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Transient testing of perforated plate matrix heat exchangers
Cryogenics 43 (2003) 101–109 www.elsevier.com/locate/cryogenics
Transient testing of perforated plate matrix heat exchangers K. Krishnakumar, G. Venkatarathnam
*
Refrigeration and Airconditioning Laboratory, Department of Mechanical Engineering, Indian Institute of Technology, Chennai 600 036, India Received 13 November 2001; accepted 7 January 2003
Abstract Till recently, perforated plate matrix heat exchangers (MHEs) were used mainly in helium liquefaction systems and Brayton cycle refrigerators. Currently they are also being used in Kleemenko or J-T refrigerators operating with a mixture of gases. It is now well understood that the effectiveness of a MHE is strongly dependent on the number of plate–spacer pairs used. It has also been shown that the traditional methods used for reducing the transient single blow method to determine the heat transfer coefficients cannot be used for MHEs. In this paper we show that the traditional methods can indeed be used for determining the heat transfer coefficients using the transient testing methods, provided certain conditions are met. 2003 Elsevier Science Ltd. All rights reserved. Keywords: Heat exchangers; Heat transfer coefficient; Transient testing
1. Introduction Conventional cryogenic heat exchangers such as the coiled tube heat exchangers of Hampson and Collins types and brazed aluminum plate fin exchangers are not suitable for small systems such as reversed Brayton cryocoolers, J-T cryocoolers, very small helium liquefiers, as well as sorption cryocoolers used in satellites, because of the large longitudinal (axial) heat conduction through the walls and the heat leak from ambient, which limit the achievable effectiveness to a maximum of about 92%. The necessity of attaining high effectiveness and high degree of compactness together in one unit led to the invention of matrix heat exchangers by McMahon et al. [1] in 1949. A matrix heat exchanger, shown schematically in Fig. 1, essentially consists of a stack of high thermal conductivity (copper or aluminum) perforated plates or wire screens, alternating with low thermal conductivity spacers (plastics, stainless steel). The packet of alternate high and low thermal conductivity materials is bonded together to form leak free passages for the streams exchanging heat between one another.
*
Corresponding author. Tel.: +91-44-445-8552/8581; fax: +91-44235-0509. E-mail address: [email protected] (G. Venkatarathnam).
The gaps in between the plates ensure uniform flow distribution (by continuous reheadering) and create turbulence which enhances heat transfer. The small flow passages (typically 0.3–1.0 mm in diameter) ensure a high heat transfer coefficient and high surface area density (up to 6000 m2 /m3 ). The spacers, being of low thermal conductivity material, also help in reducing axial conduction and consequent deterioration of performance. A number of developments have occurred during the last 20 years (see Ref. [2]). Modern bonding techniques such as diffusion bonding, vacuum brazing are used these days instead of mechanical tie rods and rubber spacers originally used by McMahon et al. It is now well understood that traditional effectiveness-NTU approach methods cannot be used for MHEs because of the discontinuity in the heat transfer surface in the axial (longitudinal) direction [2]. New rating and sizing methods have been developed [3–6] that treat the MHE as a discrete set of plate–spacer pairs. The convective heat transfer in MHE has been studied by a number of authors [1,7–12]. The single blow transient test is most suitable for determining the heat transfer coefficients of wire mesh and perforated plate MHEs. A number of methods have been developed in literature for determining the heat transfer coefficients from the single blow test data and to take into account the non-ideal conditions that exist during the testing
0011-2275/03/$ - see front matter 2003 Elsevier Science Ltd. All rights reserved. doi:10.1016/S0011-2275(03)00026-2
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Nomenclature A Ac Aplate cp C d m_ H l L M n ntu ntuf Nu p Pr Re St
overall convective heat transfer area (m2 ) flow (cross sectional) area in the plate (m2 ) 2 face face area of the plate ð¼ W H Þ (m ) specific heat of fluid (kJ/kg K) specific heat of the plate material (kJ/kg K) diameter of the perforation (m) mass flow rate (kg/s) height of the plate (m) length of plate (see Fig. 1) (m) length of the test section (see Fig. 1) (m) mass of any individual plate (kg) number of perforated plates overall ntu of test section ntu ¼ n ntuf the number of transfer units per plate Nusselt number plate porosity (open area for flow) Prandtl number Reynolds number Stanton number
T0 tin tj Tj W gfin p h v n s f f0
temperature of test section at time n ¼ 0 (K) temperature of fluid at inlet (K) temperature of fluid entering the jth plate (K) temperature of the jth plate (K) width of each plate (m) fin efficiency ¼ 1 expðntuf Þ dimensionless temperature of the stream (see Eq. (8)) 1 hout time (s) dimensionless temperature of the plate (see Eq. (8)) dimensionless time (see Eq. (8)) f=n
Subscripts in entry out exit
2. Literature review The convective heat transfer between the fluid stream and a perforated plate takes place on three surfaces: (1) the front face of the plates, (2) the tubular surface of the perforations and (3) the back face of the plates (see Fig. 2). In a perforated plate exchanger the flow cross-section changes continuously between that of the perforations and that of the spacer, the former being substantially
Fig. 1. Perforated plate matrix heat exchanger.
[13–22]. It has, however, been shown that traditional methods employed to determine the heat transfer coefficients by transient tests cannot be used for MHE [23,24]. The available knowledge on transient testing of heat exchangers therefore cannot be used directly for testing MHEs. The main objective of this paper is to derive the conditions under which the existing methods developed over the last 50 years for conventional heat exchangers can be used for MHEs also, and to make it easy for deriving heat transfer coefficients of MHE from transient tests.
Fig. 2. Cross section of a MHE showing the different convective heat transfer surfaces: (1) the front face of the plates, (2) the tubular surface of the perforations and (3) the back face of the plates.
K. Krishnakumar, G. Venkatarathnam / Cryogenics 43 (2003) 101–109
smaller than the latter. Consequently, the fluid undergoes alternate expansion and contraction as it flows through the exchanger. Because of the interruption of the boundary layer at every plate, the flow through the tubular portion can be considered as developing flow with associated high heat transfer coefficient. The length to diameter ratio is usually between 0.3 and 4. The leading face of a plate is subjected to arrays of impinging jets emanating from the plate before it. Heat transfer due to jets impinging on a blind surface has been studied in detail by several authors for different applications. It has been found in several studies that the heat transfer rates obtained with air jets impinging on blind surfaces are an order of magnitude higher than those generally associated with gaseous heat transfer media. Heat transfer rate in the back face of the plate is also high due to flow separation and resulting turbulence. For design work, as well as for comparing with experimental results, an average heat transfer coefficient may be defined as P3 h i Ai h ¼ Pi¼1 ð1Þ 3 i¼1 Ai The index i referring to the three different surfaces: the tubular surfaces of the holes and the front and the back faces of the plates. The convective heat transfer and flow
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friction in perforated plates are strongly dependent on a number of geometric parameters such as: • • • •
diameter of the perforation (d), thickness of the plates (t), thickness of the spacers (s), porosity of the plate (fraction of area open to the flow) (p).
