Ind. Eng. Chem. Res. 2001, 40, 3817-3828
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Fire Dynamics and Solar Heating Influence on an Oval-Shaped Chemical Tank Lorry Hsi-Jen Chen* and Song-Ping Lin Department of Chemical Engineering, Tamkang University, Tamsui, Taiwan 25137, R.O.C.
This paper presents a thermal response analysis of an oval-shaped, horizontal tank lorry due to fire exposure. In the accident scenario, a chemical tank lorry containing methanol, in a loading/ unloading station, caught on fire as a consequence of a loss of containment of this flammable material. To analyze the fire dynamics on the tank lorry, a mathematical model is developed for the simultaneous fluid-flow and heat-transfer problem. An application is also made to predict the thermal effect on the tank lorry exposed to solar heating for extended periods of time. 1. Introduction Transportation of hazardous materials has long been a major concern of both the proper authorities and the general public in most industrialized countries; nevertheless, transport accidents occur during the transport and storage of hazardous materials. Frequently, such accidents result in fires, explosions, and toxic releases. Both unconfined vapor cloud explosions (UVCEs) and boiling-liquid expanding-vapor explosions (BLEVEs)1 can occur as a result of transport accidents. In addition to fires and explosions, the loss of containment of toxic chemicals from a tank may give rise to a large toxic gas cloud or may pollute water supplies. The initiating cause of transport accidents may lie with the chemical, the operations, or the transporter. The chemical may catch fire, explode, or corrode the tank. Accidents can occur when operations such as charging and discharging are incorrectly executed or when tanks are accidentally overfilled. The transporter may be involved in a crash; drivers may be injured or killed. Accordingly, the events that can give rise to hazards include particularly accident impact, container failure, and loading and unloading operations. Chemicals are usually transported by tank lorries from place to place and sometimes from a long distance to local plants. It is, therefore, important to understand how a chemical tank lorry heats in the sun. Although few studies have been done on the thermal response models of cylindrical horizontal storage tanks, little attention has been devoted to the tanks with the oval shape. Many investigators have proposed mathematical models describing the thermal response of cylindrical horizontal LPG tanks,2-6 yet the simulation concepts and approaches were not identical. While Ramskill2 left out the important differential equations in his lumped-parameter model because of trade secrets and proprietary information, Beynon and co-workers3 used a distributed-parameter approach. Aydemir and co-workers4 also used a lumped approach for the tank content with two control volumes linked by the evaporation of a stratified interface. Hadjisophocleous and * Corresponding author. E-mail:
[email protected]. Fax: (+886) 2 26209887. Telephone: (+886) 2 26215656 ext. 2721.
co-workers5 used field and zone modeling techniques in studying the behavior of liquefied petroleum gas (LPG) tanks when subjected to accidental fire conditions. Thermal radiation hazard and assessment of safety spacing between tanks were discussed by Chen and coworkers6 in addition to a description of a detailed thermal response model. Tank scale effects with fire impingement were examined by Birk.7 The effects of variable physical properties of substances stored in the presence of an external heat source have been studied by Kourneta and co-workers.8 They applied to the case of an ammonia cryogenic storage tank. It must be recognized that an elliptical horizontal tank lorry is a nonpressure type, which does not provide with the conditions for BLEVE occurrence. This study aims to assess a fire hazard for an elliptical horizontal tank lorry carrying methanol. In addition, the effect of solar heating for extended periods of time on the tank lorry was assessed. We will lump the model as a single unit with four temperature nodes in the fire model. Note should be made that this model permits thermophysical properties of the chemical to be changed as the temperature varies. We obained the chemical database from CHEMCAD.9 Representative values for the chemical properties of methanol are given in Table 1. The scale effect with varying eccentricities of the ellipse and the effect of various filling capacities of the elliptical tank lorry were explored. This research may provide a valuable reference for those who are concerned with the transport safety. 2. Thermal Response Analysis Figure 1 shows a schematic of a monocompartmental, elliptical-horizontal tank lorry containing methanol. This figure shows a partially full tank lorry but is completely subjected to a pool fire engulfment due to a severe leakage. We will lump the model as a single unit with four temperature nodes: i.e., the heated vapor wall, the heated liquid wall, the mean bulk vapor, and the mean bulk liquid. The system can be described by a set of coupled differential equations with the following assumptions: (1) homogeneous thermal energy received by the tank lorry, (2) homogeneous temperature for each node, (3) negligible volumes of baffle plates inside the tank trunk, and (4) negligible solar heat received by the tank lorry as compared with the external heat input.
