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Dimensional Analysis Dimensional Analysis Fundamentals of Physics in Architecture I Degree in Architecture EPS...

Dimensional Analysis Dimensional Analysis Fundamentals of Physics in Architecture I Degree in Architecture EPS Sept, 2024 Degree in Architecture (EPS) Dimensional Analysis Sept, 2024 1 / 39 Dimensional Analysis Table of contents 1 FUNDAMENTALS. Rayleigh´s method. Buckingham´s π theorem. Quantities and Units. Tools. 2 CASES AND EXAMPLES. Degree in Architecture (EPS) Dimensional Analysis Sept, 2024 2 / 39 Dimensional Analysis FUNDAMENTALS. Table of contents 1 FUNDAMENTALS. Rayleigh´s method. Buckingham´s π theorem. Quantities and Units. Tools. 2 CASES AND EXAMPLES. Degree in Architecture (EPS) Dimensional Analysis Sept, 2024 3 / 39 Dimensional Analysis FUNDAMENTALS. Fundamentals. Quantities and Units. Tools. Rayleigh´s method. Buckingham´s π theorem. Degree in Architecture (EPS) Dimensional Analysis Sept, 2024 4 / 39 Dimensional Analysis FUNDAMENTALS. Quantities and Units. Base Physical Quantities. The seven base physical quantities from wich all the existing physical quantities derive are the following ones: Length (L). Mass (m). Time (t). Temperature (T). Electric current (I). Amount of substance (n). Luminous Intensity (I). Degree in Architecture (EPS) Dimensional Analysis Sept, 2024 5 / 39 Dimensional Analysis FUNDAMENTALS. Quantities and Units. Base Physical Quantities. Quantity Dimensions S.I. Unit Length (L) L meter (m) Mass (m) M kilogram (kg) Time (t) T second (s) Temperature (T) θ kelvin (k) Electric current (I) ∆ ampere (A) Amount of substance (n) N mole (mol) Luminous Intensity (I) I candela (cd) Degree in Architecture (EPS) Dimensional Analysis Sept, 2024 6 / 39 Dimensional Analysis FUNDAMENTALS. Quantities and Units. Derived Physical Quantities (1). The physical fundamentals in building systems are characterized by the seven base physical quantities mentioned above and the following derived physical quantities: Plane angle (θ). Solid angle (Ω). Position (r). Linear displacement (∆r). Linear velocity (v). Linear acceleration (a). Angular displacement (∆θ). Angular velocity (ω). Angular acceleration (α). Linear velocity gradient(dv/dr). Degree in Architecture (EPS) Dimensional Analysis Sept, 2024 7 / 39 Dimensional Analysis FUNDAMENTALS. Quantities and Units. Derived Physical Quantities (1). Quantity Dimensions S.I. Unit Plane angle (θ) 1 radian (rad) Solid angle (Ω) 1 steradian (sr) Position (r) L meter (m) Linear displacement (∆r) L meter (m) Linear velocity (v) LT−1 m/s Linear acceleration (a) LT−2 m/s2 Angular displacement (∆θ) 1 radian (rad) Angular velocity (ω) T−1 rad/s Angular acceleration (α) T−2 rad/s2 Linear velocity gradient(dv/dr) T−1 s−1 Degree in Architecture (EPS) Dimensional Analysis Sept, 2024 8 / 39 Dimensional Analysis FUNDAMENTALS. Quantities and Units. Derived Physical Quantities (2). The physical fundamentals in building systems are characterized by the seven base physical quantities mentioned above and the following derived physical quantities: Area (A). Volume (V). Force (F). Weight (W). Energy (E). Work (w). Heat (Q). Power (P). Efficiency (η). Degree in Architecture (EPS) Dimensional Analysis Sept, 2024 9 / 39 Dimensional Analysis FUNDAMENTALS. Quantities and Units. Derived Physical Quantities (2). Quantity Dimensions S.I. Unit Area (A) L2 m2 Volume (V) L3 m3 Force (F) MLT−2 Newton (N) Weight (W) MLT−2 Newton (N) Energy (E) ML2 T−2 Joule (J) Work (w) ML2 T−2 Joule (J) Heat (Q) ML2 T−2 Joule (J) Power (P) ML2 T−3 Watt (W) Efficiency (η) 1 Dimensionless Degree in Architecture (EPS) Dimensional Analysis Sept, 2024 10 / 39 Dimensional Analysis FUNDAMENTALS. Quantities and Units. Derived Physical Quantities (3). The physical fundamentals in building systems are characterized by the seven base physical quantities mentioned above and the following derived physical quantities: Linear density (λ). Area density (σ). Volumetric density (ρ). Specific weight (γ). Specific gravity (s). Pressure (p). Normal stress (σ). Shear stress (τ ). Axial strain (ϵ). Angular strain (γ). Degree in Architecture (EPS) Dimensional Analysis Sept, 2024 11 / 39 Dimensional Analysis FUNDAMENTALS. Quantities and Units. Derived Physical Quantities (3). Quantity Dimensions S.I. Unit Linear density (λ) ML−1 kg/m Area density (σ) ML−2 kg/m2 Volumetric density (ρ) ML−3 kg/m3 Specific weight (γ) ML−2 T−2 N/m3 Specific gravity (s) 1 Dimensionless Pressure (p) ML−1 T−2 Pascal (Pa) Normal stress (σ) ML−1 T−2 Pascal (Pa) Shear stress (τ ) ML−1 T−2 Pascal (Pa) Axial strain (1) 1 Dimensionless Angular strain (γ) 1 Dimensionless Degree in Architecture (EPS) Dimensional Analysis Sept, 2024 12 / 39 Dimensional Analysis FUNDAMENTALS. Quantities and Units. Derived Physical Quantities (4). The physical fundamentals in building systems are characterized by the seven base physical quantities mentioned above and the following derived physical quantities: Moment (M). Linear momentum (p). Angular momentum (L). Mass moment of inertia (I). First area moment of inertia (S). Second area moment of inertia (I). Degree in Architecture (EPS) Dimensional Analysis Sept, 2024 13 / 39 Dimensional Analysis FUNDAMENTALS. Quantities and Units. Derived Physical Quantities (4). Quantity Dimensions S.I. Unit Moment (M) ML2 T−2 Nm Linear momentum (p) MLT−1 kgm/s Angular momentum (L) ML2 T−1 kgm2 /s Mass moment of inertia (I) ML2 kgm2 /s First area moment of inertia (S) L3 m3 Second area moment of inertia (I) L4 m4 Degree in Architecture (EPS) Dimensional Analysis Sept, 2024 14 / 39 Dimensional Analysis FUNDAMENTALS. Quantities and Units. Derived Physical Quantities (5). The physical fundamentals in building systems are characterized by the seven base physical quantities mentioned above and the following derived physical quantities: Specific heat (Cp ). Latent heat (λ). Heat transfer rate (q). Heat flux (q/A). Thermal conductivity (k). Film coefficient (h). Thermal resistance (R). Overall heat transfer coefficient (U). Thermal gradient (dT/dr). Degree in Architecture (EPS) Dimensional Analysis Sept, 2024 15 / 39 Dimensional Analysis FUNDAMENTALS. Quantities and Units. Derived Physical Quantities (5). Quantity Dimensions S.I. Unit Specific heat (Cp ) L2 T2 θ−1 J/kgK Latent heat (λ) L2 T2 J/kg Heat transfer rate (q) ML2 T3 W Heat flux (q/A) MT3 W/m2 Thermal conductivity (k) MLT−3 θ−1 W/mK Film coefficient (h)) MT−3 θ−1 W/m2 K Thermal resistance (R) M−1 T3 θ m2 K/W Overall heat transfer coefficient (U) MT−3 θ−1 W/m2 K Thermal gradient (dT/dr) L−1 θ K/m Degree in Architecture (EPS) Dimensional Analysis Sept, 2024 16 / 39 Dimensional Analysis FUNDAMENTALS. Quantities and Units. Derived Physical Quantities (6). The physical fundamentals in building systems are characterized by the seven base physical quantities mentioned above and the following derived physical quantities: Dynamic viscosity (µ). Kinematic viscosity (ν). Volumetric flow rate (Q). Volumetric flux (Q/A). Mass flow rate (ṁ). Mass flux (ṁ/A). Hydraulic head (h). Absolute roughness (ϵ). Degree in