#
How to Reduce the Design of Disc-Shaped Heat Exchangers to a Zero-Degrees-of-Freedom Task^{ †}

^{†}

^{†}

## Abstract

**:**

## 1. Introduction

_{zb}” is internally cooled or heated by a series of internal channels that originate from an axial inlet O and develop radially outwards in a branching fashion; the working fluid may be ejected radially or be collected by a toroidal manifold placed on the external periphery of the disc. The channels may have different cross-sections, hydraulic diameters and lengths. For the sake of simplicity, in this paper the internal channels are assumed to have a circular section and to perform the function of cooling the disc material.

_{j}and diameters d

_{j}of the channels. Another design issue is how to determine whether a larger number of branchings (i.e., a “more dendritic” structure) leads to a performance improvement, and how to quantify the correlation between the number of branches and the DSHE performance. Finally, one must investigate whether the branching angles β

_{j}(that represent so to say the “footprint” of the branchings) have an influence on the overall heat exchange characteristics.

_{j}= d

_{j+1}/d

_{j}; that of successive lengths, λ

_{j}= L

_{j+1}/L

_{j}and that of the diameter of the first branch to the disc radius, δ

_{0}= d

_{0}/R

_{zb}. Quite obviously, the more branches are etched inside of the disc, the more uniform the heat extraction from the solid material, and the better the performance: thus, both the initial number of “sectors” in which the disc is subdivided (first branches, z

_{0}) and the total number of peripheral discharge points, z

_{b}= 2

^{n}z

_{0}(n = 0, n being the number of branching levels) have an influence on the performance of the device.

_{j}the and/or the λ

_{j}for a given δ

_{0}(an allometric correlation is one in which the scaling relationship between some relevant attribute and a characteristic length of the problem depends on some power of the characteristic length itself. For example, in biology, it is established opinion that within a certain species or family the metabolic rate depends on the cube of a single characteristic body length). Only a few studies [5,8,9] have investigated the influence of the number of branchings. The few numerical studies available are usually based on a pre-assigned initial choice of z

_{0}and take advantage of the circumferential symmetry of the problem to simulate only a sector spanning 2π/z

_{0}radians (Figure 2) [10,11].

- (a)
- For a given thermal load (amount of power required by the fluid), calculate the necessary external load, or vice versa;
- (b)
- Link the thermal load to the external radius of the disc;
- (c)
- Identify the desired branching configuration;
- (d)
- Verify that the load requirements are satisfied.

_{j}of two successive branches: the choice influences the DSHE performance, but there is no a priori proof that one method is better than the other under all circumstances [12,13,14]. Referring the reader to specific comparisons among different choices of δ

_{j}[3,7,12], we shall adopt in this paper a constant Reynolds number criterion: a perusal of the procedure will though show that other choices can be immediately integrated within the calculations.

_{Gz}= L

_{j}/d

_{j}) fall well within the responsibility of the designer.

_{j}and κ

_{Gz}have been assigned their value, the DSHE design reduces to a completely deterministic engineering procedure. The procedure rests on the rigorous application of thermo-fluido-dynamics principles and on a careful analysis of the implications of the selected topology (z

_{0}, z

_{b}) on the global device performance. Section 2 presents a topological description of the disc, Section 3 demonstrates that a physically correct design procedure leads to an under-specified problem (more variables than equations), and Section 4 shows how the imposition of a correct set of common-sense engineering constraints can make the problem well-posed. Section 5 provides two examples of the advantages of a practical application of the procedure.

## 2. Classification of a Disc Heat Exchanger as to Its Design Purpose

- (a)
- The relevant topological parameters of a DSHE are:
- (b)
- The ratio of the diameters of successive branches, δ
_{j}= d_{j}_{+1}/d_{j}; - (c)
- The ratio of successive lengths, λ
_{j}= L_{j}_{+1}/L_{j}; - (d)
- The inlet shape ratio defined as the ratio of the diameter of the axial inlet tube to the disc radius, δ
_{ιν}= d_{in}/R; - (e)
- The initial number of “sectors” in which the disc is subdivided, z
_{0}; - (f)
- The total number of peripheral outlets, z
_{b}= 2^{n}z_{0}.

