Observed wastewater temperatures showed an expected overall pattern of variation in residential sewers (sites 1 and 2) and collector sewers (sites 3 and 4), where wastewater temperature patterns followed those of in-sewer air temperature. The wastewater flow rates in the residential sewers generally presented a low, stable DWF with some flow spikes that reflect the rainfall events. The collector sewers have shown different wastewater flow variations, where sites 3 and 4 presented steady drops in flow rates from February to April. This is likely to be caused by higher infiltration rates during the winter months. This type of long-term flow data, combined with temperature data, may be utilised in the future to make more accurate predictions of groundwater infiltration.
The residential sewers showed similar values of heat transfer coefficient between wastewater and in-sewer air (hFr) during March and April, which was followed by a noticeable drop, from 43 W/m2·K in April to 7 W/m2·K in May, reflecting the higher thermal resistivity between wastewater and in-sewer air (Rwa). This suggests that the behaviour of heat transfer between wastewater and in-sewer air in residential sites is season-dependent. The average in-sewer air temperature in residential sewers was around 20 °C in summer (May to July), which is approximately twice that in winter (March to April). Assuming the moisture content in sewers is constant, the lower temperature in winter can increase the relative humidity, as can be observed from a hygrometric chart. For example, the in-sewer air relative humidity of 50% at 20 °C would increase to 90% if the temperature drops to 10 °C. The higher relative humidity can increase the air thermal conductivity, which may explain the larger winter values for the calibrated parameter (fc) in residential sewers. Since Fr was almost constant in all months of each sewer category, the variation of heat transfer coefficient (hFr) is directly proportional to fc. Unlike the urban sewers, the collector sites have generally presented more consistent fc values in winter and summer. The average in-sewer air temperature in collector sewers was around 11 °C in summer, excluding June, and 9 °C in winter, which limits the variation in relative humidity and hence the closer fc values. However, June has shown a slightly lower fc value in collector sewers, which may be due to the relatively high in-sewer air temperature of around 15 °C. It is also worth noting that in-sewer air has less influence on the heat exchange in collector sewers than that in residential sewers as the ratio of wastewater depth to diameters in the former is around 40% compared to approximately 5% in residential sewers.
An additional process that may be necessary to consider when simulating wastewater temperatures is the heat transfer between the in-sewer air and the ambient air. This could particularly improve the modelling accuracy in April at residential sewers, which resulted in the largest modelling discrepancy in the 2015, 2020 and Elías-Maxil models. Preliminary calibration of the thermal resistivity between in-sewer air and ambient air (Raa) in April, using the 2020 model at urban sewers, was performed to investigate the impact of the heat transfer within the air on the modelling performance. It was estimated that if Raa was around 0.001 m.K/W and the temperature difference between in-sewer air and ambient air was 3 °C, the 2020 model accuracy, represented by the bias, can be improved by around 50%. However, more details, e.g., local ambient temperatures and in-sewer air pressure, is needed to perform a more in-depth calibration of Raa. Unfortunately, this level of detail was not available and is beyond the scope of the paper. A simplified heat transfer model in a partially filled pipe can be created in the future, e.g., using a computational fluid dynamic (CFD) software package, to investigate impact of heat exchanged between in-sewer and ambient air.
The calibrated ratios of soil thermal conductivity to penetration depth (ks
) were often larger in colder months in residential and collector sewers. The high ks
(or low ds
) ratios result in low thermal resistivity between wastewater and soil (Rws
) (Equation (10)). Higher soil thermal resistivity was observed in residential sewers during warmer months (Table 2
). This is likely since the temperature difference between wastewater and soil was generally higher, by 3 °C than that between wastewater and in-sewer air. Hence, the energy balance (Equation (7)) would imply higher soil thermal resistivity. Nevertheless, key parameters related to soil thermal resistivity, e.g., soil structure, type, void size and saturation level, were unavailable which made it difficult to confirm the exact physical causes of soil thermal resistivity variations. The high flow rate in collector sewers indicates high thermal energy, which means more heat needs to be dissipated, thus the lower thermal resistivity in collector sites.
