Heat (mass) transfer between an impinging jet and a rotating disk
Y.-M. Chen, W.-T. Lee, S.-J. WuAbstract The aim of this experimental study is to in-vestigate the heat (mass) transfer of a rotating disk with an impinging circular jet. To facilitate the experiments, the naphthalene sublimation technique was employed. In or-der to analyze the results, measurements of the heat (mass) transfer of a stationary disk with an impinging jet and a rotating disk without jet impingement were also made. From the experimental results, it is found that the heat (mass) transfer are precisely divided into three regimes, namely the impingement dominated regime; the mixed regime and the rotation dominated regime. Correlation of Sherwood number of a rotating disk with jet impingement is also proposed in the present work.
List of symbols
A constant (301.6247), dimensionless
As A1 791.4937; A2 )8.2536; A3 0.4043,
di-mensionless
C correlation constant, dimensionless D Diameter of circular jet (m)
Df Diffusion coef®cient of Naphthalene into air (m2/s)
Es(x) x (2T ) 574)/117; E1(x) x; E2(x) 2x2 )1;
E3(x) 4x3 ) 3x, dimensionless
lsb the rate of change in naphthalene thickness due to
sublimation (m/s)
h local heat transfer coef®cient W/(m2K)
hj mass transfer coef®cient of a rotating disk caused
by jet (m/s)
hrj mass transfer coef®cient of rotating disk with jet
impingement (m/s)
hr mass transfer coef®cient of a disk rotating in still air
(m/s)
hm local mass transfer coef®cient (m/s)
H jet tube-to-disk spacing (m) Nu Nusselt number, dimensionless
Nur pure rotational Nusselt number (hR/k),
dimen-sionless
Nurj jet-Rotational Nusselt number, dimensionless
Pr Prandtl Number (v/a), dimensionless
PV the vapor pressure of the Naphthalene at TW(N/m2)
Qj the jet volumetric ¯ow rate (m3)
Qp the rotationally induced pumping volumetric ¯ow
rate passing the impingement radius (m3)
r Radius of disk (m)
R Impingement radial position (m) Rm Ideal gas constant (joul/moleáK)
Re Reynolds number, dimensionless
Rer Rotational Reynolds number (xR2=m),
dimension-less
Rej Jet Reynolds number (UD/m), dimensionless
Sc Schmidt number (v/Df), dimensionless
Sh Sherwood number (hmR/Df), dimensionless
Shr Pure rotational Sherwood number (hrR/Df),
di-mensionless
Shj Pure jet Sherwood number (hjR/Df), dimensionless
Shrj Jet-Rotational Sherwood number, dimensionless
T Temperature (K)
TW is the surface temperature of the naphthalene (K)
U Average exit velocity of the jet (m/s) Greek symbols
a Thermal diffusivity (m2/s)
x Rotational speed of disk (rad/s) m Kinematic viscosity of air (m2/s)
q0 the density of the local naphthalene vapor density
in the free stream (kg/m3)
qs the density of solid naphthalene (kg/m3)
qvw the local naphthalene vapor density on the disk
(kg/m3)
1
Introduction
Heat transfer on rotating disks is a commonly occurring topic in convection heat transfer, especially in rotating machinery. Jet impingement is known to be attractive as means of intensifying convective processes. When im-pinging jets are applied to the rotating disks, they can produce very high local heat or mass transfer rates. The rotating disk induces on its surface an axisymmetric wall jet which interacts with the impinging jet to produce a cross-¯ow effect, deforming the jet trajectories and consequently deforming the distribution of impingement heat/mass transfer rates. Applications involving impinge-ment cooling of rotating surfaces are found in many cooling processes, such as in bearing and gear cooling and in the cooling of gas turbine disks. In most of these ap-plications, it is important to have knowledge of the disk temperature and the ability to control it. This means also a necessity of accurate and detailed knowledge of the
con-Originals Heat and Mass Transfer 34 (1998) 195±201 Ó Springer-Verlag 1998
195
Received on 12 January 1998
Y.-M. Chen, W.-T. Lee, S.-J. Wu Department of Mechanical Engineering
National Taiwan University, Taipei, Taiwan 106, ROC Correspondence to: Y.-M. Chen
vection heat transfer between an impinging jet and a ro-tating disk.