10.1021/ie000619+ CCC: $20.00 © 2001 American Chemical Society Published on Web 07/25/2001
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Table 1. Properties of Methanol9 properties of chemical name UN number chemical formula molecular weight [kg/kmol] boiling point [K] critical temperature [K] flash point [K] density gas/vapor density liquid
vapor pressure
viscosity gas/vapor viscosity liquid specific heat gas/vapor
specific heat liquid heat of evaporation surface tension liquid
methanol
a constant
1230 CH3OH 32.04 337.85 512.64 284 0
b constant a constant b constant c constant d constant a constant b constant c constant d constant a constant b constant c constant a constant b constant c constant a constant b constant c constant d constant e constant a constant b constant c constant a constant b constant a constant b constant c constant
1 2.288 0.2685 512.64 0.2453 81.768 -6876 -8.7078 7.1926 × 10-6 3.0663 × 10-7 0.69655 205 -25.317 1789.2 2.069 3.9252 × 104 8.979 × 104 1.9165 × 103 5.3654 × 104 8.967 × 102 1.05801 × 105 -3.6223 × 102 9.3790 × 10-1 5.239 × 107 3.682 × 10-1 3.513 × 10-2 -7.04 × 10-6 -1.216 × 10-7
Under such conditions, the energy balance equation between nodes and the fluids yields
dT1 ) h1aA1(τ) (Ta - T1) - h13A1(τ) (T1 dτ T3) - h12A12(τ) (T1 - T2) + QE(τ) A1(τ) (1)
FwV1(τ) CPw
dT2 FwV2(τ) CPw ) h2aA2(τ) (Ta - T2) - h24A2(τ) (T2 dτ T4) - h12A12(τ) (T2 - T1) + QE(τ) A2(τ) (2) FgVg(τ) CPg
FlVl(τ) CPl
dT3 ) h13A1(τ) (T1 - T3) dτ h34A34(τ) (T3 - T4) (3)
dT4 ) h24A2(τ) (T2 - T4) dτ h34A34(τ) (T4 - T3) (4)
Note that eqs 1 and 2 assume uniform heat flux into the tank walls. It is more likely that the flux would be highly dependent on the angular relation between the fire and the tank wall. Also, in some cases, the fire would be under the tank, so heat flux on lower surfaces would be much greater than that on upper surfaces. In light of these considerations, the results of the present model are useful for estimating approximate times to failure; i.e., the worst-case scenario results. Also, strickly speaking, eqs 3 and 4 should be written as d[FgVgCPg(T3 - Tref)]/dτ and d[FlVlCPl(T4 - Tref)]/dτ, where Tref is the reference temperature for enthalpy change, but here we have chosen Tref as zero, as is often the most convenient value to choose. In a pullaway accident, the tank lorry driver fails to disconnect the hose before
temperature correlation where T ) temperature [K]
PM/RT/[1 - a(PM/RT)]b where P ) pressure [N/m2], M ) molecular weight, and R ) universal gas constant [kg/m3] a/b(1 + (1 - T/c)d) [kmol/m3]
ea+b/T+c ln(T)+dT2 [Pa]
aTb/(1 + c/T) [Pa‚s] ea+b/T+c ln(T) [Pa‚s] a + b[(c/T)/sinh(c/T)]2 + d[(e/T)/cosh(e/T)]2 [J/kmol‚K]
a + bT + cT2 [J/kmol‚K] a(1 - T/Tc)b [J/kmol] a + bT + cT2 [N/m]
driving away. The methanol is released and subsequently ignited by sparks or a hot component. This results in a pool fire which engulfs the tank lorry. Under such conditions, the rate of change mass within the tank is
dM ) -FlACox2(Pg/Fl + ghl) dτ
(5)
The right-hand side of eq 5 is the instantaneous mass flow rate due to a hole of area A, in which Co is a constant orifice discharge coefficient, typically near 0.60, and Pg is the gauge pressure of the tank. The total mass of liquid in the tank above the leak is M ) FlAthl. Then, dM/dτ ) Fl (dVl/dτ) ) Fl (dhl/dτ) At. Thus, we can obtain a differential equation representing the change in the fluid height:
x(
CoA dhl )dτ At
2
)
Pg + ghl Fl
(6)
Note that the cross-sectional area of an elliptic tank at any fluid height, At, is a function of hl.
