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Stochastic modeling of heat exchanger response to data uncertainties
Applied Mathematical Modelling 26 (2002) 715–726 www.elsevier.com/locate/apm
Stochastic modeling of heat exchanger response to data uncertainties R.C. Prasad
a,*
, Karmeshu b, K.K. Bharadwaj
b
a
b
Department of Engineering, University of New Brunswick, Saint John, NB, Canada E2L 4L5 School of Computer and System Sciences, Jawaharlal Nehru University, New Delhi 110067, India Received 24 August 1998; received in revised form 10 August 2001; accepted 8 October 2001
Abstract This paper analyses the performance of a counterflow concentric tube heat exchanger and estimates the uncertainties in the temperature response of the heat exchanging fluids caused by uncertainties in their inlet temperatures and overall heat transfer coefficients between the fluid streams. The analysis is based upon the two-point distribution technique. This paper describes the application of this technique to a heat exchanger for which the exact solution for the steady state temperature response is available. Results show that the uncertainty in inlet temperature has a stronger influence. The effect of data uncertainty in heat transfer coefficients, which generally have very high level of uncertainty, have only weak influence. The accuracy of predicted temperature response can, thus, be significantly improved by accurate measurement of the inlet fluid temperatures. Ó 2002 Elsevier Science Inc. All rights reserved. Keywords: Stochastic modeling; Heat exchanger; Two-point distribution
1. Introduction The performance analysis of heat exchangers is essential for their safe and economical design as well as for the design of necessary controls [1] for their safe operation under steady state as well as transient conditions which are encountered during start up and shut down. The predicted performance of a heat exchanger may, at times, significantly differ on account of inherent variability present in the model parameters and the initial values of the variables. One major
*
Corresponding author. Tel.: +1-506-648-5518; fax: +1-506-648-5513. E-mail address: [email protected] (R.C. Prasad).
0307-904X/02/$ - see front matter Ó 2002 Elsevier Science Inc. All rights reserved. PII: S 0 3 0 7 - 9 0 4 X ( 0 1 ) 0 0 0 8 2 - 8
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Nomenclature A B c D L m N P P1 , P2 p sd(x) T T U U x Greek b c k h h_ h€ r r2
a constant a constant specific heat of fluid, J kg1 K1 diameter of pipe, m length of pipe, m mass flow rate, kg s1 , kg min1 defined in Eq. (7d) perimeter of pipe, m parameter in Eq. (1) probability density function standard deviation of x ¼ rx temperature, °C, K mean value of T, °C, K overall heat transfer coefficient, W m1 K1 mean value of U, W m1 K1 distance along the length of pipe, m symbols defined in Eq. (7f) defined in Eq. (7e) defined in Eq. (7c) incremental temperature T ðxÞ T0 , °C, K dh/dx d2 h/dx2 standard deviation variance
Subscripts e exit i inlet o outside or outer pipe 1 inner pipe or fluid in inner pipe 2 fluid in the annular space U for heat transfer coefficient h for temperature
source of variability arises from the measurement uncertainty. It is, therefore, necessary to develop mathematical models in order to accurately predict their performance under actual conditions where various parameters and initial/boundary conditions are subject to measurement uncertainties.
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717
In heat exchangers, the overall heat transfer coefficient has a large measurement uncertainty (over 50%) [2]. Also the inlet fluid temperature for the heat exchanging fluids is subject to minor measurement uncertainties (1–5%). In the presence of these uncertainties, the exact deterministic solution [3] alone does not provide any inkling to the uncertainty in the response of the heat exchanger. A rational framework for describing these measurement uncertainties associated with the parameters and initial/boundary values treats them as random variables with specified first few moments or probability distribution [4]. The purpose of this paper is to investigate the effect of data uncertainty on the performance of a concentric tube heat exchanger. The analysis provides quantitative estimates of the uncertainties in the variables of interest (in this case, the temperature of the heat exchanging fluids), due to data uncertainties. The analysis is carried out by employing a method based on Rosenblueth’s twopoint distribution (TPD) [5,6].
2. Statistical description of data uncertainty Data concerning variability in the parameter values and initial conditions are usually too limited to specify the details of the shape of their probability distributions. However, in a number of problems, specification of the first two moments suffices to estimate the moments of the errors propagated through the model. The approach widely used by engineers for such types of problems is the first order second moment analysis (SMA) [7]. However, the TPD [5,6] which provides a simple and efficient technique has been employed in this work.
3. Two-point distribution Consider a random variable X with specified first two moments––mean X and standard deviation rx . Rosenblueth [5,6] proposed the representation of the random variable X in terms of two concentrations P1 and P2 placed at x ¼ x1 and x ¼ x2 . Using this representation, expressions for the expected value (Y ), standard deviation (rY ) of another random variable Y ¼ f ðX Þ can be obtained. The probability density function pðxÞ is written as: pðxÞ ¼ P1 dðx x1 Þ þ P2 dðx x2 Þ
ð1Þ
where d( ) is the Dirac delta function. For a symmetrical TPD, concentrations P1 ¼ P2 ¼ 1=2 and Eq. (1) reduces to [6] pðxÞ ¼ 12d½x ðx rÞ þ 12d½x ðx þ rÞ
ð2Þ
Using this, it is found that EðY Þ ¼ Y ¼ 12½f ðx rÞ þ f ðx þ rÞ
ð3aÞ
EðY 2 Þ ¼ 12½f 2 ðx rÞ þ f 2 ðx þ rÞ
ð3bÞ
and
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it is shown [4] that for small r, such that terms 0(r3 ) can be neglected, SMA and TPD give nearly the same result.
4. The physical problem Fig. 1 shows the schematic diagram of a counterflow concentric-tube heat exchanger. This system is represented by the following mathematical model [3]: Inner pipe: P1 U1 h2 þ m1 c1 h_1 P1 U1 h1 ¼ 0
ð4Þ
Annulus: m2 c2 h_2 þ ðP1 U1 þ P0 U0 Þh2 P1 U1 h1 ¼ 0
ð5Þ
Boundary conditions: h2 ð0Þ ¼ h2i ; h1 ðLÞ ¼ h1i
ð6Þ
The solution of Eqs. (4) and (5) yields [3]. h1 ðxÞ ¼ T1 ðxÞ T0 ¼ A ek1 x þ B ek2 x
ð7aÞ
h2 ðxÞ ¼ T2 ðxÞ T0 ¼ Að1 k1 NÞek1 x þ Bð1 k2 N Þek2 x
ð7bÞ
where, k1 ; k2 ¼ ½c ðc2 4bÞ1=2 =2 N¼ c¼
m1 c1 P 1 U1
ð7cÞ ð7dÞ
P 0 U0 P 1 U1 P 1 U1 þ m2 c2 m2 c2 m1 c1
Fig. 1. Schematic representation of a concentric counterflow heat exchanger.
