The development of a dynamic single effect, lithium bromide absorption chiller model with enhanced generator fidelity

Energy Conversion and Management 150 (2017) 574–587

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Energy Conversion and Management journal homepage: www.elsevier.com/locate/enconman

The development of a dynamic single effect, lithium bromide absorption chiller model with enhanced generator fidelity

MARK

⁎

Corey T. Misenheimer , Stephen D. Terry Department of Mechanical and Aerospace Engineering, North Carolina State University, Raleigh, NC 27695, USA

A R T I C L E I N F O

A B S T R A C T

Keywords: Absorption Lithium Bromide Chiller Steam Refrigerant

Single effect, lithium bromide absorption chillers offer the ability to utilize low-pressure steam to produce chilled water for satisfying various comfort cooling needs. Previous attempts have been made to characterize dynamic and steady-state absorption chiller operation. Though these models perform adequately, they are based on hot water driven absorption chillers. Commercially available absorption chillers often can run on both hot water and low-pressure steam. In this paper, the mathematical framework for a dynamic single effect, lithium bromide absorption chiller model capable of using low-pressure steam is presented. The transient thermodynamic FORTRAN model is grounded on mass, energy, and species balances, and builds on prior modeling efforts. Well-known correlations for heat transfer coefficients are used to describe both tube-side and shell-side heat transfer rates in each primary chiller component. To account for the absorption chiller unit receiving steam, a heat transfer model for condensation inside horizontal tubes based on distinct internal condensation flow regimes is incorporated within the generator. This heat transfer model is used with two-phase flow pressure drop equations to establish steam temperature, quality, and pressure along the generator tube bundle. Steam consumption trends are established as a function of fluctuating external conditions. These trends reasonably align with information made available online by the manufacturer, though some deviation does occur at low chiller capacities and cooling water temperatures. Additionally, the transient response of internal and external parameters from a step increase in heat input supplied to the generator mimics results of other dynamic absorption chiller models found throughout literature.

1. Introduction Electric chillers have been predominantly used to produce chilled water for satisfying cooling demands due in large part to inherent high vapor-compression cycle Coefficients of Performance (COP), flexible unit placement, and fast transients. Fundamentally, compressing a vapor necessitates a large energy input, which electric chillers receive in the form of electricity. Single effect, lithium bromide (LiBr) absorption chillers, on the other hand, harness energy from hot water or lowpressure steam less than 205 kPa (15 psig) and the affinity between an absorbent and a refrigerant to create a chilling effect. Though COPs of single effect absorption chillers are significantly less than those of equivalent size electric chillers, absorption chillers offer a niche in that they can utilize low-pressure steam or hot water that might otherwise be rejected to a low-temperature sink or the environment. The miniscule electrical power requirement relative to the heat input necessary to drive absorption chillers makes them particularly attractive in waste heat and solar thermal applications, especially given

⁎

Corresponding author. E-mail address: [email protected] (C.T. Misenheimer).

http://dx.doi.org/10.1016/j.enconman.2017.08.005 Received 14 March 2017; Received in revised form 31 July 2017; Accepted 3 August 2017 0196-8904/ © 2017 Elsevier Ltd. All rights reserved.

present-day concerns of carbon emissions. This heightened interest has spurred a plethora of analyses predicated on the steady first and second laws of thermodynamics. For example, Pongtornkulpanich et al. [1] applied basic relations to design solar-driven LiBr absorption chillers for buildings. Agyenim et al. [2] conducted a similar study on solardriven LiBr absorption chillers. Chen et al. [3] designed the framework of a LiBr absorption chiller powered via a supercritical CO2 solar collector. Gomri [4] performed a second law of thermodynamics comparison of single effect and double effect, LiBr absorption chillers. Lastly, Bakhtiari et al. [5] developed a steady-state model of a 14-kW single effect, LiBr absorption chiller. A comparison of the steady-state simulation results and experimental measurements revealed good agreement [5]. While these studies reveal important steady-state absorption chiller trends, understanding part-load and dynamic operation is vital for describing real absorption chiller performance. Large absorption chiller thermal masses coupled with temperature-driven mass transfer and deposition translate into long absorption chiller transients compared to

Energy Conversion and Management 150 (2017) 574–587

C.T. Misenheimer, S.D. Terry

α Δ ∅2 θ

Nomenclature Latin variables T M cp ṁ t UA ΔTlm h X H P Cd y g Q̇ D R″ k Re Pr Nu Ga Ja z L n N V̇ x f G Xtt c1, c2 C r gc A V NR NTU

temperature (°C) mass (kg) specific heat (kJ kg−1 K−1) mass flow rate (kg s−1) time (s) overall heat transfer coefficient (kW K−1) log mean temperature difference (°C) specific enthalpy (kJ kg−1) water-LiBr mass fraction (%) height between upper and lower shell components (m) pressure (kPa) discharge coefficient (–) fluid level in heat exchanger level (m) gravity (m s−2) heat transfer rate (kW) diameter (m) thermal resistance (m2 K kW−1) thermal conductivity (kW m−1 K−1) Reynolds number (–) Prandtl number (–) Nusselt number (–) Galileo number (–) Jakob number (–) axial distance (m) length (m) number of tubes (tubes) number of (–) volumetric flow rate (m3 s−1) quality (–) friction factor (–) mass flux (kg s−1 m−2) Lockhart-Martinelli parameter (–) constants (–) correction factor (–) radius (m) gravitational constant (kg m N−1 s−2) area (m2) volume (m3) average number of tubes per row (tube row−1) number of transfer units (–)

σ Subscripts i o in out eva con abs gen w l v IHX strat f lo ct ch st fo nom calc weak strong fi x atm wb db tot met recirc pan avg t sol vo ls no max min air

Greek variables ε ρ ζ μ Γ

heat transfer coefficient (kW m−2 K−1) difference (–) two-phase flow multiplier (–) angle from top of tube to condensate level in bottom of tube (Ra) vapor void fraction (–)

effectiveness (–) density (kg m−3) loss coefficient (–) dynamic viscosity (kg m−1 s−1) film flow rate (kg s−1 m−1)

inner tube surface outer tube surface inlet outlet evaporator condenser absorber generator wall liquid vapor intermediate heat exchanger stratified friction liquid only cooling tower chilled water steam fouling nominal calculated dilute water-LiBr solution mixture concentrated water-LiBr solution mixture film cross-sectional atmospheric wet-bulb dry-bulb total metal recirculation condenser pan average tube solution vapor only superficial nodes maximum minimum air

unit warmup and shutdown procedures. Later, Kohlenbach and Zielger [7] developed an absorption chiller model based on external and internal steady-state enthalpy balances. Despite assuming constant waterLiBr properties, constant overall heat transfer coefficient (HTC) values, and that evaporation and solution enthalpy are constant, simulation results reasonably agreed with experimental data obtained from a 10kW absorption chiller [8]. Borg and Kelly [9] modeled dynamic absorption chiller behavior using a series of interrelated control volumes with lumped heat exchanger masses and experimentally calibrated performance maps. Li et al. [10] simulated dynamic absorption chiller performance in tropical climates using local energy and mass balances

their mechanical-driven vapor-compression cycle counterparts; thus, it is advantageous to accurately depict dynamic absorption chiller behavior. However, the mass, temperature, and species transport phenomena within the chiller dictates establishing and solving a highly non-linear system of time-dependent conservation equations at each time-step, which can be computationally expensive. Several models aim to characterize dynamic single effect, LiBr absorption chiller performance while implementing significant steadystate simplifications. Anand et al. [6] led early absorption chiller research efforts in which they modeled isolated absorption chiller components as well as an entire 10.55-kW chiller in order to gain insight on 575

