Environment node

Given the process topology, an unoriented graph is built where nodes correspond to units and arcs represent process streams. The process graph contains an environment node from which the process receives feeds, and to which it supplies products. 

Figure 3 shows the locations of the nodes in a sensor field. Also, Figure 3 shows how the network station is gathering data from the current area of observation. Nodes 1 and 4 sense the environment. Nodes 2 and 5 relay the data. Node 3 receives two raw data streams and then aggregates them into a single stream. Furthermore, node 3 transmits the aggregated data to node 6. Nodes 6 and 7 relay the data onward without any processing. 

The solid in Prob. 3-43 is initially uniform in temperature at 10°C. At time zero the right face is suddenly changed to 38°C and the left face exposed to the convection environment. Nodes 3 and 6 remain at 10 C. Select an appropriate value for At and calculate the temperatures of nodes 1, 2, 4, and S after 10 time increments. Carry the calculation forward to verify the steady-state distribution. Take p = 3000 kg/m3 and c = 840 J/kg C. 

Thus in Fig. 3-5, the input/output streams into/from separate plants G, and Gj are incident to environment node Hq, in addition the plants G, and G2 are connected via certain energy stream(s), and some energy stream(s) can connect directly G2 with n.Q. We can obtain connected graph G whose node set consists of N , (units of Gj), N 2 (units of G2), and /%. The arc set J of G consists of arcs between the units of G,, arcs between the units of G2, and the arcs connecting G, and Gj with environment node as drawn. In the set of mass- and energy balance equations as considered later in Chapter 5, we can take the whole G[NJ] as the graph of material streams. 

The reader can certainly imagine the case where there are more than two cooperating plants interconnected by energy streams only, with arbitrary numbers of streams connecting the plants with the environment. We obtain again a partition such as in (3.1.9), with several matrices C, C2, and row vectors a, 0C2, . Each of the matrices (3.1.10) is that of a connected graph (node n is formally split into two or more environment nodes). 

Irrespective of how we have obtained the spanning tree G°, we choose a reference node in the mass balance problem, it will be clearly the environment node, thus C is the matrix in (3.1.6). We now can classify the nodes (units) n N = N - o according to their distance from Mq. Thus N is partitioned... 

To begin with an example, the reader is requested to turn now to Section 4.6, Fig. 4-2. First, the steady-state convention requires excluding from the set of nodes subject to balancing the inventories of sulphur and produced acids they are regarded as parts of the environment node. Now for instance the multicomponent balance of node B (sulphur burning unit)... 

Neglecting the change of accumulation in the nodes, the individual component balances of nodes n e N (units of the system, without the environment node Hq) are of the form "output minus input = integral production rate" for any component C,< see Section 4.2. Here, the integral production rate of C,( (in mass units) in node n equals, by (4.2.11)... 

We have neglected the change of accumulation in the nodes n e N in particular nodes such as inventories have been included in the environment node rto- It is not formally difficult to extend the set of variables by the accumulation terms see Section 4.7. In certain cases (such as stirred reactors), the accumulation terms can be identified by volume and concentrations measurement. In the extended balance, we then replace the definition (4.8.8) of n n) by (4.7.10) thus the components of vector n(n) are... 

Let us consider again a system of units such as in Chapter 3, thus the oriented graph G with incidence matrix (C j) we have again the equality (4.2.7). N is again the node set of the graph (environment node o included), N the set of units, J the set of arcs, thus of material streams. If Q(n) is the quantity Q ... 

The matrix C of elements C j (n e N , 7 e J) is the reduced incidence matrix of graph G, with the environment node as reference node see Chapter 3. Let us now admit the presence of energy distributors. Then the matrix D of... 

If G is not connected then at least one of the connected components does not contain the environment node and represents a subsystem connected with the remaining nodes only via certain arcs (streams) j e J (5.2.2) recall that the whole graph G of material streams is connected. Thus all the input streams are outlet streams from some splitter, and all the output streams from the subsystem have some splitter as the other endpoint. Although such scheme is conceivable, it can be regarded as unlikely. 

In our simple examples according to Fig. 8-1, we have considered the graph of the technological system restricted to one technological unit (node), plus the environment node. The presence of other node balances can change the classification. To Fig. 8-1, let us add a splitter and another heat exchanger. 

The subgraph Gy has two connected components the component contains reaction nodes (B and R), contains the environment node. The reader can himself make the reduction (restriction to and deletion of splitters) in the case when C is sulphur trioxide. According to Fig. 4-2 and the list (4.6.1), he will find node SI isolated in the first step, in the second step = B,R and = D, Al, A2, 0 (stream 13 connects D with A2). If Q = O2 or N2, and N° will be the same as in the preceding example (with different structures of the subgraphs), while if C, = H2O, G , will remain connected and contain the node 0. Finally if C is elemental sulphur then G k has nodes 0 and B, and is connected. Alas, the scheme is simplified, for example concerning the flow scheme of water and acids. Adding further streams and splitters, it can happen that the decomposition of Guk will comprise more components. 

Recall Chapter 3, Fig. 3-1 in Section 3.1. If N is again the set of rmits (without the environment node), let us consider the subset... 

Let us consider the graph reduction. The corresponding matrix operation is summation of rows of the full incidence matrix over each subset of nodes constituting a connected component of G . The connected component containing the reference (environment) node is further not considered and the Af (= Af-1) rows of the submatrix (A/, A ) are thus sums of rows of (C, ) over the respective nodes of the remaining... 

To summarise, besides mass streams, there are two utilities, electric energy and steam. The System is connected to the environment which consists of several environment nodes. The introduction of several environment nodes may look confusing. The reason for that is the need for keeping information about detailed source or destination of streams connecting the system with the... 

Number of streams and nodes fluctuates according to the operation of the system. It can be surprising that the number of nodes is greater than the number of streams in some cases. It is caused by nodes (mostly environment nodes) which are active only occasionally (imagine for example the set of customers represented by nodes where products are shipped). Some values of flows and inventories are set as "fixed" which means that they are not subjected to reconciliation. 

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