Latest / Mechanical Engineering Made Simple / (#96) Gas Laws Fluid Flow and Thermal Management

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(#96) Gas Laws Fluid Flow and Thermal Management · Mechanical Engineering Made Simple

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  Welcome back, design junkies, to Mechanical Engineering Made
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  Simple. Today we are opening what is
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  probably the most essential chapter in any engineering
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  handbook. We're talking about the
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  fundamental laws governing flow, pressure and thermal transfer.
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  These are the invisible forces, pneumatics, the incredible power
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  density of hydraulics, safe pressure, vessel design, and of
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  course, critical thermal management.
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  Yeah, and this deep dive is really our shortcut past all the
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  introductory definitions. Our mission here is to take the
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  source material and show you the key, you know, the non
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  negotiable relationships, the equations that dictate how air
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  compresses, how fluid delivers that mechanical muscle, and
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  crucially, where all the heat goes, right.
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  If you were designing anything that moves or holds pressure or
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  needs cooling, these core physics relationships are well,
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  they're your foundation. Absolutely.
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  So if we're sketching A robotic arm, sizing a massive
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  compressor, or designing a cooling loop for a server bank,
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  these are the physics we just cannot violate.
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  We'll start with the invisible workhorse air and gases.
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  Then we'll get into liquid power, move to the safety
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  calculations for air handling and pressurized vessels, and
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  then finish up with thermal dynamics.
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  OK, let's unpack this, starting with the gas law.
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  So pneumatics, it's the study of air and gases under pressure.
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  And it all starts with defining a standard A.
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  Baseline, yeah. When we talk about standard
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  atmospheric air, we're talking about very specific reference
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  conditions, 32°F and an atmospheric pressure of 14 7/10
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  PSI I at sea level. Which you'll also see expressed
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  as 29 and 92 hundredths inches of mercury.
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  Exactly. And that standard sets the
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  baseline for the weight of air, which is about 8 and £73,000 per
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  cubic foot. But the real insight, the thing
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  that connects it all, is the equation of state.
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  Foundational relationship. It's pressure V / T = 53 and 33
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  hundredths. Now wait for the listeners
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  tracking along with the source material.
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  This is a critical moment in this specific equation.
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  We are not using the units we see everyday.
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  No, you're not. P here is absolute pressure in
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  pounds per square foot, not pounds per square inch.
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  V is the specific volume per pound of error and T is absolute
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  temperature in rankings. Why are we using pounds per
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  square foot in rankings here? I mean, isn't that a commonplace
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  for mistakes? It is the most common beginner
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  mistake. Yeah, mixing up pounds per
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  square inch. Absolute Asia with the
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  requirement for pounds per square foot A's fee in this
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  specific gas constant application, the value that 53
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  and 33 hundredths it's derived specifically for the units used
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  in the US standard system. OK.
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  And using rankings, which is just Fahrenheit plus 459 and 67
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  hundredths keeps the math simple because it starts at 20
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  temperature, which is essential when you're analyzing the
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  volumetric behavior of A. Gas, And this is where getting
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  those units right really pays off.
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  Understanding compression efficiency.
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  The contrast between adiabatic and isothermal compression is, I
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  mean, it's vital for anyone signing off on a massive
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  compressor purchase. Oh, this is a huge financial and
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  physical design decision. Eightyback compression happens
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  without any heat transfer to or from the air during the process,
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  right? You put the energy in, the
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  pressure goes up, and the temperature just absolutely
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  skyrockets. It's fast, it's inefficient, and
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  it requires robust large components to handle that heat
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  and pressure surge. And isothermal compression is
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  the theoretical Unicorn it is. It's a compression that happens
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  while transmitting sufficient heat away to maintain a constant
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  temperature. Precisely, and the physics is
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  clear. The energy requirements for
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  isothermal compression are considerably less than for
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  adiabatic. In the real world, every
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  compressor you design is trying to get as close to isothermal as
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  possible. You know, with cooling jackets,
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  inner coolers after coolers. So the cooling system can be
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  just as important as the compressor itself.
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  Oh yeah, the cooling system often dictates the cost and size
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  of the compressor package just as much as the piston or the
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  scroll mechanism. That's a fantastic transition.
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  If adiabatic compression generates all this massive
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  wasted heat, let's see how efficiently we can transfer
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  power by minimizing waste in the liquid realm.
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  Let's talk about liquid muscle. Basic hydraulics.