Besides, the heat transfer and flow friction is dependent on the relative position of the perforations in the adjacent plates (inline or offset holes) and the shape of the perforations [8]. If the plates are made by mechanical means, the direction in which the metal is removed also becomes a factor. A number of authors [1,7–10,12] have studied the heat transfer through perforated plate heat exchangers, and have expressed their results as Nu ¼ C Ren . The correlations obtained by different authors are summarized in Table 1 and Fig. 3. It is quite evident that the nature of dependence of the flow on different geometrical parameters has not been well understood as yet. It is therefore, necessary to determine both the heat transfer coefficient and the flow friction characteristics experimentally for each of the surfaces used individually. The convective heat transfer characteristics of any heat exchanger surface can be determined using (a) steady state, (b) dynamic or (c) periodic test techniques.
Table 1 Convective heat transfer correlations for flow through a perforated plate matrix heat exchangers Correlation n
h ¼ a Re Nu ¼ C Ren C ¼ 4:93p 0:11 n ¼ 0:77 1:12p Nu ¼ 0:2 Re0:64
Nu ¼ 0:22 Re0:69
Nu ¼ 0:05 Re0:74 ðs=dÞ0:21
Nu ¼ 0:45 Re0:87 ðs=dÞ0:50
StPr2=3 ¼ 1:2 Re0:62 StPr2=3 ¼ C Ren C ¼ 0:00036½ð1 pÞp 0:2 2:07 n ¼ 4:36 102 p2:34 Nude ¼ 13:022 Re0:574 Pr1=3 de ¼ A=Xp Ntu =plate ¼ b Rem b ¼ 2:1 to 3:7 m ¼ 0:37 to 0:57
Limitations
Source
800 < Re < 4000 0:09 < p < 0:35
[1] [7]
Shifted holes 0:27 < p < 0:35 70 < Re < 2100 Shifted slots 0:47 < p < 0:5 30 < Re < 1600 Aligned holes 200 < Re < 1000 0:11 < s=d < 1 Aligned slots 200 < Re < 1400 0:075 < s=a < 0:88 p ¼ 0:3246 300 < Re < 3000 300 < Re < 3000 0:3 < p < 0:6
[8]
0:38 < l=d < 1:47
[12]
Re < 100 0:09 < p < 0:245
[11]
[8]
[8]
[8]
[9] [10]
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Fig. 3. Comparison of correlations for C, n in Nu ¼ C Ren as a function of porosity of the plates by different authors.
2.1. Steady state test techniques A known surface is employed for one channel and the unknown surface, whose characteristics are to be determined are used for the second channel. Generally air is used for the unknown side and steam, hot/cold water is used for the known stream. The temperature of streams entering and leaving the heat exchanger as well as the flow rates are measured. The heat transfer coefficient of the unknown side is determined from the overall heat transfer coefficient. 2.2. Transient test techniques In this method, the heat exchanger core is either heated or cooled till a steady state is achieved. The temperature of the fluid entering the exchanger is suddenly changed (lowered or increased) in a systematic fashion, and the temperature of the fluid leaving the exchanger core is continuously recorded. The heat transfer coefficient can be determined directly from the temperature– time history of fluid leaving the core. Different methods can be used for data reduction viz., maximum slope method, zero intercept method, direct curve matching method or the first moment area method. 2.3. Periodic test techniques This is a variant of the transient method. In this method, the temperature of the fluid entering the exchanger is continuously varied, and the temperature– time history of the fluid at the outlet is continuously recorded. Since a very complicated arrangement is required to vary fluid temperature in a periodic fashion, this method is more difficult to implement. Amongst the three test techniques, the steady state methods are most difficult to conduct and will necessi-
tate use of a number of equipment (viz., steam generators, condensate separators, safety devices). On the other hand, the transient methods are easy to perform, but the data reduction procedures are much more complex. The transient methods are cheaper and require less time and effort and are, in general, preferred for regenerator matrices as well as matrix heat exchangers surfaces. Several methods are available in literature for deriving the heat transfer coefficients from the variation of output temperature of the stream with time during a single blow test. One of the simplest is the maximum slope method. Locke [13] showed analytically that in the absence of axial conduction the ntu of the surface ðntu ¼ hA=mCp Þ is a function of the maximum slope of the fluid temperature response at the exit. i.e., dv ð2Þ ntu ¼ f df where hout is the dimensionless temperature at the exit of the section, f is the dimensionless time and v ¼ 1 hout . In the presence of axial conduction, closed form analytical expressions cannot be obtained, and one must resolve to numerical methods for evaluating the ntu from the maximum slope ðdv=dfÞ [14]. The maximum slope method has been studied by many workers [14– 22]. Rodrigues and Mills [23] recently analyzed the single blow transient test method for matrix heat exchangers. They showed that traditional maximum slope methods which neglect the discontinuity in the heat transfer surface in the axial direction could not be used for matrix heat exchangers. They presented the maximum slope and ntu values for different values of axial conduction and the number of plate–spacer pairs. Rodrigues and MillsÕ results clearly demonstrate the need to consider the heat exchanger as a discrete set of plate– spacer pairs in transient tests also. Farhani et al. [24] recently derived the relationship between the effect of number of plates when the ntu of each plate is infinite. It was shown analytically that the maximum slope is finite even when the number of heat transfer units per plate ðntuf ¼ ntu=nÞ is infinite for each plate and is a function of the number of plate– spacer pairs as follows: ðn1Þ dv ðn 1Þ eðn1Þ ¼ df max ðn 1Þ! 1
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi for n > 10 ð3Þ 2pðn 1Þ The above equation is valid only when the per plate ntu is infinite. Under this condition, the maximum slope occurs at a dimensionless time f ¼ ðn 1Þ. While Farhani et al. [24] clearly bring out the effect of finite number of plate–spacer pairs the results presented are only a special case when the mass flow rate tends to
K. Krishnakumar, G. Venkatarathnam / Cryogenics 43 (2003) 101–109
zero or per plate ntu being infinite, and is, in general, not valid in real life situations. In this paper we study the effect of finite-plate–spacer pairs on the transient response of MHEs for situations where the ntu or the fluid flow rate is finite. Also, we derive the conditions under which the effect of the finite number of plates becomes negligible, and an MHE can be treated as a conventional heat exchanger so that the large number of methods available in literature for reducing the transient test data can be used for MHEs as well.
