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TABLE OF CONTENTS
4099
Evolution of Temperature Distributions
in a Full-Scale Stratified Chilled-Water
Storage Tank with Radial Diffusers
Amy Musser
William P. Bahnfleth, Ph.D., P.E.
Student Member ASHRAE
Member ASHRAE
ABSTRACT
Temperature profiles in a full-scale, naturally stratified,
chilled-water thermal storage tank are described. Tests were
performed using a 1.4 million gallon (5,300 m 3), 44.5 ft (13.56
m) water depth cylindrical tank with radial diffusers. Nine
charge and discharge cycle tests were performed for various
flow rates, covering and extending beyond the normal operating range of the system. A method for obtaining thermocline
thickness from field data was derived, and a relationship
between inlet flow rate and initial thermocline thickness was
established. Significant differences between profiles obtained
for charge and discharge cycles at similar flow rates suggest
that the free surface at the top of the tank allows more mixing
to occur near the upper diffuser. A study of thermocline growth
compares measured temperature profiles with those predicted
by a numerical conduction model that uses temperature
profiles measured early in the cycle as an initial condition.
Comparison with the numerical study shows that, for high flow
rate tests, large-scale mixing induced by the inlet diffuser can
have significant effects on thermocline development, even
after the thermocline has moved away from the inlet diffuser.
INTRODUCTION
Chilled-water thermal energy storage in naturally stratified tanks is used to reduce the operating costs and refrigeration plant capacity of many large cooling systems. A recent
national survey (Potter 1994) reported an average chilledwater storage system capacity of 14,291 ton-h (50,261
kWh) with an average storage tank volume of 1,269,245 gal
(4,804 m3). Chilled-water storage was estimated to account
for 34% of the total capacity of all cool storage systems in the
United States. Sixty-one percent of the surveyed chilled-water
systems used naturally stratified storage tanks. The percentage
of new systems using stratified tanks is much higher. Potter’s
survey includes systems built between 1966 and 1994. During
the 1970s and 1980s, a variety of chilled-water storage technologies were proposed and tested, but by the end of this
period, stratified storage had become the technology of choice
because of its simplicity and low cost (Dorgan and Elleson
1993).
In a naturally stratified storage tank, warm and cool
volumes of water are stored without an intervening physical
barrier. The warm and cool volumes of water are prevented
from mixing by the stable density gradient that exists when
denser cool water lies below less dense warm water. This is
achieved by varying the direction of flow through the tank.
During a charge cycle, cool water is introduced at the bottom
of the tank through a diffuser designed to minimize mixing
while warm water is withdrawn at an equal rate through a similar diffuser at the top of the tank. During discharge, cool water
is removed from the bottom of the tank while warm water is
introduced at the top. The tank remains full at all times while
the interface between warm and cool water moves upward
during charge and downward during discharge.
Most stratified chilled-water storage tanks are cylindrical
vessels of the type used for potable water storage. One of the
most common diffuser types employed in this geometry is the
radial parallel plate design, shown in Figure 1. A radial
diffuser consists of a circular plate mounted parallel to a
spreading surface to create a channel through which inlet or
outlet water may flow. In the case of the lower diffuser, the
spreading surface is the floor of the tank. The free surface of
the water in the tank serves this function for the upper diffuser.
During storage system operation, water flows into the center
of the inlet diffuser and flows radially outward into the tank
through the channel created by the diffuser plate and spreading
surface, while an equal flow rate exits the tank at the outlet
diffuser.
Although stable buoyancy forces suppress large-scale
mixing between the warm and cool water in a stratified tank,
the separation achieved is not perfect. Figure 2 shows a typical
Amy Musser is a graduate student and William P. Bahnfleth is an assistant professor in the Department of Architectural Engineering, Pennsylvania State University, University Park.
THIS PREPRINT IS FOR DISCUSSION PURPOSES ONLY, FOR INCLUSION IN ASHRAE TRANSACTIONS 1998, V. 104, Pt. 1. Not to be reprinted in whole or in
part without written permission of the American Society of Heating, Refrigerating and Air-Conditioning Engineers, Inc., 1791 Tullie Circle, NE, Atlanta, GA 30329.
Opinions, findings, conclusions, or recommendations expressed in this paper are those of the author(s) and do not necessarily reflect the views of ASHRAE. Written
questions and comments regarding this paper should be received at ASHRAE no later than February 6, 1998.
BACK TO PAGE ONE
Figure 1 Stratified storage tank with radial parallel
plate diffusers.
Figure 2 Temperature profile in a naturally stratified
chilled-water storage tank.
2
temperature profile in a naturally stratified tank. A thin thermal transition layer, called a thermocline, forms between the
warm and cool volumes as a result of heat conduction across
the interface and mixing near the inlet diffuser. In the vicinity
of the inlet diffuser, complex three-dimensional flow patterns
may enhance mixing by several orders of magnitude relative
to pure conduction (Zurigat et al. 1991). At sufficiently large
distances from the inlet diffuser, one-dimensional “plug” flow
is established and heat transfer occurs primarily by conduction
across the thermocline and through the walls of the tank. The
thickness of the thermocline may vary over a wide range,
depending upon operating temperatures, flow rate, and
diffuser design. Thicknesses of 3 ft to 6 ft (1 m to 2 m) are not
uncommon, but thermocline thicknesses of as little as 1.5 ft
(0.5 m) have been observed in full-scale, working systems
(Bahnfleth and Joyce 1994).
In stratified storage tanks, useful energy is lost by three
primary mechanisms: conduction between the tank and
surroundings, conduction across the thermocline, and mixing
near the inlet. For steel tanks, heat loss by conduction to the
surroundings can be effectively eliminated by the application
of a small amount of exterior insulation. Concrete tanks typically require little or no insulation, particularly if they are
buried or bermed. Conduction across the thermocline is
beyond the control of the designer, but it makes a relatively
small contribution to loss of nominal capacity, as has been
shown by analytical studies from which mixing is omitted
(e.g., Homan et al. 1996). Mixing near the inlet diffuser has
more significant effects on the temperature distribution in the
tank and is a strong function of diffuser characteristics (Wildin
1990).
Wildin’s experiments in relatively shallow tanks of less
than 16 ft (4.9 m) water column indicate that two parameters,
the densimetric inlet Froude number (Fri) and the inlet
Reynolds number (Rei), have significance for diffuser design.
