ENGINEERING RESEARCH INSTITUTE UNIVERSITY OF MICHIGAN ANN - ARBOR CALCULATIONS ON DROP SIZE GROWTH AND SUPERSATURATION ON AIR IN AN ICING WIND TUNNEL By M. TRIBUS:J KIEIN Project M992-3 WRIGHT AIR DEVELOPMENT CENTER, U. S. AIR FORCE CONTRACT AF 18(600)-51, E. 0. NO. 462 Br-1 March, 1953

SUMMARY Calculations have been made for one case on the growth of droplets and the amount of supersaturation occurring in an icing wind tunnel. The results show that condensation on the droplets is negligible but that a large degree of supersaturation is present at the test section of the tunnel. ii

ENGINEERING RESEARCH INSTITUTE ~ UNIVERSITY OF MICHIGAN CALCULATIONS ON DROP SIZE GROWTH AND SUPERSATURATION OF AIR IN AN ICING WIND TUNNEL STATEMENT OF THE PHYSICAL PROBLEM In the operation of an icing wind tunnel it is often desirable to suck saturated air laden with water drops through the test section. In passing through the wind tunnel the air is expanded and cooled, and the resulting thermodynamic state of the air at the test section is often of interest. In particular it is important to know whether the air is supersaturated and also whether the water drops have grown to a size different from those present at the inlet to the wind tunnel. In the calculations which are presented in this report, only one case has been considered. This case was computed by hand and illustrates the possibility of supersaturation in wind tunnels at rather modest velocities. The system analyzed consists of the University of Michigan icing wind tunnel and a cloud of 5-micron-radius water drops with a concentration of 1 gram/cubic meter at the inlet and an initial temperature of 320F. The air is considered as ac-celerated to a velocity of 478 feet/second (Mach number = 0.45). -During the condensation process certain conservative idealizations were made which permit the calculations to be accomplished with less labor; in each case these assumptions tend to enhance condensation on the water drops. The important simplifications consist in setting both the surface tension and the heat of condensation equal to zero. In each case, therefore, the water drops are treated as though they had lower vapor pressure than is expected to occur in the actual situation. SYSTEM UNDER ANALYSIS For the physical problem described above the following mathematical system is analyzed. It is desired to determine certain variables as functions

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN of the distance x (in feet) from the tunnel inlet. These variables are: l A cross-sectional area of tunnel ft2 B pressure of air lbs/ft2 T temperature of air ~F U velocity of air ft/sec p density of saturated water vapor lbs/ft5 Pwv density of water vapor actually present lbs/ft3 Wwv weight of water vapor passing a given point in 1 sec lbs/sec r radius of water drops ft D diffusivity of water vapor ft2/sec The tunnel under consideration is of square cross section, so that A is determined from its profile as given in reference 1. Since the air flow is assumed to be isentropic, B, T, and U are determined from standard tables (ref-erence 2) and when T is known Pwv and D may be found (reference 3). This leaves v, Wwv, and r as the unknown variables, with relations between these quantities determined from consideration of continuity of flow and the laws of condensation.'EUATiIONS The continuity of flow gives at once Ww AUpwv (1) and from the laws of condensation we have r dr (Pwv - Pwv,s)D B (2) Pliq B U Bo dx dWwv = -4trn (pwv - Pwv,.s)D - - (3) B U where Pliq is the density of liquid water (62.4 lbs/ft3), n is the number of drops passing a given point per second, which for the conditions described above is 2.35 x 1010 drops/sec, and Bo is 1 atmosphere (2116 lbs/ft2). Hence 2

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN there is a system of one algebraic and two first-order differential equations to solve for the three. unknown functions. METHOD OF SOLUTION If Equation 2 is solved for (Pwv Pwv,s)D and the result sub,-"'stituted' into Equation 3, we have B U dWwv = -4nr2 Pliq dr, which is immediately integrable to give W 4 nxr3 Pliq+ C2: (4) where C2 represents the total weight of water in both liquid and vapor forms passing any point of the tunnel in 1 second. This constant is readily computed from the initial conditions. Solving Equation 12 for Pwv and substituting for WwV as given by Equation 4 results in 1 4 = 4 (C2 — nnr3 Pliq) * (5) This quantity may now be substituted into Equation 2 to give 4 (C2 3 - r3 iq AU Pwv )D Bo liq AUdx (6) which can be rewritten as dxr (C2 -- nr3 Pliq AU PW.s )D (B dx r Pliq AU2 B In Equation 7 all the variables in the right-hand expression are known functions of x or r and hence this equation is solvable numerically by the Runge-Kutta method (reference 4). These calculations have been made using formulas for second-order accuracy and increments in x graduated in accordance with the curvature of the tunnel profile. This method gives an accuracy comparable to the data obtained from the tables and the use of a third-. or higher-order accuracy procedure would not result in any improvement in overall accuracy. Once the function r is known Wwv and pwv are computed from Equations 4f and 5 respectively to complete the solution of the problem. _ ~~~~~~~~~~~~~~~~~~~~~~~3

ENGINEERING RESEARCH INSTITUTE ~ UNIVERSITY OF MICHIGAN RESULTS The computations show that the radius of the drops increases monotonlically to 5.156 microns at the test section of the tunnel. Hence the drops will always be within a few per cent of their original size at any point in the tunnel, and of the water originally present in vapor form only a relatively small amount condenses on the drops. To analyze the degree of supersaturation present the quantity (Pwv - Pwv,s)/(pwv,s) was computed. This quantity also increases monotonically as a function of x and reaches a value of 1.118 at the test section. This implies that at this point the weight of water vapor present is over twice that required for saturation. Approximate calculations show that changing the initial liquid water content within the values which can be obtained experimentally gives results only slightly different from those obtained for this case. Hence for a tunnel of this type operating approximately under the conditions described here, it may be stated that the radius of the drops changes by at most a few per cent and a large degree of supersaturation exists over most of the tunnel length. Figure 1 illustrates the idealized system treated here and gives the pertinent results of the analysis. REFERNCES 1. Nicholls, J. A., et al.., Desgn of an Icing Wind Tunnel, University of Michigan, ERI report, Contract No. AF 18(600)-51, E.O. No. 462 Br-l, June 1952. 2. nEmmons, H. W., Gas 2ynamics Tables for Air, Dover, 1943, pp. 17-18. 3. Perry, J. H., Editor, Chemical Engineers Handbook, McGraw-Hill, 1950, pp. 768-770. 4. Hildebrand, F. B., Advanced Calculus for Engineers, Prentice-Hall, 1949, pp. 103-106. 4

4 CHAMBER INLET NOZZLE PROFILE TEST 3 w To 320F T= 12 F U = 827ft/sec U 478ft/sec B2 = I ATM. MACH. No=0.45 z 3 t= 5/B B= 0.87 ATMOSPHERE t WATER VAPORr = 5.156 MICRONS PRESSURE APOR.992 GRAMS LIQUID WATER z SATURATION METER3 WATER VAPOR PRESSURE D 3.,=_02.118 TIMES SATURATION -L r | PRESSURE 0 1 2 3 4 5 6 7 8 9 DISTANCE IN FEET FIG.I. THE IDEALIZED SYSTEM AND PERTINENT RESULTS OF THE ANALYSIS