/MA"fP\'\iif'] 
 
 NATIONAL ADVISORY COMMITTEE 
 FOR AERONAUTICS 
 
 TECHNICAL MEMORANDUM 1367 
 
 HEAT TRANSFER, DIFFUSION, AND EVAPORATION 
 
 By Wilhelm Nusselt 
 
 Translation 
 
 "Warmeiibergang, Diffusion und Verdunstung. " Z.a.M.M. , Bd. 10, 
 
 Heft 2, Apr. 1930. 
 
 Washington 
 
 March 1954 UNIVERSOY OF H OOm 
 
 DOCUMr-r-: 
 
 "jgovFTinr 
 
^^^n^rt ^ircpo-, 
 
 NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS 
 
 TECHNICAL MEMDRANDUM 1367 
 
 HEAT TRANSFER, DIFFUSION, AND EVAPORATION"'- 
 
 By Wilhelm Nusselt 
 
 Although it has long been known that the differential equations of 
 the heat-transfer and diffusion processes are identical, application to 
 technical problems has only recently been made. In 1916 it was shown 
 (ref . l) 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 heat-transfer 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 T^ is cooled 
 
 by an air stream having a speed wq and a temperature Tq. According 
 to Fourier, a quantity of heat 
 
 d^Q = -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 
 
 -'-"Warmeubergang, Diffusion und Verdunstung. " Z.a.M.M. , Bd. 10, 
 Heft 2, Apr. 1930, pp. 105-121. 
 
2 NACA TM 1367 
 
 St 
 
 equation, -r— is the instantaneous temperature gradient at the surface 
 
 in the direction of the normal, and X is the thermal conductivity of 
 
 air. If equation (l) 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: 
 
 TCp § = Xv't (2) 
 
 p 
 
 where 
 
 y density of air 
 
 Cp specific heat of unit mass 
 
 X thermal conductivity of air 
 
 Therefore, the Navier-Stokes 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 exan^ile, 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 water-vapor concen- 
 tration at the surface of the drop is accordingly dependent upon the 
 water temperature. If p^ is the partial pressure of the diffusing 
 
 gas at the surface of the body and pq that in the air stream, then 
 
 the driving force of the diffusion stream is the partial-pressure 
 difference p^ - pq. 
 
 ^The following nomenclature is used herein in the case of variables 
 dependent upon time and the three coordinates x, y, and z: 
 
 ^=^+u$+v^+w^ 
 dt ot ox oy oz 
 
 Sx^ Sy2 Sz2 
 
 V^T = 
 
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; 
 
 d^G. = -k 1^ dF dt (3) 
 
 -■- on 
 
 in which 
 
 c vapor concentration, kg/cu m 
 
 Sc/Sn 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: 
 
 ^ = kV^c (4) 
 
 dt 
 
 The agreement between equations (l) 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 
 T^ and Tg of the heat transfer correspond to the gas concentrations 
 
 c^ and cq 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 gas-velocity component 
 normal to the body surface. This difference is easy to see if a one- 
 dimensional diffusion process is studied in a tube under steady-state 
 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 G-|_ kilograms per hour of ammonia 
 
 diffuses through the tube in one direction, and a quantity of air Gg 
 in the other direction. If x is a running coordinate, c-|_ the 
 ammonia, and cg the air concentration, then 
 
 ^1 = -^1 dF (5) 
 
 and 
 
 G2 = ^2 ^ (5a) 
 
 dc2 
 
4 NACA TM 1367 
 
 where k^^ and kg are the diffusion coefficients. If the partial 
 pressures of ammonia and of air are p-, and pg, respectively, then 
 
 dpi 
 GiRiT = -ki -^ (6) 
 
 and 
 
 dp 2 
 GgRgT = ^2 -^ ^g^j 
 
 Integration of equations (6) and (6a) yields 
 
 GiRiT = -kipi + Ci (7) 
 
 and 
 
 G2R2T = kgpg + Cg (7a) 
 
 hence a linear variation of partial pressures. If gravitational in- 
 fluence is now ignored, the total pressure pg in the tube is constant. 
 
 Hence, 
 
 Pi + P2 = PO (8) 
 
 It then follows from equations (7) and (7a) that 
 
 ki = k2 = k (9) 
 
 and 
 
 G^RiT = G2R2T = Vpo (10) 
 
 that is, the same volume, evaluated at the total pressure, diffuses in 
 both directions. "^ Moreover, with the notation of figure 1, 
 
 V=| (pi- -Pi") = I (P2" -V2') (11) 
 
 and 
 
 ^1 = I (ci' - ci") (12) 
 
 3 
 
 These are obviously in error; x lacking. 
 4 
 ki and k2 are equal, but it does not follow from this argument. 
 
