Ind. Eng.
done numerically in real time, say, by a Runge-Kutta scheme. This is, of course, a major computational burden, but there is, practically speaking, the whole sample interval available. One could also design the reconstructor directly in discrete time, but then the nonlinear process model must be integrated in real time resulting in the same computational requirements. Because of the relatively large computional requirements in the nonlinear case, the reduced-order reconstructor is comparatively more attractive in the nonlinear case than in the linear case. High-quality (low noise content) measurements are therefore desirable for nonlinear state reconstruction. Even with the required real-time integration, the proposed nonlinear reconstructor is considered a practical solution to the problem of unmeasured quantities in dynamic processes. Acknowledgment The author expresses his appreciation for the financial assistance given to him by the Academy of Finland. Nomenclature A = state matrix AF = reconstructor state matrix = A - FFeC B = input matrix for u Bd = input matrix for d B F = reconstructor input matrix for u C = output matrix for measurements y C = dimensionless reactor concentration normalized by Cref = steady-state feed concentration Cret = reference concentration = steady-state feed concentration c = heat capacitance d = unknown disturbances (Figure 1) FB = feedback controller gain matrix FF = reconstructor gain matrix for measurement feedback term FX = state weighting matrix in performance criterion J FU = control effort weighting factor in performance criterion
J f() = nonlinear state-space model, eq 5a fi(), f2()= partitioned f(),eq 8a,b
gz() = measurement feedback function for reduced-order reconstructor h ( ) = nonlinear output function, eq 5b
Chem. Fundam., Vol. 18, No. 4, 1979 333
J = quadratic performance functional N = model order OBS = observability matrix, eq 5 Q = rate of heat removal q = dimensionless rate of heat removal = Q/(feed rate)-Cr,,&l T = dimensionless temperature = [c.(temp in K)]/C,,fAH t = time normalized by fluid residence time = [(time).(feed rate)]/ (volume) u = manipulated inputs, known disturbances (Figure 1) x = state vector x1 = unmeasured part of x in eq 8 y = available process measurements (Figure 1) z = nonmeasurable process variables (Figure 1) Greek Letters AH = enthalpy of reaction 7 = state vector of reduced-order reconstructor Subscripts A = augmented (by unknown disturbance) ss = steady-state value (constant) Other Markings tj = (d/dt)O ^ = estimate ' = deviation from steady-state value Literature Cited Aris, R., Amundson, N. R., Chem. Eng. Sci., 12, 121-131 (1958). Astrom, K. J., "Introduction to Stochastic Conkd Theory", pp 229, 267, 275-276, Academic Press, New York, N.Y., 1970. Brosilow, C. B., Tong, M., AIChE J., 24, 492-500 (1978). Joseph, B., Brosilow, C. B., AIChE J., 24, 485-492, 500-509 (1978). Luenberger, D. G., I€€€ Trans. Autom. Control, AC-16, 596-602 (1971). Lynch, E. B., Ramirez, W. F., AIChE J., 21, 799-804 (1975). Rhodes, I.B., I€€€ Trans. Autom. Control, AC-16, 688-706 (1971). Schweppe, F. C., "Uncertain Dynamic Systems", pp 402-408, Prentlce-Hall, Inc., Englewood Cliffs, N.J., 1973. Smith, H. W., Davlson, E. J., Proc. I€€, 119, 1209-1216 (1972). Takahashi, Y., Rabins, M. J., Auslander, D. M., "Control and Dynamic Systems", pp 84, 536-540, Addison-Wesley, Reading, Mass., 1970. Thau, F. E., Int. J. Control, 17, 471-479 (1973). Thau, F. E., Int. J. Control, 19, 143-148 (1974). Walknan, P. H., Foss, A. S., to be submitted to Ind. €ng. Chem. Fundarn. (1979). Wallman, P. H., Silva, J. M., Foss, A. S. "Multivariable Integral Controls for Fixed-Bed Reactors", presented at the AIChE National Meeting in Miami, Fla., 1978. Wells, C. H., AIChE J., 17, 966-973 (1971).
Receiued for review September 11, 1978 Accepted June 7, 1979
A Flow Calorimeter for Condensable Gases at Low Temperatures and High Pressures. 1. Design and Evaluation Plet H. 0. van Kasteren' and Hans Zeldenrust Koninklijke/Shell-Laboratorium,Amsterdam (Shell Research B. V.), Amsterdam-Noord, The Netherlands
A flow calorimeter has been developed for measuring enthalpy-temperature curves of condensable gases. Measurements can be carried out over small temperature intervals and at low temperatures and high pressures. Rather low mass flow rates are used so that recirculation of the sample is unnecessary. A correction for heat losses is made. The absolute error of the final results is within 1% .
