Validation of an Enhanced Drinking Water Temperature Model during Distribution

Abstract

Drinking water temperatures are expected to increase in the Netherlands due to climate change and the installation of district heating networks as part of the energy transition. To determine effective measures to prevent undesirable temperature increases in drinking water, a model was developed. This model describes the temperature in the drinking water distribution network as a result of the transfer of heat from the climate and above and underground heat sources through the soil. The model consists of two coupled applications. The extended soil temperature model (STM+) describes the soil temperatures using a two-dimensional finite element method that includes a drinking water pipe and two hot water pipes coupled with a micrometeorology model. The extended water temperature model (WTM+) describes the drinking water temperature as a function of the surrounding soil temperature (the boundary temperature resulting from the STM+), the thermal sphere of influence where the drinking water temperature influences the soil temperature, and the hydraulics in the drinking water network. Both models are validated with field measurements. This study describes the WTM+. Previous models did not consider the cooling effect of the drinking water on the surrounding soil, which led to an overestimation of the boundary temperature and how quickly the drinking water temperature reaches this boundary temperature. The field measurements show the improved accuracy of the WTM+ when considering one to two times the radius of the drinking water pipe as the thermal sphere of influence around the pipe.

1. Introduction

During transport and distribution, drinking water temperature can increase because of the relatively high temperatures of soil surrounding drinking water distribution networks (DWDNs). Temperature is especially relevant to the microbial quality of drinking water in a DWDN [1,2]. A drinking water temperature below 25 °C at the tap is required to meet Legionella prevention and/or drinking water standards [3]. Soil temperatures are affected by climatic factors and anthropogenic heat sources above and below ground level [4,5]. With climate change, urbanization, and the energy transition, which—in the Netherlands, in particular—comes with an expected increased number of district heating networks (DHNs), urban subsurfaces will heat up even further. Currently, the drinking water temperature at the tap sporadically exceeds the norm, but, with these developments, more exceedances are expected [3].

As pipes in a DWDN can typically last for more than 100 years and modifications during their lifetimes are expensive, measures to prevent high drinking water temperatures should be taken in the coming years, well in advance of temperatures rising to problematic levels. To understand the effectiveness of various measures to keep drinking water temperature at the tap below the threshold, a modeling approach is followed. Such a model requires the possibility of modeling a highly meshed DWDN (a) in which the flows are constantly changing; (b) with various boundary conditions of the produced (incoming) drinking water temperature, which depend on the season; (c) with various boundary conditions of the soil temperature around the DWDN, which depend on the season and underground heat sources, such as the DHN; and (d) that leads to an accurate simulation of drinking water temperatures, thus allowing for the determination of effective measures to keep the drinking water temperature below the threshold of 25 °C. Blokker and Pieterse-Quirijns [4] developed an approach with a one-dimensional soil temperature model (STM) and a drinking water temperature model (WTM). The WTM calculates the drinking water temperature at each customer’s location based on a hydraulic network model (requirement a) and heat exchange between the soil around the pipe wall and the drinking water (requirement b), where the soil temperature is calculated by the STM. The STM was first developed (and validated) for soil temperatures in shallow underground, urban environments (i.e., in sand and under tiles). It was then expanded to include weather forecasts [6], other soil types (peat and clays, with their specific heat exchange properties), other soil covers (grass and trees), and generic anthropogenic heat sources [5]. The STM does not allow for modeling a specific anthropogenic heat source, such as a DHN (requirement c). The WTM was validated in a DWDN at locations with long residence times, allowing for validation of the temperature of drinking water reaching the soil temperature but not the rate at which this happens. Therefore, it is unclear if temperatures in the entire DWDN are determined accurately (requirement d). When one wants to estimate the importance of specific influences on the drinking water temperature (e.g., climate change, DHN, or countermeasures), it becomes important to validate the WTM over its temporal scale.

The WTM assumes a constant temperature at the outer pipe wall, disregarding that—in reality—heat exchange between the drinking water and the soil also affects the soil temperature within a certain sphere of influence. Various authors have suggested adding an insulation layer of soil around the DWDN in a WTM-like approach [7,8,9]. Each suggested a different approach to determine the insulation layer’s thickness. The validity of their approaches is unclear, as two [7,9] were only theoretical. The third [8] validated the model against measured water temperatures. Still, as the model included heat loss at consumers’ locations, the impact of the conductive and convective heat resistances of the soil is unclear. Hypolite et al. [10] showed that good estimates of drinking water temperature can be obtained for a system of large transport mains when the role of the soil in heat transport is considered. Models for water temperature in underground networks are also used to determine heat losses in a DHN [11,12,13,14,15,16]. For a DHN, a simple modeling approach without explicitly incorporating an insulation layer of soil around the network suffices. Firstly, the heat losses at the consumers’ locations are more substantial than the heat loss in the network and, secondly, the temperature differences between the soil and the water in the DHN are large. In order to obtain a good estimate of the order of magnitude of the heat losses, the heat exchange with the soil does not need to be known with high accuracy.

The time it takes for the temperature of the soil around a drinking water main to change is in the order of days to weeks. The time it takes for the temperature of drinking water to change is in the order of minutes to hours. As these temporal scales differ significantly, the approach of two models was used. Both models were improved, leading to (1) an enhanced STM called STM+ [17], which calculates the soil temperature profile, as it is influenced by both drinking water temperatures and external influences, such as the DHN and the weather; (2) an enhanced WTM (called WTM+), which calculates the drinking water temperature, as it is influenced by the soil temperature, as calculated by the STM+. The time scale of the WTM+ is typically hours; the time scale of the STM+ is typically months. The result of the STM+ is a soil temperature used as a boundary condition by the WTM+.

The STM+ is described and validated; the reader is referred to van Esch [17]. In the present paper, we describe the WTM+, in particular, how the thermal sphere of influence is modeled. To show that the WTM+ has improved with respect to the WTM, proper validation is essential. We validate the model with two cases. The first case is a single pipe of ca. 1 km long with a variation in residence times. The second case is a DWDN, where half of the network is potentially influenced by a DHN, and the other half is not.

