July,
1911
T H E J O U R N A L OF I , V D U S T R I A L AlYD E.VGINEERI.YG CHE.1IISTR Y .
face condenser, t . e., heat flow from condensing steam t o circulating water, but recent tests of feed-water heaters show the same peculiarity as was mentioned in the cas' of double pipe brine coolers--a rise of the coefficient with water speed to values close to U = 1000, although few designers would think of exceeding U = 3 j o for these heaters. Passage of heat from condensing liquid t o boiling liquid a t lower temperature is a characteristic of evaporators of single and miiltiple effect, and for these U = 300 is a fair average value. It is interesting t o note that this same value also applies t o the generators of absorption refrigerating machines where the he'xt of condensing steam is given up t o rich aqua ammonia liquor. There is probably no single class more interested, consciously or unconsciously, in increasing the effectiveness of heat transfer surfaces than chemical manu-
455
iacturers, and there 15 likewise no class with the opportunities for securing the great mass of data necessary for the formulation of laws of design with equal ease. There is presented in this issue a paper on the subject, discussing the various physical constants involved, which, while it leaves much to be said on the subject, will serve t o open a thorough investigation of it in these pages. I t is hoped that all the users of heat transfer apparatus will send in letters and criticisms of this paper, possibly preparing additional papers, but more important than all else, large quantities of data on every form of heat transfer apparatus in their establishments, all of which we will undertake t o print as part of a campaign of improvements that is needed a t least as much, if not more, than any other single thing common to all interests. C. E. LUCKE.
ORIGINAL PAPERS. the true conductivity of the fluid. Beyond this, heat flow results not only from true conductivity, but also from actual transportation of portions of Received March 22, 1911. the fluid in the direction of heat flow. This is called I t is the purpose of this paper to evolve and preconvection and it is the heat-transporting power sent a rational method of analyzing problems in of the fluid. The convectivity of a fluid a t any point heat transmission along the lines laid down by such depends on: ( I ) the increase in density of the fluid eminent and practical authorities as Hausbrand,' per unit change of temperature, (2) the density, Mollier,z and Berlowitz.3 The study of the flow or (3) the viscosity, ( 4 ) the mean distance from the transmission of heat has important bearings in mefluid boundary, and ( 5 ) the velocity. I t may further chanical, electrical, and chemical engineering, and be observed that in the plate, the heat flow is constant, it is a subject which merits R more systematic treatwhereas in the fluid, the heat flow is greatest a t the ment than is usually accorded t o it. surface of the plate and diminishes to nothing a t a The usual technical case of heat transmission is point or points farthest from the plate. where heat flows from a relatively hot fluid through From a practical point of view, i t is usual to consider a separating plate t o a cold fluid. The fluids are an abrupt drop in temperature from the plate surface usually air, hot gases, water, or steam; in chemical to the fluid to exist, and to assume that the f l * i d work, the number is almost without limit. The plate has the same temperature throughout. This temis usually thin and of metal, and may be flat., spherical, perature is the mean effective temperature of the or tubular in shape. Either or both fluids may be fluid, and, by definition, i t is the temperature i t would moved parallel or perpendicular to or against the plate, attain if completely mixed without gain or loss of or portions of the plate. I n heat flow, the temperature gradient may be heat. This is a temperature which is different for different parts of an apparatus, and while practically defined as the rate of change in temperature with its initial and final values may be readily measured, respect to distance in the direction that the heat is its intermediate values are obtained with difficulty. moving; in ordinary units, it is the drop in Fahrenheit For example, the temperature of water circulating degrees per linear inch in the direction of heat flow. through heating coils can be measured before i t When heat is moving uniformly from fluid t o fluid goes in and after i t comes out, but the mean effective through a plate, the temperature gradient is constant temperature a t any cross section of the coils is not throughout the pla.te. Here the temperature gradient so easily determined. is determined b y the rate of heat flow which is conDiagrammatically, this discussion may be made stant, and the heat conductivity (or resistivity) of clearer by the two following distance-temperature the plate material. I n the fluid, the temperature plots. Actually the temperature follows the curve gradient is not constant, it is a maximum a t the boundshown in A, but for convenience of treatment, it is ary surface and d.rops off as the distance from the boundary or plate surface increases. I n a n infinitesimal assumed t o follow the broken curve in B. This differentiates heat conductivity into two disfilm just next t o the plate, the temperature gradient tinct forms: ( I ) solid heat conductivity where heat is determined sole1.y by the rate of heat flow and by traverses a solid or a fluid a t rest; and ( 2 ) surface * Verdimpfen, Kondmsieren, and Kiihlen, 1909. heat conductivity where heat passes between a solid * Zeilschriff des Vereimes deufsche I n g e n i e w e , 1897. and a fluid. The first is the internal conductivity, Zeifschriii iiir Ap@aratenku?tde,1908. HEAT TRANSMISSION.
