4. Precise control of temperature field

#### 4.1. Heat source distribution model

In the analysis of temperature fields on workpiece surface, a banding heat source model with continuous equivalent distribution is usually used for heat source in grinding zone to replace the effect of disperse point heat sources to simplify the model [57].

During grinding process, there are three states—scratching, plowing and cutting—between the abrasive grains and workpiece. As shown in Figure 4 [58], grinding heat sources are under rectangular distribution (heat flux is ξqw) due to scratching and plowing effects of abrasive grains and triangular distribution (peak heat flux is uqw) due to cutting effect of abrasive grains, where qw is the average heat flux transferred into workpiece, and ξ and u are, respectively, heat flux coefficients under rectangular heat source distribution and triangular heat source distribution. Abrasive grains exert scratching and plowing effects in OA segment to generate rectangular heat source distribution, and length is a; they exert cutting effect in AL segment to generate triangular heat source distribution [59]; OL is grinding contact length and its value is l. Point B is the position where peak heat flux of triangular heat sources is located, and OB length is b. On this basis, Zhang and Mahdi [59] established a shape functional equation of triangular heat source distribution:

$$s(\mathbf{x}) = \begin{cases} 0 & \mathbf{x} \in (-\infty, 0) \\ \xi & \mathbf{x} \in (0, a) \\ \frac{\xi(b-\mathbf{x}) + u(\mathbf{x}-a)}{b-a} & \mathbf{x} \in (a, b) \\ \frac{u(\mathbf{x}-1)}{b-l} & \mathbf{x} \in (b, l) \\ 0 & \mathbf{x} \in (l, +\infty) \end{cases} \tag{1}$$

distribution model as shown in Figure 5(a). If grinding wheel is dressed very sharp and the lubricating performance of grinding fluids is very good, then abrasive grains mainly exert cutting effect, lengths of scratching and plowing effects are small, and heat source intensity is quite low due to few cutting materials in front end of contact zone. In rear end of contact zone, cutting depth is large, there are many cutting materials and heat source intensity is great, it can be approximate to a = 0, b = l and ξ = 0, u = 2. Comprehensive heat source model can be approximated to right triangular heat source distribution model as shown in Figure 5(b). When a = 0, b = l/2 and ξ = 0, u = 2, namely, abrasive grains have strong cutting effect in the middle part of contact zone, generated heat source intensity is great, and comprehensive heat source model can be approximated to isosceles triangular heat source distribution model as shown in Figure 5(c).

Figure 5. Heat source distribution models in grinding. (a) Rectangular heat source, (b) Right triangular heat source, (c)

Thermodynamic Mechanism of Nanofluid Minimum Quantity Lubrication Cooling Grinding and Temperature Field…

For grinding energies consumed during grinding process, except that a small part of them are consumed on newly generated surface to form needed surface energy, strain energy left on grinding surface layer and kinetic energy for grinding debris to fly out, most part of them are converted into heat energy within contact zone, and these heat energies can be transferred into workpiece, grinding wheel, debris, and grinding fluids in ways of heat conductivity and heat convection. Eqs. (2)–(4) represent the amount of energy transferred into the workpiece (Ew),

> θmb<sup>0</sup> 2 kð Þ rc <sup>w</sup>vwlg � �<sup>1</sup>

> > Ar A � �

� �<sup>1</sup>

s vslg

θmb<sup>0</sup> 2 φð Þ krc <sup>n</sup> þ ð Þ 1 � φ ð Þ krc <sup>f</sup>

2

h ivslg n o<sup>1</sup>

θmb<sup>0</sup> 2ð Þ krc <sup>g</sup>

where θ<sup>m</sup> is the maximum temperature rise; k, r, c stand for thermal conductivity, density, and specific heat, respectively; the subscript g, w, n, f stand for the properties of grain, workpiece,

<sup>2</sup> (2)

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2

(3)

(4)

4.2. Thermal distribution coefficient

Isosceles triangular heat source.

