The initial cyclic condition of this new energy-saving cooling system is previously displayed in Fig. 5.2. In this stable working condition, piston H locates at the lower position in the compressive chamber and free piston F is at the right end of the expansive chamber. TheP–Vcyclic diagram of expansive chamber is shown in Fig.5.3.
Because the gaseous media with high pressure (PẳPH) enters chamber Vmcvia a tiny hole at the end time period of last functioning cycle, gaseous media with low pressure (PẳPL) in chamber Vmcwill be mixed with high-pressure gaseous media.
The mixed gaseous pressure in chamber VmcisPmix. When piston H starts moving upward in compressive chamber Vc, gaseous media in chamber Vmcis assumed not entering chamber V0mc since the traveling speed of free piston is fast and the diameter of hole is very small. Because the left-side gaseous pressure of free piston F is higher than the right-side gaseous pressure of F (PmcẳPave>Pe), free piston F does not move and the gaseous pressurePein expansive chamber reaches high pressurePHand gaseous volume Vein expansive chamber increases from zero to V2as piston H continues moving up. This process can be represented by curve 1–2 in Fig. 5.3. Then the gaseous media in chamber Vmc flows into chamber V0mc. Since the gaseous pressure at the left and right sides of free piston F is same, F travels to the left with constant speed. When crankshaft rotates to the angle (θ) near 180, the gaseous volumeVein expansive chamber increases fromV2toV3which can be represented by the line 2–3. During initial period that piston H moves downwards in chamber Vc, free piston F does not move since the gaseous media in chamber V0mc is assumed not to flow into chamber Vmc, and pressure Pe in expansive chamber reduces from P3 to P4 which is represented by line 3–1.
When piston H continues traveling downwards in compressive chamber Vc, gaseous media in chamber V0mcenters chamber Vmcvia tiny hole and free piston F moves to the right end in the expansive chamber with constant speed which can be represented by line 4–1. This concludes the full functioning cycle of this cooling system.
Fig. 5.3 P–Vcyclic diagram in expansive chamber
5.2 Computer-Aided Simulation on Energy-Saving Cooling System 67
The computer-aided simulations can be applied to verify the function of this new cooling system. The piston H in compressive chamber moves in sine law (Kundu and Cohen 2008):
Vcð ị ẳθ MVcoẵ1ỵcosð ịθ
2 ð5:1ị
VẳVkỵVcỵVe ð5:2ị Here,Vkis the clearance volume:
PẳRXMi Vk
Tk
þVe
Te
þVc
Tc
ð5:3ị
As piston H in compressive chamber travels upwards with crankshaft rotating to angleω,
PePme
ð ị AẳFF ð5:4ị
Pð ị ẳω RXMi Vk
Tk
ỵ0ỵVcð ịω Tc
ð5:5ị
AssumingWẳTTck,LẳVVeok, andQẳTTce
PẳPð ị ω hLWỵVVcð ịeoωi LWþVVeoe
h i QỵVcð ịω Veo
ð5:6ị
The differential equation of free piston motion can be derived based on the second Newton law:
m d2 Veð ịθ dt2
" #
ẳ PePmc
S
FF ð5:7ị
Combine the above equations:
d2 VVeð ịθ
eo
h i
dθ2 ẳPð ị ω hLWỵVVcð ịeoωi
mw2Yo
A
1
LWỵQVVeoeð ịθ ỵVVcð ịeoθ A
PLþPH
2mw2Yo
ð5:8ị
68 5 Energy-Saving Cooling System
Here, pressureP(ω) can be determined by the following equation:
d2Vcð ịθ
dθ2 ẳ0 ð5:9ị
The computer-aided simulation determines the stress and deflection profiles shown in Figs.5.4,5.5,5.6,5.7,5.8,5.9,5.10,5.11,5.12,5.13,5.14,5.15,5.16, 5.17, and5.18.
The computer-aided simulation and analysis in Figs.5.5and5.6show the stress and deflection profiles of compressive piston link assembly in newly designed energy-saving cooling system. The analytic results exhibit that the maximum stress of 25,698.25 psi in this piston link assembly is less than the material yield strength of 36,300 psi and maximum deflection of 0.00169 in. is within material allowable deformation limit.
The computer-aided simulation and analysis in Figs. 5.8 and 5.9present the stress and deflection profiles of compressive chamber in newly designed energy- saving cooling system. The analytic results demonstrate that the maximum stress of 26,088.02 psi in this compressive chamber is less than the material yield strength of 36,300 psi and maximum deflection of 0.00017 in. is within material allowable deformation limit.
The computer-aided simulation and analysis in Figs.5.11and5.12indicate the stress and deflection profiles of piston link in newly designed energy-saving cooling system. The analytic results state that the maximum stress of 23,242.99 psi in this piston link is less than the material yield strength of 36,300 psi and maximum deflection of 0.00147 in. is within material allowable deformation limit.
Fig. 5.4 Piston and link assembly in compressive chamber
5.2 Computer-Aided Simulation on Energy-Saving Cooling System 69
Fig. 5.5 Stress profile of compressive piston link assembly
Fig. 5.6 Deflection profile of compressive piston link assembly
The computer-aided simulation and analysis in Figs.5.14and5.15display the stress and deflection profiles of crankshaft in newly designed energy-saving cooling system. The analytic results show that the maximum stress of 20,667.27 psi in this crankshaft is less than the material yield strength of 36,300 psi and maximum deflection of 0.00019 in. is within material allowable deformation limit.
Fig. 5.7 Piston head in compressive chamber
Fig. 5.8 Stress profile of piston head in compressive chamber
5.2 Computer-Aided Simulation on Energy-Saving Cooling System 71
Fig. 5.9 Deflection profile of piston head in compressive chamber
Fig. 5.10 Piston link in compressive chamber
72 5 Energy-Saving Cooling System
Fig. 5.11 Stress profile of piston link in compressive chamber
Fig. 5.12 Deflection profile of piston link in compressive chamber
5.2 Computer-Aided Simulation on Energy-Saving Cooling System 73
Fig. 5.13 Crankshaft in compressive chamber
Fig. 5.14 Stress profile of crankshaft in compressive chamber
74 5 Energy-Saving Cooling System
Fig. 5.15 Deflection profile of crankshaft in compressive chamber
Fig. 5.16 Piston in expansive chamber
5.2 Computer-Aided Simulation on Energy-Saving Cooling System 75
Fig. 5.17 Stress profile of piston in expansive chamber
Fig. 5.18 Deflection profile of piston in expansive chamber
The computer-aided simulation and analysis in Figs.5.17and5.18exhibit the stress and deflection profiles of piston in newly designed energy-saving cooling system. The analytic results present that the maximum stress of 7,884.44 psi in this piston is less than the material yield strength of 36,300 psi and maximum deflection of 0.00006 in. is within material allowable deformation limit.
The above computational simulation results demonstrate that the maximum stresses on these critical components are all less than the material yield stress and maximum material deflections are all within material allowable deformation limits.
The computational solutions confirm the good and reliable function of this newly developed energy-saving cooling system.