Tamkang Journal of Science and Engineering, Vol. 3, No. 3, pp. 157-164
(2000)
157
Cool Thermal Discharge from Ice Melting with Time-Velocity
Variation of Flowing Air by Complete Removal of Melt
C. D. Ho, H. M. Yeh and W. P. Wang
Department of Chemical Engineering,
Tamkang University
Tamsui, Taipei, Taiwan 251, R.O.C.
E-mail: cdho@mail.tku.edu.tw
Abstract
A device of cool-thermal discharge system with performance
improved has been developed by complete removal of melt with
time-velocity variation of flowing air during on-peak power
consumption. The simulation of estimating the thickness of ice
melted and the thermal penetration distance in ice layer region with
convective transfer by controlling air velocity with specified
discharging fluxes on free surface has been derived. Three examples
of different fluxes of cool-thermal discharges with constant inlet
temperature of flowing air have been illustrated.
Key Words: cool thermal discharge, melt removal, time-velocity
variation, moving boundary
1. Introduction
Cool-thermal storage system is a well known
process as a demand-side management technology
for shift air-conditioning needs from peak daytime
periods to off peak nighttime and weekend period,
while cool-thermal discharge system is that of ice
melting to supply on-peak daytime cooling
demands. The process with both operating systems
of using of water associated with lowering on-peak
daytime energy demand and increasing off-peak
night-time energy demand has enabled users to
significantly reduced their electricity costs, and
hence the enhancement in electric utility
profitability could be achieved.
The mathematical statements of the capacity
for cool-thermal storage systems with a constant
mass rate and a constant volume rate of sublimation
were derived [14,15]. The theoretical formulation of
specified heat fluxes on the boundary and with
convective boundary in cool-thermal discharge
processes has been derived [8,9,16]. The
improvements of energy transfer efficiency for
cool-thermal storage and cool-thermal discharge
systems were presented by several researchers
[2,3,6,7,11,12]. It is evident that the maximum
temperature gradient between the free surface and
the ambient air, and hence the higher transfer rate of
energy will be an important factor to design the
cool-thermal discharge systems. There are two
purposes in this work: first, to propose a new design
of cool-thermal discharge system with air flowing
over melting ice by complete removal of melt;
second, to present an analytical approach by the use
of the integral boundary-layer analysis. The present
device of cool-thermal discharge system is to
modify the previous works [8,9,16] for reducing the
thermal resistance during ice melting by removing
the melt completely and immediately, and the
maximum temperature gradient and hence the heat
transfer from ambient to the free surface is
achievable. The process of heat transfer with
moving boundary is so complicated and only a few
of literature exists [1,4,13].
2. Mathematical Statement
A semi-infinite domain ice layer in
cool-thermal discharge systems with uniform
temperature (T∞ ) initially, constant physical
properties of ice and no density change on melting
were assumed. Before the free surface starts
melting, the thermal penetration distance in ice
layer region is presented in Figure 1(a). After taking
a differential energy balance within the thermal
penetration region, the temperature distribution may
be obtained as
158
C. D. Ho et al.
∂ T / ∂ t = α i (∂ 2T / ∂ 2 x),
0 ≤ x ≤ δ (t ). (1)
where δ (t ) is the thermal penetration distance. If
an excess temperature ψ = T − T ∞ is introduced,
Eq. (1) becomes
∂ψ / ∂ t = α i (∂ 2ψ / ∂ x 2 ),
Ψ (0,t)=Ψ
0 ≤ x ≤ δ (t ). (2)
Solid
p
Ψ (x,t)
V(t), hm, Ψ air
δ( t)
Ψ ∞ =0
ψ (x , t ) =
x
Figure 1(a). A semi-infinite ice layer before
melting.
Ψ (0,t)=Ψ p
δ =
Solid
6α i t .
δ( t)-X(t)
ψ (0 , t ) =
Ψ ∞ =0
x
Figure 1(b). A semi-infinite ice layer after melting.
