Estimation of inlet temperature of a developing fluid flow in a parallel plate channel

International Journal of Thermal Sciences 57 (2012) 126e134

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Estimation of inlet temperature of a developing fluid flow in a parallel plate channel Ajit K. Parwani, Prabal Talukdar*, P.M.V. Subbarao Department of Mechanical Engineering, Indian Institute of Technology Delhi, Hauz Khas, New Delhi 110016, India

a r t i c l e i n f o

a b s t r a c t

Article history: Received 3 October 2011 Received in revised form 16 January 2012 Accepted 6 February 2012 Available online 14 March 2012

A numerical model is developed for an inverse problem of estimation of inlet temperature in a parallel plate channel. A hydrodynamically and thermally developing laminar flow is considered. The upper plate of the channel is maintained at constant heat flux and the lower plate is insulated. The Lagrange multiplier method is applied for this constrained optimization problem. The minimization of the objective function is performed using the conjugate gradient method. The adjoint and sensitivity equations along with the energy equation are discretized using the finite volume method and a code is developed for the solution of the same. The momentum equations are solved using an in-house CFD source code and are coupled with the developed numerical code for the inverse method. The direct problem is first solved with known inlet and boundary conditions and the temperature field of the domain is determined. Inverse method is then applied to predict the inlet temperature with some of the additional temperature data inside the solution domain obtained from the direct problem. The prediction of inlet temperature by the present algorithm is found to be quite reasonable. Ó 2012 Elsevier Masson SAS. All rights reserved.

Keywords: Inverse heat transfer Finite volume method Conjugate gradient method Developing flow CFD Parallel plate channel

1. Introduction A direct heat transfer problem is about the solution of the heat equation depicting a model of heat transfer e conduction, convection, radiation or a combination of more than one of them. These mathematical models are governed by various input factors e.g. surface temperature, heat source strength, heat flux, dimension of the conducting body/convecting media, etc. and their goal is to predict the temperature profile of the body either independent of (steady) or depending on the time (transient). But in the real life situations, where it is not possible to measure the causal factors (boundary temperature or flux) directly, however the temperature can be measured inside by a sensor/thermocouple, a natural problem becomes to predict the unknown causal factor. This leads to the inverse problem. The direct problem of the convective flow inside a duct has been investigated for various boundary conditions and duct shapes in great detail [1]. Several researchers carried out inverse convection problem of estimation of wall heat flux or inlet temperature in a channel for hydrodynamically fully developed flow. Moutsoglou [2] investigated the steady state inverse forced convection problem between two parallel plates. The wall heat flux of the top wall was

* Corresponding author. Tel.: þ91 11 26596337; fax: þ91 11 26582053. E-mail address: [email protected] (P. Talukdar). 1290-0729/$ e see front matter Ó 2012 Elsevier Masson SAS. All rights reserved. doi:10.1016/j.ijthermalsci.2012.02.009

estimated from measured temperature data taken at the bottom wall using straight inversion and whole domain regularization schemes. Huang and Ozisik [3] determined space wise variation of the steady state heat flux along a parallel plate channel with laminar flow field. They used the conjugate gradient method (CGM) for finding the unknown flux from the temperature measurements at various locations within the flow field. Park and Lee [4] estimated the wall heat flux for steady laminar flow inside a parallel plate channel based on the Karhunen Loeve Galerkin procedure. Raghunath [5] estimated the inlet temperature of a steady hydrodynamically developed laminar flow in a parallel plate channel from measurement of temperature at the downstream using quasiNewton conjugate gradient method. There can be seen quite a lot of works for estimation of transient boundary data. Bokar and Ozisik [6] estimated the time and space wise variations of the inlet temperature for laminar flow between parallel plates by the CGM. Machado and Orlande [7] and Li and Yan [8] applied the CGM with an adjoint equation to estimate the time and space wise variations of wall heat flux of a parallel plate channel for a laminar forced convection flow. Kim et al. Ref. [9] also estimated the time and space wise variations of wall heat flux of a parallel plate channel using sequential gradient method. The function specification method was then incorporated to stabilize the sequential estimation. Hong and Baek [10] considered two phase transient laminar flow in a parallel plate channel and estimated the inlet temperature by using the CGM and the Tikhonov