Despite the recognized need for detailed heat transfer information on rotating surfaces, its acquisition has been slowed and is still incomplete because of the expense and complexity involved with making local heat transfer measurements on rotating surfaces. Conventionally such measurements involve mounting heat ¯ux gauges, or spot heaters and thermocouples, on the disk surface, and transmitting electrical power and measurement signals from and to the rotating apparatus through slip rings. The expense and time factors involved usually result in testing of only limited geometric arrangements and ¯ow condi-tions.
Owen and Rogers [1] provide a concise review of lit-erature addressing both the ¯uid mechanics and heat transfer aspects of the subject through 1989. There is only sparse local heat transfer information available, which can be found in studies by Popiel et al. [2], Metzger & Grochsowsky [3], Metzger et al. [4], Bogdan [5], Popiel & Boguslawski [6], Metzger et al. [7]. By using the transient technique Bogdan [5] made measurements of local heat transfer. In his paper, correlation equations for local Nu-sselt number in laminar and turbulent regions were pro-posed. However, in his equations, the effect of the ratio of the radius of impingement to the jet diameter (R/D as shown in Fig. 4 and Fig. 5) was not considered. Moreover, his results were obtained on the nonisothermal surface of the disk; therefore, the uncertainty of his results must be high. The measurements of local heat transfer done by Popiel & Boguslawski [6] were carried out using a ring-shaped h-calorimeter. With the aid of smoke ¯ow visual-ization, three regimes of the impinging round jet and ro-tationally induced disk pumping ¯ow interaction have been distinguished: Jet impingement dominated regime; mixed regime; and rotation-dominated regime. In their study, heat transfer correlating equation has been given for the jet impingement dominated regime only. Recently, Brodersen & Metzger [8], Brodersen et al. [9] [10] concentrated their studies on the ¯uid ¯ow.
From the above literature review, it is found that proper heat transfer correlation, covering the three ¯ow regions are still lacking. Also, criterion for the ¯ow regime tran-sition proposed in the literature need further elucidation. The objective of the present work has been to ful®l these tasks. In this study, experimental investigation has been carried out by means of the naphthalene sublimation technique. By using this technique, the dif®culties and uncertainties that may occur in the conventional heat transfer measuring system can be circumvented. To ana-lyze the results, experiment on a rotating disk without impinging jet and experiment on a stationary disk with jet impingement were also conducted.
2
Experimental apparatus and procedure
Figure 1 shows schematically the main parts of the ex-perimental equipment employed in the present study. A test disk with a diameter of 240 mm was used in the measurements. A sheet of naphthalene with thickness of 5 mm was coated on the upper surface of the disk, and
then, the disk was attached to the end of a vertical shaft and driven by a variable-speed electric motor with speed range from 60 to 4000 rpm. The rotating speeds were measured by means of a stroboscope. The nozzle, which af®xed perpendicularly to the test disk at a desired posi-tion, was a circular jet of 7.5 mm inner diameter and was ®tted with a thermocouple to indicate the jet temperature. The aluminum jet nozzle was designed according to the hand-book of ASME (Long Radius Nozzle)[11]. Com-pressed air, after ®ltering and drying, was supplied to the rotating disk.
All the measurements ran in a suf®ciently large and air-conditioned room. By continuously measuring the room temperature with thermocouples, the variation of tem-perature during each test run had been strictly kept within 1°C. Before and after the run, naphthalene surface level was measured by use of an automated two-axis data acquisition system. This system was designed and con-structed to ful®ll the requirements of mass transfer mea-surements, summarized as precise positioning, accurate surface elevation reading and fast data acquisition. This system consists of a two-axis positioning stand, a depth gauge, a PC and a motor controller system. A schematic diagram of the con®guration is shown in Fig. 2. A stan-dard IEEE-488 bus interface was used to transmit data between the components of the system. Since the local mass transfer rate was determined from the difference in naphthalene surface pro®les measured before and after exposure to the ¯ow, the test object should be precisely installed on the positioning mechanism. The positioning mechanism consisted of an X-Y table. Stepping motors drove the movement of the two-axis stand. Each motor was equipped with an encoder and was driven by a step-ping motor driver. The travel distance in both X and Y directions was 130 mm. The X-Y table had positional re-peatability over this range of 0.46 lm. Since the average sublimation depth for the mass transfer measurement was about 200 lm, a high resolution depth gauge, linear vari-able differential transformer (LVDT), was used. Its re-peatability, according to the manufacturer's speci®cation, Fig. 1. Schematic of apparatus for the experiment of the plate rotating with the impinging jet. 1 Screw compressor, 2 Air tank 3 Air drier, 4 Filter, 5 pressure regulator, 6 honeycomb and screens, 7 impinging jet, 8 test disk, 9 thermocouple, 10 motor, 11 frequency regulator, 12 recorder, 13 stoboscope, 14 ¯owmeter 196
was 0.1 lm and its measurement range was 0.5 mm. The electronic signal in response to the gauge tip displacement was ampli®ed and converted into a DC voltage that was read then by a digital multimeter.