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et al. [19] looked at steam mass diffusion based on Nusselt’s film theory in their custom 15-kW absorption chiller. Moreover, Xu et al. [20] investigated different control schemes for a single effect, LiBr absorption chiller. Like these studies, the single effect, LiBr absorption chiller model presented in this paper builds on previous work while providing enhanced generator fidelity. In studies referenced earlier, hot water driven absorption chillers allowed authors to use widely accepted HTCs for sensible heat transfer in horizontal tubes to describe the heat addition in the generator. Commercially available absorption chillers can often run on both hot water and steam. These large chillers are rarely outfitted with their own steam boiler, condenser, and condensate pump, as that requires that the price of fuel falls well below the price of electricity (i.e. the spark spread). A more conventional layout used in industry involves absorption chillers receiving bypass steam when a reduction in demand allows for some low-pressure steam to be diverted from the main process to drive the absorption-refrigeration cycle. Steam traps located in the condensate return line leaving the generator necessitate that all steam sent to the absorption chillers is condensed. Thus, absorption chillers can only consume as much steam as they can condense. None of the models referenced previously are able to account for fluctuating steam inlet conditions. The HTC for ideal, gravity-driven condensation implemented by Yin et al. [12] is not valid for large, commercially available absorption chillers because the steam flow rates are usually too high. Also, Yin [12] did not account for the axial accumulation of condensate along the generator tube bundle, therefore, the amount of steam consumed at variable chiller operating conditions cannot be resolved. The model proposed in this paper remedies issues the aforementioned models have regarding steam consumption. To do this, a comprehensive heat transfer model for condensation inside horizontal tubes that differentiates between flow regimes is implemented. Steam temperature, quality, and pressure are established along the generator tube bundle when the heat transfer model is combined with relevant twophase flow pressure drop equations. Intrinsically, the continuous updating of quality along the generator tube bundle enables condensate accumulation in the axial direction within generator tubes to be accounted for. These additions allow for specific steady-state and transient steam consumption trends to be established based on changing steam conditions behind a control valve or other external parameters.

in each absorption chiller component. Saleh and Mosa [11] optimized performance of low capacity absorption chillers receiving hot water from solar plate collectors. Finally, Yin [12] implemented condensation HTC relations for ideal, gravity-driven steam condensation in double effect absorption chillers. The double effect absorption chiller model was subsequently used in a parametric study [13]. Though these models sufficiently align with collected experimental data in some cases, one or more steady-state assumption utilized to predict dynamic behavior results in a failure to capture all physics inside the chiller. Hence, deviations between predicted and experimental values occur during large transients. More recent studies have attempted to resolve the issues described above. This necessitates mass, energy, and species conservation equations are solved simultaneously in each absorption chiller main component. Shin et al. [14] solved the system of non-linear equations to describe dynamic behavior of a commercial, direct-fired, double effect absorption chiller. Ochoa et al. [15] developed a dynamic single effect, LiBr absorption chiller model in Matlab using an explicit finite difference method to solve the aforementioned conservation equations. The model demonstrated good agreement between predicted and measured water outflow temperatures despite assuming the vapor mass flow rate leaving the generator is equivalent to the vapor mass flow rate leaving the evaporator as well as neglecting mass deposition in the condenser [15]. In a follow-up study, Ochoa et al. [16] uncovered that relative error between measured and predicted external outlet water temperature values is considerably reduced when variable overall HTCs are used instead of constant overall HTCs. Evola et al. [17] solved the nonlinear system of equations using a Newton-Raphson scheme with a forward finite difference approximation. Error between measured and simulated external flow rate temperature values reached a maximum of 5% over the course of a day [17]. They further validated the dynamic model with a step change of 10 °C in the driving temperature [17]. Likewise, Marc et al. [18] used continuous real external temperature data over the course of two distinct days to validate their dynamic absorption chiller model. Relative error between experimental and simulated absorption chiller component thermal capacities was ± 10% [18]. Additional dynamic models were developed that combine the solving of mass, energy, and species governing equations with further refinement regarding a particular chiller component. For instance, Zinet

Fig. 1. Simplified single effect, LiBr absorption chiller model.

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C.T. Misenheimer, S.D. Terry

The combination of this increased generator fidelity and the resolving of the mass, species, and temperature transport equations in each chiller component sets the absorption chiller model in this paper apart from every other absorption chiller model found throughout literature. The steady-state and transient steam consumption trends are invaluable to industry users of large, commercially available absorption chillers that use bypass steam in combined heat and power (CHP) systems.

2.1. Condenser and evaporator The governing equations for the evaporator and condenser include the conservation of mass and energy equations. Because the refrigerant is assumed to be pure water, the species conservation equation for the evaporator and condenser is not needed. A condenser mass balance is given by Eq. (1):

dMcon = ṁ v,7−ṁ l,8 dt

2. Dynamic model description

A similar mass balance can be performed on the evaporator. The shell-side conservation of energy equation for the condenser is provided in Eq. (2):

Fig. 1 illustrates the basis from which the model presented in this paper is formed [21]. Main single effect, LiBr absorption chiller components include four shell-and-tube heat exchangers, an intermediate heat exchanger, throttle valves, and pumps. The evaporator and absorber are housed in the lower shell, and the condenser and generator are housed in the upper shell. A heat input in the generator boils-off some refrigerant from a water-LiBr mixture making the LiBr concentration of the mixture more “strong”. Conversely, the affinity between water and LiBr causes the strong solution to dilute itself as it falls over cooling tubes by absorbing vapor refrigerant in the absorber. The diluted, or “weak”, LiBr solution is then pumped back to the generator. Thus, only two working LiBr concentrations at any given instant are considered. Further assumptions are made based on those applied in previous studies to complete the thermodynamic model: a. b. c. d. e. f.

g.

h.

i.

j.

(1)

Mcon cp,con

dTcon = ṁ v,7 h7−ṁ l,8 h8−UAcon ΔTlm,con dt

(2)

The thermal mass term on the left side of Eq. (2) incorporates both the thermal mass of the liquid in the heat exchanger sump and the metal of the heat exchanger including the shell, tubes, drift eliminators, and other solid components. The total thermal mass is shown in Eq. (3):

Mcon cp,con = Ml,con cp,l,con + Mcon,met cp,con,met

(3)

An additional energy balance equation is required for the volume of water within the heat exchanger tubes. The tube-side energy balance, given by Eq. (4), is established about the exit temperature, and not the average tube-side temperature, so that the exit temperature may be resolved at subsequent time-steps. Furthermore, knowing the tube-side exit temperature enables the use of the logarithmic mean temperature difference (LMTD) method, denoted as ΔTlm . The definition of ΔTlm is given by Eq. (5).