A.K. Parwani et al. / International Journal of Thermal Sciences 57 (2012) 126e134

Nomenclature b B cp d De k J l L M Pe Pr q Q Re S Tref T To uin U

height of duct [m] dimensionless height specific heat capacity of fluid [J/(kg K)] direction of descent, defined by Eq. (15) hydraulic diameter [m] thermal conductivity [W/(m K)] objective function, defined by Eq. (11) length of duct [m] dimensionless length number of measured data Peclet number Prandtl number heat flux [W/m2] dimensionless heat flux Reynolds number Lagrangian, defined by Eq. (13) reference temperature [K] temperature [K] inlet temperature [K] inlet velocity of flow [m/s] dimensionless velocity in X coordinate

regularization method (TRM). Yang [11] estimated the time dependent contact heat and mass transfer coefficients using the CGM. They used the temperature and moisture history data of the measurements in a double-layer hollow cylinder for the inverse analysis. Chen et al. [12] applied CGM to estimate the unknown space and time dependent convective heat transfer coefficient of an annular fin. Lin et al. [13] applied CGM to estimate the heat flux at one of the boundaries for an unsteady laminar forced convection flow in a parallel plate channel considering wall conduction effect. Liu and Ozisik [14] estimated the time dependent wall heat flux and Li and Yan [15] estimated the space and time dependent wall heat flux for turbulent forced convection problem inside a parallel plate channel using CGM. Some literature are also available for the estimation of the space and time dependent wall heat flux for unsteady forced convection problem in annular duct [16,17]. Apart from the CGM, a plenty of other optimization methods [2,4,9,18e21] have been used in the recent times for inverse heat transfer problems. Recently, Das et al. [18,19] solved inverse transient conductioneradiation problem for estimation of scattering albedo, conductioneradiation parameter and boundary emissivity. They used the lattice Boltzmann method for conduction and the finite volume method (FVM) for radiation in conjunction with the genetic algorithm for optimization. Mossi et al. [20] considered an inverse problem of radiation and convection heat transfer to find out the heat input of a heater located at the top and side surfaces of the enclosure using the truncated singular value decomposition (TSVD) regularization method. Lin and Wang [21] estimated the unknown temperature at the boundary of a two dimensional complex shape heat conduction problem based on the sequential method and the concept of future time combined with the finiteelement method. As seen from the literature review that CGM is one of the most efficient methods used for inverse convective heat transfer problems and hence, in the present work, the same method is considered. It is also noticed from the literature review cited above that study of inverse problem with hydrodynamically developing flow has not received sufficient attention despite its practical importance. To the authors’ knowledge, most of the literature deal with hydrodynamcially developed flow field. This motivates the present

V X, Y

127

dimensionless velocity in Y coordinate dimensionless coordinates

Greek symbols search step size, defined by Eq. (17) Dirac delta function 3 convergence criteria g conjugation coefficient, defined by Eq. (16) l Lagrange multiplier r density of fluid q dimensionless temperature q0 dimensionless inlet temperature qs dimensionless temperature measured by sensors x space variable converting final value problem to initial value

b d

Superscripts n iteration number Subscripts M measurement location for a single sensor Ref reference value

investigation. In the present work, a hydrodynamically and thermally developing flow between two parallel plates is considered. The flow is considered to be laminar. A constant heat flux condition is imposed on the upper plate and lower plate is kept insulated (adiabatic). The inlet temperature of the fluid is unknown and is to be estimated. The considered boundary condition is also novel for an inverse problem of estimation of inlet temperature and is not taken up even for hydrodynamically developed flow. This is also worth to mention here that even with a small change in the type of boundary condition, the whole formulation for the inverse problem needs to be modified. Hence, every inverse problem is different depending on variations of either of the boundary condition, flow condition or the unknown parameter to be estimated and hence necessitates an extra effort for each of such problems. As mentioned before, in the present work, in order to minimize the objective function, the CGM is used. The CGM is also called an iterative regularization method, which means the regularization procedure is performed during the iterative processes and thus the determination of optimal regularization conditions is not needed. The conjugate gradient method derives from the perturbation principles and transforms the inverse problem to the solution of three problems, namely, the direct, sensitivity and the adjoint problem. These three problems are solved by the FVM and the calculated values are used in CGM for inverse calculations. The tests have been performed for varying inlet temperature profile. For finding out the inlet temperature in this inverse problem using the least square method, additional knowledge of temperature data inside the solution domain is needed. In a practical problem this temperature data is measured experimentally in order to estimate the inlet temperature. In the present work, instead of experimental measurement, this temperature data is calculated using a direct problem of known inlet temperature. To make it more realistic, errors have been added to these internal temperature data in one of the test cases considered in this work. The inverse method is then applied using these internal temperature data to predict the inlet temperature of the fluid previously imposed in the direct problem as an inlet condition. So, both direct and inverse problem is required to be solved for this analysis where no experimental data is available.

