Thermal Network

Thermal Network General Description

Thermal network entities are typically used to model;

• heat flow through solids, liquids or gases,
• heat flow from a surface to another surface via electromagnetic waves
• heat flow applied directly at a point

Flow Simulator enables to integrate thermal resistances of conduction, convection and radiation, and direct heat application either as a standalone network or as a thermal network coupled with flow network to generate combined results. Moreover, Steady-State and Transient analyses are available for both options.

The flow solver and thermal network solver are segregated: The thermal network and flow network carry out their runs separately and reach convergence independently of each other. They both must reach convergence for the run to complete successfully.

Quick Guide for Thermal Network Model Creation in the GUI

Flow Simulator allows users either to couple one (or more) Thermal Network with their Flow Network, or to model standalone Thermal Networks. The Thermal Network Section is placed after the Flow Section in the Element Library. Moreover, all the Thermal Network elements used in the models are listed in Thermal Model Browser.

A Standalone Thermal Network Model

In the standalone thermal network models, the thermal network elements are segregated from any flow elements. Simply, place boundary and internal nodes and then connect resistors (Conductor, Convector, Radiator and Heat Flow elements) to thermal nodes. The figure shows all the Thermal Nodes and Resistors connected to each other.

Figure: Standalone Thermal Network Example

“Node Temperature” must be set for all Boundary Temperature nodes. Currently, Flow Simulator does not have an initialization for the thermal network, thus “Node Temperature” as an initial guess must be defined for Internal Temperature Nodes in the Property Editor as well. And, if there exists a constant Heat input at the Internal Boundary location, this should be given as “Nodal Heat” input.

All the entities in the Property Editor should be filled for all used resistors:

There exist 2 subtypes of “Conductor” resistor. The conductivity can either be fixed value or be selected from the enabled materials, and Flow Simulator uses conductivity of that material. The geometric inputs of Cross-Section Area and Thickness for Simple Conductor and Inner – Outer Radiuses and Length of the Radial Conductor can be given as value or can automatically be obtained by selecting Tnodes or by measuring from GUI view.

The “Convector” element requires 2 inputs. If the Convector element is not associated with a Flow Element, then the only enabled HTC Relation is the Fixed HTC and its value should be given. The second input is Surface Area.

The “Radiator” resistor has 2 subtypes. Simple Radiation, which calculates the heat transfer on a single surface due to radiation, requires 3 inputs: Surface Area, Emissivity, and View Factor LR. However, when the radiation occurs between 2 parallel surfaces, “Radiation Between two Surfaces” subtype should be selected, and parameters of Surface Area and Emissivity should be given for both surfaces. The View Factor is given based on the View Factor Option. If the View Factor is from Left-to-Right then View Factor LR should be given, and View Factor RL should be filled if vice a versa.

The Heat Flow resistor must be positioned before or after a Boundary Temperature node, since, for internal heat addition, the Internal Tnode will be added with an input Q option as “Nodal Heat”. Surface Area and Heat Input should be set in the Property Editor for the Heat Flow resistor.

Figure: Solver > Analysis Setup > Analysis Type Selection for a Standalone Thermal Network

Once a Standalone Thermal Network has generated, the Analysis Type options are:

• “Steady State Thermal” for steady analysis
• “Transient Thermal” for transient analysis.

If needed, thermal network convergence tolerance and damping factors can be adjusted from Solvers Convergence and Damping Controls panels respectively.

Figure: Thermal Network Tolerances and Damping Factors

Thermal Networks Combined with the Flow Network

There are 2 possible ways to combine a thermal network with a flow network.

Thermal Networks Coupled with the Flow Network

For the plain models, the thermal network can be coupled to a flow network on the same GUI view.

Figure: Thermal Network coupled with Flow Network

In the above example, the Boundary Temperature Nodes are associated with the Tube element and each Convector Resistor is linked to one of the tube segments. And the Convector Resistor is coupled with a Momentum Chamber.

When a Boundary Temperature Node is associated with an element, the temperature of the Boundary Tnode will be equivalent to the temperature of the fluid. Then, the associated Element ID and the element’s Axial Segment should be defined.

If the Convector resistor is coupled to an element, there are 2 ways to define heat transfer. The first way is to “Use Element Heat Transfer Setting”, in which the heat transfer calculations will be done based on the element’s settings. The second way is to set it to a fixed value or select one of the enabled HTC relations.

Note: the HTC calculation methods are different for coupling with an element then coupling with a chamber.

All other resistors are constructed as stated in the previous section.

Once the model construction completed, the Analysis Type can be selected as

• Steady State Flow + Steady State Thermal
• Transient Flow + Transient Thermal
• Quasi-Steady Flow + Transient Thermal (If the flow model does not have transient components such as tanks or accumulators this option can be used)

Figure: Analysis Type Selection for Thermal Network coupled with Flow Network Models

Thermal Networks Attached to the Flow Network

If both thermal network and flow network consist of many resistors and elements, respectively, it is also possible to attach thermal networks to flow network through TN Box. The TN box allows the user to group thermal items in an orderly fashion.

