Jacket-Side Nusselt Number Isaac H. Lehrer Department of Chemical Engineering, Monash University, Victoria, Australia
Experimentally derived values of the over-all heat transfer coefficient and published d a t a for the vessel side were used to derive a jacket-side Nusselt number. This was
+
compared with the Prandtl analogy relation N u = (f Re Pr)/2[1 4(Pr - 1)1 in 4 have empirically determined values. Published data were used for
which f and
f and the equivalent diameter. Three velocities that follow from apparatus dimensions, flow rate, and temperature rise were combined to yield a characteristic velocity for Re. Water was injected radially or tangentially into a vertical cylindrical jacket with height/diometer
1. Flow rates varied from 0.12 to 0.57 Ib,/sec-sq
ft of heat
transfer area. Experimentally derived and calculated results were in reasonable agreement. The friction factor and equivalent diameter modifications, together with the characteristic velocity, can be used also in other appropriate heat transfer equations, with similar agreement.
HEAT
transfer in jacketed vessels is generally limited by the vessel-side resistance to a greater extent than by the jacket-side resistance, particularly when the jacketside fluid changes phase and the vessel contains a liquid with relatively high viscosity. However, there are processes in which the vessel-side heat transfer coefficient is high, and an a t least equally high jacket-side coefficient is desirable. Often, water is the most economic heat transfer medium. Methods of increasing heat transfer rates in jackets containing liquids have been discussed and referenced (Uhl and Gray, 1966)-e.g.. provision of spiral guides, and the injection of liquid a t high speed, often via nozzles that discharge tangentially. Properly installed spiral guides provide something approaching plug flow and, therefore, provide a higher temperature difference driving force compared with injection into plain jackets where considerable backmixing occurs. However, this advantage is often outweighed by the additional cost of spiral guides. I n this report, results of experiments in a plain jacket are discussed. Both radial and tangential injection have been considered. Ex perimen ta I
Tests were carried out in a concentrically jacketed vertical cylindrical vessel with the following relevant dimensions: Inside diameter of vessel 24 inches Outside diameter of vessel 24.625 inches Inside diameter of jacket 26.625 inches Water level in vessel a t start of each test 24 inches The vessel contents were agitated by a two-bladed vertical flat paddle stirrer, coaxially mounted. Four radial vertical baffles of 2-inch width were provided a t 90" intervals. The baffles extended down to within 2.4 inches
above the vessel floor, and this was also the height of the lower edge of the paddle above the vessel floor. Paddles with 12-inch and 16-inch swept diameter were used; paddle width was 2.75 inches for both lengths. Cooling water entered the jacket through a radial branch near the bottom, and left via a 2-inch bore radial branch near the top, diametrically opposite the inlet. For tests using radial entry of cooling water, the 0.75inch bore inlet branch was used bare; for tests with tangential injection, a 0.5-inch bore tube, with an end piece machined to provide tangential discharge, was fitted through this branch. Cooling water flow rate through the jacket was adjusted t o the desired value. Water was heated by direct injection of steam, with agitator running a t desired speed, but without cooling water circulation in the jacket. When the temperature reached nearly 100" C steam injection and then agitation were stopped, and the liquid level was adjusted to 24-inch height by draining. Agitation was resumed and cooling water circulation through the jacket was started. Cooling water flow rate and agitator