罗茨式蒸汽动力机的参数设计及性能分析

肖艳军，李春霞，王大源河北工业大学机械工程学院，天津 300130

1．Introduction

At present，waste heat is common in industrial production，which accounts for about 60%of the total waste heat resources［1］．But the low-temperature recovery device is limited by unstable factors，which lead to hinder the development of low-temperature waste heat recovery technology［2-5］．The current minimum steam turbine is 750 kW，so most of enterprises cannot be good for recycling a large number of low temperature waste heat resources(under 230℃，1 000 kW)in all kinds of industries．At home and abroad，there are a lot of research on screw expander，but its own high machining accuracy and higher investment cost，so the screw expander has not on a large scale to enter the market and only makes the hundreds kilowatt［6］．In order to meet the demand of market，the new device of low investment and high efficiency is imperative．

Roots-steam engine is a low-temperature recovery device，which uses compressed gas energy to convert into mechanical energy．Roots type steam engine is inverse application to roots blower，both of them are basically the same in structure．So it has simple structure and low cost．The working principle of roots-steam engine is similar to gear motor［7］，which uses compressed gas energy to form opposite direction torque and final driving load．In recent years，a lot of industries produced in the form of flue gas，steam and hot water，such as steel mills，nonferrous metal factories and petrochemical engineering，architectural material plants and light manufacturing．Especially the low-temperature waste heat has not been fully used．Therefore，the further theoretical analysis of roots-steam engine is conductive to recycle low-temperature waste heat．

2．Roots-steam engine

2．1．Structure and working principle

The roots-steam engine consists of two parts:the external shell and power transmission device，which is shown in Figure 1．Figure 1(a)shows a closed space，including chassis，front wallboard and back wallboard，main and auxiliary fuel tanks．Figure 1(b)shows a power transmission device，which consists of meshing rotor，output and synchronous rotor shaft，synchronous gear and bearing sleeve． The front and back wallboard are used for fixing bearing sleeve so that power transmission device can be con-nected to the shell．For low-temperature application field of roots-steam engine，some special treatments are made in corrosion resistance and fatigue strength．

Figure 1．The structure of roots-steam engine

The working principle of roots-steam engine is similar to gear motor［8］．When pressure air from the inlet into chamber，can produce thrust(as shown in the arrow in Figure 2)on the rotor．The thrust can generate torque relative to axis．Under the action of torque，two rotors gradually rotate as shown in the Figure 2，so the load can run through output shaft driving．

Figure 2．The working principle of roots-steam engine

2．2．The design of rotor shape

The design steps of circular arc rotor shape are described as follows［9］:firstly we should confirm the rotor vane number and center distance between both of rotors according to the requirement．Choosing appropriate span ratio determine the radius of rotor tip circle and rotor peak circle，the distance between rotor peak center and rotor center．Finally the equation of circular arc rotor is determined based on working clearance of roots-steam engine．

Main parameters of roots-steam engine are listed as follows:the rotor vane number is 2，the center distance between both of rotors is202 mm，the radius of rotor tip circle is 150 mm，the radius of rotor peak circle is 72 mm，the distance between rotor peak center and rotor center is 78 mm，rotor working clearance is 2 mm．The equation of circular arc rotor is shown as follows．

The equation of rotor peak:

The equation of rotor valley:

θ—the parameter of parametric equation;α—the angle of the rotor turning;β—the angle between meshcircle line and horizontal line．

Figure 3 shows the 3D circular arc rotor model using parameter equation．

Figure 3．Model of arc rotor

The design steps of circular arc-involutes rotor shape are as follows［10］:firstly according to the requirement we should confirm the rotor vane number and center distance between both of rotors．Choosing appropriate span ratio determine the radius of rotor tip circle，the radius and pressure angle of pitch-circle and the radius of base-circle and so on．Finally the equation of circular arc-involutes rotor is determined based on working clearance of roots-steam engine．

Main parameters of roots-steam engine as follows:the rotor vane number is 2，the center distance between both of rotors is 202 mm，the radius of rotor tip circle is 150 mm，the radius of rotor pitch-circle is 101 mm，the pressure angle of rotor pitch-circle is 51．68°，the radius of base-circle is 63 mm，rotor working clearance is 2 mm．The equation of circular arc-involutes rotor is as follows．

The equation of rotor involutes:

φ—the generating angle of involute

The equation of rotor valley:

