Applied error estimation and propagation method

The error propagation method proposed for in-situ U-value measurements in this thesis research builds on the ISO-9869 standard and Baker's error propagation method as discussed in the previous Section 3.3.4.2. The same ISO-9869 estimated intrinsic and extrinsic errors were accepted for this thesis research - see Table 8. As previously discussed, the natural variability of in-situ heat flux measurements under changing, dynamic environmental conditions are listed in ISO-9869 as a ±10% uncertainty, while Baker (2011b) uses the sd of moving average U-values obtained from the ratio of means, as an attempt to capture this natural variability 'swing' (Section 3.3.4.2.).

Instead of either of these methods, the natural variability during the monitoring period could be estimated by using a statistical approach of one of the properties of the data distribution, such as the sd around the mean, obtained from the mean of ratios. As per - Equation 47. or Equation 48., depending on use of internal surface or air temperatures respectively.

- Equation 47., mean of ratios where internal surface temperatures are used or,

- Equation 48., mean of ratios where internal air temperatures are used, where Umean_surface and Umean_air are the U-values obtained from internal surface and internal air temperatures respectively; TSi is the surface temperature of the floor in the room; RSi is the internal surface thermal resistance, taken to be 0.17 m2KW-1 in accordance with BSI (2007), while RSe is set to zero if external air temperatures (Tea) are used, as is the case in this study. Tia is the internal air temperature and q (Wm^-2) is the heat-flux density, derived from Equation 39. Index j identifies individual measurements and n is the number of measurements.

Justification for the proposed error estimation and propagation method

Table 9. gives a summary of the included errors.

Instrument error (Intrinsic)

Measuring condition/equipment

set-up errors (Extrinsic) Inherent property (not a measurement error)

± 5% Accuracy heat-flux and temperature sensors

±3% Operational/deflection error ±sd (% Natural variability U)

±5% Contact error

±5% Temperature location measurement error Total error

- Equation 49. and is the total estimated uncertainty for each individual location point measured in the environmental chamber; where sd is the natural variability of the daily or hourly U-value and sd is based on daily data for field data and hourly data for steady-state data collected

Table 9. Summary of proposed estimated measurement uncertainties; intrinsic and extrinsic errors obtained from ISO-9869, see Section 3.3.4.1.

In the proposed error estimation analysis, the natural variability of the mean U-value is represented by one standard deviation (sd) of hourly obtained U-values for an environmental chamber and daily data for in-situ field data as per Equation 49. below and is justified as follows:

1. For field measurements, daily U-values were used to estimate the final mean U-value over the monitoring period, with the sd of the daily values as an estimate of the natural variability in U between each day over the monitoring period. Daily data is used as suspended timber ground floor structures might typically be subject to a 24 hr periodic day/night cycle as suggested by Isaacs (1985b), though it is unclear whether this is really the case and what the short-term or (long-term) seasonal thermal mass time-lag of suspended ground floors is. Others have also used daily averaged U-values for field studies, e.g. Wingfield (2009) and Wingfield (2010a) for masonry and timber-framed cavity wall studies (without declared error analysis).

2. Mean of ratios: the proposed error propagation technique needs to be applied to the mean of ratios as per Equation 47. or Equation 48. and not the summation technique (or ratio of means) for reasons set out in Section 3.3 and 3.3.5.2. In the proposed technique, each day is treated as an independent datapoint. Given that environmental conditions do not vary at random but in a gradual and correlated way through time, each datapoint is an average that captures a daily time interval and the environmental variation associated with it. This approach is likely to minimise autocorrelation effects.

3. For thermal chambers, in the absence of 24-hour day/night or heating pattern cycles, smaller than 24hr time intervals could be used. However, time intervals which are too small may contain a lot of noise caused by researcher influence (e.g.

opening/closing doors, touching of sensors during data collection) and might also show the cycling of the thermostat to provide the required heat input (Isaacs, 1985a).

Hence some kind of averaging of the raw data (collected usually at 1- 5 minute intervals) over a longer time period may be useful to get a better estimate of the natural variability with other influences averaged out over a longer time period; for this reason a one-hour interval was used. Chapters 4.3.5. and 5.2.4. compare the mean U-value estimated from raw data (at 1 to 5 minute intervals) with the mean U-value of the hourly or daily data, which were very close and hence were good approximations of the estimated mean U-values for the case-studies used.

4. Moving average techniques were excluded here due to lack of independence of data and due to biasing the natural variation component low - as discussed in Section 3.3.4.2., section B.

5. The sd is a property of the distribution of the in-situ measured U-values, which reflects the spread of the data around the mean. Because the monitoring period is over a snapshot in time, using the sd provides a good estimate of the intrinsic variability observed in the observed data over the monitoring period, and would not need any further adjustment.

6. Where surface temperatures are used for U-value estimation, it could be argued that the ±5% temperature sensor location error (see Table 8.) does not apply.

However in this research, external air temperatures were used, as discussed in Section 3.3.3.3. Additionally, where surface temperature are used, uncertainty is created by the addition of a constant RSi in the final U-value estimates (see 3.3.3.2.).

