## Abstract

The transient thermochromic liquid crystal (TLC) method is applied to determine the distribution of the local heat transfer coefficients using a configuration with parallel cooling channels at an engine-relevant Reynolds number. The rectangular channels with a moderate aspect ratio and a high length-to-diameter ratio are equipped with one-sided oblique ribs with high blockage, which is a promising configuration for turbine near-wall cooling applications. In this arrangement, the three inner channels should experience same flow and thermal conditions. Numerical simulations are performed to substantiate this assumption. The symmetric single channels are sprayed with narrowband TLC with various indication temperatures. Multiple experiments were conducted. All start at ambient conditions before the fluid is heated up to several temperatures between 46 °C and 73 °C. The results show that the determined local heat transfer coefficients and therefore the Nusselt numbers vary significantly for the different experimental conditions especially at locations of high heat transfer coefficient behind the ribs. A simplified procedure with respect to measurement uncertainties is applied to enable an easy and fast valuation of the data quality. This might be used within the data reduction analysis for such experiments directly. The approach is illustrated using the obtained experimental data.

## Introduction

*T*

_{ref}, which is relative to the solid initial isothermal temperature

*T*

_{0}, assuming one-dimensional conduction into the semi-infinite wall with constant solid body thermal properties and a constant heat transfer coefficient, the wall surface temperature response can be analytically determined by

With measurements of *T*_{w}(*t*) (indication temperature of the TLC at determined time), *T*_{0}, *T*_{ref} and given thermal properties of the solid wall material (*kρc*), the local heat transfer coefficient is determined. Uncertainties in the individual measures and their effects on the uncertainty on the determined heat transfer coefficient under the given assumptions are generally determined using the methods of Coleman and Steele [4] and Moffat [5]. Thereby Yan and Owen [6] showed that the uncertainties of determining *h* are minimized when Θ ≈ 0.52 assuming equal uncertainties in the temperature measurements and neglecting uncertainties in time and thermal properties. Furthermore, the variation in uncertainties is relatively small in the range 0.3 < Θ < 0.7. This analysis provides the experimenter a very well background for qualified experimental settings.

*T*

_{ref}(

*t*) is often approximated by many small ideal steps using Duhamel’s superposition principle for linear heat conduction problems

This is especially to be considered in long internal cooling channels using the transient TLC technique, where the fluid reference temperature changes locally with position and time due to the upstream heat exchange. The analysis by Yan and Owen [6] might still be used to value the experimental conditions using the fluid flow temperature at the inlet of a test section, but higher uncertainties are to be expected for very short indication times related to high heat transfer coefficients [7]. With a single type of liquid crystal and therefore a fixed wall temperature, it can be inevitable to reach areas, where the uncertainties become intolerable (e.g., Poser et al. [8]), especially for more downstream positions with slower fluid temperature transients. Different approaches can be taken to solve this issue. One idea is to execute multiple experiments with different thermal settings, meaning changing the fluid temperature or using multiple TLCs. Multiple TLCs can be applied in layers or within mixtures (e.g., Refs. [9–16]) also in combination with multiple fluid flow temperature changes. This has been used, e.g., for multiple parameter determinations such as heat transfer coefficient and adiabatic wall temperature in impingement cooling and film cooling situations [17,18]. In both cases, the experimenter receives multiple TLC indication times during the experiment, which have to be assigned to the correct kind of TLC for data evaluation assuring a clear sequence and appearance of all TLC indications at all positions to be evaluated.

Using a single TLC coating, multiple experiments with different temperature steps can be used to adopt the conditions to individual areas of the experiments. However, each test run marks a new and unique experiment, which might differ from the others by slightly changing boundary conditions that are not in the hand of the experimenter, such as the ambient temperature, pressure, and temperature-dependent fluid flow thermal properties.

Beside these experimental adaptations, the local time dependency of the fluid reference temperature needs to be considered. For specific *T*_{ref}(*t*) distributions, analytical solutions were developed for the surface wall temperature history with the given assumptions. In Ref. [1], some of them are summarized including an exponential function [19]. This was extended by Newton et al. [20] for an exponential series with time constants to be determined from local fluid flow temperature histories. Other solutions are given by von Wolfersdorf et al. [21] using a simplified model for fluid temperature histories in long internal cooling channels or by Kwak [22] fitting an *n*th order polynomial to the fluid flow temperature history.

