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I'm not sure if my question qualifies as coursework, but here it is anyway:

I'm taking ENGR-101 this term, and for a group project my group chose the design of a solar water heater. I'm trying to find a formula to calculate our design's efficiency, and when I search for thermodynamic equations I can't find what I'm looking for.

## Homework Statement

An insulated 30 ga. water tank has a 3/4" dia. , 40' long copper pipe connected to it (the pipe leaves the tank and returns to it). A 1 ga./min. pump moves the water through the pipe. The air around the pipe is hotter than the water, so the water in the tank gets hotter as it circulates around the pipe. If the initial temperature of the water in the tank is 45 degrees fahrenheit, and the air temperature around the pipe remains constant at 100 degrees fahrenheit, how hot will the water in the tank be after 1 hour?

thermal conductivity (k) of copper:

**400**

wall thickness of pipe (l): 1 mm =

**0.001 m**

Surface area of pipe (A): (2.54 cm * 0.75 * pi)/100 * 12.2 m =

**0.73 m^2**

Heat capacity of water (Cp):

**4.2**

mass flow rate (m-dot): 1 ga/min = 3.8 l/min = 3.8 kg/min =

**0.06 kg/s**

Mass of water: 30 ga. = 113.6 l =

**113.6 kg**(mass of water in pipe negligible)

Initial difference in temp: 37.8 C - 7.2 C = 30.56 C =

**30.56 K**

## Homework Equations

The only equation I have found, which deals only with the change in temp between the ends of the pipe, is:

Tf - Ti = q * A / m-dot * Cp

q = k * (T outside - T water) / l

## The Attempt at a Solution

When I punched in my values in the equation above, I got for an answer millions of kelvins difference between Tf and Ti!

I'm also taking Calculus II this term, so I'm prepared (hopefully) to deal with the integrals that should pop up as the water in the tank gets gradually hotter as a function of time.

Thanks!