654:. 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).
2382:
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.
188:
2346:
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.
1447:), 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.
91:
2052:
647:
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).
753:
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.
36:
567:. 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
1692:
4062:
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.
2389:
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
2385:
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
2337:
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
743:
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,
2586:
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
646:
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
752:
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
571:
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
686:
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
2381:
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
2345:
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
4078:
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
211:
representation of a physical system or circuit that assumes all components are concentrated at a single point and their behavior can be described by idealized mathematical models. The lumped-element model simplifies the system or circuit behavior description into a
1041:
1247:
1436:
4063:
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.
4197:
Cooper, S.J.G., Hammond, G.P., McManus, M.C., Ramallo-Gonzlez, A. & Rogers, J.G., 2014. Effect of operating conditions on performance of domestic heating systems with heat pumps and fuel cell micro-cogeneration. Energy and
Buildings, 70,
3984:
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.
3919:
2047:{\displaystyle {\dot {Q}}={\frac {T_{i}-T_{o}}{R_{i}+R_{1}+R_{2}+R_{o}}}={\frac {T_{i}-T_{1}}{R_{i}}}={\frac {T_{i}-T_{2}}{R_{i}+R_{1}}}={\frac {T_{i}-T_{3}}{R_{i}+R_{1}+R_{2}}}={\frac {T_{1}-T_{2}}{R_{1}}}={\frac {T_{3}-T_{o}}{R_{0}}}}
682:
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.
2509:
4207:
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
2386:
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 interestâthe "lump" described above).
860:
3766:
3513:
4178:
Ramallo-GonzĂĄlez, A.P., Eames, M.E. & Coley, D.A., 2013. Lumped
Parameter Models for Building Thermal Modelling: An Analytic approach to simplifying complex multi-layered constructions. Energy and Buildings, 60,
3015:
3993:
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
343:
1110:
2333:
388:
3211:
904:
3608:
2274:
2215:
2156:
2104:
3936:(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.
1121:
2365:. 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
4066:
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.
3805:
701:
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
1293:
1287:
4016:. 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.
3641:
3551:
2390:
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.
502:
4117:
893:
2905:
2795:
756:
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:
54:
650:
An early-discovered example of a lumped-capacitance system which exhibits mathematically simple behavior due to such physical simplifications, are systems which conform to
3800:
2401:
3255:
3396:
549:
2732:
3082:
3052:
3355:
3308:
3112:
2829:
1684:
1657:
1630:
1603:
1576:
1549:
1522:
1495:
529:
2858:
2665:
4059:
A simplifying assumption in this domain is that all heat transfer mechanisms are linear, implying that radiation and convection are linearised for each problem.
3650:
3404:
3328:
3278:
2878:
2692:
2636:
2616:
1468:
2910:
698:
equations for "transient heat conduction" will be required to describe the time-varying and non-spatially-uniform temperature field within the material body.
4005:
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
678:
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
2734:. Differentiating this equation with regard to time gives the identity (valid so long as temperatures in the object are uniform at any given time):
781:
298:
2373:. 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.
666:(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
413:
288:
4151:
4121:
4075:
108:
2307:
705:
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.
208:
350:
4009:. This approximation relies on the shortest wavelength of interest being significantly larger than the longest dimension of the port.
694:
A Biot number greater than 0.1 (a "thermally thick" substance) indicates that one cannot make this assumption, and more complicated
174:
72:
575:
Real-world components exhibit non-ideal characteristics which are, in reality, distributed elements but are often represented to a
155:
2595:
If the entire body is treated as lumped-capacitance heat reservoir, with total heat content which is proportional to simple total
127:
3401:
The solution of this differential equation, by standard methods of integration and substitution of boundary conditions, gives:
1047:
259:
213:
112:
134:
1036:{\displaystyle {\dot {Q}}={\frac {T_{\rm {surf}}-T_{\rm {envr}}}{\left({\frac {1}{h_{\rm {conv}}A_{\rm {surf}}}}\right)}}}
255:
4257:
555:. Otherwise, when the circuit length is on the order of a wavelength, we must consider more general models, such as the
240:
or models in which the behaviour is distributed spatially and cannot be considered as localized into discrete entities.
237:
3117:
4188:
Ramallo-GonzĂĄlez, A.P. 2013. Modelling
Simulation and Optimisation of Low-energy Buildings. PhD. University of Exeter.
4116:
Anant
Agarwal and Jeffrey Lang, course materials for 6.002 Circuits and Electronics, Spring 2007. MIT OpenCourseWare (
3556:
141:
401:
4221:
2300:
The rate of heat loss of a body is proportional to the temperature difference between the body and its surroundings.
4262:
4242:
2354:
2284:
2220:
2161:
688:
667:
409:
101:
4080:
2528:
1242:{\displaystyle {\dot {Q}}={\frac {T_{\rm {surf}}-T_{\rm {surr}}}{\left({\frac {1}{h_{r}A_{\rm {surf}}}}\right)}}}
576:
556:
417:
123:
587:
connected in parallel even though the leakage is, in reality distributed throughout the dielectric. Similarly a
2109:
2057:
3770:
This same solution is almost immediately apparent if the initial differential equation is written in terms of
687:
approximation and heat transfer analysis. The mathematical solution to the lumped-system approximation gives
1431:{\displaystyle h_{r}=\epsilon \sigma (T_{\rm {surf}}^{2}+T_{\rm {surr}}^{2})(T_{\rm {surf}}+T_{\rm {surr}})}
642:
The lumped-capacitance model is a common approximation in transient conduction, which may be used whenever
3281:
735:
For arbitrary shapes, it may be useful to consider the characteristic length to be volume / surface area.