2. Materials and Methods

2.1. The Water Temperature Model (WTM+)

The change in water temperature, T_water, due to heat transfer (during contact time t) between the surroundings and the flowing water in a pipe can be described as follows [4,18]:

$${\displaystyle \frac{d{T}_{water}}{d\tau }}=k\left(T_{boundary}-T_{water}\right)$$

(1)

where T_boundary is the temperature that the drinking water will eventually reach and k is the heat transfer rate (1/s). In the modeled system, T_boundary is assumed to occur at some distance from the pipe wall (Section 2.2 discusses how to determine T_boundary) and k depends on that distance and the characteristics of the water, pipe material, pipe surroundings, and the hydraulic characteristics of the water flow. Specifically, k is given by the following [4,18]:

$$k={\displaystyle \frac{4·{\alpha }_{water}}{{D_1}^2\left(\frac{1}{Nu}+\frac{{\lambda }_{water}·ln\left(\frac{D_2}{D_1}\right)}{2{\lambda }_{pipe}}+\frac{{\lambda }_{water}·ln\left(\frac{D_3}{D_2}\right)}{2{\lambda }_{soil}}\right)}}$$

(2)

where D₁ is the inner pipe diameter (m), D₂ is the outer pipe diameter (m), and D₃ is the diameter where the boundary condition T = T_boundary is applied (m). The dimensionless Nusselt number, Nu, summarizes the hydraulic conditions of the system and—for pipes—can be described as a function of the dimensionless Reynolds number (Re) and Prandtl number (Pr): Nu = 0.027 Re^0.8 Pr^0.33 for turbulent flows (Re > 10000) and Nu = 3.66 for laminar flows (Re < 2300). Furthermore, α_water is the thermal diffusion coefficient [0.14 × 10⁻⁶ m²/s]; α_water = λ_water/ρ_water C_p,water; λ_water is the thermal conductivity of water [0.57 W·m⁻¹·K⁻¹]; ρ_water is the density of water [1000 kg·m⁻³]; C_p,water is the heat capacity of water [4.19 × 10³ J·kg⁻¹·K⁻¹], with the parameter values given at 20 °C. In this paper, we only consider plastic pipes, for which λ_pipe is the thermal conductivity of PVC [=0.16 W·m⁻¹·K⁻¹]. We typically consider pipes that are installed in sand, for which λ_soil is the thermal conductivity of dry, sandy soil [=1.6 W·m⁻¹·K⁻¹].

With the time-dependent boundary conditions T_water(τ = 0) = T_water,0 and T(τ = ∞) = T_boundary, the analytical solution is as follows:

$$T_{water}\left(t,\tau \right)=T_{boundary}\left(t\right)+\left(T_{water,0}\left(t\right)-T_{boundary}\left(t\right)\right)exp\left(-k\tau \left(t\right)\right)$$

(3)

where t is time and τ is the contact time (in the drinking water context, this is the residence time). It is important to note that k can also be time-dependent, e.g., when hydraulic conditions change over time. Equation (1) to Equation (3) derive from Fourier’s heat equation, describing how T_water changes over time during transport through the network, with the local difference between T_water and T_boundary as the driving force for local heat transfer. A central assumption is that the temperature profiles in the pipe wall and the surrounding soil are developed into a steady state. This assumption is reasonable if the drinking water temperature changes much faster than the soil temperature profiles.

Figure 1 visualizes how the equations and boundary conditions of the current model (Figure 1b,c) and those of its predecessor (Figure 1a) approximate the temperature profile in the modeled system. Figure 1a (the left panel) shows the situation modeled by the original WTM, where a constant soil temperature is applied directly onto the outer pipe wall (green dot), i.e., R₃ = R₂. No gradient in the soil temperature profile is considered.

Figure 1c (the right panel) shows one first possible approximation of the presented approach. When heat is exchanged between soil and water, a soil temperature profile develops with a gradient that attenuates with increasing distance from the pipe. The blue lines and green dots show the profile that develops around a pipe in which the water temperature is kept constant for a long time, as modeled with the STM+. This profile could be used as input for the WTM+ by using the T_soil and R of any green dots to determine T_boundary and D₃ (dotted red lines). If a T_boundary of 26.5 °C is used, like in the other panels, that requires a D₃ corresponding to an R of approximately 1400 m.

Figure 1b (the center panel) shows a presumably more realistic approximation of the presented approach. Contrary to the assumptions of the previous two figures, the temperatures of both the water and the pipe surroundings may change over time at any one specific location along the network. As a result, it is unrealistic to expect the fully developed temperature profiles of Figure 1c. Instead, a dynamic profile between fluctuating soil and water temperatures should be expected, represented by the dotted blue lines. If a semi-steady state can be assumed for this profile, it could be modeled with WTM+ by applying the temperature T_boundary at some distance R₃ (the radius corresponding to D₃ in the equations) of the pipe (green dot) in a way that the slope of the profile is best approximated. The true temperature profile will not be discontinuous at R₃, so the temperature profile that follows from Equation (1) to Equation (3) (red lines) does not perfectly describe the realistic situation.

In each panel of Figure 1, the model behavior is shown for three different values of T_water. These three temperatures can be imagined as occurring over the length of the pipe as the drinking water travels through the network. This means that, after a certain time t, the drinking water will have traveled a certain distance x, and the drinking water temperature will then have increased under the influence of T_boundary assumed for the surrounding soil (26.5 °C in this example). For Figure 1a, the temperature gradient for any given water temperature is much steeper (and thus heat exchange is faster) than for Figure 1b. The example temperatures will therefore be reached at different moments in time, which is why different suffixes are used to indicate the time. The situation in Figure 1a—the old WTM—will lead to an overestimation of how quickly drinking water will heat up compared to the actual situation because heat transfer is never decreased by the development of a less steep temperature gradient in the soil. The situation in Figure 1c will lead to an underestimation of how quickly drinking water will heat up compared to the actual situation because the exaggerated removal of heat for the soil by water with a constant temperature leads to overly attenuated temperature gradients and heat transfer in the soil.

Applying a T_boundary at a distance D₃ in the WTM+ should be considered a mathematical approximation to compensate for the fact that we have not built one integrated model but two coupled models (WTM+ and STM+), while the boundary conditions of the two are not fully independent. Equation (2) assumes that D₃ is constant over time, and Equation (3) assumes that D₃ and T_boundary are constant over the residence time t (i.e., constant over the pipe length). T_boundary is determined with the STM+; see Section 2.2.

In practice, the value of D₃, the distance at which a constant T_boundary can reasonably be applied, is not known a priori. In this study, we try to estimate the value of D₃ from measurements of T_water at various distances and residence times (t₀, t_y, t_y’) and estimates of T_boundary (using the STM+) in two case studies (Section 3 and Section 4). Various authors [7,8,9] have suggested values for D₃ related to the pipe installation depth. However, it is unclear which approach works best, partly due to a lack of validation. We expect that the value of D₃ is most likely larger for large pipe diameters than for small pipe diameters, with a large body of water having more potential to influence the soil around the pipe. To make this scaling influence of the pipe diameter more explicit, we introduce the thermal sphere of influence (TSoI) as an explicit scaling factor: D₃ = D₂ + 2 × TSoI × D₁.

