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Thermodynamic Calculations of Two-Stage Vapor Compression Refrigeration Cycle with Flash Chamber and Separate Vapor Mixing Intercooler

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Thermodynamic Calculations of  Two-Stage Vapor Compression Refrigeration Cycle with Flash Chamber and Separate Vapor Mixing Intercooler

Introduction
 

In the event that a high COP of a refrigeration cycle is of greater importance compared to other factors, it is possible to significantly increase the COP of a basic cycle through the use of a multistage vapor compression cycle. This is especially true when the pressure ratio between the heat rejection and heat absorption pressures is large 5 or more.
Multistaging involves one or more intermediate pressures between the heat rejection and heat absorption pressures, and a series of compressors operating between successive pressure intervals.
One type of multi-compressor vapor compression cycle includes a mixing chamber where saturated vapor from the flash chamber mixes with the vapor leaving the low pressure stage compressor. This vapor-mixing chamber acts as a regenerative intercooler since it cools the superheated vapor leaving the low-pressure stage compressor using lower temperature saturated workin fluid, mixing the two prior to the next stage of compression.

This type of refrigeration cycle is analysed in the following calculation.

Creation functions on properties and processes of working fluids
 
> 

restart
> 

with(ThermophysicalData); with(Units[Standard]); with(plots)

 

Pressure in the saturated region as a function of temperature, quality and working fluid

> 

PSTXwf := proc (T, X, wf) options operator, arrow; Property(P, temperature = T, Q = X, wf) end proc

 

Temperature in the saturated region as a function of pressure, quality and working fluid

> 

TSPXwf := proc (p, X, wf) options operator, arrow; Property(T, pressure = p, Q = X, wf) end proc

 

Specific enthalpy in the saturated region as a function of temperature, quality and working fluid

> 

HSTXwf := proc (T, X, wf) options operator, arrow; Property(enthalpy, temperature = T, Q = X, wf) end proc

 

Specific enthalpy in the saturated region as a function of pressure, quality and working fluid

> 

HSPXwf := proc (p, X, wf) options operator, arrow; Property(enthalpy, pressure = p, Q = X, wf) end proc

 

Specific entropy in the saturated region as a function of temperature, quality and working fluid

> 

SSTXwf := proc (T, X, wf) options operator, arrow; Property(entropy, temperature = T, Q = X, wf) end proc

 

Specific entropy in the saturated region as a function of pressure, quality and working fluid

> 

SSPXwf := proc (p, X, wf) options operator, arrow; Property(entropy, pressure = p, Q = X, wf) end proc

 

Density in the saturated region as a function of temperature, quality and working fluid

> 

DSTXwf := proc (T, X, wf) options operator, arrow; Property(D, temperature = T, Q = X, wf) end proc

 

Density in the saturated region as a function of pressure, quality and working fluid

> 

DSPXwf := proc (p, X, wf) options operator, arrow; Property(D, pressure = p, Q = X, wf) end proc

 

Temperature as a function of pressure, specific entropy and working fluid

> 

TPSwf := proc (p, s, wf) options operator, arrow; Property(T, pressure = p, entropy = s, wf) end proc

 

Temperature as a function of pressure, specific enthalpy and working fluid

> 

TPHwf := proc (p, h, wf) options operator, arrow; Property(T, pressure = p, enthalpy = h, wf) end proc

 

Specific enthalpy as a function of pressure, temperature and working fluid

> 

HPTwf := proc (p, T, wf) options operator, arrow; Property(H, pressure = p, temperature = T, wf) end proc

 

Specific entropy as a function of pressure, temperature and working fluid

> 

SPTwf := proc (p, T, wf) options operator, arrow; Property(S, pressure = p, temperature = T, wf) end proc

 

Density as a function of pressure, temperature and working fluid

> 

DPTwf := proc (p, T, wf) options operator, arrow; Property(D, pressure = p, temperature = T, wf) end proc

 

Specific entropy as function of pressure and specific enthalpy and working fluid

> 

SPHwf := proc (p, h, wf) options operator, arrow; Property(S, pressure = p, enthalpy = h, wf) end proc

Density as function of pressure and specific enthalpy and working fluid

> 

DPHwf := proc (p, h, wf) options operator, arrow; Property(D, pressure = p, enthalpy = h, wf) end proc