By HAROLD P. GURNEY,
,
454
T H E J O U R N A L OF I N D U S T R I A L A N D E N G I N E E R I N G C H E M I S T R Y .
and the second is the boundary conductivity or external conductivity when referred to a plate separating two fluids. The first form of conductivity is a physical characteristic or property of matter. The second form depends not a t all on the solid from
July, 19x1
The external resistance is the resultant and preceding sum of the boundary resistances of the two sides of the plate. The resistance of a square foot on one side is c1 and on the other side is Cll;separate resistances of both sides are C1/A and CII/A, and
= Ail + A-.
511
20
The external resistance of a square foot is zo 20 =
ZoA
C1
and
+
C11t
2=2 i + -Pa+ i-l l A A A' The resistance of a square foot of plate both internal and external is z. Then,
z
=
ZA
=
c1 + pa
+ cI1,
and AAT
Q = 1, + 96 + ill.
B
A
which heat is passing, but on the fluid and especially on such factors as ( I ) the conductivity, ( 2 ) the density and viscosity, (3) the expansiveness to heat, (4) the mean hydraulic radius of the space occupied b y the fluid, and ( 5 ) the mean fluid viscosity. I n order to investigate quantitatively the flow of heat, a system of notation is here used which is consistently adhered to. Q is the heat units in British thermal units which flow through a heat-transmitting plate in a time T hours, where the temperature difference between the two fluids on either side is A degrees Fahrenheit. Z is the heat resistance of the plate. A strict proportionality between the rate of heat flow and the drop in temperature may be assumed. The ratio of the temperature drop to the rate of heat flow may be defined as the resistance, the reciprocal of conductance.
These same facts may be expressed in terms of conductances and conductivities instead of resistances and resistivities, but i t must be observed that where resistances in series are additive, conductances in series-arernot additive. The resultant of conductances in series is the reciprocal of the sum of the reciprocals of the separate conductances. This results from the fact that conductance is the reciprocal of resistance and is the ratio of heat flow t o the temperature drop. I P G be used to designate conductance, then, Q,
G
--T
rate of heat flow temperature drop
I ) -
A
hence, Q = ATG.
The conductance G is the resultant-of the internal conductance GI and the external conductance G,. G
=
I
-1 + - Go1 GI The resistance Z is made up of two separate component resistances: ( I ) the internal resistance Zl of the plate, and ( 2 ) the external resistance Zo, the sum and resultant of the boundary resistance of the plate. = 2, zo The resistance possessed by a portion of the plate of one square foot area and one inch in thickness is the specific internal resistance or the internal resistivity a n d is designated b y p. The area of the plate is A square feet, the thickness is 6 inches, and the internal resistance is 2,. The plate resistance is proportional t o its thickness and t o the inverse of its area.
z
+
n.
The internal resistance of a square foot of plate of thickness d inches is zl. Z, =
Z,A
= p6
The specific internal conductance or the internal conductivity is the conductance of a portion of plate of one square foot area and one inch thickness and is designated b y A' the reciprocal of p . Since the internal conductance is proportional t o the area of the plate and inversely as its thickness, then, Al G, =-. 6
The internal conductance per square foot is g,. G,
1
gl=G= s
The boundary conductances per square foot, or the boundary conductivities are r1 = r/C1 and rll = r/Cl, and Gois the external conductance. A G o = __
-I = _1 7-1
7-11
The external conductance of a square foot is g,.
TH.E J O URhTAL OF I N D U S T R I A L A N D E N G I N E E R I N G C H E M I S T R Y .