ð7eÞ
R.C. Prasad et al. / Appl. Math. Modelling 26 (2002) 715–726
b¼
P1 U1 P0 U0 m1 c1 m2 c2
719
ð7fÞ
A¼
h2i ek2 L h1i ð1 k2 N Þ ðk2 ek1 L k1 ek2 L ÞN þ ðek2 L ek1 L Þ
ð7gÞ
B¼
h2i ek1 L h1i ð1 k1 N Þ ðk2 ek1 L k1 ek2 L ÞN þ ðek2 L ek1 L Þ
ð7hÞ
In this system, the boundary temperatures h1i and h2i as well as the overall heat transfer coefficients U0 and U1 are subject to measurement uncertainties.
5. Uncertainties in temperature response––TPD analysis We are given the mean values of the inlet fluid temperature and the overall heat transfer coefficient, assumed to be independent, h1i , h2i , U 0 and U 1 with the corresponding variances as r2h1i , r2h2i , r2u0 and r2u1 respectively. The assumption of independence seems plausible from the consideration that randomness in the parameters concerned may be attributed to independent sources. In the following sections TPD analysis for the following three cases are presented: 5.1. Case 1: Uncertainties in fluid temperatures h1i and h2i at the boundaries Inner fluid: The symmetrical TPD [6] in this case results in a probability density function pðh1i ; h2i Þ given by: pðh1i ; h2i Þ ¼ ½12dðh1i ðh1i rh1i ÞÞ þ 12dðh1i ðh1i þ rh1i ÞÞ½12dðh2i ðh2i rh2i ÞÞ þ 1dðh2i ðh2i þ rh ÞÞ 2
2i
ð8Þ
It is assumed that the random variables h1i and h2i are independent. The first moment about the origin, referred to as the mean, is given by: Z 1 Z 1 h1 ðxÞpðh1i ; h2i Þ dh1i dh2i ð9Þ Eðh1 ðxÞÞ ¼ 1
1
Since h1i and h2i are random variables, A and B occurring in Eq. (7a) are random variables and indicating this fact explicitly we have: h1 ðxÞ ¼ Aðh1i ; h2i Þek1 x þ Bðh1i ; h2i Þek2 x
ð10Þ
From Eqs. (9) and (10) we obtain: Eðh1 ðxÞÞ ¼
4 1X ½Aðh1i ; h2i Þek1 x þ Bðh1i ; h2i Þek2 x 4 k¼1
where h1i and h2i for k ¼ 1; 2; 3; 4 are indicated below:
ð11Þ
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k ¼ 1; k ¼ 2;
h1i ¼ h1i rh1i ; h1i ¼ h1i rh ;
h2i ¼ h2i rh2i h2i ¼ h2i þ rh
h1i ¼ h1i þ rh1i ; h1i ¼ h1i þ rh1i ;
h2i ¼ h2i rh2i h2i ¼ h2i þ rh2i
1i
k ¼ 3; k ¼ 4;
2i
ð12Þ
The variance varðh1 ðxÞÞ and standard deviation rh1 ðxÞ of h1 ðxÞ are obtained from the following: varðh1 ðxÞÞ ¼ r2h1 ðxÞ ¼ Eðh1 ðxÞÞ2 ½Eðh1 ðxÞÞ2 sdðh1 ðxÞÞ ¼ rh1 ðxÞ ¼
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi varðh1 ðxÞÞ
ð13Þ ð14Þ
The term Eðh; ðxÞÞ in Eq. (13) is obtained from Eq. (11) and Eðh21 ðxÞÞ is obtained from: Eðh1 ðxÞÞ2 ¼ E½A2 ðh1i ; h2i Þe2k1 x þ 2Aðh1i ; h2i ÞBðh1i ; h2i Þeðk1 þk2 Þx þ B2 ðh1i ; h2i Þe2k2 x Eðh1 ðxÞÞ2 ¼
4 1X ½A2 ðh1i ; h2i Þe2k1 x þ 2Aðh1i ; h2i ÞBðh1i ; h2i Þeðk1 þk2 Þx þ B2 ðh1i ; h2i Þe2k2 x 4 k¼1
ð15Þ ð16Þ
where h1i and h2i for k ¼ 1; 2; 3; 4 are indicated in Eq. (12). Outer fluid: The analytical solution for h2 ðxÞ is obtained from Eq. (7b): h2 ðxÞ ¼ Aðh1i ; h2i Þð1 k1 NÞek1 x þ Bðh1i ; h2i Þð1 k2 NÞek2 x
ð17Þ
Expected value of h2 ðxÞ is given by: Eðh2 ðxÞÞ ¼
4 1X ½Aðh1i ; h2i Þð1 k1 N Þek1 x þ Bðh1i ; h2i Þð1 k2 NÞek2 x 4 k¼1
ð18Þ
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi varðh2 ðxÞÞ
ð19Þ
varðh2 ðxÞÞ ¼ Eðh2 ðxÞÞ2 ½Eðh2 ðxÞÞ2
ð20Þ
sdðh2 ðxÞÞ ¼
2
Eðh2 ðxÞÞ ¼
4 1X 2 ½A2 ðh1i ; h2i Þð1 k1 N Þ e2k1 x 4 k¼1
þ 2Aðh1i ; h2i ÞBðh1i ; h2i Þð1 k1 NÞð1 k2 N Þeðk1 þk2 Þx þ B2 ðh1i ; h2i Þð1 k2 NÞ2 e2k2 x
ð21Þ
Eq. (12) indicates h1i and h2i for k ¼ 1; 2; 3; 4 in Eqs. (18) and (21). 5.2. Case 2: Uncertainty in heat transfer coefficients U0 and U1 Inner fluid: TPD analysis carried out following a similar procedure as in Case 1 results in the following: h1 ðxÞ ¼ AðU0 ; U1 Þek1 ðU0 ;U1 Þx þ BðU0 ; U1 Þek2 ðU0 ;U1 Þx
ð22Þ
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721
h2 ðxÞ ¼ AðU0 ; U1 Þð1 k1 ðU0 ; U1 ÞN ðU1 ÞÞek1 ðU0 ;U1 Þx þ BðU0 ; U1 Þð1 k2 ðU0 ; U1 ÞN ðU1 ÞÞek2 ðU0 ;U1 Þx Eðh1 ðxÞÞ ¼
4 1X ½AðU0 ; U1 Þek1 ðU0 ;U1 Þx þ BðU0 ; U1 Þek2 ðU0 ;U1 Þx 4 k¼1
ð23Þ ð24Þ
where U0 and U1 for k ¼ 1; 2; 3; 4 are indicated below: k ¼ 1;
U0 ¼ U 0 ru0 ;
U1 ¼ U 1 ru1
k ¼ 2;
U0 ¼ U 0 ru0 ;
U1 ¼ U 1 þ ru1
k ¼ 3;