2.2. T_boundary from the Soil Temperature Model (STM+)

T_boundary in Equation (1) is the temperature that the drinking water will eventually reach, and this takes place when the soil and drinking water temperature are in an (dynamic) equilibrium. This equilibrium temperature (T_boundary) may differ from the undisturbed soil temperature and therefore is determined with the STM+. For a thorough description of the STM+, including its validation, the reader is referred to work by van Esch [17].

The STM+ input parameters are a time series of atmospheric circumstances (temperatures, radiation, etc., a so-called micrometeorology model), type of surface (tiles, grass, and trees), and type of soil (dry or wet sand and clay) and evaluate heat transfer for a 2-dimensional geometry, which includes a drinking water pipe. The STM+ calculates soil temperatures in a large grid around the drinking water pipe. The temperatures around the pipe are not constant. Instead, the average temperatures over the circumference of circles around the pipe wall (at a distance of one to two times D₂) were reported as relevant for determining T_boundary. The STM+ average temperature is referred to as “the” soil temperature at a certain distance from the pipe wall (T_soil).

The STM+ can be operated in the so-called “passive” and “active” modes. In the passive mode, the drinking water temperature is a simulation result and not a boundary condition, i.e., T_soil influences the drinking water temperature, but not vice versa. It was shown that T_soil is practically constant over the distance from the pipe wall (up to 3 × D₂, or TSoI = 0.2 to 1) and equal to the calculated T_water. This, by definition, is T_boundary. van Esch [17] showed that T_boundary from the STM+ passive mode is equal to T_soil from the STM, so equal to the undisturbed soil temperature. It is important to note that this mode is used in the left panel of Figure 1. In the active mode, a constant drinking water temperature is imposed, which means that the drinking water acts as an infinite heat source. van Esch [17] showed that, in this case, T_soil can be described as a function of the distance from the pipe wall (up to TSoI = 1) and T_water,0. In addition, T_boundary, defined as the temperature reached when the soil and drinking water temperature are balanced, can easily be determined and is practically independent of this distance from the pipe wall (up to TSoI = 1). This mode is used in the right panel of Figure 1.

This means that the STM (or STM+ passive mode) combined with the WTM (or WTM+ with D₃ = D₂) reflects the situation of stagnant drinking water. The STM+ active mode combined with the WTM+ reflects the situation where the drinking water at a certain location in the pipe is continuously refreshed and kept at a constant temperature, and there is an equilibrium with the soil temperature. This mimics water flowing at a constant flow rate, where the soil temperature along the pipe is constant. In a typical drinking water distribution main, the flow rate changes over the day with (almost) stagnant water during the night and high flow rates during peak hours. The conditions are, therefore, never equal to those in the passive or active mode. Furthermore, the two modes lead to a different value for T_boundary; van Esch [17] showed that, for summer conditions, the passive mode leads to a slightly higher value than the active mode. This means that, in the case studies in this study, the coupling between STM+ and WTM+ is not obvious, i.e., both D₃ and T_boundary are unknown.

2.3. Model Normalization for Validation

In case study 1, the measurements are conducted over a few weeks, and the boundary conditions T_boundary and T_water,0 are not constant over this period, with a maximum change of 1.0 °C per 24 h. We assume that k in Equation (3) is constant during each test at a specific (constant) flow rate and, from it, we want to determine the value for D₃. In order to achieve this, we rewrite Equation (3), assuming that, at each time t during each test, the equation is valid. The model is normalized across tests with $T_{water}\left(\tau \right)=T_{water, 0}+∆TN·\left(T_{boundary}-T_{water, 0}\right)$ and $0<∆TN<1$:

$$\mathsf{\Delta }TN(\tau ,t)=1-exp\left(-k\tau \left(t\right)\right)={\displaystyle \frac{T_{water}\left(\tau ,t\right)-T_{water,0}\left(t\right)}{T_{boundary}\left(t\right)-T_{water,0}\left(t\right)}}$$

(4)

The normalized temperature difference ΔTN represents the change in the temperature of the drinking water along the pipe length relative to the difference in drinking water temperature and the surrounding soil, which we refer to as the driving force. The result is graphically shown in Figure 2 for various values of D₁ and D₃, where ΔTN = 0 represents T_water(τ) = T_water,0 and ΔTN =1 represents T_water(τ) = T_boundary, even when the boundary conditions are not constant.

2.4. Case Studies

The validity of the approach of the WTM+ to model the drinking water temperature over the length of a pipe with T_boundary from the STM+ and a layer of soil as an insulator (TSoI) needs to be verified. This will be achieved in two case studies: the first case study is a single pipe with an internal diameter of 152 mm and the second one is a DWDN, where most pipes have an internal diameter of 100 mm. Case study 1 had limited spatial variation, where many parameters were well defined but had significant temporal variation; case study 1 had limited temporal variation but significant spatial variation, with some unknown parameter values. As the value of the TSoI is not yet known, the case studies will also be used to estimate its value. This means that the case studies are used both for validation and calibration.

For both case studies, a sensitivity analysis was performed (reported in Appendix B.3 and Appendix C.3), and measurements were taken for validation and calibration. The details are described per case study.

2.5. Description Case Study 1—Single Pipe, Ø160 mm PVC

A 925 m long Ø160 mm PVC pipe (D₁ = 152 mm, D₂ = 160 mm) in Rotterdam with only a few connections along the pipe stretch was selected by Evides water company and KWR to validate the WTM+ (and the STM+). This pipe is fed from a surface water PS (pumping station) through 3 km of large-diameter mains (PVC internal diameter > 1350 mm) and a smaller branch of 570 m (internal diameter of 150.6 mm). It is assumed that the temperature at the start of this branch is equal to the temperature from the PS (ΔTN < 0.05 after 4 h for Nu = 100 (turbulent flow), PVC DN1400, and D₃ = 0). The end of this branch is the first location (L1) of the case study. The water in the case study pipe flows from location L1 to L2 (415 m) to L3 (510 m). For this pipe, taking the example of Figure 1, the required residence times (for turbulent flows) can be estimated as t_x ≈ 1 h, t_x′ ≈ 8 h, t_y ≈ 3 h, t_y’ ≈ 24 h, t_z ≈ 9 h, t_z’ ≈ 61 h. This means that measurements are focused on the residence times between 1 and 24 h.

For the single pipe, a sensitivity analysis for $\Delta TN$ was performed analytically (see Appendix B.3). It shows that ${(T}_{boundary}-T_{water,0})$ and ${\lambda }_{soil}$ are the most influential parameters. There is a distinct difference between turbulent and laminar flows.

