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Correction of systematic errors in measuring convection coefficients in a liquid-fluidized bed
Ckm&t~IEn#tneaIyl Seiencc~Vol. 48, No. 3,pp. SZl-S26, 1993. in Glrpt Britain.
ooo9_2m!9/93 S5.00+0.00 ~1992PergamonPress Ltd
Primal
CORRECTION CONVECTION
OF SYSTEMATIC ERRORS IN MEASURING COEFFICIENTS IN A LIQUID-FLUIDIZED BED
ROBERT C. BROWN Department of Mechanical Engineering, Iowa State University, (Received
18 June 1991; accepted
16 March
Ames, IA 50011, U.S.A. 1992)
Absmct-Several studies of liquid-fluid&d beds have measured convection coefficients based on electrical power dissipation from surfaces that are closely wrapped with resistance. wire. In this technique, convection co&Gents are determined by plotting dissipated power vs average wire temperature, the latter quantity determined from me’asurements of wire resistivity. Studies in our laboratory have revealed significant systematic errors in convection coefficients measured in this manner. The errors are of two types. First, the technique assumes that the surface being tested is at a uniform temperature equal to the wire temperature. Seumd, the technique assumes that all the heat is dissipated directly to the fluidized bed from the wirewrapped region of the surface. Undei the experimental conditions we investigated, these assumptions
yielded underestimates of convection mcients by t&35%. We present a methodology that accounts for these errors, thus preserving the utility of the heated-wire technique.
KVI-RODUCTION
Richardson and coworkers [l-6] have described a technique to measure convection coefficients in fluidizcd beds based on electrical power dissipation from surfaces wrapped with resistance wire. We have employed this technique in our own work with liquidfluid&d beds. In the course of these studies, we have noted significant systematic errors in convection coefficients measured in this manner. This paper investigates the nature of these systematic errors and describes a method to correct them, thereby preserving the utility of the electrically heated wire technique. The use of this methodology is demonstrated for water-fluid&d beds of 5 mm glass beads. The method in question determines the convection coefficient between an electrically heated surface and a liquid or liquid-fluid&d bed. The electrically heated surface consists of an object wrapped by a length of resistive wire. Richardson and coworkers Cl-61 have used this method to measure convection coefficients for a variety of geometrical shapes in fluid&d beds. In this technique, the measured convection coefficient, h,,,, between a heated surface and a fluid is related to the dissipated electric power, P, by the equation P = h,A(T,
- T/)
(1)
where A is the heat-transfer area, T, is the wire tcmpcratuie, and Tf is the temperature of the fluid. The power supplied to the wire can be determined from measurements of voltage drop across the wire and current flow through the wire. The resistance of the wire can also be found from these measurements through Ohm’s law. An ins-ion of eq. (1) reveals a linear relationship between power dissipated from the heated surface and the temperature of the surface. If power is plotted against surface temperature and h,,,, A, and T, are 521
constants, then a straight line is obtained. The fluid temperature can be determined from the x-intercept of the plot. The slope of the line is equal to the product h,A; if the heat exchange surface area is known, the convection coefficient, h,, can be found. The temperature of the wire, which is assumed equal to the temperature of the surface, can bc determined from measurements of the wire resistance. The resistance of the wire, R, and the temperature of the wire, T,, are related by the linear relation R = R,[l
+ u(T, - To)]
(2)
where R, is the resistance measured at a reference temperature, To, and a is the temperature coefficient of resistance of the wire. The temperature coefficient of resistance for the wire can be found by calibrating the wire in a constant temperature bath at several different temperatures. There are two potential shortcomings in evaluating the convection coefficients by eq. (1). First, it assumes that the surface being tested is at a uniform temperature equal to the measured wire temperature, T,,,. In fact, when heat convection to the fluid&d bed is very high, temperature gradients may exist between the wires on the surface, which results in an average surface temperature less than T,. The result is underestimation of convection coefficients. Equation (1) also assumes that all electric power is dissipated as heat, directly to the fluidized bed. Under some experimental conditions, heat can be conducted away from the wire-wrapped section of the heat-transfer surface before it is convected into the bed. The result, in this case, is overestimation of convection coefficients. The systematic errors resulting from these two situations can be accounted for by the following procedure. Energy enters an element of surface area in the form of electrical power, P. Energy leaves the element by both conduction, Qcandrout of the ends of the element
522
ROBERTCBROWN
and by convection, Q,,,,, into the fluid&d energy balance is expressed by P = Qcon.i+
Q.ow
bed. This (3)
The convection term is determined from the average surface temperature, T, by the relation
Q
CO””
=
it,
A(T,
-
T,)
where ia. is the actual convection coefficient at the surface. Substituting eqs (1) and (4) into eq. (3) yields the ratio of actual convection coefficient to measured convection coefficient which we call the correction factor CF:
The first term on the right-hand side of eq. (5) corrects for temperature gradients on the heat-transfer surface, while the second term corrects for conduction out of the ends of the,wire-wrapped heat-transfer region. The parameters T,, Tf, P, and h, in eq. (5) are determined from experiments. To determine h,, we must also know T, and Qcand,which we have calculated by a finite-difference computer model of the heat-transfer element. This model is discussed in the next section after the geometry of the heat-transfer element used in our experiments is described. EXPERIMENTAL
PROCEDURE
