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Application of a linear input/output model to tankless water heaters.


In the US, energy factor (EF) is currently used as the metric for the energy efficiency of residential water heaters (ASHRAE 2006; DOE 2001). This test was developed and evolved as a compromise between testing complexity and the goal of being representative of typical field conditions. The EF is a direct input/output test--it is essentially the ratio of the energy output, in hot water, to the fuel and/or electrical energy input integrated over a 24-hour period. During this time, hot water is produced in six draws over a 6-hour period. The balance of the 24-hour period is the idle time and energy used during this time is taken as part of the daily input. Each of the six draws is 10.7 gal. (40.6 L) for a total daily draw of 64.3 gal (243.7 L). The target temperature of the produced hot water is 135[degrees]F (57.2[degrees]C).

Increasingly, the effectiveness of the EF test in evaluating energy use to meet domestic hot-water demand and providing a basis for comparison of different technologies is being questioned. One of the specific concerns is the draw pattern. Field studies have shown that hot water is actually drawn in many short draws. While this impacts the performance of tank-type storage water heaters, it may be more important for newer tankless (also sometimes termed instantaneous or demand) water heaters that have "near zero" volumes and higher input rates (Hoeschele and Springer 2006). These units are designed to meet domestic hot-water loads based on continuous, modulating input with very low standby or idle losses.

Thomas et al. (2006) did a study of domestic hot-water use patterns in Toronto and found an average daily draw of about 43.8 gal (166 L). In addition, they showed that the actual draw patterns involve many very short draws and that the difference between actual draw patterns and the draw patterns that are used in the EF test can significantly affect efficiency. Based on this, they developed alternative draw patterns that could be used in a modified test method.

In the present study, we have explored the use of a linear input/output model to represent the performance of a range of water heater types over a wide load range. We then show how such a model could be used to calculate the impact that draw patterns and total daily consumption have on water heater efficiency. Relative to the current test method, the key advantage of the input/output approach is the ability to use one set of measurements to evaluate performance of a specific unit over a range of draw patterns.

The concept of the linear input/output method is straight-forward--for a given appliance there is a linear (or nearly linear) relationship between energy output and energy input at least over the range that is relevant for the application. For time periods over which the load is intermittent, output and input are averaged over the entire period. The line is unique to the inlet and outlet water temperatures.

Decicco (1990) used a linear boiler model to evaluate field performance data for commercial boilers and showed how results could be used to evaluate annual performance and the potential of energy saving upgrades. Rosa and Tostato (1990) showed the linear model could be applied to condensing boilers if the impact of seasonal water temperature changes imposed by the control system is included.

Currently, ASHRAE Special Project Committee 155 (SPC 155) is developing a method of test for commercial boilers that implements the linear model (Hewett 2005). With a relatively simple set of measurements, the characteristic performance characteristics can then be combined with system configuration and control characteristics and building load profiles to produce an application seasonal efficiency (ASE).


The objectives of this study are to evaluate the applicability of the linear input/output model to tankless water heaters and to use the results of characterization tests to evaluate impacts of draw patterns and total daily draw volume on efficiency.


All water heaters tested were setup for direct input/output measurements using the arrangement shown in Figure 1. Energy input was measured using a dry test meter for gas flow with a pulse output sensor. The resolution of this system is 1000 pulses/[ft.sup.3], and total pulses each second are recorded using a local pulse logger. Natural gas fuel composition and relevant properties are measured periodically using an "online" gas chromatograph. Energy output is measured using inlet and outlet thermocouples and a weigh scale that communicates with the lab's measurement and control system. The scale is located on the second floor balcony of the lab and drains down during the periods between hot-water draws under control of the lab's central system. Draining of this tank is stopped 15 seconds before each draw and for a 15-second period after the end of the draw to allow readings to stabilize. Temperatures and scale mass are recorded at 5-second intervals. For some very short draw tests, a 1-second temperature measurement interval was used. A 40 gal (151 L) preconditioning tank was used to heat or cool the inlet water. Inlet and outlet thermocouples were located 4 in. from the appliance.


