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Measurements of particle-liquid heat transfer in systems of varied solids fraction
humul
of Food Enginerring 31 (1907) Y-33
Copyright 0 1997 Elsevier Science Limited Printed in Great Britain. All rights reserved 0260.8774197 $17.00 t O.OlI
Pll:SO260-8774(96)00055-6 ELSEVIER
Measurements
of Particle-Liquid Heat lkansfer in Systems of Varied Solids Fraction
S. Mankad,” K. M. Nixorf’* & P. J. Fryer” “Dept of Chemical Engineering, Pembroke St, Cambridge, ‘School of Chemical Engineering, University of Birmingham, Edgbaston, (Received
UK Birmingham,
IJK
10 March 1996; accepted 24 June 1996)
ABSTRACT The heat-transfer coefjicient between particle and liquid in a flowing food mixture is an important parameter which controls the heating and cooling rate of the particles. Experiments have measured the heat-transfer coeficient, h,, between a stationary particle and a fluid stream over a wide range of slip Reynolds numbers (I < Re ~600) and local solids @action. The results have been correlated in terms of Reynolds numbel; Prandtl number and local voidage, and compared with the previously published correlations of Ram & Marshall (1952) and Agarwal (1988). This comparison showed that the Ram and Marshall correlation was quite acceptable for predicting $, for singleparticle data, though for higher solids fractions only the predictions at lower Reynolds numbers, less than ca. 100, were accurate. Howevel; at these reduced voidages Agarwal’s correlation, based on accurate single-particle data, proved to be more successfil in predicting hp and as such is more applicable to food flows than the correlation of Ranz and Marshall. 0 1997 Elsevier Science Limited. All rights reserved.
NOTATION
zl bP C
2 4
Constant Particle Constant Constant Specific Constant Particle
in correlation surface area (m’) in correlation in correlation heat capacity (J/kg K) in correlation diameter (m)
*To whom correspondence
should be addressed. 9
10
tr 4 1
a Tt
Uf Uslip
Bi Nu NUph Nus, Pr Re Re part Re local Reslip 6X E P P
AT
S. Mankad et al. Particle convective heat-transfer coefficient Fluid thermal conductivity (W/m K) Particle thermal conductivity (W/m K) Distance between particles (m) Heat exchanged (W) Volumetric fluid flow rate (m3/s) Tube radius (m) Temperature (K) Superficial fluid velocity (m/s) Slip velocity (m/s) Biot number Nusselt number Particle Nusselt number in packed bed Particle Nusselt number for single particle Prandtl number Reynolds number Particle Reynolds number ( = pd+u$p) Local Reynolds number ( = pdpuf/pE) Slip Reynolds number ( = pdpUsliplC()
(W/m2 K)
Fractional error in quantity x Bed voidage Dynamic viscosity (Pa S) Fluid density (kg/m’) Temperature difference (K)
INTRODUCTION:
SENSITIVITY
OF HEATING COEFFICIENT
RATE TO HEAT-TRANSFER
Much industrial continuous food-processing equipment treats a two-phase mixture of solids and fluid. The carrier fluid is typically non-Newtonian (such as starch solutions) and, because of the variety and complexity of foods and recipes produced, the solids exhibit a size, shape and physical property distribution. The solids are carried through the system by the fluid and are free to translate, rotate and interact with one another. In some cases, such as the continuous ohmic heater (Zhang & Fryer, 1994; Fryer, 1994), or more commonly a scraped surface heat exchanger (Lee & Singh, 1990), flow occurs simultaneously with heat transfer. To understand and model heat transfer the temperatures and heat-transfer coefficients must be known. All aspects of free-particle motion contribute to the convective heat-transfer coefficient around the particle (Mwangi et al., 1992). This heat-transfer coefficient is crucial in determining the thermal response of a particle in a flcwing food mixture. Previous research investigating convective heat-transfer coefficients for particles in food flows have mainly used stationary single particles in isolation (Chandarana & Gavin, 1989; Zuritz et al., 1987). Measurements using stationary particles will not determine effects due to particle rotation, which will increase the heat-transfer coefficient (Sastry, 1992). Studies of particles in flow have been conducted (Sastry et al., 1989, 1990; Zitoun & Sastry, 1994a, b) but the flow round the particle is difficult
Purticle-liquid heat transfer
to measure and so results are difficult to correlate. particles, the Ranz and Marshall correlation:
11
For flow around
stationary
Nu = 2+0.(jPr”3Re!‘.’blip
(1) may apply, where the velocity in Reali,, is the relative (slip) velocity between particle and fluid. This equation was, however, derived for much higher Reynolds numbers than for food flows, and for different systems; its applicability to food processing is uncertain. In any case, values from single-particle experiments may not apply to the high solids fractions found in practice: the effect of multiple particles is also unclear. Mankad (1995) describes the range of data correlations available in the literature (e.g. Ranz & Marshall, 1952; Heppel, 1985; Zuritz et al., 1987; Sastry et al., 1989; Chang & Toledo, 1989, 1990; Chandarana et al., 1989; Mwangi, 1992) and notes that they predict heat-transfer coefficients which differ by up to two orders of magnitude for the same conditions. The most recently published work in this area is that of Kelly et al. (1995) who considered the stationary case in terms of the effects of particle geometry in Newtonian and non-Newtonian fluids. To simulate particle interactions the wake effects of a two-particle system were investigated, though multi-particle flow was neglected. Some information on particle heat transfer is, however, available from the general process engineering literature, such as Agarwal (1988) who modelled heat and mass transfer in a multi-particle system based on a boundary-layer formulation for individual particles in the sphere assemblage. The predictions of this model were then favourably compared with a wide range of previously published experimental data, but the work refers only to process engineering data, and does not appear to be widely known in the field of food engineering. It is important to find the heat-transfer coefficient values in food flows if the Biot number of food particles is such (less than 10; Kreith & Bohn, 1986) that the rate of heat transfer is sensitive to the heat-transfer coefficient. An estimate of this can be made, assuming that food solids have the physical properties of water. In the flows of interest here, particle sizes are in the range 5-25 mm, and slip velocities between solid and liquid are 1O-s-1O-2 m/s (Lareo, 1996). Since the Biot number is related to the Nusselt number by the ratio of the thermal conductivities, it is possible to relate it directly to the slip Reynolds number using the Ranz and Marshall correlation as a first approximation:
Bi
_
hPdP_ 44 2k,
k,
kL = -Nu= kL
2k, = $
2k,
-$
{2+0.6Pr”“Rei$,}
(2)
P
{ 1+0.3Pr”“Re$,} P
To estimate the range of Biot highest practicable slip velocity ty = 999 kg/m”, viscosity = 1.1 viscosity = 138 cP), i.e. extreme At 15”C, the Prandtl numbers (typical for food fluids), then Re,li, = O-85, Bi = 4.3.
numbers found in practice, a l-cm particle and the of 0.01 m/s will be considered, using water (densicP) and glycerol solutions (density = 1178 kg/m-‘, cases of the fluid viscosities found in food systems. are 7.3 for water and 1730 for glycerol. If kLzk, for water: Reslip = 91, Bi = 6.5 and for glycerol:
S. Mankad et al.
12
At lower slip velocities, even lower Biot numbers will be found. Even for larger particles (2.5 cm, and a slip velocity of 0.001 m/s) Biot numbers of 3.8 (water) and 2.7 (glycerol) are obtained. Biot numbers are thus sufficiently low that the convective heat-transfer coefficient will be significant in thermal processing of these flows. Mankad et al. (1995) developed a model for food flows, using the Ranz and Marshall correlations as a first approximation, to demonstrate the importance of particle-fluid heat-transfer coefficient and thus particle-fluid slip velocity. RanzMarshall was chosen in the absence of better data. This paper reports a series of measurements of heat-transfer coefficient in various geometries of relevance to food flows, and presents correlations for the data sets obtained.
EQUIPMENT
AND METHOD
Range of experiments Any experiment should be representative of the industrial process so that results are immediately relevant to a practical system. Ideally, the temperature of a moving food particle within a flowing swarm of particles would be measured; however, the problems associated with finding the temperature of such a particle, and measuring the flow field around it, are very large. Even if particle temperature could be measured, the flow around the particle will be unknown, making it very difficult to determine the heat-transfer coefficient due to slip velocity. The equipment was thus designed around a stationary particle system. The following experiments were carried out, as shown schematically in Fig. 1: (i) Single-particle experiments [Fig. l(a)]. This work was performed to compare results with the work of Ranz & Marshall (1952), enabling the accuracy of the experiment to be confirmed. The applicability of the Ranz and Marshall expression to low Reynolds numbers can also be checked. (ii) Two-particle experiments [Fig. l(b)]. Here, the influence of particle wake effects on the convective heat-transfer coefficient was studied. The heat-transfer coefficient to a single particle, at varying distances behind another similar sized particle in the tube, was measured. (iii) Packed bed experiments [Fig. l(c)]. Measurement of the heat-transfer coefficient from a sphere in a packed bed gave an indication of the heat-transfer coefficient possible for the highest solids fraction in food flows. These experiments, together with the single-particle ones, set bounds on heat-transfer coefficients for the maximum (packed beds) and minimum (single particles) concentrations possible. (iv) VZzriable voidage bed experiments [Figure 1 (d)]. Measurement of the heattransfer coefficient from a sphere in a partially packed arrangement of spheres, shows the variation of heat-transfer coefficient with voidage, and indicates the heat-transfer coefficients possible within a continuous food-processing plant. Apparatus The experimental apparatus consisted of a flow loop with a section in which particle behaviour could be examined [Fig. l(e)]. The tube section contained a number of
Particle-liquid heat transfer
13
(a) Single particle
0
flow
(b) Double particle
I
1
I
1
c3 flow
(c) Packed bed
Fig. 1. Schematic
diagrams of experimental
equipment
(continued overleaf).
14
S. Mankad et al.
(d) Partially packed bed I
I ,oooooo,
(e) Plow loop
thermocouple
I
V
L
flowmeter
I
L
J water tank
f
control valve P-P
Fig. 1. Continued.
