Approximate calculation of breakage parameters from batch grinding tests

Pergamon PIh

Chemical Engineerin# Science, Vol. 51, No. 19, pp. 4509 4516. 1996 Copyright © 1996 Elsevier Science Ltd Printed in Great Britain. All rights reserved S001D-2509(96)00275-8 0009 2509/96 $15.00 + 0.00

APPROXIMATE CALCULATION OF BREAKAGE PARAMETERS FROM BATCH GRINDING TESTS H E N R I B E R T H I A U X and C H R I S T E L L E V A R I N O T Laboratoire des Sciences du G6nie Chimique, CNRS-ENSIC, 1 rue Grandville, 54001 Nancy, France and JOHN DODDS* Ecole des Mines d'Albi-Carmaux, Campus Jarlard, 81013 Albi Cedex 09, France

(Received 16 June 1995; accepted 15 February 1996) Al~tract--A mathematical treatment, based largely on the work of Kapur, has been developed to obtain breakage and selection matrices from batch grinding tests which are valid for the initial period of the process. The method is illustrated using experimental results for the grinding of hydrargillite and carbon in a laboratory scale stirred bead mill for 15 min and it is shown that this restriction still allows application of the method to continuous grinding processes in the mill. The breakage and selection functions determined by the method are shown to give a good representation of the grinding kinetics and to lead to a normalized breakage function. Copyright © 1996 Elsevier Science Ltd R6sum~-A partir d'un raisonnement purement math~matique et bas6 en grande partie sur les travaux de Kaput, nous montrons qu'il est tr+s facile d'extraire un couple de matrices de broyage et de s61ectiondont la validit6 est assur6e durant les premiers instants d'une experience de broyage discontinu. Ces notions sont appliqu6es au broyage de suspensions d'hydrargillite et de carbone dans un broyeur ~ billes de laboratoire, pour lequel le temps de s6jour moyen des particules est suffisamment faible, ce qui permet de restreindre l'6tude de la cin6tique de broyage aux quinze premieres minutes de la fragmentation. Le calcul de ces fonctions de s61ection et de broyage approch~es conduit ~. une bonne repr6sentation des courbes exp6rimentales, et montre que l'on peut consid6rer que cette derni6re fonction est normalisable. Copyright © 1996 Elsevier Science Ltd

Keywords: Breakage, selection, grinding, stirred bead mill.

1. I N T R O D U C T I O N

The determination of the breakage and selection functions is an important problem in grinding practice for two main reasons. Firstly, because knowledge of these functions allows the calculation of the change in partide-size distribution with time in grinding processes. Secondly, because these functions can be used to evaluate the breakage phenomena at particle level and allow a diagnosis of the mechanisms in play in a given process. Techniques for measuring these functions have been abundantly reported over the last 30 years (Austin and Bhatia, 1971; Gardner and Sukanjnajtee, 1972). Methods include the following changes to a given marked particle-size class, following changes to a 'monosize' class, or extracting the functions from results of grinding tests by numerical optimization. The first two are suitable for coarse or medium grinding (particles > 100 ttm), especially when coupled with

* Corresponding author.

the third technique. However, none of the methods are universally applicable. Finding a suitable marker is not easy and, when dealing with very small particles involved in fine grinding processes, mono-class methods become inoperative as it is difficult, and sometimes impossible, to prepare monosize samples by sieving. Furthermore, numerical optimization techniques cannot be guaranteed to give a stable solution in all cases (Williams et al., 1994). These practical limitations, and the scientific interest of the use of breakage and selection functions, suggest that approximate solutions to the problem of determining breakage and selection functions could be useful. Here we propose such a method and test it on previously published results concerning batch and continuous mode grinding in a laboratory-scale stirred bead mill (Heitzmann, 1992; Berthiaux et al., 1994). 2. T H E O R E T I C A L

2.1. Matrix analysis of dynamic population balances The conventional method of representing the elementary steps involved in comminution processes 4509

4510

H . BERTHIAUX

(Reid, 1965; Austin, 1971) consists in dividing a particle-size distribution into several size classes indexed from 1 (coarse) to n (fine). Then, two functions are defined: the selection function Si which is the probability of breakage of particles in class i, and the breakage function bi~ which is the mass fraction of particles ofclassj which is found in class i after a short period of grinding. In batch grinding the variation of the mass fraction of class w~ between time t and time t + dt can be written in terms of these functions as dw i

dt

i

-

Siwl + ~ bljSjWj. j=a

(1)