A design criterion of Fr i = 1 is universally accepted, but
proposed criteria for Re i are controversial (Dorgan and Elleson 1993). A strong Rei effect is apparent in Wildin’s data for
inlet diffusers with equal values of Fri. Recommended design
values of Rei range from 200 for shallow tanks to 2,000 for
“deep” tanks with water depth greater than 40 ft (12.2 m), but
anecdotal evidence suggests that tall tanks with Rei on the
order of 10,000 stratify well and experience little loss of nominal capacity as a result of diffuser performance (Andrepont
1992). The questionable validity of existing design guidance
justifies further study of full-scale stratified tank performance.
This paper reports temperature distribution data taken
from a full-scale, naturally stratified cylindrical chilled-water
thermal storage tank with radial parallel plate diffusers. A
method of calculating thermocline thickness appropriate for
use with field data is proposed and compared with other thermocline definitions found in the literature. Thermocline thicknesses calculated by this technique are compared at a time
early in the cycle over a range of charge and discharge flow
rates, and the subsequent development of these temperature
BACK TO PAGE ONE
profiles is described. Comparisons of observed temperature
profile development with numerical estimates based on pure
conduction are used to assess the influence of mixing late in
charge and discharge cycles. Based on a comparison of the
thermocline thicknesses produced at high inlet Reynolds
numbers in the present study with estimates based on
published data at low Reynolds numbers, the appropriateness
of current design guidelines for tall tanks is discussed.
BACKGROUND
Prior Studies of Temperature Profile Development
Wildin and Truman (1985b), among others, have
observed that stratification is achieved when the flow from an
inlet diffuser forms a gravity current. Wildin (1990) noted
there are two distinct mixing processes in naturally stratified
chilled-water storage tanks: mixing near the inlet diffuser at
the beginning of a charge or discharge cycle and larger-scale
circulation on the inlet side of the thermocline once it has
moved away from the inlet. Inlet mixing is most significant
during the first pass of the gravity current across the width of
the tank and plays a key role in determining thermocline thickness. After escaping the region of three-dimensional flow near
the inlet, the thermocline moves up or down the tank in plug
flow, undergoing relatively little change in the course of a typical charge or discharge cycle.
Yoo (1986) showed that the thickness of a two-dimensional gravity current is a function of inlet Froude number
(Fri), inlet Reynolds number (Rei), and the distance from the
diffuser outlet to the tank wall. According to Wildin (1990),
large-scale mixing appears to vary most with Rei but also
depends upon Fri. Inlet mixing is most apparent in its effect on
the initial thickness of the thermocline. Large-scale circulation manifests itself through bulk temperature change on the
inlet side of the thermocline and may not be accompanied by
an increase in thermocline thickness.
The performance of naturally stratified thermal storage
tanks has been studied extensively using one-dimensional
analytical and numerical models. These models accurately
predict behavior in the plug flow regime that exists remote
from the inlet diffuser. However, they either neglect the effects
of multidimensional flow and heat transfer near the inlet
diffuser or approximate inlet mixing by various empirical
methods. In one-dimensional models, mixing is most often
simulated by assuming that perfect mixing occurs in a specified region near the inlet diffuser or by diffusivity enhancement factors.
STRATUNM (Truman and Wildin 1989), is a one-dimensional finite-difference model that approximates inlet mixing
by blending a fixed, user-specified number of cells adjacent to
the inlet at each time step. A similar model has been implemented in the BLAST energy analysis program (Gretarsson
1992). Hussain (1989) improved the STRATUNM mixing
model by allowing the height of the mixed region to vary with
time. Based on analysis of experimental data from a 16 ft (4.9
m) deep rectangular tank with linear diffusers, he concluded
that the ratio of mixing height to thermocline height could be
correlated with ReiFri. This model was incorporated in STRATUNM and was found to improve its agreement with experimental measurements.
In the one-dimensional analysis of Homan et al. (1996),
the effects of inlet mixing are approximated through the use of
a constant effective thermal diffusivity, which is added to the
actual molecular diffusivity to obtain better agreement
between predicted and measured temperature profiles. The
primary deficiency of this approach is the demonstrably incorrect assumption of spatially and temporally uniform mixing.
The same authors present an analysis indicating that
effective diffusivity (and, therefore, thermocline thickness)
increases with PeH for a given tank. The significance of this
finding is limited, given that Rei would increase in direct
proportion to PeH as flow varies. It seems as likely, perhaps
more so, that the observed behavior is due to change in Rei and
may not be directly related to PeH.
Other studies have used more sophisticated enhanced
diffusivity models. For example, Oppel et al. (1986) calculated an eddy conductivity factor that was spatially correlated
to an inverse hyperbolic function of the ratio of Reynolds
number to Richardson number (Re/Ri). This function permits
diffusivity to be large near the inlet diffuser and to vanish in the
interior of the tank, which is the expected behavior.
A significant practical limitation of one-dimensional
models is their dependence on empirical descriptions of inlet
behavior. For design of a new tank, inlet mixing parameters
must be selected on the basis of the designer’s experience or
data found in the literature. Few data of this type have been
published for full-scale chilled-water storage systems. In addition, the validity of applying parameters derived from data for
one tank to the simulation of another tank having different
geometry, flow, or temperature conditions is questionable.
Documentation for both the STRATUNM and BLAST
models recommends use of a mixing height that is 3% to 8%
of the tank height (Truman and Wildin 1989; Gretarsson
1992). This recommendation is based on experiments
performed in a 16 ft (4.9 m) water depth rectangular prototype
tank using linear and distributed nozzle diffusers (Wildin and
Truman 1985a). Few other data regarding either thermocline
thickness or the characteristics of temperature profiles in fullscale tanks can be found in the open literature.
Definitions of Thermocline Thickness
Yoo et al. (1986) estimated thermocline thickness by
extrapolating ends of the region of linear temperature variation near the center of the thermocline to meet asymptotes at
the initial tank and the average inlet temperature. Using this
definition, the thermocline thickness was found to decrease
with increasing Fri (obtained by increasing the inlet flow rate).