Architecture (EPS) Dimensional Analysis Sept, 2024 17 / 39 Dimensional Analysis FUNDAMENTALS. Quantities and Units. Derived Physical Quantities (6). Quantity Dimensions S.I. Unit Dynamic viscosity (µ) ML−1 T−1 Pa s Kinematic viscosity (ν) L2 T−1 m2 /s Volumetric flow rate (Q) L3 T−1 m3 /s Volumetric flux (Q/A) LT−1 m/s Mass flow rate (ṁ) MT−1 kg/s Mass flux (ṁ/A) MT−1 L−2 kg/(sm2 ) Hydraulic head (h) L m Absolute roughness (ϵ) L m Degree in Architecture (EPS) Dimensional Analysis Sept, 2024 18 / 39 Dimensional Analysis FUNDAMENTALS. Quantities and Units. Derived Physical Quantities (7). The physical fundamentals in building systems are characterized by the seven base physical quantities mentioned above and the following derived physical quantities: Emissivity (ϵ). Reflectivity (ρ). Absorptivity (α). Wavelength (λ). Period of oscillation (T). Frequency (f). Degree in Architecture (EPS) Dimensional Analysis Sept, 2024 19 / 39 Dimensional Analysis FUNDAMENTALS. Quantities and Units. Derived Physical Quantities (7). Quantity Dimensions S.I. Unit Emissivity (ϵ) 1 Dimensionless Reflectivity (ρ) 1 Dimensionless Absorptivity (α) 1 Dimensionless Wavelength (λ) L m Period of oscillation (T) T s Frequency (f) T−1 s−1 Degree in Architecture (EPS) Dimensional Analysis Sept, 2024 20 / 39 Dimensional Analysis FUNDAMENTALS. Quantities and Units. Derived Physical Quantities (8). The physical fundamentals in building systems are characterized by the seven base physical quantities mentioned above and the following derived physical quantities: Electric charge (q). Electric current (I). Electromotive force (e.m.f.). Electrical voltage (V). Electrical resistance (R). Electrical resistivity (ρ). Luminous flux (Φ). Degree in Architecture (EPS) Dimensional Analysis Sept, 2024 21 / 39 Dimensional Analysis FUNDAMENTALS. Quantities and Units. Derived Physical Quantities (8). Quantity Dimensions S.I. Unit Electric charge (q) T∆ Coulomb(C) Electric current (I) ∆ Ampere(A) Electromotive force (e.m.f.) ML2 T−3 ∆−1 Volt(V) Electrical voltage (V) ML2 T−3 ∆−1 Volt(V) Electrical resistance (R) ML2 T−3 ∆−2 Ohm(Ω) Electrical resistivity (ρ) ML3 T−3 ∆−2 Ωm Luminous flux (Φ) I Lumen(Lm) Degree in Architecture (EPS) Dimensional Analysis Sept, 2024 22 / 39 Dimensional Analysis FUNDAMENTALS. Tools. Tools. Rayleigh´s method of dimensional analysis is a tool used in Physics. It expresses a functional relationship of some variables in the form of an exponential equation. Buckingham´s π theorem is a tool used in Physics. It determines the number of dimensionless groups needed to describe a phenomenon. Degree in Architecture (EPS) Dimensional Analysis Sept, 2024 23 / 39 Dimensional Analysis FUNDAMENTALS. Tools. Tools. Rayleigh´s method: The number of exponents obtained when applying Rayleigh´s method may or may not be equal to the number of equalities obtained. If the number of equations equals the number of unknowns, the problem is solved. When there are fewer equations than unknwons, some exponents must be expressed as functions of the oders. When susbtituting them into the main equation, those variables having the same exponents can be grouped. Degree in Architecture (EPS) Dimensional Analysis Sept, 2024 24 / 39 Dimensional Analysis FUNDAMENTALS. Tools. Tools. Buckingham´s π theorem: The number of dimensionless groups obtained usig Buckingham´s π theorem equals the total number of variables minus the base quantities involved. These dimensionless groups can be called π1 , π2 , π3 , etc. The dependent variable can be expressed as a function of such dimensionless groups. Degree in Architecture (EPS) Dimensional Analysis Sept, 2024 25 / 39 Dimensional Analysis FUNDAMENTALS. Tools. Rayleigh´s method. Steps. 