A refrigerated space is maintained at −20°C, and cooling water is available at 21°C. Refrigeration capacity is 120,000 kJ.h^−1. The evaporator and condenser are of sufficient size that a 5°C minimum-temperature difference for heat transfer can be realized in each. The refrigerant is 1,1,1,

Question

A refrigerated space is maintained at −20°C, and cooling water is available at 21°C. Refrigeration capacity is [latex]120,000 kJ.h^−1[/latex]. The evaporator and condenser are of sufficient size that a 5°C minimum-temperature difference for heat transfer can be realized in each. The refrigerant is 1,1,1,2-tetrafluoroethane (HFC-134a), for which data are given in Table 9.1 and Fig. F.2 (App. F).

Table 9.1: Properties of Saturated 1,1,1,2-Tetrafluoroethane [latex](R134A)^†[/latex]

		[latex]Volume m^{3}⋅kg{−1} [/latex]		[latex]Enthalpy kJ⋅kg^{−1}[/latex]		[latex]Entropy kJ⋅kg^{−1} ⋅K^{−1}[/latex]
T(°C)	P (bar)	[latex]V^l[/latex]	[latex]V^v[/latex]	[latex]H^l[/latex]	[latex]H^v[/latex]	[latex]S^l[/latex]	[latex]Sv[/latex]
−40	0.512	0.000705	0.36108	148.14	374	0.796	1.764
−35	0.661	0.000713	0.28402	154.44	377.17	0.822	1.758
−30	0.844	0.00072	0.22594	160.79	380.32	0.849	1.752
−25	1.064	0.000728	0.18162	167.19	383.45	0.875	1.746
−20	1.327	0.000736	0.14739	173.64	386.55	0.9	1.741

−18	1.446	0.00074	0.13592	176.23	387.79	0.91	1.74
−16	1.573	0.000743	0.12551	178.83	389.02	0.921	1.738
−14	1.708	0.000746	0.11605	181.44	390.24	0.931	1.736
−12	1.852	0.00075	0.10744	184.07	391.46	0.941	1.735
−10	2.006	0.000754	0.09959	186.7	392.66	0.951	1.733

−8	2.169	0.000757	0.092422	189.34	393.87	0.961	1.732
−6	2.343	0.000761	0.085867	191.99	395.06	0.971	1.731
−4	2.527	0.000765	0.079866	194.65	396.25	0.98	1.729
−2	2.722	0.000768	0.074362	197.32	397.43	0.99	1.728
0	2.928	0.000772	0.069309	200	398.6	1	1.727

2	3.146	0.000776	0.064663	202.69	399.77	1.01	1.726
4	3.377	0.00078	0.060385	205.4	400.92	1.02	1.725
6	3.62	0.000785	0.056443	208.11	402.06	1.029	1.724
8	3.876	0.000789	0.052804	210.84	403.2	1.039	1.723
10	4.146	0.000793	0.049442	213.58	404.32	1.049	1.722

12	4.43	0.000797	0.046332	216.33	405.43	1.058	1.721
14	4.729	0.000802	0.043451	219.09	406.53	1.068	1.72
16	5.043	0.000807	0.04078	221.87	407.61	1.077	1.72
18	5.372	0.000811	0.038301	224.66	408.69	1.087	1.719
20	5.717	0.000816	0.035997	227.47	409.75	1.096	1.718

22	6.079	0.000821	0.033854	230.29	410.79	1.106	1.717
24	6.458	0.000826	0.031858	233.12	411.82	1.115	1.717
26	6.854	0.000831	0.029998	235.97	412.84	1.125	1.716
28	7.269	0.000837	0.028263	238.84	413.84	1.134	1.715
30	7.702	0.000842	0.026642	241.72	414.82	1.144	1.715

35	8.87	0.000857	0.023033	249.01	417.19	1.167	1.713
40	10.166	0.000872	0.019966	256.41	419.43	1.191	1.711
45	11.599	0.000889	0.017344	263.94	421.52	1.214	1.709
50	13.179	0.000907	0.015089	271.62	423.44	1.238	1.7
55	14.915	0.000927	0.01314	279.47	425.15	1.261	1.705

60	16.818	0.00095	0.011444	287.5	426.63	1.285	1.702
65	18.898	0.000975	0.00996	295.76	427.82	1.309	1.699
70	21.168	0.001004	0.008653	304.28	428.65	1.333	1.696
75	23.641	0.001037	0.007491	313.13	429.03	1.358	1.691
80	26.332	0.001077	0.006448	322.39	428.81	1.384	1.685

(a) What is the value of ω for a Carnot refrigerator?

