Satuan tak berdimensi

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Dalam analisis dimensional, satuan tak berdimensi adalah satuan yang tidak memiliki unit fisis melainkan hanyalah bilangan. Bilangan itu pada umumnya didefinisikan sebagai produk atau rasio atau satuan yang memiliki unit.

Contoh yang lebih mudah untuk dipahami adalah ketika seorang penyortir buah-buahan di suatu industri mengatakan bahwa setiap dua puluh buah apel terdapat satu apel busuk. Maka rasio apel busuk dengan apel secara keseluruhan adalah 1/20. Bilangan tersebut adalah satuan tak berdimensi. Contoh lainnya dalah ilmu keteknikan dan fisika adalah pengukuran sudut bidang miring. Sudut umumnya diukur menggunakan rasio panjang dan tinggi yang selalu spesifik setiap sudut. Rasio tersebut, panjang dibagi tinggi, adalah satuan tak berdimensi.

Satuan tak berdimensi digunakan secara luas dalam bidang matematika, fisika, teknik, dan ekonomi dalam kehidupan sehari-hari.

Satuan tak berdimensi tidak memiliki unit fisis yang berhubungan. Namun kadang-kadang penulisan rasio unit yang saling meniadakan, seperti g/kg, di mana keduanya adalah satuan massa, hal itu cukup membantu untuk menjelaskan bahwa suatu bilangan sedang dihitung dengan proses demikian.

Nama Simbol Bidang aplikasi

Bilangan Abbe V Optik; Tingkat dispersi material optik Klimatologi, astronomi (reflektivitas permukaan suatu

Albedo α benda)

Bilangan Archimedes Ar Gerakan fluida akibat dari perbedaan massa jenis

Berat atom M Kimia

Bilangan Bagnold Ba Aliran material solid seperti pasir

Bilangan Biot Bi konduktivitas antara permukaan dan volume benda solid

Bilangan Bodenstein Distribusi waktu diam

Bilangan Bond Bo Kapilaritas yang dikendalikan oleh gaya apung Transfer kalor akibat konduksi dari permukaan ke fluida

Bilangan Brinkman Br kental

Bilangan Brownell Katz Kombinasi dari bilangan kapilaritas dan bilangan Bond

Bilangan kapilaritas Ca Aliran fluida akibat dari tegangan permukaan

Koefisien gesek statik μs Gesekan dua permukaan solid pada keadaan diam

Koefisien gesek kinetis μk Gesekan dua permukaan solid pada gerakan translasi

Faktor Colburn J Koefisien transfer kalor tak berdimensi Bilangan Courant- ν Persamaan numerik dari hyperbolic PDE Friedrich-Levy

Bilangan Damkohler Da Skala reaksi waktu terhadap fenomena perpindahan Faktor gesekan Darcy Cf or f Aliran fluida

Bilangan Dean D Aliran fluida pada pipa atau selat bengkok

Bilangan Deborah De rheologi dari fluida viskoelastik

Desibel dB rasio dua intensitas suara

Koefisien gerak Cd resistansi aliran

Bilangan Euler e Matematika

Bilangan Eckert Ec Transfer kalor konvektif

Bilangan Ekman Ek geofisika (gaya gesek (viskositas)) Digunakan untuk mengukur bagaimana respon permintaan

Elastisitas (ekonomi) E dan penawaran terhadap perubahan harga

Bilangan Eötvös Eo ???

Bilangan Ericksen Er Perilaku aliran kristal cair

Bilangan Euler Eu hidrodinamika (tekanan terhadap inersia)

Faktor gesekan Fanning f Aliran fluida di pipa

Konstanta Feigenbaum α,δ Teori chaos Konstanta kualitas α elektrodinamika kuantum struktur

bilangan-f f optik, fotografi Bilangan Foppl von – Penekukan lapisan tipis

Karman

Bilangan Fourier Fo Transfer kalor

Bilangan Fresnel F difraksi celah

Bilangan Froude Fr Perilaku gelombang dan permukaan

Gain elektronik (sinyal output terhadap sinyal input)

Bilangan Galilei Ga Aliran kekentalan yang dikendalikan oleh gravitasi

Rasio Golden matematika dan estetika

Bilangan Graetz Gz Aliran panas

Bilangan Grashof Gr Konveksi bebas

Bilangan Hatta Ha Peningkatan adsorpsi akibat dari reaksi kimia

Bilangan Hagen Hg Konveksi yang dipaksa

Gradien hidrolik i Aliran air tanah

Bilangan Karlovitz pembakaran turbulensi Bilangan Keulegan– rasio gaya perpindahan terhadap inersia benda keras dalam KC

