MAIN SOURCE AREA MODEL

Data on the urban catchment are often regrouped by city. 18 household areas can be identified. In most cases, those areas are spread over large surface with extensive sewer network. Thus, using those areas directly as source elements in the model would be an over-simplification of the reality as it would not represent the complexity of the system. The same difficulty applies to the hospital. Rooms are spread on a complex and large sewer network within the hospital. To consider that all the rooms are at the same point would be an over-simplification.

However, there is not much data to model this level of complexity. That is why the “Main source area” model is proposed. It is a generic assembling of fundamental elements (sources and pipes) that spreads the discharges in order to represent the complexity of the sources. It requires two additional parameters for the source fundamental element: the average length of pipes between the household outlet and the associated standard deviation.

Graphic symbol:

In: -

Out: wastewater flow and pharmaceutical loads.

Parameters: number of households 𝑁ℎ𝑜𝑢𝑠𝑒, number of workers 𝑁𝑤𝑜𝑟𝑘𝑒𝑟, number of hospital beds 𝑁𝐻−𝑏𝑒𝑑

present in the main source area, average length of sewers between discharge points and the outlet of the area 𝐿̅ (m) and standard deviation associated to this distribution 𝜎(𝐿) (m).

Goal: to generate the wastewater flow and pharmaceuticals loads of all population types inside a main source area.

5.4.1.1 GENERATION OF THE MAIN SOURCE AREA

The main source area model arbitrarily consists of 20 consecutives pipes with 20 sources, one at each pipe input (figure 33). Households, workers or hospital beds are distributed in the 20 sources.

Figure 33: Main source area structure.

To generate a main source area, one must follow the following steps:

 Lengths: each household, worker and hospital bed is affected with a length of pipe necessary to reach the output of the main area source. Those lengths are randomly picked following a lognormal distribution of parameters 𝜇 and 𝜎.

𝑙_𝑖= 𝑟𝑎𝑛𝑑𝑜𝑚_{𝑙𝑜𝑔𝑛𝑜𝑟𝑚}(log (𝐿̅), log (1 +𝜎(𝐿) 𝐿̅ )) With:

𝑖: index of the household or worker or hospital bed

𝑙𝑖: length of pipe to the outlet of the main source area for the household, worker or hospital bed 𝑖 (m) 𝑟𝑎𝑛𝑑𝑜𝑚𝑙𝑜𝑔𝑛𝑜𝑟𝑚(𝜇, 𝜎): return a random value with a lognormal distribution of parameters 𝜇 and 𝜎 log (𝑥): return the natural logarithm of the strictly positive real number 𝑥

𝐿̅: average length of sewers between the discharge points and the outlet of the area (m)

𝜎(𝐿): standard deviation of the length of sewers between the discharge points and the outlet of the area (m)

 Last pipe: the length of the last pipe, the one directly linked to the outlet of the main source area is set equal to the smallest length randomly picked on the previous step (no discharges point is directly connected to the outlet).

𝐿₂₀= min (𝑙) With:

𝐿20: length of the 20^th pipe of the main source area (m) min (𝑑): return the smallest value of a list of values 𝑑

𝑙: list of all the lengths picked for households, worker and hospital beds on the previous step (m)

 Other pipes: the length of each of the 19 remaining pipes is equal to the difference between the largest and the smallest length randomly picked on the previous step divided by 20.

𝐿𝑛=max(𝑙) − min (𝑙) 20 With:

𝑛: index of the pipe (1 ≤ 𝑛 ≤ 19)

𝐿_𝑛: length of the n^th pipe of the main source area (m) max (𝑑): return the largest value of a list of values 𝑑 min (𝑑): return the smallest value of a list of values 𝑑

𝑙: list of all the lengths picked for households, worker and hospital beds on the previous step (m)

 Sources: the 20 sources are linked to the 20 inputs of the 20 pipes. Their number of households, workers and hospital beds is determined by the length of pipe to the outlet of the main source area.

𝒊𝒇 𝑙𝑖≠ max(𝑙) 𝒕𝒉𝒆𝒏 𝑗 = 𝑓𝑙𝑜𝑜𝑟 (𝑙𝑖− 𝐿20

𝐿1→19 ) + 1 𝒆𝒍𝒔𝒆 𝑗 = 20

With:

𝑖: index of the household or worker or hospital bed

𝑙𝑖: length of pipe to the outlet of the main source area for the household, worker or hospital bed 𝑖 (m) max (𝑑): return the largest value of a list of values 𝑑

𝑙: list of all the lengths picked for households, worker and hospital beds on the previous step (m) 𝑗: index of the source that contain the household, worker or hospital bed 𝑖

𝑓𝑙𝑜𝑜𝑟(𝑥): return the greatest integer smaller or equal to the real number 𝑥 𝐿20: length of the 20^th pipe of the main source area (m)

𝐿1→19: length of the pipes 1 to 19 of the main source area (m)

5.4.1.2 HYPOTHESES AND CHOICES DISCUSSION

The current structure of the main source area was chosen mainly to allow different wastewater travel times.

This way, wastewater discharged at the same time in two different households reaches the WWTP at different times.

To keep the model as simple as possible, the amount of pipes and sources is set at 20. It provides enough complexity for the present case, but keeps computation needs low. Another approach was considered but not kept. It consisted of determining the number of pipes and sources according to the average length of sewers between discharge points and the outlet of the area 𝐿̅, the standard deviation associated to this distribution 𝜎(𝐿) and the targeted spatial discretization length ∆𝑥𝑡𝑎𝑟𝑔𝑒𝑡 of the pipe element.

As the model does not take into account transformations of the pharmaceuticals loads in the sewer system and as the Muskingum model used for the pipe is linear, it is not worth proposing a more complex structure for the main source area, such as a tree structure in comparison to a series of consecutive pipes.

Estimating the average length of sewers between the discharge points and the outlet of the area 𝐿̅ and the standard deviation associated to this distribution 𝜎(𝐿) is done by studying the map of the sewer network of the catchment.

The lognormal distribution of the lengths of sewers between the discharge points and the outlet of the area is chosen because it guaranties positive values.

5.4.2 URBAN SITE