Biotechnol. Prog. 1990, 6, 292-306
292
Design of Continuous Sterilization Systems for Fermentation Media Containing Suspended Solids Piero M. Armenante* and Mark A. Leskowicz Department of Chemical Engineering, Chemistry, and Environmental Science, New Jersey Institute of Technology, Newark, New Jersey 07102
A mathematical model was developed t o analyze the performance of a continuous sterilizer fed with a liquid fermentation medium containing suspended solids. This type of medium is commonly used in the industrial practice of a number of fermentation processes. T h e unsteady-state energy balance equations for the liquid and the suspended solids passing through each component of the system were solved analytically. A computer model was then used to calculate the temperature profiles in the liquid and in the solid particles as they move through the sterilizer, as well as the level of sterility achieved under different operating conditions. It appears that the level of sterility is always lower when solids are present in the system, as one may anticipate. T h e temperature gradients produced in the solids as a result of the unsteady-state heat transfer occurring as the media moves through the heat exchangers in the system result in lower temperatures in the solid cores. This reduces the effectiveness of the process by protecting the microorganisms against thermal death. In some cases it was found t h a t the level of sterility was so low that operating under such conditions would fail t o produce a sterile product. T h e model was also used t o assess the sensitivity of the sterilization process to changes in the most significant design and operating parameters for the case in which solids are present in the medium. The sterilization temperature, steam temperature, heat transfer coefficients in the heat exchangers, particle size, and residence time in the sterilizer holding section appear to be the most important of these parameters. The model obtained in this work can be used t o design continuous sterilizers processing slurry media or to evaluate the performance of existing sterilizers.
1. Introduction Sterilization can be defined as the process of inactivating undesirable contaminating microorganisms that would otherwise interfere with a subsequent process or lead to the devaluation of a product. In many biological processes, such as those utilized in the pharmaceutical industry, contaminating organisms can severely affect the fermentation of the desired product, such as an antibiotic. Heat treatment processing, or thermal sterilization, is the most widely practiced method of sterilization and is used extensively in the food, drug, and fermentation industries. Industrial thermal sterilization can be carried out either batchwise or continuously. Because of the problems associated with batch sterilization (such as long turnaround time for the fermenters and extensive labor requirements), the latter process has gained increased popularity. In addition, the continuous process has the advantage of minimizing the degradation of nutrients in the medium because of the lower activation energies associated with the degradation reactions of the nutrients vs that for the thermal death of the microorganisms (Lin, 1975; Feliciotti and Esselen, 1957). In continuous sterilization, the fermenter feed is sterilized in line as it is being charged to the presterilized fermenter vessel. Therefoie, the temperature of the medium is raised and lowered quickly, with the holding time in between also being very short but at a much higher
* To whom correspondence should be addressed.
temperature than those considered in batch sterilization processes (Deindoerfer and Humphrey, 1959b). Because of the short residence time of continuous sterilization, variations of only a few degrees in any of the sections of the sterilizer (especially the holding section, which operates a t the maximum temperature) can result in significant differences in the remaining viable fraction of the organism of many orders of magnitude. This has an even greater impact on sterility if suspended solids are present in the medium, since the heat transfer through the solid particles proceeds a t a slower rate than that in the bulk of the fluid. This results in a different temperature profile vs time inside the particle and, ultimately, in a different remaining viable fraction of microorganisms (Bailey and Ollis, 1986, p 589). This effect is more significant in continuous sterilization rather than batch processes because of the much shorter sterilization time required in the former case. In many situations, thermal equilibrium inside the particle may be far from being reached even when the particles are deep inside the holding section. This can have a severe impact on the final sterility level of the medium.
2. Objective of This Work In the industrial practice, large-scale industrial continuous sterilization processes can be under- or overdesigned precisely because of the so far unaccountable effect on sterility of the solid particles in the medium. In spite of the significant amount of previous work on continuous sterilization (Deindoerfer, 1957; Deindoerfer
8756-7938/90/3006-0292$02.50/0 0 1990 American Chemical Society and American Institute of Chemical Engineers
293
Biotechnol. Prog., 1990, Vol. 6, No. 4
and Humphrey, 1959a,b; Blakebrough, 1968; Lin, 1975, 1979),the influence of the suspended solid particles present in typical industrial fermentation media on the continuous sterilization process has received only very limited attention. The earlier models proposed (de Ruyter and Brunet, 1973; Manson and Cullen, 1974) examined some of the main features of the sterilization process in the presence of solids but did not fully account for some important aspects of the process, such as the role of the particle-liquid heat transfer coefficient, the temperature distribution inside the particles, the effects of the design parameters (e.g., sterilization temperature, steam temperature, and exchanger heat transfer coefficients), or the characteristics of specific types of heat exchangers (such as spiral heat exchangers) commonly employed in fermentation media sterilizers. Therefore, the objective of this work is t o model continuous sterilization processes in which the presence of solid particles in the fermentation medium is accounted for and to obtain suitable design equations for the sterilizer.
3. Previous Work on Sterilization of Homogeneous Liquids Previous work on continuous sterilizers has been mainly focused on sterilization of homogeneous fluids. Several models for different types of homogeneous fluid sterilization processes can be found in the literature (Deindoerfer and Humphrey, 1959a,b; Lin, 1975,1979; Ashley and Mooyman, 1981). In all these models, the rate of thermal death for the contaminating organisms is assumed to follow first-order kinetics where the specific reaction kinetic rate, Kd, is a function of the Arrhenius constant KdO, the activation energy E d (both independent of temperature), and the sterilization temperature T , according to the equation
Table I. Temperature-Time Profiles in Continuous Sterilization Systems electric heating Tb(t)= To +
+ PbCpbl
steam heating
countercurrent heat exchange
where a = and Fb’,
F@bCpb ~
Fb)~blC~bl pb’, and cpb’ refer to the second fluid
countercurrent heat exchange (with a = 1) Tb(t)= To + ( T i- T J t ‘Pbcpb1
For a generic sterilization process, a balance for the final number of surviving microorganisms per unit volume of fluid, Nf, can be written as (Bailey and Ollis, 1986, pp 587-588)
temperature homogeneous fluid systems, such as the holding section of the sterilizer, before the mathematics becomes too involved and their application of limited usefulness. Furthermore, as the flow becomes increasingly turbulent, the ideal plug flow conditions are more closely approached than if the flow is laminar. Although it is, in general, difficult to obtain an ideal plug flow in a complex system such as a sterilizer, in many largescale pieces of equipment the Peclet number, typically used to describe the degree of axial dispersion in a reactor (Brenner, 1962; Levenspiel, 1958, 1972, p 284), and the Reynolds number are so large that the flow conditions may be regarded as approximately plug flow (Aiba et al., 1973, p 265). For this reason, and also to simplify the mathematics of the proposed model, these additional refinements of the theory will not be considered in the remainder of this work.