NACA TM 1367 
 
 as well as 
 
 G2 = f (C2" - C2') (12a) 
 
 Therefore, according to this example, the partial pressure and the con- 
 centration of ammonia vary linearly along the tuhe 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 counter diffusing. 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 
 
 dcp 
 Gg = = UC2 + k -^ (13) 
 
 or, after introduction of the partial pressure, 
 
 <iP2 , , 
 
 UP2 = - k -g^ • (13a} 
 
 For the ammonia stream the corresponding equations are 
 
 dc-|_ 
 Gl = uci - k ^3^ (14) 
 
 and 
 
 GiRiT = upi - k -^ (14a) 
 
6 NACA TM 1367 
 
 Now again 
 
 Pi + P2 = PO (8) 
 
 It follows then from equations (l3a) and (l4a) that 
 
 k ^Pl 
 
 u = 
 
 ^0 ^1 
 
 p dx 
 
 (15) 
 
 If equation (l5) is substituted for u in equation (l4a), the following 
 expression is obtained for the partial pressure of ammonia: 
 
 kPo fiPi 
 GlRlT = - 5^—^ d^ (14b) 
 
 the solution of which, with regard to the boundary condition Pi = Pi' 
 at X = 0, is 
 
 .-^ -^ X 
 
 Po - Pi '""'^ 
 
 -2 L. = e (16) 
 
 Po - Pi' 
 
 Therefore, no linear variation of partial pressure along the tube axis 
 occurs in this case, but rather a logarithmic. If P3_ =: p^" at x = I, 
 
 kpo Pq - Pi" 
 ^1 = RiTI 1^ i^^^ (1-7) 
 
 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- 
 pressiire gradient. First, if p-|_ is small compared with Pq, it is 
 again true tnat 
 
 From equations (l5) and (l6) it follows that 
 
 p m 
 
 ^ = Gi ;^ = Vo (19) 
 
 Po 
 
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 p^.. 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 G-i , hence at low 
 partial pressures of ammonia, can u be taken as zero. Only then does 
 similarity of the boundary conditions aljo exist. This will be dis- 
 cussed later. Similarity requires, however, identity of the hydro- 
 dynamic equations as well. In the experiment of Thoma, the 'temperature, 
 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 
 
 w _ ^w 
 
 To ro 
 
 (20) 
 
 In equation (20) y"w ^^^ Tn ^^^ ^^^ densities of the ammonia-air 
 
 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 
 heat-transfer 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 the air evaporates. 
 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. 
 
 Stated precisely as written by Nusselt, 
 
NACA TM 1367 
 
 I. MODEL EXPERIMENT OF THOMA 
 
 A body having a surface F and temperature T is cooled by an 
 air stream having a speed Wq and a temperature Tq. 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 Wq and is mixed with ammonia to the concen- 
 tration CQ- Through the diffusion process, which occurs at the body 
 surface, the ammonia concentration has already attained the value c^. 
 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 = \L(T^ - Tq) f \-^ , -Y- (21) 
 
 where 
 
 X thermal conductivity of air 
 
 L dimension of body 
 
 f initially unknown function of two dimensionless fractions, 
 dependent on shape of body 
 
 X density of air 
 
 T\ viscosity of air 
 
 Cp specific heat of air 
 
 For diffusion, correspondingly. 
 
 ^WqT 
 
 G = kL (CO - cj f (_^ , M^ (22) 
 
NACA TM 1367 
 
 From these is obtained the relation sought: 
 
 Q = X \ gn X y ^v " ^' 
 
 G k ^ /Lvpr r^\ Cq - c 
 
 (23) 
 
 It follows from most of the experimental work that the function f may 
 be represented as a product of two functions, that is, 
 
 n 
 
 f = b --^ -^ (24) 
 
 where the constants are dependent on the shape of the body. Reference 
 11 shows that, for the flow of gases throiigh a cylindrical tube, 
 
 m = 0.786 
 
 and y (25) 
 
 n = . 85 
 
 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 Wq. In this equation, the fraction 
 
 wye /x as a factor can be taken out. In the function f it must 
 
 then follow that m = n. Since ilc 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 \/i]c^y. 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 = l/2 = 0.50. Lately Schiller 
 and Burbach (ref. 15) have again grappled with this question. They find 
 
10 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 
 
 G k V X / CO - Cv 
 If the heat-transfer coefficient a is calculated with the use of 
 
 Q = aF(Tw - To) (26) 
 
 then 
 
 G X /" rcpk ^ 1^ 
 
 '0 
 
 "Hl^^HT^ (^^) 
 