2.1 Before Melting, 0 ≤ t ≤ t i
The temperature of ice surface reaches its
melting point at t = t i , i.e. ψ = ψ p = T p − T ∞ .
Before the temperature of ice surface reaches its
melting point, the initial and boundary conditions
are
at
t = 0;
for IC,
(3)
∂ψ
− ki
= q at
∂x
x = 0;
for BC1,
(4)
ψ ( x , t ) = 0,
x = δ (t ); for BC2,
(5)
x = δ (t ); for BC3,
(6)
∂ψ
= 0,
∂x
(7)
(8)
δ (t)
X(t)
δ (t ) = 0 ,
qδ
x
x
[1 − 2 ( ) + ( ) 2 ].
2k i
δ
δ
Equation (8) is the expression for estimating the
thermal penetration distance in ice layer, and the
surface temperature can be obtained from Eqs. (7)
and (8) as
Ψ (x,t)
V(t), hm, Ψ air
where q = hm (ψ air − ψ ( 0, t )) , denoting the flux
of energy absorption or the flux of cool thermal
discharge at the free surface which is constant
during
the
discharging
period,
and
ψ air = Tair − T∞ , Tair is the temperature of
flowing air at the free surface. The solution of Eq.
(2) coupled with the use of Eqs. (3)-(6), was
obtained by the following approximation method of
integral boundary-layer analysis [5]. By following
the same mathematical treatment performed in our
previous work [16], except the temperature
distribution in the ice layer region, the temperature
distribution as well as the thermal penetration
distance with constant q were obtained by
representing ψ ( x, t ) in the quadratic expression
with the use of Eqs. (4)-(6) as follows:
at
at
qδ
q
=
2ki
k
3
α it .
2
(9)
Accordingly, the time, t i , needed for the surface
temperature to reach the melting point and the
corresponding penetration distance, δ i , may be
calculated, respectively, by
ti =
2 k i2ψ
3α i q
2
p
2
(10)
and
δi =
6α t i =
− 2 k iψ
q
p
(11)
The hourly cool-thermal discharge flux with
time-velocity variation of flowing air before
starting to melt may be calculated from
Cool Thermal Discharge from Ice Melting with Time-Velocity Variation of Flowing Air Complete Removeal of Melt
∂ψ
∂x
q = −k i
x =0
= hm (ψ air − ψ (0, t )), (12)
while the total amount of cool thermal discharge per
unit area during the time interval 0 to t i is
Q (t ) =
∫
ti
0
=
qdt
= [ h m (ψ
air
∫
ti
− ki
0
∂ψ
∂x
dt
x=0
− ψ ( 0 , t ))] t i
(13)
2.2 After Melting, t ≥ t i
∂ψ / ∂t = α i (∂ 2ψ / ∂x 2 ), X (t ) ≤ x ≤ δ (t ). (14)
at t = t i , for IC,
ψ ( x, t ) = ψ p ,
at
x = X;
(15)
for BC1, (16)
− k (∂ψ ∂x ) + ρ Q m dX dt = q ,
x = X ; for BC1’,
at
ψ = 0,
∂ψ
= 0,
∂ x
(17)
for BC2, (18)
at x = δ (t );
for BC3. (19)
Integrating Eq. (14) with respect to x over the range,
X to δ , gives
d δ
dδ
dX
ψ dx − ψ (δ , t )
