Environmental Engineering Reference
In-Depth Information
A transient momentum balance on the four components comprising the thermosy-
phonic loop, assuming one-dimensional incompressible flow, gives
L
r
A
p
+
∂ m
c
∂t
L
2
A
p
+
L
s
A
s
+
L
d
A
p
g
.
.
ρ
w
sin(
θ
)
dy
=
−
P
m
f
r
L
r
ρ
w
D
r
A
r
N
2
+
m
c
2
f
u
L
u
ρ
w
D
p
A
p
+
f
u
L
u
ρ
w
D
p
A
p
f
d
L
d
ρ
w
D
p
A
p
−
+
.
(4.4.14)
Equations expressed in finite difference forms are solved simultaneously using their
appropriate boundary conditions. When simultaneous equations are solved using a
Gauss-Seidel iterative method in order to find
T
(
x
,
y
,
t
+
t
)
for each node, the solution
is stable unconditionally and the size of the time step is only limited by the accuracy
required.
4.4.6 Solar water heater design
From a transient heat balance on a generic directly heated thermosyphon solar-energy
water heater, the following dimensionless parameters
Y
,
Z
and
X
, designated the
Heywood, Yellot and Brooks numbers respectively, have been identified (Hobson and
Norton, 1988b):
F
AV
A
C
(
τα
)
e
H
M
S
C
W
(
T
a
Heywood number:
Y
=
(4.4.15)
−
T
m
)
[
F
AV
A
C
U
L
(
UA
)
S
]
t
(
M
S
C
W
)
+
Yellot number:
Z
=
(4.4.16)
fQ
tot
M
L
C
W
(
T
a
−
Brooks number:
X
=
(4.4.17)
T
m
)
X
Y
Z
+
Z
)
=
1
(4.4.18)
1
−
exp (
−
The mean daily circulation number is
Np and
M
L
M
S
Specific load:
W
=
(4.4.19)
The Bailey number,
K
, which represents the system parameters effecting flow
within the system is defined as:
ρβgT
ref
[
h
3
−
h
2
/
2]
Bailey number:
K
=
(4.4.20)
v
.
m
ref
[
L
r
/N
(
D
r
)
+
L
ρ
/D
ρ
)]
where
T
ref
and
m
ref
are given the values 10
◦
C and 10
−
1
kg
−
1
respectively.
The relationships between the Yellot, Z, Bailey, K, Heywood, Y, Brooks, K,
numbers and the specific load, W, may be summarised as a nomogram shown in Figure
4.4.3. The Heywood, Y, and Yellot, Z, numbers and the specific load, W, are functions
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