Chapter 26. Condensation
Learning Objectives
After completing this chapter, you will be able to:
Understand the Nusselt model for prediction of the heat transfer coefficient for film condensation.
Calculate the condensation rate of a pure vapor.
Model the local rate for condensation of a mixture of vapor and an inert gas.
Identify the conditions for fog formation in general and in condensing equipment in particular.
Model the heat and mass transfer processes in the condensation of a binary mixture of two condensing species.
Analyze the condensation of a ternary mixture consisting of two condensing species with an non-condensing gas.
Write equations for calculation of the performance of condensing equipment based on a stagewise description of the condenser.
Condensation refers to a process where a species (or multiple species) from a vapor phase is transported to a cold surface or cold liquid and condenses and forms a liquid phase (usually called the condensate) at the surface. The direction of mass transfer is from the vapor to the liquid. This is the opposite of drying, where the mass transfer is from the liquid to the vapor phase. A temperature difference between the vapor and the condensing phase is needed for the process to occur. Hence a simultaneous mass and heat transfer from the vapor to the liquid is involved. Thus condensation represents another example of a system where an analysis of simultaneous heat and mass transfer is required.
Various types of equipment can be used for condensation. Commonly a shell and tube type of arrangement is used similar to that for heat exchangers. The vapor is usually on the shell side with the coolant flowing on the tube side. Either vertical or horizontal arrangements can be used. In both cases, it is necessary to find the local rate of condensation at any point in the system. It is the goal of this chapter to examine the local mass and heat transfer effects and calculate the rate of condensation for a given vapor composition and for a given condition of the condensing film or the surface. The local model can then be incorporated as a submodel in the design of the equipment. Thus the basic learning objective is to understand the effects of simultaneous heat and mass transfer in a condensing system and use these to calculate the rate of condensation for a given condition. A simple example of coupling the local model to the equipment-level model is given, but the complete design of condensation equipment is not considered here.
Four cases are analyzed here: condensation of a pure vapor, condensation of a vapor and inert gas (non-condensible) mixture, condensation of a binary gas mixture, and a ternary system with two condensing species and an inert gas. The last case is modeled best by Stefan-Maxwell relations.
A related phenomena is fog formation, which is important per se in environmental applications. It has also been observed in industrial condensing equipment and we briefly discuss the role of mass transfer in fog formation based on the film model and examine conditions under which fog formation can be expected during the condensation of a vapor with an inert non-condensing component.
26.1 Condensation of Pure Vapor
This section is more on the heat transfer aspects of condensation and does not include mass transfer considerations since a pure vapor is involved. The condensing temperature depends only on the pressure and is treated as a constant.
Condensation can be of two types: dropwise or filmwise. Dropwise condensation refers to a process where the condensed vapor forms drops on the surface rather than a continuous film. Since a large portion of the surface is exposed to the gas, the condensation rates are rather high. Heat transfer coefficients as high as 250,000 W/m2 can be achieved. However, there are two difficulties in using this effectively: it is difficult to maintain for a sustained period of time and the resistance on the coolant side is often the controlling resistance, which means the heat transfer rate is then determined by how fast the heat can be removed on the coolant side. The high value of the heat transfer coefficient from vapor to drop no longer matters!
In filmwise condensation the surface over which condensation occurs is blanketed by a layer of liquid film. This liquid film provides a resistance to heat transfer and therefore the heat transfer coefficients are much lower compared to dropwise condensation. The film thickness also increases as a function of the height or the distance along the condensing surface. Hence the heat transfer coefficient decreases along the height. In practical applications the filmwise condensation is usually encountered due to difficulties in maintaining the dropwise condensation for a prolonged period of operation. Further discussion in this section pertains to filmwise condensation and dropwise condensation is not discussed.
Condensate forms a film on the condensing surface and the liquid flows down as a thin film. It is useful to characterize the various flow patterns of film flow at this stage to give students a clearer understanding of the range of applications of the various correlations and/or models that are commonly used in practice. The flow regimes are laminar, wavy, and turbulent. The transition from laminar to wavy is around a Reynolds number (defined for film flow as Re = 4 〈v〉 δ/νL) of 30 while the flow is turbulent beyond a Reynolds number of 1500. Note: The Reynolds number for film is usually written as 4Γ/μL, where Γ is the mass flow rate per unit perimeter of the condensing surface (since Γ = δ 〈v〉 ρL per unit perimeter).
26.1.1 Laminar Regime: Nusselt Model
The problem of laminar condensation was analyzed by Nusselt (1916). The schematic basis of the model is shown in Figure 26.1 where a condensate film of thickness δ is assumed to form on the condensing wall.