+ψ ( X , t)
∫
X
dt
dt
dt
⎡ ∂ψ
⎤
∂ψ
= αi ⎢
−
⎥⋅
⎣ ∂x x =δ ∂x x= X ⎦
(20)
δ
Assume that the temperature distribution in the ice
layer can be represented by the following quadratic
expression which satisfies Eqs. (16), (18) and (19)
ψ (x , t ) ⎛ x − δ ⎞
=⎜
⎟
ψp
⎝ X −δ ⎠
2
(23)
dX
∫X ψ dx + ψ ( X , t ) dt
⎡ ∂ψ
= α i ⎢−
⎣ ∂ x
x= X
⎤
⎥,
⎦
(21)
or, further using Eqs. (16) and (17) results in
d ⎡ψ p δ ⎛ 2
+ ⎜⎜ ψ
⎢
dt ⎣ 3
⎝3
p
+
⎤ α q
⎟⎟ X ⎥ = i ,
ki
⎠ ⎦
αiρ Qm ⎞
ki
(24)
Integration from t i to t results in
δ =−
α q(t − t) ⎤ ψ pδ i
3 ⎡⎛ αi ρ Qm 2 ⎞
+ ψ p ⎟⎟ X − i i
⎢⎜⎜
⎥+
3 ⎠
ki
3
ψ p ⎣⎝ ki
⎦
(25)
dX
1
=
dt
ρQm
2ki ψ p
⎛
⎜⎜ q +
X −δ
⎝
⎞
⎟⎟
⎠
(26)
Substitution of Eq. (25) into Eq. (26) gives
dX 1
=
×
dt ρ Qm
⎧
⎫
⎪
⎪
2 ki ψ p
⎪
⎪
⎨q +
⎬
3 ⎡⎛αi ρ Qm 2 ⎞ αi q(ti − t)⎤ ψ pδi ⎪
⎪
⎟
⎜
⎪ X +ψ ⎢⎜ k + 3ψp ⎟X − k ⎥ − 3 ⎪
p ⎣⎢⎝ i
⎠
i
⎦⎥
⎩
⎭
(27)
After applying the boundary conditions, Eqs. (18)
and (19), Eq. (20) becomes
d
dt
(22)
Substitution of Eq. (23) into Eq. (17) gives
x = δ (t );
at
α ρ Q m ⎞ ⎤ αq
d ⎡ δ
⎛
.
ψ ( x , t ) dx + ⎜ψ p + i
⎟X ⎥ =
⎢
∫
dt ⎣ X
k
k
⎝
⎠ ⎦
Substitution of Eq. (23) into Eq. (22) yields
Once the surface temperature reaches the
melting point, the ice is starting to melt and the
melted ice is removed immediately, as shown in Fig.
1(b). The governing equation and the initial
boundary conditions now become
X = 0,
159
Once X is calculated from Eq. (27),
ψ ( x, t ) and δ can then be calculated from Eqs.
(23) and (25), respectively.
Similarly, the hourly cool-thermal discharge
flux and the total amount of cool thermal discharge
per unit area during the time interval t i to t can be
expressed respectively as follows:
160
C. D. Ho et al.
∂ψ
x = X = − k i ( ∂ψ / ∂x ) + ρ Q m dX / dt
∂x
= h m (ψ air − ψ p ),
q = −k i
(28)
and
Q (t ) =
=
∫
t
∫
t
ti
ti
V ( t ) = 2 . 268 ( µ g / L ρ g )[ Nu m ( t ) / Pr 1 / 3 ] 2
for laminar flow
V (t ) = 66.06( µ g / Lρ g )[ Nu m (t ) / Pr 1 / 3 + 836]5 / 4
for turbulent flow
qdt
air
− ψ p )]( t − t i ),
(29)
As an illustration, we consider here the
following three examples of discharging fluxes:
q = 1000 kJ/m 2 − h ,
The value of hm may be calculated by Eq.
(12) with the use of Eq. (8), the expression for
estimating the thermal penetration distances of ice
layer, and with the inlet air temperature
Tair = 32 o C , that is
3. Numerical Examples
1500 kJ/m 2 − h
and
hm =
q
ψ
ait
The average convection heat transfer
coefficient hm , in Eqs. (12) and (28), may be
estimated by Eqs. (30) and (31), which derived by
Incropera and DeWitt [10].
