Review Special Issues

Stability, thermophsical properties of nanofluids, and applications in solar collectors: A review

  • Received: 20 April 2021 Accepted: 12 July 2021 Published: 03 September 2021
  • Recently, renewable energies have attracted the significant attention of scientists. Nanofluids are fluids carrying nano-sized particles dispersed in base fluids. The improved heat transfer by nanofluids has been used in several heat-transfer applications. Nanofluids' stability is very essential to keep their thermophysical properties over a long period of time after their production. Therefore, a global approach including stability and thermophysical properties is necessary to achieve the synthesis of nanofluids with exceptional thermal properties. In this context, the objective of this paper is to summarize current advances in the study of nanofluids, such as manufacturing procedures, the mechanism of stability assessment, stability enhancement procedures, thermophysical properties, and characterization of nanofluids. Also, the factors influencing thermophysical properties were studied. In conclusion, we discuss the application of nanofluids in solar collectors.

    Citation: Omar Ouabouch, Mounir Kriraa, Mohamed Lamsaadi. Stability, thermophsical properties of nanofluids, and applications in solar collectors: A review[J]. AIMS Materials Science, 2021, 8(4): 659-684. doi: 10.3934/matersci.2021040

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  • Recently, renewable energies have attracted the significant attention of scientists. Nanofluids are fluids carrying nano-sized particles dispersed in base fluids. The improved heat transfer by nanofluids has been used in several heat-transfer applications. Nanofluids' stability is very essential to keep their thermophysical properties over a long period of time after their production. Therefore, a global approach including stability and thermophysical properties is necessary to achieve the synthesis of nanofluids with exceptional thermal properties. In this context, the objective of this paper is to summarize current advances in the study of nanofluids, such as manufacturing procedures, the mechanism of stability assessment, stability enhancement procedures, thermophysical properties, and characterization of nanofluids. Also, the factors influencing thermophysical properties were studied. In conclusion, we discuss the application of nanofluids in solar collectors.



    The problem of scheduling uniform parallel batch machines with processing set restrictions can be defined as follows. Let J={1,2,,n} be a set of jobs and M={M1,M2,,Mm} be a set of uniform parallel batch machines. Job j (j=1,2,,n) becomes available at its release time rj0 and requires pj0 units of processing called its length. For each job j, let MjM be the set of the eligible machines which are capable of processing the job, called its processing set. Each job will be assigned to exactly one machine and job preemption is not allowed. Machine Mi (i=1,2,,m) has a speed vi1 and a capacity Ki<n. The impact of the speed is that Mi can carry out vi units of processing in one time unit. That is, if job j is assigned to machine Mi, then it requires pj/vi processing time to be completed. Machine Mi can process several jobs as a batch simultaneously as long as the total number of these jobs does not exceed Ki. The length of a batch is the maximum of the lengths of the jobs belonging to it. Jobs in the same batch have a common start time and a common completion time. The goal is to schedule the jobs on the machines in a manner that optimizes one or more objective functions.

    Two classes of objectives are considered: the min-sum objective and the min-max objective. Specifically, let fj:[0,+)[0,+) (j=1,2,,n) be a non-decreasing function. Additional parameters that may be included for job j are its due date dj and its weight wj. For a particular schedule σ, let Cj(σ) denote the completion time of job j in σ. Let Tj(σ)=max{Cj(σ)dj,0} denote the tardiness of job j in σ. Let Uj(σ)=1 if Cj(σ)>dj and Uj(σ)=0 otherwise. (In the rest of this paper, we safely ignore σ in the notations without causing confusion.) The objectives of minimizing nj=1fj(Tj) and maxj=1,2,,n{fj(Tj)} will be considered. Following [1,2], the models can be denoted as Q|rj,dj,Mj,pbatch,Ki|fj(Tj) and Q|rj,dj,Mj,pbatch,Ki|max{fj(Tj)}.

    Many popular scheduling objectives are covered by the two models, such as total weighted completion time (wjCj) minimization, total weighted tardiness (wjTj) minimization, weighted number of tardy jobs (wjUj) minimization, makespan (Cmax=maxCj) minimization, and maximum weighted tardiness (max{wj(Tj)}) minimization. As shown in [3,4,5], most of such problems are NP-hard even for the special cases where all vi=1, all Ki=1 and all Mj=M. Thus, we are interested in polynomial time exact algorithms for some important special cases of the problems [6].

