Introduction

‘Building is cooperating with the Earth.’

Marguerite Yourcenar.

Motivation

Activating senses

Sun, wind, water, earth and life touch our living senses immediately, always, everywhere and without any intervention of reason. They simply are there in their unmatched variety, moving us, our moods, memories, imaginations, intentions and plans.

Mathematics next to senses

However, the designer transforming sun into light, air into space and water into life, touches pure mathematics next to senses. Mathematicians left alone destroy mathematics releasing it from senses, losing their unmatched beauty and relief, losing their sense for design. To restore that intimate relation, the most freeing part of our European cultural heritage my great examples are Feynman’s lectures on physics, D’Arcy Thomson’s ‘On Growth and Form’ and Minnaert’s ‘Natuurkunde van het vrije veld’ (‘Outdoor physics’). Minnaert elaborated the missing step from feeling to estimating.

I am sitting in the sun. How much energy do I receive, how much I send back into universe?

I am walking in wind. How much pressure do I receive and how much power my muscles have to overcome? It is the same pressure giving form to the sand I walk on or giving form and movement to the birds above me! I am swimming in the oldest landscape of all ages, the sea. How can I survive?

Re-constructing behaviours

No longer can I escape from reasoning, from looking for a formula, a behaviour that works. But this reasoning is next to senses and once I found a formula I can leave the reasoning behind going back into senses and sense. The formula takes its own path in my Excel sheet as a living thing. It ‘behaves’. Look! Does it take the same path as the sun, predicting my shadow? Put a pencil and a ruler in the sun. Measure, compare, lose or win your competition with the real sun as Copernicus did.

Mathematics have no longer much to do with boring calculations. Nowadays computers do the work, we do the learning. They sharpen our reasoning and senses. We see larger contexts and smaller details than ever before discovering scale. Discovering telescopic and microscopic scale we find the multiple universe we live in, freeing us from boredom forever, producing images no human can invent. We do not believe our eyes and ears, we discover them. It challenges our imagination in strange worlds no holiday can equal. Life math is a survival journey with excitement and suspense.

Science as design

But do we understand the sun? No, according to Kant (1976) we design a sun behaving like the sun we feel and see from our position and scale of time and space we live in. We never know for sure whether it will behave tomorrow in the same way as our sheet does now. But we have made something that works here and now.

‘Yes! It works.’ That is a designer’s joy.

How to use this book

This book is not a reader. It contains original texts by the authors from our school and one civil engineer to understand how specialists think, supporting our profession as urban designers.

Systematic encyclopaedia

It is ordered in an systematic encyclopaedic style. It is accessible by its table of contents (elaborated in more detail at the beginning of each chapter), and by a key word list containing some 6000 key words at the end of the book, including other authors we refer to. Full references to other authors are given in a reference list, also to be found via the key word list. Direct references into publications and websites to look up immediately as a result of reading are given as foot notes (^a) indicated by letters in the text and listed at the bottom of the page. Questions for exercise are indicated as numbered end notes (¹) by numbers in the text listed at the end of the book (see page 711). However, these questions don not yet cover the whole content of the book.

A conditional sequence: physics first

The chapter titles start as the title of the book indicates: Sun, Wind, Water, Earth, Life, Living and Legends for design. These subjects are ordered this way, because it is the conditional sequence we experience them directly outdoor and gradually can understand them best.

The sequence of the chapters follows the range of abiotic, biotic and conceptual phenomena with apparently increasing complexity. The simulation of these phenomena is firstly approached by supposing a causal sequence (effect follows cause: c Þ e) usual in physics. Even life, living and legends for design obey the boundary conditions of physics. So, we firstly try to simulate these phenomena by purely causal simulation. After all, we can not imagine living systems (B) without an abiotic environment (A), as we can not imagine conceptual systems (C) without a living environment (B). Let us call that ‘ABC-model’ (see Fig. 1).