632:
564:
405:
280:
244:
1253:
4097:
2543:
is the temperature of the object's surface and interior (since these are the same in this approximation)
702:
636:
628:
596:
465:
433:
393:
187:
3613:
3523:
627:
difference inside each lump is negligible. This approximation is useful to simplify otherwise complex
474:
4032:
866:
449:
2883:
2737:
4143:
745:
722:
588:
572:
depends to a certain extent on how accurately the signal needs to be known in a given application.
148:
4252:
4247:
3773:
2350:
1444:
671:
445:
429:
258:(PDEs) of the continuous (infinite-dimensional) time and space model of the physical system into
217:
3216:
3360:
670:
resistance across the object's boundary with a uniform bath of different temperature. When the
4147:
3989:
A rigid-walled cavity containing air (or similar compressible fluid) may be approximated as a
2668:
2358:
560:
534:
3914:{\displaystyle {\frac {dT(t)}{dt}}={\frac {d\Delta T(t)}{dt}}=-{\frac {1}{t_{0}}}\Delta T(t)}
2697:
4135:
3929:
3057:
3020:
749:
568:
225:
3333:
3286:
3090:
2800:
1662:
1635:
1608:
1581:
1554:
1527:
1500:
1473:
507:
3981:
3964:
2834:
2641:
643:
248:
4136:
4006:
3956:
3313:
3263:
2863:
2677:
2621:
2601:
2398:
Newton's law is mathematically stated by the simple first-order differential equation:
1453:
620:
1443:
In cases where there is heat transfer through different media (for example, through a
4236:
4050:. The values depend on the specifications of the unit and the wavelength of interest.
3258:
3085:
2596:
1689:
Using the thermal resistance concept, heat flow through the composite is as follows:
695:
229:
2504:{\displaystyle {\frac {dQ}{dt}}=-h\cdot A(T(t)-T_{\text{env}})=-h\cdot A\Delta T(t)}
17:
2293:
4226:
420:, while still not requiring the direct application of the full Maxwell equations.
2349:
As Newton's law of cooling states, the rate of cooling of an object â whether by
4092:
4020:
3998:
3980:
In this context, the lumped-component model extends the distributed concepts of
674:
to heat transferred into the object is larger than the resistance to heat being
663:
624:
437:
221:
90:
3054:
is a positive constant characteristic of the system, which must be in units of
191:
Representation of a lumped model consisting of a voltage source and a resistor.
3949:
2370:
855:{\displaystyle {\dot {Q}}={\frac {T_{1}-T_{2}}{\left({\frac {L}{kA}}\right)}}}
592:
552:
441:
4043:
4028:
3990:
580:
457:
251:
233:
3761:{\displaystyle \Delta T(t)=\Delta T(0)\ e^{-rt}=\Delta T(0)\ e^{-t/t_{0}}.}
3508:{\displaystyle T(t)=T_{\mathrm {env} }+(T(0)-T_{\mathrm {env} })\ e^{-rt}.}
760:
Equations for different heat transfer modes and their thermal resistances.
583:
for example, we can model the non-ideal capacitor as having a large lumped
4229:, the Open Source IMTEK Mathematica Supplement (IMS) for lumped modelling
4047:
4039:
4013:
4002:
3010:{\displaystyle {\frac {dT(t)}{dt}}=-r(T(t)-T_{\text{env}})=-r\Delta T(t)}
2579:
675:
600:
584:
461:
453:
412:. The third assumption is the basis of the lumped-element model used in
27:
Simplification of a physical system into a network of discrete components
2296:(1642â1727). This law stated in non-mathematical form is the following:
392:
Signal timescales of interest are much larger than propagation delay of
4035:
3968:
2342:
of cooling â how many degrees' change in temperature per unit of time.
4024:
728:
432:
makes the simplifying assumption that the attributes of the circuit,
347:
The change of the charge in time inside conducting elements is zero.
295:
The change of the magnetic flux in time outside a conductor is zero.
4222:
Advanced modelling and simulation techniques for magnetic components
3084:, and is therefore sometimes expressed in terms of a characteristic
3932:
to analyze the time of death of humans. Also, it can be applied to
2519:
186:
4031:
drive unit) may be approximated as a series connection of a zero-
2831:
in the first equation which begins this section, above. Then, if
1450:
As an example, consider a composite wall of cross-sectional area
3933:
2366:
2361:â is approximately proportional to the temperature difference Î
599:
distributed along its length but we can model this as a lumped
715:
631:
heat equations. It was developed as a mathematical analog of
84:
29:
2292:
is an empirical relationship attributed to
English physicist
4012:
Certain types of damping material can be approximated as a
338:{\displaystyle {\frac {\partial \phi _{B}}{\partial t}}=0}
1105:{\displaystyle {\frac {1}{h_{\rm {conv}}A_{\rm {surf}}}}}
2907:
is the temperature of the environment around the body:
2587:
equation in nonhomogeneous or poorly conductive media.
1551:
long paper faced fiber glass, with thermal coefficient
50:
3955:
all interactions between rigid bodies take place via
3808:
3776:
3653:
3616:
3559:
3526:
3407:
3363:
3336:
3316:
3289:
3266:
3219:
3120:
3093:
3060:
3023:
2913:
2886:
2866:
2837:
2803:
2740:
2700:
2680:
2644:
2624:
2604:
2404:
2328:{\displaystyle {\text{Rate of cooling}}\sim \Delta T}
2310:
2223:
2164:
2112:
2060:
1695:
1665:
1638:
1611:
1584:
1557:
1530:
1503:
1476:
1456:
1296:
1256:
1124:
1050:
907:
869:
784:
623:
to a number of discrete âlumpsâ and assumes that the
537:
510:
477:
353:
301:
1605:
and exposed to air with a convective coefficient of
708:
Some characteristic lengths of thermal systems are:
3280:
of a system may be further represented by its mass-
115:. Unsourced material may be challenged and removed.
