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NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS TECHNICAL MEMORANDUM 1367 HEAT TRANSFER, DIFFUSION, AND EVAPORATION1 By Wilhelm Nusselt Although it has long been known that the differential equations of the heattransfer and diffusion processes are identical, application to technical problems has only recently been made. In 1916 it was shown (ref. 1) that the speed of oxidation of the carbon in iron ore depends upon the speed with which the oxygen of the combustion air diffuses through the core of gas surrounding the carbon surface. The identity previously referred to was then used to calculate the amount of oxygen diffusing to the carbon surface on the basis of the heat transfer be tween the gas stream and the carbon surface. Then in 1921, H. Thoma (ref. 2) reversed that procedure; he used diffusion experiments to de termine heattransfer coefficients. Recently Lohrisch (ref. 3) has extended this work by experiment. A technically very important appli cation of the identity of heat transfer and diffusion is that of the cooling tower, since in this case both processes occur simultaneously. A relation obtained in the course of such an analysis was given by Lewis (ref. 4) and checked by Robinson (ref. 5), Merkel (ref. 6), and Wolff (ref. 7). In the following it will be shown that more accurate equations must be substituted for those used in the previous investigations of the relation between the quantity of matter exchanged by diffusion and the quantity of heat transferred by conduction. A rigid body having a uniform surface temperature Tw is cooled by an air stream having a speed w0 and a temperature TO. According to Fourier, a quantity of heat d2Q = X dF dt (1) then flows into the air stream in the direction n normal to the sur face element through a surface element dF in the time dt. In this 1'"WrmeUbergang, Diffusion und Verdunstung." Z.a.M.M., Bd. 10, Heft 2, Apr. 1930, pp. 105121. NACA TM 1367 6T equation, n is the instantaneous temperature gradient at the surface in the direction of the normal, and X is the thermal conductivity of air. If equation (1) is applied to an element of space in a gas stream, the following differential equation is obtained for the temperature field in an air stream: dT 2 Tcp = XV2T (2) where2 r density of air Cp specific heat of unit mass X thermal conductivity of air Therefore, the NavierStokes equations of motion involving u, v, and w are to be understood. For the diffusion problem, air can be treated approximately as a homogeneous body, since the molecular weights of its components nitrogen and oxygen are only slightly different. Air is now considered to be mixed with the diffusing gas, for example, ammonia or water vapor. The concentration of the water vapor at an arbitrary point in the air stream is designated c; that is, for example, there are c kilograms of ammonia in 1 cubic meter of air. In the similarity ex periment of Thoma (ref. 2), the surface of the rigid body was made of blotting paper saturated with concentrated phosphoric acid. Ammonia is absorbed very actively by such a surface, so that the partial pressure and, hence, the concentration of ammonia is very small. In a cooling tower the diffusion stream proceeds in an outward direction from the water drops. The vapor pressure and, hence, the watervapor concen tration at the surface of the drop is accordingly dependent upon the water temperature. If pw is the partial pressure of the diffusing gas at the surface of the body and pg that in the air stream, then the driving force of the diffusion stream is the partialpressure difference pw pO. 2The following nomenclature is used herein in the case of variables dependent upon time and the three coordinates x, y, and z: dT T + u T v T + T V2T = 2T + 2T 2T ;x2 oy2 oz2 NACA TM 1367 If now, again, n is a running coordinate representing the normal to the surface element dF, then the quantity of vapor diffusing through the surface dF is given by the basic relation of Fick: d2G1 = k dF dt (3) in which c vapor concentration, kg/cu m oc/pn gradient of vapor concentration in direction normal to surface k diffusion coefficient, sq m/sec If an element of space is taken in the air stream, exchange of vapor between such an element and its environment occurs partly through diffusion and partly through streaming (sensible motion). This fact leads to the differential equation: d = kV2c (4) dt The agreement between equations (1) and (3) as well as between (2) and (4) is immediately recognizable. Accompanying a calculation of temperature field from diffusion field or vice versa, there must also exist, however, an equivalence of boundary conditions. The temperatures Tw and TI of the heat transfer correspond to the gas concentrations cw and cO of diffusion. But, while during heat transfer the gas velocity is zero at the surface of the cooled body, there exists at that point, in the case of diffusion, a finite gasvelocity component normal to the body surface. This difference is easy to see if a one dimensional diffusion process is studied in a tube under steadystate conditions. Suppose a tube I meters long is filled with air and ammonia. Suppose also that by certain experimental means the concen trations at the ends of the tube are held at different, although con stant, values. Then a quantity G1 kilograms per hour of ammonia diffuses through the tube in one direction, and a quantity of air G2 in the other direction. If x is a running coordinate, cl the ammonia, and c2 the air concentration, then dce GI = k1 x (5) and dc = k2 5a) 02 = k2 d (5a) NACA