 Relation of Thoma-Lohrisch 
 
 In the derivation of his relation whereby the heat-transfer 
 coefficient may be calculated from a diffusion experiment, Thoma pro- 
 ceeds on the basis that the following condition holds: 
 
 k = ^ (28) 
 
 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 
 
 Sn ^0 " ^w 
 
 Sc <=Q - c^ 
 an 
 
 (29) 
 
 holds at any arbitrary point in the fields. From equations (l) and (3) 
 the relation sought follows immediately: 
 
 Q ^ X ^w ~ '^0 
 G ~ k Co - c^ 
 
 (30) 
 
NACA TM 1367 11 
 
 which, using equation (28), can be also written 
 
 a = cpr l2Ll^ (30a) 
 
 G ^ cq - c^ 
 
 If the heat-transfer coefficient a is calculated with equation (26), 
 there is obtained 
 
 a = - yc^ (31) 
 
 F ' P Cq - c^ ^ ^ 
 
 Equation (28) is not valid in the diffusion experiment with ammonia in 
 air, however. 
 
 According to Thoma, it is considered that 
 
 ^ = 1.25 (28a) 
 
 TCpk 
 
 In this case, Thoma uses equation (30a) as well as (3l). As a 
 correction factor he multiplies their right sides by the constant of 
 equation (28a) . He therefore obtains 
 
 ^ = -^ re ^^ " '^Q (32) 
 
 G rcpk P Cq - c^ 
 
 from which equation (30) is obtained. Then it follows that 
 
 a = 11 " (32a) 
 
 F k Co - c^ 
 
 Equation (27) assumes this form if n =0. 
 
 On the basis of this equation (32a j, Thoma derives, from similarity 
 and impulse considerations^ the equation for heat transfer: 
 
 Q= gCpTlKT, _To)4^j (33) 
 
12 NACA TM 1367 
 
 If this equation is written in the form 
 
 C T) 
 
 it is recognized as a special case of equation (21), to the extent that 
 
 \gn X / X *l gT] i ^ ' 
 
 is valid. From equation (33) it follows that 
 
 gc TiL /lw r\ 
 
 which may also be written 
 
 gc„Ti . /lw r \ 
 a= X-^i cpl-^ J (35a) 
 
 Thoma now uses equation (35a) to derive equation (32a) from (31). The 
 author cannot follow Thoma 's reasoning, because if f is replaced in 
 equation (23) by Thoma 's relation of equation (34), the following is 
 obtained: 
 
 ^ = rcp ^lJ-!2 (30a) 
 
 G ^ Cq - c^ 
 
 and 
 
 ^ = F ^^P -^z^rrz (21) 
 
 Thoma 's superaddition leads back, therefore, via equation (34), 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 13 
 
 dQ = a.dF{ly - T) (36) 
 
 wherein T is the temperature in the center of the stream cross section 
 that encounters the body sui-face element dF. 
 
 Moreover, the equation t 
 
 Q = VQTCp(T2 - Tq) (37) 
 
 applies, in which Vq is the air volume swept out by the body in unit 
 time, and To - Tq the temperature rise of that air caused by the hot 
 
 body. From equations (36) and (37) it follows" that 
 
 If 
 
 (39) 
 
 (38a) 
 
 ^2 ■ 
 
 -To 
 
 4 
 
 Tw ■ 
 
 - To 
 
 vorcp ^^ 
 
 1 
 
 then 
 
 ^= F - 1 - K 
 
 In the diffusion work for the same air volume Vq, there is 
 obtained 
 
 Co - c 
 t = 
 
 "2 • (40) 
 
 =0 - ^w 
 
 provided Co 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 
 
 C = e (41) 
 
 and hence the heat-transfer coefficient sought: 
 
 It follows only if certain additional assumptions are made. 
 
14 NACA TM 1367 
 
 a = ^Q^^'P m ^ (42) 
 
 F 1 - e 
 
 In the general case, Thoma and Lohrisch, as in the preceding, follow 
 equation (28a) and put 
 
 a = ^i_ !0I!p m _L_ (43) 
 
 re k F 1 - e 
 
 hence. 
 
 a = Z2 ^ In ^ (43a) 
 
 F k 1 - e ' 
 
 The correct relation is obtained as in the following. The 
 temperature To is eliminated between equations (37) and (38), and 
 
 a.!°ftln ^ («) 
 
 ^ " VorCplT„ - To) 
 
 Now, according to equation (23a) 
 
 If it is noted that 
 
 G = Vq(co - eg) (45) 
 
 then 
 
 -0 " '^w 
 
 tTT^'eMx ) 57^17 '*^' 
 
 is obtained, and, therefore, from equations (44) and (40), 
 
 a=!2l!Pin L___ (47) 
 
 UCpkj 
 
NACA TM 1367 15 
 
 This equation should 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 c^ = 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. 
 