Smaller thicknesses were measured at higher Fri because the
temperature gradient near the center of the thermocline was
steeper. However, the apparently beneficial effect of a thin
3
BACK TO PAGE ONE
thermocline was offset by an increase in
average temperature below the thermocline
due to inlet side mixing. Given these results,
the appropriateness of this definition of
thermocline thickness and the relevance of
thermocline thickness per se as an indicator
of stratified tank thermal performance must
be questioned.
Calculating thermocline thickness by
linear extrapolation of the central gradient
ignores the upper and lower extremities of
the thermocline. Visual inspection of
temperature profiles in stratified tanks indicates that the size of the thermal transition
region tends to increase as flow rate (and
therefore, Fr i and Rei) increases. A quantitative definition that explicitly accounts for
the nonlinear upper and lower fringes of the
thermocline should produce a variation of
thickness with flow rate that is consistent
with observation. This can be accomplished
by defining thermocline thickness as the
distance over which a specified change in Figure 3 Sensor locations and tank geometry.
temperature occurs.
Homan et al. (1996) proposed one such
design Fr i of the diffusers is 0.35, which is substantially less
definition. They identified the lower edge of the thermocline
than the recommended design value of 1 (Dorgan and Elleas the point where T = Tmax, the highest usable temperature for
son 1993). With a design Rei of 6,200, however, the diffuser
the application. The upper edge is assumed to be located at the
design violates even the “tall” tank limit suggested by
point where T = Th (Tmax Tc), where Th is the average
ASHRAE (Dorgan and Elleson 1993) by roughly a factor of
system return temperature during discharge and Tc is the averthree.
age inlet temperature during charging. This definition is independent of the size of the linear region at the center of the
Existing instrumentation, monitoring, and control
thermocline. However, it depends on the essentially arbitrary
systems were used to operate the system and to measure and
difference in temperature between Tmax and Tc. Using this defirecord data. The system is controlled by a microprocessornition, the same temperature profile would have two different
based system capable of logging all thermal storage points on
values of “thermocline thickness” in applications with differtime intervals shorter than one minute. Instrumentation
ent usable temperature limits. A more consistent definition
includes temperature sensors accurate to ±.3°F (.17°C) in the
informed by boundary layer concepts would, perhaps, define
chilled-water supply and return mains, temperature sensors
thermocline thickness using a specified fraction of Th
Tc.
accurate to ±.75°F (.42°C) distributed vertically through the
This approach is adopted in the present study.
tank at approximately 5 ft (1.5 m) intervals, and an impacttype flowmeter nominally accurate to 3% of full scale (per
DESCRIPTION OF THE TEST SITE
manufacturer’s data).
Tests were performed using a 1.4 million gallon
Due to space restrictions, the installation of the existing
(5,300 m 3 ), 44.5 ft (13.6 m) water depth, steel thermal storage
flowmeter has less than the manufacturer’s recommended
tank with a radial diffuser system. The tank is part of a
straight upstream and downstream piping runs. The meter was
privately owned chilled-water system serving a university
calibrated prior to testing by comparison with a factory-calimedical center in central Pennsylvania. Figure 3 shows the
brated ultrasonic meter while the system was in discharge
tank dimensions and interior temperature sensor locations.
configuration. The calibration curve resulting from these tests
The system operates between nominal supply and return
was checked independently after testing by comparison with
temperatures of 41.5 F (5.3 C) and 55.5 F (13.1 C), respecthermocline translational velocities calculated using tempertively, and has a capacity of approximately 14,000 ton-h
ature data recorded by sensors in the tank. It was found that,
(49,238 kWh). Thermal storage flow rate is controlled by two
because of the poor installation of the meter, indicated flow
variable-speed pumps. Design flow rates are 2,400 gpm (151
rates could be in error by as much as 15% during discharge
L/s) for charge and 1,900 gpm (120 L/s) for discharge, but the
cycles and by 30% during charge cycles.
maximum flow rate exceeds 3,000 gpm (189 L/s). The
4
BACK TO PAGE ONE
Error analysis demonstrated that flow measurement
based on thermocline transit time between two sensors was
very accurate for the constant flow tests conducted in this
study. Therefore, this method of measuring flow was used in
preference to the use of the in situ flowmeter. Error in the flow
rate calculated by thermocline transit time arises from uncertainties in sensor location, time increment, and temperature
reading at the two sensors. Assuming very conservative values
for the accuracy of the distance and time variables involved in
computing the flow rate, error analysis indicates that the accuracy of this flow measurement technique is 3% or better for
each of the flow rates tested.
DESCRIPTION OF TESTS
Tests were performed during the summer of 1996. Each
test consisted of one complete half cycle (a charge or discharge
cycle) at a constant flow rate, beginning with the tank at essentially uniform temperature and continuing until the temperature of water leaving the tank was within 2 F (1.1 C) of the
entering water temperature. To the extent possible, tank inlet
temperature was held constant during each test. Inlet temperature control was more reliable during charging, when inlet
flow comes from the chilled-water plant, than during
discharge, when it comes from the chilled-water system
return. Five complete charge cycles and four complete
discharge cycles were recorded. Flow rates were varied over
the range of values possible for the system to achieve a range
of parameter values. Table 1 shows the ranges of parameters
covered.
Half cycles were studied rather than complete cycles for
two reasons. The first was the desire to minimize disruption of
operations at the test site and to decrease the likelihood that a
test would be aborted out of operational necessity. The staff of
the chilled-water plant had full responsibility for running the
system according to the specified test protocol while maintaining continuity of operations, and there was concern prior
to the start of testing that it would be difficult to run tests over
periods as long as a full day. The second reason for using half
cycles was the investigators’ interest in comparing the performance of upper and lower diffusers over complete charge and
discharge cycles. This objective was in conflict with the goal
of evaluating full-cycle performance by thermal efficiency or
figure of merit (FoM) measures (see Wildin and Truman
[1985a, 1985b] and Tran et al. [1989] for definitions).
However, performance metrics analogous to the full-cycle
FoM can be used to evaluate thermal performance over a half
cycle. Initial concerns that the test program would disrupt
normal operation proved to be unfounded, and full-cycle tests
will be conducted at this site in the next phase of this research.
