# HEAT TRANSFER 1 INTRODUCTION AND BASIC CONCEPTS 5 2 CONDUCTION

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1 HEAT TRANSFER 1 INTRODUCTION AND BASIC CONCEPTS 5 2 CONDUCTION 11 Fourier s Law of Heat Conduction, General Conduction Equation Based on Cartesian Coordinates, Heat Transfer Through a Wall, Composite Wall (Materials in Series), Composite Wall (Materials in Parallel), Plane Wall With Internal Heat Generated, Overall Heat Transfer Coefficient, General Conduction Equation Based on Polar Cylindrical Coordinates, Heat Transfer From a Circular Duct, Critical Radius of Insulation, Unsteady State Heat Conduction, Lumped Capacity System 3 CONVECTION 55 Newton s Law of Cooling, Film Heat Transfer Coefficients, Heat Transfer to a Flowing Fluid, Types of Convection, Natural Convection, Mechanisms of Natural Convection, Natural Convection Over Surfaces, Forced Convection, Mechanism of Forced Convection, Velocity Boundary Layer, Thermal Boundary Layer 4 RADIATION 78 Black Body Radiation, Radiation Laws, Planck s Law, Stefan Boltzmann Law, Wien s Displacement Law, Radiation Properties, Emissivity, Receiving Properties, Kirchhoff s Law, The View Factor, View Factor Relation, The Reciprocity Rule, The Summation Rule, Radiation Heat Transfer, Net Radiation Between Two Surfaces, 5 BOILING AND CONDENSATION 100 Heat Transfer to Boiling Liquids, Flow Boiling Regimes, Heat Transfer to Condensation, Film Wise Condensation, Drop Wise Condensation 6 HEAT EXCHANGERS 112 Classification of Heat Exchangers, Construction of Heat Exchangers, Plate Heat Exchangers, Shell And Tube Heat Exchangers, Types of Shell And Tube Heat Exchangers Front And Rear Heads, Types of shell And Tube Heat Exchangers, Description of Internal Parts of Shell And Tube Heat Exchangers, Tubes, Shells, Nozzles, Front And Rear End Heads, Baffles, Tube Sheets, Double Pipe Heat Exchangers, Fluidized Bed Heat Exchangers, Heat Exchangers Analysis, Fluid Flow, Balance And Effectiveness, Exchanger Equation, Fouling 6 EVAPORATION 153 Evaporators, Classification of Evaporators, Batch Pan Evaporators, Natural Circulation Evaporators, Rising Film Evaporators, Falling Film Tubular Evaporators, Rising / Falling Film Tubular Evaporators, Forced Circulation Evaporators, Wiped Film Evaporators, Plate Type Evaporators, Evaporators Analysis, Capacity, Evaporator Economy CAREER AVENUES/HT 1

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3 CHAPTER 1 INTRODUCTION TO HEAT TRANSFER BASIC CONCEPTS CONDUCTION, CONVECTION AND RADIATION 1.1 INTRODUCTION In describing heat transfer problems, students often make the mistake of interchangeably using the terms heat and temperature. Actually, there is a distinct difference between the two. Temperature is a measure of the amount of energy possessed by the molecules of a substance. It is a relative measure of how hot or cold a substance is and can be used to predict the direction of heat transfer. Heat is energy in transit. The transfer of energy as heat occurs at the molecular level as a result of a temperature difference. Heat is capable of being transmitted through solids and fluids by conduction, through fluids by convection, and through empty space by radiation. The symbol for heat is Q. Heat v/s Work Distinction should also be made between the energy terms heat and work. Both represent energy in transition. Work is the transfer of energy resulting from a force acting through a distance. Heat is energy transferred as the result of a temperature difference. Neither heat nor work are thermodynamic properties of a system. Heat can be transferred into or out of a system and work can be done on or by a system, but a system cannot contain or store either heat or work. Heat into a system and work out of a system are considered positive quantities Thermodynamics provides the following information about a system: Energy can be transferred between a system and its surroundings. A system interacts with its surroundings by exchanging work and heat It deals with equilibrium states (Ideal conditions). But Thermodynamics does not give information about the rates at which energy is transferred, mechanisms through with energy is transferred, factors affecting the energy transfer and real time problems & applications In this chapter, we will learn: What is heat transfer, how heat is transferred, relevance and importance of the subject. CAREER AVENUES/HT 3

