Ejector Modeling in HYSYS Bruce Eng April 27, 2009 DRAFT Background An ejector is a piece of equipment which combines a low pressure stream (called the suction stream) and a high pressure stream (called the motive stream) to form a stream of intermediate pressure (called the discharge stream). When first encountered the idea of an ejector might seem trivial. After all, the analogous operation with temperature, adding a hot stream and a cold stream to form a medium temperature stream, only requires a pipe. However pressure difference determines flow direction, so without some equipment, a low pressure stream will not flow towards a higher pressure region. An ejector solves this problem by changing the velocity of the stream. Actually pressure difference doesn’t determine flow direction, but rather it represents a force, which as stated by Newton, causes acceleration. This is the idea behind Bernoulli’s principle: P1
ρ
+
v12 P v2 + h1 g = 2 + 2 + h2 g 2 ρ 2
From this equation, it is evident that P2 can be larger than P1 if the initial point is higher than the destination (flow does an inclined pipe), or if the outlet velocity is lower than the inlet velocity.
A typical ejector is shown above. The top flange is the inlet for the high pressure motive stream. This stream is forced into a small nozzle and then expanded so the pressure decreases to a low pressure which will be referred to as the mixing pressure, Pmix. At the same time, the velocity of
the stream increases. From the bottom flange, a low pressure suction stream is introduced. Assuming that its pressure is slightly higher than Pmix it will flow towards the high velocity motive stream. As it makes contact with the high pressure stream, it will be pulled along and accelerate. As its velocity increases, its pressure will decrease to Pmix. The two streams will also combine as the suction pressure stream becomes entrained in the motive stream. Then, after the two streams have mixed, the ejector begins to open out in what is known as the diffuser section. In this section, the velocity decreases and the pressure increases to an intermediate value. An ejector is an integral part of many processes. The paper industry commonly uses ejectors. The milk industry relies on them to help drive evaporation at vacuum pressure. It has also been proposed to use ejectors in an efficient form of air conditioning. For all of these applications, the process engineer would like to be able to calculate the possible discharge pressure and temperature for given suction and motive streams. For the process engineer, the detailed dimensions, materials, and controllability of the ejector are of secondary importance to understanding how the ejector will affect the flows, temperatures, and pressures of the process. Therefore this paper will ignore any of the sizing or controllability issues of the ejector and instead focus on the process.
One Dimensional Model The ejector process is simplified into a one dimensional model consisting of a couple of steps. Step 1: Expanding the Motive Stream The motive stream is expanded from Pmotive to Pmix. Preferably this expansion would be isentropic, but in practice that is not achievable. Instead the expansion occurs with a given adiabatic efficiency ηmotive. To find the expanded condition: Smotive = fEOS(Hmotive, Pmotive)
The equation of state is used to look up the initial entropy.
Hideal = fEOS(Smotive, Pmix)
The EOS is used to look up ideal enthalpy.
v motive_ out
2
= η motive ⋅ ( H motive − H ideal ) +
v motive_ in
2
2 → v motive_ out = 2 ⋅ η motive ⋅ ( H motive − H ideal ) + v motive_ in Most
2 2 of the energy from the expansion goes to increase the velocity.
Hmotive_real = (1- ηmotive)·(Hmotive-Hideal) + Hideal The energy which doesn’t increase the velocity increases the enthalpy. Step 2: Expanding the Suction Stream The suction stream is also expanded with an efficiency of ηsuction from Psuction to Pmix. Hsuction_real and vsuction_out are found using the same method as in step 1. Step 3: Constant Pressure Mixing After both the motive stream and the suction stream have been expanded to the mixing pressure, both streams combine forming an outlet stream with a new velocity and a new enthalpy. It is assumed that momentum and energy are both conserved in this process. m motive vmotive _ out + m suction v suction _ out = ( m motive + m suction ) ⋅ v mix → v mix =
m motive v motive _ out + m suction v suction _ out
( m motive + m suction )
conservation of momentum
1 2
m m o tivve m2 o tive_ o u t + 12 m su ctiovn su2 ctio _n o u t + m m o tivH e m o tiv e+ msu c tio H n su ctio n=
( m m o tiv e+ m su c tio n) ⋅ vm2 ix + ( m m o tiv e+ m su ctio n) ⋅ H m ix → 2 2 2 1 1 1 n su c tio n 2 mm o tivve m o tiv_e o u t + 2 ms u ctiovn su ctio _n o u t − 2 ( mm o tiv e+ msu c tio n) ⋅ v m ix + mm o tiveH m o tive+ msu c tio H H m ix = ( m m o tive+ m su c tio n) 1 2
conservation of energy Step 4: Supersonic Shock In many processes the optimal ejector design has internal supersonic flows. If the mixed velocity is supersonic then the flow will pile up somewhere along the length of the mixing section and then form a supersonic shock wave. After the shock wave the flow will be subsonic at a higher pressure and different temperature. The reality of this process is complex, but for modeling purposes, we will assume that it is governed by four equations: conservation of energy, conservation of momentum, continuity, and a relationship between density and enthalpy and pressure given by an equation of state. v mix ⋅ ρmix ⋅ A = v postshock ⋅ ρ postshock ⋅ A
Continuity (conservation of mass) v mix ( ρmix ⋅ A ⋅ v mix ) + Pmix ⋅ A = v postshock ( ρ postshock ⋅ A ⋅ v postshock ) + Ppostshock ⋅ A
Conservation of momentum at constant cross sectional area
( m m o tiv+e m s u c tio) n⋅ ( 12 vm2 ix+ H m i x) = ( m m o tiv+e m s u c tio) n⋅ ( 12 v 2p o s ts h o +c k H p o s t s h o)c k Conservation of energy ρ postshock = f EOS ( H postshock , Ppostshock
) Equation of state
These four equations can be solved by guessing a value of vpostshock and then using the continuity equation to find ρpostshock and then using the momentum equation to find Ppostshock and then using the energy equation to find Hpostshock and then calculating another value for ρpostshock using the equation of state. If the two calculated densities are not equal then the guess for vpostshock must be updated. There should be multiple solutions, but the one with vmix = vpostshock is only correct if the original velocity was subsonic. To determine if the flow was subsonic, the speed of sound can be calculated as v sound =
∂P ∂ρ
which is easily done with an equation of state.
