Effects of the internal structures of monolith ceramic substrates on thermal and hydraulic properties: additive manufacturing, numerical modelling and experimental testing

Rigorous emission regulations call for more efficient passive control catalysts for exhaust gas aftertreatment without affecting the internal combustion process and CO2 emissions. Although the state-of-art ceramic honeycomb substrate designs provide high surface area and a degree of flexibility for heat and mass transfer adaptations, additional emission reduction benefits can be achieved when more flexible designs to provide effective thermal management are introduced. The conventional cordierite honeycomb substrates are manufactured by extrusion; therefore, only substrates with straight channels can be fabricated. This study aims to highlight any design limitations of conventional substrates by employing additive manufacturing as the main method of manufacturing diamond lattice structures using DLP (digital light processing) technology. Both conventional substrates and diamond lattice structures are studied numerically and experimentally for flow through resistance and temperature distribution. Numerical predictions and experimental results showed good agreement. The results show the increase of the axial temperature distribution for diamond lattice structures and a significant decrease of the pressure drop (38–45%) in comparison with the benchmark honeycomb with similar surface area.

Temperature of the gas at the inlet of the substrate, K T gas,out Temperature of the gas at the outlet of the substrate, K T wall,ave Average temperature of the outer walls of the substrate, K V bcc Volume of the solid substrate, m 3 V cylinder Volume of the entire cylinder substrate, m 3 ε Porosity of the substrate Φ Specific length ratio, the ratio of pore diameter and strut thickness

