Parametric Study of Single-Pipe Diffusers in Stratified Chilled Water ...

VOLUME 10, NUMBER 3

HVAC&R RESEARCH

JULY 2004

Parametric Study of Single-Pipe Diffusers in Stratified Chilled Water Storage Tanks (RP-1185) Jing Song

William P. Bahnfleth, Ph.D., P.E.

Student Member ASHRAE

Member ASHRAE

John M. Cimbala, Ph.D. A parametric study was performed of the charging thermal performance of a full-scale pipe diffuser in a single cylindrical stratified chilled water storage tank by applying factorial experimental theory to the results of simulations performed with a validated computational fluid dynamics (CFD) model. Dimensional parameters having the potential to influence charging inlet performance were identified and formed into dimensionless groups using the method of repeating variables. Parameters included: the inlet Richardson number based on inlet slot width (Ril), inlet Reynolds number (Rei), ratio of inlet width to diffuser height (l/hi), ratio of inlet diffuser height to tank radius (hi/RW), and ratio of diffuser radius to tank radius (RD/RW). Thermal performance was measured in terms of equivalent lost tank height (ELH). A full 2k factorial experiment of thirty-two simulations was performed and analyzed. Parameter ranges were: 0.05-2 for Ril, 500-5000 for Rei, 0.1-1 for l/hi, 0.005-0.05 for hi/RW, and 0.707-0.866 for RD/RW. Within these ranges, Ril, l/hi, and hi/RW were found to be of first-order significance, while Rei and RD/RW were not. Two-factor interactions involving Ril, l/hi, and hi/RW were also significant. Regression models of equivalent lost tank height as functions of Ril, l/hi, and hi/RW were developed. The predictive capabilities of the regression models were tested against the results of five additional CFD simulations having parameter values different from the 2k factorial experiment cases. On average, regression models predicted factorial experiment data to within 10% with maximum error of 30% to 60%, depending on the model.

INTRODUCTION Sensible cool thermal storage with chilled water as the storage medium is a widely used and effective energy management technique that reduces energy cost through load shifting. Chilled water is stored in tanks that vary in design as dictated by a variety of factors, including architectural, site, and economic constraints. The single stratified tank is the thermal storage device of choice due to its simplicity, low cost, reliability, and potential for high performance (Dorgan and Elleson 1993). In a stratified tank, thermal separation between warmer and cooler water is maintained by the action of buoyancy forces. A sharply defined transition layer, or thermocline, forms between the warm and cool bodies of water. Water enters and leaves the tank through diffusers at the top and bottom designed to reduce mixing and promote good stratification. Four main mechanisms contribute to the degradation of stored cooling capacity in a stratified tank: heat gain from the tank surroundings, vertical conduction in the tank wall, thermal diffusion from the warmer water to the cooler water, and mixing caused by diffuser inlet jets. Of these, the last is not only the most complex and least understood but also the most important. The single-pipe diffuser is one of the two diffuser types most commonly used in today's commercial chilled water storage tanks, the other being the radial disk diffuser. Figure 1 shows section and plan views of a typical cylindrical stratified storage tank with two-ring single-pipe octagonal diffusers. A slotted pipe lower diffuser introduces water into the tank through transJing Song is a graduate research assistant and William P. Bahnfleth is an associate professor in the Department of Architectural Engineering, and John M. Cimbala is a professor in the Department of Mechanical and Nuclear Engineering, The Pennsylvania State University, University Park, Pa.

345

346

HVAC&R RESEARCH

Figure 1. Stratified storage tank with octagonal slotted-pipe diffusers.

verse slots distributed uniformly along the bottom of the pipe. Water exits the pipe in small jets directed toward the tank floor. These jets impinge on the tank floor and eventually coalesce to form a thermocline. The matching upper diffuser is slotted on the top so that inlet jets impinge on the free surface of water in the storage tank. For a given diffuser type, the thermal performance of a stratified tank depends on many parameters, including tank and diffuser dimensions, water temperatures, and water flow rate. The objective of the research reported in this paper was to develop a first-order correlation between these parameters and the charging thermal performance of a single-ring, slotted single-pipe diffuser. The present study is parallel and complementary to a previous investigation of radial disk diffusers (Bahnfleth and Musser 1999; Musser and Bahnfleth 2001a, 2001b).

EXISTING DIFFUSER DESIGN GUIDANCE The present-day approach to the design of stratified chilled water storage tanks documented in design guides and handbooks (Dorgan and Elleson 1993) is based largely on the research of Wildin and his collaborators at the University of New Mexico. The key findings of this research were that: 1. Flow near an inlet diffuser is a complex process affected by the interaction of buoyancy, inertial, and viscous forces. Good stratification with a relatively thin thermocline occurs when the inlet diffuser creates a gravity current. The thermocline forms during the first few passes of the gravity current across the tank (Wildin and Truman 1985). 2. A gravity current tends to form when the inlet Froude number (Fri) of the jet created by the inlet diffuser is low (Yoo et al. 1986). A value of unity is recommended for design purposes. 3. The inlet Reynolds number (Rei) plays an important secondary role in stratification and should be limited to some maximum value. This conclusion was based on experiments in

VOLUME 10, NUMBER 3, JULY 2004

347

which diffusers with higher Rei created poorer stratification than diffusers with lower Rei but the same Fri (Wildin and Truman 1989a, 1989b; Wildin and Sohn 1993). The inlet Froude number, which represents the ratio of characteristic inertial and buoyancy forces, is defined as q Fr i = ---------------------, 1⁄2 3 ( h i g' )

(1)

where hi is the inlet height, g′ is reduced gravity (g∆ρ /ρi, where g is gravity, ∆ρ is the density difference between incoming fluid and the fluid in the tank at the level of the inlet diffuser, and ρi is the density of inlet fluid), and q is the flow rate per unit of diffuser pipe length. The inlet Richardson number (Ri) is the inverse square of Fri and is also used to describe stratified flows in the literature. 3

h i g' 1 Ri = -------2 = --------2 Fr i q

(2)

For a radial diffuser, hi is clearly the thickness of the jet at the diffuser edge, defined by the distance between the diffuser plate and the spreading surface. For a pipe diffuser, hi has been interpreted as the height of inlet openings above the spreading surface. This definition is open to criticism since both the size of the diffuser openings and the distance of the opening from the spreading surface could influence stratification. A more appropriate choice may be the characteristic dimension of an individual opening in the diffuser, l, so that q -, Fr i = -------------------1⁄2 3 ( l g' )

