A Condensation Curve, Figure 1, is a plot of temperature against specific enthalpy, h, or cumulative heat removal rate, , for a pure vapor or a mixture. A superheated vapor at A is cooled, becoming saturated at B, the dew point Tdew by removal of heat at rate . Condensation takes place along the curve BC until the bubble temperature, Tbub, is reached at C. Thereafter, a subcooled liquid is produced at D, corresponding to an overall heat removal rate, . EF shows a cooling curve for the coolant.
Figure 1 is the Integral Condensation Curve. It describes condensation in a device where vapor and condensate follow a parallel path with complete mixing so that equilibrium is maintained. For binaries, the curve ABCD of Figure 1 is derived from the temperature-composition diagram (Figure 2), which gives the state (vapor, liquid or two-phase), the dew and bubble temperatures and the vapor fraction, θ. Thus at E, one mole of feed separates into θ moles of vapor and (1–θ) moles of liquid, of molar composition and , respectively.θ is given by the division of the tie line.
Other condensation curves may be defined. A Differential Condensation Curve arises when vapor and condensate are separated within the condenser, and depart from overall equilibrium. Such a curve is also derived from Figure 2. Tie lines are constructed to give a number of equilibrium stages. Each stage divides the mixture into a condensate and vapor, which are then considered separately. The next separation is applied to the remaining vapor. Figure 2 shows a process giving 10 equal condensates with saturated vapor. An auxiliary curve might be constructed to show how the condensates are mixed and cooled. The curve CoG shows all prior condensate fully-mixed and cooled to the vapor temperature. A differential curve is the limit of an infinite number of stages.
Such cooling curves are convenient because they do not depend on condenser geometry. They may sometimes approximate the true cooling curve of a real condenser, Fig. 3. Vertical condensers with turbulent condensate films are likely to approximate the integral curve, while the differential curve, CoG, might be a good description of a horizontal condenser, where condensate forms a layer in the bottom of the unit, or a vertical condenser with condensate in laminar flow. It is clear that differential condensation must be avoided, because of the rectification involved. The vapor becomes richer in the more volatile component with a fall in saturation temperature and driving force so that a larger area is required for a given condensation.
The condensation curve defines the mean temperature driving force (Figure 1). Consider the interval , small enough that the slope of the cooling curve is constant. The appropriate mean temperature driving force is the log mean of the terminal temperature differences between hot and cold streams (see Mean Temperature Difference):
Further, the slope of the cooling curve determines the ratio of the sensible to total heat transfer rates in an interval, Z, which fixes the gas side heat transfer resistance, Z/αg,
The overall and component mass balances show the split of total flow, , composition , into condensate and vapor flows, , and , compositions , and , respectively,
with definition of K values, ,
A quantity, , which is a decreasing function of θ, is determined. When S = 0, the mass balance, Eq. (5), summed over all components, guarantees that the individual and sum to unity. For various θ at a given temperature, S is calculated from , by Eqs. (6) and (7). Figure 4 then allows diagnosis of the state of the mixture—all liquid, all vapor or at a two-phase equilibrium. In the latter case, the particular θ which makes S = 0 has to be found.
The mixture enthalpy and, hence, the cooling curve may now be determined. At any specified temperature, T,
Figure 5 shows how the mean temperature difference, ΔTm, may be calculated for an ‘E’ shell with one-coolant pass.
The energy balance maps the cooling curve, T = T(h), of the cold fluid to T = T(h´). The area required follows when the cooling curve is split into intervals where ΔT is linear with h´,
With two passes, Figure 6 shows the essentials of a procedure used to calculate the appropriate ΔTm.
The coolant curve, T(h) is mapped to T(h´) by energy balance with the assumption that the heat transfer rates to each pass, δhI and δhII, are in the ratio of the temperature differences. The arithmetic average, (TI+TII)/2, is the effective coolant temperature. These relationships are expressed as:
With more than two passes, the temperature change in a pass is so small that the mean temperature is independent of h´. With and the effective temperatures of the two streams, the mean temperature driving force can be found by solving