Ef <sup>¼</sup> <sup>1</sup> 2

grinding wheel (Eg), and grinding fluid (Ef), respectively [39]:

θmb<sup>0</sup> 2ð Þ krc <sup>f</sup> vslg h i<sup>1</sup>

Ew <sup>¼</sup> <sup>1</sup> 2

<sup>E</sup><sup>g</sup> <sup>¼</sup> <sup>1</sup> 2

If abrasive grains are quite pure, there are many abrasive grains exerting scratching and plowing effects under dry grinding state or lubricating performance of grinding fluids is poor, and then comprehensive heat source distribution model is approximate to rectangular heat source

Figure 4. Distribution of heat source [58].

Thermodynamic Mechanism of Nanofluid Minimum Quantity Lubrication Cooling Grinding and Temperature Field… http://dx.doi.org/10.5772/intechopen.74969 69

Figure 5. Heat source distribution models in grinding. (a) Rectangular heat source, (b) Right triangular heat source, (c) Isosceles triangular heat source.

distribution model as shown in Figure 5(a). If grinding wheel is dressed very sharp and the lubricating performance of grinding fluids is very good, then abrasive grains mainly exert cutting effect, lengths of scratching and plowing effects are small, and heat source intensity is quite low due to few cutting materials in front end of contact zone. In rear end of contact zone, cutting depth is large, there are many cutting materials and heat source intensity is great, it can be approximate to a = 0, b = l and ξ = 0, u = 2. Comprehensive heat source model can be approximated to right triangular heat source distribution model as shown in Figure 5(b). When a = 0, b = l/2 and ξ = 0, u = 2, namely, abrasive grains have strong cutting effect in the middle part of contact zone, generated heat source intensity is great, and comprehensive heat source model can be approximated to isosceles triangular heat source distribution model as shown in Figure 5(c).

#### 4.2. Thermal distribution coefficient

4. Precise control of temperature field

equation of triangular heat source distribution:

Figure 4. Distribution of heat source [58].

s xð Þ¼

8

>>>>>>>>>>><

>>>>>>>>>>>:

the effect of disperse point heat sources to simplify the model [57].

In the analysis of temperature fields on workpiece surface, a banding heat source model with continuous equivalent distribution is usually used for heat source in grinding zone to replace

During grinding process, there are three states—scratching, plowing and cutting—between the abrasive grains and workpiece. As shown in Figure 4 [58], grinding heat sources are under rectangular distribution (heat flux is ξqw) due to scratching and plowing effects of abrasive grains and triangular distribution (peak heat flux is uqw) due to cutting effect of abrasive grains, where qw is the average heat flux transferred into workpiece, and ξ and u are, respectively, heat flux coefficients under rectangular heat source distribution and triangular heat source distribution. Abrasive grains exert scratching and plowing effects in OA segment to generate rectangular heat source distribution, and length is a; they exert cutting effect in AL segment to generate triangular heat source distribution [59]; OL is grinding contact length and its value is l. Point B is the position where peak heat flux of triangular heat sources is located, and OB length is b. On this basis, Zhang and Mahdi [59] established a shape functional

> 0 x ∈ð Þ �∞; 0 ξ x ∈ð Þ 0; a

<sup>b</sup> � <sup>l</sup> <sup>x</sup> <sup>∈</sup>ð Þ <sup>b</sup>; <sup>l</sup> 0 x ∈ð Þ l; þ∞

If abrasive grains are quite pure, there are many abrasive grains exerting scratching and plowing effects under dry grinding state or lubricating performance of grinding fluids is poor, and then comprehensive heat source distribution model is approximate to rectangular heat source

x ∈ð Þ a; b

(1)

