Heat Transfer Fundamentals Concerning Quenching Materials in Cold Fluids

Abstract

The current chapter discusses three principles of heat transfer related to transient nucleate boiling process when any film boiling during quenching is completely absent. They include the duration of nucleate boiling establishment, surface temperature behavior during self-regulated thermal process, and generalized fundamental equation for evaluation of transient nucleate boiling length. In fact, majority of authors in their analytical and experimental investigations always considered three classical heat transfer modes: film boiling, nucleate boiling, and convection. It is shown in the chapter that the absence of film boiling process simplifies numerical and analytical calculations. It is very important to know that mentioned principles can be used for calculation of temperature fields and stresses and development of new technologies without performing costly experiments. It is stated that universal correlation for heating and cooling time evaluation, modified by proposed principles, can be widely used for recipes development when exploring new intensive quenching technologies. Examples of calculations are provided.

1. Introduction

This chapter considers recently discovered unusual characteristics of transient nucleate boiling process. It was possible to do that terminating any film boiling process during quenching. Since it was a widely distributed opinion on three stages of cooling (film boiling, nucleate boiling, and convection), scientists paid the main attention to studying film boiling processes. The results of investigation obtained by scientists are really very important and very interesting since they allowed to decrease distortion. Four types of heat transfer modes discovered [1, 2] were important for the practice. It was established that vapor film behavior depends on size and form of the quenched steel part. These accurate investigations brought success and lifted the heat-treating technology to the next level in progressing scale. Nobody considered cooling curves during quenching without considering the film boiling process because existing theory and accurately performed experiments in many cases showed vapor films during quenching in liquid media. Engineers used this concept in practice. Even thermal scientists were sure that film boiling during hardening steel in cold fluids always exists. However, accurate experiments of French [3], which were performed in cold 5% water slow agitated solution of sodium hydroxide (NaOH), clearly displayed the absence of film boiling process when quenching spherical steel samples of different diameters from 875°C in the mentioned fluid. Accurate experiments of French were published in many papers and books [3]. Scientists did not discuss them and did not use them for solving the inverse problem to investigate heat transfer when any film boiling process is completely absent during quenching. Probably, it was impossible to explain the absence of film boiling process during quenching probes from 875°C in a slow agitated cold fluid. The current short overview presents results of investigations which were obtained using long-lasting experiments of different authors including experiments of French. As a result, the three of the most important principles were formulated, which create the fundament for intensive quenching processes. These new principles are discussed in the current short chapter.

2. Incorrect heat transfer coefficient evaluation during transient nucleate boiling

Historically in the heat-treating industry, the heat transfer coefficient (HTC) during transient nuclear boiling process is evaluated as α_eff = q/(T – T_m), which is called the effective HTC. In fact, the real HTC is evaluated as α_real = q/(T – T_s). As known, the critical radius $R_{\mathit{cr}}$ of a bubble growth depends on the overheat of a boundary layer $\mathrm{\Delta }{T}_S=T_{\mathit{sf}}-T_S$, which is determined as [4]:

$$R_{\mathit{cr}}=\frac{2\sigma T_S}{r^{\ast }{\rho }^{"}\mathrm{\Delta }{T}_S},$$E1

Note that $R_{\mathit{cr}}$ is critical radius of a bubble that is capable of growing and functioning; $\sigma$ is surface tension (N/m); $T_S$ is saturation (boiling) temperature; r* is latent heat of evaporation (J/kg); ${\rho }^{"}$ is vapor density (kg/m³); and $\mathrm{\Delta }{T}_S=T_{\mathit{sf}}-T_S$ is wall overheat. HTC related to difference $\mathrm{\Delta }{T}_S=T_{\mathit{sf}}-T_S$ is known as a real HTC and it is called unreal (effective) if it is related to $\mathrm{\Delta }{T}_m=T_{\mathit{sf}}-T_m.$ Here, T_sf is wall temperature and T_m is bath temperature. Effective HTC is used only for approximate core cooling rate and core cooling time evaluation and cannot be used for correct temperature field calculation. There is a big difference between the real and effective HTCs. Table 1 shows the difference between real and effective HTCs depending on bath temperatures T_m.

