Chiller Models | Table of Contents | Results

Cooling Tower and Water Pump Models

In addition to the chiller, the cooling tower and condenser water pump and associated piping must also be modeled in order to calculate the total energy costs of the chiller systems. The following discussion details the modeling of these components.

Cooling Tower Model

The cooling tower model used in this study was adapted from the model developed by Weber (1988) but includes the modification of using a counterflow NTU-effectiveness model used by Liu (1997) and based on the work of Braun (1988). The reader should consult these works for complete development of the mass, energy, and diffusion equations used to model the cooling tower's performance.

Liu's study discusses a possible source of error, using an effectiveness model for a counterflow cooling tower rather than a crossflow cooling tower, which is the type of tower considered in his and in the current study. Liu attempted a new effectiveness model based on a crossflow tower but the crossflow effectiveness model was less accurate in modeling the tower manufacturer's data than the counterflow effectiveness model. Liu verified that the use of Braun's counterflow model produced very accurate results.

The following assumptions were made in the modeling of the cooling tower.

  1. Heat and mass transfer only in the direction normal to water and air flow direction.
  2. Negligible heat and mass transfer through tower walls to the environment.
  3. Constant water and dry air specific heats.
  4. Negligible heat transfer from the tower fans to the air or water streams.
  5. The mass fraction of water vapor in the air/vapor mixture is approximately equal to the humidity ratio.
  6. Uniform temperature throughout the water stream at any cross section.
  7. Uniform cross-sectional area of the tower.
Using a steady state energy balance, mass balance, and mass diffusion relations on a differential volume, Braun and Weber developed several differential equations. These differential equations must be numerically integrated over the entire tower volume from air inlet to exit. This analysis can be simplified with the use of Merkel's assumptions. Merkel neglected the effect of water loss due to evaporation and assumed a Lewis number of 1. The equations simplify to:

(4.1)

and

(4.2)

where Tw is the water temperature, is the mass flow rate of air through the cooling tower, is the mass flow rate of water through the tower, ha is the enthalpy of the moist air per pound of dry air, hsw is the enthalpy of saturated air at the water conditions, dV is a differential volume, VT is the total volume, and NtuCT is the number of transfer units for the cooling tower. Again, as for the condenser, Ntu is the dimensionless parameter used for heat exchanger analysis.

Braun defines a saturation specific heat, Cs, as the derivative of the saturated air enthalpy with respect to temperature evaluated at the water temperature.

(4.3)

He then rewrites Equation 4.1 in terms of air enthalpies and Cs.

(4.4)

Braun then states that Equations 4.2 and 4.3 can be solved analytically for the exit conditions if the saturation enthalpy is linear with respect to temperature, making Cs constant. Although air saturation enthalpy does not vary linearly with temperature, an appropriate average slope between the inlet and outlet water conditions is chosen. An effectiveness relationship can now be derived in terms of Cs. This air side effectiveness is defined as the ratio of the actual heat transfer to the maximum possible air side heat transfer that would occur if the exiting stream were saturated at the temperature of the incoming water.

The actual heat transfer from the tower can then be given in terms of this effectiveness as:

(4.5)

where is the enthalpy of saturated air at inlet water conditions,is the enthalpy of the incoming air, and e, the air side effectiveness is

(4.6)

where

(4.7)

The average value for the saturation specific heat is estimated as the average slope between outlet water states:

(4.8)

where  is the enthalpy of saturated air at exit water conditions. From overall energy balances, the outlet air enthalpy can be determined.

(4.9)

The exiting water temperature is then given by:

(4.10)

Tower Ntu

The tower NtuCT is given by a relationship as developed by Lowe and Christie (1960) as:

(4.11)

where c and n are empirical constants specific to a particular cooling tower box. The tower manufacturer offers towers of different box sizes, each having its own dimensions and cooling capacity. To calculate the tower coefficients, c and n, and to verify the model's accuracy, Liu (1997) correlated the tower manufacturer's data for different box sizes. The manufacturer's catalog gives the approach and the tower water outlet temperature for specified water and air flow rates, entering wet bulb temperature, and tower water inlet temperature. The reader should consult Liu for the complete methodology used to calculate the tower coefficients for each box size. These coefficients were used for the current study.

Cooling Tower Fan Model

Several fan motors are available for each tower box size, and these motors vary the air flow rate in the towers. The manufacturer gives the tower fan shaft power for each air flow rate offered with each tower box size. Liu shows the relationship between the fan horsepower and the air flow rates. He also shows curve fits for each box size used in the tower model to translate the desired air flow rate to its corresponding fan power consumption. The equation for the fan horsepower, hp, is:

(4.12)

where A and B are constants which correspond to a particular tower box size and cfm is the air flow rate in the tower.

In order to maintain an entering condenser water set point, the tower fan must be cycled on and off. Past studies, Joyce (1990) and Liu (1997), have found the optimum tower water set point to be in the range of 65 oF to 75 oF. For this study, the entering condenser water set point was taken to be 68oF. It was assumed that if the water exiting the tower was greater than or equal to 68oF, then the fan was on with a resulting temperature above 68 oF. If the water exiting the tower was less than 68oF, then the fan would cycle on and off to maintain a 68 oF entering condenser water temperature. To account for the times that the fan was off, a tower duty, or ratio of time the tower fan was on to the total time for each temperature period, was calculated. This value was multiplied by the fan's power, given in Equation 4.12 above, to yield the average power consumed by the fan.

Tower Size Selection

In order to select a tower and fan size for each chiller, each chiller model was run to determine which tower and fan combination produced a 85oF entering condenser water temperature at outside design conditions of 98oF dry bulb temperature and 78oF wet bulb temperature. This is in accordance with conventional design practices (Trane, 1989). At this point, each chiller was fully loaded.

