Application of the Numerical Technique

 

Software Tools

Engineering Equation Solver

Engineering Equation Solver (EES) is a software package developed by Dr. Sanford Klein of the University of Wisconsin. EES incorporates the programming structures of C and FORTRAN with a built-in iterator, thermodynamic and transport property relations, graphical capabilities, numerical integration, and many other useful mathematical functions. By grouping equations that are to be solved simultaneously, EES is able to function at a high rate of computational speed. EES can also be used to perform parametric studies.

EES was chosen for this research due to its ability to seamlessly incorporate fluid property calls. Steam tables, air tables, JANAF data, psychrometric functions, and property tables for ammonia, methane, carbon dioxide, and other fluids are built into EES. Rather than store each possible data point, the Martin-Hou equation of state (EOS) is used. Water is one exception to this, as several equations of state are used for each phase. Ammonia-water mixture properties are calculated in EES using the correlation developed by Ibrahim & Klein (1993). To expand the number of available refrigerants, an interface has been developed by Dr. Klein that allows EES to utilize the National Institute of Standards and Technology's Thermodynamic Properties of Refrigerants and Refrigerant Mixtures Database (REFPROP).

Refrigerant Database

REFPROP is a FORTRAN-based program that allows a user to look up thermodynamic properties for most refrigerants currently in use or under study. In addition to pure refrigerant properties, REFPROP uses either the Carnahan-Starling-DeSantis (CSD) hard sphere equation of state or the modified Bennedict-Webb-Rubin (MBWR) equation of state to calculate the properties of refrigerant mixtures (up to five components per mixture are possible). The CSD EOS requires only six adjustable parameters per fluid, making it suitable for fluids where there are limited experimental observations, or where only a narrow range of study is of interest. The MBWR EOS is better for wide ranges of data, but requires a large amount of experimental information in order to fit its thirty-two interaction coefficients. The parameters for a number of mixtures have already been found and entered into REFPROP, but the user is responsible for matching the calculated points to experimental data for new combinations. The EES interface calls a DOS version of REFPROP in order to increase the speed of its calculations, but a Windows version is also available in a beta version still under development. The refrigerants that are available in both the DOS and the Windows versions can be found in Appendix A.

Calculation of the UA

 

In order to find the UA derived in equation 3.3, the total heat transferred and the temperature difference between the water and mixture streams must be calculated at a number of points in the heat exchanger. The heat transferred can be found by applying the energy equation to the water flowing through the heat exchanger:

 

. Q   Tot = . m   water (hwater,entrance - hwater,exit)

(4.1)

or for the refrigerant:

 

. Q   Tot = . m   mixture (hmix,exit - hmix,entrance)

(4.2)

where $\dot{m}$ is the mass flow rate and h is the enthalpy. By equating 4.1 and 4.2, a mass flow ratio can be stated.

. m   ratio =

(4.3)

The end point states for both the water and the mixture must now be determined. A pinch point (PP) and concentration must also be chosen. For air-conditioning applications, a typical condenser entrance temperature for the water side is 29 °C with a 5 °C rise, so these are the values that will be used. The pressure in the heat exchanger (P_hi) is then found by evaluating the mixture at its exit, where the concentration, the quality (saturated liquid), and the temperature (29 °C + PP) are known. With two independent properties (pressure and temperature) given, the endpoint enthalpies of the water stream can be found. Subcooling is not considered, so the mixture is entirely in the two-phase region, and its endpoint properties can be found using the pressure, concentration, and qualities of zero and one.

In order to better visualize the transfer of heat between the water and the mixture streams, the total heat transferred will be redefined on a per refrigerant mass flow rate basis. The heat exchanger can then be divided into an arbitrary number of sections (n) of equal $\delta \dot{Q}/\dot{m}_{mix}$. The heat transfer per unit mass flow rate of the mixture is directly related to the enthalpy, so an enthalpy step shall be defined as

h_mix,step = h_mix,exit - h_mix,entrance n

The initial temperature difference between the refrigerant and the water and the amount of heat transferred are known from the endpoint specifications, and the values at each successive point in the heat exchanger (n_i = 1,2,3...) are calculated using the following equations.

From equation 4.2:

(4.4)

(4.5)

hwater = qcum

(4.6)

From fluid properties:

(4.7)

For each desired pinch point and refrigerant, the values of q_cum and DT_mw calculated in equations 4.4-4.7 are numerically integrated by EES to find the actual UA (equation 4.4 is an implicit equation in EES). When performing the numerical integration, EES decides how many sections the heat exchanger should be divided into. This stepsize is either chosen using an automatic stepsize adjustment algorithm, or is directly specified by the user. For this application, one hundred steps were deemed to be sufficient, as smaller heat exchanger segments increased the computational time without increasing the accuracy. A variant of the trapezoidal rule is used to examine the preceding values and to calculate the next step. In order to evaluate the integral, EES uses a second order predictor-corrector algorithm. This algorithm was designed to solve problems where the integrand is a complex function of other variables. It estimates what the value of the integrand should be at each variable step, iterates until convergence, and then moves to the next step. When each step has been evaluated, the integral is solved and the UA is known.

Comparison of the UAs: Development of Error Scales

Once the UA is known, it can be compared with the heat exchanger area that is found using the traditional log mean temperature difference method (UA_LMTD). This can be calculated using a slightly altered version of equation 2.15:

LMTD = dT₂ - d T₁

UALMTD =

(4.8)

Two separate error scales were developed to account for both major and minor discrepancies between the UA and the UA_LMTD. For small differences, a standard error (SE) is adequate:

SE = | UA - UA_LMTD | UA
×100

In this study, however, standard errors ranged from twenty percent to over three thousand percent for some mixtures. In order to provide a meaningful scale for evaluation of errors over such large ranges, a log error ratio (LER) is defined as

LER = -log( UA_LMTD UA
)

By using either the SE or the LER, every possible range of refrigerant UA error can be analyzed.