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The lumped-element model (also called lumped-parameter model, or lumped-component model) simplifies the description of the behaviour of spatially distributed physical systems, such as electrical circuits, into a
topology
In mathematics, topology (from the Greek language, Greek words , and ) is concerned with the properties of a mathematical object, geometric object that are preserved under Continuous function, continuous Deformation theory, deformations, such ...
consisting of discrete entities that approximate the behaviour of the distributed system under certain assumptions. It is useful in
electrical systems
Electricity is the set of physics, physical Phenomenon, phenomena associated with the presence and motion of matter that has a property of electric charge. Electricity is related to magnetism, both being part of the phenomenon of electromagne ...
(including
electronics
The field of electronics is a branch of physics and electrical engineering that deals with the emission, behaviour and effects of electrons using electronic devices. Electronics uses active devices to control electron flow by amplification ...
), mechanical
multibody system
Multibody system is the study of the dynamic behavior of interconnected rigid or flexible bodies, each of which may undergo large translational and rotational displacements.
Introduction
The systematic treatment of the dynamic behavior of inte ...
s,
heat transfer
Heat transfer is a discipline of thermal engineering that concerns the generation, use, conversion, and exchange of thermal energy (heat) between physical systems. Heat transfer is classified into various mechanisms, such as thermal conduction, ...
,
acoustics
Acoustics is a branch of physics that deals with the study of mechanical waves in gases, liquids, and solids including topics such as vibration, sound, ultrasound and infrasound. A scientist who works in the field of acoustics is an acoustician ...
, etc. This may be contrasted to
distributed parameter system
In control theory, a distributed-parameter system (as opposed to a lumped-parameter system) is a system whose state space is infinite-dimensional. Such systems are therefore also known as infinite-dimensional systems. Typical examples are system ...
s or models in which the behaviour is distributed spatially and cannot be considered as localized into discrete entities.
Mathematically speaking, the simplification reduces the
state space
A state space is the set of all possible configurations of a system. It is a useful abstraction for reasoning about the behavior of a given system and is widely used in the fields of artificial intelligence and game theory.
For instance, the toy ...
of the system to a
finite
Finite is the opposite of infinite. It may refer to:
* Finite number (disambiguation)
* Finite set, a set whose cardinality (number of elements) is some natural number
* Finite verb, a verb form that has a subject, usually being inflected or marked ...
dimension
In physics and mathematics, the dimension of a Space (mathematics), mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any Point (geometry), point within it. Thus, a Line (geometry), lin ...
, and the
partial differential equation
In mathematics, a partial differential equation (PDE) is an equation which imposes relations between the various partial derivatives of a Multivariable calculus, multivariable function.
The function is often thought of as an "unknown" to be sol ...
s (PDEs) of the continuous (infinite-dimensional) time and space model of the physical system into
ordinary differential equation
In mathematics, an ordinary differential equation (ODE) is a differential equation whose unknown(s) consists of one (or more) function(s) of one variable and involves the derivatives of those functions. The term ''ordinary'' is used in contrast w ...
s (ODEs) with a finite number of parameters.
Electrical systems
Lumped-matter discipline
The lumped-matter discipline is a set of imposed assumptions in
electrical engineering
Electrical engineering is an engineering discipline concerned with the study, design, and application of equipment, devices, and systems which use electricity, electronics, and electromagnetism. It emerged as an identifiable occupation in the l ...
that provides the foundation for lumped-circuit abstraction used in
network analysis Network analysis can refer to:
* Network theory, the analysis of relations through mathematical graphs
** Social network analysis, network theory applied to social relations
* Network analysis (electrical circuits)
See also
*Network planning and ...
. The self-imposed constraints are:
# The change of the magnetic flux in time outside a conductor is zero.
# The change of the charge in time inside conducting elements is zero.
# Signal timescales of interest are much larger than propagation delay of
electromagnetic waves
In physics, electromagnetic radiation (EMR) consists of waves of the electromagnetic (EM) field, which propagate through space and carry momentum and electromagnetic radiant energy. It includes radio waves, microwaves, infrared, (visible) lig ...
across the lumped element.