Fig. 1 Simulating reality by different approaches according to the ‘ABC-model’

Biotic feed-backs included

However, biotic phenomena (including humans) and some human artifacts seem to take the effects of earlier behaviour into account, adapting next behaviour (‘empirical cycle’[a]). A one way causal simulation of such a phenomenon should contain its history from second to second including the evolutionary history of its ancestors from the very beginning. It should not exclude details that might have been crucial. That long description to predict behaviour would require too many gradually changing cycles finally solving chicken-and-egg questions typical for biology. But you can understand the pattern and process of an egg in a shorter way if you suppose what will come out (for convenience, without additional teleological assumptions). In that approach the effect also ‘precedes’ the cause (see Fig. 1). The main ‘experience’ of a species is stored in its genes and in other chemical substancies steering action, completed by increasing ‘experiences’ of a specimen born in a specific context. We still do not understand much of all feed-back loops in any organism. But, we can simplify the description of its behaviour by drawing a black box and looking what is going in (input) and what is coming out (output) in a determined period. That is called ‘systems approach’.[b] By a systems approach you design a model with the same input and output as observed to predict behaviour. In the algorithm of such a model many ‘if … then …’ statements will appear connecting the possible branches of causal behaviour in different circumstances. If the behaviour of the model is much the same as observed we are inclined to suppose the model represents reality, which is not the case.

Conceptual projection added

For our purpose, the most satisfying description of the difference between humans compared to other animals is their ability to represent a larger range of activities beforehand^{^[c]}. It is the very basis of making artifacts serving further purposes (if I will do this first, then I can do that later) and the very basis of task division (if you do this, I can do that). So, humans are supposed to simulate internally a longer range of ‘causes’ (actions) and ‘effects’ before they come into action (‘look before you leap’) than routinous animals. As soon as action and utilising its effect are connected by an intermediate (interfunctional) action, such as making an instrument, the whole range can be noted as an algorithm. Designing is such an intermediate activity in a range of activities ‘planned’ beforehand. That kind of ‘conceptual’ behaviour completes many unconscious components of behaviour stored in an organism as biotic routines. That is why in this paper we leave out the supposed ‘cognitive’ part of human behaviour as long as we can simulate (understand) it sufficiently by a black box. But, there comes a time these biotic simulations do not fit reality any more. Then, we have to add new suppositions about the ‘plan’ humans have in mind before they act. Many ‘plans’ (earning a living, finding a partner, getting children) look the same. But the question is, if these are really ‘plans’ or simply the ‘conceptualisation’ of predictable biological inclinations afterwards to justify them socially. What we can simulate by less suppositions we will do (‘razor of Ockham’). Interpreting humans as mere animals clarifies an increasing amount of behaviour^{^[d]}. But, there are still unpredictable behaviours apparently following a ‘plan’. The question is, if we ever could predict that kind of behaviour. In that case we have to give up our supposition of free will (supposed in democracy) after all. In this paper we will not do so, because it is the core of design to find unexpected possibilities (necessary in an ecological crisis). If these possibilities could be expected it would be predictions, not designs. In Fig. 1 is expressed that conceptual projection can not be used to simulate abiotic and biotic phenomena.

Levels of scale

A principle of ordering we aimed for in any separate chapter is the level of scale. So, you can choose the sub-chapter concerning the level of scale you focus on in your study. We have tried to start every chapter on the highest level of scale. There are arguments to start with the lowest level, most directly related to our senses, but we chose the other way round, because lower levels of scale are better understood knowing their context. This way, you may get a feeling for contextual factors determining a particular environment and its mathematical modelling with parameters stemming from that context. In design practice you can reason the reverse way or both ways. But, to know how to design ‘throught the scales’ you have to be aware of scales, the frame and grain of legend units, the scale specific inferences and the danger of using conclusions from an ather scale.

