Thermal Network

Description

Use Flow Simulator 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 a flow network to generate combined results. 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.

Thermal Network Model Creation in the GUI

Use Flow Simulator to couple one (or more) thermal network with a flow network, or model standalone thermal networks. The Thermal Network section is placed after the Flow section in the Element Library. Moreover, all thermal network elements used in the models are listed in Thermal Model Browser.

Standalone Thermal Network Model

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

Figure 1. 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. If there exists a constant Heat input at the Internal Boundary location, this should be given as Nodal Heat input.

All entities in the Property Editor must be filled for all used resistors.

There are two subtypes of the Conductor resistor. The conductivity can either be a fixed value or be selected from the enabled materials; 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 the GUI view.

The Convector element requires two 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 two subtypes, Simple Radiation, which calculates the heat transfer on a single surface due to radiation, requires three inputs: Surface Area, Emissivity, and View Factor LR. However, when the radiation occurs between two parallel surfaces, select the Radiation Between two Surfaces subtype, 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 View Factor is Right-to-Left.

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

Figure 2. Solver > Analysis Setup > Analysis Type Selection for a Standalone Thermal Network

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

Figure 3. Thermal Network Tolerances and Damping Factors

Thermal Networks Combined with the Flow Network

There are two ways to combine a thermal network with a flow network.

ThermalNetworks Coupled with the Flow Network

For plain models, couple the thermal network to a flow network on the same GUI view.

Figure 4. 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. 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 is equivalent to the temperature of the fluid. Then, define the associated Element ID and the element’s Axial Segment.

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

Figure 5. Analysis Type Selection for Thermal Network Coupled with Flow Network Models

Thermal Networks Attached to the Flow Network

If the thermal network and flow network consist of many resistors and elements, respectively, it is also possible to attach thermal networks to a flow network through the TN Box. Use the TN box to group thermal items.

Figure 6. Thermal Network Attached to the Flow Network through the TN Box

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

Thermal Network Element Input Variables

Thermal Network Theory

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

(1)

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.
   (2)

   Where:

   k: Conductivity

   A: Surface Area

   L: Thickness

2. Radial Conductor: Heat is carried from Left Node to Right Node along a radial conductor.
   (3)

   Where:

   r₂: Outer Radius

   r₁: Inner Radius

Convector

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

(4)

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

T: Temperature

If the convector is attached to a fluid chamber, the convector uses a relative total temperature from the fluid chamber. The relative total temperature applies to models with rotating surfaces, as found in a gas turbine secondary air systems. The swirl of the fluid in the chamber and the rotational speed of the surface (set on the thermal node) are used to calculate the relative total temperature.

Figure 7.

Radiators

1. Simple Radiator: Heat is carried from Left Node to Right Node through a simple radiator.
   (5)

   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.
   (6)

   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.

(7)

Where:

Q: user inserted heat

Numerical Method

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

(8)

And we try to obtain:

(9)

Therefore, the Newton’s method can be employed as

(10)

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:

(11)

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.