 

> 

HcompPTPEFFwf := proc (p1, T1, p2, eff, wf) local h1, s1, s2_is, T2_is, h2_is; if TSPXwf(p1, 1, wf) < T1 then h1 := HPTwf(p1, T1, wf); s1 := SPTwf(p1, T1, wf); s2_is := s1; T2_is := TPSwf(p2, s2_is, wf); h2_is := HPTwf(p2, T2_is, wf) else h1 := HSTXwf(T1, 1, wf); s1 := SSTXwf(T1, 1, wf); s2_is := s1; T2_is := TPSwf(p2, s2_is, wf); h2_is := HPTwf(p2, T2_is, wf) end if; return h1+(h2_is-h1)/eff end proc

 

 

> 

ScompPTPEFFwf := proc (p1, T1, p2, eff, wf) local h2, T2; h2 := HcompPTPEFFwf(p1, T1, p2, eff, wf); T2 := TPHwf(p2, h2, wf); return SPTwf(p2, T2, wf) end proc

 

> 

TcompPTPEFFwf := proc (p1, T1, p2, eff, wf) local h2; h2 := HcompPTPEFFwf(p1, T1, p2, eff, wf); return TPHwf(p2, h2, wf) end proc

 

> 

DcompPTPEFFwf := proc (p1, T1, p2, eff, wf) local h2, T2; h2 := HcompPTPEFFwf(p1, T1, p2, eff, wf); T2 := TPHwf(p2, h2, wf); return DPTwf(p2, T2, wf) end proc

 

 

Critical temperature as a function of working fluid

NULL

> 

TCRwf := proc (wf) options operator, arrow; Property(T_critical, water, useunits) end proc

 

Critical pressure as a function of working fluid

NULL

> 

PCRwf := proc (wf) options operator, arrow; Property(pcritical, water, useunits) end proc

 

Input Data
 

Temperature of heat source

> 

T__in := ((-10)+273.15)*Unit('K')

263.15*Units:-Unit(K)

 (3.1)

Temperature of heat consumption

> 

T__out := (50+273.15)*Unit('K')

323.15*Units:-Unit(K)

 (3.2)

Temperature difference in the evaporator

> 

`&delta;T__EV` := 2*Unit('K')

2*Units:-Unit(K)

 (3.3)

Temperature difference in the condenser

> 

`&delta;T__CD` := 2*Unit('K')

2*Units:-Unit(K)

 (3.4)

Isentropic efficiency of the compressor

> 

`&eta;__comp` := .8

.8

 (3.5)

Pressure increase in compressor 1

> 

`&pi;__comp,1` := 3

3

 (3.6)

Working fluid

> 

wf := R407c

R407c

 (3.7)

 

Calculations
 

Temperature of  the working fluid at the evaporator outlet

> 

T[1, L] := T__in-`&delta;T__EV`

261.15*Units:-Unit(K)

 (4.1)

Pressure of  the working fluid at the evaporator outlet

> 

p[1, L] := PSTXwf(T[1, L], 1, wf)

296142.9008*Units:-Unit(Pa)

 (4.2)

Specific enthalpy of  the working fluid at the evaporator outlet

> 

h[1, L] := HSTXwf(T[1, L], 1, wf)

403311.4461*Units:-Unit(J/kg)

 (4.3)

Specific entropy of  the working fluid at the evaporator outlet

> 

`s__1,L` := SSTXwf(T[1, L], 1, wf)

1791.745337*Units:-Unit(J/(kg*K))

 (4.4)

Specific volume of  the working fluid at the evaporator outlet

> 

`v__1,L` := 1/DSTXwf(T[1, L], 1, wf)

0.7794419292e-1*Units:-Unit(m^3/kg)

 (4.5)

 

Pressure of  the working fluid at the compressor 1 outlet

> 

p[2, L] := p[1, L]*`&pi;__comp,1`

888428.7024*Units:-Unit(Pa)

 (4.6)

Pressure of  the working fluid at the flash chamber outlet

> 

p[3, L] := p[2, L]

 

888428.7024*Units:-Unit(Pa)

 (4.7)

 