TM 1367 where kI and k2 are the diffusion coefficients. If the partial pressures of ammonia and of air are pi and p2, respectively, then dpl GIR1T = kI pg dp2 G2R2T = k2 d7 (6a) Integration of equations (6) and (6a) yields3 G1RIT = klP1 + C1 G2R2T = k2P2 + C2 (7a) hence a linear variation of partial pressures. fluence is now ignored, the total pressure pO Hence, Pl + P2 = PO If gravitational in in the tube is constant. It then follows from equations (7) and (7a) that kl = k2 = k and G1RIT = G2R2T = VpO (10) that is, the same volume, evaluated at the total pressure, diffuses in both directions.4 Moreover, with the notation of figure 1, V = (PI' P1 "= (P" P2') I(1' 'P(1k 3 Gl1 (el' el") 3These are obviously in error; x lacking. (11) (12) 4 and k2 are equal but it does not follow from this arg nt. k1 and k2 are equal, but it does not follow from this argument. and and NACA TM 1367 as well as G2 = ( (c2" c2') (12a) Therefore, according to this example, the partial pressure and the con centration of ammonia vary linearly along the tube axis. Now, the experimental situation in the case of the cooling tower and in the diffusion research of Thoma (ref. 2) was otherwise. In the case of the tube, the boundary conditions are different at one tube end. If that end is closed with blotting paper saturated with phosphoric acid, there is set up, for the case of linear (axial) diffusion, the equivalent of Thoma's experiment; or, if that end is closed and the bottom covered with water, a situation corresponding to that of the cooling tower is obtained. Further, Stefan (ref. 8) and Winkelmann (ref. 9) determined the diffusion coefficient k for the diffusion of water vapor in air with this arrangement. It will now be shown that in this instance linear variation of the partial pressure along the tube axis is not attained in the case of the stationary diffusion stream. Once again, air and ammonia, or water vapor, are counterdiffusing. Since, however, one end is impermeable to air, no transport of air along the tube can take place. Because of the gradient of partial pressure of air, air must diffuse in the axial direction. This molecular air transport must work against a convective air transport; that is, a sensible flow of gas in the direction of the tube axis must occur. This flow will be represented by the symbol u. Then, for the flow of air the equation is dc2 G2 = 0 = uCe2 + k g (13) or, after introduction of the partial pressure, P2 = k id (13a) For the ammonia stream the corresponding equations are dc1 G1 = uc1 k d (14) and dp1 G1RIT = up1 k d (14a) NACA TM 1367 Now again P1 + P2 PO (8) It follows then from equations (13a) and (14a) that k dpl ( PO Pi dx (15) If equation (15) is substituted for u in equation (14a), the following expression is obtained for the partial pressure of ammonia: kpo dp1 GlRIT = P dx (14b) the solution of which, with regard to the boundary condition pi = PI' at x = O, is G1RIT x PO P kpo = e (16) PO PI' Therefore, no linear variation of partial pressure along the tube axis occurs in this case, but rather a logarithmic. If pl = Pl" at x = 1, kpo PO PI" G = Rkp ln PO P (17) This equation, given earlier by Stefan (ref. 8), shows that the rate of gas diffusion through the tube is no longer proportional to the partial pressure gradient. First, if pi is small compared with pO, it is again true that k P1' P1" cl' cl G, = 1T = k (1) From equations (15) and (16) it follows that RIT u = G1 p = V0 (19) PC NACA TM 1367 That is, the gas velocity u is identical with the volume of ammonia gas streaming through a unit of surface of the tube cross section per unit time, as measured at the total pressure pO. Hydrodynamically, therefore, there exist in the ammonia problems sinks of strength u corresponding to evaporation from the source of water. Since, in the case of heat transfer, the velocity is at that point zero, an exact similarity between heat transfer and diffusion is nonexistent in the experiment of Thoma. Only at very small values of G1, hence at low partial pressures of ammonia, can u be taken as zero. Only then does similarity of the boundary conditions also exist. This will be dis cussed later. Similarity requires, however, identity of the hydro dynamic equations as well. In the experiment of Thoma, thetemperature, in the case of the diffusion work, is uniform throughout; while, in the case of heat transfer, it varies. In this instance, the air density varies with temperature. In the case of diffusion, the gas densities vary with the concentrations. Similarity obtains, therefore, only if Tw w (20) To TO In equation (20) w and y0 are the densities of the ammoniaair mixture at the wall and in the free gas stream. Further, the variation of viscosity with temperature and ammonia concentration must still be considered. Since, however, as was shown previously, similarity is possible only at small ammonia concentrations, the experiment gives the heattransfer coefficients only at very small temperature differences. The similarity conditions are more favorable in the case of the cooling tower. In this instance, a hot water surface is cooled by a cold dry air stream. Accordingly, heat flows from the water to the air, and simultaneously the water that diffuses into thd air evaporates.5 In this instance, heat transfer and diffusion are, accordingly, super posed on each other under the same stream conditions. The same hydro dynamic equations and boundary conditions are associated with equations (2) and (4). In the following discussion, the relations that follow from similarity considerations in the case of Thoma's experiment and in the case of the cooling tower are derived. 