Design related use

So, you do not have to read everything before you can use it making inventories for design (like a local atlas of thematic maps), while designing or reflecting on your designs. Reflecting on your design work is what we ask in the assignments of the course accompanying this book: how did you apply Sun in your earlier design work, what could you have done, how do you apply Sun in your actual design work and what could you do with it in the future? The same is asked for Wind, Water and so on. A growing number of computer programs for experiments and calculations per section is downloadable from http://team.bk.tudelft.nl publications 2008.

Non-disciplinary combinations like Sun, energy and plants

The chapter ‘Sun’ contains sub-chapters on energy, entropy, temperature, light, the history of our territory dependent on solar fluctuations, man-made plantation (written by Prof.dr.ir.C.M. Steenbergen and Drs. M.J. Moens), shadow and vision as well. These subjects are often related in design or better comprehensible in the offered context. Perhaps in your design you can connect things in another way than the usual scientific and specialist’s distinctions of disciplines suggest. For the same reason we did not aim for a distinction between natural and man-made phenomena in the sequence of chapters. It is rather a conditional sequence of growing complexity in cycles of inductive observing, deductive understanding and practical application. So, any chapter is better understood knowing something about the subject of the preceding chapter.

Wind, sound and noise

The chapter ‘Wind’ contains sound and noise as well, because both are movements of air. These flows are more complex than those of mere energy and light.

Water, networks and crossings

The chapter ‘Water’ is primarily based on the lecture notes Prof.dr.ir. C. van den Akker offered us for use when he retired from the Faculty of Civil engineering. Ir.D. de Bruin, drs. M.J. Moens and ir. M.W.M. van den Toorn added many subjects relevant for design. However, it contains traffic as well, based on the book of ir. B. Bach[e], because the combination of these different flows on the Earth’s surface and their resulting networks are an important part of urban and regional design. So, we did not primarily make a distinction between natural and man-made networks. The comparison of their characteristics is interesting, instructive, and may be a source of new design ideas.

Earth and site preparation

The chapter ‘Earth’, primarily written by Drs. M.J. Moens and elaborated by ir. M.W.M. van den Toorn , is better understood if you know something about wind and water. The division of its sub-chapters starts strictly with levels of scale, but then sub-chapters follow about soil pollution and preparing a site for development.

Life, ecology and nature

The ecological chapter ‘Life’ supposes sun, wind, water and earth. These conditions are discussed earlier in the book, so the chapter can focus on the distribution and abundance of life itself. Biology is physics with numerous feed-back mechanisms, not te be modelled so easily in a mathematical sense. However, it introduces approaches of system-dynamics, demography, useful in human environments as well. Life contains human life. So, this chapter tries to consider man as a species between other species (syn-ecology), while the next chapter ‘Human Living’ concentrates on human species only (aut-ecology). However, there are sub-chapters on valuing and mananging nature by man in your plan, and on the role of an urban ecologist.

The subject of this chapter is not very familiar to designers. So, you can think it is not very relevant. But in my opinion ecology, the science of distribution and abundance of species, is the very core of urban and regional design. Design changes predictable distributions. Local vegetaton and wild life clarifies much about what designers feel as a mysterious ‘genius loci’. Ecology is a neglected source of local identity. Evolution of life has something in common with design thinking: its course of trial and error into diversity and order. The evolutionary taxonomy of plants and animals, types of life, their distribution and adapation into different environments, accommodating and modifying them, give examples of the same problems any design task stands for. Your typological repertoire of design solutions selects environments and the reverse different environments select different types of design.

Living, human density and environment

The chapter ‘Living’ shows the history of human occupation in general and in The Netherlands in particular. That piece of land in between France, Belgium, Germany and Great Britain contains both lower and higher grounds, combining many characteristics of its neighbours. Its delta gives an impression of a development known from many densely populated lowlands in the world, the spatial composition of ecological, technical, economic, cultural and administrative components. A sub-chapter is devoted to urban density on different levels of scale. The sub-chapter ‘Environment’ discusses some consequences of living in high densities like environmental problems, environmental norms, gains and losses.

