Modelling on Cooling Of Hot Sand for Metal Casting

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MODELLING ON COOLING OF HOT SAND FOR METAL CASTING H. Nomura , K. Terashima and T. Banno j Jrm / ll rf;fltl .\\"\11'111.\ J:"'I.~/tIt'n;lIg. TO.Wlhwlll

['lIiI 'fn;'.,· 01 Tt't"luudo,10. Tnll/mJw -rIHJ. TOWlhw/ii -I-IfJ, ja/Hi"

Abstract . In order to establish the optimum control system of hot sand cooling in foundry, a mathematical model for cooling process has been studied , The present study is especially directed towards modelling on dynamical behavior of the p r ocess in one pass fluidized bed . The model includes linear differential equations for phase - tophase heat transfer . Computer simulations on both transient and steady- state tempera tures of the bed are provided and compared with the experimental results Sensitivity analysis of the coo ling model has been carried out . A special interest is in the vari ation of steady- state bed temperature owing to the change of inlet sand temperature. The results of computer simulat ion are presented and comparison with the experiments is done. The proposed model is shown useful in predicting process characteristics of sand coo ling system and in designing the optimum coo l ing system Keywords . Modelling; Temperature control ; Sensitivity analysis; Foundry processing; Fluidization.

INTRODUCTION

time. Temperature of sand and air in the bed is and e" ' respectively. In steady state, discharge rate of sand is equal to the feed rate, 'is ' Air with mass flow rate qa is heated from e ,,' to e" in the bed , and flows out of the tower . It" is assumed that bed sand is forced to be coo led both by flowing air and by radia l heat transfer through the bed wall (temperature ew) towards surrounding atmosphere (temperature eoo ) '

el'

With advance in foundry technology, the problem of sand cooling is becoming of considerable importance . It is necessary to control cooling of hot returned sand properly in order to establish trouble - free mou l ding operation and yield sound castings. The first step for designing control system is to comp l ete the mathematical model of the cooling process and to get informations on dynamical behavior of the system o

'lathematical model describing the cooling process in the fluidized bed can be described as follows:

In this study model experiments of continuous cooling of sand have been made using one - pass type f l uidizing tower in which continuous feed and dis charge of solid particles are done . Heat transfer model has been constructed and computer simulations have been carried out. Identification of the sys tem model has been performed both through comparing computed results with experimental ones , and through sensitivity analyses of the model. Studies of sand cooling using batch- type fluidized bed were done previously by Onaka, et al . (1971 , 1974) and Ozaki , et al (198 1 ). However no study has yet been found for con tinuous sand cooling close to the actual process.

de

cs --.!!... = dt

0

ps

de

ca ---.E. = dt

-0

- h

q S

(e . - e ) + h

q (e pa a a A

S

S1-

- e .) + h a1-

A

sa sa

A

(e - e )

sa sa

a

s

(7)

(e - e ) s

a

(e - ew)

awawa

(2 )

de

cw~ = /j A (e - e ) - h A (9 - e ) dt aw aw a w Woo Woo W

(3 )

in which hsa and haw are heat transfer coefficients for forced convection , whil e hw~ for natural con vection occuring around the cooling apparatus . The correlations presented by Sugiyama, et al. (1969) , Cooper, et al. (1958) and Bird , et al . (1960) are adopted for hsa > /jaw and hwoo ' respectively.

The present research is fo c used on establishment of the model for cooling of hot sand without moisture addition. In this case the model containing heat transfer between phases can be treated in a simple manner . From practical point of view , therefore , the findings obtained may be related rather to cooling of dry state sand than that of usual returned sand which is achieved exclusively by transfer of latent heat of vaporization. The study in the latter case will be done in the near future .

Equations (1) to (3) are linear differential equa tions resulted from heat balance on bed sands , air and bed wall. The formulation shown above can be applied to both one - pass type and batch - type flu i dized beds , qs ~2ing excl'JGea fr or.\ the above equations in the latter case. Transient temperature orofi l e of the bed , es ' and air temperature, ea ' are computed using this mod el . Further sensitivity equations can be derived to predict stability of the cooling system against the variation of its parameters .

r.10DEL OF COOLING PROCESS Figure 1 shows a schematic il l ustration of one-pass type fluidized bed . Hot sand of temoerature eo,' is supplied to cooling tower with mass flow rate In the tower sand i s fluidized by incoming air," and discharged from the bed after a certain residence

:,c: .

