HYDRODYNAMICS OF PUMPS

by Christopher Earls Brennen     © Concepts NREC 1994

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Nomenclature

ROMAN LETTERS
a	Pipe radius
A	Cross-sectional area
Aijk	Coefficients of pump dynamic characteristics
[A]	Rotordynamic force matrix
Ar	Cross-sectional area ratio
B	Breadth of passage or flow
[B]	Rotordynamic moment matrix
c	Chord of the blade or foil
c	Speed of sound
c	Rotordynamic coefficient: cross-coupled damping
cb	Interblade spacing
cPL	Specific heat of liquid
C	Compliance
C	Rotordynamic coefficient: direct damping
CD	Drag coefficient
CL	Lift coefficient
Cp	Coefficient of pressure
Cpmin	Minimum coefficient of pressure
d	Ratio of blade thickness to blade spacing
D	Impeller diameter or typical flow dimension
Df	Diffusion factor
DT	Determinant of transfer matrix [T]
e	Specific internal energy
E	Energy flux
E	Young's modulus
f	Friction coefficient
F	Force
g	Acceleration due to gravity
gs	Component of g in the s direction
h	Specific enthalpy
h	Blade tip spacing
hp	Pitch of a helix
hT	Total specific enthalpy
h*	Piezometric head
H	Total head rise
H(s,θ,t)	Clearance geometry
I	Acoustic impulse
I,J	Integers such that ω/Ω=I/J
IP	Pump impedance
j	Square root of -1
k	Rotordynamic coefficient: cross-coupled stiffness
kL	Thermal conductivity of the liquid
K	Rotordynamic coefficient: direct stiffness
KG	Gas constant
ℓ	Pipe length or distance to measuring point
L	Lift
L	Inertance
L	Axial length
	Latent heat
m	Mass flow rate
m	Rotordynamic coefficient: cross-coupled added mass
mG	Mass of gas in bubble
mD	Constant related to the drag coefficient
mL	Constant related to the lift coefficient
M	Moment
M	Mach number, u/c
M	Rotordynamic coefficient: direct added mass
n	Coordinate measured normal to a surface
N	Specific speed
N(RN)	Cavitation nuclei number density distribution function
NPSP	Net positive suction pressure
NPSE	Net positive suction energy
NPSH	Net positive suction head
p	Pressure
pA	Radiated acoustic pressure
pT	Total pressure
pG	Partial pressure of gas
pS	Sound pressure level
pV	Vapor pressure
P	Power
	Vector of fluctuating quantities
Q	Volume flow rate (or heat)
	Rate of heat addition
r	Radial coordinate in turbomachine
R	Radial dimension in turbomachine
R	Bubble radius
R	Resistance
RN	Cavitation nucleus radius
Re	Reynolds number
s	Coordinate measured in the direction of flow
s	Solidity
	Surface tension of the saturated vapor/liquid interface
S	Suction specific speed
Si	Inception suction specific speed
Sa	Fractional head loss suction specific speed
Sb	Breakdown suction specific speed
Sf	Slip factor
t	Time
T	Temperature or torque
Tij	Transfer matrix elements
[T]	Transfer matrix based on ,
[T*]	Transfer matrix based on ,
[TP]	Pump transfer matrix
[TS]	System transfer matrix
u	Velocity in the s or x directions
ui	Velocity vector
U	Fluid velocity
U∞	Velocity of upstream uniform flow
v	Fluid velocity in non-rotating frame
V	Volume or fluid velocity
w	Fluid velocity in rotating frame
	Rate of work done on the fluid
z	Elevation
ZCF	Common factor of ZR and ZS
ZR	Number of rotor blades
ZS	Number of stator blades
GREEK LETTERS
α	Angle of incidence
αL	Thermal diffusivity of liquid
β	Angle of relative velocity vector
βb	Blade angle relative to cross-plane
γn	Wave propagation speed
Γ	Geometric constant
δ	Deviation angle at flow discharge
δ	Clearance
ε	Eccentricity
ε	Angle of turn
η	Efficiency
θ	Angular coordinate
θc	Camber angle
θ*	Momentum thickness of a blade wake
Θ	Thermal term in the Rayleigh-Plesset equation
	Inclination of discharge flow to the axis of rotation
κ	Bulk modulus of the liquid
μ	Dynamic viscosity
ν	Kinematic viscosity
ρ	Density of fluid
σ	Cavitation number
σi	Cavitation inception number
σa	Fractional head loss cavitation number
σb	Breakdown cavitation number
σc	Choked cavitation number
σTH	Thoma cavitation factor
Σ	Thermal parameter for bubble growth
Σ{1,2,3}	Geometric constants
τ	Blade thickness
φ	Flow coefficient
ψ	Head coefficient
ψ0	Head Coefficient at zero flow
ω	Radian frequency of whirl motion or other excitation
ωP	Bubble natural frequency
Ω	Radian frequency of shaft rotation
SUBSCRIPTS
On any variable, Q:
Qo	Initial value, upstream value or reservoir value
Q1	Value at inlet
Q2	Value at discharge
Qa	Component in the axial direction
Qb	Pertaining to the blade
Q∞	Value far from the bubble or in the upstream flow
QB	Value in the bubble
QC	Critical value
QD	Design value
QE	Equilibrium value
QG	Value for the gas
QH1	Value at the inlet hub
QH2	Value at the discharge hub
Qi	Components of vector Q
Qi	Pertaining to a section, i, of the hydraulic system
QL	Saturated liquid value
Qm	Meridional component
QM	Mean or maximum value
QN	Nominal conditions or pertaining to nuclei
Qn,Qt	Components normal and tangential to whirl orbit
QP	Pertaining to the pump
Qr	Component in the radial direction
Qs	Component in the s direction
QT1	Value at the inlet tip
QT2	Value at the discharge tip
QV	Saturated vapor value
Qx	Component in the x direction
Qy	Component in the y direction
Qθ	Component in the circumferential (or θ) direction
SUPERSCRIPTS AND OTHER QUALIFIERS
On any variable, Q:
	Mean value of Q or complex conjugate of Q
	Complex amplitude of oscillating Q
	Time derivative of Q
	Second time derivative of Q
Q*	Rotordynamics: denotes dimensional Q
Re{Q}	Real part of Q
Im{Q}	Imaginary part of Q

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Last updated 12/1/00.

Christopher E. Brennen