A probability density function for the clearness index,with applications

This paper uses classical probability theory to derive expressions for the expected (or mean) value of quantities such as the irradiation on inclined surfaces, collector output, and net gain through windows. The random variables are the clearness index k_t and the diffuse fraction k, whose means are $\text{k}{}_{\mathrm{t}}$ and $\text{k}$, respectively. The probability function for k_t is assumed to depend only upon $\text{k}{}_{\mathrm{t}}$ and is represented by a simple function chosen so that its corresponding distribution function is a close fit to the one originally graphed by Liu and Jordan. The probability function for k is represented, for a given k_t, by an impulse function centred at the $\text{k}$ observed at that k_t; the Orgill-Hollands functional form is used for the function $\text{k}\text{(k}{}_{\mathrm{t}}\text{)}$. A general purpose integral expression is presented for the expected value of any quantity which can be expressed as a functionof k, k_t and time of day. The integral is evaluated for several such quantities: the solar irradiation on an inclined surface, the output of both flat plate and concentrating collectors, and the net heat gain through a window with an automatic shutter. Example calculations are included.     

Structure of the transient cooling of a slab

This paper reviews solutions to the classical problem of a slab of homogeneous material (conductivity λ, density ?, specific heat c), initially at temperature t_i throughout and at time t = 0 subjected to a step change of temperature at its exposed face. The roles of the dimensionless time variables $\text{ζ}{}_0\text{=}\text{λt}\text{?cy}{}^2$ and $\text{ζ}{}_2\text{=}\text{h}{}^2\text{t}\text{λ?c}$ are discussed (y is the depth below the exposed surface, h is the surface heat transfer coefficient). At large depths of $\text{b ( =}\text{hy}\text{λ}\text{)}$, a thermal disturbance is propagated at a rate determined mainly by ζ₀, but for smaller values of b it travels relatively slower.The temperature anywhere in a slab, thickness X, insulated on its rear surface is initially independent of X and at the exposed surface depends on ζ₂ alone. After some interval of ζ₀, explainable in terms of the rate of propagation o the thermal signal, temperature everywhere falls exponentially. Values for temperature at the front and back surfaces are given in terms of ζ₂ and $\text{B =}\text{hX}\text{λ}$.Values of ζ₀ are given relating to the time at which the surface temperature of a finite thickness slab starts to fall more quickly than that of an infinitely thick slab. Values of ζ₀ are also given relating to the time at which exponential cooling is established. Approximate polynomial forms are given for cooling in its early and later stages. The response time for thick and thin slabs is discussed.     

Droplet evaporation: Effects of transients and variable properties

A numerical study of the effects of transients and variable properties on single droplet evaporation into an infinite stagnant gas is presented. Sample calculations are reported for octane droplets, initially at 300°K with R_o = 0·1, 0·5, 2·5 × 10^?4m, evaporating into air at temperatures and pressures in the ranges 600–2000°K and 1–10 atm, respectively. It is found that initial size R_o is eliminated from the problem on scaling time with respect to R²₀ and that the evaporative process becomes quasi-steady with ($\text{R}\text{R}{}_0\text{)}{}^2\text{= (}\text{R1}{}_0\text{R}{}_0\text{)}{}^2\text{?}\text{βt}\text{R}{}^2{}_0$, as suggested by experiment. Comparisons of solutions using various reference property schemes with those for variable properties show that best agreement obtains using a simple $\text{1}\text{3}$ rule wherein properties are evaluated at $\text{T}{}_{\mathrm{r}}\text{= T}{}_{\mathrm{s}}\text{+}\text{(T}{}_{\mathrm{e}}\text{?T}{}_{\mathrm{s}}\text{)}\text{3}$ and $\text{m}{}_{\mathrm{1,r}}\text{= m}{}_{\mathrm{1,s}}\text{+}\text{(m}{}_{\mathrm{1,e}}\text{? m}{}_{\mathrm{1,s}}\text{)}\text{3}$. The effects of temporal storage of mass species, energy, etc. and radial pressure variations in the vapor phase prove to be negligible, the early transient behavior being solely due to sensible heat effects within the droplet and related variations in vapor-side driving forces.     