The first two assumptions result in
Kirchhoff's circuit laws
Kirchhoff's circuit laws are two equalities that deal with the current and potential difference (commonly known as voltage) in the lumped element model of electrical circuits. They were first described in 1845 by German physicist Gustav Kirchhof ...
when applied to
Maxwell's equations
Maxwell's equations, or Maxwell–Heaviside equations, are a set of coupled partial differential equations that, together with the Lorentz force law, form the foundation of classical electromagnetism, classical optics, and electric circuits.
...
and are only applicable when the circuit is in
steady state
In systems theory, a system or a Process theory, process is in a steady state if the variables (called state variables) which define the behavior of the system or the process are unchanging in time. In continuous time, this means that for those p ...
. The third assumption is the basis of the lumped-element model used in
network analysis Network analysis can refer to:
* Network theory, the analysis of relations through mathematical graphs
** Social network analysis, network theory applied to social relations
* Network analysis (electrical circuits)
See also
*Network planning and ...
. Less severe assumptions result in the
distributed-element model
: ''This article is an example from the domain of electrical systems, which is a special case of the more general distributed-parameter systems.''
In electrical engineering, the distributed-element model or transmission-line model of electrical ...
, while still not requiring the direct application of the full Maxwell equations.
Lumped-element model
The lumped-element model of electronic
circuits makes the simplifying assumption that the attributes of the circuit,
resistance,
capacitance
Capacitance is the capability of a material object or device to store electric charge. It is measured by the change in charge in response to a difference in electric potential, expressed as the ratio of those quantities. Commonly recognized are ...
,
inductance
Inductance is the tendency of an electrical conductor to oppose a change in the electric current flowing through it. The flow of electric current creates a magnetic field around the conductor. The field strength depends on the magnitude of the ...
, and
gain
Gain or GAIN may refer to:
Science and technology
* Gain (electronics), an electronics and signal processing term
* Antenna gain
* Gain (laser), the amplification involved in laser emission
* Gain (projection screens)
* Information gain in de ...
, are concentrated into idealized
electrical component
An electronic component is any basic discrete device or physical entity in an electronic system used to affect electrons or their associated fields. Electronic components are mostly industrial products, available in a singular form and are not ...
s;
resistor
A resistor is a passive two-terminal electrical component that implements electrical resistance as a circuit element. In electronic circuits, resistors are used to reduce current flow, adjust signal levels, to divide voltages, bias active el ...
s,
capacitor
A capacitor is a device that stores electrical energy in an electric field by virtue of accumulating electric charges on two close surfaces insulated from each other. It is a passive electronic component with two terminals.
The effect of ...
s, and
inductor
An inductor, also called a coil, choke, or reactor, is a passive two-terminal electrical component that stores energy in a magnetic field when electric current flows through it. An inductor typically consists of an insulated wire wound into a c ...
s, etc. joined by a network of perfectly
conducting
Conducting is the art of directing a musical performance, such as an orchestral or choral concert. It has been defined as "the art of directing the simultaneous performance of several players or singers by the use of gesture." The primary duti ...
wires.
The lumped-element model is valid whenever
, where
denotes the circuit's characteristic length, and
denotes the circuit's operating
wavelength
In physics, the wavelength is the spatial period of a periodic wave—the distance over which the wave's shape repeats.
It is the distance between consecutive corresponding points of the same phase on the wave, such as two adjacent crests, tro ...
. Otherwise, when the circuit length is on the order of a wavelength, we must consider more general models, such as the
distributed-element model
: ''This article is an example from the domain of electrical systems, which is a special case of the more general distributed-parameter systems.''
In electrical engineering, the distributed-element model or transmission-line model of electrical ...
(including
transmission line
In electrical engineering, a transmission line is a specialized cable or other structure designed to conduct electromagnetic waves in a contained manner. The term applies when the conductors are long enough that the wave nature of the transmis ...
s), whose dynamic behaviour is described by
Maxwell's equations
Maxwell's equations, or Maxwell–Heaviside equations, are a set of coupled partial differential equations that, together with the Lorentz force law, form the foundation of classical electromagnetism, classical optics, and electric circuits.
...