The drinking water temperature was measured at PS, L1, and L3, with a resolution of ±0.1 °C. The flow between L1 and L3 was controlled with a hydrant at L3 and measured. During a two-week period (18 May 2020–3 June 2020), the flow rate was controlled such that residence times of the water in the pipe from 1 to 24 h were obtained (see

Table A1. Test information: start and end time (2020), flow rates, residence time, and calculated Reynolds and Nusselt numbers (at 16 °C, average of the measured incoming water), with Nu = 3.66 for Re < 5000; Nu = 0.027 Re^0.8 Pr^0.33 for Re > 5000.

ID	Start Time	End Time	Flow Rate [m3/h]	Residence Time [h]	Re [103]	Nu [-]
1	19 May 6:30 (18 May 12:00)	20 May 16:00	2.1	8.0	4.4	3.66
2	20 May 16:15	22 May 12:30	1.4	12.0	2.9	3.66
3	22 May 13:00	25 May 16:30	0.7	24.0	1.5	3.66
4	25 May 16:45	27 May 08:30	1.4	12.0	2.9	3.66
5	27 May 09:15	27 May 20:00	16.7	1.0	35.0	221.49
6	28 May 08:00	28 May 15:00	8.5	2.0	17.8	129.04
7	28 May 15:30	29 May 18:45	1.4	12.0	2.9	3.66
8	29 May 19:00	2 June 07:45	0.9	18.6	1.9	3.66
9	2 June 08:15	3 June 10:00	2.8	6.0	5.9	53.08

). At the same time, soil temperatures were measured at various distances from the pipe wall at locations L1, L2, and L3, which were used to validate the STM+ [17]. The values for input parameters used in the WTM+ (Equations (2) and (4)) are shown in

Table A2. Parameters values of Equation (2) and (4) for case study 1. Constants for water are used (${\rho }_{water}=1000 \mathrm{k}\mathrm{g}{\mathrm{m}}^{-3}$, $C_{p,water}=4.19 \mathrm{J}\mathrm{k}{\mathrm{g}}^{-1}{\mathrm{K}}^{-1}$,${\lambda }_{water}$ = 0.57 Wm⁻¹K⁻¹, ${\alpha }_{water}$ = 0.14 m²s⁻¹).

Parameter	Unit	Value	Uncertainty	Reference
D1	mm	152	-	Information from water utility Evides
D2	mm	160	-
Nu	-	Table A1	Unclear beforehand where transition between turbulent and laminar flow is found	For turbulent flows: Nu = 0.027 Re0.8 Pr0.33 For laminar flows: Nu = 3.66 [4]
λpipe	Wm−1K−1	0.16	-	PVC [4]
D3	mm	D2 + TSoI × D1		To be determined
λsoil	Wm−1K−1	1.6	± 0.3 °C	Measured [17]
Twater,0	°C	Time series, Figure 4.	± 0.1 °C	Measured (L1)
Twater	°C		± 0.1 °C	Measured (L3)
Tboundary	°C		± 0.1 °C	For flowing water, from STM+

. Appendix B.1 has more information on the test location, including a schematic overview.

During the measurements, there was practically no demand from customers along the pipe, as there were only one residential customer and some sports facilities, which were closed during the COVID-19 pandemic. The demand at the customer location was measured, and it had a negligible effect on the flow at L1. The logging frequency for flow and temperature measurements was once per 15 min. The residence time between L1 and L3 was calculated from the pipe length, pipe diameter, and flow.

T_boundary was estimated with the STM+ [17] results for L3, where detailed info on the actual soil parameters was used. The STM+ was validated in this particular case study, and then the model was rerun in the so-called passive mode and active mode (see Section 2.2), both for laminar and turbulent flows. T_boundary for stagnant water (STM+ passive mode) was 0.3 to 0.9 (on average 0.7) °C lower than T_boundary for turbulent flowing water (STM+ active mode, Nu = 3.66) and 0.2 to 0.7 (on average 0.5) °C lower than T_boundary for laminar flowing water (STM+ active mode, Nu = 100).

During each test, the flow was kept constant. After a certain time (and with constant incoming drinking water temperatures), a dynamic equilibrium, as in the situation simulated in the STM+ active mode, could be expected. However, the constant flow was only imposed for a relatively short time, viz. 5 h to max 66 h (Table A1), and the incoming water temperature was not constant. This means that it is not obvious which STM+ (passive or active mode for turbulent or laminar flows) would be applicable. The small differences in T_boundary for this case study will have a significant effect because the difference between incoming and outgoing water temperatures and soil temperatures is relatively small (<2.0 °C). As it is unclear which is the “true” T_boundary, both STM+ modes are tested.

Data processing is required to validate the WMT+ with the measurements (Appendix B.2). A normalization step was performed because none of the parameters in Equation (4) are constant over time. Also, due to measurement accuracies, some data were excluded with the error estimation of Appendix A (22 data points where the drinking water temperature between the beginning and the end of the pipe seemed to change in the opposite direction compared to the soil temperature; 93 data points where the uncertainty was too large). As a last step, the individual data points with similar residence times were grouped and shown in boxplots.

2.6. Description Case Study 2—Drinking Water Distribution Network

Case study 2 for validating the WTM+ consisted of drinking water temperature measurements performed on 31 August 2020 in the DWDN of Almere. The Almere DWDN, owned and operated by Vitens, is ca. 700 km long and mainly consists of PVC pipes, with nominal diameters of Ø63 (8%), Ø110 (51%), Ø160 (15%), Ø200 (5%), Ø310 (6%), and > Ø400 mm (9%) (more info in [19]). The DWDN is fed by two feeding reservoirs in the northern and southern parts of the system, referred to as F1 and F2 (Figure 3). The drinking water temperature was $T_{water, F1}$ = 13.7 °C and $T_{water, F2}$ =13.3 °C on 31 August 2020, respectively. Limited variation in these temperatures is expected throughout the test (less than a day), as this is drinking water from a groundwater source.

The drinking water temperature was measured, with a resolution of ±0.05 °C, by Vitens employees at 35 locations on hydrants (see Figure 3) at two moments of the day (one between 8:00 and 12:30 and one between 12:45 and 16:30), leading to a total of 70 data points. During the tests, the hydrants were opened with a low flow. Two areas (areas A and B; see Figure 3) were selected for the temperature measurements. According to Figure 2, these times should show a distinction in drinking water temperature changes for 110 mm pipes. The two areas have a similar year of installation and thus a similar design philosophy (with similar pipe diameters, materials, and network layout) and a similar number of residents (ca. 10,000 and 7000, respectively). They have a comparable residence time from the sources. The two areas likely differ in soil temperatures (see

Table A8. The parameter values of Equation (2) and (4) for case 2. Constants for water are used (${\rho }_{water}=1000 \mathrm{k}\mathrm{g}{\mathrm{m}}^{-3}$, $C_{p,water}=4.19 \mathrm{J}\mathrm{k}{\mathrm{g}}^{-1}{\mathrm{K}}^{-1}$,${\lambda }_{water}$ = 0.57 Wm⁻¹K⁻¹, ${\alpha }_{water}$ = 0.14 m²s⁻¹).

