PSI - Issue 8

G. Fargione et al. / Procedia Structural Integrity 8 (2018) 566–572 Author name / Structural Integrity Procedia 00 (2017) 000–000

571

6

3. Application

By way of example, the application of the proposed method in the case of a shell-and-tube heat exchanger is reported. In particular, the focus will be on one of its main components, the tubes of the bundle, in order to search for an efficient and responsive solution in the choice of the material and the sizing of the tubes, that meet all the design requirements, and taking into account the system context of the tube bundle, and the resulting constraints imposed by other components. Taking into account the configuration shown in Fig. 2a, defined according the reference construction standard of Tubular Exchanger Manufacturers Association TEMA (2007), and acquired the results of the thermodynamic dimensioning of the system, the characterization of the component under analysis (Fig. 2b) allows to define objectives, requirements to be constrained, fixed and variable geometric parameters, constraints imposed by other components, as follows: • Objective {PF} – Maximize heat exchange efficiency per unit mass. • Requirements {R} – Pressure difference ∆ p, temperature difference ∆ T, bending load on the free span of tubes F. • Geometric parameters {G F } and {G V } – Fixed: outer diameter d o , length L t (from the thermodynamic dimensioning). Variable: tube wall thickness t. • Constraints between components {CbC} – Relation between tubes and shell thermal linear expansion.

Fig. 2. Shell-and-tube heat exchanger: a) system configuration; b) detail of a tube of the bundle

Equation type (1) expresses the objective function:

ρ λ ⋅ ⋅

PF q T

T

m t = = 2

∆

(5)

where q T is the heat flow, m is the mass of tube, λ is the thermal conductivity, ρ is the density of tubes material. With regard to the constraint on pressure difference requirement ∆ p, it can be expressed as Constraint Ratio (3):

t

d 2

σ

y

CR

1

=

≥

(6)

p

∆

o

where σ y is the yield strength of the material, d o and t are the outer diameter and the thickness of the tube, respectively. The corresponding Efficiency Function EF can be formulated by equation (4). Analogously, other constraint ratios and efficiency functions can be formulated for the other requirements to be constrained ( ∆ T, F). Finally, the constraint on thermal linear expansion between tubes and shell can be expresses in the form ∆ L t ≤ k ts ∆ L s (L s is the shell length), where k ts > 1 is a fixed coefficient.