An improved fluid convection solution in conventional grinding

Abstract

It is important in grinding for the fluid to remove heat from the grinding contact zone to avoid thermal damage to the workpiece surface and/or subsurface layers. The cooling effect of grinding fluid can be quantified by a convection heat convective heat transfer coefficient acting in the grinding zone. This article presents values of the convection heat transfer coefficient based on published experimental results for measured grinding temperatures. This article also presents a new convective heat transfer model based on principles of applied fluid dynamics and heat transfer. Predicted values for the convection heat transfer coefficient calculated from the model are compared with results from experiments obtained under a range of grinding conditions and with experimental data obtained from published literature. The results demonstrate that the new convection heat transfer coefficient model improves the accuracy of prediction and helps explain the variation in the value of convection heat transfer coefficient under varying process conditions. The results also show that convection efficiency strongly depends on the grinding wheel speed, grinding arc length and fluid properties. This new model can be readily introduced into current thermal models of the process used to predict and control grinding temperatures.

Introduction

Cooling by process fluids has an important influence on the removal of heat from the grinding zone and reduces the risk of thermal damage to the workpiece. The cooling efficiency is quantified by a convection heat transfer coefficient (CHTC) acting in the grinding zone. Heat is also removed from the grinding zone via conduction to the grinding chips, conduction to the wheel grains and conduction to the workpiece.^1–3 Determination of the proportion of heat energy removed by each has been the subject of numerous studies in grinding thermal analyses.^4,5 Convection to the fluid varies the proportions of heat available for distribution to other sinks, and hence, correctness in CHTC modelling is needed for accuracy in temperature prediction.

A model for the CHTC is developed based on applied fluid dynamics and heat transfer. The model is referred to as the laminar flow model (LFM). The effect of varying process conditions on the value of CHTC is investigated. The key parameters varied were depth of cut, wheel speed and work speed, fluid type and fluid delivery conditions, wheel type and diameter. The major process considered was plane surface grinding. Results of this analysis are compared with results obtained from experiments and case studies in the literature. The CHTC from case studies was obtained from inverse solution of the measured grinding temperatures reported and from application of a fluid wheel model (FWM). The LFM predicts lower CHTC values than the FWM and is described below in more detail. In the narrow range of maximum grinding temperatures between 110 °C and 165 °C, the experimental values of CHTC were closer to predictions from the LFM. However, this is the temperature range where boiling of water would be expected and could be a cause of reduced CHTC values. At lower grinding temperatures, higher values of CHTC were found.

Thermal studies

Thermal modelling is studied as a predictive process control tool^6,7 to predict the grinding temperature. Control is exercised frequently in real time to avoid the risk or occurrence of heat damage to the workpiece.

Heat partition model

It is common for heat transfer to be expressed in terms of heat flux as this enables convenient calculation of conduction and convection quantities. The grinding total heat q_t generated in the contact zone is distributed to four heat sinks: workpiece, wheel (abrasive), chips and fluid^3,8

$${\text{q}}_{\text{t}}={\text{q}}_{\text{w}}+{\text{q}}_{\text{s}}+{\text{q}}_{\text{ch}}+{\text{q}}_{\text{f}}$$

(1)

where ${\text{q}}_{\text{w}}$ is the grinding heat transferring into the workpiece, ${\text{q}}_{\text{s}}$ is the grinding heat transferring into the abrasive, ${\text{q}}_{\text{ch}}$ is the grinding heat transferring into the chips, ${\text{q}}_{\text{f}}$ is the grinding heat convective transferring into the grinding fluid and ${\text{q}}_{\text{t}}$ is the total grinding heat flux.

Flux values are defined as the rate of heat flow divided by the grinding contact area as follows. The grinding total heat, obtained from power or force measurement, is

$${\text{q}}_{\text{t}}=\frac{{\text{F}}_{\text{t}}{\text{v}}_{\text{s}}}{\text{b}{\text{l}}_{\text{c}}}$$

(2)

where ${\text{F}}_{\text{t}}$ is the tangential grinding force, ${\text{v}}_{\text{s}}$ is the wheel speed, $\text{b}$ is the grinding width and ${\text{l}}_{\text{c}}$ is the real grinding contact length.

