ENGINEERING AND PROCESS DEVELOPMENT
Analogy among Heat, Mass, and Momentum Transfer DEMONSTRATED BY DESCRIPTIVE DIMENSIONAL ANALYSIS RICHARD F. KAYSER Linde Air Products Co., Tonawanda,
N. Y .
T
IIE system of dimensions conunonly used imparts a physically meaningless dimension to many quantities. The object of this paper is to demonstrate that the nature of a quantity can be determined by meam of dimensional analysis if the physical properties of the dimensions are included in the analysis. Three different quantities-thermal conductivity and the coefficients of viscosity and diffusion-are analyzed dimensionally and related to one another; the analogy among heat, mass, and momentum transfer is demonstrated through this analysis.
not be separated from their differential operators if mauiniuni information is desired from the dimensional analysis. That many quantities may be physically described by means of descriptive dimensional analysis is shown by the following rxamples. Example 1. Relation of Coefficient of Viscosity to Mass and Momentum Transfer by Descriptive Dimensional Analysis. Figure 1 illustrates the quantities involved in the coefficient of viscosity and the dimensions used in the remainder of the paper. By definition
Many Quantities Are Physically Described by Descriptive Dimensional Analysis
Descriptive dimensional analysis was oliginated by Tour ($', 3), it differs from conventional dimensional analysis by imposing more restrictions on the properties of the dimensions of any quantity. As a simple example the concept of area may be considered. In conventional dimensional analysis area has the dimensions L2,so that in this system each L is regarded as the same quantity, neglecting the property of direction of length. In descriptive dimensions the dimensions of area aie L,L,, clearly indicating that the concept of area involves the product of two lengths a t right angles. In asimilar manner volume implies three orthogonally directed lines, so that its descriptive dimensions are L,L,L, where L,, L y ,and L, are separate and distinct lengths and must be manipulated as such in any subsequent operation on this quantity. The conventional dimensions of volume are L3, demonstrating that in a conventioiial dimensional analysis each L is regarded as the same quantity. Including the property of direction of length in a dimensional analysis results, in general, in a more complete analysis. In conventional dimensional analysis the trigonometric functions of angles are considered dimensionless, while in descriptive dimensional analysis these functions are treated precisely as ratios of lengths in different dimensions. For example, the tangent of an angle has the dimensions L,/L,. By applying descriptive dimensional analysis t o the Reynolds number, it is found that this quantity is not dimensionless but, having dimensions of a tangent, is of the physical significance of a tangent, as shoun through another method by von KBrm%n ( 1 ) . The physical significance of the Reynolds number cannot he determined by conventional dimensional analysis because the property of direction of length is neglected. Furthermore, in descriptive dimensional analysis, the system of fundamental units into which any quantity is broken dovin must have physical significance, either as natural phenomenon or by definition for the problem at hand. Thus length divided by time, L J T , L,/T, or L J T , is average velocity and may appear only where motion is involved in direction x, y, or z; otherwise it is meaningless. If instantaneous velocity is considered, its definition is dL,/dT, dL,/dT, or dL,/dT, and the length or time should
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F dli =
&y
Descriptive and conrcntional analysis applied t o the qunntities involved in Equation 1 yields the following: Physical Quantities
Descriptive Dimensions
Conventional Dimensions
F = &la
x
I'
2
d I, tl7'
In descriptive dimensions
In conventional dimensions, where separation of properties from their differential operators is permitted,
It is a t once seen that conventional dimensional analvsic asciilws to the coefficient of viscosity the dimension M / L 2 ' . which i.i physically meaningless owing t o the cancellation of lengths ii respective of direction and also of differentials By imposing the pioperty of direction upon the lengths involved, and not Teparating length or time from their differential operators, more information is gained from the analyris. As velocity in the direction is defined as dL,/dT and momentum is the product of mass and velocity-Le., momentum has the dimensions M d L / d 2 ' , -and as area has the dimensions of the product of perpendicularly directed lengths, L,L,, Equation 2 clearly states that viscosity is the momentum transferred normal t o the direction of
INDUSTRIAL AND ENGINEERING CHEMISTRY
Vol. 45, No. 12
ENGINEERING AND PROCESS DEVELOPMENT fluid flow per unit of cross-sectional area parallel to the direction of fluid flow. By means of descriptive dimensional analysis viscosity can be physically described as a momentum transfer through a unit area. Furthermore, for a gas, where the molecular velocities dLu/dT are fixed, the viscosity is a measure of the rate of transfer of mam by molecular diffusion between the parallel areas per unit of cross section through a unit length. Example 2. Relation of Thermal Conductivity to Mass and Momentum Transfer by Descriptive Dimensional Analysis. The analogy, or better the identity, among heat, mass, and momentum transfer can be shown dimensionally by applying a descriptive dimensional analysis t o the thermal conductivity, k. The rate of heat transfer by conduction is given by Fourier’s equation
tion 7 ; however, it has been found that values of c,p/k of less than unity must be used t o correlate heat transfer data. It is seen from Equations 2 and 7 that although viscosity and thermal conductivity are separate and distinct quantities, both involve the identical physical phenomena of momentum transfer per unit area. By means of descriptive dimensional analysis, the physical and mathematical relationship between these quantities can be determined. Such relationships cannot be determined by conventional dimensional analysis and heretofore have been determined by the application of the kinetic theory of gases.
