a thermal analysis of grinding_1

i ' ..

ELSEVIER Wear 216 ( IW}X} 77-86

WEAR

A thermal analysis of grinding

C.C . Chang " '* . A .Z . Szer i h ,, Departnwnt t~f Mechanical l:.'n,qineering. Kmtg-.han Inslimlc oI Techm~h;gy. Taimm 701(k'~. "rai.'an

h Del~artmen I +~l+Mech

"/8 CC Chang. A.Z S:eri / Wear 216 ( l~VSJ 77-i~6

thrown off the wheel joins the fluid that is coming directly from the nozzle: together they create a ram pressure effect. We consider the upstream edge of the accumulated fluid to be the upstream boundary of a hydrodynamic film that forms between the rotating porous cylinder and the sliding plane. This fluid film is thin and we may apply lubrication theory to calculate the pressures developed in it. The hydrodynamic pressure, enhanced by the ram pressure at the upstream boundary of the film, forces the fluid into the pores (the interstices in the grit layer) of the wheel. The fluid that fills the pores gets transported into the grinding zone by the wheel where it is heated. On exiting the grinding zone, the fluid, now carrying excess heat, is discharged from the pores of the wheel by centrifugal forces.

As we have reasoned above, not all of the applied fluid is u~fui in transporting heat. The useful flow rate, i.e., the flow rate across the grinding zone, is but a fraction of the applied flow rate as the pores of the wheel are only partially filled with coolant [ 7 I. The depth of penetration of the coolant into the pores of the wheel, and therefore the useful flow rate, depends on the material properties of both coolant and wheel, on the rate of supply of the coolant and on the prevailing kinetic conditions [ 5.6.8.9.7 ].

By making two far reaching assumptions, viz.. that local thermal equilibrium exists between fluid and grit and that the porous wheel is completely saturated with coolant. Lavine 141 constructed a simple thermal model for predicting maximum grinding zone temperature. She compared predictions for both water and oil coolant to the experimental data of Ohishi and Fumkawa [ I0 ]. who investigated workpiece temperature and the mechanism of grinding-bum in creep-feed grinding. Good agreement with experimental data was reported by Lavine only after accepting unrealistically large values, up to 0.978, for bulk porosity. Further adjustments had to be made, to the numerical values of the material properties, in order to reach agreement lot oil.

Due to the shortness of the time of residence for both grit and fluid and to the high rates of heat generation, it is unlikely that local thermal equilibrium between fluid and grit can be reached within the grinding zone. Distinct temperatures for fluid and grit should, therefore, be maintained by the analysis. Unfortunately, multi lemperature treatment for multi phase systems in porous media is fraught with difficulties. Though the concept of using effective properties for predicting overall temperature has been advocated by various researchers [ 1 I - 15], it is far from obvious, how to actually compute these under the widely differing conditions encountered in practice.

In this paper we apply the method of volume averaging to determine the effective conductivity and the effective speciiic heat capacity of fluid impregnated porous media. The system of equations obtained in this manner is not cloud. To provide closure, the general closure formula of Hadley I I5[ is applied: this includes both upper and lower limits tbr effective thermal properties, at the specified bulk porosity. Wc then couple our previous inlet flow analysis, by means of the u~ful flow rate. to the thermal analysis of the grinding zone and,

thus, arrive at a self-contained model. The model provides predictive capability for grinding temperatures. We are able to demonstrate excellent agreement with the experimental data of Ohishi and Furukawa [ 10] for creep-feed grinding, and with that of Yasui and Tsukuda [ 16] for conventional grinding.

2. Analysis

We consider the grit covering of the grinding wheel, and perhaps the entire wheel itself in some cases, to be a fluid- saturated-" porous matrix. To study heat conduction in this two-phase system we apply the technique of local volume averaging.

2.1, laJc'al volume averaging

Based on an order-of-magnitude analysis, Carbonell and Whitaker [ 17 ] found the following criteria tbr local thermal equilibrium in two-phase flow.

( i ) The time-scale, 7. must satisfy the condition

b(PCr),l"( ! + k )

and

(I-~b)(pCp)21"-(l~_ ' ,~ + kl--)

C C Chang. A,Z S=eri / Wear 216 ~ 19~ } 77-86 79

Solid volume Solkt-flwid imerf~-ial amea

A* Avetagkt$ s~tr~ Fluid volume i V=Vr+

Fig. I. A repre~ntalivc clcmcnlary wdumc.