Temperature Tm,${}^{o}C$	15	20	30	40	60	80	90	95
${\alpha }_{\mathit{\text{real}}}/{\alpha }_{\mathit{eff}}$	7.07	6.71	6.00	5.29	3.86	2.43	1.71	1.26
${\alpha }_{\mathit{\text{real}}}/{\alpha }_{\mathit{eff}}$, %	707	671	600	529	386	243	171	126

Table 1.

Ratio of real HTC to effective HTC versus temperature of water.

As seen from Table 1, effective HTC is decreased up to 700%, as compared with the real HTC. That is why the cooling process during transient nucleate boiling was considered as the slow cooling that requires the use of powerful pumps or powerful rotating propellers to make the cooling process more intensive via strong agitation of the quenchant.

3. Finite initial heat flux density to be compared with its critical value q_cr1

It was accepted by worldwide community that during quenching steel from high temperatures in cold fluid, three stages of cooling are always present: film boiling, transient nucleate boiling, and convection. It is supported by the conventional law of Fourier (2):

$$q=-\lambda \frac{\partial T}{\partial r}$$E2

During immersion of a heated steel part into cold liquid, at the very beginning of cooling $q\to \infty .$ It means that at the very beginning, initial heat flux density always accedes to the first critical heat flux density q_cr1, resulting in developed film boiling process. However, Lykov [5] considered modified law of Fourier (3) resulting in finite heat flux density:

$$q=-\lambda \frac{\partial T}{\partial r}-{\tau }_r\frac{\partial T}{\partial \tau }$$E3

It was shown by him that modified law of Fourier generates hyperbolic heat conductivity eq. (4) [5]:

$$\frac{\partial T}{\partial \tau }+{\tau }_r\frac{{\partial }^{2}T}{\partial {\tau }^2}=\mathit{\text{adiv}}\left(\mathit{\text{gradT}}\right)$$E4

Here, $\lambda$ is thermal conductivity of solid material in W/mK and $\tau$_r is relaxation time.

For the first time, hyperbolic heat conductivity Eq. (4) with the appropriate boundary and initial conditions was solved analytically by authors [6] and it was shown that initial heat flux density is finite value which can be far below the first critical heat flux density q_cr1. It means that the film boiling process is absent completely when quenching steel parts from high temperatures in cold fluids. Such conclusion was formulated from the point of view of mathematics. One can formulate the same conclusion from the point of view of physics. Really, at the very beginning of cooling the cold fluid must be heated to the boiling point of a liquid and then overheated to initiate the nucleate boiling process. During this period of time, the surface of the steel part decreases, thereby creating a temperature gradient that results in finite initial heat flux density. At that moment of time, heat flux density is compared with the first critical heat flux density. If it is below q_cr1, any film boiling process is completely absent. To support such conclusion, it makes sense to consider experiments of French [3] presented by Table 2.

Size, Inches, (mm)				Time (s)
	700°C	600°C	500°C	400°C	300°C	250°C	200°C	150°C
0.25″; (6.35)	0.027	0.037	0.043	0.051	0.09	0.15	0.29	0.69
0.5″; (12.7)	0.028	0.042	0.058	0.071	0.11	0.15	0.26	0.60
4.75″; (120.6)	0.043	0.066	0.09	0.12	0.17	0.21	0.29	0.95

Table 2.

Time required for the surface of steel spheres of different sizes to cool to different temperatures when quenched from 875°C in 5% water solution of NaOH at 20°C agitated with 0.914 m/s [3].

As seen from Table 2, surface cooling curves for spherical probes of sizes such as 6.35 mm, 12.7 mm, and 120 mm are practically similar and cooling short time for all of them is almost the same.

Similar results on drastic decrease of surface temperature, which drops within short time almost to the boiling point of a quenchant, were obtained by different authors (see Figures 1–3).

Figure 1.