Particle-liquid heat transfer
15
flow-calming sections upstream of the particle(s) under observation. A total tube pipe length of 25 m, internal diameter O-1 m was used, giving particle-to-tube internal diameter ratios similar to those in aseptic processing (i.e. between a ratio of 1:4 and 1:s; Lareo, 1996). A 0.015 m hollow copper particle was used to measure the fluid-particle heattransfer coefficient. The sphere consisted of a thin copper shell (0.5 mm) and contained a germanium resistor, which generated heat when a current flowed through it. Heat flux paste (RS components) was used to ensure good thermal contact between the heater and the particle surface. The particle was supported by a 4-mm diameter Perspex support stem, 25 mm in length, located at the rear stagnation point to minimise flow disturbance (Shallcross & Wood, 1987). The particle contained two K-type thermocouples, one at the front stagnation point and one at 90” to this, to monitor surface temperature and detect variations in heattransfer coefficient with radial position (Gillespie et al., 1968). Thermocouples were calibrated at 0 and 100°C. A range of systems was used to simulate different voidages. Packed beds were supported at their ends by thin wire mesh, whilst beds of known intermediate voidage were created by suspending particles on flexible wires between the end supports. Four fluids were used; water, polyacrylamide (PAA) solution and glycerol solution of 67 and 83%, covering a range of viscosities up to ca. 100 times that of water. These test fluids are all Newtonian to allow better characterisation and definition of Reynolds number, for descriptive purposes.
Experimental procedure The following experimental
procedure
was used:
(9 The power supplied to the particle heater was set to a constant value, chosen so the temperature difference between the fluid and the particle surface was ca. 10°C. (ii) The fluid flow rate was then set. (iii) Once steady-state conditions were achieved, particle and fluid thermocouple readings were recorded. (iv) Steps (ii) and (iii) were repeated for different fluid flow rates. coefficients were measured. Fluid and For each flow rate seven heat-transfer particle thermocouple readings, the power supplied to the particle heater (voltage and current), and the fluid flow rate were recorded; the heat-transfer coefficient and particle Reynolds number were then calculated from this data. Ten flow rates were used in all experiments between 0.6 and 5 m’/h; these correspond to slip Reynolds numbers between 05 and 5000. All experiments used a fluid temperature range of 13-18°C; physical properties for the fluids used, over this temperature range, are presented in Table 1. These data were used for all calculations. Changes in the specific heat capacity, thermal conductivity and density of the fluids were negligible over the temperature range 13-18°C. The solid surface temperature ranged between 23 and 35°C.
S. Mankad et al.
16
TABLE 1
Fluid Physical Property Range used in Experiments (Perry & Green, 1984; Newman, 1968) Property
Temperature range (“C) Concentration (%) Viscosity (Pa s) Thermal conductivity (W/m K) Specific heat capacity
(J/kgK)
Density (kg/m’)
Water
PAA soln
Glycerol soln I
Glycerol soln II
13-18 n/a 0.0011 0.628
13-18
13-18
13-18
0.5%
83%
67%
0.0027 0.628
0.111-0.161 0.320
0.027-0.020 0.365
4184
4184
-
-
999.1
999.1
1220
1176
Accuracy of results A number of problems were encountered with the construction and the running of the equipment. The particle was frequently cleaned to remove oxide films and to reduce places for air bubble adhesion: both of these lead to low heat-transfer readings. Inherent errors in the thermocouples and electronic-meter readings can be developed into a quantitative error analysis. In the studied temperature range the thermocouple accuracy was +0.75% of the temperature reading (in “C) indicated (RS thermocouple data sheet). Thermocouples were calibrated using pure melting ice and the steam above pure boiling water. Successive calibrations were performed for each new particle. The error due to the difference in calibration readings was found to be a maximum of f0*36% of the temperature reading. Voltages were measured using an Iso-Tech IDM 96 multimeter, whose specification states the displayed value contains an error of *O-5% of the voltage measured; the experimental readings ranged between 1.2 and 3.7 V. Power was supplied via a Weir 4000 supply. The voltage was read from a digital display, and the current was recorded using a Fluke 8050A digital multimeter. The accuracy of the power-supply voltage reading was &-O-5% of the reading. For the Fluke multimeter the accuracy was f0*3% (Fluke 805OA data sheet) of the current indicated. The convective heat-transfer coefficient was calculated from:
where q is the heat transferred from the particle to the fluid, A, is the surface area of the particle and AT is the temperature difference between the fluid and the particle surface. The fractional error in the heat-transfer coefficient was thus the sum of those in the individual measurements. The total fractional error in AT is: 6AT= and in q is:
f
0*75+0*36+0.5 100
= f0.0161
(4)
Particle-liquid heat transfer
6q=
+
l
0.3 +0*5 100
I
= f0.008
17
(5)
The fractional error in h,, 6hb, was thus kO.0241 (i.e. 0~0161+0*008), which corresponds to a percentage error of +2*41%; an error in h, of + 2.5% was assumed throughout. Hydrodynamic heat transfer correction It was necessary to support the particle in the flowing stream. A correction to the value of the heat-transfer coefficient was required to compensate for the loss in heat transfer due to the supporting stem (Gillespie et al., 1968). Using this method, a multiplication factor of 1.087 (Mankad, 1995) was used for flows above a particle Reynolds number of 2200, or a multiplication factor of l-057 for flows below 2200. This is standard practice when correcting for hydrodynamic effects (Shallcross & Wood, 1987); the errors occurring through neglect of these effects are well documented in the process-engineering literature (Ranz, 1952; Rowe et al., 1965). Correlation of results For each fluid used at least seven sets of data points were recorded for each particle/local Reynolds number combination. The arithmetic mean of these data points is shown. Included on the graphs are error bars which represent the standard deviation in the set of results. Some plots include lines showing k 10% deviation from the values predicted by a correlation. The original Ranz and Marshall correlation was claimed accurate to within + 10%. These lines, and standard deviation data, help to determine the association between measurements and correlations. Results were manipulated to correlate Nusselt number as a function of the independent variable(s) used in the experiments. Two Reynolds numbers were used. The particle Reynolds number, defined as:
where uf is defined as: UfZ
-
Q 7CR:
where Q is the volumetric fluid flow rate and R, is the tube radius, can be used to correlate data for low solids fractions, where it is identical to the slip Reynolds number. To model high solids fractions another Reynolds number is needed. In a system of voidage E through which Q m”/s of fluid flows, the mean fluid velocity around the stationary particle will be Q/nR2 f c:. The local Reynolds number is defined as:
S. Mankad et al.
18
as the particle(s) used in the experiment are stationary and the fluid velocity over the particle (u$E) is the axial slip velocity. For single- and two-particle experiments, the local and particle Reynolds number are equal. Experimental results for packed beds and partially voided beds have been plotted against the local Reynolds number, which is equal to the slip Reynolds number. A form of equation relating the Nusselt number to the independent variable(s) for the system under consideration was selected by inspection of correlations used by previous workers for similar systems. In each case correlations contained a number of unknown constants; a spreadsheet was used to manipulate the unknown constants to minimise the total sum of the square of the errors between the experimental values and the values predicted by the correlation (Himmelblau, 1970). RESULTS
AND ANALYSIS
Table 2 summarises the data for the different experiments. Marshall correlation [eqn (l)] is used widely to determine coefficient for food flows, it has not been tested for food experiments is to determine the level of applicability of correlation to food flow configurations, and to determine use of the equation is acceptable.
Although the Ranz and the particle heat-transfer systems. An aim of our the Ranz and Marshall the conditions for which
Single particle in isolation Results and discussion
As expected, the Nusselt number increases with increasing particle Reynolds number, due to the increasing fluid velocity around the particle. The Ranz and Marshall equation predicts most of the data well although, with the exception of the lowest Reynolds numbers in each group, the experimental results obtained seem lower than predicted. Figure 2(a) shows the percentage deviation of the experimental results from Nusselt numbers calculated from the Ranz and Marshall correlation, as a function TABLE 2
Summary of Fluid/Particle Property Single-particle experiments: Reynolds number Nusselt number Twoparticle experiments: Reynolds number Nusselt number Packed bed experiments: Reynolds number Nusselt number Voidage aperiments: Reynolds number Nusselt number
Experimental
Data
PAA
6 7% Glycerol
83% Glycerol
292-2433 20-60
114-893 20-48
18-140 16-36
5-97 9-34
292-2433 20-60
114-893 20-48
5-97 9-34
5-97 9-34
712-5937 44-163
278-2178 36-126
44-341 24-77
1.2-22 5-39
324-3477 44-163
127-1276 36-126
20-200 19-66
1-13 5-39
Water
Particle-liquid
19
heat transfer
loo m
I
o-1O.1
1
1
I
lllllli
1
10
1
,.,,.,I
s
.,,,.aoI
loo
I
1ooo
I
I.1
10000
particle Reynolds number Fig. 2. (a) Difference between single-particle heat-transfer coefficients and values predicted by the Ranz and Marshall correlation. (b) Comparison of single-particle heat-transfer coefficients with particle Reynolds number. (c) Difference between single-particle heat-transfer coefficients and values predicted by derived correlation n, 83% glycerol; x , 67% glycerol; o, PAA; o, water (continued overleaf).
20
S. Mankad et al.
of particle Reynolds number. The data represents the percentage difference calculated from the arithmetic mean of the results, at each particle Reynolds number; error bars show the standard deviation of each data point expressed as a percentage. Difference is calculated as: Percentage Experimental
difference
=
Nusselt number - Ranz and Marshall Nusselt number
x 100%
Ranz and Marshall Nusselt number
(9) The maximum percentage difference shown in Fig. 2(a) was f 15%. Nusselt numbers for water, PAA and 67% glycerol solution crossed over the values predicted by Ranz and Marshall; each high particle Reynolds numbers gave lower Nusselt numbers than the Ranz and Marshall predictions, but this trend was reversed at low particle Reynolds numbers. The experimental results for 83% glycerol solution were consistently below those predicted by Ranz and Marshall. Included in Fig. 2(a) and all subsequent graphs of this type are lines representing values 10% above and below the correlation; this was the maximum error margin estimated in the original paper (Ranz & Marshall, 1952). Figure 2(a) shows that the majority of the experimental results lie below values predicted by the Ranz and Marshall correlation (mean difference ca. -6%), and with the exception of one data point for PAA and three for 67% glycerol, all the results lie within +lO% of
20 15 10 5 0 -5 -101 t
I rrB T
1
1 1 ’
-
83% Glycerol
I 1 I 1 ..,
111
.
water
-T I
-15F ' ''ntnnt' ' ''nknBn' ' ''11111' ' ''a1111' ' 'J 0.1 1 10 100 1000 10000 particle Reynolds number Fig. 2. Continued.