4 Thermodynamics v/s Heat Transfer Thermodynamics tells about Heat Transfer tells about how much heat is transferred(dq) how heat(dq) is transferred (different modes of heat transfer) how much work is done(dw) At what rate heat(dq) is transferred Final state of the system Temperature difference inside the body It is more like a Test Cricket. Final Scores are Important. It is more like ODI. Run rate is Important. Definition of Heat Transfer Heat transfer deals about study of thermal energy transfer that is induced by a temperature difference (or gradient) Modes of heat transfer Conduction heat transfer: Occurs when a temperature gradient exists through a solid or a stationary fluid (liquid or gas). Convection heat transfer: Occurs within a moving fluid, or between a solid surface and a moving fluid, when they are at different temperatures Thermal radiation: Heat transfer between two surfaces (that are not in contact), often in the absence of an intervening medium Conduction Transfer of energy from the more energetic to less energetic particles of a substance by contact of particles of matter. Mechanisms of conduction: Lattice Vibration & free electron theory Particle collisions CAREER AVENUES/HT 4

5 Lattice Vibration (seen in solids): 1) Network of atoms 2) Vibrate hot side 3) Whole structure Particle collision (observed in solids & fluids): 1) Particle from hot side migrates to the cold side 2) hot particle collides with cold particle 3) two particles have similar energy levels, both are warm Thus metals(particle collision) are good conductors of heat (Eg: Copper) as they have lot of free electrons and ceramic materials, inert gases etc., doesn t exhibit good conducting nature as they have very less free electrons. These materials are called Insulators Convection Energy transfer by random molecular motion (as in conduction) plus bulk (macroscopic) motion of the fluid. Relative motion between the surfaces is essential for convection. In a typical convective sequence hot surface heat the surrounding fluid (at the interface) thus the interfacial fluid gets heated and becomes warmer. A fluid movement gets created which replaces the warmer fluid (at the interface) with cooler fluid (in the main stream), which can draw more heat away from the surface. Based on nature of fluid movement creation, convection is classified as: Natural (free) convection: flow induced by buoyancy forces, arising from density differences arising from temperature variations in the fluid CAREER AVENUES/HT 5

6 In this case, the fluid movement is created by the warm fluid itself. The density of fluid (at the interface) decrease as it is heated; thus, hot fluids become lighter than cool fluids. A warm fluid surrounding a hot objects rises, and is replaced by cooler fluid resulting in the self-induced fluid movement. Example: Boiling of water in a kettle 1) hot surface heats surrounding air 2) warm air rises 3) air is replaced by cool air 4) circulation begins Forced convection: flow movement caused by external means like fan, blower etc. Problem: What makes a windy, winter day feel much colder than a calm day with same temperature? Solution: Due to high forced convection is the heat loss from your body is increased due to the constant replenishment of cold air by the wind Radiation Heat transfer between two surfaces (that are not in contact), often in the absence of an intervening medium. Salient features involved in radiation Thermal radiation is energy emitted by matter Does not require presence of a medium Can occur from solid surfaces, liquids and gases. Energy is transferred in form of electromagnetic waves Anybody with temperature above absolute 0 Kelvin emits radiation CAREER AVENUES/HT 6

7 In Radiation we define, Emissive power E as the rate at which energy is released per unit area (W/m 2 ) (radiation emitted from the surface) Irradiation G as the rate of incident radiation per unit area (W/m 2 ) of the surface (radiation absorbed by the surface), originating from its surroundings 1. 5 Combined modes of heat transfer Several real time heat transfer situations would involve more than one mode of heat transfer. In such cases, the most convenient form of expressing the heat flow is in form of overall heat transfer coefficient which includes all modes of heat transfer. CAREER AVENUES/HT 7

8 1. 6 Phase change heat transfer All the above cases discussed so far involves sensible heat (internal energy) of the media. However, even the latent heat of the media could be exploited for the heat transfer purposes which includes. Boiling Heat transfer Condensation Heat transfer Such phase change heat transfer is present in evaporators, cooling towers, condensers and refrigerators. Problem: Identify the heat transfer processes that is involved in the figure below CAREER AVENUES/HT 8