(3)
(4) where dg = quantity of heat that passes a boundary of cross section S in time dt when the temperature gradient is dO/dy, distance ?J being measured in the direction of transfer. That the transfer of the heat quantity dp is in reality a transfer of mass and momentum can be shown by applying descriptive dimensions to the quantities involved in Equation 4: Physical Quantities
Descriptive Dimensions d(B.t.u.)
$2 LZLZ
s
dT
at
Conventional Dimensions d(B.t.u.1 dL de LZ dT
rl
Figure 1.
of Viscosity Example 3. Relation of Coefficient of Diffusion to Momentum Transfer by Descriptive Dimensional Analysis. The rate of mas8 txansfer by diffusion is given by Fick’s law:
Substituting these quantities in Equation 4: d(B.t.u.) dT dL’ k =
u d e
d(B.t.u.)
- de
Quantities Involved in Coefficient
dL
.;iT” L,L*
(5)
The quantity d(B.t.u.)/dtJ is related to the mass A4 possessing i t by the relationship
where dm = quantity of solute that passes a boundary of crosssectional area S in time ds when the concentration gradient is d c l d y , distance y being measured in the direction of diffusion.
d(B.t.u.) MdO
c p = ___
or
t
That the transfer of the quantity of mass dm involves a transfer of momentum can be shown by applying descriptive dimensions t o the quantities of Equation 9: Physical Quantities
Thus Equation 5 becomes
im
(7)
dt dc
For gases c, is a fixed number, so that k has the value and significance of that number multiplied by the momentum transferred in the direction of the heat transfer per unit area as shown in Equation 7 . The identity between heat and momentum transfer is obvious. Because for a gas the molecular velocities dL,/dT are also fixed, the thermal conductivity is a measure of the rate of transfer of mass by molecular diffusion between the bounding areas. The analogy among heat, mass, and momentum transfer by molecular diffusion is apparent from this simple dimensional treatment, This analogy was predicted by Osburne Reynolds, The Reynolds analogy has been defined mathematically by several workers, whose quantitative results have always been limited by the value of the ratio c , ~ / k . The Reynolds analogy predicts that this ratio has the value of unity as predicted by Equa-
dY
December 1953
Descriptive Dimensions (EM LZL, dT dM L,L,L* dLu
-
L,L,dTdM
(EL
dMdL, dT LzLs
-
dM (dLg)LzLuLc
D =
Conventional Dimension6 dM L2 dT (EM L3
=-
dM
L,L,L,
I n conventional dimensions the coefficient of diffusion has the dimensions Le/T-i.e., area divided by time, an ambiguous quantitywithout physical significance for the problem a t hand. Equation 10 ascribes to the coefficient of diffusion the precise physical significance of the momentum carried by the mass transferred per unit area divided by the concentration a t that area, clearly defining the physical phenomena existing.
INDUSTRIAL AND ENGINEERING CHEMISTRY
2635
ENGINEERING AND PROCESS DEVELOPMENT F = force S = area V = velocity A4 = mass a = acceleration t , T = time y = length Q = heat 0 = temperature c p = specific heat a t constant pressure p = coefficient of viscosity D = coefficient of diffusion ic = thermal conductivity
Summarv
By imposing more restrictions on thc properties of the dimensions of any quantity, a more complete analysis of that quantity can be obtained and the physical phenomena involved in many quantities can be determined. In this paper the coefficientP of viscosity and diffusion and thermal conductivity were given precise significance and in so doing the analogy among mass, heat, and momentum transfer was demonstrated. Acknowledgment
The author wishes to express his appreciation to William Licht, acting head of the Department of Chemical Engineering, University of Cincinnati, who checked the manuscript and off erect valuable suggestions. Nomenclature 5 , !/, z = three-dimensional Cartesian coordinates L,, L,, L, = lengths in the 3 , 7 ~ z, directions, respectively
literature Cited (1) KBrmBn, T. van, T i a m . Am. SOC.Mech. Engis., 61, 705 (1939).
(2) Swift, P. F., “Descriptive Dimensional Analysis,” unpublished thesis directed by R. S.Tour. (3) Tour, R. S.,University of Cincinnati, unpublished papers. RECEIVED for review November 1, 1952.
ACCEPTEDJuly 24, 1953.
Velocity Distribution between Parallel Plates W. G. SCHLINGER
AND
B. H. SAGE
California lnstitute o f Technology, Pasadena, Calif.
Y
ELOCITY distribution in air flowing between parallel plates has been studied by many investigators. Laufer ( 5 ) , Skinner (0), and Wattendorf and Kuethe (10) measured such velocity distributions. I n addition, Xikuradse (6) and Deissler ( 2 ) determined the velocity distribution tor water and for air in
10
Figure 1.
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20 DISTANCE
50 PARAMETER
100
cylindrical tubes. ,More recently, measurements of the velocity distribution for the flow of air between parallel plates (8) were made a t relatively Ion- Reynolds numbers. This work was directed to the study of eddy viscosity and eddy conductivity and resulted in fair agreement with the work of Laufer, Skinner, and Wattendorf. It is the purpose of the present discussion to present a correlation of the velocity distribution for Reynolds numbers between 2000 and 35,000 based upon the recent measurements (8) a t Reynolds numbers up t o 50,000. The correlation of experimental data in a form useful for the prediction of the velocity distribution in turbulent flow has been the subject of many efforts. The similarity theory proposed by I