 Boundary -Layer 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, 
 
 « - ^'^" - ^°' (43) 
 
 and for diffusion 
 
 ^^•=0 - ^w) 
 G = — (49) 
 
16 NACA TM 1367 
 
 Accordingly, there is obtained on the basis of elementary boundary- 
 layer theory 
 
 T.. - T, 
 
 a = i '^ " '0 (50) 
 
 and also 
 
 G k CQ - c^ 
 
 G \ 1 
 
 a= "1 L i (51) 
 
 F k Cq - c^ 
 
 which is identical with the equation (32b) of Thoma and with equation 
 (27) when n = 0. 
 
 Impulse Theory 
 
 Impulse theory is a carry-over of gas -kinetic considerations into the 
 domain of turbulent motions of a fluid. In its simplest form it is as- 
 sumed, as an explanation of heat transfer, that a volume of gas V having 
 the temperature Tq moves from the turbulent fluid stream to the wall, 
 where it is heated to a temperature T^ and then brought back to the core 
 of the fluid stream. It therefore takes from the wall the heat 
 
 Q = VrCp(T^ - Tq) (52) 
 
 which is given up to the fluid. It must at the same time be true that 
 
 a = P (53) 
 
 F 
 
 The volume V contains in the case of diffusion c„ kilograms of 
 
 ammonia. On impact against the body surface, ammonia is absorbed until 
 the concentration is c^. Hence, 
 
 G = V(co - c^) (54) 
 
 From equations (52) and (54) it follows that 
 
 Q '^w " ^0 f^^s 
 
 5 = r^P J^^^ (=^' 
 
NACA TM 1367 17 
 
 7 
 and, accordingly , 
 
 a = I re i (56) 
 
 F P Cq - c^ 
 
 which follows also from equation (27) when n = 1. 
 
 Comparison of Boundary-Layer 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 
 T and concentration Cg are then, respectively, different from 
 
 Tq and Cq. Within the boundary layer, exchange occurs in accordance 
 
 with boundary-layer theory. In the free gas stream, impulse theory 
 applies . 
 
 Then, in the boundary layer, 
 
 Q = ^ F(Tv - Te) (57) 
 
 and 
 
 y 
 
 Hence, 
 
 G =1 F(ce - c^) (58) 
 
 Q ^ X ( \ - ^e 
 G ^ k \cg - c^^ 
 
 (59) 
 
 In the turbulent stream, on the other hand, 
 
 Q = VrCp(Te - To) (60) 
 
 and 
 
 G = V(co - Cg) (61) 
 
 therefore, 
 
 Q "^e ~ '^O /^„v 
 
 = rc^ — — (62) 
 
 G - ' "P CO - Cg 
 
 7 
 Equation (56) is identical with (31), 
 
^^ NACA TM 1367 
 
 It can now be assumed that, approximately, 
 
 
 = a 
 
 that is, is equal to a constant.^ Also, 
 
 =0 - ^w 
 From equations (63) and '(64) it follows that 
 
 and 
 
 ^0 - ^e 
 
 CO - c^ 
 In that case, equations (59) and (62) become 
 
 Q _ X a Ty - Tq 
 G ~ k b cq - c^ 
 
 and 
 
 Q ,^ 1 - a ^w ■ 
 
 - To 
 
 G ^-^ 1 - b CO ■ 
 
 - Cy 
 
 (63) 
 
 = b n, (64) 
 
 l^ ~ l^ = 1 - a (63a) 
 
 Ty - To 
 
 = 1 - b (64a) 
 
 (59a) 
 
 (62a) 
 
 Recently, Prandtl (ref . 16) has given for this quantity the value 
 
 gCr)!! 
 
 a = /. . ^Odr 
 
 8| 
 
 P^ e '\'-ii- 
 
 (^->fi 
 
 .Qdr 
 
 in which the value of the parameter e is uncertain. It lies between 
 1.1 and 1.75. 
 
