Transient response of multi-pass plate heat exchangers considering the effect of flow maldistribution

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Chemical Engineering and Processing 47 (2008) 695–707

Transient response of multi-pass plate heat exchangers considering the effect of flow maldistribution N. Srihari, Sarit K. Das ∗ Heat Transfer and Thermal Power Laboratory, Department of Mechanical Engineering, Indian Institute of Technology-Madras, Chennai 600036, India Received 1 May 2006; received in revised form 3 November 2006; accepted 12 December 2006 Available online 14 January 2007

Abstract Present study depicts the transient response of multi-pass plate heat exchangers (PHEs) considering flow maldistribution from port to channels. Apart from the flow maldistribution, fluid axial dispersion is also considered to take care of the fluid backmixing and other deviations from plug flow. It is assumed that each multi-pass PHE is a combination of single-pass PHEs and in each heat exchanger the fluid is distributed non-uniformly amongst channels. The fluid velocity varies from channel to channel within each module of the heat exchanger so also the heat transfer coefficient. The solution techniques have been presented here for 1-2 pass and 2-2 pass arrangements. The first one is solved by analysing successive modules of the heat exchanger and the next one by iterating the responses between two heat exchanger modules. The solution for each module has been obtained analytically by using Laplace transform followed by numerical inversion from frequency domain. The results show the effect of flow maldistribution and its effect combined with the conventional heat exchanger parameters in the transient regime. It is observed that the transient characteristics such as response delay, asymptotic value and time constant are strongly dependent on the multi-pass flow arrangement, maldistribution and backmixing characterised by axial dispersion. © 2007 Elsevier B.V. All rights reserved. Keywords: Plate heat exchanger; Transient response; Multi-pass; Maldistribution; Axial dispersion

1. Introduction Plate heat exchangers (PHEs) have become more popular in power and process industries even though they were specially designed for hygienic applications, such as dairy, brewery, food processing, and pharmaceutical industries. In these heat exchangers, both the fluids flow through corrugated channels alternately. Due to this it produces higher turbulence at comparatively lower flow rates, and hence high heat transfer coefficient. In addition to this, these exchangers have advantages like compactness, smaller hold up volume and hence quicker response to control operations, flexibility, less fouling, etc. The PHEs have overcome their limitations in pressure and temperature ranges by improvement of gasket materials and their design to establish themselves as the fastest growing member of the heat exchanger family. This kind of growth demands an accu-

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Corresponding author. Tel.: +91 44 2257 4655; fax: +91 44 2257 4652. E-mail address: [email protected] (S.K. Das).

0255-2701/$ – see front matter © 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.cep.2006.12.011

rate thermal model for plant simulation, control and safety measures. The literature available in the area of plate heat exchangers is substantial. Most of these studies were devoted to steady state simulation and experiments. They are not so significant in number for transient behaviour and extremely scarce in the case of multi-pass PHEs. Multi-pass heat exchangers are widely used in industries where the heat capacity rates are unequal on the two sides with nearly equal allowable pressure drops of each fluid. Initially, uniform flow distribution from port to channels and plug flow inside the channels [1–4] were assumed for modelling thermal performance of plate heat exchangers. Transient performance of the plate heat exchangers was first studied by McKnight and Worley [5] using feed back control to high velocity flow. Simulation on the dynamic performance for co-current plate heat exchangers were presented by Zaleski and Tajszerski [6]. For counter-current plate heat exchanger, Khan et al. [7] presented the experimental and analytical studies using the sinusoidal and pulse inputs. A ‘cinematic model’ was presented by Lakshmanan and Potter [8] to predict the dynamic

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behaviour of single pass heat exchanger numerically. Dynamic analysis of single pass PHE was presented by Das and Roetzel [9] considering axial dispersion in both the fluids. In this model, an axial dispersive Peclet number was introduced to take care of all the deviations from plug flow (flow maldistribution and fluid backmixing) and phase lag effect, to account for delay at the distribution port. Transient response of multipass plate heat exchangers with thermal dispersion in fluid was presented by Das and Murugesan [10], which is an extension of the single pass model of Das and Roetzel [9]. Here also the axial dispersion term was assumed to take care of both flow maldistribution as well as backmixing. On the other hand, it is confirmed by Roetzel and Na Ranong [11,12] and Sahoo and Roetzel [13] that it is more appropriate to use axial dispersion for fluid backmixing rather than flow maldistribution. Thus, it seems to be better to consider the flow maldistribution separately instead of flow maldistribution along with fluid backmixing. Transient studies, which are discussed above, mostly assumed that the flow is equally distributed to all the channels and as a consequence the heat transfer coefficient is equal in each channel. Even where it was considered to have maldistribution, the effect was lumped into an axial dispersion term rather than isolating its effect channel by channel. In reality, the flow is distributed non-uniformly from port to channels, thus the velocity varies from channel to channel and hence the heat transfer coefficient as well. This affects performance of the heat exchanger both thermally and hydraulically. Therefore, for better design, there is a need for good knowledge of flow distribution and the effect of this distribution on the thermal and hydraulic performance. Bassiouny and Martin [14,15] developed the equations to explain the flow distribution of fluid from port to channel using the analytical technique for both Utype and Z-type configurations. A numerical work on effect of non-uniform flow distribution presented by Datta and Majumdar [16] and another numerical model presented by Thonon et al. [17] proved that pressure drop is significantly affected by flow maldistribution. Yang and Wang [18] presented the thermal performance and flow maldistribution in multi-pass PHEs. A steady state analysis was presented on the effect of flow distribution to the channels on the thermal performance of PHE by Rao et al. [19]. In this model, heat transfer coefficient inside the channel is incorporated as a function of the velocity of the fluid stream in that particular channel. Same work was extended to multi-pass also, using numerical modelling by Rao and Das [20]. From the above review of literature it is clear that analytical works on the steady state analysis of PHEs in presence of flow maldistribution have been carried out satisfactorily but the transient response has not been studied in detail. Recent work by Srihari et al. [21] on transient response for single pass PHE considered the flow maldistribution separately and dispersive Peclet number was used to take care of fluid backmixing and other deviations from plug flow. Also, transient studies on multi-pass PHE considering flow maldistribution has rarely been studied so far. This is the main motivation of the present work. In the present analysis, the scope is limited to 1-2 pass and 2-2 pass

