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Order of Magnitude > Equations & Units > Flashcards

Equations & Units Flashcards

1
Q

force

A

N (mass*accel)

kg.m / s^2

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2
Q

energy

A

J
N.m
kg.m^2 / s^2

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3
Q

power

A

W
J/s
kg.m^2 / s^3

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4
Q

pressure (or stress)

A

Pa
N/m^2
J/m^3
kg / m.s^2

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5
Q

mass flux [units]

A

kg / m^2.s

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6
Q

dynamic viscosity [units]

A

kg / m.s

Pa.s

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7
Q

kinematic viscosity [units]

A

m^2 / s (dynamic visc. / density)

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8
Q

diffusion; time as a function of κ [eq]

A

t = d^2 / κ

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9
Q

thermal energy, E [eq, x2]

A

E ~ (3/2)kT (k = Boltzmann)

E ~ m c_p ΔT

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10
Q

gas diffusivity, κ [eq]

A

κ ~ vλ

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11
Q

sound speed, v [eq]

A

v ~ (E_c / m)^1/2
where m ~ molar mass / N_Av
(E_c = binding energy)

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12
Q

diffusivity [units]

A

m^2 / s

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13
Q

energy of a photon, E [eq]

A

E ~ hf

h = Planck’s C, 10^-33

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14
Q

Boltzmann distribution for T, P(T) [eq]

A

P ~ e^-(E/kT)

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15
Q

thermal conductivity, k [eq]

A

k ~ κρ c_p

κ = thermal diffusivity

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16
Q

thermal conductivity, k [units]

A

W / m.K

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17
Q

latent heat as a fxn of binding energy

units: J/kg

A

L ~ E / (μ / N_Av) ~ E * N_Av / μ

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18
Q

specific heat, c_p [eq]

A

3R/μ

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19
Q

diffusion as a fxn of sound speed [eq]

A

κ ~ vL

gases: L ~ λ (1e-7 m)
solids/liquids: L ~ a (atomic spacing, 1e-10 m)

(Note: this doesn’t work for mass diffusion in liquids due to translational v. rotational, etc. kinetic energy… it’s ~100x less, around κ ~ 10^-9 m2/s)

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20
Q

adiabatic cooling [eq, plus process]

A

-ΔT/Δz ~ g / c_p

from equating PE & TE: mgΔz~m c_p ΔT

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21
Q

specific heat, c_p [units]

A

J/kg.K

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22
Q

conductive heat flux (via Fourier’s law)

A

F ~ -k(ΔT/Δx)

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23
Q

heat flux [units]

A

W/m^2 (or J/m^2.s)

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24
Q

heat production [eq]

A

H ~ ρ c_p (ΔT/Δt)

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25
Q

heat production [units]

A

W/m^3

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26
Q

equation relating heating rate, conduction, heat production [eq]

A

ρ c_p (T/t) ~ k(T/d^2) + H

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27
Q

advective heat flux

A

F ~ u ΔT c_p ρ

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28
Q

equation relating heating rate, advection, conduction, heat production [eq]

A

(ΔT/Δt) + u(ΔT/Δd) ~ κ (ΔT / Δd^2) + H/(ρ c_p)

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29
Q

radiative heat flux (with constants)

A

F ~ εσT^4

ε: emissivity ~ 1
σ: Stefan-Boltzmann ~ 6e-8 W/m^2.K^4

30
Q

generic mass flux equation

A

dc/dt ~ -dF/dx + sources + sinks

accumulation, storage ~ spatial variation in flux +…

31
Q

Fick’s law of diffusion (mass) [eq]

A

F_m ~ -κ(dc/dx)

32
Q

Fourier’s law of conduction [eq]

A

F ~ -k(dT/dx) ~ -κ(d (ρ c_p T) / dx)

33
Q

Flux of a solute [eq]

A

F ~ uc

34
Q

gas mass diffusion [eq]

A

κ ~ vλ

Same as thermal diffusion for gases, because all thermal energy is translational KE for gases

35
Q

particle diffusivity as a function of diameter [eq]

A

κ ~ kT/3πηd (Stokes-Einstein)

36
Q

dynamic viscosity via sound speed, etc. [eq]

