Equations & Units Flashcards
force
N (mass*accel)
kg.m / s^2
energy
J
N.m
kg.m^2 / s^2
power
W
J/s
kg.m^2 / s^3
pressure (or stress)
Pa
N/m^2
J/m^3
kg / m.s^2
mass flux [units]
kg / m^2.s
dynamic viscosity [units]
kg / m.s
Pa.s
kinematic viscosity [units]
m^2 / s (dynamic visc. / density)
diffusion; time as a function of κ [eq]
t = d^2 / κ
thermal energy, E [eq, x2]
E ~ (3/2)kT (k = Boltzmann)
E ~ m c_p ΔT
gas diffusivity, κ [eq]
κ ~ vλ
sound speed, v [eq]
v ~ (E_c / m)^1/2
where m ~ molar mass / N_Av
(E_c = binding energy)
diffusivity [units]
m^2 / s
energy of a photon, E [eq]
E ~ hf
h = Planck’s C, 10^-33
Boltzmann distribution for T, P(T) [eq]
P ~ e^-(E/kT)
thermal conductivity, k [eq]
k ~ κρ c_p
κ = thermal diffusivity
thermal conductivity, k [units]
W / m.K
latent heat as a fxn of binding energy
units: J/kg
L ~ E / (μ / N_Av) ~ E * N_Av / μ
specific heat, c_p [eq]
3R/μ
diffusion as a fxn of sound speed [eq]
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)
adiabatic cooling [eq, plus process]
-ΔT/Δz ~ g / c_p
from equating PE & TE: mgΔz~m c_p ΔT
specific heat, c_p [units]
J/kg.K
conductive heat flux (via Fourier’s law)
F ~ -k(ΔT/Δx)
heat flux [units]
W/m^2 (or J/m^2.s)
heat production [eq]
H ~ ρ c_p (ΔT/Δt)
heat production [units]
W/m^3
equation relating heating rate, conduction, heat production [eq]
ρ c_p (T/t) ~ k(T/d^2) + H
advective heat flux
F ~ u ΔT c_p ρ
equation relating heating rate, advection, conduction, heat production [eq]
(ΔT/Δt) + u(ΔT/Δd) ~ κ (ΔT / Δd^2) + H/(ρ c_p)
radiative heat flux (with constants)
F ~ εσT^4
ε: emissivity ~ 1
σ: Stefan-Boltzmann ~ 6e-8 W/m^2.K^4
generic mass flux equation
dc/dt ~ -dF/dx + sources + sinks
accumulation, storage ~ spatial variation in flux +…
Fick’s law of diffusion (mass) [eq]
F_m ~ -κ(dc/dx)
Fourier’s law of conduction [eq]
F ~ -k(dT/dx) ~ -κ(d (ρ c_p T) / dx)
Flux of a solute [eq]
F ~ uc
gas mass diffusion [eq]
κ ~ vλ
Same as thermal diffusion for gases, because all thermal energy is translational KE for gases
particle diffusivity as a function of diameter [eq]
κ ~ kT/3πηd (Stokes-Einstein)
dynamic viscosity via sound speed, etc. [eq]
η ~ κρ
η ~ vλρ
linear momentum [eq]
p ~ mv
linear momentum [units]
kg.m / s
intensive linear momentum [units] (use in budgets)
m/s (kg/m.s/kg—normalized by mass)
momentum flux [eq]
F ~ η(du/dy)
viscosity*intensive momentum gradient
Newton’s Law of Viscosity
advective mass flux [eq]
F ~ uc (u = m/s, c = kg/m3)
advective momentum flux [eq]
F ~ u_fluid * u_stuff * ρ_fluid
u_fluid = u_stuff if momentum diffusion and flow are in the same (x) direction
Reynolds number [eq]
Re ~ uL/ν (nu—kinematic visc.)
~ ρuL/μ (mu = dynamic viscosity)
(essentially advection over diffusion)
Coriolis force [eq]
2uωm sin(latitude)
Bernoulli equation
P1/ρ + 1/2 u1^2 + gz1 ~ P2/ρ + 1/2 u2^2 + gz2
ε, energy dissipation rate [units]
W/kg or m^2/s^3
t, characteristic eddy turnover time [eq]
l (length scale) / u_l
E_l, characteristic turbulent kinetic energy [units]
J/kg
E_l, characteristic turbulent kinetic energy [eq, x2]
u_l ^2
εl)^(2/3
ε, energy dissipation rate [eq]
u_l ^3 / l
~ u_L ^3 / L
Every eddy length scale has the same ε!
u_l, characteristic eddy velocity scale
(εl)^(1/3)
u_L relative to u, mean flow [eq]
u_L ~ 0.2u
turbulent diffusivity [eq]
u_L * L * 0.4
ω, angular frequency, GENERAL [eq]
2π/T (T ~ period [s])
spring equation (solve for restoring force)
F ~ -kx
x = distance from equilibrium; k = spring constant
spring constant, k [units]
N/m
= kg/s^2
spring velocity [eq]
v ~ ωx
ω, angular frequency, SPRING [eq]
sqrt(k/m) (k ~ spring constant)
ω, angular frequency, PENDULUM [eq]
sqrt(g/L) (L = length scale of pendulum)
horizontal length scale of a pendulum [eq]
d ~ Lθ
elastic modulus [eq, x2]
E ~ stress/strain
E ~ E_c / a^3 (E_c = binding energy; a = atomic spacing)
damping timescale for an oscillator [eq]
t ~ k/fω^2
~ m/f
(k = spring constant; f = “damping effectiveness”, units kg/s)
ω, oscillation frequency, for continuous materials [eq]
ω ~ sqrt(E/ρL^2)
E = elastic modulus
k, spring constant, for continuous materials [eq]
k ~ EA / L
E = elastic modulus
phase velocity from elastic modulus [eq]
v_ph ~ sqrt(E/ρ)
surface area and volume of a sphere [eqs]
A = πd^2 or 4πr^2 V = 4/3 π r^3 ~ 4r^3 ~ (1/6)πd^3
drag force [eq, x2]
1/2 c_D ρAu^2 (turbulent flow)
uLη (laminar flow)
c_D = coefficient of friction η = dynamic viscosity (Pa.s)
buoyant force [eq]
ρ_fluid Vg
Peclet number
uL/κ
Navier-Stokes equation
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