ξð Þþ b � x u xð Þ � a b � a

u xð Þ � l

4.1. Heat source distribution model

68 Microfluidics and Nanofluidics

For grinding energies consumed during grinding process, except that a small part of them are consumed on newly generated surface to form needed surface energy, strain energy left on grinding surface layer and kinetic energy for grinding debris to fly out, most part of them are converted into heat energy within contact zone, and these heat energies can be transferred into workpiece, grinding wheel, debris, and grinding fluids in ways of heat conductivity and heat convection. Eqs. (2)–(4) represent the amount of energy transferred into the workpiece (Ew), grinding wheel (Eg), and grinding fluid (Ef), respectively [39]:

$$E\_w = \frac{1}{2} \Theta\_{\mathfrak{m}} b' \left[ \mathbf{2} (\mathbf{k} \rho c)\_w v\_w l\_{\mathfrak{g}} \right]^\frac{1}{2} \tag{2}$$

$$E\_{\rm g} = \frac{1}{2} \theta\_{\rm m} b' \left[ 2(k\rho c)\_{\rm g} \left( \frac{A\_r}{A} \right)\_s v\_s l\_{\rm g} \right]^{\frac{1}{2}} \tag{3}$$

$$E\_f = \frac{1}{2} \Theta\_\mathbf{m} b' \Big[ 2(k\rho c)\_f v\_s l\_\mathbf{g} \Big]^{\frac{1}{2}} = \frac{1}{2} \Theta\_\mathbf{m} b' \Big\{ 2 \left[ \varphi(k\rho c)\_\mathbf{n} + (1-\varphi)(k\rho c)\_f \right] v\_s l\_\mathbf{g} \Big\}^{\frac{1}{2}} \tag{4}$$

where θ<sup>m</sup> is the maximum temperature rise; k, r, c stand for thermal conductivity, density, and specific heat, respectively; the subscript g, w, n, f stand for the properties of grain, workpiece, nanoparticles, and base fluid, respectively; Ar <sup>A</sup> is the ratio between the actual and nominal contact areas of grinding wheel and workpiece; vs is the peripheral speed of grinding wheel; φ is the volume fraction of nanoparticles; b<sup>0</sup> is the width of grinding wheel; and lg is the geometric contact arc length.

During grinding, the temperature of the workpiece surface is an important factor to be considered, which is reflected in the thermal distribution coefficient (R) attributed to the workpiece. According to the single abrasive grain model, the amount of heat eliminated by abrasive debris and diffused in convection are too limited to be neglected. During grinding, R can be expressed as follows:

$$R = \frac{E\_{\rm w}}{E\_{\rm w} + E\_{\rm g} + E\_{f}} = \frac{1}{1 + \sqrt{\frac{(k\rho c)\_{\rm g} v\_{\rm s}}{(k\rho c)\_{\rm w} v\_{\rm w}} \left(\frac{A\_{\rm r}}{A}\right)} + \sqrt{\frac{\wp (k\rho c)\_{\rm n} + (1 - \wp)(k\rho c)\_{f}}{(k\rho c)\_{\rm w}} \frac{v\_{\rm s}}{v\_{\rm w}}}} \tag{5}$$

#### 4.3. Heat transfer coefficient

As NMQLC method can be both heat transfer of normal-temperature gas and boiling heat transfer of nanofluid drop, so this method is the sum of two heat convection methods. According to heat transfer state of single nanofluid drop [60, 61], heat transfer coefficient under NMQLC condition can be solved in three stages: natural convection, nucleate boiling, transition boiling, and film boiling as shown in Figure 6 [62].

#### 4.3.1. Natural-convection heat transfer stage (I)

When workpiece surface temperature is Tn<sup>1</sup> (lower than boiling point of nanofluid), heat transfer surface will not generate boiling heat transfer and heat transfer enhancement is realized mainly through convective heat transfer of normal-temperature air and convection of nanofluids, mainly being the convective heat transfer of nanofluids [63]. According to Yang's study [28], the nonboiling heat transfer coefficient:

$$h\_{n1} = \frac{N\_l \mathfrak{c}\_{nf} \mathfrak{o}\_{nf} V\_l}{\pi r\_{surf}^2 \cdot t} + h'\_a \tag{6}$$

Heat transfer coefficient of critical heat flux density point hn<sup>2</sup> [64]:

where Ts is the saturation temperature; q<sup>0</sup>

Figure 6. Heat transfer coefficient of grinding surface.