Flux ${\text{q}}_{\text{f}}$ initially enters the workpiece and then re-emerges from the workpiece in the convection cooling process, giving rise to the partition between the work and grain of

$${\text{R}}_{\text{ws}}=\frac{{\text{q}}_{\text{wg}}}{{\text{q}}_{\text{wg}}+{\text{q}}_{\text{s}}}=\frac{{\text{q}}_{\text{w}}+{\text{q}}_{\text{f}}}{{\text{q}}_{\text{w}}+{\text{q}}_{\text{f}}+{\text{q}}_{\text{s}}}$$

(3)

Thermal model

From equation (1)

$${\text{q}}_{\text{t}}-{\text{q}}_{\text{ch}}={\text{q}}_{\text{w}}+{\text{q}}_{\text{s}}+{\text{q}}_{\text{f}}$$

(4)

So that by equations (3) and (4)

$${\text{R}}_{\text{ws}}({\text{q}}_{\text{t}}-{\text{q}}_{\text{ch}})={\text{q}}_{\text{w}}+{\text{q}}_{\text{f}}=({\text{h}}_{\text{w}}+{\text{h}}_{\text{f}})({\text{T}}_{\text{max}}-{\text{T}}_{\text{a}})$$

(5)

Then

$${\text{T}}_{\text{max}}=\frac{{\text{R}}_{\text{ws}}({\text{q}}_{\text{t}}-{\text{q}}_{\text{ch}})}{{\text{h}}_{\text{w}}+{\text{h}}_{\text{f}}}+{\text{T}}_{\text{a}}$$

(6)

where T_max − T_a represents temperature rise above ambient. T_max is taken to be the maximum background temperature rise in the grinding contact zone. Flash temperatures also occur at individual grain contacts that may be much higher than the background temperature rise. However, the area under the flash points is very much smaller, typically 1% of the total contact area. The grinding fluid is assumed to contact the entire grinding contact area. It is therefore the background temperature rise that is always employed for studies of fluid convection.

The work–grain partition ratio according to Rowe et al. ⁹ is expressed as

$${\text{R}}_{\text{ws}}={[1+\frac{0.974·{\text{k}}_{\text{g}}}{{\beta }_{\text{w}}·\sqrt{{\text{r}}_0·{\text{v}}_{\text{s}}}}]}^{-1}$$

(7)

where ${\text{r}}_0$ is the effective contact radius of the abrasive grain, ${\beta }_{\text{w}}=\sqrt{{\text{k}}_{\text{w}}·{\rho }_{\text{w}}·{\text{c}}_{\text{w}}}$ is the thermal property of the workpiece, ${\text{k}}_{\text{w}}$ is the thermal conductivity of the workpiece, ${\rho }_{\text{w}}$ is the density of the workpiece, ${\text{c}}_{\text{w}}$ is the specific heat capacity of the workpiece and ${\text{k}}_{\text{g}}$ is the conductivity of the abrasive.

A conduction factor for the net heat flow into the workpiece ${\text{h}}_{\text{w}}$ is expressed as

$${\text{h}}_{\text{w}}=\frac{{\beta }_{\text{w}}}{\text{C}}.\sqrt{\frac{{\text{v}}_{\text{w}}}{{\text{l}}_{\text{c}}}}$$

(8)

The factor C has a maximum value of 1.13 for a uniform heat flux distribution or 1.06 for a triangular heat flux, which is the preferred solution to match the distribution of material removal rate in the contact zone. The factor C is usually close to 1 for conventional work speeds and shallow cuts, ⁸ although lower values apply for deep grinding. ³