In this expression the temperature is a function of time since it changes during the heating and cooling periods. The type of function depends on the design of the sterilizer and on the type of heat source used (Bailey and OKs, 1986, p 588). Deindoerfer and Humphrey (1959a,b) have associated with each of these methods of heat transfer a time-temperature profile that can be derived from simple energy balances. When these profiles are substituted into the right side of eq 2, the logarithmic ratio of final to initial surviving microorganism concentrations, In (Nf/No),can be obtained. Equations 1 and 2 can be used to yield an analytical expression for In (Nf/No) also for the case of continuous sterilization of liquid media, provided that ideal plug flow through the sterilizer is assumed (Deindoerfer and Humphrey, 1959b). More complex sterilization models incorporating the effect of axial and backward diffusion on reactor design (Levenspiel, 1972, pp 272-290; Charm, 1971, pp 214-217) have also been proposed in the past (Aiba and Sonoyama, 1965; Aiba et al., 1973; Lin, 1975, 1976, 1979). These models can be of practical validity mainly for constant-
4. Approach Used in This Work In this work the temperature profiles for both the fluid and the solid spheres were obtained as functions of time and, in the latter case, also of the position within the sphere. The remaining viable fraction of bacteria was then calculated by substituting these profiles into the temperature-dependent death rate equation. The equations describing the temperature profile of the fermentation fluid during continuous sterilization have been considered for four types of heat transfer process: electrical heat, steam heat, countercurrent fluid heat exchange (general), and countercurrent exchange (preheating or heat recovery process) where the heating fluid has the same flow rate and physical properties as the cooling fluid. These expressions are reported in Table I. However, only the second and fourth expressions were actually used in our model since (with the exception of the cooler) these are the types of heat transfer processes most commonly found in typical continuous sterilizers. These equations were used to obtain the initial and boundary conditions required for the integration of the energy balance equations for the solid particles, which then yielded the temperature profiles in the particles. These analytical expressions were then used to numerically calculate the values for the temperature of the fluid
204
Biotechnol. frog., 1990, Vol. 6, No. 4 Coollng Water
Drain
Exchanger
Tster
1
Recovery Exchanger i---:
Ho 1 ding Section Tster 0
Raw
Medium
Figure 1. Schematic diagram of continuoussterilization system.
and temperature profiles in the particles vs time. Finally, the temperature profiles for the fluid and the particles were used t o calculate t h e remaining viable fraction of microorganisms, Nf/No, and determine the sterilizer efficiency.
5. Derivation of the Temperature P r o f i l e in a Spherical Solid Particle Moving t h r o u g h a C o n t i n u o u s Sterilizer The sterilizer was modeled as a series of heat exchangers and a holding tube, as shown in Figure 1. The fermentation medium first passes through a heat recovery section where its temperature is increased by cooling the sterile medium leaving the holding tube. The preheated medium is then heated up to its sterilization temperature in an indirect steam heater and kept at this temperature as it flows in the holding section where sterilization takes place. The sterile medium is precooled in the heat recovery exchanger and cooled to its final temperature in a cooler using cooling water. This medium is then fed to the fermenter. In the model, each exchanger was analyzed independently. However, the temperature of the fluid leaving a specific heat exchanger and the temperature profiles inside the particles moving with it were taken as the initial temperatures for the medium entering the next exchanger in line. A number of assumptions were made in the development of the model. The most important are as follows: (a) The feed to be sterilized is made up of a homogeneous liquid containing solid particles suspended in it by the turbulent action of the fluid. (b) The solid particles are spherical and have a uniform radius R. In addition, they have uniform density, heat capacity, and thermal conductivity. This assumption is reasonable for most feeds containing one type of solid impurities. If the solids are characterized by a size distribution curve, the size of the largest solid particle can be used in the calculation. This will produce a conservative design of the sterilizer. (c) The fluid moves
in piston flow through the exchanger train. (d) The solid particles move along with the liquid medium in the continuous sterilizer. This assumption is especially valid if the fluid moves in piston flow (following the previous assumption) and if assumption e below is valid. (e) The particles are considered to be neutrally buoyant or nearly buoyant so that sedimentation in the heat exchange tubes or channels does not occur. This is the case for most organic material commonly found in sterilization feeds. Even in those cases in which the particles are heavier than the liquid feed, one can still assume that the particles are fully entrained in the liquid, provided that the particles are small and the fluid highly turbulent. If the particles do not closely follow the liquid flow, their residence time in the sterilizer will increase and the resulting design equations will produce conservative but still valid results. (f) The amount of particles in the fluid is small enough so that the particles play a negligible role in the energy balance of the fluid (i.e., the temperature of the fluid is not affected by the presence of the particles). Because of the small clearance of the heat exchangers typically used in fermentation broth sterilization, the maximum solid size is quite limited (at least in comparison to some applications in the food industry). This, in turn, implies that the temperature difference between the solid core and the surrounding liquid, although often significant for sterilization purposes, is not usually large enough to significantly affect the overall energy balance. Here it was estimated that even for the case of a 10% solid fraction the reduction of sterility in the medium will be minimal. (g) The initial temperature of the particles in the nonsterile raw feed to the sterilizer is identical with that of the liquid. In addition, in the model proposed here only the sterility produced by the residence of the slurry media in the heat recovery exchanger, heater exchanger, and holding section was considered, thus disregarding the possible contribution of those exchangers in which the medium is cooled. 5.1. Derivation of the Temperature Profile i n a Particle Moving i n the Heat Recovery Exchanger. The temperature of the liquid medium being heated up in the heat recovery exchanger can be obtained from an elementary differential energy balance along the direction in which the fluid is moving (Kern, 1950, pp 85-93) as (Deindoerfer and Humphrey, 1959b)
Tbr(t) = To + bi(T1’- T0)t (3) which is also reported in Table I, where b,’ = A,U,/ (S,p,C,&) is a function of the system. The time, t , is counted from the instant the fluid element enters the heat exchanger. For a double-pipe countercurrent heat exchanger, b,’ becomes (4)
A similar equation can be obtained for the case of spiral heat exchangers (one of the most common types of exchanger used in industrial continuous sterilizers), as shown below in greater detail. The subscripts in the above equations refer to the liquid bulk (b) in the heat recovery exchanger (r). The numerical value of TI’ depends on the percentage of heat recovered in the exchanger and is one of the design variables of the system.
295
Biotechnol. Prog., 1990,Vol. 6, No. 4
recovery exchanger is
[
TO) Rr 2 b ( P a ) - [sin (PJ/Pnll cr-1~;
R
- Pa2 sin ( P n ) COS
1- ex.(
-5 t )
(PJI
1)
(10) with a=-
k,
(11)
PSCPS
and where the eigenvalues Pn come from the solution of the transcendental equation
p, cot p, Figure 2. Diagram of the spherical particle used in the energy balance and sterilization calculations.
A separate energy balance must also be written for the solid. With reference to Figure 2, an unsteady-state energy balance for an infinitesimally thin shell inside a solid sphere takes the form (Bird et al., 1960, p 318) (5)
+ L, - 1 = 0
(12)
The Biot number for the heat recovery exchanger, Lr, is defined as
3 L, = h k,
A simplified version of this solution for a less general case agrees with the solution found by Crank (1975, p 93) for a similar system. At the center of the sphere, equation 10 reduces to the indeterminate form 010. However, since
A first boundary condition at the sphere surface can be written as
indicating that the heat entering the surface of the solid sphere by conduction must equal the convective heat transferred to the sphere by the external fluid.
