Shell balances in cylindrical coordinates

In summary: Then, I'll give my opinion on which one is better.Please show me how you think the answer will be different.
  • #1
Muhammad Waleed Khan
4
0
I have a question regarding writing a shell balance for a cylindrical system with transport in one direction (in any area of transport phenomena). When we set up the conservation equation(say steady state), we multiply the flux and the area at the surfaces of our control volume and plug them into the eqn. Afterwards I believe that we divide the resulting equation by the volume of the control volume before taking the limit as DelX,DelZ or DelR approaches 0. In cartesian coordinates we divide this by delXdelYdelZ, but why in cylindrical CV do we divide by 2.pi.DelR.L instead of 2.pi.R.DelR.L (which is the volume of our CV)?
 
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  • #2
Muhammad Waleed Khan said:
I have a question regarding writing a shell balance for a cylindrical system with transport in one direction (in any area of transport phenomena). When we set up the conservation equation(say steady state), we multiply the flux and the area at the surfaces of our control volume and plug them into the eqn. Afterwards I believe that we divide the resulting equation by the volume of the control volume before taking the limit as DelX,DelZ or DelR approaches 0. In cartesian coordinates we divide this by delXdelYdelZ, but why in cylindrical CV do we divide by 2.pi.DelR.L instead of 2.pi.R.DelR.L (which is the volume of our CV)?
Of course, you could also divide by ##2\pi r \Delta r L##. That won't change the final result. But, the authors must of thought it was more convenient and aesthetically pleasing to do it their way. Both ways are right.
 
  • #3
Chestermiller said:
Of course, you could also divide by ##2\pi r \Delta r L##. That won't change the final result. But, the authors must of thought it was more convenient and aesthetically pleasing to do it their way. Both ways are right.
Yeah but it changes the final answer you get from integrating the resulting ODE.
 
  • #4
Muhammad Waleed Khan said:
Yeah but it changes the final answer you get from integrating the resulting ODE.
Please show me how you think the answer will be different. Start out by showing me the two versions of the ODE that you get.
 

1. What is a "Shell balance" in cylindrical coordinates?

A "Shell balance" in cylindrical coordinates refers to a mathematical equation used in fluid mechanics to describe the conservation of mass, momentum, or energy within a cylindrical region. It takes into account the flow and distribution of fluids within a cylindrical space, such as a pipe or cylinder.

2. How is a shell balance represented in cylindrical coordinates?

A shell balance in cylindrical coordinates is typically represented by the Navier-Stokes equations, which are a set of partial differential equations that describe the motion of fluid substances. These equations take into account the effects of pressure, viscosity, and external forces on the fluid.

3. What are the advantages of using cylindrical coordinates for shell balances?

Cylindrical coordinates offer several advantages for solving shell balances. They are particularly useful for systems with cylindrical symmetry, as they simplify the equations and make them more manageable. They also allow for the use of specialized techniques, such as separation of variables, to solve complex problems.

4. How are cylindrical coordinates converted to Cartesian coordinates?

Cylindrical coordinates can be converted to Cartesian coordinates using the following equations:

x = rcosθ, y = rsinθ, z = z

Where r is the distance from the origin to the point in cylindrical coordinates, θ is the angle from the positive x-axis to the point, and z is the height or depth from the xy-plane.

5. What are some practical applications of shell balances in cylindrical coordinates?

Shell balances in cylindrical coordinates have many practical applications in engineering and science. They are commonly used in the design and analysis of pipes, pumps, and turbines in the oil and gas industry. They are also used in the study of weather patterns, ocean currents, and atmospheric circulation. Additionally, they are essential in the development of fluid-based technologies, such as rocket engines and hydraulic systems.

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