A schematic diagram of the apparatus used in the experiments is given in Fig. 1. The water is pumped from a holding tank by a centrifugal pump. The water can then flow back to the tank through a bypass valve or to the fluidized bed. Two rotameters in parallel allow for a wide range of flow rates to bc metered into the fluidized bed. The water enters the fluidized bed from a plenum chamberThe plenum chamber allows the water flow to be evenly distributed across the cross section of the test section. The plenum contains a packed bed of 3 mm diameter beads to assist in distributing the flow evenly over the entire cross
I
section. The fluidized bed consists of a 5.72 cm diameter plexiglass tube arranged vertically. The plexiglass tube is 45 cm in length. Plexiglass was chosen to allow visual access to the fluidizcdbed. A wire screen, located just below the test section, acts as a distributor to support the particles in the bed. After passing through the bed, the water exits through a tube that returns to the holding tank. Convection coefficients are measured for a 1.9 cm diameter Pyrex cylinder oriented coaxially to the fluidized bed. Resistance wire wrapped around the Pyrex cylinder for heat-transfer measurements was an MWS 120 alloy. The temperature coefficient of resistance was found to be O.O0412”C-r, which compares favorably with the manufacturer’s specification of O.O045”C- ‘. The total resistance of this wire was 46.86 k-2at 30°C. The resistance wire was soldered to low-resistance leads that passed through the center of the Pyrex cylinder. A Variac transformer supplied power to the wire coil. The voltage across the length of resistance wire and the current through the wire were measured by use of digital multimeters. The equipment used in the experiments allowed voltages to be. read to 0.01 V and the current to be read to 0.001 A. For the lowest input power levels, the uncertainty in wire temperature was less than 1°C; at the higher input power levels, this uncertainty was less than 0.2”C. For the lowest power levels, the input power was read to within 3%, while the input power was read to within 1% for the higher power levels. The water used in the tests was distilled and deionized to assure that the resistance wire could not short-circuit when immersed. A test was carried out in which two exposed wires were held approximately 1 mm from each other in distilled and deionized water. A voltage was applied across the gap between the wires, and the measured current flowing between the wires was only 5 to 10 PA for voltages up to 50 V, thus, the wires would not electrically short out when immersed in the liquid-fluidized bed. A cooling coil was immersed in the water holding tank to maintain the water at a constant temperature during the tests.
u
CENTRIFUGAL PUMP
Fig. 1. Schematic diagram of the experimental apparatus.
Correction
of systematic
errors in measuring
Thermocouples were placed at the inlet and outlet of the fluidized bed to monitor the water temperature. The resistance wire used was 0.1 mm in diameter. The cylinder was wrapped with 27 turns of resistance wire, with 0.4 mm spaces between the wires. Three 1 mm wide strips of adhesive tape were attached to the surface of the cylinder under the wire to aid in keeping the wraps of resistance wire from contacting each other. The tape covered less than 5% of the total surface area and, thus, was considered to have negligible effect on the heat-transfer measurements. The length of the cylinder that was covered by wire was 18 mm. The overall heat-transfer area for the 19 mm diameter cylinder was 1.07 x 10e3 m2. The computer model consists of a two-dimensional mesh of nodes 203 x 4 in sire. A convection boundary condition was used for the outer surface of the Pyrex tube. An adiabatic boundary condition was assumed at the interior surface of the Pyrex cylinder: natural convection at the interior of the cylinder is at least two orders of magnitude less than the convection at the outer surface in contact with the fluidized bed. The ends of the grid were also assumed to be insulated since temperature gradients disappear at these locations. Parameters used as inputs to the computer model were the convection coefficient at the outer surface of the element, h,, the power input to the wires, P, and the ambient temperature of the water, T,-, Provisions were also made for including contact resistances between the heated wires and the surface upon which they were wrapped. However, contact resistance proved to be negligible for the cylindrical surface employed in this study. A more conventional heat-transfer probe was also constructed for comparison with the heated-wire probe. This 152 mm long probe consisted of a com-
__
eenvection coefkients
in a liquid-fluid&d
bed
mercial cylindrical heater element enclosed by a 16 mm diameter copper tube. Silver solder was run between the heater element and the inner diameter of the copper tube to assure good thermal contact between the two. Along one side of the tube, small slots were machined just large enough to contain thermocouple beads. The thermocouples were sealed into the grooves by solder which was ground flush with the copper surface. Measurement of average surface temperature, T,, obtained from the four thermocouples, fluid temperature, T,, from a thertnocouple in the fluidized bed, and probe power, P, allowed calculation of heat-transfer coefficients at the surface of this probe according to
PIA
h=(c-
T,)
where A is the surface area of the probe. RESULTS
The finite-difference computer model was used to simulate heat transfer from the wire-wrapped surface for several power settings and convection coefficients. An example of the resulting temperature profile is shown in Fig. 2. From the figure, it is clear that temperature gradients exist along the surface between the wires for our experimental conditions; thus, the assumption of uniform wall temperature is a poor one in liquid-fluidized beds and convection coefficients will be underestimated by the heated-wire technique. For each computer run, the resulting temperature profile was used to calculate the average surface temperature Ts for the heat-transfer surface and also the amount of heat conducted out the ends of the heat-transfer surface, Qmnd. The ratios Q,,JP and (T, - TJ/( T, - Tf) are plotted against input power
--__-----_
24
50
0
100
150
L 200
Node-
Fig. 2. Simulated
CES 43:3-F.
