Mean temperature difference, MTD, is defined as the area weighted average temperature difference between hot and cold streams in the heat exchanger:
Th = Local hot fluid temperature, °F
tc = Local cold fluid temperature, °F
MTD depends on flow arrangement and has been calculated in dimensionless form for commonly used flow arrangements, with the assumptions that U is constant and the heat content of each stream varies linearly with temperature.
Several different forms of MTD charts are available in the literature. The most useful form involves cross plots of four dimensionless parameters (R, N, P and q) defined as follows.
R = Relative Heat Capacity Ratio
= Cc/Ch = (Ti-To)/(to-ti)
N = Number of Transfer Units = U×A/Cc
P = Thermal Effectiveness = (to-ti)/(Ti-ti)
q = Dimensionless MTD = MTD/(Ti-ti)
Ch = (M × Cp)h = Hot stream heat capacity rate, Btu/hr × °F
Cc = (M × Cp)c = Cold stream heat capacity rate, Btu/hr × °F
M = Mass flow Rate, lb/hr
Cp = Specific heat, Btu/lb × °F
Ti = Hot stream inlet temperature, °F
To = Hot stream outlet temperature, °F
ti = Cold stream inlet temperature, °F
to = Cold stream outlet temperature, °F
Graphs relating R, N, P, and q are presented in Appendix A for commonly used flow arrangements. These graphs are useful for design of individual exchangers, evaluation of exchanger performance at nondesign conditions, evaluation of heat exchanger network performance, and evaluation of field performance data, as described below.
For design, terminal temperatures (and therefore P and R) are known, and required area can be calculated from N. For evaluation of networks or alternative design conditions, N, R and inlet temperatures are known, and outlet temperatures can be calculated from P. For evaluation of field performance data, terminal temperatures (R and P) are known, and U can be calculated from N.
A more common presentation of MTD information is F-factor graphs, where F is defined as the actual mean temperature difference (MTD) divided by the “log mean temperature difference” (LMTD). LMTD is the actual MTD for counterflow (F=1), and is calculated as follows:
LMTD = [(Ti-to)-(To-ti)] / ln [(Ti-to)/(To-ti)]
Using the definition of F, Equation 200-1 becomes:
Q = U × A × F × (LMTD)
F-graphs are plots of F against P with R as a parameter. They provide the same information as q on the more general graphs. Both the general graphs and F-graphs are provided in Appendix A for each flow arrangement. Only the general graph is given for counterflow (because Fº1 for counterflow). F=1 for pure component (isothermal) boiling and condensing regardless of flow arrangement.
MTD graphs described above are based on the assumption that the heat content of each stream varies linearly with temperature. This may not be correct where phase change is involved. For example, cooling a superheated vapor may involve a variable temperature desuperheating zone, an isothermal condensing zone, and a variable temperature subcooling zone. Such cases should be analyzed in segments, as indicated in Figure 200-1, so that the linear assumption is valid for each segment.