Legends for design

The chapter ‘Legends for design’ stimulates to consider these phenomena of urban physics as innovative components, legend units, spatial types given form in a design composition. It raises philosophical questions on unusual types, their suppositions, combinations and consequences.

Simulators accompanying the book

Every chapter is accompanied by Excel sheets[f] programmed with Visual Basic Language to exercise mathematical relations described in this book. These simulators show the hidden suppositions of specialists in yellow sliders by which you can change the model and see the results without own calculations. By doing so, you can ask the right questions if specialists criticize your design with mathematical certainty. They often show counter-intuitive results. If you do not believe them, then Excel allows you to show the formulas en their relations to criticize their inference. That will make you less vulnerable in the company of many specialists you will meet in practice.

1 Sun, energy and plants

Contents...................................................................................................................................... 11

1.1 Energy............................................................................................................................... 12

1.1.1.... Physical measures........................................................................................................... 12

1.1.2.... Entropy............................................................................................................................ 14

1.1.3.... Energetic efficiency........................................................................................................... 19

1.1.4.... Global energy................................................................................................................... 22

1.1.5.... National energy................................................................................................................ 28

1.1.6.... Local energy storage......................................................................................................... 33

1.2 Sun, light and shadow................................................................................................... 35

1.2.1.... Looking from the universe (a, b and latitude l)..................................................................... 35

1.2.2.... Looking from the Sun (declination d)................................................................................... 37

1.2.3.... Looking back from Earth (azimuth and sunheight)................................................................ 38

1.2.4.... Appointments about time on Earth..................................................................................... 41

1.2.5.... Calculating sunlight periods............................................................................................... 43

1.2.6.... Shadow........................................................................................................................... 45

1.3 Temperature, geography and and history............................................................... 50

1.3.1.... Spatial variation................................................................................................................ 50

1.3.2.... Long term temporal variation.............................................................................................. 55

1.3.3.... Seasons and common plants............................................................................................. 61

1.4 Planting by man............................................................................................................... 67

1.4.1.... Introduction...................................................................................................................... 67

1.4.2.... Planting and Habitat.......................................................................................................... 83

1.4.3.... Tree planting and the urban space...................................................................................... 90

1.4.4.... Hedges.......................................................................................................................... 101

1.1 Energy

1.1.1 Physical measures

The internationally accepted SI system of units defines energy and power according to Newton by distance, time and mass as follows. As long as a force ‘f’ causes an acceleration ‘a’, a distance ‘d’ is covered in a time interval ‘t’. Multiplying f by d produces the yielded energy f · d, expressed in joules.
Energy per time interval t produces the performed power f · d / t expressed in watts (see Fig. 2).[1]

Velocity ‘v’ and acceleration ‘a’ suppose distance d and time interval t:
d (distance)	d		d
	¾	= v (velocity)	¾	= a (acceleration)
t (time)	t		t²
Linear momentum ‘i’ and force ‘f’ suppose mass m, velocity v and acceleration a:
	d		d
m (mass)	¾	m = i (momentum)[2]	¾	m = ma = f (force)[3]
	t		t²
	times distance = energy ‘e’		divided by time = power ‘p’
	d²		d²
	¾	m = e (energy)[4]	¾	m = e/t = p (power)[5]
	t²		t³
Energy is expressed in joules (J), power (energy per second) in watts (W)[6]
	J=kg*m²/sec²		W = J/sec
Old measures should be replaced as follows:
k= kilo(*10³)	kWh = 3.6 MJ		kWh/year = 0.1142W
M= mega(*10⁶)	kcal = 4.186 kJ		kcal/day = 0.0485W
G= giga(*10⁹)	pk.h = 2.648 MJ		pk = hp = 735.5 W
T= tera(*10¹²)	ton TNT = 4.2 GJ		PJ/year = 31.7 MW
P= peta(*10¹⁵)[7]	MTOE = 41.87 PJ		J/sec = 1 W
E= exa(*10¹⁸)	kgfm = 9.81 J
	BTU = 1.055 kJ		W (watt) could be read as watt*year/year.
	watt*sec = 1 J
The equivalent of 1 m³ natural gas (aeq)[8], roughly 1 litre petrol[9], occasionally counts 1 watt*year:
Occasionally:	m³ aeq = 31.6 MJ and		aeq/year = 1 W, or
	Wa = watt*year = 31.6 MJ		1 W = 1 watt*year/year
	1 MJ = 0.0316888 Wa 1 GJ = 31.7 Wa 1 TJ = 31.7 kWa 1 PJ = 31.7 MWa		‘a’ from latin ‘annum’ (year) Wa is watt during a year ‘k’ (kilo) means 1 000x ‘M’ (mega) means 1 000 000x

