CONTRIBUTIONS TO THE STUDY OF THE ACOUSTIC PROPERTIES OF POROUS

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Ruido

Ruido,

que turbado por el frío de la oscura noche resopla,

con palabras que nadie entiende grita,

desorientado busca cobijo en mil rincones,

y confuso en su odisea pierde lozanía,

qué arduo sendero de soledad el que transita,

triste desventura la que le apaga,

silenciosa le quiebra en mil pedazos,

que cayendo en su lecho de muerte resuenan,

ahogando la voz ya enmudecida,

en la que un día fue su más amarga cruzada.

J. C.

Resumen

Los materiales absorbentes porosos son de gran interés en el campo de la acústica debido a su amplia aplicabilidad a muchos problemas de ingeniería de control de ruido. Algunos de estos problemas son la contaminación acústica excesiva en las infraestructuras de transporte (carreteras, vías de tren,…) o la falta de sistemas efectivos de reducción de ruido en los edificios (residenciales, oficinas,…). Ambos se han convertido en un problema de gran preocupación en la sociedad moderna no solo por los efectos nocivos sobre la salud humana sino también por el impacto negativo en el medio ambiente y su conservación. De hecho, el ajetreado ritmo de vida en nuestra sociedad actual ha hecho del ruido una de las principales fuentes de estrés y convertido éste en el principal enemigo del descanso, induciendo a su vez una mayor despreocupación hacia el cuidado y preservación de dicho medio ambiente. Con el fin de concienciar a nuestra sociedad y a los agentes que la conforman hacia un mejor uso de los recursos naturales, autoridades públicas nacionales e internacionales siguen trabajando en el desarrollo de normativas que ayuden a reducir este impacto negativo en el entorno. De hecho, la creciente necesidad de dispositivos que mejoren el aislamiento acústico ha motivado que comunidad científica e industria centren parte de sus esfuerzos en el desarrollo de nuevas e innovadores soluciones que reduzcan el ruido y ayuden a mejorar el bienestar de las personas. Así, se persigue que dichas soluciones garanticen una reducción de los niveles de ruido en los distintos ámbitos sociales siendo a su vez una condición de mayor peso que estas tengan un carácter sostenible y respetuoso con el medio. Algunos ejemplos de ello son los equipos de protección auditiva contra el ruido de maquinaria en áreas industriales, la reducción del ruido del tráfico en áreas urbanas e interurbanas utilizando barreras acústicas, o el aislamiento al ruido en edificios mediante soluciones constructivas mejoradas. En este contexto, el uso de materiales absorbentes porosos se ha convertido en una de las soluciones pasivas más extendidas para el control del ruido hasta la fecha. Por lo tanto, el estudio del comportamiento acústico de estos materiales porosos no solo constituye una prioridad en la etapa de diseño de dispositivos contra el ruido, sino también en la evaluación de su rendimiento como absorbente sonoro. Los principales mecanismos de pérdidas por los que se rigen los materiales absorbentes porosos son la fricción viscosa y la conducción de calor que se produce en el fluido (normalmente aire) que satura dicho material cuando por éste se propaga una onda

vi RESUMEN

acústica. Este amortiguamiento viscotérmico es especialmente notable en materiales tales como la lana de roca, fibras, medios granulares, placas perforadas,… siendo por ello buenos candidatos para el desarrollo de dispositivos de absorción sonora. Entre estos, existen dos tipos de materiales ampliamente utilizados en el ámbito de la ingeniería: los paneles perforados y los medios granulares. El primer tipo se puede encontrar en diversas aplicaciones, tales como silenciadores (tubos de escape, turbinas,…) o sistemas resonantes (plenum, techos,…), siendo estos últimos frecuentemente utilizados en acústica de salas o también como parte de una barrera acústica. El segundo tipo se usa comúnmente en la fabricación de estructuras de hormigón o pavimento superficiales, siendo el uso de partículas de origen natural o resultante de los procesos de reciclado una tendencia cada vez mayor a raíz de lo previamente comentado. Por lo tanto, el estudio de las propiedades acústicas de ambos tipos de materiales constituye un campo de investigación de gran interés, tanto desde el punto de vista ingenieril como para adoptar medidas sostenibles contra el ruido. Esta tesis comprende una serie de contribuciones al estudio de las propiedades acústicas de materiales porosos, más específicamente paneles perforados y medios granulares, con el objetivo de incentivar el uso de diferentes modelos y métodos para su análisis. Como se ha mencionado anteriormente, el uso de estos materiales en los ámbitos de edificación e ingeniería civil es cada vez más extendido, estando por tanto dicha investigación justificada tanto por su aplicabilidad práctica como por el impacto social esperado de la misma. Para este propósito, se han analizado cuatro casos de estudios diferentes que van desde la propuesta de modelos de predicción hasta la concepción de nuevos sistemas de absorción sonora. Previamente, y para dar una idea de la física subyacente que gobierna la acústica en un medio poroso, se ha llevado a cabo una revisión del estado del arte sobre el modelado y la caracterización de materiales porosos que permita hacer frente a dichos casos. Este trabajo preliminar comprende una revisión de los fundamentos de propagación de ondas acústicas en medios fluidos, seguido por el caso de la propagación sonora en tubos como los que conforman un panel perforado, haciendo extensiva la misma a la descripción de medios porosos convencionales como un fluido equivalente. Estos conceptos sirven de base para poder profundizar en el análisis de medios porosos más complejos como los granulares, los medios de doble porosidad y los sistemas multicapa. Así, se revisan los modelos simplificados generalmente elegidos para estudiar medios porosos genéricos y granulares, junto con los métodos numéricos más extendidos para estudiar el resto de medios complejos; así como aquellas técnicas y métodos experimentales que sirven para determinar sus parámetros físicos intrínsecos y sus propiedades acústicas, siendo estos necesarios para predecir teóricamente y evaluar experimentalmente el comportamiento acústico de materiales porosos, respectivamente. A continuación se explican cada uno de los cuatro casos de estudio analizados, haciendo especial énfasis en los resultados obtenidos más relevantes y una discusión de los mismos.

vii RESUMEN

Caso de estudio I - Modelo en elementos finitos de un panel perforado absorbente incluyendo los efectos viscotérmicos

El primer caso de estudio propone un modelo en elementos finitos para el análisis de paneles perforados absorbentes heterogéneos. Dado el uso extendido de estos sistemas resonantes como absorbentes sonoros en acústica de la edificación, se encontró interesante el proponer una metodología alternativa a los enfoques teóricos existentes, ya que estos no pueden modelar paneles no homogéneos. El motivo de ello es que estos modelos normalmente asumen una distribución periódica de orificios circulares a lo largo del panel. De esta forma, obteniendo la impedancia acústica de transferencia de un solo orificio, es posible obtener la impedancia del panel completo sin más que utilizar la porosidad o tasa de perforación del mismo. Dicho enfoque asume a su vez que las perforaciones están equidistantes, lo cual no es cierto en el caso de paneles heterogéneos en los que la distribución de las perforaciones puede no seguir siquiera un patrón. Esta suposición puede dar lugar a significantes discrepancias a la hora de predecir el comportamiento acústico real del panel perforado absorbente, dado que el efecto de interacción entre orificios diferirá de una región del panel a otra. Al emplear un modelo en elementos finitos como el propuesto, no solo se tienen en cuenta los efectos de geometría finita del panel, sino que también es posible contemplar de forma explícita la mencionada interacción entre orificios. De esta forma, es posible predecir el comportamiento acústico de este tipo de paneles perforados absorbente teniendo en cuenta el carácter heterogéneo de sus perforaciones. El modelo en elementos finitos propuesto implementa las ecuaciones linealizadas de Navier-Stokes, permitiendo así un modelado riguroso de los fenómenos viscotérmicos que tienen lugar en las perforaciones y las inmediaciones del panel cuando una onda acústica se propaga por el mismo. Dicha implementación se llevó a cabo con el software comercial de elementos finitos COMSOL® Multiphysics, que incorpora un módulo de acústica para el modelado de problemas de acústica viscotérmica. El principal inconveniente de esta metodología es que tiene un alto coste computacional en términos tiempo de procesado y requisitos de memoria, siendo descartada en la mayoría de problemas de acústica. Esto es debido principalmente a que para poder modelar las capas viscosas y térmicas de las paredes interiores de las perforaciones del panel, es necesario realizar un mallado muy fino de dichas regiones, con el consecuente aumento del número de elementos totales. Además, para problemas tridimensionales (como es el caso bajo estudio), cada nodo del problema comprende cinco grados de libertad: uno de presión, uno de temperatura y tres para las componentes cartesianas de la velocidad de partícula. Hasta la fecha, la opción más práctica era el uso de los modelos simplificados anteriormente mencionados, no obstante, el rápido desarrollo de los ordenadores en las últimas décadas hace que cada día sea más factible abordar problemas numéricos de estas dimensiones utilizando formulaciones como la propuesta. Para verificar el modelo numérico, se realizaron una serie de simulaciones que permitieron por una parte ilustrar las limitaciones asociadas al uso de modelos teóricos simplificados y por otra validar experimentalmente la metodología propuesta comparando

viii RESUMEN

los resultados de simulación con ensayos de absorción en tubo de impedancia. El setup numérico empleado en ambos casos reproduce el procedimiento normativo empleado para determinar la impedancia acústica y el coeficiente de absorción sonora utilizando el método de la función de transferencia en tubos de impedancia [ISO 10534-2:98]. Previamente, se realizó un análisis de convergencia del modelo numérico evaluando a su vez dos configuraciones de panel perforado homogéneo cuyas características geométricas eran las mismas. De este primer análisis se extrajeron dos conclusiones: (1) la metodología propuesta converge a una solución estable al aumentar el número de grados de libertad y (2) la disposición del panel en el tubo de impedancia influye en el resultado obtenido. Esta última observación viene a corroborar lo anteriormente discutido, y sirvió para motivar la fabricación de una serie de especímenes, esta vez heterogéneos, donde estas diferencias fuesen aún más evidentes. Dichas muestras fueron testeadas en tubo de impedancia siguiendo el procedimiento normativo referido, permitiendo así comparar resultados de reales con los obtenidos utilizando el modelo numérico. Los resultados de las simulaciones mostraron una buena correlación con los experimentos en términos del coeficiente de absorción del sonido para los diferentes especímenes fabricados. Cabe destacar que tanto en el análisis preliminar como en estas últimas simulaciones, los resultados se compararon con un modelo teórico de uso extendido para el estudio de paneles perforados [Maa 1987], confirmando dicha comparativa las discrepancias anteriormente mencionadas. Además, esta metodología permite ilustrar fenómenos como el de interacción entre orificios del panel al disponer de la información del campo acústico en todos los puntos del dominio del problema. En general, el modelo propuesto no solo sirve para estudiar paneles cuya distribución de sus perforaciones sea heterogénea, sino que también puede extenderse a paneles cuyas perforaciones tengan distintos tamaños o formas. La metodología propuesta es una alternativa genérica y de gran potencial para el estudio de paneles perforados cuyas características geometría no son triviales, resultando por tanto una herramienta alternativa de gran interés para el diseño y análisis de este tipo de dispositivos.

Caso de estudio II - Propiedades acústicas de hormigón poroso elaborado a partir de agregados ligeros de arlita y vermiculita

En el segundo caso de estudio, se analizaron las propiedades acústicas de un hormigón poroso elaborado a partir de agregados ligeros de arcillas expandidas: arlita y vermiculita. El uso de recursos naturales sostenibles para la fabricación de hormigón se ha convertido en una práctica cada vez más común en el ámbito de la ingeniería civil, siendo por tanto el uso de este tipo de agregados una interesante alternativa dado que no generan residuos peligrosos para el medio ambiente y son reciclables. Además, su baja densidad los hace idóneos para utilizarse en hormigones ligeros ofreciendo a su vez interesantes propiedades térmicas y una adecuada resistencia estructural. Con el fin de incorporar el comportamiento acústico de este tipo de materiales en herramientas de predicción, también se implementó un modelo teórico ya existente para materiales granulares basado en parámetros intrínsecos del material [Horoshenkov and Swift 2001]. El propósito fue

