Free Convection Film Flows and Heat Transfer

In this chapter, the basic conservation equations related to laminar free fluid flow conservation equations are introduced. For this purpose, the related general laminar free conservation equations on continuity equation, momentum equation, and energy equation are derived theoretically. On this basis, the corresponding conservation equations of mass, momentum, and energy for steady laminar free convection boundary layer are obtained by the quantities grade analysis.

In this chapter, the traditional Falkner–Skan type transformation for laminar free convection boundary layer is reviewed. The typical two-dimensional basic conservation equations for laminar free convection boundary layer are taken as example for derivation of the related similarity variables for Falkner–Skan type transformation. By means of the stream function and the procedure with the method of group theory, the similarity intermediate function variable \(f(\eta )\) is induced. Then, the velocity components are transformed to the related functions with the similarity intermediate function variable (\(f\eta )\). On this basis, partial differential momentum equation of the free convection boundary layer is transformed to related ordinary equation. At last, the limitations of the Falkner–Skan type transformation are analyzed in detail.

A new similarity analysis method with a new set of dimensionless similarity variables is provided for complete similarity transformation of the governing partial differential equations of laminar free convection and two multi-phase film flows. The derivation of the Reynolds number together with the Nusselt number and Prandtl number is reviewed by means of Buckingham \(\pi \)-theorem and dimension analysis, where the Reynolds number is taken as the one of the new set of dimensionless analysis variables. The essential work focuses on derivation of equations for the dimensionless velocity components and the dimensionless coordinate variable, by means of a detailed analysis of quantity grade of the governing conservation partial differential equations of laminar free convection. On this basis, the new similarity analysis method is produced for complete similarity transformation of the conservation partial differential equations of laminar free convection and its film flows. With the novel dimensionless velocity components devoted in this chapter, the new similarity analysis method has obvious advantages compared with the Falkner–Skan transformation. These advantages are (i) more convenient for consideration and treatment of the variable physical properties, (ii) more convenient for analysis and investigation of the two-dimensional velocity field, and (iii) more convenient for satisfaction of the interfacial mass transfer matching conditions in the numerical calculation and for rigorous investigation of mass transfer for two-phase film flows with three-point boundary problem. These advantages will be found from the successive chapters.

The advanced method reported in this chapter for treatment of fluid variable physical properties involves temperature parameter method for treatment of temperature-dependent physical properties of gases, theoretical equation method for treatment of concentration- and temperature-dependent density of vapour-gas mixture, weighted sum method for treatment of other concentration- and temperature-dependent physical properties of vapour-gas mixture and polynomial method for treatment of temperature-dependent physical properties of liquids. These methods are taken as a theoretical foundation of this book for extensive investigation of hydrodynamics and heat transfer of free convection of gases, free convection of liquids, free convection film boiling of liquid and free convection film condensation of pure vapour or vapour-gas mixture with consideration of coupled effects of variable physical properties. For the temperature parameter method based on the simple power-law of the temperature-dependent physical properties of gases, a system of the temperature parameters such as \(n_\mu \), \(n_\lambda \) and \({n_{c}}_{p}\) are reported. From these temperature parameters, it is seen that the specific heat parameter is much small, and then, it follows that the variable temperature will have more obvious effects on viscosity, thermal conductivity and density of gases than that of the specific heat. Since the determination of the temperature parameter is based on the typical experimental data, with the provided temperature parameters, the temperature-variable physical properties of gases can be stimulated very well by using the temperature parameter method. Furthermore, with the temperature parameter method the treatment of variable physical properties of vapour or gas becomes very simple and convenient. Taking water as an example, the temperature-dependent polynomials of the density, thermal conductivity and viscosity are introduced for liquid variable physical properties, while the specific heat at constant pressure is so small that it can be disregarded generally with variation of temperature. These polynomials are reliable, since the related typical experimental data. The concentration-dependent density equations of vapour-gas mixture are reported through the rigorously theoretical derivation, while the other concentration-dependent physical properties of vapour-gas mixture are expressed as the weighted sum of the physical properties of the involved vapour and gas with their concentrations (mass fraction). Since the involved vapour and gas are temperature-dependent, the physical properties of the vapour-gas mixture are concentration- and temperature-dependent.