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2. Problem statement and formulation 2.1. Direct problem First the formulation of the direct problem is presented. A hydrodynamically and thermally developing flow between two parallel plates is considered. The schematic of the problem is illustrated in Fig. 1. While the lower plate is insulated, the upper plate is exposed to known heat flux. Neglecting the axial diffusion, the governing transport equations for the laminar, non-dissipative forced convective flow of a Newtonian fluid are:

Continuity :

(1)


vu vu vP v2 u v2 u ¼  þm þ x  Momentum : r u þ v vx vy vx vx2 vy2 vv vv y  Momentum : r u þ v vx vy

q ¼ T=Tref  ; Y ¼ y=ðPeDe Þ; X ¼ x=ðPeDe Þ; U ¼ u=uin ; V ¼ v=uin Q ¼ qb= k Tref ; De ¼ 2b; Pe ¼ Re Pr; L ¼ b=ðPeDe Þ; Re ¼ uin De =v; B ¼ 1=ð2PeÞ

vu vv þ ¼ 0 vx vy




Equations Simulating Turbulence) [22,23]. The code is based on the finite volume method and is written in curvilinear coordinates. SIMPLE algorithm is used to solve the momentum equations. Once the velocity distributions are computed, the energy equation is solved with the developed Fortran code. The energy equations with the boundary conditions are non-dimensionlized by introducing the following variables:



vP v2 v v2 v þ ¼  þm vy vx2 vy2

! (2a)

!


 vT vT v2 T þv ¼ k 2 Energy : rcp u vx vy vy

(2b)

(3)

where u and v are the velocity in x and y direction, T and P are temperature and pressure respectively, r, m, k and cP are the density, viscosity, thermal conductivity and specific heat respectively and these properties are assumed to be constant. The boundary conditions are: Bottom boundary (at y ¼ 0):

Momentum : No slip : u ¼ v ¼ 0

(4)

vT Energy : Adiabatic : ¼ 0 vy

(5)

Inlet boundary (at x ¼ 0):

Momentum : Specified uniform velocity : u ¼ uin ; v ¼ 0

(6)

Energy : Specified temperature : T ¼ T0 ðyÞ

(7)

Top boundary (at y ¼ b):

Momentum : No slip : u ¼ v ¼ 0

(8)

Energy : Constant flux : kðvT=vyÞ ¼ q

(9)

In this article, the momentum equations are solved using an ‘inhouse’ CFD code FASTEST3D (Flow Analysis by Solving Transport

where Tref is a reference temperature value, Pe, Pr and Re are Peclet, Prandtl and Reynolds number. The energy equation can be expressed in dimensionless form as


v2 q vq vq 2 U ¼ Pe þ V vx vY vY 2

(10a)

vq=vY ¼ 0 at Y ¼ 0;

(10b)

vq=vY ¼ Q at Y ¼ B;

(10c)

q ¼ q0 ðYÞ at X ¼ 0

(10d)

Solution of the above equation gives temperature profile q(X, Y) inside the channel. This is referred as direct problem. 2.2. Inverse problem In the inverse problem, temperature at the inlet, q0(Y) is unknown and is to be estimated. The additional information which requires for solution of this inverse problem is some temperature data inside the solution domain. In a practical problem, these temperature data are found by measuring the temperature inside the channel at different locations using some sensors. In this work, instead of measured data, some temperature data are taken from the solution of the direct problem described in section 2.1. To predict the unknown inlet temperature, a quite natural hit and trial solution could be to guess the inlet temperature and see how best the measured temperature match with its direct problem solution profile. By the theory of statistical estimation, the optimal estimator (in general settings) is the one which minimizes the mean square norm of prediction errors and is called minimum mean square error estimator. Therefore, the steps involved in the solution of an inverse heat transfer problem (IHTP) are as below: 1. Start with an initial guess of the unknown inlet temperature. 2. Solve the direct heat equation to formulate the temperature profile. 3. Find the prediction error. 4. Minimize the mean square norm and hence optimize the guessed inlet temperature in a number of iterations.