Kinetics of the reaction between Mg2Ni and hydrogen

The reaction between Mg₂Ni and hydrogen was investigated by volumetric method. The reaction was divided into two stages; the initial stage was a very rapid reaction whose rate could not be measured, the later stage was the slower step whose rate was expressed by the equation

$\text{dn/DT}\text{=}\text{K′}\text{(}\text{P}\text{?}\text{P}{}_{\mathrm{eq}}\text{/}\text{t}$

where k′ is the constant, P_eq and P are hydrogen pressures at equilibrium and at time t. The reaction rate of the later stage did not depend upon temperature, content of hydrogen in the alloy, and directions of the reaction, desorption and absorption. The amount of reacted hydrogen, Δn, in the initial stage was expressed by

$\text{Δ}\text{n}\text{=k(}\text{P}{}_{\mathrm{o}}\text{?}\text{P}{}_{\mathrm{eq}}\text{)(}\text{n}{}_{\mathrm{s}}\text{?}\text{n}{}_{\mathrm{o}}$

where P₀ is the initial hydrogen pressure, n_s is a constant around 4, n₀ is a ratio of H to Mg₂Ni at the initiation of the run, and k is a constant. The apparent activation energy of the reaction was nearly zero. It is considered that the reaction between the alloy and gaseous hydrogen takes place on metallic Mg₂Ni in the initial stage and in the later stage reaction proceeds on the deactivated site.     

Thermal insulation provided by plain,horizontal annular cavities containing atmospheric pressure air

For the range 3 × 10³ ≤ Gr_di ≤ 10⁸ and $\text{1·3 ≤}\text{r}{}_0\text{r}{}_{\mathrm{<mtext>i</mtext>}}\text{≤ 7·5}$, it is suggested that for the steady-state rate of heat transfer outwards by combined laminar, free convection and conduction through the atmospheric pressure air contained within horizontal concentric annuli. This simple correlation, evolved from an analysis of published, as well as new, experimental information, will enable designers to predict the combined convective/conductive resistance provided by the contained air for the range of concentric pipes likely to be encountered in practice.An optimal eccentricity of 0·24 (the inner cylinder being moved vertically upwards relative to the outer cylinder from the concentric position) corresponds to the maximum combined convective/conductive resistance configuration. For the systems tested in the temperature range 18°C ≤ T ≤ 150°C, this optimal eccentricity is not significantly affected by changes in the surface emissivities.     

A kinetic model of hydrogen absorption in CeMg12

Comparing kinetic equations derived from a theoretical model with experimental data published, the kinetic mechanism of hydriding reaction of CeMg₁₂ was analysed.At the initial stage the reaction is controlled by chemisorption of hydrogen on the metal surface and the reacted fraction (F) is expressed as a function of time (t) and temperature (T)

$\text{F=(1.19 × 10}{}^2\text{)}\text{(P}{}_{\mathrm{o}}\text{? P}{}_{\mathrm{eq}}\text{)}\text{T}\text{1}\text{2}\text{exp}\text{(?3560}\text{cal/RT}\text{)t}$

in the range of 0 ? F ? 0.4. The later stage of the reaction is controlled by another mechanism of metal/hydride interface chemical reaction or hydrogen diffusion in the hydride phase which cannot be clearly distinguished at the moment.     

The maximum power,heat demand and efficiency of a heat engine operating in steady state at less than Carnot efficiency

An imperfect, Carnot-like engine operating in steady state and receiving heat through conductance k₁ from a source at T₁ and discharging heat only through a conductance k₂ to a sink at T₄ has an efficiency at maximum net power output of $\text{η}{}_{\mathrm{m}}\text{= (}\text{?}\text{g9}\text{){1 ? √(1 ? ?(1 ?}\text{T}{}_4\text{T}{}_1$)}, where ? is a non-Carnot efficiency and $\text{? =}\text{(?k}{}_1\text{+ k}{}_2\text{)}\text{(k}{}_1\text{+ k}{}_2\text{)}$.     