NlQ<sup>0</sup>

the surface temperature of the workpiece is Tn.

hn<sup>3</sup> ¼

hn<sup>2</sup> <sup>¼</sup> hfa <sup>þ</sup> cl Ts � Tnf � � � � <sup>Q</sup><sup>0</sup>

the drop specific heat capacity; and Q<sup>0</sup> is the nanofluid supply during grinding time.

4.3.3. Transition boiling heat transfer and film boiling heat transfer stages (III and IV)

� � � <sup>0</sup>:027<sup>e</sup>

stage, namely, at starting point in film boiling stage is as follows [65]:

r<sup>l</sup> hfa þ clð Þ Ts � Tl

4.4. Temperature field control equation and boundary condition

heat transfer quantity; hfa is the latent heat of vaporization; Tnf is temperature of nanofluid; cl is

Thermodynamic Mechanism of Nanofluid Minimum Quantity Lubrication Cooling Grinding and Temperature Field…

Computational process of heat transfer coefficient hn3 at the end point in transition boiling

b<sup>0</sup> � l � ð Þ Tn<sup>3</sup> � Tl

Therefore, the heat transfer coefficient hn can be obtained by the interpolation calculation when

As shown in Figure 7, temperature field model in grinding zone is established and the grinding temperature field can be simplified into 2D heat transfer analysis. Field variable T in transient temperature field meets the equilibrium differential equation of heat conduction [66]:

rl

<sup>0</sup>:<sup>08</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffiffi lnð Þ We=35þ<sup>1</sup> <sup>p</sup>

� �

<sup>a</sup> is the normal-temperature atmospheric-convection

<sup>B</sup>1:<sup>5</sup> <sup>þ</sup> <sup>0</sup>:21kdBe �<sup>90</sup> Weþ<sup>1</sup>

<sup>a</sup> (7)

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71

þ h<sup>0</sup>

<sup>a</sup> (8)

<sup>π</sup>rsurf <sup>2</sup> Tn<sup>2</sup> � Tnf � � <sup>þ</sup> <sup>h</sup><sup>0</sup>

where Nl is the total number of droplets; cnf is the liquid drop specific heat capacity; rnf is the density of nanofluids; Vl is the volume of a single droplet; rsurf is the spreading radius of a single liquid drop; t is the total time of grinding process; and h<sup>0</sup> <sup>a</sup> is the convective heat transfer coefficient of air at normal temperature.

#### 4.3.2. Nucleate boiling heat transfer and transition boiling heat transfer stages (II and III)

At the end point of nucleate boiling heat transfer and starting point of transition boiling heat transfer, namely, at critical heat flux point, heat transfer coefficient reaches the maximum value hn2. At the end of transition boiling heat transfer stage and in the initial stage of film boiling heat transfer, heat transfer coefficient reaches the minimum value hn<sup>3</sup> and computational formula of heat transfer coefficient is as follow:

Thermodynamic Mechanism of Nanofluid Minimum Quantity Lubrication Cooling Grinding and Temperature Field… http://dx.doi.org/10.5772/intechopen.74969 71

Figure 6. Heat transfer coefficient of grinding surface.

nanoparticles, and base fluid, respectively; Ar

<sup>R</sup> <sup>¼</sup> <sup>E</sup><sup>w</sup>

4.3.1. Natural-convection heat transfer stage (I)

study [28], the nonboiling heat transfer coefficient:

coefficient of air at normal temperature.

formula of heat transfer coefficient is as follow:

E<sup>w</sup> þ Eg þ Ef

transition boiling, and film boiling as shown in Figure 6 [62].

single liquid drop; t is the total time of grinding process; and h<sup>0</sup>

geometric contact arc length.

