Relationship to Thermodynamics

1. Relationship to Thermodynamics Chapter One Section 1.3 Lecture 2

2. CONSERVATION OF ENERGY (FIRST LAW OF THERMODYNAMICS) Alternative Formulations • An important tool in heat transfer analysis, often • providing the basis for determining the temperature • of a system. • Alternative Formulations Time Basis: At an instant or Over a time interval Type of System: Control volume Control surface

3. (1.12c) Volumetric Phenomena • Over a Time Interval (1.12b) Each term has units of J. APPLICATION TO A CONTROL VOLUME CV at an Instant and over a Time Interval Note representation of system by a control surface (dashed line) at the boundaries. • At an Instant of Time: Surface Phenomena Conservationof Energy Each term has units of J/s or W.

4. Special Cases (Linkages to Thermodynamics) • Transient Process for a Closed System of Mass (M) Assuming Heat Transfer • to the System (Inflow) and Work Done by the System (Outflow). For negligible changes in potential or kinetic energy (1.12a) Internal thermal energy At an instant Closed System Over a time interval

5. At an Instant of Time: • Steady State for Flow through an Open System without Phase Change or Generation: (1.12d) Open System

6. (1.13) Conservation of Energy (Instant in Time): Surface Energy Balance THE SURFACE ENERGY BALANCE A special case for which no volume or mass is encompassed by the control surface. • Applies for steady-state and transient conditions. • With no mass and volume, energy storage and generation are not pertinent to the energy • balance, even if they occur in the medium bounded by the surface. Consider surface of wall with heat transfer by conduction, convection and radiation.

7. Methodology METHODOLOGY OF FIRST LAW ANALYSIS • On a schematic of the system, represent the control surface by • dashed line(s). • Choose the appropriate time basis. • Identify relevant energy transport, generation and/or storage terms • by labeled arrows on the schematic. • Write the governing form of the Conservation of Energy requirement. • Substitute appropriate expressions for terms of the energy equation. • Solve for the unknown quantity.

8. SECOND LAW OF THERMODYNAMICS Second law • An important tool to determine how heat transfer affects • the efficiency of energy conversion. For a reversible heat engine neglecting heat transfer effects between the heat engine and large reservoirs, the Carnot efficiency is (1.15, 1.16) where and are the absolute temperatures of large cold and hot reservoirs, respectively. For an internally reversible heat engine with heat transfer to and from the large reservoirs properly accounted for, the modified Carnot efficiency is (1.17) where and are the absolute temperatures seen by the internally reversible heat engine. Note that and are heat transfer rates (J/s or W).

9. Second law (cont.) • Heat transfer resistances associated with, for example, walls separating the • internally reversible heat engine from the hot and cold reservoirs relate the heat transfer rates to temperature differences: (1.18 a,b) In reality, heat transfer resistances (K/W) must be non-zero since according to the rate equations, for any temperature difference only a finite amount of heat may be transferred. The modified Carnot efficiency may ultimately be expressed as (1.21) • From Eq. 1.21,

10. Methodology for Solving Heat Transfer Problems • Known • Find • Schematic • Assumptions • Properties • Analysis • Comments

11. Methodology Ice of mass M at the fusion temperature (Tf=0 C) is enclosed in a cubical cavity of width W on a side. The cavity wall is of thickness L and thermal conductivity k. If the outer surface is at a temperature T1>Tf,obtain an expression for the time required to completely melt the ice.

12. Methodology • Known:State briefly and concisely what is known about the problem. Don’t repeat problem statement. Known:Mass and temperature of ice. Dimension, thermal conductivity and surface temperature of containing wall.

13. Methodology 2. Find:State briefly and concisely what must be found. Find:Expression for time needed to melt the ice.

14. Methodology • Schematic:Draw a schematic of physical system. Identify control volume and heat transfer processes if necessary. Schematic:

15. Methodology 4. Assumptions:List all permissible simplifying assumptions. Assumptions: 1. Inner surface of wall is at Tf; 2. Constant properties; 3. Steady-state, 1-D conduction through each wall; 4. Conduction area of one wall may be approximated as W2 (L<<W).

16. Methodology 5. Properties:Compile property values needed. • Analysis:Applying conservation laws and introduce rate equations. Perform complete analysis before substituting numerical values.

17. Methodology Analysis: Apply 1st law over time interval t=tm (melting time): where the increase in energy stored within the control volume is due exclusively to the change in latent heat energy associate with conversion from the solid to liquid state. Heat is transferred to the ice by means of conduction through the container wall, and since the temperature difference across the wall is assuming to remain at (T1-Tf) throughout the melting process, the wall conduction rate is a constant:

18. Methodology Analysis: Apply 1st law over time interval t = tm (melting time): and the amount of energy inflow is: the mount of energy required to effect such phase change per unit mass of solid is termed the latent heat of fusion hsf. Hence the increase in energy storage is:

19. Methodology By submitting into the first law expression, it follows that: 7. Comments: Discuss your results. Comments: 1. If the ice was subcooled, the storage term would have to include sensible heat change. 2. Note that the units of K and C cancel each other in the foregoing expression for tm. Such cancellation occurs in heat transfer driven by linear temperature difference. Lecture 2

20. Problem: Silicon Wafer Problem 1.57: Thermal processing of silicon wafers in a two-zone furnace. Determine (a) the initial rate of change of the wafer temperature and (b) the steady-state temperature. SCHEMATIC:

21. Problem: Silicon Wafer (cont.) or, per unit surface area <

22. Problem: Silicon Wafer (cont.) <

23. 535 J/kg· Problem: Cooling of Spherical Canister Problem 1.64: Cooling of spherical canister used to store reacting chemicals. Determine (a) the initial rate of change of the canister temperature, (b) the steady-state temperature, and (c) the effect of convection on the steady-state temperature.

24. Problem: Cooling of Spherical Canister (cont.) SCHEMATIC: <

25. Problem: Cooling of Spherical Canister (cont.) < <

26. Problem: Cooling of Spherical Canister (cont.)