speed were kept a t desired values. Temperature of water in the vessel and cooling water inlet temperature were measured a t regular intervals, and temperature difference between water in the vessel and cooling water exit temperature was recorded. This was continued until temperature of vessel contents was less than 33°C. With the experimental conditions that existed, temperature gradients in the vessel contents were not evident during test periods. Test periods covered vessel contents temperatures between 90" and 35" C. Water temperatures were measured with thermocouples, and the measuring error of the thermocouple-recorder system was less than 1.4"C; the response time in the experimental range was less than 0.055 second per '(2. A more accurate thermocouple-precision potentiometer Ind. Eng. Chem. Process Des. Develop., Vol. 9, No. 4, 1970
553
system was used to measure water temperature in the vessel. Cooling water flow rates between 80 and 376 lb, per minute were used, estimated energy dissipation in the jacket ranging from 0.164/103 t o 0.032 hp per sq foot of heat transfer area. Flow measurement was by calibrated rotameter. T o obtain some indication of flow conditions in the jacket, pulse injections of tracer were made a t the water inlet. Tracer concentrations were measured upstream of this inlet and in the jacket exit branch; the concentration difference between these two locations provided the input to a recorder via two conductivity meters. The tracer input tube restricted flow through the inlet fitting, so that the highest flow rates attainable with the given equipment were lower than the highest flow rates used in heat transfer tests. General Calculations
The jacket-side heat transfer coefficient, h,, was obtained from known and experimentally derived values of
Q
=
UAAT,
l - -l _ - - d_ h, U k
(1) l h,
The vessel side coefficient, h,, was estimated from the dimensional equation h, = 3.14 Re'"j8 for water a t 128"F, Re' = L 2 N / v , L = 1 2 inches (Lehrer, 1968), making adjustments for physical properties a t the vessel temperatures during test periods. h, values for the 16-inch paddle were deduced from U values obtained for identical vessel temperatures and jacket water flow rates-Le.,
The temperature difference, ATrn,was evaluated on a log mean basis for long test intervals, and on a mean basis for short test intervals. Other simplifying assumptions were: No allowance for a dirt factor was necessary in the clean stainless steel vessel and jacket. Heat losses to surroundings could be neglected in calculations. With h, values of the same order of magnitude as h, values, the variation of h, with h, could be neglected. A constant value of d h = 0.0026 (hr)(sq ft)('C)/Chu could be used. The nominal heat transfer area A was 11 sq feet.
2tc
-
t, =
~
ZC
The second moment about the mean, 0 2 , is a measure of the spread of the distribution about the mean, and for a closed vessel (Levenspiel, 1962)
The jacket may be regarded as a closed vessel. The magnitude of the group ( D / u L ) is an indication of dispersion in the vessel, varying from D l u L = 0 for ideal plug flow to D / u L = m for ideal backmixing. The ratio t may also provide an indication of dead space within the vessel-Le., dead space existing when Z f T < 1. Table I lists some experimental observations and derived results. The ratio f/Tfor the tangential inlet is significantly less than 1.0, yet observation on an improvised transparent apparatus showed rotary motion throughout the jacket; the simple indication of dead space is not valid for the tangential case with diametrically opposite outlet. For o', there are no significant differences a t the 95% level between the two types of inlet, nor are differences due to flow rate variation significant a t this level. Dispersion is widespread if judged by Figure 10 (Chapter 9 of Levenspiel, 1962).
