Engineering task and calculation objective
This module calculates direct-contact (spray and jet) condensation according to Section J4 of the VDI Heat Atlas (VDI-Wärmeatlas, 12th edition 2019), the standard German reference work for heat transfer. Here the vapor condenses not on a cooled wall but directly on injected subcooled liquid — on compact jets, liquid sheets, or atomized droplets. Because no transfer wall with film resistance and fouling stands in the way, direct-contact condensers achieve very high volumetric duties with a simple, robust construction.
Direct-contact condensation is used in barometric or spray condensers of vacuum plants, in desuperheaters, in vapor knock-down duty, and in safety systems (condensation chambers). The design answers the question of how far the cooling medium heats up on contact with the vapor: for this, the module calculates dimensionless heating and outlet temperatures for jets and droplets via Stanton numbers, residence times, and Fourier numbers, as well as the droplet size distribution of the spray.
The result is the heating of the cooling water between inlet and outlet relative to the saturation temperature of the core flow — from which the required cooling water flow and the condensation duty of the apparatus follow.
Standard and calculation basis: VDI-Wärmeatlas, 12. Auflage 2019
Calculation workflow
- Choose the condensation mode and geometry: First, it is established whether the vapor condenses on compact jets (round or plane) or on a droplet spray; the governing geometric quantities are the jet length, nozzle diameter or slot width, and the surface-to-cross-section ratio F/A of the jet.
- Calculate the heating number of the jet: For the jet, the dimensionless heating number is formed from the inlet and outlet temperatures of the cooling medium and the saturation temperature of the core flow; via the Stanton number, which characterizes the heat transfer at the jet surface, the dimensionless outlet temperature follows as a function of run length and F/A.
- Determine the droplet heating via the Fourier number: For spray condensation, the transient heating of the individual droplets is calculated: from the droplet radius, residence time, thermal diffusivity, and the relative velocity between droplet and vapor, the Fourier number is obtained, and from it the mean temperature of the droplet at impact or outlet.
- Account for the droplet size distribution: Since real sprays are polydisperse, the maximum droplet radius, distribution parameters, and radial volume distribution are used; the heating is averaged over the distribution, because large droplets heat through much more slowly than small ones.
- Balance the cooling water demand and duty: From the achieved heating of the cooling medium and the energy balance (condensation enthalpy of the condensed vapor equals the enthalpy uptake of the cooling water), the required cooling water mass flow and the condensation duty of the apparatus follow.
Input quantities
| Quantity | Symbol | Unit |
|---|---|---|
| Outlet temperature (cooling medium) | TAus | °C |
| Inlet temperature (cooling medium) | TEin | °C |
| Saturated temperature (core flow) | TS | °C |
| Dimensionless temperature Eq. (4) | Θ | – |
| Heat transfer coefficient (liquid) | α | W/(m²·K) |
| Density of liquid | ρl | kg/m³ |
| Specific heat capacity (liquid) | cp | J/(kg·K) |
| Velocity of cooling water | w | m/s |
| Stanton-number Eq. (5) | St | – |
| Jet length | L | m |
| Diameter / slit width | d | m |
| Ratio F/A (circular jet) | F/A | – |
| Ratio F/A (flat jet) | F/A | – |
| Ratio F/A Eq. (6) | F/A | – |
| Dimensionless outlet temperature Eq. (6) | Θ | – |
| Stanton-number Eq. (9) (estimation) | St | – |
| Temperature for Eq. (10) | T | °C |
| Dimensionless outlet temperature Eq. (10) | Θ | – |
| Thermal conductivity | λ | W/(m·K) |
| Residence time | t | s |
| Droplet radius | r | mm |
| Fourier-number Eq. (12) | Fo | – |
| Mean temperature Eq. (13) | Θm | – |
| Value of velocity difference | v | m/s |
Frequently asked questions
When is a direct-contact condenser preferable to a surface condenser?
When mixing of condensate and cooling medium is permissible — for example with steam and water as coolant in vacuum plants or in vapor knock-down duty. The direct-contact condenser is cheaper, insensitive to fouling, and achieves small terminal temperature differences. If the condensate must remain pure (product recovery, separate circuits), it is ruled out and a surface condenser is required.
What physically limits the heating of the cooling water?
The saturation temperature of the vapor in the core flow: the cooling water can only approach it asymptotically. The dimensionless outlet temperature describes exactly this degree of approach. How close one gets depends on contact area and contact time — for jets, on jet length and F/A; for droplets, on droplet size and residence time. In practice the apparatus is designed so that the residual difference is small but the height still economical.
Why is the droplet size so decisive?
The heat-through time of a droplet grows quadratically with the radius (Fourier number). A droplet twice as large needs four times the residence time for the same heating. Since atomizers always deliver a droplet spectrum, the large droplets of the spectrum determine the required fall height or residence time — which is why the droplet size distribution enters the calculation explicitly.
What influence do inert gases have in direct-contact condensation?
Even without a wall, non-condensable gases hinder the vapor transport to the liquid surface and lower the effective condensation rate; in addition, they must be extracted at the cold end (vacuum pump, steam ejector). The basic correlations of the module apply to pure vapor; for appreciable inert gas fractions, the additional mass transfer resistance must be taken into account.