NACA TM 1367 
 
 19 
 
 From the last equations, 
 
 a ^'^p^ 
 ^ = a - (1-a) -T 
 
 (65) 
 
 There is obtained, therefore, the desired relation: 
 
 G 
 
 X 
 
 k 
 
 rc„k 
 a + (i-a) — ^ 
 
 w 
 
 -0 
 
 -w 
 
 (66) 
 
 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/\. Since n > and < a< 1, both relations lead 
 to increasing values [of Q/G] with increasing values of ircpk/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 expression^ 
 
 n = 1 
 
 (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 air-streaming 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 
 
 This, from a + (l-a) (l+A) = (l+A)'^, 
 
 ora+l+A-a-aA= l+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 xq and the specific humidity of which is Cq. At 
 the water surface the humidity content of the air is c^. 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 
 
 (68) 
 
 where |j. is the apparent molecular weight of dry air and \i-^ that of 
 
 damp air. For the diffusion field, diffusion equation (4) then applies, 
 The Navier -Stokes 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, 
 
 ^0 dw _ 
 
 ■g" dt - ^ 
 
 + TlV^W 
 
 (69) 
 
 In this expression, the air density Xq can be assumed constant. 
 A similarity consideration leads to the expression for the concentration 
 gradient at the water surface: 
 
 dc 
 dz 
 
 ^w - Cq 
 
 $ 
 
 3 fJL 
 
 ^ A^l 
 
 - 1 
 
 )(cw 
 
 Co) 
 
 BT 
 
 0' gTl 
 
 (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 (l9) and the following value of c-. 
 
 ^ ~ RT 
 
 (VI) 
 
NACA TM 1367 21 
 
 are substituted in equation (14), then 
 
 -kF dc 
 
 G = 
 
 where p is the total pressure. 
 
 Pw dz ("72) 
 
 ~ P~ 
 
 If the concentration gradient is now replaced by its equivalent 
 according to equation (70), the rate of vapor evaporation becomes 
 
 G = k -; r «- 
 
 F^ 
 
 
 1) (cw • 
 
 - co) 
 
 Tig 
 
 ' kr 
 
 
 g.^ 
 
 
 (73) 
 
 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 Wq. 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 
 
 w 
 
 
 G=kL(c„ - Cq) f (^, ^) (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 50° C, and having a dimension on a side of 0.5 meter is 
 cooled by an air stream having a temperature of 20° C. For w > 5 
 meters per second, the following value was obtained: 
 
 7ft 
 a= 6.14WQ ' kcal/(m2)(hr)(°C) (74) 
 
 It follows that in equation (24), if m = n, 
 
 .0.78 
 
 (^) 
 
 f = 0.065 I ^ I (75) 
 
22 NACA TM 1367 
 
 In equation (22) there is obtained, accordingly, 
 
 0.78 
 
 m 
 
 f = 0.065 [-T- I (75a) 
 
 Therefore, the expression for the amount of water evaporating from 
 a water area of F square meters in an hour becomes 
 
 , .0.22 
 
 fk\ 0.78 , , , , , 
 
 G = 39F [j^j Wq (cw - Cq) kg/hr (76) 
 
 where k is the diffusion coefficient according to the research of 
 Mache (ref. 18): 
 
 k = 
 
 Z8(_T_j- 3,,/,, („) 
 
 in which the total pressure p is to he used. 
 
 Evaporation with Heat Exchange in Still Air 
 
 It will now be assumed that the temperature T of the water sur- 
 face is different from the temperature Tq 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 air-lifting 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 = ror(T - Tq) + (c - Cq)^^ - ij 
 
 (68a) 
 
 Here T and c are the temperature and specific humidity at any place 
 in the field, Xq is the specific gravity of dry air at a great dis- 
 tance from the water surface at which the conditions pg and Tq 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 = 1/Tq (78a) 
 
NACA TM 1367 
 
 23 
 
 approximately. Then 
 
 -^0 
 
 To) + (^ - Co) 
 
 
 
 ik - ^) 
 
 (68b) 
 
 "becomes the value of the body force in the equation of motion (69). 
 Moreover, equation (2) of heat conduction and equation (4) of diffusion 
 must be used. With the abbreviations 
 
 and 
 
 E = 
 
 ■D 
 
 L-'tq 
 
 '(Tw - 
 
 To) 
 
 
 
 c = 
 
 gCpTl 
 
 
 
 'o 
 
 (^ 
 
 
 C 
 
 w 
 
 =o' 
 
 gTl 
 
 D = 
 
 kTo 
 
 > 
 
 similarity considerations lead to the relations 
 
 (79) 
 
 G = kL(c^ - Cq) #1 (B,E,C,D) 
 
 (80) 
 
 and 
 
 Q = XL(T„ - Tq) $2 (B,E,C,D,) 
 
 (80a) 
 
 where $ -^ and $2 are initially unknown functions of the variables 
 B, E, C, and D. 
 
 The formulation is significantly simpler and clearer if it is 
 assumed that equation (28) applies, that is. 
 
 k = 
 
 p' 
 
 for then 
 
 C = D 
 
 (28) 
 (28a) 
 
24 NACA TM 1367 
 
 In this case T and c are proportional to each other. Equation (29) 
 applies and, then. 
 