DATA ACQUISITION AND ANALYSIS
Thermal storage system flow rate, storage tank water
level, and all thermal storage temperature points (inlet, outlet,
and ten sensors inside the tank) were logged at one-minute
intervals throughout each of the nine half-cycle tests. Plots of
TABLE 1
Charge and Discharge Cycle Test Parameters
(
(
Test Type
F (C)
Tin
F (C)
Tout
Flow rate
gpm (L/s)
Elapsed time, h
Rei
Fri
PeH
Charge
41.44
(5.24)
55.38
(12.99)
675
(42.6)
36.2
1,732
0.11
9,972
Charge
41.65
(5.36)
55.66
(13.14)
1,096
(69.2)
22.3
2,822
0.18
16,188
Charge
41.47
(5.26)
55.32
(12.96)
1,418
(89.5)
17.2
3,639
0.23
20,942
Charge
41.55
(5.31)
55.70
(13.17)
1,835
(115.8)
13.3
4,719
0.30
27,114
Design)
Charge
41.38
(5.21)
55.26
(12.92)
2,274
(143.5)
10.8
5,828
0.37
33,592
Discharge
55.56
(13.09)
41.44
(5.24)
1,171
(73.9)
21.0
3,778
0.14
17,298
Design)
Discharge
55.51
(13.06)
41.39
(5.22)
1,805
(113.9)
13.5
5,822
0.26
26,674
Discharge
55.75
(13.19)
41.17
(5.09)
2,329
(147.0)
10.5
7,538
0.28
34,415
Discharge
55.67
(13.15)
41.50
(5.28)
2,974
(187.7)
8.2
9,614
0.35
43,940
5
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Initial Tank Bulk
Temperature
60
Temperature (F)
55
50
Average Inlet
Temperature
45
40
35
0
20
40
60
80
100
120
% Volume
(a)
Average Inlet
Temperature
Temperature (F)
60
55
50
Initial Tank
Bulk
Temperature
45
40
35
0
20
40
60
80
100
120
% Volume
(b)
Figure 4 Typical inlet and outlet temperature profiles: a)
charge cycle (Rei = 3,639, Fri = 0.232); b)
discharge cycle (Rei = 3,778, Fri = 0.137).
inlet and outlet temperature as a function of percentage of tank
volume for typical charge and discharge cycles are shown in
Figure 4. The relatively smooth inlet and outlet temperature
histories are representative of those observed in all tests. As
illustrated in Figure 4b, inlet temperature control was excellent, even for the discharge tests, with the inlet temperature
varying by no more than 1°F in all cases.
The passage of the thermocline through the outlet diffuser
is clearly visible in both Figures 4a and 4b. The thermocline
reaches the outlet after total cycle flow through the tank has
reached 80% to 90% of tank volume and has not completely
cleared the tank until total flow reaches 110% to 120% of tank
volume. The outlet profiles show quite clearly the thermocline
characteristics discussed previously, in particular, a central
region of relatively constant temperature gradient and asymmetrical “tails” on the inlet and outlet sides. The tail of the
distribution on the inlet side is longer than that on the outlet
side as predicted by one-dimensional theory (Yoo and Pak
1996).
Instantaneous vs. Time Series Temperature Profiles
In order to study the development of the thermocline, it is
necessary to obtain accurate records of its shape at various
points during a charge or discharge cycle. The vertical array of
temperature sensors within the tank seemingly provides a
convenient means for obtaining instantaneous profiles, which
can then be compared. However, in order for these instantaneous profiles to be useful, vertical resolution must be suffi6
ciently fine to place several sensors within the thermocline at
any given time. If the sensor spacing is equal to or greater than
the thermocline thickness, the exact extent of the thermocline
cannot be measured accurately by this method. As shown in
Figure 3, the nominal spacing of the sensors in the test tank is
approximately 5 ft (1.5 m). Typical thermocline thicknesses
were in the range 2 ft to 6 ft (0.6 m to 1.8 m), too small to be
sharply defined by the available grid. The installation of sufficient additional temperature sensors was not feasible; therefore, an alternative procedure had to be developed.
The method adopted for producing temperature profiles
with the existing instrumentation was the use of time series
recordings at individual temperature sensors. For one-dimensional flow, it is a conceptually straightforward matter to
derive spatial profiles from time series by converting the time
increment between measurements into distance using the
equivalent plug flow velocity. This method has both advantages and potential disadvantages relative to the instantaneous
profile method. Time series measurements can be taken at
very small time increments relative to the translational velocity of the flow; therefore, a high degree of resolution can be
achieved and accurate measurements of thermocline thickness
can be made without adding additional measurement points.
An additional advantage of this method over the instantaneous
profile method is that, since a series profile is produced by a
single sensor, the error due to differences in calibration
between sensors inherent in the instantaneous profile method
is eliminated. However, the spreading of the thermocline by
diffusion and by mixing as the temperature profile passes a
sensor is a potentially significant source of error. The magnitude of this error must be evaluated before time series profiles
can be said to be equivalent (for practical purposes) to instantaneous spatial profiles.
To this end, a one-dimensional transient finite-difference
model of the temperature distribution in the tank was developed. The model utilizes the standard explicit method, which
is second-order accurate in space and first-order accurate in
time (Smith 1978). Using a measured time series profile as the
initial condition, a projected time series profile was generated
for a period equal to the time necessary for the thermocline to
pass a sensor. The differences in the initial and final profiles
were compared to obtain a bound for the error inherent in
assuming time series and instantaneous profiles to be equivalent. This estimate accounts only for changes in the thermocline caused by conduction; therefore, this test provides an
accurate bound for error only in regions of the tank sufficiently
far removed from the inlet diffuser so that conduction is the
primary means of temperature blending.
Trials were conducted using the time series profiles passing the sensor at 25% of tank height as the initial condition.
The transit time for one thermocline thickness, t1, is calculated
by dividing the thermocline thickness by the plug flow velocity of water moving through the tank:
BACK TO PAGE ONE
T–T
Θ = ----------------cTh – Tc
60
Temperature History of
sensor E
Instantaneous tank
temperature profile
Temperature (F)
55
50
45
40
35
0
5
10
15
20
25
30
35
40
45
50
Height or Velocity X Time (ft)
Figure 5 Comparison of instantaneous and time series
charge cycle temperature profiles (Rei = 4,719, Fr
= 0.30).
ht
t 1 = ---V
(1)
Clearly, greater error should occur in cycles with greater
values of t1. Of the cases studied, the charge cycle with the
lowest flow rate (675 gpm, 42.6 L/s) was found to have the
greatest value of t1 due to the low value of plug flow velocity,
despite having a relatively small thermocline thickness. For
this cycle, the greatest difference between the initial and
projected values of any point in the profile was 0.33 F
(0.18 C), which is within the range of accuracy of the sensors
located inside the tank. Therefore, it was concluded that in the
conduction-dominated region of the tank, the time series
profile could be used in lieu of an instantaneous spatial profile
without significant error. A comparison between instantaneous, spatially distributed, and time series, single-point
profiles is shown in Figure 5. Figure 5 illustrates the good
correspondence between the two methods and shows that a
much more refined profile can be obtained when using time
series measurement.