Here, and in the sequel, we employ index notation to designate component: ( ) p refers to the fluid and ( )2 refers to the solid.

On the basis of our previous finding that. to first approxi- matiou, the velocities of coolant and grit are equal m one another [5,61, vl and v., in Eqs. (2a) and (2b) will both be replaced by V, the peripheral speed of the wheel.

The boundary conditions on the solid-fluid interfaces, A i_, or A2,, are continuity of temperature and continuity of heat flux

Tt=T 2, on AI2 (3)

-nL'ktVTi =-n 2 .k ~VT2. on ,4 ,~_ (4) The above system of equations and boundary conditions

cannot, of course, be solved but for the simplest of regular, known boundaries. If the boundaries are complex or if they are known only in a statistical sense, the best one can hope for is to solve the conduction problem on the average." We will use local volume-averaging techniques for this purpose. The fundamental tool of local averaging is provided by a theorem of Whitaker [21 ] and Slattery [22], which relates the average of a gradient to the gradient of the average. The averaged energy equation is a system equation "and contains system parameters, called effective heat conductivity and effective heat capacity, in contrast to the component equations, Eqs. (2a) and (2b), which contain component conductivities and heat capacities.

The system energy equation is derived by first local-averaging and then combining the component F_,qs. ( la) and ( Ib)Eqs. ( Ic) and( Id)Eqs. (2a) and (2b). Local averaging will yield good result, according to Whitaker [ 21 ], provided that

d= ~

8n C.C, Chang. A.Z. S :vd /Weur 216 (1~98J 77-86

and as the temperature is continuous across all boundaries. the integrals in Eq. (9) cancel. We are thus left with the important relationship

v= ~ = ~ 4,,'" (io) i - I I='l

where d~, = V,/V repre~nts the volume fraction of the ilh phase. For fully saturated porous medium ~, + d,, = I and we have ~f = =b, and ~5~= 1}- d,). where & is the porosity. Porosity is defined as the ratio of the volume of the voids to the total votume.

Assuming invariance of the thermal conductivities and the heat capacities over the small averaging volume V, we volume-average Eq. ( I ) and (2) with the results

V...(pC~),(VT, )=k,(V.VT, ) 1 t I )

V,..(pC,,),_(VT~_}=k:(V.VT~) ( 12 )

By using the spatial averaging theorem 118.23.24l on the right-hand side of Eqs. ( I I ) and (12). we obtain

I t~..(pCp),(VT,)=k,V.(VT,}+ ~ Jn,./gVT, an ( 13)

.41_,

'I V,.'lpCo):(VT,)=k ,V'(V7";)+ -~ n,.I,_VT, dA ( 141 ..t , I

By combining Eqs. 1131 and (14) and noting again that A~,=A:~ audn~ = -n , , we have

t~.'l(pCp),(VT'i )+(pCph(VTi)l=V'lk :(VT':)

+k : (VT , ) I+ / f,h.(/,,Vr,-~:VT,)dA (15) ,It 2

The integral in Eq. ( 15 ) vanishes on account of the I~)undary condition ( Eq. (4) ). By using the relation between average temperature and intrinsic average temperature from Eq. (10). Eq. (15) becomes

V,..Id,(pC,,),('CT,)' " +( I-~b)(pCp)..(VT._)' "-']

=V-id~k,(VT, }' "+( ] -d,)/~ ..(VT.y-" 1 (16)

Following Hadley [ 15] we now deline the efi'~cfive conductivity, k,.. of the mixture as

k,.V(T)=&k~(V7"~)' "+( I -d , )~' . . , /VT:} '2' (17)

The effective speciiic heat capacity. ( t.,Cr,),., of the mixture is defined through

( pC'p ) ,-~( 7" ) :6 ( p ( 'p )l (V7',)'"

+(I-&)(pC,,)z(V7":) 'z' (18)

With definitions (Eqs. (17) and (18)) for effective material properties. Fal. (16) becomes

V,..I(pCr),.V(T)I=V.(k,.V(T)) (19)

Eq. (19) isouraveraged energy equation for tbe two-phase system of grit and fluid. This equation contains three unknowns, (VT), k~, and (pCv)~. To solve Eq. (19) for (VT). values for effective conductivity and effective heat capacity must be provided. Eqs. (17) and (18) can be used for this purpose. However, these equations introduce two additional unknowns (VTI) t~ and (VT.,y", raising the number of unknowns to five. We must find two more equations, unfortunalely Eq. (I0) is the only additional equation available at present.