21
Particle-liquid heat transfer
Correlations
TABLE 3 for Single-Particle
Fluid
Experiments
Equation
Water
PAA soln 67% Glycerol soln 83% Glycerol soln
Nu Nu Nu Nu
= = = =
2+1.41 2+3.20 2+4.98 2+567
Correlation probability
Re”‘47 part Re”:” part Re?part Rei:r’;
0.98 0.99 0.99 0.99
the Ranz and Marshall predictions. The data suggest that the apparatus is accurate and that the Ranz and Marshall correlation may be used to estimate acceptable heat-transfer coefficients for single particles in flowing food systems, down to Reynolds numbers of unity. Correlation
Although the Ranz and Marshall equation lies close to the data, correlations made to see if a better fit could be obtained using equations of the form: NU = 2+aRei;,,,
were (10)
where a and b are constants. Table 3 shows the correlations obtained. Correlated particle Reynolds number exponents compare well with values derived by Ranz & Marshall (1952). For water and 83% glycerol solution the deviation was minimal (exponents of 0.47 and 0.49, respectively); for PAA and 67% glycerol solutions the association with the Ranz and Marshall correlation was poorer (0.38 and 0.38). The equations for each fluid were combined to form one equation for all the single particle experiments: Nu = 2+0*97Re~~,Pr”‘*’
(11)
Figure 2(b) compares the percentage difference between the experimental results and those predicted by eqn (11). The majority of the experimental data points lie within & 10% of the derived correlation. This correlation compares well with the Ranz and Marshall equation, as shown in Fig. 2(c). The data suggest that the Ranz and Marshall correlation gives an excellent prediction for heat-transfer coefficients for single particles in horizontal tubes at particle Reynolds numbers above 200, a significant distance from the particle Reynolds value range over which it was first derived. Equation (11) is a better representation of the data at low Reynolds number. lb-o-particle
systems
Results and discussion
Fluid-particle heat transfer within a packed bed will differ from that to a sphere due to distortions of the flow and temperature fields in the bed due presence of particles. The flow will be affected by the presence of particles; stream of a particle, the flow will be disturbed and unsteady, and may become turbulent. As a first step, ‘double particle’ experiments were carried out
single to the downmore and a
22
S. Mankad et al.
15-mm solid polyethylene sphere was positioned in front of the 15-mm heated copper particle to quantify the effect of flow disruption. The configuration is shown in outline in Fig. l(b); the inter-particle distance was varied between 1 and 5 cm, measured between closest contact points on the two particles. Graphs of Nusselt number against particle Reynolds number for varying inter-particle distances show that the Nusselt number increases with increasing particle distance (on average ca. 5%); however, the degree of increase is related to the particle Reynolds number (Mankad, 1995). The majority of Nusselt numbers recorded lie close to values predicted by the Ranz and Marshall correlation. However, the following may be observed: (i) For Reynolds numbers above 100, the heat-transfer coefficients, for all interparticle distances, were up to 25% higher than for a single particle in isolation. (ii) Below a particle Reynolds number of 100, the results for different particle distances were within the bounds of quoted experimental error for the Ranz and Marshall expression, i.e. effectively identical to Nusselt numbers for a single particle. (iii) Generally, for particle Reynolds numbers above 100, the convective heattransfer coefficient increased with increasing inter-particle distance. Experiments show that, at certain flow rates, particle interactions can enhance particle heat-transfer coefficients by up to 25% from those measured for single particles. The major enhancements occur in the transitional flow region between laminar and turbulent. Even a single particle can enhance the heat-transfer coefficient around a second particle, demonstrating why particle concentration has such a strong effect on the heat-transfer coefficient. A general correlation for the entire data set was found as: NU = 2+ 1.215(l/d,)“05RepOg~ro’*
(12) Figure 3(a) shows the fit to this data and Fig. 3(b) compares the percentage difference between the experimental results and those predicted by eqn (12). Almost all the data points for water and PAA experiments lie within f 10% of the derived correlation; however all results for glycerol lie well below their predicted values (ca. -35% difference), suggesting that low Re behaviour is different from that at high Re.
Packed beds Results and discussion
The convective heat-transfer coefficient around a particle in a packed bed was studied. Experiments measured the heat-transfer coefficient around a sphere within a packed bed of spheres. The 15-mm internally heated copper sphere was situated at the centre of a randomly packed bed of 15-mm diameter hollow polypropylene spheres [Fig. l(c)]. The bed dimensions were 0.1 m in diameter and 0.25 m in length. The bed voidage, E, was calculated as 0.41, and was constant for all experiments. Table 2 includes experimental data. Results are plotted against local Reynolds number, allowing comparisons to be made for similar local fluid velocities, i.e. identical flow velocities around particles.
Particle-liquid heat transfer
23
67%
1L 1
’
nllllll’
10
I
I
u
100
1000
10000
particle Reynolds number
20 10 0
-10 : -20 L -30 : -40
7
-50 1
10
1000
particle Reynolds number Fig. 3. Two-particle experimental data.
experiments. (a) Comparison of derived correlation with entire range of (b) Difference between experimental results and values predicted by derived correlation.
24
S. Mankad et al.
Figure 4(a) plots the experimental data against those predicted by Ranz and Marshall. The deviation of experimental Nusselt numbers from those estimated by the Ranz and Marshall correlation is substantial at high Re. There is a theoretical basis for this; Schlunder (1978) stated that heat-transfer coefficients for packed beds should be three times higher than those for a single particle. Ranz (1952) developed a model for packed beds which compared well with experimental values, and showed that packed bed heat-transfer rates were a factor of 2.6 higher than those for single particles. The results shown here suggest that these effects are also found in food flows. Correlation Table 4 shows correlations. For water and PAA the exponent number is significantly greater than for single-particle experiments 3 and 4), highlighting the enhancement in heat transfer in packed for the case of the two glycerol solutions. Data for all fluids may be
in the Reynolds (compare Tables beds. This is less combined as:
Nu = 0+0.5Re:~~~,Pr0-28 (13) As for the two-particle data, the equations fit high Re data well, but predict results at low Re poorly. Figure 4(b) compares the percentage difference between the experimental results and the corresponding values predicted by eqn (13) and shows a high deviation between experimental results for 83% glycerol and the predicted values. When the data are plotted with the Ranz and Marshall correlation [Fig. 4(c)], the graph shows a cross-over point. Above Re ca. 700, Ranz and Marshall underpredicts significantly, whilst below that value the correlation slightly overpredicts the data. Where the local Reynolds number is less than ca. 100, the Ranz and Marshall correlation may be satisfactorily used to predict heat-transfer coefficients around particles in packed beds to ca. f 10%. Variable voidage beds Results and discussion Two-phase solid-liquid food flow mixtures typically have solid fractions between 5 and 40% (Lareo, 1996), an intermediate stage between a close packed bed and a single particle. The processes occurring at these intermediate voidages may be unrelated to those occurring at extreme cases of packing due to channelling in the bed. Experiments were conducted using the 15-mm internally heated copper particle situated in the centre of a partially packed arrangement of 15-mm polypropylene spheres [Fig. l(d)]. The distended bed was 0.25-m long and housed in a 0.1-m internal diameter tube. The distended bed was made using helical coils of wire (l/16 in diameter), along which 15-mm polypropylene spheres had been threaded. A number of these coils were then placed within the tube section to give an approximately regular packing arrangement around the copper particle. The bed voidages used were 90, 80 and 70%, i.e. solids fractions of lo-30%. Plots of Nusselt number versus local Reynolds number (Mankad, 1995) displayed the following common features for all fluids and voidages: (i) At any particular voidage, Nusselt number increases with local Reynolds number. (ii) For a particular local Reynolds number, Nusselt number increases with decreasing voidage. This effect increases with local Reynolds number.
Particle-liquid heat transfer
25
Water CR do PAA *s”*” 67% Glycerol
_= n!5
i
local Reynolds number 60
L
Fig. 4. Packed bed. (a) Difference between experimental results and values predicted by the Ram and Marshall correlation. (b) Difference between experimental results and values predicted by derived correlation. (c) Correlation of experimental data for heat transfer with Ranz and Marshall and empirically derived equations (continued overle@).
S. Mankad
26
10
Ram and Marshall NH4 I
et al.
.
67% Glycerol
local Reynolds number Fig. 4. Continued.
voidages are greater than those (iii) The Nusselt numbers for intermediate recorded for single-particle experiments. The degree of increase was again dependent on the local Reynolds number, and was more notable for water than for glycerol. Experimental results are compared with predictions from the Ranz and Marshall correlation in Fig. 5(a). The percentage difference between the two values increases as voidage decreases. Nusselt numbers recorded using PAA and water show the highest deviation from the Ranz and Marshall prediction, starting at ca. -20% difference for 90% voidage and rising to ca. +20% at 70% voidage. The 83% glycerol results were consistently lower than the predictions of Ranz and Marshall, and varied less significantly with voidage compared with the other two fluids, ca.
Correlations
TABLE 4 for Packed Bed Experiments Equation
Fluid
Water PAA soln 67% Glycerol soln 83% Glycerol soln
Nu Nu Nu Nu
= = = =
0+042 ReEzd, 0+1.22 Rezzz, O-89+3.73 Reads, 2*57+3.18 Revs3 local
Correlation probability
0.99 0.99 0.99 0.99
27
Particle-liquid heat transfer I
I ’ ’ ““‘I
0 r ’ ““‘I
67% Glycerol
’ ’ ’ ““‘I
’ ’ “‘I”
,T
Water
83% Glycerol
4
10
100
1000
10000
local Reynolds number
67% Glycerol
Water
local Reynolds number Fig. 5. Expanded beds. (a) Difference between experimental heat-transfer coefficients and values predicted by the Ranz and Marshall correlation. (b) Variation between experimental heat-transfer coefficients and predicted values from derived correlation. (c) Variation between experimental heat-transfer coefficients and values predicted from Agarwal’s correlation (Agarwal. 1988). (d) Comparison of experimental Nusselt numbers with existing and derived correlations (continued overleaf).
28
S. Ma&ad
et al.
-10 -20 0.1
1
10
100
1000
10000
1000
10000
particle Reynolds number
100
Ranz and Marshall (1952) (assuming voidage = 1)
Ol..........,..... . 0.1 1 10
100
particle Reynolds number Fig. 5. Continued.