Introduction In recent years interest in the liquefaction of natural gas, in particular for transportation purposes, has grown considerably. The concomitant increase in demand for economic design of liquefaction (and regasification) plants has created a need for improved basic thermodynamic
data. A particular need is for accurate enthalpy-temperature relationships (cooling curves) for condensable gas mixtures at the low temperatures (down to 115 K) and high pressures (up to 70 bar) encountered in such plants. In addition to their importance in design, such data are also useful for checking and improving the accuracy of existing
0019-7874/79/1018-0333$01.00/0 0 1979 American Chemical Society
334
Ind. Eng. Chem. Fundam., Vol. 18, No. 4, 1979
Table I. Survey of Flow Calorimeters
company or institute University of Michigan
max temp pressure, range, K bar 120-275 140
Union Carbide (Linde Co.)
77-300
100
Institute of Gas Techn. (IGT)
77-300
Air Products and Chemical Inc. PVT Inc. Shell Research (KSLA)
energy level WT:
J s'
flow rate, L h-'
accuracy, 5%
2000-4000
1
40
200
70
5-60
100-600
77-500
200
20-60
150-600
1.5 noncond. 3 condensable 0.4 at WT = 30 1.0 at WT = 3 5 at W , = 0.3 0.5
150-300 100-300
70 70
0.1-0.3
25
1 (see Table 11) 1
calculation procedures for prediction of cooling curves. We have accordingly developed a flow calorimeter to determine accurate cooling curves for both of these purposes. In part 2 of this series we present the basic data obtained with the flow calorimeter and compare them with predicted values. In this first part, we deal with design and operation of the calorimeter itself. When a gas is liquefied in discrete stages, such as cooling of the vapor, condensation, and subsequent cooling of the liquid, it is important to know the power requirements for each stage. As these requirements depend not only on the enthalpy change to be made but also on the temperature at which this change has to be realized, a detailed cooling curve is of prime importance. We set the following design specifications for our flow calorimeter. It must: (a) operate in the temperature range from 100 to 300 K; (b) be capable of handling condensable gas mixtures, including a twophase gas/liquid system; (c) operate at a constant pressure, within the whole temperature range, up to 70 bar irrespective of the vapor pressure of the gas mixture; (d) provide a detailed enthalpy-temperature relation (cooling curve) with an accuracy of about 1'70 at any temperature; (e) operate at rather low mass flow rates so that recirculation of the sample is not necessary. Survey of Flow Calorimeters for Condensable Gases at High Pressure and Low Temperature In essence, flow calorimeters consist of two heat exchangers connected in series, which are kept at an initial temperature Ti and a final temperature Tf, respectively. The enthalpy change of the sample is determined from the amount of heat that has to be removed from or supplied to one of the heat exchangers to maintain its temperature when the sample is passed through. A flow calorimeter does not operate adiabatically, since the sample inlet and outlet lines are connected to it. Therefore the heat exchange with the surroundings will be a function of the temperature, the flow rate, etc. Consequently, to obtain accurate enthalpy data, the heat loss W,at the operating conditions should either be accurately known or negligible relative to the total energy W T to be supplied (enthalpy change of the sample + heat loss). A survey of the operating conditions of a number of flow calorimeters is given in Table I. Most of the data have been taken directly from the literature (Jones et al., 1962; Mather et al., 1966; Manker et al., 1964; Jenkins and Berwaldt, 1963; Dolan et al., 1968; Clark and McKinley, 1967; Wilson and Barton, 1967; Saghal et al., 1965). However, in some cases the energy level WT and the flow rates were not specified, so we have derived these values from the experimental data given. All these flow calorimeters operate at a high energy level WT, which is obtained by using either high flow rates or large temperature differences, or both. The heat losses are then negligible,
reference Jones e t al.. Mather et al., Manker et al. Jenkins and Berwaldt Dolan et al. Clark and McKinley, Saghal et al. Wilson and Barton
which can be confirmed by experiments at different flow rates, the results of which should be identical. For comparison, the characteristics of the flow calorimeter to be described in the following sections are included. Enthalpy-Temperature Relation The enthalpy-temperature relation can be derived from the individual enthalpy changes measured experimentally. Two approaches are possible, dependent on the capacity of a calorimeter. (a) From Enthalpy Measurements for Large Temperature Changes. First the enthalpy change for a large temperature difference Ti - Tf is measured. Then one of the temperatures is changed, while the other is kept constant. For example, if the final temperature is shifted from Tn to T,, the enthalpy changes are given by