Then the combined conductance G of the plate, both with respect to internal and external conductances is given by the following expression : A
= -
: :1
1
rl
_ _3 + -1 r rll
The conductance per square foot is G G
I
-
>
- _ _-rI1 rl
"A'
1
0
'
-~
1
7
and
a=
I
r1
Am
+ 1 + -rll 6
1
I n most heat-transmitting apparatus, the temperature drop is not the same a t all points of the drop. The equations deduced hold for infinitesimal plate areas, but b y adopting a mean temperature drop in the place of A , they may be applied t o finite areas. The maximum and minimum temperature differences may always be obtained from measurements on the temperatures of both fluids both before and after transit through the apparatus. The simplest mean tepperature difference would be either the arithmetic average, or the geometric mean of the maximum and minimum, but the former is too high, while the latter is too low. The most rational mean is the logarithmic mean and i t is obtained b y considering the temperature difference to vary relative t o time, or distance traversed a t a rate proportionate t o its instantaneous value. If A, is the minimum temperature difference, A , the maximum temperature difference and Am. the mean temperature difference, then the latter may be expressed in terms of the two former b y the following expression: 1,- A0 Am
=
Let the ratio of A, t o
In A,-ln
A
4,be a variable, x . I-x
1 ,
=
A-1
--In x
An arithmetic mean would be A ,
+-I,
while
a
-
geometric mean would be A,\,'x. The following table brings out the relations between the three means
- +1
1:
x-1 ~~~
X.
2
In .z
G.
1.000 0.900 0,800 0.700 0.600 0.500 0.400 0.300 0,200 0.100 0.050 0.020 0.010
1.000 0.950 0.900 0.850 0.800
1.000 0.949 0.896 0.841 0.783
0.750 0,700
0.722
1.000 0 949 0.895 0.836 0.775 0.707 0.632 0.518 0.447 0.316 0.223 0.141 0.100 0.000
0 . OOQ
0.650 0,600 0,550 0,525 0,510 0.505 0,500
For design, the use of the geometric mean gives safer values than the logarithmic mean ; for investigation, the logarithmic mean should be used. Occasionally, the arithmetic mean may be used. Where both sides of a plate have the same area as in the case of a flat plate, no doubt arises as to the proper value of A ; but where the areas of both sides of a plate differ as in a pipe, the problem of obtaining a mean area presents itself for investigation. Since the mean heat-transmitting area is nearer in value to the area of the side where there is the greatest heat resistivity, a single mean area may be obtained by weighting the areas with the respective resistivities. The area A, has a heat resistivity, c l ; A,, has a heat resistivity, ill,and A,, is the mean area.
AAT
-
0.636 0.583 0.497 0.382 0.317 o.zi 0.215 0.000
457
C1-%
=
+
51 +
CllAll 511
For a pipe A, and A, being the internal and external areas, Mollier gives the following formula:
-A.
A1 ' A,,
3
&
A, ' .41,
A rough rule is to employ as mean area the area whose resistivity is greatest, except where the areas have about equal resistivities, and then a n average is quite close. Apparatus for transmitting heat between fluids may be classified under four types: ( I ) counter-current, ( 2 ) parallel current, (3) perpendicular current, and (4) single current. I n counter-current apparatus, the fluids move in opposite directions, and the temperature difference does not generally vary greatly. I n parallel current apparatus, the fluids move in the same direction; the temperature difference is a t first large and rapidly diminishing, later it is small and slowly diminishing. I n perpendicular current apparatus, the fluids move in directions a t right angles to each other, and in this respect stands midway between counter-current and parallel current apparatus. I n practice, perpendicular current and counter-current apparatus are sometimes placed in series and so become essentially parallel current or counter-current in effect. I n single current apparatus, one fluid, and usually the fluid whose temperature i t is desired t o modify, remains in the apparatus during the entire operation while the heating or cooling fluid moves through a t a constant rate. The use of the single current apparatus makes the process intermittent; with other types, the process is continuous. For purposes of investigation and design, all four types of apparatus may be treated in essentially the same manner. When investigating a heating or cooling apparatus, the initial and final temperatures of both fluids are measured. From this the maximum and minimum, and consequently the mean temperature difference may be obtained. The amount of heat transmitted can be found by multiplying the weight of either fluid into its specific heat, then this into its change in temperature. If necessary, the heat lost in radiation
458
T H E J O U R N A L OF I N D U S T R I A L A N D E N G I N E E R I N G C H E M I S T R Y .
