MMAN2600 Formula Summary November 8, 2012 1. Hydrostatic Pressure (Static Fluids only) ∆P = P 2 − P 1 = ρgh ρg h
N Pa = 2 m
2. Stagnation Pressure (Pressure of fluid when it is brought to a stop) ρv2 P stag = P + stag = P 2 3. Shear Stress τ (Force τ (Force tangential to surface) F v du τ = = µ = µ A L dy
N Pa = 2 m
4. Viscosity (a) Dynamic Dynamic (or absolute): absolute): µ
kg m.s
2
(b) Kinematic: Kinematic: ν =
µ m ρ
s
5. Capillary Effect h =
2σs ρgR
6. Flow Visualisation (a) Stream Stream line: The curve curve that is tangent tangent to the instan instantane taneous ous local velocit velocity y. They show the instantaneous direction of fluid motion. (b) Streak Streak line: The locus of fluid particl particles es which have have passed passed sequential sequentially ly through through a prescribed point in the flow. (c) Path line: The path traced out by an individual fluid particle particle over over time. (d) Time line: line: Locus of the set of adjacent fluid particles particles that were released released at the same instant in time (a velocity profile).
1
7. Force on submerged planes F R = P C A = ρgyc A yP = y c +
I xx,C [yc + ρgP ]A sin θ 0
For a rectangular plate: (yc = b/2, I xx,c = ab 3 /12) b F R = [P 0 + ρg(s + )sin θ]ab 2 b b2 yP = s + + 2 12[s + 2b +
P 0 ] ρg sin θ
Usually P 0 = P atm so you can take P 0 = 0. 8. Bernoulli’s Equation (dynamic fluids only) (a) Head form P 1 v12 P 2 v22 + + z 1 + hpump = + + z 2 + hturbine + hloss = constant = H [m] ρg 2g ρg 2g (b) Energy form (per unit mass) P v2 emech = + + gz ρ 2
J kg
(c) General Energy form ˙ m
v12 P 1 v12 ˙ ˙ turbine + E ˙ mech loss + + gz 1 + W pump = m ˙ + + gz 1 + W ρ 2 ρ 2
P
1
(d) Assumptions: steady, incompressible flow, negligible viscous effects, negligible heat transfer, irrotational flow (no vorticity) and negligible friction only if head losses aren’t considered. (e) Some Notes: •
P H is the total head of flow (also energy grade line value), ρg is the pressure head, v2 2g
is the velocity head (difference between energy grade line and hydraulic grad line) and z is the elevation head. 2
P is the static pressure, ρ2v is the dynamic pressure and ρgz is the hydrostatic pressure. • The kinetic energy correction factor α, which is the coefficient of the velocity term has been ignored, as it is usually taken to be 1. •
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9. Continuity and Flow Rates (a) Mass flow rate
kg
˙ ˙ m = ρ V
s
(b) Volume flow rate ˙ = Q = Av V
3
m s
(c) Continuity A1v1 = A 2v2 10. Power ˙ = mgh ˙ W ˙ pump = V ∆P = F v
J s
=W
11. Pressure in accelerating systems tan θ =
ax dz isobar =− (slope of isobars) g + az dx P z = ρ(g + az )h
12. Buoyancy F B = m displacedg = ρ fluidV displacedg 13. Linear Momentum Equation
F = β m˙ v β m˙ v −
out
[N]
in
β is the dimensionless momentum-flux correction factor due to the fact that velocity across most inlets/outlets are not uniform. Usually equal to 1. 14. Reynold’s Number Re ReD =
inertial forces ρv D vD = = [dimensionless] viscous forces µ ν
Laminar flow for Re < 2300, where viscous forces dominate. Turbulent flow is characterised by disordered fluid motion and high velocity fluctuations. 15. Losses due to pipe friction (major loss) (a) Pressure Loss L ρv2 ∆P L = f = ρghL [Pa] D 2 3
(b) Head Loss ∆P L L v2 hL = = f ρg D 2g
[m]
(c) Don’t forget to change to hydraulic diameter for non-circular pipes, 4Ac Dh = = 4 × Cross sectional Area / Perimeter p 16. Darcy Friction Factor f (a) f laminar: Use table for different shapes (Table 8-1) 64 (b) f laminar and circular: Re 96 (c) f laminar and thin rectangle: Re (d) f turbulent: Use Moody chart (will need relative roughness
ε D
and Re)
17. Minor Loss
∆P L v2 hL = = K L [m] ρg 2g Due to valves, bends, expansions, contractions, etc in pipes. K L is the loss coefficient. Look it up. (Table 8-4)
18. Piping Networks (a) Series: • Flow rate is
the same • Total head loss is the sum of the head losses (b) Parallel • Head loss is
the same • Total flow rate is the sum of the flow rates 19. Turbomachinery (a) Torque = T = r × F
r
− ×m ˙ V
out
r
×m ˙ V
[Nm]
in
(b) Brake Power ˙ shaft = ωT = ωρV (r ˙ 2v2,t − r1v1,t ) = ρ V ˙ gH e W (c) Euler Head H e =
ω (r2v2,t − r1v1,t ) g 4
J W= s
(d) Output Power
(e) vi,n =
˙ shaft P = η W
˙ V 2πri b
(f) ω = (speed in rpm)
rad
2π 60
×
s
v1,t = 0
(g) No swirl condition ⇒ 20. External Flow
(a) Drag Coefficient (usually projected area) C D =
2F D ρv2 A
(b) Lift Coefficient (usually platform area) C L =
2F L ρv2 A
(c) Reynold Number Rex =
ρv x vx = µ ν
(d) C D can be obtained from Table 11-1 (for 2D objects), Table 11-2 (for 3D objects), Figure 11-34 and 11-35 for flow over cylinders and spheres. (e) C L can be obtained from Table 11-43, 11-45, 11-47 (magnus effect). 21. Parallel flow over flat plates (a) Force on the flat plate:
1 F f = C f Aρv 2 2
(b) Local coefficient of friction C f,x, C f,x,
laminar
turbulent
=
=
0.664 , Re0.5
0.059 , Re0.2
(c) Average coefficient of friction
5
Rex < 5 × 105
5 × 105
≤
Rex
≤
107
• Calculating
1 C f = L
from local coefficient: L
C f,x
0
1 dx = L
x
L
cr
C f,x, laminar dx +
C f,x,
x
0
cr
turbulent
dx
xcr is the position on the plate for which the flow transitions from laminar to turbulent (found by solving Rex = 5 × 105 ). • Calculated value for a fully laminar or turbulent plate: C f, C f,
laminar
turbulent
=
=
1.33 , Re0.5
0.074 , Re0.2
6
Rex < 5 × 105
5 × 105
≤
Rex
≤
107