(3)

i.e., 3

g'l -, Ri l = ------2 q

(4)

where l is the inlet slot characteristic dimension and other parameters are as defined previously. The inlet Reynolds number, which represents the ratio of inertia and viscous forces, is defined in terms of the flow rate per unit length q and the kinematic viscosity ν: q Re i = --v

(5)

Wildin (1990) proposed a diffuser design methodology based on target values of Fri and Rei. A design value of 1.0 for inlet Froude number is recommended with an upper limit of 2.0. Guidelines for Rei are more complex and controversial. Current ASHRAE design guidance (Dorgan and Elleson 1993) taken from a variety of sources recommends an upper limit of inlet Reynolds number of 850 for tanks 4.6 m (15 ft) or taller and a maximum of 2000 for tall tanks 12.2 m (40 ft) or more in depth. However, field measurements of full-scale tanks (Musser and Bahnfleth 1998, 1999; Stewart 2001) and anecdotal evidence (Andrepont 1992; Bahnfleth and Joyce 1994) have shown that successful operation of stratified tanks is possible at Rei of 10,000 or more.

348

HVAC&R RESEARCH

It could be conjectured that differences in the apparent significance of the inlet Reynolds number found in the literature are due in part to the difference between experiments, in which a new thermocline is generated in a uniform temperature tank, and typical operating conditions in the field, in which a partial thermocline remains at the end of discharging/beginning of charging. However, the field data of Musser and Bahnfleth were obtained with initial conditions of nearly complete discharge, which suggests that this is not the source of the discrepancy. Bahnfleth and Musser further investigated the performance characteristics of radial diffusers using computational fluid dynamics (CFD) simulations with a validated model (Bahnfleth and Musser 1999; Musser and Bahnfleth 2001a, 2001b). They modeled charging at constant flow rate and temperature into an initially uniform temperature tank. They concluded that Rei is not of first-order importance for this diffuser type within typical full-scale tank operating parameter ranges that included Rei as large as 12,000. They also demonstrated that effects attributed to Rei could be explained to some extent by other parameters that are not represented in the current design methodology (Musser and Bahnfleth 2001b). The most significant parameters influencing radial diffuser performance were found to be Ri, RD/RW, and RD/hi, where Ri is the Richardson number, defined as in Equation 2, RD is the radius of the diffuser disk, and RW is the tank radius. It was also found that increasing Ri beyond the recommended design value of 1.0 (equivalent to reducing Fri to values less than 1.0) yielded additional improvement in stratification.

METHODOLOGY The approach of the present study parallels that of ASHRAE RP-1077. A parametric study of the charging of a stratified tank with a single-ring, single-pipe, continuously slotted diffuser was performed using a CFD model. Parametric simulations followed a 2k factorial test plan using dimensionless parameters obtained by the method of repeating variables, and thermal performance metrics were calculated for each case. These were subjected to statistical analysis to determine which parameters and interactions were of first-order significance. Regression models were developed using the significant parameters. These models were tested against both the parametric study data and several additional cases with arbitrary parameter combinations.

Problem Formulation The geometry of the domain, boundary conditions, and initial conditions is shown in Figure 2. The tank was initially filled with warm, uniform temperature fluid and charged at constant flow rate by fluid at a constant, cooler temperature. Charging continued until a fully formed thermocline was established. Only the inlet region and a region above it sufficient to accommodate the thermocline were modeled. The use of axisymmetric geometry rather than a full three-dimensional simulation was dictated by limitations of available computational resources.

CFD Model Simulations were performed using a commercial finite volume CFD program (Fluent 2001). Reynolds-averaged turbulent forms of the governing equations were solved using a realizable k-ε turbulence model with an intensity-hydraulic diameter specification. A turbulence intensity of 10% was used in all cases. The realizable k-ε turbulence model constrains the Reynolds stresses to positive values so that the model is consistent with the physics of the turbulent flow (i.e., “realizable” in the sense of being physically possible). The modeling approach was validated with field data obtained from a full-scale tank, as described in a previous paper (Bahnfleth et al. 2003a).


349

Figure 2. Domain, boundary conditions, and initial conditions.

Parameterization The objective of parameterization relates dimensionless performance measures to potentially significant dimensionless groups of independent variables. The first step in this process is to identify a complete set of dimensional variables describing the problem of interest. Two thermal performance metrics were considered: thermocline thickness (ht) and equivalent tank lost height (ELH), each of which has the dimension of length (L). Thermocline thickness is the thickness of a thermal boundary layer between warm and cool fluid. Equivalent lost tank height represents total mixing and conduction losses in terms of the thickness of a layer of fluid that would have an equal capacity if it underwent the maximum possible temperature change for a given system (Musser and Bahnfleth 2001a). For example, if 90% of the theoretical capacity available in a 10 m deep tank charged at 4°C and receiving 14°C water during discharge is recovered during a discharge process, the lost height would be 1 m because it would be as if 1 m of the actual tank height was “lost” while full theoretical capacity was recovered from the other 9 m. An extensive discussion of these and other performance measures and their relative merits can be found in previously published sources (Bahnfleth et al. 2003a). As many as seven dimensional variables may affect charging performance for the process defined by Figure 2: inlet height hi (L); reduced gravity g (L/t2); inlet width l (L); flow per unit length of diffuser q (L2/t); kinematic viscosity of the fluid ν (L2/t); tank radius RW (L); and diffuser ring radius RD (L). In axisymmetric geometry, the slot in a pipe diffuser is continuous, and l is defined as the radial distance from one edge of the slot to the other, as shown in Figure 2. In reality, pipe diffusers are generally three dimensional and, rather than being continuously slotted, have multiple inlets distributed at regular intervals along the diffuser pipe (Figure 1). The

