Flow rate calculation of coating application

[bertcoen]

Dear all,

I'm currently working on a basic coating application and I'm struggling to get the flow rate stable.
I'm trying to figure out how to calculate the theoretical flow rate in the application.

The coating is stored in a pressure tank, which we put on an overpressure of 15psi.
From the pressure tank to the coating dispensing valve we use a flexible hose with an inner diameter of 1/8" (3,2mm).
The dispensing valve is a standard dispensing valve with a needle actuator which opens a certain set distance, creating an opening for the coating to spray out of the dispensing valve.

To calculate the flowrate (grams/volume per minute), I think I should use the Hagen-Poiseuille equation to calculate the pressure drop over the 1,5m flexible hose.
But how should I calculate the flow rate out of the dispensing valve?
Due to the angled needle actuator and the angled spray cap the opening is not a constant over a certain distance, so I believe the Hagen-Poiseuille equation doesn't apply. Maybe I should look into the Bernoulli equation?

Here's a quick sketch of the set up:

 

[Lnewqban]

That is very difficult to calculate.
Experimetation is your friend in this case.
The manufacturer of the valve may have some information regarding flow versus pressure.

 

[erobz]

You should be using the "Energy Equation", if the flow is approximately incompressible, steady (constant tank pressure - fixed valve setting), Newtonian fluid (approx. constant viscosity). You will need information about the valve and all other plumbing components. In practice it will be a bit of computation, but it still could be useful to make sure you are not completely undershooting/overshooting your target flow rate with the design.

If you are still interested and can fill in some of the missing details, we can get try to get through it here.

 

[bertcoen]

Thanks for the response.

I'm indeed into an experimental approach, but I wanted to understand my results a bit better, hence the theoretical counter calculations.

What I'd like to know are the relationships, and what theory they should be based on:

• Flowrate vs pressure = linear
  • this I'm sure of
• Flowrate vs radius of dispensing valve = quadratic?
  • The quadratic factor in my equations based on my experimental results is significant but only slightly.
  • I wonder what theoretical relationship there should be? And is this than based on Bernouilli or something else?
• Flowrate vs viscosity = linear or no influence?
  • Unfortunately the viscosity is a factor I can not really experiment with. Due to the reactiveness of my coating, it's also really hard to measure the actual viscosity. But I do know it's fluctuating, it should be, because something is influencing my flowrate over time.
  • I wonder what theoretical relationship there should be? Some of my data shows a linear effect but I'm not 100% sure this is due to a viscosity change or something I don't understand yet.

Just trying to understand what is going on, thanks for the help.

 

[erobz]

But I do know it's fluctuating, it should be, because something is influencing my flowrate over time.

Is the pressure in the tank constant, is the fluid depth varying significantly over time?

 

[bertcoen]

Hi Erobz,

Thanks a lot for the energy equation tip. I'll look a bit deeper into that, but it seems very related to Bernouilli.

The pressure in the tank, and the material pressure in the flexible hose just outside the tank is measured with a pressure gauge and we don't notice any fluctuations there.
I did several test were I only adapt the material pressure in the tank. This also gives me the pressure drop over the system. Which is pretty close to the pressure drop calculations I did with the Hagen-Poiseuille equation.

The fluid depth in the tank is only dropping very slowly, we only use 2-5% of the total tank volume each day. And the fluctuation I'm encountering is going up and down from day to day.

So even more specific. I did a test on a monday with different material pressures, I repeated a similar test on tuesday, no parameters changed intentionally, and for some reason the gradient of the graph changed.
Can this be related to viscosity change?
Can this be related to radius change (e.g. coating contamination)? This we tried to check and seems less likely.

 

[erobz]

Flowrate vs pressure = linear

• this I'm sure of

With a constant viscosity flow I wouldn't expect this. I would expect $Q \propto \sqrt{ P - c}$ in this system, but I'm most familiar with turbulent flow. I would have to look deeper into the effect of laminar flow (or someone else will chip in there).

 

[erobz]

So even more specific. I did a test on a monday with different material pressures, I repeated a similar test on tuesday, no parameters changed intentionally, and for some reason the gradient of the graph changed.
Can this be related to viscosity change?

