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1 fundamentals of heating engineering CALCULATION OF CONVECTIVE HEAT TRANSFER COEFFICIENT (basic criterion equations) Guidelines to carry out practical and laboratory classes Ivanovo

2 Compiled by Editor V.V. Bukhmirov D.V. Rakutina Methodological instructions are intended for students studying in heat engineering specialties, both full-time and part-time departments and studying the course Heat and Mass Transfer or Heat Engineering. The guidelines contain the most tested criterion formulas for calculating the convective heat transfer coefficient in single-phase media and when changing state of aggregation substances. The appendix shows the physical properties of some liquids and gases used in thermal power engineering. The methodological instructions can be useful for students when solving problems on the topic “Convective Heat Transfer” during practical and laboratory classes, as well as when completing tests and homework. The guidelines were approved by the TEF cycle methodological commission. Reviewer department theoretical foundations heat engineering of Ivanovo State Energy University 2

3 1. Convective heat transfer during free movement of the fluid Nu f(gr,pr), Pr 0, Heat transfer during free convection near vertical plates and vertical pipes (criteria formulas by V.P. Isachenko) Local (local) and average heat transfer coefficients for laminar liquid flow mode (10 3< Ra 10 9) рассчитывают по формулам : при T w = const Nu f,x 0,25 f,x 0,55 Ra ; (1.1) t Nu f,h 0,25 f 0,73 Ra ; (1.2) t при q w = const Nu f,x 0,25 f,x 0,60 Ra ; (1.3) t Nu f,h 0,25 f 0,75 Ra. (1.4) В формулы (1.1) (1.4) входит поправка, учитывающая зависимость физических свойств текучей среды от температуры: 0,25 Prf t Pr, (1.5) w где критерий Прандтля Prf принимают по справочным данным для текучей среды при определяющей температуре флюида, а критерий Прандтля Pr w принимают по справочным данным для текучей среды при температуре стенки. Определяющие параметры: R 0 = x локальная координата по высоте для формул (1.1) и (1.3); R 0 = h высота vertical plate or the height of the vertical pipe for formulas (1.2) and (1.4); T 0 = T f temperature of the fluid away from the heat exchange surface (outside the thermal boundary layer). t 3

4 Local (local) and average heat transfer coefficients for a developed turbulent fluid flow regime (Ra) at T w = const and at q w = const are found using the formulas: Nu f,x 0.333 f,x 0.15 Ra; (1.6) t Nu 0.333 f 0.15 Ra f. (1.7) t Defining parameters: R 0 = x local coordinate in height for formula (1.6); R 0 = h height of a vertical plate or vertical pipe for formula (1.7); T 0 = T f temperature of the fluid away from the heat exchange surface (outside the thermal boundary layer). Comment. The correction, which takes into account the change in the physical properties of the medium depending on temperature, is calculated using the formula (1.5) The transition regime of fluid flow occurs at Rayleigh numbers 9 f, x Ra 610 is characterized by instability of the flow. In approximate calculations of heat transfer during transition mode, V.P. Isachenko recommends using formulas (1.6) and (1.7) for the turbulent flow regime. Heat transfer during free convection near horizontal plates (criteria formulas by V.P. Isachenko) The average heat transfer coefficient on the surface of horizontal plates can be approximately calculated using formulas for a vertical surface (1.2), (1.4) and (1.7) with the subsequent introduction of corrections for the location of the heat-transfer surface: for the heat exchange surface facing upward 1, (1.8) hori, 3 calculation for the heat exchange surface facing down 0, (1.9) hori, 7 calculation 4

5 where the calculation is the heat transfer coefficient calculated using one of the formulas (1.2), (1.4) or (1.7). Determining parameters: R 0 min(a, b), where a and b are the dimensions of the rectangular plate; T 0 = T f temperature of the fluid far from the heat exchange surface (outside the thermal boundary layer) Heat transfer during free movement of the fluid at low Rayleigh numbers (Ra md 1) This kind of heat exchange occurs around thin wires and the flow regime in this case is called film flow. To calculate the average heat transfer coefficient for film flow, we recommend using the following criterion formulas: a) according to the data, the film flow regime occurs at Rayleigh numbers d 10 2 Ram: Nu m,d 0.058 m,d 0.675 Ra ; (1.10) b) according to L.S. Eigenson film flow regime on thin heated wires (d = 0.22 mm) exists at Rayleigh numbers Ram, d 1: Nu m, d 0.5; (1.11) c) according to M.A. Mikheeva. the film mode exists at numbers d 3 Ra m 10 and only in this case can formula (1.11) be used. In the Ra 3 2 m,d range, a transitional flow regime from film to laminar is observed, for which M.A. Mikheev recommends the formula: Nu m,d 1/8 m,d 1.18 Ra. (1.12) Determining parameters: T T 0.5 (T T) average temperature boundary layer; 0 m f R outside diameter wire. 0 d n w 5

6 1.4. Heat transfer during free convection near horizontal cylinders (pipes) (criteria formula by I.M. Mikheeva) Average heat transfer coefficient under laminar flow regime 3 8 (Raf,d) according to I.M. Mikheeva is equal to: Nu f,d 0.25 f,d 0.5 Ra (1.13) t Determining parameters: T0 T f temperature of the fluid far from the heat exchange surface (outside the thermal boundary layer); R is the outer diameter of the pipe (cylinder). 0 d n Note. The correction t, which takes into account the change in the physical properties of the medium depending on temperature, is calculated using the formula (1.5) Heat transfer during free convection near vertical plates, vertical pipes, horizontal plates, horizontal pipes and balls (criteria formula by M.A. Mikheev) According to academician M.A. Mikheev, the average heat transfer coefficient for free movement of a fluid around the bodies indicated in the section title can be calculated using a single formula: Nu n m CRa m, (1.14) where the coefficients C and n, depending on the flow regime, are given in Table 1.1. Values ​​of coefficients C and n in formula (1.14) Ra m Gr Pr Flow mode C n m m<10-3 Пленочный 0, Переходный от пленочного к ламинарному 1,18 1/ Ламинарный и переходный к турбулентному 0,54 1/4 >Turbulent 0.135 1/3 6

7 Determining parameters: T T 0.5 (T T) average temperature of the boundary layer; 0 m f w R 0 d n outer diameter of horizontal pipes and balls; R 0 = h height of vertical plate or height of vertical pipe; R 0 min(a, b), where a and b are the dimensions of the rectangular plate. In this case, depending on the location of the heat-transmitting (heat-receiving) surface, the heat transfer coefficient is either increased by 30% or reduced by 30% (see formulas (1.8) and (1.9)) Heat transfer during free movement of a fluid in a limited space In narrow slots, flat and ring channels, interlayers various shapes density heat flow q is calculated using the formulas for stationary thermal conductivity in a flat wall, introducing the concept of equivalent thermal conductivity coefficient: eq (Tw1 Tw); (1.15) q 2 where eq is the equivalent thermal conductivity coefficient; thickness of the slot or narrow channel; T w1 and T w2 are the temperatures on the walls of the narrow layer. The equivalent thermal conductivity coefficient is determined by the formula: (1.16) eq f to where f is the thermal conductivity coefficient of the fluid; ε to the convection coefficient is a correction that takes into account the increase in heat flow due to free convection in the gap. The convection coefficient depends on the Rayleigh criterion: 3 a) for values ​​of Raf 10: k 1; (1.17) b) for values ​​of 10 Ra 10: 3 0.3 to 0.105 Ra f f 6 ; (1.18) 7

8 6 f 0.2 to 0.40 Ra f 10 c) with values ​​of 10 Ra 10:. (1.19) In approximate calculations, instead of two equations (1.18) and (1.19) for the entire range of values ​​of the arguments Raf 10, you can use the dependence: 0.25 to 0.18 Ra f. (1.20) Determining parameters: T 0.5 (T T) average temperature of the fluid in the gap; T0 f w1 w2 R slot width Convective heat transfer at forced movement fluid in pipes and channels Nu f(re,gr,pr), Pr 0, Heat transfer when fluid moves in straight lines smooth pipes When liquids and gases move in pipes and channels, there are laminar (Re f, d 2300), turbulent (Ref, d 10) and transitional from laminar to turbulent (2300 Re f 10) fluid flow regimes. Determining parameters for calculating the Reynolds criterion: T 0 T f 0.5 Tf,in Tf,out average fluid temperature in the pipe; R0 d internal pipe internal diameter; G/f is the average fluid velocity over the pipe cross-section. w Heat transfer in laminar mode of fluid movement in pipes (Re 2300) Heat transfer in pipes with stabilized flow and stabilized heat transfer can be calculated at T w = const and at q w = const using the approximate formula: 8,d 4 4

9 Nu 4, (2.1) t where the correction t is calculated using formula (1.5). The defining parameters in formula (2.1): T 0 T f 0.5 Tf,in Tf,out average fluid temperature in the pipe; R0 d internal internal diameter of the pipe; G/f is the average fluid velocity over the pipe cross-section. w 0 In the laminar mode of motion in straight smooth pipes and the presence of sections of hydrodynamic and thermal stabilization, for a more accurate approximation of experimental data, two subregimes are distinguished: laminar viscous and laminar viscous-gravity. Laminar viscous flow occurs at Rayleigh numbers Ra< 810 5, а ламинарный вязкостногравитационный режим при числах Рэлея Ra При этом определяющие параметры для расчета критерия Рэлея находят по формулам: T0 0,5 Tw T f T f 0,5 Tf,вх Tf,вых; R0 d вн внутренний диаметр трубы., где Теплоотдача при ламинарном вязкостном режиме движения текучей среды в трубах (R e 2300; Ra <) Средний по внутренней поверхности трубы длиной коэффициент теплоотдачи рассчитывают по формуле Б.Г.Петухова 3, которая получена при (Ped) и: 0 w f 1 3 Nu 1,55(Ped) (). (2.2) Определяющие параметры: T 0 0,5 Tw T f вн 9 f w 0.14, где T 0,5 T T f ; f,вх f,вых R0 d вн внутренний диаметр трубы; G / f средняя по сечению трубы скорость движения флюида. w 0 Замечание. Значение w выбирают для флюида при температуре стенки T w. Величина поправка, учитывающая влияние на теплоотдачу гидродинамической стабилизации потока на начальном участке теплообмена:

