A system of capacity 70 kW operates between -8°C and 42°C respectively. Refrigerant is subcooled by 5°C and superheated by 10°C before entering the compressor. If a four cylinder single acting reciprocating compressor with bore to stroke ratio 1:1.2 and operating at 1200 rpm is used. Determine:

i. Ideal COP

ii. COP of system

iii. Mass flow rate of refrigerant

iv. Theoretical piston displacement per min

v. Bore and stroke of compressor

Assume volumetric efficiency of compressor = 80%

Cpl = 1.24 kJ/kgK, Cpv=0.74 kJ/kgK. Refrigerant properties given below:-

$Q_{a}=70 kW,$ c=4 L=$1.2\times D, N=1200rpm$

$T_{E}=T_{sat_{Pe}}=-8^{\circ}, T_{E}=T_{sat_{Pc}}=42^{\circ} Cpl=1.24\frac{kj}{kgK}$

Cpv=0.74 $\frac{kJ}{kgk}$

h1=$h g_g^\circ+Cpv(Tsup-Tsat)$

=402.3+0.74(2-8)

h1=409.7kJ/kg

h2= $h \ g_{42^{\circ}}+CPv(Tsup-Tsat)$

h2=416.8+0.74(T2-42)...(1)

For T2,

S1=S2

$S \ g _{-g^{\circ}}+CPv\times ln\frac{Tsp}{Tsat}=s \ g_{42^{\circ}}+Cpv \times ln\frac{Tsup}{Tsat}$

1.764+$0.74\times$ ln$\frac{275}{265}=1.697+0.74\times ln\frac{T2}{315}$

Assuming compression to be isentropic

T2=357.86 K=$84.86 °C$

Put in (1),

h2=448.52 $\frac{kJ}{kg}$

h2=448.52 $\frac{kJ}{kg}$

h3=$h_(f_42℃)+Cpl(Tsub-Tsat)$

=252.4+1.24(37-42)

h3=h4=246.2 kJ/kg

We know, ideal COP =$ \frac{T_E}{(T_C-T_E )}$= 5.3

COP of given system = $\frac{R.E.}{Wc}$=$\frac{(h1^{'}-h4)}{(h2-h1)}$

=402.3 $\frac{kJ}{kg}$

COP=4.02

Also, $Q ̇_a=m ̇_R×(h1^{'}-h4)$

mass flow rate,$m ̇_R=0.45 kg/sec$

Also,

$m ̇_R×v1$=theoretical piston displacement in m3 per min×$\frac{η_vol}{60}$

v1=$v_(g_{-8}℃)×\frac{Tsup}{Tsat}=0.064×\frac{275}{265}=0.066 \frac{m3}{kg}$

Therefore, theoretical piston displacement in $m^{3}/min=2.24 m^{3}/min$

We know, theoretical piston displacement in m3 per min=$c.\frac{\pi}{4} D^2$ LN

2.24=$4×\frac{\pi}{4}×D^{2}×1.2 D×1200$

D= 0.079 m = 7.9 cm

L=9.5 cm