Clase parabolico
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Transcript of Clase parabolico
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Autor: Ing. William Chauca Nolasco
MÉTODOS NUMÉRICOS II
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ECUACIÓN DIFERENCIAL PARCIAL
PARABÓLICO
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INTRODUCCIÓN
Las ecuaciones que rigen la difusión de partículas en movimiento o la conducción de calor, son ecuaciones diferenciales parciales (EDP) de tipo parabólico. Es por esto que los métodos de solución numérica de las EDP parabólicas son importantes en campos como difusión molecular, la transferencia de calor, el análisis de reactores nucleares y el flujo de fluidos. Puesto que las EDP parabólicas representan procesos de difusión que dependen del tiempo, usualmente utilizaremos las letras t y x como variables independientes, donde t es el tiempo y x es la coordenada del espacio unidimensional.
Para las EDP parabólicas en espacios bidimensionales utilizaremos x y y para las coordenadas espaciales y t como el tiempo. Los siguientes son ejemplos de EDP parabólicas:
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Conducción transitoria de calor, con la dimension espacial igual a uno [Incorpera/DeWitt]
Ecuación de difusión transitoria de neutrones, con la dimension espacial igual a uno [Hetric]:
donde f es el flujo de neutrones
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Transporte convectivo de una sustancia química con difusión [Brodkey/ Hershey]:
Donde f es la densidad de la sustancia, u(x) es la velocidad del flujo y D es la constante de difusión.
Las EDP parabólicas para dos y tres dimensiones se pueden escribir mediante la ampliación de la variable espacial a dos y tres dimensiones del espacio. Por ejemplo, la ecuación de conducción transitoria de calor, en dimensiones espaciales iguales a dos, es:
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En las clases anteriores tratamos las EDP en estado estacionario. Ahora veremos las ecuaciones parabólicas que se emplean para caracterizar problemas que varían con el tiempo. también se desarrollan estos problemas en dos dimensiones espaciales para la placa calentada. Antes, mostraremos cómo se aborda el caso unidimensional más simple.
De manera similar a la deducción de la ecuación de Laplace , se puede utilizar la conservación del calor para desarrollar un balance de calor del elemento diferencial, en la barra larga, delgada y aislada que se muestra en la figura. Sin embargo, en lugar de examinar el caso en estado estacionario, este balance también considera la cantidad de calor que se almacena en el elemento en un periodo Δt. El balance tiene la forma, entradas – salidas = acumulación, o
LA ECUACION DE CONDUCCION DE CALOR
Una barra delgada y aislada en todos los puntos excepto n sus extremos
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FIGURA 30.2Una malla utilizada para la solución por diferencias finitas de las EDP parabólicas con dos variables independientes, por ejemplo la ecuación de conducción del calor. Observe como, a diferencia de la figura 29.3, la malla está abierta en los extremos en la dimensión temporal.
FIGURA 29.3Malla usada para la solución por diferencias finitas de las EDP elípticas en dos variablesindependientes, como la ecuación de Laplace.
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que tiene un error de O[(Δx)2]. Observe que el ligero cambio en la notación de los superíndices se utiliza para denotar tiempo. Esto se hace para que un segundo subíndice pueda usarse para designar una segunda dimensión espacial cuando el método se extiende a dos dimensiones espaciales.
Una diferencia dividida finita hacia adelante sirve para aproximar a la derivada con respecto al tiempo
METODOS EXPLICITOS
la cual tiene un error de O(Δt).
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Sustituyendo las ecuaciones (30.2) y (30.3) en la ecuación (30.1), se obtiene
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Figura. 30.3
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Solucion explicita para la ecuación de conducción de calor unidimensional
X=0 T=100ºC X=10T=50ºC
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i=1 l=0
i=2 l=0i=3 l=0i=4 l=0
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X=0, T=200ºC X=10T=10ºC
t=0.1 s
t=0.2 s
TAREA: Con los datos del ejemplo anterior resolver el problema con las condiciones dadas, hasta 4 segundos con intervalos de 0.1 seg.
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CONVERGENCIA Y ESTABILIDAD
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FIGURA 30.5Ilustración de la inestabilidad. Solución del ejemplo 30.1, pero con l = 0.735
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METODOS IMPLICITO SIMPLE
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Ejemplo: Solución implícita simple de la ecuación de conducción del calor Planteamiento del problema. Con la aproximación por diferencias finitas implícita simple resuelva el problema anterior
30.10
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t=0.1 s
t=0.2 s
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−𝝀𝑻 𝒊−𝟏𝒍+𝟏+𝟐 (𝟏+𝝀)𝑻 𝒊
𝒍+𝟏−𝝀𝑻 𝒊+𝟏𝒍+𝟏=𝝀𝑻 𝒊−𝟏
𝒍 +𝟐 (𝟏−𝝀)𝑻 𝒊𝒍+𝝀𝑻 𝒊+𝟏
𝒍(30.14)
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Ejemplo: Solución de Crank-Nicolson para la ecuación de conducción del calor. Planteamiento del problema. Con el método de Crank-Nicolson resuelva el mismo anterior.
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I=1 l=0
I=2 l=0
I=3 l=0
I=m=4 l=0
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FUENTE NAKAMURA-MATHEEWS
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Vamos a considerar el modelo unidimensional del flujo del calor en un alambre aislado de longitud L (figura 10.2) La ecuación de calor , que nos da la temperatura u(x,t) en la posición x del alambre y en el instante t, es:
Figura 10.2, La ecuación del calor modela la temperatura de una alambre aislado
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ECUACIÓN DE CALOR
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MÉTODO DE CRANK - NICHOLSON
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EJEMPLO
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TAREA COMPLETAR EL CUADRO DESDE T=0.08 HASTA T = 0.20 Y GRAFICARLO
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OTRA FORMA DE ANALISIS DE EDP PARABOLICA
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FUENE- NIEVES
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