Parameter	Unit	Value	Uncertainty	Reference
D1	mm	50–300	-	Hydraulic network model Almere, from Vitens
D2	mm	1.052 × D1	-	Outside diameter of PVC estimated with multiplication factor to inside diameter. AC pipes treated as PVC.
λpipe	Wm−1K−1	0.16	Some pipes are AC, but modeled as PVC	AC pipes treated as PVC [4]
Nu	-		Unclear where transition between turbulent and laminar flow is, transition at Re = 5000 from case study 1	For Re > 5000: Nu = 0.027 Re0.8 Pr0.33 [4], Re from hydraulic network model Almere, Pr = 7. For Re <= 5000 Nu = 3.66
D3	mm	D2 + TSoI × D1	Unknown	To be determined
λsoil	Wm−1K−1	1.6	Unknown	Dry sand [17]
Twater,0	°C	$T_{water,F1}$= 13.7 $T_{water,F2}$ = 13.3	± 0.05 °C	Vitens
Twater	°C	13.0–21.0	± 0.05 °C	Measured at hydrants, Vitens
Tboundary	°C	TDM = 18.7 TDM_DHN = 19.5 TDM_NE = 20.5 TTM = 18 TTM_DHN = 20.1	Unknown	For stagnant water, from STM (Table A7).

). The pipes in areas A and B have different installation depths and soil types. Also, in area A, a DHN is installed, potentially influencing part of the DWDN. In area B, there is no DHN.

The drinking water temperatures are expected to increase from the reservoirs during distribution as, in August, the soil temperatures are higher than the source water temperatures. A variation in temperature rise is expected due to differences in residence time and local soil temperature. Regarding the relevant pipe diameters, a residence time of a maximum of 15 h in the distribution areas (i.e., excluding the transport mains) is sufficient to capture the influence of the residence time (the red and yellow lines in Figure 2). The hydraulic model of the area (provided by Vitens) shows a residence time between 5 to 24 h for the whole network and a maximum of ca. 15 h in areas A and B (

Table A9. The residence times of study areas in the Almere DWDN.

	${\mathit{T}}_{\mathit{w}\mathit{a}\mathit{t}\mathit{e}\mathit{r},0}$	Residence Time (h)	Conditions for Tboundary
A	13.3	5–15	TDM, TDM_DHN
B	13.7	10–15	TDM_NE
C	13.3	>24	TDM, TDM_DHN
D	13.3	>10	TDM
E	13.7	10–15	TDM_NE

).

Five groups of pipes are distinguished: transport mains with and without DHN crossings, distribution mains in the older parts of the network with and without DHN crossings, and distribution mains in the newer parts of the network (only without DHN crossings). For these subgroups, a specific value of T_boundary was determined with the STM for 31 August 2020 (see Table A8). In this case, study T_boundary for stagnant water was used, based on the STM and not STM+. The reasons are (a) there are many uncertainties in the (STM) model parameters, e.g., soil type (λ_soil), which means that it was not possible to reliably model this case study site in STM+; (b) the difference between T_water,0 and the estimated T_boundary is relatively large compared to the difference between T_boundary in the active and passive mode, and the added value of modeling this case study with the STM+ seemed limited.

To simulate the drinking water temperature of Almere, the WTM+ (Equations (1) and (2)) was integrated into an Epanet-MSX model [20]; the parameters are given in Table A8. The results are temperatures at each time step and each node. For comparison with the measurements, the results at the nodes closest to the hydrant locations around the measurement time were taken from the model.

A sensitivity analysis for the WTM+ in a DWDN was performed on the case study (see Appendix C.3). It shows that the model is most sensitive to T_boundary and TSoI. The larger the area where the (local) T_boundary is imposed, the larger the number of customer nodes that are affected. For the WTM+ in a DWDN, ${\lambda }_{soil}$ is not a very important parameter to estimate accurately, with the note that T_boundary, as estimated with the STM, is quite sensitive to ${\lambda }_{soil}$. The sensitivity to TSoI depends on the location in the network because of the residence time.

For model validation, the base case of

Table 1. The settings for the scenarios used for validation and calibration. Blanks are equal to the base case. Other parameter values are reported in Table A8. * The T_boundary in the optimal and extra cases is applied with Equation (A11).

Scenario	TSoI	TDM °C	TDM_DHN °C	TDM_NE °C	TTM °C	TTM_DHN °C
Base case	1	18.7	19.5	20.5	18.0	20.1
Optimal case	2	20.1 *		20.1 *
Extra cases	0.1–10	18.0–21.0 *		19.0–22.0 *

was used, for which the parameter values in Equation (2) are the best guesses from a priori knowledge of the area (see Appendix C.1). Because the sensitivity analysis shows that the WTM+ is very sensitive for TSoI and T_boundary, extra cases were introduced to calibrate the values for case study 2. Next, an optimization case was introduced, where the TSoI and T_boundary were changed to find the best match to the measurements (minimum RMSE, maximum R²). The optimal case is the best fit from the extra cases.

3. Results

3.1. Case Study 1—Single Pipe

Figure 4 shows the measured and modeled temperatures. It shows that the incoming water temperature and soil temperatures were not constant over time.

Figure 5 shows ΔTN versus the residence time with the measurement in boxplots and theoretical lines for various values of TSoI. The measurements at 1 and 2 h residence times were mostly discarded because of large measurement uncertainties. After filtering, there are no data points left with negative values, but there are still a few data points larger than 1. These are mainly found in test 4 (with a 12 h residence time).

The theoretical lines depend on the Nusselt number, with a low Nu in the laminar flow regime and a high Nu in the turbulent flow regime. Typically, the literature suggests laminar flows at Re < 2100–2300, but turbulent flows can start anywhere at Re > 3500–10,000 [21]. Figure 4 shows that, at the start of tests 5, 6, and 9, the incoming water temperature changes very quickly, about 0.5 °C in the first few hours; this is not the case for the other tests. This may suggest that, for the high-flow tests, there is turbulent flow in the pipe upstream of L1 and therefore (the same pipe diameter, but slightly different flows) probably also between L1 and L3. For residence times of 8 h and more, we may assume laminar flows. In this case study, a residence time of 7 h means Re ≈ 5000.

Figure 5 shows that the variability is large compared to the average values. It can be roughly estimated that the data correspond with values of TSoI between 0.0 (blue line) and 1.0 (purple line). As the variability of the data covers multiple theoretical lines, it is impossible to determine with enough certainty the correct value of TSoI.