The refrigerant is pure water [18]. The refrigerant leaves the condenser as a saturated liquid [20]. The refrigerant leaves the evaporator as a saturated vapor [21]. Valves are isenthalpic devices. The water-LiBr solution leaving the absorber and generator is presumed saturated [15]. The water-LiBr solution leaving the absorber and generator is in phase equilibrium with the refrigerant temperature in the evaporator and condenser, respectively [21]. The temperature of the refrigerant vapor desorbed in the generator is in phase equilibrium with the temperature of the refrigerant in the condenser and the dilute LiBr mass fraction [21]. The temperature of the cooling tower water leaving the absorber is equal to the temperature of the cooling tower water entering the condenser (T14 = T15) [18]. The absorption chiller is a two-pressure system, as refrigerant temperatures in the evaporator and condenser govern pressures in the lower and upper shell, respectively [16]. Heat losses to the environment and fluid transport delays between components are neglected [15].

ρi,con cp,i,con Vi,con

dT16 + ṁ ct cp,i,con (T16−T15) = UA con ΔTlm,con dt

(4)

The shell-side and tube-side energy balances for the condenser, given by Eqs. (2) and (4), respectively, can be extended to evaporator. For brevity, the evaporator governing equations are excluded.

ΔTlm =

(To,in−Ti,out )−(To,out−Ti,in) ln[(To,in−Ti,out )/(To,out−Ti,in)]

(5)

2.2. Absorber Unlike the evaporator and the condenser, the absorber contains a water-LiBr solution. The species conservation equation is required along with mass and energy conservation equations. An absorber mass balance is represented by Eq. (6):

dMabs = ṁ strong −ṁ weak + ṁ v,10 dt

A majority of Assumptions a through j are established throughout literature. Assumption g dictates that the vapor desorbed in the generator is superheated. The other simplifications are reasonable to facilitate the completion of the thermodynamic model. In Fig. 1, temperature terms defined as Teva, Tcon, Tabs, and Tgen represent the average shell-side temperatures in the evaporator, condenser, absorber, and generator, respectively. For both the condenser and evaporator, the shell-side refrigerant temperature is assumed to be homogeneous in accordance with the saturation pressure within the upper and lower shell, respectively. Alternatively, shell-side temperatures in the absorber and generator are defined as the average temperature between shell-side inlet and outlet mass flow rates. Mass terms, denoted as Meva, Mcon, Mabs, and Mgen, correspond to the total mass of each respective heat exchanger, including sump liquid, heat exchanger tubes, drift eliminators, and shell walls. Finally, ṁ v,7 and ṁ v,10 signify the vapor mass flow rates leaving the generator and evaporator, respectively, whereas ṁ l,8 is the saturated liquid flow rate draining from the condenser to the evaporator.

(6)

The species and shell-side energy conservation equations for the absorber are shown in Eqs. (7) and (8), respectively:

d (Mabs Xweak ) = ṁ strong Xstrong −ṁ weak Xweak dt

Mabs cp,abs

dTabs = ṁ v,10 h10 + ṁ strong h6−ṁ weak h1−UAabs ΔTlm,abs dt

(7) (8)

Like the evaporator and the condenser, a tube-side energy balance is implemented to update the temperature of the water leaving the absorber and entering the condenser tubes. The absorber tube-side energy balance is provided by Eq. (9):

ρi,abs cp,i,abs Vi,abs

dT14 + ṁ ct cp,i,abs (T14−T13) = UAabs ΔTlm,abs dt

(9)

The ΔTlm term in Eq, (8) and Eq. (9) employs both T1 and T6 rather than a uniform shell-side absorber temperature like the evaporator and condenser [21]. 577

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solution level falls below the opening, the mass flow rate from the generator to the absorber naturally decreases more than it would if based purely on generator sump level. The ζ term that is allowed to move based on heat exchanger sump level captures this reduction in flow rate. The condenser pan that collects the condensate is assumed to have a uniform rectangular cross-sectional area. In contrast, the height of the water-LiBr solution in the generator sump is a function of the semi-circle geometry of the upper shell minus the volume displaced by the generator tubes.

2.3. Generator The conservation equations for the generator are similar to that of the absorber with an exception in the conservation of energy equation given by Eq. (12):

dMgen

= ṁ weak −ṁ strong −ṁ v,7

dt

d (Mgen Xstrong ) dt

Mgen cp,gen

dTgen dt

(10)

= ṁ weak Xweak −ṁ strong Xstrong

(11) 3. Definition of parameters

= ṁ weak h3−ṁ strong h4−ṁ v,7 h7 + Qȯ ,gen

(12)

The single effect, LiBr absorption chiller model in this paper is based off of York’s YIA14F3 1377 nominal ton (4843-kW) chiller [23]. Heat transfer surface areas, tubes per heat exchanger, solution masses, sump volumes, materials, and other physical properties can be directly calculated or inferred based on listed dimensions and information readily available in online documents [24].

In Eq. (12), the Qȯ ,gen term replaces the UAΔTlm term found in energy balance equations for the other primary heat exchangers. This term is expanded on later in this paper. 2.4. Intermediate heat exchanger An intermediate heat exchanger (IHX) uses the hot concentrated water-LiBr solution draining from the generator to the absorber in order to heat the dilute water-LiBr solution being pumped from the absorber to the generator. This increases cycle efficiency since less of a heat input is required. The epsilon-NTU method, given by Eq. (13), is used to describe IHX performance [22].

εIHX =

T4−T5 T4−T2

3.1. Methods Nominal external operating conditions along with typical internal design temperatures and LiBr concentrations allow for a steady-state LMTD analysis to be performed. With steady-state heat flows across each main heat exchanger known, a nominal UA value can be solved for via Eq. (16). This nominal UA value is calibrated to match a UA value calculated using Eq. (17) for each primary shell-and-tube heat exchanger. The HTCs used are given in subsequent paper subsections. Because a variety of assumptions are made to complete the model, values for external HTCs are given a correction factor to ensure steadystate calibration at nominal conditions, similar to the technique used by Goodheart [25]. The correction factors for external HTCs in the absorber and condenser are small (about 10%) and can be attributed to any of the simplifications or large amounts of error inherent in all HTC correlations. A larger correction factor in the evaporator is likely the result of the presence of unspecified external enhancements on evaporator tubes (i.e., fins) [23]. Correction factors can be found with other nominal absorption chiller values in Table 1.

(13)

Because the details of the IHX are proprietary and not available for this study, the effectiveness of the IHX is assumed to be constant in all simulations. Constant IHX effectiveness is an assumption made in previous studies with a high-degree of success [7]. 2.5. Other devices A solution pump moves the dilute water-LiBr solution from the absorber to the generator. The change in enthalpy of an incompressible fluid across a pump is a function of pump isentropic efficiency, pressure difference, and pump work. Since the pressure difference between the upper shell and lower shell is relatively small (around 8.6 kPa during nominal chiller operation), the change in enthalpy of the dilute waterLiBr solution across the pump is not taken into account in this model [7]. Furthermore, the pump is considered a constant volumetric flow rate device. An additional constant volumetric flow rate device in the evaporator pumps refrigerant that fails to evaporate on the evaporator tube bundle back to the evaporator sprays. As illustrated in Fig. 1, two valves throttle the refrigerant and concentrated water-LiBr solution draining from the upper shell to the lower shell. Fluid flow is induced by a combination of shell-side pressure differences as well as gravity. The valves in this study are modeled based on the framework presented by Evola et al. [17]. The mass flow rate of a fluid draining by pressure difference and gravity from the condenser to the evaporator is given by Eq. (14):

UAnom =

UAcalc =

ṁ l,8 = Cd

+

R″fo,i Di

+

ln(Do / Di) 2kt

+

R″fo,o Do

+

1 ⎤ αo Do

⎦

(17)

Sensible heat transfer occurs inside the tubes in the evaporator, condenser, and absorber, in which the HTC relations are well established. A larger number of tubes in the evaporator and absorber results in lower tube-side Reynolds numbers. Gnielinski’s [26] relation for transitional and turbulent internal duct flow is suitable for calculating internal HTCs in the evaporator and absorber. Conversely, DittusBoelter’s [27] HTC correlation for the larger tube-side Reynolds numbers in the condenser is used, as there are fewer tubes in the condenser. Applicable HTCs for shell-side external flows are available in literature. The HTCs for a fluid evaporating and condensing on a tube bundle are given by Eqs. (18) and (19), respectively [28].

