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Method to compute the enthalpy difference of a liquid stream in the absence of an EoS-based function.

INTRODUCTION

The heat transfer rate to or from a liquid moving through a heat exchanger is fundamentally computed as the product of the mass flow rate and the change in enthalpy of the liquid,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1)

In many cases, this is simplified to

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2)

However, the enthalpy of a liquid is also a function of pressure and, therefore, the heat transfer rate depends on the pressure drop of the liquid as it goes through the heat exchanger as well as its change in temperature.

A number of ASHRAE Standards describing methods of test for determining the performance of components or systems that involve a liquid stream are moving to include pressures along with temperatures when computing the liquid stream enthalpies and heat transfer rates. For example, the current version of ASHRAE Standard 22-2007 (ASHRAE, 2007) already states in section 5.2.2 that "The heat rejected from the refrigerant to the water is the product of the water flow rate and the enthalpy difference [author's emphasis] of the leaving and entering water." However, in section 5.2.4, it states that "The enthalpy difference between the leaving and entering water shall be determined from temperature [author's emphasis] measurements." with no mention of pressure. The committee working to revise and update Standard 22 is proposing, at the time of this writing, to include the pressure measurement in the calculation of enthalpy. A similar tack is being taken for ASHRAE Standard 181P, "Methods of Testing for Rating Liquid to Liquid Heat Exchangers", also under development at the time of this writing.

In the recently revised AHRI Standard 550/590 (AHRI, 2011) for rating the performance of water chillers, Appendix C now directs that the calculation of heat transfer rates include the effect of pressure drop. However, these "gross" heat transfer rates are to be used only for checking the system energy balance. "Net" heat transfer rates based only on temperature differences are still used for rating purposes. (1)

The calculation of enthalpy as a function of temperature and pressure is straightforward when an equation of state is available for the liquid. For example, the thermodynamic properties of water (both liquid and vapor) are thoroughly described by equations of state (Wagner and Pruss, 2002). This description is available in REFPROP (Lemmon, et al, 2010) and other sources (IAPWS, 2012). However, a full equation of state-based description does not exist for many of the secondary liquids used in HVAC&R and industrial applications, e.g., glycol solutions, brine solutions, hydrocarbon-based liquids, and silicone-based liquids. In many cases, simplified equations are used for water with very satisfactory results. For example, AHRI Standard 550/590 (AHRI, 2011) provides equations to be used for computing the density and specific heat of water as functions of temperature.

Formulations for liquid enthalpies involving both temperature and pressure do exist in the literature. For example, Gorgy and Eckels (2010) used an enthalpy-based method for calculating heat transfer rates and showing the impact of pressure drop on heat transfer coefficients. However, this paper and most others are missing a term involving the "volume expansivity" of the liquid, likely unintended through a simplified derivation. Fortunately, this term tends to be of small consequence in most cases. However, as will be shown below, the volume expansivity of certain fluids, such as hydrocarbon blends and silicone oils, is large enough to be of importance.

The objective of this paper is to provide a derivation of the equations for computing the enthalpy and enthalpy differences of a liquid in the absence of a full equation of state. In the end, the equations are easy to implement in the analysis of data or predictions of component performance. This paper also displays the relative importance of pressure drop to change in temperature when computing heat transfer rates. It will be shown here that for certain cases, especially those with a large pressure drop in conjunction with a small temperature difference, the error in the computed heat transfer rate becomes significant (say, relative to system energy balance closure) when the pressure drop is ignored.

DERIVATION OF AN ESTIMATE FOR ENTHALPY DIFFERENCE

The basis for this derivation can be found in any textbook on classical thermodynamics; see, for example, Van Wylen and Sonntag (1973) or Klein and Nellis (2012). See also ASHRAE (2009).