temperature
523
profile on the surface of the heated-wire P = 12Wm-I).
probe (h = 2ooO W m-* K-l,
ROBERT C. BROWN
524
in Fig 3(a) and (b), respectively, for several different values of h,. From these plots it is evident that these ratios are not functions of power and, for any given experimental arrangement, will depend only on the convection coefficient at the surface. Accordingly, a plot of the correction factor given by eq. (5) vs the measured convection coefficient, h,, can be used to correct systematic errors in the experimental data arising from assumptions inherent to eq. (1). Such a plot, in a range of h, appropriate to water-fluidixed beds, is given in Fig. 4. The resulting curve has large linear ranges. Convection coefficients were measured for the wirewrapped probe in a bed of 5 mm glass beads for superficial flow velocities from 0.01 to 0.15 m/s. For each superficial velocity, a series of voltage and current measurements were taken. Using eq. (2), the resulting resistances were converted into tempcratures. Plots of power vs wire temperature were extremely linear, with linear regression analysis giving correlation coefficients generally exceeding 0.99. Accordingly, convection coefficients were constant over large power ranges; this result is in agreement with the finite-difference computer model. Furthermore, since the predicted wire temperatures were always higher than the measured wire temperatures, the use of contact resistances in the model did not improve model
predictions and were deemed unimportant in the present experiments. It is evident that the wire-wrapped surface is rougher than a bare Pyrex surface, and it is possible that the wires will affect convection coefficients. In order to determine the effect, additional tests were conducted on a plexiglass tube of the same dimensions as the previous Pyrex tube. The plexiglass tube was first wrapped with the wires on the surface, just as the Pyrex tube was wrapped. Heat transfer from the surface to a bed of glass beads fluid&d at a superficial velocity of 0.0615 m/s was measured. The surface of the plexiglass tube was then grooved on a lathe to allow the wires to be recessed. The tube was grooved to a depth equal to the wire diameter. The tube surface then closely approximated a smooth surface. The heat transfer from the surface with recessed wires was then measured. The results of the test are shown in Fig. 5. The two curves are separated from one another because of different bulk water temperatures in the two tests; however, the slopes of the two curves are nearly equal. The slope of the curve for the nongrooved surface deviates from the slope for the grooved surface by 8%. This good agreement suggests that the wires present on the surface of the tube do not significantly affect the convection coefficient within the fluid&d bed. Figure 6 compares convection coefficients obtained with our heated-wire probe with values obtained with
E
1.6
f
::;I
,
,
h ; 8000 Wly*tK
1
8
1.4
.I
1.2
s 100
200
600
300 400 Power, W
600
11
l .O,__L
.
Measured
(b) 1.8------l
Fig. 4. Correction
Heat-Transfer
factor vs measured
Caeftident,
convection
hm
coefficient.
h = 8000 W/mz/K
+ ” F
1.4
lz
1.2
’
--b
Wire Nonrecessed Wire Recessed
t
,.J
I 100
,
,
200
306
p =2ooy Wlm2l~ 400
600
j
600
700
0 0
Power. W
20
Fig. 3. Correction factor as a function of power and actual convection coefficient (convection coefficients are in Wm2K-I):
(a) contribution
from
conduction
losses;
contribution from temperature gradients between
(b)
wires.
