Ind. Eng. Chem. Res. 1995,34, 4571-4576
4671
Heat-Transfer Coefficients in Agitated Vessels. Sensible Heat Models Enio Kumpinskyt R&D Department, Ashland Chemical Company, P.O. Box 2219, Columbus, Ohio 43216
Transient models for sensible heat were developed to assess the thermal performance of agitated vessels with coils and jackets. Performance is quantified with the computation of heat-transfer coefficients by introducing vessel heating and cooling data into model equations. Of the two model categories studied, differential and macroscopic, the latter is preferred due to mathematical simplicity and lower sensitivity to experimental data variability.
Introduction In a previous publication (Kumpinsky, 19921, we discussed the concept of using dynamic energy balances to determine heat-transfer coefficients in agitated vessels. The method was applied to vessels equipped with a single straight jacket or a single internal or external coil. Since many vessels operate with multiple coils and jackets, there is a need to develop more general models. Our initial approach was to follow the principles of the previous work, by generating analytical solutions for the differential energy balances. These in turn may be used to calculate heat-transfer coefficients with experimental data. It is not efficient to manually solve for systems having more than either one jacket or one coil. We used Maple V software to generate analytical solutions by means of Laplace transforms. However, a system with more than four heat-exchange units yields extremely long solutions, and it becomes impractical even with symbolic math software. It is important to realize that many vessels have more than four heat-transfer units. Hence, there is a need t o develop an alternative method to calculate heat-transfer coefficients for any number of jackets and coils. Furthermore, analytical solutions are prone to diverge when numerically iterating for heattransfer coefficients. Indeed, small deviations in the consistency of the experimental data can lead t o failure in convergence for even a single jacket or coil. Since the likelihood of such deviations occurring in large-scale experiments is high, we need to develop a more robust method. Macroscopic energy balances, obtained from term by term integration of the differential equations, provide excellent means to compute heat-transfer coefficients without major numerical problems. The models of this work have two main applications. First, the consistency of heat-transfer capabilities with time, that is, we can compare today’s results with those of the past. Second, the verification of cooling capabilities a t peak heat generation to ensure that the vessel can safely handle specific exothermic processes. These tests do not require the loss of production time. In batch processes in chemical reactors, for example, they can be done with the raw materials being heated to reaction temperature and the product being cooled at the end. To compare the results from time to time, the requirement is consistency from test to test. The experiments can also be carried out with water or organic solvents during vessel cleaning, if the heat-transfer coefficients are not too different from those of the actual processes. The instrumentation for the tests is minimal if we are E-mail address: compuserve.com. +
enio-kumpinsky%ashchem@notesgw.
0888-5885/95/2634-4571$09.00/0
seeking a combined coefficient for all heat-transfer units: two temperature sensors, one a t the inlet and another at the exit of the common lines to all heattransfer units. The knowledge of flow rates of the service fluid allows us to obtain information on thermal consistency, but average heat-transfer coefficients can be calculated even if such flow rates are unknown. More instrumentation is required if we want to determine individual coefficients for all heat-transfer units: each unit requires a flow meter and temperature sensors at the inlet and exit of the vessel. The calculation of individual coefficients is useful, among other reasons, to identify the units that are responsible for a decay in the overall heat-transfer performance of the vessel. An experimental technique to determine process-side film coefficients was developed by Haam et al. (1992). The method was later applied to a high-efficiency impeller (Haam et al., 1993). The authors showed that the Reynolds number is the major influence on heat transfer and that vessel geometry, with good design, is of secondary importance. They also concluded that the heat-transfer coefficient varied little with angular position with respect to the baffles. On the basis of this information, it makes sense to use the surface-averaged heat-transfer coefficients of the present work for welldesigned vessels with good mixing.