2.2. E.t]&'tive material properties

The coupled set of Eq, ( I0) and Eqs, ( 17)-(19) is ill posed: four equations contain five unknowns. One more equation is needed to provide closure. We will consider now two simple choices for closure.

By equating average temperature gradients* for fluid and solid

(V7", }' "= (VT,) ' -" (20)

we are prescribing, in the language of networks, parallel cou- pling between the phases. Eqs. (I0) and (17), (18) and (20) yield the following formulae for effective properties:

.~ =d,+( I-~blkr 121)

( PCr)" =d,+( I -',b)(pC v )r (22) (pC,,),

where

kr = ~, (PCi, b - (PCL,),

These formulae represent the highest realizable heal conductivity and specific heat capacity, respectively, at the specified porosity, &. Eqs. ( 21 ) and (22) were obtained by Lavine t4l through assuming local thermal equilibrium, and were employed in her analysis.

II: on the other hand. we employ the closure assumption of equal average [teat flow-rates

(VT',) ' "= k ,( VT., )' :~ ( 23 )

we assume a series arrangement ol'the phases. Eqs. (10) and ( 17 ), ( 18 ) and ( 23 ) yield eftixtive properties as

k,. k, 124) k, ,bk, + I-d,

(p(..'r)~. ~,

('.(7. Chang. A.Z Szeri / Wear 216 I~1~ 77-P,6 81

air + grit

Coolant + grit

Vw ~- - - -

Porous wheel

Coolant z x V ;~bloz ;~

wo~ece ~ ~' '~ ~- GriMing zone Inlet Zone

Fi~. 2. Schematic ~.'iev,' of grinding meehimir.m l t klan cut ).

ductivity and specific heat capacity, respectively, at the specified porosity, ~b.

These upper and lower bounds tbr effective conductivity were combined into a single, continuous, one-parameter formula by Hadley I 151 through

(VT,)'~'={f(t~,.~b)+k,II-I ' Ik,.~b)II(VTz) 'z' (26)

Any closure model which depends only upon 6 and k, may be represented by this general closure relation 1151. Hadley calls f(k~.~b) the structure parameter of the matrix.

Employing the structure parameter../, the effective conductivity and the effective heat capacity can he written as

/,,, k,( i -~f )+,~j -- (27) k, ~bk , ( I - f )+( I -b+~[)

(pC r L. = d)J'+k ~( 1:1")+( 1-4~)( pC r L ~ 28 ) (pCr) , k,d~( I - I )+( I -~+~I )

It is easily shown that the parallel structure. Eqs. (22) and (23). is represented by f= 1,0 in Eqs. (27) and (28). while the series structure. Eqs. (24) and ( 25 t. can be obtained with .I'= 0.0.

Maxwell I I I I treated two special systems: ( i ) a dilute suspension of spherical panicles in an infinite uniform fluid. and (it) a solid I'aMy containing a ditute "suspension" of fluid- filled voids. One may verily that the efli:ctive conductivity Ibrmula ( which we wilt call the lower Maxwell Iormula) that was obtained by Maxwell in case (i). can be acquired from Eq. ( 27 ) by setting/= 2/3. Maxwell's second system yielded what we now refer to as the upper Maxwell formula: it can also be obtained from Eq. 127). by oh/rasing f=2k , / 12k,+ I ) .

In this paper we investigate the two limiting cases./'= 0 and f= I of el'fee/ire material properties, in application to thermal effects during grinding. We will also search ftw the optimal values of the matrix structure parameter.f, which could pre-

cisely predict actual situations in creep-Iced and conventional grinding.

2.3. The energy equation/or the wheel

To simplify the energy equation, we assume negligible heat conduction in the direction of motion. This is permissible &s the Peclet number. Pe = V,.L/a,.. is of order I0'. We also neglect viscous dissipation and pressure work. and choose not to consider boiling. We view the onset of boiling a.s system failure, and feel less compelled to predict heat transfer during ix)(ling than to predict the critical conditions for its (~CUITeflCO.