- (i)
- The thermal flux q
_{in}the DSHE receives by conduction/convection or electrical input; - (ii)
- The inlet and outlet coolant temperatures T
_{in}and T_{ou}_{t}; - (iii)
- The final ΔT between the disc and the fluid: T
_{D}− T_{out}; - (iv)
- The average temperature T
_{ext}of the immediate surroundings; - (v)
- The density ρ
_{f}, specific heat c_{p}_{,f}, viscosity µ_{ϕ}of the coolant; - (vi)
- The density ρ
_{D}and specific heat c_{p}_{,D}of the disc material; - (vii)
- An average heat transfer coefficient for the convection on the outside surface of the disc, h
_{ext}(if present).

#### 2.1. Thermal Load and DSHE Efficiency

_{des}depends on the type of operation, and the formulae related to some illustrative examples of operation are provided in Table 1. Once Q

_{des}is known, the coolant mass flowrate can be calculated from the global energy balance:

_{DSHE}can be approximated as follows:

_{DSHE}is not essential for the design procedure outlined here and is reflected only in the ∆T of the working fluid. Notice that Equation (2) shows that the DSHE efficiency depends mainly on the ratio s/R, “s” being the disc thickness: the higher this ratio, the higher the convection losses on the disc peripheral surface. This dependence is not considered in this study, in the sense that s is not considered a relevant variable.

_{des}, or alternatively the required load given the mass flow rate. In either case, both the efficiency η

_{DSHE}and the external radius R can be directly calculated as well.

_{D}is not a constant but a function of R: if more accuracy is required, the value of R can be calculated iteratively using the final results of the procedure specified below.

#### 2.2. Selection of the Inlet Pipe

_{in}is specified:

_{0}of the first level of branching and with the required coolant mass flowrate: notice that the diameter d

_{in}of the inlet tube is not a relevant parameter in the DSHE design, since it depends on the feeding arrangements. On this basis, an “intrinsically feasible”design procedure is described in the next section. There is no “optimization” proper here, because the proposed design procedure uses only fluid-thermodynamic constraint (the constancy of Re and the Graetz assumption, see below), and the feasibility arises simply out of symmetry and of the geometric features of the branchings. But the emerging solution is a very good start for a configuration optimization.

## 3. The Geometric/Fluid Dynamic Design Is an Underspecified Problem

_{0}, in which the disc is divided, select a Reynolds number Re

_{0}and then use Equation (4): both choices are obviously arbitrary and may depend on technological issues.

_{j}as possible, a good design choice is to control the slenderness λ

_{j}= d

_{j}/L

_{j}of the channels in such a way that the flow in all of the j branches is within the respective Graetz entry lengths (Figure 5): in other words, L

_{j}= κ

_{g}d

_{j}, with κ

_{g}falling in the shaded portion of the graph. On the other hand, the friction losses are—under the posited assumptions—also a growing function of Re:

_{j}= Re

_{0}, which leads to:

_{j}can be accommodated by the procedure, but of course it modifies the resulting configuration.

_{b}has an upper bound, posed both by Equation (6) and by possible technological limitations on the attainable surface roughness. The Graetz ratio must be checked at each level to make sure that it remains in the high-Nu region (Figure 5):

_{Gr}= 5 ÷ 15), depending on the boundary conditions on the tube wall and the flow structure. In most practical applications, a value k

_{g}≈ 8–10 is satisfactory, see Section 5 below.

_{b}circumferences identified by each splitting level are given by a recursive formula:

_{in}, R

_{0}, R

_{zb}and d

_{0}are known (Equations (2) and (4)), but it contains z

_{b}unknowns, namely, the splitting angles β

_{j}. To close the problem, z

_{b−}

_{1}auxiliary conditions must be specified.