29
Particle-liquid heat transfer
-30% difference for 90% voidage, and rising to ca. - 15% for 70% voidage. This increase in deviation, in all cases, can be attributed to the higher degree of eddy formation and/or channelling arising in low voidage beds. Overall, results for the highest voidages were closest to Ranz and Marshall. The results show that deviation from the Ranz and Marshall correlation is related to the local Reynolds number. At high flow rates, channelling and eddies are likely to occur, causing non-uniformity of flow and enhancing heat transfer; these effects are further increased at lower voidage (higher solids fraction). For low flow rates, eddy formation and channelling is small, resulting in a smoother flow profile; conditions under which Ranz and Marshall performed their original experiments. Recent work on heat transfer to particles in non-packed bed flows, with application to the food industry, was performed by Mwangi (1992). Mwangi measured the heat transfer from free-flowing spheres in a tube for high void fractions (95%). No correlation was derived from the experimental data, but Mwangi showed from his results that the Nusselt numbers estimated for these systems, on a particle Reynolds number basis, were between 80 and 200% higher than those evaluated for single spheres, which corroborates the range of values estimated through our experiments. The most substantial work predicting heat-transfer rates in expanded beds is that of Agarwal (1988). This work developed a relationship between heat transfer to a single particle and one in a bed, for the same superficial fluid velocity: NC+ = I:- “YvSP
(14) where NC+, is the Nusselt number around the particle in a packed or expanded bed, Nu,, is that around a single isolated particle, and c is the bed voidage. This correlation has been used, together with the Ranz-Marshall equation, to correlate experimental data. Correlation
A mathematical correlation expressing Nusselt number as a function of voidage and local Reynolds number, of the form: Nu =a + b Re&tllocal 6
(15) where a, b and c are constants, and E is the fractional voidage, was fitted to the data. Table 5 shows correlations for each fluid. The correlations vary markedly for each fluid used and no resemblance to correlations for packed beds or single particles is seen. The Nusselt number dependence on voidage decreases appreciably from water to glycerol. The equations can be combined as: NU = 2.7+1.1r:~“‘Rel~~~,pr’~2’
Empirical
Correlations
TABLE 5 for Variable
Water PAA soln 67% Glycerol soln 83% Glycerol soln
Voidage Experiments
Equation
Fluid Nu = Nu = NM = Nu =
(16)
14-+0.42 Ret:?, 1: ‘-4 79+2.2 ReiLz, I’ ~ “” 0+5..57 Reii;z!, ,‘~0-7q 3.5+5.6 Re@:f lO“lII‘’ -0.55
Correlation probability 0.99 0.57 0.94 0.99
30
S. Mankad et al.
The correlation probability for this equation is 0.57 and, due to the low range of Prandtl numbers and voidages used in the experiments, should be used cautiously. Figure 5(b) compares the percentage difference between the experimental results and the values predicted by eqn (16); the majority of values lie within f 10% of the values predicted by eqn (16). Comparison of the experimental data, for all combinations of fluid and bed voidage, with the values predicted by eqn (14) Agarwal’s correlation, is shown in Fig. 5(c). Single-particle experimental data is used to generate Nu,, in eqn (14). The difference between the normalised expression, Nu~~/E-‘.*~ (where Nupb is obtained from experimental data), and the corresponding experimental single-particle results for the same superficial fluid velocity are shown as percentages with error bars. This allows the data from all three sets of expanded bed experiments to be shown on one graph. The general agreement between the data and the predictions of eqn (14) is good. However, the degree of agreement varies with particle Reynolds number, being better at low Re,,, where the deviation for 83% glycerol was +8%, than at high Re,, where the average deviation for water is +20%. This is the trend which would be expected as the voidage is less influential at lower flow rates. Equation (16) is plotted, together with all the experimental results, the Ranz and Marshall correlation, and that of Agarwal, on Fig. 5(d). From this graph eqn (16) can be seen to be a constant factor of ca. 1.25 higher than the Ranz and Marshall correlation. In addition, the graph demonstrates the high degree of agreement between the experimental correlation and that of Agarwal, particularly at low particle Reynolds numbers; both are better for use in predicting heat-transfer coefficients in comparison with the Ranz and Marshall correlation. This work shows that the Ranz and Marshall correlation may be used to give a conservative estimate of the heat-transfer coefficients for particles in partially packed beds, for local Reynolds numbers below 100. However, the correlation of Agarwal, combined with the experimental single-particle data, gives a better prediction over the entire range of Reynolds numbers and voidages. Thus, together with our experimentally derived correlation, Agarwal’s equation, given accurate singleparticle data, is superior to the Ranz-Marshall equation. DISCUSSION
AND CONCLUSIONS
Experimental work has measured the convective heat-transfer coefficient around a spherical particle for scenarios of varying voidage. A step-wise experimental procedure was adopted so that the influence of different parameters on the convective heat-transfer coefficient could be measured. The principal aims were: (i) to measure the heat-transfer coefficient around particles in a system that was representative of food flow; and (ii) to test the degree of acceptability of the Ranz and Marshall correlation, which is widely used for predicting heat-transfer coefficients in food flow, but has not been verified for this use. Experiments were conducted using water, PAA solution, and glycerol solutions of 67 and 83%. Results for each stage may be summarised in terms of the Nusselt number dependence upon the parameters measured: (i) Single-particle experiments. The Nusselt number was found to be approximately proportional to the square root of the particle Reynolds number. A good agree-
Purticle-liquid heat transfer
31
ment was achieved between the experimental results and the Ranz and Marshall correlation, over the entire particle Reynolds number range. (ii) Two-pa&cl e experiments. The experimental Nusselt numbers were found to be up to 25% greater than the equivalent experimental values for single-particle experiments; however the average difference was consistently ca. + 10%. A very weak dependence between Nusselt number and inter-particle distance was observed. The Ranz and Marshall correlation was found to be acceptable for predicting heat-transfer coefficients, except in the transitional flow region, where the highest deviation between the experimental results and Ranz and Marshall predictions was observed. (iii) Packed bed experiments. Nusselt numbers for packed beds were found to be higher than the equivalent values for single particles. The increase depended on the local Reynolds number at which the measurements were being made; results for water (high local Reynolds number) showing a far greater difference between packed beds and single particles than those for glycerol (low Reynolds number). The Ranz and Marshall correlation is only suitable for prediction of Nusselt numbers in packed beds where the local Reynolds number is less than ca. 100. (iv) Variable voidage experiments. Nusselt numbers were observed to increase with decreasing voidage at any particular flow rate. The dependence of Nusselt number on voidage varied depending on the Reynolds number (local and particle) at which the correlation was derived; with a high dependence for water (high Reynolds), and lower for glycerol (low Reynolds). The applicability of the Ranz and Marshall correlation for predicting these results was better for low than for high local Reynolds numbers, where only the results for the highest voidage were compatible. However, results showed better correlation with the equation developed by Agarwal (1988) in combination with the experimental single-particle data. Correlations have been derived for each section of the work. Those for individual fluids fit the data well. However, correlations derived for the entire data set, encompassing all four fluids, generally have a low correlation probability due to a large difference, in a few cases, between the correlated predictions and experimental results. Overall, the Ranz and Marshall correlation, for any particular voidage, was only suitable for predicting particle heat-transfer rates at low (< 100) local Reynolds numbers. Above local Reynolds numbers of 100 the Ranz and Marshall correlation should be used with caution particularly if the bed voidage is low and/or the flow regime is in the transitional zone. The equation proposed by Agarawal (1988) however, proved a better predictor of heat-transfer coefficient for all voidages and flow rates used; in this respect it supersedes the equation of Ranz and Marshall for application to food flow, although accurate single-particle data is essential. These data have been obtained for a stationary sphere in a flowing fluid stream, and thus do not take account of the effects of particle rotation and translation which would be seen in practice. The results will be a lower bound on the higher heattransfer coefficients which might be found in practice; however, if fluid channelling does occur in food flows, there may be local areas of low relative motion where the heat transfer is very low, and these values are an overprediction. Care should therefore be taken in the use of this data.
32
S. Mankad et al.
ACKNOWLEDGEMENTS This publication has been produced from a collaborative research programme involving the Campden and Chorleywood Food and Drink Association (CCFDA), Food Process Engineering Department; the University of Cambridge, Chemical Engineering Department; APV Baker Limited; H. J. Heinz Company Limited; Master Foods; Nestle UK Limited; Alfa Lava1 Pumps Limited; and Unilever UK Central Resources Limited. Financial support from the MAFF/DTI LINK Food Processing Sciences Programme, EPSRC and BBSRC are gratefully acknowledged, together with the technical support of both Cambridge and Birmingham University Chemical Engineering Departments. REFERENCES Agarawal, P. K. (1988). Transport phenomena in multiparticle systems II: Particle-fluid heat and mass transfer. Chem. Engng Sci., 43(9), 2501-2510. Chandarana, D. I. & Gavin, A. (1989). Establishing thermal processes for heterogeneous foods to be processed aseptically: a theoretical comparison of process development methods. J. Food Sci., 54(l), 198-204. Chandarana, D. I., Gavin, A. & Wheaton, F. W. (1989). Particle/fluid interface heat transfer under UHT conditions at low particle/fluid relative velocities. J. Food Proc. Engng, 13, 193-206.
Chang, S. Y. & Toledo, R. T. (1989). Heat transfer and simulated sterilisation of particulate solids in a continuously flowing system. J. Food Sci., 54(4), 1017-1030. Chang, S. Y. & Toledo, R. T. (1990). Simultaneous determination of thermal diffusivity and heat transfer coefficient during sterilisation of carrot dices in a packed bed. J. Food Sci., 55(l), 199-205. Fryer, P. J. (1994). Electrical resistance heating of foods. In New Methods of Food Preservation, ed. G. Gould. Chapman and Hall, Edinburgh. Gillespie, B. M., Crandall, E. D. & Carberry, J. J. (1968). Local and average interphase heat transfer coefficients in a randomly packed bed of spheres. AIChE, 14(3), 483-490. Heppel, N. J. (1985). Measurement of the liquid-solid heat transfer coefficient during continuous sterilisation of foodstuffs containing particulates. 4th Znt. Congress on Engineering and Food, Edmonton, Canada. Himmelblau, D. M. (1970). Process Analysis by Statistical Methods. Wiley, New York. Kelly, B. P., Magee, T. R. A. & Ahmad, M. N. (1995). Convective heat transfer in open channel flow: effects of geometric shape and flow characteristics. Trans. IChemE, Part C, 73(4), 171-182.
Kreith, F. & Bohn, M. S. (1986) Principles ofHeat Transfer. Harper & Row, New York. Lareo, C. (1996). The vertical flow of solid-liquid food mixtures. Ph.D. Thesis, University of Cambridge. Lee, J. H. & Singh, R. K. (1990). Mathematical models of scraped surface heat exchangers in relation to food sterilisation. Chem. Engng. Commun., 87, 21-51. Mankad, S. (1995). Heat transfer in solid-liquid flows. Ph.D. Thesis, University of Cambridge. Mankad, S., Branch, C. A. & Fryer, P. J. (1994). The effect of slip velocity on the sterilisation of solid-liquid food mixtures. Chem. Engng Sci., 50, 1311-1321. Mwangi, J. M. (1992). Heat transfer to particles in shear flow at high Reynolds number: application to aseptic processing. Ph.D. Thesis, Cornell University. Mwangi, J. M., Datta, A. K. & Rizvi, S. S. H. (1992). Heat transfer in aseptic processing of particulate foods. In Advances in Aseptic Processing Technologies, eds R. K. Singh & P. E. Nelson. Elsevier Science, Barking, UK.