Fig. 2 Dimensions of energy

A happy coincidence

A year counts 365.24 · 24 · 60 · 60 = 31 556 926 seconds or 31.6 Msec, since M means ‘·million’.

So, the power of 1 watt during a year: 1 watt·year = 31.6 MW·sec = 31.6 MJ =

1 Wa (‘a’ derived from latin ‘annum’, year), which is energy. [10]

Occasionally the equivalent of 1 m³ natural gas (‘aeq’) or 1 litre petrol or 1 kg coal energy counts for approximately 31.6 MJ = 1 Wa energy as well.[11]

So: m³ natural gas (‘aeq’) ≈ watt·year = Wa (energy)
and m³ natural gas per year ≈ watt = W (power).

So, read ‘Wa’ and think             ‘1 m³ natural gas’, ‘1 litre petrol’ or ‘1 kg coal’ (energy);
read ‘W’ and think                     ‘1 m³ natural gas per year’ (power);
read ‘kW’ and think                   ‘1000 m³ natural gas per year’ (power);
read ‘kWh’ and think                  ‘1000 m³ natural gas per year during an hour’ (again energy).

Easy calculating kilowatthours (kWh) and joules (J) by heart

Since there are 365.24 · 24 = 8 766 hours in a year: 1 Wa (watt·year) = 8 766 watt·hour (Wh) or

8.766 kilowatt·hour (kWh), because ‘k’ means ‘·thousand’.

Since there are 31 556 926 seconds in a year: 1 Wa = 1watt·year = 31 556 926 Ws (J) or

31 557 kJ, 31.557 MJ or 0.031557 GJ, because k = ·1 000, M = ·1 000 000 and G = ·1 000 000 000.[12]

This Wa measure is not only immediately interpretable as energy content of roughly 1 m³ natural gas, 1 litre petrol or 1 kg coal, but via the average amount of hours per year (8 766) it is also easily transferable by heart into electrical measures as kWh and then via the number of seconds per hour (3 600) into the standard energy measure W·s=J (joule).

Moreover, in building design and management the year average is important and per year we may write this unit simply as W (watt). So, in this chapter for power we will use the usual standard W, known from lamps and other electric devices while for energy we will use Wa. If we know the average use of power, energy costs depend on the duration of use. So, we do not pay power (in watts, joules per second), but we pay energy (in joules, kilowatthours or wattyears): power x time.

Watts in everyday life

A quiet person uses approximately 100 W, that is during a year the equivalent of 100 m³ natural gas. That power of 100 W is the same as the power of a candle or pilot light or the amount of solar energy/m² at our latitude[g]. That is a lucky coincidence as well. The power of solar light varies from 0 (at night) to 1000 W (at full sunlight in summer) around an average of approximately 100 W.

Burning a lamp of 100 W during a year takes 100 Wa as well, but electric light is more expensive than a candle.[13] Crude oil is measured in barrels of 159 litres. So, if one barrel costs € 80, a litre costs € 0.50. However, a litre petrol (1 Wa) from the petrol station after refining and taxes costs more than € 1. Natural gas requires less expensive refinary.