At ) 2a sin φL
(7)
where φ ) cos-1(hl/b - 1). To calculate the external heat source, QE, it is necessary to calculate the area of the pool. One commonly used equation is10
Ap ) V/hm
(8)
where V is the discharge volume and hm is a pool depth factor; hm ) 0.005 m for concrete, and hm ) 0.01 m for gravel. The mass burning rate of liquids on ground pools
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Figure 1. Sketch of an oval-shaped tank trunk with an accidental liquid discharge and resulting in a pool fire.
Figure 2. Four temperature nodes of a solar-radiated, elliptical-horizontal tank trunk.
in windless conditions can be estimated by11
m′′ ) m∞(1 - e-κλdp)
(9)
where m∞ is the mass burning rate of a large pool and dp is the pool diameter. The external heat input to the tank truck can then be estimated by12
QE )
χm′′Hc
(10)
1 + 4(hf/dp)
minor axis of the ellipse. Note that once the liquid volume, Vl, is known, φ can be calculated. It can also be shown that the heated-vapor transfer area is, after considerable effort:
A1 ) 2(a + t)E(ψ,c) L + 2(a + t)(b + t)ψ - ab sin 2ψ (13) where E(ψ,c) is the incomplete elliptic integral of the second kind, ψ is the “outer” liquid-level angle, and t is the tank thickness. Specifically,
in which χ is the radiation fraction, Hc is the heat of combustion, and the flame height, hf, has been correlated as13
[
]
hf m′′ ) 42 dp Faxgdp
0.61
(11)
∫0ψx1 - c2 sin2 θ dθ
in which the quantity c is denoted as the modulus of the elliptic integral; in this case it is represented as c )
x(a+t)2-(b+t)2/(a + t). Note that ψ and φ are related
to
It can be shown that the relationship between liquid volume and liquid level angle is as follows:
1 Vl ) πab - abφ + ab sin 2φ L 2
(
E(ψ,c) )
)
(12)
where φ is the “inner” liquid-level angle, L is the tank length, a is the major axis of the ellipse, and b is the
ψ ) cos-1
(bbcos+ tφ)
Because the total tank surface area equals A1 + A2 ) 4(a + t)E(c) L + 2π(a + t)(b + t), the heated-liquid transfer area (A2) can be calculated. Note that E(c) is the complete elliptic integral of the second kind. Specif-
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Figure 3. Profiles for four node temperatures due to a pool fire.
Figure 4. Mass in tank and liquid level versus drain time.
