Critical heat fluxes in boiling of liquid nitrogen underheated to saturation point in forced-flow conditions

This paper gives the results of experimental determinations of the critical heat fluxes in the boiling of Liquid nitrogen in forced-flow conditions in the mass velocity range 2 · 10³-40 · 10³ kg/m² · sec, pressure range 29 · 10⁴–245 · 10⁴ N/m², and at underheatings corresponding to the onset of normal boiling crises.Notation q₀ critical heat flux - r heat of vaporization - i enthalpy of flow corresponding to saturation point - i enthalpy of flow corresponding to liquid temperature - surface tension - density of liquid - density of saturated vapor - C _f friction factor - W_g mass velocity - Fr_* Froude number - g acceleration due to gravity     

Theoretical analysis of convective heat transfer enhancement of microencapsulated phase change material slurries

This paper analyzes the convective heat transfer enhancement mechanism of microencapsulated phase change material slurries based on the analogy between convective heat transfer and thermal conduction with thermal sources. The influence of each factor affecting the heat transfer enhancement for laminar flow in a circular tube with constant wall temperature is analyzed using an effective specific heat capacity model. The model is validated with results available in the literature. The analysis and the results clarify the heat transfer enhancement mechanism and the main factors influencing the heat transfer. In addition, the conventional Nusselt number definition of phase change slurries for internal flow is modified to describe the degree of heat transfer enhancement of microencapsulated phase change material slurries. The modification is also consistent evaluation of the convective heat transfer of internal and external flows.c volumetric concentration of microcapsules - c_m mass concentration of microcapsules - c_p specific heat, kJ kg^–1 K^–1 - h_fs phase change material heat of fusion, kJ kg^–1 - h_m* modified convective heat transfer coefficient, W m^–2 K^–1 - k thermal conductivity, W m^–1 K^–1 - k_e effective thermal conductivity of slurry, W m^–1 K^–1 - k_b slurry bulk thermal conductivity, W m^–1 K^–1 - ML dimensionless initial subcooling - Mr dimensionless phase change temperature range - Nu conventional Nusselt number - Nu* improved Nusselt number - q_wⁿ wall heat flux, Wm^–2 - Pe Peclet number - Pr Prandtl number - Re Reynolds number - r radial coordinate, m - r₀ duct radius, m - r₁ dimensionless radial coordinate - Ste Stefan number - T temperature, K - T₁ lower phase change temperature limit, K - T₂ upper phase change temperature limit, K - T_i slurry inlet temperature, K - u axial velocity, m/s - v radial velocity, m/s - x axial coordinate, m - x₁ dimensionless axial coordinate - thermal diffusivity, m²/s - dimensionless temperature - dynamic viscosity, N·s/m² - kinematic viscosity, m²/s - _t width of thermal boundary, m - degree of heat transfer enhancement, = h_m*/(h_m*)_single - b bulk fluid (slurry) - b0 slurry without phase change - l liquid - m mean - s solid - f suspending fluid - p microcapsule particles - w wall - single single-phase fluid     

The effects of span position of winglet vortex generator on local heat/mass transfer over a three-row flat tube bank fin

The naphthalene sublimation method was used to study the effects of span position of vortex generators (VGs) on local heat transfer on three-row flat tube bank fin. A dimensionless factor of the larger the better characteristics, JF, is used to screen the optimum span position of VGs. In order to get JF, the local heat transfer coefficient obtained in experiments and numerical method are used to obtain the heat transferred from the fin. A new parameter, named as staggered ratio, is introduced to consider the interactions of vortices generated by partial or full periodically staggered arrangement of VGs. The present results reveal that: VGs should be mounted as near as possible to the tube wall; the vortices generated by the upstream VGs converge at wake region of flat tube; the interactions of vortices with counter rotating direction do not effect Nusselt number (Nu) greatly on fin surface mounted with VGs, but reduce Nu greatly on the other fin surface; the real staggered ratio should include the effect of flow convergence; with increasing real staggered ratio, these interactions are intensified, and heat transfer performance decreases; for average Nu and friction factor (f), the effects of interactions of vortices are not significant, f has slightly smaller value when real staggered ratio is about 0.6 than that when VGs are in no staggered arrangement. A cross section area of flow passage [m²] - A _mim minimum cross section area of flow passage [m²] - a width of flat tube [m] - b length of flat tube [m] - B _pT lateral pitch of flat tube: B _pT = S ₁/T _p - d _h hydraulic diameter of flow channel [m] - D _naph diffusion of naphthalene [m²/s] - f friction factor: f = pd _h/(Lu ² _max/2) - h mass transfer coefficient [m/s] - H height of winglet type vortex generators [m] - j Colburn factor [–] - JF a dimensionless ratio, defined in Eq. (23) [–] - L streamwise length of fin [m] - L _PVG longitudinal pitch of vortex generators divided by fin spacing: L _pVG = l _VG/T _p - l _VG pitch of in-line vortex generators [m] - m mass [kg] - m mass sublimation rate of naphthalene [kg/m²·s] - Nu Nusselt number: Nu = d _h/ - P pressure of naphthalene vapor [Pa] - p non-dimensional pitch of in-line vortex generators: p = l _VG/S ₂ - Pr Prandtl number [–] - Q heat transfer rate [W] - R universal gas constant [m²/s²·K] - Re Reynolds number: Re = ·u _max·d _h/ - S ₁ transversal pitch between flat tubes [m] - S ₂ longitudinal pitch between flat tubes [m] - Sc Schmidt number [–] - Sh Sherwood number [–]: Sh = hd _h/D _naph - Sr staggered ratio [–]: Sr = (2Hsin – C)/(2Hsin) - T _p fin spacing [m] - T temperature [K] - u _max maximum velocity [m/s] - u average velocity of air [m/s] - V volume flow rate of air [m³/s] - x,y,z coordinates [m] - z sublimation depth[m] - heat transfer coefficient [W/m²·K] - heat conductivity [W/m·K] - viscosity [kg/m²·s] - density [kg/m³] - attack angle of vortex generator [°] - time interval for naphthalene sublimation [s] - fin thickness, distance between two VGs around the tube [m] - small interval - C distance between the stream direction centerlines of VGs - p pressure drop [Pa] - 0 without VG enhancement - 1, 2, I, II fin surface I, fin surface II, respectively - atm atmosphere - f fluid - fin fin - local local value - m average - naph naphthalene - n,b naphthalene at bulk flow - n,w naphthalene at wall - VG with VG enhancement - w wall or fin surface     

[0, k i]1 m -Factorizations Orthogonal to a Subgraph

Let G be a graph,k₁,…,k_m be positive integers. If the edges of graph G can be decomposed into some edge disjoint. [0,k₁]-factor. F₁,…,[0,k_m]-factor F_m, then we can say F={F₁,…,F_m, is a [0,k_i]_m¹-factorization of G. If H is a subgraph with m edges in graph G and |E(H)∩E(F_i)|=1 for all 1≤i≤m, then we can call that F is orthogonal to H. It is proved that if G is a..[0,k₁+…+k_m-m+1]-graph, H is a subgraph with m edges in G, then graph G has a. [0,k_i]₁^m-factorization orthogonal to H.     