A second boundary condition at the center of the sphere is
one can easily show that the solution in this case becomes
T,(r=O,t)= To+ b,’(Tl’- To)t+
GI
2b,’(T1’-
(7) The initial condition is obtained by imposing that the initial temperature of the particles entering the heat recovery exchanger is the same as that of the liquid in which they are suspended. Following assumption g, this temperature is equal t o that of the surroundings, TO,which is a constant. Then
T,(r,t=O)= To
(8)
These boundary conditions can be made linear and homogeneous by using the appropriate variable substitution (Farlow, 1979, p 43), Le., Y,(r,t) = r[T,(r,t) - Tbr(t)l
5.2. Derivation of the Temperature Profile in a Particle Moving through the Steam Heater. From an energy balance on the cold side of a heat exchanger heated with condensing steam, the expression for the cold temperature of the fluid on the cold side (Kern, 1950, pp 85-93) is found to be (Deindoerfer and Humphrey, 1959b) Tbh(t) = Tw exP[-bh’(t - t,)l(Tl - Tw) (16) where bh’ (for the heater) is defined similarly to b,’ (for the heat recovery exchanger). For a double-pipe, countercurrent heat exchanger, bh’ can be expressed as
(9) uh bh’ = -
where Tbr(t) comes from eq 3. The resulting PDE can be solved by using the eigenfunction expansion method to express the nonhomogeneous term arising in the PDE after the substitution (Farlow, 1979, p 64). The final solution for the temperature inside a particle traveling in the heat
PbrhCpb
which is very similar to the corresponding expression for the case of a spiral exchanger heater, as shown below. The subscript (h) refers to the heater. In eq 16, the time is counted from the moment the fluid enters the sterilizer
296
BiOteChnOl. Prog., 1990, Vol. 6, No. 4
(i.e., from t h e moment i t enters the heat recovery exchanger), and t , is the time spent by the particle in the heat recovery exchanger, i.e., before the particles enter the heater. The temperature inside a particle flowing in the steam heater can be obtained from the integration of the equation
with boundary conditions
where
and initial condition Th(r,t=tr) = @,(r)
(211
The function M r ) represents the temperature distribution in the solid particles, as they emerge from the heat recovery exchanger, as a function of the radial position. This function is obtained by substituting the value t , (which is uniquely defined once the heat recovery fraction and the characteristics of the exchanger are specified), in eqs 10 and 15. In eq 21, the function @,(r)is taken to be the initial temperature distribution in the particles entering the heater. The solution to this PDE can be found by using the substitution
Tbr(t=tr)]sin
2
6,- dr
(30)
The eigenvalues Pn can be obtained from the solution of the equation
0, cot 0, + Lh - 1 = 0 (31) with the Biot number for the heater, Lh, defined as
For r
-
Lh
=
hhR k ,
0, eq 29 can be rewritten as
Th(r=O,t) = T ,
+ exp[-bh’(t - t,)](T,- T,) +
which results in the following equation with linear homogeneous boundary conditions
The eigenfunction expansion method can be used to express the second term on the right side of eq 23 as
+
-rb,’(Tl - T,) exp[-b{(t - t,)]= fll(t)Xl(r) fl,(t)X,(r) + ... + O,(t)X,(r) + ... (27) where the Xi(r) terms are the eigenvectors of the corresponding homogeneous Sturm-Liouville problem, which can be obtained by replacing eq 23 with the equation
The functions & ( t )are found by imposing the initial condition. Additional details on the method used can be found in Leskowicz (1989). The temperature profile inside the particles moving through the heater is given by
which can be obtained by using eq 14. 5.3. Derivation of the Temperature Profile in a Particle Moving through the Holding Section. The temperature of the fluid in the holding section of the sterilizer, Tster,is a constant given by Tskr = Tbh(t=th) (34) since the sterilization temperature Tshr must be equal to the temperature of the fluid at the exit of the heater (th is the sum of the residence times in the heat recovery exchanger and in the heater). The energy balance in the particles flowing through the holding section is given by
(35)
297
Biotechnol. Prog, 1990. Vol. 6, No. 4
with boundary conditions
which can be integrated with respect to time to give (37)
and initial condition Tq(r,t=th)= @h(r)
(38)
where @h(r)is the temperature distribution in a particle a t the exit of the heater. Equation 35 can be solved (Carslaw and Jaeger, 1959, p 233) to yield the final solution
(39)
where
The eigenvalues are calculated from
p, cot p,
+ L, - 1 = 0
(41)
with the Biot number for the holding section, L,, defined as
The corresponding temperature a t the center of the sphere is given by the equation
(43)
6. Derivation of the Sterilization Equation for Particles Moving through the Continuous Sterilizer With reference to Figure 2, the number of surviving microorganisms present in a thin shell of thickness Ar inside a spherical particle undergoing sterilization is An, = $,(r,t)4?rr2Ar
(44)
where +,(r,t) is the local concentration of microorganism a t a radius r inside the particle, expressed as the number of viable organisms per unit solid volume. Within the shell, an unsteady-state mass balance for the viable microorganisms can be written as dAn, --
dt
- -KdAns
(45)
Assuming that the inactivation process is described by a first-order death kinetic rate similar to that represented by eq 1,it is possible to substitute eq 44 into eq 45 to get
(47)
where +,&) is the local density of the microorganisms a t r in correspondence with the initial condition. This equation for a thin shell is analogous to eq 2 for sterilization of a homogeneous fluid. It should be noticed, however, that +,(r,t=tf) in eq 47 is a function not only of t f (as in eq 2) but also of the radial position inside the sphere. An expression for the ratio of the overall concentration of surviving microorganisms in the particle to initial concentration a t a generic time t f can be obtained by substituting eq 47 into eq 44, dividing by Ar, taking the limit for Ar 0, and integrating the resulting equation with respect to r. The final equation is
-
where the subscripts refers to the solids. In this equation, the temperature profiles inside the particle can be obtained, for each individual exchanger, from the equations derived in the previous sections. Rather than integrating eq 48 directly from t = 0 to the final time tf, it is more convenient t o consider one exchanger at a time and calculate separate integrals from 0 to t , for the heat recovery exchanger, from t , to th for the heater, and so on. This applies also to the calculation of $*. The function $,(r) calculated in correspondence with the exit of the particles from one particular exchanger can be used as the initial concentration distribution a t the entrance of the next exchanger in line. For the first exchanger of the series (heat recovery exchanger), the initial density distribution of microorganisms in the particles, $&, can be taken to be a constant equal to NO.
7. Determination of the Sterilization Parameters Used in the Model The use of the proposed model requires the knowledge of a number of parameters, namely sterilization capacity, types and dimensions of heat exchangers, heat recovery efficiency of the sterilizing system, heat transfer coefficients in the exchangers, physical properties of the liquid fermentation medium and of the solid particles, particleliquid medium heat transfer coefficient, thermal death kinetic constants for the microorganism, and required degree of sterility. The determination of these parameters is discussed below. 7.1. SterilizationCapacity. For media with potential for formation of scale, a typical requirement consists in assuming that the sterilizer must be able to feed from two to three industrial-size fermenters (each 80-120 m3 in volume) before undergoing the periodic cleaning-inplace cycle. Other systems may require less frequent cleaning. In our case, we selected a conventional industrial sterilization system having a sterilization capacity of 40
298
Biotechnol. Prog., 1990, Vol. 6, No. 4
Table 11. Characteristics of the Continuous Sterilizer Used in the Numerical Simulation recovery exchanger type volumetric flow rate (m3/h) linear velocity in exchanger (m/s) clearance between plates (mm) inlet fluid temperature ("C) heat recovery efficiency ('36 ) steam heater type volumetric flow rate (m3/h) linear velocity in exchanger (m/s) clearance between plates (mm) steam temperature ("C) holding section type linear velocity (m/s) diameter (mm)
given by
spiral
since both spirals contribute to the generation of the exchange area. In this equation the approximation was made that the spirals begin a t the origin, although this is not obviously the case in a real exchanger. Because of the presence of the square in the radius, the error introduced this way is minimal (typically less than 1%). Knowing the flow rate of the fluid, it is then possible to write the equations
40.0
0.93 12.0 25.0 0-63 spiral 40.0
0.93 12.0
145-200 Pipe
0.93
and
123.0
m3/h as the basis for our calculations. The main characteristics of such system are given in Table 11. 7.2. Types and Dimensions of Heat Exchangers. Different types of heat exchangers are commonly used in continuous sterilizers. Concentric double-pipe heat exchangers offer a number of advantages (e.g., high tolerance to flow rate variations and ease of cleaning) but are not typically used in large-scale industrial applications because of their cost. Shell-and-tube heat exchangers are uncommon because of the difficulties associated with cleaning them and maintaining sterility. Instead, plate heat exchangers and spiral heat exchangers are more typical (Vogel, 1983, pp 95-99). In large-scale applications, spiral exchangers are probably the most common alternative. Unlike the plate exchanger types, they offer the advantage of having uniform cross-sectional areas for both fluid flows, therefore minimizing the possibility of solid accumulation in dead spaces. These exchangers are manufactured by rolling thin steel sheets, kept at a close distance from each other by stubs, so that spiraling channels are formed (Lamb, 1962; Tangri and Jayaraman, 1962). The fluids move in these channels in near countercurrent paths. The use of spiral exchangers is especially common for media characterized by medium or low suspended solid concentration (Vogel, 1983, p 97). Therefore, spiral heat exchangers were used in this work to analyze the performance of a typical industrial continuous sterilizer. In the numerical model shown below, the clearance, 6, in each of the flow channels for the hot and the cold fluids was arbitrarily fixed at 1 2 mm. Once 6 is known, it is possible to calculate the total length of the spiral paths described by the fluid in their motion through the exchanger and the dimensions of the exchanger for any given duty, provided that the flow rates and the heat exchange area are also known. Mathematically, the length of an Archimedes spiral starting at the origin is given by the equation (49) where p' is the distance from the origin of the final point on the spiral and 6' is a constant equal to the radial distance between two points separated by a single turn (2a rad). Since in a spiral heat exchanger two concentric spirals are present (each spiral winding a t a distance 6, equal to the clearance, from the other), the spiral path described by each fluid will be approximately given by the equation (50) If we, is the width of the exchanger and pex its maximum radius, then the total heat exchange area available will be
(53) which can be used to calculate the width and the radius of the spiral heat exchanger, respectively, as a function of the fluid velocity and the clearance between the spirals. In these equations the heat transfer area can be determined from the energy balance for each exchanger, as shown below. The velocity should be chosen in such a way as to produce a turbulent regime in the fluid flow inside the exchangers. 7.3. Heat Recovery Efficiency of the Sterilizing System. The heat recovery efficiency for the system, u, is defined here as the amount of heat recovered in the heat recovery exchanger out of the total amount of heat required to raise the temperature of the feed from room temperature to sterilization temperature. In this work, the efficiency was assumed to be within the range 0-70 74 and was kept as one of the variable design parameters during the calculations. With reference to Figure 1, an energy balance around the heat recovery exchanger gives (54)
It is common practice to assume that the mean temperature difference for the design of spiral exchangers is equal to the logarithmic mean temperature difference (LMTD) used in countercurrent heat exchangers (Rohsenow et al., 1985, pp 4-109; Hargis et al., 1967; Baird et al., 1957). In this case, since the fluids are identical in flow rates and properties, the heat exchange area for the heat recovery exchanger can be expressed as (55) The residence time in the heat recovery exchanger can be obtained by recalling the definition of residence time for fluid moving in plug flow, i.e., At, = t, =
1 "I
By using eqs 50-52, this equation can be rewritten as
4%
At, = -
(57) 2F where the subscript r has been used, again, to indicate the heat recovery exchanger.