Heat to the chip is expressed as

$${\text{q}}_{\text{ch}}={\rho }_{\text{w}}·{\text{c}}_{\text{w}}·{\text{T}}_{\text{ch}}·\left(\frac{{\text{a}}_{\text{e}}·{\text{v}}_{\text{w}}}{{\text{l}}_{\text{c}}}\right)$$

(9)

where ${\text{v}}_{\text{w}}$ is the table speed, ${\text{T}}_{\text{ch}}$ is the material melting temperature, ${\text{a}}_{\text{e}}$ is the real cutting depth, ${\rho }_w$ is the density of the workpiece, ${\text{c}}_{\text{w}}$ is the specific heat capacity of the workpiece and ${\text{T}}_{\text{a}}$ is the ambient temperature.

Grinding arc length

In predicting the temperature of the workpiece, it is very important to determine the effective length l_c of the heat source over which the energy transfers into the workpiece. It has been found that the real contact length in grinding for vitrified and organic bonded wheels is much greater than the geometric contact length, l_g. Theoretical solutions for real contact length were derived by Rowe et al. ¹⁰

The real contact length can be expressed as

$${\text{l}}_{\text{c}}=\sqrt{8{\text{R}}_{\text{r}}^2{\text{F}’}_{\text{n}}({\text{K}}_{\text{s}}+{\text{K}}_{\text{w}}){\text{d}}_{\text{s}}+{\text{a}}_{\text{e}}{\text{d}}_{\text{s}}}$$

(10)

where ${\text{F}’}_{\text{n}}$ is the unit normal grinding force, ${\text{d}}_{\text{s}}$ is the wheel diameter, ${\text{R}}_{\text{r}}$ are the values of the roughness factor, K_s are the elastic properties of the wheel in the contact, ${\text{K}}_{\text{s}}=((1-{\nu }_{\text{s}}^2)/\pi {\text{E}}_{\text{s}})$ , K_w are the elastic properties of the material in the contact, ${\text{K}}_{\text{w}}=((1-{\nu }_{\text{w}}^2)/\text{π}{\text{E}}_{\text{w}})$ , ${\text{E}}_{\text{w}}$ and ${\text{E}}_{\text{s}}$ are Young’s moduli of the material and wheel, respectively, and ${\nu }_{\text{w}}$ and ${\nu }_{\text{s}}$ are Poisson’s ratios of the material and wheel, respectively.

From equation (6), it is evident that the convection heat transfer h_f is a critical variable of the process, and knowledge of its value is important for accuracy in workpiece surface temperature prediction.

Convection heat transfer model

In earlier studies, Rowe et al. ⁴ estimated fluid convection based on an ‘FWM’. A key assumption in this case is that a layer of fluid covers the contact zone and moves across the work surface at wheel speed. The CHTC of the FWM based on the sliding heat source is expressed as

$${\text{h}}_{\text{f}}=\frac{{\beta }_{\text{f}}}{\text{C}}\sqrt{\frac{{\text{v}}_{\text{s}}}{{\text{l}}_{\text{c}}}}$$

(11)

where the thermal property of fluid ${\beta }_{\text{f}}=\sqrt{{\text{k}}_{\text{f}}·{\rho }_{\text{f}}·{\text{c}}_{\text{f}}}$ . The factor C has a maximum value of 1.13 for a uniform heat flux distribution or 1.06 for a triangular heat flux. C = 1.06 is preferred for the same reason given for workpiece conduction. C = 1 is a reasonable approximation for purposes of estimation. A slug-flow model by Lavine and Malkin ¹¹ for creep-feed grinding came to a similar conclusion giving a factor of $2/\sqrt{\pi }=1.13$ assuming a uniform heat flux.

Equation (11) suggests that the values of CHTC for fluid cooling depend on wheel speed, real contact length and thermal properties of the fluid. However, it is well known from theory of fluid dynamics ¹² that the dynamic viscosity plays a role in convection, and this is excluded in equation (11). In the analysis of Lin et al., ¹³ a model to estimate CHTC was developed based on the same fluid dynamics and principles of heat conduction. However, in this case, conductivity was omitted.

