Please help it’s due tomorrow and I’m completely lost!!!

Starting wih an energy balance on a rectangular volume element, derive the one- dimensional transient heat conduction equation for a plane wall with constant thermal conductivity and no heat generation.

A21-tooth spur pinion mates with a 28-tooth gear. the diametral pitch is 3 teeth/in and the pressure angle is 20°, find the following values: the addendum, dedendum, circular pitch, tooth thickness, and base circle diameters.

Aheat pump with refrigerant-134a as the working fluid is used to keep a space at 25°c by absorbing heat from geothermal water that enters the evaporator at 500c at a rate of 0.065 kg/s and leaves at 40°c. the refrigerant enters the evaporator at 20°c with a quality of 23 percent and leaves at the inlet pressure as saturated vapor. the refrigerant loses 300 w of heat to the surroundings as it flows through the compressor and the refrigerant leaves the compressor at 1.4 mpa at the same entropy as the inlet. determine: (a) the degrees of subcooling of the refrigerant in the condenser, b)-the mass flow rate of the refrigerant . (c) the heating load and the cop of the pump, and d)-the theoretical minimum power input to the compressor for the same heating load.