Screw Compressor Basics
Screw Compressor Modelling, Design and Use
Professor N. Stosic Chair in Positive Displacement Compressor Technology Centre for Positive Displacement Compressor Technology City University London, U.K.
INTRODUCTION Screw Compressor Today Highly competitive market, specially in air compression and refrigeration Continuous improvement: more compact, efficient and cost effective compressors New rotor generation, rotors optimized for certain compressor duty, specialized design Scope for innovation, improvement and development
INTRODUCTION
Basics
View from Front and Top
Top and front: Admission Bottom: Change of volume Bottom and Rear: Discharge
View from Bottom and Rear
INTRODUCTION
Basics
INTRODUCTION
Swedish company SRM, pioneer and leader Screw compressor profiles, Symmetric, Asymmetric, ‘D’ and ‘G’ Screw compressor design Screw compressor technology Licence system left many screw compressor manufacturers at margins of research and development
INTRODUCTION
Many companies started their own development Gardner Denver, Atlas Copco, Compair, Kaeser, GHH Trane, Ingersol-Rand, Hitachi, Fu Sheng, Hanbel, Refcomp Holroyd Many more or less successful rotor profiles Many more or less successful screw compressor designs Need exists to concentrate efforts in R&D
INTRODUCTION
Centre for Positive Displacement Compressor Technology
Improved methods of analysis Experimental validation Design of critical components Complete machine design Product development Training in machine design
INTRODUCTION
Methods Applied Advanced Computerized design tools Machine process modelling 2-D and 3-D Computational Fluid Dynamics Modern experimental technique Computerized data acquisition
Users Renown and new companies, large and small manufacturers in the U.K and abroad
SCREW COMPRRESSOR GEOMETRY Before modelling the physical process, the rotor lobe profiles must be defined together with the remaining parameters with which the rotor and housing geometry can be fully specified.
Rotor profile: x and y coordinates, pressure angle Helix/lead angle, rotor length Interlobe, end and radial clearance Suction/discharge ports
SCREW COMPRESSOR GEOMETRY
General case: non-parallel and non-intersecting
Given profile r1 = r1 (t ,θ ) = [ x1 , y1 , z1 ] = [ x01 cosθ − y01 sin θ , x01 sin θ + y01 cos θ , p1θ ]
∂y ∂x ∂y ∂r1 ∂x1 ∂y1 ∂x01 = , ,0 = cosθ − 01 sin θ , 01 sin θ + 01 cosθ ,0 ∂t ∂t ∂t ∂t ∂t ∂t ∂t ∂r1 ∂x1 ∂y1 = , ,0 = [− y01 , x01 ,0] ∂θ ∂θ ∂θ
SCREW COMPRESSOR GEOMETRY
General case: non-parallel and non-intersecting Meshed profile r2 = r2 (t ,θ , τ ) = [ x2 , y2 , z2 ] = [ x1 − C , y1 cos Σ − z1 sin Σ, y1 sin Σ + z1 cos Σ ] = [ x02 cos τ − y02 sin τ , x02 sin τ + y02 cos τ , p2τ ] ∂r2 = [− y2 , x2 , p2 ] = [ x02 sin τ + y02 cos τ , x02 cos τ − y02 sin τ , p2 ] = ∂τ p1θ sin Σ − y1 cos Σ, p2 sin Σ + ( x1 − C ) cos Σ, p2 cos Σ − ( x1 − C ) sin Σ
Meshing condition ∂r1 ∂r1 ∂r1 ∂r1 ∂r1 ∂r2 ∂ × ∂θ ⋅ ∂τ = − ∂ × ∂θ ⋅ ∂τ = 0 t t
∂y ∂y ∂x ∂x C − x1 + ( p1 − p2 ) cot Σ x1 1 + y1 1 + p1 p1θ 1 + ( p2 − C cot Σ ) 1 = 0 ∂t ∂t ∂t ∂t
SCREW COMPRESSOR GEOMETRY