3
Data reduction procedure
The rotational Reynolds number (Rer) and jet Reynolds
number (Rej) are the governing parameters in the present
study. They are de®ned as Rer xR2=m and Rej UD/m,
where R is the distance between the jet center and the disk center, x is the rotating speed of the disk, D is the di-ameter of the circular jet, and U is the average exit velocity of the jet. The values of U were obtained from the volu-metric ¯ow rate measured by a ¯ow meter. A LDA system was also used to check the value of U. The experimental parameters used in the present study are listed in Table 1.
In the present study naphthalene sublimation technique was used to measure the mass transfer coef®cient instead of the heat transfer coef®cient. The value of the heat transfer coef®cient was calculated by applying the heat/ mass transfer analogy. The validity of this analogy was discussed in Goldstein & Karni [12]. The mass transfer coef®cient is de®ned as
hm lsbqs= qmwÿq0 1
where hmis the local mass transfer coef®cient, lsbis the
rate of change in naphthalene thickness due to sublima-tion, qsis the density of solid naphthalene, qmwis the local
naphthalene vapor density on the disk, and q0 is the
density of the local naphthalene vapor density in the free stream (which is equal to zero in the test condition). The naphthalene vapor density on the disk in Eq. (1) can be obtained by applying the ideal gas law
Pm qmwRTw 2
The equation that proposed by Ambrose et al. [13] is used to determine the vapor pressure of naphthalene, Pm.
T log Pm 0:5A
X3
s 1AsEs x 3
This equation has been widely used in other mass transfer measurement, like Souza [14], Goldstein & Cho [15]. The mass transfer coef®cient can be nondimensionalized by the following equation:
Sh hm D=Df 4
where Dfis the diffusivity of naphthalene vapor and D is
the nozzle diameter. Sherwood number can be trans-formed to Nusselt number (Nu) for heat transfer by using the analogy between the two processes as
Nu Sh Pr=Scn 5
At T 298.15 K, the value of Schmidt number is 2.28 as suggested by Chen et al. [6], and the value of Prandtl number is 0.7. The naphthalene sublimation technique makes use of the heat and mass transfer analogy to de-termine heat transfer coef®cients in convection ¯ow by measuring mass transfer coef®cients. This is discussed in detail in Souza [14] and Cho et al. [15].
The Rer and Rejthus obtained were estimated to have
uncertainly less than 2.24% and 2.08%. The maximum uncertainty of the Sherwood number is estimated to be less then 5.02% for rotating Reynolds number greater than 1000 according to the method suggested by Kline & McClintock [17].
4
Results and discussion 4.1
Heat/mass transfer of a disk rotating in still air As a jet impinging on a disk, the dominant heat/mass transfer mechanisms conducted by the impinging jet or the rotating disk should be made out. Hence, the behavior of heat/mass transfer with a jet striking on a stationary disk and a disk rotating in still air should be investigated separately. The experimental work with a jet impinging on a stationary disk was also carried out in the present study. Because of the space limitation of the paper, the results are not presented here.
Figure 3 is a plot of Sherwood number as a function of rotational Reynolds number while the test disk rotating in still air. Obviously, values of the Sherwood number against the rotational Reynolds number indicate the existence of three regions: laminar, transition and turbulent. Correla-tion between Sherwood number and rotaCorrela-tional Reynolds number in the three regions, is made form the present experimental results. The correlation equations can be expressed as:
Fig. 2. Automatic data acquisition system. 1 IEEE - 488, 2 IBM PC/XT, 3 LVDT, 4 voltage meter, 5 Probe, 6 stage ®xed frame, 7 MC4 stepping motor controller, 8 MC4 controller, 9 measurement plate, 10 stepping motor
Table 1. The list of parameters used in the present study
x (rpm) Rej R/D H/D 500 2000 0 5 1000 4000 3 1500 6000 6 2000 11000 9 2500 16500 12 3000 25000 3500 40000 4000 60000 80000 100000 197
Shr 0:59Re0:5r Rer 2:0 105 in laminar region (6a) Shr 2 10ÿ19Re4r 2:0 105 Rer 2:5 105 in transition region (6b) Shr 0:0512Re0:8r Rer 2:5 105 in turbulent region (6c) The three distinct regions were also given by Bogdan [5] and Popiel et al. [2] from the values of Nusselt number as a function of rotational Reynolds number. The critical ro-tational Reynolds number to divide these three regions in the present study are close to that reported by Bodgan [5] and Popiel et al. [2].
