lab7 - MEEN 260 Texas A&M University Laboratory Manual...

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MEEN 260 Laboratory Manual 1 T EMPERATURE M EASUREMENT IN A L UMPED C APACITANCE S YSTEM Last Updated Dec. 11, 2007, M. Lucas Purposes of the Experiment 1) To demonstrate how to measure temperature using a thermo-electric sensor and a computer aided data acquisition system. 2) To demonstrate how to estimate the heat transfer coefficient based on transient temperature response and using regression analysis techniques 3) To compare a theoretical model of heat transfer using free and forced convection Theory The Lumped Capacitance Model: The Lumped Capacitance Model is a tool that is widely used by engineers to estimate the transient temperature response of an object after it is suddenly submerged in a fluid of a different temperature. Consider a sphere that is initially at some temperature T i , and is submerged in a cold-water bath that is at some temperature T . out E T T Figure 1: A Lumped Capacitance System We would like to know the temperature of the sphere at some time t after it is dropped into the water bath. In order to do this, we can write the energy balance equation for the sphere:

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MEEN 260 Laboratory Manual 2 stored gen out in E E E E = + (1) Since no energy is coming into the sphere from the outside, and no energy is generated in the sphere, the energy balance reduces to the following: stored out E E = (2) The following equation gives the convective heat transfer between an object and the surrounding fluid: ) ( = T T hA E s out , (3) Where h is the convective heat transfer coefficient, and A s is the area of contact between the object and the surrounding fluid (air is considered a fluid in this context). The convective heat transfer coefficient, h , depends on a number of factors, including the shape and orientation of the submerged body and the fluid flow conditions around the submerged body. The rate of energy storage in the sphere is dt dT c E stored V ρ = , (4) where is the density of the sphere, V is its volume, and c is its specific heat. Of course, since the sphere is being cooled the rate of energy storage will be negative. Equation 2 then becomes dt dT c T T hA s V ) ( = (5) We can rearrange Equation 5 and integrate both sides: = t s T T dt c hA T T dT i 0 V ) ( (6) t c hA T T T T s i V ) ln( ) ln( = (7) By taking the exponential of both sides of this equation, we arrive at the following formula for the temperature as a function of time: t t i e T T T T τ = ) ( ) ( (8)
MEEN 260 Laboratory Manual 3 where s hA c V t ρ τ = has units of time, and can be viewed as a thermal time constant of the

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lab7 - MEEN 260 Texas A&M University Laboratory Manual...

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