Low-Flow Appliances to Conserve Water - Concepts of Unit Vector - Engineering Assignment Help

Assignment Task:

Using Low-Flow appliances to conserve water

The textbook uses the surface integral of a vector field over a solid to define the idea of flux, and during this point of the semester, we find that students can work through the integration without understanding what the result of their integration represents or means. The idea behind this project is to build students? intuitive understanding of flux and surface integrals of vector fields by examining the concept of flux out of a surface. The aim of a surface integral is to find the flux of a vector field through a surface. It helps, therefore, to begin what asking, ”what is flux”?

Learning Goals:
1. Review and apply concepts of unit vector of given surface, dot product, scalar projection, in context of flux.
2. Develop understanding of flux as the total flow rate across a surface.
3. Solve the total rate of flow across various surfaces.
4. Generalize their work to calculate flux across any continuous, smooth surface along a vector field.
5. Relate flux across a curve boundary to flux across a surface boundary.
6. Apply Divergence Theorem to the concept of flux across a surface boundary.

When studying surface integrals over vector fields, we often use the word flux. To develop our understanding of flux, we will consider the more intuitive definition of flux as the total rate of flow across a surface.

Warm-Up & Review: 
Set up an iterated integral that would give the flux of F(x, y, z) = xze^yi - xze^yj + zk across the surface S which is the part of the plane x + y + z = 9 in the first octant and has upward orientation.

1. Parametrize the surface
2. Find the normal vector
3. Find the Jacobian
4. Set the surface integration in terms of u and v.

Nebia Shower - Better experience, 70% less water
Atomizing nozzles use pressure and geometry to control the dispersion and break up of fluid streams. These nozzles have been developed over the last century for specific technical applications, such as injecting fuel, powderizing metal, and efficient agricultural irrigation. While regular shower nozzles, called plain orifice nozzles, produce a straight stream of water with large droplets, our atomizing nozzles produce 100s of droplets dispersed into precise patterns.

Nebia’s greater dispersion of droplets yields a larger volume of air surrounded by - and intermixed with - warm droplets of water. Therefore, the warm droplets heat a larger volume of air faster and more efficiently. The nozzle geometries are engineered to release droplets at different sizes and speeds. At any given cross-section of air between the nozzles and you, Nebia comprises 100s more droplets dispersed over 5 times the area compared to a regular shower. Translation: water completely surrounds and gets you wet all while using 70% less water. It’s not magic; it’s science! Let?s look at examples to compare.

(c) What do the numbers represent in part a and b?

(d) How do you interpret the difference between these two numbers? Is the flow out 70% less as it has been claimed?

(e) Suppose the equation of your head s be the portion of the paraboloid z = -1 - x² - y² lying above the z = -2. Calculate the rate flow outward through your head with velocity vector field of part a (atomized nozzle) and b (regular nozzle) then compare the values of he rate flow outward through your head.

(f) Can we use divergence theorem to compute part e.

There was no information about how engineers have designed this shower head online. If you can find better function to get 70 %.

Use the following link to check your answers. Flux out of a parametric surface

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