from the exterior of the apparatus may be taken into account. Knowing the area of the heat-transmitting plate or tubes, all data necessary to compute the heat-transmitting capacity of the walls is a t hand. When designing an apparatus, the heat-transmitting capacity of the plates must be previously estimated from data available or assumed. A preliminary assumption on this point t o determine within certain limits the velocities of the fluids, and from this a closer final assumption may be made. The rate a t which heat is t o be transmitted is obtainable from the weight, specific heat, initial and final temperatures of the fluid whose temperature is t o be raised or lowered, and the length of time assigned t o the operation. With respect to the fluid supplying heat or cold, its initial temperature and specific heat may be regarded as fixed, but the amount delivered per hour and final temperature are interdependent variables. The most logical starting point in design is t o make x a variable ratio of the variable temperature difference t o the fixed temperature difference. For any value of x the rate of supply of heating or cooling fluid and the required surface, hence approximate size of apparatus, is easily computed. As x increases, that is, as a larger temperature difference and a smaller temperature change is obtained, the fluid 'supply rate becomes greater but the size of the apparatus becomes smaller. Against x as abscissae may then be plotted as ordinates the total amount of depreciation, interest, maintenance, and rental on apparatus plus the cost of fluid together with the pumping and storing of the same. The curve obtained will have a minimum, and the best value of x t o adopt should be as near t o this minimum as possible, a t the same time taking into consideration other points of plant economy t h a t would tend to shift this value. In the design of single current and perpendicular current apparatus, two variables similar t o x may be chosen instead of one; a simple expedient in these cases is t o employ, as abscissae, the products of the two variables and then later try the effect of shifting the ratio of these variables. Owing t o the variety of factors which may come into play, it is not advisable to lay hard and fast rules for the solution of problems in the design of heat-transmitting apparatus. A perfectly general mathematical method might not be a t once soundly
July, 1911
theoretical or practical, but a brief outline of the simple mathematical aspects ought t o be presented. Two fluids move counter-current. A weight of hot fluid M, of specific heat u1 enters a t IO,, and leaves a t '0, undergoing a cooling degrees, while a weight of cold fluid M,, of specific heat u,, enters a t I r O O and leaves a t I 1 O 1 , changing in temperature by lcll degrees. When the object is t o heat the cold fluid, '8, is a variable and it may be made a function of x b y placing ' 0 - 1 1 0, = A and 10,- 1 1 0 ~= A x . Then, x-I or A - In x
A%= A&
'
Q is the total heat to be transferred and Q
=
Mlul.;, = M 1 l u l l ~ I neglecting I outside radiation losses.
When the object is t o cool the hot fluid,
110,
is the
Then, Am
=
At';
or A
x-I
--.In x
4
Parallel current heat transfer is graphically presented ab0ve.r When the object is to heat the cold fluid, '8, is variable; t o cool the hot fluid, 1'8, is variable. Then, A,,,
M, =
Q
u,(A
- Az-T~,) MI1
-
=
A
x-I --,
In x
or
-
Q Az-7,)
and A -
Q -
' AmTg' I n perpendicular current heat exchange, a hot fluid of weight M,, specific heat ulr enters a t ' 0 , and is cooled t o a temperature ranging from I O l l t o 10, and undergoing a mean change of temperature 7 , . A cold fluid of weight MI,, specific heat ull, enters a t 1 1 O 0 and is heated t o temperatures ranging from ~ 1 0 t, o~ I r e l and undergoing a mean temperature increase of cI1 degrees. As before, Q = M,a,.r, = Mll 011711. Then,
ull( A
July,
1911
T H E J O U R N A L OF I N D U S T R I A L A N D ENGIhrEERING C H E M I S T R Y .
Glycerin. Alcohol
. .. , . . . .
... . . . . . . .
&I
“e,,
~.
I distance I In a single current apparatus for cooling, a hot fluid is cooled from ‘ 8 , to I O , . It is here assumed t h r t the temperature of all parts of the fluid that remains in the apparatus during the operation is the same a t any time, owing to the use of good stirring devices. The cold fluid enters a t I r e O and is heated a t first to “ O , , finally to l * B l l undergoing a mean temperature rise of T~~degrees. Again, (2 = M,U,.;~= M l l u l l ~ , l . Assuming the curve 1’8,I 1 O l 1 a downward parabola, then,
For an extknded mathematical study of problems Berlowitz, “Beitrag zur Berechnung der Heizflachen,” Zeitschrift fur Apparatcnkundc, 1908. In the design of apparatus, no data as t o conductivity values is as valuable t o the engineer as those obtained by experiment on similar apparatus under exactly the same working conditions. Such data may, however, be found rare, and if all conditions do not coincide, i t may be misleading. Such data, if used, ought to be carefully analyzed into its essential parts. In the lack of such data, resource must
.,.