9 CHAPTER 2 CONDUCTION Conduction heat transfer is defined as heat transfer in solids and fluids without bulk motion. Heat conduction generally takes place in solids, though it may occur in fluids without bulk motion. In fluids, conduction is due to the collusions of the molecules during their random motion. In solids, it is due to the combination of vibrations of molecules in the lattice and the energy transport by free electrons Fourier Law of Conduction: Rate of heat conduction is proportional to the area measured normal to the direction of heat flow and to the temperature gradient in that direction. Thus the rate of heat conduction through a wall (q x ) as shown in figure with constant thickness is proportional to the temperature difference (T 2 T 1 ) between the surfaces and the area normal to the heat flow direction (A) and is inversely proportional to the thickness of the wall (L). CAREER AVENUES/HT 9

10 Fourier s law,. (1) Thermal conductivity: Thermal conductivity is defined as the ability of a substance to conduct heat. It is defined as the amount of heat conducted by a material per unit length per unit temperature difference. It is a transport property. In general, k solid > k liquid > k gas. Thermal conductivity varies with temperature. Thermal insulations consist of low conductivity materials Consider a wall as shown; we know that the rate of heat transfer through the wall increases, when: The temperatures difference between the left and right surfaces increase, The wall surface area normal to the direction of heat flow increases, The wall thickness reduces, The wall is change from brick to aluminum. If we measure temperatures of the wall from left to right and plot the temperature variation with the wall thickness, we get: CAREER AVENUES/HT 10

11 Relative to the heat flow direction, the slope of the temperature line is negative as the temperature decrease with the heat flow direction. Problem Determine the steady state rate of heat transfer per unit area through a 4. 0 cm thick homogeneous slab with its two faces maintained at uniform temperatures of 38 C and 21C. The thermal conductivity of the material is W/m. K. A) B) C) D) Solution: L =. 04m; k = W/mk; T1 = (38+273)K; T2 = (21+273)K. Substituting in equation1 we get q x = W/m 2 (per unit area mentioned in question) Ans. B 2.2 GENERAL CONDUCTION EQUATION BASED ON CARTESIAN COORDINATES Consider a control volume as shown in figure below CAREER AVENUES/HT 11

12 Applying the first law of thermodynamics to the element, we get: By combining above equations, we get CAREER AVENUES/HT 12

13 . (2) It is a general three dimensional heat conduction equation in Cartesian coordinates where, α = Thermal diffusivity = (k / ρ C p ) m 2 /s Thermal diffusivity is nothing but how fast heat is diffused through a material during changes of temperature with time. The larger the diffusivity, the faster the propagation of heat into the material. A small value of thermal diffusivity means that heat is mostly absorbed by the material and a small amount of heat will be conducted further. Case 1: No heat generation. (3) This equation is known as diffusion equation (or) Fourier s equation. Case 2: Steady-state conditions In steady state condition, the temperature does not change with time. So, (δt/δt) = 0. The heat conduction equation 2 reduces to-. (4) This equation is known as Poisson s equations. In the absence of internal heat generation, equation 4 becomes- This equation is known as Laplace equation. (5) CAREER AVENUES/HT 13

14 Case 3: One dimensional steady state heat conduction If the temperature varies only in the x direction, the equation 2 reduces to-. (6) In absence of internal heat generation (q/k) term in equation 6 becomes zero and equation gets further reduced. Case 4: Two dimensional steady state heat conduction If the temperature varies only in the x and y directions, the equation 2 becomes-. (7) In absence of internal heat generation (q/k) term in equation 7 becomes zero and equation gets further reduced. Case 5: Unsteady state, one dimensional, without internal heat generation In unsteady state, the temperature changes with time, i. e., (δt/δt) 0. So, the general conduction equation 2 reduces to-. (8) GATE Tip Reduction of general conduction equation based on Cartesian coordinates into different forms should be completely understood. CAREER AVENUES/HT 14

15 2.3 HEAT TRANSFER THROUGH A WALL For this case, the process is steady-state, no internal heat generated, and one dimensional heat flow, therefore equation (6) can be used with (q/k) term zero. By integrating equation (9):. (9). (10) by differentiating equation (10) and applying to a Fourier s law of heat conduction, the heat transfer rate through the wall is then obtained from:. (11) Then equation (11) may be written as:. (12) CAREER AVENUES/HT 15

16 2.4 COMPOSITE WALL (MATERIALS IN SERIES) Steady State Conduction: If the rate of heat conduction does not change with time, it is said to be steady state conduction. If convection on inner side and outer side of wall is considered than the equation modifies to: CAREER AVENUES/HT 16

