Estimating Total Thermal Resistance of Pot of Water on Top of Stove

Transcription

1 Appendix C Heat Plate Time-to-Boil Calculation using MathCAD Estimating Total Thermal Resistance of Pot of Water on Top of Stove d pot 8.5in Pot diameter h side 0in Estimated vertical distance from pot bottom where hot gases flow A side_pot d pot h side A side_pot 0 Area along side of pot where hot gases flow d pot A bottom_pot A 4 bottom_pot 0.037m Area of pot bottom T 4surr 819K T 1surr 35.5K Surrounding flue gas temperature inside stove Surrounding water temperature inside pot T 4surr T 1surr q total Definition of Heat Transfer in terms of overall temperature difference and total thermal resistance. R conv R cond_pot R cond_plate R contact Total thermal resistance defined in terms of convective resistance, conductive resistance, and contact resistance.

2 1 R conv ha bottom_pot Thermal resistance due to convection of hot gases R cond_pot L pot k pot A bottom_pot Thermal resistance due to conduction through pot bottom R cond_plate L plate k plate A plate Thermal resistance due to conduction through plate R contact 0.0 m K W Contact resistance between bottom of pot and plate (estimated through experimentation). d plate 8.5in Plate diameter d plate A plate A 4 plate 0.037m Area of heat exchanger plate k castiron 1. W m K Thermal conductivity of plate material k stainless 16 W m K Thermal conductivity of pot material L pot.011in thickness of pot L plate.15in thickness of plate h h value from plane W 1 wall calculations R conv m h K 0.08 m K W R cond_pot L pot m K k stainless W R cond_plate L plate m K k castiron W R conv R cond_pot R cond_plate R contact m K W

3 Relative Thermal Resistances as a Percentage of the Total Resistance R cond_pot R cond_plate R contact R conv Heat transfer which is "absorbed" by the additional contact thermal resistance of the plate/pot interface m i 5.00kg mass of water at beginning of test m f T f 4.968kg 363K mass of water at end of test Final water temp at end of test T i 88K h fg J kg C p 4180 J kgk t 35min Initial water temp at beginning of test Enthalpy of evaporation of water at 363K Specific heat of water at constant pressure time required to boil 5L of water U water m i C p T f T i m i m f h fg U water J Total Internal Energy required to bring 5L of water to a boil in 35 minutes. U water Q total_nom t W Total rate of Heat Transfer from stove to a 5L pot of water, without heat exchanger plate.

4 q total_nom Q total_nom W A bottom_pot m Total rate of Heat Transfer per unit area from stove to a 5L pot of water, without heat exchanger plate. q adjust W m Adjusted heat transfer rate which corrects for the case if there is zero contact resistance. T 4surr T 1surr q total_plate W m Total rate of heat transfer to pot sitting on top of a flat, unfinned, plate with a prescribed contact resistance. q absorb q total_nom q total_plate q absorb_adj q total_nom q total_plate q adjust q absorb W m q absorb_adj W m heat transfer which is "absorbed" by contact resistance heat transfer which is "absorbed" by contact resistance, adjusted with correction factor. Calculate intermediate temperatures T 4 T 1surr q total_plate R cond_pot R cond_plate R contact Definition of heat transfer written to solve for T 4 T 4 q total_plate R cond_pot R cond_plate R contact T 1surr T K Bottom plate temperature T 3 T 1surr q total_plate R cond_pot R contact Definition of heat transfer written to solve for T 3 T 3 q total_plate R cond_pot R contact T 1surr T 3 T 4 T K T Top plate temperature plate K Average plate temperature

5 Heat transfer gains from extended fins on the bottom of pot Individual fin efficiency calculation The dimensions shown below simplify the fins shown above by assuming rectangular cross-section w 69.6mm L fin 3.3mm fin width (this is the active width of the fin which would change depending on the diameter of the pot which is placed on top of the plate). Fin length t tip 1.4mm fin tip thickness t base 3.mm fin base thickness t tip t base t ave m average fin thickness L fin max 0.61 maximum fin effectiveness value P w P 0.144m Perimeter of each individual fin

6 A c w A c m Average cross-sectional area of fin L c L fin 0.04m Fin length corrected by assuming adiabatic tip conditions instead of accounting for convective losses along the length. hp m s m k castiron A s c m simplifying expression for the above variables f tanh m s L c m s L f c individual fin efficiency A p L c corrected fin profile area 1 3 h L c k castiron A p verification with Figure 3.18 in text b T 4surr T 4 b K A f L fin w L fin A f m Total surface area of each individual fin q fin f h A f b 6.814W heat transfer rate of an individual fin fin q fin ht base w b effectiveness of an individual fin (must satisfy fin > )

7 Overall fin array efficiency calculation and resulting heat flux from finned plate surface N fin 44 Total number of fins placed on plate o q t ha t b q t q max Overall fin array efficiency d plate A b wt 4 base N A fin b 0.07m A t N fin A f A b A t 0.174m Prime surface area: Exposed portion of the base plate area which is not occupied by fins. Total exposed surface area associated with both the fin array and prime surface. N fin A f o 1 1 A f o t Overall fin array efficiency q finned o h A t b A bottom_pot q finned W m Rate of heat transfer from finned plate surface q total_finned q finned q absorb W m Total rate of heat transfer to pot sitting on finned heat exchanger plate, accounting for contact resistance. Improvement q total_finned q total_nom q total_nom Percent improvement over pot without finned plate. t boil_finned U water q total_finned A bottom_pot 6.135min Expected TTB from finned plate, without accounting for internal energy required to bring plate up to temperature. castiron 6.95 gm 3 kg cm m 3 average density of plate material

8 C p_castiron 506 J kgk average specific heat capacity of plate material v baseplate A plate L plate m 3 total volume of the baseplate v fins L fin wn fin m 3 total collective volume of all fins m plate v fins v baseplate castiron 1.948kg total mass of baseplate U plate m plate C p_castiron T plate T i J U water U plate t boil_finned_castiron q total_finned A bottom_pot 9.963min Expected TTB from a finned plate, accounting for internal energy required to bring plate up to temperature. Bi h k castiron Biot number. If Bi<<1, then the error associated with using the lumped capacitance method is small. This assumes there is almost zero temperature gradient inside the fin perpendicular to its surface. u L fin_opt Bi definition of optimal dimensionless volume provided by Razelos u n Optimal dimensionless volume for a rectangular profile, longitudinal fin (Razelos). L fin_opt_n0 u n0 Bi 36.90mm Optimal fin length