Conduction

Question 1

General equation for conduction, Q.

Answer 1

Q. = - k A dT/dt
Q. = - k A (T1-T2)/Δx

Question 2

What does k stand for (W/mK)

Answer 2

K = thermal conductivity

• measure of materials ability to conduct heat
• material property
• temperature dependent
• find using data books (LBOTF)

Question 3

Definition of diffusivity, α (m^2/s)

Answer 3

α = heat conduction / heat storage = k / ρc

the larger α, the faster the propagation of heat into the medium

Question 4

Q. =

q. =

Answer 4

Q. = heat transfer rate (W)
q. = heat flux (W/m^2)

q. = Q. / A

Question 5

Topics thought about when engineering HX (Heat exchangers)

Answer 5

1. Experimental approach
\+ Actual physical system
\+ Real measurements are used
- Expensive, time consuming
- Often impractical

2. Analytical approach
   + Fast
   + inexpensive assumptions and approximations

Question 6

Definition of transient

Answer 6

implies variation with time

Question 7

Heat generation equation

Answer 7

E. gen = ∫ e. gen dV

therefore volumetric phenomenon

Question 8

Energy conservation equation for 1D plane wall

Answer 8

d/dx(k dT/dx) + e.gen = ρc dT/dt

for steady state, no heat gen - d2T/dx2 = 0
pg.47 LBOTF

Question 9

Equation used to find exact temperature at any point of a material.

Answer 9

Tx = (T2-T1)x + T1

L

Question 10

Equation for resistance

Answer 10

R = L / kA

Question 11

For materials with varying k, how to find q.cond

Answer 11

q.cond * L = - (aT^4/4 + bT^3/3 + cT^2/2 + d*T)

a, b, c, d all vary for each material

Question 12

Equation for energy used to change state

solid liquid

Answer 12

ρ * dδ/dt * Δh(fus)

where dδ/dt measured in m/s

Question 13

Energy conservation equation for 1D long cylinder

Answer 13

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

Question 14

Equation for resistance of a cylinder

Answer 14

Rcyl = ln(r(out) / r(in))

k2πL

Question 15

Energy conservation equation for 1D sphere

Answer 15

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

Question 16

Definition of adiabatic system

Answer 16

system where no heat or matter is transferred.

Question 17

Equation for U - overall heat transfer coefficient

Answer 17

1
U = ——————————————–
(1/h(conv)A) + Δx/kA + 1/h(rad)A)

Q. = UAΔT(overall)

Question 18

What is thermal contact resistance, and equation for it

Answer 18

Resistance seen between two different materials meeting.
Due to surface roughness, material properties, temp and pressure interface.

R(c) = 1 / h(c)A

Question 19

What does a lumped system approach imply about heat conductivity

Answer 19

It implies high k, which would mean a uniform temperature inside a body

Question 20

Lumped system approach equation for temp variation

Answer 20

T(t) - T(∞)
————- = e^(-bt)
T(i) - T(∞)

where b = hA(s) / ρVc

Question 21

how does the temp of lumped system approach the system temp?

How does this vary with ‘b’ value

Answer 21

exponentially

larger b, means faster approach

Question 22

Definition and equation for Biot number

Answer 22

Definition:
the ratio between heat transfer resistances. i.e.

Bi = Convection resistance / Conduction resistance

Bi = hL(c) / k where L(c) = V / A(s)

Question 23

When can a lumped system analysis be applied, with respect to Bi

Answer 23

Bi ≤ 0.1

Question 24

Definition and equation for Fourier number

Answer 24

Definition:
characterizes transient heat conduction

Fo = αt / L(c)^2 where: t = time
α = k / ρc

Question 25

Lumped system approach equation for temp variation, using Bi and Fo

Answer 25

T(t) - T(∞)
————- = e^(-Bi * Fo)
T(i) - T(∞)

Question 26

For fin analysis, what assumptions are taken

Answer 26

steady state
no heat generation in fin
constant k 
constant h around fin
constant h along fin

Question 27

For the case that:
fin in very long and temperature of end of fin same as T(∞)
Equation for Q.

Answer 27

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

Question 28

For the case that:
fin is insulated (adiabatic)
Equation for Q.

Answer 28

Q.longfin = -kAdT/dx = √(hpkA(c)) * (Tb - T∞) * tanh(mL)

found page 47 LBOTF

Question 29

Equation to calculate fin efficiency

Answer 29

η = Qfin / Qfin,max

Qfin,max = hA(fin)(Tb - T∞)

Question 30

Equation for fin effectiveness, εfin

Answer 30

εfin = η * Afin / Ab

Question 31

For what value of εfin is it justifiable to use a fin

Answer 31

εfin > 1 fin justifiable

if εfin < 1 this implies fin insulates the surface

Question 32

Equation for overall fin effectiveness, εoverall

Answer 32

εoverall = Aunfin + ηfinAfin
————————
Anofin

Question 33

Equation for shape factor for plane wall

Answer 33

S = A/L

Question 34

How to calculate shape factor of complex shape

Answer 34

divide into smaller shapes and then add these shape factors up.
All use look up tables

Question 35

How to use shape factor to calculate heat loss of shape

Answer 35

Q. = Sk(T1-T2)