VASCULAR STRUCTURES FOR SMART FEATURES: SELF-COOLING AND SELF-HEALING

Here we show how smart features of self-cooling and self-healing can be gained to mechanical systems with embedded vascular structures. Vascular structures mimic the circulatory system of animals. Similar to blood distribution from heart to the animal body, vascular channels provide the distribution of coolant and/or healing agent from a point to the entire body of a mechanic system. Thus the mechanic system becomes capable of cooling itself under unpredictable heat attacks and capable of healing itself as cracks occur due to applied mechanical loads. These smart features are necessary for advanced devices, equipment and vehicles. The essential design parameter is vascularization in order to provide smart features. There are distinct configurations for vascularization such as radial, tree-shaped, grid and hybrids of these designs. In addition, several theories are available for the shape optimization of vascular structures such as fractal theory and constructal theory. Unlike fractal theory, constructal theory does not include constraints based on generic algorithms and dictated assumptions. Therefore, constructal theory approach is discussed in this paper. This paper shows how smart features can be gained to a mechanical system while its weight decreases and its mechanical strength increases simultaneously.

In summary, constructal theory unifies and connects distinct fields, and it is valid for animate and inanimate. The best performing design for given conditions and constraints is the constructal design, and this constructal design changes as conditions and constraints change, i.e. the design is dynamic. Furthermore, the design is not restricted by generic algorithms and dictated design assumptions in constructal theory, i.e. the design is morphed freely in order to minimize resistances.

RESULTS AND DISCUSSION
Self-healing and self-cooling applications in engineered systems are biologically inspired. Similar to distribution of blood in the human body for keeping its temperature uniform and providing healing when a cut occurs on the body, distributing coolant and/or healing agent to the entire volume of a mechanical system provides self-cooling and/or self-healing capability to it. There are two main methods for self-healing: vascularization and microcapsules. White et al. showed that a damaged structure can be healed with embedded microcapsules which are filled with healing agents [9]. As cracks damages the microcapsules, microcapsules crack and the healing agent inside fills the gaps and polymerizes. Therefore, this healing method can be used on time for the lifetime of the mechanical system. In addition, literature shows that the mechanical strength of a structure drops after healing occurs [10][11]. Moreover, literature also shows that after the structure is damaged the applied mechanical force is unloaded for structure to be healed during experiments [12][13]. In addition, literature shows that the conductivity of a self-healing structure can be restored [14][15].
Unlike embedding microcapsules filled with healing agents, embedding a vascular structure in which healing agent flows in engineered structure enables the structure of healing countless time [16][17][18]. This embedded structure is similar to the blood veins in circulatory system of warm blooded animals. There are different kinds of vascular architectures in the literature such as: radial, grid, hybrid and tree-shaped designs. Each architecture has its positives and negatives. Radial and tree-shaped designs provide the smallest pressure drop in comparison with grid and hybrid structures [19][20][21][22]. However, grid and hybrid structures bathe the entire volume more uniformly in comparison with radial and tree-shaped designs [19][20][21][22]. In self-healing and self-cooling, it is essential to bathe the entire volume with coolant and/or healing agent due to random and unpredictable characteristics of heating and mechanical loads. The pressure drop of hybrid of grid and tree-shaped designs is smaller than grid designs and greater than tree-shaped architectures. In addition, this hybrid architecture bathes the entire volume almost as good as grid designs [22]. Therefore, hybrid of grid and tree-shaped designs became the best option for smart features.
Literature also shows that the vascularization gains self-cooling capability to a structure [16][17][18][19][20][21][22][23][24][25]. Similar to healing agent flow inside the vascular channels, a coolant flows through the embedded vascular channels. The cooling performance of the vascularized structure is affected by the volume fraction, the complexity of the design, the pressure difference which governs the flow and the flow direction [19][20][21][22]. For a given set of conditions (such as boundary conditions) and constraints (such as volume fraction), there is an optimal design which provides the smallest peak temperature. This optimal design should be morphed to the new optimal design as conditions and constraints change. Therefore, the biomimetic designs not necessarily the best performing desing, i.e. if the objectives of the biological design is different than the engineered design. Cetkin [26] shows that a better performing cooling channel configuration exists than the sinusoidal channels which are inspired from the vascular channels in epidermis. This dynamic behavior of design is in accord with constructal theory. Therefore, the optimal design for a given set of conditions and constraints can be called the constructal design. As time passes, the structure should be morphed into the next constructal design if not it cannot survive. This trend is valid for animate and inanimate (natural or engineered).