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  First things first, we need to nail down those pressure
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  notations. Again, we discussed absolute
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  pressure, but most people work with sig pounds per square inch
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  gauge. That's just what your local
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  gauge reads above the ambient air pressure.
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  Right. And C absolute is always gauge
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  pressure plus that ambient atmospheric pressure, which we
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  said is usually 14 7/10 PSI PSI at sea level.
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  Exactly. So when dealing with hydraulic
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  pumps, the power required to drive it gives us a really clear
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  metric for efficiency. The horsepower is calculated as
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  P multiplied by GPM divided by 1714 times E, where E is the
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  pump's efficiency. That looks like a bit of a mess
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  of constants, but what's the quick take away for a designer?
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  If I'm just starting to spec out a system?
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  What's the rule of thumb? The quick approximation is
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  incredibly useful, actually. For every one unit of horsepower
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  you put in, you can get the equivalent of 1 gallon per
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  minute at 1500 PSI. Wow, that's a powerful metric.
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  It is a 20 horsepower unit will comfortably drive 20 GPM at that
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  1500 the PSI rating. That kind of power density is
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  why hydraulics just dominates applications like heavy
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  construction equipment or industrial presses.
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  And the simple pay off of that pressure comes when the fluid
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  acts on a cylinder. The thrust force T sub F is just
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  a piston area a sub 1 multiplied by the pressure P.
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  It's direct. A super direct.
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  It's the application of pressure right into mechanical work.
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  And that ties directly into flow speed.
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  The piston speed S sub P relates the oil flow C sub M and cubic
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  inches per minute directly to the piston area A.
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  So you control the speed of the output by controlling the flow
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  rate into the cylinder. Simple as that.
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  Yeah, but the smooth operation of that system depends entirely
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  on how the fluid itself is behaving inside the pipes.
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  Which brings us to one of the most important dimensionless
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  numbers in engineering. The Reynolds number.
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  The Reynolds number. It's the ratio of inertial
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  forces to viscous forces. It's the physics of what happens
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  when you push thick syrup versus, say, water through a
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  pipe. It's calculated as the product
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  of DV and rho all divided by MU. So that's pipe diameter,
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  velocity and density divided by the dynamic viscosity.
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  You got it. And this is really the So what
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  moment for any pipe and valve design we hit a threshold at
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  2000. If the Reynolds number is less
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  than 2000, you have smooth, predictable, low loss laminar
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  flow. It's all very orderly.
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  But if our goes over 4000, you have fully chaotic, high loss,
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  unpredictable, turbulent flow. And that's bad.
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  Oh, it's very bad. That turbulence causes massive
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  friction losses, erosion, inefficiency.
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  Every designer spends their time trying to keep their system out
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  of that chaotic zone, which usually just means lowering flow
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  speed or increasing the pipe diameter.
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  And that band between 2004 thousand, that's the
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  transitional zone. You just don't want to be there.
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  You don't. And since viscosity is such a
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  major part of this equation, we have to mention the headache of
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  measurement. Oh yeah, we see poise sent to
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  poise or CP. There's the Krebs unit, which
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  you see a lot in paints, and of course Sable Universal.
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  The variety is a legacy issue, but you have to know which one
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  you're dealing with before you order the fluid or apply any of
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  these equations. You'll be referencing conversion
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  charts constantly if you work across different industries.
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  Right. And we should briefly note the
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  Manning equation is used primarily for calculating water
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  flow, especially in open channels, but also in some
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  closed pipes. OK, Now let's pivot sharply from
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  the flow inside the pipes to the containment of that pressure.
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  This is a critical safety component designing pressurized
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  vessels. Absolutely.
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  We've established that internal pressure exerts an equal force
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  per unit area on the walls, but how you calculate the stress
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  depends entirely on the thickness of the metal relative
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  to the diameter. And that relationship that
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  dictates safety. We basically have three main
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  stress equations for low pressure thin walled cylinders
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  where the wall thickness is small compared to the diameter.
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  Stress S equals the inside diameter multiplied by P / 20
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  where it is wall thickness. OK, that's straightforward.
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  But once the pressure rises and the wall thickness becomes
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  significant, the stress distribution changes completely.
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  It does for high pressure thick walled cylinders.
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  The simplified equation changes to stress S equals the outside
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  diameter multiplied by P / 2 T. So the reference shifts from the
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  inside to the outside diameter. That's a crucial shift.