This may be put in matrix form as follows: dW - -

dt

=

(B-

I)SW

=

(2)

AW

where S is a diagonal matrix containing the values of Si, B is a lower triangular matrix containing the values of bij, W is a column matrix of the values of W~, and I is the unit matrix. Assuming that matrix A is invariant with time, eq. (2) can be solved by diagonalization. Since A is a lower triangular matrix it can be diagonalized and its characteristic roots are its diagonal elements (i.e. - $ 3 . There is therefore a matrix P which has an inverse such that

S = - P - ' AP.

(3)

In addition, this can be put into a form equivalent to eq. (2) by making the substitution W = PZ such that dZ dt

- -

=

-

(4)

SZ.

As S is a diagonal matrix, eq. (4) can be integrated to give Z =/~o EXP(t)

(5)

where/~o is a constant of integration and EXP(t) is a column matrix of the elements e x p ( - S~- t). W is obtained by multiplying by the transfer matrix T = P#o such that W = T EXP(t).

(6)

This matrix multiplication leads to the following system of equations which gives the change in the mass fractions w~ of each particle-size class with time:

wa(t) = zll e x p ( - $1 t) w2(t )

=

"62a

e x p ( - Sat) +

w3(t)

=

"63a

e x p ( - Sat) + "632exp (-- SEt)

+ "633 e x p ( -

"622 e x p ( -

S2t)

(7)

et al.

A more condensed set of equations can be obtained in terms of the cumulative oversize mass fraction, R,-(t):

Ri(t) =

(8)

rkj exp(-- Sit). j=l

Several useful properties of the matrices used above, in particular the elements of P and Po, are relegated to the Appendix. 2.2. Kapur's solution Use of the equations given above pre-supposes knowledge of the functions bi~ and Si which are all the more difficult to obtain the higher the precision required, as this implies a larger number of classes. This lead Kapur (1970) to demonstrate that eq. (1) had the following approximate solution in terms of the cumulative fraction oversize:

R,(t) = R,(O)exp(KlaJt + Kl2) ~ )

(9)

with i- 1

I,:l '~ = - s, + 2 (sj+a

~ B,,j+,

-

, R j(0)

Sj~,.j~ R--~

j=l i-1 K!2) E (SJ +1 = j=l

(10)

Bi . J+ 1 - - SjBi , j) ;[..-(1) - {1}, Rj(O) tl"-J - - / ~ i )R~"

Berthiaux (1994) has shown how this can be generalized to any number (p) of terms:

Ri(t) n~i(O)= ~ Klk' t~.

(11)

k=l

Furthermore, by considering that Ri(t)= R(xi), eq. (11) can be also extended to any particle size (x): R (x; t) v tk lnR(x;0) = y~ K ~ ( x ) ~ . . k=l

(12)

The main interest in this polynomial solution is the simplicity of application to experimental results. A fit with results of grinding experiments, expressed in terms of In Ri (t), will give a reasonably faithful representation of grinding kinetics. However, the approach is not merely a data fit, as it is ultimately based on the selection and breakage functions which have a physical validity. Nevertheless, it is clear that information on these functions is effectively lost by this change in parameters as their extraction from equations of the complexity of eq. (10) would be very hazardous.

Sat)

2.3. Approximate calculation of the breakage and se-

lection parameters w.(t) = z.x e x p ( - Sit) + "6.2exp ( - SEt) + "6.3exp(-- Sat) + . . . . . . . . + % . e x p ( - S.t).

Noting that Kapur's approximate solution is based on the breakage and selection functions, though not in an accessible form, poses the question of developing another approximate solution which will allow the

4511

Breakage parameters from batch grinding tests breakage and selection functions to be obtained. Furthermore, in continuous processing, the grinding kinetics are often only required to be known over a relatively short period corresponding to 4 or 5 times the mean residence time of particles in a mill. Thus, the hypothesis that knowledge of grinding kinetics over only a short period can be adequate to describe a process, combined with Kapur's approach therefore suggests extracting the breakage and selection parameters from the following: Ri(t) ~ Ri(O) exp (KI 1) t).

where P ( X ) is the polynomial: P ( X ) = X i-1 + tril-l X i-2 + tri2-1 X i-3 + ... + t r ii-- 1I .

