ADRIAAN ICLINKENBEPPG iV. V. De Batuafsche Petroleum Maatschappij (Royal D u t c h Shell T h e exact solutions of the equations describing t.i*mnsient heat and mass transfer phenomena in packed solids under given conditions can he approximated for all practical purposes by error functions. These may be read from a nomograph.
HE heating or cooling of a bed of solid particles by a current of fluid has been calculated by Anzelius ( I ) , Kusselt ( 6 ) . Hausen ( 3 ) ,Schumann ( O ) , and Furnas (2). 'Chis transient form of operation is not limited to the unit operation of heat transfer, but is applicable aleo to a number or other unit operations. It is conimonly employed in mass transfer processes involving a solid-for example in adsorpt,ion (recovery of solvent vapors, gas masks, drying of air, percolation, chromatography) and .in leaching. It has been used also for liquid-liquid extraction. .1 survey of the literature has recently been given by Thiele ( 7 ) . Klotz (4)has discussed the application t o gas adsorplion. The authors mentioned in the firsf paragraph have made the following assumptions:
G r o u p ) , The Hague, HoEEamd
Heat transfer U = heat transfer coeficient (heat,/ares. time. temperatme difference) cf = specific heat fluid phase cB = specific heat solid phase p i = density fluid phase ps = density solid phase Mass transfer mass transfer coefficient (moles/area. time. mole/vd.; K, calculated on concentration difference in moving phase R == partition coeficient (equilibrium ratio of concentration in moving phase to concentration in stationary phase) The dimensionless ratios ~ h o nare: For heat transfer:
For mass transfei I
The shape of the bed is such thai thr p'roblein is one-diniensional. There is no conduction of heat in the direction of flovv. In anv cross section of the bed the rate of transfer of heat is proportional t o the difference betveen t,he mean tempcratures of fluid and solid, Tf and !l'8. The effect of temperature on volume and specific heat of Huid and solid may be neglected. Their discussion is applicable also to such xnass t,ransfer processes There the amounts of solute, moving and stagnant, Then in equilibrium are proportional-for example, to adsorption processes There the amount of solute adsorbed is proportional to the concentration in the fluid. The differential equations describing this heat transfer process are simplified by introducing a dimensionless height Y and a dimensionless time corrected for displacemcnt, Z , as- various authors have shown. The follon-ing symbols are introduced: General H = height ofbed t = t h e u = linear velocity of fluid above packing FJ = fraction by volume fluid or inoving phase P, = fraction bv volume solid or stationary phase; in manr systems Ff 4-F , is 1 a = surface area of packing ppr unit of .zolume of the bed 2' = temperature I' = dimensionlcss height Z = dimensionless time e r f ( p ) = error function of p (Equation 2) [n d i c e s 0 = initial f = fluidormoving s = solid or stationarv
z Figure 1. C & T ~ I Iof ~ E temperatures of fluid and solid 111 transient heat transfer as depumdent on dimensionlea* height and dimensionless time oorreoted for dkqhcementl
1992
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0
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588660
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66660
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66660
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100
860
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590
510
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S60
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1993
INDUSTRIAL A N D ENGINEERING CHEMISTRY
1994
According to the authors cited the heat transfer process is described by the equations:
These same equations apply to the mass transfer process. The simplest set of initial and boundary condiTions is starting with a uniform temperature, 0, of bed and flo\virig fluid and changing the inlet temperature at zero time to To: The solutions for this case, integrals containing Bessel functions, have been evaluated by numerical or graphical methods. Schumann (6) and Furnas ( 2 ) have given tv-o sets of curves
Equations 3 and 1,although approximations, are considerably more accurate than the drawings given by Furnas ( 2 ) ; the ~ d i nates of the Furnas dravings are sometimes subject to errors over 0.01. ;\loreover, the Furnas curves do not allow easy interpolation for intermediate Y values, nhich problem does not# arisr any longer. The insufficiency of rhe Furnas curves for u8c in regions of l o w concentratioii is stressed by IClotz (4). X nomog~aph(Figure 2) is based on Equation 3. It has parallel and equidistant’ scales on which are indicated
Tj T and -’
as functions of Z for a numbtir of values of Y To ranging from 1to 500. Figure 1 s h o w such curves for Y = 2, 4, and 8 (full lines) The
sholving
Vcl. 40, No. 1,o
To
~
most rapid change of the ordinates takes place in the neighborhood of Z = Y . Also shon-n in Figure 1 is the curve (brolwn line) representing
!r To l/z
[I
1
dr
iz
- .iY
J
inverse
r
\
.
.
of the rrroi functlon of
As is cviderit from Equations 3 and 4 the same noinogrqh 7‘ can be uwd l o give the --!values---that is, by jiiterehangi~igZ 2’0
=
e-‘d2&
-a
+ erf(v‘z
the
- -d/r)i
(2)
and Y and subtracting the integral from unity. This is done by turning Figure 2 upside doim arid reading the other set of smbscripts. Thus, Y = 10 and Z = 12 correspoiids to 7 2 j = 0.70, but also
where e r f ( p ) represents the error function or prohability integral
To 1 ‘e
an approximation nhich has been used by Walter (8).
T/ T - and curves are very similar to this To To latter curve and about 112 unit’of 2 at either side of it. It is seen tha,t the
rr,
Z = 1 0 a n d Y =12to--Oo.3Cu. TQ For values of Y aiid Z , higher than those indicated OIL Figure 2 the reader may easily calculate further scales to t,he nomograph. However, the _ _error ~ _ _integral vrith the simpler upper limit. of integration d.Z - 4 Y or l / Z .- 41’ thvn should be 7’ 7‘ a good appros.matiori for 2 arid ;