45:
may be too technical for most readers to understand
3913:
3794:
3760:
3635:
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3545:
3507:
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3322:
3302:
3272:
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3205:
3106:
3076:
3046:
3009:
2899:
2872:
2852:
2823:
2789:
2726:
2686:
2667:. It is expected that the system will experience
2659:
2630:
2610:
2503:
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2268:
2209:
2150:
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1678:
1651:
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523:
496:
383:{\displaystyle {\frac {\partial q}{\partial t}}=0}
382:
337:
3206:{\displaystyle t_{0}=1/r=-\Delta T(t)/(dT(t)/dt)}
2537:is the surface area of the heat being transferred
531:denotes the circuit's characteristic length, and
4142:(6th ed.). John Wiley & Sons. pp.
3944:The simplifying assumptions in this domain are:
3643:is the initial temperature difference at time 0,
3603:{\displaystyle T(t)-T_{\mathrm {env} }\ ,\quad }
2298:
1659:and exposed to air with convective coefficient
1497:long cement plaster with a thermal coefficient
680:spatially uniform temperature within the object
635:, although it also includes thermal analogs of
579:by lumped elements. To account for leakage in
408:and are only applicable when the circuit is in
2269:{\displaystyle R_{2}={\frac {L_{2}}{k_{2}A}}}
2210:{\displaystyle R_{1}={\frac {L_{1}}{k_{1}A}}}
8:
4134:Incropera; DeWitt; Bergman; Lavine (2007).
4074:Fluid systems can be described by means of
3802:, as the single function to be solved for.
3647:then the Newtonian solution is written as:
748:. The heat transferred is analogous to the
563:), whose dynamic behaviour is described by
471:The lumped-element model is valid whenever
262:(ODEs) with a finite number of parameters.
3928:This mode of analysis has been applied to
2860:is the temperature of such a body at time
3888:
3879:
3841:
3809:
3807:
3775:
3747:
3738:
3731:
3694:
3652:
3615:
3580:
3579:
3558:
3525:
3490:
3467:
3466:
3428:
3427:
3406:
3377:
3371:
3362:
3341:
3335:
3315:
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3288:
3265:
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3166:
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3125:
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3022:
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2914:
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2810:
2802:
2797:. This expression may be used to replace
2773:
2747:
2739:
2713:
2699:
2679:
2643:
2623:
2603:
2591:Solution in terms of object heat capacity
2462:
2405:
2403:
2311:
2309:
2254:
2243:
2237:
2228:
2222:
2195:
2184:
2178:
2169:
2163:
2151:{\displaystyle R_{o}={\frac {1}{h_{o}A}}}
2136:
2126:
2117:
2111:
2099:{\displaystyle R_{i}={\frac {1}{h_{i}A}}}
2084:
2074:
2065:
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515:
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482:
476:
416:. Less severe assumptions result in the
354:
352:
312:
302:
300:
175:Learn how and when to remove this message
73:Learn how and when to remove this message
57:, without removing the technical details.
4166:Heat Transfer â A Practical Approach by
2671:with time in the temperature of a body.
758:
464:, etc. joined by a network of perfectly
4109:
428:The lumped-element model of electronic
4138:Fundamentals of Heat and Mass Transfer
1632:. The right surface of the wall is at
662:To determine the number of lumps, the
4122:Massachusetts Institute of Technology
2674:From the definition of heat capacity
2552:is the temperature of the environment
1578:. The left surface of the wall is at
55:make it understandable to non-experts
7:
4076:lumped-element cardiovascular models
400:The first two assumptions result in
291:. The self-imposed constraints are:
113:adding citations to reliable sources
1282:{\displaystyle {\frac {1}{h_{r}A}}}
603:in series with the ideal resistor.
279:is a set of imposed assumptions in
4227:IMTEK Mathematica Supplement (IMS)
3896:
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2638:, the temperature of the body, or
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1392:
1389:
1386:
1363:
1360:
1357:
1354:
1334:
1331:
1328:
1325:
1224:
1221:
1218:
1215:
1182:
1179:
1176:
1173:
1158:
1155:
1152:
1149:
1093:
1090:
1087:
1084:
1072:
1069:
1066:
1063:
1018:
1015:
1012:
1009:
997:
994:
991:
988:
965:
962:
959:
956:
941:
938:
935:
932:
448:, are concentrated into idealized
365:
357:
320:
305:
25:
3636:{\displaystyle \Delta T(0)\quad }
3546:{\displaystyle \Delta T(t)\quad }
2531:between the surface and the fluid
739:Thermal purely resistive circuits
497:{\displaystyle L_{c}\ll \lambda }
283:that provides the foundation for
551:denotes the circuit's operating
89:
34:
3632:
3599:
3542:
888:{\displaystyle {\frac {L}{kA}}}
260:ordinary differential equations
243:The simplification reduces the
100:needs additional citations for
3908:
3902:
3859:
3853:
3824:
3818:
3789:
3783:
3721:
3715:
3684:
3678:
3666:
3660:
3629:
3623:
3569:
3563:
3539:
3533:
3480:
3456:
3450:
3444:
3417:
3411:
3200:
3186:
3180:
3171:
3163:
3157:
3004:
2998:
2980:
2964:
2958:
2952:
2929:
2923:
2900:{\displaystyle T_{\text{env}}}
2847:
2841:
2790:{\displaystyle dQ/dt=C(dT/dt)}
2784:
2764:
2582:between environment and object
2578:is the time-dependent thermal
2498:
2492:
2468:
2452:
2446:
2440:
2278:
1470:. The composite is made of an
1425:
1377:
1374:
1316:
256:partial differential equations
236:, etc. This is in contrast to
1:
238:distributed parameter systems
3330:, so that the time constant
3213:. Thus, in thermal systems,
4055:Heat transfer for buildings
3795:{\displaystyle \Delta T(t)}
4279:
4001:may be approximated as an
3250:{\displaystyle t_{0}=C/hA}
2282:
396:across the lumped element.
