1. Heat transfer
Conduction – Convection Systems
Al-Mustaqbal University College
Chemical and Petroleum Industries
Engineering Department
Ass. Prof. Dr. Malik Al-Taweel
2.
3.
4.
5.
6.
7. Conduction-ConvectionSystems
In heat-exchanger applications a finned-tube arrangement might be used to remove heat from
a hot liquid. The heat transfer from the liquid to the finned tube is by convection. The
extended-surface problem is a simple application of a conduction-convection systems.
Consider the one-dimensional fin exposed to a surrounding fluid
shown in Figure below. The temperature of the base of the fin is T0.
at a temperature T∞ as
Let P = perimeter = 2(Z + t)
The energy balance on an element of the fin of thickness dx is
dT d 2
T
dT
kA kA dx hPdx(T T ),
dx dx
dx
dx2
8. dT d
let T T ,
dx dx
d 2
T d 2
dx2
dx2
Let m2
= hP/kA
2
d
m2
0
dx2
(D-Operator)
D2
1 0 D 1
The roots of the equation above are:
m1 1, m2 1
The general solution is
Several cases may be considered:
CASE1: The fin is very long.
at x 0 T T0 0
at x T T 0
B.C.1
B.C.2
By applying the boundary conditions on the general equation above, we get
CASE2: The end of the fin is insulated so that dT/dx = 0 at x = L.
at x 0 T T0 0
at x L dT / dx 0 d / dx 0
B.C.1
B.C.2
From B.C.1 0 C1 C2 C1 0 C2
dT
dx
mL mL
From B.C.2 mC1e mC2 e 0
Subs. B.C.1 in B.C.2, we get
9. emL
emL
0
0
C
C2 1 0
emL
emL
emL
emL
Subs. C1&C2 in the general equation, we obtain
CASE3: The fin is of finite length and loses heat by convection from its end.
at x 0 T T0 0
B.C.1
k
d
dx
h
B.C.2 at x L
From B.C.1 0 C1 C2 C1 0 C2
k mC e mL
mC emL
hC e mL
C
emL
From B.C.2 1 2 1 2
By applying C1&C2 in the general equation, we get,
CalculationofHeatLostbytheFin
The heat lost by the fin can be calculated either by
or by
10. FORCASE1:
d
dx
d
dx
mx mx
m0 e , m0
e ,
0 x0
hP
kA( m0 ) kAm0 , For q kA 0
kA
q hPkA 0
FORCASE2:
cosh[m(L x)] d me mx
memx
, 0
2mx
2mx
0 cosh mL
dx 1 e 1 e
emL
e mL
d
1 1
0 m 0 m
mx mL
2mx
2mx
1 e 1 e dx e e
x0
mL
mL
q kA m
e e
emx
emL
0
q hPkA 0 tanh mL
FORCASE3:
The heat flow for this case is
d
dx