Temperature of  the working fluid at the flash chamber outlet

> 

`T__3,L` := TSPXwf(p[3, L], 0, wf)

287.7067379*Units:-Unit(K)

 (4.8)

Specific enthalpy of  the working fluid at the flash chamber outlet

> 

h[3, L] := HSTXwf(`T__3,L`, 0, wf)

221044.0888*Units:-Unit(J/kg)

 (4.9)

Specific entropy of  the working fluid at the flash chamber outlet

> 

`s__3,L` := SSTXwf(`T__3,L`, 0, wf)

1074.098978*Units:-Unit(J/(kg*K))

 (4.10)

Specific volume of  the working fluid at the flash chamber outlet

> 

`v__3,L` := 1/DSTXwf(`T__3,L`, 0, wf)

0.8468535493e-3*Units:-Unit(m^3/kg)

 (4.11)

 

Pressure of  the working fluid at the evaporator intlet

> 

p[4, L] := p[1, L]

296142.9008*Units:-Unit(Pa)

 (4.12)

Specific enthalpy of  the working fluid at the evaporator intlet

> 

h[4, L] := h[3, L]

221044.0888*Units:-Unit(J/kg)

 (4.13)

Temperature of  the working fluid at the evaporator intlet

> 

`T__4,L` := TPHwf(p[4, L], h[4, L], wf)

255.9553455*Units:-Unit(K)

 (4.14)

Specific enthalpy of saturated liquid of  the working fluid at the evaporator inlet

> 

h[sl, 4, L] := HSPXwf(p[4, L], 0, wf)

174250.2741*Units:-Unit(J/kg)

 (4.15)

Quality of the working fluid at the evaporator intlet

> 

x[4, L] := (h[4, L]-h[sl, 4, L])/(h[1, L]-h[sl, 4, L])

 

.2042852322

 (4.16)

 

Specific entropy of saturated liquid of  the working fluid at the evaporator inlet

> 

`s__sl,4,L` := SSPXwf(p[4, L], 0, wf)

903.2121325*Units:-Unit(J/(kg*K))

 (4.17)

Specific volume of saturated liquid of  the working fluid at the evaporator inlet

> 

`v__sl,4,L` := 1/DSPXwf(p[4, L], 0, wf)

0.7688926948e-3*Units:-Unit(m^3/kg)

 (4.18)

 

Specific entropy of  the working fluid at the evaporator inlet

> 

s[4, L] := `s__sl,4,L`+x[4, L]*(`s__1,L`-`s__sl,4,L`)

1084.726344*Units:-Unit(J/(kg*K))

 (4.19)

Specific volume of  the working fluid at the evaporator inlet

> 

v[4, L] := `v__sl,4,L`+x[4, L]*(`v__1,L`-`v__sl,4,L`)

0.1653466682e-1*Units:-Unit(m^3/kg)

 (4.20)

 

Specific entropy of  the working fluid at the compressor 1 outlet  after isentropic compression

> 

s[2, L, is] := `s__1,L`

 

1791.745337*Units:-Unit(J/(kg*K))

 (4.21)

 

Temperature of  the working fluid at the compressor 1 outlet  after isentropic compression

> 

T[2, L, is] := TPSwf(p[2, L], s[2, L, is], wf)

 

304.3683693*Units:-Unit(K)

 (4.22)

 

Specific enthalpy of  the working fluid at the compressor 1 outlet  after isentropic compression

> 

h[2, L, is] := HPTwf(p[2, L], T[2, L, is], wf)

 

429859.6298*Units:-Unit(J/kg)

 (4.23)

 

Enhtalpy change in the compressor 1 after isentropic compression

> 

l[comp, 1, is] := h[2, L, is]-h[1, L]

 

26548.1837*Units:-Unit(J/kg)

 (4.24)

 

Enhtalpy change in the compressor 1 after actual compression

> 

l[comp, 1] := (h[2, L, is]-h[1, L])/`&eta;__comp`

 

33185.22962*Units:-Unit(J/kg)

 (4.25)

 

Specific enthalpy of  the working fluid at the compressor 1 outlet  after actual compression

> 

h[2, L] := h[1, L]+l[comp, 1]

 

436496.6757*Units:-Unit(J/kg)

