u/Anxious_Strike_2931

Non-newtonian cylindrical flow

(Graduate) (ChemE) (Transport) (Fluid mech)

Solving cauchy equation for non newtonian fluid in cylindrical coordinates. I am finding the minimum radius where flow occurs which is the condition where T = T0

Rheology is given by: Sqrt(T)=Sqrt(T0)+eta*Sqrt(dv/dr)

Cauchy after initial simplification (dv/dtheta =0, steady state, etc.) Reduces to

dP/dr = -((1/r)d/dr(rT) + pg

Here's where I get absolutely lost and end up off by a factor of 2. I have a constant expression for dP/dr (combined with the constant pg term I'll call it alpha) and am allowed to substite it in just fine. But if I assume T to be constant T0 (which I believe is fine since that is what my prof plugs in at a later time than I) and plug it in, I get:

1. alpha = -(1/r)d/dr(rT0)

alpha = -(1/r)T0 (since d/dx of cx is c)

T0/(-alpha) = r

The problem is that the answer is off by a factor of 2. I'm apparently supposed to keep T as an unknown and solve like this:

2. alpha = -(1/r)d/dr(rT)

-alpha*r = d/dr(rT)

int -alpha*r dr = int d(rT) (treat it as a 1st order ode, from a BC I know the constant of integration is 0)

-alpha (r^2/2)= rT

(-alpha*r)/2 = T

Then I plug in knowing T is a constant T0 to get

r= (2*T0)/(-alpha)

What fundamental general math rule/not so obvious assumption is broken by doing it the way I did? I'm fine plugging dP/dr in early but can't for the embedded T? Why? From what I see I should have no problem evaluating what d/dr(rT0) is but it's wrong.

It's breaking my reality of math because outside of order of operations math should always work out to the same answer regardless of how you got there. Apparently there must be a rule for when you are plugging things in since I'm off by 2 otherwise.

I want math to be fluent for me, not an algorithm, and I keep being tripped up by these seemingly unspoken rules.

Solving cauchy equation for non newtonian fluid in cylindrical coordinates. I am finding the minimum radius where flow occurs which is the condition where T = T0

Rheology is given by: Sqrt(T)=Sqrt(T0)+eta*Sqrt(dv/dr)

Cauchy after initial simplification (dv/dtheta =0, steady state, etc.) Reduces to

dP/dr = -((1/r)d/dr(rT) + pg

Here's where I get absolutely lost and end up off by a factor of 2. I have a constant expression for dP/dr (combined with the constant pg term I'll call it alpha) and am allowed to substite it in just fine. But if I assume T to be constant T0 (which I believe is fine since that is what my prof plugs in at a later time than I) and plug it in, I get:

1. alpha = -(1/r)d/dr(rT0)

alpha = -(1/r)T0 (since d/dx of cx is c)

T0/(-alpha) = r

The problem is that the answer is off by a factor of 2. I'm apparently supposed to keep T as an unknown and solve like this:

2. alpha = -(1/r)d/dr(rT)

-alpha*r = d/dr(rT)

int -alpha*r dr = int d(rT) (treat it as a 1st order ode, from a BC I know the constant of integration is 0)

-alpha (r^2/2)= rT

(-alpha*r)/2 = T

Then I plug in knowing T is a constant T0 to get

r= (2*T0)/(-alpha)

What formal fundamental general math rule/not so obvious assumption is broken by doing it the way I did? I can plug in dP/dr as a constant just fine but not for the embedded T. Why?

It's breaking my reality of math because outside of order of operations math should always work out to the same answer regardless of how you got there. Apparently there must be a rule for when you are plugging things in since I'm off by 2 otherwise.

I want math to be fluent for me, not an algorithm, and I keep being tripped up by these seemingly unspoken rules.

Hello, I am a molten salt researcher and my reactors run at ~1050 C. I can't look inside my reactor as it operates sealed, but I am attempting to differentiate metal droplets that may be floating on the surface from the molten salt using viewing ports, or briefly opening it up and using mirrors to observe from a distance.

The inside of my reactors glow orange at this temperature and as of now, even with them opened up, it is difficult to know if I am looking at anything interesting as they are bright. I hope to provide contrast and make it easier to see if anything is changing at the liquid surface.