    In this paper, we focus on an important special case where all jobs have equal lengths (and equal release times). The problems under study can be denoted as Q|pj=p,dj,Mj,pbatch,Ki|fj(Tj) (the special case of Q|rj,pj=p,dj,Mj,pbatch,Ki|fj(Tj) where all rj=0) and Q|pj=p,dj,Mj,pbatch,Ki|max{fj(Tj)} (the special case of Q|rj,pj=p,dj,Mj,pbatch,Ki|max{fj(Tj)} where all rj=0). Li [7] presented polynomial time algorithms for uniform parallel machine scheduling problems Q|pj=p,dj,Mj|fj(Tj) and Q|pj=p,dj,Mj|max{fj(Tj)} (the special cases of Q|pj=p,dj,Mj,pbatch,Ki|fj(Tj) and Q|pj=p,dj,Mj,pbatch,Ki|max{fj(Tj)} where all Ki=1). We extend the results obtained in [7] to uniform parallel batch machines, allowing the machines to have non-identical capacities. Moreover, for minimizing makespan, we allow unequal release times and get an algorithm for arbitrary processing set restrictions.

    Although the above problem setting appears simple, it captures important aspects of a wide range of applications. There are many problems arise as its special or similar cases in networking and information systems. For example, Low [8] studied the problem in the context of retrieving data blocks from disks in video on demand systems. Suri et al. [9] considered the problem in peer-to-peer systems. The problem was also studied for workload balancing among packet queues [10], for data aggregation in wireless sensor networks [11]. Recently, Champati and Liang [12] studied a very similar problem where each machine has its own convex cost functions, aiming to minimize the sum cost and the maximum differential cost of the machines.

    The remainder of this paper is organized as follows. In Section 2, the related researches are reviewed. In Section 3, we consider the case of equal release times. We present an algorithm with running time O(n3m+n2mlog(mn)) for Q|pj=p,dj,Mj,pbatch,Ki|fj(Tj), as well as an algorithm with running time O(n5/2m3/2log(mn)) for Q|pj=p,dj,Mj,pbatch,Ki|max{fj(Tj)}. In Section 4, we consider the case of unequal release times. We present an algorithm with running time O(n5/2m3/2log(mn)) for Q|rj,pj=p,Mj,pbatch,Ki|Cmax. Section 5 presents the conclusions and future directions of research.

    Leung and Li [13] discussed several special cases of processing set restrictions. The processing sets of the jobs are inclusive, if for any two jobs j1 andj2, either Mj1Mj2, or Mj2Mj1. The processing sets are nested, if for any two jobs j1 andj2, either Mj1Mj2=, or Mj1Mj2, or Mj2Mj1. The processing sets are interval, if for any job j, Mj={Maj,Maj+1,,Mbj} for some 1ajbjm. The processing sets are tree-hierarchical, if each machine is represented by a tree node, and each job j is associated with a tree node Maj, such that Mj is exactly the set of the machines consisting of all the nodes on the unique path from Maj to the root of the tree.

    The problem studied in this paper combines two important sub-fields of scheduling theory: scheduling with processing set restrictions and parallel batch scheduling. The two sub-fields have received intense study in the literature, see the survey papers [13] and [14,15,16] respectively. There are also a few papers which combined the two-subfields into a unified framework [17,18,19,20,21,22,23]. In the problems studied in these papers except the last two, each job has a size and a machine can process several jobs simultaneously as a batch as long as the total size of these jobs does not exceed its capacity. Any machine cannot process the jobs whose sizes are larger than its capacity. Thus, for each job, the machines whose capacities are not less than its size form its processing set. Clearly, the processing sets of the jobs are inclusive. In [22], Li presented two algorithms with approximation ratios 3 and 9/4 for the problem of minimizing makespan on parallel batch machines with inclusive processing set restrictions, where the jobs have arbitrary lengths and the machines have the same speed. In [23], Li studied parallel batch scheduling with nested processing set restrictions to minimize makespan, and presented a (31/m)-approximation algorithm for the case of equal release times and a polynomial time approximation scheme (PTAS) for the case of unequal release times.