Optimizing energy transition paths in CO2 emission reduction strategies

In the consideration of energy use scenarios designed to avoid excessive build-up of atmospheric carbon dioxide and the effects of the associated climatic changes it can be anticipated that difficulties will ensue if needed rates of increased deployment of non-fossil fuel burning energy plants become very large. This may result if allowable maximum CO₂ concentrations are set sufficiently low, or if acceleration of non-fossil fuel energy use is sufficiently delayed.In a broad ranging numerical investigation of this problem, Perry et al.¹ have suggested that difficulty will be experienced in the transition to a non-fossil based economy, not only if the rate of growth of non-fossil fuel energy use rate, $\text{E}\text{?}{}_{\mathrm{N}}$ is excessive, but also if the second derivative, $\text{E}\text{?}{}_{\mathrm{N}}$, becomes too large. This observation has important ramifications, since $\text{E}\text{?}{}_{\mathrm{N}}$ may exceed tolerable limits, whilst $\text{E}\text{?}{}_{\mathrm{N}}$ still remain modest in size. In making the estimates an empirical method for joining early and asymptotic energy use rates was employed. In order to make sure that the conclusions reached were not sensitive to the ad hoc nature of this technique, we have introduced a more rigorous approach by using a variational interpolation approach that minimizes the peak value of $\text{E}\text{?}{}_{\mathrm{N}}$ (or a combination of $\text{E}\text{?}{}_{\mathrm{N}}$ and $\text{E}\text{?}{}_{\mathrm{N}}$.$\text{E}\text{?}{}_{\mathrm{N}}$ We find that such an optimization procedure can reduce the maximum values of $\text{E}\text{?}{}_{\mathrm{N}}$ by as much as 50% from those calculated by Perry et al. This indicates that the energy transition process should be easier that concluded by them, at least as far as a criterion based upon $\text{E}\text{?}{}_{\mathrm{N}}$ variation is concerned, but can still create difficulties, especially if a 500 ppmv CO₂ ceiling is considered hazardous and if actions to reduce fossil fuel use are delayed too long.     

Surface renewal theory for turbulent mixed convection in vertical ducts

A heat transfer correlation for opposing mixed turbulent convection in vertical ducts was obtained utilizing surface renewal theory. The correlation was found to be Nu_Db = 0.0115Re_Dh^0.8Pr^0.5 The correlation fit data to within 7% over a parameter range of 0.7 < Pr < 7, 1 × 10⁴ < Re_Dh < 2 × 10⁴, and 1 × 10⁶ < Gr_Dh < 2 × 10⁹. The mean residence time, characterizing the time a clump of fluid resides on the wall, was found to decrease as both and Re_Dh increase. This explains the enhanced heat transfer due to buoyancy in opposing mixed turbulent flows. This heat transfer enhancement was also reflected in a decreasing thermal boundary layer thickness with increasing Re_Dh, Gr_Dh or Pr.     

Water splitting reaction on a polynaphthoquinone catalyst a polynaphthoquinone-SO2-I2 system for H2O decomposition

The water splitting reaction in a polynaphthoquinone-SO₂-I₂ system under mild conditions is reported. One mole H₂O was decomposed to form 2 moles HI and 1 mole H₂SO₄ by the following two successive hydrogen transfer reactions in conjunction with the redox cycle of quinones(Q)-hydroquinones(QH₂) in the polynaphthoquinone catalyst:

$\text{2H}{}_2\text{O +SO}{}_2\text{+ Q ? QH}{}_2\text{+ H}{}_2\text{SO}{}_4\text{, QH}{}_2\text{+ I}{}_2\text{? 2HI + Q}$

The hydrogen transfer reaction from H₂O to Q was accelerated by a factor of 3–5 by light irradiation and hence this system could work also as a hybrid thermochemical cycle. When the catalytic hydrogen transfer reactions on the polynaphthoquinone are combined with the thermal decompositions of HI(2 HI ? H₂ + I₂) and H₂SO₄ ($\text{H}{}_2\text{SO}{}_4\text{? H}{}_2\text{O + SO}{}_2\text{+}\text{1}\text{2}\text{O}{}_2$), a closed water splitting cycle for the hydrogen production could be constructed. The reaction mechanism is also discussed.     

