Question 1 of 9: Two-Stage Compression Refrigeration with Flash Chamber & Direct-Contact Heat Exchanger (R-134a)
Nivaar worked solution (AI-drafted; not reviewed by a licensed engineer)
Notes on this paper
National Exam 04-BS-10, Thermodynamics — December 2013. 3 hours, Closed-Book Exam
(approved calculator and one double-sided 8.5x11-inch aid sheet permitted; property tables and
charts supplied in an appendix). Part A: answer 2 of Questions 1-3 (20 marks each). Part B: answer
4 of Questions 4-9 (15 marks each), for a 100-mark paper. Only the first two Part-A and first four
Part-B questions as they appear in the answer book are marked. All nine questions (Part A complete,
Part B complete) are solved below for completeness.
Reference texts: Cengel & Boles, Thermodynamics: An Engineering
Approach, 8th ed.; Moran, Shapiro, Boettner & Bailey, Fundamentals of Engineering
Thermodynamics, 8th ed. All state properties (R-134a, water/steam, moist air, N₂, CO₂,
air) were computed from high-accuracy equations of state in place of printed property-table
interpolation; every boxed numeric result.
Given. R-134a two-stage compression system with a flash chamber whose vapor
outlet is combined, in a direct-contact heat exchanger (DCHX), with the superheated discharge of
Compressor I — the DCHX is sized so its outlet (state 3) is exactly saturated vapor at the
intermediate pressure, the standard idealization for this flash-intercooled arrangement. Evaporator
inlet/outlet saturation temperature $T_1=-30\ ^\circ\text{C}$; flash-chamber/DCHX pressure
$P_{mid}=400$ kPa; condenser pressure $P_{cond}=1.2$ MPa. Condenser exit is saturated liquid;
evaporator exit is saturated vapor. Both compressors are isentropic. Refrigerating capacity
$\dot Q_{evap}=10$ tons $=10\times211/60=35.167$ kW.
State
Description
P
1
Evaporator exit, sat. vapor, $-30\ ^\circ$C
84.38 kPa
2
Compressor I exit (isentropic)
400 kPa
3
DCHX/flash-chamber vapor exit, sat. vapor
400 kPa
4
Compressor II exit (isentropic)
1.2 MPa
5
Condenser exit, sat. liquid
1.2 MPa
6
After high-pressure exp. valve, 5→6
400 kPa
7
Flash-chamber liquid, sat. liquid
400 kPa
8
After low-pressure exp. valve, 7→8
84.38 kPa
Find. (a) $\dot W_{c,I}$ and $\dot W_{c,II}$; (b) COP.
Fig. Q1 — T–s state points for the two-stage
R-134a cycle (2→3 and 6→7 are mixing/separation steps at the flash chamber+DCHX, drawn as
straight connectors for continuity rather than single-stream process lines; 5→6 and 7→8
are irreversible isenthalpic throttling).
Approach
Fix states 1 and 3 from the saturation conditions, compress each stage isentropically to get
states 2 and 4, find the evaporator mass flow rate directly from the given capacity, then close the
flash-chamber/DCHX combined control volume with a mass-and-energy balance to get the high-side mass
flow rate (the mass balance is automatically satisfied because the same evaporator flow returns
through the low-pressure expansion valve and evaporator, so only the energy balance is independent);
compressor work and COP follow directly.
Fix states 1 and 3 (saturation lines). State 1 is saturated vapor at
$-30\ ^\circ\text{C}$: $P_1=84.38$ kPa, $h_1=380.32$ kJ/kg, $s_1=1.7515$ kJ/kg·K. State 3 is
saturated vapor at 400 kPa ($T_3=8.93\ ^\circ\text{C}$): $h_3=403.72$ kJ/kg,
$s_3=1.7226$ kJ/kg·K.
Compressor I (1→2), isentropic. At 400 kPa with $s_2=s_1$:
$$h_2=411.99\ \text{kJ/kg}\quad(T_2=17.82\ ^\circ\text{C}).$$
Compressor II (3→4), isentropic. At 1.2 MPa with $s_4=s_3$:
$$h_4=426.50\ \text{kJ/kg}\quad(T_4=50.07\ ^\circ\text{C}).$$
Evaporator mass flow rate, from the given capacity. Condenser exit is saturated
liquid at 1.2 MPa, $h_5=265.95$ kJ/kg; throttling 5→6 into the flash chamber is isenthalpic,
$h_6=h_5$. Flash-chamber liquid is saturated liquid at 400 kPa, $h_7=212.11$ kJ/kg; throttling
7→8 into the evaporator is isenthalpic, $h_8=h_7=212.11$ kJ/kg. The evaporator flow follows
directly from the stated capacity:
$$\dot m_{evap}=\frac{\dot Q_{evap}}{h_1-h_8}=\frac{35.167}{380.32-212.11}=\boxed{0.2091\ \text{kg/s}}.$$
Flash-chamber + DCHX energy balance ⇒ high-side mass flow rate. The
combined control volume receives the Compressor-I discharge ($\dot m_{evap}$ at $h_2$) and the
throttled condenser liquid ($\dot m_{high}$ at $h_6$), and delivers saturated vapor to Compressor II
($\dot m_{high}$ at $h_3$) and saturated liquid to the low-pressure expansion valve ($\dot m_{evap}$
at $h_7$, since that liquid stream is what ultimately supplies the evaporator). The mass balance
$\dot m_{evap}+\dot m_{high}=\dot m_{high}+\dot m_{evap}$ is automatically satisfied, so only the
energy balance is independent:
$$\dot m_{evap}h_2+\dot m_{high}h_6=\dot m_{high}h_3+\dot m_{evap}h_7$$
$$\dot m_{high}=\dot m_{evap}\,\frac{h_2-h_7}{h_3-h_6}=0.2091\times\frac{411.99-212.11}{403.72-265.95}=\boxed{0.3033\ \text{kg/s}}.$$
Power input to each compressor (part a).
$$\dot W_{c,I}=\dot m_{evap}(h_2-h_1)=0.2091\times(411.99-380.32)=\boxed{6.622\ \text{kW}}.$$
$$\dot W_{c,II}=\dot m_{high}(h_4-h_3)=0.3033\times(426.50-403.72)=\boxed{6.908\ \text{kW}}.$$
Coefficient of performance (part b).
$$\text{COP}=\frac{\dot Q_{evap}}{\dot W_{c,I}+\dot W_{c,II}}=\frac{35.167}{6.622+6.908}=\boxed{2.599}.$$