Introduction
Environmental concern and ever-changing emission regulations have raised the significance of improving the performance of the exhaust gas aftertreatment system for stationary and mobile applications [1,2]. In the 1970s, initial trials of exhaust gas purification incorporated placing a catalytic converter in all petrol vehicles [3]. The key part of the converter, honeycomb monolith, typically manufactured by extrusion, consists of thousands of straight, parallel, flow-through channels coated with noble metals accountable for catalytic reactions [4][5][6]. Due to the restrictions of the extrusion process, the channel design is of a simple shape with thin walls separating the individual channels [7,8]. Only limited geometric shapes of the channels can be manufactured, such as square, hexagonal or triangular [9,10]. Research shows that hexagonal and square channels show similar performance and pressure drop under the same experimenting conditions. Contrarily, triangular channels show the poorest performance [11]. Hotspots and non-isothermal temperature profiles in axial and radial directions present additional issues associated with the substrate design [12,13]. The laminar flow inside of the honeycomb channels decreases the heat and mass transfer [14]. The awareness of the inefficient use of the catalyst full volume led to research of new possible designs of the substrate following the improvement of the manufacturing opportunities. One way of achieving the necessary requirements to increase the catalyst performance is to cautiously direct the parameters of the substrate structure to achieve the desired pressure drop and improved heat transfer. An interesting approach to enhanced monolith designs is non-stochastic structures that include 3D lattice structures of strut-and-node arrangement and custom unit cells. Non-stochastic lattice structures consist of unit cells periodically arranged in all three dimensions with the most common octet, cube, hexagon and tetrakaidekahedron, all inspired by foam designs [14,15]. Topology-optimized lattice structures have been utilized in automotive industry for weight reduction and improvement of the strength of bulk parts [16]. For catalytic applications, modified regular structures manufactured by robocasting provided improved mass transfer compared with the honeycomb sample, with a fairly higher pressure drop [17]. The tortuous path in the channels upsurges the transport between the catalyst surface and the bulk of reactant and product [18,19]. Recent research has revealed rotated cubic cells as a replacement for honeycombs [14]. Samples with 3 mm and 4 mm cell size and strut thickness of 0.5 mm were manufactured and experimentally tested [14,15]. Due to the manufacturing limitations, the samples had a low specific surface area which proved to be a disadvantage [14].
Numerical simulations can give detailed insights into specifics that are not possible or are difficult to capture throughout experimental trials to aid the design of the new catalytic supports. The simplest modelling method for honeycombs is a single-channel method, where all channels are assumed to behave alike; however, this assumption is not correct. The flow maldistribution and heat losses to the surroundings are discarded in the single-channel models [10,[20][21][22][23]. Multichannel models, such as the subgrid-scale model with a couple of representative channels, have the same drawbacks as the single-channel method [24]. Continuum models, where the pressure drop of the monolith is implemented into the model from calculations, need to have assigned a correct heat flux at boundary [22,25]. The thermal conditions, in reality, cannot be set as isothermal due to the temperature variations along the length and cross-section of the monolith. Likewise, numerical simulations have been utilized, to some extent, to study the fluid dynamics and heat transfer properties of foams and nonstochastic lattice structures. A comprehensive analysis of pressure drops in virtually reconstructed open cell foams, combining numerical and experimental studies, showed the open-cell foams outperform honeycomb monoliths in the range of low Reynolds numbers [26]. Alternatively, foams can be reconstructed by Laguerre-Voronoi tessellations, with the main drawback being the increase of the computational load [27]. Previous studies have indicated that foam-based catalysts demonstrate greater performance in comparison with the honeycomb in terms of conversion, but might exhibit higher pressure drops [3,26,28]. While foams seem to show increased performance, their non-controllable structure makes it challenging to foresee the flow behaviour [15]. To simplify the modelling of the foams in the past, the foam structure had been represented with regular cells, such as Kelvin cells. The regular structures have actually proved to have a superior trade-off between the mass transfer and pressure drop [29]. Moreover, their superior properties led to study the regular structures as potential catalyst substrates [14,30]. Modelling of complex non-stochastic structures is particularly challenging; thus, short domains consisting of only a couple of cells in the flow direction are generally used for numerical simulations [30][31][32]. However, the short domains could affect the simulation results as the flow might not be fully developed [31].
Advanced manufacturing technologies (AMT) are progressively used in many different industries. Precise and complex components can be fabricated by machining processes such as ball end milling [33] and turning [34]. Furthermore, laser hardening offers short processing time and higher performance for 3D complex shapes [35]. Additive manufacturing eases the manufacturing of complex geometrical shapes by layer by layer addition of material. The improvements in the AM technologies have allowed direct fabrications of lattice structures. Selective laser melting (SLM) is a widely employed additive manufacturing process that can be used for printing of intricate three-dimensional (3D) structures for various applications [30][31][32]. Ti-6Al-4V alloy-based 3D meshes with diamond unit cells were manufactured with concept laser M2 for optical applications [36]. Moreover, the same technology was utilized to manufacture Ti-6Al-4V optical lattice mesh structure that can be used as an oil sensor [37]. In addition, the studies on the effect of SLM process parameters and their optimization were performed for manufacturing of biomedical implants [38]. Additive manufacturing of ceramic materials has been an ongoing topic of research due to their challenging processing requirements [39]. One of the challenges is the reduced cure depth in ceramic suspensions due to scattering, resulting in a decrease of the bonding between the layers. However, reducing the layer thickness to increase the bonding of the layers as a consequence increases the building time. Correspondingly, the viscosity of the suspension must be optimised to ensure uniform layer recoating during the build [40]. Nevertheless, AM can overcome the traditional ceramic manufacturing methods limitations, such as complex geometry production. Net shape YSZ/Al 2 O 3 nanocomposite micro-components were successfully manufactured through a process based on soft lithography and powder dispersion [41]. Techniques involving polymerization of the photoactive resin can generally be divided into stereolithography (SLA) based on spot scanning and digital light processing (DLP) based on mask-image-projection [42]. Lattice structures have been manufactured by stereolithography in the past; however, the disadvantages of this process include the need of support for manufacturing and the stair-case effect which can be highly detrimental under load [15,43]. Digital light processing AM technology generally has faster manufacturing speed and is advantageous for building small components [42]. Thus, digital light processing (DLP) additive manufacturing technology has been used in this study due to the significant advantages of high precision and low cost for advanced ceramic manufacturing [41].
Research conducted so far on the improvement of the monolith designs has merely explored unit cells derived from foams, such as Kelvin cell or a simple cubic cell [14,17]. The dimensional limitations of the manufacturing technique used to manufacture the substrates cause the aforementioned to have large strut diameters and inadequate substrate dimensions resulting in low geometrical surface areas [14]. Short domains chosen for the experiments and numerical modelling of lattice structures can cause inaccuracies in pressure drop prediction and overall performance as the flow might not be fully developed [14,31]. Morphological parameters of honeycomb and lattice design need to be investigated in conditions other than isothermal. In this study, hot gas is used as the heat source rather than isothermal conditions on the outside wall. The numerical model contains longer length of the lattice structure with a higher aspect ratio. The effect of geometrical structure between lattice and traditional monolith substrate in terms of heat transfer and flow resistance is investigated by employing a steady-state three-dimensional conjugate heat transfer model. The aim of this paper is to develop and manufacture a lattice monolithic ceramic substrate. Computational fluid dynamic model is developed to aid the design and is validated for both designs and used to compare between them. The performance of the two substrate designs is tested in terms of temperature profile and pressure drop in a non-reactive environment. Finally, the effect of lattice design on the specific surface area, pressure drop and thermal management is investigated.