Fig. 1. Schematic of the problem with boundary conditions.

From the above description of solution method, an IHTP gets converted to an optimization problem with objective as the minimization of the mean square norm of prediction errors and the constraint as the direct heat Eq. (10). A formal description of the solution methodology of IHTP is as follows. The temperature is measured by sensor at a number of locations qs(Xm,Ym) and if they are free from measurement errors, then q0(Y) which satisfies the equation q(Xm,Ym; q0(Y)) ¼ qs(Xm,Ym), is the exact

A.K. Parwani et al. / International Journal of Thermal Sciences 57 (2012) 126e134

inverse solution. This implies that as the total mean square difference for all the sensors, given by,

XM

Jðq0 ðYÞÞ ¼

2

½qðXm ; Ym ; q0 ðYÞÞ  qs ðXm ; Ym Þ m¼1

(11)

decreases, the guessed q0(Y) approaches the exact solution. Hence the inverse heat transfer problem becomes the following optimization problem

minimum Jðq0 ðYÞÞ 
v2 q vq vq 2 U ¼ Pe subject to þ V vX vY vY 2

(12)

To solve this optimization problem, we may adopt any method of constrained optimization e.g. Barrier Method, Penalty Function Method, etc. However, one simple strategy to solve this constrained optimization problem is to convert the problem into an unconstrained optimization problem and then solve it using any method of unconstrained optimization e.g. Steepest Decent Method, CGM, Quasi-Newton Method, etc. In the present work, CGM us used to solve the above optimization problem. For a detailed description of CGM, one may refer to [24]. The optimization problem (12) is an equality constrained problem. To convert the constrained problem into an unconstrained one, we need to multiply the constraint equation with Lagrange multiplier (function in same space (X, Y)) and write the Lagrangian as

Sðq0 ðYÞÞ ¼

XM

2

½qðXm ; Ym ; q0 Þ  qs ðXm ; Ym Þ ! 2 vq vq 2 v q lPe dX dY U V vX vY vY 2

0

0

3

is a small predefined constant.

Now in order to proceed to the optimization procedure, the aim is to derive an expression for bn and d(Y)n for which Dq(X, Y) and VS (q0(Y)) is needed. To get Dq, the quantities in direct Eq. (10) are perturbed. Suppose that q0(Y) is perturbed by Dq0(Y) then the temperature q(X, Y) undergoes a perturbation Dq(X, Y). Substituting in the direct equations q by q þ Dq, q0(Y) by q0(Y) þ Dq0(Y), perturbed equations are obtained. Then, subtracting the original equations from the perturbed equations and neglecting the terms of the second order of smallness leads to the following sensitivity equations,

v2 Dq vDq vDq 2 U ¼ Pe þ V vX vY vY 2

(19a)

vDq=vY ¼ 0 at Y ¼ 0 and Y ¼ B;

(19b)

Dq ¼ Dq0 ðYÞ at X ¼ 0

(19c)

Now it remains to find out the expression for VS (q0(Y)). For that perturbing in Eq. (13), q0(Y) by q0(Y) þ Dq0(Y) and q by q þ Dq and subtracting the resulting expression from the Eq. (13) and neglecting the second order terms, we find

DSðq0 ðYÞÞ ¼

m¼1 ZL ZB

þ

where

M X

þ 0 0

where the conjugate direction (direction of descent), d(Y) is of the form

  dðYÞn ¼ VS q0 ðYÞn þ gn d½Yn1

where d(.) is the Dirac delta function. Subsequently integrating by parts and utilizing the boundary conditions of the sensitivity problem, we get

DSðq0 ðYÞÞ ¼

M X

gn ¼

ZL ZB "