290
Biotechnol. Prog., 1990, Vol. 6, No. 4
Table 111. Physical Properties of the Medium and Solid Particles Used in the Model Simulation (T= 93 "C) liquid medium solid particle P (kg/ms) 962 790 C p [ J / ( k "C)l 4229 1880 k [W/(m "C)1 0.672 0.445 0.307 P (CP)
Similar equations can be written for the heater (subscript h) :
(59)
7.4. Heat Transfer Coefficients in Heat Exchangers. The heat transfer coefficients were kept as variable design parameters in the model. The singleside heat transfer coefficients for the medium on either side of the heat recovery spiral exchanger were calculated by use of the equation proposed by Hargis et al. (1968) (also reported by Rohsenow et al., 1985, pp 4-log), where the equivalent hydraulic diameter was taken to be equal to twice the spiral plate clearance 6, since the width of the exchanger, wex,is typically much larger than the clearance in spiral exchangers. The resulting value was found to be hrb = 5020 W/(m2 "C). However, a more conservativevalue of 3750 W/(m2 "C) was used in most of the calculations shown below. This latter value is equal to that which can be obtained by using equations for flow in circular pipes (Kern, 1950, pp 103-104). Fouling was accounted for by a fouling factor, ff, equal to 0.0008 m2 "C/W for each side. Although fouling can occasionally be severe during a single run, periodic cleaning-in-place operations are typically carried out after a number of sterilization runs have taken place. The overall heat transfer coefficient in the heat recovery exchanger, Uf, was calculated from
which yielded a value of 467 W / ( m 2 "C). In most calculations this figure was rounded to a more conservative value of 400 W/(m2 "C). The overall heat transfer coefficient in the heater was also considered to be a design parameter of the model. Following an approach similar to that outlined for the coefficient in the recovery exchanger, the average heat transfer coefficient for the heater, u h (in which the heat transfer resistance of the steam side can be considered to be negligible), was taken to be 1000 W/(m2 "C). These agree well with the typical values values for U, and reported in the literature (Perry and Green, 1986,pp 10-45) for comparable systems. 7.5. Physical Properties of the Liquid Fermentation Medium and of Solid Particles. The physical properties of the fermentation medium and the solid particles, as far as heat transfer is concerned, were taken to be equal to those of water and a lump of flour (such as corn meal, a common component of many industrial fermentation media), respectively. The corresponding numerical values (Geankoplis, 1983, pp 832-834; Perry and Green, 1986, pp 3-260) are reported in Table 111. Even if it is assumed that the flour becomes wet with water, the heat transfer properties of the solid do not change appreciably. In fact, only the group (Y = ka/psCps appears in the equations representing the energy balance in the solid (such as eq 10).
The particle size was kept as a design parameter. In the worst assumption case, the particle radius was taken to be 6 mm. 7.6. Particle-Liquid Medium Heat Transfer Coefficient. The estimation of the particleliquid medium heat transfer coefficient proved to be more complex to obtain. Because of the lack of information in the literature, the heat transfer coefficient was obtained by rearranging the corresponding mass transfer equations available for solids suspended in liquids flowing in vertical and horizontal pipes (Ohashi et al., 1979,1981b; Kikuchi et al., 1983). Using Kolmogoroffs theory of isotropic turbulence, several authors (Armenante and Kirwan, 1989; Ohashi et al., 1981a;Levins and Glastonbury, 1972) have shown that the solid-liquid mass or heat transfer coefficient in turbulent systems is largely independent of the specific system but is predominantly a function of the energy dissipation per unit mass, t. In addition, the analogy between the mass and heat transport mechanisms has extensively been used in the past for different systems (Sherwood et al., 1975, pp 159-171; Brian et al., 1969). Therefore, the equation used in this work to calculate the particle-liquid mass transfer coefficient, h,, for the particles moving in the heat recovery exchanger is
Nu, =
h 2R jZb
=2
+ 0.44[
,1/3(2R)4/3]0.63 Pr1/3 (61) V
which was modified from the original equation of Ohashi et al. (1979) for mass transfer, where the Nusselt number and the Prandtl number were substituted for the Sherwood number and the Schmidt number, respectively. The power dissipation per unit slurry mass, t, was obtained by first calculating the pressure drop in each exchanger according to (62) where DH, le,, and f are the equivalent hydraulic diameter, length in the direction in which the fluid moves, and Fanning factor for each exchanger, respectively. Thus, t can be obtained from a mechanical energy balance in the fluid as (63) where the denominator is just the mass of one fluid in the heat exchanger. These equations can be rearranged to give 6 = 2fv3/DH (64) In our case, the value o f f was taken to be 0.0055 (Bird et al., 1960, p 186). The corresponding values of t, Nu,, and h, were found to be 0.32 m2/s3,172, and 9650 W/(m2 "C), respectively. From the knowledge of h,, the value of L , defined in eq 13 could be obtained and used to determine the eigenvalues by using eq 12. A similar procedure was adopted to obtain hh in the heater. Since all the parameters involved in the calculation were the same as in the case of h,, the two heat transfer coefficients h, and hh were found to be identical. 7.7. Thermal Death Kinetic Constants for the Microorganism. Bacillus stearothermophilus was assumed to be the contaminating microorganism because of its known resistance to heat sterilization. The associated thermal death kinetic constants for its spores are independent of temperature and are available in the literature (Aiba et al., 1973, pp 242, 255; Wang et al., 1979, p 144). They are as follows: &O = 1.53 X min-' and E d = 68 700 cal/(g mol).