10 1 7 0.6 (Re d) 1 2.5 Re d at (Re d) 0, 1 ; (2.3) at (Re d) 0, 1 1, (2.4) where is the length of the pipe. The defining parameters in formulas (2.3) and (2.4): T 0 T f 0.5 Tf,in Tf,out average fluid temperature in the pipe; R0 d internal internal diameter of the pipe; G/f is the average fluid velocity over the pipe cross-section. w 0 Heat transfer in the laminar viscous-gravitational flow regime of the fluid in pipes (Re 2300; Ra) The average heat transfer coefficient in the laminar viscous-gravitational flow regime can be calculated using the criterion equation of M. A. Mikheev: 0.33 f,d 0.33 f Nu f,d 0.15 Re Pr (Gr Pr) f,d f 0.1 t. (2.5) Determining parameters: T T 0.5 T average fluid temperature in the pipe; 0 f f,in Tf,out R0 d out internal diameter of the pipe; G/f is the average fluid velocity over the pipe cross-section. w 0 Note. The correction t, which takes into account the change in the physical properties of the medium depending on temperature, is calculated using formula (1.5). The correction factor, which takes into account the effect on heat transfer of the process of hydrodynamic stabilization of the flow at the initial stage of heat transfer, is equal to: at d 50, the value is found according to the data in Table 2.1; at d

11 Table 2.1 Value in the viscous-gravitational regime of fluid flow d ,9 1.7 1.44 1.28 1.18 1.13 1.05 1.02 1, Heat transfer in the turbulent regime of fluid movement in pipes (Re 10 4 ) The average heat transfer coefficient for turbulent fluid flow in straight smooth pipes is calculated using the formula of M.A. Mikheev: 0.8 f,d 0.43 f Nu f,d 0.021 Re Pr t. (2.6) Remark. The correction t, which takes into account the change in the physical properties of the medium depending on temperature, is calculated using formula (1.5). The correction factor taking into account the effect on heat transfer of the process of hydrodynamic stabilization of the flow in the initial section of heat transfer is equal to: at d< d ; при d >50 = 1. More exact values depending on the Reynolds criterion are given in table Table 2.2. Value for turbulent fluid flow l/d Re ,65 1.50 1.34 1.23 1.17 1.13 1.07 1.51 1.40 1.27 1.18 1.13 1.10 1, 05 1.34 1.27 1.18 1.13 1.10 1.0 8 1.04 1.28 1.22 1.15 1.10 1.08 1.06 1.03 1.14 1.11 1.08 1.05 1.04 1.03 1.02 1.01 11

12 Determining parameters: T 0 T f 0.5 Tf,in Tf,out average fluid temperature in the pipe; R0 d internal internal diameter of the pipe; G/f is the average fluid velocity over the pipe cross-section. w Heat transfer during transient mode of fluid movement in pipes (2300< Re < 10 4) Переходный режим течения характеризуется перемежаемостью ламинарного и турбулентного течений. В этом случае коэффициент теплоотдачи можно рассчитать по формуле : 0,43 f,d Nu K Pr, (2.7) f,d 0 t где комплекс K 0 зависит от числа Рейнольдса (см. табл. 2.3.), а поправку рассчитывают также как и при турбулентном режиме течения флюида. Таблица 2.3. Зависимость комплекса К 0 от числа Рейнольдса Re ,2 2,3 2,5 3,0 3,5 4, K 0 2,2 3,6 4,9 7,2 16, Замечание. Поправку t, учитывающую изменение физических свойств среды в зависимости от температуры, рассчитывают по формуле (1.5). Определяющие параметры: T 0 T f 0,5 Tf,вх Tf,вых средняя температура флюида в трубе; R0 d вн внутренний диаметр трубы; G / f средняя по сечению трубы скорость движения флюида. w 0 Переходный режим течения флюида в прямых гладких трубах также можно рассчитать по методике, изложенной в учебнике : 12

13 Nu Nu (1), (2.8) turb Nu lams where Nu lams and Nu turb are Nusselt numbers, calculated using formulas (2.1) and (2.6) for stabilized laminar and turbulent flow regimes, respectively, γ intermittency coefficient equal to: 1exp(1 Re / 2300). (2.9) Heat transfer during the movement of gases in pipes For gases, the Prandtl criterion Pr f 0.7 1, 0 and practically does not depend on temperature, therefore the temperature correction t (Prf / Prw) 1. Taking this into account, formulas (2.5), (2.6) and (2.7) can be simplified and written as: 0.25 laminar mode Nu f,d 0.33 f,d 0.1 f,d 0.146 Re Gr ; (2.10) turbulent regime Nu f,d 0.8 f,d 0.018 Re ; (2.11) transition mode Nu f,d 0.86 K0. (2.12) Note. In the presence of large temperature differences and turbulent gas flow, heat transfer coefficients may differ from the values ​​calculated using equations (2.10), (2.11) and (2.12). In this case, the calculation must be carried out according to formulas (2.5), (2.6) and (2.7), taking as a temperature correction the expression: where m Tf t, (2.13) T w T f the average gas temperature in the pipe, Kelvin; 13 T w average pipe wall temperature, Kelvin; m 0, 4 if T w > T f and m 0 if T w< T f.

14 Determining parameters: T 0 T f 0.5 Tf,in Tf,out average gas temperature in the pipe; R0 d internal pipe internal diameter; G/f is the average fluid velocity over the pipe cross-section. w Heat transfer during fluid movement in channels of arbitrary cross section All the above criteria formulas for calculating heat transfer in round pipe also applicable for calculating the heat transfer coefficient for the flow of liquids and gases in channels of a different (non-circular) cross-sectional shape (rectangular, triangular, annular, etc.), for longitudinal washing of pipe bundles enclosed in a channel of arbitrary cross-section, as well as for movement of liquid that does not fill the entire cross-section of the channel. In this case, the equivalent or hydraulic diameter of the channel should be used as a characteristic size: R d d 4f P, (2.14) 0 "eq g pipes When fluid moves in curved pipes (elbows, coils), additional turbulization occurs and, as a result, an increase in the heat transfer coefficient. To calculate heat transfer in curved pipes, the Nusselt number calculated using formula (2.6) must be multiplied by. correction factor: 11.8 d R, (2.15) g int where d int is the internal diameter of the pipe, and R is the bending radius. g 14

15 3. Convective heat transfer during forced external flow around bodies Nu f(re, Pr), Pr 0, Longitudinal flow around a plate and outer surface pipe Thickness of the hydrodynamic boundary layer at a distance x from the leading edge of the plate (pipe) when a liquid or gas flows with constant physical properties along the plate or along the outer surface of the pipe is equal to: at Rex.5 / x 4.64 / Re x; (3.1) at Rex .2 / x 0.376 / Re x. (3.2) Determining parameters: T 0 = T f temperature of the fluid far from the heat exchange surface (outside the thermal boundary layer); R 0 x longitudinal coordinate; w 0 speed of undisturbed flow (outside the hydrodynamic boundary layer) Local and surface-average heat transfer coefficients at laminar flow fluid (Re<) вдоль пластины или внешней поверхности трубы по данным и равны: при T w =const 0, 25 Nu x 0,332Re Pr Pr Pr ; (3.3) x f w Pr Pr 0, 25 Nu 0,664Re Pr ; (3.4) 0,5 1 3 при q w =const 0, 25 Nu x x f 0,46Re Pr Pr Pr ; (3.5) 0,5 1 3 f w w Pr Pr 0, 25 Nu 0,69 Re Pr. (3.6) f w 15

16 Local and average heat transfer coefficients for turbulent fluid flow (Re) along the plate or outer surface of the pipe according to the data are equal to: Nu x 0.8 x 0.43 Pr Pr 0. 25 0.0296 Re Pr ; (3.7) f w 0.8 0.43 Pr Pr 0. 25 Nu 0.037Re Pr (3.8) f w Determining parameters: T 0 = T f temperature of the fluid away from the heat exchange surface (outside the thermal boundary layer); R 0 x longitudinal coordinate in formulas (3.3), (3.5) and (3.7); R 0 is the length of the plate or pipe in formulas (3.4), (3.6) and (3.8); w 0 speed of undisturbed flow (outside the hydrodynamic boundary layer) Heat transfer during transverse flow around a single pipe The average heat transfer coefficient over the surface of a pipe or cylinder according to the data is: 0.4 1 Re 40, Nu 0.76 Re Pr t q ; (3.9) 3 0.5 40 Re 10, Nu 0.52 Re Pr t q ; (3.10) 3 5 0.6 10 Re 210, Nu 0.26Re Pr t q ; (3.11) 5 7 0, Re 10, Nu 0.023 Re Pr t q, (3.12) 0.37 0.37 0.37 0.4 Remarks. 1. Correction t, which takes into account changes in the physical properties of the medium depending on temperature, is calculated using formula (1.5). 2. The correction q, taking into account the narrowing of the flow in the narrowest section of the channel (see Fig. 1.3), is calculated using the formula: q 2 1 d H 0. 8 (3.13) 16