Using Equations (1) and (2), the TSoI was calculated for each data point throughout each test. Figure 6 shows that the value for TSoI sometimes changes gradually over time, and sometimes the change is more abrupt. The spikes in the changes are explained by the inaccuracies of the measurements. To illustrate this, we examined test 3. Figure 4 (or more clearly in Figure A3) shows that T_water,0 increases gradually from 15.8 to 17 °C, although, due to the 0.1 °C accuracy, it does not look very gradual. The same can be said for T_water (t) increasing from 15 to 15.5 °C. T_boundary increases from 14.8 to 15.3 °C. The difference between T_water (t) and T_boundary is typically less than 0.1 °C, or, given the measurement inaccuracies, the difference is between 0 and 0.2 °C. In Equation (4), this means that the denominator and the numerator are the same plus or minus 0.2 °C, or ΔTN ≈ 1. At first, T_water (t) − T_water,0 = 1.0 °C, or ΔTN = 0.8–1.2, and, after 24 h, T_water (t) − T_water,0 = 1.5 °C or ΔTN = 0.9–1.1. This explains the relatively large spikes in TSoI, with a decrease over time. The values smaller than 0 have no physical meaning and can also be explained by the uncertainty in the parameters of Equation (4). Figure 6 thus clearly illustrates the effect of measurement uncertainty. As (T_water,0 − T_boundary) ≈ 0.6 to 1.6 °C during the test period (Figure A3), it is impossible to have highly accurate results where (T_water − T_boundary) > 0.2 °C. In the calculations, it follows that, if D₃ = D₂ (TSoI = 0), (T_water − T_boundary) would be smaller than 0.1 °C for all tests, except for test 1 (0.13 °C). Even if D₃ >> D₂ (e.g., TSoI = 20), (T_water − T_boundary) > 0.1 °C only for tests 1, 2, 4, 7, and 9. It is important to note that (T_water,0 − T_water) > 0.5 °C for all tests. This means that, for TSoI, we should consider the average of the calculated values; the instantaneous values are not reliable enough.

Figure 6 also shows a decrease in TSoI over time. The tests at an 18.5 and 24 h residence time have a TSoI of 1 to 3 at the start of the test and a TSoI of 0 to 1 at the end. The test at 8 h (test 1) and the first test at 12 h (test 2), i.e., the first two tests, show higher values (TSoI = 1–2) than the later tests (tests 4, 7, and 9 with TSoI ≈ 0).

The same data processing was performed with T_boundary from the STM+ passive mode calculations. The results are shown in Figure A5. In these calculations, fewer data points were discarded because ${(T}_{boundary}-T_{water,0})$ was larger. However, the uncertainty due to ${(T}_{water}-T_{water,0})$ is still significant. This exercise leads to estimates of TSoI = 5–10, where under none of the imposed residence times does T_water (almost) equal T_boundary. The results from the passive mode are located in Appendix B.4, because, in case study 2, TSoI values are much closer to those from the results of the active mode.

3.2. Case Study 2—DWDN

Figure 7a shows the base case with TSoI = 1, the simulated drinking water temperature, at the model nodes, as well as the time corresponding to the measurements versus the measured drinking water temperature. As the morning and afternoon measurements showed a maximum variation of ca 1 °C, all data are considered as a single dataset [22]. Areas A and B are distinguished as they have different values for T_boundary. There are two locations (H1330 and H1357 in area B) where the measured drinking water temperature is ca. 3.5 °C lower than the model results and one location (H5870 in area A) where the measured drinking water temperature is ca. 2.5 °C higher than the model results. The deviations were found for both the morning and afternoon measurements, so a measurement error is not a likely cause. There are no known local hot or cold spots that may cause the deviations. It may be that one or more valves in the DWDN are different from what the model assumes, leading to longer or shorter residence times (and thus higher or lower temperatures) than what the model would calculate. This would mean that, for these locations, the hydraulic model does not calculate the correct residence time, and these locations would not be suitable for validating the WTM+. The validation results are shown for all 70 measurements and the subset of 64 measurements where the measurements of these three locations are excluded (referred to as outliers). It is important to note that the optimal case shows the same data points as potential outliers.

The maximum drinking water temperature measured in area A is 20 ± 0.05 °C, and, in area B, it is 20.6 ± 0.05 °C. The maximum drinking water temperature simulated for the measurement locations in area A is 19.5 °C (corresponding to T_{DM_DHN}), and, in area B, it is 20.5 °C (corresponding to T_{DM_NE}). The uncertainty in the model is unknown. For the data in Figure 7a, the model fits the measurements better in area A than in area B with respect to the root mean square error (RMSE) and the Pearson correlation coefficient R² (

Table A11. The goodness of fit parameters for the base case, best fit base case (best TSoI), and best fit extra cases (combination of best fit TSoI and T_boundary).

Figure, Line Color	Area	Case	Tboundary (°C)	TSoI	Excluding Outliers		Including Outliers
					RMSE (°C)	R2	RMSE (°C)	R2
Figure 7a, red	A	Base	19.5	1	0.75	0.82	0.92	0.72
Figure 7a , blue	B	Base	20.5	1	1.01	0.31	1.49	0
Figure A8a,b, magenta	Total	Base	19.5/20.5	1	0.92	0.72	1.25	0.48
n.a.	A	Change in TSoI	19.5	2	0.80	0.82	1.03	0.68
n.a.	B	Change in TSoI	20.5	2	0.76	0.66	1.29	0.08
Figure A8a, cyan	Total	Change in TSoI	19.5/20.5	2	0.77	0.80	1.17	0.57
n.a.	A	Change in Tboundary	19.0	1	0.88	0.71	1.06	0.56
n.a.	B	Change in Tboundary	20.0	1	0.80	0.50	1.24	0
Figure A8b, cyan	Total	Change in Tboundary	19.0/20.0	1	0.83	0.72	1.15	0.49
n.a.	A	Best fit TSoI	19.5	1	0.75	0.82	0.92	0.72
n.a.	B	Best fit TSoI	20.5	3	0.71	0.71	1.22	0.21
Figure A8a, cyan	Total	Best fit TSoI	19.5/20.5	2	0.77	0.80	1.17	0.57
Figure 7b, red	A	Optimal, best fit (RMSE) Tboundary AND TSoI	20.1	2	0.70	0.88
n.a.		Best fit (R2) Tboundary AND TSoI	20.5	3	0.74	0.89
Figure 7b, blue	B	Optimal, best fit (RMSE) Tboundary AND TSoI	20.1	2	0.70	0.67
n.a.		best fit (R2) Tboundary AND TSoI	21.4	7	0.78	0.74
n.a	Total	best fit (RMSE) Tboundary AND TSoI	20.1	2	0.70	0.83
n.a.		best fit (R2) Tboundary AND TSoI	20.5	3	0.72	0.84

).

