Conduction Flashcards
General equation for conduction, Q.
Q. = - k A dT/dt Q. = - k A (T1-T2)/Δx
What does k stand for (W/mK)
K = thermal conductivity
- measure of materials ability to conduct heat
- material property
- temperature dependent
- find using data books (LBOTF)
Definition of diffusivity, α (m^2/s)
α = heat conduction / heat storage = k / ρc
the larger α, the faster the propagation of heat into the medium
Q. =
q. =
Q. = heat transfer rate (W) q. = heat flux (W/m^2)
q. = Q. / A
Topics thought about when engineering HX (Heat exchangers)
1. Experimental approach \+ Actual physical system \+ Real measurements are used - Expensive, time consuming - Often impractical
- Analytical approach
+ Fast
+ inexpensive assumptions and approximations
Definition of transient
implies variation with time
Heat generation equation
E. gen = ∫ e. gen dV
therefore volumetric phenomenon
Energy conservation equation for 1D plane wall
d/dx(k dT/dx) + e.gen = ρc dT/dt
for steady state, no heat gen - d2T/dx2 = 0
pg.47 LBOTF
Equation used to find exact temperature at any point of a material.
Tx = (T2-T1)x + T1
L
Equation for resistance
R = L / kA
For materials with varying k, how to find q.cond
q.cond * L = - (aT^4/4 + bT^3/3 + cT^2/2 + d*T)
a, b, c, d all vary for each material
Equation for energy used to change state
solid liquid
ρ * dδ/dt * Δh(fus)
where dδ/dt measured in m/s
Energy conservation equation for 1D long cylinder
1/r * d/dr( r k dT/dr) + e.gen = ρc dT/dt
for steady state, no heat gen:
d/dr(r dT/dr) = 0
Therfore:
T = C1 ln(r) + C2
pg.47 LBOTF
Equation for resistance of a cylinder
Rcyl = ln(r(out) / r(in))
k2πL
Energy conservation equation for 1D sphere
1/r^2 * d/dr( r^2 k dT/dr) + e.gen = ρc dT/dt
for steady state, no heat gen:
d/dr(r^2 dT/dr) = 0
or
r d^2T/dr^2 + 2 dT/dr = 0
pg.47 LBOTF
Definition of adiabatic system
system where no heat or matter is transferred.
Equation for U - overall heat transfer coefficient
1
U = ——————————————–
(1/h(conv)A) + Δx/kA + 1/h(rad)A)
Q. = UAΔT(overall)
What is thermal contact resistance, and equation for it
Resistance seen between two different materials meeting.
Due to surface roughness, material properties, temp and pressure interface.
R(c) = 1 / h(c)A
What does a lumped system approach imply about heat conductivity
It implies high k, which would mean a uniform temperature inside a body
Lumped system approach equation for temp variation
T(t) - T(∞)
————- = e^(-bt)
T(i) - T(∞)
where b = hA(s) / ρVc
how does the temp of lumped system approach the system temp?
How does this vary with ‘b’ value
exponentially
larger b, means faster approach
Definition and equation for Biot number
Definition:
the ratio between heat transfer resistances. i.e.
Bi = Convection resistance / Conduction resistance
Bi = hL(c) / k where L(c) = V / A(s)
When can a lumped system analysis be applied, with respect to Bi
Bi ≤ 0.1
Definition and equation for Fourier number
Definition:
characterizes transient heat conduction
Fo = αt / L(c)^2 where: t = time
α = k / ρc
Lumped system approach equation for temp variation, using Bi and Fo
T(t) - T(∞)
————- = e^(-Bi * Fo)
T(i) - T(∞)
For fin analysis, what assumptions are taken
steady state no heat generation in fin constant k constant h around fin constant h along fin
For the case that:
fin in very long and temperature of end of fin same as T(∞)
Equation for Q.
Q.longfin = -kAdT/dx = √(hpkA(c)) * (Tb - T∞)
where:
p = perimeter
Tb = temperature at beginning of fin
T∞ = surrounding temp (end of fin too)
found page 47 LBOTF
For the case that:
fin is insulated (adiabatic)
Equation for Q.
Q.longfin = -kAdT/dx = √(hpkA(c)) * (Tb - T∞) * tanh(mL)
found page 47 LBOTF
Equation to calculate fin efficiency
η = Qfin / Qfin,max
Qfin,max = hA(fin)(Tb - T∞)
Equation for fin effectiveness, εfin
εfin = η * Afin / Ab
For what value of εfin is it justifiable to use a fin
εfin > 1 fin justifiable
if εfin < 1 this implies fin insulates the surface
Equation for overall fin effectiveness, εoverall
εoverall = Aunfin + ηfinAfin
————————
Anofin
Equation for shape factor for plane wall
S = A/L
How to calculate shape factor of complex shape
divide into smaller shapes and then add these shape factors up.
All use look up tables
How to use shape factor to calculate heat loss of shape
Q. = Sk(T1-T2)