Also, V1, V2, and Vg equal A1t, A2t, and [abφ - (1/2)ab sin 2φ]L, respectively. The convective heat-transfer coefficients, h1a and h2a, are calculated by the following empirical relation:5
ically,
E(c) )
∫0π/2x1 - c2 sin2 θ dθ
The interfacial wall area between bulk vapor and liquid, A12, is written as
Nu ) γReδPrε
A12 ) 2tL + 4(a sin φ)t
where Nu ) Nusselt number, Re ) Reynolds number, Pr ) Prandtl number, γ ) 0.26, δ ) 0.6, and ε ) 0.36. Because of physical properties involved in the dimensionless terms of Nu, Re, and Pr, h1a and h2a are not constant but essentially a function of the fluid temperature. The heat-transfer coefficients between the tank
(14)
and the boiling heat-transfer area, A34, is written as
A34 ) 2(a sin φ)(L - 2t)
(15)
(16)
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Figure 5. Variation in heated-vapor-wall temperature with varying filling extents.
Figure 6. Variation in mass in tank and liquid level with varying filling extents.
wall and the fluids, h13 and h24, are calculated using14
Nu ) C(Ra)n
(17)
where Ra ) gβ∆TLc3/Rν, the Rayleigh number, in which Lc ) As/P, the characteristic length. For hot surface faces up, C ) 0.54 if laminar; C ) 0.15 if turbulent. For hot surface faces down, C ) 0.27 if laminar. For hot surface faces up, n ) 0.25 if laminar; n ) 0.33 if turbulent. For hot surface faces down, n ) 0.25 if laminar. The physical properties of the fluid are evaluated at the arithmetic mean of the fluid and wall temperatures. The effective heat-transfer coefficient within the tank wall, h12, is calculated using2
h12 ) kw/(kwt)1/2
[
1 1 + (h1a + h13)1/2 (h2a + h24)1/2
]
-1
(18)
where kw is the thermal conductivity of the tank wall and t is the tank thickness. Finally, dependent on the heat flux received, the heat-transfer coefficient (h34) is calculated according to whether the heated liquid is by free convection, nucleate boiling, or film boiling. For the case of free convection, eq 17 is used; for the case of boiling, the Rohsenow’s correlation15 can be used. 3. Effect of Solar Heating Figure 2 depicts a methanol tank trunk with incident solar radiation. The mathematical model can then be
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Figure 7. Variation in heated-vapor-wall temperature with varying eccentricities.
derived with the following assumptions: (1) homogeneous thermal energy received by the tank lorry, (2) homogeneous temperature for each node, (3) negligible volumes of baffle plates inside the tank trunk, (4) negligible volume changes of vapor and liquid with time inside the tank trunk, and (5) external surface of the tank trunk modeled as a gray body. Under such conditions, the system can be described by a set of coupled differential equations:
FwV1CPw
dT1 ) h1aA1(Ta - T1) - h13A1(T1 - T3) dτ h12A12(T1 - T2) + QSA1 (19)
dT2 FwV2CPw ) h2aA2(Ta - T2) - h24A2(T2 - T4) dτ h12A12(T2 - T1) + QSA2 (20) FgVgCPg
dT3 ) h13A1(T1 - T3) - h34A34(T3 - T4) (21) dτ
QS ) ηQA - σ(Tw4 - Ts4)
(24)
where η is the absorptivity, is the emissivity, σ is the Stefan-Boltzmann constant, Tw is the tank-wall temperature, and Ts is the effective sky temperature. Although we have considered that the solar altitude varies with time in the model, the solar heating on the tank assumes uniform solar energy input over the entire tank surface. Therefore, the results of the present model are useful for estimating the approximate heat-up effect. Note that eq 16, for the convective heat-transfer coefficients (h1a and h2a) with incident solar radiation, is replaced by using17
Nu ) 0.224(ufDc/ν)0.612
(25)
where uf is the free-stream velocity of air and Dc is the characteristic diameter. The physical properties of air are evaluated at the arithmetic mean of the free-stream and wall temperatures. 4. Safety Design Aspect
dT4 FlVlCPl ) h24A2(T2 - T4) - h34A34(T4 - T3) (22) dτ The incident solar radiation, QA, can be modeled using the approximation derived by16
QA ) 1.111(1 - 0.0071CI2)(sin φs - 0.1)
(23)
where the “solar constant” is 1.111 kW/m2, CI is a dimensionless cloudiness index (0 for clear day and 10 for complete cover), and φs is the solar altitude (degrees above the horizon). Equation 23 is valid only for sin φs greater than 0.1; i.e., at low sun angles where sin φs is less than 0.1, the incident solar radiation can be assumed to be zero. The net solar heat flux, QS, to the tank surface, neglecting convection effects, can be written as
