Blofechnol. Rw. 1994, 10, 230-230
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Thermal Time Distributions in Tubular Heat Exchangers during Aseptic Processing of Fluid Foodst S. Bhamidipati and R. K. Singh' Department of Food Science, Purdue University, West Lafayette, Indiana 47907-1160 ~~~~~
The objectives of this study were to develop a mathematical model for thermal time distributions (TTD) in tubular heat exchangers and to perform a computer simulation to study the role of TTD in food process design. A computer simulation of heat transfer to a pseudoplastic fluid with variable thermophysical properties heated in a tubular heat exchanger was carried out. A power law model, with the flow behavior consistency indices modeled as temperature-dependent properties, is used to described the flow behavior of the fluid. The coupled continuity, momentum, and energy equations with corresponding boundary conditions for constant wall temperature and heat flux are solved numerically by a Dufort-Frankel finite difference technique. Thermal processing models are then applied to obtain T T D by tracking fluid elements along the flow path in a heating section. T T D and F curves are used to evaluate process effectiveness. Two tube radii (0.0254 and 0.0382 m) and two wall temperatures (125 and 130 "C)were studied. Increasing the wall temperature from 125 to 130 "C reduced the length of the heat exchanger by 11.35%. Increasing the tube radius by 150% caused an 84.2% increase in the length of the heating section.
Introduction Continuous sterilization has been used favorably as an economical and efficient means of destruction of microorganisms in the food industry. Potential advantages include less damage to the product, short processing periods, uniform and improved product quality, reduced energy consumption,and ready adaptability to automatic control. Residence time distribution (RTD) has been studied in the past to analyze isothermal systems in the field of chemical reaction engineering. Several RTD studies have albe been applied in the field of food engineering. However, RTD studies do not give information about the thermal history of the fluid elements and are only relevant to the studies of isothermal systems, but they currently are used by the regulatory agencies as part of process filing. When fluids are heated in a continuous system, there is, in addition to a residence time distribution, a distribution of thermal treatments (times)because of differences in heat treatment received by different fluid elements at different spatial locations within the reactor. Thermal time distribution (TTD) is defined as the distribution of 3'0 values. TTDs have complete information on the temperature history, in addition to the duration of heating. Nauman (1977)has discussed the theory and application of thermal time distributions and has given examples of TTDs in the laminar flow of constant-property Newtonian fluids in heat exchangers and extruders. Literature on this topic is meager since very little work has been done in this area. In addition to microbial destruction, TTD studies can provide information on nutrient destruction. Paulsson and Tragardh (1984)have simulated TTDs in a tubular heat exchanger with turbulent flow using a stochastic combined flow reactor model. Recently, Ellborg (1989),Datta (1991),andMwangietal. (1992)haveapplied this concept to food processing. Ellborg (1989) has
* Author to whom correspondence should be addressed.