(15)

þ 0 0

ZL
þ

VS q0 ðYÞ

n  2

0

dY

0

ZB n h io2 VS q0 ðYÞn1 dY

þ

(16)



Search step size b can be determined by linearization of q(Xm, n Ym;q0(Y)n  bnd(Y)n), followed by setting vS½qðYÞnþ1 =vb ¼ 0. It follows that

b ¼

m¼1

  ZL
vlðX;0Þ vlðX;BÞ DqðX;0ÞdX  DqðX;BÞdX vY vY

Pe2 lð0;YÞUð0;YÞDq0 ð0;YÞdY

0

ZB

n

PM

# v2 l vl vl 2 DqðX;YÞdX dY þ Pe U þ V vX vY vY 2

0

ZB

0

n

2½qðq0 Þ  qs Dq$dðX Xm ÞdðY  Ym Þ

m¼1

where, the conjugate coefficient g is set as 0 for initial state, otherwise is given by the FletchereReeves expression as:



(20)

(14) n



# v2 Dq vDq vDq 2 l dX dY U  Pe þ V vX vY vY 2

Z L ZB "

(13)

q0 ðYÞnþ1 ¼ q0 ðYÞn bn dðYÞn

ZB

2Dq½qðq0 Þ  qs $dðX  Xm Þ$dðY  Ym Þ

m¼1

The optimization problem thus becomes a minimization of the above equation. Starting with an initial guess q0(Y)n and assuming n ¼ 0 where n refers to the number of iterations, the iterative process begins. The iteration is determined by the formula

129

 



  q Xm ; Ym ; q0 ðYÞn  qs ðXm ; Ym Þ $ Dq Xm ; Ym ; dðYÞn PM 

n 2 m¼1 Dq Xm ; Ym ; dðYÞ (17)

Pe2 lðL;YÞUðL;YÞDqðL;YÞdY

ð21Þ

0

Now by the definition of directional derivative,

DS½q0 ðYÞ ¼

ZB

VS½q0 ðYÞDq0 ðYÞdY

(22)

0

The traditional stopping criteria condition is specified as

Jðq0 ðYÞÞ< 3

(18)

Comparing Eqs. (22) and (21) and hence allowing terms containing Dq(X,Y) of Eq. (21) to vanish gives adjoint equation as:

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v2 l 2fqðq0 Þ  qs g dðX  Xm ÞdðY  Ym Þ þ 2 vY

vl vl ¼ 0 þ Pe2 U þV vX vY

(23a)

vl=vY ¼ 0 at Y ¼ B and Y ¼ 0;

(23b)

l ¼ 0 at X ¼ L;

(23c)

We note that the adjoint problem involves a condition at the outlet of channel at X ¼ L (Eq. (23c)), instead of the inlet condition at X ¼ 0 of regular direct problem. However the final value problem can be converted into an initial value problem by defining a new space variable given by x ¼ L  X. In the process of obtaining the adjoint problem, the following integral term is left:

DSðq0 ðYÞÞ ¼

ZB

Pe2 lð0; YÞUð0; YÞDq0 ðYÞdY

(24)

0

Comparing (24) and (22), we can obtain gradient equation for the functional as

VS½q0 ðYÞ ¼ Pe2 lð0; YÞUð0; YÞ

(25)

