Two fluids flow into a cylindrical mixing chamber where they...

Two fluids flow into a cylindrical mixing chamber where they are mixed at steady state by a paddle wheel that rotates at 50 revolutions per second, Fig. P8.30. It is suggested to scale up the process to twice the amounts of liquids mixed. Because the original mixing is quite satisfactory, complete similarity is favorable. One design suggestion was to increase all piping to twice their cross sections and to keep the cylindrical mixing chamber height but to increase its diameter by $\sqrt{2}$. Another suggestion was to keep the old structure but to have the flowrates in the piping increased by 2. Find if these suggestions can satisfy similarity, and if they can, find the size and the number of revolutions of the paddle wheel. Figure P8.30 can't copy Paddle-wheel mr.er.

Key Concepts

Geometric Similarity

Geometric similarity refers to the condition in which all corresponding lengths in two systems are in the same ratio, meaning the shapes remain identical even as the size changes. In engineering and fluid mechanics, ensuring geometric similarity during scale-up means that every component of the system, such as the mixing chamber, paddles, and piping, must be resized proportionately so that the spatial relationships remain constant. This is essential because many dimensionless numbers that characterize a system's behavior—such as the Reynolds number—are functions of geometric parameters, and preserving them is key to replicating performance at different scales.

Dynamic Similarity

Dynamic similarity occurs when two systems not only have the same geometric proportions but also the same ratios of forces acting within them. In fluid dynamics, this means that the conditions of the flow, particularly the inertial forces to viscous forces ratio represented by the Reynolds number, must be identical between the model and the prototype. Achieving dynamic similarity ensures that the mixing, circulation, and turbulence observed in the small-scale system will be present in the scaled-up system, leading to predictable and reliable performance.

Scaling Laws

Scaling laws provide the mathematical relationships that dictate how different physical quantities change with size. In the context of scaling up a mixing process, these laws help determine how variables such as flow rates, pump speeds, and rotational speeds of mixing devices must be adjusted to maintain the same operational characteristics. Understanding these laws is crucial because even small geometric changes can significantly affect forces like inertia and viscosity, thereby altering the performance if the speed or flow conditions are not correctly scaled.

Reynolds Number

The Reynolds number is a dimensionless quantity used in fluid mechanics to characterize the relative importance of inertial forces to viscous forces in a flow. It is defined as the product of a characteristic velocity and length divided by the kinematic viscosity of the fluid. In mixing applications, maintaining the same Reynolds number during scale-up is vital because it directly affects flow patterns, turbulence levels, and mixing efficiency. Any change in the Reynolds number may lead to noticeably different flow regimes, undermining the desired complete similarity between the original and the scaled-up process.

Transcript

00:01 In this video we're going to go through the answer to question number 45 from chapter 9 .5.

00:05 So we're given a model of interconnected tanks and asked to find how the quantities of salt in each of the tanks changes through time.

00:16 Okay, so we need to write this or formalize this model mathematically.

00:22 So x1 is the value or the mass of salt in tank a.

00:29 Okay.

00:29 So how can we? we say that yeah what can we say about the way that that quantity changes well there's three liters per minute of pure water coming in the left but that's there's no salt in that water so that's not going to change the mass of salt in tank a but there's four liters per minute leaving poorly four liters of water a minute leaving tank a okay so it's going to be leaving yeah, whatever quantity is in tank a already, which is x1, we're going to have four over 50 kilograms leaving per minute, because that's what's going into tank b.

01:22 But then we're also going to have some amount coming back in from tank b, and by the same logic, that's going to be 1 over 50, because tank b has also got 50 litres and is coming into tank a at 1 ,000.

01:38 Per minute.

01:40 That's x2.

01:43 Similarly, x2.

01:48 So the quantity, the mass of salt in tank b changes by, well, four litres are coming in from tank a.

02:00 So that's 50 litres from tank a.

02:09 And then we're leaving, we're having one litre leaving into tank a of three liters of the salty water in tank b leaving on the right.

02:20 So we're losing four litres out of the 50 that we have per minute from.

02:32 Okay, so how can we write this in a matrix normal form? well, if x is the vector x1, x2, then we can take a factor.

02:45 Of 1 over 50 out the front and we're going to have 4, oops, minus 4, 1, 4 minus 4 times by x.

02:58 Okay, so let's call this guy without the 1 over 50.

03:05 Let's call that a prime or a star.

03:11 And then a is 1 over 50, a star.

03:17 Okay, so let's first work out what the eigenvalues of matrix a star.

03:36 So that's going to be minus 4 minus r, 1, minus 4 minus 4 minus r.

03:56 That's 4 plus r squared plus 4.

04:10 So that's going to be 60.

04:22 I've written that down wrong, so this shouldn't be a minus here, which makes that a minus rather than a plus.

04:39 Okay, that makes more sense.

04:41 So this is going to be r squared minus plus 8r, and that plus 16 minus 4 is plus 12...

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