Fig. 5. Comparison
I 22
I 24 Temperature,
of heat transfer _
I 26
28
Oc
recessed wires to a surtace wlthout
from a surface .
recessed wires.
with
Correction
of systematic
errors in measuring
convection
beating-element probe. The data designated as “uncorrected heat-wire” were obtained directly from eq. (I). The data designated as “corrected heat-wire” were calculated by multiplying h, from eq. (1) by the CF obtained from Fig 4. Note that convection coefficients obtained with our heated-wire probe do not
our
coeiikients
in a liquid-fluidized
bed
525
agree with our heating-element probe unless the CF is applied. This result experimentally validates our computer model prediction that temperature gradients between heated wires will result in an underestimation of convection coefficients between the surface and the fluidized bed when using eq. (1).
5000 0
n 4500
0
.
.
0
0
1
.
n
.
.
.
.
.
I .
.
1sooJ 0.010
0.020 9
0.030
Uncorrected
Fig. 6. Convection
coefficients
0.040 0.050 Superficial 0 Corrected Wire
0.000 0.070 velocity (m/s) l Heating Wiie
in liquid-fiuidized beds: Comparing element probe.
heated-wire
0.080
O.OQO
I
0.100
Element techniques
to a heating-
5000 -
___________.___-----’
. 2000.. 1500.. _/_*-’
_A
/-
_____________________-.----~*-----_____--_____-.__.-.-___---_,__---__-_._-*I __--
I
10001
0.000 l
0.040
0.020 Corrected
W&a
0.080 0.100 0.060 Superficial velocity (m/s) Kate et al. ---- Juma
Fig. 7. Convection eoeftkients in liquid-fluid&d corrections
beds comparing of other researchers.
0.120
0.140
k Richardson
the cmrccted
heated-wire data to
ROBERT C. BROWN
526
We compare our (corrected) heated-wire data to correlations by Juma and Richardson [4] and Kato et al. [7] in Fig. 7. Juma and Richardson [4] measured convection coefficients in a liquid-fluidized bed of paraffin oil and glass beads using the (uncorrected) heated-wire technique. Their correlation yields convection coefficients substantially smaller than those obtained in our experiments. There are differences between the present experiments and those of Juma and Richardson [4]. Their probe is of larger diameter and is horizontally rather than vertically oriented in the flow. Furthermore, the effect of Prandtl number on Nusselt number was not thoroughly investigated by Juma and Richardson [4] and caution is required in applying their correlation to a water-fluidized bed. These differences may account for part of the discrepancy between their correlation and our data. However, we suspect that the larger source of discrepancy is temperature gradients between wires and its influence on measured convection coefficients. Kato et al. [A obtained their correlation for a heated wall in a water-fluid&d bed using an experimental technique similar to that employed for our heating-element probe. Their correlation agrees well with more recent studies [8, 91. Figure 7 shows that our (corrected) heated-wire data follows closely the correlation of Kato et al. [7]. Without the use of the correction factor for the heated-wire technique, our data would have underestimated convection coefficients by 2CL35%.
underpredicted convection coefficients for liquidfluidized beds. We have devised a methodology that accounts for the errors arising from temperature gradients between wire wrappings and heat loss from the ends of the wrapped region, thus preserving the utility of the heated-wire technique. Nevertheless, the large correction factors required for systems with convey tion coefficients in excess of 1000 W m-‘K-t may make alternative experimental techniques more attractive than the heated-wire approach.
Acknowledgments-The support of the Iowa State University Power AfEliate Program and the Iowa State University Achievement Foundation is gratefully acknowledged for making this research possible. I appreciate the assistance of Larry Burken, Rod Oathout, and Jeff Delfs in performing some of the experiments and analysis described in this paper.
REFERENCES
Cl1 Latif, B. A. J. and Richardson, J. F., 1972, Chem. Eagng
c33 c41 CSI C61 c71
CONCLUSIONS
Our work suggests that the heated-wire technique is flawed by systematic errors that result in greatly
Sci. 27, 1933. J. F. and Romani, M. N., 1974, Lest. Heat Mass Translkr 1. 55. Romani, M: N.,~Richardson, J. F. and Shakiri, K. J., 1976, Chem. Ennnn Sci. 31, 619. Juma,A. K. A. and Richardson, J. F., 1982, Chem Engng Sci. 37, 1681. Khan, A. R., Juma, A.K.A. and Richardson, J. F., 1983, Chem Engng Sci. 38,2053. Juma, A. K. A. and Richardson, J. F., 1985, Chem. Engng Sci. 40, 687691.. Kato, Y., U&da, K., Kago, T. and Morooka, S., 1981, Powder Technol. 28, 173. Chiu, T. M. and Ziegler, E. N., 1983, A.1.Ch.E. J. 29,677. Muroyama, K., Fukuma, M. and Ysshunishi, A., 1986, Can. J. them. Engng 64,399.