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experimentally determined the TTD in a turbulent flow reactor using the polymerization characteristicsof dextran. The postprocessing molecular weight distribution of dextran (which is dependent on the amount of polymerization) was related to the thermal treatment received. Datta (1991)applied the concept of TTD to processing a fluid with constant thermophysical properties in a can. Datta and Liu (1992)studied TTDs for microwave and conventional heating of liquids and foods. TTDs were developed for conductive heating of solids and heating of liquids (batch and continuous). For continuous heating of foods, axial and radial variations in velocities and temperature-dependent properties were not considered. Work in the area of food process simulation, where a fluid is heated or cooled along the length of the heat exchanger, has been based on two major assumptions: a homogeneous fluid and isothermal conditions (radial direction) throughout the reactor. In the case of liquid foods,the first assumption is valid, and chemicalreactions throughout the reactor (chemical reactions or microbial, enzymatic, and nutrient destruction) can be described by a first-order reaction kinetics. However, the second assumption is invalid. The fluid gains or loses heat along the length of the heat exchanger, and therefore different parta of the fluid (radial and axial elements) receive different thermal treatments. This causes a variation in velocity and temperature in the radial and axial directions. This problem has received little attention (Kumar and Bhattacharaya, 1991;Simpson and Williams, 1974). The equations developed by Nauman (1977)and Mwangi et al. (1992)for TTD evaluation do not account for these variations in velocity and temperature profiles in the axial direction. Fluid elementa undergoing different thermal treatments result in different final concentrations of a parameter (nutrients or microorganisms). This article aims to overcome these shortcomings by developing a numerical model to predict temperature distributions in a non-Newtonian fluid along the length of a heat exchanger and then predict the TTDs. The major difficulty is accounting for the effects of temperature-dependent
0 1994 American Chemlcai Society and American Institute of Chemical Engineers
Bbtechnol. Rug., 1994, Vol. 10, No. 2
properties and non-Newtonian behavior of the fluid on the TTDs. Of these, the temperature dependence of consistency and flow behavior indices is of primary importance as this affects the viscosities and, hence, the velocity and temperature profiles of the fluid. Design of heat exchangers involves the optimization of the different process parameters, such as temperature in the heat exchanger, flow rate, tube length and diameter, and properties of the process fluid. The objectives of this research are (1) to develop a mathamatical formulation for TTDs and (2) to evaluate their role during the heating of a non-Newtonian fluid with temperature-dependent properties in a tubular heat exchanger. The approach to the problem requires a simultaneous finite difference solution of the coupled continuously, momentum, and energy balance equations along with the equation for microbial destruction.
Problem Formulation The major consideration is the flow of a pseudoplastic fluid with variable thermophysical properties in a tubular heat exchanger subjected to a constant wall temperature. A constant wall temperature boundary condition is described in this analysis. The fluid has a fully developed velocity profile at the inlet of the heat exchanger. However, due to property variations by heating, the fluid will undergo a variation in velocity profile at the entrance of the heat exchanger. The problem is to determine the temperature and velocity profile development of the fluid at various locations along the axis of the tube. The fluid temperature profile is then used to calculate the TTD. There is a lack of agreement on the effects of free convection in laminar flow of non-Newtonian fluids. It may be significant (Oliver and Jenson, 1964; Rodriguez et al., 1987) or insignificant (Joshi and Bergles, 1981;Pereiraet al., 1989). Nonetheless, this term has been neglected in the present analysis. Assumptions. The following assumptions are made: steady laminar flow, constant wall temperature, negligible viscous dissipation,small axial conduction relative to radial conduction, and insignificant free convection effects arising due to variations in density (p). Hence, the buoyancy term in the momentum equation is neglected. The fluid is at a constant temperature and has a fully developed profile at the inlet. Standard boundary layer approximations (Schlichting, 1955) are made. Using the velocitv boundarv" laver au au JU au approximations (u >> u, - >> - - -), the radial ar ax' ay' ax momentum equation can be dropped. Hence, the pressure (P(x))in the boundary layer depends only on the axial position ( x ) and is equal to the presence in the free stream outside the boundary layer. P(x) may then be obtained from a separate consideration of flow conditions in the flow stream (global continuity equation). The use of correct experimental correlations for temperature-dependent properties is extremely important, especially in the case of apparent viscosity, consistency, and flow behavior indices, because of the nonlinear relationship of shear stress with shear rate. Therefore, a non-Newtonian constitutive equation is necessary in modeling heat transfer to such fluids. Most fluid foods can be described as shear thinning (pseudoplastic), and so the flow behavior is modeled by a power law model as On the basis of these assumptions, the results in independent of the orientation of the tube, and so the governing equations (in cylindrical coordinates) become two-dimen-
231
sional andaxisymmetricand are expressed as follows (Bird et al., 1960). Governing Equations. continuity equation
momentum equation
energy equation
global continuity equation J'pu
d~ = constant
(5)
The boundary conditions are u(x,R) = 0, u(x,R) = 0, aT = 0. The initial (x,o) = 0, T(x,R)= T,, and r (1.0) conditions are u(0,r) = u(r), T(O,r)= TO,and P(0) = PO. The variations in consistency index ( K ) and flow bihavior index (n)due to changes in the temperature are of importancein the heat-transfer calculations. Therefore, these were modeled by the Arrhenius model (Bhamidipati and Singh, 1990) as
(g)
(a)
n(T) = 0.20 exp(8.08 X 104T) (6) K(T)= 3.60 exp(4.025T) (7) The temperature effects on fluid heat capacity and thermal conductivity variations are usually very small and, hence, are neglected. Although the effect of free convection was neglected, the effect of temperature on density was considered. The density of water obtained from Perry and Chilton (1973) was used: p(T)
= (1.0005 - 1.4825 X
lob T -
0.27069 X lo4 ~ ) l O O O(8) Fluid (tomato juice) specific heat (3676.0 J/kg-K) and thermal conductivity (0.55 W/cmK) were assumed to be constant and were obtained for a temperature of 120 O C (Choi and Okos, 1983). Formulation of Fluid Particle Motion. There are two approaches, (a) Lagrangian and (b) Eulerian, for the determination of particle motion in a heat exchanger. The Lagrangian approach treats the fluid elements as discrete entities, and their temperature histories are calculated by tagging each element along its path. The Eulerian approach treats the particles as a continuum, and the appropropriate governing equations in differential form are solved to determine the property values at spatial locations at each instant of motion. The former involves extensive computational work and is, therefore, not commonly employed by researchers. The present analysis deals with the latter approach. To obtain the thermal treatment received by individual particles, the temperature and velocity values at all grid points are first evaluated by solving eqs 2-5 at all of the grid points at any given instant. Once the initial location of the element and the velocity field are known,the fluid particle is tracked along its path to determine its new
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location. The position, 2,of a fluid element (Tragardh and Paulsson, 1985) is given by Thermal Processing of Foods. Food processes are designed on the basis of two criteria: (1) to maximize nutrient retention and (2) to achieve the desired sterility (minimizing microbial survival). A vulnerable factor (microbial spore or nutrient element) traveling with a fluid element is subjected to different temperatures as it is heated and cooled along its flow path in a heat exchanger. A term FOis defined as the time in minutes, necessary for processing, which gives the same final concentration of microbes or nutrients as when the temperature of the fluid is held at a reference temperature (T,,f) (Kumar and Bhattacharaya, 1991).
the basis of probability of survival of a factor (nutrient) for a canning process. This is modified for a continuous process as follows. Once the temperature history of fluid elements is known, the integrated lethality value, (Fo)s, at the end of the heating section is estimated from a mean concentration of a surviving vulnerable factor, MSF, as follows: MSF =
$Jv
(N/No)dV
@o)s = -Dreflog(MSF)
Equation 10 is used to obtain the FOvalues of any fluid element at any radial location if a fluid element is tracked along its flow path and its temperature history is known. Of interest to the food industry is the Fovalue at the slowest heating point in the tube, because this assures that every other element in the heat exchanger receives adequate thermal treatment. The microbial survival at the end of a process for the center of the tube is given by
[ N / N O ]=, 10-Fo",O'/Dld (11) The industry uses a 1 2 0 count (NINo = at a T,t'of 121.1 O C as a safe value from the standpoint of microbial safety (Clostridiumbotulinum) for low-acid foods (pH >4.6).