xf
Radial Inlet
T o find a relation that predicts the heat transfer coefficient, one has to envisage flow around a vertical jacketed cylinder of length-diameter ratio = 1, with liquid entering a t a point near the bottom and leaving a t a diametrically opposite point near the top. I n analysis or comparison with known relations, the definition of meaningful characteristic velocity and length is difficult. With regard to velocity, there is not only forced convection, but also a buoyancy contribution that has increasing significance with increasing temperature difference and decreasing throughput. The Martinelli-Boelter relation for combined natural and forced convection in vertical tubes as quoted (McAdams, 1954) may be applicable to jackets with low water throughput and high length-diameter ratios, but
Table I. Residence Time Distribution Data -
Inlet
Radial
Flow Rate,
t,
W
Seconds
5.060
13.5
3.133
22.3
1.333
51.2
3.133
22.3
1.333
51.2
Tracer Information
An indication of flow in a vessel can be obtained from tracer information (Levenspiel, 1962). The nominal residence time in a vessel of volume V, through which a. fluid passes at a volumetric flow rate V , is T = V / V . The actual mean residence time of fluid leaving a vessel, which can be established by measurement of tracer concentration c a t the exit is 554
Ind. Eng. Chem. Process Des. Develop., Vol. 9, No. 4, 1970
Tangential
f , = (Ztc/
Zc) Seconds
U
1
12.8 13.0 22.1 21.0 22.3 46.7 51.0 48.6
0.126 0.066 0.062 0.095 0.090 0.177 0.151 0.143
15.5 16.8 16.2 15.4 34.2 34.8 35.6
0.166 0.152 0.046 0.114 0.184 0.202 0.186
fails t o yield the right answers for the conditions discussed here with respect to both magnitude and relativity. I t was thought appropriate t o try a simple approach for the turbulent case--Le., the Prandtl analogy that supposed simultaneous resistance to transfer in a turbulent zone and in a laminar sublayer. This model results in the equation for smooth tubes (Grober et al., 1961)
Nu =
f Re P r 2 ( 1 + 4[Pr - 11) 4 = 1.74 Re-'
(7)
and the Blasius equation yields f i 2 5 0.04 Re-' (8) The Reynolds group R e = d,u u can be evaluated as follows: For the estimation of the equivalent diameter d,, similarity to flow between parallel flat plates has been assumed. Then (Lohrenz and Kurata, 1960), de = (h/?)'" s, where s is the perpendicular distance between parallel plates. Thus in this case,
de =
(4 3)"5
( I 1.)
f t = 0.136 f t
(9)
One can envisage two calculable characteristic velocities pertaining to forced convection in the jacket; one is the nozzle exit velocity
UB A
(14)
(gd1T)"
The characteristic velocity, u, in the Reynolds group thus becomes I:
=
(U,,04)05
+
(15)
1'8
and the group itself is
Values of Re based on Equation 16 with values of uti and P r are listed in Table I11 together with the applicable test conditions. Stirrer case 1 refers to a 16-inch paddle a t 150 rpm, stirrer case 2- to 12-inch paddle a t 84 rpm, and stirrer case 3- to a 12-inch paddle a t 355 rpm. A suitable friction factor can be evaluated by considering Equation 8 together with the friction factor plot shown in Figure 7 of Lohrenz and Kurata (1960). For the range of R e values met here-Le., 9000 to 40,000-the ratio
Parallel plate friction factor Blasius friction factor
+ ?
(17)
4
Substituting in 8
f / 2 = 0.03 Re-'
(18)
and substituting 18 in 6 yields
Nu =
0.03 Reo7'Pr 1 + @(Pr- 1)
the other is the rise velocity in the annulus. Table Ill. Temperature Intervals, Flow Rates, Parameters, and Velocity Correction for Radial Inlet Vessel-Side
Jacket-Side Calculation Values
Temp.
