Turbomachinery Lecture 4a Pi Theorem Pipe Flow Similarity Flow, Head, Power Coefficients Specific Speed

Introduction to Dimensional Analysis Thus far course has shown elementary fluid mechanics – now one can appreciate Dimensional Analysis Dimensional Analysis Identifies significant parameters in a process not completely understood. Useful in analyzing experimental data. Permits investigation of full size machine by testing smaller version Predicts consequences of off-design operation Useful in preliminary design studies for sizing machine for optimal performance Useful in sizing pumps & blowers based on performance maps Geometric similarity: assumes all linear dimensions are in constant proportion, all angular dimensions are same

Dimensional Analysis Buckingham -Theorem Basic Premise Physical process involving dimensional parameters, Q's and f(Q) is unknown. Q1 = f(Q2,Q3,...Qn) Group the n variables into a smaller number of dimensionless groups, each having 2 or more variables Physical process can be expressed as: 1 = g(2, 3,...n-k)

Dimensional Analysis Buckingham -Theorem Each  is a product of the primary variables, Q's raised to various exponents so that 's are dimensionless. where n = no. primary variables k = no. physical dimensions [L,M,T] n-k = no. 's

Dimensional Analysis Dimensional analysis requires postulation of proper primary variables judgement, foresight, good luck Dimensional analysis cannot give form of 1 = g(2, 2,...n-k) prevent omission of significant Q’s exclude an insignificant Q’s

Dimensional Analysis Basic Units Mass M Length L Time T Force is related to basic units by F=ma Force ML/T2

Example: Pressure Drop in Pipe P = f(V,,,l,d,) Pick V, , d as the 3 Q’s which will be used with each of the remaining Q’s to form the 7 - 3 = 4  terms. Pick M, L, T as the 3 primary dimensions Result

Example: Pressure Drop in Pipe P = f(V,,,l,d,) Therefore Moody Diagram turbulent laminar smooth What happens when there are several length scales: D, L, …?

Dimensional Analysis of Turbomachines Primary Variables - Q’s

Background: Head, Power, and Viscosity Q’s Head - work per unit mass - fluid dynamic equivalent to enthalpy Recalling Gibbs Equation: So head in "feet" is clearly erroneous.

Background: Head, Power, and Viscosity Q’s Power - Work per unit time - Mass Flow Rate Work per unit Mass

Background: Head, Power, and Viscosity Q’s Newtonian Fluid: Shear stress  Velocity gradient Viscosity is  - with units:

Dimensional Analysis of Turbomachines Since there are 10 Q's & 3 Dimensions we can identify 7 's. Each  contains 4 Q's, Q1, Q2, Q3, and Qn. The parameters chosen for 1, 2 & 3 were chosen carefully. Task is to find exponents of primary variables to make dimensionless groups.

Dimensional Analysis of Turbomachines The system of equations is:

Dimensional Analysis of Turbomachines Each  has 3 linear equations: M 0a + 0b + 1c +0 = 0  c = 0 L 0a + 1b - 3c + 3 = 0  b = -3 T -a + 0b + 0c - 1 =0  a = -1

Dimensional Analysis of Turbomachines M 0a + 0b + 1c + 0 = 0  c = 0 L 0a + 1b - 3c + 2 = 0  b = -2 T -1a + 0b + 0c - 2 = 0  a = -2

Dimensional Analysis of Turbomachines Aside: What is meaning of H=head? Hydraulic engineers express pressure in terms of head Static pressure at any point in a liquid at rest is, relative to pressure acting on free surface, proportional to vertical distance from point to free surface.

Dimensional Analysis of Turbomachines M 0a + 0b + 1c + 1 = 0  c = -1 L 0a + 1b - 3c + 2 = 0  b = -5 T -1a + 0b + 0c - 3 = 0  a = -3

Dimensional Analysis of Turbomachines M 0a + 0b + 1c + 1 = 0  c = -1 L 0a + 1b - 3c - 1 = 0  b = -2 T -1a + 0b + 0c - 1 = 0  a = -1

Dimensional Analysis of Turbomachines M 0a + 0b + 1c + 0 = 0  c = 0 L 0a + 1b - 3c + 1 = 0  b = -1 T -1a + 0b + 0c - 1 = 0  a = -1

Dimensional Analysis of Turbomachines M 0a + 0b + 1c + 0 = 0  c = L 0a + 1b - 3c + 1 = 0  b = -1 T -1a + 0b + 0c + 0 = 0  a = 0

Turbomachinery Non-Dimensional Parameters Derived 7 s from 10 Qs in first part of class Now ready to - develop physical significance of s - relate to traditional parameters - discuss general similitude

Flow Coefficient

Head Coefficient

Hydraulic Pump Performance Geometric similarity: all linear dimensions are in constant proportion, all angular dimensions are same Performance curves are invariant if no flow separation or cavitation BEP= best efficiency point [max] or operating point

Head Curve

Example: Changing Level of Performance for a Given Design Pressure rise

Example: Changing Level of Performance for a Given Design Same fan but different size / speed

Scaling for Performance[limited by M, Re effects]

Example

Example

Define New Variable: Vary More Than One Parameter later

Similarity – Compressible Flow - Engine

Similarity – Compressible Flow

Nondimensional Parameters