The difference between these two (large) quantities represents the enthalpy change between Tfl and TB, being a detail of the enthalpy-temperature curve to be constructed. (b) From Enthalpy Measurements for Small Temperature Changes. In this case the enthalpy change on going from Tn to Tn is measured directly. Then both temperatures are stepped up so that the adjacent part of the enthalpy curve is measured, and so on. Clearly, if these procedures are carried out at a relative accuracy of, say, 1% , both will give total enthalpy change values with the same degree of accuracy, procedure (a) because the total change is measured directly and procedure (b) because the relative error in the enthalpy change over a large temperature interval will be at least as low as that for the individual measurements. For details of a cooling curve, however, only procedure (b) can guarantee an accuracy of 1%; in the case of procedure (a), unless the large enthalpy changes are measured extremely accurately, the relative error in the change for a small temperature interval Tf2- Tf, can be rather large. To illustrate the latter point, we have collected some published data (for pure methane) of Wilson and Barton (1967) and of Jones et al. (1962) in Table IIA. It can be seen that the differences between the two sets of results are indeed about 1%, which agrees with the accuracy claimed for large temperature changes. However, when these data are used to calculate values for smaller temperature intervals, as given in Table IIB, the resultant figures show much greater discrepancies. Description of the Design Principle (See Figure 1). The flow system consists in essence of three heat exchangers, wound from copper tubing (2 mm internal diameter, 0.2 mm wall thickness) and connected in series, each provided with an electric
Ind. Eng. Chem. Fundam., Vol. 18, No. 4, 1979
Table 11. Enthalpy of Methane at 47.7 Bar
A. Large Temperature Intervals enthalpy change, J g-' temp, K . Wilson/ Jones difference, Ti Tf Barton et al. % 301.1 300.4 299.3 295.0
316.9 558.2 635.3 684.7
199.8 185.9 172.0 158.2 B.
312.6 562.6 635.3 680.4
+1.4 -0.8 0.0
+0.6
Small Temperature Intervals enthalpy change, J g-'
temp, K
Ti
Wilson/ Barton
Jones et al.
difference,
Ti 199.8 185.9 172.0
185.9 172.0 158.2
243.1 77.0 60.3
251.8 72.7 56.0
-3.5 +6.0 +7.7
NON-RETURN VALVE SAMPLE (GAS)
HEAT EXCHANGER I VACUUM CHAMBER
%
' PUXlLlARY LINES
ANNULAR SPACE -CENTRAL POST
HEAT SINK
HEAT EXCHANOER I[ SHIELD 1 THERMOMETER
THERMOMETER
The electrical energy supplied to H E I11 in order to maintain the desired temperature difference AT (= 21' 11 - TII)is measured continuously. The change in enthalpy of the sample, due to this temperature increase, can thus be calculated from the energy supplied to HE 111, the heat loss from HE I11 to the surroundings, and the sample flow rate. Temperature Measurements. The temperatures of H E I1 and H E I11 are measured with four terminal platinum-in-glass thermometers (Degussa Type P4) via a five decade Diesselhorst potentiometer (Tinsley Type 3589 R) equipped with a galvanometer-photocell amplifier in series with a light-spot galvanometer. The overall sensitivity amounts to IO-@ V per millimeter deflection, corresponding with a temperature change of a few millidegrees. The potentiometer and standard resistors are thermostated at 20 f 0.02 "C. Flow-Rate Capacity. The crucial point of the whole design is whether the flow rates permitted offer an acceptable ratio between the losses of H E I11 and the enthalpy change of the sample. If they do not, small changes in the losses will reduce the accuracy of the final results appreciably. Further, for a self-contained calorimeter, where the sample can be fed in at room temperature, the heat sink should be capable of draining the heat to be removed to the bath. At the maximum operating temperature of 300 K, no heat has to be removed from the sample. Thus any heat loss through the thermal resistance to the heat sink (at 77 K) has to be compensated by the electrical heater of HE I. Therefore the capacity of this heater determines the minimum value for the thermal resistance. This value in turn, however, limits the cooling capacity at low temperatures and so only permits flow rates which are below the acceptable level (determined by the ratio of losses to enthalpy change). To overcome these contradictory aims we have introduced a variable thermal resistance between HE I and the heat sink as follows. HE I is would on thin-walled stainless-steel tube, and thereby connected with the heat sink. This connection constitutes a high thermal resistance. A central post on the heat sink protrudes into this stainless-steel tube, thereby creating an annular space. In the high-temperature range this annular space is evacuated, but at lower temperatures helium gas is introduced. Because of the high thermal conductivity of helium and the large surfaces enclosing