Definition of Thermocline Thickness
(2)
For charge cycles, Tc represents the average inlet temperature
and Th represents the initial tank bulk temperature. For
discharge cycles, Tc represents the initial tank bulk temperature and Th represents the average inlet temperature.
Two problems can arise when defining thermocline thickness using Tmax. The first is that of identifying a cutoff point
in the “tails” of the profile when Tmax is very close to Tc. As
shown in Figure 6, the tails of the thermocline experience little
change in temperature over large distances. Therefore, small
errors in temperature measurement and normal fluctuations in
temperature of a working system may cause large variations in
calculated thermocline thickness. The accuracy of thermocline thickness measurements obtained using Homan's
method becomes questionable when the maximum allowable
outlet temperature is very close to the average inlet temperature because several widely spaced points may have the same
value of .
The second problem occurs when Tmax is significantly
higher than the average inlet temperature. The “linear” center
regions of the profiles shown in Figure 6 are essentially identical. If Tmax falls within this center region, then the calculated
thickness of the thermocline would be identical for the two
cases shown, although the tails of the profile generated at the
higher flow rate are significantly longer.
For the charge cycles shown in Figure 6, these two problems become significant when the limiting dimensionless
temperature difference, max, is less than 0.05 or greater than
0.15. For the test site, the maximum allowable outlet temperature, Tmax, is 45 ( max= 0.286). This would place the boundary of the thermocline close to the center of the temperature
distribution and would disregard the effect of the tails in determining thermocline thickness. Consequently, an alternative
definition of thermocline thickness was developed for
analysis of the data from this study.
In this study, thermocline thickness was defined without
reference to Tmax. Rather, a dimensionless cutoff temperature
Dimensionless Temperature, Θ
It has been noted that the central gradient definition of
thermocline thickness leads to estimates that do not accord
with either visual observation or intuition. The
definition proposed by Homan et al. (1996)
avoids fundamental problems associated with
1
this approach. However, difficulties arise
0.9
when applying this method because maxi0.8
0.7
mum usable outlet temperature (Tmax) is used
0.6
to determine thermocline thickness. This is
Re i = 1732
0.5
Θ m ax = 0.286
illustrated in Figure 6, which shows two
0.4
Re i = 5828
0.3
profiles taken from a sensor located at 25% of
0.2
tank height (11.25 ft, 3.4 m) for the lowest
0.1
(675 gpm, 42.6 L/s) and highest (2274 gpm,
0
-3
-2
-1
0
1
2
3
4
5
143.5 L/s) flow rates tested. Since the initial
Dista nce (ft)
tank bulk temperature and the average inlet
temperature for the two cycles compared are
not identical, profiles are plotted in dimen- Figure 6 Comparison of thermoclines formed during highest and lowest
sionless form. Dimensionless temperature, ,
flow rate charge cycles (measured at 25% of tank height).
is defined as
7
BACK TO PAGE ONE
Thermocline thickness (ft)
Thermocline thickness (ft)
8
Sensor c, H=11.25 ft (3.43 m)
7
6
Sensor g, H=31.5 ft (9.60 m)
5
Sensor i, H=41.25 ft (12.57 m)
4
3
2
1
0
10
9
8
7
6
5
4
3
2
1
0
Charge cycles
Discharge cycles
0
0
2.5
5
7.5
10
12.5
2000
4000
6000
8000
10000
15
Inlet Reynolds number (Rei)
Θmax (% of maximum temperature difference)
Figure 7 Dependence of thermocline thickness on cutoff
dimensionless temperature difference (charge
cycle, Rei = 1,732, Fri = 0.11).
Figure 8 Dependence of initial thermocline thickness on
inlet Reynolds number (evaluated at 25% of tank
height).
was chosen that included most of the region of temperature
change but did not extend so far into the tails of the profile that
physical significance was lost. For the data in Figure 6, it
appears that a dimensionless cutoff temperature between 0.05
and 0.15 will allow a meaningful comparison of thermocline
thickness for profiles generated at different flow rates.
A further test of this hypothesis was conducted to verify
that valid comparisons of profiles measured at different
sensors in the same cycle also would be possible. Data from
the lowest flow rate charge cycle (675 gpm, 42.6 L/s) were
selected because they exhibited a very smooth profile both
early and late in the cycle. Thermocline thicknesses were
calculated for dimensionless cutoff temperatures of 0.025,
0.05, 0.075, 0.1, 0.125, and 0.15 for sensors at three different
heights within the tank.
The results of these calculations are compared in Figure
7. As a larger portion of the thermocline is discarded (i.e., as
the cutoff
increases), less growth of the thermocline—in
absolute terms and as a percentage of the original thickness—
is indicated as it passes from one sensor to another. Thermocline thicknesses at all three sensors are practically indistinguishable at a cutoff temperature difference of 0.15. On the
basis of these results, it was decided to compare thermocline
thicknesses using a cutoff
of 0.1. This value avoids the
sensitivity of thermocline thickness to noise for small cutoff Θ
and the insensitivity to the tails of large cutoff Θ. For these
tests, this criterion allows meaningful comparisons of profiles
generated during different cycles and measured at various
sensors during the same cycle.
shape. At earlier points in the cycle, it is assumed that the thermocline was still sufficiently close to the inlet diffuser to be
strongly influenced by mixing.
