An improved method for determining heat transfer fin efficiencies for dehumidifying cooling coils (RP-1194).INTRODUCTION
Finned-tube heat exchangers heat exchanger
Any of several devices that transfer heat from a hot to a cold fluid. In many engineering applications, one fluid needs to be heated and another cooled, a requirement economically accomplished by a heat exchanger. are widely applied for cooling and dehumidifying moist moist
having a moderate moisture content, slightly wet to the touch.
see moist dermatitis of rabbits.
moist grain storage
grain stored at about 30% moisture in airtight silos. air, and there is much literature related to modeling their steady-state performance. For instance, the 2000 ASHRAE ASHRAE American Society of Heating, Refrigerating & Air Conditioning Engineers Handbook-HVAC Systems and Equipment (ASHRAE 2000) and ARI ARI Acute respiratory infection, see there Standard 410-2001: Forced-Circulation Air-Cooling and Air-Heating Coils (ARI 2001) present the same model for predicting the total and sensible energy transfer rates for wetted surface cooling coils. For the air side, the total (sensible and latent Hidden; concealed; that which does not appear upon the face of an item.
For example, a latent defect in the title to a parcel of real property is one that is not discoverable by an inspection of the title made with ordinary care. ) energy transfer rate for wetted surfaces is proportional proportional
values expressed as a proportion of the total number of values in a series.
the patient is a miniature without disproportionate reductions or enlargements of body parts. to the enthalpy enthalpy (ĕn`thălpē), measure of the heat content of a chemical or physical system; it is a quantity derived from the heat and work relations studied in thermodynamics. difference between the airstream and saturated saturated /sat·u·rat·ed/ (sach´ah-rat?ed)
1. denoting a chemical compound that has only single bonds and no double or triple bonds between atoms.
2. unable to hold in solution any more of a given substance. air at the temperature of the coil surface, while the sensible (convective) heat transfer is proportional to the temperature difference between the airstream and coil surface. A log mean temperature difference The log mean temperature difference (LMTD) is used to determine the temperature driving force for heat transfer in flow systems (most notably in heat exchangers). The LMTD is a logarithmic average of the temperature difference between the hot and cold streams at each end of the approach is utilized for estimating heat transfer between the coolant coolant (kōō´lnt),
n in the tubes (e.g., water) and the tube surface. Elmahdy and Mitalas (1977) and Braun et al. (1989) developed nearly equivalent and simpler models that transform the water stream to an equivalent airstream saturated with water vapor vapor /va·por/ (va´por) pl. vapo´res, vapors [L.]
1. steam, gas, or exhalation.
2. an atmospheric dispersion of a substance that in its normal state is liquid or solid. . With this transformation, the driving potential for the total energy transfer across the entire wetted section of the heat exchanger is written in terms of the enthalpy difference between the airstream and saturated air at the water temperature. The sensible heat Sensible heat is potential energy in the form of thermal energy or heat. The thermal body must have a temperature higher than its surroundings, (also see: latent heat). The thermal energy can be transported via conduction, convection, radiation or by a combination thereof. transfer is determined using the model presented in the 2000 ASHRAE Handbook--HVAC Systems and Equipment (ASHRAE 2000) and ARI Standard 410 (2001).
Each of these models uses fin efficiency to simplify calculation of steady-state heat and mass transfer for the air side. They typically employ separate fin efficiencies for convective (sensible) heat transfer and combined heat and mass transfer. For convective heat transfer Convective heat transfer is a mechanism of heat transfer occurring because of bulk motion (observable movement) of fluids. This can be contrasted with conductive heat transfer, which is the transfer of energy molecule by molecule through a solid or fluid, and radiative heat , fin efficiency accounts for the effect of the fin temperature distribution on total convective heat transfer to the fins. For combined heat and mass transfer, fin efficiency characterizes the impact of the distribution of wetted surface conditions on total energy transfer to the fins. Analytical expressions In mathematics, an analytical expression (or expression in analytical form) is a mathematical expression, constructed using well-known operations that lend themselves readily to calculation. and correlations for heat transfer fin efficiencies have been developed for a wide variety of fin geometries based on a dry analysis, and these same expressions are typically applied for combined heat and mass transfer using modified properties (Kuehn et al. 1998; Xia and Jacobi 2005).
All of the existing cooling coil models that employ the fin efficiency concept utilize relationships developed for dry fins (i.e., no condensation) in determining sensible (convective) heat transfer. However, the fin temperature distribution is different for wet and dry fins and, therefore, the use of a dry fin efficiency relationship for convective heat transfer is not strictly correct under wet conditions. This paper develops a correction factor for existing fin efficiency relationships that allows a better estimate of convective heat transfer fin efficiencies under wet conditions. The improved method is validated val·i·date
tr.v. val·i·dat·ed, val·i·dat·ing, val·i·dates
1. To declare or make legally valid.
2. To mark with an indication of official sanction.
3. using results obtained from a two-dimensional numerical analysis numerical analysis
Branch of applied mathematics that studies methods for solving complicated equations using arithmetic operations, often so complex that they require a computer, to approximate the processes of analysis (i.e., calculus). and experiments performed on an eight-row cooling coil.
This section develops a method for calculating sensible (convective) heat transfer fin efficiency under wet conditions for a classical straight fin geometry geometry [Gr.,=earth measuring], branch of mathematics concerned with the properties of and relationships between points, lines, planes, and figures and with generalizations of these concepts. with uniform cross section. However, the method can be applied to any fin geometry where fin efficiency equations exist. Figure 1 illustrates the fin geometry and boundary conditions boundary condition
The set of conditions specified for behavior of the solution to a set of differential equations at the boundary of its domain. . In order to provide the proper background, classical fin efficiency relations for heat transfer under dry conditions and heat and mass transfer under wet conditions are first presented.