For each appliance tested, a basic protocol was established which included steady state tests covering inlet water temperatures from 40[degrees]F to 70[degrees]F (4.4[degrees]C to 21.1[degrees]C) and outlet temperatures from 105[degrees]F to 133[degrees]F (40.6[degrees]C to 56.1[degrees]C). The protocol also includes a wide range of cycling conditions intended to replicate cold, warm, and hot conditions at the start of the draw. Table 1 shows the test matrix followed for each unit. All tests were done with the lab at a nominal temperature of 70[degrees]F.
Table 1. Planned Test Conditions for Each Water Heater

 Cyclic Tests

Test No. Volume, gal Volume, L T cold in, T cold in, [degrees]C

 Cyclic Tests at 7.6 L/min (2.0 gal/min)

1 1 3.8 60 15.6

2 1 3.8 60 15.6

3 1 3.8 60 15.6

4 2 7.6 60 15.6

5 2 7.6 60 15.6

6 2 7.6 60 15.6

7 3 11.4 60 15.6

8 3 11.4 60 15.6

9 3 11.4 60 15.6

10 4 15.1 60 15.6

11 4 15.1 60 15.6

12 4 15.1 60 15.6

13 5 18.9 60 15.6

14 5 18.9 60 15.6

15 5 18.9 60 15.6

16 10 37.9 60 15.6

17 10 37.9 60 15.6

18 10 37.9 60 15.6

 Cyclic Tests at less than 3.8 L/min (<1 gal/min)

19 2 7.6 60 15.6

20 2 7.6 60 15.6

21 2 7.6 60 15.6

22 10 37.9 60 15.6

23 10 37.9 60 15.6

24 10 37.9 60 15.6

 Cyclic Tests at 15.1 L/min (4 gal/min)

25 2 7.6 60 15.6

26 2 7.6 60 15.6

27 2 7.6 60 15.6

28 10 37.9 60 15.6

29 10 37.9 60 15.6

30 10 37.9 60 15.6

 Cyclic Tests

Test No. T out, [degrees]F T out, [degrees]C Idle time, min

 Cyclic Tests at 7.6 L/min (2.0 gal/min)

1 133 56.1 2

2 133 56.1 4

3 133 56.1 45

4 133 56.1 2

5 133 56.1 4

6 133 56.1 45

7 133 56.1 2

8 133 56.1 4

9 133 56.1 45

10 133 56.1 2

11 133 56.1 4

12 133 56.1 45

13 133 56.1 2

14 133 56.1 4

15 133 56.1 45

16 133 56.1 2

17 133 56.1 4

18 133 56.1 45

 Cyclic Tests at less than 3.8 L/min (<1 gal/min)

19 133 56.1 2

20 133 56.1 4

21 133 56.1 45

22 133 56.1 2

23 133 56.1 4

24 133 56.1 45

 Cyclic Tests at 15.1 L/min (4 gal/min)

25 133 56.1 2

26 133 56.1 4

27 133 56.1 45

28 133 56.1 2

29 133 56.1 4

30 133 56.1 45

 Steady-State Tests

 Flow, Flow, T cold in, T cold in, T out, T out,
 gal/min L/min [degrees]F [degrees]C [degrees]F [degrees]C

31 1.5 5.7 60 15.6 133 56.1

32 2.5 9.5 60 15.6 133 56.1

33 max max 60 15.6 133 56.1

34 2 7.6 60 15.6 133 56.1

35 2 7.6 60 15.6 105 40.6

36 2 7.6 60 15.6 115 46.1

37 2 7.6 60 15.6 125 51.7

38 2 7.6 40 4.4 133 56.1

39 2 7.6 70 21.1 133 56.1

Cyclic testing was done under computer control. A series of cyclic test conditions were defined in an input file and this typically contained combinations of draw patterns with the total test period as long as 20 hours. For each specific draw pattern, multiple draw/idle cycles were imposed and this ranged from 3 to 20. Short draws required more cycles for repeatability. During these cyclic tests, all data were recorded in multiple files, and these were analyzed later to determine average conditions and results for each pattern. During this work, eight different tankless, gas-fired water heaters were tested. Results with two representative units are discussed in this paper.


Detailed results are presented here for one condensing unit (Unit A) and one non-condensing (Unit B). Both of these represent typical tankless gas-fired water heaters with a nominal maximum input of 199,999 Btu/h (58.6 kW).

For Unit A, Figure 2 shows the measured thermal efficiency vs. flow for a 60[degrees]F (15.6[degrees]C) inlet and 133[degrees]F (56.1[degrees]C) outlet. The reduction in efficiency with decreased flow results from the burner's air/fuel ratio control system. At low water flows and low input rates, burner excess air is very high. For the cyclic runs with unit A at nominal 60[degrees]F (15.6[degrees]C) inlet/133[degrees]F (56.1[degrees]C) outlet temperatures, Figure 3 shows the input/output relationship. This is apparently linear over the entire range, and the linear regression expression for all of these data is

Input = 1.073 * output + 211.95 (1)

where input and output are in units of Btu/h averaged over the draw and idle periods.