A new analytical model for CHTC in grinding is proposed to include the following fluid properties: dynamic viscosity, conductivity, density and specific heat capacity. The aim of this work is to study and compare the predictive accuracy of the new model.

Achievable useful flow criterion

The flow of fluid through the grinding zone determines the cooling and lubrication efficiency. Useful flow is defined as the flow that actually enters the grinding contact area. Useful flow is lower than achievable if nozzle flow is low or if the jet is not well positioned. A method of calculating achievable useful flow is required to determine whether useful flow is high enough to remove sufficient grinding heat.

A method was proposed by Morgan et al. ¹⁴ for the calculation of achievable useful flow $\stackrel{·}{\text{Q}}$ based on wheel porosity and grit size. Allowing for the wheel pores to be only half-filled, achievable useful flow may be estimated by

$$\stackrel{·}{\text{Q}}={\text{v}}_{\text{s}}\text{b}\varphi \text{f}\left(\frac{15.2}{\text{M}}\right)$$

(12)

where v_s is the wheel speed, b is the grinding contact width, $\varphi$ is the bulk wheel porosity, f = 0.5 is the filling factor and M is the grit size from the wheel specification.

Equation (12) can be used to predict the value of achievable useful flow for a given wheel and process. This can be used to give guidance on supply flow.

CHTC

If by application of the achievable useful flow criterion, the supply flow is sufficient to remove heat from the grinding contact area, assumptions can be made in order to simplify and facilitate the CHTC analysis.

For conventional operations, fluid velocity ${\text{v}}_{\text{f}}$ is assumed to attain an average wheel velocity through the contact region, for example, ${\text{v}}_{\text{f}}\approx {\text{v}}_{\text{s}}$ . In the case where the wheel is adequately dressed and has uniform surface form, convection of useful flow will occur in the grinding area.

As the flow drags on the workpiece surface, it is slowed at the wall to zero speed. When the fluid progresses in the grinding direction, the depth at which the fluid experiences a drag effect increases due to Newton’s law of viscous shear. The depth builds to a value that increases with grinding direction, and there is a laminar boundary layer. There are inaccuracies built into this physical representation, for example, the flow is almost certainly turbulent within the pores. However, for simplicity of comparison with experiments, the LFM is retained, and it is assumed that the fluid enters the grinding contact zone in a laminar stream. This is an approximation to the physical situation. In the following discussion, temperatures are given relative to ambient temperature for the sake of simplicity.

Assuming steady flow in the main stream at speed v_s, speed at the wall is 0 and increases to v_s at a distance δ from the wall. Similarly, temperature T increases from 0 at the wall and is assumed to attain a maximum T_max at a distance δs from the wall, as shown in Figure 1. In the grinding situation, these boundary temperatures are reversed. The wall temperature in the grinding contact is the temperature T_max, and the stream temperature T_s is assumed to be the ambient temperature T_a = 0. This does not affect the derivation of the basic equation.

Figure 1.

Development of the flow speed and temperature boundary layers along a flat plate.

Heat transfer in steady laminar flow past a flat surface can be modelled as

$${\displaystyle h_x\approx \frac{1}{3}{\text{Pr}}^{1/3}{\mathrm{Re}}_x^{1/2}\frac{k_f}{x}}$$

(13)

In textbooks, the factor is given as 0.332 rather than1/3 = 0.333. The insignificant rounding is introduced here for convenience in the final expression given below as equation (19)

$${\text{Re}}_{\text{x}}=\frac{{\rho }_{\text{f}}{\text{v}}_{\text{s}}\text{x}}{{\text{η}}_{\text{f}}}$$

(14)

$$\text{Pr}=\frac{{\text{C}}_{\text{f}}{\text{η}}_{\text{f}}}{{\text{k}}_{\text{f}}}$$