. Another way of viewing the validity of the lumped-element model is to note that this model ignores the finite time it takes signals to propagate around a circuit. Whenever this propagation time is not significant to the application the lumped-element model can be used. This is the case when the propagation time is much less than the
period
Period may refer to:
Common uses
* Era, a length or span of time
* Full stop (or period), a punctuation mark
Arts, entertainment, and media
* Period (music), a concept in musical composition
* Periodic sentence (or rhetorical period), a concept ...
of the signal involved. However, with increasing propagation time there will be an increasing error between the assumed and actual phase of the signal which in turn results in an error in the assumed amplitude of the signal. The exact point at which the lumped-element model can no longer be used depends to a certain extent on how accurately the signal needs to be known in a given application.
Real-world components exhibit non-ideal characteristics which are, in reality, distributed elements but are often represented to a
first-order approximation
In science, engineering, and other quantitative disciplines, order of approximation refers to formal or informal expressions for how accurate an approximation is.
Usage in science and engineering
In formal expressions, the ordinal number used b ...
by lumped elements. To account for leakage in
capacitor
A capacitor is a device that stores electrical energy in an electric field by virtue of accumulating electric charges on two close surfaces insulated from each other. It is a passive electronic component with two terminals.
The effect of ...
s for example, we can model the non-ideal capacitor as having a large lumped
resistor
A resistor is a passive two-terminal electrical component that implements electrical resistance as a circuit element. In electronic circuits, resistors are used to reduce current flow, adjust signal levels, to divide voltages, bias active el ...
connected in parallel even though the leakage is, in reality distributed throughout the dielectric. Similarly a
wire-wound resistor
A resistor is a passive two-terminal electrical component that implements electrical resistance as a circuit element. In electronic circuits, resistors are used to reduce current flow, adjust signal levels, to divide voltages, bias active el ...
has significant
inductance
Inductance is the tendency of an electrical conductor to oppose a change in the electric current flowing through it. The flow of electric current creates a magnetic field around the conductor. The field strength depends on the magnitude of the ...
as well as
resistance distributed along its length but we can model this as a lumped
inductor
An inductor, also called a coil, choke, or reactor, is a passive two-terminal electrical component that stores energy in a magnetic field when electric current flows through it. An inductor typically consists of an insulated wire wound into a c ...
in series with the ideal resistor.
Thermal systems
A lumped-capacitance model, also called lumped system analysis, reduces a
thermal system
A thermodynamic system is a body of matter and/or radiation, confined in space by walls, with defined permeabilities, which separate it from its surroundings. The surroundings may include other thermodynamic systems, or physical systems that are ...
to a number of discrete “lumps” and assumes that the
temperature
Temperature is a physical quantity that expresses quantitatively the perceptions of hotness and coldness. Temperature is measured with a thermometer.
Thermometers are calibrated in various temperature scales that historically have relied o ...
difference inside each lump is negligible. This approximation is useful to simplify otherwise complex
differential heat equations. It was developed as a mathematical analog of
electrical capacitance
Capacitance is the capability of a material object or device to store electric charge. It is measured by the change in charge in response to a difference in electric potential, expressed as the ratio of those quantities. Commonly recognized are ...
, although it also includes thermal analogs of
electrical resistance
The electrical resistance of an object is a measure of its opposition to the flow of electric current. Its reciprocal quantity is , measuring the ease with which an electric current passes. Electrical resistance shares some conceptual parallels ...
as well.
The lumped-capacitance model is a common approximation in transient conduction, which may be used whenever
heat conduction
Conduction is the process by which heat is transferred from the hotter end to the colder end of an object. The ability of the object to conduct heat is known as its ''thermal conductivity'', and is denoted .
Heat spontaneously flows along a te ...
within an object is much faster than heat transfer across the boundary of the object. The method of approximation then suitably reduces one aspect of the transient conduction system (spatial temperature variation within the object) to a more mathematically tractable form (that is, it is assumed that the temperature within the object is completely uniform in space, although this spatially uniform temperature value changes over time). The rising uniform temperature within the object or part of a system, can then be treated like a capacitative reservoir which absorbs heat until it reaches a steady thermal state in time (after which temperature does not change within it).