Results of mass transfer obtained from Kreith et al. [18] and Sparrow & Chaboki [19] in laminar region are also displayed in Figure 3 for comparison. The correlation equation attained from Kreith et al. [18] has the form, Shr 0:6Re0:5r , and the equation obtained by Sparrow &
Chaboki [19] is expressed as Shr 0:625 Re0:5r .
Compar-ing the equations in the literature with the present exper-imental data, the maximum deviation is 5.9%. The present results show good agreement to that reported by Kreith et al. [18] and Sparrow & Chaboki [19]. As also can be seen in Fig. 3, the Sherwood number obtained in the present study is in proportion to fourth power of rotational Rey-nolds number in transition region, and is in proportion to 0.8th power of rotational Reynolds number in turbulent region. Similar proportion between Nusselt number and rotational Reynolds number in these two regions were also reported by Bogdan [5], Metzger et al. [4] and Popiel et al. [2].
4.2
Heat/mass transfer of a rotating disk with jet impingement Figures 4 and 5 show the measured Sherwood numbers as a function of R/D and Rer with a jet striking on the
rotating test disk. The distributions of Sherwood
num-ber at different values of R/D are shown in Figure 4. In this case, Rej is ®xed at 60 000. Obviously, as compared
with the result without jet impingement, there is a sud-den increase in Sherwood number as Rer 2:0 105
(the laminar region discussed previously) for the presence of an impinging jet. In the region as
Rer 2:0 105, the Sherwood number remains
differ-ent constants at differdiffer-ent values of R/D. The result im-plies that the increase in rotational Reynolds number causes insigni®cant in¯uence on the Sherwood number. On the contrary, variations of values R/D lead to con-sequential effect on the Sherwood number. A smaller value of R/D gives rise to a larger value of Sherwood number as Rer 2:0 105. It is noteworthy that the
transition from laminar region to turbulent region as the disk rotating in still air is ambiguous for the presence of jet. From the value of Shrj against Rer shown in Fig. 4,
the value of Shrj almost matches that of a disk rotating
in still air as the rotational Reynolds number is greater than a critical value. The critical value of rotational Reynolds number is slightly different from the various values of R/D.
Figure 5 is a plot of Sherwood number with jet im-pingement against rotational Reynolds number at various jet Reynolds number. In this case, the value of R/D is kept at 6. The variations of Shrj against Rerdisplay much
similarity to that shown in Fig. 4. The value of Shrjnearly
remains constant as Rer 2:0 105at different values of
Rej. A larger value of Rejleads to a larger value of Shrjas
Rer 2:0 105. As can be seen from Fig. 5, the value of
Sherwood number with jet impingement comes close to that of a disk rotating in still air as rotational Reynolds number is greater than a critical value. The result indicates that the impinging jet produces trivial in¯uence on heat/ mass transfer, as rotational Reynolds number is greater than a critical value. Obviously, in¯uence of jet on laminar heat/mass transfer is much greater than in turbulent. Progress in laminar heat/mass transfer may be attributed to introduction of new amount of fresh air into the disk
Fig. 3. Local heat/mass transfer of a rotating disk in still air Fig. 4. Local heat/mass transfer of rotating disk with impingingjet at Rej 60 000
boundary and promotion of turbulent level in the boun-dary caused by jet.
From the results shown in Figs. 4 and 5, there can exist three regimes of heat/mass transfer. The regime where the Sherwood number remains constant can be called as jet dominated regime since the rotational Reynolds number gives negligible in¯uence on Sherwood number. As rota-tional Reynolds number is greater than a critical value, the Sherwood numbers measured at various jet Reynolds number come very close to that measured for a disk ro-tating in still air. The regime can be named as rotation dominated regime since the impingement jet generates incidental in¯uence on heat/mass transfer. A narrow re-gime, where both rotational disk and impinging jet produce important effect on heat/mass transfer, survives between these two distinct regimes de®ned previously and can be speci®ed as mixed regime. Precise de®nition and classi®cation of these three regimes will be made below.
