938
Ind. Eng. Chem. Res. 1996, 35, 938-942
Heat-Transfer Coefficients in Agitated Vessels. Latent Heat Models Enio Kumpinsky† R&D Department, Ashland Chemical Company, P.O. Box 2219, Columbus, Ohio 43216
Latent heat models were developed to calculate heat-transfer coefficients in agitated vessels for two cases: (1) heating with a condensable fluid flowing through coils and jackets; (2) vacuum reflux cooling with an overhead condenser. In either case the mathematical treatment, based on macroscopic balances, requires no iterative schemes. In addition to providing heat-transfer coefficients, the models predict flow rates of service fluid through the coils and jackets, estimate the percentage of heat transfer due to latent heat, and compute reflux rates. Introduction In an earlier publication (Kumpinsky, 1992), we discussed the use of dynamic energy balances to determine heat-transfer coefficients in agitated vessels. Later (Kumpinsky, 1995) we compared differential and macroscopic balances, showing that the latter are more advantageous to determine heat-transfer coefficients. Indeed, they can be used with any number of jackets and coils, they are less susceptible to fail due to data inconsistencies, and the mathematical treatment is simpler. In the current work we apply the concept of macroscopic balances to processes involving latent heat. Suzuki (1986) summarized the theory to calculate heattransfer coefficients in condensation processes. That treatment is very useful when designing new equipment. However, contamination of the surfaces can have a marked effect on the actual heat-transfer coefficient [for example, see Kaminski (1986)]. In the present work we present methods to obtain the coefficients using experimental data, thus taking into account any contamination that might affect heat transfer in the vessel. Two cases are considered in this paper, i.e., heating with vapor and cooling with vacuum reflux using an overhead condenser. When working with chemical reactors, it is many times imperative to maintain consistent cycle times to avoid product variability. Slight and persistent reductions in heat-transfer capabilities may lead to longterm shifts in product quality. The methods of this work can be used to detect such changes. In addition to applications in distillation, overhead condensers can be used to cool the contents of chemical reactors. A reflux solvent is continuously boiled out of the reactor and returned as a liquid, thus utilizing latent heat of vaporization as the cooling source. This technique can be employed to keep the temperature constant during an exothermic process or to quickly cool the charge under vacuum to terminate a chemical reaction. An overhead reflux condenser, usually of the shell-and-tube type, can foul outside and inside the tubes, thus reducing heat-transfer capabilities. It fouls outside as a result of corrosion or contamination from the cooling fluid. It fouls inside as the process fluid corrodes the surfaces, and the reflux solvent carries over mists containing solids, which deposit on the tube walls. It is desirable to periodically test the overhead condenser to ensure that it can properly perform. Examples are provided to illustrate the use of the techniques proposed in this paper. The examples use water vapor as the latent heat source. To make the †
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mathematical treatment easier, the tabulated data of Liley et al. (1984) was converted into SI units and curve fitted to simple forms (see eqs 11 and 19). The density and specific heat of the liquid phase were assumed to be constant. Energy Balances Heating with Vapor. In our previous publication (Kumpinsky, 1995) we discussed the advantages of macroscopic balances over the solution of differential equations when computing heat-transfer coefficients. Hence, in this paper we will follow the macroscopic approach. The ensuing energy balance is written for a vessel heated by a saturated vapor flowing through coils and jackets with a certain inlet quality, as shown in Figure 1: M
t Ajm{fjm∫t [Tjim(t) ∑ m)1 f
FvCpvVv[Tv(tf) - Tv(t0)] ) U{ Tv(t)] dt + (1 - fjm)
0
∫tt (∆Tjm)ln dt} + f
0
M
t Acn{fcn∫t [Tcin(t) - Tv(t)] dt + ∑ m)1 f
0
(1 - fcn)
∫tt (∆Tcn)ln dt}} - FvCpvqv∫tt [Tv(t) t Tvi(t)] dt + ∫t h(t) dt f
f
0
0
f
0
(1)
where the mean logarithmic temperature differences are given by:
( (
(∆Tjm)ln ) [Tjim(t) - Tjem(t)]/ln
(∆Tcn)ln ) [Tcin(t) - Tcen(t)]/ln
) )
Tjim(t) - Tv(t)
Tjem(t) - Tv(t)
Tcin(t) - Tv(t)
Tcen(t) - Tv(t)
(2)
Note that we are using a constant overall heattransfer coefficient for all surfaces. Furthermore, eq 1 assumes that the coefficient for the condensing vapor is the same as that for the condensate. Both simplifications are obviously not true on a microscopic scale, but the key idea is that the heat-transfer coefficient of eq 1 provides the means to assess, on average, how well heat is transferred through the walls. As discussed before (Kumpinsky, 1995), the definition of U is arbitrary and it depends on the form of the temperature difference. Definition consistency from test to test is required for comparison purposes. For each © 1996 American Chemical Society
Ind. Eng. Chem. Res., Vol. 35, No. 3, 1996 939
{
fjm )
}
∫ [T (t) - T (t)] dt ∫ [T (t) - T (t)] dt ∫ λ [T (t)]χ (t) dt∫ (∆T ) dt
Cpjm 1+
tf
tf
jim
t0
jem
tf
v
tf
jm
t0
m′jm ) UAjm
jim
t0
jm
jim
jm ln
t0
-1
(7)
{
∫tt λjm[Tjm(t)]χjim(t) dt + ∫tt [Tjim(t) - Tv(t)] dt t Cpjm∫t [Tjim(t) - Tjem(t)] dt ∫tt (∆Tjm)ln dt f
0
f
}
0
f
0
f
0
-1
(8)
Coils (n ) 1 to N): Figure 1. Representation of an agitated vessel with coils and jackets for heating with a vapor.
