Lumped capacitance model
Encyclopedia
A lumped capacitance model, also called lumped system analysis, reduces a thermal system to a number of discrete “lumps” and assumes that the temperature
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, although it also includes thermal analogs of electrical resistance
as well.
The lumped capacitance model is a common approximation in transient conduction, which may be used whenever heat conduction
within an object is much faster than heat conduction across the boundary of the object. The method of approximation then suitably reduces one aspect of the transient conduction system (spacial 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 spacially-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 (see below for detail).
(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
resistance across the object's boundary with a uniform bath of different temperature. When the thermal resistance to heat transferred into the object is larger than the resistance to heat being diffused 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 will be 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.
A Biot number greater than 0.1 (a "thermally thick" substance) indicates that one cannot make this assumption, and more complicated heat transfer 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
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:
For arbitrary shapes, it may be useful to consider the characteristic length to be volume / surface area.
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
), 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 
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 in the introduction).
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.
Newton's law is mathematically stated by the simple first-order differential equation:
Thermal energy in joule
s
Heat transfer coefficient
Surface area of the heat being transferred
Temperature of the object's surface and interior (since these are the same in this approximation)
Temperature of the environment
is the time-dependent thermal gradient
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 mediums.
, and 
, the temperature of the body, or 
. It is expected that the system will experience exponential decay 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
given by: 
. Thus, in thermal systems, 
. (The total heat capacity
of a system may be further represented by its mass-specific heat capacity 
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:
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.
'
(heating, ventilating and air-conditioning, or building climate control), to ensure more nearly instantaneous effects of a change in comfort level setting.
Temperature
Temperature is a physical property of matter that quantitatively expresses the common notions of hot and cold. Objects of low temperature are cold, while various degrees of higher temperatures are referred to as warm or hot...
difference inside each lump is negligible. This approximation is useful to simplify otherwise complex differential
Differential equation
A differential equation is a mathematical equation for an unknown function of one or several variables that relates the values of the function itself and its derivatives of various orders...
heat equations. It was developed as a mathematical analog of electrical capacitance, although it also includes thermal analogs of electrical resistance
Electrical resistance
The electrical resistance of an electrical element is the opposition to the passage of an electric current through that element; the inverse quantity is electrical conductance, the ease at which an electric current passes. Electrical resistance shares some conceptual parallels with the mechanical...
as well.
The lumped capacitance model is a common approximation in transient conduction, which may be used whenever heat conduction
Heat conduction
In heat transfer, conduction is a mode of transfer of energy within and between bodies of matter, due to a temperature gradient. Conduction means collisional and diffusive transfer of kinetic energy of particles of ponderable matter . Conduction takes place in all forms of ponderable matter, viz....
within an object is much faster than heat conduction across the boundary of the object. The method of approximation then suitably reduces one aspect of the transient conduction system (spacial 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 spacially-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 (see below for detail).
Method
To determine the number of lumps the Biot numberBiot number
The Biot number is a dimensionless number used in non-steady-state heat transfer calculations. It is named after the French physicist Jean-Baptiste Biot , and gives a simple index of the ratio of the heat transfer resistances inside of and at the surface of a body...
(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
Convective heat transfer
Convective heat transfer, often referred to as convection, is the transfer of heat from one place to another by the movement of fluids. The presence of bulk motion of the fluid enhances the heat transfer between the solid surface and the fluid. Convection is usually the dominant form of heat...
resistance across the object's boundary with a uniform bath of different temperature. When the thermal resistance to heat transferred into the object is larger than the resistance to heat being diffused 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 will be 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.
A Biot number greater than 0.1 (a "thermally thick" substance) indicates that one cannot make this assumption, and more complicated heat transfer 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
Characteristic length
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.Examples:* Reynolds number* Biot number...
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
- FinFinA fin is a surface used for stability and/or to produce lift and thrust or to steer while traveling in water, air, or other fluid media, . The first use of the word was for the limbs of fish, but has been extended to include other animal limbs and man-made devices...
: thickness/2 - Long cylinderCylinder (geometry)A cylinder is one of the most basic curvilinear geometric shapes, the surface formed by the points at a fixed distance from a given line segment, the axis of the cylinder. The solid enclosed by this surface and by two planes perpendicular to the axis is also called a cylinder...
: diameter/4 - SphereSphereA sphere is a perfectly round geometrical object in three-dimensional space, such as the shape of a round ball. Like a circle in two dimensions, a perfect sphere is completely symmetrical around its center, with all points on the surface lying the same distance r from the center point...
: 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. The heat transferred is analogous to the electrical current 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:
| Transfer Mode | Rate of Heat Transfer | Thermal Resistance |
|---|---|---|
| Conduction |  |
 |
| Convection |  |
 |
| Radiation |  |
 , where |
In cases where there is heat transfer through different media (for example, through a composite material
Composite material
Composite materials, often shortened to composites or called composition materials, are engineered or naturally occurring materials made from two or more constituent materials with significantly different physical or chemical properties which remain separate and distinct at the macroscopic 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: example of a thermal circuit with one resistive and one capacitative element
Newton's law of cooling 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. This law stated in non-mathematical form is that the rate of heat loss of a body is proportional to the difference in temperatures between the body and its surroundings. (see below for the equivalent mathematical equation). 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 in the introduction).
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.
Newton's law is mathematically stated by the simple first-order differential equation:
Thermal energy in jouleJoule
The joule ; symbol J) is a derived unit of energy or work in the International System of Units. It is equal to the energy expended in applying a force of one newton through a distance of one metre , or in passing an electric current of one ampere through a resistance of one ohm for one second...
s
Heat transfer coefficient Surface area of the heat being transferred Temperature of the object's surface and interior (since these are the same in this approximation) Temperature of the environment is the time-dependent thermal gradientGradient
In vector calculus, the gradient of a scalar field is a vector field that points in the direction of the greatest rate of increase of the scalar field, and whose magnitude is the greatest rate of change....
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 mediums.
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 capacityHeat capacity
Heat capacity , or thermal capacity, is the measurable physical quantity that characterizes the amount of heat required to change a substance's temperature by a given amount...
, and , the temperature of the body, or . It is expected that the system will experience exponential decay 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 constantTime constant
In physics and engineering, the time constant, usually denoted by the Greek letter \tau , is the risetime characterizing the response to a time-varying input of a first-order, linear time-invariant system.Concretely, a first-order LTI system is a system that can be modeled by a single first order...
given by: . Thus, in thermal systems, . (The total heat capacityHeat capacity
Heat capacity , or thermal capacity, is the measurable physical quantity that characterizes the amount of heat required to change a substance's temperature by a given amount...
of a system may be further represented by its mass-specific heat capacity 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 sciences to analyze the time of death of humans. Also it can be applied to HVACHVAC
HVAC refers to technology of indoor or automotive environmental comfort. HVAC system design is a major subdiscipline of mechanical engineering, based on the principles of thermodynamics, fluid mechanics, and heat transfer...
(heating, ventilating and air-conditioning, or building climate control), to ensure more nearly instantaneous effects of a change in comfort level setting.