5Stated precisely as written by Nusselt. NACA TM 1367 I. M)DEL EXPERIMENT OF THOMA A body having a surface F and temperature Tw is cooled by an air stream having a speed wO and a temperature TO. The heat loss of the body has the value Q kilocalories per hour. A diffusion experi ment is now carried out by using a body of the same shape. The air stream has the same speed w0 and is mixed with ammonia to the concen tration cO. Through the diffusion process, which occurs at the body surface, the ammonia concentration has already attained the value cy. The diffusion experiment shows that the surface absorbs G kilograms of ammonia per hour. What relation exists between Q and G? This question is discussed in the following on the basis of different theories. Similarity Theory The similarity theory of reference 10 leads to the following ex pression for the rate of heat exchange: Q = XL(T TO) f 1YT ) (21) where X thermal conductivity of air L dimension of body f initially unknown function of two dimensionless fractions, dependent on shape of body y density of air I viscosity of air Cp specific heat of air For diffusion, correspondingly, G = kL (co w) f (1 (22) \_T g 1 :F' NACA TM 1367 From these is obtained the relation sought: f(^T c Stlw oy cp g TcT Q_. X 1g v 0 (23) G k ,fIy Ojg Cw It follows from most of the experimental work that the function f may be represented as a product of two functions, that is, m n f = b (24) where the constants are dependent on the shape of the body. Reference 11 shows that, for the flow of gases through a cylindrical tube, m = 0.786 and (25) n = 0.85 j Two reasons then occur to set the two exponents equal to each other. If it is assumed that the velocity profile over the cross section is independent of density, the velocity components u, v, and w are proportional, in the differential equation of heat conduction, to the stream velocity at the center wO. In this equation, the fraction wrCp/X as a factor can be taken out. In the function f it must then follow that m = n. Since qcp g/X varies only slightly among the different gases, the influence of the magnitude of n on Q is slight. On that account an m = n was chosen, and in that manner a very simple kind of equation was obtained. Such a choice was expressly limited to gases, because it appeared, since the experimentation of Stanton with water had given a greater value for m than Nusselt had found for gases, that the power form is valid only in a narrow range of values of X/'cpy. From the more recent work of Sonnecken and Stender (see ref. 12) were obtained, depending on the experimental conditions, values of m between 0.72 and 0.91 and values of n be tween 0.35 and 0.50. Merkel (ref. 13) gives for the same research m = 0.87 and n = m/2 = 0.435. Rice (ref. 14) proposes for flow in smooth tubes m= 5/6 = 0.83 and n = 1/2 = 0.50. Lately Schiller and Burbach (ref. 15) have again grappled with this question. They find NACA TM 1367 that the Nusselt relation for gases with m = n is also experimentally confirmed for water and, further, support theoretically the equality of the exponents. This important question will be discussed later. Equations (23) and (24) lead to n Q = Cpk Tw TO G k X co cw (23a) If the heattransfer coefficient a is calculated with the use of Q = aF(TV TO) n G X (rcpk I F k X ) C0 Cy (26) (27) Relation of ThomaLohrisch In the derivation of his relation whereby the heattransfer coefficient may be calculated from a diffusion experiment, Thoma pro ceeds on the basis that the following condition holds: (28) k  Ck p CpT In that circumstance the two differential equations (2) and (4) are interchangeable. The temperature field is then proportional to the concentration field, and it follows that the relation T Tw cO Cw (29) holds at any arbitrary point in the fields. From equations (1) and (3) the relation sought follows immediately: Q X T, To S(o 30) G k C0 cw then NACA TM 1367 11 which, using equation (28), can be also written T TO Q= epy (30a) G CO Cw If the heattransfer coefficient a is calculated with equation (26), there is obtained G 1 a= r yCp (31) F 9P cO C Equation (28) is not valid in the diffusion experiment with ammonia in air, however. According to Thoma, it is considered that X = 1.25 (28a) In this case, Thoma uses equation (30a) as well as (31). As a correction factor he multiplies their right sides by the constant of equation (28a). He therefore obtains Q= YcP w T"o (32) G ycpk CO Cw from which equation (30) is obtained. Then it follows that GX 1 a=G X (32a) F k co cw Equation (27) assumes this form if n = 0. On the basis of this equation (32a), Thoma derives, from similarity and impulse considerations, the equation for heat transfer: Q = gcpTL(T To) (pgO (33) (N NACA TM 1367 If this equation is written in the form Q = XL(Tw TO) c9 \ / (33a) it is recognized as a special case of equation (21), to the extent that T(wor Scpg\ Tcpg /0y x X X (34 ) is valid. From equation (33) it follows that gce 7L Lw(r a = ) (35) which may also be written gc p7 L L a L = ) (35a) X ITF P (T) Thoma now uses equation (35a) to derive equation (52a) from (31). The author cannot follow Thoma's reasoning, because if f is replaced in equation (23) by Thoma's relation of equation (54), the following is obtained: Q w T (30a) G Co Cw and G 1 a= ap CO (31) F P cO cw Thoma's superaddition leads back, therefore, via equation (54), to equation (31) and does not give equation (32a), which was used by Thoma. Hitherto it has always been assumed that an alteration of a field is effective only in the vicinity of a body. If