An early-discovered example of a lumped-capacitance system which exhibits mathematically simple behavior due to such physical simplifications, are systems which conform to ''Newton's law of cooling''. This law simply states that the temperature of a hot (or cold) object progresses toward the temperature of its environment in a simple exponential fashion. Objects follow this law strictly only if the rate of heat conduction within them is much larger than the heat flow into or out of them. In such cases it makes sense to talk of a single "object temperature" at any given time (since there is no spatial temperature variation within the object) and also the uniform temperatures within the object allow its total thermal energy excess or deficit to vary proportionally to its surface temperature, thus setting up the Newton's law of cooling requirement that the rate of temperature decrease is proportional to difference between the object and the environment. This in turn leads to simple exponential heating or cooling behavior (details below).
Method
To determine the number of lumps, the
Biot number
The Biot number (Bi) is a dimensionless quantity used in heat transfer calculations. It is named after the eighteenth century French physicist Jean-Baptiste Biot (1774–1862), and gives a simple index of the ratio of the thermal resistances ''ins ...
(Bi), a dimensionless parameter of the system, is used. Bi is defined as the ratio of the conductive heat resistance within the object to the
convective heat transfer
Convection (or convective heat transfer) is the heat transfer, transfer of heat from one place to another due to the movement of fluid. Although often discussed as a distinct method of heat transfer, convective heat transfer involves the combin ...
resistance across the object's boundary with a uniform bath of different temperature. When the
thermal resistance
Thermal resistance is a heat property and a measurement of a temperature difference by which an object or material resists a heat flow. Thermal resistance is the reciprocal of thermal conductance.
* (Absolute) thermal resistance ''R'' in kelvin ...
to heat transferred into the object is larger than the resistance to heat being
diffused
Molecular diffusion, often simply called diffusion, is the thermal motion of all (liquid or gas) particles at temperatures above absolute zero. The rate of this movement is a function of temperature, viscosity of the fluid and the size (mass) of ...
completely within the object, the Biot number is less than 1. In this case, particularly for Biot numbers which are even smaller, the approximation of ''spatially uniform temperature within the object'' can begin to be used, since it can be presumed that heat transferred into the object has time to uniformly distribute itself, due to the lower resistance to doing so, as compared with the resistance to heat entering the object.
If the Biot number is less than 0.1 for a solid object, then the entire material will be nearly the same temperature, with the dominant temperature difference being at the surface. It may be regarded as being "thermally thin". The Biot number must generally be less than 0.1 for usefully accurate approximation and heat transfer analysis. The mathematical solution to the lumped-system approximation gives
Newton's law of cooling
In the study of heat transfer, Newton's law of cooling is a physical law which states that
The rate of heat loss of a body is directly proportional to the difference in the temperatures between the body and its environment.
The law is frequently q ...
.
A Biot number greater than 0.1 (a "thermally thick" substance) indicates that one cannot make this assumption, and more complicated
heat transfer
Heat transfer is a discipline of thermal engineering that concerns the generation, use, conversion, and exchange of thermal energy (heat) between physical systems. Heat transfer is classified into various mechanisms, such as thermal conduction, ...
equations for "transient heat conduction" will be required to describe the time-varying and non-spatially-uniform temperature field within the material body.
The single capacitance approach can be expanded to involve many resistive and capacitive elements, with Bi < 0.1 for each lump. As the Biot number is calculated based upon a
characteristic length
In physics, a characteristic length is an important dimension that defines the scale of a physical system. Often, such a length is used as an input to a formula in order to predict some characteristics of the system, and it is usually required by ...
of the system, the system can often be broken into a sufficient number of sections, or lumps, so that the Biot number is acceptably small.
Some characteristic lengths of thermal systems are:
* Plate: thickness
*
Fin
A fin is a thin component or appendage attached to a larger body or structure. Fins typically function as foils that produce lift or thrust, or provide the ability to steer or stabilize motion while traveling in water, air, or other fluids. Fin ...
: thickness/2
* Long
cylinder
A cylinder (from ) has traditionally been a three-dimensional solid, one of the most basic of curvilinear geometric shapes. In elementary geometry, it is considered a prism with a circle as its base.
A cylinder may also be defined as an infin ...
: diameter/4
*
Sphere
A sphere () is a Geometry, geometrical object that is a solid geometry, three-dimensional analogue to a two-dimensional circle. A sphere is the Locus (mathematics), set of points that are all at the same distance from a given point in three ...
: diameter/6
For arbitrary shapes, it may be useful to consider the characteristic length to be volume / surface area.