(14)

The resistance to flow from the line and valve is lumped into a single term (ζ). The term ζ is defined by Eq. (15):

ζ con = ζnom,con (ynom,con / ycon )2

(16)

πLt n ⎡ 1 αi Di ⎣

g

2Ax2,con gc ρavg ⎡ΔP + ρavg g (Hcon + ycon ) ⎤ c ⎣ ⎦ ζ con

̇ Qnom ΔTlm,nom

1/3

2 3 ⎛ ρl gkl ⎞ αo,eva = 0.821Refi−,0.22 eva ⎜ 2 ⎟ ⎝ μl ⎠

(15)

(18) 3 1/3

As noted in [17], the ζ term corresponds to typical control schemes for absorption chillers in the refrigerant loop. Moreover, the drain for the concentrated water-LiBr solution in the generator is often located on the side of the shell, not in the bottom of the sump. As the generator

ρ (ρ −ρv ) gkl ⎞ 1/3 ⎛ l l αo,con = 1.51Re− ⎟ fi,con ⎜ μl2 ⎠ ⎝

The film Reynolds number is found according to Eq. (20): 578

(19)

Energy Conversion and Management 150 (2017) 574–587

C.T. Misenheimer, S.D. Terry

Table 1 Nominal chiller parameters.

P For a Δ T< 7.1 °C: αo,gen = 1042(Tw−Tgen)1/3⎛ con ⎞ ⎝ Patm ⎠ ⎜

0.4

⎟

(24)

Parameter

Value

Parameter

Value

Capacity (kW)

4843

0.0158

COP (–)

0.7097

̇ Vweak (m3 s−1) ̇ Vrecirc (m3 s−1)

P11 (kPa) x11 (–) T11 (°C) ṁ st (kg s−1)

163.41 0.99

Xweak (%) Xstrong (%)

59.29 64.3

113.94 3.0914

Lt (m) ζnom,gen (–)

8.5 2.0037

3.2. Generator

0.31248

ζnom,con (–)

13.1189

29.44

ynom,gen (m)

0.3899

Single effect, LiBr absorption chillers can utilize low-pressure steam or hot water as a heat source. In order to determine accurate steam temperatures, HTCs, and quality along the generator tube bundle, Dobson and Chato’s model for heat transfer for a fluid simultaneously flowing and condensing is applied [31]. This heat transfer model divides condensation flow into annular, stratified-wavy, and stratified flow regimes. The Nusselt number for annular condensation is represented by Eq. (26) [31]:

Vcṫ (m3 s−1) T13 (°C)

34.45

ynom,con (m)

0.2587

T16 (°C) ̇ (m3 s−1) Vch T17 (°C) T18 (°C)

38.44 0.2082

Cd (–) Ax ,gen (m2)

0.60 0.00456

12.22 6.67

Ax ,con (m2) Hgen (m)

0.002 1.8288

Teva (°C) Tcon (°C) Tabs (°C) Tgen (°C)

3.26 44.6 44.2 94.9

ṁ v,7 (kg s−1) ṁ l,8 (kg s−1) ṁ v,10 (kg s−1) ṁ strong (kg s−1)

2.095 2.095 2.095 24.83

Pcon (kPa) Ceva (–) Ccon (–)

9.377 2.660 1.115

0.78 0.8779 2.62

Cabs (–)

0.904

Peva (kPa) Hcon (m) Ax ,pan,con (m2) rgen (m)

Cgen (–)

1.00

Mtot ,met (kg)

28,894

R″fo,i,eva (m2 K kW−1)

0.00514

Mi,tot ,l (kg)

4242

R″fo,i,con (m2 K kW−1)

0.0327

Mo,tot ,l (kg)

7701

R″fo,i,abs (m2 K kW−1)

0.0327

Mtot (kg)

40,837

0.00

εIHX (–)

0.70

R″fo,o (m2 K kW−1)

0.00

neva (tubes)

476

ncon (tubes) nabs (tubes) ngen (tubes)

212 732 200

(m)

0.0178 0.0241 0.0179

34 26

ρt ,gen (kg m−3)

8857.3

52

Patm (kPa)

101.353

Do,gen (m)

0.01905

NR eva (tubes row-1) NR con (tubes row−1) NRabs (tubes row−1)

Refi =

4Γo μ

0.510

The term NR in the above equation represents the average number of tubes per row in each respective heat exchanger. Note that the flow regime for an evaporating fluid falls within the wavy flow regime according to Eq. (22):

kl

0.25

= 0.23

0.12 Re vo ⎡ GaPrl ⎤ 1 + 1.11Xtt0.58 ⎢ ⎣ Jal ⎥ ⎦

θ + ⎛1− strat ⎞ Nustrat π ⎠ ⎝

(28)

0.25

ρ (ρ −ρ ) gh kl3 αi,gen = 0.728σ 0.75 ⎡ l l v lv ⎤ ⎢ μ (Tst −Tw ) Di ⎥ ⎣ l ⎦

(29)

Gnielinski’s HTC for sensible heat transfer is used once all the steam is condensed. Local tube-side HTCs are established based on local properties and quality. In horizontal tubes, the two-phase flow pressure drop is a combination of pressure losses due to friction and acceleration, as shown in Eqs. (30) and (31). An appropriate two-phase flow 2 multiplier (∅lo ) correlation is used in conjunction with Dobson and Chato’s heat transfer model in order to establish steam temperature, quality, and pressure along the generator tube bundle [33]. It is assumed that local steam saturation temperatures are governed by local steam pressures.