φ—the generating angle of involute

The equation of rotor peak:

φ—the generating angle of involute

Figure 4 shows the 3D circular arc-involutes rotor model using parameter equation．

Figure 4．Model of arc-involutes rotor

2．3．The theoretical calculation of roots-steam engine

According to the working principle of rootssteam engine，pressure air from inlet can produce torque on the rotor．This is the theoretical input torque of roots-steam engine．In Figure 2，the rotor tip circle line is A1C1and A2C2;the rotor hypothetical meshing points are B1and B2．The effective pressure arc length of rotor1 is．The effective pressure arc length of rotor1 is．The torque of rotor 1 and 2 are as follows．

O1B1，O2B2—meshing radius of rotor，m;P1—inlet pressure of roots-steam engine，Pa;P2—outlet pressure of roots-steam engine，Pa;b—rotor length of roots-steam engine，m．

The Figure 2 shows:

Rm—addendum circle radius of rotor，m

The input torque of roots-steam engine is as follows:

That formula shows that the input torque of roots-steam engine is proportional to the difference between inlet pressure and outlet pressure．And it is also proportional to the rotor length of roots-steam engine．When the engine is rotating，the meshing radius of rotor is changing．So the meshing radius should take average value to calculate torque．

The Figure 5 shows theoretical input torque of roots-steam engine in low-temperature conditions．Main parameters of roots-steam engine as follows:the rotor vane number is 2，the center distance between both of rotors is 202 mm，the radius of rotor tip circle is 150 mm，rotor length of roots-steam engine is 355 mm，rotational speed around the center is 98．125 r/min．And we can see that input torque of roots-steam engine is proportional to the difference between inlet pressure and outlet pressure．

Figure 5．Input torque of roots-steam engine

3．The simulation

The working principle of roots-steam engine using waste heat is complexity，so we need to carry on the internal flow analysis of engine to reflect the flow in the body，and now the research is less．In order to further analysis analyze performance of engine，dynamic grid technique is adopted to simulate the flow process of the roots-steam engine．

3．1．Fluid domain

Look at circular arc-involutes rotor，for example．The 2D model of roots-steam engine is established using software．Figure 6 shows fluid domain between chassis and rotors．

Figure 6．The fluid domain model of roots-steam engine

3．2．Grid model

Then import fluid domain into mesh modules．Considering the grid computing，the model use triangular mesh division．Figure 7 shows that element number is 42976 and node number is 21750 of mesh model．That mesh model not only meets calculation precision，but also guarantees the calculation speed．

Figure 7．The grid model of roots-steam engine

3．3．Outline files

The movement of grid is rotation of two rotors around their central axis．The left rotor is spinning counterclockwise and the right rotor is spinning clockwise．The rotational speed around the center is 98．125 r/min．The outline file is as follows．

(rotating_left 3 point)

(time 0 1 60)

(omega_z-98．125-98．125-98．125)

(rotating_right 3 point)

(time 0 1 60)

(omega_z 98．125 98．125 98．125)

3．4．Results analysis

The roots-steam engine selects 0．6 MPa condition to simulate working performance at CFD environment．The inlet boundary condition is 0．6 MPa，5%turbulence intensity and 0．05 Hydraulic Diameter．The output boundary condition is 0 MPa，5%Reflux turbulence intensity and 0．05 Reflux Hydraulic Diameter．The rotor boundary condition is moving wall and 98．125 r/min rotational speed around the center．That boundary condition load to the model，and then we obtain speed vector diagram(Figure 8)of roots-steam engine and stress diagram(Figure 9)．

Figure 8．The velocity vector diagram

Figure 9．The cloud picture of stress

From the analysis results，we can see that the speed vector diagram of engine in low-temperature conditions produce phenomenon of impact and vortex，which making the rotor hit and will reduce the life of the rotor．The maximum stress value of fluid domain is 0．6 MPa，and the roots-steam complies with the design specification．

4．Experimental study

In order to prove that roots-steam can run smoothly under low temperature resources，we should do experimental test in different conditions．Table 1 shows electricity of roots-steam engine in low temperature resources conditions．

Table 1．Electricity of roots-steam engine

5．Conclusion

Using parameter design of rotor and dynamic grid technique，this paper establish movement boundary to simulate fluid domain of roots-steam engine．It is not only capture the instantaneous flow changing，but also can understand performance of engine at the low-temperature conditions．That result can provide reference for the design and structure optimization．

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