ix RESUMEN

validar la aplicabilidad del mismo a este particular tipo de materiales, pudiendo así utilizarlo a modo de herramienta de diseño de los mismos como absorbente acústico. El estudio se ha centrado en el análisis de diferentes muestras para cuya elaboración se emplearon distintas dosificaciones agua/cemento y tamaño de árido. En total, se evaluaron cinco tipos de mezclas: dos de arlita y tres de vermiculita (dos de un tipo y una de otro), siendo el tamaño de árido menor de 4 mm para la mitad de muestras de arlita y mayor para la otra mitad. Las tres mezclas de vermiculita se prepararon con áridos menores de 1 mm, mayores de 1 mm, y entre 0.5 y 4 mm. En ambos casos las densidades totales de las muestras ya preparadas rondaban entre los 550 y 700 kg/m3, siendo las relaciones de agua-cemento utilizadas entre el 50 % y el 80 %. Por cada una de las mezclas se rellenaron moldes para seis muestras de cada tipo, considerándose este número suficientemente representativo de cada mezcla. Previamente al estudio de sus propiedades acústicas, se llevó a cabo un proceso de caracterización de aquellos parámetros físicos que sirven como datos de entrada al modelo teórico: porosidad, resistividad al flujo, tamaño de poro y tortuosidad. Cabe destacar que se para obtener este último se hizo uso de una metodología inversa basada en la minimización de la diferencia entre los datos de absorción acústica medidos experimentalmente y los predichos con el modelo. El resto de parámetros físicos se obtuvieron siguiendo diferentes procedimientos experimentales bien establecidos. La porosidad se obtuvo siguiendo el método de saturación de agua, pesando las muestras antes y después de ser saturadas. La resistividad al flujo se determinó utilizando un método acústico indirecto basado en medidas en tubo de impedancia. El tamaño de poro se calculó a partir de la distribución de tamaño de poro obtenida utilizando un método de extracción de agua. Las medidas experimentales de absorción acústica se llevaron a cabo utilizando el montaje de medida en tubo de impedancia tradicional descrito en la normativa existente. Los resultados experimentales indican que aunque todas las muestras tienen porosidades similares en torno al 39%, aquellas que tienen partículas de menor tamaño ofrecen una mayor resistividad al flujo, siendo el amortiguamiento de las ondas acústicas mayor en estos casos. Estas observaciones se confirman en las mediciones en un tubo de impedancia, al ser estas muestras las que presentan un mayor ancho de banda efectivo de absorción, mientras que la frecuencia máxima es muy similar en todos los casos. Por otra parte, el ensayo de extracción de agua confirma la distribución estadística de tipo logarítmico que presenta el tamaño de poro de este tipo de materiales granulares, obteniéndose desviación de dicho tamaño de entre 0.15 y 0.25. Además, los resultados experimentales del coeficiente de absorción acústica muestran una buena correlación con los obtenidos a partir del modelo de predicción, siendo menos preciso en el caso de las mezclas de vermiculita. Esto era de esperar al tratarse de un tipo de árido que ofrece una alta absorción de agua en comparación con la arlita. Aunque las imágenes tomadas utilizando microscopio electrónico de barrido (abreviado como SEM, en inglés) muestran una buena adhesión en el interfaz agregado-cemento, la precisión de los resultados en los ensayos con agua es algo menor, y por tanto las predicciones a partir de estos datos menos fidedignos.

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En resumen, este estudio demuestra que este tipo de hormigones son una alternativa ecológica en términos de absorción acústica a las utilizadas tradicionalmente en la industria de la ingeniería civil. Este tipo de hormigón puede a su vez adherirse a otros elementos constructivos ya existentes que desempeñen una función estructural, ofreciendo el primero las prestaciones acústicas. Por tanto, y aunque es necesario seguir investigando otras dosificaciones, los resultados preliminares obtenidos alientan a trabajar en el desarrollo de hormigón poroso elaborado a partir de este tipo de áridos para su aplicación en los sectores de la edificación y la ingeniería civil.

Caso de estudio III – Modelado acústico de hormigón perforado utilizando la teoría de doble porosidad

El tercer caso de estudio aúna los conceptos de material perforado y hormigón poroso para desarrollar un novedoso material: hormigón perforado. Esta innovadora solución consiste en un compuesto de hormigón sobre el que se practican una serie de orificios o perforaciones, lo que proporciona una reducción significativa del peso total, conservando buenas características estructurales, pudiendo además convertirlo en una opción atractiva debido a la mejora de su impacto visual. Si bien existe algún antecedente en lo que a hormigón perforado se refiere, los trabajos realizados hasta la fecha se centran principalmente en la componente visual y estética, sin abordar el comportamiento acústico del mismo al tratarse de hormigones de muy baja porosidad (no ligeros). Esta idea surge mayormente de la necesidad de proponer nuevas soluciones en el ámbito de la ingeniería civil, donde buena parte de las construcciones se realizan a partir de hormigón, que reúnan características para su uso con fines absorbentes acústicos. Dado que en el Caso de estudio II se demostró que el hormigón poroso elaborado a partir de agregados de arcilla expandida reúne las condiciones para ser utilizado como absorbente acústico, se tomó dicha base experimental para el nuevo estudio, específicamente las mezclas elaboradas a partir de arlita. Además, y con el fin de poder utilizar la información resultante del estudio en herramientas de predicción, se ha modelado dicho material utilizando la teoría de doble porosidad. Esta teoría permite modelar dicho material a partir de los parámetros físicos de la matriz “sólida” (los determinados para el hormigón poroso en al caso de estudio anterior) y las dimensiones de las perforaciones que sobre ésta se practican. En primer lugar, se elaboraron dos conjuntos de muestras a partir del mismo tipo de mezcla: no perforadas y perforadas. Las primeras se emplearon para caracterizar la matriz sólida (es decir, obtener sus parámetros físicos y acústicos), mientras que las segundas sirvieron tanto para evaluar el rendimiento absorbente del hormigón perforado como para validar la aplicabilidad del modelo de doble porosidad [Olny and Boutin 2003]. La caracterización de los parámetros físicos y de las propiedades acústicas se realizó siguiendo los mismos procedimientos que en el caso de estudio anterior. Para la caracterización acústica de las muestras sin perforar se utilizó un método basado en el setup en tubo de impedancia que permite obtener las propiedades acústicas intrínsecas del material: impedancia característica y número de onda. Este procedimiento, también llamado método de las dos cavidades [Utsuno et al. 1989], requiere la medición de la impedancia acústica del material bajo estudio siguiendo el mismo procedimiento descrito

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en la normativa correspondiente pero repitiendo la medida dejando detrás de él una cavidad de aire. Dichas propiedades servirán junto con las características geométricas de las perforaciones como dato de entrada en el modelo anterior. En cuanto a la elaboración de las muestras de hormigón perforado, se utilizaron moldes de silicona y cilindros metálicos sólidos con base circular plana para conseguir los distintos radios de perforación circular: 6, 10 y 15 mm. Para facilitar el proceso de extracción de las muestras, previamente se añadió una silicona de desmolde en dicho molde. Una vez extraídas, se procedió a su caracterización acústica siguiendo el procedimiento en tubo de impedancia normalizado. Los resultados obtenidos del proceso de caracterización de las muestras sin perforar se corresponden en su mayor parte a los ya discutidos en el caso de estudio anterior. Las muestras perforadas revelan que un mayo radio de perforación se traduce en un desplazamiento de la frecuencia de máxima absorción acústica hacia altas frecuencias con un ligero ensanchamiento de dicha curva y una disminución del valor pico. Esta observación se confirma en las predicciones del modelo de doble porosidad, donde nuevamente se utiliza el modelo para materiales granulares propuesto por Horoshenkov y Swift [Horoshenkov and Swift 2001] para la región de hormigón poroso, mientras que la región perforada se modela utilizando la teoría de propagación acústica en tubos. Los resultados correspondientes a las propiedades acústicas intrínsecas del material muestran una buena correlación con los obtenidos a partir del método de las dos cavidades, apreciándose una menor velocidad de fase (parte real del número de onda) para aquellas muestras cuyo tamaño de grano es mayor. En resumen, se propuso una nueva solución basada en hormigón poroso que además de ofrecer prestaciones de carácter estético y reducción del uso de material, presenta condiciones para ser utilizado con fines acústicos. A partir de un proceso de caracterización de muestras sin perforar, se puede modelar el comportamiento acústico de este tipo de materiales utilizando dicha información en un modelo de doble porosidad empleado habitualmente en el ámbito de materiales porosos. Además, los resultados de las mediciones para diferentes radios de perforación muestran que la frecuencia de máxima absorción del hormigón perforado puede ajustarse eligiendo adecuadamente el tamaño de estas perforaciones. Esta particularidad permitiría la fabricación de dispositivos de control de ruido elaborados a partir de hormigón cuyas características absorbentes se ajustasen a condiciones de diseño específicas.

Caso de estudio IV - Modelado acústico de paneles ranurados

El último caso de estudio propone un modelo teórico simple y eficiente para predecir las propiedades acústicas de los paneles acústicos ranurados absorbentes mediante el uso conjunto de un enfoque de fluido equivalente y el Método de la Matriz de Transferencia (TMM, en inglés). Un panel acústico ranurado comprende un sustrato con una superficie frontal ranurada y una superficie posterior con orificios circulares perforados, siendo comúnmente utilizados como resonadores absorbentes (dejando un plenum o cavidad de aire posterior) en acústica de la edificación por su estética y eficacia acústica. Algunas aplicaciones comunes de este tipo de paneles son los auditorios, centros de congresos,

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gimnasios,… Si bien existen modelos analíticos en la literatura para el análisis de paneles con ranuras o perforaciones circulares, éstos no son aplicables a soluciones híbridas que comprenden ambos tipos de geometría. Para superar esta limitación puede utilizarse el TMM, comúnmente aplicado al análisis de estructuras multicapa como es el caso bajo estudio. El modelo propuesto utiliza un enfoque de fluido equivalente utilizando el modelo clásico de Johnson-Champoux-Allard (JCA) para materiales porosos [Johnson et al. 1987, Champoux et al. 1991] para describir la propagación acústica a través de éste. Para distinguir las regiones ranuradas y con perforaciones circulares, se utilizaron expresiones específicas de la resistividad al flujo para este tipo de geometrías, así como términos de corrección de orificio para tener en cuenta los efectos de geometría finita del panel y el interfaz entre ambas regiones. A este último respecto cabe mencionar que, si bien no existe un término de corrección riguroso para dicho interfaz, tomando una aproximación propuesta por Ingard [Ingard 1953] se obtuvieron resultados satisfactorios para los casos analizados. Para obtener la respuesta total del sistema (ranura + perforado + cavidad), previamente se describe cada uno de estos “subsistemas” con una matriz de transferencia. Dichas matrices se definen a partir de las propiedades acústicas (impedancia característica y número de onda) obtenidas con el modelo JCA para cada región. Así, obteniendo la matriz correspondiente a cada subsistema, es posible obtener la matriz global del absorbente utilizando el TMM, y a partir de ésta el coeficiente de absorción del sistema completo. Las predicciones del modelo se compararon con un modelo en elementos finitos que implementa el método normativo de medida en tubo de impedancia. En dicho modelo, las regiones correspondientes a las ranuras frontales del panel y las perforaciones circulares traseras se modelaron nuevamente siguiendo el modelo de fluido equivalente JCA anteriormente mencionado. En este caso no es necesario emplear términos de corrección de ningún tipo al estar estos implícitos en la resolución del problema numérico en todo el dominio del resonador. Además, y con el propósito de comprender mejor el comportamiento acústico de este tipo de dispositivos, se llevó a cabo un estudio paramétrico modificando diferentes características geométricas del panel mientras el resto se dejaban constantes. Este tipo de análisis resultó de gran utilidad para evaluar la influencia de las características geométricas del panel en el rendimiento absorbente del resonador y establecer así una guía práctica en la fase de diseño previa a su desarrollo. Los resultados del modelo propuesto fueron comparados con los resultados en términos del coeficiente de absorción sonora obtenidos en las simulaciones numéricas con el modelo en elementos finitos, obteniendo una excelente correlación entre los resultados de uno y otro. Dicha verificación precedió el estudio paramétrico ulterior, en el que se analizó la influencia de modificar las dimensiones características de las regiones ranuradas (ancho y espesor) y perforadas (radio y espesor). Los resultados mostraron que una mayor relación ancho/radio resulta en un desplazamiento del máximo de absorción hacia baja frecuencia a la vez que una disminución de dicho valor. Por otra parte, si bien al aumentar la relación de espesores entre la región ranurada y perforada también se producía dicho desplazamiento del pico de máxima absorción hacia baja frecuencia, esta vez su valor aumentaba.

xiii RESUMEN

En resumen, se propuso un modelo simple basado en un enfoque de fluido equivalente y el TMM para estimar las propiedades acústicas de paneles ranurados absorbentes. La verificación de dicho modelo a partir de simulaciones en elementos finitos lo convierte en una herramienta sencilla y directa de gran utilidad para el diseño de estos dispositivos. Es más, al utilizarse una metodología genérica para dicho modelado, es posible extender el mismo a sistemas análogos en los que por ejemplo se incorporen materiales porosos en la cavidad del resonador sin cambios significativos en dicho proceso de modelado.