In the food industry, lethality achieved in the heating section is neglected, and only that achieved in the holding section (where the temperature remains constant) is accounted for. Due to a lack of accurate mathematical tools to validate the lethality caused by heating, it is still disregarded by the regulatory agencies. But, it is beyond doubt (Simpson and Williams, 1974; Kumar and Bhattacharaya, 1991) that the thermal treatment received in a heater affects the vulnerable factors (microbes or nutrients). In continuous processing, eq 11 has been used under the safe assumptions that (1) fluid elements follow streamlines, and so the coldest point always remains at the center of the tube, and (2) fluid velocity at different radial locations remains constant along the length of the heat exchanger (because of constant thermophysical properties). If the fluid elementsdo not follow streamlines, such an assumption causes overdesigning and, hence, overprocessing. Therefore, to evaluate FO values accurately, temperatures at all spatial locations in the heating section should be obtained, and then eq 9 should be used to track the particle to obtain the thermal histories and finally eq 10 (with accurate Z values) used to evaluate cumulative Fo. One method of studying process effectiveness is based on the mass average sterilizing value, (PO), obtained as
Eo =
(14)
Equation 14 is useful in terms of determining both microbial destruction and nutrient retention over the entire volume. Thermal Time Distribution. To obtain distributions of thermal times, temperatures andFovalueswere obtained for fluid elements at different locations along the radius of the tube. Equation 9 is used to track the particles and eq 10 to evaluateFovaluesusing C.botulinum as the target microorganism. Equation 14, being an integrated d u e , will result in larger numbers and so is not used in process design; it is more useful in describing the nutritional quality. From the standpoint of the center-point destruction of microbes, eq 10 is used for process design. The distributions of temperature were obtained in a manner similar to that of residence time distributions (RTDs) by evaluating the E curve (Levenspiel, 1989): E m = -(dQ/Q) (15) dT In a similar manner, TTDs were obtained as
Here, d Q is the fraction of liquid having a lethality value between FO and FO + dFo. A check was made to ensure that the area under the curve was always equal to 1.0 (i.e., J-o"E(F0)d F 0 = 1.0). The integrated curves, or the F curves, for temperature and FOdistributions were obtained, respectively, as
Solution The governing equations were written in their nondimensional form using the following dimensionless parameters:
JRF0(L,r)u(L,r)r dr (12)
J:u(L,r) r dr POgives an average value of sterilization at the outlet based on the final POvalues (of fluid elements) at the exit of the tube (and thus does not account for thermal histories of individual elements). - Although not useful in process design calculations, Fo does provide an estimate of overprocessing. Hayakawa (1978) has defined a bulk lethality value on
The finite difference formulation of the nondimensional equations is shown in the Appendix. The coupled continuity, momentum, and energy equations were solved explicitly using the Dufort-Frankel finite difference method (DuFort and Frankel, 1953). It has been used successfully in simulating heat transfer to non-Newtonian fluids (Joshi and Bergles, 1981). The Dufort-Frankel method is more stable than the ordinary second-order central difference method for the
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40
Simulation
so20
d
233
-
$ 3
x=o 0 1o(L
500
100
5WO
10W
0.2
0
lw00
0.6
04
Figure 1. Comparison of heat-transfer simulation results for laminar isothermal flow of pseudoplastic fluid with’ eq 23 for constant n and p. Table 1. Parameters Used in Simulations value parameter 0.07746 m/s initial average velocity of fluid initial u component velocity 0.0 m/s initial temperature of fluid 70.0 “C radius of tube and corresponding 2.547 X 10-2 and 3.821 X flow rates 10-2 m; 1.579 X 10-4 and 3.231 X 1 P m*/s 125 and 130 OC wall temperature
explicit scheme. Here, the unknown variable at the i + 15 level is written in terms of known variables at the previous two streamwise (axial) steps, namely, i and i - 1. uij+l+ uij-1- ui+lj
yij=
08
I
C(= 5)
NQ.
- ui-lj
A?
Figure 2. Temperature profile along the length of the heat exchanger for a wall temperature T, = 125 OC and initial fluid temperature 2’0 = 70 OC. Plot of dimensionless temperature (8) versus dimensionlessradius (I)for different dimensionless axial distances ( x = X/NR&+).