## 4. The Proper Additional Constraints

_{0}, and the total number of branchings z

_{b}are specified, a series of geometric constraints can be derived that completely defines the configuration by providing an equation for each of the β

_{j}. The line of reasoning is as follows: the circular arc $\widehat{AB}$ at the external radius R

_{zb}defined by γ

_{zb}will contain (Figure 6) 2

^{zb}terminal points, each one being the outlet of a single channel. For all of the terminal points on the circumference to be equispaced, the central angle γ

_{zb}spanned by two adjacent terminal points $\u27e8C,D\u27e9$ (with $\widehat{CD}$=$\widehat{EF}$) must be equal to $\frac{\pi}{{z}_{0}{2}^{\left({z}_{b}-1\right)}}$. Then by simple trigonometric considerations, and making use of Equations (7) and (8):

_{b}equations, that together with the second equations in (9) and (2) make the problem position complete (via the solution of a set of trigonometric equation in R

_{j}). Consider that the angles γ

_{0}γ

_{j}γ

_{zb}are given by the recursive formula:

## 5. Examples of Application

#### 5.1. Comparison of Possible DSHE Configurations for a Glycol/Water Heater

- (a)
- The three branches at level 0 are far too short, and therefore the temperature of the portion of the disc within the radius R
_{0}(refer to Figure 6) is bound to be higher-than-optimal, because the heat exchange area z_{0}πd_{0}L_{0}is too small, even if the Nu_{0}is here rather high; - (a)
- Since the branches at levels 1 and 2 do not respect the constant-represcription (the design choice was in this case $\delta =\sqrt{2}/2$, i.e., constant velocity in successive branches), there is a disuniformity in the heat transfer (and therefore in the disc bulk temperature) between R
_{1}and R_{2}; - (c)
- The κ
_{GZ}is not the same in branches 1 and 2, and this adds up to the non-constant re effect, increasing the disuniformity in the disc body.

#### 5.2. Selection of the DSHE Configuration for the Cooling of an Electronic Chip

- (a)
- The level 0 branch is too long: the Graetz number is relatively low, and the (calculated) Nu is about 8;
- (b)
- The branches at level 1 are, on the contrary, too long, and the flow does not become fully developed at the end of the channels (Figure 9, right): the Nu
_{1}is high but the fluid displays a strong radial temperature disuniformity at the outlet; - (c)
- In spite of the above shortcomings, the temperature of the disc is approximately constant, validating the assumption made in Section 2 as to the external convection loss.

## 6. Conclusions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Nomenclature

A: m^{2} | Disc area | p, bar | Pressure |

c_{p}, J/(kg × K) | Specific Heat | q, W/m^{2}; Q, W | Specific and total heat load |