[FIGURE 1 OMITTED]
Fin efficiency for a dry fin is developed from the solution to a one-dimensional heat conduction Heat conduction or thermal conduction is the spontaneous transfer of thermal energy through matter, from a region of higher temperature to a region of lower temperature, and hence acts to even out temperature differences. problem along the fin height direction. Assuming the fin tip is adiabatic ad·i·a·bat·ic
Of, relating to, or being a reversible thermodynamic process that occurs without gain or loss of heat and without a change in entropy. , uniform heat conductivity conductivity /con·duc·tiv·i·ty/ (kon?duk-tiv´i-te) the capacity of a body to transmit a flow of electricity or heat; the conductance per unit area of the body.
1. for the fin, [k.sub.f], and uniform convection coefficient coefficient /co·ef·fi·cient/ (ko?ah-fish´int)
1. an expression of the change or effect produced by variation in certain factors, or of the ratio between two different quantities.
2. across the fin, [h.sub.a-f], the fin efficiency for heat transfer only, [[eta].sub.f], is expressed as
[[eta].sub.f] = [[tan h(m*[H.sub.f])]/[m*[H.sub.f]]], (1)
m = [square root of [[h.sub.[a - f]]/[[k.sub.f]*t]]] (2)
and where [H.sub.f] is the fin height and t is half of the fin thickness (Kuehn et al. 1998).
Fin efficiency for a wet fin is developed using an analogous analogous /anal·o·gous/ (ah-nal´ah-gus) resembling or similar in some respects, as in function or appearance, but not in origin or development.
adj. approach by adding the additional assumptions that the air specific heat, [C.sub.p,a], is constant and Lewis number is unity (Kuehn et al. 1998; Xia and Jacobi 2005). The fin efficiency for combined heat and mass transfer, [[eta]*.sub.f]<io> is
[[eta].sub.f.sup.*] = [[tan h([m.sup.*]*[H.sub.f])]/[[m.sup.*]*[H.sub.f]]], (3)
[m.sup.*] = [square root of [1/[[[[C.sub.p,a]/[[h.sub.[a - f].sup.*]*[C.sub.s]]] + [[t.sub.w]/[k.sub.w]]]/[[k.sub.f]*t]]]] (4)
and where [k.sub.w] and [t.sub.w] are the thermal conductivity thermal conductivity
A measure of the ability of a material to transfer heat. Given two surfaces on either side of the material with a temperature difference between them, the thermal conductivity is the heat energy transferred per unit time and per unit and thickness of the condensate condensate, matter in the form of a gas of atoms, molecules, or elementary particles that have been so chilled that their motion is virtually halted and as a consequence they lose their separate identities and merge into a single entity. water film; [h*.sub.a - f] is the convection coefficient for a wet fin, which is somewhat higher than for a dry fin; and [C.sub.s] is the air saturation saturation, of an organic compound
saturation, of an organic compound, condition occurring when its molecules contain no double or triple bonds and thus cannot undergo addition reactions. specific heat (Braun et al. 1989) defined as the derivative derivative: see calculus.
In mathematics, a fundamental concept of differential calculus representing the instantaneous rate of change of a function. of the saturation air enthalpy with respect to temperature. In practice, is evaluated at the base temperature of the fin so that
[C.sub.s] = [[[d[h.sub.s]]/dT]|.sub.T = [T.sub.b]], (5)
where is [h.sub.s] saturated air enthalpy and [T.sub.b] is the fin base temperature.
Equation 4 was developed assuming film condensate and uniform film thickness along the fin. However, the film thickness is not uniform in practice, and drop-wise condensate occurs as well. It is common to exclude the condensate conduction conduction, transfer of heat or electricity through a substance, resulting from a difference in temperature between different parts of the substance, in the case of heat, or from a difference in electric potential, in the case of electricity. term and utilize an air-to-fin convection heat transfer coefficient The heat transfer coefficient is used in calculating the convection heat transfer between a moving fluid and a solid in thermodynamics. The heat transfer coefficient is often calculated from the Nusselt number (a dimensionless number). that accounts for the effects of condensate. Thus, Equation 4 becomes
[m.sup.*] = [square root of [[[h.sub.[a - f].sup.*]*[[C.sub.s]/[C.sub.p,a]]]/[[k.sub.f]*t]]]. (6)
In modeling a wetted fin surface, it is also necessary to determine fin efficiency for heat transfer only in order to determine sensible heat transfer. In previous modeling approaches, dry fin equations, such as Equation 1, were applied for this purpose. However, a temperature profile for a wet fin is not identical to that for a dry fin. An improved method for determining sensible heat transfer fin efficiency under wet conditions is developed by utilizing the fin temperature profile determined from the solution to the combined heat and mass transfer problem. This analysis provides a distribution of saturation air enthalpy along the fin ([T.sub.f]), which also uniquely determines the fin temperature distribution ([h.sub.s,f]). The relationship between the saturated air enthalpy and fin temperature profiles is approximated using the saturation specific heat determined with Equation 5. Assuming [C.sub.s] is constant over the relatively small temperature range within a fin,
[C.sub.s] = [[[h.sub.s,f](y) - [h.sub.s,b]]/[[T.sub.f](y) - [T.sub.b]]], (7)
where [T.sub.f] is the local fin temperature, [h.sub.s,f] is the local saturated air enthalpy evaluated at [T.sub.f], and [h.sub.s,b] is hs at the fin base temperature, [T.sub.b]. The temperature distribution at any location can then be determined from knowledge of the saturation air enthalpy profile using
[T.sub.f](y) - [T.sub.b] = [[[h.sub.s,f](y) - [h.sub.s,b]]/[C.sub.s]]. (8)
Rearranging Equation 8 gives
[[[T.sub.f](y) - [T.sub.a] + [T.sub.a] - [T.sub.b]]/[[T.sub.b] - [T.sub.a]]] = [[[h.sub.s,f](y) - [h.sub.a] + [h.sub.a] - [h.sub.s,b]]/[[C.sub.s]*([T.sub.b] - [T.sub.a])]], (9)
leading to the dimensionless fin temperature profile relation
[[[T.sub.f](y) - [T.sub.a]]/[[T.sub.b] - [T.sub.a]]] = 1 + [1/[C.sub.s]]*[[[h.sub.s,b] - [h.sub.a]]/[[T.sub.b] - [T.sub.a]]]*[[[h.sub.s,f](y) - [h.sub.a]]/[[h.sub.s,b] - [h.sub.a]]] - [1/[C.sub.s]]*[[[h.sub.s,b] - [h.sub.a]]/[[T.sub.b] - [T.sub.a]]]. (10)
The convective (sensible) heat transfer fin efficiency is defined as the ratio of the actual fin heat transfer to the heat transfer that would occur if the entire fin were at the base temperature. Under wetted surface conditions, the heat transfer fin efficiency is determined from the dimensionless temperature distribution as
[MATHEMATICAL EXPRESSION A group of characters or symbols representing a quantity or an operation. See arithmetic expression. NOT REPRODUCIBLE re·pro·duce