Otros trabajos

Aunque los cuatros casos de estudio anteriores conforman el núcleo de trabajo de esta tesis, el desarrollo de la misma permitió abordar otros el estudio de casos similares de gran interés para la comunidad de materiales acústicos. Brevemente, comentar un par de estos casos. Por ejemplo, el estudio de la resistencia al flujo de paneles perforados con distintas geometrías de orificio. Si bien la mayoría de paneles perforados comúnmente empleados en la práctica presentan perforaciones de geometría circular o ranurada por su facilidad de fabricación, resulta de especial interés el estudio de otro tipo de geometrías para su posible uso con fines estéticos en aplicaciones de alta gama. Utilizando una metodología numérica de dinámica de fluidos, se calculó la impedancia acústica de transferencia para distintas geometrías (circular, ranurada, triangular, cuadrada), comparando éstas también al ser utilizadas en un panel perforado absorbente. Otro de estos trabajos, incluido éste en la sección de otras publicaciones en los Apéndices, llevó a cabo una evaluación de diferentes metodologías para estudiar las propiedades acústicas de paneles perforados heterogéneos. Como se veía en el Caso de Estudio I, no existen modelos sencillos para predecir de forma rigurosa el comportamiento acústico de este tipo de absorbentes. Aunque en esta ocasión el tipo de paneles estudiados sí presentaban una distribución periódica de sus perforaciones, siguen considerándose heterogéneos porque el tamaño de las perforaciones difiere entre regiones superficiales del mismo. En dicho trabajo se comparan distintas metodologías analíticas y numéricas para su modelado, discutiendo las ventajas y limitaciones de cada una de ellas, así como su potencial para ser utilizadas en herramientas de predicción. En general, los cuatro casos de estudio anteriores y los trabajos adicionales contribuyen a (1) proponer nuevas metodologías y modelos que permitan estimar las propiedades acústicas de los materiales porosos; (2) desarrollar materiales innovadores de reducción de ruido que puedan ser adoptados en sistemas prácticos reales; y (3) fomentar la investigación en materiales porosos y su aplicación en aplicaciones de control de ruido. De estas contribuciones se extrajeron las siguientes conclusiones generales:

Un modelo de elementos finitos que implementa la formulación lineal completa de Navier Stokes (Full Linearized Navier Stokes, en inglés) permite predecir el rendimiento de absorción acústica de los paneles perforados absorbentes heterogéneos para los que los enfoques analíticos generales podrían no ser adecuados.

xiv RESUMEN

El hormigón poroso elaborado a partir de áridos ligeros de arlita y vermiculita puede convertirse en una alternativa sostenible en términos de absorción sonora a otras soluciones comúnmente utilizadas en aplicaciones prácticas.

Las propiedades acústicas del hormigón perforado pueden predecirse utilizando la teoría de doble porosidad basada en la teoría de homogeneización de estructuras periódicas siempre que se tengan en cuenta los criterios de longitud de onda y la relación entre las distintas escalas porosas del medio en cuestión.

El funcionamiento como absorbente de los paneles acústicos ranurados se pueden modelar fácilmente utilizando la teoría existente de medios porosos concibiendo éste como un sistema multicapa en el que las diferentes capas se describen utilizando un enfoque de fluido equivalente.

Además de estas conclusiones generales, cabe destacar las siguientes observaciones adicionales:

La principal ventaja del enfoque FLNS es que es genérico y se puede utilizar para estudiar cualquier geometría de orificio de un panel perforado sin la necesidad de procedimientos de ajuste o formulaciones modificadas. Además, el uso de una metodología de elementos finitos permite ilustrar fenómenos físicos que ocurren en las proximidades del panel (por ejemplo, el efecto de interacción entre orificios) para una mejor comprensión de su comportamiento acústico. Además, puede usarse para proponer sistemas perforados absorbentes optimizados cuando no existe una solución analítica, o para obtener nuevos términos de corrección de orificio.

Los procedimientos experimentales ampliamente utilizados en el campo de la caracterización de medios porosos se pueden utilizar para evaluar las propiedades acústicas y no acústicas del hormigón poroso ligero. Además, las predicciones para el coeficiente de absorción acústica utilizando un modelo teórico que se basa en los parámetros físicos determinados experimentalmente mostraron una excelente correlación con las medidas experimentales.

Es posible “sintonizar” la frecuencia de máxima absorción acústica de un elemento de hormigón perforado ajustándose a las limitaciones del diseño escogiendo adecuadamente el tamaño de sus mesoporos. Esta característica particular ofrece una amplia gama de posibilidades relacionadas con la aplicabilidad de tales soluciones a diferentes dispositivos de control de ruido comúnmente utilizados en ingeniería civil (por ejemplo, barreras acústicas, muros de aislamiento,...).

La combinación del modelo JCA (Johnson-Champoux-Allard) y del Método de la Matriz de Transferencia (TMM) ha demostrado ser una herramienta potencial para estudiar el comportamiento acústico de paneles acústicos ranurados para los que fallan los enfoques de geometría de poro único. Se encontró que un estudio paramétrico de los paneles acústicos ranurados absorbentes es muy útil para una mejor comprensión

xv RESUMEN

del rol que desempeñan las características geométricas del panel en su rendimiento de absorción sonora, sirviendo a su vez para establecer una guía práctica de diseño del mismo.

En resumen, los casos de estudio analizados sirven tanto como una contribución al estudio de las propiedades acústicas de materiales absorbentes porosos, como para alentar el uso de algunos de los enfoques propuestos en problemas reales de ingeniería. Aunque aún queda trabajo por hacer para establecer los modelos y metodologías presentados, y por qué no, para hacerlos extensivos para su aplicabilidad en un marco multidisciplinario, queda patente su aportación en aras del bienestar social y el desarrollo humano.

Abstract

Porous absorbent materials are of great interest in the field of acoustics because of their broad applicability to many noise control engineering problems. Some of these problems are the excessive noise pollution in transport infrastructure or the lack of effective noise reduction systems in buildings. In this context, the growing need for improved noise isolation devices has focused the attention of the scientific community and industry partners on the development of new and innovative solutions. Therefore, the study of their acoustic behavior not only constitutes a priority in the design stage of these devices but also in the assessment of their sound absorption performance. This thesis comprises a series of contributions to the study of the acoustic properties of porous materials with the aim of fostering the use of different approaches and methods to analyse these. For this purpose, four different case studies ranging from the proposal of prediction models to the conception of novel sound absorption systems are presented. Previously, and in order to give an insight of the underlying physics governing the porous acoustics, a review of the background theory necessary to cope with these case studies is carried out. This preliminary work let also tackle the different models and methods commonly used to investigate the acoustic properties of porous media. First, a finite element model for the analysis of heterogeneous perforated panel absorbers was proposed. Given the extended use of these resonator systems as sound absorbers in building acoustics, it was found interesting to propose an alternative methodology to existing theoretical approaches, as these fail to model non-homogeneous panels. Simulation results showed a good agreement when compared with experiments in terms of sound absorption coefficient for different manufactured specimens. Next, the acoustic properties of porous concrete made from arlite and vermiculite lightweight aggregates were analysed. By following well-established experimental procedures, a comprehensive characterization process was undertaken for differently prepared samples. A simple theoretical approach based on these measured physical parameters was successfully used to predict the sound absorption performance of these materials, as confirmed through measurements in an impedance tube. Further, these were proven to be an environmentally friendly alternative to those traditionally utilised in the civil engineering industry, besides offering potential properties when used both for structural purposes and as an acoustically effective system. This characterization process served to subsequently validate the

ABSTRACT xviii

applicability of the double porosity theory on the design of an innovative solution, so-called perforated concrete. This innovative concept consists on a compound of concrete on which a series of holes or perforations are made, thus providing a significant reduction of the overall weight while preserving good structural and acoustic features, apart from turning it into an attractive option due to its enhanced visual impact. Moreover, results of measurements for different specimens confirmed that the peak absorption frequency of perforated concrete can be tuned by suitably choosing the perforation radii. Finally, a simple and efficient theoretical model was implemented to predict the acoustic properties of grooved acoustic panels by jointly using an equivalent fluid approach and the Transfer Matrix Method (TMM). These systems comprise a substrate with a slotted front surface and a back surface with perforated circular holes, being specifically used as resonant absorbers in building acoustics because of their aesthetics and acoustical effectivity. The proposed model was verified by comparison between theoretical and numerical finite element results for the sound absorption coefficient of such devices, a parametric study being of great utility to evaluate the influence of its geometrical characteristics on the absorption performance and to establish a guide for their practical design. On the whole, these case studies contributed to (1) propose new methodologies and models that let estimate the acoustic properties of porous materials; (2) develop innovative noise reduction materials likely to be adopted in real practical systems; and (3) encourage further research into porous materials and their application in noise control applications. It is the great desire of the author of this thesis this latter point to be indeed accomplished in the nearly future for the sake of social welfare and human development.

Acknowledgments

So this is it, the thesis ended. In my opinion, this should not only be an occasion for celebration but also to realize what is left behind and what comes next. Taking a look back to recent years, I see lots of working hours dedicated to what I have tried to summarize in this document. Nevertheless, I can also see many people with which I have spent time gaining knowledge and sharing rewarding experiences. I would like to dedicate this thesis to all of them, but more especially to all of these researchers whose valuable contributions to the field of acoustics have made my work my passion. To all of them, thank you very much. However, this journey is not just about work. It is about personal growth in the path of life too. For this reason, I feel committed to thanking all those people that matter to me. To my beloved family, particularly to my father for his patience and wise advice; to my mother for her love and strength; and to my brother and sister for putting up with me over the years. I love you. To my friends: Jaime, César, Juanlu, Fran, Jorge,… Thank you for the funny moments spent together and those still to come. To all those that come and go, leaving in me brief moments of happiness, including my heartbreakers, I love you too. To all those restless students whose feedback certainly contributed to remain motivated and eager to learn every passing day. To my work colleagues Pedro and Enrique, thank you for being there whenever I needed for your help. To all those researchers and industry partners that supported me one way or another: Salvador and Miguel Ángel (UA), Jesús and Romina (UPV), Nuria and Ramón (UMH), José (BEYMA), Izaskun (ORONA), Javier (SAES)…Special thanks also to people from Coimbra, Luis and Paulo, you should be for many an example to follow because of you great humility and undeniable capacity. Last but not least, to my mentor and friend Jaime Ramis Soriano, thank you for letting me be part of your life. Once the acknowledgments are finished, I would like to stand out that I feel fortunate for having the chance to do what pleases me the most. I hope so for a long time and that the effort in the writing of this thesis proves it.

Contents

v

xvii

xix

xxi

Resumen

Abstract

Acknowledgments

Contents

List of symbols xxiii

Chapter 1 Introduction 1

1.1 Background of the thesis 1

1.2 Scope of research 2

1.3 Structure of the thesis 4

Chapter 2 Acoustic wave propagation in fluid media 5

2.1 Wave equation in fluids 5

2.2 Acoustic impedance 6

2.3 Plane waves in fluids. Reflection and absorption coefficients 7

Chapter 3 Perforated panels 9

3.1 Perforated panel absorber 9

3.2 Acoustic plane wave propagation in cylindrical tubes 10

3.3 Finite size effects 11

Chapter 4 Porous media 13

4.1 Macroscopic description of porous media 13

4.2 Physical parameters 14

4.3 Fluid equivalent to a porous material 15

xxii CONTENTS

Chapter 5 Complex porous media 17

5.1 Granular media 17

5.2 Double porosity media 18

5.3 Multilayered systems 19

Chapter 6 Prediction tools 23

6.1 Johnson-Champoux-Allard (JCA) approach for porous media 23

6.2 Horoshenkov-Swift model for granular media 24

6.3 Finite element models 24

Chapter 7 Porous media characterization 27

7.1 Non-acoustic properties 27

7.2 Acoustic properties 32

7.3 Inverse methodology 36

Chapter 8 Conclusions 39

8.1 Concluding remarks 39

8.2 Further work 40

8.3 Future research and perspectives 41

Appendices List of publications 43

43

53

67

77

A.1 A finite element model of perforated panel absorbers includingviscothermal effects

A.2 Acoustic properties of porous concrete made from arlite and vermiculite lightweight aggregates

A.3 Acoustic modeling of perforated concrete using the dual porositytheory

A.4 Modeling of grooved acoustic panels

A.5 Other publications 85

References 103

List of symbols

α Absorption coefficient

Estimated absorption coefficient

α∞ Tortuosity of a porous material

γ Ratio of specific heats

η Dynamic viscosity coefficient [Pa·s]

θi Pore shape factors i = 1, 2 in the Horoshenkov-Swift model for granular media [Horoshenkov and Swift 2001]

κ Heat conduction coefficient [W/(m·K)]

λ Second viscosity coefficient (Chapter 2) [Pa·s]

Λ Viscous characteristic length [m]

Λ’ Thermal characteristic length [m]

ρ (Complex dynamic) density of a/(an equivalent) fluid [kg/m3]

ρ0 Quiescent (air) density [kg/m3]

ρw Water density [kg/m3]

σ Flow resistivity of a porous absorbent material [N·s/m4]

σp Standard deviation of the pore size

ϕ Open area ratio or porosity

xxiv LIST OF SYMBOLS

ψ Fok function

ω Angular frequency [rad/s]

a Half-width of a slit [m]

A Complex amplitude of an acoustic wave propagating in positive direction [Pa]

b Distance between perforations in a perforated panel [m]

B Complex amplitude of an acoustic wave propagating in negative direction [Pa]

c0 Speed of sound in air [m/s]

Cp Specific heat at constant pressure [J/(kg·K)]

CF Cost function to be minimized in the Nelder-Mead direct search optimization method [Nelder and Mead 1965]

d Thickness of a perforated panel or porous material [m]

D Backing air cavity depth of a perforated panel absorber [m]

F Viscosity correction function used in the Horoshenkov-Swift model for granular media [Horoshenkov and Swift 2001]

g Gravitational constant [m/s2]

H Height to a reference position in the water suction method proposed by Leclaire et al. [Leclaire et al. 1998] [m]

H12 Complex transfer function measured in the procedure described in the ISO 10534-2:1998 [ISO 10534-2:98]

I Identity tensor used in the weak integral form of the FLNS equations

k Complex wave number [rad/m]

k0 Wave number in air [rad/m]