U0 ¼ U 0 þ ru0 ;
U1 ¼ U 1 ru1
k ¼ 4;
U0 ¼ U 0 þ ru0 ;
U1 ¼ U 1 þ ru1
sdðh1 ðxÞÞ ¼
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi varðh1 ðxÞÞ
ð26Þ
varðh1 ðxÞÞ ¼ Eðh1 ðxÞÞ2 ½Eðh1 ðxÞÞ2
ð27Þ
ð25Þ
Eðh1 ðxÞÞ in Eq. (27) can be obtained from Eq. (24) and Eðh1 ðxÞÞ2 is obtained from the following equation: Eðh1 ðxÞÞ2 ¼
4 1X ½A2 ðU0 ; U1 Þe2k1 ðU0 ;U1 Þx þ 2AðU0 ; U1 ÞBðU0 ; U1 Þeðk1 ðU0 ;U1 Þþk2 ðU0 ;U1 ÞÞx 4 k¼1
þ B2 ðU0 ; U1 Þe2k2 ðU0 ;U1 Þ
ð28Þ
where U0 and U1 for k ¼ 1; 2; 3; 4 are indicated in Eq. (25). Outer fluid: Using Eq. (23), the expected value, variance and the standard deviation of h2 ðxÞ are obtained and expressed by the following equation: 4 1X ½AðU0 ; U1 Þð1 k1 ðU0 ; U1 ÞN ðU1 ÞÞek1 ðU0 ;U1 Þx Eðh2 ðxÞÞ ¼ 4 k¼1
þ BðU0 ; U1 Þð1 k2 ðU0 ; U1 ÞN ðU1 ÞÞek2 ðU0 ;U1 Þx
ð29Þ
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi varðh2 ðxÞÞ
ð30Þ
varðh2 ðxÞÞ ¼ Eðh2 ðxÞÞ2 ½Eðh2 ðxÞÞ2
ð31Þ
sdðh2 ðxÞÞ ¼
Eðh2 ðxÞÞ2 ¼
4 1X ½A2 ðU0 ; U1 Þð1 k1 ðU0 ; U1 ÞN ðU1 ÞÞ2 e2k1 ðU0 ;U1 Þx 4 k¼1
þ 2AðU0 ; U1 ÞBðU0 ; U1 Þð1 k1 ðU0 ; U1 ÞN ðU1 ÞÞð1 k2 ðU0 ; U1 ÞN ðU1 ÞÞ e½k1 ðU0 ;U1 Þþk2 ðU0 ;U1 Þx þ B2 ðU0 ; U1 Þð1 k2 ðU0 ; U1 ÞN ðU1 ÞÞ2 e2k2 ðU0 ;U1 Þx
Eq. (25) indicates the values of U0 and U1 for k ¼ 1; 2; 3; 4 in Eqs. (29) and (32).
ð32Þ
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5.3. Case 3: Uncertainty in fluid temperatures h1i and h2i at the boundaries and the heat transfer coefficients U0 and U1 The analytical solution for h1 ðxÞ and h2 ðxÞ from Eqs. (7a) and (7b) can be written as: h1 ðxÞ ¼ Aðh1i ; h2i ; U0 ; U1 Þek1 ðU0 ;U1 Þx þ Bðh1i ; h2i ; U0 ; U1 Þek2 ðU0 ;U1 Þx
ð33Þ
and h2 ðxÞ ¼ Aðh1i ; h2i ; U0 ; U1 Þð1 k1 ðU0 ; U1 ÞN ðU1 ÞÞek1 ðU0 ;U1 Þx þ Bðh1i ; h2i ; U0 ; U1 Þð1 k2 ðU0 ; U1 ÞN ðU1 ÞÞek2 ðU0 ;U1 Þx
ð34Þ
An analysis similar to that presented in Case 1 earlier leads to the following expressions for the expected values of h1 ðxÞ and h2 ðxÞ due to combined uncertainties in h1i , h2i , U0 and U1 : 16 1 X Eðh1 ðxÞÞ ¼ ½Aðh1i ; h2i ; U0 ; U1 Þek1 ðU0 ;U1 Þx þ Bðh1i ; h2i ; U0 ; U1 Þek2 ðU0 ;U1 Þx 16 k¼1
Eðh2 ðxÞÞ ¼
ð35Þ
16 1 X ½Aðh1i ; h2i ; U0 ; U1 Þð1 k1 ðU0 ; U1 ÞNðU1 ÞÞek1 ðU0 ;U1 Þx 16 k¼1
þ Bðh1i ; h2i ; U0 ; U1 Þð1 k2 ðU0 ; U1 ÞNðU1 ÞÞek2 ðU0 ;U1 Þx
ð36Þ
The standard deviations and variances of h1 ðxÞ and h2 ðxÞ are obtained from the following: pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sdðh1 ðxÞÞ ¼ varðh1 ðxÞÞ ð37Þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sdðh2 ðxÞÞ ¼ varðh2 ðxÞÞ ð38Þ varðh1 ðxÞÞ ¼ Eðh1 ðxÞÞ2 ½Eðh1 ðxÞÞ2
ð39Þ
varðh2 ðxÞÞ ¼ Eðh2 ðxÞÞ2 ½Eðh2 ðxÞÞ2
ð40Þ
2
2
where Eðh1 ðxÞÞ and Eðh2 ðxÞÞ are obtained from: 16 1 X 2 ½A2 ðh1i ; h2i ; U0 ; U1 Þe2k1 ðU0 ;U1 Þx Eðh1 ðxÞÞ ¼ 16 k¼1 þ 2Aðh1i ; h2i ; U0 ; U1 ÞBðh1i ; h2i ; U0 ; U1 Þeðk1 ðU0 ;U1 Þþk2 ðU0 ;U1 ÞÞx þ B2 ðh1i ; h2i ; U0 ; U1 Þe2k2 ðU0 ;U1 Þx
ð41Þ
and Eðh2 ðxÞÞ2 ¼
16 1 X ½A2 ðh1i ; h2i ; U0 ; U1 Þð1 k1 ðU0 ; U1 ÞN ðU1 ÞÞ2 e2k1 ðU0 ;U1 Þx 16 k¼1
þ 2Aðh1i ; h2i ; U0 ; U1 ÞBðh1i ; h2i ; U0 ; U1 Þð1 k1 ðU0 ; U1 ÞNðU1 ÞÞ
ð1 k2 ðU0 ; U1 ÞN ðU1 ÞÞeðk1 ðU0 ;U1 Þþk2 ðU0 ;U1 ÞÞx 2
þ B2 ðh1i ; h2i ; U0 ; U1 Þð1 k2 ðU0 ; U1 ÞN ðU1 ÞÞ e2k2 ðU0 ;U1 Þx
ð42Þ
R.C. Prasad et al. / Appl. Math. Modelling 26 (2002) 715–726
723
h1i , h2i , U0 and U1 for k ¼ 1; 2; 3; . . . ; 16 in Eqs. (35), (36), (41) and (42) are obtained by the combination of the following: h1i ¼ h1i rh1i h2i ¼ h2i rh2i U0 ¼ U 0 r u0
ð43Þ
U1 ¼ U 1 r u1
6. Results, validation and discussion Solutions for uncertainties (as visualized by standard deviation) of 5% in mean fluid temperatures T1i and T2i at the boundaries and uncertainties of 10% in the heat transfer coefficients U0 and U1 have been utilized here. Numerical calculations are carried out to determine the expected values and the standard deviations for T1 ðxÞ and T2 ðxÞ for a counterflow heat exchanger with the following data:
Fig. 2. Plot of uncertainty in response temperature due to uncertainty in inlet fluid temperatures.