(18)

As the - Sj are the zeros of this same polynomial, then eq. (17) can be written as i-1

Wi(t) = Ri(O) I-[ (KI 1~ + S j ) e x p ( K I 1)t) j=l

(13)

i-1

- R i - l ( 0 ) 1-[ (Kl~-)t + Sflexp(Kl~-)lt). (19)

The first step is to form the following summations involving the mass fraction in each size class:

j=l

Equations (15) and (19) then lead to Wl (t) = wl (t) i-1

ril H (sj - si) exp ( - si t)

dw 2 W2(t) = - - ~ + Slw2(t)

j=l i-1

= R,(0) I~ (KI 1) + S j ) e x p ( K I 1)t)

dw3 S dw3 W3(t) = ' - ~ T +($1 + 2 ) - ~ - - ~ - S 1 S 2 w 3 .

j=l i-1

Continuing in this manner leads to an equation which includes the sums a~,-1 of the products of the parameters Sj, two by two, three by three, etc., up to index i - 1 such that

_~_ $ 1 S 3

..1- $ 2 S 4

_~_ S I S 4

+

-~- . . . -}- S I S i _ I

"~ $ 2 S 3 S 4 + $ 2 S 3 S 5 +

-~- S 2 S i _ l

~- $ 2 S 3 S i _

.....~

For i = 2, eq. (20) leads to

de- . . .

r22($1 - S2)exp ( - S2t) = R2(0)(K~ 1) + Sl)

... SIS2Si_

1

1 d- . . ,

di- l wi tril_ 1 di- Z wi ._ d i- 3 wi -~-['[7i + dti_ 2 + a'2 1 d t i - 3 + ... i 1 + ffi- 1 wi.

(14)

Replacing each wi by its expression in terms of S~ and z~j [eq. (7)] allows eq. (14) to be written more simply as follows: i-1

Wi(t)

= "Gii H

(Sj -

x exp (-- K~1~t) - Rx (0)(K~1) + $1) exp ( - K~1) t). As the second term of the second part of this equation equals zero it can then only be satisfied if

which then leads to Wi(t)-

For i = 1, it is clear that

K~1) = - S1.

.,. + 5253

0"i3- 1 = $ 1 S 2 S 3 + S 1 S 2 S 4 -~ $ 1 S 2 S 5 +

(20)

j=l

" f l l ~ R 1 (0)

tTil- 1 = S1 _}_ $ 2 ..}_ S 3 _~_ .....1_ Si - 1 ffi2 1 = S I S 2

- R~-I (0) I~ ( K I ~ I + Sj)exp(KI~-~I t).

S i ) e x p ( - Sit).

(15)

R 2 (0)

2"22 =

K~21~= - $2. Equations (7) and (16) then lead to 1721 =

-- R 1 (0).

As I-[~---~(KI~x + Sfl in eq. (19) involves the term ( K I ~ + Si-1) which is zero, it is clear that, by recurrence:

j=l

r i / = Ri(0)

But since wi (t) = Ri(t) - R i - 1 (t) = Ri (0) exp (K 11)t) - R i - 1 (0) exp ( K ! ' I t)

(21)

KI 1~ = - Si.

(16)

the successive derivatives of the w~, and then Wi can be determined analytically leading to

Equations (7) and (16) then lead to "fii- 1 = -"~ij
R i - 1 (0)

1 = O.

Wi(t) = Ri(O)P(KI 1') exp (KI 1~t) - Ui_~(O)P(Kll)l)exp(Kl~_~l t)

(17)

Under these conditions the elements of the column matrix EXP(t) I-eq. (6)] are exp( + KI 1)' t) and the

4512

H. BERTHIAUX et al.

transfer matrix T is -

Rl(O)

--Rl(O) 0

0

'•"

R2(0)

0

-R2(0)

"'"

0

R3(0)

0

-- R 3 (0)

T=

0

0 Rn-2(0) 0

-Rn-2(0)

0

-

1

-1 0

-R~_,(O)

0

The matrix PD being composed of elements of the diagonal of T (see the Appendix) allows easy calculation of the passage matrix P since P = T#o ~. 0

''"

1

0

-1

0 Rn-l(0)

0

R~(O) _

Thus, following the variation of particle size distribution with time in a grinding experiment, expressing the results in terms of the cumulative fraction under•..