K Complex bulk modulus of an equivalent fluid [Pa]

L Distance between the porous sample and the rigid backing in the method proposed by Utsuno et al. [Utsuno et al. 1989] [m]

l0 Added length correction for a finite tube [m]

lm Microscopic characteristic size in a double porosity media [m]

lM Mesoscopic characteristic size in a double porosity media [m]

n Normal unit vector of a boundary [m]

N Number of layers of a multilayered system

LIST OF SYMBOLS xxv

NPr Prandtl number

p Acoustic pressure [Pa]

pc Pressure boundary condition [Pa]

P Pressure applied in the water suction method proposed by Leclaire et al. [Leclaire et al. 1998] [Pa]

P0 Atmospheric pressure [Pa]

PDF Probability density function of a log-normal distribution

rc Electric resistivity of a fluid-saturated porous material [Ω·m]

rf Electric resistivity of a fluid [Ω·m]

R Reflection coefficient; radius of a tube [m]

RS Surface resistance of a perforated panel [Pa·s/m]

s Microphone spacing in the procedure described in the ISO 10534-2:1998 [ISO 10534-2:98] [m]; pore size in standard units [m]

T Temperature [ºC]

[T] Transfer matrix of a porous layer or perforated panel

v Particle velocity vector [m/s]

v Particle velocity along one direction or through a porous material [m/s]

vc Velocity boundary condition [m/s]

V Normal velocity flow through a porous material [m/s]

Vf Fluid volume in the open pores of a porous absorbent material [m3]

Vt Total volume of a porous absorbent material [m3]

w Distance between the porous sample and the rigid termination in the flow resistivity test proposed by Ingard and Dear [Ingard and Dear 1985] [m]

x1 Distance from the microphone farthest to the porous sample in the procedure described in the ISO 10534-2:1998 [ISO 10534-2:98] [m]

x Vector with the physical parameters to be fit in the Nelder-Mead direct search optimization method [Nelder and Mead 1965]

Z Acoustic impedance boundary condition [Pa·s/m]

Z0 Characteristic impedance in air [Pa·s/m]

Zc Complex characteristic impedance [Pa·s/m]

xxvi LIST OF SYMBOLS

ZL Acoustic impedance of a close tube of thickness L [Pa·s/m]

Zpp Acoustic transfer impedance of a perforated panel [Pa·s/m]

Zt Acoustic transfer impedance [Pa·s/m]

ZS Surface impedance [Pa·s/m]

Chapter 1

Introduction

1.1 Background of the thesis

Noise pollution has become a problem of significant concern in modern society not only because of the harmful effects on the human health but due to the negative impact on the environment and its preservation [WHO 1999]. For these reasons, scientific community together with industrial partners and public authorities are making efforts on the development of solutions that reduce noise and help improve the well-being of people. Some good examples are protection against machinery noise in industrial areas, reduction of traffic noise in urban and interurban areas, and noise isolation in buildings. In this context, the use of porous absorbent materials has become one of the most extended passive solutions for noise control so far. Although there exist numerous types of porous absorbent materials, and many other new ones are developed each passing day under the name of metamaterials [Cummer et al. 2016], they all share the same loss mechanism principle: thermos-viscous dissipation (i.e. viscous and thermal damping of acoustic waves). The acoustic waves attenuate due to viscous friction and heat conduction when propagating through porous absorbent media. Thus, materials ranging from rock wool, fibres, granular media, perforated plates, are good candidates on the development of sound absorption devices. Among these, two types of materials are widely used in engineering practice: perforated panels and granular media. The first type can be found in diverse applications such as mufflers or resonator systems, these latter frequently used in room acoustics or as part of a noise barrier. The second is commonly used in the manufacturing of concrete structures or pavement surfaces, the use of particles of natural origin or resulting from recycling processes being an increasing trend in this regard. Therefore, the study of the acoustic properties of these materials constitutes a research field of great interest, both from the engineering design point of view and also to adopt sustainable measures against noise.

2 INTRODUCTION

1.2 Scope of research

The main aim of this thesis was to make a series of contributions to the study of the acoustic properties of porous materials, more specifically perforated panels and granular media. As mentioned above, the use of these materials in buildings and civil engineering is widespread. For this reason, the choice of this scope of research is justified regarding the practical applicability and of expected social impact. Hence, once the research framework was settled, a comprehensive analysis of state of the art concerning modelling and characterisation of porous materials was carried out in the first place. Thus, while the fundamentals of porous acoustics were being assimilated, four different case studies were analysed, to list:

Case study I. Finite element modelling of perforated panel absorbers

Perforated panel absorbers have become an environmentally friendly alternative to fibres and foams when used as sound absorbers in noise control applications because of their higher durability and enhanced low-frequency absorption. There exist many analytical models in the literature that let predict their acoustic properties, allowing the design and analysis of acoustic devices that contain them; however, these are not valid in the case of not homogeneously distributed perforations. Although some authors have derived modified formulations from experimental data on inhomogeneous samples, this procedure is far from generic, with the added disadvantage of requiring the prior manufacturing of the design to analyse. For this reason, we explored an alternative modelling technique that consisted in a finite element model that implements the linearised Navier Stokes equations in the frequency domain. The proposed methodology was found to give a better estimation of the absorption performance of these resonator systems than existing models when compared to experimental results. Besides, it was helpful for the analysis of the sound field in the perforated panel and to better understand the roles of finite size and interaction effect.

Case study II. Acoustic properties of lightweight porous concrete made from sustainable materials

The increasing social awareness on the preservation of the environment has prompted the building and civil engineering industries towards the use of ecological or sustainable materials, also called “green materials”, in the manufacturing of lightweight porous concrete. This widespread practice motivated the development of porous concrete made from arlite and vermiculite lightweight aggregates. These materials are not only recyclable and of natural origin, but are also lightweight, have adequate structural strength, and offer good thermal performance among other interesting features. Consequently, the characterisation of this type of material to assess its acoustic properties was considered of great interest. Preliminary results showed its good sound absorption capacity, making them a promising sustainable alternative to traditional solutions. Additionally, a simple model for granular media together with an inverse methodology was used to predict the acoustic

INTRODUCTION 3

properties of the prepared samples. In general, measured and estimated values of sound absorption and acoustic impedance showed a good agreement and confirmed that this model could be successfully used to design this type of porous concrete.

Case study III. Perforated concrete: modelling and analysis

The two previous case studies raised up the following question: what if we mix the concepts of perforations and porous concrete? The resulting media, herein referred as perforated concrete, can be considered a dual or double porosity material as it is composed of two porous networks of different characteristic sizes: that of the granular host media and that of the mesoperforations. Insomuch as there was no previous research relating to the acoustic modelling of such systems, we proposed the application of a double porosity approach to study their acoustic properties. Experimental characterisation data of different non-perforated specimens along with mesopore size data was incorporated in the model and then used to predict the acoustic behaviour of perforated concrete samples. These predictions were compared to experimental sound absorption results, obtaining a good correlation. In particular, it was found that the peak sound attenuation frequency could be tuned depending on the design constraints by using different perforation radii. This preliminary study was primarily a seed for the conception of new concrete-based devices for noise reduction, being nowadays an ongoing research topic in the framework of the Cost Action DENORMS [DENORMS].

Case study IV. Modelling of grooved acoustic panels

The last case study dealt with the modelling of grooved acoustic panels, this type of perforated panels consisting of an interconnected network of periodically arranged slits and circular holes. The decorative and aesthetic effect of a panel of this type turns it into an attractive choice for multiple indoor applications (workspaces, gymnasiums, auditoriums, etc.) in which a positive visual impact is as significant as having an acoustically effective system. The lack of work concerning the acoustic modelling of these devices and potential applicability of predictive models in the design stage thereof motivated this research. Using the classical Johnson-Champoux-Allard (JCA) equivalent fluid model, the acoustic wave propagation in the slits and circular holes regions was described, and then the Transfer Matrix Method employed to derive the acoustic properties of the whole absorber. A Finite Element (FE) procedure was used to verify the utility of the model, the comparison between numerical and theoretical results for the sound absorption coefficient showing that the proposed approach provides right predictions. Furthermore, a parametric study served to investigate the influence of the geometrical parameters of the panel on its acoustic behaviour and discuss the limits of the model. The four case studies above described constituted the central axis of the research work carried out throughout the development of the current thesis, all of them resulting in papers published in indexed scientific journals (see Appendices A.1 to A.4). Accordingly, the subsequent chapters are focused either on setting the theoretical background necessary

4 INTRODUCTION

to undertake these case studies or on extending some of the outcomes derived from the entire research work.

1.3 Structure of the thesis

The structure of this thesis follows to a great extent a work plan developed to progressively assimilate the fundamentals of acoustic wave propagation in porous media simultaneously to the analysis of different case studies, some of which constitute a novelty in the field of porous materials. In Chapter 2, fundamentals of acoustic plane wave propagation in fluid media are first reviewed, followed by the case of sound propagation in straight tubes as those that make up a perforated panel absorber in Chapter 3. A description of conventional porous media, its physical parameters, and the use of a fluid equivalent at the macroscopic scale are made in Chapter 4, to then deepen into the analysis of granular media, double porosity media, and multilayered systems in Chapter 5. A review of the simplified models usually chosen to study generic porous and granular media, along with the most extended numerical methods to explore more complex systems, is carried out in Chapter 6. In Chapter 7, those experimental techniques and methods that serve to determine the intrinsic physical parameters and the acoustic properties of porous media are briefly introduced, these being necessary to theoretically predict and experimentally assess the acoustic behaviour thereof, respectively. Thereby, once the background theory is presented, the four different case studies analysed are explained (Appendices A.1 to A.4). Finally, the main conclusions of this thesis along with further work and future research work are summarised in Chapter 8.

Chapter 2

Acoustic wave propagation in fluid media

2.1 Wave equation in fluids

An acoustic wave can be understood as a disturbance that propagates in a fluid (e.g. air) causing a pressure variation. This variation occurs due to the forward and backward movement of the fluid molecules from their equilibrium position along the direction of propagation, originating a series of compressions and expansions that affect the pressure, density and temperature of the fluid. The deviation from the ambient pressure is also called acoustic pressure and the speed with which these compressions and rarefactions occur, particle velocity [Morse and Ingard 1986]. The acoustic wave propagation in a fluid can be described from the Navier-Stokes equations, while the majority of problems in general acoustics can be studied using the well-known Helmholtz equation. Nevertheless, some important phenomena such as thermal conductivity and viscous friction are not accounted for by the latter. These viscothermal effects are especially significant and should not be neglected when studying the acoustic wave propagation through narrow geometries such as the porous network of absorbent material since they dampen and reduce the speed of the acoustic waves propagating through it. Therefore, these “loss mechanisms” must be regarded when analysing porous absorbent media. The Navier-Stokes equations can be linearised for small acoustic perturbations so that the resulting continuity equation, momentum equation, and entropy equation can be written as [Kampinga et al. 2011]

0 0j v (2.1)

0j p v v v (2.2)

0 pj C T j p T + (2.3)

where ρ is the density of the fluid, v the particle velocity vector, p the acoustic pressure, and T the temperature; ρ0 is the quiescent density, ω the angular frequency, λ the second viscosity coefficient, η the dynamic viscosity coefficient, Cp denotes the specific heat at

6 ACOUSTIC WAVE PROPAGATION IN FLUID MEDIA

constant pressure, and κ is the heat conduction coefficient. These equations can be solved for a given problem by prescribing the appropriate boundary conditions in each of the boundary regions. As mentioned above, for those cases in which the viscothermal effects can be neglected, if the fluid is assumed to be homogeneous, isotropic and perfectly elastic, the constitutive equation that relates pressure and density in the fluid can be expressed as

20p cr= (2.4)

where c0 is the speed of sound in air. This expression, combined with the previous Equations (2.1) and (2.2) for the case when the effects of dissipation are not taken into account (λ = η = 0), allows obtaining the acoustic Helmholtz equation [Kinsler et al. 1999]

2 20 0p k p + = (2.5)

where k0 = ω/c0 is the wave number in air. Usually, three different types of boundary conditions are used to solve the previous Helmholtz equation: the Dirichlet boundary condition, the Neumann boundary condition and the Robin (or impedance) boundary condition, these corresponding to the case when the acoustic pressure pc, the particle velocity vc, and the acoustic impedance (the ratio pressure/normal velocity) Z, are imposed on a boundary, respectively

cp p= (2.6)

0 c

pj v

nwr

¶= -

¶ (2.7)

c

c

pZ

v= (2.8)

where n represents the normal unit vector. The impedance boundary condition is useful when studying sound absorption problems, as it is frequently used to represent a porous absorbent material, the acoustic impedance being a concept that is worth recalling next.

2.2 Acoustic impedance

The acoustic impedance can be defined as the complex ratio of the acoustic pressure to the particle velocity in a fluid medium. As a rule, this parameter is used to study the transmission of acoustic waves between different media and their interaction in terms of sound reflection and absorption. For freely progressive plane waves, the acoustic impedance may be referred to as the characteristic impedance, being an intrinsic property of the medium itself.