The new similarity analysis method is used to replace the traditional Falkner-Skan type transformation for creating similarity governing models of laminar free convection. With this method, the velocity components are directly transformed into the corresponding dimensionless velocity components. Then, it is more convenient to equivalently transform the governing partial differential equations into the related ordinary differential ones, without inducing stream function and the intermediate function variable required by the traditional Falkner-Skan type transformation. Furthermore, with this method, it is more convenient for treatment of variable physical properties. The temperature parameter method is used for treatment of variable physical properties of gases. With this method the physical property factors coupled with the governing ordinary differential equations are transformed to the functions of the Prandtl number, temperature parameters, and the boundary temperature ratio for simultaneous solution. There are obvious effects of variable physical properties on velocity and temperature fields, as well as heat transfer of free convection of gas. Based on the heat transfer analysis and related rigorous numerical results, the prediction equations of gas free convection heat transfer is created. Since the Prandtl number and temperature parameters of gases are based on the experimental data, these prediction equations of gas free convection heat transfer are reliable and then have practical application value.

Based on the study of Chap. 6, the temperature-dependent specific heat is further considered for investigation of laminar free convection of polyatomic gases with consideration of variable physical properties. The viscosity, thermal conductivity, and specific heat parameters are provided for a series of polyatomic gases. The governing energy ordinary differential equation is further derived out for consideration of temperature-dependent specific heat, by using the new similarity analysis method. A system of numerical solutions are obtained for variation of the temperature parameters \(n_\upmu , n_\lambda \), and \(n_{c_\mathrm{p} } \), Prandtl number, and the boundary temperature ratio . It is seen from the numerical results that there are obvious effects of variable physical properties on velocity and temperature fields, as well as heat transfer of free convection of polyatomic gases. The theoretical equations of heat transfer of polyatomic gas free convection created based on the heat transfer analysis contain a only one no-given variable, the wall temperature gradient. Based on the system of numerical solutions on the wall dimensionless temperature gradient, the prediction equation of the wall temperature gradient is created by means of a curve-fitting method, and then, the theoretical equations on heat transfer are available for prediction of heat transfer. It is found that the gas temperature parameters, Prandtl number, and the boundary temperature ratio dominate the heat transfer of laminar free convection of polyatomic gases. Because the temperature parameters are based on the typical experimental data, these equations on heat transfer are reliable for engineering prediction of laminar free convection of polyatomic gas.

The new similarity analysis method is used to transform the governing partial differential equations of laminar free convection of liquid into the corresponding governing dimensionless system, which are identical to the corresponding governing dimensionless system of gas laminar free convection, except different treatment of variable physical properties. Due to the different variable physical properties from gases, the polynomial approach is suggested for treatment of temperature-dependent physical properties of liquid. Taking water as an example, the polynomial approach is applied for expressions of temperature-dependent density, thermal conductivity, and viscous. These expressions are reliable because they are based on the typical experimental values of the physical properties. By means of the equations of the physical property factors coupled with the governing ordinary differential equations of liquid laminar free convection created by the new similarity analysis method, the non-linear governing equations with corresponding boundary conditions are simultaneously solved numerically. The effect of variable physical properties on water laminar free convection along an isothermal vertical plate is investigated. It is found that the wall temperature gradient is the only one no-given condition for prediction of heat transfer. Compared with wall temperature, the bulk temperature dominates heat transfer of laminar free convection. By means of the curve-fitting equation on the wall temperature gradient, the heat transfer analysis equations based on the new similarity analysis model become those with the practical application value for heat transfer prediction.

Experimental investigations were carried out to verify the results of the previous chapters for effects of variable physical properties on laminar free convection of air and water. By increasing the wall temperature for the liquid laminar free convection or increasing the boundary temperature ratio for gas laminar free convection of gas, the velocity component of the free convection increases, and the velocity profile moves to the direction of the flat plate. Consequently, the thickness of the velocity boundary layer decreases. With an increase of the plate height x, the velocity component of water or air free convection increases, and the velocity profile moves toward to the fluid bulk. As a result, the thickness of velocity boundary layer increases. It is found that the agreement between the measured and calculated velocity fields is good, thus it is confirmed that the results in Chaps. 6–8 are reliable.

