Article pubs.acs.org/jced
Isobaric Heat Capacity of Boric Acid Solution Maogang He,*,† Chao Su,† Xiangyang Liu,† and Xuetao Qi† †
Key Laboratory of Thermo-Fluid Science and Engineering, Ministry of Education, Xi’an Jiaotong University, Xi’an, Shaanxi Province 710049, P. R. China ABSTRACT: The isobaric heat capacities of boric acid solution are crucial for the design of boron recovery systems in nuclear power plants. In this work, the isobaric heat capacities of boric acid solution containing 0.648 mol/kg and 3.700 mol/kg 10B from T/K = (303.16 to 394.25) and up to 5.11 MPa are measured with a flow calorimeter. The expanded uncertainties in temperature, pressure, and isobaric heat capacity were ± 0.02 K, ± 3.20 kPa, and ± 0.0098, respectively. The isobaric heat capacity of boric acid solution decreases when the 10B concentration and pressure increase, and increases as the temperature increases. A correlation equation is presented for the isobaric heat capacities of boric acid solution, the absolute relative deviation between the correlation and experiment is less than 0.6 %.
1. INTRODUCTION In a pressurized water reactor (PWR), boric acid is dissolved in the reactor coolant and used as a neutron absorber to control the speed of nuclear fission.1 Boron has two stable isotopes, namely 10 B and 11B. Normally, 10B is chosen for PWR because 10B has much higher absorbing capacity than 11B.2 However, using boric acid to control the reactivity in the PWR results in high quantities of boric acid in the liquid wastes from the primary circuit of the PWR.3 For example, a 106 kW power unit in the Daya Bay Nuclear Power Plant discharges 132 m3 liquid waste (the 10B concentration of boric acid solution is 0.648 mol/kg) preprocessed or 23 m3 concentrated liquid waste (the 10B concentration of boric acid solution is 3.700 mol/kg) every year.4 Recovery and recycling of boric acid are necessary for environmental reasons and for reducing the operational costs of nuclear power plants.5 The evaporation method is a simple but technically and economically viable method for separating boric acid.6 Basic thermophysical properties, such as heat capacity and viscosity, are indispensable for the design of the recovery system. However, to our best knowledge, no experimental data for heat capacity of boric acid solution have been reported until now. The aim of our study is to provide accurate heat capacity data of boric acid solution for designing the evaporation equipment for recovering boron from the liquid waste of nuclear power plants. In this work, the isobaric heat capacities of boric acid solution containing 0.648 mol/kg or 3.700 mol/kg 10B from T/K = (303.16 to 394.25) and up to 5.11 MPa are presented. On the basis of the experimental results, a correlation equation is presented for the isobaric heat capacity of boric acid solutions.
Figure 1. Experimental apparatus for measuring the isobaric heat capacity. 1, sample; 2, filter; 3, plunger type pump; 4, damper; 5, preheater; 6, experimental cell; 7, absolute pressure transmitter; 8, globe valve; 9, condenser; 10, back-pressure valve; 11, three way valve; 12, container for measurement; 13, container for waste.
elsewhere.7−9 The apparatus is shown in Figure 1. The experimental cell consisted mainly of heaters, thermometers, and vacuum cylinders, as shown in Figure 2. To reduce heat loss, double vacuum cylinders were used, and the fluid was heated with heaters inside the tube through which the fluid flowed. The temperature in the experimental cell was controlled by two electrical heaters and two platinum resistant thermometers (PRT). Two PRTs were inserted into the copper blocks to obtain the temperatures of fluid before and after heating. All the PRTs were purchased from Fluke Corporation with an uncertainty ± 0.01 K. The pressure of the experimental cell was measured by two pressure
2. EXPERIMENT SECTION 2.1. Apparatus. The isobaric heat capacity of boric acid solution was measured using a flow method, as described © 2014 American Chemical Society
Received: September 26, 2014 Accepted: November 13, 2014 Published: November 25, 2014 4200
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Table 2. Experimental Uncertainties of Temperature, Pressure, Power of Heater, and Isobaric Heat Capacity factor of uncertainty temperature
pressure
Figure 2. Schematic diagram of the experimental cell. 1, copper heater; 2, heater; 3, sealed tube; 4, entrance for fluid; 5, for vacuum; 6, PRT; 7, outlet of wire; 8, exit for fluid; 9, heater; 10, support.