A change in TSoI has the largest effect on the temperatures at lower residence times, and a change in T_boundary has the largest effect on the temperatures at higher residence times. Changing both parameters simultaneously (the extra cases) may lead to a smaller RMSE or larger R². Table A11 shows that the RMSE decreases from 1.5 °C (without outliers) to 0.76 °C with an increase in the TSoI from 0.05 to 2 and then increases again for TSoI > 2. At the same time, the R² increases from <0 (without outliers) to 0.80 and then decreases.

The result with the lowest RMSE and highest R² is found for T_boundary = 20.1 °C and TSoI = 2 (Table A11, both areas together; this is referred to as the optimal case). Figure 7b shows this graphically. The fit for area A is better than for area B. For areas A and B, the optimum is not found at the same T_boundary—which can be expected as the pipe installation depth and soil types differ in the two areas—nor at the same TSoI, which has not been explained. In the validation for case study 2, the a priori estimates of the soil temperatures (base case) and TsoI = 1–2 lead to the best fit. In the calibration for case study 2 with the a posteriori estimates of the soil temperatures (extra case), TsoI = 2–3 leads to the best fit.

4. Discussion

4.1. Reflection on the Model Approach

In general, the two case studies show a reasonable match between modeled and measured temperatures, as shown in Figure 5 and Figure 7. In both cases, however, the spread in the measured temperatures also reveals plenty of case-by-case deviations. At least in part, these deviations are caused by the approximations inherent to the modeling approach.

The most crucial approximations involve introducing a two-step approach through the coupled STM and WTM models. In the first step—the STM+—a characteristic, steady-state interaction between pipe and environment is determined based on a simulation of a given, two-dimensional cross-section of the system perpendicular to the water flow. This characteristic situation is then transferred to the second step—the WTM+—to evaluate the temperature of the water when it travels through this characteristic situation for a prolonged duration. This approach keeps the model practically manageable, unlike a fully detailed, dynamic model in three spatial dimensions of a complete distribution network and its environment. However, the approach also leads to the simplification of several aspects.

The first aspect is that, in the STM+, general assumptions about the soil, built environment, local weather, etc., must be made, which are then taken to represent large parts of the network. In other words, the WTM+ assumes a limited number of values of T_boundary over space and time. Blokker and Pieterse-Quirijns [4] established that the soil temperature 1 m below ground level varies no more than 1 °C per day. As typical drinking water residence times (hours) are in the same order of magnitude, it seems a reasonable approximation to ignore temporal variations in T_boundary. The spatial variation most likely should not be ignored.

A second aspect concerns the assumption that T_boundary in Equation (1) is not influenced by the drinking water temperature. Figure 1b illustrates that we assume that, over (the residence) time (over the length of the pipe), the tangent to the temperature gradients in the soil is such that there is a constant T_boundary at a specific value of D₃. A third aspect, following the second, concerns the assumption that the temperature gradients in the soil around the drinking water pipes are well-developed, semi-steady profiles. This assumption is unlikely to be true as the drinking water temperature and flow rate in a real DWDN are never constant. The temperature of the drinking water source may change, and the flow velocity of the water changes with daily demand patterns. Both these aspects are related to how the STM+ and WTM+ are coupled. As case study 1 indicates that TSoI may not be constant (Figure 6), this assumption of coupling the STM+ and WTM+ may not be valid. An extensive three-dimensional modeling approach is required to validate this assumption or determine a better way of coupling.

4.2. Estimation of TSoI

The WTM+ does not describe the real heat exchange through the soil around the pipe. Therefore, the thickness of the soil layer is not a real physical entity but serves the purpose of mimicking a decrease in the heat exchange rate. This also means that the fact that T_boundary from the STM+ is more or less constant over TSoI = 0.2 to 1 (not tested for larger values) does not help to determine the value of D₃. It is expected that a temperature change in the drinking water will mean that, after a few hours, only a small layer of the soil around the drinking water pipe will be affected. The STM+ can be used to determine the value of this penetration depth. However, the meaning of this for the value of D₃ is unclear. This means that the value of D₃ (or of the TSoI) is not known a priori. In this study, we tried to deduce the value from measurements. An alternative would be to deduce it from an integrated model.

The values for D₃ that we found in the literature can be converted into equivalent values of TSoI with the help of Equation (1). Hubeck-Graudal et al. [8] used the total heat transmission coefficient between soil and drinking water. They included the conductive heat resistance of the soil plus a correction factor for the convective resistance at the soil surface. We found D₃ ≈ 4(H + 0.1), with H equal to the burial depth. For case study 1, with D₁ = 160 mm, D₂ = 150 mm and H = 1 m, TSoI = (4(H + 0.1) − D₂)/2D₁ = 13.9. De Pasquale et al. [7] used $D_3=D_2+2l=D_2+2\sqrt{{\displaystyle \frac{{\alpha }_{soil}}{\omega }}}=D_2+2\sqrt{{\displaystyle \frac{8.0E-7}{1.992E-7}}}$ , so $TSoI={\displaystyle \frac{\sqrt{\frac{{\alpha }_{soil}}{\omega }}}{D_1}}=13.2$, with ω being 2π divided by the number of seconds in one year. Hypolite et al. [10] developed the Barletta method, which uses the soil temperature at the pipe burial depth as a boundary condition and a shape factor to define D₃. If 2H >> D₂, then D₃ ≈ 4H, and TSoI = (4H − D₂)/2D₁ = 12.6. Díaz et al. [9] assumed that D₃ = H, and thus TSoI = (H − D₂)/2D₁ = 2.8. For case study 2, for the most dominant diameters, D₁ = 110 mm, D₂ = 100 mm, and H = 1 m, the TSoI values are 21.5, 20.0, 19.5, and 4.5, respectively. The values of the first three [7,8,10] are comparable, on average, to TSoI = 13.2 and 20.3 for case studies 1 and 2, respectively; they seem to largely overestimate our results of TSoI = 0.0–3.0. The values found by Díaz et al. [9]—2.8 and 4.5—are closer to what we found. It is remarkable that the approaches from these authors all assume a thermal sphere of influence independent of the pipe diameter.

Case study 1 (Figure 6) shows that the TSoI was not constant throughout the test. In the first few hours after a sudden change in residence time, TSoI equals 2 to 3 and then decreases to 0 to 1. For turbulent flows, the low TSoI value is reached faster than for laminar flows. Case study 2 (Table A11) shows that the optimal TSoI differs for areas A and B, with TSoI = 1 to 2 and 2 to 3, respectively. In a real network, the flows change constantly, and the conditions are more equal to the first phase of the tests in case study 1. This means that a TSoI = 2 is recommended for laminar flows. For turbulent flows, a smaller TSoI (0 to 1) seems more appropriate.

