1. Introduction and Fundamental Concepts [1]

1. Introduction and Fundamental Concepts [1] • System-Surroundings-Interaction • Physical Phenomena, Physical Quantities, and Physical Relations • Physical Quantity • Dimensions • Numerical value • Unit of Measure • Dimensions – A simple key to some physical understanding of fluid mechanics • Systems of Dimensions and Units • Physical Quantity • Units of Measure and Principle of Absolute Significance of Relative Magnitude (PASRM) • Dimension is a power-law monomial • Physical Relation and Principle of Dimensional Homogeneity • Dimensionless Variables and ‘Measuring/Scaling’

Universe / Isolated System (Physical) System Fundamental Concept: System-Surroundings-Interaction • The very first task in any one problem: • Identify the system • Identify the surroundings • Identify the interactions between the system and its surroundings, e.g., • Mechanics - Force (identify all the forces on the system by its surroundings) • Thermodynamics - Energy and Energy Transfer (identify all forms of energy and energy transfer between the system and its surroundings) Surroundings • Interaction • Mechanical interaction (force) • Thermal interaction (energy and energy transfer) • Electrical, Chemical, etc.

Q Example: Thermodynamics - Heating Water • Water 1 liter in a container in atmosphere. • Add heat of the amount Q. [Assume no heat loss elsewhere.] QUESTION: How much is the temperature rise?

System 1: Water + Container • How much is the energy transfer as heat into the System 1? Q • System 2: Water only • How much is the energy transfer as heat into the System 2? Energy transfer as heatQ into the two systems are not the same. Two different systems have two different (energy) interactions with their own surroundings. Obviously, at least the mass of the two systems are not the same.

System 1: Water stream in the pipe only, exclude the solid pipe and flange. System 2: Water stream in the pipe, and the solid pipe and flange (cut through the bolts). Example: (Fluid) Mechanics - Flow in Pipe

System 1: • Pressure and shear stress distributions on the surfaces only. • No force at the solid bolt. • System 2: • There are forces at the solid bolts acting on the system. F External forces on the two systems are not the same. Two different systems have two different (mechanical) interactions with their own surroundings. Obviously, at least the mass of the two systems are not the same.

Key Point: Define your system first before you apply an equation. • Since the application of basic principles / equation is always to a specific system, • define your system first before you apply an equation.

Classification of Systems

Isolated system: Insulated hot water bottle (approximately isolated over a short period of time, no energy absorption due to radiation, etc.) Closed system Open system (part of the mass is evaporated out of the system) Example: Various types of systems

Boeing 747-400 Cruising speed Mach Number = 0.85 (Compressible Flows). (From http://www.boeing.com/companyoffices/gallery/images/commercial/747400-06.html) Physical Phenomena, Physical Quantities, and Physical Relations • physical phenomena • physical quantities • physical relations[relations among physical quantities]

Physical QuantityDescribing A Physical Quantity • Physical quantity is a concept. • A quantifiable/measurable attribute we assign to a particular characteristic of nature that we observe. • We must find a way to ‘quantify’ it. 1. Describing a physical quantity. We need 3 things: • Dimension • Numerical value with respect to the unit of measure • Unit of measure

In fact, we can change these numerical values to any numerical value that we want so long as we choose the corresponding unit of measure Q. • aQ and Q must go together. • Change the unit of measureQ, the numerical value aQ must be changed accordingly.

Key Point: aQ and Q must go together. • Always write the corresponding unit Q for the corresponding numerical value aQ of a physical quantity. [Except when that quantity is dimensionless.] m = 5 5 what? 5 kg 5 lbm 5 ton? m = 5 ton

In fact, we can change these numerical values to any numerical value that we want so long as we choose the corresponding unit of measure Q. Fundamental Concept: Quantification and Measure(ment) • (Any sort of) Quantification is always based on measure, unit of measure, measurement. • There is a degree of arbitrariness in choosing a unit of measure.

Key Point: Dimensions - A simple key to gain some understanding of fluid mechanics (or rather physics in general) • The followings cannot be emphasized enough. • To gain some physical understanding of fluid (mechanics), pay attention to the dimensions of the physical quantity/relation of interest. • Choose any dimension that you can relate physically, not necessarily – and often not - MLtT. • Enthalpy h has the dimension of • L2t-2 “What is this?” • Energy/Mass O.k. This, I can relate.

More Example:Dimensions - A simple key to gain some understanding of fluid mechanics (or rather physics in general) • Specific heat C has the dimension of • L2t-2T-1 “What is this?” • O.k. This, I can relate. Note reads “Energy per unit mass per unit (change in) temp” • I can guess that C should somehow be related to the amount of energy per unit mass per unit (change in) temperature.