v. re·pro·duced, re·pro·duc·ing, re·pro·duc·es
1. To produce a counterpart, image, or copy of.
2. Biology To generate (offspring) by sexual or asexual means. IN ASCII ASCII or American Standard Code for Information Interchange, a set of codes used to represent letters, numbers, a few symbols, and control characters. Originally designed for teletype operations, it has found wide application in computers. (11)
Defining a heat transfer fin efficiency correction factor as
[C.sub.F] = [1/[C.sub.s]]*[[[h.sub.s,b] - [h.sub.a]]/[[T.sub.b] - [T.sub.a]]], (12)
[[eta].sub.f] can be related to the heat and mass transfer fin efficiency according to according to
1. As stated or indicated by; on the authority of: according to historians.
2. In keeping with: according to instructions.
[[^.[eta]].sub.f] = 1 - [C.sub.F]*(1 - [[^.[eta]].sub.f.sup.*]). (13)
Equation 13 is a general expression for estimating heat transfer fin efficiency from combined heat and mass transfer fin efficiency and operating conditions when condensation occurs.
Figure 2 shows saturated air enthalpy as a function of temperature, along with example air states at the fin base (point b) and in the local bulk airstream away from the fin (point a). The correction factor in Equation 12 is the ratio of the slope of the line connecting points b and a and the slope at point b of the saturation line ([C.sub.s]). It should be clear from this depiction that the correction factor is always less than or equal to one, approaching one as the air nears thermal equilibrium thermal equilibrium
The condition under which two substances in physical contact with each other exchange no heat energy. Two substances in thermal equilibrium are said to be at the same temperature. See also thermodynamics.
Noun 1. with the fin base (i.e., saturated air at the fin base temperature). For this limiting case, the heat transfer fin efficiency is equal to the heat and mass transfer fin efficiency. At other conditions, the heat transfer fin efficiency is greater than the heat and mass transfer fin efficiency.
[FIGURE 2 OMITTED]
In order to apply the modified heat transfer fin efficiency of Equation 13 within a cooling coil model, such as those presented by ARI (2001), Elmahdy and Mitalas (1977), Braun et al. (1989), and Kuehn et al. (1998), it is necessary to specify representative values for the tube surface and air conditions for use in Equation 12. For any individual control volume within an overall coil model, the local temperature of tube material, which is considered to be uniform, is used to evaluate [T.sub.b] and [h.sub.s,b]. The air condition varies within the control volume, and it was found by numerical numerical
expressed in numbers, i.e. Arabic numerals of 0 to 9 inclusive.
a numerical code is used to indicate the words, or other alphabetical signals, intended. study that simply using the local air inlet inlet /in·let/ (-let) a means or route of entrance.
pelvic inlet the upper limit of the pelvic cavity.
thoracic inlet the elliptical opening at the summit of the thorax. conditions for [T.sub.a] and [h.sub.a] works well.
NUMERICAL VALIDATION See validate.
validation - The stage in the software life-cycle at the end of the development process where software is evaluated to ensure that it complies with the requirements.
The simple geometry depicted de·pict
tr.v. de·pict·ed, de·pict·ing, de·picts
1. To represent in a picture or sculpture.
2. To represent in words; describe. See Synonyms at represent. in Figure 3 was studied numerically nu·mer·i·cal also nu·mer·ic
1. Of or relating to a number or series of numbers: numerical order.
2. Designating number or a number: a numerical symbol. in order to evaluate improvements associated with the new approach for determining heat transfer fin efficiency. It is a single finned finned
Having a fin, fins, or finlike parts. Often used in combination: single-finned; multifinned. tube with a counter-flow arrangement. Table 1 gives geometrical ge·o·met·ric also ge·o·met·ri·cal
a. Of or relating to geometry and its methods and principles.
b. Increasing or decreasing in a geometric progression.
2. and heat transfer parameters associated with the numerical study.
[FIGURE 3 OMITTED]
The parameters of Table 1 were chosen to be consistent with an existing eight-row cooling coil that has been studied extensively in a laboratory environment. A detailed description of the coil tested in the laboratory is provided in the section "Experimental Validation." Two different models were developed for this geometry in order to evaluate the improvement associated with the modified heat transfer fin efficiency for wet surfaces: (1) a two-dimensional, finite-volume reference model and (2) a model that incorporates fin efficiencies for both heat transfer and heat and mass transfer. The models employ the same basic assumptions utilized in developing the fin efficiencies in order to isolate isolate /iso·late/ (i´sah-lat)
1. to separate from others.
2. a group of individuals prevented by geographic, genetic, ecologic, social, or artificial barriers from interbreeding with others of their kind. the impact of the improved method for determining heat transfer fin efficiency.
Table 1. Characteristics of the Straight Finned-Tube Heat Exchanger Parameter Value Fin material Aluminum Fin material 237 W/m K thermal conductivity, [k.sub.f] Fin length, 0.3 m [L.sub.f] Fin height, 0.02 m [H.sub.f] Fin thickness, 2t 0.0002 m Dry fin convection 45.9 coefficient, W/[m.sup.2] K [h.sub.a-f] Wet fin convection 49.8 coefficient, W/[m.sup.2] K [h*.sub.a-f] Water-to-tube heat 0.31 m K/W transfer resistance per unit length, [R.sub.w-t.sup.']
