fundamentals of engineering supplied-reference handbook - Ventech!

Use the cylinder diameter in the evaluation of the Nusselt
and Reynolds numbers.
Re n c
1 – 4
4 – 40
40 – 4,000
4,000 – 40,000
40,000 – 250,000
0.330
0.385
0.466
0.618
0.805
0.989
0.911
0.683
0.193
0.0266
For flow past a constant-temperature sphere.
Nu = 2.0 + 0.60Re 0.5 Pr 1/3
(1 < Re < 70,000, 0.6 < Pr < 400)
Use the sphere diameter in the evaluation of the Nusselt and
Reynolds numbers.
Conductive Heat Transfer
Steady Conduction with Internal Energy Generation
For one-dimensional steady conduction, the equation is
2
2
d T/dx + Qgen
k = 0 � , where
Qgen � = the heat generation rate per unit volume, and
k = the thermal conductivity.
For a plane wall:
T
Q�
Q�
Q�
L
2
⎛ x
−
⎝
2
⎞
⎛ T − T
⎞⎛
x ⎞
⎛ T + T
gen
( ) ⎜ ⎟ s2
s1
s1
s2
x = 1 + ⎜ ⎟⎜
⎟ + ⎜ ⎟
2k ⎜ 2
L ⎟ 2 ⎝ L ⎠ 2 ⎠
" "
Q Q 2Q L � � � + = , where
1
2
( dT dx)
L
k(
dT dx)
L
"
1 = k −
"
2 = −
⎠
gen
For a long circular cylinder:
⎝
⎠
⎝
⎞
70
1
r
T
d
dr
dT Q�
⎛ ⎞ gen
⎜r
⎟ + = 0
⎝ dr ⎠ k
Q�
2
r ⎛ 2
r ⎞
−
k ⎜ 2 4 r ⎟
⎝ 0 ⎠
gen 0
() r = ⎜1
⎟ + Ts
r Q�
′ = π � , where
2
0 gen Q
HEAT TRANSFER (continued)
Q′ � = the heat-transfer rate from the cylinder per unit
length.
Transient Conduction Using the Lumped Capacitance
Method
If the temperature may be considered uniform within the
body at any time, the change of body temperature is given
by
� hA T − T = −ρc
V dT dt
( ) ( )
Q = s ∞ p
The temperature variation with time is
−(
hAs / ρcpV
)t
T – T∞ = (Ti – T∞) e
The total heat transferred up to time t is
Qtotal = ρcPV (Ti – T), where
ρ = density,
V = volume,
cP = heat capacity,
t = time,
As = surface area of the body,
T = temperature, and
h = the heat-transfer coefficient.
The lumped capacitance method is valid if
Biot number = Bi = hV/kAs