Design of lattice structures
Lattice can be defined as a structure which consists of a regular network of struts connected by nodes in a threedimensional space. The unit cell size, type, strut size and cell orientation have a considerable influence on the surface area, thermal properties of the structure and fluid flow behaviour. Diamond lattice cell has been selected as the representative model as it has shown great performance for other applications and it is self-supportive, which is of great importance for successful additive manufacturing [37,44]. The CAD model of the investigated diamond lattice substrate is shown in Fig.  1a, with the corresponding internal structure in Fig. 1b and a diamond unit cell in Fig. 1c and d, illustrating the main properties of the diamond lattice structure. The porosity was calculated from the volume distribution of the solid and total volume of the cylindrical samples as shown in Eq. 1.
The specific surface area was calculated as the geometrical surface area of the whole substrate divided by the volume of the cylinder as shown in Eq. 2.
Honeycomb monoliths are available in different cell density and shapes. Nevertheless, the most common commercial honeycomb is a square channelled 400 CPSI [14,26,45,46]. Therefore, a square channelled honeycomb monolith with 400 CPSI and 0.2 mm thick walls is used as the benchmark. The CAD model of the honeycomb substrate is illustrated in Fig. 2a. Figure

Numerical modelling
In order to aid the design of the lattice structures, a numerical approach using computational fluid dynamics (CFD) is used to predict the effect of the internal structure of the substrate on fluid dynamics, pressure drop and temperature distribution. The main focus of the numerical study is on the physical behaviour of the flow within the substrate; therefore, the chemical reactions are not taken into the account for simplification and to place the interest strictly on the flow behaviour driven by the geometry change, which is a common practice when investigating fluid flow in substrates [1,30,47]. The main idea is that, with the lattice design, the temperature distribution in radial and axial direction of the substrate will be considerably different than of the traditional honeycomb substrate, concluding that the geometry can influence the temperature distribution. A finite-volume-based commercial code ANSYS CFX enables to solve the conjugate heat transfer problems with two separate domains, where the thermal energy of solid and fluid is exchanged in their interface [48]. The equations required are momentum balances and energy balances for the solid and fluid phases. The transport of enthalpy through the  fluid and heat transfer is modelled with thermal energy model. Steady-state Navier-Stokes equations were solved in the laminar model for honeycomb substrate and in shear stress transport (SST) k-ω model for the lattice substrate. It is important to note that two CFD models have been developed and validated; one for the lattice structure and one for the honeycomb (this is because the lattice structure has different dimensions due to the manufacturing limitation). Second-order upwind biased discretization is used to calculate the advection terms in the discrete finite volume equations. Residuals are the most important measure of convergence; hence, RMS residuals are set as the convergence criteria with the magnitude of limit 10 −6 . Pressure drop was monitored as the value of interest and it was ensured that the steady-state value was reached with the overall imbalance less than 1%. The boundary conditions are constructed with mass flow rate and temperature at the inlet, zero-gauge pressure at the outlet, and constant heat transfer coefficient at the walls. The computational domain is represented as a quarter of the full substrate with lateral boundaries as symmetry for the honeycomb and periodic for the lattice. The dimensions of the honeycomb and lattice domain are matched with the dimensions of the substrates in the experiments. The fluid domain was extended by 3 mm before and after the solid. The boundary conditions and the computational domains are shown in Fig. 3. Table 1 summarizes in detail the computational domains and the boundary conditions for the numerical models. Thermal conductivity of 2.5 W/mK is used for the cordierite honeycomb in the present model [1,49]. Zirconia has a reported thermal conductivity in the range of 2.2-2.9 W/mK [50]. The value chosen was the same as for the cordierite sample as in this range significant differences in the temperatures were not observed.
The fluid is considered as an ideal gas; thus, the equation of state pV = nRT is valid. The temperature of the incoming exhaust gas in automotive exhaust systems can be as high as 1173 K [51]. Relatively high air temperature of 923 K was chosen for model validation for higher experimental measurement accuracy. The heat transfer coefficient was calculated from the average sample surface temperature and inlet and outlet temperatures obtained from the experiment [52]. Radiation heat transfer was not taken into the account since the outer tube temperatures were low [53]. Total heat losses Q were obtained with Eq. 3: Total heat transferred from the substrate wall, through the wrapping and the pipe is calculated from Eq. 4: Equating Eqs. 3 and 4, heat transfer coefficient can be derived as Eq. 5: The laminar flow (Re = 125-1250) in the honeycomb due to the very small channel diameter (1.05 mm) justifies the use of the laminar model. The determination of the flow behaviour in the lattice includes calculation of Reynolds number with appropriate characteristic length. Various definitions of the characteristic length for lattice structures are found in the literature. Strut diameter, mean cell diameter or hydraulic diameter can be used to find the Reynolds number [14,31,54]. In this study, the Reynolds number was calculated with the strut diameter as the characteristic length as suggested in [14].The Reynolds number based on the strut diameter in this study lies in the range of 2-25 suggesting laminar to early transition regime [14]. The simulations for the lattice structure were performed both by laminar and SST-k-ω model. The simulation results were compared with the experimental results, resulting in a smaller discrepancy when using SST-k-ω model. Consequently, the SST-k-ω model is selected, given that it is able to deal with transitional flows and performs well in the proximity of solid walls. The mesh is constructed using Ansys software package built-in mesher. The fluid domain of the honeycomb channels is meshed with hexahedral elements with the mesh details shown in Fig. 4 (left), whereas the inlet and outlet sections are meshed with unstructured mesh. Mesh independence is reached at element count of 8,162,045. The solid channels of the monolith substrate are meshed using adaptive hexahedral mesh with a global size of 0.2 mm. It is assumed that there are no considerable thermal gradients across the structure due to very fine thickness of the channel walls, which is confirmed through a set of additional simulations with only 0.3% error between temperatures at the interface when using mesh with 350,373 (element size 0.2 mm) and 6,849,920 elements (element size 0.04 mm). For the lattice structure, different meshing approach is used. Owing to the complex structure, unstructured mesh is generated for both solid and fluid domain. The details of the mesh for the fluid domain are presented in Fig. 4 (right). Curvature-based mesh refinement with fine mesh around the lattice strut surface is used, with mesh size gradually increasing away from the surface at growth rate of 1.3 until the maximum value. The mesh independent solution is reached with

