Maths/Physics/Engineering calculation problem - can you check my maths please

[saltynay]

I am building a fluid bed filter which basically provides the maximum surface area for bacterial colonisation which is chemical filtration for an aquarium. The shape of this will be a reverse cone and the sand needs to be kept in a constant state of suspension to allow for the optimum flow of water and surface area of the sand. The water pump I will be using is 670-700 litres per hour.

Maths bit
Simple key to my algebra:
x = unknown value or multiply
r = radius
a = area
d = diameter

Work through of problem:
So I have found that I need 1.13m/min flow to lift 1mm sand which is standard medium sized blasting sand. I also know 2000 litres per hour lifts 0.033 cubic metres a minute of sand, so I converted this to 650L/h as an understatement of my pump to account for pipe resistance for the flow. So if 650 = 2000/x then 2000/650 = x, x = 3.0769230769231, so if I apply this to the original known value 2000/x = 0.033/x this gives me 0.010725 cubic metres a minute is lifted by my pump.

So combine this with the water velocity required to lift the sand we get 0.010725 x 1.13 = 0.01211925 so to suspend the sand my cone must have a cross sectional area at the bottom of less then 0.01211925 cubic metres.

So to convert this to a circle I used a=pi r^2, a/pi = 0.00385767709 then re transferred to the previous equation 0.00385767709^(1/2) = r, r = 0.0621102012, d = 2(0.0621102012), d = 0.124220402 x 100 (this value is in m so convert to cm), d = 12.4220402cm so the base of my cone must be smaller then this value for sufficient lift and wider then this value to allow for sufficient dropping of the particles so that they are not blown out of the top.

Can you please check through that this is correct as I have a gut feeling that it is wrong somewhere.

 

[saltynay]

Shame where are all the maths geeks tonight?

 

[oldskool]

Salty, when I get a chance later I will work on this, if Luke doesn't get to it first. He sounds like a whiz kid at math. Anyway, I will check this later when I can.

EDIT: Salty, as far as your math goes, it looks good, but I have a question about this :

so if I apply this to the original known value 2000/x = 0.033/x this gives me 0.010725 cubic metres a minute is lifted by my pump.

The part I am trying to understand is how did you come up with the 0.010725 number ? Up to and after that it all looked good to me. I just can't figure that particular section.

 

[saltynay]

I know the set figure that a 2000 Litres per hour pump will lift 0.033 cubic metres of sand into a suspended state. I have a 670-700L/h pump so I know that 2000=0.033 and so allowing an estimate of pipe resistance and external variables I understate my pump to 650L/h, 650L/h = 2000L/h divided by x. I then re-arranged this equation to get x, so x = 2000/650 = 3.0769230769231, now that I know what my added variable is, I am able to convert the original equation 2000=0.033, so 2000/x = 0.033/x as they are both divided by the same number they are both reduced at the same ratio and 2000/x = 650 and 0.033/x = 0.010725 therefore I know that 650L/h = 0.010725 cubic metres of sand lifted.

 

[patonb]

It looks okay BUT watch your sig figs. "3.0769230769231" doesn't exist in real life, as "0.033" biggest value.

 

[saltynay]

I am going to be working from rounded down anyway to be on the safe side. Here is the plan stand made 100mm tall from metal of some sort, filled with polystyrene for noise reduction as it will disguise the pump. Then funnel (made from acrylic) converting the 10mm pump outlet into a 50mm containing a diffusion plate to spread the flow relatively evenly which will be 40mm tall and wrapped in the same metal as the stand with a couple of leds and an airline to provide extra oxygen to the water flow for the bacteria growth. The cone which will be 210mm tall, 50mm at bottom and 150mm at exit made from transparent acrylic and the leds will make this area glow as the sand is thrown around within the reactor. The cone will then have a funnel joined which will convert the 150mm to a 15mm line back up to the tank. Check valves will be placed before and after the device and on the airline so that it doesn't create a leak line.

Anyone have a lava lamp with metal stand that they could quickly measure up? I think if I used the frame for one it would be easier to make my reactor stand provide I can find one with right dimensions or close to.