The performance of water chillers
The capacity of reciprocating compressors is most commonly controlled in steps by cylinder — unloading, with corresponding steps in the chilled water flow temperature, when using a conventional thermostatic expansion valve. Control from a temperature sensor, located in the flowline from a chiller, is not a safe proposition and it should always be done from a thermostat in the chilled water return. In the simpler case, variations in suction pressure may be used to unload cylinders, some variation in the chilled water flow temperature being tolerated. It is vital that the chilled water flow temperature is not allowed to approach too closely to freezing point.
EXAMPLE 12.5
A plant has a design duty of 235 kW when chilling water from 13°C to 7°C with an evaporating temperature of 3°C. The compressor has six cylinders which are unloaded in
Evaporating temperature (°C) Fig. 12.8 Performance diagram showing how a back pressure valve and a hot gas valve may be used to control evaporating temperature. Back pressure valves should not be used with hermetic and Semihermetic compressors. 
Pairs through a stepcontroller from a thermostat immersed in the return chilled water line. Establish the temperature settings for the stepcontroller to unload and load the cylinders, within the following restraints: maximum allowable chilled water flow temperature, 7°C;
Minimum allowable evaporating temperature, 1°C; setpoint of the low water temperature cutout, 3.5° ± 1°C.
The compressor has the following performance when running at constant speed on six cylinders and at a constant condensing pressure:
Evaporating temperature (°C): 0 3 6 9 12
Refrigeration duty (kW): 210 235 260 285 310
Answer
Since the machine runs at a constant speed the volumetric flow rate of refrigerant is directly proportional to the number of loaded cylinders. Hence, because the suction density is constant for a given evaporating temperature, we can say the mass flow rate, and so the refrigeration capacity, is also directly proportional to the number of loaded cylinders at any given evaporating temperature. The following table can be compiled:
No. of cylinders 
Evaporating temperature 

0°C 
3°C 
6°C 
9°C 
12°C 

Duty in kW 

6 
210 
235 
260 
285 
310 
4 
140 
156.7 
173.3 
190 
206.7 
2 
70 
78.3 
86.7 
95 
103.3 
Three compressor characteristics can now be drawn (Figures 12.9(a) and (b)). As with air cooler coils the water chilling evaporator has a characteristic performance that can be expressed in two ways: as kW of refrigeration against evaporating temperature or chilled water flow temperature. Thus in Figure 12.9(a) we can plot a pair of points, A and B, at 3°C and 7°C, respectively, representing the evaporating and chilled water flow temperatures at a design duty of 235 kW. These points lie on the refrigerantside and waterside characteristics of the evaporator for a return water temperature of 13°C and they share a common zero load point at C, when the flow and return water temperatures and the evaporating temperature are each 13°C. In a fashion similar to that adopted earlier for air cooler coils, additional pairs of characteristics may be drawn, parallel to the design pair, for other return water temperatures.
As the load on the chiller diminishes the return water temperature drops and the pair of evaporator characteristics shifts to the left (Figure 12.9(a)). When the evaporator refrigerant characteristic cuts the compressor characteristic for six cylinders at 1 °C (point D) the load is 219.3 kW and water is being chilled to 4.6°C (point E) from a return water temperature of 10.2°C (point F). We have now reached one of the restraints imposed for safe operation and the machine must be unloaded from six to four cylinders. The return water temperature to effect this is 10.2°C. After unloading, the balance moves from D to G, with evaporation at 3.5°C (point G) and a chilled water flow temperature of 6.1°C (point H). Upon further fall in load the return water temperature reduces to 7.1°C (point K) with an evaporating temperature of 1°C (point I) and a chilled water flow temperature of 3.4°C (point J). Two more cylinders are unloaded when the thermostat in the return chilled water line senses 7.1°C (point K) and a new balance is struck at an evaporating temperature of 3.8°C (point
Temperature, °C (a) 
Temperature, °C (b) Fig. 12.9 (a) Chiller unloading, (b) Chiller loading. 
L) and a chilled water flow temperature of 5.1°C (point M). As the load continues to decline the return water temperature continues to fall and we see that if it dropped to 4.1°C (point P) the evaporating temperature would be 1°C (point N) but the chilled water flow temperature would be 2.3°C (point O), at a value less than the low water temperature cutout point of 2.5°C (= 3.5° — 1°). We may observe here that if we adopt a return water temperature of 5°C (point S) for unloading the last two cylinders and switching off the compressor, evaporation will be at 1.8°C (point Q) and the chilled water flow temperature will be 3.1 °C (point R), giving us a margin of 0.6°C above the low water temperature cutout value.
When the system load starts to increase and the return water temperature rises to 7°C (point 1 in Figure 12.9(b)) the machine is started with two cylinders operating and balances at an evaporating temperature of 3.6°C (point 2) and a water flow temperature of 5.0°C (point 3). As the load continues to rise the pair of evaporator characteristics shifts to the right (Figure 12.9(6)) until the chilled water flow temperature reaches 7°C (point 6) when the evaporating temperature is 5.6°C (point 5) and the return water temperature is 9.1°C (point 4). Two more cylinders are then loaded (four in all now operating) and the balance moves to evaporation at 2.5°C (point 7) with a water flow temperature of 5.15°C (point 8). With a progressive load increase the return chilled water temperature rises to 11.1 °C (point 9) when the system is evaporating at 4.1°C (point 10) and the water is being produced at 7°C (point 11). The compressor then loads fully, working with six cylinders, balancing anew at evaporating and flow temperatures of 1.5°C (point 12) and 5.4°C (point 13), respectively. If the load goes on rising to its design duty the balance will eventually return to points A and B, chilled water being produced at 7°C when the evaporating temperature is 3°C, from a return water temperature of 13°C (point C).
The loading and unloading setpoints for the stepcontroller can now be summarised:
Table 12.2 Loading and unloading setpoints for example 12.5