350

HVAC&R RESEARCH

definitions of inlet width for the axisymmetric and three-dimensional cases are consistent if l is formally defined as the ratio of inlet area to inlet perimeter. The seven independent variables, together with one of the dimensional performance measures, comprise a set of eight dimensional parameters involving two dimensions (length and time) that can be subjected to a formal dimensional analysis. The method of repeating variables, with the tank radius as the repeating variable, reduces the dimensional variables to six independent groups (a nondimensional performance metric and five independent parameters). 3

h ⎛R l h R W g' ------t- = func ⎜ ------D- , ------- , ------i- , -----------, R RW R ⎝ W W RW q2

v⎞ ---⎟ q⎠

(6)

or 3

⎛R l h R W g' ELH ------------ = func ⎜ ------D- , ------- , ------i- , -----------, RW ⎝ RW RW RW q2

v⎞ ---⎟ q⎠

(7)

A drawback of the mechanical application of this method is that the groups generated may not be commonly used dimensionless parameters or may not be defined in a way that makes the greatest physical sense. However, any dimensionless parameter generated by Pi analysis can be taken to a power or multiplied by any other parameter to obtain a replacement parameter. This property is useful for making the results more physically relevant or to put them in terms of standard dimensionless groups. After several transformations, Equations 6 and 7 can be put into the following forms: R h l h ----t = func ⎛ ------D- , ---- , ------i- , Ri l , Re i⎞ ⎝ RW hi RW ⎠ hi

(8)

R l h ELH ------------ = func ⎛ ------D- , ---- , ------i- , Ri l , Re i⎞ ⎝ RW hi RW ⎠ hi

(9)

or

The independent parameter sets in Equations 8 and 9 include the inlet Richardson number, as defined in Equation 4, and the inlet Reynolds number. These parameter sets are complete if applied to a particular fluid of constant thermal diffusivity. Thermal diffusivity was omitted from the analysis because it would have added an additional parameter (Prandtl number) that was held constant in the present study.

Factorial Experiment Design Some parameters generated by dimensional analysis may not be significant. The objective of parametric analysis is to identify which of the parameters in Equations 8 and 9 (or their combinations) significantly affect stratification and inlet performance. The magnitude of these effects was estimated to first order by applying factorial experimental design theory (Montgomery 2000). The factorial design theory will not be reviewed in detail here, but the following discussion provides a brief review of the terminology and methods used in this study. Factorial design is used in experiments involving several parameters when it is desired to determine which parameters (also called factors) or their combinations have significant effects on a dependent parameter (also called a response variable). Experiments constructed according


351

to factorial design theory investigate all possible combinations of the parameters of interest. The application of factorial methods provides the most efficient means of identifying parametric relationships, i.e., it defines the smallest number of tests that can be performed for a given order of model and number of parameters. Variables in a factorial experiment can be continuous or discrete. Each independent variable can be tested at any number of values if it is continuous. The number of values tested for each independent variable in an experiment determines the highest order of the regression correlation that can be constructed from the results. A special case called a 2k factorial design is an efficient method for weeding out insignificant parameters, as it requires the minimum number of experiments with which k factors can be completely analyzed. In a 2k design, two values (also called levels) of each factor are selected. These levels of the parameters can be arbitrarily called “low” and “high.” The total number of treatment combinations (unique combinations of high and low values of factors) is 2k, where k is the number of independent parameters. The effect of a factor is the change in response produced by a change in the level of that factor averaged over the levels of the other factors. As a simple example, suppose that a dependent variable (response variable) X is a function of two independent variables (factors) y and z. The effect of y is the average of the change in X as y changes from its low value (level) to its high value when z is at its low value and the change in X as y changes from low to high while z is at its high value. An effect is a main effect if it refers to the primary factors of interest in the experiment. In some experiments, the difference in response between the values of one factor is not the same at all values of the other factors. In other words, the effect of a factor depends on the value chosen for other factors. When this occurs, it is said that there is an interaction between these factors. The interaction effect is defined as the average difference between the effects of factors that have interactions. An interaction effect can also be classified by its order. The more factors involved, the higher the order is. For example, a two-factor interaction is called a second-order interaction, while a five-factor interaction is called a fifth-order interaction. For a 2k experiment, response is modeled as linear over the parameter ranges chosen because there are only two values for each parameter. Higher order correlation requires three or more values of each significant parameter, which greatly increases the number of experiments needed. In a 2k experiment, proper selection of the high and low factor values is critical. The range must be sufficiently large to identify the important parameters for realistic cases. However, use of a very wide range over which variation of the response variable is nonlinear can give erroneous results.

Factorial Test Plan The low and high factor values chosen for this study were selected after reviewing the dimensions and operating parameters of a number of existing full-scale tanks. Table 1 lists the low and high values of the five independent nondimensional parameters in Equations 8 or 9. A 2k factorial design with 5 parameters (k = 5) requires 32 simulations to cover all possible combinations (called treatment combinations in the experimental design literature) of high and low values for each parameter. The complete list of treatment combinations is shown in Table 2. The high value of each parameter is noted by a “+” and the low value by “–.” In order to simulate the 32 cases shown in Table 2, it was necessary to develop dimensional CFD models with the correct nondimensional parameter values for each treatment combination. Dimensional parameters for each simulation case were calculated inversely from nondimensional factor values using equation-solving software (Klein and Alvarado 2002) after arbitrarily fixing certain dimensions. Tank radius was limited to avoid a domain size that would require an impractically long computing time. Fluid viscosity was permitted to vary from case to case, but the Prandtl number was held constant.

352

HVAC&R RESEARCH

Table 1. Nondimensional Parameter Ranges RD / RW

l / hi

hi / RW

Ril

Rei

Low (–)

0.707

0.1

0.005

0.05

500

High (+)

0.866

1

0.05

2

5000

Table 2. Treatment Combinations for 2k Factorial Experiment Case RD/RW (A) 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

+ + + + + + + + -

l/hi (B)

hi/RW (C)

Ril (D)

Rei (E)

Case

RD/RW (A)

l/hi (B)

hi/RW (C)

Ril (D)

Rei (E)