0 -

1

0

-1 0

P=

0 0

1

_

:

0

0

• ""

-1

1 0

The grinding matrix B can then be obtained from eqs (2) and (3): B = I - PSP-

(22)

1 S- 1

0

-1

1_

size R i ( t ) and fitting these to a simple exponential with the argument K! ~), leads to values of the breakage and selection functions valid for short grinding times:

m

0 51 -- 5 2

$1 $2-$3 S1

$3-$4 S1

.--

0

0 $2-$3

0

$2

$3-$4 S2

$3-$4

0

S3

B=

0 Sn - 2 -- an - I

0

an - 2

Sn- x - Sn

$I

Sn- ~ -

S,

S._ ~ -

S~ 1

Sn- 2

Sn- |

4513

Breakage parameters from batch grinding tests

(a) 16 A 14 A A

I~ 12 :L

A A

O

.

.~ ~0

I:)U ;;g

A AAAAAAAAAA

AA

A

8

A

Ii

6

Fig. 1. Schematic diagram of the Dyno-Mill KDL.

0

I

I

I

I

I

5

10

15

20

25

30

t (min)

(b) 24

22 S i =

bij --

--

K mi - 1

K~1)

(23)

_ K~I)

20 =L

(1)

Kj

16

"~ 14

It would seem a priori that the breakage function determined in this way is not normalizable. That is, it does not necessarily depend on the class size ratio xi/xj. However, and as is in agreement with many results from the literature (Koka and Trass 1985; Gotsis et al., 1985), if the evolution of the selection function can be expressed as a power law such as Si = kx7 and if the class intervals are in geometric progression (of factor r) then the breakage function is normalizable:

F l - r ~ l . Fxil~=(1-r~)r~(i-J-').

bij= L

r~

J LxjJ

(24)

3. A P P L I C A T I O N T O W E T G R I N D I N G I N A L A B O R A T O R Y SCALE STIRRED BEAD MILL

3.1. Experimental equipment Wet phase grinding experiments have been performed in a Dyno-Mill K D L laboratory-scale stirred bead mill as shown in Fig. 1. This comprises a 600 ml grinding chamber filled to 80% with zirconium oxide beads (diameter 1500 #m) stirred by four disk-shaped slotted blades. The temperature in the chamber is maintained constant by a jacket through which cooling water is circulated. A separating slot of 0.1 mm is located at the outlet end of the chamber to allow the ground particles to exit and to retain the grinding beads. Continuous grinding experiments have been performed by pumping the suspension to be ground through the chamber using a constant rate peristaltic pump, and taking samples at the inlet and the outlet for particle-size analysis using a Malvern Mastersizer. Batch experiments were performed by blocking the inlet and outlet of the mill and taking samples from the centre of the mill with a syringe fitted with a plastic tube.

12 --

A AAA

10 --

AAAAA

I

5

I

10

AAA

I

15

AAA

I

I

20 25 t (rain)

A

I

30

i

35

40

Fig. 2. Attainment of steady state conditions based on media particle size for (a) hydrargillite and (b) carbon.

3.2. Experimental results Two types of suspension have been ground in the mill: hydrargillite or Gibbsite (chemical formula AI(OH)3) and carbon using the experimental conditions given in Table 1. In the continuous grinding experiments, the mill was started containing only the grinding beads and water before continuous introduction of the suspension to be ground. The approach to the steady state regime for both of the materials was followed by the median volume diameter at the outlet of the mill. Figures 2(a) and (b) show stability is reached in both cases after about 15 min. In the batch experiments, the mill was loaded with the grinding beads and the suspension to give an initial uniform distribution in the mill, and after starting the mill, samples were taken at times between 0 and 15 min. The changes in the cumulative weight fraction oversize during this period were then fitted to the firstorder Kapur expression to describe the grinding kinetics. The details of this procedure have been fully described elsewhere, see Varinot et al. (1995):

KI 1) = kx~.

