Lecture 4 Conduction

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4364

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Mechanical Engineering

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Oct 30, 2023

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8/29/23 1 MECE 4364 Heat Transfer Dr. Dong Liu Department of Mechanical Engineering University of Houston 1 Lecture 4 1 Last Lecture ¨ For single mode problems, you may identify the heat transfer process and apply the rate equations directly to find the answer ¨ For multi-mode problems, you may have to 1) Analyze the heat transfer processes to determine which modes must be considered 2) Define the control volume 3) Apply the energy conservation equation and correlate the respective terms to the rate equations of conduction, convection or radiation 2 ࠵? = −࠵?࠵? ࠵?࠵? ࠵?࠵? = ࠵?࠵? ∆࠵? ࠵? ∆࠵? = ࠵? ! − ࠵? " Fourier’s law Newton’s law of cooling ࠵? = ℎ࠵? ࠵? ! − ࠵? " ࠵? #$% = ࠵? & ࠵?࠵? & ࠵? & ’ − ࠵? ( ࠵?࠵? ( ࠵? ( ’ Radiation heat transfer Stefan-Boltzmann’s law ࠵? = ࠵?࠵?࠵?࠵? ! ’ ̇ ࠵? ./ − ̇ ࠵? 012 + ̇ ࠵? 34/ = ̇ ࠵? 52 2

8/29/23 2 Example 5: Multimode Effects ¨ A thin electrical heating element provides a uniform heat flux ࠵? 6 77 to the outer surface of a duct through which air flows. The duct wall is 10-mm thick and a thermal conductivity of 20 W/m K. At a particular location, the air temperature is 30℃ and the convection heat transfer coefficient between the air and inner surface of the duct is 100 W/m 2 K. a) What heat flux ࠵? 0 77 is required to maintain the inner surface of the duct wall at ࠵? . = 85℃ ? b) For the conditions of part (a), what is the temperature (To) of the duct surface next to the heater? 3 3 Example 5 4 4

8/29/23 3 Example 5 5 5 Problem-Solving Technique ¨ Step 1: Problem Statement ¤ Key information given ¤ Quantities to be found ¨ Step 2: Schematic ¤ Control volume ¤ Key energy and mass transfer processes ¨ Step 3: Assumptions and Approximations ¤ Simplifications ¨ Step 4: Physical Laws ¤ Single mode or multimode ¤ Conservation equations and rate equations ¨ Step 5: Properties ¤ Property tables ¨ Step 6: Calculations ¤ Solve equations and compute numbers 6 6

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8/29/23 4 Conduction (Chapter 2) 7 7 Conduction q Fourier ’ s law : discovered by Joseph Fourier in 1822 1) Heat transfer area A is always normal to the direction of heat transfer ( A does not have to be a constant ) 2) k is the thermal conductivity ( k does not have to be a constant ) 3) Rate of conduction in a given direction is proportional to the temperature gradient in that direction 4) Negative sign ensures that heat is conducted in the direction of decreasing temperature q In heat flux form q It has a magnitude and a direction … 8 ࠵? 9 = −࠵?࠵? ࠵?࠵? ࠵?࠵? Jean Fourier (1768-1830) ࠵? 9 77 = ࠵? ࠵? = −࠵? ࠵?࠵? ࠵?࠵? dT/dx is simply the slope of the temperature curve on a T-x diagram. 8

8/29/23 5 Fourier’s Law ¨ Heat flux is a vector! ¨ In 3D, Fourier’s law becomes ¤ In the Cartesian coordinates ¤ Knowing the temperature field ࠵? = ࠵? ࠵?, ࠵?, ࠵? , heat flux can be calculated 9 ⃗࠵? %% = −࠵?∇࠵? ⃗࠵? %% = −࠵?∇࠵? = −࠵? ࠵?࠵? ࠵?࠵? ⃗ ࠵? + ࠵?࠵? ࠵?࠵? ⃗ ࠵? + ࠵?࠵? ࠵?࠵? ࠵? ࠵? & %% = −࠵? ’( ’& ࠵? ) %% = −࠵? ’( ’) ࠵? * %% = −࠵? ’( ’* ⃗࠵? %% = ࠵? & %% ⃗ ࠵? + ࠵? ) %% ⃗ ࠵? + ࠵? * %% ࠵? 9 Example 1 10 10

