M3 Conduction Lumped system analysis PDF

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Lumped system analysis...

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MEE 2005

HEAT TRANSFER MODULE -3 CODUCTION -II Course Instructor Nataraj G Assistant Professor (Sr.) School of Mechanical Engineering Vellore Institute of Technology, Vellore

Unsteady state or Transient state analysis

Dr. G.Nataraj, SMEC, VIT University

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Over view of Module III Conduction II – Un-Steady state 1D heat conduction o Unsteady state heat transfer, - Systems with negligible internal resistance – (lumped heat capacity analysis) ; - Infinite bodies – flat plate, cylinder and sphere; - Semi-infinite bodies; - chart solutions.

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Systems with negligible internal resistance – (lumped heat capacity analysis)

Dr. G.Nataraj, SMEC, VIT University

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Transient state analysis- lumped heat capacity method In heat transfer analysis,  some bodies are observed to behave like a “lump” - whose interior temperatures remains essentially uniform at all times during a heat transfer process.  The temperature of such bodies can be taken to be a function of time only T(t).  Consider a small hot copper ball coming out of an oven (Fig). Measurements indicate the temperature of the copper ball changes Copper Ball With Uniform Temperature with time, but it does not change with position at any given time.  Thus the temperature of the ball remains uniform at all times, and we can talk about the temperature of the ball with no reference to a specific location.  Now let us consider a large potato put in a vessel with boiling water. After few minutes, if you take out the potato, you must have noticed that the temperature distribution within the potato is not even close to being uniform.  Thus, lumped system analysis is not applicable in this case.

Dr. G.Nataraj, SMEC, VIT University

Potato With Different Temperatures At Different Locations 5

Transient state analysis- lumped heat capacity method  Before presenting a criterion about applicability of the lumped system analysis, we develop the formulation associated with it.  Consider a hot metal forging that is initially at a uniform temperature Ti and is quenched by immersing it in a liquid of lower temperature 0 , until it eventually reaches.  This reduction is due to convection heat transfer at the solid-liquid interface.  The essence of the lumped system analysis is the assumption that the temperature of the solid is uniform within the body at all times and changes with time only, T=T(t).  This assumption implies that temperature gradients within the solid are negligible.

Dr. G.Nataraj, SMEC, VIT University

Cooling Of A Hot Metal Forging

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Transient state analysis- lumped heat capacity method

During a differential time interval dt , the temperature of the body rises by a differential amount dT. An energy balance of the solid for the time interval dt can be expressed as,

 Heat transfer into   The increase in the energy      the body during dt    of the body during dt 

hA(T  T ) dt  VC pdT

(1)

Introducing the temperature difference,

  T  T And recognizing that

(2) (d  / dT)  (dT/ dt) ,it follows that

VC p d   hAs dt

(3)

where a body of volume V, surface area As, density, and specific heat Cp is initially at a uniform temperature Ti (Fig). Separating variables and integrating from the initial condition, for which t=0 and T(0)=Ti, we then obtain t VC p  d     dt hAs i  0

(4)

Dr. G.Nataraj, SMEC, VIT University

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Transient state analysis- lumped heat capacity method Where,  i  Ti  T (5) Evaluating the integrals it follows that,

 VCp hAs

ln

 t i

(6)

(Or)

  VC p    T  T   exp   t hA  i Ti  T s    

(7)

 Equation 6, may be used to determine the time required for the solid to reach some temperature T , or, conversely,  Equation 7 may be used to compute the temperature reached by the solid at some time t .  The foregoing results indicate that the difference between the solid and fluid temperatures must decay exponentially to zero as t approaches infinity. This behavior is shown in Figure.

Transient Temperature Response Of Lumped Capacitance Solids Corresponding To Different Thermal Time Constants t

Dr. G.Nataraj, SMEC, VIT University

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Transient state analysis- lumped heat capacity method  From (Equation 7) it is also evident that the quantity  VCp  may be interpreted as a  hA s 

thermal time constant. This time constant may be expressed as,

 1    (8)  VC p   R t C t hA  s  Where Rt is the resistance to convection heat transfer and Ct is the lumped thermal capacitance of the solid.  Any increase in Rt or Ct will cause a solid to respond more slowly to changes in its thermal environment and will increase the time required to reach thermal equilibrium (q=0).

 Once the temperature T(t) at any time t is available from Equation 7, the rate of convection heat transfer between the body and its environment at that time can be determined from Newton's law of cooling as,

Q  hA s [T(t)  T ]

Dr. G.Nataraj, SMEC, VIT University

(W)

(9)

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Lumped heat capacity method - problems

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2.

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Lumped heat capacity method - problems 3) A mild steel sphere of 15 mm diameter is planned to be cooled by an air flow at 20o C. The convective heat transfer co-efficient is 110 W/m2K. Calculate the following a) Time required to cool the sphere from 700oC to 150o C b) Instantaneous heat transfer rate at 150o C c) Total energy transferred up to 150o C Take for mild steel ρ = 7850 kg /m3, Cp = 474 J/kg K, α = 0.044 m2/h, k = 43 W/mK

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