With this, each approach requires a specific uncertainty assessment for the measured heat transfer coefficients. Owen et al. [23] extending the analysis given by Yan and Owen [6] for the exponential series showed the increasing uncertainties with larger time constants (slower fluid temperature rise) for given Θ, where *T*_{ref} is taken as the time-wise asymptotic value for the fluid temperature. They have shown that minimal relative uncertainty is still obtained in the range 0.5 < Θ < 0.6. Kwak [22] analyzed different fluid flow temperature histories and showed that especially for high heat transfer coefficients (short TLC indication times for given conditions), the uncertainty in measuring time plays an important role. Uncertainty analysis as guidance for the experimenter becomes, therefore, more demanding and specific on the modeling of, e.g., the fluid flow *T*_{ref} history as well as for the application of multiple TLC indications [24].

The current study uses a simplified equivalent step approach for the fluid temperature, meaning that after the determination of the local heat transfer coefficient with the specific fluid flow temperature history using Eq. (3), a step function in flow temperature is considered which would lead to the same heat transfer coefficient at the determined TLC indication time applying Eq. (2). Therewith a local Θ value can be assigned for each position and used in a preliminary valuation of the experimental data directly within the data analysis process, although a detailed uncertainty analysis at given specific conditions is still required. The approach is demonstrated by experiments using different single TLC in a symmetric cooling channel setup for several inlet fluid temperature settings at given flow conditions. This allows to investigate those cooling passages at one time and, therefore, at the same flow and thermal conditions.

The results reveal that specially controlled fluid flow temperature histories as suggested, e.g., by Ma et al. [25] should be considered for improved measurement accuracy, as will be discussed later.

## Experimental Approach

### Geometry of the Investigated Cooling Configuration.

The geometry of the investigated Perspex model is shown in Fig. 1. It consists of five similar individual channels separated by thin Perspex walls. With a width of 32.5 mm and a height of 13 mm, each single channel has the hydraulic diameter *d*_{h} = 18.57 mm and a moderate aspect ratio of 2.5. Each passage length is 812.5 mm which results in the length- to-diameter ratio *l*/*d*_{h} = 43.75.

On a length of 422.5 mm, the top wall of the cooling channels is featured by 13 oblique ribs each. The ribbed section is joined by a 195 mm (10.5 *d*_{h}) long smooth area on both sides. The square ribs with height *e* = 3.25 mm and pitch *P* = 32.5 mm are angled by 45 deg with an altering orientation from channel to channel. The five individual passages are supplied by a common plenum with constant 13 mm height. In this setup, the inner three channels are regarded as symmetrical.

### Experimental Setup.

Figure 2 shows a model of the test rig. Air at ambient conditions is sucked through a dust filter. The flow field is then reduced to 13 mm × 201.5 mm. This cross section remains for 260 mm before the flow is separated into the five single cooling channels. At the end of these passages, there is another 130 mm plenum with equal dimensions as before. This plenum is then connected to the vacuum pump by pipes in which the mass flow is measured.

The camera has a frame rate of 15 fps and is positioned above the Perspex model and records the inner top wall through the Perspex wall. From the five single channels of this setup, the inner three are investigated. They are sprayed with different narrowband TLC as depicted in Fig. 1. The indication temperatures are 32.2 °C, 38.1 °C, and 45.0 °C and will be later referred to as TLC32, TLC38, and TLC45, respectively. The TLC was calibrated at a steady-state calibration facility [26]. The calibrated indication temperature is the temperature value, where the green signal reached its intensity maximum. In order to increase the contrast, a thin layer of black paint is added on the top of the TLC. Furthermore, there are four lamps positioned above the Perspex model to assure an appropriate illumination.

A total of 28 thermocouples is used in the measurement section of this setup (see Fig. 1). The thermocouples are installed through the bottom wall and extend 6.5 mm into the Perspex model. All thermocouples are mounted in the centerline of each cooling channel. They were calibrated using an isothermal dry block and are read out with a sampling rate of 10 Hz.

At the beginning of each measurement, the desired mass flow is set using the vacuum pump. The measurements start when the heater with presumably set heating power is turned on. A signal current is emitted by the heater that switches on an LED and is logged by the temperature measuring instrument to synchronize the recorded video with the temperature measurements. Depending on the set heating power, the test air heats up to a certain level. In this setup, four different temperature levels were set for the time-asymptotic fluid temperature at the channel inlets. Table 1 lists the Reynolds numbers of each run and the therewith obtained dimensionless temperatures Θ_{in} as given in Eq. (2) using *T*_{ref} = *T*_{in} for the three investigated channels.