ACOUSTIC WAVE PROPAGATION IN FLUID MEDIA 7

In fact, a fluid medium is usually described from its complex characteristic impedance, Zc, and wave number, k

cZ K (2.9)

kK

(2.10)

where ρ and K represent the complex dynamic density and bulk modulus of that media, respectively. These complex quantities let consider the dissipative processes resulting from viscosity and heat conduction in the fluid. On the other hand, the acoustic transfer impedance is used when studying the sound transmission through a specific finite fluid medium, this being defined as the ratio of the acoustic pressure drop across this fluid and the particle velocity through it, v, the latter being assumed to be constant

t

pZ

v

D= (2.11)

2.3 Plane waves in fluids. Reflection and absorption coefficients

Plane waves are those in which the main acoustic variables hitherto described (i.e. acoustic pressure and particle velocity) have constant amplitude and phase in all planes perpendicular to the direction of propagation. Let consider a harmonic plane wave propagating along the x-axis in fluid media 1 encounters a fluid media 2. In this case, a solution to the Helmholtz equation for the acoustic pressure at any point can be expressed as the superposition of two waves that propagate in opposite directions along that axis with different amplitude but the same wave number [Fahy 2000]

( ) jkx jkxp x Ae Be-= + (2.12)

where A and B are the complex amplitudes of the waves propagating in positive and negative directions, respectively. The particle velocity is given by

( ) ( )1 jkx jkx

c

v x A e B eZ

-= - (2.13)

Zc being the characteristic impedance in the corresponding propagation media.

8 ACOUSTIC WAVE PROPAGATION IN FLUID MEDIA

Thus, the acoustic impedance in the x position can be then derived from

( ) ( )( )

jkx jkx

c jkx jkx

p x A e B eZ x Z

v x A e B e

-

-

+= =

- (2.14)

According to the scheme of Figure 2.1, the acoustic impedances in the positions x = 0 and x = d can be related by the expression

( )( ) ( )

( ) ( )2 ,2

,2,2 2

co t0

cotc

cc

jZ d k d ZZ Z

Z d jZ k d

- +=

- (2.15)

which constitutes the basis of the Impedance Transfer Method (ITM) [Pierce 1981]; Zc,2 and k2 being now the characteristic impedance and the wave number in fluid 2, respectively. Thereby, the reflection coefficient for normal incidence plane waves in x = 0 can be defined as the ratio of the acoustic pressure of the waves propagating in the positive and negative direction in that position, which yields

( )( )( )

,1

,1

00

0c

c

Z ZR

Z Z

-=

+ (2.16)

Zc,1 being the characteristic impedance in fluid 1. The sound absorption coefficient α (0) is related to the reflection coefficient R (0) as follows

( ) ( )2

0 1 0Ra = - (2.17)

As it will be discussed in Chapter 4, a porous media may be replaced by an equivalent fluid, the impedance boundary condition commonly used to study its sound absorption performance implying it to be rigidly backed (i.e. Z (d) = ∞). The method for determining the acoustic impedance and the reflection and absorption coefficients in such scenario will be described in detail in Chapter 7.

Figure 2.1 Plane waves propagating along the x-axis in fluid media 1 encounter a fluid media 2.

0x = x d=

jkxBe

jkxAe-

x

Fluid 1 Fluid 2

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10 PERFORATED PANELS

0 0cot( )ppZ Z jZ k D= - (3.1)

where Zpp is the acoustic transfer impedance of the perforated panel, Z0 the characteristic impedance in air, and D the backing air cavity depth. Many authors [Maa 1987, Beranek and Vér 1992, Atalla and Sgard 2007] have developed impedance models to predict the acoustic behaviour of such devices, this being mainly determined by the size of the perforations, the open area ratio (i.e. ratio of perforated to total area), the panel thickness and the air cavity depth. In doing so, the perforations are considered to be narrow cylindrical tubes so that the underlying theory of sound propagation may be used, as will be shown below. When the perforations are reduced to the submillimetre size, higher acoustic resistance and broader absorption bandwidth are achieved, along with a positive visual for indoor applications. Maa studied these so-called Micro-Perforated Panel (MPP) absorbers and presented an approximate theory for their design [Maa 1975]. Since then, a lot of research has been carried out to study those solutions that improve their sound absorption features, as for instance the arrangement of multiple perforated layers [Maa 1987], the orifice design [Randeberg 2000], or the parallel combination of different MPP absorbers [Sakagami et al. 2009]. Likewise, similar expressions were developed for micro slit panels [Maa 2001], which despite offering less acoustic attenuation than the former, are sometimes preferred for aesthetics and for being less expensive. Next, the fundamentals of acoustic wave propagation in cylindrical tubes having a circular cross-section and in slits are briefly reviewed.

3.2 Acoustic plane wave propagation in cylindrical tubes

The acoustic plane wave propagation in cylindrical tubes is to some extent analogous to the plane wave propagation in a fluid bounded with rigid walls, the main difference laying in the fact that for narrow tubes, energy loss mechanisms associated to the viscous friction and heat conduction of the fluid cannot be neglected. Some pioneer works carried out by Kirchhoff [Kirchoff 1868] and Rayleigh [Rayleigh 1940] considered in a first approach these narrow tubes to have a uniform circular cross-section and to be of infinite length. Later on, Zwikker and Kosten [Zwikker and Kosten 1949] proposed a simplified model to study both viscous and thermal effects for such case. In general, when the size of the tubes is much smaller than the wavelength of the acoustic wave propagating throughout it, the effective density and bulk modulus of the inner air can be written as

0

1

0

21

J s j

s j J s j

(3.2)

PERFORATED PANELS 11

0

1 Pr

Pr 0 Pr

21 1

PK

J N s j

N s j J N s j

(3.3)

where s = R(ωρ0/η)1/2, R being the radius of the tube, and J0 and J1 are Bessel functions of the first kind and zeroth and first order, respectively, γ is the ratio of specific heats, P0 is the atmospheric pressure, and NPr is the Prandtl number; the attenuation associated with the viscous and thermal losses in a tube being frequency dependent and linked to its size. On the other hand, although this model assumes the tubes to have circular cross-section, other approaches can be found in the literature to deal with slits [Biot 1956] or tubes of arbitrary cross-sectional shape [Stinson 1991]. As a matter of fact, given that some perforated panels commonly used in practice exhibit a slit-like perforation shape, this approach is occasionally more suitable for prediction purposes. The effective density and the bulk modulus of the air in slits are defined by [Biot 1956]

0

tan h '1

'

s j

s j

(3.4)

0

Pr

Pr

tanh '1 1

'

PK

N s j

N s j

(3.5)

where s’ = a(ωρ0/η)1/2, a being the half-width of the slit. Both geometries, the circular tube and the slit, are depicted in Figure 3.2.

(a) (b)

Figure 3.2 Tube geometries: (a) circular; (b) slit.

3.3 Finite size effects

When the length of these tubes is much smaller than the acoustic wavelength in the air, thermal effects are considered negligible. Thus, one may determine the effective density of

2R 2a

12 PERFORATED PANELS

a single cylindrical tube from Equation (3.2) (Equation (3.4) for slits), and then use the open area ratio to determine the acoustic transfer impedance of a perforated panel as

1ppZ j dwr

f= (3.6)

where d is the panel thickness and ϕ the open area ratio. Nevertheless, the aforedescribed theory assumed the tubes to be of infinite length, which is not strictly true for the case of a perforated panel. As a result, the air inside the tubes behaves like an oscillating rigid piston whose total length is larger than the thickness of the panel. Moreover, the viscous dissipation due to the flow distortion at the front and rear surface of the panel must also be considered. Hence, additional corrections must be done to account for the finite size effects. Some authors [Rayleigh 1940, Ingard 1953, Beranek 1992] have suggested correction terms to account for these dissipative and inertial effects, also referred to as resistive and reactive end-corrections, respectively. For the former, Ingard [Ingard 1953] defined the surface resistance RS = (ηρ0ω/2)1/2, for the latter, Rayleigh [Rayleigh 1940] proposed the impedance of the panel be corrected with an additional length l0 = 2(8R/(3π)) for the case of circular tubes. The first correction must be applied twice to account for both the inlet and outlet surface effects. Further, if the tubes (or perforations) are close enough from each other (e.g. high open area ratio perforated panels), it is necessary to make use of Fok’s function in the reactance correction term due to the holes interaction effect [Rschevkin 1963]. The resulting acoustic transfer impedance of the perforated panel can then be rewritten

( )0 01

2p p S

lZ j d R j

w rw r

f y x

æ ö÷ç ÷= ç + + ÷ç ÷÷çè ø (3.7)

where ψ (ξ) is the Fok function, which is given by

3 5

16 7 8

1 1.40925 0.33818 0.06793

0.02287 0.03015 0.01641 ...

(3.8)

with ξ = 0.88(2R)/b, being b the distance between perforations. It should be noted that these expressions are valid for circular tubes, different corrections available in the literature being necessary for the case of slits [Randeberg 2000]. In general, the above expressions are reasonably accurate in most cases. In this regard, it is worth pointing out that these expressions are satisfied in the linear regime, where the sound pressure levels of the impinging wave are low, extra corrections being necessary otherwise [Melling 1973].

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14 POROUS MEDIA

models constitutes in most cases the preferred choice to study its acoustic properties. A summary of models for porous media can be found in [Attenborough 1982] or in the book by Allard and Atalla [Allard and Atalla 2009]. These approaches make a macroscopic description of the porous media that relies only on prior knowledge of their intrinsic physical parameters, which will be defined below. Again, as in the case of sound propagation in cylindrical tubes, the acoustic wavelength must be much larger than the characteristic size of the pores. Thus, this methodology is found to be not just a valuable tool for predicting the acoustic behaviour of porous media, but also to avoid the use of more complex modelling methods.

4.2 Physical parameters

When studying the acoustic wave propagation in porous media, some physical parameters are worth to be determined as these are directly linked to its acoustic behaviour. In fact, at least one or more of these parameters are used by prediction models to estimate their acoustic properties. Because of that, a brief definition of those being more frequently obtained is put forth next.

Porosity

The porosity ϕ is a parameter that provides information regarding the amount of fluid volume inside a porous absorbent material and is expressed as the ratio of the fluid volume in its open pores Vf and its total volume Vt

f

t

V

Vf = (4.1)

It must be remarked that only open porosity is considered for this calculation, since are the open pores those that intervene in the dissipation mechanisms when acoustic waves propagate through the material. Even though a good porous absorbent material usually has a high porosity (e.g. most fibrous materials have values very close to unity), there is another useful parameter related to sound attenuation: the flow resistivity.

Flow resistivity

The flow resistivity gives a measure of the resistance that an air-saturated porous media offers to steady flow of air. Therefore, it provides an idea of how much sound energy would be lost due to the microstructure of the pore network. Nevertheless, too high flow resistivity values may cause a significant impedance mismatch with the adjacent fluid, the acoustic energy transferred from this fluid being reflected back by the porous media, thus reducing its sound absorption capability.

POROUS MEDIA 15

The flow resistivity σ is defined as the ratio of the pressure differential, Δp, across a porous absorbent material of thickness d to the normal flow velocity V through it

p

Vds

D= (4.2)

Tortuosity

As noted above, the classical theory of sound propagation in narrow tubes can be applied to study uniform straight pores. Unfortunately, this is not the case for most porous media, whose complex pore geometry urges the need of additional parameters such as tortuosity. The tortuosity is an indicator of the geometrical complexity of a pore network, which relates the actual distance that an acoustic wave travels through a porous medium and the thickness of it (i.e. values larger than 1 imply dispersion of the acoustic wave). Represented by α∞, it represents thus the effect that the relative orientation of the pores to the incident sound field has on the wave propagation. In brief, the tortuosity of the propagation path plays a key role in the absorption features of the material.

Viscous and thermal characteristic lengths

The characteristic lengths of a porous material are related to the size of its pores, whose effects on the sound propagation and therefore on its absorption are evident. Different pore sizes involve different pore surfaces and volumes, its associated viscothermal effects being described from these. For the viscous effects, Johnson et al. [Johnson et al. 1987] defined the viscous characteristic length, which is related to the smallest size of the pores and is denoted as Λ. Similarly, a thermal characteristic length was proposed by Champoux and Allard [Champoux and Allard 1991] for the thermal effects, this latter being related to the largest size of the pores and represented by Λ’. Accordingly, these characteristic lengths also depend on the shape of the pores, which although not having a simple geometry are as a rule assumed to be circular or are fitted from experimental measurements. While the above parameters constitute the input of most prediction models, some authors have introduced additional ones such as mean pore size and its standard deviation for granular materials, which obey a log-normal statistical distribution [Horoshenkov and Swift 2001]. All the same, given the complex microstructure of porous media, this latter and most of the hitherto defined parameters cannot be directly determined. For this reason, these are habitually measured using the experimental procedures described in Chapter 7.

4.3 Fluid equivalent to a porous material

Using the previously defined physical parameters to study the sound propagation in porous absorbent materials on a macroscopic scale usually presents severe difficulties due to the complicated internal geometry of their pore network. Instead, an average is carried out over a representative elementary volume of the porous media, whose dimensions must be much smaller than the acoustic wavelength of interest. Hence, given that the solid skeleton

16 POROUS MEDIA

is typically assumed to be motionless, the porous material can be replaced on a macroscopic scale by an equivalent fluid described by a complex characteristic impedance and wave number, likewise by a complex dynamic density and bulk modulus. These properties can be either estimated using a prediction model as those described in Chapter 6 or experimentally determined using the measurement method described in Chapter 7. Thus, the acoustic impedance at normal incidence of a porous absorbent material of thickness d when backed by a rigid wall (this being a typically analysed configuration), can be obtained from

cot( )S cZ jZ kd=- (4.3)

which is also known as the surface impedance, ZS.