power of heater
transmitters (Rosemount, 3051S; 0−10 MPa) with an uncertainty ± 0.025 %. At the beginning of the measurement, the pressure and flow of the fluid was regulated by a plunger-type pump (Scientific Systems, series 1500) and a back-pressure valve. The temperature of the fluid was controlled by a preheater. The fluid flowed into a waste container after passing though the experimental cell. Once the flow was stable, the fluid flowed into the container for measurement by switching a three-way valve. The flow of fluid was checked by mass with an analytical balance (ME204, Mettler Toledo, uncertainty ± 0.2 mg). When the inlet and outlet temperatures of fluid in the experimental cell were stable and identical, the heater inside the cell was open to heat the experimental fluid. After the temperatures of inlet and outlet stabilized with a temperature difference of about 3 K, the temperatures of the inlet and outlet, the pressure, and heating power were recorded by a Keithley 2002 multimeter. 2.2. Sample. In this work, boric acid was provided by Tianjin Fuchen Chemical Reagent Factory with the purity higher than 0.995 in mass fraction. The boric acid solution was prepared by weighing. Because the boric acid cannot be completely dissolved in water below T/K = 353.15 at atmosphere pressure when the 10B concentration is 3.700 mol/kg, we added NaOH after preparing the 3.700 mol/kg 10B boric acid solution. The molar proportion of Na+ and B3+ in the boric acid solution is 1:4. This method is commonly used to pretreat the concentrated liquid waste containing boric acid in the nuclear industry.10,11 NaOH was provided by Hebei Binhai Investment Company with a purity higher than 0.99 in mass fraction. The purity and the suppliers of the chemicals are listed in Table 1.
isobaric heat capacity
chemical
mass fraction
supplier
> 0.995 > 0.990
Tianjin Fuchen Chemical Reagent Factory Hebei Binhai Investment Company
± 0.05 K ± 0.01 ppm ± 0.01 K ± 0.01 K ± 1.25 kPa ± 0.01 ppm ±1 kPa ± 1.60 kPa ± 1 mV ± 1 mV ± 20 ppm ± 0.0001 w ± 0.0001 w ± 0.001 g/s
the temperature difference between experimental temperature and room temperature uΔT combined standard uncertainty uc
± 0.01 K ± 0.0049
experiment, the heat from the heater cannot be completely absorbed by the fluid. The heat loss includes several parts, such as heat transfer from the sample and the calorimeter to surroundings,12 and some heat loss occurs through the heater lead-in wires.13 The real isobaric heat capacity can be given by cp = cp(ob) −
P0 P0 P = − qm ·ΔT qm ·ΔT qm ·ΔT
(2)
where cp(ob) is the observed isobaric heat capacity, cp is the real isobaric heat capacity, and P0 is heat loss of heater per unit time. With the same ΔT, the influence of the heat loss on the experimental result for cp gets smaller as mass flow rate qm increases, because the heating power P rises as qm increases.8 Therefore, we determined the influence of the heat loss by measuring cp at different qm (2−10 g/min, interval is 1 g/min) with the same ΔT at the same temperature and pressure. When the relative deviation between the isobaric heat capacities at qm and above qm is smaller than 0.1 %, we thought cp(ob) at qm is cp. 2.4. Assessment of Uncertainties. The expanded uncertainties of temperature, pressure, and power of heater in the measurement can be given by14 U = kuc = k
