Jet Pump (ejector/eductor) Sizing and Rating (fluids.jet_pump)¶
This module contains a model for a jet pump, also known as an eductor or an ejector.
For reporting bugs, adding feature requests, or submitting pull requests, please use the GitHub issue tracker or contact the author at Caleb.Andrew.Bell@gmail.com.
Interfaces¶
- fluids.jet_pump.liquid_jet_pump(rhop, rhos, Kp=0.0, Ks=0.1, Km=0.15, Kd=0.1, d_nozzle=None, d_mixing=None, d_diffuser=None, Qp=None, Qs=None, P1=None, P2=None, P5=None, nozzle_retracted=True, max_variations=100)[source]¶
Calculate the remaining two variables in a liquid jet pump, using a model presented in [1] as well as [2], [3], and [4].
There is no guarantee a solution will be found for the provided variable values, but every combination of two missing variables are supported.
- Parameters
- rhop
float
The density of the primary (motive) fluid, [kg/m^3]
- rhos
float
The density of the secondary fluid (drawn from the vacuum chamber), [kg/m^3]
- Kp
float
,optional
The primary nozzle loss coefficient, [-]
- Ks
float
,optional
The secondary inlet loss coefficient, [-]
- Km
float
,optional
The mixing chamber loss coefficient, [-]
- Kd
float
,optional
The diffuser loss coefficient, [-]
- d_nozzle
float
,optional
The inside diameter of the primary fluid’s nozle, [m]
- d_mixing
float
,optional
The diameter of the mixing chamber, [m]
- d_diffuser
float
,optional
The diameter of the diffuser at its exit, [m]
- Qp
float
,optional
The volumetric flow rate of the primary fluid, [m^3/s]
- Qs
float
,optional
The volumetric flow rate of the secondary fluid, [m^3/s]
- P1
float
,optional
The pressure of the primary fluid entering its nozzle, [Pa]
- P2
float
,optional
The pressure of the secondary fluid at the entry of the ejector, [Pa]
- P5
float
,optional
The pressure at the exit of the diffuser, [Pa]
- nozzle_retractedbool,
optional
Whether or not the primary nozzle’s exit is before the mixing chamber, or somewhat inside it, [-]
- max_variations
int
,optional
When the initial guesses do not lead to a converged solution, try this many more guesses at converging the problem, [-]
- rhop
- Returns
- solution
dict
Dictionary of calculated parameters, [-]
- solution
Notes
The assumptions of the model are:
The flows are one dimensional except in the mixing chamber.
The mixing chamber has constant cross-sectional area.
The mixing happens entirely in the mixing chamber, prior to entry into the diffuser.
The primary nozzle is in a straight line with the middle of the mixing chamber.
Both fluids are incompressible, and have no excess volume on mixing.
Primary and secondary flows both enter the mixing throat with their own uniform velocity distribution; the mixed stream leaves with a uniform velocity profile.
If the secondary fluid is a gas, it undergoes isothermal compression in the throat and diffuser.
If the secondary fluid is a gas or contains a bubbly gas, it is homogeneously distributed in a continuous liquid phase.
Heat transfer between the fluids is negligible - there is no change in density due to temperature changes
The change in the solubility of a dissolved gas, if there is one, is negigibly changed by temperature or pressure changes.
The model can be derived from the equations in
liquid_jet_pump_ancillary
and the following:Conservation of energy at the primary nozzle, secondary inlet, and diffuser exit:
The mixing chamber loss coefficient should be obtained through the following expression, using the mixing chamber exit velocity to obtain the friction factor.
Continuity of the ejector:
References
- 1
Karassik, Igor J., Joseph P. Messina, Paul Cooper, and Charles C. Heald. Pump Handbook. 4th edition. New York: McGraw-Hill Education, 2007.
- 2
Winoto S. H., Li H., and Shah D. A. “Efficiency of Jet Pumps.” Journal of Hydraulic Engineering 126, no. 2 (February 1, 2000): 150-56. https://doi.org/10.1061/(ASCE)0733-9429(2000)126:2(150).
