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Newton's law of cooling says that the rate of heat loss from a surface is proportional to the...

Question:

Newton's law of cooling says that the rate of heat loss from a surface is proportional to the difference in the temperature between the surface and the air, i.e.{eq}\frac{dT}{dt}= -k(T - T_a) {/eq} where T is temperature, t is time, {eq}T_a {/eq} is the ambient (air) temperature and k is a rate constant. With the continuous casting of steel sheeting, the steel can not be rolled into bundles until the temperature drops beneath a certain temperature, {eq}TR = 200^\circ {/eq}C. If the initial sheet temperature is {eq}T(0)= 650^\circ {/eq} C and the air temperature is {eq}T_a= 35^\circ {/eq} C, solve the differential equation for T(t).

Newtons Law of Cooling:


The rate of change of temperature {eq}T(t) {/eq} of a body with respect to time is directly proportional to the difference between {eq}T \ \ \text{and} \ \ T_{a} {/eq} where {eq}T_{a} {/eq} is the temperature of the surrounding medium. Therefore

{eq}\hspace{30mm} \displaystyle{ \dfrac{dT}{dt} \propto (T - T_{a}) \\ \dfrac{dT}{dt} = -k (T - T_{a}) } {/eq}

where {eq}k {/eq} is the rate of constant.

Therefore the solution of this model is:

{eq}\hspace{30mm} \displaystyle{ T(t) = T_{a} + \left( T_{0} - T_{a} \right) e^{-kt} } {/eq}

where {eq}T_{0} {/eq} is the initial temperature.


Answer and Explanation: 1


Now by using Newton's Law of Cooling:

{eq}\hspace{30mm} \displaystyle{ \dfrac{dT}{dt} = -k (T - T_{a}) \\ \dfrac{dT}{dt} = -k(T - 35) \\ T(t) = 35 + (T_{0} - T_{a}) e^{-kt} \\ T(t) = 35 + 615 e^{-kt} } {/eq}

Now the at time {eq}R {/eq} the temperature drops to {eq}200^{ \circ }, {/eq} that is

{eq}\hspace{30mm} \displaystyle{ T(R) = 200 \\ 35 + 615 e^{-kR} = 200 \\ k = -\dfrac{1}{R} \ln \left( \dfrac{165}{615} \right) \\ k \approx \dfrac{1.31}{R} } {/eq}

Hence the solution of the given model is:

{eq}\hspace{30mm} \displaystyle{ \boxed{ T(t) = 35 + 615 e^{-\frac{1.31}{R} t} } } {/eq}



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Types of Heating & Cooling Systems

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Chapter 10 / Lesson 9
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You can choose from a number of different heating and cooling systems to make your home comfortable to live in. Watch this video to learn about the advantages and disadvantages of forced air, radiant heat, baseboard heat and geothermal systems.


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