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COMPLEX POROUS MEDIA 19

11 1dp

M

m M

K

K K

(5.2)

where ϕM is the meso-porosity. Hence, one may obtain the effective properties of the mesoporous (ρM, KM) and the microporous (ρm, Km) mediums, and thereupon use a double porosity approach to predict the acoustic behaviour of the double porosity material. However, when the interscale ratio is low (lm/lM ≈ 10-3), pressure diffusion effect phenomena occur in the microporous medium due to the high permeability contrast, a pressure diffusion function being necessary in Equation (5.2). Based on previous works, several authors have developed double porosity media with enhanced sound absorption properties. An example is a meso-perforated material, which results from drilling a microporous medium. Apart from reducing the weight, Sgard et al. [Sgard et al. 2005] showed that a remarkable improvement of the absorption performance might be achieved if the meso-perforations are appropriately designed. Thus, for this double porosity material, ρm and Km can be obtained from a porous media model or experimental measurements on the microporous material; whereas ρM and KM can be calculated, for the case of mesopores having a circular cross-section, using Equations (3.2) and (3.3). It should be pointed out that, in the high permeability contrast situation, the pressure diffusion function accounts for the finite size effects discussed in Chapter 3 as well.

5.3 Multilayered systems

Multilayered systems consist of a series arrangement of multiple acoustic elements intended to achieve a broader absorption frequency band than single layer absorbers. Such systems may be composed either of air-spaced perforated panels or porous absorbent materials with different characteristics, the combination thereof being also a configuration usually found in many noise control applications (e.g. mufflers, isolation walls,…). These assemblies give rise to a wide range of possibilities that prove the potential of using appropriate layer combinations when designing a sound absorber. It is therefore of great interest to establish a methodology that helps to study the acoustic properties of these absorbers for the sake of their design and development. The absorption performance of a multilayered system can be predicted using the theory described so far [Dunn and Davern 1986]. Let consider for instance the multilayered system depicted in Figure 5.3, which is composed of N layers having different depths, di (i = 1...N).

20 COMPLEX POROUS MEDIA

Figure 5.3 Multilayered system composed of N layers. Under plane wave incidence, assuming each layer (e.g. perforated panel, air cavity, porous layer,…) to be laterally infinite, continuity of pressure and particle velocity exists at each layer interface. Thus, the absorption properties of the whole system can be derived by progressively using the ITM (see Equation (2.15)). An extended alternative and still plane wave based is the Transfer Matrix Method (TMM). In this method, each layer is represented using a generic transfer matrix that relates the sound pressure and particle velocity at the upstream (Mi) and downstream (Mi’) of the layer

( )( )

[ ]( )( )

( )( )

' '

,11 ,12

' ',21 ,22

i ii ii

ii ii i i

p M p Mt tp MT

t tv M v M v M

é ù é ùé ù é ùê ú ê úê ú ê ú= =ê ú ê úê ú ê úê ú ê úë ûë û ë û ë û (5.3)

where [T]i is the transfer matrix corresponding to the ith layer [Mechel 2008]

[ ]( ) ( )

( ) ( )

,

,

cos sin

sin cos

i i c i i i

ii i i i

c i

k d jZ k d

T jk d k d

Z

é ùê úê ú= ê úê úë û

(5.4)

where Zc,i and ki are the complex characteristic impedance and the wave number in the ith layer, respectively. By multiplying the individual transfer matrices, the overall transfer matrix of the multilayered system can be obtained

[ ] [ ] [ ] [ ] 11 12

1 221 22

...M N

t tT T T T

t t

é ùê ú= = ê úë û

(5.5)

The surface impedance of the whole absorber is then calculated as ZS = t11/t21.

1 2 N

( )( )

1

1

p M

v M

ì üï ïï ïí ýï ïï ïî þ

( )( )

'

'N

N

p M

v M

ì üï ïï ïí ýï ïï ïî þ. . . . . . .

COMPLEX POROUS MEDIA 21

An analogous matrix to that of Equation (5.4) may be retrieved for thin perforated panels, where the upstream and downstream particle velocities are assumed to be the same [Wu 1997]

[ ]1

0 1pp

pp

ZT

é ùê ú= ê úë û

(5.6)

where Zpp is the acoustic transfer impedance of the perforated panel defined in Chapter 3. This latter matrix has been used to estimate the sound absorption performance of multiple layer perforated panel systems [Lee and Kwon 2004], showing a good agreement when compared to experimental results. In summary, both methods can be applied in designing multilayered systems, the TMM greatly simplifying their analysis since it also allows for the calculation of acoustic field quantities.

Chapter 6 Prediction tools

6.1 Johnson-Champoux-Allard (JCA) approach for porous media

In 1987, Johnson et al. [Johnson et al. 1987] proposed a semi-phenomenological model to study the dissipative visco-inertial effects in fluid-saturated rigid porous media with pores of arbitrary cross-section. This model describes the acoustic behaviour of porous media in the whole frequency range from its asymptotic limits at low and high frequencies. The resulting expression for the complex dynamic density is given by

20 0

2 2 20

41 1JCA

j

j

(6.1)

Based on previous work, Champoux and Allard [Champoux and Allard 1991] described the dissipative effects occurring because of the thermal exchanges in the fluid filling this porous media, an expression for the complex bulk modulus being derived

11

20 P r 0

2P r 0

8 '1 1 1

' 16JC A

P j NK

j N

(6.2)

On the whole, the air in the porous media is replaced by an equivalent fluid whose effective properties are obtained from Equations (6.1) and (6.2). The physical parameters used in these expressions are those previously defined in Chapter 4, viz., porosity, flow resistivity, tortuosity, viscous and thermal characteristic lengths. Thus, the so-called Johnson-Champoux-Allard (JCA) approach fully describes the dissipative viscothermal effects in porous media and let successfully predict its acoustic properties. Unlike well-known empirical models [Delany and Bazley 1970, Voronina and Horoshenkov 2003], this still simple model is more refined from the microstructural point of view, apart from applying to most porous absorbents. Even though some authors have shown the limitations of the above expressions and have suggested improved formulations [Pride et al. 1993, Lafarge 1993], the need for additional parameters turns the JCA approach into the up to date most widespread model to study the absorption performance of rigid porous media.

24 PREDICTION TOOLS

6.2 Horoshenkov-Swift model for granular media

Granular media have pores of a different shape whose size typically obeys a particular statistical distribution. Although the variation of the pore shape does not seem to be that important [Attenborough 1993], the pore size distribution has a noticeable effect on the acoustic properties of the material. By assuming a simple pore shape and a pore size distribution close to log-normal (often found in granular materials), Horoshenkov and Swift [Horoshenkov and Swift 2001] derived a simple model for the prediction of the acoustic properties of consolidated and nonconsolidated granular media. The model is based on four physical parameters: porosity, flow resistivity, tortuosity and pore size deviation; all of them being measurable by following easily reproducible experimental tests. The expressions proposed for the complex dynamic density and bulk modulus of the fluid equivalent to the granular media can be written as follows

0H S j F

(6.3)

1

00

0 P rP r

1H S

PK

j F NN

(6.4)

where F (ω) stands for a viscosity correction function defined by

2

1 2

1

1

1

a aF

b

(6.5)

with a1 = θ1/θ2, a2 = θ1, and b1 = a1, being θ1 = (4/3)e4ξ – 1 and θ2 = e3ξ/2/(√2) when circular pore shape is assumed, where ξ = (σpln2)2 and σp is the standard deviation in the log-normally distributed pore size, and ε is a dimensionless parameter. Horoshenkov and Swift showed that this practical model yields satisfactory results when compared to experimental data. Moreover, some recent works proved this model to be trustworthy to predict the acoustic properties of different sort of concrete-based granular media made from alumina cement [Horoshenkov et al. 2003], expanded clays [Asdrubali and Horoshenkov 2002, Vasina et al. 2006] or hemp [Glé et al. 2013].

6.3 Finite element models

The Finite Element Method (FEM) is used to numerically solve partial differential equations as those governing the acoustic wave propagation in fluid media. Given a specific boundary value problem, a weak form of these partial differential equations is derived including the boundary conditions to be imposed. This weak form is then discretised by subdividing the problem domain into finite elements and assembled into a

POROUS MEDIA CHARACTERISATION 25

system matrix whose numerical solution yields approximate values of the acoustic field variables at discrete points over this domain. The use of this method for the analysis of acoustic problems is becoming each passing day a more usual practice among acoustic engineers. In fact, the above models for porous and granular media can be easily incorporated into a finite element procedure by describing the porous domain as an equivalent fluid in terms of its effective properties. Thereby, it is straightforward to analyse the acoustic field in the porous medium by numerically solving the Helmholtz equation in that domain, which is of great interest for the better understanding of its acoustic behaviour. Besides, it enables taking into account the finite geometry effects and let analyse the influence of a porous material in real noise control systems. Many works in the literature make use of an equivalent fluid approach for the analysis of simple or complex porous media such as multilayered systems [Panneton and Atalla 1996] or double porosity media [Sgard et al. 2005]. In these works, the absorption performance of the porous material was assessed by following a power balance method under plane wave incidence conditions, as in the TMM (see Chapter 5). Figure 6.1 depicts the corresponding general scheme, which consists of a rigidly backed porous material inserted into a rigid walls waveguide. The fluid in the waveguide was modelled as lossless air and the porous material as an equivalent fluid following the JCA approach. A plane incidence pressure wave was imposed at the opposite side of the waveguide, the weak integral forms associated with the coupled waveguide and porous media domains being discretised using finite elements and solved for the acoustic pressure and particle velocity. It was straightforward then to compute the normal incidence absorption coefficient from the power dissipated through viscous and thermal effects. As will be seen in Chapter 7, this scheme resembles the impedance tube setup commonly used to determine the sound absorption of porous materials experimentally.

Figure 6.1 Numerical scheme of a rigidly backed porous material in a waveguide. Unfortunately, the equivalent fluid approach used in this procedure presents some limitations, especially for the analysis of some complex porous media (e.g. perforated panels whose tubes have a non-uniform cross-section or present a heterogeneous distribution thereof). While some authors [Bolton and Kim 2010] have proposed an alternative but still finite element based simplified CFD model for these cases, a Full Linearized Navier Stokes (FLNS) formulation is required to capture both viscous and thermal dissipation effects in the fluid media. For some time, the finite element implementation of this formulation was unfeasible due to the high computational cost

Waveguide

Porous material

Rigid wall

Plane wavecondition

26 PREDICTION TOOLS

associated with the high-density mesh required to appropriately resolve the dampening viscous and thermal boundary layers [Morse and Ingard 1986]. Nonetheless, high development of computers over past decades has paved the way for such implementations and nowadays becomes possible to solve models with a large number of degrees of freedom. Kampinga et al. [Kampinga et al. 2011] proposed an efficient finite element model for the analysis of viscothermal acoustics. In their model, the used particular set of equations lead to a complex symmetric finite element system matrix that speeded up the calculation time and reduced the memory usage. The weak integral form of the fluid governing equations (Equations (2.1), (2.2) and (2.3)), reads

0 0

, , , 0vw w w

j jp p p p T

p T

(6.6)

0 , 2 , , ,

, 2

v v v v v

v v I I n

w w w w

w

j p

p

(6.7)

0 , , , , np w w w wj C T T T T j T p T T

(6.8)

where vw, Tw and pw are weighting functions, ε = 0.5( v + ( v)T) is the symmetric part of the velocity gradient, εw is defined like ε by replacing v by vw, and I is the identity tensor. The discretised form of Equations (6.6) to (6.8) lead to the following system matrix

1,1 1,3

2 ,2 2 ,3

3 ,1 3 ,2 3 ,3

0

0

0

fvM M

M M

M M M

T Q

p

ì üï ïì üé ù ï ï ï ïï ï ï ïê ú ï ï ï ïï ï ï ïê ú =í ý í ýê ú ï ï ï ïï ï ï ïê ú ï ï ï ïê úë û ï ï ï ïî þ ï ïî þ

(6.9)

where v

, T

and p

are the vectors of the nodal values; Mi,j the sub-matrices entries and

the right-hand side of Equation (6.9), which contains the natural boundary conditions, are derived from Equations (6.6) to (6.8). For the three-dimensional implementation, five degrees of freedom per node were necessary: one for the acoustic pressure, one for the temperature, and three for the particle velocity components. The former system matrix can be solved using COMSOL Multiphysics® finite element modelling software. Contrary to the equivalent fluid approach, the FLNS model is generic and can handle any arbitrary geometry. In most cases, this methodology may not only result in a much more accurate approach but also serve to explain underlying phenomena (e.g. holes interaction effect in the case of perforated panels). Additionally, it can be used to cross-check the adequacy of applying a simplified model to study a specific porous material. To sum up, both methodologies are complementary, the application of one or the other depending on the complexity of the porous material to be analysed and the accuracy desired on the predictions.