{
fcn ) Cpcn
1+
jacket we have two equations, one for the saturated vapor phase and one for the liquid phase.
∫tt λjm[Tjm(t)]χjim(t) dt ) t UfjmAjm∫t [Tjim(t) - Tv(t)] dt;
vapor: m′jm
f
0
m ) 1 to M (3)
∫t [Tjim(t) - Tjem(t)] dt ) t U(1 - fjm)Ajm∫t (∆Tjm)ln dt; m ) 1 to M
liquid: Cpjmm′jm
∫tt λcn[Tcn(t)]χcin(t) dt ) t UfcnAcn∫t [Tcin(t) - Tv(t)] dt;
(4)
n ) 1 to N (5)
∫tt [Tcin(t) - Tcen(t)] dt ) t U(1 - fcn)Acn∫t (∆Tcn)ln dt; n ) 1 to N Cpcnm′cn
0
f
t0
cn ln
∫tt λcn[Tcn(t)]χcin(t) dt + ∫tt [Tcin(t) - Tv(t)] dt t Cpcn∫t [Tcin(t) - Tcen(t)] dt ∫tt (∆Tcn)ln dt
(9)
f
0
f
}
0
0
-1
(10)
With fjm and fcn known from eqs 7 and 9, we go back to eq 1 to calculate the overall heat-transfer coefficient U. We can then evaluate the mass flow rates m′jm and m′cn using eqs 8 and 10. The enthalpy of vaporization of the service fluid is obtained from the literature. For water, the following correlation is generated by curve fitting, based on steam tables in Liley et al. (1984):
(6)
The symbols λ[T(t)] mean that the latent heat of vaporization is a function of the vapor temperature. Equations 1 and 3-6 is a set of 1 + 2M + 2N equations with unknowns U, m′jm, fjm, m′cn, and fcn. Equation 1 provides an overall heat-transfer coefficient, averaged over all coils and jackets. If mass flow rates m′jm and m′cn are known, heat-transfer coefficients can be calculated for individual coils and jackets. If the fluid leaving a heat-transfer unit is not fully condensed, its inlet and exit temperatures are the same. In this case, fjm and fcn are equal to 1, and all the heat transfer to the process side is due to vapor condensation. If full vapor condensation and subcooling do not occur, eqs 4 and 6 are eliminated, thus simplifying the calculations. Combining eqs 1 and 3-6, assuming condensation and subcooling, we find the unknowns for the jackets (2M) and coils (2N):
Jackets (m ) 1 to M):
cin
-1
λ(T) ) 2323 - 1.669T - 4.342 × 10-3T2; 35 < T < 205 °C (11)
f
0
cn
f
f
f
{
v
tf
cn
f
o
0
liquid:
m′cn ) UAcn
cin
t0
0
For the coils we also have two equations:
m′cn
cen
tf
0
o
tf
cin
t0
tf
f
vapor:
tf
t0
f
0
}
∫ [T (t) - T (t)] dt ∫ [T (t) - T (t)] dt ∫ λ [T (t)] χ (t) dt∫ (∆T ) dt
where λ is in kJ/kg and T is in °C. In our work on sensible heat (Kumpinsky, 1995) we wrote a simplified equation for U of a batch process with no heat input. The treatment is valid for latent heat as well. This calculation is based only on vessel process data and inlet and exit temperatures of the service fluid. It can be used as a quick check in the field, as the mathematical treatment is very simple. The equation is the following:
FvCpvVv
U)
(
ln
M
(
×
N
∑ Ajm + n)1 ∑ Acn)(tf - t0)
m)1 M
N
M
N
)