Re | T_{0} (°C) | T_{in} (°C) | Θ_{in,TLC32} | Θ_{in,TLC38} | Θ_{in,TLC45} |
---|---|---|---|---|---|

19,511 | 23.6 | 46.3 | 0.38 | 0.64 | 0.94 |

19,395 | 23.9 | 51.0 | 0.30 | 0.52 | 0.78 |

19,293 | 23.7 | 60.5 | 0.23 | 0.39 | 0.58 |

18,714 | 24.0 | 73.1 | 0.17 | 0.29 | 0.43 |

Re | T_{0} (°C) | T_{in} (°C) | Θ_{in,TLC32} | Θ_{in,TLC38} | Θ_{in,TLC45} |
---|---|---|---|---|---|

19,511 | 23.6 | 46.3 | 0.38 | 0.64 | 0.94 |

19,395 | 23.9 | 51.0 | 0.30 | 0.52 | 0.78 |

19,293 | 23.7 | 60.5 | 0.23 | 0.39 | 0.58 |

18,714 | 24.0 | 73.1 | 0.17 | 0.29 | 0.43 |

### Numerical Simulations.

In order to verify the flow conditions, numerical Reynolds-averaged Navier–Stokes (RANS) simulations were performed. The numerical domain corresponds to the Perspex model and starts right after the transition piece from the heater cross section to the 13 mm × 201.5 mm plenum with 260 mm length. As in the Perspex model, each of the single channels is featured by 13 oblique ribs and has a total length of 812.5 mm. The outlet is placed 130 mm behind the cooling channels.

The numerical domain is shown in Fig. 3. At the inlet, the fluid enters at 1 bar pressure, with the temperature *T*_{inlet} = 60 °C and medium turbulence (5%). A mass flowrate of $m\u02d9=45.5g/s$ is set at the outlet. All walls are simulated as smooth no-slip walls at isothermal conditions (*T*_{w} = 20 °C). The fluid is air as ideal gas with temperature-dependent properties, and the shear stress transport (SST) turbulence model [27] is used. Steady-state simulations were performed using ansys cfx 17.1. The simulation converged to a residual level of 10^{−4}. A mesh convergence study was performed with three differently refined block-structured grids. They have a total number of 4.3 million, 8 million, and 19 million nodes and a high near-wall resolution $(y1+<1)$. Following the procedure of Celik and Karatekin [28], the grid convergence index (GCI) was calculated and revealed only minor grid dependencies. For the bulk temperatures and pressures at multiple positions in the channel as exemplary results, the GCI of the finest grid was below $1%$. This mesh was used for the simulation with some details shown in Fig. 4.

### Flow and Temperature Conditions.

Equal flow conditions are the basis for further assessment and comparison of the three investigated channels. Table 2 lists some results of the numerical simulations at this test rig. The mass flowrate between the considered channels varies by less than $0.1%$. Larger deviations can be found in the outer channels. This stems from the boundary layer development on the upstream channel sidewalls. A temperature and velocity boundary layer is formed before the fluid enters the single channels and remains in them. In contrast to that, the boundary layer of the inner passages starts to form at the bridges between them. The differences in the mass flow between the left and right outer channels might occur from the orientation of the ribs. Still, these differences have a negligible influence on the three inner channels.

Channel | $m\u02d9(g/s)$ | T_{x/d2.6} (°C) | T_{x/d7.9} (°C) | T_{x/d16.1} (°C) |
---|---|---|---|---|

1 | 9.087 | 51.10 | 48.68 | 43.31 |

2 (left) | 9.105 | 51.79 | 49.29 | 43.75 |

3 (middle) | 9.101 | 51.80 | 49.30 | 43.75 |

4 (right) | 9.103 | 51.79 | 49.29 | 43.76 |

5 | 9.077 | 51.10 | 48.68 | 43.20 |

Channel | $m\u02d9(g/s)$ | T_{x/d2.6} (°C) | T_{x/d7.9} (°C) | T_{x/d16.1} (°C) |
---|---|---|---|---|

1 | 9.087 | 51.10 | 48.68 | 43.31 |

2 (left) | 9.105 | 51.79 | 49.29 | 43.75 |

3 (middle) | 9.101 | 51.80 | 49.30 | 43.75 |

4 (right) | 9.103 | 51.79 | 49.29 | 43.76 |

5 | 9.077 | 51.10 | 48.68 | 43.20 |

The simulated temperatures show symmetrical results as expected. While the outer passages are also influenced by the Perspex model sidewalls, the three inner channels show almost identical temperatures. In Table 2, the bulk temperatures at three downstream positions are listed. These positions correspond to the sections of the thermocouple measuring points in the experiment (see Fig. 1).

