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)

Calculate the remaining two variables in a liquid jet pump, using a model presented in [1] as well as [2], [3], and [4].

$$N = \frac{2R + \frac{2 C M^2 R^2}{1-R} - R^2 (1+CM) (1+M) (1 + K_m + K_d + \alpha^2) - \frac{CM^2R^2}{(1-R)^2} (1+K_s)} {(1+K_p) - 2R - \frac{2CM^2R^2}{1-R} + R^2(1+CM)(1+M)(1+K_m + K_d + \alpha^2) + (1-j)\left(\frac{CM^2R^2}{({1-R})^2} \right)(1+K_s)}$$

$$P_1 - P_2 = \frac{1}{2}\rho_p \left(\frac{Q_p}{A_n}\right)^2 \left[(1+K_p) - C(1+K_s) \left(\frac{MR}{1-R}\right)^2\right]$$

$$\text{Pressure ratio} = N = \frac{P_5 - P_2}{P_1 - P_5}$$

$$\text{Volume flow ratio} = M = \frac{Q_s}{Q_p}$$

$$\text{Jet pump efficiency} = \eta = M\cdot N = \frac{Q_s(P_5-P_2)}{Q_p(P_1 - P_5)}$$

$$R = \frac{A_n}{A_m}$$

$$C = \frac{\rho_s}{\rho_p}$$

There is no guarantee a solution will be found for the provided variable values, but every combination of two missing variables are supported.

Parameters

rhopfloat

The density of the primary (motive) fluid, [kg/m^3]

rhosfloat

The density of the secondary fluid (drawn from the vacuum chamber), [kg/m^3]

Kpfloat, optional

The primary nozzle loss coefficient, [-]

Ksfloat, optional

The secondary inlet loss coefficient, [-]

Kmfloat, optional

The mixing chamber loss coefficient, [-]

Kdfloat, optional

The diffuser loss coefficient, [-]

d_nozzlefloat, optional

The inside diameter of the primary fluid’s nozle, [m]

d_mixingfloat, optional

The diameter of the mixing chamber, [m]

d_diffuserfloat, optional

The diameter of the diffuser at its exit, [m]

Qpfloat, optional

The volumetric flow rate of the primary fluid, [m^3/s]

Qsfloat, optional

The volumetric flow rate of the secondary fluid, [m^3/s]

P1float, optional

The pressure of the primary fluid entering its nozzle, [Pa]

P2float, optional

The pressure of the secondary fluid at the entry of the ejector, [Pa]

P5float, 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_variationsint, optional

When the initial guesses do not lead to a converged solution, try this many more guesses at converging the problem, [-]

Returns

solutiondict

Dictionary of calculated parameters, [-]

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:

$$P_1 = P_3 + \frac{1}{2}\rho_p V_n^2 + K_p\left(\frac{1}{2}\rho_p V_n^2\right)$$

$$P_2 = P_3 + \frac{1}{2}\rho_s V_3^2 + K_s\left(\frac{1}{2}\rho_s V_3^2\right)$$

$$P_5 = P_4 + \frac{1}{2}\rho_d V_4^2 - K_d\left(\frac{1}{2}\rho_d V_4^2\right)$$

The mixing chamber loss coefficient should be obtained through the following expression, using the mixing chamber exit velocity to obtain the friction factor.

$$K_m = \frac{4f_d L}{D}$$

$$K_d = \frac{P_4 - P_5}{0.5 \rho_d V_4^2} = 1 - \left(\frac{A_4}{A_5} \right)^2 - C_{pr}$$

$$K_s = \frac{P_2 - P_3}{0.5\rho_s V_3^2} - 1$$

$$K_p = \frac{P_1 - P_n}{0.5\rho_p V_n^2} - 1$$

Continuity of the ejector:

$$\rho_p Q_p + \rho_s Q_s = \rho_d Q_d$$

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)

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.

$$P_1 - P_2 = \frac{1}{2}\rho_pV_n^2(1+K_p) - \frac{1}{2}\rho_s V_3^2(1+K_s)$$

Rearrange to express V3 in terms of Vn, and using the density ratio C, the expression becomes:

$$P_1 - P_2 = \frac{1}{2}\rho_p V_n^2\left[(1+K_p) - C(1+K_s) \left(\frac{MR}{1-R}\right)^2\right]$$

Using the primary nozzle area and flow rate:

$$P_1 - P_2 = \frac{1}{2}\rho_p \left(\frac{Q_p}{A_n}\right)^2 \left[(1+K_p) - C(1+K_s) \left(\frac{MR}{1-R}\right)^2\right]$$

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

rhopfloat

The density of the primary (motive) fluid, [kg/m^3]

rhosfloat

The density of the secondary fluid (drawn from the vacuum chamber), [kg/m^3]

Kpfloat

The primary nozzle loss coefficient, [-]

Ksfloat

The secondary inlet loss coefficient, [-]

d_nozzlefloat, optional

The inside diameter of the primary fluid’s nozle, [m]

d_mixingfloat, optional

The diameter of the mixing chamber, [m]

Qpfloat, optional

The volumetric flow rate of the primary fluid, [m^3/s]

Qsfloat, optional

The volumetric flow rate of the secondary fluid, [m^3/s]

P1float, optional

The pressure of the primary fluid entering its nozzle, [Pa]

P2float, optional

The pressure of the secondary fluid at the entry of the ejector, [Pa]

Returns

solutionfloat

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)

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)

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

Vfloat

Vessel volume, [m^3]

Pfloat

Vessel actual absolute operating pressure - less than P_atm!, [Pa]

P_atmfloat, optional

The atmospheric pressure surrounding the vessel, [Pa]

Returns

mfloat

Air leakage flow rate, [kg/s]

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)

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:

$$W = 0.026P^{0.34}V^{0.6}$$

if P < 100 torr:

$$W = 0.032P^{0.26}V^{0.6}$$

else:

$$W = 0.106V^{0.6}$$

In the above equation, the units are lb/hour, torr (vacuum), and cubic feet; they are converted in this function.

Parameters

Vfloat

Vessel volume, [m^3]

Pfloat

Vessel actual absolute operating pressure - less than P_atm!, [Pa]

P_atmfloat, optional

The atmospheric pressure surrounding the vessel, [Pa]

Returns

mfloat

Air leakage flow rate, [kg/s]

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)

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

Pfloat

Vessel actual absolute operating pressure - less than P_atm!, [Pa]

P_atmfloat, optional

The atmospheric pressure surrounding the vessel, [Pa]

conservativebool

Whether to use the high values or low values in the table, [-]

Returns

mfloat

Air leakage flow rate, [kg/s]

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)

Calculates an estimated leakage of air into a vessel using a correlation from Seider [1].

$$W = 5 + \left[ 0.0298 + 0.03088\ln P - 0.0005733(\ln P)^2 \right]V^{0.66}$$

In the above equation, the units are lb/hour, torr (vacuum), and cubic feet; they are converted in this function.

Parameters

Vfloat

Vessel volume, [m^3]

Pfloat

Vessel actual absolute operating pressure - less than P_atm!, [Pa]

P_atmfloat, optional

The atmospheric pressure surrounding the vessel, [Pa]

Returns

mfloat

Air leakage flow rate, [kg/s]

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