Thermal purely resistive circuits
A useful concept used in heat transfer applications once the condition of steady state heat conduction has been reached, is the representation of thermal transfer by what is known as thermal circuits. A thermal circuit is the representation of the resistance to heat flow in each element of a circuit, as though it were an
electrical resistor
A resistor is a passive two-terminal electrical component that implements electrical resistance as a circuit element. In electronic circuits, resistors are used to reduce current flow, adjust signal levels, to divide voltages, bias active el ...
. The heat transferred is analogous to the
electric current
An electric current is a stream of charged particles, such as electrons or ions, moving through an electrical conductor or space. It is measured as the net rate of flow of electric charge through a surface or into a control volume. The moving pa ...
and the thermal resistance is analogous to the electrical resistor. The values of the thermal resistance for the different modes of heat transfer are then calculated as the denominators of the developed equations. The thermal resistances of the different modes of heat transfer are used in analyzing combined modes of heat transfer. The lack of "capacitative" elements in the following purely resistive example, means that no section of the circuit is absorbing energy or changing in distribution of temperature. This is equivalent to demanding that a state of steady state heat conduction (or transfer, as in radiation) has already been established.
The equations describing the three heat transfer modes and their thermal resistances in steady state conditions, as discussed previously, are summarized in the table below:
In cases where there is heat transfer through different media (for example, through a
composite material
A composite material (also called a composition material or shortened to composite, which is the common name) is a material which is produced from two or more constituent materials. These constituent materials have notably dissimilar chemical or ...
), the equivalent resistance is the sum of the resistances of the components that make up the composite. Likely, in cases where there are different heat transfer modes, the total resistance is the sum of the resistances of the different modes. Using the thermal circuit concept, the amount of heat transferred through any medium is the quotient of the temperature change and the total thermal resistance of the medium.
As an example, consider a composite wall of cross-sectional area
. The composite is made of an
long cement plaster with a thermal coefficient
and
long paper faced fiber glass, with thermal coefficient
. The left surface of the wall is at
and exposed to air with a convective coefficient of
. The right surface of the wall is at
and exposed to air with convective coefficient
.
Using the thermal resistance concept, heat flow through the composite is as follows:
where
,
,
, and
Newton's law of cooling
Newton's law of cooling is an empirical relationship attributed to English physicist
Sir Isaac Newton
Sir Isaac Newton (25 December 1642 – 20 March 1726/27) was an English mathematician, physicist, astronomer, alchemist, theologian, and author (described in his time as a "natural philosopher"), widely recognised as one of the great ...
(1642–1727). This law stated in non-mathematical form is the following:
Or, using symbols:
An object at a different temperature from its surroundings will ultimately come to a common temperature with its surroundings. A relatively hot object cools as it warms its surroundings; a cool object is warmed by its surroundings. When considering how quickly (or slowly) something cools, we speak of its ''rate'' of cooling – how many degrees' change in temperature per unit of time.
The rate of cooling of an object depends on how much hotter the object is than its surroundings. The temperature change per minute of a hot apple pie will be more if the pie is put in a cold freezer than if it is placed on the kitchen table. When the pie cools in the freezer, the temperature difference between it and its surroundings is greater. On a cold day, a warm home will leak heat to the outside at a greater rate when there is a large difference between the inside and outside temperatures. Keeping the inside of a home at high temperature on a cold day is thus more costly than keeping it at a lower temperature. If the temperature difference is kept small, the rate of cooling will be correspondingly low.
As Newton's law of cooling states, the rate of cooling of an object – whether by
conduction
Conductor or conduction may refer to:
Music
* Conductor (music), a person who leads a musical ensemble, such as an orchestra.
* Conductor (album), ''Conductor'' (album), an album by indie rock band The Comas
* Conduction, a type of structured f ...
,
convection
Convection is single or multiphase fluid flow that occurs spontaneously due to the combined effects of material property heterogeneity and body forces on a fluid, most commonly density and gravity (see buoyancy). When the cause of the convec ...
, or
radiation
In physics, radiation is the emission or transmission of energy in the form of waves or particles through space or through a material medium. This includes:
* ''electromagnetic radiation'', such as radio waves, microwaves, infrared, visi ...