FEM application in phase change exchangers

Application of FEM in the analysis of condenser and vaporizer has been illustrated taking into account property variation with temperature. Accurate shell side pressure drop in a condenser has been determined by the present method taking into account the gradual reduction in vapour flow due to condensation from inlet to outlet. As the present method analyses the exchanger in small elements, analysis of an evaporator working under the conditions of partial vapour blanketing is also possible.

---

Anwendung der Finite Elemente Methode bei Wärmetauschern mit Phasenwechsel

Zusammenfassung Die Verwendung der Finite Elemente Methode zur Berechnung von Kondensator und Verdampfer unter Berücksichtigung von temperaturabhängigen Stoffwerten ist hier dargestellt worden. Mit dem dargestellten Verfahren ist der genaue Druckverlust im Rohrraum eines Kondensators bestimmt worden, wobei die schrittweise Verminderung der Dampfströmung aufgrund der Kondensation von Ein-zu Auslaß mit berücksichtigt wurde. Mit der gegenwärtigen Methode, die einen Wärmeaustausch mittels kleiner Elemente berechnet, ist ebenso eine Auslegung eines Verdampfers mit einer partiellen Dampfabdeckung möglich.

---

Nomenclature A heat transfer area in an elment upto any section, m² - A _c elemental heat transfer area, m² - B weightage - C UA/2 - D _e characteristic dimension, m - D _s shell diameter, cm - F _t temperature correction factor - f friction factor, m²/cm² - G mass velocity, kg/m² sec - G W/LN ^2/3 - g acceleration of gravity, m/sec² - h condensing coefficient, W/m²°C - h ₀ boiling coefficient, W/m²°C - k ₁ thermal conductivity of condensate atT _f , W/m°C - L length of tubes in the element, m - l distance of the fluid stream traverses in the element, m - LMTD log mean temperature difference, °C - N ₁,N ₂ shape functions - N _t number of tubes effective for condensation - P pressure drop, N/cm² - P _shell shell side vapour pressure drop, N/cm² - P _tube tube side stream pressure drop, N/cm² - q heat flux in the element, W/m² - S specific gravity of vapour - T _c tube side stream temperature, °C - T _c2 tube side stream exit temperature, °C - T _f (T _w +T _s )/2 - T _s saturation temperature, °C - T _w mean wall temperature in an element, °C - T ₁,T ₂ temperatures at nodes 1 and 2, °C - U overall heat transfer coefficient, W/m²°C - W ₁ tube side fluid thermal capacity rate, W/°C - W ₂ vapour mass flow rate, kg/hr - W mass of vapour condensed in any element, kg/sec - _f viscosity of condensate atT _f , N sec/m² - _f density of condensate atT _f , kg/m³     

The mathematical modelling of thermal radiation in high temperature fluidized bed

The aim of this study is composed of two parts. One of them is to calculate the radiation heat flux and the other is to determine the overall heat transfer coefficient for the gas-fluidized bed. The radiative heat transfer model is developed for predicting the total heat transfer coefficients between submerged surfaces and fluidized beds for several working temperatures. The role of radiation heat transfer in the overall heat transfer process at an immersed surface in a gas-fluidized bed at high temperatures is investigated. Analytical results are compared with the previously done experiments and a good agreement between the two, is obtained.

---

Bestimmung der Wärmeübertragungs-Koeffizienten in Gas-Wirbelschichten

Zusammenfassung Diese Untersuchung besteht aus folgenden zwei Teilen: 1. Kalkulation des Radiationswärmeübergangs in Gas-Wirbelschichten. 2. Bestimmung des Wärmeübergangs-Koeffizienten in Gas-Wirbelschichten. Dieses Radiationswärmeübergangsmodell wurde entwickelt, um die Wärmeübertragungs-Koeffizienten zwischen der eingetauchten Oberfläche und der Wirbelschicht bei verschiedener Wärme schätzungsweise zu bestimmen. Es wurde das Verhältnis der Radiationswärmeübertragung in Gas-Wirbelschichten zum totalen Wärmeübergang untersucht. Die Meßwerte wurden mit theoretischen Resultaten verglichen.

---

Nomenclature c (x) specific heat capacity of packet [J/kg K] - c _p specific heat capacity of particle [J/kg K] - c _pg specific heat capacity of gas [J/kg K] - d _p average diameter of the bed particles [m] - f ₀ the fraction of time that a unit surface exposed to the bubble phase - 1–f ₀ the fraction of time that a unit surface exposed to the packet phase - g acceleration due to gravity [m/s²] - h _b heat transfer coefficient for the surface in contact with bubble [W/m² K] - h _bc conduction heat transfer coefficient for the surface/bubble [W/m²K] - h _br radiation heat transfer coefficient for the surface/bubble [W/m²K] - h _p heat transfer coefficient for the surface in contact with packet [W/m²K] - h _pc conduction heat transfer coefficient for the surface/packet [W/m² K] - h _pr radiation heat transfer coefficient for the surface/packet [W/m² K] - h _T total heat transfer coefficient between bed and surface [W/m² K] - k ⁰ thermal conductivity of the emulsion phase for fixed bed [W/m K] - k(x) thermal conductivity of packet [W/m K] - k _e the logarithmic mean of conductivity for first layer in packet [W/m K] - k _g the logarithmic mean of conductivity for the first layer in packet [W/m K] - K extinction coefficient [1/m] - m mass [kg] - n number of layers - p air pressure [pa] - q _pc mean local conduction heat transfer for packet [kW/m²] - q _pr mean local radiation heat transfer for packet [kW/m²] - Q _p average heat flux during packet contact with surface [kW/m²] - Q _b average heat flux during bubble contact with surface [kW/m²] - R gas constant [287.04 J/kg K] - t time [s] - t _g residence time for gas bubble [s] - t _k residence time for packet [s] - T temperature [K] - T _b bed temperature [K] - T _W surface temperature [K] - V _mf minimum fluidization velocity [m/s] - v _t terminal velocity [m/s] - x distance [m] Greek symbols t time increment - x thickness of the layer - emissivity - thermal diffusivity [m²/s] - (x) voidage of fluidized bed - _mf void ratio of the bed at minimum fluidization - ₀ voidage of fixed bed - _g dynamic viscosity of gas [kg/m s] - _g kinematic viscosity of gas [m²/s] - (x) density of packet [kg/m³] - _p density of particles [kg/m³] - _g density of gas [kg/m³] - Stefan-Boltzmann constant [5.66·10^–8 W/m²K⁴] - geometric shape factor for particles Dimensionless numbers Ar Archimedes numberAr=g d _p ³ ( _p – _g ) _g / _g ² - Nu Nusselt numberNu=h·d/k - Re Reynolds numberRe=d _p ·V _mf / _g - Pr Prandtl numberPr=C _{pg g} /k _g      