Biotechnol. Prog., 1990, Vol. 6,No. 4
300
7.8. Required Degree of Sterility. In this work, the initial concentration of microorganisms in the liquid medium as well as in the solid particles was assumed to be No = lo5 microorganisms/mL (Aiba et al., 1973, p 264; Wang et al., 1979, p 146). It was further assumed that a safe margin of operation would be that in which the probability of contaminating a lOO-m3 batch is only i.e., one batch in a thousand (Aiba et al., 1973, p 263). This means that the sterilizer must be able to reduce the total number of microorganisms initially in the batch (equal to 1013), t o only Therefore, the sterilization ratio was calculated to be Nf/ N~ = 10-3/1013 = 10-18. Following a common design practice, we also assumed that this sterilization level must be attained in the liquid counting only the contribution of the holding section and disregarding any sterilization that may take place in other parts of the system, such as the heater or the heat recovery exchanger (Ashley and Mooyman, 1981). However, the level of sterility achieved in each of the heat exchangers prior to the holding section was also calculated in our model and is reported below.
8. Numerical Determination of the Temperature Profiles and Sterilization Effectiveness inside a Particle The level of complexity of the analytical expressions for the degree of sterilization (eq 48) and for the temperature profiles (eqs 10, 15, 29, 33, 39, and 43) is such that a numerical approach was used for their evaluation. A FORTRAN program was developed for this purpose. The design parameters required by the model (i.e., heat recovery efficiency, initial medium temperature, sterilization temperature, steam temperature, heat transfer coefficients of exchangers, clearances in spiral plate exchangers, particle-liquid heat transfer coefficients, particle radius, and overdesign factor of the holding section) were given as input. The temperature profiles a t various increments of time (typically 1 s or less) were obtained. The calculation of the temperature profile in the sphere required the numerical calculation of the eigenvalues, Le., the roots of eqs 12, 31, and 41. Typically, 20 of the infinite number of zeros of these equations were determined numerically by using a modified regula falsi algorithm. An equal number of terms was used in the temperature equations as an approximation of the infinite series. In all cases, it was found that this was more than adequate to insure convergence of the series (in most instances, convergence occurred by using just a few terms). The temperatures inside the spherical particles were calculated a t 11equally spaced positions along the radius by using eqs 10,29, and 39 for r # 0 and eqs 15,33, and 43 for r = 0. The temperature profile for a spherical particle leaving one exchanger was used as the initial condition for the same sphere entering the next exchanger. The temperature profiles were used to determine the fraction of remaining viable microorganisms a t the end of each heat exchanger by using eq 48, appropriately modified to be used in numerical computation. All the results shown below were obtained by assuming that the holding section alone reduces the number of microorganisms originally present in the liquid by a factor of 10-l6. The actual level of sterility was then calculated in the solids as well as in the liquid. 9. Results and Discussion In addition to the determination of the temperature profiles and the degree of sterility, the computer model
TemDerature ( ' C ) 48""R n C
1
120
20
,1
'1
LoglO(No/N) 2oo t.. I
L
- Llquld Temperature
1
-
1
Solid Sterllity
r
c
25
~
I
0 0
20
40
60
80
100
120
140
160
0 180
Time (s) Figure 3. Temperature-time profiles and sterilization effectiveness for liquid and solid particles moving inside a continuous sterilizer (T,= 150 "C). T,,,,= 145 "C; u = 63%; U, = 400 W/(mz "C); u h = 1000 W/(m2 "C); L = 130;R = 6 mm;
x = 1.
was used to determine the sensitivity of the sterilization process to changes in the design parameters. The role of these parameters is examined in greater detail below. 9.1. Effect of Steam Temperature. The temperature profiies in the liquid bulk and at the center of a solid sphere having a radius of 6 mm and moving along with the liquid are reported in Figure 3 as functions of time (where time is counted from the moment the fluid first enters the sterilizer) for a fixed set of parameters. The largest fraction of time is spent by the slurry in the recovery exchanger (approximately 104 s). Only less than 3 s is spent in the holding section. However, because of the high temperature, this short residence time is still sufficient to ensure that a reduction of 16 orders of magnitude in the number of microorganisms in the liquid occurs in the holding tube, as required. The curves show that a significant temperature difference exists between the core of the sphere and the fluid. In the heat recovery section, this difference is about 14 "C but becomes as high as 23 "C in the heater before beginning to decrease. Even at the exit of the holding section, the difference is still some 7 oc. Figure 3 also shows the level of sterilization, expressed in terms of the logarithm (base 10) of the ratio of the initial number to the final number of viable microorganisms for the bulk liquid and for the solid. The time spent by the slurry in the heat recovery exchanger (first 104 s) does not practically contribute to the overall sterilization. By contrast, significant sterilization in the liquid bulk occurs in the heater and in the holding section, with reductions of 103 and 16 orders of magnitude, respectively, in the number of surviving microorganisms. Figure 3 shows that sterilization also occurs in the solid but to a much lesser extent. The overall solid sterilization level is only lO-l8.e7, of which a reduction of 13 orders of magnitude is achieved in the heater and less than 6 in the holding section. Although this level of solid sterility would still be more than acceptable, it is much lower than that achieved in the liquid. Figure 4 shows the corresponding results for the case in which the sterilization temperature is still 145 "C, as before, but high-temperature steam a t 180.3 "C is used (corresponding to a saturated steam pressure of 10 atm). The residence times in the heat recovery exchanger and in the holding section are the same as in the previous case, but the time required by the heater to bring the fluid to its sterilization temperature is now only 20 s instead of 1 min. This reduces the time available for the solid core to
Biotechnol. Prog., 1990,Vol. 6, No. 4
301
Temperature ('C) - Llquld Temperature
140 t
lZo
04 0
---
Solld Center Temp Llquld S t e r l l l t y
Log;;ro/Ni
7o
LoglO(No/N)
140 1
I
/r
60
Solld S t e r l l l t y
J --0 20
40
60
80
100
120
60 140
Time (s) Figure 4. Temperature-time profiles and sterilization effectiveness for liquid and solid particles moving inside a continuous sterilizer (T,= 180.5 "C). Tster= 145 "C; u = 63%;
40 -
x = 1.
20
Ur=400W/(m2"C);Uh=lOOOW/(m2"C);L=130;R=6mm;
equilibrate its temperature with the surrounding fluid. The temperature gap between the solid center and the bulk fluid is now as high as 36.5 "C in the heater and 33 "C a t the end of the sterilizing section. This has a very serious impact on the degree of sterility. The number of surviving microorganisms in the liquid is reduced by about 37 orders of magnitude (of which 16 are lost in the holding tube, as required). However, the ratio N / N o in the solids leaving the holding section drops to only about which is very far from the value of 10-'6 that is considered acceptable here for complete sterilization. A plot of the overall sterility level in the liquid and in the solid as a function of the steam temperature, T,, is given in Figure 5. The increase in steam temperature has the effect of increasing the temperature gradient across the exchanger wall, thus reducing the size of the exchanger and the residence time of the fluid in it. This, in turn, reduces the time available for the particles to raise their internal temperature to values high enough for sterilization to occur. This plot shows that at high steam temperatures the presence of solids may reduce the degree of sterility to levels that may become unacceptable, in spite of the fact t h a t sterilization in the liquid is still achieved. Depending on the type of feed, the higher steam temperature may also result in increased fouling and, consequently, in a reduction of the heat transfer efficiency. This, in turn, may reduce the sterilization level in the solids. The designer can compensate for this by using more conservative design values for the fouling factor in the heater or by obtaining actual fouling factors for the specific feed from preliminary experimental data in pilot equipment. 9.2. Effect of Sterilization Temperature. The effect of sterilization temperature is shown in Figure 6. In this figure the steam temperature was kept constant at 150 "C. The level of sterilization obtained in the solids at low sterilization temperature is not acceptable. The reason for this can be attributed to the fact that, by lowering the sterilization temperature, the residence time of the medium in the heater is reduced because of the increased temperature difference between the steam and the medium. This reduces the core temperature of the solid a t the entrance of the holding section. 9.3. Effect of Particle Size. The particle size plays a fundamental role in the sterilization process and can be considered the single most important factor affecting the overall level of sterility. Unfortunately, in many practical
0 140
150
160
170
190
180
200
Steam Temperature, Tw ("C) Figure 5. Sterilization effectiveness vs sterilizing steam temperature. T,,,,= 145 "C; u = 63%; U, = 400 W/(m2 "C); Uh = 1000 W/(m2 "C); L = 130; R = 6 mm; X = 1.
