Network Representation and Component Characteristics

The flow network of an electronic cooling system is constructed by graphically representing the paths followed by the flow streams as they pass through different components of the system. There are no restrictions placed on the interconnections of the components in the network and the size of the network. Prediction of the system-wide flow and temperature distributions requires specification of the flow and heat transfer characteristics of the components used in the network model. The pressure loss in a component can be represented as a function of flow rate with the following equation:

where: K = loss coefficient
         ρ= fluid density
         Q = volumetric flow rate
         A = flow area

The loss coefficients for standard components (screens, ducts, bends, etc.) are available from handbooks.For card arrays, the loss coefficient can be determined using the Moody chart with corrections to account for the blockage effects of heat sinks and electronic components. For nonstandard components, supplier data, CFD analysis, or testing can be used to get the flow characteristics. The performance characteristics of fans and pumps are specified in terms of pressure rise as a function of the flow rate. The bulk temperatures in the different cooling streams are determined from the heat dissipated into these paths as well as mixing of the streams in different parts of the system. The average surface temperatures are determined from the surface heat transfer coefficients. Empirical correlations for the Nusselt number (dimensionless heat transfer coefficient) have the following form.

where: A  = constant
           Re = Reynolds number
           Pr = Prandtl number
           m = constant
           n = constant

Solution of the Conservation Equations

Each component in the system is represented by a combination of links and nodes. Pressure and temperature are calculated at each node while the flow rates are associated with links. The flow characteristics of each link, given by Eq. (1), constitute the momentum equations. Mass conservation is imposed at each node of the network. The forms of the discretized momentum and continuity equations are given below.

Momentum Equation for a Link

The quantities SCR and R are determined by linearizing Eq. (1). Thus, R is the slope of Dp-Q curve while SCR represents the departure of this curve from a linear variation.

Mass Conservation at a Node

Here n denotes the total number of links at that node.

The calculation of the heat loss/gain in each link in combination with the imposition of energy balance at each node determines the temperature distribution in the network.

An efficient method to solve momentum and mass conservation equations is to use the "SIMPLE" algorithm of Patankar. It involves the following steps.

1. Assume a distribution of flow rates over the links and pressures at the nodes.
2. Calculate the flow rates for the links from the momentum equation with the existing nodal pressures.
3. Construct a pressure correction equation by combining the corrected momentum and continuity equations. Solve the matrix of the pressure correction equations using a direct method and update the pressures and the flow rates.
4. Solve the discretized energy equations at all nodes using a direct solution to determine the temperatures at all nodes.
5. Repeat steps 2 to 4 till convergence.

The resulting algorithm is fast and robust.

Practical electronics cooling systems can be considered as networks of flow paths through components such screens, filters, fans, ducts, bends, heat sinks, power supplies, and card arrays.

The characteristics of these components in terms of their flow and thermal resistances can be obtained from handbooks, vender specs, or in-house testing. Their use in the FNM approach provides a fast and accurate prediction of the flow distribution and the resulting thermal performance of the system. Unlike CFD (Computational Fluid Dynamics), FNM employs overall characteristics of components instead of attempting to calculate a detailed distribution of velocity and temperature within a component.

As a result, FNM is very fast in terms of model definition, computation, and examination of results.