transient heat conduction (chap# 4)

Heat and Mass Transfer II(ME-315)

Sidra Zahid.(Lecturer)E-mail:- [email protected]

Department of Engineering SciencesNational University of Sciences and TechnologyPN Engineering College, PNS Jauhar, Karachi

Heat Transfer By J. P. Holman.

Heat and Mass Transfer By F. P. Incropera and D. P. DeWitt

Heat Transfer: A Practical Approach By Yunus A Cengel (Main book)

Books

AssessmentOne-hour tests (2)30%Quizzes (6) 10% Assignments (2) 10%Final50%

ExamsNote: All exams will be closed books and closed notes Only one sheet of formulae will be provided

Ch. 1 Transient Heat ConductionTransient Conduction:

Unsteady state conduction

Fourier equation

Lumped-capacity method

Fourier and Biot number

Use of Heisler charts

Unsteady State Conduction The temperature of a body, in general, varies with time as well as position. In rectangular coordinates, this variation is expressed as T(x, y, z, t), where (x, y, z) indicates variation in the x, y, and z directions, respectively, and t indicates variation with time. The change of temperature in time which occurs when a body is heated or cooled by its environment is to be determined from the constant property diffusion equation

Models for Transient Conduction Lumped thermal capacity model

The semi-infinite solid model and

The finite-sized model.

Lumped Parameter AnalysisA small copper ball can be modeled as a lumped system, but a roastbeef cannot.

Lumped Parameter Analysis

Lumped Parameter Analysisis a positive quantity whose dimension is (time). The reciprocal of b hastime unit (usually s), and is called the time constant.

Lumped Parameter AnalysisRate of convection heat transfer between the body and its environment at that time can be determined from Newtons law of cooling asThe total amount of heat transfer between the body and the surrounding medium over the time interval t = 0 to t is simply the change in the energy content of the body:The amount of heat transfer reaches its upper limit when the body reaches the surrounding temperature T.

Lumped Parameter AnalysisSolve Examples 4.1 and 4.2 (Cengel)

Lumped Parameter AnalysisWhen the convection coefficient h is high and k is low, large temperaturedifferences occur between the inner and outer regions of a large solid.

Example

TRANSIENT HEAT CONDUCTION IN LARGEPLANE WALLS, LONG CYLINDERS, ANDSPHERES WITH SPATIAL EFFECTS

Dimensionless Groups

2. Infinite Solid Model

One Term Approximation Solutionwhere the constants A1 and 1 are functions of the Bi number only, and their values are listed in Table 41 against the Bi number for all three geometries.The function J0 is the zeroth-order Bessel function of the first kind (see Table 42).

One Term Approximation Solution

Heisler ChartsThe transient temperature charts in figures for a largeplane wall, long cylinder, and sphere were presented by M. P. Heisler in 1947 and are called Heisler charts.

They were supplemented in 1961 with transient heat transfer charts by H. Grber. There are three charts associated with each geometry:

1. determine the temperature To at the center of the geometry at a given time t. 2. determine the temperature at other locations at the same time in terms of To. determine the total amount of heat transfer up to the time t. These plots are valid for > 0.2.

Midplane temperature (from M. P. Heisler)

Temperature distribution (from M. P. Heisler)

Heating of Large Brass Plates in an OvenIn a production facility, large brass plates of 4 cm thickness that are initially at a uniform temperature of 20C are heated by passing them through an oven that is maintained at 500C (Fig). The plates remain in the oven for aperiod of 7 min. Taking the combined convection and radiation heat transfercoefficient to be h = 120 W/m2 C, determine the surface temperature of the plates when they come out of the oven.

Heating of Large Brass Plates in an OvenAssumptions:

1 Heat conduction in the plate is one-dimensional. 2 The thermal properties of the plate and the heat transfer coefficient are constant. 3 The Fourier number is > 0.2 so that the one-term approximate solutions are applicable.

Properties:The properties of brass at room temperature are k = 110 W/m C, pho = 8530 kg/m3, Cp = 380 J/kg C, and alpha=33.9 x 10^-6 m2/s (Table A-3).