Additive manufacturing
Stereolithography-based method with DLP light source is used to polymerize the photocurable resin and form the green bodies in this study. The aforementioned is a promising additive manufacturing technology for complex shape ceramic part production [42]. The DLP technology might overcome the manufacturing limitations observed in previous research, where the resulting substrates had low specific surface area, requiring higher overall catalyst volume [14]. The samples were manufactured on Admaflex 130 (Admatec Europe BV, The Netherlands) with a x,y resolution of 40 μm. The printing area consists of a transparent glass surface at the bottom and the building platform, which moves vertically up and down, at the top. The light source with DMD chip comprising an array of several thousand microscopic mirrors that rotate according to the pixels in the image is located under the glass. The Admaflex 130 is equipped with a rotating foil system which transports the slurry on the foil from the reservoir to the manufacturing zone and to the pump where the excess slurry is pumped back to the reservoir. Photopolymerisable light sensitive resin is exposed to the light (405 nm), which subsequently cures and hardens the resin. The process of curing layer by layer is repeated until the final three-dimensional green body is built [55]. The commercial ceramic suspension (Z-130, Admatec Europe BV, The Netherlands) was used for manufacturing of the representative diamond lattice sample. The Z-130 feedstock contains a mixture of photosensitive resins and a solid loading of zirconia powder. The layer thickness was set as 25 μm and the exposure time was varied between 1.5 and 2 s. The remaining slurry on the surface of the sample was cleaned by immersing the sample in ultrasonic bath filled with ethanol. Two consequent thermal treatments were performed to achieve a dense ceramic substrate. The debinding in a tube furnace (TSH/1S/75/450, Elite Thermal Systems Ltd, UK) and sintering in a chamber furnace (HTF 17/27, Carbolite Gero, UK) were performed using cycles recommended by the Admatec company. The final sintered product is shown in Fig. 6a, with a Fig. 6b; similar approach was used by [14,56]. The benchmark cordierite honeycomb used for the experimental validation has square channels with a cell diameter of 1.05 mm, separated by 0.2 mm walls. The diameter of the sample is 25 mm and the length is 57 mm. The geometric specific surface area of the sample is 2870 m −1 with the porosity of 0.68.

Experimental setup
A bespoke test facility was designed and constructed to allow for heat transfer characterization and pressure drop measurements of both the conventional honeycomb support and the novel additive manufactured lattice support. The main functions of the test facility are to measure the inlet and outlet air temperature, surface temperature of the substrate, and lastly to measure the pressure drop over the substrate. The schematic of the experimental setup is shown on Fig.  7a. Compressed air is fed into variable flow meter (Omega, FL-2517, accuracy: ± 5% of full-scale reading, Omega Engineering, UK) with a valve used to control the flow rate. The air flow is led through a heated line and stainless steel spiral placed in the furnace to heat the air to a high temperature. The sample is wrapped with insulation mat and placed into the testing section. Once the air before the substrate (T gas, in ) reaches 923 K and the values of interest do not change over time, the system is being considered in steady state and the temperatures and differential pressure values are captured. The vertical testing section shown in Fig. 7b is equipped with two openings for the differential pressure sensor (NXP MPX5010DP piezoresistive transducers with 5.0% maximum error, range 0 to 10 kPa, RS Components Ltd, UK) connection