(15)

$${\text{h}}_{\text{x}}=\frac{{\text{q}}_{\text{x}}}{{\text{T}}_{\text{max}}}$$

(16)

Assuming a constant wall temperature, the convection factor is

$${\text{h}}_{\text{f}}=\frac{1}{{\text{l}}_{\text{c}}}\underset{0}{\overset{{\text{l}}_{\text{c}}}{\int }}{\text{h}}_{\text{x}}\text{dx}=2{\text{h}}_{\text{l}}$$

(17)

Therefore

$${\displaystyle h_f\approx \frac{2}{3}{\text{Pr}}^{1/3}{\mathrm{Re}}_l^{1/2}\frac{k_f}{l_c}}$$

(18)

The value of l_c is the maximum value of x measured along the surface of the plate, that is, the grinding arc length. The wall temperature distribution in grinding arc length is not constant. An approximate expression for average temperature rise is ${\text{T}}_{\text{average}}=\frac{2}{3}{\text{T}}_{\text{max}}$ . Thus, the average temperature rise must replace the maximum temperature rise in equation (16), thereby introducing a further factor of 2/3 to equation (18). Then, the CHTC LFM is

$${\text{h}}_{\text{f}}=\frac{4}{9}{\rho }_{\text{f}}^{1/2}{\text{c}}_{\text{f}}^{1/3}{\text{η}}_{\text{f}}^{-1/6}{\text{k}}_{\text{f}}^{2/3}\sqrt{\frac{{\text{v}}_{\text{s}}}{{\text{l}}_{\text{c}}}}$$

(19)

where the fluid properties are as follows: h_f is the convection coefficient, ${\rho }_{\text{f}}$ is the density, c_f is the specific heat capacity, η_f is the dynamic viscosity and k_f is the thermal conductivity.

The factor 4/9 = 0.444 in equation (19). An almost similar model is obtained assuming an average heat flux along the contact zone rather than an average contact temperature. This assumption was mentioned by Lavine in a review of this article. The factor of 0.444 in equation (19) is then increased to 0.453.

Comparison of predicted CHTC with data from experiments

Experimental data from Morgan et al.,^15,16 Barczak et al. ¹⁷ and Lin et al. ¹³ were entered into the CHTC model and also into the thermal model described above. The real contact length based on equation (10) is used for the thermal analysis. The experimental conditions are listed in Table 1. The thermal properties of steels (EN31, EN8 and M2), cast iron, water-based emulsion and abrasive are listed in Table 2.

Table 1.

Experimental conditions.

Source of data	Morgan et al. 16	Lin et al. 13
Grinding wheel	Aluminium oxide	Aluminium oxide
Coolant	Water-based emulsion	Water-based emulsion
Material	EN31, EN8, M2	Cast iron, M50
Wheel speed vs (m/s)	25, 45	36
Real depth of cut ae (µm)	2.8–7	15–30
Work speed vw (m/s)	0.108, 0.25	0.27
Grinding width b (mm)	20	5

Table 2.

Thermal properties of materials

Parameters	EN31	EN8	M2	CI	Al2O3	Emulsion
Thermal conductivity k (W/(m K))	34	51	24	54	35	0.56
Density ρ (kg/m3)	7815	7845	7860	7300	3980	1000
Specific heat capacity c (J/(kg K))	506	486	506	511	765	4200
Thermal diffusivity α (m2/s)	8.6e−6	13.4e−6	6e−6	14.4e−6	11.5e−6	0.13e−6
Thermal property β (J/(m2 K s0.5))	11,595	13,944	9770	14,150	10,323	1534
Poisson’s ratio	0.29	0.29	0.29	0.26	0.22	N/A
Modulus of elasticity E (N/m2)	210e9	200e9	205e9	160e9	49.6e9	N/A
Viscosity η ((N s)/m2)	N/A	N/A	N/A	N/A	N/A	0.001

CI: cast iron.

