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Tuesday, July 9, 2013

2-K (Hooper’s) method



K is a dimensionless factor defined as the excess head loss in pipe fitting, expressed in velocity heads. K does not depend on the roughness of the fitting (or the attached pipe) or the size of the system, but it is a function of the Reynolds number and the exact geometry of the fitting. The 2-K method accounts for these dependencies by the following equation.

Kf= K1/Re + Koo(1 + 1/IDin.)

Or in SI units

Kf= K1/Re + Koo(1 + 25.4/IDmm.)

where
K1 = K for the fitting at Re = 1
Koo = K for a large fitting at Re= infinity
ID = Internal pipe diameter, in. (mm).

The ID correction in the two K expression accounts for the size differences. K is higher for small sizes, but nearly constant for larger sizes. However, the effect of pipe size (e.g., 1/ID) does not accurately reflect data over a wide range of sizes for valves and fittings. Further
Hooper’s scaling factor is not consistent with the Crane values at high Reynolds numbers and is especially inconsistent for larger fitting sizes.

New Crane Method



The latest version of the equivalent length method given in Crane Technical Paper 410 (Crane Co. 1991) requires the use of two friction factors. The first is the actual friction factor for flow in the straight pipe (f ), and the second is a standard friction factor for the particular fitting(fT)
hL = Kfv2/2g
where
Kf = 4f(Leq/D)

The value of f is determined from the Colebrook equation
f =0.0625/[log(3.7D/E)]2
where
E is the pipe roughness (0.0018 in., 0.045mm) for commercial steel pipe.

The Crane paper gives fT for a wide variety of fittings, valves, etc. This method gives satisfactory results for high turbulence levels (e.g., high Reynolds number) but is
less accurate at low Reynolds number. This method provides a better estimate for the effect of geometry, but does not reflect any Reynolds number dependence.

Monday, June 17, 2013

K-factor Method



The excess loss in a fitting is normally expressed in a dimensionless “K” factor. The excess head loss (dH) is less than the total by the amount of frictional loss that would be experienced by straight pipe of the same physical length. The loss coefficient Kf , depends on the Reynolds number of the flow. The values of Kf at low Re can be significantly greater than those at high Re. Additionally, valves and fittings do not scale exactly. e.g., the loss coefficient for a 1/4 in. valve is not the same as that for a 4 in. valve.

Wednesday, June 12, 2013

The Equivalent length (L/D) Method



The equivalent length adds some hypothetical length of pipe to the actual length of the fitting.
However, the drawback is that the equivalent length for a given fitting is not constant, but depends on Reynolds number, pipe roughness, pipe size, and geometry.
 
Kf = f(Leq/D)

Leq = Kf .(D/f)

Every equivalent length has a specific friction factor. The method assumes that
  • Sizes of fittings of a given type can be scaled by the corresponding pipe diameter.
  • The influence of Reynolds number on the friction loss on the fitting is the same as the pipe loss.
However, neither of the above assumption is accurate.
Furthermore, the nature of the laminar or turbulent flow field within a valve or a fitting is generally quite different from that in a straight pipe. Therefore, there is an uncertainty when determining the effect of
Reynolds number on the loss coefficients. This method does not properly account for the lack of exact scaling for valves and fittings.

Saturday, May 25, 2013

Jaw Crushers

In a jaw crusher feed is admitted between two jaws, set to from a V open at the top. One jaw, the fixed, or anvil, jaw, is nearly vertical and does not move; the other, the swinging jaw, reciprocates in a horizontal plane. It makes an angle of 20 to 30 degrees with the anvil jaw. It is driven by an eccentric so that it applies great compressive force to lumps caught between the jaws. The jaw faces are flat or slightly bulged; they may carry shallow horizontal grooves. Large lumps caught between the upper parts of the jaws are broken, drop into the narrower space below, and are recrushed the next time the jaws close. After sufficient reduction they drop out the bottom of the machine. the jaws open and close 250 to 400 times per minute.
the most common type of jaw crusher is the Blake crusher. In this machine an eccentric drives a pitman connected to two toggle plates, one of which is pinned to the frame and the other to the swinging jaw. The pivot point is at the top of the movable jaw or above the top of the jaws on the centerline of the jaw opening. The greatest amount of motion is at the bottom of the V, which means that there is little tendency for a crusher of this kind to choke.

Friday, May 14, 2010

Bond’s law

Bond postulated that work required to form particles of size Dp from very large feed is proportional to the square root of the surface-to-volume ratio of the product, Sp/Vp. By relation Sp/Vp = 6/ɸsDp, from which it follows that
P/ ṁ = Kb/(Dp)^0.5
Where Kb is a constant that depends on the type of machine and on the material being crushed. To use this equation, a work index Wi is defined as the gross energy requirement in kilowatt-hours per ton of feed needed to reduce a very large feed to such a size that 80% of the product passes a 100-µm screen. This definition leads to a relation between Kb and Wi. If Dp is in millimeters, P in kilowatts, and ṁ in tons per hour,
Kb = 0.3162 Wi
If 80% of the feed passes a mesh size of Dpa millimeters and 80% of the product a mesh of Dpb millimeters, it follows
P/ ṁ = 0.3162 Wi [1/(Dpb)^0.5 - 1/(Dpa)^0.5]

Rittinger’s and Kick’s laws

Rittinger's law states that work required in crushing is proportional to the new surface created. In other words, crushing efficiency is constant and for a given machine and feed material is independent of the sizes of feed and product. Rittinger’s law is written as-
P/ ṁ = Kr(1/Dsb – 1/Dsa)
Kick proposed another law based on stress analysis of plastic deformation within the elastic limit, which states that the work required for crushing a given mass of material is constant for the same reduction ratio, that ia, the ration of the initial particle size to the final particle size. This leads to the relation
P/ ṁ = Kk ln(Dsa/Dsb)