For a decay process the geometric mean is appropriate, so that the characteristic throughflow velocity can be taken as (u,ua)" ', Values corresponding to test conditions are shown in Table 11. The velocity contribution arising from buoyancy-Le., temperature effects-should be independent of throughput, and could be estimated by using p, the cubical expansion coefficient per temperature degree, and A T , the temperature rise of water in the jacket, taken from Q = uc,,AT. Thus upward acceleration is g p A T , and for a jacket height Z L * 2 feet,
If we assume the mean velocity is appropriate, the buoyancy effect correction is
Interval,
O
C
W
Re
Pr
VH
5.10 5.40 6.65 5.20 5.80 7.10 5.60 6.00 7.10
0.314 0.264 0.167 0.264 0.181 0.110 0.214 0.147 0.093
5.00 5.70 6.65 5.30 6.00 6.90 5.50 6.00 6.65
0.364 0.253 0.169 0.241 0.176 0.124 0.198 0.136 0.091
5.10 5.60 6.30 5.10 5.70 6.65 5.60 6.00 6.65
0.297 0.263 0.177 0.244 0.187 0.124 0.217 0.148 0.103
Stirrer Case 1 90-85 90-40 45-40 90-85 90-40 45-40 90-85 90-40 45-40
1.333 1.333 1.333 3.333 3.333 3.333 6.266 6.266 6.266
13,483 12,177 9,240 25,135 21,599 17,806 39,092 36,047 31,141
Stirrer Case 2 90-85 90-40 45-40 90-85 90-40 45-40 90-85 90-40 45-40
1.333 1.333 1.333 3.333 3.333 3.333 6.266 6.266 6.266
14,960 11,620 9,133 24,496 19,938 18,315 40,143 35,875 32,813 Stirrer Case 3
Table II. Calculation Values of Forced Convection Velocities Jacket Water Flow Rate, w , Lb, /Second
Nozzle Velocity, v r i , FtjSecond
Annulus Rise Velocity, V A , Ft/Second
Ft/Second
1.333 3.333 6.266
6.99 17.45 32.74
0.0381 0.0957 0.1799
0.516 1.290 2.370
(V~V~)"',
90-85 90-40 45--40 90-85 90-40 45-40 90-85 90-40 45-40
1.333 1.333 1.333 3.333 3.333 3.333 6.266 6.266 6.266
13,484 12,039 9,617 25,442 22,319 18,853 39,981 36,047 32,973
Ind. Eng. Chem. Process Des. Develop., Vol. 9,No. 4, 1970 555
Table IV shows Nu evaluated by Equation 19, using 4 and Re as defined by 7 and 16, respectively, in juxtaposi-
tion to Nu values derived from experimental data-i.e., where Xu = h,>d,/k,and h, was obtained by use of Equations l, 2, and 3. Summarizing, the ratios of
Nusselt number by calculation Nusselt number derived from experiment x u 19 Nuexperiment are average 1.014; minimum 0.794; maximum 1.258. For the various vessel-side temperature intervals, the averages are 90-85" C: 0.946; 90-40°C: 1.033; 4 5 4 0 ° C : 1.064. More uniform ratios could be achieved by introducing the usual viscosity correction ( k / p & ) "1 4 , which by itself would result in 0.968 instead of 0.946 for the 90" to 85°C interval average and by allowing for the considerable backmixing in the jacket, which means actual water temperatures that are higher than the mean values used in calculation, in turn entailing physical property values that result in relatively higher Nu values with increasing temperature.
Table IV. Comparison of Nusselt Groups, Jacket-Side Vessel-Side Temp. Interval, ' C
Jacket Flow Rate, w
Nu from Experiment
Nufrom Eq i p / Nuexperimen+
RADIAL INLET Stirrer Case 1 90-85 90-40 45-40 90-85 90-40 45-40 90-85 90-40 45-40
1.333 1.333 1.333 3.333 3.333 3.333 6.266 6.266 6.266
90-85 90-40 45-40 90-85 90-40 45-40 90-85 90-40 45-40
1.333 1.333 1.333 3.333 3.333 3.333 6.266 6.266 6.266
61.98 56.07 45.30 101.31 88.77 79.95 148.80 140.79 128.22
65.42 55.22 55.22 98.87 89.35 69.90 167.28 128.52 113.15
0.947 1.015 0.820 1.025 0.994 1.144 0.890 1.095 1.133
