Question 1. The van der Waals equation of state is given by the equation: a

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1 Kopiereg voorbehou Universiteit van retoria University of retoria Copyright reserved Departement Chemiese Ingenieurswese Department of Chemical Engineering CHEMICAL ENGINEERING CIR EKSAMEN Volpunte: Tydsduur: ure Interne Eksaminator: rof Walter Focke Eksterne Eksaminator: Dr S rins Mei Ho to know that you are an engineer: If you have no life - and you can ROVE it mathematically. CHEMIESE INGENIEURSWESE CIR EXAM Full marks: Time: hours Internal Examiner: rof. Walter Focke External Examiner: Dr S rins May Open book exam: Only the prescribed textbook (or two alternative materials textbooks) and class notes allowed! lease state and justify all assumptions you make! Question. The van der Waals equation of state is given by the equation: a V b V (a) Give a physical interpretation for the constants a and b (b) Show that the fugacity coefficient can be calculated as follows: z n d (c) Derive an expression for the fugacity of a van der Waals gas described for the specific situation where the constant a =. (d) For binary gas mixtures the mixing rule for the constant b is given by: b = b y + b y y + b y However, the following combining rule applies to the interaction parameter: b = (b + b )/ Obtain an expression for n () Question. It is proposed to manufacture low-cost geysers from pipe-grade polypropylene. These geysers will form part of a novel pressurized solar heating system. The water temperature will not exceed 7 C and the pressure relieve valve will be set at bar. The geyser design is that of a cylindrical vessel with spherical end-caps. The length of the cylinder is m and the diameter is 6 mm. The design wall thicknesses of the vessel at the spherical end caps and in the cylindrical section are allowed to differ. Use the data in Figure and calculate the required wall thicknesses subject to a % safety factor assuming that the geysers must last at least years. lease recommend a suitable quality control test that will provide compelling evidence that the design and manufacturing process for the proposed geyser is reliable. lease explain your calculations and proposals by appropriate annotations on Figure. Note: lease assume that the thin-wall approximation can be used in the geyser design. () Question. A polymer test bar is reinforced with long cylindrical fibres (modulus E f ) placed in alternating layers. Consider three possibilities: (a) The fibres are all placed parallel to the length of the bar (b) The fibres are all placed perpendicular to the bar axis, and (c) One layer of the glass fibres is oriented parallel and the next is oriented perpendicular to the axis of the test bar. Assume strong that Hooke s law holds, good interfacial bonding and that the modulus of the matrix equals E m. For each of the three possibilities stated above:

2 Derive expressions for the modulus of the bar as a function of the fibre loading expressed in terms of volume fraction fibres (). Calculate the maximum fibre loading physically possible. Make sure to provide geometric sketches supporting your computations. (6) ******************************* Vraag. Die van der Waals toestandsvergelyking word gegee deur: a V b V (e) Gee fisiese interpretasies vir die konstantes a en b (f) Toon dat die fugasiteitskoëffisiëent as volg bereken kan word: z n d (g) Lei n vergelyking af vir die fugasiteit van n van der Waals gas vir die spesifieke geval waar a =. (h) Vir binêre gasmengsels is die mengrëel vir die konstante b as volg: b = b y + b y y + b y Die volgende kombinasierëel geld egter vir die interaksieparameter: b = (b + b )/ Verkry nou n uitdrukking vir n () Question. Daar is voorgestel dat lae-koste geysers van pyp-graad polypropileen vervaardig moet word. Hierdie geysers vorm deel van n nuwe hoë-druk sonverwarmingsstelsel. Die water temperatuur sal nie 7 C oorskry nie en die veiligheidsklep word gestel op bar. Die geyser ontwerp is n silindriese tenk met sferiese doppe. Die lengte van die silinder is m en die diameter is 6 mm. Die ontwerpwanddiktes van die drukvat in die sferiese dop gedeelte en in die silindriese gedeeltes mag verskil. Gebruik die data in Figuur en bereken die benodigde wanddiktes vir n veiligheidsfaktor van % as die geysers ten minste jaar moet hou. Stel asseblief ook n goeie kwaliteitstoets voor wat oortuigende bewys sal lewer dat die ontwerp sowel as die vervaardigde geyser wel betroubaar is. Verduidelik u berekeninge en voorstelle deur middel van annotasies op Figuur. Nota: Aaanvaar aub dat die dun-wand benadering gebruik kan word in die geyser ontwerp. () Question. n olimeer toetsstaaf is versterk met lang silindriese vesels (modulus E f ) wat lag-vir-laag neergelê is. Beskou drie moontlikhede: (a) Die vesels is parallel to die lengte van die staaf neergelê (b) Die vesels lê loodreg tov die lengte van die staaf, en (c) Die vesels lê afwisselend loodreg en parallel tov die langas van die staaf. Aanvaar dat Hooke se wet geld, dat die binding tussen die vesels en die matriks uitstekend is, en dat die modulus van die matriks gelyk is aan E m is. Vir elk van die moontlikhede hierbo aangedui: Lei vergelykings af wat die modulus van die staaf beskryf as n funksie van die vesellading bereken volgens volumefraksie vesels (). Bereken die maksimum vessel lading wat dfisies moontlik is. Maak aub gebruik van sketse om die geometriese uitleg van die veselplasings te verduidelik. (6)

3 Name:. Student No.: Wall stress, Ma C 7 C 8 C 9 C C Time to failure, h Figuur / Figure Wall stress, Ma C 7 C 8 C 9 C C Time to failure, h Figuur / Figure

4 Answer : (a) Give a physical interpretation for the constants a and b a and b represent the effects of the attractive forces between the molecules and their finite volume respectively (b) Show that the fugacity coefficient can be calculated as follows: z n d dnf V d z d zdn dn f zdn f dn z dn z dn n z d Because as. (c) Derive an expression for the fugacity of a van der Waals gas described for the specific situation where the constant a =. But along the isotherm V V z V b V b b z V b d dv dv V T V V bt dv dv V b V b f b n V b V b b b dv V b V b dv V b V V b f b b n V b V (d) For binary gas mixtures the mixing rule for the constant b is given by:

5 b = b y + b y y + b y However, the following combining rule applies to the interaction parameter: b = (b y + b y )/ Obtain an expression for n b by b y y b y b b b y y y b y b y yy b y yy b y y y b y y y b y b y ( nb) ( bn bn ) n n Tn,, b ˆ b n nb T n n,, This result is reminiscent of the truncated virial EOS. Indeed, rearrangement shows that z b V V b V b