_flush( )

...\functions.php:5279

1.4449

28405280

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/a-mathematical-study-of-the-evolution-of-fouling-a', 9 )

...\functions.php:5279

1.4454

28408248

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...\autoptimizeMain.php:591

1.4455

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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

1.4455

28408656

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