– is approximately proportional to the temperature difference Δ''T''. Frozen food will warm up faster in a warm room than in a cold room. Note that the rate of cooling experienced on a cold day can be increased by the added convection effect of the
wind
Wind is the natural movement of air or other gases relative to a planet's surface. Winds occur on a range of scales, from thunderstorm flows lasting tens of minutes, to local breezes generated by heating of land surfaces and lasting a few hou ...
. This is referred to as
wind chill
Wind chill or windchill (popularly wind chill factor) is the lowering of body temperature due to the passing-flow of lower-temperature air.
Wind chill numbers are always lower than the air temperature for values where the formula is valid. When ...
. For example, a wind chill of -20 °C means that heat is being lost at the same rate as if the temperature were -20 °C without wind.
Applicable situations
This law describes many situations in which an object has a large thermal capacity and large conductivity, and is suddenly immersed in a uniform bath which conducts heat relatively poorly. It is an example of a thermal circuit with one resistive and one capacitative element. For the law to be correct, the temperatures at all points inside the body must be approximately the same at each time point, including the temperature at its surface. Thus, the temperature difference between the body and surroundings does not depend on which part of the body is chosen, since all parts of the body have effectively the same temperature. In these situations, the material of the body does not act to "insulate" other parts of the body from heat flow, and all of the significant insulation (or "thermal resistance") controlling the rate of heat flow in the situation resides in the area of contact between the body and its surroundings. Across this boundary, the temperature-value jumps in a discontinuous fashion.
In such situations, heat can be transferred from the exterior to the interior of a body, across the insulating boundary, by convection, conduction, or diffusion, so long as the boundary serves as a relatively poor conductor with regard to the object's interior. The presence of a physical insulator is not required, so long as the process which serves to pass heat across the boundary is "slow" in comparison to the conductive transfer of heat inside the body (or inside the region of interest—the "lump" described above).
In such a situation, the object acts as the "capacitative" circuit element, and the resistance of the thermal contact at the boundary acts as the (single) thermal resistor. In electrical circuits, such a combination would charge or discharge toward the input voltage, according to a simple exponential law in time. In the thermal circuit, this configuration results in the same behavior in temperature: an exponential approach of the object temperature to the bath temperature.
Mathematical statement
Newton's law is mathematically stated by the simple first-order differential equation:
where
*''Q'' is thermal energy in
joule
The joule ( , ; symbol: J) is the unit of energy in the International System of Units (SI). It is equal to the amount of work done when a force of 1 newton displaces a mass through a distance of 1 metre in the direction of the force applied ...
s
*''h'' is the
heat transfer coefficient
In thermodynamics, the heat transfer coefficient or film coefficient, or film effectiveness, is the proportionality constant between the heat flux and the thermodynamic driving force for the flow of heat (i.e., the temperature difference, ). ...
between the surface and the fluid
*''A'' is the surface area of the heat being transferred
*''T'' is the temperature of the object's surface and interior (since these are the same in this approximation)
*''T''
env is the temperature of the environment
*Δ''T''(''t'') = ''T''(''t'') − ''T''
env is the time-dependent thermal
gradient
In vector calculus, the gradient of a scalar-valued differentiable function of several variables is the vector field (or vector-valued function) \nabla f whose value at a point p is the "direction and rate of fastest increase". If the gradi ...
between environment and object
Putting heat transfers into this form is sometimes not a very good approximation, depending on ratios of heat conductances in the system. If the differences are not large, an accurate formulation of heat transfers in the system may require analysis of heat flow based on the (transient) heat transfer equation in nonhomogeneous or poorly conductive media.
Solution in terms of object heat capacity
If the entire body is treated as lumped-capacitance heat reservoir, with total heat content which is proportional to simple total
heat capacity
Heat capacity or thermal capacity is a physical property of matter, defined as the amount of heat to be supplied to an object to produce a unit change in its temperature. The SI unit of heat capacity is joule per kelvin (J/K).
Heat capacity i ...
, and
, the temperature of the body, or
. It is expected that the system will experience
exponential decay
A quantity is subject to exponential decay if it decreases at a rate proportional to its current value. Symbolically, this process can be expressed by the following differential equation, where is the quantity and (lambda) is a positive rate ...
with time in the temperature of a body.