Figure 26.1 Schematic for the model used in Nusselt analysis for filmwise condensation in a laminar falling film.
The film thickness increases as a function of x, the vertical distance along the wall. The local heat transfer coefficient, hx, depends on the local film thickness and can be calculated as
where kl is the thermal conductivity of the liquid.
The film thickness can be calculated using the mass and momentum balance if the operating conditions are in laminar mode:
The model is called the Nusselt model for condensation. Details leading to this equation are left as an exercise problem. Note that ρL – ρv is usually approximated as ρL. This approximation is used from here forward.
Hence the local heat transfer coefficient is given as
The average rate over a length of a tube L is given as
Integrating, the following relation can be derived:
The equation is valid up to a Reynolds number of 30. (Note: The factor (4/3)/41/4 arises from the integration and has a value of 0.943, which is the leading constant in the above equation for hL.)
The condensate flow rate (per unit perimeter) at the end of the tube is of interest and can be calculated from an overall heat balance:
Γ = hL(Tsat – Tw)L/ĥgl
The Reynolds number, 4Γ/μL, corresponding to this has to be calculated in order to check if the flow is laminar and has to be less than 30. The flow becomes wavy if the Reynolds number is larger than 30.
Corrections for Subcooling and Superheat
The Nusselt model described above does not account for the subcooling of the liquid. A correction is usually applied. The latent heat of vaporization term, ĥgl, in Equation 26.2 is usually modified to include the liquid subcooling and is replaced by , which is defined as
If the vapor enters as a superheated fluid at a temperature of Tv instead of at Tsat, a further correction to the latent heat is applied:
26.1.2 Wavy and Turbulent Regime
The wavy regime occurs for Re > 30. The effect of waves on the surface is to increase the rate of condensation. However, a detailed model-based analysis is complicated and analytical solutions are not possible unlike the case of laminar flow. The increase in heat transfer is on the order of 20% but can be as high as 50%. The laminar equation can still be used for a conservative design as it will underpredict the heat transfer coefficient. A correction factor of 0.8Re0.11 was recommended by McAdams (1954). The vapor velocity is assumed to small in the Nusselt analysis and its effect is neglected.
Turbulent Regime
For the turbulent regime, empirical correlations are commonly used. A simple correlation of McAdams (1954) is often used:
The local and average heat transfer coefficients are the same in turbulent films. Wavy and turbulent regimes can be modeled in detail by using computational fluid dynamic tools.
26.2 Condensation of a Vapor with a Non-Condensible Gas
In this section we consider the condensation of a species present in the gas along with an inert non-condensing gas. The species has to diffuse from the bulk gas to the interface and hence mass transfer considerations are to be included in addition to heat transfer models. The film model is widely used for local mass and heat transfer analysis. The species mole fraction and temperature profiles are shown in Figure 26.2. The mole fraction varies only in the vapor film since the condensate is pure liquid. The temperature varies in both the vapor film and liquid and also on the coolant side as shown in Figure 26.2.
Figure 26.2 Film model for condensation showing the mole fraction profile in the vapor film and the temperature profiles in the liquid and the coolant. Adapted from Thermopedia, A–Z Index.
26.2.1 Mass Transfer Rate
The mass transfer in the vapor film is modeled as a UMD (unimolecular diffusion) process and the rate of mass transport is given as
yAg is the local mole fraction in the bulk gas. yAi is the mole fraction at the interface and will depend on the interface temperature, which has to be calculated based on heat transfer consideration. The equation for the calculation of the interface temperature will be developed shortly in this section.
26.2.2 Heat Transfer Rate and Ackermann Correction Factor
The heat transfer rate is enhanced by mass transfer due to diffusion-induced convection. An augmentation factor ø is used to calculate the enhancement. Hence the heat transfer rate in the vapor film is
where ϕ is the blowing factor caused by mass transfer and is defined as
The heat transfer rate from the bulk gas to the interface is often written in the literature in the following alternate form:
The first term on the right-hand side accounts for the conduction part of the flux (now enhanced due to mass transfer) while the second term accounts for the sensible heat change across the film. The factor εh is the augmentation factor for the (film) conductive part of the flux only and not for the total heat transfer. This factor is called the Ackermann correction factor. Comparing the two expressions, Equations 26.8 and 26.10, we find that
The use of this factor and the use of Equation 26.10 is more common in some of the older works on condensation heat transfer. The preceding discussion shows that the two formulations are entirely equivalent. Either factor can be used.
The limiting cases of the Ackermann correction factor should also be noted:
For low ϕ, εh equals one and the heat transfer coefficient is not enhanced by mass transport.
For large ϕ, ε h equals zero and the heat transfer rate depends only on the sensible heat change from Tg to Ti.