∑ ui 2
(3)
where ui is the uncertainty of each influencing factor, uc is the combined standard uncertainty, and k is the confidence coefficient which is usually taken to be 2 or 3. When k = 2, the degree of confidence is 95 %; when k = 3, the degree of confidence is 99 %. In this study, the confidence coefficient of the compound uncertainty is taken to be 2. Uncertainties in the isobaric heat capacity are associated with uncertainties of the measured quantities in the working equation 1 which were used to compute the isobaric heat capacity. The expanded uncertainty in the measurements is given by
Table 1. Purity of Boric Acid and NaOH boric acid NaOH
uncertainty
platinum resistance thermometer u1 data collection u2 temperature stability u3 combined standard uncertainty uc pressure transmitter u1 data collection u2 pressure stability u3 combined standard uncertainty uc voltage of standard resistance u1 voltage of heater u2 resistance of standard resistance u3 combined standard uncertainty uc power of heater up mass flow uqm
2.3. Work Equation. For flow calorimeters, the isobaric heat capacity cp of fluid at a constant pressure p can be calculated by7−9 P P cp(T , p) = = qmΔT qm(T2 − T1) (1)
2 ⎛ ∂cp ⎞2 ⎛ ∂cp ⎞2 2 k ⎛ ∂cp ⎞ 2 2 ⎜ ⎟ Ucp = k·uc = ⎜ ⎟ uΔT + ⎜ ⎟ uq + ⎜ ⎟ uP cp ⎝ ∂ΔT ⎠ ⎝ ∂qm ⎠ m ⎝ ∂P ⎠
where qm is the mass flow of fluid through the calorimeter; P is the heat flow obtained from the heater; T1 and T2, are the inlet and outlet temperature; T = (T1 + T2)/2. During the
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Table 3. Isobaric Heat Capacities of Pure Watera T
p
cp,exp −1
cp,IAPWS‑95 −1
−1
−1
K
MPa
J·g ·K
J·g ·K
296.19 296.20 296.23 296.26 296.26 296.29 296.31
0.10 1.02 2.01 3.02 4.02 5.04 6.02
4.189 4.189 4.209 4.210 4.157 4.128 4.133
4.182 4.180 4.177 4.174 4.171 4.168 4.165
were measured. Table 3 compares the calculated values from the IAPWS-95 formulation and experimental data in our work; the relative deviations are smaller than ± 1 %. Our experimental results agree with the IAPWS-95 formulation.15 The isobaric heat capacities of boric acid solution with 0.648 mol/kg and 3.700 mol/kg 10B were measured in the temperature range from T/K = (303.16 to 394.25) and in the pressure range from p/MPa = (0.1 to 5.11). The experimental results are presented in Table 4 and Table 5. Figure 3 shows the
RDb % 0.16 0.23 0.78 0.87 −0.33 −0.95 −0.77
Table 5. Isobaric Heat Capacities of Boric Acid Solution with 3.700 mol/kg 10B (with NaOH, Na+:B3+ = 1:4)a
Expanded uncertainties U(T) = ± 0.02 K, U(p) = ± 3.20 kPa, Ur(cp, exp) = ± 0.0098. bRD = (cp,exp − cp,IAPWS‑95)/cp,IAPWS‑95; cp,exp is experimental result, cp,IAPWS‑95 is the calculated value from the IAPWS-95 Formulation.