17 GATE Tip As the system is steady-state and no internal heat generated, the heat flows enter and exit each layer are equal. Therefore: Q1 = (T a -T 1 )/R 1 Q2 = (T 1 -T 2 )/R 2 Q3 = (T 2 -T 3 )/R 3 Q4 = (T 3 -T 4 )/R 4 Q5 = (T 4 -T b )/R 5 And, Q 1 = Q 2 = Q 3 = Q 4 = Q 5 (due to steady state) CAREER AVENUES/HT 17

18 Problems (On conduction through a composite wall) The wall of a storage room is composed of three layer. The outer layer is stone lined 23 cm thick. The middle layer is brick 15 cm thick, the inside layer is cement 10 cm thick. The temperatures of the outside air is 25 o C and on the inside air is 10 o C. The film co-efficient for outside air and brick is 50 W/m 2 K. Film co-efficient for inside air and cement is 15 W/m 2 K. Find heat flow rate. Given: k for stone = 2. 1 W/mK; k for brick = 2. 7 W/mK; k for cement = W/mK; Thickness of cement, L1 = 10 cm = m Thickness of cork, L2 = 15 cm = m Thickness of brick, L3 = 23 cm = m Inside air temperature, Ta = 10 o C = 263 K Outside air temperature, Tb = 25 o C = 298 K Film co-efficient for inner side, h a = 15 W/m2K Film co-efficient for outside, h b = 50 W/m2K Q = ΔT overall / R = [T a T b ] / [(1/h a A) + (L1/k1A) + (L2/k2A) + (L3/k3A) + 1/h b A)] Find Q/A using the above relation. The negative sign of the answer indicates that the heat flows from the outside into the storage room). Problem: A heating reactor wall is made up of three layers, inside layer with thermal conductivity 8. 5 W/m K, the middle layer with conductivity W/m K, the outer layer with conductivity W/m K. The respective thickness of the inner, middle and outer layers is 25 cm, 5 cm and 3 cm respectively. The inside and outside wall temperatures are 600 o C and 50 o C respectively. Find interface temperatures. k1 = 8. 5 W/m K k2 = W/m K k3 = W/m K L1 = m L2 = m L3 = m T1 = 600 o C = 873 K T4 = 50 o C = 323 K T T T T 4 Inner lay er middle lay er outer lay er K 1 K 2 K 3 L 1 L 2 L 3 As the system is steady-state and no internal heat generated, the heat flows enter and exit each layer are equal. Convective heat transfer coefficient h a and h b are not given. So, neglect those terms. Therefore: CAREER AVENUES/HT 18

19 Q1 = (T 1 -T 2 )/R 1.. (a) Q2 = (T 2 -T 3 )/R 2 Q3 = (T 3 -T 4 )/R 3 And, Q1/A = Q2/A = Q3/A = (Q/A)Overall heat transfer (using below relation)due to steady state Q/A = [T1 T4] / [(L1/k1) + (L2/k2) + (L3/k3)] Q/A = W/m 2 Using relation (a)- Q1 = (T1-T2)/R1 = (873-T2)/R1 where R1 = L1/Ak 1 R1 = 0. 25/(A*8. 5) Q1/A = Q/A = W/m 2 Thus T2 = 846K Similarly find T3. COMPOSITE WALL (MATERIALS IN PARALLEL) A parallel composite of two materials is shown below. L R 2 L K2 A2 T 1 2 T 2 T 1 T 2 1 R1 L K1 A1 The heat transfer rate in the network is q x T1 T2 where Rtot Rtot 1 1 R11 R2 Alternately, the heat transfer rate can be calculated as the sum of heat transfer rates in the individual materials, i. e. q x q1x q 2x T1 T2 R1 T1 T2 R2 CAREER AVENUES/HT 19

20 Past GATE Questions on Concept of composite walls 1. At steady state, the temperature variation in a plane wall made of two different solids I and II is shown below. Then, the thermal conductivity of material I A) is smaller than that of II B) is greater than that of II C) is equal to that of II D) can be greater than or smaller than that of II (Note: As temperature drop in material I is higher than II, it has lower thermal conductivity than II) 2. The variation of thermal conductivity of a metal with temperature is often correlated using an expression of the form k = k 0 + at where k is the thermal conductivity, and T is the temperature (in K). The units of a in the SI system will be: A) W/m K B) W / m C) W/m K 2 D) None; a is just a number (Note: Term at has same dimensions as that of k i:e. (W/mK) CAREER AVENUES/HT 20

