Conductive Heat Transfer

Heat transfer by conduction

Libraries:
Simscape / Foundation Library / Thermal / Thermal Elements

Description

The Conductive Heat Transfer block represents heat transfer by conduction between two layers of the same material. For a flat surface, the Fourier law describes the transfer,

$Q = k \cdot \frac{A}{D} (T_{A} - T_{B}),$

where:

Q is the heat flow.
k is the thermal conductivity of the material.
A is the area normal to the heat flow direction.
D is the distance between layers, that is, the thickness of material.
T_A is the temperature of layer A.
T_B is the temperature of layer B.

Heat conduction through a round pipe wall is

$Q_{c y l} = 2 π k \cdot \frac{L}{\ln (\frac{d_{o u t}}{d_{i n}})} (T_{A} - T_{B}),$

where:

L is the pipe length.
d_in is the inner diameter.
d_out is the outer diameter.

You can specify the thermal conductivity by using the Conductivity type parameter:

Constant — Thermal conductivity remains constant during simulation. You specify the thermal conductivity by using the Thermal conductivity parameter.
Variable input — You specify the thermal conductivity using the input physical signal at port K, which can vary during simulation. The Minimum thermal conductivity parameter specifies the lower bound for the physical signal.
Tabulated data — You specify the thermal conductivity by using a lookup table based on temperature. In this case, thermal conductivity can also vary during simulation.

The Tabulated data option uses the average temperature of the block to find the thermal conductivity. For planar wall geometry, the average temperature is

$T_{a v g} = \frac{T_{A} + T_{B}}{2} .$

For cylindrical wall geometry, the average temperature is

$T_{a v g} = (T_{A} - T_{B}) \cdot (\frac{d_{o u t}}{d_{o u t} - d_{i n}} - \frac{1}{\ln (\frac{d_{o u t}}{d_{i n}})}) + T_{B},$

which assumes that T_B is at the inside of the cylinder.

A and B are thermal conserving ports associated with the material layers. Because the block positive direction is from port A to port B, the heat flow is positive if it flows from A to B.

Variables

To set the priority and initial target values for the block variables prior to simulation, use the Initial Targets section in the block dialog box or Property Inspector. For more information, see Set Priority and Initial Target for Block Variables.

Nominal values provide a way to specify the expected magnitude of a variable in a model. Using system scaling based on nominal values increases the simulation robustness. Nominal values can come from different sources, one of which is the Nominal Values section in the block dialog box or Property Inspector. For more information, see Modify Nominal Values for a Block Variable.

Examples

House Heating System

Model a simple house heating system. The model contains a heater, thermostat, and a house structure with four parts: inside air, house walls, windows, and roof.

Open Model

Lithium-Ion Battery Pack with Fault

Simulate a battery pack consisting of multiple series-connected cells in an efficient manner. It also shows how a fault can be introduced into one of the cells to see the impact on battery performance and cell temperatures. For efficiency, identical series-connected cells are not just simply modeled by connecting cell models in series. Instead a single cell is used, and the terminal voltage scaled up by the number of cells. The fault is represented by changing the parameters for the Cell 10 Fault subsystem, reducing both capacity and open-circuit voltage, and increasing the resistance values.

Open Model

Optimal Pipeline Geometry for Heated Oil Transportation

Model shows the opposing effects of viscous warming and conductive cooling on the temperature of a buried pipeline segment for heated oil transportation. A requirement for these systems is for the crude oil to stay well above the pour point to provide a continuous steady flow. The system is modeled using the Thermal Liquid Foundation Library. Crude oil thermodynamic properties are stored in workspace matrices and are used to populate the Thermal Liquid Settings block mask parameters.

Open Model

Two-Phase Fluid Refrigeration

Models a vapor-compression refrigeration cycle using two-phase fluid components. The compressor drives the R-134a refrigerant through a condenser, an expansion valve, and an evaporator. The hot gas leaving the compressor condenses in the condenser via heat transfer to the environment. The pressure drops as the refrigerant passes through the expansion valve. The drop in pressure lowers the saturation temperature of the refrigerant. This enables it to boil in the evaporator as it absorbs heat from the refrigerator compartment. The refrigerant then returns to the compressor to repeat the cycle. The controller turns the compressor on and off to maintain the refrigerator compartment temperature within a band around the desired temperature.