the annular space a low thermal resistance results. With this construction the system can handle flow rates up to about 25 gf h over the whole temperature range, which rates are still low enough to make recirculation of the sample unnecessary. Extra cooling capacity, if required, can be obtained by circulating a refrigerant through the annular space via the auxiliary lines. Sample Injection System (see Figure 2). In order to maintain a constant sample flow through the calorimeter, two injection systems were constructed, one for use at atmospheric pressure for liquids with normal boiling points above room temperature and another for use at pressures up to 70 bar for gases. The injection pump consists of two pistons covered with rolling membranes within a housing. By moving the two pistons towards each other a constant flow is obtained. Depending upon whether the sample is liquid or gaseous at room temperature, different gearboxes are mounted on the pump motor to obtain the required mass flow rate. A constant-pressure valve at the outlet of the calorimeter maintains the preset pressure in the system. The pressure
IIIT-~,!
GAS RETURN
Figure 1. Schematic diagram of flow calorimeter.
heater. It is accommodated in a vacuum chamber to eliminate heat exchange with the surroundings due to conduction and convection. The vacuum chamber is placed in a Dewar vessel, which can be filled with any coolant suitable for the temperature range of the measurements. In general, liquid nitrogen (77 K) is used for this purpose. A constant sample flow is maintained through the heat exchangers by means of a sample-injection system operating at a constant pressure adjustable up to 70 bar. The first heat exchanger, HE I, cools the gas from room temperature down to approximately the desired initial temperature. The heat to be removed is drained to the refrigerant bath (Dewar vessel) via a clamp, the heat sink, between one end of the H E I and the wall of the vacuum chamber. Heat exchanger I can be kept at any temperature between that of the coolant and room temperature. The second heat exchanger, HE 11, functions as a stabilizer, after the enormous temperature change of the gas mixture, and is set at a temperature which only differs by a few degrees from that of H E I. The third heat exchanger, HE 111, can be set at a temperature 0 to 20" above that of HE 11. T o reduce the radiation losses-and hence the temperature gradients-to an acceptable level, HE I1 and HE I11 are surrounded by shields kept at the same temperature as the corresponding heat exchangers.
335
336 Ind. Eng. Chem. Fundam., Vol. 18, No. 4, 1979
vi0
ul
vie
T
VENT
&
I
LOW-EOILIM
I
LIPUIDS AND OASES AT LOW OR HlOH PRESSURE
HIOH-(IOILlNO
LIQUIDS AT LOW PRESSURE
Figure 2. Flow sheet for liquids and gases.
difference across the rubber membranes is kept to a minimum by maintaining a constant nitrogen back pressure on the outsides. The actual pressures can be read from pressure gauges with an accuracy of 0.2 7%. The total sample volume of the pump is 0.75 L. Sample Flow Rate Measurement. For liquids a t atmospheric pressure and room temperature the outlet of the calorimeter is connected to a receiver. As soon as the calorimeter is in equilibrium and stationary operation, the outlet is switched to a special weighed receiver. After some time, usually 20 min, the sample stream is switched back again. The average mass flow rate is calculated directly from the amount of sample collected in the measured time interval. In the case of gases, the gas leaving the constant-pressure valve at atmospheric pressure is saturated with water before its volume is measured using a wet-gas meter. This gas meter is provided with a photocell device. Each time the indicator passes this photocell, viz. after passage of 1 L, this is recorded on the same recorder where the energy registration occurs. The average flow rate is determined from the total time elapsed for a given number of revolutions. After correction for the water content, barometric pressure and room temperature, the volume flow rate a t normal temperature and pressure of the sample is known. The mass flow rate is calculated from this value and the molecular weight of the sample. Design of the Heat Exchangers. The most important requirement for the heat exchanger design is that the residence time for all components of the mixture under
