Computer Modelling and Simulation of a Smart Water Heater
Maria Kathleen Ellul
University of Malta
ellul_maria@yahoo.com
Adrian Muscat
University of Malta
adrian.muscat@um.edu.mt
Abstract
A computational model for a typical domestic water
heater is developed. The model updates the simulated
temperature of the mass of water periodically where the
rate of change in temperature depends on the system
inputs, outputs and losses. The water heater model is
validated against measured data. A user model is added
to simulate the times and type of hot-water usage. A
simple basic prediction algorithm based on statistics
and probabilities is then developed to check the viability
of such a system. Hot-water usage is detected by
processing the sampled temperature. The electrical
energy consumed by the water heater over a period of
time is calculated to demonstrate the usefulness of a
smart water heater
Index Terms
Smart, Water Heater, Modelling, Simulation
1. Introduction
The low-cost commonly available water heater
is inherently a lossy device and this means that a
good percentage of the energy consumed heating
water ends up being lost as heat to the
surroundings. Informal measurements and practice
show that if a family of four persons switches on
and off the water heater just when required savings
of up to 80% are recorded. In reality some people
switch the water heater permanently ‘ON’, others
switch it ‘ON’ say 8 hours before it being used and
switch it off after each use and yet others install a
timer. All these techniques suffer either, increased
energy consumption or flexibility, and may require
the user to learn how to adjust the timer.
In this paper the viability of a ‘smart’ water
heater is studied, where the user simply installs the
water heater and then the embedded computer
determines when to switch on the heater or not. To
investigate this problem a computational model for
a typical domestic water heating and usage system
is developed. The system consists of the water
heater, the users and the prediction algorithm.
The water heater model simulates the
temperature variations over time. The users model
determines when users make use of hot water and
the prediction algorithm calculates the next time
that hot water is in demand.
2. Water Heater Physics
An electric water heater consists of (a) an
inner steel tank, that holds the water being heated,
(b) insulation that surrounds the tank so as to
decrease the amount of heat lost to the
surroundings, (c) dip tube to allow cold water to
enter the tank , (d) pipe to allow hot water to leave
the tank, (e) thermostat that reads and controls the
temperature of the water inside the tank, (f)
heating element that heats the water by means of
electricity, (g) drain valve to drain water during
maintenance periods, (h) a pressure relief valve
for safety reasons, and (i) a sacrificial anode rod
that decreases corrosion of the steel tank.
Water temperature inside the heater is
controlled by the mechanical thermostat. The
temperature may usually be set by the user
somewhere in between 49 and 70OC.
The heater makes use of the convection
heating principle to heat the water in the tank. Hot
water is less dense than cold water. So, cold water
moves to the bottom of the tank while the hot
water rises to the top.
The heat gained or lost by the water is given as
in [1]:
∆Q = mc∆θ + losses
where
Q is the heat gained by the
water in Joules (J), m is the mass of the water, c is
the specific heat capacity of the water, 4200J/kgK,
∆Q is the change in temperature experienced by
the object, and losses are the heat loss experienced
through the insulation and supporting studs. Also,
∆Q = Phe ∆t
Where Phe is electrical power, and t is the time
taken for the water to gain an energy Q. For
simplicity, it will be assumed that the energy loss
is negligible next to the energy gained when the
water is being heated. Therefore by substitution:
Phe ∆t = mc∆θ
Hence the relationship between a change in
temperature and a change in energy is linear.
Taking into consideration heat losses from the
tank, the expected temperature vs time graph is
approximately exponential .
Although the water heater has good insulation,
heat loss is inevitable since even so the insulator
has a small value of heat conductivity.
Newton’s Law of cooling states: “…the rate of
loss of heat is proportional to the access
temperature over the surroundings”, [1], p.636)
As the heat inside the water heater decreases, the
temperature difference between the temperature of
the water inside the water heater and the
temperature of the surrounding air (ambient
temperature) decreases and hence gives an
exponential decreasing graph when considering
temperature of the water inside the tank vs. time.