The two-dimensional reference model for the fin neglects conduction across the fin thickness but considers conduction along the fin height (y) and length (x). Because of symmetry symmetry, generally speaking, a balance or correspondence between various parts of an object; the term symmetry is used both in the arts and in the sciences. , only the fin above the tube is modeled. For any point within the fin, the steady-state energy equation is
[k.sub.f]*2t*([[[partial derivative partial derivative
In differential calculus, the derivative of a function of several variables with respect to change in just one of its variables. Partial derivatives are useful in analyzing surfaces for maximum and minimum points and give rise to partial differential ].sup.2][T.sub.f]]/[[partial derivative][x.sup.2]] + [[[partial derivative].sup.2][T.sub.f]]/[[partial derivative][y.sup.2]]) - [[q.sup.?].sub.[a - f]] = 0, (14)
where [q".sub.a-f] is the local energy transfer flux flux
In metallurgy, any substance introduced in the smelting of ores to promote fluidity and to remove objectionable impurities in the form of slag. Limestone is commonly used for this purpose in smelting iron ores. from the airflow stream to the fin at any point with coordinates x and y.
The edges of the fins are assumed to be adiabatic, and the thermal resistance of the tube attached to the fin base is neglected. With these assumptions, the boundary conditions for the fin model are
x = 0,[[[partial derivative][T.sub.f]]/[[partial derivative]x]] = 0 and x = [L.sub.f],[[[partial derivative][T.sub.f]]/[[partial derivative]x]] = 0 (15)
and y = 0,[k.sub.f]*[[[partial derivative][T.sub.f]]/[[partial derivative]y]]*2t - [[q.sup.'].sub.[t - w]]/2 = 0 and y = [H.sub.f],[[[partial derivative][T.sub.f]]/[[partial derivative]y]] = 0, (16)
where [q'.sub.t-w] is the heat transfer rate from the tube to the water flow stream per unit length in the x direction. One half of [q'.sub.t-w] is from the fin base above the tube, and the other half is from the fin base below the tube.
The airflow is assumed to have a uniform velocity velocity in which the same number of units of space are described in each successive unit of time.
See also: Velocity throughout the entire fin heat exchanger, and the water flow is assumed to be incompressible in·com·press·i·ble
Impossible to compress; resisting compression: mounds of incompressible garbage.
in with a constant specific heat. The energy equations for the air and water are written as
[m.sup.'].sub.a]*[[d[h.sub.a]]/dx] = - [[q.sup.?].sub.[a - f]] (17)
[C.sub.w]*[[dT.sub.w]/dx] = [[q.sup.'].sub.[t - w]], (18)
where [m'.sub.a] is the air mass flow rate per unit height of fin, [h.sub.a] is the local air enthalpy, [C.sub.w] is the water heat capacitance capacitance, in electricity, capability of a body, system, circuit, or device for storing electric charge. Capacitance is expressed as the ratio of stored charge in coulombs to the impressed potential difference in volts. rate, and [T.sub.w] is the local bulk water temperature. The water temperature has a one-dimensional variation in the direction of water flow, whereas the air temperature varies in two dimensions because of a dependence of the heat flux on fin temperature.
Conservation of mass for the water vapor within the airstream leads to
[[m.sup.'].sub.a]*[[d[W.sub.a]]/dx] = - [[m.sup.'].sub.c], (19)
where [W.sub.a] is the local air humidity humidity, moisture content of the atmosphere, a primary element of climate. Humidity measurements include absolute humidity, the mass of water vapor per unit volume of natural air; relative humidity (usually meant when the term humidity ratio away from the surface and .m".sub.c] is the mass flow rate of condensate per unit surface area of the fin.
The boundary conditions for the air and water flows are
x = [L.sub.f],[h.sub.a] = [h.sub.a,in,HX],[W.sub.a] = [W.sub.a,in,HX] (20)
x = 0,[T.sub.w] = [T.sub.w,in,HX], (21)
where [h.sub.a,in,Hx] and [W.sub.a,in,Hx] are the heat exchanger air inlet enthalpy and humidity ratios, and [T.sub.w,in,Hx] is the inlet water temperature to the heat exchanger.
The heat transfer rate between tube and water is
[[q.sup.'].sub.[t - w]] = [([T.sub.t] - [T.sub.w])/[[R.sup.'].sub.[t - w]]], (22)
where [T.sub.t] is the local tube temperature, which is considered to be equal to the fin base temperature, and [R'.sub.t-w] is the convection heat transfer resistance between tube and water per unit length.
The energy flux from the air to a particular point on the fin depends upon whether moisture condenses or not. If the local fin surface temperature is above the local dewpoint of the air, then the surface is dry and the condensate term in Equation 19 is zero. In this case, the local heat flux to the fin from the air for Equations 14 and 17 is due to convective heat transfer only and is determined as
[[q.sup.?].sub.[a - f]] = [h.sub.[a - f]]*([T.sub.a] - [T.sub.f]), (23)
where [T.sub.a] is the local air inlet temperature, which is a function of [h.sub.a] and [W.sub.a], and [h.sub.a-f] is a global heat transfer coefficient for the heat exchanger under dry conditions.
