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As an example, consider a composite wall of cross-sectional area <math>A</math>. The composite is made of an <math>L_1</math> long cement plaster with a thermal coefficient <math>k_1</math> and <math>L_2</math> long paper faced fiber glass, with thermal coefficient <math>k_2</math>. The left surface of the wall is at <math>T_i</math> and exposed to air with a convective coefficient of <math>h_i</math>. The right surface of the wall is at <math>T_o</math> and exposed to air with convective coefficient <math>h_o</math>.
Using the thermal resistance concept, heat flow through the composite is as follows:
<math display="block">\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}</math>
where <math>
====Newton's law of cooling====
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Or, using symbols:
▲:<math>\text {Rate of cooling} \sim \!\, \Delta T</math>
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.
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====Mathematical statement====
Newton's law is mathematically stated by the simple first-order differential equation:
where
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.
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If the entire body is treated as lumped-capacitance heat reservoir, with total heat content which is proportional to simple total [[heat capacity]] <math>C</math>, and <math>T</math>, the temperature of the body, or <math>Q = C T</math>. It is expected that the system will experience [[exponential decay]] with time in the temperature of a body.
From the definition of heat capacity <math>C</math> comes the relation <math>C = dQ/dT</math>. Differentiating this equation with regard to time gives the identity (valid so long as temperatures in the object are uniform at any given time): <math>dQ/dt = C (dT/dt)</math>. This expression may be used to replace <math>dQ/dt</math> in the first equation which begins this section, above. Then, if <math>T(t)</math> is the temperature of such a body at time <math>t</math>, and <math>T_\text{env}</math> is the temperature of the environment around the body:
where <math>r = hA/C</math> is a positive constant characteristic of the system, which must be in units of <math>s^{-1}</math>, and is therefore sometimes expressed in terms of a characteristic [[time constant]] <math>t_0</math> given by: <math>t_0 = 1/r = -\Delta T(t)/(dT(t)/dt)</math>. Thus, in thermal systems, <math>t_0 = C/hA</math>. (The total [[heat capacity]] <math>C</math> of a system may be further represented by its mass-[[specific heat capacity]] <math>c_p</math> multiplied by its mass <math>m</math>, so that the time constant <math>t_0</math> is also given by <math>mc_p/hA</math>).▼
▲:<math> \frac{d T(t)}{d t} = - r (T(t) - T_{\mathrm{env}}) = - r \Delta T(t)\quad </math>
▲<math>r = hA/C</math> is a positive constant characteristic of the system, which must be in units of <math>s^{-1}</math>, and is therefore sometimes expressed in terms of a characteristic [[time constant]] <math>t_0</math> given by: <math>t_0 = 1/r = -\Delta T(t)/(dT(t)/dt)</math>. Thus, in thermal systems, <math>t_0 = C/hA</math>. (The total [[heat capacity]] <math>C</math> of a system may be further represented by its mass-[[specific heat capacity]] <math>c_p</math> multiplied by its mass <math>m</math>, so that the time constant <math>t_0</math> is also given by <math>mc_p/hA</math>).
The solution of this differential equation, by standard methods of integration and substitution of boundary conditions, gives:
▲: <math> T(t) = T_{\mathrm{env}} + (T(0) - T_{\mathrm{env}}) \ e^{-r t}. \quad </math>
If:
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then the Newtonian solution is written as:
▲: <math> \Delta T(t) = \Delta T(0) \ e^{-r t} = \Delta T(0) \ e^{-t/t_0}. \quad </math>
This same solution is almost immediately apparent if the initial differential equation is written in terms of <math>\Delta T(t)</math>, as the single function to be solved for.
▲:<math> \frac{d T(t)}{d t} = \frac{d \Delta T(t)}{d t} = - \frac{1}{t_0} \Delta T(t)\quad </math>
===Applications===
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