2.3. Solution procedure The momentum equations are solved using an in-house CFD code FASTEST3D. This code solves the governing equations in dimensional form. The grid is generated using commercial software Ansys ICEM CFD. Uniform flow of air at inlet velocity of uin ¼ 2.5 cm/s (Peclet number, Pe ¼ 397) is considered to flow through a duct of height b ¼ 12.8 cm and length l ¼ 63.5 cm. The resulting velocity distributions from the solution of the momentum equations are shown in Fig. 2. Both u and v velocity distributions are needed in the energy equation as well as the equations which governs the inverse problem. The properties are assumed to be constant and hence the momentum equations are decoupled from the energy equation. The upper wall heat flux q is taken as 1000 W/m2. The energy equation, sensitivity equation and the adjoint equations are discretized using the FVM. The convective terms are discretized using the upwind scheme [25]. The dimensionless step sizes are DX ¼ 0.0000625 and DY ¼ 0.0000125 in the axial and transversal directions respectively. A grid independency study is carried out with the direct problem. For this study, the temperature at inlet boundary is assumed constant at 500 K. The upper wall is maintained at a constant heat flux and the lower wall is adiabatic. Four uniform and one non-uniform grids are considered for this study. The numbers of grid points for uniform grid are 22  22 (grid 22), 52  52 (grid 52), 101  101 (grid 101) and 152  152 (grid 152). The numbers of grid points for the non-uniform grid is 101  101 with mesh density concentrated toward the entrance and both the walls. The grid independency results are shown in Fig. 3(a)e(b). Due to the existence of large temperature gradient near to the top boundary, the grid independency study is carried out at y ¼ 0.125 m apart from the y centerline (y ¼ 0.065 m). It can be seen from these figures that uniform grid of 101  101 mesh is fine enough to generate a grid independent solution for the present study. Quantitatively, the average % temperature difference (at y ¼ 0.125 m) of the uniform grid 101 from the finer grid 152 and the non-uniform grid is 0.21% and 0.55% respectively and hence in all the simulations

Fig. 2. u (a) and v (b) velocity contours for inlet velocity of uin ¼ 2.5 cm/s.

the 101 grid is considered to be accurate enough for the present work. The reason of considering a uniform grid for the entire (direct and inverse) problem is that the subroutine for the inverse problem is applicable to uniform grid only at the current stage. The computational procedure for the solution of this inverse problem using the CGM may be summarized as follows: Suppose q0(Y) is available at nth iteration. Step 1.1. Solve the continuity and momentum equations (Eqs. 1 and 2) using the CFD code. Step 1.2. Solve the energy equation (Eq. (10)) with a known inlet temperature and find out the temperature field. The aim is to find the inlet temperature profile to be known in step 1.2 from the information of some of the interior temperature data calculated in step 1.2. These interior temperature data are referred as data measured by sensors in the next section. Step 2.1. Assume inlet temperature as zero and solve the direct problem given by Eq. (10) to obtain dimensionless temperature field, q(X,Y). Step 2.2. Examine the stopping criterion 3 (taken as 1  104) given by Eq. (18). If satisfied, stop the calculation and assumed inlet temperature is the estimated inlet temperature. If not satisfied, continue to step 2.3. Step 2.3. Solve the adjoint problem given by Eq. (23) for l(X,Y). Step 2.4. Compute the gradient of the functional S(q0(Y)) from Eq. (25). Step 2.5. Compute the conjugate coefficient and direction of descent from Eqs. (16) and (15), respectively.

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131

3. Results and discussion The objective of this work is to estimate the inlet temperature of the duct without any prior information on the functional form of the inlet temperature distribution. Several inlet profiles, including a smooth function and a step function are examined. The effects of numbers of measurement M, distribution of measurement in the transversal direction and functional form of unknown inlet temperature on the accuracy of estimations are investigated. Three different functions of the unknown inlet temperature are tested in this work to examine the accuracy of the algorithm. Different cases considered in this work can be summarized as follows: i. The inlet temperature profile assuming to be a sine function Case 1: Different number of sensors arranged transversely at a distance of X ¼ 20DX ¼ 0.00125. Case 2: Nineteen sensors arranged transversely at different X locations. Case 3: Different number of sensors arranged axially along the centerline at Y ¼ 50DY ¼ 0.000625. Case 4: Different number of sensors arranged in XeY plane. Case 5: Different number of sensors arranged transversely at a distance of X ¼ 20DX with error introduced. ii. The inlet temperature profile assuming to be a triangular function Case 6: Different number sensors arranged transversely at a distance of X ¼ 20DX. iii. The inlet temperature profile assuming to be a step function. Case 7: Different number of sensors arranged transversely at a distance of X ¼ 20DX.

Fig. 3. Grid independency study; temperature variation near centerline at y ¼ 0.065 m (a) and near top boundary at y ¼ 0.125 m (b) for the different grid considered.

% error ¼

Step 2.6. Set Dq0(Y) ¼ d(Y)n, and solve the sensitivity problem given by Eq. (19) for Dq(X,Y). Step 2.7. Compute the search step size from Eq. (17). Step 2.8. Compute the new estimation for q0(Y) from Eq. (14) and return to Step 2.1.