17 3. The correction ε φ, taking into account the influence of the angle of attack of the oncoming flow (angle of attack is the angle between the velocity vector and the axis of the pipe) on the heat transfer coefficient, is taken according to the data in Table. 3.1 given in the problem book: Correction for the angle of attack of the oncoming flow Table 3.1. φº ε φ 1.0 1.0 0.99 0.93 0.87 0.76 0.66 For an approximate calculation of ε φ, formulas that approximate experimental data are proposed: according to data according to data 2 1 0.54 cos ; (3.14) sin. (3.15) Determining parameters: T 0 = T f temperature of the fluid far from the heat exchange surface (outside the thermal boundary layer); R is the outer diameter of the pipe; w 0 d n 0 w max G / f min the maximum flow velocity in the narrowest cross-section of the channel in a limited flow (Fig. 3.1.a) or the inflow velocity of an unlimited flow (Fig. 3.1.b). 17

18 w max d d H w 0 w max a) b) Fig. 3.1. Transverse flow around a single pipe in limited (a) and unbounded flow (b) 3.3. Heat transfer during transverse flow around a tube bundle The average heat transfer coefficient α 3 for the third row of a pipe bundle and all subsequent rows of pipes in the bundle in the direction of fluid movement at 10 3 Re210 5 according to the data is equal to: n 1 3 0.25 Prf w s Nu 3 CRe Pr Pr ( 3.16) where C 0.26 and n 0.65 for a corridor arrangement of pipes in a bundle (Fig. 3.2.a); C 0.41 and n 0.60 with a staggered arrangement of pipes in the bundle (Fig. 3.2.b). Notes. 1. The correction ε φ, which takes into account the influence of the angle of attack of the oncoming flow (angle of attack is the angle between the velocity vector and the axis of the pipe) on the heat transfer coefficient, is calculated using formula (3.14) or formula (3.15). More accurate values ​​of the correction ε φ for a tube bundle depending on the angle of attack φ are given in Table. 3.2 given in the problem book. Table 3.2. Correction for the angle of attack of the oncoming flow in a tube bundle φº ε φ 1.0 1.0 0.98 0.94 0.88 0.78 0.67 0.52 0.42 18

19 2. The correction ε s, taking into account the relative position of pipes in the bundle, is calculated using the formulas: for deep rows of pipes in a corridor bundle d 0.15; (3.17) s S 2 for deep rows of staggered bundle pipes s S S 1 6, if S 1 /S 2 2, (3.18) 1 2 s = 1.12, if S 1 /S 2 2; (3.19) where S is the transverse pitch of the pipes in the bundle; S 2 is the longitudinal pitch of the pipes in the bundle. Determining parameters: 0 T 0.5 Tf,in Tf,out T average fluid temperature in the bundle; f R 0 d n outer diameter of the pipe; w 0 w max G / f min maximum flow velocity in the narrowest cross section of the bundle The average heat transfer coefficient for pipes of the first row in the direction of flow in corridor and staggered bundles is equal to: 0. (3.20) 1, 6 3 Average heat transfer coefficient for pipes of the second row in corridor and chess beams, respectively, it is equal to: corridor beam 2 0, 93; (3.21) checkerboard bundle 2 0, 7 3, (3.22) where 3 heat transfer coefficient for pipes of the third row of the bundle Average heat transfer coefficient for the entire bundle when it flows around liquid or gas (Re=) depending on the number of rows along the direction of fluid movement (n3 ) is equal to: 1 2 n n 2, (3.23) where n 2 is the number of rows of pipes in the direction of fluid movement (liquid or gas). 19

20 T f, in d T f, out w s 1 w s 2 a) T f, in d T f, out w w s 1 s 2 b) Fig. 3.2. Geometric parameters of checkerboard (a) and corridor (b) beams. 4. Convective heat exchange when the aggregate state of a substance changes Depending on the phase state of the fluid, convective heat exchange in a single-phase medium is distinguished and convective heat exchange during phase transformations, which includes heat exchange during condensation (transition of vapor into liquid) and heat exchange during boiling (transition of liquid into vapor) . 20

21 4.1. Heat transfer during film condensation of vapors The average heat transfer coefficient during film condensation of vapors on a vertical surface is calculated using the Nusselt formula: 2 3 gr 4 pl pl 0.943, (4.1) pl T n Tw H where g 9.8 m/s 2 acceleration of gravity; r latent heat of vaporization, J/kg; pl thermal conductivity coefficient of film - dynamic viscosity coefficient of condensate, W/(m K); condensate pl, Pa s; pl film density, kg/m 3 ; Tn saturation temperature at a given pressure; T w wall temperature; H is the height of the vertical surface. Determining parameters: T saturation temperature at a given pressure; 0 T n R 0 H vertical plate height or pipe height The average heat transfer coefficient for film condensation on an inclined surface is calculated by the formula: 4 incline vertical cos, (4.2) where vertical is the heat transfer coefficient calculated by formula (4.1) for a vertical surface; the angle between the direction of gravity and the axis directed along the heat exchange surface The average heat transfer coefficient for film condensation on a horizontal pipe with laminar flow of the condensate film is calculated using the Nusselt formula: 2 3 gr pl pl 0.728 4, (4.3) pl (Tn Tw) dtr where g 9.8 m/s 2 free fall acceleration; r latent heat of vaporization, J/kg; pl thermal conductivity coefficient film - 21

22 ki of condensate, W/(m K); pl dynamic coefficient of condensate viscosity, Pa s; pl film density, kg/m 3 ; Tn saturation temperature at a given pressure; T w wall temperature; d tr is the outer diameter of the pipe, m. Formula (4.3) is valid for the laminar mode of film flow, which exists when the condition is met: where pl d tr 0.5 pl 20 g, (4.4) pl surface tension force of the film, N/m; g 9.8 m/s 2 acceleration of free fall; pl film density, kg/m 3. Determining parameters: T saturation temperature at a given pressure; 0 T n R outer diameter of the pipe. 0 d tr Formulas for calculating local heat transfer coefficients, heat transfer during wave and turbulent film flow, as well as the thickness of the condensate film are given in the literature 1-3, Heat transfer during boiling of liquids Bubble boiling in a large volume To calculate heat transfer during boiling of water in a large volume, use the following formulas 2,3, 8: 2.33 0.5 n 38.7 T p (4.5) 0.7 0.15 n 3.0 q p, (4.6) where p n saturation pressure, bar; q heat flux density, W/m 2. T T w T n superheating of the liquid in the boundary layer. 22

23 Film boiling in a large volume Using the analogy of the processes of condensation and film boiling, the following formulas can be used to calculate the heat transfer coefficient during film boiling: boiling on a vertical surface 3 gr 4 p w p p 0.943; (4.7) p T H boiling on a horizontal pipe where p and p p, 3 gr p f p p 0.728 4, (4.8) p T dtr density, thermal conductivity coefficient and dynamic coefficient of vapor viscosity; g fluid density; r latent heat of vaporization. The saturation temperature at a given pressure is taken as the determining temperature in formulas (4.7) and (4.8). 23

24 List of basic designations a thermal diffusivity coefficient, m 2 /s; c specific mass heat capacity, J/(kg K); d diameter, m; F heat exchange surface area, m 2 ; f cross-sectional area, m2; g gravity acceleration, m/s 2 ; G mass flow, kg/s; h height, m; specific enthalpy, J/kg; P perimeter, m; l linear size, m; length, m; p pressure, Pa; p pressure drop, Pa; q surface heat flux density, W/m2; q l linear heat flux density, W/m; Q heat flow, W; r radius, m; latent heat of vaporization, J/kg; T temperature, 0 C or K; w speed, m/s; x coordinate, m; degree of steam dryness; α heat transfer coefficient, W/(m 2 K); coefficient of volumetric expansion, K -1 ; wall thickness, m; thickness of the boundary layer, m; thermal conductivity coefficient, W/(m K); dynamic viscosity coefficient, Pa s; kinematic viscosity coefficient, m 2 /s; density, kg/m3; surface tension coefficient, N/m. Similarity criteria (numbers) R Nu 0 Nusselt criterion (number); 3 g Gr R 0 T 2 0 Grashof criterion; 24

25 s p Pr Prandtl criterion; a Ra Gr Pr Rayleigh criterion; w 0 R 0 w 0 R 0 Re Reynolds criterion; w 0 R 0 Pe Re Pr Peclet criterion. a Indices w wall; f fluid fluid (liquid or gas); kr critical; eq equivalent; g hydraulic; turbulent tour; laminar lam; averaging sign; 0 refers to the defining parameter; input; out exit. Defining (characteristic) quantities R 0 defining (characteristic) size, m; T 0 defining (characteristic) temperature, 0 C; w 0 defining (characteristic) speed, m/s; T 0 defining (characteristic) temperature difference, 0 C (K); 25