In the Netherlands 2008, 1 m³ natural gas (1Wa) costs approximately € 0,70[h]. However, an electric Wa costs approximately € 1.80. That is more than 2 times as much. Why?

Conversion of fuel into a useful kind of energy

Electric energy is usually expressed in ‘kWh_e’ (‘e’ = electrical), heat energy in ‘kWh_th’ (‘th’ = thermal).

A kWh_eelectricity is more expensive than a kWh_th of heat by burning gas, petrol or coal, because a power station can convert only approximately 38% from the energy content of fossile fuels into electricity (efficiency h=0.38). The rest is necessarily produced as heat, mainly dumped in the environment ‘cooling’ the power station like any human at work also looses heat.[14] That heat content could be used for space heating, but the transport and distribution of heat is often too expensive.

However, enterprises demanding both heat (Q) and work (W) at the same spot, could gain a profit by generating both locally (cogeneration, in Dutch ‘warmte-kracht-koppeling’ WKK).

Necessary heat loss

The necessary heat loss is described by two main laws of thermodynamics: no energy gets lost by conversion (first law of thermodynamics), but it always degrades (second law of thermodynamics).

By any conversion only a part of the original energy can be utilised by acculumation and direction at one spot of application. The rest is dispersed as heat content Q (many particles moving in many directions), to concentrate a minor useful part W (work) on the spot where the work has to be done. The efficiency h of the conversion is W/(W+Q). In the case of electricity production it is 38kWh_e/100kWh or 38%. Once the work W is done, even the energy of that work is transformed into heat. However, according to the first law of thermodynamics both energy contents are not lost, they are degraded, dispersed, less useful. However it could still be useful for other purposes.

For example, the temperature of burning gas is ample 2000^oC, much too warm for space heating. If you would use the heat from burning fuels firstly for cooking, then for heating rooms demanding a high temperature and at last for heating rooms demaning a low temperature, the same heat content is used three times at the same cost in a ‘cascade’. To organise that is a challenge of design.

Exergy

Theoretically any difference in temperature can be used to extract some work, but the efficiency of a small temperature difference DT is lower than that of a large temperature difference (see Fig. 3).

Fig. 3 The %maximum amount of work (W) retrievable from a temperature difference DT

The amount of work you can get out of heat (W/Q) per temperature difference available is called exergy. Apparently, chemical energy like fossile fuels do have a higher ‘quality’ than work; work has a higher quality than heat; high temperature heat has a higher quality than low temperature heat.

So, using high quality energy where low quality would be enough, leaves unused the opportunity to use the same energy several times in a cascade of uses.

The ‘quality’ of energy can be expressed in a single quantity. That quantity is called ‘entropy’.

1.1.2 Entropy

The ‘quality’ of energy

The ‘quality’ of heat (Q) and work (W) is apparently different, though both are ‘energy’.

In the same way high temperature (T) energy has a higher ‘quality’ than the same energy at low T.

Converting fossile fuels into heat, the ‘state’ of energy changes. But how to describe that ‘state’ and its ‘quality’? To introduce that ‘state’ in energy calculations the term ‘entropy’ S is invented by Clausius ca. 1855. In a preliminary approach one could think S = Q/T, but it concerns change, forcing us into differentials. It is often translated as ‘disorder’, but it is a special kind of disorder as Boltzmann showed in 1877. What we often perceive as ‘order’, a regular dispersion in space, is ‘disorder’ in thermodynamics. Let us try to understand that kind of thermodynamic disorder to avoid confusion of both kinds of ‘order’.

‘Disorder’ in thermodynamics

In Fig. 4 all possible distributions of n = {1,2,3,4} particles in two rooms are represented.

If you mark every individual particle by A, B, C, D, you can count the possible combinations producing the same distribution k over the rooms numbered as k = {0,1 …n).

Fig. 4 k Distributions of n particles in two rooms

Fig. 5 The decreasing probability of concentration with a growing number of particles

The numbers n and k determine the probability P(n,k) that this combination will occur[i].