The differential change in enthalpy as a function of temperature and pressure can be expanded as follows,

dh = [([partial derivative]h/[partial derivative]T).sub.P] dT + [([partial derivative]h/[partial derivative]P).sub.T] dP (3)

From the Gibbs relation

Tds = dh - [upsilon]dP (4)

it follows that

[([partial derivative]h/[partial derivative]P).sub.T] = [upsilon] + T [([partial derivative]s/[partial derivative]P).sub.T]

Substituting in the definition of specific heat and the Maxwell relation,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (6)

results in the following equation,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (7)

where [[alpha].sub.P] is the "volume expansivity" or "coefficient of thermal expansion" of the liquid. (Note that the T in front of [[alpha].sub.P] is an absolute temperature.) The volume expansivity can also be written as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (8)

It is typical in the literature to see Equation (7) in which T[[alpha].sub.p] is missing; more on this below.

The change in enthalpy from one state to another can be found by integrating Equation (7). Some possible paths of integration are shown in Figure 1. Although following path A might seem like the most direct route, it is likely that the specific heat is known as a function of temperature at only one pressure. If we take this as the reference pressure (state 0), then the integration of Equation (7) should be done along the three segments shown for path B,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (9)

[FIGURE 1 OMITTED]

If density is independent of pressure, and therefore volume expansivity is also independent of pressure (that is, the fluid is "incompressible"), Equation (9) simplifies to

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (10)

If state 1 is taken to be the reference point, Equation (10) can then be used to provide a value for the enthalpy of a liquid at a given temperature and pressure,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (11)

VOLUME EXPANSIVITY

The volume expansivity is readily calculated from the density function according to Equation (8). Values of T[[alpha].sub.p] obtained from REFPROP (2) for water and various higher boiling point hydrocarbons are shown in Figure 2. Figure 3 shows Tap as a function of temperature for water as reported by REFPROP. Volume expansivity goes to zero as the temperature decreases toward water's maximum density point and then becomes negative as density begins to decrease as the freeze point is reached. The impact of volume expansivity on enthalpy is relatively small for water, 0 < T[[alpha].sub.p] < 0.15 for most HVAC&R applications with T < 50[degrees]C (122[degrees]F). However, 0.3 < T[[alpha].sub.p] < 0.5 for the higher boiling hydrocarbon liquids in Figure 2.

Volume expansivity as functions of temperature and concentration are shown for various other common heat transfer fluids as follows (3): Figure 4a) propylene glycol aqueous solutions as DOWFROST[R] (DOW, 2008), Figure 4b) potassium formate aqueous solutions as Dynalene HC (Dynalene, 2001), Figure 4c) a hydrocarbon blend as Dynalene HF-LO (Dynalene, n/d), and Figure 4d) silicone heat transfer fluid as Syltherm[TM] HF (DOW, 2001). In each case, density "data" obtained from the references were fitted with polynomial equations in temperature. These equations were then manipulated according to Equation (8) to determine the volume expansivity for each fluid and concentration.

[FIGURE 2 OMITTED]

The vertical scales in Figures 4a through 4d were kept the same to easily compare the various fluids. The volume expansivity of glycols is just slightly larger than of water. Salt solutions (brines) as represented by potassium formate appear to have a smaller volume expansivity than water; Tap is also less sensitive to temperature for brines than for water: 0.1 < T[[alpha].sub.p] < 0.2 for-50[degrees]C (-58[degrees]F) < T < 200[degrees]C (392[degrees]F). Hydrocarbon-based heat transfer fluids have a larger volume expansivity than water, comparable to the other hydrocarbons shown in Figure 2. Silicone oils, especially when used at high temperatures, have a rather large volume expansivity, T[[alpha].sub.p] > 0.5 for T > 100[degrees]C (212[degrees]F).

Including the volume expansivity in the derivation leading to Equation (11) tends to reduce the effect of pressure difference on enthalpy (except in those rare situations where the volume expansivity is negative). The impact of the volume expansivity on the pressure drop effect is quite small for water, glycols, and brines. The accuracy of heat transfer analyses is impacted only minimally when pressure drop is included, but the volume expansivity neglected, for fluids such as water, glycols, and brines. On the other hand, the effect of pressure drop on heat transfer rate will be overstated if the volume expansivity is ignored when applying a silicone oil at high temperatures.