26 Appendix Table 1. Physical properties of dry air (B=1, Pa) T, 0 C, c p, kg/m 3 kJ/(kg K). 10 2, W/(m K). 10 6, Pa c. 10 6, m2/s a 106 m2/s Pr 7 16 .2 16.7 9.23 10.04 10.80 12.79 12.43 14.6 15.2 15.7 16.2 16.7 0.728 0.728 0.723 0.716 0.293 1.247 1.205 1.165 1.128 1.005 1.005 1.005 1.005 1.005 2 .44 2.51 2.59 2.67 2.76 17.2 17.6 18.1 18.6 19.1 13.28 14.16 15.06 16.00 16.96 17.2 17.6 18.1 18.6 19.1 0.707 0.705 0.703 0.701 0.093 1.060 1.029 1.000 0.972 1.005 1.005 1.009 1.009 1.009 2.83 2.90 2.96 3.05 3.1 3 19.6 20.1 20.6 21.1 21.5 17.95 18.97 20.02 21.09 22.10 19.6 20.1 20.6 21.1 21.5 0.698 0.696 0.694 0.692 0.946 0.898 0.854 0.815 0.779 1.009 1 009 1.013 1.017 1.022 3.21 3.34 3.49 3.64 3.78 21.9 22.8 23.7 24.5 25.3 23.13 25.45 27.80 30.09 32.49 21.9 22.8 23, 7 24.5 25.3 0.688 0.686 0.684 0.682 0.746 0.674 0.615 0.566 0.524 1.026 1.038 1.047 1.059 1.068 3.93 4.27 4.60 4.91 5.21 2 6.0 27.4 29.7 31.4 33, 0 34.85 40.61 48.33 55.46 63.09 26.0 27.4 29.7 31.4 33.0 0.680 0.677 0.674 0.676 0.456 0.404 0.362 0.329 0.301 1.093 1.114 1.135 1.156 1.172 5.74 6, 22 6.71 7.18 7.63 36.2 39.1 41.8 44.3 46.7 79.38 96.89 115.4 134.8 155.1 36.2 39.1 41.8 44 .3 46.7 0.687 0.699 0.706 0.713 0.277 0.257 0.239 1.185 1.197 1.210 8.07 8.50 9.15 49.0 51.2 53.5 177.1 199.3 233.7 49.0 5 1.2 53, 5 0.719 0.722 0.724 26

27 Table 2. Physical parameters of carbon dioxide CO 2 (B= Pa) T, 0 C, c p, kg/m 3 kJ/(kgk) 10 2, W/(mK) 10 6, 10 6, Ns/m 2 m 2 /s a10 6 m 2 /s Pr Table 3. Physical parameters of nitrogen N 2 (B= Pa) T, 0 C, kg/m 3 c p, kJ/(kgk) 10 2, W/(mK) 10 6 , 10 6, Ns/m 2 m 2 /s a10 6 m 2 /s Pr Table 4. Physical parameters of hydrogen H 2 (B= Pa) T, 0 C, kg/m 3 c p, kJ/(kgk) 10 2, W/(mK) 10 6, 10 6, Ns/m 2 m 2 /s a10 6 m 2 /s Pr

28 Table 5. Physical properties of methane CH 4 (B= Pa) T, 0 C, s p, kg/m 3 kJ/(kg K) 10 3, W/(m K) 10 6, Pa s 10 6, m 2 /s a10 6 m 2 /s Pr Table 6. Physical properties of ethane C 2H 6 (B= Pa) T, 0 C c, p, kg/m 3 kJ/(kg K) 10 3, W/(m K) 10 6, Pa s 10 6, m 2 / s a10 6 m 2 / s Pr Table 7. Physical properties of gaseous propane C 3H 8 (B= Pa) T, 0 C, s p, kg/m 3 kJ/(kg K ) 10 3, W/(m K) 10 6, Pa s 10 6, m 2 /s a10 6 m 2 /s Pr

29 Table 8. Physical properties of water at the saturation line T, p 10-5, c, p, a C Pa kg/m 3 kJ/ W/ ,. 10 6, m (kg K) (m K) 2 /s Pa. s m 2 /s. 10 4,. 10 4, K -1 N/m Pr ,013 1.013 1.013 1.013 1.013 1.013 1.013 1.013 1.013 1.013 1.013 1.43 999.9 999.7 998.2 995.7 992.2 988.1 983 .2 977.8 971 .8 965.3 958.4 951.0 4.212 4.191 4.183 4.174 4.174 4.174 4.179 4.187 4.195 4.208 4.220 4.223 55.1 57.4 59.9 61.8 63.5 64.8 65.9 66.8 67.5 68.0 68.3 68.5 13.1 13.7 14.3 14.9 15.3 15.7 16.0 16.3 16.6 16.8 16.9 17.5 653.3 549, 4,469.9 406.1 355.1 314.9 282.5 259.0 1.789 1.306 1.006 0.805 0.659 0.556 0.478 0.415 0.365 0.326 0.295 0.272-0.63 +0.7 1.8 2 3.21 3.87 4, 49 5.11 5.70 6.32 6.95 7.52 8.08 756.4 741.6 726.9 712.2 696.5 676.9 662.2 643.5 625.9 607.2 588 .6 569.0 13.67 9.52 7.02 5.42 4.31 3.54 2.93 2.55 2.21 1.95 1.75 1.98 2.7 3.61 4.76 6.18 7.92 10.03 12.55 15.55 19.08 23.20 27.98 943.1 934.8 926.1 917.0 907.4 897.3 886.9 876.0 863, 0 852.8 840.3 823.3 4.250 4.266 4.287 4.313 4.346 4.380 4.417 4.459 4.505 4.555 4.614 4.681 68.6 68.6 68.5 68.4 68.3 67.9 67.4 67.0 66.3 65 .5 64.5 63.7 17.1 17.2 17.2 17.3 17.3 17.3 17.2 17.1 17.0 16.9 16.6 16.4 237.4 217.8 201.1 186.4 173.6 162.8 153.0 144.2 136.4 130.5 124.6 119.7 0.252 0.233 0.217 0.203 0.191 0.181 0.173 0.165 0.158 0.153 0, 148 0.145 8.64 9.19 9, 72 10.3 10.7 11.3 11.9 12.6 13.3 14.1 14.8 15.9 548.4 528.8 507.2 486.6 466.0 443.4 422.8 400 .2 376.7 354.1 331.6 310.0 1.74 1.36 1.26 1.17 1.10 1.05 1.00 0.96 0.93 0.91 0.89 0.48 39.78 46.94 55.05 64.19 74.45 85.92 98.70 112.9 128.65 146.08 165.37 186.74 210.53 813.6 799.0 784.0 767, 9,750.7 732.3 512.5 691.1 667.1 640.2 610.1 574.4 528.0 450.5 4.766 4.844 4.949 5.070 5.230 5.485 5.736 6.071 6.574 7.244 8.165 9.504 13.984 40.321 62.8 61, 8 60.5 59.0 57.4 55.8 54.0 52.3 50.6 48.4 45.7 43.0 39.5 33.7 16.2 15.9 15.6 15.1 14 .6 13.9 13.2 12.5 11.5 10.4 9.17 7.88 5.36 1.86 114.8 109.9 105.9 102.0 98.1 94.2 91.2 88.3 85.3 81.4 77.5 72.6 66.7 56.9 0.141 0.137 0.135 0.133 0.131 0.129 0.128 0.128 0.128 0.127 0.127 0.126 0.126 0.126 8 18.1 19.1 21.6 23.7 26.2 29.2 32.9 38.2 43.3 53.4 66.5 261.9 237.4 214.8 191.3 168.7 144.2 120.7 98.10 76.71 56, 70 38.16 20.21 4.709 0.87 0.86 0.87 0.88 0.90 0.93 0.97 1.03 1.11 1.22 1.39 1.60 2.35 6.79 29

30 30 Table 9. Physical properties of water vapor in a saturated state T, 0 C p 10-5, Pa, kg/m 3 r, kJ/kg c p, kJ/ (kg K) 10 2, W/(m K) 10 6, Pa s 10 6, m 2 /s Pr 0.0061 0.0123 0.0234 0.0424 0.0738 0.1233 0.1992 0.3116 0.4736 0.7011 1.013 1.43 1.98 2 .7 3.61 4.76 6.18 7.92 10.03 12.55 15.55 19.08 23.20 27.98 33.48 39.78 46.94 55.05 64.19 74.45 85.92 98.70 112.9 128.65 146.08 165.37 186.74 210.53 0.1302 0.1981 0.2932 0.4232 0.598 0.826 1.121 1.496 1.966 2.547 3.258 4.122 5.157 6.394 7.862 9.588 11, 62 13.99 16.76 19.98 23.72 28.09 33.19 39.15 46.21 54.58 64.72 77.10 92.76 113.6 144.0 203.0 2202.8 2174 .3 2145.0 2114.4 2082.6 2049.5 2015.2 1978.8 1940.7 1900.5 1857.8 1813.0 1765.6 1715.8 1661.4 1604.4 1542.9 1476.3 1404.3 1325.2 1238.1 1139.7 1027.1 893.1 719.7 438.4 1.861 1.869 1.877 1.885 1.895 1.907 1.923 1.942 1.967 1.997 2.135 2.1 77 2.206 2.257 2.315 2.395 2.479 2.583 2.709 2.856 3.023 3.199 3.408 3.634 3.881 4.158 4.468 4.815 5.234 5.694 6.280 7.118 8.206 9.881 12.35 16.24 23.03 56.52 1.697 1.770 1.824 1.883 1.953 2.034 2.122 2.214 2, 309 2.407 2.372 2.489 2.593 2.686 2.791 2.884 3.012 3.128 3.268 3.419 3.547 3.722 3.896 4.094 4.291 4.512 4.803 5.106 5.489 5.827 6.268 6.838 7.513 8.257 9.304 10.70 12.79 17.10 9.156 9.493 9.746 9.989 10.270 10.586 10.921 11.272 11.620 11.960 11.97 12 ,46 12.85 13.24 13.54 13.93 14.32 14.72 15, 11 15.60 15.99 16.38 16.87 17.36 17.76 18.25 18.84 19.32 19.91 20.60 21.29 21.97 22.86 23.94 25.21 26 .58 29.14 33.7 328.9 200.7 127.5 83.88 56.90 39.63 28.26 20.02 15.07 11.46 8.85 6.89 5.47 4.39 3.57 2.93 2.44 2.03 1.71 1.45 1.24 1.06 0.913 0.794 0.688 0.600 0.526 0.461 0.403 0.353 0.310 0.272 0.234 0.202 0.166 1.00 1.00 1.00 1 .00 0.99 0.99 0.99 0.99 0.99 1.08 1.09 1.09 1.11 1.12 1.16 1.18 1.21 1.25 1.30 1.36 1.41 1.47 1.54 1.61 1.68 1.75 1.82 1.90 2.01 2.13 2.29 2.50 2.86 3.35 4.03 5.23 11, 10

