Ben Wolfe 11/3/14. Figure 1: Theoretical diagram showing the each step of heat loss.

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1 Condenser Analysis Water Cooled Model: For this condenser design there will be a coil of stainless steel tubing suspended in a bath of cold water. The cold water will be stationary and begin at an ambient temperature of 25C. As the steam flows through the condensing coil, the heat will be transferred from the steam into the surrounding cold water bath. The cold water bath is assumed to have adiabatic boundary conditions, so all of the heat loss form the condensing pipe will remain in the water bath, causing the water to gradually heat up over the 9 hour cycle. Assumptions: The water bath will have an adiabatic boundary condition meaning it will not lose any heat to the environment. Radiation will be neglected. Steam flows into the condenser at a constant rate. Steam enters the condenser at 200C. The temperature of the outside of the pipe wall will be isothermal and equal to the temperature of the water bath. The water bath will heat up uniformly, meaning the bulk temperature is constant throughout the entire water bath. The temperature of the inside surface of the condensing pipe is equal to the temperature of the water bath. Approach: The approach being used to solve this heat transfer problem is to break down the condenser into three different sections that all require a different set of heat transfer equations. Step 1: Steam enters the condenser at 200C and must be cooled to 100C Step 2: The steam remains at 100C as enough heat is removed to condense steam into liquid water. Step 3: The liquid water cools below 100C and exits the condenser. The length of pipe required to complete each step will be calculated. The goal of this condenser design is to minimize the length of pipe required, because that will lower the total cost of the system.

2 Figure 1: Theoretical diagram showing the each step of heat loss. Outer Diameter inches meters Wall thickness 0.02 inches meters Inner Diameter inches meters Pipe Length 240 inches meters Surface Area inner pipe meters^2 Temperature of cold water bath 25 Celsius 298 Kelvin Initial Temperature of steam 200 Celsius 473 Kelvin Volumetric Flow rate (from boiler-liquid) 1.08 gal/9 hours E-07 m^3/s Volumetric Flow rate (from boiler-steam) m^3/s Mass flow rate (m dot) kg/s velocity of steam (in condenser tube) m/s velocity of water (after condensation) m/s Density of water 1000 kg/m^3 Density of steam kg/m^3 viscosity of steam 1.23E-05 kg/m*s thermal conductivity (steam) W/m*K Prandtl number (steam) 1 dimensionless Cp (Specific heat of steam) 1970 J/kg*K viscosity of water (at 75C) kg/m*s thermal conductivity (water) 0.67 W/m*K Cp (Specific heat of water) 4186 J/kg*K Hv (heat of vaporization of water) J/kg Table 1: Material Properties

3 Equations used: Valid for isothermal wall condition Re D = ρ v D μ Step 1 Calculations: The convection coefficient for internal flow can be calculated using the Nusselt number, which depends on the Reynolds number. The Nusselt number increases with the Reynolds number, so a high Reynolds number is more desirable. To achieve a high Reynolds number, the velocity of the moving fluid must be increased. The fluid velocity increases with a decreasing pipe diameter, so a small pipe diameter is chosen. Re turbulent Nu turbulent h bar (convection coefficient) W/m^2*K Find: how much pipe length it takes for the steam to reach 100 C L1 h π D ΔT lm = m C p (T 0 T i ) Where T 0 = 100C and T i = 200C Rearranging this equation to solve for L1 yields: L1 = ln ( ΔT i C P ) m ΔT o h π D L1 = 4.3 inches =.011 meters

4 Step 2 Calculations: Find: how much pipe length it takes to condense steam at 100 C I can find how much power is needed to condense the steam by knowing the mass flow rate and the heat of vaporization of water. Then by setting this value equal to the convection inside the pipe the length of pipe required for condensing can be found. Power needed from heat of vaporization q 2 = m H v q 2 = W = h A s (100 C T c ) = h π D L 2 (100 C T c ) Where Tc is the temperature of the water bath, and h is the convection coefficient from step 1. L 2 = 78 inches = 1.98 meters Step 3 Calculations: Finally, the steam is fully condensed and cools in the remainder of the pipe. There is no requirement for the exiting temperature of the fluid, but it is calculated because heat is still being added to the water bath. The Reynolds number, Nusselt number, and convection coefficient must be recalculated because the fluid is no longer steam, it is liquid water. Re laminar Nu 3.66 laminar h bar (convection coefficient) W/m^2*K pipe length (L3) inches meters As (Surface area inside pipe) meters^2 To = Ts - (Ts-Ti)/(e^(h*As/(mdot*Cp)) 25 Celsius A total pipe length of 20 ft is assumed to be submerged in the water bath so L3 is simply the remaining length of pipe. The pipe used for this condenser can only be purchased in 25ft length increments, so it would be best to avoid going over 25 feet of required pipe for cost reasons. 20 is being used for this analysis rather than 25ft, because some of the pipe will be used to connect to the boiling chamber and will therefore not be submerged in the water bath. L3 = 20ft L2 L1 L3 = inches = 4.0 meters Liquid water should exit the condenser at 25C

5 Transient Analysis The previous calculations were only valid when the cold water bath is at 25C, but over time this water will heat up and a transient analysis is necessary to understand the full functionality of this condenser. Assuming all of the heat loss from the steam enters the water bath and assuming there are 14 gallons of available water the following chart shows the results of this transient analysis. Time (hours) Temp of water bath Step 1 Step 2 Step 3 T out Liquid water q 1 + q 2 + q 3 Tc L1 (m) q1 (W) L2 (m) q2 (W) L3 (m) q3 (W) Temp (C) qtotal (W) Table 2: Transient analysis of with 14 gallons of water and no heat loss to the environment From this data it can be seen that the length of pipe required for step 2 increases substantially as the temperature of the water bath increases. After 9 hours of continuous operation, the water bath will reach a temperature of 74.3C and the required pipe length will be approximately 19.5 ft before all of the steam condenses.

6 Conclusions: This condenser is much cheaper because there is less stainless steel tubing required than the air cooled condenser. As a result it is a more feasible design. This analysis has been conservative in its approach for several reasons. Heat losses to the environment were neglected entirely, meaning the water bath should be losing heat constantly and the temperature shouldn t reach as high as predicted. Also, radiation was neglected entirely which should again benefit the condenser in the real world because of these heat losses. Bill of Materials: Description Cost source 25ft 1/8" tubing $ le_1?ie=utf8&psc=1&smid=a192ls2fevgcgz 18 gallon tote $ in-x in-Storage-Tote-in-Green / Compression Fitting -1/8" $ epoxy sealing $0.00 RIT machine shop sealing grommet 1/8" (pkg Grainger Item # 3MPL2 of 50) $1.74 Hex locknut 1/8" NPT $ _1?ie=UTF8&psc=1&smid=ATVPDKIKX0DER Total Cost of condenser = $73.19 Sources: Properties of Stainless Steel Properties of Steam