In general, the design of tank lorries should follow the Codes and Regulations. The Codes usually require an emergency shut-off valve in each of the liquid transfer and the vapor line. The emergency shut-off valves are also actuated by a heat-sensing element located near the hose connection point and by manual actuation at the rear end of the tank lorry. If bursting of a hose connection due to improper attachment or a defective hose occurs, a built-in excess flow valve should be tripped to block the leakage. Depressuring systems are used extensively for protecting tanks from overpressure. Overpressure has many causes, such as external fire, exothermic reactions, inadequate cooling, blocked lines, equipment failure, or human error. Proper sizing of a relief system is therefore essential. This requires the determination of proper relieving conditions (for example, flows, pressures, and temperatures) such that
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Figure 8. Variation in mass in tank and liquid level with varying eccentrities.
Figure 9. Daily course of mean air temperature in Taiwan in Aug 1999.
relief valves and their associated pipework can be sized for the worst scenario. 5. Results and Discussion The modeling ordinary differential equations were programmed in Fortran and solved numerically using the adaptive algorithm of Runge-Kutta-Fehlberg.18 The numerical integration involved in the elliptic integrals was carried out by using an adaptive quadrature scheme based on the Gauss-Kronrod algorithm.19 To simulate the failure phenomena of a tank lorry, several varied process variables were programmed in.
The process variables include the tank-filling capacity and tank size. Note that methanol has a flash point of 284 K; therefore, it may easily give off enough vapor to form an ignitable mixture with air once it is released. Figure 3 shows the profiles for four node temperatures due to a pool fire. Note that simulation stops at the boiling point of methanol (338 K). In the fire dynamics, methanol is not yet heated to the state of vaporizing, as can be seen from Figure 3. This is probably due to a heat-sink effect of large quantities of liquid methanol. Note also that the tank metal is likely to fail because of tensile stress beyond a certain temperature, say 773 K. Figure 4 shows the profiles of liquid mass in the tank
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Figure 10. Profiles for four node temperatures with incident solar radiation from 6 a.m. to 6 p.m.
Figure 11. Variation in heated-vapor-wall temperature with varying wind velocities.
and liquid height versus drain time. Figure 5 shows the variation in the heated-vapor-wall temperature (node 1) for varying time lapse at different filling capacities. Figure 6 shows the changes in liquid mass and liquid height versus time lapse at different filling capacities. To investigate the scale effect of the elliptic tank lorry, we kept the major axis, a, fixed and vary the eccentricity of the ellipse (xa2-b2/a). The scale effect, with a constant filling extent of 90%, on the heated-vapor-wall temperature is shown in Figure 7. As seen from Figure 7, larger eccentricity of the tank results in a faster rise of temperature when exposed to fire. Also, the scale effect on the liquid mass and liquid height in the tank
can be observed as shown in Figure 8. Depending on the heat intensity, quantities of the discharged liquid, and eccentricity of the tank, the explosion of the methanol tank will take place in about 17-21 min. This is in good agreement with the current records. Similar evidence was validated by Shebeko and co-worker’s study20 on the Alma-Ata, Russia, LPG tank-car accident. In that accident, a tank explosion with fireball formation took place in 15-17 min after fire engulfed the entired tank. To simulate the thermal response of a methanol tank lorry subjected to solar exposure, several varied process variables were programmed in. The process variables
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Figure 12. Variation in heated-vapor-wall temperature with varying eccentricities of elliptical tank trunk.