General case: non-parallel and non-intersecting corresponds to the rotor – hobbing tool relation Special cases: p2=0, rotor - plate milling tool, grinding tool relation Σ=0, screw compressor rotors
SCREW COMPRESSOR GEOMETRY
Rotor profile, Σ=0, i=p2/p1, k=1-1/i Meshing condition dy01 C C ky01 − sin θ + kx01 + cosθ = 0 dx01 i i
Meshed profile
θ i θ y02 = x01 sin kθ + y01 cos kθ + C sin i
x02 = x01 cos kθ − y01 sin kθ − C cos
Rack profile x0 r = x01 cos θ − y01 sin θ y0 r = x01 sin θ + y01 cos θ − r1θ
Numerical solution of the meshing condition
Task: to find θ for a zero function Simple iteration method: Fast and reliable, but valid only for certain function Additional complication In certain areas two or more θ are the zero function Only one is valid, additional values found by half interval method
SCREW COMPRESSOR GEOMETRY
Demonstrator profile
‘N’ ROTOR PROFILE
SCREW COMPRESSOR GEOMETRY
- Rack generation procedure - Straight line on the rack - involute rotor contact - Small torque transmitted - Large displacement - Short sealing line - Strong gate rotor,
SCREW COMPRRESSOR THERMODYNAMICS Differential approach: Set of differential equations solved simultaneously Equations of continuity, momentum and energy Preintegrated equations inadequate and inaccurate if high leakage rate and heat transfer is involved
SCREW COMPRESSOR THERMODYNAMICS
Internal Energy dU dV ω = m! in hin − m! out hout + Q − ω p dθ dθ m! in hin = m! suc hsuc + m! l , g hl , g + m! oil hoil m! out hout = m! dis hdis + m! l ,l hl ,l
Continuity ω
dm = m! in − m! out dθ
m! in = m! suc + m! l , g + m! oil
m! out = m! dis + m! l ,l
Leakage Flow Momentum wl2 dx dp + f =0 wl dwl + ρ 2 Dg m! l = ρl wl Ag = Ag
p22 − p12 p2 a ζ + 2 ln p1 2
m! = ρ wA
SCREW COMPRESSOR THERMODYNAMICS
Oil injection
dToil ho Ao (T − Toil ) = ω moil coil dθ
k=
ω moil coil ω d S coil = 6ho ∆θ ho Ao ∆θ
Toil =
T − kToil , p 1+ k
Numerical solution, Runge-Kutta IV order solver U − ( mcT )oil V U (θ ), m(θ ), V (θ ), v = , U = ( mu ) + ( mu )oil , u = m m Ideal Gas T = (γ − 1)
u R
p=
RT v
Real gas p=f1(T,v) u=f2(T,v)
Wet vapour
u = (1 − x ) u f + xug
v = (1 − x ) v f + xvg
SCREW COMPRESSOR THERMODYNAMICS
Compressor integral parameters m = min − mout
Wind =
m! = mz1n / 60 V! = 60m / ρ 0 F1n + F2 n ) Lnz1 ρ ( m! t = 60 m! ηv = m! t
Psin d
P = V!
Wt ηt = Wind Wt = RT1 ln
∫
Vdp
cycle
Wind z1 n Pind = 60 V Wsind = ∫ dp m cycle
Wa ηa = Wind p2 p1
Wa =
γ R (T2 − T1 ) γ −1
SCREW COMPRESSOR THERMODYNAMICS
Calculation of pressure loads On compressor rotors Radial forces B
Rx = − p ∫ dy = − p ( yB − y A ) A
B
Ry = − p ∫ dx = − p ( xB − x A ) A
Rotor torque B
B
(
T = p ∫ xdx + p ∫ ydy − 0.5 p xB − x A + yB − y A A
A
2
2
2
2
)
SCREW COMPRESSOR THERMODYNAMICS
Bearing reactions and rotor deflections d 2δ M = 2 EI dz
SCREW COMPRESSOR OPTIMIZATION
Optimization variables and target function Single stage: Rotor variables: r0 Female rotor addendum r1 Male rotor lobe radius r2 Male rotor tip radius r3 Female rotor tip radius Compressor variables: Built-in volume ratio Operation variables: Shaft speed Oil flow Injection position Oil temperature 9 Variables