Surface (1) and core (2) temperature curves versus time during quenching cylindrical probe 20 mm in diameter in low concentration of water polymer solution at 20°C [7].

Figure 2.

Cooling curves obtained in Idemitsu Kosan Co., Ltd. Lab (Japan) for cylindrical specimen of 28 mm diameter and 112 mm length when quenching in water flow of 1.5 m/s at 20°C [8].

Figure 3.

Cooling curves versus time during quenching cylindrical probe 50 mm diameter in 14% NaCl water solution at 23°C [9].

In Figures 1–3, the surface temperature of probes after initial drastic decrease maintains at the level of boiling point of a liquid and is called the self-regulated thermal process (SRTP).

Figure 4 presents a comparison of experimental core cooling curve with core cooling curve obtained for constant surface temperature during self -regulated thermal process.

Figure 4.

A comparison of experimental cooling curve with calculated cooling curve when surface temperature during quenching 50 mm probe in water salt solution is constant.

Table 3 presents initial temperature $T_I$ and temperature T_II at the end of self-regulated thermal process (SRTP) versus thickness of the stainless probes.

Diameter or thickness in mm	$T_I$, ${}^{o}C$	$T_{II}$, ${}^{o}C$	$\left(T_I+T_{II}\right)/2$
20	126	107.2	117
40	121	107.2	114
50	120	107.2	113.6
60	119	107.2	113
80	117.3	107.2	112
100	116	107.2	111.5
120	115.4	107.2	111

Table 3.

Initial temperature ${\mathrm{T}}_{\mathrm{I}}$ and temperature T_II at the end of self-regulated thermal process (SRTP) versus thickness of the stainless probes.

The average surface temperature for a probe 50 mm in diameter during numerical calculation within the self-regulated thermal process was approximately equal to 114°C (see Figure 3 and Table 3). Error due to averaging of surface temperature is rather small and is equal to −0.47% (see Table 4).

${\tau }_{nb},s$	Tcenter, ${}^{o}C$; experiment	$\overline{T}$center,${}^{o}C$; calculation	$\epsilon ,\%$
0	850	850	0
5	847.5	847.5	0
10	797.2	797.06	−0.016
15	708.2	707.6	−0.085
20	620	619	−0.16
25	541	539.5	−0.3
30	472.5	470.8	−0.36
35	414	412.3	−0.41
40	364.7	362.9	−0.49
45	323	321.2	−0.56
50	287.7	286.3	−0.47
55	258	257.23	−0.30
60	233.7	233	−0.30
65	210.8	212.9	+0.99
70	195.7	196.1	+0. 22
75	181	182.5	+0.83

Table 4.

Errors at the core of a cylindrical probe 50 mm diameter generated by replacing the real surface temperature during SRTP with average constant surface temperature.

Based on unusual characteristics of SRTP, intensive quenching processes IQ-2 and IQ-3 were developed. The IQ-2 technological process explores transient nucleate boiling (see Figure 5a), while transient nucleate boiling process is absent while performing IQ-3 technology (see Figure 5b). It is very important to know how much technological process IQ-3 differs from technological process IQ-2, which is essentially cheaper.

Figure 5.

The temperature difference between IQ-2 and IQ-3 processes when quenching a cylindrical steel probe 50 mm in diameter in an agitated water solution at 20°C and in condition when HTC is infinity: a) it is an IQ-2 process and b) it is a IQ-3 process.

As one can see from Figure 5, the temperature difference between IQ-2 and IQ-3 processes in core cooling curve change is insignificant (see Table 5).

Temperature interval	850–500°C	850–400°C	850–300°C	700–400°C
Cooling time for IQ-3 process in seconds	25.8	32.8	37	18
Cooling time for IQ-2 process in seconds	27.6	36	46.8	20.7

Table 5.

Core cooling time difference between IQ-2 and IQ-3 processes when transient nucleate boiling and direct convection occur during quenching.

This fact opens the possibility of approximate core cooling time calculation of different steel parts using average values of Kondrta’ev numbers Kn (see Figure 6).