3. The analytical model Let the test matrix consist of n perforated plates, separated from each other by ðn 1Þ insulating spacers as shown in Fig. 4. At time n ¼ 0 , the fluid and the matrix are at the same temperature ðT0 Þ. At time n ¼ 0, a step change Dt ¼ tin T0 is introduced in the fluid temperature. Thereafter, (at n ¼ 0þ ), the temperature of the fluid at the inlet is maintained constant at tin . Let Tj represent the temperature of the jth plate and tj , tjþ1 , the temperature of the fluid at the inlet and exit of the jth plates respectively as shown in Fig. 1. The governing equations and initial condition for the fluid and solid matrix can be expressed as follows: Governing equations hjþ1 ¼ hj þ p ðsj hj Þ
ð4Þ
dsj þ p sj ¼ p hj df
ð5Þ
Initial condition and boundary condition The plate temperature is uniform till a step change is introduced in the fluid temperature. The initial condition can therefore be written as follows:
Fig. 4. Cross section of a perforated plate showing the notation followed for the stream and plate temperatures.
sj ¼ 1
at f ¼ 0 and 1 6 j 6 n
h1 ¼ 0 at f > 0
105
ð6Þ ð7Þ
In the above expressions, the dimensionless time, temperature and effectiveness are defined as follows: ! t tin T tin m_ cp s¼ f¼ n h¼ MC T0 tin T0 tin p ¼ 1 entuf
ntu ¼
hAgfin m_ cp
ð8Þ
It is convenient to define the dimensionless time in terms of the mass of a single plate in the case of MHEs, whereas for a conventional heat exchanger the mass of the entire heat exchanger is normally used. For comparing the results with the available results of conventional heat exchanger, the dimensionless time f0 is defined as follows: f0 ¼ f=n
ð9Þ
In order to make the analysis tractable, the following assumptions have been made: 1. Axial conduction through the spacers is negligible. 2. Thermo physical properties of the fluid and the matrix are independent of temperature. 3. The total heat capacity of the fluid (mass specific heat) is small compared to that of the matrix. 4. The test section is well insulated, and no heat transfer occurs between the section and the surroundings. 5. There is no temperature gradient in the lateral direction. While axial or longitudinal heat conduction plays a very important role in the performance of a MHE, during the transient testing, the longitudinal heat conduction can be made very negligible and close to zero, by using strips of a low conductivity material such as paper or plastic to separate the perforated plates from each other at three or four places along the circumference. The whole assembly can be placed inside a paper or thin plastic pipe [25]. In most cases the maximum temperature difference between the section and the fluid is maintained at less than 10–15 K. The variation of thermo physical properties is very small over this temperature range, and can therefore be neglected. Because of the small difference between the temperature of the matrix and the ambient, the heat loss to ambient will be very small. It can be reduced even further, by insulating the test section. The heat capacity of the test matrix can be made much larger than the tube used to hold the matrices by using either a paper tube or a very thin plastic tube. Because of the small thickness of the plate, the Biot number will be small (<0.01) and the plate temperature will be constant in the flow direction [2]. (In the lateral direction, a small
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temperature gradient will exist between the walls and the center of the plate only where the axial conduction is large.)
fmax ¼ n 1
3.1. Solution of governing equations
n ¼ 2:
p ¼ 23
ðfor fmax ¼ 0Þ
ð15Þ
The dimensionless temperature of the jth plate ðsj Þ and the fluid leaving the jth plate ðhÞ can be obtained by solving the governing equations and the initial condition as follows:
Z f sj ðfÞ ¼ expðp fÞ 1 þ p expðp fÞhj df ð10Þ
n ¼ 3:
p ¼ 24
ðfor fmax ¼ 0Þ
ð16Þ
n ¼ 4:
p ¼ 25
ðfor fmax ¼ 0Þ
ð17Þ
0
hjþ1 ðfÞ ¼ hj ð1 p Þ þ p expðp fÞ
Z f expðp fÞhj df 1 þ p
ð11Þ
While analytical solutions can be obtained when the total number of plates are few, by successive substitution, the above equations can only be solved numerically for n > 5.
4. Results and discussion The governing equations along with the boundary and initial conditions were solved numerically by successive substitution, starting from the plate at the entry of the fluid to that at the exit of the fluid. The program was validated by comparing the results obtained with that of Farhani et al. [24] for large values of ntu (Eq. (3)), and with the numerical results presented by Rodriguez and Mills [23]. Kohlmayr [15] showed that the maximum slope does not occur when the overall ntu of the heat exchanger is less than 2.0 in conventional heat exchangers. The results obtained by us do show that we approach the limit obtained by Kohlmayr analytically for conventional heat exchangers. The same result can also be shown analytically. The time at which the maximum slope occurs can be obtained by differentiating Eq. (11) with respect to the dimensionless time f twice, and solving for time f as follows: n ¼ 2:
fmax ¼
2 3p 2p 3 þ 4p þ
n ¼ 3:
fmax ¼
ð14Þ
The condition for which no maximum slope occurs (or fmax ¼ 0) can be generalized for few plates as follows:
The relationship between p and the number of plates for the condition of no maximum slope can be generalized by the method of induction as follows: p ¼
0
for p ¼ 1
2 nþ1
ðfor fmax ¼ 0Þ
ð18Þ
If we have a large number of plates, or ntuf , the per plate ntu is small, then Eq. (18) can be rewritten as follows: nþ1 ntuf > ln ð19Þ ðfor fmax > 0Þ n1 Eq. (19) can be rewritten in terms of the overall ntu of the test section as follows: nþ1 ntuf > n ln ð20Þ ðfor fmax > 0Þ n1 Fig. 5 shows the variation of the right hand side of the above equation as a function of number of plates. It can be observed that minimum ntu for observing maximum slope approaches 2 as the number of plates is larger than about 15. Fig. 6 shows the variation of f0max for a test section with 30 plate–spacer pairs. The results obtained with different plates show similar trend, and not distinguishable from that for 30 plates when shown together in Fig. 6. The dimensionless times were obtained by solving Eqs. (10) and (11) numerically for 30 plates. It can be observed that the dimensionless time at which
ð12Þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 6p þ 42p 2p
ð13Þ
The equations become quite complex when the number of plates is greater than 5. For the special case of p ¼ 1 or ðntuf ¼ 1Þ, the time at which maximum slope occurs can be generalized as follows:
Fig. 5. Minimum number of heat transfer units (RHS of Eq. (20)) required for observing a maximum slope.
K. Krishnakumar, G. Venkatarathnam / Cryogenics 43 (2003) 101–109
Fig. 6. Dimensionless time at which maximum slope is observed for 30 plate–spacer pairs.
maximum slope ðdv=df0 Þmax occurs tends to zero at overall ntu close to two, as predicted by Eq. (20). Table 2 and Fig. 7 show the variation of maximum slope ðdv=df0 Þmax when the overall ntu is greater than 2.0, for different number of plates. The corresponding values for a conventional heat exchanger given by Kays and London [26] is also shown. It can be seen that the maximum slope obtained by us for different plates is same as that given by Kays and London [26] for an overall ntuðn ntuf Þ of 4. The difference, however, increases with an increase in ntu. The results clearly show that the conventional methods for reducing the single blow data should not be used for ntu > 4, irrespective of the number of plates used in the test. Kohlmayr [15] has presented the variation of maximum slope for smaller number of heat transfer units. Table 3 shows a comparison of results obtained by us with that presented in Ref. [15]. It is evident that the values obtained by us considering the discrete nature of MHE are same as that obtained by Kohlmayr for a
Table 2 Comparison of maximum slope with number of plates and overall ntu for a MHE with that of a conventional heat exchanger at high values of ntu [26] ntu 4 6 8 10 20 30 40 60 80 90
Maximum slope n ¼ 10
n ¼ 20
n ¼ 30
[26]
0.632 0.738 0.829 0.907 1.152 1.254 1.294 1.314 1.317 1.317
0.632 0.741 0.837 0.923 1.246 1.455 1.591 1.734 1.789 1.800
0.632 0.741 0.838 0.926 1.268 1.512 1.695 1.935 2.067 2.109
0.632 0.741 0.840 0.929 1.284 1.564 1.797 2.196 2.533 2.683
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Fig. 7. Comparison of maximum slope with number of plates and overall ntu for a MHE with that of a conventional heat exchanger (n ¼ 1) [26].