Table 4.1 lists the manufacturer's data for the towers considered in this study. The table lists a tower identification number and its corresponding air flow rate in cubic feet per minute (cfm), fan motor horsepower (hp), and dimensions (length (l), width (w), and height (h)).


Table 4.1: Cooling Tower Data
Tower
cfm
hp
l
w
h
1
2
70,800
10
7'9 5/8"
18'0 1/2" 3/4"
10'2" 3/4"
2
80,750
15
7'9 5/8"
18'0 1/2" 3/4"
10'2" 3/4"
3
88,300
20
7'9 5/8"
18'0 1/2" 3/4"
10'2" 3/4"
4
95,000
25
7'9 5/8"
18'0 1/2" 3/4"
10'2" 3/4"
5
99,400
15
9'9 1/4"
20'0 1/2" 3/4"
10'2" 3/4"
6
108,700
20
9'9 1/4"
20'0 1/2" 3/4"
10'2" 3/4"
7
116,550
25
9'9 1/4"
20'0 1/2" 3/4"
10'2" 3/4"
8
123,550
30
9'9 1/4"
20'0 1/2" 3/4"
10'2" 3/4"
9
124,450
20
11'9 3/4"
20'6 1/2" 3/4"
10'2" 3/4"
10
133,450
25
11'9 3/4"
20'6 1/2" 3/4"
10'2" 3/4"
11
142,200
30
11'9 3/4"
20'6 1/2" 3/4"
10'2" 3/4"
12
143,800
30
11'9 3/4"
20'6 1/2" 3/4"
11'6"
13
157,550
40
11'9 3/4"
20'6 1/2" 3/4"
11'6"
14
166,050
30
11'9 3/4"
22'0 1/2" 3/4"
13'3 5/8"
15
181,800
40
11'9 3/4"
22'0 1/2" 3/4"
13'3 5/8"
16
195,000
50
11'9 3/4"
22'0 1/2" 3/4"
13'3 5/8"
17
193,800
40
11'9 3/4"
22'0 1/2" 3/4"
15'11 5/8"
18
207,800
50
11'9 3/4"
22'0 1/2" 3/4"
15'11 5/8"
19
220,150
60
11'9 3/4"
22'0 1/2" 3/4"
15'11 5/8"
20
221,550
50
11'9 3/4"
22'0 1/2" 3/4"
18'7 5/8"
21
235,000
60
11'9 3/4"
22'0 1/2" 3/4"
18'7 5/8"
22
255,400
50
13'11 1/8" 3/4"
24'0 1/2" 3/4"
18'7 5/8"
23
270,700
60
13'11 1/8" 3/4"
24'0 1/2" 3/4"
18'7 5/8"
24
290,050
75
13'11 1/8" 3/4"
24'0 1/2" 3/4"
18'7 5/8"

Water Pump Model

The model for the condenser water pumping power used in this study is similar to that found in Weber's study (1988). It is modeled to reflect the actual performance of a closely coupled cooling tower-condenser connection.

Weber developed the baseline parameters using typical values for current design practices at a condenser water flow rate of 3 gpm/ton. These typical values for the water piping system used as guidelines for Weber's model were:

The pressure drop across the condenser varied according to the chiller selected. These values were either obtained directly from the chiller manufacturer or calculated in a process outlined in the manufacturer's catalog (Trane, 1989,1993; Tecochill, 1997). Table 4.2 lists the pressure drops across the condenser for each of the chiller types considered for this study.

Table 4.2: Pressure Drop Across Condenser for Different Chillers
Chiller Type and Size
Pressure Drop Across Condenser (Ft)
1000 Ton Electric Chiller
20
1000 Ton Single Stage Absorption Chiller
30
1000 Ton Double Stage Absorption Chiller
55
1000 Ton Gas Engine Driven Chiller
20
500 Ton Electric Chiller
20
500 Ton Single Stage Absorption Chiller
28
500 Ton Single Stage Absorption Chiller
20
500 Ton Gas Engine Driven Chiller
8

Using the recommended design practice for sizing the condenser pipe (ASHRAE, 1996), these values led to a system design using twelve inch black iron pipe with 450 diameters equivalent length for all the valves and fittings. The strainer head was converted at design conditions to 250 diameter equivalent length.

The pressure rise that must be supplied by the condenser water pump at a given flow rate can be expressed in terms of the required total dynamic head calculated in feet:

(4.13)

where Hpiping is all dynamic piping and friction losses due to piping, valves, strainers, and fittings; Htower is the required static head due to tower lift (11 feet); and Hcondenser is the dynamic head loss due to water flow in the condenser (listed in Table 4.2). The head loss in the pipe and fittings is calculated using:

(4.14)

where s is the velocity of the water through the condenser tubes in ft/sec, g is the gravitational constant in ft/sec2, d is the pipe diameter in ft, Leq is the total equivalent length of straight pipe including all piping, valves, fittings, and strainers, and f is the friction factor. The friction factor can be calculated from White (1986) for traditionally turbulent and completely turbulent rough pipes as:

(4.15)

where ep, the average roughness of black iron pipe, is assumed to be 0.00015 feet, and Red is the Reynolds number calculated using the pipe inside diameter.

The power in kW that must be applied to the pump shaft can then be calculated.

(4.16)

where gpm is the flow rate through the condenser, and the pump efficiency, hpump, was assumed to be 0.65. Weber (1988) assumed this electricity to water pump efficiency for a constant speed pump.

The required input power to the pump motor is calculated from:

(4.17)

where hmotor , the motor efficiency, was assumed to be 0.85 by Weber.

The above cooling tower, pump, and fan relationships allow the calculations of the tower water outlet temperature, pump power, and average fan power for the chiller system under any specified chiller loading and ambient conditions.