Figure 6.

Real Kn_nb and effective Kn_conv Kondrat’ev numbers versus time when quenching a 50 mm cylindrical sample in a low concentration of water salt solution at 20°C.

Figure 6 presents real Kn_nb and effective Kn_conv Kondrat’ev numbers as a result of the calculation of core cooling curves for a cylindrical probe 50 mm diameter quenched in still fluid with convective HTC 548 W/m²K. Convective biot number for such condition is equal to 0.6 and duration of transient nucleate boiling, according to [10], is equal to 72.6 s. Kondrat’ev numbers Kn_nb and Kn_conv were calculated by the well-known equation presented in [10, 11] on the basis of solving the inverse problem for calculating HTCs as α_r = q/(T – T_s) and as α_eff = q/(T – T_m). Note that effective HTC is used only for core cooling time evaluation (see Eq. (5)):

$$\tau =\left[\frac{{\mathit{kBi}}_V}{2.095+3.867{\mathit{Bi}}_V}+ln\frac{T_o-T_m}{T-T_m}\right]\cdot \frac{K}{\mathit{aKn}}$$E5

Here,

$\tau$ is cooling time in seconds; k = 1, 2, 3 for plate, cylinder accordingly; Bi_V is generalized Biot number; T_o is initial temperature; T_m is bath temperature; K Kondrat’ev form factor; a is thermal diffusivity of steel; and Kn is dimensionless Kondrat’ev number.

It is also used for cooling rate v evaluation (see Eq. (6)):

$$v=\frac{\mathit{aKn}}{K}\left(T-T_m\right)$$E6

According to Figure 6, the average effective Kn_conv = 0.625 while real Kn_nb = 0.93. It means that real generalized Biot number Bi_V = 10 [12].

There are many experimental data and databases related to effective HTCs, which are mainly used in the heat-treating industry for recipes development. That is why this issue is discussed here to be able to use cost-effective HTCs obtained by different authors. The real HTCs and real Kondrat’ev numbers are used for calculation of temperature field and residual stresses. In the last decade, it was possible due to the absence of film boiling process. The problem of elimination of film boiling process is solved effectively by the next three main approaches:

• The use of water salt solutions of optimal concentration as a quenchant where the first critical heat flux density is maximal [11].

• The use of low concentration in water of inverse solubility polymers to decrease initial heat flux density due to creation of the thin surface insulating layer (see Eq. (7)).

$$q_{\mathit{\text{in}}}=\frac{q_o}{\left(1+2\frac{\delta }{R}\frac{\lambda }{{\lambda }_{\mathit{\text{coat}}}}\right)}$$E7

Here, q_in is initial heat flux density of cylindrical probe covered by polymeric layer; q_o is initial heat flux density of cylindrical probe free of polymeric layer; $\delta$ is thickness of polymeric layer; R is radius of cylindrical probe; $\lambda$ is thermal conductivity of steel; and ${\lambda }_{\mathit{\text{coat}}}$ is thermal conductivity of insulating layer.

• The use of resonance effect generated by hydrodynamic emitters to destroy any film boiling process during quenching steel in cold fluids [13].

4. Fundamentals of transient nucleate boiling processes

When any film boiling is completely absent due to q_in < q_cr1, one can formulate three very important for the practice characteristics of the transient nucleate boiling process. They are:

1. For a given condition of cooling in cold fluid, the duration of establishing developed nucleate boiling is almost the same, independently of the form and size of the steel part. It can be explained by an extremely high heat exchange during shock boiling which lasts for a very short time, thereby creating a huge temperature gradient in a very thin surface layer. Core temperature during this time is not affected by shock boiling at all and cooling process at the very beginning is considered as a cooling of semi-infinity domain. It is happening because the speed of heat distribution is a finite value and during this short time of cooling, the heat does not reach layers located at the core of steel parts (it follows from Eq. 3 and Eq. 4).

