O n a Unified Approach to the Study of Transfer Processes he article "A Unified Approach to the Study of Transfer 'Processes," by George D. Fulford and David C. T. Pei (2) is most interesting. However, it seems that another term should be included with their Equation 2. The resulting equation should be:
T
ax
- -+-+-
ay
-
G
[Qz
+ + Qg
+ Qz1
(1)
in which Q may be defined as the rate of interphasial property transfer per unit volume. I n the differential balance of mass transfer of a single component, these Q's can be defined as the rates of interphasial mass transfer per unit volume. They are the losses or gains of mass across element surfaces by the phenomenon other than bulk flow, molecular motion, and internal generation or consumption. I n the x direction, this rate is represented by: Q, = K,a(C,*
- C,)
(2) where K, is the mass transfer coefficient in the x-direction in mass per unit time per unit surface per unit concentration; a, is the interphasial surface per unit volume; C* is the concentration of A in x-direction at equilibrium in mass per unit volume. Hence, substituting Equation 2 into Equation 1 with the meaning of the symbol from the article by Fulford and Pei gives :
bCA at
-[a):(
I
a(',CA)
3Y
RA - [K,a(C,* - C,)
I
a(uZcA)]
+
az
+ Kd C , *
+
- c,> &a(Cz* - CZ)l (3)
I n the differential balance of heat transfer, these Q's are the heat losses or gains from each element of volume a t the surface to the surrounding areas and can be represented as QZ = h,b(T - T") (4) in which h, is the heat transfer coefficient in x-direction in heat energy per unit time per unit surface per degree; b is the interphasial surface per unit volume; T* is the surrounding temperature in degree. Substituting Equation 4 into Equation l yields
dT
at
90
- T*),
+ h,b(T
-
T*),
+
1+
!!?'
h,b(T - T"),)
PCP
(5)
I n the overall balance, similar equations can be developed from the fundamental law of conservation of matter or energy as follows :
alll [Y ; I:: a,
MI, a11 p+t-,+---2
P?C P L(h,b(T
I N D U S T R I A L AND E N G I N E E R I N G CHEMISTRY
Net rate of accumulation = Net rate of bulk and molecular in volume element transport across element surface Net rate of interphasial transport across element surface Net rate of generation or consumption in volume element
+ +
(6) For mass transfer of component A, the equation is
(7) in which Q A is the interphasial mass flow rate of component A . For energy transfer, the corresponding equation becomes :
in which E is the total energy of the system which is the sum of the internal energy, kinetic energy, and potential energy; Q is the rate of heat transfer to the system per unit time; W is the rate of work done by the system per unit time; Q" is the interphasial rate of energy transfer; and AHT is the rate of generation of the energy per unit time. The justification of the addition of this interphasial rate of transfer of matter or energy to Equation 2 of Fulford and Pei has been shown in some articles (7, 3, 4). Nomenclature a
= interphasial surface per unit volume in differential mass
transfer interphasial surface per unit volume in differential heat transfer ca = concentration of component A in mass per unit volume CA* = concentration of component A a t equilibrium in mass per unit volume
6
=
is associated with the Department of Chemical and Paper Engineering at Lowell Technological Institute, Lowell, Mass. 07854. G. D. Fulford and D. C. T. Pei are the authors of the review discussed by Dr. Chen and are associated with the Department of Chemical Engineering, University of Waterloo, Waterloo, Ont., Canada. AUTHORS Ning Hsing Chen
C ,
= heat capacity at constant pressure in energy per unit mass
per degree DAB= diffusion coefficient of component A in square feet per hour E = total energy of the system = internal energy 4- kinetic energy potential energy F = rate of generation of property P p e r unit surface area G = rate of generation per unit volume of property P in bulk of system g = gravity acceleration H = enthalpy per unit mass h = heat transfer coefficient in energy per unit time per unit surface area per degree K = mass transfer coefficient in mass per unit time per unit surface area per unit concentration k = thermal conductivity in energy per unit time per unit surface area per degree per unit length M A = mass of component A m = mass flow rate in mass per unit time P = intensive property Q = rate of heat transfer per unit time RA = reaction rate of component A in differential mass balance in mass per unit time per unit volume TA = reaction rate of component A in overall mass balance in mass per unit time per unit volume
+
Reply:
S
= cross-sectional area normal to the flow
T = temperature T* = temperature in the surrounding areas t
= time
= average velocity = work done by the system to the surrounding areas = height above reference plane I1 = molecular flux of property P p = density fi = viscosity q5” = viscous dissipation AH, = rate of internal generation of energy per unit time
I I
W Z
Subscripts
1
= flowin.