Unloading

The letters and numbers in brackets in the table refer to Figures 12.9(a) and (b). 
Within reason, it does not matter how frequently cylinders are loaded or unloaded except for the last step, which cycles the compressor motor onoff. It is most important that the motor is not switched on too often because the inrush of electric current on starting causes the windings to overheat and repeated starts at short intervals will burn them out. Typical allowable starts per hour are: 4 to 6 for hermetic, reciprocating compressors, 8 to 10 for open, reciprocating machines and 2 for hermetic centrifugals.
When the cooling load is a little less than the maximum refrigerating capacity the plant runs for a lot of the time and when it is almost zero the compressor will be mostly off. If the cooling load is half the average cooling capacity of the last step of refrigeration, the machine will start and stop most frequently and the following analysis may be considered.
Average cooling capacity of last step of refrigeration: 
4r 
KW 
Average cooling load on system: 
0.5 qr 
KW 
Surplus capacity available for cooling down the water in the system: 
0.5 qt 
KW 
Mass of water in the system: 
M 
Kg 
Specific heat capacity of water: 
4.187 
KJ kg 
Allowable variation in return chilled water temperature: 
At 
K 
Time taken to cool down the water in the system: 
0 
S 
Hence
0. 5 qr = (4.1SlmAt)/Q
And the cooling down time is therefore
0 = (2 x 4.181 mAt)/qT (12.3)
If the cooling load is assumed as a constant and equals 0.5qT, the time taken to warm up the water in the system by At after the compressor has shut down will equal that taken to cool it down when the compressor was running, for all practical purposes. The time between successive starts will then be 20 and if the allowable frequency of starting is / times per hour we can write
/= 3600/20. By substitution in equation (12.1) we have / = (36009r)/(2 x 2 x 4.181mAt) = (215qr)/(mAt) (12.4)
M= (2l5qr)/(fAt) (12.5)
EXAMPLE 12.6
For the case of example 12.5 and Figures 12.9(a) and (b), determine the minimum amount
Of water allowable in the system if the number of starts per hour for the compressor is
Limited to six.
Answer
When the return chilled water temperature falls to 5°C the last step of the chiller switches off, immediately prior to which the refrigeration capacity was 74.4 kW (point Q). The return water temperature then starts to rise and when it reaches 7°C (point 1) the compressor is started on its first step and the capacity is initially 78.3 kW (point 3). The return chilled water then starts to fall again towards 5°C. Thus
Qs = (74.4 + 18.3)12 = 76.35 kW
When the load on the cooler coil and the heat gain to the chilled water pipes in the system is half this value the greatest number of starts per hour will prevail and since this must be limited to six we can use equation (12.5) to establish the necessary mass of water in the system, m, to ensure this:
M = (215 x 76.35)/(6 x (7 — 5)) = 1368 kg
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