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HeatFluxValue

represents a thermal heat flux boundary condition for PDEs with predicate pred indicating where it applies, with model variables vars and global parameters pars.

HeatFluxValue[pred,vars,pars,lkey]

represents a thermal heat flux boundary condition with local parameters specified in pars[lkey].

Details

HeatFluxValue specifies a boundary condition for HeatTransferPDEComponent and is used as part of the modeling equation:

HeatFluxValue is typically used to model heat flow through a boundary caused by a heat source or sink outside of the domain.

A flow rate is the flow of a quantity like energy or mass per time. Flux is the flow rate through the boundary and is measured in the units of the quantity per area per time.
HeatFluxValue models the rate of thermal energy flowing through some part of the boundary with dependent variable temperature in [], independent variables in [] and time variable in [].
Stationary variables vars are vars={Θ[x₁,…,x_n],{x₁,…,x_n}}.
Time-dependent variables vars are vars={Θ[t,x₁,…,x_n],t,{x₁,…,x_n}}.
The non-conservative time dependent heat transfer model HeatTransferPDEComponent is based on a convection-diffusion model with mass density , specific heat capacity , thermal conductivity , convection velocity vector and heat source :

In the non-conservative form, HeatFluxValue with heat flux in [ $TemplateBox[{InterpretationBox[, 1], {"W", , "/", , {"m", ^, 2}}, watts per meter squared, {{(, "Watts", )}, /, {(, {"Meters", ^, 2}, )}}}, QuantityTF]$ ] or [ $TemplateBox[{InterpretationBox[, 1], {"J", , "/(", , {"m", ^, 2}, , "s", , ")"}, joules per meter squared second, {{(, "Joules", )}, /, {(, {{"Meters", ^, 2}, , "Seconds"}, )}}}, QuantityTF]$ ] and boundary unit normal models:

Model parameters pars as specified for HeatTransferPDEComponent.
The following additional model parameters pars can be given:
parameter default symbol

"BoundaryUnitNormal" Automatic

"HeatFlux" 0 , heat flux [ $TemplateBox[{InterpretationBox[, 1], {"W", , "/", , {"m", ^, 2}}, watts per meter squared, {{(, "Watts", )}, /, {(, {"Meters", ^, 2}, )}}}, QuantityTF]$ ]
All model parameters may depend on any of , and , as well as other dependent variables.
To localize model parameters, a key lkey can be specified and values from association pars[lkey] are used for model parameters.
HeatFluxValue evaluates to a NeumannValue.
The boundary predicate pred can be specified as in NeumannValue.
If the HeatFluxValue depends on parameters that are specified in the association pars as …,keyp_i…,p_iv_i,…, the parameters are replaced with .

Examples

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Basic Examples (2)

Set up a thermal heat flux boundary condition:

Wolfram Language code: HeatFluxValue[x ≥ 0, {Θ[t, x], t, {x}}, <|"HeatFlux" -> q[t, x]|>]

Model a temperature field and a thermal insulation and a thermal heat flux boundary with:

Set up the heat transfer model variables :

Wolfram Language code: vars = {Θ[t, x], t, {x}};

Set up a region :

Wolfram Language code: Ω = Line[{{0}, {1 / 5}}];

Specify heat transfer model parameters mass density , specific heat capacity and thermal conductivity :

Wolfram Language code: pars = <|"MassDensity" -> 1.2, "SpecificHeatCapacity" -> 1006.14, "ThermalConductivity" -> 0.026|>;

Specify boundary condition parameters for a heat flux of :

Wolfram Language code: pars["BC1"] = <|"HeatFlux" -> 3|>;

Specify the equation:

Wolfram Language code:

eqn = HeatTransferPDEComponent[vars, pars] == 
	HeatInsulationValue[x == 0, vars, pars] + HeatFluxValue[x == 1 / 5, vars, pars, "BC1"]

Wolfram Language code:

Inactive[Div][{{-0.026}} . Inactive[Grad][Θ[t, x], {x}], {x}] + 1207.37 Θ^(1, 0)[t, x] == NeumannValue[0, x == 0] + NeumannValue[3, x == (1/5)]

Set up initial conditions:

Wolfram Language code: ics = Θ[0, x] == 0;

Solve the PDE:

Wolfram Language code: Tfun = NDSolveValue[{eqn, ics}, Θ, {t, 0, 600}, x∈Ω];

Visualize the solution:

Wolfram Language code: Manipulate[Plot[Tfun[t, x], {x}∈Ω, ...], {{t, 140}, 0, 600, 20}, Rule[...]]