a
T
Table 4. Isobaric Heat Capacities of Boric Acid Solution with 0.648 mol/kg 10Ba T
p
cp,exp −1
−1
K
MPa
J·g ·K
303.16 303.22 303.29 303.38 303.57 303.82 313.23 313.41 313.55 313.72 313.76 313.84 323.61 323.69 323.77 323.85 323.94 324.01 333.19 333.25 333.28 333.36 333.41 333.59 343.47 343.61 343.79 343.87 343.93 344.16
0.10 1.04 2.09 3.07 4.01 5.06 0.10 1.02 2.06 3.11 4.04 5.07 0.10 1.05 2.03 3.01 4.07 5.06 0.10 1.04 2.01 3.08 4.11 5.09 0.10 0.99 2.03 3.01 4.07 5.11
3.993 3.987 3.972 3.959 3.951 3.939 4.002 3.994 3.986 3.973 3.962 3.949 4.013 4.016 4.003 3.991 3.978 3.972 4.019 4.012 4.007 4.001 3.996 3.983 4.031 4.022 4.018 4.013 4.005 4.001
T
p
cp,exp
K
MPa
J·g−1·K−1
353.21 353.33 353.52 353.64 353.83 353.89 363.35 363.38 363.45 363.66 363.73 363.92 373.29 373.37 373.51 373.58 373.74 373.92 383.46 383.63 383.79 383.91 384.07 384.22 393.15 393.29 393.47 393.66 393.83 393.93
0.10 1.02 2.05 3.03 4.10 5.09 0.31 1.04 2.04 3.09 4.07 5.10 0.54 1.03 2.14 3.05 4.11 5.06 0.63 1.02 2.05 3.09 4.15 5.11 0.64 1.03 2.16 3.08 4.06 5.02
4.045 4.040 4.036 4.022 4.015 4.013 4.056 4.048 4.042 4.039 4.034 4.027 4.062 4.055 4.048 4.042 4.037 4.032 4.069 4.061 4.055 4.051 4.044 4.032 4.082 4.078 4.072 4.067 4.064 4.055
p
cp,exp −1
−1
K
MPa
J·g ·K
303.24 303.33 303.57 303.64 303.88 303.91 313.21 313.39 313.41 313.63 313.88 313.21 323.55 323.63 323.85 323.97 324.14 324.33 333.27 333.32 333.58 333.78 333.92 334.22 343.43 343.55 343.74 343.96 344.21 344.35
0.10 1.07 2.14 3.09 4.03 5.04 0.10 1.03 2.05 3.05 4.08 5.06 0.10 1.05 2.00 3.01 4.06 5.09 0.10 1.04 2.05 3.05 4.00 5.03 0.10 1.01 2.05 3.04 4.01 5.03
3.946 3.939 3.933 3.921 3.904 3.901 3.953 3.951 3.942 3.937 3.930 3.926 3.958 3.954 3.967 3.964 3.955 3.951 3.962 3.955 3.952 3.947 3.939 3.933 3.969 3.961 3.951 3.945 3.937 3.933
T
p
cp,exp
K
MPa
J·g−1·K−1
353.32 353.51 353.66 353.86 353.93 354.11 363.39 363.52 363.68 363.85 364.01 364.13 373.26 373.31 373.54 373.67 373.82 373.99 383.51 383.78 383.93 384.07 384.15 384.32 393.47 393.61 393.85 393.98 394.12 394.25
0.10 1.03 2.01 3.05 4.03 5.02 0.10 1.04 2.03 3.04 4.01 5.01 0.52 1.04 2.02 3.03 4.00 4.98 0.49 1.03 2.01 3.06 4.02 5.02 0.53 1.02 2.01 3.06 3.97 5.02
3.978 3.972 3.966 3.962 3.954 3.945 3.985 3.983 3.975 3.969 3.958 3.955 3.996 3.993 3.989 3.982 3.973 3.967 4.008 4.006 3.996 3.987 3.981 3.976 4.021 4.013 4.008 4.002 3.996 3.989
Expanded uncertainties U(T) = ± 0.02 K, U(p) = ± 3.20 kPa, Ur(cp, exp) = ± 0.0098.
a
Expanded uncertainties U(T) = ± 0.02 K, U(p) = ± 3.20 kPa, Ur(cp, exp) = ± 0.0098.
a
where uqm, uP, and uΔT are uncertainties of qm, P, and ΔT, respectively. The expanded uncertainties in temperature, pressure, and isobaric heat capacity in this work are estimated to be less than ± 0.02 K, ± 3.20 kPa, and ± 0.0098, respectively, as shown in Table 2.