d, m | Tube diameter | R, m | Disc radius |

Gz = RePr/κ_{Gz} | Graetz number | Re | Reynolds number |

k, W/(m × K) | Thermal conductivity | s, m | Disc thickness |

L | Channel length | T | Temperature |

m | Mass flowrate | U | Equivalent heat transfer coefficient |

n | Number of levels (“splits”) | z_{0} | Number of initial branchings |

Nu = hd/k | Nusselt number | z_{b} | Number of outlets at R_{ext} |

Greek Symbols | |||

β | Branching angle | λ = L_{j+1}/L_{j} | Channel length ratio |

γ | Central spanning angle | µ, kg/(ms) | Dynamic viscosity |

δ = d_{j+1}/d_{j} | Diameter ratio | ν, m^{2}/s | Kinematic viscosity |

η | Disc efficiency | p, kg/m^{3} | density |

κ_{Gz} = L/d | Graetz factor |

## References

- Sherman, T.F. On connecting large vessels to small: The meaning of Murray’s Law. J. General. Physiol.
**1981**, 78, 431–453. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Bejan, A. Entropy Generation through Heat and Fluid Flow; J. Wiley & Sons: New York, NY, USA, 1982. [Google Scholar]
- Capata, R.; Gagliardi, L. Experimental investigation on the Reynolds dependence of the performance of branched heat exchangers working with organic fluids. Int. J. Heat Mass Transf.
**2019**, 140, 129–138. [Google Scholar] [CrossRef] - Miguel, A.F. A study of entropy generation in tree-shaped flow structures. Int. J. Heat Mass Transf.
**2016**, 92, 349–359. [Google Scholar] [CrossRef] - 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. [Google Scholar] [CrossRef] [Green Version] - Sciubba, E. Entropy generation minimization as a design tool. Part 1: Analysis of different configurations of branched and non-branched laminar isothermal flow through a circular pipe. IJoT
**2011**, 4, 11–20. [Google Scholar] - Sciubba, E. A critical reassessment of the hess–murray law. Entropy
**2016**, 18, 283. [Google Scholar] [CrossRef] [Green Version] - Shah, R.K.; Sekulić, D.S. Fundamentals of Heat Exchangers Design; J. Wiley & Sons: New York, NY, USA, 2003. [Google Scholar]
- Bejan, A.; Alalaimi, M.; Sabau, A.S.; Lorente, S. Entrance-length dendritic plate heat exchangers. Int. J. Heat Mass Transf.
**2017**, 114, 1350–1356. [Google Scholar] [CrossRef] - Cancellario, A. Waste Heat Recovery System at Nanoscales: Process Simulation, Optimization, Design of Heat Exchanger. Master’s Thesis, University of Roma Sapienza, Rome, Italy, 2017. [Google Scholar]
- Wechsatol, W.; Lorente, S.; Bejan, A. Dendritic convection on a disc. Int. J. Heat Mass Transf.
**2003**, 46, 4381–4391. [Google Scholar] [CrossRef] - Capata, R.; Beyene, A. Experimental evaluation of three different configurations of constructal disc-shaped heat exchangers. Int. J. Heat Mass Transf.
**2017**, 115, 92–101. [Google Scholar] [CrossRef] - Sciubba, E. Shape from function: The exergy cost of viscous flow in bifurcated diabatic tubes. Energy
**2020**, 213, 118663. [Google Scholar] [CrossRef] - Silva, A.K.; Bejan, A. Dendritic counterflow heat exchanger experiments. Int. J. Thermal Sci.
**2006**, 45, 860–869. [Google Scholar] [CrossRef] - Bejan, A. Constructal-theory network of conducting paths for cooling a heat generating volume. Int. J. Heat Mass Transf.
**1997**, 40, 799–816. [Google Scholar] [CrossRef]

**Figure 1.**Sketch of a disc-shaped heat exchanger (

**a**): highly-branched; (

**b**): assembly; (

**c**) single-branched configuration.

**Figure 2.**“Typical” CFD of a DSHE. Reprinted with permission from Ref. [12]. Copyright 2017 Elsevier.

**Figure 3.**Fundamental modes of operation of a DSHE: (

**a**) Convection both sides; (

**b**) Conduction both sides; (

**c**) Conduction and convection.

**Figure 9.**Ultra-micro DSHE: Results of a RANS CFD simulation (Adapted from Ref. [10]).

Type of Operation (Heater) | Needed Design Specifications | Q_{DES} |
---|---|---|

Heat input by convection on both sides | h_{ext}, T_{ext}, T_{D} | ${Q}_{des}=2\pi {h}_{ext}{R}^{2}\mathit{\u2206}{T}_{D}$ |

Electrical or conduction heating on both sides | q_{in}, s_{D}, k_{D} | ${Q}_{des}=2\pi {q}_{in}{R}^{2}$ |

Electrical or conduction heating on one surface, cooling by free convection on the opposite one | h_{ext}, T_{ext}, T_{D}, q_{in}, s_{D}, k_{D} | ${Q}_{des}=2\pi {R}^{2}\left({q}_{in}-{h}_{ext}\mathit{\u2206}{T}_{D}\right)$ |

**Table 2.**An Al/Mn alloy DSHE cooled by a glycol/water mixture (Data adapted from [3]).

Disc Radius, m | 0.075 | Glycol Mix Mass Flowrate, kg/s | 0.181 |

Disc Thickness, m | 0.015 | Glycol Mix Density at Inlet, kg/m^{3} | 1088 |

Number of Root Splits, Z_{0} | 3 | Glycol Mix Viscosity at Inlet, m^{2}/s | 5.97 × 10^{−6} |