[FIGURE 3 OMITTED]

[FIGURE 4 OMITTED]

ESTIMATE FOR THE CHANGE IN ENTHALPY ACROSS A HEAT EXCHANGER

Through a series of algebraic manipulations, Equation (10) can be rearranged in terms of the pressure difference of interest across a heat exchanger, [P.sub.2] - [P.sub.1],

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (12)

If the specific heat can be approximated as a linear function of temperature or a constant and changes in density are small over the relatively small changes in temperature and pressure typically encountered in an HVAC&R heat exchanger, the last term is found to be negligible and Equation (12) simplifies to (4)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (13)

where the average temperature, [bar.T] = ([T.sub.1] + [T.sub.2])/2, is used to evaluate properties. Equation (13) can be rearranged as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (14)

The second term in the final parentheses of Equation (14) can be considered as the relative impact on the change in enthalpy or heat transfer rate of the change in pressure of the fluid stream, or as the error if the pressure drop is ignored,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (15)

The error in computed heat transfer rate according to Equation (15) if pressure drop is ignored for water over a range of temperature and pressure differences is shown in Figure 5a. As noted above, there is little additional error by ignoring T[[alpha].sub.p], especially at lower temperatures. As an example, the error in computed heat transfer rate will be greater than 1% for temperature differences less than 2.2[degrees]C[delta] (4[degrees]F[delta]) when pressure drops of 100 kPa (15 psid) are neglected. Because the product of specific heat and density is a weak function of temperature for water, the results in Figure 5a are relatively insensitive to operating temperature.

On the other hand, the impact of pressure drop and volume expansivity on the computed heat transfer rate can be more significant for other liquids. For example, the impact of pressure drop and volume expansivity on computed heat transfer rate for a silicone oil at -40[degrees]C (-40[degrees]F) is shown in Figure 5b. The impact decreases as temperature increases. The much lower specific heat of the silicone oil relative to water (densities are similar) results in the pressure drop having a stronger influence on its enthalpy change. The larger value of volume expansivity, however, offsets this somewhat. At very high temperatures, silicone oil's large values of Tap nearly negate the effect of pressure drop on heat transfer rate.

[FIGURE 5 OMITTED]

Note that [Q.sub.[DELTA]P] can be additive or subtractive depending on the relative changes in temperature versus pressure. If state 2 represents the outlet state and state 1 the inlet state, then [P.sub.2] - [P.sub.1] < 0. In an evaporator, [T.sub.2] - [T.sub.1] < 0 also, so [Q.sub.[DELTA]P] is greater than zero (additive). In a condenser, [T.sub.2] - [T.sub.1] > 0, so [Q.sub.[DELTA]P] is less than zero (subtractive). When computing the energy balance closure for systems with both a liquid chilling evaporator and a liquid cooled condenser, the absolute values of the pressure drop effects are additive, so ignoring pressure drop doubly impacts the accuracy of the energy balance closure.

As an example, the energy balance closure, EBC, of heat transfer rate and power measurements made on a water-cooled chiller is presented here (5)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (16)

Four methods of computing the evaporator and condenser heat transfer rates are considered:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

using full-fledged enthalpy equation (17)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

temperatures only, ignoring pressure drop (18)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

[Q.sub.[DELTA]P] per Equation (15), with T x [[alpha].sub.P] = 0 (19)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

[[??].sub.3] with [Q.sub.[DELTA]P] per Equation (15) (20)

In this case, the chiller was being tested to determine evaporator bundle heat transfer coefficients. Tests with both water and a 30% propylene glycol/water mixture were run. At nominal conditions, the chilled liquid pressure drop through the evaporator tube bundle was about 52 kPa (7.5 psid); see Figure 6. At full capacity, the chilled liquid temperature difference was about 2.8[degrees]C[delta] (5[degrees]F[delta]).6 Tests were also run at full capacity over a range of flow rates. The elevated flow rates resulted in large pressure drops and small temperature differences that accentuate the impact of pressure drop on computed heat transfer rate. The condenser cooling water was run at a nominal flow rate, resulting in a pressure drop of about 47 kPa (6.8 psid) and temperature differences ranging downward from 2.8[degrees]C[delta] (5[degrees]F[delta]) as load was varied.