2. The surface temperature of a steel part, beginning from the start of full nucleate boiling establishing, maintains at the level of boiling point of the fluid insignificantly differing from it. The so-called self-regulated thermal process is established. The initial temperature of self-regulated thermal process T_I can be evaluated as:

   $$T_I=T_S+{\vartheta }_I{,}^{o}C$$E8

   where

   $${\vartheta }_I=0.293\cdot {\left[\frac{2\lambda \left({\vartheta }_o-{\vartheta }_I\right)}{R}\right]}^{0.3}$$E9

   The temperature of self-regulated thermal process T_II at the end of nucleate boiling is evaluated as:

   $$T_{\mathrm{II}}=T_S+{\vartheta }_{\mathit{II}}{,}^{o}C$$E10

   where

   $${\vartheta }_{\mathit{II}}=0.293\cdot {\left[{\alpha }_{\mathit{\text{conv}}}\left({\vartheta }_{\mathit{II}}+{\vartheta }_{\mathit{uh}}\right)\right]}^{0.3}$$E11

   ${\vartheta }_I{,}^{o}C$ is overheat of a boundary layer at the beginning of SRTP; ${\vartheta }_{\mathit{II}}$ is overheat of a boundary layer at the end of SRTP.

   It is possible to use average temperature $\left(T_I+T_{\mathit{II}}\right)/2$ for approximately calculating temperature fields and residual stresses during quenching of steel parts in fluids when film boiling is completely absent with the accuracy <1% .

3. As a result of long-lasting accurate experiments and appropriate analytical solutions, for the fixed initial temperatures T_o and T_m, the author [14] has formulated very important for the practice the main characteristic of the transient nucleate boiling process. It says that duration of transient nucleate boiling is directly proportional to squared thickness of steel part, inversely proportional to thermal diffusivity of material, depends on the form of steel part, and convective Biot number Bi. The generalized equation for such statement can be mathematically formulated as [14]:

   $${\tau }_{\mathit{nb}}=\mathrm{\Omega }{k}_F\frac{D^2}{a}$$E12

Here, ${\tau }_{\mathit{nb}}$ can be considered as a width of noise generated by vapor bubbles, which is equal to its duration measured in seconds; $\mathrm{\Omega }$ is dimensionless parameter depending on convective HTC; $k_F$ is dimensionless form coefficient; D is thickness of steel part in m; and a is thermal diffusivity of steel in m² s⁻¹.

For the fixed T_o = 850°C and T_m = 20°C, the dimensionless coefficient $\mathrm{\Omega }$, presented by Figure 7, depends only on convective Biot number Bi (Table 6).

Figure 7.

Value Ω versus convective Biot number Bi [15].

Form of sample	$k_F$
Slab	0.1013
Cylinder	0.0432
Sphere	0.0253
Cylinder Z = D	0.0303
Cube	0.0338

Table 6.

Dimensionless form coefficients $k_F$ for different forms of steel parts.

The heat transfer coefficients (HTCs) during transient nucleate boiling process were evaluated using Tolubinsky’s equation [4]:

$$\frac{\alpha }{\lambda }\sqrt{\frac{\sigma }{g\left(\rho -{\rho }^{"}\right)}}=75\cdot {\left(\frac{q}{r^{•}{\rho }^{"}{W}^{"}}\right)}^{0.7}\cdot {\left(\frac{\nu }{a}\right)}^{-0.2}.$$E13

According to Tolubinsky equation [4], the real HTC during nucleate boiling is calculated from the rewritten Eq. (14):

$$\alpha =75\lambda {\left[\frac{g\left({\rho }^{\text{'}}-{\rho }^{″}\right)}{\sigma }\right]}^{0.5}{\left(\frac{a}{v}\right)}^{0.2}{\left(\frac{1}{r^{\ast }{\rho }^{″}{w}^{″}}\right)}^{0.7}\cdot q^{0.7}$$E14

or

$$\alpha ={\mathit{cq}}^{0.7}$$E15

where.