(25)

Values of k and ~ for hydrargillite and carbon are given in Table 1. 3.3. Approximate breakage and selection functions As described in the theoretical section, the breakage and selection functions are given by eqs (23) and the

4514

H. BERTHIAUXet aL

Table 1. Milling conditions in batch and continuous mode Bead filling (% vol)

Slurry conc (% vol)

Mill speed (rpm)

Flow rate (ml s- 1)

c~ (- )

k (s- 1#m - ~)

80 80

15 15

3385 3385

1.00 0.17

1.39 1.18

- 6.2 x 10- s - 9.0 x 10 -5

Hydrargillite Carbon

~-~ 1.00

-", 1.00

--- • Hydrargillite - . Carbon

O

e-

~ f '

:

.~**~"

gi''ite

g o.~o

", 0.10

e~

/¢ t~ O.Ol o.I

I

I

I

I

1

Relative particle size x/xj

I I I

0.01

1.0

/ / I

I

values found are shown in Figs 3 and 4 for both materials. These diagrams are in terms of the variation of these functions with particle size and it can be seen that, as demonstrated, the breakage function is in fact normalizable for both materials. To further illustrate these notions let us consider an example using the results for hydrargillite based on eight particle-size classes in geometric progression with r = 0.68 as shown in Table 2. The breakage and selection matrices valid for grinding times less than or equal to 15 min are therefore as follows:

I

I I III

I00

Fig. 4. Variation of the selection function with particle size for hydrargillite and carbon. These two matrices have been used to predict the grinding kinetics to compare with the experimental results. Figure 5 shows that the agreement with experimental results is very good for a batch grinding test with hydrargillite. 4. CONCLUSION By restricting analysis to the initial period of grinding, i.e. less than 15 min, it has been shown possible to determine the breakage and selection functions from batch grinding tests without making any

m

B =

I

I0 Particle size x (p.m)

Fig. 3. Variation of the cumulative breakage function with particle size for hydrargillite and carbon.

S=

I I11[I

0.608 0 0 0 0 0 0 0

0 0.358 0 0 0 0 0 0

0 0 0.210 0 0 0 0 0

0 0 0 0.123 0 0 0 0

0 0 0 0 0.073 0 0 0

0 0 0 0 0 0.043 0 0

0 0 0 0 0 0 0.025 0

0 0 0 0 0 0 0 0.000

0 0.412 0.242 0.142 0.084 0.049 0.010 0.061

0 0 0.412 0.242 0.142 0.084 0.049 0.071

0 0 0 0.412 0.242 0.142 0.084 0.120

0 0 0 0 0.412 0.242 0.142 0.204

0 0 0 0 0 0.412 0.242 0.346

0 0 0 0 0 0 0.421 0.588

0 0 0 0 0 0 0 1.000

0 0 0 0 0

0 0 1.000

( m i n - 1)

(min-

l)

Breakage parameters from batch grinding tests Table 2. Size intervals chosen for both slurries Size class i

dmax -- drain (pln)

1

>39.1 39.1-26.7 26.7 18.2 18.2-12.4 12.4-8.84 8.48 5.79 5.79--3.95 <3.95

2 3 4 5 6 7 8

I i,j,k KI ") P Pi Ri(t)

S Si T t Wi

100

¢-

.o

v-v.,,v~

~-

~/~

W

o 39.1 I~m • 18.2 p,m r, 8.48 I~m

~'v.-

v'v~-.~__

Wi

• 3.95 p,m

60 q

X, X i

Z

4515

unit matrix, dimensionless size indexes, dimensionless nth K a p u r parameters, s - " passage matrix, dimensionless polynomial [eq. (5)], s - i + 1 cumulative weight fraction oversize xi, dimensionless ratio for geometric progression, dimensionless selection matrix, sselection parameter, s- 1 transfer matrix, dimensionless time, s weight fraction in size interval i, dimensionless vector with components wi, dimensionless sum including the selection parameters and wi [eq. (12)], s -i+ ' particle sizes, #m intermediate vector, dimensionless

O

"~,

_o0~e~'e'e,,~e 20

=l

s o

5

l0 Time, t (rain)

15

20

Fig. 5. Comparison between the cumulative weight fraction oversize as a function of time for various particle sizes and the predictions using the breakage and selection matrices determined using the proposed method.