Step 5: Compression in the Diffuser Section Finally the post shock subsonic flow is expanded in the diffuser section. The pressure increases as the velocity decreases. The process deviates from isentropic compression by an efficiency ηdiffuser.
To find the outlet condition: Spostshock = fEOS(Hpostshock, Ppostshock)
The equation of state is used to look up the initial entropy.
Hideal = Hpostshock + ηdiffuser·1/2·(vpostshock2-voutlet2) Poutlet = fEOS(Spostshock,Hideal) Most of the energy from the velocity goes into compression Houtlet_real = (1- ηmotive)· 1/2·(vpostshock2-voutlet2) + Hideal The energy which doesn’t cause compression increases the enthalpy. At this point the outlet pressure and enthalpy is known which was the goal of this model.
HYSYS Simulation An ejector model can be created in HYSYS using the following flowsheet:
All the conditions including flowrate of the motive stream and suction stream are set. In addition the pressure of the expanded motive stream is set (to Pmix) and a set block is used to set the expanded suction stream to this same pressure. The “ejector calc” spreadsheet does all of the mixing and supersonic shockwave calculations.
The motive velocity (cell B2) is calculated as v motive = 2 ⋅
wexp anded motive m
. This comes from the
formula for kinetic energy solved for v. The term wexpanded is imported from the HYSYS expander and represents the work output by an expander operating with a given adiabatic efficiency. This is equivalent to step 1 of the 1-D model. The suction velocity (cell B6) is calculated in a similar fashion. This is equivalent to step 2 of the 1-D model. The mixed velocity (cell B7) is calculated from the formula m motivev motive _ out + m suctionv suction _ out vmix = . Then the incoming kinetic energy (cell B8) and outgoing ( m motive + m suction ) kinetic energy (cell B9) are calculated and their difference (cell B10) is assumed to be dissipated as heat which is added by the heater “E-mix”. This is equivalent to step 3 of the 1-D model. The majority of the spreadsheet is devoted to finding the post shock wave velocity using the procedure described in step 4 of the 1-D model. Guesses for v are made using bisection with the maximum velocity assumed to be 99.5% of the original velocity and the minimum velocity assumed to be 10% of the original velocity. Each guessed v is the average of the maximum and the minimum velocity. Depending on the sign of the error between the two density calculations, either vmax_next = vguess or vmin_next = vguess. The depicted sheet uses 12 iterations to solve the equations
of step 4. Each iteration needs an equation of state calculation which is the reason for the multiple streams (iteration1 – iteration12). This approach takes quite a bit of typing in the HYSYS spreadsheet. It is left as an exercise to the reader to develop a HYSYS flow sheet which uses a recycle to perform this iterative calculation. Finally the kinetic energy left for compression in the diffuser section is calculated as 1/2·(msuction+mmotive)·(vpostshock2-voutlet2) where voutlet is assumed to be 0. This is used to set the power for the last HYSYS compressor. This is equivalent to step 5 of the 1-D model.
Discussion Rating Cases This HYSYS ejector simulation does a decent job of modeling a possible ejector process. However this ejector is only applicable is the design case. It does not include sizing data which is necessary for rating calculations. Ejectors are known to be particularly poor at operating away from design conditions. This is because the velocity and pressure profile inside the ejector is determined by the pressure of the feeds and the sizing of the ejector internals. When the velocities differ from design values, there is no guarantee that constant pressure mixing will occur. In addition, entrained fluids may reseparate leading to instability. Many times multiple ejectors are placed in parallel if heavy turn down is going to be required. Consult a vendor for more information. Parameters This ejector simulation has four parameters that need to be specified in addition to the inlet stream conditions. These are the mixing pressure (Pmix), the motive fluid expansion efficiency (ηmotive), the suction fluid expansion efficiency (ηsuction), and the diffuser compression efficiency, (ηdiffuser). A good guess for the efficiencies is between 80% - 90% with 83% being a typical value. The mixing pressure is process dependent. Typically the mixing pressure is chosen (by trial and error) in order to maximize the outlet pressure.
References Chemical Engineering and Processing 41 (2002) 551–561 Evaluation of steam jet ejectors Hisham El-Dessouky*, Hisham Ettouney, Imad Alatiqi, Ghada Al-Nuwaibit Energy Conversion & Management 41 (2000) 1841-1849 Investigation of ejector design at optimum operating condition E.D. Rogdakis*, G.K. Alexis Desalination 222 (2008) 639–645 Experience of design and optimization of multi-effects desalination systems in Iran R.K. Kamali*, S. Mohebinia Fundamentals of Compressible Fluid Dynamics By P. Balachandran Section 9.3.3