∑ Ajm[Tv(t0) - Tjma] + n)1 ∑ Acn[Tv(t0) - Tcna] m)1 ∑ Ajm[Tv(tf) - Tjma] + n)1 ∑ Acn[Tv(tf) - Tcna] m)1
where the average temperatures are given by:
(12)
940
Ind. Eng. Chem. Res., Vol. 35, No. 3, 1996 ν
[Tjim(ti) + Tjem(ti)], ∑ i)1
Tjma ) 2ν/
m ) 1, M (13)
ν
[Tcin(ti) + Tcen(ti)], ∑ i)1
Tcna ) 2ν/
n ) 1, N
The summation symbols vanish in eq 12 if the temperature sensors are located in the common lines, to and from the vessel, in which case only a single temperature in and a single temperature out are available. The logarithmic term is reduced to a simple ratio of Tv(to) Ta and Tv(tf) - Ta. Ta is the mean temperature of the service fluid, calculated with the equivalent form of eq 13, using the global inlet and exit temperatures of the service fluid. Vacuum Cooling with Reflux. Figure 2 shows the vessel configuration for this case. The energy balance for the vessel is given by:
dTv - h(t) ) UdAd(∆T)ln ) -r′(t) {λ[Tg(t)] + FvCpvVv dt Cpv[Tgi(t) - Tge(t)]} (14)
reflux rate for the test is obtained from
FvCpvVv[Tv(tf) - Tv(t0)] -
and for the condenser,
FdCpdVd
Figure 2. Representation of an agitated vessel equipped with an overhead condenser for reflux.
r′mean )
dTd ) -UdAd(∆T)ln - FdCpdqd[Tde(t) - Tdi(t)] dt (15)
∫tt h(t) dt f
0
∫tt {λ[Tgi(t)] + Cpv[Tgi(t) - Tge(t)]} dt t -FdCpdqd∫t [Tde(t) - Tdi(t)] dt ∫tt {λ[Tgi(t)] + Cpv[Tgi(t) - Tge(t)]} dt )
f
0
f
0
f
(18)
0
where the mean logarithmic temperature difference is given by
(∆T)ln ) [Tde(t) - Tdi(t)]/ln
(
)
Tde(t) - Tgi(t)
Tdi(t) - Tge(t)
(16)
If only one among Tde(t) or Tdi(t) is known, then the logarithmic mean can be replaced with an arithmetic difference, i.e., Tde(t) - [Tge(t) + Tgi(t)]/2 or Tdi(t) - [Tge(t) + Tgi(t)]/2. To obtain macroscopic balances, we integrate eqs 14 and 15 term by term. The term on the left-hand side of eq 15 is much smaller than each term on the right-hand side, so we can neglect the former. The condenser overall heat-transfer coefficient for vacuum cooling is then:
FvCpvVv[Tv(tf) - Tv(t0)] Ud )
Ad
∫tt h(t) dt
Again we can check for thermal consistency by obtaining the ratio of the two right-hand side terms of eq 18. In order to use eq 11 to determine the latent heat of vaporization of water, we must know the vapor/liquid temperature in the condenser, Tg(t). The vessel or condenser absolute pressure during reflux experiments can be easily measured, and from the pressure data we obtain Tg(t). For water we use saturated steam tables, and for solvents, vapor pressure data. The correlation below, extracted from the data in Liley (1984), covers most practical applications involving water:
Tg(P) ) 9.848(log P)2 + 24.62(log P) + 11.12; 6 < P < 200 kPa absolute (19) where Tg(t) is in degrees Celsius and P is in kPa absolute.