Ideas I have:

1. Two opposing viewing ports, at an angle, cut into the sealed reactor so that I can observe the surface of the liquid. One port will have a light source shining into the reactor, and the other is a viewing port where I intend to filter out different colors of light using colored glass, and reduce the total amount of light coming through to enhance contrast. From what I understand, green light is most visible to the human eye and is far enough from the orange glow that I can safely try to filter all light that is not close to the green wavelengths. It seems reasonably easy to get a high power light source for reasonable prices (even just a laser that is incredibly out of focus).

How well this would work to create contrast, I am not sure, but I intend to take advantage of the metal's reflective nature. I have thought of blue light as well but notice that blue/black lights tend to have this fuzziness associated with it, likely making picking out details too hard for me.

2. I have a cheap digital camera and could attempt to remove the IR filter from the lens. Then filter out the visible light with multiple thin black trash bags which should allow a lot of IR through, but little visible light. Then I would add layers and layers of glass (should block IR) until I have achieved enough dimming to hopefully see some contrast between metal and salt. The idea being that they have different emissivity therefore radiating different amounts of IR. I am aware of filters but the budget is low so those may break the budget quickly if I'm not right about what wavelength light is best.

3. Welding goggles. While simple, I believe most of my light is closer to red-orange than blue-UV which I believe welding goggles tend to be more geared towards.

4. I can also attempt to briefly open the reactor mid-operation and use mirrors to observe it at a distance with whatever contrasting method that is suitable. I do believe this would be more difficult to see a clear image and also believe this may be an issue as white smoke can no longer be blown away as easily compared to when the reactor is sealed.

I am open to any other ideas as this is well outside of my breadth of expertise. The budget is fairly low (max 50-150 bucks for everything) as this is just an idea I want to try to make qualitative analysis easier and wouldn't necessarily get direct funding.

Hello, I am a molten salt researcher and my reactors run at ~1050 C. I can't look inside my reactor as it operates sealed, but I am attempting to differentiate metal droplets that may be floating on the surface from the molten salt using viewing ports, or briefly opening it up and using mirrors to observe from a distance.

The inside of my reactors glow orange at this temperature and as of now, even with them opened up, it is difficult to know if I am looking at anything interesting as they are bright. I hope to provide contrast and make it easier to see if anything is changing at the liquid surface.

Ideas I have:

1. Two opposing viewing ports, at an angle, cut into the sealed reactor so that I can observe the surface of the liquid. One port will have a light source shining into the reactor, and the other is a viewing port where I intend to filter out different colors of light using colored glass, and reduce the total amount of light coming through to enhance contrast. From what I understand, green light is most visible to the human eye and is far enough from the orange glow that I can safely try to filter all light that is not close to the green wavelengths. It seems reasonably easy to get a high power light source for reasonable prices (even just a laser that is incredibly out of focus).

How well this would work to create contrast, I am not sure, but I intend to take advantage of the metal's reflective nature. I have thought of blue light as well but notice that blue/black lights tend to have this fuzziness associated with it, likely making picking out details too hard for me.

2. I have a cheap digital camera and could attempt to remove the IR filter from the lens. Then filter out the visible light with multiple thin black trash bags which should allow a lot of IR through, but little visible light. Then I would add layers and layers of glass (should block IR) until I have achieved enough dimming to hopefully see some contrast between metal and salt. The idea being that they have different emissivity therefore radiating different amounts of IR. I am aware of filters but the budget is low so those may break the budget quickly if I'm not right about what wavelength light is best.

3. Welding goggles. While simple, I believe most of my light is closer to red-orange than blue-UV which I believe welding goggles tend to be more geared towards.

4. I can also attempt to briefly open the reactor mid-operation and use mirrors to observe it at a distance with whatever contrasting method that is suitable. I do believe this would be more difficult to see a clear image and also believe this may be an issue as white smoke can no longer be blown away as easily compared to when the reactor is sealed.

I am open to any other ideas as this is well outside of my breadth of expertise. The budget is fairly low (max 50-150 bucks for everything) as this is just an idea I want to try to make qualitative analysis easier and wouldn't necessarily get direct funding.