For case 1, schematic of the arrangements of four sensors (M ¼ 4) are shown in Fig. 4(a) with dimensionless spacing of 20DY between each two sensors. Similar arrangements are assumed for M ¼ 19, 9 and 6 with corresponding dimensionless spacing of 5DY, 10DY and 15DY, respectively. The results of such arrangements are shown in Fig. 4(b). The results illustrate the effects of the number of measurements on the accuracy of the estimation, for sensors located at X ¼ 20DX. The results show that the accuracy of estimation improves by increasing the number of sensors and with either of the 19, 9 and 6 sensors, the present algorithm estimates the exact inlet temperature profile almost accurately. It is also seen that measurement with four sensor (M ¼ 4) are insufficient to produce good results. Quantitatively, ignoring a few prediction near the boundary, the average percentage error between actual and estimated temperature is 1.07%, 1.13%, 2.03% and 11.39%, with M ¼ 19, 9, 6 and 4 respectively. The initial estimation for the inlet temperature is taken as zero for all the simulations. The average percentage error is calculated as:

Average of ðexact temperature  estimated temperatureÞ  100 Average exact temperature

The effect of measurements along the axial direction on the accuracy of the estimation of inlet temperature is presented in Fig. 5. In this case (case 2), nineteen measurements along y axis are assumed to take either at one of the locations of X ¼ 40DX, 20DX, 10DX and 3DX. It should be noted that measurements taken at

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Fig. 4. (a) Schematic location of four sensors arranged in transverse direction at X ¼ 20DX distance. (b) Plots of estimated inlet temperature profile for a sine input with variations in number of sensors embedded at transverse position at X ¼ 20DX.

X ¼ 20DX and 10DX, produce accurate prediction whereas the measurements taken at X ¼ 40DX and 3DX produce comparatively poorer prediction of inlet temperature. Sensors positioned at X ¼ 3 DX is too near to boundary and possess fluctuating gradients near the middle, and sensors positioned at X ¼ 40DX is too far from the boundary and possess relatively low sensitivity coefficients therefore the average percentage error for both is higher than sensors positioned at X ¼ 20DX and 10DX. Quantitatively, with X ¼ 40DX,

Fig. 5. Plots of estimated inlet temperature profile for an inlet sine profile with variations in axial locations of sensors embedded.

20DX, 10DX and 3DX, the average percentage errors between actual and estimated temperature are 2.33%, 1.07%, 0.52% and 1.62%, respectively. Fig. 6(a) refers to test case 3 where all sensors are assumed to be embedded at the centerline (Y ¼ 50DY) along X direction. Six sensors (M ¼ 6) are shown in the schematic corresponding to dimensionless spacing of 15DX. Similar arrangements are assumed with M ¼ 19 and 9 corresponding to dimensionless spacing of 5DX and 10DX, respectively. The results are shown in Fig. 6(b). The results show that at boundary the estimated profile deviates much from exact profile. The reason is that the inlet temperature originated at both regions is unlikely to diffuse into the centerline region given that the temperature and flow fields are still developing. That is, there is almost no functional relationship between the measurement and estimated quantities. Therefore the estimations are not as good as compared to estimations of sensors embedded at transverse locations. This is also expected as the estimation is performed for a transverse inlet temperature profile and hence, the measurements taken in the same transverse direction are a better choice. With M ¼ 19, the average percentage error between actual and estimated temperature is 5%, while with M ¼ 9 and 6, it is approximately 5.6%. Fig. 7 refers to case 4 where sensors are arranged at various locations in the solution domain. The numbers of measurement considered here include M ¼ 27, 18, 9, 6 and 5. With M ¼ 9, three columns of three sensors are arranged at dimensionless vertical spacing of 25DY and dimensionless horizontal spacing of 10DX as shown in Fig. 7(a).Similar types of arrangements are made with M ¼ 27 where three columns of nine sensors are arranged at dimensionless vertical spacing of 10DY and dimensionless horizontal spacing of 10DX. Similarly with M ¼ 18, three columns of six sensors are arranged at dimensionless vertical spacing of 15DY and dimensionless horizontal spacing of 10DX. The arrangement of sensors with M ¼ 6 and M ¼ 5 are arranged in two columns as shown in Fig. 7(b) and (c) respectively. The test results for all these cases are plotted in Fig. 7(d) and it is seen that more the number of

Fig. 6. (a) Schematic showing the location of six sensors arranged in axial direction at Y ¼ 50DY distance. (b) Plots of estimated inlet temperature profile for an inlet sine profile with variations in number of sensors embedded at axial position at Y ¼ 50DY.

