Approximate method for computation of glass cover temperature and top heat-loss coefficient of solar collectors with single glazing

An improved equation form for computing the glass cover temperature of flat-plate solar collectors with single glazing is developed. A semi-analytical correlation for the factor f—the ratio of inner to outer heat-transfer coefficients—as a function of collector parameters and atmospheric variables is obtained by regression analysis. This relation readily provides the glass cover temperature (T_g). The results are compared with those obtained by numerical solution of heat balance equations. Computational errors in T_g and hence in the top heat loss coefficient (U_t) are reduced by a factor of five or more. With such low errors in computation of T_g and U_t, a numerical solution of heat balance equations is not required. The method is applicable over an extensive range of variables: the error in the computation of U_t is within 2% with the range of air gap spacing 8 mm to 90 mm and the range of ambient temperature 0°C to 45°C. In this extended range of variables the errors due to simplified method based on empirical relations for U_t are substantially higher.     

Effect of absorption of solar radiation in glass-cover(s) on heat transfer coefficients in upward heat flow in single and double glazed flat-plate collectors

Absorption of solar radiation in the glass cover(s) of a flat plate solar collector increases the temperature of cover(s) and hence changes the values of convective and radiative heat transfer coefficients. The governing equations for the case of single as well as double glazed collector have been solved for inner and outer surface temperatures of glass cover(s) with/without including the effect of absorption of solar radiation in the glass cover(s), with appropriate boundary conditions. The effects of absorption of solar radiation on inner and outer surface temperatures and consequently on convective and radiative heat transfer coefficients have been studied over a wide range of the independent variables. The values of glass cover temperatures obtained from numerical solutions of heat balance equations with and without including the effect of absorption of solar radiation in the glass cover(s) are compared. For a single glazed collector the increase in glass cover temperature due to absorption of solar radiation could be as high as 6°. The increase in temperatures of first and second glass covers of a double glazed collector could be as high as 14° and 11°, respectively. The effect on the convective heat transfer coefficient between the absorber plate and the first glass cover is substantial. The difference in the values of the convective heat transfer coefficients between the absorber plate and the first glass cover (h_cp1) of a double glazed collector for the two cases: (i) including the effect of absorption and (ii) neglecting the effect of absorption in glass cover, could be as high as 49%. Correlations for computing the temperatures of inner and outer surfaces of the glass cover(s) of single and double glazed flat plate collectors are developed. The relations developed enable incorporation of the effect of absorption of solar radiation in glass cover(s) in the relations for inner and outer surface temperatures in a simple manner. By making use of the relations developed for inner and outer surface temperatures of glass cover(s) the convective and radiative heat transfer coefficients can be calculated so close to those obtained by making use of surface temperatures of glass cover(s) obtained by numerical solutions of heat balance equations that numerical solutions of heat balance equations are not required.     

KI–T estimations for embedded flaws in pipes – Part I: Axially oriented cracks

This study reports a numerical investigation on the linear-elastic K_I and T-stress values over the front of elliptical cracks axially embedded in the wall of a pipe/cylindrical structure, under a uniform pressure applied on the inner surface of the pipe. The numerical procedure employs an interaction integral approach to compute the linear-elastic stress intensity factor (SIF) K_I and T-stress values from very detailed crack-front meshes. The verification study confirms the accuracy of the adopted numerical procedure in computing the K_I values based on existing results for external axial surface cracks in the wall of a cylindrical structure. The parametric investigation covers a wide range of geometric parameters including: the wall thickness to the inner radius ratio of the pipe (t/R_i), the crack depth over the wall thickness ratio (a/t), the crack aspect ratio (a/c) and the crack location measured by the ratio of the distance from the centerline of the crack to the outer surface of the pipe over the pipe wall thickness (e_M/t). Subsequent efforts develop, from a nonlinear curve-fitting procedure, a new set of equations to estimate the T-stress and K_I values at three critical front locations of the axial elliptical cracks: the crack-front point O nearest to the outer surface of the pipe, the crack-front point I nearest to the inner surface of the pipe and the crack-front point M on the centerline of the axial crack. These equations combine a second-order polynomial with a power-law expression to predict the pronounced variations in the T-stress and K_I values with respect to the geometric parameters. The coefficients of the new K_I and T-stress equations either take a constant value or incorporate the linear variation with respect to the pipe wall thickness over the inner radius ratio, t/R_i. The proposed equations demonstrate a close agreement with the finite element (FE) results, which indicate very strong dependence of the T-stress and K_I values at point O and point I on the corresponding ligament lengths, e_O and e_I.     

