INSTRUCTOR: PM DR. MAZLAN ABDUL WAHID TEXT: Heat Transfer A Practical Approach by Yunus A. Cengel Mc Graw Hill

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M 792: IUO: M D. MZL BDUL WID http://www.fkm.utm.my/~mazlan X: eat ransfer ractical pproach by Yunus.

1 M 792: IUO: M D. MZL BDUL WID X: eat ransfer ractical pproach by Yunus. engel Mc Graw ill hapter ransient eat onduction ssoc rof Dr. Mazlan bdul Wahid aculty of Mechanical ngineering Universiti eknologi Malaysia 1

2 ransient onduction Many heat transfer problems are time dependent hanges in operating conditions in a system cause temperature variation with time, as well as location within a solid, until a new steady state (thermal equilibrium) is obtained. In this chapter we will develop procedures for determining the time dependence of the temperature distribution eal problems may include finite and semi-infinite solids, or complex geometries, as well as two and three dimensional conduction olution techniques involve the lumped capacitance method, exact and approximate solutions, and finite difference methods. We will focus on the Lumped apacitance Method, which can be used for solids within which temperature gradients are negligible (ections ) Lumped apacitance Method onsider a hot metal that is initially at a uniform temperature, i, and at t=0 is quenched by immersion in a cool liquid, of lower temperature he temperature of the solid will decrease for time t>0, due to convection heat transfer at the solid-liquid interface, until it reaches ( x,0) = i x 2

3 Lumped apacitance Method If the thermal conductivity of the solid is very high, resistance to conduction within the solid will be small compared to resistance to heat transfer between solid and surroundings. emperature gradients within the solid will be negligible, i.e.. the temperature of the solid is spatially uniform at any instant. ( x,0) = i x Lumped apacitance Method tarting from an overall energy balance on the solid: h ( s ) = ρvc d dt he time required for the solid to reach a temperature is: t ρvc θ ln h s θ = i where θ = θ = he temperature of the solid at a specified time t is: θ θ i (5.1) = i h s = exp t ρvc he total energy transfer, Q, occurring up to some time t is: t 0 0 t ( ρvc) θ [ 1 exp( t τ )] Q = q dt = h θ dt = / i i i & = & t out st (5.2) (5.3) 3

4 ransient emperature esponse Based on eq. (5.2), the temperature difference between solid and fluid decays exponentially. Let s define a thermal time constant 1 τ t = Vc tt h ( ρ ) = s t is the resistance to convection heat transfer, t is the lumped thermal capacitance of the solid Increase in t or t causes solid to respond more slowly and more time will be required to reach thermal equilibrium. Validity of Lumped apacitance Method eed a suitable criterion to determine validity of method. Must relate relative magnitudes of temperature drop in the solid to the temperature difference between surface and fluid. solid (due to conduction) solid / liquid (due to convection) ( L / k) = = (1/ h) cond conv hl = Bi k What should be the relative magnitude of solid versus solid/liquid for the lumped capacitance method to be valid?

5 Biot and ourier umbers he lumped capacitance method is valid when hlc Bi = < 0.1 where the characteristic length: k L c =V/ s =Volume of solid/surface area We can also define a dimensionless time, the ourier number: q. (5.2) becomes: θ θ i o = αt 2 L c = = exp [ Bi o] (5.) i xample he heat transfer coefficient for air flowing over a sphere is to be determined by observing the temperature-time history of a sphere fabricated from pure copper. he sphere, which is 12.7 mm in diameter, is at 66 before it is inserted into an air stream having a temperature of 27. thermocouple on the outer su rface of the sphere indicates 55, 69 s after the sphere is ins erted in the air stream. alculate the heat transfer coefficient, assuming that the sphere behaves as a spacewise isothermal object. Is your assumption reasonable? 5

6 Other transient problems When the lumped capacitance analysis is not valid, we must solve the partial differential equations analytically or numerically xact and approximate solutions may be used abulated values of coefficients used in the solutions of these equations are available ransient temperature distributions for commonly encountered problems involving semi-infinite solids can be found in the literature LUMD YM LYI Interior temperature of some bodies remains essentially uniform at all times during a heat transfer process. he temperature of such bodies can be taken to be a function of time only, (t). eat transfer analysis that utilizes this idealization is known as lumped system analysis. small copper ball can be modeled as a lumped system, but a roast beef cannot. 12 6