c21 Richardson,
::I
ooo9_2m!9/93 S5.00+0.00 ~1992PergamonPress Ltd
Primal
CORRECTION CONVECTION
OF SYSTEMATIC ERRORS IN MEASURING COEFFICIENTS IN A LIQUID-FLUIDIZED BED
ROBERT C. BROWN Department of Mechanical Engineering, Iowa State University, (Received
18 June 1991; accepted
16 March
Ames, IA 50011, U.S.A. 1992)
Absmct-Several studies of liquid-fluid&d beds have measured convection coefficients based on electrical power dissipation from surfaces that are closely wrapped with resistance. wire. In this technique, convection co&Gents are determined by plotting dissipated power vs average wire temperature, the latter quantity determined from me’asurements of wire resistivity. Studies in our laboratory have revealed significant systematic errors in convection coefficients measured in this manner. The errors are of two types. First, the technique assumes that the surface being tested is at a uniform temperature equal to the wire temperature. Seumd, the technique assumes that all the heat is dissipated directly to the fluidized bed from the wirewrapped region of the surface. Undei the experimental conditions we investigated, these assumptions
yielded underestimates of convection mcients by t&35%. We present a methodology that accounts for these errors, thus preserving the utility of the heated-wire technique.
KVI-RODUCTION
Richardson and coworkers [l-6] have described a technique to measure convection coefficients in fluidizcd beds based on electrical power dissipation from surfaces wrapped with resistance wire. We have employed this technique in our own work with liquidfluid&d beds. In the course of these studies, we have noted significant systematic errors in convection coefficients measured in this manner. This paper investigates the nature of these systematic errors and describes a method to correct them, thereby preserving the utility of the electrically heated wire technique. The use of this methodology is demonstrated for water-fluid&d beds of 5 mm glass beads. The method in question determines the convection coefficient between an electrically heated surface and a liquid or liquid-fluid&d bed. The electrically heated surface consists of an object wrapped by a length of resistive wire. Richardson and coworkers Cl-61 have used this method to measure convection coefficients for a variety of geometrical shapes in fluid&d beds. In this technique, the measured convection coefficient, h,,,, between a heated surface and a fluid is related to the dissipated electric power, P, by the equation P = h,A(T,
- T/)
(1)
where A is the heat-transfer area, T, is the wire tcmpcratuie, and Tf is the temperature of the fluid. The power supplied to the wire can be determined from measurements of voltage drop across the wire and current flow through the wire. The resistance of the wire can also be found from these measurements through Ohm’s law. An ins-ion of eq. (1) reveals a linear relationship between power dissipated from the heated surface and the temperature of the surface. If power is plotted against surface temperature and h,,,, A, and T, are 521
constants, then a straight line is obtained. The fluid temperature can be determined from the x-intercept of the plot. The slope of the line is equal to the product h,A; if the heat exchange surface area is known, the convection coefficient, h,, can be found. The temperature of the wire, which is assumed equal to the temperature of the surface, can bc determined from measurements of the wire resistance. The resistance of the wire, R, and the temperature of the wire, T,, are related by the linear relation R = R,[l
+ u(T, - To)]
(2)
where R, is the resistance measured at a reference temperature, To, and a is the temperature coefficient of resistance of the wire. The temperature coefficient of resistance for the wire can be found by calibrating the wire in a constant temperature bath at several different temperatures. There are two potential shortcomings in evaluating the convection coefficients by eq. (1). First, it assumes that the surface being tested is at a uniform temperature equal to the measured wire temperature, T,,,. In fact, when heat convection to the fluid&d bed is very high, temperature gradients may exist between the wires on the surface, which results in an average surface temperature less than T,. The result is underestimation of convection coefficients. Equation (1) also assumes that all electric power is dissipated as heat, directly to the fluidized bed. Under some experimental conditions, heat can be conducted away from the wire-wrapped section of the heat-transfer surface before it is convected into the bed. The result, in this case, is overestimation of convection coefficients. The systematic errors resulting from these two situations can be accounted for by the following procedure. Energy enters an element of surface area in the form of electrical power, P. Energy leaves the element by both conduction, Qcandrout of the ends of the element
522
ROBERTCBROWN
and by convection, Q,,,,, into the fluid&d energy balance is expressed by P = Qcon.i+
Q.ow
bed. This (3)
The convection term is determined from the average surface temperature, T, by the relation
Q
CO””
=
it,
A(T,
-
T,)
where ia. is the actual convection coefficient at the surface. Substituting eqs (1) and (4) into eq. (3) yields the ratio of actual convection coefficient to measured convection coefficient which we call the correction factor CF:
The first term on the right-hand side of eq. (5) corrects for temperature gradients on the heat-transfer surface, while the second term corrects for conduction out of the ends of the,wire-wrapped heat-transfer region. The parameters T,, Tf, P, and h, in eq. (5) are determined from experiments. To determine h,, we must also know T, and Qcand,which we have calculated by a finite-difference computer model of the heat-transfer element. This model is discussed in the next section after the geometry of the heat-transfer element used in our experiments is described. EXPERIMENTAL
PROCEDURE