21 3. A composite flat wall of a furnace is made of two materials A and B. The thermal conductivity of A is twice of that of material B, while the thickness of layer of A is half of that of B. If the temperatures at the two sides of the wall are 400 and 1200 K, then the temperature drop (in K) across the layer of material A is A) 125 B) 133 C) 150 D) 160 (Note: Draw the diagram of composite walls and apply basic equations) 4. A composite wall consists of two plates A and B placed in series normal to the flow of heat. The thermal conductivities are k A and k B and the specific heat capacities are C pa and C pb, for plates A and B respectively. Plate B has twice the thickness of plate A. At steady state, the temperature difference across plate A is greater than that across plate B when A) C pa > C pb B) C pa < C pb C) k A < 0. 5 k B D) k A > 2 k B (Note: Draw the diagram of composite walls and apply basic equations) 5. The composite wall of an oven consists of three materials A, B and C. Under steady state operating conditions, the outer surface temperature T so is 20 C, the inner surface temperature T si is 600 C and the oven air temperature is T = 800 C. For the following data thermal conductivities k A = 20 W/(m K) and k C = 50 W/m K), thickness L A = 0. 3 m, L B = m and L C = m inner-wall heat transfer coefficient h = 25 W/m 2 K), the thermal conductivity k B W/(mK) of the material B, is calculated as A) 35 B) C) D) CAREER AVENUES/HT 21

22 6. Two plates of equal thickness (t) and cross-sectional area, are joined together to form a composite as shown in the figure. If the thermal conductivities of the plates are k and 2k then, the effective thermal conductivity of the composite is A) 3k/2 B) 4k/3 C) 3k/4 D) 2k/3 7. For the composite wall shown below (case 1), the steady state interface temperature is 180 C. If the thickness of layer P is doubled (Case 2), then the rate of heat transfer (assuming 1-D conduction) is reduction by A) 20% B) 40% C) 50% D) 70% CAREER AVENUES/HT 22

23 Answer key 1-A 2-C 3-D 4-C 5-B 6-B 7-B Explanatory Answers 3. Ans: (D) Explanation: Let T be the temperature of wall connecting layers A and B. Given, K A = 2K B x A = 1 2 x B T1T T T2 We have, q = x A x B KAA KBA KA K B (T 1 T) = (T T 2 ) x x 2K 1 x 2 (T 1 T) B A B K B = (T T 2 ) x (400 T) 4 = T 1200 T = 560 K Temperature drop across layer of material A, T A = T A = 160 K B B 4. Ans: (C) Explanation: The heat flux through plate A, TA Q A = K A x QAx T A = K A The heat flux through plate B, TB Q B = K B 2x QB2x T B = K B (i) (ii) CAREER AVENUES/HT 23

24 Since, both plates are in series, Q A = Q B = Q For T A > T B, Q x Q2x > K K 1 A 2 K > A K B B K B K A < 2 K A < 0. 5 K B 5. Ans: (B) Explanation: Equating the heat flux due to conduction and convection, Tsi Tso = h (T T si ) R R R A B C T si T so LA LB L K K K C A B C = h (T T si ) LA LB LC 1 Tsi T so (i) KA KB KC h T Tsi Given, L A = 0. 3 m, L B = m, L C = m K A = 20 W/ (m K), K C = 50 W/ (m K), K B =? h = 25 W/ (m 2 K), T si = C T so = 20 0 C, T = C Putting values in Eq. (i), KB = K B K B = W/ (m K) 6. Ans: (B) Explanation: Overall heat transfer resistance, R = R 1 + R 2 2t t t K A KA 2KA Or eff K K 2K 2K eff K eff = 4K 3 CAREER AVENUES/HT 24

25 7. Ans: (B) Explanation: Case 1 The rate of heat transfer q T T A x / K R Here, q A R1 R2 Or R 1 = 2R 2 q1 160 A R2 (i) Case 2 If the thickness of layer P is doubled, the resistance will also be doubled to 2R 1. q A 2R R 1 2 q (ii) A 5R 2 R 2 Percentage reduction in rate of heat transfer Q = q 1/ A q 2 / A 100 q / A R2 R2 = R 2 = 40% 1 CAREER AVENUES/HT 25

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