 G = k(c, - Cq) = ^O^P^^T^ (81) 
 
 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), 
 
 $-^ = $2 (81a) 
 
 which is valid only if equation (28) obtains. For the determination of 
 the function * there is introduced, as a matter of expediency, a new 
 dependent variable: 
 
 To /n \ 
 
 S = T - Tq + — ^- -ij (c - Co) (82) 
 
 which then leads to the differential equations 
 
 To dw To 2 
 Tdt = T^^ + ^'^ 
 
 and y (83) 
 
 rocp ^ = XV2S2 
 
 from which, through consideration of similarity involving the gradient 
 at the water surface, it follows that 
 
 ,) 
 
 ^ T„ - To + ^ (^ - ^) (=v - co^ 
 
 This gradient can also be calculated by the use of equation (82), from 
 which it follows that 
 
 aa dT , ^0 
 
 dK ^ dT _^ ;^ /M_ _ -j^\ dc 
 5n Sn Y'o \^-\ J ^ 
 
 (85) 
 
MCA TM 1367 
 
 25 
 
 If the divergence ("speed of expansion") u at the water surface is 
 considered negligible, equations (l) and (3) apply: 
 
 Q = -XT T- 
 on 
 
 and 
 
 G = -KF 
 
 5^ 
 
 (1) 
 (3) 
 
 From equations (l), (3), (81), (84), and (85 
 
 l^(cw - Co) 
 
 G = 
 
 f (B+E), D 
 
 and 
 
 ^ ^ XF(T^^_-Co) ^ 
 
 b+E), cl 
 
 (86) 
 
 The second of the equations (86) must also apply for the case 
 c^ - cq = 0, and therefore when E = 0. The equation then goes over 
 
 into the usual form for heat transfer. If the latter is known, the 
 rate of evaporation G can thus be calculated. 
 
 On the basis of heat-transfer research (ref. 19), for large values 
 of B/c, the following can be written: 
 
 Q = ClX I (§) (T^ 
 
 F 
 
 "L5r^Cp(T^ - Tq)^" 
 
 1/4 
 
 XTlT, 
 
 
 
 (87) 
 
 in which the coefficient C-, is dependent upon the form and orientation 
 with respect to gravity. 
 
 With equation (87), one now obtains from equation (86) 
 
 ^ , F ^ Ib + E , s 
 
 L^ro(T, -To) l'(^- i)K - Cq) 
 
 kTiT 
 
 
 
 Tlk 
 
 (c^ - Cq) (88) 
 
26 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, j, t]j 
 and Cp pertain to the vapor-air mixture and, indeed, are to be taken 
 as mean values over the whole field. The term k is given by equation 
 (77). The thermal conductivity of air X is in the following form in 
 reference 10: 
 
 0.00167(1 + 0.000194T) Vt' , , , ^ 
 
 X = ^ 117 ^ ^ kcal/ (m) (hr ) (°C ) (89 ) 
 
 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 vapor-air mixture can be assumed equal to that of 
 air. The terms x and Cp are to be calculated according to the 
 
 relation appropriate for a mixture of gases. Thus, there are obtained 
 
 at t = 50° C 
 
 (90) 
 
 ^ =0.87 0.84 
 
 kyc 
 
 P 
 
 This fraction is different from unity for this vapor-air mixture. 
 Accordingly, equations (80) and (80a) apply. Since equations (30) and 
 (80a) become equations (87) and (88) in the limiting case C = D, in 
 general, in the first approximation, the following can be written: 
 
 G = 
 and 
 
 CikF(c„ - cq) 
 
 f^ (91) 
 
 XF(T^ - Tq) 
 
 -1/¥ 
 
 Q = Ci L y — (92) 
 
 Then the following is obtained: 
 
 4 
 
 l=^xtl^ (-) 
 
 or 
 
 
 X ^w - ^0 
 
 k^^p^O T„ - To 
 
 (94) 
 
NACA TM 1367 27 
 
 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 water-vapor 
 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 Tq is greater than the temperature of the 
 water surface T^, it is possible, for finite values of Tq - T-^^ and 
 Cw " cq, 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 
 
 there is obtained 
 
 and 
 
 ^^^=^f^-l) (95a) 
 
 G = CnkL(c^ - Crs) (96) 
 
 Q = C2XL(Tq - T^) (97) 
 
 Therefore, 
 
 G k cw - cq 
 
 (98) 
 
 Q ^ Tq - Tv 
 The coefficient Co 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 Wq flows by a water surface. However, 
 
 the water temperature T^ and the air temperature Tq 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, 
 
28 NACA TM 1367 
 
 at 50° 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: 
 
 Lworo 
 Re = —^ (79a) 
 
