Menu planning for the athletes

The energy needs of the athletes vary widely, depending on the activity. For instance, it is not uncommon for Tour de France cyclists to require more than 5000 kcal/d. The recent guideline suggests that women need approximately 1800–2200 kcal/d and men need approximately 2400–3000 kcal/d^{(Reference Kerksick, Harvey and Stout39)}. Energy expenditure is affected by the body composition and duration, intensity and frequency of exercise^{(Reference Kerksick, Arent and Schoenfeld40)}. Approximately 2000 kcal/d of calories is needed by the general public, an additional 1100 kcal/d is needed by a runner and endurance athletes usually require more calories than weight lifters^{(Reference Kerksick, Arent and Schoenfeld40)}. In the present study, diets were planned to provide 3000–3500 kcal/d for an exercise day (Table 2). We also assumed that when an athlete reaches a certain age, because of the lack of intense exercise he/she starts to eat as recommended for healthy individuals, i.e., 19–59 years old adults consume 1600–3000 kcal/d⁽³⁸⁾. Data employed in this study were openly available for use and adapted from the references providing dietary recommendations by citing the source. We provided the restrictions, which apply to our calculations by making some assumptions. Lifespan is indeed a long time, and there will be inevitably some changes in the nutritional preferences of the athletes in this period. We believe that after the publication of this study, many other people will utilise the same method in their research; therefore, we wanted to provide as much detail to them as possible.

Table 2. Total calories and composition of the diet plans for different groups of athletes and a 50 years old healthy retired person.

Total calorific uptake, carbohydrates, proteins and fats supplied by the diet to the athletes are expressed per kg of the bodyweight of the athlete.

There is an ideal weight and height range for athletes to be successful in any sports. Wenzel Coaching⁽⁴¹⁾ indicated that the height range of the cyclist is 157–193 cm, the weight range of the male climbers is between 50 and 69 kg and those of the male sprinters is 60–90 kg. The average height and the weight of these athletes were 1⋅80 m and 68⋅8 kg, and in the present study, calculations are performed for an athlete with 67 kg body weight (Table 2). Wilk's score measures the strength in powerlifting against other powerlifters. A weightlifter with 90 kg of body weight with Wilk's score of 450 and 500 can compete in the national-level meet or at world-class, respectively^{(Reference Nuckols42)}; calculations presented in Table 2 are valid for such a weightlifter. In the 2012/2013 season, the height and weight of the athletes of New Zealand's Blues rugby team were varying between 202–178 cm and 80–129 kg, respectively⁽⁴³⁾. In Table 2, assessments have been done for an average-weight rugby player.

People with varying heights and weights may be professional golfers. Irish golfer Rory McIlroy was a good PGA Tour golfer in 2010 and rose to No. 1 in the world by 2012, he was 178 cm tall and 73 kg. Tiger Woods was 188 cm tall and 70 kg when he turned pro in 1996.

He is widely regarded as one of the greatest golfers of all time and one of the most famous athletes in history and elected to the World Golf Hall of Fame in 2020. Gary Woodland was voted the most athletic golfer on the PGA tour in 2011. He was 185 cm tall and 91 kg^{(Reference Rose44)}. The golfer, for whom the calculations are performed in Table 2, had exactly the same bodyweight as Tiger Woods and similar body weight as Rory McIlroy.

Thermodynamic considerations

Fig. 1 illustrates the athlete's body as a thermodynamically open system. While calculating the entropy generation by the athletes the same procedure is employed as Öngel et al. ^{(Reference Öngel, Yildiz and Akpınaroğlu20)}, a healthy person was assumed to digest 99 % of the carbohydrates, 95 % of the lipids and 92 % of the proteins. We calculated entropy generation during the metabolism of carbohydrates, lipids and proteins in terms of glucose, palmitic acid and the average of 20 amino acids based on the following equations:

(1)$${\rm C}_6{\rm H}_{12}{\rm O}_6 + 6{\rm O}_2\to 6{\rm H}_2{\rm O} + 6{\rm C}{\rm O}_2$$