2 = flowout
REFERENCES (1) C.arslaw, H. S. and Jaeger, J. C., “Conduction of Heat in Solids,” Oxford University Press, 1957. (2) Fulford, G. D., and Pei, D. C. T., IND. END.CHEM.,61 (5), 47 (1969). (3) Hi,mmelblau, D. M., “Basic Principles and Calculations in Chemical Engineering,” 2nd ed., Prentice-Hall, Englewood Cliffs, N. J., 1967. (4) Smoot, L. D., Chem. Eng., 68, (18) 126 (August 21, 1961).
N. H. Chen
Boundary Conditions and the Conservation Equations
above proposal to incorporate a rate of interfacial Thetransfer into the general conservation equation
[Equation 2 of our review (I), IND.ENG.CHEM.,61 (5), 47 ( 1 9 6 9 ) ] appears attractive. However, it must be remembered that such an equation represents the behavior within a phase; it was derived, as we stressed, on the assumption that the medium possesses continuous properties. Conditions a t the interfaces (or discontinuities) of each phase are taken into account by boundary conditions imposed on the differential conservation equations a t the interfaces, as indeed we pointed out in the review on page 50. The boundary conditions should not be included within the differential equations before solution, and so general terms representing rates of interfacial transfer do not seem desirable in the differential conservation equations. From a practical point of view, the reason may perhaps best be seen by considering the heat transfer case. The definition of the local heat transfer coefficient, Equation 4 above, is usually in terms of the difference between a temperature at the interface (or in the adjacent phase) T* and the mean temperature over the cross section of the phase in question at that position. This mean temperature, p, is not the same as the point temperature T, and the two should be clearly distinguished in Equation 5 , which is misleading as it stands. I n a two- or three-dimensional temperature field, f’ can usually be found from the general heat transfer equation only by first solving the latter for the local temperature T = T[x,y,(z),(t)] and then carrying out the appropriate averaging operation to get f’. I t is therefore not helpful to introduce the additional unknown itself into the differential equation to be solved. How then has this approach apparently been used successfully in Ref. (4) above (and again in a later
part of the same series)? The answer lies in the particular problems which were considered. I n both cases, the assumptions made in simplifying the general mass transfer equation before substituting into it the mass transfer coefficient type interphase transfer term were such that no variation in concentration across the phase normal to the main direction of flow remained. Under these conditions, the equations of motion have been effectively volume-averaged, though this is not explicitly stated. The same result is obtained without ambiguity by carrying out formal volume averaging, with the same assumptions, and writing the simplified result in differential form (by placing the inlet and outlet control surfaces, S6and So, a differential distance apart). Once the equations have been volume-averaged-Le., integrated-the objection to incorporating boundary conditions of various types of course disappears, and it is then only necessary to identify and define appropriately the contributions from the various surfaces in the volumeaveraged equations. For instance, in the volumeaveraged mass transfer equation (Equation 12b of I) PA@)may be entirely or partly due to a mass-transfercoefficient type of surface transfer; Equation 12b therefore includes Equation 7 above. Similarly, our Equation 12d includes Equation 8 above. Because a great deal of applied chemical engineering is based on the use of experimentally determined friction factors and transfer coefficients, the importance of the volume-averaging procedure in providing a bridge between the “theoretical” and “applied” aspects of transfer processes cannot be overemphasized. We thank Dr. Chen for bringing u p this important topic which was somewhat slighted in our review.
G. D. Fulford VOL. 6 1
NO. 1 1
NOVEMBER 1 9 6 9
91