Energy Balances The energy balance for a single jacket or coil (Kumpinsky, 1992) can be extended to a vessel equipped with multiple heat-transfer units. The assumptions are as follows: well-mixed vessel; constant physical properties; constant flow rates; and the heat-transfer coefficient is averaged over time for the experimental data under scrutiny, and it is uniform over the surface of each heattransfer unit. Figure 1is a sketch of the equipment of this work. For the interior of a well-mixed vessel, we can write:
N
For jacket m, m = 1 to M , the equation is
For coil n, n = 1 t o N , be it internal or external, we must start with a partial differential equation in time
0 1995 American Chemical Society
4572 Ind. Eng. Chem. Res., Vol. 34, No. 12, 1995
is the most reasonable approach to estimate cooling capabilities at peak exotherm, as only one service temperature needs to be guessed (see Example 3 in the Examples and Discussion section). One could think that this is a conservative approach for jackets, since the temperature difference between the vessel and the coil is minimal. This is not the case, however. When calculating the heat-transfer coefficient, a lower (conservative) AT leads to a higher U. Then, when estimating peak exotherms, the higher U is combined with the conservative AT, thus compensating for each other. The key point is consistency. If one form of AT is used to calculate U,then the same form of AT must be employed in combination with U when performing other heattransfer calculations. We now integrate eqs 1-3 term by term, from initial time to to final time tr. For the liquid inside the wellmixed vessel, we obtain:
'r
4">T(f)
Figure 1. Representation of an agitated vessel with its heattransfer components.
t and distance x along the coil length [see eq 10 in Kumpinsky (199211. This energy balance is integrated in x to yield Any parameter or property that varies with time can be easily incorporated into the integrals. For jacket m and coil n: The initial conditions are
T"(t)= T"(t,), Tj,(t) = Tj,(t0), T&) = Tc,(to) at t = t o (4) The heat-transfer coefficients Ujm and U,, are arbitrarily defined process parameters that depend on the form of the mean temperature differences AFm and ATm. These are functions of the inlet and exit temperatures of the service fluids in the coils and jackets, the vessel temperature, and flow patterns. As a guideline for heat exchangers, Mueller (1973) suggested that usually, but not always, the arithmetic difference is employed for laminar flow and the logarithmic difference is used for turbulent flow. Once the form of the temperature difference is established, all calculations that follow must be carried out with that particular definition. The two types of temperature differencesare presented here for comparison purposes. (ATjm)aeth= T,(t) - Tj,,(t)
The arithmetic mean for the jacket in eq 5 assumes complete back-mixing, that is, the temperature at the exit is the same as the uniform temperature inside the jacket. In this work, we chose the arithmetic mean for two reasons, even though the flow through the heattransfer units is usually turbulent. First, it allows us t o generate analytical solutions for eqs 1-4. Second, it
ejmCPjmym[Tjm(tf) - Tjem(tO)l = U,,JjmA:[T"(t) T,,(t)l dt - ej,Cpjmqj,~~[Tje,(t)- Tj&)l dt (7) e,C,,V,tTm(t,)
- TC"(t&1 =
ucdmJp"(t) -
T,(t)l dt - e,C,,q,A:[T,.,(t)
- T,,(t)l
dt (8)
Over any reasonable period of testing time, the terns on the left-hand side of eqs 7 and 8 are much smaller than each of the individual terms on the right-hand side, so that the former can be neglected. T&, the mean temperature in the coil, is not readily measurable. We could arbitrarily set T&) as the average between the so that inlet and exit temperatures, Ti&) and Tce,,(t), the value of U,, would be tied to this defmition. Here we chose a different approach. We follow a fluid element from the inlet to the exit of each coil. It is reasonable to assume that, during its travel along the coil, the fluid element sees a very small change in T&). This is because in most cases the time scale of fluid circulation through the coil is much smaller than the time scale of temperature change in the vessel. For instance, let us consider a typical industrial-scale situation of a 20-m-long coil, 2-in. nominal size, made of Schedule 40 steel, with a liquid flowing at 25 m3/h. For this example, the residence time of a fluid element in the coil is about 6 s. For a vessel with three of these coils and excellent heat-transfer capabilities, a Tdt) change of no more than 0.2 "C is expected during this time frame. Hence,
Ind. Eng. Chem. Res., Vol. 34, No. 12, 1995 4573 The integration of eq 9 in x and average of the result over the length L, to obtain the mean heat-transfer coefficient over the entire coil results in
ucAcn
QcnCpcnqcn
(10)
It is now possible to write expressions that can be readily used to calculate individual jacket and coil heattransfer coefficients. Appropriate instrumentation must be available, such as flow meters and inlet and exit temperature sensors. If the temperature sensors are located exclusively at the inlet and exit jacket and coil manifolds, then only average coefficients can be generated for the combination of coils and the combination of jackets. Note that eq 12 incorporates eq 10.
