Heat transfer models
Library of 1-dimensional heat transfer with lumped elements
These components are modeled after the Modelica.Thermal.HeatTransfer library.
This package contains components to model 1-dimensional heat transfer with lumped elements. This allows especially to model heat transfer in machines provided the parameters of the lumped elements, such as the heat capacity of a part, can be determined by measurements (due to the complex geometries and many materials used in machines, calculating the lumped element parameters from some basic analytic formulas is usually not possible).
Note, that all temperatures of this package, including initial conditions, are given in Kelvin.
HeatCapacitor
Lumped thermal element storing heat
HeatCapacitor(hp::HeatPort, C::Signal)
Arguments
hp::HeatPort
: heat port [K]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 used that here as in Modelica. You need to define the starting temperature at the top level for the HeatPort you define.
Sims/src/../lib/heat_transfer.jl:74
ThermalConductor
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]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
Sims/src/../lib/heat_transfer.jl:139
Convection
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]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
Sims/src/../lib/heat_transfer.jl:225
BodyRadiation
BodyRadiation(port_a::HeatPort, port_b::HeatPort, Gr::Signal)
Arguments
port_a::HeatPort
: heat port [K]port_b::HeatPort
: heat port [K]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)
Sims/src/../lib/heat_transfer.jl:316
ThermalCollector
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]
Sims/src/../lib/heat_transfer.jl:341
Sources
FixedTemperature
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]T::Signal
: temperature at port [K]
Sims/src/../lib/heat_transfer.jl:380
PrescribedTemperature
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]T::Signal
: temperature at port [K]
Sims/src/../lib/heat_transfer.jl:410
FixedHeatFlow
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 Q_flow is given as a parameter. The heat flows into the component to which the component FixedHeatFlow is connected, if parameter Q_flow is positive.
If parameter alpha is > 0, the heat flow is mulitplied by (1 + alpha*(port - T_ref)) in order to simulate temperature dependent losses (which are given an reference temperature T_ref).
(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) FixedHeatFlow(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]
Sims/src/../lib/heat_transfer.jl:446
PrescribedHeatFlow
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 Q_flow is given as a parameter. The heat flows into the component to which the component PrescribedHeatFlow is connected, if parameter Q_flow is positive.
If parameter alpha is > 0, the heat flow is mulitplied by (1 + alpha*(port - T_ref)) in order to simulate temperature dependent losses (which are given an reference temperature T_ref).
(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) 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]