Linear stability analysis of double-diffusive solar pond

In this communication, the stability of the double-diffusive solar ponds has been investigated in the linear approximation. The corresponding linearized system of equations of motion is reduced to a single integro-differential equation using the Green-function technique. In contrast to the conclusions of Veronis that, initially, the instability occurs as an oscillatory mode and at a value of R_T (Rayleigh number for temperature) greater than R_S the motion becomes steady, the present analysis shows that, initially, as R_T increases from zero but remains considerably less than R_S, exponentially growing and decaying modes (steady motion) occur first; for a value of R_T more than a critical value R_T^c, the motion becomes oscillatory. This oscillatory motion may, due to the basic non-linear dynamics of the system, even become aperiodic. Further, for R_S → ∞, the minimum value of R_T for which steady motions can occur tends to $\text{K}{}^{\mathrm{<mtext>?</mtext><mtext>1</mtext><mtext>2</mtext>}}\text{·R}{}_{\mathrm{S}}$, where $\text{K}\text{= K}{}_{\mathrm{S}}\text{/K}{}_{\mathrm{T}}$ where K_S and K_T are diffusivity coefficients for salt and temperature, respectively; as a contrast, according to Veronis, $\text{R}{}_{\mathrm{T}}\text{a}\text{?}\text{σ}{}^{\mathrm{?1}}\text{R}{}_{\mathrm{S}}\text{; σ = v/K}{}_{\mathrm{T}}\text{, v}$ being the kinematic viscosity.     

Performance study of an OTEC system

This paper describes a series of studies carried out to analyse the performance of an Ocean Thermal Energy Conversion (OTEC) system. The objective function, $\text{A}\text{W}{}_{\mathrm{net}}$, where A is the total heat exchanger area and W_net is the net work output of the system, was used for the parametric and optimisation studies. By using $\text{A}\text{W}{}_{\mathrm{net}}$ the heat exchangers were directly related to the remaining OTEC components. Since changes in one component of the system invariably affect the rest of the system, it was thus possible to evaluate the combined effects on the OTEC power plant.The effects of the following parameters on system performance were investigated: ocean fluid velocity through the exchanger, log-mean temperature differences of the heat exchangers, heat transfer enhancement and cold seawater pipe diameter. It was concluded that for a 1 MW_e OTEC power plant, the net output of the plant becomes zero when ΔT (the temperature difference between the hot and cold ocean streams) approaches 12·80°C. The power cycle used in this study was a simple closed Rankine cycle with ammonia as the working fluid.     

Stages in the transient cooling of a slab

The classical solutions to the one dimensional transient heat flow situation are examined and criteria are set up to map stages in the cooling process in a parallel sided slab following step changes at the exposed surface. The temperature θ, at a point within the slab, undergoes a perceptible deviation from the undisturbed state, and from the state determined by cooling from the nearer side alone, at times ζ₀ (the Fourier number) found from a relation of type $\text{d}{}^3\text{θ}\text{dζ}{}_0{}^3\text{= 0}$ in the appropriate progressive wave solution. Cooling becomes virtually exponential at a time ζ₀ when $\text{d}{}^2\text{θ}\text{dζ}{}_0{}^2$ for the first term in the standing wave solution equals the sum of the higher terms. The response times are conveniently expressed in terms of ζ₀, ζ₀B or ζ₀B² (B is the Biot number) according to the model involved. Examples are given illustrating these results in a building context.     