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  It is. And if you're dealing with ultra
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  high pressure things, something approaching, I don't know, deep
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  sea submersible hull or industrial synthesis vessels,
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  you have to move to Lame's equation and.
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  That one's much more complex. Far more complex because it uses
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  the ratio of the squares of the outside radius R and the inside
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  radius R It really emphasizes how dramatically stress
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  concentrates near that inner wall at extreme pressures.
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  And the structural design doesn't stop at the walls
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  cylinder ends. They're normally made
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  hemispherical or curved. Right.
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  And that's done very precisely so that the stress exerted on
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  the ends is equal to or even less than the stress in the
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  cylindrical wall. You eliminate a weak point by
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  distributing the force smoothly. Exactly.
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  So moving to industrial environments, let's talk about
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  air handling, the controlled movement of large volumes of
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  air, OK, We all know CFM cubic C per minute, that's the volume
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  flow rate. But when we talk about forcing
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  air through ductwork, we hit static pressure FTP this.
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  Is the measure of resistance. It's caused by elbows, filters,
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  registers, anything in the way, and it's measured in inches of
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  water gauge or WG. And what's truly fascinating and
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  often really expensive is the power relationship.
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  Here. Static pressure varies is the
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  square of the flow rate. Yeah, that's a big one.
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  You double your required flow rate.
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  Your static pressure resistance quadruples.
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  And on the human factor side, we have the sound rating right?
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  Unlike the decibel scale, which is logarithmic, the sound scale
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  is a linear unit of perceived sound loudness.
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  So if something is rated at six sounds, it is perceived as twice
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  as loud as 3 sounds. Which is invaluable when you are
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  designing a system for, you know, a place where people
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  actually have to work. Totally.
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  Now let's talk about the infamous air moving law,
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  commonly known as the cube law for horsepower.
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  Ah yes, this relationship governs how dramatically the
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  required input power increases with flow rate.
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  This is the equation that catches companies out when they
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  try to retrofit or upgrade ventilation without enough
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  analysis. The new horsepower required is
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  equal to the cube of the ratio of the new flow rate to the
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  existing flow rate multiplied by the existing horsepower.
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  So why is it Q back? Right, that dramatic jump.
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  Because you are fighting friction, velocity, and pressure
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  all at the same time, You're moving a larger mass of air,
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  which is a linear increase. You're moving it faster.
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  That's a squared increase, and you're overcoming exponentially
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  greater resistance that static pressure squared.
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  The power requirement, it just explodes.
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  It does. Let's run the example from the
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  source material because it illustrates this perfectly.
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  Say you have a system moving 7000 cubic feet per minute and
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  it runs on A1 horsepower motor. OK.
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  But your new safety requirements demand you increase the flow to
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  10,000 cubic feet per minute? So the ratio of the flows is
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  about 1 and 429 thousandths. When you cube that ratio, the
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  required new horsepower doesn't just go up by 40%, it jumps to 2
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  and 9200. So basically 3 horsepower.
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  So simply increasing the flow by about 40% almost triples the
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  power requirement and it triples the associated energy costs.
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  This is why air handling design needs to be perfect from day
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  one. And these laws dictate things
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  like industrial air quality standards for design.
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  A typical office needs an air change every six to 12 minutes.
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  But an industrial environment, like a foundry, requires a much
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  faster change, say every 4 to 8 minutes.
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  So if you have an 800,000 cubic foot foundry and you target a 5
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  minute air change, you need a system capacity of 160,000 cubic
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  feet per minute. And that cube law is running in
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  the background of every single one of those calculations.
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  Which brings us to our final segment, the heat equation.
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  Thermal management is critical because all that power we've
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  just discussed in the compressors, the hydraulic
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  pumps, the fans, it eventually turns into waste heat.
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  It has to go somewhere. Let's start with basic sensible
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  heat transmission through a surface.
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  The formula is H equals AU times the difference between T sub
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  naughty outside temperature and key sub I inside temperature.
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  Where a is the area and U is the coefficient of heat
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  transmission. And that U value is where
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  material selection provides the ultimate payoff.
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  Look at the stark contrast. A single pane window has AU
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  value of 1 and 13 hundredths, right?
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  But 3/4 of an inch of drywall with three inches of insulation
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  reduces that U value to a mere 7 hundredths.
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  That's an order of magnitude difference in heat retention.
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  It just shows the non negotiable value of proper insulation for
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  energy efficiency. Absolutely.
