Refrigeration Product Test Procedure Theoretical Analysis of Potential Measurem ent Error for Incomplete Cycling Products 2/5/13

Incomplete cycling products have compressor cycles that last longer than 12 hours. For these products, the DOE test procedure calls for a test period that is 24 hours long (see, for example, 10 CFR 430, subpart B, appe ndix A, section 4.1).

Assuming that the duration of compressor on-cycles 1 and off-cycles 2 remains consistent and does not vary, the long-term average energy use of th e product should be equal to the average energy use associated with a full compressor cycle, consis ting of an on-cycle and an off-cycle. This is because the energy use is dominated by the energy consumed during the on-cycles and the total time duration is equal to the number of cycles multiplied by the duration of a single cycle. For example, after 1000 cycles the remaining ener gy use associated with compressor operation beyond the 1000 th cycle, referred to here as dE, is much smaller than the energy use associated with the 1000 on-cycles. Likewi se, the remaining time associated with operation beyond the 1000 th cycle, referred to here as dT, is much sm aller than the duration of 1000 on-cycles. Hence, the average energy use of a single cycle is approximated as

݁ݏܷ	ݕ݃ݎ݊݁ܧ
ܶ݅݉݁ ൌ ܧ݀ ൅ ܧ1000 ݐ݀ ൅ 1000ܶ ൎ ܧ1000
1000ܶ ൌ ܧ
ܶ
where E is the energy used during a compressor on-cycle and T is the duration of a full compressor cycle.

Because long-term average energy use is equal to the average energy use for a full compressor cycle, stopping the energy test at 24 hours can introduce error into the measurement. This analysis illustrates the potential magnitude of this error.

The analysis assumes that the duration of compressor on-cycles and off-cycles remains constant (i.e. that the unit has reached steady state, as required by th e DOE test procedure prior to measuring energy use, see Appendix A, sections 2.9 and 4), power input is constant at a level P (with units of Watts) throughout the on-cycle, and power input is zero during the off-cycle. The duration of a full compressor cycle is designa ted T (with units of hours). The duty cycle  is defined as the ratio of the duration of a comp ressor on-cycle to the duration of the complete compressor cycle. Hence, th e duration of the on-cycle is  T and the duration of the off-cycle is (1-  )T.

As discussed above, the long-term average energy us e of the product is equal to E/T, where E is the energy use associated with the full compressor. In terms of the parameter P defined above, this average energy use is equal to:

ܧ ௚ ௩௔ ൌ ݁ݏܷ	ݕ݃ݎ݊݁ܧ
ܶ݅݉݁ ൌ ܧ
ܶ ൌ ݅݉݁ݐ	݈ܿ݁ݕܿ ∙ ܱ݊ ൈ ܲ
ܶ ൌ ܲ߶ܶ
ܶ ൌ ܲ߶

1 Time periods when the compressor is operating.
2 Time periods when the compressor is not operating.2

The error associated with use of a 24-hour test period depends on the time at which the test period starts. This analysis compares test period s that begin at the start of a compressor on-cycle and test periods that begin at the end of a compressor on-cycle. Measurement error is generally positive for the former case and negative for the latter case. Figure 1 and Figure 2 below illustrate the refrigeration product power input as a function of time for each of these cases. The figures also illustrate four regions of time for each of these cases when a 24-hour test period could end. The equation for measurement error is different for each of these eight cases.

Figure 1: Power Input Trace for Test Peri od Start at the Start of an On-Cycle

Figure 2: Power Input Trace for Test Pe riod Start at the End of an On-Cycle 

### T

0 T  T T(1+  ) 2T P

12 34On‐cycleOff‐cycle 

### Power Input

Time

Four possible time ranges for the end of the test period (24 hours).



### T

0 T(1

‐  ) T T(2

‐  ) 2T P

12 34On‐cycleOff‐cycle



### Power Input

Time

Four possible time ranges for the end of the test period (24 hours).3

The measured energy use for the 24-hour test pe riod is equal to the power input P times the duration of compressor operation during the 24-h our period. To determine average energy use per unit of time, this value is divided by the te st period duration, 24 hours. Table 1 below shows calculations for the duration of compressor operation and the aver age energy use per unit of time for the 24-hour test period for the eight cases id entified by Figures 1 and 2. The table also shows the fraction of error in the m easurement—this value can be converted to pe rcent error by multiplying it by 100%.

Table 1: Assessment of Measured Energy Use and Measurement Error Test Period Start Time Time Range of Test Period End Duration of Compressor

Operation T on (hours) Average Energy Use E m = P×T on /24 (Watts) Fraction of Error in Measurement Err = ( ா௠ ா௔௩௚ െ 1 )

Start of On-Cycle 1 24 ܲ ൬ 1െ߶ ߶ ൰

2 ܶ߶ ܲܶ߶
24 ൬ ܶ
24 െ 1 ൰

3 24 െ ܶ ሺ 1െ߶ ሻ ܲ
24 ሾ 24 െ ܶ ሺ 1െ߶ ሻሿ ൬ 1െ߶ ߶ ൰ ൬ 1 െ ܶ
24 ൰

4 2ܶ߶ ܲܶ߶
12 ൬ ܶ
12 െ 1 ൰

End of OnCycle 1 0 0 -1

2 24 െ ܶ ሺ 1െ߶ ሻ ܲ
24 ሾ 24 െ ܶ ሺ 1െ߶ ሻሿ

െ ൬ 1െ߶ ߶ ൰ ൬ ܶ
24 െ 1 ൰

3 ܶ߶ ܲܶ߶
24 െ ൬ 1 െ ܶ
24 ൰

4 24 െ 2ܶ ሺ 1െ߶ ሻ ܲ
24 ሾ 24 െ 2ܶ ሺ 1െ߶ ሻሿ

െ ൬ 1െ߶ ߶ ൰ ൬ ܶ
12 െ 1 ൰

In order to illustrate the potenti al magnitude of the error in th e 24-hour test period, the percent error is plotted in Figure 3 for compressor cycl e durations from 12 to 36 hours and duty cycles equal to 25%, 50%, and 75%. The plot shows that the error can be quite large depending on the compressor cycle duration and the duty cycle.4

Figure 3: Percent Error Usi ng a 24-Hour Test Period as a Function of Compressor Cycle Duration and Duty Cycle ‐ 100%

‐ 80% ‐ 60% ‐ 40% ‐ 20%

0% 20% 40% 60%

12 16 20 24 28 32 36PercentErrorUsing24‐hourTestPeriod

Compressor Cycle Duration (hours) 0.25
0.5
0.75
0.25
0.5
0.75 Duty Cycle

Test Period Start with On

‐ Cycle Start

Test Period Start with On

‐ Cycle End