+ + + + + + + + -

+ + + + + + + + -

+ + + + + + + + -

+ + + + + + + + + + + + + + + +

17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32

+ + + + + + + + -

+ + + + + + + + -

+ + + + + + + + -

+ + + + + + + + -

-

“+”: high values, “-“: low values

Thermal Performance Measures It was assumed, based on prior experience with the modeling of radial disk diffusers (Bahnfleth and Musser 1999) and field data from tanks with both disk and pipe diffusers (Musser and Bahnfleth 1999), that the radially averaged vertical temperature distribution in the tank would reach a relatively stable shape once it had moved away from the inlet diffuser, after which only slow change due to diffusion would occur. A simulation was considered finished when the bottom end (“tail”) of the thermocline reached a dimensionless temperature (Θ) value of 0.01. Dimensionless temperature is defined as the fraction of the difference between reference warm and cool fluid temperatures, typically the nominal charge inlet and discharge inlet temperatures. By this definition, fluid at the charge inlet temperature has a Θ value of zero, and fluid at the discharge inlet temperature has a Θ value of one. Metrics of thermal performance considered in this study included thermocline thickness (ht) and equivalent lost height (ELH). Thermocline thickness is essentially a thermal boundary layer thickness. The definition of thermocline thickness proposed for use in evaluating simulation results was the distance between the points in the profile at which Θ values of 0.01 and 0.99 were observed. In other words, the height over which 98% of the overall temperature change occurred. Equivalent lost tank height, as previously defined, is the thickness of a layer of water representing the capacity loss during the cycling of a stratified tank.


353

Identification and Modeling of Significant Effects An indication of the significance of parameters in an experiment can be obtained by a t-test. The t-test can be performed graphically by plotting normal scores vs. effects on probability coordinates. Normal scores of effects that are not statistically significant will form a straight line. Effects that do not fall on this line are more likely to be significant. The degree of importance of an effect is indicated by the distance of its point from the line. The farther away from the normal line, the greater is the effect. The importance of interactions between the main factors in an experiment is given by interaction plots, which graphically display the change in response of one factor as another is varied. Both types of plots were used in the analysis of the factorial experiment and are discussed further during the presentation of results. After significant main effects and interactions are identified, regression models can be constructed and used to predict the response for arbitrary combinations of factor values within the ranges of the experimental design. In most cases, a linear regression model can be used for a two-level 2k factorial design. The method of least-squares is typically used to estimate the regression coefficients in a multiple linear regression model with k parameters. For a 25 factorial experiment involving parameters A, B, C, D, and E, there are 32 possible effects: 5 primary effects (A, B, C, D, and E); 10 two-factor effects (AB, AC, BC, AD, BD, CD, AE, BE, CE, and DE); 10 three-factor effects (ABC, ABD, ACD, BCD, ABE, ACE, BCE, ADE, BDE, and CDE); 5 four-factor effects (ABCD, ABCE, ABDE, ACDE, and BCDE), and 1 five-factor effect (ABCDE). A linear regression model that takes all effects into account has the form, yˆ = β 0 + β A x A + β B x B + β C x C + β D x D + β E x E + β AB x A x B + β AC x A x C + β BC x B x C + β AD x A x D + β BD x B x D + β CD x C x D + β AE x A x E + β BE x B x E + β CE x C x E + β DE x D x E + β ABC x A x B x C + β ABD x A x B x D + β ACD x A x C x D + β BCD x B x C x D + β ABE x A x B x E

(10)

+ β ACE x A x C x E + β BCE x B x C x E + β ADE x A x D x E + β BDE x B x D x E + β CDE x C x D x E + β ABCD x A x B x C x D + β ABCE x A x B x C x E + β ABDE x A x B x D x E + β ACDE x A x C x D x E + β BCDE x B x C x D x E + β ABCDE x A x B x C x D x E ,

where yˆ is the predicted response, β0 is the intercept, βi is the regression coefficient for ith effect, and xi is the value of the ith effect; i denotes A, B, C, D, E and their combinations. Typically, the regression model can be simplified by discarding the terms that have negligible effects. The success of a regression model can be tested by using it to reproduce the data from which it was generated. A more demanding test is to predict the result of arbitrary combinations of parameters that were not part of the data set used to produce the model. Mean and maximum deviation are of interest, as is the distribution of residuals. The residuals ( y – yˆ ) can be plotted against predicted response values ( yˆ ) on normal probability coordinates. If residuals fall on or near the normal line, the distribution of residuals ( y – yˆ ) is homogeneous and the regression model represents the data well. When several higher order effects appear to have significance or residuals are not normally distributed, a transformed response (for example, a regression of the square root of the response variable) may provide better predictions. A transformation is considered to be good if the total number of significant effects is reduced or if the data points of significant effects become closer to the normal line even though the number of significant effects remains the same. Another indication of the goodness of a transformed model is the normal probability plot of the residuals. Good transformation helps to linearize data and randomize the residual distribution.

354

HVAC&R RESEARCH

Statistical analysis software (Minitab 2000) was used to analyze the results of the CFD simulation factorial experiments. This package was used to perform variance analysis, residual analysis, and regression modeling and to generate normal probability and interaction plots.

RESULTS The flow and temperature characteristics of CFD solutions varied significantly depending on the parameter combination. In some cases, intense mixing occurred over a large vertical region, resulting in a thick thermocline and significant bulk temperature increase below the thermocline. Other cases had weaker mixing, and the thermocline was sharp and well formed after only a few passes of the initial gravity current. A review of the characteristics of factorial experiment simulation results will be the topic of a separate paper. However, one aspect of these results is of significance to the analysis of the factorial experiment. It was found that thermocline thickness was not a very practical or informative means of interpreting results. By the definition of thermocline thickness used in analysis of the data, a very large range of thicknesses was obtained that did not relate particularly well to the capacity loss associated with the profile shape. The cause of this problem was generally the presence of a long tail on the inlet side of the temperature profile with a value of Θ that was small but greater than 0.01 over a large distance. This is illustrated in Figure 3. Virtually all of the temperature change in the thermocline occurs over a vertical distance of approximately 0.2 × Z/RW, where Z is the distance above the tank floor. However, using a thermocline definition based on the region bounding 99% of the temperature change, the thermocline thickness for the profile shown is at least 3 × Z/RW. Consequently, detailed analysis of results was confined to investigation of the relationship between ELH and the various independent parameters.

Figure 3. Fully developed radially mass averaged temperature profile of Case 11, flow time at 259 minutes.


355

Both inlet height (hi) and storage tank radius (RW) were tested as scale factors to obtain a nondimensional form of ELH. The results of the two approaches were not strikingly different. Therefore, for brevity, discussion of results here is limited to consideration of the ELH/hi dimensionless performance parameter. Discussion of thermocline thickness results and results using ELH/RW as the performance measure may be found in the final report of the project (Bahnfleth et al. 2003b).