285:lumped-circuit abstraction
4081:ventricular assist device
3391:{\displaystyle mc_{p}/hA}
2529:heat transfer coefficient
2369:. This is referred to as
577:first-order approximation
557:distributed-element model
418:distributed-element model
4023:drive unit (typically a
668:convective heat transfer
613:lumped-capacitance model
544:{\displaystyle \lambda }
402:Kirchhoff's circuit laws
277:lumped-matter discipline
271:Lumped-matter discipline
3310:multiplied by its mass
2727:{\displaystyle C=dQ/dT}
2290:Newton's law of cooling
2285:Newton's law of cooling
2279:Newton's law of cooling
689:Newton's law of cooling
652:Newton's law of cooling
3915:
3796:
3762:
3637:
3604:
3547:
3509:
3392:
3351:
3324:
3304:
3282:specific heat capacity
3274:
3251:
3207:
3108:
3078:
3077:{\displaystyle s^{-1}}
3048:
3047:{\displaystyle r=hA/C}
3011:
2901:
2874:
2854:
2825:
2791:
2728:
2688:
2661:
2632:
2612:
2505:
2394:Mathematical statement
2329:
2302:
2270:
2211:
2152:
2100:
2048:
1680:
1653:
1626:
1599:
1572:
1545:
1518:
1491:
1464:
1432:
1283:
1243:
1106:
1037:
889:
856:
768:Rate of Heat Transfer
633:electrical capacitance
617:lumped system analysis
545:
525:
498:
384:
339:
281:electrical engineering
205:lumped-component model
201:lumped-parameter model
192:
124:"Lumped-element model"
4098:Model order reduction
3916:
3797:
3763:
3638:
3605:
3553:is defined as :
3548:
3510:
3393:
3352:
3350:{\displaystyle t_{0}}
3325:
3305:
3303:{\displaystyle c_{p}}
3275:
3252:
3208:
3109:
3107:{\displaystyle t_{0}}
3079:
3049:
3012:
2902:
2875:
2855:
2826:
2824:{\displaystyle dQ/dt}
2792:
2729:
2689:
2662:
2633:
2613:
2518:is thermal energy in
2506:
2377:Applicable situations
2330:
2271:
2212:
2153:
2101:
2049:
1681:
1679:{\displaystyle h_{o}}
1654:
1652:{\displaystyle T_{o}}
1627:
1625:{\displaystyle h_{i}}
1600:
1598:{\displaystyle T_{i}}
1573:
1571:{\displaystyle k_{2}}
1546:
1544:{\displaystyle L_{2}}
1519:
1517:{\displaystyle k_{1}}
1492:
1490:{\displaystyle L_{1}}
1465:
1433:
1284:
1244:
1107:
1038:
890:
857:
744:as though it were an
703:characteristic length
637:electrical resistance
546:
526:
524:{\displaystyle L_{c}}
499:
450:electrical components
394:electromagnetic waves
385:
340:
190:
3806:
3774:
3651:
3614:
3557:
3524:
3405:
3361:
3334:
3314:
3287:
3264:
3217:
3118:
3091:
3058:
3021:
2911:
2884:
2864:
2853:{\displaystyle T(t)}
2835:
2801:
2738:
2698:
2678:
2660:{\displaystyle Q=CT}
2642:
2622:
2602:
2402:
2308:
2221:
2162:
2110:
2058:
1693:
1663:
1636:
1609:
1582:
1555:
1528:
1501:
1474:
1454:
1294:
1254:
1122:
1048:
905:
867:
782:
535:
508:
475:
424:Lumped-element model
351:
299:
197:lumped-element model
109:improve this article
18:Lumped element model
4258:Electronic circuits
2694:comes the relation
2304:Or, using symbols:
1373:
1344:
771:Thermal Resistance
761:
746:electrical resistor
589:wire-wound resistor
565:Maxwell's equations
406:Maxwell's equations
247:of the system to a
4093:System isomorphism
3940:Mechanical systems
3911:
3792:
3758:
3633:
3600:
3543:
3505:
3388:
3347:
3320:
3300:
3270:
3247:
3203:
3104:
3074:
3044:
3007:
2897:
2870:
2850:
2821:
2787:
2724:
2684:
2657:
2628:
2608:
2501:
2325:
2266:
2207:
2148:
2096:
2044:
1676:
1649:
1622:
1595:
1568:
1541:
1514:
1487:
1460:
1445:composite material
1428:
1348:
1319:
1279:
1239:
1102:
1033:
885:
852:
759:
672:thermal resistance
561:transmission lines
541:
521:
494:
380:
335:
266:Electrical systems
218:electrical systems
216:. It is useful in
193:
4263:Electronic design
4243:Conceptual models
4153:978-0-471-45728-2
3930:forensic sciences
3894:
3871:
3836:
3726:
3689:
3595:
3485:
3357:is also given by
3323:{\displaystyle m}
3273:{\displaystyle C}
2977:
2941:
2894:
2873:{\displaystyle t}
2687:{\displaystyle C}
2669:exponential decay
2631:{\displaystyle T}
2611:{\displaystyle C}
2465:
2423:
2314:
2264:
2205:
2146:
2094:
2042:
2000:
1958:
1888:
1831:
1789:
1705:
1463:{\displaystyle A}
1441:
1440:
1277:
1237:
1231:
1134:
1100:
1031:
1025:
917:
883:
850:
844:
794:
372:
327:
226:multibody systems
185:
184:
177:
159:
83:
82:
75:
16:(Redirected from
4270:
4209:
4205:
4199:
4195:
4189:
4186:
4180:
4176:
4170:
4164:
4158:
4157:
4141:
4131:
4125:
4114:
3948:all objects are
3920:
3918:
3917:
3912:
3895:
3893:
3892:
3880:
3872:
3870:
3862:
3842:
3837:
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3810:
3801:
3799:
3798:
3793:
3767:
3765:
3764:
3759:
3754:
3753:
3752:
3751:
3742:
3724:
3705:
3704:
3687:
3642:
3640:
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3601:
3593:
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3477:
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3353:
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2942:
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2896:
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2796:
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2788:
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2717:
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2609:
2510:
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2502:
2467:
2466:
2463:
2424:
2422:
2414:
2406:
2334:
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2331:
2326:
2315:
2312:
2294:Sir Isaac Newton
2275:
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2199:
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1496:
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1400:
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1343:
1338:
1337:
1306:
1305:
1288:
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1285:
1280:
1278:
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1258:
1248:
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1245:
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1227:
1208:
1207:
1194:
1188:
1187:
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1185:
1163:
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1161:
1141:
1136:
1135:
1127:
1111:
1109:
1108:
1103:
1101:
1099:
1098:
1097:
1096:
1077:
1076:
1075:
1052:
1042:
1040:
1039:
1034:
1032:
1030:
1026:
1024:
1023:
1022:
1021:
1002:
1001:
1000:
977:
971:
970:
969:
968:
946:
945:
944:
924:
919:
918:
910:
894:
892:
891:
886:
884:
882:
871:
861:
859:
858:
853:
851:
849:
845:
843:
832:
826:
825:
824:
812:
811:
801:
796:
795:
787:
762:
750:electric current
712:Plate: thickness
591:has significant
550:
548:
547:
542:
530:
528:
527:
522:
520:
519:
503:
501:
500:
495:
487:
486:
414:network analysis
404:when applied to
389:
387:
386:
381:
373:
371:
363:
355:
344:
342:
341:
336:
328:
326:
318:
317:
316:
303:
289:network analysis
180:
173:
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117:
93:
85:
78:
71:
67:
64:
58:
38:
37:
30:
21:
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4161:
4154:
4133:
4132:
4128:
4115:
4111:
4106:
4089:
4072:
4057:
3982:acoustic theory
3978:
3957:kinematic pairs
3942:
3926:
3884:
3863:
3843:
3828:
3811:
3804:
3803:
3772:
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3743:
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3522:
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2600:
2599:
2593:
2577:
2551:
2458:
2415:
2407:
2400:
2399:
2396:
2379:
2313:Rate of cooling
2306:
2305:
2287:
2281:
2250:
2249:
2239:
2224:
2219:
2218:
2191:
2190:
2180:
2165:
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2159:
2132:
2131:
2113:
2108:
2107:
2080:
2079:
2061:
2056:
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2032:
2021:
2008:
2007:
1990:
1979:
1966:
1965:
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1934:
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1920:
1909:
1896:
1895:
1877:
1864:
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1579:
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1531:
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1504:
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1471:
1452:
1451:
1404:
1380:
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1292:
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1199:
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1167:
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1142:
1120:
1119:
1078:
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1056:
1046:
1045:
1003:
982:
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972:
950:
926:
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903:
902:
875:
865:
864:
836:
827:
816:
803:
802:
780:
779:
741:
660:
644:heat conduction
609:
607:Thermal systems
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532:
511:
506:
505:
478:
473:
472:
426:
364:
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348:
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51:help improve it
48:
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4216:External links
4214:
4211:
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4200:
4190:
4181:
4171:
4168:Yunus A Cengel
4159:
4152:
4126:
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4085:
4083:implantation.
4071:
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4017:
4010:
4007:end correction
3995:
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3202:
3199:
3196:
3192:
3188:
3185:
3182:
3179:
3176:
3173:
3169:
3165:
3162:
3159:
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3150:
3147:
3144:
3140:
3136:
3133:
3128:
3124:
3101:
3097:
3071:
3068:
3064:
3043:
3039:
3035:
3032:
3029:
3026:
3006:
3003:
3000:
2997:
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2988:
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2973:
2969:
2966:
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2960:
2957:
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2936:
2931:
2928:
2925:
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2890:
2869:
2849:
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2840:
2820:
2817:
2813:
2809:
2806:
2786:
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2776:
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2763:
2760:
2757:
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2723:
2720:
2716:
2712:
2709:
2706:
2703:
2683:
2656:
2653:
2650:
2647:
2627:
2607:
2592:
2589:
2584:
2583:
2575:
2553:
2549:
2544:
2538:
2532:
2522:
2500:
2497:
2494:
2491:
2488:
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2479:
2476:
2473:
2470:
2461:
2457:
2454:
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2439:
2436:
2433:
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2427:
2421:
2418:
2413:
2410:
2395:
2392:
2378:
2375:
2324:
2321:
2318:
2283:Main article:
2280:
2277:
2262:
2257:
2253:
2246:
2242:
2236:
2231:
2227:
2203:
2198:
2194:
2187:
2183:
2177:
2172:
2168:
2144:
2139:
2135:
2130:
2125:
2120:
2116:
2092:
2087:
2083:
2078:
2073:
2068:
2064:
2039:
2035:
2028:
2024:
2020:
2015:
2011:
2004:
1997:
1993:
1986:
1982:
1978:
1973:
1969:
1962:
1954:
1950:
1946:
1941:
1937:
1933:
1928:
1924:
1916:
1912:
1908:
1903:
1899:
1892:
1884:
1880:
1876:
1871:
1867:
1859:
1855:
1851:
1846:
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1835:
1828:
1824:
1817:
1813:
1809:
1804:
1800:
1793:
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1777:
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1755:
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1726:
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1673:
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1592:
1588:
1565:
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1538:
1534:
1511:
1507:
1484:
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1459:
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1427:
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1407:
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1397:
1394:
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1371:
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1322:
1318:
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1304:
1300:
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1249:
1235:
1226:
1223:
1220:
1217:
1212:
1206:
1202:
1197:
1192:
1184:
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1178:
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1170:
1166:
1160:
1157:
1154:
1151:
1146:
1139:
1133:
1130:
1117:
1113:
1112:
1095:
1092:
1089:
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1081:
1074:
1071:
1068:
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1043:
1029:
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1017:
1014:
1011:
1006:
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996:
993:
990:
985:
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964:
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953:
949:
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937:
934:
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916:
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878:
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862:
848:
842:
839:
835:
830:
823:
819:
815:
810:
806:
799:
793:
790:
777:
773:
772:
769:
766:
765:Transfer Mode
740:
737:
733:
732:
726:
719:
713:
659:
656:
621:thermal system
615:, also called
608:
605:
540:
518:
514:
493:
490:
485:
481:
425:
422:
398:
397:
390:
379:
376:
370:
367:
362:
359:
345:
334:
331:
325:
322:
315:
311:
307:
272:
269:
267:
264:
224:), mechanical
183:
182:
97:
95:
88:
81:
80:
42:
40:
33:
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
4275:
4264:
4261:
4259:
4256:
4254:
4251:
4249:
4246:
4244:
4241:
4240:
4238:
4228:
4225:
4223:
4220:
4219:
4215:
4204:
4201:
4194:
4191:
4185:
4182:
4175:
4172:
4169:
4163:
4160:
4155:
4149:
4145:
4140:
4139:
4130:
4127:
4123:
4119:
4113:
4110:
4103:
4099:
4096:
4094:
4091:
4090:
4086:
4084:
4082:
4077:
4070:Fluid systems
4069:
4067:
4064:
4060:
4054:
4049:
4045:
4041:
4037:
4034:
4030:
4026:
4022:
4018:
4015:
4011:
4008:
4004:
4000:
3996:
3992:
3988:
3987:
3986:
3983:
3975:
3970:
3966:
3962:
3958:
3954:
3951:
3947:
3946:
3945:
3939:
3937:
3935:
3931:
3923:
3921:
3905:
3899:
3889:
3885:
3881:
3876:
3873:
3867:
3864:
3856:
3850:
3844:
3838:
3832:
3829:
3821:
3815:
3812:
3786:
3780:
3768:
3755:
3748:
3744:
3739:
3735:
3732:
3728:
3718:
3712:
3706:
3701:
3698:
3695:
3691:
3681:
3675:
3669:
3663:
3657:
3626:
3620:
3596:
3576:
3572:
3566:
3560:
3536:
3530:
3520:
3519:
3518:
3515:
3502:
3497:
3494:
3491:
3487:
3463:
3459:
3453:
3447:
3441:
3424:
3420:
3414:
3408:
3399:
3385:
3382:
3378:
3372:
3368:
3364:
3342:
3338:
3317:
3295:
3291:
3283:
3267:
3260:
3259:heat capacity
3257:. (The total
3244:
3241:
3237:
3233:
3230:
3225:
3221:
3197:
3194:
3190:
3183:
3177:
3174:
3167:
3160:
3154:
3148:
3145:
3142:
3138:
3134:
3131:
3126:
3122:
3099:
3095:
3087:
3086:time constant
3069:
3066:
3062:
3041:
3037:
3033:
3030:
3027:
3024:
3001:
2995:
2989:
2986:
2983:
2971:
2967:
2961:
2955:
2949:
2946:
2943:
2937:
2934:
2926:
2920:
2917:
2888:
2867:
2844:
2838:
2818:
2815:
2811:
2807:
2804:
2781:
2778:
2774:
2770:
2767:
2761:
2758:
2755:
2752:
2748:
2744:
2741:
2721:
2718:
2714:
2710:
2707:
2704:
2701:
2681:
2672:
2670:
2654:
2651:
2648:
2645:
2625:
2605:
2598:
2597:heat capacity
2590:
2588:
2581:
2574:
2570:
2566:
2562:
2558:
2554:
2548:
2545:
2542:
2539:
2536:
2533:
2530:
2526:
