CALCULATION OF CENTER-LINE TEMPERATURES IN TUBULAR HEAT EXCHANGERS FOR PSEUDOPLASTIC FLUIDS IN STREAMLINE FLOW S.
E. C H A R M
Department of Nulrition, Food Science, and Technology, Massachusetts Institute of Technology, Cambridge 39, Mass.
An equation was developed for determining the center-line temperature of pseudoplastic fluids by extending Eckert’s consideration of Newtonian fluids. It was experimentally difficult to achieve fully developed thermal conditions. The equation checked satisfactorily with experimental results both where fully developed thermal flow did not exist and also where fully developed thermal flow did exist.
is commonly used in evaporators, sterilizers, and various other heat-processing equipment. I n certain of these processes, it is useful or necessary to know the temperature a t the center line of a tube, where the greatest temperature lag exists. This is especially important in sterilization or pasteurization processes. When the flow in the tube is turbulent, the temperature is essentially uniform across the radius and is equal to the “mixing cup” average temperature. However, if the flow is streamline, then a nonuniform temperature distribution exists across the radius. The center line exhibits the lowest temperatures in the tube when a fluid is heated and the highest temperatures when it is cooled. I t is possible to calculate the center-line temperature and, under certain conditions. the complete temperature profile for pseudoplastic fluids, from a knowledge of the mixing cup average temperature anld the pseudoplasticity constants.
T
HE TUBULAR HEAT EXCHASGER
Determination of the Temperature Profile for Streamline Flow ond Fully Developed Thermal Conditions
I n 1950, Eckert ( 2 ) suggested a method for calculating the temperature profile of a Newtonian fluid in streamline flow. T h e development of Eckert’s method is as follows: Consider a fluid in streamline flow being heated in a tube. T h e heat flow through the layer of fluid immediately adjacent to the wall must be independent of y, since this layer has a negligibly small velocity. and therefore no heat is carried away by convection. For a cylinder of radius, Tu, the equation expressing the inward rate of heat transfer a t the wall is
Solving for the temperature gradient, and differentiating with respect toy, gives (3)
Determination of Type of Flow
A pseudoplastic fluid is defined as one which has a shear stress-rate of shear relationship expressed by T
=
b
(z)’
The type . . of flow in a t.ube may be established from a consid8D.P-s
eration of the generalized Reynold’s number, gc b(,,s+3)a.
If
the Reynold‘s number is less than 2000, the flow is streamline ( 5 ) . At Reynold’s numbers greater than 2000, the flow becomes turbulent. For Kewtonian fluids, this occurs a t Reynold’s numbers greater than 3000, but for pseudoplastics, the Reynold’s number at which turbulence begins is indefinite
At the wall, y = 0 and Equation 3 becomes (4)
When a fluid is being heated in a tubular heat exchanger, a t the entrance to the tube, the temperature profile is rectangular. According to Eckert ( 3 ) ,as the fluid moves through the tube, the shape of the temperature profile changes gradually from a rectangle to that of a cubic parabola when fully developed thermal conditions exist (Figure l ) . Fully developed thermal conditions exist when the thermal boundary layers meet a t the center line. At this point the center-line temperature \vi11 start to rise. Where fully developed thermal conditions do not exist in a tube, the centerVOL. 1 NO. 2 M A Y 1 9 6 2
79
An energy balance shows that
Merrill (4)has shown the point velocity for a pseudoplastic fluid to be
The volumetric rate of flow is
Therrnolly Undeveloped
Assuming T , to be constant and substituting Equations 6, 8, and 9 in Equation 7 , this becomes
J
I
T, -
R-0-R Distance
Along
7-11
- a(fw-
7)
x
Radius
Figure 1. Temperature distribution as a function of distance along pipe
[[I
+ fr
(r%r)
- 2
(*)'I/ r
- r
dr
=
Upon integrating Equation 10, and simplifying
line temperature may be assumed to be equal to the initial temperature of the fluid. The equation for the cubic parabola temperature profile, assumed to apply at fully developed thermal conditions, is given by
T
=
T , - a ( r , - r) - b(r, - r ) ? - c(rw - r ) 3
(5)
If one differentiates Equation 5 and substitutes in Equation 4 for conditions at the wall, one obtains
Since at the center line, d T / d r must be zero, then c = - - -
2 a 3 TU.2
Therefore, Equation 5 becomes
If T , is determined from Equation 12 and is less than the initial temperature T I although the fluid is being heated, it indicates that fully developed thermal conditions do not exist a t this point. Therefore the center-line temperature must still be equal to the initial temperature, until fully developed thermal flow does exist. Equation 12 applies only to fully developed thermal conditions. In Figure 1 a t L = 0, the effect, assuming a parabolic temperature profile a t a point where fully developed thermal flow does not exist, is depicted graphically. The point at which fully developed thermal flow begins may be found by noting where TI = T , as determined by Equation 1 L.