26.2.3 Interface Temperature Calculations
The condensation equipment design requires the estimation of the local mass flux, NA, and the local rate of heat transport in the vapor film, qv. The value for the interface temperature is required; this is determined by a heat balance at the interface.
The total heat released at the interface is equal to the rate of heat transport in the gas film plus the energy released at the interface due to the latent heat of condensation. The latter depends on the rate of mass transfer, the rate at which condensation is taking place. The total heat released is therefore equal to
where ΔHc is the molar latent heat of condensation. A positive value for ĥgl is implied here.
The heat transferred at the interface is then equal to the heat transfer in the liquid to the walls of the condenser:
Hence the interfacial temperature is determined by equating the two equations for qi:
where hL is the heat transfer coefficient in the liquid film.
Again the wall temperature may not be known and only the temperature of the cooling liquid may be known. In this case Equation 26.12 is replaced by
where Ui is the overall heat transfer coefficient from the interface to the cooling liquid and Tc is the (local) temperature of the coolant at the point under consideration.
Sensible Heat Correction
The analysis does not include the sensible heat change of the condensed liquid from Ti to Twall. An additional term has to be added to account for this. The term can be neglected as a first approximation. More precisely, the calculation of the sensible heat loss requires the calculation of the average temperature of the liquid film. A simple way to account for this is to augment the value of the latent heat of vaporization by adding a correction term as shown below:
ΔHc = Hc + (3/8)CpL(Ti – Twall)
The augmented value, ΔHc, is then used in place of ΔHc to account for the sensible heat change in the liquid. The factor 3/8 can be viewed as the contribution from the average temperature of the liquid. The condensation rate calculation is illustrated in Example 26.1
Example 26.1 Rate of Condensation from a Water + Air Mixture
Water is condensing on a surface at 310 K. The gas mixture has 65% water vapor and is at a temperature of 370 K. The total pressure is 1 atm. Calculate the rate of condensation.
Use the following data for the physical properties: specific heat of water vapor Cpg,L = 37.06 J/mol K; specific heat of liquid water = 75.13 J/mol K; heat of vaporization = 43,000 J/mol.
The Antoine equation constants values for water are A = 18.3036, B = 3816.44, and C = 46.13 with temperature in °C.
Use the following values for the transport parameters: gas-side heat transfer coefficient = 12 W/m2K; gas-side mass transfer coefficient = 0.1 m/s; liquid-side heat transfer coefficient = 400 W/m2K.
Note that the liquid side mass transfer coefficient is not needed since the liquid is a pure component.
Solution
The interface temperature is dictated by the relative rates of heat and mass transfer. This is unknown. Once the interface temperature is found the problem is essentially solved. Hence the solution procedure is simply solving for the interface temperature in an iterative manner. The steps are outlined here.
We start with an assumed interface temperature, Ti. Let Ti = 314 K. The vapor pressure at the interface is calculated using the Antoine equation. Note that the units are in mm Hg and need to be converted to pascals. Pvap = 58.325 mm Hg. This sets the mole fraction of water at the gas side of the interface: yi = pvap/P = 58.325/760 = 0.0768.
The mass transfer rate across the gas film can now be calculated using Equation 26.7. The total concentration needed here is calculated using the ideal gas law at an average film temperature: Tf = (Ti + Tg)/2 = 342 K.
The total concentration based on the bulk gas temperature would be 32.93. Hence no significant errors would be expected in using any of these values for the temperature. Now using Equation 26.7, we have NA = 0.2979 mol/s.m2.
The heat transfer to the interface can now be calculated using Equations 26.9 and 26.8. The following values are found; students should verify these: Peclet factor, , and finally .
The rate of total heat release at the interface can now be calculated. This is the sum of the rate of heat transfer to the interface plus the heat released by the latent heat of condensation. The latter is proportional to the mass transfer rate: q(total) = 1698 W/m2.
The heat released at the interface is equal to the heat transferred across the liquid film to the cold walls: q(total) = hL(Ti – Twall).
The interface temperature can therefore be calculated as
The calculated value is 314.2 K, which differs from the assumed value by only 0.2 °C. The calculations can be repeated if needed if the initial guess of the interface temperature is different from the calculated value.
Note: The effect of subcooling of the liquid was neglected here. It can be shown that it adds only an additional 2% to the total heat flux at the interface and hence does not introduce significant errors in the interface temperature.
26.2.4 Condenser Model
The local film transport model can be incorporated into the models for the vapor phase, molar flow rate in the liquid, and coolant temperature. It is easier to set up the mass and energy balance in finite difference models for each differential section of the condenser and solve these sequentially. The model equations are similar to those for humidification shown in Chapter 25. An additional equation for coolant temperature is included. The model equations are presented in Section 26.5 in a more general setting. It is fairly straightforward to set up a MATLAB program to use these equations and march in the z-direction. A local model to calculate NA, qv (which in turn needs Ti) and qi is set up separately and used in the marching scheme.