31 Table 10. Physical properties of MK oil T, 0 C, c p, kg/m 3 kJ/(kg K), W/(m K) 10 4, Pa s 10 6, m 2 /s 10 4, K -1 Pr ,0 903.0 894.5 887.5 879.0 1.645 1.712 1.758 1.804 1.851 0.1510 0.1485 0.1461 0.1437 0.2 342.0 186.2 8.56 8.64 8.71 8.79 8.5 864.0 856.0 848.2 840.7 1.897 1.943 1.989 2.035 2.081 0.1389 0.1363 0.1340 0.1314 0.4 603.3 399.3 273.7 202.1 110.6 69.3 46.6 32.3 24.0 8.95 9.03 9.12 9.20 9.0 825.0 817.0 809.2 801.6 2.127 2.173 2.219 2.265 2.311 0.1264 0.1240 0.1214 0.1188 0.2 110.4 87.31 70.34 56.90 17.4 13.4 10.7 8.7 7.1 9.37 9.46 9.54 9, 65 9.3 113.5 Table 11. Physical properties of transformer oil T, 0 C, c p, kg/m 3 kJ/(kg K), W/(m K) 10 4, Pa s 10 6, m 2 /s 10 4, K -1 Pr ,5 886.4 880.3 874.2 868.2 1.549 1.620 1.666 1.729 1.788 0.1123 0.1115 0.1106 0.1008 0.8 335.5 198.2 128.5 89.4 70.5 37.9 22.5 14.7 10.3 6.80 6.85 6.90 6.95 7.1 856.0 850.0 843.9 837.8 1.846 1.905 1.964 2.026 2.085 0.1082 0.1072 0.1064 0.1056 0.3 49.5 38.6 30.8 25.4 7.58 5.78 4.54 3.66 3.03 7.05 7.10 7, 15 7.20 7.8 71.3 59.3 50.8 825.7 819.6 2.144 2.202 2.261 0.1038 0.1030 0.3 18.1 15.7 2.56 2.20 1.92 7 .30 7.35 7.40 43.9 38.8 34.9 31

32 Table 12. Physical properties of MS-20 oil depending on temperature T, 0 C, s p, kg/m 3 kJ/(kg K), W/(m K) 10 4, Pa s 10 6, m 2 /s 10 4, K -1 Pr,3 903.6 897.9 892.3 886.6 881.0 875.3 1.951 1.980 2.010 2.043 2.072 2.106 2.135 0.136 0.135 0.135 0.134 0.132 0 .131 0.24 6.24 6.31 6 , 7 91.9 58.4 39.2 27.5 20.3 6.51 6.55 6.60 6.64 6.3 835.7 830.0 824.4 818.7 2.320 2.353 2.382 2.420 2.445 0.124 0.123 0.122 0.121 0, 4 101.0 79.76 61.80 53.17 15.7 12.1 9.61 7.5 6.5 6.73 6.77 6.82 6.87 6, Table 13. Thermophysical properties of AMT- oil 300 T o C P n kpa kg/m 3 W/(mK) h" kJ/kg s r kJ/(kg K) 10 6 m 2 /s Pr,9 1.3 1.8 2.8 4.2 6 .5 10.2 15.8 24.8 30.9 66.6 90.120 0.119 0.117 0.115 0.114 0.112 0.111 0.108 0.106 0.104 0.102 0.100 0.099 0.095 0.093 0.0 91 0.088 0.086 31.2 64.0 96.5 134.5 170.0 208.2 248.0 288.0 330.0 374.0 418.0 462.0 510.0 556.0 612.0 672.0 715.0 770.0 1.60 1.68 1.73 1, 81 1.87 1.94 2.01 2.08 2.14 2.22 2.28 2.34 2.42 2.48 2.53 2.62 2.68 2.6 16.8 8.46 5 ,17 4.44 2.47 1.77 1.31 1.09 0.914 0.775 0.663 0.569 0.507 0.465 0.406 0.6 53.8 39.7 29.8 22.9 19.9 16.5 15.0 13, 1 11.8 10.8 10.1 9.3 8.5 32

33 Table 14. Physical properties of ammonia steam in the state of saturation T, 0 C p 10 5, Pa. r, kJ/kg, kg/m,7464 1.2443 1.9788 3.0253 4.2 1358.6 1554.6 1296.5 1262.5 0.645 1.038 1.604 2.390 3.396 10.776 12.133 16.1 1187 ,2 1143, 5 1100.6 4.859 6.694 9.034 12.005 Table 15. Physical properties of liquid ammonia in the saturated state T, 0 C p 10 5, Pa., c p, J/(kg K), kg/m 3 W/(m K) 10 6 , m 2 /s 10 4, K -1 Pr ,7464 1.2443 1.9788 3.0253 4.0 677.7 665.0 652.0 638.6 4.442 4.47 4.401 4.549 4.594 0.629 0.608 0.585 0, 563 0.540 0.355 0.304 0.264 0.245 17.28 18.32 19.32 20.25 21.12 1.95 1.77 1.56 1.38 1.396 10.776 12.133 16.7 610.3 595.2 579.5 4.646 4.708 4.777 4.860 0.518 0.494 0.472 0.449 0.234 0.227 0.222 0.216 22.54 23.86 25.66 33.14 1.31 1.32 1.335 1.33 33

34 Table 16. Physical properties of flue gases (B = 1, Pa; p = 0.13; p O = 0.11; CO 2 H 2 p N 2 = 0.76) T, 0 C, kg/m 3 s P, kJ/(kg K) 10 2, W/(m K) a 10 6, m 2 /s 10 6, Pa s 10 6, m 2 /s Pr ,295 0.950 0.748 0.617 0.525 0.457 0.405 0.363 0.330 0.301 0.275 0.257 0.240 1.042 1.068 1.097 1.122 1.151 1.185 1.214 1.239 1.264 1.290 1.306 1.323 1.340 2.28 3.13 4.01 4.84 5.70 6.56 7, 42 8.27 9.15 10.0 10.90 11, 75 12.62 16.9 30.8 48.9 69.9 94.3 121.1 150.9 183.8 219.7 258.0 303.4 345.5 392.4 15.8 20.4 24 .5 28.2 31.7 34.8 37.9 40.7 43.4 45.9 48.4 50.7 53.0 12.20 21.54 32.80 45.81 60.38 76.30 93.1 131.8 152.5 174.3 197.1 221.0 0.72 0.69 0.67 0.65 0.64 0.63 0.62 0.61 0.60 0.59 0, 58 0.57 0.56 34

35 T, C, W / (m K) avg, kJ / (kg K) a 10 6, m 2 /s v 10 8, m 2 /s Bukhmirov V.V. Calculation of the heat transfer coefficient (handbook)_v.6 Table 17. Physical properties of mercury and some molten metals Metal Mercury Hg T pl=-38.9 o C; T boil=357 o C; r pl=11.72 kJ/kg; r is=291.8 kJ/kg, kg/m.90 8.95 9.65 10.3 11.7 0.1390 ​​0.1373 0.1373 0.1373 0.1373 4.36 4.89 5, 30 5.72 6.64 11.4 9.4 8.6 8.0 7.1 Pr .72 1.92 1.62 1.40 1.07 Tin Sn T pl=231.9 o C; T boil=2270 o C; r pl=58.2 kJ/kg; r is=3015 kJ/kg Bismuth Bi T pl=271 o C; T boil=1477 o C; r pl=50.2 kJ/kg; r is=855.4 kJ/kg Lithium Li T pl=179 o C; T boil=1317 o C; r pl=661.5 kJ/kg; r is=19595 kJ/kg Alloy 56.5% Bi+43.5% Pb; T pl=123.5 o C; T boil=1670 o C,1 33.7 33.1 32.6 13.0 14.4 15.8 17.2 37.2 39.0 41.9 45.3 9.8 10.3 11.4 12.6 14.0 0.255 0.255 0.255 0.255 0.151 0.151 0.151 0.151 4.187 4.187 4.187 4.187 0.146 0.146 0.146 0.146 0.146 19.2 19.0 18.9 18.8 8.61 9.72 10.8 11.9 17.2 18.3 20.3 22.3 6.39 6.67 7.50 8.33 9.44 27.0 24.0 20.0 17.3 17.1 14.2 12.2 10.8 111, 0 92.7 81.7 73.4 28.9 24.3 18.7 15.7 13.6 1.41 1.26 1.06 0.92 1.98 1.46 1.13 0.91 6 .43 5.03 4.04 3.28 4.50 3.64 2.50 1.87 1.44 Alloy 25% Na+75% K T pl= -11 o C; T boil=784 o C,2 24.5 25.8 27.1 28.4 29.6 30.9 1.143 1.072 1.038 1.005 0.967 0.934 0.900 23.9 27.6 31.0 34.7 39.0 43, 6 48.8 60.7 45.2 36.6 30.8 26.7 23.7 21.4 2.51 1.64 1.18 0.89 0.69 0.54 0.44 Sodium Na T pl =97.8 o C; T boil=883 o C; r pl=113.26 kJ/kg; r is=4208 kJ/kg; .9 81.4 70.9 63.9 57.0 1.356 1.327 1.281 1.273 1.273 68.3 67.8 63.0 58.9 54.2 59.4 50.6 39.4 33.0 28.9 0 .87 0.75 0.63 0.56 0.53 35