test, whether in the gaseous or the liquid state, should be the same. If this is not so, long equilibrium times will be needed after any change in temperature or flow rate. Moreover, stable control of all parameters governing the vapor-liquid equilibrium such as flow rate, temperature, and pressure is necessary. The design has been evaluated with a two-component mixture in a glass model system. It appeared that a slight broadening of the top (inlet) of the heat exchangers was necessary to prevent small vapor bubbles, created continuously in the top section, from moving upstream and causing a phase separation. This broadening of the heat exchanger inlets creates a velocity gradient which causes the small bubbles to coalesce and form a larger one. As soon as its diameter is larger than that of the tubing, the bubble is pushed by the liquid through the heat exchanger. In this way a regular and stable chain of vapor and liquid “packages” passes through the system. The turns of the helix of the heat exchangers are soldered together to minimize temperature gradients along them. The temperature sensors, platinum-in-glass resistance thermometers, are connected to the last three windings of each heat exchanger. They are deliberately not placed in the sample stream as this would disturb the flow pattern. To eliminate temperature differences between sample and heat exchangers, the latter are twice as large as theoretically necessary. Energy Measurement on Heat Exchanger 111. Heat exchanger I11 is really the most important part of the whole calorimeter since the change in enthalpy of the sample is derived from the heat supplied to HE I11 to maintain the preset temperature difference AT (= TIII- TII).In consequence, the accuracy with which the heat consumption is measured determines the final accuracy of the whole instrument. The heat supplied to HE 111, however, varies in time on account of fluctuations in either the temperatures of HE I1 and HE I11 or the flow rate. Average values over a certain time interval, 20 min, for the power W and the flow rate 7are therefore derived from the simultaneous energy and flow measurements, given by
where R( T ) = heater resistance of HE I11 at temperature T , W = average power, 7= average flow rate, E , = instantaneous voltage across R(T), and V , = instantaneous flow rate. This means that the instantaneous voltages have to be squared and subsequently integrated. The instantaneous voltages are measured by a partial-compensation method. The greater part of the voltage is compensated by a five decade Croyden potentiometer, the last decades of which are normally set to zero. The remaining fluctuating unbalance, usually less than 5% of the actual voltage, is fed to a potentiometer recorder (Servogor Type RE 512) which also registers the time integral of this deviation voltage. The millivolt range of this recorder must be selected to cover the actual voltage-fluctuations as closely as possible. It can be shown that under practical conditions the average energy level W can be calculated with sufficient accuracy with the relation (3)
where EcR = compensating voltage, F = time integral of the deviation signal, and A = the amplitude of the voltage fluctuations. Heat Losses. The ratio of the heat losses of heat exchanger I11 to the enthalpy change of the sample to be
Ind. Eng. Chem. Fundam., Vol. 18, No. 4, 1979
g P
tm-
m
m
s20.
I-4
337
P
s
3
W
W
I
I
/
L - 9 ,
lot
I
IO '
/ -CONDUCTION
RADIATION
0
I50
200
2 50 TEMPERATURE T ,,
300 K
Figure 4. Components of the heat loss of the empty calorimeter at AT = 10 K. 0
I
200
I50
2% TEMPERATURE T,
I 300 K
Figure 3. Heat loss of the empty calorimeter.
measured determines to what degree of accuracy the losses have to be known. Since in our calorimeter the flow rate is rather low (up to 25 g/h), correction for the heat losses will be required. We therefore evaluated the heat losses of the empty calorimeter to obtain data on their magnitude and the mechanisms involved and then investigated the effect of sample flow on this behavior. The heat losses as a function of temperature of the empty calorimeter are given in Figure 3. It follows from this figure that for AT = 10 K the loss ranges from 3 to 26 mW. These values have to be compared with the enthalpy change of a sample to be measured. If we assume a specific heat of the sample of 1.5 J/g, a flow rate of 25 g/h, and a temperature change of 10 K, the extra energy to be supplied to HE I11 with sample flow amounts to about 100 mW. It is clear that proper correction for the heat loss is required. The construction of the calorimeter is such that there are three mechanisms by which heat can be exchanged between HE I11 and its surroundings, viz.: (1)conduction to heat exchanger I1 via the interconnecting sample tube and the supports between HE I11 and HE 11; the amount of this leakage varies linearly with the temperature differences AT; (2) radiation between the return sample line (outlet) and the walls of the vacuum chamber which are a t a constant temperature T M ;its contribution is proportional to the difference of the fourth powers of the temperatures involved; (3) radiation between HE I11 and the surrounding shield S 111, due to temperature gradients. The heat loss Wo can therefore be represented by the relation
As the temperature gradients are proportional to the differences in heat exchange for the various parts of a unit, they are often proportional to the total heat loss of that unit. Assuming further the temperature gradients to vary linearly with the length L of the unit, we can write for relation 4
..