(b) Calculate ω and m ∙ for a vapor-compression cycle (Fig. 9.2) if the compressor efficiency is 0.80.

Accepted Answer

(a) Allowing 5°C temperature differences, the evaporator temperature is −25°C = 248.15 K, and the condenser temperature is 26°C = 299.15 K. Thus, by Eq. (9.3) for a Carnot refrigerator,

[latex] \omega=\frac{T_C}{T_H-T_C} [/latex] (9.3)

[latex] \omega=\frac{248.15}{299.15-248.15}=4.87 [/latex]

(b) With HFC-134a as the refrigerant, enthalpies for states 2 and 4 of Fig. 9.2 are read directly from Table 9.1.

The entry at −25°C indicates that HFC-134a vaporizes in the evaporator at a pressure of 1.064 bar. Its properties as a saturated vapor at these conditions are:

[latex] H_2=383.45 \mathrm{~kJ} \cdot \mathrm{kg}^{-1} \quad S_2=1.746 \mathrm{~kJ} \cdot \mathrm{kg}^{-1} \cdot \mathrm{K}^{-1} [/latex]

The entry at 26°C in Table 9.1 shows that HFC-134a condenses at 6.854 bar; its enthalpy as a saturated liquid at these conditions is:

[latex] H_4=235.97 \mathrm{~kJ} \cdot \mathrm{kg}^{-1} [/latex]

If the compression step is reversible and adiabatic (isentropic) from saturated vapor at state 2 to superheated vapor at state 3′,

[latex] S_3^{\prime}=S_2=1.746 \mathrm{~kJ} \cdot \mathrm{kg}^{-1} \cdot \mathrm{K}^{-1} [/latex]

This entropy value and the condenser pressure of 6.854 bar are sufficient to specify the thermodynamic state at point 3′. One could find the other properties at this state using Fig. F.2, following a curve of constant entropy from the saturation curve to the condenser pressure. However, more precise results can be obtained using an electronic resource such as the NIST WebBook. Varying the temperature at a fixed pressure of 6.854 bar shows that the entropy is [latex]1.746 kJ⋅kg^{−1}⋅K^{−1} at T = 308.1 K [/latex]. The corresponding enthalpy is:

[latex] H_3^{\prime}=421.97 \mathrm{~kJ} \cdot \mathrm{kg}^{-1} [/latex]

and the enthalpy change is:

[latex] (\Delta H)_S=H_3^{\prime}-H_2=421.97-383.45=38.52 \mathrm{~kJ} \cdot \mathrm{kg}^{-1} [/latex]

By Eq. (7.17) for a compressor efficiency of 0.80, the actual enthalpy change for step 2 → 3 is:

[latex] \eta \equiv \frac{(\Delta H)_S}{\Delta H} [/latex] (7.17)

[latex] H_3-H_2=\frac{(\Delta H)_S}{\eta}=\frac{38.52}{0.80}=48.15 \mathrm{~kJ} \cdot \mathrm{kg}^{-1} [/latex]

Because the throttling process of step 1 → 4 is isenthalpic, [latex]H_1 = H_4.[/latex] The coefficient of performance as given by Eq. (9.4) therefore becomes:

[latex] \omega=\frac{H_2-H_1}{H_3-H_2} [/latex] (9.4)

[latex] \omega=\frac{H_2-H_4}{H_3-H_2}=\frac{383.45-235.97}{48.15}=3.06 [/latex]

and the HFC-134a circulation rate as given by Eq. (9.5) is:

[latex] \dot{m}=\frac{\dot{Q}_C}{H_2-H_1} [/latex] (9.5)

[latex] \dot{m}=\frac{\dot{Q}_C}{H_2-H_4}=\frac{120,000}{383.45-235.97}=814 \mathrm{~kg} \cdot \mathrm{h}^{-1} [/latex]

Question 9.1: A refrigerated space is maintained at −20°C, and cooling wat...

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