Based on the above assumptions, the energy equation ( Eq. (19) ) simplifies to

;KT) ;J~-

82 C.C, Chang. A,Z Szeri I Wear 216 (1998) 77-86

a.(x.~) is finite (34)

(T),(x,h)=(T),,(xJt) ( 35 )

a(T),.(x,h) a(T),(x,h) k , . - - k , , ~ (36)

where T,, is the temperature in the workpiece, q,(.r) is the x- distribution of the total heat generation rate. and h is the penetration depth. The preci~ form of the rate of heat generation per unit length, q,(xL is not known: this forces us to accept the approximation of replacing qt(x) in the interfacial condition (Eq. (33)) by its average value qT=l / L.fd'q, (x)d.r. In consequence, Eq. ( 33 ) is satisfied in the mean

L I .

- , . .7"-- dx - k ~ d.r

(33a) I .

= fq, (-Od.~" I |

The integral jotqdx)d.~ " = LqT equals the rate of heat generation in grinding. Though we consider q, an input parameter. we make no prior assumption as to its partitioning between wheel and workpiece, s

2.4. The energy equation fiw the ~wlrkpiece

On account of the slowness of the velocity of translation. heat conduction in the direction of motion cannot he neglected in the workpiece. In con~quence, the energy equation for the workpiece has the form

il Z [ i~ 2 T. ~J "- Z I pC~)~v~ =k I ~ + . . . . 1 ~37)

;Jx ~ ax- &"- f

7~. k,~. and V,, represent the temperature, the conductivity and the velocity of the workpiece, respectively.

The boundary and interface conditions on T,, are (of.. Fig. I)

"Experiments: re~.ull~, indk'alc IbM q~/ql is less than 0.05 in ro.'p fi:ed grinding with water. ~e, Im)~t of heal i~. carried away by the ctmlanl 13 I.

T.(O,z')=Ti., T.(~.z'), T.(x.oc) are finite. (38)

2.5. Numerical scheme

The whole domain is divided into three layers which are governed by Eq. (30) Eqs. (31 ) and (37). Numerically these layers are treated as forming a composite material. Eqs. (30) and (31) are parabolic differential equations and are solved by the Crank-Nicholson scheme. Eq. (37), on the other hand. is an elliptic differential equation and is solved by the ~heme of successive over-relaxation (SOR) by lines. The penetration depth, h, is supplied by the inlet zone analysis of Chang et al. 15,6]. The set of equations are solved iteratively, with an error criterion, e, defined as follows:

i~ u ~ i

Here m and n are mesh size in the x-direction and z-direction, respectively, and k is the iteration counter here. The iteration will stop when E is smaller than 0.0001.

3. Results and discussion

This section compares theoretical predictions with experimental data for oil and water coolants, both in conventional and in creep-teed grinding. Though we have began work on generalizing our model to include cooling with oil-in-water emulsions, for the pre~nt we restrict ourselves to single- phase coolants. Unless otherwise stated, the calculations reported here were performed with the conditions of Table I and the properties listed in Table 2.

We start the di~ussion by presenting results for creep-feed (down cut ) grinding, Fig. 3a and b displays maximum grinding zone temperature, T,,m, plotted against the grinding energy flux, qr, an input parameter to the problem. Fig. 3a is for water coolant and Fig. 3b is for oil. The variation of T, ..... with qT is very close to linear for both coolant types, with the slope depending on the structure parameter of the matrix.f, as well as on the material properties of the coolant. In these

Table I Grinding condilion~; (down cut )

Pararncters Crcep-I;:ed Conventional grinding grinding

Peripheral veh~:ity ol + wheel. V,, (m/s) 18 20 Wurkpie~:e vdt~ily. V+ (mints) 1.2 33 Diaftleler of proteus wheel, D ( men ) 305 205 Inlel temwralurc. T, (C) 30 30 Bulk Ixm~sity. tb 0.5 0.5 Nozz|e vckx:ity, r,. (m/s) 0.25 n.25 i~pth of cut. a (rum) 0.5 -

C C. Chang. A.Z S=eri ] Wear 216 ( 19()8j 77-86 83

Table 2 Material propertie~ I ha~ed on 3fFC)

k lW/m ptkg / Cp (J/kg /~, (Ns/ K) m ~ ) K) m')