− ψ (0, t )
ψ
hm L
= 0.664 Re 1L/ 2 Pr 1 / 3 , 0.6 ≤ Pr ≤ 50,
kg
(
and Re L = 2 .268 Nu m Pr
1
3
)
2
(30)
For turbulent flow
hm L
kg
= (0.035 Re
and
in
4/5
L
⎡ 0.6 < Pr < 60 ⎤
,
− 836) Pr , ⎢
5
8⎥
⎣5 × 10 < Re L ≤ 10 ⎦
1/ 3
[(
Re L = 66.06 Nu m Pr
which
hm = q /(ψ air
1
)+ 836]
5
3
4
(31)
hm = q /(ψ air − ψ ( 0, t )
and
− ψ p ) are the convective transfer
coefficients of the periods before melting and after
melting, respectively, and hm is function of time.
In order to meet the specified discharging
flux q and the temperature of flowing air, the air
velocity is determined as follows:
air
−
q
ki
3
α it
2
(34)
in which the temperature-time history at free
surface may be obtained from Eq. (9), and then the
average Nusselt number is calculated by the
following
Nu m ( t ) ≡
For laminar flow
q
=
2000 kJ/m 2 − h , with the ambient temperature,
Tair = 32 o C .
N um ≡
(33)
3.1 Before Melting, 0 ≤ t ≤ t i
[ − k i ( ∂ ψ / ∂ x ) + ρ Q m dX / dt ]dt
= [ h m (ψ
Nu m ≡
(32)
hm L
kg
(35)
substitution of Eq. (35) into Eq. (32) or (33) gives
V (t ) = 2.268( µ g Lρ g ) ×
⎤
⎡
⎥
⎢
Lq
⎥
⎢
⎥
⎢ ⎛
⎞
⎢ k g ⎜ψ air − q 3 α i t ⎟ Pr 1 / 3 ⎥
⎟
ki 2
⎥⎦
⎢⎣ ⎜⎝
⎠
2
for laminar flow
V ( t ) = 66 . 06 (µ g
Lρ
g
(36)
)×
⎡
⎢
Lq
⎢
⎢
⎛
3
q
⎢ k g ⎜ ψ air −
α it
⎜
2
ki
⎢⎣
⎝
for turbulent flow
5
⎞
⎟ Pr
⎟
⎠
1
3
⎤ 4
⎥
+ 836 ⎥
⎥
⎥
⎥⎦
(37)
3.2 After Melting, t ≥ t i
Once the surface temperature reaches the
melting point, the ice is starting to melt and the
Cool Thermal Discharge from Ice Melting with Time-Velocity Variation of Flowing Air Complete Removeal of Melt
melted ice is removed immediately. By following
the same mathematical treatment in the previous
section, except the temperature on free surface is at
melting point thereafter.
hm =
ψ ait
q
−ψ
= constant
(38)
p
and then the average Nusselt number is calculated
by the following
Nu
m
( t ) = Nu
m
≡
hm L
kg
(39)
substitution of Eq. (39) into Eq. (32) or (33) gives
V = 2 .268 (µ g
⎡
Lq
Lρ g )⎢
⎢ k (ψ − ψ
⎣ g air
p
)Pr
1
3
2
⎤
⎡
Lq
⎢
Lρ g )
+ 836⎥
⎥
⎢ k (ψ − ψ ) Pr 13
p
⎦
⎣ g air
5
4
(41)
for turbulent flow
Table 1. Physical properties of ice (Incropera, et al.,
1990)
T
ρ
c
k
( C ) (kg / m ) (kJ / kg ⋅ K ) (W / m ⋅ K )
3
o
0
-20
920
920
2.040
1.945
1.88
2.03
L = 10m , 20m and 30m . Some physical
properties of ice are given in Table l [10].
Substituting these values into the appropriate
equations with the physical properties of ice taken
at its average temperature, the equations of X , δ
and ψ were derived. The thermal penetration
distances and the temperature distributions in the
ice layer may be estimated, respectively, from Eqs.
(7) and (8) before melting and Eqs. (23) and (25)
after melting, while the air velocity is calculated
from Eq. (36) or (37) before melting and Eq. (40) or
(41) after melting. The most important assumption
in this work is that the decreases in volume due to
the ice melting is neglected.