When the fin temperature is below the dew point dew point: see dew. temperature of the inlet air locally, dehumidification occurs and the fin is wet. In this case, the condensate flux and energy flux due to heat and mass transfer are generally given as
[[m.sup.?].sub.c] = [h.sub.d]*([W.sub.a] - [W.sub.s,f]) (24)
[[q.sup.?].sub.[a - f]] = [h.sub.[a - f].sup.*]*([T.sub.a] - [T.sub.f]) + [h.sub.d]*([W.sub.a] - [W.sub.s,f])*[h.sub.fg,f], (25)
where [h.sub.d] is the mass transfer coefficient In engineering, the mass transfer coefficient is a diffusion rate constant that relates the mass transfer rate, mass transfer area, and concentration gradient as driving force: , [W.sub.s,f] is the humidity ratio for saturated air at the surface with a temperature [T.sub.f], [h.sub.fg,f] is the heat of water vaporization vaporization, change of a liquid or solid substance to a gas or vapor. There is fundamentally no difference between the terms gas and vapor, but gas is used commonly to describe a substance that appears in the gaseous state under standard conditions of at [T.sub.f], and [h*.sub.a-f] is a global heat transfer coefficient for the heat exchanger under wet conditions (Kuehn et al. 1998).
Equation 24 can be rewritten using the Lewis relation [Le = [h*.sub.a-f]/([h.sub.d] [C.sub.p,a])], the relation for mixture enthalpy of ideal gas mixtures with constant specific heats [h = [C.sub.p,a] (T -[T.sub.ref]) + W [h.sub.fg]], and the assumption that the heat of vaporization heat of vaporization
The amount of heat required to convert a unit mass of a liquid at its boiling point into vapor without an increase in temperature. has negligible This article or section is written like a personal reflection or and may require .
Please [ improve this article] by rewriting this article or section in an . dependence on temperature, so that
[[q.sup.?].sub.[a - f]] = [[h.sub.[a - f].sup.*]/[C.sub.p,a]]*([h.sub.a] - [h.sub.s,f])*[1/Le] + [h.sub.[a - f].sup.*]*([T.sub.a] - [T.sub.f])*(1 - 1/Le), (26)
where [C.sub.p,a] is the specific heat of the air-water vapor mixture.
It is most common to assume that the Lewis number is unity for air-water vapor mixtures at atmospheric pressure atmospheric pressure
or barometric pressure
Force per unit area exerted by the air above the surface of the Earth. Standard sea-level pressure, by definition, equals 1 atmosphere (atm), or 29.92 in. (760 mm) of mercury, 14.70 lbs per square in., or 101. . For this assumption, Equation 26 reduces to the following commonly used form (Kuehn et al. 1998):
[[q.sup.?].sub.[a - f]] = [[h.sub.[a - f].sup.*]/[C.sub.p,a]]*([h.sub.a] - [h.sub.s,f]) (27)
The differential equation differential equation
Mathematical statement that contains one or more derivatives. It states a relationship involving the rates of change of continuously changing quantities modeled by functions. describing the fin (Equation 14) was discretized using a second-order centered finite-difference method (Incropera and DeWitt 1996). The corresponding boundary conditions (Equations 15 and 16) were discretized using a first-order finite-difference method (Incropera and DeWitt 1996), and the equations for [q'.sub.t-w] (Equation 22) and [q".sub.a-f] (Equations 23, 26, or 27, depending on the fin condition and the assumption of ) were discretized using a first-order upwind scheme, which means that the local fluid conditions inherit To receive property according to the state laws of intestate succession from a decedent who has failed to execute a valid will, or, where the term is applied in a more general sense, to receive the property of a decedent by will.
inherit v. their inlet conditions (Murthy and Mathur 1998). The discretized equations were solved using a line-by-line tri-diagonal matrix algorithm algorithm (ăl`gərĭth'əm) or algorism (–rĭz'əm) [for Al-Khowarizmi], a clearly defined procedure for obtaining the solution to a general type of problem, often numerical. (Murthy and Mathur 1998). After solving these equations,[q".sub.a-f], [m".sub.c], [w'.sub. t-w] and were calculated for each control volume. The air and water energy equations (Equations 17 and 19 for the air and Equation 18 for the water) were discretized again using the first-order finite difference A finite difference is a mathematical expression of the form f(x + b) − f(x + a). If a finite difference is divided by b − a, one gets a difference quotient. approach and were solved directly for the local outlet conditions. An iterative it·er·a·tive
1. Characterized by or involving repetition, recurrence, reiteration, or repetitiousness.
2. Grammar Frequentative.
Noun 1. solution was necessary to couple the solution of the fin conduction problem to the solution of the air and water energy equations. At each iteration One repetition of a sequence of instructions or events. For example, in a program loop, one iteration is once through the instructions in the loop. See iterative development.
(programming) iteration - Repetition of a sequence of instructions. , it was necessary to compare all of the local surface temperatures to the local air dew-point temperatures to determine the condensate and energy fluxes. The final solution gives two-dimensional distributions for fin temperature, air temperature and humidity, and surface fluxes along with one-dimensional water temperature distributions.
Model Employing Fin Efficiencies
Typical heat exchanger and cooling coil models utilize fin efficiencies to account for temperature and enthalpy distributions along the fin height (y) direction and neglect fin conduction in the longitudinal lon·gi·tu·di·nal
Running in the direction of the long axis of the body or any of its parts. (x) direction. The fin efficiencies are derived assuming that the local air states are uniform in the y direction but vary in the x direction. In addition, the specific heat of the air is assumed to be constant. With these assumptions, the energy equation for the air at a dry location in the heat exchanger of Figure 3 can be expressed as
[m.sub.a]*[C.sub.p,a]*[[dT.sub.a]/dx] = [[eta].sub.f]*[h.sub.[a - f]]*[H.sub.f]*([T.sub.t] - [T.sub.a]), (28)
where [m.sub.a] is the air mass flow rate and [T.sub.a] represents the bulk air temperature across the entire fin in the y direction at any x. The fin efficiency accounts for the temperature distribution in the fin and allows the heat transfer rate to be expressed as a function of the base temperature of the fin, which is also the tube surface temperature.
For wet sections within the coil, the assumption of a Lewis number of unity and an overall fin efficiency for heat and mass transfer are employed. In this case, the air energy equation is
[m.sub.a]*[[d[h.sub.a]]/dx] = [[eta].sub.f.sup.*]*[h.sub.[a - f].sup.*]*[H.sub.f]*([h.sub.s,t] - [h.sub.a]), (29)
where [[eta]*.sub.f] is the heat and mass transfer fin efficiency and [h.sub.s,t] is the saturated air enthalpy at [T.sub.t]. The fin efficiency accounts for the distribution of saturated air enthalpies along the fin height.