(22)

Consenza and Vliet [29] characterized heat transfer from a waterLiBr solution falling over a tube bundle while absorbing refrigerant. Their HTC is represented by Eq. (23): −1/3

⎛ 3μΓo,abs ⎞ αo,abs = 0.3kRefi0.46 ⎟ ,abs ⎜ 2 ⎝ ρl g ⎠

(26)

The Soliman [32] transition criterion based on a local modified Froude number and superficial Reynolds number (Rels) is used to predict the transition between annular and stratified-wavy flow regimes. Soliman [32] reported annular flow exists for modified Froude numbers above 7, and stratified-wavy flow exists for modified Froude numbers below 7. However, Dobson and Chato [31] found that a better value for transition is 20. In this study, for modified Froude numbers above 20, the condensation flow regime is considered annular. Conversely, for modified Froude numbers below 7, the flow regime is presumed to be stratified-wavy. Linear interpolation between annular and stratifiedwavy flow regimes is used when the modified Froude number is between 7 and 20. Ideal, gravity-driven condensation occurs for generator inlet vapor Reynolds numbers less than 35,000. The HTC for ideal condensation is given by Eq. (29) [28]:

0.3769

(21)

Refi,eva < 5800Prl−1.06

Di,gen αi,gen

c Nustrat = 0.0195Rels0.8 Prl0.4 ⎡1.376 + 1c2 ⎤ ⎢ Xtt ⎥ ⎣ ⎦

0.4014 0.4014 0.0502

(20)

ṁ o 2(NR) Lt

2.22 = 0.023Rels0.8 Prl0.4 ⎡1 + 0.89 ⎤ ⎥ ⎢ X tt ⎦ ⎣

0.5

In Eq. (20), Γo is the film mass flow rate over the outside of the tubes. The film mass flow rate can be calculated using Eq. (21):

Γo =

kl

(25)

(27)

0.4014

0.01905 0.0254 0.01905

Di,gen αi,gen

0.4

⎟

The HTC for the stratified-wavy flow regime is the superposition of gravity-driven condensation in the upper part of tube and forced convection from the collected condensate in the bottom of the tube, as shown in Eqs. (27) and (28) [31]. Shear forces between fast moving vapor and slower moving condensate result in the formation of waves which increase local HTCs.

Nu =

kt ,eva (kW m−1 K-1) kt ,con (kW m−1 K-1) kt ,abs (kW m−1 K-1) kt ,gen (kW m−1 K-1) cp,t ,gen (kJ kg−1 K−1) cp,met (kJ kg−1 K-1)

0.0173

Do,eva (m) Do,con (m) Do,abs (m)

Nu =

0.5731

R″fo,i,gen (m2 K kW−1) (m) (m) (m)

⎜

0.0063

T14 (°C)

Di,eva Di,con Di,abs Di,gen

P For a Δ T> 7.1 °C: αo,gen = 5.56(Tw−Tgen )3⎛ con ⎞ ⎝ Patm ⎠

(23)

Finally, Yin [12] had success using Jakob and Hawkin’s [30] HTC correlation with a pressure correction for nucleate pool boiling from horizontal tubes, exhibited by Eqs. (24) and (25), in their dynamic absorption chiller model.

dP dP d ⎡ (1−x )2 x2 ⎤ = ⎛ ⎞ + Gst2 + ⎥ ⎢ dz dz dz ρ (1 σ ) ρ − ⎝ ⎠f vσ ⎦ ⎣ l 579

(30)

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C.T. Misenheimer, S.D. Terry 2 2 ⎛ dP ⎞ 2 2flo Gst ⎛ dP ⎞ = ∅lo = ∅lo ρl Di ⎝ dz ⎠ f ⎝ dz ⎠lo

reached within five iterations per time-step. Liquid water and vapor properties from XSteam [35] are stored in the FORTRAN program as callable functions. The functions contain either polynomial correlations or lookup tables based on one or multiple inputs. Similarly, water-LiBr solution properties are stored as callable FORTRAN functions [36]. Properties for heat transfer calculations include density, thermal conductivity, dynamic viscosity, and specific heat [37]. Lithium bromide equilibrium correlations based on mass fraction and refrigerant temperature along with enthalpy for aqueous LiBr solutions are obtained from ASHRAE [38]. A crystallization temperature correlation is used to monitor a LiBr solution’s proximity to solidification [36]. Two additives are employed by the vendor: one prevents corrosion of various internal steel and copper components, and the other additive (2-Ethyl-1-Hexanol) enhances heat and mass transfer inside the absorption chiller [23]. Because both additives are included in the LiBr solution in small amounts, their effect on internal heat and mass transfer calculations is not taken into account, as is customary in previous successful absorption chiller studies. Various dimensions and nominal chiller operating parameters can be found in Table 1.

(31)

A Gauss-Seidel iterative method is used to solve for steam temperature, quality, and pressure along the generator tube bundle. The generator tube bundle is divided into a fixed number of differential elements (dE) and the same amount of node locations plus one. A pseudo finite volume method uses steam temperature, quality, and pressure stored at generator tube node locations to calculate average steam properties over a dE between node locations. The average properties are used to calculate local two-phase flow multipliers, local HTCs, local heat transfer rates into the tube walls over the dE, and steam pressure and quality at the subsequent node location. Acceptable convergence is considered when the l2-norm of the steam temperature, quality, and pressure residuals drops four orders of magnitude. After convergence, local generator solution temperatures are found by interpolating between T4 and T7 along the length of the tube bundle. Local HTCs on external tube surfaces are then calculated via Eq. (24) or Eq. (25). The total heat transfer rate into the generator sump solution, given by Eq. (32), is the summation of the heat transfer rate from each tube wall dE into the water-LiBr mixture, and is thus a function of local temperature differences between tube walls and the sump solution:

⎡ Qȯ ,gen = ngen ⎢ ⎣

Nno − 1

∑ 1

⎤ αo,gen (πDo,gen Δz )(Tw−Tgen ) ⎥ ⎦

4.1. Steady-state trends The condensation heat transfer model coupled with two-phase flow pressure drop correlations allow for specific absorption chiller behavior within the generator tubes to be observed. This behavior is used to validate the absorption chiller model. Figs. 3 and 4 visually illustrate local steam temperatures, pressures, qualities, and tube wall temperatures along the generator tube bundle when the absorption chiller is operating at nominal, steady-state conditions as listed in Table 1. High vapor qualities at the generator tube bundle entrance result in large shear forces between high vapor velocities and low collected condensate velocities. These shear forces cause a pressure drop as indicated by Fig. 3. Because steam saturation temperature is a function of pressure, a slight temperature decrease is observed in Fig. 4. As the vapor qualities approach zero, the pressure drop per dE decreases due to a reduction in shear forces between the vapor and condensate. Once all of the vapor is condensed out, heat is transferred to the tube walls sensibly as seen in Fig. 4. Moreover, pressure losses become small as the pressure drop across a dE is solely a function of single-phase friction between the condensed liquid and tube walls. Per manufacturer’s data, the generator is a counterflow shell-andtube heat exchanger [23]. The temperature outside of the generator

(32)

Finally, local tube wall temperatures are updated along the generator tube bundle by solving the 1-dimensional radial heat conduction equation in the axial direction using a Crank-Nicolson discretization and Thomas’ Algorithm with Neumann boundary conditions [34]. The three-dimensional radial heat conduction equation can be reduced to purely axial conduction because of the thin-walled generator tubes. A graphical representation of generator solution strategy is shown in Fig. 2. 4. Simulation results As previously discussed, the system of governing equations is highly nonlinear. After an implicit discretization, a Newton-Raphson iterative scheme intended for systems of nonlinear equations is used to solve the system of equations introduced in Section 2. Sufficient convergence is considered when the error of unknown variables at the subsequent time-step drops four orders of magnitude. Convergence is typically

Fig. 2. Graphical generator solution strategy.