Table 3 Comparison of maximum slope with number of plates and overall ntu for a MHE with that of a conventional heat exchanger at low values of ntu [15] ntu 2.2 2.5 3.0 3.5 4.0 4.5 5.0
Maximum slope n ¼ 10
n ¼ 20
n ¼ 30
[15]
0.545 0.555 0.578 0.605 0.632 0.660 0.687
0.544 0.553 0.577 0.604 0.632 0.660 0.688
0.544 0.553 0.577 0.604 0.632 0.660 0.688
0.544 0.553 0.577 0.604 0.632 0.660 0.688
conventional heat exchanger in which there are no discontinuities in the heat transfer surface for 20 and 30 plates. A small difference, however, exists between the values obtained by us for 10 plates and that given by Kohlmayr [15]. It is evident from the above discussion that the MHE behaves exactly like a conventional heat exchanger when the number of plates is more than 15, provided the overall ntu is less than 4. Normal single blow data reduction procedures can therefore be used if the ntu of the test section lies between 2 and 4. The ntu of the test section, however, depends on the heat transfer coefficient (Reynolds number), the flow rate as well as the area (number of plates) used in the test. Let us now consider the requirements of number of plates to be used so that the ntu of the test section is between 2 and 4. The number of heat transfer units of the test section can be expressed in terms of Nusselt and Reynolds numbers as follows: ntu ¼
hA Nu A ¼ m_ cp Re Pr Ac
ð21Þ
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In the above expression A is the overall heat transfer area of all plates and Ac the flow cross section in the plates. 1 The heat transfer area in the plates can be expressed in terms of the plate porosity (fraction of open area), the flow length in each plate l and the diameter of the perforation d as follows (see Fig. 2): A ¼ n½Aperforations þ 2ð1 pÞAplate
l ¼ nAplate face 4p þ 2ð1 pÞ d
face
ð22Þ
The ratio of heat transfer area to the cross sectional area of the flow is given by the expression:
A A l 1p ¼n 4 þ2 ¼ ð23Þ Ac Aplate face p d p The ntu of the test section can be estimated using Eq. (21) as follows: ntu ¼
Nu A A ¼ St Re Pr Ac Ac
ð24Þ
Substitution of Eq. (23) into the above expression gives:
ntu l 1p ¼ St 4 þ 2 ð25Þ n d p Fig. 8 shows the variation of ntu per plate estimated using different correlations given in literature. A plate porosity of 0.3246 [9] was chosen for comparison so that a number of correlations which are valid for this porosity can be compared. A plate thickness to diameter ratio of 1.0, which is typical value in most heat exchangers, and a Prandtl number of 0.7 were assumed. The ntu per plate values are plotted only over the Reynolds number ranges where the tests were performed by the respective authors. As shown earlier, the maximum slope will occur only when total ntu is greater than 2 and conventional methods used only for ntu less than or equal to 4. For 10 plates, the two limits therefore correspond to ntu=n of 0.2 (2/10) and 0.4 (4/10) respectively. For 20 plates, the limits for ntu=n are 0.1 and 0.2 respectively. It can be seen that a large range of Reynolds numbers can be covered with 10 or 20 plates, while maintaining the total ntu to be between 2 and 4. If higher number of plates are used, the total ntu will be more than 4. In that case, methods which treat the MHE as a discontinuous heat exchanger [23] should be used for obtaining the heat transfer coefficients from the transient tests. Recently it was shown by Pavan and Venkatarathnam [5] that the Reynolds numbers of a MHE for least 1 It is customary to use the velocity inside the perforations and the diameter of the perforation as the characteristic dimension while calculating the Reynolds number of the flow through perforated plate heat exchanger.
Fig. 8. Variation of overall ntu per plate with Reynolds number for different plate porosity (p) using Nusselt number correlations available in literature. The horizontal lines show the region for an overall ntu of 2 and 4 for different plates (l=d ¼ 1).
volume should be of the order of 100 for the cold stream and about 400 for the hot stream for a MHE used in a helium refrigeration system. Most authors have obtained f and j factors for MHE either in laminar or turbulent flow, but not both. It is clear from Fig. 8 that the Reynolds numbers range of 100–2000 can be covered with only two different sets of plates, for any plate porosity.
5. Conclusions The following conclusions can be drawn from this study: • Conventional single blow data reduction procedures can be used for MHEs only if the overall ntu of the test section is less than 4, and the number of plates used is about 10–20. • No maximum slope in the time–temperature history of the fluid stream leaving the test section will occur if the overall ntu is less than 2, as in the case of a conventional heat exchanger. • Different number of plates need to be used for different Reynolds number ranges in order that the ntu is between 2 and 4.
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K. Krishnakumar, G. Venkatarathnam / Cryogenics 43 (2003) 101–109 [4] Venkatarathnam G. Effectiveness-ntu relationship in perforated plate matrix heat exchangers. Cryogenics 1996;36(4):235–41. [5] Kumar KP, Venkatarathnam G. Optimization of matrix heat exchanger geometry. J Heat Transfer 2000;122(3):579–86. [6] Ramesh P, Venkatarathnam G. Transient response of perforated plate matrix heat exchangers. Cryogenics 1999;38(12):1243–9. [7] Ornatskii AP, Perkov VV, Khudzinskii VM. Experimental study of perforated plate heat exchanger for micro cryogenic systems. Promishelnaya Teplo Tekhnika 1983;5:28–33. [8] Mikulin EI, Shevich YA, Potapov VN, Solntsev MY, Yusova GM. Study of matrix type heat exchangers made of perforated plates. Khim Neft Mashin 1980;(9):8–10. [9] Orlov VK, Shevyakova SA, Valeev GW. Heat exchange and hydraulic resistance in perforated plate heat exchangers. Khim Neft Mashin 1978;(8):10–1. [10] Shevyakova SA, Orlov VK. Study of hydraulic resistance and heat transfer in perforated plate heat exchangers. Inzh Fiz Zh 1983;45: 32–6. [11] Hubbell RH, Cain CL. New heat transfer and friction factor design data for perforated plate heat exchangers. Adv Cryog Eng 1987;33:383–90. [12] Hu J, Gong L, Zhu T, Guo T. Heat transfer characteristics of perforated plate, Part-I, mean heat transfer coefficients. In: Proceedings of International Cryogenic Eng Conference, Beijing. 1990. [13] Locke GL. Heat transfer and flow friction characteristics of porous solids. Technical Report TR No. 10, Stanford University, California, USA. 1950. [14] Pucci P, Howard CP, Piersall JCH. The single-blow transient testing technique for compact heat exchangers. J Eng Power 1967;89:29–40. [15] Kohlmayr GF. Exact maximum slopes for transient matrix heat transfer testing. Int J Heat Mass Transfer 1966;9:671–80.
