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https://www.mesubjects.net/wpadmin/post.php?post=6097&action=edit Q. ANS. Pressure Drop
PRESSURE DROP
It is difference of pressure between
two points in a fluid flow. These two
points can be at horizontal level or at
different levels.
(i)Pressure drop is the driving force for
the flow of a fluid.
(ii) Likewise temperature difference is
the driving force for heat transfer.
(iii) Voltage difference is the driving
force for current flow.
(iv) Concentration difference is the
driving force for mass transfer
(v) Momentum difference is the
driving force for momentum transfer.
Factors on which pressure loss depends
(i) Pipe material and its roughness
(ii) Shape and size of the pipe
(iii) Shape and size of the pipe fitting
(iv) Type of flow (laminar, turbulent and transition flow) which depends on Reynolds number
Thus pressure drop depends on
(i) density
(ii) velocity
(iii) viscosity
(iv) diameter
(v) length
(vi) surface roughness of the pipe and pipe fitting.
TYPES OF PRESSURE DROP
Three types of Pressure drop

Frictional pressure drop

Gravitational pressure drop (two points at different elevations)

Acceleration pressure drop in an evaporator (During phase change only i.e. liquid changing into vapor)
OR
Deacceleration pressure drop in a condenser (During phase only i.e. vapor changing into liquid)
Frictional Pressure Drop In Liquids
It is divided in two parts.
MAJOR PRESSURE LOSS IN PIPES
Frictional pressure loss in a straight length pipe
MINOR PRESSURE LOSS IN PIPES
Pressure loss caused by pipe fittings (Tees, elbow). These are minor since these are small as compared to major pressure losses.
FINDING OF MAOR PRESSURE DROP
LAMINAR FLOW
It can be found in two ways.
(i) First–Darcy – Welsbach Equation which is applicable only for liquids with fully developed and steady flow. First head loss (Δh) is calculated and then pressure drop (Δp) is found.
Frictional head loss ∆h_{f} —–Darcy – Welsbach equation
∆h_{f} = 4f (L/D) (V^{2}/2g) = 4fLV^{2}/2gD
∆h_{f} =Frictional head loss in a pipe of length ‘L’ with average velocity of flow, V
f = Fanning friction factor = 16/Re (dimensionless)
L = Length of the pipe (m)
D = internal diameter of a circular pipe (m)
D = Hydraulic diameter (4A / P) in case of a noncircular pipe (m)
A = area of flow in a non circular pipe (m^{2})
P = Perimeter of the noncircular pipe (m)
V = Average velocity of flow of fluid (m/s)
g = Local acceleration due to gravity (m/s^{2})
(ii) FANNING EQUATION
∴ Frictional pressure drop ∆p_{f}
∆p_{f} = f (ρL V^{2}/ 2D)= fLρV^{2}/ 2D
(a) For a laminar flow, friction factor is a function of only Reynolds number and f = 64 / Re
(b) FOR THE TURBULENT FLOW
DarcyWelsbach equation for turbulent flow
∆p =f L ρ V^{2}/2D
Head loss (h_{f} ρ g =∆p_{f})
h_{f} = f LV^{2}/2gD
But Darcy friction factor ‘ f ’ is found from ColebrookWhite equation
1/√f = –2 log 10 ( ε /3.7 D_{h} + 2.51/Re√f )
Where
f is the Darcy friction factor, dimensionless
ε is height of roughness, m
D is the inside diameter for circular pipes ,m (D_{h} = Di)
D_{h} is the hydraulic diameter for noncircular pipes ,m, and Dh = 4A/P
(A is the area of flow in the non circular pipe)
(P is the inside perimeter of the non circular pipe)
Re is the Reynolds number > 4000 for turbulent flow
There are number of empirical equations to find friction factor ‘f’ but most commonly used is the ColebrookWhite equation
Note
Friction factor ‘ f ’ is present on both sides of the equation. Thus it can be solved only by an iterative procedure which can be very time consuming and cumbersome. Darcy equation when plotted is Moody Diagram. From this diagram, friction factor for turbulent flow can be directly found after knowing the Reynolds number and pipe roughness.
