0.1 Module 2 - model: body ambient bondgraph model using heat flux  (Page 2/3)

Figure 1: Bond Graph Model of Heat Flow and Temperature of the Human Body

Where,

HF = Heat Flow

T = Temperature

Eva = Evaporation

Rad = Radiation

Con = Convection

Amb = Ambient

Sk = Skin

Bc = Body Core

For those new to Bondgraph models, the following explanation uses terminology specific to the model presented in this paper:

• Each element shows that there is a relationship between heat flow (HF) and temperature (T). To translate the above into generic bond graph terminology, heat flow is the bondgraph “flow” factor; temperature is the “effort” factor.
• The arrows illustrate the sign convention for direction of power flow. Power flow can be in either direction (either positive or negative)
• Strokes show causality. As shown in the system equations, heat flux is dependant upon temperature
• “R-Element”: The resistor is a 1-port an element in which the heat flux and temperature variables at the single port are related by a static function

Figure 2: Symbol for an R-Element

The half arrow means that the T and HF is positive; where T represents temperature, and HF, represents heat flow. The constitutive relationship is that heat flow is a function of temperature.

• “Transformer”: The bondgraphic transformer can represents the heat flux transducer used to measure heat flux, located on the ambient-to-body threshold on the skin. It conserves power and transmits the factors of power with proper scaling as defined by the transformer modulus.

In a departure from a standard Bondgraph transformer, the modulus for this project inserts an offset and slope to the modulus representative of the error in the measuring device.

Figure 3: Symbol for a Transformer

• The large nodes labeled “1” are called 1-junctions. They indicate a node where the heat flows sum to zero.
• Radiation, convection, and evaporation are heat transfer phenomenon affecting the ambient.

System equations

From the Bondgraph model, system equations may be generated using a step by step procedure. For the explanation of this model, generic heat transfer equations will be show to demonstrate the basis of usage in fundamental thermodynamic theory. Depending on the application, these equations can be replaced with more sophisticated equations based on heat transfer theory, or one could utilize empirically derived equations from field data.

Observe elements contributing to the system and write down equations looking at causalities.

For natural convection,

HF = h∆T [W/m²]

h = convective thermal heat transfer coefficient

∆T = temperature difference between surface (skin) and fluid (air)

For conduction,

HF = k∆T/x [W/m²]

k = thermal conductivity of the material [W/m·ºK]

∆T = temperature difference between surface (skin) and fluid (air) [ºC]

x = material thickness [m]

For radiation,

HF = ε(T)· σ ·T^4 [W/m²]

ε(T) = correction factor, (emissivity correction factor times radiation spectrum formula)

σ = Stefan-Boltzmann constant, 5.670400×10−8 [W·m-2·K-4]

T = temperature [K]

For evaporation, heat transfer equations are very complex, and now shown here. They are characterized by an s-shaped curve relating heat flux to surface temperature differences, which is the same reliance on temperature that holds for all equations in this model.