1.2
-
1-
- 0.8
s
o’61 0.4
(20)
In a similar fashion, the equations of momentum, ener , and continuity are written explicitly and solved for t e unknown values at streamwise step, i + 1, in terms of the previous streamwise steps. Once all variables at step i + 1 are known, they can be utilized to evaluate unknowns at step i + 2. At the inlet of the tube ( x = 01,information is known at only one station, i = 1. Therefore, for the first few steps, a standard explicit mehod that requires information only from the previous step is used, and then the DufortFrankel method takes over. A nonuniform grid is chosen to account for the large variations in velocity and temperature gradients near the wall. Stability and consistency requirements were checked at each step. A radial grid spacing with 31 nodes gave good results. Verification of Model. Christiansen and Craig (1962) have developed an integronumerical approach for solving the heat transfer to non-Newtonianfluids. Metzner et al. (1957) and Christiansen et al. (1966) obtained correlations to explain radial variation of fluid viscosity for a forced convection problem and presented their results as
Y
The numerical model is compared with their results for constant property. Results are presented in Figure 1as N N u versus Nh. As can be seen, the correlations are good. The maximum deviation is 6%, which is less than the mean deviation of 8% between the experimental and simulation results of Christiansen et al. (1966).
Results and Discussion The rheological data in the high-temperature range (7Ck 110“C)were obtained from Bhamidipati and Singh (1990). These results were extrapolated and used in the present
065
07
075
08
085
09
095
1
e(=H)
Figure 3. Temperature distributions (E(8))as a function of dimensionless temperature (e) at various dimensionless axial distances (x = z / N R Y , ) .
study (125 and 130 “C). Table 1 shows the values of different parameters used in the simulation. The laminar flow assumption is valid ( N G= ~ 45.33 for a temperature of 125 “C and D = 0.0254 m). The length of the heat exchanger was based on the amount of lethality required at the tube center at the end of processing. Process evaluation was carried out on the basis of the destruction of C.botulinum. A coldest point lethal value of 2.45 min was chosen for the process time (Simpson and Williams, 1974). The length of the heating tube required for this was 3.99 m. A fully developed velocity profile exists at the entrance of the tube. As the fluid enters the tube, the velocity profile undergoes a change because of the sudden change in viscosity. This exists until the entrance length is reached, and then the fluid attainsa fully developed velocity profile with no radial velocity component. Therefore, the application of eq 10 at the entrance region involves the determination of the position of the fluid element by eq 9. After the entrance length, the fluid element followed the streamlines to the end of the tube, and so evaluation of TTDs was made easier. Figure 2 is a plot of the dimensionlesstemperature profile for different positions along the length of the tube. The plots are shown for the dimensionless distances, x,of 0.0 (0.0 m), 0.7 (0.8 m), 1.5 (1.6 m), 2.2 (2.4 m), 2.9 (3.2 m), and 3.6 (3.99 m).
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No. 2
1 o.oa 1
--
0.08
r4-
4'
2i
0.06
0.04
0.02
0 0.001
0.01
0.1
LO
1
1000
100
F.(min)
20
40
60
80
100
120
100
140
180
F.(min)
Figure 5. Plot of Fcurves of Fo values for various dimensionless axial distances (x = r/NRdvpt).
0
10
20
30
40
10
1w)
Fob")
Figure 4. Comparison of thermal time distributions at different dimensionelssaxialdistances (x= z / N ~ R )Plot . of E(F0)versus FO.
0
1
50
60
70
80
90
100
Fdmm)
Figure 6. Comparison of diiferent expressions for lethality.TTD plot at x = 1.5. Figure 3 shows the distribution of temperature, as obtained by eq 15,at different lengths. The distributions shift to the right with increasing length. For x = 3.6,the distribution is close to a vertical line, showing that the fluid is close to 125 OC. As x increases, the expected value (E(8))of the distribution increases. The curve for x = 3.6 has been cut off at E(6) = 1.4 for ease of comparison. However, the actual value of E(8)is 2.07. For large x , the distribution is a straight line with zero width [E(!!'') -3, showingthat the entire fluid attains the wall temperature.