A schematic diagram of the apparatus used in the experiments is given in Fig. 1. The water is pumped from a holding tank by a centrifugal pump. The water can then flow back to the tank through a bypass valve or to the fluidized bed. Two rotameters in parallel allow for a wide range of flow rates to bc metered into the fluidized bed. The water enters the fluidized bed from a plenum chamberThe plenum chamber allows the water flow to be evenly distributed across the cross section of the test section. The plenum contains a packed bed of 3 mm diameter beads to assist in distributing the flow evenly over the entire cross
I
section. The fluidized bed consists of a 5.72 cm diameter plexiglass tube arranged vertically. The plexiglass tube is 45 cm in length. Plexiglass was chosen to allow visual access to the fluidizcdbed. A wire screen, located just below the test section, acts as a distributor to support the particles in the bed. After passing through the bed, the water exits through a tube that returns to the holding tank. Convection coefficients are measured for a 1.9 cm diameter Pyrex cylinder oriented coaxially to the fluidized bed. Resistance wire wrapped around the Pyrex cylinder for heat-transfer measurements was an MWS 120 alloy. The temperature coefficient of resistance was found to be O.O0412”C-r, which compares favorably with the manufacturer’s specification of O.O045”C- ‘. The total resistance of this wire was 46.86 k-2at 30°C. The resistance wire was soldered to low-resistance leads that passed through the center of the Pyrex cylinder. A Variac transformer supplied power to the wire coil. The voltage across the length of resistance wire and the current through the wire were measured by use of digital multimeters. The equipment used in the experiments allowed voltages to be. read to 0.01 V and the current to be read to 0.001 A. For the lowest input power levels, the uncertainty in wire temperature was less than 1°C; at the higher input power levels, this uncertainty was less than 0.2”C. For the lowest power levels, the input power was read to within 3%, while the input power was read to within 1% for the higher power levels. The water used in the tests was distilled and deionized to assure that the resistance wire could not short-circuit when immersed. A test was carried out in which two exposed wires were held approximately 1 mm from each other in distilled and deionized water. A voltage was applied across the gap between the wires, and the measured current flowing between the wires was only 5 to 10 PA for voltages up to 50 V, thus, the wires would not electrically short out when immersed in the liquid-fluidized bed. A cooling coil was immersed in the water holding tank to maintain the water at a constant temperature during the tests.
u
CENTRIFUGAL PUMP
Fig. 1. Schematic diagram of the experimental apparatus.
Correction
of systematic
errors in measuring
Thermocouples were placed at the inlet and outlet of the fluidized bed to monitor the water temperature. The resistance wire used was 0.1 mm in diameter. The cylinder was wrapped with 27 turns of resistance wire, with 0.4 mm spaces between the wires. Three 1 mm wide strips of adhesive tape were attached to the surface of the cylinder under the wire to aid in keeping the wraps of resistance wire from contacting each other. The tape covered less than 5% of the total surface area and, thus, was considered to have negligible effect on the heat-transfer measurements. The length of the cylinder that was covered by wire was 18 mm. The overall heat-transfer area for the 19 mm diameter cylinder was 1.07 x 10e3 m2. The computer model consists of a two-dimensional mesh of nodes 203 x 4 in sire. A convection boundary condition was used for the outer surface of the Pyrex tube. An adiabatic boundary condition was assumed at the interior surface of the Pyrex cylinder: natural convection at the interior of the cylinder is at least two orders of magnitude less than the convection at the outer surface in contact with the fluidized bed. The ends of the grid were also assumed to be insulated since temperature gradients disappear at these locations. Parameters used as inputs to the computer model were the convection coefficient at the outer surface of the element, h,, the power input to the wires, P, and the ambient temperature of the water, T,-, Provisions were also made for including contact resistances between the heated wires and the surface upon which they were wrapped. However, contact resistance proved to be negligible for the cylindrical surface employed in this study. A more conventional heat-transfer probe was also constructed for comparison with the heated-wire probe. This 152 mm long probe consisted of a com-
__
eenvection coefkients
in a liquid-fluid&d
bed
mercial cylindrical heater element enclosed by a 16 mm diameter copper tube. Silver solder was run between the heater element and the inner diameter of the copper tube to assure good thermal contact between the two. Along one side of the tube, small slots were machined just large enough to contain thermocouple beads. The thermocouples were sealed into the grooves by solder which was ground flush with the copper surface. Measurement of average surface temperature, T,, obtained from the four thermocouples, fluid temperature, T,, from a thertnocouple in the fluidized bed, and probe power, P, allowed calculation of heat-transfer coefficients at the surface of this probe according to
PIA
h=(c-
T,)
where A is the surface area of the probe. RESULTS
The finite-difference computer model was used to simulate heat transfer from the wire-wrapped surface for several power settings and convection coefficients. An example of the resulting temperature profile is shown in Fig. 2. From the figure, it is clear that temperature gradients exist along the surface between the wires for our experimental conditions; thus, the assumption of uniform wall temperature is a poor one in liquid-fluidized beds and convection coefficients will be underestimated by the heated-wire technique. For each computer run, the resulting temperature profile was used to calculate the average surface temperature Ts for the heat-transfer surface and also the amount of heat conducted out the ends of the heat-transfer surface, Qmnd. The ratios Q,,JP and (T, - TJ/( T, - Tf) are plotted against input power
--__-----_
24
50
0
100
150
L 200
Node-
Fig. 2. Simulated
CES 43:3-F.