The new similarity analysis method is applied to the similarity transformation of the governing partial differential equations of laminar free convection on inclined plate. It is seen that the transformed governing ordinary differential equations on the inclined plate are same as those on the vertical plate. Then, it follows that there are identical governing ordinary differential equations and dimensionless prediction equations on heat transfer both for inclined and vertical cases of laminar convection. In this case, the vertical case can only be regarded as a special example of the inclined case. Therefore, the numerical solutions and prediction equations on heat transfer for vertical case can be directly used for the inclined case. Finally, the simple and direct correlations for describing the transformation of the velocity components and heat transfer from the vertical case to the inclined case for the free convection are derived.

The new similarity analysis method is successfully applied for complete similarity transformation of the governing partial differential equations of laminar free film boiling of subcooled liquid with consideration of coupled effects of variable physical properties, where the laminar free film boiling of saturated liquid is only regarded as its special case. The dimensionless velocity components as the solutions for vapor and liquid films have definite physical meanings. It follows that the new similarity analysis method is appropriate for extensive investigation of the two-phase boundary layer problems with consideration of coupled effects of variable physical properties, such as the temperature-dependent density, thermal conductivity, and absolute viscosity of the medium of vapor and liquid films. The interfacial balance equations between the vapor and liquid films are considered in detail, such as mass flow rate balance, velocity component balance, shear force balance, temperature balance, and energy balance.

Physical property factors coupled with the theoretical and mathematical models of the laminar free convection film boiling of liquids are treated into the functions of dimensionless temperature, for simultaneous solutions with the three-point boundary values conditions of the two-phase film flow. Then, the numerical solutions of momentum and temperature fields at different wall superheated grades and liquid bulk subcooled grades are theoretically reliable, because the variable physical properties are treated rigorously. On this basis, a system of rigorous numerical solutions for momentum and temperature fields of the two-phase film flows are calculated with taking the film of boiling water as the example, in which the related boiling of saturated water is only the special case. The numerical procedure presented here is reliable for rigorous solutions of the theoretical models of three-point boundary value problem with the two-phase flow. The dimensionless velocity components have definite physical meanings; then, the corresponding solutions of the models can be easily understood. With increasing the wall superheated grades, the maximum of velocity field of vapor film will increase and shift far away from the plate. The velocity of vapor film will decrease with increasing the liquid subcooled grade. With increasing the liquid subcooled degree, the thickness of liquid film will increase, and the velocity profile level of liquid film will decrease slower and slower. Furthermore, with increasing wall superheated grade, the effect of wall superheated grade on the velocity field of liquid film will decrease.

By means of the heat and mass transfer analysis based on the new similarity analysis method, it is found that only the wall temperature gradient and mass flow rate parameter are no-given variables respectively, for prediction of heat and mass transfer of the film boiling. The wall temperature gradient is proportional to heat transfer, and will decrease with increasing the wall superheated grade, and increase with increasing the bulk subcooled grade. Additionally, the wall temperature gradient is steeper with higher liquid bulk subcooled grade and with lower wall superheated grade. The curve-fit equation for evaluation of the wall temperature gradient provided in this chapter agrees very well with the related rigorous numerical solutions, and useful for a reliable prediction of heat transfer of the laminar film boiling of water. From the numerical results, it is seen that vapor film thickness will increase with increasing wall superheated grade or with decreasing the water bulk subcooled grade, and in the iterative calculation it is a key work to correctly determine the suitable value. The solutions of the governing equations are converged in very rigorous values of vapor film thickness. The interfacial velocity component will increase with increasing the wall superheated grade except the case for very low liquid bulk subcooled grade, and will decrease with increasing the liquid bulk subcooled grade. The boiling mass flow rate is proportional to the induced mass flow rate parameter. The mass flow rate parameter will increase with increasing the wall superheated grade, decrease obviously with increasing the liquid subcooled grade, and decrease slower and slower with increasing the liquid subcooled grade. The mass flow rate parameter is formulated according to the numerical solutions, and then, prediction equation for boiling mass transfer is created for reliable evaluation.