Chapter 7 Porous media characterisation

7.1 Non-acoustic properties

As indicated in the previous chapter, most prediction models used to describe the acoustic wave propagation in porous absorbent media make use of physical parameters such as porosity, flow resistivity, tortuosity, and characteristic lengths. Hence, the determination of these so-called non-acoustic properties constitutes a key point to address porous media characterisation. For this purpose, several methods and experimental techniques have been developed over time, their applicability depending on the nature of the porous material to be characterised. A synthetic review of these methods can be found in the book by Cox and D’Antonio [Cox and D’Antonio 2009], some of them being described below, with particular emphasis on those performed during the development of this thesis.

Porosity test

One of the most straightforward experiments to measure the porosity of a porous material is the water saturation method, which uses the total volume of water left in the sample after soaking, Vw, and the total volume of the dry sample, Vt, to derive it as ϕ = Vw/Vt. The main drawback of this method is that both the open and closed pores may be saturated, only the open pores being accessible to the acoustic waves propagating through the material. Furthermore, it is not guaranteed that the water fills all the open pores, which may result in discrepancies when compared to other methods. To cope with these issues, Beranek [Beranek 1942] had previously proposed a method based on the isothermal compression of the air within and external to a sample introduced in a chamber connected to a manometer. This method was improved by Champoux et al. [Champoux et al. 1991] and Leclaire et al. [Leclaire et al. 2003], the use of a controlled reference chamber to reduce the influence of temperature and atmospheric pressure changes on the results being the main novelty. Afterwards, by Panneton and Gross [Panneton and Gros 2005] a much simpler method which involved the weighing of the material sample both in air and under vacuum conditions, the porosity being determined from the mass difference. Among these methods, the water saturation method is probably the least used as considered too intrusive (or even destructive) to characterize conventional porous materials (e.g. fibres, foams…). However, this is not the case of granular media, whose consistency makes it a simple and suitable approach to characterise them. In fact, Vasina et al. [Vasina

28 PO

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POROUS MEDIA CHARACTERISATION 31

Characteristic lengths test

The viscous and thermal characteristic lengths were shown to be parameters necessary for the description of porous media in the JCA model. Even though obtaining them is somehow complex and presents some uncertainties, there exist a few methods to determine them. Leclaire et al. [Leclaire et al. 1996] proposed a method to determine these parameters from the high-frequency asymptotic behaviour of ultrasonic waves when a porous material is saturated with air or with helium. Measurements with two different gases enabled to deduce both parameters from the propagation velocity experimental data. On the other hand, since the thermal characteristic length is directly linked to the specific area and volume of the pores, by application of the Brunauer-Emmett-Teller (BET) theory [Henry et al. 1995], it is possible to determine their specific area and then calculate the thermal characteristic length. Moreover, there also exist optical methods to estimate this parameter from the microstructural analysis of the pore network using a Scanning Electron Microscope (SEM). In this thesis, none of the above methods was used to estimate these parameters. Instead, they were deduced utilising an inversion methodology as that used by Panneton and Olny to determine the viscous [Panneton and Olny 2006] and thermal [Olny and Panneton 2008] characteristic lengths of porous materials from knowledge of its porosity, flow resistivity and experimental data of the complex dynamic density and bulk modulus. This inversion technique will be described at the end of this chapter.

Mean pore size test

The mean pore size and its standard deviation are physical parameters that give an insight of the pore network of a porous material. Following the water suction method proposed by Leclaire et al. [Leclaire et al. 1998], these parameters may be determined by theoretically fitting the measured cumulative pore size distribution to a log-normal distribution with probability density function

( ) ( ) ( )2 221

2p

p

PD F ej j s

js p

- -= (7.3)

where φ = -log2s, with s being the pore size in standard units determined from the experiment, and the median value is calculated as <φ> = -log2<s>. This non-acoustic method uses the experimental setup shown in Figure 7.5, which consists in determining the volume of water extracted by capillary action when applying a pressure drop to a water-saturated sample. The applied pressure, P, is obtained from the height, H, to a reference position as P = ρwgH, where ρw is the water density, and g is the gravitational constant. It is then straightforward to determine both the mean pore size and its standard deviation from the cumulative volume of water extracted data. This procedure was used by Horoshenkov and Swift [Horoshenkov and Swift 2001] to measure the standard deviation of pore size and predict the acoustic properties of granular materials,

32 POROUS MEDIA CHARACTERISATION

whose pore size distribution is close to log-normal. Accordingly, it was justified the choice of this method to characterise the granular materials studied in this thesis.

Figure 7.5 Mean pore size test: schematic of the method for the measurement of the pore size of a porous material proposed by Leclaire et al. [Leclaire et al. 1998].

7.2 Acoustic properties

Once the most widespread procedures for the determination of the non-acoustic properties of porous media have been reviewed, it turns out reasonable to describe the methods used to measure their acoustic properties. These methods are essential both to assess the acoustic behavior of porous materials and to validate the models described in Chapter 6. In what follows, a description of those methods used to determine the sound absorption coefficient of a porous material and to evaluate its intrinsic acoustic properties, namely complex characteristic impedance and wave number, is given.

Sound absorption coefficient and acoustic impedance test

One of the pioneer experimental techniques used to determine the sound absorption coefficient of porous materials under normal incident plane waves was the Standing-Wave-Ratio (SWR) method. In this method, the sound absorption coefficient of a porous sample placed at the end of the impedance tube can be determined from the standing wave pattern resulting from exciting the system at different frequencies using a speaker positioned on the other end of the tube. The main drawbacks of this method are its tedious measurement procedure (the microphone was manually moved and only a discrete frequency was evaluated at a time) and start up limitations (minimum wavelength and microphone mechanism limited on the tube dimensions). To cope with these limitations, Seybert and Ross [Seybert and Ross 1977] proposed an improved technique in which the reflection coefficient, and thus the sound absorption coefficient, was obtained from the frequency spectrums of two microphones flush-mounted in the tube for a random excitation signal. A detailed description of this technique, also known as the “transfer function method”, can be also found in the works by Chung and Blaser [Chung and Blaser 1980a, Chung and Blaser 1980b]. Even though both methods were adopted for their standardization [ISO 10534-1:96, ISO 10534-2:98], the transfer function method is nowadays by far the most used to

Sample

H

BuretteGraduatedtest tube

POROUS MEDIA CHARACTERISATION 33

determine the sound absorption performance of porous materials. Figure 7.6 shows the experimental arrangement established in the standard ISO 10534-2 [ISO 10534-2:98] for the determination of sound absorption coefficient and acoustic impedance in impedance tubes.

Figure 7.6 Experimental arrangement for the measurement of the sound absorption coefficient and acoustic impedance according to the standard ISO 10534-2 [ISO 10534-2:98].

From the acoustic pressure data measured in the two microphone positions, p1 and p2, spaced a distance s, the complex transfer function H12 = p2/p1 can be obtained. The reflection coefficient can then be related to this transfer function by

0

0 1

0

212

12

jk sj k x

jk s

H eR e

e H

--=

- (7.4)

where x1 is the distance from the microphone farthest to the porous sample. Evaluating the acoustic impedance or the sound absorption coefficient is then straightforward by substituting Equation (7.4) into Equations (2.16) and (2.17), respectively. It should be mentioned that this method requires a calibration process to correct possible phase mismatches on the microphonic responses, which consists on performing a measurements twice with the microphone positions exchanged. Further, the spacing of these microphone positions and the tube diameter determine the valid measurement frequency range to ensure plane wave propagation in the tube. For the implementation of the current method, two circular-cross section stainless steel tubes were set up: a small one for granular samples and a large one for perforated panel absorbers. The tube characteristics of the small one are those specified for the flow resistivity test in the previous section, the largest one having a thickness of 6 mm and an inner diameter of 100 mm, the microphones being spaced 85 mm (cut-off frequency around 1800 Hz). The acquisition system, speaker and post-processing software used for the experiments were the same in both experimental rigs. Some pictures of the impedance tubes and their respective appliances are depicted in Figure 7.7.

Speaker Sample

1xs

1p 2pRigidpiece

34 PO

Figure(b) Det

Charac

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POROUS MEDIA CHARACTERISATION 35

measurement procedure, this improved version of the two-cavity method adopted the transfer function method [Seybert and Ross 1977, Chung and Blaser 1980a, Chung and Blaser 1980b]. Figure 7.8 shows a schematic diagram of the measurement method proposed by Utsuno et al. [Utsuno et al. 1989]

Figure 7.8 Experimental setup for the measurement of characteristic impedance and wave number proposed by Utsuno et al. 1989 [Utsuno et al. 1989].

The porous sample of thickness d to be characterized is placed in the impedance tube as shown in previous figure. By moving the rigid piece behind the sample, the surface impedance is measured twice: with a backing air cavity of depth L and a backing air cavity of depth L’. The theoretical acoustic impedances for both backing conditions correspond to those of two closed tubes of thicknesses L and L’, respectively

( )0 0co tLZ jZ k L= - (7.5)

( )' 0 0cot 'LZ jZ k L= - (7.6)

which resemble Equation (4.3) for a porous absorbent material backed by a rigid wall. Solving Equation (2.15) for both termination conditions, and after some tedious mathematical derivations, yields the complex characteristic impedance and wave number

( ) ( )( ) ( )

1/ 2' ' ' '

' '

L L L L

c

L L

Z Z Z Z Z Z Z ZZ

Z Z Z Z

æ ö- - - ÷ç ÷ç= ÷ç ÷ç ÷- - -çè ø (7.7)

( )( )( )( )

1lo g

2c L c

c L c

Z Z Z Zk

d Z Z Z Z

æ ö+ - ÷ç ÷= ç ÷ç ÷÷ç - +è ø (7.8)

where the sign in Equation (7.7) must be chosen so as to let the real part of Zc be positive. Therefore, from the surface impedance measurement data for both termination conditions, Z and Z’, it was possible to obtain the acoustic properties of the porous material under test. In general, the two-cavity method was proven to be more advantageous than the two-thickness method as it does not require two samples of porous material to be measured, thus avoiding its associated uncertainty. Although some authors have recently proposed additional methods that let further simplify the measurement procedure (e.g. the three microphone method proposed by Salissou and Panneton [Salissou and Panneton

Speaker

Sample

1xs

1p 2p

Movablerigid piece

d (')L

Aircavity

36 POROUS MEDIA CHARACTERISATION

2010] and the transfer-matrix approach proposed by Song and Bolton [Song and Bolton 2000]), the two-cavity method is still a useful and straightforward procedure that can be easily implemented in any basic laboratory impedance tube setup. The method developed in this thesis was the one worked out by Utsuno et al. [Utsuno et al. 1989]. The measurements were performed using the custom made tube described in previous section for the flow resistivity test. Thereby, the acoustic properties of a porous material were obtained by just moving the rigid piece that lets achieve the backing air cavity, making it a versatile impedance tube to study porous media.

7.3 Inverse methodology

Experimental methods for the measurement of the physical parameters of porous absorbent materials are sometimes difficult to carry out (e.g. to guarantee the complete saturation of the sample in the methods involving water) or may present a high uncertainty for some type of materials. As an alternative, the use of inverse characterization methodologies has become very popular among acousticians community. These methodologies not only reduce the duration of the characterization process, but also let cope with limitations in laboratory equipment availability. In brief, an optimization procedure is used to minimize a cost function defined as the difference between a measured and an estimated acoustic parameter (e.g. the sound absorption coefficient). Since the estimated acoustic parameter is usually calculated from a prediction model, which in turn depends on a set of physical parameters, it is then possible to obtain the values of these parameters that best fit the measured data to the theoretical model. Besides, this methodology can be likewise used to derive new empirical models [Ramis et al. 2014] starting from a generic existing one [Delany and Bazley 1970]. The Nelder-Mead direct search optimization method [Nelder and Mead 1965] is a technique that lets minimize the aforementioned example cost function, which can be defined as

( ) ( ) ( )1

,x xN

i ii

CF f fa a=

= -å (7.9)

where α(fi) and (fi) correspond to the measured and estimated values of the sound absorption coefficient for each frequency of interest fi, respectively, and x is a vector with the physical parameters to be fit. For this purpose, an initial estimate followed by successive iterations is carried out until a desired tolerance is reached. While this technique may present some discontinuities or convergence problems, in most cases it constitutes a handy methodology to estimate the physical parameters of a porous material within an acceptable tolerance range. Numerous authors [Asdrubali and Horoshenkov 2002, Vasina et al. 2006, Geslain et al. 2012, Glé et al. 2013] have used these strategies to derive the physical parameters of a porous material from measurement data in an impedance tube. Fellah et al. [Fellah et al. 2007] used a similar procedure for ultrasonic characterization of porous absorbing material

POROUS MEDIA CHARACTERISATION 37

from pulse reflection and transmission data in time domain. These works lead to the development of an inverse method in the current thesis to obtain the physical parameters of granular materials for which some existing techniques present severe applicability difficulties.