assumptions as to their mathematical form. This method is particularly useful in the case of extension to continuous grinding with short residence time and is fully adequate to describe such a process. However, it is obvious that the method may not be valid in circumstances less favourable than the cases treated here. F o r example, when the residence time is longer and there is superposition of other phenomena affecting grinding kinetics such as internal classification in the mill or changes in pulp rheology. Nevertheless, the matrices calculated by the method presented here could be a starting point for a more complete analysis in more complicated situations. Acknowledgement~This work forms part of the coordinated research action funded jointly by the ADEME and Ecotech-CNRS. Aid by an IFPRI grant is also gratefully acknowledged and one of the authors (C.V.) would like to thank Le Carbone Lorraine for a bursary. NOTATION A B

Bij bij EXP(t)

intermediate matrix [ = (B - I)S], s-1 breakage matrix, dimensionless cumulative breakage parameter, dimensionless breakage parameter, dimensionless vector of components exp ( - Sit), dimensionless

Greek letters #o intermediate diagonal matrix, dimensionless try- 1 sum of Si 'k by k' to index i - 1, S - k zij transfer parameter, dimensionless

REFERENCES

Austin, L. G., 1971/72, Introduction to the mathematical description of grinding as a rate process. Powder Technol. 5,1 17. Austin, L. G. and Bhatia, V. K., 1971/72, Experimental methods for grinding studies in laboratory mills. Powder Technol. 5, 261-266. Berthiaux, H., 1994, Mod61isation du broyage fin dans un broyeur 5.jets d'air et 5.lit fluidis6. Doctorate Thesis, INPL Nancy. Berthiaux, H., Heitzmann, D. and Dodds, J. A., 1994, Validation of a model of a stirred bead mill by comparing results obtained in batch and continuous mode grinding. Proceedings of the 8th European Symposium on Comminution, Stockholm. Gardner, R. P. and Sukanjnajtee, K., 1972, A combined tracer and back calculation method for determining particulate breakage functions in ball milling. Part I. Rationale and description of the proposed method. Powder Technol. 6, 65-74. Gotsis, C., Austin, L. G., Luckie, P. T. and Shoji, K., 1985, Modelling of a grinding circuit with a swing-hammer mill and a twin cone classifier. Powder Technol. 42, 209-216. Heitzmann, D., 1992, Caract6risation des op6rations de dispersion broyage. Cas d'un broyeur 5. billes continu pour la dispersion des pigments. Doctorate thesis, INPL Nancy. Kapur, P. C., 1970, Kinetics of batch grinding. Part B. An approximate solution to the grinding equation. Trans. AIME 247, 309 313. Koka, V. R. and Trass, O., 1985, Analysis of the kinetics of coal breakage by wet grinding in the Szego mill. Powder Technol. 43, 287-294. Reid, K. J., 1965, A solution to the batch grinding equation. Chem. En~tng Sci. 20, 953. Varinot, C., Berthiaux, H., Heitzmann, D. and Dodds, J. A., 1995, Pr+vision de la distribution granulom6trique en broyage fin continu. Entropie (188/189), 71-76.

4516

H. BERTHIAUXet al.

Williams, M. C , Meloy, T. P. and Tarsham, M., 1994, Assessment of numerical solution approaches to the inverse problem for grinding systems: dynamic population balance model problems. Powder Technol. 78, 257-261.

The term pz; can therefore he calculated from the i - 1 previous values using the formula

APPENDIX

It may be noted that the terms p~j can take any value forj ~>i which is not surprising since the passage matrix P is not unique as it is composed of characteristic roots to within a multiplying factor. This means that Pij = 0 if j > i and p. = 1. The P matrix is therefore of the following form:

This annex presents some properties of the matrix algebra developed in the text. Construction of the P matrix In practice, the passage matrix P is obtained by an algorithm given by the system A P = PD. Writing this in an indexed form gives

n

n aikPkj = ~. Plkdkj • k=l k=l

i-I

-- aik Pij = k~=l aii _ aj-~-'-"~jPky"

P=

I

P21 :

01 "'.......

[_P.I

01 1

0

P..-I

1

Since alk = 0 for k > i, dkj = 0 for k different from j, and djj = ajj; this can be written as

or

n ~ alkPk j = a j j p i j k=l

The elements of PD The diagonal matrix #o can be defined from the matrix product T = P#o. That is from Zlj = Pij/tj.

i-1

aii Pij -.k ~ aikPk j = a j j P i j . k=l

By putting i = j in this and noting that p~j = 1 (see above), the components of/t o are clearly the elements of the diagonal of T.