Thermodynamics of pressure plateaus in metal-hydrogen systems

In binary, or pseudo-binary metal-hydrogen systems the phase transformation reaction $\text{(x}{}_2\text{? x}{}_1\text{)}{}^{\mathrm{?1}}\text{MH}{}_{\mathrm{x1}}\text{+}\text{1}\text{2}\text{H}{}_2\text{? (x}{}_2\text{? x}{}_1\text{)}{}^{\mathrm{?1}}\text{MH}{}_{\mathrm{x2}}$ gives rise to a hydrogen dissociation pressure plateau. It is shown that for this reaction, the enthalpy of transformation, ΔH₁₂, and entropy of transformation, ΔS₁₂, are not fundamental system parameters and their relation to more basic relative molar thermodynamic quantities is derived. It is shown with complete generality that ΔH₁₂ and ΔS₁₂ defined by calorimetric measurement determine the plateau pressure, P₁, temperature dependence according to the relation R ln P₁ = 2ΔH₁₂/T ? 2ΔS₁₂.It has often been observed that ΔH₁₂ and ΔS₁₂ appear to be independent of temperature. By considering the thermodynamics of a regular interstitial solution model, it is shown for both miscibility gap and structural transformation systems that ΔH₁₂ and ΔS₁₂ are not necessarily constant. On the other hand it is shown that for real systems such that the P₁ ? 1 atm at room temperature, which are convenient to study and hence account for most observations, ΔH₁₂ and ΔS₁₂ are sensibly constant.Finally it is concluded that Q = ?xΔH₁₂ is an adequate engineering approximation for the heat required to decompose a practical hydrogen storage hydride MH_x.     

Thermochemical water splitting cycles: impact of thermal burdens and kinetics

Equations are rigorously derived for evaluating the thermal efficiency of a thermochemical water splitting cycle, from which it is possible to assess the impact of each heat burden or loss separately. The equations of continuity are shown to be coupled, as a consequence, heat flow is found to be the rate determining process for the operation of a thermochemical water splitting plant. Since heat flow is rate determining, the chemical rate of reaction must be fast relative to heat flow even in the asymptotic approach to completion. The conclusion is drawn that recycling of reactants, which is required if ΔG ? 0, will probably result in an uneconomical cycle.Unlike a thermomechanical engine which can be characterized by a single parameter, the thermal efficiency η, the thermochemical engine requires two, one, η, which measures the effective use of heat and a second, τ, which measures the effective use of power. The former is defined in the conventional manner, namely the work divided by the heat. The latter is defined as the ratio of the average chemical rate for product in the reaction volume for each stage multiplied by the heat required by the cycle to split a mol of water divided by the power of the source:

$\text{τ=vR}{}^{\mathrm{\alpha }}{}_{\mathrm{i}}\text{V}{}_{\mathrm{i}}\text{Q/P=vp}{}_{\mathrm{i}}\text{Q/PδH}{}_{\mathrm{i}}\text{=1}$

ν adjusts for the stoichiometry, p_i is the power absorbed at stage i and ΔH_i is the enthalpy change in the reaction at i. To maintain a balanced plant τ must equal unity.It is, also, concluded that hybrid cycles are favoured because of the additional degree of freedom.     

An R curve approach to COD and J for an austenitic steel

This paper describes investigations of the fracture toughness of a type 316 stainless steel by means of $\text{δ}{}_{\mathrm{R}}$ and $\text{J}{}_{\mathrm{R}}$ curves. Both COD (δ) and $\text{J}$ values rise dramatically after crack initiation and reach plateau values after some crack extension. These plateaux are much higher than the value of δ and $\text{J}$ at maximum load and are considered to represent the maximum fracture toughness of the material.A load/crack mouth opening record is shown to be sufficient to obtain a $\text{J}{}_{\mathrm{R}}$ as well as a $\text{δ}{}_{\mathrm{R}}$ curve. The maximum plateau value can be obtained only in specimens with sufficient ligament depth. It is shown that this is achieved for a specimen whose depth, $\text{W}$, is four times the thickness, $\text{B}$, with a ligament length of $\text{2·8B}$.The relationship between the $\text{δ}{}_{\mathrm{R}}$ concept and the model of crack tip behaviour proposed by Green & Knott¹³ for an extending ductile tear is also investigated.     

