In most cases the differential pressure across the pump Dptot is measured and the head H is calculated by the following formula: H =
Δptot/ρ /g
(m)
where
Δptot
total pressure difference
ρ g
density of fluid in kg/cu.m free fall acceleration 9.8 sq.m/s
flow Q n
0.0025 cu m 4000 rpm
INLET INLET AREA CALCULATED AS for radial impeller 2(PI)*r1*b1 sqm A1 = r1 = b1 =
rad radial position of impeller's r's inlet edg edge (m) the blade height at the inlet (m)
for semi axial impeller
A1 =
2*(pi)*(r1 hub+ r1 shroud)/2*b1
(sq.m)
The entire flow must pass through this ring area. C1 m is then calculated C1m =
Q impel mpelller/A r/A1
(m/s) /s)
The tangential velocity U1 equals the product of radius and angular frequency: U1 =
2*pi*r1*n/60 = r1*ω r1*ω
(m/s)
n = rpm ω = angular velocity When the velocity triangle has been drawn, see figure 4.4, based on α1, C1m and U1, the relative flow angle β can be calculated. Without inlet rotation (C1=Cm)this becomes: tan β1 =
C1m/U1
β1 =
Inv tan β1
OUTLET
:
Outlet area calculated as
A2 =
2*pi*r2*b2
(sq.m)
for semi axial impeller
A2 =
2*(pi)*(r2 hub+ r2 shroud)/2*b2
(sq.m)
C2m is then calculated C2m =
Q impeller/A2
(m/s)
The tangential velocity U2 equals the product of radius and angular frequency: U2 =
2*pi*r2*n/60 = r2*ω
(m/s)
In the bignig of design phase ,β2 is assumed to have the same value as the blade angle The relative velocity can be calculated from: W2 =
C2m/sin β2
(m/S)
and C2u as: C2u=
U2 - C2m/ tan β2
(m/s)
4.2 Euler’s pump equation Euler’s pump equation is the most important equation in connection with
pump design. The equation can be derived in many different ways. The method described here moment of momentum equation which describes flow forces and velocity triangles at inlet and outlet.
A control volume is an imaginary limited volume which is used for setting up equilibrium equations. The equilibrium equation can be set up for torques, energy and other f of momentum equation is one such equilibrium equation, linking mass flow and velocities with impeller diameter. A control volume between 1 and 2, as shown in figure 4.6, is often used for an impeller. The balance which we are interested in is a torque balance. The torque (T) from the drive shaft corresponds to the torque originating from the fluid’s flow through the impeller with mass flow m=ρQ:
T=
m*(r2*C2u - r1 * Ciu)
(Nm)
By multiplying the torque by the angular velocity, an expression for the
is found. At the same time, radius multiplied by the shaft power (P2) angular velocity equals the tangential velocity, r2 w = U2 . This results in:
P2 = = = = =
T* ω (Watt) m*ω*(r2* C2u - r1 * C1u) m*(ω*r2* C2u - ω*r1 * C1u) m (U2* C2u - U1 * Ciu) Q*ρ * (U2* C2u - U1 * Ciu)
According to the energy equation, the hydraulic power added to the fluid can be written as the increase in pressure Δptot across the impeller multiplied by Phyd =
Δptot *Q
(Watt)
The Head is defined as : H =
Δptot / ρ * g
(m)
and the expression for hydraulic power can therefore be transcribed to:
Phyd = =
Q*H*ρ * g m * H *g
(Watt)
If the flow is assumed to be loss free, then the hydraulic and mechanical power can be equate: Phyd =
P2
m * H *g = m (U2* C2u - U1 * Ciu) H=
(U2* C2u - U1 * Ciu) / g
This is the equation known as Euler’s equation, and it expresses the impel ler’s head at tangential and absolute velocities in inlet and outlet. If the cosine relations are applied to the velocity triangles, Euler’s pump equation can be written as the sum of the three contributions: Static head as consequence of the centrifugal force • Static head as consequence of the velocity change through the impeller • Dynamic head H=
((U2 sq - U1 sq) / 2/g) +
(W1 sq- W 2 sq)/2/g) + (C2 sq- C1 sq)/2/g
(m)
Static head as consequence of the Centrifugal force
Static head as consequence of the Velocity change
Dynamic head
If there is no flow through the impeller and it is assumed that there is no inlet rotation, then the head is only determined by the tangential velocity based on (4.17) where C2u=U2 H0 =
U2 sq/g
(m)
When designing a pump, it is often assumed that there is no inlet rotation meaning that C 1u equeals zero H=
(U2 * C2u)/g
(m)
4.8 Specific speed of a pump As described in chapter 1, pumps are classified in many different ways for example by usage or flange size. Seen from a fluid mechanical point of view, this is, however, not very practical because it makes it almost impossible to compare pumps which are designed and used differently.
A model number, the specific speed (nq) Specific speed is given in different units
is therefore used to classify pump In Europe the following form iscommonly use
nq =
nd *sqrt (Qd)/ Hd power (3/4)
Where nd Qd = Hd =
= rotational speed in the design point [rpm) = Flow at the design point [m3/s] = Head at the design point [m]
The expression for nq can be derived from equation (4.22) and (4.23) as the speed which yields a head of 1 m at a flow of 1 m3/s
The impeller and the shape of the pump curves can be predicted based on the specific speed, see figur 4.17. Pumps with low specific speed, so-called low nq pumps, have a radial outlet with large outlet diameter compared to inlet diameter. The head cur are relatively flat, and the power curve has a positive slope in the entire flow area
On the contrary, pumps with high specific speed, so-called high nq pumps. have an increasingly axial outlet, with small outlet diameter compared to the width. Head curves are typically descending and have a tendency to create saddle points. Performance curves decreases when flow increases. Different pump sizes and pump types have different maximum efficiency
kp/kg
A1 =
0.00285 sqm
r1 = b1 =
0.03 m 0.01512 m
p1-p2
C1m
=
0.877177 m/sec
U1 =
12.56637 m/sec
tan β1 =
0.069804
β1 =
3.992969 deg
A2 =
C2m
0.002262 sq m
=
1.105243 m/sec
U2 =
18.84956 m/sec
tan β2 =
0.058635
β2 =
3.355693 deg
W2 =
18.88193 m/sec
C2u=
0 m/sec
includes a control volume which limits the impeller, the
low quantities which are of interest. The moment
he flow Q
d:
es