Heating of Large Brass Plates in an OvenAnalysis The temperature at a specified location at a given time can be determined from the Heisler charts Noting that the half-thickness of the plate isL = 0.02 m, from Fig. 413 we haveAlso

Heating of Large Brass Plates in an Ovenand

Cooling of a LongStainless Steel Cylindrical Shaft

A long 20-cm-diameter cylindrical shaft made of stainless steel 304 comes outof an oven at a uniform temperature of 600C (Fig.). The shaft is then allowedto cool slowly in an environmentchamber at 200C with an average heattransfer coefficient of h = 80 W/m2 C. Determine the temperature at the center of the shaft 45 min after the start of the cooling process. Also, determine the heat transfer per unit length of the shaft during this time period.


Assumptions 1 Heat conduction in the shaft is one-dimensional. 2 The thermal properties of the shaft and the heat transfer coefficient are constant. 3 The Fourier number is > 0.2 so that the one-term approximate solutions are applicable.Properties The properties of stainless steel- 304 at room temperature are k = 14.9 W/m C, pho = 7900 kg/m3, Cp = 477 J/kg C, and alpha= 3.95 * 10^-6 m2/s (Table A-3).


Analysis The temperature within the shaft may vary with the radial distance r as well as time, and the temperature at a specified location at a given time can be determined from the Heisler charts. Noting that the radius of the shaft is ro = 0.1 m, from Fig. 414 we have

TRANSIENT HEAT CONDUCTION INMULTIDIMENSIONAL SYSTEMSThe temperature in a shortcylinder exposed to convection from all surfaces varies in both the radial and axial directions, and thus heat is transferred in both directions.

TRANSIENT HEAT CONDUCTION INMULTIDIMENSIONAL SYSTEMSHeislers Charts charts are used to determine the temperature distribution and heat transfer in 1-D heat conduction problems associated with a large plane wall, a long cylinder, a sphere.

Using a superposition approach called the productsolution, these charts can also be used to construct solutions for the 2-D transient heat conduction problems encountered in geometries such as a short cylinder, a long rectangular bar, or a semi-infinite cylinder orplate, and even 3-D problems associated with geometries such as a rectangular prism.

TRANSIENT HEAT CONDUCTION INMULTIDIMENSIONAL SYSTEMSA short cylinder of radius ro and height a is the intersection of a long cylinder of radius ro and a plane wall of thickness a.

TRANSIENT HEAT CONDUCTION INMULTIDIMENSIONAL SYSTEMSThe solution for the two-dimensional short cylinder of height a and radius ro is equal to the product of the nondimensionalized solutions for the one-dimensional plane wall of thickness a and the long cylinder of radius ro, which are the two geometries whose intersection is the short cylinder.

TRANSIENT HEAT CONDUCTION INMULTIDIMENSIONAL SYSTEMSA long solid bar of rectangularprofile a x b is the intersectionof two plane walls of thicknesses a and b.

TRANSIENT HEAT CONDUCTION INMULTIDIMENSIONAL SYSTEMSThe solution of a 2-D problem involves the product of two 1-D solutions

The solution of a 3-D problem involves the product of three 1-D solutions.

TRANSIENT HEAT CONDUCTION INMULTIDIMENSIONAL SYSTEMS

Cooling of a Short Brass CylinderA short brass cylinder of diameter D = 10 cm and height H = 12 cm is initially at a uniform temperature Ti = 120C. The cylinder is now placed in atmospheric air at 25C, where heat transfer takes place by convection, with a heat transfer coefficient of h = 60 W/m2 C. Calculate the temperature at (a) thecenter of the cylinder and (b) the center of the top surface of the cylinder 15 min after the start of the cooling.

Cooling of a Short Brass CylinderAssumptions 1 2-D heat conduction 2 Constant thermal properties and heat transfer coefficient.3 The Fourier number is > 0.2

Properties The properties of brass at room temperature are k = 110 W/m Cand alpha= 33.9 x 106 m2/s (Table A-3). More accurate results can be obtainedby using properties at average temperature.

Cooling of a Short Brass Cylinder

Q:- Determine the total heat transfer from the short brass cylinder ( 8530 kg/m3, Cp 0.380 kJ/kg C) discussed in previous question.

Solution:- We first determine the maximum heat that can be transferred from the cylinder, which is the sensible energy content of the cylinder relative to its environment:

ExampleA dozen approximately spherical apples, 10 cm in diameter are taken from a 30C environment and laid out on a rack in a refrigerator at 5C. They have approximately the same physical properties as water,and h is approximately 6 W/m2K as the result of natural convection. What will be the temperature of the centers of the apples after 1 hr? How long will it take to bring the centers to 10C? How much heat will the refrigerator have to carry away to get the centers to 10C?

Example

Example

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transient heat conduction (chap# 4)

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