the whole stream cross section is affected, then another definition of a is necessary. Set NACA TM 1367 Q = adF(Tw T) (36) wherein T is the temperature in the center of the stream cross section that encounters the body surface element dF. Moreover, the equation Q = VoTcp(T2 TO) (37) applies, in which V0 is the air volume swept out by the body in unit time, and T2 TO the temperature rise of that air caused by the hot body. From equations (36) and (37) it follows6 that a rc n Tw In (38) F Tw T2 If T2 TO = (39) then = VOyp 1n (38a) F 1  In the diffusion work for the same air volume VO, there is obtained c C2 (40) cO ew provided c2 is the ammonia concentration behind (downstream of) the body. For the first special case, which is presupposed by the validity of equation (28), there would be 5 = E (41) and hence the heattransfer coefficient sought: 6It follows only if certain additional assumptions are made. NACA TM 1367 a=V.p In 1 (42) F 1 l In the general case, Thoma and Lohrisch, as in the preceding, follow equation (28a) and put a= VOrCp in 1 (43) yc k F 1 e hence, Va V In 1 (45a) F k 1 & The correct relation is obtained as in the following. The temperature T2 is eliminated between equations (37) and (38), and VOTCp 1 a n (44) F Q VOrcp(Tw TO) Now, according to equation (23a) n Q x fcyk\ 1 S cpk (23a) T T = CO o Cw If it is noted that G = VO(cO c2) (45) then n VT yTepk) cO c2(46) Tw TO cO cw is obtained, and, therefore, from equations (44) and (40), V0cep 1 a = In (47) F n (Tc pk NACA TM 1367 This equationshould therefore replace equation (43a) of Thoma Lohrisch. According to Thoma's research, e is in the vicinity of 0.25. If the numerical value, of equation (28a) is inserted in equation (47) and, further, the values n = m = 0.6, equation (47) yields a value of a some 11 percent smaller than Thoma's equation (43a). An additional inaccuracy arises also in the calculation of e according to equation (40). Thoma, in that equation, put cw = 0, since he assumed that in consequence of the strong absorption of am monia by phosphoric acid the partial pressure of ammonia in that re gion is zero. Since, however, the ammonia must diffuse through the boundary layer, a finite vapor pressure of ammonia must exist at the surface. It can naturally be quite small, but it is necessary, first, to measure it once. Lohrisch also employed Thoma's experimental tech nique in the case of water vapor, to the extent that he saturated with water the blotting paper comprising the body surface. In the calcu lation of e, he assumed that the vapor pressure at the body surface corresponds to the water temperature. In section II, concerning evaporation in a cooling tower, it is shown that the vapor pressure at a body surface is smaller than the saturation pressure. BoundaryLayer Theory In its most primitive and simplest form, this theory supposes that an air stream flowing past a body may be considered to consist of two contiguous but sharply demarcated portions, namely, the boundary layer adjacent to the body surface, and the balance of the air stream. The one is associated with a laminar flow, the other with a turbulent, which in the first approximation is treated as a potential flow. Then it is assumed that in the latter a complete equalization of temperature or of ammonia concentration occurs. Figure 2 exhibits this distri bution of temperature T and concentration c in the case of heat transfer or of a similarity experiment. The width of the boundary layer is indicated by y. For heat transfer, then, XF(Tw TO) Q = (48) y and for diffusion kF(c0 cw) G = (49) Y NACA TM 1367 Accordingly, there is obtained on the basis of elementary boundary layer theory Q Tw To (5) G k cO cw and also a = G 1 (51) F k CO cw which is identical with the equation (32b) of Thoma and with equation (27) when n = 0. Impulse Theory Impulse theory is a carryover of gaskinetic considerations into the domain of turbulent motions of a fluid. In its simplest form it is as sumed, as an explanation of heat transferthat a volume of gas V having the temperature TO moves from the turbulent fluid stream to the wall, where it is heated to a temperature Tw and then brought back to the core of the fluid stream. It therefore takes from the wall the heat Q = VYCp(Tw T0) (52) which is given up to the fluid. It must at the same time be true that Vrc a= p (53) F The volume V contains in the case of diffusion c0 kilograms of ammonia. On impact against the body surface, ammonia is absorbed until the concentration is cy. Hence, G = V(c0 Cw) (54) From equations (52) and (54) it follows that Q Tw TO (55) G= P Co Cw NACA TM 1367 and, accordingly7 a yc (56) F P C0 Cw which follows also from equation (27) when n = 1. Comparison of BoundaryLayer and Impulse Theories It will now be assumed that in the turbulent core neither perfect equalization of speed nor, therefore, of temperature or concentration is attained. On the core side of the boundary layer, the temperature Te and concentration ce are then, respectively, different from TO and cO. Within the boundary layer, exchange occurs in accordance with boundarylayer theory. In the free gas stream, impulse theory applies. Then, in the boundary layer, Q = 1 F(Tw Te) (57) y and G = ) F(ce cw) (58) y Hence, QX e (59) In the turbulent stream, on the other hand, Q = Vcp(Te TO) (60) and G = V(c0 ce) (61) therefore, Q Te TO (62) G 'cp cO Ce 7Equation (56) is identical with (31). NACA TM 1367 It can now be assumed that, approximately, Tw Te a (63) that is, is equal to a constant.8 Also, ce cw b O = b e (64) Co Cw From equations (63) and (64) it follows that Te = a (63a) Tw