CE427 Fall Semester, 2000 Unit Operations Lab 3 September 24, 2000 COOLING TOWER THEORY

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1 CE427 Fall Semester, 2000 Unit Operations Lab 3 September 24, 2000 Introduction COOLING TOWER THEORY Cooling towers are used etensivel in the chemical and petroleum industries. We also know from the thermodnamics course that the are often used for heat removal in power generation ccles. Undoubtedl more are used in air conditioning ccles than an place else. (Net time ou are in a tall cit building, look below and ou ll probabl see dozens of them on the rooftops of lower buildings.) As technolog goes, the are prett old stuff, but the interesting from a technical standpoint because both heat and mass transfer occur simultaneousl, just as in distillation. Most distillations are carried out in plate columns where one can visualize steps. Cooling towers usuall have grids a sort of packing, so mass and heat transfer occur essentiall continuousl up (and down) the column. Distillation often involves several components, whereas in a cooling tower, it s just a miture of water and air. Chapter 23 of McCabe, Smith, and Harriott (M, S&H) is devoted to Humidification Operations including Cooling towers. Theor from the Tetbook Read Chapter 23. You ma be somewhat acquainted with the first half or so of this chapter because the material is covered in CE212. The equation development of particular importance to us is on Pages 757 through 760. It is built around Figure 23-8(a). Thus, there is a temperature gradient in the liquid which supplies sensible heat to the air-water interface. This heat is then carried awa into the gas stream b two mechanisms, one being sensible heat and the other latent heat from the vaporization of a small amount of water at the surface. There are a large number of equations in the tet. We present an abridged version of these equations here. Equation numbers in decimal form are those from MS&H; all other equation numbers pertain to this write-up. First, Equation (23.22) below is the overall energ balance equation, which is analogous to the operating line in distillation or absorption. Here H denotes the enthalp (Btu/lb or J/g), while G denotes the mass flow rate (lb/ft 2 /hr or kg/m 2 /hr). The subscript is associated with the liquid phase, while with the gas phase. G dh d( G H ) (23.22) Assuming that onl a small fraction of the liquid evaporates in this process compared to the total amount of liquid fed into the tank, we can write G b G a G : In these equations the subscript a depicts the top and b the bottom of the Cooling tower. The datum plane (or reference temperature, T 0 value) for all enthalp calculations is arbitrar and it eventuall cancels out on both sides of the equations. For water, it is usuall taken at a temperature of T o 32 F (according to Perr s Handbook). For liquid water then, H c L (T T o ) [Equation (23.26)]. Thus, the differential form of Equation (23.22) becomes: G dh G c dt L (1) Integrating from the bottom of the column (i.e. end b) to an particular point the column, one obtains: G ( H H ) G c ( T T ) (2) b L b This is the equation of the Operating Line required for the determination of the number of transfer units (NTU) as described later.

2 2 The value of H b can be determined from the measured wet- and dr-bulb temperatures of the entering air. One can use a pschrometric chart from Perr s (Chapter 12-5) for this. The one in the tetbook (Figure 23.2) is not convenient because it doesn t contain enthalp data. The spreadsheet referred to in Part (c) also calculates the enthalp directl from the wet- and dr-bulb temperatures. The value of H a is be calculated using Equation (2) since G c L and G are known. These estimates will also be verified based on the measured dr- and wet-bulb temperatures of the eit airstream. As mentioned previousl, heat is transferred into the gas stream b two mechanisms. The first is the sensible heat transfer and is given b Equation (23.24). The second is the latent heat transfer into the gas stream. This involves the mass transfer rate [Equation (23.25)]. The equation is multiplied b the latent heat of vaporization,λ o, at the datum plane temperature, and the Lewis Relation [Equation (23.21)] is emploed together with it to obtain [this is Eq. above (23.36)]: G dh k M B a(h i H )dz (3) Here M B is the molecular weight of the liquid; a is the interfacial area between the two phases which is the same for both heat and mass transfer; k is the mass transfer coefficient in lb-mol/ft 2 /hr. or equivalent SI units; and Z is the height of the element being considered measured from the base of the Cooling tower. H i and H are the gas-phase enthalpies measured at either the interface (i) or the bulk () of the gas phase at the particular height Z in the Cooling tower. In integrated form, this equation is: H a H b H dh i H zt 0 k M G B a dz (4) It can be seen that the integral on the left has the general form of the integral defining the number of transfer units (NTU) in terms of the gas film [Equation (22.23), Page 705] i.e.: N H dh i H There are two issues to consider with respect to this equation: (5) A) In this equation the enthalp appears to be the driving force for the mass transfer rather than b humidit differences. Here the numerator depicts the change in vapor concentration, while the denominator is the average driving force. While it appears from the equation that it the enthalp that drives the mass-transfer this is in fact not the case: It is the simplifing assumptions made during the Cooling tower derivation that give this appearance. As pointed out in the Foust etr al. [Principles of Unit Operations, John Wile & Sons, Inc. (1960), pp ], enthalp is an etensive thermodnamic propert and as such, cannot be a driving force for an transfer operation. If it makes ou feel an better, ou can think of this as a pseudo driving force. See the reference for some discussion of this point if ou are interested. B) The equation (5) is difficult to use since it requires some means for determining the temperature at the air-water interface in order to calculate H i. However, there is a reasonable wa out of this complication if we assume that the resistance to mass transfer is predominantl in the gas-phase. This is a reasonable assumption for this sstem since air (in comparison to water) is a much poorer medium for heat transfer. Refer back to Figures 23.8(a) then, it appears that the temperature T in the liquid bulk is not ver different from T i. Then the enthalp at the interface H i, can be replaced b H, the enthalp of air saturated at the bulk temperature of the liquid, T. Note that the smbol, H, was earlier defined as the enthalp of the liquid, but here the prime sign emphasizes the fact that this is the saturation enthalp of the vapor phase at temperature T.