Carpenter osilasi aliran fluida

Bilangan Knudsen Kn Perkiraan kontinu dalam fluida

Kt/V Kedokteran

Bilangan Kutateladze K Aliran dua fase yang saling berlawanan Aliran konveksi bebas dalam fluida yang tak dapat

Bilangan Laplace La bercampur

Bilangan Lewis Le Rasio persebaran massa dan termal

Koefisien gaya angkat CL Gaya angkat pada airfoil pada berbagai sudut datang Parameter Lockhart- χ Aliran gas basah Martinelli Rasio resistansi waktu pada gelombang Alfven melintasi

Bilangan Lundquist S waktu dalam plasma

Bilangan Mach M Dinamika gas Bilangan magnetik

Rm magnetohidrodinamika

Reynolds Koefisien kekasaran n Aliran terbuka (aliran yang dikendalikan oleh gravitasi

Manning Aliran Marangoni akibat dari deviasi tekanan permukaan

Bilangan Marangoni Mg termal

Bilangan Morton Mo ???

Bilangan Nusselt Nu transfer kalor dengan konveksi yang dipaksa

Bilangan Ohnesorge Oh Atomisasi cairan, aliran Marangoni

Bilangan Péclet Pe adveksi–masalah difusi

Bilangan Peel adhesi dari struktur mikro dengan substrat + pH pH Kimia (kologaritma dari aktvitas ion H terlarut) matematika (rasio dari keliling lingkaran terhadap

Pi π diameternya)

Rasio Poisson ν Elastisitas (dimuat pada arah transversal dan longitudinal) elektronika (besar daya riil terhadap daya dalam

Faktor daya perhitungan)

Bilangan daya Np Konsumsi daya oleh agitator transfer kalor Konveksi (ketebalan termal dan momentum

Bilangan Prandtl Pr batas lapisan)

Koefisien Pressure CP Tekanan yang terjadi pada titik pada airfoil

Radian rad pengukuran sudut

Bilangan Rayleigh Ra Gaya apung dan gaya viskositas pada konveksi bebas

Indeks Refraktif n elektromagnetisme, optika

Bilangan Reynolds Re Perilaku aliran (inersia terhadap viskositas)

Masa jenis relatif RD hidrometer, perbandingan material

Bilangan Richardson Ri Efek gaya apung pada kestabilan aliran

Skala Rockwell Tingkat kekerasan mekanis

Bilangan Rossby Ro Gaya inersia pada geofisika Z atau

Bilangan Rouse Transpor sedimen P

Bilangan Schmidt Sc Dinamika fluida (transfer massa dan difusi)

Bilangan Sherwood Sh Transfer massa dengan konveksi yang dipaksa

Bilangan Sommerfeld Pelumasan batas

Bilangan Stanton St Transfer panas pada konveksi yang dipaksa

Bilangan Stefan Ste Transfer panas ketika terjadi perubahan fase

Bilangan Stokes Stk Dinamika partikel

Tegangan ε Sains material, elastisitas Bilangan Strouhal Sr Aliran bergelombang dan kontinu

Bilangan Taylor Ta Aliran fluida berotasi nonlinearitas dari gelombang gravitasi permukaan pada

Bilangan Ursell U lapisan fluida dangkal

Faktor van 't Hoff i Analisa kuantitatif (Kf dan Kb)

Parameter Wallis J* Kecepatan nondimensional dalam aliran multifase Bilangan kecepatan Kecepatan pembakaran berlapis relatif terhadap gas

pembentukan api hidrogen Aliran multifase dengan permukaan bergeombang yang

Bilangan Weber We kuat

Bilangan Weissenberg Wi Aliran viskoelastik

Bilangan Womersley α Aliran bergelombang dan kontinu

[sunting] Satuan tak berdimensi bernilai tetap (konstan)

Beberapa konstanta fisika dasar seperti kecepatan cahaya dalam ruang vakum, konstanta gravitasi semesta, konstanta Planck, dan lain sebagainya hanya memiliki satu nilai. Kegunaan dari satuan tak berdimensi fisis ini tidak dapat dipisahkan dari sistem, nilainya ditentukan dari hasil eksperimen.

