I
LeROY A. BROMLEY Radiation Laboratory and Department of Chemical Engineering, University of California, Berkeley, Calif.
Prediction of Performance Characteristics +ofHickmanBadger Centrifugal Boiler Compression Still
HICKMAN
(7) has described and given operating data on a centrifugal boiler still in which the liquid to be evaporated flows in a thin film outward along the inside of a rapidly rotating cone. The vapors generated are compressed and returned to the other side of the cone, where they condense to supply heat for the evaporajion taking place on the inside. Flow Regime in Film Whether the flow along the cone is viscous or turbulent can be predicted by calculation of the Reynolds number. For flow along a flat plate 4r Re = -
r
T o proceed with the derivation, the following assumptions are made. 1. Viscous flow in both evaporating and condensing films. 2. No nucleation (bubble, or drop formation). 3. No inert gas present. 4. Condensate is flung off only a t outer edge. 5. Heat flow only by conduction. G. No abrupt temperature drops a t phase boundaries.
r = P%YS %
= flow rate per unit length
normal to flow, lb./hr. ft. p = viscosity of liquid If the Reynolds number is less than about 2000, viscous flow is expected. For this case Equation 1 may be rewritten Re =
2w T7P
2w
2000
x
a
x
p
750
1000 X a X 0.55 X 2.42 =
foot
(3)
(4)
I n a centrifugal force field large compared to gravity on a cone inclined to the axis of rotation at an angle 4, g would be replaced by 4&N2 sin 4. It is implicitly assumed that the velocity profile is fully developed a t any radius. Although this cannot be exact, it is probably a good approximation and is
But as heat flow by conduction only is postulated
r2/s
= E - W1/3 -
where E, the quantity in square brackets in Equation 6, is a constant quantity in any single experiment. For the next part of the derivation the metal resistance to heat transfer will be neglected. The error introduced will be corrected later. By a heat balance at any point r, one may write for the flow, W , on the condensing side (outside) of the cone [the flow on the evaporating W)]: side is (Wp
-
dW -dr- -- -U2arAt h sin 4
-
2.rrrAt
(t + ic) x
-
sin 4
-
where W = total weight flow, pounds per hour r = any radius, feet Inspection shows that the highest Reynolds number occurs near the hub of the cone. Let us calculate this radius a t which the flow would change from turbulent to viscous. The largest feed rate reported is 1500 pounds per hour total on the two 54-inch outside diameter rotors at about 125' F. The critical radius is then rocit =
similar to the approach used by Nusselt (3)for condensation. Hence Equation 4 can be written for this case as
I
Theory
For flow down vertical walls it has been shown (2)
P
where
or about 2 inches. Thus for all reported tests the flow was viscous. It might be turbulent near the hub of a larger cone a t very high feed rates.
Residu
Centrifugal evaporator condenser or phase barrier ( I )
-
VOL. $0, NO. 2
FEBRUARY 1958
233
~~
Table I.
WD/WF
(2) f
(2)
for hc+
03
Values of f
wD for Use in Equation 1 1 (K)
0
0.01
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
0.99
1.0
1.50
1.29
1.12
1.07
1.03
1.01
1.00
0.99
0.99
0.98
0.99
1.00
1.00
1.50
1.50
1.53
1.55
1.58
1.62
1.66
1.70
1.75
1.81
1.89
1.98
2.00
where W, = feed rate to one cone of the evaporator. Integrating Equation 7 for the amount of distillate, W,, between ri and ro results in Equation 8 : W,4!3
( WF -
I
Wn)4/3
+
WDJ3
but we are interested in the average heat as defined by transfer coefficient,
n,
Solving Equation 9 for E and eliminating At and E by means of Equations 8 and 6, one obtains Rotor casing, blower, and variablespeed rotor drive of still, thermal lagging and instruments removed ( I )
or
E
= 2.18
[f
( g ) ] [ g
I):(
x
where
values for which are tabulated in Table I, along with values calculated for f (WD/PV,)for h, m-i.e., no condensate resistance or perfect dropw''ise condensation. From Table I it is apparent that for practical purposesf ( WD/Wp)= I .O over the region of most usefulness, but could be increased considerably if the condensate could be removed (as in dropwise condensation or flung off by enough centrifugal force). The function g ( r i / r o ) , which is equal to the last terms in brackets in Equation 10, is tabulated in Table 11.
-
Feed side of 54-inch copper rotors, before and after 200 hours' distillation of sea water at 150" F. ( I )
234
INDUSTRIAL AND ENGINEERING CHEMISTRY
Table 11.