For Unit B, Figure 4 shows the measured thermal efficiency vs. flow for a 60[degrees]F (15.6[degrees]C) inlet and 133[degrees]F (56.1[degrees]C) outlet. Again, the reduction in efficiency with decreased flow results from the burner's air/fuel ratio control system. For the cyclic runs with unit B at these same inlet and outlet temperature conditions, Figure 3 shows the input/output relationship. The regression equation for this is

Input = 1.2051 * output + 271.2 (2)



For both of these linear regression equations, the correlation coefficient is essentially 1.0.


The EF rating includes a simulated use test as discussed above, including six equal, large draws an hour apart and 18 hours of idle. To better reflect actual, in-field use patterns, Thomas et al. (2009) have proposed two alternative, representative draw patterns. These are as follows.

Modified 1:

1 draw of 90 L (23.8 gal) followed by 40 minutes of standby

18 draws of 2.1 L (0.55 gal) with 10 minutes standby in between

18 draws of 2.1 L (0.55 gal) with 3 minutes of standby

All draws are made at 11.4 L/min (3.0 gpm)

Modified 2:

1 draw of 90 L (23.8 gal) followed by 40 minutes of standby

18 draws of 2.1 L (0.55 gal) with 10 minutes standby in between

18 draws of 2.1 L (0.55 gal) with 3 minutes of standby First draw at 13.8 L/min (3.7 gpm), all other draws at 3 L/min (0.8 gpm).

For both of these patterns, an idle period follows the draws for the balance of the 24-hour period.

As an alternative to testing under these draw patterns, the regression equations developed from Figures 3 and 5 can be used to calculate performance over the 24-hour period. However, some careful consideration needs to be given to the applicability of the input/output method at the very lowest end of the input/output relation. In the EF test, during the one-hour idle/draw pattern combination, the average output rate over this period is approximately 6800 Btu/h (2.0 kW). The average draw rate over 24 hours is approximately 1750 Btu/h (0.51 kW). Both of these are very far on the left side of Figures 3 and 5. For Unit A, Figure 6 shows the measured input/output data at the very low end of the range and the results of the regression analysis (Equation 1) in this range. This shows that the relationship is reasonable even at this low output rate. These points show linearity for short draws (as low as 1 gallon) with relatively long (to 45 minutes) idle periods. Also, it is interesting to note that in Figures 3, 5, and 6, cyclic draws taken at different flow rates have been found to lie on the same line. This would not be expected to also be the case with different inlet and outlet temperatures.



During a very long (e.g., 18-hour) idle period, the regression equation for Unit A would predict an input rate of 212 Btu/h (0.062 kW). However, for this tankless water heater, the input rate during this period is only the standby electric power use of the electronics, which is on the order of 20 Btu/h (5.9 X [10*.sup.2] kW). Clearly, the linear relationship cannot be valid at the extreme low end of the input/output curve, although the departure from this relationship occurs at output rates much lower than are under consideration for typical home use during occupied periods. For this reason, the following approach is suggested for using the input/output relationship to predict performance under arbitrary draw patterns: (1) during active draw periods, use the linear relationship to predict input required for specific outputs, and (2) during extended idle periods (e.g., >2 hours, use the actual measured standby energy use to estimate input required. This 2-hour time period is assumed for example and should be further evaluated).

In evaluating the 24-hour performance, the procedure used here has been to apply the standby energy consumption to the entire idle period. During the active draw period, the output rate is evaluated for each draw individually combining the draw period and preceding idle period. For the first draw, it has been assumed that it follows a 1-hour idle period.

To better illustrate the use of this approach during the active draw period, the following example is provided. In the EF tests the amount of energy output during the hourly draw is

Energy out = 10.7[gal/h] x 8.329[Btu/gal * [degrees]F] x (135 - 58)[degrees]F = 6862[Btu/h] (3)

The corresponding input during this period, for Unit A from Equation 1, is

Energy in = 1.073 x 6862 Btu/h + 211.95 = 7575 Btu/h (4)

This procedure is applied to all draws during the simulated use test, combined with the power use during the extended idle period, and all inputs and outputs are summed over the 24-hour period.

Using this approach, for Unit A, an EF of 0.90 is predicted. This can be compared with an actual measured EF of 0.9. For this same unit, with the modified draw pattern 1 above, the efficiency over a 24-hour period is 0.88. The actual measured efficiency over this draw pattern is 0.87. The performance calculated over modified draw pattern 2 is similar. The key advantage of the second draw pattern is the lower flow rate and, in turn, longer draw periods, which greatly improves accuracy during testing.