36 Literature 1. Problem book on heat and mass transfer / F.F. Tsvetkov, R.V. Kerimov, V.I. Velichko; Ed. F.F. Tsvetkov. M.: Publishing house MPEI, p. 2. Isachenko V.P., Osipov V.A., Sukomel A.S. Heat transfer. - M.: Energoizdat, p. 3. Krasnoshchekov E.A., Sukomel A.S. Problem book on heat transfer. - M.: Energy, p. 4. Mikheev M.A. Basics of Heat Transfer. - M. - L.: GEI, p. 5. Galin N.M., Kirillov L.P. Heat and mass transfer (in nuclear energy). M.: Energoatomizdat, p. 6. Thermotechnical reference book/ed. V.N. Yureneva and P.D. Lebedeva. T M., Energy p. 7. Industrial furnaces. reference guide for calculations and design / Kazantsev E.I. M., Metallurgy, p. 8. Industrial heat power engineering and heating engineering: Handbook M., Chechetkin A.V. High temperature coolants. - M., Energy, Workshop on heat transfer: Proc. manual for universities / A.P. Solodov, F.F. Tsvetkov, A.V. Eliseev, V.A. Osipova; Ed. A.P. Solodova. M.: Energoatomizdat, p. 36

37 Contents 1. Convective heat transfer during free movement of a fluid Heat transfer during free convection near vertical plates and vertical pipes (criteria formulas by V.P. Isachenko) Heat transfer during free convection near horizontal plates (criteria formulas by V.P. Isachenko) Heat transfer during free movement fluid medium at low Rayleigh numbers (Ra md 1) Heat transfer during free convection near horizontal cylinders (pipes) (criteria formula by I.M. Mikheeva) Heat transfer during free convection near vertical plates, vertical pipes, horizontal plates, horizontal pipes and balls (criteria formula of M.A. Mikheev) Heat transfer with free movement of a fluid in a limited space 7 2. Convective heat transfer with forced movement of a fluid in pipes and channels Heat transfer with fluid movement in straight smooth pipes Heat transfer with laminar movement of a fluid in pipes (Re 2300 ) Heat transfer during the turbulent regime of fluid movement in pipes (Re 10 4) Heat transfer during the transition regime of fluid movement in pipes (2300< Re < 10 4) Теплоотдача при движении газов в трубах Теплоотдача при движении текучей среды в каналах произвольного поперечного сечения Теплоотдача при турбулентном течении флюида в изогнутых трубах Конвективная теплоотдача при вынужденном внешнем обтекании тел Продольное обтекание пластины и внешней поверхности трубы Теплоотдача при поперечном обтекании 37

38 of a single pipe Heat transfer during transverse flow around a tube bundle Convective heat transfer when the aggregate state of a substance changes Heat transfer during film condensation of vapors Heat transfer during boiling of liquids Bubble boiling in a large volume Film boiling in a large volume 23 List of basic notations 24 Appendix 26 Literature 36 38

39 CALCULATION OF CONVECTIVE HEAT TRANSFER COEFFICIENT (basic criterion equations) Guidelines for practical and laboratory exercises Compiled by: Vyacheslav Viktorovich BUKHMIROV Editor M.A. Ivanova License of the Republic of Lithuania from the city. Signed for seal. Format / 16. Printing is flat. Bake condition l.0.93. Circulation. Order. Ivanovo State Energy University Printed in Ivanovo, st. Rabfakovskaya, 34 39

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According to the convective heat transfer equation, also called the Newton-Richmann law, heat flow is directly proportional to the temperature difference between the wall and the liquid and the heat exchange surface area. The proportionality coefficient in this equation is called the average convective heat transfer coefficient:

, (1)

where Q is heat flow, W; q = Q/F - surface heat flux density, W/m2; - average convective heat transfer coefficient, W/(m 2 ∙K); - temperature pressure of heat transfer, o C; - temperature of the heat exchange surface (wall), o C; - liquid temperature far from the wall, o C; F - heat exchange surface area (wall), m2.

Regardless of the direction of the heat flow (from the wall to the liquid or vice versa), we will consider it positive, that is, we will use the temperature difference modulus.

The value of the heat transfer coefficient depends on a large number of different factors: a) physical properties of the liquid; b) the speed of fluid movement; c) shape, size and orientation in space of the heat exchange surface; d) the magnitude of the temperature difference, the direction of heat transfer, etc. Therefore, its theoretical definition is impossible in most cases.

Expressions (1)-(3) allow us to experimentally determine the average heat transfer coefficient by measuring the values ​​of Q, F, and:

, (4)

that is, the average heat transfer coefficient is numerically equal to the heat flux transmitted through a unit of heat exchange surface at a unit temperature difference (1 o C or 1 K).

3. Local (local) convective heat transfer coefficient

The average heat transfer coefficient is an important, but not always sufficient characteristic of heat transfer processes. In many cases, values ​​of heat transfer coefficients are required at individual points of the heat exchange surface, that is, local (local) values. Local coefficients characterize heat transfer in the vicinity of a given point (x) and are part of the local heat transfer equation:

or , (6)

where dF is the elementary (infinitesimal) heat exchange surface in the vicinity of point x, m 2 ; - elementary heat flow, W; - local heat flux density, W/m2; - local convective heat transfer coefficient, W/(m 2 ∙K); - local temperature difference, o C; - local surface (wall) temperature, o C; - temperature of the liquid far from the wall (we assume that it is constant along the entire heat exchange surface), o C.

From expressions (5) and (6) it follows that local heat transfer coefficients can, in principle, be found experimentally by measuring the values ​​, dF, and , related to the corresponding infinitesimal area:

. (7)

In practice, the required number of finite but sufficiently small sections are identified along the surface and measurements are taken for each i-th section of the surface:

, (8)

where is the average value of the heat transfer coefficient for the i-th section, W/(m 2 ∙K); - surface area of ​​the i-th section, m 2 ; - heat flow within the i-th section, W; - average value of surface temperature for the i-th section; - average heat flux density within the i-th section, W/m2; i = 1,2,…,n – number of the next section; n - number of sections.

When heat transfer occurs on a vertical surface, n sections of equal height are distinguished (see Fig. 4). If we measure the surface temperature at the boundaries of selected areas, starting from its lower edge (i=1), then the average temperature for the i-th area will be determined by the formula

. (9)

The average value of the heat transfer coefficient (8) for the small i-th section is an approximate value of the local heat transfer coefficient (7). The smaller the area, the more accurate the result.

The results of a large number of experiments to determine heat transfer coefficients (8) are summarized in the form of empirical (experimental) criterion equations (see Section 5). In the future, these equations are used in engineering calculations to determine heat transfer coefficients.

4. Nature of changes in the local heat transfer coefficient

The local heat transfer equation (5)-(6) can be written in the following form:

, (10)

where is the local thermal resistance of heat transfer, m 2 ∙K/W.

Thus, during heat transfer, the local surface heat flux density () is directly proportional to the local temperature difference and inversely proportional to the local thermal resistance of heat transfer.

Almost all the thermal resistance of heat transfer is concentrated near the wall surface within the thermal boundary layer, while the local thermal resistance is proportional to the local thickness of this layer.

During heat transfer under conditions of free convection near a heated vertical surface (Fig. 2), a boundary layer is formed along the surface along the flow. The thickness of the layer increases from bottom to top, and when the surface height is sufficient, the initially laminar boundary layer gradually transforms into a turbulent one.

In the region of laminar (layered) flow, the local heat transfer coefficient decreases along the height of the surface due to an increase in the thickness of the boundary layer and, consequently, due to an increase in its local thermal resistance (see Fig. 2).

In the transition region, an increase in the heat transfer coefficient is observed despite an increase in the thickness of the boundary layer. This occurs due to additional convective heat transfer by the resulting vortices.

In the region of developed turbulent flow, the thickness of the boundary layer continues to grow, but the eddy convective heat transfer increases to the same extent, so the thermal resistance and heat transfer coefficient remain constant, that is, they cease to change along the height of the surface (see Fig. 2).

Fig.2. Boundary layer and local heat transfer:

1 - wall (heat exchange surface); 2 - hydrodynamic boundary layer; 3 - hydrodynamic "flow core"

5. Calculation of local heat transfer coefficient

using criterion equations

With free convection, the local heat transfer coefficient on a vertical surface can be calculated using criterion empirical formulas of the following form:

, (11)

where C, n and 0.25 are empirical (determined from experience) constants; - local Nusselt number; - local Rayleigh number; Pr, are the Prandtl numbers taken at the determining temperature and at the wall temperature, respectively. For more details, see section. 6.

The values ​​of the empirical constants (Table 1) depend on the regime of free movement of the fluid. The mode of free movement at a given point x of the heat exchange surface is determined by the value of the local Rayleigh number at this point.

Table 1. Values ​​of empirical constants

For gases, the factor is close to unity, since due to the weak dependence of the Prandtl number of gases on temperature, therefore for gases formula (11) takes a simpler form:

Having calculated the local Nusselt number, the local heat transfer coefficient included in it is determined (see Section 6).

Numbers (criteria) of similarity

Each similarity criterion is dimensionless a complex (combination) made up of physical quantities that influence the process: the determining temperature (temperature difference), the determining speed (with forced convection), the determining size, and the physical properties of the liquid. As a result, each similarity criterion characterizes a certain ratio of physical effects characteristic of the phenomenon under consideration.

