La question posée (en français, avec des mots justes ; pour un calcul, l'objectif doit être clairement écrit formellement)

Transcription

1 Exercise : You have to make one ton of mayonnaise sauce using 95 % oil, 2.5 % egg yolk, 2.5 % vinegar. What is the minimum energy that you have to spend? Calculation for mayonnaise Hervé 4th October 2013 Below, I have a calculation to make. The best way to do: 1. think about strategy 2. do it. Here the strategy is to analyse that you have to calculate, so that you have to use a methode for calculation. This method is the "backbone calculation" Maple document. La question posée (en français, avec des mots justes ; pour un calcul, l'objectif doit être clairement écrit formellement) I have to make a mayonnaise sauce, and for this I have to give energy. The question is to know how much energy E should I use for a certain amount of sauce (mass M) Analyse de la question (souvent, les questions sont résolues quand elles sont analysées) On introduit les données (il s'agit de poser par écrit, de réunir les données dont on dispose) I know the mass M of sauce to be done. I know the proportions (in which unit?) to use [immediately I see that the question is poorly asked because these awful people (90 %) don't tell me whether it's mass of volume. It will be easier to calculate if it were mass. Then : Assumption 1 : the exercise is giving masses. This means that there is : - mass m[o] of oil - mass m[v] of vinegar (Assumtion 2 : this is water) - mass m[y] of egg yolk I realize that the exercise was not giving masses but rather proportions. I have to translate the proportions into masses.

2 What I know are : - p[o], p[v], p[y] What is a proportion? I don't know. And my life depends on the answer, then I HAVE to know. Let's is make it simple. They tell me that there is a proportion p[o] of oil. A proportion is a ratio. Here there only two quantities to be considered : the mass of oil, the total mass of sauce. What about the ration M[o]/M? It sounds great. I conclude that for any index : p[x]= M[x]/M which mean that I have to use Maple : (2.1.1) Modèle qualitatif (surtout, faire un schéma, en détaillant les hypothèses qui y conduisent) I have now to make a picture (this is said above, and I am very obedient). What I see : Water, oil droplets Here assumption 3 : all droplets are of the same radius r. Assumption 4 : 2-D description (I guess that there will be no change when moving from 2D to 3D, to be checked) Here the droplets are circular, but indeed I know that with 95 % of oil, they are not : assumption 5.

3 Just to finish : in my model, only oil and water are present, but the exercise is wrong because it does not tell me which surfactant can be used. And this will make the emulsion much more easy or not. Modèle quantitatif (transformer le modèle qualitatif en modèle quantitatif par description du modèle qualitatif, avec l'introduction de symboles, en vue du calcul formel) A certain number N of droplets They have a radius r. I see the total yellow part : this is oil, mass m[o] I see a blue part : vinegar+yolk, m[v]+m[y] La résolution Recherche d'une stratégie de résolution (souvent, cela devient inutile, mais quand c'est utile, il ne faut pas bâcler!) I know that this is so simple that no strategy is to be used, except one : just write down the relationship between the various parameters that were put down. La mise en oeuvre de la stratégie (attention : derrière la stratégie, il y a la tactique)

4 I "clean the desk" : Then I ask a question about energy : this energy has something to do with surface. Then it's surface tension : (3.2.1) isolate for DG Then I have to compare the initial surface and the final surface. Initial? Imagine (assumption) that oil was first out of water. Then Ai = 0. Then we need to calculate the final surface Af. (3.2.2) Not I have : (3.2.3) I now see that I am looking for Af. This area is the sum of the area for all droplets. (3.2.4) Here, a is missing. As droplets are spheres of radius r : (3.2.5) (3.2.6) (3.2.7) Here, I see that N is missing. And I realize that the mass was not used. I can guess that that the total mass of droplets M[o] will be equal to the product of N by the volume v[o] of one drop : isolate for N (3.2.8) (3.2.9)

5 (3.2.10) Here, M[o] is missing. But we now that (3.2.11) Then : (3.2.12) I see that m[o] is not given : (3.2.13) (3.2.14) (3.2.15) This does not work. Let's try : (3.2.16) isolate for M[o] (3.2.17) And here it is : I know all parameters.

6 Validation (le deuxième, troisième... calcul doit être différent du premier ; il ne s'agit pas de refaire le même chemin) I just finished making it with paper and pen, and I find the same result. Expression des résultats On recopie le résultat trouvé dans la résolution (il s'agit de mettre au net) Introduction des valeurs numériques (si possible, avec les références) I have to collect data : M : given p[o] : given r : not given, but I shall assume different values. Introduction des données dans l'expression des solutions First with r = 0.1 cm (here, this is given of course in J, because al data were in SI units) (4.3.1) Then with r=0.01 cm (4.3.2) On joue à varier les paramètres pour explorer l'espace des solutions Done above. If I wanted, I could make a curve in function of r, such as :

7 DG J r m Conclusions et perspectives Conclusions Here I used the value for oil and water, but I should have calculated using data for a particular surfactant. It is very easy to change the value in the end. SIU, then :

8 DG J I see that the energy is divided by two. Which is fine, but not much. Indeed I understand that the exercise was misleading another way: what was calculated was the energy for oil and water, but it does not tell anything of "spontaneous emulsions" (see more). Then this energy that I calculated has nothing to do with the real life. r m Perspectives I shall go on it over and over, because I did not consider the particular process which was used, and I know that this is a very important step.

9