Figures 2 and 3 show the relations of air
velocity with the traversed length of air and the
specified heat flux as a parameter for
T∞ = −20 o C , respectively. It is seen from Figs. 2
and 3 that V is increasing before ice melting and
(40)
for laminar flow
V = 66.06(µ g
⎤
⎥
⎥
⎦
161
Qm
(kJ / kg )
334
---
As an illustration, we assign the following
numerical values for the physical properties of air
and
total
operating
time
per
day:
ρ g = 115
. kg / m3 , (0o − 4 0o C) ; µg = 0067
. kg/ m-h ,
(0o − 4 0oC) . The physical properties of ice at 1 atm
are given in Table l [10]. The results are shown in
o
Figs. 2 and 3 with T∞ = −20 C .
4. Results and Discussion
As an illustration, we assign the following
o
o
numerical values: T ∞ = −20 C ; T air = 32 C ;
kept constant due to constant free temperature after
melting. Moreover, V increases with both of the
specified heat flux and the traversed length of air.
The mathematical statement of cool-thermal
discharge systems from ice melting with complete
removal of melt, has been studied and the
approximation solution has been derived with
integral boundary-layer analysis in energy balances.
The expressions for dimensionless thickness of
melting ice X , Eq. (27), dimensionless thermal
penetration distance δ , Eqs. (8) and (25), and the
temperature distribution ψ ( x, t ) , Eqs. (7) and (23),
are obtained. The results of the analysis indicated
that, the flowing air velocity increases with the
amount of cool thermal discharge and the traversed
length of air. The present paper is actually the
extension of cool-thermal discharge systems in the
previous work [9], in which there is a water layer
with thermal resistance existing while energy
transferring from the flowing air to the ice layer
surface. The maximum temperature gradient on the
free surface with complete removal of melt in
cool-thermal discharge systems has a positive
influence on energy transfer rate, and hence the
application of this concept to design cool-thermal
discharge systems is technically and economically
feasible.
162
C. D. Ho et al.
10
q = 2 0 0 0 k J /m 2 -h
9
8
7
q = 1 5 0 0 k J /m 2 -h
V , m/sec
6
5
q = 1 0 0 0 k J /m 2 -h
4
3
2
1
0
0 .0
0 .2
0 .4
0 .6
0 .8
1 .0
t/t 0
Figure 2. Time-velocity variation of air with the traversed length as a parameter, (L=20 m)
8
L =30m
7
L =20m
L =10m
6
V , m/sec
5
4
3
2
1
0
0 .0
0 .2
0 .4
0 .6
0 .8
1 .0
t/t 0
2
Figure 3. Time-velocity variation of air with the specified heat flux as a parameter. ( q = 1500 kJ/m − h )
Cool Thermal Discharge from Ice Melting with Time-Velocity Variation of Flowing Air Complete Removeal of Melt
Nomenclature
cp
specific heat of ice (kJ / kg − K )
hm convective transfer coefficient (kJ / h−m2 − K)
k g thermal conductivity of air (kJ / kg − m − K )
ki
L
thermal conductivity of ice (kJ / kg − m − K )
traversed length of air (m)
Nu m
Nusselt number (-)
p atmosphere pressure (kPa)
q cool-thermal discharge flux (kJ / m 2 − h)
Qm heat of melting (kJ / kg )
T ( x, t ) ice temperature (K )
Tp melting point of ice (K )
T∞ initial temperature of ice layer (K )
t
time (h)
t i time needed for the surface temperature to
reach the melting point (h)
V
flowing air velocity (m / sec)
X thickness of melting ice (m)
x
x axis (m)
Greek letters
α i thermal diffusivity of ice, k i / ρc p (m 2 / h)
δ thermal penetration distance (m)
δ i thermal penetration distance as the surface
temperature reaching themelting point (m)
ψ T − T∞ (K )
ψ p T p − T∞ (K )
ρ
3
ice density (kg / m )
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Tamkang Journal of Science and Engineering, Vol. 3, No. 3, pp. 157-155
(2000)