,
,
1.1
0.9 0.8 0.16 0.1
0.5 0.9 1.1 1.3 6.3 10
Another table given by Lamb and Wilson may be added, although it corroborates certain values given above.
I
1.
.
Aluminum. . . . . . . . . , . , , , . , , . . . , Brass.. . . . . . . . . . , . . , , , . , . , , , . . . .
995-1005 590-735 . . . . . . . . . . . . 2085-2095 . . . . . . . . . . . . 480-470 . . . . . . . . . . . . . . . . 245-220 Tin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 445-315 Zinc. ., , . . . . . . 880 ................ 7.9 Crranite. . . . . . , , . . . . . . . . , . , , . , , . , 14.8-16.0 Marble. , , , .. .,.., ,, , ,. ,, , 13.6-16.3 V+’hite dry sand ... , , , , . 2.7 Compact sand , . . . , , , , . , . . , . , , , 1.48 Plaster of Paris., , . , , , , , , , , , , , , , 3.7 Pasteboard. , , , , . . . , , . , , , , , , , , 1.3 0.87 Fir (across grain), . . , , . , , , , , . , , 0.26 Hair f e l t . . . . , . . . . . . , , , . . 0.31-0.42 Charcoal, , , . . , , , , , , , , ,,,,, 0.44 Silicate cotton.. , , , 0.44 0.48 0.47 Air (no baffles). , , . , , , , , . .: , . , 0.57 Pure s a w d u s t . . , , , , . . , , . , 0.70 Dry asbestos , ., . , . 0.86 Sand.. . . . . . . . . . . . . . . . . 2.15
.... . . . . . . .... .
.
.
..
. .
of this type, the reader is referred to papers by M.
.... .. . . Air ..... . . . . . . . . Carbon dioxide . . %r.,
2
459
. .
.
. . .. . . . . .... .. . .. . . . . . . . . . . .
. . . .. . . . . . . . . . . . . . .. . . . . . .
P. 0.001 0.0017-0.0014 0,00048 0.0021 0.0043 0.00225-0.004 15 0.0012 0.12 0.065 0.07 0.37 0.68 0.26 0.77 1.15 3.9 2.8 2.3 2.3 2.1 2.1 1.75 1.4 1.15 0.47
Boundary resistivity is very difficult t o express in terms of formulae or equations, owing t o the number of factors upon which i t depends. The resistivity of a water boundary is usually expressed as a function of the mean velocity of the water where that velocity 1 2
Zeztschrqt des Vereanes deutscher Iwenteure, 1876.
\V Ernst. Kazserluhe Akadamze des Wzssenscheffen,1902.
T H E J O U R N A L OF I N D U S T R I A L A N D E N G I N E E R I N G C H E M I S T R Y .
460
is artificially impressed, but where the water is “still,” i t is not so expressed. The resistivity of denser and more viscous liquids is often referred t o the resistivity t h a t water would have under like conditions as a standard. The resistivity of an air boundary is also expressed as a function of the mean air velocity, but the air density has a marked effect, close to the inverse of its cube root. As air is usually a t atmospheric pressure, this is not taken into account. The heat resistivity of steam is expressed as a function of its mean initial velocity, although its density has a n effect much the same as with air. The steam resistivity is not commonly separated from the water resistivity on the opposite side of the plate, and a favorite formula makes the external resistivity a function simply of the plate or tube area. Finally, to allow for incrustation, heat resistance of the metal, oily surface, etc., a factor is often applied to the external resistance. A more rational way is to add the heat resistance of a known thickness of metal, plus the heat resistance of a n assumed thickness of scale and oil onto the external resistance. Between metal and water, the resistivity
c
+
0.005
[r = 60 z o o + ] where v is the 0.3 L, velocity of the water in feet per second according t o Mollier.1 When the water is not artificially moved g = 0.01[ r = 1001, although this will be affected b y the temperature difference somewhat, owing t o induced currents. For water stirred by mixing devices, g = 0.0025 - 0.0013 [ y = 400 - 8001. Ser*
+
=
places
c=
~
0.002 -~ tiV
[ r = 5001. ,I,
but this gives too
high a flow of heat. The external resistance zo of a square foot of plate with moving on both sides is the sum of the separate boundary resistivities. 20
-
d