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Online documents reveal that the absorption chiller consumes 163.4 kPa steam at a rate of no more than 3.09 kg s−1 when cooling tower and chilled-water flow rates and temperatures correspond to nominal conditions found in Table 1 [23]. Figs. 3 and 4 validate this statement, as indicated by the quality extending approximately the length of the generator tube bundle. If the quality curve shifts to the left in Figs. 3 and 4, the absorption chiller can consume more steam. On the other hand, if the quality curve shifts to the right in Figs. 3 and 4, the absorption chiller can consume less steam. As mentioned earlier, all vapor must be condensed in the generator tube bundle due to the presence of steam traps in condensate lines leaving the generator. According to Figs. 3 and 4, there is little room for a rightward shift in the quality curve, signifying that any additional steam provided to the absorption chiller would yield a quality greater than zero at the tube bundle exit. Thus, the absorption chiller model accurately depicts steam consumption during nominal, steady-state conditions. Increased generator fidelity allows for the establishment of important steam consumption trends based on changing steam inlet parameters as well as other external chiller temperatures and flow rates. These trends further validate the absorption chiller model. Simulation details can be found in Table 2. The first simulations involve investigating how steam inlet conditions affect condensation lengths inside generator tubes, while keeping other external flow rates and inlet temperatures constant. Results are depicted in Figs. 5 and 6 for variable inlet steam pressures and variable steam mass flow rates, respectively. Unsurprisingly, Fig. 5 shows condensation lengths are shorter when the absorption chiller is fed higher pressure steam while maintaining constant steam mass flow rates because temperature differences between the tube walls and steam saturation temperatures are larger. The condensation curves shifted to the left at higher steam pressures indicate that the absorption chiller can consume more steam at higher pressures, while the condensation curves shifted to the right at lower steam pressures designate that the absorption chiller cannot consume more steam at lower pressures. Similarly, Fig. 6 demonstrates lower steam mass flow rates while holding steam pressure constant produces shorter condensation lengths. While these results are expected, Figs. 5 and 6 illustrate an important revelation. When feeding the absorption chiller with high mass flow rates of low-pressure steam, the absorption chiller cannot condense all of the steam, and the quality of the mixture leaving the generator tube bundle is greater than zero. In reality, a small condensate drain and steam trap behind the absorption chiller necessitate that all of the steam provided to the chiller is condensed. Therefore, the condensation curves that show the quality leaving the generator tube bundle greater than zero are not possible. Results shown in Figs. 5 and 6 further corroborate the nominal, steady-state condensation length in Figs. 3 and 4, as a slight increase in steam mass flow rate or a decrease in steam inlet pressure yields a rightward shift in the quality curve to the point where some vapor still remains at the generator tube bundle exit. In a chilled-water system, external temperatures and flow rates fluctuate. These changes in temperature and flow rate result in absorption chiller perturbations and influence how much steam can be

Fig. 3. Pressure and quality along generator tube bundle.

Fig. 4. Steam temperature, tube wall temperature, and quality along generator tube bundle.

tubes on the steam entrance side corresponds to the hot, concentrated LiBr solution draining back to the absorber. Conversely, local shell-side generator temperatures external to the condensate return portion of tubes correspond to the preheated diluted LiBr solution pumped from the absorber. Therefore, the gradually decreasing local tube wall temperature trend illustrated by Fig. 4 is expected. Furthermore, low tube wall thermal conductivities (provided in Table 1) from the nickel component in the copper-nickel alloy tubes cause a large drop in local tube wall temperate once latent heat transfer ceases and sensible heat transfer commences. A small wobble can be observed in the tube wall temperature around the 6-meter mark along the generator tubes. This occurs due to the transition between annular and stratified-wavy HTCs not being smooth in the condensation heat transfer model.

Table 2 Summary of steady-state trend simulations. Parameter

Investigation 1

Investigation 2

Investigation 3

Investigation 4

P11 (kPa) ṁ st (kg s−1) T13 (°C) ̇ (m3 s−1) Vch

Variable 3.0914 29.44 0.2082

165.5 Variable 29.44 0.2082

165.5 3.0914 Variable 0.2082

165.5 3.0914 29.44 Variable

T17 (°C) Vcṫ (m3 s−1)

12.22 0.3125

12.22 0.3125

12.22 0.3125

12.22 0.3125

2655 5

2655 6

2655 7, 8

2655 9, 10

h11 (kJ kg−1) Figure Number

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out of the water-LiBr solution in the generator sump. After several minutes, the chiller reaches steady-state operation, and the three refrigerant flow rates (ṁ v,7 , ṁ l,8 , and ṁ v,10 ) converge to roughly the same value. However, these values are now higher than their nominal, steady-state values. The evaporator temperature decreases as a consequence of the increased mass flow rates entering and leaving the evaporator because the enthalpy of the gaseous refrigerant leaving the heat exchanger is much larger than the enthalpy of the condensate arriving from the condenser. Ultimately, a reduction in the chilledwater temperature leaving the evaporator is observed. On the other hand, the steam pressure and mass flow rate provided to the generator can be reduced to keep the chilled-water temperature leaving the evaporator constant. In Fig. 8, the predicted heat input as a function of chiller capacity deviates from values made available by the manufacturers in some instances, especially at significantly reduced chiller capacities and low T13 values [23]. Inherent error in all HTC correlations and property functions as well as unknown external enhancements on the outside of evaporator tubes likely somewhat contribute to these discrepancies. Another source of potential error is the absence of the enhanced heat and mass transfer effects from the ethyl additive in the transient thermodynamic analysis. A more pronounced source of error comes from the aqueous water-LiBr solution state equation provided by ASHRAE, which starts to diverge at exceptionally low LiBr mass fractions characteristic of operating an absorption chiller at reduced loads and cooling tower return water temperatures. The model is calibrated at T13 values equal to 29.44 °C, hence, the chiller model excels at predicting the required heat input at various chiller capacities when T13 is held at this temperature. Because the exact chiller performance data is proprietary and not available for this study, the overlaid manufacturer’s data in Fig. 8 represents general chiller behavior for their entire chiller line, nearly all of which employ two tube passes in the evaporator [23]. Multiple evaporator tube passes significantly affect chiller performance in Fig. 8 since absorption chiller capacity is defined as the ratio of chilling power to nominal chilling power. Nevertheless, the chiller model does an adequate job at capturing absorption chiller performance at various chiller capacities and cooling tower return water temperatures. Another external parameter that can change in a chilled-water system is the chilled-water flow rate supplied to the evaporator tube bundle. A conventional control scheme for chilled water involves matching actual chilled-water temperature to some setpoint. Valves or variable frequency drive (VFD) pumps can move to satisfy this setpoint.

Fig. 5. Quality along generator tube bundle versus inlet steam pressure.

Fig. 6. Quality along generator tube bundle versus steam mass flow rate.

condensed. Cooling tower water pumps are typically constant flow rate devices, thereby allowing the temperature of the water entering the absorber (T13) to fluctuate based on local ambient conditions. Despite modern control systems best efforts to limit cooling tower return water temperature variability, deviations in T13 inevitably occur due to diurnal and seasonal weather changes. Condensation length as a function of variable T13 is depicted in Fig. 7. Lower cooling tower return water temperatures allow the absorption chiller to consume more steam because the water-LiBr solution temperature in the generator sump is lower. In contrast, higher entering absorber water temperatures extend the length required to condense all steam supplied to the absorption chillers. These steam consumption trends align with manufacturer’s information [23]. Reduced cooling tower water temperatures also allow for higher absorption chiller COPs. This trend is implicitly shown in Fig. 8, and reasonably complies with vendor data [23]. As T13 decreases, the amount of energy input required to achieve nominal cooling capacities is lessened. Below nominal tower return water temperatures cause the tube-side temperature leaving the absorber to decrease. This reduced tower water temperature leaving the absorber then proceeds to the condenser tube bundle, where it induces a condenser shell-side temperature drop. The drop in upper shell pressure as governed by the reduced condenser temperature allows for more refrigerant to be boiled

Fig. 7. Quality along generator tube bundle versus cooling tower return water temperature.