3 3 With this assumption, Equation (5 ) takes the form: report. N N O H a H b dh H H NTU (6) This Eq (6) will be the defining equation for the number of transfer units, NTU in the project Calculation of NTU s Three methods denoted below as (a), (b) and (c) are used to estimate the NTU. (a) Graphical Integration In graphical integration, one plots 1/(H -H ) versus H between the limits of H b and H a. To begin with, a plot of H and H might take the form (Figure 1): H a H Top of Column Bottom of Column H b Water Line H a H b Air Line H b H H a Therefore, the driving force at an point, H H, for this eample, would increase with H (Figure 2): Driving Force Sketch H - H H b H H a

4 4 The required plot for NTU would take the form shown on the graph below (Figure 3): 1/(H -H ) Area NTU H b H H a (b) Special case when the H line is essentiall straight. In Figure 1, the curvature of the upper line ( water line ) comes about mainl from the nonlinearit of the curve of water vapor pressure versus temperature. Sometimes, this curvature is not ver great. The operating line that is, the air line will be straight if the liquid water and dr air rates are constant. Even though there is some evaporation (perhaps three or four per cent) this can be ignored. Under these conditions for a cooling tower, Equation (22.19) on Page 705 applies (i.e. simple logarithmic mean equations can be used to calculate NTU just as our learnt in the heat echanger problems last semester). Hence, no graphical integration is required. (c) Analtical Integration for NTU The procedure for calculating the NTU s b graphical integration actuall begins with a plot of and H versus T rather than versus H. The cooling water inlet and eit temperatures, namel T a and T b, are the primar phsical parameters of interest and are measured. This plot takes the form given below (Figure 4): H H T b Bottom T a Top Water Temperature, T

5 5 This will be the first plot that ou will prepare for the graphical integration procedure. It is also the beginning plot for the development of the analtical method. Because the curvature of the H line is not ver great, it can be ver well represented b a second order equation: 2 H αt + βt + γ (all parameters in fps units) (7) where α (8) β (9) γ (10) GcL Also H H G T T b + ( b) (assuming G X is constant with length) (11) This is the operating line equation in terms of temperature. For simplicit, define the slope of this line as s: Gc s G L Thus Equation (11) can be epressed as: H st + f (12) (13) where f H b st b. (14) Hence, the driving force is: H H ( γ f ) + ( α s) T + βt 2 (15) It is convenient to set: a γ f (16) b Recall that: α s c β (17) (18) NTU H a H b dh H H (6) Note that H - H in Equation ( 6 ) is now in terms of T, not H. The dumm variable must now be epressed in terms of temperature. Since H st + d, dh sdt. Hence, Equation (6) becomes: NTU T a s dt a + bt + ct Tb 2 (19)