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Fig. 3. Plot of uncertainty in response temperature deviation due to uncertainty in heat transfer coefficient.
D1 ¼ 0:05 m m1 c1 ¼ 1000 J min1 K1 T1i ¼ 20 °C U1 ¼ 25:0 W m2 K1 L ¼ 1:0 m
D0 ¼ 0:075 m m2 c2 ¼ 1500 J min1 K1 T2i ¼ 100 °C U0 ¼ 10:0 W m2 K1 T0 ¼ 40 °C
Figs. 2–4 show the results for Cases 1, 2 and 3 respectively. The results are summarized in Table 1. The expected values of T1 ðxÞ and T2 ðxÞ obtained from the present analysis are found to be identical to those obtained from the deterministic solution presented in Ref. [3]. It is further observed (Figs. 2 and 3 and Table 1, Cases 1 and 2) from these results that the influence of uncertainties in U0 and U1 is significantly less than that from the uncertainties in temperatures T1i and T2i . Also, the uncertainties in T1 ðxÞ and T2 ðxÞ due to uncertainties in U0 and U1 (Fig. 3) increase from zero at the inlet to a maximum at the exit for each fluid stream. The uncertainties in T1i and T2i result in relatively similar uncertainties (Fig. 2) in T1 ðxÞ and T2 ðxÞ. These deviations are slightly larger at the high temperature end for each fluid stream. The combined effect of uncertainties in the boundary temperatures and the heat transfer coefficient is dominated by the uncertainties in temperature (Fig. 4 and Table 1, Case 3). A relatively minor influence of the uncertainty in heat transfer coefficient on the response of the heat exchanger is advantageous due to the following:
R.C. Prasad et al. / Appl. Math. Modelling 26 (2002) 715–726
725
Fig. 4. Plot of uncertainty in response temperature due to combined uncertainties in inlet fluid temperatures and heat transfer coefficients. Table 1 Summary of results Case # 1
2
3
Data uncertainty
Uncertainty in response temperature
T (°C)
%
Range (°C)
Max (%)
T1i ¼ 20 1 T2i ¼ 100 5
5 5
T1 : 1 to 1:24 T2 : 4 to 5
5 5
U (W m2 K1 )
%
U1 ¼ 25 2:5 U0 ¼ 10 1
10 10
T1 : 0 to 1:28 T2 : 0 to 0.92
3:6 1.1
T (°C), U (W m2 K1 )
% T1 : 1 to 1.8 T2 : 4.1 to 5.0
5 5
T1i T2i U1 U0
¼ 20 1 ¼ 100 5 ¼ 25 2:5 ¼ 10 1
5 5 10 10
1. The experimental and/or empirical data for heat transfer coefficients are usually subject to large uncertainties. 2. The fluid temperatures at the boundaries can be measured more accurately and the data uncertainties can be minimized.
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R.C. Prasad et al. / Appl. Math. Modelling 26 (2002) 715–726
The response of a heat exchanger, thus, can be predicted with low deviation and variance by improving the accuracy of temperature measurement.
7. Conclusion The influence of data uncertainty on the performance of a concentric counterflow heat exchanger has been investigated using a method based on TPD. The available deterministic solution [3] and TPD analysis [4] have been utilized. Expressions are presented for the expected values, variances and standard deviations due to uncertainties in fluid inlet temperatures and heat transfer coefficients separately as well as due to their combined effect. The expected values of the temperature of each fluid stream turn out to be same as given by the deterministic solution [3].
Acknowledgements The work was performed under a program of studies funded by the Natural Sciences and Engineering Research Council of Canada under grant number A5477.
References [1] S. Kakac, Y. Yener, Transient laminar forced convection in ducts, in: S. Kakac, R.K. Shah, A.E. Bergles (Eds.), Low Reynolds Number Heat Exchangers, Hemisphere, New York, 1983, pp. 205–227. [2] T.S. Ravigurarajan, A.E. Bergles, General correlations for pressure drop and heat transfer for single-phase turbulent flow in internally ribbed tubes, in: P.J. Bishop (Ed.), Augmentation of Heat Transfer in Energy Systems, vol. 52, ASME Publication, NY, 1985, pp. 9–20. [3] R.C. Prasad, Analytical solution for a double-pipe heat exchanger with non-adiabatic condition at the outer surface, Int. Comm. Heat Mass Transf. 14 (1987) 665–672. [4] Karmeshu, F. Lara-Rasano, Modelling data uncertainty in growth forecasts, Appl. Math. Model. 11 (1987) 62–68. [5] E. Rosenblueth, Point estimates for probability moments, Proc. Natl. Acad. Sci. USA 72 (1975) 3812–3814. [6] E. Rosenblueth, Two-point estimates in probabilities, Appl. Math. Model. 5 (1981) 329–335. [7] J.R. Banjamin, C. Cornell, Probability Statistics and Decisions for Civil Engineers, McGraw-Hill, New York, 1970.