configurations of PHE, further these models can be extended to any 1-n or m-n/n-n multi-pass configurations. The significant feature in the present analysis is that the unequal flow distribution is considered as suggested by the Bassiouny and Martin [14,15]. 2. Mathematical formulation The multi-pass PHEs are modelled mathematically by considering the following assumptions, which are reasonable for normal range of operation with temperature transients. Some of the assumptions are same as those made for single pass model, i.e. transient analysis of the PHE considering flow maldistribution by Srihari et al. [21]. The main challenge in the modelling of multi-pass arrangements lies in treating the fluid mixing and redistribution at the pass interface, the cumulative phase lag effect and pass interfacial heat transfer: (1) Thermo-physical properties of the fluids are independent of temperature. (2) Heat transfer takes place only between the channels and not between the channels and ports or through the seals and gaskets. (3) The heat exchanger is thermally insulated from the surroundings. (4) The axial dispersion term is taken care of fluid backmixing alone, not the flow maldistribution from channel to channel. (5) The flow maldistribution from channel to channel in individual passes has been taken into account and heat transfer coefficient is considered to be a function of flow velocity in that particular channel. (6) Heat transfer at pass boundaries is considered with appropriate flow directions. (7) The fluid at the end of each pass is adiabatically mixed. (8) The plate thickness is small so that axial conduction can be neglected. (9) The heat exchanger is started from a uniform temperature for both the fluids. In a PHE, actual fluid flow deviates from the one-dimensional plug flow. The deviations are mainly due to the fluid backmixing in the channel, flow maldistribution from channel to channel, heat leakage through gaskets and seals, etc. Care has been taken while formulating the governing equations in this analysis to consider the effects due to phase lag at channel entry and exit ports. 3. Single pass plate heat exchangers The single pass counter flow PHE for Z-type configuration is shown in Fig. 1 and parallel flow PHE is shown in Fig. 2. The coordinate system is chosen in the direction of fluid flow in the first channel. The channels are named from 1 to N (chosen to be odd) and the plates from 1 to N + 1, in which odd and even numbered channels will carry fluid 1 and fluid 2, respectively. To minimize the heat loss to the surroundings cold fluid will flow through the odd numbered channels. The two

N. Srihari, S.K. Das / Chemical Engineering and Processing 47 (2008) 695–707

697

the above assumptions. These equations are: • Equations for counter flow: C1 ∂θi ∂ 2 θi ∂θi ˙i = Ac Di 2 − (−1)i−1 w L ∂τ ∂X ∂X (hA)i + (θWi + θWi+1 − 2θi ) 2L     N +1 i = 1, 3, 5, . . . , 2 −1 2

Fig. 1. Counter flow with Z-type configuration.

end plates which are exposed to only one fluid on one side and surroundings on the other end, which is assumed to be insulated. To frame the energy balance equations for the fluid 1, fluid 2 and all the plates, small elements of fluid and plate can be considered as control volumes as shown in Fig. 3 considering

C2 ∂θi ∂ 2 θi ∂θi ˙i = Ac Di 2 − (−1)i−1 w L ∂τ ∂X ∂X (hA)i (θWi + θWi+1 − 2θi ) + 2L    N i = 2, 4, 6, ..., 2 2

(1)

(2)

• Equations for parallel flow: C1 ∂θi ∂ 2 θi ∂θi (hA)i ˙i = Ac Di 2 − w + (θWi + θWi+1 − 2θi ) L ∂τ ∂X ∂X 2L     N +1 i = 1, 3, 5, . . . , 2 −1 (3) 2

∂ 2 θi ∂θi C2 ∂θi (hA)i ˙i = Ac Di 2 − w + (θWi + θWi+1 − 2θi ) L ∂τ ∂X ∂X 2L    N i = 2, 4, 6, ..., 2 (4) 2 • Wall equations (for both counter and parallel flow):

Fig. 2. Parallel flow with Z-type configuration.

Fig. 3. Control volume for the fluid inside the channel and the plate.

CW ∂θWi (hA)i (hA)i−1 = (θi−1 − θWi ) + (θi − θWi ) L ∂τ 2L 2L (i = 2, 3, 4, . . . , N)

(5)

CW ∂θW1 (hA)1 = (θ1 − θW1 ) L ∂τ 2L

(6)

CW ∂θW,N+1 (hA)N = (θN − θW,N+1 ) L ∂τ 2L

(7)

In the above governing equations, the number of variables is large and can be minimized by using few non-dimensional terms like NTU, heat capacity rate ratio R2 , and dispersive Peclet number, Pe. All other parameters also scaled with respect to the uniform flow case. Channel to channel heat transfer coefficient variation is considered based on channel velocity raised to the power ‘n’ from conventional heat transfer equation Nu = C Ren Prr . Thus the non-dimensional form of governing equations can be reduced as follows:

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• For counter flow: (Rτ )mi+1 ∂ti ∂ti 1 ∂ 2 ti − (−1)i−1 = 2 RWi ∂z Pei ∂x ∂x NTUi (RN )mi+1 (tWi + tWi+1 − 2ti ) 2 (i = 1, 2, 3, 4, . . . , N) +

(8)

• For parallel flow: 1 ∂ 2 ti ∂ti (Rτ )mi+1 ∂ti NTUi = − + (RN )mi+1 RWi ∂z Pei ∂x2 ∂x 2 × (tWi +tWi+1 −2ti )

Fig. 4. Varying velocity in the inlet and exit ports for Z-type configuration.