Momentum and heat transfer in the entrance region of a parallel plate channel: developing laminar flow with constant wall temperature

A new integral method of solution is presented for developing laminar flow and heat transfer in the entrance region of a parallel plate channel with uniform surface temperature. Unlike earlier Karman-Pohlhausen analyses, the new analysis provides solutions which are free from jump discontinuities in the gradients of the velocity and temperature distributions throughout and at the end of the entrance region. The hydrodynamic and thermal results from the present analysis therefore join smoothly and asymptotically to their fully-developed values. The heat transfer results obtained are further found to agree well with previously published numerical solutions.Nomenclature a half width of the channel, m - D _h hydraulic diameter (=4a), m - h local heat transfer coefficient,W/(m²·K) - h _m mean heat transfer coefficient defined by Eq- (9),W/(m²·K) - k thermal conductivity, W/(m·K) - L _H axial length of the hydrodynamic entrance region, m - L _T axial length of the thermal entrance region, m - L _in,H axial length of the hydrodynamic inlet region, m - L _in,T axial length of the thermal inlet region, m - Nu _x local Nusselt number,hD _h /k, dimensionless - Nu _m mean Nusselt number defined by Eq. (9),h _mD_h/k, dimensionless - P pressure, N/m² - P _O pressure at the entrance, N/m² - Pr Prandtl number,c _p /k, dimensionless - Re Reynolds number, 4aU _o /v, dimensionless - T absolute temperature, K - T _b fluid bulk temperature, K - T _c centerline temperature, K - T _w wall temperature, K - U _c centerline velocity, m/s - U ₀ velocity of the fluid at entrance, m/s - U core velocity, m/s - u velocity component inx direction, m/s - v velocity component iny direction, m/s - x spatial coordinate, axial distance, m - y spacial coordinate measured from channel wall, m Greek Letters molecular thermal diffusivity, m²/s - hydrodynamic shape factor, dimensionless - _T thermal shape factor, dimensionless - hydrodynamic boundary layer thickness, m - * /a, dimensionless - _T thermal boundary layer thickness, m - * _{T T} /a, dimensionless - dimensionless distance,y/ ory/a - Pohlhausen's shape factor, dimensionless - dynamic viscosity coefficient, kg/(m·s) - v kinematic viscosity,/, m²/s - dimensionless axial distance,x/(a·Re) - _H dimensionless axial length of the hydrodynamic entrance region (=L _H /(a·Re)) - _T dimensionless axial length of the thermal entrance region (=L _T /(a·Re)) - _in,H dimensionless axial length of the hydrodynamic inlet region (=L _in,H/(a·Re)) - _in,T dimensionless axial length of the thermal inlet region (L _in,T /(a·Re)) - fluid density, kg/m³     

Experimental study of natural convection heat transfer between an outer horizontal cylindrical envelope and an inner concentric heated square cylinder with two slots

This paper presents the results of an experimental study of natural convection heat transfer between a horizontal cylindrical envelope and an internal concentric heated square cylinder with two slots. The internal cylinder was a hollow one with horizontal slots on its top and bottom surfaces. The ratio of slot widthS to the side heightH was 0.0612 and 0.3878. The ratio of the envelope inner diameterD _o to the side heightH was 2.653. Air was used as the working fluid. The range of Ray-leigh number was 1.77×10²8.72×10⁶ forS/H=0.0612 and 1.32×10²6.25×10⁶ forS/H=0.3878. The results show that there are three different heat transfer regimes in different Ray-leigh number regions, i.e. pure conduction regime, transition regime and convection regime. The average heat transfer results were correlated into two empirical equations. Comparison was made with the non-slotted case. It is found that slots of the internal cylinder can significantly enhance the heat transfer.

---

Experimentelle Untersuchung des Wärmeübergangs bei natürlicher Konvektion zwischen einer horizontalen zylindrischen Außenhülle und einem konzentrischen, beheizten, quadratischen Prisma mit zwei Schlitzen

Zusammenfassung In der Arbeit werden die Ergebnisse einer experimentellen Untersuchung des Wärmeübergangs bei natürlicher Konvektion zwischen einer horizontalen zylindrischen Außenhülle und einem beheizten quadratischen Prisma mit zwei Schlitzen vorgestellt. Das Prisma selbst ist hohl und weist in der oberen und unteren Begrenzungsfläche je einen horizontalen Längsschlitz auf. Das Verhältnis von SchlitzweiteS zu SeitenhöheH beträgt 0,0612 und 0,3878, das des HülleninnendurchmessersD _o zur SeitenhöheH beträgt 2,653. Als Arbeitsmedium diente Luft. Die Rayleigh-Zahlen variierten zwischen 1,7·10² und 8,72·10⁶ fürS/H=0,0612 und zwischen 1,32·10² und 6,25·10⁶ fürS/H=0,3878. Die Ergebnisse belegen die Existenz dreier unterschiedlicher Wärmeübergangsregime in den verschiedenen Rayleigh-Zahl-Bereichen, und zwar reiner Leitungsbereich, Übergangsgebiet und Konvektionsbereich. Die Ergebnisse für den Wärmeübergang werden im Vergleich mit jenen für ein Prisma ohne Schlitze durch zwei Korrelationbeziehungen dargestellt. Es zeigt sich, daß durch Anbringung von Schlitzen am Innenprisma der Wärmeübergang wesentlich verstärkt werden kann.