Open Model

Ports

Input

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K — Thermal conductivity control signal, W/(K*m)
physical signal

Input physical signal that controls the thermal conductivity. The signal saturates when the value is outside the minimum limit specified by the Minimum thermal conductivity parameter.

Dependencies

To enable this port, set the Conductivity type parameter to Variable input.

Conserving

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A — Layer A
thermal

Thermal conserving port associated with layer A.

B — Layer B
thermal

Thermal conserving port associated with layer B. For cylindrical wall geometry, layer B is the inside layer.

Parameters

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Conductivity type — How to specify thermal conductivity
`Constant` (default) | `Variable input` | `Tabulated data`

Whether to specify thermal conductivity as:

Constant — Thermal conductivity remains constant during simulation. You specify the thermal conductivity by using the Thermal conductivity parameter.
Variable input — You specify thermal conductivity using the input physical signal at port K, which can vary during simulation. The Minimum thermal conductivity parameter specifies the lower bound for the physical signal.
Tabulated data — You specify thermal conductivity by using a lookup table based on temperature. In this case, thermal conductivity can also vary during simulation.

Wall geometry — Wall shape for heat conduction
`Planar` (default) | `Cylindrical`

Wall shape for heat conduction, specified as:

Planar — Heat transfer is through a flat, rectangular wall. Specify the wall area and thickness.
Cylindrical — Heat transfer is through a round pipe wall. Specify the pipe inner diameter, outer diameter, and length.

Area — Area of heat transfer
`1e-4 m^2` (default) | positive scalar

Area of heat transfer normal to the heat flow direction.

Dependencies

To enable this parameter, set Wall geometry to Planar.

Thickness — Thickness of material
`0.1 m` (default) | positive scalar

Thickness of material, that is, the distance between layers.

Dependencies

To enable this parameter, set Wall geometry to Planar.

Inner diameter — Inner diameter of pipe
`0.05 m` (default) | positive scalar

Inner diameter of the pipe, that is, the diameter of the inner layer of material.

Dependencies

To enable this parameter, set Wall geometry to Cylindrical.

Outer diameter — Outer diameter of pipe
`0.1 m` (default) | positive scalar

Outer diameter of the pipe, that is, the diameter of the outer layer of material.

Dependencies

To enable this parameter, set Wall geometry to Cylindrical.

Length — Pipe length
`1 m` (default) | positive scalar

Length of the pipe.

Dependencies

To enable this parameter, set Wall geometry to Cylindrical.

Thermal conductivity — Thermal conductivity of the material
`401 W/(K*m)` (default) | positive scalar

Thermal conductivity of the material.

Dependencies

To enable this parameter, set Conductivity type to Constant.

Minimum thermal conductivity — Lower bound for thermal conductivity
`10 W/(K*m)` (default) | positive scalar

Lower bound for the thermal conductivity value. The input signal at port K saturates at this value to prevent the thermal conductivity from further decreasing.

Dependencies

To enable this parameter, set Conductivity type to Variable input.

Thermal conductivity vector — Vector of thermal conductivity values for use in table lookup
`[413, 401, 392, 383, 371, 357, 342] W/(K*m)` (default) | vector of positive values

Vector of thermal conductivity values that correspond to the Temperature vector parameter values. The vector size must be the same as the Temperature vector parameter. The block performs one-dimensional table lookup of thermal conductivity by using the average temperature of the block, linear interpolation, and nearest extrapolation. For more information on the interpolation and extrapolation algorithms, see PS Lookup Table (1D).

Dependencies

To enable this parameter, set Conductivity type to Tabulated data.

Temperature vector — Vector of temperature values for use in table lookup
`[200, 273, 400, 600, 800, 1000, 1200] K` (default) | strictly increasing vector

Vector of temperature values that correspond to the Thermal conductivity vector parameter values. The vector must be strictly increasing. The values can be nonuniformly spaced. The vector must contain at least two values.