(i = 1, 2, 3, 4, . . . , N) (9)

• Wall equations (for both counter and parallel flow): RC

∂tWi = Ki−1 (ti−1 − tWi ) + Ki (ti − tWi ) ∂z (i = 2, 3, 4, . . . , N)

(10)

RC

∂tW1 NTU1 = RW1 (t1 − tW1 ) ∂z 2

(11)

RC

∂tW,N+1 = KN (tN − tW,N+1 ) ∂z

(12)

where Ki = (NTUi /2)RWi (R2 RN )mi+1 , and mj = j − 2[j/2]. 3.1. Maldistribution parameter and non-dimensional volume flow rate The most important feature of the present analysis is the inclusion of proper distribution of fluid in the channels from the port. As per Bassiouny and Martin [14,15] for the flow channelling of Z-type configuration, the volumetric flow rate increases along the flow direction. Non-dimensional volume flow rate and flow distribution parameter for same inlet and exit ports were derived as: volume flow rate in the channel mean volume flow rate for uniform flow cosh my =m sinh m

v˙ c =

 m2 =

1 ζc



nAc Ap

(13)

inlet port and gets distributed into the channels 1, 2, 3, . . . the time at which the fluid stream enters the subsequent channels will increase. Hence the amount of phase lag also increase and the last channel will have maximum phase lag from the time at which the combined fluid enters the heat exchanger at point 1, as shown in Fig. 4. In the present study, flow maldistribution is included while calculating the phase lag effect. In case of Z-type configuration (Fig. 1) as the fluid flows in the inlet conduit the flow streams are distributed such that the minimum flow rate is found in the first channel and gradually the flow rate increases up to the last channel. Equations become more complex due to the effect of flow maldistribution on the phase lag. It has to be understood that the phase lag effect becomes even more complicated in a multi-pass heat exchanger. This is because after each pass the fluid gets collected and redistributed giving a different distribution pattern in each pass. As the fluid inside the port progresses, its velocity decreases as the fluid streams leaving each channel (due to decrease in volume flow rate) as shown in Fig. 4. The velocities of the fluid in the port after leaving the channels 1, 2, 3, . . . are represented by V1 , V2 , V3 , . . . , respectively. To find these velocities, applying the continuity condition at the entry of the channels, the ratios of these velocities can be obtained using the non-dimensional volume flow rate in channels. The non-dimensional volume flow rate equation for the first channel can be written as: v˙ c1 = n1

V˙ c1 V˙ g1

(15)

Substituting Eq. (14) to the continuity equation for the first channel gives:

2 (14)

where distribution parameter ‘m’, given in the expressions mainly depends on the exchanger geometry, configuration, flow frictional characteristics and number of channels. Its value increases as the square of the number of channels. 3.2. Phase lag and maldistribution effect Phase lag effect is a special feature in the transient studies of the plate heat exchangers. As the fluid enters through the

V˙ c1 V1 =1− Vg1 V˙ g1

(16)

Similarly, equation of the velocity ratio for the ith channel for both the fluids: ⎤ ⎡ i  1 ⎣ V2i−1 = 1− v˙ c,2j−1 ⎦ Vg1 n1 j=1

(i = 1, 2, 3, . . . , n1)

for fluid 1

(17)

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⎤ ⎡ i V2i 1 ⎣ v˙ c,2j ⎦ = 1− Vg2 n2

as follows: • For counter flow: ◦ At x = 0:

j=1

(i = 1, 2, 3, . . . , n2)

for fluid 2

(18) ti −

The time required for the fluid to travel the distance between the channels (‘lc ’ as shown in Figs. 2 and 3) can be calculated as:

1 ∂ti = f1 (z − φi )u(z − φi ) Pei ∂x     N +1 i = 1, 3, 5, . . . , 2 −1 2

τ1 =

l1 Vg1

(19)

τ2 =

l2 Vg2

(20)

∂ti =0 ∂x

(21)

∂tWi =0 ∂x

τ2i+1 = τ2i+2 =

lc V2i−1 lc V2i

(i = 1, 2, 3, . . . , n1 − 1) (i = 1, 2, 3, . . . , n2 − 1)

(22)

(28)

τi τra1

ti −

(i = 1, 2, 3, . . . , N)

φ2i−1 =

2i−1 

φ2j−1

(i = 1, 2, 3, . . . , n1)

(24)

j=1

φ2i =

2i 

φ2j

(i = 1, 2, 3, . . . , n2)

(25)

∂ti =0 ∂x ∂tWi =0 ∂x

φi,exit = φi ,

for a U-type plate exchanger,

(26)

and φi,exit = φn+1−i ,

for a Z-type plate exchanger.