TO and CO ce = 1 b (64a) CO C In that case, equations (59) and (62) become Q X a Tw TO( (59a) G k b cO cw and Q 1 a Tw TO = ycpl (62a) G= 1 b c0 cw Recently, Prandtl (ref. 16) has given for this quantity the value 8 Sgcpp 1 gj a 1+ X V  '1 in which the value of the parameter e is uncertain. It lies between 1.1 and 1.75. NACA TM 1367 From the last equations, a rcpk S= a (1a) (65) There is obtained, therefore, the desired relation: x L ycpk]'TW TO S= + (1a) X]T T0 (66) G k XJ0 cw If this is compared with equation (23a), it is seen that the two functions can be distinguished only with respect to the dependency on the fraction ycpk/k. Since n > 0 and 0 < a< 1, both relations lead to increasing values [of Q/G] with increasing values of ycpk/X. In the case of gases, only a small range of variation of this fraction, in the vicinity of unity, occurs and is, therefore, of significance. If agreement of the two relations is demanded, there is obtained as the connection between two constants the expression9 n = 1 a (67) Then, if n = 0.4, a = 0.6. II. EVAPORATION OF WATER Stefans should be credited with having first recognized that the evaporation of water is a problem of diffusion. At the same time, he developed the theory of diffusion. It is necessary to distinguish among several different cases in connection with diffusion. Consider first a quiet surface of water at the same temperature as the overlying air. If the relative humidity of the air is less than 100 percent, water evaporates; that is, superheated water vapor diffuses into the air from the water surface. Since, under the same conditions, water is lighter than air, an airstreaming occurs. Above the water a rising current of air develops that sucks dry air over the water sur face. Through that mechanism the evaporation is increased. If a wind blows over the water surface, a further increase in evaporation occurs as a consequence of turbulence. If the water temperature differs from 9This, from a + (1a) (1+A) = (l+A)n, or a + 1 + a a = 1 + nA +...etc. 20 NACA TM 1367 the air temperature, an intrinsic influencing of the evaporation occurs, while because of the resulting heat exchange the convection is influenced. Evaporation in Still Air and in Uniform Temperature Field Above a surface of water of area F having a representative dimension L, a layer of air exists, the density of which, at some distance away, is YO and the specific humidity of which is co. At the water surface the humidity content of the air is cw. At any point above the [water] plane the vapor content of the air increases as a consequence of evaporation; it becomes lighter and suffers a lift Z in the amount Z = (c CO) (( ) (68) where pi is the apparent molecular weight of dry air and 11 that of damp air. For the diffusion field, diffusion equation (4) then applies. The NavierStokes equations of motion are required for the determina tion of the velocity components appearing therein (eq. 4); in these equations, the lift Z appears as an external force acting in the direction of negative gravity; and, therefore, TOdw 2 (69) Sd = Z + (69) In this expression, the air density 0T can be assumed constant. A similarity consideration leads to the expression for the concentration gradient at the water surface: dc CO 1) (c c0) dz L T 2 (70) The vapor mass evaporating per unit time from the water surface G is calculated according to equation (14). If the value of u accord ing to equation (19) and the following value of c1 S= (71) cw RT NACA TM 1367 21 are substituted in equation (14), then kF dc G Pw dz (72) 1 P where p is the total pressure. If the concentration gradient is now replaced by its equivalent according to equation (70), the rate of vapor evaporation becomes (cw cO)F L yl 1) (cw C0) c g G =k 4 `  (73) pw 2 ky Evaporation in Wind and in Uniform Temperature Field Over a water surface a wind passes whose speed at some distance from water has the uniform value wo. The air and water temperatures are identical. Gravitational influences can accordingly be ignored if the airspeed exceeds several meters per second. Therefore, the ob servations and formulas of section I apply (ref. 17). It follows that, by equation (22), the rate of water evaporation is G = kL(cw co) f L\ g (22) If the assumption is now made that the velocity u arising from the evaporation normal to the water surface can be ignored, the function f of equation (22) can be taken over from the corresponding heat transfer problem in accordance with equation (21). The results of reference 17 should be considered here. A copper plate heated elec trically to 500 C, and having a dimension on a side of 0.5 meter is cooled by an air stream having a temperature of 200 C. For w > 5 meters per second, the following value was obtained: a = 6.14w0.78 kcal/(m2)(hr)(C) (74) It follows that in equation (24), if m = n, f 0.0 78 f = 0.065 0.78 (75) \ A. / NACA TM 1367 In equation (22) there is obtained, accordingly, 0.78 f = 0.065 ) (75a) Therefore, the expression for the amount of water evaporating from a water area of F square meters in an hour becomes S0.22 0.78 G = 39F ) w0 (cw co) kg/hr (76) where k is the diffusion coefficient according to the research of Mache (ref. 18): 1.89 k= 078 ( )3 sq m/hr (77) in which the total pressure p is to be used. Evaporation with Heat Exchange in Still Air It will now be assumed that the temperature Tw of the water sur face is different from the temperature TO of the air. Hence, heat transfer occurs in addition to diffusion. In this instance, both processes are coupled through the resulting air stream to the extent that airlifting is caused