How about tank substance settling?

 

[bertcoen]

That's indeed something that can be a possible cause, but we don't see changes in dry content when we see changes in material flow, which makes it slightly less realistic.

 

[erobz]

Temperature change? Moisture addition from the compressor?

 

[bertcoen]

Temperature change, but again that's viscosity change.
Is the viscosity relationship linear to the flow rate?

 

[Chestermiller]

Is the coating expected to be Newtonian, or is its viscosity expected to vary with shear rate?

Have you done any tests with a fluid of known constant viscosity, rather than your coating.

By what mechanism does the coating viscosity vary with time? Does it have to be in contact with air?

Do you have any idea what the nominal magnitude of the viscosity is?

 

[erobz]

Temperature change, but again that's viscosity change.
Is the viscosity relationship linear to the flow rate?

In believe in a Newtonian fluid the viscosity is independent of flowrate and pressure.

 

[erobz]

I believe a Newtonian fluid in the laminar flow regime we would expect $Q \propto P$. So it "appears" from this statement:

Flowrate vs pressure = linear

• this I'm sure of

We are dealing with Newtonian fluid.

What is changing the viscosity of the flow day to day, (if nothing else could have changed) is not related to the flow characteristics of the fluid. There seems to be a hidden uncontrolled external factor. IMO.

 

[bertcoen]

Is the coating expected to be Newtonian, or is its viscosity expected to vary with shear rate?

Have you done any tests with a fluid of known constant viscosity, rather than your coating.

By what mechanism does the coating viscosity vary with time? Does it have to be in contact with air?

Do you have any idea what the nominal magnitude of the viscosity is?

I do believe the coating is Newtonian indeed.

No other coatings with known constant viscosity have been tested in the same setup.

The curing mechanism of the coating is a humidity reactive system. So every possible unwanted contact with air we try to avoid.

Nominal dynamic viscosity is 1,0 Pa.s.

 

[bertcoen]

In believe in a Newtonian fluid the viscosity is independent of flowrate and pressure.

Ok, so with fluctuating viscosity I would get the same flowrate if other variables/factors remain the same? This feels counter intuitive?

 

[erobz]

Ok, so with fluctuating viscosity I would get the same flowrate if other variables/factors remain the same? This feels counter intuitive?

No, I mean this to be interpreted as this viscosity of the fluid doesn't care about the flow rate or the pressure. However, the flow rate does care about both the viscosity and the pressure.

 

[Chestermiller]

Hi Erobz,

Thanks a lot for the energy equation tip. I'll look a bit deeper into that, but it seems very related to Bernouilli.

The pressure in the tank, and the material pressure in the flexible hose just outside the tank is measured with a pressure gauge and we don't notice any fluctuations there.
I did several test were I only adapt the material pressure in the tank. This also gives me the pressure drop over the system. Which is pretty close to the pressure drop calculations I did with the Hagen-Poiseuille equation.

The fluid depth in the tank is only dropping very slowly, we only use 2-5% of the total tank volume each day. And the fluctuation I'm encountering is going up and down from day to day.

So even more specific. I did a test on a monday with different material pressures, I repeated a similar test on tuesday, no parameters changed intentionally, and for some reason the gradient of the graph changed.
Can this be related to viscosity change?
Can this be related to radius change (e.g. coating contamination)? This we tried to check and seems less likely.

View attachment 313854

I don't understand this graph. Is this done with an offline test to get higher pressure drops? Why is the flow rate zero when the pressure drop is 7 psi? If you have a rough estimate of the viscosity, what do you predict for the pressure drop/flow rate relationship of the hose alone?

My game plan would be to consider the tube and the valve separately.

 

[bertcoen]

The graph is the relationship between the pressure we put on the tank and the flow rate (the weight out of the dispensing valve in 30 seconds). The pressure drop of 7psi is indeed the pressure drop over the hoses from the tank to the dispensing valve. If I calculate this theoretically with Hagen Poiseuille I get very close to the 7psi, so that makes sense to me.