(27)

3.3. Boundary conditions According to Danckwerts [22], a sudden temperature drop will be experienced at entrance due to the axial dispersion of the fluid. Because no heat transfer is assumed in the port, it is not taken into consideration inside the port. The boundary conditions for Eqs. (8)–(12) may be written using the above considerations

N 2



(i = 1, 2, 3, . . . , N + 1)

(29) (30)



 i = 1, 3, 5, . . . , 2

  N +1 −1 2

(i = 1, 2, 3, . . . , N + 1)

(31)

(32) (33)

• For parallel flow: ◦ At x = 0: ti −

j=1

After the fluid streams leaves each channel inside the exit port, the phase lag encountered to reach the exit point can be computed in a similar way. Under the condition of dimensional symmetry in construction, as shown in Figs. 1 and 2, the relationship for this phase lag at exit can be reduced to:

 i = 2, 4, 6, . . . , 2

1 ∂ti = f2 (z − φi )u(z − φi ) Pei ∂x    N i = 2, 4, 6, . . . , 2 2

(23)

The total phase lag at the entry of each channel is the cumulated sum of the phase lags given by:



◦ At x = 1:

The dimensionless phase lag at the entrances of consecutive channels may be given as: φi =

699

1 ∂ti = f1 (z − φi )u(z − φi ) Pei ∂x     N +1 −1 i = 1, 3, 5, . . . , 2 2 (34)

ti −

1 ∂ti = f2 (z − φi )u(z − φi ) Pei ∂x    N i = 2, 4, 6, . . . , 2 2

∂tWi =0 ∂x

(i = 1, 2, 3, . . . , N + 1)

(35)

(36)

◦ At x = 1: ∂ti =0 ∂x ∂tWi =0 ∂x

(i = 1, 2, 3, . . . , N) (i = 1, 2, 3, . . . , N + 1)

(37) (38)

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be extended to m-n/n-n type PHE configuration but the analysis becomes quite complex as the number of passes increases and larger number of iterations are involved. 5. Solution for temperature distribution 5.1. Single-pass

Fig. 5. Schematic diagram of 1-2 pass arrangement.

4. Analysis of multi-pass plate heat exchangers To analyse the multi pass PHEs, the above mathematical model can be used as the subsystem. In general multi-pass arrangements can broadly be classified in two ways, first one is 1-n type PHE and the second m-n/n-n type PHE. The simplest examples of these heat exchangers are 1-2 pass and 2-2 pass as shown in Figs. 5 and 6. To analyse them conveniently, it is required to split each of these heat exchangers into two small modules like HX1 and HX2 as shown in these figures. There is a possibility of error at the interface by not considering the heat interaction there and it can be corrected by incorporating a fictitious channel to each module. Effect of this error diminishes as the number of channel increases. 4.1. 1-n pass plate heat exchanger In case of 1-2 pass PHE, hot side and cold side inputs of HX1 are known hence its response can easily be obtained independent of HX2. The outlet of the cold side of HX1 will act as input to the HX2, then, it is possible to obtain the response of HX2. The cold fluid have two passes as it leaves from all the channels of HX1 get mixed up at the port out let and enter into the HX2. The hot fluid has only one pass as it is distributed to all the channels of HX1 and HX2. In the same way, subsequently it can be extend to any number of passes, i.e. 1-n type PHE. 4.2. m-n/n-n type plate heat exchanger The 2-2 pass PHE analysis is quite complex because the performance of HX1 depends on the response of HX2 and vice versa. So the solution can be obtained by iterating the responses of both the heat exchanger modules. In the same way, it can

The mathematical model presented in the current analysis for counter and parallel flow heat exchangers is expressed by the partial differential Eqs. (8)–(12), subjected to boundary conditions, represented by Eqs. (28)–(33). The initial condition for both the fluids and the walls may be taken to be uniform, since only starting from uniform temperature (cold state) is considered here. Thus: ti,0 = tWi,0 = 0. By applying the Laplace transform to the governing equations with respect to the reduced time variable z, ordinary differential equations can be obtained. From the wall equations one has: TWi =

Ki−1 Ki Ti−1 + Ti Mi−1 Mi

where Mi = Rc s + Ki − 1 + Ki . By substituting the wall temperature Eq. (39), into transformed governing equations for fluids. • For counter flow: d 2 Ti dTi NTUi Ki−1 = (−1)i−1 Pei Ti−1 − Pei (RN )mi+1 2 dx dx 2 Mi  mi+1 Rτ NTUi + Pei s + (RN )mi+1 RWi 2   Ki Ki × 2− Ti − Pei (RN )mi+1 − Mi Mi+1 ×

NTUi Ki+1 Ti+1 2 Mi+1

(i = 1, 2, 3, . . . , N)

(40)

• For parallel flow: dTi NTUi Ki−1 d2 Ti = Pei Ti−1 − Pei (RN )mi+1 dx2 dx 2 Mi  mi+1 Rτ NTUi + Pei s + (RN )mi+1 RWi 2   Ki Ki × 2− Ti − Pei (RN )mi+1 − Mi Mi+1 ×

Fig. 6. Schematic diagram of 2-2 pass arrangement.

