#### Introduction

Cases of type 'Simulation' and 'Standard design' can be submitted as a 'Correlation based estimation'. This means that instead of performing a CFD simulation to determine the the flow and temperature parameters, correlations will be used to estimate them. The approach is based on setting up and calculating a thermal-hydraulic network model. The results of such a network are either shown on the platform (for simulation cases) or used to perform a standard design optimization.

A thermal-hydraulic network model consists of interconnected resistances, of which the values are based on correlations. A thermal-hydraulic network model is essentially a lumped model, similar to a resistive network for an electric circuit.

The first network (the hydraulic network) respresents the fluid region(s). The thermal behavior is captured by a second, purely thermal network, representing the whole case (so both fluid and solid regions). These two networks are set up and solved sequentially to estimate the flow and temperature parameters.

This article documents how both the hydraulic and thermal networks are set up and how they are based on correlations. The layout and interpretation of the thermal network on the platform is described in this article.

#### Hydraulic network

If one or more fluid regions are present in the case in question, a hydraulic network will be set up and calculated. For each fluid region, a subnetwork is generated, for which the Bernoulli's mechanical energy equation (extended for friction) will be solved: \[\sum_{inlets}\frac{P_{in}}{\rho}+\frac{\bar{U}_{in}^2}{2}+gz_{in}-\sum_{outlets}\frac{P_{out}}{\rho}+\frac{\bar{U}_{out}^2}{2}+gz_{out}=h_{l_T}=f\frac{L}{D}\frac{\bar{U}}{2}\] With:

- \(P\): static pressure
- \(\rho\): density
- \(\bar{U}\): average velocity
- \(g\): gravitational vector
- \(z\): height
- \(h_{l_T}\): total hydraulic loss
- \(f\): friction factor of the complete fluid boundary
- \(L\): equivalent flow length of the geometry
- \(D\): equivalent diameter of the geometry

The most important result for subsequent use is the pressure drop from in- to outlet and the average flow speed in the domain. The pressure drop is of use for (purely) flow optimizations, whereas the average flow speed is an important parameter for the subsequent thermal problem.

The geometrical parameters z (height), L (length) and D (diameter) are evaluated from the geometry. The boundary conditions determine either flow or pressure (or define pressure as a function of flow in case of fans and pumps), whereas a hydraulic correlation will be used to estimate the friction factor (\(f\)).

✎ Note |
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If a case doesn't contain any fluid regions, this step of the correlation is skipped. ColdStream will immediately continue with the thermal network. |

#### Thermal network

A thermal network is set up for all the fluid and solid regions present in the case. All regions will have to be interconnected in one way or another.

If a case comprises only on solid regions, i.e. a purely conduction case, this thermal network is generated immediately. If a case also comprises fluid regions, it is a conjugate heat transfer problem and the hydraulic network will be solved first.

Generally, the energy balance will be solved for each region: \[\sum\dot{Q}_i=\sum_i\frac{T_{region}-T_{boundary,i}}{R_{th,i}}=\sum\dot{Q}_{source,region}\] Where:

- \(\dot{Q}_i\): the heat transfer from a region to a boundary
- \(T_{region}\): the mean temperature of a region
- \(T_{boundary}\): the mean temperature of a boundary
- \(R_{th}\): the thermal resistance between the boundary and the region
- \(\dot{Q}_{source}\): the internal energy source of a region (if present)

The energy balance for each of the regions is coupled through the fluxes crossing the interfaces from one region to another. Both regions and boundaries are represented in the network by a node. Each node has one temperature, hence the approach being a 1D approach. The temperature of a boundary or region can be interpreted as an average temperature in the center of a boundary/region.

Whereas \(T\) and \(\dot{Q}\) are either given or calculated, the thermal resistance will be estimated based on correlations. One can distinguish three types of thermal resistances:

- Convective thermal resistance: the thermal resistance between a solid-fluid interface and a fluid region node, being: \[R_{th}=\frac{1}{hA}\]Where:
- \(h\) represents the convective heat transfer coefficient, determined by a correlation
- \(A\) represents the heat transfer area, or the area of the interface

- Conductive thermal resistance: the thermal resistance between a boundary and a solid region node, being: \[R_{th}=\frac{L}{\kappa A}\]Where:
- \(L\)represents the length of the thermal path
- \(\kappa\) represents the thermal conductivity of the material
- \(A\) represents the heat transfer area, or the area of the interface

- Radiative thermal resistance: the thermal resistance due to radiation, often in paralle with a conductive/convective resistance, being: \[R_{th}=\frac{1}{\epsilon\sigma A (T_{region}-T_{boundary,i})^3}\]Where:
- \(\epsilon\) represents the emissivity
- \(\sigma\) represents the Boltzmann constant
- \(A\) represents the heat transfer area, or the area of the interface

These three types of thermal resistances constitute the thermal network, which is then solved for the heat flux (\(\dot{Q}\)) and the temperature (\(T\)).

#### Combined network

The network that is displayed on the platform comprises the resulting information from both the hydraulic and the thermal network. For interpretation of the thermal network, please refer to this article.

In the case of a standard design, the estimation is used to optimize the performance of the design with the given set of targets. The network is internally updated every design iteration, whereas the network shown on ColdStream is the one of the final design.