Chapter 8 Conclusions

8.1 Concluding remarks

The objective of this thesis was to contribute to investigating the acoustic properties of porous materials intended for use in building and civil engineering industries. For this purpose, a comprehensive study of state of the art regarding porous media, specifically perforated panel absorbers and granular media, was previously carried out. A subsequent point was the review of the fundamentals of acoustic wave propagation in perforated panels and porous media, along with an introduction to complex porous media. These preliminary tasks also included an in-depth description of the existing modeling approaches and characterization methods that let assess the acoustic behavior of porous media. It should be noted that all this background knowledge was acquired as progress was made during the development of the thesis. Hence, once the fundamentals were settled for each of the four different case studies analyzed; the results and detailed discussion of which can be found in Appendices A.1 to A.4, the following general conclusions were outlined:

A finite element model that implements the Full Linearized Navier-Stokes (FLNS) formulation allows the prediction of the absorption performance of heterogeneous perforated panel absorbers for which a general analytical approach might not be suitable.

Porous concrete made from arlite and vermiculite lightweight aggregates can become a sustainable alternative in terms of sound absorption to other solutions commonly used in practical applications.

The acoustic properties of perforated concrete can be predicted using a double porosity approach based upon the homogenization theory of periodic structures provided the wavelength criteria and interscale ratio are taken into account.

Grooved acoustic panel absorbers can be easily modeled using porous media existing theory by conceiving it as a multilayer system in which the different regions are described using an equivalent fluid approach.

40 CONCLUSIONS

Apart from these conclusions, additional remarks were drawn:

The main advantage of the FLNS approach is that it is generic and it can be used to study any geometry case without the need for fitting procedures or modified formulations. Besides, the use of a finite element methodology lets illustrate physical phenomena occurring in the vicinities of the panel (e.g. holes interaction effect) for a better understanding of their acoustic behavior. Additionally, it can be used to propose optimized perforated absorber systems when no analytical solution exists, or to derive new correction terms.

Experimental procedures widely used in the field of porous media characterization can be used to assess the non-acoustic and acoustic properties of lightweight porous concrete. Moreover, predictions for the sound absorption coefficient using a theoretical model that relies on the experimentally determined physical parameters showed a good agreement with the measurements.

The peak absorption frequency of perforated concrete can be tuned depending on the design constraints by adequately choosing the mesopore size. This particular feature offers a wide range of possibilities relating to the applicability of such solutions to different noise control devices commonly used in civil engineering (e.g. noise barriers, isolation walls,…).

The combination of JCA (Johnson-Champoux-Allard) model and TMM (Transfer Matrix Method) have shown to be a potential tool to study the acoustic behavior of grooved acoustic panels for which single pore geometry approaches fail. A parametric study of grooved acoustic panel absorbers was found very useful for the better understanding of the role of the geometrical characteristics of the panel on its sound absorption performance and may serve to establish a guide for their practical design.

In summary, the analyzed study cases are intended to serve not only as a contribution to the study of the acoustic properties of porous media but also to encourage the use of some of the proposed approaches in real engineering problems. All the same, further work has still to be conducted to settle down the presented models and methodologies, and why not, to make them extensive for their applicability in a multidisciplinary framework.

8.2 Further work

The author considers that is worth pointing out that the current thesis was carried out while being a member of the Applied Acoustics Group of the University of Alicante. This fact made possible to enrich the thesis contents with the experience acquired on the day-to-day research work developed with many industry partners, technology centers, and research groups from different institutions. This work also included a support role in dozens of undergraduate final projects and activities linked to the academic framework. Additionally, I was given the opportunity to perform several stays of research at two worldwide

CONCLUSIONS 41

* This research work was awarded the Best Paper and Presentation Award by the European Acoustics Association (EAA) in the EuroRegio2016 conference held on 11th and 12th of June of 2016 in Porto (Portugal).

recognized foreign universities: the University of Southampton and Universidade de Coimbra, with the last one of which close friendly work collaboration has been consolidated. Apart from the fruitful discussions held with other scientific community members, I also had the chance to divulge some of the undergoing related research (some of which are not included in this document) in relevant international conferences. Some additional selected works carried out during this period are listed below:

Carbajo, J., Godinho, L., Amado-Mendes, P. and Ramis, J. (2015) Resistencia al flujo de paneles perforados. CMN2015 Congress on Numerical Methods in Engineering, Lisboa, Portugal.

Carbajo, J., Ramis, J., Godinho, L. and Amado-Mendes, P. (2016) Modelado de paneles acústicos ranurados. Revista de Acústica 48 (1-2), 3-9.*

Carbajo, J., Ramis, J., Godinho, L. and Amado-Mendes, P. (2018) Assessment of methods to study the acoustic properties of heterogeneous perforated panel absorbers. Appl. Acoust. 133, 1-7.

The conference works served both to get feedback from experts in this research field and to add value to the discussions in the succeeding peer-reviewed journal papers. Regarding the last journal paper, it should be pointed out that it resulted from a work carried out by the end of the development of this thesis document. In general, the above items constitute the additional thesis outcomes and briefly summarize the other achievements raised up from the same.

8.3 Future research and perspectives

Once the main conclusions and additional outcomes of this thesis have been referred, it is time now to give a short outline of what is coming next. From the scientific point of view, it is desired that some of the developed works encourage others for the development of new noise control devices and alternative isolation systems based on the proposed solutions. In this regard, the Applied Acoustics research group of the University of Alicante is already adopting some of these to real practical cases with an industrial partner. Moreover, the acquired knowledge may serve as a reference for undergoing related works and as a basis for future research work in this field. In particular, these are two of the ideas that are intended to be studied in the nearly future because of their expected impact:

Perforated panel absorber with perforated partitions. Preliminary work shows a significant broadening of the sound absorption bandwidth of these systems when compared to conventional ones. In fact, this solution is far from being just an innovative concept and may perhaps be adopted in a real system as it may not require sophisticated additive manufacturing or similar techniques to be made.

Metaporous concrete. A concept that arises from past and ongoing research by members of the Universidade de Coimbra and the University of Alicante on the design

42 CONCLUSIONS

of concrete-based sound absorbers. Under the framework of this collaborative work, several innovative and acoustically effective solutions are being conceived, their applicability to civil engineering noise reduction applications being the next stage to be undertaken.

Regarding the perspectives, being an active participant in the international research network Design for Noise Reducing Materials [DENORMS] may not only serve to participate in different events (workshops, training schools, brainstorming sessions…), which undoubtedly contribute to gain a valuable experience, but also constitute a spark for the definition of new research projects in the forthcoming years. Hence, I believe that having worked closely with other experts in porous media during the last years evidence a focus towards the settlement of this research field as an active working line for the Applied Acoustics group of the University of Alicante.

Appendices List of publications

A.1 A finite element model of perforated panel absorbers including viscothermal effects

Reference Carbajo, J., Ramis, J., Godinho, L., Amado-Mendes, P. and Alba, J. (2015) A finite element model of perforated panel absorbers including viscothermal effects. Applied Acoustics 90, 1-8. Abstract Most of the analytical models devoted to determine the acoustic properties of a rigid perforated panel consider the acoustic impedance of a single hole and then use the porosity to determine the impedance for the whole panel. However, in the case of not homogeneous hole distribution or more complex configurations this approach is no longer valid. This work explores some of these limitations and proposes a finite element methodology that implements the linearized Navier Stokes equations in the frequency domain to analyse the acoustic performance under normal incidence of perforated panel absorbers. Some preliminary results for a homogenous perforated panel show that the sound absorption coefficient derived from the Maa analytical model does not match those from the simulations. These differences are mainly attributed to the finite geometry effect and to the spatial distribution of the perforations for the numerical case. In order to confirm these statements, the acoustic field in the vicinities of the perforations is analysed for a more complex configuration of perforated panel. Additionally, experimental studies are carried out in an impedance tube for the same configuration and then compared to previous methods. The proposed methodology is shown to be in better agreement with the laboratorial measurements than the analytical approach. DOI 10.1016/j.apacoust.2014.10.013

List of publications

A.2 Acoustic properties of porous concrete made from arlite and vermiculite lightweight aggregates

Reference Carbajo, J., Esquerdo-Lloret, T. V., Ramis, J., Nadal-Gisbert, A. V. and Denia, F. D. (2015) Acoustic properties of porous concrete made from arlite and vermiculite lightweight aggregates. Materiales de Construcción 65 (320). Abstract The use of sustainable materials is becoming a common practice for noise abatement in building and civil engineering industries. In this context, many applications have been found for porous concrete made from lightweight aggregates. This work investigates the acoustic properties of porous concrete made from arlite and vermiculite lightweight aggregates. These natural resources can still be regarded as sustainable since they can be recycled and do not generate environmentally hazardous waste. The experimental basis used consists of different type specimens whose acoustic performance is assessed in an impedance tube. Additionally, a simple theoretical model for granular porous media, based on parameters measurable with basic experimental procedures, is adopted to predict the acoustic properties of the prepared mixes. The theoretical predictions compare well with the absorption measurements. Preliminary results show the good absorption capability of these materials, making them a promising alternative to traditional porous concrete solutions. DOI 10.3989/mc.2015.01115

ist of publications

A.3 Acoustic modeling of perforated concrete using the dual porosity theory

Reference Carbajo, J., Esquerdo-Lloret, T. V., Ramis, J., Nadal-Gisbert, A. V. and Denia, F. D. (2017) Acoustic modeling of perforated concrete using the dual porosity theory. Applied Acoustics 115, 150-157. Abstract Perforated concrete shows nowadays a high potential for many construction and building engineering applications. This work is devoted to the analysis of the acoustic properties of perforated concrete made from arlite lightweight aggregates. Concrete produced from these materials is an environmentally friendly alternative to traditional materials and offers a higher durability, excellent strength-to-weight ratio and low cost. In particular, it is shown that the acoustic behavior of perforated concrete can be modeled using a dual porosity approach based on the knowledge of the non-acoustic properties of the matrix granular material and geometrical data. To this end, various non-perforated and perforated samples were prepared and characterized in an experimental test facility, their acoustic properties being determined through the transfer function impedance tube method. Experimental and estimated results related to the acoustic properties of a number of prepared specimens are presented, showing a good agreement. Results suggest that this approach is suitable for practical design of such materials as part of noise control systems. DOI 10.1016/j.apacoust.2016.09.005

A.4 Modeling of grooved acoustic panels

Reference Carbajo, J., Ramis, J., Godinho, L. and Amado-Mendes, P. (2017) Modeling of grooved acoustic panels. Applied Acoustics 120, 9-14. Abstract Grooved acoustic panels are rigid plates consisting of an interconnected network of periodically arranged slits and circular holes. These devices are traditionally used as resonant absorbers in building acoustics. This paper presents a model to predict the acoustic properties of such absorbers. For this purpose, the classical Johnson-Champoux-Allard (JCA) model and the Transfer Matrix Method (TMM) are used together to describe the acoustic wave propagation through these systems. The proposed model was validated against Finite Element (FE) simulations in terms of sound absorption coefficient under normal incidence, for different configurations with good agreement. Additionally, a parametric study was found very useful to investigate the influence of the geometrical characteristics of the panel on the absorption performance of the absorber and in understanding its acoustic behaviour. As expected, the performance of the absorber varies with the size of the pores and thickness of the panel. The limits of the proposed approach are also discussed. The model is proven to be a useful tool to estimate the acoustic properties of grooved acoustic panel absorbers in a simple manner. DOI 10.1016/j.apacoust.2017.01.006

A.5 Other publications

Reference Carbajo, J., Ramis, J., Godinho, L. and Amado-Mendes, P. (2017) Modelado de panels acústicos ranurados. Revista de Acústica 48 (1-2), 3-9. Abstract The acoustic behaviour of common porous media is highly dependent on their open porosity, this consisting of an interconnected network of pores including kinematic and dead-end porosities. A properly chosen dead-end porosity can help enhance the sound absorption properties of such materials because of the thermal exchanges between the fluids filling each of these pores. This work presents a model to deal with previous situations for the specific case of rigid panels with periodically arranged circular holes containing slit-like dead-end pores. Analytical solutions describing acoustic wave propagation in pores of circular cross-section and slits are used together to this end. Preliminary results show the absorption capability of these systems, making them an interesting alternative to traditional perforated panel solutions. Additionally, the model is proven to be a useful tool to estimate their acoustic properties in a simple manner that can also be extended to other geometry cases. DOI -

Reference Carbajo, J., Ramis, J., Godinho, L. and Amado-Mendes, P. (2018) Assessment of Methods to study the acoustic properties of heterogeneous perforated panel absorbers. Applied Acoustics 133, 1-7. Abstract Heterogeneous perforated panels are of great interest as sound absorbers, while achieving a smart high-end decoration in meeting rooms, showrooms, conference halls, etc. However, their uneven nature may pose a difficulty when trying to predict their acoustic performance using traditional impedance models. This work explores the use of four different methodologies: Admittance Sum Method, Parallel Transfer Matrix Method, Equivalent Circuit Method and Finite Element Method, together with the Johnson-Champoux-Allard (JCA) model, to estimate the acoustic properties of heterogeneous perforated panels. The proposed approaches were compared in terms of the computed sound absorption curves for different absorber arrangements. Even though the above methodologies exhibit a good agreement on the predictions, it was found that special attention must be paid to the type of backing cavity configuration (isolated or non-isolated) to yield correct results. Additionally, equivalent circuits for configurations with multiple cavity depths were proposed and their sound absorption also analyzed. In general, preliminary results show that these methodologies may be useful in the design stage of such devices. DOI 10.1016/j.apacoust.2017.12.001

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