Particle-liquid heat transfer
33
Mwangi, J. M., Rizvi, S. S.H. & Datta, A. K. (1993). Heat transfer to particles in shear flow: application in aseptic processing. J. Food Engng, 19, 55-74. Newman, A. A. (1968). Glycerol. Morgan-Grampian, London. Perry, R. H & Green, D. W. (1984) Chemical Engineers Handbook, 6th edn. McGraw-Hill, New York. Ranz, W. E. (1952). Friction and transfer coefficients for single particles and packed beds. Chem. Engng Prog., 48(5), 247-253.
Ranz, W. E., & Marshall, 141-147,
W. R. (1952). Evaporation
from drops. Chem. Engng Prog.. 48,
173-180.
Rowe, P. N., Claxton, K. T. & Lewis, J. B. (1965). Heat and mass transfer from a single sphere to an extensive flowing field. Trans. IChemE, 43, 15-31. Sastry, S. K., Heskitt, B. F. & Blaisdell, J. L. (1989). Experimental and modelling studies on convective heat transfer at the particle-liquid interface in aseptic processing systems. Food Technol., 43(3),
132-143.
Sastry, S. K., Lima, M., Brunn, T. & Heskitt, B. F. (1990). Liquid-to-particle heat transfer during continuous tube flow: influence of flow rate and particle to tube diameter ratio. J. Food Proc. Pres., 13(3), 239-253.
Sastry, S. K. (1992). Liquid-to-particle Advances
in Aseptic Processing
Science, Barking, UK. Schhinder, E. U. (1978). Transport
heat transfer coefficient in aseptic processing. In eds R. K. Singh & P. E. Nelson. Elsevier
Technologies,
phenomena
in packed beds. Chem. React. Engng Rev:
ACS Symp. 72.
Shallcross, D. C., & Wood, D. G. (1987). Overall heat transfer around spheres: yet another correlation. Chemeca 87, Melbourne, Vol. 2, pp. 62.1-62.8. Zhang, L. & Fryer, P. J. (1994). Food sterilisation by electrical heating: sensitivity to process parameters. AZChE, 40(5), 888-898. Zitoun, K. B. & Sastry, S. K. (1994). Determination of convective heat transfer coefficient between fluid and cubic particles in continuous tube flow using non-invasive experimental techniques. J. Food Proc. Engng, 17(2), 209-228. Zitoun, K. B. & Sastry, S. K. (1994). Convective heat transfer coefficient for cubic particles in continuous flow using the moving thermocouple method. J. Food Proc. Engng, 17(2), 229-241.
Zuritz, C. A., McCoy, S. & Sastry, S. K. (1987). Convective heat transfer coefficients for irregular particles immersed in non-Newtonian fluid during tube flow. ASEA Paper, no. 87-6538.
of Food Enginerring 31 (1907) Y-33
Copyright 0 1997 Elsevier Science Limited Printed in Great Britain. All rights reserved 0260.8774197 $17.00 t O.OlI
Pll:SO260-8774(96)00055-6 ELSEVIER
Measurements
of Particle-Liquid Heat lkansfer in Systems of Varied Solids Fraction
S. Mankad,” K. M. Nixorf’* & P. J. Fryer” “Dept of Chemical Engineering, Pembroke St, Cambridge, ‘School of Chemical Engineering, University of Birmingham, Edgbaston, (Received
UK Birmingham,
IJK
10 March 1996; accepted 24 June 1996)
ABSTRACT The heat-transfer coefjicient between particle and liquid in a flowing food mixture is an important parameter which controls the heating and cooling rate of the particles. Experiments have measured the heat-transfer coeficient, h,, between a stationary particle and a fluid stream over a wide range of slip Reynolds numbers (I < Re ~600) and local solids @action. The results have been correlated in terms of Reynolds numbel; Prandtl number and local voidage, and compared with the previously published correlations of Ram & Marshall (1952) and Agarwal (1988). This comparison showed that the Ram and Marshall correlation was quite acceptable for predicting $, for singleparticle data, though for higher solids fractions only the predictions at lower Reynolds numbers, less than ca. 100, were accurate. Howevel; at these reduced voidages Agarwal’s correlation, based on accurate single-particle data, proved to be more successfil in predicting hp and as such is more applicable to food flows than the correlation of Ranz and Marshall. 0 1997 Elsevier Science Limited. All rights reserved.
NOTATION
zl bP C
2 4
Constant Particle Constant Constant Specific Constant Particle
in correlation surface area (m’) in correlation in correlation heat capacity (J/kg K) in correlation diameter (m)
*To whom correspondence
should be addressed. 9
10
tr 4 1
a Tt
Uf Uslip
Bi Nu NUph Nus, Pr Re Re part Re local Reslip 6X E P P
AT
S. Mankad et al. Particle convective heat-transfer coefficient Fluid thermal conductivity (W/m K) Particle thermal conductivity (W/m K) Distance between particles (m) Heat exchanged (W) Volumetric fluid flow rate (m3/s) Tube radius (m) Temperature (K) Superficial fluid velocity (m/s) Slip velocity (m/s) Biot number Nusselt number Particle Nusselt number in packed bed Particle Nusselt number for single particle Prandtl number Reynolds number Particle Reynolds number ( = pd+u$p) Local Reynolds number ( = pdpuf/pE) Slip Reynolds number ( = pdpUsliplC()
(W/m2 K)
Fractional error in quantity x Bed voidage Dynamic viscosity (Pa S) Fluid density (kg/m’) Temperature difference (K)
INTRODUCTION:
SENSITIVITY
OF HEATING COEFFICIENT
RATE TO HEAT-TRANSFER
Much industrial continuous food-processing equipment treats a two-phase mixture of solids and fluid. The carrier fluid is typically non-Newtonian (such as starch solutions) and, because of the variety and complexity of foods and recipes produced, the solids exhibit a size, shape and physical property distribution. The solids are carried through the system by the fluid and are free to translate, rotate and interact with one another. In some cases, such as the continuous ohmic heater (Zhang & Fryer, 1994; Fryer, 1994), or more commonly a scraped surface heat exchanger (Lee & Singh, 1990), flow occurs simultaneously with heat transfer. To understand and model heat transfer the temperatures and heat-transfer coefficients must be known. All aspects of free-particle motion contribute to the convective heat-transfer coefficient around the particle (Mwangi et al., 1992). This heat-transfer coefficient is crucial in determining the thermal response of a particle in a flcwing food mixture. Previous research investigating convective heat-transfer coefficients for particles in food flows have mainly used stationary single particles in isolation (Chandarana & Gavin, 1989; Zuritz et al., 1987). Measurements using stationary particles will not determine effects due to particle rotation, which will increase the heat-transfer coefficient (Sastry, 1992). Studies of particles in flow have been conducted (Sastry et al., 1989, 1990; Zitoun & Sastry, 1994a, b) but the flow round the particle is difficult
Purticle-liquid heat transfer
to measure and so results are difficult to correlate. particles, the Ranz and Marshall correlation:
11
For flow around
stationary
Nu = 2+0.(jPr”3Re!‘.’blip
(1) may apply, where the velocity in Reali,, is the relative (slip) velocity between particle and fluid. This equation was, however, derived for much higher Reynolds numbers than for food flows, and for different systems; its applicability to food processing is uncertain. In any case, values from single-particle experiments may not apply to the high solids fractions found in practice: the effect of multiple particles is also unclear. Mankad (1995) describes the range of data correlations available in the literature (e.g. Ranz & Marshall, 1952; Heppel, 1985; Zuritz et al., 1987; Sastry et al., 1989; Chang & Toledo, 1989, 1990; Chandarana et al., 1989; Mwangi, 1992) and notes that they predict heat-transfer coefficients which differ by up to two orders of magnitude for the same conditions. The most recently published work in this area is that of Kelly et al. (1995) who considered the stationary case in terms of the effects of particle geometry in Newtonian and non-Newtonian fluids. To simulate particle interactions the wake effects of a two-particle system were investigated, though multi-particle flow was neglected. Some information on particle heat transfer is, however, available from the general process engineering literature, such as Agarwal (1988) who modelled heat and mass transfer in a multi-particle system based on a boundary-layer formulation for individual particles in the sphere assemblage. The predictions of this model were then favourably compared with a wide range of previously published experimental data, but the work refers only to process engineering data, and does not appear to be widely known in the field of food engineering. It is important to find the heat-transfer coefficient values in food flows if the Biot number of food particles is such (less than 10; Kreith & Bohn, 1986) that the rate of heat transfer is sensitive to the heat-transfer coefficient. An estimate of this can be made, assuming that food solids have the physical properties of water. In the flows of interest here, particle sizes are in the range 5-25 mm, and slip velocities between solid and liquid are 1O-s-1O-2 m/s (Lareo, 1996). Since the Biot number is related to the Nusselt number by the ratio of the thermal conductivities, it is possible to relate it directly to the slip Reynolds number using the Ranz and Marshall correlation as a first approximation:
Bi
_
hPdP_ 44 2k,
k,
kL = -Nu= kL
2k, = $
2k,
-$
{2+0.6Pr”“Rei$,}
(2)
P
{ 1+0.3Pr”“Re$,} P
To estimate the range of Biot highest practicable slip velocity ty = 999 kg/m”, viscosity = 1.1 viscosity = 138 cP), i.e. extreme At 15”C, the Prandtl numbers (typical for food fluids), then Re,li, = O-85, Bi = 4.3.
numbers found in practice, a l-cm particle and the of 0.01 m/s will be considered, using water (densicP) and glycerol solutions (density = 1178 kg/m-‘, cases of the fluid viscosities found in food systems. are 7.3 for water and 1730 for glycerol. If kLzk, for water: Reslip = 91, Bi = 6.5 and for glycerol:
S. Mankad et al.
12
At lower slip velocities, even lower Biot numbers will be found. Even for larger particles (2.5 cm, and a slip velocity of 0.001 m/s) Biot numbers of 3.8 (water) and 2.7 (glycerol) are obtained. Biot numbers are thus sufficiently low that the convective heat-transfer coefficient will be significant in thermal processing of these flows. Mankad et al. (1995) developed a model for food flows, using the Ranz and Marshall correlations as a first approximation, to demonstrate the importance of particle-fluid heat-transfer coefficient and thus particle-fluid slip velocity. RanzMarshall was chosen in the absence of better data. This paper reports a series of measurements of heat-transfer coefficient in various geometries of relevance to food flows, and presents correlations for the data sets obtained.
