Fluids & Heat

Question 1

How do you find a velocity vector field?

Answer 1

phi_theta/r , -phi_r

Question 2

how to find stagnation points?

Answer 2

intersection of phi = 0 with streamlines

Question 3

how to show flow is irrotational?

Answer 3

show del cross velocity = 0 (curl - take determinant of derivatives)

Question 4

what law relates pressure, density and temperature?

Answer 4

ideal gas law P = rho R T where R is the ideal gas constant.

Question 5

relate dynamic and kinematic viscosity.

Answer 5

dynamic viscosity equals kinematic viscosity times density.

Question 6

relate force and shear stress

Answer 6

F = tau A

Question 7

relate shear stress and velocity

Answer 7

tau = mu du/dy

Question 8

relate height and surface tension for a liquid in a tube.

Answer 8

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.

Question 9

state fourier’s law. what is the significance?

Answer 9

q_x” = -k dT/dx. Conductive heat transfer. relates heat flux, temperature and thermal conductivity.

Question 10

state Newton’s Law of Cooling and the significance.

Answer 10

q” = h(T_s - T_inf). Convective heat transfer. Relates heat flux, surface temperature and ambient temperature. h is convection heat transfer coefficient.

Question 11

state Stefan - Boltzmann law and significance.

Answer 11

E_b = sigma T_s^4. Radiative heat transfer. Relates emissive power, surface temperature and stefan-boltzmann constant.

Question 12

Define thermal conductivity.

Answer 12

k. Property of a material to conduct heat. Measured in W/mK

Question 13

Define density.

Answer 13

rho. The degree of compactness of a substance. Measured in kg/m^3.

Question 14

Define specific heat.

Answer 14

c_p. The heat required to raise the temperature of a given substance by one degree. J/kgK.

Question 15

Define thermal diffusivity.

Answer 15

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.

Question 16

State the heat equation.

Answer 16

nabla^2T + q/k = 1/alpha dT/dt

Question 17

define critical insulation radius

Answer 17

thermal conductivity divided by convection heat transfer coefficient. This ratio allows for maximum heat transfer. r_cr = k/h

Question 18

m^2 for fins

Answer 18

m^2 = (h V)/(kA_c)

Question 19

boundary conditions for adiabatic tip

Answer 19

dtheta/dx (@x=L) = 0

Question 20

boundary conditions for convection heat transfer tip

Answer 20

htheta(L) = -kdtheta/dx(@x=L)

Question 21

boundary conditions for prescribed temperature tip

Answer 21

theta(L) = theta_L

Question 22

boundary conditions for infinite fin

Answer 22

theta(L) = 0

Question 23

view factor

Answer 23

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

Question 24

how can you relate view factors in two opposite directions? (reciprocity relation)

Answer 24

A_1F_12 = A_2F_21

Question 25

Biot number

Answer 25

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

Question 26

What are the assumptions of the lumped capacitance method?

Answer 26

1. 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.

Question 27

What happens when Bi << 1?

Answer 27

The resistance to conduction within the solid is much less than the resistance to convection which means we can assume uniform temperature distribution.

Question 28

What happens when Bi >> 1?

Answer 28

The temperature difference across the solid is much larger than between the surface, so the temperature gradient is more significant.

Question 29

What is the Fourier number?

Answer 29

Fo = alpha t/L_c^2 where L_c = V_body/A_surface.

Question 30

What is the lumped capacitance equation?

Answer 30

(rho V c_p/hA_s)ln(theta_i/theta) = t. Where theta_i=T_i - T_infty and theta = T - T_infty

Question 31

how to solve cooling problems.

Answer 31

First calculate biot number, if it is much less than 1 you can use lumped capacitance.

Question 32

what is the equation for defining wavelength?

Answer 32

lambda = c/v

Question 33

What is q from surface a to b for blackbody radiation?

Answer 33

q_ab = A_a F_ab sigma(T_b^4 - T_a^4)

Question 34

equation for irradiation

Answer 34

G = q_ab/A_b

Question 35

summation rule for view factors

Answer 35

for surfaces a, b, c, F_aa + F_ab + F_ac = 1

Question 36

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?

Answer 36

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

Question 37

Bernoulli's Equation

Answer 37

v_1^2/2 + gz_1 + P_1/rho_1 = v_2^2/2 +gz_2 + P_2/rho_2

Question 38

Navier stokes, incompressible flow

Answer 38

del u = 0 | du/dt + udelu = -1/rho delP + nu del^2u + f

Question 39

Reynolds number

Answer 39

UL/nu

Question 40

Reynolds number, pipeflow

Answer 40

4Q/(pi nu d) or U_avg d/nu

Question 41

Mach number

Answer 41

M = u/c, represents how much faster than the speed of sound

Question 42

equation for fins

Answer 42

d^2T/dz^2 = hP/kA(T-T_infty) where P is the perimeter

Question 43

Q for two black bodies

Answer 43

Q_12 = sigma A1 F_12(T_1^4 T_2^4)

Question 44

stability condition for explicit finite differences

Answer 44

the coefficient for the node of interest at the previous time must be greater than or equal to zero

Question 45

energy balance

Answer 45

dE/dt = E_in - E_out + E_gen

Question 46

what's the difference between explicit and implicit finite difference?

Answer 46

explicit uses current time step, implicit uses the next one.

Question 47

what are the units of heat flux?

Answer 47

[w/m^2]

Question 48

when is flow turbulent?

Answer 48

Re > 4000

Question 49

when is flow laminar

Answer 49

Re < 2100

Question 50

prandtl number

Answer 50

viscous diffusion rate/ thermal diffusion rate = c_p mu/k

Question 51

nusselt number

Answer 51

total heat transfer / conductive heat transfer = hL/k

Question 52

what is a thermal boundary layer?

Answer 52

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.

Question 53

thermal resistance, conduction, cartesian

Answer 53

R_th = L/(kA)

Question 54

thermal resistance, convection

Answer 54

R_th = 1/(hA)

Question 55

thermal resistance, radiation

Answer 55

R_th = 1/(h_r A)

Question 56

thermal resistance, conduction, cylindrical

Answer 56

R_th = ln(r2/r1)/(2piLk)

Question 57

thermal resistance, conduction, spherical

Answer 57

R_th = 1/(4pik) (1/r1 - 1/r2)

Question 58

laplacian in cylindrical coordinates

Answer 58

1/r d/dr (r dT/dr) + 1/r^2 d^2T/dphi^2 + d^2T/dz^2

Question 59

laplacian in spherical coordinates

Answer 59

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

Question 60

temperature distribution for a fin of infinite length

Answer 60

theta/theta_b = e^-mx

Question 61

heat transfer rate for a fin of infinite length

Answer 61

sqrt(hPkA_c) theta_b

Question 62

when can you assume a fin to be infinite

Answer 62

mL>2

Question 63

what is the difference between heat flux and heat loss

Answer 63

heat loss is heat flux times area