61.60 55.22 46.24 107.44 91.39 81.60 179.38 129.34 110.57
1.057 0.991 0.970 0.938 0.935 1.003 0.848 1.080 1.201
Stirrer Case 2 65.10 54.75 44.85 100.77 85.41 81.87 152.13 139.71 132.81 Stirrer Case 3
Tangential Inlet
Water was injected through an opening inclined approximately 40" from vertical. R e was evaluated by Equation 16; however, instead of defining u 4 as a rise velocity, the characterisiic second velocity here is the slot velocityLe., S in u s = V S is defined by
S = (height of annulus) (width of annulus) = (23.5) 144
90-85 90-40 45-40 90-85 90-40 45-40 90-85 90-40 45-40
sq feet (20)
Observations on an improvised transparent apparatus confirmed the validity of assuming rotary bulk movement in the jacket. The use of the buoyancy correction, L'B, is less valid here than in the case of radial inlet. The highest value of u8, 0.41 foot per second, occurs at the lowest flow rate, when the tangential effect is least pronounced, whereas at high flow rates the effect of U B is rather insignificant. Thus refinements such as vector addition of velocities seem hardly worthwhile in this context. As in the case of the radial inlet, Equation 19 has been used to evaluate Nu. Stirrer cases 1 and 3 are as described previously; stirrer case 4 is a 16-inch paddle a t 100 rpm. Table VI shows calculation quantities and Table IV shows Nusselt groups N u at various conditions. Ratios of
Nusselt number by calculation NU is Nusselt number by experiment NUexperlment are average 1.037, minimum 0.759, maximum 1.293. Conclusions
Jacket-side Nusselt numbers have been estimated from measurements using equipment described earlier from previously published data for paddle stirrers. The tables listing Re values down to units and Nu values to the second decimal place are not meant to indicate precision and accuracy to that extent, but are rather a list of 556
Nu from Eq. 19
Ind. Eng. Chem. Process Des. Develop., Vol. 9, No. 4,1970
1.333 1.333 1.333 3.333 3.333 3.333 6.266 6.266 6.266
60.15 55.35 47.01 102.33 95.22 89.46 151.89 140.79 133.53
75.75 55.90 49.64 89.62 84.32 71.13 174.76 132.19 120.77
0.794 0.990 0.947 1.142 1.129 1.258 0.869 1.065 1.106
TANGENTIAL INLET Stirrer Case I 90-85 90-35 40-35 90-85 90-35 40-35 90-85 90-35 40-35
1.333 1.333 1.333 3.333 3.333 3.333 4.467 4.467 4.467
90-85 90-35 40-35 90-85 90-35 40-35
1.333 1.333 1.333 4.467 4.467 4.467
115.27 106.00 98.39 211.95 199.11 191.29 261.50 247.53 233.98
111.56 88.07 81.36 163.90 191.89 157.23 344.23 305.03 278.29
1.033 1.204 1.209 1.293 1.038 1.217 0.759 0.811 0.841
126.07 97.03 93.67 296.07 270.84 235.94
0.885 1.134 1.030 0.909 0.975 1.003
104.20 93.77 105.25 183.52 167.26 165.96 230.15 235.26 222.33
1.044 1.111 0.905 1.115 1.151 1.102 1.087 1.017 1.039
Stirrer Case 3 111.90 110.08 96.44 269.13 263.98 236.74 Stirrer Case 4 90-85 90-35 40-35 90-85 90-35 40-35 90-85 90-35 40-35
1.333 1.333 1.333 3.333 3.333 3.333 4.467 4.467 4.467
108.74 104.18 95.22 204.71 192.51 182.88 250.09 239.15 230.99
Table V. Forced Convection Velocities
Table VII. Energy Dissipation and Over-All Heat Transfer Coefficients
Jacket Water Flow Rate, w , Lb, /Second
Nozzle velocity, yo, Fi/Second
Slot Velocity, v.4,
Ft/Second
(V0V?)",
Ft/Second
Calculation Values 1.333 3.333 4.467
15.67 39.18 52.50
0.1312 0.3282 0.4374
1.435 3.59 4.78
Jacket water temperatures ranged from 18" to 41° C. Representative averages indicated by the Prandtl numbers in Table I11 and VI. RADIAL INLET 6.266
U.