26.3 Fog Formation
What leads to fog formation? The vapor can cool below the dew point temperature in the film, leading to supersaturation. The vapor then condenses as tiny mist or fog rather than condensing into existing liquid film. This leads to fog formation. Thus if a hot vapor cools faster than the rate of mass transfer and maintains its humidity, it can start condensing in the vapor phase itself, leading to a mist or fog. The key parameter is the Lewis number, Le, the ratio of thermal diffusivity to mass diffusivity. If Le > 1, vapor cools faster than the rate of mass transfer and fog formation can occur.
The temperature and concentration during the process of cooling of a vapor–inert gas mixture is shown in Figure 26.3.
Figure 26.3 Concentration and temperature profiles during cooling of a vapor, indicating the conditions for fog formation.
Three possibilities can arise: the pA versus T values in the vapor film can touch the equilibrium line tangentially, which occurs for Le = 1; the line can be beneath the equilibrium curve for Le < 1 and no fog formation can be expected; and the curve can intersect and cross the equilibrium line before the interface point is reached. In this case there is a region where the vapor is supersaturated; this can lead to fog formation in the vapor film near the interface. This happens for Le > 1. For example, in condensing organic vapors from air, the Schmidt number is in general larger than the Prandtl number. This leads to Le > 1; the cooling proceeds faster than condensation, leading to the danger of fog formation.
For Le < 1 the local temperature in the film is sufficiently high to keep y below the corresponding saturation value and fog formation does not occur. For example for an air–water system, Le is 0.87 and fog formation is not likely in condensing such a mixture.
It is also useful to look at this from a film theory point of view. Recall from Chapter 25 that the Lewis number (Le) is a dimensionless number defined as the ratio of thermal diffusivity to mass diffusivity. Hence the thermal film thickness is larger than the mass transfer film thickness if the Lewis number is greater than one. The expected concentration profile is shown in Figure 26.4.
Figure 26.4 Mole fraction and temperature profile in the film for Le > 1 and Le < 1. Note the film thicknesses for heat and mass transfer are not equal when Le ≠ 1. b denotes the bulk gas and s denotes the liquid–vapor interface. Note that the temperature profile is shown assuming no enhancement in heat transfer due to mass transfer.
For Le > 1, the temperature profile in the film would be lower than that for the case of Le = 1. The vapor mole fraction at some point in the film can therefore exceed the saturation mole fraction. This can cause a local supersaturation and lead to local condensation in the gas phase as fog. Note that an excursion into the supersaturation zone does not necessarily imply fog formation. A critical supersaturation ratio has to be reached and the presence of nucleating aerosols may be needed. Thus fog formation is less likely in clean gases compared to a dusty mixture. The fine dust particles act as nucleation sites.
The most comprehensive work on fog formation in condensers is Amelin (1967) in which process calculations, nucleation, and the droplet growth of pure substances are described. Steinmeyer (1972) elucidated the industrial significance of fog formation in condensers and gave rules for minimizing fog problems. Brouwers (1992) presented detailed models.
26.4 Condensation of Binary Gas Mixture
The problem of condensation from a binary mixture of species A and B is considered next. Both species are assumed to be condensable. An example would be the condensation of a vapor leaving the top of a distillation column.
The mass transport part in the vapor film is now solved as a two-component diffusion problem. Thus both NA and NB need to be calculated. The liquid phase composition at the vapor–liquid interface is needed for this calculation (in addition to an estimate of the interfacial temperature). This is often fixed by the relative rates of condensation. Thus
The mole fraction of B is equal to 1 – xAi for a binary system.
This assumption is often referred to the unmixed film model assumption (Colburn and Drew, 1937; Taylor and Krishna, 1993) and implies a large resistance to mass transfer on the liquid side or low condensation rate. More details leading to the assumption of the unmixed model are provided in Section 26.5. The model is useful for the horizontal type of condenser where the condensate is formed from the vapor phase and there are no additional flow terms that cause concentration changes in the bulk liquid. This is also valid at the top of a vertical condenser where the downward liquid flow rate is small. Since the interfacial concentration is directly related to the condensation rates, the values of liquid-side mass transfer coefficient are not needed when using this assumption.
An alternative assumption is called the completely mixed film. Here xAi is taken as the bulk liquid value xAL and bulk liquid concentrations are specified or calculated from the overall mass balance in the bulk liquid. This would imply no resistance on the liquid side for mass transfer. Again the values of liquid-side mass transfer coefficients are not needed. We proceed with the unmixed film assumption here and derive an equation for the calculation of the condensation rates from equilibrium considerations at the interface.