The blue graph is the test we did on a monday a while back, so we put the pressure of the tank on different pressures and measured the flow rate. On tuesday we did the same, kept everything the same, but the graph has a different gradient.
Why that is, is what I'm trying to figure out.
My best guess before posting on this forum was viscosity change (due to temperature, slow precuring in material supply, ...).
That's why I'm wondering if there should be a theoretical relationship between the flow rate and the viscosity, and if so is it a linear behavior and on what theory is it based?

Thanks for the help!

 

[bertcoen]

No, I mean this to be interpreted as this viscosity of the fluid doesn't care about the flow rate or the pressure. However, the flow rate does care about both the viscosity and the pressure.

Ok that makes more sense :).

 

[erobz]

The pressure drop of 7psi is indeed the pressure drop over the hoses from the tank to the dispensing valve. If I calculate this theoretically with Hagen Poiseuille I get very close to the 7psi, so that makes sense to me.

Something is inconsistent. There must be flow to generate a differential pressure across the hose?

EDIT: unless you have a $7 \, \rm{psi}$ static elevation head, but you would read a $0 \, \rm{psi}$ at the nozzle in that case.

 

[Chestermiller]

The graph is the relationship between the pressure we put on the tank and the flow rate (the weight out of the dispensing valve in 30 seconds). The pressure drop of 7psi is indeed the pressure drop over the hoses from the tank to the dispensing valve. If I calculate this theoretically with Hagen Poiseuille I get very close to the 7psi, so that makes sense to me.

You get 7 psi for a flow rate of zero? Not likely. In these calculations, what value of viscosity did you use?

The blue graph is the test we did on a monday a while back, so we put the pressure of the tank on different pressures and measured the flow rate. On tuesday we did the same, kept everything the same, but the graph has a different gradient.
Why that is, is what I'm trying to figure out.
My best guess before posting on this forum was viscosity change (due to temperature, slow precuring in material supply, ...).

Was the valve setting constant in all cases, or did it change? What is the temperature dependence of the viscosity, and is the observed change in flow rate consistent with this?

That's why I'm wondering if there should be a theoretical relationship between the flow rate and the viscosity, and if so is it a linear behavior and on what theory is it based?

Thanks for the help!

Even for just the hose, the pressure drop will be a function of viscosity.

Are you doing tests by varying the valve opening (measured by stem height)?

 

[bertcoen]

You get 7 psi for a flow rate of zero? Not likely. In these calculations, what value of viscosity did you use?

Was the valve setting constant in all cases, or did it change? What is the temperature dependence of the viscosity, and is the observed change in flow rate consistent with this?

Even for just the hose, the pressure drop will be a function of viscosity.

Are you doing tests by varying the valve opening (measured by stem height)?

1. The graphs are the actual results of a test, so by extrapolation of the trendline I get 7psi which than indicates the pressure drop. Or is this not correct? If not, why not?

2. A. Valve setting was the same in the test out of which we made that graph.
2. B. The viscosity - temperature relationship we don't know yet.

3. Here are the results of a varying valve opening versus the flow rate.

 

[bertcoen]

But I'm most interested in the theoretical relationships which should apply?

1. Flow rate versus pressure
2. Flow rate versus viscosity
3. Flow rate versus radius = opening of valve

 

[erobz]

Is there a static elevation head between the fluid level in the tank and the nozzle jet?

 

[Lord Jestocost]

The coating is stored in a pressure tank, which we put on an overpressure of 15psi.

Is the "coating" a stabilized suspension?

 

[erobz]

For laminar flow (flow regime determined by computing the Reynolds Number) you would get something to the effect of:

$$ \frac{P}{\gamma} = z + \left( \sum_{\rm{plumbing}} \frac{32 \mu L}{ \gamma D^2 A} \right) Q + h_{nozzle} ( Q ) $$

$P$ is tank pressure - gauge
$Q$ is the volumetric flowrate
$D$ is the tube\plumbing diameter(s)
$L$ is the length of tube\plumbing with diameter $D$
$z$ is the static elevation head between free liquid surface and nozzle jet
$\gamma$ is the specific weight of the fluid
$\mu$ is the viscosity at operating Temperature
$A$ is the cross sectional area of the tube\plumbing components
$h_{nozzle} ( Q )$ is the head loss of the nozzle as a function of flowrate at any particular setting which would be given by the manufacturer.