situations in which only two temperature sensors are available, one upstream of the vessel, before the service fluid is directed into the various heat-transfer units, and one downstream of the vessel, after all lines converge. In this case, the only viable alternative is to calculate an average heat-transfer coefficient for the entire vessel. The left-hand side of eq 6 is then combined with a simplified version of eq 15 in which the summation symbols are not necessary, since all inlet temperatures as well as exit temperatures are identical and the sums of Ajm and A,, are simply Aj and A,. For a batch process (qv = 0) with no significant heat input (h = O ) , we can write a very simple equation for U if the jacket and coil temperatures remain about the same during the test. For these conditions, eq 1 can be rewritten as dTv(t)
M
dt
m=l
@vcpvvv-= -U{
CA,,[TV(t)- I?,,]
+
N
CACn[Tv(t) - T,,,Il;
Tv(t)= Tv(to) at t = to (16)
n=l
where the average temperatures in the jacket m and coil n, Tjma and Ten,, are determined from P
We now define thermal consistency, 8, as follows: P
e = I~ [ ~ v c p v ~ v v [ ~T,(~~)I v ( t f ~+ q V Q T v ( t ) Tvi(t)ld t } -
i=l
$'h(t)dtl/H~II (13)
Tcna =
to
where the total heat exchanged HT is given by
, n = l , N (17)
2P
Equation 16 can be integrated analytically to yield
M
U=
evcpvvv X
M
N
N
n=l
As the name indicates, the thermal consistency quantifies the cohesion of the experimental data. It relates the heat gaineuost by the vessel contents with the heat removedadded by the heat-transfer surfaces. The closer 0 is to unity, the more consistent the thermal data is. The lack of coherence in energy data leads the analytical solutions to diverge numerically when solving for heat-transfer coefficients. Equations 11 and 12 eliminate numerical conversion problems. We can also obtain an overall heat-transfer coefficient for the vessel by replacing the individual Ujm and Ucn values with a single coefficient U in eq 14. By using eq 12, HT
= f M
N
where HT is the left-hand side of eq 6. Th&e are
\ m=l
n=l
Equation 18 can be solved with a simple calculator in the field, especially if the number of data points is small coupled with a low count of heat-transfer units. Example 1 in the Examples and Discussion section uses experimental data for a vessel with two jackets to compare eqs 11 and 18 with the complete analytical solution. We present the analytical solution for a batch process in a vessel with two jackets and no heat input. Equation 1 with qv, h(t),and N set t o zero was combined with eqs 2 and 4. The arithmetic temperature difference of eq 5 was used, and it was assumed that both jackets have the same inlet temperature, Tji(t). This system was solved by means of the symbolic math facilities of Maple V, Version 3, followed by manipulations to set the solution in a more useful form. The result is the following (see Nomenclature):
4574 Ind. Eng. Chem. Res., Vol. 34, No. 12, 1995 2 -
1300 6
'
I
7
8
9
I I 0.98 10 11 12 13
t (min) Figure 2. Overall heat-transfer coefficient and thermal consistency for example 1, calculated with data up to time t . Table 1. Temperature Data for Example 1
with
fcA) = A3
+ (gl+ g,+ %1+ av~,+ Pjl + Pj2)A2+
[(91+ Pjlxg,
+ Pj2) +
+ a"3,)(Pj1 + PjZ) +
+ ~ Z ~ V J +I ]( 9 1 + Pjl)%dj2 + ( 9 2 +
51%~
Pj2)%1Pjl
(21)
where the three values of & are the solutions (all negative) of fA) = 0 and D, = d/?'A)/dA a t each A = A,. The solution for Tje2(t)is obtained from eq 20 by simply exchanging all subscripts 1 and 2.