 (4.26)

 

 

Temperature of  the working fluid at the compressor 1 outlet  after actual compression

> 

T[2, L] := TPHwf(p[2, L], h[2, L], wf)

 

310.8484073*Units:-Unit(K)

 (4.27)

 

 

Specific entropy of  the working fluid at the compressor 1 outlet  after actual compression 

> 

s[2, L] := SPTwf(p[2, L], T[2, L], wf)

 

1813.323123*Units:-Unit(J/(kg*K))

 (4.28)

 

 

Specific volume of  the working fluid at the compressor 1 outlet  after actual compression 

> 

v[2, L] := 1/DPTwf(p[2, L], T[2, L], wf)

 

0.2925673013e-1*Units:-Unit(m^3/kg)

 (4.29)

 

 

Pressure of  the working fluid at the vapor mixing intercooler inlet

> 

p__5 := p[3, L]

 

888428.7024*Units:-Unit(Pa)

 (4.30)

 

Temperature of  the working fluid at the vapor mixing intercooler inlet

> 

T[5] := TSPXwf(p__5, 1, wf)

 

293.4573166*Units:-Unit(K)

 (4.31)

 

Specific enthalpy of  the working fluid at the vapor mixing intercooler inlet

> 

h[5] := HSTXwf(T[5], 1, wf)

 

418291.6709*Units:-Unit(J/kg)

 (4.32)

 

Specific entropy of  the working fluid at the vapor mixing intercooler inlet

> 

s[5] := SSTXwf(T[5], 1, wf)

 

1753.034453*Units:-Unit(J/(kg*K))

 (4.33)

 

Specific volume of  the working fluid at the vapor mixing intercooler inlet

> 

v[5] := 1/DSTXwf(T[5], 1, wf)

 

0.2635126281e-1*Units:-Unit(m^3/kg)

 (4.34)

 

 

Temperature of the working fluid at the condenser outlet

> 

T[3, H] := T__out+`&delta;T__CD`

325.15*Units:-Unit(K)

 (4.35)

Pressure of  the working fluid at the condenser outlet

> 

p[3, H] := PSTXwf(T[3, H], 0, wf)

 

2319352.795*Units:-Unit(Pa)

 (4.36)

 

Specific enthalpy of  the working fluid at the condenser outlet

> 

h[3, H] := HSTXwf(T[3, H], 0, wf)

 

280642.2931*Units:-Unit(J/kg)

 (4.37)

 

Specific entropy of  the working fluid at the condenser outlet

> 

s[3, H] := SSTXwf(T[3, H], 0, wf)

 

1264.225345*Units:-Unit(J/(kg*K))

 (4.38)

 

Specific volume of  the working fluid at the condenser outlet

> 

v[3, H] := 1/DSTXwf(T[3, H], 0, wf)

 

0.9977241453e-3*Units:-Unit(m^3/kg)

 (4.39)

 

 

Pressure of  the working fluid at the compressor 2 outlet

> 

p[2, H] := p[3, H]

 

2319352.795*Units:-Unit(Pa)

 (4.40)

 

Pressure of  the working fluid at the compressor 2 inlet

> 

p[1, H] := p[2, L]

 

888428.7024*Units:-Unit(Pa)

 (4.41)

 

Pressure increase in the compressor 2

> 

`&pi;__comp,2` := p[2, H]/p[1, H]

 

2.610623439

 (4.42)

 

Specific enthalpy of  the working fluid at the flash chamber intlet

> 

h[4, H] := h[3, H]

280642.2931*Units:-Unit(J/kg)

 (4.43)

Pressure of  the working fluid at the flash chamber intlet

> 

p[4, H] := p[3, L]

888428.7024*Units:-Unit(Pa)

 (4.44)

Temperature of  the working fluid at the flash chamber intlet

> 

`T__4,H` := TPHwf(p[4, H], h[4, H], wf)

289.4442708*Units:-Unit(K)

 (4.45)

Specific enthalpy of saturated liquid of  the working fluid at the flash chamber inlet

> 

h[sl, 4, H] := HSPXwf(p[4, H], 0, wf)

221044.0888*Units:-Unit(J/kg)

 (4.46)

Specific enthalpy of saturated vapor of  the working fluid at the flash chamber inlet