---

Nomenclature C _p specific heat at constant pressure, J/(kg·K) - D _i diameter of the related circular cylinder whose circumferential area is equal to that of the unslotted square cylinder, m - D _o internal diameter of the outer circular envelope, m - F _i surface area of the inner two slot cylinder, m - g gravitational acceleration, m/s² - H distance between the opposite sides of the square cylinder with two slots, m - K _eq dimensionless equivalent thermal conductivity - L axial length of the test section, m - m ratio of the area of the unslotted square cylinder surface to that of the slotted square cylinder - P pressure in the enclosure, Pa - Q total power input to the enclosure, W - Q _cond radial heat conduction, W - Q _conv convective heat transfer, W - Q _r radiation heat transfer, W - Q _los end heat dissipation, W - R air gas constant, J/(kg·K) - Ra Rayleigh number - S slot width, m - T _i wall temperature of the inner cylinder, K - T _o wall temperature of the outer envelope, K - T _m mean temperature, K - T temperature difference=T _i –T _o , K - W maximum gap width of the test annuli=(D _o –H)/2 for the square case, m Greek symbols ₀ black body radiation constant, W/(m²·K⁴) - _s equation system emissivity - air thermal conductivity, W/(m·K) - _eq equivalent thermal conductivity, W/(m·K) - air dynamic viscosity, kg/(m·s) This work was supported by the National Natural Science Foundation of China.     

Binary laminar boundary layer on a vertical surface in the presence of combined free and forced convection

Heat and mass transfer at a vertical surface is examined in the case of combined free and forced convection. The boundary layer equations, transformed to ordinary differential equations, contain a parameter that determines the effect of free convection on the forced motion. Criteria are offered for differentiating the free-convection, forced-convection, and combined regimes.Notation x, y coordinates - u, v velocity components - g acceleration of gravity - T temperature - kinematic viscosity - coefficient of thermal expansion - a thermal diffusivity - ₁ partial vapor density - D diffusion coefficient - W₂ mass velocity of air - independent variable - _w shear stress at wall - thermal conductivity - r latent heat of phase transition - , dimensionless temperature and partial vapor density - m^* the complex (m ₁–m _1w )/(1–m(_1w ) - c_p specific heat at constant pressure - G Grashof number - R Reynolds number - P Prandtl number - S Schmidt number     

Prediction of roll temperature with a non-uniform heat flux at tool and workpiece interface

In a metal forming process, plastic deformation of the workpiece takes place at tool and workpiece interface region. Tool has been identified as one of the key parameters in controlling the productivity of any manufacturing industry. The deformation of metals and friction at the contact region produce large amount of heat, a part of that heat is conducted towards the tool where it is removed by forced convection. These cooling and heating cycles finally result in a substantial change in the temperature distribution in the roll. In this paper, an attempt is made to study the temperature and heat flux distribution in the roll by considering a non-uniform heat flux at the roll-workpiece interface for a cold rolling process. Adopting an elemental approach, a methodology has been proposed to model non-uniform heat flux at the interface. For this purpose both tool and workpiece has been considered together, thus a coupled approach is used to model both deformation and heat transfer phenomenon. It is demonstrated that the present approach of modeling is more general than that available in the literature. For example, a constant value of heat flux at the interface that is considered by several investigators is shown to be a special case of the present investigation, particularly when the deformation and relative velocity is very small. It is shown that the error in maximum temperature associated with constant heat flux assumption could be more than 5% in situations when reduction and relative velocity is high. The results are presented for temperature and heat flux distributions in the roll for different operating conditions.a thermal diffusivity, (m²/sec) - B pre-strain coefficient - C yield stress at unit strain, (N/m²) - e rate of deformation heat generation per unit volume, (W/m³) - f friction factor - h heat transfer coefficient, (W/m² °C) - k thermal conductivity, (W/m °C) - K yield stress at unit strain, (N/m²) - L bite length, (m) - n strain hardening exponent - P pressure between tool and workpiece, (N/m²) - q heat flux, (W/m²) - q_f friction heat flux, (W/m²) - heat flux entering towards the roll for any arbitrary element j (W/m²) - R roll radius, (m) - S_o yield stress in plane strain, (N/m²) - T temperature difference (T = T_r – T_o), (°C) - T surrounding temperature, (°C) - y strip thickness, (m) - V_rel relative slipping velocity, (m/sec) - V velocity, (m/sec) - Pe Peclet number - Bi Biot number - _T Total bite angle - mean effective strain - mean true stress, (N/m²) - mean strain rate - friction stress, (N/m²) - coefficient of friction - angle between heating and cooling regions - angle of cooling spray region - r, polar coordinates - x, y Cartesian coordinates - o initial value - f final value - r related to roll - s related to strip - a average value - j elemental region     

Luikov equations applicable to sublimation-drying

Present paper presents a derivation of Luikov equations applicable to sublimation-drying. The physical situation and transfer mechanism are elucidated clearly. The coefficients appearing in Luikov equations are given in a more explicit way. Some formulation mistakes in recent publications are indicated.

---

Anwendung der Luikov-Gleichungen auf die Sublimationstrocknung

Zusammenfassung Die Untersuchung bezieht sich auf eine Ableitung der Luikov-Gleichungen, mittels deren sich der Vorgang der Sublimationstrocknung analysieren läßt. Physikalische Anfangssituation und Austauschmechanismen werden klar herausgestellt und die in den Luikov-Gleichungen auftretenden Koeffizienten in expliziter Weise angegeben. Ferner erfolgt Hinweis auf Formulierungsfehler in jüngeren Veröffentlichungen.