Equation 29 is applied when the local tube surface temperature is below the local air dew-point temperature; otherwise, only Equation 28 is applied. Unlike the reference model, the models employing fin efficiencies do not have the ability to track the transition of wet-to-dry surfaces along the fin height direction. They can only track the transition along the direction of airflow.
For wet sections, it is also necessary to consider a mass balance on the water within the airstream in order to fully characterize the air state. In order to determine the air temperature variation, a water vapor mass balance ([m.sub.a] [dw.sub.a]/dx = [-m'.sub.c]).can be combined with a basic energy balance ([m.sub.a] [dh.sub.a]/dx = [[eta].sub. f] [h*.sub.a-f] [H.sub. f] ([T.sub.a] - [T.sub.t]) - [m.sub. c] [h.sub. fg]) and the mixture enthalpy relation (h = [C.sub.p,a] (T -[T.sub.ref]) + W [h.sub.fg]) to give
[m.sub.a]*[C.sub.p,a][[dT.sub.a]/dx] = [[^.[eta]].sub.f]*[h.sub.[a - f].sup.*]*[H.sub.f]*([T.sub.t] - [T.sub.a]), (30)
where [[eta].sub.f] is the heat transfer fin efficiency for a wet surface. This relationship is nearly the same as the energy balance for a dry section (Equation 28), except that the heat transfer coefficient and fin efficiency are different because of the influence of the wetted surface. The traditional models for cooling coils typically employ a heat transfer efficiency derived for dry surfaces in determining the air temperatures for wet sections of the coil.
The energy equation for the water stream and the heat exchanger boundary conditions are identical to those presented for the reference model. These relations, along with Equations 28-30 and property functions, are sufficient to determine the one-dimensional variation in water temperature, tube temperature, air enthalpy, and air temperature in the flow direction within the heat exchanger.
Existing models for cooling coils, such as those presented by ARI (2001), Elmahdy and Mitalas (1977), Braun et al. (1989), and Kuehn et al. (1998), utilize a combination of analytical analytical, analytic
pertaining to or emanating from analysis.
control of confounding by analysis of the results of a trial or test. and numerical techniques to solve the equations. They involve determination of the point for transition between dry and wet sections of the coil and use of separate analytical solutions to the basic equations for each of these sections that arise from the use of some simplifying assumptions. However, in order to be consistent with the reference model and isolate the effects of utilizing the improved heat transfer fin efficiency, a numerical solution to the modeling equations was implemented. In order to evaluate the improvements associated with the use of heat transfer fin efficiency correction presented in this paper, the model described in this section was solved for two different cases for heat transfer fin efficiency in Equation 30: (1) a dry heat transfer fin efficiency (Equation 1) and (2) a corrected heat transfer fin efficiency based on a wet analysis (Equation 13).
The differential equations were discretized and solved numerically using the first-order finite-difference method and upwind scheme (Murthy and Mathur 1998). The Gauss-Seidel iteration (Incropera and DeWitt 1996) was used to update the boundary conditions that couple the different finite-difference equations in the solution of the counter-flow arrangement. For each air control volume, it is necessary to determine whether moisture is condensing con·dense
v. con·densed, con·dens·ing, con·dens·es
1. To reduce the volume or compass of.
2. To make more concise; abridge or shorten.
a. or not. This is evaluated by comparing the surface temperature from a dry analysis with the local air inlet dew point.
Table 2 gives sample inlet conditions considered for comparisons of numerical predictions for the different models compared in this study. The inlet relative humidity relative humidity
The ratio of the amount of water vapor in the air at a specific temperature to the maximum amount that the air could hold at that temperature, expressed as a percentage. was varied between 40% and 80% with all other inlet conditions held constant. These conditions were chosen for presentation because the heat exchanger transitions from partially wet at the lowest humidity to fully wet conditions at the highest relative humidity.
Table 2. Heat Exchanger Inlet Conditions for Numerical Study [T.sub.a [RH.sub.a [M.sub.a], [T.sub.w [M.sub.w], in], in], % g/s in], g/s [degrees]C [degrees]C 26.67 40-80 0.36 4.44 0.16
Model results are compared in terms of their ability to predict total and sensible energy transfer rates for the heat exchanger as determined by
[q.sub.tot] = [C.sub.w]*([T.sub.w,out,HX] - [T.sub.w,in,HX]) (31)
[q.sub.sen] = [C.sub.a]*([T.sub.a,in,HX] - [T.sub.a,out,HX]), (32)
where the subscript (1) In word processing and scientific notation, a digit or symbol that appears below the line; for example, H2O, the symbol for water. Contrast with superscript.
(2) In programming, a method for referencing data in a table. "HX" denotes that values are for the heat exchanger instead of local control volumes.
Figure 4 gives comparisons between predictions of total energy transfer rates for the reference and fin efficiency models as a function of inlet air relative humidity. The total energy transfer rate increases with relative humidity because of increased latent energy transfer. Predictions of total energy transfer rates for the fin efficiency model are independent of the heat transfer fin efficiency used for wet sections (only the combined heat and mass transfer fin efficiency influences the total energy transfer rate in the wet section). Therefore, only a single result is presented for this approach. For the reference model, results are presented for both a Lewis number of unity and 0.9. According to Kuehn et al. (1998), air-water vapor mixtures have a Lewis number of around 0.9 at atmospheric conditions for a wide range of temperatures and humidities. A Lewis number of unity is assumed for the model that employs fin efficiencies and for most traditional cooling models.