1955

H. Nomura, K. Terashima and T. Banno

1956

nning of fluidization were sought after and

EXPERIMENTAL

assigned to each experiment ,

A vertical-type fluidizing tower was constructed in order to conduct model experiments of cooling of hot sand. Figure 2 shows a typical design of the experimental apparatus. Hot sand in the hopper was fed into the cooling tower through a rotary f~eder. The cooling tower was made of a acrylic p1pe of 100 mm in diameter. Air from the compressor was passed through a flow meter and flowed into the tower from its bottom. Bed temperature was measured

continuously by CA thermocouples inserted at three locations in the bed. Other thermocouples were installed near the inlet and the exit of the bed to measure both temperature of supplied and discharged sand. Pressure drop through the bed was checked by a manometer. Flow rate of coolant air was varied in the range of

20 to 180 Nl/min, while feed rate of hot sand 160 to 240 g/min. Cooling characteristics were examined for temperatures of feed sand: 50°C, 70°C and 90°C. Two kinds of silica sand, No 6 and No <7 were used in the experiments. Table 1 shows grain size distribution of sand specimen . In typical experiments, sand is put into the apparatus and fluidized by air. After the fluctuation of bed pressure drop becomes negligible, continuous supply of hot sand into the bed is commenced , At the same time cooled sand begins to be discharged from the exit.

Temperature monitoring is continued

till the bed temperature reaches a steady-state value.

F

e p

indi~ates

that rapid mixing states are produced,

lead1ng temperature to be uniform throughout the bed. In the figures are also shown simulation curves of temperature change computed from the present model

Figure 5 gives the result for the inlet sand temperature about 50°C, and Fig , 6 about 90°C. In each case bed temperature profiles are obtained

by varying air flow velocity , In this study to hold the inlet sand temperature constant during the experiment is difficult especially for 90°C as indicated in Fig , 6. In computer simulation, therefore, empirical expressions representing the actual

inlet temperature were introduced in Eqs. (1) to (3) to obtain agreeable numerical solutions of bed temDerature profile. It is shown in both figures that the process of temperature change up to steady state is fairly affected by air flow rate, and the simulation explains the experiments reasonably

Figure 7 shows the effect of sand feed rate on the bed temperature change ,

of computer

These experiments were

This trend can

s~multaneous

a1r

ea

simulation ~

Simulation gives the solution of temperatures of sand 8 s

and wall

ew as

'

suggested from Eqs. (1) to (3)

In the present case sand temperature is comouted to be nearly equal to air temperature, as discussed

by Lebenspiel, et al. (1977).

Therefore computed

curves are described as bed temperature in the

above figures. Figure 8 shows steady-state values of bed temperature which are plotted against air flow rate . As seen from this, the calculated temperature shown by the broken line falls quite near the measured one .,

As mentioned above, the present model predicts reasonably well both transient and steady-state temperatures for continuous cooling of hot

sand ~

It is added further that the quantitative agreement between experiments and computer simulations has

been found for batch cooling. These indicate that the cooling model can simulate the characteristics

(i) for fixed bed (T o Shirai (1956)) 2

No clear difference in

be also predicted quantitatively from the results

As yagi and Shirai (1949) reported, pressure drop through fluidized bed is kept nearly constant even when fluid velocity through the bed is changed. In this study bed condition was investigated using No.6 sand by varying air flow velocity , Bed pressure drop was measured and plotted in Fig.3. This result demonstrates the case of batch fluidization without continuous feed and discharge of sand. As seen from the figure, for three cases of bed weight, 6p rises with increase of air flow velocity and thereafter it reaches a constant level. It was observed that bed was settled in fixed one for the range of low pressure drop, while it developed into fluidized bed for the range of constant pressure drop. Solid lines in the figure show the value calculated from the following equations:

u p l g !J

experimental conditions.

bed temperatures observed from the three thermocouples was found during each experiment. This

panies the rise in bed temperature

Fluidization Characteristics

= 50r.CJa L (

Figures 5 to 7 represent typical examples of transient temperature change of the bed under various

done under the inlet sand temperature about 50°C and air flow rate 180 Nl/min , It is clearly shown that the increase of feed rate of hot sand accom-

RESULTS AND DISCUSSIONS

6p

Sand Cooling Process and Computer Simulation

)

(4 )

of sand cooling over a broad range of conditions.