temperature
523
profile on the surface of the heated-wire P = 12Wm-I).
probe (h = 2ooO W m-* K-l,
ROBERT C. BROWN
524
in Fig 3(a) and (b), respectively, for several different values of h,. From these plots it is evident that these ratios are not functions of power and, for any given experimental arrangement, will depend only on the convection coefficient at the surface. Accordingly, a plot of the correction factor given by eq. (5) vs the measured convection coefficient, h,, can be used to correct systematic errors in the experimental data arising from assumptions inherent to eq. (1). Such a plot, in a range of h, appropriate to water-fluidixed beds, is given in Fig. 4. The resulting curve has large linear ranges. Convection coefficients were measured for the wirewrapped probe in a bed of 5 mm glass beads for superficial flow velocities from 0.01 to 0.15 m/s. For each superficial velocity, a series of voltage and current measurements were taken. Using eq. (2), the resulting resistances were converted into tempcratures. Plots of power vs wire temperature were extremely linear, with linear regression analysis giving correlation coefficients generally exceeding 0.99. Accordingly, convection coefficients were constant over large power ranges; this result is in agreement with the finite-difference computer model. Furthermore, since the predicted wire temperatures were always higher than the measured wire temperatures, the use of contact resistances in the model did not improve model
predictions and were deemed unimportant in the present experiments. It is evident that the wire-wrapped surface is rougher than a bare Pyrex surface, and it is possible that the wires will affect convection coefficients. In order to determine the effect, additional tests were conducted on a plexiglass tube of the same dimensions as the previous Pyrex tube. The plexiglass tube was first wrapped with the wires on the surface, just as the Pyrex tube was wrapped. Heat transfer from the surface to a bed of glass beads fluid&d at a superficial velocity of 0.0615 m/s was measured. The surface of the plexiglass tube was then grooved on a lathe to allow the wires to be recessed. The tube was grooved to a depth equal to the wire diameter. The tube surface then closely approximated a smooth surface. The heat transfer from the surface with recessed wires was then measured. The results of the test are shown in Fig. 5. The two curves are separated from one another because of different bulk water temperatures in the two tests; however, the slopes of the two curves are nearly equal. The slope of the curve for the nongrooved surface deviates from the slope for the grooved surface by 8%. This good agreement suggests that the wires present on the surface of the tube do not significantly affect the convection coefficient within the fluid&d bed. Figure 6 compares convection coefficients obtained with our heated-wire probe with values obtained with
E
1.6
f
::;I
,
,
h ; 8000 Wly*tK
1
8
1.4
.I
1.2
s 100
200
600
300 400 Power, W
600
11
l .O,__L
.
Measured
(b) 1.8------l
Fig. 4. Correction
Heat-Transfer
factor vs measured
Caeftident,
convection
hm
coefficient.
h = 8000 W/mz/K
+ ” F
1.4
lz
1.2
’
--b
Wire Nonrecessed Wire Recessed
t
,.J
I 100
,
,
200
306
p =2ooy Wlm2l~ 400
600
j
600
700
0 0
Power. W
20
Fig. 3. Correction factor as a function of power and actual convection coefficient (convection coefficients are in Wm2K-I):
(a) contribution
from
conduction
losses;
contribution from temperature gradients between
(b)
wires.