$c=75{\lambda }^{\text{'}}{\left[\frac{g\left({\rho }^{\text{'}}-{\rho }^{″}\right)}{\sigma }\right]}^{0.5}\cdot {\left(\frac{a}{v}\right)}^{0.2}{\left(\frac{1}{r^{\ast }{\rho }^{″}{w}^{″}}\right)}^{0.7}$; $W^{"}$ = d_of .

Here, $\alpha$ is the real HTC during nucleate boiling process in W/m² K; $\lambda$ is thermal conductivity of liquid in W/mK; g is gravitational acceleration in m/s²; $\rho$ is liquid density in kg/m³; $\rho$” is vapor density in kg/m³; q is heat flux density in W/m²; $W^{"}$ is vapor bubble growth rate in m/s; is kinematic viscosity in m²/s; $a$ is thermal diffusivity of liquid in m²/s; $d_o$ is diameter of a bubble in m; and f is frequency of a bubble departure in Hz.

Calculations of HTCs were made for maximal critical heat flux density of water salt solution which was equal to 15 MW/m². Dimensionless correlations of Tolubinsky and Shekriladze [4, 16] were used for evaluation of HTCs, which are presented in Table 7.

Temperature of water solution	Tolubinsky	Shekriladze	Average
10	152,248	176,546	164,397
20	193,929	243,641	218,785
40	224,989	241,615	233,302
60	271,273	271,323	271,298

Table 7.

Real HTCs in W/m²K during nucleate boiling process depending on the temperature of water solution when heat flux density is 15 MW/m².

According to Tolubisky equation, cooling during transient nucleate boiling process is very intensive, even in still fluid if any film boiling is completely absent (see Table 7).

Heat flux density during initial time of quenching in cold fluids reaches almost 20 MW/m² when film boiling is absent (see Figure 8a and b). It is comparable with the first critical heat flux density of cold water when q_cr2/q_{cr1 =} 0.05.

Figure 8.

Heat flux density variation versus time during quenching cylindrical specimens 25 band 50 mm in water salt solution of optimal concentration at 20°C: a – diameter 25 mm.

Heat transfer coefficients during transient nucleate boiling process can be easily evaluated if heat flux density during nucleate boiling is known (see Figure 8a and b).

According to the investigation of the author [4], during the extremely fast cooling produced by shock boiling, the first critical heat flux density $q_{\mathit{cr}1}$ is very large, because the ratio (16):

$$\frac{q_{\mathit{cr}2}}{q_{\mathit{cr}1}}=0.05$$E16

which during conventional cooling is five times larger (see Eq. (17) published in [4, 17]:

$$\frac{q_{\mathit{cr}2}}{q_{\mathit{cr}1}}=0.2$$E17

According to authors [4, 17], the first critical heat flux density q_cr1 for water at 20°C is equal to 5.9 MW/m² and for water at 10°C is equal to 6.5 MW/m^2.. During shock boiling, the mentioned critical heat flux densities reach values 29.5 MW/m² and 32.5 MW/m² that follow from comparing Eq. (16) and Eq. (17). It means that any film boiling is completely absent during quenching in slow agitated cold water at 10–20°C. The conclusion made is in good agreement with the experimental data (see Figures 1–3). Since any film boiling process in many cases is completely absent, one can consider fundamental characteristics of transient nucleate boiling process as the main factors during quenching process in cold fluids. The transient nucleate boiling process exists independently of the will of people. It is happening when meteorites fall into oceans. They produce noise generated by tiny vapor bubbles. The same is happening during volcanos’ activity located in oceans and near seas. Transient nucleate boiling can be:

• Seen due to growth of bubbles and their departure.

• Heard due to noise produced by tiny bubbles.

• Smelt due to vapor that has a specific smell.

• Felt due to quench tanks’ vibration during boiling.

Discovered characteristics of transient nucleate boiling process create a stable basis for:

• Making intensive quenching process into a mass production since there is no need to design costly quench tanks equipped with the powerful propellers and tanks.

• Designing absolutely new processes called austempering and martempering technological processes produced in cold water and water solutions, instead of melted salts and alkalis [18].