by both the lesser specific gravity of the water vapor and the heating of the air. Instead of equation (68) of section II, the lift Z per cubic meter has the value Z = yor(T TO) + (c c0)( 1 (68a) Here T and c are the temperature and specific humidity at any place in the field, yT is the specific gravity of dry air at a great dis tance from the water surface at which the conditions pO and TO pre vail, and r is the coefficient of expansion of air: r = 1/T (78) In the case of excesses of temperature that are not too great, r = l/TO (78a) NACA TM 1367 approximately. Then Z = T(T TO) + (c co)(  TO F 1 becomes the value of the body force in the equation of Moreover, equation (2) of heat conduction and equation must be used. With the abbreviations motion (69). (4) of diffusion L3o2 (Tw T0) B = gr2T0 C= gc pr i i 1( 1) g12 (c c ) w 0 D g9I (79) similarity considerations lead to the relations (80) G = kL(cw CO) 41 (B,E,C,D) Q = XL(T, TO) 12 (B,E,C,D,) where 4 1 and 12 are initially unknown functions of the B, E, C, and D. (80a) variables The formulation is significantly simpler and clearer if it is assumed that equation (28) applies, that is, k = c pTO ofor then for then (28) (68b) (28a) C = D NACA TM 1367 In this case T and c are proportional to each other. Equation (29) applies and, then, Q (T TO) G k(cw c ) Tw O  TOc c0 The exchanged heat and evaporated moisture stand, therefore, in a very simple relation to each other. If the values of G and Q according to equations (80) and (80a) are substituted in equation (81), (81a) which is valid only if equation (28) obtains. For the determination of the function 4 there is introduced, as a matter of expediency, a new dependent variable: (82) = T To + o 1 (c c) which then leads to the differential equations TO dw rTO 2w g dt TO and (83) d2 v2 rocp d = from which, through consideration of similarity involving at the water surface, it follows that Tw To + (  n = L f B+E), n E L the gradient c1 (84) . .J This gradient can also be calculated by the use of equation (82), from which it follows that NQ 6T +TO 1 ) (85) 7n =n TO 1 1 o(5 (81) 41 =" 2 ! NACA TM 1367 If the divergence ("speed of expansion") u at the water surface is considered negligible, equations (1) and (3) apply: and GT G= kF a 3c From equations (1), (3), (81), (84), and (85): kF(cy CO) r + ] G = L f B+E), D XF(T, TO) Q = LT f [(B+E), C] The second of the equations (86) must also apply co = 0, and therefore when E = 0. The equation the usual form for heat transfer. If the latter of evaporation G can thus be calculated. (86) for the case then goes over is known, the On the basis of heattransfer research (ref. 19), for large values of B/C, the following can be written: 1/4 2c (T TO 1/4 S= C (I ) (Tw TO) = C1X ; XTOT in which the coefficient C1 is dependent upon the form and orientation with respect to gravity. With equation (87), one now obtains from equation (86) 4 kF B + E S= L V D (w co) 4 F L YO(Tw TO) = Clk k + L krlT0 (cw C0) (87) cw  into rate (88) NACA TM 1367 It should be observed, above all, that this equation is valid only if equation (28) is valid, and, hence, C = D. The values k, X, r, ~j and Cp pertain to the vaporair mixture and, indeed, are to be taken as mean values over the whole field. The term k is given by equation (77). The thermal condutivity of air ) is in the following form in reference 10: 0.00167(1 + 0.000194T) 1f kal/(m)(hr)(Oc) (89) S= 11 kal/(m)(hr)(C) (89) 1 +  T Since the thermal conductivity of water vapor in the germane tem perature range is only slightly less than that of air, the thermal conductivity of the vaporair mixture can be assumed equal to that of air. The terms y and cp are to be calculated according to the relation appropriate for a mixture of gases. Thus, there are obtained at t = 0 500 C (90) S= 0.87 0.84 kyep kp This fraction is different from unity for this vaporair mixture. Accordingly, equations (80) and (80a) apply. Since equations (80) and (80a) become equations (87) and (88) in the limiting case C = D, in general, in the first approximation, the following can be written: 4 SCljF(cw co) B (91) G= L (91) L FD and kF(Tw TO B+E Q = Cl L C (92) Then the following is obtained: 4 G k fC eC cO ji (93) Q X D Tw To or 4 G k AF Cw CO Q X kcpYrO Tw T9 NACA TM 1367 If the temperature within the domain falls below the dewpoint, fog formation is initiated and the equations become invalid. Equation (87), along with equations (88), (91), and (93), is applicable only at large differences of temperature and watervapor concentration. For small differences, the function f(B/C) can be expressed only by means of a graphical representation. By such a representation, it can be shown that for B + E = 0, f approaches a constant value. If the air temperature TO is greater than the temperature of the water surface Tw, it is possible, for finite values of TO Tw and cw CO, that B + E = 0. In this case, no convective streaming occurs, but rather only a molecular transport of heat and vapor. Hence, for B = E (95) or To TW To 1 (95a) cw c0 O (1 ) there is obtained G = C2kL(cw CO) (96) and Q = C2XL(TO Tw) (97) Therefore, G k cw (98) n = (98) iQ T To T The coefficient C2 is dependent upon the shape of the water surface. Evaporation with Heat Exchange and Air Flow As in the case of Evaporation in Wind and in Uniform Temperature Field, a wind having a speed w0 flows by a water surface. However, the water temperature Tw and the air temperature TO are now dif ferent. With a small partial pressure of water vapor assumed, equations (21), (22), and (23) of section I apply here as well. Since, however, NACA TM 1367 at 