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CIEG 305 DERIVATION OF THE ENERGY EQUATION 1 st Law of Thermodynamics Change in energy per time Rate at which heat is added to system Rate at which work done by system =- From Reynolds Transport Theorem, B=Energy=E. Intensive value β=dE/dm=e

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e consists of e internal + e kinetic + e potential + e other e other associated with chemical reactions and we neglect it e internal generally only of interest if thermo comes into play (temperature changes)

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NOW FOR THE WORK TERM, divide into three parts Recall that · means d/dt Shaft work isolates portion of work done by fan blades, pistons etc that stick through control surface A device adding energy to flow does NEGATIVE shaft work A device removing energy from the flow does POSITIVE shaft work

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Total pressure work is the integral over the control surface of the pressure times the normal velocity Total viscous work is the integral over the control surface of the pressure times the respective velocity

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WHEN IS SHEAR TERM NEGLIGIBLE? 1)On solid surface the no-slip condition applies, V=0 so viscous term goes to zero. 2)Surface of machine, then incorporate into the shaft work term 3)Inlet or outlet: Set up problem so flow is normal to surface area element so tangential stresses go to zero. Normal stresses would still exist but these are often small. BOTTOM LINE: quite often we can neglect the viscous work term.

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ENERGY EQUATION OR SIMPLIFYING FOR STEADY FLOW PROBLEMS, d/dt=0

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FOR STEADY 1D FLOW PROBLEMS, usually the case for us

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Along a stream tube the mass flux is constant and we have another form of the steady energy equation States that upstream head can differ from downstream head ONLY if there is heat transfer, shaft work or viscous work (ignored here)

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Common application of the steady flow equation is for flow through pipes, possibly including pumps or turbines All “h” parts on the right hand side are loss terms in units of length h friction = frictional losses, always positive in real, viscous flows h pump = adds energy to flow h turbine = extracts energy from flow H1H1 H2H2

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TURBINE H1 > H2 ENERGY PUMP H1 < H2 ENERGY

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