8/29/23 6 Example 1 11 11 Example 2 12 12

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8/29/23 7 Example 2 13 13 Example 3 14 The temperature distribution across a wall 1 m thick at a certain instant of time is given as ࠵? ࠵? = 900 − 300࠵? − 50࠵? : where T is in ℃ and x is in m. A uniform heat generation q gen = 1000 W/m 3 is present in the wall of area 10 m 2 having the properties ρ = 1600 kg/m 3 , k = 40 W/m K, and c p = 4 kJ/kg K. ¨ Determine the rate of heat transfer entering the wall (x = 0) and leaving the wall (x = 1 m). ¨ Determine the rate of change of energy storage in the wall. 14

8/29/23 8 Example 3 15 15 Example 3 16 16

8/29/23 9 Thermal Conductivity ¨ Thermal conductivity measures the ability of a material to conduct heat ¤ A high k value indicates that the material is a good heat conductor , and a low k value indicates the material is a poor heat conductor or insulator ¤ For pure copper at room temperature k = 401 W/m · K which means a 1-m-thick copper wall will conduct heat at a rate of 401 W per m 2 area per K temperature difference across the wall 17 ࠵? 9 = −࠵?࠵? ࠵?࠵? ࠵?࠵? 17 Thermal Conductivity 18 ¨ Thermal conductivity ¨ Depends on the physical structure of the matter ¨ In general, k solid > k liquid > k gas At T = 300 K k air = 0.0263 W/mK k water = 0.613 W/mK k iron = 80.2 W/mK k silicon = 148 W/mK k copper = 401 W/mK k silver = 429 W/mK k diamond = 2200 W/mK k single-wall CNT = 3000 W/mK 18

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8/29/23 10 Thermal Conductivity of Gases 19 ¨ Thermal energy is transported by kinetic energy exchange ¨ Molecules are sparsely spaced and molecular force fields are weak ¨ Interactions via collisions of molecules ¨ No bulk motion (zero average velocity) ¨ For ideal gases, the kinetic theory predicts that ¨ The higher the temperature, the faster the molecules move and the higher the number of such collisions, and the better the heat transfer ࠵? ∝ ࠵? 19 Thermal Conductivity of Liquids 20 ¨ Thermal energy transport is by kinetic energy exchange ¨ Molecules are closely spaced and intermolecular interactions are strong ¨ Interactions via collisions ¨ Thermal conductivities of liquids usually lie between those of gases and solids ¨ Unlike gases, thermal conductivities of most liquids decrease with increasing temperature, but water is exception 20

8/29/23 11 Thermal Conductivity of Solids 21 ¨ Thermal energy transport is by ¨ Lattice vibration waves (phonons) ¨ Lattice: molecules are positioned at relatively fixed positions in a periodic manner ¨ Free electron movement in metals ¨ Thermal conductivity of a solid is obtained by adding the lattice and electronic components ¨ Thermal conductivity of an alloy of two metals is usually much lower than that of either metal 21 Thermal Conductivity 22 ¨ Thermal conductivity varies with temperature It is common practice to evaluate k at the average temperature and treat it as a constant in calculations 22

8/29/23 12 Other Important Properties ¨ Specific heat ࠵? ; ¤ The heat storage capability of a material per unit mass (unit: J/kg K) ¨ Heat capacity ࠵? ࠵? ; ¤ The heat storage capability of a material per unit volume (unit: J/m 3 K) ¨ Thermal diffusivity ࠵? ࠵? ≡ < => ! = ?@.A.2B 20 >0/C1>2 D4?2 ?@.A.2B 20 520E4 4/4E3B (࠵? : /࠵?) ¤ A larger ࠵? indicates faster the propagation of heat into the medium ¤ A small ࠵? means that heat is mostly absorbed by the material and a small amount of heat is conducted further 23 ࠵?~ ࠵?࠵? Heat diffusion length 23 Property Tables ¨ How to find these properties? ¨ Table A.1 Thermophysical properties of metallic solids 24 ࠵? = 97.1×10 F ? ࠵? = 97.1×10 GF ? 24

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8/29/23 13 Property Tables ¨ Table A.2 Thermophysical properties of non-metallic solids ¤ Similar to metals ¨ Table A.3 Thermophysical properties of common materials 25 25 Property Tables ¨ Table A.4 Thermophysical properties of gases at 1 ATM (1 standard atmospheric pressure = 101325 Pa) 26 26