Stochastic modeling of heat exchanger response to data uncertainties R.C. Prasad
a,*
, Karmeshu b, K.K. Bharadwaj
b
a
b
Department of Engineering, University of New Brunswick, Saint John, NB, Canada E2L 4L5 School of Computer and System Sciences, Jawaharlal Nehru University, New Delhi 110067, India Received 24 August 1998; received in revised form 10 August 2001; accepted 8 October 2001
Abstract This paper analyses the performance of a counterflow concentric tube heat exchanger and estimates the uncertainties in the temperature response of the heat exchanging fluids caused by uncertainties in their inlet temperatures and overall heat transfer coefficients between the fluid streams. The analysis is based upon the two-point distribution technique. This paper describes the application of this technique to a heat exchanger for which the exact solution for the steady state temperature response is available. Results show that the uncertainty in inlet temperature has a stronger influence. The effect of data uncertainty in heat transfer coefficients, which generally have very high level of uncertainty, have only weak influence. The accuracy of predicted temperature response can, thus, be significantly improved by accurate measurement of the inlet fluid temperatures. Ó 2002 Elsevier Science Inc. All rights reserved. Keywords: Stochastic modeling; Heat exchanger; Two-point distribution
1. Introduction The performance analysis of heat exchangers is essential for their safe and economical design as well as for the design of necessary controls [1] for their safe operation under steady state as well as transient conditions which are encountered during start up and shut down. The predicted performance of a heat exchanger may, at times, significantly differ on account of inherent variability present in the model parameters and the initial values of the variables. One major
*
Corresponding author. Tel.: +1-506-648-5518; fax: +1-506-648-5513. E-mail address: [email protected] (R.C. Prasad).
0307-904X/02/$ - see front matter Ó 2002 Elsevier Science Inc. All rights reserved. PII: S 0 3 0 7 - 9 0 4 X ( 0 1 ) 0 0 0 8 2 - 8
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Nomenclature A B c D L m N P P1 , P2 p sd(x) T T U U x Greek b c k h h_ h€ r r2
a constant a constant specific heat of fluid, J kg1 K1 diameter of pipe, m length of pipe, m mass flow rate, kg s1 , kg min1 defined in Eq. (7d) perimeter of pipe, m parameter in Eq. (1) probability density function standard deviation of x ¼ rx temperature, °C, K mean value of T, °C, K overall heat transfer coefficient, W m1 K1 mean value of U, W m1 K1 distance along the length of pipe, m symbols defined in Eq. (7f) defined in Eq. (7e) defined in Eq. (7c) incremental temperature T ðxÞ T0 , °C, K dh/dx d2 h/dx2 standard deviation variance
Subscripts e exit i inlet o outside or outer pipe 1 inner pipe or fluid in inner pipe 2 fluid in the annular space U for heat transfer coefficient h for temperature
source of variability arises from the measurement uncertainty. It is, therefore, necessary to develop mathematical models in order to accurately predict their performance under actual conditions where various parameters and initial/boundary conditions are subject to measurement uncertainties.
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717
In heat exchangers, the overall heat transfer coefficient has a large measurement uncertainty (over 50%) [2]. Also the inlet fluid temperature for the heat exchanging fluids is subject to minor measurement uncertainties (1–5%). In the presence of these uncertainties, the exact deterministic solution [3] alone does not provide any inkling to the uncertainty in the response of the heat exchanger. A rational framework for describing these measurement uncertainties associated with the parameters and initial/boundary values treats them as random variables with specified first few moments or probability distribution [4]. The purpose of this paper is to investigate the effect of data uncertainty on the performance of a concentric tube heat exchanger. The analysis provides quantitative estimates of the uncertainties in the variables of interest (in this case, the temperature of the heat exchanging fluids), due to data uncertainties. The analysis is carried out by employing a method based on Rosenblueth’s twopoint distribution (TPD) [5,6].
2. Statistical description of data uncertainty Data concerning variability in the parameter values and initial conditions are usually too limited to specify the details of the shape of their probability distributions. However, in a number of problems, specification of the first two moments suffices to estimate the moments of the errors propagated through the model. The approach widely used by engineers for such types of problems is the first order second moment analysis (SMA) [7]. However, the TPD [5,6] which provides a simple and efficient technique has been employed in this work.
3. Two-point distribution Consider a random variable X with specified first two moments––mean X and standard deviation rx . Rosenblueth [5,6] proposed the representation of the random variable X in terms of two concentrations P1 and P2 placed at x ¼ x1 and x ¼ x2 . Using this representation, expressions for the expected value (Y ), standard deviation (rY ) of another random variable Y ¼ f ðX Þ can be obtained. The probability density function pðxÞ is written as: pðxÞ ¼ P1 dðx x1 Þ þ P2 dðx x2 Þ
ð1Þ
where d( ) is the Dirac delta function. For a symmetrical TPD, concentrations P1 ¼ P2 ¼ 1=2 and Eq. (1) reduces to [6] pðxÞ ¼ 12d½x ðx rÞ þ 12d½x ðx þ rÞ
ð2Þ
Using this, it is found that EðY Þ ¼ Y ¼ 12½f ðx rÞ þ f ðx þ rÞ
ð3aÞ
EðY 2 Þ ¼ 12½f 2 ðx rÞ þ f 2 ðx þ rÞ
ð3bÞ
and
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it is shown [4] that for small r, such that terms 0(r3 ) can be neglected, SMA and TPD give nearly the same result.
4. The physical problem Fig. 1 shows the schematic diagram of a counterflow concentric-tube heat exchanger. This system is represented by the following mathematical model [3]: Inner pipe: P1 U1 h2 þ m1 c1 h_1 P1 U1 h1 ¼ 0
ð4Þ
Annulus: m2 c2 h_2 þ ðP1 U1 þ P0 U0 Þh2 P1 U1 h1 ¼ 0
ð5Þ
Boundary conditions: h2 ð0Þ ¼ h2i ; h1 ðLÞ ¼ h1i
ð6Þ
The solution of Eqs. (4) and (5) yields [3]. h1 ðxÞ ¼ T1 ðxÞ T0 ¼ A ek1 x þ B ek2 x
ð7aÞ
h2 ðxÞ ¼ T2 ðxÞ T0 ¼ Að1 k1 NÞek1 x þ Bð1 k2 N Þek2 x
ð7bÞ
where, k1 ; k2 ¼ ½c ðc2 4bÞ1=2 =2 N¼ c¼
m1 c1 P 1 U1
ð7cÞ ð7dÞ
P 0 U0 P 1 U1 P 1 U1 þ m2 c2 m2 c2 m1 c1
Fig. 1. Schematic representation of a concentric counterflow heat exchanger.