Furthermore, the cooling requirements can be deterministic and random. The deterministic cooling requirements are due to heat sources which are known and steady, and random ones are unsteady and diverse. A structure is protected from random and deterministic heat sources via embedded vascular cooling channels [20]. Random cooling requirements are responsible of damaging the structure which is designed to work in steady state due to their unpredictable nature.
In addition of gaining smart features to a structure, vascularization provides mechanical strength with light weight which are essential for advanced vehicles [19][20][21][22][27][28]. The strength of a structure decreases due to removal of the material in order to create vascular channels. However, if the material volume is fixed, the removed material is placed outside of the structure [19,29]. It is also known from the strength of materials that the centerline of a structure is not stressed under bending. Therefore, removing the material from center and placing it around the vascular channels increase mechanical strength of the structure, i.e. the material is put where it is loaded the most. Therefore, the strength of the structure increases with vascularization. Furthermore, if the structure is heated then the effect of thermal stresses cannot be neglected. Cetkin et al. shows when the effect of thermal stresses can and cannot be neglected [29]. Coolant flow in the vascular channels decreases the peak temperature and creates a more uniform temperature distribution in the solid domain; so, thermal stresses decrease greatly. Therefore, selfcooling structures have greater mechanical strength than non-vascularized structures under great heat fluxes.
Similar to how advanced capabilities of self-healing and self-cooling require vascularization in order to bathe the entire volume with coolant fluid and/or healing agent, vascularization is also essential in order to decrease the resistances of the distribution of energy, goods and water [1][2][3][4]30]. A factory distributes all its products to the cities located around the world and collects raw materials around the world similar to distribution and collection of coolant and/or healing agent in advanced materials. These distribution and collection kinds of flows are examples of flows from a point to an area (or volume).

Vascularized Structures with Uniform and Non-uniform Heating
A structure can be bathed by coolant and/or healing agent with embedded vascular channels while its mechanical strength increases and its weight decreases. Discussion of some vascular channel configurations with uniform and non-uniform heating is required in order to uncover how the coolant and/or healing agent is distributed throughout the structure with random (non-uniform heating) and prescribed (uniform heating) boundary conditions. The objective is to find the design with the best cooling performance (i.e. uncovering the design which bathes the entire volume with coolant and/or healing agent).
Here we show that the mechanical strength and thermal performance of a heated and mechanically loaded circular plate can be increased with embedded radial and tree-shaped vascular structures in it. The diameter and thickness of the circular plate are D and H, and their ratio is D/H = 10 which is fixed, Fig. 1a [19]. The solid volume is fixed, so is the volume of the vascular channel network. The plate is subjected to uniformly distributed force and uniform heat flux, both acting from below, Fig. 1a.
The dimensionless governing equations (the conservation of the mass, the conservation of the momentum for the fluid domain, the energy equations for fluid and solid domains, the generalized Hooke's law and the conservation of momentum equations for solid domain) were solved in a finite element software [31]. Mesh test was also performed to confirm mesh independency of the results.
The heat flux and the mechanical load are applied on the bottom surface of the plate as shown in Fig. 1a. The pressure difference between inlet and outlet is non-dimensionalized as Bejan number [32,33]. where μ and α are dynamic viscosity and thermal diffusivity of the fluid. The value of ̃ represents the dimensionless overall pressure difference between the coolant inlet and outlet. The flow is laminar in all the channel configurations, Re < 2000.
The peak temperature and the peak stress is affected by the design. Therefore, the design corresponding the smallest resistance to the flow of heat, fluid and stress can be uncovered by freely morphing the design. Figure  1b shows the relation between the temperature, stress and number of ducts when is 10 7 and 10 8 . The maximum stress decreases when the number of the cooling channels increases from 6 to 8, and it increases when the number of the cooling ducts increases from 8 to 32. The reason of this behavior is that the maximum stress increases in the vicinity of the junctions of the cooling ducts. Even though the peak stress is the minimum when the number of the channels is 8, neighboring designs (6 cooling channels when = 10 7 and 12 cooling channels when = 10 8 ) offer minimum peak temperatures. In summary, when the pressure drop is prescribed, it is possible to identify one design (or a group of designs) that provides the minimum peak stress and peak temperature, or vice versa. However, there is no optimal design for all the constraints and conditions.
Next consider a square plate with length L and thickness H = 0.1L with embedded vascular channels, Fig.  2a [22]. The plate is subjected to a uniformly distributed load and uniform heat flux both acting from below. The volume of the solid and the fluid are fixed. Lg is the length scale of the square area in which the grid cooling channels are embedded as shown in Fig. 2a.