    For scheduling with processing set restrictions, the review focuses on the case of equal job lengths. Lin and Li [24] obtained an algorithm for P|pj=p,Mj|Cmax (the special case of Q|rj,pj=p,Mj|Cmax where all rj=0 and all vi=1) that runs in O(n3logn) time, and generalized the algorithm to solve Q|pj=p,Mj|Cmax in O(n3log(nvlcm)) time, where vlcm denotes the least common multiple of v1,v2,,vm. They also obtained an algorithm for P|pj=p,Mj(interval)|Cmax (the special case of P|pj=p,Mj|Cmax with interval processing set restrictions) that runs in O(m2+mn) time. Harvey et al. [25] independently developed an algorithm for P|pj=p,Mj|Cmax that runs in O(n2m) time. Brucker et al. [26] presented algorithms running in O(n2m(n+logm)) time for Q|pj=p,dj,Mj|wjTj, Q|pj=p,dj,Mj|wjUj, P|rj,pj=1,dj,Mj|wjTj and P|rj,pj=1,dj,Mj|wjUj. Li [7] presented an algorithm for Q|pj=p,dj,Mj|fj(Tj) that runs in O(n3m+n2mlog(mn)) time. For the special cases where fj(Tj)=Cj or fj(Tj)=Uj, the running time of the algorithm can be improved to O(n5/2mlogn). He also presented an algorithm for Q|pj=p,dj,Mj|max{fj(Tj)} that runs in O(n5/2mlog(mn)) time. For the special cases where fj(Tj)=Cj (i.e., Q|pj=p,Mj|Cmax), the running time of the algorithm can be improved to O(n2(m+log(nvmax))logn), where vmax=max{vj}. Lee et al. [27] showed that Q|rj,pj=p,Mj|Cmax can be solved in O(m3/2n5/2log(mn)) time, and P|rj,pj=p,Mj|Cmax(the special case of Q|rj,pj=p,Mj|Cmax where all vi=1) can be solved in O(m3/2n5/2logn) time. Shabtay et al. [28] obtained various results for several problems of scheduling uniform machines with equal length jobs, processing set restrictions and job rejection. Hong et al. [29] studied P|rj,pj=p,Mj|Cj. For the problem with a fixed number of machines, they provided a polynomial time dynamic programming algorithm. For the general case, they presented two polynomial time approximation algorithms with approximation ratios 3/5 and 5/7 respectively. Jiang et al. [30] presented a comprehensive overview (including their new findings) of ideal schedules for various scheduling problems in different machine environments and with different job characteristics. An ideal schedule is a schedule that simultaneously minimizes the total completion time and makespan. They pointed out that in most problems that are known to have an ideal schedule, the jobs have equal processing times. Along with other results, they proved that any optimal schedule for Q|pj=p,Mj|Cj also minimizes makespan. Jing et al. [31] studied the problem of scheduling high multiplicity jobs to minimize makespan on parallel machines with processing set restrictions, setup times and machine available times. High multiplicity means that jobs are partitioned into several groups and in each group all jobs are identical. Whenever there is a switch from processing a job of one group to a job of another group, a setup time is needed. They formulated the problem as a mixed integer programming and proposed a heuristic for it.

    Pinedo [32] and Glass and Mills [33] presented algorithms for P|pj=p,Mj(nested)|Cmax (the special case of P|pj=p,Mj|Cmax with nested processing set restrictions) that run in time O(nlogn) and O(m2) time respectively. Li and Li [34] presented algorithms for P|rj,pj=p,Mj(inclusive)|Cmax and P|rj,pj=p,Mj(tree)|Cmax (the special cases of P|rj,pj=p,Mj|Cmax with inclusive and tree-hierarchical processing set restrictions) that run in O(n2+mnlogn) time. For uniform machines, they showed that Q|rj,pj=p,Mj(inclusive)|Cmax and Q|rj,pj=p,Mj(tree)|Cmax can be solved in O(mn2logm) time. Later, Li and Lee [35] developed an improved algorithm for P|rj,pj=p,Mj(inclusive)|Cmax that runs in O(min{m,logn}nlogn) time, and an improved algorithm for P|rj,pj=p,Mj(tree)|Cmax that runs in O(mnlogn) time.

    For parallel batch scheduling, the review focuses on equal job lengths or uniform parallel batch machines. Liu et al. [36] presented an algorithm for P|rj,pj=p,pbatch,B|Cmax (the special case of Q|rj,pj=p,Mj,pbatch,Ki|Cmax where all Mj=M, all Ki=B and all vi=1) that runs in O(nlogn) time. Ozturk et al. [37] presented a 2-approximation algorithm for the problem of scheduling jobs with equal lengths, unequal release times and sizes on identical parallel batch machines (all Ki=B) to minimize makespan. Wang and Leung [18] studied the problem of scheduling jobs with equal lengths and arbitrary sizes on parallel batch machines (all vi=1) with non-identical capacities. The problem fits into the model of scheduling with inclusive processing set restrictions, since each job can only be processed by the machines whose capacities are not less than the size of the job. Wang and Leung [18] presented a 2-approximation algorithm, as well as an algorithm with asymptotic approximation ratio 3/2 for the problem. Li et al. [38] proposed several heuristics for the problem of scheduling jobs with unequal lengths, release times and sizes on uniform parallel batch machines with identical capacities to minimize makespan. Zhou et al. [39] presented an effective discrete differential evolution algorithm for the problem of scheduling jobs with unequal lengths and sizes on uniform parallel batch machines with non-identical capacities to minimize makespan. Both [38] and [39] have not included processing set restrictions.

