Oil is confined in a 4 -in.- -diameter cylinder by a piston having a radial clearance of 0.001 i

Oil is confined in a 4 -in.- -diameter cylinder by a piston...

Key Concepts

Lubrication Theory

Lubrication theory is a branch of fluid mechanics that deals with the flow of fluids in thin layers or films, typically between closely spaced surfaces. It simplifies the Navier-Stokes equations under the assumption that one dimension is much smaller than the others, allowing for analytical expressions for pressure, velocity, and shear stress. This theory is vital for analyzing and predicting leakage rates and frictional losses in mechanical systems involving narrow clearances.

Pressure Driven Flow

Pressure driven flow is a fundamental concept in fluid mechanics where a difference in pressure forces the fluid to move from regions of high pressure to low pressure. This principle is used to calculate the flow rate in systems where an external force generates a pressure gradient. It is particularly applicable in confined flows, where the pressure difference, fluid properties, and geometric parameters determine the fluid's rate of leakage.

Clearance Flow

Clearance flow refers to the movement of fluid through a narrow gap between two surfaces. This concept is significant in applications where precision is critical, as the small distance between the surfaces greatly influences the fluid's velocity profile and leakage rate. The analysis of clearance flow often leverages lubrication theory to simplify complex geometrical considerations and obtain useful predictions about fluid behavior.

Laminar Flow

Laminar flow describes a flow regime in which the fluid moves in smooth, parallel layers with minimal mixing between them. This concept is integral in scenarios involving narrow gaps or small clearances, where the flow is dominated by viscous forces. Understanding laminar flow allows engineers to employ simplified models, such as those based on Poiseuille’s law, which predict velocity profiles and flow rates under steady conditions.

Viscosity

Viscosity is a measure of a fluid's resistance to flow and deformation due to internal friction. It plays a crucial role in determining how easily a fluid can be displaced under an applied force, and its value is typically temperature-dependent. In engineering applications, accurate knowledge of a fluid's viscosity is essential to predict the behavior of the fluid under various flow conditions, particularly in confined geometries.

Transcript

00:01 So we have problem number nine, this is a problem number nine in which there is a piston cylinder arrangement in which all is confined and the dimensions of piston and cylinder are being given.

00:14 That it has a radial clearance.

00:19 So this should be the radial clearance, which is a.

00:22 So if you write a has radial clearance, d as diameter of the piston, sorry, diameter of the cylinder.

00:30 And a force is applied a force f is being applied on this piston so we need to find the rate of the oil leak past piston that is rate of the oil which will pass through this radio clearance okay so let us note note down all the given things d that is diameter of the cylinder is four inch okay after that a radial clearance which is a equals to 0 .001 inch and the length of piston that is this is the length of piston let us say l is 2 inches and force is being given as 4 ,500 pound okay and various things are being given such as the property we have to take as a 30 oil at 120 fahrenheit so from the table we can get the value which will be used afterwards okay so basically the rate of oil leak will pay as volume flow we can say will be let us write formula q that is volume flow is equal to pi d a q pressure difference delta p by 12 mu and l okay where this uh mu is related to the viscosity which is known as the dynamic viscosity okay and delta p is the pressure difference that is p2 minus p1 p1 minus p2 that is p1 is the inside pressure let us say this is inside pressure and atmospheric pressure will be from outside now let us start plugging things so if you look at the system a pi is constant d is being given 12 is constant l is being given so we need to find delta p and this mu so delta p will be equal to pressure difference that is p1 minus p at mostive pressure now facial is always force per unit area so force by area this is known as the pressure and this will be equal to force by area since our cross -section area will be a circular area so pi d square by four that is four f by pi f by pi d squared okay now if we are plug in the values which are being given so four and the value of f will be four thousand five hundred lbf okay into this is uh pi divided by pi so here it will divide by pi okay d square diameter square the diameter is given in diameter is given in inches so let us write four square simply and diameter is in denominator in denominator so let us write four square okay so four four will get cancelled out so we will be having and one more thing we should take care of that this should be four inch whole square okay okay so 4 by pi into 4500 lb will be f by 4 square and one more thing since these are all in that is fps that foot pound and second so we will be we have to use the pressure in term of psi so this will be 4 ,500 lbf by 16 inches squared this is four we have to use calculator for this 4 ,500 4505 divided by 4 divided by 5 divided by 5 that is 358 358 lbf by inch squared so this is known as 358 find 09 it was so 0 .1 .0 okay no problem 358 psi this is the required pressure now from the table we can get the value of mu that is coefficient of dynamic viscosity from the table mu equal to and up from the table and for the given condition the condition was condition was as a 30 oil at 120 degrees fahrenheit so at this point mu will be given as 0 .06 into 0 .0 .0209 lbf that's by feet square so if you multiply we will be getting 0 .06 into 0 .029 that's 1 .254 1 .254 into 2 10 is the power minus 3 on second by feet square okay now we have to use mass flow formula that we have written over here this formula q will be equal to q will be equal to this is pi by 12 pi d a q delta p so this is pi d d means four inches pi d a q a is 0 .001 inches or 1 into 10 to power minus 3 cube into delta p.

08:40 Delta p is given at 358 p s i so one thing to be sure that we should take care of the units with this also so let us write all the things with the units this is inch this is inch pi d and then a q that is 1 into 10 to the power minus 3 inches cube okay then then uh delta p that is 358 p s i 358 p s i divide by 12 into mu into length mu is 1 .254 point 254 into 10 to power minus 3 lb s by feet square okay now everything is in inches what else okay length length we have two inches length we have two inches now everything uh everything is in inches so this is in feet so we have to convert it in inch so 1 .254 in 10 to power minus 3 a pound second by feet square so in place of feet square we should write 12 inch square so basically we are dealing with for three times pi by three into okay by three one this is one so inch into inch into 10 to power minus nine inch cube into three 358 psi and 12 inch square will be in the numerator so 144 inch square divided by 1 .254 into 10 to power minus 3 lbs okay into 2 inches okay this is 262 oh sorry 72 72 72 now we have to use calculator to calculate everything okay okay so this is a 10 to power minus 9 358 300 38 into 72 into 72 into this pi divide by 3253 divide by 1 .255 and 4 okay 1 .25 and 4 so this comes out to be 21 ,500 25 .15 .15 .1 .17 into 10 into the power this is minus 9 and this will be minus 6 inch inch will get cancelled.

12:50 Okay so we will be having this value as 21 ,525 .125 .1707 into 10 xx0 .10 .5...

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