Heat transfer models

Lumped thermal element storing heat

HeatCapacitor(hp::HeatPort; C::Signal)

Arguments

• hp::HeatPort : heat port [K]

Keyword/Optional Arguments

• C::Signal : heat capacity of the element [J/K]

Details

This is a generic model for the heat capacity of a material. No specific geometry is assumed beyond a total volume with uniform temperature for the entire volume. Furthermore, it is assumed that the heat capacity is constant (indepedent of temperature).

This component may be used for complicated geometries where the heat capacity C is determined my measurements. If the component consists mainly of one type of material, the mass m of the component may be measured or calculated and multiplied with the specific heat capacity cp of the component material to compute C:

   C = cp*m.
   Typical values for cp at 20 degC in J/(kg.K):
      aluminium   896
      concrete    840
      copper      383
      iron        452
      silver      235
      steel       420 ... 500 (V2A)
      wood       2500

NOTE: The Modelica Standard Library has an argument Tstart for the starting temperature [K]. You really can't use that here as in Modelica. You need to define the starting temperature at the top level for the HeatPort you define.

Lumped thermal element transporting heat without storing it

ThermalConductor(port_a::HeatPort, port_b::HeatPort; G::Signal)

Arguments

• port_a::HeatPort : heat port [K]
• port_b::HeatPort : heat port [K]

Keyword/Optional Arguments

• G::Signal : Constant thermal conductance of material [W/K]

Details

This is a model for transport of heat without storing it. It may be used for complicated geometries where the thermal conductance G (= inverse of thermal resistance) is determined by measurements and is assumed to be constant over the range of operations. If the component consists mainly of one type of material and a regular geometry, it may be calculated, e.g., with one of the following equations:

Conductance for a box geometry under the assumption that heat flows along the box length:

        G = k*A/L
        k: Thermal conductivity (material constant)
        A: Area of box
        L: Length of box

Conductance for a cylindrical geometry under the assumption that heat flows from the inside to the outside radius of the cylinder:

    G = 2*pi*k*L/log(r_out/r_in)
    pi   : Modelica.Constants.pi
    k    : Thermal conductivity (material constant)
    L    : Length of cylinder
    log  : Modelica.Math.log;
    r_out: Outer radius of cylinder
    r_in : Inner radius of cylinder

Typical values for k at 20 degC in W/(m.K):

      aluminium   220
      concrete      1
      copper      384
      iron         74
      silver      407
      steel        45 .. 15 (V2A)
      wood         0.1 ... 0.2

Lumped thermal element for heat convection

Convection(port_a::HeatPort, port_b::HeatPort; Gc::Signal)

Arguments

• port_a::HeatPort : heat port [K]
• port_b::HeatPort : heat port [K]

Keyword/Optional Arguments

• Gc::Signal : convective thermal conductance [W/K]

Details

This is a model of linear heat convection, e.g., the heat transfer between a plate and the surrounding air. It may be used for complicated solid geometries and fluid flow over the solid by determining the convective thermal conductance Gc by measurements. The basic constitutive equation for convection is

   Q_flow = Gc*(solidT - fluidT)
   Q_flow: Heat flow rate from connector 'solid' (e.g. a plate)
      to connector 'fluid' (e.g. the surrounding air)

Gc is an input signal to the component, since Gc is nearly never constant in practice. For example, Gc may be a function of the speed of a cooling fan. For simple situations, Gc may be calculated according to

   Gc = A*h
   A: Convection area (e.g. perimeter*length of a box)
   h: Heat transfer coefficient

where the heat transfer coefficient h is calculated from properties of the fluid flowing over the solid. Examples:

Machines cooled by air (empirical, very rough approximation according to R. Fischer: Elektrische Maschinen, 10th edition, Hanser-Verlag 1999, p. 378):

    h = 7.8*v^0.78 [W/(m2.K)] (forced convection)
      = 12         [W/(m2.K)] (free convection)
    where
      v: Air velocity in [m/s]

Laminar flow with constant velocity of a fluid along a flat plate where the heat flow rate from the plate to the fluid (= solid.Q_flow) is kept constant (according to J.P.Holman: Heat Transfer, 8th edition, McGraw-Hill, 1997, p.270):

   h  = Nu*k/x;
   Nu = 0.453*Re^(1/2)*Pr^(1/3);
   where
      h  : Heat transfer coefficient
      Nu : = h*x/k       (Nusselt number)
      Re : = v*x*rho/mue (Reynolds number)
      Pr : = cp*mue/k    (Prandtl number)
      v  : Absolute velocity of fluid
      x  : distance from leading edge of flat plate
      rho: density of fluid (material constant
      mue: dynamic viscosity of fluid (material constant)
      cp : specific heat capacity of fluid (material constant)
      k  : thermal conductivity of fluid (material constant)
   and the equation for h holds, provided
      Re < 5e5 and 0.6 < Pr < 50

BodyRadiation(port_a::HeatPort, port_b::HeatPort; Gr::Signal)

Arguments

• port_a::HeatPort : heat port [K]
• port_b::HeatPort : heat port [K]

Keyword/Optional Arguments

• Gr::Signal : net radiation conductance between two surfaces [m2]

Details

This is a model describing the thermal radiation, i.e., electromagnetic radiation emitted between two bodies as a result of their temperatures. The following constitutive equation is used:

    Q_flow = Gr*sigma*(port_a^4 - port_b.4)

where Gr is the radiation conductance and sigma is the Stefan-Boltzmann constant. Gr may be determined by measurements and is assumed to be constant over the range of operations.

