Heat Loss

The objective of the computational study was the investigation of means to improve the performance of PCB-based microPCR in terms of duration, energy consumption, and temperature uniformity for a two-temperature (65 °C and 95 °C) PCR protocol. The geometry of the microPCR used in the computational study is shown in Figure 2. It included a PCB layer (with thickness of 1.68 mm) with an embedded microheater, i.e., a copper line. The thickness, width, and total length of the copper line were 25 μm, 100 μm, and 4.973 m. On top of the PCB layer, there was a PMMA layer (with thickness of 300 μm) with meander-shaped microchannel which was sealed with a polyolefin layer (with thickness of 50 μm). The depth, width, and total volume of the microchannel were 100 μm, 2.5 mm, and 41 μL, respectively. The footprint of the microPCR design was 54.1 × 24.9 mm². The footprint as well as the stack of materials in the PCB layer were very close to the thin chip used in the experiments (cf. Section 2.1).
The study was based on a modeling framework [19 (link)], including an energy balance in the solid domains and the fluid in the microchannel, which read: $\rho C_p\frac{dT}{dt}=\nabla \cdot \left(k\nabla T\right)+Q,$
where T, k, ρ, and C_p are the temperature, the thermal conductivity, the density, and the heat capacity of the solid or the fluid. Q is the heat generation rate at the microheater. It was zero for all domains except for the microheater.
The Joule heating mechanism and the details of the geometry of the microheaters (operating as resistances) were taken into account. The heat generation rate according to the Joule heating read: $Q=J\cdot E.$
E is the electric field in the microheater and J is the current density, which read: $J=\sigma E,$
and was calculated by the current conservation equation, i.e.,
$\nabla \cdot J=0.$
σ is the electrical conductivity of the microheater, which, for the case of copper, was linear, with the following formula
$\sigma =\frac{1}{{\rho }_0\left[1+\alpha (T-T_0)\right]},$
where ρ₀ is the electrical resistivity at temperature equal to T₀, and a is the temperature coefficient of resistivity.
Heat losses by convection and radiation were applied on all external surfaces of the device. The heat transfer coefficient is a function of the surface temperature, the latter coming from a computational study for the heat losses of microfluidic devices [29 ]. A time varying voltage was applied across the microheater in order to achieve the desired thermal cycle, resembling the functionality of a simplified temperature controller. During heating, a constant voltage was applied, during cooling the temperature controller was switched off. Finally, electrical insulation was applied to all other boundaries of the heaters.
The numerical calculations required for the study were performed by the finite element method implemented with the commercial code COMSOL (COMSOL Inc., Stockholm, Sweden).

All behavioural testing was performed during the light phase of the 12 h light/dark cycle in a sound-controlled room with diffuse overhead lighting. Testing was conducted in the mornings prior to feeding. Additionally, control testing was interspersed throughout the treatment conditions to account for any time-of-day effects and other confounding variables. Fish were randomly assigned to either a control group or one of the three treatment groups. During testing, a three-sided enclosure of white corrugated plastic encircled the testing arena to minimize exposure to external stimuli. Water temperature was maintained between 26 °C and 28 °C. The arena was placed on a pad heated to 35 °C to reduce heat loss between trials. Luminance in all testing arenas was measured at ~ 32 cd/m³ (cal SPOT photometer; Cooke Corp. CA, USA). A Basler GenICam acA1300-60gc Area Scan video camera (Basler Inc., USA) was suspended approximately 1 m above testing arenas to record zebrafish behaviour. Zebrafish movement was tracked using Noldus EthoVision XT ® tracking software (v. 11.0, Noldus, Wageningen, NL) using differencing settings. Quantification of behaviour began immediately after the fish or shoal was placed in the center of the arena. The time the fish was immobile (immobility) was quantified in EthoVision with a 5% threshold^{49 (link),50 (link)}. The individual tests, novel tank dive test (n = 80) and light/dark test (n = 81), consisted of ~ 20 fish per group for each 0, 1, 5, and 15 mg/L concentration of chlordiazepoxide. For the shoaling test (n = 250), each shoal consisted of 5 fish. There were 10 trials conducted in each of the shoaling treatment groups, however, in the control group 20 trials were conducted as they were interspersed throughout the testing trials. Data were only excluded from analyses if the tracking software did not acquire data for the total time spent in arena.

Upon exposure to the atmosphere following the hydrogen explosion, CsMP experienced rapid cooling. It was assumed convective heat loss occurred only at the surface, with conductive cooling due to the heat gradient formed between the centre and surface^{36 (link)} dominating within the particle bulk. To further simplify the model, radiative heat loss at the particle surface was taken to be negligible. The radial and temporal heat profile of the particle due to just conductive heat loss was then modelled by solving the one-dimensional, spherically symmetric heat equation $$\begin{array}{c}\hfill {\rho }_p{c}_p\frac{\partial T}{\partial t}=\frac{k_p}{r^2}\frac{\partial }{\partial r}\left(r^2,\frac{\partial T}{\partial r}\right),\end{array}$$ where ${\rho }_p$ is density of the particle, $c_p$ the heat capacity of the particle, $k_p$ the thermal conductivity and r the radial coordinate. The physical particle properties (e.g. density, heat capacity, thermal conductivity) were assumed to be constant throughout the model. The convective heat loss at the particle surface was accounted for by imposing the boundary condition $$\begin{array}{c}\hfill k_p{\left(\frac{\partial T}{\partial r}\right)}_{r=R}=-q_c,\end{array}$$ where $$\begin{array}{c}\hfill q_c=h(T_{\infty }-T_s)\end{array}$$ is the convective heat flux. h is the heat transfer coefficient, $T_s$ is the surface temperature of the particle and $T_{\infty }$ the temperature of the surroundings^{23 (link)}. The heat transfer coefficient was calculated using the equation $$\begin{array}{c}\hfill h=\frac{\text{Nu}{k}_g}{2r_p},\end{array}$$ where Nu is the Nusselt number, $k_g$ the thermal conductivity of the air surrounding the particle and $r_p$ the particle radius. Calculation of the Nusselt number required the particle Reynolds number, Re, determined by $$\begin{array}{c}\hfill \text{Re}=\frac{2|(v_g-v_p)|r_p{\rho }_g}{{\eta }_g},\end{array}$$ where $v_g$ is the velocity of the surrounding air, $v_p$ the particle velocity, ${\rho }_g$ the air density and ${\eta }_g$ the air viscosity. In previous studies^{26 (link)}, the Nusselt number was calculated for a negligible internal temperature gradient by invoking the lumped capacitance approximation. This states that for Biot numbers close to zero (Bi $\to 0$ ) , the convective heat transfer to the surrounding gas limits the surface heat flux, and internal conduction is large enough to equilibrate the internal temperature gradient of the pyroclast. This contradicts our key hypothesis that radial variations in viscosity caused the unique internal texture observed in ‘Type B’ CsMP. For this reason the Nusselt number used in this study was calculated using newer data from Moitra et al.^{37 (link)}, that is not dependent on the lumped capacitance approximation: $$\begin{array}{c}\hfill \text{Nu}=a+b{\text{Re}}^{1/2}{\text{Pr}}^{1/3}\end{array}$$ with fitting parameters $a=76$ and $b=1.9$ , and Pr is the Prandtl number, taken to be 0.71 for ambient air^{37 (link)}.Figure 5

Schematic outlining the physical processes at each scale of the model. Particle-scale: the cooling of the isotropic, spherical particle from convective heat transfer to the surrounding environment and conductive cooling within the particle. Bubble-scale: The bubble growth model in which growth is limited by viscosity and halted once the temperature is lower then the glass transition temperature.