NDLI: A “fractal” modification of Torricelli’s formula

Content Provider	Springer Nature Link
Author	Maramathas, Athanasios J. Boudouvis, Andreas G.
Copyright Year	2009
Abstract	A modification is proposed of Torricelli’s (1608–1647) formula for the velocity of water discharging from a small hole at the bottom of a large tank filled with fractal solid material. The new formula takes proper account of the mechanical energy losses due to flow in the solid matrix, thus expanding the area of validity of the classical Torricelli’s formula. Moreover, it offers a convenient alternative to Darcy’s law for estimating the discharge rate from an aquifer. The new formula was derived from laboratory experiments, with a low-Reynolds number discharge flow (Darcian flow). It was tested in a natural karst aquifer where the flow is non-Darcian, at Almiros spring on the island of Crete (Greece). In both cases, the predictive capability of the modified formula is established.Une modification est proposée pour la formule de Torricelli (1608–1647) pour la vitesse de l’eau sortant d’un petit trou au fond d’un grand réservoir rempli de matériel solide fractal. La nouvelle formule prend mieux en compte les pertes d’énergie mécanique dues au flux au sein de la matrice solide, étendant ainsi le domaine de validité de la formule classique de Torricelli. De plus cette formule offre une alternative à la loi de Darcy pour l’estimation des taux de décharges des aquifères. La nouvelle formule dérive de tests en laboratoire effectués avec un flux de sortie (flux de Darcy) à bas numéro de Reynolds. Elle a été testée pour un aquifère karstique naturel où le flux ne suit pas la loi de Darcy, à la source d’Almiros sur l’île de Crète (Grèce). Dans les deux cas, la capacité prédictive de la formule modifiée a été démontrée.Se propone una modificación de la fórmula de Torricelli (1608–1647) para la velocidad de la descarga de agua desde un agujero pequeño en el fondo de un gran tanque rellenado con material sólido fractal. La nueva fórmula toma debida cuenta de las pérdidas de energía mecánica debidas al flujo en la matriz sólida, expandiendo así el área de validez de la clásica formula de Torricelli. Más aún, ofrece una alternativa conveniente a la ley de Darcy para la estimación de los ritmos de descarga de un acuífero. La nueva fórmula fue derivada de experimentos de laboratorio para el flujo de descarga con un número de Reynolds bajo (flujo de Darcy). Ello fue probado en un acuífero kárstico natural donde el flujo es no darciano, en el manantial de Almiros en la isla de Creta (Grecia). En ambos casos, se establece la capacidad predictiva de la formula modificada.本文提出了对托里拆利(1608–1647)公式的一种改进, 用于计算以分形固体材料填充的大水箱中, 水自其底部小孔泄出时的速度。新公式恰当考虑了固体基质中流动所致的机械能损失, 因此扩展了经典托里拆利公式的应用范围。此外, 它可方便地替代达西定律用于估算含水层渗速。该公式得自低雷诺数渗流 (达西流) 实验。利用某非达西流的天然喀斯特含水层 (希腊Crete岛Almiros泉) 对其进行了检验。两种情况下, 都证实了改进公式的预测能力。Ο τύπος του Torricelli (1608–1647) για τον υπολογισμό της ταχύτητας εκροής του νερού μιάς δεξαμενής από οπή μικρού, σε σχέση με τη διατομή της δεξαμενής, εμβαδού στον πυθμένα της, μετασχηματίστηκε προκειμένου να εφαρμόζεται και στην περίπτωση που η δεξαμενή είναι γεμάτη με πορώδες ή ρωγμώδες υλικό η υφή του οποίου θα μπορούσε να περιγραφεί από τη φράκταλ γεωμετρία. Ο νέος τύπος λαμβάνει υπ’ όψιν του τις απώλειες μηχανικής ενέργειας του νερού, προσφέροντας έτσι μία εναλλακτική λύση προς το νόμο του Darcy για τον υπολογισμό της παροχής εκφόρτισης ενός υδροφόρου ορίζοντα. Η νέα σχέση εξήχθη από πειράματα σε Νταρσιανού τύπου ροές (μικρού αριθμού Reynolds) ενώ δοκιμάστηκε και σε ένα καρστικό υδροφόρο ορίζοντα (πηγή Αλμυρού Ηρακλείου Κρήτης) όπου η ροή είναι μη Νταρσιανή. Και στις δύο περιπτώσεις προέβλεψε ικανοποιητικά τις πραγματικές μετρήσεις.É proposta uma modificação da fórmula de Torricelli (1608–1647) para o cálculo da velocidade de descarga de água a partir de um pequeno buraco localizado no fundo de um grande tanque preenchido por material sólido fractal. A nova fórmula toma em consideração as perdas de energia mecânica devidas ao escoamento na matriz sólida, estendendo desta forma o domínio de validade da fórmula clássica de Torricelli. Para além disso, esta fórmula é uma alternativa à lei de Darcy na estimação do caudal de descarga de um aquífero. Esta nova fórmula foi derivada a partir de experiências laboratoriais com um fluxo de descarga caracterizado por um número de Reynolds baixo (escoamento Darciano). A fórmula foi testada num aquífero cársico onde o escoamento é não-Darciano, na nascente Almiros, na ilha de Creta (Grécia). Em ambos os casos, foi comprovada a capacidade de previsão da fórmula modificada.
Starting Page	311
Ending Page	316
Page Count	6
File Format	PDF
ISSN	14312174
Journal	Hydrogeology Journal
Volume Number	18
Issue Number	2
e-ISSN	14350157
Language	Portuguese
Publisher	Springer-Verlag
Publisher Date	2009-09-04
Publisher Institution	International Association of Hydrogeologists
Publisher Place	Berlin, Heidelberg
Access Restriction	One Nation One Subscription (ONOS)
Subject Keyword	Hydraulic properties Karst Torricelli’s formula Darcy’s law Greece Waste Water Technology Water Pollution Control Water Management Aquatic Pollution Geology Hydrogeology
Content Type	Text
Resource Type	Article
Subject	Earth and Planetary Sciences Water Science and Technology

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