Figure 7 shows the relative contributions of pressure drop to heat transfer rate for this example data set. For those test points with high evaporator flow rates (high pressure drop with small temperature differences), the pressure drop can account for 1-3% of the heat transfer rate. The condenser pressure drop subtracts from the heat transfer rate by 0.5-1.5%. Because the impact of pressure drop on the evaporator and condenser heat transfer rates are of opposite signs, they are additive when computing the energy balance closure for the chiller. The energy balance closures (EBC's) computed using Equations (17)-(20) are shown in Figure 8. Ignoring pressure drop in calculation of enthalpy differences, EBC(1), results in energy balance errors of 1-2%, rising to 3-4% for those runs with large AP's and small AT's. Accounting for pressure drop in the enthalpy differences reduces the energy balance errors to within +1% and generally centered around zero. In this case (both liquids are water), Tap values are very small (0.98-1.01) so that EBC(2) (ignoring Tap) and EBC(3) (including T[[alpha].sub.p]) are nearly indistinguishable from each other and from the calculations using an equation of state for enthalpy, EBC(0).

[FIGURE 6 OMITTED]

[FIGURE 7 OMITTED]

Figure 9 shows the pressure drops and temperature differences occurring during tests of the above chiller with a 30% propylene glycol/water mixture as the chilled liquid. The corresponding relative contributions of pressure drop to heat transfer rate are shown in Figure 10. Here, errors of 5-10% can occur if pressure drop is ignored in the calculation of the chilled liquid heat transfer rates. The improvements in chiller energy balance closure when including pressure drop, EBC(2) and EBC(3), versus ignoring pressure drop, EBC(1), are evident in Figure 11. As shown in Figure 12, the volume expansivity of the propylene glycol solution has a larger effect than for water, but the impact is still relatively mild--only 9-13% of the pressure drop effect that is at most 10% of the total heat transfer rate per Figure 10.

[FIGURE 8 OMITTED]

[FIGURE 9 OMITTED]

[FIGURE 10 OMITTED]

[FIGURE 11 OMITTED]

CONCLUSIONS

This paper presents a derivation of simplified equations that can be used to compute heat transfer rates for liquid streams that include the impact of both change in temperature and pressure drop across the heat exchanger. The equations capture the effect of the "volume expansivity" of the liquid, a term that is missing from other formulations in the literature. Fortunately, the effect of volume expansivity is small in most cases, especially when the liquid is water. However, the volume

expansivity can be of noticeable influence for certain liquids, such as hydrocarbon blends and silicone oils, especially when employed at higher operating temperatures. The volume expansivity is easily computed from typically available density information via Equation (8), and so it is recommended that it be included in formulations for enthalpy differences such as Equation (14).

It is also shown that including pressure drop in the calculation of heat transfer rates can improve the accuracy of those quantities. Method of test and performance standards such as ASHRAE Standard 22 and 181P and AHRI Standard 550/590 that typically require energy balance closures of +3% can be more easily met when pressure drop is correctly included in the calculation of the heat transfer rates; no amount of instrument accuracy can offset an inherently inaccurate calculation.

