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Energy Conversion System

(a) Determine the non-dimensional expressions and . [8 marks] Solution: There are 6 variables involved in the problem: velocity , density , viscosity , surface tension , diameter is and acceleration . Choose the three recurring variables: velocity , density and diameter . From Buckingham pi theorem we have non-dimensional groups. The choice of velocity , density and diameter as the repeating variables satisfies: They are measurable, good design parameters and in combination, contain all the dimension and . We form the three groups according to the 2nd theorem. As the groups are all dimensionless, there must have dimensions such as: , we can use the principle of dimensional homogeneity to equate the dimensions for each group. For the first group, In terms of dimensions, we have that: Thus, we have the following linear system: For For For The solutions for is given as: Giving as And a similar procedure is followed for the other groups. Thus, we have the following linear system: For For For The solutions for is given as: Giving as Finally the group: Thus, we have the following linear system: For For For The solutions for is given as: Giving as Thus the problem may be described by the following function of three non-dimensional groups, We take into account that any dimensionless group may be raised to any power and it is still a valid group. Furthermore, the reciprocal of any dimensionless group is valid, too. We concluded that: Finally the non-dimensional expressions and are: Pay attention that the group obeys manipulation rules of equivalence. Remember that: Velocity : Density : Viscosity : Surface tension : Distance : Acceleration : [] (b) Comment on the physical significance of the non-dimensional numbers formed. [6 marks] Solution: The non-dimensional expressions found for and were: The non-dimensional number given by: is the so-called Reynolds number: The Reynolds number is a non-dimensional number that gives a measure of the ratio of inertia force to viscous force. This parameter correlates the viscous behavior of all Newtonian fluids. In fluid dynamic engineer should do estimate of the Reynolds number range of the flow under study. Very low indicates viscous creeping motion, where inertia effects are negligible. Moderate implies a smoothly varying laminar flow. High probably spells turbulent flow, which is slowly varying in the time-mean but has superimposed strong random high-frequency fluctuations. The non-dimensional number given by: is the so-called Weber number: It is a non-dimensional number useful for analyzing fluid flows where there is an interface between two different fluids. The Weber number is the ratio between the inertial force and the surface tension force. Since the Weber number represents an index of the inertial force to the surface tension force acting on a fluid element, it can be useful analyzing thin films flows and the formation of droplets and bubbles. Question 2 (a) Two vessels are connected by a pipe in which there is a valve. One vessel of contains nitrogen at and , and the other of contains helium at and . The valve is opened and the two gases are allowed to mix. Assuming the system is well insulated, calculate (i) the final temperature of the mixture [8 marks] Solution: The amount of energy given off by the warmer gas equals the amount
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