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assumed that the absorption chiller receives steam at nominal conditions per Table 1 when the SCV is in its full open position. Moreover, the SCV is assumed to be isenthalpic, and the steam pressure and enthalpy upstream of the SCV are assumed to be constant throughout the entirety of the transient simulation. The complete thermal system, comprised of an absorption chiller, cooling demand, cooling towers, and SCV, is illustrated in Fig. 11. The cooling towers are modeled using Merkel’s enthalpy potential theory that attributes heat transfer within the tower to enthalpy differences between air being drawn through the tower (hair) and the enthalpy of moist air taken at the bulk water temperature falling through the tower (hair,ct) [39]. Merkel’s theory in integral form in terms of heat removed from the water by the cooling tower is given by Eq. (33) [39]:

NTUct =

∫T

T13

16

dTl,ct hair ,ct −hair

(33)

The NTU term on the left side of Eq. (33) is generally referred to as the tower demand when dealing with cooling towers. A separate program uses vendor data for a 2748-kW (625 nominal ton) counterflow cooling tower with Merkel’s theory and the Chebyshev four-point numerical integration method to calculate the tower demand at different tower flow rates and ambient wet bulb temperatures [40]. A power regression can then be performed on the data points, as illustrated in Fig. 12. This allows cooling tower performance to be calculated as a function of ambient air conditions and flow rates through the tower. ASHRAE approximations are used to calculate the enthalpy of air at various atmospheric conditions [38]. Given ambient air conditions, the water temperature entering the cooling tower (T16), and the water flow rate through the tower, the only unknown in Eq. (33) at each time-step is the outlet cooling tower water temperature (T13). Due to the nonlinearity of Eq. (33), a Newton-Raphson non-linear solver is employed in conjunction with the Chebyshev four-point numerical integration method to resolve for T13 at each time-step. For simplicity, the cooling tower water leaving the condenser of the absorption chiller is assumed to be evenly distributed among the cooling towers. Further simulation details can be found in Table 3, and the transient response of the absorption chiller model is depicted in Fig. 13 through Fig. 21. Cooling tower water and chilled-water flow rates, along with the temperature of the chilled water returning from the cooling demand, are presumed to be constant throughout the simulation. Results are qualitatively compared to similar studies in literature. As a consequence of the fast valve action, the higher steam pressure provided to the generator tube bundle translates to a higher steam temperature. Initially, the large difference between steam temperature

Fig. 8. Absorption chiller percent design energy input versus chiller capacity [23].

The effect that variable chilled-water flow rate has on condensation lengths in the generator tube bundle and overall chiller performance is shown in Figs. 9 and 10. Lower chilled-water flow rates translate to lower than nominal chilled-water temperatures leaving the absorption chiller, which prompts a shell-side evaporator temperature drop. Reduced pressures stemming from lower evaporator saturation temperatures drives the dilute LiBr mass fraction leaving the absorber up, which ultimately increases the concentrated LiBr mass fraction leaving the generator. Of note, the dilute LiBr mass fraction increases more than the concentrated LiBr mass fraction. According to Assumption f made earlier, the generator temperature will rise to reach equilibrium with the upper-shell pressure governed by the condenser and the concentrated LiBr mass fraction draining back to the absorber. Higher water-LiBr solution temperatures in the generator sump increase generator tube wall temperatures which lower local heat transfer rates from condensing steam to the tube walls. Thus, less steam is condensed out per dE, the overall condensation length necessary to condense all vapor increases, and the absorption chiller is not capable of consuming nominal steam mass flow rates. Furthermore, the smaller difference between dilute and concentrated LiBr mass fractions results in less refrigerant boiled-off in the generator, which means less vapor leaves the evaporator and the chiller runs at reduced capacities once it achieves steady-state operation. Fig. 10 illustrates the capacity penalty when using reduced chilled-water flow rates to achieve larger temperature differences across the evaporator. The chiller model reasonably agrees with the general chiller data made available by the vendor. Again, the most probable reason for divergence in Fig. 10 is the general chiller data not being fully representative of the specific absorption chiller modeled in this study. 4.2. Transient results The manner in which a dynamic, single effect absorption chiller model responds to a step-change in thermal input is pervasive throughout literature. Traditionally, the step-change involves some sudden increase in the temperature of the hot water supplied to the generator tube bundle. A similar step increase in heat input is investigated in this study. Rather than simply increasing the hot water temperature proceeding to the generator tube bundle, it is assumed that the single effect, LiBr absorption chiller model presented in this paper is operating at reduced capacity when a pseudo steam control valve (SCV) is opened quickly from some throttled position to full open. Consequently, the steam pressure and steam mass flow rate provided to the absorption chiller increase as a function of valve position. It is

Fig. 9. Quality along generator tube bundle versus chilled-water flow rate.

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Fig. 10. Absorption chiller capacity versus chilled-water temperature leaving evaporator [23].

Fig. 12. Cooling tower characteristic curve for 2748-kW (625 nominal ton) counterflow cooling tower.

and local tube wall temperature results in high local in-tube heat transfer rates into the tube wall from condensing steam. This causes large amounts of vapor to be condensed out early in the tube bundle, which according to the second group of terms in Eq. (30), yields a negative pressure drop, or a pressure rise. As the average tube wall temperature increases over time, local heat transfer rates inside the generator tubes decrease, less vapor is condensed per dE, and the distance required to completely condense all of the steam increases. The larger distances required for condensation allows shear forces between the slower moving condensate and faster moving steam vapor to prevail over the deceleration term in Eq. (30); hence, a pressure drop is observed. Predictably, the condensate return temperature increases with the overall condensation length because the steam remains more near its saturation temperature longer, allowing less time for subcooling. These trends are illustrated clearly in Figs. 13 and 14. In Fig. 14, the time values in the legend correspond to time elapsed after the SCV attains its full open position. According to online documents made

available by the manufacturer, the absorption chiller exhibits higher than nominal steam consumption upon startup or during the onset of large upward capacity transients [23]. While the chiller is not allowed higher than nominal steam mass flow rates for the purpose of this simulation, it is easy to see that this model could capture this phenomenon due to initial large differences between steam temperature and tube wall temperature. At reduced capacities, the pressure in the upper shell of the absorption chiller is less than nominal as governed by the reduced saturation temperature of the refrigerant in the condenser. A large stepchange in generator heat input, in the form of higher steam pressure and mass flow rate provided to the generator tube bundle, results in a large amount of vapor desorbed from the shell-side solution in the generator, as shown in Fig. 15. This phenomenon can also be viewed by the sudden rise in condenser temperature in Fig. 17, and confirmed by an abrupt increase in the cooling tower water temperature exiting the

Fig. 11. Schematic of thermal system for transient simulation.