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[16] Kohlmayr GF. An indirect curve matching method for transient matrix heat transfer testing in the low ntu range. Int J Heat Mass Transfer 1967;11:567–81. [17] Kohlmayr GF. Extension of the maximum slope method to arbitrary upstream fluid temperature change. J Heat Transfer 1968;90:130–4. [18] Kohlmayr GF. Analytical solution of the single-blow problem by a double Laplace transform method. J Heat Transfer 1968;90:176–8. [19] Liang CY, Yang WJ. Modified single blow transient technique for performance evaluation of heat transfer surfaces. J Heat Transfer 1975;97:16–21. [20] Huber ML, Jones MC. A frequency response study of packed bed heat transfer at elevated temperatures. Int J Heat Mass Transfer 1988;31:1001–8. [21] Mullisen RS, Loehrke RI. A transient heat exchanger evaluation test for arbitrary fluid inlet temperature variation and longitudinal core conduction. J Heat Transfer 1986;(108):370–6. [22] Luo X, Roetzel W, L€ udersen U. The single blow transient testing technique considering longitudinal core conduction and fluid dispersion. Int J Heat Mass Transfer 2001;44:121–9. [23] Rodrigues JI, Mills AF. Analysis of single-blow transient testing of perforated plate heat exchangers. Int J Heat Mass Transfer 1990;33:1969–76. [24] Farhani F, Venkatarathnam G, Sarangi S. Effect of finite number of plates on the transient response of perforated plates during a single blow transient test of matrix heat exchanger surfaces. In: Proceedings of the 18th Int Conf Cryogenic Eng, Mumbai, India. 2000. p. 307–10. [25] Kumar KK. Some studies on heat transfer and flow friction in matrix heat exchangers. PhD thesis, Indian Institute of Technology Madras (under preparation). [26] Kays WM, London AL. Compact heat exchangers. 2nd ed. McGraw-Hill; 1984.
Transient testing of perforated plate matrix heat exchangers K. Krishnakumar, G. Venkatarathnam
*
Refrigeration and Airconditioning Laboratory, Department of Mechanical Engineering, Indian Institute of Technology, Chennai 600 036, India Received 13 November 2001; accepted 7 January 2003
Abstract Till recently, perforated plate matrix heat exchangers (MHEs) were used mainly in helium liquefaction systems and Brayton cycle refrigerators. Currently they are also being used in Kleemenko or J-T refrigerators operating with a mixture of gases. It is now well understood that the effectiveness of a MHE is strongly dependent on the number of plate–spacer pairs used. It has also been shown that the traditional methods used for reducing the transient single blow method to determine the heat transfer coefficients cannot be used for MHEs. In this paper we show that the traditional methods can indeed be used for determining the heat transfer coefficients using the transient testing methods, provided certain conditions are met. 2003 Elsevier Science Ltd. All rights reserved. Keywords: Heat exchangers; Heat transfer coefficient; Transient testing
1. Introduction Conventional cryogenic heat exchangers such as the coiled tube heat exchangers of Hampson and Collins types and brazed aluminum plate fin exchangers are not suitable for small systems such as reversed Brayton cryocoolers, J-T cryocoolers, very small helium liquefiers, as well as sorption cryocoolers used in satellites, because of the large longitudinal (axial) heat conduction through the walls and the heat leak from ambient, which limit the achievable effectiveness to a maximum of about 92%. The necessity of attaining high effectiveness and high degree of compactness together in one unit led to the invention of matrix heat exchangers by McMahon et al. [1] in 1949. A matrix heat exchanger, shown schematically in Fig. 1, essentially consists of a stack of high thermal conductivity (copper or aluminum) perforated plates or wire screens, alternating with low thermal conductivity spacers (plastics, stainless steel). The packet of alternate high and low thermal conductivity materials is bonded together to form leak free passages for the streams exchanging heat between one another.
*
Corresponding author. Tel.: +91-44-445-8552/8581; fax: +91-44235-0509. E-mail address: [email protected] (G. Venkatarathnam).
The gaps in between the plates ensure uniform flow distribution (by continuous reheadering) and create turbulence which enhances heat transfer. The small flow passages (typically 0.3–1.0 mm in diameter) ensure a high heat transfer coefficient and high surface area density (up to 6000 m2 /m3 ). The spacers, being of low thermal conductivity material, also help in reducing axial conduction and consequent deterioration of performance. A number of developments have occurred during the last 20 years (see Ref. [2]). Modern bonding techniques such as diffusion bonding, vacuum brazing are used these days instead of mechanical tie rods and rubber spacers originally used by McMahon et al. It is now well understood that traditional effectiveness-NTU approach methods cannot be used for MHEs because of the discontinuity in the heat transfer surface in the axial (longitudinal) direction [2]. New rating and sizing methods have been developed [3–6] that treat the MHE as a discrete set of plate–spacer pairs. The convective heat transfer in MHE has been studied by a number of authors [1,7–12]. The single blow transient test is most suitable for determining the heat transfer coefficients of wire mesh and perforated plate MHEs. A number of methods have been developed in literature for determining the heat transfer coefficients from the single blow test data and to take into account the non-ideal conditions that exist during the testing
0011-2275/03/$ - see front matter 2003 Elsevier Science Ltd. All rights reserved. doi:10.1016/S0011-2275(03)00026-2
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K. Krishnakumar, G. Venkatarathnam / Cryogenics 43 (2003) 101–109
Nomenclature A Ac Aplate cp C d m_ H l L M n ntu ntuf Nu p Pr Re St
overall convective heat transfer area (m2 ) flow (cross sectional) area in the plate (m2 ) 2 face face area of the plate ð¼ W H Þ (m ) specific heat of fluid (kJ/kg K) specific heat of the plate material (kJ/kg K) diameter of the perforation (m) mass flow rate (kg/s) height of the plate (m) length of plate (see Fig. 1) (m) length of the test section (see Fig. 1) (m) mass of any individual plate (kg) number of perforated plates overall ntu of test section ntu ¼ n ntuf the number of transfer units per plate Nusselt number plate porosity (open area for flow) Prandtl number Reynolds number Stanton number
T0 tin tj Tj W gfin p h v n s f f0
temperature of test section at time n ¼ 0 (K) temperature of fluid at inlet (K) temperature of fluid entering the jth plate (K) temperature of the jth plate (K) width of each plate (m) fin efficiency ¼ 1 expðntuf Þ dimensionless temperature of the stream (see Eq. (8)) 1 hout time (s) dimensionless temperature of the plate (see Eq. (8)) dimensionless time (see Eq. (8)) f=n
Subscripts in entry out exit
2. Literature review The convective heat transfer between the fluid stream and a perforated plate takes place on three surfaces: (1) the front face of the plates, (2) the tubular surface of the perforations and (3) the back face of the plates (see Fig. 2). In a perforated plate exchanger the flow cross-section changes continuously between that of the perforations and that of the spacer, the former being substantially
Fig. 1. Perforated plate matrix heat exchanger.