 

[Chestermiller]

For your hose, the Reynolds number is give by $$Re=\frac{4\dot{m}}{\pi D \mu}$$where $\dot{m}$ is the mass flow rate, D is the inside diameter, and $\mu$ is the coating viscosity. For a mass flow rate of 0.12 gm/sec (3.6 gm in 30 seconds), a hose diameter of 0.32 cm, and a viscosity of 10 Poise (1.0 Pa-s), the Reynolds number is $$Re=\frac{(4)(0.12)}{\pi(0.32)(10)}=0.048$$According to this, the flow in the hose should be Laminar under all test conditions.

The shear rate at the wall at this flow rate (assuming a density of 1 gm/cc) will be given by $$\gamma=\frac{32Q}{\pi D^3}=\frac{(32)(0.12)}{\pi (0.32)^3}=37\ sec^{-1}$$The shear stress at the wall would then be $$\tau=\mu \gamma=(37)(10)=370\ dynes/cm^2$$The pressure drop due to flow in the hose would then be $$\Delta p=\frac{4L}{D}\tau=\frac{4(150)}{0.32}\tau=693750\ dynes/cm^2=10.1\ psi$$So, if M is the cumulative mass of coating in 30 seconds, the pressure drop in the hose should be given by $$\Delta p=\frac{m}{3.6}(10.1)=2.8 M\ psi$$

 

[erobz]

For your hose, the Reynolds number is give by $$Re=\frac{4\dot{m}}{\pi D \mu}$$where $\dot{m}$ is the mass flow rate, D is the inside diameter, and $\mu$ is the coating viscosity. For a mass flow rate of 0.12 gm/sec (3.6 gm in 30 seconds), a hose diameter of 0.32 cm, and a viscosity of 10 Poise (1.0 Pa-s), the Reynolds number is $$Re=\frac{(4)(0.12)}{\pi(0.32)(10)}=0.048$$According to this, the flow in the hose should be Laminar under all test conditions.

The shear rate at the wall at this flow rate (assuming a density of 1 gm/cc) will be given by $$\gamma=\frac{32Q}{\pi D^3}=\frac{(32)(0.12)}{\pi (0.32)^3}=37\ sec^{-1}$$The shear stress at the wall would then be $$\tau=\mu \gamma=(37)(10)=370\ dynes/cm^2$$The pressure drop due to flow in the hose would then be $$\Delta p=\frac{4L}{D}\tau=\frac{4(150)}{0.32}\tau=693750\ dynes/cm^2=10.1\ psi$$So, if M is the cumulative mass of coating in 30 seconds, the pressure drop in the hose should be given by $$\Delta p=\frac{m}{3.6}(10.1)=2.8 M\ psi$$

I'm getting $10.47 \, \rm{psi}$ using mathcad with the formula I quoted above for the hose.

 

[Chestermiller]

I'm getting $10.47 \, \rm{psi}$ using mathcad with the formula I quoted above for the hose.

View attachment 313862

It must be due to my rounding off the diameter to 0.32 cm vs 0.3175 cm

 

[Tom.G]

curing mechanism of the coating is a humidity reactive system.

Two possible ways I see this could cause your variations:

1) The spray nozzle is not being cleaned at end-of-shift and the residual compound is curing in it during the night.
2) Air used to pressureize the tank is moist causing partial curing. Air right out of a compressor is notoriously wet.

#1 has an obvious solution.
#2 As a test, get a tank of Dry Nitrogen with appropriate pressure regulator. Purge the reservoir,hose, nozzle, etc. and try that for a few days/weeks.

If #2 solves the problem, consider a dryer for the air line (expensive), perhaps a dessicant canister (periodic maintenance required) in the air line, or continue with the Dry Nitrogen tank.

Cheers,
Tom

p.s. As a last resort, farm out the coating process to someone else with a well-written/comprehensive quality requirement. The downside (beyond cost) is the incoming quality-control inspections needed.