This leads to the conclusion that the mathematical expressions in the present model are adequate and

(ii)

can be utilized to establish control functions in sand cooling system.

for fluidized bed (from weight balance)

(5 ) Measured values agree reasonably with calculated ones o Additional experiments show that almost the same value of pressure drop is obtained in the continuous operation as in the batch one ,

Air flow

rate in later experiments of fluidization cooling is decided using the above results. In Fig.4 the ratio of sand discharge rate to sand feed rate was plotted against time of fluidization. After 8 min , feed and discharge rate become nearly equal and weight of bed sand is changed little , Fluctuations seen in steady state are largely due to the characteristic movement of bed sand affected by air bubbles. The initial transient is caused by the deviation of sand discharge rate from the one equilibrated with given air flow rate and sand feed rate. Conditions equilibrated from the begi-

As one of the features of this model, it is noted that the model provides stable numerical solutions against irregular fluctuations of system parameters

such as bed weight and height of fluidization. As a typical example Fig 9 shows the influence of fluidization height on temperature change. Inlet sand temperature, omitted from the figure,

is the

same as that for air flow rate 120 Nl/min in Fig 5. In this case the bed height is observed to change as much as 95 - 140 mm during the experiment. So computation of bed temperature is done for two

values of fluidization height, 95 and 140 mm However it is found that the temperature profiles are not so changed for the two cases, as seen from

the figure This indicates that even if accurate parameter change is unknown for those described above, simple treatment of the model can afford proper prediction of cooling behavior. This is

Cooling of Hot Sand for Metal Casting

considered to be one merit of the present model when utilized as control model in actual systems.

f!.6

1957

(16 )

s

Sensitivity Analysis of the Cooling System The sensitivity of a system to the variations of its parameters plays an important role in the analysis and synthesis of control systems. A sys tematic survey of researches on the sensitivity of systems has been done by Kokotovic and Rutman (19 65) and Ngo (1971). The problem of investigating the supplementary motion caused by small parameter variations leads to the analysis of the following partial derivatives:

ox.1.

(6 )

aa. J

Where f!.a.,., ~ 0 (m tJ) . Here ui i is the sensitivitv function of the quantity x i relative to the parameter 0.':0 In this study the principle of sensitivity analysis has been applied to the investigation of temperature stability . In particular , the movement of steady - state bed temperature due to the change of inlet sand temperature is analyzed and compared with experiments . In steady state , left - hand sides of Eqs . (1) to (3) being equa l to zero,one obtains:

A :r=b

(7)

in which 6

Figures 10 and 11 show the shift in bed temperature profiles affected by the change of inlet sand tempera tures. In each case parameters are kept unchanged except inlet sand temperature The maximum difference values of inlet sand temperature observed in Figs.lO and 11 are approximately 6.5°C and 20°C, respectively . The corresponding variation of s teady-state bed temperature is computed using Eqs . (12) to (16) and its variation range is marked in each figure. The reasonably good agreement between experimental measurement and theoretical prediction is demons trated. In most control systems , much attention is paid to slight changes of variables from their steady-s tate values. In suc~ cases , quantitative estimation of the effect of parameter variation on the steady- state value can be done by the procedure presented here. From the above discussion it is conc luded that the sensitivity analysis gives reasonable explanation to the actually observed phenomena . It again supports the availability of the present Mathe matical model in the control system of sand coo ling process. As the following step , an opt imal control system on cooling of hot sand can be designed by modern control theory. The LQI-control with robustness for continuous disturbances may be useful approach for temperature control of hot sand The main results of c onstructing optimal control systems will be presented in the near future

(8 )

s

SUMMARY

A

(9)

h1.J~ AWoo

(1 0)

where primes in Eqs. (8) and (10) denote transpose. From Eq .(7):r is given by (11 )

Using this equation vector element 9~ is solved , and the sensitivity of steady-state bed temperature to the variation of inlet sand temperature is derived on the basis of definition , Eq (6) , as

~OMi:l'ICLIC U [,.E

aes as

si

a :'s X ~

~' S

x+

(12 )