EQUIPMENT
AND METHOD
Range of experiments Any experiment should be representative of the industrial process so that results are immediately relevant to a practical system. Ideally, the temperature of a moving food particle within a flowing swarm of particles would be measured; however, the problems associated with finding the temperature of such a particle, and measuring the flow field around it, are very large. Even if particle temperature could be measured, the flow around the particle will be unknown, making it very difficult to determine the heat-transfer coefficient due to slip velocity. The equipment was thus designed around a stationary particle system. The following experiments were carried out, as shown schematically in Fig. 1: (i) Single-particle experiments [Fig. l(a)]. This work was performed to compare results with the work of Ranz & Marshall (1952), enabling the accuracy of the experiment to be confirmed. The applicability of the Ranz and Marshall expression to low Reynolds numbers can also be checked. (ii) Two-particle experiments [Fig. l(b)]. Here, the influence of particle wake effects on the convective heat-transfer coefficient was studied. The heat-transfer coefficient to a single particle, at varying distances behind another similar sized particle in the tube, was measured. (iii) Packed bed experiments [Fig. l(c)]. Measurement of the heat-transfer coefficient from a sphere in a packed bed gave an indication of the heat-transfer coefficient possible for the highest solids fraction in food flows. These experiments, together with the single-particle ones, set bounds on heat-transfer coefficients for the maximum (packed beds) and minimum (single particles) concentrations possible. (iv) VZzriable voidage bed experiments [Figure 1 (d)]. Measurement of the heattransfer coefficient from a sphere in a partially packed arrangement of spheres, shows the variation of heat-transfer coefficient with voidage, and indicates the heat-transfer coefficients possible within a continuous food-processing plant. Apparatus The experimental apparatus consisted of a flow loop with a section in which particle behaviour could be examined [Fig. l(e)]. The tube section contained a number of
Particle-liquid heat transfer
13
(a) Single particle
0
flow
(b) Double particle
I
1
I
1
c3 flow
(c) Packed bed
Fig. 1. Schematic
diagrams of experimental
equipment
(continued overleaf).
14
S. Mankad et al.
(d) Partially packed bed I
I ,oooooo,
(e) Plow loop
thermocouple
I
V
L
flowmeter
I
L
J water tank
f
control valve P-P
Fig. 1. Continued.
Particle-liquid heat transfer
15
flow-calming sections upstream of the particle(s) under observation. A total tube pipe length of 25 m, internal diameter O-1 m was used, giving particle-to-tube internal diameter ratios similar to those in aseptic processing (i.e. between a ratio of 1:4 and 1:s; Lareo, 1996). A 0.015 m hollow copper particle was used to measure the fluid-particle heattransfer coefficient. The sphere consisted of a thin copper shell (0.5 mm) and contained a germanium resistor, which generated heat when a current flowed through it. Heat flux paste (RS components) was used to ensure good thermal contact between the heater and the particle surface. The particle was supported by a 4-mm diameter Perspex support stem, 25 mm in length, located at the rear stagnation point to minimise flow disturbance (Shallcross & Wood, 1987). The particle contained two K-type thermocouples, one at the front stagnation point and one at 90” to this, to monitor surface temperature and detect variations in heattransfer coefficient with radial position (Gillespie et al., 1968). Thermocouples were calibrated at 0 and 100°C. A range of systems was used to simulate different voidages. Packed beds were supported at their ends by thin wire mesh, whilst beds of known intermediate voidage were created by suspending particles on flexible wires between the end supports. Four fluids were used; water, polyacrylamide (PAA) solution and glycerol solution of 67 and 83%, covering a range of viscosities up to ca. 100 times that of water. These test fluids are all Newtonian to allow better characterisation and definition of Reynolds number, for descriptive purposes.
Experimental procedure The following experimental
procedure
was used:
(9 The power supplied to the particle heater was set to a constant value, chosen so the temperature difference between the fluid and the particle surface was ca. 10°C. (ii) The fluid flow rate was then set. (iii) Once steady-state conditions were achieved, particle and fluid thermocouple readings were recorded. (iv) Steps (ii) and (iii) were repeated for different fluid flow rates. coefficients were measured. Fluid and For each flow rate seven heat-transfer particle thermocouple readings, the power supplied to the particle heater (voltage and current), and the fluid flow rate were recorded; the heat-transfer coefficient and particle Reynolds number were then calculated from this data. Ten flow rates were used in all experiments between 0.6 and 5 m’/h; these correspond to slip Reynolds numbers between 05 and 5000. All experiments used a fluid temperature range of 13-18°C; physical properties for the fluids used, over this temperature range, are presented in Table 1. These data were used for all calculations. Changes in the specific heat capacity, thermal conductivity and density of the fluids were negligible over the temperature range 13-18°C. The solid surface temperature ranged between 23 and 35°C.
S. Mankad et al.
16
TABLE 1
Fluid Physical Property Range used in Experiments (Perry & Green, 1984; Newman, 1968) Property
Temperature range (“C) Concentration (%) Viscosity (Pa s) Thermal conductivity (W/m K) Specific heat capacity
(J/kgK)
Density (kg/m’)
Water
PAA soln
Glycerol soln I
Glycerol soln II
13-18 n/a 0.0011 0.628
13-18
13-18
13-18
0.5%
83%
67%
0.0027 0.628
0.111-0.161 0.320
0.027-0.020 0.365
4184
4184
-
-
999.1
999.1
1220
1176
Accuracy of results A number of problems were encountered with the construction and the running of the equipment. The particle was frequently cleaned to remove oxide films and to reduce places for air bubble adhesion: both of these lead to low heat-transfer readings. Inherent errors in the thermocouples and electronic-meter readings can be developed into a quantitative error analysis. In the studied temperature range the thermocouple accuracy was +0.75% of the temperature reading (in “C) indicated (RS thermocouple data sheet). Thermocouples were calibrated using pure melting ice and the steam above pure boiling water. Successive calibrations were performed for each new particle. The error due to the difference in calibration readings was found to be a maximum of f0*36% of the temperature reading. Voltages were measured using an Iso-Tech IDM 96 multimeter, whose specification states the displayed value contains an error of *O-5% of the voltage measured; the experimental readings ranged between 1.2 and 3.7 V. Power was supplied via a Weir 4000 supply. The voltage was read from a digital display, and the current was recorded using a Fluke 8050A digital multimeter. The accuracy of the power-supply voltage reading was &-O-5% of the reading. For the Fluke multimeter the accuracy was f0*3% (Fluke 805OA data sheet) of the current indicated. The convective heat-transfer coefficient was calculated from:
where q is the heat transferred from the particle to the fluid, A, is the surface area of the particle and AT is the temperature difference between the fluid and the particle surface. The fractional error in the heat-transfer coefficient was thus the sum of those in the individual measurements. The total fractional error in AT is: 6AT= and in q is:
f
0*75+0*36+0.5 100
= f0.0161
(4)
Particle-liquid heat transfer
6q=
+
l
0.3 +0*5 100
I
= f0.008
17
(5)
The fractional error in h,, 6hb, was thus kO.0241 (i.e. 0~0161+0*008), which corresponds to a percentage error of +2*41%; an error in h, of + 2.5% was assumed throughout. Hydrodynamic heat transfer correction It was necessary to support the particle in the flowing stream. A correction to the value of the heat-transfer coefficient was required to compensate for the loss in heat transfer due to the supporting stem (Gillespie et al., 1968). Using this method, a multiplication factor of 1.087 (Mankad, 1995) was used for flows above a particle Reynolds number of 2200, or a multiplication factor of l-057 for flows below 2200. This is standard practice when correcting for hydrodynamic effects (Shallcross & Wood, 1987); the errors occurring through neglect of these effects are well documented in the process-engineering literature (Ranz, 1952; Rowe et al., 1965). Correlation of results For each fluid used at least seven sets of data points were recorded for each particle/local Reynolds number combination. The arithmetic mean of these data points is shown. Included on the graphs are error bars which represent the standard deviation in the set of results. Some plots include lines showing k 10% deviation from the values predicted by a correlation. The original Ranz and Marshall correlation was claimed accurate to within + 10%. These lines, and standard deviation data, help to determine the association between measurements and correlations. Results were manipulated to correlate Nusselt number as a function of the independent variable(s) used in the experiments. Two Reynolds numbers were used. The particle Reynolds number, defined as:
where uf is defined as: UfZ
-
Q 7CR:
where Q is the volumetric fluid flow rate and R, is the tube radius, can be used to correlate data for low solids fractions, where it is identical to the slip Reynolds number. To model high solids fractions another Reynolds number is needed. In a system of voidage E through which Q m”/s of fluid flows, the mean fluid velocity around the stationary particle will be Q/nR2 f c:. The local Reynolds number is defined as:
S. Mankad et al.
18
as the particle(s) used in the experiment are stationary and the fluid velocity over the particle (u$E) is the axial slip velocity. For single- and two-particle experiments, the local and particle Reynolds number are equal. Experimental results for packed beds and partially voided beds have been plotted against the local Reynolds number, which is equal to the slip Reynolds number. A form of equation relating the Nusselt number to the independent variable(s) for the system under consideration was selected by inspection of correlations used by previous workers for similar systems. In each case correlations contained a number of unknown constants; a spreadsheet was used to manipulate the unknown constants to minimise the total sum of the square of the errors between the experimental values and the values predicted by the correlation (Himmelblau, 1970). RESULTS
AND ANALYSIS
Table 2 summarises the data for the different experiments. Marshall correlation [eqn (l)] is used widely to determine coefficient for food flows, it has not been tested for food experiments is to determine the level of applicability of correlation to food flow configurations, and to determine use of the equation is acceptable.
Although the Ranz and the particle heat-transfer systems. An aim of our the Ranz and Marshall the conditions for which
Single particle in isolation Results and discussion
As expected, the Nusselt number increases with increasing particle Reynolds number, due to the increasing fluid velocity around the particle. The Ranz and Marshall equation predicts most of the data well although, with the exception of the lowest Reynolds numbers in each group, the experimental results obtained seem lower than predicted. Figure 2(a) shows the percentage deviation of the experimental results from Nusselt numbers calculated from the Ranz and Marshall correlation, as a function TABLE 2
Summary of Fluid/Particle Property Single-particle experiments: Reynolds number Nusselt number Twoparticle experiments: Reynolds number Nusselt number Packed bed experiments: Reynolds number Nusselt number Voidage aperiments: Reynolds number Nusselt number
Experimental
Data
PAA
6 7% Glycerol
83% Glycerol
292-2433 20-60
114-893 20-48
18-140 16-36
5-97 9-34
292-2433 20-60
114-893 20-48
5-97 9-34
5-97 9-34
712-5937 44-163
278-2178 36-126
44-341 24-77
1.2-22 5-39
324-3477 44-163
127-1276 36-126
20-200 19-66
1-13 5-39
Water
Particle-liquid
19
heat transfer
loo m
I
o-1O.1
1
1
I
lllllli
1
10
1
,.,,.,I
s
.,,,.aoI
loo
I
1ooo
I
I.1
10000
particle Reynolds number Fig. 2. (a) Difference between single-particle heat-transfer coefficients and values predicted by the Ranz and Marshall correlation. (b) Comparison of single-particle heat-transfer coefficients with particle Reynolds number. (c) Difference between single-particle heat-transfer coefficients and values predicted by derived correlation n, 83% glycerol; x , 67% glycerol; o, PAA; o, water (continued overleaf).
