Heat transfer, diffusion and evaporation


Material Information

Heat transfer, diffusion and evaporation
Series Title:
Physical Description:
37 p. : ill. ; 27 cm.
Nusselt, Wilhelm, 1882-
United States -- National Advisory Committee for Aeronautics
Place of Publication:
Washington, D.C
Publication Date:


Subjects / Keywords:
Aeronautics   ( lcsh )
Heat -- Transmission   ( lcsh )
federal government publication   ( marcgt )
bibliography   ( marcgt )
technical report   ( marcgt )
non-fiction   ( marcgt )


The general similarity of heat and mass transfer (diffusion) processes is discussed, with particular reference to the lack of complete identity of the relations governing the two phenomena. It is indicated that, for example, the boundary conditions in the two cases at the surface of a body will not be the same. The correct equation of diffusion is given for various simple cases. Generalized relations for combined heat and mass transfer are then evolved for particular situations, comparisons being made among several different approaches to the problem. Finally, the effect of a buoyancy force field on the generalized relations is considered, with special reference to the evaporation of water.
Includes bibliographic references (p. 35-36).
Sponsored by National Advisory Committee for Aeronautics
Statement of Responsibility:
by Wilhelm Nusselt.
General Note:
"Report date March 1954."
General Note:
"Translation of "Wärmeübergang, diffusion und verdunstung." Z.a.M.M., Bd. 10, Heft 2, Apr. 1930."

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University of Florida
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All applicable rights reserved by the source institution and holding location.
Resource Identifier:
aleph - 003780007
oclc - 43313856
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Full Text
pu _~r 1(i




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. 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 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 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. 105-121.

NACA TM 1367

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)


r 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 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 water-vapor 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 partial-pressure
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

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)

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 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 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
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

GIR1T = -kI -pg

G2R2T = k2 -d7


Integration of equations (6) and (6a) yields3

G1RIT = -klP1 + C1

G2R2T = k2P2 + C2


hence a linear variation of partial pressures.
fluence is now ignored, the total pressure pO
Pl + P2 = PO

If gravitational in-
in the tube is constant.

It then follows from equations (7) and (7a) that

kl = k2 = k


G1RIT = G2R2T = VpO


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.



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.



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
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

G1 = uc1 k d- (14)


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

PO P kpo
= e (16)

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

u = G1 -p- = V0 (19)

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, 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

Tw w (20)

In equation (20) -w and y0 are the 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 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


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)


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

Q = Cpk Tw TO
G k X co cw


If the heat-transfer coefficient a is calculated with the use of

Q = aF(TV TO)

G X (rcpk I
F k X ) C0 Cy



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 -
Ck p

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


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


NACA TM 1367 11

which, using equation (28), can be also written

Q= epy (30a)

If the heat-transfer 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)

NACA TM 1367

If this equation is written in the form

Q = XL(Tw TO) c-9 \ / (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)

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

Q w T (30a)
G Co Cw


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.

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

= (39)

= VOyp 1n (38a)
F 1 -

In the diffusion work for the same air volume VO, there is

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 heat-transfer 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


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

a n (44)
VOrcp(Tw TO)

Now, according to equation (23a)

Q x fcyk\ 1
S- cpk (23a)
T T = CO o Cw

If it is noted that

G = VO(cO c2) (45)


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 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 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.

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,

Q = (48)

and for diffusion

kF(c0 cw)
G = (49)

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 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 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

a= p (53)

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 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
Te and concentration ce are then, respectively, different from
TO and cO. Within the boundary layer, exchange occurs in accordance
with boundary-layer theory. In the free gas stream, impulse theory

Then, in the boundary layer,

Q = 1 F(Tw Te) (57)

G = ) F(ce cw) (58)

Q-X e (59)

In the turbulent stream, on the other hand,

Q = Vcp(Te TO) (60)


G = V(c0 ce) (61)


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)


CO ce
= 1 b (64a)

In that case, equations (59) and (62) become

Q X a Tw TO(
G k b cO cw


Q 1 a Tw TO
-= ycpl (62a)
G= 1 b c0 cw

Recently, Prandtl (ref. 16) has given for this quantity the value


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 (1-a) (65)

There is obtained, therefore, the desired relation:

x L ycpk]'TW TO
S= + (1-a) 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.


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

9This, from a + (1-a) (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

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 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,

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)

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,
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):

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 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 = 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


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 =

gc pr

i i 1(



(c c )
w 0



similarity considerations lead to the relations


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.



The formulation is significantly simpler and clearer if it is
assumed that equation (28) applies, that is,

k- =
c pTO
ofor then
for then




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),


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:


= T To + o -1 (c c)

which then leads to the differential equations

TO dw rTO 2w
g dt TO



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


41 =" 2


NACA TM 1367

If the divergence ("speed of expansion") u at the water surface is
considered negligible, equations (1) and (3) apply:



G= -kF a-

From equations (1), (3), (81), (84), and (85):

kF(cy CO) r + ]
G = L f B+E), D

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.


for the case
then goes over
is known, the

On the basis of heat-transfer 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 -; X-TOT-

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)

kF B + E
S= L V D (w co)
= Clk k +
L krlT0

(cw C0)


cw -


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 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 condu-tivity 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 + --

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 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
S= 0.87 0.84
This fraction is different from unity for this vapor-air 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:
SCljF(cw co) B (91)
G= L (91)
kF(Tw TO B+E
Q = Cl L C (92)

Then the following is obtained:

G k fC eC cO
ji -(93)
Q- X D Tw To
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 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 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)


-To TW To 1 (95a)
cw c0 O (1 )

there is obtained

G = C2kL(cw CO) (96)


Q = C2XL(TO Tw) (97)


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:

Re = g-LU (79a)

On the basis of similarity theory one has again, first of all, the

6T Tw TO
= L f(Re,C) (99)

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),

u = l (19a)
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.


By the same reasoning, the following applies for heat transport:

= Urep (T TO) k


and, with equation (19a),

- F Q -G Cp1 (Tw TO]


From equations (99), (99a), (101), and (102), the relation sought
between Q and G is


X(T, TO)f(Re,C) P
k(cw co)f(Re,D) + r (T T)


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)

From equations (101) and (99a) there is obtained, with equation

bkF(cw cO) Rem
G =
pw ) LDn


and, at the same time, from equation (103a),


If cw = Co, no evaporation occurs and equation (105) becomes
equation (24).



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

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

= 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

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. 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 heat-transfer 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 i0-0 (115)

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 ambient-air 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 time-1 apparently missing in equation (116).
12This does not seem to be correct

NACA TM 1367

The heat-transfer 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


in which the constant C1 is dependent upon the configuration of the
thermometer well. For a cylindrical well of height H,

C1 = 0.83


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):

4 -0.000028Re
b 0.78 = 0 9 +
ab = 0.069 f Re + 0.83 f e


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 plate-shaped, wetted thermometer, the
following is obtained with equations (119), (111), and (117):

S0.78 4 B+ -0.000028 (--w

02 = 0.069 (+ 0.83 e
H .3i' D


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 heat-transfer coefficient
coefficient 02 are now evaluated according to

respect to each other,
heat transfer and
a and the evaporation
equations (26) and

Pwy YOcpk^
- X (Top k c (cw co)
k i


and, with equation (112),

a- (1


- 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


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 ix-Diagramm 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.


NACA TM 1367


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,

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






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