Values of g
n/r,
0
0.1
0.2
0.3
0.4
g(yt/ro)
1.00
1.00
1.03
1.06
1.09
(3
for Use in Equation 1 1
0.5
0.6
0.7
0.8
0.9
1.0
1.12
1.16
1.20
1.25
1.29
1.33
P E R F O R M A N C E C H A R A C T E R I S T I C S O F H I C K M A N STILL Because, in practice, ri would be made as small as possible, g(r 2000). For heating fluids in turbulent flow down vertical walls Drew (2) gives k3,,Zg 113 CnP 113 4 r 113 h = 0.01
(T-)
(3-)
(y)
(15)
If it is assumed that this would also be approximately valid for evaporation with g replaced by the centrifugal force, one obtains for the evaporation coefficient in turbulent flow
WF was used, as the flow should change little between r$ and the critical radius. The coefficient is essentially independent of radius as long as turbulence persists. The condensation coefficient between rz and roritmay be calculated from
condensing from a relatively stagnant vapor. This is allowed for in the efficiency, t. Energy is also lost through drag of the scoop(s). If the scoop is streamlined and made as small as possible consistent with handling the flow, this energy need be only 10 to 20% of the kinetic energy loss. If, on the other hand, large scoops are used, and not streamlined, this energy loss could easily be 1 to 10 times the kinetic energy loss and represent a serious loss in energy. There are also small losses due to windage and mechanical friction, but these should be small. These latter losses should be reduced to a minimum by proper design and are also allowed for by an efficiency factor, b. Work to rotor 4r2N2rO2 Wp -~ lb. ofproduct qpgc
and Work to compressor Work to rotor
(
For n rotors of radius rOn operating at the same total feed and product rates as for one rotor and at optimum speed (given by Equations 23 and 25), and assuming negligible boiling point elevation of the feed; Equation 26 applies for total work (including compressor). The optimum speed for n rotors is l / z / k times that for one rotor of the same diameter. Total work for n rotors, radius ron Total work for 1 rotor, radius ro
r
87
3
(I9)
Work Delivered To Compressor. For each pound of distillate this work is where W , is the amount condensed between ri and rcrit.
Power Required As pointed out by Hickman, power is used to turn the rotor and compress the vapor, a small amount is used in auxiliary equipment, and some is lost as heat. As the rotor and compressor use the major share of the power, the optimum speed of rotation is calculated that will ,minimize the power requirement for a desired amount of feed and product for a certain rotor. Power to Rotor. The rotor must overcome the frictional loss caused by flow of liquid over the surface; per unit mass of distillate:
as long as the temperature drop for heat transfer, At, and the mean boiling point elevation, BPE, of the evaporating liquid are small compared to the absolute temperature, T , of evaporation. J is the mechanical equivalent of heat. If the rotor metal resistance to heat transfer is neglected At may be eliminated by means of Equations 11 and 9. Addition of Equations 19 and 20 then results in the equation for total work:
Thus the total power consumed per pound of product may be reduced substantially by increasing the distillation area relative to the load. A large boiling point elevation would reduce the eff ectiveness of additisnal area. From Equation 24 it can be seen that the temperature should be as high as possible (without the formation of scale). Although it is important to improve rotor efficiency, it is even more important to improve compressor efficiency. Hickman suggests that the ratio of power supplied to the compressor to power supplied to the rotor be 3.22 for a commercial still. This is nearly that
VOL. 50, NO. 2
FEBRUARY 1958
235
-
Table 111.
Comparison of Calculated and Hickman’s Experimental Heat Transfer Coefficients Uoalod.
AP
Temp.,
in HzO
F. Still 2, 1000 r.p.m., city water
2WF9
78 80 93 100 110 125 137 135.2 144 155
0.73 0.78 1.20 1.55 1.85 2.80 3.25 4.10 6.00 7.00
5.29% NaCl
107 109
1.75 2.7
Sea water Still 4, 400 r.p.m., 54 inch O.D., city water
99 118 119 121 121 117 119 116 120 122
1.60 6.3 4.3 2.8
Qcean water
predicted by Equation 25 for optimum rotor speed.
