Let's go **step by step** through the washability problem. We'll focus on calculations for the key items and explain each step. --- # **Given Data Table** | Specific Gravity | Wt, % | % Ash | |------------------|-------|-------| | Floats 1.40 | 40.77 | 9.28 | | 1.40/1.60 | 11.26 | 14.07 | | 1.60/1.80 | 7.61 | 18.30 | | 1.80/2.00 | 1.88 | 24.97 | | 2.00/2.20 | 1.68 | 34.57 | | 2.20/2.50 | 4.88 | 46.99 | | 2.50/2.80 | 3.60 | 54.92 | | 2.80/3.00 | 3.28 | 68.20 | | Sinks 3.00 | 32.96 | 80.63 | --- ## **a. What is the ash content of the whole coal?** To find the **overall ash content**, use the weighted average: \[ \text{Overall ash} = \frac{\sum (\text{Wt,%} \times \text{Ash,%})}{100} \] Let's calculate this step by step: 1. Multiply each fraction's weight % by its ash %. 2. Sum all the results. 3. Divide by 100 (since weights are in %). | Wt, % | % Ash | Wt × Ash | |-------|-------|----------| | 40.77 | 9.28 | 378.34 | | 11.26 | 14.07 | 158.53 | | 7.61 | 18.30 | 139.23 | | 1.88 | 24.97 | 46.94 | | 1.68 | 34.57 | 58.08 | | 4.88 | 46.99 | 229.35 | | 3.60 | 54.92 | 197.71 | | 3.28 | 68.20 | 223.70 | | 32.96 | 80.63 | 2657.58 | Sum of Wt × Ash = **4089.46** \[ \text{Overall ash} = \frac{4089.46}{100} = \boxed{40.89\%} \] --- ## **b. Maximum specific gravity for clean coal with ≤14% ash, theoretical yield, and sink ash** ### **Step 1: Identify specific gravity cut point for 14% ash** - The clean coal must have **≤14% ash**. - Look for the highest specific gravity interval where the cumulative ash is **≤14%**. From the table: - Up to **Floats 1.40**: 40.77% @ 9.28% Ash (OK) - Next interval: **1.40/1.60**: 11.26% @ 14.07% Ash (OK, but just at 14%) #### **Calculate cumulative weight and ash content for floats up to 1.60:** - Total weight = 40.77 + 11.26 = **52.03%** - Ash in each part: (40.77 × 9.28) + (11.26 × 14.07) = 378.34 + 158.53 = **536.87** - Average ash = 536.87 / 52.03 = **10.32%** So, up to 1.60 sp. gr., the clean coal would have **10.32% ash** (which is less than 14%). Let's check if including the next fraction (1.60/1.80) keeps us under 14%: - Add 7.61% at 18.30% ash: - New total weight = 52.03 + 7.61 = **59.64%** - Add 7.61 × 18.30 = 139.23 - New total ash = 536.87 + 139.23 = **676.10** - New average ash = 676.10 / 59.64 = **11.34%** Still less than 14%. Let's try including the next cut (1.80/2.00): - Add 1.88% at 24.97% ash: - New total weight = 59.64 + 1.88 = **61.52%** - Add 1.88 × 24.97 = 46.94 - New total ash = 676.10 + 46.94 = **723.04** - New average ash = 723.04 / 61.52 = **11.76%** Still less than 14%. Add the next interval (2.00/2.20): - Add 1.68% at 34.57% ash: - New total weight = 61.52 + 1.68 = **63.20%** - Add 1.68 × 34.57 = 58.08 - New total ash = 723.04 + 58.08 = **781.12** - New average ash = 781.12 / 63.20 = **12.36%** Still less than 14%. Add next interval (2.20/2.50): - Add 4.88% at 46.99% ash: - New total weight = 63.20 + 4.88 = **68.08%** - Add 4.88 × 46.99 = 229.35 - New total ash = 781.12 + 229.35 = **101.47** - New average ash = 101.47 / 68.08 = **14.84%** Now, we've **exceeded** 14% ash. So, we must **stop at the previous interval** (up to 2.20 sp. gr.), which gives us a yield of **63.20%** at **12.36% ash**. ### **Step 2: Ash content of the rejects (sink product)** - Rejects = 100% - 63.20% = **36.80%** - Ash in rejects = total ash - ash in clean coal = 4089.46 - 781.12 = **3308.34** - Ash % in rejects = 3308.34 / 36.80 = **89.93%** --- ### **Summary Table** | Product | Yield (%) | Ash (%) | |----------------|-----------|----------| | Clean Coal | 63.2 | 12.36 | | Rejects (Sink) | 36.8 | 89.93 | --- ## **c. Is this an easy or difficult separation? Why?** - **Easy separation**: Large difference in ash content between clean coal and rejects, and a high yield of clean coal at required ash content. - Here, you get **63.2% yield at 12.36% ash** (well below the 14% requirement), and rejects have **very high ash (89.93%)**. - The **ash content gap** between clean coal and rejects is large, indicating a **clear separation**. ### **Conclusion:** **This would be a relatively easy separation** because the difference in ash content between the clean product and the reject is very large, and you can recover a high yield of clean coal at the required ash level. --- # **Final Answers** ### a. What is the ash content of the whole coal? **40.89%** ### b. If a clean coal product with a maximum ash content of 14% is required: - **Maximum specific gravity of separation:** 2.20 - **Theoretical yield of clean coal:** 63.2% - **Ash % of rejects:** 89.93% ### c. Would this be an easy or difficult separation and why? **Easy**, because of the high yield at low ash and a large ash difference between clean coal and rejects. --- If you want to see the washability curves or the step-by-step Excel setup, just ask!