In this chapter, the work is focused on constitution of mathematical models of the laminar free convection film condensation of superheated vapor, while, the film condensation of saturated vapor is only regarded as its special case. The new similarity analysis method is successfully applied for similarity transformation of the governing partial differential equations of laminar free convection film condensation of superheated vapor with consideration of coupled effects of variable physical properties of liquid and vapor films. In the transformed governing ordinary differential equations, the dimensionless velocity components of liquid and vapor films have definite physical meanings, and then the solutions of the governing models can be understood easily. In the analysis and similarity transformation of the mathematical models, the interfacial balance equations between the liquid and vapor films are considered in detail, such as mass flow rate balance, velocity component balance, shear force balance, temperature balance, and energy balance. Therefore, such mathematical model is serious theoretically and has its application value in practice.

The work is dealt with for solutions of velocity and temperature fields on laminar free film condensation of superheated vapor on a vertical flat plate at atmospheric pressure with consideration of various factors including variable physical properties. The film condensation of saturated vapor is only its special case. The system of ordinary differential equations is computed by a successively iterative procedure and an iterative method is adopted for the numerical solutions of the three-point boundary value problem. With increasing the wall subcooled grades, the maximum of velocity field of liquid film will increase and shift far away from the plate. In addition, the velocity of liquid film will decrease with increasing the vapor superheated grade. Compared with the effect of wall subcooled grades on the velocity of liquid film, the related effect of the vapor superheated grade is obviously weak. With increasing the wall subcooled grades, the thickness of liquid film will increase. With increasing the vapor superheated grade, the thickness of liquid film will decrease. The temperature grade of liquid film on the wall will decrease with increase in wall subcooled grade, and increase with increasing vapor superheated grade. Compared with the effect of wall subcooled grades on the temperature of liquid film, the related effect of the vapor superheated grade is obviously weak. The velocity of vapor film will increase with increasing the wall superheated grades, and decrease with increasing vapor superheated grade. With increasing wall subcooled grade, the velocity of vapor film will decrease slightly. With increasing the vapor superheated grade, the velocity of vapor film will decrease obviously.

With heat and mass transfer analysis, the theoretical equations for Nusselt number and mass flow rate are provided for the laminar free convection film condensation of vapor where only the wall temperature gradient and condensate mass flow rates are unknown variables, respectively. With increase of the wall subcooled grade, the wall temperature gradient will decrease, especially for lower wall subcooled grade. While, with increase of the vapor bulk superheated grade, the wall temperature gradient will increase. However, the effect of the wall subcooled grade on the wall temperature gradient is more obvious than that of the vapor bulk superheated grade. With increase of the wall subcooled grade, the condensate film thickness will increase, especially for lower wall subcooled grade, while with increase of the superheated grade, the condensate film thickness will decrease. However, the effect of the wall subcooled grade on the condensate film thickness is more obvious than that of the vapor bulk superheated grade. With increase of the wall subcooled grade, the velocity components will increase, especially for the small value of the wall subcooled grade. While with increase of the vapor bulk superheated grade, the velocity components will decrease. As per the results, with increase of the wall subcooled grade, the condensate mass flow rate parameter will increase, especially due to the function of condensate film thickness. While with increase of the vapor bulk superheated grade, the mass flow rate parameter will decrease. However, the effect of the wall subcooled grade on the condensate mass flow rate parameter is more obvious than that of the vapor bulk superheated grade. On the basis of the rigorous numerical solutions, the wall temperature gradient and then mass flow rate parameter are formulated, and then the formulated equations for reliable predictions of heat and mass transfer are created for heat and mass transfer application of the laminar free convection film condensation of water vapor.

In this chapter, the film condensation of saturated water vapor is taken as an example for analyzing the effects of various physical conditions on heat transfer. The effects of four physical conditions including Boussinesq approximation (i.e. ignoring variable physical properties), shear force at the liquid–vapor interface, inertia force of the condensate film, and the thermal convection of the condensate film on the heat transfer coefficient of the film condensation are deeply investigated. It is found that the variable physical properties and thermal convection cause larger effect on heat transfer of laminar free convection film condensation, meanwhile, the effect of the variable physical properties is even larger than that of the thermal convection. It follows that it is necessary to consider variable physical properties for investigation on heat transfer of free film condensation. Compared with the variable physical properties and thermal convection, the effect of the Interfacial shear force and inertia force will be much smaller on heat transfer of laminar free convection film condensation, meanwhile, the effect of the inertia force is little bit smaller than that of the interfacial shear force.

