EPG slurry

What does this mix of carrier, metal and surfactant weigh, how thick is it, does it flow smoothly at this pump rate, and what does it do to the water's permittivity?

The parts

Water optional

The carrier's base water. Sets the permittivity and the temperature the viscosity is taken at. About this part.

The drive

c Carrier fluid

The liquid the particulate is carried in.

m Metal

The particulate.

φ Metal loading

The metal's share of the volume.

s Surfactant

The surfactant's share of the volume.

D Tube inner diameter

The bore the slurry is pumped through.

Q Flow rate

L/min

What the pump delivers.

From the parts

Every number handed to the simulation that you did not type, and where it came from. Each derivation that ran a calculation links to that calculation's page with these numbers in, so it can be checked alone.

w Water profile 1

Filtered Water, as the engine's nearest named water.
T Temperature 22 °C

Filtered Water's recorded temperature.

Result

ρ Slurry density 1.342 g/cm³

What a litre weighs.

η Slurry viscosity 1.143 mPa·s

How thick it is.

Re Reynolds number 36888

Under about 2300 it flows in layers; over about 4000, in eddies.

v Mean velocity 1.256 m/s

How fast it moves down the tube — against the holding's 50 in/s for slurry.

ε Effective permittivity 90.17

The water's dielectric constant with the metal in it.

With your numbers

36888 = \frac{1.342 g/cm³ 1.256 m/s 25 mm}{1.143}

LaTeX

36888 = \frac{1.342\,\mathrm{g/cm³} \cdot 1.256\,\mathrm{m/s} \cdot 25\,\mathrm{mm}}{1.143\,\mathrm{mPa·s}}

Worth knowing

Flow regime in the tube — Re = 36888: turbulent. The medium is mixing as it moves, and the pressure the pump needs rises roughly with the square of the flow rather than linearly.

This result is a link — the address bar holds your numbers, so it can be pasted into a post and opened to the same answer.

The working, step by step

1. Slurry density
ρ 1.342 g/cm³

What the mix weighs — the volume-weighted density.
With these numbers

$1.342 g/cm³ = (1 - 5 % - 0.5 %) 998.2 kg/m³ + 5 % 7.874 g/cm³ + 0.5 % 1.05 g/cm³$
LaTeX
```
1.342\,\mathrm{g/cm³} = (1 - 5\,\mathrm{%} - 0.5\,\mathrm{%}) \cdot 998.2\,\mathrm{kg/m³} + 5\,\mathrm{%} \cdot 7.874\,\mathrm{g/cm³} + 0.5\,\mathrm{%} \cdot 1.05\,\mathrm{g/cm³}
```
Open this step on its own page, with these inputs
2. Slurry viscosity
η 1.143 mPa·s

How much the particulate thickens the carrier.
With these numbers

$1.143 mPa·s = 1.002 mPa·s (1 + 2.5 5 % + 6.2 5^{% 2})$
LaTeX
```
1.143\,\mathrm{mPa·s} = 1.002\,\mathrm{mPa·s} \left( 1 + 2.5 \cdot 5\,\mathrm{%} + 6.2 \cdot 5\,\mathrm{%}^2 \right)
```
Open this step on its own page, with these inputs
3. Flow regime in the tube
Re 36888

With that density and viscosity through this bore at this rate: laminar or turbulent, and how fast.
With these numbers

$36888 = \frac{1.342 g/cm³ 1.256 m/s 25 mm}{1.143}, 1.256 m/s = \frac{4 0.000617 m³/s}{π 25^{mm 2}}$
LaTeX
```
36888 = \frac{1.342\,\mathrm{g/cm³} \cdot 1.256\,\mathrm{m/s} \cdot 25\,\mathrm{mm}}{1.143\,\mathrm{mPa·s}}, \qquad 1.256\,\mathrm{m/s} = \frac{4 \cdot 0.000617\,\mathrm{m³/s}}{\pi \cdot 25\,\mathrm{mm}^2}
```
- Re = 36888: turbulent. The medium is mixing as it moves, and the pressure the pump needs rises roughly with the square of the flow rather than linearly.
Open this step on its own page, with these inputs
4. Permittivity of a metal-loaded fluid
ε 90.17

What the metal does to the water's permittivity — the number the cell calculations would use instead of the water's own.
With these numbers

$90.17 = 77.87 \frac{1 + 2 5 %}{1 - 5 %}$
LaTeX
```
90.17 = 77.87 \cdot \frac{1 + 2 \cdot 5\,\mathrm{%}}{1 - 5\,\mathrm{%}}
```
Open this step on its own page, with these inputs

What this looks like

Reynolds number against flow rate Flow rate swept from 18.5 L/min to 55.5 L/min with everything else held at your numbers. The dashed lines cross where you are.

The formula behind the curve

Re = \frac{ρ v D}{η}

Re: Reynolds number
ρ: Slurry density, kg/m³
v: Mean velocity, m/s
D: Tube diameter, m
η: Slurry viscosity, Pa·s

LaTeX

Re = \frac{ρ \cdot v \cdot D}{η}

What moves the answer Each input moved 10% either way, with the others held still, and the effect on reynolds number.

The formula behind the curve

Re = \frac{ρ v D}{η}

Re: Reynolds number
ρ: Slurry density, kg/m³
v: Mean velocity, m/s
D: Tube diameter, m
η: Slurry viscosity, Pa·s

LaTeX

Re = \frac{ρ \cdot v \cdot D}{η}

Open the bare numbers — the same simulation on its own page, every derived value editable.

EPG slurry

The parts

The drive

From the parts

Result

Worth knowing

The working, step by step

1. Slurry density

2. Slurry viscosity

3. Flow regime in the tube

4. Permittivity of a metal-loaded fluid

What this looks like