2523:
2521:
2517:
2514:
2513:
2512:
2495:
2489:
2483:
2480:
2477:
2474:
2471:
2459:
2455:
2449:
2443:
2437:
2434:
2431:
2428:
2425:
2419:
2416:
2411:
2408:
2393:
2391:
2387:
2383:
2376:
2374:
2372:
2368:
2364:
2360:
2356:
2352:
2347:
2343:
2341:
2335:
2322:
2316:
2301:
2297:
2295:
2291:
2286:
2276:
2260:
2255:
2251:
2244:
2240:
2234:
2229:
2225:
2201:
2196:
2192:
2185:
2181:
2175:
2170:
2166:
2142:
2137:
2133:
2128:
2123:
2118:
2114:
2090:
2085:
2081:
2076:
2071:
2066:
2062:
2037:
2033:
2026:
2022:
2018:
2013:
2009:
2002:
1995:
1991:
1984:
1980:
1976:
1971:
1967:
1960:
1952:
1948:
1944:
1939:
1935:
1931:
1926:
1922:
1914:
1910:
1906:
1901:
1897:
1890:
1882:
1878:
1874:
1869:
1865:
1857:
1853:
1849:
1844:
1840:
1833:
1826:
1822:
1815:
1811:
1807:
1802:
1798:
1791:
1783:
1779:
1775:
1770:
1766:
1762:
1757:
1753:
1749:
1744:
1740:
1732:
1728:
1724:
1719:
1715:
1708:
1702:
1699:
1687:
1671:
1667:
1644:
1640:
1617:
1613:
1590:
1586:
1563:
1559:
1536:
1532:
1509:
1505:
1482:
1478:
1457:
1448:
1446:
1405:
1401:
1381:
1369:
1349:
1345:
1340:
1320:
1313:
1310:
1307:
1302:
1298:
1273:
1268:
1264:
1259:
1250:
1233:
1210:
1204:
1200:
1195:
1190:
1168:
1164:
1144:
1137:
1131:
1128:
1118:
1115:
1114:
1079:
1058:
1053:
1044:
1027:
1004:
983:
978:
973:
951:
947:
927:
920:
914:
911:
901:
898:
897:
879:
876:
872:
863:
846:
840:
837:
833:
828:
821:
817:
813:
808:
804:
797:
791:
788:
778:
775:
774:
770:
767:
764:
763:
757:
754:
751:
747:
738:
736:
730:
727:
724:
720:
718:: thickness/2
717:
714:
711:
710:
709:
706:
704:
699:
697:
696:heat transfer
692:
690:
684:
681:
677:
673:
669:
665:
657:
655:
653:
648:
645:
640:
638:
634:
630:
626:
622:
618:
614:
606:
604:
602:
598:
594:
590:
586:
582:
578:
573:
570:
566:
562:
558:
554:
538:
516:
512:
491:
488:
483:
479:
469:
467:
463:
459:
455:
451:
447:
443:
439:
435:
431:
423:
421:
419:
415:
411:
407:
403:
395:
391:
377:
374:
368:
360:
346:
332:
329:
323:
313:
309:
294:
293:
292:
290:
286:
282:
278:
270:
265:
263:
261:
257:
253:
250:
246:
241:
239:
235:
231:
230:heat transfer
227:
223:
219:
215:
210:
206:
202:
199:(also called
198:
189:
179:
176:
168:
157:
154:
150:
147:
143:
140:
136:
133:
129:
126: â
125:
121:
120:Find sources:
114:
110:
104:
103:
98:This article
96:
92:
87:
86:
77:
74:
66:
56:
52:
46:
43:This article
41:
32:
31:
19:
4203:
4193:
4184:
4174:
4167:
4162:
4137:
4129:
4112:
4073:
4065:
4061:
4058:
3979:
3960:
3950:rigid bodies
3943:
3927:
3924:Applications
3769:
3646:
3516:
3400:
2673:
2594:
2585:
2572:
2568:
2564:
2560:
2556:
2546:
2540:
2534:
2524:
2515:
2397:
2388:
2384:
2380:
2362:
2348:
2344:
2339:
2336:
2303:
2299:
2289:
2288:
1688:
1449:
1442:
755:
742:
734:
731:: diameter/6
725:: diameter/4
707:
700:
693:
685:
679:
661:
651:
649:
641:
629:differential
619:, reduces a
616:
612:
610:
574:
470:
427:
410:steady state
399:
284:
276:
274:
242:
204:
200:
196:
194:
171:
162:
152:
145:
138:
131:
119:
107:Please help
102:verification
99:
69:
60:
44:
4179:pp.174-184.
4021:loudspeaker
3999:reflex port
899:Convection
776:Conduction
664:Biot number
625:temperature
595:as well as
559:(including
438:capacitance
245:state space
222:electronics
220:(including
165:August 2023
63:August 2019
4237:Categories
4104:References
4038:source, a
3114:given by:
2371:wind chill
2355:convection
2351:conduction
1116:Radiation
597:resistance
593:inductance
581:capacitors
553:wavelength
466:conducting
458:capacitors
442:inductance
434:resistance
254:, and the
209:simplified
135:newspapers
4253:Acoustics
4248:Mechanics
4198:pp.52-60.
4044:capacitor
4033:impedance
4029:subwoofer
3991:capacitor
3976:Acoustics
3897:Δ
3877:−
3848:Δ
3778:Δ
3733:−
3710:Δ
3696:−
3673:Δ
3655:Δ
3618:Δ
3573:−
3528:Δ
3492:−
3460:−
3152:Δ
3149:−
3067:−
2993:Δ
2987:−
2968:−
2947:−
2487:Δ
2481:⋅
2475:−
2456:−
2435:⋅
2429:−
2359:radiation
2320:Δ
2317:∼
2019:−
1977:−
1907:−
1850:−
1808:−
1725:−
1703:˙
1314:σ
1311:ϵ
1165:−
1132:˙
948:−
915:˙
814:−
792:˙
639:as well.
539:λ
492:λ
489:≪
462:inductors
454:resistors
366:∂
358:∂
321:∂
310:ϕ
306:∂
252:dimension
234:acoustics
4087:See also
4048:inductor
4040:resistor
4014:resistor
4003:inductor
2580:gradient
723:cylinder
676:diffused
601:inductor
585:resistor
504:, where
430:circuits
287:used in
214:topology
4046:and an
4036:voltage
3994:cavity.
3969:dampers
3965:springs
2527:is the
1289:, where
468:wires.
207:) is a
149:scholar
49:Please
4208:(2020)
4150:
4146:â261.
4025:woofer
3961:joints
3725:
3688:
3610:where
3594:
3484:
3017:where
2880:, and
2618:, and
2520:joules
2511:where
2217:, and
2054:where
729:Sphere
658:Method
569:period
460:, and
444:, and
249:finite
151:
144:
137:
130:
122:
2357:, or
721:Long
203:, or
156:JSTOR
142:books
4148:ISBN
4042:, a
3967:and
3934:HVAC
3517:If:
2571:) â
2563:) =
2367:wind
2340:rate
2338:its
1524:and
446:gain
275:The
195:The
128:news
4144:260
4120:),
4118:PDF
4027:or
3963:),
3398:).
2976:env
2893:env
2576:env
2550:env
2464:env
716:Fin
111:by
53:to
4239::
4019:A
3997:A
2353:,
2158:,
2106:,
1686:.
691:.
611:A
456:,
452:;
440:,
436:,
232:,
228:,
4156:.
4124:.
3971:.