Initial thermocline thickness is plotted as a function of
inlet Reynolds number in Figure 8. Because diffuser geometry
was fixed for these tests, plots of thermocline thickness as a
function of flow rate, Fri, and various combinations of Rei and
Fri do not provide additional insight and are, therefore, not
shown. What can be noted is that initial thermocline thickness,
as defined herein, increases with flow rate and, hence, with Rei
and Fri. Because both of these parameters, as well as Pe, are
linear functions of flow rate, it is not possible to separate their
TABLE 2
Thermocline Thicknesses Measured
at 11.25 ft (3.43 m) Tank Height
RESULTS
(
Thermocline Thickness
As documented in Table 2, thermocline thicknesses for
the four discharge cycles and five charge cycles recorded
varied from 2 ft to 9 ft (0.61 m to 2.7 m) after the thermocline
had moved slightly more than 10 ft (3 m) away from the inlet.
Hereafter, these are referred to as “initial” thicknesses because
they were recorded at the earliest stage of the cycle at which
the temperature profile had achieved an essentially stable
8
(
Test Type
Flow rate
gpm (L/s)
Rei
Fri
Thermocline
Thickness
ft (m)
Charge
675
(42.6)
1,732
0.11
2.42
(0.74)
Charge
1,096
(69.2)
2,822
0.18
2.86
(0.87)
Charge
1,418
(89.5)
3,639
0.23
3.15
(0.96)
Charge
1,835
(115.8)
4,719
0.30
3.57
(1.09)
Design)
Charge
2,274
(143.5)
5,828
0.37
3.59
(1.09)
Discharge
1,171
(73.9)
3,778
0.14
3.27
(1.00)
Design)
Discharge
1,805
(113.9)
5,822
0.26
6.48
(1.97)
Discharge
2,329
(147.0)
7,538
0.28
7.43
(2.26)
Discharge
2,974
(187.7)
9,614
0.35
9.22
(2.81)
BACK TO PAGE ONE
Thermocline Growth
As the thermocline moves away from the inlet diffuser, its
development should be influenced less by mixing and more by
56
54
Initial tank bulk
temperature
Temperature (F)
52
Sensor i, H=41.25 ft
(12.57 m)
Sensor g, H=31.5 ft
(9.60 m)
Sensor c, H=11.25 ft
(3.43 m)
50
48
46
44
Average inlet
temperature
42
40
-5
-4
-3
-2
-1
0
1
2
3
4
5
Distance (ft)
(a)
56
54
Initial tank bulk
temperature
52
Temperature (F)
independent effects with tests conducted at constant diffuser
height and thermal diffusivity. However, one may speculate
that the observed effects are primarily the result of Rei variation, since Fri is less than unity in all cases and varies little
while Rei varies over a wide range from moderately high to
very high.
An interesting trend illustrated by Figure 8 is the tendency
for discharge cycle thermoclines to be much thicker than
charge cycle thermoclines at the same Rei for Rei substantially
greater than 4,000. This suggests that the free surface at the
upper diffuser allows more mixing than the no-slip, rigid
boundary imposed at the lower diffuser by the tank floor. This
contradicts the findings of Didden and Maxworthy (1982),
who experimentally investigated isothermal gravity current
flow in a scale model freshwater/saltwater stratified system.
They found no difference in the development of surface and
bottom gravity currents because an immobile film that formed
at the top of the tank behaved as a no-slip boundary. Results
of the present study indicate that this is not the case for thermally stratified chilled-water storage.
Qualitative comparison of thermocline thicknesses
obtained in this study at relatively high Rei with those generated at lower Rei in prior investigations suggests that, at least
for lower radial parallel plate diffusers, the existing design
criteria for tall tanks may be quite conservative. Wildin (1989)
published inlet and outlet temperature profiles for three diffusers operating at the same Fri (1.26) but at Rei values of 240,
850, and 1700, respectively, for octagonal, larger radial, and
smaller radial designs. Although these profiles do not show
the complete thermocline exiting the tank, enough data were
reported to permit lower bound calculations for thermocline
thickness to be made. The portion of the profile recorded is
primarily on the outlet side of the thermocline. According to
theory and observation, the inlet side tail should be longer;
therefore, an estimate obtained by doubling the distance from
the cutoff point to the center of the thermocline will underestimate the total thickness.
Although the inlet height for Wildin’s radial diffusers was
only 4 in. (0.1 m), minimum charge cycle thermoclines of
roughly 2 ft (0.6 m) and minimum discharge cycle thermoclines of at least 4 ft (1.22 m) were calculated using the
procedure adopted in the present study. The thermoclines
observed in the present study are not appreciably larger than
those observed by Wildin, while the water depth of the tank is
three times larger. From a practical perspective, therefore, it
may be argued that a design Rei of 2000 would be unnecessarily conservative in this case. It is important to note that the
upper radial diffusers in Wildin’s study had full top plates,
unlike the diffuser in the present study. This may have
improved discharge cycle performance for reasons discussed
previously.
50
Sensor c, H=11.25 ft
(3.43 m)
Sensor g, H=31.5 ft
(9.60 m)
Sensor i, H=41.25 ft
(12.57 m)
48
46
44
Average inlet
temperature
42
40
-5
-4
-3
-2
-1
0
1
2
3
4
5
Distance (ft)
(b)
Figure 9 Comparison of time series thermoclines at
various sensors: a) charge cycle (Rei = 1,732,
Fri = 0.11); b) discharge cycle (Rei = 5,828,
Fri = 0.37).
conduction. Figure 9 compares time series charge cycle thermocline profiles recorded by three different temperature
sensors. Profiles were recorded by the sensors located 11.25 ft,
31.5 ft, and 41.25 ft (3.43 m, 9.60 m, and 12.57 m) above the
tank floor during the 675 gpm (42.6 L/s) and 2,274 gpm (143.5
L/s) charge cycles.
At the lower flow rate (Figure 9a), roughly 16.5 hours
elapse as the thermocline moves between the first two sensors
and an additional 8 hours elapse between the last two. As
expected, the thermocline thickens with time. On the inlet side
of the temperature profile, the “tail” of the thermocline exhibits more growth than on the outlet side. A small increase in
temperature below the thermocline with the passage of time is
suggestive of heat transfer across the thermocline, as it is
accompanied by a nearly symmetric change on the outlet side
of the distribution.