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Table 3 Transient simulation parameters and initial conditions. Parameter

Value

Parameter

Value

Simulation duration (hr)

2

2655

Time-step size (s)

0.036

Initial chiller capacity (–)

42.9%

Steam enthalpy throughout entire simulation (kJ kg−1) Steam conditions downstream of SCV at full open ̇ (m3 s−1) Vch

Time from simulation start to SCV open (min) SCV open time (s)

30

T17 (°C)

12.22

56 1.18 62.1

Vcṫ (m3 s−1) Nct (2748 nominal kW cooling tower) Twb (°C)

0.31248

Initial steam mass flow rate through SCV (kg s−1) Initial steam pressure downstream of SCV (kPa) Initial steam temperature downstream of SCV (°C)

86.8

Tdb (°C)

33.9

Nominal

0.2082

3 24

Fig. 15. Internal refrigerant flow rates.

driving force sharply increases mass flow rates from the generator and condenser to the absorber and evaporator, respectively. This translates into small initial increases in shell-side and tube-side evaporator temperatures, as seen in Figs. 17 and 18, respectively. After several minutes, the shell-side evaporator temperature and the chilled-water temperature (T18) fall to their nominal values, and refrigerant mass flow rates converge to the same value. Both of these occurrences are documented in literature [17]. Lastly, the concentrated water-LiBr solution draining from the generator initially spikes when the SCV is opened because the larger condenser pressure induces increased drainage flow rates. This causes the generator level to drop well below nominal levels and inhibits access to the entrance of the drain located on the side of the upper shell, thereby resulting in a subsequent reduction of the concentrated water-LiBr mass flow rate. This occurrence, depicted in Fig. 16, is captured by Eqs. (14) and (15). Additionally, both diluted and concentrated LiBr mass fractions increase as the absorption chiller capacity increases as shown in Fig. 19. This behavior is expected according to Assumption f, and well documented throughout literature and industry. As depicted in Fig. 16, the mass flow rate of the weak water-LiBr solution increases as the LiBr concentration goes up due to higher densities at higher mass fractions and the pumps being constant volumetric flow rate devices. Logically, the mass in the evaporator and condenser increase accordingly as more refrigerant is driven from the generator to the refrigerant loop where it

Fig. 13. Generator tube-side inlet steam pressure, outlet pressure, and outlet temperature.

Fig. 14. Overall steam condensation length propagation.

condenser tube bundle (T16) in Fig. 18. The upper shell pressure increases accordingly as a function of the temperature of the refrigerant in the condenser. Per Eq. (14), larger differences between upper shell and lower shell pressures serve as a larger driving force for inducing flow back to the lower shell. As illustrated in Figs. 15 and 16, the large

Fig. 16. Internal water-LiBr solution flow rates.

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defined as the heat removed from the chilled water in the evaporator tube bundle divided by the heat removed during nominal chiller operation, initially decreases, depicted in Fig. 21. This is expected given that the chilled-water temperature first increases before decreasing at the beginning of the large upward capacity transient shown in Fig. 18. Likewise, absorption chiller COP decreases abruptly once the SCV is opened. This is because the chiller is consuming steam at nominal pressures and mass flow rates, yet the chilled-water temperature leaving the evaporator tube bundle is still decreasing to nominal values. Eventually, the chiller capacity and COP approach their nominal values. Similar trends are observed in literature [17]. 4.3. Comments on numerical scheme As illustrated in the simulation figures, the Newton-Raphson numerical scheme performs well, converging in five iterations or less at each time-step. The implicit discretization of the governing equations presented in Section 2 of this paper allows any size time-step to be used. Furthermore, the Gauss-Seidel iteration resolves steam temperature, quality, and pressure along the generator tube bundle generally within 15 or less iterations per time-step. In Fig. 13, a slight oscillation appears in the predicted condensate temperature near the beginning of the upward capacity chiller transient. By definition, the Crank Nicolson discretization of the axial heat conduction equation yields second-order temporal and spatial error terms. These second-order leading error terms are accompanied by third-order partial derivatives. Generally, odd-numbered partial derivatives in leading error terms produce solutions that can be dispersive, or oscillatory, in nature [34]. This behavior is in contrast to a dissipative solution generated from an approximation with even-numbered partial derivatives in the leading error term. Smaller time-steps and additional elements can help suppress this initial oscillation further at the expense of increased simulation-time. Nevertheless, the brief oscillation is quickly curbed, as illustrated by Fig. 13 and confirmed by smooth quality curves in Fig. 14.

Fig. 17. Average internal shell-side heat exchanger temperatures.

5. Conclusions In this paper, a dynamic single effect, LiBr absorption chiller model is developed as a FORTRAN program. The dynamic model builds on work completed in previous successful studies that are grounded on mass, energy, and species conservation equations. A model for the description of condensation heat transfer inside generator tubes is implemented in conjunction with pressure drop equations for two-phase flow; thereby allowing steam temperature, quality, pressure, and tube wall temperature to be continuously updated along the generator tube

Fig. 18. External temperatures.

Fig. 19. LiBr mass fractions.

is deposited, as illustrated in Fig. 20. Conversely, mass in the generator and absorber drops. After the SCV is opened quickly, the absorption chiller capacity,

Fig. 20. Heat exchanger masses.

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Fig. 21. Chiller capacity and COP.

bundle. The model is capable of seamlessly switching between condensation and sensible forced convection heat transfer once the quality drops to zero inside the generator tubes, and can account for being fed steam at variable flow rates, pressures, and qualities. Increased generator fidelity allows for important steady-state steam consumption trends to be revealed as a function of changing external parameters. The trends established in this paper align with manufacturer’s figures and data in online documents reasonably well when operating the chiller at medium to high chiller capacities and warmer cooling tower return water temperatures. Deviation is observed at significantly reduced chiller capacities and low cooling water temperatures. Other absorption chiller models throughout literature often validated the dynamic behavior with a step increase in thermal input supplied to the generator. This test is mimicked by fast valve action in which an SCV is opened rapidly. All internal and external parameters exhibit an appropriate response to the sudden increase in thermal load. Moreover, the heat transfer model employed in the generator sheds light on overall condensation length propagation as well as higher than nominal steam consumption during large upward transients and chiller startup procedures. Finally, the ability to accurately portray part-load steam consumption, capture perturbations stemming from changing external parameters, and accept fluctuating inlet steam conditions makes this chiller model a good candidate for coupling it to a steam or hot water source in a CHP system. Acknowledgements Work supported through the INL Laboratory Directed Research & Development (LDRD) Program under DOE Idaho Operations Office Contract No. DE-AC07-05ID14517. This manuscript has been authorized by Battelle Energy Alliance, LLC under Contract No. DEAC07-05ID14517 with the U.S. Department of Energy. The U.S. Government retains and the publisher, by accepting the article for publication, acknowledges that the U.S. Government retains a nonexclusive, paid-up, irrevocable, world-wide license to publish or reproduce the published form of this manuscript, or allow others to do so, for U.S. Government purposes. References [1] Pongtornkulpanich A, Thepa S, Amornkitbamrung M, Butch C. Experience with fully operational solar-driven 10-ton LiBr/H2O single-effect absorption cooling system in Thailand. Renew Energy 2008;33:943–9. [2] Agyenim F, Knight I, Rhodes M. Design and experimental testing of the performance of an outdoor LiBr/H2O solar thermal absorption cooling system with a cold store. Sol Energy 2010;84:735–44. [3] Chen L, Chen YM, Sun MH, Zhang YL, Zhang XR. Concept design and formation of a lithium bromide-water cooling system powered by supercritical CO2 solar collector.

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