Numerical Solution The numerical solution of the macroscopic models is trouble-free, if a few guidelines are followed. All integrals must be evaluated by numerical methods. Newton-Cotes closed formulas were chosen here (Abramowitz and Stegun, 19721, but orthogonal polynomial integration formulas would also be suitable. For tests with less than 11 data points (P < 111, the Newton-Cotes formula with P points was employed. For P L 11,the 11-point formula was used. If the points were not equally spaced for P < 11, or if P 2 11, interpolation was required, as the Newton-Cotes formulas demand equal spacing. For interpolation, we chose the Lagrange method (Carnahan et al., 1969).The only pitfall in the numerical solution of the macroscopic balances of this work is interpolation. When the data contain rough figures, for example, 80,80,79 "C, instead of a smoother series, for example, 80.0, 79.5, 79.0 "C, numerical interpolation with higher order polynomials may yield undesirable maxima and minima near the interpolation point. Hence, the interpolated value may be quite different from what one would expect. This, of course, translates into erroneous integrations. As a general rule, linear or quadratic interpolation polynomials eliminate this problem. For the analytical solution, we used eqs 19 and 20 to calculate the heat-transfer coefficient for each jacket or eq 19 to evaluate the overall heat-transfer coefficient. The latter is obtained by making U, = U,l = UJ2 in the various a constants. A multivariable Newton-Raphson method (Press et al., 1992) can be used to compute UJl and UJz.Single-variable Newton-Raphson method may be employed to yield the coefficient U,.The convoluted integrals need to be obtained with open formulas, since
t (min) 0 1 2 3 4 5 6 7 8
9 10 11 12 13
Tji(t)("C) 17 17 17 17 18 18 18 18 18 18 18 18 18 18
Tje(t)("C) 25 25 25 24 24 24 24 24 24 24 24 24 24 24
T d t ) ("C) 65 64 63 62 61 60 59 59 58 57 56 56 55 54
the value of the integrand a t t = tf is quite different from those inside the interval. Newton-Cotes opentype formulas (Abramowitz and Stegun, 1972) were used here. With the analytical method, the thermal consistency must be very close to 1, or numerical divergence may occur.
Examples and Discussion Three examples will be presented to show the following: (1)that macroscopic balances are preferred over analytical solutions from a numerical strategy viewpoint; (2) that macroscopic balances may be used t o calculate the overall heat-transfer coefficient even if the service fluid flow rates are unknown; and (3) that reactor cooling capabilities can be assessed for peak exotherms. The vessel and service fluids in the examples use water, with C, = 4184 J/(kg-K)and p = 1000 kg/m3. Example 1. A batch reactor with a working volume of 15 m3 has two identical jackets, each with a heattransfer surface area of 8 m2 and a water flow rate of 60 m 3 k . Assume that there is no heat input from the agitator. The temperature data, the same €or both jackets, are given in Table 1. The data were used with three models: macroscopic (eq 1l),averaged differential (eq 18), and differential (eq 19). Figure 2 plots the overall heat-transfer coefficients for the three models and the thermal consistency with an increased number of data points. For example, an abscissa of 10 indicates that data up to 10 min were used in the calculations. Clearly, eq 11yields the most consistent results at any time t. Equations 18 and 19 give varying results, synchronized with the fluctuations in thermal efficiency.