• Design of original intensive quenching process in water salt solutions of optimal concentration negatively charged to guarantee absence of any film boiling process during quenching [19].

• Use of resonance effect to destroy any film boiling during quenching in cold fluids [13].

• Design the method and apparatus for control quality of hardened steel parts [20].

5. Universal correlation for heating and cooling time evaluation during steel quenching

There are numerous computer codes for numerical calculation of cooling time and cooling rate of any steel part’s configuration. The author of this chapter proposed a generalized universal equation for heating and cooling time evaluation while heat treating steel parts [21], which has a very simple form:

$${\tau }_{\mathit{eq}}={\overline{E}}_{\mathit{eq}}\frac{K}{\mathit{aKn}}$$E18

Here, $E_{\mathit{eq}}$ is specified if value N = (T_o – T_m)/(T – T_m) is known (see Table 8). Table 8 provides $E_{\mathit{eq}}$ value depending on N, which varies within 1.5–1000. Kondrat’ev form coefficients K (see Table 9) are provided by authors [11, 12, 22]. Kondrat’ev number Kn is calculated as:

						BiV$=0.1$
N	1.5	2	2.5	3	3.5	4	4.5	5	10	100	1000
Plate	0.44	0.73	0.96	1.14	1.29	1.43	1.54	1.65	2.38	4.64	6.95
Cylinder	0.49	0.77	1.00	1.18	1.33	1.47	1.58	1.69	2.42	4.68	6.99
Sphere	0.55	0.81	1.04	1.22	1.37	1.51	1.62	1.73	2.46	4.72	7.02
						${\mathit{Bi}}_V=0.3$
Plate	0.50	0.78	1.01	1.19	1.34	1.48	1.60	1.70	2.39	4.70	7.00
Cylinder	0.59	0.88	1.10	1.28	1.44	1.57	1.69	1.79	2.49	4.79	7.09
Sphere	0.68	0.97	1.19	1.37	1.53	1.66	1.78	1.88	2.58	4.88	7.18
						${\mathit{Bi}}_V=0.5$
Plate	0.53	0.817	1.04	1.22	1.38	1.51	1.63	1.73	2.43	4.73	7.03
Cylinder	0.65	0.94	1.16	1.35	1.50	1.63	1.75	1.86	2.55	4.85	7.16
Sphere	0.78	1.07	1.29	1.47	1.62	1.76	1.88	1.98	2.67	4.98	7.28
						${\mathit{Bi}}_V=1$
Plate	0.57	0.86	1.08	1.27	1.42	1.55	1.67	1.78	2.47	4.77	7.07
Cylinder	0.74	1.03	1.25	1.44	1.59	1.72	1.84	1.94	2.64	4.94	7.25
Sphere	0.91	1.20	1.42	1.60	1.76	1.89	2.01	2.11	2.80	5.11	7.41
						${\mathit{Bi}}_V=2$
Plate	0.61	0.90	1.12	1.30	1.46	1.59	1.71	1.81	2.51	4.81	7.11
Cylinder	0.81	1.1	1.32	1.50	1.66	1.79	1.91	2.02	2.71	5.01	7.33
Sphere	1.01	1.30	1.52	1.71	1.86	1.99	2.11	2.22	2.91	5.21	7.51
						${\mathit{Bi}}_V=5$
Plate	0.63	0.92	1.14	1.32	1.48	1.61	1.73	1.83	2.53	4.83	7.13
Cylinder	0.86	1.15	1.37	1.55	1.71	1.84	1.96	2.07	2.76	5.06	7.36
Sphere	1.10	1.38	1.61	1.80	1.94	2.08	2.20	2.30	3.00	5.29	7.58
						${\mathit{Bi}}_V=\infty$
Plate	0.64	0.93	1.15	1.33	1.49	1.62	1.74	1.84	2.54	4.84	7.15
Cylinder	0.87	1.16	1.38	1.56	1.72	1.85	1.97	2.08	2.77	5.07	7.38
Sphere	1.11	1.39	1.62	1.80	1.95	2.09	2.20	2.31	3.00	5.30	7.60

Table 8.