From the definition of heat capacity
comes the relation
. Differentiating this equation with regard to time gives the identity (valid so long as temperatures in the object are uniform at any given time):
. This expression may be used to replace
in the first equation which begins this section, above. Then, if
is the temperature of such a body at time
, and
is the temperature of the environment around the body:
where
is a positive constant characteristic of the system, which must be in units of
, and is therefore sometimes expressed in terms of a characteristic
time constant In physics and engineering, the time constant, usually denoted by the Greek letter (tau), is the parameter characterizing the response to a step input of a first-order, linear time-invariant (LTI) system.Concretely, a first-order LTI system is a sy ...
given by:
. Thus, in thermal systems,
. (The total
heat capacity
Heat capacity or thermal capacity is a physical property of matter, defined as the amount of heat to be supplied to an object to produce a unit change in its temperature. The SI unit of heat capacity is joule per kelvin (J/K).
Heat capacity i ...
of a system may be further represented by its mass-
specific heat capacity
In thermodynamics, the specific heat capacity (symbol ) of a substance is the heat capacity of a sample of the substance divided by the mass of the sample, also sometimes referred to as massic heat capacity. Informally, it is the amount of heat t ...
multiplied by its mass
, so that the time constant
is also given by
).
The solution of this differential equation, by standard methods of integration and substitution of boundary conditions, gives:
If:
:
is defined as :
where
is the initial temperature difference at time 0,
then the Newtonian solution is written as:
This same solution is almost immediately apparent if the initial differential equation is written in terms of
, as the single function to be solved for.
Applications
This mode of analysis has been applied to
forensic science
Forensic science, also known as criminalistics, is the application of science to criminal and civil laws, mainly—on the criminal side—during criminal investigation, as governed by the legal standards of admissible evidence and criminal ...
s to analyze the time of death of humans. Also, it can be applied to
HVAC
Heating, ventilation, and air conditioning (HVAC) is the use of various technologies to control the temperature, humidity, and purity of the air in an enclosed space. Its goal is to provide thermal comfort and acceptable indoor air quality. HV ...
(heating, ventilating and air-conditioning, which can be referred to as "building climate control"), to ensure more nearly instantaneous effects of a change in comfort level setting.
Mechanical systems
The simplifying assumptions in this domain are:
* all objects are
rigid bodies
In physics, a rigid body (also known as a rigid object) is a solid body in which deformation is zero or so small it can be neglected. The distance between any two given points on a rigid body remains constant in time regardless of external force ...
;
* all interactions between rigid bodies take place via
kinematic pair
In classical mechanics, a kinematic pair is a connection between two physical objects that imposes constraints on their relative movement (kinematics). German engineer Franz Reuleaux introduced the kinematic pair as a new approach to the study o ...
s (''joints''),
springs and
dampers.
Acoustics
In this context, the lumped-component model extends the distributed concepts of
acoustic theory Acoustic theory is a scientific field that relates to the description of sound waves. It derives from fluid dynamics. See acoustics for the engineering approach.
For sound waves of any magnitude of a disturbance in velocity, pressure, and density w ...
subject to approximation. In the acoustical lumped-component model, certain physical components with acoustical properties may be approximated as behaving similarly to standard electronic components or simple combinations of components.
*A rigid-walled cavity containing air (or similar compressible fluid) may be approximated as a
capacitor
A capacitor is a device that stores electrical energy in an electric field by virtue of accumulating electric charges on two close surfaces insulated from each other. It is a passive electronic component with two terminals.
The effect of ...
whose value is proportional to the volume of the cavity. The validity of this approximation relies on the shortest wavelength of interest being significantly (much) larger than the longest dimension of the cavity.
*A
reflex port
A bass reflex system (also known as a ported, vented box or reflex port) is a type of loudspeaker enclosure that uses a port (hole) or vent cut into the cabinet and a section of tubing or pipe affixed to the port. This port enables the sound fro ...
may be approximated as an
inductor
An inductor, also called a coil, choke, or reactor, is a passive two-terminal electrical component that stores energy in a magnetic field when electric current flows through it. An inductor typically consists of an insulated wire wound into a c ...
whose value is proportional to the effective length of the port divided by its cross-sectional area. The effective length is the actual length plus an
end correction
Whenever a wave forms through a medium/object (organ pipe) with a closed/open end, there is a chance of error in the formation of the wave, i.e. it may not actually start from the opening of the object but instead before the opening, thus resulting ...
