Pear-shaped Curve perimeter?

T

Television

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Greetings!

First things first, I am an archaelogist not mathematician, so bear with me. I need to know perimeter of pear-shaped curve as described in here http://mathworld.wolfram.com/Pear-ShapedCurve.html. This shape is typical to Roman water pipes and I try to calculate volumetric flow rate inside piping system in Pompeii. For this I need length of perimeter for this shape. I understand that this is not possible to solve computionally and question is how can I do that? Specifically I am looking for a solution that can be implemented by programming. I would be very gratefully for any help or hints to correct solution. Reasonable approximation is sufficient to my purpose, but as accurate as possible is, of course, better. If needed I can give additional information.

Thank you all!

Television
 
Television said:
Greetings!

First things first, I am an archaelogist not mathematician, so bear with me. I need to know perimeter of pear-shaped curve as described in here http://mathworld.wolfram.com/Pear-ShapedCurve.html. This shape is typical to Roman water pipes and I try to calculate volumetric flow rate inside piping system in Pompeii. For this I need length of perimeter for this shape. I understand that this is not possible to solve computionally and question is how can I do that? Specifically I am looking for a solution that can be implemented by programming. I would be very gratefully for any help or hints to correct solution. Reasonable approximation is sufficient to my purpose, but as accurate as possible is, of course, better. If needed I can give additional information.

Thank you all!

Television
Click to expand...
Does the cross-section of the pipe resemble the shape shown?

If it does, then to calculate the volumetric flow-rate - you would need the area enclosed by the shape (not the perimeter).
 
Subhotosh Khan said:
Does the cross-section of the pipe resemble the shape shown?

If it does, then to calculate the volumetric flow-rate - you would need the area enclosed by the shape (not the perimeter).
Click to expand...

Liquids flowing in a pipe will have a velocity that is affected by gravity, fluid viscosity, and the downward angle, size, shape and friction of the pipe; AND the water level within the pipe.

With your pear shaped pipe the water level will rise slowly at first and then it will rise much faster as the horizontal wall width narrows; once it is 100% full then water head pressure will play a factor.

If you can provide a few more details it may help people to offer assistance. What you're seeking can certainly be estimated with some dimensions, pipe angle, width, and also does the pipe make any radicle turns?
 
markmain said:
Liquids flowing in a pipe will have a velocity that is affected by gravity, fluid viscosity, and the downward angle, size, shape and friction of the pipe; AND the water level within the pipe.

With your pear shaped pipe the water level will rise slowly at first and then it will rise much faster as the horizontal wall width narrows; once it is 100% full then water head pressure will play a factor.

If you can provide a few more details it may help people to offer assistance. What you're seeking can certainly be estimated with some dimensions, pipe angle, width, and also does the pipe make any radicle turns?
Click to expand...

Roman pipes were quite irregular by shape. Reason for this is their manufacturing process (I won't go to details). Nearest shape I have found is that pear-shaped curve. I have studied this subject for some time now and I am well aware of physics behind the flow calculations.

Pompeii had presurized piping system and only some parts have survived to modern times. Basic assumption is that pipes were running full all the time (water was flowing through system all the time in Italy, it was different ex. in North Africa where water was not so abundant). System had one main water tower at the highest point of the town, several secondary towers, which were used to control pressure and served as distribution centers, lead pipes, leaden junction boxes and some bronzen rotary taps.

I try to calculate volumetric flow rate in selected points in piping system that served private houses. I have measurements of pipe diameters, assumed pipe line lengths and bends, heights of secondary water towers a.s.l. including assumptions of height of tanks on top of the towers, elevations of pipes where I calculate flow a.s.l. and of course known factors like viscosity, roughness of pipes etc. It is easy enough to calculate flow using basic formula for velocity and continuation equation, but I want to have more accurate result and include pressure losses. In these formulas hydraulic diameter is used instead of more normal diameter for circular pipes that are used today. Hydraulic diameter is calculated as following:

D = 4A/P whereD = hydraulic diameter, A = cross-sectional area, P = length of wetted contact line inside the pipe.

It is for this P I need to calculate perimeter of pear-shaped curve (in a side note if someone notice possible mistakes in my thought process let me know). Formula for area A is given in page I linked. Apart of coefficients for some fitting types in pipe line (and it is possible that these are impossible to get without experimenting) this is last piece of information I need before I can finalize my scripts for MatLab. I had to make many reasonable assumptions due the state of preservation in piping sytem and lacking documentation so good approximation is best I can hope for in this stage. I am very grateful for any help you can give me.

Thank you!

Television
 
Last edited:
Television said:
Roman pipes were quite irregular by shape. Reason for this is their manufacturing process (I won't go to details). Nearest shape I have found is that pear-shaped curve. I have studied this subject for some time now and I am well aware of physics behind the flow calculations.

Pompeii had presurized piping system and only some parts have survived to modern times. Basic assumption is that pipes were running full all the time (water was flowing through system all the time in Italy, it was different ex. in North Africa where water was not so abundant). System had one main water tower at the highest point of the town, several secondary towers, which were used to control pressure and served as distribution centers, lead pipes, leaden junction boxes and some bronzen rotary taps.

I try to calculate volumetric flow rate in selected points in piping system that served private houses. I have measurements of pipe diameters, assumed pipe line lengths and bends, heights of secondary water towers a.s.l. including assumptions of height of tanks on top of the towers, elevations of pipes where I calculate flow a.s.l. and of course known factors like viscosity, roughness of pipes etc. It is easy enough to calculate flow using basic formula for velocity and continuation equation, but I want to have more accurate result and include pressure losses. In these formulas hydraulic diameter is used instead of more normal diameter for circular pipes that are used today. Hydraulic diameter is calculated as following:

D = 4A/P whereD = hydraulic diameter, A = cross-sectional area, P = length of wetted contact line inside the pipe.

It is for this P I need to calculate perimeter of pear-shaped curve (in a side note if someone notice possible mistakes in my thought process let me know). Formula for area A is given in page I linked. Apart of coefficients for some fitting types in pipe line (and it is possible that these are impossible to get without experimenting) this is last piece of information I need before I can finalize my scripts for MatLab. I had to make many reasonable assumptions due the state of preservation in piping sytem and lacking documentation so good approximation is best I can hope for in this stage. I am very grateful for any help you can give me.

Thank you!

Television
Click to expand...

I forward the email to a friend who might have some ideas. Let me point you to a couple interesting links that may help:

http://math.stackexchange.com/questions/51539/a-math-function-that-draws-water-droplet-shape

http://mathworld.wolfram.com/Pear-ShapedCurve.html
 
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