> 

h[ss, 4, H] := h[5]

418291.6709*Units:-Unit(J/kg)

 (4.47)

Quality of the working fluid at the flash chamber intlet

> 

x[4, H] := (h[4, H]-h[sl, 4, H])/(h[ss, 4, H]-h[sl, 4, H])

 

.3021492262

 (4.48)

 

Specific entropy of saturated liquid of  the working fluid at the flash chamber inlet

> 

`s__sl,4,H` := SSPXwf(p[4, H], 0, wf)

1074.098979*Units:-Unit(J/(kg*K))

 (4.49)

Specific volume of saturated liquid of  the working fluid at the flash chamber inlet

> 

`v__sl,4,H` := 1/DSPXwf(p[4, H], 0, wf)

0.8468535493e-3*Units:-Unit(m^3/kg)

 (4.50)

 

Specific entropy of saturated vapor of  the working fluid at the flash chamber inlet

> 

s[ss, 4, H] := s[5]

1753.034453*Units:-Unit(J/(kg*K))

 (4.51)

Specific volume of saturated vapor of  the working fluid at the flash chamber inlet

> 

v[ss, 4, H] := v[5]

0.2635126281e-1*Units:-Unit(m^3/kg)

 (4.52)

 

Specific entropy of  the working fluid at the flash chamber inlet

> 

s[4, H] := `s__sl,4,H`+x[4, H]*(s[ss, 4, H]-`s__sl,4,H`)

1279.238807*Units:-Unit(J/(kg*K))

 (4.53)

Specific volume of  the working fluid at the flash chamber inlet

> 

v[4, H] := `v__sl,4,H`+x[4, H]*(v[ss, 4, H]-`v__sl,4,H`)

0.8552991072e-2*Units:-Unit(m^3/kg)

 (4.54)

Ratio of mass flow rates between the high pressure circuit and that of the low pressure circuit

> 

phi := (h[5]-h[3, L])/(h[5]-h[4, H])

 

1.432971113

 (4.55)

 

Specific enthalpy of  the working fluid at the compressor 2 intlet 

> 

h[1, H] := ((phi-1)*h[5]+h[2, L])/phi

 

430996.0476*Units:-Unit(J/kg)

 (4.56)

 

Temperature of  the working fluid at the compressor 2 intlet

> 

T[1, H] := TPHwf(p[1, H], h[1, H], wf)

 

305.4690036*Units:-Unit(K)

 (4.57)

 

Specific entropy of  the working fluid at the compressor 2 intlet

> 

s[1, H] := SPTwf(p[1, H], T[1, H], wf)

 

1795.472299*Units:-Unit(J/(kg*K))

 (4.58)

 

Specific volume of  the working fluid at the compressor 2 intlet

> 

v[1, H] := 1/DPTwf(p[1, H], T[1, H], wf)

 

0.2839964907e-1*Units:-Unit(m^3/kg)

 (4.59)

 

 

Specific entropy of  the working fluid at the compressor 2 outlet  after isentropic compression

> 

s[2, H, is] := s[1, H]

 

1795.472299*Units:-Unit(J/(kg*K))

 (4.60)

 

Temperature of  the working fluid at the compressor 2 outlet  after isentropic compression

> 

T[2, H, is] := TPSwf(p[2, H], s[2, H, is], wf)

 

350.5265693*Units:-Unit(K)

 (4.61)

 

Specific enthalpy of  the working fluid at the compressor 2 outlet  after isentropic compression

> 

h[2, H, is] := HPTwf(p[2, H], T[2, H, is], wf)

 

455572.2348*Units:-Unit(J/kg)

 (4.62)

 

Enhtalpy change in the compressor 2 after isentropic compression

> 

l[comp, 2, is] := h[2, H, is]-h[1, H]

 

24576.1872*Units:-Unit(J/kg)

 (4.63)

 

Enhtalpy change in the compressor 2 after actual compression

> 

l[comp, 2] := (h[2, H, is]-h[1, H])/`&eta;__comp`

 

30720.23400*Units:-Unit(J/kg)

 (4.64)

 

Specific enthalpy of  the working fluid at the compressor 2 outlet  after actual compression

