Fluids & Heat Flashcards
How do you find a velocity vector field?
phi_theta/r , -phi_r
how to find stagnation points?
intersection of phi = 0 with streamlines
how to show flow is irrotational?
show del cross velocity = 0 (curl - take determinant of derivatives)
what law relates pressure, density and temperature?
ideal gas law P = rho R T where R is the ideal gas constant.
relate dynamic and kinematic viscosity.
dynamic viscosity equals kinematic viscosity times density.
relate force and shear stress
F = tau A
relate shear stress and velocity
tau = mu du/dy
relate height and surface tension for a liquid in a tube.
h = (2 sigma cos theta)/(gamma R) where sigma is the surface tension, theta is the angle of contact, gamma is the specific weight of the liquid and R is the radius of the tube.
state fourier’s law. what is the significance?
q_x” = -k dT/dx. Conductive heat transfer. relates heat flux, temperature and thermal conductivity.
state Newton’s Law of Cooling and the significance.
q” = h(T_s - T_inf). Convective heat transfer. Relates heat flux, surface temperature and ambient temperature. h is convection heat transfer coefficient.
state Stefan - Boltzmann law and significance.
E_b = sigma T_s^4. Radiative heat transfer. Relates emissive power, surface temperature and stefan-boltzmann constant.
Define thermal conductivity.
k. Property of a material to conduct heat. Measured in W/mK
Define density.
rho. The degree of compactness of a substance. Measured in kg/m^3.
Define specific heat.
c_p. The heat required to raise the temperature of a given substance by one degree. J/kgK.
Define thermal diffusivity.
alpha. Measures the rate of transfer of heat of a material from the hot side to the cold side. Analogous to whether a material is cold to the tough. alpha = k / rho c_p.
State the heat equation.
nabla^2T + q/k = 1/alpha dT/dt
define critical insulation radius
thermal conductivity divided by convection heat transfer coefficient. This ratio allows for maximum heat transfer. r_cr = k/h
m^2 for fins
m^2 = (h V)/(kA_c)
boundary conditions for adiabatic tip
dtheta/dx (@x=L) = 0
boundary conditions for convection heat transfer tip
htheta(L) = -kdtheta/dx(@x=L)
boundary conditions for prescribed temperature tip
theta(L) = theta_L
boundary conditions for infinite fin
theta(L) = 0
view factor
geometric quantity, independent of the surface properties and temperature. ranges between 0 and 1. between two surfaces A1 and A2,
F12 = 1/A1 integral A2 integral A1 (cos theta1 costheta2)/(pi r^2) dA1 dA2
how can you relate view factors in two opposite directions? (reciprocity relation)
A_1F_12 = A_2F_21
Biot number
ratio of heat transfer resistance inside of and at the surface of a body. Bi = hL_c/k_b. Where L_c = V_body/A_surface
What are the assumptions of the lumped capacitance method?
- Temperature of the solid is spatially uniform at any given instant. 2. Resistance to conduction within the solid is small compared to the resistance between the solid and its surroundings.
What happens when Bi «_space;1?
The resistance to conduction within the solid is much less than the resistance to convection which means we can assume uniform temperature distribution.
What happens when Bi»_space; 1?
The temperature difference across the solid is much larger than between the surface, so the temperature gradient is more significant.
What is the Fourier number?
Fo = alpha t/L_c^2 where L_c = V_body/A_surface.
What is the lumped capacitance equation?
(rho V c_p/hA_s)ln(theta_i/theta) = t. Where theta_i=T_i - T_infty and theta = T - T_infty
how to solve cooling problems.
First calculate biot number, if it is much less than 1 you can use lumped capacitance.
what is the equation for defining wavelength?
lambda = c/v
What is q from surface a to b for blackbody radiation?
q_ab = A_a F_ab sigma(T_b^4 - T_a^4)
equation for irradiation
G = q_ab/A_b
summation rule for view factors
for surfaces a, b, c, F_aa + F_ab + F_ac = 1
q for two isothermal diffuse, gray surfaces that form an enclosure and are each characterized by uniform radiosity and irradiation? what happens if it’s blackbody?
q_12 = q_1 = -q_2 = [sigma(T_1^4 - T_2^4)] / [ (1-e_1)/(e_1A_1) + 1/(A_1F_12) + (1-e_2)/(e_2A_2)], for blackbody e=1
Bernoulli’s Equation
v_1^2/2 + gz_1 + P_1/rho_1 = v_2^2/2 +gz_2 + P_2/rho_2
Navier stokes, incompressible flow
del u = 0
du/dt + udelu = -1/rho delP + nu del^2u + f
Reynolds number
UL/nu
Reynolds number, pipeflow
4Q/(pi nu d) or U_avg d/nu
Mach number
M = u/c, represents how much faster than the speed of sound
equation for fins
d^2T/dz^2 = hP/kA(T-T_infty) where P is the perimeter
Q for two black bodies
Q_12 = sigma A1 F_12(T_1^4 T_2^4)
stability condition for explicit finite differences
the coefficient for the node of interest at the previous time must be greater than or equal to zero
energy balance
dE/dt = E_in - E_out + E_gen
what’s the difference between explicit and implicit finite difference?
explicit uses current time step, implicit uses the next one.
what are the units of heat flux?
[w/m^2]
when is flow turbulent?
Re > 4000
when is flow laminar
Re < 2100
prandtl number
viscous diffusion rate/ thermal diffusion rate = c_p mu/k
nusselt number
total heat transfer / conductive heat transfer = hL/k
what is a thermal boundary layer?
the thickness, delta _t is the value for y for which [(T_s -
T)/(T_s - T_inf)] = .99. it develops if fluid free streams and surface temperatures differ.
thermal resistance, conduction, cartesian
R_th = L/(kA)
thermal resistance, convection
R_th = 1/(hA)
thermal resistance, radiation
R_th = 1/(h_r A)
thermal resistance, conduction, cylindrical
R_th = ln(r2/r1)/(2piLk)
thermal resistance, conduction, spherical
R_th = 1/(4pik) (1/r1 - 1/r2)
laplacian in cylindrical coordinates
1/r d/dr (r dT/dr) + 1/r^2 d^2T/dphi^2 + d^2T/dz^2
laplacian in spherical coordinates
1/r^2 d/dr (r^2 dT/dr) + 1/(r^2sintheta) d/dtheta (sin theta dT/dtheta) + 1/(r^2sin^2theta) d^2T/dphi^2
temperature distribution for a fin of infinite length
theta/theta_b = e^-mx
heat transfer rate for a fin of infinite length
sqrt(hPkA_c) theta_b
when can you assume a fin to be infinite
mL>2
what is the difference between heat flux and heat loss
heat loss is heat flux times area