3. RESULTS AND DISCUSSION To evaluate the reliability of the apparatus, the isobaric heat capacities of pure water at T/K = 296 from p/MPa = (0.1 to 6)
Figure 3. Isobaric heat capacities of pure water and boric acid solution at p = 0.1 MPa: ○, 0.648 mol/kg 10B; ●, 3.700 mol/kg 10B (with NaOH, Na+:B3+ = 1:4); △, pure water. 4202
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solution with 3.700 mol/kg 10B, a1, a2, and a3 are 3.704 J·g−1·K−1, 7.904·10−4 J·g−1·K−2, and −6.446·10−3 J·g−1·K−1·MPa−1, respectively. The absolute relative deviation between the experimental data and eq 5 is less than 0.6 %. Figure 5 compares the experimental data with eq 5. It can be found that eq 5 agrees well with experiment.
4. CONCLUSIONS A flow calorimeter was developed to measure the isobaric heat capacities of boric acid solution. Isobaric heat capacities of boric acid solution with 0.648 mol/kg and 3.700 mol/kg 10B in the temperature range from T/K = (303.16 to 393.47) from p/MPa = (0.1 to 5.11) have been measured. The isobaric heat capacity of boric acid solution is smaller than that of pure water. With increasing 10B concentration or increasing pressure, the isobaric heat capacity of boric acid solution decreases. The isobaric heat capacity of boric acid solutions increases with increasing temperature. On the basis of the experimental data, a correlation equation for the isobaric heat capacity of boric acid solution is presented; it agrees well with experiment.
Figure 4. Isobaric heat capacities of boric acid solution and pure water at T = 303 K: ○, 0.648 mol/kg 10B; ●, 3.700 mol/kg 10B (with NaOH, Na+:B3+ = 1:4); △, pure water.
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AUTHOR INFORMATION
Corresponding Author
*Tel: 86-29-82663863. E-mail:
[email protected]. Funding
The supports provided by the National Natural Science Foundation of China (No. 51376141) and the National Basic Research Program of China (No. 2015CB251502) for the completion of the present work are gratefully acknowledged.
Figure 5. Comparison between experimental data and correlation: ○, 0.648 mol/kg; ●, 3.700 mol/kg (with NaOH, Na+:B3+ = 1:4).
Notes
isobaric heat capacities of pure water and boric acid solution with 0.648 mol/kg and 3.700 mol/kg 10B at p/MPa = 0.1. The isobaric heat capacities of pure water are calculated from the IAPWS-95 Formulation.15 Figure 4 shows the isobaric heat capacities of pure water and boric acid solution with 0.648 mol/kg and 3.700 mol/kg 10B at T/K = 303.16 from p/MPa = (0.1 to 5.06). The isobaric heat capacity of boric acid solutions is found to be smaller than that of pure water. The isobaric heat capacity of boric acid solution increases faster than that of pure water with increasing temperature, and decreases faster than that of pure water with increasing pressure. The isobaric heat capacity of boric acid solution decreases with increasing 10B concentration. What is more, the effect of acid concentration on cp values is obviously not linear according to the experimental data. After reviewing the literature regarding the isobaric heat capacity of other kinds of acid solutions, such as HCl solutions16 and H2CO2 solutions,17 we note that the heat capacities are also not linear with the effect of acid concentration. So we believe this phenomenon is normal. From Figure 3 and Figure 4, we can find that the isobaric heat capacity of boric acid solution increases linearly with temperature at a fixed pressure and decreases linearly with increasing pressure at a constant temperature. Therefore, the isobaric heat capacity of boric acid solution can be correlated as follows: cp = a1 + a 2T + a3p
The authors declare no competing financial interest.
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REFERENCES
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where a1, a2, and a3 are fitting parameters which can be obtained from the experimental data. Values for the isobaric heat capacity of boric acid solution with 0.648 mol/kg 10B are for a1, a2, and a3 3.651 J·g−1·K−1, 1.116·10−3 J·g−1·K−2, and −7.788·10−3 J·g−1·K−1·MPa−1, respectively. For the boric acid 4203
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