-
Figure 7. Effect of changing tube wall temperature on TTD. Plot of E(Fo)versus FO for two wall temperatures achieving a center-pointFOof 2.5 min for a tube diameter of 2.647 X 10-2 m. Corresponding heat exchanger lengths are shown in brackets. Figure 4 is a plot of TTDs (as obtained by eq 16)along the length of the heat exchanger. While the range of FO values, for x = 1.5,is 0.004-60min, at x = 3.6 it is 2.51-165 min. The TTDs shift to the right along the length of the tube, showing an incrase in variance (spread) and average values of the distributions or a higher destruction of vulnerable factors. This is because, initially, most of the fluid is at about the same temperature, and with increasing length, the fluid near the wall is exposed to higher temperatures for a longer time (and so, higher FOvalues) than the fluid near the center of the tube, and hence there is a larger variance in the TTD. Furthermore, there is an increase in the FOvalues with increasing tube length, which is expected. The fluid near the wall is at a higher temperature and has a longer residence time and, thus, higher Fo values. The area under the curve between any two Fo values in Figure 4represents the volume fraction of liquid having an FOvalue in that range (Datta, 1993). This is shown in Figure 5 as a plot of the F curve of the FOvalue distribution as obtained by eq 18. For x = 0.7,only 20% of the fluid has an FOvalue of 2.45 min or more. At x = 3.6, the entire fluid has an FOvalue higher than 2.45 min. While the lowest value (of an F curve for microbial destruction) is important from a sterilization viewpoint, the entire curve (of an F curve for nutrient destruction) is significant for evaluating the destruction of nutrients. Although Figures 6-9 are plotted with reference to microbial destruction, nutrient destruction (withdifferent D values and 2 values) follows similar plots and is also described. Figure 6 is a plot of a TTD curve at x = 1.5 to compare eqs 10, 12,and 14 for evaluating lethalities. While the center-point lethality, is important from a regulatory standpoint, (F0)s(of a nutrient destruction TTD curve) is important frqm a processors viewpoint. (Fo)sgives a lower value than FOand should be used in the calculation of process effectiveness(nutrient destruction). Figure 7 is a plot of lethality distribution (TTD) for two wall temperatures, 125 and 130 OC,and at corresponding heat exchanger lengths that achieve a center-pointlethality of 2.45 min. Increasing the temperature from 125 to 130 O C (a 4% increase) caused a decrease in length of 11.2% (from 3.99to 3.54m). At higher temperatures, the variance (spread) and the average of the distribution are increased, suggesting a higher destruction of vulnerable factors. Although this results in a higher destruction of microorganisms, it also results in higher destruction of nutrients. Therefore, it is better (from a nutrient retention stand-
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10
1
track an element of fluid along the length of the heat exchanger to obtain the thermal history of the fluid elements. Expressions for TTDs were developed an expressed as E curves and F curves. The analysis also showed that the heating section effectively contributes to the fluid lethality (Kumar and Bhattacharaya, 1991; Simpson and Williams, 1974)and has to be accounted for in process design. The role of TTD as a tool to evaluate process effectiveness has been described by comparing processes with different wall temperatures and tube radii for vulnerable factor destruction. A larger spread in TTD causes higher destruction, as is the case for high wall temperatures and large tube radii. Process design based on TTD evaluation gives a better understanding of the system.
100
F.(min)
Notation
Figure 8. Comparison of F curves for two wall temperatures achieving a center-pointFOof 2.45 min for a tube diameter of 2.547
X
area (m') specific heat (J/kg.K) diameter of the tube (m) D value decimal reduction time (min) activation energy for microbial destruction (kJ/kg.mol) F value (min) mass average sterilizing value (min) integrated lethality value (min) average heat-transer coefficient (W/m2*K) thermal conductivity (W/mK) consistency index (Pes") mass flow rate (kg/s) flow behavior index survival of microorganisms generalized Reynolds number
lo-*m.