[13–22]. It has, however, been shown that traditional methods employed to determine the heat transfer coefficients by transient tests cannot be used for MHE [23,24]. The available knowledge on transient testing of heat exchangers therefore cannot be used directly for testing MHEs. The main objective of this paper is to derive the conditions under which the existing methods developed over the last 50 years for conventional heat exchangers can be used for MHEs also, and to make it easy for deriving heat transfer coefficients of MHE from transient tests.
Fig. 2. Cross section of a MHE showing the different convective heat transfer surfaces: (1) the front face of the plates, (2) the tubular surface of the perforations and (3) the back face of the plates.
K. Krishnakumar, G. Venkatarathnam / Cryogenics 43 (2003) 101–109
smaller than the latter. Consequently, the fluid undergoes alternate expansion and contraction as it flows through the exchanger. Because of the interruption of the boundary layer at every plate, the flow through the tubular portion can be considered as developing flow with associated high heat transfer coefficient. The length to diameter ratio is usually between 0.3 and 4. The leading face of a plate is subjected to arrays of impinging jets emanating from the plate before it. Heat transfer due to jets impinging on a blind surface has been studied in detail by several authors for different applications. It has been found in several studies that the heat transfer rates obtained with air jets impinging on blind surfaces are an order of magnitude higher than those generally associated with gaseous heat transfer media. Heat transfer rate in the back face of the plate is also high due to flow separation and resulting turbulence. For design work, as well as for comparing with experimental results, an average heat transfer coefficient may be defined as P3 h i Ai h ¼ Pi¼1 ð1Þ 3 i¼1 Ai The index i referring to the three different surfaces: the tubular surfaces of the holes and the front and the back faces of the plates. The convective heat transfer and flow
103
friction in perforated plates are strongly dependent on a number of geometric parameters such as: • • • •
diameter of the perforation (d), thickness of the plates (t), thickness of the spacers (s), porosity of the plate (fraction of area open to the flow) (p).
Besides, the heat transfer and flow friction is dependent on the relative position of the perforations in the adjacent plates (inline or offset holes) and the shape of the perforations [8]. If the plates are made by mechanical means, the direction in which the metal is removed also becomes a factor. A number of authors [1,7–10,12] have studied the heat transfer through perforated plate heat exchangers, and have expressed their results as Nu ¼ C Ren . The correlations obtained by different authors are summarized in Table 1 and Fig. 3. It is quite evident that the nature of dependence of the flow on different geometrical parameters has not been well understood as yet. It is therefore, necessary to determine both the heat transfer coefficient and the flow friction characteristics experimentally for each of the surfaces used individually. The convective heat transfer characteristics of any heat exchanger surface can be determined using (a) steady state, (b) dynamic or (c) periodic test techniques.
Table 1 Convective heat transfer correlations for flow through a perforated plate matrix heat exchangers Correlation n
h ¼ a Re Nu ¼ C Ren C ¼ 4:93p 0:11 n ¼ 0:77 1:12p Nu ¼ 0:2 Re0:64
Nu ¼ 0:22 Re0:69
Nu ¼ 0:05 Re0:74 ðs=dÞ0:21
Nu ¼ 0:45 Re0:87 ðs=dÞ0:50
StPr2=3 ¼ 1:2 Re0:62 StPr2=3 ¼ C Ren C ¼ 0:00036½ð1 pÞp 0:2 2:07 n ¼ 4:36 102 p2:34 Nude ¼ 13:022 Re0:574 Pr1=3 de ¼ A=Xp Ntu =plate ¼ b Rem b ¼ 2:1 to 3:7 m ¼ 0:37 to 0:57
Limitations
Source
800 < Re < 4000 0:09 < p < 0:35
[1] [7]
Shifted holes 0:27 < p < 0:35 70 < Re < 2100 Shifted slots 0:47 < p < 0:5 30 < Re < 1600 Aligned holes 200 < Re < 1000 0:11 < s=d < 1 Aligned slots 200 < Re < 1400 0:075 < s=a < 0:88 p ¼ 0:3246 300 < Re < 3000 300 < Re < 3000 0:3 < p < 0:6
[8]
0:38 < l=d < 1:47
[12]
Re < 100 0:09 < p < 0:245
[11]
[8]
[8]
[8]
[9] [10]
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Fig. 3. Comparison of correlations for C, n in Nu ¼ C Ren as a function of porosity of the plates by different authors.
2.1. Steady state test techniques A known surface is employed for one channel and the unknown surface, whose characteristics are to be determined are used for the second channel. Generally air is used for the unknown side and steam, hot/cold water is used for the known stream. The temperature of streams entering and leaving the heat exchanger as well as the flow rates are measured. The heat transfer coefficient of the unknown side is determined from the overall heat transfer coefficient. 2.2. Transient test techniques In this method, the heat exchanger core is either heated or cooled till a steady state is achieved. The temperature of the fluid entering the exchanger is suddenly changed (lowered or increased) in a systematic fashion, and the temperature of the fluid leaving the exchanger core is continuously recorded. The heat transfer coefficient can be determined directly from the temperature– time history of fluid leaving the core. Different methods can be used for data reduction viz., maximum slope method, zero intercept method, direct curve matching method or the first moment area method. 2.3. Periodic test techniques This is a variant of the transient method. In this method, the temperature of the fluid entering the exchanger is continuously varied, and the temperature– time history of the fluid at the outlet is continuously recorded. Since a very complicated arrangement is required to vary fluid temperature in a periodic fashion, this method is more difficult to implement. Amongst the three test techniques, the steady state methods are most difficult to conduct and will necessi-
tate use of a number of equipment (viz., steam generators, condensate separators, safety devices). On the other hand, the transient methods are easy to perform, but the data reduction procedures are much more complex. The transient methods are cheaper and require less time and effort and are, in general, preferred for regenerator matrices as well as matrix heat exchangers surfaces. Several methods are available in literature for deriving the heat transfer coefficients from the variation of output temperature of the stream with time during a single blow test. One of the simplest is the maximum slope method. Locke [13] showed analytically that in the absence of axial conduction the ntu of the surface ðntu ¼ hA=mCp Þ is a function of the maximum slope of the fluid temperature response at the exit. i.e., dv ð2Þ ntu ¼ f df where hout is the dimensionless temperature at the exit of the section, f is the dimensionless time and v ¼ 1 hout . In the presence of axial conduction, closed form analytical expressions cannot be obtained, and one must resolve to numerical methods for evaluating the ntu from the maximum slope ðdv=dfÞ [14]. The maximum slope method has been studied by many workers [14– 22]. Rodrigues and Mills [23] recently analyzed the single blow transient test method for matrix heat exchangers. They showed that traditional maximum slope methods which neglect the discontinuity in the heat transfer surface in the axial direction could not be used for matrix heat exchangers. They presented the maximum slope and ntu values for different values of axial conduction and the number of plate–spacer pairs. Rodrigues and MillsÕ results clearly demonstrate the need to consider the heat exchanger as a discrete set of plate– spacer pairs in transient tests also. Farhani et al. [24] recently derived the relationship between the effect of number of plates when the ntu of each plate is infinite. It was shown analytically that the maximum slope is finite even when the number of heat transfer units per plate ðntuf ¼ ntu=nÞ is infinite for each plate and is a function of the number of plate– spacer pairs as follows: ðn1Þ dv ðn 1Þ eðn1Þ ¼ df max ðn 1Þ! 1
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi for n > 10 ð3Þ 2pðn 1Þ The above equation is valid only when the per plate ntu is infinite. Under this condition, the maximum slope occurs at a dimensionless time f ¼ ðn 1Þ. While Farhani et al. [24] clearly bring out the effect of finite number of plate–spacer pairs the results presented are only a special case when the mass flow rate tends to
K. Krishnakumar, G. Venkatarathnam / Cryogenics 43 (2003) 101–109
zero or per plate ntu being infinite, and is, in general, not valid in real life situations. In this paper we study the effect of finite-plate–spacer pairs on the transient response of MHEs for situations where the ntu or the fluid flow rate is finite. Also, we derive the conditions under which the effect of the finite number of plates becomes negligible, and an MHE can be treated as a conventional heat exchanger so that the large number of methods available in literature for reducing the transient test data can be used for MHEs as well.