1 Gather all the variables involved (n). 2 Relate the dependent variable to the remaining variables using a function. 3 Write the above function in the form: R = K × Ra1 × Rb2 × Rc3 × Rd4 ×......Rm n 4 Express the dimensions of the n variables involved in terms of the k base quantities. 5 Substitute the dimensions of each of the variables in the above equation. 6 Obtain a set of equations relating the arbitrary exponents. 7 Substitute the values of the exponents in the main equation and group the variables with the same exponent. Degree in Architecture (EPS) Dimensional Analysis Sept, 2024 26 / 39 Dimensional Analysis FUNDAMENTALS. Tools. Rayleigh´s method. Example. To keep and object in motion on a circumference with constant speed, a force called centripetal force is to be aplied. Determine the formula of such centripetal force supposing it is a function which depends on m (mass of the object), R (radius of teh circumference) and v (linear velocity of the object.) 1 Gather all the variables involved (n=4): the centripetal force (Fc ), the radius of the circumference (R), the mass of the object (m) and the linear velocity of the projectile (v). 2 Relate the dependent variable to the remaining variables using a function: Fc = f(R, m, v). 3 Write the above function in the form: Fc = K × Ra × mb × vc Degree in Architecture (EPS) Dimensional Analysis Sept, 2024 27 / 39 Dimensional Analysis FUNDAMENTALS. Tools. Rayleigh´s method. Example. 4 Express the dimensions of the n variables involved in terms of the k base quantities. Centripetal force: [Fc ] = MLT−2. Radius of the circumference: [R] = L. Mass of the object: [m] = M. Linear velocity: [v] = LT−1. 5 Substitute the dimensions of each of the variables in the above equation. MLT−2 = La Mb (LT−1 )c → L1 M1 T−2 = La+c Mb T−c Degree in Architecture (EPS) Dimensional Analysis Sept, 2024 28 / 39 Dimensional Analysis FUNDAMENTALS. Tools. Rayleigh´s method. Example. 6 Obtain a set of equations relating the arbitrary exponents. L: 1 = a + c → a = −1. M: 1 = b → b = 1. T: −2 = −c → c = 2. 7 Substitute the values of the exponents in the main equation and group the variables with the same exponent. mv2 Fc = KRa mb vc → Fc = KR−1 m1 v2 → Fc = K R Degree in Architecture (EPS) Dimensional Analysis Sept, 2024 29 / 39 Dimensional Analysis FUNDAMENTALS. Tools. Buckingham´s π theorem. Steps. 1 Gather all the variables involved (n). 2 Relate the dependent variable to the remaining variables using a function. 3 Express the k base quantities in terms of the most simple variables involved excluding the dependent variable. There is another way to to carry out this step if the problem is more complex. 4 Select k reference variables excluding the dependent variable to be linearly independent (i.e., they have a non-zero determinant) and multiply them by the remaining variables to the power of unknown exponents. They must contain the k dimensions. Degree in Architecture (EPS) Dimensional Analysis Sept, 2024 30 / 39 Dimensional Analysis FUNDAMENTALS. Tools. Buckingham´s π theorem. Steps. 