Minimum and maximum values of k represent the extreme concentrations in one room or the other.

The more particles there are, the more combinations are possible and the more improbable will be the two extreme cases of accumulation in one room. For example, if there are 10 particles, the probability of total sprawl is 252 possible combinations from 1024 (25%), but the probability of total accumulation in one room is 1 case from 1024 (0.1% see Fig. 5, left).

Fig. 5 (A) shows the least probable distribution of 100 particles in a cylinder, but state B is very

probable. These probabilities can be calculated as approximately 1/13·10²⁹ (A) and 1/13 (B).

So, if anything changes it will most probably change from A into B instead of from B into A.

That asymmetry of process is the core of thermodynamics.

From Fig. 5 you also can learn that by an increasing number of particles most combinations accumulate around the middle of k=0.5·n. If you would calculate the possible combinations of 1000 particles the probability of sprawl (B) between k=495 and 505 (1% of n) would be practically 1 (100%). The graph would show a vertical line rather than a gaussian ‘bell’.

Difference of entropy

Suppose now the content of the cylinder is a mole of gas (that is approximately 6·10²³ particles, Avogadro’s number n). Then the probability of state B approximates 1 (100%). The probability of state A is again 1/2ⁿ. That is nearly zero, because the number 2ⁿ is extraordinary large: a 1 with more than 10²³ zeros. An ordinary computer can not calculate all combinations of that number as done in Fig. 4. However, to determine the entropy of state A we need the natural logarithm (the exponent to ‘e’ or 2.718) of that probability: ln1/2ⁿ or ln(2^-n). And ln(2^-n) is easily written as -n·ln(2). That will save a lot of calculation, because n will disappear in the definition of entropy by Boltzmann using that probability:

Fig. 6 The statistical definition of entropy by Boltzmann in 1877

In state A and B with n = 6·10²³ particles, the number of moles is 1; n is Avogadro’s number.

R is a constant (gas constant) we will explain later.[j] So, entropy is related to probability by a constant! However, Boltzmann chose the logarithm of probability, because if you want to know the entropy of two sub systems (for example two moles), you would have to multiply the combination of each sub system. If you take the logarithm first, than you can simply add both[k].

In this case we can write the increase of entropy from stage A into B as S_B-S_A:

Fig. 7 The increase of entropy from accumulation in one room into sprawl in two rooms

The probability of state B is very near 1, and the logarithm of 1 is zero, so we can write:

Fig. 8 Simplifying the formula of Fig. 7

_{So, the entropy of stage B is R·ln(2). The natural logarithm of 2 is
0.693, but what is R?}

_{R is the gas constant per mole of gas:}

Fig. 9 Defining the gas constant R

In Fig. 9 P is the pressure (force/m²) and V is the volume (m³). So, on balance P·V is ‘force times distance’: energy (expressed in newton·m: joule). T is the temperature in degrees of Kelvin (K).

In a mole of gas the proportion between that energy and temperature in normal conditions appears to be the same[l]: 8.31472 joule/K. That constant is named ‘gas constant’ R. So, that is also valid for both stage A and B. Now we could calculate the increase of entropy as R.ln(2) = 5.8 joule/K·mole.

However, in thermodynamics the ‘probability’ of a state contains more than the distribution over two rooms. For example the reduced freedom of movements of particles in liquids and solids. That is why we limit ourselves here to complete freedom of movement (gas) to describe the states A and B. Moreover, gas plays a dominant role in energy conversion any engineer is occupied with.

Change of entropy

If a mole of gas expands from A to B, the heat content Q disperses over a doubled volume. So, the temperature tends to drop and the system immediately starts to adapt to the temperature of the environment. That causes an influx of extra heat energy DQ. So, in a slow process T could be considered as constant and the pressure will halve to keep also P·V constant at R·T (see Fig. 10).