Therefore considering all the facts discussed
above on water heaters, the expected pattern of the
temperature inside the water heater is expected to
be ‘ fig.1:
T
were carried out on a typical 80ltr home water
heater that is available from hardware stores.
Two K-type thermocouples were attached in
parallel to the EL–USB–TC Thermocouple Data
Logger. Thermocouples in parallel give an
average reading of the temperatures read at each
different thermocouple [3, 4]. Since the
temperature inside the water heater differs
depending on the level of the water, each
thermocouple was placed in two separate levels
touching the water heater tank directly so as to
read and store the average temperature values of
the water inside the tank. One thermocouple was
placed behind and held by the temperature
indicator which is at the top section of the water
heater while the other was placed in the lower part
of the water heater next to the heating element’s
terminals.
The data logger was set to store readings every
ten seconds. Every data set stored a total of
approximately three and a half days. The results
taken over a period of a few months were then
studied.
T
T
Electrical
Energy in
Heating
Element
Thermostat
T
Water
t
Fig. 1 Temperature vs. Time graph for heating
and cooling water
Furthermore the thermostat controls the power
input such that the water temperature oscillates
between a minimum and maximum temperatures
defined by the thermostat settings as well as the
hysteresis characteristics of the thermostat.
3. The Water Heater Model
The water heater model consists of updating
the water temperature by (a) energy input from the
heater (b) heat loss through insulation (c) heat loss
due to exchange of hot water with cold water. The
losses and gains incurred and the rate of change
have a big impact on the performance of the water
heater and therefore a measurements of
temperatures of the water inside the water heater
Water
Temperature
out
Cold in
Hot out
Fig. 2: The water heater model
Fig.4 shows a block diagram of the water
heater model. The model is designed such that the
electrical energy input, the cold water in and the
hot water out cause a change in the temperature.
The water temperature is also a model output that
will be used by the usage detection algorithm and
the prediction algorithm. The thermostat action
was preserved so as to act as an upper limit
constraint for the water temperature.
Five different events that cause a change in
temperature were considered: (1), Heating the
water inside the water heater, (2) Natural cooling
of the water inside the water heater, (3) Cool
water usage, (4) Warm water usage, and (5) Hot
water usage.
The heating and natural cooling of the water
inside an 80ltr hot water heater was calculated by
studying the results taken from the data logger
when the hot water heater was switched off for
approximately 8 hours and switched on again
without hot water being used. The result was then
plotted using Microsoft Excel and best fitting
equations were found through a best curve fit. The
best fit equation for heating is given below and in
fig.3.
Tnow = Tmax + 2 – (Tmax – T + 2)e-0.002n
It may be seen that the natural cooling could be
taken to be a linear function since it takes a very
long time to cool to ambient temperature were the
exponential curve would be seen.
Tnow = 0.0032n + T
heating process which is also seen to increase
exponentially as seen previously. Therefore this
process was divided into two as seen below.
The exponential decay results from an overall
averaging out of the temperature inside the water
heater. This is because the cold water that would
have just entered the water heater continues to
cool down the overall temperature through heat
convection and heat conduction. Additionally the
straight line graph was calculated from the
beginning of the exponential series found so as to
ensure continuation.
For all the following equations:
Tnow = current temperature,
Tmax = maximum allowable temperature,
n = a time step of 10 seconds and
T = starting temperature.
The slow hot water flow:
Tnow = -0.026n + T
Tnow = 3.5e-0.008n + T – 3.5
60
T e m p e r a tu r e (D e g C )
50
40
Series1
30
Series2
20
10
0
1 99 197 295 393 491 589 687 785 883 981 1079 1177 1275 1373
Time Step
Figure 3 Heating graph: Data logger reading
superimposed with equation from curve fitting;
Cool, warm and hot water usages are based on
three different hot water flows. These all force the
temperature inside the tank to decrease drastically,
but at different rates.