    In this section, we consider the case of equal release times. We will present algorithms for Q|pj=p,dj,Mj,pbatch,Ki|fj(Tj) and Q|pj=p,dj,Mj,pbatch,Ki|max{fj(Tj)}.

    Since fj (j=1,2,,n) is a non-decreasing function, for both Q|pj=p,dj,Mj,pbatch,Ki|fj(Tj) and Q|pj=p,dj,Mj,pbatch,Ki|max{fj(Tj)}, there is an optimal schedule in which the first batch on each machine starts at time zero, and the batches on each machine are processed successively. Moreover, we can assume that there are ni empty batches on machine Mi (i=1,2,,m) to which the jobs may be assigned, where ni denotes the smallest integer such that niKin. The k-th batch on Mi, Bk,i, completes at time kp/vi, k=1,2,,ni. To find a feasible schedule, we need only to assign the jobs to mi=1ni empty batches such that all jobs obey the processing set restrictions.

    First, we consider Q|pj=p,dj,Mj,pbatch,Ki|fj(Tj).

    If job j is assigned to Bk,i and MiMj, then the cost incurred is defined to be cjki=fj(max{kp/vidj,0}), j=1,2,,n, k=1,2,,ni, i=1,2,,m. Let C=maxj,k,icjki. By regarding each job jJ as a vertex in X, and each empty batch Bk,i as Ki vertices yk1,yk2,,ykKi in Y, we construct a bipartite graph G with bipartition (X,Y), where j is joined to yk1,yk2,,ykKi if and only if MiMj, and the incurred costs are equal to cjki, j=1,2,,n, k=1,2,,ni, i=1,2,,m. Then we use the Successive Shortest Path algorithm to solve the bipartite weighted matching problem [40] and get a matching of minimum cost that saturates every vertex in X. From this matching we can construct an optimal schedule easily.

    The Successive Shortest Path algorithm runs in O(|X|S(|X|+|Y|,|X||Y|,C)) time, where S(|X|+|Y|,|X||Y|,C) is the time for solving a shortest path problem with |X|+|Y| vertices, |X||Y| edges (these edges have non-negative costs), and maximum coefficient C. Currently, S(u,a,C)=O(a+ulogu) [41]. Note that |X|=n and |Y|2mn. We get:

    Theorem 3.1. There is an exact algorithm for Q|pj=p,dj,Mj,pbatch,Ki|fj(Tj) that runs in O(n3m+n2mlog(mn)) time.

    Next, we consider Q|pj=p,dj,Mj,pbatch,Ki|max{fj(Tj)}. Let OPT denote the objective value of an optimal schedule.

    Recall that we are focusing on an optimal schedule in which machine Mi (i=1,2,,m) processes ni batches (some batches may be empty), where ni denotes the smallest integer such that niKin. Each batch on Mi has Ki positions to accommodate jobs, and there are at most 2n positions on Mi, i=1,2,,m. Therefore, there are at most 2mn positions in total. Since each position has at most n choices of accommodating a job, there are at most 2mn2 possible values for OPT. We can sort these values in ascending order in O(mn2log(mn)) time. Then, we perform a binary search in the interval to determine OPT in O(log(mn)) iterations.

    For each value λ selected, we test whether there is a feasible schedule whose objective value is no more than λ. To this end, we construct a bipartite graph G with bipartition (X,Y) as follows. Regard each job jJ as a vertex in X, and each empty batch Bk,i as Ki vertices yk1,yk2,,ykKi in Y, where j is joined to yk1,yk2,,ykKi if and only if MiMj and fj(kp/vidj)λ, j=1,2,,n, k=1,2,,ni, i=1,2,,m. Then we use the algorithm in [42] to solve the maximum cardinality bipartite matching problem. If the obtained matching saturates every vertex in X, then the procedure succeeds and we search the lower half of the interval. Otherwise, the procedure fails and we search the upper half of the interval.