[FIGURE 4 OMITTED]
Figure 4 shows that errors associated with the assumption of a Lewis number of unity are very small for representative operating conditions. The model employing fin efficiency also gives total energy transfer rate results that are close to the reference model. They differ slightly because the reference model accounts for fin conduction in the longitudinal (x) direction and allows for partially wet and dry conditions in the fin height direction. Similar results were obtained for a wide variety of operating conditions. The heat and mass transfer fin efficiency provides a very good characterization A rather long and fancy word for analyzing a system or process and measuring its "characteristics." For example, a Web characterization would yield the number of current sites on the Web, types of sites, annual growth, etc. of the saturated air enthalpy distribution on the fin.
Figure 5 gives comparisons between predictions of sensible energy transfer rates for the reference and fin efficiency models. The sensible heat transfer rate decreases with entering air relative humidity because increased latent energy transfer causes an increase in the heat exchanger surface temperature. The reference model results were obtained for a Lewis number of unity. For the fin efficiency model, results were determined using heat transfer fin efficiencies in the wet section that were derived for both dry surfaces (Equation 1) and wet conditions (Equation 13). The use of the modified heat transfer fin efficiency gives predictions that are in much closer agreement with the two-dimensional results as compared with the use of dry heat transfer fin efficiency relations. The errors associated with using a heat transfer fin efficiency derived for dry conditions in the wet section increase as more of the heat exchanger is wetted (i.e., at higher inlet relative air humidities). For the results of Figure 5, the errors in sensible heat transfer were reduced from a maximum of about 12% to less than 1% by applying the improved heat transfer fin efficiency. Similar results were obtained for a wide variety of operating conditions where moisture was condensing on the fin.
[FIGURE 5 OMITTED]
Figure 6 shows variation in local heat transfer and heat and mass transfer fin efficiencies as a function of position along the airflow stream for an air inlet relative humidity of 60%. Heat transfer fin efficiency results are presented for both the dry analysis (original model) and the new approach that corrects for the wetted fin. The initial 20% of the fin surface is dry, so only the heat transfer fin efficiency for dry surfaces applies. After condensation starts, the heat and mass transfer fin efficiency for wet surfaces applies and increases as the surface becomes wetter when approaching the chilled-water inlet. In contrast, the heat transfer fin efficiency for the wet surface decreases in the direction of the airflow and approaches the heat and mass transfer fin efficiency as the air approaches saturation. The error associated with using a dry local fin efficiency is largest for saturated air conditions and is greater than 15% at the air outlet condition for the results of Figure 6.
[FIGURE 6 OMITTED]
The improved method of determining heat transfer fin efficiency under wet surface conditions was implemented within an overall cooling coil model for a coil that has been extensively tested in a laboratory as described by Zhou and Braun (2005). This is an eight-row coil with wavy fins and a fin spacing of eight fins per inch, which is representative of large commercial applications. Table 3 provides the parameters that characterize the coil, and Figure 7 depicts its flow circuiting.
[FIGURE 7 OMITTED]
Table 3. Characteristics of Eight-Row Test Coil Coil Physical Parameter Eight-Row Coil Fin geometry Wavy Coil depth, m 0.264 Number of fins per inch (/0.0254 m) 8 Coil face width, m 0.6096 Coil face height, m 0.6096 Tube material Copper Tube outer diameter, m 0.0127 Tube thickness, m 0.0004 Tube longitudinal pitch, m 0.033 Tube transverse pitch, m 0.0381 Fin material Aluminum Fin thickness, m 0.0002
The coil was tested using a psychrometric wind tunnel wind tunnel, apparatus for studying the interaction between a solid body and an airstream. A wind tunnel simulates the conditions of an aircraft in flight by causing a high-speed stream of air to flow past a model of the aircraft (or part of an aircraft) being tested. that allows control of the coil inlet air temperature, humidity and flow rate, and water temperature and flow rate. Sufficient instrumentation instrumentation, in music: see orchestra and orchestration.
In technology, the development and use of precise measuring, analysis, and control equipment. was installed to determine air-side and water-side heat transfer rates under steady-state conditions In telecommunication, the term steady-state condition has the following meanings:
The cooling coil model used for implementing and evaluating the improved fin efficiency approach was presented in ARI Standard 410 (2001). It separates the coil into dry and wet sections and applies the log mean temperature difference method to the dry section and the log mean enthalpy difference method to the wet section. Both the original and improved methods for heat transfer fin efficiency were implemented for comparison with experimental results.
Table 4 gives sample comparisons between model predictions and experimental results for a range of conditions where the coil was partially and fully wetted. Predictions for total energy transfer rate are within 3.2% of the values determined from experimental measurements, which is less than the measurement uncertainty and energy balance error between air-side and water-side measurements (6%) for the test setup See BIOS setup and install program. . The model predictions for total energy transfer rate are independent of the approach used to characterize the heat transfer fin efficiency in the wet section. Predictions of sensible heat transfer are within 1.2% of the measurements for the modified heat transfer fin efficiency approach, which is an improvement compared to the 8.4% difference associated with the original method.
Table 4. Comparisons with Experimental Results Case [T.sub.a, [RH.sub.a, [M.sub.a] [T.sub.w, [M.sub.w] in], in], % in], [degrees]C kg/s [degrees]C kg/s 1 27.57 45.61 1.02 2.76 0.45 2 28.38 52.68 1.02 3.18 0.46 3 28.31 60.8 1.01 3.47 0.47 4 28.32 68.63 1 3.24 0.46 5 28.23 76.03 1 2.41 0.45 [q.sub.tot,]kW [q.sub.sen,] kW Case Exp. Original Improved Exp. Original Model Model Model 1 21.62 0.93 20.93 15.46 15.9 2 23.79 23.23 23.23 14.66 15.27 3 25.28 25.2 25.2 13.34 14.03 4 27.2 27.06 27.06 12.26 13.07 5 29.27 29.61 29.61 11.38 12.33 [q.sub.sen,] kW Case Improved Model 1 15.45 2 14.74 3 13.46 4 12.39 5 11.51
Figure 8 shows comparisons for sensible heat ratio (SHR SHR Shore
SHR Spontaneously Hypertensive Rat
SHR Staff Human Resources
SHR Saskatoon Health Region (Saskatoon, SK, Canada)
SHR Shift Logical Right
SHR Sensible Heat Ratio
SHR Supplementary Homicide Report
SHR Steroid Hormone Receptor ), defined as the ratio of the sensible to total energy transfer rates. The SHR predictions by the original model are beyond the uncertainty bounds for the experimental measurements. The modified heat transfer fin efficiency model improves these predictions into or on the uncertainty bounds and provides greater relative improvements in SHR at lower values where the coil is wetter.