5 Construct a set of p=n-k dimensionless groups from the original variables (the so-called π groups). Remember to use the dependent variable, variables with the same dimensions, and dimensionless variables. Multiply the remaining π groups by the inverse of their dimensions in order to non-dimensionalize them. 6 Relate the dimensionless groups in a single equation. Degree in Architecture (EPS) Dimensional Analysis Sept, 2024 31 / 39 Dimensional Analysis FUNDAMENTALS. Tools. Buckingham´s π theorem. Example. To keep and object in motion on a circumference with constant speed, a force called centripetal force is to be aplied. Determine the formula of such centripetal force supposing it is a function which depends on m (mass of the object), R (radius of teh circumference) and v (linear velocity of the object.) 1 Gather all the variables involved (n=4): the centripetal force (Fc ), the radius of the circumference (R), the mass of the object (m) and the linear velocity of the projectile (v). 2 Relate the dependent variable to the remaining variables using a function: Fc = f(R, m, v). Degree in Architecture (EPS) Dimensional Analysis Sept, 2024 32 / 39 Dimensional Analysis FUNDAMENTALS. Tools. Buckingham´s π theorem. Example. Break down the variables and their dimensions in a table: Number Variable Dimension M L T 1 Fc MLT−2 1 1 -2 2 R L 0 1 0 3 m M 1 0 0 4 v LT−1 0 1 -1 Degree in Architecture (EPS) Dimensional Analysis Sept, 2024 33 / 39 Dimensional Analysis FUNDAMENTALS. Tools. Buckingham´s π theorem. Example. 3 Express the k base quantities in terms of the most simple variables involved excluding the dependent variable. k = 3(L, M, T) L = [R]. M = [m]. [v] = LT−1 → T = L[v]−1. 4 Construct a set of p=n-k dimensionless groups from the original variables (the so-called π groups). Remember to use the dependent variable, variables with the same dimensions, and dimensionless variables. Multiply the remaining π groups by the inverse of their dimensions in order to non-dimensionalize them. Degree in Architecture (EPS) Dimensional Analysis Sept, 2024 34 / 39 Dimensional Analysis FUNDAMENTALS. Tools. Buckingham´s π theorem. Example. p=n−k→p=4−3=1 [Π1 ] = [Fc ]M−1 L−1 T2 = 1 [Fc ][m]−1 [R]−1 ([R][v]−1 )2 = 1 [Fc ][m]−1 [R][v]−2 = 1 → Fc m−1 Rv−2 = K 5 Relate the dimensionless groups in a single equation. mv2 Fc = K R Degree in Architecture (EPS) Dimensional Analysis Sept, 2024 35 / 39 Dimensional Analysis FUNDAMENTALS. Tools. Buckingham´s π theorem. Example. There was another way to carry out the step 3. 3 Select k reference variables excluding the dependent variable to be linearly independent (i.e., they have a non-zero determinant) and multiply them by the remaining variables to the power of unknown exponents. They must contain the k dimensions. The determinant of the k variables that are linearly independent is: 0 1 0 1 0 0 ̸= 0 (1) 0 1 −1 [Π1 ] = Fc Ra mb vc = 1 → M0 L0 T0 = MLT−2 La Mb (LT−1 )c Degree in Architecture (EPS) Dimensional Analysis Sept, 2024 36 / 39 Dimensional Analysis FUNDAMENTALS. Tools. Buckingham´s π theorem. Example. M : 0 = 1 + b → b = −1. L : 0 = 1 + a + c → a = 1. T = 0 = −2 − c → c = −2. 4 Construct a set of p=n-k dimensionless groups from the original variables (the so-called π groups). [Π1 ] = Fc R1 m−1 v−2 = 1 → Fc Rm−1 v−2 = K 5 Relate the dimensionless groups in a single equation. mv2 Fc = K R Degree in Architecture (EPS) Dimensional Analysis Sept, 2024 37 / 39 Dimensional Analysis CASES AND EXAMPLES. Table of contents 1 FUNDAMENTALS. Rayleigh´s method. Buckingham´s π theorem. Quantities and Units. Tools. 2 CASES AND EXAMPLES. Degree in Architecture (EPS) Dimensional Analysis Sept, 2024 38 / 39 Dimensional Analysis CASES AND EXAMPLES. Cases and Examples. Dimensional homogeneity. Dimensionless quantities. Dimensional analysis. Degree in Architecture (EPS) Dimensional Analysis Sept, 2024 39 / 39