Log1O( No/N 1
140 I
i
100 -
80 60 -
40 -
20
t
0' 130
-i--
I
I
I
135
140
14 5
150
Ste r iIizat io n Temper at ure, Tster ( 'C) Figure 6. Sterilization effectiveness vs sterilization temperature. T, = 150 "C; u = 60%; U, = 400 W/(m2 "C); uh = 1000 W/(m2 "C);L = 130; R = 6 mm; X = 1.
cases, the particle size can hardly be considered a design parameter controlled by the designer. As one may expect, smaller particles have higher core temperatures. This results in enhanced sterilization effectiveness as clearly demonstrated in Figures 7 and 8, which show, respectively, the core temperature profiles and the levels of sterilization vs time as functions of particle radius. Figure 7 shows t h a t the average temperature difference between the liquid and the solid center is about 3.7,8.1, and 14 "C for particles having, respectively, radii of 3, 4.5, and 6 mm. Figure 8 shows that a satisfactory
Biotechnol. Prog., 1990, Vol. 6, No. 4
302
Temperature
Log1O( No/N 1
('C)
160 140
-
80
120 -
Llquld
1 I
1
50
100
0
0
R
.
4.6"
150
200
Time (s) Figure 7. Core temperaturetime profiies for particles of different radii. Tshr= 145 "C; T, = 150 "C; u = 63%; U, = 400 W/(m2 "c); uh = 1000 W/(m2 "c);L = 130; = 1.
x
140
LoQ~O(N o/N)
+ Llquld 120
# R-Bmm R
.
4.6 mm
* R=6mm 100
I
80
60
40
20
0 0
50
100
150
200
Time (s) Figure 8. Sterilization effectiveness for particles of different radii. Tster= 145 "C; T , = 150 "C; u = 63%; U, = 400 W/(m2 "C);
Uh = 1000 W/(m* "C); L = 130;X = 1. level of sterilization is achieved in all cases, a t least for the operating conditions chosen in this example (including a steam temperature equal t o 150 "C). If the steam temperature is increased t o 180.5 "C, however, t h e sterilization effectiveness in the solids is reduced to the point of being unacceptable even for the case of the 3-mm radius particles. This effect is clearly shown in Figure 9. The reason for this resides in the shorter residence time of the slurry in the heater. This drastically reduces the equilibration time available to the particles to raise their core temperature to the sterilization temperature of the holding tube. The designer may partially overcome this
2
3
4
6
6
7
R (mm) Figure 9. Sterilization effectiveness for particles of different radii at variable steam temperatures. Tster= 145 "C; u = 63%; U, = 400 W/(mz "C);uh = 1000 W/(mz "C); L = 130;X = 1.
problem by using a lower steam temperature when larger particles are present, thus allowing a "gentler" heating process to take place in the heater. 9.4. Effect of the Heat Transfer Coefficient in the Heat Exchangers. T h e value of the heat transfer coefficients affects significantly the design of the heat exchangers and the length of the residence time of the medium in them. However, since no appreciable sterilization takes place in the heat recovery exchanger even a t energy recovery factors as high as 63 %, the overall sterilization effectiveness is not greatly affected by U,. Instead, the value of uh plays a significant role in the overall sterilization by determining the residence time of the medium in the heater. This has two main effects: on one hand, it prolongs the time spent by the slurry at a relatively high temperature where presterilization may occur. On the other hand, it extends the time available to the particles to equilibrate their temperature with that of the fluid, thus enabling sterilization to occur even inside the particles as they pass through the holding section. This effect is similar to the lowering of the steam temperature described in the previous section. The role of Uh is clearly shown in Figure 10, where an increase in the overall sterilization effectiveness in the liquid of about 85% occurs as uh varies between 1000 and 500 W/(m2 "C). An even more significant effect can be observed in the solids, for which the sterilization effectiveness increases by a factor of 5 for the same range of Uh. Although the actual values of Uh cannot be easily altered by changing the operating conditions, this shows that a conservative design of the heater may greatly help reduce the risk of underdesigning the sterilizer. 9.5. Effect of Increased Sterilization Time. Sterilization effectiveness can be improved by increasing the residence time of the medium in the holding section above the level that was calculated to be sufficient to ensure complete sterilization of the liquid (i.e., a reduction of 16 orders of magnitude in the microorganism concentration). This approach is equivalent to overdesigning the holding section of the sterilizer.
Biotect,no/. Prog., 1990,Vol. 6, No. 4 250
303
LoglO(No/N)
LoglO(No/N)
160 I
/
200
l4OI 120
1
100
150
I
+ Llquld \
6ol
100
40 50
0 450
550
650
750
U, W / m '
850
950
1050
1
0
"c)
Figure 10. Sterilization effectiveness as a function of the heat transfer coefficient in the heater, Uh. Tater= 145 "C;T , = 150 "C; u = 63%; U, = 400 W/(m2 "C); L = 130; X = 1.
Figure 11 shows the effect of the overdesign factor, X, on sterility. Here X is defined as the ratio of the actual average residence time of the fluid in the holding section to the residence time required to ensure adequate sterility in the liquid. The figure shows that the sterility increases with X, although not dramatically. For the liquid, this can be attributed to the fact that the heater acts as a very effective sterilizer. However, the increase in the liquid sterility level is still directly proportional to the factor 16X, as expected. In the solid, this effect is not as pronounced since the temperature in the solid core a t the entrance of the holding section is not the same as that of the liquid. This shows that overdesigning the sterilizing section of the system may not necessarily be the most effective way to ensure complete sterilization of slurry media. In addition, overdesigning still poses the problem of reducing the nutritional content of the medium because of the degradation of some of the medium components. 9.6. Effect of Energy Recovery Efficiency. The heat (or energy) recovery efficiency, u,plays a very minor role in the sterilization process, at least for the process condition chosen in the example. This can be attributed to the fact that whenever little or no energy is recovered, the size of the heater increases accordingly (although not in direct proportion to u ) , thus compensating for the reduced size or absence of the heat recovery exchanger. In addition, the contribution of the recovery exchanger to the overall sterilization process, including equilibration of the core temperature of the particle with the liquid, is rather marginal, thus providing an additional justification for the results found here. 9.7. Effect of Particle-Liquid Heat Transfer Coefficient. The possible effects of the particle-liquid heat transfer on sterility were investigated by varying the value of the dimensionless Biot number (Lh = hhR/k,). It was assumed that the value of L h (valid for particlesolid heat transfer in the heater) had the same value as the corresponding dimensionless number (L, = h,R/k,) for the heat recovery exchanger. Figure 12 shows that the
2
3
4
x (-1 Figure 11. Sterilization effectiveness as a function of overdesign factor for the holding section. TSbr= 145 "C; T , = 150 "C; u = 63%; U, = 400 W/(m2 "C); Uh = 1000 W/(m2 "C); L = 130.
LoglO(No/N)
+Llquld
0 Solld
140 120-
\
I
I
100 80 -
60 40
1
I 0 10 20 3 0 40 5 0 6 0 70 80 90 100 110 120130140150
Biot Number, L (-1 Figure 12. Sterilization effectiveness as a function of the Biot number, L (containing t h e particle-liquid h e a t transfer coefficient). T,,, = 145 "C; T, = 150 "C; u = 63%; U,= 400 W/(m2 "C); u h = 1000 W/(mz "C); L = 130; X = 1. effect of L is rather limited even when it is changed by an order of magnitude.