Factorial Analysis of CFD Results The column “CFD Results” in Table 3 gives values of the response variable ELH/hi, calculated from the parametric CFD simulations. Figure 4 is the normal probability plot of the effects of the independent parameters on response variable ELH/hi (see Equation 9). Significant effects are labeled with letters corresponding to the independent dimensionless parameters, as shown in the legend. Statistical analysis indicated that the two main effects l/hi (B) and Ril (D) were important. The main effects RD/RW (A) and Re (E) were less than 10% of the maximum effect in all cases and were, therefore, discarded as being negligible to first order. Several two-factor effects were significant (as much as 90% of the magnitude of the main effects), but the effects of three-factor and higher interactions were one to two orders of magnitude smaller than the main effects and considered negligible. Figure 5 is the interaction matrix for ELH/hi. All of the factors (independent parameters) are arranged along the diagonal of the plot. The high and low values of a factor are shown, respectively, at the right and left of the column in which it lies. The plots in the box lying at the intersection of the row associated with one factor and the column associated with another factor describe the average interaction between those two factors. They show the average effect due to the column factor as it changes from its low value to its high value when the row factor is fixed at each of its two values. If the two lines are parallel, regardless of their slope, there is no interaction. The greater the departure of the lines from parallel, the stronger the interaction. For example, both RD/RW and Rei have trivial interactions with other parameters, but there are strong interactions between Ril and l/hi.

Linear Regression Model Development The column “Regression Model Predictions” in Table 3 gives values of the response variable ELH/hi, calculated from three linear regression models (original, transformed, and simplified), developed using the least-squares method. The “original” model approximates the response variable ELH/hi, using all ten effects identified as significant in Figure 4. These include the two main effects l/hi (B) and Ril (D) and eight other interactions, the highest of which is a four-factor interaction. The model has the following form: h l l ELH ------------ ≈ 0.448 + 19.8 ---- + 0.813Ri l – 6.88 ---- Ri l – 12.33 ------i- Ri l hi hi hi RW l hi l hi l hi – 126 ---- ------- Ri l + 165 ---- ------- + 0.00843 ---- ------- Ri l Re i hi RW hi RW hi RW hi – 0.00023 ------- Ri l Re i + 0.000012Ri l Re i RW

(11)

l – 0.000434 ---- Ri l Re i hi

Tests indicated that combined application of log and square root transformations to response variable ELH/hi produced a more accurate regression model and reduced the number of effects

356

HVAC&R RESEARCH

Table 3. Comparison of Regression Model (Equations 11, 12, and 13) ELH/hi Predictions of Factorial Experiment Cases with CFD Simulation Results Case

CFD Results

Regression Model Predictions Original Model

Transformed Model

1 2.81 2.41 2.21 2 2.17 2.41 2.21 3 1.02 1.01 1.02 4 1.02 1.01 1.02 5 3.78 3.75 4.76 6 3.62 3.75 4.76 7 2.14 2.23 2.06 8 2.29 2.23 2.06 9 27.28 27.85 29.92 10 27.92 27.85 29.92 11 2.83 3.20 2.25 12 2.25 3.20 2.25 13 24.06 20.64 18.79 14 23.97 20.64 18.79 15 3.21 2.50 3.48 16 3.25 2.50 3.48 17 2.68 2.52 2.65 18 2.49 2.52 2.65 19 1.02 1.01 1.06 20 1.06 1.01 1.06 21 6.15 7.18 6.10 22 8.48 7.18 6.10 23 2.43 2.48 2.45 24 2.50 2.48 2.45 25 28.28 27.85 30.19 26 28.45 27.85 30.19 27 2.37 3.20 2.26 28 2.05 3.20 2.26 29 18.48 20.73 18.94 30 16.05 20.73 18.94 31 3.32 2.51 3.50 32 3.57 2.51 3.50 Average Absolute Value of Difference Average Absolute Value of Difference Percentage Maximum Absolute Value of Difference Percentage

Difference %

Simplified Model

Original Model

Transformed Model

Simplified Model

2.42 2.42 1.04 1.04 5.37 5.37 2.24 2.24 30.31 30.31 2.26 2.26 18.88 18.88 3.49 3.49 2.42 2.42 1.04 1.04 5.37 5.37 2.24 2.24 30.31 30.31 2.26 2.26 18.88 18.88 3.49 3.49

-14.0% 11.1% -1.6% -1.9% -1.0% 3.4% 4.3% -2.6% 2.1% -0.3% 12.9% 41.9% -14.2% -13.9% -22.1% -23.1% -5.9% 1.5% -0.9% -4.9% 16.7% -15.4% 2.2% -0.7% -1.5% -2.1% 34.9% 56.1% 12.2% 29.1% -24.4% -29.6% 0.8 13% 56.1%

-21.2% 1.8% 0.1% -0.3% 25.7% 31.4% -3.4% -9.7% 9.7% 7.2% -20.7% -0.3% -21.9% -21.6% 8.3% 6.9% -1.2% 6.5% 3.8% -0.4% -0.8% -28.1% 0.9% -2.0% 6.7% 6.1% -4.9% 10% 2.5% 18.0% 5.3% -1.9% 0.92 9% 31.4%

-13.8% 11.5% 1.8% 1.4% 41.8% 48.1% 5.0% -2.0% 11.1% 8.6% -20.2% 0.3% -21.5% -21.2% 8.6% 7.2% -9.6% -2.6% 1.7% -2.4% -12.7% -36.7% -7.7% -10.4% 7.2% 6.5% -4.7% 10.3% 2.1% 17.6% 5.1% -2.1% 1.04 12% 48.1%


Figure 4. Normal probability plot of effects on response variable ELH/hi.

Figure 5. Interaction plot for response variable ELH/hi.