Multistage: 9 Variables x Number of stages + Interstage pressures Target function: Specific power combined with compressor price F=w1L+w2C
Box constrained simplex method for efficient and reliable multivariable optimization f ( x1 , x2 ,..., xn ) F = w1 L + w2C gi ≤ xi ≤ hi , i = 1, n gi ≤ yi ≤ hi , i = n + 1, m
yn+1,…,ym
f ( x h ) = max f ( x1 ), f ( x 2 ),..., f ( x k ) f ( x g ) = min f ( x1 ), f ( x 2 ),..., f ( x k ) 1 k i r l i l = + α − ( ) x x x x ≠ , x= x x x ∑ j k − 1 i =1 x r ( new ) = 0.5 x r ( old ) + cx + (1 − c) x h + ( x − x h )(1 − c)(2 R − 1)
nr c= 1 + − n k r r
nr + kr −1 nr
r0 [mm] r1 [mm] r2 [mm] r3 [mm] Built-in volume ratio Rotor speed [rpm] Oil flow [lit/min] Injection position [o] Oil temperature [o]
Oil Free Oil Flooded Refrigeration
Dry Oil-Flooded Refrig. 2.62 0.74 0.83 19.9 17.8 19.3 6.9 5.3 4.5 11.2 5.5 5.2 1.83 4.1 3.7 7560 3690 3570 12 8 65 61 33 32
EXAMPLE OF CALCULATION 5-6-128 mm Oil-Flooded Air Compressor
7 m3/min, max 10 m3/min at 8 bar (abs) 5-14 bar (abs), max 15 bar (max)
EXAMPLES OF CALCULATION
Rotors optimized for oil flooded operated air compression
5/6-128 mm, L/D 1.65 Displacement 1.56 l/rev Interlobe sealing line 0.13 m Blow-hole area 1.85 mm2
EXAMPLES OF CALCULATION
CAD Interface: Compressor ports
EXAMPLES OF CALCULATION
Experimental verification of the model
EXAMPLES OF CALCULATION
Compressor in the test bed
EXAMPLES OF CALCULATION
Comparison of the calculated and test results Flow-Power
EXAMPLES OF CALCULATION
EXAMPLES OF 3-D CFD CALCULATION Majority of design problems can be solved by the one-dimensional approach, some of them require the two dimensional calculation, however, there are situations where 3-D CFD must be applied Such are Oil flow distribution, Fluid-Solid Interaction
Grid generation - 1 • Grid topology - polyhedral - O - grid - H - grid - C - grid
• Cell shape
Grid generation - 2 • Grid topology strongly affects accuracy, efficiency and ease of calculation • Full structured block generated hexahedral 3D-O mesh • Screw compressor sub-domains: - Male rotor - Female rotor - End clearances Rotor connections, clearances, leakage paths - Suction port - Discharge port - Suction and discharge receivers Automatic discretization process: - The rack generating procedure - Rack - a rotor with an infinite radius - Divides working domain in two parts male and female rotor,
Comet
Screw Compressor FSI calculations
Mathematical model for screw compressor is based on conservation laws of continuity, momentum, energy, concentration and space: d ρ dV + ∫ ρ ( v − v s ) ⋅ ds = 0 dt V∫ S
ρ = ρ ( p, T ),
! − 2 µ div vI − pI T = 2µ D 3 T = 2η D + λ div wI −
d ρ vdV + ∫ ρ v( v − v s ) ⋅ ds = ∫ T ⋅ ds + ∫ fb dV dt V∫ S S V
(3λ + 2η )α (T − Tr ) I
d ρ hdV + ∫ ρ h( v − v s ) ⋅ ds = ∫ q h ⋅ ds + dt V∫ S S
∫ sh dV + ∫ ( v ⋅ grad p + S : grad v) dV − ∫ pvs ⋅ ds +
V
V
S
d ρ co dV + ∫ ρ co ( v − v s ) ⋅ ds = ∫ q co ⋅ ds + ∫ Sco dV dt V∫ S S V d dV − ∫ v s ⋅ ds = 0 dt V∫ S
e = e( p , T )
d pdV dt V∫
q h = κ grad T q co = ρ Do grad co
Closed by constitutive relations and equation of state and accompanied by turbulence model.