Dimensionless quantity

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In dimensional analysis, a dimensionless quantity is a quantity without an associated physical dimension. It is thus a "pure" number, and as such always has a dimension of 1. Dimensionless quantities are widely used in mathematics, physics, engineering, economics, and in everyday life (such as in counting). Numerous well-known quantities, such as π, e, and φ, are dimensionless.

Dimensionless quantities are often defined as products or ratios of quantities that are not dimensionless, but whose dimensions cancel out when their powers are multiplied. This is the case, for instance, with the engineering strain, a measure of deformation. It is defined as change in length over initial length but, since these quantites both have dimensions L (length), the result is a dimensionless quantity.

A dimensionless quantity is not always a ratio; for instance, the number of people N in a room is a dimensionless quantity.

Contents

[hide]

 1 Properties  2 Buckingham π theorem o 2.1 Example  3 Standards efforts  4 Examples  5 List of dimensionless quantities  6 Dimensionless physical constants  7 See also  8 References  9 External links

[edit] Properties

 Even though a dimensionless quantity has no physical dimension associated with it, it can still have dimensionless units. It is sometimes helpful to use the same units in both the numerator and denominator, such as kg/kg, to show the quantity being measured (for example, to distinguish a mass ratio from a volume ratio). The quantity may also be given as a ratio of two different units that have the same dimension (for instance, light years over meters). This may be the case when calculating slopes in graphs, or when making unit conversions. Such notation does not indicate the presence of physical dimensions, and is purely a notational convention. Other common dimensionless units are % (= 0.01), ppt (= 10−3), ppm (= 10−6), ppb (= 10−9), and angle units (radians, grad, degrees). Units of amount such as the dozen and the gross are also dimensionless.  The -dimensionless- ratio of two quantities with the same dimensions has the same value regardless of the units used to calculate them. For instance, if body A exerts a force of magnitude F on body B, and B exerts a force of magnitude f on A, then the ratio F/f will always be equal to -1, regardless of the actual units used to measure F and f. This is a fundamental property of dimensionless proportions and follows from the assumption that the laws of physics are independent of the system of units used in their expression. In this case, if the ratio F/f was not always equal to -1, but changed if we switched from SI to CGS, for instance, that would mean that Newton's Third Law's truth or falsity would depend on the system of units used, which would contradict this fundamental hypothesis. The assumption that the laws of physics are not contingent upon a specific unit system is also closely related to the Buckingham π theorem. A formulation of this theorem is that any physical law can be expressed as an identity (always true equation) involving only dimensionless combinations (ratios or products) of the variables linked by the law (e.g., pressure and volume are linked by Boyle's Law -they are inversely proportional). If the dimensionless combinations' values changed with the systems of units, then the equation would not be an identity, and Buckingham's theorem would not hold.

[edit] Buckingham π theorem

Another consequence of the Buckingham π theorem of dimensional analysis is that the functional dependence between a certain number (say, n) of variables can be reduced by the number (say, k) of independent dimensions occurring in those variables to give a set of p = n − k independent, dimensionless quantities. For the purposes of the experimenter, different systems which share the same description by dimensionless quantity are equivalent.

[edit] Example

The power consumption of a stirrer with a given shape is a function of the density and the viscosity of the fluid to be stirred, the size of the stirrer given by its diameter, and the speed of the stirrer. Therefore, we have n = 5 variables representing our example.

Those n = 5 variables are built up from k = 3 dimensions which are:

 Length: L (m)  Time: T (s)  Mass: M (kg).

According to the π-theorem, the n = 5 variables can be reduced by the k = 3 dimensions to form p = n − k = 5 − 3 = 2 independent dimensionless numbers which are, in case of the stirrer:

 Reynolds number (a dimensionless number describing the fluid flow regime)  Power number (describing the stirrer and also involves the density of the fluid) [edit] Standards efforts The CIPM Consultative Committee for Units contemplated defining the unit of 1 as the 'uno', but the idea was dropped.[1][2][3][4] [edit] Examples