The 2020 model has generally improved the accuracy of predicting wastewater temperatures, as proven by the majority of modelled values showing correlation coefficients higher than 0.9 (Figure 10
values of the 2020 model were relatively low, with around 60% being equal to or less than 0.4, compared to 40% in the case of the Elías-Maxil model. A
value of 0.4 is considered low for the purpose of predicting wastewater temperatures; for example, March at site 3 (March 3) has a
of 0.36, and a corresponding CRMSE of 0.10 °C, which is close to the measurement accuracy. Therefore, reducing the model error would require more accurate sensors. The CRMSE has the advantage of representing the absolute error of the model, which is independent of the overall bias. Thus, when neglecting the bias between modelled and observed temperatures, e.g., in May at site 1 of the 2020 model, the CRMSE was relatively low (0.3 °C) and hence showed better accuracy than that interpreted by the PDF plot (Figure 8
The RMSE of the 2020 model ranged between 0.15 and 0.5 °C, which was relatively low in comparison to the 2015 model (0.12 to 0.87 °C), Elías-Maxil (0.16 to 0.8 °C) and TEMPEST (0.8 °C to 7.8 °C). Overall, the 2020 model average RMSE (among all months) was 0.27 °C, which is slightly lower than that of previous models, e.g., 0.35 °C in the 2015 model, 0.42 °C in Elías-Maxil. Nevertheless, the Elías-Maxil model presented higher absolute errors, as shown by the high , of up to 1.5 in comparison to 1 in the 2020 model, and low correlation values that reached 0.25 in comparison to 0.72 in the 2020 model. The TEMPEST model has presented the highest overall RMSE values with an overall average of 2.0 °C. It is worth noting that TEMPEST was validated on shorter datasets and limited time periods that excluded months like April, which particularly presented higher variation in observed temperatures.
Overall, the 2020 model has presented more accurate results that showed considerable differences when predicting the potential viable heat recovery in comparison to the 2015 model. Abdel-Aal (2015) [9
] performed a sensitivity analysis on all model parameters for urban and large collector sewers. The sensitivity of the model to ds
is much less (e.g., by 50%) than that of the heat transfer coefficient between wastewater and in-sewer air. Therefore, and through close examination of the model behaviour, the reason behind the improved accuracy is the new relation developed in this paper.
The 2020 model requires few input parameters that are accessible, measurable or predictable through existing hydrodynamic models; these are mainly the upstream wastewater temperature, depth and velocity in addition to the in-sewer air temperature. This is important for the model’s practicality and deployment in large sewer networks, which can save considerable computational time. Enhancing the model’s practicality facilitates its implementation in existing hydrodynamic software packages.
Collector sewers have generally shown lower errors than residential sewers. This was expected because of the higher wastewater flow rates, around 35-fold, than that of residential sewers. The higher flow rates demand more energy to be dissipated from wastewater to drop its temperature by 1 °C. This is due to the heat capacity of wastewater, i.e., the multiplication product of specific heat capacity (cp,w) and the wastewater mass flow rate. Therefore, the model is less sensitive to the calibrated parameters when collector sewers’ data were utilised for calibration, using either model, in comparison to that of the residential sewers. This explains the lower collector sewer values and the more symmetric collector sewer PDF plots.
A novel relationship to estimate the heat transfer coefficient between wastewater and in-sewer air was developed to better predict the behaviour of heat exchange mechanisms at the wastewater-air boundary in combined sewers. Case study results predicting temperatures with the novel relationship indeed indicate the importance of including in-sewer air–water boundary heat transfer. The new relations enabled more accurate predictions of the potential heat recovery from sewers, which can result in estimating up to 25% additional viable heat recovery in winter.
The new heat transfer coefficient between wastewater and in-sewer air (hFr), which ranged between 5 and 58 W/m2·K, was estimated and calibrated as a function of the wastewater Froude number using seasonal data from small residential and large collector sewers. The model was validated on larger independent datasets for both residential and collector sewers. The calibrated parameter for the heat transfer coefficient between wastewater and in-sewer air (fc) had a stable value for periods of several months, indicating that the proposed parameterisation was an improvement on previous approaches and appeared to work well in both residential and collector sewers. Warmer weather presented lower but stable fc values to reflect the change in the heat transfer processes during warmer periods. The new parameterisation improved the modelling accuracy as presented by the absolute errors and is based on available hydrodynamic data that can facilitate the simulation process.
A normalised Taylor diagram was effective in summarising modelling errors graphically and comparing between various models, where the majority of the 2020 model results were close to the observed reference point, which demonstrates a generally high modelling accuracy. This accuracy is adequate for the purpose of the 2020 model, which is to assess the viability of heat recovery from individual sewers in large sewer networks. The PDF of the 2020 model results has shown over-prediction and underprediction of wastewater temperatures, which can overall minimise the modelling error when it is implemented to assess the viability of heat recovery from large networks.
Observed wastewater temperatures show seasonal variations in both residential and collector sewers and showed wastewater temperature patterns following air temperature patterns. Hence, considering seasonal data as well as in-sewer air–water boundary heat transfer is important when studying the process of heat transfer in sewers.
Although the modelling error in April at urban sewers was reduced by the 2020 model, it has particularly shown the largest discrepancy, and this requires further investigations. It is envisaged that additional processes, such as heat transfer between in-sewer air and ambient air, may need to be considered in order to further improve the modelling accuracy.