0.0s
1
0'm51 0.02
0.015 0.01
tc 1
R = 5.821 x IO-' (t= 7.56 m)
I
/
m
\\
10
1M)
[(pu2-W)/(2"-3K(3n+ l/n)")l
F.(min)
PI
Figure 9. Effect of changing tube diameter on TTD. Plot of E(F0)versus FOfor two tube diameters achieving a center-point FOof 2.45 min for a wall temperature of 125 O C . Corresponding heat exchanger lengths are shown in brackets.
point) to have low wall temperatures. Figure 8 shows the corresponding F curves showing the cumulative distribution of FOvalues, making it easier to directly compare the two processes. Figure 9 shows the effect of the diameter of the tube on TTD. TTDs are plotted for two radii (with the same initial fluid velocity) that achieve a center-point Fa value of 2.45 min. Increasing the radius from 0.02547 to 0.03821 m (a 150% increase) caused an increase in length of 84 % (from 3.99 to 7.34 m). A smaller radius tube gives a smaller spread and a smaller average value, showing that the smaller the radius of the tube, the higher the nutrient retention. Theoretically, the tube with the smallest radius would be the most efficient, but other factors like fouling (which causes clogging of the tube) and economic considerations limit the size of the smallest tube. These results are in agreement with those of Kumar and Bhattacharaya (1991)and Simpson and Williams (1974).
Summary and Conclusions A finite difference simulation to study heat transfer to a non-Newtonian fluid with variable thermophysical properties heated in a tubular heat exchanger has been developed and applied to evaluate the processing of liquid food in terms of microbial and nutrient destruction on the basis of the thermal time distributions (TTDs). The temperature and velocity profiles were determined and used in conjunction with thermal process equations to
NRe
P 4 dQ
Q
r R
RG t
T U
8 u
V X
x z
Graetz number ( h c kL) (evaluatedat arithmetic mean ulk temperature) Nusselt number (ItDlk) (evaluated at arithmetic mean bulk temperature) Prandtl number ( p c p / k )(evaluated at arithmetic mean bulk temperature) Reynolds number (puD/p) pressure (Pa) wall heat flux (W/m2) Cij(B*rj) (0.5)(rj+i- rj-1)(0.5)(xi+l- xi-1) (m3) total volume of liquid (ma) radial coordinate radius of the tube (m) gas constant (J/kgmol.K) time (8) temperature ( O C ) axial velocity component (m/s) Eulerian velocity (m/s) radial velocity component (m/s) volumetric flow rate (m3/s) axial coordinate position of fluid element (m) Z value for temperature dependence of reaction
Greek Characters (Y
? 6 Ma
F
thermal diffusivity (k/pcp) shear rate (5-1) correction factor [ ( 3 n + 1 ) / ( 3 n- 113 apparent viscosity (Pad dimensionless radius ( r / R )
Bbttwt~W.Prog., 1094, Vd. 10, No. 2
dimensionless axial velocity (ula) density (kg/ms) shear stress (Pa) dimensionless temperature [(Ti- To)/CTw - TO)] dimensionless axial distance
X
(XlNdPr)
Subscripts b
bulk coldest point grid point in axial direction grid point in radial direction initial value reference value tube wall tube axis
C 1
i 0
ref W 2
with or without particles in tube flow. J. Food Proc. Eng. 1990,12,275-293.
Bud, R.; Stewart,W.;Lighffmt, E. Transport Phenomena;Wiley and Sons: New York, 1960. Choi, Y.; Okos, M. R. Thermal properties of tomato juice concentrates. Trans. ASAE 1983,17 (l),305-311. Christiansen, E. B.; Craig, S. E. Heat transfer to pseudoplastic fluids in laminar flow. AIChE J. 1962, 8, 154-160. Chriitiansen,E. B.; Jensen,G.; Tao, F.Laminar flow heattransfer. AIChE J. 1966,12,1196-1202. Datta, A. K. Mathematical modeling of biochemical changes during processing of liquid foods and solutions. Biotechnol. Prog. 1991, 7,397-402.