Branching Exponent, Z_{b} | 2 | Glycol Mix Conductivity kW/(m × K) | 0.65 |

D_{feed}, M | 0.022 | Glycol Mix Specific Heat, J/(kg × K) | 870 |

D_{0}, M | 0.013 | T_{glycol}_{,In}, K | 300 |

D_{1}, M | 0.0065 | T_{glycol}_{,Out}, K | 303 |

D_{2}, M | 0.00325 | T_{external Air}, K | 293 |

L_{0}, M | 0.011 | Q_{electr}, W | 500 |

L_{1}, M | 0.048 | LMTD, K | 6.38 |

L_{2}, M | 0.024 | U_{avg}, W/(m^{2} × K) | 7227 |

Re_{CHANNELS} | 913 | η_{DSHE} | 0.91 |

Disc Radius, m | 0.084 | L_{0}, M | 0.030 |

Disc Thickness, m | 0.015 | L_{1}, M | 0.031 |

Number of Root Splits, Z_{0} | 3 | L_{2}, M | 0.033 |

Branching Levels, N | 2 | T_{GLYCOL}_{, IN}, K | 300 |

Total Number of Fluid Outlets, Z_{b} | 12 | T_{GLYCOL}_{, OUT}, K | 304 |

D_{feed}, m | 0.022 | T_{EXTERNAL AIR}, K | 293 |

D_{0}, m | 0.0059 | Q_{ELECTR}, W | 500 |

D_{1}, m | 0.0029 | LMTD, K | 6.66 |

D_{2}, m | 0.00148 | U_{avg}, W/(m^{2} × K) | 14,400 |

RE_{CHANNELS} | 2000 | η_{DSHE} | 0.91 |

Disc Radius, m | 0.052 | DMSO Mass Flowrate, kg/s | 7.84 × 10^{−4} |

Disc Thickness, m | 0.001 | DMSO Mix Density at Inlet, kg/m^{3} | 1101 |

Number of Root Splits, Z_{0} | 2 | DMSO Mix Viscosity at Inlet, m ^{2}/s | 1.8 × 10^{−6} |

Branching Exponent, N | 1 | DMSO Mix Conductivity K, W/(m × K) | 0.16 |

D_{feed}, M | 0.0012 | DMSO Mix Specific Heat, J/(kg × K) | 1960 |

D_{0}, M | 0.0011 | T_{dmso}_{,In}, K | 298 |

D_{1}, M | 0.00055 | T_{dmso}_{,Out}, K | 352 |

L_{0}, M | 0.0123 | T_{external Air}, K | 300 |

L_{1}, M | 0.0093 | T_{disc}, W | 358 |

Re_{in} | 209 | Q_{load}, W | 72.22 |

Re_{1} | 694 | LMTD, K | 35 |

Re_{2} | 968 | U_{avg}, W/(m^{2} × K) | 35,400 |

Disc Radius, m | 0.01 | L_{0}, m | 0.0055 |

Disc Thickness, m | 0.001 | L_{1}, m | 0.0043 |

Number of Root Splits, Z_{0} | 4 | L_{2}, m | 0.0034 |

Branching Exponent, N | 1 | T_{dmso, In}, K | 298 |

Total Number of Fluid Outlets, Z_{b} | 8 | T_{dmso, Out}, K | 345 |

D_{feed}, m | 0.0012 | T_{external Air}, K | 300 |

D_{0}, m | 0.0011 | Q_{electr}, W | 72.22 |

D_{1}, m | 0.00055 | LMTD, K | 31 |

D_{2}, m | 0.00027 | U_{avg}, W/(m^{2} × K) | 2973 |

RE_{CHANNELS} | 115 | R_{ein} | 839 |

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2022 by the author. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Sciubba, E.
How to Reduce the Design of Disc-Shaped Heat Exchangers to a Zero-Degrees-of-Freedom Task. *Energies* **2022**, *15*, 1250.
https://doi.org/10.3390/en15031250

**AMA Style**

Sciubba E.
How to Reduce the Design of Disc-Shaped Heat Exchangers to a Zero-Degrees-of-Freedom Task. *Energies*. 2022; 15(3):1250.
https://doi.org/10.3390/en15031250

**Chicago/Turabian Style**

Sciubba, Enrico.
2022. "How to Reduce the Design of Disc-Shaped Heat Exchangers to a Zero-Degrees-of-Freedom Task" *Energies* 15, no. 3: 1250.
https://doi.org/10.3390/en15031250