(39)

NTUi Ki+1 Ti+1 2 Mi+1

(i = 1, 2, 3, . . . , N)

(41)

The above transformed equations can be expressed, in the matrix form as: dT¯ ¯ T¯ =A (42) dx

N. Srihari, S.K. Das / Chemical Engineering and Processing 47 (2008) 695–707

where the temperature vector T¯ is given by:  T ¯T = T1 , T2 , . . . , TN , dT1 , dT2 , . . . , dTN dx dx dx

 dTi = dj un+i,j eβj x dx 2N

(43)

(i = 1, 2, 3, . . . , N) 

AN+1,1 = Pe1



1

RWN

AN+i,i−1 = −Pei (RN )mi+1  AN+i,i =Pei

F2 (s) e−φ4 s , . . . , Fk (s) e−φN s , 0, 0, 0 . . . 0]T

  NTUN KN KN − 2− 2 MN MN+1

NTUi Ki−1 2 Mi

  Rτ mi+1 Ki NTUi Ki 2− s + (RN )mi+1 − RWi 2 Mi Mi+1

AN+i,i+1 = −Pei (RN )mi+1

NTUi Ki+1 2 Mi+1

= −(−1)i+1 Pei

(i = 1, 2, 3, . . . , N)

(i = 1, 2, 3, . . . , N)

for counter flow

for parallel flow

¯ other than those described above are All elements of matrix A zero. By deriving the eigenvalues βj and eigenvectors (uj) from the ¯ the solution to Eq. (42) can be obtained. This coefficient matrix A is a simple boundary value problem with distance coordinate as the only variable. The solution is: ¯ D ¯ T¯ = U¯ B(x)

(44)

¯ where B(x) is a diagonal matrix: ¯ B(x) = diag{eβ1 x , eβ2 x , . . . , eβ2N x }

(45)

The columns of the matrix U¯ are corresponding eigenvec¯ and D ¯ is a coefficient vector, which depends tors of matrix A, on the boundary conditions, given by Eqs. (28)–(33). The fluid temperature distribution and its derivative can thus be expressed as: Ti =

2N  j=1

dj uij eβj x

(i = 1, 2, 3, . . . , N)

(46)

(49)

where k = 1 for odd N and k = 2 for even N. Here the functions F1 (s) and F2 (s) are the Laplace transform of inlet temperature functions f1 (z) and f2 (z) of fluids 1 and 2, respectively. In this sense, this formulation is general one in which any inlet temperature as a function of time can be used. This feature helps to iterate between two modules of a multi-pass heat exchangers where input to the one module comes from the output of other module. In case where heat exchanger has got hot inlet temperature in the form of temperature jump (f(z) = unit step function), F(s) is given as 1/s. For the cold inlet temperature (f(z) = 0), F(s) = 0. However, since the functions f1 and f2 can be monitored at the inlet of heat exchanger, we have to add the influence of the phase lag at the entry of that channel which appeared as exponential term in the above matrix. ¯ can be obtained from: Hence matrix D ¯ =W ¯ −1 S¯ D

AN+i,N+i

AN+i,N+i = Pei

(48)

S¯ = [F1 (s) e−φ1 s , F2 (s) e−φ2 s , F1 (s) e−φ3 s ,

NTUN KN−1 2 MN

s + (RN )mi+1

(47)

The right hand vector S¯ contains the inlet temperature functions and can be written as:

NTU1 K2 2 M2

A2N,N−1 = −PeN (RN )mi+1

A2N,N =PeN

¯D ¯ = S¯ W

  1 NTU1 K1 K1 , s+ − 2− RW1 2 M1 M2

AN+1,2 = −Pe1 RN

(i = 1, 2, 3, . . . , N)

j=1

¯ can be determined by substituting The coefficient matrix D the Eqs. (46) and (47) into the transformed form of boundary conditions Eqs. (28)–(33), to obtain the matrix equation:

¯ can be written as: and the coefficient matrix A Ai,N+i = 1

701

(50)

5.2. Response in time domain The temperature response in the frequency domain has been obtained by solving Eq. (42), which has been inverted into the time domain by Laplace inversion. To carry out the inversion analytically the expressions are too complex. Hence, the Laplace inversion by Crump’s [23] numerical algorithm using Fourier series approximation is used in the present analysis. Any function g(Z) with a Laplace transform G(s) can be expressed as:    ∞  exp(aZ) 1 ikπ k g(Z) = G(a) + Re G a + (−1) Z 2 Z k=i (51) The constant a is chosen in the domain 4 < aZ < 5 to minimize the truncation error. It can be further simplified by using Fast Fourier Transform. Substituting Z = 2nZ/M, the above equation results in: exp(aZn ) g(Zn ) = Z  M−1      2πnk 1 ikπ × Re exp i − g(a) G a+ Z M 2 k=1

(52)

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N. Srihari, S.K. Das / Chemical Engineering and Processing 47 (2008) 695–707

The term indicated by summation is obtained by Fast Fourier Transformation at every point Zn in that domain. The final temperature of the combined fluid at the outlet can be calculated by considering temperature responses at the exit of the each channel and corresponding phase lag, and using the weighted mean average as follows:

m ˙ i ti (z − φi )

t(z) = (53) m ˙i 6. Solution sequence for multi-pass plate heat exchanger 6.1. The 1-2 pass PHE The 1-2 pass PHE as shown earlier, can be divided into two single pass PHEs—one counter flow (HX1) and the other parallel flow (HX2). The coupling of the hot and cold fluid between these two modules is shown in Fig. 7a. First the response of the counter flow heat exchanger HX1 was computed using the solution procedure explained in the preceding section because both the inlet temperatures of this module are known. The resulting cold side outlet temperature is a second order response but for the convenience of computation it is approximated to the first order response using the transfer function (G) as: G=