For simple cases, Gr may be analytically computed. The analytical equations use epsilon, the emission value of a body which is in the range 0..1. Epsilon=1, if the body absorbs all radiation (= black body). Epsilon=0, if the body reflects all radiation and does not absorb any.

   Typical values for epsilon:
   aluminium, polished    0.04
   copper, polished       0.04
   gold, polished         0.02
   paper                  0.09
   rubber                 0.95
   silver, polished       0.02
   wood                   0.85..0.9

Analytical Equations for Gr

Small convex object in large enclosure (e.g., a hot machine in a room):

    Gr = e*A
    where
       e: Emission value of object (0..1)
       A: Surface area of object where radiation
          heat transfer takes place

Two parallel plates:

    Gr = A/(1/e1 + 1/e2 - 1)
    where
       e1: Emission value of plate1 (0..1)
       e2: Emission value of plate2 (0..1)
       A : Area of plate1 (= area of plate2)

Two long cylinders in each other, where radiation takes place from the inner to the outer cylinder):

    Gr = 2*pi*r1*L/(1/e1 + (1/e2 - 1)*(r1/r2))
    where
       pi: = Modelica.Constants.pi
       r1: Radius of inner cylinder
       r2: Radius of outer cylinder
       L : Length of the two cylinders
       e1: Emission value of inner cylinder (0..1)
       e2: Emission value of outer cylinder (0..1)

This is a model to collect the heat flows from m heatports to one single heatport.

ThermalCollector(port_a::HeatPort, port_b::HeatPort)

Arguments

• port_a::HeatPort : heat port [K]
• port_b::HeatPort : heat port [K]

Fixed temperature boundary condition in Kelvin

This model defines a fixed temperature T at its port in Kelvin, i.e., it defines a fixed temperature as a boundary condition.

(Note that despite the name, the temperature can be fixed or variable. FixedTemperature and PrescribedTemperature are identical; naming is for Modelica compatibility.)

FixedTemperature(port::HeatPort; T::Signal)

Arguments

• port::HeatPort : heat port [K]

Keyword/Optional Arguments

• T::Signal : temperature at port [K]

Variable temperature boundary condition in Kelvin

This model represents a variable temperature boundary condition. The temperature in [K] is given as input signal T to the model. The effect is that an instance of this model acts as an infinite reservoir able to absorb or generate as much energy as required to keep the temperature at the specified value.

(Note that despite the name, the temperature can be fixed or variable. FixedTemperature and PrescribedTemperature are identical; naming is for Modelica compatibility.)

PrescribedTemperature(port::HeatPort; T::Signal)

Arguments

• port::HeatPort : heat port [K]

Keyword/Optional Arguments

• T::Signal : temperature at port [K]

Fixed heat flow boundary condition

This model allows a specified amount of heat flow rate to be "injected" into a thermal system at a given port. The constant amount of heat flow rate Qflow is given as a parameter. The heat flows into the component to which the component FixedHeatFlow is connected, if parameter Qflow is positive.

If parameter alpha is > 0, the heat flow is mulitplied by (1 + alpha*(port - Tref)) in order to simulate temperature dependent losses (which are given an reference temperature Tref).

(Note that despite the name, the heat flow can be fixed or variable.)

FixedHeatFlow(port::HeatPort; Q_flow::Signal, T_ref::Signal = 293.15, alpha::Signal = 0.0)

Arguments

• port::HeatPort : heat port [K]

Keyword/Optional Arguments

• Q_flow::Signal : heat flow [W]

Keyword/Optional Arguments

• T_ref::Signal : reference temperature [K]
• alpha::Signal : temperature coefficient of heat flow rate [1/K]

Prescribed heat flow boundary condition

This model allows a specified amount of heat flow rate to be "injected" into a thermal system at a given port. The constant amount of heat flow rate Qflow is given as a parameter. The heat flows into the component to which the component PrescribedHeatFlow is connected, if parameter Qflow is positive.

If parameter alpha is > 0, the heat flow is mulitplied by (1 + alpha*(port - Tref)) in order to simulate temperature dependent losses (which are given an reference temperature Tref).

(Note that despite the name, the heat flow can be fixed or variable.)

PrescribedHeatFlow(port::HeatPort, Q_flow::Signal; T_ref::Signal = 293.15, alpha::Signal = 0.0)

Arguments

• port::HeatPort : heat port [K]
• Q_flow::Signal : heat flow [W]

Keyword/Optional Arguments

• T_ref::Signal : reference temperature [K]
• alpha::Signal : temperature coefficient of heat flow rate [1/K]