Evaluation of the Constant
A consideration of the enthalpy changes in each streamline, as well as the enthalpy changes in the bulk fluid between the entrance and a point where fully developed conditions exist, makes it possible to evaluate the constant a in Equation 5 .
Table I .
Material
Apple sauce
Banana puree
Ammonium alginate, 3.37%
I I E C FUNDAMENTALS
The foregoing discussion applies to cooling as well as heating fluids in tubes. However, in cooling if T , is calculated to be greater than TI, fully developed thermal flow does not exist and the center-line temperature must actually equal the initial temperature.
Flow Rates and Physical Constants of Various Fluids in Table II
Flow Rate,
80
From Equation 6, it is noted that the center-line temperature (t. = 0) is given by 5 Ts = T , - - arm (12) 6
Lb./Hr. (heating) 465.0 (cooling) 490 (heating) 536 (cooling) 394 (heating) 200
B.t.u. (ft.)
K,
C?l, B.t.u.
(Hr.)(Ft.*)(' F.) 0.314
Lb.-" F. 0,960
0.400
0.343
0.870
0.980
P
( G. / Cc .)
1.084
1.115
0.985
Ti
TP
87.6
F. 101.7
122.5
108,6
112.0
141.3
114.0
99.7
214.0
106.5
' F.
Table II.
Comparison o'f Calculated and Experimental Center-Line Temperature at 12.48 Feet from the Entrance of a Tube Heat Exchanger (1.00-inch I.D.)
Fluid
Apple sauce
Experimental, T,, F.
Experimental, T I , O F.
Calcd., ( E q . 72-15), TQ, F.
(heating) 135.2 (cooline) 70.5'
87.6
101.7
75
122.5
108.6
137.5
112.0
141.3
72
0.458
(heating) 208.0 (cooling) 66.2
114.0
99.7
128.3
(heating) 214
106.5
162.1
0.534
121 . o
S
0.645 0,645
Banana puree
Ammonium alginate (3.37%) a
0.458
Calcd. ( E q .2 2 ) , T,, F.
Condition of Flow
Thermally undeveloped Thermally undeveloped
Experimental, Tc, O F.
Final Cal;d.,"
T,,
F.
88
87.6
122
122.5
Thermally undeveloped Thermally undeveloped
113
112.0
114.0
114.0
Developed thermally
127.0
121 . o
When flow is thermally nndeueloped, T , is equal to initial temperature.
Calculations of Mixing Cup Average Temperatures, 1 2
I n order to calculate a employing Equation 11, it is first necessary to calculate T z ,the mixing cup average temperature a t the point under consideration. T h e calculations of T S may be carried out in the following manner. Charm and Merrill ( ; I ) suggested the following equation for evaluating the heat transfer coefficient of pseudoplastic fluid in streamline flow. hiD = 2.0 K
(x;L) W'iCp o.33 (b,
+
(3 l/s))O*" b,(3 - l / ~ ) 2
Also
and
required to obtain the mixing cup temperature. T h e fluid passed through a calming srction before entering the heat exchanger. I t was possible to recycle the fluid. T h e tube wall was 0.0625 inch thick. The fluids investigated included banana puree, apple sauce, and a 3.377, ammonium alginate solution. The th erma properties of the fluids were determined experimentally by Charm and Merrill ( I ) . The physical properties are summarized in Table I. T h e shear stress-shear rate characteristics of the fluids under consideration were determined in a narrow gap coaxial viscometer. Employing the generalized Reynold's number discussed previously, it was found that under all the test conditions the fluids were in streamline flow. The calculated and experimental results for center-line temperatures are compared in Table 11. Discussion of Results
By assuming a value of Tf it is possible to evaluate Equation 13 and then solve for 1JA from Equation 15. By employing this value in Equation 14, it is possible to determine whether or not Equation 14 results in a n equality for the assumed value of Tf. If a n inequality results, a new value for T Zis assumed. Thus it is possible through trial and error solution to solve for Ti employing Equations 13, 14, and 15. Experimental
I n order to check Equation 12, the center-line temperatures of several pseudoplastic fluids were measured during heating and cooling in a straight tube. The experimental apparatus employed herein has been described in detail elsewhere ( 7 ) . Briefly, the apparatus consisted primarily of a concentric-tube heat exchanger with the fluid passing through the center tube. The center tube was copper, 1.00 inch in diameter and 12.48 feet long. Thermocouple probes were located to measure the center-line temperature a t the end of the heat exchanger-the inlet temperature, Ti, and mixing cup outlet temperature, T I . Three thermocouples were placed along the tube wall to measure the variation in wall temperature along the tube length. There was substantially no \,ariation. A baffle arrangement was
The difficulty of achieving fully developed thermal conditions in streamline flow is apparent from Table 11. I n the case of ammonium alginate, fully developed thermal conditions did exist a t the point of measurement. The agreement between the experimental and calculated result was satisfactory. I n the case of the other fluids, thermally developed conditions did not exist a t the point of measurement and the center-line temperature was found to be experimentally equal to the initial temperature. This was predicted, however, from Equation 12 as in each of these heating cases T , determined from Equation 12 was less than the initial temperature, TI, while in the cooling experiments it was greater. Equation 12 appears to be useful for determining the thermal conditions of flow and the center-line temperatures for pseudoplastic fluids in streamline flow. Nomenclature