.
W I
20
d100
I
150
I
200
I
1
250 300 TEMPERATURE, K
Figure 5. Heat loss of the calorimeter with sample flow (isopentane, 20 g/L) and without sample flow, both at AT = 10 K.
The experimental heat loss of the empty calorimeter has been analyzed with this model and the results for the three contributing terms are given in Figure 4. The heat loss of the calorimeter with sample flow has been determined from measurements with isopentane for which accurate specific heat data, obtained with an adiabatic calorimeter, are available. The latter heat loss has been calculated from the relation = w,,, - AH.v, (6)
w,,
I energy supplied to HE 111, where W,l = heat loss, W I I = and VI = sample mass flow rate. The heat loss of the calorimeter with sample flow appeared to be lower than without (see Figure 51, owing to a reversed energy exchange between HE I11 and its shield. Using the same model for the heat losses the final empirical relation for the heat balance of calorimeter with sample flow is given by WIII = AH-V+ u ( T I I ~- TM4)+ b-AT*TIII- c*AH.V* TI,? (7)
338 Ind. Eng. Chem. Fundam., Vol. 18, No. 4, 1979 Table 111. Primary Data of Experiments with Methane (Pressure: 50.0 Bar)
w,
run no. 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 55 56 57 72 73 74 75 76 77 78 79 68 69 70 71 80 81 82 83 84 85
TI,, K 126.39 126.50 126.63 126.69 141.47 141.60 141.76 156.63 156.88 156.96 170.50 170.66 170.78 170.78 180.36 180.56 180.60 180.73 185.46 185.50 185.59 185.32 187.56 189.71 192.37 194.39 193.99 196.75 199.21 204.17 204.39 214.96 215.26 225.21 225.38 235.78 245.49 245.55 255.72
V,
TI15
x 103
x 103
J s-'
g s-l
115.74 115.63 115.49 115.31 130.82 130.7 2 130.59 145.93 145.66 145.48 159.65 159.53 159.39 159.16 170.81 170.74 170.50 170.36 179.98 179.85 179.75 183.04 185.25 187.32 189.7 3 190.96 191.11 194.08 196.74 198.77 198.57 204.10 203.80 214.88 214.66 225.27 235.42 235.16 245.39
103.72 141.29 185.00 221.69 105.08 145.14 189.85 109.69 204.45 248.51 131.53 178.05 230.26 275.39 133.06 183.75 235.77 292.62 124.02 158.88 200.7 4 84.62 95.53 122.38 236.08 547.78 357.76 166.33 132.85 128.27 216.77 184.63 303.29 143.91 222.27 198.12 123.09 184.81 175.63
K
2.776 3.742 4.809 5.636 2.698 3.678 4.725 2.616 4.146 5.683 2.805 3.7 40 4.747 5.612 2.776 3.7 50 4.737 5.763 3.731 ~. 4.675 5.747 5.625 5.520 5.677 5.618 4.5 61 3.588 3.493 5.601 3.529 5.649 3.640 5.791 3.695 5.643 5.780 3.798 5.777 5.735
121.1 121.1 121.1 121.0 136.1 136.2 136.2 151.3 151.3 151.2 165.1 165.1 165.1 165.0 175.6 175.7 175.6 175.5 182.7 182.7 182.7 184.2 186.4 188.5 191.0 192.7 192.6 195.4 198.0 201.5 201.5 209.5 209.5 220.0 220.0 230.5 240.4 240.4 250.6
Table IV. Primary Data of Methane at 32.0 Bar run no.
