# Fundamentals of Heat Transfer Conduction HW/Tutorial # 1 WRF Chapters 14-15; WWWR Chapters 15-16 ID Chapters 1-2 Tutorial #1 WRF#14.12, WWWR #15.26, WRF#14.1, WWWR#15.2, WWWR#15.3, WRF#15.1, WWWR # 16.2.

To be discussed on Jan. 21, 2020. By either volunteer or class list. Fundamentals of Heat Transfer Conduction, Convection, and Radiation Heat Transfer Mode Ref. ID Figure 1.1 (p 2) Heat transfer (or heat) is thermal energy in transit due to a temperature difference

Conduction Heat Transfer First mechanism - molecular interaction (e.g. gas) Greater motion of molecule at higher energy level (temperature) imparts energy to adjacent molecules at lower energy levels Second mechanism by free electrons (e.g. solid) qx q dT

k ; kT A dx A Thermal Conductivity Physical origins and rate equation (Ref. ID; Figure 1.2) Association of conduction heat transfer with diffusion energy due to molecular activities.

Thermal Conductivity of Gas Estimation of the thermal conductivity of gas Ref. WWWR pp202-203 ; WRF pp199-200 (Self Study) Derived from the gas kinetic theory: (1) Considering the summation of the energy flux associated with the molecules crossing the control surface; (2) The number of molecules involved is related to average random molecular velocity. (3) : Boltzmann constant, d: molecular diameter, m: mass per molecule.

1 k 3T / m 1.5d 2 [Unit = W/(m-K)] Thermal Conductivity of Solid

Estimation of the thermal conductivity of solid Ref. WWWR pp204; WRF pp 200 (Self Study) (1) Derived from the Wiedemann, Franz, Lorenz Equation (1872). (2) The free electron mechanism of heat conduction is directly analogous to the mechanism of electric conduction.

ke : electrical conductivity, T: absolute temperature, L: Lorenz number. L k constant 2.45 10-8W / K 2 @20o C keT dT q x ' ' k dx The proportionality constant k is a transport property known as

the thermal conductivity (W/mK) and is a characteristic of the wall material. dT T2 T1 L dx T1 T2 T q x ' ' k

k L L r Finally for the same amount of heat flow the fluxes based on The inner and out surface areas differ by approximately 42%. qr 17860

302.7kW / m2; qr 17860 212.6kW / m2 Ai 0.059 A 0.084 0 Convection: Heat transfer due to convection involves the energy exchange between a surface and an adjacent fluid Forced Convection: When a fluid is made to flow past a solid surface by an external agent such as a fan or pump

Free/Natural Convection: Warmer (or cooler) fluid next to the Solid boundary cause circulation because of the density variation Resulting from the temperature variation throughout a region of the fluid. Newtons Law of Cooling: q/A = hT q: rate of convective heat transfer (W); A: area normal to direction of heat transfer; h: convective heat transfer coefficient, T: temperature Difference between the surface and the fluid. Convective Heat Transfer Processes: Ref: ID (Figure 1.5; p7)

(a) Forced Convection, (b) Free/Natural Convection, (c) Boiling, and (d) Condensation. Boundary layer development in convection heat transfer Ref. ID (P. 6; Fig. 1.4) Radiant Heat Transfer (1) No medium is required for its propagation. (2) Energy transfer by radiation is maximum when the two Surfaces are separated by vacuum.

(3) Radiation heat transfer rate equation is given by the Stefan-Boltzmann law of thermal radiation: q T 4 A q: rate of radiant energy emission (W); A: area of emitting surface (m2); T: absolute temperature; : StefanBoltzmann Constant = 5.676 x 10-8 W/m2-K4 Radiation Exchange. Ref: ID (Figure 1.6; P. 9)

(a) At surface and (b) between a surface and large surroundings. The three modes of heat transfer have been considered separated. In real world, different modes of heat transfer are coupled. Consider the case below for steady state conduction through a plane wall with its surfaces held at constant temperature T1 and T2.

T1 T2 x L 10.27 = 2.67+3.8*2 k value for 85%

Magnesia WWWR Page 676 With interpolation The heat diffusion equation Ref. ID (P. 61; Fig. 2.8) Ref. ID (P. 62)

q x dx q x q y dy q y q z dz q z Thermal energy generation Energy storage Conservation of energy

q x dx x q y dy y q z

dz z ( 2.7 a ) ( 2.7b ) ( 2.7 c )

E g q dxdydz (2.8) T E st C p dxdydz ( 2.9) t

Ein E g E out E st (1.11a)

q x q y q z q dxdydz q x dx q y dy q z dz T C p dxdydz t (2.10)

q y q x q z T dx dy dz q dxdydz C p dxdydz

x y z t qx qy qz T

kdydz x T kdxdz y T kdxdy z ( 2.12a )

( 2.12b) ( 2.12c ) (2.11) Heat (Diffusion) Equation: at any point in the medium the rate of energy transfer by conduction in a unit volume plus the volumetric rate of thermal energy must equal to the rate of change of thermal energy stored within the volume.

T T T T k k k q C P x x y y z z t Net conduction heat flux into the controlled volume

T '' k dx q x ' ' q x dx x x (2.14) If the thermal conductivity is constant. 2

2 2 T T T q 1 T 2 2 2

k t x y z (2.15) (2.13) Where = k/( Cp) is the thermal diffusivity

Under steady-state condition, there can be no change in the amount of energy storage. T T T k k k q 0 x x y y z z If the heat transfer is one-dimensional and there is no energy generation, the above equation reduces to d

dT (k ) 0 dx dx (2.17) Under steady-state, one-dimensional conditions with no energy generation, the heat flux is a constant in the

direction of transfer. (2.16) Cylindrical coordinates (1) When the del operator is expressed in cylindrical coordinates, the general form of the heat flux vector , and hence the Fouriers

Law, is T 1 T T q" kT k (i j k ) r r Z T

k T T q " k ; q " ; q " k r r r z z

Cylindrical coordinates (2) Spherical coordinates (1) When the del operator is expressed in spherical coordinates, the general form of the heat flux vector , and hence the Fouriers Law, is T 1 T 1 T

q" kT k (i j k ) r r r sin T k T

k T q " k ; q " ; q " r r r sin r Spherical coordinates (2)

General Form of The Differential Energy Equation DT kT q C v Dt conduction

dissipation substantial derivative transient + convective Initial Conditions Values of T and v at the start of time interval of interest.

Boundary Conditions Values of T and v existing at specific positions of the boundaries of a system i.e. for given values of the significant space variables Isothermal Boundaries Insulated Boundaries Ref: ID (P. 91, Table 2.2)