Figure: Thermal Network attached to Flow Network through TN Box

The TN Box is dragged and dropped in the model area near the element to which the thermal network is attached.

In order to attach the TN Box to the Flow Element or Chamber, first hold the Shift key and select the element or chamber with Left-Click, then keep holding down the Shift key, and Double-Click on the same element or chamber. Then the TN Box will relate to the target item with a dashed line.

The next step is to create the Thermal Network. Double-click on the TN box to get inside. By dragging and dropping thermal nodes and thermal resistor elements, the Thermal Network can be formed, and the procedure explained in the previous section can be followed.

Thermal Network Elements Input Variables

Thermal Node Input Variables

Thermal Resistor Input Variables

Conductors

Convectors

Radiators

Heat Flows

Thermal Network Theory

Flow Simulator enables to model a standalone thermal network or couple thermal networks to flow network. Flow Simulator’s steady thermal solver has two assumptions:

1. Heat (Q) can be along heat resistors (branches)
2. Heat (Q) is balanced at user defined temperature nodes (TNodes)

Therefore, at a temperature node I, energy balance is written as

Flow Simulator includes four types of thermal resistors:

1. Conductors
   1. Simple Conductor
   2. Radial Conductor
2. Convectors
3. Radiators
   1. Simple Radiator
   2. Two Surface Radiation
4. Heat Flow

Thermal Resistors and Heat Flow Equations

Conductors

1. Simple Conductor: Heat is carried from Left Node to Right Node and regardless of the actual heat flow direction, (positive or negative) is carried from Left to Right.

   

   Where:

   k: Conductivity

   A: Surface Area

   L: Thickness

2. Radial Conductor: Heat is carried from Left Node to Right Node along a radial conductor.

   

   Where:

   r₂: Outer Radius

   r₁: Inner Radius

Convector

Heat flows from Left Node to Right Node along a convector.

Where:

h: Heat transfer coefficient (can either be fixed or based on a correlations). Refer Solver General theory sections for Heat Transfer Correlations.

A: Surface area

Radiators

1. Simple Radiator: Heat is carried from Left Node to Right Node through a simple radiator.

   

   Where:

   σ: Stefan-Boltzmann constant

   : Emissivity of Left Surface

   A_L: Left Surface’s Area

2. Two Surface Radiation:Heat is carried from Left Surface to Right Surface.

   

   Where:

   F_L-R: View Factor

Heat Flow

User inserted Heat is carried from Left Node to Right Node. Since the Heat Flow resistor provides a constant heat flow, the derivatives are always zero.

Where:

Q: user inserted heat

Numerical Method

At a Node I, the energy imbalance can be defined as:

And we try to obtain:

Therefore, the Newton’s method can be employed as

Jacobian Matrices

2D Jacobian Matrix for Linear Solver

The Jacobian matrix used in Newton’s method is formed by looping over all heat resistors and the residual vector is obtained for free. It’s assumed that heat is flowing from Left node to Right node. The Jacobian matrix for each type of resistor has the form:

1D Jacobian Matrix for Sparse Solver

As the 2D Jacobian matrix requires Tnodes*Tnodes memory allocation, Linear Solver becomes inadequate for larger models. Thus, 1D coefficient matrix is formed again by looping all heat resistors, and sparse matrix is obtained by using the coordinate storage format.

Thermal Network Elements Outputs

Outputs in file with “res” extension. Output units controlled by user setting in “Output Control” panel.

References

1. Chapman, A. J. (n.d.). Fundamentals of Heat Transfer. 1987: Macmillan.
2. Churchill, S. a. (1977). A Correlating Equation for Forced Convection From Gases and Liquids to a Circular Cylinder in Crossflow. J. Heat Transfer, 300-306.
3. Churchill, S. W. (1975). averaged Nusselt number for turbulent free convection around a. Int. J. Heat Mass Transfer, 1049.
4. Holman, J. P. (n.d.). Heat Transfer (8th ed.).
5. Incropera F.P., D. D. (n.d.). Fundamentals of Heat and Mass Transfer", 7th edition Eq. 8.62.
6. Incropera, F. a. (2006). Fundamentals of Heat and Mass Transfer, 6th Edition. John Wiley & Sons.
7. Kreith, F. a. (1980). Basic Heat Transfer.
8. Lapides, M. a. (1957). HEAT TRANSFER SOURCE FILE DATA. APEX 425 United States.
9. M.E.Crawford, W. a. (1976). Convective Heat and Mass Transfer.
10. R.K.Shah, M. a. (1987). Turbulent and transition flow convective heat transfer in ducts, in Handbook of Single-Phase Convective Heat Transfer Chapter 4.
11. Weise, S. (n.d.). Average turbulent Nu on short, vertical planes and cylinders, Heat Transmission, Third Edition. McAdams.