K e−τd s 1 + τc s

(54)

where K is the gain, τ d the delay time and τc is the time constant. Using the least square fit to the cold side response of HX1 computed earlier, the values of K, τ d and τ c have been obtained. These values are useful to apply the inlet temperature function on cold side to the heat exchanger HX2. This is required because for the module HX2 only the hot inlet temperature is known, the cold inlet comes from the outlet of HX1. Hence, the inlet temperature function is assigned from the approximated first order response of HX1. The cold and hot side response of the module HX2 can be obtained in the same way like HX1 for parallel flow configuration. The response on cold side for 1-2 pass PHE is same as the response of HX2 because on the cold side outlet of HX1 is

Fig. 7. Schematic representations of flow coupling in (a) 1-2 pass PHE and (b) 2-2 pass PHE.

the input to the HX2 as shown in Fig. 7a. Whereas in the hot side, the response of 1-2 pass PHE is computed by combining the outlet temperature response of HX1 and HX2 along with proper phase lag effect at the port exits. Thus, final response of 1-2 pass plate heat exchanger have been obtained by using the responses of two heat exchanger modules HX1 and HX2. In the same way, it can be extended to any 1-n type multi-pass PHE. 6.2. The 2-2 pass plate heat exchanger The interaction of the hot and cold fluids in the two counter flow modules of 2-2 pass PHE is shown in Fig. 7b. In this type of pass arrangement the response of HX1 and HX2 are mutually dependent, which make the solution more complicate. Initially the response of HX1 can be obtained using step input on hot side and cold side is zero. Then the hot response of HX1 is approximated to the first order function and used as hot side input to HX2 and zero input to cold side. The cold side response of HX2 is again approximated to the first order function and input to the HX1 to continue for the next iteration. The procedure is continued till the values of K, τ d , and τ c of both hot response of HX1 and cold response of HX2 converge. The final hot and cold responses are given by: (Tout,hot )2-2 = (Tout,hot )HX1 (Tout,cold )2-2 = (Tout,cold )HX2 The method can be extended to any n-n or m-n pass PHE. 7. Results and discussion Using the above-mentioned procedure, selected outputs are presented here to describe the nature and effects of the maldistribution on multi-pass arrangements. In this analysis, mainly the effects of flow maldistribution parameter m2 and dispersion parameter Pe on temperature transients are presented. The plate spacing and entry length from port inlet to the first plate are chosen to be 2% and 10% of the effective length of the plates, respectively. In the present analysis due to flow maldistribution from port to channel the velocity differs from one channel to another and hence, the heat transfer coefficient as well. But the variation of the Peclet number with Reynolds number is not significant within the small range of Reynolds number [24]. So, its variation from channel to channel is not taken into account and same value is considered for both the fluids. Here Peclet number of 20 is taken for all the responses, against 5 taken in the previous studies [9,10], because it indicates only the backmixing of the fluid inside the channel. Actually, all the responses obtained are ideally characterised by second order system with delay time, but it incorporates too many constants, the physical significance of which are not always obvious for overdamped systems. On the other hand, first order system response approximation to this type of response is not very inaccurate, except the mismatch of slope during the first few percentage of temperature rise. So, the same tempera-

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Fig. 8. Effect of maldistribution for HX1 of 1-2 pass PHE; N = 15, Pe = 20, NTU = 1, R = 1.

ture transient results can be well described by comparing with FOS parameters. 7.1. Effect of flow maldistribution and heat exchanger parameters for 1-2 pass In this parametric study the values considered are N = 15, Rc = 0.2, Rτ = 1.0, Rg2 = 1.0, RN = 1.0, NTU = 1.0, and Pe = 20. Overall transient response of the 1-2 pass PHE are shown in Figs. 8–10. Fig. 8 represents the effect of flow maldistribution on the cold and the hot side temperature transients of the heat exchanger HX1, which characterises for the counter flow with Ztype configuration. It can be observed that initial delay increases with maldistribution on both cold and hot sides because the fluid has to travel across all the channels to give the temperature rise at the outlet. As a result all the transient curves show parallel lines without crossover. It is also observed that initial delay is more in the hot side than in the cold side which is due to strong phase lag effect at the hot side and due to the fact that the cold fluid enters HX1 with zero temperature. The temperature response of the cold fluid at steady state represents the effectiveness of the heat exchanger, since it is assumed that initially both the fluids are at

Fig. 9. Effect of maldistribution for HX2 of 1-2 pass PHE; N = 15, Pe = 20, NTU = 1, R = 1.

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Fig. 10. Effect of maldistribution for overall 1-2 PHE; N = 15, Pe = 20, NTU = 1, R = 1.

same temperature and heat capacity rate ratio is unity. It is evident that at the steady state the non-dimensional cold and the hot fluid temperatures add up to unity to satisfy the energy balance condition for equal heat capacity rates. The outlet temperature of the cold fluid decreases at steady state with maldistribution (m2 ), which indicates that the effectiveness of the heat exchanger reduces with increase of maldistribution. Hot side transients, shows the opposite trend at the steady state, i.e. the steady state temperature increases with maldistribution, thereby, satisfying the energy balance. The responses of HX2 have been shown in Fig. 9. It is interesting to note here that for both the cold and the hot side, fluids are entering with non-zero temperature. Due to this, cold side transients are showing a gradual rise, taking longer time to reach the steady state, whereas in single-pass, transients shows the sudden rise during initial period and comes to steady state early. The combined overall transient response of 1-2 pass PHE is shown in Fig. 10. It is observed that initial delay period is more on both the cold and hot sides because on both the sides, fluids are entering with non-zero temperature, which brings out a strong phase lag effect. The cold side outlet temperature response of the whole heat exchanger is same as that of heat exchanger HX2. It shows that the temperature transients are strongly affected with the increase in flow maldistribution since the maximum difference between the responses for m2 = 3 and m2 = 6 is more than that of between m2 = 0 and m2 = 3. It is also observed that cold side steady state temperature, i.e. the effectiveness diminishes as the maldistribution increases. The effect of maldistribution and Peclet number on transient response of 1-2 pass PHE for cold and hot side, are represented by Figs. 11 and 12. Here, PHE with minimum and maximum Peclet numbers of 5 and 30 is taken into consideration. The cold side temperature transients show gradual increase in temperature and taking longer time to come to steady state due to multi-pass arrangement as mentioned in the above case. The temperature transients are strongly affected by flow maldistribution at lower and higher dispersive Peclet numbers in the transient state. It is also observed that initial delay period increases with Peclet number on both cold and hot sides, because of strong dispersion at lower Peclet number and faster response. It is observed that in the steady state region, lower Peclet number (strong dispersive

