# Entropy Generation Minima in Different Configurations of the Branching of a Fluid-Carrying Pipe in Laminar Isothermal Flow

## Abstract

**:**

_{0}and delivering a given mass flow rate m

_{0}.The optimization is performed using an objective function that prescribes the minimization of the entropy generation rate due — in this simple case — only to viscous flow effects within the tubes. Several fundamental simplifying assumptions are made to reduce the problem to a multi-variable optimization in three independent variables: the aspect ratio of the domain served by the flow, the diameter ratio of the primary and secondary branches, and the length of the secondary branch (the location of both the “source” of the fluid and the “sink”, i.e., the place of desired delivery of the fluid, being a datum).It is shown that the solution is strongly dependent both on the aspect ratio and on the diameter ratio, and that the "optimal" configurations display some resemblance to the branching patterns observed in natural structures. The study poses a challenge both to Designers and to Natural Scientists: are the optima suggested by the present procedure compatible with the structures currently used in heat exchangers and flow devices? Are they compatible with the structures observed in nature? No final answer is provided in this preliminary study, but a possible "falsification" procedure is outlined in the conclusions.

## 1. Introduction

- (a)
- In natural flow systems (both inert and biological structures), why does a flow bifurcation (of the type shown in Figure 1) occur?
- (b)
- In man-made applications, is there an “optimal” bifurcation pattern for a given design goal, and how can it be identified?

_{0}from one “source” O to two delivery “sinks” A and B (Figure 1), can an “optimal” geometry be identified a priori?

## 2. The Entropy Generation Rate in a Channel

#### 2.1. The standard derivation of the velocity profile in plane Poiseuille flow

#### 2.2. Alternative derivation of the velocity profile of plane Poiseuille flow

## 3. The Entropy Generation Rate in a Simple Bifurcation

_{r}= H/L (Figure 3). In this simple model, the fluid proceeds from left to right, and the “goal” of the device is to deliver the mass flow rate m

_{0}/2 to each one of the end points A and B. To provide a more realistic example, let us consider here that the channel is of circular section, so that instead of Equation (4) — the equivalent formulae for steady, laminar, fully developed flow in a round pipe apply.

_{0}, m

_{1}, L

_{0}, L

_{1}, d

_{0}, d

_{1}, and the (constant) fluid properties. The mass flow rate m

_{0}is prescribed, and the bifurcation is flow-wise symmetrical, so that m

_{1}= 0.5 m

_{0}. The flow is isothermal.

_{0}/L; and the general expression for the entropy generation rate is:

## 4. Results and Discussion

_{r}= H/L and different bifurcation lengths (identified by the ratio λ = L

_{0}/L), under three physically meaningful situations:

- (1)
- The Reynolds number remains constant over the entire fluid path: Re
_{0}= Re_{1}. This defines a diameter ratio δ = r_{1}/r_{0}= 0.5; - (2)
- The velocity remains constant over the entire fluid path: U
_{0}= U_{1}. This defines a diameter ratio δ = r_{1}/r_{0}= 0.707; - (3)
- The volume occupied by the fluid in the unsplit portion is equal to that occupied by the two bifurcated branches. This defines a diameter ratio δ = r
_{1}/r_{0}= $\sqrt{\frac{{\text{L}}_{0}}{2{\text{L}}_{1}}}$.

#### 4.1. The ideal case, no bifurcation losses

_{f}defined in Section 3 above), that represents the viscous dissipation in a straight (i.e., unsplit) tube of length L and radius r

_{0}. The following features are apparent:

- (a)
- For all cases, the entropy generation rate strongly depends on the aspect ratio a
_{r}of the domain in which the bifurcation occurs. With the geometry selected here, higher a_{r}lead — for the same splitting ratio λ — to a longer bifurcated stretch, of smaller diameter and therefore with higher losses; - (b)
- The constant velocity case displays a lower entropy generation rate than the constant Re case for any bifurcation length. This is due to the fact that the mass conservation constraint imposes a higher diameter ratio on the split portions of the tubes that are therefore affected by a lower dissipation rate;
- (c)
- The constant fluid volume configurations display extremely high entropy generation rates for low splitting ratios, but are the least dissipative structure for high. However, the minimum dissipation is attained with diameter ratios near unity: the short bifurcations have a larger diameter than the initial portion of the channel, and the velocities are under the specified mass flow rate constraint correspondingly lower.
- (d)
- For each physical situation (constant Re, constant U, and constant V
_{fluid}) there is indeed a rather well identifiable “optimal” configuration for each aspect ratio that displays a minimum value of the entropy generation rate. The exact values are reported in Table 1.

**Figure 5.**Dimensionless entropy generation in the bifurcation, without splitting losses. The dashed lines indicate the loci of the optimum.