[FIGURE 12 OMITTED]

This is especially true in situations with small changes in temperature relative to the pressure drop, for example, per Figure 5a. In reviewing Equation (14), the effect of pressure drop on heat transfer rate relative to change in temperature is increased for liquids with lower densities and lower specific heats such as hydrocarbon blends and silicone oils. On the other hand, the effect of pressure drop on enthalpy is diminished for liquids with higher values of volumetric expansivity, such as hydrocarbon blends and silicone oils applied at elevated temperatures. Again, the correct or accurate calculations are easy enough to do. The calculations do require that the liquid pressures be measured in collocation with the temperatures to obtain enthalpies. In many cases, the pressure drop is a required measurement anyway and so is available for use in formulations like Equation (14).

NOMENCLATURE

[c.sub.P] = specific heat or heat capacity at constant pres sure [Btu/lbm x R, J/kg x K]

EBC = energy balance closure, see Equation (16) [dimensionless]

EBC(j) = energy balance closures corresponding to the heat transfer rates calculated according to Equations (17)-(20)

h = enthalpy [Btu/lbm, J/kg]

[??] = mass flow rate [lbm/hr, kg/sec]

P = pressure [psi, Pa]

PrGl = propylene glycol/water solution

[??] = heat transfer rate [Btu/hr, W]

[??], j = 0 ... 3 = heat transfer rates calculated according to Equations (17)-(20)

[Q.sub.[DELTA]P] = non-dimensional impact of pressure drop on change in enthalpy (or heat transfer rate) relative to temperature difference; see Equation (15)

s = entropy [Btu/lbm-R, J/kg-K]

T = temperature [[degrees]F, [degrees]C; R, K]

[bar.T] = average temperature [[degrees]F, [degrees]C; R, K]

v = specific volume [ftvlbm, mvkg]

[[??].sub.Cmpr] = power (rate of work) input to compressor [Btu/hr, W]

Greek Symbols

[[alpha].sub.p] = volume expansivity, defined in Equation (8) [1/R 1/K]

[DELTA], [delta] = difference

[rho] = density [lbm/[ft.sup.3], kg/[m.sup.3]]

Subscripts

Cond = condenser

Evap = evaporator

i = in or entering

liq = liquid

o = out or leaving

0 = indicates the reference state for computing enthalpies

1, 2 = indicate the starting and ending states

REFERENCES

AHRI. 2011. AHRI Standard 550/590 (IP), 2011 Standard for Performance Rating Of Water-Chilling and Heat Pump Water-Heating Packages Using the Vapor Compression Cycle, Arlington, VA: Air-Conditioning, Heating, and Refrigeration Institute.

ASHRAE. 2007. ANSI/ASHRAE Standard 22-2007, Methods of Testing for Rating Water-Cooled Refrigerant Condensers, Atlanta: American Society of Heating, Refrigerating and Air-conditioning Engineers, Inc.

ASHRAE. 2009. 2009 ASHRAE Handbook--Fundamentals, Atlanta: ASHRAE. (See Chapter 2, pp 2.4-2.5.)

DOW. 2001. SYLTHERM HF Technical Data Sheet, accessed via http://msdssearch.dow.com/PublishedLiteratureDOWCOM/dh_0040/0901b80380040bb0.pdf? filepath=/heattrans/pdfs/noreg/176-01470.pdf&fromPage=GetDoc on 25-Jan-2012.

DOW. 2008. Engineering and Operating Guide for DOW FROST and DOWFROST HD, accessed via http://msds search.dow.com/PublishedLiteratureDOWCOM/dh_010e/ 0901b8038010e417.pdf?filepath=heartrans/pdfs/noreg/ 180-01286.pdf&fromPage=GetDoc on 25-Jan-2012.

Dynalene. 2001. Dynalene HC Heat Transfer Fluid Engineering Guide, accessed via http://www.dynalene.com/ pdf/hcguide.pdf on 25-Jan-2012.

Dynalene. n/d. Dynalene HF-LO Technical Data Sheet, accessed via http://www.dynalene.com/pdf/HFLO.pdf on 25-Jan-2012.