357

358

HVAC&R RESEARCH

that needed to be included. This is shown in Figure 6, the normal probability plot of effects on transformed response variable log ( ELH ⁄ h i ) . Only seven effects are significant, and significant effects lie closer to the normal line than in Figure 4. The resulting simplified regression model is hi hi ELH l hi l log ⎛ ------------⎞ ≈ 0.714 + 0.412 ---- – 0.0288Ri l – 3.53 ------- – 3.64 ------- Ri l + 5.66 ---- ------⎝ hi ⎠ hi RW hi RW RW l – 0.0746 ---- Re i – 0.000007Ri l Re i . hi

(12)

The goodness of fit of the regression model was tested by plotting residuals on normal probability coordinates. As shown in Figure 7, residuals of transformed model predictions fall on the normal line within a confidence interval of 95%, which indicates that the distribution of residuals is homogeneous and the regression models represent the data well. Predictions of the original and transformed linear regression models (Equations 11 and 12, respectively) are compared with factorial experiment CFD simulation results in Table 3. The differences between transformed linear regression model predictions and the CFD results are mainly smaller than the differences between original model predictions and the CFD results. On average, the transformed regression model predicted CFD model results to within 9%. The maximum absolute percentage difference was approximately 30%, nearly half as large as the maximum error for the original model. Equation 12 includes all effects indicated to be significant by normal probability plot analysis (Figure 6), including higher order effects involving Re. However, prior analysis of main effects showed that Re was not of first-order importance. This suggests that simplification of the regression models by eliminating effects involving Re should have little effect on its predictions.

Figure 6. Normal probability plot of effects on response variable log ( ELH ⁄ h i ) .


359

Figure 7. Normal probability plot of residuals on response variable log ( ELH ⁄ h i ) .

When Re effects are not modeled, two effects disappear from Equation 12, and the simplified expression becomes hi hi l ELH log ⎛ ------------⎞ ≈ 0.714 + 0.412 ---- – 0.0483Ri l – 3.53 ------- – 3.67 ------- Ri l ⎝ hi ⎠ hi RW RW l hi l + 5.66 ---- ------- – 0.0746 ---- Ri l . hi RW hi

(13)

The comparison of model predictions with CFD results was repeated for Equation 13 with the results shown in Table 3. The average difference between simplified model predictions and CFD model predictions was essentially unchanged. The worst agreement for any individual case increased to 48%, but that was still better than the more complex untransformed original model. Figure 8 is a correlation plot comparing the ELH/hi predictions of the transformed full and simplified models with the 2k factorial CFD simulation results using the data presented in Table 3. It shows that agreement between regression model predictions and CFD simulation results is reasonably good, and there is no significant difference between the transformed full model and the simplified model.

360

HVAC&R RESEARCH

Figure 8. Regression model predictions of ELH/hi vs. 2k factorial CFD simulation results.

DISCUSSION Regression Model Predictive Capabilities Field data of sufficient completeness and quality were not available, so the regression models derived from the factorial experiment simulations could not be tested directly with experimental results. Instead, the models were tested against the results of five additional CFD simulations having parameter values different from the 2k factorial experiment cases. Parameter sets for the five test cases are shown in Table 4. The Reynolds number, hi/RW, and RD/RW were fixed at the average of their respective high and low factorial experiment values in all cases. The remaining two parameters, l/hi and Ril, were also set to their average values in Case 33. In the remaining four cases, one of these two was set to its average value while the other was set to values at one-quarter and three-quarters of the range between the high and low values. Parameters l/hi and Ril were varied in preference to the others because they were shown to be the most significant through analysis of the factorial experiment. Changes in these two parameters were, therefore, expected to produce the largest changes in the response. Values of ELH/hi predicted by the original, transformed, and simplified regression models were compared with CFD results, as shown in Table 5. The ELH values predicted by the three regression models are all larger than the values obtained from CFD simulations. The error in ELH values predicted by the transformed full model (Equation 12) varies from 48% to 79%, with an average of 65%. Agreement is worst for Case 33, in which all independent parameter values were midway between the high and low values used in the factorial experiment and,


361

Table 4. Parameter Values for Additional Test Cases Case

RD/RW

l/hi

hi/RW

Ril

Rei

33 34 35 36 37

0.7865 0.7865 0.7865 0.7865 0.7865

0.55 0.55 0.55 0.325 0.775

0.0275 0.0275 0.0275 0.0275 0.0275

1.025 0.5375 1.513 1.025 1.025

2750 2750 2750 2750 2750

Table 5. Comparison of Regression Model Predictions of ELH/hi (Equations 11, 12, and 13) with CFD Simulation Results for Additional Test Cases Case

CFD Results

Regression Model Predictions Original Model

Transformed Model

Simplified Model

33 1.93 8.19 3.45 34 3.05 10.87 4.81 35 1.75 5.50 2.57 36 1.49 5.23 2.44 37 2.96 11.15 5.16 Average Absolute Value of Difference Average Absolute Value of Difference Percentage Maximum Absolute Value of Difference Percentage

3.46 4.83 2.58 2.45 5.17

Difference % Original Model

Transformed Model

Simplified Model

324% 257% 215% 252% 276% 6 265% 324%

78.8% 58.1% 47.6% 64.6% 74.0% 1.45 64.6% 78.8%

79.2% 58.5% 47.8% 64.9% 74.4% 1.47 65% 79.2%

therefore, as far removed as possible from the values used to derive the regression. Predictions using the simplified model (Equation 13) were nearly identical to those by the transformed full regression model (Equation 12), and these two models both predict ELH much better than the original model (Equation 11), for which discrepancies were as large as 324%. An effort was made to improve the accuracy of the regression model by combining the results of the five additional test cases (cases 33-37, Tables 4 and 5), with the original 32 cases of the factorial experiment and recalculating coefficients. However, this did not improve model accuracy. The likely reason that it was not beneficial, as shown by statistical model goodness-of-fit tests, is that there are strong nonlinear correlations between the main factors and interactions. Consequently, not only more data, but also a higher order regression model, are needed in order to significantly improve accuracy.

Effects of Dimensionless Parameters Analysis of the factorial experiment indicates that within the range of parameter values investigated, the equivalent lost height associated with charging by an axisymmetric slotted pipe diffuser is governed primarily by three dimensionless parameters: Ril, l/hi, hi/RW, and their interactions. Inlet performance improves as Ril increases, and deteriorates as either l/hi or hi/RW increases. The first-order effects of RD/RW and Re are negligible within the parameter ranges considered. The Richardson number clearly accounts for the stabilizing influence of a favorable density gradient. The relatively stronger stable buoyancy effects associated with a larger Richardson number tend to reduce mixing. This finding agrees with prior studies, which have shown dependence of stratification on inlet geometry and Richardson or Froude number (e.g, Wildin and Truman 1989a, 1989b; Zurigat et al. 1991). Values of Ril greater than one (equivalent to values of Froude number less than one) resulted in better stratification than the commonly recommended design value of unity.