26.4.1 Condensation Rates: Unmixed Model
The calculations start off by assuming an interface temperature. This is needed to assign the equilibrium composition at the interface and complete the mass flux calculations. The assumed interface temperature is then iteratively corrected in order to satisfy the heat balance requirement.
First the K values are calculated from the vapor pressure data based on the (assumed) interface temperature. The vapor composition at the interface is related to the interfacial liquid concentration by a suitable equilibrium relation. Raoult’s model is used here:
This is valid for ideal solutions while for non-ideal systems an activity coefficient correction is used:
yAi = γAKAxAi
where γA are the activity coefficients, which are a function of liquid composition. A similar equation holds for B. The activity coefficients are usually calculated by the Margules equation with the needed parameters obtained from Gmehling and Onken (1984) or other data sources. For more details on γ, standard texts on thermodynamics (e.g., Smith et al., 2005; Sandler, 2006) should be consulted. We use the ideal solution for our discussion.
Applying Equation 26.15 to species B we have
The sum of the mole fraction is equal to one. Hence
KAxAi + KBxBi = 1
Since xBi is equal to 1 – xAi this expression reduces to
All the compositions are known now.
The mass transfer model discussed in Section 6.3 holds where the effect of diffusion-induced convection is also included. This section should be reviewed at this point. The relevant equation is Equation 6.34 for the flux of A; there is a similar equation for the flux of B. If these are added we get the following equation for the total flux:
where yAi is equal to the equilibrium value KAxAi. Here is the dimensionless total flux, defined by .
The individual fluxes can now be calculated as xAiNt and xBiNt on the basis of the unmixed model in Equation 26.14.
Example 26.2 illustrates the calculation of the condensation fluxes through a simple example where we assume that the interface temperature is known.
Example 26.2 Rate of Condensation of a Methanol + Water Mixture
A mixture of methanol and water at 90 °C containing 40% by mole of methanol is in contact with a cold wall. Find the rate of condensation and the composition of the condensate formed at this point. Use the unmixed model for the liquid composition. Assume an interface temperature of 85 °C. Use the following data (methanol = 1; water = 2): specific heat of components in the gas phase: Cpg1 = 45 J/mol.K, Cpg2 = 34 J/mol.K; heat of condensation: ΔHgl,1 = 36000 J/mol, ΔHgl,2 = 43000 J/mol. We have the following Antoine constants with pressure in pascals and temperature in K: for methanol, A1 = 23.402, B1 = 3593.4, and C1 = –34.92; for water, A2 = 23.196, B2 = 3816.4, and C2 = –46.13. The gas-side heat transfer coefficient is 60 W/m2K. The gas-side mass transfer coefficient is 0.08 m/s.
Solution
The liquid-side composition at the interface based on the assumption of the unmixed model is proportional to the condensation rate. Hence
x1 = N1/Nt
A similar relation applies for x2.
Using the Antoine equation, the vapor pressures are computed as follows: vapor pressure of methanol = 2.153 bars; vapor pressure of water = 0.575 bars.
The corresponding K values are K1= 2.1528 and K2 = 0.5745.
The liquid composition at the interface can now be calculated using Equation 26.17. The values are x1i = 0.27786 and x2i = 1 – x1i = 0.7221.
The composition in the vapor at the interface is calculated using Raoult’s law and an ideal solution: for methanol, y1 = K1x1 = 0.5904 and for water y2 = K2x2 or 1 – y1 = 0.4096.
All the quantities needed to find the total flux are known. Thus using Equation 26.18 we have .
The individual fluxes in dimensionless units are and .
The actual fluxes are calculated by multiplying by kmC the scaling factor. Here C is the total concentration, calculated as 34.75 mol/m3. This is based on the average temperature of the gas film.
The species fluxes are now calculated: for methanol we have 0.7057 mol/m2 s, and for water we have 1.8344 mol/m2 s.
Note that the methanol flux is lower as expected. Methanol accumulates at the vapor side of the interface. The mole fraction is close to 0.6, compared to that in the vapor of 0.4. Water accumulates on the liquid side of the interface. The mole fraction at the interface is 0.4 compared to 0.6 in the bulk.
26.4.2 Calculation of the Interface Temperature
The equation for the calculation of the interfacial temperature is similar to that for a single condensing case. The heat of condensation for both A and B is now included in the interfacial heat balance; the equation is
The correction factor for heat transfer augmentation is based on ϕ, which in turn now depends on both NA and NB now:
Equation 26.19 assumes that the wall temperature is known. If only the coolant temperature (Tc) is known rather than the condensing wall temperature, the last term in that equation is replaced by U(Ti – Tc), where U is the overall (condensate side + condensing wall) heat transfer coefficient. The relation 1/U is equal to 1/hL + 1/hc is used to get U, where hc is the heat transfer coefficient on the coolant side.