Hence some allowance for uncertainty around temperature measurements seems appropriate and this temperature location measurement error has been retained, though it's actual effect is unknown.

7. Like the other error propagation methods, the method proposed here also does not account for any variation in U due to longer-term seasonal changes unless measured over a longer time period including different seasons. The same assumptions about independent errors and normally distributed data is applied here as in all other methods, though further research would be required to investigate these assumptions; these issues are beyond the scope of this thesis research.

Error propagation: Combining intrinsic and extrinsic errors with natural variability of U in the quadratic sum

Given that buildings exist in a changing environment, it could be argued that the natural variation in in-situ measured U-values is a property of a dynamic U-value and should not be combined with true sources of error such as instrument and measurement condition errors.

However, the natural variability of U over time as a result of changing environmental conditions is treated as an 'error' by ISO-9869 and Baker (2011b) in the estimation of a static U-value and by combining the natural variability with the other errors in the quadratic sum - see Equation 41. and Equation 43. respectively. The natural variability is not ideally combined with the true sources of error in the quadratic sum¹⁰ as it is not a true source of error. While all intrinsic and extrinsic sources of error could be combined in the quadratic sum and the natural variability estimate could be reported separately alongside, this would make

comparisons between U-values difficult (e.g. Umean ±dx(intrinsic & extrinsic errors) ±natural variability).

Hence for the purpose of this thesis and after ISO-9869 and Baker (2011), the true sources of error were combined in the quadratic sum with the natural variability as per Equation 49.

Doing so allowed the presentation of a final estimate of uncertainty around the mean estimated U-value which better enabled comparison between different point U-values on the floor, pre/post intervention comparisons and estimating whole floor U-values from point-U-values.

A limitation of this technique is that underestimations of the uncertainty in U might occur due to use of the quadratic sum. There might be some compensation for this as some ISO-9869 assumed errors appear conservatively estimated compared to other estimates - see Appendix 3.D.; there might also be some double-counting as some errors cannot be separated from the sd - though whether this is the case and the actual extent of the errors remains unknown. Implications of this proposed error propagation technique are that for all the data, the errors will be minimum ±9%, based on the inherent instrument and

measurement errors in Table 9., with an addition of the sd as a representation of the natural variation of the daily or hourly U-values as per Equation 49. With some exceptions,

uncertainty estimates in this thesis generally fell between Bales' (1985, p4.) error estimate range of ±5% to ±20% for in-situ field measurements - see Chapters 4, 5 and 6.

10 Nor should the natural variability of U be combined in the simple addition of the individual errors as this would

There are also situations where not all of the above individual estimated error components are required, for example when estimating differences or the impact of changes for the same sensor locations and these conditions are set out below and in Equation 50.

Point U-value comparisons between interventions

For comparison between U-value points on a floor (in the same intervention and between different floors), all of the above errors set out in Equation 49. and in Table 9. apply. However where identical instruments and measurement conditions occur (such as a sensor with the same fixing and location and use of the same sensors in the same location when pre/post intervention), contact, deflection and instrument errors need not be included to facilitate comparison of relative differences between the same locations on the floor – see Equation 50. The temperature location measurement error has been retained in all cases: even where temperature sensors remained in the same place: the intervention in itself might have had an effect on observed temperatures.

- Equation 50. is the total estimated uncertainty where a relative comparison is made for the same point location; the first term is the error estimated from temperature location measurement errors; sd is based on daily data for field data and hourly data for steady-state data collected

Whole floor U-value comparisons

Obtaining whole floor U-value estimates from observed point U-values leads to uncertainties and these are identified below:

(1.) Errors in each point measurement will influence the whole floor estimated value and can be accounted for by summing the 'average' or 'weighted' individual errors according to the adopted 'whole floor' U-value estimation technique - see Chapter 4.4.2.

(2.) Natural variation of U in each point can be accounted for by inclusion of sd for each U-value point in the final uncertainty estimate and adjusted as described in (1.) but with the same limitations as noted prior.

(3.) Spatial uncertainties:

⁃ Spatial variation with regards to the location and number of sensors (i.e.

resolution) are an unknown but expected uncertainty arising from the spatial variation of heat-flow across the floor.

These can be minimised for by taking as many point measurements as possible and use of thermal images in sensor placement as noted in Sections 3.3.2., 3.2.6.1. and Chapter 4.4.2.

⁃ There will be an unknown uncertainty in estimating techniques of the whole floor U-value; this can be minimised with use of thermal imaging to ensure appropriate point U-value averaging or weighting techniques - see Chapter 4.4.2.

For comparison between whole floor U-values, only the first 2 sources of uncertainty (error in each point measurement and variability in U) can be included and can be weighted for each individual point U-value in the final estimated uncertainty of the whole floor U-value, as illustrated in Chapter 4.4.2. The spatial uncertainties cannot be quantified and can only be minimised for by careful research design. Unknown uncertainties will be associated with the above error estimates and assumptions, use of surface temperature sensors (and RSi addition) and spatial and natural variations and whole floor averaging techniques, which would affect confidence in final estimated whole floor U-values. Transparency is required in

communication of results and where comparisons are undertaken, as discussed in the following section.