At the higher flow rate (Figure 9b) the transit time of the
thermocline is five hours between the first and second sensors
and two hours between the second and third. The much more
jagged profile at the lower sensor is indicative of more vigorous mixing during thermocline formation than in the low-flow
case. The outlet side of the temperature profile thickens as
time passes, although not to the same extent as in the low-flow
case. On the inlet side, the bulk temperature actually decreases
9
0.9
1
Dimensionless Temperature, Θ
Dimensionless Temperature, Θ
BACK TO PAGE ONE
Re=1,732
Re=2,822
Re=3,639
Re=4,719
Re=5,828
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0.7
0.6
0.5
0.4
Re=3,778
Re=5,822
Re=7,538
Re=9,614
0.3
0.2
0.1
0
0
90
92
94
96
98
100
102
104
106
108
110
%Volume
Figure 10 Dimensionless charge cycle outlet temperature
profiles.
and the gradient at the center of the thermocline becomes
larger as it travels up in the tank This behavior is symptomatic
of an inlet diffuser that is causing substantial mixing in the
interior of the tank. If mixing at the beginning of the charge
cycle is very vigorous, a large portion of the tank near the inlet
will mix to a temperature intermediate between the inlet and
ambient temperatures. As the thermocline moves away from
a charge inlet and cold water continues to flow in, the temperature of the water near the inlet will decrease steadily until it
approaches the inlet temperature.
Comparison of Outlet Profiles
Dimensionless outlet profiles representative of the final
shape of the thermocline are compared for the five charge
cycles in Figure 10. Three of the profiles are nearly indistinguishable from one another. The other two profiles are similar
in shape but appear to be shifted slightly to the left or right.
Given the similarity in shape of all five profiles, this difference
is consistent with errors in flow measurement. This indicates
that for the range of flow rates tested, the small initial thermocline thickness advantage achieved at lower flow rates may
have been offset by additional conduction during the substantially longer cycle times of those tests. These results substantiate the importance of both inlet mixing and conduction
across the thermocline in determining overall tank performance for full-scale tanks.
Dimensionless outlet temperature profiles for the four
discharge cycles are superimposed in Figure 11. Unlike charge
cycle profiles, final discharge cycle profiles are clearly differentiated by Reynolds number. This suggests that mixing may
have been significant over a larger portion of the tank during
discharge cycles. Given that the upper and lower diffuser
plates have identical dimensions, the effect of the free surface
boundary associated with the upper diffuser may have played
a role in the enhanced mixing evident in the data. The tests
conducted at lower flow rates produce steeper outlet temperature profiles, especially on the outlet side of the thermocline.
Comparing Figure 11 with Figure 10, the outlet temperature
profiles for the discharge cycles are much thicker than for
charge cycles of similar Reynolds number. Initial thermocline
thickness for these cases was on the order of 20% of the tank
water depth. In this case, the strong influence of flow rate on
10
0.8
90
92
94
96
98
100
102
104
106
% Volume
Figure 11 Dimensionless
discharge
temperature profiles.
cycles
outlet
thermocline thickness together with the larger portion of the
tank involved in the inlet region may suppress the “equalizing” effect of diffusion later in the cycle that appeared to occur
in charge cycles.
Inlet Mixing Influence on Mature
Thermocline Characteristics
The finite-difference diffusion model described previously also was used to predict growth of the thermocline
between sensors assuming that only plug flow and conduction
were at work. For each charge cycle, the actual time series
temperature profile taken by sensor c, located 11.25 ft (3.43 m)
above the floor of the tank, was used as the initial condition for
simulations that predicted profile shape at the positions of
sensors g and i, 31.5 ft (9.6 m) and 41.25 ft (12.57 m), respectively, above the tank floor. The heights of sensors c, g, and i
are roughly 10%, 70%, and 95% of the tank water depth,
respectively. Figure 12a shows a typical starting condition and
the corresponding predicted temperature profile at 95% tank
height. These projected profiles were then compared with the
actual temperature profile measured at the higher sensor.
For all flow rates, the predicted and measured profiles
were nearly identical on the outlet side of the thermocline. For
the lower half of the thermocline, the best agreement with the
numerical approximation generally occurred for lower flow
rate cases. At higher flow rates, the actual profile had a lower
temperature than predicted in the “tail” region on the inlet side
of the thermocline. This result, shown in Figure 12b, indicates
that large-scale mixing continues to be a factor in thermocline
development, even after the thermocline has reached onequarter tank height. It also was observed that the measured
profiles in the higher flow rate tests were less smooth than
those for the lower flow rate tests, even though the pure
conduction model predicts smoothing of all the profiles. This
also suggests large-scale mixing effects late in the cycle.
Discharge cycles were analyzed in a similar manner. For
these cases, sensor g at a height of 31.5 ft (9.6 m) provided the
initial condition, and the finite-difference model predicted the
temperature distribution passing sensor i at 11.25 ft (3.43 m)
above the tank floor. Comparison of the predicted and
measured result at the lower sensor (not shown for brevity)
again showed close agreement between the two profiles, with
BACK TO PAGE ONE
56.00
Sensor c, Actual Profile
Temperature (F)
54.00
52.00
Sensor i, Numerically
predicted profile
50.00
48.00
46.00
44.00
42.00
40.00
-6
-4
-2
0
2
4
6
Distance X Time (ft)
(a)
As Figure 12 illustrates, the effects of both conduction
and large-scale mixing that occur late in the cycle are minor in
comparison to the mixing that occurs near the inlet during
formation of the thermocline. This suggests the existence of
two distinct flow regions in the tank: an inlet region characterized by substantial mixing and a core region where changes
in the thermocline are minimal.
56
Sensor i, Actual profile
Temperature (F)
54
52
Sensor i, Numerically
predicted profile
50
sensors was compared to the profile predicted by the pure
conduction model at the later sensor, and a value of R2 was
obtained. The R2 values ranged from 0.9992 for the lowest
flow rate case to 0.9999 for the highest flow rate case. This
result suggests that over the range of Peclet numbers represented by these tests, the effects of conduction are small. For
every case tested, comparison with the results in Table 3 shows
mixing to have roughly the same importance as conduction in
the central region of the tank.
48
46
44
42
40
-6
-5
-4
-3
-2
-1
0
1
2
3
4
5
6
CONCLUSIONS
Distance X Time (ft)
(b)
Figure 12 Comparison of measured charge cycle
temperature
profile
development
and
conduction model prediction (Rei = 1,732, Fri =
0.11): a) initial profile at Sensor c and predicted
profile at Sensor i; b) measured and predicted
profiles at Sensor i.
some departure from the prediction on the inlet side.