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  We also talk about specific heat, which is the amount of
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  heat required to raise a substances temperature 1°
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  compared to water which has a specific heat of 1.
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  The formula there is the heat required in B2 or calories is
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  calculated regulated as WCT. So wait time specific heat times
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  the temperature. Rise, which is why water is so
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  good at carrying heat away. Exactly.
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  Water is 1.000. Aluminum, which is often used in
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  heat sinks, is only 215 thousandths and lead is a tiny
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  31 thousandths. And that's why you can touch a
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  piece of hot lead much longer than hot aluminum.
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  It just doesn't store the same amount of energy.
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  All systems lose heat through conduction, convection, and
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  radiation. Let's focus on radiation, the
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  black body effect. This heat loss is heavily
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  dependent on the materials emissivity, which is symbolized
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  by E. Emissivity is basically the
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  correction factor, a perfect black body, a theoretical object
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  that absorbs all incoming radiation and is the most
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  efficient radiator. At 600°F it loses 4 watts per
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  square inch. But if that surface is made of
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  copper with a medium oxide surface, its emissivity factor E
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  is only 4/10. Right.
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  Therefore, the actual radiation loss is 4 * 4/10, which equals
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  only 1 6/10 watts per square inch.
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  This is why polished metals are used to reflect heat and
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  enclosures, while black or dark surfaces are used on heat sinks
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  to maximize the emission. Yep, emissivity dictates whether
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  your system cooks or cools itself.
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  So let's apply this to the common problem of cooling
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  electronics or mechanical cabinets.
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  If you have an enclosed cabinet with no active cooling, you can
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  approximate the internal temperature rise delta T Delta T
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  is approximately 4 and 8 hundredths multiplied by Q / A +
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  1 and 110th. Q is the heat generated in watts
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  and A is the surface area in square feet.
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  And consider the real sticks here, a cabinet generating 1200
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  watts of heat with a surface area of 55 6/10 square feet.
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  The expected temperature rise delta T is approximately 90°F.
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  So if the ambient air is 72°. The internal temperature shoots
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  up to 162°F. That will cause almost any piece
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  of commercial electronics to fail.
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  So to save your system, you need a fan.
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  The resulting temperature drop, the new delta T drop, is
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  calculated as three and one tenths multiplied by Q / V,
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  where V is the fan flow rate in CFM.
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  And that leads to a clever design consideration for your
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  fan arrangement. If the air inside your cabinet
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  has a lot of internal resistance, you know high
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  impedance from densely packed components.
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  You want 2 smaller fans placed in series.
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  They build up pressure better. But if the air can flow easily,
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  low impedance. Then two fans in parallel are
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  better as they essentially double the flow rate V rather
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  than boosting the pressure SP. And finally, for anyone dealing
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  with facility cooling, the commercial rating of one ton of
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  air conditioning is standardized as 12,000 B2 per hour.
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  Which is equal to 3 5/10 kilowatts.
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  You need that conversion factor when you're integrating facility
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  cooling with your system's heat load queue.
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  So to wrap it up. We covered how the relationship
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  between pressure, volume and temperature dictates compression
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  efficiency and why cooling is the real driver of compressor
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  design. Right.
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  And we walked through how the Reynolds number predicts the
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  chaos of turbulent flow and why all those viscosity units are so
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  varied. And we've finished with thermal
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  transfer, where emissivity and surface area are absolutely
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  critical to preventing catastrophic failure.
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  And if we connect this all back to that initial safety
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  challenge, we spent time on how high pressure vessels require
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  the much more complex Lemay's equation for design accuracy.
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  We talk thin walls, thick walls and ultra high pressures.
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  So think about what you are designing to contain or
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  withstand in the most extreme environments, Whether it's the
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  hull of a deep sea submersible under immense external pressure
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  or a composite fuel tank in a high altitude rocket under high
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  internal pressure. Failure is just not an option,
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  right? So if those pressure equations
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  hold true, consider what unexpected material property
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  becomes the limiting design factor in those environments.
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  Is it density, fracture, toughness, or something else
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  entirely? Something for you to Mull over.
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  That is a phenomenal challenge to leave us with.
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  And if you haven't already, get over to our YouTube channel
• 17:17
  Mechanical Engineering Made Simple, where we post hands on
• 17:20
  how to videos, visual breakdowns, and practical
• 17:22
  engineering walkthroughs. You could actually use a family
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  that solves together, stays together.