500 C the partial pressure of water vapor has already attained a value of 0.125 atmosphere, they must be modified. For the Reynolds number the following will, for brevity, be used: LwoTo Re = gLU (79a) On the basis of similarity theory one has again, first of all, the relations 6T Tw TO = L f(Re,C) (99) and Cw co f(Re,D) (99a) In these, in the case of a smooth water surface such as a tank,10 L is the principal dimension; and for a water drop, it is the diameter. At the water surface there exists, further, between the rate of evaporation and the concentration gradient the following relation: G c (14 F = ucw k 7 (144 where, according to equation (19), GR1Tw u = l (19a) Fp and RI, in this equation, is the gas constant of water vapor, and p is the total pressure. There is then obtained Gp (100) ucw = FF(1 in which pw is the partial pressure of water vapor at the water surface. If the value of equation (100) is used in equation (14a), the concen tration gradient at the water surface becomes C G (1 ( on = kF 100r pond or pool. NJACA TM 1367 By the same reasoning, the following applies for heat transport: = Urep (T TO) k (14b) and, with equation (19a),  F Q G Cp1 (Tw TO] (102) From equations (99), (99a), (101), and (102), the relation sought between Q and G is Pw X(T, TO)f(Re,C) P k(cw co)f(Re,D) + r (T T) (103) If the value of f given by equation (24) is substituted in equation (103), =1 ( P) G p P X (rocpk n k\ x / Tw TO + cp(Tw TO) Ce CO From equations (101) and (99a) there is obtained, with equation bkF(cw cO) Rem G = pw ) LDn (104) and, at the same time, from equation (103a), (105) If cw = Co, no evaporation occurs and equation (105) becomes equation (24). (103a) Qn=( NACA TM 1367 Vapor Pressure at Water Surface It is natural to assume thatat the evaporating water surface, the vapor pressure is equal to the pressure at the saturation value corre sponding to the water temperature T and that, therefore, P = Ps" At the same time, the vapor concentration at the water surface then becomes S=r" (106) that is, it is equal to the saturation density. That this, however, is not the case has already been conjectured by Winkelmann (ref. 9) and then demonstrated by Mache (ref. 18), who found, on the basis of a thorough research on evaporation in a cylindrical tube, that the follow ing relation exists between the rate of evaporation and the pressure in question: = KO(Ps Pw) (107) that is, the vapor pressure over the water surface, during evaporation, is always smaller than the saturation pressure corresponding to the temperature of the water surface. The coefficient KO is a temperature function that unfortunately has not yet been precisely determined. If the density of vapor instead of the partial pressure is introduced in equation (107), G= B(c" cw) (108) where c" is the saturation concentration of water vapor at the water temperature Tw, and pl is a constant dependent upon temperature, which, according to the researches of Mache, assumes values dependent upon the water temperature as indicated in the following table of values: Unfortunately, it is precisely in the technically important temperature tw, oC KO, 1/hr I1, m/hr 92.4 0.0086 148 87.8 .0080 133 82.1 .0084 108 27.5 .084 925 NACA TM 1367 range between 00 and 500 C that only a single experimental value is available. Application to Psychrometer of August For many technical applications of the diffusion relations pre viously developed, it is appropriate to calculate the evaporation coefficient on the basis of the heattransfer analog, as has already been done in the treatment of burning and vaporization of the carbon in iron ore (ref. 1). Set G= pF(c" co) kg/hr (109) from which the dimensions of the evaporation coefficient are 3 = m/hr (110) In the case of August's psychrometer, evaporation takes place from a moist thermometer in still air. In such an instance, equation (91) applies in the calculation of the mass of water evaporating per unit time. If, further, an evaporation coefficient P2 is inserted, where = k I + E (ill) 2 = C l D ( then, equation (91) becomes G = 02F(cw C0) (91a) Further, equation (108) still applies. If the unknown concentration cw is eliminated between equations (108) and (91a), the evaporation coefficient p in equation (109) becomes 1 1 1 S (112) P 1P P2 When the heat transferred along the stem of the thermometer is ignored, the heat balance of the wetted thermometer may be expressed as Q = G [r + cp(T, Ts] (113) In this expression, Tw is the temperature of the wetted thermometer, Ts the saturation temperature corresponding to the partial pressure NACA TM 1367 p of the water vapor at the surface of the wetted thermometer, r the heat of vaporization at the pressure Pw, and cp the specific heat of the limiting curve [?] at the same pressure. If TO is the environment temperature, as measured with a dry thermometer, the heat extracted from the surroundings is Q = aF(TO Tw) (114) The heattransfer coefficient a consists of two parts, several terms of a sum at, covering the heat conveyed by thermal conduction to the thermometer, and a portion as that gives the magnitude of heat radia tion. The latter is ITO,4 4 \100/ \100 as = Cs i00 (115) TO Tw where C. is the radiation coefficient of water, that is,11 Cs = 3.35 kcal/(m2)(oC4) (116) and TO' is the mean temperature of the fixed body surrounding the wetted thermometer with which radiation is exchanged. It is certainly approximately equal to the ambientair temperature TO, yet surely not quite precisely equal. Herein, under certain conditions, exists a not unimportant source of error in psychrometry. This source of error can successfully be eliminated (as was communicated to the author by Dipl.Ing. Kaissling) by surrounding the wet thermometer by a radiation shield, which consists, as does the wet thermometer itself, of a wetted surface. If TO' = To and the attainment of room temperature is assumed, equation (115) becomes, approximately,12 as = Cs (115a) 11Dimension time1 apparently missing in equation (116). 