In Fig. 5, the measured temperatures of all experiments at selected positions are depicted. It can be found that the temperature decreases downstream of the cooling channel, which is obvious, since the fluid loses energy to the cooler channel walls. Another finding is that there is no ideal temperature step. The more downstream the thermocouples are located the slower the temperature increases over time. Small differences between the outer two channels and the middle one can be detected, whereby somewhat higher temperatures occur at the mid. The reason of these differences might be found in the used mesh heater, which does not create a fully uniform temperature field. However, the overall thermal behavior of all channels is similar. The envelope of the fluid temperatures along the models looks the same for all experiments with different temperature levels. To consider the small differences, the heat transfer distributions for each channel have been evaluated with its own measured temperatures.

### Data Reduction.

After an experiment, the recorded videos are all transformed to a uniform format without warping and with equidistant spacing. Bi-linear interpolation of the measured fluid temperatures leads to the temporally and spatially resolved temperature field in the test model. The same evaluation procedure is performed for all experiments, including filtering of the video signal, identification of green intensity maxima, and solving Eq. (3) iteratively in order to determine the local heat transfer coefficient. A detailed description of the evaluation technique is given by Poser and von Wolfersdorf [26].

*k*is determined at the time-wise averaged fluid temperature

*T*

_{m}until the local TLC indication. These Nusselt numbers are further scaled to a Nusselt number ratio by dividing them by the Dittus–Boelter correlation

*T*

_{m}used for the viscosity

*η*as also used for the thermal conductivity in Eq. (4). Both the Reynolds number and Nusselt number are based on the single channel.

Heat transfer coefficients, Nusselt number ratios, and further solution variables that are determined pixel by pixel are displayed either as contour plots or as linear averages parallel to the rib orientation. In each case, every pixel with distance of 1.5 mm or less from the walls or ribs is ignored in order to avoid wrong evaluations caused by reflections and blocked field of vision.

## Results and Discussion

### Variation of Fluid Temperature.

First, the calculated Nusselt number ratio of different experiments is compared in dependence of the fluid temperature. For each of the investigated three channels, its Nusselt number distribution is depicted in Fig. 6. The most obvious finding is that for some cases, there are no values available. This means that there has been no TLC detection, as the walls did not reach the TLC indication temperature within the maximum set test duration. The test duration is limited to fulfill safely the criteria for the semi-infinite wall assumption (see Schultz and Jones [29]) as well as to avoid effects due to lateral conduction in regions with high heat transfer variations, although they could be evaluated using post-processing methods (e.g., Refs. [30,31]). Therefore, in the further discussion, configurations TLC38-T46, TLC45-T46, and TLC46-T51 will not be considered. It can be found that the heat transfer coefficients and with them the Nusselt number ratios are strongly influenced by the rib turbulators. By breaking the boundary layer, the heat transfer in the area of the ribs is greatly increased and reaches values of Nu/Nu_{0} = 7 and higher.

Looking at a single channel, e.g., the middle one with TLC38, it can be concluded that the results are affected by the choice of the fluid temperature. Higher results are found for higher fluid temperatures. If the dimensionless temperature is calculated by Eq. (2) with the temperature at the first thermocouple position as reference temperature, the test run with T51 leads to Θ_{in} = 0.52 (compare Table 1). But in this case, it can be seen that no TLC indication took place in the rear area of the cooling channel within the set experimental duration.

The comparison of all three investigated channels in one experiment also shows considerable differences. The left, middle, and right channels of the T73 experiment should be compared since this is the only experiment in which all three passages entirely reached the indication temperature for the set conditions. Here, it can be seen that the Nusselt number ratio increases with decreasing TLC indication temperatures.

These differences become more obvious when comparing the line averaged Nusselt number ratios as shown in Fig. 7. It can be found that the highest differences between the cases are found in the beginning of the ribbed section. Between the fourth and fifth rib the peak value of the line averaged Nusselt number ratio Nu/Nu_{0} ranges from ≈4.7 in case of T46 to ≈26 for T73. For better illustration, the highest values are cut in Fig. 7. The influence of the fluid temperature on the results is clearly visible. In all cases, it can be seen that the Nu/Nu_{0} ratio increases with higher fluid temperature.