In Fig. 5, the Nusselt number obtained by Popiel & Boguslawski [6] is also exhibited for comparison. The experimental conditions in Popiel & Boguslawski [6] are close to that in the present study. From the result shown in Fig. 5, it reveals that the Nusselt number ratio of Nurj/Nur
reported by Popiel & Boguslawski [6] shows good con-sistency with the Sherwood number ratio of Shrj/Shr
obtained in the present study.
To make a further discussion of heat/mass transfer of a rotating disk with jet impingement, the following steps deduce correlation of Sherwood number:
1. Calculate hj hrj) hrat each measurement point,
where hjis the coef®cient of mass transfer caused by jet
impingement.
2. Find proper correlation equations for Shj, where Shjis
de®ned as Shj hjR/Df.
By virtue of the above procedures, the jet Sherwood number is correlated as
Shj CRe0:7r R=D0:1 7
It is noteworthy that the Sherwood number of a rotating disk with jet impingement is then expressed as
Shrj Shj+ Shr
CRe0:7
j R=D0:1 jet impinging component
0:59Re0:5 r ; Rer < 2:0 105 (8a) 2 10ÿ19Re4 r; 2:0 105 Re 2:5 105 (8b) 0:051Re0:8 r ; Rer > 2:5 105 (8c) 8 > < > : (rotational component)
From the result shown in Eq.8 (a)(b)(c), the Sherwood number of a rotating disk with jet impingement can be divided in two components. One of the components is
Fig. 5. Local heat/mass transfer of a rotating disk with impinging jet at R/D 6
Fig. 6. The correlation of the ¯ow regimes of a rotating disk with impinging jet
guided by the impinging jet, and the other is directed by rotating disk. The value of C in Eq. (7) is not a constant. Figure 7 shows the values of C as a function of R/D and Rer. The result in Fig. 7 reveals that C is a function of R/D
only as Rer < 104; a function of Rerand R/D as
104 Re
r 1:5 105, and a function of Rer only as
Rer> 1.5 ´ 105.
4.3
Heat/mass transfer regimes of a rotating disk with jet impingement
The behavior of heat/mass transfer involving a jet im-pinging on a stationary disk or a disk rotating in still air has been determined. Based on the results discussed pre-viously, ¯ow regimes associated with signi®cant heat/mass transfer mechanisms can be resolved. In order to distin-guish the ¯ow regimes, the procedures listed below have been employed:
1. Calculate Qj/Qpat each measured point, where
Qj( RejpDm/4) is the jet volumetric ¯ow rate and
Qp( 0.067prmRer) is the rotationally induced pumping
volumetric ¯ow rate passing the impingement radius. The method to calculate Qpwas proposed by Schlichting
[20].
2. Calculate the value of hrj)hrat each measured point,
where hrj is the mass transfer coef®cient of a rotating
disk with jet impingement and hris the mass transfer
coef®cient of a disk rotating in still air.
3. The following criterions are used to distinguish the ¯ow regimes:
(hrjÿhr)/hrj 10% in rotation dominated
re-gime
hr/hrj 10% in jet impingement dominated
regime
(hrjÿhr)/hrj> 10% and hr/hrj>10% in mixed
regime
Figure 6 shows the correlation attained in terms of the ratio Qjto Qpversus Rer. Each of data point represents
conditions at an observed transition point; the resulting band of points separate the rotational and impingement dominated zones. Correlation developed by Metzger & Grochowsky [3] is also presented in Fig. 7 for comparison. The results reported by Metzger & Grochowsky [3] were
obtained by means of ¯ow visualization experiments. As can be seen in Fig. 6, the left transition line obtained in the present study coincides with that attained from Metzger & Grochowsky [3], while the right transition line deviation can be attributed to the uncertainty coming from smoke ¯ow visualization and the smaller range of ¯ow conditions in their experiments. The values of Rerand Qj/Qppervade
much wider ranges than that used in Metzger & Gro-chowsky [3]. The ratio of Qj/Qpin the present study varies
from 10)2to 103, while the upper limit of Q
j/Qpin Metzger
& Grochowsky [3] is about 1. 5
Conclusions
In the present work, heat/mass transfer of a rotating disk with jet impingement was experimentally examined by virtue of naphthalene sublimation techniques. Distinct divisions of the three heat/mass transfer regimes are made from the experimental results in the present study. Cor-relation of Sherwood number of a rotating disk with jet impingement is proposed in the present study. It is con-cluded that the Sherwood number of a rotating disk with jet impingement is the sum of the two components, gov-erned by the impinging jet and the rotating disk respec-tively.
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