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Solutions_Midterm_2

# Solutions_Midterm_2 - Problem z r Consider an...

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Problem: Consider an incompressible liquid contained in a closed annular region between a tubing of radius κR and a cylindrical container of inner radius R , as shown in the figure. The liquid in the vertical annulus can be circulated by the following mechanical and thermal methods. 1. In the first method , the tubing is moving upward with a constant velocity u 0 through the fixed container. As the inner tube is pulled upwards it passes through a tight seal, and no fluid leaves with it at the top (or enters with it at the bottom). This results in an isothermal circulating flow of the liquid, which is upward along the moving tubing and downward along the fixed container wall..It is assumed that the flow is steady. In petroleum production, flows similar to this occur in annular spaces between tubing and coil tubing. Another example is the seals of some reciprocating machinery, such as in the annular space between piston and ring. a. Set up the equation of motion for the circulating flow, using the dimensionless radius r r R = ( . State clearly how assumptions are used to simplify the flow equation. (7.5 points) b. Define boundary and mass-conservation conditions that are required to solve the equation of motion. (7.5 points) c. Show that the velocity distribution in the annular region, far from the end disturbances, has the following form (15 points) 2 1 2 3 0 =C C ln C (1) z u r r u + + ( ( 2. In the second method , the tubing is fixed and heated up by steam. The temperatures at the outer surface of the tubing wall and the inner surface of the container are maintained constant at T 1 and T 2 respectively. The liquid density varies to a small degree with temperature. This results in a similar circulating flow as observed in the first method.

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Solutions_Midterm_2 - Problem z r Consider an...

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