    The algorithm in [42] runs in O(|X|+|Y||X||Y|) time. Note that |X|=n and |Y|2mn. We get:

    Theorem 3.2. There is an exact algorithm for Q|pj=p,dj,Mj,pbatch,Ki|max{fj(Tj)} that runs in O(n5/2m3/2log(mn)) time.

    In this section, we consider the case of unequal release times. We will present an algorithm for Q|rj,pj=p,Mj,pbatch,Ki|Cmax, which generalizes the one in [27] for Q|rj,pj=p,Mj|Cmax (the special case of Q|rj,pj=p,Mj,pbatch,Ki|Cmax where all Ki=1). Let OPT denote the makespan of an optimal schedule for Q|rj,pj=p,Mj,pbatch,Ki|Cmax. Let Λ= {λ|λ=rj+kp/vi;j,k{1,2,n} and i{1,2,m}}. The following lemma, adopted from [27], still holds for Q|rj,pj=p,Mj,pbatch,Ki|Cmax.

    Lemma 4.1. The set Λ, as defined above, contains all candidates for OPT.

    Since |Λ|mn2, there are at most mn2 possible values for OPT. We can sort these values in ascending order in O(mn2log(mn)) time. Then, we perform a binary search to determine OPT in O(log(mn)) iterations.

    For each value λ selected, we use the following procedure, AssignJobs, to test whether there is a feasible schedule whose makespan is no more than λ.

    AssignJobs (λ):

    Step 1. Assign bi=min{ni,λvi/p} empty batches of length p and capacity Ki to machine Mi, where ni denotes the smallest integer such that niKin. The k-th batch on Mi, Bk,i, starts at time λ(bik+1)p/vi and completes at time λ(bik)p/vi, k=1,2,,bi, i=1,2,,m.

    Step 2. Construct a bipartite graph G with bipartition (X,Y) as follows. Regard each job jJ as a vertex in X, and each empty batch Bk,i as Ki vertices yk1,yk2,,ykKi in Y, where j is joined to yk1,yk2,,ykKi if and only if MiMj and rjλ(bik+1)p/vi, j=1,2,,n, k=1,2,,bi, i=1,2,,m.

    Step 3. Use the algorithm in [42] to solve the maximum cardinality bipartite matching problem. If the obtained matching saturates every vertex in X, then the procedure succeeds and terminates. Otherwise, the procedure fails and terminates.

    Lemma 4.2. If OPTλ, then AssignJobs will generate a feasible schedule in O(n5/2m3/2) time for Q|rj,pj=p,Mj,pbatch,Ki|Cmax whose makespan is at most λ.

    Proof. Let σ denote an optimal schedule. Consider machine Mi, i=1,2,,m. Since OPTλ, in σ, Mi processes at most bi batches. Without loss of generality, assume that there are bi batches (some of which may be dummy empty batches) processed on Mi in σ, denoted as B1,i,B2,i,Bbi,i.

    Modify σ as follows. Let Bbi,i be completed at time λ, Bbi1,i be completed at time λp/vi, ..., B1,i be completed at time λ(bi1)p/vi, i=1,2,,m. Denote by ˜σ the modified schedule. Since each batch in ˜σ starts no earlier than its corresponding batch in σ, ˜σ is a feasible schedule. The makespan of ˜σ is exactly λ. Clearly, AssignJobs will generate a feasible schedule which is no worse than ˜σ.

    The algorithm in [42] runs in O(|X|+|Y||X||Y|) time. Since |X|=n and |Y|2mn, the running time of AssignJobs is O(n5/2m3/2).

    We get:

    Theorem 4.3. There is an exact algorithm for Q|rj,pj=p,Mj,pbatch,Ki|Cmax that runs in O(n5/2m3/2log(mn)) time.

    We studied the problem of scheduling jobs with equal lengths and processing set restrictions on uniform parallel batch machines with non-identical capacities. For the case of equal release times, we gave efficient exact algorithms for various objective functions. For the case of unequal release times, we gave efficient exact algorithms for minimizing makespan. The findings extend previous results for uniform machines counterparts.

    Future research should focus on extending the algorithms to the case of unequal release times for objective functions other than makespan. It would also be interesting (and difficult) to extend the techniques to other related models, for general or special cases of processing set restrictions, such as scenario-dependent processing times and/or due dates [43], two-agent scheduling problems [44], or multitasking scheduling problems [45].

    This work is supported by Natural Science Foundation of Shandong Province China (No. ZR2020MA030).

    The authors declare there is no conflict of interest.



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