Model validation
The pressure loss of substrates is highly dependent on the channel design, geometry, length, and hydraulic diameter of the channels [47]. The substrates must be designed in a way to offer high surface area for catalytic reactions, low back-pressure, and fast light-off without compromises in mechanical properties. Therefore, our aim is to understand the pressure drop and thermal behaviour of both the conventional honeycomb and diamond lattice structures. The validation of the numerical model consists of comparing the experimental results for pressure drop and temperature (T 1 , T 2 , T 3 , T 4 , T 5 , T 6 ) with the results obtained by numerical simulations. The validation has been achieved for various mass flow rates. The obtained results for the temperature distribution showed similar trends; hence, for simplification purposes, only the results for 1.1 kg/h are presented. The lattice sample is 4 mm longer than the 400 CPSI honeycomb substrate due to the manufacturing limitations. However, it is believed that the minor difference in the length of the lattice will not have a great impact on the obtained results. The validation of computation fluid dynamic models for both lattice and honeycomb design is presented in this section.   [47,57]. It can be concluded that the CFD prediction results for pressure drop agree well with experimentally measured pressure drops.
The second part of the study is to validate the numerical model against experimental data for temperature measurements. Figure 9 shows the temperature distribution along the honeycomb substrate, at the centre (T 1 , T 3 , T 5 ) as well as the surface (T 2 , T 4 , T 6 ), with mass flow rate of 1.1 kg/h. Note that the error bars lie within the point. The maximum temperature equal to the inlet gas temperature of 923 K is found at the centre of the inlet of the honeycomb substrate. The temperature gradually decreases as the flow reaches the exit of the substrate and the lowest temperature point is located at the outer surface of the honeycomb substrate with a value of 768 K. The temperature gradient between the centre and the surface of the monolith is expected due to the heat losses to the surroundings. The axial gradient from the inlet centre to the exit centre is around ΔT T1-T5 = 82 K for the honeycomb substrate highlighting the non-uniform axial temperature distribution. Very good agreement between the CFD model prediction and experimental results with a maximum of 3.5% error at the mass flow rate of 1.1 kg/h is achieved.

Validation of model for the lattice structures
Validation study with respect to the obtained experimental results is performed to determine the accuracy of the model with lattice structures. The lattice substrate has a diameter of 25.4 mm with 61.3 mm length, pore diameter of 1.85 mm, strut diameter of 0.65 mm, specific surface area of 1480 m −1 , and the porosity of 0.73. Figure 10 shows the model validation of the pressure drop for lattice substrate with varying mass flow rate (0.7-1.8 kg/h) and constant inlet gas temperature of 923 K. The pressure drop linearly increases with the flow rate and at the highest flow rate examined (1.8 kg/h) is 78 Pa as seen in Fig. 10. The linear dependence is expected in the examined mass flow rate range where the maximum inlet velocity at the highest mass flow rate is equal to 1.6 m/s. Exponential trend for lattice structures is observed in literature for higher velocities [14,58,59]. The numerical model over predicts the pressure drop values for 0.7 and 1.1 kg/h, but slightly under predicts the value at the highest flow rate (1.8 kg/h) by 1.6%. The percentage error between the experimental and numerical data is decreasing with the increase of the mass flow rate. Higher percentage error between the lowest flow rate pressure drop obtained by experiment and simulation of 13.7% is explained by possible instrument uncertainty in the given range. Namely, the value of pressure drop in the experiment is 22 Pa while the differential pressure device has a maximum error of 5% in the instrument range. This observation is assumed to be correct owing to the reduction of percentage error with the increase of the mass flow rate. The numerical results for pressure drop agree well with the experimentally measured pressure drop.
The same validation approach as in the honeycomb model has been applied for the temperature distribution. Figure 11 shows the temperature distribution along the axial length of the lattice substrate, at the centre (T 1 , T 3 , T 5 ) as well as the surface (T 2 , T 4 , T 6 ), at mass flow rate of 1.1 kg/h. Note that the error bars lie within the point. Maximum temperature is located at the inlet centre of the lattice substrate with a value of 922 K, whereas the lowest temperature is located at the exit outer surface with a value of 694 K. The predicted temperatures calculated with the numerical model follow the same trend as in the experiments. The highest temperature values are located in the inlet centre of the sample, whereas the lowest temperatures are located towards the outer surface of the sample. Heat losses to the surroundings cause the radial temperature gradient in the lattice substrate, the same as for the honeycomb substrate. The axial gradient from the inlet centre to the exit centre is around ΔT T1-T5 = 66 K for the lattice substrate, 20% lower than for the honeycomb substrate. Generally, the numerical data is closely matched with the Fig. 8 Comparison of the experimental and numerical results for pressure drop of the honeycomb monolith at various flow rates experimental data with the highest discrepancy of 9.6% between the calculated and measured temperatures.
Numerical models have been validated for the benchmark honeycomb substrate and the lattice substrate. For the honeycomb model, the pressure drop can be predicted with a maximum error of less than 10% and the predicted temperatures show a deviation of approximately 4% from the experimental results. Lattice model can predict the pressure drops with a maximum error below 14% in the studied range, while the temperatures show a maximum deviation of less than 10% from the experimental results. Both models show good agreement with the experiment.

