Question

Consider the draining problem of liquid water under gravity (g) from a cylindrical tank of diameter D with initial water height of h, above the nozzle of a much smaller diameter d (see schematic below). The height h of the water free surface above the nozzle in the tank will drop with time t. The following functional form between h and t should exit, h= f(t,d,D,g,h.) You are asked to do the following: 2.1 Express each of the variables in terms of basic dimensions. 2.2 Identify the number (r) of basic dimensions and the number of pi terms (k - r) based on the Buckingham Pi Theorem. 2.3 Derive all pi terms by using the repeating variable method with d and g as the repeating variables. D ho 60

Answer 1

Given haf(tid, D, g, ho) 12.1 variable & dimension (in MLT system): Variable Dimension h L t T d D L oo LT-2 ho L 2-2 (h&T) :, No. of basic dimensions =r=2 k=6 (no. of variables) No. of dimensioniess terms ('x'terms ) = K-ra 6-2 = 4. 2-3 Leb dfg be repeating variables Let tishda gb {a,b= constants} Since ned have same eñi is dimensionless dimension a=-1 & b=0 b=0

.. Tia h d Leti T2= t dagb writing dimensional equation: [Mºcºro] = [7] CLIOCL7-2] b equating powers of Tal on both sides: 1-2 b=0 e (T) b=% & atbao ( سال) . ..a+%=0 la=-/ 1-1/2 ghz : +2=td t īza Jgd Let: Az =D dagb since Dfd have same dimensions: a=-1 f bio. X3= old

Let tt = ho dogb since ho fd have same dimensions: a=-1&b=o. K 4 = ho d so we Can write : Ti= f (T2, T3, T4) h f(t ） ola ho d Jod t old' h=df ( (Ans.) - ſid

Please Rate,

Thank You!!!