---

Nomenclature C M _v/V _f, concentration of vapor, kg/m³ - c _pv specific heat of vapor at constant pressure, J/kg K - c _pw specific heat of adsorbed water at constant pressure, J/kg K - c _s specific heat of solid skeleton, J/kg K - C _s M _s/V _f, concentration of solid skeleton, kg/m³ - C _w M _w/V _f, concentration of adsorbed water, kg/m³ - f V _w/V _f, volumetric fraction of adsorbed water - j _F mass flux of vapor by diffusion (Fick) transfer, kg/m² s - j _D mass flux of vapor by filtration (Darcy) transfer, kg/m² s - j _v total mass flux of vapor, kg/m² s - k permeability, m² - M _s mass of solid skeleton, kg - M _v mass of vapor in pores, kg - M _w mass of adsorbed water, kg - P pressure, Pa - q heat flux, W/m² - R gas constant, J/kg K - T temperature, K - V _f volume of the framework of porous medium, m³ - V _v volume of vapor in porous medium, m³ - V _w volume of the absorbed water, m³ Greek symbols /(c _p), effective thermal diffusivity, m²/s - _m effective vapor diffusivity in porous medium, m²/s - _p R T /, Luikov pressure diffusivity, m²/s - +f, porosity of the porous medium - effective thermal conductivity of porous body, W/m K - dynamic viscosity of vapor, kg/m s - kinematic viscosity, m²/s - Ck/=k/, Luikov filtration motion coefficient, s - V _v/V _f, volumetric fraction of vapor - density of absorbed water, kg/m³ - (c _p) M _v c _pv+M _s c _s+M _w c _pw /V _f=Cc _pv+C _s c _s+fc _pw, effective product of density and specific heat of humid porous body, J/m³K     

Optimum rib size to enhance forced convection in a horizontal triangular duct with ribbed internal surfaces

The optimum rib size to enhance heat transfer had been proposed through an experimental investigation on the forced convection of a fully developed turbulent flow in an air-cooled horizontal equilateral triangular duct fabricated on its internal surfaces with uniformly spaced square ribs. Five different rib sizes (B) of 5 mm, 6 mm, 7 mm, 7.9 mm and 9 mm, respectively, were used in the present investigation, while the separation (S) between the center lines of two adjacent ribs was kept at a constant of 57 mm. The experimental triangular ducts were of the same axial length (L) of 1050 mm and the same hydraulic diameter (D) of 44 mm. Both the ducts and the ribs were fabricated with duralumin. For every experimental set-up, the entire inner wall of the duct was heated uniformly while the outer wall was thermally insulated. From the experimental results, a maximum average Nusselt number of the triangular duct was observed at the rib size of 7.9 mm (i.e. relative rib size ). Considering the pressure drop along the triangular duct, it was found to increase almost linearly with the rib size. Non-dimensional expressions had been developed for the determination of the average Nusselt number and the average friction factor of the equilateral triangular ducts with ribbed internal surfaces. The developed equations were valid for a wide range of Reynolds numbers of 4,000 < Re _D < 23,000 and relative rib sizes of under steady-state condition. A Inner surface area of the triangular duct [m²] - A _C Cross-sectional area of the triangular duct [m²] - B Side length of the square rib [mm] - C _P Specific heat at constant pressure [kJ·kg^–1·K^–1] - C ₁, C ₂, C ₃ Constant coefficients in Equations (10), (12) and (13), respectively - D Hydraulic diameter of the triangular duct [mm] - Electric power supplied to heat the triangular duct [W] - f Average friction factor - F View factor for thermal radiation from the duct ends to its surroundings - h Average convection heat transfer coefficient at the air/duct interface [W·m^–2 ·K^–1] - k Thermal conductivity of the air [W·m^–1 ·K^–1] - L Axial length of the triangular duct [mm] - Mass flow rate [kg·s^–1] - n ₁, n ₂, n ₃ Power indices in Equations (10), (12) and (13), respectively - Nu _D Average Nusselt number based on hydraulic diameter - P Fluid pressure [Pa] - Pr Prandtl number of the airflow - _c Steady-state forced convection from the triangular duct to the airflow [W] - _l Heat loss from external surfaces of the triangular duct assembly to the surroundings [W] - _r Radiation heat loss from both ends of the triangular duct to the surroundings [W] - Re _D Reynolds number of the airflow based on hydraulic diameter - S Uniform separation between the centre lines of two consecutive ribs [mm] - T Fluid temperature [K] - T _a Mean temperature of the airflow [K] - T _ai Inlet mean temperature of the airflow [K] - T _ao Outlet mean temperature of the airflow [K] - T _s Mean surface temperature of the triangular duct [K] - T Ambient temperature [K] - U Mean air velocity in the triangular duct [m·s^–1] - _r Mean surface-emissivity with respect to thermal radiation - Dynamic viscosity of the fluid [kg·m^–1·s^–1] - Kinematic viscosity of the airflow [m²·s^–1] - Density of the airflow [kg·m^–3] - Stefan-Boltzmann constant [W·m^–2·K^–4]     

Heat,Water, and Solution Transfer in Unsaturated Porous Media: I -- Theory Development and Transport Coefficient Evaluation

A detailed theory describing the simultaneous transfer of heat, water, and solute in unsaturated porous mediais developed. The theory includes three fully-coupledpartial differential equations. Heat, water, andsolute move in the presence of temperature, T; matricpressure head, _m ; solution osmotic pressure head _o ; and solute concentration C gradients. Thetheory can be applied to describe the mass and energyin radioactive waste repositories, food processing,underground energy storage sites, buried electriccables positions, waste disposal sites, and inagricultural soil. Several transport coefficients forheat, water, and solute are included in the theory. The coefficients are evaluated for a silty clay loamsoil to clarify their dependence on water content (),T, and C. The thermal vapor diffusivity D _Tv first increased as increased to0.22 m³/m³ then decreased with furtherincreases in . D _Tv was 3 orders of magnitudegreater than either isothermal vapor D _mv orosmotic vapor D _ov , diffusivities at of0.20~m³/m³, T of 50°C, and C of 0.001mol/kg. All of the liquid and vapor water transport coefficients increased with increasing T. D _Tv decreased with increasing C to a greater extent thanD _mv and D _ov . The effective thermalconductivity decreased slightly with increasing C. Thesolute diffusion coefficient D _d was 6 to 7orders of magnitude greater than the thermal soluteand salt sieving diffusion coefficients at of0.20~m³/m³, T of 50°C, and C of 0.001 mol/kg.     