The maximum and minimum values of the interfacial temperature are the dew point and the bubble point of the binary gas mixture at the bulk gas composition. Hence these should be estimated and the trial value should be chosen within this range.
This completes the local model for the condensation rate. The local model predicts the mass flux of both components and the quantity of heat to be removed from the system at a given location in the condenser. This model is then incorporated into a global model for the condenser and used for design or performance evaluation of the equipment. The global model equations to be solved are presented in the next section.
26.5 Condenser Model
A model for a condenser with a binary mixture of two condensible gases is presented in this section. This provides the equations needed to simulate the condensation equipment. Also, the material balance equations on the liquid indicate more clearly the basis of the two limiting assumptions of mixed and unmixed liquid.
A condenser is divided into a number of mixing cells connected in series. Each cell is referred to here as a stage. A schematic diagram is shown in Figure 26.5. This forms the basis for the model equations shown next.
26.5.1 Liquid and Vapor Phase Balances
The balance in the liquid for species A for any stage s leads to
where we use the following notations:
Total liquid molar flow rate entering the sth stage of the condenser, Ls–1
Mole fraction of A in the liquid entering the jth stage, xA,s–1
Perimeter of the condensing surface, P
Length element for the stage, Δz
Flux of A entering the bulk liquid, which is equal to the local condensation rate, NA
Total liquid molar flow rate leaving the sth stage of the condenser, Ls
Mole fraction of A in the liquid leaving the sth stage, xA,s
The total mass balance is used to relate Ls and Ls–1:
where ns is the number of components condensing and Nj is the condensing flux of the jth component. Using this in Equation 26.21 we have
which can be rearranged to
The two limiting cases examined in the literature arise out of some simplifying assumptions based on this general mole balance.
Unmixed Model
The first assumption is when Ls–1 is small and the term on the left-hand side can be neglected. This leads to
This is referred to as the completely unmixed model by Taylor et al. (1986) and other sources. Thus the unmixed model applies when there is no flow contribution to the mole balance in the liquid or when the flow contribution is small.
Mixed Model
A second limiting case occurs if the flux entering the bulk liquid is small compared to the flow rate of liquid entering the stage. Here we write Equation 26.21 in terms of component flow rate as
where l is the component flow rate, defined as
lA,s–1 = Ls–1xA,s–1
If the flow rate is large compared to mass transfer rate the second term on the left-hand side of Equation 26.26 is dropped and we have
The composition in the liquid is now fixed by the flow rate considerations and not by the rate of condensation. This is the basis for the mixed liquid model used in the condensation literature. The term “mixed” should not be interpreted as a completely backmixed model and implies that the flow from the previous stage gets mixed with the condensate and provides the composition for the next stage.
Equation 26.23 or 26.24 is more general and includes both limiting cases. The equations are supplemented by the vapor balance and the energy balance.
Vapor Phase Mass Balance
The vapor balance is as follows and is similar to the liquid balance. For species balances A, B, C, and so on, the following balance equation can be derived:
Here yA is the mole fraction in the bulk vapor. V is the total vapor molar flow with s denoting the stage number.
Energy Balances
The energy balance for the vapor phase is
The following equation holds for the energy balance for the liquid:
Here qi is the heat transferred to the liquid at the interface, which is given as
qw is the heat flux across the tube wall to the coolant.
Coolant energy balance leads to the following relation:
The plus sign is used if the coolant is in cocurrent flow with the vapor; the minus sign is used otherwise.
qw is given as
qw = hc(Tw – Tc)
Here Tw is the wall temperature, which is given as
hL(TL – Tw) = hc(Tw – Tc)
It is fairly straightforward to set up a MATLAB program to use these equations and march in the z-direction. A local model to calculate NA, qv (which in turn needs Ti), and qi is set up separately and used in the marching scheme. It may be also noted that the stagewise model in this section is equivalent to an implicit Euler method applied to a differential model for the condenser. Taylor et al. (1986) and Furno et al. (1986) also provide more details on the numerical aspects of the model and provide illustrative results.
26.6 Ternary Systems
Binary condensation in the presence of an inert gas is of importance in many practical applications. This is a ternary system for mass transfer modeling. Sardesai and Webb (1982) provide experimental data for a number of systems. The modeling of such a system can be completed using the Stefan-Maxwell equations as shown by Krishna and Panchal (1977). In this section we summarize the model equations and also show an interesting example where a chemical reaction occurs at the interface.