12This does not seem to be correct NACA TM 1367 The heattransfer coefficient ab is dependent upon the flow conditions in the vicinity of the thermometer. If the psychrometer is hanging in a region in which the air is quiet, then, according to equation (92), b = Cl + a = 0E L CC (117) in which the constant C1 is dependent upon the configuration of the thermometer well. For a cylindrical well of height H, C1 = 0.83 (118) In equation (116 [117]), L is replaced by H. If the wetted thermometer is placed in a current of air, there is obtained, for example, for a plateshaped thermometer, the following relation (ref. 20): 4 0.000028Re b 0.78 = 0 9 + ab = 0.069 f Re + 0.83 f e (119) wherein it is supposed that the wind flows along the thermometer well in a horizontal direction. If, in the energy equation (109) is inserted, and that of chrometer formula is obtained: (113), the value of G from equation Q from (114), the following psy r + cp(Tw Ts) ( Ts) ) = ( + C() 1 + L) ,L bJ + C) (0 2 (To TW) (120) For the diffusion constant of a plateshaped, wetted thermometer, the following is obtained with equations (119), (111), and (117): 0.78 S0.78 4 B+ 0.000028 (w 02 = 0.069 (+ 0.83 e H .3i' D (121) Equation (120) gives a decrease of the psychrometer constant with increase of airspeed, which has been well confirmed by the investi gations of Edelmann, Sworykin, and Recknagel. NACA TM 1367 Application to Theory of Cooling Tower In this [apparatus], finely divided warm water trickles downward and is cooled by a rising current of cold air. If, at some point, w0 is the relative speed of the water and air with equations (103), (104), and (105) apply for the evaporation. If the heattransfer coefficient coefficient 02 are now evaluated according to (92a), respect to each other, heat transfer and a and the evaporation equations (26) and Pwy YOcpk^  X (Top k c (cw co) k i (122) and, with equation (112), a (1 b= bX Oli  k T op + l cp(cw co) + S n n (gep epk(c CO) t g_ m 1 (I ) 7Re Material on the technical applications of the formulas sented will soon be published elsewhere.13 (123) here pre Translated by H. H. Lowell National Advisory Committee for Aeronautics 1Since the transmission of the original manuscript to the editor's office on May 16, 1929, the following papers have appeared: E. Schmidt, Verdunstung und Warmeibergang, Gesundheitsing., 1929, p. 525.; R. Mollier, Das ixDiagramm fur Dampfluftgemische, Stodolafestschr., Zurich, 1929, p. 438; H. Thiesenhusen, Untersuchungen fiber die Wasserverdunstungsgeschwindigkeit in Abhangigkeit von der Temperatur des Wassers, der Luftfeuchtigkeit und Windgeschwindigkeit, Gesundheitsing., 1930, p. 113. J NACA TM 1367 REFERENCES 1. Nusselt, W.: Die Verbrennung und die Vergasung der Kohle auf dem Rost. Z.V.D.I., 1916, p. 102. 2. Thoma, H.: Hochleistungskessel. Berlin, Springer, 1921. 3. Lohrisch, H.: Bestimmung von Warmeubergangszahlen durch Diffusions versuche, Diss., Munchen, 1928. 4. Lewis: The Evaporation of a Liquid into a Gas. Mech. Eng., 1922, p. 445. 5. Robinson: The Design of Cooling Towers. Mech. Eng., 1923, p. 99. 6. Merkel: Verdunstungskihlung. Forsch.Arb., VDI, Heft 275, 1925. 7. Wolff: Untersuchungen uiber die Wasserickkthlung in kiinstlich belifteten.Kuhlwerken. Minchen, Oldenbourg, 1928. 8. Stefan: Versuche 5ber die Verdampfang. Wiener Ber., Bd. 68, 1874, p. 385. 9. Winkelmann, A.: Uber die Diffusion von Gasen und Dampfen. Ann d. Phys., Bd. 22, 1884, p. 1. 10. Nusselt: Das Grundgesetz des WArmeuberganges. Gesundheitsing., 1915, p. 477. 11. Nusselt: Der Wgrmeoibergang im Rohr. Forsch.Arb., VDI, Heft 89, 1910. 12. Nusselt: Die Wirmeibertragung an Wasser im Rohr, Festschrift anlisslich des 100 jahrigen. Bestehens des Tech. H. S. Fridericiana zu Karlsruhe, Karlsruhe, 1925, p. 366. 13. Merkel: Hiitte, des Ingenieurs Taschenbuch, 25 Auflage, Bd. 1, Berlin, 1925, p. 454. 14. Rice: Free and Forced Convection in Gases and Liquids, II. Phys. Rev., vol. 33, 1924, p. 306. 15. Schiller und Burbach: Warmeibertragung stromender Flussigkeit in Rohren. Phys. Zs., Bd. 29, 1928, p. 340. 16. Prandtl: Bemerkung fiber den Wgrmeiibergang im Rohr. Phys. Zs., Bd. 29, 1928, p. 487. 36 NACA TM 1367 17. Nusselt und Jiirges: Die Kthlung einer ebenen Wand durch einen Luftstrom. Gesundheitsing., 1922, p. 641. 18. Mache, H.: Nber die Verdunstungsgeschwindigkeit des Wassers in Wasserstoff und Luft. Wiener Sitzungsber., Bd. 119, 1910, p. 1399. 19. Nusselt: Die Warmeabgabe eines wagrecht liegenden Rohres oder Drahtes in Flussigkeiten und Gasen. Z.V.D.I., 1929, p. 1475. 20. Nusselt: Die Gasstrahlung bei der Str8mung im Rohr. Z.V.D.I., Bd. 70, 1926, p. 763. NACA TM 1367 0 f' 0  E*I a. C% .. pp < Po /r. 4   ^ __ A X M 5H NACALangley 32654 1000 c r w L 6Hi cd CS"2z ,62021 7 ^a^ 3>>rt < o>0m g, e *" a 16 E0 ~ L W 0 Cl C', I .M^ 6 *ss ~'s  < O Su5 ...gL g0 = 0iggg c i i.2 'orI SO (D I 1. 1 8 E41~~~~. go a~ WI LWI w i go" 40 b as a > ~ 0 XZZNd sa o cc 4 0 Al0 o co 5  lu 000 0.cpl a z Z im 4 0 a; c 41 0 t3" m o 0 0r u 'd ba Vnp N d 3 GM ZZ Is w* u E 2 2Ns.4 b Ez tI SM .4 ; i 4 5a. a 2 a d Mid 9z aC WI I.. ho o4~ y4j e v4 3 w C'l Sa g ~Io "a 1 ^1 3fll Sgl. ll ^ei U SSZ U^ ZKBEi&^ a) 0 0 ol so. 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