I am following a policy of giving a surplus of data from which must be picked the useful items. I am assuming the intelligent reader will question the data and be trying to qualify their answers so that they are giving correct accurate statements based on the data supplied. The skill to demonstrate here is that the answers are made to make sense, despite the fog which often follows from presentation of information. DJS 20211203

1. Nationally, we use 150 litres per day per person and the total amounts to about 55% of all water use. Your household water bill for six months and a family of four comes to 70m³.

(i) Express the family consumption in litres per day and compare this with the national average.

(ii) Convert 150 litres per day per person to a total consumption for a population of 56 million (just England; the situation is different in the other nations). Express this in a sensible unit.

(iii) Use that last result to estimate to 2sf, the total water demand for England.

2. The biggest domestic waste of water is flushing the toilet. Apparently we do this five times a day (really? so little?). A low-flow toilet flushes 1.6 gallons, while an older and conventional toilet will use 5 to 7 gallons per flush.

(i) Convert those gallon measures to litres to 3sf

(ii) Suppose you convert your 5 gallon flush to a low-flush unit. How much water is saved per day per person? How would that reduce your daily use if it was previously 150 litres per day? Why is this a false assumption?

(ii) Suppose your old toilet has a 4-gallon flush and that you convert it to a 1.6 litre flush. Suppose that you also subscribe to the 'yellow - mellow' line of thinking, which is to flush solids always and urine less frequently. This might well reduce you to say three flushes per day. Calculate your new use per day if you had been on 150 litres per day before these changes.

(iv) Q1 said the household use is 55% of the whole. If everyone could reduce their water demand to 60% of what it was, what would be the percentage change nationally?

3. From a US magazine, Public Services Magazine [1] Each person’s urine emits, on average, 4,000 grams of nitrogen and 365 grams of phosphorus every year. When urine is flushed it goes to a sewage treatment plant, which does little to remove nutrients. [...] Of all the nitrogen flowing into wastewater treatment plants, 75 percent comes from urine; of all the phosphorus, that figure is 55 percent. (Wastewater plants in large urban areas typically have systems for removing the nutrients, but those systems represent a huge capital investment; plants in smaller communities often can’t afford to install the systems.)

From Gov.UK [2] • Every day in the UK about 347,000 kilometres of sewers collect over 11 billion litres of waste water.

• Sewage treatment is essentially about removing polluting organic material from waste water. This can be removal of solid waste and other substances, for example ammonia, which can be harmful to fish. The result is much cleaner water being returned to the environment, and sludge that contains the organic matter and dead bacteria from the treatment process.

(i) UK population 67.22 million (2020) and 58.68 million (1999). Convert the 'about 20kg per person' to the right to a 2sf number for 1990.

(ii) Predict the 2020 sludge total based on the 1990 figures.

4. By 2020 the situation in the UK has moved significantly and this site [3] says on its first page: In the UK, approximately 12 million tonnes of wastewater is processed each day = 140 tonnes per second (on average). Further to this a considerable amount of rainfall enters the wastewater network. Around 53 million tonnes per annum of untreated sludge (i.e. sewage sludge) is collected from about 8,500 wastewater treatment works servicing small villages to large cities. Later, I found Currently around 87% of sludge is recycled to agricultural land as biosolids, 4% is incinerated, 3% goes for industrial use (e.g. as a fuel for cement production) and 6% is used for land reclamation or restoration. Further on, I found Around 3.5 million tonnes per annum (or 170,000 truckloads) of biosolids are recycled to agricultural land in the UK. This is applied to about 150,000 hectares per annum, or 1.3% of the UK’s agricultural land. The financial value to UK agriculture of nutrients in biosolids is around £25 million per annum – mainly as phosphate and nitrogen as well as sulphur, potash and magnesium. There is a strong demand from farmers as it is worth about £170 per hectare in nutrients alone.

(i) Draw the new pie chart for 2020, to match the one above right.

(ii) If publication [2] is dated 2002 and publication [3] is 2020 then, working only to 2sf does the move from 11 to 12 billion litres of waste water makes sense, if the 2002 UK population was 59.37 million? The good student will find more than one calculation to make here and might add comment.

(ii) It implies above that 3.5 million tonnes of biosolids is 87% of the whole. If so, what is the whole 100%?

(iii) Around 53 million tonnes per annum of untreated sludge is collected. Using your previous answer, if what has been extracted is the solids element and if we assume that the remainder is water (though we ought to ask where it goes), how much water is that? What percentage of the daily wastewater is this?