TIII,
K
TII, K
110 llOA 111 105 106 112 113 114 115 116 109 123 124 117 118 119 120 121 122
149.45 149.37 149.67 151.26 151.66 160.06 159.85 169.95 170.07 178.48 181.08 181.78 182.78 182.91 188.15 198.10 197.98 207.73 207.83
139.28 139.21 139.21 141.23 141.32 149.75 149.69 159.85 159.70 170.07 175.67 178.29 178.20 178.31 182.88 188.24 188.28 198.05 198.00
x
W,
io3
Js-'
116.53 116.15 159.24 110.88 159.86 163.21 133.25 138.44 187.30 104.37 461.20 290.83 516.23 518.80 98.76 147.08 116.92 103.26 129.01
x
V,
io3
-g s - '
3.006 2.958 4.024 2.815 4.022 3.918 3,219 3.014 4.021 2.158 1.61 1 1.037 1.867 1.859 3.986 4.011 3.196 3.218 4.037
T,
3.440 3.432 3.427 3.438 3.574 3.578 3.568 3.819 3.805 3.800 4.229 4.228 4.242 4.222 4.898 4.930 4.906 4.929 5.898 5.895 5.900 6.348 7.246 8.752 15.664 34.828 34.263 17.432 9.313 6.480 6.478 4.557 4.557 3.594 3.606 3.161 2.952 2.949 2.804
Table V. Enthalpy of Methane at 50.0 Bar (in J g - ' ) temp, K this research Jones et al. Keesom e t al. 120 130 140 150 160 170 180 185 190 19 5 20 0 210 220 230 240 250 25 5
0.0 34.56 70.15 107.2 146.2 188.4 237.1 266.4 307.0 451.4 506.7 562.6 600.8 633.9 664.4 693.1 707.1
0.0 34.7 70.3 107.1 146.4 188.3 237.7 266.5 308.1 437.2 504.6 561.9 602.1 635.6 666.5 695.4 707.9
0.0 33.9 68.6 103.8 142.7 184.1 231.8 259.8 302.9 465.7 513.8 562.3 600.0 633.5 663.2 691.2 705.8
Table VI. Enthalpy of Methane at 32.0 Bar (in J g-') temp, K this research Jones e t al. Keesom e t al. 140 150 160 170 17 5 180 185 190 200 210
0.0 37.2 77.4 121.8 148.5 436.4 462.3 483.2 518.8 547.2 0 32bar 0 50 bar
0.0 37.2 77.4 123.0 149.0 435.1 459.4 480.3 516.3 546.4
0.0 36.0 75.3 119.2 144.3 446.0 466.9 485.8 518.8 548.1
THIS RESEARCH
cp,J
K
g-I K-'
144.4 144.3 144.4 146.2 146.5 154.9 154.8 164.9 164.9 174.3 178.4 180.0 180.5 180.6 185.5 193.2 193.1 202.9 202.9
3.731 3.783 3.738 3.840 3.796 3.989 3.998 4.461 4.445 5.558 52.792 79.598 60,213 60.514 4.521 3.613 3.620 3.128 3.117
Measurements To examine the performance of our calorimeter we have made measurements using pure methane under both supercritical and subcritical conditions. The primary data are given in Tables I11 and IV. The specific heats were calculated from eq 7 using the parameter values derived from the isopentane measurements.
150
200
250 TEMPERATURE, K
Figure 6. Deviation of the enthalpies of methane found in this research and by Keesom et al. from those of Jones et al.
The derived enthalpy-temperature relations together with literature data are given in Tables V and VI. The differences between the data from the three sources are plotted in Figure 6. Jones (1962) estimated his results to be accurate within 1% or 2 J/g; the disagreement with our results falls within these uncertainty limits except for one point. This point, however, lies in the temperature range where the specific heat changes dramatically with temperature. Since our calorimeter is specifically designed to measure enthalpy changes over small temperatures intervals, Le., the specific heats, it is interesting to compare these more stringent quantities with those derived from the enthalpy data in the literature. The results are given in Table VII. In view of the experimental error of 1% in both Jones' and our results the agreement is again satisfactory except for one point.