10. Design Considerations for Continuous Sterilization in the Presence of Suspended Solids The equations given above enable the designer to calculate the remaining fraction of surviving microorganisms in the liquid and in the solids as they emerge from the holding section of the sterilizer. To obtain the corre-
Biotechnol. Prog., 1990,Vol. 6, No. 4
304
sponding sterility level for the medium as a whole, the following expression can be used:
where & represents the volume fraction of solids in the medium. However, unless the solid concentration is very low, it is advisable to ignore this correction and consider the sterility of the solids as the most critical design factor. In this model the sterilizing effect of the other heat exchangers past the holding tube has been neglected. This adds an additional safety margin to the design. The probability that large lumps of material enter the sterilizer with the medium can be minimized by using protective filters or screens. However, this may prove to be difficult in a number of industrial situations and does not eliminate the possibility that lumps of materials may be formed downstream from the protective system as a result of agglomeration or heat effects. In our model the assumption was made that the velocity profiles in the exchanger and holding section are flat. However, even in turbulent flow this approximation may not be sufficiently correct for most designs. In such cases a conservative design can be obtained by imposing that the residence time in each piece of equipment is increased by multiplying it by the ratio of the maximum velocity in the equipment (typically in correspondence with the center line of the fluid cross section) to the average velocity used for the other calculations (Charm, 1971, p 215). In most cases this approach also eliminates the need for other axial flow dispersion corrections, since the axial dispersion in the forward direction at the center line is usually negligible. The examples shown above have been obtained by assuming that only one type of material (agglomerated flour) forms lumps resulting in solid particles. Should this not be the case, the numerical values reported above may change as a result of the different material properties. This may significantly affect the results in many cases and require an even more conservative design. Any such effects, however, can be predicted by the proposed model when different values of the numerical parameters are used. The particle size is the most important factor affecting the sterility of the system. However, of all the design parameters actually available t o the designer, t h e temperature difference between the heating fluid in the heater (typically steam) and the medium appears to be one of the most critical. High temperature differences occur when the steam temperature is high or the sterilization temperature is low, or a combination of both. This results in a short residence time in the heater, thus preventing the solids from reaching a temperature high enough for a period of time long enough to ensure an adequate level of sterilization in the holding section. The experimental validation of the present model for slurry media sterilization as well as the validation of any of the other models previously proposed for continuous sterilization of homogeneous fluids is very difficult. This originates from the difficulty of measuring the number of surviving microorganisms when their number is exceedingly low, as required by sterilization procedures. In addition, the detection of any viable microorganisms in concentrations of the order of 10-11 organisms/mL, as in our case, requires an extremely large volume of sample to analyze (of the order of 100 000 m3 on the average for the detection of j u s t one viable microorganism or spore). [This sometimes poses some problems, as far as validation of the system is concerned, even when equipment is purchased from vendors, as one of us (P.M.A.) previously exper-
ienced.] It should also be noticed that microorganism thermal death becomes dominated by probability laws when the numbers of organisms is very small (Blakebrough, 1968, p 37). This implies that any validation should be carried out with multiple, statistically designed experiments on large-scale equipment. This is not only very impractical but is also associated with formidable costs. It should be remembered that the main purpose of the development of the present model is not so much the exact determination of the number of viable microorganisms at the end of the process but the derivation of equations that can be used by process engineers to correctly design a continuous sterilizing system for slurry media.
11. Conclusions T h e results shown above indicate t h a t under the appropriate conditions the presence of solids may produce a nonsterile medium at the exit of a continuous sterilizer designed to handle only liquid media. The level of sterility of the product is a function of a number of parameters, which have to be accounted for during the design of the system. The most important of these parameters are those that reduce the heat flux to the center of the particles (such as the size of the particles or their physical properties) and those that may produce a shorter residence time in the heating section (such as the heat transfer coefficients of the heat exchangers and the temperature gradients across the exchanger wall). In general, those designs that produce a gradual increase of the medium temperature are more suited to handle slurry media. This results from the extra time available to the particles to raise their internal temperature to that of the surrounding liquid before the medium enters the sterilizing section. Allowing the medium to reside longer in the holding section may also be an alternative but not necessarily the most valid. The model presented here can be used to incorporate the effects of these parameters in the design, thus providing a reliable method to predict the level of sterility achieved by a continuous sterilizer for such complex media.
Notation surface area of heat exchanger, m2 surface area of a spiral heat exchanger, m2 surface area of heat recovery exchanger, m2 surface area of heater exchanger, m2
DH
Ed
f
ff F hh
AU/(Sp8Cphl), value of b' for heater exchanger, s-l value of b' for heat recovery exchanger, s-l heat capacity, J/(kg "C) heat capacity of bulk fluid, J/(kg "C) heat capacity of solid particle, J/(kg "C) coefficient defined in either eq 30 or eq 40, "C m hydraulic diameter in exchanger (equals 4 times the cross-sectional area divided by the wet perimeter), m activation energy for microorganism thermal death reaction, cal/(g mol) Fanning factor for flow in smooth pipes, dimensionless fouling factor, "2 'C/W volumetric flow rate, m3/s particle-liquid heat transfer coefficient when particle moves through heater exchanger, W/ (m2 "C)
Biotechnol. Prog., 1990, Vol. 6, No. 4
particle-liquid heat transfer coefficient when particle moves through holding section, W/(m2 "C) particle-liquid heat transfer coefficient when - particle moves through heat recovery exchanger, W/(m2 "C) single-side heat transfer coefficient in heat recovery exchanger, W/(m2 "C) heat conductivity, W/(m "C) heat conductivity of bulk fluid, W/(m "C) heat conductivity of a solid particle, W/(m "C) specific kinetic rate for microorganism thermal death reaction, s-l Arrhenius constant for microorganism thermal death reaction, s-l path length of a fluid moving inside a heat exchanger, m length of an Archimedes spiral, m length of spiral path described by a fluid moving in a spiral heat exchanger, m path length of a fluid moving inside a heat recovery exchanger, m Biot number for particles in heater exchanger, dimensionless (see eq 32) Biot number for particles in holding section, dimensionless (see eq 42) Biot number for particles in heat recovery exchanger, dimensionless (see eq 13) dummy variable, dimensionless number of microorganisms contained in the spherical shell between r and r + Ar inside a solid particle, dimensionless number concentration of viable microorganisms or spores, mL-1 initial number concentration of viable microorganisms or spores, mL-1 final number concentration of viable microorganisms or spores, mL-1 Nusselt number = hD/k, dimensionless pressure drop in a heat exchanger, Pa Prandtl number, C,p/k, dimensionless electric energy input rate, W radial position within a solid spherical particle, m radius of pipe in heater exchanger for the case of double-pipe configuration, m radius of pipe in heat recovery exchanger for the case of double-pipe configuration, m radius of a solid particle, m ideal gas constant, cal/(g mol K) cross-sectional area available for fluid flow in heat exchangers, m2 cross-sectional area available for fluid flow in heat recovery exchanger, m2 time, s time at which the fluid leaves the heater exchanger, S
