A Numerical Method for Calculating the Mean Temperature

 

For zeoptropic refrigerants, some of the assumptions from the classical LMTD derivation must be discarded. Beginning with the first equation of Chapter 2, the mass flow rate through the heat exchanger is still assumed to be constant, but the specific heats for both the cold and the hot streams are now not constant. In the classical LMTD derivation, it was assumed that the fluids would not undergo a phase change and that they would have constant specific heats (so that dh = c_p dT). In this investigation, however, the more general case is studied, allowing a two-phase region of mixtures with nonlinear temperature-enthalpy curves. As discussed in the introduction, this nonlinearity results in varying specific heats. In fact, the specific heat is a function of both temperature and area. Returning then instead to the second definition in Chapter 2 for the heat transferred between two streams in a heat exchanger,

 

(3.1)

where DT_mw is the temperature difference between the zeotropic refrigerant mixture and the water at any given point. To find the total heat transferred, equation 3.1 is integrated over the entire area of the heat exchanger.

ó
õ Q₀

0 
dQ DT_mw
= ó
õ A₀

0 
U dA

It is still reasonable to assume that the overall heat transfer coefficient will be fairly constant over the area (hereafter denoted as U₀). The area is nondimensionalized so that A' = A/A₀, and the above equation becomes

 

ó õ Q0 0  dQ DTmw = U0 A0 ó õ 1 0  dA' = U0 A0

(3.2)

It should be noted that, by definition, UA is inversely proportional to the total thermal resistance. This resistance is obviously not strictly constant over the heat exchanger area; the condensation or evaporation of the fluids and the changes in temperature will affect both the thermal conductivity and the convection coefficients. However, the magnitude of these variations is assumed to be small, so as to only insignificantly affect the overall heat transfer coefficient. If this is not the case, the right hand side of equation 3.2 could become quite complicated, and would require as parameters the exact specifications of the heat exchanger under consideration.

An examination of equation 3.2 reveals its similarity to equation 2.15. However, because the temperature can be radically nonlinear, the above integral cannot be evaluated in closed form (recalling that dQ is a function of enthalpy). Instead, it must be replaced by a numerical integration in which the temperature difference between the streams is found at incremental steps of the amount of heat transferred in the heat exchanger.

 

U0 A0 = Q0 å i = 0  1 DTmw dQ

(3.3)

As the number of steps approaches infinity, equation 3.3 becomes identical to equation 2.15.

  

Figure 3.1: A Counterflow Heat Exchanger: Temperature Examined at Four Points

  

Figure 3.2: A Counterflow Heat Exchanger: Temperature Examined at Eleven Points

Calculations like those above would once have required hours of expensive computer time to solve. Given current computational power and accurate properties in computer-readable form, however, they can now be performed in minutes on a desktop PC. The number of steps, then, can be made sufficiently large so as to give a reasonably accurate solution without demanding a significant increase in solution time. As a graphical representation of this improvement, compare the obvious inaccuracy of Figure 3.1 with the increased precision of Figure 3.2.

Two other numerical approaches to finding a mean temperature have been suggested by Granryd & Conklin (1990). In the first method, the temperature profiles are linearized in small segments, from which an effective specific heat can be calculated. The second method also focuses on the specific heat. A polynomial approximation for c_p is developed from a curve fit, and then integrated where appropriate. A disadvantage to both these methods is the complexity of the calculations that must still be performed once c_p is found in order to determine the heat exchanger size.