Ind. Eng. Chem. Fundam., Vol. 18, No. 4, 1979
Table VII. Comparison of Specific Heats (C, ) of Methane at 50.0 Bar A. Jones et al. B. Keesom et al. temp, C,: J K -c!125 3.47 135 3.56 145 3.68 155 3.93 165 4.19 175 4.94 185 7.04 195 19.65 205 5.73 215 4.02 225 3.35 235 3.09 245 2.89
C,: J
-g-
3.39 3.47 3.52 3.89 4.14 4.77 7.11 21.1 4.85 3.77 3.35 2.97 2.80
B -A, 7% -2.3 -2.5 -4.3 -1.0 -1.2 -3.5
+1.0 +7.5 -15 -6.2 0.0 -4.0 -3.0
K and at pressures up to 70 bar with an absolute error a t any temperature of less than 1%. Literature Cited
C. This work
C,:J -Q 3.46 3.56 3.70 3.90 4.22 4.87 6.99 19.97 5.59 3.82 3.31 4.05 2.87
339
Clark, R. G., McKinley, C., Paper 11 in "Proceedings of the Forty-Sixth Annual Convention of the Natural Gas Processors Associatm", Houston, March 1967. Dolan, J. P., Eakln, B. E., Bukacek, R. F., Ind. Eng. Chem. Fundam., 7 , 647 (1968). Jenkins, A. C., Berwaldt, 0. E., Ind. Eng. Chem. Process Des. D e v . , 2 , 193 (1963). Jones, M. L., Mage, D. T., Faulkner, R. C., Kak, D. L., Chem. Eng. hog. Symp. Ser., No. 4 4 , 59, 52 (1962). Keesom, W. H., Bijl, A., Monte, L. A. J., Appl. Sci. Res. A , 3 , 261 (1952). Manker, E. A., Mage, D.T., Mather, A. E., Powers, J. E., Katz, D.L., Paper 3 In "Proceedings of the Forty-Third Annual Convention of the Natural Gas Processors Association", New Orleans, March 1964. Mather, A. E., Yesavage, V. F., Powers, J. E., Kak, D. L., Paper 12 in " P r d n g s of the Forty-Fifth Annual Convention of the Natural Gas Processors Association", 1966. Sa@al, P. N., Wt,J. M., Jambhekar, A., Wkm, G. M.. in "International Advances in Cryogenic Engineering. Proceedings of the 1964 Cryogenic Engineering Conference held at the University of Pennsylvania", Vol. 10, p 224, K. D. Timmerhaus, Ed., Plenum Press, New York, N.Y., 1965. Wilson, G. M., Barton, S. T., Paper 18 in "Proceedings of the Forty-Sixth Annual Conference of the Natural Gas Processors Association", Houston, March 1967.
C-A %
-0.3 0.0 +0.6 -0.7 +0.7 -1.4 -0.7 +1.5 -2.4 -5.0 -1.2 -1.3 -0.7
Conclusion
A flow calorimeter has been developed which can measure enthalpy-temperature relations of small amounts of condensable gases at temperatures between 100 and 300
Received for review July 17, 1978 Accepted July 23, 1979
A Flow Calorimeter for Condensable Gases at Low Temperatures and High Pressures. 2. Compilation of Experimental Results and Comparison with Predictions Based on a Modified Redlich-Kwong Equation of State Plet H. G. van Kasteren' and Hans Zeldenrust Koninklijke/Shell-Laboratorium,Amsterdam (Shell Research B. V.) Amsterdam-Noord, The Netherlands
The enthalpy-temperature relationships of methane, ethane, propane, nitrogen, and binary mixtures thereof have been determined. These basic thermodynamic data have been used to improve the predictions of a computer program based on a modified Redlich-Kwong equation of state. The agreement between experiments and final predictions is in general good except for the liquid phase of some binaries containing nitrogen. This discrepancy is postulated to be due to the existence of two liquid phases.
Introduction
For the economic design of liquefaction and regasification plants (for natural gas transportation purposes) accurate enthalpy data are required. The total enthalpy change from room temperature down to 110 K and also the enthalpies at intermediate temperatures are of interest. Therefore detailed isobaric enthalpy-temperature reiationships are needed. These cooling curves depend on the composition of the gas mixture and on the pressure. In view of the great difference in composition of natural gases, the number of cryogenic refrigerants and the variety of operating pressures encountered in liquefaction plants, a calculation procedure for the prediction of cooling curves is indispensable. For this prediction of thermodynamic properties a large number of equations of state have been proposed in the literature and a survey of these was made by Van Aken et al. (1976). We have used a computer program to predict 0019-7874/79/10 18-Q339$01.QQ/Q
cooling curves which is based on a modified RedlichKwong equation of state (MRK). Judging from the small amount of experimental data available on systems at low temperatures, we suspected that the enthalpies predicted using this program would diverge seriously from experiment, particularly for multicomponent systems a t temperatures below 170 K. However, the accuracy of experimentally determined enthalpy changes a t low temperatures is also doubtful, since they are often derived from the difference between two large enthalpy changes over very large temperature intervals. These uncertainties with respect to both the available experimental data and the predicted values led us to carry out a program in which a number of accurate, detailed cooling curves were determined with the aid of the lowtemperature flow calorimeter and the data so obtained used to improve the predictions of the MRK-based computer program. The results of this program are @ 1979 American Chemical Society