By means of the new similarity analysis method, the governing partial differential equations of laminar free convection film condensation of vapor–gas mixture are transformed into the complete dimensionless mathematical models. The transformed complete governing mathematical models are equivalent to the system of dimensionless governing equations, which involve (1) the continuity, momentum, and energy equations for both liquid and vapor–gas mixture films, as well as species conservation equation with mass diffusion in the vapor–gas mixture film, (2) a set of interfacial physical matching conditions, such as those for two-dimensional velocity component balances, shear force balance, mass flow rate balance, temperature balance, heat transfer balance, concentration condition, as well as the balance between the condensate mass flow and vapor mass diffusion. On the other hand, the transformed complete similarity mathematical models of the film condensation of vapor–gas mixture are very well coupled with a series of physical property factors, such as the density factor, absolute viscosity factor, thermal conductivity factor, of the medium of liquid film and the vapor–gas mixture film. Thus, the transformed complete similarity mathematical models are advanced ones for consideration of variable physical properties.

A set of physical matching conditions at the liquid–vapor interface are considered and rigorously satisfied for getting reliable solutions related to the three-point boundary value problem on the laminar free convection film condensation of vapor–gas mixture. With the example on the laminar free convection film condensation of water vapor–air mixture, a system of the interfacial vapor saturation temperature \(T_{{s},\mathrm{int}}\) is found out, which only depends on the bulk vapor mass fraction for a special bulk temperature. The numerical solutions of the interfacial vapor saturation temperature \(T_{{s},\mathrm{int}}\) are further formulated into an equation for its reliable prediction. A system of rigorous numerical results is successfully obtained, including velocity and temperature fields of the condensate liquid film, as well as the velocity, temperature, and concentration fields of the vapor–gas mixture film. With increasing the vapor mass fraction (or decreasing the gas mass fraction) in the bulk, the condensate liquid film thickness, the condensate liquid velocity, and vapor–gas mixture velocity at the liquid–vapor interface will increase at an accelerative pace. It proved that the noncondensable gas in the vapor–gas mixture has a decisive effect on the laminar free convection film condensation from vapor–gas mixture. The wall temperature has also a decisive effect on the laminar free convection film condensation from vapor–gas mixture. With increasing wall temperature, the condensate liquid film thickness, the condensate liquid velocity, as well as velocity of the vapor–gas mixture at the liquid–vapor interface will decrease. However, with increasing the wall temperature, the thicknesses of the momentum, temperature, and concentration boundary layers of the vapor–gas mixture will increase.

The theoretical equations on heat and mass transfer are set up for laminar free Convection film condensation of vapor–gas mixture. In the theoretical equations only dimensionless wall temperature gradient and condensate mass flow rate parameter are no-given variables respectively for prediction of heat and mass transfer rates. The laminar free Convection film condensation of water vapor in presence of air on a vertical flat plate is taken as example for the numerical solutions on condensate heat and mass transfer, including those on the dimensionless temperature gradient and mass flow rate parameter. Both by decreasing the bulk vapor mass fraction and the reference wall subcooled grade, the wall dimensionless temperature gradient will increase at accelerative pace. Both decreasing the bulk vapor mass fraction and the reference wall subcooled grade will cause decreasing the condensate mass flow rate parameter at accelerative pace. These phenomena demonstrate the decisive effect of the non-condensable gas on condensate heat and mass transfer of the laminar forced film condensation of vapor–gas mixture. The system of the rigorous key solutions on the wall dimensionless temperature gradient and the condensate mass flow rate parameter is formulated to the simple and reliable equations for the laminar free Convection film condensation of water vapor–air mixture. Coupled with these formulated equations, the theoretical equations on the condensate heat and mass transfer can be respectively used for reliable and simple prediction of heat and mass transfer rate on laminar free Convection film condensation of water vapor–air mixture. Additionally, it is found that the condensate heat transfer rate is dominated by the wall subcooled temperature\(t_\text{w}-t_{\text{s},\,\text{int}}\) and the wall temperature gradient, the condensate mass flow rate is dominated by the condensate mass flow rate parameter, and the condensate heat transfer rate is identical to the condensate mass flow rate. Due to the quite different condensate mechanisms, the condensate heat and mass transfer rate of the laminar free Convection film condensation from vapor in presence of non-condensable gas is quite different from that of pure vapor, even for \(C_{\text{mv},\infty } \rightarrow 0.\)