For Unit B, using the regression results in Equation 2, the predicted EF is 0.80. The listed EF is 0.82. The predicted performance of this unit over modified draw pattern 1 is 0.79.

This approach can be used to illustrate the impact that total daily water use has on performance as measured with any draw pattern. Figure 7 shows the results, for example, of the EF draw pattern (i.e., six equal draws, one hour apart, followed by an 18-hour idle period) with varied daily water use using Unit B. To obtain the results shown in Figure 7, the pattern of the EF or modified draw pattern 1 was held the same, but the total volume during the day was varied by changing the volume in each draw, e.g., to reduce the total daily volume to 80% of the standard value, the volume of each draw would be multiplied by 0.8. For each draw, energy output divided by time of the draw and preceding idle period was used to calculate average output rate, Btu/h (kW). The linear relation for the specific water heater considered is then used to calculate the energy input for that draw and these are summed for the entire draw period.

The results shown in Figure 7 could be considered relative to the results of Hoeschele and Springer (2006), in which it was shown that the performance of tankless water heaters is lower under conditions of frequent short draws relative to the long draws of the EF test. The input/output relationship method discussed here also yields a strongly reduced efficiency under conditions where the cycling pattern produces a low average input and output rate, i.e., on the left side of the line in Figure 3. How this translates to a daily average efficiency depends strongly upon the details of the assumed draw pattern. For this reason it is very important that draw patterns used for comparing actual in-field efficiency be realistic.


As another illustration of the potential use of this approach, Figure 8 shows the impact of electrical power draw during the long standby period on the EF and EF from the modified draw pattern 1. In the case of the modified draw pattern, the impact of the standby electrical power use is greater because the 24-hour total load is lower and the idle period is larger.


The data used to develop the input output relationships presented here are based on actual cyclic draw patterns and, in this sense, capture the effects of cycling that are built into draw patterns. The results, however, may not be accurate for draw patterns that differ significantly from the range of patterns evaluated in this study.


Results of laboratory tests have shown that with cyclic draw patterns over a very wide range of load, a linear input/output model reasonably represents performance.

For predicting energy use during extended standby periods with these units, the extrapolation of the linear input/output relationship should not be used; instead, use of results of direct measurement of standby power is recommended.

This approach can be used to evaluate the performance of specific tankless water heaters over a wide range of use conditions.


This work has been sponsored by the Minnesota Department of Commerce, Office of Energy Security, and New York State Energy Research and Development Authority through the U.S. Department of Energy State Technology Advancement Collaboration (STAC) program.


ASHRAE. 2006. ANSI/ASHRAE Standard 118.2-2006, Method of Testing for Rating Residential Water Heaters. Atlanta: American Society of Heating, Refrigerating and Air-Conditioning Engineers, Inc.

DeCicco, J.M. 1990. Applying a linear model to diagnose boiler fuel consumption. ASHRAE Transactions 96(1):296 * 304.

DOE. 2001. Final rule regarding test procedures and energy conservation standards for water heaters. 10CFR430, Federal Register 55(201), U.S. Department of Energy.

Hewett, M. 2005. Inside ASHRAE Standard 155: A new measure of commercial boiler system performance. Boiler Systems Engineering.

Hoeschele, M., and D. Springer. 2006. Field and laboratory testing of gas tankless water heater performance. ASHRAE Transactions 114(2).

Rosa, L., and R. Tosato. 1990. Experimental evaluation of the seasonal efficiency of condensing boilers, Energy and Buildings 14:237 * 41.

Thomas, M., S. Hayden, K. Wittich, D. MacKenzie, and H. Lomax. 2009. Hot water use in Canada and the implications for performance test standards. Presentation at ACEEE Water Heater Forum, Asilomar, California, June 2009. American Council for an Energy Efficient Economy.

Thomas A. Butcher, PhD


Ben Schoenbauer

Associate Member ASHRAE

Thomas A. Butcher is deputy chair of the Sustainable Energy Technologies Department at Brookhaven National Laboratory, Upton, NY. Ben Schoenbauer is a research engineer at the Center for Energy and Environment, Minneapolis, MN.
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Author:Butcher, Thomas A.; Schoenbauer, Ben
Publication:ASHRAE Transactions
Article Type:Report
Geographic Code:1USA
Date:Jan 1, 2011
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