Comparison between the honeycomb and lattice designs with the same cell density
CFD model has been validated for a lattice with pore diameter (d p ) of 1.85 mm, although the DLP technology is superior to SLA for manufacturing of complex geometries, the ceramic slurry formulation needs to be optimised for a successful print with thinner struts and smaller cell sizes. However, to have a fair comparison between the honeycomb and the lattice design, the same cell density is investigated with the validated numerical model. Square channelled honeycomb monolith with 400 CPSI and 0.2 mm thick walls was chosen as the benchmark. The investigated lattice has identical pore diameter of 1.05 mm as the honeycomb, with strut diameter of 0.2 mm, porosity of 0.87, and specific surface area of 2062 m −1 . The diameter and the length of both models were kept the same to rule out any dimensional effects. The corresponding values are 12.5 mm and 57 mm for diameter and length, respectively.

Velocity
In the actual honeycomb channels, the velocity is affected by the flow maldistribution, consequently affecting the residence time and conversion efficiency, resulting in utilization of only a part of the substrate [10,60]. Research has found that the reactions do not influence the flow distribution as much as the heat loss to the surroundings [10]; therefore, it is feasible to study the flow distribution in non-reactive conditions. It has similarly been determined that the heat losses cause variations in temperature profile in the monolith [61]. The velocity flow field is likely to present some essential differences between honeycomb monoliths and lattices. Below, we investigate the differences in the air flow velocity field between the honeycomb and the lattice. Figures 12a and b show the air velocity contours at mass flow rate of 1.1 kg/h for both honeycomb and lattice substrates. According to the Fig. 12a, the incoming air flow velocity increases instantly after entering the honeycomb channels due to the reduction in the cross-sectional area. As  [10]. Channels close to the periphery of the substrate show higher maximum velocity than the channels in the centre of the sample due to the fluid's physical properties at the periphery. Since the air is treated as compressible fluid, the specific volume decreases along the channel length due to the decrease in temperature causing the decrease in the average velocity in axial direction in the substrates. Figure 12b presents the velocity field contour for the lattice. The flow distribution can be explained as a combination of external flow around cylinders and internal flow between the lattice cells. Flow path in the lattice becomes tortuous where the velocity of the fluid changes rapidly as the fluid comes in contact with the lattice struts [59]. The struts of the lattice present resistance to the fluid flow with forward stagnation point and wake region behind the strut. Maximum velocity is observed in between the lattice cells with a value of 3.08 m/s. The overall lower velocity values in the lattice can benefit the efficiency of the substrate due to the increase of the residence time of the flow in the substrate and increased flow to solid interaction. Moreover, the reduction of the overall velocity could have an influence on the thermal degradation of the substrate brick and pressure drop decrease across the entire substrate [62].

Pressure drop
Reliable comparison of the back-pressure between the lattice and honeycomb sample is achieved using the values of normalized pressure drop presented as pressure drop over length of the sample. The comparison of the pressure drop for a lattice with the same pore density as the benchmark honeycomb is shown in Fig. 13. In the mass flow rate range of interest, the pressure drop of the honeycomb varies linearly with the mass flow rate [47,56,62]. Considering the flow in the honeycomb channels is fully laminar, the viscous effects are the main contributor to the pressure drop in the studied mass flow rate range [60]. Furthermore, the pressure drop follows the same linear trend in the examined mass flow rate range for the lattice structure. In contrast to the honeycomb substrate, the pressure drop across the d p = 1.05 mm lattice is about 38-45% lower. The lower pressure drop is a consequence of the higher porosity of the lattice. Namely, the higher porosity significates that the void fraction, which is available for the fluid flow, in the lattice structure is higher. Similarly, lower pressure drop with respect to the honeycomb, at low mass flow rates, has been observed in literature for open cell substrates with a porosity of 0.95 [14].