3.333
1.333
Energy dissipation in jacket per sq foot of nominal heat transfer area,
Table VI. Temperature Intervals, Flow Rates, and Parameters for Tangential Inlet Vessel-Side Temp., Interval, C
Re
Pr
76,900 71,021 64,257 60,109 54,414 50,980 29,696 26,067 23,228
5.6 6.1 6.8 5.4 6.1 6.7 4.9 5.7 6.5
W
4.467 4.467 4.467 3.333 3.333 3.333 1.333 1.333 1.333 4.467 4.467 4.467 1.333 1.333 1.333
80,320 76,670 65,504 29,800 27,600 22,558
5.3 5.9 6.7 5.2 5.3 6.7
Stirrer Case 4 90-85 90-36 40-35 90-85 90-35 40-35 90-85 90-35 40-35
4.465 4.465 4.465 3.333 3.333 3.333 1.333 1.333 1.333
71,776 66,802 63,056 56,806 51,694 47,864 26,966 25,226 22,132
1100 g,A
5.9 6.5 7.1 5.8 6.4 7.1 5.4 5.9 7.0
calculation quantities. Considerable errors may occur in the shorter temperature and time intervals, particularly over the 90" to 85" C range; however, the over-all ranges 90" t o 40" C and 90" to 35" C are relatively reliable. Table VI1 shows broadly some additional items that may be of interest. For the 17 different combinations of jacket flow rate and vessel stirrer, 37 runs were made; experimental values are averages of a t least two values for each condition. For the radial inlet, the average ratio of ( N u l 9/ Nuexperiment) = 1.033 for the 90" to 40" C interval, and for the tangential inlet (Nu 1 3 /Nuexperiment) = 1.055 for the 90' to 35°C interval, extremes being 0.935, 1.129 and 0.81 1, 1.151, respectively. I n the experiments reported here, energy dissipation in the jacket ranged from far below t o well above the 0.01 hp per sq foot of heat transfer area, mentioned as a usually satisfactory figure (Pfaudler Permutit, Inc., 1957). Equation 19 provides a reasonable estimate of the Nusselt group, considering the lack of data on fluid friction and heat transfer in curved ducts, the short length-
0.00260
0.000164
Over-all heat transfer coefficients U over vessel contents, temperature interval 90° t o 40" C, derived from experiment
Stirrer case 1 Stirrer case 2 Stirrer case 3
146 120 141
122 104 115
93 81 90
TANGENTIAL INLET 4.467
U'
3.333
1.333
Energy dissipation in jacket per sq foot of nominal heat transfer
.
area
E-
Stirrer Case 3 90-85 90-35 40-35 90-85 90-35 40-35
WL'd ~
0.0172
Jacket-Side Calculation Values
Stirrer Case 1 90-85 90-35 40-35 90-85 90-35 40-35 90-85 90-35 40-35
E-
~
Leu,' 1100 g,A
0.0316
0.01314
0.00084
Over-all heat transfer coefficient U over vessel contents, temperature intervai 9 P to 35O C, derived from experiment
Stirrer case 1 Stirrer case 3 Stirrer case 4
196 178 175
170
121 123 122
157
diameter ratio of bounding surfaces, one-sided heat transfer, and the absence of possible calculation refinements such as viscosity and AT corrections. The geometry and flow conditions discussed here do not suggest an obvious representative velocity that can be used in the well-known heat transfer equations. Averages, and values based on dissipation assuming isotropic turbulence, are not representative and did not yield calculated values that resembled experimentally derived results. However, consideration of fluid packets resulting from breakup of the entering jet suggested the decay relation (-udu) c: [ ( u 2 / L )d z ) , leading to the geometric mean as the representative velocity. The characteristic velocity in the Reynolds group was then defined in terms of the geometric mean of two limiting velocities, u, and 0 4 , as well as the buoyancy correction U B , as shown in