On effect of wind speed on passive solar still performance based on inner/outer surface temperatures of the glass cover

The thermal performance of a passive single basin solar still was investigated by computer simulation using the climatic conditions of Jeddah (lat. 21° 42′ N, long. 39° 11′ E), Saudi Arabia under the two conditions: (a) the temperatures of the inner T_gi and outer T_go surfaces of the still cover are equal and (b) the temperatures of the inner T_gi and outer T_go surfaces of the still cover are not equal. Effect of wind speed V on the daily productivity P_d of the still for these conditions was studied. It was indicated that for the condition T_gi = T_go, there is a critical mass (depth) of basin water beyond which P_d increases as V increases until a typical velocity V_t. For basin water masses less than the critical mass, P_d was found to decrease with increasing V until V_t. After V_t, the change in P_d becomes insignificant. When T_gi ≠ T_go, P_d was found to be less dependent on wind speed V for all investigated values of mass of basin water m_w in the range 0 < m_w ≤ 100 kg. The rate of heat transfer by forced convection due to wind should be estimated on the basis of the temperature of the upper surface of the still cover.     

A two-stage Lie-group shooting method (TSLGSM) to identify time-dependent thermal diffusivity

We consider an inverse problem for identifying a time-dependent thermal diffusivity α(t) in a heat conduction equation T_t(x, t) = α(t)T_xx(x, t), with the aid of an extra measurement of temperature at an internal point of a rod. Because the data are acquired at an inner point we require to develop a two-stage Lie-group shooting method (TSLGSM) to solve this inverse problem. The present approach is novel and is examined through some numerical tests. By comparing the calculated results with exact ones we can assure that the TSLGSM is an accurate and efficient method, whose estimation error is small even for the identification of a discontinuous and oscillatory thermal diffusivity. Under noise, the identified solutions are also acceptable.     

KI–T estimation for embedded flaws in pipes – Part II: Circumferentially oriented cracks

This paper, in parallel to the investigation on axially embedded cracks reported in the companion paper, presents a numerical study on the linear-elastic K_I and T-stress values over the front of elliptical cracks circumferentially embedded in the wall of a pipe/cylindrical structure, under a uniform pressure applied on the inner surface of the pipe. The numerical procedure employs the interaction-integral approach to compute the linear-elastic stress-intensity factor (SIF) K_I and T-stress values for embedded cracks with practical sizes at different locations in the wall of the pipe. The parametric study covers a wide range of geometric parameters for embedded cracks in the pipe, including: the wall thickness to the inner radius ratio (t/R_i), the crack depth over the wall thickness ratio (a/t), the crack aspect ratio (a/c) and the ratio of the distance from the centerline of the crack to the outer surface of the pipe over the pipe wall thickness (e_M/t). The parametric investigation identifies a significant effect of the remaining ligament length on both the T-stress and K_I values at the crack-front location (denoted by point O) nearest to the outer surface of the pipe and at the crack-front location (denoted by point I) nearest to the inner surface of the pipe. The numerical investigation establishes the database to derive approximate functions from a nonlinear curve-fitting procedure to predict the T-stress and K_I values at three critical front locations of the circumferentially embedded crack in a pipe: points O, I and M. The proposed T-stress and K_I functions utilize a combined second-order polynomial and a power-law expression, which presents a close agreement with the T-stress and K_I values computed from the very detailed finite element models. The comparison between the circumferentially embedded crack and the axially embedded crack indicates that both the T-stress and K_I values at crack-front points O and I in a circumferential crack equal approximately 50% the T-stress and K_I values at the corresponding front locations in an axial crack with the same crack depth ratio, the same crack aspect ratio and the same pipe wall thickness to the inner radius ratio.     