Coefficients $E_{\mathit{eq}}$ depending on dimensionless value N that varies from 1.5 to 1000, taking into account different values of generalized Biot numbers Bi_V [19].

Form of sample	Kondrat’ev form factor K
Slab	$K=\frac{L^2}{9.87}$
Cylinder	$K=\frac{L^2}{5.783}$
Sphere	$K=\frac{R^2}{9.87}$
Cylinder Z = D	$K=\frac{R^2}{8.2505}$
Cube	$K=\frac{L^2}{29.61}$

Table 9.

Kondrat’ev form factor K for different forms of steel parts.

$$\mathit{Kn}=\frac{{\mathit{Bi}}_V}{{\left({{\mathit{Bi}}_V}^2+1.437{\mathit{Bi}}_V+1\right)}^{0.5}}$$E19

Here

$${\mathit{Bi}}_V=\frac{\alpha }{\lambda }K\frac{S}{V}$$E20

Example 1: Cylindrical sample 50 mm diameter, made of AISI (American Iron and steel institute) 1040 steel, is quenched from 860°C in low concentration of still water salt solution at 20°C [23, 24]. Maximal surface compression residual stress reaches its maximal value at the moment when core temperature of cylinder reaches 430°C. Calculate cooling time from 860–430°C to provide further self-tempering process of surface layers and obtain high surface compression residual stress. For the given process, N = (860–100°C)/(430–100°C) = 2.3. According to Table 7, for Bi_V = 10 and N = 2.3, E_eq = 1.36. Kondrat’ev form factor K = R²/5.783 = 108.1 × 10⁻⁶ m². Average value of thermal diffusivity of AISI 1040 steel a = 5.4 × 10⁻⁶ m²/s. According to Eq. (18), $\tau$ = (1.3 × 108.1 × 10⁻⁶ m²)/(5.4 × 10⁻⁶ m²/s × 0.93) = 29.3 s. Approximately $\tau$= 30 s.

Example 2: Cylindrical forging 50 mm at its end, made of AISI 1040 steel, has a temperature 950°C. It is quenched intensively in spray water salt solution of low concentration creating condition when ${\mathit{Bi}}_V$ tends to infinity. Of still water salt solution at 20°C. Calculate cooling time from 970–480°C to provide further self-tempering process of surface layers and obtain high surface compression residual stress and improved mechanical properties due to high temperature thermomechanical treatment. For given condition, N = (970–20°C)/(480–20°C) = 2. According to Table 7, for Bi_V equal infinity and N = 2, E_eq = 1.16. Kondrat’ev form factor K = R²/5.783 = 108.1 × 10⁻⁶ m². Average value of thermal diffusivity of AISI 1040 steel for interval temperatures from 500–950°C is a = 5.6 × 10⁻⁶ m²/s. According to Eq. (18), $\tau$ = (1.16 × 108.1 × 10⁻⁶ m²)/(5.6 × 10⁻⁶ m²/s × 1) = 22.4 s.

Forging requires a shorter cooling time because it is sensitive to crack formation due to overheating.

6. Conclusions

1. For given condition of cooling, the time of transient nucleate boiling establishment is approximately the same independently of the form and size of the quenched steel part. It is explained by an extremely forced heat transfer process which can be considered as a cooling of semi-infinity domain.

2. Transient nucleate boiling is considered as a self-regulated thermal process when surface temperature maintained at the level of boiling point of a fluid insignificantly differs from its boiling point. In this case, the real surface temperature can be replaced by its average value.

3. Length of transient nucleate boiling process is directly proportional to squared thickness of steel part, inversely proportional to thermal diffusivity of material, and depends directly on the form of steel part and convective Biot number Bi if initial temperatures T_o and T_m are fixed.

4. The formulated principles are the basis for design of new technologies and their accurate control. The mentioned principles provide correct temperature fields’ calculation of quenched steel parts without performing, in many cases, painstaking costly experiments.