Fluid Water 0 .613 997 -I 1"79 0D00855 O i l (minera l o i l ) 0 .13 881) 1860 0 .045

Air 0.0262 lag I(Hk~ - Workp ieee

Stee l ( p la in carbon ) 60 .5 7854 434 -

Wheel Aluminum ox ide 46 3970 765 -

figures f= 1.0 yields the highest possible conductivity and specific heat capacity al the speci6ed bulk viscosity a,d is the result of equating average mean temperature gfadiem~, Eq. (20). to obtain closm. The valnef= 0.0 mVvesenls the smallest possible conductivity and heat capacity consistcm with the given porosity, it is obtained from the clostm: ~ssumplion of equal mean heat flow rams, Eq. (23). I1 is Ihus the upper limit of T,.,. at given grinding energy flux.

The slruclure function depends on ~ propeffier, of bOllb matrix and coolant. Dependence on the loner is throngh k,. the ratio of component heat conductivities. But examiaation of the Ohishi and Furukawa data for two different coolm~s.

~' IiO

J

C 300. %

J

~L +oo.

=+

~j

mo

~ - (a) I f '0 .0

: ExF~imenml DatalIOI / /

/ /

/ _" - Bue l l

/ / f-O.$

/ 1 / ~ 1 / ~ i - i ] I " f " l.o

/ /1 . . . . . t

/ l / J / / ,1~ "" "" ~ 50. / ~ ' /

~0 ~ I l l l l l I l [ l , l , t l l l l I I l l t l41 l l l l l l t I I I I I I I I I I I I+F , I ,+ , I I l l , I O10 210 *.0 [ 0 [O I O10 I 2.0

Grindi+!g E .ergy Flux, qr(l.VJ mm ~ )

Itb)'" ]]~f-o.o / f-o.s

I / ' . - Bum

/ / / / ; .w

iiii/" / /~ / / F1 : Expenmenlatl~lall0l

- - ~ - f - I ,O

o ~, , , , , ' l - l~r , i i+ ; i i l , , , , , i , i i , , , i i , i i i i +, ,+ i , , o .c 2.0 +.o I.,~ I.o ' +o.o

(3finding Energy Flux q r(W ! mm P )

Fig. 3+ (a) Maximum grinding zone lempera+urc v~. grinding heal flax I~r creep-feed grinding with water. ( b ) Maximum grinding zone teml'~Jralurc vs, g r ind ing heat f lux t'or ,,:rccp-I'ced gr ind in l : w i th oi l .

ino

C ?.+

d 2

++ - 105-

$

84 C.C. Chang, A.Z. S:eri / Wear 216 ( i~93~ 77--86

(a) -

Waler, I D = 305 mm, V,= 18 m/s

V,,-- 0.6 ram/s, a = 2 mm i

qT = 4.8x 10e WIm2, f=0.8

+ : Experimental Data [10] Vw ~- I I I I i ! I I I I I I I t 70 6:$ 60 $$ 50 4:$ 40 3~ 30 25 [ 2tl 15. l0

I , " - I , , t s ~__-~-~.. / / / / /A - p~.. 65 $

~o ~ 3~

4U

aS

I

5 o

x-direction {mnO

to

& m 20 ~,

D = 305 ram, V,= 18 m/s 3~

V,, = 0+6 ram/S, a = 2 mm i 40

qT =7.45 x l0 ~ W/m 2, f= 0.8 vw ~]

70 6~ G0 S:$ ~0 45 4O :$~ 30 '251 20 |$ to 5 i

x-direct ion (ram) Fig. 6. { a ) Tcml~ratur ct~nt~ur ~ff the ~'orkpi~.'c" below burning limit. { h ~ Tcn|p~.ralurc ~:cmlour o1" the workpi~.'e ov,2r burning limit.

viz.. water and oil. suggests thai the dependence of f on k, is weak. For either water or oil. in creep-teed grinding we find f= 0.8. Our theoretical results in Fig. 3a and b were obtained with this value of the structure parameter. The plots show

excellent agreement for both water and oil coolants, with the experimental data of Ohishi and Furukawa I IOl.