Complete heat transfer solutions of an insulated regular polyhedron by using an RPSWT model

The heat transfer characteristics for an insulated regular polyhedron (including sphere) are analyzed by using the same RPSWT model in the present study as that used by Wong and Chou previously [Energy Convers. Manage. 44 (2003) 3015]. The thermal resistance of the inner convection term and the wall conduction term in the heat transfer rate are not neglected in the present study. Thus, the complete heat transfer solution will be obtained. The present results can be applied more extensively to practical situations, such as an insulated gas tank. The results of the critical thickness t_cr and the neutral thickness t_e are independent of the values of J (generated by the effect of the inner convection term and the wall conduction term). However, the heat transfer rates are dependent on the values of J. The present study shows that the thermal resistance of the inner convection term and the body conduction term cannot be neglected in the heat transfer equation in situations of low to medium inner convection coefficients h_i and/or low to medium wall conductivities K and/or great wall thickness t₁, especially in situations with small body sizes or/and great outer convection coefficients h_o.     

Calculation of tubular absorber heat loss factor

Although several empirical equations exist for heat loss factor in flat-plate collectors, no similar relation is available for the heat loss factor of a tubular absorber with a concentric glass cover, employed as the target of a linear solar concentrator. A semi-empirical equation for the heat loss factor as a function of the various variables involved is developed. A relatively simple equation for the factor f has also been proposed. The present equation predicts the overall heat loss factor, U_L, to within ±5% of the value obtained by exact solution of the simultaneous equations, in the range of variables—absorber temperature, 60°C to 220°C, emittance of the black coating, 0.1 to 0.95, and wind velocity, 1.5 m/s to 10 m/s. The proposed correlation also takes into account the effect of property variations.     

Kinetic model and prediction for coal hydrogasification

Coal hydrogasification is a key component of zero emission coal (ZEC) power generation system which discharges little CO₂ and other pollutants at a thermal efficiency close to 70%. In addition, coal hydrogasification itself has many advantages. A hydrogasification kinetic model including ten homogeneous reactions and four heterogeneous reactions is established in this work and is validated against experiment data available in literatures. The validated model is then used to predict the effects of different reaction conditions including the reaction temperature T, the reaction pressure p_t, the H₂/coal mass ratio U and the reaction time t on coal hydrogasification properties. The results indicate that coal hydrogasification is facilitated by the increased p_t and t. When T is not higher than 1273 K, the gasification process is promoted with T increment. Increasing U can promote the coal hydrogasification process on the whole. When U is larger than 0.5, however, the coal conversion ratio (x_coal) will slightly decrease with U increment.     

Top heat-loss factor of double-glazed box-type solar cooker from indoor experiments

The top heat-loss factor (U_t) of a box-type solar cooker varies with plate temperature, wind heat-transfer coefficient and ambient temperature. A method for correlating U_t with these variables is presented for a cooker with double glazing. A set of equations is developed for correlating data obtained in indoor experiments at different plate temperatures and wind speeds.     

Slip flow and heat transfer in axially moving micro-concentric cylinders

The hydrodynamics and thermal behaviors of fluid flow in axially moving micro-concentric cylinders are investigated analytically. Effects of Knudsen number, velocity and radius of the cylinders on the microchannel hydrodynamics and thermal behaviors are investigated. It is found that as Kn increases the slip in the hydrodynamic and thermal boundary condition increases. The slip and the jump at the inner surface are much larger than that of the outer one. When the outer cylinder velocity approaches the inner cylinder one, the slip velocity vanishes. Also, the effect of the variation of U₁ on the temperature jump for adiabatic outer surface is insignificant.     