Figure 13. Variation in heated-vapor-wall temperature with varying cloudiness indexes.
include the intensity of solar radiation, free-stream velocity of air, tank fill level, and tank size. Note that the value of the sky effective temperature depends on atmospheric conditions, ranging from a low of 230 K under a cold, clear sky to a high of approximately 285 K under warm, cloudy conditions. The thermophysical properties of air must be evaluated at various temperatures in the simulation. The discrete data of air can be found from work by Incropera and DeWitt,14 and the properties are computed using the algorithm of cubic interpolatory spline.21 The daily course of mean air temperature in Taiwan in August of 1999 (Figure 9) was obtained from the Central Weather Bureau (http:// www.cwb.gov.tw). Note also that the intensity of solar radiation is a function of cloudiness index and solar
altitude. In this work, the simulation period lasts from 6 a.m. to 6 p.m. In response to this time lapse, solar altitudes considered are in the range of evenly divided intervals of 0°-90°-0° angles. Figure 10 shows the thermal response profiles for four node temperatures of the methanol tank lorry exposed to incident solar radiation, with a cloudiness index of 1, when the tank is 90% full. Note that in the beginning T1 and T2 exhibit negative net solar flux due to zero solar irradiation. The heated-vapor-wall temperature reaches peak around noontime (at 360 min). Figure 11 shows the thermal responses of the heated-vapor-wall temperature with various wind velocities. Additionally, we vary the eccentricity of the elliptic tank lorry to observe the scale effect. Figure 12 shows this scale effect on the heated-
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Figure 14. Variation in heated-vapor-wall temperature with varying filling extents.
Figure 15. Variation in mean-bulk-liquid temperature with varying filling extents.
vapor-wall temperature. As seen in Figure 12, smaller eccentricity of the tank results in a slower rise of temperature when exposed to solar radiation. By varying the cloudiness indexes, we are able to investigate the changes of the heated-vapor-wall temperature as shown in Figure 13. Also, by varying the filling extent, we are capable of observing the changes of the heatedvapor-wall temperature and the mean-bulk-liquid temperature as shown in Figures 14 and 15, respectively. As can be seen from Figures 14 and 15, the bulk liquid plays a crucial role in receiving the solar energy like a
heat sink; the larger the quantity, the slower the temperature rise. Also, this bulk liquid affects the rise of the heated-vapor-wall temperature. Note that different scenarios can be simulated when necessary. 6. Conclusions This paper has presented a fire dynamics for an elliptical horizontal tank lorry with an accidental liquid discharge that results in a pool-fire impingement. Dependent on the heat intensity, quantities of the
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discharged liquid, and eccentricity of the tank, the explosion of the methanol tank will take place in about 17-21 min, which is in good agreement with the current records. In addition, an application is made to predict the thermal impact on the tank lorry exposed to incident solar radiation for extended periods of time. The results from the solar heating model are valuable to show how a chemical tank lorry heats in the sun. It is clear that the modeling approach for the present topic is not unique; however, we should point out that we are more concerned with worst-case scenarios. Although the results obtained from the predictive solar-heating model seem reasonable, further