Ind. Eng. Chem. Res., Vol. 34, No. 12, 1995 4676 Table 2. Temperature Data for Example 2 t (min) 0
T d t )("C) 18 18 19 19 19 19 18 18 18 19
1
2 3 4 5 6 7 8 9
Tcei(t) ("C) 23 23 23 23 23 23 23 22 22 22
Ted) ("C)
24 24 24 24 24 24 24 23 23 23
Tdt)("C) 67 66 65 64 63 61 60 59 58 57
This does not mean that the differential methods are not acceptable. The maximum deviation between the results obtained with eqs 11 and 19 is 12%, which is well within the realm of experimentatal error. However, the macroscopic balances are preferred in view of low fluctuations with the number of data points and easier mathematical treatment. The reason for this robustness is that the macroscopic models cover all values of vessel temperature, from T&) to T&), as seen in the denominator of eq 11. The integration of the differential models, on the other hand, leads to equations that contain only T,(to) and T&), as observed in eqs 18 and 19, making the solution susceptible to small variations in thermal consistency. Example 2. In this example, we show that it is possible to perform heat calculations even if the flow rates of service fluid are unknown. The thermal consistency cannot be calculated under these circumstances. Only the overall heat-transfer coefficient and the total flow rate can be determined. Table 2 presents experimental data for a vessel in a batch process with two coils. No heat is generated internally. The cooling water has an inlet temperature Tci(t)for both coils, and the exit temperatures for each coil are Tc,l(t) and Tce2(t).The heat-transfer surfaces of the coils are A,1 = 6.0 m2 and Ac2 = 7.5 m2. We use eq 15 for two coils and no jackets, i.e., M = 0 and N = 2. With the thermal consistency assumed to be unity, HTis given by the lefthand side of eq 6 with q v and h(t)set to zero. The total flow rate of service fluid in a vessel with jackets and coils is calculated from eq 22, , which is derived from eqs 11 and 12:
L
(22)
+
For this example, u = ucl uC2. The solution is U = 1397W/(m2*K>Bndq = i36.2 m3k.Hence, by assuming 8 = 1, we can calculate the overall heat-transfer coefficient and the total flow rate of the service fluid through the entire vessel. The overall heat-transfer coefficient calculated by the quick method of eq 18 agrees very well with the preceding results: 1403 W/(m2-K). Example 3. The overall heat-transfer coefficient obtained for the chemical reactor of example 1 can be
used to assess the feasibility of safely running exothermic processes in that equipment. Via eq 11,a U value of 1600 W/(m2*K)was determined. For a n exothermic process at 60 "C, assuming a jacket temperature of 21 "C and a reaction mixture density of 1000 kg/m3, the maximum heat removal rate is [1600 W/(m2*K)x 16 m2 x (60-21) KY15 000 kg = 66.56 Wkg. The advantage of the arithmetic mean can be realized with this example. Only one temperature needs to be assumed for the jacket. The logarithmic mean requires the guessing of two temperatures, one at the inlet and one at the exit of the jacket, and it shows no numerical advantage over the arithmetic mean. For example, if for the service fluid we assume inlet and exit temperatures of 17 and 25 "C, the logarithmic mean is 38.86 "C, compared to 39 "C for the arithmetic mean. The heat generation at peak exotherm for a specific composition can be determined experimentally with lab equipment, such as the RC1 Reaction Calorimeter by MettlerToledo. There are many publications on the usage of the RC1, such as Grob et al. (19871, Jacobsen (19901, and Landau and Blackmond (1994). The RC1 data can be used to calculate the cooling demand at peak exotherm in wattskilogram, which must be safely lower than the cooling capacity of the reactor, as determined earlier. If the cooling test in the plant equipment is done with water or an organic solvent, experience will dictate how the U value of the test liquid relates to the coefficient of the actual reaction mixture. U values for the test liquid and reaction mixture can be obtained in the reaction calorimeter, from which a relationship is established on the basis of a number of experiments.