Fig. 5. Comparison
I 22
I 24 Temperature,
of heat transfer _
I 26
28
Oc
recessed wires to a surtace wlthout
from a surface .
recessed wires.
with
Correction
of systematic
errors in measuring
convection
beating-element probe. The data designated as “uncorrected heat-wire” were obtained directly from eq. (I). The data designated as “corrected heat-wire” were calculated by multiplying h, from eq. (1) by the CF obtained from Fig 4. Note that convection coefficients obtained with our heated-wire probe do not
our
coeiikients
in a liquid-fluidized
bed
525
agree with our heating-element probe unless the CF is applied. This result experimentally validates our computer model prediction that temperature gradients between heated wires will result in an underestimation of convection coefficients between the surface and the fluidized bed when using eq. (1).
5000 0
n 4500
0
.
.
0
0
1
.
n
.
.
.
.
.
I .
.
1sooJ 0.010
0.020 9
0.030
Uncorrected
Fig. 6. Convection
coefficients
0.040 0.050 Superficial 0 Corrected Wire
0.000 0.070 velocity (m/s) l Heating Wiie
in liquid-fiuidized beds: Comparing element probe.
heated-wire
0.080
O.OQO
I
0.100
Element techniques
to a heating-
5000 -
___________.___-----’
. 2000.. 1500.. _/_*-’
_A
/-
_____________________-.----~*-----_____--_____-.__.-.-___---_,__---__-_._-*I __--
I
10001
0.000 l
0.040
0.020 Corrected
W&a
0.080 0.100 0.060 Superficial velocity (m/s) Kate et al. ---- Juma
Fig. 7. Convection eoeftkients in liquid-fluid&d corrections
beds comparing of other researchers.
0.120
0.140
k Richardson
the cmrccted
heated-wire data to
ROBERT C. BROWN
526
We compare our (corrected) heated-wire data to correlations by Juma and Richardson [4] and Kato et al. [7] in Fig. 7. Juma and Richardson [4] measured convection coefficients in a liquid-fluidized bed of paraffin oil and glass beads using the (uncorrected) heated-wire technique. Their correlation yields convection coefficients substantially smaller than those obtained in our experiments. There are differences between the present experiments and those of Juma and Richardson [4]. Their probe is of larger diameter and is horizontally rather than vertically oriented in the flow. Furthermore, the effect of Prandtl number on Nusselt number was not thoroughly investigated by Juma and Richardson [4] and caution is required in applying their correlation to a water-fluidized bed. These differences may account for part of the discrepancy between their correlation and our data. However, we suspect that the larger source of discrepancy is temperature gradients between wires and its influence on measured convection coefficients. Kato et al. [A obtained their correlation for a heated wall in a water-fluid&d bed using an experimental technique similar to that employed for our heating-element probe. Their correlation agrees well with more recent studies [8, 91. Figure 7 shows that our (corrected) heated-wire data follows closely the correlation of Kato et al. [7]. Without the use of the correction factor for the heated-wire technique, our data would have underestimated convection coefficients by 2CL35%.
underpredicted convection coefficients for liquidfluidized beds. We have devised a methodology that accounts for the errors arising from temperature gradients between wire wrappings and heat loss from the ends of the wrapped region, thus preserving the utility of the heated-wire technique. Nevertheless, the large correction factors required for systems with convey tion coefficients in excess of 1000 W m-‘K-t may make alternative experimental techniques more attractive than the heated-wire approach.
Acknowledgments-The support of the Iowa State University Power AfEliate Program and the Iowa State University Achievement Foundation is gratefully acknowledged for making this research possible. I appreciate the assistance of Larry Burken, Rod Oathout, and Jeff Delfs in performing some of the experiments and analysis described in this paper.
REFERENCES
Cl1 Latif, B. A. J. and Richardson, J. F., 1972, Chem. Eagng
c33 c41 CSI C61 c71
CONCLUSIONS
Our work suggests that the heated-wire technique is flawed by systematic errors that result in greatly
Sci. 27, 1933. J. F. and Romani, M. N., 1974, Lest. Heat Mass Translkr 1. 55. Romani, M: N.,~Richardson, J. F. and Shakiri, K. J., 1976, Chem. Ennnn Sci. 31, 619. Juma,A. K. A. and Richardson, J. F., 1982, Chem Engng Sci. 37, 1681. Khan, A. R., Juma, A.K.A. and Richardson, J. F., 1983, Chem Engng Sci. 38,2053. Juma, A. K. A. and Richardson, J. F., 1985, Chem. Engng Sci. 40, 687691.. Kato, Y., U&da, K., Kago, T. and Morooka, S., 1981, Powder Technol. 28, 173. Chiu, T. M. and Ziegler, E. N., 1983, A.1.Ch.E. J. 29,677. Muroyama, K., Fukuma, M. and Ysshunishi, A., 1986, Can. J. them. Engng 64,399.
c21 Richardson,
::I