4.3. Heat transfer coefficient

expressed as follows:

70 Microfluidics and Nanofluidics

<sup>A</sup> is the ratio between the actual and nominal

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi φð Þ krc <sup>n</sup>þð Þ 1�φ ð Þ krc <sup>f</sup> ð Þ krc <sup>w</sup>

vs vw <sup>r</sup> (5)

<sup>a</sup> (6)

<sup>a</sup> is the convective heat transfer

contact areas of grinding wheel and workpiece; vs is the peripheral speed of grinding wheel; φ is the volume fraction of nanoparticles; b<sup>0</sup> is the width of grinding wheel; and lg is the

During grinding, the temperature of the workpiece surface is an important factor to be considered, which is reflected in the thermal distribution coefficient (R) attributed to the workpiece. According to the single abrasive grain model, the amount of heat eliminated by abrasive debris and diffused in convection are too limited to be neglected. During grinding, R can be

<sup>¼</sup> <sup>1</sup>

As NMQLC method can be both heat transfer of normal-temperature gas and boiling heat transfer of nanofluid drop, so this method is the sum of two heat convection methods. According to heat transfer state of single nanofluid drop [60, 61], heat transfer coefficient under NMQLC condition can be solved in three stages: natural convection, nucleate boiling,

When workpiece surface temperature is Tn<sup>1</sup> (lower than boiling point of nanofluid), heat transfer surface will not generate boiling heat transfer and heat transfer enhancement is realized mainly through convective heat transfer of normal-temperature air and convection of nanofluids, mainly being the convective heat transfer of nanofluids [63]. According to Yang's

hn<sup>1</sup> <sup>¼</sup> Nlcnf <sup>r</sup>nf Vl

4.3.2. Nucleate boiling heat transfer and transition boiling heat transfer stages (II and III)

where Nl is the total number of droplets; cnf is the liquid drop specific heat capacity; rnf is the density of nanofluids; Vl is the volume of a single droplet; rsurf is the spreading radius of a

At the end point of nucleate boiling heat transfer and starting point of transition boiling heat transfer, namely, at critical heat flux point, heat transfer coefficient reaches the maximum value hn2. At the end of transition boiling heat transfer stage and in the initial stage of film boiling heat transfer, heat transfer coefficient reaches the minimum value hn<sup>3</sup> and computational

<sup>π</sup>rsurf <sup>2</sup> � <sup>t</sup> <sup>þ</sup> <sup>h</sup><sup>0</sup>

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð Þ krc <sup>g</sup>vs ð Þ krc <sup>w</sup>vw

� � <sup>r</sup>

Ar A

þ

1 þ

Heat transfer coefficient of critical heat flux density point hn<sup>2</sup> [64]:

$$h\_{n2} = \frac{\left[h\_{\rm fu} + c\_l \left(T\_s - T\_{\rm nf}\right)\right] Q' \rho\_l}{\pi r\_{\rm surf} 2 \left(T\_{n2} - T\_{\rm nf}\right)} + h'\_a \tag{7}$$

where Ts is the saturation temperature; q<sup>0</sup> <sup>a</sup> is the normal-temperature atmospheric-convection heat transfer quantity; hfa is the latent heat of vaporization; Tnf is temperature of nanofluid; cl is the drop specific heat capacity; and Q<sup>0</sup> is the nanofluid supply during grinding time.

#### 4.3.3. Transition boiling heat transfer and film boiling heat transfer stages (III and IV)

Computational process of heat transfer coefficient hn3 at the end point in transition boiling stage, namely, at starting point in film boiling stage is as follows [65]:

$$h\_{n3} = \frac{N\_l Q' \rho\_l \left[ h\_{\hat{\rm fl}} + c\_l (T\_s - T\_l) \right] \cdot \left[ 0.027 e^{\frac{0.08\sqrt{\ln(0.05 + 1)}}{\delta^{1.5}}} + 0.21 k\_d B e^{\frac{-60}{\delta^{10}}} \right]}{b' \cdot l \cdot (T\_{n3} - T\_l)} + h'\_a \tag{8}$$

Therefore, the heat transfer coefficient hn can be obtained by the interpolation calculation when the surface temperature of the workpiece is Tn.