f
0
∫t (∆T)ln dt t -FdCpdqd∫t [Tde(t) - Tdi(t)] dt t Ad∫t (∆T)ln dt
Numerical Methods
)
tf
0
f
0
f
(17)
0
The ratio of the two right-hand sides of eq 17 yields the thermal consistency of the experimental data. Deviations from unity will almost invariably occur in practice, and they can be tolerated to a certain extent. A good guideline is to aim for a thermal consistency between 0.9 and 1.1. The experimental data should be questioned outside this range. The instantaneous reflux rate r′(t) is calculated from eq 14 once Ud is known. The mean
The solution of the equations in this work does not require iterations. The same considerations on macroscopic balances made in our previous publication (Kumpinsky, 1995) concerning numerical integration and interpolation are valid here. Newton-Cotes closed formulas were used for integration (Abramowitz and Stegun, 1972). The equally-spaced integrands were obtained through Lagrange interpolation (Carnahan et al., 1969). Examples Two examples will be presented here: one for heating with steam in a vessel with one coil and one jacket; another for the cooling with an overhead reflux con-
Ind. Eng. Chem. Res., Vol. 35, No. 3, 1996 941
Figure 3. Overall heat-transfer coefficient for example 1, calculated with data up to time t using the macroscopic balance (eq 1) and the averaging method (eq 12); χ(t) ) 1.
Figure 5. Percentage of the heat-transfer surface area of the jacket and coil of example 1, subjected to latent-heat exchange, and contribution of latent heat to the total heat transferred to the vessel fluid. Table 2. Temperature Data for Example 2
Figure 4. Mass flow rate of service fluid through the coil and jacket of example 1. Table 1. Temperature Data for Example 1 t (min)
Tji(t), Tci(t) (°C)
Tje(t) (°C)
Tce(t) (°C)
Tv(t) (°C)
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
121 121 122 123 124 125 125 125 125 125 125 125 125 125 125 125
65 65 67 67 68 70 70 71 72 74 75 76 78 79 81 81
73 74 74 76 77 78 79 81 82 83 84 86 87 88 90 91
50 51 52 53 55 56 57 58 59 61 62 63 65 66 67 68
denser under vacuum. In all cases the fluid in the vessel and in the heat-transfer units is water, with Cp ) 4184 J/(kg K) and F ) 1000 kg/m3. Heat added by agitation can be neglected, i.e., h(t) ) 0. Example 1. A reactor with a working volume of 7.5 m3 has a standard jacket with a heat-transfer surface area of 17.8 m2 and an internal coil with a surface area of 12.3 m2. The time-temperature data for this example is provided in Table 1. Results. Figure 3 shows the overall heat-transfer coefficient with data up to time t, using the macroscopic balance, eqs 1 and 7-10, and the averaging model, eq 12. There is an excellent agreement between the models for this case. Equation 12 is more convenient if the user is only interested in the overall heat-transfer coefficient. If more information is required (see Figures 4 and 5), the macroscopic balance must be employed. Figure 4 exhibits the estimated steam flow rates through the jacket and coil using eqs 8 and 10, respectively. The information on flow rates provides the process engineer with the background required to size flow meters and to resize pipes to increase capacity. Figure 5 shows the time-averaged percentage of the surface area where
t (min)
Tdi(t) (°C)
Tde(t) (°C)
Tv(t) (°C)
P (kPa abs)
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14
28 28 28 28 28 28 28 28 28 28 28 28 28 28 28
38 38 38 37 37 37 37 37 36 36 36 36 36 36 36
99 97 94 92 90 87 85 83 81 79 77 75 74 72 71
94 86 77 73 67 60 56 50 48 43 39 35 33 30 28
latent-heat exchange takes place, up to time t. It also displays the contribution of latent heat to the thermal gain by the vessel fluid. As the vessel temperature rises, fj and fc go up as a result of the increase in servicefluid exit temperature. The % Latent curve in Figure 5 represents the percentage of latent heat (fjm and fcn terms) in eq 1, with respect to the sum of latent (fjm and fcn) and sensible [(1 - fjm) and (1 - fcn)] heat terms. As expected, the latent-heat contribution is dominant and it increases as the vessel temperature rises. This analysis would be greatly enhanced if fluid flow rates could be measured. Individual heat-transfer coefficients could be calculated and data accuracy might be checked. However, expenses with the installation of metering equipment may not be justifiable. Example 2. A well-mixed chemical reactor with a working volume of 10 m3 is equipped with a shell-andtube reflux condenser with heat-transfer surface area of 40 m2. A total of 130 m3/h of cooling water flows through the condenser shell. Sensible heat in the condenser and heat input from the agitator can be neglected. Table 2 provides the time-dependent data for this example. Results. Using eq 17 one finds a Ud value of 708 W/(m2 K) based on vessel data and 696 W/(m2 K) based on condenser data. Here we define the thermal consistency as the ratio, based on the experimental data, of the heat removed from the vessel and the heat gained by the cooling fluid in the overhead condenser, i.e., the two numerators of eq 17. The numerical result is 1.016; that is, there is a difference of only 1.6% between the two values. The mean reflux rate can be calculated by eq 18 using steam data from the combination of eqs 11 and 19. It is equal to 0.65 kg/s based on the vessel data and 0.64 kg/s based on the condenser data. Once the overall heat-transfer coefficient is known, we obtain the instant reflux rate by means of eq 14, as follows. Tg(t)
942
Ind. Eng. Chem. Res., Vol. 35, No. 3, 1996 T ) temperature U ) heat-transfer coefficient V ) volume Greek Symbols (∆T)ln ) mean logarithmic temperature difference λ ) latent heat of vaporization ν ) number of data points collected for each variable, e.g., time and temperature F ) liquid density χ ) quality of the service fluid at the inlet of the jackets or coils
Figure 6. Instant reflux rate of example 2.