362

HVAC&R RESEARCH

Figure 9. ELH (m) as a function of hi with l = 0.04 m (0.13 ft), RW = 8 m (26.2 ft).

Other parametric dependencies implied by the regression models are not as clear-cut. Reliance on the main effects without consideration of their interactions can, in fact, be misleading. For example, the main effect of increasing either l/hi or hi/RW is to increase ELH/hi. In a typical application, these parameters might vary as a result of changing the inlet height, hi. Clearly, varying hi changes l/hi and hi/RW in opposing directions, so the overall effect of l/hi and hi/RW (whether it will cause ELH/hi to increase or decrease) will depend on which effect is dominant. An example of parameter interactions is given in Figure 9, which shows ELH as a function of hi for several values of Ril. ELH is calculated using Equation 13 for a tank radius of 8 m (26.2 ft) and inlet width of 0.04 m (0.13 ft). In all cases, ELH decreases to a minimum as hi decreases from its maximum value and then increases as hi becomes smaller. The value of the inlet height at which the minimum ELH occurs decreases as the Richardson number increases. Wildin (1996) reported improvement in stratification as the inlet height of a pipe diffuser decreased, but not the existence of an optimal height, which could be missed in an experimental study with other objectives. The predicted behavior can be explained in physical terms. Increasing hi should increase entrainment into the inlet jet from the surrounding fluid, thereby reducing the temperature difference between the jet and its surroundings. This would decrease the magnitude of stabilizing buoyancy forces, tending to increase mixing and thicken the thermocline. Conversely, decreasing the inlet height should decrease mixing due to entrainment. However, moving the inlet closer to the tank floor eventually will cause an interaction between jet and floor that results in a net increase in mixing. The shifting of the inlet height at which minimum ELH occurs to smaller values as Ril increases could be attributed to the strengthening of buoyancy forces acting to suppress viscous and inertial effects due to the jet-floor interaction that tends to increase mixing.


363

Although RD/RW has a significant effect on the performance of radial diffusers (Bahnfleth and Musser 1999), it had little influence on the equivalent lost tank height for the single-ring slotted-pipe diffusers investigated in this study. The likely reason for this difference is that a radial diffuser creates a single, outward-flowing gravity current, while an octagonal pipe diffuser creates two currents of roughly equal mass flow—one flowing outward toward the tank shell and one flowing inward toward the tank center. Changing the diffuser radius has an adverse effect on one of these currents, it will tend to have a beneficial effect on the other, with the result that the overall effect is small. Typical design guidance recommends that a single-ring pipe diffuser should be placed at the radius that divides the floor area of a cylindrical tank into equal parts. The findings of this study indicate both that this is a reasonable choice and that stratification is not very sensitive to RD/RW over the range considered. Previous studies of radial disk diffusers found that Rei did not significantly affect charging performance over a range from 1000 to 12000 (Bahnfleth and Musser 1999). The present study found the same to be true for single-pipe diffusers in a range from 500 to 5000. As noted previously, evidence from both full-scale tanks and CFD analysis supports this finding (Musser and Bahnfleth 1998, 1999; Stewart 2001; Andrepont 1992; Bahnfleth and Joyce 1994). The most likely reasons for the minor influence of Rei are that diffusers are intentionally designed to introduce fluid into a stratified tank at a low velocity through relatively small inlets, which results in relatively small values of Rei and the strong stabilizing effect of buoyancy forces during thermocline formation.

SUMMARY AND CONCLUSIONS A parametric study of charging with a single-ring axisymmetric diffuser was performed. The method of repeating variables was used to identify dimensionless parameters describing the geometry and flow near a lower diffuser during charging. Six dimensionless parameters were identified: a nondimensional thermal performance metric and five independent parameters that determine its value. A 25 factorial experiment consisting of thirty-two transient axisymmetric CFD simulations was performed. Simulations were performed with a CFD model based on a model tested in a study of a full-scale cylindrical chilled water storage tank with double-ring diffusers (Bahnfleth et al. 2003a). First-order linear regression models of thermal performance, as represented by equivalent lost tank height, were developed from CFD results. Model accuracy was tested by comparing predictions with results of the 25 factorial test simulations and five additional simulations with parameter combinations different from the factorial experiment cases. The primary conclusions derived from the results of this study are: • The parameters most strongly affecting the formation of the thermocline are the inlet Richardson number based on inlet slot width (Ril) and the ratio of inlet width to diffuser height (l/hi). The effect of the ratio of diffuser height to tank radius (hi/RW) is also significant, although apparently to a lesser degree than the effects of Ril and l/hi. Parameter interactions involving Ril, l/hi, and hi/RW are also significant. • Over the range investigated, increasing Richardson number based on slot width (Ril) improves stratification, as has been documented by numerous prior studies. Increasing Richardson number beyond unity significantly improves stratification. • Thermal performance deteriorates as l/hi increases from 0.1 to 1 or hi/RW increases from 0.005 to 0.05 when these parameters are varied independently. However, since l/hi is inversely proportional to hi and hi/RW is directly proportional to hi, the net effect of changing inlet height depends on the relative strength of these two effects.

364

HVAC&R RESEARCH

• Neither inlet Reynolds number (Rei) nor the ratio of diffuser radius to tank radius, RD/RW, has significant effects on stratification over the ranges considered (500 to 5000 and 0.707 to 0.866, respectively). • The inlet length scale that should be used in the Richardson number representing a single-pipe diffuser is different from that most representative of a radial diffuser. Consequently, the validity of using the same design procedure for both diffuser types is questionable. Results of the present work can be used to roughly estimate parameter combinations consistent with good thermal performance. However, the differences between the continuously slotted axisymmetric diffuser modeled in the parametric study and typical three-dimensional diffusers, as well as the common use of pipe diffusers with multiple rings, warrant further study with consideration given to parameters such as inlet jet spacing, whose significance could not be assessed within the scope of this investigation.