Consider this example: Sarah says, "Out of every 10 apples I gather, 1 is rotten.". The rotten- to-gathered ratio is (1 apple) / (10 apples) = 0.1 = 10%, which is a dimensionless quantity. Another more typical example in physics and engineering is the measure of plane angles. An angle is measured as the ratio of the length of a circle's arc subtended by an angle whose vertex is the centre of the circle to some other length. The ratio, length divided by length, is dimensionless. When using radians as the unit, the length that is compared is the length of the radius of the circle. When using degree as the units, the arc's length is compared to 1/360 of the circumference of the circle. [edit] List of dimensionless quantities

All numbers are dimensionless quantities. Certain dimensionless quantities of some importance are given below:

Standard Name Definition Field of application symbol

Abbe number V optics (dispersion in optical materials)

chemistry (Proportion of "active" molecules

Activity coefficient γ or atoms)

climatology, astronomy (reflectivity of

Albedo α surfaces or bodies)

Archimedes number Ar motion of fluids due to density differences

[5] Arrhenius number α Ratio of activation energy to thermal energy

Atomic weight M chemistry

[6]

Bagnold number Ba flow of bulk solids such as grain and sand.

the ratio of heat transfer irreversibility to Bejan number Be total irreversibility due to heat transfer and (thermodynamics) fluid friction[7]

Bejan number dimensionless pressure drop along a Be (fluid mechanics) channel[8]

[5]

Bingham number Bm Ratio of yield stress to viscous stress

Biot number Bi surface vs. volume conductivity of solids

Bodenstein number residence-time distribution

[9]

Bond number Bo capillary action driven by buoyancy

heat transfer by conduction from the wall to a

Brinkman number Br viscous fluid

Brownell-Katz combination of capillary number and Bond number number

Capillary number Ca fluid flow influenced by surface tension

Coefficient of static μs friction of solid bodies at rest friction

Coefficient of kinetic μk friction of solid bodies in translational motion friction

Colburn j factor dimensionless heat transfer coefficient

Courant-Friedrich- ν numerical solutions of hyperbolic PDEs [10] Levy number

Damkohler number Da reaction time scales vs. transport phenomena

Damping ratio ζ the level of damping in a system

Darcy friction factor Cf or f fluid flow

Dean number D vortices in curved ducts

Deborah number De rheology of viscoelastic fluids

Decibel dB ratio of two intensities, usually sound

Drag coefficient Cd flow resistance

ratio of electric surface conductivity to the

Dukhin number Du electric bulk conductivity in heterogeneous systems Euler's number e mathematics

Eckert number Ec convective heat transfer

Ekman number Ek geophysics (frictional (viscous) forces)

Elasticity widely used to measure how demand or E (economics) supply responds to price changes

Eötvös number Eo determination of bubble/drop shape

Ericksen number Er liquid crystal flow behavior

hydrodynamics (pressure forces vs. inertia

Euler number Eu forces)

Fanning friction f fluid flow in pipes [11] factor

Feigenbaum α,δ chaos theory (period doubling) [12] constants

Fine structure α quantum electrodynamics (QED) constant

f-number f optics, photography

Foppl–von Karman thin-shell buckling number

Fourier number Fo heat transfer

[13]

Fresnel number F slit diffraction

Froude number Fr wave and surface behaviour

Gain electronics (signal output to signal input)

Galilei number Ga gravity-driven viscous flow

Golden ratio mathematics and aesthetics

Graetz number Gz heat flow

Grashof number Gr free convection

Hatta number Ha adsorption enhancement due to chemical

reaction

Hagen number Hg forced convection

Hydraulic gradient i groundwater flow

Karlovitz number turbulent combustion turbulent combustion

Keulegan–Carpenter ratio of drag force to inertia for a bluff object KC number in oscillatory fluid flow

ratio of the molecular mean free path length

Knudsen number Kn to a representative physical length scale

Kt/V medicine

Kutateladze number K counter-current two-phase flow

Laplace number La free convection within immiscible fluids

ratio of mass diffusivity and thermal

Lewis number Le diffusivity

lift available from an airfoil at a given angle of

Lift coefficient CL attack

Lockhart-Martinelli χ flow of wet gases [14] parameter

ratio of a resistive time to an Alfvén wave

Lundquist number S crossing time in a plasma

Mach number M gas dynamics

Magnetic Reynolds

Rm magnetohydrodynamics number

Manning roughness n open channel flow (flow driven by gravity) [15] coefficient

Marangoni flow due to thermal surface

Marangoni number Mg tension deviations

Morton number Mo determination of bubble/drop shape

Nusselt number Nu heat transfer with forced convection