HEAT & MASS TRANSFERBMCT 3143

CHAPTER 1HEAT TRANSFER-INTRODUCTION

CHAPTER 2ONE-DIMENTIONAL CONDUCTION

CHAPTER 3MULTIPLE DIMENSION CONDUCTION

CHAPTER 4UNSTEADY STATE CONDUCTION

HEAT & MASS TRANSFERBMCT 3143

CHAPTER 4CHAPTER 4UNSTEADY STATE CONDUCTION

LUMPED SYSTEM ANALYSIS•There are many problems encountered in engineering practice where the temperature varies with respect to time. Such systems are known as unsteady state heating or cooling.

•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)-known as lumped system analysis- systems that can be considered uniform in

3

analysis- systems that can be considered uniform in temperature.

Consider a small hot copper ball coming out of an oven.Measurements indicate the temperature of the copperball changes with time, but it does not change withposition at any given time. Thus the temperature of theball remains uniform at all times, and we can talk aboutthe temperature of the ball with no reference to aspecific location.

A small copper ball can be modeled as a lumped system, but a roast beef cannot.

LUMPED-HEAT-CAPACITY SYSTEM: Uniform temperature through the solid

Hot steel ball

qCold water

Equation:

Boundary Conditions:

Solution:

Thermal Network for the single-capacity system

Thermal resistance:

Thermal capacitance:

Time constant:

Applicability:

Criteria for Lumped System Analysis

Lumped system analysis is applicable if

When Bi ≤ 0.1, the temperatures within the body relative

Characteristiclength/dimension

Biot number

5

When Bi ≤ 0.1, the temperatures within the body relative to the surroundings (i.e., T −T∞) remain within 5 percent of each other.

Assumption on Lumped Capacitance method

Surface convection resistance is large compared with the internal conduction resistance

Example:

A person is found dead at 5pm in a room whose temperature is 20oC. The temperature of the body is measured to be 25oC when found and the heat transfer coefficient is estimaed to be 8 W/m2.K. Modeling the body as a 30 cm be 8 W/m2.K. Modeling the body as a 30 cm diameter, 1.70 m long cylinder and using the lumped system analysis as a rough approximation, estimate the time death of that person.

Assumptions 1 The body can be modeled as a 30-cm-diameter, 1.70-m-long cylinder. 2 The thermal properties of the body and the heat transfer coefficient are constant.3 The radiation effects are negligible. 4 The person was healthy(!) when he or she died with a body temperature of 37°C.

PropertiesThe average human body is 72 percent water by mass, and thus we can assume the body to have the properties of water at the average temperature of (37 + 25)/2 = 31°C; k = 0.617 W/m · °C, ρ = 996 kg/m3, and cp = 4178 J/kg·°C (Table A–9).

Analysis The characteristic length and the Biot number areAnalysis The characteristic length and the Biot number are

Example:

A steel ball (c = 0.46kJ/Kg.K, k=35W/mK, ρ= 7800 kg/m3) , 5 cm in diameter and initially at a uniform temperature of 450 0C is suddenly placed in a controlled environment in which the temperature is maintained at 100 0C. The convection heat transfer coefficient is 10W/m2.K. Calculate the time required for coefficient is 10W/m2.K. Calculate the time required for the ball to obtain a temperature of 150 0C.

TRY THIS..

1) A copper plate of 2 mm thick is heated up to 400oC and then quenched into water at 20oC. Find the time required for the plate to reach the temperature of 40oC. Assume the following data:

h= 80 W/m2.KPlate dimensions = 30 cm x 30 cm.Cp =0.4 kJ/ kg.Kρ= 8800 kg/m3

2) One thousand spheres made of copper of diameter 10 mm are annealed in the annealing furnaceInitial temperature of the spheres = 20oC. Temperature of the annealing furnace = 400oC. The properties of copper are given as

h= 30 W/m2.KCp =0.4 kJ/ kg.Kρ= 8200 kg/m3Find the time required for the spheres to reach the temperature of 300oC

TRANSIENT HEAT CONDUCTION IN SEMI-INFINITE SOLIDS

Semi-infinite solid: An idealized body that has a single plane surface and extends to infinity in all directions.

The earth can be considered to be a semi-infinite medium in determining the variation of temperature near its surface.

11

Schematic of a semi-infinite body.

temperature near its surface.

A 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.

For 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.

TRANSIENT HEAT FLOW IN A SEMI-INFINITE SOLID

Differential Equation: Boundary Conditions:

Solution:

Analytical solution for the case of constant temperature Ts on the surface

13

Error function

Complementary error function

Error function is a standardmathematical function, just like thesine and cosine functions, whose valuevaries between 0 and 1.

14

Example

A highway made of asphalt is initially at a uniform temperature of 55oC. Suddenly the highway surface temperature is reduced to 25oC by rain. Determine the temperature at the depth of 3 cm from the highway surface and the heat flux transferred from the highway after 60 minutes. Assume the highway surface temperature is minutes. Assume the highway surface temperature is maintained at 25oC.