Calculation of heat transfer coefficients for nucleate boiling in binary mixtures of refrigerant-oil

The heat transfer coefficient for nucleate boiling of pure liquids can be determined in many cases by the simple relation $\text{h = C · q}{}^{\mathrm{n}}$. In nucleate boiling of mixtures with widely varying properties, the concentration gradient close to the heating surface strongly affects the heat transfer. As the composition of the mixture is difficult to obtain there, it is tried to develop relations as simple as the one mentioned above. The following form is chosen $\text{h = C (Y) · q}{}^{\mathrm{n(Y)}}$ with Y being a function of both, the kind of mixture A_K and the concentration w: $\text{Y}\text{= f (}\text{A}{}_{\mathrm{K}}\text{, w}\text{)}$. Based on experimental values for four different refrigerant-oil mixtures in concentrations of $\text{w = 0.005}\text{to}\text{0.20}$, the following relation renders best results: $\text{h}\text{= 0.085·[}\text{exp. (b}{}_1\text{w}\text{) +}\text{exp.(b}{}_2\text{w}\text{)],}\text{q}\text{(0.89-}\text{Bw}\text{)}$ For each kind of oil, however, different values of b₁, b₂ and B have to be used; these are given.     

Heat transfer of tubes closely spaced in an in-line bankTransfert thermique de tubes serres dans un reseau en ligneWärmeübergang in einem fluchtenden rohrbündelTeплoпepeнoc oт тpyб в кopидopнoм лyKhcyкe

An experimental investigation of heat transfer around four cylinders closely spaced in a cross-flow of air has been conducted. The cylinders are settled in tandem with equal distances between centers. Their inline pitch ratio is in the range of $\text{1.15 ≤}\text{c}\text{d ≤ 3.4}$ (c = distance between cylinders' centers, d = cylinder diameter); the Reynolds number ranges from 10⁴ to 5 × 10⁴. It is found that there exists a critical Reynolds number Red_c at which the heat transfer behavior changes drastically, and is correlated with the in-line pitch ratio by $\text{Red}{}_{\mathrm{c}}\text{= 1.14 × 10}{}^5\text{(}\text{c}\text{d)}{}^{\mathrm{?5.84}}$.Variations of characteristic features of the mean and local Nusselt numbers are discussed in relation to the length of the vortex formation region behind the cylinder.     

Thermochemical hydrogen preparation—Part V. A feasibility study of the sulfur iodine cycle

A sulfur-iodine cycle consists of the following three reactions:

$\text{2H}{}_2\text{O + SO}{}_2\text{+ I}{}_2\text{→ H}{}_2\text{SO}{}_4\text{+ 2HI,}$

$\text{H}{}_2\text{SO}{}_4\text{→ H}{}_2\text{O + SO}{}_2\text{+}\text{1}\text{2O}{}_2\text{,}$

$\text{2 HI → H}{}_2\text{+ I}{}_2\text{.}$

It was found that the first reaction can be performed as a cell reaction without the addition of external energy. The sulfuric acid and the hydriodic acid are produced separately in the anode and cathode compartments, respectively. The second and third reactions can be carried out as catalytic thermal decompositions. A process flow sheet of this cycle and its mass balance was based on experimental results, and the heat balance for this cycle was made. It was found that internal heat exchange for this cycle was very large (about 2600 kcal/mol H₂), due mainly to the low yield of the decomposition reaction of hydrogen iodide. Theoretical and experimental studies were made to improve the yield of this reaction. The following three methods seem to be promising for this purpose: (1) continuous removal of the hydrogen produced in the reaction zone; (2) performance of the reaction at low temperature (185–250°C) and high pressure (100 atml; and (3) substitution of the benzene-cyclohexane cycle (6HI ? C₆H₆ → C₆H₁₂ + 3I₂; C₆H₁₂ → C₆H₆ ? 3H₂) for the hydrogen iodide decomposition step.