Skin friction and heat transfer in power-law fluid laminar boundary layer along a moving surface

Analytical and numerical solutions are presented for momentum and energy laminar boundary layer along a moving plate in power-law fluids utilizing a similarity transformation and shooting technique. The results indicate that for a given power-law exponent n(0<n?1) or velocity ratio parameter ξ, the skin friction σ decreases with the increasing in ξ or n. The shear force decreases with the increasing in dimensionless tangential velocity t. When Prandtl number N_Pr=1, the dimensionless temperature w(t) is a linear function of t, and the viscous boundary layer is similar to that of thermal boundary layer. In particular, w(t)=t if ξ=0, i.e., the velocity distribution in viscous boundary layer has the same pattern as the temperature distribution in the thermal boundary and δ=δ_T. For N_Pr?1, the increase of viscous diffusion is larger than that of thermal diffusion with the increasing in N_Pr, and δ_T(t)<δ(t). The thermal diffusion ratio increases with the increasing in n(0<n?1) and ξ.     

Lattice Boltzmann simulation of natural convection around a horizontal elliptic cylinder inside a square enclosure

Natural convection between a square outer cylinder and a heated elliptic inner cylinder has been studied numerically. The inner and outer walls are maintained at temperatures T_h and T_c, respectively, with T_h > T_c. Lattice Boltzmann method (LBM) has been used to investigate the hydrodynamic and thermal behaviors of the fluid at various vertical positions of the inner cylinder for different Rayleigh numbers ranging from 10³ to 10⁶. The results show that streamlines, isotherms, and the number, size and formation of the cells strongly depend on the Rayleigh number and the position of inner cylinder. The changes in heat transfer quantities have also been presented.     

Numerical study on in-tube laminar heat transfer of supercritical fluids

Numerical calculations were carried out for supercritical carbon dioxide flowing in miniature tubes with Re less than 1000. The heat transfer coefficient α and friction factor f were numerically studied for different values of the tube diameter, pressure, mass flux, and heat flux. When compared with the constant property flow, where Nu = 4.364 and f = 64/Re for a circular tube under a constant heat flux condition, a large divergence from the constant value was obtained for both Nu and f·Re in the vicinity of the pseudocritical temperature T_m. When cooled under a constant heat flux, Nu attained its peak value when T_b > T_m and its minimum value when T_b < T_m, while f·Re attained its peak value at T_b = T_m. With regard to the heating process, the reverse tendencies were confirmed. The variations of the specific heat with temperature were found to be the dominant factor for Nu. In addition, empirical correlations that considered the cross-sectional distribution of thermophysical properties were proposed to predict the values of Nu and f both in the near-pseudocritical temperature region and in the thermal entrance region of the tube. The proposed correlations were also verified by comparing the predicted results with numerical results obtained by using supercritical water.     

Heat transfer on a power law stretching sheet subjected to free stream pressure gradient

The present work deals with the numerical study of temperature distribution in the laminar boundary layer driven by the stretching boundary surface subjected to pressure gradient. The similarity transformation obeying the same power law based on composite reference velocity (union of velocities of the stretching boundary and free stream) has been employed that leads to a single set of equations, irrespective of the condition whether U_w > U_∞ or U_w < U_∞, containing three parameters: β measuring the stretch rate of the moving boundary, ε is the ratio of free stream velocity to composite reference velocity and Pr is the Prandtl number of the ambient fluid. The numerical solutions of the thermal boundary layer equations are obtained for three Prandtl numbers 0.7, 1.0 and 10 for 0 ? ε ? 1 and for 0 ? β ? 2. The heat transfer coefficient show appreciable dependence on the ratio of free stream velocity to union of velocities of the stretching surface boundary and free stream.     

Identification of spacewise and time dependent source terms in 1D heat conduction equation from temperature measurement at a final time

Inverse problems of identifying the unknown spacewise and time dependent heat sources F(x) and H(t) of the variable coefficient heat conduction equation u_t = (k(x)u_x)_x + F(x)H(t) from supplementary temperature measurement (u_T(x)?u(x, T_f)) at a given single instant of time T_f > 0, are investigated. For both inverse source problems, defined to be as ISPF and ISPH respectively, explicit formulas for the Fréchet gradients of corresponding cost functionals are derived. Fourier analysis of these problems shows that although ISPF has a unique solution, ISPH may not have a unique solution. The conjugate gradient method (CGM) with the explicit gradient formula for the cost functional J₁(F) is then applied for numerical solution of ISPF. New collocation algorithm, based on the piecewise linear approximation of the unknown source H(t), is proposed for the numerical solution of the integral equation corresponding to ISPH. The proposed two numerical algorithms are examined through numerical examples for reconstruction of continuous and discontinuous heat sources F(x) and H(t). Computational results, with noise free and noisy data, show efficiency and high accuracy of the proposed algorithms.     