#### 4.4. Temperature field control equation and boundary condition

As shown in Figure 7, temperature field model in grinding zone is established and the grinding temperature field can be simplified into 2D heat transfer analysis. Field variable T in transient temperature field meets the equilibrium differential equation of heat conduction [66]:

Figure 7. Two-dimensional heat conduction model with nodal network [66].

$$\begin{cases} \frac{\partial^2 T}{\partial x^2} + \frac{\partial^2 T}{\partial z^2} = \frac{1}{\alpha\_w} \frac{\partial T}{\partial t} \\\\ \alpha\_w = \frac{k\_w}{\rho\_w c\_w} \end{cases} \tag{9}$$

fixed heat generated on the grinding surface [66]. As the thermal conductivity of the air is very low, more heat is concentrated in the grinding zone, thus increasing the grinding temperature.

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Figure 8. Distributions of the grinding temperature and temperature curves in different states [68].

This chapter has presented a review of published researches in the application of NMQLC during grinding. The following conclusions may be drawn from the present literature review: 1. The amount of nanofluids using NMQLC is very small (7.5–350 mL/h based on published literatures) compared with flood lubrication (usually 60 L/h), so this technique is an

2. NMQLC can improve the lubrication condition in grinding area and reduce the friction coefficient effectively, thus reducing the grinding force and specific grinding energy,

3. NMQLC can strengthen the heat transfer in the grinding zone that NMQLC could realize a

4. Based on published literatures, nanoparticles that have effective lubrication properties are TiO2, SiO2, Al2O3, MoS2, ZnO, and nanodiamond; nanoparticles that have effective cooling properties are CuO, NiO, CNTs, and SiC. A mixed use of nanoparticles with good lubricated properties and nanoparticles with good cooling properties can obtain lower grinding

reducing workpiece surface roughness, and improving the life of grinding wheel.

environmentally friendly lubrication-cooling method.

lubrication-cooling effect close to that of flood lubrication.

force, grinding temperature, and better surface quality.

5. Conclusions

where α<sup>w</sup> is the thermal diffusivity.

Difference in the equation of various nodes in internal grids can be obtained based on (9):

$$T\_{t + \Delta t}(i, j) = \left[1 - \frac{2\Delta t (k\_x + k\_z)}{\rho\_w c\_w \Delta l^2}\right] T\_l(i, j) \cdot \frac{\Delta t \{k\_x \cdot [T(i, j + 1) + T(i, j - 1)] + k\_z \cdot [T(i + 1, j) + T(i - 1, j)]\}}{\rho\_w c\_w \Delta l^2} \tag{10}$$

As for boundary conditions analysis in grinding zone, coordinate node (i, j) on the workpiece surface is taken as an example. According to energy conservation law [67, 68], the temperature at the node (i, j) after Δt:

$$T\_{t+\Delta t}(i,j) = \frac{\Delta t}{\rho\_w \mathbb{C}\_w \Delta t} \left\{ \mathbb{k} \cdot \left[ T(i-1,j) + T(i+1,j) + T(i+1,j) \cdot \mathfrak{T}(i,j) \right] + \left[ q - \mathbb{h}(T\_t - T\_d) \right] \cdot \Delta l \right\} + T(i,j) \tag{11}$$

#### 4.5. Precise control of temperature field

The temperature field at different times during the steady process can be obtained by solving the difference Eq. (11). Figure 8 shows the temperature isoline under NMQLC at different times and the corresponding time-space distribution of surface temperature. It can be seen that the grinding process can be divided into three stages, namely, cut-in, steady state, and cut-out [66]:

Cut-in: when abrasive grains start to contact and cut the workpiece, the undeformed chip thickness increases gradually and the heat generated on the grinding interface begins to be transmitted into the workpiece surface.

Steady state: the undeformed chip thickness kept at the mean value and workpiece surface temperature stops increasing. The temperature field reaches the steady state.

Cut-out: the undeformed chip thickness decreases gradually in the cut-out region. According to the theory of heat transfer, the heat conduction in the cut-out region is reduced considering the Thermodynamic Mechanism of Nanofluid Minimum Quantity Lubrication Cooling Grinding and Temperature Field… http://dx.doi.org/10.5772/intechopen.74969 73

Figure 8. Distributions of the grinding temperature and temperature curves in different states [68].

fixed heat generated on the grinding surface [66]. As the thermal conductivity of the air is very low, more heat is concentrated in the grinding zone, thus increasing the grinding temperature.