- 3
Elmore, Emily, Khalid Al-Mutairi, Bilal Hussain, and A. Sheriff El-Gizawy. “Development of Analytical Model for Predicting Dual-Phase Ejector Performance,” November 11, 2016, V007T09A013.
- 4
Ejectors and Jet Pumps. Design and Performance for Incompressible Liquid Flow. 85032. ESDU International PLC, 1985.
Examples
>>> ans = liquid_jet_pump(rhop=998., rhos=1098., Km=.186, Kd=0.12, Ks=0.11, ... Kp=0.04, d_mixing=0.045, Qs=0.01, Qp=.01, P2=133600, ... P5=200E3, nozzle_retracted=False, max_variations=10000) >>> s = [] >>> for key, value in ans.items(): ... s.append('%s: %g' %(key, value)) >>> sorted(s) ['M: 1', 'N: 0.293473', 'P1: 426256', 'P2: 133600', 'P5: 200000', 'Qp: 0.01', 'Qs: 0.01', 'R: 0.247404', 'alpha: 1e-06', 'd_diffuser: 45', 'd_mixing: 0.045', 'd_nozzle: 0.0223829', 'efficiency: 0.293473']
Objective Function¶
- fluids.jet_pump.liquid_jet_pump_ancillary(rhop, rhos, Kp, Ks, d_nozzle=None, d_mixing=None, Qp=None, Qs=None, P1=None, P2=None)[source]¶
Calculates the remaining variable in a liquid jet pump when solving for one if the inlet variables only and the rest of them are known. The equation comes from conservation of energy and momentum in the mixing chamber.
The variable to be solved for must be one of d_nozzle, d_mixing, Qp, Qs, P1, or P2.
Rearrange to express V3 in terms of Vn, and using the density ratio C, the expression becomes:
Using the primary nozzle area and flow rate:
For P, P2, Qs, and Qp, the equation can be rearranged explicitly for them. For d_mixing and d_nozzle, a bounded solver is used searching between 1E-9 m and 20 times the other diameter which was specified.
- Parameters
- rhop
float
The density of the primary (motive) fluid, [kg/m^3]
- rhos
float
The density of the secondary fluid (drawn from the vacuum chamber), [kg/m^3]
- Kp
float
The primary nozzle loss coefficient, [-]
- Ks
float
The secondary inlet loss coefficient, [-]
- d_nozzle
float
,optional
The inside diameter of the primary fluid’s nozle, [m]
- d_mixing
float
,optional
The diameter of the mixing chamber, [m]
- Qp
float
,optional
The volumetric flow rate of the primary fluid, [m^3/s]
- Qs
float
,optional
The volumetric flow rate of the secondary fluid, [m^3/s]
- P1
float
,optional
The pressure of the primary fluid entering its nozzle, [Pa]
- P2
float
,optional
The pressure of the secondary fluid at the entry of the ejector, [Pa]
- rhop
- Returns
- solution
float
The parameter not specified (one of d_nozzle, d_mixing, Qp, Qs, P1, or P2), (units of m, m, m^3/s, m^3/s, Pa, or Pa respectively)
- solution
Notes
The following SymPy code was used to obtain the analytical formulas ( they are not shown here due to their length):
>>> from sympy import * >>> A_nozzle, A_mixing, Qs, Qp, P1, P2, rhos, rhop, Ks, Kp = symbols('A_nozzle, A_mixing, Qs, Qp, P1, P2, rhos, rhop, Ks, Kp') >>> R = A_nozzle/A_mixing >>> M = Qs/Qp >>> C = rhos/rhop >>> rhs = rhop/2*(Qp/A_nozzle)**2*((1+Kp) - C*(1 + Ks)*((M*R)/(1-R))**2 ) >>> new = Eq(P1 - P2, rhs) >>> solve(new, Qp) >>> solve(new, Qs) >>> solve(new, P1) >>> solve(new, P2)
References
- 1
Ejectors and Jet Pumps. Design and Performance for Incompressible Liquid Flow. 85032. ESDU International PLC, 1985.