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Fig. 11. Effect of maldistribution and Peclet number for 1-2 pass PHE (cold fluid); N = 15, NTU = 1, R = 1.

Fig. 14. Effect of maldistribution for 2-2 pass PHE (cold fluid); N = 15, Pe = 20, NTU = 1, R = 1.

Fig. 13 depicts the effect of maldistribution and number of channels of 1-2 pass PHE on cold side. In this case, it is observed that as the number of channels increases, initial delay period of response increases because for higher number of channels it takes longer time to give temperature response at the outlet due to larger port length. In the transient regime the temperatures shows the similar kind of responses as in the cases of 1-2 pass heat exchangers. It is observed that maldistribution strongly affects the responses at both lower and higher number of channels. At the steady state cold fluid temperature is more for higher number of channels because the effectiveness increases with the number of channels due to reduction in end effects of PHE.

Fig. 12. Effect of maldistribution and Peclet number for 1-2 pass PHE (hot fluid); N = 15, NTU = 1, R = 1.

effect) curve shows minimum temperatures on cold side, which indicates that the effectiveness of the heat exchanger decreases with increasing dispersion. At the same time it is also observed that the cold side temperature, i.e. effectiveness, decreases with maldistribution. Hence, it can be easily concluded that the performance of the heat exchanger will decrease not only with the dispersion in the fluid but also with flow maldistribution.

Fig. 13. Effect of maldistribution and number of channels for 1-2 pass PHE (cold fluid); N = 15, Pe = 20, R = 1.

7.2. Effect of flow maldistribution and heat exchanger parameters for 2-2 pass In this case, the values considered are N = 15, Rc = 0.2, Rτ = 1.0, Rg2 = 1.0, RN = 1.0, NTU = 1.0, and Pe = 20 and analysis is not like previous one—both hot and cold fluid of both heat exchangers HX1 and HX2 are non-zero at entry. The effect of flow maldistribution on cold side for 2-2 pass PHE is shown in Fig. 14. It is observed that some sudden change in slope of responses occur after a certain period, due to the collection and redistribution of the fluid at the exit of each pass. Then a slow reduction in slope is observed over a longer time than the 1-2 pass PHE, taking longer time to come to steady state due to multi-passing of cold fluid. The transient responses are difficult to distinguish because of the closeness, so these can be approximated by first order system (FOS) response. In order to do that, a first order response curve is fitted to simulated result by non-linear regression analysis as shown in Fig. 15. It must be seen that although there lies a discrepancy between a simulated and approximated curves, features such as delay time, time constant and asymptotic value of response are depicted quite accurately by the approximated curve. Fig. 16 shows the temperature transients of multi-pass PHEs with the single-pass PHE for m2 = 6 having N = 30, NTU = 2, Pe = 20 and R = 1. The reduction in cold side steady state temperature with maldistribution indicates the decrease in effectiveness, which is more for 2-2 pass arrangement when

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Fig. 15. Curve fitting to the simulation data for approximation as a first order system.

Fig. 16. Effect of maldistribution and NTU for 2-2 pass PHE (cold fluid); N = 15, Pe = 20, R = 1.

compared to 1-2 pass and single-pass PHEs. In case of 1-2 pass configuration, decrease in effectiveness is lower than the single-pass PHE due to the parallel flow in the second module of the heat exchanger. This also reconfirms the fact that 1-2 pass arrangement is a poor choice for equal heat capacity rates of the two fluids. Fig. 17 shows the comparison of time constant

Fig. 17. Comparison of multi-pass PHEs with single-pass configuration for time constant; N = 30, Pe = 20, R = 1.

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Fig. 18. Comparison of experimental temperature response with the theoretical model for 1-2 flow arrangement; Re = 1110, N = 31, NTU = 2.0, Rg2 = 0.9 for flow distribution, m2 = 3.8.