residence time of the fluid in the heater exchanger, S
residence time of the fluid in the holding section, S
time a t which the fluid leaves the heat recovery exchanger, s residence time of the fluid in the heat recovery exchanger, s temperature, K (or "C where indicated) bulk fluid temperature at the entrance of the heat recovery exchanger, K (or "C where indicated)
305
cold-side bulk fluid temperature at the exit of the heat recovery exchanger, K (or "C where indicated) hot-side bulk fluid temperature at the exit of the heat recovery exchanger, K (or "C where indicated) bulk fluid temperature, K (or "C where indicated) bulk fluid temperature in the heater exchanger, K (or "C where indicated) bulk fluid temperature in the heat recovery exchanger, K (or "C where indicated) temperature inside solid particle when the particle moves through the heater exchanger, K (or "C where indicated) temperature inside solid particle when the particle moves through the holding section, K (or "C where indicated) temperature inside solid particle when the particle moves through the heat recovery exchanger, K (or "C where indicated) sterilization temperature for bulk fluid, K (or "C where indicated) steam temperature, K (or "C where indicated) energy recovery efficiency of sterilizing system, dimensionless overall heat transfer coefficient, W/(m2 "C) overall heat transfer coefficient in heater exchanger, W/(m2 "C) overall heat transfer coefficient in heat recovery exchanger, W/(mz "C) velocity, m/s velocity of fluid in the exchanger, m/s velocity of fluid in heater exchanger, m/s width of a spiral heat exchanger unit, m overdesign factor in holding section, dimensionless eigenvector (see eq 27) auxiliary variable, "C m (see eq 9) auxiliary variable, "C m (see eq 22) Greek Symbols heat diffusivity in solid particle, mz/s nth eigenvalue, dimensionless distance between metal spirals in spiral heat exchanger, m radial distance separating two points on a spiral in correspondence with one turn ( = 2 rad), ~ m distance between metal spirals in heater exchanger, m distance between metal spirals in heat recovery exchanger, m mechanical power dissipation per unit fluid mass, m2/s3 temperature profile function in particle as particle leaves the heater exchanger, K (or "C where indicated) temperature profile function in particle as particle leaves the heat recovery exchanger, K (or "C where indicated) function used in eigenfunction expansion method (see eq 27) viscosity, kg/ (m s) kinematic viscosity, mz/s volumetric fraction of solids in the medium, dimensionless radial distance of a generic point on a spiral, m density of fluid, kg/m3 radius of a spiral heat exchanger unit, m density of solid particle, kg/m3
Biotechnol. frog., 1990,Vol. 6, No. 4
306 $8
*SO
local concentration of microorganisms at position r inside a particle and at time t , mL-l initial value of $B, mL-1
Literature Cited Aiba, S.; Sonoyama, T. Residence time distribution of a microbial suspension in a straight pipe. J . Ferment. Technol. 1965,43, 534-539. Aiba, S.; Humphrey, A. E.; Millis, N. F. Biochemical Engineering, 2nd ed.; Academic Press: New York, 1973. Armenante, P. M.; Kirwan, D. J. Mass transfer to microparticles in agitated systems. Chem. Eng. Sci. 1989,44,2781-2796. Ashley, M. H. J.; Mooyman, J. G. 1981, Continuous sterilization of media. 182nd National Meeting of the American Chemical Society, New York, August 1981. Bailey, J. E.; Ollis, D. F. Biochemical Engineering Fundamentals, 2nd ed.; McGraw-Hill: New York, 1986. Baird, M. H. I.; McCrae, W.; Rumford, F.; Slesser, C. G. M. Some considerations on heat transfer in spiral plate heat exchangers. Chem. Eng. Sci. 1957, 7,112-115. Bird, R. B.; Stewart, W. E.; Lightfoot, E. N. Transport Phenomena; John Wiley & Sons: New York, 1960. Blackebrough, N. Biochemical and Biological Engineering Science; Academic Press: London, 1968; Vol. 2. Brenner, H. The diffusion model of longitudinal mixing in beds of finite length. Numerical values. Chem. Eng. Sci. 1962,17, 229-244. Brian, P. L. T.; Hales, H. B.; Sherwood, T. K. Transport of heat and mass between liquids and spherical particles in an agitated tank. AZChE J. 1969, 15, 727-733. Carslaw, H. S.; Jaeger, J. C. Conduction of Heat in Solids, 2nd ed.; Clarendon Press: Oxford, England, 1959. Charm, S. E. The Fundamentals of Food Engineering, 2nd ed.; Avi Publishing Company: Westport, CT, 1971. Crank, J. The Mathematics of Diffusion, 2nd ed.; Oxford University Press: New York, 1975. de Ruyter, P. W.; Brunet, R. Estimation of process conditions for continuous sterilization of food containing particulates. Food Technol. 1973,27,44-51. Deindoerfer, F. H. 1957, Calculation of heat sterilization times for fermentation media. Appl. Microbiol. 1957, 5 , 221-228. Deindoerfer, F. H.; Humphrey, A. E. Analytical methods for calculating heat sterilization times, Appl. Microbiol. 1959a, 7,256-264. Deindoerfer, F. H.; Humphrey, A. E. Principles in the design of continuous sterilizers. Appl. Microbiol. 1959b, 7, 264-270. Farlow, S. J. Partial Differential Equations; John Wiley and Sons: New York, 1979. Feliciotti, E.; Esselen, W. B. Thermal destruction rates of thiamine pureed meats and vegetables, Food Technol. 1957,11,77-84. Geankoplis, C. J. Transport Processes and Unit Operations, 2nd ed.; Allyn and Bacon: Boston, MA, 1983. Hargis, A. M.; Beckman, A. T.; Loiancono, J. J. Application of spiral plate heat exchangers. Meeting of the American Institute of Chemical Engineers, Houston, T X , February 1967.
Kern, D. Q. Process Heat Transfer;McGraw-Hill Kogakusha: Tokyo, Japan, 1950. Kikuchi, K.; Sugawara, T.; Ohashi, H. Correlation of mass transfer between particles and liquid in liquid fluidized beds. J . Chem. Eng. Jpn. 1983,16,426-428. Lamb, B. R. The Rosenblad spiral heat exchanger. Chem. Eng. 1962, A-108-111. Leskowicz, M. A. Mathematical model of a continuous sterilization system for fermentation media containing suspended solids. Master’s Thesis, New Jersey Institute of Technology, Newark, NJ, 1989. Levenspiel, 0. Longitudinal mixing of fluids flowing in circular pipes. Ind. Eng. Chem. 1958,50,343-346. Levenspiel, 0. Chemical Reaction Engineering, 2nd ed.; John Wiley & Sons: New York, 1971. Levins, D. M.; Glastonbury, J. R. Particle-liquid hydrodynamics and mass transfer in a stirred vessel. Part 11-Mass transfer. Trans. Inst. Chem. Eng. 1972,50,132-146. Lin, S. H. A theoretical analysis of thermal sterilization in a continuous sterilizer. J . Ferment. Technol. 1975, 53, 92-98. Lin, S. H. Continuous high temperature/short time sterilization of liquid foods with steam-injection heating, Chem. Eng. Sci. 1976,31, 77-82. Lin, S. H. Residence time distribution of flow in a continuous sterilization process. Process Biochem. 1979, 14 (7), 23-27. Manson, J. E.; Cullen, J. F. Thermal process simulation for aseptic processing of food containing discrete particulate matter. J . Food Sei. 1974,39, 1084-1089. Ohashi, H.; Sugawara, T.; Kikuchi, K.; Henmi, T. Mass transfer between particles and liquid in solid-liquid two-phase upflow in vertical tubes. J . Chem. Eng. Jpn. 1979, 12, 190195. Ohashi, H.; Sugawara, T.; Kikuchi, K.; Konno, H. Correlation of liquid-side mass transfer coefficient for single particles and fixed beds. J . Chem. Eng. Jpn. 1981a, 14,433-438. Ohashi, H.; Sugawara, T.; Kikuchi, K. Mass transfer between particles and liquid in solid-liquid two-phase upflow in the low-velocityregion through vertical tubes. J. Chem. Eng. Jpn. 1981b, 14,489-491. Perry, R. H.; Green, D. W. Perry’s Chemical Engineers’ Handbook, 6th ed.; McGraw-Hill: New York, 1984. Rohsenow, W. M.; Hartnett, J. P.; GaniE, E. N. (Eds.) Handbook of Heat Transfer with Applications, 2nd ed.; McGraw-Hill: New York, 1985. Sherwood, T. K.; Pigford, R. L.; Wilke, C. R. Mass Transfer; McGraw-Hill: New York, 1975. Tangri, N. N.; Jayaraman, R. Heat transfer on a spiral plate heat exchanger. Trans. Inst. Chem. Eng. 1962,40,161-167. Vogel, H. C. (Eds.) Fermentation and Biochemical Engineering Handbook; Noyes Publications: Park Ridge, NJ, 1983. Wang, D. I. C.; Cooney, C. L.; Demain, A. L.; Dunnill, P.; Humphrey, A. E.; Lilly, M. D. Fermentation and Enzyme Technology; John Wiley & Sons: New York, 1979. Accepted May 30, 1990.