3. The analytical model Let the test matrix consist of n perforated plates, separated from each other by ðn 1Þ insulating spacers as shown in Fig. 4. At time n ¼ 0 , the fluid and the matrix are at the same temperature ðT0 Þ. At time n ¼ 0, a step change Dt ¼ tin T0 is introduced in the fluid temperature. Thereafter, (at n ¼ 0þ ), the temperature of the fluid at the inlet is maintained constant at tin . Let Tj represent the temperature of the jth plate and tj , tjþ1 , the temperature of the fluid at the inlet and exit of the jth plates respectively as shown in Fig. 1. The governing equations and initial condition for the fluid and solid matrix can be expressed as follows: Governing equations hjþ1 ¼ hj þ p ðsj hj Þ
ð4Þ
dsj þ p sj ¼ p hj df
ð5Þ
Initial condition and boundary condition The plate temperature is uniform till a step change is introduced in the fluid temperature. The initial condition can therefore be written as follows:
Fig. 4. Cross section of a perforated plate showing the notation followed for the stream and plate temperatures.
sj ¼ 1
at f ¼ 0 and 1 6 j 6 n
h1 ¼ 0 at f > 0
105
ð6Þ ð7Þ
In the above expressions, the dimensionless time, temperature and effectiveness are defined as follows: ! t tin T tin m_ cp s¼ f¼ n h¼ MC T0 tin T0 tin p ¼ 1 entuf
ntu ¼
hAgfin m_ cp
ð8Þ
It is convenient to define the dimensionless time in terms of the mass of a single plate in the case of MHEs, whereas for a conventional heat exchanger the mass of the entire heat exchanger is normally used. For comparing the results with the available results of conventional heat exchanger, the dimensionless time f0 is defined as follows: f0 ¼ f=n
ð9Þ
In order to make the analysis tractable, the following assumptions have been made: 1. Axial conduction through the spacers is negligible. 2. Thermo physical properties of the fluid and the matrix are independent of temperature. 3. The total heat capacity of the fluid (mass specific heat) is small compared to that of the matrix. 4. The test section is well insulated, and no heat transfer occurs between the section and the surroundings. 5. There is no temperature gradient in the lateral direction. While axial or longitudinal heat conduction plays a very important role in the performance of a MHE, during the transient testing, the longitudinal heat conduction can be made very negligible and close to zero, by using strips of a low conductivity material such as paper or plastic to separate the perforated plates from each other at three or four places along the circumference. The whole assembly can be placed inside a paper or thin plastic pipe [25]. In most cases the maximum temperature difference between the section and the fluid is maintained at less than 10–15 K. The variation of thermo physical properties is very small over this temperature range, and can therefore be neglected. Because of the small difference between the temperature of the matrix and the ambient, the heat loss to ambient will be very small. It can be reduced even further, by insulating the test section. The heat capacity of the test matrix can be made much larger than the tube used to hold the matrices by using either a paper tube or a very thin plastic tube. Because of the small thickness of the plate, the Biot number will be small (<0.01) and the plate temperature will be constant in the flow direction [2]. (In the lateral direction, a small
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K. Krishnakumar, G. Venkatarathnam / Cryogenics 43 (2003) 101–109
temperature gradient will exist between the walls and the center of the plate only where the axial conduction is large.)
fmax ¼ n 1
3.1. Solution of governing equations
n ¼ 2:
p ¼ 23
ðfor fmax ¼ 0Þ
ð15Þ
The dimensionless temperature of the jth plate ðsj Þ and the fluid leaving the jth plate ðhÞ can be obtained by solving the governing equations and the initial condition as follows:
Z f sj ðfÞ ¼ expðp fÞ 1 þ p expðp fÞhj df ð10Þ
n ¼ 3:
p ¼ 24
ðfor fmax ¼ 0Þ
ð16Þ
n ¼ 4:
p ¼ 25
ðfor fmax ¼ 0Þ
ð17Þ
0
hjþ1 ðfÞ ¼ hj ð1 p Þ þ p expðp fÞ
Z f expðp fÞhj df 1 þ p
ð11Þ
While analytical solutions can be obtained when the total number of plates are few, by successive substitution, the above equations can only be solved numerically for n > 5.
4. Results and discussion The governing equations along with the boundary and initial conditions were solved numerically by successive substitution, starting from the plate at the entry of the fluid to that at the exit of the fluid. The program was validated by comparing the results obtained with that of Farhani et al. [24] for large values of ntu (Eq. (3)), and with the numerical results presented by Rodriguez and Mills [23]. Kohlmayr [15] showed that the maximum slope does not occur when the overall ntu of the heat exchanger is less than 2.0 in conventional heat exchangers. The results obtained by us do show that we approach the limit obtained by Kohlmayr analytically for conventional heat exchangers. The same result can also be shown analytically. The time at which the maximum slope occurs can be obtained by differentiating Eq. (11) with respect to the dimensionless time f twice, and solving for time f as follows: n ¼ 2:
fmax ¼
2 3p 2p 3 þ 4p þ
n ¼ 3:
fmax ¼
ð14Þ
The condition for which no maximum slope occurs (or fmax ¼ 0) can be generalized for few plates as follows:
The relationship between p and the number of plates for the condition of no maximum slope can be generalized by the method of induction as follows: p ¼
0
for p ¼ 1
2 nþ1
ðfor fmax ¼ 0Þ
ð18Þ
If we have a large number of plates, or ntuf , the per plate ntu is small, then Eq. (18) can be rewritten as follows: nþ1 ntuf > ln ð19Þ ðfor fmax > 0Þ n1 Eq. (19) can be rewritten in terms of the overall ntu of the test section as follows: nþ1 ntuf > n ln ð20Þ ðfor fmax > 0Þ n1 Fig. 5 shows the variation of the right hand side of the above equation as a function of number of plates. It can be observed that minimum ntu for observing maximum slope approaches 2 as the number of plates is larger than about 15. Fig. 6 shows the variation of f0max for a test section with 30 plate–spacer pairs. The results obtained with different plates show similar trend, and not distinguishable from that for 30 plates when shown together in Fig. 6. The dimensionless times were obtained by solving Eqs. (10) and (11) numerically for 30 plates. It can be observed that the dimensionless time at which
ð12Þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 6p þ 42p 2p
ð13Þ
The equations become quite complex when the number of plates is greater than 5. For the special case of p ¼ 1 or ðntuf ¼ 1Þ, the time at which maximum slope occurs can be generalized as follows:
Fig. 5. Minimum number of heat transfer units (RHS of Eq. (20)) required for observing a maximum slope.