By definition By law Systems of Dimensions and Units • Primary Quantities and Primary Dimensions • Choose a set of primary quantities (and consider their dimensions to be independent). • Three systems of common use areMLtT, FLtT, FMLtT • Units of Measure for Each Primary Quantity • Choose a unit of measure for each primary quantity. MLtT: In SI: M – kg, L – m, t – sec, T – K • Derived Quantities/Dimensions • Through physical relations, we have derived quantities and their dimensions • e.g., [Velocity] = L/t, [Acceleration] = L/t2 [Force] = ML/t2, etc.

Key Point: There is some arbitrariness in choosing primary quantities and dimensions. • Conceptually, for example, we can choose ELtT Energy-Length-Time-Temp as a set of primary quantities and primary dimensions in place of MLtT

From Physics Laboratory, The National Institute of Standards and Technology (NIST)’s web page: http://physics.nist.gov/cuu/Units/SIdiagram2.html Note that there can be some characters missing from this diagram due to font and file related issues during the making of the presentation slide. Go to the NIST’s web page given above for the original. Example of Systems of Units: SI

The ratio A2/A1should be the same regardless of whether the numerical values of A1and A2are expressed in m2 or ft2. A2 A1 What about Fundamental Concepts: Physical Quantity: Chosen System of Units and The Principle of Absolute Significance of Relative Magnitude (PASRM) • To have some physical sense, (we prefer to use and/or) we require(of systems of units to be used) that the ratio of magnitudes of any two concrete physical quantities should not depend on the system of units used.

The dimensions of a derived quantity q must be in the form of a power-law monomial e.g., • It cannot be, e.g., • In other words, the argument of these functions must be dimensionless, e.g., if we know that we then know that Dimension (function) is a power-law monomial

Dimensionless quantityq has the dimension of Efficiency Angle (radian) Dimensionless Quantity Example

Independent Dimensions: Physical quantities q1,…, qrare said to have independent dimensions if none of these quantities has a dimension function that can be written in terms of a power-law monomial of the dimensions of the remaining quantities. (Barenblatt, 1996) By definition By law Independent Dimensions • In MLtT system, since the dimension of velocity V can be written as the power-law monomial of L and t, V = L1t-1 the physical quantities (Velocity, Length, time) do not have independent dimension. • Similarly, (Length, Time, Mass, and Force) do not have independent dimension since, according to Newton’s Second law of motion, F = MLt-2

Physical Relation Dimensionally homogeneous ~ Physical Relation (We derive something wrong somewhere.) ~ Dimensionally homogeneous Physical Relation and Principle of Dimensional Homogeneity (PDH) • Requirement/Premise: Any equation that describes a physical relation cannot be dependent upon an arbitrary choice of units (within a given class of systems of units). • Principle of Dimensional Homogeneity (PDH): All physically meaningful equations, i.e., physical relation/equation, aredimensionally homogeneous. (Smits, 2000) If is a physical equation, then have the same dimension, i.e., Useful for checking our derived result: [We shall deal only with physical equations.]

QUESTION: Without the knowledge of mechanics, can you tell whether this result/equation is wrong: The result is not correct. Example: The Use of The Principle of Dimensional Homogeneity in Checking Our Results • We can use PDH to tell whether something is definitely wrong. • However, we cannot use PDH to tell whether something is definitely correct. Even though our result is dimensionally homogeneous, we cannot tell whether it is correct by PDH alone.

Dimensionless Variables: From if we divide through by one of the term, say Xn, we obtain The new variables, e.g., then have no dimension. We call these variables dimensionless variables. Scaling/Measuring: Xi is Zifraction ofXn. Dimensionless Variables and ‘Measuring/Scaling’ • Scaling: • One can think of the above process as themeasurement/scalingof the variables Y, X1,…,Xn-1, with Xn. • In other words, wemeasure, e.g., Xi as a fraction (or per cent) of Xn,or we measure Xi relative to Xn

Unit of measure L cm D (= 10 cm) D • The pipe is 300 cm long - unit of measure = cm • The pipe is 30 D long - unit of measure = D Example: Dimensionless Variables and ‘Measuring/Scaling’

Unit of measure L cm D (= 10 cm) D Key Idea: Use the units/scales of measure that are inherent in the problem itself, not the man-made one irrelevant to the problem. • In order to understand physical phenomena better, we prefer to use the units/scales of measure that exist in the problem itself, not the man-made one irrelevant to the problem.

Key Point: The numerical value of dimensionless variable does not depend on the (appropriate) system of units used. • While the numerical values of power output in the units of Watt and hp are not the same, 2,000 W VS 2,000/746 = 2.68 hp the numerical value of the dimensionless variables efficiency is the same regardless of whether we use W or hp: Other examples of dimensionless variables are • Reynolds number Re • Mach number M

1. Introduction and Fundamental Concepts [1]