The equations used in the simulation for the
three different hot water flows were found by
filling a 15 litre container with hot water using 3
different flows from the same outlet point. The
time taken to fill the 15 litre container was
recorded and the effect of the water flow on the
temperature of the water inside the water heater
was stored in the temperature data logger. The
results were then studied and mathematical
equations were formed using Microsoft Excel
through best curve fitting.
Through the studies of the three graphs
resulting from the three flows, it could be seen that
while the water flows, there is an approximate
straight line graph, but once the flow stops, the
graph decays exponentially and then starts the
The medium hot water flow:
Tnow = -0.029n + T
Tnow = 1.75e-0.019n + T – 1.75
The high hot water flow:
Tnow = -0.065n + T
Tnow = 3e-0.02n + T – 3
The seasonal ambient temperature of the water
going into the tank was not considered. This
affects the results for forced cooling when winter
and summer incoming cold water temperatures
are taken into account.
The tests were made on a single indoor 80ltr
water heater used by a family of three people in
Malta. The results found will change according to
the insulation of different types of water heaters,
their size and where they are installed i.e. outdoor
or indoor and if it is installed in a cold or warm
country.
Three different hot water flows were taken
into account. In reality more different types of
water flows exist depending on how hot the user
likes his or her bath or shower.
The water flows were only tested in a shower
that is always used by the owners of the tested
water heater. Changing the place of hot water
usage in the same home or using a different type
of shower head may affect the flow of the water
hence the hot water may cool down faster or
slower.
different times and may also turn off the water
during different times of their washing
experience.
53.6
T e m p e ra tu re (D e g C )
53.4
53.2
Series1
53
Series2
52.8
52.6
52.4
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23
Time Step
Figure 4: Medium hot water flow graph 1
4. Model Validation
54
53.5
T e m p e r a tu r e (D e g C )
The results of the forced cooling were taken at
different times and on different days. This may
result on the input water temperatures being
different and hence resulting in an error as my
simulation is only based on these results. This was
also tested at only one basic internal temperature
and not at different inside temperatures. If the
water heater temperature was raised or lowered,
this may result in different readings with different
overall equations.
53
52.5
Series1
52
Series2
The water heater model was validated against
the measured data. Fig.8. shows the steady state
temperature for the case of a thermostat controlled
heater.
Temperature (Deg C)
51.5
51
50.5
1 19 37 55 73 91 109 127 145 163 181 199 217 235 253 271 289 307
Time Step
Figure 5 Medium hot water flow graph 2:
Reading from data logger superimposed with
equation from curve fitting
The testing was also done in a shower one
floor directly below the water heater. If the testing
was done in the upstairs bathroom adjacent to the
water heater, the water pressure for the hot water
would have been less, since height is a factor of
pressure, and hence the hot water flow would have
been less.
The heating element may not have the same
properties as another heating element inside a
different water heater.
The heat lost from the pipes leading to the
shower was not taken into account.
The temperature data logger used to measure
and record the temperature inside the heater is
accurate to ± 0.5 degC. It was also not inserted
directly inside the water heater but touched the
tank in two separate places to take an average of
the temperature inside the tank. This would have
been more accurate if the temperature was taken
directly from inside the water tank.
A usage of hot water was considered for an
average of between 1 and 10 minutes with water
always running. This is not true as people take
Tim
e
Figure 6 Heating and Natural Cooling Graph.
Measured data can be superimposed on this.
From the above graph, it can be seen that the
temperature is kept between a minimum and
maximum temperature which is taken from the
average temperature set by the thermostat. The
heating part is exponential while the natural
cooling part is linear. This graph is a close match
to measured data.
The behaviour of the model during usage was
also tested. The descent gradients in the graphs
are steeper than that of the natural cooling. That
is as expected and this feature is used to locate
usage in time.
The model was then tested to see how it would
react when a number of usages run one after the
other. A graph of one single run is seen in fig.9.
