# حل تحلیلی معادله انتقال آلودگی با ضرایب متغیر در رودخانه با استفاده از تبدیل لاپلاس

نوع مقاله : مقاله پژوهشی

نویسندگان

1 دانشجوی کارشناسی ارشد، گروه مهندسی و مدیریت آب، دانشکده کشاورزی، دانشگاه تربیت مدرس، تهران، ایران.

2 دانشیار، گروه مهندسی و مدیریت آب، دانشکده کشاورزی، دانشگاه تربیت مدرس، تهران، ایران.

3 استاد، گروه مهندسی و مدیریت آب، دانشکده کشاورزی، دانشگاه تربیت مدرس، تهران، ایران.

چکیده

رودخانه‌ها یکی از مهمترین منابع طبیعی آب در جهان به‌شمار می‌روند. مدل‌سازی انتقال آلودگی در رودخانه‌ها توسط معادله دیفرانسیل جزئی جابه‌جایی-پراکندگی-واکنش (‌ADRE) انجام می‌گیرد. در پژوهش حاضر، با استفاده از تبدیل لاپلاس که یک ابزار قدرتمند و مفید در حل معادلات دیفرانسیل می‌باشد، پاسخ تحلیلی معادله ADRE در دامنه محدود با ضرایب متغیر به‌ازای شرایط مرزی بالادست و پایین‌دست دیریکله در رودخانه به‌دست آمد. به‌منظور استفاده از حل تحلیلی موجود در این پژوهش، سه مثال ارائه گردید که در هرکدام رودخانه مد نظر، به ‌بازه‌های دو، چهار و هشت قسمتی تقسیم شده‌است، که ضمن متغیر بودن پارامترهای جریان، آلاینده و هندسه رودخانه در هر سه مثال، برای هر کدام از مثال‌ها، دقت حل تحلیلی موجود در زمانی‌که تقسیم‌بندی بازه‌ها بیشتر می‌شود در مقایسه با حل عددی بررسی شده است. با مشخص کردن ماتریس‌های سرعت، ضریب پراکندگی، سطح مقطع و ... به‌عنوان ورودی مسئله، ماتریس انتشار محاسبه و به تبع آن دستگاه معادلات پیچیده‌ای ایجاد می‌شود که پیچیدگی کار را دوچندان می‌کند. با حل دستگاه معادلات مذکور، مقدار غلظت آلاینده محاسبه می‌گردد. به‌منظور اعتبارسنجی حل تحلیلی موجود، از حل عددی استفاده می‌شود، نتایج نشان داد که هرچه تعداد تقسیم‌بندی‌های رودخانه بیشتر باشد، دقت حل بالاتر می‌رود و دو حل تحلیلی و عددی دارای انطباق خوبی با یکدیگر خواهند بود. باتوجه به توانایی و عملکرد حل تحلیلی موجود، می‌توان اذعان داشت که، حل تحلیلی موجود در این پژوهش می‌تواند به‌عنوان ابزاری جهت صحت‌سنجی و اعتبارسنجی حل‌های عددی و سایر حل‌های تحلیلی برای ضرایب متغیر معادله مورد استفاده قرار گیرد.

کلیدواژه‌ها

موضوعات

عنوان مقاله [English]

### Analytical solution of the pollution transport equation with variable coefficients in river using the Laplace Transform

نویسندگان [English]

• Mehdi Mazaheri 2
• Jamal Mohammad Vali Samani 3
1 Master Student, Department of Water Engineering and Management, Faculty of Agriculture, Tarbiat Modares University, Tehran, Iran.
2 Associate Professor, Department of Water Engineering and Management, Faculty of Agriculture, Tarbiat Modares University, Tehran, Iran.
3 Professor, Department of Water Engineering and Management, Faculty of Agriculture, Tarbiat Modares University, Tehran, Iran.
چکیده [English]

Rivers are one of the most important natural water resources in the world. Pollution transport modeling in rivers is performed by the partial advection-dispersion-reaction equation (ADRE). In the present study, using the Laplace transform, which is a powerful and useful tool in solving differential equations, the analytical solution of the ADRE equation was obtained in a finite domain with variable coefficients for the upstream and downstream Dirichlet boundary conditions and the initial zero condition in the river. To use the analytical solution in this study, three examples are presented, each of which, the river are divided into two, four, and eight parts, which, while the parameters of flow, pollution, and river geometry are variable in all three examples, for each of the examples, the accuracy of the analytical solution available when the segmentation of the intervals increases as compared to the numerical solution. By specifying the matrices of velocity, dispersion coefficient, cross-section, etc. as input to the problem, the diffusion matrix is calculated and, consequently, a complex system of equations is created that doubles the complexity of the work. The amount of pollutant concentration is calculated by solving the system of the above equations. The numerical solution is used to validate the existing analytical solution, the results showed that the greater the number of river divisions, the higher the accuracy of the solution, and the two analytical and numerical solutions will be well compatible with each other. Given the ability and performance of the existing analytical solution, it can be acknowledged that the analytical solution in this study can be used as a tool to validate and verification numerical solutions and other analytical solutions for the coefficients of the equation.

کلیدواژه‌ها [English]

• Concentration distribution function
• Finite domain
• Mathematical modeling

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