Thermal conductivity variation on natural convection flow of water–alumina nanofluid in an annulus

This work is focused on the numerical modeling of steady laminar natural convection flow in an annulus filled with water–alumina nanofluid. The inner surface of the annulus is heated uniformly by a uniform heat flux q and the outer boundary is kept at a constant temperature T_c. Two thermal conductivity models namely, the Chon et al. model and the Maxwell Garnett model, are used to evaluate the heat transfer enhancement in the annulus. The governing equations are solved numerically subject to appropriate boundary conditions by a penalty finite-element method. A parametric study is conducted and a selective set of graphical results is presented and discussed to illustrate the effects of the presence of nanoparticles, the Prandtl number and the Grashof number on the flow and heat transfer characteristics for both nanofluid models. It is found that significant heat transfer enhancement can be obtained due to the presence of nanoparticles and that this is accentuated by increasing the nanoparticles volume fraction and Prandtl number at moderate and large Grashof number using both models. However, for the Chon et al. model the greatest heat transfer rate is obtained.     

Power-optimization of non-ideal energy converters under generalized convective heat transfer law via Hamilton-Jacobi-Bellman theory

A multistage irreversible Carnot heat engine system operating between a finite thermal capacity high-temperature fluid reservoir and an infinite thermal capacity low-temperature environment with generalized convective heat transfer law [q∝m(ΔT)] and the irreversibility of heat resistance and internal dissipation is investigated in this paper. Optimal control theory is applied to derive the continuous Hamilton-Jacobi-Bellman (HJB) equations, which determine optimal fluid temperature configurations for maximum power output under the conditions of fixed duration and fixed initial temperature of the driving fluid. Based on general optimization results, the analytical solution for the case with Newtonian heat transfer law (m=1) is further obtained. Since there are no analytical solutions for the other heat transfer laws (m≠1), the continuous HJB equations are discretized and dynamic programming (DP) algorithm is adopted to obtain complete numerical solutions of the optimization problem, and the relationships among the maximum power output of the system, the process period and the fluid temperature are discussed in detail.     

Fully developed forced convective heat transfer in an annulus partially filled with metallic foams: An analytical solution

An analytical solution for fully developed forced convective heat transfer in an annulus partially filled with metallic foam was proposed. The inner surface attached with an annular metallic foam layer was exposed to constant heat flux while the outer surface was adiabatic. In the metallic foam region, the Brinkman–Darcy equation was used to describe the fluid flow and the thermal non-equilibrium model was employed to establish the heat transfer equations. At the porous-fluid interface, no-slip coupling conditions were utilized to couple flow and heat transfer of the porous and open regions. A closed-form analytical solution was obtained for velocity and temperature profiles. The explicit form of friction factor and the Nusselt (Nu) number were also provided. The solutions were validated by two extreme cases: the empty annulus and the annulus fully filled with metallic foam. The effects of key parameters on friction factor, Nu number, and j/f^1/3 were examined. The relationship between flow heterogeneity and heat transfer was also discussed by introducing the flow heterogeneity coefficient. The porosity, pore density, and foam thickness for engineering applications were recommended. In the present analytical solution, a benchmark was also established for improving discretizing schemes in numerical works.     

Momentum and heat transfer in laminar boundary layer behind shock wave

Analytical and numerical solutions are established for momentum and energy laminar boundary layer induced by a shock wave. The results indicated that skin friction σ decreases with increasing in velocity ratio ξ(1≤ξ< 6). For each specified ξ(1≤ξ< 6), temperature w(t) increases with increasing of Tw but decreases with Te , and for a range of t ∈[1,ξ], w(t) decreases with the increasing of t. Thermal diffusion increases with increasing of uw but decreases with increasing Ue.