Gorgy, E., and S. Eckels. 2010. "Average Heat Transfer Coefficient for Pool Boiling of R-134a and R-123 on Smooth and Enhanced Tubes (RP-1316)", HVAC&R Research, vol 16, no 5, pp 657-676.

IAPWS. 2012. Website of The International Association for the Properties of Water and Steam, accessed on 02-Apr-2012, http://www.iapws.org/.

Klein, S., and G. Nellis. 2012. Thermodynamics, Cambridge University Press. (See Chapter 10.)

Lemmon, E.W., and M.L. Huber, and M.O. McLinden. 2010. NIST Standard Reference Database 23: Reference Fluid Thermodynamic and Transport Properties-REFPROP, Version 9.0, National Institute of Standards and Technology, Standard Reference Data Program, Gaithersburg.

Van Wylen, G.J., and R.E. Sonntag. 1973. Fundamentals of Classical Thermodynamics, Second Edition, John Wiley and Sons. (See Chapter 10.)

Wagner, W., and A. Pruss. 2002. "The IAPWS Formulation 1995 for the Thermodynamic Properties of Ordinary Water Substance for General and Scientific Use", J. Phys. Chem. Ref. Data, vol 31, pp 387-535.

K.J. Schultz, PhD

Member ASHRAE

DISCUSSION

Phillip Johnson, Director of Engineering, McQuay International, Staunton, VA: A well written and useful paper for the industry; thanks for this contribution! I don't quite understand the author's comment in footnote 1, since the net capacity used by AHRI Standard 550/590-2011 for published chiller ratings does not give credit for higher capacity with higher pressure drop. It is the gross capacity that includes an additive term in the evaporator due to pressure drop (and a subtractive term in the condenser). This A P/p term is intended to more closely capture the total heat exchanged between water and refrigerant in order to provide for a more accurate energy balance. It is still a simplification of the thermodynamics, but it avoids the complexity of requiring enthalpy values at the inlet and outlet. Considering the pressure drop due to frictional losses as a source of heat generation, it is somewhat intuitive to think of the net capacity in an evaporator as what is available for useful cooling of a building load or process load. Considering different evaporator designs with varying frictional losses, but equal net capacity and equal temperature change, as the frictional losses increase the gross capacity increases in order to dissipate the additional energy losses from the water to the refrigerant, but the end user of the cooled fluid receives no additional benefit or cooling capacity. The penalty for higher frictional losses is reflected in a system analysis as increased pumping energy.

Ken Shultz: Thanks for the comment. I agree entirely. The footnote was meant to convey an admittedly nontechnical rationale (as a way to be brief) for not using the "gross" capacity to rate a chiller, instead using the "net" capacity as defined in AHRI 550/ 590-2011. Your comment clarifies my footnote in a much more technical manner. The calculation of gross capacity in AHRI 550/ 590-2011 is an example of a formulation that neglects the small adjustment of 1 - T[[alpha].sub.p] in the [[DELTA].sub.P/[rho]] term in Equations 13 and 14. As shown in the paper, this has little consequence when the chilled fluid is water.

(1.) It just feels wrong that, for the same change in temperature, an evaporator bundle with larger pressure drop should be credited with having higher capacity.

(2.) REFPROP includes the function "VolumeExpansivity" for direct reporting of [[alpha].sub.P].

(3.) Brand-name fluids are used in the following as examples only. It is not the author's intent to endorse these fluids over other comparable competitive fluids. These fluids were chosen for inclusion here simply because properties appropriate for this analysis were readily accessible via the internet.

(4.) Equation (13) is a heuristic approximation to Equation (12), not the true result of a formal algebraic manipulation. By example, it can be shown that Equation (13) very closely matches Equation (12).

(5.) A plus sign is used in front of [[??].sub.Cond] in the numerator to be consistent with the convention used in Equations (1) and (17) (20).

K.J. Schultz is a Senior Development Engineer for Trane, a division of Ingersoll Rand, La Crosse, WI, USA.
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Date:Jul 1, 2013
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