One of the similarity criteria in the equation is determined(searched for), all others are defining criteria, that is, they play the role of independent variables affecting heat transfer.

Let's consider local numbers (criteria) of similarity.

Nusselt number: , (12)

where is the local coefficient of convective heat transfer, W/(m 2 ∙K); x – coordinate in which the local heat transfer coefficient is sought, m ​​(see Section 7); - coefficient of thermal conductivity of the liquid, W/(m∙K).

This is a defined similarity criterion, since it includes the desired heat transfer coefficient. The Nusselt number can be thought of as a relative heat transfer coefficient: , where is the reference scale, which has the same dimension as the heat transfer coefficient. That is, the Nusselt number characterizes the intensity of heat transfer or, more precisely, the ratio of the intensity of heat transfer and thermal conductivity of the liquid. If the Nusselt number is found, for example, using (11) or (11 a), then

This is the main determining criterion of similarity. Based on its numerical value, the mode of free movement of the liquid is determined: laminar, transitional, turbulent. Different modes of motion correspond to different physical mechanisms of heat transfer, which is expressed in different values ​​of the empirical constants C and n in equations like (11) and (11a) (see also Section 9).

The Rayleigh number can be thought of as the ratio of the lift force of the thermal boundary layer to the friction force due to viscosity.

Grashof number: , (16)

where g is the acceleration of gravity, m/s 2 ; - thermal coefficient of volumetric expansion of the liquid, 1/K; - local temperature difference, o C ( - local temperature of the surface (wall), o C; - liquid temperature far from the wall, o C). This page violates copyright

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The coefficients of convective heat transfer in this case are of the order of 10 kcal/m2 h deg. It has been discovered that the coefficients of radiant heat transfer at temperatures approximately equal to the atmospheric temperature are of the order of 2 kcal / m2 - h - deg. This means that under such conditions no accurate measurement with a conventional thermometer is possible.  

The convective heat transfer coefficient a is a function of thermophysical properties, temperature and speed of the coolant, as well as the configuration and size of the heat exchange surface.  

Coefficients of convective heat transfer on the internal surfaces of windows and windows: P 3 etc. 4 kcal / m1 hour grid.  

The coefficients of convective heat exchange between gases and pipes in heat exchangers or packing in regenerators are determined using formulas given in reference books and special manuals. A number of them are given in the relevant sections of this book. In all cases, to increase the intensity of convective heat transfer, it is necessary to strive for the greatest uniformity of flushing of all heating surfaces with gases, reduce to optimal sizes the cross-sections of channels formed by the material in the layer through which the coolant flows, and increase the flow rate to values ​​justified by technical and economic calculations.  

The coefficient of convective heat transfer in the layer of air (outside) is significantly less than in the layer of water or steam (inside the device), therefore the resistance to external heat transfer RH for the heating device is relatively high. Therefore, to increase the heat flow it is necessary to develop the outer surface of the heating device. In devices this is done by creating special protrusions, bosses and fins. However, this reduces the heat transfer coefficient.  

The coefficient of convective heat exchange between the medium and the body placed in it at the same speeds of movement for liquids is many times greater than for gases. Liquids are opaque to heat rays, gases are transparent. Therefore, when measuring the temperature of gases, it is necessary to take into account the influence on the temperature of the meter of radiant heat exchange between the surface of the meter and the walls of the pipe.  

The coefficients of convective heat exchange between the nozzle and hot gas or air are determined from experimental data.  

The convective heat transfer coefficient strongly depends on the fiber diameter and the relative velocity of the medium due to a sharp change in the thickness of the laminar boundary layer, comparable to the fiber diameter.  

The coefficients of convective heat exchange between the nozzle and hot gases or air are determined from experimental data.  

The coefficient of convective heat exchange between the walls of the room and the air contained in it is 11 36 W / m2 - deg.  

Consequently, the coefficient of convective heat transfer depends on the method of heat supply, and with complex heat transfer (convection and radiation) it is significantly higher compared to only convective heat transfer, all other things being equal.  

The average values ​​of the coefficient of convective heat transfer on the vertical surfaces of enclosures in a room can be determined without much error using formula (1.64), since the temperature differences and geometric dimensions of heated and cooled surfaces that occur in reality usually correspond mainly to a turbulent regime. All the formulas considered, including (1.64), are written for a vertical freely located surface.  

To determine the convective heat transfer coefficient, criterion equations are usually used. These equations under heat transfer conditions characteristic of the room are given in Table. 5 for forced and free convection. They refer to the conditions of motion near the surface of the plate. They are characterized by unidirectionality and uniformity, in a word, orderly movement.  

The average value of the convective heat transfer coefficient сс, (sometimes denoted oc) ranging from 0 to an arbitrary section / can be determined based on the theorem on the integral mean.  

α – characterizes the intensity of convective heat transfer and depends on the speed of the coolant, heat capacity, viscosity, surface shape, etc.

[W/(m 2 deg)].

The heat transfer coefficient is numerically equal to the heat flow power transferred to one square meter of surface with a temperature difference between the coolant and the surface of 1°C.

The main and most difficult problem in calculating convective heat transfer processes is finding the heat transfer coefficient α . Modern methods for describing the coefficient process. thermal conductivity theory based boundary layer, allow one to obtain theoretical (exact or approximate) solutions for some fairly simple situations. In most cases encountered in practice, the heat transfer coefficient is determined experimentally. In this case, both the results of theoretical solutions and experimental data are processed using methods theoriessimilarities and are usually presented in the following dimensionless form:

Nu=f(Re, Pr) - for forced convection and

Nu=f(Gr Re, Pr) - for free convection,

Where
- Nusselt number, - dimensionless heat transfer coefficient ( L- characteristic flow size, λ - coefficient of thermal conductivity); Re=- Reynolds number characterizing the ratio of inertial forces and internal friction in the flow ( u- characteristic speed of movement of the medium, υ - kinematic viscosity coefficient);

Pr=- the Prandtl number, which determines the ratio of the intensities of thermodynamic processes (α is the thermal diffusivity coefficient);

Gr=
- Grasshof number, characterizing the ratio of Archimedean forces, inertial forces and internal friction in the flow ( g- acceleration of gravity, β - thermal coefficient of volumetric expansion).

• What does the heat transfer coefficient depend on? The order of its magnitude for various cases of heat transfer.

Convective heat transfer coefficient α the higher the thermal conductivity coefficient λ and flow rate w, the lower the coefficient of dynamic viscosity υ and the higher the density ρ and the smaller the reduced channel diameter d.

The most interesting case of convective heat transfer from the point of view of technical applications is convective heat transfer, that is, the process of two convective heat exchanges occurring at the interface between two phases (solid and liquid, solid and gaseous, liquid and gaseous). In this case, the calculation task is to find the heat flux density at the phase interface, that is, a value showing how much heat a unit of phase interface receives or releases per unit time. In addition to the above factors influencing the process of convective heat transfer, the heat flux density also depends on the shape and size of the body, on the degree of surface roughness, as well as on the surface temperatures and the heat-releasing or heat-receiving medium.

To describe convective heat transfer, the formula is used:

q ct = α(T 0 -T st ) ,

Where q ct - heat flux density on the surface, W/m 2 ; α - heat transfer coefficient, W/(m 2 °C); T 0 And T st- temperature of the medium (liquid or gas) and surface, respectively. Size T 0 - T st often denoted by Δ T and is called temperature difference . Heat transfer coefficient α characterizes the intensity of the heat transfer process; it increases with an increase in the speed of movement of the medium and during the transition from a laminar mode of motion to a turbulent one due to the intensification of convective transfer. It is also always greater for those media that have a higher thermal conductivity. The heat transfer coefficient increases significantly if a phase transition occurs on the surface (for example, evaporation or condensation), always accompanied by the release (absorption) of latent heat. The value of the heat transfer coefficient is strongly influenced by mass transfer on a surface.

CONVECTIVE HEAT TRANSFER (HEAT TRANSFER)

The process of heat exchange between the surface of a solid and a liquid having different temperatures is called heat transfer. Heat transfer is usually accompanied by thermal conductivity. The combined process of convection and heat conduction is called convective heat transfer.

According to the Newton-Richmann law, the heat flow in the heat transfer process is proportional to the heat transfer coefficient, the heat exchange surface area and the temperature difference between the surface of the body and the liquid.

Q = (t s – t l)F , 2.17

In calculations, the temperature difference t c – t l is taken in absolute value. The heat transfer coefficient α W/(m 2 K) characterizes the intensity of the heat transfer process and depends on a large number of factors:

= ƒ (t f, t st, d, λ, ν, ω, ℓ, ġ, β Х…….) 2.18

where: tf - liquid temperature, 0 C; t st – wall temperature, 0 C; d – pipe diameter, m;

λ – thermal conductivity of the liquid, W/ (m K): ω – liquid flow speed, m/s; ℓ – defining size (for pipes – diameter), m; g – free fall acceleration, 9.8 m/s 2 ;

β – coefficient of volumetric expansion, 1/K; X is the nature of the fluid flow; ν – kinematic viscosity coefficient, m 2 /s.

From formula 2.18 it is clear that the heat transfer coefficient is difficult to determine, because it depends on a large number of variables.

There are two ways to solve problems of convective heat transfer: analytical and using similarity theory.

When analytically solving problems of convective heat transfer, differential equations are compiled that take into account thermal and dynamic phenomena in the process under consideration. The derivation of such equations is considered in the specialized literature.

Convective heat transfer in an incompressible single-phase medium is described by the following equations.