Thermal properties
It is expected that the tortuous pathway of the lattice substrate can enhance the heat transfer and minimize the manifestation of hot spots that cause catalyst deactivation. Interconnected solid struts in the internal structure of the lattice structure allow cross-mixing of the flow leading to an augment of the heat transfer [63]. The studies demonstrate that flow distribution within the honeycomb has an immense influence on the distribution of temperature and species concentration in the substrate [64]. Therefore, thermal properties of the honeycomb and lattice were investigated. Figure 14 shows the temperature distribution contours with 1.1 kg/h mass flow rate for honeycomb and lattice substrate. In case of the honeycomb, as seen in Fig. 14a, the area with the peak temperature, equal to the temperature of the incoming gas, is located in the central region of the substrate. The contour shown for honeycomb implies that by increasing the axial distance, the temperature gradient from the inlet face to the outlet face increases. Decrease of the temperature radially to the direction of the surface of the substrate is a consequence of the heat losses to the ambient [1,10,64]. The shape of the peak temperature region shows a downward slope in axial direction indicating the highest heat transfer in the inlet central region of the honeycomb. On the other hand, as shown in Fig.  14b, the diamond lattice has a more uniform temperature distribution at the whole core region in the axial direction indicating the higher heat transfer caused by less thermal mass. The peak temperature region has a downward slope trend from the inlet to the exit region.
The channel flow and the presence of the solid walls in the honeycomb cause limitations in radial heat transport. In contrast, the solid material of the honeycomb substrate increases thermal resistance and causes higher temperatures towards the substrate's outer surface. The same temperature distribution trend is expected at lower inlet temperatures. Further It can be concluded that the lattice substrate performed better thermally and hydraulically. The predictions of the pressure gradient presented the same trend in which the pressure drop across the lattices is lower regardless of the increase of the mass flow rate as compared with the benchmark honeycomb in the studied range. This pressure drop study indicates that utilizing the lattice design is significantly more efficient due to the lower pressure drop and lower average velocity in comparison with the 400 CPSI HC. The d p = 1.05 mm lattice substrate showed improvement in the thermal performance showing lesser axial temperature gradients in the core of the substrate. The lower temperature at the periphery of the lattice is a consequence of the lower amount of solid phase causing higher heat dissipation to the surroundings. Hence, the lattice design should be optimized to achieve higher temperatures at the outer surface.

Effect of the lattice design on the substrate performance
The main parameters of the lattice are the strut diameter (d s ), strut length (L s ), cell length (L c ), pore diameter (d p ), porosity (ε), and the specific surface area (S v ). The strut diameter affects the porosity, the specific surface area of the substrate as well as the mechanical strength. If the volume of the substrate is to be constant, the feasible increase of the lattice porosity is achievable by increasing the pore size. Should the pore size be double, samples with porosity up to 0.94 are achievable. Doubling the pore size, on the other hand, decreases the surface area by half. The specific surface area (S v ) of the lattice substrate has a large influence on the resulting temperature profile [65]. Current improvements in the ceramic additive manufacturing could allow fabrication of strut diameters as low as 0.2 mm. The strut diameter in this study was varied from 0.2 to 0.65 mm, covering a wide range of substrate porosities. Table 2 presents the main geometrical properties of the studied lattice structures. Figures 15a and b show the effect of the strut diameter on the specific surface area and the effect of porosity of the diamond lattices on the specific length ratio. Specific surface area increases with the increase of the strut diameter as pictured in Fig. 15a. As an overall trend, the specific surface increased with increasing the cell size and strut thickness at constant pore diameter. The specific length ratio, Φ, representing the ratio of pore diameter and strut thickness is plotted against porosity in Fig. 15b. As the specific length ratio increases, the porosity of the sample increases because the strut diameter becomes smaller compared with the pore diameter [13]. Figure 16 shows the effect of the lattice porosity (x-axis) on the specific surface area (blue bars) and pressure drop (yellow bars). With the decrease of the porosity, specific surface area and pressure drop values increase. The lowest pressure drop of 269 Pa/m is observed for lattice porosity of 0.94, whereas the highest pressure drop of 579 Pa/m is observed for the lattice porosity of 0.73. The results show that the pressure drop is inversely proportional to the porosity, which is in agreement with the literature [27]. On the other hand, the specific surface area of the lattice increases by 29 % with the decrease of the porosity from 0.94 to 0.73. The increase of the surface area caused by the increase of the strut diameter at constant pore diameter results in a decrease of the porosity [27]. As a drawback of the increase of the surface area and the decrease of the porosity, the pressure drop increases. From the results presented in Fig. 16, it can be concluded that the pressure drop is directly proportional to the specific surface area.
Since the increase of the strut diameter increases the amount of the solid phase, the strut diameter increase is expected to have an influence on the heat transfer. Therefore, a study of the strut thickness increase on the temperature distribution is conducted in the following section. The strut diameter was varied from 0.2 to 0.65 mm resulting in a decrease of the substrate porosity at constant pore diameter as shown in Table 2.
In Fig. 17, one can see the result of the increasing strut thickness on temperature distribution in lattice substrate at the inlet mass flow rate of 1.1 kg/h. Figure 17a illustrates the temperature contour for the lattice with a strut diameter of 0.2 mm. This design has the least surface area of 950 m −1 and the highest porosity of 0.94 among all substrates. The temperature distribution in the solid is uniform throughout the whole core area with steep radial gradients due to the heat loss boundary on the outer surface wall of the lattice. The lowest temperatures are present in the periphery of the substrate as a result of less solid material in the radial direction. Similarly, low axial temperature gradients in the core of the substrate are seen for the lattice with d s = 0.3 mm (Fig. 17b), with sharp radial gradients towards the substrate's periphery. In comparison, the temperatures are slightly higher at the periphery of the substrate with d s = 0.3 mm than with d s = 0.2 mm due to the thicker outer mat providing more resistance to the external heat loss. Further increase of the strut diameter to 0.5 mm (Fig. 17c) causes the increase of the axial temperature gradients. On the other hand, the temperature at the periphery of the sample is higher when compared with substrates with d s = 0.2 mm and d s = 0.3 mm. The lowest uniformity of temperature in the core of the substrate is apparent when the strut thickness value is 0.65 mm (Fig. 17d). The substrate with d s = 0.65 mm has the lowest porosity of 0.73, hence the highest amount of solid material present in the substrate volume. The increase of the strut diameter results in the decrease of the void to solid ratio. Additionally, the higher the amount of solid, the poorer the heat transfer to the solid resulting in higher temperature gradients. The highest axial temperature gradients are thereupon observed with the highest strut thickness. The velocity in between the lattice cells for the larger strut diameter is higher resulting in a decrease of the residence time in the lattice following a decrease in total heat transfer. From the presented contours, it can be concluded that the increase of the heat exchange surface formed by the increase of the strut diameter increases the temperature gradient over the core region. Nevertheless, the increase of the solid phase has a positive influence on the temperatures in the substrate's periphery. This improvement indicates a possibility of more effective utilization of the heat within the lattice by tailoring the lattice design and strut thickness where necessary. Fig. 15 a Effect of the strut diameter on the specific surface area. b Effect of the porosity on the specific length ratio Fig. 16 Effect of the porosity of the lattice on the pressure drop and specific surface area As described earlier, one of the key design requirements for the monolith substrates is high surface area and low pressure drop. When judging the efficiency of the substrate, priority can be given to higher surface area to increase the conversion efficiency, however with a penalty of increased pressure loss. The increase of the strut diameter results in the increase of the specific surface area due to the additional solid material present in the substrate; however, the additional exposed surface slightly increases the pressure drop and the temperature gradients. The judgement on the optimal strut thickness must be made considering the environment in which the substrate will be used. The substrate with thicker struts and lower pore density could be retaining more heat during cooling periods because of the higher thermal inertia, whereas the substrate with thinner strut diameter and higher pore density possesses higher surface area and possibility of a quicker warm up.