> 

h[2, H] := h[1, H]+l[comp, 2]

 

461716.2816*Units:-Unit(J/kg)

 (4.65)

 

Specific enthalpy of  the working fluid at the compressor 2 outlet  after actual compression

> 

T[2, H] := TPHwf(p[2, H], h[2, H], wf)

 

355.4757017*Units:-Unit(K)

 (4.66)

 

Specific entropy of  the working fluid at the compressor 2 outlet  after actual compression 

> 

s[2, H] := SPTwf(p[2, H], T[2, H], wf)

 

1812.878289*Units:-Unit(J/(kg*K))

 (4.67)

 

Specific volume of  the working fluid at the compressor 2 outlet  after actual compression 

> 

v[2, H] := 1/DPTwf(p[2, H], T[2, H], wf)

 

0.1148144181e-1*Units:-Unit(m^3/kg)

 (4.68)

 

 

Heat rejection in the condenser referred to 1 kg of refrigerant in the low pressure circuit

> 

q[out] := (h[2, H]-h[3, H])*phi

259473.7948*Units:-Unit(J/kg)

 (4.69)

Heat addition in the evaporator referred to 1 kg of refrigerant in the low pressure circuit

> 

`#msub(mi("q",fontstyle = "normal"),mo("in"))` := h[1, L]-h[4, L]

182267.3573*Units:-Unit(J/kg)

 (4.70)

The total work of compressors referred to 1 kg of the working fluid in the low pressure circuit

> 

l[c] := phi*l[comp, 2]+l[comp, 1]

77206.43753*Units:-Unit(J/kg)

 (4.71)

Coefficent of performance of a refrigerator

> 

COP[R] := `#msub(mi("q",fontstyle = "normal"),mo("in"))`/l[c]

 

2.360779270

 (4.72)

 

Coefficent of performance of a heat pump

> 

COP[HP] := q[out]/l[c]

3.360779269

 (4.73)

 

Functions for plotting Ts-diagram
 

Ploting the Refrigeration Cycle on a Ts-diagram
 
 
> 

plots:-display(Ts_sl, Ts_ss, Ts_1_2_L, Ts_2_1H_L, Ts_4H_3_L, Ts_3_4_L, Ts_4_1_L, Ts_1_2_H, Ts_2_2_sv_H, Ts_2_sv_3_H, Ts_3_4_H, Ts_4_5_H, Ts_5_1_H, labels = [s*[kJ/(kg*K)], T*[K]], size = [800, 500], gridlines)

 

Functions for plotting ph-diagram
 

Ploting the Refrigeration Cycle on a ph-diagram
 
> 

plots:-display(ph_sl, ph_ss, ph_1_2_L, ph_2_1H_L, ph_4H_3_L, ph_3_4_L, ph_4_1_L, ph_1_2_H, ph_2_2s_H, ph_2s_3_H, ph_3_4_H, ph_4_5_H, ph_5_1_H, labels = [h*[kJ/(kg*K)], pressure*[MPa]], size = [800, 500], gridlines)

 

Functions for plotting hs-diagram
 

Ploting the Refrigeration Cycle on a hs-diagram
 
> 

plots:-display(hs[sl], hs[ss], hs[1, L-2, L], hs[2, L-1, H], hs[4, H-3, L], hs[3, L-4, L], hs[3, L-4, L], hs[4, L-1, L], hs[1, H-2, H], hs[2, H-ss, 2, H], hs[ss, 2, H-3, H], hs[3, H-4, H], hs[4, H-5], hs[5-1, H], labels = [s*[kJ/(kg*K)], h*[kJ/kg]], size = [800, 500], gridlines)

 

Functions for plotting pv-diagram
 

Ploting the Refrigeration Cycle on a pv-diagram
 
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plots:-display(pv[sl], pv[ss], pv[1, L-2, L], pv[2, L-1, H], pv[4, H-3, L], pv[3, L-4, L], pv[3, L-4, L], pv[4, L-1, L], pv[1, H-2, H], pv[2, H-ss, 2, H], pv[ss, 2, H-3, H], pv[3, H-4, H], pv[4, H-5], pv[5-1, H], labels = [v*[m3/kg], p*[MPa]], size = [800, 500], gridlines)

 
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