Numerical analysis of simulation accuracy for hypersonic heat transfer in subsonic jets of dissociated nitrogen

One-dimensional problems of the flow in a boundary layer of finite thickness on the end face of a model and in a thin viscous shock layer on a sphere are solved numerically for three regimes of subsonic flow past a model with a flat blunt face exposed to subsonic jets of pure dissociated nitrogen in an induction plasmatron [1] (for stagnation pressures of (10⁴–3·10⁴) N/m² and an enthalpy of 2.1·10⁷ m²/sec²) and three regimes of hypersonic flow past spheres with parameters related by the local heat transfer simulation conditions [2, 3]. It is established that given equality of the stagnation pressures, enthalpies and velocity gradients on the outer edges of the boundary layers at the stagnation points on the sphere and the model, for a model of radius R_m=1.5·10^–2 m in a subsonic jet the accuracy of reproduction of the heat transfer to the highly catalytic surface of a sphere in a uniform hypersonic flow is about 3%. For surfaces with a low level of catalytic activity the accuracy of simulation of the nonequilibrium heat transfer is determined by the deviations of the temperatures at the outer edges of the boundary layers on the body and the model. For this case the simulation conditions have the form: dU_e/dx=idem, p₀=idem, T_e=idem. At stagnation pressuresP ₀2·10⁴ N/m² irrespective of the catalycity of the surface the heat flux at the stagnation point and the structure of the boundary layer near the axis of symmetry of models with a flat blunt face of radius R_m1.5·10^–2 m exposed to subsonic nitrogen jets in a plasmatron with a discharge channel radius R_c=3·10^–2 m correspond closely to the case of spheres in hypersonic flows with parameters determined by the simulation conditions [2, 3].Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 135–143, March–April, 1990.     

A parametric study of boiling in a porous bed

The boiling process in a granular porous medium creates several different layers of the bed — a capillary layer [fixed bed], a cracked layer with horizontal cracks, and a chimney layer with vertical channels. Underneath those layers the bed becomes dry. An analysis is presented to calculate the heights of these different layers.

---

Eine parametrische Studie des Siedevorganges in einer körnigen Schichtung

Zusammenfassung Der Siedevorgang in einer körnigen Schichtung bewirkt oft, daß sich in ihr mehrere Schichten ausbildeneine unveränderte kapillare Schicht, darüber eine Zwischenschicht mit horizontalen Sprüngen und darüber eine Schicht durchsetzt von vertikalen Kanälen. Unter diesen Schichten trocknet eine genügend tiefe Schichtung aus, d. h. ist frei von Flüssigkeit und mit überhitztem Dampf erfüllt. Die vorliegende Arbeit berichtet über eine analytische Studie mit dem Ziel, die Höhen der verschiedenen Schichten zu ermitteln.

---

Nomenclatsre b channel width, m - c specific heat, J/kg K - f porosity, m³/m³ - g gravitational acceleration, m/s² - h _lv heat of evaporation, J/kg - k _t thermal conductivity, W/K m - Kg gravitational moisture transport coefficient, s - K _p single phase permeability, m² - K_w moisture transport coefficient, m²/s - l height, m - m mass flow, kg/m²s - N number of channels per unit cross-sectional area, l/m² - p pressure, N/m² - q heat flux from walls per unit area, W/m² - q volume heat source, W/m³ - q₀ heat flux conducted upward atx=0 per unit area - r, R effective particle radius, m - R radius of channels in chimney region, m - t temperature, K - velocity, m/s - W moisture content, kg water/kg dry bed - x coordinate, m Greek Symbols contact angle - mass generation rate of vapor, kg/m³s - tortuosity + vapor injection effect - two-phase to the permeabilityK _p - viscosity, kg/s m - density, kg/m³ - surface tension, N/m - wall shear stress, N/m² Indices d dry substance - e exit - i inlet - 1 liquid - s saturation - v vapor - c capillary - cr cracked - ch chimney - 0 atx=0 - d dry - f fluidized - t total=dry+cracked+chimney We dedicate this paper to Fran Bonjakovi on his 80th birthday. His early work enhanced our understanding of boiling     

Condensation and isothermal water transfer in cement mortar: Part II - transient condensation of water vapor

The transient wetting of a mortar sample swept by a flow of humid air is experimentally studied at temperatures of 30 and 55°C. The water content profile shape and evolution are found to be very different from those which were observed during imbibition. The boundary condition on the exposed wall of the sample is examined. A convenient evolution of the coefficient of diffusion with water content is explored. This coefficient is interpreted in terms of pure vapor diffusion, even at relatively high water contents. But its values at low water content and its temperature dependence are inconsistent. Additional explanations are then considered with the assumption that the vapor condensation in the medium is not an equilibrium process between vapor and liquid phases. The physical origin of such a nonequilibrium process is discussed. A tentative set of transfer and phase change coefficients is proposed in order to describe the experimental data by means of numerical simulation. Then, some aspects of the imbibition processes are re-examined, taking into account the consequences of a nonequilibrium condensation.Nomenclature volumic rate of phase change - D ₀ coefficient of free diffusion of the water vapor in air - D _hv vapor diffusion coefficient of the medium - E, E equivalent air thickness - h relative humidity of gaseous phase - h _c relative humidity at the capillary condensation threshold - h _a relative humidity of the flowing air - h ₀ relative humidity at the air-material interface - h _E equilibrium relative humidity at a given water content - J global massic flux - M molar mass of water - R gas constant - T temperature - t time - x distance from the interface - ₀ total porosity - volumetric water content - _h condensation coefficient (see Equation (8)) - _L mass density of liquid water - vs mass density of saturated water vapor     

Wärmeübergang bei freier Konvektion unter dem Einfluß von elektrolytisch erzeugten Wasserstoffblasen

Zusammenfassung Die Arbeit befaßt sich mit dem Wärmeübergang von einer waagrechten Heizfläche an wäßrige NaOH-Lösungen, um zu klären, ob der Wärmeübergang bei einphasiger freier Konvektion und beim Blasensieden durch an der Heizfläche elektrolytisch erzeugte Gasblasen verbessert werden kann. Hierzu wurden Messungen zwischen 60°C und Siedetemperatur bei Umgebungsdruck ausgeführt, wobei sich die Wärmestromdichte von 4,78·10⁴ W/m² bis 2,10·10⁵ W/m² und die elektrische Stromdichte von 0 bis 2100 A/m² erstreckten. Um Stoffwerte des Wassers durch Zugabe von NaOH nicht wesentlich zu beeinflußen, wurde die Lösungskonzentration bis höchstens 0,25 mol/l entsprechend 10 g/l variiert. Die Messungen ergaben eine Verbesserung des Wärmeübergangs durch elektrolytisch erzeugte Gasblasen im Vergleich zu dem ohne elektrolytische Blasenbildung bis zum Faktor 6. Die Verbesserung nimmt mit steigender Lösungstemperatur und steigender elektrischer Stromdichte zu. Eine höhere Wärmestromdichte führt zwar zu einer Zunahme des Wärmeübergangskoeffizienten . Gleichzeitig nimmt jedoch das Verhältnis /₀ ab, wenn ₀ der Wärmeübergangskoeffizient ohne elektrolytische Blasenbildung ist. Der Einfluß der Lösungskonzentration auf den Wärmeübergang ist im untersuchten Konzentrationsbereich vernachlässigbar klein.