A.K. Parwani et al. / International Journal of Thermal Sciences 57 (2012) 126e134

133

Fig. 9. Plots of estimated inlet temperature profile for an inlet triangular profile with variations in number of sensors embedded at transverse position.

sensors, more accurate is the prediction. Quantitatively, with M ¼ 27, 18, 9, 6 and 4, the average percentage errors between the actual and estimated temperature are 1.07%, 1.4%, 2.4%, 3% and 9.5%, respectively. Case 5 is similar to case 1 but in this case, the effect of the measurement errors on the accuracy of estimation is studied. The measurement error in a range of 1  15 K with an average error of 7 K is randomly added to the exact temperatures which are

Fig. 7. Schematic showing the location of nine (a), six (b) and five (c) Sensors arranged in XeY plane. (d) Plots of estimated inlet temperature profile for an inlet sine profile with variations in number of sensors embedded in XeY plane.

Fig. 8. Plots of estimated inlet temperature profile for an inlet sine profile with sensor embedded at transverse positions taking into account of errors in the sensors reading.

Fig. 10. Plots of estimated inlet temperature profile for a step input with variations in number of sensors embedded (a) single discontinuity, (b) double discontinuity.

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calculated from direct problem with known inlet temperature. Fig. 8 shows results for this case. The number of transversal measurements considered here are same as with case 1which includes M ¼ 19, 9, 6 and 4 correspond to dimensionless spacing of 5DY, 10DY, 15DY and 20DY, respectively. Since some errors are introduced in each sensor measurement, therefore, as the number of measurements increase prediction in the estimation of inlet temperature becomes poorer. The result shows better estimation with M ¼ 6 measurements. However with M ¼ 4 the result is very poor because of insufficient measurement data. Fig. 9 shows the estimation of inlet temperature for a triangular profile (case 6). It illustrates the effect of the number of measurements on the accuracy of the estimation, for sensor located at X ¼ 20DX. The number of transversal measurements considered here are same as with case 1which includes M ¼ 19, 9, 6 and 4 correspond to dimensionless spacing of 5DY, 10DY, 15DY and 20DY, respectively. Result shows similar trend as seen with case 1. Fig. 10(a) and (b) show the result of step functional form with single and double discontinuity, respectively of inlet temperature profile (Case 7). It illustrates the effect of the number of measurements on the accuracy of the estimation, for sensor located at X ¼ 20DX. The number of transversal measurements considered here are M ¼ 24 and 9 correspond to dimensionless spacing of 4DY and 10DY, respectively. This case presents a very difficult case for an inverse analysis because of the discontinuities involved in this problem. Even with twenty-four errorless measurements, the exact inlet temperature profile could not be fully recovered. 4. Conclusions The conjugate gradient method (CGM) with adjoint problem was successfully applied for the solution of an inverse forced convection problem to determine the unknown inlet temperature profile in a parallel plate channel with hydrodynamically and thermally developing laminar flow. The formulation of the inverse problem is presented in detail. Several test cases involving different profiles of inlet temperature distribution, different numbers and locations of sensors and measurement with artificial error are considered. More is the number of accurate sensor embedded, higher is the accuracy of estimation of inlet temperature is noted. With the errors in sensors included, the lesser numbers of measurements are preferred. Present geometry suggests that four numbers of measurements are insufficient for the present numerical model. If the measurements are taken quite close to the inlet boundary which temperature is to be estimated, the results of estimation deteriorate. The profiles with discontinuities present a very difficult case for estimation. The overall results show that the inverse solution obtained by the CGM remains stable and found to be quite reasonable. Acknowledgment This work is supported by a fund from the SERC division of Department of Science and Technology, Government of India. This financial contribution to this research is greatly appreciated.

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