5. Wastewater is not necessarily treated at all. From wikipedia [4]: Few reliable figures exist on the share of the wastewater collected in sewers that is being treated in the world. A global estimate by UNDP and UN-Habitat in 2010 was that 90% of all wastewater generated is released into the environment untreated.[54] A more recent study in 2021 estimated that globally, about 52% of sewage is treated.[55] However, sewage treatment rates are highly unequal for different countries around the world. For example, while high-income countries treat approximately 74% of their sewage, developing countries treat an average of just 4.2%.[55]

Even in the UK, there are (too) many incidents which result in untreated sewage being released into watercourses or the sea. source [5] cites a count of 400,000 events in England in 2020, which is only true if the water companies are corrrectly reporting. That doesn't sound right, does it?

Q5. Paris has a plan to make the River Seine clean enough to swim in by the 2024 Olympics. To do that, the city is building a large subterranean water tank with enough capacity to hold 12 million U.S. gallons. The tank will store stormwater and prevent sewage from spilling into the river when drains overflow during heavy rain. Tubes connecting it to the sewer allow the tank to pump potentially contaminated rainwater back into the system after the rain passes. It’s not guaranteed, however, to make the river completely free of sewage. from https://www.bloomberg.com/news/articles/2021-12-04/how-paris-plans-to-make-the-seine-swimmable-by-2024?cmpid=BBD120721_CITYLAB&utm_medium=email&utm_source=newsletter&utm_term=211207&utm_campaign=citylabdaily.

(i) Express the tank capacity in cubic metres. There are 3.78541 litres to a US gallon and (you already know) 4.54609 litres to an imperial (UK) gallon.

(ii) Now assume that the tank is a cuboid with sides in the proportions 3:2:1 and give these in metres to 2sf.

(iii) One of the objectives is to be able to use this tank or these tanks for swimming events in the 2024 Olympics. An Olympic pool is 50x25x2m, so how many such pools might the original volume represent?

(iv) Later in the same article: With a capacity of 46,000 cubic meters — sufficient to hold over 12 million U.S. gallons — it could comfortably swallow enough water to fill an Olympic-sized swimming pool about 30 times over. The tank is an “extraordinary project of exceptional size ....Check this content against your answers so far, and express N to 2sf, assuming that the French do *not* work in US units. if your answer disagrees, make comment.

(v) My answers disagree with the count of 30 Olympic swimming pools. So I worked backwards to find the assumed voilume of a single such pool and then decided that the pool must be 25m wide and 25 or 50 m in length, giving a depth between 2 and 3 metres. If we assume that we can have exactly 30 identical pools (which wouldn't happen), give the depth of these pools to the nearest decimetre.

(vi) Thinking like a town planner might, if you built two pools 50x25x3, the remainder, which I think is 38500 m³, might make a rowing course. For Olympics, the width is 108m and the length 2150m. Work out the depth - it is too shallow.

Depth is critical as well; the depth of a body of water can effect a boat’s speed. For this reason, a race course must be at least three meters deep throughout all racing lanes at the shallowest point, if the depth over the course is unequal. From https://www.row2k.com/features/2796/Anatomy-of-a-Race-Course/

Assume this 3m depth and assume that lanes must be 13.5m wide plus at least half a lane each side to avoid bank effects. Juggle the number of lanes and the total length to make a suggestion about a rowing facility.

Consider instead building a boating lake, for the body of water (after you have subtracted two Olympic 50m swimming pools) of mean depth 1.5m. What area could you have? I suggest a hectare, 100x100m is a sensible unit to use.

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[1] https://psmag.com/news/liquid-gold-why-flushing-a-toilet-is-a-colossal-waste,

[3] https://assuredbiosolids.co.uk/about-biosolids/

[4] https://en.wikipedia.org/wiki/Sewage_treatment

[5] https://www.theguardian.com/commentisfree/2021/nov/01/sewage-england-rivers-seas-outcry-tories-effluent-dumping-companies You might wonder whether this report is balanced.

[6] https://www.gov.uk/government/news/water-companies-could-face-legal-action-after-investigation-launched-into-sewage-treatment-works I read this and wondered whether OfWat and Defra have enough money, enough power or are simply lazy. The government is supposed to protect its population and seems to be failing in this.

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**A1 ** 70m³ for four for six months => 35m³ each, 35000 litres, or a bit under 100 litres a day, 96 litres/day to 2sf. This is ⅔ or 64% of the national mean and suggests that you don't have many baths and have efficient toilets (the biggest waste of water is flushing).