Examples
Calculating primary fluid nozzle inlet pressure P1:
>>> liquid_jet_pump_ancillary(rhop=998., rhos=1098., Ks=0.11, Kp=.04, ... P2=133600, Qp=0.01, Qs=0.01, d_mixing=0.045, d_nozzle=0.02238) 426434.60314398
Vacuum Air Leakage Estimation¶
- fluids.jet_pump.vacuum_air_leakage_HEI2633(V, P, P_atm=101325.0)[source]¶
Calculates an estimated leakage of air into a vessel using fits to a graph of HEI-2633-00 for air leakage in commercially tight vessels [1].
There are 5 fits, for < 1 mmHg; 1-3 mmHg; 3-20 mmHg, 20-90 mmHg, and 90 mmHg to atmospheric. The fits are for maximum air leakage.
Actual values may be significantly larger or smaller depending on the condition of the seals, manufacturing defects, and the application.
- Parameters
- Returns
- m
float
Air leakage flow rate, [kg/s]
- m
Notes
The volume is capped to 10 ft^3 on the low end, but extrapolation past the maximum size of 10000 ft^3 is allowed.
It is believed
vacuum_air_leakage_Seider
was derived from this data, so this function should be used in preference to it.References
- 1
“Standards for Steam Jet Vacuum Systems”, 5th Edition
Examples
>>> vacuum_air_leakage_HEI2633(10, 10000) 0.001186252403781038
- fluids.jet_pump.vacuum_air_leakage_Ryans_Croll(V, P, P_atm=101325.0)[source]¶
Calculates an estimated leakage of air into a vessel using a correlation from Ryans and Croll (1981) [1] as given in [2] and [3].
if P < 10 torr:
if P < 100 torr:
else:
In the above equation, the units are lb/hour, torr (vacuum), and cubic feet; they are converted in this function.
- Parameters
- Returns
- m
float
Air leakage flow rate, [kg/s]
- m
Notes
No limits are applied to this function.
References
- 1
Ryans, J. L. and Croll, S. “Selecting Vacuum Systems,” 1981.
- 2
Coker, Kayode. Ludwig’s Applied Process Design for Chemical and Petrochemical Plants. 4 edition. Amsterdam ; Boston: Gulf Professional Publishing, 2007.
- 3
Govoni, Patrick. “An Overview of Vacuum System Design” Chemical Engineering Magazine, September 2017.
Examples
>>> vacuum_air_leakage_Ryans_Croll(10, 10000) 0.0004512
- fluids.jet_pump.vacuum_air_leakage_Coker_Worthington(P, P_atm=101325.0, conservative=True)[source]¶
Calculates an estimated leakage of air into a vessel using a tabular lookup from Coker cited as being from Worthington Corp’s 1955 Steam-Jet Ejector Application Handbook, Bulletin W-205-E21 [1].
- Parameters
- Returns
- m
float
Air leakage flow rate, [kg/s]
- m
References
- 1
Coker, Kayode. Ludwig’s Applied Process Design for Chemical and Petrochemical Plants. 4 edition. Amsterdam ; Boston: Gulf Professional Publishing, 2007.
Examples
>>> vacuum_air_leakage_Coker_Worthington(10000) 0.005039915222222222
- fluids.jet_pump.vacuum_air_leakage_Seider(V, P, P_atm=101325.0)[source]¶
Calculates an estimated leakage of air into a vessel using a correlation from Seider [1].
In the above equation, the units are lb/hour, torr (vacuum), and cubic feet; they are converted in this function.
- Parameters
- Returns
- m
float
Air leakage flow rate, [kg/s]
- m
Notes
This formula is rough.
References
- 1
Seider, Warren D., J. D. Seader, and Daniel R. Lewin. Product and Process Design Principles: Synthesis, Analysis, and Evaluation. 2nd edition. New York: Wiley, 2003.
Examples
>>> vacuum_air_leakage_Seider(10, 10000) 0.0018775547