ð7eÞ
R.C. Prasad et al. / Appl. Math. Modelling 26 (2002) 715–726
b¼
P1 U1 P0 U0 m1 c1 m2 c2
719
ð7fÞ
A¼
h2i ek2 L h1i ð1 k2 N Þ ðk2 ek1 L k1 ek2 L ÞN þ ðek2 L ek1 L Þ
ð7gÞ
B¼
h2i ek1 L h1i ð1 k1 N Þ ðk2 ek1 L k1 ek2 L ÞN þ ðek2 L ek1 L Þ
ð7hÞ
In this system, the boundary temperatures h1i and h2i as well as the overall heat transfer coefficients U0 and U1 are subject to measurement uncertainties.
5. Uncertainties in temperature response––TPD analysis We are given the mean values of the inlet fluid temperature and the overall heat transfer coefficient, assumed to be independent, h1i , h2i , U 0 and U 1 with the corresponding variances as r2h1i , r2h2i , r2u0 and r2u1 respectively. The assumption of independence seems plausible from the consideration that randomness in the parameters concerned may be attributed to independent sources. In the following sections TPD analysis for the following three cases are presented: 5.1. Case 1: Uncertainties in fluid temperatures h1i and h2i at the boundaries Inner fluid: The symmetrical TPD [6] in this case results in a probability density function pðh1i ; h2i Þ given by: pðh1i ; h2i Þ ¼ ½12dðh1i ðh1i rh1i ÞÞ þ 12dðh1i ðh1i þ rh1i ÞÞ½12dðh2i ðh2i rh2i ÞÞ þ 1dðh2i ðh2i þ rh ÞÞ 2
2i
ð8Þ
It is assumed that the random variables h1i and h2i are independent. The first moment about the origin, referred to as the mean, is given by: Z 1 Z 1 h1 ðxÞpðh1i ; h2i Þ dh1i dh2i ð9Þ Eðh1 ðxÞÞ ¼ 1
1
Since h1i and h2i are random variables, A and B occurring in Eq. (7a) are random variables and indicating this fact explicitly we have: h1 ðxÞ ¼ Aðh1i ; h2i Þek1 x þ Bðh1i ; h2i Þek2 x
ð10Þ
From Eqs. (9) and (10) we obtain: Eðh1 ðxÞÞ ¼
4 1X ½Aðh1i ; h2i Þek1 x þ Bðh1i ; h2i Þek2 x 4 k¼1
where h1i and h2i for k ¼ 1; 2; 3; 4 are indicated below:
ð11Þ
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k ¼ 1; k ¼ 2;
h1i ¼ h1i rh1i ; h1i ¼ h1i rh ;
h2i ¼ h2i rh2i h2i ¼ h2i þ rh
h1i ¼ h1i þ rh1i ; h1i ¼ h1i þ rh1i ;
h2i ¼ h2i rh2i h2i ¼ h2i þ rh2i
1i
k ¼ 3; k ¼ 4;
2i
ð12Þ
The variance varðh1 ðxÞÞ and standard deviation rh1 ðxÞ of h1 ðxÞ are obtained from the following: varðh1 ðxÞÞ ¼ r2h1 ðxÞ ¼ Eðh1 ðxÞÞ2 ½Eðh1 ðxÞÞ2 sdðh1 ðxÞÞ ¼ rh1 ðxÞ ¼
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi varðh1 ðxÞÞ
ð13Þ ð14Þ
The term Eðh; ðxÞÞ in Eq. (13) is obtained from Eq. (11) and Eðh21 ðxÞÞ is obtained from: Eðh1 ðxÞÞ2 ¼ E½A2 ðh1i ; h2i Þe2k1 x þ 2Aðh1i ; h2i ÞBðh1i ; h2i Þeðk1 þk2 Þx þ B2 ðh1i ; h2i Þe2k2 x Eðh1 ðxÞÞ2 ¼
4 1X ½A2 ðh1i ; h2i Þe2k1 x þ 2Aðh1i ; h2i ÞBðh1i ; h2i Þeðk1 þk2 Þx þ B2 ðh1i ; h2i Þe2k2 x 4 k¼1
ð15Þ ð16Þ
where h1i and h2i for k ¼ 1; 2; 3; 4 are indicated in Eq. (12). Outer fluid: The analytical solution for h2 ðxÞ is obtained from Eq. (7b): h2 ðxÞ ¼ Aðh1i ; h2i Þð1 k1 NÞek1 x þ Bðh1i ; h2i Þð1 k2 NÞek2 x
ð17Þ
Expected value of h2 ðxÞ is given by: Eðh2 ðxÞÞ ¼
4 1X ½Aðh1i ; h2i Þð1 k1 N Þek1 x þ Bðh1i ; h2i Þð1 k2 NÞek2 x 4 k¼1
ð18Þ
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi varðh2 ðxÞÞ
ð19Þ
varðh2 ðxÞÞ ¼ Eðh2 ðxÞÞ2 ½Eðh2 ðxÞÞ2
ð20Þ
sdðh2 ðxÞÞ ¼
2
Eðh2 ðxÞÞ ¼
4 1X 2 ½A2 ðh1i ; h2i Þð1 k1 N Þ e2k1 x 4 k¼1
þ 2Aðh1i ; h2i ÞBðh1i ; h2i Þð1 k1 NÞð1 k2 N Þeðk1 þk2 Þx þ B2 ðh1i ; h2i Þð1 k2 NÞ2 e2k2 x
ð21Þ
Eq. (12) indicates h1i and h2i for k ¼ 1; 2; 3; 4 in Eqs. (18) and (21). 5.2. Case 2: Uncertainty in heat transfer coefficients U0 and U1 Inner fluid: TPD analysis carried out following a similar procedure as in Case 1 results in the following: h1 ðxÞ ¼ AðU0 ; U1 Þek1 ðU0 ;U1 Þx þ BðU0 ; U1 Þek2 ðU0 ;U1 Þx
ð22Þ
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721
h2 ðxÞ ¼ AðU0 ; U1 Þð1 k1 ðU0 ; U1 ÞN ðU1 ÞÞek1 ðU0 ;U1 Þx þ BðU0 ; U1 Þð1 k2 ðU0 ; U1 ÞN ðU1 ÞÞek2 ðU0 ;U1 Þx Eðh1 ðxÞÞ ¼
4 1X ½AðU0 ; U1 Þek1 ðU0 ;U1 Þx þ BðU0 ; U1 Þek2 ðU0 ;U1 Þx 4 k¼1
ð23Þ ð24Þ
where U0 and U1 for k ¼ 1; 2; 3; 4 are indicated below: k ¼ 1;
U0 ¼ U 0 ru0 ;
U1 ¼ U 1 ru1
k ¼ 2;
U0 ¼ U 0 ru0 ;
U1 ¼ U 1 þ ru1
k ¼ 3;