(2)$${\rm C}_6{\rm H}_{32}{\rm O}_2 + 23{\rm O}_2\to 16{\rm C}{\rm O}_2 + 16{\rm H}_2{\rm O}$$

(3)$$\eqalign{&{\rm C}_{4\cdot 57}{\rm H}_{9\cdot 03}{\rm N}_{1\cdot 27}{\rm O}_{2\cdot 25}{\rm S}_{0\cdot 046} + 4\cdot 75\,{\rm O}_2 \cr & \quad \to 3\cdot 245\,{\rm H}_2{\rm O} + 3\cdot 935\,{\rm C}{\rm O}_2 + 0\cdot 635\,{\rm C}{\rm H}_4{\rm N}_2{\rm O}}$$

Fig. 1. Description of the human body as a thermodynamically open system.

Describing the metabolism with equations (1–3) is an oversimplification, during exercise lactic acid accumulates in the muscles of the athletes and then it is reused. Energy, e.g., enthalpy, extracted from carbohydrates will not be affected by lactic acid accumulation and its reuse in the muscles, since enthalpy is a state function and its value depends on the initial, carbohydrates and the oxygen, and the final chemical species, e.g., carbon dioxide and water^{(Reference Mahan45)}. Ulu et al. ^{(Reference Ulu, Semerciöz and Özilgen46)}, while discussing energy storage and reuse in biological systems, reported that entropy generation associated with this phenomenon is extremely small and does not contribute to the lifespan entropy accumulation; therefore, entropy generation during lactic acid utilisation and reuse in the muscles is neglected.

Metabolic waste after digestion and absorption processes was removed with the urine and the faeces. Following the same procedure as Öngel et al. ^{(Reference Öngel, Yildiz and Akpınaroğlu20)}, the amount of urine was determined to limit its urea content as 20 g/l and 65 % of urea is excreted. The amount of faeces excretion was calculated by equation (4). It was assumed that the athletes consume 3⋅0 litres/d water and non-athlete healthy people drink 2⋅0 litre water daily.

(4)$$\mathop \sum \limits_{{\rm in}} \dot{m}_{{\rm in}}-\;\mathop \sum \limits_{{\rm out}} \dot{m}_{{\rm out}} = \displaystyle{{dm_{{\rm system}}} \over {dt}}$$

where dm _system/dt, i.e., water retention in the body, was assumed zero.

The amounts of the inhaled O₂, excreted CO₂, H₂O and the wastes excreted from the body for each athlete are shown in Table 3.

Table 3. The amounts of the inhaled O₂, excreted CO₂ and H₂O, urine and faces and metabolic heat production, work performance and the total heat loss from the body during the active years and after retirement

In the present study, calorie uptake with each food is calculated in terms of their enthalpies exactly the same way as Öngel et al. ^{(Reference Öngel, Yildiz and Akpınaroğlu20)} In biological systems, the molecular structures are usually so sophisticated that their thermodynamic properties are not readily available in the tables and usually calculated with the group contribution method^{(Reference Özilgen and Sorgüven47)}. The best-known group contribution methods were suggested by Joback and Reid^{(Reference Joback and Reid48)}. According to this method, a complex biological structure is separated into its contributing chemical sub-groups, then thermodynamic properties of each sub-group are evaluated individually. The thermodynamic property of the complex chemical structure is calculated by summing up the thermodynamic properties of its sub-groups. For instance, sucrose is a disaccharide, composed of two monosaccharides glucose and fructose. The standard enthalpies of the formation of glucose and fructose are −1271 and −1265 kJ/mol; when we add up these numbers, we estimate the enthalpy of the formation of sucrose according to the group contribution method as −2536 kJ/mol. In the standard thermodynamic tables, the enthalpy of formation of sucrose is given as −2226 kJ/mol. The difference between the estimate and the experimentally determined value is approximately 12 %. While carrying out such calculations, only metabolisable chemicals are employed. Any non-digestible chemical, which is available in the food is not included in the calculations. Sucrose, glucose and fructose have relatively simple structures, and their thermodynamic properties are available in the tables; on the other hand, with the extremely sophisticated structures, referring to the group contribution method is the only way to proceed^{(Reference Özilgen and Sorgüven47)}.