Discharge cycles exhibited much more mixing than charge
cycles during formation of the thermocline, but large-scale
mixing in the core of the tank later in the cycle was not
observed. However, the much longer “tails” developed during
the formation of the thermocline were evident in the much
thicker thermoclines exiting the tank at the end of the cycle.
Linear regression was used to provide a more quantitative
comparison of the predicted and measured profiles. Table 3
shows R2 values for each of the various tests. The quantity R2
varies from zero to one, with a value of one corresponding to a
perfect match between the two sets of data. The R2 values for
these tests were all quite high, indicating that once the thermocline moved away from the inlet diffuser the effects of mixing
on the temperature profile were small. Generally, predicted and
measured profiles become more similar (R2 increases) with
decreasing flow rate, but there is significant scatter in the data, and
even at high flow rates the correlation is good. The results shown
in Table 3 also indicate that large-scale mixing effects do not seem
to have a significantly greater impact on discharge cycles than on
charge cycles at the same flow rate.
Relative Influence of Conduction
and Mixing Far from the Inlet Diffuser
Linear regression also was used to quantify the effects of
conduction as the thermocline moved between sensor c
located at 11.25 ft (3.43 m) and sensor g at 31.5 ft (9.6 m)
during charge cycles. The measured profile at the first of these
Field-measured temperature profile data taken from a tall,
naturally stratified chilled-water storage tank have been
presented and discussed. Profiles were obtained for complete
charge and discharge cycles at a variety of constant flow rates.
The tank studied has radial diffusers, which are designed to operate in the recommended range of Fri but at values of Rei substantially higher than are recommended by existing design guidance.
Several conclusions may be drawn from the results of these tests:
•
A method for determining thermocline thickness that is
appropriate for analysis of this type of field data has
been defined. This method addresses problems encountered with previously proposed measures of thermocline
thickness, the most serious of which is neglecting the
tails of the profile and subsequently underestimating the
effects of mixing.
•
Using the definition of thermocline thickness adopted in
this study, charge cycle thermocline thicknesses measured
TABLE 3
Linear Regression Analysis of Predicted
and Measured Thermoclines
Test Type
Rei
R2
Charge
1,732
0.9991
Charge
2,822
0.9927
Charge
3,639
0.9985
Charge
4,719
0.9925
Charge
5,828
0.9916
Discharge
3,778
0.9962
Discharge
5,822
0.9935
Discharge
7,538
0.9486
Discharge
9,614
0.9936
11
BACK TO PAGE ONE
•
•
•
•
near the tank outlet are small to moderate (5% to 8% of tank
height), indicating that good stratification can be achieved
in full-scale tanks at higher Reynolds numbers than are
currently considered acceptable.
Eisenhauer, and Mr. Roger Zimmeran are gratefully acknowledged.
The thickness of thermocline produced by the inlet diffuser
was found to be an increasing function of inlet flow rate.
Although thermoclines produced in the lower flow rate
charge cycle tests initially were slightly smaller than thermoclines produced in higher flow rate tests, outlet profiles
for the charge cycles in this study were nearly identical to
one another. Outlet profiles for high flow rate discharge
cycles, however, remained thicker than those for low flow
rate discharge cycles.
A
= plan area of the tank
g
= gravitational acceleration
For equal Fri and Rei, the upper diffuser produced thicker
thermoclines during discharge cycles than did the lower
diffuser during charge cycles. This enhanced mixing may
be due to the influence of the free surface at the top of the
tank on gravity wave formation. The thermocline formed at
design discharge occupied roughly 15% of the tank volume
and probably would be considered marginally acceptable,
at best. The thicker initial thermoclines formed in discharge
cycles resulted in thicker outlet profiles, which tend to
reduce the overall thermal performance of the tank. The
greatly enhanced mixing apparent at the upper diffuser in
this tank indicates that the upper diffuser may have a more
significant influence on tank performance over a full cycle
than the lower diffuser. As it has been commonly assumed
that upper and lower diffusers will behave in the same
manner, these results provide justification for more focused
study of the performance of the upper diffuser.
q
NOMENCLATURE
g'
hi
∆ρ
= reduced gravitational acceleration, g' = g ------ρi
= height of the inlet diffuser
ht
H
= thermocline thickness
= tank water depth
L
= characteristic length of inlet diffuser
(perimeter for radial diffusers)
= inlet flow rate per unit length of diffuser,
Q
q = ---L
Q
t1
= tank inlet volumetric flow rate
= time required for the thermocline to translate
one thermocline thickness
T
Tc
= temperature
= average inlet temperature (charge), initial tank
bulk temperature (discharge)
Th
T max
= average system return temperature
(discharge), initial tank bulk temperature
(charge)
= maximum acceptable discharge temperature
Tin
Tout
= tank inlet temperature
= tank outlet temperature
V
= plug flow velocity of water through the tank,
Comparison of actual temperature profile development
with predictions based on a finite-difference conduction
model indicated that large-scale mixing effects are significant relative to conduction; however, most of this mixing
appears to occur when the thermocline is near the inlet
diffuser. This suggests that limiting flow rate at the start of
the cycle, a practice already observed by many system operators, will result in a thinner thermocline being established
and maintained throughout the length of the cycle.
α
= thermal diffusivity of water
Comparison of temperature profiles for high Rei operation (1732 to 9614) obtained in the present study with
profiles generated by radial diffusers at low Rei (850 to
1700) observed by Wildin (1989) suggest that a design
Rei criterion of 2000 for tanks over 40 ft (12.2 m) water
depth may be unnecessarily conservative, particularly
with respect to diffusers with a fixed spreading surface
(i.e., lower diffusers).
Inlet Reynolds number, Re i = q---
Q
V = ---A
= kinematic viscosity of water
ρ
ρi
= density of water
= density of inflow water
Θ
Θ max
= dimensionless temperature difference
= dimensionless maximum acceptable discharge
temperature
v
q
Inlet densimetric Froude number, Fr i = -----------------------1⁄2
( g' h i3 )
VH
Peclet number, Pe H = -------α
1
Richardson number, Ri = --------2Fr i
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ACKNOWLEDGMENTS
The authors wish to express their sincere appreciation to
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12
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13