A

η ~ κρ

η ~ vλρ

37
Q

linear momentum [eq]

A

p ~ mv

38
Q

linear momentum [units]

A

kg.m / s

39
Q

intensive linear momentum [units] (use in budgets)

A

m/s (kg/m.s/kg—normalized by mass)

40
Q

momentum flux [eq]

A

F ~ η(du/dy)
viscosity*intensive momentum gradient
Newton’s Law of Viscosity

41
Q

advective mass flux [eq]

A

F ~ uc (u = m/s, c = kg/m3)

42
Q

advective momentum flux [eq]

A

F ~ u_fluid * u_stuff * ρ_fluid

u_fluid = u_stuff if momentum diffusion and flow are in the same (x) direction

43
Q

Reynolds number [eq]

A

Re ~ uL/ν (nu—kinematic visc.)
~ ρuL/μ (mu = dynamic viscosity)
(essentially advection over diffusion)

44
Q

Coriolis force [eq]

A

2uωm sin(latitude)

45
Q

Bernoulli equation

A

P1/ρ + 1/2 u1^2 + gz1 ~ P2/ρ + 1/2 u2^2 + gz2

46
Q

ε, energy dissipation rate [units]

A

W/kg or m^2/s^3

47
Q

t, characteristic eddy turnover time [eq]

A

l (length scale) / u_l

48
Q

E_l, characteristic turbulent kinetic energy [units]

A

J/kg

49
Q

E_l, characteristic turbulent kinetic energy [eq, x2]

A

u_l ^2

εl)^(2/3

50
Q

ε, energy dissipation rate [eq]

A

u_l ^3 / l
~ u_L ^3 / L
Every eddy length scale has the same ε!

51
Q

u_l, characteristic eddy velocity scale

A

(εl)^(1/3)

52
Q

u_L relative to u, mean flow [eq]

A

u_L ~ 0.2u

53
Q

turbulent diffusivity [eq]

A

u_L * L * 0.4

54
Q

ω, angular frequency, GENERAL [eq]

A

2π/T (T ~ period [s])

55
Q

spring equation (solve for restoring force)

A

F ~ -kx

x = distance from equilibrium; k = spring constant

56
Q

spring constant, k [units]

A

N/m

= kg/s^2

57
Q

spring velocity [eq]

A

v ~ ωx

58
Q

ω, angular frequency, SPRING [eq]

A

sqrt(k/m) (k ~ spring constant)

59
Q

ω, angular frequency, PENDULUM [eq]

A

sqrt(g/L) (L = length scale of pendulum)

60
Q

horizontal length scale of a pendulum [eq]

A

d ~ Lθ

61
Q

elastic modulus [eq, x2]

A

E ~ stress/strain

E ~ E_c / a^3 (E_c = binding energy; a = atomic spacing)

62
Q

damping timescale for an oscillator [eq]

A

t ~ k/fω^2
~ m/f
(k = spring constant; f = “damping effectiveness”, units kg/s)

63
Q

ω, oscillation frequency, for continuous materials [eq]

A

ω ~ sqrt(E/ρL^2)

E = elastic modulus

64
Q

k, spring constant, for continuous materials [eq]

A

k ~ EA / L

E = elastic modulus

65
Q

phase velocity from elastic modulus [eq]

A

v_ph ~ sqrt(E/ρ)

66
Q

surface area and volume of a sphere [eqs]

A 
A = πd^2 or 4πr^2
V = 4/3 π r^3 ~ 4r^3 ~ (1/6)πd^3
67
Q

drag force [eq, x2]

A

1/2 c_D ρAu^2 (turbulent flow)
uLη (laminar flow)

c_D = coefficient of friction
η = dynamic viscosity (Pa.s)
68
Q

buoyant force [eq]

A

ρ_fluid Vg

69
Q

Peclet number

A

uL/κ

70
Q

Navier-Stokes equation

A

du/dt ~ ν(d^2u/dx^2…) - (u du/dx + v du/dy…) - 1/ρ dP/dx

OoM: u/t ~ ν(u/x^2) - (u*u/x) - 1/ρ P/x

Order of Magnitude (2 decks)

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