Comparison of CHTC values with experimentaldata for water-based emulsion

A key aim of this study is to compare CHTC values calculated from both the newly developed LFM and the FWM with values obtained inversely using data from experiments (EXPT). This is achieved by matching the theoretical and measured grinding temperatures using equation (6) and published case studies data. The CHTC values calculated for the LFM and FWM and the value of CHTC (EXPT) for the cases of grinding cast iron and steel^13,16 are given in Tables 3 and 4, respectively.

Table 3.

Calculated and experimental results of CHTC values.

Source of data	Lin et al. 13 and Morgan et al. 16
Grinding wheel	Aluminium oxide
Coolant	Water-based emulsion
Material	Cast iron
Wheel speed (m/s)	36	36	36	36	36
Work speed (m/s)	0.27	0.27	0.27	0.27	0.27
Real depth of cut (µm)	15	18	23	27	30
Real contact length lc (mm)	3.99	4.36	4.8	5.19	5.47
Tangential force (N)	13	15	18	21	23
Normal force (N)	27	32	37	43	48
Temperature measurement (°C)	126	127	136	147	157
EXPT (W/(m2 K))	37,146	47,217	49,827	51,289	47,519
LFM (W/(m2 K))	46,276	44,269	42,191	40,575	39,523
FWM (W/(m2 K))	145,760	139,360	132,820	127,730	124,420

FWM: fluid wheel model; LFM: laminar flow model.

Table 4.

Calculated and experimental results of CHTC values.

Source of data	Morgan et al. 16
Grinding wheel	Aluminium oxide
Coolant	Water-based emulsion
Material	EN31		EN8	M2
Wheel speed (m/s)	25	45	25	25	45
Work speed (m/s)	0.108	0.25	0.25	0.108	0.25
Real depth of cut (µm)	7	4.8	2.8	7	4.8
Real contact length lc (mm)	3.2	3.0	4.1	3.3	3.2
Tangential force (N)	34	32	82	40	36
Normal force (N)	94	86.5	186	102	100
Temperature measurement (°C)	123	130	160	113	140
EXPT (W/(m2 K))	33,324	72,243	42,840	70,141	84,887
LFM (W/(m2 K))	43,175	59,917	37,927	42,430	58,034
FWM (W/(m2 K))	135,910	188,610	119,390	133,570	182,690

FWM: fluid wheel model; LFM: laminar flow model.

It is very interesting to note that taking the temperature measurement values to calculate the CHTC values (EXPT) for the water-based emulsion, the LFM exhibits better agreement than the FWM over the narrow range of conditions presented. In these conditions, the value of real contact length was 3 mm<l_c<5.5 mm.

The values for CHTC (EXPT) are shown in Figure 2(a) with the predicted values of CHTC for the LFM and FWM. A linear fit is shown through each set of data points. For comparison with experimental results, temperatures are stated relative to 0 °C.

Figure 2.

(a) Comparison of CHTC for EXPT, LFM and FWM. (b) Convection factors based on published ¹³ temperature measurements.

The values and trend line for the LFM agree better with EXPT than those for the FWM. It is evident in this range that convection is reduced as temperature is increased. The CHTC average value is approximately 60,000–70,000 W/m². It is noted that the boiling of fluid can substantially reduce the cooling effectiveness when occurring at a threshold temperature.

Taking a wider range of temperatures, the agreement is less clear. Figure 2(b) shows results for cast iron and M50 steel. At higher temperatures, there is increased evidence of fluid burnout, and the CHTC reduces to approximately zero. This is expected of course. The thermal model requires that h_f is made equal to zero to take account of fluid boiling at temperatures exceeding 180 °C. At low temperatures, it is possible that the CHTC value may be increased compared to the LFM line. However, it is suggested that this temperature range requires further research to test the accuracy of the measurements.