Moody diagram
Applicable for turbulent flow only, Re > 4000, However laminar and transition region has also been shown on Moody Diagram. (Moody diagram is a graphical form of ColebrookWhite equation). Refer for Moody Diagram.
X coordinate —– Reynolds number
Y coordinate—–Darcy friction factor ‘ f ’
There are number of curves for various values of pipe roughness
Roughness of various pipe materials is also mentioned on this diagram.
GRAVITATIONAL PRESSURE DROP
∆p = ρg(Z_{1} –Z_{2})
ACCELERATION PRESSURE DROP DUE TO PHASE CHANGE
∆p = G^{2} ∆v
Where ∆v is the change in specific volume between the two points under consideration
G is the mass flux = mass flow rate per unit area
NOTE: PRESSURE DROP DUE TO DIFFERENT ATMOSPHERIC PRESSURES AT THE TWO LOCATIONS
In most practical problems, the change in elevation is not extremely large. So atmospheric pressure is assumed to be constant. Otherwise there will be another pressure drop due to difference in atmospheric pressure between the two points. If the two points under consideration are widely separated. For example, pipe line is from an Arab country to India.
Finding Minor Pressure losses
Pressure drop takes place with fluids moving in pipes/channels. There are three methods for finding minor pressure loss in pipe fittings.
1. The equivalent length method
Equivalent length method is applicable for each type of fitting. It is calculated by equation given below:
h_{f} = f (l_{eq} /D_{h}) (v^{2}/2g)
2. The valve flow coefficient (Cv) method
3. The resistance coefficient (K) method
Most commonly used method.
K Method
Under this method, mathematically, minor head loss for every situation is given as
h_{L} = K (v1^{2}/2g)
where v1 is the velocity in the SMALLER pipe irrespective of enlargement or contraction of the pipe
K is called minor loss coefficient. It has no units (dimensionless). The value of K for various pipe fittings is given below:
MINOR LOSS COEFFICIENT ‘K’ FOR MOST COMMONLY USED PIPE FITTING
Sr. No. Type of Pipe fitting Value of ‘K’
1. Tee, Flanged, Dividing Line Flow 0.2
2. Tee, Threaded, Dividing Line Flow 0.9
3. Union, Threaded 0.08
4. Elbow, Flanged Regular 90^{o} 0.3
5. Elbow, Threaded Regular 90^{o }1.5
6. Elbow, Threaded Regular 45^{o} 0.4
7. Elbow, Flanged Long Radius 90^{o} 0.2
8. Elbow, Threaded Long Radius 90^{o} 0.7
9. Elbow, Flanged Long Radius 45^{o} 0.2
10. Globe Valve, Fully Open 10
11. Angle Valve, Fully Open 2
12. Gate Valve, Fully Open 0.15
13. Gate Valve, 1/4 Closed 0.26
14. Gate Valve, 1/2 Closed 2.1
15. Gate Valve, 3/4 Closed 17
16. Ball Valve, Fully Open 0.05
17. Ball Valve, 1/3 Closed 5.5
18. Ball Valve, 2/3 Closed 200
19. Diaphragm Valve, Open 2.3
20. Water meter 7.0
NOTE: Values of factor K for many other fittings not covered here but can be found in literature.
Total friction head loss
h _{total loss} = Σ h _{major loss} + Σ h _{minor loss}
PRESSURE DROP IN GASES AND VAPORS
In compressible fluids (gases and vapors), there is significant increase in velocity due to pressure decrease (Expansion). Hence it requires a different equation as compared to that of liquids.
(p_{1}^{2}—p2^{2}) / 2p1 = f (L/D) ρ1(V_{av}^{2}/ 2) (T av / T1)
p1 = pressure at entrance, is known
p2 = pressure at exit, to be found
T av = (T1 + T2) / 2
Use DARCY EQUATION to find the pressure drop in gases and vapors. Use it when density difference between two points is less than 2 %. It is used for liquids as liquids being incompressible.
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