d ρ kdV + ∫ ρ k ( v − v s ) ⋅ ds = ∫ q k ⋅ ds + ∫ ( P − ρε )dV , S S V dt ∫V ε ε2 d ρε dV + ∫ ρε ( v − v s ) ⋅ ds = ∫ qε ⋅ ds + ∫ (C1 P − C2 ρ + C3 ρε ∇ ⋅ v)dV ∫ V S S V dt k k
Thermodynamic properties of real fluids - p-v-T equation compressibility factor z - z is assumed to change linearly with pressure err<2% - Antoine equation for saturation temperature
p = z ⋅ RT = z ( p) ⋅ RT ρ
z = p ⋅ B1 + B2 Tsat =
A2 A1 − log p
- Clapeyron equation for latent heat
1 1 dPsat hL = T ⋅ − ⋅ ρ v ρl dTsat
- Specific heat for constant pressure
c pv = C0 + C1 ⋅ T + C2 ⋅ T 2 + C3 ⋅ T 3 ρ=
- Density of mixture - Coefficient in the pressure correction equation
1 1 − co2 co2 − ρv ρl
dρ 1 ρ v ⋅ b1 ρ v − Cρ = = ⋅ dp zRT z ρ T
Multiphase flow
Euler-Lagrange approach
d (mo ho ) dh dmo ! = mo o + hol = Qcon + Q! mass dt dt dt
- Oil is assumed to be a passive ‘species’ - Mass calculated from the concentration Oil drag force influence concentration
mo = m ⋅ Co 1 f drag = − ρ Ao Cdrag v o − v ( v o − v) 2
- Energy source due to heat transfer between working fluid and oil is:
dT T k − T k −1 ! ≈ mo C po Qcon = mo C po δt dt
- Liquid phase is assumed to be an active ‘species’ m ⋅ C pm ⋅ (T − Ts )
- Mass source evaporated/condensed mass
m! L =
- Energy source energy of evaporation/condensation
m − mol dmL Q! mass = hL ≈ hL ol = hL m! L δt dt
hL s
Boundary conditions - Wall boundaries with wall functions are introduced on the housing and rotors. - Compressor positioned between suction and discharge receivers of small volume - Inlet & outlet receivers and oil port are treated as boundary domains:
- Mass equation corrected by mass source to maintain constant pressure
− p V ⋅ρ p dm ≈ const ⋅ m! add = δt pconst dt p =const
- Energy equation corrected by energy source to update energy balance
dm! = m! add ⋅ hadd Q! add = hadd add dt p = const
Screw Compressor performance - Volume flow (inlet and outlet)
tend
∑
V! = 60 ⋅
t =tstart
- Mass flow (inlet, outlet and oil)
m! =
tend
∑ V!
t = t start
- Boundary forces
(t ) f
⋅ ρ (t )
[kg
i =1
sec ]
Fx = pb * Axb ; Fy = pb * Ayb ; Fz = pb * Azb I
- Restraint Forces and Torque
I
(t ) V!f = ∑ v fi S fi
(t ) V!f m3 min ,
FrS = ∑ FrS (i ),[ N ]; i =1
I
Fa = ∑ Fa (i ),[ N ]; i =1
I
FrD = ∑ FrD (i ), [ N ] i =1
I
T = ∑ T (i ), [ Nm] i =1
- Compressor shaft power
P = 2 ⋅ π ⋅ n ⋅ (TM + TF ) [W ]
- Specific power
Pspec = P ! V ⋅1000
- Efficiency Volumetric and adiabatic
! ηv = V
Vd
;
kW 3 m min
ηi = Pad
P
Oil injected - Pressure in axial section
Oil injected - Pressure and velocity
Oil injected - Pressure 3D view
Real fluid - Ammonia – pressure
Experimental verification – P-α diagram
Oil injected – Deformation Pressure1
Pinl=1 b Pout=7 b n=5000 rpm tinl=20 oC tout=40 oC
Oil injected – Deformation Pressure-1
Pinl=1 b Pout=7 b n=5000 rpm tinl=20 oC tout=40 oC
Oil injected – Deformation Pressure-2
Pinl=1 b Pout=7 b n=5000 rpm tinl=20 oC tout=40 oC mag=20,000x
Oil injected – Deformation Temperature
Pinl=1 b Pout=3 b n=5000 rpm tinl=20 oC tout=150oC mag=1,000x
Oil injected – Deformation Pressure+Temperature
Pinl=30 b Pout=90 b tinl=0 oC tout=40 oC
n=5000 rpm mag=2,000x
DESIGN EXAMPLES
Oil-free air compressor Oil-flooded air compressor Retrofit rotors of an air compressor Refrigeration compressor
DESIGN EXAMPLES
Compressors for Oil-Free Air Delivery Design aims: Delivery: 350-700 and 700-1000 m3/h Working pressure: 1-2.5 (2.7) bar Volumetric efficiency 90 % + Low specific power Simple, reliable and compact machine
DESIGN EXAMPLES