Heat transfer equation:

α = -(λ/θ) (∂t / ∂n) n=0, where θ = t – t 0 . 2.19

The differential equation of thermal conductivity (continuity) has the form:

∂t /∂τ = а 2 t = [∂ 2 t / ∂x 2 +∂ 2 t / ∂y 2 + ∂ 2 t / ∂z 2 ] λ /с ρ 2.20

where: ∂t /∂τ – temperature field of the object under study, which depends on the temperature change along the axes, i.e. from the Laplace operator,

2 t = ∂ 2 t /∂x 2 + ∂ 2 t / ∂y 2 + ∂ 2 t /∂ z 2 , 2.21

and on thermophysical properties: thermal diffusivity coefficient – A(m 2 /s), specific heat capacity - s (kJ/(kg K) and density ρ (kg/m 3)

Differential equation of motion:

∂ω/ ∂τ = gβ – 1/ρ ( ρ) + ν 2 ω. 2.22

Differential continuity equation:

∂ω x / ∂х + ∂ω y / ∂у + ∂ω z / ∂z = 0 or div = 0 2.23

The given differential equations of convective heat transfer 2.19 – 2.22 describe countless processes. To solve a specific problem, the conditions of uniqueness should be added to the given equations. Uniqueness conditions provide a mathematical description of special cases. The uniqueness conditions are:

1) from geometric conditions characterizing the shape and size of the body or system in which the process occurs;

2) physical conditions characterizing the physical properties of the environment;

3) boundary conditions that determine the features of the process at the boundaries of the liquid medium;

4) temporary or initial conditions characterizing the features of the process at the initial moment of time; for stationary processes these conditions disappear.

The solution of the given systems of differential equations and uniqueness conditions with a large number of variables turns out to be complex. Therefore, the experimental path of research and the application of similarity theory are of great importance.

Similarity theory is based on three theorems.

The first theme of similarity: for similar phenomena, the similarity numbers are numerically the same.

Second similarity theorem: if a physical phenomenon is described by a system of differential equations, then it is always possible to represent them in the form of similarity equations.

Third similarity theorem: those phenomena are similar whose uniqueness conditions are similar, and the similarity numbers made up of the uniqueness conditions are numerically the same.

The essence of the similarity theory is that dimensional physical quantities that influence convective heat transfer are combined into dimensionless complexes, and in such a way that the number of complexes is less than the number of quantities from which these complexes are composed. Complexes or similarity numbers are assigned the names of scientists who have made a great contribution to the study of heat transfer processes and hydrodynamics

The resulting dimensionless complexes are considered as new variables. They reflect not only the influence of single factors, but also their combination, which simplifies the description of the process under study. The theory of similarity is the theoretical basis of the experiment and facilitates the analysis of processes. Let us consider the application of similarity theory to study convective heat transfer processes.

From formula 2.17 it is clear that the intensity of convective heat transfer is characterized by a heat transfer coefficient, which depends, in particular, on the determining size, heat transfer surface area, thermal diffusivity, thermal conductivity, temperature pressure, fluid velocity, kinematic viscosity coefficient, etc.

From these quantities, dimensionless complexes are composed - similarity numbers (similarity criteria).

Nusselt number Nu = αℓ / λ 2.24

Reynolds number Re = ωℓ / ν 2.25

Grashof number Gr = g β Δt ℓ 3 / ν 2 2.26

Prandtl number Рr = ν / A 2.27

The Nusselt number is a definable number because it includes the desired heat transfer coefficient. Reynolds, Grashof, and Prandtl numbers are decisive. They consist of quantities known before solving the problem. In general

Nu= ƒ (Re, Gr, Pr) 2.28

To solve problems, the above equation is written in power form:

Nu = c Re m Gr n Pr r 2.29

There are natural (free) and forced flow of fluid.

Natural convection occurs due to the difference in densities of cold and hot liquid particles near the heating surface. The intensity of thermal expansion is characterized by the temperature coefficient of volumetric expansion β For gases, which in most cases can be considered ideal, the coefficient of volumetric expansion is determined by the equality

With natural convection, equation 2.28 simplifies:

Nu= c (Gr, Pr) n 2.31

Forced convection is created by an external source (pump, fan). For forced convection, equation 2.28 is:

Nu = c Re m Pr n 2.32

The objective of the experiment is to determine the specific type of functional connection in the similarity equation, i.e. you should find the numerical values ​​of coefficients, exponents, etc.

Nuℓ/λ 2.33

As experimental studies have shown, the flow regime is determined by the flow velocity.

O. Reynolds experimentally established that when a fluid moves, there are two types of flow that obey different laws. In one type of flow, all particles move only along parallel trajectories and the movement coincides for a long time with the direction of the entire flow. The liquid moves calmly, without pulsations. This movement is called laminar. With laminar flow in a pipe, the Reynolds number is less than 2300.

In the second type of flow, continuous mixing of all layers of liquid occurs. A flow is a disordered mass of chaotically moving particles. This type of flow is called turbulent. In turbulent flow, the Reynolds number is more than 10 4 .

For Reynolds numbers greater than 2000 but less than 1. 10 4 fluid movement is unstable. The flow regime is called transitional.

The theoretical study of convective heat transfer problems is based on the boundary layer theory developed by L. Prandtl.

The concepts of thermal and dynamic boundary layers are introduced.

If the temperatures of the wall and the liquid are not the same, then a thermal boundary layer is formed near the wall, in which a temperature change occurs. Outside the boundary layer, the fluid temperature is the same and equal to the flow temperature.

A thin boundary layer of liquid near the surface, in which the velocity changes from the value of the velocity of the undisturbed flow far from the wall to zero directly at the wall, is called a dynamic boundary layer.

Fig.2.4 Temperature and velocity distribution in thermal

and dynamic boundary layer

With increasing viscosity, the thickness of the dynamic layer increases; with increasing flow speed, the thickness of the dynamic layer decreases. The flow in a dynamic layer can be either laminar or turbulent and is determined by the Reynolds number.

The thicknesses of the thermal and boundary layers may not coincide. The ratio of the thicknesses of the dynamic and thermal boundary layers is determined by the dimensionless Prandtl number. For viscous liquids, for example oils, Pr>1. For viscous liquids, such as oils, the thickness of the dynamic boundary layer is greater than the thickness of the thermal boundary layer. For gases Pr ≈ 1 and the layer thicknesses are approximately the same. For liquid metals Pr< 1, толщина теплового пограничного слоя больше толщины динамического пограничного слоя.

If the movement inside the thermal boundary layer is laminar, then heat transfer is carried out by conduction. With an increase in velocity in the boundary layer and the appearance of turbulence, the intensity of fluid mixing should be taken into account.

In the process of longitudinal flow around any body by an unlimited flow of liquid with a constant flow velocity in the immediate vicinity of the surface of the body, the flow speed must drop to zero.

When solving problems of convective heat transfer, you should pay attention to what temperature is taken as the determining temperature for a given similarity equation, because the physical parameters of liquids and gases change with temperature.

For the simplest cases, when the flow temperature varies within small limits, the average temperature of the liquid can be defined as the arithmetic mean at the entrance to the channel t 1 and the exit from the channel t 2: t l = 0.5 (t 1 – t 2).

For more accurate calculations, use the formula

t f = 0.5 (t 1 – t 2) (∆t b - ∆t m)/ ℓn (∆t b /∆t m), 2.34

where ∆ t b and ∆ t m are the temperature pressures in the initial and final sections of the pipe or channel.

Some similarity numbers include a linear size, and take the size that determines the development of the process. For pipes, the determining size for liquid flow inside the pipe is the internal diameter, for external flow - the outer diameter of the pipe, for channels of non-circular cross-section - the equivalent diameter deq = 4F / S is accepted, where F is the cross-sectional area of ​​the channel, S is the total (wetted) perimeter channel. When flowing around a plate, its length in the direction of flow is taken as the determining size.

Attention should be paid to the analogy between the processes of heat and mass transfer.

The heat conduction equation discussed above - Fourier's law (equation 2.3) is similar to the basic law of the diffusion process (molecular mass transfer) - Fick's law.

m = - D grad c i 2.35

where m is the mass flow density, kg / (m 2 s); D – diffusion coefficient, m 2 / s; c i is the mass concentration of the component under consideration per unit volume of the substance, kg/m 3 . Let's compare these laws:

Q = -λgrad t F m = - D grad c i F

The same mathematical notations for Fourier's and Fick's laws reflect the analogy of mass and heat transfer. For example, in gases the carriers of mass and heat are the same: Each molecule, along with its own mass, also carries energy. A thin boundary layer is formed near the surface, in which the concentration of the substance will change from the saturation state at the surface to the concentration of the substance in the flow.

Equation of mass transfer in direction at(across the flow) has the form

β = (D / c 0 - c f) (∂с / ∂у) 2.36

Equation of mass transfer by diffusion and concentration

ω x (∂с/∂х) + ω у (∂с/∂у) = D [(∂ 2 c/∂х 2) + (∂ 2 с/∂у 2) 2.37

The equations of continuity and motion (2.20 and 2.22) will remain unchanged.

The numbers Nu and Pr are similar in writing

Nu =αℓ/λ Nu d = βℓ/ D - sometimes called the Sherwood number 2.38

Pr = ν/ a Pr d = ν/ D - sometimes called the Schmitd number 2.39

Nu = Nu d; Рr = Рr d 2.40

The same dimensionless equations under the same boundary conditions will give the same solutions suitable for describing the processes of both heat transfer and mass transfer.

βℓ/D = α ℓ/λ , then 2.41

β/D= α /λ2.42

With large differences in temperature or concentration, the analogy between the processes of heat and mass transfer is violated, because The dependences of thermophysical properties on temperature and concentration are not the same.