C. E. Gall
and R. R. Hudgins . University of Waterloo Waterloo, Ontario, Canada
I
I
Teaching Dimensionless Groups in Chemical Engineering
T h e traditional approach to the derivation of dimensionless groups involves two steps"guessing" appropriate variables describing the system, and employing the Buckingham Pi Theorem1 to collect them into groups. By contrast, the approach discussed in this article is from a more rigorous engineering science viewpoint. A paradigm is presented to show visually the st,ructure of chemical engineering dimensionless groups.l To the student approaching this subject for the first time, the traditional approach tends to obscure a number of features of the structure of dimensionless groups. I n the first place, the central importance of convective fluid motion may not he obvious. In addition, it is possible to overlook the fact that there are a number of dimensionless groups that relate the same transport processes as a result of diierent mechanisms. Such a case is exemplified by the Reynolds, Grashof, Galileo, and Taylor numbers (defined in the table), which all relate convective and molecular momentum transport under the influence of various predominant mechanisms. As a background to further discussion, it must first of all be assumed that the mechanisms of transport are identical on the molecular level; i.e., molecular motion is the basis of transport of heat, mass, and momentum. Neglected will be any discussion of radiant heat transfer and electronic conduction of heat, as in liquid metals. A general diffusivity, E, may he defined in the following manner: E =
Transfer rste of s quantity of concentration € Area across which transfer occurs Driving force expressed as the change in concentration € with distance in the direction of transfer
As an exam~le.consider Fourier's law of heat transfer by conducti&' The diffusivity for heat transfer is d e fined at constant density and heat capacity as: k PC,
QIA q__ -pCP(dTldZ) -pCn(dTldz)
where a = thermal diffusivity, k = thermal conductivity, p = density, C , = heat capacity at constant pressure, Q = rate of heat transport, A = area, T = temperature, q = heat flux, and z = axial coordinate. This eauation is more commonly seen as which is the molecular transport flux equation for heat. SILBEBBERG, I. H..AND MCKEWA,J. J., JR., Petrol. Refiner, 32, No. 4, 179; N o . 5, 147; N o . 6 , 101 (1953).
A mare comprehensive summary of the structure of dimensionless groups has been made from a different point of view by ELINKENBERG, A,, A N D MOOY,H. H., Chern. Eng. Progr., 44,17 (1948).
The molecular transport flux equations have counterparts written for convective and total transport. I n the case of total transport of mass, heat, and momentum, two forms of the flux equations may be writteneither in terms of the point gradient of the transported quantity or in terms of the overall gradient existing in the system. I n an analogous fashion, convective transfer can be considered as a product of a diiusivity and concentration gradient. The diiusivity in this case is apparent from the following example for convected heat transfer. The heat transfer rate per unit area cross section by the convection of a fluid of velocity v, from point 1 to point 2, is by a simple heat balance equal to:
"
where D is the distance from point 1 to point 2. The diffusivity in this expression is recognized as Dv. The same diffusivity, Dv, is to be recognized as that for transport by convection of heat, mass, or momentum. I n the case of heat transfer to the walls of a cylinder by fluid flowing with velocity v in the direction of the axis, the convective transfer is proportional to a radial velocity of v~ of the eddies within the fluid and a heat concentration difference &(,T 'b - T,) between the hulk and the wall. The turbulent eddy velocity in the radial direction is proportional in some manner to the average axial velocity; therefore, the heat flux by the turbulent Vorious Forms of the Reynolds Number Symbol v for identified
r.mlln
no/"
II$
Reynolds
Nn.
u
Reynolds
Nna
1
~
~
l
i
lNo.~
- )
~D 2g~
Comments on v
Average velocity far channel or pipe flow Average velocity of stream inside a packed bed Velocity resulting from gravitational forces in "i"rona l. l i~-. rl~ . ....-.l.i."-
Grashof
Taylor Dean Power
1
Nor
D a B ( A T ) g ~ Velocity resulting fmm buoyancy forces in viscous liquid Velocity for flow in curved channels Velocity for power input to pmpeller with grhvitataonal and drag forces present
v = kinematic viscosity, r = orosity of bed, g~ = gravitational constant, 0 = thermal CoefRcient of expansion, L = length, P = power input to propeller, g, = conveeion factor, and n = rotational speed of propeller.