The new similarity analysis method has been applied to extensively study the gravity-driven flow of a non-Newtonian liquid film along inclined surface. The partial differential equations governing the hydrodynamics of the power-law fluid are transformed exactly into a set of two ordinary differential equations, which can be calculated numerically to an arbitrary degree of accuracy. The non-linearity of the momentum boundary layer problem for power-law fluid increases with increasing pseudo-plasticity \(\left| 1- n \right| \) and the variable grid spacing is therefore increasingly important. The solutions of the system of dimensionless ordinary differential equations depends only on the single parameter \(n\), and all other parameters, like the streamwise location \(x\), the fluid properties \(K/\rho \), and the component of the gravitational acceleration along the surface \({g}\cdot \text{ cos }\alpha \) have been combined into a generalized local Reynolds number Re\(_{x }\) and dimensionless velocity \(W_x \) and \(W_y \). Various flow characteristics can thus be expressed only in term of n and Re\(_{x}\). In order to determine \(x_{0}\) the particular position \(x_{0}\), at which the entire freestream has been entrained into the momentum boundary layer, and the associated critical film thickness \(\delta _\mathrm{l} (x_0 )\), knowledge about the total mass flow rate\(\rho Q\) within the film is also required, together with the new dimensionless mass flux parameter \(\phi \). The latter quantity, which depends on the dimensionless boundary layer thickness\(\eta _{\delta _\mathrm{l} }\) and the velocity components \(W_{x,\delta _\mathrm{l} } \) and \(W_{y,\delta _\mathrm{l} } \) at the edge of the boundary layer, is generally obtained as the numerical solution of the transformed problem and turned out to be function only of the power-law indexn. However, to facilitate rapid and accurate estimate of \(\phi \), polynomial curve-fit formulas have been developed on the basis of the new similarity analysis model.

The pseudo-similarity solutions of the thermal boundary layer of a falling film flow of power-law fluids are presented. Based on a proposed “local Prandtl number”, the dependence of the thickness of the momentum boundary layer and thermal boundary layer on the power-law index and local Prandtl number are discussed. Their variations with power-law index and local Prandtl number are also presented. The momentum layer thickness depends only on the power-law index, while the thermal boundary layer thickness depends both on the power-law index and the local Prandtl number. The momentum boundary layer thickness decreases significantly with the increase of the power-law index. While the thermal boundary layer thickness decreases slightly with increasing the power-law index and decreases with increasing the parameter local Prandtl number. With the introduction of the “local Prandtl number”, it is found that the heat transfer problem turned out to involve only two independent parameters, the power-law index and the local Prandtl number. The pseudo-similarity solution and the assumed true-similarity solution are presented for the investigation of non-similarity thermal boundary layer. The degree of non-similarity of thermal boundary layer has been determined for various values of power-law indices and local Prandtl numbers.

A deep study is done on heat transfer from an inclined plane surface to an accelerating liquid film of a non-Newtonian power-law fluid. The new similarity analysis method for the accompanying hydrodynamic problem was adopted in combination with a local pseudo-similarity method. The resulting transformed problem turned out to involve only two independent parameters, namely the power-law index and the local Prandtl number. All other related physical properties and parameters are combined into the induced local Reynolds number, and the dimensionless velocity components. Accurate numerical results are obtained for combinations of local Prandtl number from 0.001 to 1000 and the power-law index n in the range \(0.2\le n\le 2\). Special treatment for the low and high local Prandtl number cases is essential in order to maintain the numerical accuracy. The calculated results obtained both by using local similarity and local pseudo-similarity methods are practically indistinguishable for \(n = 1\) over the entire local Prandtl number range. Furthermore, it is found that the wall temperature gradient which depends on local Prandtl number and power-law index is the only one no-given condition for evaluation of heat transfer. With increasing the local Prandtl number, the heat transfer coefficient increases, but with increasing the power-law increase, the heat transfer coefficient decreases. A set of accurate curve-fit formulas for the wall temperature gradient is provided, so that the rapid estimates of the heat transfer rate for any combination of the local Prandtl number and power-law index within the parameter ranges considered are realized.