6 6 The integral has the following form: where δ 4ac-b 2 d 2 1 2c + b tan 2 a + b + c δ δ (19a) (19b) There are several alternate forms of the integral in integral tables, but this is the form we will use. Thus, the NTU s are given b the equation: NTU tan δ 2cT + b tan δ 2s 1 a 1 2 ct b + b δ (20) This whole procedure has been set up in the analsis Ecel spreadsheet. Height of a Transfer Unit (HTU) Since the height of the cooling tower is Z, the height of a transfer unit, HTU, is simpl Z divided b the number of transfer units, NTU: HTU ZT / NTU (21) In designing a cooling tower, the rate of incoming hot water is known as well as the condition of the air that will be used to cool it that is, the wet- and dr-bulb air temperatures. Some optimization of the air rate to be used is required. If the air rate is high, then the power requirements will be high, and a relativel short tower can be used. On the other hand, if the air rate is low, then the power requirement is low, but the tower will be relativel high. The air rate affects the slope of the operating line and therefore the separation of the water and air lines in Figure 1. At high air rates, the separation of these lines is large, and the number of transfer units decreases, and vice versa. The idea that ou should get from this is that the NTU s required are a measure of the difficult of the cooling job to be done. The larger the NTU s, the more difficult the job is. Once the NTU requirement is known, one can calculate the height of tower required b multipling the NTU s b the HTU. But what determines the height of a transfer unit? Answer: HTU is determined b the fundamental features of the sstem: the interfacial area for heat/mass transfer, a, and the mass-transfer coefficient, k. Since these two factors cannot be measured individuall, at least in our equipment, we prefer to look at their product k a. It is important to note that not though Equation (21) relates NTU and HTU to the height of the Cooling tower (Z); the parameters (i.e. NTU and HTU) are in fact independent of each other: while NTU is directl dependent on the driving force for mass-transfer, HTU depends on the phsical features of the mass transfer via a and k. Refer back to Equation (4). We assume that the product, k M B a/g, is constant. Combining the forms and ideas of Equations (3), (4), and (6), one finds: NTU k am ZT B G 0 dz k am G B Z T (22) In view of Equation (21), then HTU is given b: G HTU k am B (23)

7 7 Here, M B k a (or also called K a in data sheet and elsewhere) is of more fundamental interest than HTU and will be calculated from the results of the eperiment. Evaporation Rate There is one more item of interest, namel, the evaporation rate of water. The theor development here assumes that the evaporation is negligible (at least in the liquid phase). This assumption leads to a straight operating line and relativel simple epressions for the NTU. Let us eamine of this is indeed the case. The rate of evaporation, G e, is given b the equation: G G (# # ) (24) e a b where the smbols, # a and # b, represent the humidities at the top and bottom of the column, respectivel. (Same as cursive H in MS&H) The measured inlet wet- and dr-bulb temperatures are used to get # b. This equation is used to get H a. This value, in combination with Equation (23.8) leads to a value of # a. The evaporation rate can then be calculated using Equation (24). Another wa to get # a is to use the eit dr and wet bulb temperature. Unlike previous ears, where there was a tremendous spra from the eit port of this column, we have made engineering efforts to reduce the spra. The dr and wet-bulb temperatures measured should be reasonable and the should be used in conjunction with Eq. (24) above, to get a second estimate of the evaporation rate. Lab Instructions: 1) Zero the air manometer. At the start of the eperiments, under no-flow conditions, the manometer should read zero, else adjust it using the black knob at the right-top corner of the manometer. 2) Fill up the inlet water tank (made of copper: does not have a heating coil) with hot water. To do this, drain the inlet water for a few minutes/seconds until ou are sure that ou are getting "hot" water. Then fill the copper tank with the water. You will need to do this ever 1-2 runs since the eit water that comes out of the Cooling tower does not recirculate in the sstem, it drains out of the sstem. 3) Ensure that the wicks of the wet-bulb temp thermocouples are completel wet at all times. Climb up to the top of the tower if necessar to confirm this before each run. 4) The power panel for the eperiment has three switches: Tower (to power up the sstem) followed b Air (for the air blower) and Pump (for water flow). The air flow-rate ma be adjusted b changing the damper setting from 1 through 5. Pick an four damper settings for our eperiment. Simultaneousl, the water flow rate can be adjusted using the valve below the manometer. Use 4, 8 and 12 liters/min for ou eperiment. Therefore, ou will perform eperiments under 12 conditions (4 air flow rates X 3 water flow rates). 5) Ensure that the sstem has reached stead state before starting acquisition of temperature 1-6 data using the Labview program as we did in the previous eperiment. Note that the data in the first column of our output file is Time (irrelevant in this case since we are onl interested in stead-state results). The second to seventh column have temperature values (in degree C) at stead state. The eighth column is packed with zero. Jot down what column corresponds to what temperature. Caution: Ensure that the eperiment is clean and the place is dr. This wa ou will get good data. Good Luck!

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