Conclusions Sensible heat models were developed to determine heat-transfer coefficients of agitated vessels equipped with coils and jackets. Differential and macroscopic methods were presented. The macroscopic balances are preferred because their mathematical treatment is simpler and they are much less sensitive to experimental data variability. In addition, we can generate macroscopic balances for any number of jackets and coils. Microscopic balances, on the other hand, are specific to the number of heat-transfer units. Four units is the maximum workable number, and the multipage solution is extremely sensitive to thermal consistency deviations from unity. It is possible to calculate individual heat-transfer coefficients if each heat-transfer unit is equipped with a flow meter and temperature sensors, upstream and downstream of the reactor. It is not uncommon, however, to have unknown service fluid flow rates in an industrial setting. The macroscopic balances may be used to calculate overall heat-transfer coefficients even under this circumstance. The methods of this work can be used to assess chemical reactor cooling capabilities at peak exotherm, in combination with laboratory calorimetric data. Finally, a simple equation was written (eq 18)that can be employed to do quick heattransfer calculations in the field, providing reasonable estimates for U.
Nomenclature A = heat-transfer surface area of coil or jacket C , = specific heat h = heat input into the vessel due to reaction andlor
agitation
4576 Ind. Eng. Chem. Res., Vol. 34,No. 12, 1995
HT= defined in eq 14 L = coil length m = index for jackets, m = 1,M M = number of jackets n = index for coils, n = 1,N N = number of coils P = number of data points collected for each variable, e.g., time and temperature q = volumetric flow rate of service fluid in the jackets and coils t = time T = temperature U = surface-averaged heat-transfer coefficient V = volume x = length coordinate in the coils, 0 at the inlet and L at the exit
Greek Letters aj,
= (UjmAjdl(pjmCpjmVyn)
adm = ( U , m A j d pjm
= q j m 1VJY
y = defined in
l(pvCpvVv)
eq 10
6 = thermal consistency, eq 13 p = density
Subscripts c = coil cen = exit of coil n cin = inlet of coil n cn = coil n cna = average for coil n f = final, end of test j =jacket jem = exit of jacket m jim = inlet of jacket m jm =jacket m jma = average for jacket m
0 = initial, beginning of test v = vessel
Literature Cited Abramowitz, M.; Stegun, I. A. Handbook of Mathematical Functions; Dover: New York, 1972 (10th printing); pp 886-887. Carnahan, B.; Luther, H. A.; Wilkes, J. 0. Applied Numerical Methods; John Wiley: New York, 1969; pp 27-34. Grob, B.; Riesen, R.; Vogel, K. Reaction Calorimetry for the Development of Chemical Reactions. Thermochim. Acta 1987, 114,83-90. Haam, S.; Brodkey, R. S.; Fasano, J. B. Local Heat Transfer in a Mixing Vessel Using Heat Flux Sensors. Znd. Eng. Chem. Res. 1992,31,1384-1391. Haam, S.; Brodkey, R. S.; Fasano, J. B. Local Heat Transfer in a Mixing Vessel Using a High-Efficiency Impeller. Znd. Eng. Chem. Res. 1993,32,575-576. Jacobsen, J. P. Reaction Calorimeter. A Useful Tool in Chemical Engineering. Thermochim. Acta 1990,160,13-23. Kumpinsky, E. Experimental Determination of Overall Heat Transfer Coefficient in Jacketed Vessels. Chem. Eng. Commun. 1992,115,13-23. Landau, R. N.; Blackmond, D. G. Scale Up Heat Transfer Based on Reaction Calorimetry. Chem. Eng. Prog. 1994,November, 43-48. Mueller, A. C. Heat Exchangers. In Handbook of Heat Transfer; W. M. Rohsenow, J. P. Hartnett, Eds.; McGraw-Hill: New York, 1973; Sect. 18, p 2. Press, W. H.; Vetterling, W. T.; Teukolsky, S. A.; Flannery, B. P. Root Finding and Nonlinear Sets of Equations. In Numerical Recipes in FORTRAN, 2nd ed.; Cambridge University Press: Port Chester, NY,1992; Chapter 9, pp 372-375.
Received for review April 26, 1995 Accepted August 8, 1995@
IE9 5026 50 Abstract published in Advance A C S Abstracts, November 15, 1995. @