20
S. Mankad et al.
of particle Reynolds number. The data represents the percentage difference calculated from the arithmetic mean of the results, at each particle Reynolds number; error bars show the standard deviation of each data point expressed as a percentage. Difference is calculated as: Percentage Experimental
difference
=
Nusselt number - Ranz and Marshall Nusselt number
x 100%
Ranz and Marshall Nusselt number
(9) The maximum percentage difference shown in Fig. 2(a) was f 15%. Nusselt numbers for water, PAA and 67% glycerol solution crossed over the values predicted by Ranz and Marshall; each high particle Reynolds numbers gave lower Nusselt numbers than the Ranz and Marshall predictions, but this trend was reversed at low particle Reynolds numbers. The experimental results for 83% glycerol solution were consistently below those predicted by Ranz and Marshall. Included in Fig. 2(a) and all subsequent graphs of this type are lines representing values 10% above and below the correlation; this was the maximum error margin estimated in the original paper (Ranz & Marshall, 1952). Figure 2(a) shows that the majority of the experimental results lie below values predicted by the Ranz and Marshall correlation (mean difference ca. -6%), and with the exception of one data point for PAA and three for 67% glycerol, all the results lie within +lO% of
20 15 10 5 0 -5 -101 t
I rrB T
1
1 1 ’
-
83% Glycerol
I 1 I 1 ..,
111
.
water
-T I
-15F ' ''ntnnt' ' ''nknBn' ' ''11111' ' ''a1111' ' 'J 0.1 1 10 100 1000 10000 particle Reynolds number Fig. 2. Continued.
21
Particle-liquid heat transfer
Correlations
TABLE 3 for Single-Particle
Fluid
Experiments
Equation
Water
PAA soln 67% Glycerol soln 83% Glycerol soln
Nu Nu Nu Nu
= = = =
2+1.41 2+3.20 2+4.98 2+567
Correlation probability
Re”‘47 part Re”:” part Re?part Rei:r’;
0.98 0.99 0.99 0.99
the Ranz and Marshall predictions. The data suggest that the apparatus is accurate and that the Ranz and Marshall correlation may be used to estimate acceptable heat-transfer coefficients for single particles in flowing food systems, down to Reynolds numbers of unity. Correlation
Although the Ranz and Marshall equation lies close to the data, correlations made to see if a better fit could be obtained using equations of the form: NU = 2+aRei;,,,
were (10)
where a and b are constants. Table 3 shows the correlations obtained. Correlated particle Reynolds number exponents compare well with values derived by Ranz & Marshall (1952). For water and 83% glycerol solution the deviation was minimal (exponents of 0.47 and 0.49, respectively); for PAA and 67% glycerol solutions the association with the Ranz and Marshall correlation was poorer (0.38 and 0.38). The equations for each fluid were combined to form one equation for all the single particle experiments: Nu = 2+0*97Re~~,Pr”‘*’
(11)
Figure 2(b) compares the percentage difference between the experimental results and those predicted by eqn (11). The majority of the experimental data points lie within & 10% of the derived correlation. This correlation compares well with the Ranz and Marshall equation, as shown in Fig. 2(c). The data suggest that the Ranz and Marshall correlation gives an excellent prediction for heat-transfer coefficients for single particles in horizontal tubes at particle Reynolds numbers above 200, a significant distance from the particle Reynolds value range over which it was first derived. Equation (11) is a better representation of the data at low Reynolds number. lb-o-particle
systems
Results and discussion
Fluid-particle heat transfer within a packed bed will differ from that to a sphere due to distortions of the flow and temperature fields in the bed due presence of particles. The flow will be affected by the presence of particles; stream of a particle, the flow will be disturbed and unsteady, and may become turbulent. As a first step, ‘double particle’ experiments were carried out
single to the downmore and a
22
S. Mankad et al.
15-mm solid polyethylene sphere was positioned in front of the 15-mm heated copper particle to quantify the effect of flow disruption. The configuration is shown in outline in Fig. l(b); the inter-particle distance was varied between 1 and 5 cm, measured between closest contact points on the two particles. Graphs of Nusselt number against particle Reynolds number for varying inter-particle distances show that the Nusselt number increases with increasing particle distance (on average ca. 5%); however, the degree of increase is related to the particle Reynolds number (Mankad, 1995). The majority of Nusselt numbers recorded lie close to values predicted by the Ranz and Marshall correlation. However, the following may be observed: (i) For Reynolds numbers above 100, the heat-transfer coefficients, for all interparticle distances, were up to 25% higher than for a single particle in isolation. (ii) Below a particle Reynolds number of 100, the results for different particle distances were within the bounds of quoted experimental error for the Ranz and Marshall expression, i.e. effectively identical to Nusselt numbers for a single particle. (iii) Generally, for particle Reynolds numbers above 100, the convective heattransfer coefficient increased with increasing inter-particle distance. Experiments show that, at certain flow rates, particle interactions can enhance particle heat-transfer coefficients by up to 25% from those measured for single particles. The major enhancements occur in the transitional flow region between laminar and turbulent. Even a single particle can enhance the heat-transfer coefficient around a second particle, demonstrating why particle concentration has such a strong effect on the heat-transfer coefficient. A general correlation for the entire data set was found as: NU = 2+ 1.215(l/d,)“05RepOg~ro’*
(12) Figure 3(a) shows the fit to this data and Fig. 3(b) compares the percentage difference between the experimental results and those predicted by eqn (12). Almost all the data points for water and PAA experiments lie within f 10% of the derived correlation; however all results for glycerol lie well below their predicted values (ca. -35% difference), suggesting that low Re behaviour is different from that at high Re.
Packed beds Results and discussion
The convective heat-transfer coefficient around a particle in a packed bed was studied. Experiments measured the heat-transfer coefficient around a sphere within a packed bed of spheres. The 15-mm internally heated copper sphere was situated at the centre of a randomly packed bed of 15-mm diameter hollow polypropylene spheres [Fig. l(c)]. The bed dimensions were 0.1 m in diameter and 0.25 m in length. The bed voidage, E, was calculated as 0.41, and was constant for all experiments. Table 2 includes experimental data. Results are plotted against local Reynolds number, allowing comparisons to be made for similar local fluid velocities, i.e. identical flow velocities around particles.
Particle-liquid heat transfer
23
67%
1L 1
’
nllllll’
10
I
I
u
100
1000
10000
particle Reynolds number
20 10 0
-10 : -20 L -30 : -40
7
-50 1
10
1000
particle Reynolds number Fig. 3. Two-particle experimental data.
experiments. (a) Comparison of derived correlation with entire range of (b) Difference between experimental results and values predicted by derived correlation.
24
S. Mankad et al.
Figure 4(a) plots the experimental data against those predicted by Ranz and Marshall. The deviation of experimental Nusselt numbers from those estimated by the Ranz and Marshall correlation is substantial at high Re. There is a theoretical basis for this; Schlunder (1978) stated that heat-transfer coefficients for packed beds should be three times higher than those for a single particle. Ranz (1952) developed a model for packed beds which compared well with experimental values, and showed that packed bed heat-transfer rates were a factor of 2.6 higher than those for single particles. The results shown here suggest that these effects are also found in food flows. Correlation Table 4 shows correlations. For water and PAA the exponent number is significantly greater than for single-particle experiments 3 and 4), highlighting the enhancement in heat transfer in packed for the case of the two glycerol solutions. Data for all fluids may be
in the Reynolds (compare Tables beds. This is less combined as:
Nu = 0+0.5Re:~~~,Pr0-28 (13) As for the two-particle data, the equations fit high Re data well, but predict results at low Re poorly. Figure 4(b) compares the percentage difference between the experimental results and the corresponding values predicted by eqn (13) and shows a high deviation between experimental results for 83% glycerol and the predicted values. When the data are plotted with the Ranz and Marshall correlation [Fig. 4(c)], the graph shows a cross-over point. Above Re ca. 700, Ranz and Marshall underpredicts significantly, whilst below that value the correlation slightly overpredicts the data. Where the local Reynolds number is less than ca. 100, the Ranz and Marshall correlation may be satisfactorily used to predict heat-transfer coefficients around particles in packed beds to ca. f 10%. Variable voidage beds Results and discussion Two-phase solid-liquid food flow mixtures typically have solid fractions between 5 and 40% (Lareo, 1996), an intermediate stage between a close packed bed and a single particle. The processes occurring at these intermediate voidages may be unrelated to those occurring at extreme cases of packing due to channelling in the bed. Experiments were conducted using the 15-mm internally heated copper particle situated in the centre of a partially packed arrangement of 15-mm polypropylene spheres [Fig. l(d)]. The distended bed was 0.25-m long and housed in a 0.1-m internal diameter tube. The distended bed was made using helical coils of wire (l/16 in diameter), along which 15-mm polypropylene spheres had been threaded. A number of these coils were then placed within the tube section to give an approximately regular packing arrangement around the copper particle. The bed voidages used were 90, 80 and 70%, i.e. solids fractions of lo-30%. Plots of Nusselt number versus local Reynolds number (Mankad, 1995) displayed the following common features for all fluids and voidages: (i) At any particular voidage, Nusselt number increases with local Reynolds number. (ii) For a particular local Reynolds number, Nusselt number increases with decreasing voidage. This effect increases with local Reynolds number.
Particle-liquid heat transfer
25
Water CR do PAA *s”*” 67% Glycerol
_= n!5
i
local Reynolds number 60
L
Fig. 4. Packed bed. (a) Difference between experimental results and values predicted by the Ram and Marshall correlation. (b) Difference between experimental results and values predicted by derived correlation. (c) Correlation of experimental data for heat transfer with Ranz and Marshall and empirically derived equations (continued overle@).
S. Mankad
26
10
Ram and Marshall NH4 I
et al.
.
67% Glycerol
local Reynolds number Fig. 4. Continued.
voidages are greater than those (iii) The Nusselt numbers for intermediate recorded for single-particle experiments. The degree of increase was again dependent on the local Reynolds number, and was more notable for water than for glycerol. Experimental results are compared with predictions from the Ranz and Marshall correlation in Fig. 5(a). The percentage difference between the two values increases as voidage decreases. Nusselt numbers recorded using PAA and water show the highest deviation from the Ranz and Marshall prediction, starting at ca. -20% difference for 90% voidage and rising to ca. +20% at 70% voidage. The 83% glycerol results were consistently lower than the predictions of Ranz and Marshall, and varied less significantly with voidage compared with the other two fluids, ca.
