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17-Phys-B6 Applied Thermodynamics and Heat Transfer · December 2018

Question 4 of 8: Ammonia Ice-Making Refrigeration Plant — Power, COP, Condenser Water Flow

Nivaar worked solution (AI-drafted; not reviewed by a licensed engineer)

Notes on this paper

Paper format. 17-Phys-B6 Applied Thermodynamics and Heat Transfer, National Examination December 2018 — a three-hour open-book examination; candidates are expected to bring both a thermodynamics text and a heat-transfer text to make use of the property tables and graphs. A complete examination is five questions — either three from Part A (Thermodynamics, Q1–Q4) and two from Part B (Heat Transfer, Q5–Q8), or two from Part A and three from Part B — every question carrying equal value; all eight are solved below as a complete study set. Question 2 is solved as one connected narrative: the wet steam whose quality is measured by the throttling calorimeter in part (a) is the same steam entering the turbine in part (b), which is what makes part (c)'s "isentropic despite heat loss" observation checkable. Question 3 gives every cycle temperature directly from the printed diagram but no pressures, so it is solved purely from energy balances (constant specific heat, cold-air-standard) rather than isentropic pressure ratios — the intended reading, since no compressor/turbine pressure ratio is given anywhere on the page.

Reference texts. Y. A. Çengel and M. A. Boles, Thermodynamics: An Engineering Approach, 8th ed. (ideal-gas mixtures, air-standard Otto and Brayton cycles, throttling calorimeters, steam turbines, vapour-compression refrigeration); F. P. Incropera and D. P. DeWitt, Fundamentals of Heat and Mass Transfer, 7th ed. (composite cylindrical conduction with convection at both surfaces, heat generation in a solid cylinder, internal/external convection combined via an overall coefficient, effectiveness–NTU heat-exchanger analysis). Ammonia and steam saturation/superheat property values were computed (Bell et al., IAPWS-95 / REFPROP-quality equations of state) and cross-checked against the printed appendix tables on pages 5–6 of the source exam and standard steam tables, which they matched to 3–4 significant figures throughout.

Question 4: Ammonia Ice-Making Refrigeration Plant — Power, COP, Condenser Water Flow

Question text not reproduced: the examination questions are © Engineers and Geoscientists BC. Open the official past paper (linked at the top of this page) to read the question, then follow the worked solution below.

Given. An ideal vapour-compression cycle with ammonia: saturated vapour leaves the evaporator at $-15\,{}^{\circ}\text{C}$ (state 1), an actual (95%-efficient) compression to the condensing pressure at $31\,{}^{\circ}\text{C}$ (state 2), saturated liquid leaves the condenser at $31\,{}^{\circ}\text{C}$ (state 3), and an isenthalpic throttle back to the evaporator (state 4). Refrigeration capacity 70 kW; condenser cooling water enters at 20°C, leaves at 27°C.

Given data
QuantitySymbolValue
Refrigeration capacity$\dot Q_{evap}$70 kW
Evaporator temperature$T_{evap}$$-15\,{}^{\circ}\text{C}$
Condenser temperature$T_{cond}$$31\,{}^{\circ}\text{C}$
Compressor efficiency$\eta_c$95%
Cooling water inlet / outlet$T_{w1}/T_{w2}$$20\,{}^{\circ}\text{C}/27\,{}^{\circ}\text{C}$

Find. Compressor power per kW of refrigeration; the COP; and the cooling-water flow rate.

Enthalpy h (kJ/kg)ln PAmmonia vapour-compression cycle — P–h diagram (schematic dome)12s234
Vapour-compression cycle on $P$–$h$ coordinates (schematic dome): 1 saturated-vapour evaporator exit, 2s ideal (isentropic) compression end, 2 actual compression end (95% efficiency), 3 saturated-liquid condenser exit, 4 throttle exit back to the evaporator.

Approach. Read $h_1,s_1$ (saturated vapour, $-15\,{}^{\circ}\text{C}$) and $h_3$ (saturated liquid, 31°C) from the saturated ammonia table; find the ideal compression end state 2s by matching $s_{2s}=s_1$ in the superheated ammonia table at the condensing pressure; apply the compressor efficiency to get the ACTUAL work and hence $h_2$; then apply energy balances to the evaporator, compressor and condenser in turn.

  1. State 1 — saturated vapour at $-15\,{}^{\circ}\text{C}$ (interpolated). $$h_1=1425.7\text{ kJ/kg},\qquad s_1=5.5452\text{ kJ/kg}\cdot\text{K}$$
  2. State 2s — isentropic compression to the 31°C condensing pressure ($\approx1.2$ MPa table). Matching $s_{2s}=s_1=5.5452\text{ kJ/kg}\cdot\text{K}$ in the superheated table at 1.2 MPa (between 100°C and 120°C): $$h_{2s}=1663.0\text{ kJ/kg}\qquad(T_{2s}\approx102.1\,{}^{\circ}\text{C})$$ $$w_{isen}=h_{2s}-h_1=1663.0-1425.7$$ $$\boxed{w_{isen}=237.3\text{ kJ/kg}}$$
  3. Actual compressor work and state 2. $$w_{actual}=\frac{w_{isen}}{\eta_c}=\frac{237.3}{0.95}$$ $$\boxed{w_{actual}=249.8\text{ kJ/kg}}$$ $$h_2=h_1+w_{actual}=1425.7+249.8=1675.5\text{ kJ/kg}$$
  4. State 3 — saturated liquid at 31°C (interpolated), throttled to state 4. $$h_3=327.75\text{ kJ/kg}=h_4\quad(\text{isenthalpic throttle})$$
  5. Refrigeration effect, power per kW of refrigeration, COP. $$q_{evap}=h_1-h_4=1425.7-327.75$$ $$\boxed{q_{evap}=1098.0\text{ kJ/kg}}$$ $$\frac{\dot W}{\dot Q_{evap}}=\frac{w_{actual}}{q_{evap}}=\frac{249.8}{1098.0}$$ $$\boxed{0.2275\text{ kW per kW of refrigeration}}$$ $$COP=\frac{q_{evap}}{w_{actual}}=\frac{1098.0}{249.8}$$ $$\boxed{COP=4.396}$$
  6. Cooling-water flow rate. Mass flow of ammonia, then heat rejected at the condenser, then the water balance: $$\dot m_{Ar}=\frac{\dot Q_{evap}}{q_{evap}}=\frac{70}{1098.0}=0.06376\text{ kg/s}$$ $$\dot Q_{cond}=\dot m_{Ar}(h_2-h_3)=0.06376\times(1675.5-327.75)$$ $$\boxed{\dot Q_{cond}=85.9\text{ kW}}$$ $$\dot m_w=\frac{\dot Q_{cond}}{c_{p,w}(T_{w2}-T_{w1})}=\frac{85.9}{4.186\times(27-20)}$$ $$\boxed{\dot m_w=2.93\text{ kg/s}}$$
Question 4 — results
QuantityValue
Actual compressor work $w_{actual}$249.8 kJ/kg
Power per kW of refrigeration0.2275 kW/kW
Coefficient of performance, COP4.396
Ammonia mass flow rate0.0638 kg/s
Condenser heat rejection $\dot Q_{cond}$85.9 kW
Cooling-water flow rate $\dot m_w$2.93 kg/s