26.6.1 Stefan-Maxwell Model
The mass transfer of the species in the gas film is modeled by the Stefan-Maxwell model studied in Chapter 15. Note that this is an inverted model and the mole fractions are the explicit variables. The following equation applies for species i with i = 1 to 3 for a ternary system. Extension to systems with more than three components is straightforward:
The boundary conditions are known in the bulk gas. The boundary condition at the interface has to be set up. The mole fraction in the liquid is needed for this purpose. The unmixed model is commonly used for species 1 and 2, which are assumed to be condensable species. (Species 3 is the inert non-condensable component of the mixture.)
The mole fraction in the liquid side of the interface is therefore equal to Ni/(N1 + N2). The corresponding mole fraction in the vapor near the interface is given as Kixi. Hence yi,s = KiNi/(N1 + N2) where the subscript s is used for the interface.
The determinacy conditions have to be used as an additional equation. Species 3 is non-condensing and hence N3 is set to zero. The zero divergence of N1 and N2 provides the additional equations. The computational procedure shown in Chapter 15 can be used to solve for both the mole fraction profiles in the vapor film as well as the condensation rate. In particular direct solution using BVP4C is quite useful. Details are left as an exercise problem.
26.6.2 Condensation with Reaction
In some cases a chemical reaction of the condensed species can occur in the liquid film. A model for this was developed by Ramaswamy and Ramachandran (2013) for the reaction scheme:
B + C ⇌ A
where all the species are present in the vapor. C (species 3) is treated as a non-condensing species while A and B are condensing. A three-component Stefan-Maxwell model was used for the vapor film.
Species C is assumed to react instantaneously at the vapor–liquid interface. Hence N3 is not set to zero but is set to ksCys,3. The rate of reaction of C at the interface produces a flux condition at the interface. An industrially important example is the condensation of a gas mixture consisting of acetic acid (B), ketene (C), and acetic anhydride (A). In this application, the reaction of ketene during the condensation process has to be reduced since the ketene has to be recovered as the vapor phase product. The models provided some basis for the design of condensers to mitigate the ketene loss due to reaction.
Summary
For problems with simultaneous heat and mass transfer, the convection due to mass transfer can cause a flow and thereby provide an additional mode for heat transfer. The heat transfer rate is then enhanced (or retarded if mass transfer is in opposite direction to heat transfer) due to mass transfer. An augmentation factor can be derived and used to predict these effects. Such models are useful in the design of condensation equipment and in many other applications.
For a vapor consisting of a pure component, the temperature is equal to the dew point, which is constant. There are no mass transfer limitations and the process is governed by the rate of heat transfer in the condensing liquid film.
Condensation can be classified as dropwise or filmwise. Heat transfer coefficients are an order of magnitude higher in dropwise condensation. However, industrial applications usually operate in the filmwise region since dropwise condensation is difficult to maintain.
The liquid flow pattern in filmwise condensation can be classified into a laminar, wavy, or turbulent regime depending on the range of the Reynolds number. For laminar flow a theoretical analysis of Nusselt is useful to predict the heat transfer coefficient. For wavy and turbulent flow, empirical correlations or CFD models are useful.
Practical problems involve condensation of a vapor in the presence of a non-condensible vapor or simultaneous condensation of two components, for example, in the condenser in a distillation column. Heat transfer considerations alone are not sufficient to design these systems and a study of simultaneous heat and mass transfer is needed.
The local model for the heat and mass transfer rate can be derived on the basis of the film model for a vapor plus an inert gas mixture. The mass transfer rate is modeled as a UMD model. This includes diffusion-induced convection. The heat transfer rate is also enhanced due to diffusion-induced convection. An Ackermann correction factor is used to calculate this enhancement.
The film thicknesses for heat and mass transfer are generally assumed to be equal to each other. However, this is valid only if the Lewis number is close to one. For organic vapors the Lewis number can be greater than one. This can cause a faster rate of cooling compared to the rate of mass transfer, causing supersaturation in the film. This can lead to fog formation in the condensing equipment, which is undesirable. Models using different film thickness values can provide conditions leading to fog formation and can also suggest preventive strategies.
Condensation of two components follows a similar methodology to that for a single component. But since the liquid composition depends on the relative rate of condensation, some assumptions are needed in order to fix the local liquid composition. Two assumptions are that the liquid is well mixed and the liquid is segregated.
Models for condensing equipment can be set up using mass and heat balances over a differential control volume. If the control volume is discretized by an implicit finite difference, a stagewise model is obtained, which is easier to solve. The model equations are summarized in Section 26.5.