Dependencies

To enable this parameter, set Conductivity type to Tabulated data.

Extended Capabilities

C/C++ Code Generation
Generate C and C++ code using Simulink® Coder™.

Version History

Introduced in R2007b

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R2022b: Model variable thermal conductivity and heat transfer through a round pipe wall

The new Conductivity type parameter lets the thermal conductivity be either constant, which you specify by the Thermal conductivity parameter, or variable. You can specify variable thermal conductivity either as a physical signal at port K or as a lookup table based on temperature.

Additionally, you can set the new Wall geometry parameter to either Planar or Cylindrical. Planar refers to a flat, rectangular wall. Cylindrical refers to a round pipe wall.

In the default configuration, with Conductivity type set to Constant and Wall geometry set to Planar, the block functions as in previous releases.

Conductive Heat Transfer

Description

Variables

Examples

House Heating System

Lithium-Ion Battery Pack with Fault

Optimal Pipeline Geometry for Heated Oil Transportation

Two-Phase Fluid Refrigeration

Ports

Input

K — Thermal conductivity control signal, W/(K*m) physical signal

Dependencies

Conserving

A — Layer A thermal

B — Layer B thermal

Parameters

Conductivity type — How to specify thermal conductivity Constant (default) | Variable input | Tabulated data

Wall geometry — Wall shape for heat conduction Planar (default) | Cylindrical

Area — Area of heat transfer 1e-4 m^2 (default) | positive scalar

Dependencies

Thickness — Thickness of material 0.1 m (default) | positive scalar

Dependencies

Inner diameter — Inner diameter of pipe 0.05 m (default) | positive scalar

Dependencies

Outer diameter — Outer diameter of pipe 0.1 m (default) | positive scalar

Dependencies

Length — Pipe length 1 m (default) | positive scalar

Dependencies

Thermal conductivity — Thermal conductivity of the material 401 W/(K*m) (default) | positive scalar

Dependencies

Minimum thermal conductivity — Lower bound for thermal conductivity 10 W/(K*m) (default) | positive scalar

Dependencies

Thermal conductivity vector — Vector of thermal conductivity values for use in table lookup [413, 401, 392, 383, 371, 357, 342] W/(K*m) (default) | vector of positive values

Dependencies

Temperature vector — Vector of temperature values for use in table lookup [200, 273, 400, 600, 800, 1000, 1200] K (default) | strictly increasing vector

Dependencies

Extended Capabilities

C/C++ Code Generation Generate C and C++ code using Simulink® Coder™.

Version History

R2022b: Model variable thermal conductivity and heat transfer through a round pipe wall

See Also

K — Thermal conductivity control signal, W/(K*m)
physical signal

A — Layer A
thermal

B — Layer B
thermal

Conductivity type — How to specify thermal conductivity
`Constant` (default) | `Variable input` | `Tabulated data`

Wall geometry — Wall shape for heat conduction
`Planar` (default) | `Cylindrical`

Area — Area of heat transfer
`1e-4 m^2` (default) | positive scalar

Thickness — Thickness of material
`0.1 m` (default) | positive scalar

Inner diameter — Inner diameter of pipe
`0.05 m` (default) | positive scalar

Outer diameter — Outer diameter of pipe
`0.1 m` (default) | positive scalar

Length — Pipe length
`1 m` (default) | positive scalar

Thermal conductivity — Thermal conductivity of the material
`401 W/(K*m)` (default) | positive scalar

Minimum thermal conductivity — Lower bound for thermal conductivity
`10 W/(K*m)` (default) | positive scalar

Thermal conductivity vector — Vector of thermal conductivity values for use in table lookup
`[413, 401, 392, 383, 371, 357, 342] W/(K*m)` (default) | vector of positive values

Temperature vector — Vector of temperature values for use in table lookup
`[200, 273, 400, 600, 800, 1000, 1200] K` (default) | strictly increasing vector

C/C++ Code Generation
Generate C and C++ code using Simulink® Coder™.