7 Integrating with = i at t = 0 = (t) at t = t he geometry and parameters involved in the lumped system analysis. ime consta nt 13 he temperature of a lumped system approaches the environment temperature as time gets larger. his equation enables us to determine the temperature (t) of a body at time t, or alternatively, the time t required for the temperature to reach a specified value (t). he temperature of a body approaches the ambient temperature exponentially. he temperature of the body changes rapidly at the beginning, but rather slowly later on. large value of b indicates that the body approaches the environment temperature in a short time. 1 7

8 eat transfer to or from a body reaches its maximum value when the body reaches the environment temperature. he rate of convection heat transfer between the body and its environment at time t he total amount of heat transfer between the body and the surrounding medium over the time interval t = 0 to t he maximum heat transfer between the body and its surroundings 15 riteria for Lumped ystem nalysis haracteristic length Biot number Lumped system analysis is applicable if When Bi 0.1, the temperatures within the body relative to the surroundings (i.e., ) remain within 5 percent of each other. 16 8

eat ransfer in Lumped ystems When the convection coefficient h is high and k is low, large

9 17 mall bodies with high thermal conductivities and low convection coefficients are most likely to satisfy the criterion for lumped system analysis. eat ransfer in Lumped ystems When the convection coefficient h is high and k is low, large temperature differences occur between the inner and outer regions of a large solid. nalogy between heat transfer to a solid and passenger traffic to an island. 18 9

I ODUIO I LG L WLL, LOG YLID, D WI IL We will

time and position in one-dimensional problems

exposed to convection from its surfaces for i

10 I ODUIO I LG L WLL, LOG YLID, D WI IL We will consider the variation of temperature with time and position in one-dimensional problems such as those associated with a large plane wall, a long cylinder, and a sphere. ransient temperature profiles in a plane wall exposed to convection from its surfaces for i >. chematic of the simple geometries in which heat transfer is one-dimensional. 19 ondimensionalized One-Dimensional ransient onduction roblem 20 10

11 ondimensionalization reduces the number of independent variables in one-dimensional transient conduction problems from 8 to 3, offering great convenience in the presentation of results. 21 xact olution of One-Dimensional ransient onduction roblem 22 11

12 23 he analytical solutions of transient conduction problems typically involve infinite series, and thus the evaluation of an infinite number of terms to determine the temperature at a specified location and time. 2 12

13 pproximate nalytical and Graphical olutions he terms in the series solutions converge rapidly with increasing time, and for τ > 0.2, keeping the first term and neglecting all the remaining terms in the series results in an error under 2 percent. olution with one-term approximation

uniform temperature i subjected to convection from both sides to an

14 (a) Midplane temperature ransient temperature and heat transfer charts (eisler and GrÖber charts) for a plane wall of thickness 2L initially at a uniform temperature i subjected to convection from both sides to an environment at temperature with a convection coefficient of h. 27 (b) emperature distribution 28 1

15 (c) eat transfer 29 15

16 16

specified surface temperature corresponds to the case of

17 17 3 he dimensionless temperatures anywhere in a plane wall, cylinder, and sphere are related to the center temperature by he specified surface temperature corresponds to the case of convection to an environment at with a convection coefficient h that is infinite.

18 I ODUIO I MI-III OLID chematic of a semi-infinite body. or short periods of time, most bodies can be modeled as semi-infinite solids since heat does not have sufficient time to penetrate deep into the body. emi-infinite solid: n idealized body that has a single plane surface and extends to infinity in all directions. he earth can be considered to be a semi-infinite medium in determining the variation of temperature near its surface. thick wall can be modeled as a semi-infinite medium if all we are interested in is the variation of temperature in the region near one of the surfaces, and the other surface is too far to have any impact on the region of interest during the time of observation. 35 nalytical solution for the case of constant temperature s on the surface rror function omplementa ry error ransformation of variables in the derivatives of the heat conduction equation by the use of chain rule

19 rror function is a standard mathematical function, just like the sine and cosine functions, whose value varies between 0 and nalytical solutions for different boundary conditions on the surface 38 19

20 Dimensionless temperature distribution for transient conduction in a semi-infinite solid whose surface is maintained at a constant temperature s

21 1 2 Variation of temperature with position and time in a semi-infinite solid initially at

21 Variation of temperature with position and time in a semi-infinite solid initially at temperature i subjected to convection to an environment at with a convection heat transfer coefficient of h.

Chapter 4 TRANSIENT HEAT CONDUCTION

Heat and Mass Transfer: Fundamentals & Applications Fourth Edition Yunus A. Cengel, Afshin J. Ghajar McGraw-Hill, 2011 Chapter 4 TRANSIENT HEAT CONDUCTION LUMPED SYSTEM ANALYSIS Interior temperature of