---

Heat transfer in free convection under the influence of electrolytically generated hydrogen bubbles

The paper deals with heat transfer from a horizontal heating surface to weak aqueous solutions of NaOH in order to explain whether the heat transfer in natural convection and pool boiling can be enhanced by hydrogen bubbles generated electrolytically at the heating surface. Measurements were made at liquid temperatures between 60°C and saturation temperature at atmospheric pressure. The heat flux density ranged from 4.78·10⁴ W/m² to 2.10·10⁵ W/m² and the current density from 0 to 2100 A/m². In order not to essentially change the physical properties of water by addition of NaOH, the concentration of the solution was varied only up to 0.25 mol/l. The experiments showed an enhancement of heat transfer up to a factor of 6 due to the electrolytically produced hydrogen bubbles. The enhancement of heat transfer increases with increasing solution temperature and with increasing current density. An increasing heat flux density leads to an increase of the heat transfer coefficient . At the same time the ratio /₀ decreases, where ₀ is the heat transfer coefficient without hydrogen evolution. The effect of concentration on heat transfer coefficients can be neglected in the concentration range covered by the experiments.

---

Formelzeichen a Temperaturleitfähigkeit [m²/s] - C Konstante - E ₀ reversible Zersetzungsspannung von Wasser [V] - i elektrische Stromdichte [A/m²] - g Fallbeschleunigung [m/s²] - Gr Grashof-Zahl (Gr= g l ³/ ²) - K ₁ Kennzahl (K ₁=i E ₀/q) - K ₂ Kennzahl (K ₂=p _s/p) - l charakteristische Länge [m] - m, n Exponenten - Nu Nußelt-Zahl (Nu= l/) - p Systemdruck [MPa] - p _s Dampfdruck [MPa] - Pr Prandtl-Zahl (Pr=/a) - q Wärmestromdichte [W/m²] - Wärmeübergangskoeffizient [W/m² K] - ₀ Wärmeübergangskoeffizient ohne Elektrolyse [W/m² K] - räumlicher Wärmeausdehnungskoeffizient [1/K] - Wärmeleitfähigkeit [W/K m] - Temperatur [°C] - treibende Temperaturdifferenz [K] - kinematische Viskosität [m²/s] Herrn Prof. Dr.-Ing., U. Grigull zum 80. Geburtstag gewidmet     

Convective heat transfer in the thermal entrance region of parallel-flow noncircular duct heat exchanger arrays

Convective heat transfer properties of a hydrodynamically fully developed flow, thermally developing flow in a parallel-flow, and noncircular duct heat exchanger passage subject to an insulated boundary condition are analyzed. In fact, due to the complexity of the geometry, this paper investigates in detail heat transfer in a parallel-flow heat exchanger of equilateral-triangular and semicircular ducts. The developing temperature field in each passage in these geometries is obtained seminumerically from solving the energy equation employing the method of lines (MOL). According to this method, the energy equation is reformulated by a system of a first-order differential equation controlling the temperature along each line.Temperature distribution in the thermal entrance region is obtained utilizing sixteen lines or less, in the cross-stream direction of the duct. The grid pattern chosen provides drastic savings in computing time. The representative curves illustrating the isotherms, the variation of the bulk temperature for each passage, and the total Nusselt number with pertinent parameters in the entire thermal entry region are plotted. It is found that the log mean temperature difference (T _LM), the heat exchanger effectiveness, and the number of transfer units (NTU) are 0.247, 0.490, and 1.985 for semicircular ducts, and 0.346, 0.466, and 1.345 for equilateral-triangular ducts.

---

Konvektiver Wärmeübergang im thermischen Einlaufgebiet von Gleichstromwärmetauschern mit nichtkreisförmigen Strömungskanälen

Zusammenfassung Die Untersuchung bezieht sich auf das konvektive Wärmeübertragungsverhalten eines Gleichstromwärmetauschers mit nichtkreisförmigen Strömungskanälen bei hydraulisch ausgebildetet, thermisch einlaufender Strömung unter Aufprägung einer adiabaten Randbedingung. Zwei Fälle komplizierter Geometrie, nämlich Kanäle mit gleichseitig dreieckigen und halbkreisförmigen Querschnitten, werden bezüglich des Wärmeübergangsverhaltens bei Gleichstromführung eingehend analysiert. Das sich entwickelnde Temperaturfeld in jedem Kanal von der eben spezifizierten Querschnittsform wird halbnumerisch durch Lösung der Energiegleichung unter Einsatz der Linienmethode (MOL) erhalten. Dieser Methode entsprechend erfolgt eine Umformung der Energiegleichung in ein System von Differentialgleichungen erster Ordnung, welches die Temperaturverteilung auf jeder Linie bestimmt.Die Temperaturverteilung im Einlaufgebiet wird unter Vorgabe von 16 oder weniger Linien über dem Kanalquerschnitt erhalten, wobei die gewählte Gitteranordnung drastische Einsparung an Rechenzeit ergibt. Repräsentative Kurven für das Isothermalfeld, den Verlauf der Mischtemperatur für jeden Kanal und die Gesamt-Nusseltzahl als Funktion relevanter Parameter im gesamten Einlaufgebiet sind in Diagrammform dargestellt. Es zeigt sich, daß die mittlere logarithmische Temperaturdifferenz (T _LM), der Wärmetauscherwirkungsgrad und die Anzahl der Übertragungseinheiten (NTU) folgende Werte annehmen: 0,247, 0,490 und 1,985 für halbkreisförmige Kanäle sowie 0,346, 0,466 und 1,345 für gleichseitig dreieckige Kanäle.