 On the basis of similarity theory one has again, first of all, the 
 relations 
 
 and 
 
 In these, in the case of a smooth water surface such as a tank,-*-^ L is 
 the principal dimension; and for a water drop, it is the diameter. At 
 the water siirface there exists, further, between the rate of evaporation 
 and the concentration gradient the following relation: 
 
 F = uc^ - k 3^ (14^ 
 
 where, according to equation (l9), 
 
 u = -^ (19a) 
 
 and R-i , in this equation, is the gas constant of water vapor, and p is 
 the total pressure. There is then obtained 
 
 uc^ = ^ (100) 
 
 in which p^ is the partial pressure of water vapor at the water surface. 
 If the value of equation (lOO) is used in equation (l4a), the concen- 
 tration gradient at the water surface becomes 
 
 . ^ . ^ ^\- ^ ^ (101) 
 
 dn kF 
 
 lOor pond or pool. 
 
MCA TM 1367 
 
 29 
 
 By the same reasoning, the following applies for heat transport: 
 
 Q 
 
 = UTc„ (T^ - To) - X 
 
 3t 
 
 (14b) 
 
 and, with equation (19a), 
 
 -i = xT[^-^^^p(^w-To)] 
 
 (102) 
 
 From equations (99), (99a), (lOl), and (102), the relation sought 
 between Q and G is 
 
 Q _ / Pw \ X(T. - To)f(Re,C) ^^ 
 
 G- [^ p j Mo^ - Co)f(Re,D) ^ n^ ^P^^w '^q) U03) 
 
 If the value of f given by equation (24) is substituted in 
 equation (103), 
 
 (24), 
 
 From equations (lOl) and (99a) there is obtained, with equation 
 
 G = 
 
 bkF(c^ - Cq) k, 
 
 m 
 
 (i-2?) ">" 
 
 and, at the same time, from equation (l03a), 
 
 (104) 
 
 Q = 
 
 bF(T^ - Tq) 
 
 2_ ^^gp^(cw - ^o) 
 
 Re 
 
 m 
 
 (105) 
 
 If c-^f = Cqj no evaporation occurs and equation (105) becomes 
 equation (24). 
 
30 
 
 NAG A TM 1367 
 
 Vapor Pressure at Water Surface 
 
 It is natural to assume that, at 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, Pw "= Ps* 
 
 At the same time, the vapor concentration at the water surface then 
 becomes 
 
 c - r - 
 ^w ' Trn 
 
 (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: 
 
 |T = Kb(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 Kq 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), 
 
 F=Pi(^"- 
 
 -w) 
 
 (108) 
 
 where 
 
 is the saturation concentration of water vapor at the water 
 
 temperature T^j and |3-|^ 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; 
 
 t,, °c 
 
 Ko, 1/hr 
 
 Pi, m/hr 
 
 92.4 
 87.8 
 82.1 
 27.5 
 
 0.0086 
 .0080 
 .0084 
 .084 
 
 148 
 133 
 108 
 925 
 
 Unfortunately, it is precisely in the technically important temperature 
 
NACA TM 1367 31 
 
 range between 0° and 50° 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 heat-transfer analog, as has already 
 been done in the treatment of burning and vaporization of the carbon in 
 iron ore (ref. l) . Set 
 
 G = pF(c" - cq) kg/hr (109) 
 
 from which the dimensions of the evaporation coefficient are 
 
 p = 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 p_ is inserted, where 
 
 4 
 k 
 
 P2 = CiJa5^ (111) 
 
 ^— 
 
 then, equation (91) becomes 
 
 G = p2F(c^ - Cq) (91a) 
 
 Further, equation (108) still applies. If the unknown concentration 
 c-jj is eliminated between equations (l08) and (91a), the evaporation 
 coefficient p in equation (109) becomes 
 
 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 fr + Cp(T^ - Ts)J (113) 
 
 In this expression, T^ is the temperature of the wetted thermometer, 
 Ts the saturation temperature corresponding to the partial pressure 
 
32 NACA TM 1367 
 
 p of the water vapor at the surface of the wetted thermometer, r the 
 heat of vaporization at the pressure p^, and c the specific heat of 
 the limiting curve [?] at the same pressure. 
 
 If Tq is the environment temperature, as measured with a dry 
 thermometer, the heat extracted from the surroundings is 
 
 Q = aF(To - T^) (114) 
 
 The heat-transfer coefficient a consists of two parts, several terms 
 of a sum a^, 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 
 
 ^s = Cs -^ i -T — (115) 
 
 where Cg is the radiation coefficient of water, that is, 
 
 Cg = 3.35 kcal/(m2)(°C*) (116) 
 
 and Tq ' is the mean temperature of the fixed body surrounding the 
 
 wetted thermometer with which radiation is exchanged. It is certainly 
 approximately equal to the ambient-air temperature Tq, 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 Tq ' = Tq and the attainment of room temperature is assumed, 
 equation (115) becomes, approximately, 
 
 ag = Cg (115a) 
 
 11 
 "This does not seem to be correct. 
 