The grid channels are connected to the periphery with radial channels. Coolant enters or exits from the center of the grid, and it is driven by the pressure difference maintained between the inlet and outlet boundaries. The results were obtained by solving the governing equations numerically. Figure 2b shows that the minimum peak temperatures plotted against the peak stresses as Lg/L varies. The effect of the flow direction on the peak temperature and stress is weak. Tpeak and decreases when Lg/L < 0.25, and the peak temperature increases as Lg/L increases even though the peak stress decreases and increases. The peak stress is the minimum when the design is a hybrid of grid and trees. However, the peak temperature is the minimum with radial channels.
Consider that the heating is concentrated in a small region on the vascularized solid domain. The area of the heated spot is 1/16 of the square area of length scale Lg, Fig. 2 [22]. The total heating rate of the concentrated heat generation is fixed, i.e. volumetric heating rate increases as the heat generating region size decreases in order to conform fixed heating rate. Figure 3a shows how the peak temperature changes as Lg/L increases. When the concentrated heating is located in the center of the slab, Tpeak decreases as Lg/L. When the concentrated heating is located in the corner of the grid, Tpeak increases as Lg/L increases. Tpeak is the lowest when Lg/L = 0.25 with the concentrated heating is in the corner, and when Lg/L = 0.625 with the concentrated heating is in the center. In addition, when Lg/L = 0.375 the peak temperature becomes almost as low as the lowest peak temperature obtained when the concentrated heating is located in the center or in the corner. Figure 3b shows the temperature distribution in the mid-plane of square domain when the heat generation is concentrated in the center of the slab and in the corner of the grid for Lg/L = 0.25 and 0.5. The effect of flow direction on the temperature distribution is also shown in Fig. 3b. Consider a square plate of length scale L, and thickness of H = 0.1L as shown in Fig. 4 [20]. A vascular channel network is embedded in the plate in order to keep it under an allowable temperature ceiling while the plate is heated with a concentrated and moving heat flux. The length scale of the square footprint of the heating spot is 0.1L and it moves with the constant speed of W from one edge of the plate to another. Four possible beam paths are discussed as shown in Fig. 4. The volume of the solid is fixed, so is the volume of the fluid. Coolant enters or exits from the center of the slab while the pressure difference between the inlet and exit boundaries is constant. The flow is incompressible with constant properties, and the dimensionless governing equations are time dependent, the dimensionless equations can be found in Ref. [20]. Figure 4 shows the average peak temperature of a solid structure (without embedded vascular channels) and the average peak temperature in four competing designs with embedded vascular channel configurations. The error bars indicate the maximum and minimum peak temperatures when the dimensionless time is greater than 0.1, i.e. after the entire concentrated heat flux enters the plate surface. Figure 4 also shows that a plate heated by a moving beam with an unpredictable path can be cooled by embedding vascular cooling channels in the plate. The effect of changing from no cooling to vascular cooling is dramatic with random cooling requirements similar to with prescribed cooling requirements.
Next, consider the plate with uniform heating load applied on its surface has embedded channels configured as radial, tree-shaped and their hybrid, Ref [34]. Figure 5 shows how the temperature distribution changes as the design and pressure drop (the difference between the inlet and outlet pressures) are altered. The resistance to the fluid flow is smaller in tree-shaped designs in comparison with the radial designs (for instance mass flow rate is 9 to 19% greater for the same pressure drop and volume fraction with tree-shaped design in comparison to the radial design Ref [34]). However, Figure 5 also shows that the thermal resistances in tree-shaped designs are greater than the radial designs, therefore, the peak temperature value is greater in tree-shaped designs when the pressure drop increases, for example the comparison of radial and tree-shaped designs with 290 Pa   30 Pa, (f) 290 Pa [34] pressure drop value. Overall, the tree-shaped and radial designs promise to minimum resistances to the fluid flow and heat flow, respectively. The novel idea is to combining these two in one design, i.e. the hybrid design as shown in Fig. 5. Hybrid design performs almost as good as radial design and tree-shaped designs in terms of resistances to the heat and fluid flow. Both resistances are slightly greater (several per cent) than the corresponding minimum value of the best performing design. The result of Figure 5 uncovers that in some cases the conductive resistances are in great importance. In order to increase the overall thermal conductance of a solid material high-conductivity inserts can be placed. Figure  6 shows how the locations of this high-conductivity inserts affect the thermal conductance by minimizing the peak temperature for fixed boundary conditions, Ref. [35]. Figure 6 shows that high-conductivity inserts should be embedded non-equidistantly in order to minimize thermal resistances. However, Figure 6 shows that there are family of best options, i.e. similar performing designs. Figure 6. The effect of the second level high-conductivity insert locations on maximum temperature [35]