for various arrangements of PHEs against flow maldistribution, it represents how fast the heat exchanger responds with respect to its asymptotic value. 1-2 pass heat exchanger responds faster than the 2-2 pass exchanger and single pass heat exchanger responds faster than the two multi-pass arrangements. The time constant increases substantially at higher flow maldistribution for 2-2 pass arrangement, but for 1-2 and single pass arrangements, its increase is very nominal and it takes place only at higher flow maldistribution. It is clear that the overall variation of time constant with maldistribution is very much dependent on pass arrangement. Although the present study is analytical one which does not require any validation, the assumptions made in the study need to be justified. For this purpose the results of the present model is compared to the experimental data of Srihari [25] as shown in Fig. 18. It can be seen that the actual hot inlet temperature is a steep rise rather than an ideal jump because it is not possible to create an ideal jump experimentally. In the computation a fitted function to the actual hot inlet temperature is used. It can be observed that the model in general agrees with experimental observation quite well keeping the experimental uncertainties in mind. 8. Conclusion A detail analysis has been presented here for the temperature transients of multi-pass PHEs considering flow maldistribution from port to channels. In this analysis 1-2 and 2-2 pass models are presented which can be extended to analyse 1-n and m-n/n-n type PHEs. These heat exchangers are divided into smaller single pass heat exchanger modules for the purpose of analysis. In case of 1-2 pass PHE, the solution have been obtained in a straight forward way by analysing the successive modules. The 2-2 pass PHEs have been simulated by iterating between the responses of each module of the heat exchanger. The solution to each module is obtained by solving the governing equations using Laplace transformation followed by numerical Laplace inversion to the time domain, using Fourier series approximation. The main contribution in the present analysis is that, in case of multi-pass PHEs, the effect of flow maldistribution and fluid backmixing are separated while their signatures on heat

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exchangers are studied. Here flow maldistribution from port to channel is considered through an analytical model available in literature and the axial dispersive Peclet number is used to take care of fluid backmixing within the channels. Moreover, while calculating the responses also the fluid mixing and redistribution at the pass interface, the cumulating effect of phase lag and pass interfacial heat transfer have been considered. First order approximation has been used in the iteration process, to find the input for the successive passes and results are presented for a unit step rise in the hot side. The temperature responses are analysed in transient regime for various pass arrangements. It is observed that initial delay period is more on both cold and hot sides due to strong phase lag effect and multi-passing. A gradual temperature rise is observed in case of 2-2 pass PHE due to successive delays in the passes during multi-passing of both the fluids whereas in single pass and 1-2 pass arrangements the response is faster due to the presence of a single pass fluid. In both multi-pass cases, the effect of flow distribution along with dispersive Peclet number shows similar behaviour. However, the temperature response deviations from uniform flow distribution are more predominant in 2-2 pass PHE when compared to 1-2 pass PHE. In general, it is observed that the nature of the response in transient regime are dependent on the flow maldistribution, flow arrangements, number of channels and fluid backmixing represented by axial dispersion. The time required to reach the steady state is more for multi-pass arrangements when compared to single-pass PHE. It is also observed that asymptotic value of these transients on the cold side diminishes with the increase in flow maldistribution indicating the performance reduction of the heat exchanger. It is observed that effect of flow maldistribution on 2-2 pass PHE is predominant than that of the 1-2 pass PHE. Thus the present method offers an efficient transient multi-pass PHE analysis considering flow maldistribution.

K lc

gain in the first order system path traversed by fluid particle between two consecutive channels (m) L fluid flow length in channel (m) m maldistribution parameter mj j − 2[j/2], where j is an integer m ˙ mass flow rate in the conduit n number of channels on one side NTU1 number of transfer units of the heat exchanger ˙ Pe axial dispersive Peclet number, wL/A cD ˙ a2 /w ˙ a1 R2 capacity rate ratio in the channels, w Rc wall heat capacity rate ratio, CW /C1 ˙ g2 /w ˙ g1 Rg2 capacity rate ratio of the combined flow, w Rgτ characteristic rate ratio of the combined flow, τ rg2 /τ rg1 RN Ua2 /Ua1 Rτ characteristic time ratio in channels, τ ra2 /τ ra1 s transformed time variable in Laplace domain S¯ matrix with inlet fluid function t dimensionless temperature, (θ − θ g1,in )/(θ g2,in − θ g1,in ) T temperature obtained by Laplace transformation of temperature t T¯ temperature matrix u(z − φ) a unit step function in z with a phase lag of φ ˙ a ]1(2) Ua1(2) [hA/w U¯ matrix of eigenvectors of the matrix A v˙ c non-dimensional channel volume flow rate velocity of the combined fluid (m s−1 ) Vg Vi the velocity of the fluid in the port after ith channel volume flow rate in the channel (m3 s−1 ) V˙ c V˙ g volume flow rate of the combined fluid (m3 s−1 ) ˙ w thermal capacity rate of the fluid in channel (W K−1 ) x dimensionless space coordinate along the channel, X/L X space coordinate (m) y dimensionless space coordinate along the port z dimensionless time, τ/τ a1

Acknowledgements The authors gratefully acknowledge the financial support from SIDA-Swedish research links and Swedish research council (VR). They are also grateful to Prof. B. Sunden and Dr. B.P. Rao for their suggestions and support. Appendix A. Nomenclature

A ¯ A Ac ¯ B C Cp D ¯ D f(Z) F(s) h i

heat transfer area per effective plate (m2 ) coefficient matrix for system of differential equations free flow area in a channel (m2 ) diagonal matrix heat capacity of resident fluid (J K−1 ) specific heat capacity of fluid (J/(kg K)) axial dispersion coefficient (W/(m K)) matrix resulting form boundary condition inlet temperature function Laplace transform of f(Z) heat transfer coefficient (W/(m2 K)) square root of −1

Greek letters βj jth eigenvalue of matrix A θ temperature (K) τ time (s) τc dimensionless time constant τd dimensionless delay time τ ra1(2) residence time in case of uniform distribution in fluid ˙a 1 (or 2), C/w τ time of travel in the port (s) φ dimensionless phase lag (cumulative value) φ dimensionless phase lag (discrete value), τ i /τ ra1 Subscripts a the case of uniform flow distribution g combined flow before departing into channels i ith channel W plate Wi ith plate 0 initial 1 the fluid in odd channels 2 the fluid in even channels

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