By assuming that the nutrients intake into the body is at 25°C, and H₂O and CO₂ exit at 37°C, energy balance was calculated via equation (5). The energy balance around the human is:

(5)$$\Delta \dot{H} = \sum \dot{n}_p( {h_f^{-^\circ } + h^-{-}h^{-^\circ }} ) _p-\sum \dot{n}_r( {h_f^{-^\circ } + h^-{-}h^{-^\circ }} ) _r$$

where $\Delta \dot{H}$ is the enthalpy release via equation (5), $\dot{n}_p$ and $\dot{n}_p$ represent the mole number rates of the products output from and the reactants input to the system, respectively; parameters $\;h_f^{-^\circ }$, h ⁻ and $h^{-^\circ }$ are the formation enthalpies at the standard conditions. The thermodynamic properties of the compounds are shown in Table 4. It was assumed that 32, 106 and 8 moles of ATP were created by the metabolism of one mole of each of glucose, palmitic acid and the average 20 amino acids, respectively. Total work achieved by the formation of the ATP molecules was found with equation (6); 34⋅6, 32⋅2 and 10⋅4 % represent the metabolic efficiency (η) of glucose, palmitic acid and the average of 20 amino acids, respectively^{(Reference Silva and Annamalai17,Reference Annamalai and Silva18)} .

(6)$$\dot{W}_{{\rm ATP}} = \eta \Delta \dot{H}$$

where $\dot{W}_{{\rm ATP}}$ represents the total work performance rate via ATP utilisation.

Table 4. Thermodynamic properties of the nutrients and the products of the metabolism at 1 atm (adapted from Kuddusi)^{(Reference Kuddusi19)}

Total heat loss from the skin and through respiration was calculated from equations (7) and (8) with the conversion rate recommended by Hall^{(Reference Hall49)} and presented in Table 3.

(7)$$ \eqalign{ \mathop \sum \limits_{{\rm in}} & [ \dot{m}\;( h + e_p + e_k) ] _{{\rm in}}-\mathop \sum \limits_{{\rm out}} [ \dot{m}\;( h + e_p + e_k) ] _{{\rm out}} \cr & + \mathop \sum \limits_i \dot{Q}_i-\dot{W} = 0} $$

(8)$$\dot{Q}_{{\rm total}} = 3\dot{W}_{{\rm ATP}}$$

Entropy balance

In accordance with the descriptions in the literature, the weights of a male cyclist, weightlifter, rugby player and golfers were assumed 67, 85, 105 and 70 kg, respectively. The entropy generation rate of the athletes was calculated depending on their diets by equation (9) and is given in Table 5.

(9)$$\left({\sum ns} \right)_{{\rm out}}-\;\left({\sum Nns} \right)_{{\rm in}}-\sum \displaystyle{{Q\;} \over T} = \Delta s_{{\rm gen}}$$

where n describes the mole number of chemicals taken in or excreted from the system, s is the entropy of the chemicals, T is the human body temperature as 37°C and Q is the heat generated by the body. Daily consumption of the amounts of carbohydrates, lipids and proteins uptake are shown in Table 2 made it possible to calculate the amounts of the O₂ breathed in O₂, respired and CO₂ and H₂O exhaled and the excreted waste (Table 3).

Table 5. The total entropy generation rate and the expected lifespan of the athletes

The athletes were assumed to retire when they become 50 years old and start consuming a special diet prepared for them. The lifespan entropy generation limit was assumed to be 11 404 kJ/K kg^{(Reference Silva and Annamalai6,Reference Silva and Annamalai17)} , and the athletes die when they come to this limit. Diet plans have been designed in the present study by referring to the nutritional guidelines and scientific articles to provide sufficient nutrients for each athlete, and then the lifespan entropy has been calculated as presented in Table 5. These calculated lifespans are then compared to the observed longevities to be sure that the thermodynamic model is correct.