C

heat capacity ( J/K )

q

mass flow rate ( k /s ) g heat transfer coefficient

y + 3

s ['C'cifi c heat

where

x

As a first step to establishment of the optimum system of sand cooling process , the mathematical model of the process including heat transfer equations was proposed . Computer s imulations were done and the results were compared with those obtained from the experiments of continuous coo lin g of hot sand in the one - pass type fluidized bed . The results are summarized as : (1) Computer simulation predicted reasonably well the obse r ved temperature profile at both the transient and steady- state periods (2) Sens itivity analysis was done and the variation of steady - state bed temperature caused by the change of inlet sand temperature was discussed Reasonably good agreement between measurements and predictions was attained . (3) The present model of cooling process was backed up by the experimental findings. It is possible to design the optimum cooling system on the basis of the present model

h =

.J.rea (

( ,T/kg · K

bed heiq:lt ( (14 )

Cl

(15 )

-

fluid densitj ( k / m grdvitati~!lal

)

3

)

conversi o n fa c tor

particle G ia~eter ( m ) Neg le c ting high order terms of Taylor series expansion , one can describe the deviation ASs caused by the variation of inlet sand temperatur e

toe . 3 1.

·K

)

q

z

2

IT?

drag ccefficic!1t

i

'w/m

bed weight ( kg 2 cross sectional area of bed ( m

~

-

)

1958

H. Nomura, K. Terashima and T. Bann o

SUBSCR I PT

s

sand

Q

a ir

i.!

wall

.~;-

sand - a ir

· -1

air - wall

".: .;l

wall - atmosphere REFERPJCES

Bird , R B , W E. Stewart and E . N. Lightfoot ( 1960) . Tr ansport Phenomena I John \-li ley and Sons , In e . , New York , ?p 414 . Kokotovic , P . V., and R . S Rutman (1965) . Sensiti ~ity of Automatic Control Svstems (Survey) , Automation and Remote Control , 26 , pp . 727 . Kunii , 0 , and O. Leve n spie l ( 1977) . Fluidizati on Engineering , R. E . Krieger Pu b . Camp , New Yo rk, pp 8 . Ngo , N T . (1~ 71' . Sensi tivity of Automatic Contr ol Systems (Review) , Automat ika Telemekhanika , No . 5 May. pp . 53 . Onaka, 1. , T Uemura and K. Chiji iwa (1 97 11 . Coo li ng of Foundry Sand hy Counter Air Flow , Imono(J Japan Foundrymen ' s Soc . ) , ~ , No . l O, pp . 893 . Onaka , I . , Y Kita and K Chij iiwa ( 1974 ) . Cooling of Molding Sand by Vib ratin g Fl u idiz ed Bed , Imono (J Japan Foundrymen ' s Soc . ) , ~ , No . 2 , pp . 124

TABLE 1
~0.6

silica sand No . 7 sil i ca sa nd

Ozaki , M , and R. Isomura ( 1981) . Study on the M.ol.: ir.g Sand Co0 1i nq , ':"oyoda '.2e.::hnic...ll Review , No 3 , Apr ., pp . 9 Shirai , T . (1956) . Fluid F l ow throug h Fi x ed Beds; Reoresentation of the Fluid Friction with the Drag Coefficient of Single Parti cle , CD ' Kagaku - Kogaku (Chemical Eng Japan) , 20 , No . 8 , pp . 434 . Su giyama, S ., 11 Kasatani and N. Arai ( 1969) . Heat transfer in In clillcd Fluidized Bed , Kagaku - Kogaku (Chem i ca l Eno Japa n), li , No . 5 , pp . 435. We nde r , L , and G. T . Coope r (1958) . Heat Tr ansfer between Fluidized - Solids Beds and Boundary Su rfaces ---- Correlation of Dat a , A. I . Ch . E Journal , 4 , pp . 15 Yagi , S . , and T . Shi~ai (1949) F l uidization Veloc ity of Powder (Studies on Fluid Cata ly st 1) , Kagaku Kikai (J . Chemical Mash ine , Japanese) , ~ , pp .l80 .

Grain Size Distribu t ion of Specimen ( % )

28 ( 59 0)

35 (420)

48 (297 )

70 (2 10)

0.5

3.4

20 . 1

33 . 8

tr.

tr.

t r.

2.4

100 (149) 27 . 2 20. 0

14 5 (105) 11. 5 55.0

2 00 (74)

250 (53 )

pan

2. 9

0.5

0 .1

19 .6

2.0

0.8

1959

Coo l ing of Ho t San d fo r Me t a l Casting

Hopper

~nl~te s~~, ~ s

~ qa

' ea

11 __________ _

~'N.j- . . . . . ' . . ...