Datta, A. K.Error estimatesfor approximatekinetic parameters used in food literature. J. Food Eng. 1993,18, 181-199. Datta, A. K.; Liu, J. Thermal time distributions for microwave and conventional heating of food. Trans. Inst. Chem. Eng. 1992, 7 0 , 8 3 4 .
E. C.; Frankel, S. P. Stability conditions in numerical treatment of parabolic differential equations. Math. Tables
Equations Ax+ = xi+1-xi Ax-= xi - xi-1 AY+ = Yj+l - Yj PY+ = Yj - Yj-1
Mort,
Aids Comput. 1953, 7,135-153.
Acknowledgment This research was supported in part by the NRI competitive grants program of the United States Department of Agriculture under award number 92-37500-8016. Appendix Nondimensionalized Governing Equations. continuity equation
momentum equation
Ellborg, A. Thermal time distribution in a tube heat exchanger. A study with application to sterilizationof liquid foods. PbD. Dissertation, Lund University, Lund, Sweden, 1989. Hayakawa, K. A critical review of mathematical procedures for determiningproperheat eterilizationproce. Food Technol. 1978,3, 59-65. Joehi, S. D.; Berglea, A E. Heat transfer to laminar in-tube flow of pseudoplastic fluids. AIChE J. 1981,27 (5), 872-875. Kumar, A.; Bhattacharaya, M. Analysis of aseptic processing of
a non-Newtonian liquid food in a tubular heat exchanger. Chem. Eng. Commun. 1991,103,27-51.
Levenspiel, 0. Chemical Reaction Engineering, 2nd ed.; John Wiley and Sons, Inc.: New York, 1989. Metzner, A.; Vaughn, R.; Houghton, G. Heat transfer to nonNewtonian fluids. AIChE J. 1957,3,92-100. Mwangi, J. M.; Datta, A. K.; Rizvi, S. S. H. Heat transfer in asepticprocessingofparticulatafoods.In Advances in Aseptic Processing Technologies; Singh, R. K., Nelson, P. E., Eds.; Elsevier: London, 1992; pp 73-102. Nauman, E. B. Nonisothermal reactors. Chem. Eng. Sci. 1977, 32,359-367.
Oliver, D. R.; Jenson, V. G. Heat transfer to pseudoplastic fluids in laminar flow in horizontaltubes. Chem. Eng. Sci. 1964,19, 115-121.
Paulsson, B. 0.;Tragardh, C. Calculation of sterilizationeffect distribution in heat exchangers in turbulent flow. In Engineering and Food; McKenna, B. M., Ed.; Elsevier: London, 1984, Vol. 1, pp 151-164. Pereira, E. C.; Bhattacharya, M.;Morey, R. V. Modeling heat transfer to non-Newtonian fluids in a double tube heat exchanger. Trans. ASAE 1989,32 (l),256-262. Perry, R. H.; Chilton, C. H. Chemical Engineer's Handbook, 5tb ed.; McGraw-Hill Book Co.: New York, 1973. Rodriguez-luna, G.; Segurajauregui, J. S.; Torres, J.; Brtio, E. Heat transfer to non-Newtonian fluid foods under laminar flow conditions in horizontal tubes. J. Food Sci. 1987, 52, 97-79.
Schlicting,H. Boundary Layer Theory;McGraw-Hik New York, 1955.
Simpson,S. G.; Williams, M. C. An analysis of high temperature/ short timesterilizationduringlaminar flow. J.Food Sci. 1974, 39,1047-1054.
Tragardh, C.; Paulsson, B. Heat transfer and sterilization in continuous flow heat exchangers. In Developments in Food Preservation, 3; Thome, S., Ed.; Elsevier: London, 1986; pp global continuity equation
245-272.
Accepted October 12, 1993.6
Literature Cited Bhamidipati, S.; Singh, R. K. Flow behavior of tomato sauce
6
Abstract published in Advance ACS Abstracts, December 16,
1993.