### Equivalent Step Temperature.

Figure 6 shows that setting the experimental parameters based solely on Θ_{in} is not sufficient for the whole passage. If the reference temperature is measured near the entrance, areas at the rear part of the test model are not described appropriately since the fluid temperature is considerably lower in that location (compare Fig. 5).

As in long internal cooling channels, the local fluid flow temperature depends on the actual heat transfer distribution upstream of the position of interest, the non-dimensional parameter Θ will vary locally as well as with time. Furthermore, with strong variations in heat transfer coefficients, it is desirable to determine the uncertainty on local pixel level obtaining thereby a distribution of measurement uncertainties. The uncertainty analysis is then to be based on the outcome of the experiment, e.g., for the local fluid flow temperature, their modeling in the data analysis and the TLC indication times reflecting the local heat transfer coefficient.

*T*

_{eq}is determined using Eq. (2) for the given TLC indication time

*t*

_{ind}, its temperature, and the heat transfer coefficient determined from Eq. (3). Therewith a local value of Θ

_{eq}is obtained using

With this, a first valuation of the data quality with respect to the Θ criterion can locally be made. The obtained distribution of Θ_{eq} is depicted in Fig. 8 as contour plots for all experiments considered for evaluation. First, the case TLC38-T51 for the middle channel will be discussed, in which the temperature level at the inlet was optimal (Θ_{in} = 0.52). Figure 8 reveals that Θ_{eq} near the channel entrance is in the range of this optimum as well, when it is calculated with the equivalent step temperature as reference. In the middle of the channel, Θ_{eq} is high which indicates that the fluid temperature was just slightly higher than the TLC indication temperature. As mentioned before, the wall at the rear part of this channel did not reach the TLC temperature within the duration of the experiment. The other two cases (T61 and T73) for the middle channel show generally values in the range 0.3 < Θ_{eq} < 0.7. The cases T46 and T51 for the left channel are similar to the cases T61 and T73 for the middle channel. For the right channel, the consideration of the Θ_{eq} distribution shows that only the T73 test should be considered for further detailed analysis. As the various experiments are evaluated with their local Θ_{eq} distribution, the above-mentioned cases are assumed to provide results with the least uncertainties.

However, in order to explain the different values for Θ_{eq} and the significance of this value for the experiment’s quality valuation, it is worth taking a closer look at the left channel with T73. This case is chosen since Θ_{eq} appears in it as both small and large values. While Θ_{eq} is low at the entrance, it reaches high values near the ribs. The reason for the high values near the rib turbulators is the high heat transfer coefficient. Here, the TLC indications already happened within the flow heating process. Compare the bottom plot of Fig. 5 where the circle symbols at *x*/*d*_{h} = 16.1 show the fluid temperature measured between the third and fourth rib of that channel. When the indication takes place just a little later but still within the steep rise of the fluid temperature, Θ_{eq} decreases. Small values of Θ_{eq} mean that the TLC indication was at a time when the fluid temperature already came close to its maximum level. This shows that there is a large impact of the time resolution.

As with the used approach of an equivalent step temperature a preliminary data quality evaluation can be made, the temporal uncertainties are still to be considered since using Eq. (2) does not value the fast TLC indication times *t*_{ind} for high heat transfer coefficients. In order to take into account the temporal uncertainties, the TLC indication time will be discussed. Figure 9 shows the distribution of *t*_{ind} as contour plots. Comparing Figs. 8 and 9 for the above-discussed case TLC32-T73 confirms the previous finding that the areas of highest heat transfer coefficients behind the ribs are associated to short indication times and high Θ_{eq} values. The obvious finding is that higher fluid temperatures reduce the indication times while they are increased with higher TLC indication temperatures. Both too short and too long indication times can cause poor results. With short experimental times, the measuring sampling rate, synchronization, and inertia of the thermocouples become increasingly relevant. On the other hand, long test times can lead to wrong results because the condition of the semi-finite wall and neglected lateral heat conduction might be violated.

### Selected Experiments.

The combination of the equivalent step temperature distribution and the TLC indication time allows a fast and easy valuation of the thermal setting for the conducted experiments, where both temperature uncertainties and temporal uncertainties are considered. The best cases determined with the described assessment will be summarized shortly. For the left channel with TLC32, the cases with T46 and T51 were found to be best since the indication time in the other cases is too short, namely below 2 s in small areas near the ribs and below 5 s in large areas overall. The middle channel should be evaluated with T61. Short indication times for T73 and poor thermal conditions expressed by high Θ_{eq} values in case of T51 disqualify them. Since a complete TLC indication took place in the right channel only for T73, it should be evaluated at this temperature.

