In a solid sphere, a heat flow is produced by a source that generates energy at a rate of S W/m3 . generates energy at a rate of S W/m3 throughout its volume. The radius of the sphere is R, its thermal conductivity k and the surface of the sphere is maintained at a temperature TR. Derive an expression between the temperature and the distance from the center, in the steady state, for: (a) k = k0, constant. (b) k = k0 (1 + B T) (T = temperature, k0 and B = constants).

A food product containing 85% moisture content is being frozen. Estimate the specific heat of the product at -4 ° C when 62% of the water is frozen. The specific heat of the dry product is 2.5 kJ / (kg ° C). it is assumed that the specific heat of water at -10 ° C is the same as the specific heat of water at 0 ° C, and the specific heat of ice follows the function Cp es = 0.0062 Tfrozen + 2.0649. Cp of frozen product =      kJ / kg ° C.

A firearm can be modeled as a kind of heat engine, where the projectile acts as a piston that separates from the rest of the system during expansion. Consider a rifle with a 1.70 kg barrel made of iron [specific heat = 448 J/(kg °C)]. The rifle fires a 4.00 g bullet that exits the barrel with a speed of 350 m/s. When the propellant is ignited, 1.10% of the energy released goes into propelling the bullet (this is the thermal efficiency of the "engine"). The other 98.9% can be approximated as being entirely absorbed by the barrel, which increases in temperature uniformly for a short time before this energy is dissipated into the surroundings. What is this temperature increase (in °C)? (Round your answer to at least one decimal place.) °℃

A heat engine using air starts with p1 = 3 atm and V1 of 5 L. Its pressure increases at constant volume to p2. It then expands adiabatically to V3 = 12 L. Finally it returns at constant pressure to its original state. Can someome please explain how to calculate Q12 and Q31 ? Please show all work so I can understand the process! Thank you!