Equations 14, 15, and 16. In using the Prandtl analogy Equation 6, the Blasius friction factor Equation 8 was modified according to the plot for parallel plates Equation 17, resulting in Equation 19. The equivalent diameter was evaluated from Equation 9. Being a simple, yet usable, early model involving f explicitly, the Prandtl analogy has been used t o demonstrate calculation procedure. The same modifications of velocity, equivalent diameter, and friction factor (or equivalent constant) values can be applied to other wellknown relations appropriate to the Re range, such as
N u = 0.026 ReosP r '
'
0 14
with similar results. Ind. Eng. Chem. Process Des. Develop., Vol. 9, No. 4, 1970 557
GREEKLETTERS
Nomenclature
A = heat transfer area, sq ft c = concentration, lb,/cu f t c, = specific heat a t constant pressure, Chu/lb,C = Btu/lb,-” F D = mass or molecular diffusivity, ft’/sec d = thickness, ft di = jacket inner diameter, ft d, = jacket outer diameter, ft d, = equivalent diameter, ft do = nozzle diameter, f t E = energy dissipation/ A , hpisq ft f = Fanning friction factor g = gravitational acceleration, ft. /secL g, = conversion factor, lb,-ft/lbi-sec’ h, = vessel side heat transfer coefficient, Chuihrsq ft-”C = Btuihr-sq F h, = jacket-side heat transfer coefficient, Chu/ hrsq ft-”C = Btu/hr-sq F k = thermal conductivity, Chu/hr-ft-”C = Btu/ hr-ft-”F L = paddle swept diameter, characteristic length, ft N = stirrer speed, revolutions/sec Q = heat transfer rate, Chu/hr Rpm = revolutions/ minute s = cross-sectional flow area, sq ft s = [(d2- d1)/2] = jacket spacing, ft T = temperature, C t = time, sec t = mean residence time, sec t, = tracer mean residence time, sec 7J = overall heat transfer coefficient, Chu/ hr-sq C = Btu/hr-sq ft F u = characteristic velocity in Peclet group y = volume, cu f t volumetric flow rate, cu ft/sec U A = characteristic jacket velocity ft/sec L“ = velocity due to thermal expansion, ftisec U(, = nominal nozzle exit velocity, ft/sec v = characteristic jacket velocity, ft/sec u = mass flow rate, lb,/sec height, ft 2 = distance, f t f t - O
8 = coefficient of expansion,/OC A = difference 0 = t t=reduced time, dimensionless p = viscosity, lb,/ft-sec u = momentum diffusivity, sq ft/sec $ = velocity ratio
SUBSCRIPTS
i = vessel-side j = jacket-side m = mass, mean 1~ = a t wall FUNCTIONS AND DIMENSIONLESS GROUPS Nu = (hodelk)= Nusselt number
uL’D = Peclet number (diffusion and mixing) P r = (c,l/k) = Prandtl number Re = (d,v,v) = Reynolds number
f t - O
O
~
f t - O
-O
v =
z=
558
Ind. Eng. Chem. Process Des. Develop., Vol. 9, No. 4, 1970
where E is the exit age distribution function, derived from measurements of concentration c by the relations C
E ( t ) = __ BCAt
’
and E = FE ( t ) .
Literature Cited
Grober, H., Erk, S., Grigull, U., “Fundamentals of Heat Transfer,” Chap. 11, p. 244, McGraw-Hill, New York, 1961. Lehrer, I. H., IND.ENG.CHEM.PROCESS DESIGNDEVELOP. 7, 226 (1968). Levenspiel, O., “Chemical Reaction Engineering,” Chap. 9, pp. 260, 252, Wiley, New York, 1962. Lohrenz, J., Kurata, F., Ind. Eng. Chem. 52, 703 (1960). McAdams, W. H., “Heat Transmission,” Chap. 9, p. 233, McGraw-Hill, New York, 1954. Pfaudler Permutit, Inc., Rochester 3, N . Y., “Pfaudler Agitating Nozzle,” Bull. 950 (1957). Uhl, V. W., Gray, J. B., “Mixing,” Vol. 1, Chap. 5 , p. 111, Academic Press, New York, 1966. RECEIVED for review March 5 , 1969 ACCEPTED March 4, 1970