Table I11 compares the heat transfer coefficients reported by Hickman to those calculated by use of Equation 11. The values range from f 7 1 to -32%. The measured values with the greatest deviation are those with either very low At or very low W D , which tend to magnify experimental errors. On the whole the agreement is satisfactory, indicating that the proposed mechanism, viscous flow, is probably correct. At least part of the deviation a t low pressures (low temperatures) could be explained by the neglect in the derivation of the heat transfer resistance a t the vapor-liquid phase boundary. This error would be especially large if the accommodation (or sticking) coefficient of water molecules striking the saline water surface is much less than unity.
F
=
Do
E
236
Sq. Ft.,
4150 4080
3550 3180
12.9 378 265 174 91 510 327 462 198 42
4000 1990 2070 2120 2240 1980 2020 2020 2 140 2320
2720 2360 2440 24810 2520 2350 2450 2250 2480 3380
acceleration of gravity, feet ’ hr .z = gravitational constant, 4.18 X lbm. ft. 108lbf. hr.2
U
=
(2)
G
see Equation 9 and Table TI
same as above with ro replaced by rorit = coefficient of heat transfer, B.t.u. hr. ft.2 F. = coefficient of heat transfer for B.t.u. evaporation, hr. sa. ft. F = coefficient of heat transfer for B.t.u. condensation, hr. sa. ft. a F. = mechanical equivalent of heat, ft., Ibf. 778 ___ B.t.u. = thermal conductivity of liquid, B.t.u. hr. ft. O F . = thermal conductivity of rotor B.t.u. metal, ___ hr. ft. O F. = rate of rotor rotation, rev. per hour (or minute) = optimum rate of rotor rotation, rev. per hour (or minute) = pres. difference between condensing and evaporating sides of rotor, Ibf./sq. ft. = radius, feet = inside and outside radius, respectively, feet = radius a t which flow changes from turbulent to viscous, feet =
+71 50 53 +44 +27 39 24 $ 8 + 3
11 17.8
at
% Error Based on Experiment
2120 2410 2550 2740 3180 3070 3650 3560 3760 4180
T
g
F.
3620 3620 3900 3940 4050 4270 4510 3850 3890 4000
= see equation 11 and Table I
=
O
u s x , t I.
B.t.u./Hr. Sq. Ft., ’ F.
13.5 15 18 20.5 22 23 24 31 38 41
Re
(2)
INDUSTRIAL AND ENGINEERINO CHEMISTRY
B.t.u./Hr.
friction loss on rotor per unit mass distillate, ft. Ibf./lbrn.
g
of evaporating liquid, a F. heat capacity of vapor, B.t.u./ lb. F. = outside diameter of rotor, feet = constant parameter in Equation 5
Lb./Hr.
(2)
gc
Nomenclature
C,
50 50 50 50 80 80 80 44 50 44 958 967 962 968 968 946 931 938 962
=
g
On the basis of the derived equations it is possible to predict the operating characteristics of the Hickman-Badger still and the optimum conditions of operation.
= mean boiling point elevation
50
1.5
Conclusions
BPE
50 50
8.5 6.5 9.3 4.3 1.6
f Comparison of Heat Transfer Coefficients with Experiment
Lb./Hr.
2wDt
++ + +
- 5
++28 17
+- 4716 - 15 - 15 - 11
- 16 - 18
- 10 - 14 - 32
Reynolds number total temperature drop for heat transfer, O F. = absolute temperature of evaporation, R. = over-all heat transfer coefficient (see Equation 13), B.t.u./hr. sq. ft. a F. = over-all heat transfer coefficient (not including metal resistance), B.t.u./hr. sq. ft. = =
F. Udd, Uexptl= calculated and experimental values of I/ = liquid flow on a rotor (condensate flow after Equation 6, lbm./hr.) = feed flow to a rotor (one cone only), lbm./hr. = distillate rate from one rotor cone, lbm./hr. = distillate rate a t rcrit, lbm./hr. = metal wall thickness of rotor feet = liquid film thickness, feet Y = mass flow per unit periphery r normal to flow, lbm./hr. ft. = efficiency of rotor and com9r, i?c pressor, respectively x = latent heat of vaporization a t temperature T , B.t.u./lb. ft. = viscosity of liquid, lbm./hr. ft. J/ P, P v = density of liquid and vapor, respectively, lbm./cu. ft. = angle of rotor to its axis 4
w
literature Cited Hickman, K. C . D., IND. ENG.CHEM. 49,786 (1957). McAdams, W. H., “Heat Transmission,” 3rd ed., McGraw-Hill, New York, 1954. Nusselt, W., 2. Ver. Deut. Ing. 60, 541, 549 (1916). RECEIVED for review August I, 1957 ACCEPTED October 19, 1957