The usages are circled in red. This test was
done a number of times with different
combinations to make sure all different
combinations give good results. The results
obtained matched the measured data.
T e m p e r a tu r e (d e g C )
70
60
50
40
30
20
10
0
6:00:00 a one is seen in the hot water usage tables
more than once then there is a good probability
that there will be a usage at that time and a usage
is predicted. The predicted table would then be
used to determine when the water should be
heated or not depending on the series of ones and
zeros.
Series1
0 0 :0 0 :1 0
0 1 :2 3 :0 0
0 2 :4 5 :5 0
0 4 :0 8 :4 0
0 5 :3 1 :3 0
0 6 :5 4 :2 0
0 8 :1 7 :1 0
0 9 :4 0 :0 0
1 1 :0 2 :5 0
1 2 :2 5 :4 0
1 3 :4 8 :3 0
1 5 :1 1 :2 0
1 6 :3 4 :1 0
1 7 :5 7 :0 0
1 9 :1 9 :5 0
2 0 :4 2 :4 0
2 2 :0 5 :3 0
2 3 :2 8 :2 0
0 0 :5 1 :1 0
0 2 :1 4 :0 0
Separate tables were used for different days. A
time window of 4 weeks is used when calculating
the probability of occurrence.
Time
Figure 98 Three usages of hot water in series
5. Users’ Model
The users’ model is required to generate
instances of hot water uses in any one of the three
intensities described in section 3.0. Informal
research showed that many people follow patterns
throughout the day and this includes hot water
usage for example when taking showers. For
example for most people interviewed hot water
usage is predictable during weekdays, however
during weekends hot water usage occurs at
random time and is less deterministic. The users
were therefore modelled by adding to the event
calendar the usages at the appropriate times.
A pseudo code of how this table method was
implemented is given in [5].
The hot water usage detection was tested by
observing that the ones and zeros in the output
vector correlate to the time domain temperature
plot.
The operation of the basic time series
prediction algorithm was observed to verify that it
operated as expected. A weekday graph followed
by a weekend graph with random usage times
may be seen below.
On weekdays, hot water usage is consistent
and deterministic, therefore the adding of the
‘ON’ and ‘OFF’ events work perfectly. This may
be seen in the weekday graph where the water is
heated to the right temperature just before
someone uses the hot water and then the water
was allowed to cool after the usage when no water
was going to be used within the next two hours.
6. Prediction Method
The
table
method
involves
storing
temperatures of the hot water heater throughout
the day at equal intervals. The temperatures are
then studied and another table is derived, by
gradient comparisons, to show when there is hot
water usage. A hot water usage is indicated by a
one and a zero indicates that no hot water is being
used.
The hot water usage table is stored and
compared to other hot water usage tables to find a
pattern of hot water usage. If, for example, at
T e m p e r a tu r e (d e g C )
70
60
50
40
30
20
10
0
Series1
0 0 :0 0 :0 0
0 1 :1 3 :0 0
0 2 :2 6 :0 0
0 3 :3 9 :0 0
0 4 :5 2 :0 0
0 6 :0 5 :0 0
0 7 :1 8 :0 0
0 8 :3 1 :0 0
0 9 :4 4 :0 0
1 0 :5 7 :0 0
1 2 :1 0 :0 0
1 3 :2 3 :0 0
1 4 :3 6 :0 0
1 5 :4 9 :0 0
1 7 :0 2 :0 0
1 8 :1 5 :0 0
1 9 :2 8 :0 0
2 0 :4 1 :0 0
2 1 :5 4 :0 0
2 3 :0 7 :0 0
The smart water heater is required to learn the
times when hot water is demanded. In doing so the
heater can be automatically switched on say 1 or 2
hours before the time of demand. In this way the
minimum required electrical energy is consumed.
In this paper a simple algorithm based on statistics
is used to demonstrate the feasibility of the smart
water heater. This time series prediction algorithm
is termed the table method in this paper.