The condensation of a ternary mixture can be modeled along similar lines as the binary case. The mass transfer part of the model is formulated now as a multicomponent diffusion problem and the Stefan-Maxwell model described in Chapter 15 can be used to compute the mass flux of all the condensing components. Again the assumption of an unmixed liquid or completely mixed liquid is needed to set the interface vapor mole fractions. The unmixed assumption is simpler to use. Calculation of the interface temperature is done by a heat balance similar to the binary case. The Ackermann correction is used to calculate enhancement in heat transfer due to mass transfer.
Review Questions
26.1 What is meant by filmwise condensation?
26.2 What is dropwise condensation?
26.3 Which type of condensation is common in industry?
26.4 What is the basis for the Nusselt theory of condensation?
26.5 How does the local heat transfer coefficient in a liquid film vary with distance along a vertical condensing surface?
26.6 What is meant by modified latent heat of condensation? Where is it used?
26.7 Explain why heat transfer rate can be enhanced by simultaneous mass transfer.
26.8 What is the augmentation factor for heat transfer?
26.9 What is the Ackermann correction factor?
26.10 State the equations to calculate the rate of mass transfer to condense a vapor with some non-condensable gas.
26.11 State the equations to calculate the rate of heat transfer in the vapor film.
26.12 How is the interface temperature calculated?
26.13 State the conditions that can lead to fog formation.
26.14 If Le > 1, which film (heat or mass) is thicker?
26.15 What would be the design implications if fog forms in the vapor space of the condenser?
26.16 What additional assumptions are needed to model condensation of a mixture of two condensable gases?
26.17 How does the assumption of an unmixed liquid fix the liquid composition for condensation of two components?
Problems
26.1 Derivation of the Nusselt model for condensation. Consider a vapor condensing on a wall and forming a liquid film. Show that the change in the mass flow rate per unit perimeter, Γ, is related to the film thickness as per the following equation:
From a heat balance show that the change in mass flow rate is given as
Equate the two expressions for the change in Δand derive an expression for dδ/dx. Integrate this expression and derive the expression given by Equation 26.2 for δ as a function of x. Find an expression for the local heat transfer coefficient and verify that it is proportional to x–1/4. Find an expression for the average heat transfer coefficient for a wall of height L by integration of the local value and hence Equation 26.5.
26.2 Flow regimes in a condenser. Saturated steam at 356 K condenses on a 5 cm vertical tube whose surface is maintained at 340 K. Find the height at which the flow becomes wavy. Calculate and plot the flim thickness and the heat transfer coefficient up to this height and also find the average heat transfer coefficient. At what height would the film become turbulent?
26.3 Condensation of steam: design for tube length. Saturated steam at 55 °C is to be condensed at a rate of 10 kg/hr on the outside of a 3 cm diameter vertical tube by maintaining the surface at 45 °C. What is the operating pressure? Find the tube length required for this purpose.
26.4 Condensation of pure ethanol. Ethanol is condensing at 1 atm pressure on the outside of a tube that is 3 m long with a 2.5 cm diameter. The tube is maintained at 25 °C by circulating cooling water inside the tube. Calculate the quantity of ethanol condensed.
26.5 Concentration and temperature profiles in the vapor film. Warm air at 90% humidity at 5 °C is diffusing into cold air at 5 °C and 70% humidity. Sketch the compositions of the y – T diagram for Le = 0.5, Le = 1.0, and Le = 2. Indicate if fog formation is likely.
26.6 Condensation of a vapor with a non-condensable gas. Benzene vapor with 25% benzene and the rest air is condensing on a tube at a temperature of 300 K. Find the rate of condensation. Use hg = 10 W/m2 K for the heat transfer coefficient and km = 0.08 m/s for the mass transfer coefficient.
26.7 Condensation of two components. Methanol and water are condensing on a tube of diameter of 25 mm with a methanol content of 50%. The vapor inlet temperature is 360 K. The coolant temperature at the top is 308 K with a coolant flow rate of 0.06 kg/s. Estimate the interface temperature and the condensation rate of methanol and water at the top of the condenser. Assume the unmixed case. The calculations should be put in an m-file or subroutine for use in the next part of the problem.
26.8 Simulation of a condenser for methanol-water mixture. Simulate the condenser for the data in previous example. Use the stagewise model shown in Section 26.5.
26.9 Condensation of two components with a third inert gas. Krishna and Panchal (1977) simulated the condensation of a methanol–water system in the presence of air. The composition at the inlet was 0.7, 0.2, and 0.1 for methanol, water, and air, respectively. Assume as a interface temperature of 310 K. Set up the model for the mole fraction of the species as a function of distance along the diffusion film using the Stefan-Maxwell model. Find the flux of the components. Use a film thickness of 2 mm. Use the unmixed model for the ratio of fluxes.