---

Nomenclature A cross sectional area [m²] - a characteristic length [m] - C _c specific heat of cold fluid [J kg^–1 K^–1] - C _h specific heat of hot fluid [J kg^–1 K^–1] - C _p specific heat [J kg^–1 K^–1] - C _r specific heat ratio,C _r=C _c/C_h - D _h hydraulic diameter of duct [m] - f friction factor - k thermal conductivity of fluid [Wm^–1 K^–1] - L length of duct [m] - m mass flow rate of fluid [kg s^–1] - N factor defined by Eq. (20) - NTU number of transfer units - Nu _{x, T} local Nusselt number, Eq. (19) - P perimeter [m] - p pressure [KN m^–2] - Pe Peclet number,RePr - Pr Prandtl number,/ - Q _T total heat transfer [W], Eq. (13) - Q ideal heat transfer [W], Eq. (14) - Re Reynolds number,D _h/ - T temperature [K] - T _b bulk temperature [K] - T _e entrance temperature [K] - T _w circumferential duct wall temperature [K] - u, U dimensional and dimensionless velocity of fluid,U=u/u - , dimensional and dimensionless mean velocity of fluid - w generalized dependent variable - X dimensionless axial coordinates,X=D _h ² /a ² x* - x, x* dimensional and dimensionless axial coordinate,x*=x/D _hPe - y, Y dimensional and dimensionless transversal coordinates,Y=y/a - z, Z dimensional and dimensionless transversal coordinates,Z=z/a Greek symbols thermal diffusivity of fluid [m² s^–1] - * right triangular angle, Fig. 2 - independent variable - T _LM log mean temperature difference of heat exchanger - effectiveness of heat exchanger - generalized independent variable - dimensionless temperature - _b dimensionless bulk temperature - dynamic viscosity of fluid [kg m^–1 s^–1] - kinematic viscosity of fluid [m² s^–1] - density of fluid [kg m^–3] - heat transfer efficiency, Eq. (14) - generalized dependent variable     

Simulation und Regelung von Verdunstungskühlern

Zusammenfassung Aus den Bilanzgleichungen für Masse und Energie wurde ein instationäres, eindimensionales Modell für einen Verdunstungskühler mit Wasserumlauf hergeleitet. Die Lösung der Modellgleichungen erfolgte mit Hilfe der Linienmethode. Die Übereinstimmung der berechneten Ergebnisse mit stationären und Instationären Versuchswerten ist gut. Als Anwendungsbeispiel wurden ein 2-Punkt-Regler und ein PI-Regler für die Kühlmittel-Austrittstemperatur verglichen.

---

Simulation and control of evaporative coolers

A dynamic, one-dimensional model for an evaporative cooler with water recirculation has been derived from mass and energy balances. The solution of the model equations employs the method of lines. Calculated results are in good agreement with both stationary and non-stationary experimental data. As an application example a comparison of a 2-point-controller and a PI-controller for the cooling medium outlet temperature has been carried out.

---

Formelzeichen A Austauschfläche m² - B Breite der Trennwand m - c _p spez. isob. Wärmekapazität J/(kg·K) - g Erdbeschleuningung m/s² - i Zeiger - M Molmasse kg/mol - m Masse kg - m Massenstrom kg/s - P Pumpenleistung W - p Druck, Partialdruck Pa - r Verdampfungsenthalpie J/kg - Re Reynoldszahl - t Temperatur °C - x Wasserdampfbeladung der Luft (kg H₂O)/(kg tr. Luft) Griechische Zeiche Wärmeübergangskoeffizient W/(K·m²) - Differenz - Dicke m - Dynamische Viskosität Pa·s - V Kinematische Viskosität m²/s - P Dischte kg/m³ - Stoffübergangskoeffizient kg/(m²·s) - Zeit s 2|Indizes (tiefgestellt) A Austritt - E Eintritt - F Film - K Kühlmittel - L Luft - S Sättigung - T Trennwand - Umg Umgebung - UW Umlaufwasser - V verdunstet - W Wasser - W, ab Abschlämmwasser - W, zu Zusatzwasser 3|Indizes (hochgestellt) g gasförmig - 0 trocken Wir danken Herrn W. Gohl von der Fa. E. W. Gohl GmbH, Singen/Hohentwiel für seine freundliche Unterstützung und die bereitwillige Überlassung der Meßwerte der Verdunstungskühler.     

Treatment of moisture condensation on fins

An analysis of natural convection from a vertical plate fin when the fin base temperature is below the dew point of the surrounding air is presented in this paper. The analytical solution derived is based upon a constant heat and mass transfer coefficient and is also valid for forced convection. The results of this simplified theory are compared with a numerical solution where the coupling of convection and conduction is taken into account. An experimental verification of the results is also shown.

---

Aus Kondensation von Feuchtigkeit an Rippen

Zusammenfassung Es wird eine Analyse der freien Konvektion an einer vertikalen plattenförmigen Rippe dargestellt, bei der die Temperatur im Anfangsbereich der Rippe unterhalb des Taupunktes der umgebenden Luft liegt. Die abgeleitete analytische Lösung beruht auf einem konstanten Wärme- und Stoffübergangskoeffizienten und gilt auch für die erzwungene Konvektion. Die Resultate dieser vereinfachten Theorie werden mit einer numerischen Lösung verglichen, in der die Verbindung von Konvektion und Wärmeleitung in Betracht gezogen wird. Angeführt wird auch eine experimentelle Bestätigung der Resultate.

---

Nomenclature a _f thermal diffusivity of air - A, B constants in Eq. (7) - c constant defined in Eq. (3) - D diffusion coefficient - f an arbitrary function ofT andx in Eq. (12) - F ₁,F ₂ coefficients in differential Eq. (13) - g gravitational acceleration - h heat transfer coefficient - h _m mass transfer coefficient - k thermal conductivity of fin - k _f thermal conductivity of air - l latent heat of moisture condensation - L total length of fin - L _w length of wet fin - m parameter, (h/kt)^1/2 - m _l dimensionless parameter, 1+ B/T _r - m _y parameter,m m _l ^1/2 - p pressure of surrounding air - p _ws saturation pressure of water vapor - p _w partial pressure of water vapor in air - Pr Prandtl number,/a _f - q total heat fluxl - q _c convective heat flux - q _m heat flux - q _r radiative heat flux - R parameter in Eq. (14) - R _w specific gas constant of water vapor - t half thickness of fin - T temperature - T _b base temperature of wet fin - T _c base temperature of dry fin=saturation temp. of vapor - T _r reference temperature defined in Eq. (15) - T temperature of surrounding air - T temp, difference between fin surface and surroundings - v initial temperature for quasilinearization - x vertical coordinate, see Fig. 1 - y horizontal coordinate, see Fig. 1 - coefficient of thermal expansion - emissivity - dimensionless parameter in Eq. (14) - ø _d heat flux of dry fin - ø _tot total heat flux of dry-wet fin - kinematic viscosity - Stefan-Boltzman coefficient - relative humidity of air