Design of Cooler Condensers for Mixtures of Vapors with Noncondensing Gases

A. P. COLBURN, E. I. du Pont de Nemours & Company, Inc., Wilmington, Del., AND 0. A. HOUGEN, University of Wisconsin, Madison, Wis.

I n conahsing vapors from mixtures of vapors with noncondensing gases, the gas f i lm and over-all heat transmission coeficients vary widely from point to point in the apparatus, and also the change in heat content of the gaseous mixture is not propor- tional to the change in temperature. For these reasons no simple relationship expressing the mean temperature difference between the gas-vapor and cooling-water stream can be used.

This paper outlines a method of computing the required condenser surface in which values of 1 / U At are determined at a suficient number of points along the path of gas flow to permit calculation of a correct average value of U A t by graphical in- tegration. The value of UAt at any point in the

HE design of a surface condenser for the condensation of vapors from a mixture of vapors with noncondens- T ing gases presents several very unusual conditions and

complications in heat transmission not encountered in the condensation of pure vapors or in the cooling or heating of homogeneous fluid streams.

In the condensation of vapors from their admixture with noncondensing gases, all the properties of the gas stream vary greatly as the water vapor is removed. For example, the heat transmission coefficient of the gas film, the rate of gas flow, and the heat capacity of the gas stream per pound mole of inert gas decrease enormously as condensation pro- ceeds. The condensation of the vapors depends upon the diffusion of the vapor molecules through the gas mixture; hence, mass transmission as well as heat transmission co- efficients must be considered ; the problem involves prin- ciples of diffusion as well as of heat transmission. Also the condensate loses heat as it flows over the condenser surface so that more heat passes through the film of cooling water than through the adjoining gas film.

For these various reasons no method of calculating mean temperature differences based upon terminal conditions is applicable. Indeed, the true average temperature differ- ence between the two fluid streams interchanging heat may be greater than the temperature difference at either end of the condenser. An accurate method of calculation for this type of design problem has long been sought.

Nevertheless, attention has heretofore been given prin- cipally to methods of estimating for design purposes the mean temperature difference between the vapor-gas mixture and the cooling liquid which are interchanging heat, since i t is realized that the usual logarithmic mean of the terminal tem- perature differences of the heat interchanger is entirely in- applicable for the reasons mentioned above. The method, for example, of Haug and Mason (6) allows for the rapidly changing heat capacity of the vapor-gas mixture and pro- vides for the estimation of the average temperature differ- ence by calculating the temperatures on the two sides of the condenser surface from heat balances, thus tacitly assuming a uniform rate of heating of the water stream.

condenser is obtained, through trial and error, by equating the heat transferred through the condensate, the tube wall, and the cooling-water film, to the sum of the heat transferred by the sensible cooling of the uncondensed gas and the latent heat equivalent of the vapor transferred by diflmion and condensed. The necessary surface area is obtained by multi- plying the heat transferred per hour by the in- tegrated average value of 1/UAt . Coeficients for the transfer of the vapor are estimated on the basis of a correlation previously published. I n a n ex- ample presented, the over-all coeficient, U, varies from 294 to 60, whereas the effective heat trans- mission coeflcient of the condensing vapor-gas film varies from 5600 to 75.

I t has been recently pointed out by Tymstra (8) that the Haug and Mason method errs in that the temperature differ- ences should be averaged according to the length of the ex- changer rather than according to the heat content of the water stream since the inference of a uniform rate of heating the cooling water is inadmissible. Tymstra has proposed an approximate method for estimating an average tempera- ture difference which consists in adding to the logarithmic mean temperature difference one-half of the “arithmetic mean temperature difference between a straight line connecting the gas inlet and outlet temperatures and the gas heat-con- tent curve.” This method is in error in that it presupposes a uniform over-all heat transmission coefficient over the entire length of condenser. Actually the over-all heat transfer coefficient varies from point to point in a cooler condenser, being high where the vapors are relatively concentrated and low where most of the vapor has been removed. It is, therefore, obvious that no simple mean temperature differ- ence and no simple average heat transfer coefficient is ap- plicable.

A basic statement of the problem is given by the equation:

d A = dp/UAt (1)

where At = difference in temperature between gas and cool- ing medium.

As pointed out above, the usual formula,

A = q / U A t , (2)

does not apply to this case. For the present case, to integrate Equation 1 analytically

it would be necessary to find expressions for both U and At as functions of q. Such expressions were not obtained since the difficulties involved are too great, as can be seen later by studying, for example, the variables affecting U and At. Some approximations can doubtless be made to effect this solution, a t least for specific cases, but the authors have pre- ferred for the general case the exact integration of Equation 1 by a graphical method. While it is difficult to find ex- pressions for U and At in terms of q, i t is quite straightfor- ward for any given problem to calculate the product, UAt,

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