Correlations
TABLE 4 for Packed Bed Experiments Equation
Fluid
Water PAA soln 67% Glycerol soln 83% Glycerol soln
Nu Nu Nu Nu
= = = =
0+042 ReEzd, 0+1.22 Rezzz, O-89+3.73 Reads, 2*57+3.18 Revs3 local
Correlation probability
0.99 0.99 0.99 0.99
27
Particle-liquid heat transfer I
I ’ ’ ““‘I
0 r ’ ““‘I
67% Glycerol
’ ’ ’ ““‘I
’ ’ “‘I”
,T
Water
83% Glycerol
4
10
100
1000
10000
local Reynolds number
67% Glycerol
Water
local Reynolds number Fig. 5. Expanded beds. (a) Difference between experimental heat-transfer coefficients and values predicted by the Ranz and Marshall correlation. (b) Variation between experimental heat-transfer coefficients and predicted values from derived correlation. (c) Variation between experimental heat-transfer coefficients and values predicted from Agarwal’s correlation (Agarwal. 1988). (d) Comparison of experimental Nusselt numbers with existing and derived correlations (continued overleaf).
28
S. Ma&ad
et al.
-10 -20 0.1
1
10
100
1000
10000
1000
10000
particle Reynolds number
100
Ranz and Marshall (1952) (assuming voidage = 1)
Ol..........,..... . 0.1 1 10
100
particle Reynolds number Fig. 5. Continued.
29
Particle-liquid heat transfer
-30% difference for 90% voidage, and rising to ca. - 15% for 70% voidage. This increase in deviation, in all cases, can be attributed to the higher degree of eddy formation and/or channelling arising in low voidage beds. Overall, results for the highest voidages were closest to Ranz and Marshall. The results show that deviation from the Ranz and Marshall correlation is related to the local Reynolds number. At high flow rates, channelling and eddies are likely to occur, causing non-uniformity of flow and enhancing heat transfer; these effects are further increased at lower voidage (higher solids fraction). For low flow rates, eddy formation and channelling is small, resulting in a smoother flow profile; conditions under which Ranz and Marshall performed their original experiments. Recent work on heat transfer to particles in non-packed bed flows, with application to the food industry, was performed by Mwangi (1992). Mwangi measured the heat transfer from free-flowing spheres in a tube for high void fractions (95%). No correlation was derived from the experimental data, but Mwangi showed from his results that the Nusselt numbers estimated for these systems, on a particle Reynolds number basis, were between 80 and 200% higher than those evaluated for single spheres, which corroborates the range of values estimated through our experiments. The most substantial work predicting heat-transfer rates in expanded beds is that of Agarwal (1988). This work developed a relationship between heat transfer to a single particle and one in a bed, for the same superficial fluid velocity: NC+ = I:- “YvSP
(14) where NC+, is the Nusselt number around the particle in a packed or expanded bed, Nu,, is that around a single isolated particle, and c is the bed voidage. This correlation has been used, together with the Ranz-Marshall equation, to correlate experimental data. Correlation
A mathematical correlation expressing Nusselt number as a function of voidage and local Reynolds number, of the form: Nu =a + b Re&tllocal 6
(15) where a, b and c are constants, and E is the fractional voidage, was fitted to the data. Table 5 shows correlations for each fluid. The correlations vary markedly for each fluid used and no resemblance to correlations for packed beds or single particles is seen. The Nusselt number dependence on voidage decreases appreciably from water to glycerol. The equations can be combined as: NU = 2.7+1.1r:~“‘Rel~~~,pr’~2’
Empirical
Correlations
TABLE 5 for Variable
Water PAA soln 67% Glycerol soln 83% Glycerol soln
Voidage Experiments
Equation
Fluid Nu = Nu = NM = Nu =
(16)
14-+0.42 Ret:?, 1: ‘-4 79+2.2 ReiLz, I’ ~ “” 0+5..57 Reii;z!, ,‘~0-7q 3.5+5.6 Re@:f lO“lII‘’ -0.55
Correlation probability 0.99 0.57 0.94 0.99
30
S. Mankad et al.
The correlation probability for this equation is 0.57 and, due to the low range of Prandtl numbers and voidages used in the experiments, should be used cautiously. Figure 5(b) compares the percentage difference between the experimental results and the values predicted by eqn (16); the majority of values lie within f 10% of the values predicted by eqn (16). Comparison of the experimental data, for all combinations of fluid and bed voidage, with the values predicted by eqn (14) Agarwal’s correlation, is shown in Fig. 5(c). Single-particle experimental data is used to generate Nu,, in eqn (14). The difference between the normalised expression, Nu~~/E-‘.*~ (where Nupb is obtained from experimental data), and the corresponding experimental single-particle results for the same superficial fluid velocity are shown as percentages with error bars. This allows the data from all three sets of expanded bed experiments to be shown on one graph. The general agreement between the data and the predictions of eqn (14) is good. However, the degree of agreement varies with particle Reynolds number, being better at low Re,,, where the deviation for 83% glycerol was +8%, than at high Re,, where the average deviation for water is +20%. This is the trend which would be expected as the voidage is less influential at lower flow rates. Equation (16) is plotted, together with all the experimental results, the Ranz and Marshall correlation, and that of Agarwal, on Fig. 5(d). From this graph eqn (16) can be seen to be a constant factor of ca. 1.25 higher than the Ranz and Marshall correlation. In addition, the graph demonstrates the high degree of agreement between the experimental correlation and that of Agarwal, particularly at low particle Reynolds numbers; both are better for use in predicting heat-transfer coefficients in comparison with the Ranz and Marshall correlation. This work shows that the Ranz and Marshall correlation may be used to give a conservative estimate of the heat-transfer coefficients for particles in partially packed beds, for local Reynolds numbers below 100. However, the correlation of Agarwal, combined with the experimental single-particle data, gives a better prediction over the entire range of Reynolds numbers and voidages. Thus, together with our experimentally derived correlation, Agarwal’s equation, given accurate singleparticle data, is superior to the Ranz-Marshall equation. DISCUSSION
AND CONCLUSIONS
Experimental work has measured the convective heat-transfer coefficient around a spherical particle for scenarios of varying voidage. A step-wise experimental procedure was adopted so that the influence of different parameters on the convective heat-transfer coefficient could be measured. The principal aims were: (i) to measure the heat-transfer coefficient around particles in a system that was representative of food flow; and (ii) to test the degree of acceptability of the Ranz and Marshall correlation, which is widely used for predicting heat-transfer coefficients in food flow, but has not been verified for this use. Experiments were conducted using water, PAA solution, and glycerol solutions of 67 and 83%. Results for each stage may be summarised in terms of the Nusselt number dependence upon the parameters measured: (i) Single-particle experiments. The Nusselt number was found to be approximately proportional to the square root of the particle Reynolds number. A good agree-
Purticle-liquid heat transfer
31
ment was achieved between the experimental results and the Ranz and Marshall correlation, over the entire particle Reynolds number range. (ii) Two-pa&cl e experiments. The experimental Nusselt numbers were found to be up to 25% greater than the equivalent experimental values for single-particle experiments; however the average difference was consistently ca. + 10%. A very weak dependence between Nusselt number and inter-particle distance was observed. The Ranz and Marshall correlation was found to be acceptable for predicting heat-transfer coefficients, except in the transitional flow region, where the highest deviation between the experimental results and Ranz and Marshall predictions was observed. (iii) Packed bed experiments. Nusselt numbers for packed beds were found to be higher than the equivalent values for single particles. The increase depended on the local Reynolds number at which the measurements were being made; results for water (high local Reynolds number) showing a far greater difference between packed beds and single particles than those for glycerol (low Reynolds number). The Ranz and Marshall correlation is only suitable for prediction of Nusselt numbers in packed beds where the local Reynolds number is less than ca. 100. (iv) Variable voidage experiments. Nusselt numbers were observed to increase with decreasing voidage at any particular flow rate. The dependence of Nusselt number on voidage varied depending on the Reynolds number (local and particle) at which the correlation was derived; with a high dependence for water (high Reynolds), and lower for glycerol (low Reynolds). The applicability of the Ranz and Marshall correlation for predicting these results was better for low than for high local Reynolds numbers, where only the results for the highest voidage were compatible. However, results showed better correlation with the equation developed by Agarwal (1988) in combination with the experimental single-particle data. Correlations have been derived for each section of the work. Those for individual fluids fit the data well. However, correlations derived for the entire data set, encompassing all four fluids, generally have a low correlation probability due to a large difference, in a few cases, between the correlated predictions and experimental results. Overall, the Ranz and Marshall correlation, for any particular voidage, was only suitable for predicting particle heat-transfer rates at low (< 100) local Reynolds numbers. Above local Reynolds numbers of 100 the Ranz and Marshall correlation should be used with caution particularly if the bed voidage is low and/or the flow regime is in the transitional zone. The equation proposed by Agarawal (1988) however, proved a better predictor of heat-transfer coefficient for all voidages and flow rates used; in this respect it supersedes the equation of Ranz and Marshall for application to food flow, although accurate single-particle data is essential. These data have been obtained for a stationary sphere in a flowing fluid stream, and thus do not take account of the effects of particle rotation and translation which would be seen in practice. The results will be a lower bound on the higher heattransfer coefficients which might be found in practice; however, if fluid channelling does occur in food flows, there may be local areas of low relative motion where the heat transfer is very low, and these values are an overprediction. Care should therefore be taken in the use of this data.
32
S. Mankad et al.
ACKNOWLEDGEMENTS This publication has been produced from a collaborative research programme involving the Campden and Chorleywood Food and Drink Association (CCFDA), Food Process Engineering Department; the University of Cambridge, Chemical Engineering Department; APV Baker Limited; H. J. Heinz Company Limited; Master Foods; Nestle UK Limited; Alfa Lava1 Pumps Limited; and Unilever UK Central Resources Limited. Financial support from the MAFF/DTI LINK Food Processing Sciences Programme, EPSRC and BBSRC are gratefully acknowledged, together with the technical support of both Cambridge and Birmingham University Chemical Engineering Departments. REFERENCES Agarawal, P. K. (1988). Transport phenomena in multiparticle systems II: Particle-fluid heat and mass transfer. Chem. Engng Sci., 43(9), 2501-2510. Chandarana, D. I. & Gavin, A. (1989). Establishing thermal processes for heterogeneous foods to be processed aseptically: a theoretical comparison of process development methods. J. Food Sci., 54(l), 198-204. Chandarana, D. I., Gavin, A. & Wheaton, F. W. (1989). Particle/fluid interface heat transfer under UHT conditions at low particle/fluid relative velocities. J. Food Proc. Engng, 13, 193-206.
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