K. Krishnakumar, G. Venkatarathnam / Cryogenics 43 (2003) 101–109
Fig. 6. Dimensionless time at which maximum slope is observed for 30 plate–spacer pairs.
maximum slope ðdv=df0 Þmax occurs tends to zero at overall ntu close to two, as predicted by Eq. (20). Table 2 and Fig. 7 show the variation of maximum slope ðdv=df0 Þmax when the overall ntu is greater than 2.0, for different number of plates. The corresponding values for a conventional heat exchanger given by Kays and London [26] is also shown. It can be seen that the maximum slope obtained by us for different plates is same as that given by Kays and London [26] for an overall ntuðn ntuf Þ of 4. The difference, however, increases with an increase in ntu. The results clearly show that the conventional methods for reducing the single blow data should not be used for ntu > 4, irrespective of the number of plates used in the test. Kohlmayr [15] has presented the variation of maximum slope for smaller number of heat transfer units. Table 3 shows a comparison of results obtained by us with that presented in Ref. [15]. It is evident that the values obtained by us considering the discrete nature of MHE are same as that obtained by Kohlmayr for a
Table 2 Comparison of maximum slope with number of plates and overall ntu for a MHE with that of a conventional heat exchanger at high values of ntu [26] ntu 4 6 8 10 20 30 40 60 80 90
Maximum slope n ¼ 10
n ¼ 20
n ¼ 30
[26]
0.632 0.738 0.829 0.907 1.152 1.254 1.294 1.314 1.317 1.317
0.632 0.741 0.837 0.923 1.246 1.455 1.591 1.734 1.789 1.800
0.632 0.741 0.838 0.926 1.268 1.512 1.695 1.935 2.067 2.109
0.632 0.741 0.840 0.929 1.284 1.564 1.797 2.196 2.533 2.683
107
Fig. 7. Comparison of maximum slope with number of plates and overall ntu for a MHE with that of a conventional heat exchanger (n ¼ 1) [26].
Table 3 Comparison of maximum slope with number of plates and overall ntu for a MHE with that of a conventional heat exchanger at low values of ntu [15] ntu 2.2 2.5 3.0 3.5 4.0 4.5 5.0
Maximum slope n ¼ 10
n ¼ 20
n ¼ 30
[15]
0.545 0.555 0.578 0.605 0.632 0.660 0.687
0.544 0.553 0.577 0.604 0.632 0.660 0.688
0.544 0.553 0.577 0.604 0.632 0.660 0.688
0.544 0.553 0.577 0.604 0.632 0.660 0.688
conventional heat exchanger in which there are no discontinuities in the heat transfer surface for 20 and 30 plates. A small difference, however, exists between the values obtained by us for 10 plates and that given by Kohlmayr [15]. It is evident from the above discussion that the MHE behaves exactly like a conventional heat exchanger when the number of plates is more than 15, provided the overall ntu is less than 4. Normal single blow data reduction procedures can therefore be used if the ntu of the test section lies between 2 and 4. The ntu of the test section, however, depends on the heat transfer coefficient (Reynolds number), the flow rate as well as the area (number of plates) used in the test. Let us now consider the requirements of number of plates to be used so that the ntu of the test section is between 2 and 4. The number of heat transfer units of the test section can be expressed in terms of Nusselt and Reynolds numbers as follows: ntu ¼
hA Nu A ¼ m_ cp Re Pr Ac
ð21Þ
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K. Krishnakumar, G. Venkatarathnam / Cryogenics 43 (2003) 101–109
In the above expression A is the overall heat transfer area of all plates and Ac the flow cross section in the plates. 1 The heat transfer area in the plates can be expressed in terms of the plate porosity (fraction of open area), the flow length in each plate l and the diameter of the perforation d as follows (see Fig. 2): A ¼ n½Aperforations þ 2ð1 pÞAplate
l ¼ nAplate face 4p þ 2ð1 pÞ d
face
ð22Þ
The ratio of heat transfer area to the cross sectional area of the flow is given by the expression:
A A l 1p ¼n 4 þ2 ¼ ð23Þ Ac Aplate face p d p The ntu of the test section can be estimated using Eq. (21) as follows: ntu ¼
Nu A A ¼ St Re Pr Ac Ac
ð24Þ
Substitution of Eq. (23) into the above expression gives:
ntu l 1p ¼ St 4 þ 2 ð25Þ n d p Fig. 8 shows the variation of ntu per plate estimated using different correlations given in literature. A plate porosity of 0.3246 [9] was chosen for comparison so that a number of correlations which are valid for this porosity can be compared. A plate thickness to diameter ratio of 1.0, which is typical value in most heat exchangers, and a Prandtl number of 0.7 were assumed. The ntu per plate values are plotted only over the Reynolds number ranges where the tests were performed by the respective authors. As shown earlier, the maximum slope will occur only when total ntu is greater than 2 and conventional methods used only for ntu less than or equal to 4. For 10 plates, the two limits therefore correspond to ntu=n of 0.2 (2/10) and 0.4 (4/10) respectively. For 20 plates, the limits for ntu=n are 0.1 and 0.2 respectively. It can be seen that a large range of Reynolds numbers can be covered with 10 or 20 plates, while maintaining the total ntu to be between 2 and 4. If higher number of plates are used, the total ntu will be more than 4. In that case, methods which treat the MHE as a discontinuous heat exchanger [23] should be used for obtaining the heat transfer coefficients from the transient tests. Recently it was shown by Pavan and Venkatarathnam [5] that the Reynolds numbers of a MHE for least 1 It is customary to use the velocity inside the perforations and the diameter of the perforation as the characteristic dimension while calculating the Reynolds number of the flow through perforated plate heat exchanger.
Fig. 8. Variation of overall ntu per plate with Reynolds number for different plate porosity (p) using Nusselt number correlations available in literature. The horizontal lines show the region for an overall ntu of 2 and 4 for different plates (l=d ¼ 1).
volume should be of the order of 100 for the cold stream and about 400 for the hot stream for a MHE used in a helium refrigeration system. Most authors have obtained f and j factors for MHE either in laminar or turbulent flow, but not both. It is clear from Fig. 8 that the Reynolds numbers range of 100–2000 can be covered with only two different sets of plates, for any plate porosity.
5. Conclusions The following conclusions can be drawn from this study: • Conventional single blow data reduction procedures can be used for MHEs only if the overall ntu of the test section is less than 4, and the number of plates used is about 10–20. • No maximum slope in the time–temperature history of the fluid stream leaving the test section will occur if the overall ntu is less than 2, as in the case of a conventional heat exchanger. • Different number of plates need to be used for different Reynolds number ranges in order that the ntu is between 2 and 4.
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