- (1)
- Use of an “optimal” diameter ratio (the basic assumption of all allometric models) is avoided, since this parameter is uniquely specified once the physical flow type has been assigned (constant Re, constant U, constant fluid volume);
- (2)
- A suitable physical correlation is introduced in the above mentioned models to link the “optimal” splitting ratio λ to the diameter ratio δ. in such a way that the minimum entropy generation — which is the most reasonable indicator of “optimal performance” — is at least one of the components of the objective function;
- (3)

Case: Re_{0} = Re_{1} | δ =r_{1}/r_{0} | Minimum $\frac{\dot{S}}{\dot{{S}_{ref}}}$ | |

a_{r} = 0.2 | 0.5 | 1 | 1.80 |

a_{r} = 1 | 0.5 | 0.9 | 4.98 |

a_{r} = 2 | 0.5 | 0.9 | 8.94 |

Case: U_{0} = U_{1} | δ =r_{1}/r_{0} | Minimum $\frac{\dot{S}}{\dot{{S}_{ref}}}$ | |

a_{r} = 0.2 | 0.707 | 0.9 | 1.18 |

a_{r} = 1 | 0.707 | 0.7 | 1.87 |

a_{r} = 2 | 0.707 | 0.4 | 2.73 |

Case: V_{0} = 2V_{1} | δ =r_{1}/r_{0} | Minimum $\frac{\dot{S}}{\dot{{S}_{ref}}}$ | |

a_{r} = 0.2 | 1.15 | 0.8 | 0.86 |

a_{r} = 1 | 1.15 | 0.8 | 0.95 |

a_{r} = 2 | 1.36 | 0.9 | 1.05 |

#### 4.2. A more realistic case with added bifurcation losses

_{1}in Equation (9) is multiplied by a factor (1 + n

_{D}): the constant n

_{D}can be derived from one of the many semi-empirical correlations for pressure losses in sudden restrictions, and the values assumed here are displayed in Figure 6. The results are shown in Figure 7a-c. The values of the entropy generation rates are made dimensionless like in the previous case.

- (a)
- For all cases, the entropy generation rate displays a marked increase: for the same aspect ratio a
_{r}, same splitting length and same diameter ratio, the dissipation is increased of a factor between 1.5 and 4. This confirms the importance of real-flow effects on the optimal configuration; - (b)
- For each physical situation (constant Re, constant U, constant V
_{fluid}) there is still an “optimal” configuration that displays a minimum value of the entropy generation rate, but the minimum is markedly shifted towards lower splitting ratios (earlier bifurcation), except for the constant volume case, in which it is very near the “T” configuration (λ≈1). The exact values are reported in Table 2.

Case: Re_{0} = Re_{1} | δ = r_{1}/r_{0} | λ | Minimum $\frac{\dot{S}}{\dot{{S}_{ref}}}$ |

a_{r} = 0.2 | 0.5 | 0.9 | 2.71 |

a_{r} = 1 | 0.5 | 0.4 | 9.81 |

a_{r} = 2 | 0.5 | 0 | 18.07 |

Case: U_{0} = U_{1} | δ = r_{1}/r_{0} | λ | Minimum $\frac{\dot{S}}{\dot{{S}_{ref}}}$ |

a_{r} = 0.2 | 0.707 | 0.9 | 1.80 |

a_{r} = 1 | 0.707 | 0.4 | 4.85 |

a_{r} = 2 | 0.707 | 0 | 9.03 |

Case: V_{0} = 2V_{1} | δ = r_{1}/r_{0} | λ | Minimum $\frac{\dot{S}}{\dot{{S}_{ref}}}$ |

a_{r} = 0.2 | 1.15 | 0.8 | 0.88 |

a_{r} = 1 | 1.36 | 0.9 | 1.14 |

a_{r} = 2 | 1.50 | 1 | 1.40 |

## 5. Conclusions

- (a)
- For all examined situations, a configuration exists that displays the lowest viscous entropy generation rate compatible with the imposed constraints;
- (b)
- In all cases, the diameter ratio δ can be derived from purely phenomenological considerations: there is no a priori optimal value for this parameter;
- (c)
- The entropy generation rate appears to be a consistent Lagrangian for the identification of the “optimal” configuration, and furthermore, the optima thus derived appear different from those suggested by both allometric and arithmetic/geometric models.

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**MDPI and ACS Style**

Sciubba, E.
Entropy Generation Minima in Different Configurations of the Branching of a Fluid-Carrying Pipe in Laminar Isothermal Flow. *Entropy* **2010**, *12*, 1855-1866.
https://doi.org/10.3390/e12081855

**AMA Style**

Sciubba E.
Entropy Generation Minima in Different Configurations of the Branching of a Fluid-Carrying Pipe in Laminar Isothermal Flow. *Entropy*. 2010; 12(8):1855-1866.
https://doi.org/10.3390/e12081855

**Chicago/Turabian Style**

Sciubba, Enrico.
2010. "Entropy Generation Minima in Different Configurations of the Branching of a Fluid-Carrying Pipe in Laminar Isothermal Flow" *Entropy* 12, no. 8: 1855-1866.
https://doi.org/10.3390/e12081855