For a TSoI > 0, the heat transfer rate k for the WTM+ is smaller than for the WTM (D₃ = 0). For a Ø110 PVC pipe, a TSoI = 2 means that, for high flow rates (Nu = 100), k is reduced by 36%, and by 60% for low flow rates (Nu = 3.66); for a Ø160 PVC pipe, the values are 27% and 57%, respectively. This means that a considerably longer contact time is needed before the drinking water will have reached the soil temperature than was assumed by Blokker and Pieterse-Quirijns [4]. For a distribution network of Ø110 PVC pipes, a residence time of 12 h is needed instead of 4 h. It is important to note that, as, typically, residence times are longer, the conclusion by Blokker and Pieterse-Quirijns [4] still holds: most customers will experience drinking water temperatures equal to the soil temperature at a 1 m depth.

4.3. Recommendations for Validation Measurements

Extra measurements are recommended for model validation, especially to better understand the TSoI over time and if the TSoI is independent of the pipe diameter. This means that measurements over a range of pipe diameters, including one larger than 160 mm, are required.

We encountered various challenges in the validation process; the learning points are summarized into recommendations for measuring (

Table 2. An overview of case studies and recommendations for measurements.

	Case Study 1	Case Study 2	Recommendation for Measurements
Trajectory	Single pipe	DWDN	Single pipe, measured at 3 or more locations over pipe length
D1 (mm)	152	101 (51% of length)	Range of diameters
Length (km)	0.9	Ca. 700	Result from Q and τ
τ (h)	1, 2, 8, 12, 18, 24	5–24	Use model to estimate the range of interest (depends on D1) and measure at three or more τ, plus at very large τ
Q (m3/h)	Controlled (1, 4 to 0.06)	Result of demands	Use flows both in laminar and turbulent regimes
Duration	2 weeks (2 h to 2 days per test)	1 day (morning, afternoon)	2–5 days per test
# data points	870 (41 to 266 per test)	72	>40 per test; duplicate each test
Water source	Surface water	Groundwater	Groundwater
Season	Spring	Summer	Summer, winter
Parameters that are known accurately	D1, D2, λsoil, λwater, λpipe, αwater, τ (Q, Re), Tboundary, Twater, Twater,0	D1, D2, λwater, λpipe (but PVC was assumed for all pipes in the model) αwater, Twater, Twater,0	D1, D2, λsoil, λwater, λpipe, αwater, τ, Nu, Tboundary, Twater, Twater,0
Strength	Many parameters are known accurately. Parameters are constant over space	Absolute temperature differences are significant. Parameters are constant over time	Many parameters are known accurately. Absolute temperature differences are significant. Parameters are constant over space or time
Weakness	Absolute temperature differences are small. Flow is enforced, not natural, and there may not be a natural equilibrium Nusselt not known	Residence time and Tboundary are not accurately known Nusselt is not known

). In both case studies, the many boundary conditions are continuously changing, are unknown, or are difficult to assess. The two case studies have different strengths and weaknesses [22]. In case study 1, many parameters were well known and approximately stable over the pipe length. However, the differences in the three temperature parameters were relatively small, which meant that the accuracy of the thermal sensor was not high enough. In case study 2, various parameters were not well known. Here, the differences between drinking water and soil temperature were large enough to hardly be affected by measurement uncertainties, and measurements were duplicated (morning and afternoon).

A single pipe stretch allows for the best estimate of all parameters. To measure temperature after a residence time under both turbulent and laminar flow regimes, sensors should be placed at several locations along the pipe length. It is recommended to measure during summer (or winter) in a groundwater-fed system to ensure measurement uncertainties of T_water and T_water,0 that are small relative to the temperature differences.

It is important to note that T_boundary is a model parameter from the STM+. By definition, T_boundary can only be measured when the drinking water and soil temperature are in equilibrium. For the STM+ passive mode, T_boundary can be validated in stagnant water by measuring the temperature of the drinking water. For the STM+ active mode, T_boundary can be validated in flowing water by measuring the drinking water at a very high residence time (the value depends on pipe diameter). Be aware that, as the weather conditions are never constant, T_boundary is, at best, a dynamic equilibrium.

4.4. Practical Application

The 25 °C norm for drinking water in the Netherlands is threatened by, e.g., the installation of a DHN parallel to the DWDN and climate change. Large-scale replacement programs for an aging DWDN and urban developments in greening urban areas allow for distinctive choices with respect to DWDN installation, e.g., enforcing a minimum distance between the DHN and DWDN or installing the DWDN under grass. As a DWDN typically can last for more than 100 years, the time to take measures to prevent high drinking water temperatures is in the coming years. The developed models allow us to quantify the effect of various scenarios and countermeasures. For example, the modeling approach supported determining the minimum distances between DHN and DWDN pipes. First, it became clear that, even without the DHN, the drinking water temperatures may sometimes exceed 25 °C. Therefore, the acceptance of a maximum of a 1 °C increase in the drinking water temperature due to the DHN was decided. It also became clear that, to ensure this maximum increase in the entire DWDN, the distances between the DHN and DWDN would have to be very large, leading to a situation where, in practice, the installation of a DHN could not occur. Instead, the maximum increase was imposed on 95% of the customer connections. Based on a large set of realistic scenarios (WTM+ on the Almere DWDN, with T_boundary from the STM+ for various distances between the DWDN and DHN of various diameters and temperatures), the minimum distance was determined as 1.5 m (wall to wall), with the option of 1.0 m for a maximum of 25% of the DWDN length or 0.5 m for a maximum of 5% of the DWDN length [23].

5. Conclusions

In this study, we enhanced the water temperature model from Blokker and Pieterse-Quirijns [4] into a WTM+ that includes a thermal sphere of influence (TSoI), which accounts for the soil not having an infinite heat capacity. The soil temperatures are affected by climate factors, soil type, soil cover, and (subsurface) anthropogenic heat sources, and the boundary conditions for the WTM+ are modeled with the STM+. We applied the WTM+ on a single pipe and a DWDN over a range of relevant residence times. We used measurements in these case studies to show the validity of the WTM+ and its coupling to the STM+ and determine the value of the TSoI.

Case study 1 showed that using the WTM+ with T_boundary from the STM+ and a TSoI of ca. 1 improves the old WTM. Calibration was a challenge because some boundary conditions and model parameters could not be accurately determined and had to be estimated. The Reynolds number, above which conductive and convective heat exchange between the pipe wall and the drinking water were found, was estimated at 5000.

Case study 2 allowed for a validation in a DWDN. Given all uncertainties, an RMSE of under 1.0 °C and R² > 0.7 is quite good, and using a TSoI of 2 improved the results.

In the two case studies, applying the STM+ and WTM+ together can accurately describe the drinking water temperature change over time throughout a network.