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= 3 ◦ C ∼

temperature T w =100 ◦ C, air flow Prandtl Pr = 0.7

√
τ w /ρ, friction temperature T ∗
= q w /(ρc p u ∗ ), pipe wall
10.3. Consider the Von Ka´ rman–Martinelli heat–momentum analogy
for a turbulent pipe flow:
a. If a two-layer universal velocity profile will be employed,
that is,
+
+
+
u ¯ = y
for 0 < y < 10
+
+
u ¯ = 5.0 + 2.5 � n y + for 10 < y
For a constant wall heat flux, determine the universal temperature profiles at the corresponding two-layer region. Then
determine the Nu D where Nu D = function (Re D , Pr, f ).
b. If Re D = 10 4 , Pr = 1, calculate Nu D from (a) and then compare it with correlation Nu D = 0.023 Re 0.8 · Pr 0.4 . If velocity
D
increases, the laminar sublayer thickness will be increased or
decreased? Why? How about Nu D ?
10.4. Consider a fully developed turbulent flow between two parallel plates with a gap of b and a uniform wall heat flux (q/A) w .
Using the Ka´ rman–Martinelli analogy, determine the turbulent
heat transfer coefficient. The result should be in a format such as
h2b
Nu =
= function of (Re, Pr, f )
k
If Re = (V2b/υ) = 2 × 10 3 , 2 × 10 4 , 2 × 10 5 , and Pr = 0.7, compare your result of Nu to those of semiempirical correlations, such
that Nu = (h2b/k) = 0.023Re 0.8 Pr 0.4
10.5. Consider the turbulent flow heat transfer.
a. Derive the following momentum and energy equations
for a turbulent boundary-layer flow, a 2-D flat plate,
incompressible, constant properties:
∂u
∂u
∂
∂u
u
+ v
=
(ν + ε M )
∂x
∂y
∂y
∂y
∂T
∂T
∂
ν
ε M ∂T
u
+ v
=
+
∂x
∂y
∂y
Pr
Pr t ∂y
b. Derive the following momentum and energy equations for a
fully developed turbulent flow in a circular tube, incompressible, constant properties:
1 d
du
1 dP
r(v + ε M )
=
r dr
dr
ρ dx
∂T
1 ∂
∂T
u
=
r(α + ε H )
∂x
r ∂r
∂r
10.6. Consider a fully developed turbulent pipe flow in a circular tube
with a 5.0 cm I-D, constant properties.
217
Turbulent Flow Heat Transfer