 Dimension time"! apparently missing in equation (116), 
 
 12n 
 
NACA TM 1367 33 
 
 The heat-transfer coefficient a. 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), 
 
 x' 
 
 a-b = Ci ^ /^ 
 
 K^ (117) 
 
 in which the constant C-j^ 1^ dependent upon the configuration of the 
 thermometer well. For a cylindrical well of height H, 
 
 Ci = 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 plate-shaped thermometer, the following 
 relation (ref. 20): 
 
 ^ ^„ ^, 0.000028Re 
 
 X 0-^8 X A B + E , , 
 
 a^ = 0.069 L Re + 0.83 jj '\ I — q — e (119) 
 
 wherein it is supposed that the wind flows along the thermometer well 
 in a horizontal direction. 
 
 If, in the energy equation (113), the value of G from equation 
 (109) is inserted, and that of Q from (114), the following psy- 
 chrometer formula is obtained: 
 
 ^r + Cp(T^ - Tj] (c" - Cq) = (c^ + Cg) ^^ + ^ j (Tq - T^) (120) 
 
 For the diffusion constant of a plate-shaped, wetted thermometer, the 
 following is obtained with equations (119), (ill), and (117): 
 
 0.78 
 
 /wol\ 
 
 .0.78 4 -0.000028 [-^ J 
 
 32 = 0.063 Ht) * 0.83 I /\[iire (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. 
 
34 
 
 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, Wq 
 
 is the relative speed of the water and air with respect to each other, 
 equations (103), (104), and (105) apply for the heat transfer and 
 evaporation. If the heat-transfer coefficient a and the evaporation 
 coefficient Pg ^^^ now evaluated according to equations (26) and 
 
 (92a), 
 
 e-'^ 
 
 and, with equation (112), 
 
 
 (122) 
 
 n 
 a / Pw\ X f TqC k ^ 
 
 hX 
 
 3lL 
 
 rgCpTjA ^Cpk(Cy - Cq) / gT^ \ 
 
 Re 
 
 m 
 
 (123) 
 
 Material on the technical applications of the formulas here pre- 
 sented will soon be published elsewhere.-'-'^ 
 
 Translated by H. H. Lowell 
 National Advisory Committee 
 for Aeronautics 
 
 13 
 
 Since the transmission of the original manuscript to the editor's 
 
 office on May 16, 1929, the following papers have appeared: E. Schmidt, 
 Verdunstung und Warmeiibergang, Gesundheitsing. , 1929, p. 525.; R. 
 Mollier, Das ix-Diagramm fiir Dampf luf tgemische, Stodolafestschr . , 
 Zurich, 1929, p. 438; H. Thiesenhusen, Untersuchungen uber die 
 
 Wasserverdunstungsgeschwindigkeit in Abh'dngigkeit von der Temperatur 
 des Wassers, der Luf tf euchtigkeit und Windgeschwindigkeit, 
 Gesundheitsing., 1930, p. 113. 
 
NACA TM 1367 35 
 
 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: Verdunstungskiihlung . Forsch. -Art. , VDI, Heft 275, 1925. 
 
 7. Wolff: Untersuchungen uber die Wasseruckkilhlung in kiinstlich 
 
 beliifteten. Kiiiilwerken. Munchen, Oldenbourg, 1928. 
 
 8. Stefan: Versuche uber 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 Warmeubergang im Rohr. Forsch. -Arb. , VDI, Heft 89, 
 
 1910. 
 
 12. Nusselt: Die Warmeiibertragung an Wasser im Rohr, Festschrift 
 
 anlasslich 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: Warmeiibertragung stromender Flussigkeit in 
 
 Rohren. Phys. Zs., Bd. 29, 1928, p. 340. 
 
 16. Prandtl: Bemerkung iiber den W&meubergang im Rohr. Phys. Zs., Bd. 
 
 29, 1928, p. 487. 
 
36 NACA TM 1367 
 
 17. Nusselt und Jiirges: Die Kutilung einer ebenen Wand durch einen 
 
 Luftstrom. Gesundheitsing. , 1922, p. 641. 
 
 18. Mache, H.: Uber 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 Stromung im Rohr. Z.V.D.I., 
 
 Bd. 70, 1926, p. 763. 
 
NACA TM 1367 
 
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