The line averaged Nusselt number ratios parallel to the ribs are shown in Fig. 10 for the selected experiments. The results of the different cases are fairly consistent. High heat transfer takes place in the ribbed section (11.4 < *x*/*d*_{h} < 32.4), whereas the Nusselt number ratio in the smooth area is close to 1. The greatest disparities between the individual cases can still be found near the ribs, where the highest heat transfer coefficients were determined and the shortest TLC indication times occurred.

These effects on high measurement uncertainties are well known [22,32]. Especially when applying the transient TLC technique to high-speed supersonic flows (e.g., Mee et al. [33]) or blade tip heat transfer investigations for gas turbines (e.g., Kwak and Han [34]). A good discussion and analysis on these aspects is given in Ma et al. [25] proposing “ramp-heating” of the fluid flow temperature instead of “step-heating” for high heat transfer coefficients with given disadvantages of this approach for low heat transfer coefficients. In long internal cooling channels, downstream positions will experience a kind of slow “ramp-heating” type of fluid flow temperature inherently due to the given upstream heat exchange already for a “step-heating” at the inlet (see Fig. 5). A slow ramping at the inlet will then lead to even slower ramps at downstream positions. This should be considered in view of the duration of the experiment and the semi-finite wall assumption and lateral heat conduction.

With this, rather than adopting the temperature conditions for the fluid temperature “steps” or the wall by TLC selection, multiple experiments should be performed with different controlled fluid temperature histories at the inlet of the investigated cooling configuration.

With different local variations of the measured fluid temperature history, Duhamel’s principle can still be used for first data evaluation. Thereby the errors induced by this “series-of steps” model should be considered, as pointed out by Kwak [22] in the view of the applied fluid flow temperature histories. It might be used for first data reduction after an experiment at chosen conditions. The presented approach for an “equivalent step” can be used to value qualitatively the chosen conditions and suitable adaptations.

## Conclusion

Transient TLC heat transfer measurements have been performed in cooling channels having a large length-to-diameter ratio. A symmetric cooling channel test configuration applying different narrowband TLC for the individual channels at varying inlet fluid temperature settings was used. Therewith a relatively wide range of dimensionless temperature levels has been investigated and their effects on the obtained local heat transfer distributions discussed. As the fluid flow temperature changes significantly in stream-wise direction in such tests as well as TLC indication times vary strongly due to large differences in heat transfer, an attempt is made to separately rate the effects of temperature and time conditions on a local basis. A simplified approach is used based on a local equivalent fluid flow temperature step. This allows an evaluation of the applied experimental temperature settings directly in the data analysis. Additionally, the indication time is considered to address also the contribution of time uncertainty as especially important for high heat transfer coefficients. The analysis should support the identification of appropriate experimental conditions for such tests, but is not to be considered as a substitute for a detailed uncertainty analysis based on actual local conditions.

## Acknowledgment

The authors gratefully acknowledge Ansaldo Energia for permission to publish this paper. This project has received funding from the European Union’s Horizon 2020 research and innovation programme under the Grant Agreement No. 764545—TURBO-REFLEX.

## Conflict of Interest

There are no conflicts of interest.

## Data Availability Statement

The authors attest that all data for this study are included in the paper.

## Nomenclature

*c*=specific heat capacity (J/(kg K))

*e*=rib height (m)

*h*=heat transfer coefficient (W/(m

^{2}K))*j*=time-step (–)

*k*=thermal conductivity (W/(m K))

*l*=length (m)

*p*=pressure (bar)

*t*=time (s)

*u*=velocity (m/s)

*x*=position in flow direction (m)

*y*=lateral coordinate (m)

*A*=cross-sectional area (m

^{2})*N*=number of total time-steps (–)

*P*=rib pitch (m)

*T*=temperature (K)

- $m\u02d9$ =
mass flowrate (kg/s)

- $q\u02d9$ =
heat flux (W/m

^{2})*d*_{h}=hydraulic diameter (m)

- $y1+$ =
dimensionless wall distance of the first node (–)

- Nu =
Nusselt number (–)

- Pr =
Prandtl number (–)

- Re =
Reynolds number (–)

### Greek Symbols

*η*=viscosity (kg/(m s))

- Θ =
dimensionless temperature (–)

*ρ*=density (kg/m

^{3})

### Subscripts

- eq =
equivalent step

- in =
single channel entrance

- ind =
indication of TLC

- inlet =
numerical inlet

- m =
time-averaged

- ref =
reference

- w =
wall

- 0 =
initial condition