	P·V = R·T (see Fig. 9), so P = R·T/V (see the graph left). _{If at any moment Q := P·V, any small change dQ equals}P·dV and a larger change DQ from stage 1 into 2 is the sum of these small changes: [m] so, . Remember nowFig. 8: if , then also . So,

Fig. 10 Extending 1 mole of gas (22.42 liter at 1 atmosphere) from 10 to 20 liter keeping T at 0^oC or 273.26K.

The heat energy Q is equal to P·V, but if it increases P itself is dependent on V.

So, every infinitely little increase of V (dV) has to be multiplied by a smaller P. Summing these products P·dV between V = 1 and V = 2 is symbolised by the first ‘definite integral’ sign in Fig. 10. However, that formula can not be solved if we do not substitute P by R·T/V (see Fig. 9) in the next formula. In that case the mathematicians found out that definite integral is equal to R·T·ln(2).

Now we have a real quantity for DQ, because R·T·ln(2) = 1574 joule.

So, DQ/T = R·ln(2), and R·ln(2) reminds us of Fig. 8: it is DS, the change of entropy!

A few steps according to Fig. 7 takes us back to the statistical definition of Boltzmann in Fig. 6, but now it is related to heat content Q and temperature T, the variables used in any engineering.

If DS = DQ/T, then also dS = dQ/T and now we can write the famous integral of Clausius:

Fig. 11 The thermodynamic definition of entropy

This formula shows that an increasing heat content increases entropy, but a higher temperature decreases it. If we now keep the heat content the same (closed system) and increase volume, then accumulation, pressure and temperature decrease (Boyle-Gay Lussac, see Fig. 9), so entropy will increase.

So, accumulation (storage, difference between filled and empty) decreases entropy, increases order.

Design and the conception of order, specialists’ conceptions

The explanantion of entropy above is extended, because of two reasons.

Firstly, while defending a concept of order, arrangement in design, designers often refer to low entropy and that is not always correct. Perceptual order could refer to a regular dispersion of objects in space and just that means sprawl, entropy. In thermodynamics an irregular dispersion with local accumulations has a lower entropy (disorder) than complete sprawl. However, in fluids and solids rectangular or hexagonal patterns with low entropy appear, due to molecular forces. But in general, if the particles have freedom of movement, sprawl is much more probable than accumulation.

It reminds us of the avoidance of urban sprawl. Thermodynamically accumulation is possible, but very improbable. So, if thermodynamics has any lessons for designers: sprawl is not the task of design, if there is freedom of movement, than it very probably happens without intention.

Secondly, energy and entropy are basic concepts in any engineering. To understand specialists in their reasoning and to be able to criticise them demands some insight by designers. The impact of the industrial revolution, the accumulation of population in cities can not be understood without understanding the manipulation of sprawl on another level of scale as has happened in the development of the internal-combustion engine. The internal-combustion engine is extensively used in industry and traffic. So, I would like to proceed with some explanation of that engine, the main application of sunlight stored in fossile fuels in human society.

Forced concentration

The (change of) force by which a piston is pushed out of a cylinder is equal to the proportion of (change of) energy and entropy Fig. 12. In a cylinder engine, alternating states of dispersion are used to convert imported disordered energy (heat) partly into directed movement. It is only possible by exporting part of the heat in an even more dispersed form (cooling). The necessary event of cooling makes an efficiency of 100% impossible and increases entropy in a larger environmental system. The reverse, adding rotating energy to this engine the principle that can be used for heating (heat pump) and cooling (refrigerator).

Fig. 12 Carnot-engine

1.1.3 Energetic efficiency

The proportion of the applicable part from total energy content of a primary source is the efficiency of the conversion.[15] In Fig. 13 some conversion efficiencies are represented.

Device or process	chemical->thermic	thermic->mechanisal	mechanical->electric	electric->mechanical	electric->radiation	electric->chemical	chemical->electric	radiation->electric	thermic->electric	efficiency
100%
electric dynamo
electric motor
90%
steam boiler
HR-boiler
80%
c.v.-boiler
electric battery
70%