Across 56 million, 150 litres / day each is 56x10⁶ x 365 x 150 /1000 = 3.066 x 10⁹ m³ / year = 3.1 m³/yr to 2sf, though I'm happy with calling that just three gigalitres, 3 Gl/year.

That last answer is 55% of the total, so 3.066 / 0.55 = 5.57 Gl/yr to 2sf

**A2 ** Litres per gallon 4.54(609), very similar to grams per pound 453.592 (so 454 to 3sf, like it says on an older jamjar). 1.6gals= 7.28, and 5-7 gals = 22.7 - 31.8 litres all to 3sf.

(ii) 5 flushes at (22.735-7.275) litres per event is 77.3 litres per person per day, about half the 150 value. The 150 value is the national average, which inlcudes those with 'old' toilets and those with 'new', so the 150 figure does not apply to an individual case.

(iii) For any one day, the saving per flush is (4-1.6)x4.54609 = 10.9(106) litres per flush. But this calculation would be easier to look at what you used before, 5x4x4.54 = 90.09 litres/day and what you use 'now', 3x1.6x4.54=21.82 litres/day, a saving of 68.3 (3sf in the final result) litres per day 68.3/150 is 45.5% saving; you've reduced your previous value to 55% of what it was.

(iv) 60% of 55% is 33%; the 150 litres a day would be more like 100 litres per day (to 1sf, because we're using very approximate figures)

**A3 **(i)** **UK total sludge 1990 1130066 tonnes so divide by 58.68 million = 19.258 kg / person of sludge solids

(ii) 1130066 x 67.22/58.68 = 1294530 tonnes or last answer x population. 3sf would be appropriate, 1.30 Mt.

**A4 ** (ii) If the 11 billion was right and nothing changed, we'd expect growth to be in line with population growth, so 11x67.22/59.37 = 12.45, so perhaps, as that is 12 billion to 2sf. If the population growth was not supported by other growth, then the extra would apply to only the 55% used in Q2(iv) taking the 55% of 11billion from 6.05 to 6.85 an additional 0.8 billion litres, a new total of 11.8 billion, which is also 12 billion to 2sf.

(iii) 3.5/0.87 = 4.023 million tonnes of biosolids, which leaves, from the 53 million tonnes per annum, some 49 million tonnes **per year** of water. Compared to 12 million tonnes **per day**, this is small, 49/(12x365) = 49/4380 = 1.1%

**A5 **** ** 12x10⁶x 3.78541 =4.5425 x 10⁷ litres = 45425 m³. 3x2x1 is 6 units³, so one unit³ is 4.5x10⁷/6 = 7570 m³ and one linear unit is ₃√7570.82x10⁶ = 19.6 m to 3sf, so the tank would be 59x39x20m to 2sf. It is tempting to move to 3sf but the initial information, 12 million US gallons was already rounded to 2sf. The move from 45 million cubic units to a linear unit will sort out the thinkers (or those who ask others, take or steal answers); the check in (iv) ought to sort all of this out

(iii) 45425 / (50x25x2) = 18.1 pools. Nothing like as big as you might expect.

(iv) 46000 m³ / (50x25x2) = 18.4 pools, not the 30 in the text. I think the author has used a 25m pool.

(v) 46000 m³ / 30 = 1533.3m³ per pool. 153x10⁶ / (25x25) = 2.453. Dividing by 25x50 fails to give an adequate depth, so the author *has* used 25m square pools and they would be 2.45m deep, which is 2.5m to the nearest decimetre.

(vi) 46000 - 2x(50x25x3) = 38500 m³ which is an olympic rowing course 108m wide, 2150m long and 16.5 cm deep.

38500 m³ / (58x1150) = 57.7 cm deep not good enough.

38500 m³ / (3x4x13.5) = 238 m to the nearest whole metre for three race lanes (so four lane widths). Hence one good lane gives a length of 475m, two lanes 317m, three lanes 238m, four lanes 190m. An eight is 18m long, so at least 25m width is required to turn around reliably. In practice a boat takes about 8m width and wants more than 10m, so the 'one good lane' gives an opportunity for practice between two boats, or for staggered starts. It seems to me that a boating lake would be better.

38500 / 1.5 = 25667 m², 2.57 ha. That's a decent size, 6 acres. The Serpentine (Hyde Park, London) is 16ha and 5m deep; my local lake is 22ha, and in practice we use about a third of that for boating. The size of the boating lake in Regent's Park (London) is a challenge to discover.