Effect of contact length on thermal predictionin LFM

Real contact length is an important factor in the calculation of grinding temperature. It is difficult to determine the real contact length in any given case as the theoretical models of contact length are derived employing a wide range of variables, the values of which are not available in most of the cases reported in the literature. However, the upper and lower limits of contact length can be analysed. In this context, two contact length values l_c = l_g and l_c = 3l_g are used to predict temperatures and to determine the limiting range for contact length wherein the values of CHTC predicted by the LFM correlate with CHTC (EXPT).

The predicted temperatures for the LFM using the two different contact lengths, l_g and 3l_g, are obtained from equations (6) and (19), and the results are shown in Figure 3. Most of the temperature measurements lie between the temperature predictions for contact lengths of l_g and 3l_g.

Figure 3.

Measured and predicted maximum temperatures.

Case study comparing water- and oil-based grinding fluids

The following are typical calculations for water and oil coolants according to the two models for a wheel speed of 30 m/s and a contact length of 10 mm.

For water:

• k_f = 0.56 W/(mK), ρ_f = 1000kg/m³,c_f = 4200 J/(kgK), η_f = 0.001 (N s)/m²

• β_f = 1600 J/(m² K s^0.5) and G_f = 510.5 J/(m² K s^0.5)

• h_f(FWM) = 87,700 W/(m² K)

• h_f(LFM) = 27,960 W/(m² K)

For oil:

• k_f = 0.14 W/(mK), ρ_f = 900 kg/m³, c_f = 2100 J/(kg K) and η_f = 0.02 (N s)/m²

• β_f = 514.4 J/(m² K s^0.5) and G_f = 87.7 J/(m² K s^0.5)

• h_f(FWM) = 28,175 W/(m² K)

• h_f(LFM) = 4804 W/(m² K)

These values are of the right order of magnitude but rather lower than some experimental estimates based on temperature measurements given below.

Other published convection results for water- and oil-based grinding fluids

Values of convection factor obtained inversely depend on the thermal model employed. Other researchers’ results must be adjusted to match the particular thermal model employed.

Values of fluid convection factor have been estimated from grinding experiments and adjusted as follows. Rowe and Jin ¹⁸ reported average values of 290,000 W/(m²K) for water-based emulsions. Stephenson et al. ¹⁹ reported 23,000 W/(m²K) for neat oil. Applying adjustments for the difference in thermal model employed and the application of real contact length reduces these values to approximately 174,000 and 13,800 W/(m²K). More recent measurements by Jin and Stephenson ⁷ were for grinding 51CrV4 steel with a B252 cubic boron nitride (CBN) wheel at 0.4mm depth of cut. These values similarly adjusted to match the refined model in this article gave the following results:

• Water-based (1): 104,000 W/(m²K) at 50 m/s wheel speed

• Water-based (2): 144,000 W/(m² K) at 50 m/s wheel speed

• Water-based (3): 84,000 W/(m² K) at 50 m/s wheel speed

• Mineral oil: 26,200 W/(m² K) at 50 m/s wheel speed

• Mineral oil: 78,100 W/(m² K) at 146 m/s wheel speed

Results were all obtained in grinding at temperatures well below fluid boiling. Unfortunately, insufficient data are available to conclude whether the new model is an improvement for oil fluids. However, experiments confirmed that the fluid convection factors depend on wheel speed as might be expected.

Conclusions

CHTCs of fluid in the grinding zone have been predicted through principles of applied fluid dynamics and heat transfer. The values of convection coefficient strongly depend on grinding wheel speed, grinding arc length and fluid properties. Experimental results show that the values of the CHTC can be high and of the order of 60,000–70,000 W/(m² K).

The predicted CHTC values of the LFM showed better agreement than the values of FWM with experimental data in the region approaching fluid boiling. Contact length limits l_g and 3l_g can be used to provide boundaries on predictions of maximum grinding temperatures. The preliminary case study shows that the LFM is a useful analytical tool for prediction of the value of the CHTC in the grinding zone.