U0 ¼ U 0 þ ru0 ;
U1 ¼ U 1 ru1
k ¼ 4;
U0 ¼ U 0 þ ru0 ;
U1 ¼ U 1 þ ru1
sdðh1 ðxÞÞ ¼
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi varðh1 ðxÞÞ
ð26Þ
varðh1 ðxÞÞ ¼ Eðh1 ðxÞÞ2 ½Eðh1 ðxÞÞ2
ð27Þ
ð25Þ
Eðh1 ðxÞÞ in Eq. (27) can be obtained from Eq. (24) and Eðh1 ðxÞÞ2 is obtained from the following equation: Eðh1 ðxÞÞ2 ¼
4 1X ½A2 ðU0 ; U1 Þe2k1 ðU0 ;U1 Þx þ 2AðU0 ; U1 ÞBðU0 ; U1 Þeðk1 ðU0 ;U1 Þþk2 ðU0 ;U1 ÞÞx 4 k¼1
þ B2 ðU0 ; U1 Þe2k2 ðU0 ;U1 Þ
ð28Þ
where U0 and U1 for k ¼ 1; 2; 3; 4 are indicated in Eq. (25). Outer fluid: Using Eq. (23), the expected value, variance and the standard deviation of h2 ðxÞ are obtained and expressed by the following equation: 4 1X ½AðU0 ; U1 Þð1 k1 ðU0 ; U1 ÞN ðU1 ÞÞek1 ðU0 ;U1 Þx Eðh2 ðxÞÞ ¼ 4 k¼1
þ BðU0 ; U1 Þð1 k2 ðU0 ; U1 ÞN ðU1 ÞÞek2 ðU0 ;U1 Þx
ð29Þ
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi varðh2 ðxÞÞ
ð30Þ
varðh2 ðxÞÞ ¼ Eðh2 ðxÞÞ2 ½Eðh2 ðxÞÞ2
ð31Þ
sdðh2 ðxÞÞ ¼
Eðh2 ðxÞÞ2 ¼
4 1X ½A2 ðU0 ; U1 Þð1 k1 ðU0 ; U1 ÞN ðU1 ÞÞ2 e2k1 ðU0 ;U1 Þx 4 k¼1
þ 2AðU0 ; U1 ÞBðU0 ; U1 Þð1 k1 ðU0 ; U1 ÞN ðU1 ÞÞð1 k2 ðU0 ; U1 ÞN ðU1 ÞÞ e½k1 ðU0 ;U1 Þþk2 ðU0 ;U1 Þx þ B2 ðU0 ; U1 Þð1 k2 ðU0 ; U1 ÞN ðU1 ÞÞ2 e2k2 ðU0 ;U1 Þx
Eq. (25) indicates the values of U0 and U1 for k ¼ 1; 2; 3; 4 in Eqs. (29) and (32).
ð32Þ
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5.3. Case 3: Uncertainty in fluid temperatures h1i and h2i at the boundaries and the heat transfer coefficients U0 and U1 The analytical solution for h1 ðxÞ and h2 ðxÞ from Eqs. (7a) and (7b) can be written as: h1 ðxÞ ¼ Aðh1i ; h2i ; U0 ; U1 Þek1 ðU0 ;U1 Þx þ Bðh1i ; h2i ; U0 ; U1 Þek2 ðU0 ;U1 Þx
ð33Þ
and h2 ðxÞ ¼ Aðh1i ; h2i ; U0 ; U1 Þð1 k1 ðU0 ; U1 ÞN ðU1 ÞÞek1 ðU0 ;U1 Þx þ Bðh1i ; h2i ; U0 ; U1 Þð1 k2 ðU0 ; U1 ÞN ðU1 ÞÞek2 ðU0 ;U1 Þx
ð34Þ
An analysis similar to that presented in Case 1 earlier leads to the following expressions for the expected values of h1 ðxÞ and h2 ðxÞ due to combined uncertainties in h1i , h2i , U0 and U1 : 16 1 X Eðh1 ðxÞÞ ¼ ½Aðh1i ; h2i ; U0 ; U1 Þek1 ðU0 ;U1 Þx þ Bðh1i ; h2i ; U0 ; U1 Þek2 ðU0 ;U1 Þx 16 k¼1
Eðh2 ðxÞÞ ¼
ð35Þ
16 1 X ½Aðh1i ; h2i ; U0 ; U1 Þð1 k1 ðU0 ; U1 ÞNðU1 ÞÞek1 ðU0 ;U1 Þx 16 k¼1
þ Bðh1i ; h2i ; U0 ; U1 Þð1 k2 ðU0 ; U1 ÞNðU1 ÞÞek2 ðU0 ;U1 Þx
ð36Þ
The standard deviations and variances of h1 ðxÞ and h2 ðxÞ are obtained from the following: pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sdðh1 ðxÞÞ ¼ varðh1 ðxÞÞ ð37Þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sdðh2 ðxÞÞ ¼ varðh2 ðxÞÞ ð38Þ varðh1 ðxÞÞ ¼ Eðh1 ðxÞÞ2 ½Eðh1 ðxÞÞ2
ð39Þ
varðh2 ðxÞÞ ¼ Eðh2 ðxÞÞ2 ½Eðh2 ðxÞÞ2
ð40Þ
2
2
where Eðh1 ðxÞÞ and Eðh2 ðxÞÞ are obtained from: 16 1 X 2 ½A2 ðh1i ; h2i ; U0 ; U1 Þe2k1 ðU0 ;U1 Þx Eðh1 ðxÞÞ ¼ 16 k¼1 þ 2Aðh1i ; h2i ; U0 ; U1 ÞBðh1i ; h2i ; U0 ; U1 Þeðk1 ðU0 ;U1 Þþk2 ðU0 ;U1 ÞÞx þ B2 ðh1i ; h2i ; U0 ; U1 Þe2k2 ðU0 ;U1 Þx
ð41Þ
and Eðh2 ðxÞÞ2 ¼
16 1 X ½A2 ðh1i ; h2i ; U0 ; U1 Þð1 k1 ðU0 ; U1 ÞN ðU1 ÞÞ2 e2k1 ðU0 ;U1 Þx 16 k¼1
þ 2Aðh1i ; h2i ; U0 ; U1 ÞBðh1i ; h2i ; U0 ; U1 Þð1 k1 ðU0 ; U1 ÞNðU1 ÞÞ
ð1 k2 ðU0 ; U1 ÞN ðU1 ÞÞeðk1 ðU0 ;U1 Þþk2 ðU0 ;U1 ÞÞx 2
þ B2 ðh1i ; h2i ; U0 ; U1 Þð1 k2 ðU0 ; U1 ÞN ðU1 ÞÞ e2k2 ðU0 ;U1 Þx
ð42Þ
R.C. Prasad et al. / Appl. Math. Modelling 26 (2002) 715–726
723
h1i , h2i , U0 and U1 for k ¼ 1; 2; 3; . . . ; 16 in Eqs. (35), (36), (41) and (42) are obtained by the combination of the following: h1i ¼ h1i rh1i h2i ¼ h2i rh2i U0 ¼ U 0 r u0
ð43Þ
U1 ¼ U 1 r u1
6. Results, validation and discussion Solutions for uncertainties (as visualized by standard deviation) of 5% in mean fluid temperatures T1i and T2i at the boundaries and uncertainties of 10% in the heat transfer coefficients U0 and U1 have been utilized here. Numerical calculations are carried out to determine the expected values and the standard deviations for T1 ðxÞ and T2 ðxÞ for a counterflow heat exchanger with the following data:
Fig. 2. Plot of uncertainty in response temperature due to uncertainty in inlet fluid temperatures.