. This approximation relies on the shortest wavelength of interest being significantly larger than the longest dimension of the port.
*Certain types of damping material can be approximated as a
resistor
A resistor is a passive two-terminal electrical component that implements electrical resistance as a circuit element. In electronic circuits, resistors are used to reduce current flow, adjust signal levels, to divide voltages, bias active el ...
. The value depends on the properties and dimensions of the material. The approximation relies in the wavelengths being long enough and on the properties of the material itself.
*A
loudspeaker
A loudspeaker (commonly referred to as a speaker or speaker driver) is an electroacoustic transducer that converts an electrical audio signal into a corresponding sound. A ''speaker system'', also often simply referred to as a "speaker" or " ...
drive unit (typically a
woofer
A woofer or bass speaker is a technical term for a loudspeaker driver designed to produce low frequency sounds, typically from 50 Hz up to 1000 Hz. The name is from the onomatopoeic English word for a dog's bark, " woof" (in contrast to th ...
or
subwoofer
A subwoofer (or sub) is a loudspeaker designed to reproduce low-pitched audio frequencies known as bass and sub-bass, lower in frequency than those which can be (optimally) generated by a woofer. The typical frequency range for a subwoofer is ...
drive unit) may be approximated as a series connection of a zero-
impedance voltage
Voltage, also known as electric pressure, electric tension, or (electric) potential difference, is the difference in electric potential between two points. In a static electric field, it corresponds to the work needed per unit of charge to m ...
source, a
resistor
A resistor is a passive two-terminal electrical component that implements electrical resistance as a circuit element. In electronic circuits, resistors are used to reduce current flow, adjust signal levels, to divide voltages, bias active el ...
, a
capacitor
A capacitor is a device that stores electrical energy in an electric field by virtue of accumulating electric charges on two close surfaces insulated from each other. It is a passive electronic component with two terminals.
The effect of ...
and an
inductor
An inductor, also called a coil, choke, or reactor, is a passive two-terminal electrical component that stores energy in a magnetic field when electric current flows through it. An inductor typically consists of an insulated wire wound into a c ...
. The values depend on the specifications of the unit and the wavelength of interest.
Heat transfer for buildings
A simplifying assumption in this domain is that all heat transfer mechanisms are linear, implying that radiation and convection are linearised for each problem.
Several publications can be found that describe how to generate lumped-element models of buildings. In most cases, the building is considered a single thermal zone and in this case, turning multi-layered walls into lumped elements can be one of the most complicated tasks in the creation of the model. The dominant-layer method is one simple and reasonably accurate method. In this method, one of the layers is selected as the dominant layer in the whole construction, this layer is chosen considering the most relevant frequencies of the problem. In his thesis,
Lumped-element models of buildings have also been used to evaluate the efficiency of domestic energy systems, by running many simulations under different future weather scenarios.
Fluid systems
Lumped-element models can be used to describe fluid systems by using voltage to represent pressure and current to represent flow; identical equations from the electrical circuit representation are valid after substituting these two variables. Such applications can, for example, study the response of the human cardiovascular system to
ventricular assist device
A ventricular assist device (VAD) is an electromechanical device for assisting cardiac circulation, which is used either to partially or to completely replace the function of a failing heart. The function of a VAD differs from that of an artifici ...
implantation.
[Farahmand M, Kavarana MN, Trusty PM, Kung EO. "Target Flow-Pressure Operating Range for Designing a Failing Fontan Cavopulmonary Support Device" IEEE Transactions on Biomedical Engineering. DOI: 10.1109/TBME.2020.2974098 (2020)]
See also
*
System isomorphism
*
Model order reduction Model order reduction (MOR) is a technique for reducing the computational complexity of mathematical models in numerical simulations. As such it is closely related to the concept of metamodeling, with applications in all areas of mathematical model ...
References
{{Reflist
External links
Advanced modelling and simulation techniques for magnetic componentsIMTEK Mathematica Supplement (IMS) the Open Source IMTEK Mathematica Supplement (IMS) for lumped modelling
Conceptual models
Mechanics
Acoustics
Electronic circuits
Electronic design