Conclusions
In this study, an additively manufactured ceramic diamond lattice substrate has been proposed to overcome the limitations of the conventional honeycomb substrate. DLP additive manufacturing technology has been successfully used to manufacture the diamond lattice substrate. The comparison of the honeycomb design and the diamond lattice has been achieved by pairing numerical simulations and experimental studies with additive manufacturing. The effect of the cell size and strut diameter on the geometric characteristics as well as pressure drop and temperature distribution were studied through numerical simulations.
The key findings of this study may be summarized as follows: & DLP additive manufacturing technology enables fabrication of self-supported ceramic diamond lattice substrates. & As an overall trend, the specific surface area of the lattice increases with increasing the cell size and strut thickness at constant pore diameter. As a consequence, the increase of the surface area results in a decrease of the porosity which results in the increase of the pressure drop. & The pressure drop across the diamond lattices is lower regardless of the lattice pore size as compared with the honeycomb, with a decrease of up to 80%, but with a penalty of lower surface area. & The lattice with a pore size of 1.05 mm and strut thickness of 0.2 mm with a similar surface area as the honeycomb substrate, demonstrates a decrease of pressure drop by 38-45% at studied flow rates. & Diamond lattices exhibit lower axial temperature gradients for all porosities examined. The lattices display homogeneous temperature distribution at the core area relative to the conventional honeycomb design, indicating the higher heat transfer properties caused by the tortuosity of the flow and the ability of flow exchange in both radial and axial direction. This improvement indicates a possibility of more effective utilization of the heat within the lattice.
The reduction of the back-pressure could have a positive impact on the fuel economy, while the improved core temperature distribution allows to conclude that the lattice structure could have better thermal performance in the exhaust system. The findings present potential for improvement of the design of the ceramic catalytic supports by utilization of DLP additive manufacturing technology. The diamond lattice could be specifically tailored to yield desired temperature distribution and pressure drop by controlling the strut diameter and cell size. Further design improvement of the diamond lattice is needed to obtain higher temperatures at the periphery of the substrate, as well as coating and testing with the actual exhaust gas to get the insights on the conversion efficiency and warmup/cool down behaviour of the diamond lattice structures. Data availability Data will be available upon request.

Compliance with ethical standards
Conflict of interest The authors declare that they have no conflict of interest.
Ethical approval Not applicable.
Consent to participate Not applicable.

Consent to publish Not applicable.
Code availability The commercial Ansys CFX code was used for the numerical model.
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.