ACKNOWLDEGMENTS This research was conducted with the sponsorship of ASHRAE Research Project 1185, Thermal Performance of Single-Pipe Diffusers in Stratified Chilled Water Storage Tanks. The authors appreciate both the sponsorship of this work by ASHRAE and the guidance of the Project Monitoring Subcommittee: Dr. Chang Sohn (Chair), Prof. Maurice Wildin, Prof. Kelly Homan, Dr. William E. Stewart, Jr., and Mr. Richard Kooy. NOMENCLATURE

Symbols g g′ hi k

= = = =

l q

= =

r RW RD x

= = = =

y

=

yˆ z β

= = =

gravitational constant, m/s2 (ft/s2) reduced gravity (g∆ρ /ρ), m/s2 (ft/s2) diffuser inlet height, m (ft) number of independent parameters in a factorial experiment characteristic slot dimension, m (ft) inlet flow rate per unit inlet length, m2/s (ft2/s) radial coordinate, m (ft) tank radius, m (ft) diffuser ring radius, m (ft) value of an effect in a regression model of factorial experiment results response variable in a factorial experiment predicted value of a response variable elevation, m (ft) regression coefficient

Dimensionless Parameters Fr Rei Ri Ril

= = = =

3

ν

= kinematic viscosity, m2/s (ft2/s)

ρ ρi

= fluid density, kg/m3 (lbm/ft3)

∆ρ

= density difference between inlet fluid and the initial fluid in the tank at the level of the inlet diffuser, kg/m3 (lbm/ft3)

ELH = equivalent lost tank height, ∑ mC p ( T h – T ) ∆t ELH = ------------------------------------------ , m (ft) AρC p ( T h – T c ) Cp = fluid specific heat, J/kg⋅°C (Btu/lbm⋅°F) T

= fluid temperature, °C (°F)

Tc

= average inlet temperature, °C (°F)

Th

= average initial bulk temperature, °C (°F)

A m·

= tank plan area, m2 (ft2)

∆t

= charging time increment, s

Θ

sional) or arc length (axisymmetric two-dimensional) = nondimensional temperature, T – Tc θ = ----------------Th – Tc

1⁄2

inlet Froude number, Fr i = q ⁄ ( h i g' ) inlet Reynolds number, Re i = q ⁄ v 2 3 Richardson number, Ri = g'h i ⁄ q inlet Richardson number based on ratio of inlet area to inlet perimeter (three-dimen-

= inlet fluid density, kg/m3 (lbm/ft3)

= mass flow rate during a time increment, kg/s (lbm/s)


365

REFERENCES Andrepont, J.S. 1992. Chilled water storage case studies: central plant capacity expansions with O&M and capital cost savings. International District Heating and Cooling Association Fifth Annual College/University Conference. Bahnfleth, W., and W. Joyce. 1994. Energy use in a district cooling system with stratified chilled-water storage. ASHRAE Transactions 100(1):1767-1778. Bahnfleth, W., and A. Musser. 1998. Thermal performance of a full scale stratified chilled water storage tank. ASHRAE Transactions 107(2):377-388. Bahnfleth, W., and A. Musser. 1999. Parametric study of charging inlet diffuser performance in stratified chilled water storage tanks with radial diffusers. Final report, ASHRAE research project 1077. Bahnfleth, W., J. Song, and J. Cimbala. 2003a. Measured and modeled charging of a stratified chilled water thermal storage tank with slotted pipe diffusers. HVAC&R Research 9(4):467-491. Bahnfleth, W., J. Song, and J. Cimbala. 2003b. Thermal performance of single-pipe diffusers in stratified chilled water storage tanks. Final Report, ASHRAE Research Project 1185. Dorgan, C.E., and J. Elleson. 1993. Design Guide for Cool Thermal Storage. Atlanta: American Society of Heating, Refrigerating and Air-Conditioning Engineers, Inc. Fluent. 2001. Fluent User Manual. Lebanon, NH: Fluent Inc. Klein and Alvarado. 2002. Engineering Equation Solver User Manual. Madison, Wisc.: F-Chart Software Inc. Minitab. 2000. Minitab User Manual. State College, Pa.: Minitab Inc. Montgomery, D.C. 2000. Design and Analysis of Experiments, 5th ed. New York: John Wiley & Sons, Inc. Musser, A., and W. Bahnfleth. 1998. Evolution of temperature distributions in a full-scale stratified chilled water storage tank. ASHRAE Transactions 104(1):55-67. Musser, A., and W. Bahnfleth. 1999. Field-measured performance of four full-scale cylindrical stratified chilled-water thermal storage tanks. ASHRAE Transactions 105(2):218-230. Musser, A., and W. Bahnfleth. 2001a. Parametric study of charging inlet diffuser performance in stratified chilled water storage tanks with radial diffusers: Part 1—Model development and validation. HVAC&R Research 7(2):52-65. Musser, A., and W. Bahnfleth. 2001b. Parametric study of charging inlet diffuser performance in stratified chilled water storage tanks with radial diffusers: Part 2—Dimensional analysis, parametric simulations and simplified model development. HVAC&R Research 7(2):205-222. Stewart, W.E. 2001. Operating characteristics of five stratified chilled water thermal storage tanks. ASHRAE Transactions 107(2):12-21. Wildin, M. 1990. Diffuser design for naturally stratified thermal storage. ASHRAE Transactions 96(1):1094-1102. Wildin, M. 1996. Experimental results from single-pipe diffusers for stratified thermal energy storage. ASHRAE Transactions 102(2):123-132. Wildin, M., and C. Sohn. 1993. Flow and temperature distribution in a naturally stratified thermal storage tank. USACERL Technical Report FE-94/01. Wildin, M., and C. Truman. 1985. A summary of experience with stratified chilled water tanks. ASHRAE Transactions 91(1b):956-975. Wildin, M., and C. Truman. 1989a. Performance of stratified vertical cylindrical thermal storage tanks, Part I: Scale model tank. ASHRAE Transactions. 95(1):1086-1095. Wildin, M. and C. Truman. 1989b. Performance of stratified vertical cylindrical thermal storage tanks, Part II: Prototype tank. ASHRAE Transactions 95(1):1096-1105. Yoo, J., M.W. Wildin, and C.R. Truman. 1986. Initial formation of a thermocline in stratified thermal storage tanks. ASHRAE Transactions 92(2):280-292. Zurigat, Y., P. Liche, and A. Ghajar. 1991. Influence of inlet geometry on mixing in thermocline thermal energy storage. International Journal of Heat and Mass Transfer 34:115-125.