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Fig. 3. Plot of uncertainty in response temperature deviation due to uncertainty in heat transfer coefficient.
D1 ¼ 0:05 m m1 c1 ¼ 1000 J min1 K1 T1i ¼ 20 °C U1 ¼ 25:0 W m2 K1 L ¼ 1:0 m
D0 ¼ 0:075 m m2 c2 ¼ 1500 J min1 K1 T2i ¼ 100 °C U0 ¼ 10:0 W m2 K1 T0 ¼ 40 °C
Figs. 2–4 show the results for Cases 1, 2 and 3 respectively. The results are summarized in Table 1. The expected values of T1 ðxÞ and T2 ðxÞ obtained from the present analysis are found to be identical to those obtained from the deterministic solution presented in Ref. [3]. It is further observed (Figs. 2 and 3 and Table 1, Cases 1 and 2) from these results that the influence of uncertainties in U0 and U1 is significantly less than that from the uncertainties in temperatures T1i and T2i . Also, the uncertainties in T1 ðxÞ and T2 ðxÞ due to uncertainties in U0 and U1 (Fig. 3) increase from zero at the inlet to a maximum at the exit for each fluid stream. The uncertainties in T1i and T2i result in relatively similar uncertainties (Fig. 2) in T1 ðxÞ and T2 ðxÞ. These deviations are slightly larger at the high temperature end for each fluid stream. The combined effect of uncertainties in the boundary temperatures and the heat transfer coefficient is dominated by the uncertainties in temperature (Fig. 4 and Table 1, Case 3). A relatively minor influence of the uncertainty in heat transfer coefficient on the response of the heat exchanger is advantageous due to the following:
R.C. Prasad et al. / Appl. Math. Modelling 26 (2002) 715–726
725
Fig. 4. Plot of uncertainty in response temperature due to combined uncertainties in inlet fluid temperatures and heat transfer coefficients. Table 1 Summary of results Case # 1
2
3
Data uncertainty
Uncertainty in response temperature
T (°C)
%
Range (°C)
Max (%)
T1i ¼ 20 1 T2i ¼ 100 5
5 5
T1 : 1 to 1:24 T2 : 4 to 5
5 5
U (W m2 K1 )
%
U1 ¼ 25 2:5 U0 ¼ 10 1
10 10
T1 : 0 to 1:28 T2 : 0 to 0.92
3:6 1.1
T (°C), U (W m2 K1 )
% T1 : 1 to 1.8 T2 : 4.1 to 5.0
5 5
T1i T2i U1 U0
¼ 20 1 ¼ 100 5 ¼ 25 2:5 ¼ 10 1
5 5 10 10
1. The experimental and/or empirical data for heat transfer coefficients are usually subject to large uncertainties. 2. The fluid temperatures at the boundaries can be measured more accurately and the data uncertainties can be minimized.
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The response of a heat exchanger, thus, can be predicted with low deviation and variance by improving the accuracy of temperature measurement.
7. Conclusion The influence of data uncertainty on the performance of a concentric counterflow heat exchanger has been investigated using a method based on TPD. The available deterministic solution [3] and TPD analysis [4] have been utilized. Expressions are presented for the expected values, variances and standard deviations due to uncertainties in fluid inlet temperatures and heat transfer coefficients separately as well as due to their combined effect. The expected values of the temperature of each fluid stream turn out to be same as given by the deterministic solution [3].
Acknowledgements The work was performed under a program of studies funded by the Natural Sciences and Engineering Research Council of Canada under grant number A5477.
References [1] S. Kakac, Y. Yener, Transient laminar forced convection in ducts, in: S. Kakac, R.K. Shah, A.E. Bergles (Eds.), Low Reynolds Number Heat Exchangers, Hemisphere, New York, 1983, pp. 205–227. [2] T.S. Ravigurarajan, A.E. Bergles, General correlations for pressure drop and heat transfer for single-phase turbulent flow in internally ribbed tubes, in: P.J. Bishop (Ed.), Augmentation of Heat Transfer in Energy Systems, vol. 52, ASME Publication, NY, 1985, pp. 9–20. [3] R.C. Prasad, Analytical solution for a double-pipe heat exchanger with non-adiabatic condition at the outer surface, Int. Comm. Heat Mass Transf. 14 (1987) 665–672. [4] Karmeshu, F. Lara-Rasano, Modelling data uncertainty in growth forecasts, Appl. Math. Model. 11 (1987) 62–68. [5] E. Rosenblueth, Point estimates for probability moments, Proc. Natl. Acad. Sci. USA 72 (1975) 3812–3814. [6] E. Rosenblueth, Two-point estimates in probabilities, Appl. Math. Model. 5 (1981) 329–335. [7] J.R. Banjamin, C. Cornell, Probability Statistics and Decisions for Civil Engineers, McGraw-Hill, New York, 1970.