Fluid iz e d bed

. ...

,

,

,

"

'

Ambi e n t t e mp, ( e~ )

Flo .... meter

Fig , 1 .

Air fi Her

Schematic illustration of fluidized bed.

T

Air compre Ssor

Fig . 2. ~o.

Va 1ve

6 sand

Exp erimental apparatus.

3, 5,r-------------------------------------~ Calculated

100

No.

3 • Fluidized bed

Fi xed bed

~

100

7 sand

Ai r flow rdte

600 9

o 20 Nl /min • 100 c. 150

5a

... 180 Sdnd feed rate

).-_ _ _~~~~-..e..:l>l>

f

:; 10

Bed we ig ht

0,003

0 . 15

0.01

0.1

0. 5

1.0

3.0

6

Air flow velocity (m/sec)

8

10

rime (min)

Fig Fig. 3.

4.

~60,--------__________________

g. <1J

ratio in continuous

fluidization.

~

0- - - 0. - - - - - - - - - c;.

40

rl,./7 /

6 et

_' er - -

0

.,

--

0- - -0 - -

"

....os- - -<1: - - o~

0-- - -0 - --0

0·"0'

~

90

.0-'

J::f

--

SO

11

Change of discharge - to- feed

Relation between bed pressure drop and flow velocity .

t'. 100

...

40 9!m;n

100g

~

o.

p._,D

~==--~

80

8

---

g~-~'~

.,

.,

f/

70

~

60

~ 30r-----~---~----~---~---"--

:-~~=:= Sdnd feed rate Caled. ~

a:)

10

Obsvd .

L -___

~

______

200 9/min[

I

Air flow rate 6:) Nl/mln

:~6

No. 7 sand ~

______

~

6

______

~

50

___

Ii

~_

;'

~

40

Calcd

l

10 dJ

10

Time (min)

10

Fig . 5 .

Sand feed rate

30

280 g/ rnin

~

Change of bed temperature with time .

Obsvd

No. 7 sand

Air flow rdte 60 Nl/min

110 180

6 Time (min)

~ig .

I WC4-J *

6.

Change of bed time

tem~erature

10

with

1960

H. Nomu r a , K. Te r as hima a nd T. Banno

~60 r-----------------------

--------~

~

I

(a)

Inlet sand temp .

Sand feed rate

lac

Ca l cd.

Sand feed rClte

o

Nl / mi n

200 g/min

la)

Calcd . Qbsvd .

60

240 g/ min

100 160

SO

:;'

4 6 Time (min)

10

cl. 40

! D

Change of bed tempera ture with time.

~

60

.

~ ~

Fig . 7 .

(c)

l sooc

70

Air flow rate Obsvd.

I

(b)

90°C 1 70°C

(D)

SO

Ca l ed . Obsvd .

,;

~ ~ ~

.,

)0 SO

Ic) 60.--------------------------,

o

Obsvd.

Height o f fluidizdti on

SO

40

~

)0

~

~

20

. o·

V9~et ;~~~

10

Fig . 8 .

120 Nl /m in

4 6 Time (min)

Fi g . 9 .

60

110

130

Air flo w rate (Nl/mi n )

="

sand temp. 50"C feed rate: 200 g/min

Air fl ow rate

-0

10 0

~ 30

--0. -"

95 "'" 140

Ca lcd .

CalCd. Obsvd .

40

Steady - state bed tempera ture plotted against air flow rate .

10

Effe c t o f fl u i d izati o n height

:;'30

on tempe ratu r e change .

r------ ------- ------- ------- ------,

.': SO

fJ

40

o

~ )0

~

~

40

f ~--~--~--~----~--~--

- ~­ fu.-BL - -- - ---r

o

~ 20

- - - - -i1i -

Ai r (low r
1, 0 Nl / min

S,wd f eed rHe

160 IJ / min

~jo .

6

10 .

ro

10

Air flow rate

12() Nl /min

Sand feed rate

200 g/min

No . 7 sand

6 Time (mi n)

7 sa nd

Time (min)

F ig

~

10

Effect o f inlet s and tempera ture on bed tempera ture .

Fig . 11 .

10

Effect of inlet sand tempera ture on bed tempera ture .