3959:(
3952:;
3909:)
3906:t
3903:(
3900:T
3890:0
3886:t
3882:1
3874:=
3868:t
3865:d
3860:)
3857:t
3854:(
3851:T
3845:d
3839:=
3833:t
3830:d
3825:)
3822:t
3819:(
3816:T
3813:d
3790:)
3787:t
3784:(
3781:T
3756:.
3749:0
3745:t
3740:/
3736:t
3729:e
3722:)
3719:0
3716:(
3713:T
3707:=
3702:t
3699:r
3692:e
3685:)
3682:0
3679:(
3676:T
3670:=
3667:)
3664:t
3661:(
3658:T
3630:)
3627:0
3624:(
3621:T
3597:,
3588:v
3585:n
3582:e
3577:T
3570:)
3567:t
3564:(
3561:T
3540:)
3537:t
3534:(
3531:T
3503:.
3498:t
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3488:e
3481:)
3475:v
3472:n
3469:e
3464:T
3457:)
3454:0
3451:(
3448:T
3445:(
3442:+
3436:v
3433:n
3430:e
3425:T
3421:=
3418:)
3415:t
3412:(
3409:T
3386:A
3383:h
3379:/
3373:p
3369:c
3365:m
3343:0
3339:t
3318:m
3296:p
3292:c
3268:C
3245:A
3242:h
3238:/
3234:C
3231:=
3226:0
3222:t
3201:)
3198:t
3195:d
3191:/
3187:)
3184:t
3181:(
3178:T
3175:d
3172:(
3168:/
3164:)
3161:t
3158:(
3155:T
3146:=
3143:r
3139:/
3135:1
3132:=
3127:0
3123:t
3100:0
3096:t
3070:1
3063:s
3042:C
3038:/
3034:A
3031:h
3028:=
3025:r
3005:)
3002:t
2999:(
2996:T
2990:r
2984:=
2981:)
2972:T
2965:)
2962:t
2959:(
2956:T
2953:(
2950:r
2944:=
2938:t
2935:d
2930:)
2927:t
2924:(
2921:T
2918:d
2889:T
2868:t
2848:)
2845:t
2842:(
2839:T
2819:t
2816:d
2812:/
2808:Q
2805:d
2785:)
2782:t
2779:d
2775:/
2771:T
2768:d
2765:(
2762:C
2759:=
2756:t
2753:d
2749:/
2745:Q
2742:d
2722:T
2719:d
2715:/
2711:Q
2708:d
2705:=
2702:C
2682:C
2655:T
2652:C
2649:=
2646:Q
2626:T
2606:C
2573:T
2569:t
2567:(
2565:T
2561:t
2559:(
2557:T
2555:Î
2547:T
2541:T
2535:A
2525:h
2516:Q
2499:)
2496:t
2493:(
2490:T
2484:A
2478:h
2472:=
2469:)
2460:T
2453:)
2450:t
2447:(
2444:T
2441:(
2438:A
2432:h
2426:=
2420:t
2417:d
2412:Q
2409:d
2363:T
2323:T
2261:A
2256:2
2252:k
2245:2
2241:L
2235:=
2230:2
2226:R
2202:A
2197:1
2193:k
2186:1
2182:L
2176:=
2171:1
2167:R
2143:A
2138:o
2134:h
2129:1
2124:=
2119:o
2115:R
2091:A
2086:i
2082:h
2077:1
2072:=
2067:i
2063:R
2038:0
2034:R
2027:o
2023:T
2014:3
2010:T
2003:=
1996:1
1992:R
1985:2
1981:T
1972:1
1968:T
1961:=
1953:2
1949:R
1945:+
1940:1
1936:R
1932:+
1927:i
1923:R
1915:3
1911:T
1902:i
1898:T
1891:=
1883:1
1879:R
1875:+
1870:i
1866:R
1858:2
1854:T
1845:i
1841:T
1834:=
1827:i
1823:R
1816:1
1812:T
1803:i
1799:T
1792:=
1784:o
1780:R
1776:+
1771:2
1767:R
1763:+
1758:1
1754:R
1750:+
1745:i
1741:R
1733:o
1729:T
1720:i
1716:T
1709:=
1700:Q
1672:o
1668:h
1645:o
1641:T
1618:i
1614:h
1591:i
1587:T
1564:2
1560:k
1537:2
1533:L
1510:1
1506:k
1483:1
1479:L
1458:A
1426:)
1420:r
1417:r
1414:u
1411:s
1406:T
1402:+
1396:f
1393:r
1390:u
1387:s
1382:T
1378:(
1375:)
1370:2
1364:r
1361:r
1358:u
1355:s
1350:T
1346:+
1341:2
1335:f
1332:r
1329:u
1326:s
1321:T
1317:(
1308:=
1303:r
1299:h
1274:A
1269:r
1265:h
1260:1
1234:)
1225:f
1222:r
1219:u
1216:s
1211:A
1205:r
1201:h
1196:1
1191:(
1183:r
1180:r
1177:u
1174:s
1169:T
1159:f
1156:r
1153:u
1150:s
1145:T
1138:=
1129:Q
1094:f
1091:r
1088:u
1085:s
1080:A
1073:v
1070:n
1067:o
1064:c
1059:h
1054:1
1028:)
1019:f
1016:r
1013:u
1010:s
1005:A
998:v
995:n
992:o
989:c
984:h
979:1
974:(
966:r
963:v
960:n
957:e
952:T
942:f
939:r
936:u
933:s
928:T
921:=
912:Q
880:A
877:k
873:L
847:)
841:A
838:k
834:L
829:(
822:2
818:T
809:1
805:T
798:=
789:Q
517:c
513:L
484:c
480:L
378:0
375:=
369:t
361:q
333:0
330:=
324:t
314:B
178:)
172:(
167:)
163:(
153:¡
146:¡
139:¡
132:¡
105:.
76:)
70:(
65:)
61:(
47:.
20:)
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