is obtained from eq 19, since the absolute pressure in the condenser is known. Then, using eq 11, we calculate the latent heat of vaporization, where T ) Tg(t). The results are presented in Figure 6. The instant reflux rate provides useful information for equipment redesign and quantifies the reduction in heat-transfer rates as the temperature in the vessel drops. Conclusion Mathematical models were developed to determine heat-transfer coefficients in agitated vessels that utilize latent heat as the cooling or heating source. They can be used to assess heat-transfer capabilities in processes that involve heating with a condensable vapor in coils and jackets and vacuum cooling with an overhead condenser. The model for heating also allows the user to determine the flow rate of vapor through the coils and jackets, as well as the fraction of heat transfer due to latent and sensible heats. The contribution from latent heat is expected to be several times greater than the one from sensible heat. An average-temperature model was provided (eq 12). It can be used to calculate the overall heat-transfer coefficient when heating the charge with a vapor in the coils and jackets. The agreement of the average-temperature model with the macroscopic model is excellent in example 1. The model for reflux vacuum cooling can be employed to determine the overall heat-transfer coefficient and the reflux rate, both of which are indicators of heat-transfer performance. The instant reflux rate shows the decay in heat transfer as the charge is cooled, and it can be applied to process improvement, for example, in the redesign of the condenser riser and return lines. Nomenclature A ) heat-transfer surface area Cp ) specific heat f ) fraction of surface area, time-averaged, where latentheat exchange takes place h ) heat input into the vessel due to reaction and/or agitation m′ ) mass flow rate of service fluid through the coils or jackets P ) absolute pressure in the vessel and condenser r′ ) reflux rate t ) time
Subscripts c ) coil cen ) exit of coil n cin ) inlet of coil n cn ) coil n cna ) average for coil n d ) condenser de ) condenser exit (service fluid) di ) condenser inlet (service fluid) f ) final ge ) fluid at the condenser exit (return to the vessel) gi ) vapor of vessel fluid at the condenser inlet j ) jacket jem ) exit of jacket m jim ) inlet of jacket m jm ) jacket m jma ) average for jacket m m ) index for jackets n ) index for coils M ) number of jackets N ) number of coils 0 ) initial
Literature Cited Abramowitz, M.; Stegun, I. A. Handbook of Mathematical Functions; Dover: New York, 1972; 10th print, pp 886-887. Carnahan, B.; Luther, H. A.; Wilkes, J. O. Applied Numerical Methods; John Wiley: New York, 1969; pp 27-34. Kaminski, D. A., Ed. Conduction. In General Electric Heat Transfer Data Book; The General Electric Co.: Schenectady, NY, Dec 1986. Kumpinsky, E. Experimental Determination of Overall Heat Transfer Coefficient in Jacketed Vessels. Chem. Eng. Commun. 1992, 115, 13-23. Kumpinsky, E. Heat-Transfer Coefficients in Agitated Vessels. Sensible Heat Models. Ind. Eng. Chem. Res. 1995, 34, 45714576. Liley, P. E.; Reid, R. C.; Buck, E. Physical and Chemical Data. In Perry’s Chemical Engineers’ Handbook; Green, D. W., Ed.; McGraw-Hill: New York, 1984; Section 3, p 237. Suzuki, K. Heat Transfer and Hydrodynamics of Two-Phase Annular Flow. In Encyclopedia of Fluid Mechanics; Cheremisinoff, N. P., Ed.; Gulf: Houston, TX, 1986; Vol. 3, pp 1356-1391.
Received for review August 11, 1995 Revised manuscript received November 17, 1995 Accepted December 5, 1995X IE950507W
X Abstract published in Advance ACS Abstracts, February 1, 1996.