A
= surface area associated with over-all heat transfer
coefficient A,, A I = outside and inside surface areas of tube, respectively = constant fixing cubic parabola a b = a pseudoplastic constant = pseudoplastic constant ?valuated a t bulk average 6, temperature VOL. 1
NO. 2
MAY 1962
81
2
dV dr
= pseudoplastic constant evaluated a t tube wall tem-
W
perature = specific heat of fluid = diameter of tube
X
= = = = =
Q
r
=
To
S
TI T2 U V
I’
= distance from wall
7
= shear stress
= shear rate = = = = =
Q
= mass flow rate = tube wall thickness
= =
= = =
gravitational constant inside heat transfer coefficient outside heat transfer coefficient thermal conductivity thermal conductivity of tube wall length along tube pressure rate of heat transfer amount of heat transferred distance from center tube radius pseudoplastic constant entering temperature mixing cup average temperature a t distance L over-all heat transfer coefficient velocity
literature Cited
(1) Charm, S. E., Merrill, E. W., Food Research 24, 3 (1959). (2) Eckert, E. R. G., “Introduction to the Transfer of Heat and
r)
Mass,” p. 99, McGraw-Hill, New York, 1950. (3) Zbid., p. 103. M errill, E. W., J . Colloid Sci.2, 7 (1956). 5) Metzner, A. E., “Advances in Chemical Engineering,” Vol. I, p. 114, T. B. Drew, J. W. Hoopes, eds., Academic Press, New York, 1956. (6) Shaver, R. G., “Pseudoplastic Fluids in Turbulent Flow,” Sc.D. thesis, Dept. of Chemical Engineering, M.I.T., 1956. RECEIVED for review November 21, 1961 ACCEPTED March 12, 1962 Contribution No. 398 from the Department of Nutrition, Food Science, and Technology, Massachusetts Institute of Technology, Cambridge. Mass.
CYCLIC MIGRATION OF BUBBLES IN VERTICALLY VIBRATING LIQUID COLUMNS R. H. B U C H A N A N , G R A E M E J A M E S O N , ‘ A N D D O E R I A M A N O E D J O E
The University of New South Wales, Kenrington, Australia
A new procedure has been developed for bringing gas, liquid, or solid phases into contact to promote exchange phenomena. When liquid columns are subjected to high-amplitude low-frequency vertical vibrations, bubbles migrate cyclically from the surface to the bottom, where they aggregate and then move back up the column. The result is violent and rapid mixing of the gas with the liquid and with suspended solids. An equation is derived which permits accurate prediction of the minimum frequency for cyclic migration for the air-water system a t atmospheric pressure. Some deviation occurs with liquids whose properties differ markedly from water and a t pressures other than atmospheric.
in liquid-liquid extraction has become an established chemical engineering operation. This report describes another type of pulse column which appears to have considerable promise for mixing and mass transfer. In this case no perforated plates are used in the vertically vibrating liquid column. At the high vibrational intensities involved, bubbles of gas migrate alternately from the top of the column to the bottom, and then return to the top. The over-all effect is one of violent and rapid mixing of the gas with the liquid and with the suspended solids if present. This phenomenon provides a good method of gas-liquid and gasliquid-solid contacting when countercurrent flow is unnecessary. I t could, for example, be used to promote difficult gas absorptions in either the absence or presence of suspended catalysts, or to clean or extract solids and produce emulsions. The pulsating bubble phenomenon was first described by Pielmeier ( 5 ) , who was mainly concerned with its cleaning HE USE OF PULSE COLUMNS
1 Present
a2
address, Cambridge University, Cambridge, England.
I&EC FUNDAMENTALS
action. H e wrongly attributed this cyclic action to cavitation. His paper goes into no detail on the bubble motion except to state that, by using a stroboscope, it is found that small suspended bubbles have minimum volumes when the liquid is at the bottom of its stroke, and maximum volume when at the top. T h r downward migration of bubbles is not elucidated. This new process is described in general terms, and the derivation and experimental verification of an equation which relates the limiting parameters of the process are presented. Liquid columns 5 to 15 cm. in diameter and 15 to 100 cm. high are used with amplitudes of 0.05 to 1.0 cm.and frequencies of 1200 to 2900 r.p.m. Two papers on other aspects of this work are in preparation, one on oxygen absorption in sodium sulfite solution and the other on cavitation. Description of Process
A typical case is described here, in which the liquid height and density, the amplitude of vibration, and the external pressure are fixed, and the frequency of the vibration is varied (Figure 1). The liquid is placed in a glass or Perspex cylinder,