Scope (5)

Basic Uses (3)

Define model variables vars for a transient acoustic pressure field with model parameters pars and a specific boundary condition parameter:

Wolfram Language code:

vars = {Θ[t, x, y], t, {x, y}};
pars = <|"MassDensity" -> 1.2, "SpecificHeatCapacity" -> 1006.14, "ThermalConductivity" -> 0.026, "BoundaryCondition1" -> <|"HeatFlux" -> q1|>|>;
HeatFluxValue[x == 1, vars, pars, "BoundaryCondition1"]

Define model variables vars for a transient heat field with model parameters pars and multiple specific parameter boundary conditions:

Wolfram Language code:

vars = {Θ[t, x, y], t, {x, y}};
pars = <|"MassDensity" -> 1.2, "SpecificHeatCapacity" -> 1006.14, "ThermalConductivity" -> 0.026, "BoundaryCondition1" -> <|"HeatFlux" -> q1|>, "BoundaryCondition2" -> <|"HeatFlux" -> q2|>|>;

Evaluate the first boundary condition:

Wolfram Language code: HeatFluxValue[x == 0, vars, pars, "BoundaryCondition1"]

Evaluate the second boundary condition:

Wolfram Language code: HeatFluxValue[x == 1, vars, pars, "BoundaryCondition2"]

Model a temperature field and a thermal insulation and a thermal heat flux boundary with:

Set up the heat transfer model variables :

Wolfram Language code: vars = {Θ[t, x], t, {x}};

Set up a region :

Wolfram Language code: Ω = Line[{{0}, {1 / 5}}];

Specify heat transfer model parameters mass density , specific heat capacity and thermal conductivity :

Wolfram Language code: pars = <|"MassDensity" -> 1.2, "SpecificHeatCapacity" -> 1006.14, "ThermalConductivity" -> 0.026|>;

Specify boundary condition parameters for a heat flux of :

Wolfram Language code: pars["BC1"] = <|"HeatFlux" -> 3|>;

Specify the equation:

Wolfram Language code:

eqn = HeatTransferPDEComponent[vars, pars] == 
	HeatInsulationValue[x == 0, vars, pars] + HeatFluxValue[x == 1 / 5, vars, pars, "BC1"]

Set up initial conditions:

Wolfram Language code: ics = Θ[0, x] == 0;

Solve the PDE:

Wolfram Language code: Tfun = NDSolveValue[{eqn, ics}, Θ, {t, 0, 600}, x∈Ω];

Visualize the solution:

Wolfram Language code: Manipulate[Plot[Tfun[t, x], {x}∈Ω, ...], {{t, 140}, 0, 600, 20}, Rule[...]]

Time Dependent (1)

Model a temperature field and a thermal heat flux through part of the boundary with:

Set up the heat transfer model variables :

Wolfram Language code: vars = {Θ[t, x], t, {x}};

Set up a region :

Wolfram Language code: Ω = Line[{{0}, {1 / 5}}];

Specify heat transfer model parameters mass density , specific heat capacity and thermal conductivity :

Wolfram Language code: pars = <|"MassDensity" -> 1.2, "SpecificHeatCapacity" -> 1006.14, "ThermalConductivity" -> 0.026|>;

Specify a thermal heat flux of applied at the left end for the first 300 seconds:

Wolfram Language code: pars["HeatFlux"] = Piecewise[{{3, t ≤ 300}, {0, t > 300}}];

Set up initial conditions:

Wolfram Language code: ics = Θ[0, x] == 0;

Set up the equation with a thermal heat flux of applied at the left end for the first 300 seconds:

Wolfram Language code:

eqn = HeatTransferPDEComponent[vars, pars] == 
	HeatFluxValue[x == 0, vars, pars]

Solve the PDE:

Wolfram Language code: Tfun = NDSolveValue[{eqn, ics}, Θ, {t, 0, 600}, x∈Ω];

Visualize the solution:

Wolfram Language code: Manipulate[Show[Plot[Tfun[t, x], ...], Graphics[...]], {{t, 160}, 0, 600, 20}, Rule[...]]

Time-Dependent Nonlinear (1)

Model a temperature field with a nonlinear heat conductivity term with:

Set up the heat transfer model variables :

Wolfram Language code: vars = {Θ[t, x], t, {x}};

Set up a region :

Wolfram Language code: Ω = Line[{{0}, {1 / 5}}];

Specify heat transfer model parameters mass density , specific heat capacity and a nonlinear thermal conductivity :

Wolfram Language code:

pars = <|"MassDensity" -> 1.2, "SpecificHeatCapacity" -> 1006.14, "ThermalConductivity" -> {{0.026 + 10 ^ -2Θ[t, x] }}|>;

Specify a thermal heat flux of applied at the left end for the first 300 seconds:

Wolfram Language code: pars["HeatFlux"] = Piecewise[{{3, t ≤ 300}, {0, t > 300}}];

Set up initial conditions:

Wolfram Language code: ics = Θ[0, x] == 0;

Set up the equation with a thermal heat flux of applied at the left end for the first 300 seconds:

Wolfram Language code:

eqn = HeatTransferPDEComponent[vars, pars] == 
	HeatFluxValue[x == 0, vars, pars]

Solve the PDE:

Wolfram Language code: TfunNonlinear = NDSolveValue[{eqn, ics}, Θ, {t, 0, 600}, x∈Ω];

Solve a linear version of the PDE:

Wolfram Language code:

parsLinear = pars;
parsLinear["ThermalConductivity"] = 0.026;
TfunLinear = NDSolveValue[{HeatTransferPDEComponent[vars, parsLinear] == 
	HeatFluxValue[x == 0, vars, parsLinear], ics}, Θ, {t, 0, 600}, x∈Ω];

Visualize the solutions:

Wolfram Language code: Manipulate[Show[Plot[{TfunNonlinear[t, x], TfunLinear[t, x]}, ...], Graphics[...]], {{t, 200}, 0, 600, 20}, Rule[...]]