Oil changing the porous matrix to that of a conventional grinding wheel, we find that the value o f f= O.96 is repre~n-

C C Chang. A.Z S:eri / Wear 216 (1~,98) 77--86 g5

2=0

~ ~-~

15.0

~ aO.O

". 0 : Exlx'Vlng'ntal Da~ i t Ol

"" "" ~ ~ ~ f - t .e

f -e . l 5 .o

- . . . . - . . . . . . . . . 1-),0

0,00.~) . . . . , .+ . . i . . . . . . . . i ' , . . , . i . . . . . . . . . I ' '~ ' ' ' ' ' '~ . . . . . . , . . i 0.~' 1.0 |.S ZO 2- 't ~.0 ~L5 Depth of era. a (ram)

Fig. 7. Grinding energy flux reached to bum vs. depth of cut for creep-Iecd

gr inding with water.

tative. Fig. 4 displays our theoretical prediction for Tin., vs. qT for both water and oil coolants, and compares the~ plots with experimental data* by Yasui and Tsukuda 1161. The agreement is, once more, excellent.

Due to a lack of published data on conventional grinding, the remaining results of this paper are for creep- feed grinding. Fig. 5 is for the fraction of grinding energy that is conducted into the workpiece. The plot shows the ratio qw/qT tO b~ almost independent of the total grinding energy flux. qT. but that this ratio is only ~ 0.05 for water, while it is ,,- 0.3 for oil. The only relevant experimental result we are aware of is by Shafto et ai. [ 3 ], who asserted that q~/qT < 0.05 for water in creep-feed grinding. All that can be concluded here is that our results do not contradict with the finding of Shafto ~t al.

Fig. 6a and b displays temperature distributions in the workpiece at two values ofq, 4.8 W/ram 2 and 7.45 W/ram 2. respectively. The latter value repre~nts conditions c lo~ to bum. There is good agreement between the theoretical temperature contours and the experimental points of Ohishi and Furukawa [ I01.

Creep-feed grinding is usually done in one pass. at a larger depth of cut on a slow moving workpiece. The depth of cut has strong effect on the total energy flux, i.e.. the rate of energy relea~. If the rate of energy relea~ is too high. coolant vaporization or burn occurs. We define that value of the total energy flux that initiates burn, its critical value, and designate it by qt*. Fig. 7 plots ql* as function of the depth of cut, a . for creep-feed grinding with water coolant. 1~. is indicated in this plot that q,* decreases as the depth of cut increases. If the rate of energy of release is too high. coolant vaporization or burn occurs. This prediction is in good agreement with the experimental data of Ohishi and Furukawa I I0].

* The conditions of Yasui and Tsukuda data are listed in Tables I and 2.

4. Cemlmkms

( I ) It is unlikely that local thermal equiliMium Ik-tween grit and coolant can exist within the contact zing, dee tothe fast transients and the high rates of heat generation which characterize grinding. "rberefme. in onr analysis of the thermal process of grinding, we maintain distinct component temperatures. The pnncipal theoretical tool we employ is local volume averaging; this enables us to derive an averaged energy equation and define average (effective) thermal properties for the two-pha~e system af grit and coolant.

(2) By making use of the stmctme parameter of the matrix. which has the value o f f= 0.8 for creep-feed grinding and the valtle off= 0.96 for conventional grinding, we can accurately predict thermal performances for both water an oil coolams.

5. Nmaeldatare

A interracial area inside porous media A,, nominal area of contact b width of grinding zone C r specific heat at constant Wessure D wheel diameter d pore diameter f matrix smgture parameter h depth of cut k thermal conductivity L length of grinding zone, I linear dimension of the averaging volume n normal vector on interfacial surface q heal flux T temperature V volume V~ peripheral velocity of porous wheel v velocity v,, nozzle velocity x coordinate in the direction of porous wheel mntion : coordinate normal m wheel or workpiece surface

Greek letters and symbols

,~ thermal diffusivity d, porosity d~ volume fraction p density ( ) average ( . ) ' intrinsic average

Sub.scripts

I air-matrix system 2 coolant-matrix system e effective I fluid 2 .solid

86 c,c. Clumg. A.Z Szeri / Wear 216 [ 19~8) 77-.86

t2, 21 f lu id-sol id interface T total

e wheel w workpiece

in inlet

max max imum

r relative

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