3/5 lobe rotors Large displacement Convenient gear ratio 5/3=1.67
DESIGN EXAMPLES
Features of 3/5 ‘N’ Rotors - Highest possible displacement - Higher delivery - Better volumetric efficiency - Better adiabatic efficiency - Stronger gate rotor - Long durability - High reliability - Easy manufacturing - Easy compressor assembly - Reasonable noise
DESIGN EXAMPLES
XK18 Screw Compressor General Arrangement
DESIGN EXAMPLES
New design, fully customized by the manufacturer
New rotors, New, improved compressor New concept, better screw compressor compared with competition
DESIGN EXAMPLES
Comparison of Test Results R1- GHH C80 R2- Drum D9000 R3- Mouvex Typhoon R4- GHH CS-1000
DESIGN EXAMPLES
Screw Compressor Family for Oil-Flooded Operation Design aims: Delivery: 0.6-60 m3/min Working pressure: 5-13 bar Volumetric efficiency 90 % + Low specific power Simple, reliable and compact machine
DESIGN EXAMPLES
4/5 Rotors
Reasonable efficiency for moderate pressure ratios, at least the same as for 4/6 rotors Improved capacity and efficiency Lower power consumption Reduced manufacturing cost
Five compressors, rotor diameters: 74, 102, 159, 225 and 285 mm, L/D 1.55
DESIGN EXAMPLES
Performance of the Compressor Family
DESIGN EXAMPLES
Proven design, fully customized to the the manufacturer’s needs
New rotors, New, improved compressor
DESIGN EXAMPLES
Test Results, Compressors 73 and 159 mm
DESIGN EXAMPLES
Retrofit ‘N’ Rotors
5.4 % more displacement 6.5 % higher delivery 4 % better volumetric efficiency 2.5 % better adiabatic efficiency 75 % less torque on the gate rotor
DESIGN EXAMPLES
Asymmetric Rotors, the most common screw compressor rotors
‘N’ Rotor retrofit for more efficient screw compressors
DESIGN EXAMPLES
Design of a semihermetic compressor for air-conditioning and refrigeration based on ‘N’ rotors • • • • •
Semihermetic, convertible to open All existing refrigerants Modern, better than competition 5/6-102 mm L/D=1.55 Efficient rotors for air condition and refrigeration
DESIGN EXAMPLES
5/6-102 mm L/D=1.55 compressor
• • • • • • • •
Theoretical displacement 0.72 l/rev Gear-box 1500-10000 rpm Delivery 0.920-5.80 m3/min R-22 Air condition 5/40 oC 60-380 kW COP 4.20 at 3000 rpm Refrigeration -5/30 oC 33-210 kW COP 3.95 at 3000 rpm
DESIGN EXAMPLES
Refrigeration Application
Application of Design Optimization: A Family of Two-Stage Compressors
Range: 22-250 kW 8-16 bar (abs)
2nd
Two-stages: 19 Variables Target Function: Minimum specific power And only specific power 1st
Number of frames VFD vs gearbox Stage size Stage speed Stage rotor profile
CONCLUSIONS The screw compressor is a mature product at the millenium meeting point. Orchestrated efforts of a large number of companies driven by market forces resulted in the compact and efficient compressor machine. Every detail counts today. A small difference will give a small, but distinctive improvement which may be used as an individual advantage. Optimization opens hidden spots still left for better compressors. They result in stronger but lighter rotors with higher displacement, more compact and more efficient compressor machines at all areas of their application.
EPILOGUE The Centre for Positive Displacement Compressor Technology carries out research and provide a service to manufacturing companies in all aspects of compressor design and development. Through the everyday activity the Centre is gaining experience in the compressor research, development and design. Some of the recent experience is given in this presentation.