Applications (2)

Time Dependent (1)

Model a temperature field and a thermal heat flux through part of the boundary with:

Set up the heat transfer model variables :

Wolfram Language code: vars = {Θ[t, x], t, {x}};

Set up a region :

Wolfram Language code: Ω = Line[{{0}, {1 / 5}}];

Specify heat transfer model parameters mass density , specific heat capacity and thermal conductivity :

Wolfram Language code: pars = <|"MassDensity" -> 1.2, "SpecificHeatCapacity" -> 1006.14, "ThermalConductivity" -> 0.026|>;

Specify a thermal heat flux of applied at the left end for the first 300 seconds:

Wolfram Language code: pars["HeatFlux"] = Piecewise[{{3, t ≤ 300}, {0, t > 300}}];

Set up initial conditions:

Wolfram Language code: ics = Θ[0, x] == 0;

Set up the equation with a thermal heat flux of applied at the left end for the first 300 seconds:

Wolfram Language code:

eqn = HeatTransferPDEComponent[vars, pars] == 
	HeatFluxValue[x == 0, vars, pars]

Solve the PDE:

Wolfram Language code: Tfun = NDSolveValue[{eqn, ics}, Θ, {t, 0, 600}, x∈Ω];

Visualize the solution:

Wolfram Language code: Manipulate[Show[Plot[Tfun[t, x], ...], Graphics[...]], {{t, 160}, 0, 600, 20}, Rule[...]]

Time-Dependent Nonlinear (1)

Model a temperature field with a nonlinear heat conductivity term with:

Set up the heat transfer model variables :

Wolfram Language code: vars = {Θ[t, x], t, {x}};

Set up a region :

Wolfram Language code: Ω = Line[{{0}, {1 / 5}}];

Specify heat transfer model parameters mass density , specific heat capacity and a nonlinear thermal conductivity :

Wolfram Language code:

pars = <|"MassDensity" -> 1.2, "SpecificHeatCapacity" -> 1006.14, "ThermalConductivity" -> {{0.026 + 10 ^ -2Θ[t, x] }}|>;

Specify a thermal heat flux of applied at the left end for the first 300 seconds:

Wolfram Language code: pars["HeatFlux"] = Piecewise[{{3, t ≤ 300}, {0, t > 300}}];

Set up initial conditions:

Wolfram Language code: ics = Θ[0, x] == 0;

Set up the equation with a thermal heat flux of applied at the left end for the first 300 seconds:

Wolfram Language code:

eqn = HeatTransferPDEComponent[vars, pars] == 
	HeatFluxValue[x == 0, vars, pars]

Solve the PDE:

Wolfram Language code: TfunNonlinear = NDSolveValue[{eqn, ics}, Θ, {t, 0, 600}, x∈Ω];

Solve a linear version of the PDE:

Wolfram Language code:

parsLinear = pars;
parsLinear["ThermalConductivity"] = 0.026;
TfunLinear = NDSolveValue[{HeatTransferPDEComponent[vars, parsLinear] == 
	HeatFluxValue[x == 0, vars, parsLinear], ics}, Θ, {t, 0, 600}, x∈Ω];

Visualize the solutions:

Wolfram Language code: Manipulate[Show[Plot[{TfunNonlinear[t, x], TfunLinear[t, x]}, ...], Graphics[...]], {{t, 200}, 0, 600, 20}, Rule[...]]

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HeatFluxValue

Details

Examples

Basic Examples (2)

Scope (5)

Basic Uses (3)

Time Dependent (1)

Time-Dependent Nonlinear (1)

Applications (2)

Time Dependent (1)

Time-Dependent Nonlinear (1)

Text

CMS

APA

BibTeX

BibLaTeX

parameter	default	symbol
"BoundaryUnitNormal"	Automatic
"HeatFlux"	0	, heat flux [ $TemplateBox[{InterpretationBox[, 1], {"W", , "/", , {"m", ^, 2}}, watts per meter squared, {{(, "Watts", )}, /, {(, {"Meters", ^, 2}, )}}}, QuantityTF]$ ]

HeatFluxValue

Details

Examples

Basic Examples (2)

Scope (5)

Basic Uses (3)

Time Dependent (1)

Time-Dependent Nonlinear (1)

Applications (2)

Time Dependent (1)

Time-Dependent Nonlinear (1)

See Also

Tech Notes

Related Guides

History

Text

CMS

APA

BibTeX

BibLaTeX