full-scale experiments may be warranted to confirm its validity. The result of this research might help those who are concerned with the safety of transportation, particularly the safety aspect in designing inherently safer tank lorries. Nomenclature A ) hole area (m2) A1(τ) ) heated-vapor-transfer area at time τ (m2) A2(τ) ) heated-liquid-transfer area at time τ (m2) A12(τ) ) interfacial-wall-transfer area between bulk vapor/ liquid at time τ (m2) A34(τ) ) heat-transfer area between nodes 3 and 4 at time τ (m2) Ap ) pool area (m2) As ) surface area (m2) At ) cross-sectional area of elliptic tank at any fluid height (m2) a ) major axis of elliptic trunk (m) b ) minor axis of elliptic trunk (m) C ) constant in eq 17 c ) modulus of elliptic integral Co ) orifice discharge constant in eqs 5 and 6 ()0.61) CI ) dimensionless cloudiness index CPg ) specific heat of vapor (kJ/kg‚K) CPl ) specific heat of liquid (kJ/kg‚K) CPw ) specific heat of tank wall (kJ/kg‚K) Dc ) characteristic diameter (m) dp ) pool diameter (m) E(c) ) complete elliptic integral of the second kind E(ψ,c) ) incomplete elliptic integral of the second kind g ) acceleration due to gravity (9.81 m/s2) Hc ) heat of combustion (kJ/kg) hf ) flame height (m) hl ) fluid height (m) hm ) pool depth factor (m) h1a ) heat-transfer coefficient from ambient to heated vapor surface (kW/m2‚K) h2a ) heat-transfer coefficient from ambient to heated liquid surface (kW/m2‚K) h12 ) heat-transfer coefficient from heated vapor wall to heated liquid wall (kW/m2‚K) h13 ) heat-transfer coefficient from heated vapor wall to bulk vapor (kW/m2‚K) h24 ) heat-transfer coefficient from heated liquid wall to bulk liquid (kW/m2‚K) h34 ) heat-transfer coefficient from bulk vapor to bulk liquid (kW/m2‚K) kl ) liquid thermal conductivity (W/m‚K) kw ) tank wall thermal conductivity (W/m‚K) L ) tank-trunk length (m) Lc ) characteristic length (m) M ) liquid mass in tank (kg) m′′ ) mass burning rate of liquids (kg/m2‚s) n ) constant in eq 17 Nu ) Nusselt number P ) perimeter (m)
Pg ) gauge pressure of tank (kPa) Pr ) Prandtl number QA ) incident solar radiation (kW/m2) QE(τ) ) external heat input due to pool fire at time τ (kW/ m2) QS ) net solar heat flux (kW/m2) Ra ) Rayleigh number T1 ) heated-vapor-wall temperature (K) T2 ) heated-liquid-wall temperature (K) T3 ) bulk-vapor temperature (K) T4 ) bulk-liquid temperature (K) Ta ) ambient temprature (K) Ts ) effective sky temprature in eq 24 () 273 K) Tw ) tank wall temprature (K) t ) tank-wall thickness (m) uf ) free-stream velocity of air (m/s) V ) discharge volume (m3) V1 ) volume of heated-vapor-wall (m3) V2 ) volume of heated-liquid-wall (m3) Vg ) vapor volume (m3) Vl ) liquid volume (m3) V1(τ) ) heated-vapor-wall volume evaluated at time τ (m3) V2(τ) ) heated-liquid-wall volume evaluated at time τ (m3) Vg(τ) ) vapor volume evaluated at time τ (m3) Vl(τ) ) liquid volume evaluated at time τ (m3) Greek Symbols R ) thermal diffusivity (m2/s) β ) volumetric expansion coefficient (K-1) γ ) constant in eq 16 δ ) constant in eq 16 ∆T ) temperature difference (K) ε ) constant in eq 16 ) emissivity in eq 24 ()0.21) η ) absorptivity in eq 24 ()0.5) κ ) extinction coefficient (m-1) λ ) mean beam length corrector ν ) kinematic viscosity (m2/s) Fg ) vapor density (kg/m3) Fl ) liquid density (kg/m3) Fw ) tank-wall density (kg/m3) σ ) Stefan-Boltzmann constant (5.67 × 10-11 kW/m2‚K4) τ ) time lapse (min) φ ) “inner” liquid-level angle (rad) φs ) solar altitude (rad) χ ) radiation fraction ()0.17 for methanol) ψ ) “outer” liquid-level angle (rad)
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Received for review July 6, 2000 Accepted May 29, 2001 IE000619+