Volume 42, Number 1 I, ~ovember1965 / 61 1
mechanisms varies as
Thus, the ratio of heat flux by turbulent mechanism to heat flux by molecular mechanism is
where r is the radial coordinate. If fpCp(dT/dr)dr is replaced by an average gradient, i.e.
where R is the radius of the cylinder, we can imagine that it is possible to characterize the dimensionless ratio Dv/m as the ratio of the average heat flux by the turbulent transport to the average flux by molecular transport. The dimensionless groups, therefore, are seen to arise as ratios of the diffusivities of various quantities under different transport mechanisms, and can be regarded as a measure of the ratio of migration rates of quantities by their respective mechanisms. The figure shows, within the circles, the characteristic diffusivity related to the flux of the transported quantity given by the coordinates of the diagram. The dimensionless groups are formed by taking ratios of the various pairs of diffusivities, as shown on the appropriate connecting lines. The central importance of convective momentum transport is shown by the large
TRANSPORT Molecular
Convective
Total "Point "Overall Gradient" Gradient"
Heat PCJ
Momentum DV
Mass C
The structure of dimensionless groups. = concentration .a= eddy diffurivily o f heat Ea = total diffurivity for heat NN,, = Nusselt number h = heat transfer coefficient Np., = Peslet number for heat
Solid lines connect groups commonly I~sed; the dashed liner connect less common gmupr
E
tmnrfer, commonly Groeh number Np, = Prondtl number Nsa = Schmidt number B = diffvrivity for molecular transport o f moss 9)K = Knvdren diffurivity Nxn = Knvdsen number Npmm= Peclet number for moss tronrfor
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Journal o f Chemical Education
called
Nsb = Shewood number em = eddy diffusivity o f moss e r = eddy diffurivity o f momentum Em = total diffurivily for mas, k, = overall mar, transfer coef6sient N s =~ Stanton number for mass transfer f = Fanning friction factor Et = total diffwivily for momentum N q = Stanton number for heot transfer
A = mean free poth P = vis~osity
G =
average speed o f molecules
d = capillary diameter
number of dimensionless groups which are derived from it. Why this occurs is also implied in the figure. The eddy diffusivity of heat and mass transport are defined only by analogy with the corresponding molecular diffusivities. Since they are dependent on fluid motion for their existence, any meaningful dimensionless group relating molecular and convective heat or mass transport will have to be related to the convective momentum transport. The analogous relationship of heat and mass transport variables, and of the dimensionless groups relating them, may be seen from the symmetry of the diagram in the figure about the momentum axis. From this diagram, it is also clear how such quantities as the modified Peclet and Stanton numbers relate fluxes of the same transported quantities based on different gradients. For every situation in which convective momentum transport occurs, the diffusion coefficient Dv will have to be identified specifically. D will be a characteristic
length of the system, such as a length of pipe, vessel diameter, particle diameter, etc. Likewise v will require identification. For the Reynolds number, this velocity is identified for a number of special cases, given in the table. Thus, it can be seen from the figure that each of the fundamental dimensionless groups covers a number of other dimensionless groups arising out of specific situations. Characterizing the various transported quantities by means of diffusivities in the manner shown here has certain advantages of presentation. From the overall symmetry of the figure, a student should be able to appreciate quickly the analogy between various types of heat, mass, and momentum transport. He should also be able to grasp the central importance of the convective momentum transport in formulating dimensionless groups. Fially, he should he able to see from this scheme the similarity of many groups of different names which relate identical transported quantities in different physical situations.
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