Time
Figure 9 Weekday hot water usage including
prediction
T e m p e r a tu r e (d e g C )
70
60
50
40
30
20
10
0
Series1
are given in [5].The system is simulated as a
discrete event system in the event view as defined
in [5]. The event calendar is initialised with a first
event to start heating the water in the hot water
heater and seven days of daily hot water usages.
0 0 :0 0 :0 0
0 1 :1 3 :0 0
0 2 :2 6 :0 0
0 3 :3 9 :0 0
0 4 :5 2 :0 0
0 6 :0 5 :0 0
0 7 :1 8 :0 0
0 8 :3 1 :0 0
0 9 :4 4 :0 0
1 0 :5 7 :0 0
1 2 :1 0 :0 0
1 3 :2 3 :0 0
1 4 :3 6 :0 0
1 5 :4 9 :0 0
1 7 :0 2 :0 0
1 8 :1 5 :0 0
1 9 :2 8 :0 0
2 0 :4 1 :0 0
2 1 :5 4 :0 0
2 3 :0 7 :0 0
9. Conclusions
Time
Figure 10 Weekend hot water usage including
prediction
On the other hand, since the weekends are
inconsistent the results are not that good. Firstly it
seems that a usage was predicted in the early hours
of the morning as the water was heated. This area
is circled in black. The actual next usages are seen
circled in white. In this case there was hot water
there for these usages. There is another usage
circled in red that was not predicted and in this
case the water was not warm as it should have
been if it was predicted.
Nonetheless the simple prediction algorithm is
still useful in determines the viability of a ‘smart’
water heater. Better algorithms can improve the
‘smartness’
7. Energy Saving Calculations
The energy consumed by the water heater over
a period of time is calculated by integrating the
power supplied to the heater over time.
Simulations show that for the case of a family
of three people one can save about 2 hours of
electricity input daily (3 kWhr units) with the table
method compared to the case of manually
switching the heater on and off when required.
The above was calculated for a weekday and
not a weekend. This is because the weekend does
not yield significant savings because of the
irregular times of hot water usage. When
compared to the case of using a timer the energy
consumed is approximately the same with the
added luxury of the ‘smart’ feature.
8. Software Development
The design of the Smart Water Heater was
developed using the JAVA language in the Eclipse
Platform Version 3.3.0. Details on this software
In this paper a water heater model was
developed and validated with measured data. To
this model a users’ model and a prediction
algorithm were added so as to carry out a
feasibility study of the concept of a smart water
heater.
The simulation results show that with the use
of a simple prediction algorithm and one
temperature sensor the energy savings are
comparable to that of using a timer, with the
added convenience of not having to reset the timer
due to weekly and seasonal changes.
Nevertheless there is ample space to improve
on the prediction method used resulting in more
savings. The prediction algorithm should strive to
supply the right amount of hot water at the right
time and in the right quantity. More studies can
therefore be carried out in this respect. For
example the learning algorithm can also vary the
maximum temperature set resulting in higher
savings and studies that consider various usage
patterns can be included, to be able to determine
the benefits for different lifestyles and conditions.
Another feature that may be desired is a
dissatisfaction input through which the water
heater will learn that it either missed a usage event
or the water was not heated to the required
temperature.
[1] Eastop and McConkey, Applied Thermodynamics
for Engineering Technologists, 5th Edition, 1993
(Pearson Education Ltd., Harlow)
[2] Nelkon and Parker, Advanced Level Physics, 7th
Edition, 1997 (Heinemann, Oxford)
[3] Retrieved on 15th October 2007 from
http://rabi.phys.virginia.edu/HTW/supplements/the
rmometers_and_thermostats.pdf
[4] Retrieved on 17th November 2007 from
http://www.allaboutcircuits.com/vol_1/chpt_9/5.ht
ml
[5] Design of a Plug ‘n Play Smart Water Heater,
Maria Kathleen Ellul, University of Malta, 2008