[FIGURE 8 OMITTED]
This paper develops and presents an improved method for determining heat transfer fin efficiency for wetted fins. The heat transfer fin efficiency is determined using a correction factor applied to the heat and mass transfer fin efficiency (Equation 13). Detailed numerical and experimental results show that the method provides improved predictions of sensible heat transfer rates compared to the use of dry surface fin efficiency relations, especially for conditions where cooling coils are fully wetted. The improved method is readily implemented in existing models and does not result in additional complexity.
This study was funded by ASHRAE (ASHRAE RP-1194). Their support is gratefully appreciated.
NOMENCLATURE nomenclature /no·men·cla·ture/ (no´men-kla?cher) a classified system of names, as of anatomical structures, organisms, etc.
C = specific heat, kJ/kg K
C = heat capacitance rate, kW/K
t = half of fin thickness, m
h = convection coefficient, W/[m.sup.2] K or specific enthalpy, kJ/kg
H = height, m
[h.sub.d] = mass transfer coefficient, kga/s [m.sup.2]
[h.sub.fg] = heat of water vaporization, kJ/kgw
k = heat conductivity, W/m K
L = length, m
Le = Lewis number
m = mass flow rate, kg/s
q = heat transfer rate, W
R = heat resistance, K/W
RH = relative humidity, %
T = temperature, [degrees]C
W = humidity ratio, [kg.sub.w]/[kg.sub.a]
[[eta].sub.f] = heat transfer fin efficiency for dry surface
[[eta].sub.f] = heat transfer fin efficiency for wet surface
[[eta]*.sub.f] = heat and mass transfer fin efficiency for wet surface
a = air
b = fin base
c = condensate
f = fin
in = inlet
out = outlet
s = saturation
sen = sensible
t = tube
tot = total
w = water
* = wet fin
' = per unit length
" = per unit surface area
ARI. 2001. ARI 410-2001: Forced-Circulation Air-Cooling and Air-Heating Coils. Arlington, VA: Air-Conditioning and Refrigeration refrigeration, process for drawing heat from substances to lower their temperature, often for purposes of preservation. Refrigeration in its modern, portable form also depends on insulating materials that are thin yet effective. Institute.
ASHRAE. 2000. 2000 ASHRAE Handbook-HVAC Systems and Equipment. Atlanta: American Society of Heating, Refrigerating re·frig·er·ate
tr.v. re·frig·er·at·ed, re·frig·er·at·ing, re·frig·er·ates
1. To cool or chill (a substance).
2. To preserve (food) by chilling. and Air-Conditioning Engineers, Inc.
Braun, J.E., S.A. Klein, and J.W. Mitchell. 1989. Effectiveness models for cooling towers and cooling coils. ASHRAE Transactions 95(2):164-74.
Elmahdy, A.H., and G.P. Mitalas. 1977. A simple model for cooling and dehumidifying coils for use in calculating energy requirements for buildings. ASHRAE Transactions 83(2):103-17.
Incropera, F.P., and D.P. DeWitt. 1996. Fundamentals of Heat and Mass Transfer. New York New York, state, United States
New York, Middle Atlantic state of the United States. It is bordered by Vermont, Massachusetts, Connecticut, and the Atlantic Ocean (E), New Jersey and Pennsylvania (S), Lakes Erie and Ontario and the Canadian province of : John Wiley John Wiley may refer to:
Kuehn, T.H., J.W. Ramsey, and J.L. Threlkeld. 1998. Thermal Environmental Engineering. Upper Saddle River Saddle River may refer to:
Murthy, J.Y., and S.R. Mathur. 1998. Numerical methods in heat, mass, and momentum transfer. Lecture notes.
Xia, Y., and A.M. Jacobi. 2005. Air-side data interpretation and performance analysis for heat exchangers with simultaneous heat and mass transfer: Wet and frosted surfaces. International Journal of Heat and Mass Transfer 48:5089-102.
Zhou, X., and J.E. Braun. 2005. Dynamic modeling of chilled chill
1. A moderate but penetrating coldness.
2. A sensation of coldness, often accompanied by shivering and pallor of the skin.
3. water cooling Water cooling is a method of heat removal from components. As opposed to air cooling, water is used as the heat transmitter. Water cooling is commonly used for cooling internal combustion engines in automobiles and electrical generators. coils. ASHRAE 1194-RP Final Report, American Society of Heating, Refrigerating and Air-Conditioning Engineers, Inc., Atlanta.
Xiaotang Zhou, PhD Associate Member ASHRAE James E. Braun, PhD Fellow ASHRAE Qingfan Zeng, PhD
Xiaotang Zhou is an Engineering Leadership Program associate for the Carrier Corporation, Syracuse, NY. James E. Braun is a professor of mechanical engineering for Ray W. Herrick Laboratories, Purdue University Purdue University (pərdy`, -d`), main campus at West Lafayette, Ind. , West Lafayette West Lafayette, city (1990 pop. 25,907), Tippecanoe co., W Ind., a suburb of Lafayette, on the Wabash River; inc. 1924. A primarily residential city, it is the seat of Purdue Univ. , IN. Qingfan Zeng is a senior engineer for Carrier China, Shanghai Shanghai (shăng`hī`, shäng`hī`), city (1994 est. pop. 12,980,000), in, but independent of, Jiangsu prov., E China, on the Huangpu (Whangpoo) River where it flows into the Chang (Yangtze) estuary. , China.
Received May 26, 2006; accepted June 11, 2007
This paper is based on findings resulting from ASHRAE Research Project RP-1194.